Underactuation

The motors can't directly produce acceleration in every direction, typically because there are fewer actuators than degrees of freedom.

Underactuation means a mechanical system cannot be commanded to track an arbitrary trajectory directly. MIT's Russ Tedrake defines it this way in Underactuated Robotics: write the dynamics as q̈ = f₁(q, q̇) + f₂(q, q̇)u, where q is the vector of generalized coordinates and u is the control input; the system is underactuated whenever f₂ has rank less than the dimension of q, meaning the motors cannot directly produce acceleration in certain directions. The most common cause is simply having fewer actuators than degrees of freedom, as in a cart-pole system or a quadrotor. Legged robots and humanoids are underactuated too: N motors control only N joints, while the body's six degrees of freedom in space have no motor of their own and can only be controlled indirectly, through the contact forces between the feet and the ground — this is the root reason walking is hard. Underactuated grippers turn the same property to their advantage, using a small number of motors to drive more joints that passively wrap around an object.

ExampleThe Acrobot is a two-link arm with a motor only at the elbow: the shoulder joint has no actuation at all, so swinging it up from hanging straight down to a handstand can only be done by rocking the elbow motor back and forth, 'pumping' energy into the system a little at a time.

Also called

Underactuated System, Underactuated Robot

Related

Floating Base · Degrees of Freedom (DoF) · Active DoF / Passive DoF · Inverted Pendulum Model (IPM) · Adaptive (Underactuated) Gripper · Nonholonomic Constraint

Sources

Underactuated Robotics (Russ Tedrake, MIT): Fully-actuated vs Underactuated Systems
Underactuation - Wikipedia