Chinese Journal of Aeronautics, (2016), 29(2): 441–447
Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics
[email protected] www.sciencedirect.com
Obstacle avoidance for multi-missile network via distributed coordination algorithm Zhao Jiang *, Zhou Rui School of Automation Science and Electrical Engineering, Beihang University, Beijing 100083, China Received 23 July 2015; revised 31 August 2015; accepted 30 September 2015 Available online 23 February 2016
KEYWORDS Cooperative guidance; Distributed algorithms; Impact time; Missile guidance; Multiple missiles; Obstacle avoidance; Proportional navigation
Abstract A distributed coordination algorithm is proposed to enhance the engagement of the multi-missile network in consideration of obstacle avoidance. To achieve a cooperative interception, the guidance law is developed in a simple form that consists of three individual components for target capture, time coordination and obstacle avoidance. The distributed coordination algorithm enables a group of interceptor missiles to reach the target simultaneously, even if some member in the multi-missile network can only collect the information from nearest neighbors. The simulation results show that the guidance strategy provides a feasible tool to implement obstacle avoidance for the multi-missile network with satisfactory accuracy of target capture. The effects of the gain parameters are also discussed to evaluate the proposed approach. Ó 2016 Chinese Society of Aeronautics and Astronautics. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction As the need for highly adaptive guidance and control approaches is increasing, obstacle avoidance techniques have been proposed for the unmanned aerial vehicles,1–3 ground vehicles,4,5 unmanned surface vehicles,6,7 autonomous underwater vehicles,8,9 and mobile robots.10,11 Considering that the group of interceptor missiles is guided in the complex environment, the threat avoidance and geopolitical restrictions are also indispensable to the development of guidance and control * Corresponding author. Tel.: +86 10 82316849. E-mail addresses:
[email protected] (J. Zhao),
[email protected] (R. Zhou). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
systems. For this reason, some studies have recently focused on the design of the reference routes12 and guidance laws13 for obstacle avoidance. However, only the single interceptor missile was considered in the above work. To improve the performance in detecting the targets and penetrating the defense systems, the implement of cooperative engagement for multimissile network is required.14–17 The difficult problem is an achievement of obstacle avoidance for multi-missile network with satisfactory accuracy of target capture as well as effective coordination of impact time between each member.18–20 In the current literature, many advanced cooperative guidance laws have been proposed for the multi-missile networks. The first class of approaches investigates the design of the impact-time constraints to achieve a simultaneous interception against the given target. In Ref.21, the closed form of impact time control guidance law is developed on the basis of the linear formulation, which can guide a group of missiles to the target at a desirable time. Then, the time-varying navigation gain is used to coordinate the impact time of the multi-missile net-
http://dx.doi.org/10.1016/j.cja.2016.01.011 1000-9361 Ó 2016 Chinese Society of Aeronautics and Astronautics. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
442 work.22 The extensions of the time-constrained guidance law are also presented to control both the impact time and impact angle.23,24 The above guidance strategies typically require that the global information of time-to-go is available to each interceptor missile. For this reason, the distributed control architecture of the impact-time constraint is proposed to enhance the cooperative engagement of multiple missiles.25,26 The discrete topology model is also used to feature the desired impact time using the consensus theory.27 Another class of approaches employs the leader–follower model to formulate the cooperative guidance problem for the multi-missile network. In Refs.28,29, a nonlinear state tracking controller and a state regulator are developed to solve the time-constrained guidance, respectively. Then, the consensus protocols are applied to the design of leader–follower strategy which guarantees that the impact time of each follower can converge to the leader in finite time.30 To facilitate the heterogeneous multi-missile engagement, a distributed leader–follower model is proposed on the basis of proportional navigation (PN) guidance law.31 Furthermore, the virtual leader scheme is also employed to achieve the impact time control by transforming the constrained guidance problem to the nonlinear tracking problem.32 Although the aforementioned design of the impact-time constraints21–27 and leader–follower strategies28–32 have promoted the development of the coordination algorithms for the multi-missile networks, the obstacle avoidance is not taken into account in these cooperative guidance approaches. Therefore, this paper presents an extension of the PN-based distributed guidance algorithm to enhance the engagement of the multi-missile network in consideration of obstacle avoidance. The contribution of the manuscript is summarized as follows: (1) the PN-based guidance law is developed in a simple form that consists of three individual components for target capture, time coordination and obstacle avoidance; (2) each member in the multi-missile network only requires the information of time-to-go from neighbors to perform a cooperative engagement; (3) the obstacle avoidance is achieved with satisfactory accuracy of target capture as well as effective coordination of impact time. The rest of paper is organized as follows. Section 2 presents the basic assumptions and the engagement geometry. Section 3 describes the formulation of cooperative strategy in detail. In Section 4, the feasibility of the proposed guidance law is demonstrated by numerical simulation. Section 5 presents the performance evaluation by discussing the effect of gain parameters. Finally, concluding remarks are presented in Section 6.
