Translational parallel manipulators and methods of operating the same

Abstract

In one aspect, a translational parallel manipulator is provided and includes a fixed platform including three guide members. The three guide members include first ends and second ends, and the first ends of the three guide members are all coupled to each other and the second ends of the three guide members are all spaced-apart from each other. The manipulator also includes a movable platform spaced-apart from the fixed platform and three serial subchains coupled between the three guide members and the movable platform. In one aspect, a translational parallel manipulator is provided and includes a fixed platform, a movable platform spaced-apart from the fixed platform, and a plurality of subchains coupled between the fixed platform and the movable platform. At least one of the plurality of subchains includes no more than four one degree-of-freedom joints.

Claims

1. A translational parallel manipulator comprising: a fixed platform comprising three guide members, wherein: the three guide members comprise first ends and second ends; and the first ends of the three guide members are coupled to each other and the second ends of the three guide members are spaced apart from each other; a movable platform spaced apart from the fixed platform; and three serial subchains coupled between the three guide members and the movable platform, wherein at least one of the three serial subchains comprises: a first prismatic joint; a first revolute joint coupled to the first prismatic joint; a second prismatic joint coupled to the first revolute joint; and a second revolute joint coupled to the second prismatic joint and parallel to the first revolute joint; wherein the first prismatic joint, the first revolute joint, the second prismatic joint, and the second revolute joint comprise one degree-of-freedom joints.

2. The translational parallel manipulator of claim 1, wherein the three guide members are co-planar with each other.

3. The translational parallel manipulator of claim 1, wherein the three guide members are non-planar with each other.

4. The translational parallel manipulator of claim 1, wherein the first ends of the three guide members are disposed above the second ends of the three guide members.

5. The translational parallel manipulator of claim 1, wherein the first ends of the three guide members are disposed below the second ends of the three guide members.

6. The translational parallel manipulator of claim 1, wherein the three serial subchains are identical in structure.

7. The translational parallel manipulator of claim 1, wherein: the first prismatic joint is coupled to one of the guide members; and the second revolute joint is coupled to the movable platform.

8. The translation parallel manipulator of claim 1, wherein the first prismatic joint comprises a driven prismatic joint.

9. The translational parallel manipulator of claim 8, wherein the first revolute joint, the second prismatic joint, and the second revolute joint comprise passive joints.

10. The translational parallel manipulator of claim 1, wherein at least one serial subchain comprises four one degree-of-freedom joints.

11. The translational parallel manipulator of claim 1, wherein: at least one of the first prismatic joint, the first revolute joint, the second prismatic joint and the second revolute joint comprises a driven joint; and at least one of the first prismatic joint, the first revolute joint, the second prismatic joint and the second revolute joint comprises a passive joint.

12. The translational parallel manipulator of claim 11, wherein the driven joint comprises the first prismatic joint.

13. The translational parallel manipulator of claim 12, wherein the first revolute joint, the second prismatic joint, and the second revolute joint comprise a passive joint.

14. The translational parallel manipulator of claim 1, wherein the three subchains comprise symmetric subchains.

15. The translational parallel manipulator of claim 1, wherein an intersection angle between the first prismatic joint and the second prismatic joint is less than ninety degrees.

16. The translational parallel manipulator of claim 1, wherein an intersection angle between the first prismatic joint and the second prismatic joint is greater than ninety degrees.

17. A translational parallel manipulator comprising: a fixed platform comprising three guide members, wherein: the three guide members each comprise a first end and a second end; and the first ends of the three guide members are coupled to each other and the second ends of the three guide members are spaced apart from each other; a movable platform spaced apart from the fixed platform; and three serial subchains, each of the three serial subchains coupled to one of the three guide members and coupled to the movable platform, wherein each of the three serial subchains comprises: a first prismatic joint coupled to one of the guide members; a first revolute joint coupled to the first prismatic joint; a second prismatic joint coupled to the first revolute joint; and a second revolute joint coupled to the second prismatic joint and coupled to the movable platform; wherein the first prismatic joint, the first revolute joint, the second prismatic joint, and the second revolute joint comprise one degree-of-freedom joints.

18. The translational parallel manipulator of claim 17, further comprising three linear actuators, each linear actuator coupled to the fixed platform and the first prismatic joint of each serial subchain.

19. A translational parallel manipulator comprising: a fixed platform comprising three guide members, wherein: the three guide members comprise first ends and second ends; and the first ends of the three guide members are coupled to each other and the second ends of the three guide members are spaced apart from each other; a movable platform spaced apart from the fixed platform; and three serial subchains coupled between the three guide members and the movable platform, wherein at least one of the three serial subchains comprises: a first prismatic joint; a first revolute joint coupled to the first prismatic joint; a second prismatic joint coupled to the first revolute joint; and a second revolute joint coupled to the second prismatic joint and parallel to the first revolute joint; wherein at least one serial subchain comprises four one degree-of-freedom joints.

20. The translation parallel manipulator of claim 19, wherein the first prismatic joint comprises a driven prismatic joint.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The present disclosure can be better understood with reference to the following drawings and description. The components in the figures are not necessarily to scale, emphasis instead being placed upon illustrating the principles of the disclosure.

(2) FIG. 1 is a bottom perspective view of one example of a translational parallel manipulator, in accordance with one aspect of the present disclosure.

(3) FIG. 2 is a top perspective view of a portion of the manipulator shown in FIG. 1, in accordance with one aspect of the present disclosure.

(4) FIG. 3 is top perspective view of the manipulator shown in FIG. 1, in accordance with one aspect of the present disclosure.

(5) FIG. 4 is a top perspective view of the manipulator shown in FIG. 1 illustrating examples of structural parameters of the manipulator, in accordance with one aspect of the present disclosure.

(6) FIG. 5 is a bottom perspective of another example of a translational parallel manipulator, in accordance with one aspect of the present disclosure.

(7) FIG. 6 is a top perspective view of a portion of the manipulator shown in FIG. 5, in accordance with one aspect of the present disclosure.

(8) FIG. 7 is a top perspective view of the manipulator shown in FIG. 5, in accordance with one aspect of the present disclosure.

(9) FIG. 8 is a bottom perspective view of another example of a translational parallel manipulator, in accordance with one aspect of the present disclosure.

(10) FIG. 9 is a top perspective view of a portion of the manipulator shown in FIG. 8, in accordance with one aspect of the present disclosure.

(11) FIG. 10 is a top perspective view of the manipulator shown in FIG. 8, in accordance with one aspect of the present disclosure.

(12) FIG. 11 is a bottom perspective view of another example of a translational parallel manipulator, in accordance with one aspect of the present disclosure.

(13) FIG. 12 is a top perspective view of a portion of the manipulator shown in FIG. 11, in accordance with one aspect of the present disclosure.

(14) FIG. 13 is a top perspective view of the manipulator shown in FIG. 11, in accordance with one aspect of the present disclosure.

(15) FIG. 14 is a bottom perspective view of another example of a translational parallel manipulator, in accordance with one aspect of the present disclosure.

(16) FIG. 15 is a top perspective view of a portion of the manipulator shown in FIG. 14, in accordance with one aspect of the present disclosure.

(17) FIG. 16 is a top perspective view of the manipulator shown in FIG. 14, in accordance with one aspect of the present disclosure.

(18) FIG. 17 is a diagram illustrating one example of a three-dimensional (3D) workspace associated with the manipulator shown in FIG. 1 with .sub.i equal to 45 degrees, in accordance with one aspect of the present disclosure.

(19) FIG. 18 is a diagram of several x-y cross-sections taken from FIG. 17 at various z values, in accordance with one aspect of the present disclosure.

(20) FIG. 19 is a diagram illustrating another example of a three-dimensional (3D) workspace associated with the manipulator shown in FIG. 1 with .sub.i equal to 30 degrees, in accordance with one aspect of the present disclosure.

(21) FIG. 20 is a diagram of several x-y cross-sections taken from FIG. 19 at various z values, in accordance with one aspect of the present disclosure.

(22) FIG. 21 is a diagram illustrating another example of a three-dimensional (3D) workspace associated with the manipulator shown in FIG. 1 with .sub.i equal to 60 degrees, in accordance with one aspect of the present disclosure.

(23) FIG. 22 is a diagram of several x-y cross-sections taken from FIG. 21 at various z values, in accordance with one aspect of the present disclosure.

(24) FIG. 23 is a top perspective view of another example of a translational parallel manipulator illustrating a forward kinematic singularity, in accordance with one aspect of the present disclosure.

(25) FIG. 24 is a top perspective view of another example of a translational parallel manipulator, in accordance with one aspect of the present disclosure.

(26) FIG. 25 is a diagram of one example of a three-dimensional distribution of kinematic manipulability of one example of a translational parallel manipulator such as the example shown in FIG. 1, in accordance with one aspect of the present disclosure.

(27) FIG. 26 is a diagram of several x-y cross-sections taken from FIG. 25 at various z values, in accordance with one aspect of the present disclosure.

(28) FIG. 27 is a top perspective view of one example of a translational parallel manipulator, such as the manipulator shown in FIG. 1, illustrating examples of loads on the manipulator, in accordance with one aspect of the present disclosure.

