-, 22 ^ 3, SOLVING THE INVERSE KINEMATICS USING THE 1 NEWTON-...

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-, 22 ^ 3, SOLVING THE INVERSE KINEMATICS USING THE 1 NEWTON-RAPHSON METHOD d, Q2 di Consider the manipulators of figure 1 with do = 30cm, d1 = 30em. d2 25cm. We want to use the Newton-Raphson method to solve the a inverse kinematics problem. We consider two cases: The task space is the end-effector's location (FerBer 2c). A, The task space is (x. Penzel. where O2 is the end- effector's orientation The desired homogenous transformation for the end-effector is given by 0.50 -0.87 39.69 0.87 0.50 34.33 1.00 20.00 (1) Flg. 1.00 The Newton-Raphson method is a numerical method used to c) Write code to implement the Newton-Raphson solve nonlinear equations of the general form f(r) 0. The method and solve the inverse kinematics problem method is iterative where the solution can be calculated using d) What are the joint variables that allow to put the end-effector at the desired location re, and (2) orientation de? c) Verify your results using the forward kinematics where J is the Jacobian matrix calculated at equations 1) Find the direction and angle of rotation from the homo- geneous transformation HO. In both cases, feel free to compare your result with built-in function fsolve or equivalent functions. 2) Case I PROBLEM 2: KINEMATICS AND INVERSE KINEMATICS OF a) Write the equations relating Te- to the con- RRP MANIPULATOR figuration variables 01,02 ds and the link lengths do,d1,d2- Figure 2 shows a simple RRP manipulator with B1 = 50cm b) Find the Jacobian matrix of the system and B2 = 30cm. The configuration shown in the figure c) Write code to implement the Newton-Raphson (so/lzs) corresponds to 01 = 90°. A suggested representation of the axes is shown in the figure. Feel free to use other method and solve the inverse kinematics problem d) What are the joint variables that allow to put the representations. end-effector at the desired location? 1) What are the configuration variables? e) Verify your results using the forward kinematics 2) Write the equations for the position of the end-effector equations in the base reference frame. 3) Write equations for the configuration variables in terms of and the lengths of the links. You should Case 2 be able to simplify the equations and solve analytically. 3) a) Write the equations relating is Parze to the con- 4) We want to put the end-effector at position figuration variables 01,02, d3 and the link lengths (Ocm. 34. 33cm, (50cm). find the configuration variables. do,d1,d2- For this question, use the results of the previous b) Find the Jacobian matrix of the system question to solve analytically by hand. 2% 2, a 0, 23 shoulder 4 R² H., B, yo waist Flg. 5) Use fsolve function or your Newton-Raphson code to solve for the configuration of the robot that puts the end-effector at the following locations a) Zr) = (0cm, 34.33cm, 50cm), b) Ze) = (30cm, 50cm, 50cm), c) = (30cm, 10cm, 10cm). 6) Use Matlab or equivalent tools to represent the robot at each one of these configurations. Represent the prismatic joints by squares and the revolute joints by circles. 7) Now we want to study the velocity kinematics of the manipulator, find Je such that r. He (3) ze 8) Calculate J, at configuration (@1,02.d) = 33cm) 9) Determine the speed of the configuration variables so that Yes 2.) Sem/s 3cm/s, 2cm/6) at the con- figuration of the previous question. 2

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