guidance law with impact time and impact angle constraints

Accepted Manuscript Guidance Law with Impact Time and Impact Angle Constraints Zhang Youan, Ma Guoxin, Liu Aili PII: DOI: Reference: S1000-9361(13)00088-5 http://dx.doi.org/10.1016/j.cja.2013.04.037 CJA 95 To appear in: Received Date: Revised Date: Accepted Date: 20 March 2012 28 April 2012 July 2012 Please cite this article as: Z Youan, M Guoxin, L Aili, Guidance Law with Impact Time and Impact Angle Constraints, (2013), doi: http://dx.doi.org/10.1016/j.cja.2013.04.037 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain Guidance Law with Impact Time and Impact Angle Constraints ZHANG Youan*, MA Guoxin, LIU Aili Department of Control Engineering, Naval Aeronautical and Astronautical University, Yantai 264001, China Received 20 March 2012; revised 28 April 2012; accepted July 2012 Abstract A novel closed-form guidance law with impact time and impact angle constraints is proposed for salvo attack of anti-ship missiles, which employs missile’s normal acceleration (not jerk) as the control command directly Firstly, the impact time control problem is formulated as tracking the designated time-to-go (the difference between the designated impact time and the current flight time) for the actual time-to-go of missile, and the impact angle control problem is formulated as tracking the designated heading angle for the actual heading angle of missile Secondly, a biased proportional navigation guidance (BPNG) law with designated heading angle constraint is constructed, and the actual time-to-go estimation for this BPNG is derived analytically by solving the system differential equations Thirdly, by adding a feedback control to this constructed BPNG to eliminate the time-to-go error—the difference between the standard time-to-go and the actual time-to-go, a guidance law with adjustable coefficients to control the impact time and impact angle simultaneously is developed Finally, simulation results demonstrate the performance and feasibility of the proposed approach Keywords: guidance law; proportional navigation guidance; feedback control; impact time; impact angle; missiles Introduction1 The impact angle and impact time are important constraints for missile’s homing problem There have been a lot of studies and applications on impact angle control for decades, from the biased proportional navigation guidance (BPNG) law with impact angle constraint [1, 2] to optimal control guidance law with impact angle constraint [3-6], from taking Lyapunov method [7] to the backstepping method [8], too numerous to mention one by one While the impact angle control is widely used to increase the lethality of warheads, the impact time control is employed to carry out a salvo attack for anti-ship missiles against close-in weapon system The studies on the impact time control are rare relatively, because it is difficult for the missile to adjust its flight time by the normal force which only changes the velocity direction Assuming that the heading angle is small, Jeon et al [9] derive the closed-form solution based on the linear formulation An impact time control guidance law is proposed which is a combination of the well-known proportional navigation guidance law with the navigation constant of and the feedback of the impact time error Based on Ref.[9]: Sang et al [10] propose a guidance law switching logic for maintaining the seeker lock-on condition and a time-to-go calculation method for the missile with the limitations of maneuvering and seeker’s field-of-view, Zhao et al [11] propose a time-cooperative guidance law using cooperative variables, and Zou et al [12] propose a decentralized time-cooperative guidance law using decentralized consensus algorithms The impact time control can also be achieved by dynamic inverse method [13], and the time-cooperative guidance also achieved by leader-follower strategy [14] In contrast, the studies on the impact angle control and impact time control simultaneously are hard to find Jeon et al expand their approach from [9] by including impact angle constraints to analysis, and then propose a guidance law to control both impact time and impact