(29) FIG. 28 is a top perspective view of a portion of one example of a translational parallel manipulator, such as the manipulator shown in FIG. 1, illustrating one example of an equivalent spring structure of the portion, in accordance with one aspect of the present disclosure.

(30) FIG. 29a is a top perspective view of a portion of one example of a translational parallel manipulator, such as the manipulator shown in FIG. 1, illustrating one example of a deformation of actuation of the portion, in accordance with one aspect of the present disclosure.

(31) FIG. 29b is a top perspective view of a portion of one example of a translational parallel manipulator, such as the manipulator shown in FIG. 1, illustrating one example of deformation of constraints of the portion, in accordance with one aspect of the present disclosure.

(32) FIG. 30a is a diagram of one example of a distribution of a maximum eigenvalue of a stiffness matrix at an x-y section taken at a z value of 1432, in accordance with one aspect of the present disclosure.

(33) FIG. 30b is a diagram of another example of a distribution of a maximum eigenvalue of a stiffness matrix at an x-y section taken at a z value of 1582, in accordance with one aspect of the present disclosure.

(34) FIG. 30c is a diagram of another example of a distribution of a maximum eigenvalue of a stiffness matrix at an x-y section taken at a z value of 1732, in accordance with one aspect of the present disclosure.

(35) FIG. 30d is a diagram of one example of a distribution of a minimum eigenvalue of a stiffness matrix at an x-y section taken at a z value of 1432, in accordance with one aspect of the present disclosure.

(36) FIG. 30e is a diagram of another example of a distribution of a minimum eigenvalue of a stiffness matrix at an x-y section taken at a z value of 1582, in accordance with one aspect of the present disclosure.

(37) FIG. 30f is a diagram of another example of a distribution of a minimum eigenvalue of a stiffness matrix at an x-y section taken at a z value of 1732, in accordance with one aspect of the present disclosure.

(38) FIG. 31a is a diagram illustrating one example of a three-dimensional (3D) workspace associated with a translational parallel manipulator with .sub.i equal to 30 degrees, in accordance with one aspect of the present disclosure.

(39) FIG. 31b is a diagram of one example of a stiffness distribution taken at an x-y section of a maximum eigenvalue at a z value of 1050 (of FIG. 31a) and having .sub.i equal to 30 degrees, in accordance with one aspect of the present disclosure.

(40) FIG. 31c is a diagram of one example of a stiffness distribution taken at an x-y section of a minimum eigenvalue at a z value of 1050 (of FIG. 31 and having .sub.i equal to 30 degrees, in accordance with one aspect of the present disclosure.

(41) FIG. 31d is a diagram illustrating another example of a three-dimensional (3D) workspace associated with a translational parallel manipulator with .sub.i equal to 60 degrees, in accordance with one aspect of the present disclosure.

(42) FIG. 31e is a diagram of one example of a stiffness distribution taken at an x-y section of a maximum eigenvalue at a z value of 1975 (of FIG. 31d) and having .sub.i equal to 60 degrees, in accordance with one aspect of the present disclosure.

(43) FIG. 31f is a diagram of one example of a stiffness distribution taken at an x-y section of a minimum eigenvalue at a z value of 1975 (of FIG. 31d) and having .sub.i equal to 60 degrees, in accordance with one aspect of the present disclosure.

(44) FIG. 32 is a top perspective view of another translational parallel manipulator, in accordance with one aspect of the present disclosure.

(45) FIG. 33 is a top perspective view of a portion of the manipulator shown in FIG. 32, in accordance with one aspect of the present disclosure.

(46) FIG. 34 is a top perspective of the manipulator shown in FIG. 32 illustrating examples of structural parameters of the manipulator, in accordance with one aspect of the present disclosure.

(47) FIG. 35 is a top perspective view of the manipulator shown in FIG. 32 illustrating examples of loads on the manipulator, in accordance with one aspect of the present disclosure.

(48) FIG. 36 is a top perspective view of a portion of the manipulator shown in FIG. 32 illustrating one example of an equivalent spring structure of the portion, in accordance with one aspect of the present disclosure.

(49) FIG. 37a is a top perspective view of a portion of the manipulator shown in FIG. 32 illustrating one example of a deformation of actuation of the portion, in accordance with one aspect of the present disclosure.

(50) FIG. 37b is a top perspective view of a portion of the manipulator shown in FIG. 32 illustrating one example of a deformation of constraints of the portion, in accordance with one aspect of the present disclosure.

(51) FIG. 38 is a diagram illustrating one example of a three-dimensional (3D) workspace associated with the manipulator shown in FIG. 32, in accordance with one aspect of the present disclosure.

(52) FIG. 39 is a diagram of several x-y cross-sections taken from FIG. 38 at various z values, in accordance with one aspect of the present disclosure.

(53) FIG. 40a is a diagram associated with the manipulator shown in FIG. 32 of one example of a distribution of a maximum eigenvalue of a stiffness matrix at an x-y section taken at a z value of 900, in accordance with one aspect of the present disclosure.

(54) FIG. 40b is a diagram associated with the manipulator shown in FIG. 32 of another example of a distribution of a maximum eigenvalue of a stiffness matrix at an x-y section taken at a z value of 1000, in accordance with one aspect of the present disclosure.

(55) FIG. 40c is a diagram associated with the manipulator shown in FIG. 32 of another example of a distribution of a maximum eigenvalue of a stiffness matrix at an x-y section taken at a z value of 1100, in accordance with one aspect of the present disclosure.

(56) FIG. 40d is a diagram associated with the manipulator shown in FIG. 32 of one example of a distribution of a minimum eigenvalue of a stiffness matrix at an x-y section taken at a z value of 900, in accordance with one aspect of the present disclosure.

(57) FIG. 40e is a diagram associated with the manipulator shown in FIG. 32 of another example of a distribution of a minimum eigenvalue of a stiffness matrix at an x-y section taken at a z value of 1000, in accordance with one aspect of the present disclosure.

(58) FIG. 40f is a diagram associated with the manipulator shown in FIG. 32 of another example of a distribution of a minimum eigenvalue of a stiffness matrix at an x-y section taken at a z value of 1100, in accordance with one aspect of the present disclosure.

(59) FIG. 41a is a diagram illustrating another example of a three-dimensional (3D) workspace associated with the manipulator shown in FIG. 32 with .sub.i equal to 40 degrees, in accordance with one aspect of the present disclosure.

(60) FIG. 41b is a diagram of one example of a stiffness distribution taken at an x-y section of a maximum eigenvalue at a z value of 1000 and having .sub.i equal to 40 degrees, in accordance with one aspect of the present disclosure.

(61) FIG. 41c is a diagram of one example of a stiffness distribution taken at an x-y section of a minimum eigenvalue at a z value of 1000 and having .sub.i equal to 40 degrees, in accordance with one aspect of the present disclosure.

(62) FIG. 41d is a diagram illustrating another example of a three-dimensional (3D) workspace associated with the manipulator shown in FIG. 32 with .sub.i equal to 50 degrees, in accordance with one aspect of the present disclosure.

(63) FIG. 41e is a diagram of one example of a stiffness distribution taken at an x-y section of a maximum eigenvalue at a z value of 1000 and having .sub.i equal to 50 degrees, in accordance with one aspect of the present disclosure.

(64) FIG. 41f is a diagram of one example of a stiffness distribution taken at an x-y section of a minimum eigenvalue at a z value of 1000 and having .sub.i equal to 50 degrees, in accordance with one aspect of the present disclosure.

(65) FIG. 42 is a diagram of one example of a control system for one example of a translational parallel manipulator, in accordance with one aspect of the present disclosure.

(66) Other features and advantages of the present disclosure will become apparent from the following detailed description. It should be understood, however, that the detailed description and the specific examples, while indicating preferred examples, are given by way of illustration only, because various changes and modifications within the spirit and scope of the present disclosure will become apparent to those skilled in the art.

DETAILED DESCRIPTION

(67) As indicated above, existing translational parallel manipulators have numerous drawbacks. The present disclosure includes translational parallel manipulators that may alleviate one or more, and other, drawbacks of existing translational parallel manipulators. For example, manipulators of the present disclosure may address one or more of the following conditions: The manipulator may be designed as a symmetric form (subchains have the same/identical structure) and arrangement of the subchains may be of a non-Cartesian form to eliminate orthogonality of the constraints; driven joints or actuators should be placed on or part of a fixed platform of the manipulator and, in some examples, actuation may be linear to assure high resolution inputs; the manipulator may be designed with a minimal number of 1-DOF joints (e.g., no U or S joints should be included); prismatic joints of the manipulator may not be directly connected to the movable platform and the movable platform should be as small as possible in size; decoupled motions, over-constraint elimination, singularity free workspace, and isotropic configuration may be included in the topological constraint design.

(68) In order to realize the above design objectives and overcome drawbacks of conventional manipulators, topological structures of 3-DOF translational parallel manipulators (TPMs) are provided. In some examples of topological constraint designs of serial joints, an inclined planar displacement subset will be defined based on displacement group theory. A triangular pyramidal constraint is also provided and applied in designs of constraints between different subchains of the manipulator.

