Kinematics of the spherical parallel robot.

Staicu, Stefan ; Rugescu, Radu

1. INTRODUCTION

When compared with serial mechanisms, the parallel robot is a spatial mechanical structure with specific features: smaller in weight and with improved stiffness, good positioning accuracy, suitable arrangement of actuators and a low workspace volume for a similar target area.

Considerable effort has been devoted to the kinematics of full parallel manipulators. Among these robots, the class of manipulators known as Stewart-Gough platforms had focused special attention. The Star parallel manipulator (Herve & Sparacino, 1992) and the Delta parallel robot (Clavel, 1988; Tsai & Stamper, 1996) are equipped with three motors that drive the moving platform in a general translational motion. The general direct kinematics of the Agile Wrist spherical parallel robot was previously developed in a tedious manner (Gosselin & Angeles, 1989).

Spherical parallel manipulators provide high stiffness and accuracy due to their mechanical arrangement in which all the actuators are anchored to the base. Three kinematical chains support the end-effector. Accuracy and precision in the execution of the task of the wrist are essential since the robot is intended to operate on fragile objects; where positioning errors of the tool could end in costly damages.

The matrix approach here presented largely simplifies the computations and transforms them into a highly reliable tool.

2. INVERSE GEOMETRIC MODEL

Some recursive matrix relations for the kinematics of a three-degree-of-freedom spherical parallel robot are first established. This mechanism presents a structure with nine axes, concurrent in a common centre of rotation, namely the fixed point O (Fig. 1). Let [Ox.sub.0][y.sub.0][Z.sub.0]([T.sub.0]) be a fixed Cartesian frame, about which the manipulator moves. It possesses three legs of known size and mass. The architecture consists of two main elements, the base and the moving platform, which is free to undergo arbitrary rotations around its centre and three identical legs, each of these composed of two links coupled by revolute joints.

The first element [T.sub.1] of leg A is called proximal link and is one of the three driving parts of the robot. It is a homogenous rod of radius r, rotating around the [Oz.sup.A.sub.1] axis with the angular velocity [[omega].sup.A.sub.10] = [[??].sup.A.sub.10] and the angular acceleration [[epsilon].sup.A.sub.10] = [[??].sup.A.sub.10]. The distal link [T.sub.2] of the same radius r is connected to the [A.sub.2] [x.sup.A.sub.2] [y.sup.A.sub.2] [z.sup.A.sub.2] frame and manifests a relative rotation with the angle [[phi].sup.A.sub.21] and angular velocity [[omega].sup.A.sub.21] = [[??].sup.A.sub.21].

The platform of the robot is a three arms star with length equal to l = r [square root of (2 + [sin.sup.2] [delta]/3], which rotates with an angular velocity [[omega].sup.A.sub.32] = [[??].sup.A.sub.32] with respect to the neighboring body. The link angles of the manipulator are securely the same on each of the legs connecting the end-effectors to the base. They are denoted by [beta] = [pi]/2 for the proximal link and [gamma] = [[pi]/2, for the distal link. The angles [[alpha].sub.A] = 0, [[alpha].sub.B] = 2[pi]/3, [[alpha].sub.C] =-2[pi]/3, [delta]=[pi]/6 and sin[theta] = 1/[square root of 3], cos[theta] = [square root of 2/3] give the geometry of the base and the initial position of the manipulator's arm. Each leg of the robot is a serial spherical wrist: the first joint couples the base with the proximal link, the second joint couples the proximal link with the distal link and the last one couples the distal link with the platform (see Fig. 1).

Let us consider the rotation angles [[phi].sup.A.sub.10], [[phi].sup.B.sub.10], [[phi].sup.C.sub.10] of the three actuators [A.sub.1], [B.sub.1], [C.sub.1] as the variables that give the instantaneous position of the mechanism. In the inverse geometric problem however, it can be considered that the three Euler angles [[alpha].sub.1], [[alpha].sub.2], [[alpha].sub.3] give the absolute orientation of the mobile platform. Pursuing the leg A along the path O[A.sub.1] [A.sub.2] [A.sub.3], we obtain the successive transformation matrices (Staicu, 2007; Rugescu, 2003)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

with the displacement matrices given in (2).

