Inertial Navigation Notes

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References

Pitch, Roll, Yaw, and Attitude

Pitch Roll and Yaw of a plane
^{Image from https://atadiat.com/en/e-towards-understanding-imu-frames-vpython-visualize-orientation}

Roll: Rotation about the forward body axis. It describes tilting left or right.

Pitch: Rotation about the lateral body axis. It describes tilting up or down.

Yaw: Rotation about the vertical body axis. It describes heading or turning left or right.

Attitude: The orientation of the sensor body relative to a chosen reference frame. It describes how the body axes are rotated with respect to a navigation frame such as North East Down or Earth centred Earth fixed. For exampl, an aircraft is flying straight east, wings level, with its nose raised slightly. Its attitude is:

Roll = 0 degrees, no left or right tilt.
Pitch = 5 degrees nose up.
Yaw = 90 degrees, pointing east relative to north.

This single set of angles fully describes the aircraft orientation relative to the navigation frame.

Attitude can be represented by the vector [roll, pitch, yaw], but attitude itself is a three dimensional rotation, not inherently a vector. It is a rotation between coordinate frames.

Rotating body frame to reference fame image

A vehicle lets say, moving at a constant attitude.

2D Example

A gyro measures angular velocity, $ω$ , so get get the angle of the base frame w.r.t to the navigation frame we integrate $ω$ w.r.t. time:

$θ (t) = \int_{t_{0}}^{t} ω (τ) d τ$

Where $τ$ is a dummy variable for integration ( $t$ used for the integration limits).

Once we have $θ$ that will rotate the force from the body to reference frame we can do the rotation - well i think it is more of a projection of the body force vectors onto the nagivation frames axis:

Project body frame force vectors onto navigation frame E, or x, axis

So we get:

$x_{i} = | z_{b} | \sin (θ) + | x_{b} | \cos (θ)$

And using similar reasoning:

$z_{i} = | - x_{b} | \sin (θ) + | z_{b} | \cos (θ)$

These forces measured by the IMU are accelleration. Integrate to get velocity.

$\begin{aligned} V_{x_{i}} (t) & = \int_{t_{0}}^{t} x_{i} (τ) d τ + V_{0} \\ = \int_{t_{0}}^{t} | z_{b} (τ) | \sin (θ (τ)) + | x_{b} (τ) | \cos (θ (τ)) d τ + V_{0} \\ = [| z_{b} (t) | - \cos (θ (t)) + | x_{b} (t) | \sin (θ (t)) + K_{1}] - [| z_{b} (t_{0}) | - \cos (θ (t_{0})) + | x_{b} (t_{0}) | \sin (θ (t_{0})) + K_{2}] + V_{0} \\ = [| z_{b} (t_{0}) | \cos (θ (t_{0})) - | z_{b} (t) | \cos (θ (t))] + [| x_{b} (t) | \sin (θ (t)) - | x_{b} (t_{0}) | \sin (θ (t_{0}))] + V_{0} + K \end{aligned}$

Integrate again to get distance.

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Inertial Navigation Notes

Page Contents

References

Pitch, Roll, Yaw, and Attitude

Rotating Body Frame To Reference (Navigation) Frame

A 2D navigation example