Bezier-curve Navigation Guidance for Impact Time and Angle Control

Transcription

1 Bezier-curve Navigation Guidance or Impact Time and Angle Control Gun-Hee MOON 1, Sang-Wook SHIM 1, Min-Jea TAHK*,1 *Corresponding author 1 Korea Advanced Institute o Science and Technology, Daehakro 91 KAIST, Yeuseunggu, Daejeon, Republic o Korea, ghmoon@dcl.kaist.ac.kr, swshim@dcl.kaist.ac.kr, mjtahk317@gmail.com* DOI: / Received: 0 November 017/ Accepted: 17 January 018/ Published: March 018 Copyright 018. Published by INCAS. This is an open access article under the CC BY-NC-ND license ( Aerospace Europe CEAS 017 Conerence, 16 th -0 th October 017, Palace o the Parliament, Bucharest, Romania Technical session Guidance, Navigation & Control I Abstract: This paper addresses a novel impact time and angle control guidance law using a Bézier curve. The nd order Bézier curve consists o one control point and two boundary points; initial point P0, middle point P1 and end point P. The curve is tangent to the line P0-P1 and the line P1-P, respectively, and always exists in the convex hull o the control points. Proposed Bézier curve navigation guidance, makes the missile ollow the Bézier curve with the eedback orm o guidance command so that the missile hits the target on the desired time in the desired direction. We conducted numerical simulations on several terminal conditions to demonstrate the perormance o proposed method. Key Words: Bézier-curve navigation guidance, impact time, impact angle, anti-ship missiles 1. INTRODUCTION In modern warare, as high valuable targets can protect themselves eectively against the guided missile, the guided missile has evolved to use some smart strategies. For example, recent anti-tank missiles have top-attack mode; the turret on the tank is vulnerable in vertical direction. The anti-ship missiles attack the vessel simultaneously, SALVO attack, to saturate the target's deense resources. These strategies are achievable with special guidance laws which controls impact time or angle on the terminal phase. The impact angle control (IAC) guidance law steers the missile in a vulnerable direction so that the target get atal damage. Ryoo proposed an IAC guidance law minimizing time-to weighted energy through the optimal control theory in [1]. Manchester developed an IAC guidance that ollows the circular path in the homing phase in []. In the space launch vehicle area, the explicit guidance is oten used or trajectory shaping. Ohlmeyer created the generalized vector explicit guidance (GENEX) which is available or the 3D IAC guidance [3]. A mixture guidance method employing the PN guidance or IAC has been proposed by Park in [4]. The impact time control (ITC) guidance makes the missile hit the target on a desired time. Through the ITC guidance, the SAVLO attack can be accomplished in an implicit way. A biased proportional navigation scheme [5], [6], and some nonlinear control theory based method [7] have been developed or ITC. INCAS BULLETIN, Volume 10, Issue 1/ 018, pp (P) ISSN , (E) ISSN

2 Gun-Hee MOON, Sang-Wook SHIM, Min-Jea TAHK 106 It is more successul i one can hit the target simultaneously in the vulnerable direction o the target. Impact time and angle control (ITAC) guidance can do this. There were some attempts to control the impact time and angle together. J.-I. Lee, and et al., used jerk command scheme or ITAC guidance in [8]. T.-H. Kim proposed a polynomial guidance method or ITAC in [9]. In this paper, we propose a novel Bézier-curve navigation guidance impact time and angle control guidance law that uses a Bézier curve as a reerence trajectory. BNG/ITAC consists o two phases; deployment phase and terminal BNG phase. On the deployment phase, guidance law makes the missile adjust the light path to decrease the impact time error that is expected or the terminal BNG phase. In the terminal BNG phase, a eedback orm o Bézier-curve navigation guidance law is employed to meet the desired impact angle. This paper composed with ollowing items; chapter is about preliminary background, chapter 3 is BNG/ITAC guidance law, and chapter 4 is numerical simulation and the conclusion. Problem Description. PRELIMINARY BACKGROUND Consider the ollowing two dimensional anti-ship missile engagement kinematics with a static target in Fig. 1. The vehicle can accelerate perpendicular to the light path direction. x V cos, y V sin, a / V (1) In this paper, we assume that the vehicle has a constant speed through the speed controller. Anti-ship missiles have a designed cruise speed so that they nominally ly with the designed speeds. By satisying the ollowing boundary condition, the missile can hit the target in the desired γ direction. The terminal time is speciied as t. x( t ) x, y( t ) y, ( t ) x( t ) x, y( t ) y, ( t ) The missile position is described in the inertial coordinate system. The guidance command is calculated on a guidance rame, where the x-axis is aligned in the light path direction. Then the x- and y- are calculated on the G-rame as ollows, () R ( x x) ( y y) atan( y y, x x) LOS (3) x y R cos( ) LOS R sin( ) LOS (4) (5) INCAS BULLETIN, Volume 10, Issue 1/ 018

