Specifying a sensor¶
Specifying a sensor's performance characteristics is a prerequisite to simulating its behavior. Users have the option to create a custom specification or to use a sensor from the devices library.
Model equation¶
A common model equation is used for both accelerometers and gyros (and by extension IMUs). This is consistent with the model found in Standard [05] and Standard [06] (Sections 8.3).
\[
O = Q\left(\frac{L(\mathbf{M}\mathbf{I}) + B + N}{1+\epsilon}\right)
\]
Where the terms have the following meanings:
- \(O\) is the measured output along a single sensing axis.
- \(Q\) is quantization due to digitization.
- \(L\) clamps the input between minimum and maximum sensing limits.
- \(\mathbf{M}\) is a matrix of misalignment terms that relate output axes to the case reference axes.
- \(\mathbf{I}\) is the true 3-axis orthogonal inertial input.
- \(B\) are measurement biases.
- \(N\) are measurement noise sources.
- \(\epsilon\) is scale factor error.
When specifying a sensor, care should be taken to reference this equation. It is also important to ensure that other tools use consistent definitions. For example, scale factor is a divisor here but may equivalently be specified as a multiplier with the inverse value.
Custom specifications¶
Sensor specifications are regular Python files. Each specification is an
instance of a GyroSpecification,
AccelerometerSpecification,
or IMUSpecification object. IMU
specifications are composites that include both gyro and accelerometer
specifications. Anywhere a gyro or accelerometer specification can be used, an
IMU specification can be substituted. For obvious reasons, the reverse is not
true.
Default values¶
Congratulations, you now have a sensor specification! Specifications consist of parameters grouped into logical categories: data interface, random noise, bias, scale factor, etc. All parameters default to zero (or identity) values. Users are only required to set parameters that matter in their application. The number of independent sensing axes of the sensor defaults to 3.
from inertialsim.sensors.gyro import GyroSpecification
gyro = GyroSpecification()
print(gyro.noise.random_walk.value)
print(gyro.noise.random_walk.units)
[[[0.]
[0.]
[0.]]]
rad/sqrt(s)
random_walk.value is an array with 3 elements because axes = 3 is the
default for all sensor specifications.
Setting values¶
Manufacturers often use non-standard units of measurement on their data sheets. Converting these manually is error prone, so InertialSim supports automatic conversion of common units to SI (or metric) units. Outside of sensor specifications, InertialSim exclusively uses SI units.
Values are set using Parameter objects, which
take a value and units input.
In a simple example, we demonstrate the equivalence between using SI units and US customary units in an accelerometer bias specification:
from inertialsim.sensors import Parameter
from inertialsim.sensors.accelerometer import AccelerometerSpecification
accelerometer = AccelerometerSpecification(axes=2)
# Set the parameter using meters per second per second
accelerometer.bias.fixed = Parameter(0.0003048, "m/s/s")
print(accelerometer.bias.fixed.value)
print(accelerometer.bias.fixed.units)
# Reset the parameter using feet per second per second
accelerometer.bias.fixed = Parameter(0.001, "ft/s/s")
print(accelerometer.bias.fixed.value)
print(accelerometer.bias.fixed.units)
[[[0.0003048]
[0.0003048]]]
m/s/s
[[[0.0003048]
[0.0003048]]]
m/s/s
The US customary units are converted to metric units behind the scenes in the
Parameter object. In this case 1 international foot = 0.3048 meters.
International foot
In this example the conversion used the international foot. There is also a US Survey foot which should no longer be in use but is sometimes found in older reference. This is just one of many reasons why SI units are the default in InertialSim.
In a more complex example, common but non-standard velocity random walk units
are converted to standard units by the Parameter object.
accelerometer = AccelerometerSpecification(axes=1)
accelerometer.noise.random_walk = Parameter(0.1, "g/sqrt(Hz)")
print(accelerometer.noise.random_walk.value)
print(accelerometer.noise.random_walk.units)
[[[0.980665]]]
m/s/sqrt(s)
Velocity random walk results from integrating random noise in acceleration
signals. 1 g here is a standard unit of gravity equivalent to 9.80665
meters per second per second (see Standard [01]) and sqrt(Hz) is
a derived unit used often in spectral analysis. These are converted to base
units of meters and seconds.
