Lab 8: Python Code for All the Math from Lesson 3 up to Lesson 6

Objective of lab 8

In this lab, we will develop functions in Python that can compute all the math that we learned from lesson 3 up to lesson 6. We will use these functions when studying the next parts of the course.

Required Software

• Visual Studio Code or any IDE for Python

Instructions

Based on the lessons that we have learned starting from Lesson 3: Orientation in Robotics (Rotation Matrices) to Lesson 6: Pose in Robotics (Exponential Coordinates of Robot Motions), develop functions in Python that can calculate the following. One function

• gets the rotation matrix R and returns its inverse (6 points).

• gets a 3-vector $\hat{\omega}$ and gives back the $3 \times 3$ skew-symmetric matrix representation of it (7 points).

• gets the angle $\theta$ and the unit axis of rotation $\hat{\omega}$ and returns the rotation matrix representing the rotation about $\hat{\omega}$ by $\theta$. This function should be able to handle the special cases of rotation operators about $\hat{x}$, $\hat{y}$, and $\hat{z}$. Use Rodrigues' formula to write this function (7 points).

• gets the $3 \times 3$ skew-symmetric matrix representation of $\hat{\omega}$ and returns the 3-verctor $\hat{\omega}$ (7 points).

• gest the 3-vector exponential coordinates for rotation $\hat{\omega}\theta$ and extracts the rotation axis $\hat{\omega}$ and the rotation amount $\theta$ (7 points).

• gets a $3 \times 3$ rotation matrix R and finds the axis $\hat{\omega}$, and angle $\theta$ representation of it (7 points).

• gets the rotation matrix $R \in SO(3)$ and a position vector $p \in \mathbb{R}^3$ and returns the $4 \times 4$ homogenous matrix T coressponding to them (7 points).

• gets the $4 \times 4$ homogenous transformation matrix T and extract the $3 \times 3$ rotation matrix and $3 \times 1$ position vector from it (7 points).

• gets the $4 \times 4$ homogenous transformation matrix T and computes the inverse of it (7 points).

• gets a 3-vector and returns its homogenous coordinates (6 points).

• gets the $4 \times 4$ homogenous transformation matrix T and computes its $6 \times 6$ adjoint representation $[Ad_{T}]$ (7 points).

• gets the 6-vector exponential coordinates of motion $\mathcal{S}q$ and extracts the normalized screw axis $\mathcal{S}$ and the distance traveled along the screw q (7 points).

• gets the 6-vector screw axis and computes the matrix representation of it (6 points).

• gets the screw axis $\mathcal{S}$ and q and calculates the corresponding homogenous transformation matrix $T \in SE(3)$ (6 points).

• gets the $4 \times 4$ homogenous transformation matrix and computes the screw axis and q (6 points).

Guidelines for Lab 8 report

• Submit one report per person (disclose all your collaborators, including the AI ones) through Canvas. It will be graded based on the above rubric. Please see above for points for each part.
• The report should include an example for each function that receives pertinent input and produces accurate output. Provide this in written format.
• Reports are due one week after the class that they completed. You can submit the report one week later than the due date for half credit.

Pat yourself on the back for completing Lab 8! You've done a fantastic job.