# Three-Axis Accelerometer

Implement three-axis accelerometer

## Description

The Three-Axis Accelerometer block implements an accelerometer on each of the three axes. The ideal measured accelerations (${\overline{A}}_{imeas}$) include the acceleration in body axes at the center of gravity (${\overline{A}}_{b}$), lever arm effects due to the accelerometer not being at the center of gravity, and, optionally, gravity in body axes can be removed.

`${\overline{A}}_{imeas}={\overline{A}}_{b}+{\overline{\omega }}_{b}×\left({\overline{\omega }}_{b}×\overline{d}\right)+{\stackrel{˙}{\overline{\omega }}}_{b}×\overline{d}-\overline{g}$`

where ${\overline{\omega }}_{b}$ are body-fixed angular rates, ${\stackrel{˙}{\overline{\omega }}}_{b}$ are body-fixed angular accelerations and $\overline{d}$ is the lever arm. The lever arm ($\overline{d}$) is defined as the distances that the accelerometer group is forward, right and below the center of gravity.

`$\overline{d}=\left[\begin{array}{l}{d}_{x}\\ {d}_{y}\\ {d}_{z}\end{array}\right]=\left[\begin{array}{c}-\left({x}_{acc}-{x}_{CG}\right)\\ {y}_{acc}-{y}_{CG}\\ -\left({z}_{acc}-{z}_{CG}\right)\end{array}\right]$`

The orientation of the axes used to determine the location of the accelerometer group (xacc, yacc, zacc) and center of gravity (xCG, yCG, zCG) is from the zero datum (typically the nose) to aft, to the right of the vertical centerline and above the horizontal centerline. The x-axis and z-axis of this measurement axes are opposite the body-fixed axes producing the negative signs in the lever arms for x-axis and z-axis.

Measured accelerations (${\overline{A}}_{meas}$) output by this block contain error sources and are defined as

`${\overline{A}}_{meas}={\overline{A}}_{imeas}×{\overline{A}}_{SFCC}+{\overline{A}}_{bias}+noise$`

where ${\overline{A}}_{SFCC}$ is a 3-by-3 matrix of scaling factors on the diagonal and misalignment terms in the nondiagonal, and ${\overline{A}}_{bias}$are the biases.

Optionally discretizations can be applied to the block inputs and dynamics along with nonlinearizations of the measured accelerations via a Saturation block.

## Parameters

Units

Specifies the input and output units:

UnitsAccelerationLength
`Metric (MKS)`Meters per second squaredMeters
`English`Feet per second squaredFeet

Accelerometer location

The location of the accelerometer group is measured from the zero datum (typically the nose) to aft, to the right of the vertical centerline and above the horizontal centerline. This measurement reference is the same for the center of gravity input. The units are in selected length units.

Subtract gravity

Select to subtract gravity from acceleration readings.

Second order dynamics

Select to apply second-order dynamics to acceleration readings.

The natural frequency of the accelerometer. The units of natural frequency are radians per second.

Damping ratio

The damping ratio of the accelerometer. A dimensionless parameter.

Scale factors and cross-coupling

The 3-by-3 matrix used to skew the accelerometer from body axes and to scale accelerations along body axes.

Measurement bias

The three-element vector containing long-term biases along the accelerometer axes. The units are in selected acceleration units.

Update rate (sec)

Specify the update rate of the accelerometer. An update rate of 0 will create a continuous accelerometer. If noise is selected and the update rate is 0, then the noise will be updated at the rate of 0.1. The units of update rate are seconds.

If you:

• Update this parameter value to 0 (continuous)

• Configure a fixed-step solver for the model

• Do not have a Control System Toolbox™ license

you must also select the Automatically handle rate transition for data transfer check box in the Solver pane. This check box enables the software to handle rate transitions correctly.

Noise on

Select to apply white noise to acceleration readings.

Noise seeds

The scalar seeds for the Gaussian noise generator for each axis of the accelerometer.

Noise power

The height of the PSD of the white noise for each axis of the accelerometer. The units are:

• (m/s2)/Hz for ```Metric (MKS)```

• (ft/s2)/Hz for `English`

Lower and upper output limits

The six-element vector containing three minimum values and three maximum values of acceleration in each of the accelerometer axes. The units are in selected acceleration units.

## Inputs and Outputs

InputDimension TypeDescription

First

Three-element vectorContains the actual accelerations in body-fixed axes, in selected units.

Second

Three-element vectorContains the angular rates in body-fixed axes, in radians per second.

Third

Three-element vectorContains the angular accelerations in body-fixed axes, in radians per second squared.

Fourth

Three-element vectorContains the location of the center of gravity, in selected units.

Fifth (Optional)

Three-element vectorContains the gravity in body axis, in selected units.

OutputDimension TypeDescription

First

Three-element vectorContains the measured accelerations from the accelerometer, in selected units.

## Assumptions and Limitations

Vibropendulous error and hysteresis effects are not accounted for in this block. Additionally, this block is not intended to model the internal dynamics of different forms of the instrument.

 Note:   This block requires the Control System Toolbox product for discrete operation (nonzero sample time).

## Reference

Rogers, R. M., Applied Mathematics in Integrated Navigation Systems, AIAA Education Series, 2000.