Reference no: EM131102123
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 = ((νR + νL)r/2) cosθ
y· = x·R sin θ = ((νR + νL)r/2) sinθ
θ· = ((νR - νL)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, νR and νL, 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?
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