VARIABLE STIFFNESS MECHANISM AND SYSTEM

Cross-Reference to Related Applications

This application is based upon and claims priority to U.S. Provisional Patent Application Serial No. 61/698,105, entitled "VARIABLE STIFFNESS MECHANISM", filed September 7, 2012, the contents of the disclosure of which is hereby incorporated by reference in its entirety.

Field of the Invention

The present invention generally relates to the field of suspension systems, and more particularly relates to variable geometry suspension systems.

Background of the Invention Springs are used in many common applications such as in automotive suspension systems. Typically, a spring has an unchanging spring constant, which is a ratio of an applied external force to a resulting change in length of the spring.

Attempts at improving passive suspension designs have utilized one of three techniques:

adaptive, semi-active, and fully active suspension. An adaptive suspension utilizes a passive spring and an adjustable damper with slow response to improve the control of ride comfort and road holding. A semi-active suspension is similar, except that an adjustable damper has a faster response and a damping force is controlled in real-time. A fully active suspension replaces the adjustable damper with active elements, such as hydraulic, pneumatic and electromagnetic damper control, which can achieve optimum vehicle control, but at a high cost due to design complexity. As a result of research in semi-active suspension systems, a gap in capabilities between semi-active and fully active suspension systems has been narrowed. Semi-active suspension systems are widely used in the automobile industry due to their small weight and volume, as well as low energy consumption compared to fully active suspension systems. A semi-active suspension systems may include one or both of a fast response magneto-rheological (MR) damper and a fast response electro-rheological (ER) damper. ER and MR fluids comprise a suspension of polarized solid particles dispersed in a non-conducting liquid.

In known passive suspension systems, both a viscous damping coefficient and a spring rate of suspension elements are usually used as optimization arguments.

Theoretically, a semi-active suspension system involves both damping modulation and stiffness modulation. However, most practical implementations of semi-active suspension systems only control a viscous damping coefficient of a shock absorber while keeping stiffness constant. This is partly due to a relatively lower energy requirement for damping modulation. Another reason is due to unavailability of pragmatic low-power stiffness modulation methods. Furthermore, it has been shown that a combined variation of stiffness and damping achieves a better performance than a variation of damping or stiffness alone.

Semi-active suspension systems fall into a general class of variable damper, variable lever arm, and variable stiffness. Variable damper semi-active suspension systems are capable of varying the damping coefficients across their terminals. Initial practical implementations were achieved using a variable orifice viscous damper. By closing or opening the variable orifice, damping characteristics change from soft to hard and vice versa. With time, use of ER and MR fluids replaced the use of variable orifices. When an electric field (or a magnetic field for MR) is imposed, the polarized solid particles align along a direction of the electrical field. When this happens, a yield stress of the ER or MR fluid changes, hence a damping effect occurs.

Controllable rheological properties make ER and MR fluids suitable for use as smart materials for active devices, transforming electrical energy to mechanical energy.

Variable lever arm semi-active suspension systems conserve energy between the suspension and spring storage. Such suspension systems are characterized by controlled force variation which consumes minimal power. The main idea behind their operation is a variation of a force transfer ratio, which is achieved by moving a point of force application. If the point of force application moves orthogonally to an acting force, then, theoretically, no mechanical work is involved in the control. This phenomenon is called reciprocal actuation.

The variable stiffness feature of known semi-active suspension systems is achieved either by changing a free length of a spring or by a mechanism that changes a stiffness of the suspension system using one or more moving parts. In one known variable stiffness semi-active suspension system, a hydro-pneumatic spring with a variable stiffness characteristic is used. In another known variable stiffness semi-active suspension system, a desired stiffness variation is achieved by augmenting a variable lever arm type system with a traditional, or conventional, passive suspension system.

The control of semi-active suspension systems has been a subject of much research. An initial aim of a controlled suspension was solely centered on ride comfort. One of the initial control concepts developed is a sky-hook damper. The sky-hook damper is a fictitious damper between a sprung mass and an inertial frame (fixed in the sky). A damping force of the sky-hook damper reduces vibration of the sprung mass. A similar concept, called ground-hook, has also been developed for road-friendly suspension systems. These control concepts have also been applied to semi-active suspensions. Other control concepts that have been applied to semi-active and fully active suspension systems include: optimal control, robust control, and robust optimal control.

A known variable geometry actuator that reduces power consumption of active suspension elements is a 3D concept called a Delft Active Suspension (DAS). The DAS concept includes a wishbone that can be rotated over an angle and is linked to a pretensioned spring at a variable length. The spring pretension generates an effective actuator force at an end of the wishbone. A force is controlled by varying a position of the pretensioned spring via an electric motor. While the DAS has inherent variation in stiffness, it is disadvantageous because it may be a source of discomfort to a passenger in a vehicle and/or may lead to instability in an absence of a secondary spring. Also, there is a possibility of fail-safe issues as with most active suspension systems. A modified fixed spring design with optimized geometry, called electromechanical low-power active suspension (eLPAS), may overcome some challenges posed by the DAS concept, in that a moving mass is reduced. Packaging and stiffness variation issues were overcome by optimized geometry. However, ensuring a good fail-safe behavior remains an open problem.

A "suspension system" is a collective term given to system of springs, damper and linkages that isolates a vehicle body (sprung mass) from a wheel assembly (unsprung mass). An "independent suspension system" is a suspension system that has no linkages between two hubs of a same axle.

Summary of the Invention

In one embodiment, a variable stiffness mechanism is disclosed. The variable stiffness mechanism includes: a cantilever arm with one end rotatably attachable to a sprung mass, the cantilever arm including a receiving location attachable to an unsprung mass; a linear spring having an upper end attachable to a point of attachment on the sprung mass and having a lower end that engages the cantilever arm; and a device, attachable to the sprung mass, for relocating a position on the sprung mass of the point of attachment of the upper end of the linear spring. A stiffness of the variable stiffness mechanism depends on the position on the sprung mass of the point of attachment of the upper end of the linear spring.

In another embodiment, a variable stiffness mechanism for a land vehicle is disclosed. The variable stiffness mechanism for a land vehicle includes: a lower wishbone with one end rotatably attachable to a land vehicle and another end attachable to a wheel assembly; a horizontal rail attachable to the land vehicle; a sleeve that engages the horizontal rail by a plain bearing; a vertical strut having an upper end attached to the sleeve and having a lower end that engages the lower wishbone; and an actuator, attachable to the land vehicle, for relocating a position, along a length of the horizontal rail, of the sleeve. A stiffness of the variable stiffness mechanism depends on the position, along the length of the horizontal rail, of the sleeve.

In still another embodiment, a stabilization platform for a camera is disclosed. The stabilization platform for a camera includes a platform and a variable stiffness mechanism attached to the platform. The variable stiffness mechanism includes: a cantilever arm with one end rotatably attachable to the platform, the cantilever arm including a receiving location attachable to an unsprung mass; a linear spring having an upper end attachable to a point of attachment on the platform and having a lower end that engages the cantilever arm; and a device, attachable to the platform, for relocating a position on the platform of the point of attachment of the upper end of the linear spring. A stiffness of the variable stiffness mechanism depends on the position on the platform of the point of attachment of the upper end of the linear spring.

Brief Description of the Drawings

The accompanying figures, in which like reference numerals refer to identical or functionally similar elements throughout the separate views and which together with the detailed description below are incorporated in and form part of the specification, serve to further illustrate various embodiments and to explain various principles and advantages all in accordance with the present invention, in which:

FIG. 1 is a sketch of a variable stiffness mechanism concept.

FIG. 2 is a perspective view of a variable stiffness mechanism in accordance with one embodiment of the invention.

FIG. 3 is an illustration of a quarter car model of a passive case of one embodiment of the variable stiffness mechanism shown in FIG. 2.

FIG. 4 is a plot of car body acceleration for the passive case.

FIG. 5 is a plot of tire defection acceleration for the passive case. FIG. 6 is a graph of results of a drop test: car body acceleration for the passive case. FIG. 7 is a graph of results of a drop test: tire deflection acceleration for the passive case.

FIG. 8 is a graph of a time domain simulation: car body acceleration for the passive case.

FIG. 9 is a graph of a time domain simulation: suspension deflection for the passive case. FIG. 10 is a graph of a time domain simulation: tire deflection for the passive case.

FIG. 11 is a graph of a time domain simulation: control mass position for the passive case.

FIG. 12 is a graph of a frequency domain simulation: car body acceleration for the passive case.

FIG. 13 is a graph of a frequency domain simulation: suspension deflection for the passive case.

FIG. 14 is a graph of a frequency domain simulation: tire deflection for the passive case. FIG. 15 is an illustration of a model of an orthogonal nonlinear energy sink of an active case of a variable suspension mechanism.

FIG. 16 are graphs of variance gain for the active case.

FIG. 17 are graphs of frequency response: medium amplitude for the active case.

FIG. 18 are graphs of frequency response: high amplitude for the active case. FIG. 19 is an illustration of a quarter car model of an active case of one embodiment of the variable stiffness mechanism shown in FIG. 2.

FIG. 20 are graphs of car body acceleration for the active case.

FIG. 21 are graphs of suspension travel for the active case.

FIG. 22 are graphs of tire deflection for the active case. FIG. 23 are graphs of control mass displacement for the active case.

FIG. 24 are graphs of actuator forces for the active case.

FIG. 25 are graphs of variance gain: car body acceleration and rattle space displacement for a semi-active case of a suspension system. FIG. 26 is an illustration of a quarter car model of a semi-active case of one embodiment of the variable stiffness mechanism shown in FIG. 2.

FIG. 27 is an illustration of a nonparametric MR-damper model for a semi-active suspension system. FIG. 28 are graphs of polynomial approximation for the semi-active case. FIG. 29 is a graph of car body acceleration for the semi-active case. FIG. 30 is a graph of suspension travel for the semi-active case. FIG. 31 is a graph of tire deflection for the semi-active case. FIG. 32 is a graph of control mass displacement for the semi-active case. FIG. 33 are graphs of parameter estimates for the semi-active case. FIG. 34 are graphs of control currents for the semi-active case. FIG. 35 are graphs of MR damper forces for the semi-active case.

FIG. 36 is an illustration of a half car model of one embodiment of roll stabilization using the variable stiffness mechanism shown in FIG. 2. FIG. 37 is a modeling schematic for roll stabilization using the variable stiffness mechanism.

FIG. 38 is an illustration of a bicycle model for roll stabilization.

FIG. 39 is an illustration of an idealized half car model for roll dynamics modeling.

FIG. 40 is a schematic of a hydraulic system.

FIG. 41 is a graph of lateral tire force approximation. FIG. 42 are graphs showing results of a data collection experiment.

FIG. 43 are graphs of parameter estimation validation.

FIG. 44 are graphs of NTSHA fishhook maneuver.

FIG. 45 are graphs of fishhook responses. FIG. 46 are graphs of fishhook responses: comparison of cases.

FIG. 47 are graphs of voltage commands and spool valve displacements. FIG. 48 are graphs of actuator forces and power.

FIG. 49 is a front view of the variable stiffness mechanism shown in FIG. 2 at low stiffness. FIG. 50 is the front view of the variable stiffness mechanism shown in FIG. 2 at high stiffness and a tire.

FIG. 51 is the front view of the variable stiffness mechanism shown in FIG. 2 at high stiffness with the tire shown in FIG. 50.

FIG. 52 is a top view of the variable stiffness mechanism shown in FIG. 2 and a tire. FIG. 53 is the top view of the variable stiffness mechanism shown in FIG. 2 without the tire shown in Fig. 52.

FIG. 54 is a side view of the variable stiffness mechanism shown in FIG. 2 and a tire.

FIG. 55 is the side view of the variable stiffness mechanism shown in FIG. 2 without the tire shown in FIG. 54. FIG. 56 is a cut view of the variable stiffness mechanism shown in FIG. 2 and a tire.

FIG. 57 is the cut view of the variable stiffness mechanism shown in FIG. 2 without the tire shown in FIG. 56.

Detailed Description

As required, detailed embodiments of the present invention are disclosed herein. However, it is to be understood that the disclosed embodiments are merely examples of the invention, which can be embodied in various forms. Therefore, specific structural and functional details disclosed herein are not to be interpreted as limiting, but merely as a basis for the claims and as a representative basis for teaching one skilled in the art to variously employ the present invention in any appropriately detailed structure and function. Furthermore, the terms and phrases used herein are not intended to be limiting; but rather, to provide an understandable description of the invention.

The illustrations of examples described herein are intended to provide a general understanding of the structure of various embodiments, and they are not intended to serve as a complete description of all the elements and features of apparatus and systems that might make use of the structures described herein. Many other embodiments will be apparent to those of skill in the art upon reviewing the above description. Other embodiments may be utilized and derived therefrom, such that structural and logical substitutions and changes may be made without departing from the scope of this disclosure. Figures are also merely representational and may not be drawn to scale. Certain proportions thereof may be exaggerated, while others may be minimized. Accordingly, the specification and drawings are to be regarded in an illustrative rather than a restrictive sense. This disclosure describes research on the design, analysis, experimentation and application of a high efficient, low-power variable stiffness suspension system. The central concept is based on a recently designed variable stiffness mechanism which comprises a horizontal control strut and a vertical strut. The horizontal strut is used to vary the load transfer ratio by controlling the location of the point of attachment of the vertical strut to the car body. This movement is controlled either passively using the horizontal strut, actively using a hydraulic actuator, or semi-actively using a magneto-rheological (MR) damper. All the three cases are considered. The system is analyzed using an £ ₂ -gam analysis based on the concept of energy dissipation. The analyses, simulation, and experimental results show that the variable stiffness suspension achieves better performance than the constant stiffness counterpart. The performance criteria used are; ride comfort, characterized by the car body acceleration, suspension deflection, and road holding, characterized by tire deflection. Moreover, the variable stiffness architecture is used in the suspension system to counteract the body roll moment, thereby enhancing the roll stability of the vehicle. To this effect, the lateral dynamics of the system is developed using a bicycle model. The accompanying roll dynamics are also developed and validated using experimental data. The positions of the left and right control masses are optimally allocated to reduce the effective body roll and roll rate. Simulation results show that the resulting variable stiffness suspension system has more than 50% improvement in roll response over the traditional constant stiffness counterparts. The simulation scenarios examined are the fishhook and the double lane change maneuvers. A variable stiffness mechanism (1000) includes: a cantilever arm (1010) with one end (1005) rotatably attachable to a sprung mass (1020), the cantilever arm including a receiving location (1030) attachable to an unsprung mass (1035); a linear spring (1040) having an upper end (1045) attachable to a point of attachment (1050) on the sprung mass and having a lower end (1046) that engages the cantilever arm; and a device (1060), attachable to the sprung mass, for relocating a position on the sprung mass of the point of attachment of the upper end of the linear spring. A stiffness of the variable stiffness mechanism depends on the position on the sprung mass of the point of attachment of the upper end of the linear spring.

VARIABLE STIFFNESS SUSPENSION SYSTEM: PASSIVE CASE An overview of the design, analyses and experimentation of the passive case of the variable stiffness suspension system is now presented.

The design is based on the concept of a variable stiffness mechanism. The system is analyzed using an £ ₂ -gam analysis based on the concept of energy dissipation. The analyses, simulation, experimental results, show that the variable stiffness suspension achieves better performance than the constant stiffness counterpart. The performance criteria used are; ride comfort, characterized by the car body acceleration, suspension deflection, and road holding, characterized by tire deflection. The variable stiffness mechanism concept is shown in the Figures. The idea is to vary the overall stiffness of the system by letting d vary passively under the influence of a horizontal spring-damper system. The horizontal force can also be generated by either a semi-active or an active device.

An experiment was performed. It is a quarter car test rig scaled down to a ratio of 1: 10 compared to an average passenger car. The quarter car body is allowed to translate up-and- down along a rigid frame. This was made possible through the use of two pairs of linear motion ball-bearing carriages, with each pair on separate parallel guide rails. The guide rails are fixed to the rigid frame and the carriages are attached to the quarter car frame. The quarter car frame is made of 80/20 aluminum framing and then loaded with a solid steel cylinder weighing approximately 801bs. The horizontal and vertical struts are the 2011 Honda PCX scooter front suspensions. The road generator is a simple slider-crank mechanism actuated by Smartmotor SM3440D geared down to a ratio of 49: 1 using CMI gear head P/N 34EP049. Three accelerometers are attached, one each to the quarter car frame, the wheel hub, and the road generator. Data acquisition was done using the MATLAB data acquisition toolbox via NI USB-6251. Experiments were performed for the passive case, where the horizontal strut is just a passive spring-damper system, and also for the fixed stiffness case, where the top of the vertical strut is locked in a fixed position. This position is the equilibrium position of the unexcited passive case.

Two tests were carried out; sinusoidal, and drop test. For the sinusoidal test, the road generator is actuated by a constant torque from the DC motor. As a result, the quarter car frame moves up and down in a sinusoidal fashion. Here the ground is non-accelerating as against the sinusoidal test where the ground simulates the road signal. For the drop test, the suspension system was dropped to the ground from a fixed height (6 inches from the equilibrium position and the wheel was not in contact with the ground). The resulting quarter car body acceleration and tire deflection accelerations were recorded. This test examines the response of the system to initial conditions.