Specification classes will raise an error if inputs contain invalid or unsupported units.
Device library¶
InertialSim comes pre-packaged with a library of specifications. These specifications are populated from publicly available data sheets and user manuals.
Example IMU specification¶
An example IMU specification is available which documents all the available gyro, accelerometer, and IMU options and supported units.
inertialsim/devices/example_imu.py
# -----------------------------------------------------------------------------
# Copyright (c) 2023-2025, Inertial Simulation LLC.
# All rights reserved.
# Do not use or redistribute without permission.
# Email: info@inertialsim.com
# -----------------------------------------------------------------------------
"""Example IMU specification.
This example contains all available specifications and documents their usage and
supported inputs.
All specifications have a standard default when the SensorSpecification object
is initialized. Default values result in an ideal sensor (e.g. errors = 0.0,
scales = 1.0, etc.). If a parameter is unknown or irrelevant to the
application, it should be left at the default value.
There is significant variation in how sensor manufacturers specify their
devices. This example uses actual values from a variety of sensors to
illustrate this variety.
"""
from inertialsim.sensors import Parameter
from inertialsim.sensors.imu import IMUSpecification
# Create the specification object.
example_imu = IMUSpecification()
# ----------------------------
# Sensor identifiers.
# ----------------------------
# These are strings that identify the sensor being specified.
example_imu.manufacturer = "Acme Inc."
example_imu.model = "IMU-1234"
example_imu.version = "V23.2"
# ----------------------------
# Sensor interface parameters.
# ----------------------------
# The data interface of the sensor. The sensor interface includes the data
# frequency, which is the frequency at which the sensor transmits new data, and
# the data type, which is one of rate or delta.
#
# For IMU sensors the interface is shared by gyro and accelerometers so it is
# not possible to set those separately. Every sensor has one interface by
# default but additional interfaces can be added.
# data_rate: The frequency at which new data is output from the sensor. Units
# must be Hz. Default value = Parameter(100, "Hz"). The Honeywell HG4930 IMU has
# a 600Hz "control" message which outputs rate data.
example_imu.data_interface.sample_rate = Parameter(600, "Hz")
# quantization: Outputs will be quantized at simulated digital levels. Use this
# when the parameters of the analog to digital conversion are known. If the
# sensor to simulate has been characterized experimentally, e.g. by the Allan
# deviation method, use Noise.quantization instead. Do not use both. Inputs
# are a tuple of 2 parameters: the first for the gyros and the second for the
# accelerometers.
#
# Gyro units must be one of: rad/s/LSB, or deg/s/LSB. Default value =
# Parameter(None, "rad/s/LSB"). Accelerometer units must be one of: m/s/s/LSB,
# g/LSB, or ft/s/s/LSB. Default value = Parameter(None, "m/s/s/LSB"). The Analog
# Devices ADIS16495 supplies a "Sensitivity" parameter in units of LSB/deg/s so
# we need to invert it to get supported units. The Honeywell HG4930 supplies
# scale factors in feet/s/s.
example_imu.data_interface.quantization = (
Parameter(1 / 655360, "deg/s/LSB"),
Parameter(600 * pow(2, -14), "ft/s/s/LSB"),
)
# The Honeywell HG4930 IMU also has a 100Hz "inertial" message which outputs
# delta data.
example_imu.data_interface.delta_sample_rate = Parameter(100, "Hz")
example_imu.data_interface.delta_quantization = (
Parameter(pow(2, -33), "rad/LSB"),
Parameter(pow(2, -27), "ft/s/LSB"),
)
# -------------
# Input limits
# -------------
# The extreme values of the input (negative and positive) within which
# performance is of the specified accuracy. Simulated outputs outside of this
# range are truncated to the limits.
# minimum: Minimum negative input. Simulated outputs less than this value will
# be truncated to this value.
#
# Gyro units must be one of: rad/s or deg/s.
# Default value = Parameter([-Inf,-Inf,-Inf], "rad/s").
# In this example, -200 deg/s will be applied to all 3 axes.
example_imu.gyro.input_limits.minimum = Parameter(-450.0, "deg/s")
