Determination of Workspace Volume of Parallel Manipulators Using Monte Carlo Method

Abstract

In this paper, we present a Monte Carlo simulation based method to determine the workspace of spatial parallel and hybrid manipulators. The method does not need the solution of the forward kinematics problem which is often difficult for spatial multi-degree-of-freedom parallel and hybrid manipulators. The method uses the solution of the inverse kinematics problem, which is often much simpler. The method can also readily incorporate joint limits and obtain the well-conditioned workspace. The approach is illustrated with a six-degree-of-freedom hybrid parallel manipulator which is a model for a human hand with three fingers. A typical human hand geometry and the range of motion at the joints are incorporated and the inverse kinematics equations for each finger is derived and used to obtain the volume of the hybrid parallel manipulator.

A Appendix I: Solution of the IK Problem of the Proposed Manipulator

For the most general case, the position vector of the point $S_1$ (see Fig. 2a) is given as the expressions of X, Y and Z below. From the expressions in Eqs. 2, 3 and 4 we obtain $E_1=X^2+(Y+d)^2+(Z-h)^2$ as given in Eq. 5. Using the expressions for $E_1$ and Z from Eqs. 4 and 5, in Sylvester’s dialytic method we can obtain the eliminant for $\psi _1$ as a quartic function of the angular variable. The value of $\theta _1$ may be obtained by solving the expression for $-X+(Y+d)$ symbolically and the value of $\phi _1$ is obtained by using terms from the expressions of Z and $E_1$ as discussed in [6].

(2)

$$\begin{aligned}&Y=\frac{1}{2}( l_{{11}}\sin \left( \psi _{{1}}+\theta _{{1}} \right) +l_{{11}}\sin \left( \psi _{{1}}-\theta _{{1}} \right) +l_{{12}}\sin \left( \phi _{{1} }+\psi _{{1}}+\theta _{{1}} \right) \nonumber \\&\qquad -\,l_{{12}}\sin \left( \phi _{{1}}-\psi _{{1}}+\theta _{{1}} \right) +l_{{13}}\sin \left( { \gamma _i}+\phi _{{1}}+ \psi _{{1}}+\theta _{{1}} \right) -l_{{13}}\sin \left( { \gamma _i}+\phi _{{ 1}}-\psi _{{1}}+\theta _{{1}} \right) )\nonumber \\&\qquad -\,d \end{aligned}$$

(3)

$$\begin{aligned} Z=-\sin \left( \phi _{{1}}+{ \gamma _i}+\theta _{{1}} \right) l_{{13}}-\sin \left( \theta _{{1}}+\phi _{{1}} \right) l_{{12}}-\sin \left( \theta _{{ 1}} \right) l_{{11}}+h \end{aligned}$$

(4)

(5)