# imuSensor

IMU simulation model

## Description

The `imuSensor`

System object™ models receiving data from an inertial measurement unit (IMU). You can specify
the reference frame of the block inputs as the `NED`

(North-East-Down) or
`ENU`

(East-North-Up) frame by using the
`ReferenceFrame`

argument.

To model an IMU:

Create the

`imuSensor`

object and set its properties.Call the object with arguments, as if it were a function.

To learn more about how System objects work, see What Are System Objects?

## Creation

### Syntax

### Description

returns a System object, `IMU`

= imuSensor`IMU`

, that computes an inertial measurement unit reading
based on an inertial input signal. `IMU`

has an ideal accelerometer and
gyroscope.

`IMU = imuSensor(`

returns an
`'accel-gyro'`

)`imuSensor`

System object with an ideal accelerometer and gyroscope. `imuSensor`

and
`imuSensor('accel-gyro')`

are equivalent creation syntaxes.

`IMU = imuSensor(`

returns an
`'accel-mag'`

)`imuSensor`

System object with an ideal accelerometer and magnetometer.

`IMU = imuSensor(`

returns an
`'accel-gyro-mag'`

)`imuSensor`

System object with an ideal accelerometer, gyroscope, and magnetometer.

`IMU = imuSensor(___,"ReferenceFrame",`

returns an `RF`

)`imuSensor`

System object that computes an inertial measurement unit reading relative to the reference
frame `RF`

.

`IMU = imuSensor(___,`

sets one or more properties using name-value arguments in addition to any of the previous
input arguments.`Name=Value`

)

### Input Arguments

`RF`

— Reference frame of sensor inputs

`'NED'`

(default) | `'ENU'`

Reference frame of the sensor inputs, specified as either `'NED'`

(North-East-Down) or `'ENU'`

(East-North-Up).

**Note**

If you choose the NED reference frame, specify the sensor inputs in the NED reference frame. Additionally, the sensor models the gravitational acceleration as [0 0 9.81] m/s

^{2}.If you choose the ENU reference frame, specify the sensor inputs in the ENU reference frame. Additionally, the sensor models the gravitational acceleration as [0 0 −9.81] m/s

^{2}.

**Data Types: **`char`

| `string`

## Properties

Unless otherwise indicated, properties are *nontunable*, which means you cannot change their
values after calling the object. Objects lock when you call them, and the
`release`

function unlocks them.

If a property is *tunable*, you can change its value at
any time.

For more information on changing property values, see System Design in MATLAB Using System Objects.

`IMUType`

— Type of inertial measurement unit

`'accel-gyro'`

(default) | `'accel-mag'`

| `'accel-gyro-mag'`

Type of inertial measurement unit, specified as a `'accel-gyro'`

,
`'accel-mag'`

, or `'accel-gyro-mag'`

.

The type of inertial measurement unit specifies which sensor readings to model:

`'accel-gyro'`

–– Accelerometer and gyroscope`'accel-mag'`

–– Accelerometer and magnetometer`'accel-gyro-mag'`

–– Accelerometer, gyroscope, and magnetometer

You can specify `IMUType`

as a value-only argument during creation
or as a `Name,Value`

pair.

**Data Types: **`char`

| `string`

`SampleRate`

— Sample rate of sensor (Hz)

`100`

(default) | positive scalar

Sample rate of the sensor model in Hz, specified as a positive scalar.

**Data Types: **`single`

| `double`

`Temperature`

— Temperature of IMU (^{o}C)

`25`

(default) | real scalar

Operating temperature of the IMU in degrees Celsius, specified as a real scalar.

When the object calculates temperature scale factors and environmental drift noises,
25 ^{o}C is used as the nominal temperature.

**Tunable: **Yes

**Data Types: **`single`

| `double`

`MagneticField`

— Magnetic field vector in local navigation coordinate system (μT)

`[27.5550 -2.4169 -16.0849]`

(default) | three-element real-valued vector

Magnetic field vector in microtesla, specified as a three-element row vector in the local navigation coordinate system.

The default magnetic field corresponds to the magnetic field at latitude zero, longitude zero, and altitude zero.

**Tunable: **Yes

**Data Types: **`single`

| `double`

`Accelerometer`

— Accelerometer sensor parameters

`accelparams`

object (default)

Accelerometer sensor parameters, specified by an `accelparams`

object.

**Tunable: **Yes

`Gyroscope`

— Gyroscope sensor parameters

`gyroparams`

object (default)

Gyroscope sensor parameters, specified by a `gyroparams`

object.

**Tunable: **Yes

`Magnetometer`

— Magnetometer sensor parameters

`magparams`

object (default)

Magnetometer sensor parameters, specified by a `magparams`

object.

**Tunable: **Yes

`RandomStream`

— Random number source

`'Global stream'`

(default) | `'mt19937ar with seed'`

Random number source, specified as a character vector or string:

`'Global stream'`

–– Random numbers are generated using the current global random number stream.`'mt19937ar with seed'`

–– Random numbers are generated using the mt19937ar algorithm with the seed specified by the`Seed`

property.

**Data Types: **`char`

| `string`

`Seed`

— Initial seed

`67`

(default) | nonnegative integer scalar

Initial seed of an mt19937ar random number generator algorithm, specified as a real, nonnegative integer scalar.

#### Dependencies

To enable this property, set `RandomStream`

to
`'mt19937ar with seed'`

.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

## Usage

### Syntax

### Description

`[`

generates accelerometer and gyroscope readings from the acceleration and angular
velocity inputs.`accelReadings`

,`gyroReadings`

] = IMU(`acc`

,`angVel`

)

This syntax is only valid if `IMUType`

is set to
`'accel-gyro'`

or `'accel-gyro-mag'`

.

