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Title:
A MULTI-DOF SENSOR FOR AN INDUSTRIAL ROBOT
Document Type and Number:
WIPO Patent Application WO/2009/106407
Kind Code:
A3
Abstract:
The invention relates to a multi-DOF sensor for an industrial robot, which robot has at least two DOF. The multi-DOF sensor is arranged for sensing at least one force and is constituted by a plurality of one-DOF sensors (11, 12). A first end face of each one-DOF sensor (11, 12) is connected to a surface of a first body (31 a, 31 b) and a second end face of each one-DOF sensor (11, 12) is connected to a surface of a second body. The invention also relates to an industrial robot comprising the invented multi-DOF sensor.

Inventors:
BROGAARDH TORGNY (SE)
Application Number:
PCT/EP2009/051213
Publication Date:
December 23, 2009
Filing Date:
February 04, 2009
Export Citation:
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Assignee:
ABB TECHNOLOGY AB (SE)
BROGAARDH TORGNY (SE)
International Classes:
G01L5/16
Foreign References:
JP2005091106A2005-04-07
US5821431A1998-10-13
JPH07140025A1995-06-02
Attorney, Agent or Firm:
JOHANSSON WEBJÖRN, Ingmari (Intellectual PropertyIngenjör Bååths Gata 11, Västerås, SE)
Download PDF:
Claims:

CLAIMS

I . A multi-DOF sensor for an industrial robot, the robot having at least two DOF, the multi-DOF sensor being arranged for sensing at least force in 2 DOF, wherein the multi-DOF sensor is constituted by a plurality of one-DOF sensors (1 ,

I I , 12), each one-DOF sensor having a first end face (2) connected to a first surface (21 a, 21 b) of a first element and a second end face (3) connected to a surface (31 a, 31 b) of a second element.

2. A multi-DOF sensor according to claim 1 and having two DOF sensing two force DOF or one force DOF and one torque DOF.

3. A multi-DOF sensor according to claim 1 and having three DOF, two or three are sensing force DOF and the other(s) torque DOF.

4. A multi-DOF sensor according to claim 1 and having four DOF of which two or three are sensing force DOF and the other(s) torque DOF.

5. A multi-DOF sensor according to claim 1 and having five DOF of which two or three are sensing force DOF and the other(s) torque DOF.

6. A multi-DOF sensor according to claim 1 and having six DOF of which three are sensing force DOF and three are sensing torque DOF.

7. A multi-DOF sensor according to any of claims 2-6 and including an element having at least one planar surface to which at least two one-DOF sensors (1 ) are connected with their respective first face (2).

8. A multi-DOF sensor according to any of claims 2-7 and including an element having two or three orthogonal surfaces to which each at least one one- DOF sensor (1 ) is connected with its first face (2).

9. A multi-DOF sensor according to any of claims 2-8 and including an element having at least one V-shaped portion and wherein in each said portion

one pair of one-DOF sensors (11 , 12) is arranged such that the first face (2) of each one-DOF sensor (11 , 12) in the pair is connected to a respective surface (21 a, 21 b) of the V-shaped portion.

10. A multi-DOF sensor according to any of claims 3-9 and including an element having a pyramid shaped portion, on which a group of three one-DOF sensors is arranged such that the first face (2) of each one-DOF sensor (1 ) in the group is connected to a respective surface (21 c, 21 d) of the pyramid shaped portion.

11. A multi-DOF sensor according to claim 10, wherein the element has three such V-shaped portions.

12. A multi-DOF sensor according to any of claims 1 -11 , wherein at least one of said surfaces (21 a, 21 b, 31 a, 31 b, 21 c, 31 c) is resiliently arranged on the respective element.

13. A multi-DOF sensor according to claim 12, wherein said resiliently arranged surface (21 a, 21 b, 31 a, 31 b, 21 c, 31 c) is the surface of a membrane.

14. A multi-DOF sensor according to claim 6, wherein five of the one-DOF sensors (1 ) are connected to a respective surface (21 d, 21 e, 21 f) of an element that is a rod, on which all five said surfaces are located.

15. A multi DOF-sensor according to claim 6, characterized in that it has three pairs of one-DOF sensor corresponding to a kinematic coupling arrangement.

16. A multi DOF-sensor according to claim 6, characterized in that it is arranged as a Delta-coupling or a modified Delta-coupling, based on the kinematics structure of the Delta parallel robot.

