2015 American Control Conference Palmer House Hilton July 1-3, 2015. Chicago, IL, USA
A Nonlinear Guidance Law for Impact Time Control Abdul Saleem1 and Ashwini Ratnoo2
proportional navigation [6]. The guidance command was based on the error between estimated and designated impact time. Lee et al. presented a guidance law to control impact time and angle [7]. The method uses an optimal solution satisfying the condition that the length of the estimated trajectory divided by the speed equals to the desired time. A sliding mode based guidance law for impact time and angle control was presented by Harl and Balakrishnan [8]. A polynomial guidance command with constraints on impact time and impact angle was proposed by Kim et al. [9]. The time-to-go was estimated in terms of predicted trajectory, which was expanded using Taylor series approximation. An optimal control problem was proposed by Arita and Ueno [10] for impact time and angle control. The problem was formulated as a two point boundary value problem and was solved numerically. Sliding mode control based guidance laws for impact time control was presented by Kumar and Ghose [11], [12], wherein the time-to-go estimates was found using approximate methods. In [11], the time-to-go was estimated using a formula in terms of heading error for proportional navigation guidance law and in [12], it was based on the curvature of the trajectory towards the interception point. In contrast to recursive and numerical solutions, present work proposes a guidance law with closed-form solution for the impact time. Further, a feedback form of the guidance law is presented for a robust performance against autopilot delay. Based on the guidance law, a cooperative methodology is presented for salvo attack. The remainder of this paper is organized as follows. In Section II, the guidance problem for impact time control is stated. Section III derives the guidance command and the closed-form expression for the impact time. Section IV presents the implementation of the proposed method with feedback control. The simultaneous impact of multiple interceptors is discussed in Section V. A detailed simulation study is presented in Section VI. Finally, the concluding remarks are presented in Section VII.
Abstract— This paper discusses the problem of impact time control of an interceptor against a stationary target. A nonlinear guidance law is proposed with the interceptor heading angle variation as a function of the range to target. Closed-form expressions for the design parameters are derived for an exact analysis of the impact time. A feedback form of the guidance law is presented for addressing realistic implementation in the presence of autopilot lag. Using the closed-form expressions of the impact time, a cooperative guidance scheme is presented for simultaneous impact in a salvo attack. Extensive simulation studies are presented validating the analytic findings.
I. INTRODUCTION Many interceptor guidance laws are aimed to satisfy constraints such as desired impact angle, predefined impact time, or minimum control effort in addition to the achievement of final collision with the target. If the engagement time is shorter, then the reaction time of the target against the interceptor is reduced. Impact time control also finds an application in salvo attacks where multiple interceptors simultaneously hit a target with a greater warhead effect. System delays like autopilot lag can adversely affect the accurate control of impact time. In such situations, a prospective guidance law which is accurate in achieving a desired impact time and robust with respect to autopilot delays is desired. Tahk et al. [1] proposed a recursive method for timeto-go estimation. The time-to-go estimate were updated at each instant and a compensation method for time-to-go error was obtained. Shin et al. [2] proposed a method for timeto-go estimation using time histories of optimal guidance command. Dhananjay and Ghose [3] came up with a method using interpolation to estimate time-to-go of proportional navigation guidance(PN) law. Cho et al. [4] developed a closed-form guidance command involving time-to-go estimation for missiles with time-varying velocity. The time-to-go estimation was done numerically. The authors [1]–[4] used time-to-go in the guidance command and methods for control over the impact time was not presented. Jeon et al. [5], presented an approach for impact time control. They used a linearized model of engagement kinematic with small angle assumption for the heading angle. The guidance command in Ref. [5] consisted of two terms, one for reducing the miss distance and the other for adjusting the impact time. Jung and Kim presented a guidance law using
II. P ROBLEM S TATEMENT Consider a planar interceptor-target engagement geometry as shown in Fig. 1. Here the target is assumed to be stationary and the interceptor is moving with a constant speed V . The kinematic equations for the engagement geometry are given as
1 Assistant Professor, Department of Applied Electronics and Instrumentation Engineering, Government Engineering College, Kozhikode, India
R˙ = −V cos δ −V sin δ θ˙ = R
[email protected] 2
Assistant Professor, Autonomous Vehicles Laboratory, Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India
[email protected] 978-1-4799-8684-2/$31.00 ©2015 AACC
651
(1) (2)
y
δ
Target V
x am
R
δ0
δ αm
θ Reference
Interceptor
Fig. 1: Engagement geometry.
tc
R = f R0
where R is the range, θ is the line of sight (LOS) angle. The heading error δ is defined as δ = αm − θ
Fig. 2: Proposed heading error variation.