In order to study the behavior of the quarter car system at full scale as well as responses like suspension deflection, which were difficult to measure experimentally, and excitation scenarios that are difficult to implement experimentally, realistic simulations were carried out using

MATLAB SimMechanic, Second Generation. First, the system was modeled in Solidworks®. Next, the SimMechanic model was developed. The mass, vertical strut and tire damping and stiffness used are the ones given in the "Renault Megane Coupe" model.

In the time domain simulation, the vehicle traveling at a steady horizontal speed of 40mph was subjected to a road bump of height 8cm. The Car Body Acceleration, Suspension Deflection, and Tire Deflection responses were compared between the constant stiffness and the passive variable stiffness cases. For the constant stiffness case, the control mass was locked at three different locations (d = 40cm, d = 45.56cm and d = 50cm). The value d = 45.56cm is the equilibrium position of the control mass. Next, a simulation was performed for the passive case. A more detailed explanation of the passive case of the variable stiffness suspension system is now presented as follows.

This portion of the disclosure presents, in more detail, the passive case of the variable stiffness suspension system. This disclosure presents the design, analysis, and experimental validation of the passive case of a variable stiffness suspension system. A central concept is based on a recently designed variable stiffness mechanism. The variable stiffness mechanism comprises a horizontal control strut and a vertical strut. A main idea is to vary the load transfer ratio by moving a location of a point of attachment of a vertical strut to a car body. This movement is controlled passively using the horizontal strut. The system is analyzed using an £ ₂ -gam analysis based on the concept of energy dissipation. The analyses, simulation and experimental results show that the variable stiffness suspension achieves better performance than a constant stiffness counterpart. The performance criteria used are: ride comfort, characterized by the car body acceleration; suspension deflection; and road holding, characterized by tire deflection. Nomenclature

II v II Euclidean norm of the vector v

y _u Vertical displacement of the unsprung mass

y _s Vertical displacement of the sprung mass

K Half distance between points C and D

Is Vertical strut length

Natural length of vertical strut

Length of the lower wishbone

H Height of the control mass from the pivot point of the lower wishbone x Distance between points O and A along the lower wishbone

k _t , b _t Tire spring constant and damping coefficient

k _s , b _s Vertical Strut stiffness and damping coefficient

Control(Horizontal) Strut stiffness and damping

m _s , m _u , m _d Sprung, unsprung and control masses

Moment of inertia of control arm

The minimum eigenvalue of the matrix A

The maximum eigenvalue of the matrix A

The minimum singular value of the matrix A

σηιαχ } The maximum singular value of the matrix A

Ai:j,k l The sub-matrix of matrix A formed by rows i to j and columns k to I

A,j The sub-matrix of matrix A formed by rows i to j and all columns tr{A] The trace of the matrix A

det{A] The determinant of the matrix A

-Cs (91. 92) The set of points that lie on the line segment joining the vectors q ^r ₁ andq ^r ₂ / Identity matrix

ei,n The ith column of the identity matrix of dimension n

31 The set of real numbers

Re{ } The real part of the complex number

Improvements over passive suspension designs is an active area of research. Past approaches utilized one of three techniques: adaptive, semi-active or fully active suspension. An adaptive suspension utilizes a passive spring and an adjustable damper with slow response to improve control of ride comfort and road holding. A semi-active suspension is similar, except that an adjustable damper has a faster response and damping force is controlled in real-time. A fully active suspension replaces the damper with a hydraulic actuator, or another type of actuator such as an electromagnetic actuator, which can achieve optimum vehicle control, but at the cost of design complexity, expense, etc. The fully active suspension is also not fail-safe in the sense that performance degradation results whenever the control fails, which may be due to either mechanical, electrical or software failures. Recently, research in semi-active suspensions has continued to advance with respect to capabilities, narrowing the gap between semi-active and fully active suspension systems. Today, semi-active suspensions (e.g., using Magneto- Rheological (MR), Electro-Rheological (ER), etc.) are widely used in the automobile industry due to their small weight and volume, as well as low energy consumption compared to purely active suspension systems. However, most semi-active suspension systems are designed to only vary a damping coefficient of a shock absorber while keeping stiffness constant. Meanwhile, in suspension optimization, both the damping coefficient and a spring rate of the suspension elements are usually used as optimization arguments. Therefore, a semi-active suspension system that varies both the stiffness and damping of the suspension element could provide more flexibility in balancing competing design objectives. Suspension designs that exhibit variable stiffness phenomenon are few in literature considering the vast amount of researches that has been done on semi-active suspension designs. One type of a variable geometry actuator for vehicle suspension called the Delft active suspension (DAS). Although, the intention of the DAS design was not to vary the stiffness of the suspension system, the DAS design used a variable geometry concept to vary the suspension force by effectively changing the stiffness of the suspension system. The basic idea behind the DAS concept is based on a wishbone that can be rotated over an angle and is connected to a pretensioned spring at a variable location. The spring pretension generates an effective actuator force, which can be manipulated by changing the position. This was achieved using an electric motor. A variable stiffness suspension system that includes two springs connected in series in known. One of the springs is stiffer than the other. Under normal load conditions, the softer spring is responsible for keeping a good ride comfort. Upon the imposition of heavier load forces, the vehicle is supported more stiffly and primarily by the stronger spring. Conversion between the two conditions was done automatically by engagement under heavy load conditions of a pair of stop shoulders acting to limit the compression of the light spring. Similarly, upon excessive extension of the springs, an additional set of stop shoulders are engaged automatically to limit the amount of extension of the softer spring and causes the stiffer spring to resist further extension. A variable stiffness system has been proposed to suppress a building's responses to earthquakes, which had an aim to achieve a non- stationary and non-resonant state during earthquakes. Others have used an air spring in a suspension system to vary the stiffness among three discrete values. Still others have proposed a suspension system that uses two controllable dampers and two constant springs to achieve variable stiffness and damping. A Voigt element and a spring in series are used to control system stiffness. The Voigt element is comprised of a controllable damper and a constant spring. The equivalent stiffness of the whole system is changed by controlling the damper in the Voigt element.

This disclosure presents the design and analysis of the passive case of a variable stiffness suspension system. The variation of stiffness concept used herein utilizes "reciprocal actuation" to effectively transfer energy between a vertical traditional strut and a horizontal oscillating control mass, thereby improving the energy dissipation of the overall suspension. Due to the relatively few number of moving parts, the concept can easily be incorporated into existing traditional front and rear suspension designs. An implementation with a double wishbone is shown herein.

SYSTEM DESCRIPTION Variable Stiffness Concept

The variable stiffness mechanism concept is shown in Fig 1. The Lever arm OA, of length L, is pinned at a fixed point O and free to rotate about O. The spring AB is pinned to the lever arm at A and is free to rotate about A. The other end B of the spring is free to translate horizontally as shown by the double headed arrow. It is also free to rotate about point B. Without loss of generality, the external force F is assumed to act vertically upwards at point A. d is the horizontal distance of B from O. The idea is to vary the overall stiffness of the system by letting d vary passively under the influence of a horizontal spring-damper system (not shown in the figure). Let k and Z ₀ be the spring constant and the free length of the spring AB respectively, and Δ the vertical displacement of the point A. The overall free length Δ ₀ of the mechanism is defined as the value of Δ when no external force is acting on the mechanism.

Mechanism Description

The suspension system considered is shown in Fig 2. The schematic diagram is shown in Fig 3. The model comprises a quarter car body, wheel assembly, two spring-damper systems, road disturbance, and lower and upper wishbones. The points O, A and B are the same as shown in the variable stiffness mechanism of Fig 1. The horizontal control force u controls the position d of the control mass m _d which, in turn, controls the overall stiffness of the mechanism. The tire is modeled as a linear spring of spring constant k _t .

The assumptions adopted in Fig. 3 are summarized as follows:

1. The lateral displacement of the sprung mass is neglected, i.e., only the vertical displacement y _s is considered. 2. The wheel camber angle is zero at the equilibrium position and its variation is negligible throughout the system trajectory.

3. The springs and tire deflections are in the linear regions of their operating ranges. Equations of Motion

Let be defined as the generalized coordinates. The equations of motion, derived using Lagrange's method, are then given by

M(9)q + C(9, 9) + B(9)q - K(q) + G(9) e ₃ , ₃ " + W _d (0 d _r (2) where m _s + m _u + m _d m _u l _D cos6 0

M{9) m _u l _D cos9 I _c + m _u lp os ² 0 0

0 0 m _d

C(9, 9) = -m _u l _D 9 ² sm9w(9),

+ b _s g _e - g _de b _s 9d

, , _α 2l _A (d-l _A cos9)(dsin9-Hcos9)

9d6\ > J— _H 2 _+d 2 ₊ i^_ _2lAdcos g_ _2HlAsin g'

, , „ i¾(dsin0-/icos0) ²

H ² +d ² +l _A -2l _A dcose-2Hl _A sine' k _t (p _t - 1) ¾ + l _D sm6

K{q) = k _t (p _t - 1)/ _D cos0(y _s + l _D s e)

k _s (p _s — l)(d— l _A cos9)

G l _D cos9 r(t) is the road displacement signal. It is a function of the road profile and the vehicle velocity. The terms p _s andp _t characterize the compression of the vertical strut and tire springs respectively. They are defined as the instantaneous length divided by its free length.

Properties

The following properties of the dynamics given in (2) are exploited in subsequent analyses:

1. The inertia matrix M(9) is symmetric, positive definite. Also, because each element of M(9) can be bounded below and above by positive constants, it follows that the eigenvalues, hence the singular values of M(9) can also be bounded by constants. Thus, there exists m , m ₂ GR " ⁺ such that m^xW ² ≤x ^T M(9 x≤m ₂ \\x\\ ² and (3)

— V eS ² (4) 2. C(9, Θ) can be upper bounded as follows

Also, there exist a matrix V _m (9, Θ) such that C(9, Θ) = V _m (9, 9)q and

Ι ^τ (½(Θ)-¾Θ,Θ))Ϊ = 0, v e s ² (6) The property in (6) is the usual skew symmetric property of the Coriolis/centripetal matrix of Lagrange dynamics.

3. The damping matrix B (9) is symmetric and positive semi definite. Also, there exists positive definite matrices B_ and B such that

0 < x ^T Bx≤ χ ^τ Β {θ)χ < x ^T Bx, Vx G Έ ² . (7) 4. The stiffness vector K(q) is Lipschitz continuous, i.e., there exists a positive constant k ₂ such that

\\ K(q ₁ - K(q ₂ \\≤k ₂ \\ q ₁ - q ₂ \\. (8)

5. The unique static equilibrium point q ₀ = [y _So θ ₀ d ₀ ] ^T of the undisturbed system is known and is given by K(q ₀ ) - G(e ₀ ) + e _3>3 u ₀ = 0. (9)

SYSTEM ANALYSIS

A finite-gain stability analysis of the system described above is now presented. The disturbance d _r in (2) is assumed to be unknown a priori but bounded in the sense that d _r G £ ₂ . As a result, robust optimal control is considered in which the gain of the system is optimized under worst excitations: The following definition describes the notion of stability used in the subsequent analyses.

Consider a nonlinear system x = f x, w) z = h(x) (10) where x G l ⁿ , w G R " ^v , z G l ^m are the state, input, and output vectors respectively. The system in (10), with the mapping M _H : £ ^v _e → £™, is said to be finite-gain instable if there exist real constants y, β≥ 0 such that l|Af„(w) || _£ ≤y||w|| _£ + ? _< (11) where \\. \\ _£ denotes the £ norm of a signal, and £ is the extended £ space defined as £ϊ = {χ\χ _τ Ε £ ^η , \/τ Ε [0,∞)} (12) where χ _τ is a truncation of χ given as

For the purpose of this paper, the £ ₂ -space is considered, hence the finite-gain ^-stability condition in (11) is rewritten as l|Af„(w) || ₂ ≤Y\\w\\ ₂ + , (14) where ||. || ₂ denotes the £ ₂ norm of a signal given by

IMI ₂ = (J _o " X ^T {t) _X {t)dtf. (15)

7 ^* = inf{y| || _H (w) || ₂ < y||w|| ₂ + β] is the gain of the system, and, in the case of linear quadratic problems, is the H _∞ norm of the system. Given an attenuation level γ > 0, and the corresponding system dynamics, the objective is to show that (14) is satisfied for some β > 0. This solution is approached from the perspective of dissipative systems. The following definition describes the concept of dissipativity with respect to the system in (10).

The dynamics system (10) is dissipative with respect to a given supply rate s(w, z) G R " , if there exists an energy function V(x)≥ 0 such that, for all x(t ₀ ) = ^x o ^an d t ≥ t ₀ , V(x(t _f )) < V(x(t ₀ )) + J*' s(w, z)dt, Vw G £ ₂ . (16)

If the supply rate is taken as

S(W, Z) = Y ² II w II ² -II z II ² , (17) then the dissipation inequality in (16) implies finite-gain ^-stability, and the system is said to be y-dissipative. The dissipativity inequality is then written as V≤ γ w II ² - II z II ² . (18)

Performance Objective

As usual with suspension systems designs, the performance criterion is expressed in terms of the ride comfort, suspension deflection, and dynamic tire force. The performance vector z = (19) characterizes the ride comfort, suspension deflection, and road holding performances, where ω ₁₍ ω ₂ , anda>3 are the respective user specified performance weights for car body acceleration C a , suspension deflection y _st j, and dynamic tire force The ride comfort is characterized by the car body acceleration y _s which is approximated using the following high gain observer: εή = Αη + by _s , η 0 = 0

(20)

Ycbs — c ti

ε where

The £ ₂ - ^norm °f the car body acceleration can be upper bounded as llycba ll ₂ < c ₁ ||y _s || ₂ < c ₁ ||e|| ₂ , (21) where

2A¾ _lax (P)Hb|| ₂ l|c|| ₂

Ci =

^min O) and P is the solution of the Lyapunov equation PA + A ^T P + 1 = 0, which is obtained as

The suspension deflection is given as {d(0) ² - d(t) ² - 2tfx(sin0(O) - sin0(t))

-2x(d(O)cos0(O) - d(t)cos0(t))} ² (22)

\yo _s -y _s \

< [0 k ₄₁ k ₄₂ ] \Θ-Θ ₀ \ (23)

\d - d ₀ \

Using the Cauchy-Schwarz inequality, y _sd ( can be upper bounded as y _sd (t) < k ₄ II e II, (24) where / ₄₁ , k ₄₂ , and/ ₄ are positive constants, and k ₄ ≥ *Jk ₄₁ + k\

The dynamic tire force is characterized using the tire deflection and is given by o, ^~ y _s + ' _D (sin0 _o -sin^) (25) where k ₅ is a positive constant. Using the Cauchy-Schwarz inequality, _dt f( can be upper bounded as y _dtf < jl + k II e ||= k ₆ II e \\. (27)

Finally, the £ ₂ - ^norm °f ^tne performance vector in (19) can be upper bounded as ll z|| ₂ ≤0i ll e ll ₂ +02 H e ll ₂ (28) where

0 _! and

0 ₂ = ω ₂ Α: ₄ + ω ₃ /ο ₆ .

Constant Stiffness Case

Now, consider the constant stiffness case in which the control mass is locked at a given position d. As a result, the overall stiffness is constant for the entire trajectory of the system. For this case, the dynamics in (2) reduces to

M ₁ (9)q ₁ + 0^9, Θ) + B ₁ (9)q ₁ - K^q + G^O) = w, (29) where

Mi = Af _1:2 ,i _:2 , Ci = Cv.2,

Kl— Kl 2> Bl— ^1:2,1:2' ^an ^ w = W _di d _r , W _di = W _di:2:i:2

Here, the corresponding dynamics of the control mass has been eliminated.

Let ei = qi - q _0l (30) where yso

o ₁ = (3 D θο be the equilibrium value of the reduced state vector q _x . After using the Mean Value Theorem, the closed-loop dynamics in (29) is expressed as

M ₁ e ₁ + V _m e ₁ + K ₁ e ₁ + B ₁ e ₁ = w (32) where dG ₁

dq ₁ + dq ₁

<?i=<i ι=ζ

The matrix

is positive defi

Proof. Let λ be an eigenvalue of P. It follows that G S, since P is symmetric. The characteristic polynomial of P is given by ρ(λ) = det{AI - P) (34) det{(A - 1)(λΙ— M)— mil] (35)

Now, λ = 1 which implies that λ = 1 is not an eigenvalue of P. Suppose without loss of generality that λ≠ 1, then λ ² -λ- ρ(λ) = (A-l) ² det{ ^' ^'Ι - Μ] (36)

Thus, there exists an eigenvalue _m of M such that which implies that λ =

+ _m ± (l + _m ) ² - 4U _m - ml)), (38) from which it follows that λ > 0. Because P is symmetric, the conclusion follows.

