Homework #2 Due 1/30/08

Read: Skaar and DelCastillo, Intro and Chap. 1.

1. 1. Craig, problem 1.4.

2. Recall the results from HW#1 concerning the dynamics or kinetics of the two-link robot pictured to the left:

d²q₁/dt²=f₁(q₁,q₂,dq₁/dt,dq₂/dt,t₁,t ₂)

d²q₂/dt²=f₂(q₁,q₂,dq₁/dt,dq₂/dt,t₁,t ₂)

(a) We seek to simulate the response of the two

links, q₁(t), q₂(t) to a particular reference input q_1r(t) and q_2r(t), given

“PID control” of each of the actuators at each of the two joints. As with

the simulation shown in lecture 3 the reference input will connect

an initial pose of q_1r(0)=p/4 and q_2r(0)=0 with a terminal pose

q_1r(t_f) = q_1f = p/2 and

q_2r(t_f) = q_2f = -p/4

(Note that these two poses might have been “taught” by a human using a “teach pendant”.) Determine

your particular value of the reference-maneuver terminal instant, t_f, in accordance with your last

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

(b) Complete and plot over the interval 0<t<2t_f your two joints’ reference paths by fitting the values above

to the general form q_ir(t) = C_o + C₁t +C₂t² + C₃t³, i=1,2, 0<t<t_f; q_ir(t) = q_if, i=1,2, t_f<t<2t_f. Calculate the eight coefficients

-- four for each link’s desired-movement description -- in such a way that the arm would be at rest both at t=0

and t= t_f.

(c) We will neglect herein the actuator dynamics, and assume that the two control torques and can be developed

directly using PID control. Recall that PID control entails use of the error at any instant e₁(t)= q_1r(t)-q₁(t) and

e₂(t)= q_2r(t)-q₂(t), respectively for each of our two joints, in accordance with the control laws below. Defining

the six elements of the state as indicated write out the six state equations dx_i/dt=g_i(x₁ … x₆, t), i=1, 2 … 6.

Apply arbitrary physical and gain parameters, L₁, L₂, m₁ … K_i2.