Suppose we have a robot moving in a 2D plane with dynamics x(t+1)=x(t) +v(t), v(t+1)=0.5v(t) + a(t), where æ(t), v(t) and a(t) denote its location, velocity and acceleration, respectively. Let the robot be initially at rest at a starting location (1) = [33] It has to reach a final location (T) 3 at T = 50, at which it comes to a full stop. The goal is to design an optimization problem that moves the robot to its desired location while minimizing the tal acceleration ||a(t)||2, satisfying the above dynamics, and with an additional constraint that the velocity magnitude (2-norm) is no larger than 0.15. (a) Formulate the above optimization problem. Is it convex? What category does it fall into? Solve it using CVX. Plot the trajectory of the robot in the 2D plane, plot ||v(t)||2 and ||a(t)||2 versus t. (b) Now suppose we would like the robot to avoid a certain obstacle represented by a region in the 2D plane. Specifically, the path of the robot should not go through a circle of radius 1 centered around the origin. Such problem is generally non-convex, so we will solve it iteratively as follows: Save the solution you got from Part (a) as xold (t). Re-solve the problem in Part (a) with the additional constraint:

Suppose we have a robot moving in a 2D plane with dynamics x(t+1)=x(t) +v(t), v(t+1)=0.5v(t) + a(t), where æ(t), v(t) and a(t) denote its location, velocity and acceleration, respectively. Let the robot be initially at rest at a starting location (1) = [33] It has to reach a final location (T) 3 at T = 50, at which it comes to a full stop. The goal is to design an optimization problem that moves the robot to its desired location while minimizing the tal acceleration ||a(t)||2, satisfying the above dynamics, and with an additional constraint that the velocity magnitude (2-norm) is no larger than 0.15. (a) Formulate the above optimization problem. Is it convex? What category does it fall into? Solve it using CVX. Plot the trajectory of the robot in the 2D plane, plot ||v(t)||2 and ||a(t)||2 versus t. (b) Now suppose we would like the robot to avoid a certain obstacle represented by a region in the 2D plane. Specifically, the path of the robot should not go through a circle of radius 1 centered around the origin. Such problem is generally non-convex, so we will solve it iteratively as follows: Save the solution you got from Part (a) as xold (t). Re-solve the problem in Part (a) with the additional constraint:

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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