Sensor Fusion Based Missile Guidance

Transcription

1 Sensor Fusion Based Missile Guidance Pubudu N Pathirana School of Engineering and Technology, Deakin University, Geelong, Victoria 217, Australia pubudu@deakin.edu.au Andrey V Savkin School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney 252, Australia savkin@unsw.edu.au Abstract In this paper, we improve the guidance system performance via sensor fusion techniques. Vision based guidance systems can be improved in performance via radar tacking or employing video tracking by unmanned flying vehicles. We also introduce an image texture gradient based image segmentation technique to identify the target in a typical surface-to-air type application with the proposed Robust Extended Kalman Filter based state estimation technique for the implementation of the Proportional Navigation guidance controller. Keywords: Sensor fusion, Robust Extended Kalman Filter, Missile Guidance. 1 Introduction The need for improvement in the missile guidance systems was brought about as a result of recent developments in weapon system and sub-system technologies as well as a shift in guided weapon system deployment and operational philosophies. In the past, due to real-time computing constraints, major simplification of engagement kinematics model, performance index and constraints had to be implemented in order to render the solution suitable for mechanization of a real system. With recent technological advances, particularly in computing, the past constraints do not apply. It is now feasible to look at guidance strategies that are not only more efficient in achieving miss distances in order to maximize lethality, but also economical and simpler in production. Onboard video cameras can be used for missile guidance replacing the onboard radar systems specially in the terminal phase of a surface to air missile. The passive video imaging is less expensive while it consumes lesser energy and hence simplifies the onboard electronics considerably. In missile systems, not all the states are available for measurement. Images from a video signal can be used to This work is supported by the Australian Research Council estimate the target in the image plane and consequently this information can be used to estimate the state variables in the guidance system. A moving target captured by an onboard camera mounted on a SAM typically consists a target moving in a background uniformly moving in a specific direction. Considering the ideas of perspective projection of object on to the image plane via simple geometrical optic concepts, the resulting nonlinear state estimation problem can be addressed by Robust Extended Kalman FilterREKF) which is based on new theoretical developments presented in [1, 2]. The systems performance can be improved by incorporating additional radar measurement or employing additional cheaper unmanned flying vehiclesufv) or multiple missiles[]. In this application we consider the Proportional NavigationPN) guidance as the controller for its wide usage although a robust H type controller taking account of uncertain target maneuver can also be relevant[4]. Firstly, we introduce the missile/target kinematics in D case. Two example have been chosen for simulation to demonstrate the advantages of information fusion and multiple missiles in the missile guidance problem. Firstly, we compare the performance of a video guided missile together with such a missile assisted by a ground based radar station. Secondly, we consider the improvement of a video guided missile by using video captured by additional unmanned flying vehicles. Here, we consider the terminal phase of the Surface to air missile and assume non-rotational flight of the missile for the sake of simplicity which is a fair assumption for most cases in the considered flight phase. The same ideas can be extended for the flight with consideration for the rotational kinematics. 2 Missile Target Kinematic Model Let the position vector of the missile and the target with respect to the earth frame be x M 1, x M 2, x M ) and x T 1, x T 2, x T ) respectively. With the target/missile state defined as x T/M E = [x T/M 1 x T/M 2 x T/M ẋ T/M 1 ẋ T/M 2 ẋ T/M ],