3 107 Bezier-curve Navigation Guidance or Impact Time and Angle Control Y G V X G T M 0 Y I 0 O X I Bézier Curve Fig. 1 - Planar Engagement Geometry A Bézier curve is described with control points and two boundary points. This curve penetrates both boundary points in the tangential direction between the boundary point and the consecutive control point. Bézier curves always lie in the convex hull that is sequentially made with the boundary points and control points. The nd order Bézier curve, quadratic Bézier curve, has two boundary points, and one control point so that the curve lies in a triangle. In general, this curve is expressed with the Bernstein polynomial orm as ollows, x (1 ) x (1 ) x x 0 c y (1 ) y (1 ) y y (0 1) 0 c The Bézier curve independent variable τ varies rom zero to one. This equation can be reormulated as ollowing nd order polynomials or each coordinate variable. x a a a 0 1 y b b b 0 1 (0 1) The quadratic Bézier curve is deterministic, i the two boundary points and two slopes one the points are given. Especially, the length o the quadratic Bézier curve is obtainable analytically as ollows, (6) (7) where B (8) L x y d c c c d c 4( a b ) c 4( a a b b ) c a b (9) INCAS BULLETIN, Volume 10, Issue 1/ 018

4 Gun-Hee MOON, Sang-Wook SHIM, Min-Jea TAHK 108 The indeinite integral o above equation gives the length, L( ) c c c d c c 1 3/ c ( c c1 ) c c1 c0 (4 cc0 c1 )ln c c1 c 0 8c c L L(1) L(0) B 1 c c1 c 0 c c 3/ ( c c1 ) c c1 c0 (4 cc0 c1 )ln c c1 c 0 8c c c 0 (10) Acceleration Command on the Bézier Curve Consider that missiles ly along the path which is unction o τ, and the path can be expressed as, The velocity o the missile is By the rule o chain, we obtain x x( ), y y( ) (11) V x y (1) V ( x y ) (13) And the light path angle and the slope o the path are related as ollows, Dierentiating above equation gives, From Eq. (14), we have y tan (14) x 1 xy xy sec x sec 1 tan y x x Substituting Eq. (13), (15), and (16) to γ = a/v gives the acceleration on the curve, a C V x y x y ( ) x y I the curve is a quadratic Bézier curve, the command becomes, I τ = 0, a () BNG 3/ V a b b1 a a1 b 3/ c c1 c0 ( ( ) ( ) ) (15) (16) (17) (18) INCAS BULLETIN, Volume 10, Issue 1/ 018

5 109 Bezier-curve Navigation Guidance or Impact Time and Angle Control a BNG (0) V ( ab1 a1b ) 3/ b1 a1 (19) 3. BÉZIER-CURVE NAVIGATION GUIDANCE In this paper, we propose a composite guidance scheme impact time control guidance law. It consists o two phases o guidance law. The deployment phase adjusts the light path so that the remaining time to ly in the terminal phase is to be the predicted time-to-. The terminal phase employs the Bézier-curve navigation guidance so that the missile hit the target in desired direction. Bézier-Curve Navigation Guidance Bézier-curve navigation guidance control the missile to ollow a quadratic Bézier curve, which begins at the current point and ends at the target point. Then the Bézier curve should satisy ollowing conditions. x(0) a x, y(0) b y 0 0 x(1) a a a 0, y(1) b b b Let the slope o the curve at the starting boundary point be same with the current light path angle, then, x(0) a, y(0) b 1 1 y(0) / x(0) tan In the same manner, let the slope o the curve at the terminal boundary point be same with the desired impact angle. x(1) a a, y(1) b b 1 1 y(1) / x(1) tan Solving the system o equations; Eq. (0), (1) and (), determines the coeicients o the Bézier curve as, tan tan a ( b a tan ) / a ( a tan b a tan ) / b tan ( b a tan ) / b ( b tan b tan a tan tan ) / I the curve is described on a guidance rame which is aligned to the missile velocity direction, the current light path angle is zero, and the desired terminal angle becomes γ-. again, (0) (1) () (3) 0, (4) INCAS BULLETIN, Volume 10, Issue 1/ 018

