CS 545: Introduction to Robotics Fall 2019HW 3: Inverse KinematicsIn this assignment, you will compute numerically the inverse kinematics of a planarrobot with n = 3 rotational joints. Specifically, our goal is to compute the joint-positionsthat bring the end-effector to a desired position pd. In this exercise we will ignorethe orientation of the end-effector. We will formulate the problem as an optimizationprogram as follows:minimize qkFK(q) − pdk2subject to li ≤ qi ≤ ui, i = {1, . . . , n}.The objective is to find the values of the joints q that minimize the difference betweenthe actual position of the end-effector computed using the forward kinematics FK(q),and the desired position pd, subject to the joint limits li, ui.1. First, implement the forward kinematics function in the file fk.py. The functionshould receive the joint values q and the lengths of the 3 links L, and it shouldreturn the position of the robot’s end-effector. Report the results for the followingvalues:(a) q = [0.0, 0.0, 0.0], L = [1.0, 1.0, 1.0](b) q = [0.3, 0.4, 0.8], L = [0.8, 0.5, 1.0](c) q = [1.0, 0.0, 0.0], L = [3.0, 1.0, 1.0]Hint: Express the pose of the first link relative to the reference frame using a rigidtransform with an elementary rotation about the z-axis, then the pose of the secondlink relative to the first link and so on up to the end-effector.2. For the inverse kinematics computation, you will use the scipy.optimize package.1We will numerically compute a solution through the following function call:solution = minimize(objective, q0, method=‘SLSQP’, bounds=bnds)CS 545: Introduction to Robotics Fall 2019objective refers to the objective function that the optimizer attempts to minimize.q0 is the initial vector of values and bounds specify the range of allowable values.The scipy.minimize function has problems handling type promotion, so makesure all your numerical values are of type np.float64.We specify the following parameters for the IK prCS 545代写、代做Robotics、Python代写、Poblem:pd = [0.1, 1.33, 0.0], L = [0.7, 1.0, 1.0], q0 = [0, 0, 1.86],(li, ui) = (−π, π) ∀iWe provide the file ik-a.py, which includes also code for plotting. The initial con-figuration of the robot should show up as follows, with the base of robot in redand the desired end-effector position in green. To compute the inverse kinematics,fill in the objective function. Visualize the result with the given plotting function,and save as ik-a solution.png.Figure 1: Example visualization. A solution state should resultin the arm touching the green sphere.3. You will now add an obstacle that the robot should avoid. You will approximate theobstacle with a sphere, with circle c = (cx, cy, cz) and radius r. To detect whetherthe robot collides with the obstacle, the easiest way is to implement a line-sphere2CS 545: Introduction to Robotics Fall 2019intersection algorithm2. If there is no real solution, then there is no intersectionbetween a line and the sphere. In the provided file collision.py, fill in the functionline sphere intersection. It should implement the algorithm and return thevalue under the square-root (the discriminant).4. To ensure that the robot does not collide with the obstacle, we will add three obstaclecollision constraints, one for each robot link. The constraints are providedas input to the optimization algorithm, as shown in the provided file ik-b.py. Fillin the code for the constraints, using the collision detection algorithm from theprevious exercise. The parameters for the collision sphere are as follows:c = (0.6, 0.5, 0),r = 0.2Note that if the constraints are satisfied, they should return a non-negative value.Using the following parameters for the obstacle, solve and visualize the solution.Save your visualization as ik-b solution.png.5. Try increasing the radius of the obstacle r. What do you observe? Also experimentwith different starting configurations q0. Discuss the results in a file namedanswers.pdf.转自：http://www.3daixie.com/contents/11/3444.html
