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).

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

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

(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.

(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.

(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.

(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?

(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?

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.

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}:

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 $R_{04}$?

(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 $R_{04}$?

Question 4 (Programming):

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.

Submission:

• Submit your answers via Canvas in a PDF format. It is OK to write the math and draw the drawings by hand if you are short of time.
• For the coding section, give a link to your GitHub page so that I can check the code.

Good luck!