1
3
Optimal three-dimensional impact time guidance
4
with seeker’s field-of-view constraint
5
Shaoming HE
a
, Chang-Hun LEE
b,*, Hyo-Sang SHIN
c, Antonios TSOURDOS
c6 aSchool of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China
7 bDepartment of Aerospace Engineering, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea 8 cSchool of Aerospace, Transport and Manufacturing, Cranfield University, Cranfield MK43 0AL, United Kingdom
9 Received 12 December 2019; revised 1 February 2020; accepted 28 March 2020 10
12 KEYWORDS
13
14 Applied optimal control;
15 Field-of-view constraint;
16 Impact time control;
17 Missile guidance;
18 Optimal error dynamics
Abstract This paper proposes a new three-dimensional optimal guidance law for impact time con-trol with seeker’s Field-of-View (FOV) constraint to intercept a stationary target. The proposed guidance law is devised in conjunction with the concept of biased Proportional Navigation Guid-ance (PNG). The guidGuid-ance law developed leverages a nonlinear function to ensure the boundedness of velocity lead angle to cater to the seeker’s FOV limit. It is proven that the impact time error is nullified in a finite-time under the proposed method. Additionally, the optimality of the biased com-mand is theoretically analyzed. Numerical simulations confirm the superiority of the proposed method and validate the analytic findings.
Ó 2020 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
19
20 1. Introduction
21 The ability to satisfy the desired impact time of anti-ship 22 missiles is often favorable for increasing the survivability since 23 this strategy enables penetrating ship-board self-defense 24 systems by using multiple missiles to intercept a target simulta-25 neously.1–4. For this reason, extensive research activities have 26 been performed to develop impact time control guidance for 27 various guided weapons in recent years.
28 The first paradigm of impact time control in missile
guid-29 ance was presented in Ref.1, in which the optimal control
the-30 ory and linear kinematic model were utilized in deriving the
31 guidance law. The resultant guidance command is given by
32 an optimal form of PNG command with a feedback command
33 that reduces the impact time error. In Ref.5, an guidance law
34 to control impact time, similar to Ref. 1, was developed by
35 using the nonlinear engagement kinematics model. Extension
36 of Ref.1was also reported in Ref.6, where the impact angle
37 control constraint was also considered. In this study, the jerk
38 command, i.e., the time derivative of acceleration, was selected
39 as the control input with the purpose of providing an
addi-40 tional degree-of-freedom for impact time synthesis. Recently,
41 a general approach was proposed in Ref.7to convert existing
42 guidance laws into their impact time control versions. This
43 method can be applied to any guidance law that has a reliable
44 time-to-go estimation. The work in Ref. 8 analyzed the
* Corresponding author.
E-mail address:lckdgns@kaist.ac.kr(C.-H. LEE).
Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier Chinese Journal of Aeronautics, (2020), xxx(xx): xxx–xxx
Chinese Society of Aeronautics and Astronautics
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45 optimality of error dynamics in impact time synthesis for mis-46 sile guidance. Different from previous works, the authors in 47 Ref.9suggested a new PNG that relies on a time-varying guid-48 ance gain with the purpose of adjusting the interception time 49 gap of multiple missiles. Following the guidance design philos-50 ophy proposed in Ref. 8, a guidance law to control both 51 impact time and angle was devised in Ref.10. A detailed anal-52 ysis of the minimum and maximum range of feasible impact 53 times was also investigated in this work.
54 Impact time control guidance laws have also been devel-55 oped from the trajectory shaping standpoint. Assuming that 56 a desired Line-of-Sight (LOS) angle profile is a polynomial 57 function, trajectory shaping guidance laws were proposed in 58 Refs.11,12. The impact time constraint in these guidance laws 59 can be satisfied by optimizing the polynomial coefficients. Dif-60 ferent from Refs.11,12, a guidance law to control impact time 61 was accomplished in a way to shape the velocity lead angle in 62 Refs. 13–15. The advantage of trajectory-shaping-based 63 impact time control is that this strategy does not requires the 64 information on the time-to-go, thus simplifying the 65 implementation.
66 It is known that impact time control requires to make a 67 detour to adjust the impact time error, which might cause a 68 seeker’s loss of target due to limited FOV constraint. For this 69 reason, it is meaningful to design impact time control guidance 70 laws that not violate the seeker’s FOV condition. The earliest 71 work that considers the seeker’s FOV limit under impact time 72 constraint was reported in Ref.16. This method was based on 73 the switching of guidance command from the original guidance 74 command providing the desired impact time to the guidance 75 command maintaining the constant look angle. This switching 76 strategy, unfortunately, suffers from a sudden abrupt change 77 of the acceleration command. A similar idea was also utilized 78 in Ref.17by switching the guidance gain to cater to the see-79 ker’s FOV limit. To enforce a continuous command, the 80 authors in Refs.18,19proposed a bias type of PNG for con-81 trolling impact time and utilized the nonlinearity of cosine 82 function to constrain the seeker’s FOV. The seeker’s FOV con-83 straint was also considered in Refs. 15–20 through velocity 84 lead angle shaping, but these works fail to address the optimal-85 ity issue. Note that guidance laws proposed in Refs. 15–20 86 ensures a monotonic decrease of the velocity lead angle from 87 the initial point. Therefore, these two guidance laws cannot 88 fully exploit the seeker’s available look angle to achieve a lar-89 ger desired impact time if the initial lead angle is small. 90 Notice that most guidance laws providing the desired 91 impact time are studied in a Two-Dimensional (2D) engage-92 ment kinematics. In order to fully utilize the synergism effect 93 between horizontal and vertical planes, the authors in Refs. 94 2–23 suggested impact time guidance laws formulated in a 95 Three-Dimensional (3D) engagement scenario for tactical mis-96 siles. However, these works never consider the seeker’s FOV 97 limit in designing of guidance laws, thus posing some difficul-98 ties in practical implementation. As a remedy, this paper aims 99 to suggest a new optimal guidance law to control impact time 100 while satisfying the seeker’s FOV limit in 3D space. Similar to 101 Refs.1,5,8the biased PNG concept is utilized in deriving the 102 guidance command. More specifically, the proposed guidance 103 command consists of two terms: PNG as the baseline com-104 mand and the feedback command of impact time error. The 105 error feedback term is determined by utilizing the concept of 106 the optimal error dynamics, developed in our previous work.
