Fusion

Fusion is a sensor fusion library for Inertial Measurement Units (IMUs), optimised for embedded systems. Fusion is a C library but is also available as the Python package imufusion. Two example Python scripts, simple_example.py and advanced_example.py are provided with example sensor data to demonstrate use of the package.

AHRS algorithm

The Attitude And Heading Reference System (AHRS) algorithm combines gyroscope, accelerometer, and magnetometer data into a single measurement of orientation relative to the Earth. The algorithm also supports systems that use only a gyroscope and accelerometer, and systems that use a gyroscope and accelerometer combined with an external source of heading measurement such as GPS.

The algorithm is based on the revised AHRS algorithm presented in chapter 7 of Madgwick's PhD thesis. This is a different algorithm to the better-known initial AHRS algorithm presented in chapter 3, commonly referred to as the Madgwick algorithm.

The algorithm calculates the orientation as the integration of the gyroscope summed with a feedback term. The feedback term is equal to the error in the current measurement of orientation as determined by the other sensors, multiplied by a gain. The algorithm therefore functions as a complementary filter that combines high-pass filtered gyroscope measurements with low-pass filtered measurements from other sensors with a corner frequency determined by the gain. A low gain will 'trust' the gyroscope more and so be more susceptible to drift. A high gain will increase the influence of other sensors and the errors that result from accelerations and magnetic distortions. A gain of zero will ignore the other sensors so that the measurement of orientation is determined by only the gyroscope.

Initialisation

Initialisation occurs when the algorithm starts for the first time and after an acceleration rejection timeout. During initialisation, the acceleration and magnetic rejection features are disabled and the gain is ramped down from 10 to the final value over a 3 second period. This allows the measurement of orientation to rapidly converges from an arbitrary initial value to the value indicated by the sensors.

Acceleration rejection

The acceleration rejection feature reduces the errors that result from the accelerations of linear and rotational motion. Acceleration rejection works by comparing the instantaneous measurement of inclination provided by the accelerometer with the current measurement of inclination of the algorithm output. If the angular difference between these two inclinations is greater than a threshold then the accelerometer will be ignored for this algorithm update. This is equivalent to a dynamic gain that deceases as accelerations increase.

Prolonged accelerations from linear and rotational motion may result in the accelerometer being unusable as a measurement of inclination. This is detected by the algorithm as an acceleration rejection timeout. An acceleration rejection timeout occurs when the number of algorithm updates that have ignored the accelerometer exceeds ten times the number that have used the accelerometer within a defined period. If an acceleration rejection timeout occurs then the algorithm will reinitialise.

Magnetic rejection

The magnetic rejection feature reduces the errors that result from temporary magnetic distortions. Magnetic rejection works using the same principle as acceleration rejection operating on the magnetometer instead of the accelerometer and by comparing the measurements of heading instead of inclination. A magnetic rejection timeout will not cause the algorithm to reinitialise. If a magnetic rejection timeout occurs then the heading of the algorithm output will be set to the instantaneous measurement of heading provided by the magnetometer.

Algorithm outputs

The algorithm provides three outputs: quaternion, linear acceleration, and Earth acceleration. The quaternion describes the orientation of the sensor relative to the Earth using the North-West-Up (NWU) convention. This can be converted to a rotation matrix using the FusionQuaternionToMatrix function or to Euler angles using the FusionQuaternionToEuler function. The linear acceleration is the accelerometer measurement with the 1 g of gravity removed. The Earth acceleration is the accelerometer measurement in the Earth coordinate frame with the 1 g of gravity removed.

Algorithm settings

The AHRS algorithm settings are defined by the FusionAhrsSettings structure and set using the FusionAhrsSetSettings function.

Setting	Description
gain	Determines the influence of the gyroscope relative to other sensors. A value of 0.5 is appropriate for most applications.
accelerationRejection	Threshold (in degrees) used by the acceleration rejection feature. A value of zero will disable this feature. A value of 10 degrees is appropriate for most applications.
magneticRejection	Threshold (in degrees) used by the magnetic rejection feature. A value of zero will disable the feature. A value of 20 degrees is appropriate for most applications.
rejectionTimeout	Acceleration and magnetic rejection timeout period (in samples). A value of zero will disable the acceleration and magnetic rejection features. A period of 5 seconds is appropriate for most applications.