It follows from Rayleigh-Ritz Inequality that

ViWxW ² ≤X ^T PX≤ P2\\X\\ ² , (39) where p ₁ = _min {P), and p ₂ = _max {P).

Theorem 1. If the matrix

-i?! - -^! ⁷ - m ₁ M^ ¹ B ₁

(40)

¹ 2 -K-^-m^M^B^ -2B ₁ where

(41) is negative definite along the entire trajectory of the closed-loop error system in (32), then the £ ₂ -norm of the performance vector in (19) can be upper bounded as l|z|| ₂ ≤7ilM| ₂ +/?i, (43) where

Pi ^ft i'

and = max{ _{1( 2} } (46) mi _x ¹

σ = σ„ (47)

Proof. Consider the energy function

V(e _i ,e _i =-y ^T ₁ P _Xi , (49) where

Taking time derivative of (49) and using the skew symmetric property in (6) yields

—m ₁ e M ₁ ¹ V _m e ₁ — m e[ M ₁ ⁱ B ₁ e ₁ — m ₁ e[ M ₁ ⁱ K ₁ e ₁ . (51) Using the property in (5) yields v m ₁ M ₁ ¹

≤χ ^τ ₁ Η ₁ χι + χ ^τ ι w, (52)

I

which after using the negative definiteness of H ₁ yields

^ < - *ιΙΙ ² + σ||*ιΙΙΙΜ|. (53)

Take W(t) = JVfa). When 7^)≠ 0, W = V/(2^JV) yields

(54)

2p ₂ 2 PT " When V( i) = 0, it can be verified that where D ⁺ denotes the upper right hand differentiation operator. Hence for all values of Next using comparison yields W(t)≤lV(0)exp (- £)

₊ ^j ||exp (-M=2) *. (57) which implies that + ^ _J || _w ||exp (- ^) _dT . (58)

Thus

Lastly, after using the inequality in (28), the L ₂ -noxm of the performance vector can be upper bounded as

The £ ₂ -g ^am °f the system decreases with increasing h-^. This means that the more the negative definiteness of H _{l t} the more the disturbance rejection achievable by the system.

The following theorem gives the bounds on achievable 7.

Theorem 2. Given an attenuation level γ, and provided that the performance weights are selected to satisfy the sufficient condition = max{ ₁₍ Φ2} < (60) then the closed loop error system in (32) is y-dissipative with respect to the supply rate s(w, z) = y ² II w II ² -|| z II ² (61) if γ ^≥ τ^ - ⁽⁶²⁾ Proof. Consider the energy storage function in (49). Taking first time-derivate, and adding and subtracting the supply rate yields

V≤ χ ^τ Η _ί χ + x ^T Lw ≤Y ² \\w\\ ² - \\z\\ ² + x ^T H _lX

-γ ² w—

2γ ² I + ^X ^T LL ^T x + <p ² \\x\\ ² < 7 ² ||w|| ² - llzll ² + X ^t (H ₁ + (φ ² + ) l) x

≤ Y ² \\w\\ ² - ||z|| ² - (h, - Φ ² - ^) llzll ² (63) After using the inequality in (62)

V≤ y ² II w II ² -II z || ² , (64) which implies that the closed loop error system in (32) is y-dissipative. The inequality in (62) shows that the level of performance achievable is limited by the amount of damping and stiffness available in the system. It is shown hereinbelow that this limit can be pushed further by using a variable stiffness architecture. The lower bound in (62) is termed "best-case-gain". It defines the smallest gain achievable by the system.

The stiffness and damping matrices K _{l t} and B ₁ contain bounded functions of state and uncertain dynamic parameters which range between bounded values. Thus the best-case gain of the system with constant stiffness can be lower bounded as

(65) where is the smallest positive number larger than the smallest singular value of H ₁ , and γ _ί is termed the "robust best-case gain".

Passive Variable Stiffness Case Additional Nomenclature eig{A] Set of the eigenvalues of matrix

A U Union of sets

Π Intersection of sets

Here, the control mass is allowed to move under the influence of a restoring spring and damper forces. There is no external force generator added to the system. As a result, the system response is purely passive. Let k _u and b _u be the spring constant and damping coefficient of the restoring spring and damper respectively. The control force u is then given by u = -b _u d - k _u (d - l _0d ), (66) and the resulting dynamics of the control mass is given by m _d d + b _u d + k _u (d— l _0d ) + k _s (p _s — l) (d— xcos9)

+ ^ g _M e + b _s g _d d = 0, (67) and the static equilibrium equation for the control mass is given by (d ₀ - l _0d ) + k _s (p _So - l)(d ₀ - xcos0 _o ) = 0, (68) where d ₀ is the equilibrium position of the control mass, and l _0d is the free length of the restoring spring. Let e _d = d - d ₀ (69) be the displacement of the control mass from its equilibrium position. Substituting (69) into (67) and using the Mean Value Theorem yields m _d e _d + B _d ^T e + K _d ^T e = 0, (70) where e = £] ^■ < ⁷¹ >

Now, consider the energy function

V ₂ (e,e) =.χ ₂ 'Ρ ₂ χ ₂ , (74) where, and

I ml

(76) ml M - is positive definite, with m ² < _min {M}. Taking the first time derivative of (74), and following a similar procedure as followed in the constant stiffness case yields

V ₂ ≤Y ² \\w\\ ² -\\z\\ ² +χ ^τ ₂ Η ₂ χ _2ι (77) where

K = mM- ¹ K-- ^c ^I, (79) and

Now, the robust best-case gain of the system with a passive variable stiffness is given by 0.5σ

(82) where h ₂ ^* is the smallest positive number larger than smallest singular value of H ₂ . Here, the spring constant k _u , and the damping coefficient b _u of the control mass restoring spring-damper system can be chosen such that γ ₂ < Y _\ . Thus, a better performance can be achieved just by letting the stiffness vary naturally using a spring-damper system. This assertion is supported subsequently by experimental and simulation results. This is a very appealing result due to its practicability. No additional electronically controlled or force generating device is required, only mechanical elements like the spring and damper are used.

EXPERIMENT An experimental setup was constructed. The experimental setup comprised a quarter car test rig scaled down to a ratio of 1:10 compared to an average passenger car in the year 2004. The quarter car body is allowed to translate up-and-down along a rigid frame. This was made possible through the use of two pairs of linear motion ball-bearing carriages, with each pair on separate parallel guide rails. The guide rails are fixed to the rigid frame and the carriages are attached to the quarter car frame. The quarter car frame is made of 80/20 aluminum framing and then loaded with a solid steel cylinder weighing approximately 801bs. The horizontal and vertical struts are the 2011 Honda PCX scooter front suspensions. The road generator is a simple slider-crank mechanism actuated by Smartmotor SM3440D geared down to a ratio of 49:1 using CMI gear head P/N 34EP049. Three accelerometers are attached, one each to the quarter car frame, the wheel hub, and the road generator. Data acquisition was done using the MATLAB data acquisition toolbox via NI USB-6251. Experiments were performed for the passive case, where the horizontal strut is just a passive spring-damper system, and also for the fixed stiffness case, where the top of the vertical strut is locked in a fixed position. This position is the equilibrium position of the passive case when the system is not excited. Two tests were carried out; sinusoidal, and drop test. For the sinusoidal test, the road generator is actuated by a constant torque from the DC motor. As a result, the quarter car frame moves up and down in a sinusoidal fashion. To facilitate a good comparison of the observations, the "approximate gain" of the system defined as where z(t) is the signal of interest, and r(t) is the road acceleration signal, is numerically computed. The signals of interest are the frame acceleration and tire deflection acceleration signals. The experimental procedure was repeated multiple times in order to verify the repeatability of the experiment. Figures 4 and 5 show the box plots of the approximate gain distributions for the fixed stiffness and passive variable stiffness cases. It is seen that the worst and best case gains for the fixed stiffness are higher than those of the passive variable stiffness case, thereby confirming the analytical result obtained earlier that the variable stiffness achieves better dissipation.

For the drop test, the suspension system was dropped to the ground from a fixed height (6 inches from the equilibrium position and the wheel was not in contact with the ground). Here the ground is non- accelerating as against the sinusoidal test where the ground simulates the road signal. The resulting quarter car body acceleration and tire deflection accelerations were recorded. This test examines the response of the system to initial conditions. Figures 6 and 7 show the car body acceleration responses and tire deflection acceleration responses for the fixed and variable stiffness cases. Table 1 shows the approximate gains for the sinusoidal and the rms values of the drop test. The approximate gains of the sinusoidal test given in the table are the mean values of the multiple experiments.

Table 1: RMS/APPROXIMATE GAIN VALUES OF EXPERIMENTAL RESULTS CBA: Car Body Acceleration. TDA: Tire Deflection Acceleration

Fixed Passive

*Drop (RMS) 0.4543 0.3710 0.2746 0.2396

*Sinusoidal (G 0.6220 0.5170 1.3316 1.2944

Simulation

In order to study the behavior of the quarter car system at full scale as well as responses like suspension deflection, which were difficult to measure experimentally, and excitation scenarios that are difficult to implement experimentally, realistic simulations were carried out using MATLAB SimMechanics Second Generation. First, the system was modeled in Solidworks®. Next, the SimMechanics model was developed. The mass/inertia properties used are the ones generated from the Solidworks model. The vertical strut and tire damping and stiffness used are the ones given in a "Renault Megane Coupe" model. The values are given in Table 2.

Table 2: DYNAMIC PARAMETER VALUES

Parameter Value m _s 315 kg

™u 37.5 kg b _s 1500 N/m/s k _s 29500N/m k _t 210000 N/m

Time Domain Simulation In the time domain simulation, the vehicle traveling at a steady horizontal speed of 40mph is subjected to a road bump of height 8cm. The Car Body Acceleration, Suspension Deflection, and Tire Deflection responses are compared between the constant stiffness and the passive variable stiffness cases. For the constant stiffness case, the control mass was locked at three different locations (d = 40cm, d = 45.56cm and d = 50cm). The value d = 45.56cm is the equilibrium position of the control mass. Next, a simulation is performed for the passive case. The results are reported in Figures 8, 9 and 10 which are the the car body acceleration, suspension deflection, and tire deflection responses, respectively. Figure 11 shows the position history of the control mass for the passive variable stiffness case.

Frequency Domain Simulation For the frequency domain simulation, an approximate frequency response from the road disturbance input to the performance vector given in (19), is computed using the notion of variance gain. The approximate variance gain is given by GO) = (84)

^J ,4 ² sin ² (a t) dt where z denotes the performance measure of interest which is taken to be car body acceleration, suspension deflection, and tire deflection. The closed loop system is excited by the sinusoid r = i4sin(a>t), t £ [0, 2πΝ/ω] , where N is an integer big enough to ensure that the system reaches a steady state. The corresponding output signals were recorded and the approximate variance gains were computed using (84). Figures 12, 13 and 14 show the variance gain plots for the car body acceleration, suspension deflection, and tire deflection respectively. The figures show that the variable stiffness suspension achieves better vibration isolation in the human sensitive frequency range (4-8Hz), and better handling beyond the tire hop frequency (>59Hz). Using a detailed £ ₂ -gam analysis based on the concept of energy dissipation, it is shown that inclusion of a variable stiffness mechanism in the suspension design yields an improvement in the performance of the traditional system in terms of ride comfort, suspension deflection, and road holding. The analysis claims are supported by both experimental and simulation results.

The description of the passive case of the variable stiffness suspension system has now been disclosed.

It is foreseeable that the passive spring/damper system will be replaced with a semi-active element such as the MR damper. It is also foreseeable to use nonlinear passive elements in the horizontal strut. It is also foreseeable to use nonlinear paths for the control mass as well.

VARIABLE STIFFNESS SUSPENSION SYSTEM: ACTIVE CASE An overview of the design, analyses and experimentation of the active case of the variable stiffness suspension system is now presented.

In one embodiment, the horizontal strut used in the passive case is replaced with a force generator comprising hydraulic or pneumatic actuators. The horizontal strut is used to vary the load transfer ratio by actively controlling the location of the point of attachment of the vertical strut to the car body. The control algorithm, effected by a hydraulic actuator, uses the concept of nonlinear energy sink to effectively transfer the vibrational energy in the sprung mass to a control mass, thereby reducing the transfer of energy from road disturbance to the car body at a relatively lower cost compared to the traditional active suspension using the skyhook concept. The analyses and simulation results showed that a better performance can be achieved by subjecting the point of attachment of a suspension system, to the chassis, to the influence of a horizontal nonlinear energy sink system.

Nonlinear Energy Sinks (NES) are essentially nonlinear damped oscillators which are attached to a primary system (this refers to the main system whose vibration is intended to be absorbed) for the sake of vibration absorption and mitigation. Such attachments have been used extensively in engineering applications, particularly in vibration suppression or aeroelastic instability mitigation. The motivation for the use of NES is primarily due to their proven capability to achieve one-way irreversible energy pumping from the linear primary system to the nonlinear attachment. A goal, therefore, was to achieve a one-way irreversible energy pumping of the road disturbance to the secondary system whose vibration is orthogonal to the car body motion. A fairly general nonlinear function was used in this work, instead of cubic nonlinearity that is generally used.

The control development was done using a Lyapunov based adaptive method. First, the error dynamics was reduced using time scale decomposition and Tichonov's Theorem. Next, the update law was designed, and the proof of stability of the error dynamics given using Lyapunov technique. The resulting control and update laws are summarized below:

Desired Force (NES) Fd {IQ ₄ — d) - k-2 si»h(a(¾ _a - d}) - ¾<

Tracking Error e

U date Law Θ = -FYe

Fictitious Control S =

Slow Control ¾½ Final Control u =

Similar to passive case, the simulation models were developed using MATLAB SimMechanic, second generation. Also, another very interesting result obtained from this work is that, by designing the active suspension system this way, the power requirement was cut down by 40%. This is because the direction of actuation is nearly orthogonal to the direction of excitation.

There are Constant Stiffness Passive (CSP), Constant Stiffness Active (CSA), Variable Stiffness Passive, and Variable Stiffness NES (NES) cases. For the CSA case, the vertical strut is replaced by a hydraulic actuator, controlled to track the skyhook damping force. more detailed explanation of the active case of the variable stiffness suspension system is now presented as follows.

This portion of the disclosure presents, in more detail, the active case of a variable stiffness suspension system. The central concept is based on a recently designed variable stiffness mechanism which comprises a horizontal control strut and a vertical strut. The horizontal strut is used to vary the load transfer ratio by actively controlling the location of the point of attachment of the vertical strut to the car body. The control algorithm, effected by a hydraulic actuator, uses the concept of nonlinear energy sink to effectively transfer the vibrational energy in the sprung mass to a control mass, thereby reducing the transfer of energy from road disturbance to the car body at a relatively lower cost compared to the traditional active suspension using the skyhook concept. The analyses and simulation results show that a better performance can be achieved by subjecting the point of attachment of a suspension system, to the chassis, to the influence of a horizontal nonlinear energy sink system.

Nomenclature y _u Vertical displacement of the unsprung mass

y _s Vertical displacement of the sprung mass

Is Vertical strut length

Natural length of vertical strut

H Height of the control mass from the pivot point of the lower wishbone k _t , b _t Tire spring constant and damping coefficient

k _s , b _s Vertical strut stiffness and damping coefficient

Control (horizontal) strut stiffness and damping

m _s , m _u , m _d Sprung, unsprung and control masses

IMI The Euclidean norm of the vector v

31 The set of real numbers

A goal of a vehicle suspension system is to isolate the vehicle from road disturbances while keeping good contact with the road. An ideal suspension should be able to minimize the car body accelerations, dynamic tire forces, and energy consumption while satisfying the constraints imposed on the rattle space. A traditional passive suspension comprises appropriately selected spring, or spring set, and viscous dampers. The characteristics of these passive elements are selected in order to enhance the ride comfort and road holding performances. However, these are competing objectives due to their complementary behavior across the frequency spectrum. This is a major drawback affecting the design of passive suspensions. Improvements over passive suspension designs is an active area of research. Past approaches utilize one of three techniques; adaptive, semi-active or fully active suspension. An adaptive suspension utilizes a passive spring and an adjustable damper with slow response to improve the control of ride comfort and road holding. A semi-active suspension is similar, except that the adjustable damper has a faster response and the damping force is controlled in real-time.

On the other hand, Nonlinear Energy Sinks (NES) are essentially nonlinear damped oscillators which are attached to a primary system for the sake of vibration absorption and mitigation. The primary system refers to the main system whose vibration is intended to be absorbed. Such attachments have been used extensively in engineering applications, particularly in vibration suppression or aeroelastic instability mitigation. The vibration of systems with essential

(strongly or weakly) coupled nonlinearity has been studied extensively in literature. It has been shown in that such attachments can be designed to act as a sink for unwanted vibrations generated by external impulsive excitations. The underlying dynamical phenomenon governing the passive energy pumping from a primary vibrating system to the attached nonlinear energy sink has be shown to be a resonance capture on a 1:1 manifold. It has been shown that under certain conditions, vibration energy gets passively pumped from directly excited primary system to the nonlinear secondary system in a one-way irreversible fashion. Nonlinear passive absorbers can be designed with far smaller additional masses than the linear absorbers, thanks to the energy pumping phenomenon. This corresponds to a controlled one-way channeling of the vibration energy to a passive nonlinear structure where it localizes and diminishes in time due to damping dissipation. This allows nonlinear energy pumping to be used in coupled mechanical systems, where the essential nonlinearity of the attached absorber enables it to resonate with any of the linearized modes of the substructure.

