Meaning of the columns of a rotation matrix, and testing a candidate matrix (20 marks): 22-Mec-B12 Robotics, December 2018, Q1

Question 1: Meaning of the columns of a rotation matrix, and testing a candidate matrix (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A single candidate array offered as the rotation of frame $\{B\}$ relative to frame $\{A\}$, with entries $\sqrt{3}/2 = 0.8660$ and $1/2 = 0.5$.

Find. (i) the physical meaning of the three columns, and (ii) whether the array is a legitimate rotation matrix, stating the tests that decide it.

Figure 1.1 — the candidate matrix seen in the $Y$–$Z$ plane of $\{A\}$ (the $X$ axes point out of the page and coincide). The second and third columns are the $\{B\}$ axes drawn in $\{A\}$.

Approach. Interpret the columns as direction cosines, then apply the two-part membership test for the rotation group: orthonormal columns ($R^{T}R = I$) and a determinant of $+1$.

Read the columns as the axes of $\{B\}$ drawn in $\{A\}$. By definition $$ {}^{A}_{B}R = \begin{bmatrix} {}^{A}\hat{X}_B & {}^{A}\hat{Y}_B & {}^{A}\hat{Z}_B \end{bmatrix}, $$ so column 1 is the unit vector along $\hat{X}_B$ written with $\{A\}$ components, column 2 is $\hat{Y}_B$ and column 3 is $\hat{Z}_B$. Every entry is a direction cosine — the cosine of the angle between one axis of $\{B\}$ and one axis of $\{A\}$. The rows carry the dual reading: row $i$ is the axis $\hat{X}_A,\hat{Y}_A,\hat{Z}_A$ expressed in $\{B\}$, which is why ${}^{B}_{A}R = {}^{A}_{B}R^{T}$.
Test 1 — each column must be a unit vector. The columns are $c_1 = (1,0,0)$, $c_2 = (0,\sqrt{3}/2,-1/2)$ and $c_3 = (0,1/2,\sqrt{3}/2)$: $$ \|c_1\| = 1, \qquad \|c_2\| = \sqrt{\tfrac{3}{4}+\tfrac{1}{4}} = 1, \qquad \|c_3\| = \sqrt{\tfrac{1}{4}+\tfrac{3}{4}} = 1. $$ A frame axis cannot stretch, so any column of length other than one immediately disqualifies the array.
Test 2 — the columns must be mutually perpendicular. Taking the three dot products, $$ c_1\cdot c_2 = 0, \qquad c_1\cdot c_3 = 0, \qquad c_2\cdot c_3 = \tfrac{\sqrt{3}}{4}-\tfrac{\sqrt{3}}{4} = 0. $$ Tests 1 and 2 together are exactly the statement $R^{T}R = I$, i.e. $R$ is orthogonal and its inverse is simply its transpose.
Test 3 — the determinant must be $+1$. Expanding along the first column, $$ \det R = 1\cdot\left[\left(\tfrac{\sqrt{3}}{2}\right)\left(\tfrac{\sqrt{3}}{2}\right)-\left(\tfrac{1}{2}\right)\left(-\tfrac{1}{2}\right)\right] = \tfrac{3}{4}+\tfrac{1}{4} = \boxed{+1}. $$ Orthogonality alone only forces $\det R = \pm 1$; the value $-1$ describes a reflection, which turns a right-handed triad into a left-handed one and is not a physically realisable rigid rotation. The check $\hat{X}_B\times\hat{Y}_B = \hat{Z}_B$ is the same test done by hand.
Conclude, and identify the rotation. All three tests pass, so the array is a valid rotation matrix. Comparing it with the standard $X$-axis rotation $$ R_X(\theta) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & c\theta & -s\theta \\ 0 & s\theta & c\theta \end{bmatrix} $$ gives $c\theta = \sqrt{3}/2$ and $s\theta = -1/2$, hence $\boxed{{}^{A}_{B}R = R_X(-30^\circ)}$ — frame $\{B\}$ is frame $\{A\}$ turned $30^\circ$ about $-\hat{X}_A$. The equivalent-angle formula $\theta = \arccos\!\big[(\operatorname{tr}R-1)/2\big] = 30^\circ$ agrees.

Question 1 — results
Item	Value	Verdict
Meaning of the columns	${}^{A}\hat{X}_B$, ${}^{A}\hat{Y}_B$, ${}^{A}\hat{Z}_B$ (direction cosines)	—
Column norms	$1,\;1,\;1$	pass
Column dot products	$0,\;0,\;0$	pass
$\det R$	$+1$	pass
Overall	valid rotation matrix	$R_X(-30^\circ)$

Topic. Membership test for the rotation group $SO(3)$.

Key relations. $R^{T}R = I$ (orthonormal columns) and $\det R = +1$; equivalently $R^{-1} = R^{T}$ and $\hat{X}\times\hat{Y} = \hat{Z}$.

Why this works. A rotation matrix is nothing more than the three unit vectors of one frame written in another. Unit length preserves scale, mutual perpendicularity preserves angles, and a positive determinant preserves handedness — together they are exactly the properties of a rigid body turned in space.

Common pitfall. Checking only the determinant. A matrix can have $\det = +1$ and still have non-unit or skewed columns, in which case it distorts the body it is supposed to rotate.

Source. Craig, J. J., Introduction to Robotics: Mechanics and Control, 4th ed., Pearson, 2018, §2.2–2.3 (description of an orientation; properties of rotation matrices).

Question 1 of 7: Meaning of the columns of a rotation matrix, and testing a candidate matrix (20 marks)

Question 1: Meaning of the columns of a rotation matrix, and testing a candidate matrix (20 marks)