Question

Question 3 a) For the 3-DoF robot in Figure 3, draw the frames if the D-H convention is used b) Using the D-H frames, express rotation matrix R as a function of 8,8... c) Solve the inverse kinematics problem, that is, 0.0... given a desired orientation defined by the rotation matrix: - Jq4. where d) Express the Jacobian matrix for the relationship velocity of frame 3 expressed within frame o. is the absolute angular Figure 3: 3-Dof robot with the joint space 4 -19, e) Solve for the pose(s) (-10 ) in which singularities occur. For these poses) draw the robot configuration and explain why it is a singularity. 1) For inverse kinematics, using a 2-Dof planar robot, briefly explain using just text diagrams (no mathematics required) why the following occurs: i. There exists no solutions to the inverse kinematics ii. There exists multiple solutions to the inverse kinematics

Answer 1

ra Let us consider a 300F Soind 0.,02,9, has been used. The digital encoders are used to measure for proper Positioning d the end effector along xg and z axes • The frame convention and Denavit - Hartenberg convention is shown in big untz Link 3 Link) The kinematic analysis d any robotic system is performed In 2 ways le forward kinematics and Inverse kinematis The forward kinematics a multi DOF robotic manpula tor is an easy task due to availability d Denant - Hartenberg Conventor (x(t) =f com] The calculation I rant vaniables to bring end ellector d to the Yobotic manipulator and vequired position Ovientations defined by inverse kinemati jot) = f'((t)] / Bley The muevse kinematics - multi-Dof robotic manipuator can be obtained usina 3 different techniques each iterative approach algebric approach, geometric approach, By substuting the Denavit - Harte. 6, d;,a;,& i) in the general ma the transformation matrices All 9 the Denavit Hartenberg parameters the general matrix given in the ea, ze mabrices A to Az are obtained a below

conti Sonti o Santi Canti Soni Santi contiCanto - Con Sant Santi Ctnt anni conti Anton Antil * rc, O-s, o OoTo ol A, - sa o Lo 3 ó - Slogin) o La Corin)? Coz/e) o 2₂ (6₂ ) Cogna) 3 S0;- ) Mathematically forward kinematics ean free spins can be obtainer by multiplying A to A matrices as given below A = A... A which results to After applying the above steps forward kmematis equation tou 3 DOF is obtained as qiven below Poc a le cose, coson + L2 Coso, cos cos (@3-1/2)-LZ.COSO, samo, svoll 1.7.) Pusl, sino, coso+12 sing, cos On Cos (03-T/2) - Ląsino, smo, sen (05M.) PZ=L, Sin Q, - [2 Sin Q2 Cos@,-"72) - COSO, sn@s-11,2)

The analytical inverse kinematic analysis for 3- DOF robotic manipulator has been performed using geometrical Qorach. The inverse kinematic equations are oblaine geometrically ! O, = tan'65/30) 26, C2 O 3 = 31 - cost (2,²4 L₂²-K2) on - 0-1 where k= Voc?ty² + 7² d = cos (l) for 250 d=-cos" (d/k) for Z 70 da verty? ja sint (La Sin (1 - 0)] O 2