Redundant sensors

In this example, we extend the Simple gyro example to a sensor with redundant sensing axes. Four (or more) sensors arranged in a non-orthogonal configuration can provide fault tolerance. It has been shown that this tetrad arrangement is superior to redundant 3-axis orthogonal systems (six total sensors). See Reference [21] for more details.

Sensor geometry

We simulate a 4-axis sensor with each sensing axis normal to the face of a tetrahedron. The sensing axes are illustrated in red in the image below (the fourth axis on the rear face is not shown for clarity). The x,y,z input reference axes are shown. It is apparent that one sensing axis is aligned with the negative z input reference axis while the other three axes measure some component of all the input reference axes.

Create and visualize the input

We reuse the input from the Simple gyro example.

# -----------------------------------------------------------------------------
# Copyright (c) 2023-2025, Inertial Simulation LLC.
# All rights reserved.
# Do not use or redistribute without permission.
# Email: info@inertialsim.com
# -----------------------------------------------------------------------------
%matplotlib widget

import numpy as np

from inertialsim import plot
from inertialsim.geometry import Vector
from inertialsim.time import span

# Create a time 0:60s time vector sampled at 10Hz
time = span(start=0.0, end=60.0, dt=0.1)

# Create a 3-axis sinusoidal signal with arbitrary frequency and phase.  This is
# applied to the input reference axes (IRA).  See Figure above.
angular_rate = np.zeros((time.size, 3, 1))
angular_rate[:, 0:1] = 0.001 * time - 0.02 * np.sin(0.3 * time) + 0.02
angular_rate[:, 1:2] = -0.002 * time + 0.015 * np.sin(-0.15 * time)
angular_rate[:, 2:3] = 0.0015 * time - 0.03 * np.sin(0.05 * time)

rate_vector = Vector.from_xyz(angular_rate, time)

rate_plot = plot.TimeSeries(title="Input Angular Rate", ylabel="Angular Rate (rad/s)")
rate_plot.line(time, angular_rate, xlimits=[0, 60])
rate_plot.legend(["X", "Y", "Z"])

Figure

Simulate a redundant gyro measuring the input

Next we create and simulate an ideal 4-axis gyro (no noise, bias or other errors) in order to visualize the redundancy. The GyroSpecification has a misalignment.fixed field which maps the input reference axes (typically marked on the case of the physical device) to the internal sensing axes.

from inertialsim.sensors import Parameter, SensorModel
from inertialsim.sensors.gyro import Gyro, GyroSpecification

# Create a gyro model and keep the input sampling rate (10Hz).
model = SensorModel()
model.data_interface.simulate_sample_rate = False

# Create a default specification and set the input axes
specification = GyroSpecification(axes=4)
# Each row maps the input reference axes (IRA) to the measurement axes.  The
# components are functions of the tetrahedron external angles (120 degrees) and
# dihedral angle (~70.529 degrees).
specification.misalignment.fixed = Parameter(
    [
        [np.sqrt(2 / 3), 0, np.sqrt(1 / 3)],
        [-np.sqrt(1 / 6), np.sqrt(1 / 2), np.sqrt(1 / 3)],
        [-np.sqrt(1 / 6), -np.sqrt(1 / 2), np.sqrt(1 / 3)],
        [0, 0, -1],
    ],
    "dimensionless",
)
gyro = Gyro(model, specification)
output = gyro.simulate(angular_rate=rate_vector)

rate_plot = plot.TimeSeries(title="Gyro Measurements", ylabel="Angular Rate (rad/s)")
rate_plot.line(time, output.angular_rate.data, xlimits=[0, 60])
rate_plot.legend(["Axis 1", "Axis 2", "Axis 3", "Axis 4"])

Figure

Analyzing the result

As expected the fourth sensing axis measures the (negative) z-axis input while the other axes measure components of all the inputs. The first sensing axis lies in the x-z input plane and does not measure any component of the y input. This examples demonstrates InertialSim's ability to simulate any combination of input and sensing geometry.