ELEC-848: Control Systems Design
for Robots and Telerobots
Assignment # 2
Robot Control I: Consider a two-link planar manipulator, as shown in figure below, with
dynamics
1 = ( ) 1 2 1
2 2
2 1
2
1 2 1 1 2 2
I I m m 2 c q c c c + + + + +
+ ( ) 1 2 2
2
2 2 2 2
I m c q c c + + – 2 1 2 2 1 2
2 q q s m c
2
2 1 2 2 2
q s m c − + ( ) 1 1 2 1 1 2 2 12 1 1
m m gc m gc b q c c + + +
2 = ( ) ( ) 2
2
2 1 2 1 2 2
2
2 2 2 2
I m c q I m q c c c + + + +
+ 2 12 2 2
2
2 1 2 2 1 2
m s q m gc b q c c + +
and dynamic parameters l1=l2=1m, lc1=lc2=0.5m, m1=50kg, m2=40kg, I1=I2=10kg.m2
b1=b2=10, and g=9.8. Also assume similar motor and drive train for each joint with
Ra1=Ra2=1 Km1=Km2=0.2N.m/A, Im1=Im2=0.2kg.m2, Bm1=Bm2=0.08, and Kb1=Kb2=0.1
V.s/rad.
1. Find the effective joint inertia and damping parameters I1eff, I2eff, b1eff and b2eff for both
r1 = r2 = 1 and r1 = r2 =0.01.
Note that some of the parameters are function of 1 or 2 so make sure to include the
full range of the effective parameters.
2. Design the following trajectory tracking controllers
• Decentralized PD controller + Feed forward
• Decentralized PD controller + Feed forward + Computed torque
• Inverse dynamics + PD controller + Feed forward
• The above inverse dynamics controller with 50% uncertainty in M and h
such that (q1,q2) start from the resting position (0,0) and follow trajectory (/3.sin(t) , /6.sin(t)). The
desired damping factor and the natural frequency of the closed-loop systems are = 1, n = 3 rad/s and
r1 = r2 = 1 (Make any assumptions, only if necessary).
3. Repeat the above simulations for an indirect drive actuated manipulator, that is with r1 = r2 = 0.01.
U
NOTES:U
1. Do not use Simulink blocks to generate the derivatives of the inputs; instead find the derivatives mathematically and use separate source blocks for them.
2. Run the simulation for 25 seconds.
3. Use the 3×2 subplot command to display 6 plots in a single figure for each controller:
a. top left = desired and actual joint 1 angles
b. top right = desired and actual joint 2 angles
c. middle left = joint 1 angle error
d. middle right = joint 2 angle error
e. bottom left = joint 1 control command
f. bottom right = joint 2 control command
4. Use the 2×1 subplot command to display 2 plots in a single figure for each controller:
a. top = joint 1 angle error for all methods
b. top right = joint 2 angle error for all methods
5. Compare the performance of the above controllers and provide a discussion on the effect of gear ratio. Support your analysis with theory (mathematical equations).
Due date: 91:00 pm Tuesday, March 24, 2020
Delivery: In Class
NOTE: Please email your commented .m and .mdl MATLAB files.
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