Derive the equations of motion quml1 f1q1 q2 q1 q2 and

Question

Question

Consider the two­-link planar manipulator shown below with link length l1, l2 mass m1, m2 the distance of the center of mass of the links from the joints l_c1, l_c2 and centroid moment of inertia I₁, I₂. The coordinates describing the system are q₁ , q₂ and the respective joint torques τ₁, τ₂.

1. Derive the equations of motion (q¨₁ = f₁(q₁ , q₂ , q^.₁ q^.₂), and q¨₂ = f₂(q₁ , q₂ , q^.₁ , q^.₂ ) ) by hand using Newton­Euler's method.

2. If torsional springs of stiffness k₁, k₂ are added to the joints, how do the equations get modified?

3. Using the skeleton code provided in myOde.m, solve the dynamic equations from part (1) over 5 seconds. Assume I_c1 = 0.5 I_c2 = 0.5 and all other parameters are 1.0 in their own consistent MKS units. The initial parameters of the systems are:

q1 (0) = Π/3, q2 (0) = Π/4

q^.₁ (0) = 0 , q_^.2 (0) = 0

4. Using function getTotalEnergy.m provided, calculate the kinetic, potential, and total energy. Create a plot showing how these energies change during the simulation time, make sure to add a title and label the axes correctly. A common practice to check if the integration was done correctly is to plot the energy of the system. Explain how this information could be used to prove the numerical solution is correct.

Use File testCodeMain.m to test your ODE, but you can change the inputs and outputs of the functions as you see fit.