`[`

generates accelerometer and gyroscope readings from the acceleration, angular velocity,
and orientation inputs.`accelReadings`

,`gyroReadings`

] = IMU(`acc`

,`angVel`

,`orientation`

)

This syntax is only valid if `IMUType`

is set to
`'accel-gyro'`

or `'accel-gyro-mag'`

.

`[`

generates accelerometer and magnetometer readings from the acceleration and angular
velocity inputs.`accelReadings`

,`magReadings`

] = IMU(`acc`

,`angVel`

)

This syntax is only valid if `IMUType`

is set to
`'accel-mag'`

.

`[`

generates accelerometer and magnetometer readings from the acceleration, angular
velocity, and orientation inputs.`accelReadings`

,`magReadings`

] = IMU(`acc`

,`angVel`

,`orientation`

)

This syntax is only valid if `IMUType`

is set to
`'accel-mag'`

.

`[`

generates accelerometer, gyroscope, and magnetometer readings from the acceleration and
angular velocity inputs.`accelReadings`

,`gyroReadings`

,`magReadings`

] = IMU(`acc`

,`angVel`

)

This syntax is only valid if `IMUType`

is set to
`'accel-gyro-mag'`

.

`[`

generates accelerometer, gyroscope, and magnetometer readings from the acceleration,
angular velocity, and orientation inputs.`accelReadings`

,`gyroReadings`

,`magReadings`

] = IMU(`acc`

,`angVel`

,`orientation`

)

This syntax is only valid if `IMUType`

is set to
`'accel-gyro-mag'`

.

### Input Arguments

`acc`

— Acceleration of IMU in local navigation coordinate system
(m/s^{2})

*N*-by-3 matrix

Acceleration of the IMU in the local navigation coordinate system, specified as a
real, finite *N*-by-3 array in meters per second squared.
*N* is the number of samples in the current frame. Do not include
the gravitational acceleration in this input since the sensor models gravitational
acceleration by default.

To specify the orientation of the IMU sensor body frame with respect to the local
navigation frame, use the `orientation`

input argument.

**Data Types: **`single`

| `double`

`angVel`

— Angular velocity of IMU in local navigation coordinate system (rad/s)

*N*-by-3 matrix

Angular velocity of the IMU in the local navigation coordinate system, specified
as a real, finite *N*-by-3 array in radians per second.
*N* is the number of samples in the current frame. To specify the
orientation of the IMU sensor body frame with respect to the local navigation frame,
use the `orientation`

input argument.

**Data Types: **`single`

| `double`

`orientation`

— Orientation of IMU in local navigation coordinate system

*N*-element quaternion column vector | 3-by-3-by-*N*-element rotation matrix

Orientation of the IMU with respect to the local navigation coordinate system,
specified as a `quaternion`

*N*-element column vector or a 3-by-3-by-*N*
rotation matrix. Each `quaternion`

or rotation matrix represents a frame
rotation from the local navigation coordinate system to the current IMU sensor body
coordinate system. *N* is the number of samples in the current
frame.

**Data Types: **`single`

| `double`

| `quaternion`

### Output Arguments

`accelReadings`

— Accelerometer measurement of IMU in sensor body coordinate system
(m/s^{2})

*N*-by-3 matrix

Accelerometer measurement of the IMU in the sensor body coordinate system,
specified as a real, finite *N*-by-3 array in meters per second
squared. *N* is the number of samples in the current frame.

**Data Types: **`single`

| `double`

`gyroReadings`

— Gyroscope measurement of IMU in sensor body coordinate system (rad/s)

*N*-by-3 matrix

Gyroscope measurement of the IMU in the sensor body coordinate system, specified
as a real, finite *N*-by-3 array in radians per second.
*N* is the number of samples in the current frame.

**Data Types: **`single`

| `double`

`magReadings`

— Magnetometer measurement of IMU in sensor body coordinate system (μT)

*N*-by-3 matrix (default)

Magnetometer measurement of the IMU in the sensor body coordinate system,
specified as a real, finite *N*-by-3 array in microtelsa.
*N* is the number of samples in the current frame.

**Data Types: **`single`

| `double`

## Object Functions

To use an object function, specify the
System object as the first input argument. For
example, to release system resources of a System object named `obj`

, use
this syntax:

release(obj)

### Specific to `imuSensor`

`loadparams` | Load sensor parameters from JSON file |

`perturbations` | Perturbation defined on object |

`perturb` | Apply perturbations to object |

## Examples

### Create Default `imuSensor`

System object

The `imuSensor`

System object™ enables you to model the data received from an inertial measurement unit consisting of a combination of gyroscope, accelerometer, and magnetometer.

Create a default `imuSensor`

object.

IMU = imuSensor

IMU = imuSensor with properties: IMUType: 'accel-gyro' SampleRate: 100 Temperature: 25 Accelerometer: [1x1 accelparams] Gyroscope: [1x1 gyroparams] RandomStream: 'Global stream'

The `imuSensor`

object, `IMU`

, contains an idealized gyroscope and accelerometer. Use dot notation to view properties of the gyroscope.