angle [15] To improve the control precision of the guidance law of [15] when the terminal angle is large, Cheng et al [16] propose a compensation method against the linearized error Assuming the position of the target is known beforehand, Harl et al [17] present a sliding mode based impact time and angle guidance law by introducing a line-of-sight (LOS) rate shaping process, in which the parameter must be tuned by hand or by off-line iterative routine Huang et al [18] transform the missile’s nonlinear kinematical model by using the heading angle as independent variable, and design a mid-course guidance law with impact angle and impact time constraints by using optimal control theory The results in [17,18] may both contain singular solutions To date, the representative closed-form guidance law to achieve the impact time and impact angle control is only seen in [15], in which “jerk” is used as the control command (the missile’s normal acceleration command must be produced by time integration of this command) and the calculation for the control command is very complex Relative to the approach in [15], this paper proposes a new simple form of guidance law with impact time and impact angle constraints, which uses missile’s normal acceleration as the control command directly The proposed guidance law is composed of a constructed BPNG with impact angle constraint and a feedback control of the impact time error Simulations show that the proposed guidance law has superior performance, especially for a moving or maneu*Corresponding author Tel.: +86-0535-6635655 E-mail address: zhangya63@sina.com vering target Problem Formulation Consider a two-dimensional homing scenario shown in Fig.1 where the missile M has a constant speed V and the target T is stationary R, q, θ and ϕ denote the range-to-go, the LOS angle, the heading angle and the lead angle in the inertial reference frame, respectively The designated impact time and impact angle are represented as Td and θd The equations of this homing guidance problem are given by R = −V cos ϕ , Rq = V sin ϕ , θ = an / V , q = θ + ϕ (1) where an is the missile’s normal acceleration, i.e the control command Y T R ϕ V θ M O q X Fig Homing guidance geometry The designated time-to-go tgo is the difference between the designated impact time and the current flight time, so we have tgo = Td − t , tgo = −1 (2) The objective of controlling the impact time and impact angle simultaneously can be described as t go → tgo , θ →θd where tgo is the actual time-to-go of missile Biased Proportional Navigation Guidance Law with Impact Angle Constraint To obtain the analytical solution of time-to-go conveniently from guidance law with impact angle constraint, we construct a new form of BPNG law as following an = NVq − KV [θ − Nq + ( N −1)θd ]/ R (3) where the coefficients are chosen as N ≥3, K ≥1 Note that, when N =3, K =1, under the assumption of small angle, we have an = 3Vq − V [θ − 3q + 2θ d ] / R = 3Vq + 3V (q − θ ) / R + 2V (θ − θ d ) / R ≈ 3Vq + 3Vq + 2V (θ − θ d ) / R = 6Vq + 2V (θ − θ d ) / R This is accordant to the results in [3,15] Define α = θ − Nq + ( N −1)θd (4) R = −V cos ϕ (5a) Substituting Eq.(3) to Eq.(1) results in ϕ = −( N −1)V sin ϕ / R + KVα / R α = −KVα / R (5b) (5c) Note that Eq.(5c) indicates that α →0 and Eq.(5b) reveals that ϕ →0 when α →0 From α =θ −Νq + (N−1)θd = −Ν (q−θ )−(N−1)(θ −θd) = −Ν ϕ −(N−1)(θ − θd), we have θ →θd finally Thus, the guidance law (3) can achieve the desired impact angle requirement The time-to-go estimation of the guidance law (3) is derived in the following Eliminating the time variable from Eqs.(5b) and (5c) yields dϕ N −1 sin ϕ = −1 dα K α (6) By using the approximation sinϕ ≈ϕ (known from the above, this approximation is rational because ϕ goes to zero gradually during the homing guidance), the solution of the differential equation (6) is obtained ϕ = Cα N −1 K + K α N −1 − K (7) where C is a constant, which is given from the initial condition C = [ϕ (0) − K α (0)]/ α (0) N −1 − K N −1 K (8) Eliminating the time variable from Eqs.(5a) and (5c) with substitution by Eq.(7) yields cos(Cα dR = R N −1 K + K α) N −1 − K dα Kα (9) Integration of Eq.