2 Target Missile UAV Target Z Z Y Y Missile UAV Ground based Radar X X Figure 2.1: Missile navigation with ground radar station Figure 2.2: Missile navigation with unmanned aerial vehiclesuav) the target missile relative state x can be stated as : x = x T M = x T E x M E 2.1) x,x,x ) 1 2 T T T x,x,x ) E E E for system ẋ = Ax + B 1 u + B 2 w 2.2) with system matrices 1 1 A = 1 B 1 = 1, B 2 = ) with u = [a x a y a z ] and w being the control input vector and the target maneuver respectively. The system given in equation 2.2 represent the missile target kinematic model for both cases that we consider in this paper. Measurement Equation Perspective projection reflects image formation using an ideal pinhole camera according to the principals of geometrical optics[5]. The following algebraic relations describing the perspective transformation can be given. Let the point-wise target location in the image plane be x 1, x 2 ) with the focal length of the camera f. Assuming x f x 1,x 2 ) Figure.1: Projection of a point wise target on to the image plane a non rotating frame attached to the missile, the measurements [x 1 x 2 ẋ 1 ẋ 2 ] are : x 1 = x 2 = fx1 f x fx2 f x.1) With primarily a video based sensing system, here we consider two instances. In case I, the video based missile performance is improved via employing a ground radar stationfigure 2.1). In case II, the video based missile performance is enhanced by the employing two unmanned flying vehicles for video capturingfigure 2.2) with enhanced spacial diversity. The measurement equation for each case is in the form of : y = β xt)) + v.2)

3 where [ βx) = ] fx 1 t) f x t)) fx 2t) f x t)) fx 1t) f x t)) fx 2t) f x t)) x 1 t) x 2 t) x t) fx 1 t) f x t)) fx 2 t) f x t)) fx 1t)+x M 1 t) x1 1 t) f x t)+x M t) x1t))) fx 2 t)+x M 2 t) x1 2 t) f x t)+x M t) x1t))) fx 1t)+x M 1 t) x2 1 t) f x t)+x M t) x2t))) fx 2 t)+x M 2 t) x2 2 t) f x t)+x M t) x2t))), single missile, with video and radar.) missile with UFV and x is the state of the missile target dynamic system and x M is the missile state and x i is the state of the i th UAV. Here, we assume the rotational kinematics of the missile is negligible for the sake of simplicity and the same ideas can be extended for the case with rotational dynamics accounted..1 Target Localization Optical flow is the apparent motion of the brightness/intensity patterns observed when the camera is moving relative to the objects being imaged[5, 6]. Let Ix 1, x 2, t) denote the image intensity function at time t at the image point x 1, x 2 ). Assuming that the overall intensity of the image is time independent, the well known Optical flow equation can be written as I x 1 dx 1 dt + I x 2 dx 2 dt + I t =.4) If the optical flow, v = v 1, v 2 ) = dx 1 /dt, dx 2 /dt) the equation.4 can be written as u 2 = mu 1 + c.5) where m = I/ x1 I/ x 2 and c = 1 I I/ x 2 t. The texture gradient m can be used to segment the video stream to localize the target in the image plane. In a practical system, as the computational efficiency is crucial, we propose using this texture gradient rather than calculating computationally taxing the overall optical flowmagnitude and direction)[7, 8, 9]. Local and global techniques in calculating the optical flow for the ill-posed problem are presented in [9] and in [8] respectively. The resultant is low pass filtered to isolate the target. 4 Proportional Navigation Since the first successful test of the Lark missile in December 195, Proportional NavigationPN) has come to be widely employed by homing missiles. Specially in the case of intercepting airborne targets in the case of SAMs PNG has been proved as a useful guidance scheme[1]. Under this scheme, the missile is commanded to turn at a rate proportional to the angular velocity of the line of sitelos). The missile normal acceleration is given by : u N = NV c λ 4.1) where, N is the navigation constant, λ is the LOS angle and V c is the closing velocity. PNG works ideally for constant speed maneuver targets. Extensions to PNG has been proposed when the target acceleration is significant: augmented PNGAPNG)[11], biased PNGBPNG)[12] to hit with a desired impact angle can be used. A detailed description on Pure PNGPPN)[1, 14] and True PNGTPN)[1, 14, 1]. In our implementation, we use the APNG with effective navigation ratio of, presented[15] in the form of u = t 2 [I t g I ] ˆx 4.2) g where, t g is the estimated time to go, and ˆx is the estimated state. 5 Set-Value state estimation with a non-linear signal model We consider nonlinear uncertain system of the form ẋ = Ax, u) +B 2 w z = Kx, u) 5.1) y = Cx) +v, as a general form of the system given by equation 2.2, and defined on the finite time interval [, s]. Here, xt) R n denotes the state of the system yt) R l is the measured output and zt) R q is the uncertainty output. The uncertainty inputs are wt) R p and vt) R l. Also, ut) R m is the known control input. We assume that all of the functions appearing in 5.1) are with continuous and bounded partial derivatives. Additionally, we assume that Kx, u) is bounded. This was assumed to simplify the mathematical derivations and can be removed in practice[2][16]. The matrix B 2 is assumed to be independent of x, and is of full rank.