107 8
A nonlinear function is leveraged to shape the error feedback 108 term, ensuring the boundedness of the velocity lead angle.
The-109 oretical analysis uncovers that the impact time error converges
110 to zero as the time goes and the velocity lead angle constraint is
111 always satisfied the proposed guidance law. By analyzing the
112 optimality of the biased command, a theoretical way to choose
113 the guidance gain is also proposed to guarantee finite and zero
114 terminal guidance command. Up to the best of our knowledge,
115 no 3D impact time control guidance law that considers the
see-116 ker’s field-of-view limit is available in the open reference.
117 The remaining of this paper is constructed as follows.
Sec-118 tion 2 provides the problem formulation. The details of the
119 proposed guidance law is presented in Section3. The
theoret-120 ical analysis of the proposed method is provided in Section4.
121 Nonlinear engagement simulations are conducted in Section5.
122 Finally, the concluding remarks are provided in Section6.
123 2. Problem formulation
124 In this section, the guidance problem to be solved in this study
125 is formulated. To derive the engagement kinematics, four basic
126 assumptions are introduced as follows:
127 Assumption 1. The target is stationary or slowly moving.
128 Assumption 2. The missile is treated as a point-mass with an
129 ideal dynamics.
130 Assumption 3. The missile speed is assumed to be constant.
131 Assumption 4. The seeker’s look angle is approximated by the
132 velocity lead angle.
133 Here, it is worth noting that the above assumptions have
134 been widely considered for developing impact time guidance
135 laws for anti-ship missiles: (Assumption 1) A target such as
136 battle ship is slowly moving compared to anti-ship missiles.
137 Therefore, the target speed can be negligible when designing
138 a guidance law. (Assumption 2) Most missile guidance and
139 control systems have been designed using the separated
guid-140 ance and control structure, based on the fact that the
band-141 widths of the guidance loop and the control loop are
142 considerably different. In this perspective, the guidance
prob-143 lem can be considered as the kinematics problem with ideal
144 missile dynamics. (Assumption 3) The missile’s speed is
typi-145 cally slowly varying compared to other state variables.
There-146 fore, it could be treated as a piece-wise constant at each time
147 step. (Assumption 4) For typical anti-ship missiles, the
angle-148 of-attack and sideslip angle are very small during the endgame.
149 Therefore, the seeker’s look angle can be approximated by the
150 velocity lead angle.
151 Based on the above assumptions, we consider the
engage-152 ment geometry in the 3D space, as depicted inFig. 1. In this
153 figure, the notation Xð ; Y; ZÞ represents the inertial reference
154 frame for describing the positions of the missile and the target.
155 The relative range between the missile and the target is
156 expressed by the notation R. InFig. 1, the parameters# and
157 u, as shown inFig. 1, denote the LOS angles between the
mis-158 sile and the target in the inertial reference frame. To be more
159 specific, # is the LOS angle in azimuth direction, and the u
160 is the LOS angle in elevation direction. The parametersh and 161 / represent the velocity lead angles of the missile in the LOS 162 frame as shown inFig. 1.h is the velocity lead angle in pitch 163 direction, and / is the velocity lead angle in yaw direction. 164 Note that these angles can be calculated using the information 165 of the gimbal angles measured by an onboard seeker, as stated 166 in Ref.24. InFig. 1,r represents the total velocity lead angle 167 of the missile in the engagement plane. It can be considered as 168 the heading error to achieve the interception condition for the 169 stationary target. Finally, the notation V represents the missile 170 speed. The nonlinear engagement kinematics in the 3D space 171 can be written as24 172 _R ¼ V cos h cos / ð1Þ 174 174 175 _# ¼ V Rsinh ð2Þ 177 177 178 _u ¼ V
Rcos#cosh sin / ð3Þ
180 180 181 _h ¼ az Vþ V Rcosh sin 2/ tan # þV Rsinh cos / ð4Þ 183 183 184 _/ ¼ ay Vcosh V
Rsinh sin / cos / tan # þ V Rcosh sin2h sin / þV Rcosh sin / ð5Þ 186 186
187 where ayand azrepresent the acceleration components of the 188 missile in yaw direction and pitch direction, respectively. 189 Additionally, fromFig. 1, the relationship between the pro-190 jected velocity lead angles and the look angle can be deter-191 mined as follows
192
cosr ¼ cos h cos / ð6Þ
194 194
195 The ultimate goal of a guidance is to make the heading 196 error or zero-effort-miss become zero such that the missile 197 can follow the desired collision course to intercept a target. 198 This condition can be mathematically formulated as
199
r tð Þ ¼ 0f ð7Þ
201 201
202 where tfdenotes the final impact time.
203 To enable simultaneous attack against a stationary target,
204 this paper also considers the impact-time constraint, which is
205 given by
206
tf¼ td ð8Þ 208208
209 where, the notation tfrepresents the final time, and the
nota-210 tion tdrefers to the impact time to be achieved. Here, td is a
211 kind of design parameter which is selected by a designer to
212 accomplish a mission objective.
213 In order to maintain target lock-on, i.e., within the seeker’s
214 FOV limit, the seeker’s look angle should not exceed its
max-215 imum permissible value. Under Assumption 4, the seeker’s
216 FOV constraint can be approximated as
217
jrj 6 rmax< 90 ð9Þ 219219
220 where rmax denotes the maximum permissible value of the
221 velocity lead angle.
