MTRN4230 - Assignment 3 - Solved

 

 Robotics 
 

AIMS
1.       Kinematics

2.       DH parameters

3.       Jacobians



ACTIVITIES
 

1. Answer to the questions below
a.       A vector 𝐴𝑝   is rotated about 𝑍𝐴 axis by 𝜃  degrees and then rotated about 𝑋𝐴 asis by 𝜙  degrees. Give the rotation matrix considering the orders given. (0.5)

b.       Frame {B} initially coincident with frame {A}. Now rotate {B} about 𝑍𝐵 axis by 𝜃 degrees and rotate the resulting frame about 𝑋𝐵 𝑎𝑥𝑖𝑠 𝑏𝑦 𝜙 degrees. Find rotation matrix for vectors 𝐵𝑝 𝑡𝑜 𝐴𝑝   .   (0.5)

c.       Given below frames  

  

Calculate 𝐵𝐶𝑇   when 𝑈𝐴𝑇  , 𝐵𝐴𝑇    and 𝑈𝐶𝑇    are given. (1)

d.       Proof that inverse of a rotation matrix must be equal to its transpose and rotation matrix is orthonormal. Show it with the help of two vectors embedded in a rigid body so no matter how the body rotates, the geometric angel between them (two vectors) preserve. (1)

e.       Show the link frames for the below manipulators schematically (0.5 + 0.5)

  

 

   

f.        A 2DOF positioning table is used to help welding (two rotary joints 𝜃1, 𝜃2). The forward kinematics from based (link 1) to the bed of the table (link 2) is  

  

Unit vector fixed in frame of link 2 is 2𝑉  . 𝐹𝑖𝑛𝑑 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 − 𝑘𝑖𝑛𝑒𝑚𝑎𝑡𝑖𝑐 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑓𝑜𝑟  

𝜃1,𝜃2) when this unit vector is aligned with 0𝑍 𝑎𝑥𝑖𝑠.    Are there multiple solutions and is there a singular condition? (2)

2. A manipulator shown below that is known as SCARA when d4= 0.1, a1 = 0.4 and a2 = 0.3