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ENGG 5402
Advanced Robotics Time Allowed: 2 weeks February 8, 2024 Notes: 1. ALL questions should be answered. 2. Full marks is 100. 3. If you need use your own variables and abbreviations, please define them properly. 1 (15%) (Quaternion Orientation) Please prove the following equations. 1. (5%) The unit quaternion Q = {η, ϵ} is invariant with respect to the rotation matrix R and its transpose RT . R(η, ϵ)ϵ = RT (η, ϵ)ϵ = ϵ (1) 2. (10%) The quaternion product is expressed by: Q1 ∗ Q2 = { η1η2 − ϵT1 ϵ2, η1ϵ2 + η2ϵ1 + ϵ1 × ϵ2 } (2) 2 (25%) (Rotation Matrix) Please answer the following questions. 1. (10%) Find the rotation matrix corresponding to the set of Euler angles ZXZ. 2. (10%) Discuss the inverse solution for the Euler angles ZY Z in the case sθ = 0. 3. (5%) Explain the singularity situation while using Euler angles to represent rotations. (e.g., ZY X presentation) 3 (30%) (Direct Kinematics) 1. (15%) Derive the DH table for the cylindrical arm in Fig. 1. 2. (15%) Find the direct kinematics for the cylindrical arm in Fig. 1. Figure 1: Cylindrical Arm
(30%) (Jacobian and Singularity) For the 6R spatial open chain of Fig. 2, 1. (15%) Determine its Jacobian J(θ). 2. (15%) Find its kinematic singularities.