## Calculating angle of attack without an air data boom

Angle of attack ($\alpha$) and angle of sideslip ($\beta$) have always been some of the most difficult parameters to measure precisely. A number of methods has been proposed in the past for measuring those angles with the aid of an inertial reference (IRS/INS) [1], most of which include -to various extent- aerodynamic modelling of the vehicle under test.

The present post reviews a method initially developed and used for the YF-16 [2] where $\alpha$ and $\beta$ values can be derived for both static and dynamic test techniques. Considering the high accuracy inertial systems available today, this is a particularly interesting method as only an INS is required without any aerodynamic modelling and can be used in evaluations of aircraft with no available $\alpha$ and $\beta$ data acquisition -common in some TPS exercises.

Attention: By no means does this suggest that an air data boom is not required, it just reviews a method in case one is not available. There are a number of assumptions involved, which -despite being reasonable- can have an effect on the accuracy of the result.

For this method the data needed are inertia velocities (with reference to Earth axes), wind information and the aircraft Euler angles and rates. Basic assumptions of the simplified steps presented below include that aircraft is rigid and the wind is acting only on the horizontal plane.

First the Earth axes INS velocities ($E$) are converted to velocities relative to the surrounding air mass ($am$) by correcting for wind. (Accurate wind knowledge might be another challenge, but it can either be calculated using other techniques, or derived by the onboard air data computer.)

$V_{x_{am}}=V_{x_{E}}+V_w\cos\psi_w \\ V_{y_{am}}=V_{y_{E}}+V_w\sin\psi_w \\ V_{z_{am}}=V_{z_{E}}$

Then the conversion to INS sensed body axes can be applied using the following matrix.

$\left| \begin{array}{ccc} V_{x\prime_B}\\ V_{y\prime_B}\\ V_{z\prime_B} \end{array} \right| = \left| \begin{array}{ccc} \cos\theta \cos\psi & \cos\theta \sin\psi & -\sin\theta \\ -\cos\phi \sin\psi+ \sin\theta\sin\phi\cos\psi & \sin\theta\sin\phi\sin\psi+\cos\phi\cos\psi & \cos\theta\sin\phi \\ \sin\theta\cos\phi\cos\psi+\sin\phi\sin\psi & \sin\theta\cos\phi\sin\psi-\sin\phi\cos\psi & \cos\theta\cos\phi \end{array} \right| \left| \begin{array}{ccc} V_{x_{am}}\\ V_{y_{am}}\\ V_{z_{am}} \end{array} \right|$

The actual body axis velocities can be calculated by correcting for the displacement of the sensed INS axes from the aircraft CG ($l_x, l_y, l_z$) using the body axes rotational rates ($p,q,r$). The latter can either be derived from the Euler angle rates, or can be read directly from onboard rotational gyros.

$p=\dot{\phi}-\dot{\psi}\sin\theta\\ q=\dot{\theta}\cos\phi-\dot{\psi}\cos\theta\sin\phi\\ r=\dot{\psi}\cos\theta\cos\phi-\dot{\theta}\sin\phi$

$V_{x_B}=V_{x\prime_B}-ql_y+rl_p \\ V_{y_B}=V_{y\prime_B}-rl_r+pl_y \\ V_{z_B}=V_{z\prime_B}-pl_p+ql_r$

Using the corrected body velocities $\alpha$ and $\beta$ can then be derived from the following relations:

$\alpha=\arctan\frac{V_{z_B}}{V_{x_B}}\\ \beta=\arctan\frac{V_{y_B}}{\sqrt(V_{x_B}^2+V_{z_B}^2)}$

The accuracy of the method depends highly in the accuracy of the measured data and would require an error analysis. From similar studies it is estimated that an accuracy of 0.5 deg could be feasible.

References
[1] Zeis, J.E., “Angle of attack and sideslip estimation using an inertial reference platform”, MSc Thesis, AFIT, 1988.
[2] Olhausen. J. “Use of a Navigation Platform for Performance Instrumentation on the YF-16”. AIAA 13th Aerospace Sciences Meeting. AIAA-75-32., Pasedena, CA, Jan 75.

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