IMU

IMU simulation model

Since R2020a

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Libraries:
Sensor Fusion and Tracking Toolbox / Multisensor Positioning / Sensor Models
Navigation Toolbox / Multisensor Positioning / Sensor Models

Description

The IMU Simulink^® block models receiving data from an inertial measurement unit (IMU) composed of accelerometer, gyroscope, and magnetometer sensors. 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 Reference Frame parameter.

Examples

IMU Sensor Fusion with Simulink

Generate and fuse IMU sensor data using Simulink®. You can accurately model the behavior of an accelerometer, a gyroscope, and a magnetometer and fuse their outputs to compute orientation.

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Ports

Input

Linear Acceleration — Acceleration of IMU in local navigation coordinate system (m/s²)

N-by-3 matrix of real scalar

Acceleration of the IMU in the local navigation coordinate system, specified as an N-by-3 matrix of real scalars 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 port.

Data Types: single | double

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

N-by-3 matrix of real scalar

Angular velocity of the IMU sensor body frame in the local navigation coordinate system, specified as an N-by-3 matrix of scalars 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 port.

Data Types: single | double

Orientation — Orientation of IMU in local navigation coordinate system

N-by-4 array of real scalar | 3-by-3-by-N-element rotation matrix

Orientation of the IMU sensor body frame with respect to the local navigation coordinate system, specified as an N-by-4 array of real scalars or a 3-by-3-by-N rotation matrix. Each row the of the N-by-4 array is assumed to be the four elements of a quaternion (Sensor Fusion and Tracking Toolbox). N is the number of samples in the current frame.

Data Types: single | double

Temperature — Temperature of IMU (℃)

N-by-1 vector of real scalar

Temperature of IMU, specified as an N-by-1 vector of real scalars in degrees Celsius.

Dependencies

To enable this port, in the Parameters tab, select Specify temperature from input port.

Data Types: single | double

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

N-by-3 matrix of real scalar

Magnetic field vector in the local navigation coordinate system, specified as an N-by-3 matrix of scalars in microteslas.

Dependencies

To enable this port, in the Parameters tab, select Specify magnetic field from input port.

Data Types: single | double

Output

Accel — Accelerometer measurement of IMU in sensor body coordinate system (m/s²)

N-by-3 matrix of real scalar

Accelerometer measurement of the IMU in the sensor body coordinate system, returned as an N-by-3 matrix of real scalars in meters per second squared. N is the number of samples in the current frame.

Data Types: single | double

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

N-by-3 matrix of real scalar

Gyroscope measurement of the IMU in the sensor body coordinate system, returned as an N-by-3 matrix of real scalars in radians per second. N is the number of samples in the current frame.

Data Types: single | double

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

N-by-3 matrix of real scalar

Magnetometer measurement of the IMU in the sensor body coordinate system, returned as an N-by-3 matrix of real scalars in microtesla. N is the number of samples in the current frame.

Data Types: single | double

Parameters

Parameters

Reference frame — Navigation reference frame

NED (default) | ENU

Navigation reference frame, specified as 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.

Specify temperature from input port — Specify temperature from input port

off (default) | on

Select this check box to enable the input of temperature by using the Temperature input port.

Temperature (^oC) — Operating temperature of IMU (^oC)

25 (default) | real scalar

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

When the block calculates temperature scale factors and environmental drift noises, 25 ^oC is used as the nominal temperature.

Dependencies

To enable this parameter, unselect Specify temperature from input port.

Data Types: single | double

Specify magnetic field from input port — Specify magnetic field from input port

off (default) | on

Select this check box to enable the input of magnetic field by using the Magnetic field input port.

Magnetic field (NED) — Magnetic field vector expressed in NED navigation frame (μT)

[27.5550, -2.4169, -16.0849] (default) | 1-by-3 vector of scalar

Magnetic field vector expressed in the NED navigation frame, specified as a 1-by-3 vector of scalars.

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

Dependencies

To enable this parameter, set Reference frame to NED and unselect Specify magnetic field from input port.

Data Types: single | double

MagneticField (ENU) — Magnetic field vector expressed in ENU navigation frame (μT)

[-2.4169, 27.5550, 16.0849] (default) | 1-by-3 vector of scalar

Magnetic field vector expressed in the ENU navigation frame, specified as a 1-by-3 vector of scalars.

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

Dependencies

To enable this parameter, set Reference frame to ENU and unselect Specify magnetic field from input port.