222 In summary, the problem considered in this study is to
223 devise an optimal guidance law to satisfy constraints Eqs.(7)
224
and (6) without violating condition Eq. (9). It should be 225 pointed out that the desired impact time tdcannot be set
arbi-226 trarily large due to the seeker’s FOV limit. With seeker’s FOV
227 constraint Eq.(9), we haveV 6 _R 6 V cos rmax from Eq.
228
(1). Therefore, the feasible range of achievable impact time 229 under the seeker’s FOV limit can be obtained using the
for-230 mula, i.e., range over velocity, as
231 td2 R0 V; R0 Vcosrmax ð10Þ 233 233 234 where R0 denotes the initial value of the relative range.
235 Remark 1. Note that the term R0=V cos rmax in Eq.(10) is a
236 conservative estimation for the achievable impact time as the
237 maximum velocity lead angle cannot be maintained during the
238 flight. However, this information provides some insights into
239 the initial design of the desired impact time. The feasible region
240 of impact time given the interception scenario will be
numer-241 ically analyzed during simulation studies inSection 5.
242 3. Impact time guidance law design
243 In order to intercept the target with the desired impact time td,
244 the concept of biased PNG is utilized in this paper. That is, the
245 proposed guidance command is composed of a baseline PNG
246 part for target capture and a biased command for impact time
247 error regulation.
248 3.1. Baseline guidance law
249 The optimal 3D PNG command is given by24
250
aPNG¼ NX V ð11Þ 252252
253 where N indicates the proportional navigation constant of
254 PNG, which is a positive constant. The variable X is the
255 LOS rate vector, andV is the missile velocity vector. These
256 vectors can construct the interception engagement plane in
257 the 3D space as mentioned in Ref.25. As the relative range
258 is generally not controllable during the engagement, the
259 PNG command in the 3D space is typically applied in two
per-260 pendicular planes which are defined in the velocity frame as24
T R σ C Y M X A B V Z θ φ
Fig. 1 Three-dimensional homing engagement geometry.
261
aPNG
y ¼ NV_kysinh sin / þ NV_kzcosh ð12Þ
263 263 264 aPNG z ¼ NV_kycos/ ð13Þ 266 266
267 where _kyand _kzrefer to the components of the LOS rate vec-268 tor, which are defined in the LOS frame. Those angles can be 269 directly measured by the onboard seeker. Note that
270 _ky¼ V Rsinh ð14Þ 272 272 273 _kz¼ V Rcosh sin / ð15Þ 275 275
276 Substituting Eqs.(14) and (15)into Eqs.(12) and (13)yields 277 aPNG y ¼ NV2 R sin/ ð16Þ 279 279 280 aPNGz ¼ NV 2 R sinh cos / ð17Þ 282 282
283 3.2. Biased guidance command
284 Extending the results of 2D scenario5to a 3D engagement, the 285 interception time under 3D PNG can be easily determined as 286 tf¼ t þ R V 1þ sin2r 2 2Nð 1Þ ð18Þ 288 288
289 To realize the impact time control, we propose the follow-290 ing biased PNG command
291 ay¼ aPNGy þ absin/ ¼ NV2 R þ ab sin/ ð19Þ 293 293 294 az¼ aPNGz þ absinh cos / ¼ NV2 R þ ab sinh cos / ð20Þ 296 296
297 where abindicates the feedback term of the impact time error. 298 This term will be determined in the later.
299 Here, let us define the impact time error at the current time 300 aset¼ td tf. Taking time-derivative ofetprovides
301 _et ¼ _tf ¼ _R V _R V sin2r 2 2N1ð ÞRV sinr cos rð Þ_r 2N1 1 ¼ cos r 1 þ sin2r 2 2N1ð Þ h i Rcosr 2N1 ð ÞV2 N1 ð ÞV2 R sin 2r þ a bsin2r h i 1 ¼ cos r 1 þ sin2r 2 2N1ð Þ h i Rcosr sin2r 2N1 ð ÞV2 N1 ð ÞV2 R þ ab h i 1 ¼ cos r 1 þ sin2r 2 2N1ð Þ h i þðN1Þ cos r sin2r 2N1 Rcosr sin 2r 2N1 ð ÞV2 ab 1 ð21Þ 303 303
304 where Eq.(6)is utilized to derive the derivative of velocity lead 305 angler.
306 In Eq.(21), the initial value of velocity lead angler is small 307 in practice since the terminal homing guidance phase generally 308 follows a precise guidance handover from a proper mid-course 309 guidance law. The velocity lead angle also gradually decreases 310 as the time goes during the flight. Therefore, the velocity lead 311 angle is kept small during the flight. Hence, we have 312 sinr r; cos r 1 r2=2. By utilizing the small angle 313 approximation of r and by ignoring the higher-order terms 314 regardingr, the dynamics of the impact time error as provided 315 in Eq.(21)can be simplified as follows
316 _et¼ Rsin2r 2N 1 ð ÞV2ab ð22Þ 318 318 319 According to the previous study8, the optimal desired error
320 dynamics for a tracking error e is given as
321 _e þq tð Þ tgo e¼ 0 ð23Þ 323 323 324 where tgo¼ tf t denotes the time-to-go and q tð Þ > 0 is a
non-325 linear function to be designed. Based on this result, consider
326 the following desired error dynamics for the system Eq.(16)
327 to handle the seeker’s FOV constraint.
328 _etþ K/ r=rð maxÞ tgo et¼ 0 ð24Þ 330 330 331 where K represents a positive guidance gain that governs the
332 convergence pattern of et./ xð Þ is a user-defined function to
333 shape the velocity angle and satisfies the following condition.
334 Condition 1. The function / xð Þ is defined on 1; 1½ and
335 satisfies / 1ð Þ ¼ 0; / 1ð Þ ¼ 0 and / 0ð Þ ¼ 1. Furthermore,
336 function / xð Þ monotonically increases when x 2 1; 0½ and
337 monotonically decreases when x2 0; 1ð .
