imufilter

Orientation from accelerometer and gyroscope readings

Description

The imufilter System object™ fuses accelerometer and gyroscope sensor data to estimate device orientation.

To estimate device orientation:

1. Create the imufilter 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 = imufilter

FUSE = imufilter('ReferenceFrame',RF)

FUSE = imufilter(___,Name=Value)

Description

FUSE = imufilter returns an indirect Kalman filter System object, FUSE, for fusion of accelerometer and gyroscope data to estimate device orientation. The filter uses a nine-element state vector to track error in the orientation estimate, the gyroscope bias estimate, and the linear acceleration estimate.

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

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

Example: FUSE = imufilter('SampleRate',200,'GyroscopeNoise',1e-6) creates a System object, FUSE, with a 200 Hz sample rate and gyroscope noise set to 1e-6 radians per second squared.

Input Arguments

RF — Reference frame of filter

'NED' (default) | 'ENU'

Reference frame of the filter, 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 — Sample rate of input sensor data (Hz)

100 (default) | positive finite scalar

Sample rate of the input sensor data in Hz, specified as a positive finite scalar.

Tunable: No

Data Types: single | double | uint8 | uint16 | uint32 | uint64 | int8 | int16 | int32 | int64

DecimationFactor — Decimation factor

1 (default) | positive integer scalar

Decimation factor by which to reduce the sample rate of the input sensor data, specified as a positive integer scalar.

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

Tunable: No

Data Types: single | double | uint8 | uint16 | uint32 | uint64 | int8 | int16 | int32 | int64

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 | uint8 | uint16 | uint32 | uint64 | int8 | int16 | int32 | int64

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 | uint8 | uint16 | uint32 | uint64 | int8 | int16 | int32 | int64

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 | uint8 | uint16 | uint32 | uint64 | int8 | int16 | int32 | int64

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 | uint8 | uint16 | uint32 | uint64 | int8 | int16 | int32 | int64

LinearAcclerationDecayFactor — 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 LinearAccelerationDecayFactor to a lower value. If linear acceleration changes slowly, set LinearAccelerationDecayFactor to a higher value. Linear acceleration drift is modeled as a lowpass-filtered white noise process.

Tunable: Yes

Data Types: single | double | uint8 | uint16 | uint32 | uint64 | int8 | int16 | int32 | int64

InitialProcessNoise — Covariance matrix for process noise

9-by-9 matrix

Covariance matrix for process noise, specified as a 9-by-9 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

  Columns 7 through 9

                   0                   0                   0
                   0                   0                   0
                   0                   0                   0
                   0                   0                   0
                   0                   0                   0
                   0                   0                   0
   0.009623610000000                   0                   0
                   0   0.009623610000000                   0
                   0                   0   0.009623610000000

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

Data Types: single | double | uint8 | uint16 | uint32 | uint64 | int8 | int16 | int32 | int64

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)

Description

[orientation,angularVelocity,interData] = FUSE(accelReadings,gyroReadings) fuses accelerometer and gyroscope readings 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

Output Arguments

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

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

Orientation that can rotate quantities from a global 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 global coordinate 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 imufilter

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

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 from IMU data

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.mat' sensorData Fs

accelerometerReadings = sensorData.Acceleration;
gyroscopeReadings = sensorData.AngularVelocity;

Create an imufilter System object™ with sample rate 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 = imufilter('SampleRate',Fs,'DecimationFactor',decim);

Pass the accelerometer readings and gyroscope readings to the imufilter 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);

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

imufilter fusion correctly estimates the change in orientation from an assumed north-facing initial orientation. However, the device's x-axis was pointing southward when recorded. To correctly estimate the orientation relative to the true initial orientation or relative to NED, use ahrsfilter.

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

plot(time,eulerd(q,'ZYX','frame'))
title('Orientation Estimate')
legend('Z-axis', 'Y-axis', 'X-axis')
xlabel('Time (s)')
ylabel('Rotation (degrees)')

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)')

Remove Bias from Angular Velocity Measurement

This example shows how to remove gyroscope bias from an IMU using imufilter.