17. A multi DOF-sensor according to claim 6, characterized in that it it is arranged as a Tau-coupling or a modified Tau-coupling, based on the structure of the TAu parallel robot.

> 18. An industrial robot including a multi-DOF sensor according to any of claims 1 -17.

Description:

A MULTI-DOF SENSOR FOR AN INDUSTRIAL ROBOT

Field of the invention

The present invention relates to a multi-DOF sensor for an industrial robot, the robot having at least two DOF, the multi-DOF sensor being arranged for sensing at least one force DOF.

DOF is the abbreviation of Degrees Of Freedom.

Background of the invention

Industrial robots usually require control of forces and torques appearing in the joints of the robot. Such control includes measuring the forces and torques for which sensors are required.. Since usually the forces to be measured are located at a distance from the sensor, these forces are usually calculated from torques- and force measurement signals from the sensor. This means that the sensor usually needs more DOFs than the number of DOFs to be measured. How many DOFs that will be needed in a particular application depend on how many DOFs that need to be measured and the placement of the sensor. There will also be restrictions with respect to the number of DOF of the robot. In the general case when using a six-DOF robot a six-DOF sensor being capable of sensing three force DOFs and three torque DOFs is used.

Thus various kinds of multi-DOF sensors have been developed for this purpose. Examples on such sensors can be found e.g. in the following disclosures: EP 1218652688 discloses a force sensor chip having a multiaxial force sensing function. The force sensor chip comprises a base member having an operating part provided with an external force application area, a supporting part for supporting the operating part, connecting parts for connecting the operating part and the supporting part, and strain resistance elements disposed in the connecting parts or within boundaries between the connecting parts and the operating part.

EP 1327870 discloses six axis force sensor for detecting force on robot, in which multiple connecting portions have multiple strain resistant devices consisting of active layer formed on one of front and rear surfaces.

EP 1653208 discloses a multi-axis force sensor chip for e.g. robot, has strain resistance devices provided on deforming areas of connecting parts, and temperature-compensating devices provided on non-deforming areas of action part. JP 04134231 discloses a semiconductor three-axis force sensor, which has silicon single crystal plate forming diffusion strain gauge and detects X, Y-axis moments through separate diaphragms.

JP 04249728 discloses a semiconductor triaxial force sensor with counterweight mounting shaft. A detecting arm of a force sensor extended in a direction passing through a diaphragm. A counterweight is fixed to an end of the mounting shaft. The counterweight is arranged not to be larger than the outer diameter of the force sensor.

JP 2001264198 discloses a multi-axis force sensor for cell, has elastic beam formed with P-type diffusion area doped with N-type impurity. JP 1223626 discloses a sensor in which center and outside peripheral supporting bodies and beams connecting tama are constituted into one body with common semiconductor materials, and semiconductor gauge resistances are buried on sides of supporting bodies of the beam.

US 5526700 discloses a six component force gage for measuring displacement between two bodies along orthogonal axes (x, y and z) including a semiconductor substrate carrying on its surface a plurality of piezoresistors supported by a structure attached to the two bodies that transforms the force and moment into deformation components applied in the plane of the substrate surface there enabling the sensors to be fabricated economically en masse using IC manufacturing techniques.

WO 2000023778 discloses a position detector for a detecting a mobile element using an inexpensive detecting cell made of micromachined silicon comprising: a micromachined monolithic chip having a fixed part and a mobile part subject to displacements, arms subject to deformation according to the displace- ments of the mobile part; electric means for measuring the deformations; coupling means between the mobile element and the mobile part of the monolithic chip capable of moving the mobile part according to the mobile displacement.

WO 8808521 discloses a force and moment detector using resistor and has elements on semiconductor substrate forming bridge circuit across which voltage is read to determine 3-d forces.

WO 9000735 discloses a multidimensional force sensor in single integral unit. The sensor has flexible beams and response elements acting on sensor which give electrical outputs proportional to number of independent forces.

Many of the prior art multi-DOF sensors for industrial robots are expensive to manufacture and sensitive to disturbances. It is important to hold the costs for the sensor at a reasonable level. It is also necessary that a multi-DOF in an industrial robot is highly reliable and accurate.

The object of the present invention therefore is to meet this demand and thus solve the problem to provide a multi-DOF that is cheap and reliable.