338 From Eqs.(24) and (22), the feedback term for nullifying
339 the impact time error can be determined as follows
340 ab¼ K/ r=rð maxÞ 2N 1ð ÞV2 Rtgosin2r et ð25Þ 342 342 343 Then, the proposed impact time control guidance law can
344 be obtained as 345 a ¼ NV2 R sinr þ K/ r=rð maxÞ 2N 1ð ÞV2 Rtgosinr et ea ð26Þ 347 347 348 where ea¼ 0; sin /= sin r; sin h cos /= sin r½ T represents the
349 unit vector that indicates the commanded acceleration
direc-350 tion in the velocity frame.
351 It is clear that the nonlinear function/ xð Þ provides a way
352 to shape the impact time error feedback command: the
capac-353 ity in regulating the impact time error reduces when the
mag-354 nitude of the velocity lead angle approaches its maximum
355 permissible value rmax. Furthermore, the proposed guidance
356 law degrades into classical PNG whenjrj ¼ rmax. Note that,
357 when the proposed guidance law becomes PNG, the proposed
358 guidance law can keep satisfying seeker’s FOV limit condition
359 since the velocity lead angle starts to gradually decrease under
360 PNG for a stationary target26.
361 4. Analysis of proposed guidance law
362 4.1. Optimality and convergence of impact time error
363 In this subsection, the convergence pattern of the impact time
364 error under the proposed guidance law as well as the
optimal-365 ity of the proposed guidance law will be analyzed theoretically.
366 Proposition 1. Assuming that the error feedback gain
367 K/ r=rð maxÞ is instantaneous constant for the analysis purpose,
368 then the biased guidance command ab is optimal in minimizing
369 the following performance index
370 J¼1 2 Z tf t R2sin4r tf s ð ÞK/ r=rð maxÞ1a 2 bds ð27Þ 372 372
373 Furthermore, the impact time error will approach 0 as t! tf, 374 i.e.,e tð Þ ¼ 0 under the proposed guidance law.f
375 Proof. Denote 376 gðtÞ ¼ Rsin 2r 2N 1 ð ÞV2 ð28Þ 378 378
379 Then, according to Theorem 1 in Ref.8, the biased com-380 mand ab, shown in Eq. (25), is the solution of the following 381 finite-time optimal regulation problem
382 min ab J¼1 2 Z tf t Wð Þas 2 bð Þdss ð29Þ 384 384
385 with the dynamic constraints as 386
_etð Þ ¼ g tt ð Þabð Þ et tð Þ ¼ 0tf ð30Þ 388
388
389 where the weighting function W tð Þ satisfies 390 K/ r rmax ¼RtgoW1ð Þgt 2ð Þt tf t W 1ð Þgs 2ð Þdss ð31Þ 392 392
393 Assuming that the error feedback gain K/ r=rð maxÞ is 394 instantaneous constant for the analysis purpose, solving Eq. 395 (31)gives the weighting function W tð Þ as
396 WðtÞ ¼ R2sin 2r tK/ r=rð maxÞ1 go ð32Þ 398 398
399 Since the optimal error dynamics, as shown in Eq.(24), is 400 given in the form of Cauchy-Euler equation, it can be readily 401 shown that the time history ofetis obtained as
402 et¼ et;0 tgo tf K/ r=rð maxÞ ð33Þ 404 404
405 whereet;0represents the impact time error at the initial time. 406 In the following section, we will show that / r=rð maxÞ is 407 lower bounded by a small positive constant during the homing 408 phase. Then, it follows from Eq.(33)that the impact time error 409 et will be nullified before the time of interception, thus 410 accomplishing the impact time control as desired. Addition-411 ally, the convergence speed ofetis decided by the selection of 412 K.
413 InProposition 1, the performance index as shown in Eq. 414 (27) can give us general insights into the command pattern 415 of the proposed guidance law and the behavior of guidance 416 laws under the circumstance of impact time control. From 417 Eq. (27), we can readily observe that the magnitude of the 418 weight function decreases as R and r decrease for a given 419 time-to-go. Therefore, the resultant guidance command might 420 increase as R orr decrease as shown in Eq.(27). This is a gen-421 eral phenomenon of impact-time-control guidance laws. For a 422 stationary target, the parameterr can be considered as a head-423 ing error. Therefore, decreasing R andr means that the missile 424 approaches a target, converging to a collision course. For the 425 impact time control, the missile needs to make a detour away 426 from the desired collision triangle, hence adjusting the flight 427 time as desired. By the geometric rule, we readily know that
428 changing the flight trajectory is getting easy as the missile
devi-429 ates from the collision course. Namely, once the missile makes
430 the collision course to a target, more control effort is required
431 to correct the flight time as desired. This is a general
character-432 istic of impact time control guidance laws, and the above
per-433 formance index implies this fact. Additionally, based on the
434 above performance index, we can determine the command
pat-435 tern of the proposed guidance law. A more detailed analysis
436 will be provided inSection 4.3.
437 It is worth noting that the result ofProposition 1is based
438 on the approximation of the instantaneous constant for the
439 purpose of theoretical analysis. Therefore, the results obtained
440 might not be exactly the same as the actual ones. However, this
441 assumption is widely-utilized in guidance law analysis Refs.
442
27–29and we would say that the results obtained are enough 443 to grasp the overall properties of the proposed method.
444 4.2. Velocity lead angle analysis
445 In this subsection, we will analyze the properties of velocity
446 lead angler under the proposed guidance law to show that
447 condition Eq.(9)is always satisfied. The results are presented
448 in the following proposition.
449 Proposition 2. If the initial velocity lead angle r0 satisfies
450 jr0j 6 rmax, then the proposed guidance law ensures that
451 jrj < rmaxfor all t> 0.
452 Proof. In the following,Proposition 2 will be proved in two
453 steps. Step 1 will show jrj 6 rmax holds for all t> 0 if
454 jr0j 6 rmaxand Step 2 will subsequently demonstrate the
veloc-455 ity lead angler will never achieve the maximum permissible
456 value rmax for all t> 0 under the proposed guidance law if
457 jr0j 6 rmax.
