ahrsfilter

Orientation from accelerometer, gyroscope, and magnetometer readings

Description

The ahrsfilter System object™ fuses accelerometer, magnetometer, and gyroscope sensor data to estimate device orientation.

To estimate device orientation:

1. Create the ahrsfilter 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

FUSE = ahrsfilter

FUSE = ahrsfilter('ReferenceFrame',RF)

FUSE = ahrsfilter(___,Name=Value)

Description

FUSE = ahrsfilter returns an indirect Kalman filter System object, FUSE, for sensor fusion of accelerometer, gyroscope, and magnetometer data to estimate device orientation and angular velocity. The filter uses a 12-element state vector to track the estimation error for the orientation, the gyroscope bias, the linear acceleration, and the magnetic disturbance.

FUSE = ahrsfilter('ReferenceFrame',RF) returns an ahrsfilter System object that fuses accelerometer, gyroscope, and magnetometer data to estimate device orientation relative to the reference frame RF.

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

Input Arguments

RF — Local navigation coordinate system

'NED' (default) | 'ENU'

Local navigation coordinate system, specified as either 'NED' (North-East-Down) or 'ENU' (East-North-Up).

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.

SampleRate — Input sample rate of sensor data (Hz)

100 (default) | positive scalar

Input sample rate of the sensor data in Hz, specified as a positive scalar.

Tunable: No

Data Types: single | double

DecimationFactor — Decimation factor

1 (default) | positive integer

Decimation factor by which to reduce the input sensor data rate as part of the fusion algorithm, specified as a positive integer.

The number of rows of the inputs –– accelReadings, gyroReadings, and magReadings –– must be a multiple of the decimation factor.

Data Types: single | double

AccelerometerNoise — Variance of accelerometer signal noise ((m/s²)²)

0.00019247 (default) | positive real scalar

Variance of accelerometer signal noise in (m/s²)², specified as a positive real scalar.

Tunable: Yes

Data Types: single | double

MagnetometerNoise — Variance of magnetometer signal noise (μT²)

0.1 (default) | positive real scalar

Variance of magnetometer signal noise in μT², specified as a positive real scalar.

Tunable: Yes

Data Types: single | double

GyroscopeNoise — Variance of gyroscope signal noise ((rad/s)²)

9.1385e-5 (default) | positive real scalar

Variance of gyroscope signal noise in (rad/s)², specified as a positive real scalar.

Tunable: Yes

Data Types: single | double

GyroscopeDriftNoise — Variance of gyroscope offset drift ((rad/s)²)

3.0462e-13 (default) | positive real scalar

Variance of gyroscope offset drift in (rad/s)², specified as a positive real scalar.

Tunable: Yes

Data Types: single | double

LinearAccelerationNoise — Variance of linear acceleration noise (m/s²)²

0.0096236 (default) | positive real scalar

Variance of linear acceleration noise in (m/s²)², specified as a positive real scalar. Linear acceleration is modeled as a lowpass-filtered white noise process.

Tunable: Yes

Data Types: single | double

LinearAccelerationDecayFactor — Decay factor for linear acceleration drift

0.5 (default) | scalar in the range [0,1)

Decay factor for linear acceleration drift, specified as a scalar in the range [0,1). If linear acceleration is changing quickly, set LinearAcclerationDecayFactor to a lower value. If linear acceleration changes slowly, set LinearAcclerationDecayFactor to a higher value. Linear acceleration drift is modeled as a lowpass-filtered white noise process.

Tunable: Yes

Data Types: single | double

MagneticDisturbanceNoise — Variance of magnetic disturbance noise (μT²)

0.5 (default) | real finite positive scalar

Variance of magnetic disturbance noise in μT², specified as a real finite positive scalar.

Tunable: Yes

Data Types: single | double

MagneticDisturbanceDecayFactor — Decay factor for magnetic disturbance

0.5 (default) | positive scalar in the range [0,1]

Decay factor for magnetic disturbance, specified as a positive scalar in the range [0,1]. Magnetic disturbance is modeled as a first order Markov process.

