Dynamics and Control

In this exercise you are going to perform system identification and implement a PD

controller for a $1$ degree$-$of$-$freedom robot manipulator.

Again you are provided with a C library containing a dynamic simulator and

a file in which you should implement the controller. For this part of the assign$-$

ment, download the corresponding code and uncompress it$.$ The subdirectory $dyn$

contains the following files:

Imakefile This file is used to create a machine specific Makefile by typing xmkmf$.$

PD$\_$control.c This is the file you have to edit in order to implement the controller

and to perform the experiments required for system identification.

lib$/$libDyn$.$a This library contains the dynamic simulator.

In the file control.c make sure that you fill in the UTA$\_$ID variable with your

UTA student ID number. The solution you will obtain depends on this number and

thus submissions using a different a ID number will likely yield wrong answers.

We will not accept answers derived from an incorrect student ID number.

The Dynamic Simulator

To generate the simulator you have to type make. This creates the simulator exe$-$

cutable Dynamics. The simulator looks as follows:

The Slider on the bottom allows you to set the reference angle configuration

for your controller $($the reference velocity is always set to $0).$

Determine the dynamic parameters of the robot. $(100)$

Here you are supposed to perform system identification to determine the

parameters required later to perform model compensation in your PD control

loop. The dynamic model of this single joint manipulator has the form

$=I{}^{}+B{}^{}+G()$

These are caused by the inertia of the arm $(I),$ the viscous friction in the joint

$(B)$ and by gravity $(G).$

To determine these elements of the dynamics you have to perform experi$-$

ments with the simulated robot. Your interface here is given in the function

PD$\_$control$()$ which receives all relevant information from the robot system

and is called at a rate of $500$ Hz $.$

double PD$\_$control$($theta$,$ theta$\_$dot, theta$\_$ref, theta$\_$dot$\_$ref$)$

double theta, theta$\_$dot, theta$\_$ref, theta$\_$dot$\_$ref;

$\{$

$...$

return$(...)$;

$\}$

This function receives the current joint angle $($theta$),$ the corresponding ro$-$

tational velocity $($theta$\_$dot$),$ the desired reference angle $($theta$\_$ref$)$ set using

the slider, and the reference velocity $($theta$\_$dotref$).$ The output of this func$-$

tion should be the amount of torque you want to apply to the joint.

$($HINT: To perform system identification you want to collect this data and

analyze it in order to determine the system parameters$).$

For this part of the assignment you should hand in the system parameters

found as well as a description of the method used to determine them.

$2.*$ Implement a PD controller with model compensation for the $1$ degree$-$of$-$

freedom robot. $(20)$

Using the parameters found above you should implement a PD controller

with model compensation for this robot. Use of the model will give the

system the appearance of a unit$-$mass system without friction or gravity and

critically damped behavior should therefore be achieved by a choice of gains

where $K_v=2\sqrt[\phantom{2}]{K_p}.$

For this part of the assignment you have to hand in the controller code you

implemented in the function $P{D}_{-}$control in the file $P{D}_{-}$control.c$.$

PD$\_$control.c

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