(69) The translational parallel manipulators included herein are capable of being used in a wide variety of applications and functions including, but not limited to, forming, cutting, grinding, metrology, laser EDM, any other machining or manufacturing process or application, pick-and-place operation, haptic controller, motion simulator, or any other application or function utilizing a manipulator. The manipulator may include at least one tool holder and/or tool for performing any desired operation. The tool holder and/or tool may be located on or part of the movable platform. The movable platform may also include a plurality of tool holders and/or tools.

Topological Constraint and Structural Design

(70) As far as 3-DOF TPMs are concerned, in some examples, three actuations may be required for 3-DOF output motions and, hence, three actuators should be arranged in different subchains to build a closed-loop form of the parallel manipulator. Thus, in this example, the manipulator includes three subchains connecting a movable platform to a fixed platform for a basic topological arrangement of the 3-DOF TPM. To generate three translational output motions, rotations of the movable platform may be simultaneously constrained by the three connected subchains, namely intersection motions of the three subchains may have no rotational output motions. This 3-DOF TPM may include three orthogonal prismatic joints serially connected in each subchain, thereby resulting in a total of nine prismatic joints involved to eliminate all rotations, and commonly constrain the movable platform to generate three orthogonal translational output motions. However, due to orthogonal constraints and redundant weight of guide members of the prismatic joints, low load/weight ratio, stiffness and accuracy may be induced.

(71) By considering the displacement group theory, in one example, the general method for the design of the topological structure of the subchains of a 3-DOF TPM is to include a planar displacement subgroup and a translational displacement subgroup into a subchain. The planar displacement subgroup contains two non-parallel translational motions in a plane and a rotational motion that is parallel to a normal of the plane. The planar displacement subgroup can be realized by a planar joint, Pl, that is equivalent to a serial connection of 1-DOF joints in three different ways: 1) Two non-parallel prismatic joints and a revolute joint (an axis of the revolute joint is perpendicular to moving directions of the two prismatic joints). 2) A prismatic joint and two parallel revolute joints (axes of the revolute joints are perpendicular to moving direction of the prismatic joint). 3) Three parallel revolute joints.

(72) The general translational displacement subgroup can be generated by: 1) A prismatic joint, P. 2) A parallelogram joint, Pa, namely, four parallel revolute joints are connected in a parallelogram in a closed-loop form. 3) Two parallel revolute joints, RR.

(73) Thus, there are three types of structural combinations for the subchain design for a 3-DOF TPM, namely: 1) Subchain Pl-P: A planar joint with a prismatic joint (a direction of the prismatic joint is not perpendicular to a normal of the planar joint). 2) Subchain Pl-Pa: A planar joint with a parallelogram joint (axes of the revolute joints of the parallelogram joint are not parallel to a normal of the planar joint). 3) Subchain Pl-RR: A planar joint with two parallel revolute joint (axes of the two parallel revolute joints are not parallel to a normal of the planar joint).

(74) A serial combination of planar and translational displacement subgroups can build a subchain of the 3-DOF TPM, while the closed-loop connection of three subchains with non-parallel revolute joints between the different subchains yields a 3-DOF TPM.

(75) By considering the output motions of the three types of subchains, subchains Pl-P and Pl-Pa generate a Schoenflies displacement subgroup (three orthogonal translations and a rotation), respectively. The subchain Pl-RR generates two non-parallel planar displacement subgroups (three orthogonal translations and two non-parallel rotations). In some examples of a 3-DOF TPM, three rotations of the movable platform should be constrained and eliminated from the closed-loop connections of three subchains Pl-P or Pl-Pa. Similarly, in other examples, there are six rotations that should be eliminated from the closed-loop connection of three subchains Pl-RR. Thus, in accordance with design constraint that requires a minimal number of 1-DOF joints, subchains with only a Schoenflies motion will be used in the proposed design. In the following description, a plurality of 3-DOF TPMs are introduced to illustrate aspects of the present disclosure.

Four General Structures of 3-DOF TPMs

(76) In FIGS. 5-7, one example of a TPM is illustrated. In this example, the manipulator includes a subchain Pl-P having a planar joint and a prismatic joint. The planar joint is configured with two revolute joints and a prismatic joint, namely R.sub.1P.sub.1R.sub.2. A normal of the planar joint is n.sub.1. A planar motion {G(n.sub.1)} is generated in plane K.sub.1, and a Schoenflies motion {X(n.sub.1)} can be obtained by connection of a planar joint and of a prismatic joint P.sub.0. By combining three subchains Pl-P in a star constraint (the constraint planes K.sub.1, K.sub.2 and K.sub.3 of the subchains are arranged in a star shape), a 3Pl-P TPM is obtained. The intersection operation of the displacement subsets between the subchains is given by:

(77) $\begin{array}{cc}\begin{array}{c}{Z}_m=\stackrel{3}{\underset{i=1}{.Math.}}\left\{X\left(n_i\right)\right\}=\stackrel{3}{\underset{i=1}{.Math.}}\left[\left\{G\left(n_i\right)\right\}\left\{T\left(n_i\right)\right\}\right]\\ =\underset{i=1}{\stackrel{3}{.Math.}}\left[\left\{R\left(n_i\right)\right\}\left\{T\left(K_i\right)\right\}\left\{T\left(n_i\right)\right\}\right]\\ =\stackrel{3}{\underset{i=1}{.Math.}}\left[\left\{T\left(K_i\right)\right\}\left\{T\left(n_i\right)\right\}\right]\\ =\left\{T\right\}\end{array}& \left(1\right)\end{array}$
where Z.sub.m represents the output motions of the movable platform. {T(n.sub.i)}, {T(K.sub.i)} and {T} respectively represent a translation along vector n.sub.i, two translations in plane K.sub.i and three orthogonal translations. {R(n.sub.i)} represents a rotation around the vector n.sub.i. Three non-parallel rotations of the three subchains are eliminated by the star constraint. The output motions of the 3Pl-P are three translations. The actuated joint is determined in the planar displacement subgroup of each subchain, because there is a common translational motion (along an intersection line of the constraint planes K.sub.1, K.sub.2 and K.sub.3) in the three planar displacement subgroups. The 3Pl-P TPM has moving actuators.

(78) In a similar way, the planar joint can be configured with three parallel revolute joints. Thus, a 3Pl-P TPM with configuration RRR of the planar joint can be constituted as shown in FIGS. 8-10.

(79) To avoid the generation of intersection motions between the three planar displacement subgroups that are arranged in the star constraint configuration above, or conversely to design a structural configuration that places the actuators on the fixed platform, the serial connection of a parallelogram joint and of two parallel revolute joints can be used to replace the planar joint of the subchain Pl-P to generate a Schoenflies motion (FIGS. 11-13). In FIGS. 11-13, axes of joints P.sub.0, R.sub.1 and R.sub.2 are parallel to axis n.sub.1. An axis of the parallelogram joint Pa, is perpendicular to the axis n.sub.1. A spherical translation {T(S.sub.1)} is generated on spherical surface S.sub.1 by serial joints R.sub.1Pa.sub.1R.sub.2, where a spherical center and radius are S.sub.O and S.sub.r respectively. A rotation {R(n.sub.1)} is obtained from the parallel joints R.sub.1 and R.sub.2. Thus, a Schoenflies motion {X(n.sub.1)} is generated by the subchain P-RPaR. By combining the three subchains P-RPaR in a star constraint configuration, a 3P-RPaR TPM is obtained. The intersection operation of the displacement subsets between the subchains is:

(80) $\begin{array}{cc}\begin{array}{c}{Z}_m=\stackrel{3}{\underset{i=1}{.Math.}}\left\{X\left(n_i\right)\right\}=\underset{i=1}{\stackrel{3}{.Math.}}\left[\left\{R\left(n_i\right)\right\}\left\{T\left(S_i\right)\right\}\left\{T\left(n_i\right)\right\}\right]\\ =\stackrel{3}{\underset{i=1}{.Math.}}\left[\left\{T\left(S_i\right)\right\}\left\{T\left(n_i\right)\right\}\right]\\ =\left\{T\right\}\end{array}& \left(2\right)\end{array}$

(81) It can be seen that the output motions of the 3P-RPaR are three translations, while the three non-parallel rotations are eliminated by the star constraint. The actuated joint may be selected as the prismatic joint P.sub.0 of each subchain since there are no intersection motions between the three spherical translations and the three non-parallel rotations. A manipulator has fixed actuators and the guide members of the three linear actuators are arranged in a triangular form to build the triangular 3P-RPaR TPM. Additionally, the guide members may be arranged in a star form to build a star 3P-RPaR TPM as shown in FIGS. 14-16. A size of the movable platform of the star 3P-RPaR may be reduced by the star connection of the guide members, because the movable platform is connected by three revolute joints whose axes intersect at a center point of the movable platform. A size of the movable platform is dependent on the cross-section of the connected revolute joints instead of their longitudinal section. Compared to the triangular 3P-RPaR TPM, a higher load/weight ratio can be achieved in the star 3P-RPaR TPM with a movable platform of smaller size. In other examples, the axes of the three revolute joints may intersect each other at a point of the movable platform other than the center point. In further examples, the axes of the three revolute joints may intersection each other, but not necessarily at the same point.

(82) Although the star 3P-RPaR TPM illustrated in FIGS. 11-13 is designed with fixed actuators and a reduced size of the movable platform, this manipulator includes a large quantity of 1-DOF joints and the input motions are coupled by spherical translations of the parallelogram joints. To reduce the number of joints and decouple the motions of the star 3P-RPaR TPM, a prismatic joint may be used to replace the parallelogram joint in each subchain as illustrated in FIGS. 1-3.