(9) on [0, t] yields α ∫R(0) R dR = ∫α (0) R cos(Cα N −1 K + K α) N −1 − K dα Kα (10) Taylor series expansion of Eq.(10) over Cα (Ν−1)/Κ +K α / (N−1−K ), i.e ϕ with higher order terms neglected yields R = Dα K eh(α ) (11) where D = R (0)α (0) − K e − h (α (0)) , 2( N −1) N −1 +1 C2 KC K K α K − α − α2 2 4( N − 1) ( N − 1) − K 4( N − − K ) Substituting Eq.(11) into Eq.(5c) yields h(α ) = − −1 Dα K eh(α ) dα = −KVdt (12) Integration of Eq.(12) on [0, tf ] (tf represents the terminal time) yields ∫α (0) −1 tf Dα K eh(α ) dα = −∫ KVdt (13) h(α ) ≈1+h(α ) (the accuracy about this approximation is discussed in Remark in detail) with substitution By using e of C, D, the estimation of tf is obtained from Eq.(13) as tˆf = R(0)eC1ϕ (0) +C2 [ϕ (0) +α (0)]2 × [1 + C3ϕ (0) + C4ϕ (0)α (0) + C5α (0)2 ]/ V where C1 = , 4( N −1 + K ) C2 = K , 4( N −1)( N −1 + K ) C3 = − , 4( N − 1)(2 N − 1) (14) C4 = − K (3 N − + K ) , 2( N − 1)(2 N − 1)( N − + K )( N + K ) C5 = − K2 + 4( N − 1)(2 N − 1)( N − − K ) K2 − [( N − 1) − K ]( N + K )( N − − K ) K 4( N − − K ) (1 + K ) tˆf can be regarded as the time-to-go estimation of missile at the initial time, so the time-to-go estimation at the current time t can be denoted as tˆgo = R eC1ϕ +C2 (ϕ +α )2 (1 + C3ϕ + C4ϕα + C5α ) / V (15) Remark 1: The accuracy of the approximation e h(α ) ≈1+h(α ) depends on the absolute value of h(α ) From the expression of h(α ), we obtain h(α (0)) = −C1ϕ (0) − C2 [ϕ (0) + α (0)]2 N −1 1V K K + ( α ) = ( α α )2 ≥ h C K N ( − 1) 2 N −1− K 2R =− [q (0) − θ d ] ≤ 0, ϕ (0) − 4( N − + K ) 4( N − + K ) We find that h(α(0)) is composed of two terms: the first denotes the initial heading error of missile; the second denotes the initial difference between initial LOS angle and the designated impact angle of missile Notice that the absolute value of h(α ) is maximal at the initial time, and later it goes to zero with time Thus, the maximal error of this approximation locates at the initial time, and the error goes to zero with time Guidance Law Design with Impact Time and Impact Angle Constraints Now we find the solution to achieve the desired impact time requirement by adding a feedback control to the BPNG law (3) The time-to-go estimation tˆgo of guidance law (3) is considered to be accurate, i.e tˆgo = t f − t When tˆgo = tgo , i.e tf = Td, the guidance law (3) can satisfy the impact time and impact angle constraints simultaneously, and no additional feedback control is needed From Eq.(5c) we have dα /d(tf −t) = dα /d tˆgo =KVR /α, then R tˆgo = α KVα (16) When tˆgo ≠ tgo , we add a feedback control to Eq.(3) to drive tˆgo → tgo Thus, we devise the closed-form guidance law as follows an = NVq − KV 2α / R + aε (17) where aε is the additional feedback control, designed to eliminate the error between the designated time-to-go and the time-to-go of BPNG, which is defined by ε = tgo − tˆgo (18) α = − KV α / R + aε / V (19) From Eqs.(17) and (1), we have In control designing, α is taken as the slow variable and α as the fast variable Substituting Eq.(19) into Eq.(16) yields tˆgo = −1 + R aε KV 2α (20) Combining Eqs.(18) and (20), we have ε =− R aε KV 2α (21) The feedback control is selected as aε = k1Kαε (22) which results in ε = −k1Rε / V , and ε →0 is ensured when k1>0 Along with ε→0 and aε →0, the proposed guidance law (17) and (22) reduces to the BPNG law, and finally θ →θd is ensured Thus, the proposed guidance law (17) and (22) can achieve the impact time and impact angle requirements simultaneously Remark 2: The feedback control (22) maintains the dynamics of α as α = − KV α / R + k1 K εα / V In general salvo attack scenarios, the designated impact time Td is selected to ensure ε >0 So the term of k1K εα / V provides α a tendency of divergence if ε ≠0 Remark 3: When the limitation of missile’s normal acceleration is given, the feasible interval of Td can be calculated geographically by reference to Section in [10] Using the proposed guidance law (17) and (22) as the original guidance law, the switching logic of [10] can be also used to maintain the seeker lock-on condition Simulation Results and Analysis Let us first consider an engagement scenario in which the missile has a