Use kinematicTrajectory to create a trajectory with two parts. The first part has a constant angular velocity about the y- and z-axes. The second part has a varying angular velocity in all three axes.

duration = 60*8;
fs = 20;
numSamples = duration * fs;
rng('default') % Seed the RNG to reproduce noisy sensor measurements.

initialAngVel = [0,0.5,0.25];
finalAngVel = [-0.2,0.6,0.5];
constantAngVel = repmat(initialAngVel,floor(numSamples/2),1);
varyingAngVel = [linspace(initialAngVel(1), finalAngVel(1), ceil(numSamples/2)).', ...
    linspace(initialAngVel(2), finalAngVel(2), ceil(numSamples/2)).', ...
    linspace(initialAngVel(3), finalAngVel(3), ceil(numSamples/2)).'];

angVelBody = [constantAngVel; varyingAngVel];
accBody = zeros(numSamples,3);

traj = kinematicTrajectory('SampleRate',fs);

[~,qNED,~,accNED,angVelNED] = traj(accBody,angVelBody);

Create an imuSensor System object™, IMU, with a nonideal gyroscope. Call IMU with the ground-truth acceleration, angular velocity, and orientation.

IMU = imuSensor('accel-gyro', ...
    'Gyroscope',gyroparams('RandomWalk',0.003,'ConstantBias',0.3), ...
    'SampleRate',fs);

[accelReadings, gyroReadingsBody] = IMU(accNED,angVelNED,qNED);

Create an imufilter System object, fuse. Call fuse with the modeled accelerometer readings and gyroscope readings.

fuse = imufilter('SampleRate',fs, 'GyroscopeDriftNoise', 1e-6);

[~,angVelBodyRecovered] = fuse(accelReadings,gyroReadingsBody);

Plot the ground-truth angular velocity, the gyroscope readings, and the recovered angular velocity for each axis.

The angular velocity returned from the imufilter compensates for the effect of the gyroscope bias over time and converges to the true angular velocity.

time = (0:numSamples-1)'/fs;

figure(1)
plot(time,angVelBody(:,1), ...
     time,gyroReadingsBody(:,1), ...
     time,angVelBodyRecovered(:,1))
title('X-axis')
legend('True Angular Velocity', ...
       'Gyroscope Readings', ...
       'Recovered Angular Velocity')
ylabel('Angular Velocity (rad/s)')

figure(2)
plot(time,angVelBody(:,2), ...
     time,gyroReadingsBody(:,2), ...
     time,angVelBodyRecovered(:,2))
title('Y-axis')
ylabel('Angular Velocity (rad/s)')

figure(3)
plot(time,angVelBody(:,3), ...
     time,gyroReadingsBody(:,3), ...
     time,angVelBodyRecovered(:,3))
title('Z-axis')
ylabel('Angular Velocity (rad/s)')
xlabel('Time (s)')

Algorithms

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

The imufilter uses the six-axis Kalman filter structure described in [1]. The algorithm attempts to track the errors in orientation, gyroscope offset, and linear acceleration 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:

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

where x_k is a 9-by-1 vector consisting of:

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

• b_k –– 3-by-1 gyroscope zero angular rate bias vector, in radians per second, at time k

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

• w_k –– 9-by-1 additive noise vector

• F_k –– 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 and gyroReadings inputs are chunked into 1-by-3 frames and DecimationFactor-by-3 frames, respectively. The algorithm uses the most current accelerometer readings corresponding to the chunk of gyroscope readings.

Detailed Overview

Step 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:

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

where N is the decimation factor specified by the DecimationFactor property, and fs is the sample rate specified by the SampleRate property. The brackets indicate the matrix dimensions.

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

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

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

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

During the first iteration, the orientation estimate, q⁻, is initialized by ecompass with an assumption that the x-axis points north.

Estimate Gravity from Orientation

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

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

See ecompass 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:

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

Error Model

The error model is the difference between the gravity estimate from the accelerometer readings and the gravity estimate from the gyroscope readings: $$z=g-gAccel$$.

Kalman Equations

The Kalman equations use the gravity estimate derived from the gyroscope readings, g, 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 state, g, into the 3-by-9 true state, H.

