Homework: optimal control without a reference trajectory A disadvantage of the formulation that we developed in class is that it requires a reference trajectory r. It would be nice if we could get rid of this vector in our cost function. Suppose that we want to move a limb to targets that can appear anywhere. We want an optimal control law that can generate the motor commands no matter where the goal of the movement might be. To approach the problem, let us extend the state vector to include information about where the goal is. Below, g represents location of the goal. We want an optimal control law of this form. Where: Because we got rid of r, we now have:

Using the model of the elbow described in the last lecture, here we intend to simulate control of the arm when the target changes midway into the movement. Start with x=[pos, velocity] and the parameter values for k, b, and m and compute matrix A and C for delta_t of 0.01 sec. Now let us extend x so it is [pos, velocity, goalpos]. Add a row to A and C so that goal position remains invariant in time and motor commands do not affect the goal position. Using B from the last slide, we have a y that is a 2x1. Suppose we want a movement time of 0.45 sec with the following cost characteristics: Solve for the sequence of W matrices. You now have a control law.

1.Suppose we see a target at 30 deg (you will need to do your simulation in radians). The initial conditions are: x=[ ]. Simulate the movement and plot position and the motor commands. To do the simulation, on each iteration you will need to start with a measure of state and generate a motor command. Based on the motor command, predict the next state, and continue with the loop. In this simulation, the goal remains the same. 2.Now suppose that during the movement, at 70ms into the movement the target jumps from 30 deg to 15 deg. Simulate this. Note that the system nearly completely compensates for the perturbation. 3.Let’s investigate how the compensation depends on the timing of the goal change. Suppose that the goal changes at 150ms into the movement. Simulate this. You should see that the compensation is partial. 4.Finally, change the goal at 220ms. You should see that there is very little compensation. People’s movements show similar behavior. When target of a reach changes very early in the movement, there is complete compensation. However, a late change in the movement is often under-compensated.