constant speed of 250m/s and the target is a stationary ship Initial position of the missile and the target are set to be (0, 0) and (-10, 0.5)km The missile’s normal acceleration is limited within 5g The parameters are taken as N =3, K =1, k1 =7 The missile is guided by the proposed guidance law (17) and (22) The simulation step is 0.01s To investigate the performance of the proposed law, three cases of simulations are carried out: (1) the designated impact time is set to 50s, designated impact angle is set to 10°, the initial heading angle is taken as 0°, ±30°, and ±60° respectively, the simulation results are shown as Fig.2; (2) the designated impact angle is set to 10°, initial heading angle is taken as 30°, the designated impact time is set to 45s, 48s, 50s, and 55s respectively, the simulation results are shown as Fig.3; (3) the designated impact time is set to 50s, initial heading angle is taken as 30°, the designated impact angle is set to 0°, ±10°, ±30° respectively, the simulation results are shown as Fig.4 θ0=0° θ0=±30° θ0=±60° Y /km t f =50 s θf =10° -1 -2 -3 -4 -10 -8 -6 -4 -2 X /km Fig Trajectories for multiple initial heading angles Td=45s Td=48s Td=50s Y /km Td=55s θf =10° -1 -2 -10 -8 -6 -4 -2 X /km Fig Trajectories for multiple designated impact times 4 θd=0° θd=±10° θd=±30° Y /km t f =50 s -1 -2 -3 -10 -8 -6 -4 -2 X /km Fig Trajectories for multiple designated impact angles Simulation results show that, the proposed guidance law can satisfy the requirements for multiple initial heading angles, multiple designated impact time and multiple designated impact angles of missiles Fig.5 shows the concerned results of the first set of simulation correspondingly Known from Fig.5c, the guidance command reaches maximal in the early stage, which drives the missile to turn slightly away from the target (reflected by the large changes of ϕ and α in Fig.5a in the early stage) to adjust the time-to-go Along with the time-to-go error goes to zero, the proposed guidance law becomes the BPNG law with impact angle constraint, and θ and q tend to the designated angle 10°(as shown in Fig.5b) The impact time control and the impact angle control are both achieved finally φ /deg 100 -100 10 20 30 40 α /deg t /s 50 θ0=0o 200 θ0=±30o 100 θ0=±60o -100 10 20 30 40 50 t /s (a) Lead angle and α angle θ /deg 100 -100 10 20 q /deg 100 θ0=±30o 50 θ0=±60o 30 40 50 30 40 50 t /s θ0=0o -50 10 20 t /s (b) Heading angle and LOS angle 10 an /g -5 -10 10 20 30 40 t /s 50 θ0=0o 10 θ0= ±30o ε /s θ0= ±60o -5 10 20 30 40 50 t /s (c) Normal acceleration and time-to-go error Fig Simulation results of the proposed guidance law for multiple initial heading angles Fig.6 illustrates the comparing results between the proposed guidance law and the guidance law in Ref.[15] for the first case of the initial heading angle 30° and 60° From Fig.6(d), we know that the time-to-go error with the proposed guidance law vanishes faster than that with Ref.[15] To achieve the impact time control, missile is needed to maneouver during the homing guidance From Fig.6, we can visualize the proposed guidance law as large-maneouvering first and straight-flight second, while the guidance law in Ref.[15] as straight-flight first and large-maneouvering second (normal acceleration command even diverging finally) In actually application, the former is very valuable for missiles to implement the impact Y /km θ0=30o proposed θ0=30o Ref.[15] t f =50 s θ0=60o proposed -1 -2 -10 θ0=60o Ref.[15] -8 -6 -4 -2 X /km (a) Trajectories 50 φ /deg θ0=30o proposed θ0=30o Ref.[15] -50 θ0=60o proposed -100 θ0=60o Ref.[15] 10 20 30 40 50 40 50 t /s (b) Lead angle θ0=30o proposed θ0=30o Ref.[15] θ0=60o proposed an /g θ0=60o Ref.[15] -2 -4 -6 10 20 30 t /s (c) Normal acceleration ε /s 10 θ0=30o proposed θ0=30o Ref.[15] θ0=60o proposed θ0=60o Ref.[15] -2 10 20 30 40 50 t /s (d) Time-to-go error Fig Simulation results of the proposed guidance law and the guidance law in Ref.