Kinematics modelling of the 3-R[P.bar]R planar parallel robot.

Staicu, Stefan ; Carp-Ciocardia, Daniela Craita ; Codoban, Alexandru 等

Abstract: Recursive matrix relations for the kinematics of the commonly known 3-R[P.bar]R planar parallel robots are established in this paper. The robot has three identical legs connecting to the moving platform. The legs are located in the same plane. Knowing the motion of the platform, in the inverse kinematics problem, several relations and graphs for the positions, velocities and accelerations of the manipulator are determined.

Key words: kinematics, matrix, planar parallel robot, platform

1. INTRODUCTION

Parallel manipulators are closed-loop mechanisms that consist of separate serial chains connecting the fixed base to a moving platform.

A mechanism is said to be a planar robot if all the moving links perform planar motions that are situated in parallel planes. In a planar linkage the direction of translation of a prismatic joint must be parallel to the plane of motion.

Aradyfio and Qiao (1985) examined the inverse kinematics solution for the three different 3-DOF planar parallel robots. Gosselin, Angeles and Sefrioui (1988, 1995), Pennock and Kassner (1990), each presents an interesting study of a planar parallel robot, where a moving platform is connected to a fixed base by three legs consisting of two binary links and three parallel revolute joints. Williams et al. (1988) analysed at Ohio University the control of a planar parallel manipulator.

[FIGURE 1 OMITTED]

2. KINEMATICS ANALYSIS

A recursive method is introduced in the present paper, to reduce significantly the number of equations and computation operations by using a set of matrices for the kinematics of the 3-RPR planar parallel robots (Fig. 1).

Having a closed-loop structure, the planar parallel robot is a special symmetric mechanism composed of three planar kinematical chains of variable length, with identical topology, all connecting the fixed base to the moving platform. Each leg consists of two links, with two revolute joints and one prismatic joint in-between. In order to analyse this mechanisms, we attach a Cartesian frame [0x.sub.0][y.sub.0][z.sub.0] to the fixed base, while another reference frame [Gx.sub.G][y.sub.G][Z.sub.G] is attached to the moving platform (Fig. 2).

One of the three active legs consists of a fixed revolute joint, a moving cylinder 1 of length [l.sub.1], which has a rotation with the angle [phi].sup.A.sub.10]. We consider also a system of reference [A.sub.2][x.sup.A.sub.2][y.sup.A.sub.2][z.sup.1.sub.2] attached to the piston 2 of length [l.sub.2]. This system has a relative motion with the displacement [[lambda].sup.A.sub.21]. Finally, a revolute joint is introduced on a planar platform, which is schematised as an equilateral triangle with the edge l = r[square root of 3].

[FIGURE 2 OMITTED]

Pursuing the first leg A on the O[A.sub.1] [A.sub.2] [A.sub.3], we obtain the following transformation matrices:

[a.sub.10] = [a.sup.[phi].sub.10] [a.sup.A.sub.[alpha]], [a.sub.21] = [theta], [a.sub.32] = [a.sup.[phi].sub.32] [a.sub.[beta]][[theta].sup.T], (1)

where [a.sup.A.sub.[alpha]] [a.sub.[beta]], [theta] are constant matrices and where [a.sup.[theta].sub.k,k-1] is an orthogonal rotation matrix (Staicu et al., 2003, 2006).

Within the inverse geometric problem, the position of the mechanism is given by the coordinates [x.sup.G.xub.0], [y.sup.G.sub.0] of the mass centre of the platform and by the orientation angle [phi], which are expressed by the analytical functions

[x.sup.G.sub.0] / [x.sup.G*.sub.0] = [y.sup.G.sub.0]/[y.sup.G*.sub.0] = [phi] / [phi]* = 1 - cos [pi] / 3 t. (2)

We obtain the following relations between angles: from the rotation conditions of the moving platform:

[[phi].sup.A.sub.10] + [[phi].sup.A.sub.32] = [[phi].sup.B.sub.10] + [[phi].sup.B.sub.32] = [[phi].sup.C.sub.10] + [[phi].sup.C.sub.32] = [phi]. (3)

The six variables [[phi].sup.A.sub.10], [[lambda].sup.A.sub.21], [[phi].sup.B.sub.10] [[lambda].sup.A.sub.21] [[phi].sup.C.sub.10] [[lambda].sup.C.sub.21] will be determined by several vector-loop equations, as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where one denoted:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Recursive relations express the absolute angular velocities [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the velocities [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the joints. [A.sub.k].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

The derivatives with respect to the time of the equations (3) and (4) lead to the following matrix conditions of connectivity:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a skew-symmetric matrix associated to the unit vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. From these equations, we obtain the complete Jacobian matrix and the relative velocities [[omega].sup.A.sub.10, [v.sup.A.sub.21], [[omega].sup.A.sub.32].

As for the accelerations [[epsilon].sup.A.sub.10], [[gamma].sup.A.sub.21], [[epsilon].sup.A.sub.32] of the robot, the derivatives of the conditions (7) give the relations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The relationships (7) and (8) represent the inverse kinematics model of the planar parallel robot. As application let us consider a mechanism having the following characteristics:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

[FIGURE 5 OMITTED]

Using the MATLAB software, a computer program was developed to solve the studied inverse kinematics problem. Finally, the displacements (Fig. 3), the velocities (Fig. 4) and the accelerations (Fig. 5) of the three prismatic actuators are plotted versus time, using the program mentioned above.

3. CONCLUSIONS

Within the inverse kinematical analysis, some exact relations that give the time-history evolution of the displacements, velocities and accelerations of each element of the parallel robot have been established in the present paper.

The simulation by the presented program certifies that one of the major advantages of the current matrix recursive formulation is a reduced number of additions or multiplications and consequently a smaller processing time of numerical computation. Also, the proposed method can be applied to various types of complex robots when the number of components of the mechanism is increased.

4. REFERENCES

Aradyfio, D.D. & Qiao, D. (1985). Kinematic Simulation of Novel Robotic Mechanisms Having Closed Chains, ASME Mechanisms Conference, Paper 85-DET-81

Gosselin, C. & Angeles, J. (1988). The optimum kinematic design of a planar three-degree-of-freedom parallel manipulator, ASME Journal of Mechanisms, Trans. and Automation in Design, 110, 1, pp. 35-41

Sefrioui, J. & Gosselin, C. (1995). On the quadratic nature of the singularity curves of planar three-degree-of-freedom parallel manipulators, Mechanism and Machine Theory, 30, 4, pp. 533-551

Pennock, G.R. & Kassner, D.J. (1990). Kinematic Analysis of a Planar Eight-Bar Linkage: Application to a Platform-type Robot, ASME Mechanisms Conference, Paper DE - 25, pp. 37-43

Williams II, R.I. & Reinholtz, C.F. (1988). Closed-Form Workspace Determination and Optimisation for Parallel Mechanisms, The 20th Biennial ASME Mechanisms Conference, Kissimmee, Florida, DE, Vol. 5-3, pp. 341-351

Staicu, S.; Zhang, D. & Rugescu, R. (2006). Dynamic modelling of a 3-DOF parallel manipulator using recursive matrix relations, Robotica, Cambridge University Press, 24, 1, pp. 125-130

Staicu, S. & Carp-Ciocardia, D.C. (2003). Dynamic analysis of Clavel's Delta parallel robot, Proceedings of the IEEE International Conference on Robotics & Automation ICRA'2003, Taipei, Taiwan, pp. 4116-412