## 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.

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

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.

## Trapped in a Gimbal lock

Describing aircraft attitude with Euler angles has probably been the most common way to do so, especially when considering the kinematic equations that describe aircraft motion. However solving the Euler based kinematic equations for simulation purposes contains a mighty trap… singularities! Those singularities are due to the vertical tangent of pitch angle (theta) which result to the phenomenon of gimbal lock. In short gimbal lock is encountered when the grand-parent and grand-child axis of a  three-set gimbals align and that’s the point where you lose one degree of freedom and cannot rotate as desired anymore. Your system of equations blows up and you are trapped in the gimbal lock!

There are a couple of solutions to escape a gimbal lock. The first and cheapest one is to keep yourself away from that point – not really a realistic option especially when you are looking to simulate aerobatic maneuvers. The second and much preferred solution is to transfer yourself out of the Euler angles into the dark side of quaternions. Quaternions are another way of rotating an Euclidean vector in a more compact way and without singularities. They are composed from a scalar and a vector part where practically the vector part describes the rotation axis and he scalar part the angle of rotation.

$q=(r,\vec{v}), q\in \mathbf{H}, r \in \mathbf{R}, \vec{v} \in \mathbf{R^{3}}$

The penalty you have to pay to transit to the quaternion formulation of the kinematic equations, beside the conversion of your solver code and the developing of the understanding behind it, is the addition of one extra equation, as now instead of 3 you have 4 equations.

Math are great… you just need to get used to them :-)

Posted in Aviation Science | 1 Comment

## My Greek Hurricane Mk.IIc skin ~ 2000

Back in the days (~2000) we were flying the Jane’s WW2 Fighters simulator on the dawn of the online flight simulation era. Even though the Hurricane was not one of the main aircraft I flew, historical reasons imposed me to create a skin as flown by Royal Hellenic Air Force pilots in Egypt during the 2nd World War with Greek markings on the fuselage and RAF markings on the wings. Minimum tribute to those heroic pilots…

## MIL-STD-1797 FQ Criteria & Ghost Aircraft Project

MIL-STD-1797 is one of the most interesting Flying Qualities documents out there. Providing the extra steps that move away of the classical modal requirements found in MIL-F-8785C, 1797 contains criteria which can be much more sophisticated in order to cover the non-conventional FBW aircraft.

In short, some of those criteria (like LOES and Bandwidth) along with HQR tasks, were used to evaluate a number of different FCS laws for the Air-to-air refueling role as part of an exercise. Use of MATLAB and its toolboxes has made the calculation of the criteria parameters much less troublesome than it used to be few years ago.

The exercise under the name “Ghost Aircraft Project” was much enjoyed by TPS students and one of its humorous presentation slides is seen below (I was not among the presenters by the way, they just put my name there).

## PIO Rating Scales

While the standard Cooper-Harper rating scale is well documented and the way that it should be used is very clear, the case is not the same with all rating scales. The PIO rating scale is one such example.

The most common (standard) PIO rating scale used is the one found in MIL-STD-1797 initially created by Calspan (1981); however the ratings 1-6 are not accompanied by any descriptions, and no extensive guide is found for this scale. Veridian Engineering modified the standard scale (1999) combining it with some descriptions and changing slightly one of its decision tree questions. According to my limited experience, this version of the scale (seen below) can provide more consistent results as the ratings are clearly defined.

The question which arises though, for both the standard scale and for the suggested modified version is the exact definition of “oscillation” and “undesirable motion”. What is the difference?
While (according to my knowledge) it is not officially documented , during the Calspan Variable Stability Training course some interesting points are made on this point and generally on the scale:

1. An undesired motion is an simple overshoot or a small quickly damped cycle.
2. An oscillation is more than half a cycle or one overshoot.
3. The answer to whether an undesirable motion compromised task performance or not is a judgement the test pilot must make.
4. The PIO rating scale is not to replace CHR, but it is an other communication aid between the pilot and the engineers.

An interesting part of a GARTEUR report describing the evolution of the PIO scale and the versions which are currently used is found attached.

PIO Rating Scales – Garteur

## Boeing 777-300ER, Flight Control System evaluation

A number of test points including HQ tasks, pitch captures, offset approaches as well as response type identification using basic step inputs took place in Toronto CAE simulation facilities. Primary, Secondary and Direct laws were evaluated and envelope protections were verified for each case. Really good HQs even in Direct law for an inexperienced airline pilot.

The last time I flew in the B-777 sim was 10 years ago in London, UK. Time goes by fast…