MEEG 6614: Advanced Robotics Spring 2023 Home Work 1: Due Date: Jan 24, 20221. The rotational motion of a rigid body is described by two alternative representations: Space three 1-2-3 sequence...

Need two files. One file for the solutions of each problems and another file is .zip code of Matlab code. Make sure mention each problem number and each part number of the problem. e.g. Problem 1, Problem 2: (i) (ii) (iii)



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 1 1 2 2 3 3, ,φ φ φa a a and Body two 1-2-1 sequence 1 1 2 2 3 1, ,θ θ θb b b . At a given instant, 1 2 330 , 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 1 2 31, 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 1 1 2 2 3 3, ,φ φ φa a a and Body two 1-2-1 sequence 1 1 2 2 3 1, ,θ θ θb b b . Let the angular velocity of body B be described by 1 1 2 2 3 3 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 run 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 run 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.