Kinematics modeling of the planar 3-PRP parallel robot.

Staicu, Stefan ; Carp-Ciocardia, Daniela Craita

1. INTRODUCTION

Compared with serial robots, potential advantages of the parallel manipulators are higher kinematical precision, lighter weight, better stiffness, greater load bearing, stabile capacity and suitable positional actuator arrangements.

A mechanism is said to be a planar robot if all the moving links perform planar motions that are situated in parallel planes.

Aradyfio and Qiao (1985) examined the inverse kinematics solution for the three different 3-DOF planar parallel robots. Each of Gosselin, Angeles and Sefrioui (1988, 1995), Pennock and Kassner (1990), presented 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. Merlet (2000) solved the forward posed kinematic problem for a broad class of planar parallel robots.

2. KINEMATICS ANALYSIS

In the present paper a recursive method is developed for deriving the inverse kinematics of the 3-PRP planar parallel robot (Fig.1.) in a numerically efficient way.

The planar parallel robot with seven links (T, k=1,2, ..., 7) is a special symmetric closed-loop structure composed of three kinematical chains with identical topology, all connecting the fixed base to the moving platform (Fig.1.). Each leg consists of two links, with two prismatic joints and one revolute joint in-between. We attach a Cartesian frame [x.sub.0] [y.sub.0] [z.sub.0] ([T.sub.0]) to the fixed base, while another reference frame [x.sub.G] [y.sub.G] [z.sub.G] is located at the centre G of the moving triangle (Fig.2.).

One of three active legs (for example leg A) consists of a prismatic joint, which is as well as a linear drive 1 linked at the [x.sup.A.sub.1] [y.sup.A.sub.1] [z.sup.A.sub.1] frame, having a rectilinear motion of displacement [[lambda].sup.A.sub.10] velocity [v.sup.A.sub.10] and acceleration [[lambda].sup.a.sub.10]. Following link is a rigid body 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], velocity and [[omega].sup.A.sub.21] acceleration [[epsilon].sup.A.sub.21] prismatic passive joint is introduced at a planar moving platform as an equilateral triangle with the edge l = [l.sub.0] [square root of 3], which relatively translates at the displacement [[lambda].sup.A.sub.32], the velocity [v.sub.A.sub.32] and the acceleration [[gamma].sup.A.sub.32] along [z.sup.A.sub.3] axis.

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

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

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

In relations (1) [a.sup.A.sub.[alpha]], [[theta].sub.1], [[theta].sub.2] are constant matrices and [a.sup.[phi].sub.21] is an orthogonal rotation matrix (Staicu, 2009).

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

From the rotation conditions of the moving platform, we obtain the following relations between angles:

[[phi].sup.A.sub.21] = [[phi].sup.B.sub.21] = [[phi].sup.C.sub.21] = [phi]. (3)

Six independent variables, [[lambda].sup.A.sub.10], [[lambda].sup.A.sub.32], [[lambda].sup.B.sub.10], [[lambda].sup.A.sub.32], [[lambda].sup.C.sub.10], [[lambda].sup.A.sub.32] will be determined by several vector-loop equations, as follows:

[[??].sup.1.sub.10] + [2.summation over (k=1)] [q.sup.T.sub.k0] [[??].sup.i.sub.k+1,k] + [q.sup.T.sub.30] [[??].sup.Gi.sub.3] = [[??].sup.G.sub.0], (4)

where one denoted:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The following recursive relations give the absolute angular velocities [[??].sup.A.k0] and the velocities [[??].sup.A.sub.k0] of the joints [A.sub.k]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

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 [[??].sub.3] is a skew-symmetric matrix associated to the unit vector [[??].sub.3]. From these equations, we obtain the complete Jacobian matrix and the relative velocities [v.sup.A.sub.10], [[omega].sup.A.sub.21], v.sup.A.sub.32].

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The relations (7) and (8) represent the inverse kinematics model of the planar 3-PRP parallel robot.

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using the MATLAB software, a computer program was developed to solve the studied inverse kinematic 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.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

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

Merlet, J-P. (1996). Direct kinematics of planar parallel manipulators, Proceedings of the IEEE International Conference on Robotics & Automation, Minneapolis, Minnesota, pp 3744-3749

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

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

Staicu, S. (2009). Inverse dynamics of the 3-PRR planar parallel robot, Robotics and AUtonomoUs Systems, 57, 5, pp 556