Recently, the concept of "reciprocal actuation" was used to design a variable stiffness suspension system for isolating a car body from road disturbance. The system is essentially a passive vibration isolation system in which the motion of the secondary linear attachment is made orthogonal to the primary system. The primary and secondary systems are coupled through the traditional suspension system. In this paper, the above concept is extended by using an active linear generator, controlled to mimic a nonlinear energy sink, to drive a orthogonal secondary system. The orthogonal secondary system refers to a vibration absorber or isolator. The motivation for the use of NES is primarily due to their proven capability to achieve one-way irreversible energy pumping from the linear primary system to the nonlinear attachment. The goal therefore is to achieve a one-way irreversible energy pumping of the road disturbance to the secondary system whose vibration is orthogonal to the car body motion. A fairly general nonlinear function is used in this work, instead of cubic nonlinearity that is generally used.

SYSTEM DESCRIPTION

Variable Stiffness Concept

The variable stiffness mechanism concept is shown in Fig 1. The lever arm OA, is pinned at a fixed point O and free to rotate about O. The spring AB is pinned to the lever arm at A and is free to rotate about A. The other end B of the spring is free to translate horizontally as shown by the double headed arrow. Without loss of generality, the external force F is assumed to act vertically upwards at point A. d is the horizontal distance of B from O. An idea is to vary the overall stiffness of the system by letting d vary actively under the influence of a horizontal linear force generator (not shown in the figure). The variables k and Z ₀ denote the spring constant and the free length of the spring AB respectively, and Δ the vertical displacement of the point A. The overall free length Δ ₀ of the mechanism is defined as the value of Δ when no external force is acting on the mechanism.

Orthogonal Nonlinear Energy Sink Fig. 15 shows the orthogonal nonlinear energy sink (NES) considered in this deisclosure. The term orthogonal NES is used to describe the concept because the direction of motion of the secondary system is orthogonal to the primary system. This is suitable, structurally, for the application considered in this paper. The direction of motion of the secondary system is made orthogonal to the primary system for the following reasons: it makes the system conformable to the concept of the variable stiffness mechanism described above, and it prevents direct gravity compensation by the actuator, thereby reducing the power consumption of the system. The subsystems 5 ₁₍ C and S ₂ constitute the primary subsystem, and are allowed to slide vertically together as a unit of total sprung mass m _s + m _d . The subsystem C is termed the control mass (or control subsystem). It, together with the nonlinear spring and the dashpot of damping coefficient b _d , constitute the secondary subsystem. The nonlinear function is defined as

F = g(d) = -/£i(/ _0d - d - - d)), (lb) where l _0d is the free length of the idealized nonlinear spring. The nonlinear function used is fairly more general compared to the pure cubic nonlinearity that have been used in the past. The Taylor series expansion is fc _2i _ ₁ (a ₁ ) > 0 (2b) from which it is seen that the considered nonlinearity contains all the odd powers of the deflection l _0d — d. The mass labeled U is the unsprung mass, whose displacement y _u is used as the source of disturbance to the system.

An approximate frequency response from the input y _u to the sprung mass acceleration y _s and the rattle space deflection y _s —y _u , is computed using the notion of variance gain. The approximate variance gain is given by

/ ₀ ^2πίν/ω ζ2 at

GO) = where z denotes the performance measure of interest (sprung mass acceleration and rattle space deflection in this case). The system is excited by the sinusoid r = i4sin(a>t), t £ [0, 2πΝ / ω] , where N is an integer big enough to ensure that the system reaches a steady state. The corresponding output signals were recorded and the approximate variance gains were computed using (3b). The system parameter values used are given in Table 1. The resulting variance gain responses are shown in Figs. 15 and 16 for the constant stiffness suspension (CSS), the variable stiffness suspension with linear energy sink (VSS:LES), and the variable stiffness suspension with nonlinear energy sink (VSS:NES). For a traditional constant stiffness suspension , the position of the control mass is fixed. For the variable stiffness suspension with linear energy sink, the control mass is allowed to move under the influence of a linear horizontal spring and damper. The figures show that the variable stiffness suspension achieves better vibration isolation, with a significant improvement from the linear energy sink case to the nonlinear energy sink case. As shown in Fig. 16, the improvement gained in vibration isolation results in a corresponding performance degradation in the rattle space deflection. However, when compared with the improvement in the vibration isolation, there is an overall improvement in performance associated with the use of the variable stiffness suspension with nonlinear energy sink. This agrees with the usual trade-off in suspension design. The performance improvement can further be increased by transitioning from LES in low frequency range (< 8Hz) to NES in high frequency range (> 8Hz). In general, as pointed out by one of the reviewers, the frequency characteristics of nonlinear systems vary with the amplitude of the input signal. As a result, in order to show that the above observation is preserved with varying amplitude of the input signal, two additional frequency characteristics are shown for two different amplitudes of the input signal: one corresponding to medium amplitude (A = 5cm), and the other corresponding to high amplitude (A=10cm). Active Variable Stiffness Suspension System

The quarter car model of the suspension system considered is shown in Fig 19. It comprises a quarter car body, wheel assembly, two spring-damper systems, road disturbance, and lower and upper wishbones. The points O, A and B are the same as shown in the variable stiffness mechanism of Fig 1. The horizontal control force, exerted by the hydraulic actuator H, controls the position d of the control mass m _d which, in turn, controls the overall stiffness of the mechanism. The control force is designed (see below) to mimic the orthogonal NES introduced hereinabove. The tire is modeled as a linear spring of spring constant k _t .

The assumptions adopted in Fig. 19 are summarized as follows: 1. The lateral displacement of the sprung mass is neglected, i.e., only the vertical displacement y _s is considered.

2. The wheel camber angle is zero at the equilibrium position and its variation is negligible throughout the system trajectory.

3. The springs and tire deflections are in the linear regions of their operating ranges. Control Masses and Actuator Dynamics

The hydraulic system comprises a source of hydraulic pressure, a spool valve, and a hydraulic cylinder, as shown in Fig. 19. The hydraulic pressure is supplied by a hydraulic pump (not shown ), which is typically augmented with accumulators to reduce pressure fluctuations and supply additional fluid for peak demands. The hydraulic cylinder is a double acting cylinder. The piston motion is obtained by modulating the oil flow into and out of the cylinder chambers, which are connected to the spool valve through cylindrical ports. The modulation is proved by the spool valve. The dynamics of the hydraulic actuator, as well as the spool valve. aAv _p — βΡι + γχ _ν ]Ρ ₅ — sgn(x (4b)

F = AP, (6b) where A is the pressure area in the actuator, P _L is the load pressure, v _p = d is the actuator piston velocity, F is the output force of the actuator, , β and γ are positive parameters depending on the actuator pressure area, effective system oil volume, effective oil bulk modulus, oil density, hydraulic load flow, total leakage coefficient of the cylinder, discharge coefficient of the cylinder, and servo valve area gradient, x _v is the spool valve position, τ is the actuator electrical time constant, K is the DC gain of the four-way spool valve, and u is the input current to the servo valve. Control Development

In terms of the output force, F, exerted by the actuator, the actuator dynamics in (4b) is written as

F = - F - αΑ ² ά + γΑΰ, (7b) where u = x _v ^P _s - sgn(x _v ) - ^F (8b) is a fictitious control variable, from which the slow component (or envelop) of the control is obtained, after singular perturbation of the valve dynamics. Let the actuator force tracking error be defined as e = F - F _d , (9b) where

Pa = ~ ^k i( ^l o _d - d) - k ₂ sm (a(l _0d - d)) - b _d d, (10b) is the desired force to be tracked by the actuator force dynamics in (7b). Taking the derivative of (9b) yields the actuator force tracking error dynamics e = - F - aA ² d + γΑΰ - F _d (1 lb)

= -pe - F _d + YA(u - Y ^T &), (13b) where the regression matrix Y and the unknown parameter vector Θ are given by

^Y = [d F _d F _d ] (14b) Ητ £ rl < ^15b > and F _d is an approximation of the desired force F _d obtained using a high gain observer

¾P = ^A hgV + b _hg F _d (16b) = sat Q- cl _g p, a, b) (17b) where the saturation function sat(.. . ) is given by and

^A 9 = [Z{ I] · ^b hg = [¾ . _hg = [J] , ε ₂ « 1.

This is done because , as can be seen in (10b), F _d contains the unmeasurable signal d. It can be shown that the estimation error, F _d = F _d — F _d decays, in the fast time scale, to the ball \F _d \ < 0(ε ₂ ). The saturation function is used to overcome the peaking phenomenon associated with high gain observers. The fictitious control u is then designed as follows u = Y ^T Q - k e - CiSgnCe), (19b) where Θ is an adaptive estimate of Θ, the adaptation law will be designed hereinbelow, and k^nd c ₁ are control gains. Thus the closed loop error dynamics, obtained by substituting (19b) into (13b), is given by έ = -(β + k yK)e - F _d - c^sgnCe) - γΑΥ ^τ Θ, (20b) where the parameter estimation error Θ is given by ø = ø— ø. (21b) In order to simplify the controller design for the actuators, the spool valve dynamics is reduced, using a singular perturbation technique. The control input is designed as u =—KfX _v + ^1+KKf u _si (22b) where u _s is a slow control in time and Kf is a positive design control gain. Consequently, the valve psuedo-closed loop dynamics is given by εχ _ν + x _v = u _s , (23b) where

(24b) l+KK _f is the perturbation constant. The pseudo-closed loop in (23b) has a quasi-steady state solution, ^x _v = 0) = x _v ., given by x _v = u _s . (25b)

Using the fast time scale v = ^ and Tichonov's Theorem, the valve dynamics is decomposed into fast and slow time scales as follows x _v = x _v + η + 0 (ε), (26b) £ = -,, (27b) where η (v) is a boundary layer correction term. It is seen that η (v) decays exponentially in the fast time scale. Typically, the time constant τ in the actual system is designed to satisfy 0 < ε « 1. Therefore, by choosing the control gain Kf large enough, the perturbation constant can be made as small as possible. As a result, η + 0 (ε) becomes negligibly small, and the fictitious control becomes

Assuming sufficient pressure for the hydraulic pump, the term inside the square root operator is taken as positive. Thus sgn(iZ) = sgn(u _s ), (29b) which implies that

1

u _s = U (p _s - sgn(u) ) ² . (30b) Stability Analysis

A Lyapunov-based stability analysis of the closed loop error dynamics in (20b) is now presented. The adaptation law for the parameter estimation is designed. It is also shown that if the control gains are chosen to satisfy certain sufficient conditions, then the actuator force tracking error will approach zero asymptotically. Given the adaptive update law g = -TYe, (31b) where Γ is a positive definite adaptation gain matrix. If the control gain c ₁ is chosen to satisfy the following sufficient conditions

> J£i > Jf d, (32b)

¹ yA yA ' then the actuator tracking error in (9b) approaches zero asymptotically, i.e., e(t)→ 0, ast→ oo. Proof. Consider the following positive definite Lyapunov function candidate y = I _e ² + ^ 0 ⁷ T- ¹ 0. (33b) Taking the first time derivative and substituting the closed loop error dynamics in (20b) yields V = ee - γΑψΥ ^'1 ^ (34b)

= e (-(/? + k _x yA)e - F _d - c^sgnCe) - γΑΥ ^τ θ} - γΑ Τ ^'1 ^, (35b) which, after applying update laws, becomes

Ϋ≤-(β + k _lY A)e ² + \F _d \ \e\ - c _lY A\e\. (36b) Using the sufficient condition in (32b), the inequality in (36b) yields V≤ -(/? + k yti)e ² < 0. (37b)

(33b) and (37b), it follows that V(t) is bounded, which also implies that e(t), and0(t) are bounded. Using the boundedness of F _d (t), from (17b), it follows from (20b) that e(t) is bounded, which implies that the signal e(t) is uniformly continuous. Integrating (37b) yields

t _→∞ ^{J J} β+krfA °° '

Thus, using Barbalat's Lemma, it can be shown that e(t)→ 0, ast→ oo.

Simulation

In order to study the behavior of the quarter car system to different road excitation scenarios, as well as measure responses such as suspension deflection, realistic simulations were carried out using MATLAB SimMechanics. First, the system was modeled in Solidworks®, and then translated to a SimMechanic model. The vertical strut and tire damping and stiffness used are the ones given in the "Renault Megane Coupe" model. The values are given in Table 1. The values of the hydraulic parameters were obtained empirically, and are given in Table 2.

Table 1: DYNAMIC PARAMETER VALUES

Parameter Value m _s 315 kg

™u 37.5 kg

™-d 5 kg b _s 1500 N/m/s b _d 1500N /m/s k _s 29500N/m k _t 210000 N/m k, 5000N/m k ₂ 15000N/m lo _d 0.25m

0.5m 5mm a 2

Table 2: HYDRAULIC PARAMETER VALUES

In the simulation, the vehicle traveling at a steady horizontal speed of 40mph is subjected to a road bump of height 10cm. The Car Body Acceleration, Suspension Travel, and Tire Deflection responses are measured. The suspension travel is defined as the vertical distance between the centers of mass of the sprung and unsprung masses, and the tire deflection as the difference between the center of mass of the unsprung mass and the road height. Simulations were carried out for the constant stiffness and the variable stiffness suspension systems. For the constant stiffness suspension, the control mass was locked at a fixed position corresponding to the equilibrium position of the control mass for the variable stiffness system. Moreover, for each stiffness type, both passive and active cases were considered. The passive case of the constant stiffness suspension is the traditional passive suspension, while in the active case, the passive spring damper is replaced with a hydraulic actuator controlled to track a skyhook suspension force. The skyhook concept stipulates that the active suspension force be designed to track the force produced by a fictitious damper attached between the sprung mass and the inertial frame. A fictitious damper between the sprung mass and the inertial frame is called the skyhook damper. On the other hand, the passive case of the variable stiffness suspension corresponds to the LES, while the active case corresponds to the NES. The results obtained are reported in Figures 20-24. Table 3 shows the variance gains for the different responses. Fig. 20 shows the car body acceleration, which is used here to describe the ride comfort. The lower the car body acceleration, the better the ride comfort. As seen in the figure, the NES is the most "ride friendly" suspension, outperforming the skyhook control. As shown in Fig. 20, associated with this improvement is a corresponding degradation in the suspension travel. This agrees with the observation made hereinabove, as well as the well know tradeoff between ride comfort and suspension deflection. Fortunately, the degradation in suspension deflection is not as much as the improvement gained in the ride comfort, resulting in an overall better performance.

Moreover, the suspension travel performance can be improved by designing a gain scheduled controller, using an observed frequency of the sprung mass as the scheduling variable. As a result, the NES can be turned on and off depending on the frequency, as described previously. Fig. 23 shows the position history of the control mass for the variable stiffness suspension, from which the boundedness of the motion of the control mass is seen. The maximum displacement of the control mass from the equilibrium position is about 7cm. This implies that the space requirement for the control mass is small, which further demonstrates the practicality of the system. Fig. 22 shows that there is no significant reduction in the tire deflection. Thus, the suspension systems are approximately equally "road friendly". It is also seen, in Fig. 24, that the hydraulic force from the NES is about 60% of that from the skyhook counterpart. This translates to a lower power requirement for the proposed system.

Table 3: VARIANCE GAIN VALUES

Constant Constant Variable Variable Stiffness Stiffness Stiffness Stiffness Passive Active Passive NES

Car Body Acceleration 109.0389 64.2818 65.6127 42.9737

Suspension Deflection 80.8817 80.8725 84.3834 82.6723

Tire Travel 1.0562 1.0100 1.0188 1.0152

The hydraulic actuator is controlled using the concept of nonlinear energy sink. Simulation results show that resultant system effectively reduces the transferred vibrational energy from the road disturbance to the sprung mass. This is done at a reduced cost (in terms of actuation energy), when compared with the traditional vertical actuation using the skyhook concept.

The description of the active case of the variable stiffness suspension system has now been presented. It is foreseeable that the hydraulic actuator will be replaced with a semi-active element like the MR damper. It is foreseeable that an online frequency observer can be designed to observe the time dependent frequency of oscillation of the sprung mass. This frequency will then be used to design a gain scheduled controller between the LES and NES, with the LES active at low frequency and the NES active at high frequency. This would improve the suspension travel performance at frequencies below the natural frequency of the sprung mass.

VARIABLE STIFFNESS SUSPENSION SYSTEM: SEMI-ACTIVE CASE

An overview of the design, analyses and experimentation of the semi-active case of the variable stiffness suspension system is now presented.

