[SOLVED] Cmpsci 403: introduction to robotics: perception, mechanics, dynamics, and control hw 06

Using the example given in class as a guide, derive the equations of motion (in Matlab) for the double
pendulum with parameter definitions as in the figure below. A torque τ1 with vector τ1
ˆk (where ˆk = ˆi × ˆj)
acts between the base and body 1, and a torque τ2 with vector τ2
ˆk acts between body 1 and body 2. Assume
gravity g = 9.81 m/s2
in the −ˆj direction. Write a function to simulate the double pendulum. Provide a
copy of your working code.
✓1
✓2
m1,I1
m2,I2
(0, 0)
ˆj
ˆi
B
C
l1
l2
c1
c2
Figure 1: Double pendulum and parameter definitions.
1. With τ1 = τ2 = 0, solve the initial boundary value problem from the initial condition
θ1 = 3 rad, θ2 = 0 rad, ˙θ1 = ˙θ2 = 0 rad/s
on the time interval t = [0 s, 7 s]. Use parameters
m1 = m2 = 1 kg, I1 = I2 = 0.05 kg·m2
, l1 = 1 m, l2 = 0.5 m, c1 = 0.5 m, c2 = .25 m.
Plot θ1(t) and θ2(t). Does the solution display any repetitive and predictable patterns?
2. Plot the total system energy (T + V ) over the same interval. Verify energy conservation (You will see
almost constant energy having a small drift).
3. (Optional) Derive the equations again considering the addition of three springs with potential energies
Ve1 =
1
2
κ1(θ1 − θ1,0)
2 Ve2 =
1
2
κ2(θ2 − θ2,0)
2
Ve3 =
1
2
k3(∥rC − r0∥ − l0)
2
The vector r0 = [rx ry]
T
represents the attachment point for the last spring.
4. (Optional) After verifying energy conservation of your equations, simulate the system with
κ1 = κ2 = 10 Nm/rad, k3 = 50 N/m
θ1,0 = θ2,0 = 0, l0 = 0, rx = 0 m, ry = 0.5 m
Apply τ1 = − ˙θ1 and τ2 = − ˙θ2 and use the same initial state as in step 2.
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