EAS401: Figure Q1 Shows a Coordinate Frame 𝐴 Here is Pointing out of Page According to the Righthand Rule: Robotics Mechanics and Control Assignment, SUSS, Singapore

Question 1

Figure Q1 shows a coordinate frame {𝐴}. Here is pointing out of page according to the righthand rule. The frame {𝐴𝐴} is rotated by 𝜃𝜃 = 20° about 𝑌 𝐴𝐴 into a new coordinate frame {𝐵}

(a) Plot a figure to schematically illustrate the coordinate frame {𝐵𝐵} based on the illustration given in Figure Q1.

(b) List formula to calculate the rotation matrix 𝐵𝐵𝑅𝑅𝐴𝐴 and present the number result.

(c) Demonstrate that the 𝐵𝐵𝑅𝑅𝐴𝐴 is orthonormal by calculation.

(d) A position vector is expressed in {𝐴𝐴} as 𝐴𝑃 = [1 0 2]𝑇𝑇. Calculate its position in {𝐵}, i.e., 𝐵𝑃.

(e) A position vector is expressed in {𝐵} as 𝐵𝑃 = [1 0 2]𝑇. Calculate its position in {𝐴}, i.e., 𝐴𝑃.

(f) Calculate determinant of 𝐵𝐴 , i.e., |𝐵𝑅 | .

Question 2

A coordinate frame {𝐵𝐵} is located at the base of a robot manipulator. Frame {𝐶} describes the position and orientation of a depth camera that was originally coincident with {𝐵} then translated 7 units in 𝑋𝐵, translated −2 units in 𝑌 𝐵 translated 5 units in 𝑍𝐵; the resulting frame is denoted by {𝐶′}. The frame is then rotated about 𝑍𝐶′ by −20° into a new frame {𝐶′′}. The final transform is rotated about 𝑌𝐶′′ by −110° into the frame {𝐶}. The camera detects an object having coordinates 𝐶𝐶𝑃𝑃 = [0.5 0.2 3.2]𝑇.

(a) Formulate the homogeneous transformation.
(b) Calculate the object coordinates in frame {𝐵𝐵}, i.e.
(c) Determine 𝐵𝑃𝐶

Problem 3

Consider the presentation of the orientation of a frame {𝐵𝐵} via the following successive rotations:

Start with the frame coincident with a known frame {𝐴𝐴}. Rotate {𝐵𝐵} first about 𝑌𝐵 by an angle 𝛽, then about 𝑋𝐵 by an angle 𝛾𝛾, and finally, about 𝑍̂ 𝐵 by an angle𝛼.

(a) Derive the equivalent rotation matrix 𝐵𝐴.
(b) If the rotation matrix is expressed in terms of 𝑟𝑟11, 𝑟𝑟12, …, 𝑟𝑟33, i.e.

Problem 4

The arm with three degrees of freedom shown in Figure Q4 is like the 3R planar arm, except that Joint 1’s axis is not parallel to the other two. Instead, there is a twist of 90 degrees in magnitude between Joints (Axes) 1 and 2. Note that no 𝑙𝑙3 need to be defined.

(a) Copy Figure Q4 to your submission and show the defined link frames in the figure.
(b) Determine the link parameters and list the results in the following table.

(c) Derive the kinematic equations for 3𝑇𝑇0.

Problem 5 

A three-link RRP manipulator is shown in Figure Q5. Perform forward kinematic analysis for the mechanism.

(a) Copy Figure Q5 to your submission and show the defined link frames in the figure.
(b) Determine link parameters and list the results in the following table.

(c) Derive the kinematic equations for 3𝑇0.

Problem 6

The PUMA 560 manipulator is schematically shown in Figure Q6. The robot arm is programmed to insert the tool into the socket (or Goal) at location 𝐺𝑇𝑆 . The pose (position and orientation) of the robot wrist, 𝑊𝑇𝐵 , can be determined by reading the joint angle sensors. It is also assumed that 𝑆𝑇𝐵 and 𝐺𝑇𝑆 are known.

(a) Using force control, the robot feels around with the tooltip until it inserts the tool into the socket. At this “calibration” configuration, frame {𝐺} and frame {𝑇} are coincident.

Determine the unknown tool frame 𝑇𝑊 based on the given conditions.

(b) With the tool frame pose solved, how can the robot arm be controlled to perform the task?

Problem 7

Develop a PYTHON program to compute the forward kinematics of PUMA 560. The DH parameters are listed in the following table:

The following values are adopted for the specification:

The workspace of the robot arm is controlled by the following range of joint variables:

Requirements:

• The code should be able to take the joint variables and the pose (position and orientation) of a position vector expressed in frame {6} (i.e., 6𝑃𝑃) as inputs, compute the forward kinematics mechanism, and output the pose of the position vector expressed in frame {0}.

• Verify (Test) the code by using the following input values: [𝜃𝜃1, 𝜃𝜃2, 𝜃𝜃3, 𝜃𝜃4, 𝜃𝜃5, 𝜃𝜃6] = [0, 30°, 20°, −30°, −45°, 45°] and 6𝑃𝑃 = [0.05, 0.05, 0.05]𝑇𝑇.

• List the code in the submission and provide the source code (.py) as an accompanying submission.

Problem 8

At the Tesla AI Day event on 19 August 2021, Elon Musk, the company’s co-founder and chief executive officer, said that “Tesla is basically the world’s largest robotics company”. Comment on this claim by performing a literature review on textbook knowledge, the robotics industry, the robotics applications, the latest developments in robotics, and relevant technologies (e.g., computer vision, sensor, communication, data analytics, machine learning, and self-driving vehicle technologies).

Requirements:

• This essay question is an open-ended question.
• Use your own words to summarize the literature review.
• Concisely present your answer. The maximum length of the answer is 1000 words.