IMU.Gyroscope

ans = gyroparams with properties: MeasurementRange: Inf rad/s Resolution: 0 (rad/s)/LSB ConstantBias: [0 0 0] rad/s AxesMisalignment: [3⨯3 double] % NoiseDensity: [0 0 0] (rad/s)/√Hz BiasInstability: [0 0 0] rad/s RandomWalk: [0 0 0] (rad/s)*√Hz NoiseType: "double-sided" BiasInstabilityCoefficients: [1⨯1 struct] TemperatureBias: [0 0 0] (rad/s)/°C TemperatureScaleFactor: [0 0 0] %/°C AccelerationBias: [0 0 0] (rad/s)/(m/s²)

Sensor properties are defined by corresponding parameter objects. For example, the gyroscope model used by the `imuSensor`

is defined by an instance of the `gyroparams`

class. You can modify properties of the gyroscope model using dot notation. Set the gyroscope measurement range to 4.3 rad/s.

IMU.Gyroscope.MeasurementRange = 4.3;

You can also set sensor properties to preset parameter objects. Create an `accelparams`

object to mimic specific hardware, and then set the IMU `Accelerometer`

property to the `accelparams`

object. Display the `Accelerometer`

property to verify the properties are correctly set.

SpecSheet1 = accelparams( ... 'MeasurementRange',19.62, ... 'Resolution',0.00059875, ... 'ConstantBias',0.4905, ... 'AxesMisalignment',2, ... 'NoiseDensity',0.003924, ... 'BiasInstability',0, ... 'TemperatureBias', [0.34335 0.34335 0.5886], ... 'TemperatureScaleFactor', 0.02); IMU.Accelerometer = SpecSheet1; IMU.Accelerometer

ans = accelparams with properties: MeasurementRange: 19.62 m/s² Resolution: 0.00059875 (m/s²)/LSB ConstantBias: [0.4905 0.4905 0.4905] m/s² AxesMisalignment: [3⨯3 double] % NoiseDensity: [0.003924 0.003924 0.003924] (m/s²)/√Hz BiasInstability: [0 0 0] m/s² RandomWalk: [0 0 0] (m/s²)*√Hz NoiseType: "double-sided" BiasInstabilityCoefficients: [1⨯1 struct] TemperatureBias: [0.34335 0.34335 0.5886] (m/s²)/°C TemperatureScaleFactor: [0.02 0.02 0.02] %/°C

### Generate IMU Data from Stationary Input

Use the `imuSensor`

System object™ to model receiving data from a stationary ideal IMU containing an accelerometer, gyroscope, and magnetometer.

Create an ideal IMU sensor model that contains an accelerometer, gyroscope, and magnetometer.

`IMU = imuSensor('accel-gyro-mag')`

IMU = imuSensor with properties: IMUType: 'accel-gyro-mag' SampleRate: 100 Temperature: 25 MagneticField: [27.5550 -2.4169 -16.0849] Accelerometer: [1x1 accelparams] Gyroscope: [1x1 gyroparams] Magnetometer: [1x1 magparams] RandomStream: 'Global stream'

Define the ground-truth, underlying motion of the IMU you are modeling. The acceleration and angular velocity are defined relative to the local NED coordinate system.

numSamples = 1000; acceleration = zeros(numSamples,3); angularVelocity = zeros(numSamples,3);

Call `IMU`

with the ground-truth acceleration and angular velocity. The object outputs accelerometer readings, gyroscope readings, and magnetometer readings, as modeled by the properties of the `imuSensor`

System object. The accelerometer readings, gyroscope readings, and magnetometer readings are relative to the IMU sensor body coordinate system.

[accelReading,gyroReading,magReading] = IMU(acceleration,angularVelocity);

Plot the accelerometer readings, gyroscope readings, and magnetometer readings.

t = (0:(numSamples-1))/IMU.SampleRate; subplot(3,1,1) plot(t,accelReading) legend('X-axis','Y-axis','Z-axis') title('Accelerometer Readings') ylabel('Acceleration (m/s^2)') subplot(3,1,2) plot(t,gyroReading) legend('X-axis','Y-axis','Z-axis') title('Gyroscope Readings') ylabel('Angular Velocity (rad/s)') subplot(3,1,3) plot(t,magReading) legend('X-axis','Y-axis','Z-axis') title('Magnetometer Readings') xlabel('Time (s)') ylabel('Magnetic Field (uT)')

Orientation is not specified and the ground-truth motion is stationary, so the IMU sensor body coordinate system and the local NED coordinate system overlap for the entire simulation.

Accelerometer readings: The

*z*-axis of the sensor body corresponds to the Down-axis. The 9.8 m/s^2 acceleration along the*z*-axis is due to gravity.Gyroscope readings: The gyroscope readings are zero along each axis, as expected.

Magnetometer readings: Because the sensor body coordinate system is aligned with the local NED coordinate system, the magnetometer readings correspond to the

`MagneticField`

property of`imuSensor`

. The`MagneticField`

property is defined in the local NED coordinate system.

### Model Rotating Six-Axis IMU Data

Use `imuSensor`

to model data obtained from a rotating IMU containing an ideal accelerometer and an ideal magnetometer. Use `kinematicTrajectory`

to define the ground-truth motion. Fuse the `imuSensor`

model output using the `ecompass`

function to determine orientation over time.

Define the ground-truth motion for a platform that rotates 360 degrees in four seconds, and then another 360 degrees in two seconds. Use `kinematicTrajectory`

to output the orientation, acceleration, and angular velocity in the NED coordinate system.

```
fs = 100;
firstLoopNumSamples = fs*4;
secondLoopNumSamples = fs*2;
totalNumSamples = firstLoopNumSamples + secondLoopNumSamples;
traj = kinematicTrajectory('SampleRate',fs);
accBody = zeros(totalNumSamples,3);
angVelBody = zeros(totalNumSamples,3);
angVelBody(1:firstLoopNumSamples,3) = (2*pi)/4;
angVelBody(firstLoopNumSamples+1:end,3) = (2*pi)/2;
[~,orientationNED,~,accNED,angVelNED] = traj(accBody,angVelBody);
```

Create an `imuSensor`

object with an ideal accelerometer and an ideal magnetometer. Call `IMU`

with the ground-truth acceleration, angular velocity, and orientation to output accelerometer readings and magnetometer readings. Plot the results.

