DESCRIPTION

The present invention relates to a flexible joint for rotary couplings.

Flexible couplings were born with the idea of replacing traditional revolving couplings due to the advantages they can provide. They exploit the large deflections of flexible elements to obtain the required motion. Compared to traditional mechanisms, mechanisms based on flexible joints have less friction and play, no need for lubrication, possibility of manufacturing at the macro and micro scale (using the technologies of microelectronics and micro electromechanical systems).

Among the various flexible joints, the so-called Cross Axis with rectilinear beams is known, described, for example, in J. Paros and L. Weisbord, «How to design flexure hinges», Machine Design, p. 151-156, 1965. It can be generically schematized with two beams joined at a base and crossed at their midpoint; the beams then connect through a joint to an ideally rigid element that acts as a mobile platform. In literature this mechanism has been studied from different points of view, for example geometric, kinematic, precision and static in order to meet the requirements of different applications. For example, one can mention its implementation in robotic limbs or its application in the prosthetic field as described, for example, in P. Bilancia, M. Baggetta, G. Berselli, L. Bruzzone and P. Fanghella, «Design of a bio-inspired contact-aided compliant wrist», Robotics and Computer Integrated

Manufacturing, 2021.

On the other hand, although reliable, flexible joints in general, and Cross Axis with rectilinear beams in particular, unlike traditional kinematic couplings, do not guarantee a fixed relative rotation center between the bodies they connect, an aspect which, above all in precision fields such as robotics, makes their application more problematic.

The object of the present invention is, therefore, to provide a selective compliance joint configured to increase the accuracy of the known selective compliance joints, in particular of the Cross Axis with straight beams, in terms of rotation or to limit the displacement of the axis of relative rotation between the connected bodies, a phenomenon which in the literature is also defined as "axis drift", "center shift" or "parasitic motion".

The invention achieves the object with a flexible joint comprising two beams of fixed length crossed and interlocked at one end to a fixed base and at the opposite end to a rigid mobile platform which moves with respect to the fixed base. The beams are initially curved, have an inclined axis of symmetry and are initially arranged symmetrically so that the centers of gravity coincide.

In this way, by acting on just two parameters, the radius of curvature or, equivalently, the semi- opening angle (given that the length of the beams is fixed) and the inclination of the axis of symmetry of the beams, it is possible to optimize the displacement of the center of rotation, in particular reducing this displacement to a minimum value which makes the flexible joint very similar to a revolving couple.

Further characteristics and improvements are the subject of the subclaims.

The characteristics of the invention and the advantages deriving from it will become clearer from the following detailed description of the attached figures, in which: fig. 1 shows a three-dimensional model of a flexible joint with rectilinear beams according to the prior art. fig. 2 shows the joint of the previous figure at rest and with the mobile platform AB rotated with the pole of the displacements of the mobile platform with respect to the fixed frame highlighted. fig. 3 shows the parametric model of a joint according to an embodiment of the invention. fig. 4 shows how the center of rotation can be graphically determined when a rigid mobile element moves from the AB position to the A’B’ position. fig. 5 shows a comparison of the deformation of the Cross Axis with straight beams (a) and of the Cross Axis with curvilinear beams (b) in a first configuration obtained for a specific value of the radius of curvature R with angle β equal to 45 degrees or equal to arctan (b/a) of fig. 3. fig. 6 shows the module of the displacement of the center of rotation as the radius of curvature R varies with angle £ equal to the previous case. fig. 7 shows the position of the center of rotation as β and R vary in a joint according to an embodiment of the invention. The Cross Axis Flexural Pivot (CAFP) is a flexible joint having the purpose to provide relative rotation of the parts it connects. It comprises yielding elements that exploit large deflections and displacements to obtain the desired motion. The yielding elements are two beams superimposed and fixed to the two bodies which rotate relatively. In the generalized configuration of such a joint, the beams overlap at their midpoint and have a rectilinear axis as shown in Fig. 1.

In these joints, the center of rotation, known as the "pole of displacements" or "center of finite plane rotation", moves as the imposed rotation or momentum applied to the moving platform increases. This phenomenon is defined in the literature as axis drift, center drift or parasitic motion as shown in Fig. 2.

The inventors have surprisingly realized that by introducing some geometric modifications to the known joints it is possible to obtain a considerable impact on the accuracy of the joint quantified by the displacement module of its center of rotation. With reference to Fig. 3, this modification can be summarized in the following points:

• introduction of an initial curvature in the flexible beams, defined by the semi-angle a subtended by the axis of the beams AC, BD;

• relative positioning of the beams with respect to the corresponding centers of gravity G, placed in coincidence with the ideal rotation center of the joint.

• variation of the inclination angle β of the symmetry axis OG of the flexible beams; By making these modifications to the geometry of the straight-sided CAFP, the joint behavior approaches that of an ideal revolving pair, whose center of rotation, ideally, does not move.