458 Step 1. Differentiatingr with respect to time and
substitut-459 ing Eq.(26)into it yields
460 _r ¼ ðN 1ÞV R sinr þ K/ r=rð maxÞ 2N 1ð ÞV Rtgosinr et ð34Þ 462 462 463 For system Eq.(34), consider W¼ 0:5r2as a Lyapunov
func-464 tion candidate. Taking the time derivative of W results in
_ W ¼ ðN1ÞVr sin rR þK/ r=rð maxÞ 2N1ð ÞVr Rtgosinr et ¼ ðN1ÞVr sin r R þ abr V ð35Þ 466 466 467 Since/ 1ð Þ ¼ 0 and / 1ð Þ ¼ 0, we have
_
Wjr¼rmax¼
N 1
ð ÞVrmaxsinrmax
R < 0 ð36Þ 469469
470 which reveals thatR ¼ r r 6 rf j maxg is a positively invariant
471 set. That is,jrj 6 rmax holds for all t> 0 if r02 R.
472 Step 2. Without loss of any generality, assume that
473 jr0j 6 rmax c1 with c1 being a small constant. Note that
474 if jr0j ¼ rmax, there exists a certain time t such that
475 jr tð Þj ¼ rmax c1 since _W jr¼rmax < 0. Next, the following
476 two cases are considered.
477 Case 1. abr < 0. Under this condition, it follows from
478 Eq.(35)that _W < 0, which means jrj < rmaxis true.
479 Case 2. abr P 0. If _W > 0, there exists a small constant
480 c2 such that _W jjrj¼rmaxc2 ¼ 0 since function _W is
481 uous with respect tor and _W jr¼rmax< 0. Therefore, we
482 can readily conclude that the velocity lead angler is con-483 strained by ½rmaxþ c2; rmax c2 for all t > 0. This 484 means that the inequality jrj < rmax holds when
485 abr P 0.
486 487 488
489 Finally, Combining the results of C ases 1 and 2 leads to the 490 proof ofProposition 2.
491 Proposition 2reveals that the velocity lead angle constraint 492 Eq.(9)is satisfied under the proposed guidance law, demon-493 strating that the proposed approach can handle the seeker’s 494 FOV constraint. Furthermore, Proposition 2 also indicates 495 that the error feedback gain K/ r=rð maxÞ is lower bounded by 496 a positive constant during the homing phase. According to 497 Proposition 2, one can imply that cosr P cos rmax> 0. Then, 498 it follows from Eq.(1)that R decreases monotonically. This 499 reveals that there exists a certain time tf such that R tð Þ ¼ 0.f 500 SinceProposition 1shows thatetð Þ ¼ 0, it can be concludedtf 501 that R tð Þ ¼ R td ð Þ ¼ 0. Therefore, the proposed guidance lawf 502 enables intercepting a target with desired impact time. The 503 convergence of the velocity lead angle is given in the following 504 proposition.
505 Proposition 3. Under the proposed guidance law, the velocity 506 lead angle r will converge to zero at the time of impact, i.e., 507 r tð Þ ¼ 0.f
508 Proof. Assume that r tð Þ – 0, it follows from Eq.f (35) that 509 Wj_ t¼tf ¼ N 1ð ÞVr sin r=R since etð Þ ¼ 0. As we discussedtf
510 before, R tð Þ ¼ R td ð Þ ¼ 0.f Therefore, the condition 511 Wj_ t¼tf ¼ 1 holds. Notice that the function _Wis continuous
512 with respect to time. Thus, there exists a certain time t1; t1< tf, 513 such that _W< 0 holds for all t 2 t½1; tf. FromProposition 2, 514 we know that the lead angle r is uniformly bounded with 515 jrj 6 rmax for all t> 0, which means that W tð Þ is bounded.1 516 With this in mind, it follows from the fact of _Wjt¼ tf¼ 1 517 and the continuity of W_ that there exists a certain time 518 t2; t1< t2< tf, such that the condition _W6 W tð Þ= t1 ðf t1Þ 519 holds for all t2 t½2; tf. Therefore, we have
520 W tð Þ ¼ W tf ð Þ þ1 Rtf t1Wdt_ 6 W tð Þ 1 W tð Þ1 tft1ðtf t1Þ ¼ 0 ð37Þ 522 522 523 By the definition of W, it is clear that W tð Þ is non-negative,f
524 i.e., W tð Þ 0. Then, it can be concluded that W tf ð Þ ¼ 0.f
525 Therefore, the velocity lead angle r will converge to zero at
526 the time of impact, i.e.,r tð Þ ¼ 0.f
527 4.3. Guidance command analysis
528 From Eq.(27), it can be noted that the time-to-go power order
529 K/ r=rð maxÞ 1 will become negative when / r=rð maxÞ is close
530 to zero. It is not desirable in terms of ensuring a finite guidance
531 command when the missile approaches the target. However,
532
Proposition 3 shows that the velocity lead angle r will 533 approach zero value as t! tf, i.e.,r tð Þ ¼ 0, which means thatf
534 / r tð ð Þ=rf maxÞ ¼ 1. This condition enables the possibility of
535 choosing K to guarantee a finite and even zero final guidance
536 command, thus providing enough operational margins to react
537 to undesired external disturbances or unexpected situations
538 when the missile is near the target. From Proposition 3, one
539 can conclude that the performance index of the biased
com-540 mand can be approximated as
541 J1 2 Z tf t R2sin4r tf s ð ÞK1a2bds ð38Þ 543 543 544 when the missile approaches the target. From Eq.(38), it can
545 be observed that the weighting term R2sin4r=tK1
go decreases 546 as the parameters R orr decrease. Because of this tendency,
0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 ϕ1 (x) ϕ2 (x) ϕ3 (x) |x| ϕ ( x) 0 0.5 1.0 0 0.5 1.0 0 0.2 0.4 0.6 0.8 1.0 0.5 1.0 n=1 n=3 n=5 n=1 n=3 n=5 n=1 n=3 n=5 ϕ1 (x ) ϕ2 (x ) ϕ3 (x )
(a)Profiles of candidate functions with n=5 (b) Profiles of candidate functions with different n |x|
Fig. 2 Examples of candidate function/ xð Þ.
