Rotation matrix

From TheAnalogThing

This page describes Electronics to implement 3-dimensional coordinate transformations in terms of a rotation matrix. The most typical use case is some analog circuit which computes the 3-dimensional particle trajectory (x,y,z). This three-dimensional space is typically projected on the two dimensional oscilloscope screen by just showing some plane, for instance the z-plane, so that (x,y) are visible. By means of a physical rotation matrix, you can rotate the (x,y,z) space so that you can see a two-dimensional image (projection) from an arbitrary direction.

A 24-step switch can be used for angle adjustment in 15° steps. These 4x24 switches are available on Aliexpress for less than 10€. Search for "4*24 Rotary Switch".

4x24 rotary switch.jpg

The switch must be disassembled to remove the limiter, so that the axis can be rotated over the full 360° range.

Limiter in switch.jpg

In those planes that make the sin() and -sin() signals, the contact must be removed from the A position and inserted in position B, which is 90° phase shifted from A. For the -sin() signal, the phase shift is in the other direction.

Contact in switch.jpg

For yaw, pitch and roll rotations you need three switches.

3 switches.jpg

This is the schematic for half of the switch. It has an input A and two outputs A*sin(alpha) and A*cos(alpha). The schematic for the other half is identical, but the contact in the switch is phase shifted in the other direction so that the outputs are -A*sin(alpha) and A*cos(alpha).

Schematic rotation switch.jpg

This is the main schematic:

Schematic rotator.jpg

It doesn't matter that the outputs X' and Y' are inverted. Just rotate the switch 180° to invert the signs.

Three rotators can be combined for making yaw, pitch and roll rotations:

Yaw pitch roll.jpg