hx19 Ultrasonic IMU drift correction, initialization and xyz computation

The hexamite ultrasonic xyzlab setup can be useful to cancel IMU yaw drift. It can also be used to find the initial condition and provide absolute coordinates for IMU extrapolation and interpolation. Hx19lab is very precise, and smoothly stable for slow moving objects. IMU can be used to achieve reality sensation by increasing the sampling to over 43 s/s. The xyzLab setup below can be placed by every wall to create a surround positioning. Although hexamite 3d ultrasonic positioning requires only 3 receivers and 1 tag, for computational simplicity the xyzLab standard is to use 4 units, spaced at equal distances for 3d Cartesian coordinate results.

Simplifying Spatial Configuration

Receiver 0 at (0,0,0) and Receiver 1 at (X₁,0,0) and Receiver 2 at (0,Y₂,0)

Given the above configuration, the following is true:

X = (R₀²- R₁²+ X₁²) / (2*X₁)

similarly

Y = (R₀²- R₂² + Y₂²) / (2*Y₂)

and

Z²= R₀²- X²- Y²

Hx19 receiver 0 measures and returns R₀ as mm, and similarly receiver 1 returns its distance R₁ to the tag. In a clockwise enumeration receiver 3 returns R₂.

Receiver 2 is therefore redundant, and can be used to compute another set of X,Y,Z for averaging purposes.

Example:

Let the receivers on the cross be 1m apart in the x and y direction see illustration below. The green Z line is orthogonal on the yellow plane. The blue square represents the tag moving away and towards the yellow plane. Blue lines are the distances from tag to receivers (shown as blue circles at specific coordinate points). Blue line lengths is embedded in the serial string streaming from the receivers. Red lines are points on the yellow plane.

Let X₁=1m and Y₂=1m, then:

Receiver 0 at (0,0,0) and Receiver 1 at (1,0,0) and Receiver 2 at (0,1,0) then

X = (R₀²- R₁² + 1) / 2

Y = (R₀²- R₂² + 1) / 2

The above spatial arrangement makes positioning math efficient, simple and within grasp of those who don't have time for math elaboration. And it reduces processor load significantly, i.e. sets this operation within realm of most Microcontrollers.

Hx19 measurement is in millimeters. The distances between receivers on the hx19xyzLab cross is always 1000mm, so the axes should be in terms of mm instead of m.

HEXAMITE WILL EMBED ULTRASONIC CORRECTION INTO YOUR IMU UPON REQUEST