Table 1 Initial conditions for homing engagement.
Parameter Value
Missile initial position (m) ð6000; 6000; 0Þ
Missile flight velocity (m/s) 250
Missile initial velocity lead angleh0; /0() 10, 10
Target position (m) ð0; 0; 0Þ
547 it is required to select sufficiently large value of K such that the 548 term tK1go is greater than R
2sin4r intended to react to the 549 decreasing of the weighting value. However, it follows from 550 Eqs. (19) and (20)that the biased command ab is multiplied 551 by either sin/ or sin h cos /. Since / and h are two velocity 552 lead angles in pitch and yaw directions, we can then choose 553 proper K such that tK1go is larger than R
2sin2r to guarantee 554 finite terminal guidance command. It is worth noting that 555 the decreasing patterns of R andr are mainly determined by 556 the PNG command in the proposed guidance law. Since 557 R Vtgo when the interceptor approaches the target, the 558 decreasing rate of R is proportional to tgo. For the PNG with 559 constant N, the closed-form solution ofr is given as r ¼ CtN1go 560 where C is a constant determined by the initial condition30 561 Considering the fact that r is small, the decreasing rate of 562 R2sin2r is, therefore, proportional to t2Ngo. With this in mind, 563 a suitable choice of K that guarantees a zero final guidance 564 command is K 1 > 2N.
565 From Eq.(26), it is found that the conditionr ¼ 0 is con-566 sidered as a singularity in terms of generating the guidance 567 command. This condition might result in an abrupt guidance 568 command. But, it can be readily verified that this singular con-569 dition is trivial becauser does not become zero (i.e., r – 0) 570 except for the final interception point for the stationary target. 571 To see this, assume that there exists a certain time
572 t3; 0 < t3< tf such that r ¼ 0 holds for all t 2 t½3; tf, then,
573 the missile will fly along a straight line to hit the target when
574 tP t3. This implies thatet becomes zero at t¼ t3. However,
575 it follows from Proposition 1 that et becomes zero only at
576 the moment of interception. Therefore, it can be readily
con-577 cluded that t3¼ tf, meaning that the pointr ¼ 0 is trivial.
578 4.4. Selection of/ xð Þ
579 From the previous analysis, one can note that the function
580 / xð Þ is the key factor in the proposed method because the
con-581 vergence pattern of the impact time error is decided by this
582 function. Also, this function ensures the boundedness of the
583 velocity lead angle. For these reasons, this subsection will
pro-584 vide the details of how to choose this specific function. Any
585 function of/ xð Þ that satisfies Condition 1, shown in Section3,
586 can be utilized for implementing the proposed guidance law.
587 Candidates of/ xð Þ could be 588 /1ð Þ ¼ 1 jxjx n ð39Þ 590 590 591 /2ð Þ ¼x 1 1 e1 e xj jn e1 ð40Þ 593593 594 /3ð Þ ¼ cosx pxn 2 ð41Þ 596 596 100 50 0 50 100 0 5 az (m/s 2) ay (m/s 2) 10 15 20 25 30 35 40 45 0 20 40 60 80 100 td=35 s td=37 s td=40 s td=43 s td=35 s td=37 s td=40 s td=43 s Flight time (s) 0 5 10 15 20 25 30 35 Flight time (s) 40 45 1 0 1 2 3 4 5 6 7 8 9
Impact time error (s)
td=35 s td=37 s td=40 s td=43 s 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 Flight time (s) V
elocity lead angle (°) td=35 s
td=37 s td=40 s td=43 s 0 2000 4000 2000 4000 0 500 1000 1500 2000 Target td=35 s td=37 s td=40 s td=43 s Z (m) 6000 6000 X (m) Y (m) 0
(a) Interception trajectory (b) Impact time error
(c) Acceleration command (d) Velocity lead angle
Fig. 3 Simulation results of proposed guidance law with various impact times.
597 where n> 0 is a design parameter to control the curvature of 598 / xð Þ.Fig. 2shows the profiles of these three different candi-599 date functions.
600 Remark 2. With larger n; / xð Þ is closer to 1 when jxj < 1 but 601 / xð Þ converges to zero more sharply when jxj approaches 1. 602 Therefore, the parameter n is a trade-off design between 603 desired impact time and velocity lead angle margin: larger n 604 provides the possibility of achieving longer desired impact time 605 while smaller n generates more operational margins for the 606 velocity lead angle.
607 Remark 3. Note the form of the function / xð Þ suggested in 608 Eqs.()()()(39)–(41)are proposed heuristically and many other 609 meaningful candidate functions that satisfy Condition 1 can be 610 utilized. It is worth pointing out the design of function/ xð Þ 611 depends on the capability of the interceptor. For example, 612 the candidate/3ð Þ, shown in Eq.x (41), is more favorable to
613 achieve longer desired impact time as this function provides
614 longer during to explore the maximum look angle in the
hom-615 ing phase, compared with functions/1ð Þ and /x 2ð Þ. This canx
616 be clearly confirmed byFig. 2(a). However, if the
maneuver-617 ability of the missile is limited, candidates/1ð Þ and /x 2ð Þ arex
618 more favorable to save energy consumption.
619 5. Numerical simulations
620 In this section, we conduct numerical simulations for an
anti-621 ship missile to show the effectiveness of the proposed method.
