Trajectory Tracking of a Planer Parallel Manipulator by Using Computed Force Control Method

Abstract

Despite small workspace, parallel manipulators have some advantages over their serial counterparts in terms of higher speed, acceleration, rigidity, accuracy, manufacturing cost and payload. Accordingly, this type of manipulators can be used in many applications such as in high-speed machine tools, tuning machine for feeding, sensitive cutting, assembly and packaging. This paper presents a special type of planar parallel manipulator with three degrees of freedom. It is constructed as a variable geometry truss generally known planar Stewart platform. The reachable and orientation workspaces are obtained for this manipulator. The inverse kinematic analysis is solved for the trajectory tracking according to the redundancy and joint limit avoidance. Then, the dynamics model of the manipulator is established by using Virtual Work method. The simulations are performed to follow the given planar trajectories by using the dynamic equations of the variable geometry truss manipulator and computed force control method. In computed force control method, the feedback gain matrices for PD control are tuned with fixed matrices by trail end error and variable ones by means of optimization with genetic algorithm.

Appendices

Appendix 1. Rate of the Passive Joint Variables

$s_{1}$, $s_{2}$ and $s_{3}$: the independent joint variables

$\alpha$, $\beta$, $\gamma$ and $\theta$: the passive (dependent) joint variables

The rate of the passive joint variables is given in terms of the independent joint variables as follows.

$$\dot{\alpha } = a_{\alpha } \,\dot{s}_{1} + b_{\alpha } \,\dot{s}_{2} + c_{\alpha } \,\dot{s}_{3} ,$$

$$\dot{\beta } = a_{\beta } \,\dot{s}_{1} + b_{\beta } \,\dot{s}_{2} + c_{\beta } \,\dot{s}_{3}$$

$$\dot{\gamma } = a_{\gamma } \,\dot{s}_{1} + b_{\gamma } \,\dot{s}_{2} + c_{\gamma } \,\dot{s}_{3} ,$$

$$\dot{\theta } = a_{\theta } \,\dot{s}_{1} + b_{\theta } \,\dot{s}_{2} + c_{\theta } \,\dot{s}_{3}$$

The coefficients are defined as

• $$a_{\alpha } = \frac{{\cot \left( {\alpha - \theta } \right)}}{{s_{1} }}$$
  $$b_{\alpha } = - \frac{{\csc \left( {\beta - \gamma } \right)\,\,\csc \left( {\alpha - \theta } \right)\;\sin \left( {\beta - \theta } \right)}}{{s_{1} }}$$
  $$c_{\alpha } = \frac{{\csc \left( {\beta - \gamma } \right)\,\,\csc \left( {\alpha - \theta } \right)\;\sin \left( {\gamma - \theta } \right)}}{{s_{1} }}$$
• $$a_{\beta } = 0,\;b_{\beta } = - \frac{{\csc \left( {\beta - \gamma } \right)\,\,}}{{s_{3} }},\;c_{\beta } = \frac{{\cot \left( {\beta - \gamma } \right)\,\,}}{{s_{3} }}$$
• $$a_{\gamma } = 0,\;b_{\beta } = - \frac{{\cot \left( {\beta - \gamma } \right)\,\,}}{{s_{2} }},\;c_{\gamma } = \frac{{\csc \left( {\beta - \gamma } \right)\,\,}}{{s_{2} }}$$
• $$a_{\theta } = \frac{{\csc \left( {\alpha - \theta } \right)}}{2d},\;b_{\theta } = - \frac{{\csc \left( {\beta - \gamma } \right)\,\,\csc \left( {\alpha - \theta } \right)\;\sin \left( {\alpha - \beta } \right)}}{2d},\;c_{\theta } = \frac{{\csc \left( {\beta - \gamma } \right)\,\,\csc \left( {\alpha - \theta } \right)\;\sin \left( {\alpha - \gamma } \right)}}{2d}$$

Appendix 2. Features of the Manipulator

The masses: $m_{2} = m_{4} = 0.3\;{\text{kg}}$, $m_{3} = m_{5} = 0.135\;{\text{kg}}$; $m_{6} = 0.456\;{\text{kg}}$, $m_{7} = 0.181\;{\text{kg}}$, $m_{8} = 1.772\;{\text{kg}}$

The inertias: $I_{2} = I_{4} = 0.82 \times 10^{ - 3} \;{\text{kg}} . {\text{m}}^{2}$, $I_{3} = I_{5} = 0.33 \times 10^{ - 3} \;{\text{kg}} . {\text{m}}^{2}$,$I_{6} = 3.2 \times 10^{ - 3} \;{\text{kg}} . {\text{m}}^{2}$, $I_{7} = 0.63 \times 10^{ - 3} \;{\text{kg}} . {\text{m}}^{2}$, $I_{8} = 18.42 \times 10^{ - 3} \;{\text{kg}} . {\text{m}}^{2}$.

The mass center of gravity: $\rho_{c2} = \rho_{c4} = 0.0812\;{\text{m}}$, $\rho_{c3} = \rho_{c5} = 0.0741\;{\text{m}}$, $\rho_{c6} = 0.1418\;{\text{m}}$, $\rho_{c7} = 0.0946\;{\text{m}}$, $\rho_{c8} = 0.142\,{\text{m}}$.

The viscous friction coefficient: $c = 50\;\;Ns/m$

The norm of disturbance force: $\eta = \left[ {\begin{array}{*{20}c} 2 \\ 2 \\ \end{array} } \right]\quad \left( N \right)$

The parameters of the Genetic algorithm:

The crossover type: uniform crossover

The probability of crossover: 0.4

The mutation probability (per bit): 0.05

The selection type: The Stochastic Universal Sampling method

The size of the population: 30

The maximum number of generations: 30

The crossover type: uniform crossover

The probability of crossover: 0.4

The mutation probability (per bit): 0.05

The selection type: The Stochastic Universal Sampling method

The size of the population: 30

The maximum number of generations: 30.