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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
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
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1. 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 x(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) = [3] 1] 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 total acceleration ₁ ||a(t)||2, satisfying the above dynamics, and with an additional constraint that the velocity magnitude (2-norm) is no larger than 0.15. It has to reach a final location x (T) = at T = 50, at which (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:
Transcribed Image Text:1. 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 x(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) = [3] 1] 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 total acceleration ₁ ||a(t)||2, satisfying the above dynamics, and with an additional constraint that the velocity magnitude (2-norm) is no larger than 0.15. It has to reach a final location x (T) = at T = 50, at which (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:
x(t) xold (t) ≥ ||*old(t)||2, Vt. Ignore the velocity magnitude constraint for this part (problem may be infeasible if you do not). • Save the (new) solution you get as old (t). Repeat the previous step until convergence i.e., until the new solution is the same as the old solution. (May take around 10-15 steps to converge.) Plot the trajectory of the robot in the 2D plane. On the same figure, plot a circle in red denoting the obstacle region to be avoided. Also plot ||v(t)||2 and ||a(t)||2 versus t.
Transcribed Image Text:x(t) xold (t) ≥ ||*old(t)||2, Vt. Ignore the velocity magnitude constraint for this part (problem may be infeasible if you do not). • Save the (new) solution you get as old (t). Repeat the previous step until convergence i.e., until the new solution is the same as the old solution. (May take around 10-15 steps to converge.) Plot the trajectory of the robot in the 2D plane. On the same figure, plot a circle in red denoting the obstacle region to be avoided. Also plot ||v(t)||2 and ||a(t)||2 versus t.
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