# Accelerometer units must be one of: m/s/s, g, or ft/s/s.
# Default value = Parameter([-Inf,-Inf,-Inf], "m/s/s").
example_imu.accelerometer.input_limits.minimum = Parameter(-20.0, "g")
# maximum: Maximum positive input. Simulated outputs greater than this value
# will be truncated to this value.
#
# Gyro units must be one of: rad/s or deg/s.
# Default value = Parameter([Inf,Inf,Inf], "rad/s").
# In this example, each axis has a different maximum input. The Analog Devices
# ADIS16495 IMU has distinct minimum and maximum input rates (-450, 480). Each
# axis (x,y,z) is the same but could be specified individually if needed.
example_imu.gyro.input_limits.maximum = Parameter([480.0, 480.0, 480.0], "deg/s")
# Accelerometer units must be one of: m/s/s, g, or ft/s/s.
# Default value = Parameter([Inf,Inf,Inf], "m/s/s").
example_imu.accelerometer.input_limits.maximum = Parameter(20.0, "g")
# ------------
# Noise terms
# ------------
# Common gyro noise terms describing quantization, (angle) random walk, bias
# instability, rate random walk, and rate ramp. See Reference [20] and Standard
# [05] for details.
# Each of these noise terms can be determined experimentally from Allan
# variance/deviation plots or from data sheets and direct knowledge of the
# sensor. Not all terms will be applicable to all sensors.
# quantization: Uniformly distributed random noise arising from digital
# quantization of the input (default = 0.0). Use this parameter when the sensor
# to simulate has been characterized experimentally, e.g. by the Allan deviation
# method. If the parameters of the analog to digital converter are known
# explicitly, use ScaleFactor.quantization instead. Do not use both.
#
# Gyro units must be one of: rad or deg.
# Default value = Parameter([0.0,0.0,0.0], "rad").
# When a data sheet is available the ScaleFactor.quantization value is known
# precisely and this field should be left at the defaults.
example_imu.gyro.noise.quantization
# Accelerometer units must be one of: m/s or ft/s.
# Default value = Parameter([0.0,0.0,0.0], "m/s").
example_imu.accelerometer.noise.quantization
# random_walk: Random white noise in the angular rate signal that when
# integrated results in angle error buildup (angle random walk). Also known
# equivalently as noise density.
#
# Gyro units must be one of: rad/sqrt(s), rad/s/sqrt(Hz), deg/sqrt(h), or
# deg/s/sqrt(Hz).
# Default value = Parameter([0.0,0.0,0.0], "rad/sqrt(s)").
# The VectorNav VN-100 IMU supplies "Noise Density" parameters which are
# equivalent to random walk parameters.
example_imu.gyro.noise.random_walk = Parameter(0.0035, "deg/s/sqrt(Hz)")
# Accelerometer units must be one of: m/s/sqrt(s), m/s/s/sqrt(Hz), m/s/sqrt(h),
# g/sqrt(Hz), ft/s/sqrt(s), ft/s/s/sqrt(Hz), or ft/s/sqrt(h).
# Default value = Parameter([0.0,0.0,0.0], "m/s/sqrt(s)").
# The Analog Devices ADIS16495 IMU supplies both random walk and noise density
# parameters which are not close in value but not identical.
example_imu.accelerometer.noise.random_walk = Parameter(0.008, "m/s/sqrt(h)")
# bias_instability: The random variation in bias as computed over specified
# finite sampling time and averaging time intervals. Also known equivalently as
# bias in-run stability and flicker noise.
#
# Gyro units must be one of: rad/s, deg/s, or deg/h.
# Default value = Parameter([0.0,0.0,0.0], "rad/s").
# The Safran STIM300 IMU supplies a bias instability and a drift rate stability
# number (of unknown meaning) with the same units but significantly different
# values.
example_imu.gyro.noise.bias_instability = Parameter(0.3, "deg/h")
# Accelerometer units must be one of: m/s/s, g, or ft/s/s.
# Default value = Parameter([0.0,0.0,0.0], "m/s/s").
# The Safran STIM300 IMU supplies a bias instability parameter from Allan
# Variance analysis and a 1-year stability number which is significantly larger.
example_imu.accelerometer.noise.bias_instability = Parameter(0.02 * 1e-3, "g")
