E28: Mobile Robotics - Fall 2015 - HOMEWORK 6

Pure pursuit overview

In class, we showed that the pure pursuit algorithm for driving along a line can be straightforward to implement if we have access to the transformation T^L_R from the robot frame to the line frame. We assume the line frame is defined such that the line lies on the x-axis of the coordinate frame.

We also assume the robot-to-line transformation is given by

So the point (x, y) specifies the position of the robot in the line frame.

To run the pure pursuit controller, then:

1. Compute the point p_c, the closest point on the line to the robot. In the line frame, the coordinates are simply (x, 0).

2. March ahead of p_c along the line by some distance α to obtain p_d, the pursuit point for the robot. The line-frame coordinates of p_d are (x + α, 0).

3. Issue the controls

x^· _R = k_x,                θ^· = k_θ(c_y/c_x)

where the point (c_x, c_y) are the coordinates of pd, expressed in the robot frame - that is, the coordinates obtained by mapping the point (x + α, 0) through the inverse of the transformation T^L_R.

Simulated pure pursuit

Write a simulator to evaluate some control strategies for pure pursuit. Assume you have a differential drive robot whose body-frame velocities x^·_R and 9^·_R can be commanded directly (note: you can probably re-use some code, and throw away the parts that deal with v_L and v_R).

a. Start the robot at (x, y, θ) = (0, -0.5, 0). Set the gains to α = 0.2, k_x = 0.1 and k_θ = 2.0, and simulate for 30 seconds using Euler's method with ?t = 0.01 second.

Graph the motion of the robot in the world frame on an (x, y) plot. Make sure your plot has equal scaling on the x and y axes (in MATLAB, for instance, use the axis equal command).

b. Simulate what happens with a look ahead distance too small (α = 0.05) and too large (α = 1.0). Submit plots for each.

c. Restore α = 0.2 and simulate what happens when you clamp the angular velocity to be in the ±0.15 rad/s range (i.e. enforce a maximum rotational velocity limit). You should see some oscillation in the robot's motion. Modify the α and/or k_θ gains to fix the behavior, and submit plots of both the overshooting and fixed behavior, with the second plot labeled with the new gains.