J. Zhao, R. Zhou (2) The seeker and autopilot dynamics of each interceptor missile are much faster in comparison with the guidance loop. (3) The velocity of each interceptor missile is constant and the acceleration input only changes its direction.
2.2. Engagement geometry Suppose that n missiles participate in the multi-missile network to intercept a stationary target simultaneously. Under the prescribed assumptions, the two-dimensional geometry on manyto-one engagement is depicted in Fig. 1. Let Mi denote each interceptor missile and T denote the target, and then, the pursuit situation can be described by the following equations of motion33,34 8 r_i ¼ Vi cos hi > > > > Vi sin hi > > < k_ i ¼ ri ð1Þ A > i > > _ ¼ c i > > Vi > : hi ¼ ci ki where the subscript (i = 1, 2, . . ., n) represents each member in the multi-missile network; ri is the missile-to-target range; Vi is the total velocity of each missile; ki is the line-of-sight angle; the terms ci and hi represent the heading angles in the inertial reference frame and line-of-sight frame, respectively; the acceleration command is defined as Ai. The problem studied herein is to find the coordination algorithm that can guide the group of missiles to the given target at the same time without obstacle collision, even if the initial conditions of each member are different. 3. Distributed coordination algorithm Based on the traditional PN guidance law, this section focuses on the design of the coordination algorithm for the multimissile network in the distributed formulation. The proposed cooperative guidance law consists of three individual components as Ai ¼ Api þ Ani þ Aai
where the terms Api , Ani and Aai are used for the target capture, time coordination and obstacle avoidance, respectively.
2. Preliminaries 2.1. Basic assumptions To simplify the nonlinear dynamics of the missile-target engagement, the pursuit situation in planar plane is considered in this paper. We assume some common conditions in the following part to facilitate the formulation of distributed coordination algorithm for the multi-missile network. (1) Both the interceptor missile and the target are considered as the geometric points in the planar plane.
ð2Þ
Fig. 1
Geometry on many-to-one engagement.
Obstacle avoidance for multi-missile network via distributed coordination algorithm The PN component Api is derived from the traditional PN guidance law in the form of33 Api ¼ Ni Vi k_ i ðtÞ
ð3Þ
where Ni is the navigation constant. In order to coordinate the impact time of each member in the multi-missile network, the component Ani is included into the proposed coordination algorithm Eq. (2) which has a simple form as follows26: Ani ¼ Kn ri ðtÞni ðtÞ
ð4Þ
where Kn is the gain parameter. The term ni (t) represents the relative error of the time-to-go between each missile: ni ðtÞ ¼ tgo;i ðtÞ t^go;i ðtÞ
ð5Þ
where the estimation of the time-to-go for each member tgo;i ðtÞ in the multi-missile network is given in the form of t^go;i ðtÞ ¼
ri ðtÞ Vi
ð6Þ
Note that, the term Ani ? 0 m/s2 as ri (t) ? 0 km if ni (t) ? 0 s as ri (t) ? 0 km, i.e., the effect of the coordination component Ani will be weakened when the relative error of the timeto-go gradually decreases as the group of missiles approach the given target. To guide the interceptor missiles to the target at a same time, the design of the coordination variable tgo;i ðtÞ will be discussed in the following part. The centralized coordination algorithms usually consider the case that each missile communicates with all the other members in the multi-missile network. However, the guidance laws would be out of work when some interceptor missile is only able to obtain the effective information from its nearest neighbors. Therefore, it is necessary to select the coordination variable tgo;i ðtÞ in the distributed formulation for an agreement on the impact time. Fig. 2 illustrates an example of the communication limit in the multi-missile network. As shown in Fig. 2, the effective communication region of each missile is marked with Si, in which Mi can obtain the information of the time-to-go from its neighbors. Thus, there are totally n communication regions with the group of missiles. Considering the limit of these communication regions, the coordination variable tgo;i ðtÞ in the component Ani is defined in distributed form as 1 X tgo;i ðtÞ ¼ ð7Þ t^go;j ðtÞ si 1 j2Si ;j–i where si is the total number of the missiles in the region Si. It also means that Mi can obtain the information of the time-togo from its si –1 neighbors in the distributed communication regions. Thus, the component Ani will try to adjust the timeto-go of each member in the multi-missile network for a simul-
Fig. 2
Communication limit in multi-missile network.
