Robot Path Finding (Part l): Finding the Gradient to poimt In this problem, we will use optimization to help a robot find the most efficient path between two given points. There is a catch though - someone put some obstacles in its way. In this question, your task is to help the robot find the most efficient path between two points, while avoiding the those obstacles as much as possible. We will make use of penalty function to reduce this constrained nonlinear optimization problem to unconstrained optimization. You are given an initial path matrix, X, comprised of n + 1 vectors (each with 2 coordinates) xo, xi, X2, X3..Xn. The start and end points of your path are fixed, and are given by xo andx You are also given certain point obstacles (vectors with 2 coordinates), ri, in the obstacle matrix R. In order to optimize the robot's path, you will be using the Steepest Descent algorithm. The objective function you will be using in order to carry out the optimization is given by F(x0, X1-Xp) defined as l- F(xo,xi....xn)- ci The symbols in this formula mean the following CI and c2 are "weights" (constants) that determine the effect of each term on the value of F(h) o c1 controls the 'loss' the objective function suffers for getting close to the obstacles o C2 controls the 'loss' the objective function suffers for having very long line segments k is the number of obstacles your robot must avoid . Adding in the denominator prevents division by zero In this subproblem, do the following a. Derive an expression for the gradient VFof the objective function F with respect to xi Your final formula should be in terms of the xjs, the rs, e, n and k (not necessarily all of them) It is probably easiest to take the derivative with respect to each entry of the vector xi separately and to then combine the result. Remember to show your work b. Mathematically describe the steps you will follow in order to implement the steepest descent algorithm to minimize the objective function F. Use Golden Section Search to perform a line search For the use of golden section search, assume that F is unimodal (although it isn't) and that you are able find an initial bracket on the minimum Robot Path Finding (Part l): Finding the Gradient to poimt In this problem, we will use optimization to help a robot find the most efficient path between two given points. There is a catch though - someone put some obstacles in its way. In this question, your task is to help the robot find the most efficient path between two points, while avoiding the those obstacles as much as possible. We will make use of penalty function to reduce this constrained nonlinear optimization problem to unconstrained optimization. You are given an initial path matrix, X, comprised of n + 1 vectors (each with 2 coordinates) xo, xi, X2, X3..Xn. The start and end points of your path are fixed, and are given by xo andx You are also given certain point obstacles (vectors with 2 coordinates), ri, in the obstacle matrix R. In order to optimize the robot's path, you will be using the Steepest Descent algorithm. The objective function you will be using in order to carry out the optimization is given by F(x0, X1-Xp) defined as l- F(xo,xi....xn)- ci The symbols in this formula mean the following CI and c2 are "weights" (constants) that determine the effect of each term on the value of F(h) o c1 controls the 'loss' the objective function suffers for getting close to the obstacles o C2 controls the 'loss' the objective function suffers for having very long line segments k is the number of obstacles your robot must avoid . Adding in the denominator prevents division by zero In this subproblem, do the following a. Derive an expression for the gradient VFof the objective function F with respect to xi Your final formula should be in terms of the xjs, the rs, e, n and k (not necessarily all of them) It is probably easiest to take the derivative with respect to each entry of the vector xi separately and to then combine the result. Remember to show your work b. Mathematically describe the steps you will follow in order to implement the steepest descent algorithm to minimize the objective function F. Use Golden Section Search to perform a line search For the use of golden section search, assume that F is unimodal (although it isn't) and that you are able find an initial bracket on the minimum