Fixing a Skewed Motion Sensor

A robot's motion sensor measures along two axes that should be perpendicular, but manufacturing leaves them slightly skewed. Gram–Schmidt builds a true perpendicular frame from them: keep the first axis, subtract from the second its part along the first, and rescale both to length 1.

How it works

The Gram–Schmidt process

In one sentence

Given a basis $\mathbf{x}_1, \ldots, \mathbf{x}_p$ , set $\mathbf{v}_k = \mathbf{x}_k - \sum_{j<k}\frac{\mathbf{x}_k\cdot\mathbf{v}_j}{\mathbf{v}_j\cdot\mathbf{v}_j}\mathbf{v}_j$ ; then $\mathbf{v}_1, \ldots, \mathbf{v}_p$ is an orthogonal basis of the same subspace, and dividing by lengths makes it orthonormal.

General formula

{\htmlData{sym=2}{\mathbf{u}_1}} = \frac{{\htmlData{sym=0}{\mathbf{v}_1}}}{\|{\htmlData{sym=0}{\mathbf{v}_1}}\|}, \qquad {\htmlData{sym=3}{\mathbf{w}_2}} = {\htmlData{sym=1}{\mathbf{v}_2}} - ({\htmlData{sym=1}{\mathbf{v}_2}}\cdot{\htmlData{sym=2}{\mathbf{u}_1}})\,{\htmlData{sym=2}{\mathbf{u}_1}}, \qquad {\htmlData{sym=4}{\mathbf{u}_2}} = \frac{{\htmlData{sym=3}{\mathbf{w}_2}}}{\|{\htmlData{sym=3}{\mathbf{w}_2}}\|}

Each new direction keeps only what's new about it: its projection onto the directions already chosen is subtracted away.

What the symbols mean

$\mathbf{v}_1$
First measured axis
The sensor's first axis, as measured.
$\mathbf{v}_2$
Second measured axis
Should be perpendicular to the first, but isn't quite.
$\mathbf{u}_1$
First true axis
The first axis scaled to length 1.
$\mathbf{w}_2$
Corrected second axis
v₂ with its part along u₁ removed.
$\mathbf{u}_2$
Second true axis
w₂ scaled to length 1: perpendicular to u₁.

Normalize the first measured axis: $\mathbf{u}_1 = \mathbf{v}_1/\|\mathbf{v}_1\|$ .
The second axis leans toward the first by its projection $(\mathbf{v}_2\cdot\mathbf{u}_1)\mathbf{u}_1$ ; subtract it to get $\mathbf{w}_2$ , now perpendicular to $\mathbf{u}_1$ .
Normalize: $\mathbf{u}_2 = \mathbf{w}_2/\|\mathbf{w}_2\|$ . The pair $\mathbf{u}_1, \mathbf{u}_2$ is orthonormal and spans the same plane.
With more axes, keep going: subtract from each new vector its projections onto all the earlier ones.

Learn the math

The Gram–Schmidt ProcessLin Alg · Lesson 34, Orthogonality and Least Squares