Data Types: single | double

Seed — Initial seed for randomization

67 (default) | nonnegative integer

Initial seed of a random number generator algorithm, specified as a nonnegative integer.

Data Types: single | double

Simulate using — Type of simulation to run

Interpreted Execution (default) | Code Generation

• Interpreted execution — Simulate the model using the MATLAB^® interpreter. This option shortens startup time. In Interpreted execution mode, you can debug the source code of the block.

• Code generation — Simulate the model using generated C code. The first time that you run a simulation, Simulink generates C code for the block. The C code is reused for subsequent simulations if the model does not change. This option requires additional startup time.

Accelerometer

Maximum readings (m/s²) — Maximum sensor reading (m/s²)

inf (default) | real positive scalar

Maximum sensor reading in m/s², specified as a real positive scalar.

Data Types: single | double

Resolution ((m/s²)/LSB) — Resolution of sensor measurements ((m/s²)/LSB)

0 (default) | real nonnegative scalar

Resolution of sensor measurements in (m/s²)/LSB, specified as a real nonnegative scalar.

Data Types: single | double

Constant offset bias (m/s²) — Constant sensor offset bias (m/s²)

[0 0 0] (default) | real scalar | real 3-element row vector

Constant sensor offset bias in m/s², specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.

Data Types: single | double

Axis skew (%) — Sensor axes skew (%)

diag([100 100 100]) (default) | scalar | 3-element row vector | 3-by-3 matrix

Sensor axes skew in percentage, specified as a scalar, a 3-element row vector, or a 3-by-3 matrix. The diagonal elements of the matrix account for the misalignment effects for each axes. The off-diagonal elements account for the cross-axes misalignment effects. The measured state v_measure is obtained from the true state v_true via the misalignment matrix as:

$$v_{measure}=\frac{1}{100}M{v}_{true}=\frac{1}{100}\left[\begin{array}{ccc}{m}_{11}& m_{12}& m_{13}\\ m_{21}& m_{22}& m_{23}\\ m_{31}& m_{32}& m_{33}\end{array}\right]v_{true}$$

• If you specify the property as a scalar, then all the off-diagonal elements of the matrix take the value of the specified scalar and all the diagonal elements are 100.

• If you specify the property as a vector [a b c], then m₂₁ = m₃₁ = a, m₁₂ = m₃₂ = b, and m₁₃ = m₂₃ = c. All the diagonal elements are 100.

Data Types: single | double

Velocity random walk (m/s²/√Hz) — Velocity random walk (m/s²/√Hz)

[0 0 0] (default) | real scalar | real 3-element row vector

Velocity random walk in (m/s²/√Hz), specified as a real scalar or 3-element row vector. This property corresponds to the power spectral density of sensor noise. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.

Data Types: single | double

Bias Instability (m/s²) — Instability of the bias offset (m/s²)

[0 0 0] (default) | real scalar | real 3-element row vector

Instability of the bias offset in m/s², specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.

Data Types: single | double

Bias instability filter numerator coefficients — Bias instability filter numerator coefficients

fractalcoef.Numerator (default) | real-valued row vector

Bias instability filter numerator coefficients, specified as a real-valued row vector. To specify coefficients for fractal noise, use the fractalcoef (Sensor Fusion and Tracking Toolbox) function.

Data Types: single | double

Bias instability filter denominator coefficients — Bias instability filter denominator coefficients

fractalcoef.Denominator (default) | real-valued row vector

Bias instability filter denominator coefficients, specified as a real-valued row vector. To specify coefficients for fractal noise, use the fractalcoef (Sensor Fusion and Tracking Toolbox) function.

Data Types: single | double

Acceleration random walk ((m/s²)(√Hz)) — Acceleration random walk ((m/s²)(√Hz))

[0 0 0] (default) | real scalar | real 3-element row vector

Acceleration random walk of sensor in (m/s²)(√Hz), specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.

Data Types: single | double

Noise Type — Type of random noise

double-sided (default) | single-sided

Select the type of random noise as:

• double-sided — Random noise coefficients have a scale factor of 2.

• single-sided — Random noise coefficients have a scale factor of 1.

Data Types: single | double

Bias from temperature ((m/s²)/℃) — Sensor bias from temperature ((m/s²)/℃)

[0 0 0] (default) | real scalar | real 3-element row vector

Sensor bias from temperature in (m/s²)/℃, specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.