Tunable: Yes

Data Types: single | double

InitialProcessNoise — Covariance matrix for process noise

12-by-12 matrix

Covariance matrix for process noise, specified as a 12-by-12 matrix. The default is:

  Columns 1 through 6

   0.000006092348396                   0                   0                   0                   0                   0
                   0   0.000006092348396                   0                   0                   0                   0
                   0                   0   0.000006092348396                   0                   0                   0
                   0                   0                   0   0.000076154354947                   0                   0
                   0                   0                   0                   0   0.000076154354947                   0
                   0                   0                   0                   0                   0   0.000076154354947
                   0                   0                   0                   0                   0                   0
                   0                   0                   0                   0                   0                   0
                   0                   0                   0                   0                   0                   0
                   0                   0                   0                   0                   0                   0
                   0                   0                   0                   0                   0                   0
                   0                   0                   0                   0                   0                   0

  Columns 7 through 12

                   0                   0                   0                   0                   0                   0
                   0                   0                   0                   0                   0                   0
                   0                   0                   0                   0                   0                   0
                   0                   0                   0                   0                   0                   0
                   0                   0                   0                   0                   0                   0
                   0                   0                   0                   0                   0                   0
   0.009623610000000                   0                   0                   0                   0                   0
                   0   0.009623610000000                   0                   0                   0                   0
                   0                   0   0.009623610000000                   0                   0                   0
                   0                   0                   0   0.600000000000000                   0                   0
                   0                   0                   0                   0   0.600000000000000                   0
                   0                   0                   0                   0                   0   0.600000000000000

The initial process covariance matrix accounts for the error in the process model.

Data Types: single | double

ExpectedMagneticFieldStrength — Expected estimate of magnetic field strength (μT)

50 (default) | real positive scalar

Expected estimate of magnetic field strength in μT, specified as a real positive scalar. The expected magnetic field strength is an estimate of the magnetic field strength of the Earth at the current location.

Tunable: Yes

Data Types: single | double

OrientationFormat — Output orientation format

'quaternion' (default) | 'Rotation matrix'

Output orientation format, specified as 'quaternion' or 'Rotation matrix'. The size of the output depends on the input size, N, and the output orientation format:

• 'quaternion' –– Output is an N-by-1 quaternion.

• 'Rotation matrix' –– Output is a 3-by-3-by-N rotation matrix.

Data Types: char | string

Usage

Syntax

[orientation,angularVelocity,interData] = FUSE(accelReadings,gyroReadings,magReadings)

Description

[orientation,angularVelocity,interData] = FUSE(accelReadings,gyroReadings,magReadings) fuses accelerometer, gyroscope, and magnetometer data to compute orientation and angular velocity measurements. The algorithm assumes that the device is stationary before the first call.

Input Arguments

accelReadings — Accelerometer readings in sensor body coordinate system (m/s²)

N-by-3 matrix

Accelerometer readings in the sensor body coordinate system in m/s², specified as an N-by-3 matrix. N is the number of samples, and the three columns of accelReadings represent the [x y z] measurements. Accelerometer readings are assumed to correspond to the sample rate specified by the SampleRate property.

Data Types: single | double

gyroReadings — Gyroscope readings in sensor body coordinate system (rad/s)

N-by-3 matrix

Gyroscope readings in the sensor body coordinate system in rad/s, specified as an N-by-3 matrix. N is the number of samples, and the three columns of gyroReadings represent the [x y z] measurements. Gyroscope readings are assumed to correspond to the sample rate specified by the SampleRate property.

Data Types: single | double

magReadings — Magnetometer readings in sensor body coordinate system (µT)

N-by-3 matrix

Magnetometer readings in the sensor body coordinate system in µT, specified as an N-by-3 matrix. N is the number of samples, and the three columns of magReadings represent the [x y z] measurements. Magnetometer readings are assumed to correspond to the sample rate specified by the SampleRate property.

Data Types: single | double

Output Arguments

orientation — Orientation that rotates quantities from local navigation coordinate system to sensor body coordinate system

M-by-1 array of quaternions (default) | 3-by-3-by-M array

Orientation that can rotate quantities from the local navigation coordinate system to a body coordinate system, returned as quaternions or an array. The size and type of orientation depends on whether the OrientationFormat property is set to 'quaternion' or 'Rotation matrix':

• 'quaternion' –– the output is an M-by-1 vector of quaternions, with the same underlying data type as the inputs

• 'Rotation matrix' –– the output is a 3-by-3-by-M array of rotation matrices the same data type as the inputs

The number of input samples, N, and the DecimationFactor property determine M.

You can use orientation in a rotateframe function to rotate quantities from a local navigation system to a sensor body coordinate system.

Data Types: quaternion | single | double

angularVelocity — Angular velocity in sensor body coordinate system (rad/s)

M-by-3 array (default)

Angular velocity with gyroscope bias removed in the sensor body coordinate system in rad/s, returned as an M-by-3 array. The number of input samples, N, and the DecimationFactor property determine M.

