Lecture 33 :
Actuator dynamics and PD, PID control for robots.
Some information about Control Strategies
We will take the same example of sliding block & will explain control strategies but physically.
Consider a sliding block which we have to move on a frictionless floor from AA to BB
Figure 33.11
PROPORTIONAL CONTROL ACTION:-
Control strategy iswhere
V= Command,= Proportional gain ,= Desired position , = Actual position
When we use this control strategy for moving the block from one position to other, it is just like adding the spring of stiffness to the system. By this we are introducing springing effect in the system.When we use spring in mechanical system it exert a restoring force which is always directed towards mean position. We will use this analogy & build an equivalent spring-mass system & we will study its behaviour & then correlate it with actual one.
Figure 33.12 Equivalent Spring-Mass system
This task of positioning block by using proportional control action is just like releasing the spring-mass system after disturbing it from mean position(which is desired position) by a distance equal to difference between two positions.
Now analyse the behaviour of the standard spring-mass system.
As the spring will exert a force on block which is always directed towards the mean position & which is proportional to deflection of spring which is given by Force exerted by spring =where , are measured from some reference.
One thing to be noted is that what we say command in Control system is analogous to spring force exerted on the mass . This force will move the block towards BB'. But as this force is proportional to deflection of spring, its magnitude will go on decreasing as it move towards BB' . Hence the block will be subjected to force whose magnitude reduces as it moves towards BB'. At BB' its magnitude will be zero, but as the block is having some velocity it will move beyond BB'. From control point of view this can be termed as OVERSHOOTING (crossing the desired position ) Now the spring starts getting extended. It will exert a retarding force on the block whose magnitude is proportional to deflection of spring. Note that force exerted by spring is always directed towards mean position. Because of this force block will come to Momentarily rest at some position beyond BB'. But now spring force will do the job of accelerating the block & move block towards BB'. In this way Block will perform oscillations about mean position BB' of amplitude equal to ( - ). It will never come to rest at BB'.
This same will happen if we incorporate only proportional control action for position control system.
= Proportional gain ,
= Desired position , 
= Actual position 
where 