(83) In FIGS. 1-3, prismatic joint P.sub.1 connects two parallel revolute joints R.sub.1 and R.sub.2 with an intersection angle .sub.i. Joints P.sub.1, R.sub.1 and R.sub.2 generate a rotation around axis n.sub.1 and two translations in plane K.sub.1. Thus, the resulting new displacement subset can be defined as an inclined planar displacement subset {I(n.sub.1,K.sub.1)} that can be generated by serial joints R.sub.1P.sub.1R.sub.2. The difference between motions of the {I(n.sub.1,K.sub.1)} and the general planar displacement subgroup {G(n.sub.1)} is that the axis of rotation of {I(n.sub.1,K.sub.1)} is not perpendicular to the moving directions of the two translations. This type of inclined constraint between the rotation and translations can be used for a new structural design of the Schoenflies motion. By combining a prismatic joint P.sub.0 with joints R.sub.1P.sub.1R.sub.1 of the inclined planar displacement subset (P.sub.0 is not perpendicular to the normal of the plane K.sub.1), a new structure of the Schoenflies motion can be obtained as a subchain P-RPR. Three P-RPR subchains can be combined in a closed-loop form, where the three guide members of the joints P.sub.0 are connected in a star form. The three constraint planes K.sub.1, K.sub.2 and K.sub.3 of the subchains are arranged in a triangular pyramidal constraint form that intersect at point O.sub.m. Consequently, a new 3P-RPR TPM can be obtained. The intersection operation of the displacement subsets between the subchains is:

(84) $\begin{array}{cc}\begin{array}{c}{Z}_m=\stackrel{3}{\underset{i=1}{.Math.}}\left\{X\left(n_i\right)\right\}=\stackrel{3}{\underset{i=1}{.Math.}}\left[\left\{I\left(n_i,K_i\right)\right\}\left\{T\left(n_i\right)\right\}\right]\\ =\stackrel{3}{\underset{i=1}{.Math.}}\left[\left\{R\left(n_i\right)\right\}\left\{T\left(K_i\right)\right\}\left\{T\left(n_i\right)\right\}\right]\\ =\stackrel{3}{\underset{i=1}{.Math.}}\left[\left\{T\left(K_i\right)\right\}\left\{T\left(n_i\right)\right\}\right]\\ =\left\{T\right\}\end{array}& \left(3\right)\end{array}$

(85) Three non-parallel rotations are eliminated by the triangular pyramidal constraint. In one example, the prismatic joint P.sub.0 of each subchain (the joints R.sub.1, P.sub.1, R.sub.2 are passive joints) may be selected as the driven or actuated joint because there are no intersection motions between the three inclined planar displacement subsets in the triangular pyramidal constraint, namely:

(86) $\begin{array}{cc}\begin{array}{c}{Z}_m=\stackrel{3}{\underset{i=1}{.Math.}}\left\{I\left(n_i,K_i\right)\right\}\\ =\underset{i=1}{\stackrel{3}{.Math.}}\left[\left\{R\left(n_i\right)\right\}\left\{T\left(K_i\right)\right\}\right]\\ =\left\{E\right\}\end{array}& \left(4\right)\end{array}$
where {E} is a fixed displacement subgroup. There are no output motions between the three parallel subchains R.sub.1P.sub.1R.sub.2 when the actuators of the three prismatic joints P.sub.0 are locked.

(87) By arbitrarily assuming that two of the actuators are locked, the intersection operation of the displacement subsets between the remaining structures is expressed as:

(88) $\begin{array}{cc}\begin{array}{c}{Z}_m=\left\{X\left(n_1\right)\right\}.Math.\left\{I\left(n_2,K_2\right)\right\}.Math.\left\{I\left(n_3,K_3\right)\right\}\\ =\left[\left\{R\left(n_1\right)\right\}\left\{T\left(K_1\right)\right\}\left\{T\left(n_1\right)\right\}\right].Math.\\ \left[\left\{R\left(n_2\right)\right\}\left\{T\left(K_2\right)\right\}\right].Math.\left[\left\{R\left(n_3\right)\right\}\left\{T\left(K_3\right)\right\}\right]\\ =\left[\left\{T\left(K_1\right)\right\}\left\{T\left(n_1\right)\right\}\right].Math.\left\{T\left(K_2\right)\right\}.Math.\left\{T\left(K_3\right)\right\}\\ =\left\{T\right\}.Math.\left\{T\left(K_2\right)\right\}.Math.\left\{T\left(K_3\right)\right\}\\ =\left\{T\left(\stackrel{.fwdarw.}{O_m{O}_{f2}}\right)\right\}\end{array}& \left(5\right)\end{array}$
where {T({right arrow over (O.sub.mO.sub.f2)})} is a translational output motion of the movable platform along the intersection line {right arrow over (O.sub.mO.sub.f2)} of the planes K.sub.2 and K.sub.3. Similarly, if one of the actuators is locked, the output motions of the movable platform are:

(89) $\begin{array}{cc}\begin{array}{c}{Z}_m=\left\{X\left(n_1\right)\right\}.Math.\left\{X\left(n_2\right)\right\}.Math.\left\{I\left(n_3,K_3\right)\right\}\\ =\left[\left\{R\left(n_1\right)\right\}\left\{T\left(K_1\right)\right\}\left\{T\left(n_1\right)\right\}\right].Math.\left[\left\{R\left(n_2\right)\right\}\left\{T\left(K_2\right)\right\}\left\{T\left(n_2\right)\right\}\right].Math.\\ \left[\left\{R\left(n_3\right)\right\}\left\{T\left(K_3\right)\right\}\right]\\ =\left[\left\{T\left(K_1\right)\right\}\left\{T\left(n_1\right)\right\}\right].Math.\left[\left\{T\left(K_2\right)\right\}\left\{T\left(n_2\right)\right\}\right].Math.\left\{T\left(K_3\right)\right\}\\ =\left\{T\right\}.Math.\left\{T\right\}.Math.\left\{T\left(K_3\right)\right\}\\ =\left\{T\left(K_3\right)\right\}\end{array}& \left(6\right)\end{array}$
where {T(K.sub.3)} represents two translational outputs in the plane K.sub.3. Thus, the three linear input motions of the manipulator are decoupled. They correspond to the three output motions of the movable platform, respectively.

(90) Based on the general Grubler-Kutzbach criterion, mobility of the manipulator is zero. Thus, the structure of the manipulator may be an over-constrained mechanism. However, in overconstrained mechanisms, large reaction moments and variable friction of the joints can affect the mobility of the movable platform due to unavoidable manufacturing or assembly errors. As a consequence, related output kinematic errors may be induced. Thus, by adding a revolute joint in each subchain, the manipulator may be converted into a non-over-constrained mechanism. An axis of the added revolute joint R.sub.3 is parallel to a moving direction of the prismatic joint P.sub.1. Equivalently, joint P.sub.1 may be replaced by a cylindrical joint to eliminate the overconstrained condition. A 3P-RCR TPM can be obtained by the replacement. The mobility of the non-overconstrained 3P-RCR TPM is three according to the general Grler-Kutzbach criterion. Based on displacement group theory, the additional three non-parallel rotations (R.sub.3) can also be eliminated by the triangular pyramidal constraint. Thus, the added joints have no influence on the mobility and kinematic properties of the manipulator. They are only active when some constraint errors are encountered during the continuous motions.

(91) In some examples, the manipulator illustrated in FIGS. 1-3 offers a number of features that were achieved as a consequence of overcoming the drawbacks of conventional TPMs. These, satisfying at the same time the design objectives, include, for example: 1) Three identical subchains are designed and arranged symmetrically in a non-Cartesian constraint. 2) All actuators provide linear actuation and are placed on the fixed platform. 3) Only four 1-DOF joints are included in each subchain, no Uor S joints are involved. 4) The movable platform is connected with three revolute joints in a star form. A small size of the movable platform is realized. 5) The three input-output translations are decoupled and the manipulator is a non-over-constrained manipulator. Elimination of the over-constrained mechanism characteristics does not influence the mobility and kinematic properties of the manipulator. 6) The manipulator is singularity free and is characterized by an isotropic configuration (described in more detail below).

(92) As a consequence, based on the above features (i.e., non-Cartesian constraint, fixed linear actuation, minimal number of 1-DOF joints, smaller size of the movable platform, decoupled motions) and the drawback analysis of conventional 3-DOF TPMs, the manipulator can achieve a higher load/weight ratio, stiffness and accuracy. In order to evaluate kinematic performances of the manipulator, related position solutions, workspace, velocity, load, singularity, isotropic and dexterity analysis are performed and described below.