Data Types: single | double

interData — Intermediate data

structure

Since R2024a

Intermediate data, returned as a structure containing residual and residual covariance.

Data Types: struct

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 ahrsfilter

tune	Tune ahrsfilter parameters to reduce estimation error
residual	Compute residual and residual covariance for ahrsfilter

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

Estimate Orientation Using ahrsfilter

Load the rpy_9axis file, which contains recorded accelerometer, gyroscope, and magnetometer sensor data from a device oscillating in pitch (around y-axis), then yaw (around z-axis), and then roll (around x-axis). The file also contains the sample rate of the recording.

load 'rpy_9axis' sensorData Fs
accelerometerReadings = sensorData.Acceleration;
gyroscopeReadings = sensorData.AngularVelocity;
magnetometerReadings = sensorData.MagneticField;

Create an ahrsfilter System object™ with SampleRate set to the sample rate of the sensor data. Specify a decimation factor of two to reduce the computational cost of the algorithm.

decim = 2;
fuse = ahrsfilter('SampleRate',Fs,'DecimationFactor',decim);

Pass the accelerometer readings, gyroscope readings, and magnetometer readings to the ahrsfilter object, fuse, to output an estimate of the sensor body orientation over time. By default, the orientation is output as a vector of quaternions.

q = fuse(accelerometerReadings,gyroscopeReadings,magnetometerReadings);

Orientation is defined by angular displacement required to rotate a parent coordinate system to a child coordinate system. Plot the orientation in Euler angles in degrees over time.

ahrsfilter correctly estimates the change in orientation over time, including the south-facing initial orientation.

time = (0:decim:size(accelerometerReadings,1)-1)/Fs;

plot(time,eulerd(q,'ZYX','frame'))
title('Orientation Estimate')
legend('z-axis', 'y-axis', 'x-axis')
ylabel('Rotation (degrees)')

Simulate Magnetic Jamming on ahrsFilter

This example shows how performance of the ahrsfilter System object™ is affected by magnetic jamming.

Load StationaryIMUReadings, which contains accelerometer, magnetometer, and gyroscope readings from a stationary IMU.

load 'StationaryIMUReadings.mat' accelReadings magReadings gyroReadings SampleRate

numSamples = size(accelReadings,1);

The ahrsfilter uses magnetic field strength to stabilize its orientation against the assumed constant magnetic field of the Earth. However, there are many natural and man-made objects which output magnetic fields and can confuse the algorithm. To account for the presence of transient magnetic fields, you can set the MagneticDisturbanceNoise property on the ahrsfilter object.

Create an ahrsfilter object with the decimation factor set to 2 and note the default expected magnetic field strength.

decim = 2;
FUSE = ahrsfilter('SampleRate',SampleRate,'DecimationFactor',decim);

Fuse the IMU readings using the attitude and heading reference system (AHRS) filter, and then visualize the orientation of the sensor body over time. The orientation fluctuates at the beginning and stabilizes after approximately 60 seconds.

orientation = FUSE(accelReadings,gyroReadings,magReadings);

orientationEulerAngles = eulerd(orientation,'ZYX','frame');
time = (0:decim:(numSamples-1))'/SampleRate;

figure(1)
plot(time,orientationEulerAngles(:,1), ...
     time,orientationEulerAngles(:,2), ...
     time,orientationEulerAngles(:,3))
xlabel('Time (s)')
ylabel('Rotation (degrees)')
legend('z-axis','y-axis','x-axis')
title('Filtered IMU Data')

Mimic magnetic jamming by adding a transient, strong magnetic field to the magnetic field recorded in the magReadings. Visualize the magnetic field jamming.

jamStrength = [10,5,2];
startStop = (50*SampleRate):(150*SampleRate);
jam = zeros(size(magReadings));
jam(startStop,:) = jamStrength.*ones(numel(startStop),3);

magReadings = magReadings + jam;

figure(2)
plot(time,magReadings(1:decim:end,:))
xlabel('Time (s)')
ylabel('Magnetic Field Strength (\mu T)')
title('Simulated Magnetic Field with Jamming')
legend('z-axis','y-axis','x-axis')

Run the simulation again using the magReadings with magnetic jamming. Plot the results and note the decreased performance in orientation estimation.

reset(FUSE)
orientation = FUSE(accelReadings,gyroReadings,magReadings);

orientationEulerAngles = eulerd(orientation,'ZYX','frame');

figure(3)
plot(time,orientationEulerAngles(:,1), ...
     time,orientationEulerAngles(:,2), ...
     time,orientationEulerAngles(:,3))
xlabel('Time (s)')
ylabel('Rotation (degrees)')
legend('z-axis','y-axis','x-axis')
title('Filtered IMU Data with Magnetic Disturbance and Default Properties')

The magnetic jamming was misinterpreted by the AHRS filter, and the sensor body orientation was incorrectly estimated. You can compensate for jamming by increasing the MagneticDisturbanceNoise property. Increasing the MagneticDisturbanceNoise property increases the assumed noise range for magnetic disturbance, and the entire magnetometer signal is weighted less in the underlying fusion algorithm of ahrsfilter.