Data Types: single | double

Temperature scale factor (%/℃) — Scale factor error from temperature (%/℃)

[0 0 0] (default) | real scalar in the range [0,100] | real 3-element row vector in the range [0,100]

Scale factor error from temperature in %/℃, specified as a real scalar or real 3-element row vector with values ranging from 0 to 100. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.

Data Types: single | double

Gyroscope

Maximum readings (rad/s) — Maximum sensor reading (rad/s)

inf (default) | real positive scalar

Maximum sensor reading in rad/s, specified as a real positive scalar.

Data Types: single | double

Resolution ((rad/s)/LSB) — Resolution of sensor measurements ((rad/s)/LSB)

0 (default) | real nonnegative scalar

Resolution of sensor measurements in (rad/s)/LSB, specified as a real nonnegative scalar.

Data Types: single | double

Constant offset bias (rad/s) — Constant sensor offset bias (rad/s)

[0 0 0] (default) | real scalar | real 3-element row vector

Constant sensor offset bias in rad/s, specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.

Data Types: single | double

Axis skew (%) — Sensor axes skew (%)

diag([100 100 100]) (default) | scalar | 3-element row vector | 3-by-3 matrix

Sensor axes skew in percentage, specified as a scalar, a 3-element row vector, or a 3-by-3 matrix. The diagonal elements of the matrix account for the misalignment effects for each axes. The off-diagonal elements account for the cross-axes misalignment effects. The measured state v_measure is obtained from the true state v_true via the misalignment matrix as:

$$v_{measure}=\frac{1}{100}M{v}_{true}=\frac{1}{100}\left[\begin{array}{ccc}{m}_{11}& m_{12}& m_{13}\\ m_{21}& m_{22}& m_{23}\\ m_{31}& m_{32}& m_{33}\end{array}\right]v_{true}$$

• If you specify the property as a scalar, then all the off-diagonal elements of the matrix take the value of the specified scalar and all the diagonal elements are 100.

• If you specify the property as a vector [a b c], then m₂₁ = m₃₁ = a, m₁₂ = m₃₂ = b, and m₁₃ = m₂₃ = c. All the diagonal elements are 100.

Data Types: single | double

Bias from acceleration ((rad/s)/(m/s²) — Sensor bias from linear acceleration (rad/s)/(m/s²)

[0 0 0] (default) | real scalar | real 3-element row vector

Sensor bias from linear acceleration in (rad/s)/(m/s²), specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.

Data Types: single | double

Angle random walk ((rad/s)/(√Hz)) — Angle random walk ((rad/s)/(√Hz))

[0 0 0] (default) | real scalar | real 3-element row vector

Angle random walk of sensor in (rad/s)/(√Hz), specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.

Data Types: single | double

Bias Instability (rad/s) — Instability of the bias offset (rad/s)

[0 0 0] (default) | real scalar | real 3-element row vector

Instability of the bias offset in rad/s, specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.

Data Types: single | double

Bias instability filter numerator coefficients — Bias instability filter numerator coefficients

fractalcoef.Numerator (default) | real-valued row vector

Bias instability filter numerator coefficients, specified as a real-valued row vector. To specify coefficients for fractal noise, use the fractalcoef (Sensor Fusion and Tracking Toolbox) function.

Data Types: single | double

Bias instability filter denominator coefficients — Bias instability filter denominator coefficients

fractalcoef.Denominator (default) | real-valued row vector

Bias instability filter denominator coefficients, specified as a real-valued row vector. To specify coefficients for fractal noise, use the fractalcoef (Sensor Fusion and Tracking Toolbox) function.

Data Types: single | double

Noise Type — Type of random noise

double-sided (default) | single-sided

Select the type of random noise as:

• double-sided — Random noise coefficients have a scale factor of 2.

• single-sided — Random noise coefficients have a scale factor of 1.

Data Types: single | double

Rate random walk ((rad/s)(√Hz)) — Integrated white noise of sensor ((rad/s)(√Hz))

[0 0 0] (default) | real scalar | real 3-element row vector

Integrated white noise of sensor in (rad/s)(√Hz), specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.

Data Types: single | double

Bias from temperature ((rad/s)/℃) — Sensor bias from temperature ((rad/s)/℃)

[0 0 0] (default) | real scalar | real 3-element row vector

Sensor bias from temperature in (rad/s)/℃, specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.

Data Types: single | double

Temperature scale factor (%/℃) — Scale factor error from temperature (%/℃)

[0 0 0] (default) | real scalar in the range [0,100] | real 3-element row vector in the range [0,100]

Scale factor error from temperature in %/℃, specified as a real scalar or real 3-element row vector with values ranging from 0 to 100. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.