# rate_random_walk: Random white noise in the angular rate derivative (angular
# acceleration) that results in angular rate error buildup (rate random walk).
#
# Gyro units must be one of: rad/s/sqrt(s), deg/s/sqrt(s), or deg/h/sqrt(h).
# Default value = Parameter([0.0,0.0,0.0], "rad/s/sqrt(s)").
# Few sensors supply rate random walk values so we leave the defaults in place.
# The Safran STIM300 IMU supplies Allan Variance graphs but with insufficient
# resolution to accurately determine values.
example_imu.gyro.noise.rate_random_walk
# Accelerometer units must be one of: m/s/s/sqrt(s), g/sqrt(h), or
# ft/s/s/sqrt(s).
# Default value = Parameter([0.0,0.0,0.0], "m/s/s/sqrt(s)").
example_imu.accelerometer.noise.rate_random_walk
# rate_ramp: Linear increase in the angular rate signal over long averaging time
# intervals.
#
# Gyro units must be one of: rad/s/s, deg/s/s, or deg/h/h.
# Default value = Parameter([0.0,0.0,0.0], "rad/s/s").
# Very few sensors supply rate ramp values so we leave the defaults in place.
# The Analog Devices ADIS16495 IMU supplies Allan Variance graphs but with
# insufficient resolution to accurately determine values.
example_imu.gyro.noise.rate_ramp
# Accelerometer units must be one of: m/s/s/s, g/h, or ft/s/s/s.
# Default value = Parameter([0.0,0.0,0.0], "m/s/s/s").
example_imu.accelerometer.noise.rate_ramp
# -----------
# Bias terms
# -----------
# Bias is the non-zero output of a sensor at rest, or equivalently, the output
# that has no correlation with the input. It is comprised of a fixed component
# (that can be calibrated), a random component (that may vary from turn-on to
# turn-on or with changing conditions), and a temperature dependent component.
# fixed: A fixed bias that is constant across all conditions. For many sensors,
# this may be removed by factory calibration in which case this parameter can
# represent any known residual error in that calibration.
#
# Gyro units must be one of: rad/s, deg/s, or deg/h.
# Default value = Parameter([0.0,0.0,0.0], "rad/s").
# The majority of sensors are factory calibrated so no fixed bias remains. We
# leave the parameter at its default.
example_imu.gyro.bias.fixed
# Accelerometer units must be one of: m/s/s, g, or ft/s/s.
# Default value = Parameter([0.0,0.0,0.0], "m/s/s").
example_imu.accelerometer.bias.fixed
# repeatability: The standard deviation of a random bias component that varies
# with each turn-on of the sensor and/or varies across external conditions. Also
# known as bias repeatability. This parameter may represent unknown residuals
# from the factory calibration of a fixed bias.
#
# Gyro units must be one of: rad/s, deg/s, or deg/h.
# Default value = Parameter([0.0,0.0,0.0], "rad/s").
example_imu.gyro.bias.repeatability = Parameter(20, "deg/h")
# Accelerometer units must be one of: m/s/s, g, or ft/s/s.
# Default value = Parameter([0.0,0.0,0.0], "m/s/s").
# The Safran STIM300 IMU supplies a min and max repeatability number (-7.5mg,
# 7.5mg) so we assume a normal distribution and using the Empirical Rule we
# approximate this as a 3-sigma value.
example_imu.accelerometer.bias.repeatability = Parameter(7.5 * 1e-3 / 3, "g")