622 The initial condition considered scenario are summarized in
623
Table 1. To implement the proposed method, the proportional 624 navigation constant of the PNG in the 3D space is set to
625 N¼ 3. The guidance gain for the feedback term is selected
626 as K¼ 8 > 2N þ 1 in all simulations. The nonlinear function
627 / xð Þ is set as / xð Þ ¼ cos px5
2
without any further tuning. 0 2000 4000 0 2000 4000 0 500 1000 1500 2000 Target σmax=40° σmax=45° σmax=50° σmax=55° Z (m) 6000 6000 5 10 15 20 25 30 35 40 45 0 10 20 30 40 50 60 Flight time (s) V
elocity lead angle (°)
σmax=40°
σmax=45°
σmax=50°
σmax=55°
0 5 10
Impact time error (s)
15 20 25 30 35 σmax=40° σmax=45° σmax=50° σmax=55° 40 45 1 0 1 2 3 4 5 6 7 8 Flight time (s) 50 0 50 100 0 5 10 15 20 Flight time (s) 25 30 35 40 45 0 20 40 60 80 100 σmax=40° σmax=45° σmax=50° σmax=55° σmax=40° σmax=45° σmax=50° σmax=55° az (m/s 2) ay (m/s 2) X (m) Y (m)
(a) Interception trajectory (b) Impact time error
(c) Acceleration command (d) Velocity lead angle
Fig. 4 Simulation results of proposed guidance law with various velocity lead angle constraints.
Table 2 Summary of initial conditions.
Scenario Scenario 1 Scenario 2 Scenario 3 Scenario 4
h0() 10 39 0 10
/0() 10 0 39 10
628 Since the missile achievable acceleration is always bounded 629 due to physical constraint, the maximum magnitudes of ay 630 and azare chosen as 100 m/s2in all simulations.
631 5.1. Performance with different impact times
632 In this section, the proposed guidance law is tested with vari-633 ous impact times td¼ 35; 37; 40; 43 s. Note that td¼ 43 s is 634 the approximated maximum desired impact time for the con-635 sidered scenario through numerical analysis. The maximum 636 permissible lead angle is set asrmax¼ 40.Fig. 3(a) compares 637 the interception trajectory obtained from different desired 638 impact times, showing that a longer and more curved path is 639 required for a larger impact time. The impact time error under 640 various conditions obtained from the proposed guidance law is 641 depicted inFig. 3(b), demonstrating that the impact time error 642 becomes zero at the final time. The impact time errors recorded 643 in simulations under the considered four different conditions 644 are less than 0.05 s. This result indicates that the proposed 645 method can successfully intercept the target with an accurate 646 impact time as desired. The missile acceleration command is 647 presented in Fig. 3(c). Clearly, more energy consumption is 648 observed at the beginning of the homing phase as tdincreases. 649 Therefore, the duration of the guidance command saturation is 650 longer with a larger desired impact time. Additionally, the
651 guidance commands converge to zero at the time of
intercep-652 tion in all the considered scenarios. FromFig. 3(c), we can
653 also note that the magnitude of acceleration command
654 increases significantly during the terminal homing phase when
655 the desired impact time approaches its maximum feasible
656 value. It is caused by the fact the available time for driving
657 the velocity lead angle to zero relatively short once the desired
658 impact time approaches its maximum permissible value.Fig. 3
659 (d) shows the profile of the velocity lead angle when employing
660 the proposed guidance law. The results inFig. 3 (d) reveals
661 that the magnitude of the velocity lead angle initially increases
662 intended to reduce the impact time error, and finally converges
663 to zero at the final time to ensure target interception.
664 5.2. Performance with different velocity lead angle constraints
665 In this subsection, the performance of the proposed guidance
666 law is evaluated with various velocity lead angle constraints
667 rmax¼ 40; 45; 50; 55. The desired impact time is set as
668 td¼ 42 s. The simulation results, including interception
trajec-669 tories, impact time error, missile acceleration commands, and
670 velocity lead angle profiles, are presented inFig. 4,
demon-671 strating that the missile successfully intercepts the target at
672 the specific time with various lead angle constraints.Fig. 4also
673 reveals that the proposed guidance law tries to leverage the 0 2000 4000 0 2000 4000 0 500 1000 1500 2000 2500 Target Scenario 1 Scenario 2 Scenario 3 Scenario 4 Y (m) Z (m) 6000 6000 15 10 5 0 20 25 30 35 40 1 0 1 2 3 4 5 6 Flight time (s) Scenario 1 Scenario 2 Scenario 3 Scenario 4
Impact time error (s)
100 50 0 50 100 15 10 5 0 20 25 30 35 40 Scenario 1 Scenario 2 Scenario 3 Scenario 4 50 0 50 100 Scenario 1 Scenario 2 Scenario 3 Scenario 4 Flight time (s) ay (m/s 2) az (m/s 2) 0 5 10 15 20 25 30 35 40 45 5 10 15 20 25 30 35 40 45 Flight time (s) V
elocity lead angle (°)
Scenario 1 Scenario 2 Scenario 3 Scenario 4
X (m)
(a) Interception trajectory (b) Impact time error
(c) Acceleration command (d) Velocity lead angle
Fig. 5 Simulation results of proposed guidance law with various initial conditions.
674 maximum available lead angle to reduce the impact time error 675 during the initial period of the flight. For this reason, the inter-676 ception trajectory is more curved and more control energy is 677 required during the initial period with a largerrmax. Therefore, 678 the impact time error under the proposed guidance law with a 679 larger FOV constraint converges to zero faster, as shown in 680 Fig. 4(b). With smallerrmax, however, a slightly sharper turn 681 of the acceleration command can be noted from Fig. 4 (c) 682 when the missile approaches the target. The reason for this fact 683 is that the interceptor with smallerrmaxhas a shorter time in 684 regulating the lead angle to zero.
685 5.3. Performance with different initial conditions
686 Now, let us examine the performance of the proposed method 687 with various initial conditions, as shown inTable 2. The max-688 imum permissible lead angle is set as rmax¼ 40 and the 689 desired impact time is set as td¼ 40 s. Note that scenario 1
690 is the same as the scenario studied in previous subsections.