6 Gun-Hee MOON, Sang-Wook SHIM, Min-Jea TAHK 110 a ( z x tan ) / tan 1 a (z x tan ) / tan b 0 b 1 z (5) Substituting these coeicients to Eq. (19) gives the acceleration command to ollow the Bézier curve as, a BNG V z tan V z z x tan z cot x Here, we deine this guidance law as Bézier-curve navigation guidance (BNG). Fig. shows the easible impact angle region in the guidance rame. I the terminal light path exists out o this region, there is no easible Bézier curves that satisy the given boundary condition. This region can be represented by ollowing inequalities. (6) (7) LOS easible impact angle region T y G M x G Fig. -Terminal guidance phase, Bézier-curve navigation guidance easible impact angle region schematic Deployment Phase or Impact Time and Angle Control The deployment phase control the light path o the missile to reduce the time-to- error o the terminal phase. The inner loop controls the light path angle, and the outer loop controls the time to error. The FPA error is deined as ollows, and it is attenuated in proportional control scheme as, err cmd a V N V 1 AP err err The light path angle time constant, τ γ, is reciprocal to the navigation constant, N γ. I γ cmd is constant, the light path angle will converges to the command similar to a irst order system. In the outer loop, FPA command is updated by the t- error. The t- prediction is obtained as, (8) tˆ L ( x, y, ) / V (9) B cmd INCAS BULLETIN, Volume 10, Issue 1/ 018

7 111 Bezier-curve Navigation Guidance or Impact Time and Angle Control Here the predicted t- is calculated with FPA command which will be meet when the terminal guidance phase starts. Then the time-to- error is calculated as, T tˆ (30) The light path angle command is controlled as ollows, cmd 1 dlb cmd V d T The Bézier curve derivative respect to the FPA is calculated numerically, so it leads the FPA to converge a proper initiative FPA o the terminal guidance phase. When the FPA error decrease less than the tolerance values, it initializes the terminal guidance phase. At the phase, BNG is used to meet both the impact angle and the impact time. Fig. 3 the shows block diagram o the deployment phase. In this scheme the t- Loop should be aster than the FPA loop. (31) T 1 B dl d V T cmd err VN γ AF L B /V xy, Fig. 3 - Deployment Phase Controller (Inner loop-fpa control, Outer loop-t error compensation) Simulation Scenarios 4. NUMERICAL SIMULATION The proposed guidance law is simulated to demonstrate its perormance. Table 1 shows the simulation scenarios. For each scenario, it is perormed with the dierent impact time constraints. Table 1 Table o simulation scenarios Scenario # 1 Scenario # X Pos (m) (initial/terminal) Y Pos (m) (initial/terminal) FPA (deg) (initial/terminal) 0 / / 0 0 / / -60 Nγ 1 1 t (sec) INCAS BULLETIN, Volume 10, Issue 1/ 018

8 Gun-Hee MOON, Sang-Wook SHIM, Min-Jea TAHK 11 Simulation Results Fig. 4 shows the trajectories o scenario 1. As shown in the igure, BNG guided the missile toward the desired target along the proper Bézier curve. Fig. 5 shows the FPA o scenario 1. The FPAs at the inal moment are almost -30 deg or each case, and each case satisies desired impact time closely. Fig. 6 shows the acceleration command o scenario 1. At the beginning large guidance command is applied at the deployment phase. Ater then, a bell shaped acceleration command is exerted by BNG. At the end point, the sharp jump is related to the cross over between the missile and target. Fig. 7 shows the trajectories o scenario. Fig. 8 shows FPAs o scenario. It shows that all desired impact angle and impact time is generally meet the reerence command. Fig. 9 shows the guidance commands or scenario. Table shows the summary o the simulation results. It says that the given constraints are satisied with the proposed method. Fig. 4 - trajectories comparison to dierent impact time (Scenario 1) Fig. 5 - FPA comparison to dierent impact time (Scenario 1) INCAS BULLETIN, Volume 10, Issue 1/ 018