IMU = imuSensor('accel-mag','SampleRate',fs); [accelReadings,magReadings] = IMU(accNED,angVelNED,orientationNED); figure(1) t = (0:(totalNumSamples-1))/fs; subplot(2,1,1) plot(t,accelReadings) legend('X-axis','Y-axis','Z-axis') ylabel('Acceleration (m/s^2)') title('Accelerometer Readings') subplot(2,1,2) plot(t,magReadings) legend('X-axis','Y-axis','Z-axis') ylabel('Magnetic Field (\muT)') xlabel('Time (s)') title('Magnetometer Readings')

The accelerometer readings indicate that the platform has no translation. The magnetometer readings indicate that the platform is rotating around the *z*-axis.

Feed the accelerometer and magnetometer readings into the `ecompass`

function to estimate the orientation over time. The `ecompass`

function returns orientation in quaternion format. Convert orientation to Euler angles and plot the results. The orientation plot indicates that the platform rotates about the *z*-axis only.

orientation = ecompass(accelReadings,magReadings); orientationEuler = eulerd(orientation,'ZYX','frame'); figure(2) plot(t,orientationEuler) legend('Z-axis','Y-axis','X-axis') xlabel('Time (s)') ylabel('Rotation (degrees)') title('Orientation')

### Model Rotating Six-Axis IMU Data with Noise

Use `imuSensor`

to model data obtained from a rotating IMU containing a realistic accelerometer and a realistic magnetometer. Use `kinematicTrajectory`

to define the ground-truth motion. Fuse the `imuSensor`

model output using the `ecompass`

function to determine orientation over time.

Define the ground-truth motion for a platform that rotates 360 degrees in four seconds, and then another 360 degrees in two seconds. Use `kinematicTrajectory`

to output the orientation, acceleration, and angular velocity in the NED coordinate system.

```
fs = 100;
firstLoopNumSamples = fs*4;
secondLoopNumSamples = fs*2;
totalNumSamples = firstLoopNumSamples + secondLoopNumSamples;
traj = kinematicTrajectory('SampleRate',fs);
accBody = zeros(totalNumSamples,3);
angVelBody = zeros(totalNumSamples,3);
angVelBody(1:firstLoopNumSamples,3) = (2*pi)/4;
angVelBody(firstLoopNumSamples+1:end,3) = (2*pi)/2;
[~,orientationNED,~,accNED,angVelNED] = traj(accBody,angVelBody);
```

Create an `imuSensor`

object with a realistic accelerometer and a realistic magnetometer. Call `IMU`

with the ground-truth acceleration, angular velocity, and orientation to output accelerometer readings and magnetometer readings. Plot the results.

IMU = imuSensor('accel-mag','SampleRate',fs); IMU.Accelerometer = accelparams( ... 'MeasurementRange',19.62, ... % m/s^2 'Resolution',0.0023936, ... % m/s^2 / LSB 'TemperatureScaleFactor',0.008, ... % % / degree C 'ConstantBias',0.1962, ... % m/s^2 'TemperatureBias',0.0014715, ... % m/s^2 / degree C 'NoiseDensity',0.0012361); % m/s^2 / Hz^(1/2) IMU.Magnetometer = magparams( ... 'MeasurementRange',1200, ... % uT 'Resolution',0.1, ... % uT / LSB 'TemperatureScaleFactor',0.1, ... % % / degree C 'ConstantBias',1, ... % uT 'TemperatureBias',[0.8 0.8 2.4], ... % uT / degree C 'NoiseDensity',[0.6 0.6 0.9]/sqrt(100)); % uT / Hz^(1/2) [accelReadings,magReadings] = IMU(accNED,angVelNED,orientationNED); figure(1) t = (0:(totalNumSamples-1))/fs; subplot(2,1,1) plot(t,accelReadings) legend('X-axis','Y-axis','Z-axis') ylabel('Acceleration (m/s^2)') title('Accelerometer Readings') subplot(2,1,2) plot(t,magReadings) legend('X-axis','Y-axis','Z-axis') ylabel('Magnetic Field (\muT)') xlabel('Time (s)') title('Magnetometer Readings')

The accelerometer readings indicate that the platform has no translation. The magnetometer readings indicate that the platform is rotating around the *z*-axis.

Feed the accelerometer and magnetometer readings into the `ecompass`

function to estimate the orientation over time. The `ecompass`

function returns orientation in quaternion format. Convert orientation to Euler angles and plot the results. The orientation plot indicates that the platform rotates about the *z*-axis only.

orientation = ecompass(accelReadings,magReadings); orientationEuler = eulerd(orientation,'ZYX','frame'); figure(2) plot(t,orientationEuler) legend('Z-axis','Y-axis','X-axis') xlabel('Time (s)') ylabel('Rotation (degrees)') title('Orientation')

`%`

### Model Tilt Using Gyroscope and Accelerometer Readings

Model a tilting IMU that contains an accelerometer and gyroscope using the `imuSensor`

System object™. Use ideal and realistic models to compare the results of orientation tracking using the `imufilter`

System object.