This results in a significant increase in the accuracy of the joint.

Now let’s see some mathematical expressions that help to understand such a result.

Fig. 3 shows a straight beam cross axis NMEF. This cross axis is formed by a rigid platform EF which moves with respect to a fixed frame NM. The movement takes place through the deflection of the straight beams ME and NF, both of length (1), which overlap in their midpoint (G).

The same figure shows a curvilinear beam cross axis composed of a platform AB moving with respect to a fixed frame DC. The motion takes place thanks to the deflection of the curved beams DB and CA. Curved beams have the same length as straight beams, i.e. (1). Unlike the rectilinear cross axis, the curved beams do not overlap at their midpoint, but are arranged symmetrically in such a way that their center of gravity coincides with point (G).

The positioning constraint of the center of gravity, combined with the fixed length (1), determines the possibility of creating infinity with two cross axes by varying the semi-angle of opening of the beams (alpha) and the inclination of their axis of symmetry (beta).

Let us now consider the arch on the left shown in Fig. 3. The characterization of the one on the right will then take place by considering the symmetry of the structure. Given an arc of a circle, its length 1 can be described by:

Where R is the radius and a is the semi-aperture angle of arc AC. Therefore, once the length 1 of the arc is known, its curvature can be expressed indistinctly with R or a.

Calling 0 the center of curvature of the arc and G the center of gravity of the arc, placing a global reference system XY in G, it is possible to demonstrate that the distance of the center of gravity G from the center 0 is

Calling β the angle of inclination of the axis of symmetry of the arc with respect to the global X axis, the coordinates of the point 0 can be written as

The Cartesian coordinates (x, y) of the ends (A,

C) of the curved beam are given by

Given that the length of the beams of the two models must be the same, it is possible to define a value of β for which, if the radius R tends to infinity, the arches tend to transform into homologous beams belonging to the rectilinear case. This value of β, called βeq is described by the following equation:

Where b and a are the lengths defined in figure 2.

The displacement of the center of gravity of the free section of a curvilinear beam fixed at one end has been studied in the literature. For example, refer to Verotti, M. (2016) "Analysis of the center of rotation in primitive flexures: Uniform cantilever beams with constant curvature". Mechanism and Machine Theory, 97, 29-50 to be considered an integral part of this description.

It is possible to graphically interpret the position of the center of rotation as shown in Fig. 4. Assume that the rigid body S identified by the segment undergoes a finite rotation of value φ. This segment is supposed to be integral with the curved beam in U and perpendicular to its axis at that point. The center of rotation can be found:

1.Drawing the joining lines and of ends A and B of the segment;

2.After point 1, drawing the perpendiculars to the segments and in their midpoints MA and MB;

3.The intersection point at point 2 is the rotation center Cr for the deformed configuration .

It can be shown that, if the ratio between the bending moment and the stiffness is constant along the axis of the fixed beam, the center of rotation during deformation:

• lies on the axis of symmetry of the arc that describes the beam in the undeformed configuration (the position of this axis in space is controlled by the angle β);

• tends to the center of gravity G of the arc that describes the beam in the undeformed configuration, when the ratio between the bending moment and the flexural stiffness tends to zero;

• tends to infinity when the free section in U tends to a complete rotation.

The complete mathematical steps are beyond the scope of the present description. What is important to underline here is that, referring for example to the cross axis of fig.3, given the length 1 of the beams and the initial position of their center of gravity, only two parameters (α,β) are sufficient for a complete characterization of the joint according to the invention.

We will see in the following pages what is the behavior of the rotation center of the joint according to the invention as a (or equivalently R given the relationship) and β varies as determined by numerical simulations in comparison with the corresponding straight beam joint.

Behavior of the center of rotation when R varieas

The results of the analyzes described below are obtained for values of β = β _eq , in order to compare the displacement of the center of rotation of the two cases in a manner solely dependent on the presence of curvature of the beams.

The analyzes were performed with the following numerical values of the characteristic parameters of the system:

• a = 40 mm is the base of the Cross Axis with straight beams;

• b = 40 mm is the height of the Cross Axis with straight beams;

• 1 = 56.57 mm is the length of the beams (deducible from a and b)

• E = 2500 MPa is Young’s modulus

• t1 = 3 mm is the side parallel to the axis of rotation of the section during bending;

• t2 = 5 mm is the side generating the moment of inertia of the section;

• β = β _eq =45°;

• R = 50 mm is the radius of curvature of the beams of the curvilinear beam model;

During all the analyzes performed, it was preferred rather than applying a pure moment, to apply a rotation to the platform; in this way the displacements of the rigid body are independent of the applied stress. To compare the displacement of the center of rotation between the case with straight beams and the case with curved beams, an anticlockwise rotation of φ==5° was applied.

