Feedback Linearization
- Also called: exact linearization, nonlinear dynamic inversion, NDI
- Category: Control › Feedback control
- Tags: nonlinear-control, model-based-control, manipulator-control, dynamics, trajectory-tracking
A nonlinear control technique that uses a model of the system to cancel its nonlinearities through state feedback, so that standard linear control design can be applied to the result.
Definition
Feedback linearization is a method for controlling nonlinear systems by algebraically transforming them into linear ones. The controller uses a model of the dynamics to compute an input that cancels the nonlinear terms and introduces a new virtual input, through which the closed-loop system behaves like a simple linear system, often a chain of integrators. A conventional linear controller, such as a proportional-derivative law or pole placement, is then designed for this linear system. Unlike linearization around an operating point, the cancellation is exact wherever the model holds.
In robotics the most familiar instance is computed torque control for manipulators. The rigid-body dynamics of an arm are M(q)q̈ + C(q, q̇)q̇ + g(q) = τ. Choosing the torque τ = M(q)v + C(q, q̇)q̇ + g(q) cancels the inertia, Coriolis and gravity terms and leaves q̈ = v for every joint. The new input v can be a PD law on joint position and velocity errors plus a desired acceleration, giving decoupled linear error dynamics with gains chosen directly for the desired response. Operational space control applies the same idea in end-effector coordinates.
The theory distinguishes input-state linearization, where the whole state is made linear, from input-output linearization, where only the relationship between the input and a chosen output is linearized. In the input-output case, the output is differentiated until the input appears, and the number of differentiations is the relative degree. If the relative degree is lower than the system order, part of the dynamics, the internal or zero dynamics, is hidden from the output and must be stable for the approach to work; systems where it is unstable are called non-minimum phase.
The main weakness is model dependence. Cancellation is exact only if the model is exact; errors in mass, friction or actuator behavior leave residual nonlinearities that the linear outer loop must absorb. Practical controllers add robust or adaptive terms, integral action, or learned models of the residual dynamics. Input saturation can also break the cancellation when the required torque exceeds what the actuators can deliver.
Why it matters
Feedback linearization connects nonlinear robot dynamics with the large toolbox of linear control, which is why computed torque control remains a standard method for manipulators and why many legged and aerial controllers use the same idea. Understanding its assumptions explains both why model-based controllers can track precisely and why they degrade when the model is wrong.
Key applications
- Computed torque control — trajectory tracking for robot manipulators
- Operational space control — controlling end-effector motion and force in task coordinates
- Quadrotor control — canceling nonlinear attitude and thrust dynamics for tracking
- Bipedal walking — linearizing selected outputs in hybrid zero dynamics controllers
- Mobile robot path following — linearizing kinematic models of wheeled vehicles
In practice
A team controlling a robot arm usually builds its dynamic model from the URDF with a rigid-body dynamics library such as Pinocchio or RBDL, identifies inertial and friction parameters from test motions, and implements computed torque control in a real-time loop at the joint torque level. They then tune the outer PD gains for each joint to get the desired stiffness and damping, and compare commanded and measured torques to find where the model is inaccurate. When tracking degrades at high speed or under load, they refine the model, add friction compensation or add a learned residual term rather than simply raising the feedback gains.
Common misconceptions
- Feedback linearization is the same as linearizing around an operating point — it cancels nonlinearities exactly over a region using the full model rather than approximating near one point
- The result is robust because the closed loop is linear — it depends on an accurate model, and model errors leave uncanceled nonlinear terms
- Every nonlinear system can be feedback linearized — conditions on the system structure must hold, and non-minimum-phase systems cannot be handled directly
- Higher feedback gains fix model errors — they can mask errors but amplify noise and excite unmodeled flexibility; improving the model is usually more effective
Who should know this
- Control engineers working on manipulators and legged robots
- Robotics students learning nonlinear and model-based control
- Drone flight controller developers
- Researchers combining learned models with model-based control
Related technologies
- Computed Torque Control
- Operational Space Control
- Pinocchio
- Differential Flatness
- Adaptive Control
- Sliding Mode Control
Related terms
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
- frax: JAX library for fast, differentiable robot kinematics, dynamics and inverse kinematics
Videos
- Easy Introduction to Feedback Linearization - Control Engineering Tutorials: Aleksandar Haber PhD