4.1 Statics

Statics deals with relating the forces at the robots joints when the robot is in static (mechanical) equilibrium - i.e. the acceleration of the robot and all its components are zero.

Say we are operating in 6 DoF space ($m=6$)

$f=\left[\begin{array}{c}{f}_x\\ f_y\\ f_z\end{array}\right]$ describes the forces on the three axes $\mu =\left[\begin{array}{c}{\mu }_x\\ {\mu }_y\\ {\mu }_z\end{array}\right]$ describes the angular force component on the three axes $F={\left[\begin{array}{cc}f& \mu \end{array}\right]}^T$ describes the spatial force or Wrench

Now we would like an equation that relates our $n$x1 vector of torques $\tau$ to F.

Mechanical power is $W=F^T\cdot V$ (Force * Velocity)

We know our motors are the source of Power. Therefore

W = \tau^{T} \cdot \dot{q} = F^{T} \cdot V$$$$tau^{T} \cdot \dot{q} = F^{T} \cdot J(q) \cdot \dot{q} $\tau =J(q{)}^T\cdot F$ There we go!

This essentially say that to counteract an external wrench $F_e=-F$, the robot applies torques $\tau$ at its joints.