Mechanics of particles and systems Definition

Underactuated system

Builds on Control system Generalized coordinates

Definition

Let $n, m \ge 1$ . Let $q \in \R^n$ be generalized coordinates and $u \in \R^m$ the inputs. Consider a second-order control system of the form $\ddot q = f_1(q, \dot q, t) + f_2(q, \dot q, t)\, u,$ where $f_1$ takes values in $\R^n$ and $f_2$ takes values in the $n \times m$ real matrices. The system is fully actuated in the state $(q, \dot q)$ at time $t$ if $\rank f_2(q, \dot q, t) = n$ . It is underactuated in that state at time $t$ if $\rank f_2(q, \dot q, t) < n$ . A system is called underactuated if it is underactuated in some state at some time.

Remarks

For a mechanical system $M(q) \ddot q + c(q, \dot q) = B(q) u$ with invertible inertia matrix $M(q)$ , one has $f_2 = M(q)^{-1} B(q)$ , so $\rank f_2 = \rank B(q)$ . In particular, a system with $m < n$ inputs is underactuated in every state.

Example: for the pendulum on a cart, $q = (x, \theta)$ and the only input is a force on the cart, so $B = (1, 0)^T$ and $\rank f_2 = 1 < 2$ .

Non-example: a planar two-link arm with a motor at each joint has $B = I_2$ , so $\rank f_2 = 2 = n$ in every state, and it is fully actuated.

A system can be fully actuated in some states and underactuated in others, for instance when $B(q)$ drops rank at some configurations.

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Tag	`YQHWF17P6Y`
URI	`m2://YQHWF17P6Y`
Kind	definition
Writing	W0 stub
MSC	70Q05, 93C10, 70E60
Defines	Underactuated system
Uses	Control system, Generalized coordinates
Changed	3 November 2026

Underactuated system

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