An overview of the design, analyses and experimentation of the semi-active case of the variable stiffness suspension system is now presented. The semi-active case of the variable stiffness suspension system uses two MR dampers, one in the vertical direction and the other in the horizontal direction. The nonlinear, nonparametric model of the MR damper is also shown schematically in the drawings.

The control for the vertical MR damper was designed to track the skyhook damping force, while the control for the horizontal MR damper was designed to track the NES. One of the challenges encountered in the control design of the horizontal MR damper is that, while the model of the MR damper is dissipative, the desired NES force is conservative. This means that NES can only be tracked in the passive sub-cycle and not in the active sub-cycle. This problem was resolved by "clipping" the reference NES force in the passivity region of the MR damper. The conservativeness of the NES implies that energy is absorbed from the system and stored during a half-cycle (termed the passive sub-cycle), and supplied back to the system during the next half- cycle (termed the active sub-cycle). Because MR dampers are primarily dissipative, they cannot supply energy to the system. Consequently, "clipping" in the passive region means that the resultant desired force was designed such that energy is dissipated from the system as much as possible, according to the specification of the NES, during the passive sub-cycle, and nothing is done during the active sub-cycle. This was done to ensure a "trackable" desired force for the horizontal MR damper.

The developed control and update laws are summarized in the following algorithm:

Algorithm 4.1 : CO TROL UP AT

comment: Clipped Desired Force comment: Compute tracking error

e I- — p.-i

comment; Contpiite control enrrei

i _f , = < ( >< <! F ÷ )P ₂ (i} )

J

comment; Parameter Update

returii Θ}

Similar to the passive and active cases, the simulation models were developed using MATLAB SimMechanic, second generation. Simulations were carried out for the constant stiffness and the variable stiffness suspension systems. For the constant stiffness suspension, the control mass was locked at a fixed position corresponding to the equilibrium position of the control mass for the variable stiffness system.

The lower the car body acceleration is, the better the ride comfort is. The variable stiffness suspension is a more "ride friendly" suspension, outperforming the traditional vertical skyhook control. Associated with this improvement is a corresponding degradation in the suspension travel. This agrees with the observation made hereinabove, as well as the well-known trade-off between ride comfort and suspension deflection. Fortunately, the 12% degradation in suspension deflection is not as much as the 30% improvement gained in the ride comfort, resulting in an overall better performance. The maximum displacement of the control mass from the equilibrium position is less than 15cm. This implies that the space requirement for the control mass is small, which further demonstrates the practicality of the system. There is no significant reduction in the tire deflection. Thus, the suspension systems are approximately equally "road friendly". A more detailed explanation of the semi-active case of the variable stiffness suspension system is now presented as follows. This portion of the disclosure presents, in more detail, the semi-active case of the variable stiffness suspension system. This disclosure presents the semi-active case of a variable stiffness suspension system. The central concept is based on a recently designed variable stiffness mechanism which comprises a horizontal strut and a vertical strut, both of which are semi- actively controlled by MR dampers. The vertical MR damper force is designed to track a Skyhook damper force, while the horizontal strut is used is to vary the load transfer ratio by semi-actively controlling the location of the point of attachment of the vertical strut to the car body. The control algorithm, effected by the horizontal MR damper, uses the concept of a nonlinear energy sink to effectively transfer the vibrational energy in the sprung mass to a control mass, thereby reducing the transfer of energy from road disturbance to the car body. The analyses and simulation results show that a better performance can be achieved by subjecting the point of attachment of a suspension system, to the chassis, to the influence of a horizontal nonlinear energy sink system.

The original concept of semi-active suspension was introduced as an alternative to the costly, highly complicated, and power-demanding active systems. While fully active suspension systems are theoretically unrestricted energy wise, semi-active elements must be either dissipative or conservative in their energy demand. So far, semi-active designs fall into one of a general class of variable damper, variable lever arm, and variable stiffness.

Variable damper type semi-active devices are capable of varying the damping coefficients across their terminals. Initial practical implementations were achieved using a variable orifice viscous damper. By closing or opening the orifice, the damping characteristics change from soft to hard and vice versa. With time, the use of electro-rheological (ER) and magneto-rheological (MR) fluids replaced the use of variable orifices. ER and MR fluids comprise a suspension of polarized solid particles dispersed in a nonconducting liquid. When an electric (or magnetic for MR) field is imposed, the particles become aligned along the direction of the imposed field. When this happens, the yield stress of the fluid changes, hence the damping effect. The controllable rheological properties make ER and MR fluids suitable for use as smart materials for active devices, transforming electrical energy to mechanical energy.

Variable lever arm type semi-active suspensions conserve energy between the suspension and spring storage. They are characterized by controlled force variation which consumes minimal power. The main idea behind their operation is the variation of the force transfer ratio which is achieved by moving the point of force application. If this point moves orthogonally to the acting force, theoretically no mechanical work is involved in the control. This phenomenon has been called "reciprocal actuation".

Variable stiffness semi-active suspensions exhibit a variable stiffness feature. This is achieved either by changing the free length of a spring or by a mechanism which changes its effective stiffness characteristics as a result of one or more moving parts. A hydro-pneumatic spring with a variable stiffness characteristic is known. A desired stiffness variation achieved by augmenting a variable lever arm type system with a traditional passive suspension system is also known.

The control of semi-active suspension systems has gained much research interest over the years. The initial aim of a controlled suspension was solely centered on ride comfort. One of the initial control concepts developed is the sky -hook concept. A sky-hook damper is a fictitious damper between the sprung mass and the inertial frame (fixed in the sky). The damping force of the sky-hook damper reduces the sprung mass vibration. A similar concept called ground-hook has also been developed for road friendly suspensions. These control concepts have also been applied to semi-active suspensions. Other control concepts that have been applied to semi- active and active suspensions include: optimal control, robust control, and robust optimal control, etc.

Recently, research in semi-active suspensions has continued to advance with respect to their capabilities, narrowing the gap between semi-active and fully active suspension systems.

Meanwhile, most semi-active systems only control the viscous damping coefficient of the shock absorber while keeping the stiffness constant.

On the other hand, Nonlinear Energy Sinks (NES) are essentially nonlinear damped oscillators which are attached to a primary system for the sake of vibration absorption and mitigation. The primary system refers to the main system whose vibration is intended to be absorbed. Such attachments have been used extensively in engineering applications, particularly in vibration suppression or aeroelastic instability mitigation. The vibration of systems with essential

(strongly or weakly) coupled nonlinearity has been studied extensively. It has been shown that such attachments can be designed to act as a sink for unwanted vibrations generated by external impulsive excitations. The underlying dynamical phenomenon governing the passive energy pumping from a primary vibrating system to the attached nonlinear energy sink has been shown to be a resonance capture on a 1 : 1 manifold. It has been shown that under certain conditions, vibration energy gets passively pumped from the directly excited primary system to the nonlinear secondary system in a one-way irreversible fashion. Nonlinear passive absorbers can be designed with far smaller additional masses than the linear absorbers, thanks to the energy pumping phenomenon. This corresponds to a controlled one-way channeling of the vibration energy to a passive nonlinear structure where it localizes and diminishes in time due to damping dissipation. This allows nonlinear energy pumping to be used in coupled mechanical systems, where the essential nonlinearity of the attached absorber enables it to resonate with any of the linearized modes of the substructure.

Recently, the concept of "reciprocal actuation" was used to design a variable stiffness suspension system for isolating a car body from road disturbance. The system is essentially a passive vibration isolation system in which the motion of the secondary linear attachment is made orthogonal to the primary system. The primary and secondary systems are coupled through the traditional suspension system. In this paper, the above concept is extended by using two semi-active devices, one controlled to mimic a skyhook damper and the other controlled to mimic a nonlinear energy sink and used to drive an orthogonal secondary system. The orthogonal secondary system comprises a vibration absorber or isolator. The motivation for the use of NES is primarily due to their proven capability to achieve one-way irreversible energy pumping from the linear primary system to the nonlinear attachment. The goal therefore is to achieve a one-way irreversible energy pumping of the road disturbance to the secondary system whose vibration is orthogonal to the car body motion. A fairly general nonlinear function is used in this work, instead of cubic nonlinearity that is generally used. Due to a few number of moving parts, the concept can easily be incorporated into existing traditional front and rear suspension designs. An implementation with a double wishbone is disclosed herein.

SYSTEM DESCRIPTION

Variable Stiffness Concept

The variable stiffness mechanism concept is shown in Fig. 1. The Lever arm OA, is pinned at a fixed point O and free to rotate about O. The spring AB is pinned to the lever arm at A and is free to rotate about A. The other end B of the spring is free to translate horizontally as shown by the double headed arrow. Without loss of generality, the external force F is assumed to act vertically upwards at point A. d is the horizontal distance of B from O. The idea is to vary the overall stiffness of the system by letting d vary actively under the influence of a horizontal linear force generator (not shown). The variables k and Z ₀ denote the spring constant and the free length of the spring AB respectively, and Δ the vertical displacement of the point A. The overall free length Δ ₀ of the mechanism is defined as the value of Δ when no external force is acting on the mechanism.

Orthogonal Nonlinear Energy Sink

Fig. 15 shows the NES considered in this disclosure. The term orthogonal NES is used to describe the concept because the direction of motion of the secondary system is orthogonal to the primary system. This is suitable, structurally, for the application described in this disclosure. The subsystems 5 ₁₍ C and S ₂ constitute the primary subsystem, and are allowed to slide vertically together as a unit of total sprung mass m _s + m _d . The subsystem C is termed the control mass (or control subsystem). It, together with the nonlinear spring and the dashpot of damping coefficient b _d , constitute the secondary subsystem. The nonlinear function is defined as

F = g(d) = -k ₁ (l _0d - d) - ^sm ia^l^ - d)), (lc) where l _0d is the free length of the idealized nonlinear spring. The nonlinear function used is fairly more general compared to the pure cubic nonlinearity that has been used in the past. The Taylor series expansion is from which the generality is obvious. The mass labeled U is the unsprung mass, whose displacement y _u is used as the source of disturbance to the system.

An approximate frequency response from the input y _u to the sprung mass acceleration y _s and the rattle space deflection y _s —y _u , is computed using the notion of variance gain. The approximate variance gain is given by where z denotes the performance measure of interest (sprung mass acceleration and rattle space deflection in this case). The system is excited by the sinusoid r = i4sin(a>t), t G [0, 2πΝ/ω] , where N is an integer big enough to ensure that the system reaches a steady state. The corresponding output signals were recorded and the approximate variance gains were computed using (3c). The resulting variance gain responses are shown in Figs. 15 and 25 for a constant stiffness suspension (CSS), a variable stiffness suspension with linear energy sink (VSS:LES), and the variable stiffness suspension with nonlinear energy sink (VSS:NES). With a traditional constant stiffness suspension, a position of the control mass is fixed. With the variable stiffness suspension with linear energy sink, the control mass is allowed to move under the influence of a linear horizontal spring and damper. The figures show that the variable stiffness suspension achieves better vibration isolation, with a significant improvement from the linear energy sink case to the nonlinear energy sink case. As shown in Fig. 25, the improvement gained in vibration isolation results in a corresponding performance degradation in the rattle space deflection. However, when compared with the improvement in the vibration isolation, there is an overall improvement in performance associated with the use of the variable stiffness suspension with nonlinear energy sink. This agrees with the usual trade-off in suspension design. The performance improvement can further be increased by transitioning from LES in low frequency range (< 8Hz) to NES in high frequency range (> 8Hz).

Semi-active Variable Stiffness Suspension System The quarter car model of the suspension system considered is shown in Fig. 26. The quarter car model comprises a quarter car body, wheel assembly, two spring - MR damper systems, road disturbance, and lower and upper wishbones. The points O, A and B are the same as shown in the variable stiffness mechanism of Fig. 1. The motion of the control mass, which in turns determine the effective stiffness of the suspension system, is influenced by the MR1. The MR1 damper force is designed (see below) to mimic the orthogonal NES introduced above, and the MR2 damper force is designed to mimic the traditional skyhook damping force. The tire is modeled as a linear spring of spring constant k _t .

The assumptions adopted in Fig. 26 are summarized as follows:

1. The lateral displacement of the sprung mass is neglected, i.e., only the vertical displacement y _s is considered.

2. The wheel camber angle is zero at the equilibrium position and its variation is negligible throughout the system trajectory.

3. The springs and tire deflections are in the linear regions of their operating ranges.

The damping characteristics of the considered semi-active device can be changed by a control current. However, there is no corresponding energy input into the system as a result of the control current. This implies a passivity constraint on the MR-damper model. The control current is designed to mimic a desired force as close as possible, while enforcing the passivity constraint. This approach has been used in the past for semi-active control design.

MR-damper Modeling

The relationship between the MR-damper control current and the damping force exhibit a nonlinear phenomenon, and as a result, MR-damper based vibration control is a challenging task. Different damper models have been developed to capture the behavior of MR-dampers. Generally, the approaches that exists in literature can be grouped into parametric and nonparametric. The parametric modeling technique characterizes the MR-damper device as a collection of (linear and/or nonlinear) springs, dampers and other physical elements. A number of studies have addressed the parametric modeling of MR-dampers. One of the early models is the Bouc-Wen model which was derived from a Markov- vector formulation to model nonlinear hysteric systems. Later, a Bingham viscoelastic-plastic model was described. Others have developed a phenomenological model that accurately portrays the response of an MR-damper in response to cyclic excitations. This is a modified Bouc-Wen model governed by ordinary differential equations. Bouc-Wen based models in semi-active seismic vibration control have proven to be easy to use and numerically amenable. Others have studied the parametric model of MR-dampers, emphasizing the difference between the pre-yield viscoelastic region and the post- yield viscous region as a key aspect of the damper. One such model is where the damper force is modeled using the nonlinear static semi-active damper model. This allows for fulfilling the passivity constraint of the MR-damper.

On the other hand, nonparametric modeling employs analytical expressions to describe the characteristics of the modeled device based on both testing data analysis and device working principle. Although parametric models effectively characterizes the MR-dampers at fixed values of the control current, they do not include the magnetic field saturation that is inherent in MR- dampers. The representation of the magnetic field saturation is crucial in accurately using the MR-damper model for design analysis and control development. Recently, a nonparametric model has been proposed where the characteristics of a commercial MR-damper are represented by a series of continuous functions and differential equations, which are tractable using numerical simulation techniques. This model is used herein to represent the dynamics of the MR-damper. The nonparametric model is shown schematically in Fig. 27. The input to the model is the relative velocity, v(t), across the damper terminals, and the output is the damper force, F(t). The modeled aspect of the MR-damper are: Maximum Damping Force, f^C^C )

This is described using a polynomial function of the control current, i(t), as

P ₂ (0 = A ₀ + A _t i + A ₂ i ² + A ₃ i ³ + A ₄ i ⁴ , (4c) where A _j ,j = 0— 4 are the polynomial coefficients with appropriate units. Shape Function S _b (v(t))

This is used to preserve the wave-shape correlation between the damper force and the relative velocity across the damper, and is given by where v _r = v— v ₀ , (6C) and b ₀ > 0, b ₁ > 0, b ₂ > 0, v ₀ are constants. Delay Dynamics G(s, i(t))

A first-order filter is used to create the hysteresis loop observed in experimental data. It is given in state space form as x = - i (i)x + F _s = - i + P ₂ (i S _b (17)

F _h = P _{l (0} x, ^(7C) where x is the internal state of the filter and Pi(i(t)) is a polynomial function of the control current given by where h _j ,j = 0— 2 are polynomial coefficients with appropriate units. It is worth noting that the condition is imposed on P ₁ (i(t)) in order to guarantee a decaying solution. Offset Function, F _bias In some cases, the damping force is not centered at zero because of the effect of the gas-charged accumulator in the damper. The force bias F _bias is included in the model to capture this effect, and as result, the overall damper force is given by

F(t) = F _h + F _hias . (10c) Table 1 shows the optimal values of the MR-damper model obtained from experimental data via an optimization process. In terms of the input v(t) and output F(t), the overall dynamics of the MR-damper is given by

Fh = ^~ i (i)F _h + Pi (i)S _b (v)P ₂ (i) (11c) F(t = F _h + F _bias (12c) Table 1 : MR DAMPER PARAMETER VALUES

Control Development

The schemes used for the desired damper forces are the NES and skyhook based control forces f _dH = k _i smh(oc ₁ (l _0d - d , (13c) fd _v = b _sky y _s , (14c) where f _d ., i = {H, V] are the corresponding horizontal and vertical desired forces respectively, and <x are positive constants used to tune the performance of the NES control, and b _sky is the damping coefficient of the skyhook damper which is a fictitious vertical damper between the sprung mass and inertial frame.