Set the MagneticDisturbanceNoise to 200 and run the simulation again.

The orientation estimation output from ahrsfilter is more accurate and less affected by the magnetic transient. However, because the magnetometer signal is weighted less in the underlying fusion algorithm, the algorithm may take more time to restabilize.

reset(FUSE)
FUSE.MagneticDisturbanceNoise = 20;

orientation = FUSE(accelReadings,gyroReadings,magReadings);

orientationEulerAngles = eulerd(orientation,'ZYX','frame');

figure(4)
plot(time,orientationEulerAngles(:,1), ...
     time,orientationEulerAngles(:,2), ...
     time,orientationEulerAngles(:,3))
xlabel('Time (s)')
ylabel('Rotation (degrees)')
legend('z-axis','y-axis','x-axis')
title('Filtered IMU Data with Magnetic Disturbance and Modified Properties')

Track Shaking 9-Axis IMU

This example uses the ahrsfilter System object™ to fuse 9-axis IMU data from a sensor body that is shaken. Plot the quaternion distance between the object and its final resting position to visualize performance and how quickly the filter converges to the correct resting position. Then tune parameters of the ahrsfilter so that the filter converges more quickly to the ground-truth resting position.

Load IMUReadingsShaken into your current workspace. This data was recorded from an IMU that was shaken then laid in a resting position. Visualize the acceleration, magnetic field, and angular velocity as recorded by the sensors.

load 'IMUReadingsShaken' accelReadings gyroReadings magReadings SampleRate
numSamples = size(accelReadings,1);
time = (0:(numSamples-1))'/SampleRate;

figure(1)
subplot(3,1,1)
plot(time,accelReadings)
title('Accelerometer Reading')
ylabel('Acceleration (m/s^2)')

subplot(3,1,2)
plot(time,magReadings)
title('Magnetometer Reading')
ylabel('Magnetic Field (\muT)')

subplot(3,1,3)
plot(time,gyroReadings)
title('Gyroscope Reading')
ylabel('Angular Velocity (rad/s)')
xlabel('Time (s)')

Create an ahrsfilter and then fuse the IMU data to determine orientation. The orientation is returned as a vector of quaternions; convert the quaternions to Euler angles in degrees. Visualize the orientation of the sensor body over time by plotting the Euler angles required, at each time step, to rotate the global coordinate system to the sensor body coordinate system.

fuse = ahrsfilter('SampleRate',SampleRate);
orientation = fuse(accelReadings,gyroReadings,magReadings);

orientationEulerAngles = eulerd(orientation,'ZYX','frame');

figure(2)
plot(time,orientationEulerAngles(:,1), ...
     time,orientationEulerAngles(:,2), ...
     time,orientationEulerAngles(:,3))
xlabel('Time (s)')
ylabel('Rotation (degrees)')
title('Orientation over Time')
legend('Rotation around z-axis', ...
       'Rotation around y-axis', ...
       'Rotation around x-axis')

In the IMU recording, the shaking stops after approximately six seconds. Determine the resting orientation so that you can characterize how fast the ahrsfilter converges.

To determine the resting orientation, calculate the averages of the magnetic field and acceleration for the final four seconds and then use the ecompass function to fuse the data.

Visualize the quaternion distance from the resting position over time.

restingOrientation = ecompass(mean(accelReadings(6*SampleRate:end,:)), ...
                              mean(magReadings(6*SampleRate:end,:)));

figure(3)
plot(time,rad2deg(dist(restingOrientation,orientation)))
hold on
xlabel('Time (s)')
ylabel('Quaternion Distance (degrees)')

Modify the default ahrsfilter properties so that the filter converges to gravity more quickly. Increase the GyroscopeDriftNoise to 1e-2 and decrease the LinearAccelerationNoise to 1e-4. This instructs the ahrsfilter algorithm to weigh gyroscope data less and accelerometer data more. Because the accelerometer data provides the stabilizing and consistent gravity vector, the resulting orientation converges more quickly.