Data Types: single | double

Magnetometer

Maximum readings (μT) — Maximum sensor reading (μT)

inf (default) | real positive scalar

Maximum sensor reading in μT, specified as a real positive scalar.

Data Types: single | double

Resolution ((μT)/LSB) — Resolution of sensor measurements ((μT)/LSB)

0 (default) | real nonnegative scalar

Resolution of sensor measurements in (μT)/LSB, specified as a real nonnegative scalar.

Data Types: single | double

Constant offset bias (μT) — Constant sensor offset bias (μT)

[0 0 0] (default) | real scalar | real 3-element row vector

Constant sensor offset bias in μT, specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.

Data Types: single | double

Axis skew (%) — Sensor axes skew (%)

diag([100 100 100]) (default) | scalar | 3-element row vector | 3-by-3 matrix

Sensor axes skew in percentage, specified as a scalar, a 3-element row vector, or a 3-by-3 matrix. The diagonal elements of the matrix account for the misalignment effects for each axes. The off-diagonal elements account for the cross-axes misalignment effects. The measured state v_measure is obtained from the true state v_true via the misalignment matrix as:

$$v_{measure}=\frac{1}{100}M{v}_{true}=\frac{1}{100}\left[\begin{array}{ccc}{m}_{11}& m_{12}& m_{13}\\ m_{21}& m_{22}& m_{23}\\ m_{31}& m_{32}& m_{33}\end{array}\right]v_{true}$$

• If you specify the property as a scalar, then all the off-diagonal elements of the matrix take the value of the specified scalar and all the diagonal elements are 100.

• If you specify the property as a vector [a b c], then m₂₁ = m₃₁ = a, m₁₂ = m₃₂ = b, and m₁₃ = m₂₃ = c. All the diagonal elements are 100.

White noise PSD (μT/√Hz) — Power spectral density of sensor noise (μT/√Hz)

[0 0 0] (default) | real scalar | real 3-element row vector

Power spectral density of sensor noise in μT/√Hz, specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.

Data Types: single | double

Bias Instability (μT) — Instability of the bias offset (μT)

[0 0 0] (default) | real scalar | real 3-element row vector

Instability of the bias offset in μT, specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.

Data Types: single | double

Bias instability filter numerator coefficients — Bias instability filter numerator coefficients

fractalcoef.Numerator (default) | real-valued row vector

Bias instability filter numerator coefficients, specified as a real-valued row vector. To specify coefficients for fractal noise, use the fractalcoef (Sensor Fusion and Tracking Toolbox) function.

Data Types: single | double

Bias instability filter denominator coefficients — Bias instability filter denominator coefficients

fractalcoef.Denominator (default) | real-valued row vector

Bias instability filter denominator coefficients, specified as a real-valued row vector. To specify coefficients for fractal noise, use the fractalcoef (Sensor Fusion and Tracking Toolbox) function.

Data Types: single | double

Noise Type — Type of random noise

double-sided (default) | single-sided

Select the type of random noise as:

• double-sided — Random noise coefficients have a scale factor of 2.

• single-sided — Random noise coefficients have a scale factor of 1.

Data Types: single | double

Random walk ((μT)*√Hz) — Integrated white noise of sensor ((μT)*√Hz)

[0 0 0] (default) | real scalar | real 3-element row vector

Integrated white noise of sensor in (μT)*√Hz, specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.

Data Types: single | double

Bias from temperature (μT/℃) — Sensor bias from temperature (μT/℃)

[0 0 0] (default) | real scalar | real 3-element row vector

Sensor bias from temperature in μT/℃, specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.

Data Types: single | double

Temperature scale factor (%/℃) — Scale factor error from temperature (%/℃)

[0 0 0] (default) | real scalar in the range [0,100] | real 3-element row vector in the range [0,100]

Scale factor error from temperature in %/℃, specified as a real scalar or real 3-element row vector with values ranging from 0 to 100. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.

Data Types: single | double

Algorithms

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 Simulink® Coder™.

Version History

Introduced in R2020a

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

You can use these new parameters in the Accelerometer, Gyroscope, or Magnetometer tab to specify the coefficients of the transfer function used to generate the bias instability noise and specify the scale factor of the noise coefficients.

• Bias instability filter numerator coefficients

• Bias instability filter denominator coefficients

• Noise type