Underactuated system
- Also called: underactuated robot, underactuated mechanical system
- Category: Control › Feedback control
- Tags: dynamics, nonlinear-control, legged-robots, aerial-robots, control-theory
A mechanical system whose control inputs cannot command arbitrary accelerations in all of its degrees of freedom, such as a quadrotor, a cart-pole or a legged robot.
Definition
A mechanical system is underactuated when its control inputs cannot produce an arbitrary instantaneous acceleration in every degree of freedom. For a system written in the standard manipulator form, M(q)q̈ + C(q, q̇)q̇ = τ_g(q) + Bu, this is the case when the actuation matrix B has lower rank than the number of generalized coordinates q. A fully actuated robot arm, with a motor on every joint, can follow any joint trajectory within its torque limits; an underactuated system cannot, and some motions are only possible by exploiting its natural dynamics.
Examples are common. The cart-pole and the Acrobot, a two-link pendulum with a motor only at the elbow, are the textbook cases. A quadrotor has six degrees of freedom but four rotor inputs, so it must tilt to accelerate sideways. A legged robot has a floating base with six unactuated coordinates: the body can only be moved through contact forces at the feet, which are limited by friction and can only push, and during a flight phase the center of mass follows a ballistic path whatever the joint motors do. Cars, boats and anything with passive joints or elastic elements are also underactuated.
Underactuation does not mean the system is uncontrollable. The cart-pole and the quadrotor are controllable, but their controllers have to work with the dynamics instead of cancelling them. Common techniques include partial feedback linearization, energy shaping for swing-up, differential flatness for quadrotors, trajectory optimization with feedback stabilization, and reduced-order models such as the linear inverted pendulum for walking. In hardware design the same idea is used deliberately: an underactuated hand or gripper has fewer motors than joints and relies on springs, tendons or linkages to distribute motion.
Why it matters
Most robots that move through the world, including legged robots, drones and balancing robots, are underactuated. Recognizing this changes the control approach: arbitrary trajectories can no longer be tracked, so planning must respect the dynamics, and stability depends on what the robot can and cannot push against.
Key applications
- Legged locomotion and balance
- Quadrotor and fixed-wing flight control
- Balancing robots and inverted pendulum systems
- Swing-up and acrobatic maneuvers
- Underactuated grippers and hands
In practice
A team building a two-wheeled self-balancing robot has two degrees of freedom in the pitch plane, wheel rotation and body pitch, but only one input, the wheel torque. To drive forward, the controller cannot simply spin the wheels: it first rolls them slightly backward so the body tips forward, then accelerates under it. This non-minimum-phase response surprises engineers used to fully actuated systems. The team linearizes the dynamics around the upright position, designs an LQR controller on the full state, and limits commanded accelerations so that the body stays within the controller's region of attraction and the motors stay within their torque limits.
Common misconceptions
- Underactuated means uncontrollable — many underactuated systems are fully controllable; they just cannot follow arbitrary trajectories
- Underactuation is only about counting motors — what matters is whether the inputs can accelerate every degree of freedom; a legged robot with a motor on every joint is still underactuated because its floating base is not directly actuated
- Underactuation is a design flaw — it often comes with lighter, simpler and more efficient machines, and is chosen deliberately in grippers and passive walkers
Who should know this
- Legged robot and humanoid control engineers
- Drone flight control engineers
- Researchers in nonlinear and optimal control
- Robotics students learning robot dynamics
Related technologies
- Cart-Pole and Acrobot Benchmarks
- Linear Quadratic Regulator (LQR)
- Partial Feedback Linearization
- Energy Shaping Control
- Drake
Related terms
- Feedback Linearization
- Differential Flatness
- DOF
- Linear Inverted Pendulum
- Region of attraction
- Underactuated Hand
Related projects
- Drake: MIT robotics toolbox for model-based dynamics, optimization, and control system design
- DART: C++ physics engine for articulated rigid body dynamics and robotics simulation
- Extreme Parkour with Legged Robots: RL system for legged robot parkour trained in under 20 hours
- PACE Sim2Real: Systematic sim-to-real transfer via actuator dynamics identification
- frax: JAX library for fast, differentiable robot kinematics, dynamics and inverse kinematics
Videos
- Lecture 1 | MIT 6.832 (Underactuated Robotics), Spring 2020 | Why study dynamics?: underactuated
- Lecture 1 | MIT 6.832 Underactuated Robotics, Spring 2009: MIT OpenCourseWare