Reset the filter, fuse the data, and plot the results.

fuse.LinearAccelerationNoise = 1e-4;
fuse.GyroscopeDriftNoise     = 1e-2;
reset(fuse)

orientation = fuse(accelReadings,gyroReadings,magReadings);

figure(3)
plot(time,rad2deg(dist(restingOrientation,orientation)))
legend('Default AHRS Filter','Tuned AHRS Filter')

Algorithms

Note: The following algorithm only applies to an NED reference frame.

The ahrsfilter uses the nine-axis Kalman filter structure described in [1]. The algorithm attempts to track the errors in orientation, gyroscope offset, linear acceleration, and magnetic disturbance to output the final orientation and angular velocity. Instead of tracking the orientation directly, the indirect Kalman filter models the error process, x, with a recursive update:

$$x_k=\left[\begin{array}{c}{\theta }_k\\ b_k\\ a_k\\ d_k\end{array}\right]=F_k\left[\begin{array}{c}{\theta }_{k-1}\\ b_{k-1}\\ a_{k-1}\\ d_{k-1}\end{array}\right]+w_k$$

where x_k is a 12-by-1 vector consisting of:

• θ_k –– 3-by-1 orientation error vector, in degrees, at time k

• b_k –– 3-by-1 gyroscope zero angular rate bias vector, in deg/s, at time k

• a_k –– 3-by-1 acceleration error vector measured in the sensor frame, in g, at time k

• d_k –– 3-by-1 magnetic disturbance error vector measured in the sensor frame, in µT, at time k

and where w_k is a 12-by-1 additive noise vector, and F_k is the state transition model.

Because x_k is defined as the error process, the a priori estimate is always zero, and therefore the state transition model, F_k, is zero. This insight results in the following reduction of the standard Kalman equations:

Standard Kalman equations:

$$\begin{array}{l}{x}_k^{-}=F_k{x}_{k-1}^{+}\\ P_k^{-}=F_k{P}_{k-1}^{+}{F}_k^T+Q_k\\ \\ y_k=z_k-H_k{x}_k^{-}\\ S_k=R_k+H_k{P}_k^{-}{H}_k{}^T\\ K_k=P_k^{-}{H}_k^T{\left(S_k\right)}^{-1}\\ x_k^{+}=x_k^{-}+K_k{y}_k\\ P_k^{+}=P_k{}^{-}-K_k{H}_k{P}_k^{-}\end{array}$$

Kalman equations used in this algorithm:

$$\begin{array}{l}{x}_k^{-}=0\\ P_k^{-}=Q_k\\ \\ y_k=z_k\\ S_k=R_k+H_k{P}_k^{-}{H}_k{}^T\\ K_k=P_k^{-}{H}_k^T{\left(S_k\right)}^{-1}\\ x_k^{+}=K_k{y}_k\\ P_k^{+}=P_k{}^{-}-K_k{H}_k{P}_k^{-}\end{array}$$

where:

• x_k⁻ –– predicted (a priori) state estimate; the error process

• P_k⁻ –– predicted (a priori) estimate covariance

• y_k –– innovation

• S_k –– innovation covariance

• K_k –– Kalman gain

• x_k⁺ –– updated (a posteriori) state estimate

• P_k⁺ –– updated (a posteriori) estimate covariance

k represents the iteration, the superscript ⁺ represents an a posteriori estimate, and the superscript ⁻ represents an a priori estimate.

The graphic and following steps describe a single frame-based iteration through the algorithm.

Before the first iteration, the accelReadings, gyroReadings, and magReadings inputs are chunked into DecimationFactor-by-3 frames. For each chunk, the algorithm uses the most current accelerometer and magnetometer readings corresponding to the chunk of gyroscope readings.

Detailed Overview

Walk through the algorithm for an explanation of each stage of the detailed overview.

Model

The algorithm models acceleration and angular change as linear processes.

Predict Orientation

The orientation for the current frame is predicted by first estimating the angular change from the previous frame:

$$\Delta {\phi }_{N\times 3}=\frac{\left(gyroReading{s}_{N\times 3}-gyroOffse{t}_{1\times 3}\right)}{fs}$$

where N is the decimation factor specified by the DecimationFactor property and fs is the sample rate specified by the SampleRate property.