Open Loop Tracking Error Development

Although the MR damper parameter values given in Table 1 were determined experimentally, they can change over time due to usage and other causes. As a result, an adaptive tracking control for the MR damping force is developed. To this effect, it is assumed that the coefficients of the polynomial P ₂ (i) ^are unknown. Also, the desired control force may not generally satisfy the passivity constraint at a given instant. At the instances when the passivity constraint is violated, the desired damping force lies outside the "trackable" passivity region of the MR damper. In order to ensure a valid tracked desired damping force, the force f _d . given in (13c) is "clipped" in the passivity region. Using the Final Value Theorem, the steady state MR damper force, from (11c), is given by

F _ss = S _b v)P ₂ (i). (15c)

Thus, the tracked desired damper force is obtained by "clipping" f _d as follows

(S _b (y P2 if v _r f _d < v _r S _b {y ^~ )P ₂

Fd ifdf V = I fdt if v _r S _b (v)P ₂ < v _r f _d < v _r S _b (v)P ₂ (16c)

\ S _b (v)P ₂ if v _r f _d ≥v _r S _b (v)P ₂ where = min fP ₂ ( ) (17c)

P ₂ = max {P ₂ (i)}, (18c)

[0 Iraax] and i _max is the maximum current that can be sent to the MR damper. Now, let e = F - F _d (19c) be the tracking error of the damper force. Taking the first time derivative of (19c) yields e = F = -PiF + S^P;, (20c)

= -Pi(e + F _d ) + S^P,. (21c)

The response of MR dampers are very fast compared to the vibrating mechanical system.

Hence, the commanded desired force F _d (e, e, v) is assumed to be fairly constant compared to the dynamics of the MR damper. Adding and subtracting the term Pi(—F _d + S _b P ₂ ) yields the open loop error system e = -P ₁ e + P ₁ S _b (P ₂ -P ₂ ) + P ₁ a(i), (22c) where a(i) = -F _d +S _b P ₂ , (23c) and P ₂ is an adaptive estimate of the polynomial P ₂ (i). The update law is designed subsequently.

Closed Loop Error System Development

First, a close approximation of the polynomial P ₂ (i) is given within the operating interval. Given the bounds f^andP^ the polynomial P ₂ (i) is approximated in the interval [0 i _max ] as

^(0 = E2 + Γ~(?2 - &) + β ΐ), (24 ^C ) where β(ί) = ί(ί- i _max )(6» ₀ + ίθ ₁ + ί ² θ ₂ ), (25c) is chosen to satisfy the constraints ?(0) = ?(i _max ) = 0, which implies that P ₂ (0) = P2 and ^('max) ⁼ Pi- Th ^e approximation is largely dependent on the monotonicity and the onto properties of P ₂ (i).

Lemma 1. There exists unique ideal parameters Pj, P ₂ , θ ₀ , θ ₁₍ and0 ₂ such that the approximated polynomial given in (24c) matches the original polynomial in (4c) exactly.

Proof. The approximated polynomial is written in an expanded form as follows p ₂ (i) = Ez + i (^-e ₀ i _max ) + ί ² (θ ₀ - ^ )

+i ³ (0 ₁ -0 ₂ i _max ) + i ⁴ 0 ₂ . (26c)

Comparing (26) with (4) yields the system of linear equation

The determinant of the coefficient matrix in (27c) is—1, which implies that the coefficient matrix is full ranked. As a result, there exists a unique vector r — I ^7*

[P^ P ₂ θ ₀ θ θ ₂ \ that satisfies (27c).

Fig. 28 shows the plot of the actual and the approximated polynomials with θ ₀ , θ , θ ₂

determined using least square method, given and P ₂ .

The polynomial in (24c) is linear in the unknown parameters 9j,j = 0— 2. Thus (22c) becomes e = -P e + S _b P ₁ Y ^T {Q - Θ) + P^i) (28c) = -P e + SuP^Q + P^i , (29c) where is the parameter vector to be estimated, with a corresponding parameter estimation error vector ø = ø— ø, (31c)

Y ί(ί imax) (32c) is the current dependent regression matrix, and 2 (i) = E2 + -^ (P2 - E2) + Y(i) ^T & (33c)

The following lemma is used to guarantee the existence of a valid control current in the interval

[0 'max] -

Lemma 2. If the parameter update law is designed such that the estimate Θ is continuous, then the polynomial a(i) given in (23c) has at least one root in the operating interval [0 i _max ].

Proof. From (16c), it is seen that the clipped desired damping force satisfies the following passivity constraint

(l7 - V ₀ )S _b (v)P2 < (l7 - V ₀ F _d (u, 17) < (17 - v ₀ S _b ( P ₂ . (34c) Also, from (5c), it can be shown that the term (i7— v ₀ )S _b (v) is positive. Thus, dividing through by (i7— v ₀ )S _b (v) in (34) yields < i < p (35c) which implies that

¾ (36c) Because Θ is continuous by the hypothesis, it implies that P ₂ (i) is continuous. Also, because P ₂ (0) = f^and ^ _max ) = P ₂ , using the Intermediate Value Theorem, it follows that there exists at least one i _c G [0 i _max ] such that which implies that a(i _c ) = F _d (f _d , 17) - S _b (v)P ₂ (i _c ) = 0. (38c)

Thus, i _c G roots(a(i)) and, because i _c G [0 i _max ] , the proof is complete.

Next, suppose that Θ is continuous, then, using Lemma 2, it follows that there exists a control current i _c G [0 i _max ] such that a(i _c ) = 0. Consequently, the closed loop error system is given by e = -P e + S^ ^ _c ) ⁷ ©. (39)

Stability Analysis Theorem 1. Given the update law

§ = LeS _b (y)Y(i _c ), 0(0) = Θ ₀ , (40c) where L is a positive constant adaptation gain, and the control law i _c = arg min roots( (r)), (41c)

[0 'max] the closed-loop error dynamics in (39c) is stable, and the tracking error e(t) approaches zero asymptotically. Also, the parameter estimate Θ is continuous, thus satisfying the hypothesis of Lemma 2.

Proof. Consider the positive definite Lyapunov candidate function

V _L = e ² + ^ Θ ^Τ Θ. (42) Taking the first time derivative of (42c) along the closed loop trajectory in (39c) yields

V _L = ee - Θ§ (43c)

Substituting the update law in (40c) yields

V _L = -Ρ _ιβ ² . (45c)

Because Pi(i) > 0, it implies that V _L is negative semi-definite, and because V _L is positive definite, it follows that V _L G £ _∞ . From (42c), it follows that e, Θ G £ _∞ , which also implies that Θ G £ _∞ -since Θ is a constant. Integrating (45c) yields from which it follows that e G £ ₂ . Also from (39), it follows that e G £ _∞ which implies that e is uniformly continuous. Thus, because e G £ ₂ and uniformly continuous, it can be shown using Barbalat's lemma that e(t)→ 0 asymptotically. The following algorithm summarizes the control and update laws developed above:

[shadowbox]Control/Updatef_d,v, Clipped Desired Force

F_d F_d(f_d,v)

Compute tracking error

e F-F_d

Compute control current i_c = _[0i_max]roots(-F_d+S_b(v)P_2(i))

Parameter Update

L_0 ^A te()S_b(v)Y(i_c) d+_0

i_c, where

roots(a(i)) is the set of roots of the polynomial a(i),

{ } is Empty set. Simulation Results

In order to study the behavior of the quarter car system to different road excitation scenarios, as well as measure responses like suspension deflection, realistic simulations were carried out using MATLAB SimMechanics. First, the system was modeled in Solidworks®, and then translated to a SimMechanics model. The vertical strut and tire damping and stiffness used are the ones given in the "Renault Megane Coupe" model. The values are given in Table 2.

Table 2: DYNAMIC PARAMETER VALUES

Parameter Value m _s 315 kg

™u 37.5 kg b _s 1500 N/m/s k _s 29500N/m

210000 N/m

In the simulation, the vehicle traveling at a steady horizontal speed of 40mph is subjected to a road bump of height 10cm. The Car Body Acceleration (CBA), Suspension Travel (ST), and Tire Deflection (TD) responses are measured. The suspension travel is defined as the vertical displacement of the center of mass of the sprung mass with respect to the unsprung mass, and the tire deflection as the vertical displacement of the unsprung mass with respect to the road level. Simulations were carried out for the constant stiffness and the variable stiffness suspension systems. For the constant stiffness suspension, the control mass was locked at a fixed position corresponding to the equilibrium position of the control mass for the variable stiffness system. The results obtained are reported in Figs. 29-35. Table 3 shows the variance gains for the different responses. Fig. 29 shows the car body acceleration, which is used here to describe the ride comfort. The lower the car body acceleration, the better the ride comfort. As seen in the figure, the variable stiffness suspension is a more "ride friendly" suspension, outperforming the traditional vertical skyhook control. As shown in Fig. 30, associated with this improvement is a corresponding degradation in the suspension travel. This agrees with the observation made hereinabove, as well as the well-known tradeoff between ride comfort and suspension deflection. Fortunately, the 12% degradation in suspension deflection is not as much as the 30% improvement gained in the ride comfort, resulting in an overall better performance. Fig. 32 shows the position history of the control mass for the variable stiffness suspension, from which the boundedness of the motion of the control mass is seen. The maximum displacement of the control mass from the equilibrium position is less than 15cm. This implies that the space requirement for the control mass is small, which further demonstrates the practicality of the system. Fig. 31 shows that there is no significant reduction in the tire deflection. Thus, the suspension systems are approximately equally "road friendly". Fig. 33 shows the parameter estimates. The upper sub-figure shows that the parameters are not updated in the control of the horizontal MR damper. This is because the corresponding control current is bang-bang, switching from i _c = 0 to i _c = i _max . As a result, the elements of the regression matrix given in (32c) are zeros, which further implies, from (40c), that 0 = 0. Thus, the parameter estimate will remain constant. Fig. 35 shows the horizontal and vertical damper forces.

Table 3: VARIANCE GAIN VALUES

Constant Constant Variable Variable Stiffness Stiffness Stiffness Stiffness Passive Active Passive NES

Car Body Acceleration (s ^_1 ) 109.0389 64.2818 65.6127 42.9737

Suspension Deflection 80.8817 80.8725 84.3834 82.6723

Tire Travel 1.0562 1.0100 1.0188 1.0152 The description of the semi-active case of the variable stiffness suspension system has now been presented. The vertical MR damper force is designed to track a Skyhook damper force, while the horizontal MR damper control algorithm uses the concept of nonlinear energy sink to effectively transfer the vibrational energy in the sprung mass to a control mass. Simulation results show that the system in accordance with the invention shows as much as 30% improvement in ride comfort over the traditional vertical Skyhook controller. Associated with the ride improvement is a 12% degradation in the suspension deflection performance but the suspension limit is not exceeded. There is no significant change in the road holding

performance. Thus, a better semi-active suspension performance overall can be achieved by using an additional semi-active device to control the horizontal location of the point of attachment of the vertical strut to the car body, and controlled to mimic a fictitious nonlinear energy sink attached between the control mass and the vehicle body.

The description of the semi-active case of the variable stiffness suspension system has now been presented. It is foreseeable that an online frequency observer can be designed to observe the time dependent frequency of oscillation of the sprung mass. This frequency will then be used to design a gain scheduled controller between the LES and NES, with the LES active at low frequency and the NES active at high frequency. This would improve the suspension travel performance at frequencies below the natural frequency of the sprung mass. ROLL STABILIZATION USING VARIABLE STIFFNESS SUSPENSION SYSTEM

An overview of the design, analyses and experimentation of roll stabilization using the variable stiffness suspension system is now presented.

Roll dynamics is critical to the stability of road vehicles. A loss of roll stability results in a rollover accident. Typically, vehicle rollovers are very dangerous. Research by the National Highway Traffic Safety Administration (NHTSA) shows that rollover accidents are the second most dangerous form of accidents in the United States, after head-on collision. In 2000, approximately 9,882 people were killed in the United States in a rollover accident involving light vehicles. Rollover crashes kill more than 10,000 occupants of passenger vehicles each year. As part of its mission to reduce fatalities and injuries, since model year 2001, the National Highway Traffic Safety Administration (NHTSA) has included rollover information as part of its New Car Assessment Program (NCAP) ratings. One of the primary means of assessing rollover risk is the static stability factor (SSF), a measurement of a vehicle's resistance to rollover. The higher the SSF, the lower the rollover risk. Roll stability, on the other hand, refers to the capability of a vehicle to resist overturning moments generated during cornering that is to avoid rollover. Several factors contribute to roll stability, among which are Static Stability Factor (SSF), kinematic and compliance properties of the suspension system etc. In this disclosure, the variable stiffness architecture discussed previously is used in the suspension system to counteract the overturning moment, thereby enhancing the roll stability of the vehicle. The proposed system can be used in conjunction with existing roll stabilization methods; provided that there is no significance interferes with the suspension system.

The model is comprises a half car body (sprung mass), two identical wheel assemblies

(unsprung masses), two vertical spring-damper systems, left and right lower and upper wishbones, hydraulic actuators etc. The main idea of the design is to vary the effective vertical reactive forces of the left and right suspensions to counteract the body roll moments. This is achieved by an appropriately designed control for the hydraulic actuators.

During cornering, a vehicle experiences a radially outwards lateral acceleration acting at the center of mass, as well as corresponding lateral tire forces acting at the tire/road contacts. This results in a roll moment which causes the vehicle to lean outwards. To counteract this roll moment, the outside suspension should become stiffer while the inside suspension should become softer. This generates a counter moment to improve the stability of the roll dynamics.

The variable suspension system was modeled. The yaw dynamics of the vehicle was effectively decoupled from the roll dynamics by modeling it as a rigid bicycle in a planar motion. The model has three degrees of freedom. As a result, the yaw dynamics were given by a set of three coupled first order ordinary differential equations. To capture the effect of the nonlinear tire forces at large slip angles, the well-known Pacejka "Magic Formula" was used to model the tire lateral forces. The corresponding longitudinal tire forces were obtained by enforcing the friction cone constraint. This was done in order to keep the total tire forces from exceeding the maximum frictional force. The effect of longitudinal load transfer was captured by summing forces in the vertical direction, and taking moments about the body lateral axis, while neglecting pitch dynamics. The roll dynamics was obtained using the free body diagram of an idealized half car model of the system, where the suspension forces have been replaced with their horizontal components, M _L , M _R , and vertical components N _L , N _R . The assumptions adopted for the roll dynamic model are summarized as follows:

1. The half car body is symmetric about the mid-plane, and as a result the center of mass is located on the mid-plane at a height h above the base of the chassis.

2. The road is level and the points of contact of the tires are on the same horizontal plane.

3. The springs and damper forces are in the linear regions of their operating ranges.

4. The compliance effects in the joints are negligible. In order to validate the obtained model, as well as ensure realistic simulations subsequently, the parameters of the roll dynamics are estimated so that the resultant roll dynamics matches the data obtained experimentally. The vehicle used for the data collection is a Toyota Highlander Hybrid 2007 equipped with Inertial Measurement Unit, during one of the maneuvers. Two sets of data were collected. The first is termed the Circle Data, in which the car is driven around cones arranged on in a circular fashion. The second is termed the Eight Data. Here, the vehicle is driven several times along an eight-shaped path. The data collected for each experiment includes the longitudinal and lateral velocities, lateral acceleration, roll angle and roll rate. The parameters of the model are estimated using the trust-region-reflective method in MATLAB.

The control development was done hierarchically. First for the vehicle body roll, then for the control masses, and finally for the hydraulic actuators. The desired actuator forces required to achieve a desired roll behavior were designed using a model reference adaptive control and sliding mode techniques, then the necessary servo current commands to the spool valve were designed from the actuator dynamics using an adaptive singular perturbation approach. Next, a Lyapunov-based stability analysis was carried out for the overall closed loop error dynamics to guarantee the convergence of the tracking error and boundedness of the system states.

The performance of the proposed control was examined via simulation, using the National Highway Traffic Safety Administration (NHTSA) fish hook and the ISO 3888 double lane change maneuvers. The results show that by using the actuated variable stiffness mechanism together with the developed control, the roll angle and roll rates are reduced by more than 50%. The Fish hook maneuver, by NHTSA, is a very useful test maneuver in the context of rollover, in that it attempts to maximize the roll angle under transient conditions. The procedure is outlined as follows, with an entrance speed of 50 mph (22.352m/s):

1. The steering angle is increased at a rate of 720 deg/s up to 6.55stat, where 5stat is the steering angle which is necessary to achieve 0.3g stationary lateral acceleration at 50mph. 2. This value is held for 250ms. 3. The steering wheel is turned in the opposite direction at a rate of 720deg/s up to -6.55stat.

By using the variable stiffness mechanism together with the developed control algorithm, the roll angle and roll rates are reduced by more than 50%. It is also seen that the control allocation exhibit some ganging phenomenon. The ISO 3888 Part 2 Double Lane Change course was developed to observe the way vehicles respond to hand wheel inputs drivers might use in an emergency situation. The course requires the driver to make a sudden obstacle avoidance steer to the left (or right lane), briefly establish position in the new lane, and then rapidly return to the original lane. From the corresponding control masses and roll responses, it can be concluded that the variable stiffness systems has much better behavior during the severe obstacle avoidance maneuver.