Load a struct describing ground-truth motion and a sample rate. The motion struct describes sequential rotations:

yaw: 120 degrees over two seconds

pitch: 60 degrees over one second

roll: 30 degrees over one-half second

roll: -30 degrees over one-half second

pitch: -60 degrees over one second

yaw: -120 degrees over two seconds

In the last stage, the motion struct combines the 1st, 2nd, and 3rd rotations into a single-axis rotation. The acceleration, angular velocity, and orientation are defined in the local NED coordinate system.

load y120p60r30.mat motion fs accNED = motion.Acceleration; angVelNED = motion.AngularVelocity; orientationNED = motion.Orientation; numSamples = size(motion.Orientation,1); t = (0:(numSamples-1)).'/fs;

Create an ideal IMU sensor object and a default IMU filter object.

IMU = imuSensor('accel-gyro','SampleRate',fs); aFilter = imufilter('SampleRate',fs);

In a loop:

Simulate IMU output by feeding the ground-truth motion to the IMU sensor object.

Filter the IMU output using the default IMU filter object.

orientation = zeros(numSamples,1,'quaternion'); for i = 1:numSamples [accelBody,gyroBody] = IMU(accNED(i,:),angVelNED(i,:),orientationNED(i,:)); orientation(i) = aFilter(accelBody,gyroBody); end release(aFilter)

Plot the orientation over time.

figure(1) plot(t,eulerd(orientation,'ZYX','frame')) xlabel('Time (s)') ylabel('Rotation (degrees)') title('Orientation Estimation -- Ideal IMU Data, Default IMU Filter') legend('Z-axis','Y-axis','X-axis')

Modify properties of your `imuSensor`

to model real-world sensors. Run the loop again and plot the orientation estimate over time.

IMU.Accelerometer = accelparams( ... 'MeasurementRange',19.62, ... 'Resolution',0.00059875, ... 'ConstantBias',0.4905, ... 'AxesMisalignment',2, ... 'NoiseDensity',0.003924, ... 'BiasInstability',0, ... 'TemperatureBias', [0.34335 0.34335 0.5886], ... 'TemperatureScaleFactor',0.02); IMU.Gyroscope = gyroparams( ... 'MeasurementRange',4.3633, ... 'Resolution',0.00013323, ... 'AxesMisalignment',2, ... 'NoiseDensity',8.7266e-05, ... 'TemperatureBias',0.34907, ... 'TemperatureScaleFactor',0.02, ... 'AccelerationBias',0.00017809, ... 'ConstantBias',[0.3491,0.5,0]); orientationDefault = zeros(numSamples,1,'quaternion'); for i = 1:numSamples [accelBody,gyroBody] = IMU(accNED(i,:),angVelNED(i,:),orientationNED(i,:)); orientationDefault(i) = aFilter(accelBody,gyroBody); end release(aFilter) figure(2) plot(t,eulerd(orientationDefault,'ZYX','frame')) xlabel('Time (s)') ylabel('Rotation (degrees)') title('Orientation Estimation -- Realistic IMU Data, Default IMU Filter') legend('Z-axis','Y-axis','X-axis')

The ability of the `imufilter`

to track the ground-truth data is significantly reduced when modeling a realistic IMU. To improve performance, modify properties of your `imufilter`

object. These values were determined empirically. Run the loop again and plot the orientation estimate over time.

aFilter.GyroscopeNoise = 7.6154e-7; aFilter.AccelerometerNoise = 0.0015398; aFilter.GyroscopeDriftNoise = 3.0462e-12; aFilter.LinearAccelerationNoise = 0.00096236; aFilter.InitialProcessNoise = aFilter.InitialProcessNoise*10; orientationNondefault = zeros(numSamples,1,'quaternion'); for i = 1:numSamples [accelBody,gyroBody] = IMU(accNED(i,:),angVelNED(i,:),orientationNED(i,:)); orientationNondefault(i) = aFilter(accelBody,gyroBody); end release(aFilter) figure(3) plot(t,eulerd(orientationNondefault,'ZYX','frame')) xlabel('Time (s)') ylabel('Rotation (degrees)') title('Orientation Estimation -- Realistic IMU Data, Nondefault IMU Filter') legend('Z-axis','Y-axis','X-axis')

To quantify the improved performance of the modified `imufilter`

, plot the quaternion distance between the ground-truth motion and the orientation as returned by the `imufilter`

with default and nondefault properties.

qDistDefault = rad2deg(dist(orientationNED,orientationDefault)); qDistNondefault = rad2deg(dist(orientationNED,orientationNondefault)); figure(4) plot(t,[qDistDefault,qDistNondefault]) title('Quaternion Distance from True Orientation') legend('Realistic IMU Data, Default IMU Filter', ... 'Realistic IMU Data, Nondefault IMU Filter') xlabel('Time (s)') ylabel('Quaternion Distance (degrees)')

## Algorithms

**Tip**

In the following algorithm description, variables in italic fonts are inputs or outputs
of the `imuSensor`

object. Variables in bold fonts are properties of the
`imuSensor`

. Variables in normal fonts are properties of the `accelparams`

, `gyroparams`

, or `magparams`

object.

### Accelerometer

The following algorithm description assumes an NED navigation frame. The accelerometer model
uses the ground-truth orientation and acceleration inputs and the `imuSensor`

and `accelparams`

properties to model accelerometer readings.

