[SOLVED] MTRN4230  Assignment 3

AIMS

1. Kinematics
2. DH parameters
3. Jacobians

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

DESCRIPTION:

• derivations, the steps need to be elaborated with details in your report and please comment the steps. Without a full derivation your MATLAB code won’t be evaluated
• simulation in MATLAB / SIMULINK

NOTE: Please DO NOT use any pre-written MATLAB code or toolbox e.g. Peter Corke Robotics Toolbox. Code should be completely written by you.

DELIVERABLES (a, b, c, d, e, f):

1. Use DH convention find the forward kinematics (MATLAB)
   • Forward kinematics should be parametric (use syms in MATLAB)

(0.5)[derivation] (1)[MATLAB code]

• Small decimal should be considered zero with the below condition

|number|<0.0001 is considered zero in calculation of H matrix. (0.5)

[MATLAB code]

1. Find inverse kinematics (1) [derivation] (1) [MATLAB code]
2. Create kinematics verification mechanism as below and show the error as defined below in SIMULINK (use the trajectory provided) (2) [Simulink file and error plots]

• error = output – input Note: Use SIMULINK file given that can generate trajectory

1. Plot workspace when −180^𝑜 ≤ 𝜃_1,𝜃₂ ≤ 180^𝑜 𝑎𝑛𝑑 0 ≤ 𝑑₃ ≤ 0.1 in MATLAB(1) [ code and plot]
2. Calculate Jacobian and provide a function in MATLAB to calculate it (1) [ derivation],

(1) [ code]