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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.