Homework #3                   Due 2/6//08

 

Read:  Skaar and DelCastillo, Chap. 2.

 

Recall the results from HW#2 with your particular reference path for the two joint angles:

qir(t) = Co + C1t +C2t2 + C3t3, i=1,2, 0<t<tf; qir(t) = qif(tf), i=1,2, t>tf

where this “rest-to-rest” reference input connects

an initial pose of q1r(0)=p/4 and q2r(0)=0 with a terminal pose

q1r(tf) = q1f =   p/2        and

q2r(tf) = q2f =  -p/4

Where tf is based on  your last

name:  tf=[10+n]/20 where n=1 if your last name begins with “A”, n=2 if “B”, and so on. 

We next want to determine the actual simulated response for q1  and q2  to this reference input,

given your state equations based upon the PID control and six-element state definition given below. 

 

  1. Given the following initial values of the state, and assuming, as discussed in

class, an initial state of equilibrium, determine the corresponding initial values of x3(0) and x6(0).  Do this for

arbitrary physical and control parameters.  Use x1(0)=p/4 x2(0)=0 x4(0)=0 x5(0)=0

  1. Given the following physical and control parameters (all units consistent), numerically integrate from your

initial conditions of 1  to determine the response to your reference input over time.  Superimpose in a single

plot q1r(t)  q2r(t) . x1(t) and x4(t).   Advance your plot as far in time as necessary to indicate a coming to rest

at the desired terminal joint angles.  Use m1=m2=1.5; L1=L2=1; g=1; Kp1=20; Kd1=40; Ki1=10; Kp2=3; Kd2=2; Ki2=1.