Feedback Linearization and Computed Torque

A robot arm's dynamics are nonlinear, which makes simple controllers track poorly when the arm moves fast. Computed torque control uses a model of the arm to cancel those nonlinear terms, leaving a simple linear system that a basic controller can handle perfectly.

Controller Model error Trajectory speed

Cancel, then control

With the arm model $M(\mathbf{q})\ddot{\mathbf{q}} + \mathbf{h}(\mathbf{q}, \dot{\mathbf{q}}) = \boldsymbol\tau$, where $\mathbf{h}$ holds Coriolis and gravity, choose

$$\boldsymbol\tau = \hat M(\mathbf{q})\left(\ddot{\mathbf{q}}_d + K_d\dot{\mathbf{e}} + K_p\mathbf{e}\right) + \hat{\mathbf{h}}(\mathbf{q}, \dot{\mathbf{q}})$$

If the model is exact, the error obeys $\ddot{\mathbf{e}} + K_d\dot{\mathbf{e}} + K_p\mathbf{e} = 0$, a simple linear system for each joint.

What to notice

With an exact model, computed torque tracks almost perfectly, even fast. PD control with the same gains lags behind and sags under gravity. Add model error and computed torque gets worse, since its cancellation is no longer exact, but it still usually beats PD. This idea of using the model to make a nonlinear system look linear is called feedback linearization, and it is the basis of most precise arm controllers.