Generation of artificial IMU data from simple waypoint trajectory

Hello,

I would like to generate some synthetic IMU (accelerometer and gyroscope only) data given a sampled 3D trajectory using the imuSensor system object from MATLAB's navigation toolbox.

The IMU sensor object https://www.mathworks.com/help/nav/ref/imusensor-system-object.html requires acceleration (in the local navigation coordinate system, e.g., NED) and angular velocity (also in the local navigation coordinate system). I only have waypoints (positions in 3D) and assume that regarding orientation, there is no auto-roll or auto-pitch, but just "auto-yaw" meaning that the nose points always in the direction of movement.

I started with a simple trajectory, a circle in the plane: (cos(t), sin(t), 0) and sampled it (e.g., with 10 Hz one full revolution) to obtain "waypoints". See figure below:

Using the waypointsTrajectory object https://www.mathworks.com/help/fusion/ref/waypointtrajectory-system-object.html, I obtain via [position,orientation,velocity,acceleration,angularVelocity] = trajectory(), the acceleration and angularVelocity (in the local navigation coordinate system), which are required by the IMU sensor object to output accel and gyro readings in the sensor body coordinate system. As stated above, I use no auto-pitch, no auto-roll, only the auto-yaw, a setting which is implicitly used by the waypointTrajectory object.

The resulting accel data (in the sensor body coordinate system) confuse me though. See below:

On the body z-axis, as expected we measure gravitational acceleration, but on the x and y we have a sinusoidal acceleration. If my device is moving with "auto-yaw" along a circle (I verified that speed is constant) shouldn't I just see a constant acceleration on the x or y axis (namely the centripetal acceleration)?

Thanks!

[Brian Fanous]

Can you explain your setup a bit more? Are the plots shown in your post the outputs of the imuSensor or the outputs of the waypointTrajectory? Can you post the code you used?

[Effesian]

Hello,

Figure 1 (showing dots forming a circle) are the waypoints obtained by sampling r(t) = (cos(t), sin(t)) at regular time intervals.

The bottom two figures are the outputs of the imuSensor, that is, accelerometer readings (therefore you see accel_z = 9.81 m/s^2) and gyroscope readings.

Code snippet used to obtain acceleration and angular velocity in local navigation coordinate system from waypoints (circle in 2D and z component is 0):

trajectory = waypointTrajectory(waypoints', t, 'SampleRate', fs, 'AutoPitch', false, 'AutoBank', false);

count = 1;
while(~isDone(trajectory))
   [position(count,:),orientation(count),velocity(count,:),acceleration(count,:),angularVelocity(count,:)] = trajectory();
   count = count + 1;
end

Code snippet for getting (noise-free therefore I multiplied the nose terms by zero) accelerometer and gyroscope readings:

% Accel params
ACCEL_BIAS = 0*[0.1, 0.2, 0.3]; % Units [m/s^2]
imuError.accel_velocity_random_walk = 0*3e-3; % Units [m/(s^2*sqrt(Hz))]
accelParams = accelparams('ConstantBias', ACCEL_BIAS, 'NoiseDensity', imuError.accel_velocity_random_walk);

% Gyro params
GYRO_BIAS = 0*[0.1, 0.2, 0.3]; % Units [m/s^2];
imuError.angle_random_walk = 0*8e-4;
gyroParams = gyroparams('ConstantBias', GYRO_BIAS, 'NoiseDensity', imuError.angle_random_walk);

% Generate IMU readings
imu = imuSensor('accel-gyro', 'SampleRate', fs, 'Accelerometer', accelParams, 'Gyroscope', gyroParams);
[accelReadings, gyroReadings] = imu(acceleration, angularVelocity, orientation');

accelReadings and gyroReadings are then plotted. As written above, what confuses me is the accel values on the x and y axis.

Thanks!

Answer 1

[Brian Fanous]

The last line of your code:

[accelReadings, gyroReadings] = imu(acceleration, angularVelocity, orientation');

has a sublte but very important bug. Just like for complex numbers in MATLAB, when you use the ' operator on a quaternion array you are taking the conjugate transpose. So

orientation'

is actually conjugating all the quaternions in the array and transposing the array. You should just use the transpose operator here:

orientation.'

which does not conjugate. Try that and you'll see the sinusoidal artifact in the accelerometer signal disappear.

[Effesian]

@Brian Fanous: Thanks! That fixed the issue. Now, besides gravity the simulated accel readings are constant on the y-axis and zero on the x-axis. If I am not mistaken this means that the x-axis points along the direction of movement and the y-axis in the opposite direction of the center of the circle such that is captures the full centripetal acceleration (with a minus sign).

[Effesian]

I have a follow up question. If it's better, I can ask it as a new question.

Using

trajectory = waypointTrajectory(waypoints', t, 'SampleRate', fs, 'AutoPitch', false, 'AutoBank', false, 'Velocities', velocityWp');

and

[position(count,:),orientation(count),velocity(count,:),acceleration(count,:),angularVelocity(count,:)] = trajectory();

to obtain acceleration and agular velocity values and feeding them into the imuSensor model

[accelReadings, gyroReadings] = imu(acceleration, angularVelocity, orientation.');

with waypoints forming a circle as shown above, now all works fine. However, if I provide the orientation explicitly as in

trajectory = waypointTrajectory(waypoints', t, 'SampleRate', fs, 'Orientation', orientation_atWp, 'Velocities', velocityWp');

such that the nose points along the trajectory (as in auto-yaw) and we have no pitch and no roll, then I get "jumps" in the gyro signal at the beginning and end.

I observed that the angular velocity signal is not constant, but has jumps at the beginning and end.

Oddly, the orientation that I provide to the waypoint trajectory object as well as the one I obtain as a return value from the waypoint trajectory object is smooth:

Any ideas where the jumps in the angular velocity returned by the waypoint trajectory object could come from?

Thanks!

[Greg Dionne]

When you use the Orientation property, waypointTrajectory decouples the orientation from the path. It instead takes the orientation from the supplied values. It will assume an initial and final angular velocity of zero which can be confirmed in your first plot.

Hope that helps

[Effesian]

Ok thanks! Since in my simulation the numer of provided waypoints and orientation samples should be the same as the numer of acceleration and angular velocity samples I get back from that function, I ended using lookupPose() instead of waypointTrajectory(). WaypointTrajectory does some interpolation and therefore leads to some undesired effects (e.g., the zero initial and final values you described.).

See

https://www.mathworks.com/matlabcentral/answers/1782040-how-is-the-number-of-sample-points-returned-by-waypointtrajectory-determined