Question

*If MatLab is used for matrix calculation please include .m file or text from your code in case I still have any doubt. Thank you!!!!

1. In Figure 1 below is a Polar (Planar R-P) manipulator, with an orange block moving along the blue track by a variable length di’. On the orange black a motor is fixed at height ai'. The end-effector (yellow dot) is fixed to the motor shaft using a link-2 of fixed length a2' and it rotates at angle 01. Solve the following for the given manipulator: - a. Show the kinematic diagram with reference frames. b. Perform a forward kinematics and determine the homogenous transformation matrix (C1H). Show the expressions for the position of the end-effector. c. For a given position of the end-effector (x, y) in the global reference frame, perform the inverse kinematics and determine the expression for 0. YO Y1 x1 a2 01 fa XO Figure 1:Planar Polar(R-P) manipulator

Answer 1

given Polar Planar manipulator (a) Kinematic diagram Yo az yi Xi 1 x" oi Та, di I oh (b) Homogeneous transformation matrix H = Rot (zo, Oi) Trans (x, di) Trans и H OH H - OHS Trans (xo,d,) il - Trons (Y) 9,) HI = Rot (Z", 01) Translx", A2) H

iha o 1? "J O o с со b ng об 0 a 14 H caso, oli G -sino, o H - sino coso, X 0 OOO 1 5 o- 0 Trams (",) 0 d iH coso, sino, - sinoi coso, 0 ggo 0- oo Coso, -sindi az co.so, tdi sino, coso, asino, ta, coordinates of end effector are given by so acoso, tdi azsino, ta, Y 1

(c) Now inverse kinematics given given x,y, a, and want to find o, and di 2 we from the derived coordinates in can write we 2 a, tazsino, = y o, can be known so known a, az, y are sinn in we can also write a dit acoso, - aa, oi and X are known so di can be found d = x 892 az dia ST-(2-a) ² coso, Si- sinzo sino, = yra, as az