Rigid-Body Dynamics: Inertia and the Mass Matrix

With the kinematics in place, we can ask what force does. For a single rigid body the answer is Newton for translation and Euler for rotation,

$$f = m\,\dot v, \qquad \tau = I\,\dot\omega + \omega \times (I\,\omega),$$

where $I$ is the $3\times3$ inertia tensor, symmetric and positive definite, encoding how mass is spread out. The $\omega \times I\omega$ term is why a spun object with unequal axes tumbles: rotation couples into itself.

Kinetic energy and the mass matrix

The clean way to see where a whole mechanism's inertia lives is energy. Kinetic energy of one body is

$$T = \tfrac12 m\,\lVert v\rVert^{2} + \tfrac12\,\omega^{\top} I\,\omega .$$

For an articulated system in generalized coordinates $q$, every body's velocity is a linear function of $\dot q$, so summing the energies gives a quadratic form,

$$T = \tfrac12\,\dot q^{\top} M(q)\,\dot q .$$

That defines the mass matrix (joint-space inertia) $M(q)$: symmetric, positive definite, and, importantly, a function of configuration. It is not a constant the way a point mass is.

The mass matrix is configuration dependentinteractive

q₁ (shoulder) = 35°q₂ (elbow) = 60°

M(q)

3.57	1.15
1.15	0.72

Move the elbow: every entry changes. Move the shoulder: nothing does. M depends on q₂, not q₁. The off-diagonal is joint coupling.

The equation everything else hangs on

Put the inertia ($M\ddot q$), the velocity-dependent terms that the $\omega \times I\omega$ coupling generalizes to ($C(q,\dot q)\dot q$, the Coriolis and centrifugal forces), and gravity together, and you get the equation of motion for the whole mechanism,

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

Forward dynamics (given torques, find accelerations) means solving it for $\ddot q$. The naive route is to build $M(q)$ explicitly and invert it, which is where the Composite Rigid Body Algorithm and, more cleverly, Featherstone's $O(n)$ Articulated-Body Algorithm come in. That is Part 2.