Inverse and Forward Position Solutions

(93) The inverse and forward position solutions of the manipulator illustrated in FIGS. 1-3 define position relationships between the three linear input motions of actuations and the three translational output motions of the movable platform, to provide the basic constraint relations for the practical position control system. With reference to FIG. 4, a fixed reference frame O.sub.f-x.sub.fy.sub.fz.sub.f is placed at a center of the fixed platform and a parallel moving reference frame O.sub.m-x.sub.my.sub.mz.sub.m is defined at a center of the movable platform. The guide members O.sub.fG.sub.i (i=1, 2, 3) of the three linear actuators are perpendicular to axis z.sub.f. The guide members are arranged along axis x.sub.f with intersection angles .sub.1, .sub.2 and .sub.3, respectively. The prismatic joints and revolute joints P.sub.i,0, R.sub.i,1, P.sub.i,1 and R.sub.i,2 are connected in sequence. The basic structural parameters are .sub.i, .sub.i, g.sub.i, and where .sub.i is an intersection angle between an axis of R.sub.i,1, and a moving direction of P.sub.i,1, g.sub.i represents a size of a slide block between points A.sub.i and B.sub.i which is parallel to the axis z.sub.f, while r.sub.i is a size of the movable platform between points O.sub.m and C.sub.i which is perpendicular to the axis z.sub.f. The joint variables of P.sub.i,0, R.sub.i,1, P.sub.i,1 and R.sub.i,2 are l.sub.i, .sub.i,u, h.sub.i and .sub.i,d, respectively. The input variables are the lengths of travel l.sub.i of the linear actuators between points O.sub.f and A.sub.i which takes place along the three guide members, while an output variable is a position vector r.sub.O (x, y, z) of the movable platform between points O.sub.f and O.sub.m. In a given subchain, the kinematic constraint of the closed loop is:
l.sub.i+g.sub.i=r.sub.O+r.sub.i+h.sub.i(7)
where the vectors are defined as:

(94) $\begin{array}{cc}\{\begin{array}{c}{l}_i=\stackrel{.fwdarw.}{O_f{A}_i}={\left[.Math.l_i.Math.\cos {}_i.Math.l_i.Math.\sin {}_{i}0\right]}^T\\ g_i=\stackrel{.fwdarw.}{A_i{B}_i}=\stackrel{.fwdarw.}{O_f{B}_i}-\stackrel{.fwdarw.}{O_f{A}_i}={\left[\begin{array}{cc}0& 0\end{array}-.Math.g_i.Math.\right]}^T\\ r_O=\stackrel{.fwdarw.}{O_f{O}_m}={\left[\begin{array}{cc}x& y\end{array}z\right]}^T\\ r_i=\stackrel{.fwdarw.}{O_m{C}_i}=\stackrel{.fwdarw.}{O_f{C}_i}-\stackrel{.fwdarw.}{O_f{O}_m}={\left[.Math.r_i.Math.\cos {}_i.Math.r_i.Math.\sin {}_{i}0\right]}^T\end{array}& \left(8\right)\end{array}$
Thus, the vector h.sub.i can be deduced and expressed as:

(95) $\begin{array}{cc}\begin{array}{c}{h}_i=\stackrel{.fwdarw.}{C_i{B}_i}=l_i+g_i-r_O-r_i\\ ={\left[\begin{array}{ccc}\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\cos {}_i-x& \left(.Math.l_i.Math.-.Math.r_i.Math.\right)\sin {}_i-y& -.Math.g_i.Math.-z\end{array}\right]}^T\end{array}& \left(9\right)\end{array}$

(96) In accordance with the structural constraint condition, namely the intersection angle between the variable vectors l.sub.i and h.sub.i is the constant .sub.i the corresponding kinematic constraint equation can be obtained by a dot product of l.sub.i and h.sub.i as

(97) $\begin{array}{cc}\begin{array}{c}{l}_i.Math.h_i=.Math.l_i.Math.\cos {}_i\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\cos {}_i-x\right)+.Math.l_i.Math.\sin {}_i\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\sin {}_i-y\right)=.Math.l_i.Math..Math.h_i.Math.\cos {}_i\\ =.Math.l_i.Math.\sqrt{{\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\cos {}_i-x\right)}^2+{\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\sin {}_i-y\right)}^2+{\left(.Math.g_i.Math.+z\right)}^2}\cos {}_i\end{array}& \left(10\right)\end{array}$
To solve Eq. (10) with respect to unknown parameter l.sub.i, an inverse position solution of the manipulator can be defined as:
l.sub.i=r.sub.i+x cos .sub.i+y sin .sub.icot .sub.i{square root over ((x sin .sub.iy cos .sub.i).sup.2+(g.sub.i+z).sup.2)}(11)
where the symbol is selected as + for a larger value of l.sub.i, because l.sub.i is a non-negative number, and initial positions of the slide blocks are placed outward from the symmetric center point O.sub.m.

(98) The forward position solution is obtained by solving the equations of the inverse position solution when the output variables x, y, and z are unknown parameters. By writing Eq. (11) three times (once for each i=1, 2, 3), three non-linear equations are obtained:

(99) 0$\begin{array}{cc}\{\begin{array}{c}\begin{array}{c}\cot {}_1\sqrt{{\left(x\sin {}_1-y\cos {}_1\right)}^2+{\left(.Math.g_1.Math.+z\right)}^2}+\\ x\cos {}_1+y\sin {}_1+.Math.r_1.Math.-.Math.l_1.Math.=0\end{array}\\ \begin{array}{c}\cot {}_2\sqrt{{\left(x\sin {}_2-y\cos {}_2\right)}^2+{\left(.Math.g_2.Math.+z\right)}^2}+\\ x\cos {}_2+y\sin {}_2+.Math.r_2.Math.-.Math.l_2.Math.=0\end{array}\\ \begin{array}{c}\cot {}_3\sqrt{{\left(x\sin {}_3-y\cos {}_3\right)}^2+{\left(.Math.g_3.Math.+z\right)}^2}+\\ x\cos {}_3+y\sin {}_3+.Math.r_3.Math.-.Math.l_3.Math.=0\end{array}\end{array}& \left(12\right)\end{array}$

(100) In accordance with the computational results of symbolic mathematical software, an analytical form of the solutions for x, y, and z exists, but the expression is complex and long. Thus, to solve the forward position configuration of the manipulator illustrated in FIGS. 1-3, numerical evaluations of Eq. (12) can also be used.

(101) To illustrate the developments in this paper, a hypothetical manipulator with structural parameters listed in Table 1 will be assumed. Moving ranges of all joints are represented by l.sub.i.sub.min, l.sub.i.sub.max, h.sub.i.sub.min, h.sub.i.sub.max, .sub.i,min and .sub.i,max. Numerical forward position solutions, based on Eq. (12), of the manipulator in eight configurations are listed in Table 2.

(102) TABLE-US-00001 TABLE 1 Structural parameters of the Manipulator Parameter Value Parameter Value Parameter Value .sub.1 90 g.sub.i 600 mm h.sub.i.sub.min 1800 mm .sub.2 210 r.sub.i 215 mm h.sub.i.sub.max 900 mm .sub.3 330 l.sub.i.sub.min 600 mm .sub.i, min 60 .sub.i 45 l.sub.i.sub.max 1800 mm .sub.i, max 60

(103) TABLE-US-00002 TABLE 2 Forward position solutions of the Manipulator Config- uration l.sub.1, l.sub.2, l.sub.3 (mm) x (mm) y (mm) z (mm) Solutions 1 800, 800, 800 0 0 1185 1 0 0 15 2 2 900, 1000, 1100 61.58 98.14 1380.7 1 61.58 98.14 180.7 2 3 1000, 1150, 1200 30.8 113.36 1498 1 30.8 113.36 297.8 2 4 1150, 1300, 1250 30.08 81.91 1616 1 30.08 81.91 416.5 2 5 1350, 1400, 1200 113.89 30.6 1699 1 113.89 30.6 498.5 2 6 1500, 1450, 1250 110.8 99.49 1780 1 110.8 99.49 580.3 2 7 1550, 1400, 1500 56.25 66.93 1867 1 56.25 66.93 666.8 2 8 1600, 1600, 1600 0 0 1985 1 0 0 785 2

(104) In Table 2, the eight configurations, defined by l.sub.1, l.sub.2, l.sub.3, were arbitrarily assumed within the moving ranges of the linear actuators. In accordance with Eq. (12), all possible solution results for x, y, and z are listed. In each configuration given by l.sub.1, l.sub.2, l.sub.3, there are two solutions for x, y, and z. The two solutions for x and y are equal in each configuration. The first solution value for z in each configuration is smaller than the second solution value, which means that two possible position solutions of the movable platform and structure exist. These solutions are symmetric with respect to the plane B.sub.1B.sub.2B.sub.3. Thus, the first solution can be determined as the real solution because of physical interference and of the initial positions of the subchains as shown in FIG. 4.

Workspace Evaluation

(105) The workspace of the manipulator illustrated in FIGS. 1-3 is a basic property that reflects its working capacity. The shape and size of the workspace provides basic boundary information for related trajectory planning of applications. As indicated above, the manipulator is capable of being used in a variety of applications with a variety of functionalities and operations. In connection with the manipulator shown in FIGS. 1-3, the reachable workspace of the movable platform may be defined as the space that can be reached by the three translations of the point O.sub.m in the fixed reference frame O.sub.f-x.sub.fy.sub.fz.sub.f. The boundary of the reachable workspace is simultaneously constrained by moving ranges, physical interference and geometric constraints of all joints in the closed-loop form.