Sce-691 narios 2 and 3 are two extreme scenarios in which the initial
692 velocity lead angle approaches its maximum feasible value.
693 The interceptor in scenario 4 has opposite initial flight path
694 angle in the horizontal plane as scenario 1. The interception
695 trajectories of all scenarios are shown in Fig. 5 (a), which
696 reveals that the missile flight trajectory of scenario 2 is located
697 in the vertical plane and the interception trajectory of scenario
698 3 locates in the horizontal plane. This can be attributed to the
699 fact that the proposed 3D guidance law converges to a 2D
700 impact time control guidance law when the initial flight path
701 angle of one plane becomes zero. This can be easily verified
702 by the guidance command Eq.(26).Fig. 5(b) compares the
703 impact time errors of all scenarios under the proposed
guid-704 ance law. It follows from this figure that scenarios 1 and 4 have
705 the same impact time error dynamics since these two scenarios
706 have the same initial velocity lead angle. Similarly, scenarios 2
707 and 3 also have the same impact time error dynamics. The
708 guidance commands in both pitch and yaw planes for all
sce-709 narios are presented inFig. 5(c). Since the proposed guidance
710 law in scenarios 2 and 3 reduces to 2D guidance, the
com-711 manded acceleration in one plane becomes zero. We can also
712 note that the scenarios 1 and 4 share the same guidance
com-713 mand in the pitch plane while having the opposite value in the
714 yaw plane. The reason is that these two scenarios have
sym-715 metric interception trajectories with respect to a vertical lpane,
716 as confirmed byFig. 5(a).Fig. 5(d) shows the time histories of
100 60 20 20 60 100 Existing method Proposed method 0 10 20 30 40 50 Flight time (s) Acceleration command (m/s 2) 2 0 2 4 6 8 10 0 10 20 30 40 50 Flight time (s) Existing method Proposed method
Impact time error (s)
0 10 20 30 40 50 10 20 30 40 50 60 70 Flight time (s) V
elocity lead angle (°)
Existing method Proposed method 0 2000 4000 6000 8000 10000 1000 2000 3000 4000 X (m) Z (m) Existing method Proposed method
(a) Interception trajectory (b) Impact time error
(c) Acceleration command (d) Velocity lead angle
Fig. 6 Comparison results between proposed method and existing method.
Table 3 Quantitative comparisons of control effort.
Method Existing method Proposed method
Control effort (m2/s3) 25473 11671
717 look angle under the proposed guidance law for all scenarios. 718 The result inFig. 5(d) reveals that the velocity lead angle can 719 be constrained with its feasible region and converges to zero 720 under the proposed guidance law with various initial 721 conditions.
722 5.4. Comparison with other guidance laws
723 To further show the superiority of the proposed guidance law, 724 comparison simulation is conducted in this subsection. As 725 mentioned before, up to the best of the author’s knowledge, 726 no impact-time-control guidance law with seeker’s FOV con-727 straint in a 3D space is available in the open literature. There 728 have been only some studies on some impact-time-control 729 guidance laws considering the seeker’s FOV limit in a 2D 730 space. Because of this reason, there is a limitation to compare 731 the proposed method to existing methods directly. Therefore, 732 in the comparison study, the proposed method is compared 733 to existing methods, in a vertical engagement plane only. For 734 the purpose of comparison, the impact-time-control guidance 735 considering the seeker’s FOV limits based on the backstepping 736 control technique31is implemented. The guidance command of 737 this method is given as
738 a¼ V_kzþ _VR sinh k2ðV cos h VRÞ sinh ð42Þ 740 740 741 where 742 VR¼ V þ k1sgmf V tð ðd tÞ RÞ ð43Þ 744 744
745 and the sigmoid function sgmf is selected as 746 sgmf xð Þ ¼ 1 2g3x 3þ3 2gx if xj j6 g sgn xð Þ else ( ð44Þ 748 748
749 In the above guidance command, k1; k2, and g represents 750 the design parameters. For comparison, the engagement sce-751 nario and the design parameters given in Ref. 31 are used: 752 td¼ 50 s; rmax¼ 60; k1¼ 125; k2¼ 1, and g ¼ 200. More 753 detail information can be found in Ref. 31. The simulation 754 results obtained from these two different guidance laws are 755 shown inFig. 6, which clearly indicates that all cases can suc-756 cessfully intercept the target while satisfying the desired impact 757 time. However, the difference between these two guidance laws 758 lies in the fact that the existing method shows an abrupt 759 change of the guidance command, as can be confirmed by 760 Fig. 6(b). This characteristic is not desirable for the guidance 761 operation because it might result in the instability of guidance 762 loop.32The control energy consumption, defined as
Rtf
0 a 2dt, is 763 recorded as inTable 3. From this result, it is concluded that 764 the proposed guidance law consumes less control energy, com-765 pared with the existing method. This is because the proposed 766 method is developed based on the concept of optimal error dy-767 namics.8 Last but not least, the proposed method can be 768 applied to a 3D engagement scenario, unlike previous methods 769 including Ref.31.
770 6. Conclusions
771 In this paper, the impact time control guidance problem for a 772 tactical missile with seeker’s FOV constraint against a station-773 ary or a slowly-moving target in a 3D engagement scenario is
774 suggested. The proposed guidance law consists of two
com-775 mand terms: a baseline 3D PNG command term and a
feed-776 back command term of an impact time error. Theoretical
777 analysis uncovers that the impact time error will be nullified
778 as the missile approaches the target. The optimality of the
779 biased feedback term and the behavior of the velocity lead
780 angle are also discussed to provide better insights into the
pro-781 posed guidance law. Although the proposed method is
devel-782 oped for a stationary target or a slowly-moving target, the
783 guidance algorithm developed can also be leveraged for a
784 non-maneuvering target by utilizing the
predicted-785 interception-point concept.
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