# temperature: A linear coefficient relating change in sensor temperature to a
# change in bias.
#
# Gyro units must be one of: rad/s/C, deg/s/C, deg/h/C,
# deg/s/F, or deg/h/F. Note: for change in temperature (delta), the
# Celsius (C) and Kelvin (K) scales are equivalent, so units per Kelvin
# are implicitly supported.
# Default value = Parameter([0.0,0.0,0.0], "rad/s/C").
# This Analog Devices ADIS16495 supplies a deviation of the bias over the full
# temperature range (-40C to 85C) which cannot be transformed into our linear
# model so we leave the default units.
example_imu.gyro.bias.temperature
# Accelerometer units must be one of: m/s/s/C, g/C, ft/s/s/C,
# g/F, or ft/s/s/F.
# Default value = Parameter([0.0,0.0,0.0], "m/s/s/C").
example_imu.accelerometer.bias.temperature
# -------------
# Scale factor
# -------------
# Scale factor is the ratio of change in output to change in input. It has two
# components: the analog response of the sensing elements and the digitization
# (if any) of the result.
# A perfect sensor has a linear analog response with a scale of 1.0. Imperfect
# sensors may have a non-unit scale and non-linear deviations from the linear
# trend.
# If the sensor output is digitized it will exhibit quantization where the
# analog input is sampled at discrete digital levels.
# fixed: A fixed offset from the ideal unit scale factor (1.0) that is constant
# across all conditions. For many sensors, this may be removed by factory
# calibration in which case this parameter can represent any known residual
# error in that calibration.
#
# Units are ratios of similar quantities so must be one of: dimensionless, %, or
# ppm.
# Default value = Parameter([0.0,0.0,0.0], "dimensionless").
# The majority of sensors are factory calibrated so no fixed scale factor error
# remains. We leave the parameter at its default.
example_imu.gyro.scale_factor.fixed
example_imu.accelerometer.scale_factor.fixed
# repeatability: The standard deviation of a random scale factor error that
# varies with each turn-on of the sensor and/or varies across external
# conditions. This parameter may represent unknown residuals from the factory
# calibration of fixed scale factor.
#
# Units are ratios of similar quantities so must be one of: dimensionless, %, or
# ppm.
# Default value = Parameter([0.0,0.0,0.0], "dimensionless").
# The Safran STIM300 IMU supplies a "nominal" scale factor accuracy. We assume
# this is a single standard deviation of a normal distribution.
example_imu.gyro.scale_factor.repeatability = Parameter(500, "ppm")
example_imu.accelerometer.scale_factor.repeatability = Parameter(0.2, "%")
# -----------------
# Gyro misalignment
# -----------------
# Misalignment is the offset between the ideal orthogonal input reference axes
# (IRA) and the actual sensitive input axis (IA) of the sensor. It is comprised
# of a fixed component (that can be calibrated) and a random component (that may
# vary from turn-on to turn-on or with changing conditions).
# See Standard [05] and [06] for details.
# fixed: A unit-norm row vector specifying the sensitive input axis (IA) in the
# ideal orthogonal input reference axes (IRA). For a perfectly aligned
# three-axis sensors this results in a 3x3 identity matrix. For a real sensor
# with misalignments the result will be a rotation matrix that is not
# orthogonal. For many sensors, this fixed misalignment may be removed by
# factory calibration in which case this parameter can represent any residual
# error in that calibration. An example of a misaligned x-axis is:
# misalignment = Parameter(np.array([0.999,0.02,-0.04]),"dimensionless").
#
# Units must be "" or None.
# Default value = Parameter([[1,0,0],[0,1,0],[0,0,1]], "dimensionless").
# The majority of sensors are factory calibrated so no fixed misalignment error
# remains. We leave the parameter at its default.
example_imu.gyro.misalignment.fixed
example_imu.accelerometer.misalignment.fixed
# repeatability: Standard deviation of a (small) random angle offset between the
# sensitive input axis (IA) and the corresponding ideal input reference axis
# (IRA). The random offset will be applied around both of the mutually
# perpendicular input reference axes. For example, an x-axis offset of 0.01
# degrees will be treated as rotations by a normally distributed random angle
# with deviation of 0.01 degrees applied around the y and z input reference
# axes. This parameter may represent an unknown residual from the factory
# calibration of a fixed bias or an offset that varies with changing conditions
# like packaging stress and shock.
# Units must be one of: rad or deg.
# Default value = Parameter([0.0,0.0,0.0], "rad").
# The VectorNav VN-100 sensor supplies a "cross-axis sensitivity" parameter.
example_imu.gyro.misalignment.repeatability = Parameter(0.05, "deg")
# The Safran STIM300 IMU supplies a misalignment and an orthogonality parameter
# in units of milli-radians so we add them together and scale to a supported
# unit (rad).
example_imu.accelerometer.misalignment.repeatability = Parameter(1.2 * 1e-3, "rad")