Robotic Systems · Lecture 6

Lagrange equations and manipulator dynamics

Simeon Nedelchev

Lagrange equations

\frac{d}{dt}\frac{\partial L}{\partial\dot q} -\frac{\partial L}{\partial q}=\tau

L=T-V

M(q)\ddot q+c(q,\dot q)=\tau

For independent hinge coordinates, $v=\dot q$ .

Quaternion configurations require tangent-space care.

Point masses at link ends.

Angles measured from downward vertical.

The second angle is relative to the first link.

M(0)=\begin{bmatrix}5&2\\2&1\end{bmatrix}

Symmetric and positive definite.

Off-diagonal terms represent coupling.

$c=Cv+g$ in a common textbook decomposition.

MuJoCo stores the bias vector in qfrc_bias.

Passive damping remains separate.

Match inertias before comparing engines.

Check symmetry, energy, and force balance.