The Manipulator Equation

Part 1 ended with the equation of motion. Here is where it comes from, what each term does, and the two directions you can run it.

One equation, two derivations

The Lagrangian route starts from energy. With $L = T - V$ (kinetic minus potential), the Euler-Lagrange equations $\frac{d}{dt}\frac{\partial L}{\partial \dot q} - \frac{\partial L}{\partial q} = \tau$ grind out, after substituting $T = \tfrac12\dot q^{\top}M(q)\dot q$, exactly

The Newton-Euler route writes $f = m\,a$ and $\tau = I\dot\omega + \omega\times I\omega$ for every body and projects onto the joint axes. It produces the same equation, and done recursively over the tree it is the RNEA algorithm of the next chapter. Both give the standard manipulator form, with $M$ symmetric positive definite, $C\dot q$ the Coriolis and centripetal terms, and $g$ gravity.¹

Reading the three terms

• $M(q)\ddot q$: inertia felt at the joints. Configuration dependent, as the mass matrix demo showed.
• $C(q,\dot q)\dot q$: Coriolis and centrifugal forces. Quadratic in velocity, the bookkeeping cost of describing motion in a rotating, moving frame.
• $g(q)$: gravity.

Run the arm along a prescribed motion and watch inverse dynamics split the required torque into those three parts:

Two directions

• Forward dynamics. Given torques, find the acceleration, $\ddot q = M^{-1}(\tau - C\dot q - g)$. This is what a simulator does every step.
• Inverse dynamics. Given a desired acceleration, find the torque that produces it. This is what a controller does, and what the demo computes.

Both look like they need $M^{-1}$, an $O(n^{3})$ operation if you form and factorize $M$. You do not have to. Featherstone's articulated-body algorithm computes forward dynamics in $O(n)$, and its inverse-dynamics sibling RNEA is also $O(n)$. That is the next chapter. It is also worth knowing that the derivatives of this equation, $\partial\ddot q/\partial q$ and $\partial\ddot q/\partial\tau$, have closed-form $O(n)$ algorithms too,² which is what makes minimal-coordinate engines like Drake differentiable and usable inside trajectory optimization.