Dynamics

$L(q,\dot q) = KE - PE$

$\dfrac{d}{dt}\left(\dfrac{\partial L}{\partial \dot q}\right) - \dfrac{\partial L}{\partial q} = \tau$

Write kinetic and potential energy in terms of the joint coordinates, then this recipe mechanically produces the equations of motion — no free-body diagrams needed.

$M(q)\ddot q + C(q,\dot q)\dot q + g(q) = \tau$

Every rigid robot's dynamics reduce to this form: an inertia term, a velocity-dependent term, and a configuration-dependent gravity term, all balanced against the applied joint torques.

$M(q)$: how much torque it takes to accelerate the joints — couples every joint to every other through the arm's changing geometry.

$C(q,\dot q)\dot q$: Coriolis (cross-joint velocity coupling) and centripetal ($\dot q^2$) effects — zero when the arm isn't moving.

$g(q)$: the torque needed just to hold the arm still against gravity, which changes with pose.

Forward dynamics: given $\tau$, solve for $\ddot q$ — used to simulate how the robot moves.

$\ddot q = M(q)^{-1}\big(\tau - C(q,\dot q)\dot q - g(q)\big)$

Inverse dynamics: given a desired $\ddot q$ (and $q$, $\dot q$), solve for the required $\tau$ — used directly in model-based control.

Computes inverse dynamics in $O(n)$ time for $n$ joints, without ever forming $M(q)$ explicitly.

Forward pass: propagate velocities and accelerations outward from the base to the tip, link by link.

Backward pass: propagate forces and torques back from the tip to the base, accumulating each joint's required torque.

Use the inverse dynamics model to cancel the robot's own nonlinearities:

$\tau = M(q)\big(\ddot q_{des} + \text{feedback}\big) + C(q,\dot q)\dot q + g(q)$

With a perfect model this turns the nonlinear arm into a simple linear system for the feedback term to stabilize. Gravity compensation is the special case that only cancels $g(q)$, holding a pose without drift.