The angular change is converted into quaternions using the rotvec quaternion construction syntax:

$$\Delta Q_{N\times 1}=\mathrm{quaternion}(\Delta {\phi }_{N\times 3},\text{'}rotvec\text{'})$$

The previous orientation estimate is updated by rotating it by ΔQ:

$$q_{1\times 1}^{-}=\left(q_{1\times 1}^{+}\right)\left({\displaystyle \prod _{n=1}^N\Delta Q_n}\right)$$

During the first iteration, the orientation estimate, q⁻, is initialized by ecompass.

Estimate Gravity from Orientation

The gravity vector is interpreted as the third column of the quaternion, q⁻, in rotation matrix form:

$$g_{1\times 3}={\left(rPrior(:,3)\right)}^T$$

See [1] for an explanation of why the third column of rPrior can be interpreted as the gravity vector.

Estimate Gravity from Acceleration

A second gravity vector estimation is made by subtracting the decayed linear acceleration estimate of the previous iteration from the accelerometer readings:

$$gAcce{l}_{1\times 3}=accelReading{s}_{1\times 3}-linAccelprio{r}_{1\times 3}$$

Estimate Earth's Magnetic Vector

Earth's magnetic vector is estimated by rotating the magnetic vector estimate from the previous iteration by the a priori orientation estimate, in rotation matrix form:

$$mGyr{o}_{1\times 3}={\left(\left(rPrior\right)\left(m^T\right)\right)}^T$$

Error Model

The error model combines two differences:

• The difference between the gravity estimate from the accelerometer readings and the gravity estimate from the gyroscope readings: $$z_g=g-gAccel$$

• The difference between the magnetic vector estimate from the gyroscope readings and the magnetic vector estimate from the magnetometer:$$z_m=mGyro-magReadings$$

Magnetometer Correct

The magnetometer correct estimates the error in the magnetic vector estimate and detects magnetic jamming.

Magnetometer Disturbance Error

The magnetic disturbance error is calculated by matrix multiplication of the Kalman gain associated with the magnetic vector with the error signal:

$$mErro{r}_{3\times 1}={\left(\left(K{(\text{10:12},:)}_{3\times 6}\right){\left(z_{1\times 6}\right)}^T\right)}^T$$

The Kalman gain, K, is the Kalman gain calculated in the current iteration.

Magnetic Jamming Detection

Magnetic jamming is determined by verifying that the power of the detected magnetic disturbance is less than or equal to four times the power of the expected magnetic field strength:

$$tf=\{\begin{array}{c}\text{true}\\ \text{false}\end{array}\begin{array}{c}\text{if}\\ \text{else}\end{array}\begin{array}{c}{{\displaystyle \sum \left|mError\right|}}^2>\left(4\right){\left(\text{ExpectedMagneticFieldStrength}\right)}^2\\ \end{array}$$

ExpectedMagneticFieldStrength is a property of ahrsfilter.

Kalman Equations

The Kalman equations use the gravity estimate derived from the gyroscope readings, g, the magnetic vector estimate derived from the gyroscope readings, mGyro, and the observation of the error process, z, to update the Kalman gain and intermediary covariance matrices. The Kalman gain is applied to the error signal, z, to output an a posteriori error estimate, x⁺.

Observation Model

The observation model maps the 1-by-3 observed states, g and mGyro, into the 6-by-12 true state, H.

The observation model is constructed as:

$$H_{3\times 9}=\left[\begin{array}{cccccccccccc}0& g_z& -g_y& 0& -\kappa g_z& \kappa g_y& 1& 0& 0& 0& 0& 0\\ -g_z& 0& g_x& \kappa g_z& 0& -\kappa g_x& 0& 1& 0& 0& 0& 0\\ g_y& -g_x& 0& -\kappa g_y& \kappa g_x& 0& 0& 0& 1& 0& 0& 0\\ 0& m_z& -m_y& 0& -\kappa m_z& -\kappa m_y& 0& 0& 0& -1& 0& 0\\ -m_z& 0& m_x& \kappa m_z& 0& -\kappa m_x& 0& 0& 0& 0& -1& 0\\ m_y& -m_x& 0& -\kappa m_y& \kappa m_x& 0& 0& 0& 0& 0& 0& -1\end{array}\right]$$

where g_x, g_y, and g_z are the x-, y-, and z-elements of the gravity vector estimated from the a priori orientation, respectively. m_x, m_y, and m_z are the x-, y-, and z-elements of the magnetic vector estimated from the a priori orientation, respectively. κ is a constant determined by the SampleRate and DecimationFactor properties: κ = DecimationFactor/SampleRate.

See sections 7.3 and 7.4 of [1] for a derivation of the observation model.

