Estimation of Roll Damping Coefficient from Roll and Trajectory Data Using Extended Kalman Filter


 In this paper, a new method to estimate roll aerodynamic characteristics of a rolling projectiles is proposed. It is estimated from the measured roll rate and trajectory positional data using Extended Kalman Filter. Modified point mass model of trajectory modelling, in state space form, is used to represent system dynamics of Extended Kalman Filter. The roll and position data at every time step constitutes the measurement vector. Along with positions and velocities, roll damping coefficient is included as a state variable. As roll damping coefficient depends on projectile configurations and Mach number. Roll damping coefficients are estimated for two configurations viz. roll stabilized shell and fin stabilized rocket. The measurements are simulated for full flight regime to cover complete Mach regime. Estimated values are compared with known results for various Mach numbers. In both the cases estimation is in close agreement with known results.

Well-designed projectiles will have an equilibrium roll rate that avoids both the yawing frequency and their first natural frequency of vibration [2]. The roll rate is directly related to stability of projectile hence the roll damping coefficient is crucial input for trajectory modelling. It should be known during design and can be refined during development.
In order to design a projectile, the aerodynamic coefficients must be predicted to within engineering accuracy.
Initial estimates of aerodynamic coefficients are usually obtained using semi-empirical engineering codes and computational methods. Analytical and semi-empirical methods are used in the literature to estimate C l p and C l δ .
Prediction of rolling moment coefficient is relatively easy and straightforward while only approximate methods are used for estimating roll damping coefficient. Difficulties arise because of the assumptions and limitations of each analysis or approximate method. Also, very often, only supersonic speed regimes are considered because of the relative ease in using the linearized supersonic theory [3]. Thus, estimation of roll damping coefficient in transonic and subsonic applications remain grey area.
Most of initial estimates of aerodynamic coefficients are refined further using wind tunnel tests and / or flight tests. However, the roll-damping moment coefficient is difficult to measure and requires a free rolling rig in association with a sting balance [4] for its measurement. As the magnitude of bearing friction of rolling rig under test conditions is difficult to estimate, friction effects on the measured aerodynamic characteristics are usually not adequately evaluated [5]. Thus, there is a need of estimating accurate roll-damping moment by other method.
An estimation of unknown aerodynamic coefficients from flight data has been of interest for many years. The coefficients are used to provide final verification flight [6]. Nowadays the position and roll during flight can be obtained using tracking radar or telemetry. Parameter estimation techniques allow an engineer to form an empirical model of a system using measured data. Filtering techniques are generally employed to remove noise and estimate unobserved state.
Kalman Filter is an accepted and well-known technique for state estimation. The EKF estimation method can deal with both process and measurement noise. It gives an optimal state estimate even in the presence of noise and un-modelled dynamics. In Extended Kalman filter, nonlinearity of system is taken care by employing instantaneous linearization at each time step. It assumes Gaussian distribution for the uncertainties in system dynamics. Furthermore, unknown variables can be included as part of state variables and parameter estimation problem can be transformed into state estimation problem [7].
In [8], Kalman filtering technique is used for trajectory estimation of a maneuverable target utilizing the information acquired from ground-based radar. Use of different version of EKF for projectile identification and impact point prediction is also reported in [9]. Neural Kalman filter approach to target tracking is presented as a technique to improve the motion model of the target while it is being tracked in flight [10]. EKF is successfully used to estimate ballistics coefficient [11], drag coefficient [12][13] and atmospheric conditions [14] from trajectory data. Flight state parameters of guided projectile are obtained using Extended Kalman filter (EKF) and Linearized Kalman filter (LKF) [15].
The aim of this paper is to present an approach to extract information from flight test data using Extended Kalman Filter. An estimation approach is applied to predict the roll damping coefficients. Problem of estimation of roll damping coefficient is transformed into the state estimation problem. The roll damping coefficients of a projectile is obtained from the measured roll rate profile and positional trajectory data using Extended Kalman Filter. The simulation is carried out for roll stabilized and fin stabilized projectiles to cater different configuration and roll rate so that proposed method can be applied for real-world problem. There are limitations of methods of estimating roll damping coefficient during design.
When trajectory data is available the method proposed in this paper will be a better tool to refine the estimates.