4 A : Launch phase image B: Mid-course image Image Y axis 15 1 Image Y axis Image X axis Image X axis C :Texture gradient -m A) D: Texture gradient - m B) Filtered directional optical flow Angels in degrees) Filtered directional optical flow Angles in degrees) Image Y axis Image X axis Image Y axis Image X axis C :Filtered texture gradientm) A) D: Filtered texture gradientm) B) C :Target localizeda) D: Target localizedb)

5 The uncertainty in the system is defined by the following nonlinear integral constraint[1, 2, 17, 18, 19] : Φ x)) + s L 1 wt), vt)) dt d + s L 2 zt)) dt, 5.2) where d is a positive real number. Here, Φ, L 1 and L 2 are bounded non-negative functions with continuous partial derivatives satisfying growth conditions of the type φx) φx ) β ) 1 + x + x x x 5.) where is the euclidian norm with β >, and φ = Φ, L 1, L 2. Uncertainty inputs w ), v ) satisfying this condition are called admissible uncertainties. We consider the problem of characterizing the set of all possible states X s of the system 5.1) at time s which are consistent with a given control input u ) and a given output path y ) ; i.e., x X s if and only if there exists admissible uncertainties such that if u t) is the control input and x ) and y ) are resulting trajectories, then xs) = x and yt) = y t), for all t s. 5.1 The State Estimator The state estimation set X s is characterized in terms of level sets of the solution V x, s) of the PDE t V + max w R m { xv. Ax, u ) + B 2 w ) L 1 w, y Cx) ) + L 2 Kx, u ) ) } = V, ) = Φ. 5.4) The PDE 5.4) can be viewed as a filter, taking observations u t), y t), t s and producing the set X s as a output. The state of this filter is the function V, s) ; thus V is an information state for the state estimation problem. Theorem 5.1 Assume the uncertain system 5.1), 5.2) satisfies the assumptions given above. Then the corresponding set of possible states is given by X s = {x R n : V x, s) d}, 5.5) where V x, t) is the unique viscosity solution of 5.4) in C R n [, s]). proof see [2]. 5.2 A Robust Extended Kalman Filter Here we consider an approximation to the PDE 5.4) which leads to a Kalman filter like characterization of the set X s. Petersen and Savkin in [2] presented this as a Extended Kalman filter version of the solution to the Set Value State Estimation problem for a linear plant with the uncertainty described by an Integral Quadratic Constraint IQC). This IQC is also presented as a special case of equation 5.2. We consider uncertain system described by 5.1) and an integral quadratic constraint of the form s x) x ) X x) x ) ) wt) Qt)wt) + vt) Rt)vt)dt d s zt) zt)dt. 5.6) where N >, Q > and R >. For the system 5.1), 5.6), the PDE 5.4) can be written as t V + xv.ax, u ) xv B 2 Q 1 B 2 x V y Cx) ) R y Cx) ) Kx, u ) Kx, u ) =. V x, ) = x x ) N x x ). 5.7) Considering a function ˆxt) defined as ˆxt) arg min x V x, t), with the following equations 5.8),5.9) and 5.1) define our approximate solution to the PDE 5.7) : xt) = A xt), u ) + X 1 [ x C xt)) R y C xt)) ) + x K xt), u ) K xt), u ) ]. xt) = x, 5.8) Xt) is defined as the solution to the Riccati Differential Equation RDE) and Ẋ + x A x, u ) X + X x A x, u ) +XB 2 Q 1 B 2X x C x) R x C x) φt) x K x, u ) x K x, u ) =. t X) = N 5.9) [ y C x) ) R y C x) ) K x, u ) K x, u ) ]dτ. 5.1) The function V x, t) was approximated by a function of the form Ṽ x, t) = 1 2 x xt)) Xt) x xt)) + φt). Hence, it follows from Theorem 5.1 that an approximate formula for the set X s is given by X s = { x R n : 1 } 2 x xs)) Xs) x xs)) d φs) This amounts to the so called Robust Extended Kalman Filter generalization presented in[2].