[FIGURE 1 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Let us suppose for example, that the absolute motion of the platform is a 3-D rotation around the fixed point O, defined by the Euler angles

[[alpha].sub.l] = [[alpha].sup.*.sub.l](1 - cos [pi]/3 t), (l = 1,2,3). (3)

From the orientation of the platform in the fixed frame, successive relative rotations around axes Ox, Oy, Oz lead to the rotation matrix a = [a.sub.3] [a.sub.2] [a.sub.1], with the following identities

[a.sup.[omicron]T.sub.30] [a.sub.30] = [b.sup.[omicron]T.sub.30] [b.sub.30] = [c.sup.[omicron]T.sub.30] [c.sub.30] = a, (4)

[a.sup.[omicron].sub.30] = [a.sup.AT.sub.[alpha]] [a.sup.T.sub.[theta]] [[theta].sup.T.sub.1] [[theta].sup.T.sub.2] [[theta].sup.T.sub.1] [a.sup.T.sub.[delta]] [[theta].sup.T.sub.1] (5)

From these relations we obtain the real time evolution of nine characteristic angles [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3. VELOCITIES AND ACCELERATIONS

The kinematics of the elements of leg A , for example, is characterized by the absolute angular velocities given by the recursive relations (6) and the velocity the joint [A.sub.k] by (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

the the velocity the joint [A.sub.k]. Knowing the rotation of the platform through relations (3), we develop the inverse kinematics model (Staicu, 2007) and determine the velocities [[??].sup.S.sub.k0], [[??].sup.A.sub.k0] and accelerations [[??].sup.A.sub.k0], [[??].sup.A.sub.k0] of each moving joint [A.sub.k] by differentiating (6) and (7). The matrix conditions of connectivity is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

with

[[??].sbu.1] = [[1 0 0].sup.T], [[??].sbu.2] = [[0 1 0].sup.T], [[??].sbu.3] = [[0 0 1].sup.T] (9)

These relations give the relative angular velocities [[omega].sup.A.sub.10] [[omega].sup.A.sub.21], [[omega].sup.A.sub.32] as a function of the angular velocities [[??].sub.1], [[??].sub.2], [[??].sub.3] of the platform. The relative angular accelerations [[epsilon].sup.A.sub.10], [[epsilon].sup.A.sub.21], [[epsilon].sup.A.sub.32] are given by the conditions of connectivity resulting from (8):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where [[??].sub.1], [[??].sub.2], [[??].sub.3] are three skew symmetric matrices.

4. RESULTS AND CONCLUSIONS

Consider, for example, a spherical robot that has the following characteristics: [[alpha].sup.*.sub.1] = [pi]/36, [[alpha].sup.*.sub.2] = [pi]/18, [[alpha].sup.*.sub.3] = [pi]/12, r = 0.074 m, [DELTA]t = 3 s. The graphs for angular velocities [[omega].sup.j.sub.10] (Fig. 2) and angular accelerations [[epsilon].sup.j.sub.10] (Fig. 3) of the three revolute actuators result. The recursive matrix method proves superior, avoiding any analytical resolution of tedious equations. It gives directly the motion of any point of the robot, easy, accurate, free of algebraic errors and fast. The computational program is elementary and the method perfectly suited for a real time automatic control of robots.

5. REFERENCES

Clavel, R. (1988) Delta: a fast robot with parallel geometry, Proceed. 18th Int. Symp. Industrial Robots, Lausanne

Gosselin, C. & Angeles, J. (1989), The optimum kinematic design of a planar three-degree-of-freedom parallel manipulator, ASME J. Mech. Trans. Aut. Des. 110(1)p.35-41

Herve, J-M. & Sparacino, F. (1992) Star. A New Concept in Robotics, Proceedings of the Third International Workshop on Advances in Robot Kinematics, Ferrara

Rugescu, R. (2003), Bound Vectors & Spectra of Rigid Rotation Matrices, Proc. SISOM-2003, Bucharest, Romania, pp.134-139.

Staicu, S., Liu, X-J. & Wang, J. (2007) Inverse dynamics of the HALF parallel manipulator with revolute actuators, Nonlinear Dynamics, Springer, 50, 1-2, pp. 1-12

Tsai, L-W. & Stamper, R. (1996) A parallel manipulator with only translational degrees of freedom, ASME Design Engineering Technical Conferences, Irvine, CA.