A more detailed explanation of roll stabilization using the variable stiffness suspension system is now presented as follows.

This portion of the disclosure presents, in more detail, roll stabilization using the variable stiffness suspension system. A variable stiffness architecture is used in the suspension system to counteract the body roll moment, thereby enhancing the roll stability of the vehicle. The variation of stiffness concept uses the "reciprocal actuation" to effectively transfer energy between a vertical traditional strut and a horizontal oscillating control mass, thereby improving the energy dissipation of the overall suspension. The lateral dynamics of the system is developed using a bicycle model. The accompanying roll dynamics are also developed and validated using experimental data. The positions of the left and right control masses are sequentially allocated to reduce the effective body roll and roll rate. Simulation results show that the resulting variable stiffness suspension system has more than 50% improvement in roll response over the traditional constant stiffness counterparts. The simulation scenarios examined is the fishhook maneuver.

Nomenclature

δ Front wheel steering angle

Φ Vehicle body roll angle

Vehicle yaw angle

r Yaw rate

17 X. Longitudinal velocity

y Lateral velocity d _L Left control mass displacement

d _R Right control mas displacement

Is Vehicle roll moment of inertia

Iz Vehicle yaw moment of inertia

Distance of front axle from the center of mass

l _r Distance of rear axle from the center of mass

k _s Stiffness of suspension spring

b _s Suspension damping coefficient

Natural length of suspension spring

k _S b Roll stiffness due to anti-roll bar

b _S b Roll damping coefficient due to anti-roll bar

m _s Sprung mass of half car

Is Roll moment of inertia

Iz Yaw moment of inertia

31 Set of real numbers

Roll dynamics is critical to the stability of road vehicles. A loss of roll stability results in a rollover accident. Typically, vehicle rollovers are very dangerous. Research by the National Highway Traffic Safety Administration (NHTSA) shows that rollover accidents are the second most dangerous form of accidents in the United States, after head-on collision. In 2000, about 9,882 people were killed in the United States in a rollover accident involving light vehicles. Rollover crashes kill more than 10,000 occupants of passenger vehicles each year. As part of its mission to reduce fatalities and injuries, since model year 2001, the National Highway Traffic Safety Administration (NHTSA) has included rollover information as part of its New Car Assessment Program (NCAP) ratings. One of the primary means of assessing rollover risk is the static stability factor (SSF), a measurement of a vehicle's resistance to rollover. The higher the SSF, the lower the rollover risk. Roll stability, on the other hand, refers to the capability of a vehicle to resist overturning moments generated during cornering, that is to avoid rollover. Several factors contribute to roll stability, among which are Static Stability Factor (SSF), kinematic and compliance properties of the suspension system etc.

A number of rollover prevention and roll stability enhancement methods exist in literature that are based on one or more of differential braking, steer-by-wire, differential drive torque distribution, and active steering. A rollover prevention system using a combination of steer-by- wire and differential braking is known. A differential braking based anti-rollover control algorithm based on a time-to-rollover metric was proposed for sport utility vehicles and was evaluated using human-in-the-loop simulations. A method of identifying real-time predictive lateral load transfer ratio for rollover prevention systems has been proposed.

Mechanical/electromechanical systems to improve the roll stability of road vehicles are known. One the earliest mechanical systems is an anti-roll bar (or sway bar or stabilizer bar). A sway bar is usually in the form of a torsional bar connecting opposite (left/right) wheels together. It generally helps in resisting vehicle body roll motions during fast cornering or road irregularities by increasing the suspension's roll stiffness, independent of the vertical spring constants. Anti- roll systems are one of: passive, semi-active, or active, by design.

The embodiment disclosed herein uses a variable stiffness architecture in the suspension system to counteract the overturning moment, thereby enhancing the roll stability of the vehicle. The proposed system can be used in conjunction with existing roll stabilization methods, provided that there is no significance interfere with the suspension system. The variation of stiffness concept uses the "reciprocal actuation" to effectively transfer energy between a vertical traditional strut and a horizontal oscillating control mass, thereby improving the energy dissipation of the overall suspension. Due to the relatively fewer number of moving parts, the concept can easily be incorporated into existing traditional front and rear suspension designs. An embodiment with a double wishbone is disclosed.

SYSTEM DESCRIPTION

Mechanism Description An underlying concept of the variable stiffness suspension is shown in Fig. 1. The lever arm

OA, is pinned at, and free to rotate about the fixed point O. The end point A of the spring AB is pinned at, and free to rotate about A. The other end B of the spring is free to translate horizontally as shown by the double headed arrow. The external force F is assumed to act vertically upwards at point A. The resistive force due to the spring, as a result of the force F, depends on the horizontal distance d of B from O. In other words, the effective stiffness of the system is a function of d.

The schematic diagram of the half car model of the variable stiffness suspension system considered is shown in Fig. 36. The subsystems comprising the lower wishbone, suspension spring-damper system, control mass, and the hydraulic actuator are the implementations of the variable stiffness concept described above. Two of them are implemented for the right and left suspensions respectively, with d _L and d _R being the implementations of horizontal distance d shown in the variable stiffness concept, for the left and right suspensions respectively. These horizontal displacements are controlled by hydraulic actuators. The half-car model of Fig. 36 comprises a half car body (sprung mass), two identical wheel assemblies (unsprung masses), two vertical spring-damper systems, left and right lower and upper wishbones, hydraulic actuators. During cornering, a vehicle experiences a radially outwards lateral acceleration acting at the center of mass, as well as corresponding lateral tire forces acting at the tire/road contacts. This results in a roll moment which causes the vehicle to lean outwards. To counteract this roll moment, the outside suspension should become stiffer while the inside suspension should become softer. This generates a counter moment to improve the stability of the roll dynamics. This is achieved by controlling the hydraulic actuators to change the horizontal distances d _L and d _R to vary the effective vertical reactive forces of the left and right suspensions appropriately. This method provides added roll stability to vehicle irrespective of other methods, such as the passive or active stabilizer bars, etc. As a result, the proposed system can be used either as a standalone, or in conjunction with other methods. As will be shown by simulation later, it is seen that this system can improve roll stability up to 50% , even with passive sway bar already in place.

Modeling

Fig. 37 shows a schematic of the modeling aspects of the system. Yaw Dynamics The yaw dynamics of a vehicle may effectively decoupled from the roll dynamics by modeling it as a rigid bicycle in a planar motion as shown in Fig. 38. The model has three degrees of freedom. As a result, the yaw dynamics are given by a set of three coupled first order ordinary differential equations. However, since the maneuvers considered in this paper are constant speed maneuvers, the corresponding forward velocity dynamic is remove and the remaining yaw dynamics are given as follows: x = VySmip (Id) y = VxSimjj + v _y cosijj (2d) φ = τ (3d) v _y =— (F _X fSinS + Fyf CosS + F _yr )— v _x r (4d) r = γ (lf (F _X fSinS + F _y ^cos(5)— Z _r F _yr ), (5d)

To capture the effect of the nonlinear tire forces at large slip angles, the well-known Pacejka "Magic Formula" is used to model the tire lateral forces. The lateral forces are expressed as

Fyj = -μμ _ν] Ρ _ζ] , (j = f, r), (6d) where μ is the maximum friction coefficient of the road surface, F _z - is the normal load at each tire, and μ^- is the tire-road interaction coefficient given by the Magic Formula y - = MF(Syj = sin (ctan ^"1 ^^)), (7d) where s _y j are the lateral slip ratios, given respectively for the front and rear tires as v δ -v _x smS +r If cos δ

v _x cos5 +rlf sinS

Syr = ^JL —- (9d)

Here, v _x is the constant vehicle forward speed. In order to keep the total tire forces from exceeding the maximum frictional force, the friction cone constraint is enforced as follows

¾ + F _y ² - = μ¾ (lOd) which implies that

(l id)

The effect of longitudinal load transfer is captured by summing forces in the vertical direction, and taking moments about the body lateral axis, while neglecting pitch dynamics, as follows

P _Z f + Pzr = mg (13d) l/F _Z f — l _r F _zr = z(F _X f CosS— FyfSinS + F _xr ), (14d) where z is the height of the body center of mass from the ground. After some algebraic manipulations, and using (6d) and (11), equations (13d) and (14d) yield the expressions for the respective normal loads at the front and rear tires as

Roll Dynamics The free body diagram of an idealized half car model of the system is shown in Fig. 39, where the suspension forces have been replaced with their horizontal components, M _L , M _R , and vertical components N _L , N _R .

The assumptions adopted for the subsequent dynamic model are summarized as follows:

1. The half car body is symmetric about the mid-plane, and as a result the center of mass is located on the mid-plane at a height h above the base of the chassis.

2. The road is level and the points of contact of the tires are on the same horizontal plane.

3. The springs and damper forces are in the linear regions of their operating ranges.

4. The compliance effects in the joints are negligible. 5. The compliance of the tires/unsprung masses are neglected.

The instantaneous lengths of the left and right suspensions are given respectively as l\ = (7cos0— d _L cos0— Hsin0 + T _L ) ²

+ (z— d _L sin0 + /i ₂ cos0) ² , (17d) IR = (— 7cos0 + d _R cos0— Hsin0— T _R ² + (z + d _R sin0 + /i ₂ cos0) ² , (18d) and the corresponding suspension forces are given by

Fs _L = k _s (lo _s - ) - b _s i _L (19d) F _SR = k _s (l _0s - l _R ) - b _s l _R . (20d) Thus, the horizontal and vertical components of the left, and right suspension forces are given by

M _L =— (7cos0 - d _L cos0 - Hsin0 + T _L ), (21d)

M _R = ^ (-7cos0 + d _R cos0 - Hsin0 - T _R ), (22d)

IR

p

N _L =—(z - d _L sin0 + /i ₂ cos ), (23d) p

N _R = (z + d _R sin0 + /i ₂ cos ). (24d)

Following the assumptions above, and neglecting the lateral dynamics, the roll equations of motion of the system are given by the following set of differential algebraic equations:

^N L + ^N R - ^m s9 - ^m _s z = °< (25d)

M _c - I _s > - b _sb <j> - k _sb (p = 0, (26d) T + (z - 7sin0 - /icos0) ² - Z ² = 0, (27d)

Ύΐ + (z + 7sin0 - /icos0) ² - Z ² = 0, (28d) where

M _c = g _L (N _L , 0)d _L + # _R (NR, _R , 0)d _R (N _L , N _R , M _L , M _Rl <t>)h ₂ + Fy _j z, (29d) with

M _l , φ = -N _L cos0 + _L sin0, (30d) g _R (N _R , M _R , (p) = N _R cos0 - _R sin0, (3 Id)

Y(N _L , N _R , M _L , M _R , φ = (N _L + N _R )sin0 + ( _L + _R )cos0, (32d) and k _sb , b _sb are the stiffness and damping due to the sway bar and other compliance and damping elements that have indirect or direct influence on the roll dynamics. The total ground force F _L + F _R is equivalent to the lateral tire forces F _yj from the yaw dynamics.

Control Masses and Actuator Dynamics A hydraulic system considered is shown schematically in Fig. 40. The hydraulic system comprises a source of hydraulic pressure, a spool valve, and a hydraulic cylinder. The hydraulic pressure is supplied by a hydraulic source which is essentially a hydraulic pump, typically augmented with accumulators to reduce pressure fluctuations and supply additional fluid for peak demands. The hydraulic cylinder is a double acting cylinder. The piston motion is obtained by modulating the oil flow into and out of the cylinder chambers, which are connected to the spool valve through cylindrical ports. The modulation is provided by the spool valve.

Fig. 41 is a graph of lateral tire force approximation.

The dynamics of the hydraulic actuator, as well as the spool valve, are given by

P _L = -aAv _p - P _L + γχ _ν /Ρ ₅ - sgn(x _v )P _Ll (33d)

F = AP, (35d) where A is the pressure area in the actuator, P _L is the load pressure, v _p = d is the actuator piston velocity, F is the output force of the actuator, a, β and γ are positive parameters depending on the actuator pressure area, effective system oil volume, effective oil bulk modulus, oil density, hydraulic load flow, total leakage coefficient of the cylinder, discharge coefficient of the cylinder, and servo valve area gradient, x _v is the spool valve position, τ is the actuator electrical time constant, K is the DC gain of the four-way spool valve, and u is the input current to the servo valve. After summing forces along the line of action of the actuators on the control masses, the equations of motion of the left and right control masses, together with the actuator model, are given by m _d di = F?— j Cos0— NjSin0 (36d) τχ _ν . =—x _Vi + Ku _t . (38d)

The subscript i = (L, F) is used to indicate left and right quantities respectively. Control Design

Details of the design of control design for the hydraulic actuators geared toward improvement of the body roll dynamics are now presented. First, the control laws are designed, and the resulting closed loop error system given. The desired actuator forces required to achieved a desired roll behavior are designed using a model reference adaptive control and sliding mode techniques, then the necessary servo current command to the spool valve is designed from the actuator dynamics using an adaptive singular perturbation approach. Next, a Lyapunov-based stability analysis is carried out for the overall closed loop error dynamics to guarantee the convergence of the tracking error and boundedness of the system states. The control development is done hierarchically. First for the vehicle body roll, then for the control masses, and finally for the hydraulic actuators.

Vehicle Body Roll

Here, the desired reference roll model is given by l _s (p _m + k ₂ (p _m + k^ _m = 0, (39d) where = ω ₁₍ (40d) fc ₂ = + 2a½ (41d) were designed to minimize

J = C (ω? 0m( ² + wl0 _m (t) ² + Oi0m(O + k ₂ <p _m (t)f) dt (42d) subject to (39d), where ω ₁ , ando> ₂ are performance weights used to penalize the performance index with respect to roll and roll rate respectively. The performance index in (42d) is chosen to ensure smooth and bounded roll dynamics of the vehicle body, with the performance weights specifying a trade-off between achieved boundedness (controlled by and smoothness (controlled by k ₂ ) of the ride. Let e(t) = 0(t) - 0m( (43d) be the tracking error defining how well the roll dynamics in (26d) tracks the reference model in (39d). The objective is then to drive the tracking error to as small as possible using the actuator forces. Taking the first and second derivatives of (43d) and subtracting (39d) from (26d) yields l _s e + k ₂ e + k _x e = M _c - (k ₂ - b _sb )<p - (/ - fc _s6 )0. (44d) To facilitate subsequent control design and analyses, the nonlinear lateral force given by the Pacejka formula is approximated as

Fyj Qi iSj) (45d)

= L(s _j ) ^T Q, (46) where the regression matrix R (s _j ) and the constant coefficient vector Q are given by L(Sj) = [L^s _j ) L ₂ (s _j ) L _n (s _j )] ^T , (47d) with

L^s _j ) = sin ((2£ - l)tan ^"1 (5 )) , i = l, 2, ... , n (49d) being the set of bases functions. Other bases functions can be used (e.g., polynomial, rational function, etc.). The functions in (49d) are used as basis for the lateral tire force approximation because they preserve the form given in the Magic formula. Fig. 41 shows the resulting approximation for n = 10, where the ideal weight vector Q was obtained using a least square approach.