[15] Table Initial parameters in simulation Initial positions (km) Initial heading angles (°) Designated impact angles (°) Missile (-10, 0.5) 30 10 Missile (-6, 6) 15 -30 Missile (-3, -10) 20 60 Then let us consider the salvo attack scenario for multi-missiles Suppose that three missiles cooperatively attack a single target which locates at (0, 0), the speed of each missile is 250m/s, the designated impact time is 50s, and the other initial parameters are shown as Table.1 The results for this simulation are shown in Fig.7, which indicates that the proposed guidance law can be applied to the cooperative attack for multi-missiles M1 M2 M3 Y /km t f =50 s -2 -4 -6 -8 -10 -10 -8 -6 -4 -2 X /km Fig Trajectories of cooperative attack for multi-missiles Maneuvering of the target ship will bring about some error in impact time and impact angle, and even cause failure for the impact in the cooperative guidance field To test the feasibility of the proposed law for the case of target maneuvering, the first missile in Table.1 is taken as an example Suppose that the target speed is 20m/s with different initial heading angles We use (Rf, Tf, θf) to describe the miss distance, impact time and impact angle Notice that the designated impact time and impact angle are 50s and 10° respectively The simulations are stopped if R0 When the target has no normal acceleration, the results are summarized in Table.2 For the worse case of the initial heading angle of ±120°, when normal acceleration of the target is set to ±0.7m/s2, the results of this maneuvering case are summarized in Table.3 Extensive simulations demonstrate that the miss distance of the proposed approach is smaller than Ref.[15], especially when target moves against the incoming missile direction Small miss distance is the most important in the homing As for the accuracy of the impact time and impact angle, we find that the proposed approach has smaller impact angle control error while Ref.[15] has smaller impact time control error in most cases Table Simulation Results with Target Speed 20m/s and Normal Acceleration 0m/s2 Initial heading angles of the target 0° 60° -60° 120° -120° (Rf, Tf, θf) by proposed approach (2.2m, 53.8s, 8.8°) (0.4m, 50.6s, 15.1°) (0.3m, 53.4s, 3.6°) (0.4m, 50.0s, 4.6°) (16.3m, 47.4s, 7.6°) (Rf, Tf, θf) by Ref.[15] (2.6m, 52.2s, 7.6°) (1.5m, 50.8s, 19.2°) (1.3m, 51.7s, -3.3°) (85.8m, 49.9s, 14.1°) (19.9m, 50.0s, 15.0°) Table Simulation Results with Target Speed 20m/s and Normal Acceleration ±0.7m/s2 Initial heading angles and normal accelerations of the target 120° and 0.7m/s2 120° and -0.7m/s2 -120° and 0.7m/s2 -120° and -0.7m/s2 (Rf, Tf, θf) by proposed approach (19.4m, 47.5s, -3.8°) (2.2m, 50.4s, 11.0°) (2.0m, 53.5s, 7.4°) (19.8m, 47.3s, 4.2°) (Rf, Tf, θf) by Ref.[15] (106.6m, 49.7s, 35.5°) (1.7m, 51.2s, 11.3°) (2.4m, 51.9s, 4.2°) (59.9m, 49.9s, 54.5°) Conclusions To improve the performance of impact time and impact angle control, this paper proposed a novel closed-form guidance law with impact time and impact angle constraints Compared with the previous representative guidance law, the proposed guidance law in this paper has a simple form and takes missile’s normal acceleration as the control command directly and thus simplifies the calculation complexity, which leads to an easy implementation, while the previous one takes “jerk” as the control command which must be integrated to obtain the missile’s normal acceleration command, and the formula of the previous one is very complex, which leads to a difficult implementation Extensive simulations of various engagements demonstrate that the proposed guidance law provides satisfactory performance against stationary or slightly maneuvering targets The proposed guidance law is designed with the consideration of stationary target, although the examples show its validity to the slow or small-maneuver targets In future study, the design and analysis of guidance law with impact time and impact angle constraints against the moving or maneuvering targets should be considered 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University 2010; 28(2): 165-170 [in Chinese] Biography: ZHANG Youan is a professor and Ph.D supervisor at department of control engineering, Naval Aeronautical and Astronautical University, Yantai, China His research areas are missile guidance, nonlinear control and autopilot design E-mail: zhangya63@sina.com