Suppose we are trying to understand motion control for this car. For example, suppose we would like to automate (i.e., to design an automatic controller for) the parallel parking maneuver that you attempted to do manually in the first section of this lab. The question is then how to perform a sequence of forward/back translations (driving) together with clockwise/counterclockwise rotations (steering) so that a net sideways motion is produced?

To gain some intuition, we can think of the parallel parking problem as a problem of generating motion (sideways motion of the car) from shape change. The notion of using non-reciprocal cyclic shape change to generate motion was introduced in the Lecture Notes. In the case of the car, we associate one shape variable with forward/backwards driving and one shape variable with steering. One way to picture this is to represent the car as a single wheel (e.g., a unicycle). Then, one shape variable is the angle φ of the wheel about its spin axis or axle (i.e., a positive rotation of φ gives a forward displacement of the car; in our picture we only see a top view of the car, so we don't see the wheel rolling - for that we would need a sideview). A second shape variable is the angle θ of the wheel about the vertical axis (i.e., the steering angle). A non-reciprocal shape change implies, for example, a sequence of maneuvers as follows: (1) steer angle θ by α degrees, (2) turn φ by β degrees, (3) steer θ by -α degrees, (4) turn φ by -β degrees.

To see the net result of such a non-reciprocal shape change on the vehicle, i.e., to see where the vehicle ends up as a result of this sequence of maneuvers, we can examine the corresponding sequence of transformations applied to our picture of the car. That is, we associate a transformation matrix for translation with shape variable φ and a transformation matrix for rotation with shape variable θ. Then, we apply these matrices in the prescribed order (i.e., we multiply them in the prescribed order) and see what the resulting transformation matrix produces!