Algorithm internal states

The AHRS algorithm internal states are defined by the FusionAhrsInternalStates structure and obtained using the FusionAhrsGetInternalStates function.

State	Description
accelerationError	Angular error (in degrees) of the instantaneous measurement of inclination provided by the accelerometer. The acceleration rejection feature will ignore the accelerometer if this value exceeds the accelerationRejection threshold set in the algorithm settings.
accelerometerIgnored	true if the accelerometer was ignored by the previous algorithm update.
accelerationRejectionTimer	Acceleration rejection timer value normalised to between 0.0 and 1.0. An acceleration rejection timeout will occur when this value reaches 1.0.
magneticError	Angular error (in degrees) of the instantaneous measurement of heading provided by the magnetometer. The magnetic rejection feature will ignore the magnetometer if this value exceeds the magneticRejection threshold set in the algorithm settings.
magnetometerIgnored	true if the magnetometer was ignored by the previous algorithm update.
magneticRejectionTimer	Magnetic rejection timer value normalised to between 0.0 and 1.0. A magnetic rejection timeout will occur when this value reaches 1.0.

Algorithm flags

The AHRS algorithm flags are defined by the FusionAhrsFlags structure and obtained using the FusionAhrsGetFlags function.

Flag	Description
initialising	true if the algorithm is initialising.
accelerationRejectionWarning	true if the acceleration rejection timer has exceeded 25% of the rejectionTimeout value set in the algorithm settings.
accelerationRejectionTimeout	true if an acceleration rejection timeout has occurred and the algorithm is initialising.
magneticRejectionWarning	true if the magnetic rejection timer has exceeded 25% of the rejectionTimeout value set in the algorithm settings.
magneticRejectionTimeout	true if a magnetic rejection timeout has occurred during the previous algorithm update.

Gyroscope offset correction algorithm

The gyroscope offset correction algorithm provides run-time calibration of the gyroscope offset to compensate for variations in temperature and fine-tune existing offset calibration that may already be in place. This algorithm should be used in conjunction with the AHRS algorithm to achieve best performance.

The algorithm calculates the gyroscope offset by detecting the stationary periods that occur naturally in most applications. Gyroscope measurements are sampled during these periods and low-pass filtered to obtain the gyroscope offset. The algorithm requires that gyroscope measurements do not exceed +/-3 degrees per second while stationary. Basic gyroscope offset calibration may be necessary to ensure that the initial offset error plus measurement noise is within these bounds.

Sensor calibration

Sensor calibration is essential for accurate measurements. This library provides functions to apply calibration parameters to the gyroscope, accelerometer, and magnetometer. This library does not provide a solution for calculating the calibration parameters.

Inertial calibration

The FusionCalibrationInertial function applies gyroscope and accelerometer calibration parameters using the calibration model:

i_c = Ms(i_u - b)

• i_c is the calibrated inertial measurement and return value
• i_u is the uncalibrated inertial measurement and uncalibrated argument
• M is the misalignment matrix and misalignment argument
• s is the sensitivity diagonal matrix and sensitivity argument
• b is the offset vector and offset argument

Magnetic calibration

The FusionCalibrationMagnetic function applies magnetometer calibration parameters using the calibration model:

m_c = Sm_u - h

• m_c is the calibrated magnetometer measurement and return value
• m_u is the uncalibrated magnetometer measurement and uncalibrated argument
• S is the soft iron matrix and softIronMatrix argument
• h is the hard iron offset vector and hardIronOffset argument

Fast inverse square root

Fusion uses Pizer's implementation of the fast inverse square root algorithm for vector and quaternion normalisation. Including the definition FUSION_USE_NORMAL_SQRT in FusionMath.h or adding this as a preprocessor definition will use normal square root operations for all normalisation calculations. This will slow down execution speed for a small increase in accuracy. The increase in accuracy will typically be too small to observe on any practical system.