Explain why we must remove the steering angle alpha from

Question

E28: Mobile Robotics - Fall 2015 - HOMEWORK 4

1. Tricycle kinematics

Formally, the kinematic system for the differential drive robot is given by

x^· = x^·_R cos θ = ((ν_R + ν_L)r/2) cosθ

y^· = x^·_R sin θ = ((ν_R + ν_L)r/2) sinθ

θ^· = ((ν_R - ν_L)r/2d)sinθ

In matrix form, we have

We note that the above formulation yields velocities which are linear in the controls (in this case, ν_R and ν_L, but not necessarily linear in the elements of the state q = (x, y, θ). In fact, the definition of a kinematic system requires the output velocities to be linear in the controls.

a. Explain why we must remove the steering angle α from the controls of the tricycle, and place it into the tricycle's configuration (state) in order to form a valid kinematic system.

b. We still want to control steering, so we will introduce a new control α^· which is the steering velocity (how quickly we are pivoting the front wheel). Finish the discussion from class by factoring the tricycle equations of motion into matrix form:

(That is, fill in the elements of the matrix above.)

2. Broken tricycle

The tricycle has a single powered front wheel which spins at a speed of ω. Why would it be impossible to steer such a vehicle if instead the two rear wheels were powered to spin at the same exact angular velocity ν?