Summary of invention The object of the invention is met in that a multi-DOF sensor initially specified includes the specific features that it is constituted by a plurality of one- DOF sensors, each one-DOF sensor having a first end face connected to a surface of a first body and a second end face connected to a surface on a second body. One-DOF sensors are today available at very low costs. Such a sensor is also robust and reliable. It produces an output signal that distinctly represents the force applied to the sensor. Combining such sensors in a mechanically well defined relationship results in a multi-DOF sensor that maintains the accuracy and robustness of the component one-DOF sensors. The one-DOF sensors can be combined in various optimal ways for obtaining various layouts and allowing various combinations regarding force sensing and torque sensing. The multi-DOF sensor thus solves the stated problem and meets the demands for application in an industrial robot.

According to a first alternative preferred embodiment the multi-DOF sensor has two DOF and senses two force DOF or one force DOF and one torque DOF.

According to another alternative preferred embodiment the multi-DOF sensor has three DOF, of which one, two or three sense force DOF and the other(s) torque DOF.

According to still another alternative preferred embodiment the multi-DOF sensor has four DOF, of which two or three sense force DOF and the other(s) torque DOF.

According to still another alternative preferred embodiment the multi-DOF sensor has five DOF of which two or three sense force DOF and the other(s) torque DOF.

According to still another alternative preferred embodiment the multi-DOF has six DOF, of which three are sensing force DOF and three are sensing torque DOF. With the above mentioned alternative embodiments the multi-DOF sensor can be adapted to the various needs related to various kinds of applications with industrial robots of various number DOF.

According to a further preferred embodiment the multi-DOF sensor includes a body having at least one planar surface to which at least two-one DOF sensors are connected with their respective first face.

According to a further preferred embodiment the multi-DOF sensor includes a body having at least one V-shaped portion, and one pair of one-DOF sensors is arranged in each said portion such that the first face of each one-DOF sensor in the pair is connected to a respective surface of the V-shaped portion. According to a further preferred embodiment the multi-DOF sensor includes a body having a pyramid shaped portion, and a group of three one-DOF sensors is arranged on the portion such that the first face of each one-DOF sensor in the group is connected to a respective surface of the pyramid shaped portion.

The embodiments specified nearest above represent various alternatives for arranging the one-DOF sensors. The various alternatives have various advantages depending on how the multi-DOF is applied, how many of the DOF which relate to force or torque respectively and other operating conditions of the robot.

The various arrangements can also be combined in different ways for obtaining an optimized total layout of the multi-DOF sensor.

According to a further preferred embodiment the multi-DOF sensor has a body with three V-shaped portions of the kind described in an embodiment above.

Such an embodiment is particularly advantageous for a multi-DOF sensor in a robot having six DOF and through which three force DOF and three torque DOF can be measured very accurately.

According to a further preferred embodiment at least one of the surfaces to which a one-DOF sensor is connected is resiliently arranged on the respective body. Thereby the accuracy is increased in certain kinds of applications.

According to a further preferred embodiment the resiliently arranged surface is the surface of a membrane. Connecting the one-DOF sensor to a membrane is an easy and reliable way of obtaining the resiliency. According to a further preferred embodiment five of the one-DOF sensors are connected to a respective surface of a body that is a rod, on which all five said surfaces are located.

This gives the possibility to mount the sensors inside for example a handle for lead through programming.

According to a preferred embodiment the multi-DOF sensor has three pairs of one-DOF sensors corresponding to a kinematic coupling arrangement.

A traditional kinematic coupling is a very effective and precise arrangement for constraining six-DOF and when applying such a kinematic coupling concept to a six-DOF sensor the sensor will be optimal.

Also a six-links parallel kinematics manipulator with only axial forces in its links has a geometry, which applied to a six-DOF sensor will give an optimal sensor lay out. According to a further preferred embodiment the multi-DOF sensor is arranged as a Delta-coupling or a modified Delta-coupling, based on the kinematics structure of the Delta parallel robot.

A Delta-coupling is a well defined structure within the field of kinematic coupling arrangements and includes three pairs of links. According to a further preferred embodiment the multi-DOF sensor is arranged as a Tau-coupling or a modified Tau-coupling, based on the kinematics structure of the Tau parallel robot.

A Tau-coupling is a well defined structure within the field of kinematic coupling arrangements and includes three clusters of links, one cluster having three links, one cluster having two links and one cluster having one link.