The observation model is constructed as:

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

where g_x, g_y, and g_z are the x-, y-, and z-elements of the gravity vector estimated from the 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 3-by-3 matrix used to track the variability in the measurements. The innovation covariance matrix is calculated as:

$$S_{\left[3\times 3\right]}=R_{\left[3\times 3\right]}+\left(H_{\left[3\times 9\right]}\right)\left(P_{\left[9\times 9\right]}^{-}\right){\left(H_{\left[3\times 9\right]}\right)}^T$$

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:

  $$R_{\left[3\times 3\right]}=\left(\lambda \text{+}\xi \text{+}\kappa \left(\beta \text{+}\eta \right)\right)\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right].$$

  The following properties define the observation model noise variance:

  • κ –– (DecimationFactor/SampleRate)²

  • β –– GyroscopeDriftNoise

  • η –– GyroscopeNoise

  • λ –– AccelerometerNoise

  • ξ –– LinearAccelerationNoise

Update Error Estimate Covariance

The error estimate covariance is a 9-by-9 matrix used to track the variability in the state.

The error estimate covariance matrix is updated as:

$$P_{\left[9\times 9\right]}^{+}=P_{\left[9\times 9\right]}^{-}-\left(K_{\left[9\times 3\right]}\right)\left(H_{\left[3\times 9\right]}\right)\left(P_{\left[9\times 9\right]}^{-}\right)$$

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 9-by-9 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, the cross-correlation terms are assumed to be negligible compared to the autocorrelation terms, and are set to zero:

$$Q=\left[\begin{array}{ccccccccc}{P}_{1,1}^{+}+{\kappa }^2\left(P_{4,4}^{+}+\beta +\eta \right)& 0& 0& -\kappa \left(P_{4,4}^{+}+\beta \right)& 0& 0& 0& 0& 0\\ 0& P_{2,2}^{+}+{\kappa }^2\left(P_{5,5}^{+}+\beta +\eta \right)& 0& 0& -\kappa (P_{5,5}^{+}+\beta )& 0& 0& 0& 0\\ 0& 0& P_{3,3}^{+}+{\kappa }^2\left(P_{6,6}^{+}+\beta +\eta \right)& 0& 0& -\kappa (P_{6,6}^{+}+\beta )& 0& 0& 0\\ -\kappa \left(P_{4,4}^{+}+\beta \right)& 0& 0& P_{4,4}^{+}+\beta & 0& 0& 0& 0& 0\\ 0& -\kappa (P_{5,5}^{+}+\beta )& 0& 0& P_{5,5}^{+}+\beta & 0& 0& 0& 0\\ 0& 0& -\kappa (P_{6,6}^{+}+\beta )& 0& 0& P_{6,6}^{+}+\beta & 0& 0& 0\\ 0& 0& 0& 0& 0& 0& {\nu }^2{P}_{7,7}^{+}+\xi & 0& 0\\ 0& 0& 0& 0& 0& 0& 0& {\nu }^2{P}_{8,8}^{+}+\xi & 0\\ 0& 0& 0& 0& 0& 0& 0& 0& {\nu }^2{P}_{9,9}^{+}+\xi \end{array}\right]$$

where

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

• κ –– DecimationFactor/SampleRate

• β –– GyroscopeDriftNoise

• η –– GyroscopeNoise

• ν –– LinearAcclerationDecayFactor

• ξ –– LinearAccelerationNoise

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 9-by-3 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:

$$K_{\left[9\times 3\right]}=\left(P_{\left[9\times 9\right]}^{-}\right){\left(H_{\left[3\times 9\right]}\right)}^T{\left({\left(S_{\left[3\times 3\right]}\right)}^T\right)}^{-1}$$

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 estimations:

$$x_{\left[9\times 1\right]}=\left(K_{\left[9\times 3\right]}\right){\left(z_{\left[1\times 3\right]}\right)}^T$$

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:

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

where

• ν –– LinearAcclerationDecayFactor

Estimate Gyroscope Offset

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

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

Compute Angular Velocity

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

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

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

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

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: imufilter object returns residual and residual covariance

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