Kinematics of a 3-PRR planar parallel manipulator.

Staicu, Stefan ; Magheti, Ioan ; Carp-Ciocardia, Daniela Craita 等

1. INTRODUCTION

The parallel architectures have the following potential advantages in comparison with serial robots: higher kinematical precision, lighter weight and better stiffness, stabile capacity and suitable position of arrangement of actuators.

For a planar mechanism, the loci of all points of its bodies can be drawn conveniently in a same plane.

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

A recursive method is introduced in the present paper, with the purpose to reduce significantly the number of equations and computation operations. This method uses a set of matrices for the kinematics of the 3-PRR planar parallel manipulator (Fig.1).

2. KINEMATICS ANALYSIS

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 bodies, with one prismatic joint and two revolute joints. The prismatic joints of the manipulator (PRR) are in fact, three actively controlled prismatic cylinders, which can be installed on the fixed base. The whole mechanism consists of seven moving bodies, six revolute joints and three prismatic joints.

Let us attach a Cartesian frame [x.sub.0] [y.sub.0] [z.sub.0] to the fixed base with its origin located at the centre O and the [z.sub.0] axis perpendicular to the base. The origin of the central coordinate system [x.sub.G] [y.sub.G] [z.sub.G] is located at the centre G of the moving platform (Fig. 2).

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

One of the three active legs consists of a prismatic joint, which is like a piston 1 linked at the [x.sup.A.sub.1] [y.sup.A.sub.1] [z.sup.A.sub.1] frame, and has a displacement [[lambda].sup.A.sub.0], a velocity [v.sup.A.sub.10] = [[??].sup.A.sub.10] and an acceleration [[??].sup.A.sub.10]. The second element of the leg is a rigid rod 2 of length [l.sub.2] linked at the [x.sup.A.sub.2] [y.sup.A.sub.2] [z.sup.A.sub.2] frame, having a relative rotation about [z.sup.A.sub.2] axis with the angle [[phi].sup.A.sub.21], the velocity [[omega].sup.A.sub.21] = [[??].sup.A.sub.21] and the acceleration [[??].sup.A.sub.21]. Finally, a revolute joint is connected to the equilateral moving platform with the edge l = r [square root of 3], which rotates with the angle [[phi].sup.A.sub.32] and the angular velocity [[omega].sup.A.sub.32] = [[??].sup.A.sub.32] about [z.sup.A.sub.3] axis.

Pursuing the first leg A in the O [A.sub.0] [A.sub.1] [A.sub.2] [A.sub.3] way, we obtain the following matrices of transformation:

[a.sub.10] = [[theta].sub.1] [A.sup.A.sub.[alpha]], [a.sub.21] = [a.sup.[phi].sub.21] [[theta].sub.2] [[theta].sup.T.sub.1], [a.sub.32] = [a.sup.[phi].sub.32] [[theta].sub.2], (1)

where [a.sup.A.sub.[alpha]], [[theta].sub.1], [[theta].sub.2] are three constant matrices while [a.sup.[phi].sub.k,k-1] is an orthogonal rotation matrix (Staicu et al., 2007).

In the inverse geometric problem, we can consider that the position of the mechanism is completely given by the coordinates [x.sup.G.sub.0], [y.sup.G.sub.0] of the mass centre G and 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].sup.*] = 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.21] + [[phi].sup.A.sub.32] = [[phi].sup.B.sub.21] + [[phi].sup.B.sub.32] = [[phi].sup.C.sub.21] + [[phi].sup.C.sub.32] = [phi]. (3)

Other six variables [[lambda].sup.A.sub.10], [[phi].sup.A.sub.21], [[lambda].sup.B.sub.10], [[phi].sup.B.sub.21], [[lambda].sup.C.sub.10], [[phi].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 [[??].sup.A.sub.k0] and the velocities [[??].sup.A.sub.k0] of the joints [A.sub.k]:

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

Equations of geometrical constraints (3) and (4) can be derivate with respect to the time to obtain the following matrix conditions of connectivity (Staicu et al., 2008)

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

where [[??].sub.3] is a skew-symmetric matrix associated to the unit vector [[??].sub.3] directed to the positive direction of [z.sub.k] axis.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

As for the relative accelerations [[epsilon].sup.A.sub.10], [[gamma].sup.A.sub.21], [[epsilon].sup.A.sub.32] of the manipulator, the derivatives with respect to the time of the equations (7) give other following conditions of connectivity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The relationships (7) and (8) represent the inverse kinematical model of the planar parallel manipulator.

As application let us consider a mechanism having the following characteristics:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

Using the MATLAB software, a computer program was developed to solve the studied inverse kinematics problem. To illustrate the algorithm, we assume that for a period of three seconds the platform starts from a central configuration, rotates and moves along a rectilinear trajectory.

Based on the computational program, the displacements (Fig. 3), the velocities (Fig. 4) and the accelerations (Fig. 5) of the actuators are plotted versus time.

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

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

Staicu, S. & Zhang, D. (2008) A novel dynamic modelling approach for parallel mechanisms analysis, Robotics and Computer-Integrated Manufacturing, Pergamon-Elsevier, 24, 1, pp. 167-172