[Solved] COMP550 Homework 2

Algorithmic Robotics

Please read the honor code and the additions described in the course syllabus. Present your work and your work only. You must explain all of your answers. Answers without explanation will be given no credit.

1. (20 points) A set S and an operator form a group if the following properties are satisfied:

Closure For all s₁ and s₂ in S, s₁s₂ is also an element of S.

Associativity For all s₁, s₂ and s₃ in S, (s₁s₂)s₃ = s₁(s₂s₃).

Identity There exists an element of S denoted by I such that Is₁ = s₁I = s₁ for all s₁ in S.

Inverse For each s₁ there exists a s₂ in S such that s₁s₂ = s₂s₁ = I.

Let T be the set of all rigid body transformations in 2D in homogeneous coordinates. Prove that T and regular matrix multiplication form a group. That is, prove that T and regular matrix multiplication satisfy the properties listed above. In your answers, use the facts that:

• Rotation matrices with matrix multiplication form a group.
• Translation vectors and vector addition form a group.

In your answers, you can write a rigid body transformation:

T_i

2. (10 points)
   • (5 points) What is a rotation of radians about the axis [0 0 1]^T as a unit quaternion, q₁?.
   • (5 points) Given the quaternion q₂ = 0+1i+0j+0k, what is q₁ q₂? Here, is quaternion multiplication.
3. (15 points)

Figure 1: From left to right: a manipulator with two prismatic joints, a manipulator with three revolute joints, and a manipulator with two revolute joints and a prismatic joint.

For each of the three manipulators shown in Figure ??, determine the topology and dimension of the manipulators configuration space.

4. (30 points) Figure ?? shows a three-link kinematic chain in 2D. The lengths of link A₁, A₂ and A₃ are l₁, l₂ and l₃, respectively. The joint angles of the chain are ₂ and ₃, as shown in Figure ??. For each link, we attach a local frame to the base end of that link (e.g., for link A₁, the axes of frame 1, x₁ and y₁ are labeled).

Figure 2: The Three-Link Chain

• (5 points) Determine the topology and dimension of the configuration space for this manipulator.
• (5 points) Determine the Homogeneous coordinates v₃ of the point 2 in the local frame of link A₃, in terms of l₁, l₂, l₃, ₂ and ₃.
• (5 points) Determine the forward kinematics of this three-link chain. That is, calculate the homogeneous coordinates v₁ of the point 2 in the local frame of link A₁, in terms of l₁, l₂, l₃, ₂ and ₃.
• (5 points) Determine the homogeneous transformations from the local frame of A₃ to the local frame of A₁. That is, determine the transformation matrices T₂ and T₃ such that, T₂ moves A₂ from its local frame to the local frame of A₁, and T₃ moves A₃ from its local frame to the local frame of A₂. Then the transformation matrix T₂T₃ moves A₃ from its local frame to the local frame of A₁.
• (10 points) Show that v₁ = T₂T₃v₃.

5. (15 points) Consider workspace obstacles A and B. If AB 6= 0/, do the configuration space obstacles QA and QB always overlap? If A B = 0/, is it possible for the configuration space obstacles QA and QB to overlap? Justify your claims for each question.
6. (10 points) Suppose five polyhedral bodies float freely in a 3D world. They are each capable of rotating and translating. If these are treated as one composite robot, what is the topology of the resulting configuration space (assume that the bodies are not attached to each other)? What is the dimension of the composite configuration space?

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