9 113 Bezier-curve Navigation Guidance or Impact Time and Angle Control Fig. 6 - Acceleration command comparison to dierent impact time (Scenario 1) Fig. 7 - trajectories comparison to dierent impact time (Scenario 1) Fig. 8 - FPA comparison to dierent impact time (Scenario ) INCAS BULLETIN, Volume 10, Issue 1/ 018

10 Gun-Hee MOON, Sang-Wook SHIM, Min-Jea TAHK 114 Fig. 9 - Acceleration command comparison to dierent impact time (Scenario ) Table Summary o the simulation results Scenario # 1 Scenario # desired impact time (sec) terminal FPA (deg) t (sec) CONCLUSIONS This paper proposed a composite guidance style impact time and angle control guidance law, named as Bézier-curve navigation guidance impact time and angle control (BNG/ITAC). The BNG/ITAC consists o two phases; deployment phase and terminal BNG phase. On the deployment phase, the light path is controlled to reduce the t- error expected in the terminal BNG phase. At the terminal phase, the BNG guidance is applied to meet the desired impact angle. The perormance o BNG/ITAC is demonstrated through the numerical simulation. The impact angle and impact time are all satisied relatively with a little error, due to numerical reason. ACKNOWLEDGEMENTS This work has been supported by Agency or Deense Development(ADD) and Deense Acquisition Program Administration (DAPA). INCAS BULLETIN, Volume 10, Issue 1/ 018

11 115 Bezier-curve Navigation Guidance or Impact Time and Angle Control REFERENCES [1] C.-K. Ryoo, H. Cho, M.-J. Tahk, Time-to- Weighted Optimal Guidance with Impact Angle Constraints, IEEE Transactions on Control Systems Technology, vol. 14, no. 3, May 006. [] I. R. Manchester, and A. V. Savkin, Circular-Navigation-Guidance Law or Precision Missile/Target Engagements, Journal o Guidance, Control, and Dynamics, vol. 9, no., March-April 006, pp [3] E. J. Ohlmeyer, and C.-A. Phillips, Generalized Vector Explicit Guidance, Journal o Guidance, Control, and Dynamics, vol. 9, no., March-April 006. [4] B.-G. Park, B.-J. Jeon, T.-H. Kim, M.-J. Tahk, and Y.-H. Kim, Composite Guidance Law or Impact Angle Control o Tactical Missiles with Passive Seekers, 01 APISAT, Nov , Jeju, Korea [5] T.-H. Kim, C.-H. Lee, M.-J. Tahk and I.-S. Jeon, Biased PNG Law or Impact-Time-Control, Transactions o the Japan Society or Aeronautical and Space Sciences, vol. 56, no. 4, 013, pp [6] I.-S. Jeon, J.-I. Lee, and M.-J Tahk, Impact-Time-Control Guidance Law or Anti-Ship Missiles, IEEE Trans. Control Systems Technology, vol. 14, No., pp , 006 [7] S. R. Kumar and D. Ghose, Sliding Mode Control Based Guidance Law with Impact Time Constraints, in American Control Conerence (ACC) 013, pp , June 013 [8] J.-I. Lee, I.-S. Jeon, and M.-J. Tahk, Guidance Law to Control Impact Time and Angle, IEEE Tran. Aerospace and Electronic Systems, vol. 43, no. 1, January 007. [9] T.-H. Kim, C.-H. Lee, I.-S. Jeon, M.-J. Tahk, Augmented Polynomial Guidance with Impact Time and Angle Constraints, IEEE Trans. Aerospace and Electronic Systems, vol. 39, no. 4, October 013. INCAS BULLETIN, Volume 10, Issue 1/ 018