**Obtain Total Acceleration**

To obtain the total acceleration (*totalAcc*), the acceleration is
preprocessed by negating and adding the gravity constant vector (*g*=
[0; 0; 9.8] m/s^{2} assuming an NED frame) as:

$$totalAcc=-acceleration+g$$

The `acceleration`

term is negated to obtain zero
total acceleration readings when the accelerometer is in a free fall. The
`acceleration`

term is also known as the specific force.

**Convert to Sensor Frame**

Then the total acceleration is converted from the local navigation frame to the sensor frame using:

$$a=\left(orientation\right){\left(totalAcc\right)}^{T}$$

If the orientation is input in quaternion form, it is converted to a rotation matrix before processing.

**Bulk Model**

The ground-truth acceleration in the sensor frame, *a*, passes through the bulk model, which adds axes misalignment and bias:

$$b={\left(\left[\begin{array}{ccc}1& \frac{{\alpha}_{2}}{100}& \frac{{\alpha}_{3}}{100}\\ \frac{{\alpha}_{1}}{100}& 1& \frac{{\alpha}_{3}}{100}\\ \frac{{\alpha}_{1}}{100}& \frac{{\alpha}_{2}}{100}& 1\end{array}\right]\left({a}^{T}\right)\right)}^{T}+\text{ConstantBias}$$

where ConstantBias is a property of `accelparams`

, and *α*_{1}, *α*_{2}, and *α*_{3} are given by the first, second, and third elements of the AxesMisalignment property of `accelparams`

.

**Bias Instability Drift**

The bias instability drift *β*_{1} is modeled as white
noise biased and then filtered:

$$\begin{array}{l}{g}_{1}{\beta}_{1}(k)=-{g}_{2}{\beta}_{1}(k-1)-{g}_{3}{\beta}_{1}(k-2)\dots -{g}_{n+1}{\beta}_{1}(k-n)\\ +{f}_{1}x(k)+{f}_{2}x(k-1)+\dots +{f}_{m+1}x(k-m)\\ x(k)=(\text{BiasInstability})w(k)\end{array}$$

where *k* is the discrete time step index, BiasInstability is
a property of `accelparams`

,
*w* is white noise that follows a normal distribution of mean 0 and
variance of 1. The discrete time step size is the reciprocal of the SampleRate
property. [*g*_{1},
*g*_{2}, …,
*g*_{n+1}] are the
denominator coefficients specified in the
`BiasInstabilityCoefficients`

property of the `accelparams`

object.
[*f*_{1},
*f*_{2}, …,
*f*_{m+1}] are the
numerator coefficients of the `BiasInstabilityCoefficients`

property.
*n* and *m* are the orders of the denominator and
numerator coefficients, respectively.

**White Noise Drift**

White noise drift is modeled by multiplying elements of the white noise random stream by the standard deviation:

$${\beta}_{2}=\left(w\right)\left(\sqrt{\frac{\text{SampleRate}}{s}}\right)\left(\text{NoiseDensity}\right)$$

where *w* is white noise that follows a normal
distribution of mean 0 and variance of 1, SampleRate is an
`imuSensor`

property, and NoiseDensity is an
`accelparams`

property.
The scale variable *s* = 2 if the `NoiseType`

property of the `accelparams`

object is
double-sided and *s* = 1 if the `NoiseType`

property is single-sided.

**Random Walk Drift**

The random walk drift is modeled by biasing elements of the white noise random stream and then filtering:

$${\beta}_{3}(k)={\beta}_{2}(k-1)+w(k)\left(\frac{\text{RandomWalk}}{\sqrt{\frac{\text{SampleRate}}{s}}}\right)$$

where *k* is the discrete time step index, RandomWalk is a
property of `accelparams`

, SampleRate is a
property of `imuSensor`

, *w* is white noise that follows
a normal distribution of mean 0 and variance of 1. The discrete time step size is the
reciprocal of the SampleRate
property. The scale variable *s* = 2 if the
`NoiseType`

property of the `accelparams`

object is
double-sided and *s* = 1 if the `NoiseType`

property is single-sided.

**Environmental Drift Noise**

The environmental drift noise is modeled by multiplying the temperature difference from a standard with the temperature bias:

$${\Delta}_{e}=(\text{Temperature}-25)(\text{TemperatureBias})$$

where Temperature is a property of `imuSensor`

, and TemperatureBias is a property of `accelparams`

. The constant 25 corresponds to a standard temperature.

**Scale Factor Error Model**

The temperature scale factor error is modeled as:

$$scaleFactorError=1+\left(\frac{\text{Temperature}-25}{100}\right)(\text{TemperatureScaleFactor})$$

where Temperature is a property of `imuSensor`

, and TemperatureScaleFactor is a property of `accelparams`

. The constant 25 corresponds to a standard temperature.

**Quantization Model**

The quantization is modeled by first saturating the continuous signal model:

$$e=\{\begin{array}{c}\begin{array}{c}\text{MeasurementRange}\\ -\text{MeasurementRange}\end{array}\\ d\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}\text{if}\\ \text{if}\\ \text{else}\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}d>\text{MeasurementRange}\\ -d>\text{MeasurementRange}\\ \end{array}$$

and then setting the resolution:

$$accelReadings=(\text{Resolution})\left(\mathrm{round}\left(\frac{e}{\text{Resolution}}\right)\right)$$

where MeasurementRange is a property of `accelparams`

.

### Gyroscope

The following algorithm description assumes an NED navigation frame. The gyroscope model uses
the ground-truth orientation, acceleration, and angular velocity inputs, and the
`imuSensor`

and `gyroparams`

properties to model accelerometer readings.