The Delta-coupling and the Tau-coupling are well adapted for applying a multi-DOF sensor according to the invention, and result in a compact arrangement and simple manufacture.

The above described preferred embodiment are specified in the dependent claims.

According to a further aspect of the invention an industrial robot includes a multi-DOF sensor according to the present invention or any of the preferred embodiments thereof.

With an industrial robot having the invented multi-DOF sensor corresponding advantages as described above are gained.

The invention will be nearer explained in the following detailed description of its working principle and illustrative examples thereof with reference to the accompanying drawings.

Brief description of drawings

Fig. 1 is a schematic perspective view of a one-DOF force sensor. Fig. 2 is a schematic perspective view of two one-DOF force sensors in a groove.

Fig. 3 is a side view of the arrangement of fog. 2.

Fig. 4 schematically illustrates a kinematic coupling in a perspective view.

Fig. 5 is a theoretical representation of the device of fig. 4. Fig. 6a is a section through a part of fig. 4.

Fig. 6b is a physical representation analogue to fig. 6a.

Fig. 7 is a coordinate system for a six-DOF sensor.

Fig. 8 illustrates a six-DOF sensor with overload protection.

Figures 9 and 10 are two different theoretical representations of a hexapod structure.

Figures 11 and 12 are two different theoretical representations of a Delta structure.

Figures 13 and 14 are two different theoretical representations of a Tau structure.

Figures 15-20 are theoretical representations of modified Delta and Tau structures, each structure represented in two different ways.

Fig. 21 is a section through a detail of an example of the invented multi-

DOF sensor. Fig. 22 is a perspective view of a detail of an example of the invented multi-DOF sensor in a modified Tau structure.

Fig. 23 is a perspective view of the detail of fig. 22 according to an alternative example.

Figures 24-52 are theoretical representations of various examples of the multi-DOF sensor according to the invention.

Figures 53-57 are alternative theoretical representations of some of the examples illustrated in figures 24-52.

Figures 58 and 59 are physical representations of figures 55 and 57 respectively.

Detailed description of examples

A multi-DOF sensor according to the invention is constituted by a plurality of one-DOF force sensors. When just the term sensor is used in the following text, a one-DOF force sensor is meant. Such sensors are widely used for automotive applications. Initially a one-DOF sensor will be described.

Fig. 1 in a schematical perspective view illustrates a one-DOF sensor 1. The shape of the sensor can be simplified by a circular plate having a first end face 2 and a second end face 3, the two end faces being parallel.

In operation the sensor is located between two elements such that the first end face 2 is connected to a surface on the first element and the second end face to a surface of the second element. These surfaces thus also are in parallel. The connection of the sensor 1 to the respective surface of the elements can be made in various ways e.g. by screws or by glueing. The sensor measures the normal force between the two elements, i.e. the force in the Z-direction in the figure. Usually also shear forces appear between the elements, i.e. forces in the x - and y-directions. The best performance is when the sensor sensitivity for these shear

forces is as small as possible in relation to the sensitivity in the direction to be measured.

The sensor produces an output signal S that is a function of these forces, Fx, Fy and Fz. Fig 2 and 3 schematically illustrates a two-DOF sensor consisting of two one-DOF force sensors. Figure 2 is a perspective view thereof. The two sensors 11 , 12 are arranged in a V-shaped groove in one of the elements between which the two-DOF sensor operates. The groove has two planar surfaces 31 a, 31 b, to which a respective sensor 11 , 12 is attached. The surfaces meet each other at an angle αi.

In fig. 3, which is a section through the two-DOF sensor perpendicular to the intersection line of the surfaces 31 a, 31 b, the complementary surfaces 21a, 21 b of the other element can be seen.

In figures 2 and 3 the appearing forces are indicated as well as the general and local coordinate systems.

The sensor signals as functions of the groove forces can be calculated as

where:

511 = kx Fx11 + ky Fy11 + kz Fz11

512 = kx Fx12 + ky Fy12 + kz Fz12

and

a11 = 0.5 kx

a12 = ky sin(α1/2) - kz cos(α1/2) a13 = 0.5 ky cos(α1/2) + 0.5 kz sin(α1/2) a21 = 0.5 kx a22 = ky sin(α1/2) + kz cos(α1/2) a23 = 0.5 ky cos(α1/2) + kz sin(α1/2)

In fig. 4 a traditional kinematic coupling is illustrated. Such a coupling has one element with a surface to which three half balls are attached in a 120° " distribution and a second element with a surface having three V-shaped grooves matching the half balls. This represents a six-DOF arrangement for three force DOF and three torque DOF.

