Approximate Computed Torque

Introduction

In general, you do not know the dynamic model of the robot exactly. So you use estimates of the model in the computed torque law. The estimates are $\hat{M}(\theta )$, $\hat{C}(\theta ,\dot{\theta })\dot{\theta }$ and $\hat{g}(\theta )$.

How it works

The control law uses the estimates:

$$\tau =\alpha v+\beta \alpha =\hat{M}(\theta )\beta =\hat{C}(\theta ,\dot{\theta })\dot{\theta }+\hat{g}(\theta )$$

The closed loop equation is:

$$M\ddot{\theta }+C\dot{\theta }+g=\hat{M}({\ddot{\theta }}_d+K_v\dot{E}+K_{p}E)+\hat{C}\dot{\theta }+\hat{g}$$

Subtract $\hat{M}\ddot{\theta }$ from both sides and rearrange:

$$(M-\hat{M})\ddot{\theta }+(C-\hat{C})\dot{\theta }+(g-\hat{g})=\hat{M}(\ddot{E}+K_v\dot{E}+K_{p}E)$$

So the error equation is:

$$\boxed{\ddot{E}+K_v\dot{E}+K_{p}E=\delta }$$ $$\delta ={\hat{M}}^{-1}[(M-\hat{M})\ddot{\theta }+(C-\hat{C})\dot{\theta }+(g-\hat{g})]$$

• if $M=\hat{M}$, $C=\hat{C}$ and $g=\hat{g}$, then $\delta =0$ and the error goes to zero
• if the estimates are wrong, the mismatch acts as a disturbance and causes servo errors

Example

For the single link robot $m{l}^2\ddot{\theta }+mgl\cos \theta =\tau$, the actual mass is $m=2$ kg, but it is estimated as $\hat{m}=2.1$ kg. The controller uses:

$$\alpha =\hat{m}{l}^2\beta =\hat{m}gl\cos \theta$$

The closed loop equation is:

$$m{l}^2\ddot{\theta }+mgl\cos \theta =\hat{m}{l}^2({\ddot{\theta }}_d+k_v\dot{e}+k_{p}e)+\hat{m}gl\cos \theta$$

The error equation is:

$$\ddot{e}+k_v\dot{e}+k_{p}e=\frac{1}{\hat{m}{l}^2}[(m-\hat{m})l^2\ddot{\theta }+(m-\hat{m})gl\cos \theta ]$$

The error in the mass estimate, $m-\hat{m}=-0.1$ kg, drives a servo error.