**Convert to Sensor Frame**

The ground-truth angular velocity is converted from the local frame to the sensor frame using the ground-truth orientation:

$$a=\left(orientation\right){\left(angularVelocity\right)}^{T}$$

If the orientation is input in quaternion form, it is converted to a rotation matrix before processing.

**Bulk Model**

The ground-truth angular velocity in the sensor frame, *a*, passes through the bulk model, which adds axes misalignment and bias:

$$b={\left(\left[\begin{array}{ccc}1& \frac{{\alpha}_{2}}{100}& \frac{{\alpha}_{3}}{100}\\ \frac{{\alpha}_{1}}{100}& 1& \frac{{\alpha}_{3}}{100}\\ \frac{{\alpha}_{1}}{100}& \frac{{\alpha}_{2}}{100}& 1\end{array}\right]\left({a}^{T}\right)\right)}^{T}+\text{ConstantBias}$$

where ConstantBias is a property of `gyroparams`

, and *α*_{1}, *α*_{2}, and *α*_{3} are given by the first, second, and third elements of the AxesMisalignment property of `gyroparams`

.

**Bias Instability Drift**

The bias instability drift *β*_{1} is modeled as white
noise biased and then filtered:

$$\begin{array}{l}{g}_{1}{\beta}_{1}(k)=-{g}_{2}{\beta}_{1}(k-1)-{g}_{3}{\beta}_{1}(k-2)\dots -{g}_{n+1}{\beta}_{1}(k-n)\\ +{f}_{1}x(k)+{f}_{2}x(k-1)+\dots +{f}_{m+1}x(k-m)\\ x(k)=(\text{BiasInstability})w(k)\end{array}$$

where *k* is the discrete time step index, BiasInstability is
a property of `gyroparams`

,
*w* is white noise that follows a normal distribution of mean 0 and
variance of 1. The discrete time step size is the reciprocal of the SampleRate
property. [*g*_{1},
*g*_{2}, …,
*g*_{n+1}] are the
denominator coefficients specified in the
`BiasInstabilityCoefficients`

property of the `gyroparams`

object.
[*f*_{1},
*f*_{2}, …,
*f*_{m+1}] are the
numerator coefficients of the `BiasInstabilityCoefficients`

property.
*n* and *m* are the orders of the denominator and
numerator coefficients, respectively.

**White Noise Drift**

White noise drift is modeled by multiplying elements of the white noise random stream by the standard deviation:

$${\beta}_{2}=\left(w\right)\left(\sqrt{\frac{\text{SampleRate}}{s}}\right)\left(\text{NoiseDensity}\right)$$

where *w* is white noise that follows a normal
distribution of mean 0 and variance of 1, SampleRate is an
`imuSensor`

property, and NoiseDensity is an
`gyroparams`

property.
The scale variable *s* = 2 if the `NoiseType`

property of the `gyroparams`

object is
double-sided and *s* = 1 if the `NoiseType`

property is single-sided.

**Random Walk Drift**

The random walk drift is modeled by biasing elements of the white noise random stream and then filtering:

$${\beta}_{3}(k)={\beta}_{2}(k-1)+w(k)\left(\frac{\text{RandomWalk}}{\sqrt{\frac{\text{SampleRate}}{s}}}\right)$$

where *k* is the discrete time step index, RandomWalk is a
property of `gyroparams`

, SampleRate is a
property of `imuSensor`

, and *w* is white noise that
follows a normal distribution of mean 0 and variance of 1. The discrete time step size
is the reciprocal of the SampleRate
property. The scale variable *s* = 2 if the
`NoiseType`

property of the `gyroparams`

object is
double-sided and *s* = 1 if the `NoiseType`

property is single-sided.

**Environmental Drift Noise**

The environmental drift noise is modeled by multiplying the temperature difference from a standard with the temperature bias:

$${\Delta}_{e}=(\text{Temperature}-25)(\text{TemperatureBias})$$

where Temperature is a property of `imuSensor`

, and TemperatureBias is a property of `gyroparams`

. The constant 25 corresponds to a standard temperature.

**Acceleration Bias Drift**

The acceleration bias drift is modeled by multiplying the acceleration input and acceleration bias:

$${\Delta}_{a}=acceleration*\text{AccelerationBias}$$

where AccelerationBias is
a property of `gyroparams`

.

**Scale Factor Error Model**

The temperature scale factor error is modeled as:

$$scaleFactorError=1+\left(\frac{\text{Temperature}-25}{100}\right)(\text{TemperatureScaleFactor})$$

where Temperature is a property of `imuSensor`

, and TemperatureScaleFactor is a property of `gyroparams`

. The constant 25 corresponds to a standard temperature.

**Quantization Model**

The quantization is modeled by first saturating the continuous signal model:

$$e=\{\begin{array}{c}\begin{array}{c}\text{MeasurementRange}\\ -\text{MeasurementRange}\end{array}\\ d\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}\text{if}\\ \text{if}\\ \text{else}\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}d>\text{MeasurementRange}\\ -d>\text{MeasurementRange}\\ \end{array}$$

and then setting the resolution:

$$gyroReadings=(\text{Resolution})\left(\mathrm{round}\left(\frac{e}{\text{Resolution}}\right)\right)$$

where MeasurementRange is a property of `gyroparams`

.

### Magnetometer

The following algorithm description assumes an NED navigation frame. The magnetometer model
uses the ground-truth orientation and acceleration inputs, and the
`imuSensor`

and `magparams`

`magparams`

(Sensor Fusion and Tracking Toolbox)
properties to model magnetometer readings.

