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Estimation of yaw angle from flight data using extended Kalman filter

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Abstract

In any armament system, projectile yaw during flight is undesirable because it affects range. For drag calculation, the area of the cross section of the projectile perpendicular to the air flow, called reference area, is used. If yaw angle is zero, reference area depends only on the diameter of the projectile. When yaw angle is nonzero reference area is more; resulting in more drag. For a given projectile, a mathematical relation between yaw angle and corresponding reference area can be established. This paper proposes a method to estimate yaw angle at each point on the trajectory from the flight data. The method uses extended Kalman filter (EKF) and system dynamics is modeled using modified point mass trajectory model. At each instant reference area is estimated using EKF. This area is mapped to yaw angle through pre-established mathematical relation. The method is validated using simulation. In special cases, the results are validated with the help of actual flight data. The proposed method is effective and can be used for indirect measurement of yaw angle or angle of attack.

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Availability of data and material

The data used in in the paper is taken from cited reference and simulated.

Abbreviations

\({{x}}\) :

Horizontal distance (range), m

\({{y}}\) :

Lateral distance (drift), m

\({{z}}\) :

Vertical distance (height), m

\({{d}}\) :

Diameter of projectile, m

\({{m}}\) :

Mass of projectile, kg

\({{I}}_{{x}}\) :

Axial moment of inertia, kg m2

\({\rho}\) :

Density of air, kg/m3

\({{p}}\) :

Roll rate, rad/s

\({{v}}\) :

Projectile speed, m/s

\({\alpha}_{{r}}\) :

Yaw angle, rad

\({{t}}\) :

Time, s

\({{v}}_{{x}}{, }{{v}}_{{y}}{{, v}}_{{z}}\) :

Relative air velocity components, m/s

\({{u}}_{{x}}{, }{{u}}_{{y}}{{, u}}_{{z}}\) :

Projectile velocity components, m/s

\({{c}}_{{x}}{, }{{c}}_{{y}}{{, c}}_{{z}}\) :

Coriolis acceleration components, m/s2

\({{g}}_{{x}}{, }{{g}}_{{y}}{{, g}}_{{z}}\) :

Gravitational acceleration components, m/s2

\({{w}}_{{x}}{, }{{w}}_{{y}}{,}{{w}}_{{z}}\) :

Wind velocity components, m/s

\(\ddot{{x}}{, }\ddot{{y}}{,}\ddot{{z}}\) :

Projectile acceleration components, m/s2

\({{C}}_{{{D}}_{0}}\) :

Zero yaw drag force coefficient

\({{C}}_{{{D}}_{{\alpha}^{2}}}\) :

Yaw drag force coefficient

\({{C}}_{{{L}}_{\alpha}}\) :

Lift force coefficients

\({{C}}_{{\gamma}_{{p\alpha}}}\) :

Magnus force coefficient

\({{C}}_{{{M}}_{\alpha}}\) :

Pitching moments coefficient

\({{x}}\) :

System state vector

\({f(x})\) :

Nonlinear function of states

\({{w}}\) :

Random zero mean process noise

\({{z}}\) :

Measurement state vector

\({h(x)}\) :

Measurement equation

\({{v}}\) :

Random zero mean measurement noise

\(\left[\stackrel{\mathrm{\sim }}{{x}}{ , }\stackrel{\mathrm{\sim }}{{y}}{ , }\stackrel{\mathrm{\sim }}{{z}}\right]\) :

Actual positions

\(\left[{\delta}_{{x}}{ , }{\delta}_{{y}}{ , }{\delta}_{{z}}\right]\) :

Gaussian noise in the measurement

\(\upmu \) :

Exposed area of projectile at different yaw angle

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Acknowledgements

The authors gratefully acknowledge the support of Dr. VV Rao, Director ARDE, DK Joshi Director PXE, R. D. Misal Associate Director, M. Padmanabhan, A. Anandaraj and colleagues in the Ballistics Group for their valuable suggestions and encouragement during this work. Work is properly Acknowledged.

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Authors and Affiliations

  1. Armament Research and Development Establishment, Pune, Maharashtra, 411 007, India

    L. K. Gite & R. S. Deodhar

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  1. L. K. Gite
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  2. R. S. Deodhar
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Contributions

Myself LKG with Dr. RSD are worked on it. I have proposed the concept and verified through simulation. Dr. RSD supervised the complete work and suggested more simulations cases and modification.

Corresponding author

Correspondence to L. K. Gite.

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Not applicable since data used in the paper is open literature and relevant references are given. It is declared that there is no conflict of interest.

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Gite, L.K., Deodhar, R.S. Estimation of yaw angle from flight data using extended Kalman filter. AS 5, 393–402 (2022). https://doi.org/10.1007/s42401-022-00131-3

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