(3)
t f . The relation between the initial heading error δ0 and the design parameters can be derived by substituting R = R0 in (5) log(1 − sin |δ0 |) (8) b= f −1
where αm is the interceptor heading angle. The interceptor maneuvers using the lateral acceleration am , which is applied normal to its velocity. The turning rate α˙ m can be related to the lateral acceleration as am (4) α˙ m = V The impact time t f corresponds to the instant when the target is intercepted, that is
It will be shown subsequently (III-A) that the impact time can be governed by parameter f and a corresponding b can be found using (8) to satisfy initial conditions. Differentiating (5) with respect to time yields b −b( R − f ) ˙ cos δ δ˙ sgn(δ ) = e R0 R R0
lim R → 0
t→t f
sgn(δ ) =
bV δ˙ = − sgn(δ ) (1 − sin |δ |) R0
Consider an engagement with an initial heading error δ0 , as δ0 ∈ (−π/2, π/2). The two phase variation in the heading error δ , as a function of the range R, is proposed as sin |δ | = 1 − e
,
=0
f R0 ≤ R ≤ R0 R < f R0
(10)
(11)
Eq. (2) can also be written as V sin |δ | θ˙ = − sgn(δ ) R
(5)
(12)
Differentiating (3) with respect to time and substituting for δ˙ and θ˙ from (11) and (12) respectively, yields bV V sin |δ | ˙ αm = − sgn(δ ) (13) (1 − sin |δ |) + R0 R
(6)
Here R0 is the initial range and b and f are control design parameters. It can be observed that heading error δ goes to zero at R = f R0 . Hence a collision course is achieved at R = f R0 and thereafter the interceptor flies along a straight line path with δ = 0. The parameter f can be varied. However, the collision course can be delayed only till R = 0. Consequently the permissible values of f should satisfy f ∈ [0, 1)
δ |δ |
Substituting R˙ from (1 ) in (9) yields
III. C ONTROL L AW F ORMATION
0
(9)
where
The problem here is to find a prospective guidance law governing am that can guarantee a predefined impact time t f . Further, with a control over t f simultaneous impact of multiple interceptors is desired to be achieved.
−b RR − f
tf t R=0
The proposed guidance command is obtained by using (13) in (4) as 2 bV V 2 sin |δ | am = − sgn(δ ) (1 − sin |δ |) + (14) R0 R
(7)
Note that both the phases of (5) and (6) are governed by (14) as δ = 0 ⇒ sgn(δ ) = am = 0
Note that f = 1 corresponds to achieving the collision course instantaneously after launch which is physically impossible due to lateral acceleration saturation. A typical plot of the proposed variation of δ is shown in Fig. 2. The heading error reduces from its initial value to zero at a time tc corresponding to R = f R0 followed by an interception at
A. Impact Time The analytic expression of the engagement time for the two phases is derived subsequently. 652
1) Phase 1: As shown in Fig. (2), tc is the time of flight in the first phase where δ varies from δ0 (at t=0) to 0 (at t = tc ). Differentiating (5) with respect to time leads to
dδ b −b cos δ sgn(δ ) = e dt R0
R R0 − f
dR dt
IV. I MPLEMENTATION A. Ideal Engagement In an ideal engagement scenario, without autopilot delay, the open loop guidance law of (14) is used. A relation between the parameter f and the impact time t f can be obtained from (25) as t f max − t f (26) f= t f max − t f min
(15)
From (1), (5) and (15), it can be deduced that −V dt =
R0 sgn(δ ) dδ b 1 − sin |δ |
The procedure for open loop control is given in Algorithm 1.
(16)
Algorithm 1 Open loop control
Integrating both sides of (16), results in −V
Z tc 0
R0 dt = b
Z 0 sgn(δ ) δ0
1 − sin |δ |
1:
dδ
R0 1 − tan |δ0 | − sec δ0 ⇒ tc = − V b Substituting for b from (8) in (18) , leads to R0 1 − tan |δ0 | − sec δ0 (1 − f ) tc = V log(1 − sin |δ0 |)
(17) 2: 3: 4: 5:
(18)
(19)
B. Feedback Control In the presence of autopilot delay, the interceptor heading variation of (4) is no longer valid as the commanded and achieved lateral accelerations are different. Hence a feedback form is proposed for the guidance law where the control parameters b and f are computed at each guidance cycle time as t 0f max − (t f − t) f0 = 0 (27) t f max − t 0f min
2) Phase 2: In second phase, the heading error is zero as governed by (6) and hence the time of flight for this phase can be expressed as t f − tc =
f R0 V
(20)
B. Total Time of Flight Total time of flight can be written using (19) and (20) as, f R0 R0 1 − tan |δ0 | − sec δ0 (1 − f ) + (21) tf = V log(1 − sin |δ0 |) V
b0 =
1 − tan |δ0 | − sec δ0 1−
