Robotic Systems · Lecture 5

Newton–Euler equations and free rigid bodies

Simeon Nedelchev

Rigid-body dynamics

m\ddot p=F+mg^W

I_B\dot\omega^B+\omega^B\times I_B\omega^B=\tau^B

$I_B$ and $\omega^B$ use body coordinates.

L^W=R_{WB}I_B\omega^B

World momentum stays constant without external torque.

$I_B=\operatorname{diag}(1,2,3)$ , $\omega=(1,1,1)$

\dot\omega=(-1,1,-1/3)

Body components change despite zero torque.

Rotational energy.

World angular momentum.

Quaternion norm.

Each checks a different part of the implementation.

Small perturbations can grow around intermediate-axis rotation.

A qualitative physical check complements residuals.

Refine the timestep for a torque-free body.

Explain which errors decrease and which assumptions remain.