MEEG 6614: Advanced Robotics
Spring 2023
Home Work 1: Due Date: Jan 24, 2022
1. The rotational motion of a rigid body is described by two alternative representations: Space three
1-2-3 sequence given by XXXXXXXXXX, ,φ φ φa a a and Body two 1-2-1 sequence XXXXXXXXXX, ,θ θ θb b b . At a given
instant, XXXXXXXXXX , 45 , 60θ θ θ= = =
. What are the angles 1 2 3, ,φ φ φ in this configuration? Use the
handout to compute the angles. At this instant, let XXXXXXXXXX, 2, 3θ θ θ= = = in rad/s, what are the
angular rates 1 2 3, ,φ φ φ ? Use the handout to compute these angular rates.
2. Quadcopters are used today in a number of applications. Let the orientation of a flying copter,
denoted by a body B, be described in the fixed frame A by two alternative representations: Space
three 1-2-3 sequence XXXXXXXXXX, ,φ φ φa a a and Body two 1-2-1 sequence XXXXXXXXXX, ,θ θ θb b b . Let the angular
velocity of body B be described by XXXXXXXXXX
A B ω ω ω= + +ω b b b , where the expressions for 1 2 3, ,ω ω ω
in each sequence is given in the handout. It is assumed that the rotors on the quadcopter are spun
such that 1 2 31Sin , 2Sin 2 , 3Sin 3t t tω ω ω= = = in rad/sec and the initial conditions of the copter are
1 2 330 , 45 , 60θ θ θ= = =
.
(i) Integrate the kinematic differential equations arising out of Body two 1-2-1 sequence
over 10 seconds and plot the orientation angles. Animate the motion in Matlab. Do you
un into a configuration close to singularity? What happens to the simulation at this
configuration?
(ii) Integrate the kinematic differential equations arising out of Space three 1-2-3 sequence
over 10 seconds and plot the orientation angles. Animate the motion in Matlab. Do you
un into a configuration close to singularity? What happens to the simulation at this
configuration?
(iii) During the simulation with the two representations, if your configuration comes close to
a singularity, switch to the alternate representation until you come away from the
singularity. Then, switch back to the original representation and complete your
simulation. Plot and animate the solutions. Comment on your results.