Figures 5a and 5b show the two models in their deformed configurations, after applying the assumption of large displacements through the Ansys code. The numerical values of the displacements of the left end (KP1 in fig. 5a and 5b) of the rigid body were extracted. The displacement of the center of rotation is therefore evaluated considering the single beam and applying, to the analytical formulas (1) and (2) seen above, the numerical values relating to the displacement of the end of the beam integral with the rigid body (KP1).

In figures 5a and 5b it is possible to superimpose the results of the analysis; in all subsequent figures; the node E is the centerline node of the segment describing the rigid body.

The following comparison table shows the most interesting data relating to the behavior of the two models. The displacements of the center of rotation coincide with the final positions of the center of rotation, since the initial positions of the center itself coincide with the origin of the reference system shown in figures 5a and 5b (located at point G, corresponding to the coincident center of gravity of the two arches).

The magnitude of the displacement of the center of rotation, adding a curvature to the flexible beams, was reduced by 4.5%.

The analyzes also show that the center of rotation of the rigid body has a zero percentage displacement along the y axis in the case with straight beams, while in the case with curvilinear beams it is greater than in the straight case, but still limited compared to the modulus of the overall displacement (it is 3.93% of the modulus of the rotation center displacement).

Ultimately, through the non-linear numerical analyses, it was possible to ascertain that the displacement of the rotation center of the rigid body is sensitive to the presence of the curvature of the beams; if the goal is to improve the precision of this mechanism, then it is advisable to adopt the configuration with curvilinear beams.

It is now interesting to understand whether, again as R varies, there can be greater reductions in the magnitude of the displacement of the center of rotation; in particular we want to understand what is the trend of the displacement module of the center of rotation (from now on it will sometimes be called MSC for convenience) as a function of the radius of curvature .

Analyzes were performed on the same numerical model (with the parameters having the same values used for the previous analyses), iterating the search for the solution on different R values: this parameter was varied between 20 mm and 200 mm, with 50 intermediate iterations, i.e. increasing the radius by 3.6 mm at each iteration. The result obtained is explained in the graph shown in Fig. 6.

Observing the figure, it can be seen that the MSC grows very rapidly, in particular it stabilizes asymptotically at a value which in the case studied is equivalent to the MSC of the rectilinear case, i.e. 5.06 mm. The value of the MSC corresponding to the minimum radius (20 mm) is 3.71 mm. Compared to the rectilinear case there is a reduction of the MSC of 26.68%, which is by no means negligible in the design of the mechanism.

Behavior of the center of rotation when β varies

In the previous analyses, the behavior of the Cross Axis was studied as the radius of curvature varied. It was found that the presence of the curvature reduces the displacement of the center of rotation of the rigid element, thus making the mechanism more precise.

Let us now see the behavior of the center of rotation of the rigid body of the Cross Axis as the angle β of inclination (with respect to the global x axis) of the axis of symmetry of the arc that describes the beam varies. We want to understand if to improve the precision of the mechanism it is convenient to act on this angle and if this can have an influence at least comparable or greater on the MSC than that of the bending radius.

A series of analyzes was therefore carried out on a cross axis parametric model with curvilinear beams, iterating the solution for different values of β and different values of a. In particular, for each of the 50 a values ranging from 0° to 180°, 50 β values ranging from 0° to 90° were analysed, for a total of 2500 analyses. Fig 7 shows the displacement of the center of rotation of each configuration divided by the displacement of the cross axis at straight beams with β = 45° ( ). The parameters describing the model are shown below and are the same as those used in the analyzes as a function of the radius of curvature, obviously except for R and β:

• a = 40 mm is the base of the Cross Axis with straight beams;

• b = 40 mm is the height of the Cross Axis with straight beams;

• 1 = 56.57 mm is the length of the beams;

• E = 2500 MPa is Young’s modulus;

• tl = 3 mm is the side along the axis of rotation of the section during bending;

• t2 = 5 mm is the side generating the moment of inertia of the section;

With reference to Figure 7, the displacement of the center of rotation of the curvilinear cross axis, normalized with respect to the rectilinear cross axis, significantly depends on both a and β. According to the figure, there is an improvement of the accuracy when parameter d is less than one. For example, for the curvilinear cross axis with a = 132° and β = 83°, then d = 0.14. This means that the curvilinear cross axis has a displacement of the center of rotation 86% lower than the reference straight beam model. All this occurs for a 90° rotation of the mobile platform.

In general, the angle a is typically selected in the range 0°-180°, particularly 90°-180° while the angle β is typically selected in the range 0°-90°, particularly 45°-90°.

The joint optimized according to the above criteria has the typical advantages of flexible mechanisms: no friction, no wear, no lubricant needed, monolithic production at the micro- and macro-scale. Furthermore, thanks to the proposed optimization, the disadvantage relating to the displacement of the rotation axis is considerably reduced, allowing its use to be extended to all applications where high rotation precision is required.