FORC · Lecture 3

Jacobian control and differential inverse kinematics

Simeon Nedelchev

Task velocity

\dot p=J_p(q)v

The Jacobian maps generalized velocity to tool velocity in a declared frame.

b=\dot p^\star+K_p(p^\star-p)

v=J_p^\top(J_pJ_p^\top+\lambda^2I)^{-1}b

Damping trades tracking accuracy for smaller commands.

J_p(0,0)=\begin{bmatrix}0&0\\2&1\end{bmatrix}

Vertical motion is locally available.

Horizontal motion needs a change of configuration.

Damping does not impose a speed constraint.

Joint limits and collisions require explicit treatment.

mj_jacSite gives translational and rotational blocks.

mj_integratePos provides configuration perturbations for finite differences.

What happens to task error as damping grows?

Compare speed, error, and the smallest singular value.