**Convert to Sensor Frame**

The ground-truth acceleration is converted from the local frame to the sensor frame using the ground-truth orientation:

$$a=\left(orientation\right){\left(totalAcc\right)}^{T}$$

If the orientation is input in quaternion form, it is converted to a rotation matrix before processing.

**Bulk Model**

The ground-truth acceleration in the sensor frame, *a*, passes through the bulk model, which adds axes misalignment and bias:

$$b={\left(\left[\begin{array}{ccc}1& \frac{{\alpha}_{2}}{100}& \frac{{\alpha}_{3}}{100}\\ \frac{{\alpha}_{1}}{100}& 1& \frac{{\alpha}_{3}}{100}\\ \frac{{\alpha}_{1}}{100}& \frac{{\alpha}_{2}}{100}& 1\end{array}\right]\left({a}^{T}\right)\right)}^{T}+\text{ConstantBias}$$

where ConstantBias is a property of `magparams`

, and *α*_{1}, *α*_{2}, and *α*_{3} are given by the first, second, and third elements of the AxesMisalignment property of `magparams`

.

**Bias Instability Drift**

The bias instability drift *β*_{1} modeled as white noise
biased and then filtered:

$$\begin{array}{l}{g}_{1}{\beta}_{1}(k)=-{g}_{2}{\beta}_{1}(k-1)-{g}_{3}{\beta}_{1}(k-2)\dots -{g}_{n+1}{\beta}_{1}(k-n)\\ +{f}_{1}x(k)+{f}_{2}x(k-1)+\dots +{f}_{m+1}x(k-m)\\ x(k)=(\text{BiasInstability})w(k)\end{array}$$

where *k* is the discrete time step index, BiasInstability is
a property of `magparams`

,
*w* is white noise that follows a normal distribution of mean 0 and
variance of 1. The discrete time step size is the reciprocal of the SampleRate
property. [*g*_{1},
*g*_{2}, …,
*g*_{n+1}] are the
denominator coefficients specified in the
`BiasInstabilityCoefficients`

property of the `magparams`

object.
[*f*_{1},
*f*_{2}, …,
*f*_{m+1}] are the
numerator coefficients of the `BiasInstabilityCoefficients`

property.
*n* and *m* are the orders of the denominator and
numerator coefficients, respectively.

**White Noise Drift**

White noise drift is modeled by multiplying elements of the white noise random stream by the standard deviation:

$${\beta}_{2}=\left(w\right)\left(\sqrt{\frac{\text{SampleRate}}{s}}\right)\left(\text{NoiseDensity}\right)$$

where *w* is white noise that follows a normal
distribution of mean 0 and variance of 1, SampleRate is an
`imuSensor`

property, and NoiseDensity is an
`magparams`

property.
The scale variable *s* = 2 if the `NoiseType`

property of the `magparams`

object is
double-sided and *s* = 1 if the `NoiseType`

property is single-sided.

**Random Walk Drift**

The random walk drift is modeled by biasing elements of the white noise random stream and then filtering:

$${\beta}_{3}(k)={\beta}_{2}(k-1)+w(k)\left(\frac{\text{RandomWalk}}{\sqrt{\frac{\text{SampleRate}}{s}}}\right)$$

where *k* is the discrete time step index, RandomWalk is a
property of `magparams`

, SampleRate is a
property of `imuSensor`

, *w* is white noise that follows
a normal distribution of mean 0 and variance of 1. The discrete time step size is the
reciprocal of the SampleRate
property. The scale variable *s* = 2 if the
`NoiseType`

property of the `magparams`

object is
double-sided and *s* = 1 if the `NoiseType`

property is single-sided.

**Environmental Drift Noise**

The environmental drift noise is modeled by multiplying the temperature difference from a standard with the temperature bias:

$${\Delta}_{e}=(\text{Temperature}-25)(\text{TemperatureBias})$$

where Temperature is a property of `imuSensor`

, and TemperatureBias is a property of `magparams`

. The constant 25 corresponds to a standard temperature.

**Scale Factor Error Model**

The temperature scale factor error is modeled as:

$$scaleFactorError=1+\left(\frac{\text{Temperature}-25}{100}\right)(\text{TemperatureScaleFactor})$$

where Temperature is a property of `imuSensor`

, and TemperatureScaleFactor is a property of `magparams`

. The constant 25 corresponds to a standard temperature.

**Quantization Model**

The quantization is modeled by first saturating the continuous signal model:

$$e=\{\begin{array}{c}\begin{array}{c}\text{MeasurementRange}\\ -\text{MeasurementRange}\end{array}\\ d\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}\text{if}\\ \text{if}\\ \text{else}\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}d>\text{MeasurementRange}\\ -d>\text{MeasurementRange}\\ \end{array}$$

and then setting the resolution:

$$magReadings=(\text{Resolution})\left(\mathrm{round}\left(\frac{e}{\text{Resolution}}\right)\right)$$

where MeasurementRange is a property of `magparams`

.

## Extended Capabilities

### C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

The object functions, `perturbations`

and
`perturb`

, do not support code generation.

Usage notes and limitations:

See System Objects in MATLAB Code Generation (MATLAB Coder).

## Version History

**Introduced in R2019b**

### R2023b: Specify bias instability noise coefficients and noise type for accelerometer, gyroscope, and magnetometer

You can use the new `BiasInstabilityCoefficients`

property of the
`accelparams`

, `gyroparams`

, and `magparams`

objects to
specify the coefficients of the transfer function used to generate the bias instability
noise. Additionally, you can use the new `NoiseType`

property of these
objects to set the scale factor of the noise coefficients as `1`

or
`2`

.

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