A. Rolling Motion Equation
The equation of rolling motion about the body axis is usually written as where p = ∅̇ is roll rate, d is reference diameter, v is velocity, q ∞ is dynamic pressure, A ref is reference area and I xx is axial moment of inertia and For steady state roll motion (p = p s ), ∅̈ = 0 and therefore Various researchers have provided empirical relation between C l p and C l δ by assuming their ratio to be a constant during steady state motion.
For four-fin body in the plus (+) formation, i.e., cruciform fins, Adams and Dugan obtained the expression where b o is the total span of two-fin panels including the center body diameter. Eastman empirically wrote the following correlation in a form similar to the result of the analysis of Adams and Dugan C l p C l δ = -2.15 y c d (4) in which y c is the distance between the rolling body axis and the area center of one fin panel. Eastman showed that his correlation was valid not only for supersonic speeds as Adams and Dugan's analysis implied but for all speed regimes.
This approximation was extended to an arbitrary number of fins. This was further extended for curved and wraparound fins as follows Were ( It is known that C l δ is relatively easier to measure in wind tunnels or compute with analytic and empirical formulae.
Estimating the roll-damping moment coefficient from wind tunnel is much more difficult. It requires a free rolling rig in association with a sting balance. Moreover, the magnitude of bearing friction of rolling rig under test conditions is difficult to estimate, so friction effects on the measured aerodynamic characteristics are usually not adequately evaluated [4][5].
The correlations in equations (3), (4) or (5) are used to find coefficient C l p using C l δ of specific projectile. It means different projectile configurations like rolling, non-rolling and with different number of fins has different valid empirical relation.
Moreover, for constant C l p C l δ ⁄ , steady state roll rate p s must vary identically with the velocity along the trajectory. In general, velocity of projectile decreases over time in ballistics phase. Thus, p must also decrease to keep the ratio C l p C l δ ⁄ constant. Therefore, if p decreases and if the ratio of C l p C l δ ⁄ is constant along the trajectory, then this does not translate to a true steady state roll value [16]. Thus, none of these relations is direct, accurate, or valid for all speed regimes, and hence, each of these methods of estimating C l p has limitations and regime of validity. Furthermore, these calculations are effective for a particular configuration. Thus, there is need to obtain accurate roll-damping moment coefficient by other practical methods.

B. Projectile Dynamics
If the projectile has good dynamic stability modified point mass (MPM) model can be used to approximate the motion of the projectile [13]. MPM model is conventional point mass model along with axial roll and instantaneous equilibrium yaw term added to it. The MPM takes into account accelerations due to drag, lift, Magnus force, Coriolis force and gravity.
It estimates angle of yaw, drag, drift and Magnus force effects resulting from yaw of repose. Drag, lift, Magnus force coefficient and pitch and roll damping coefficients are aerodynamics inputs to the model [17]. It has been proved that modified point mass model accurately calculates trajectories of roll stabilized, slowly rolling and finned projectiles [18].
To define trajectory equations a right-handed, orthonormal, ground-fixed, Cartesian coordinate system as a frame of reference is considered and shown in Fig 1. The modified point mass model can be written in scalar form as follows [-(C D 0 +C D α 2 α r 2 ) vv y +C L α α y v 2 -d 2 C γ pα pm y ] + c y -g y (10) where r= √ x 2 +y 2 +(x+R)

C. Extended Kalman Filter
Kalman filtering is used to estimate unknown variables from a series of measurements containing statistical noise and other inaccuracies.
To apply Extended Kalman Filter (EKF) the set of non-linear differential equations representing the system must be expressed in a state space form as [11] ẋ=f(x)+w (13) where vector x represents the state of the system, f(x) is a nonlinear function of the state variables and w is a random, zero mean process noise. Presence of process noise indicates that model of real system is not precise. Covariance of process noise w is represented by a matrix Q and is called process noise matrix.
Let the relation between measurement and state vector be given by z=h(x)+v (14) where z is the measurement vector, h(x) is the measurement equation and v is the zero mean measurement noise.
For a system in which measurements are discrete we can rewrite nonlinear measurement equations as where z k is the measurement vector and v k is noise vector at instance k. The discrete measurement noise matrix R k consist of matrix of covariance between measurement noise source.
The system dynamics matrix F and the measurement matrix H at any instance are related to nonlinear system and measurement equations as The fundamental matrix Φ k required for propagating state forward can be approximated by Taylor series expansion for exp (FT s ) and is given by

3! +…
Often this series is approximated by only first two terms Where T s is the sampling time and I is the identity matrix. In EKF fundamental matrix is used for calculating Kalman gain.
Riccati equations, required for the computation of the Kalman gains are The discrete process noise matrix Q k is calculated using equation (22) With an extended Kalman filter the new state estimate is the old state estimate projected forward to the new measurement time plus gain times a residual where The time step for the integration was set to match the data rate of measurement and Runge-Kutta 4th order integration was carried out to predict the system states.
The system state variables are defined to include all the parameters of interest. The state is estimated at every measurement step and thus estimation of the parameters of interest is obtained.

A. Formulation of Problem
This study includes both formulation of appropriate models and estimation of the parameters in these models.
Estimation of parameters through the filtering method is an indirect process. Here, parameter estimation problem is transferred into a state estimation problem. Modified point mass model is chosen as system dynamics since it adequately describes projectile dynamics. The unknown parameter is defined as additional state variables in system state. In this problem, roll damping coefficient C l p is considered as unknown variable and augmented to other seven state variables namely, three projectile positions, three velocities and roll. Total number of state variable will be eight and each will be estimated using EKF from four measured variables namely range, height, drift and roll of the projectile at every instance.
The measurement vector is z=[x, y, z, sp] T and in state space form z=h(x)+v can be written as where [x , ỹ , z, sp ] T denotes actual positions and roll and[δ x , δ y , δ z , δ sp ] T is measurement Gaussian noise in respective coordinates and matrix R covariance of measurement noise.

B. Extended Kalman Filter Design and Implementation
The simulation flow chart is shown in Fig 2. The MPM model i.e equation (6)(7)(8)(9)(10)(11)(12) is used to simulate the trajectory elements and suitable noise is added to it to reflect it as measurements as described in equation (26). These measurements are passed to EKF model and all eight states variable are estimated for each time step.

C. Filter Initialization
To begin the Kalman filtering procedure an initial estimate of both, the states and the covariance matrix, are essential.
These can be initialized in different ways. For the unknown variable the most common way to do so is by a simple guess.
Here initial value of unknown variable roll coefficient is obtained from empirical method. The other states can be initialized using first few measurements [19] and simple differentiation.

A. Simulation of Measurement Data
Generally, projectiles are either roll stabilized or fin stabilized. To cover both type of configurations estimation of roll damping coefficients for roll and fin stabilized projectiles are carried out. In roll stabilized configuration there is no fin to projectile hence roll produced roll moment coefficient and fin cant angle value are zero. This configuration has very high roll rate. In fin stabilized configuration roll moment coefficients and fin cant angle are non-zero. Fin stabilized configuration is slowly rolling configuration. For simulation purpose it is assumed that these values are known.
Configurations and inputs considered are given in Table 1. The complete details on aerodynamics, mass and inertia for configuration A are referred from [20] and for configuration B from [2].

B. Estimation of Roll Damping Coefficient for Roll Stabilized Shell
In this case, the measurements are simulated for muzzle velocity 935m/s and 50 deg elevation in standard atmospheric conditions for configuration A, i.e., roll stabilized shell described in Table 1. The measured data contains x, y, z, sp for every time step of 0.1s from launch to impact point. The maximum velocity is 935m/s and minimum velocity is 325m/s during the flight it means Mach No 0.95 to 2.50 is covered.
The aim of simulations is to estimate the roll damping characteristic of the projectile. As it is roll stabilized configuration C l δ = 0 and δ = 0 and characteristics of roll will depend only on C l p . In EKF roll damping coefficient μ Clp is included as unknown variable along with other state variables. The state x= [x, y, z, sp, x, y, z, ̇μ Clp ] T is estimated for each time step T s . The initial conditions for EKF are appropriately set and process and measurement noise matrices are initialized based on noise level.
The velocity components namely horizontal (ẋ=u x ), vertical (z=u z ) and lateral (ẏ=u y ) are estimated at every time step.
The difference between estimated and actual is shown as error in estimation for all three velocities. These are plotted in

C. Estimation of Roll Damping Coefficient for Fin Stabilized Rocket
Roll damping coefficient is successfully estimated for roll stabilized shell where C l δ = 0 and δ =0. It is necessary to check same methodology for roll stabilized rockets where C l δ ≠ 0, δ ≠0 and characteristics of roll depends on C l p , C l δ and δ. In this case rocket configuration B, as described in Table 1 The convergence of three velocities are shown in Fig. 8. The difference between actual with estimated roll rate and errors in estimated roll damping coefficient with theoretical error shown in Fig. 9 Fig. 11. In this case also percentage error is less than 5%. In both cases the estimation of roll damping coefficient is within 5% after convergence. The estimated and known roll damping coefficient is plotted with respect to Mach number and shown in Fig. 12 and 13. The close agreement in estimation indicates practicability of proposed methods and mathematical model for estimation of roll damping coefficient.