[SOLVED] MTRN4230-ASSIGNMENT 3

Description

MTRN4230 Robotics ASSIGNMENT 3

AIMS

1. Kinematics
2. DH parameters
3. Jacobians

ACTIVITIES

1. Answer to the questions below

1. A vector ^𝐴𝑝  is rotated about 𝑍_𝐴 axis by 𝜃  degrees and then rotated about 𝑋_𝐴 asis by 𝜙   Give the rotation matrix considering the orders given. (0.5)
2. Frame {B} initially coincident with frame {A}. Now rotate {B} about 𝑍_𝐵 axis by 𝜃 degrees and rotate the resulting frame about 𝑋_𝐵 𝑎𝑥𝑖𝑠 𝑏𝑦 𝜙 Find rotation matrix for vectors ^𝐵𝑝 𝑡𝑜 ^𝐴𝑝 .   (0.5)
3. Given below frames

Calculate ^𝐵_𝐶𝑇   when ^𝑈_𝐴𝑇  , ^𝐵_𝐴𝑇    and _𝑈^𝐶𝑇    are given. (1)

1. 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)
2. Show the link frames for the below manipulators schematically (5 + 0.5)

1. 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