HW 3: Orientation in Robotics ‐ Rotation Matrices

This homework assignment is designed to test your understanding of rotation matrices and especially their application to represent orientation in robotics.

Note: Most of these questions are adapted from the Modern Robotics textbook by Kevin Lynch and Frank Park with some modifications.

Question 1:

In terms of the $\hat{x}_s$, $\hat{y}_s$, $\hat{z}_s$ coordinates of a fixed space frame {s}, the frame {a} has its $\hat{x}_a$ axis pointing in the direction (0,0,1) and its $\hat{y}_a$ axis pointing in the direction (-1,0,0), and the frame {b} has its $\hat{x}_b$ axis pointing in the direction (1,0,0) and its $\hat{y}_b$ axis pointing in the direction (0,0,-1) (feel free to use your coordinate frames to determine the missing axis).

(a) Draw by hand the three frames, at different locations so that they are easy to see (9 points).

(b) Write down the rotation matrices $R_{sa}$ and $R_{sb}$ (9 points).

(c) Given $R_{sb}$, how do you calculate ${R^{-1}_{sb}}$ without using a matrix inverse? Write down ${R^{-1}_{sb}}$ and verify its correctness using your drawing (9 points).

(d) Given $R_{sa}$ and $R_{sb}$, how do you calculate $R_{ab}$ (again without using matrix inverses)? Compute the answer and verify its correctness using your drawing (9 points).

(e) Let $R = R_{sb}$ be considered as a transformation operator consisting of a rotation about $\hat{x}$ by $-90^{o}$. Calculate $R_1 = R_{sa}R$, and think of $R_{sa}$ as a representation of an orientation, $R$ as a rotation of $R_{sa}$, and $R_1$ as the new orientation after the rotation has been performed. Does the new orientation $R_1$ correspond to a rotation of $R_{sa}$ by $-90^{o}$ about the world-fixed $\hat{x}_s$-axis or about the body-fixed $\hat{x}_a$-axis? Now calculate $R_2 = RR_{sa}$. Does the new orientation $R_2$ correspond to a rotation of $R_{sa}$ by $-90^{o}$ about the world-fixed $\hat{x}_s$-axis or about the body-fixed $\hat{x}_a$-axis? Draw all the coordinate frames and show the rotation using your 3D coordinate frames (9 points).

(f) Use $R_{sb}$ to change the representation of the point $p_b = (1,2,3)$ (which is in {b} coordinates) to {s} coordinates. Does this move the position of point p in the physical space? (9 points)

(g) Choose a point p represented by $p_s = (1,2,3)$ in {s} coordinates. Calculate $p' = R_{sb}p_s$ and $p" = R^{T}_{sb}p_s$. For each operation, should the result be interpreted as changing coordinates (from the {s} frame to {b}) without moving the point p or as moving the location of the point without changing the reference frame of the representation?(9 points)

Question 2:

Let p be a point whose coordinates are $p = (\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{6}}, \frac{1}{\sqrt{2}})$ with respect to the fixed frame $\hat{x}-\hat{y}-\hat{z}$. Suppose that p is rotated about the fixed-frame $\hat{x}$-axis by 30 degrees, then about the fixed-frame $\hat{y}$-axis by 135 degrees, and finally about the fixed-frame $\hat{z}$-axis by -120 degrees. Denote the coordinates of this newly rotated point by p'. Note: You can use MATLAB/Python to do the matrix multiplication. (9 points)

Question 3:

(a) Given a fixed frame {0} and a moving frame {1} initially aligned with {0}, perform the following sequence of rotations on {1}: (9 points)

1. Rotate {1} about the {0} frame $\hat{x}$-axis by $\alpha$; call this new frame {2}.

2. Rotate {2} about the {0} frame $\hat{y}$-axis by $\beta$; call this new frame {3}.

3. Rotate {3} about the {0} frame $\hat{z}$-axis by $\gamma$; call this new frame {4}.

What is the final orientation $R_{04}$ (just write in terms of the matrices in and no need for matrix expansion or multiplication)?

(b) Suppose that the third step above is replaced by the following: “Rotate {3} about the $\hat{z}$-axis of frame {3} by $\gamma$; call this new frame {4}.” What is the final orientation $R_{04}$ (same here)? (9 points)

Question 4 (Programming): (10 points)

Write a function in Python that checks to see if a given $3\times 3$ matrix is a rotation matrix. This matrix should be close enough to be a member of SO(3). You can assign a tolerance (1e-5) so that for example 0.999999 is considered to be 1 or 0.000001 is considered to be 0. Show some output example as well.

Submission:

• Submit your answers via Canvas in a PDF format.
• For the coding section, give a link to your GitHub page or submit the code directly through the canvas.

Good luck!