Innovation Covariance

The innovation covariance is a 6-by-6 matrix used to track the variability in the measurements. The innovation covariance matrix is calculated as:

$$S_{6x6}=R_{6x6}+\left(H_{6x12}\right)\left(P_{{}_{12x12}}^{-}\right){\left(H_{6x12}\right)}^T$$

where

• H is the observation model matrix

• P⁻ is the predicted (a priori) estimate of the covariance of the observation model calculated in the previous iteration

• R is the covariance of the observation model noise, calculated as:

  $$R_{6\times 6}=\left[\begin{array}{cccccc}acce{l}_{\text{noise}}& 0& 0& 0& 0& 0\\ 0& acce{l}_{\text{noise}}& 0& 0& 0& 0\\ 0& 0& acce{l}_{\text{noise}}& 0& 0& 0\\ 0& 0& 0& ma{g}_{\text{noise}}& 0& 0\\ 0& 0& 0& 0& ma{g}_{\text{noise}}& 0\\ 0& 0& 0& 0& 0& ma{g}_{\text{noise}}\end{array}\right]$$

  where

  $$acce{l}_{\text{noise}}=\text{AccelerometerNoise}\text{+}\text{LinearAccelerationNoise}\text{+}{\kappa }^2\left(\text{GyroscopeDriftNoise}\text{+}\text{GyroscopeNoise}\right)$$

  and

  $$ma{g}_{\text{noise}}=\text{MagnetometerNoise}\text{+}\text{MagneticDisturbanceNoise}\text{+}{\kappa }^2\left(\text{GyroscopeDriftNoise}\text{+}\text{GyroscopeNoise}\right)$$

  The following properties define the observation model noise variance:

  • κ –– DecimationFactor/SampleRate

  • AccelerometerNoise

  • LinearAccelerationNoise

  • GyroscopeDriftNoise

  • GyroscopeNoise

  • MagneticDisturbanceNoise

  • MagnetometerNoise

Update Error Estimate Covariance

The error estimate covariance is a 12-by-12 matrix used to track the variability in the state.

The error estimate covariance matrix is updated as:

$$P_{{}_{12\times 12}}^{+}=P_{{}_{12\times 12}}^{-}-\left(K_{12\times 6}\right)\left(H_{6\times 12}\right)\left(P_{{}_{12\times 12}}^{-}\right)$$

where K is the Kalman gain, H is the measurement matrix, and P⁻ is the error estimate covariance calculated during the previous iteration.

Predict Error Estimate Covariance

The error estimate covariance is a 12-by-12 matrix used to track the variability in the state. The a priori error estimate covariance, P⁻, is set to the process noise covariance, Q, determined during the previous iteration. Q is calculated as a function of the a posteriori error estimate covariance, P⁺. When calculating Q, it is assumed that the cross-correlation terms are negligible compared to the autocorrelation terms, and are set to zero:

$$Q=\left[\begin{array}{cccccccccccc}{P}^{+}(1)+{\kappa }^2\left(P^{+}(40)+\beta +\eta \right)& 0& 0& -\kappa \left(P^{+}(40)+\beta \right)& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& P^{+}(14)+{\kappa }^2\left(P^{+}(53)+\beta +\eta \right)& 0& 0& -\kappa \left(P^{+}(53)+\beta \right)& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& P^{+}(27)+{\kappa }^2\left(P^{+}(66)+\beta +\eta \right)& 0& 0& -\kappa \left(P^{+}(66)+\beta \right)& 0& 0& 0& 0& 0& 0\\ -\kappa \left(P^{+}(40)+\beta \right)& 0& 0& P^{+}(40)+\beta & 0& 0& 0& 0& 0& 0& 0& 0\\ 0& -\kappa \left(P^{+}(53)+\beta \right)& 0& 0& P^{+}(53)+\beta & 0& 0& 0& 0& 0& 0& 0\\ 0& 0& -\kappa \left(P^{+}(66)+\beta \right)& 0& 0& P^{+}(66)+\beta & 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& {\nu }^2{P}^{+}(79)+\xi & 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& {\nu }^2{P}^{+}(92)+\xi & 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& {\nu }^2{P}^{+}(105)+\xi & 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& {\sigma }^2{P}^{+}(118)+\gamma & 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& {\sigma }^2{P}^{+}(131)+\gamma & 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& {\sigma }^2{P}^{+}(144)+\gamma \end{array}\right]$$

where

• P⁺ –– is the updated (a posteriori) error estimate covariance

• κ –– DecimationFactor/SampleRate

• β –– GyroscopeDriftNoise

• η –– GyroscopeNoise

• ν –– LinearAccelerationDecayFactor

• ξ –– LinearAccelerationNoise

• σ –– MagneticDisturbanceDecayFactor

• γ –– MagneticDisturbanceNoise

See section 10.1 of [1] for a derivation of the terms of the process error matrix.

