Full Dynamic Modeling of the General Stewart Platform Manipulator via Kane’s Method

Abstract

Full dynamic modeling of the general Stewart platform manipulator is a complex task due to its closed-loop structure. In this paper, a complete model of inverse dynamics of the most general Stewart platform manipulator, without any simplification on dynamic properties of its components, is presented. It is noted that the equations obtained considering the comprehensive and detailed description of the Stewart system bear a computational burden considerably less than that from the more traditional methods of dynamic modeling, and this was indeed verified in the results obtained through simulations performed using the algorithm presented in this research.

Appendix

System parameters:

All parameters are in SI units.

Universal joint locations:

$$\left[ {\begin{array}{*{20}c} {\varvec{b}_{1} } & \cdots & {\varvec{b}_{6} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {0.483} & {0.354} & { - 0.354} \\ {0.129} & {0.354} & {0.354} \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} { - 0.483} & { - 0.129} & {0.129} \\ {0.129} & { - 0.483} & { - 0.482} \\ 0 & 0 & 0 \\ \end{array} } \\ \end{array} } \right]$$

Spherical joint locations:

$$\left[ {\begin{array}{*{20}c} {\varvec{p}_{1}^{\varvec{'}} } & \cdots & {\varvec{p}_{6}^{\varvec{'}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {0.068} & { - 0.068} & { - 0.197} \\ {0.188} & {0.188} & { - 0.035} \\ { - 0.05} & { - 0.05} & { - 0.05} \\ \end{array} } & {\begin{array}{*{20}c} { - 0.129} & {0.129} & {0.197} \\ { - 0.153} & { - 0.153} & { - 0.035} \\ { - 0.05} & { - 0.05} & { - 0.05} \\ \end{array} } \\ \end{array} } \right]$$

Mass of each part:

$$\begin{aligned} &m_{p} = 30 \hfill \\ &m_{e} = 2 \hfill \\ &m_{h} = 1.2 \hfill \\ \end{aligned}$$

Legs’ center of masses:

$$\begin{aligned} \varvec{e} & = \left[ {\begin{array}{*{20}c} {0.05} & {0.03} & {0.25} \\ \end{array} } \right]^{\text{T}} \\ \varvec{h} & = \left[ {\begin{array}{*{20}c} {0.03} & {0.03} & { - 0.2} \\ \end{array} } \right]^{\text{T}} \\ \end{aligned}$$

Moments of inertia of each part:

$$\begin{aligned} &{}_{{}}^{\varvec{P}} \varvec{I}_{\varvec{p}} = \left[ {\begin{array}{*{20}c} {2.5} & 0 & 0 \\ 0 & {2.5} & 0 \\ 0 & 0 & 5 \\ \end{array} } \right] \hfill \\& {}_{{}}^{\varvec{i}} \varvec{I}_{{1\varvec{i}}} = \left[ {\begin{array}{*{20}c} {0.11} & {0.15} & {0.4} \\ {0.15} & {0.2} & {0.3} \\ {0.4} & {0.3} & {0.5} \\ \end{array} } \right] \hfill \\ &{}_{{}}^{\varvec{i}} \varvec{I}_{{2\varvec{i}}} = \left[ {\begin{array}{*{20}c} {0.17} & {0.07} & {0.15} \\ {0.07} & {0.15} & {0.13} \\ {0.15} & {0.13} & {0.25} \\ \end{array} } \right] \hfill \\ \end{aligned}$$

Friction coefficients of the joints:

Prismatic joints (actuators):

$$\left[\begin{array}{ccc} c_{1} & \cdots & c_{6} \\ \end{array}\right] =\left[ \begin{array}{cccccc} 20& 120 & 95& 10 & 1150& 0.65 \\ \end{array}\right]$$

Universal joints:

$$\left[ {\begin{array}{*{20}c} {c_{1} } & \cdots & {c_{6} } \\ \end{array} } \right] = \left[{\begin{array}{*{20}c} 1 & {150} & {80} &2 & {1000} & {0.05}\\ \end{array} }\right]$$

Spherical joints:

$$\left[ {\begin{array}{*{20}c} {c_{1} } & \cdots & {c_{6} } \\ \end{array} } \right] = \left[{\begin{array}{*{20}c} 5 & {120} & {80} &3 & {860} & {0.15}\\ \end{array} } \right]$$