(106) In this example, during continuous motions of the movable platform, the two parallel revolute joints R.sub.i,1, R.sub.i,2 of each subchain are passive joints. Furthermore, vector h.sub.i has the same intersection angles with vector g.sub.i and the axis z.sub.m, where the vector g.sub.i is always parallel to the axis z.sub.m since there are no rotational outputs on the movable platform. Thus, the rotational angles .sub.i,u, .sub.i,d of the two parallel revolute joints of each subchain are equal. The angles .sub.i,u, .sub.i,d can be determined by the closed-loop constraint as:

(107) $\begin{array}{cc}{}_{i,u}={}_{i,d}=\arccos \left(\frac{\left[\begin{array}{ccc}0& 0& 1\end{array}\right]\stackrel{.fwdarw.}{B_i{D}_i}}{.Math.\stackrel{.fwdarw.}{B_i{D}_i}.Math.}\right)\mathrm{where}& \left(13\right)\\ \begin{array}{c}\stackrel{.fwdarw.}{B_i{D}_i}=\stackrel{.fwdarw.}{O_f{D}_i}-\stackrel{.fwdarw.}{O_f{B}_i}\\ =\left[\begin{array}{c}\left(.Math.r_i.Math.+.Math.h_i.Math.\cos {}_i\right)\cos {}_i+x\\ \left(.Math.r_i.Math.+.Math.h_i.Math.\cos {}_i\right)\sin {}_i+y\\ z\end{array}\right]-\left[\begin{array}{c}.Math.l_i.Math.\cos {}_i\\ .Math.l_i.Math.\sin {}_i\\ -.Math.g_i.Math.\end{array}\right]\\ =\left[\begin{array}{c}\left(.Math.r_i.Math.+.Math.h_i.Math.\cos {}_i\right)\cos {}_i+x-.Math.l_i.Math.\cos {}_i\\ \left(.Math.r_i.Math.+.Math.h_i.Math.\cos {}_i\right)\sin {}_i+y-.Math.l_i.Math.\sin {}_i\\ z+.Math.g_i.Math.\end{array}\right]\end{array}& \left(14\right)\end{array}$

(108) The translational displacements l.sub.i (P.sub.i,0) and h.sub.i (P.sub.i,1) are determined by the inverse position solution from Eqs. (11) and (9), respectively. Thus, by considering the moving ranges, physical interference and geometric constraints of all joints, the corresponding constraint equations of the reachable workspace can be obtained as:

(109) $\begin{array}{cc}\{\begin{array}{c}{.Math.l_i.Math.}_{min}.Math.l_i.Math.{.Math.l_i.Math.}_{\mathrm{ma}x}\\ {.Math.h_i.Math.}_{min}.Math.h_i.Math.{.Math.h_i.Math.}_{\mathrm{ma}x}\\ {}_{i,min}{}_{i,u}={}_{i,d}{}_{i,\mathrm{ma}x}\\ z-.Math.g_i.Math.\end{array}& \left(15\right)\end{array}$

(110) In order to evaluate the shape and size of the reachable workspace, a numerical search can be performed based on Eq. (15). In the fixed reference frame O.sub.f-x.sub.fy.sub.fz.sub.f, if a point (x, y, z) satisfies the inequalities expressed by Eq. (15), that point can be considered as part of the reachable workspace. Thus, by searching all points on each x-y section of O.sub.f-x.sub.fy.sub.fz.sub.f, the set of the valid points is obtained, and the 3D arrangement of all valid points can be considered as the reachable workspace of the manipulator.

(111) As an example, the values of the structural parameters and of the moving ranges of all joints listed in Table 1 will be utilized. Based on the numerical search results in Eq. (15), the basic 3D shape and size of the reachable workspace is shown in FIG. 17. The shape of the reachable workspace is symmetric about the axis z.sub.f Extreme values of the coordinates of the reachable workspace are: x[848,848], y[583,945], z[1873,1230] (the unit is mm). In FIG. 18, three x-y sections of the reachable workspace are generated and are taken at z=1432 mm, 1582 mm and 1732 mm. The three x-y sections have different shapes and sizes. By considering practical motion ranges of the movable platform and general applications of the manipulator, in one example, the manipulator may have a practical workspace (namely, the sub-workspace of the reachable workspace) defined in the form of a cylinder with a height of 300 mm (between two x-y sections at z=1432 mm and 1732 mm), where the maximal diameter of the cylinder is =884 mm (around a center point O.sub.f).

(112) Among all the basic structural constants .sub.i, .sub.i, g.sub.i and r.sub.i, .sub.i determines a symmetric property of the workspace, variations of g.sub.i and r.sub.i can only change a size of the workspace (instead of the shape of the workspace), because the different lengths of the g.sub.i and r.sub.i have no influence on the constraint angles between the axes of all the joints. In the inclined planar displacement subsets of the three subchains, the intersection angle .sub.i determines the inclination of the triangular pyramidal constraint that can change the constraint angles between the moving direction of joint P.sub.i,1, and the axes of joints R.sub.i,1, R.sub.i,2. Different constraint angles of the joints will generate workspaces of different shapes and sizes. Thus, by comparing the variation of all the basic structural constants, the variation of .sub.i may have the most significant influence on the workspace.

(113) The variation range of the intersection angle is .sub.i(0,90). In FIG. 17, the workspace is generated for .sub.i equal to 45. In order to illustrate variation of the workspace for different values of .sub.i, a numerical evaluation of the workspace is shown in FIGS. 19-22, for .sub.i=30 and 60. By comparing the shape and size of the three different workspaces in FIGS. 17 and 22, extreme values of the coordinates of the reachable workspace are not changed dramatically, while the maximal diameter of the cylinder of the defined practical workspace has been reduced to =570 mm (.sub.i=30) and =604 mm (.sub.i=60), while the height of the cylinder remains constant. Compared to the =884 mm (.sub.i=45) in FIGS. 17 and 18, the size of the practical workspace in FIGS. 19-22 has decreased. The coordinate z of the workspace decreases with the increase of .sub.i. Thus, by evaluating the workspace with the variation of .sub.i, the maximal size of the practical workspace can be achieved when .sub.i=49.8. The cylinder of the practical workspace has the maximal diameter of =908 mm and height 300 mm between the two x-y sections at z=1895 mm and 1595 mm.

Velocity and Load Jacobian Analysis

(114) In order to establish a stiffness model of the manipulator, all of the deformations including three rotations and three translations of the movable platform should be considered. Thus, a 66 overall Jacobian matrix is required for the stiffness modeling of this limited-DOF parallel manipulator. The overall Jacobian matrix includes the Jacobian of the constraints and actuations. Correspondingly, the stiffness of structural constraints and the stiffness of actuations can be included in the 66 overall Jacobian matrix, respectively. Based on the theory of reciprocal screws, the Jacobian of constraints and actuations are analyzed as follows.

(115) By assuming that the deformations of the output movable platform include 6-DOF motions, the vectors for the angular and linear velocities of the movable platform can be respectively expressed in the fixed reference frame O.sub.f-x.sub.fy.sub.fz.sub.f as:

(116) $\begin{array}{cc}\{\begin{array}{c}={\left[\begin{array}{ccc}{}_{\mathrm{xf}}& {}_{\mathrm{yf}}& {}_{\mathrm{zf}}\end{array}\right]}^T\\ v={\left[\begin{array}{ccc}{v}_{\mathrm{xf}}& v_{\mathrm{yf}}& v_{\mathrm{zf}}\end{array}\right]}^T\end{array}& \left(16\right)\end{array}$

(117) Based on the closed-loop constraint of each subchain in Eq. (7) and the non-overconstrained structure of the manipulator with five joints in each subchain, the instantaneous twist $.sub.Om of the movable platform is a linear combination of the five instantaneous twists of the joints, namely:
$.sub.Om=[.sup.Tv.sup.T].sup.T={dot over (l)}.sub.i{circumflex over ($)}.sub.i,1+{dot over ()}.sub.i,u{circumflex over ($)}.sub.i,2+{dot over (h)}.sub.i{circumflex over ($)}.sub.i,3+{dot over ()}.sub.i,d.sub.i,4+{dot over ()}.sub.i,c{dot over ($)}.sub.i,5(17)
where {circumflex over ($)}.sub.i,j (i=1, 2, 3, j=1, 2, 3, 4, 5) is a unit screw associated with the j.sup.th joint of the i.sup.th subchain. {dot over (l)}.sub.i, {dot over ()}.sub.i,u, {dot over (h)}.sub.i, {dot over ()}.sub.i,d, and {dot over ()}.sub.i,c represent velocities of the joints. The linear combination of the instantaneous twists of each subchain is the same because the three subchains are connected to the movable platform with identical output motions.