Kalman Gain

The Kalman gain matrix is a 12-by-6 matrix used to weight the innovation. In this algorithm, the innovation is interpreted as the error process, z.

The Kalman gain matrix is constructed as:

$$K_{12\times 6}=\left(P_{{}_{12\times 12}}^{-}\right){\left(H_{6\times 12}\right)}^T{\left({\left(S_{6\times 6}\right)}^T\right)}^{-1}$$

where

• P⁻ –– predicted error covariance

• H –– observation model

• S –– innovation covariance

Update a Posteriori Error

The a posterior error estimate is determined by combining the Kalman gain matrix with the error in the gravity vector and magnetic vector estimations:

$$x_{12\times 1}=\left(K_{12\times 6}\right){(z_{1\times 6})}^T$$

If magnetic jamming is detected in the current iteration, the magnetic vector error signal is ignored, and the a posterior error estimate is calculated as:

$$x_{9\times 1}=\left(K(\text{1:9},\text{1:3}\right){(z_g)}^T$$

Correct

Estimate Orientation

The orientation estimate is updated by multiplying the previous estimation by the error:

$$q^{+}=\left(q^{-}\right)\left({\theta }^{+}\right)$$

Estimate Linear Acceleration

The linear acceleration estimation is updated by decaying the linear acceleration estimation from the previous iteration and subtracting the error:

$$linAccelPrior=(linAccelPrio{r}_{k-1})\nu -b^{+}$$

where

• ν –– LinearAccelerationDecayFactor

Estimate Gyroscope Offset

The gyroscope offset estimation is updated by subtracting the gyroscope offset error from the gyroscope offset from the previous iteration:

$$gyroOffset=gyroOffse{t}_{k-1}-a^{+}$$

Compute Angular Velocity

To estimate angular velocity, the frame of gyroReadings are averaged and the gyroscope offset computed in the previous iteration is subtracted:

$$angularVelocit{y}_{1\times 3}=\frac{{\displaystyle \sum gyroReading{s}_{N\times 3}}}{N}-gyroOffse{t}_{1\times 3}$$

where N is the decimation factor specified by the DecimationFactor property.

The gyroscope offset estimation is initialized to zeros for the first iteration.

Update Magnetic Vector

If magnetic jamming was not detected in the current iteration, the magnetic vector estimate, m, is updated using the a posteriori magnetic disturbance error and the a posteriori orientation.

The magnetic disturbance error is converted to the navigation frame:

$$mErrorNE{D}_{1\times 3}={\left({\left(rPos{t}_{3\times 3}\right)}^T{(mErro{r}_{1\times 3})}^T\right)}^T$$

The magnetic disturbance error in the navigation frame is subtracted from the previous magnetic vector estimate and then interpreted as inclination:

$${\rm M}=m-mErrorNED$$

$$inclination=\text{atan2}({\rm M}(3),{\rm M}(1))$$

The inclination is converted to a constrained magnetic vector estimate for the next iteration:

$$\begin{array}{l}m(1)=\left(\text{ExpectedMagneticFieldStrength}\right)\left(\cos (inclination)\right)\\ m(2)=0\\ m(3)=\left(\text{ExpectedMagneticFieldStrength}\right)\left(\sin (inclination)\right)\end{array}$$

ExpectedMagneticFieldStrength is a property of ahrsfilter.

References

[1] Open Source Sensor Fusion. https://github.com/memsindustrygroup/Open-Source-Sensor-Fusion/tree/master/docs

[2] Roetenberg, D., H.J. Luinge, C.T.M. Baten, and P.H. Veltink. "Compensation of Magnetic Disturbances Improves Inertial and Magnetic Sensing of Human Body Segment Orientation." IEEE Transactions on Neural Systems and Rehabilitation Engineering. Vol. 13. Issue 3, 2005, pp. 395-405.

Extended Capabilities

C/C++ Code Generation

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

Usage notes and limitations:

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

• This System object also supports strict single-precision code generation.

Version History

Introduced in R2018b

R2024a: ahrsfilter object returns residual and residual covariance

ahrsfilter object, FUSE returns a third output argument, interData, containing residual and residual covariance stored as a structure.