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:

1. Create the imuSensor object and set its properties.

2. 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

IMU = imuSensor

IMU = imuSensor('accel-gyro')

IMU = imuSensor('accel-mag')

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

IMU = imuSensor(___,"ReferenceFrame",RF)

IMU = imuSensor(___,Name=Value)

Description

IMU = imuSensor returns a System object, IMU, that computes an inertial measurement unit reading based on an inertial input signal. IMU has an ideal accelerometer and gyroscope.

IMU = imuSensor('accel-gyro') returns an imuSensor System object with an ideal accelerometer and gyroscope. imuSensor and imuSensor('accel-gyro') are equivalent creation syntaxes.

IMU = imuSensor('accel-mag') returns an imuSensor System object with an ideal accelerometer and magnetometer.

IMU = imuSensor('accel-gyro-mag') returns an imuSensor System object with an ideal accelerometer, gyroscope, and magnetometer.

IMU = imuSensor(___,"ReferenceFrame",RF) returns an imuSensor System object that computes an inertial measurement unit reading relative to the reference frame RF.

IMU = imuSensor(___,Name=Value) sets one or more properties using name-value arguments in addition to any of the previous input arguments.

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/s².

• 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/s².

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 (^oC)

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 ^oC 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

[accelReadings,gyroReadings] = IMU(acc,angVel)

[accelReadings,gyroReadings] = IMU(acc,angVel,orientation)

[accelReadings,magReadings] = IMU(acc,angVel)

[accelReadings,magReadings] = IMU(acc,angVel,orientation)

[accelReadings,gyroReadings,magReadings] = IMU(acc,angVel)

[accelReadings,gyroReadings,magReadings] = IMU(acc,angVel,orientation)

Description

[accelReadings,gyroReadings] = IMU(acc,angVel) generates accelerometer and gyroscope readings from the acceleration and angular velocity inputs.

This syntax is only valid if IMUType is set to 'accel-gyro' or 'accel-gyro-mag'.

[accelReadings,gyroReadings] = IMU(acc,angVel,orientation) generates accelerometer and gyroscope readings from the acceleration, angular velocity, and orientation inputs.

This syntax is only valid if IMUType is set to 'accel-gyro' or 'accel-gyro-mag'.

[accelReadings,magReadings] = IMU(acc,angVel) generates accelerometer and magnetometer readings from the acceleration and angular velocity inputs.

This syntax is only valid if IMUType is set to 'accel-mag'.

[accelReadings,magReadings] = IMU(acc,angVel,orientation) generates accelerometer and magnetometer readings from the acceleration, angular velocity, and orientation inputs.

This syntax is only valid if IMUType is set to 'accel-mag'.

[accelReadings,gyroReadings,magReadings] = IMU(acc,angVel) generates accelerometer, gyroscope, and magnetometer readings from the acceleration and angular velocity inputs.

This syntax is only valid if IMUType is set to 'accel-gyro-mag'.

[accelReadings,gyroReadings,magReadings] = IMU(acc,angVel,orientation) generates accelerometer, gyroscope, and magnetometer readings from the acceleration, angular velocity, and orientation inputs.

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/s²)

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/s²)

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

Common to All System Objects

step	Run System object algorithm
release	Release resources and allow changes to System object property values and input characteristics
reset	Reset internal states of System 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:

1. yaw: 120 degrees over two seconds

2. pitch: 60 degrees over one second

3. roll: 30 degrees over one-half second

4. roll: -30 degrees over one-half second

5. pitch: -60 degrees over one second

6. 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:

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

2. 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/s² 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 α₁, α₂, and α₃ are given by the first, second, and third elements of the AxesMisalignment property of accelparams.

Bias Instability Drift

The bias instability drift β₁ 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. [g₁, g₂, …, g_n+1] are the denominator coefficients specified in the BiasInstabilityCoefficients property of the accelparams object. [f₁, f₂, …, 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 α₁, α₂, and α₃ are given by the first, second, and third elements of the AxesMisalignment property of gyroparams.

Bias Instability Drift

The bias instability drift β₁ 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. [g₁, g₂, …, g_n+1] are the denominator coefficients specified in the BiasInstabilityCoefficients property of the gyroparams object. [f₁, f₂, …, 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 magparamsmagparams (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 α₁, α₂, and α₃ are given by the first, second, and third elements of the AxesMisalignment property of magparams.

Bias Instability Drift

The bias instability drift β₁ 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. [g₁, g₂, …, g_n+1] are the denominator coefficients specified in the BiasInstabilityCoefficients property of the magparams object. [f₁, f₂, …, 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.