Lecture 34 : Trajectory tracking control ( feed forward, computed torque and inverse dynamics approach)
Recall Robot & Actuator dynamics equation.
We have assumed this seconnd order differemtial equation linear by combining all the non-linear terms & treating them as disturbance. But this is true in small range of operation. But as speed of motion increases, non-linear terms containing vary at faster rate. So there will be error. Therefore there is upper limit on speed when using linear controller for robot. Hence productivity will be very low. Instead of this if we use non-linear controller which you will see in next lectures,we will have highly accurate control of Robot & there will be large range of speed.
In case of linear control system high quality actuators & sensors are used to produce linear behaviour in specified operation range which increases cost. But less expensive components with non-linear characteristics can be used in case of non-linear control system.
Any real world system is highly non-linear in nature. Non-linear system is better approximation to real world system than linear one.
Qualititatively a system is said to be stable if starting the system somewhere near its operating point implies that it will be around the point ever after.
Consider a simple pendulum. In one case bob is vertically down & in one case it is vertically up. Now see if we perturbed the system slightly , what will happen in both cases. In one case the system will return back to its original position after some time or it will be around that point ever after but in other case it will not be around the point if we perturbed it. This can clear that the given equilibrium is stable only when if we perturb the system from equilibrium position slightly & if it is around that equilibrium point ever after then it is said to be stable.
e.g. Consider a missile moving along a particular trajectory & if there is slight perturbation from desired tarjectory due to any disturbances & if the control system is stable then it will return back to or move along desired path.
Enables analysis of nonlinear controllers
Synthesis of nonlinear, adaptive, robust controllers possible for both regulation and trajectory tracking applications
Study of some mathematical preliminaries is needed before studying Lyapunov's theorem of stability:
Locally Positive Definite Function (lpdf) :
A function is said to be a
locally positive definite function(lpdf) if
It is continuous.
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There exist a constant r>0 and a functionof class K such that
There is a ball of radius r & x should lie in that ball. In 2-dimensional this ball wll be circle , in three dimensional will be sphere & in n-dimensional it will be hypersphere.
Figure 34.5
Norm of x
(||x||) is similar to distance in geometry but it is in n-dimensional space.
.
of class K such that 