(118) In FIG. 27, unit vector s.sub.i,j that is along the j.sup.th joint axis of the i.sup.th subchain is given by:

(119) $\begin{array}{cc}{s}_{i,1}=s_{i,2}=s_{i,4}=\frac{l_i}{.Math.l_i.Math.}={\left[\begin{array}{ccc}\cos {}_i& \sin {}_i& 0\end{array}\right]}^T& \left(18\right)\\ s_{i,3}=s_{i,5}=\frac{h_i}{.Math.h_i.Math.}=\frac{\left({\left[\begin{array}{ccc}\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\cos {}_i-x& \left(.Math.l_i.Math.-.Math.r_i.Math.\right)\sin {}_i-y& -.Math.g_i.Math.-z\end{array}\right]}^T\right)}{\left(\sqrt{{\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\cos {}_i-x\right)}^2+{\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\sin {}_i-y\right)}^2+{\left(.Math.g_i.Math.+z\right)}^2}\right)}& \left(19\right)\end{array}$
where l.sub.i can be obtained from Eq. (11). Thus, the unit screws {circumflex over ($)}.sub.i,j of Eq. (17) are:

(120) $\begin{array}{cc}{\widehat{\$}}_{i,1}=\left[\begin{array}{c}0\\ s_{i,1}\end{array}\right]={\left[\begin{array}{cccccc}0& 0& 0& \cos {}_i& \sin {}_i& 0\end{array}\right]}^T& \left(20\right)\\ {\widehat{\$}}_{i,2}=\left[\begin{array}{c}{s}_{i,2}\\ r_s{s}_{i,2}\end{array}\right]=\left[\begin{array}{c}{s}_{i,2}\\ \stackrel{.fwdarw.}{O_{\mathrm{fr}}{O}_m}{s}_{i,2}\end{array}\right]=\left[\begin{array}{c}{s}_{i,2}\\ \left(\stackrel{.fwdarw.}{O_{\mathrm{fr}}{O}_f}+r_O\right)s_{i,2}\end{array}\right]={\left[\begin{array}{cccccc}\cos {}_i& \sin {}_i& 0& -\left(.Math.g_i.Math.+z\right)\sin {}_i& \left(.Math.g_i.Math.+z\right)\cos {}_i& x\sin {}_i-y\cos {}_i\end{array}\right]}^T& \left(21\right)\\ {\widehat{\$}}_{i,3}=\left[\begin{array}{c}0\\ s_{i,3}\end{array}\right]=\frac{{[\begin{array}{cccccc}0& 0& 0& \left(.Math.l_i.Math.-.Math.r_i.Math.\right)\cos {}_i-x& \left(.Math.l_i.Math.-.Math.r_i.Math.\right)\sin {}_i-y& -.Math.g_i.Math.-z\end{array}]}^T}{\sqrt{{\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\cos {}_i-x\right)}^2+{\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\sin {}_i-y\right)}^2+{\left(.Math.g_i.Math.+z\right)}^2}}& \left(22\right)\\ {\widehat{\$}}_{i,4}=\left[\begin{array}{c}{s}_{i,4}\\ 0s_{i,4}\end{array}\right]={\left[\begin{array}{cccccc}\cos {}_i& \sin {}_i& 0& 0& 0& 0\end{array}\right]}^T& \left(23\right)\\ {\widehat{\$}}_{i,5}=\left[\begin{array}{c}{s}_{i,5}\\ r_i{s}_{i,5}\end{array}\right]=\frac{\left[\begin{array}{c}\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\cos {}_i-x\\ \left(.Math.l_i.Math.-.Math.r_i.Math.\right)\sin {}_i-y\\ -.Math.g_i.Math.-z\\ -.Math.r_i.Math.\sin {}_i\left(.Math.g_i.Math.+z\right)\\ .Math.r_i.Math.\cos {}_i\left(.Math.g_i.Math.+z\right)\\ .Math.r_i.Math.\left(x\sin {}_i-y\cos {}_i\right)\end{array}\right]}{\sqrt{{\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\cos {}_i-x\right)}^2+{\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\sin {}_i-y\right)}^2+{\left(.Math.g_i.Math.+z\right)}^2}}& \left(24\right)\end{array}$

(121) In accordance with the theory of reciprocal screws, a reciprocal screw $.sub.a$.sub.b is that the wrench $.sub.a has no work along the twist of $.sub.b. In each subchain of the manipulator, a unit constraint screw {circumflex over ($)}.sub.i,c is reciprocal to all joint screws (instantaneous twist $.sub.Om of the movable platform). The wrench {circumflex over ($)}.sub.i,c cannot activate the motions of the five joints of each subchain. Thus, based on the structural constraints of the joints in each subchain, the unit constraint screw {circumflex over ($)}.sub.i,c can be identified as an infinite-pitch wrench screw, namely:

(122) $\begin{array}{cc}\begin{array}{c}{\widehat{\$}}_{i,c}=\left[\begin{array}{c}0\\ k_{i,c}\end{array}\right]=\left[\begin{array}{c}0\\ \frac{s_{i,2}{s}_{i,3}}{.Math.s_{i,2}{s}_{i,3}.Math.}\end{array}\right]\\ =\frac{{\left[\begin{array}{cccccc}0& 0& 0& -\sin {}_i\left(.Math.g_i.Math.+z\right)& \cos {}_i\left(.Math.g_i.Math.+z\right)& x\sin {}_i-y\cos {}_i\end{array}\right]}^T}{\sqrt{{\left(x\sin {}_i-y\cos {}_i\right)}^2+{\left(.Math.g_i.Math.+z\right)}^2}}\end{array}& \left(25\right)\end{array}$
which represents a unit couple of constraints imposed by the joints of the i.sup.th subchain, and the couple is exerted on the movable platform around the direction of the unit constraint vector k.sub.i,c. The constraint vector, k.sub.i,c is perpendicular to the unit vectors s.sub.i,2 and s.sub.i,3. The reciprocal product of the unit constraint screw {circumflex over ($)}.sub.i,c and of the instantaneous twist on $.sub.Om is:

(123) $\begin{array}{cc}{\widehat{\$}}_{i,c}{\$}_{\mathrm{Om}}={\left({\widehat{\$}}_{i,c}\right)}^T{\$}_{\mathrm{Om}}=\frac{\left[\begin{array}{cccccc}-\sin {}_i\left(.Math.g_i.Math.+z\right)& \cos {}_i\left(.Math.g_i.Math.+z\right)& x\sin {}_i-y\cos {}_i& 0& 0& 0\end{array}\right]{\$}_{\mathrm{Om}}}{\sqrt{{\left(x\sin {}_i-y\cos {}_i\right)}^2+{\left(.Math.g_i.Math.+z\right)}^2}}=J_{i,c}{\$}_{\mathrm{Om}}=0& \left(26\right)\end{array}$
where the reciprocal matrix is

(124) $\begin{array}{cc}=\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ 1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\end{array}\right]& \left(27\right)\end{array}$

(125) Thus, the Jacobian of constraints can be obtained from Eq. (26), which transforms the angular and linear velocities of the movable platform to 0. By considering the constraints of the three subchains, the Jacobian of the constraints is:

(126) $\begin{array}{cc}{J}_c=\left[\begin{array}{c}{J}_{1,c}\\ J_{2,c}\\ J_{3,c}\end{array}\right]=\left[\begin{array}{cccc}{k}_{1,c}^T& 0& 0& 0\\ k_{2,c}^T& 0& 0& 0\\ k_{3,c}^T& 0& 0& 0\end{array}\right]& \left(28\right)\end{array}$
where k.sub.i,c is determined in Eq. (16). Each row of J.sub.c represents a unit wrench of structural constraints of a subchain, which is exerted on the movable platform. Thus, the Jacobian of constraints determines the 3-DOF translational output motions of the movable platform.

(127) If J.sub.c is of full rank, the unique solution for J.sub.i,c$.sub.Om=0 (Eq. (26) is =0. Thus, based on Eqs. (17) and (20)-(24), a constraint equation can be obtained as:
{dot over ()}.sub.i,us.sub.i,2+{dot over ()}.sub.i,ds.sub.i,4+{dot over ()}.sub.i,cs.sub.i,5=0(29)
Since s.sub.i,2=s.sub.i,4s.sub.i,50, thus, Eq. (29) can be deduced as:

(128) 0$\begin{array}{cc}\{\begin{array}{c}{\stackrel{.}{}}_{i,u}+{\stackrel{.}{}}_{i,d}=0\\ {\stackrel{.}{}}_{i,c}=0\end{array}& \left(30\right)\end{array}$
namely, stating that rotational angles of joints and R.sub.i,1 and R.sub.i,2 are equal, and that the added revolute joint R.sub.i,3 has no rotation in the motions of the structure. Thus, the kinematic motions and constraints of the manipulator are not affected by the elimination of the overconstrained mechanism condition.