Thus, the roll error dynamics in (44d) becomes l _s e + k ₂ e + k e - L ^T Qz + Yh ₂ = / _φ< (50d) where ίφ = 9L&L + 9Rd _R - Yh ₂ + L ^T Qz ^~ (k ₂ ^{~ b} sb <P ^~ Oi - k _sb ) ⁽ f>, (5 Id) and h ₂ , Q are the adaptive estimates of the unknown system constant parameters h ₂ , Q, with the corresponding estimation errors given by h ₂ = h ₂ - h ₂ , (52d)

Q = Q - Q. (53d) The parenthesized arguments have been dropped unless otherwise required for clarity. Let d _L ^* = d ₀ + A _L , (54d) d _R ^* = d ₀ + A _R , (55d) where

A _L , A _R = argmin{|/ ₀ 1 : -Δ < A _L , A _R ≤ A], (56d) be the desired displacement of the control masses. Δ defines the physical limits on the allowable positions of the control masses. The optimization in (56d) defines a control allocation problem. Control allocation approach is generally used when different possible control choices can produce the same result. This usually happens when the number of effectors exceeds the state dimension, as the case in this paper. The general control allocation problem, as well as existing solution methods, are well expounded upon in. However, due to the special form of (5 Id), the solution to (56d) is obtained sequentially as follows

A _L = clip ( , -A, A), (57d) where the saturation function, clip(. . . ), is defined

I a, ifx < a

x, ifa < x < b

b, ifx > b

= min{max{a, x), b), (60d) and ίφ = (9L + g _R d ₀ - Yh ₂ + L ^T Qz - (k ₂ - b _sb )<p - (/ - (61d)

(Details of the optimization procedure are given in the Allocation Procedure section, below.) Consequently, let ¾ be the residual value of fS after the optimization above. Also, let ^r i( = e(t) + OL _x e{€) (62d) defines a sliding surface for the roll error dynamics. Then, the corresponding closed loop roll error dynamics is given by = ¾ - (k ₂ - a^s)^ - (fei - a _x k ₂ - ct ₁ ))e + L ^T Qz - Yh ₂ . (63d) Control Masses

In order to ensure smoothness of the ensuing motion of the control masses, the desired trajectory of the control masses is given by the following first-order low pass filter dynamics edf = -df + d\, i = {L, R}. (64d)

Let = e _d . + a ₂ e _d ., (65d) defines a sliding surface for the position tracking error of the control masses, where a ₂ is a positive control gain. Differentiating (65d) and substituting the control mass dynamics in (36d) yield the closed loop tracking error dynamics m _d ri = -(k ₃ - a ₂ m _d )ri - a ₂ m _d e _d . - N _t + e _F ., (67d) where the desired actuator force is given by

F = k ₃ ri + icos0 + (68d) and the actuator force tracking error is given by e _Fi = F? - F†. (69d) k ₃ > 0 is a control gain, and the desired position dynamics N _t = m _d df is assumed to be upper bounded as follows

\N _i \≤c _i (70d) Hydraulic Actuators In order to simplify the controller design for the actuators, the spool valve dynamics is canceled by using a singular perturbation technique. This approached has been used extensively in literature for this type of problem. Consequently, the control input is designed as u _t = -K _f x _v . + -^½u _s ., i = {L, F] (7 Id) where u _s . is a slow control in time and Kf is a positive design control gain. Consequently, the valve psuedo-closed loop dynamics is given by εχ _ν . + x _v . = u _s ., (72d) where ε =— (73d) l+KKf is the perturbation constant. The pseudo-closed loop in (72d) has a quasi-steady state solution, χ _ν . (ε = 0) = x _v ., given by x _v . = u _s .. (74d) Using the fast time scale v = ^ and Tichonov's Theorem yields xvt = x _Vi + V + (75d) dv = - ⁷ Ι· ( ^76d ) where η (v) is a boundary layer correction term. It is seen that η (v) decays exponentially in the fast time scale. Typically, the time constant τ in the actual system is designed to satisfy 0 < ε « 1. Therefore, by choosing the control gain Kf large enough, the perturbation constant can be made as small as possible. As a result, η + 0 (ε) becomes negligibly small. Thus, the actuator dynamics in (37d) becomes

F = f(F?, di + g(F , x _Vi )u _Si , (77d) where f{F , d^) = ^F - _a Ad _i (78d) g(F?,Xvt) = Y ^{A p} s - sgn(x _Vi ) (79d)

Functions /(F", d _j ) and g{F , x _Vj ), hence the dynamics in (77d), contain unknown system parameters β, ctandy. Therefore, an adaptive control approach is used to design the control u _s .. Thus, the actuator force closed loop tracking error dynamics is given by e _Fl = F?-F?, (80d)

= fi ^~ f i - 9t e _Ft -Fi +jFi +Qt +f ^f + k _u e _Fl - ). (82d) The slow control u _s . is designed as follows where f _t and g _t are the estimates of f _t and g _t respectively, and the derivative Ff of the desired force is approximated using the high gain observer

where

It can be shown that the estimation error, Ff = fif — Ff decays very fast to the ball \F†\ < 0(ε ₂ ). Thus, the actuator force closed loop tracking error system becomes =fi + (^)gi -gk _{u Fi} -F _l ^d , (87d) where fi =f _i -fi = - _i F _l ^a - _i Ad _i , (88d)

9i = 9t -§t = YiA P _s - sgn(x _v .) -†, (89d) where βι =β-βν (90d) (91d)

?i = Y- Yi. (92d) are the parameter estimation errors.

Stability Analysis

The overall closed loop error system is

Is = ¾ - (k ₂ - ^s)^ - (fc _j - a {k ₂ - a^e + L ^T Qz - Yh ₂ , (93d)

¾ = -(k ₃ - a ₂ m _a )ri - ₂ m _a e _a . - N _t + e _F ., (94d) F _i =fi + 9t - gk _u e _Fi - F*. (95d) Given the adaptive update laws

K ₂ =Proj(h ₂ , -L Y^), (96d)

Q = Proj(Q, L _Q LZT , (97d) a _i = Proj(a _i , -L _a Ae _F .d), (98d) f _i = Proj(f _i , ¾(¾* -/«) « (100d) where L _ft , L _a , hp,L _y are positive adaptation gain constants, and L _Q is a positive definite adaptation gain matrix. If the control gains k _lt and/e ₂ are chosen to satisfy the following sufficient conditions k ₂ - > p ₁ +p ₂ (lOld) (102d) k ₃ > p ₄ + -, (103d)

then the closed loop system in (93d)-(95d) is uniformly ultimately bounded with respect to the closed ball (A signal x(t) is uniformly ultimately bounded (UUB) with respect to a closed ball B _r if for all r > 0, there exists T(r) such that || x(t ₀ ) \\≤r implies that x(t) G B(r), Vt > t ₀ + T) where max il _s , p ₃ , m _d , ₂ m _d , (106d) λ ₂ = min{p ₁ ,a ₁ ,ct: ₂ ,ct: ₂ m _d , :|m _d ,p ₅ ,l}, (107d) (108d)

4p ₂ 4p ₄ ^RJ 2p ₆ with e = [h ₂ Q ^T a _L a _R h β _κ f _L f _R ] ^T (109d) satisfying

Θ II≤ eg, c _e > 0. (HOd)

Proof. Consider the candidate Lyapunov function

V = r? + ^φ ² +— h\ + ¹ Q ^T L- ₀ ¹ Q

2 ¹ 2 ^ 2L _h ^z 2 ^¾ V ^ [L,R] (rn _d r _t ² + a ₂ m _d e _d ² . + e ² . (Hid)

Its time derivative is h _{2 2}

+∑i={L,R] m _d r _t r _t + a ₂ m _d e _di (r _t - a ₂ e _d .) + e _F .e _F . - ^ - ^ - ^ , (112d) which, after substituting the closed loop error dynamics (93d)-(95d) and applying the update laws (96d)-(100d), becomes

V≤ r _x {-p _x r _x - p ₂ r _x + ¾) - α _χ φ ² (113d)

+∑i={L,R] {- c ₂ m _d rf - a m _d el. - \η\(ρ ₄ \η \ - q) - p ₅ e . - |e _F .|(p ₆ |e _F .| -

0(¾))). (114d)

Using the boundedness property of the parameter estimation error in (1 lOd), due to the projection operator, the inequality in (114d) yields

V≤ -p _x rl - α _χ φ ² Θ || ² + σ

≤ -λ ₂ \\χ \\ ² +σ, (116d) where

Using the Comparison Lemma, it follows that < ¾ σ + (y (t ₀ ) - ½ σ) exp (- ½ (t - t ₀ )). (119d) Thus, any trajectory starting outside of B _r will approach B _r monotonically, and any trajectory starting inside B _r will remain in B _r . This shows that the system is uniformly ultimately bounded.

Simulation The performance of the proposed control is examined via simulation, using the NTSHA fish hook maneuver. The simulation is carried out using MATLAB/SIMULINK. In order to ensure a realistic simulation, the dynamic parameters are estimated so that they match experimental data well. The vehicle used for the data collection is a Toyota Highlander Hybrid 2007 equipped with Inertial Measurement Unit, shown in Fig. 42 during one of the maneuvers. Two sets of data were collected. The first is termed the Circle Data, in which the car is driven around cones arranged on in a circular fashion. The second is termed the Eight Data. Here, the vehicle is driven several times along an eight-shaped path. Fig. 42 also shows the corresponding trajectories of the vehicle. The data collected for each experiment includes the longitudinal and lateral velocities, lateral acceleration, roll angle and roll rate. The parameters of the model are estimated using the trust-region-reflective method in MATLAB. Figs. 42 and 43 show validations of the estimated parameters against new datasets which were not used for the estimation process. As can be seen in the figures, there are discrepancies in the high frequency components. The reason is primarily because the compliance of the tire is neglected in the roll dynamic model. However, this does not create a concern because the high frequency roll components are not perceived by the passenger and can effectively be damped out by using an appropriate damper in the suspension elements. Moreover, the purpose of the parameter estimation is so that fairly reasonable parameter values can be obtained for the simulation.

Fig. 44 are graphs of NTSHA fishhook maneuver.

The resulting estimated system parameter values used in the subsequent simulations are given in

Table 1.

Table 1: ESTIMATED PARAMETER VALUES

Parameter Value m _s 777.4 kg 210.86 kgm ²

The values of the hydraulic parameters for the hydraulic system considered were obtained empirically, and are given in Table 2.

Table 2: HYDRAULIC PARAMETER VALUES

Parameter Value a 4.515 x 10" N/m g ^1/2

Fish hook Maneuver

The Fish hook maneuver, by NHTSA, is a very useful test maneuver in the context of rollover, in that it attempts to maximize the roll angle under transient conditions. The procedure is outlined as follows, with an entrance speed of 50mph (22.352m/s):

1. The steering angle is increased at a rate of 720 deg/s up to 6.55 _stat , where 5 _stat is the steering angle which is necessary to achieve 0.3g stationary lateral acceleration at 50mph.

2. This value is held for 250ms.

3. The steering wheel is turned in the opposite direction at a rate of 720deg/s up to -

6.55, stat-

The steering angle to the wheels, and the resultant trajectory of the vehicle, for the fish hook maneuver is shown in Fig. 44.

In the subsequent figures, the subscripts R and L are used to denote the corresponding right and left quantities respectively. Figs. 45 and 46 show the resulting control masses and roll responses respectively, where the constant and variable stiffness cases are plotted together for comparison. These results show that by using the variable stiffness mechanism together with the developed control algorithm, the roll angle and roll rates are reduced by more than 50%. As can also been seen from Fig. 45, the maximum displacement of the control masses is set as 8cm. This, compared to the track width of typical passenger car leaves enough room for the engine and other internal components. Moreover, this limit can be adjusted by changing the value of Δ in (57d) and (58d). For comparison sake, additional simulation is carried with Δ = 4cm. The result is compared with the constant stiffness case and the variable stiffness case with Δ = 8 cm, as shown in Fig. 46. It is seen that restricting the maximum displacement of the control mass to 4cm results in a 25% reduction in the roll angle and roll rates, as against 50% reduction achieved by setting Δ = 8cm. Consequently, it is stated that the value of the control parameter Δ can be chosen depending of the design constraints specified by the designer. The closer to zero the value is, the closer the resulting performance to the constant stiffness counterpart. It is also seen that the control allocation exhibit some ganging phenomenon.

The corresponding voltage input with the associated hydraulic force outputs, spool valve responses, and instantaneous fluid power from the hydraulic system are shown in Figs. 47 and 48. The voltage command is saturated by ±2V. This is done to conform with the maximum voltage requirement of the hydraulic actuator. Allocation Procedure

In light of the special structure of the allocation problem in (56), an optimization problem is defined as follows: Find the values of x, and y that minimize the function f(x, y) = \ax + by + c\, a, b, c, x, y £ R " , (120d) subject to the constraint -A≤x, y≤A. (121d)

Now, since min f(x, v) = min j min f(x, v) \, (122d)

-A≤x,y≤A ^{J J J} -A≤y≤A -A≤x≤A ^{J J )} the solution is approached sequentially. First, a minimization is carried out with respect to x. Then, the residual objective function is minimized with respect to y. Let g(y) = by + c, (123d) then f x, y) = \ax + g y \. (124d)

Minimizing with respect to x yields the minimizer x ^* given by

- αϋρ (- ^ , -Δ, Δ), (126d) and the residual objective function becomes f x ^* , y) = by + ax ^* + c. (127d) Minimizing this residual function with respect to y, subject to the constraint— Δ < y < A, yields the minimizer y ^* given by

Conclusion

The improvement of the roll dynamics of road vehicles using the variable stiffness suspension system has been described. The nonlinear yaw and roll dynamic models are developed for the system and validated using experimental data. An adaptive control is developed for the hydraulic actuators to appropriately modulate the positions of the control masses in order to generate a counter roll moment, thereby improving the roll stability of the resultant system. A Lyapunov-based stability analysis shows that the closed loop system is uniformly ultimately bounded. The effectiveness of the proposed system is demonstrated via simulations. The simulated Fishhook maneuvers shows that the proposed system has over 50% reduction in roll angle and roll rates compared to the constant stiffness counterpart.

The description of roll stabilization using the variable stiffness suspension system has now been presented. It is foreseeable that other linear actuators (such as electrical, magneto-rheological, etc.) can be used to drive the control masses. It is foreseeable that hardware-in-the-loop simulations can be used to examine the performance of the system. It is foreseeable that a combined vibration isolation/roll stabilization performance enhancement can be utilized. It is also foreseeable that a combined pitch/roll stabilization control can be developed using a full car model. Although specific embodiments have been illustrated and described herein, it should be appreciated that any arrangement calculated to achieve the same purpose may be substituted for the specific embodiments shown. The examples herein are intended to cover any and all adaptations or variations of various embodiments. Combinations of the above embodiments, and other embodiments not specifically described herein, are contemplated herein.

The Abstract is provided with the understanding that it is not intended be used to interpret or limit the scope or meaning of the claims. In addition, in the foregoing Detailed Description, various features are grouped together in a single embodiment for the purpose of streamlining the disclosure. This method of disclosure is not to be interpreted as reflecting an intention that the claimed embodiments require more features than are expressly recited in each claim. Rather, as the following claims reflect, inventive subject matter lies in less than all features of a single disclosed embodiment. Thus, the following claims are hereby incorporated into the Detailed Description, with each claim standing on its own as a separately claimed subject matter.

The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. As used herein, the singular forms "a", "an" and "the" are intended to include the plural forms as well, unless the context clearly indicates otherwise. It should be understood that the terms "comprises" and/or "comprising," when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. The term "another", as used herein, is defined as at least a second or more. The terms "including" and "having," as used herein, are defined as comprising (i.e., open language). The term "coupled," as used herein, is defined as "connected," although not necessarily directly, and not necessarily mechanically. "Communicatively coupled" refers to coupling of components such that these components are able to communicate with one another through, for example, wired, wireless or other communications media. The term "communicatively coupled" or "communicatively coupling" includes, but is not limited to, communicating electronic control signals by which one element may direct or control another. The term "configured to" describes hardware, software or a combination of hardware and software that is adapted to, set up, arranged, built, composed, constructed, designed or that has any combination of these characteristics to carry out a given function. The term "adapted to" describes hardware, software or a combination of hardware and software that is capable of, able to accommodate, to make, or that is suitable to carry out a given function.

The terms "controller", "computer", "processor", "server", "client", "computer", "computer system", "personal computing system", or "processing system" describe examples of a suitably configured processing system adapted to implement one or more embodiments herein. Any suitably configured processing system is similarly able to be used by embodiments herein, for example and not for limitation, a personal computer, a laptop computer, a tablet computer, a smart phone, a personal digital assistant, a workstation, or the like. A processing system may include one or more processing systems or processors. A processing system can be realized in a centralized fashion in one processing system or in a distributed fashion where different elements are spread across several interconnected processing systems.

The terms "computer", "computer system", and "personal computing system", describe a processing system that includes a user interface and which is suitably configured and adapted to implement one or more embodiments of the present disclosure. The terms "network",

"computer network", "computing network", and "communication network", describe examples of a collection of computers and devices interconnected by communications channels that facilitate communications among users and allows users to share resources. The terms "wireless network", "wireless communication network", and "wireless communication system" describe a network and system that communicatively couples computers and devices primarily or entirely by wireless communication media. The terms "wired network" and "wired communication network" similarly describe a network that communicatively couples computers and devices primarily or entirely by wired communication media.

The corresponding structures, materials, acts, and equivalents of all means or step plus function elements in the claims below are intended to include any structure, material, or act for performing the function in combination with other claimed elements as specifically claimed. The description herein has been given for purposes of illustration and description, but is not intended to be exhaustive or limited to the examples in the form disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the examples described or claimed. The disclosed embodiments were chosen and described in order to best explain the principles of the embodiments and the practical application, and to enable others of ordinary skill in the art to understand the various embodiments with various modifications as are suited to the particular use contemplated. It is intended that the appended claims below cover any and all such applications, modifications, and variations within the scope of the embodiments.

The terms "a" or "an", as used herein, are defined as one or more than one. The term plurality, as used herein, is defined as two or more than two. The term another, as used herein, is defined as at least a second or more. The terms including and/or having, as used herein, are defined as comprising (i.e., open language). The term coupled, as used herein, is defined as connected, although not necessarily directly, and not necessarily mechanically.

Unless stated otherwise, terms such as "first" and "second" are used to arbitrarily distinguish between the elements such terms describe. Thus, these terms are not necessarily intended to indicate temporal or other prioritization of such elements.

Although specific embodiments of the invention have been disclosed, those having ordinary skill in the art will understand that changes can be made to the specific embodiments without departing from the spirit and scope of the invention. The scope of the invention is not to be restricted, therefore, to the specific embodiments, and it is intended that the appended claims cover any and all such applications, modifications, and embodiments within the scope of the present invention.

What is claimed is: