Continuous-Discrete Quadrature Filters for

Proceedings of the Third IFAC International Conference on Advances in Control and Optimization of Dynamical Systems (ACODS 2018)

Continuous-Discrete Quadrature Filters for Intercepting a Ballistic Target on Reentry using Seeker Measurements Rahul Radhakrishnan ∗ Shovan Bhaumik ∗ Nutan Kumar Tomar ∗ ∗

Indian Institute of Technology Patna, Bihar, 801106 India (e-mail: rahul.pee13, shovan.bhaumik, nktomar@ iitp.ac.in)

Abstract: In this work, a real-life ballistic target tracking problem is solved with continuousdiscrete (CD) quadrature filters, where the measurements are obtained from an inbuilt seeker of an interceptor. To obtain a more accurate approximation of the stochastic differential equation representing the process dynamics, Ito-Taylor expansion of order 1.5 is used. Further, the estimated states are fed to a guidance block which is implemented using proportional navigation guidance (PNG) law. The performance of the closed loop interceptor is studied in terms of root mean square error (RMSE) of target-interceptor states and miss-distance. It has been observed that the RMSE and miss-distances obtained from the CD quadrature filters are less when compared to the discrete-discrete quadrature filters. Keywords: State estimation, Kalman filter, Ballistic target tracking, PNG law, Miss-distance. 1. INTRODUCTION Accurate tracking of a ballistic target on reentry is important for its successful interception. Ballistic missiles, having a predetermined trajectory is usually directed at targets of strategic importance. This tracking problem finds its application in defense scenarios where foreign objects like enemy missiles, satellite debris etc. has to be successfully intercepted and destroyed at a desired distance from the ground, preferably by a guided missile (Mehra, 1971). Tracking or estimating the states of a ballistic target on reentry is a challenging task due to the nonlinear nature of the process and measurement models. Extended Kalman filter (EKF) was the first suboptimal solution applied for this kind of problem (Mehra, 1971; Siouris et al., 1997). Since EKF suffers from poor track accuracy and divergence, filters like unscented Kalman filter (UKF) and particle filter (PF) were applied for solving this problem (Farina et al., 2002; Ristic et al., 2003). All these works made use of measurements obtained from a static ground based RADAR. Tracking of a ballistic target using EKF and divided difference filter with measurements obtained from an active seeker unit mounted on an interceptor missile was reported in (Dwivedi et al., 2006; Bhale et al., 2006). Till date, the solution is obtained from discrete-discrete filters. Thus all the works discretized the process model for the implementation of filtering algorithm. However, improper or inaccurate discretization of the process dynamics can have an adverse effect on the performance of filtering algorithms (Sarkka, 2007). This is because of the fact that most of the real-life problems are nonlinear and the process dynamics is always represented in continuous domain. By

© ACODS 2018

choosing a proper discretization method which can accurately approximate the stochastic differential equation representing the process with a step size which is much smaller than the measurement sampling time, the continuous nature of the process dynamics can be preserved up to some extent. This approach to nonlinear estimation is referred to as continuous-discrete (CD) filtering. Euler-Maruyama method was the most popular approach for deriving the stochastic difference equation from continuous process model (Kulikov and Kulikova, 2016; Sarkka, 2007; Kulikov and Kulikova, 2017). Since this method gives only a simple first order approximation of the process dynamics, Ito-Taylor expansion of order 1.5 was proposed for approximating the continuous process model more accurately (Arasaratnam et al., 2010). Until now, the EKF, UKF and cubature Kalman filter (CKF) have been extended to the CD domain (Arasaratnam et al., 2010). The advantage of using this discretization method is that it includes more terms (higher order derivatives) in the approximated process model, such that the model becomes more accurate. In this work, we consider a real-life 6DOF ballistic target tracking problem where the measurements are obtained from an active seeker installed in the interceptor missile. So the entire problem scenario can be described as an interceptor missile equipped with an on-board computer which executes the estimation algorithm as well as the guidance law to track and intercept the incoming target. This phase of the engagement scenario is termed as terminal or homing guidance phase. The measurements generated by the seeker is processed by the estimator and guidance block for generating the guidance commands. Then these guidance commands are fed to an ideal autopilot, such that necessary corrections are made in the interceptor

- 399 -

Proceedings of the Third IFAC International Conference on Advances in Control and Optimization of Dynamical Systems (ACODS 2018)

dynamics. Thus, a closed-loop target-interceptor scenario is defined so that the position of interceptor is fed back to the estimator. This kind of ballistic target-interceptor engagement scenario is not yet explored using the CD filtering approach. Contribution of this work can be summarized as follows: a) implementation of CD quadrature based filters for accurate tracking and interception of a ballistic target on reentry, where the 6DOF state dependent coefficient (SDC) model is approximated using Ito-Taylor expansion of order 1.5 b) implementation of proportional navigation guidance (PNG) law for determining interceptor accelerations and feeding them back to the estimator. All these calculations were made under the assumption that an ideal autopilot is considered, which realizes the guidance commands instantaneously. Performance of the CD quadrature filters is compared based on RMSE of target-interceptor states and miss-distance. It has been observed that CD filters performed with higher accuracy and lower miss-distance. 2. PROBLEM FORMULATION In this section, 3D behaviour of a ballistic target and interceptor is discussed, where the measurements are obtained by an active seeker mounted on an interceptor missile. Here we opt for a 6DOF SDC model for modeling the targetinterceptor dynamics (Dwivedi et al., 2006; Li et al., 2001). In SDC model, target accelerations are replaced by a single parameter, the ballistic coefficient (β) which is included as a state.

2.2 Seeker measurement model Terminal guidance phase of target-interceptor scenario consists of the interceptor mounted seeker obtaining measurements regarding the incoming ballistic target. The measurements obtained are range rate, gimbal angles along yaw and pitch plane and line of sight (LOS) rates along yaw and pitch plane. Range rate is considered in LOS frame, gimbal angles along fin-dash frame and LOS rates along inner gimbal frame. Appropriate frame transformations are done wherever required. In Fig. 1, LOS frame of reference is represented by (Xl , Yl , Zl ) axes and (Xi , Yi , Zi ) represents inertial frame of reference. It can be noted that LOS vector is along the direction of Xl . Here λa is the azimuth angle and λe , the elevation. These are expressed as (Dwivedi et al., 2006):  ∆y  k λa,k = tan−1 (2) ∆zk and   ∆xk −1 p . (3) λe,k = tan ∆yk2 + ∆zk2 Rate of change of these angles are ∆zk ∆vy,k − ∆yk ∆vz,k λ˙ a,k = (4) ∆yk2 + ∆zk2 and ∆vx,k (∆yk2 + ∆zk2 ) − ∆xk (∆yk ∆vy,k + ∆zk ∆vz,k ) p λ˙ e,k = (∆x2k + ∆yk2 + ∆zk2 ) ∆yk2 + ∆zk2 (5)

2.1 Process model

Xi

Let us assume xt , yt , zt , vtx , vty , vtz , atx , aty , atz are the positions, velocities and accelerations of the target in x, y and z direction respectively. Similarly, xm , ym , zm , vmx , vmy , vmz , amx , amy , amz are the positions, velocities and accelerations of the interceptor missile. Now the relative positions and velocities in inertial frame can be represented as ∆x = xt − xm , ∆y = yt − ym , ∆z = zt − zm , ∆vx = vtx − vmx , ∆vy = vty − vmy , ∆vz = vtz − vmz . The rate of change of relative positions and velocities can be written as (Radhakrishnan et al., 2016): ˙ = ∆vx , ∆y ˙ = ∆vy , ∆z ˙ = ∆vz , ∆v ˙ x = atx − amx , ∆x

Target Xl ∆x

LOS Interceptor Yl Zl λe ∆y

Yi

Zi ∆z λa

P rojection of LOS vector

˙ = 0. ˙ y = aty − amy , ∆v ˙ z = atz − amz , 1/β ∆v Now the stochastic differential equation representing the process can be expressed as: p dX(t) = f (X(t), t)dt + Qdς(t), (1)

Fig. 1. Representation of LOS frame w.r.t inertial frame

where X = [∆x ∆y ∆z ∆vx ∆vy ∆vz 1/β]T , f = [∆vx ∆vy ∆vz atx −amx aty −amy atz −amz 0]T , Q ∈