COMP550-Homework 2 Solved

1.   A set S and an operator · form a group if the following properties are satisfied:

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

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

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

                         Inverse         For each s1 there exists a s2 in S such that s1·s2 = s2·s1 = 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:

(a)    Rotation matrices with matrix multiplication form a group.

(b)   Translation vectors and vector addition form a group.

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

Ti  

2.    

(a)    (5 points) What is a rotation of   radians about the axis [0 0 1]T as a unit quaternion, q1?.

(b)   (5 points) Given the quaternion q2 = 0+1i+0j+0k, what is q1 ·q2? Here, “·” is quaternion multiplication.

 

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 manipulator’s configuration space.

3.   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 θ2 and θ3, 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

(a)    Determine the topology and dimension of the configuration space for this manipulator.

(b)   Determine the Homogeneous coordinates v3 of the point 2 in the local frame of link A3, in terms of l1, l2, l3, θ2 and θ3.

(c)    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, θ2 and θ3.

(d)   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 T2·T3 moves A3 from its local frame to the local frame of A1.

(e)    Show that v1 = T2·T3·v3.

5.   Consider workspace obstacles A and B. If A∩B 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.   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?1.   A set S and an operator · form a group if the following properties are satisfied:

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

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

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

                         Inverse         For each s1 there exists a s2 in S such that s1·s2 = s2·s1 = 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:

(a)    Rotation matrices with matrix multiplication form a group.

(b)   Translation vectors and vector addition form a group.

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

Ti  

2.    

(a)    (5 points) What is a rotation of   radians about the axis [0 0 1]T as a unit quaternion, q1?.

(b)   (5 points) Given the quaternion q2 = 0+1i+0j+0k, what is q1 ·q2? Here, “·” is quaternion multiplication.

 

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 manipulator’s configuration space.

3.   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 θ2 and θ3, 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

(a)    Determine the topology and dimension of the configuration space for this manipulator.

(b)   Determine the Homogeneous coordinates v3 of the point 2 in the local frame of link A3, in terms of l1, l2, l3, θ2 and θ3.

(c)    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, θ2 and θ3.

(d)   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 T2·T3 moves A3 from its local frame to the local frame of A1.

(e)    Show that v1 = T2·T3·v3.

5.   Consider workspace obstacles A and B. If A∩B 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.   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?