CS5335 Assignment 5- Transformations and Kinematics Solution

• Submissions should be made electronically via the Canvas. Please ensure that your solutions for both the written and/or programming parts are present and zipped into a single file.
• You are welcome to discuss the programming questions (but not the written questions) with other students in the class. However, you must understand and write all code yourself. Also, you must list all students (if any) with whom you discussed your solutions to the programming questions.
1. 2 points. Consider a frame {B}, which is obtained from frame {A} by first rotating about the XA axis by an angle of θ, followed by a translation of [1,2,3]⊤ (in the {A} frame).
(a) Find the homogeneous transformation matrix ATB.
(b) Compute the homogeneous transformation matrix BTA.
(c) Given and Ap = [4,5,6]⊤, compute Bp.
(d) Sketch the frames for and Ap = [4,5,6]⊤, and check that your answer in part (c) seems correct.
2. 2 points.
(a) Consider a frame {A}. It is first rotated about the YA axis by an angle θ to form frame {B}, and then rotated about the ZB axis by an angle ϕ to form frame {C}. Determine the 3 × 3 rotation matrix, ARC, which will transform the coordinates of a position vector from Cp, its value in frame {C}, into Ap, its value in frame {A}.
(b) Recall that there are multiple interpretations of the rotation matrix. Suppose ARB = Ry(θ), the rotation matrix obtained by rotating about the YA axis.
• When viewed as a transformation, multiplying the rotation matrix ARB by a vector Bp results in the vector Ap = ARBBp, specifying the same point in space in a different coordinate frame.
• When viewed as an operator, multiplying the same rotation matrix R = Ry(θ) by a vector Ap results in the vector Ap′ = Ry(θ)Ap, specifying a different point in space in the same coordinate frame. The new point Ap′ is the result of rotating the original point Ap about the YA axis by an angle of θ.
Interpreting the rotation matrix from part (a) as an operator, explain why it is equivalent to first rotating by an angle ϕ about the ZA axis, followed by rotating by an angle θ about the YA axis.

3. 2 points. (Spong, Problem 2-37) Consider the diagram above. A robot is set up 1m from a table. The table is 1m long on each side. A frame {1} : x1,y1,z1 is fixed to the edge of the table as shown. A cube measuring 20cm on a side is placed on top of the table and at the center of the table with frame {2} : x2,y2,z2 established at the center of the cube’s bottom face as shown. A camera is situated directly above the center of the block 2m above the table top with frame {3} : x3,y3,z3 attached as shown.
(a) Find the homogeneous transformations relating each of these frames to the base frame {0}.
(b) Find the homogeneous transformation relating the cube’s frame {2} to the camera frame {3}.
(c) Suppose the robot pushes the cube and the camera now detects the cube’s origin at [0.2,−0.3,2]⊤. Use the above transformations to determine the position of the cube in the base frame {0}.

4. 2 points. For the three degree-of-freedom (3-DOF) manipulators shown above, find the forward kinematics map from the robot base (center of the vertical cylinder’s bottom face) to the end-effector. For rotational joints, the manipulators are shown in their zero configuration, i.e., the diagram illustrates the arms at θ1 = 0,θ2 = 0, and θ3 = 0 for part (a). The link lengths are given by L1,L2,L3, as follows:
(a) L1 is the vertical height of the first joint, L2 is the length of the arm segment after θ1,2 and before θ3, and L3 is the length of the arm after θ3.
(b) L1 and L2 are the same as in part (a), and L3 is determined by the translational joint, θ3, which determines the additional amount by which the hand extends beyond the L2 part of the arm.
Hint: Decompose the forward kinematics map into a series of simple transformations. A suggested set of intermediate coordinate frames for part (a) are shown in the diagram above. You are free to choose the intermediate coordinate frames – it should not affect the answer.

5. 2 points. Consider the 2-DOF planar arm shown above with two rotational joints, specified by θ1 and θ2, and link lengths ℓ1 and ℓ2 respectively. We are interested in solving the inverse kinematics problem for this arm, i.e., given a desired wrist position [xw,yw]⊤, find a joint configuration [θ1,θ2]⊤ that achieves the position, if a solution exists. In this problem, we will derive a geometric solution to this problem by finding α,β,γ.
(a) Find an expression for cosβ using the law of cosines, illustrated on the right.
(b) Using your answer from part (a), find an expression for θ2.
(c) Assuming you now have θ2, find an expression for γ.
(d) Find an expression for α, and therefore also for θ1.
Hint: The diagram shows a wrist position with two inverse kinematics solutions.
The expressions for θ1 and θ2 for the other solution should be very similar to what you have found above.