(129) By assuming that the actuated joint (P.sub.i,0) in each subchain is locked, a unit actuation screw {circumflex over ($)}.sub.i,a is reciprocal to all passive joint (R.sub.i,1P.sub.i,1R.sub.i,2R.sub.i,3) screws of the i.sup.th subchain. The wrench of {circumflex over ($)}.sub.i,a has no work on all passive joints but the actuated joint. Thus, the unit actuation screw {circumflex over ($)}.sub.i,a can be identified as a zero-pitch screw along the direction of the unit actuation vector k.sub.i,a i.e.:

(130) $\begin{array}{cc}\begin{array}{c}{\widehat{\$}}_{i,a}=\left[\begin{array}{c}{k}_{i,a}\\ r_i{k}_{i,a}\end{array}\right]=\left[\begin{array}{c}\frac{k_{i,c}{s}_{i,3}}{.Math.k_{i,c}{s}_{i,3}.Math.}\\ r_i\left(\frac{k_{i,c}{s}_{i,3}}{.Math.k_{i,c}{s}_{i,3}.Math.}\right)\end{array}\right]\\ =\frac{\left[\begin{array}{c}-\cos {{}_i\left(.Math.g_i.Math.+z\right)}^2-\left(x\sin {}_i-y\cos {}_i\right)\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\sin {}_i-y\right)\\ -\sin {{}_i\left(.Math.g_i.Math.+z\right)}^2+\left(x\sin {}_i-y\cos {}_i\right)\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\cos {}_i-x\right)\\ \left(.Math.g_i.Math.+z\right)\left(-\left(.Math.l_i.Math.-.Math.r_i.Math.\right)+x\cos {}_i+y\sin {}_i\right)\\ .Math.r_i.Math.\sin {}_i\left(.Math.g_i.Math.+z\right)\left(-\left(.Math.l_i.Math.-.Math.r_i.Math.\right)+x\cos {}_i+y\sin {}_i\right)\\ -.Math.r_i.Math.\cos {}_i\left(.Math.g_i.Math.+z\right)\left(-\left(.Math.l_i.Math.-.Math.r_i.Math.\right)+x\cos {}_i+y\sin {}_i\right)\\ .Math.r_i.Math.\left(x\sin {}_i-y\cos {}_i\right)\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)-x\cos {}_i-y\sin {}_i\right)\end{array}\right]}{\sqrt{\begin{array}{c}\begin{array}{c}{\left(\begin{array}{c}\cos {{}_i\left(.Math.g_i.Math.+z\right)}^2+\\ \left(x\sin {}_i-y\cos {}_i\right)\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\sin {}_i-y\right)\end{array}\right)}^2+\\ {\left(\begin{array}{c}{\left(\sin {}_i\left(.Math.g_i.Math.+z\right)\right)}^2-\\ \left(x\sin {}_i-y\cos {}_i\right)\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\cos {}_i-x\right)\end{array}\right)}^2+\end{array}\\ {\left(.Math.g_i.Math.+z\right)}^2{\left(-\left(.Math.l_i.Math.-.Math.r_i.Math.\right)+x\cos {}_i+y\sin {}_i\right)}^2\end{array}}}\end{array}& \left(31\right)\end{array}$
which represents a unit force of actuation imposed by the actuated joint of the i.sup.th subchain that is exerted on the movable platform along the direction of the unit actuation vector k.sub.i,a. The vector k.sub.i,a is perpendicular to the unit vectors s.sub.i,3 and passes through the axes of joints R.sub.i,1 and R.sub.i,2. The reciprocal product of the unit actuation screw {circumflex over ($)}.sub.i,a and of the instantaneous twist $.sub.Om is:

(131) $\begin{array}{cc}\begin{array}{c}{\widehat{\$}}_{i,a}{\$}_{\mathrm{Om}}={\left({\widehat{\$}}_{i,a}\right)}^T{\$}_{\mathrm{Om}}\\ =\frac{-.Math.{\stackrel{.}{l}}_i.Math.\left({\left(.Math.g_i.Math.+z\right)}^2+{\left(x\sin {}_i-y\cos {}_i\right)}^2\right)}{\sqrt{\begin{array}{c}\begin{array}{c}{\left(\cos {{}_i\left(.Math.g_i.Math.+z\right)}^2+\left(x\sin {}_i-y\cos {}_i\right)\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\sin {}_i-y\right)\right)}^2+\\ {\left(\sin {{}_i\left(.Math.g_i.Math.+z\right)}^2-\left(x\sin {}_i-y\cos {}_i\right)\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\cos {}_i-x\right)\right)}^2+\end{array}\\ {\left(.Math.g_i.Math.+z\right)}^2{\left(-\left(.Math.l_i.Math.-.Math.r_i.Math.\right)+x\cos {}_i+y\sin {}_i\right)}^2\end{array}}}\\ =.Math.{\stackrel{.}{l}}_i.Math.\cos {}_i=.Math.{\stackrel{.}{l}}_i.Math.\cos \left(\frac{}{2}-{}_i\right)=.Math.{\stackrel{.}{l}}_i.Math.k_{i,a}^T{s}_{i,2}\end{array}& \left(32\right)\\ \mathrm{namely},& \\ \begin{array}{c}{J}_{i,\mathrm{ar}}{\$}_{\mathrm{Om}}=\left[\begin{array}{cc}\frac{{\left(r_i{k}_{i,a}\right)}^T}{k_{i,a}^T{s}_{i,2}}& \frac{k_{i,a}^T}{k_{i,a}^T{s}_{i,2}}\end{array}\right]{\$}_{\mathrm{Om}}\\ =\frac{{\left[\begin{array}{c}.Math.r_i.Math.\sin {}_i\left(.Math.g_i.Math.+z\right)\left(-\left(.Math.l_i.Math.-.Math.r_i.Math.\right)+x\cos {}_i+y\sin {}_i\right)\\ -.Math.r_i.Math.\cos {}_i\left(.Math.g_i.Math.+z\right)\left(-\left(.Math.l_i.Math.-.Math.r_i.Math.\right)+x\cos {}_i+y\sin {}_i\right)\\ .Math.r_i.Math.\left(x\sin {}_i-y\cos {}_i\right)\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)-x\cos {}_i-y\sin {}_i\right)\\ -\cos {{}_i\left(.Math.g_i.Math.+z\right)}^2-\left(x\sin {}_i-y\cos {}_i\right)\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\sin {}_i-y\right)\\ -\sin {{}_i\left(.Math.g_i.Math.+z\right)}^2+\left(x\sin {}_i-y\cos {}_i\right)\left(\left(.Math.l_i.Math.-.Math.r_i.Math.\right)\cos {}_i-x\right)\\ \left(.Math.g_i.Math.+z\right)\left(-\left(.Math.l_i.Math.-.Math.r_i.Math.\right)+x\cos {}_i+y\sin {}_i\right)\end{array}\right]}^T{\$}_{\mathrm{Om}}}{-{\left(.Math.g_i.Math.+z\right)}^2-{\left(x\sin {}_i-y\cos {}_i\right)}^2}\\ =.Math.{\stackrel{.}{l}}_i.Math.\end{array}& \left(33\right)\end{array}$

(132) Thus, the Jacobian of actuations is determined by Eq. (33), which transforms the angular and linear velocities of the movable platform to the linear input velocity {dot over (l)}.sub.i of the actuator. By considering the motions of the three subchains, the Jacobian of actuations assumes the form:

(133) $\begin{array}{cc}{J}_{\mathrm{ar}}=\left[\begin{array}{c}{J}_{1,\mathrm{ar}}\\ J_{2,\mathrm{ar}}\\ J_{3,\mathrm{ar}}\end{array}\right]=\left[\begin{array}{cc}\frac{{\left(r_1{k}_{1,a}\right)}^T}{k_{1,a}^T{s}_{1,2}}& \frac{k_{1,2}^T}{k_{1,2}^T{s}_{1,2}}\\ \frac{{\left(r_2{k}_{2,a}\right)}^T}{k_{2,a}^T{s}_{2,2}}& \frac{k_{2,a}^T}{k_{2,a}^T{s}_{2,2}}\\ \frac{{\left(r_3{k}_{3,a}\right)}^T}{k_{3,a}^T{s}_{3,2}}& \frac{k_{3,a}^T}{k_{3,a}^T{s}_{3,2}}\end{array}\right]=\left[\begin{array}{cc}\frac{{\left(r_1{k}_{1,a}\right)}^T}{\cos \left(\frac{}{2}-{}_1\right)}& \frac{k_{1,a}^T}{\cos \left(\frac{}{2}-{}_1\right)}\\ \frac{{\left(r_2{k}_{2,a}\right)}^T}{\cos \left(\frac{}{2}-{}_2\right)}& \frac{k_{2,a}^T}{\cos \left(\frac{}{2}-{}_2\right)}\\ \frac{{\left(r_3{k}_{3,a}\right)}^T}{\cos \left(\frac{}{2}-{}_3\right)}& \frac{k_{3,a}^T}{\cos \left(\frac{}{2}-{}_3\right)}\end{array}\right]& \left(34\right)\end{array}$
where the last three columns are dimensionless, while the first three columns are related to the length of r.sub.i. It is necessary to homogenize the units of the Jacobian of actuations for which the stiffness matrix and performance index are invariant with respect to the length unit. Thus, the dimensionally homogeneous Jacobian of actuations can be expressed as:

(134) $\begin{array}{cc}{J}_a=\left[\begin{array}{c}{J}_{1,a}\\ J_{2,a}\\ J_{3,a}\end{array}\right]=\left[\begin{array}{cc}\frac{{\left(r_1{k}_{1,a}\right)}^T}{k_{1,a}^T{s}_{1,2}.Math.r_1.Math.}& \frac{k_{1,a}^T}{k_{1,a}^T{s}_{1,2}}\\ \frac{{\left(r_2{k}_{2,a}\right)}^T}{k_{2,a}^T{s}_{2,2}.Math.r_2.Math.}& \frac{k_{2,a}^T}{k_{2,a}^T{s}_{2,2}}\\ \frac{{\left(r_3{k}_{3,a}\right)}^T}{k_{3,a}^T{s}_{3,2}.Math.r_3.Math.}& \frac{k_{3,a}^T}{k_{3,a}^T{s}_{3,2}}\end{array}\right]& \left(35\right)\end{array}$