In fig. 5 the kinematic coupling of fig. 4 is illustrated in a more theoretical way for explaining the principle. The three grooves G with the cooperating balls B and the six contact points P are thus elucidated. When forming a six-DOF sensor according to the invention each ball can be seen as replaced by two one-DOF force sensor, one for each ball contact point, which analogy is illustrated in figs. 6a and 6b. Each ball contact point thus is replaced by an area contact on each sensor.

In fig. 6b it can be seen that each of the sensors 11 , 12 are attached to a respective membrane 8 on the two elements 9, 10, which makes the coupling flexible.

Fig 7 shows the coordinate system of the whole six-DOF force/torque sensor and the coordinate systems for the three grooves. From the signals s11 , s12, s21 , s22, s31 , s32 the three forces FxO, FyO and FzO and the three torques TxO, TyO and TzO of the six-DOF sensor can be calculated by:

Fx3 Fy3 Fz3

where:

For a six-DOF overload protection a common arrangement according to fig. 8 can be used, either integrated to the two coupling elements or as a separate component. In the figure the overload protection consists of three components, each comprising a pin 7 connected to the first element 10 cooperating with a hole 6 in the second element 9. A multi-DOF sensor constituted by a number of one-DOF force sensors can be derived also from the structures of robots based on the parallel kinematics concept, i.e. having a system of links, where each link transfer only axial forces, such as a hexapod structure, a Delta structure, a Tau-structure and modified forms or hybrids of these structures. Figure 9 is a theoretical representation of a hexapod structure, commonly used in a parallel kinematics manipulator. Each connection of each of the six links L to the elements Ei, E 2 is represented by a ball B, each having two contact points P. With reference to fig 6a and 6b each ball B with its two contact point represents a ball arrangement as illustrated in fig 6a which is analogue to the arrangement on the one-DOF force sensors of fig 6b. With these analogues a hexapod as

represented in fig 9 can be considered as equivalent to two connected classical kinematic couplings.

The hexapod structure of fig. 9 is in fig 10 illustrated in an alternative representation, where only the connections at the bottom element E1 is illustrated. Each of the three connections thus is of the V-groove type. The arrangement of the six one-DOF force sensors in the hexapod structure represents a six-DOF force/torque sensor. The connections to the upper element E2 do not add any further DOF for the sensor since the links L transfer only axial forces.

Another traditional parallel kinematic structure is the Delta structure, which has three clusters of links each cluster having two links L as illustrated in fig. 11. Fig 12 illustrates a Delta structure in a representation similar to that of fig. 10 for the classical kinematic coupling. There will be no grooves when representing a Delta structure in this way. Instead each ball B will be connected to a plane A through a respective point P. The two balls in one cluster are connected to planes that are co-planar. The planes in fig 12 thus will form a pyramid of three sides facing outwards and the connection of the links to the arms will form a complementary inside pyramid. This coupling will thus when used as a six-DOF force/- torque sensor be formed by outer and inner three side pyramid structures with the one-DOF force sensors placed in between, one pair between each pair of pyramid sides.

A further parallel kinematic structure is the Tau structure, which has three clusters of links, one cluster, having one link L, another cluster having two links L and a third cluster having three links L, as illustrated in fig. 13. Fig 14 illustrates a Tau structure in a representation similar to that of fig. 10. Also in this case there will be no grooves. The geometry however will be more complex than the Delta structure. There will be six different planes, each connected to a respective ball at one single contact point.

In fig. 15 a modified Delta structure is illustrated. The difference in relation to the traditional Delta structure illustrated in fig. 11 is that the links are connected to the element in a 3D, frame instead of a 2D connection as in fig. 11.

Fig. 16 illustrates the 3D Delta structure of fig. 15 in a representation similar to that of fig. 10. Also with this structure there will be six different planes each connected to a respective ball at one single contact point.

In fig. 17 a modified Tau structure is illustrated. In this modified Tau structure five of the joints are mounted in a common line. This modified structure thus is a 1 14 D structure.

Fig. 18 illustrates the 1 14 D Tau structure of fig. 16 in a representation similar to that of fig. 10. Also with this structure there will be six different planes each connected to a respective ball at one single contact point.

