Question

1. Consider the SCARA manipulator shown in the figure. (a) Perform the forward displacement analysis for the manipulator relating the the twist T= Se in terms of the joint angles Oj, i=1,2,4 and dz. The variable (e is measured in reference to the fixed X axis. If de = 0, then the gripper is aligned with the fixed X axis. Use Homogenous Transformation Matrices. (b) Find the Jacobian J of the manipulator relating the twist T = in terms of the joint angles Oi, i=1,2,4 and d3. Z + Y Figure 1: SCARA Manipulator. (c) Suppose that the robot joint angle vector is q={0 0 1 0). i. Explain why this is a singular configuration. Find all the joint velocities that produce a zero twist (this set of joint velocities is know as the null space of J). ii. Find the end effector twist if q = (1 -1 10).

Answer 1

Page-1 solution: from the data- I A SCARA manipulator is very important for robotic assembly due to its compliance and rigidity with desired direction. → SCARA manipulator has and degree of man - freedom same a 3 Rotatory Joints (R). 3 Joint is prismat ic () ... co-ordinate system {0}, {i} will have same origin of {3} , {t} The direction of Z3 from the given figure, The link transformation matrices relating sucessive link can be obtained.

page-2 (c, -s, ool G]- si To Ci oo o to -S₂ o al S2 Cz oo ه ه الا ه OOTO The P-H parameter of SCARA manipulator. lia, aq 13 lola-d. 1-d3.1.0 . (cu -Shool 20 7 * ง 0 0 cooo 3 Tool-d₃L hooa : 'CTJ-10100 1,26) - 6oo.1 The transformation matoix ? [TJ is obtained as º[+] = [],GILI () (C2C, -SiG = 18,29 C2C, 0 0 o o 1 ac, +Q2C 1 2 1 a,S,+Q2812 -03

- Page 3 => The orientation of the [h] can be describe by the angle d and position (x,y, 2) x=a.6, +92C12 y = a, 5, +Q₂612 z=-d3 0 = 0,+O2 +04 b) Inverse of SCARA manipulator Link ai di di loil 1 ao 100* 12 al 180° oo* - 100 du ra, us, o ace]. [cz Sz o azC2] A, = si ci o as: A = S2 -S₂0 a2C2 J ñ og Lo 0 0 1 0 0

page-4 (ooo A₂ = 10 loo To old₂ le ooi (en -Su ool Ish Cu oo oodul le) A Ti = A ----Am C D 0, - a tan2 (ae, Cy) - a tan2 (Q, +az ez, Q2S) -> 0, 3a tan 2 (C+01-C2) > OH = 0, +62 - a tanz (01 812) الله - ,0 = d « » c 2 = 0e toy -a, -22 20,22 a, = length of link 1 az = length of linka C, =coso, 162 - C2 = Cos O2

page-5 S, = sino, S2 = sino2 c) Jocobian of SCARA manipulator J = 120X(-0) ZA*(04-0) 22'07 Ca,ci] Caic, +926 127 O = ais, O2 = as, +₂512 LOJ LOJ fac + a2C12 Ou = lais, +0₂ S12 Ld3-de , Similarly, => 20 = 2,= K - 22 = 2z=-k Ja,s, -a₂S2 -A₂512 Jocobian , J = la cita2612 a₂512 OoToDo OOOOOī! Please give me positive vote.