Now that we have covered the disturbance observer we can begin discussing position control. Consider an object with a mass, m, in space

$F=m\ddot x$

The position, x, of this mass can be controlled by applying a force to the object. One simple way to decide the value of such a force is to use position proportional and velocity feedback, this is otherwise known as Proportional Derivative(PD) control.

$F^{ref}=-K_p (x-x^{ref})-K_d (\dot x-\dot{x}^{ref})$

This simple feedback structure will be able to realize position control and its performance will depend on the values of the constants $K_p$ and $K_d$.

This structure can be understood intuitively as a spring&damper attached to between the object and its referential position. The greater the distance between the current and referential positions, the greater the stretch of the spring will be and therefore the spring will exert a greater force to try and bring them back together. On the other hand the higher the velocity between the current and referential positions, the harder the damper will work to bring the velocity down to zero. Unfortunately this structure is not perfect, if for example a constant disturbance is applied to the object then the object  will never converge to the desired position. This phenomenon is known as steady state error. You can imagine for yourself that if the spring damper system was allowed to hang against gravity, the distance between the object and the desired position would never reduce to zero because of the gravity. Of course the further the spring stretches the greater a restoring force it creates however as the spring approaches zero stretch, it also approaches zero force and therefore the position error can’t be corrected with this method.

Often times a integral term is used to compensate for the steady state error, however this method has the problem of having a unfavorable trade-off between the speed at which the system adapts to external disturbances and the stability of the system. A more advanced way of compensating for external disturbances is to use the disturbance observer which was discussed in the previous section. The figure below shows a position control method that uses a PD controller to generate a force reference and a disturbance observer to realize this force by canceling disturbance forces.

Next: Force Control

Previous: Disturbance Observer

### 3 Responses to “Position Control”

1. This black background kind of make my eyes hurt..any better background?

• You can have any color you want, as long as its black 🙂

2. In the above figure, the output of PD controller (Kp+sKv) is a kind of torque, right? If so, then how comes the output times Jn is torque reference as shown in above figure?

I think the relation should be: Kp+sKv=Jn*(acceleration)=torque (t^ref)? right?