Computed torque control

Computed torque control is a control scheme used in motion control in robotics. It combines feedback linearization via a PID controller of the error with a dynamical model of the controlled robot.

Let the dynamics of the controlled robot be described by

$𝐌\left(\overrightarrow{\theta }\right)\ddot{\overrightarrow{\theta }}+𝐂(\overrightarrow{\theta },\dot{\overrightarrow{\theta }})\dot{\overrightarrow{\theta }}+{\overrightarrow{\tau }}_g\left(\overrightarrow{\theta }\right)=\overrightarrow{\tau }$ where $\overrightarrow{\theta }\in {ℝ}^N$ is the state vector of joint variables that describe the system, $𝐌\left(\overrightarrow{\theta }\right)$ is the inertia matrix, $𝐂(\overrightarrow{\theta },\dot{\overrightarrow{\theta }})\dot{\overrightarrow{\theta }}$ is the vector Coriolis and centrifugal torques, ${\overrightarrow{\tau }}_g\left(\overrightarrow{\theta }\right)$ are the torques caused by gravity and $\overrightarrow{\tau }$ is the vector of joint torque inputs.

Assume that we have an approximate model of the system made up of $\widetilde{𝐌}\left(\overrightarrow{\theta }\right),\widetilde{𝐂}(\overrightarrow{\theta },\dot{\overrightarrow{\theta }}),{\tilde{\overrightarrow{\tau }}}_g\left(\overrightarrow{\theta }\right)$. This model does not need to be perfect, but it should justify the approximations $𝐌{\left(\overrightarrow{\theta }\right)}^{-1}\widetilde{𝐌}\left(\overrightarrow{\theta }\right)\approx \mathrm{𝟏}$ and ${𝐌}^{-1}(𝐂(\overrightarrow{\theta },\dot{\overrightarrow{\theta }})\dot{\overrightarrow{\theta }}+{\overrightarrow{\tau }}_g\left(\overrightarrow{\theta }\right))\approx {𝐌}^{-1}(\widetilde{𝐂}(\overrightarrow{\theta },\dot{\overrightarrow{\theta }})\dot{\overrightarrow{\theta }}+{\tilde{\overrightarrow{\tau }}}_g\left(\overrightarrow{\theta }\right))$.

Given a desired trajectory ${\overrightarrow{\theta }}_d(t)$ the error relative to the current state $\overrightarrow{\theta }(t)$ is then ${\overrightarrow{\theta }}_e(t)={\overrightarrow{\theta }}_d(t)-\overrightarrow{\theta }(t)$.

We can then set the input of the system to be

$\overrightarrow{\tau }(t)=\widetilde{𝐌}\left(\overrightarrow{\theta }\right)({\ddot{\overrightarrow{\theta }}}_d(t)+K_p{\overrightarrow{\theta }}_e(t)+K_i\underset{0}{\overset{t}{\int }}{\ddot{\overrightarrow{\theta }}}_e(t^{\prime })d{t}^{\prime }+K_d{\dot{\overrightarrow{\theta }}}_e(t))+\widetilde{𝐂}(\overrightarrow{\theta },\dot{\overrightarrow{\theta }})+{\tilde{\overrightarrow{\tau }}}_g\left(\overrightarrow{\theta }\right)$

With this input the dynamics of the entire systems becomes

$\begin{array}{cc}𝐌\left(\overrightarrow{\theta }\right)\ddot{\overrightarrow{\theta }}+𝐂(\overrightarrow{\theta },\dot{\overrightarrow{\theta }})\dot{\overrightarrow{\theta }}+{\overrightarrow{\tau }}_g\left(\overrightarrow{\theta }\right)=& \widetilde{𝐌}\left(\overrightarrow{\theta }\right)({\ddot{\overrightarrow{\theta }}}_d(t)+K_p{\overrightarrow{\theta }}_e(t)+K_i\underset{0}{\overset{t}{\int }}{\ddot{\overrightarrow{\theta }}}_e(t^{\prime })d{t}^{\prime }+K_d{\dot{\overrightarrow{\theta }}}_e(t))+\widetilde{𝐂}(\overrightarrow{\theta },\dot{\overrightarrow{\theta }})+{\tilde{\overrightarrow{\tau }}}_g\left(\overrightarrow{\theta }\right)\\ \ddot{\overrightarrow{\theta }}+𝐌{\left(\overrightarrow{\theta }\right)}^{-1}(𝐂(\overrightarrow{\theta },\dot{\overrightarrow{\theta }})\dot{\overrightarrow{\theta }}+{\overrightarrow{\tau }}_g\left(\overrightarrow{\theta }\right))=& \underset{\approx \mathrm{𝟏}}{\underbrace{𝐌{\left(\overrightarrow{\theta }\right)}^{-1}\widetilde{𝐌}\left(\overrightarrow{\theta }\right)}}({\ddot{\overrightarrow{\theta }}}_d(t)+K_p{\overrightarrow{\theta }}_e(t)+K_i\underset{0}{\overset{t}{\int }}{\ddot{\overrightarrow{\theta }}}_e(t^{\prime })d{t}^{\prime }+K_d{\dot{\overrightarrow{\theta }}}_e(t))+𝐌{\left(\overrightarrow{\theta }\right)}^{-1}(\widetilde{𝐂}(\overrightarrow{\theta },\dot{\overrightarrow{\theta }})+{\tilde{\overrightarrow{\tau }}}_g\left(\overrightarrow{\theta }\right))\\ \ddot{\overrightarrow{\theta }}=& {\ddot{\overrightarrow{\theta }}}_d(t)+K_p{\overrightarrow{\theta }}_e(t)+K_i\underset{0}{\overset{t}{\int }}{\ddot{\overrightarrow{\theta }}}_e(t^{\prime })d{t}^{\prime }+K_d{\dot{\overrightarrow{\theta }}}_e(t)\\ 0=& {\ddot{\overrightarrow{\theta }}}_e+K_p{\overrightarrow{\theta }}_e(t)+K_i\underset{0}{\overset{t}{\int }}{\ddot{\overrightarrow{\theta }}}_e(t^{\prime })d{t}^{\prime }+K_d{\dot{\overrightarrow{\theta }}}_e(t)\end{array}$

and the normal methods for PID controller tuning can be applied. In this way the complicated nonlinear control problem has been reduced to a relatively simple linear control problem.