HEXAPOD – FORCES

Engineering Fundamentals

INTRODUCTION

Hexapod or so called Steward platform mechanisms are widely used in precision engineering applications. The big advantage of this mechanism is the parallel linkage of all Degrees Of Freedom (DOF) from the moving platform to the base. In most cases this enables a much stiffer and compact design compared to a conventional mechanism where the independent DOF’s are stacked in a sequential way.

As an extension of the Precision Point sheet ‘HEXAPOD–KINEMATICS’ this sheet focuses on the forces in the system.

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Definitions

The following figure displays the relevant geometry.

Hexapod - Forces

P_B,P_P

= Hinge locations in the Base and Platform

R_B,R_P

= Radius of pitch circle of the hinge locations Base and Platform

\alpha _{B,}\alpha _P

= Angle between the hinge locations of a leg-pair

Z_P

= Z-height of the platform in relation to the base

M

= Point of interest of the platform

\overrightarrow e_#

= Orthogonal unit vectors, standard basis

\overrightarrow n_#

= Unit vector collinear with leg number #

\overrightarrow F_#

= Force in leg number #, also extracted in x,y,z

L_{Fx}, Fy, Fz

= External force load in point M

L_{Tx}, Ty, Tz

= External torque load in point M

Force components

It can be seen that the vector force \overrightarrow F_i can be extracted to three orthogonal vectors F_{ix},F_{iy} and F_{iy}. These vectors can also be derived as

\overrightarrow{F_{ix}}=\overrightarrow{e_x}\ {\cdot}\ \overrightarrow{n_i}\ {\cdot}\ F_i
F_{iy}=\overrightarrow{e_y}\ {\cdot}\ \overrightarrow{n_i}\ {\cdot}\ F_i
F_{iz}=\overrightarrow{e_z}\ {\cdot}\ \overrightarrow{n_i}\ {\cdot}F_i

Equilibrium

In a static situation, the following equilibrium equations can be derived:

L_{Fx}+\sum _{i=1}^{i=6}\overrightarrow{e_x}{\cdot}\overrightarrow{n_i}{\cdot}F_i=0
L_{Fy}+\sum _{i=1}^{i=6}\overrightarrow{e_y}{\cdot}\overrightarrow{n_i}{\cdot}F_i=0
L_{Fz}+\sum _{i=1}^{i=6}\overrightarrow{e_z}{\cdot}\overrightarrow{n_i}{\cdot}F_i=0
L_{Tx}+\sum _{i=1}^{i=6}[\overrightarrow{e_y}{\cdot}\overrightarrow{n_i}{\cdot}-(Z_M-Z_{P_i})+\overrightarrow{e_z}{\cdot}\overrightarrow{n_i}{\cdot}(Y_M-Y_{P_i})]{\cdot}F_i=0
L_{Ty}+\sum _{i=1}^{i=6}[\overrightarrow{e_x}{\cdot}\overrightarrow{n_i}{\cdot}(Z_M-Z_{P_i})+\overrightarrow{e_z}{\cdot}\overrightarrow{n_i}{\cdot}-(X_M-X_{P_i})]{\cdot}F_i=0
L_{Tz}}+\sum _{i=1}^{i=6}[\overrightarrow{e_x}{\cdot}\overrightarrow{n_i}{\cdot}-(Y_M-Y_{P_i})+\overrightarrow{e_y}{\cdot}\overrightarrow{n_i}{\cdot}(X_M-X_{P_i})]{\cdot}F_i=0

This can be rewritten as a matrix equation

Hexapod - Forces - Equilibrium Matrix

And thus

\left[\begin{matrix}\begin{matrix}\begin{matrix}L_{Fx} \\L_{Fy}\end{matrix} \\L_{Fz}\end{matrix} \\\begin{matrix}L_{Tx} \\\begin{matrix}L_{Ty} \\L_{Tz}\end{matrix}\end{matrix}\end{matrix}\right] +T_L\ {\cdot}\ \left[\begin{matrix}\begin{matrix}\begin{matrix}F_1 \\F_2\end{matrix} \\F_3\end{matrix} \\\begin{matrix}F_4 \\\begin{matrix}F_5 \\F_6\end{matrix}\end{matrix}\end{matrix}\right]= \overrightarrow 0\ \Leftrightarrow \ \begin{matrix}\overrightarrow L=-T_L\ {\cdot}\ \overrightarrow F \\{\Box} \\\overrightarrow F=-T_L^{-1}\ {\cdot}\ \overrightarrow L\end{matrix}

Note that \overrightarrow L contains both force as well as torque loads, which typically imposes the use of dimensionless values referenced to SI units.

This page uses QuickLaTeX to display formulas.