In fig. 19 a further modified Tau structure is illustrated. Also in this structure five of the joints are mounted on a common line. Three of the links merge in one joint and two links merge in one joint. This structure thus is a merging 1 14 D Tau structure.

Fig. 20 illustrates the merging Tau structure in a representation similar to that of fig. 10. The joint where three links merge can be represented by a pyramid structure, c.f. the description related to fig. 12. The joint where two links merge can be represented by a V-groove structure, c.f. the description related to figures 2 and 3. The third joint is a single plane joint.

The embodiments illustrated in figures 17-20, where most of the joints are on a common line give the possibility to mount the multi-DOF force sensors inside for example a handle for lead through programming.

The Tau structure can be modified in five different types regarding the connections of the balls to the element, namely six planes one V-groove and four planes two V-grooves and two planes one pyramid and three planes - one pyramid, one V-groove and one plane

Fig. 21 illustrates an arrangement for a pyramid type connection. This example is suitable when a compliant sensor is required. The two tree-side pyramids between which the six one-DOF force sensors are located with one pair of sensors on each side of the pyramid, c.f. fig. 12. Each side of the inner and the outer pyramid is made from a rhomboid sheet of steel. Portions thereof are cut out and bent as illustrated in fig. 21 to form the sides of the respective pyramid with the surface 21 c, 31 c to which the sensors 1 are attached. The sensors 1 thereby will be flexibly mounted.

Fig. 22 illustrates an arrangement where each one-DOF force sensor cooperates with a plane and relates to the case represented by figures 17 and 18, the 1 14 D Tau-structure. The arrangement consists of two identical bent sheets of metal forming an outer and inner part. Only the inner part is visible in fig. 22. On it's outwardly facing surfaces 21d, 21 e, 21f the sensors 1 a-1f are mounted. The outer part is mounted with its inwardly facing surfaces onto the sensors 1 a-1f. The outer part not seen in the figure can be used as a handle when for example lead through programming is performed.

Figure 23 illustrates how the arrangement of fig. 22 can be provided with springs and overload protection. Inside the inner part of the arrangement illustrated in fig. 22 an L-shaped solid bar 4 is mounted. Between the inner part and the solid bar 4 six springs 5 are provided. Overloading protection is formed by a hole 6 on each surface 21 d, 21 e, 21 f of the inner part and a cooperating pin 7 attached to the corresponding surfaces of the bar 4. The multi-DOF sensor according to the invention can also be applied to applications needing less DOF than six as in the examples described above. The invented multi-DOF sensor thus can be arranged as a five-DOF, four-DOF, three- DOF or a two-DOF sensor.

In figures 24 to 52 various alternatives are illustrated for a multi-DOF sensor of various DOF. The alternatives are illustrated in a representation similar to that of fig. 10, i.e. by balls (B) cooperating with planes (A) grooves (G) or pyramids (C) in one, two and three points (P) of contacts respectively. The examples are illustrated by this representation in order to simplify the presentation. A ball with one contact point thus represents one single one-DOF force sensor as illustrated in fig. 1. A ball with two contact points represents a groove-arrangement of two one-DOF force sensors as illustrated in figs. 2 and 3 and a ball with three contact points represents three one-DOF force sensors as illustrated in fig. 21. The examples represented in figures 24 to 52 are summarized in the below table. Type of outputs indicates the distribution of the DOF on force and torque respectively and is in the table indicated by FfT, where F represents the number of force outputs and T the number of torque outputs.

The sensor structures based on the modified Tau- and Delta-structure represented in figures 47 to 51 can be synthesized for kinematic parallel

manipulators, which would lead to linkage systems as represented in figures 53- 56.

Fig. 53 is based on the kinematics of the structure in fig. 47. The platform orientation will be position dependent, which will also reduce the work space. Fig. 54 is based on the kinematics of the structure in fig. 48. The platform orientation will be position dependent, which will also reduce the work space.

Fig. 55 is based on the kinematics of the structure in fig. 49. The platform orientations will be position dependent but the work space will not be reduced to any larger extent. Fig. 56 is based on the kinematics of the structure in fig. 50. The platform orientation will be position dependent, which will also reduce the work space.

Fig. 57 is based on the kinematics of the structure in fig. 51.

Figures 58 and 59 exemplifies a representation of the theoretical figures above with physical elements, where fig. 58 corresponds to fig.55 and fig. 59 corresponds to fig. 57.