6 Parameter Value Comments N.1 I 6 Weighting on the Initial viscosity solution Q 1 I Weighting on the uncertainty uncertainty in the driving command R.1 I 2, I 5 Weighting on the uncertainty uncertainty in the measurements Am.1 Amplitude of the target maneuver T 12s Flight time x [1 Initial state 2]m Table 6.1: Simulation parameters for case I. Parameter Value Comments N.1 I 6 Weighting on the Initial viscosity solution Q I Weighting on the uncertainty in missile driving command R 1 7 I 2, I 6 Weighting on the uncertainty in the measurements Am.1 Amplitude of the target maneuver T 12s Flight time x [1 Initial state 2]m xt) = A xt) + B 1 ut)+ X 1 t)[β 1 xt) R yt) β xt)))] xt) = x, ut) =. 6.1) Ẋ + A X + XA + XB 2 Q 1 B 2X β 1 xt) Rβ 1 xt) = X) = N, 6.2) 6.) where βx) is given in equation. with, β 1 x) = x βx). 6.4) The simulation parameters used are given in table 6.1 and the significant improvement is shown in figure 7.1. For the second system we simulate the same maneuvering target and the parameters used are given in the table 6.2. The trajectory of each ariel body for a typical case of a maneuvering target w =.5rad/sec) is shown in figure 7.2. The improvement in the system performance is shown in figure 7.. For both cases we consider, the target maneuver we employed, is given by Table 6.2: Simulation parameters for case II. wt) = Am [ t sin ωt) 2 sin ωt) ] 6.5) 6 Simulations We have carried out simulations for target localization in the video stream. Initially, the texture gradientm in equation.5) has been calculated for every consecutive imagessee A,B in figure 5.1) and then low passed filtered using a filter with a pass band less than.2 of the normalized frequency. C,D of figure 5.1 shows the directional texture gradient vectors. Location of the maximum of the filtered image corresponds to the target as shown in E,F of figure 5.1. The noise in the images are assumed only to be any bounded function of time and space and hence the estimated target locations are subjected to bounded functions in time. This robust assumptions in inline with the robust extended Kalman filter assumptions as stated in section 5. To demonstrate the advantage of using information fusion for video and radar sensors to improve the systems performance, we simulate a missile with an onboard camera assisted by radar ground station. The equation for the state estimation and the corresponding Riccati Differential equation obtained from equation 5.8 and 5.9 are as follows : for a range of maneuver frequenciesω). Table 6.1 gives the parameter values used in the simulations. 7 Conclusion In this paper we considered the improvement of missile performance of a surface to air missile using information fusion ideas. For economic and availability reasons, video based guidance is becoming a key areas of interest in the defence industry and we investigated the issue of improving performance of such systems via employing sensor fusion ideas. The two cases we considervideo imaging incorporated radar and video receiving with multiple vehicles) enhanced the video based guidance systems performance significantly. References [1] A.V Savkin and I.R Petersen. Recursive state estimation for uncertain systems with an integral quadratic constraint. IEEE Transactions on Automatic Control, 46):18 18, [2] I.R Petersen and A.V Savkin. Robust Kalman Filtering for Signals and Systems with Large Uncertainities. Birkhauser, Boston, 1999.

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