Mass Matrix

The matrix M(q) in a robot's dynamics equation that describes how hard each joint is to accelerate.

The mass matrix appears in a robot's standard dynamics equation, τ = M(q)q̈ + c(q, q̇) + g(q): τ is joint torque, q, q̇, and q̈ are joint angles and their first and second derivatives, c is the Coriolis and centrifugal term, and g is the gravity term. M(q) is an n×n matrix (n is the number of joints) that determines how much torque a given set of joint accelerations requires; kinetic energy can also be written as T = ½q̇ᵀM(q)q̇. It's symmetric, positive definite, and changes with posture: with the arm fully extended, the shoulder joint has to move more inertia. Its off-diagonal entries capture inertial coupling between joints — accelerating only the elbow joint also produces a reaction torque at the shoulder. Computed-torque control, operational-space control, and forward-dynamics simulation all need it, and libraries like Pinocchio and MuJoCo compute it directly.

ExampleFor a planar two-link arm, when the elbow is straight, the diagonal entry of the mass matrix corresponding to the shoulder joint is at its largest — the same shoulder torque produces less angular acceleration when the arm is extended than when the elbow is bent.

Also called

Inertia Matrix, Joint-Space Inertia Matrix (JSIM), M(q)

Related

Rigid-Body Dynamics · Coriolis and Centrifugal Terms · Composite Rigid Body Algorithm · Inverse Dynamics · Forward Dynamics · Euler-Lagrange Equations

Sources

Modern Robotics 8.1.3: Understanding the Mass Matrix (Northwestern)
Wikipedia: Mass matrix