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taneous arrival. However, the distributed design does not reduce the relative time-to-go error of the whole multimissile network, but reduces the error of each small group. Note that, the limit of the communication region is actually determined by the performance of the communication systems. In the current literature, the distance between interceptor missiles is typically used to represent the communication limit. Therefore, the neighboring missiles are also determined by the relative distance between each missile in this paper. The third component Aai for obstacle avoidance is based on the simple design of the potential function in the form of Aai ¼ Ka cos ai =ðLi RÞ2
ð8Þ
where Ka is the gain parameter; the term R is the radius of the obstacle; Li represents the distance between the obstacle and each interceptor missile. As shown in Fig. 3, the circleshaped obstacle O is used herein because any irregularly shaped obstacles O1 and O2 can be simply replaced with it. Note that, the angle ai 2 [p/2, +p/2] is defined in the lineof-sight frame with respect to the center of the obstacle. It can be found that the component Aai mainly depends on the distance Li and heading angle ai. To be specific, the magnitude of the input component Aai gradually increases as |Li R| ? 0 km and |ai| ? 0°, whereas some larger |Li R| and |ai| result in a decrease in the magnitude of the acceleration. Thus, the complete distributed coordination algorithm for the multi-missile network can be given by Eq. (9) on the basis of the three components. The effect of target capture is determined by the PN component Api that continues during the entire engagement. The coordination component Ani will dominate the guidance algorithm Eq. (9) when the time-to-go error of the multi-missile network increases. The third component Aai is mainly used for the obstacle avoidance and it comes into effect as the interceptor missiles approach towards the obstacle. The integration of the three components leads to a trade-off in order to achieve the obstacle avoidance with satisfactory accuracy of target capture and effective coordination of impact time. Ai ¼ Ni Vi k_ i ðtÞ þ Kn ri ðtÞni ðtÞ þ Ka cos ai =ðLi RÞ2
ð9Þ
The improved coordination algorithm is quite easy to be implemented due to the simple design in the form of Eq. (9). The performance evaluation of the proposed cooperative guidance law will be presented in the following section. 4. Numerical simulations The numerical simulation is performed in a scenario of simultaneous arrival to evaluate the proposed distributed algorithm Eq. (9). Suppose that a group of three missiles participate in
Fig. 3
Geometry on interceptor missile for obstacle avoidance.
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the multi-missile network to intercept the given target at (0, 0) km with different initial conditions as shown in Table 1. The limit of the acceleration is set to be 5.0 9.81 m/s2 and the navigation constant, 3.0, for each interceptor missile. The gain Ka is set to be 20 and the gain Kn is set to be 30/(r0tgo0), where r0 is the mean value of the initial missile-to-target ranges and tgo0 is the mean value of the initial time-to-go estimations. As shown in Fig. 4, a simple communication topology is selected for the multi-missile network, in which M1 and M3 can only obtain the information of time-to-go from the neighbor M2. Thus, the coordination variables for the group of three missiles are expressed as 8 > < tgo;1 ðtÞ ¼ t^go;2 ðtÞ ð10Þ tgo;2 ðtÞ ¼ ðt^go;1 ðtÞ þ t^go;3 ðtÞÞ=2 > : tgo;3 ðtÞ ¼ t^go;2 ðtÞ where the total numbers of missiles in regions Si (i = 1, 2, 3) are s1 = 2, s2 = 3 and s3 = 2, respectively. Figs. 5 and 6 show the numerical results of the distributed coordination algorithm. Case 1 demonstrates the results of the simplified guidance law without component Aai , whereas the results of the complete coordination algorithm Eq. (9) are illustrated by Case 2. In Case 1, the ground tracks and the time-togo of the multi-missile network show that the guidance law Fig. 6 target. Table 1
Times-to-go of interceptor missiles against a stationary
Initial conditions of multi-missile network.
Missile
ri (km)
Vi (m/s)
hi (°)
ki (°)
M1 M2 M3
24 23 22
295 300 305
60 60 60
30 15 0
without Aai can drive the group of three missiles to simultaneously intercept the target at 86.18 s. However, M3 cannot fly round the given obstacle at (14, 5) km with a radius of 2 km. In contrast, the distributed coordination algorithm Eq. (9) succeeds in obstacle avoidance and a simultaneous arrival within a dispersion of 0.1 s. Note that the final impact time increases to 94.13 s since the group of missiles move further rounds to avoid the given obstacle. 5. Discussion 5.1. Effects of gain parameters
Fig. 4
Communication topology for multi-missile network.
Fig. 5 Ground tracks of interceptor missiles against a stationary target.
Regarding the effectiveness of the cooperative guidance law Eq. (9), the component Ani for coordination of impact time and the component Aai for obstacle avoidance are largely determined by the gain parameters Kn and Ka. Therefore, a proper selection of these gains may guarantee that the multimissile network achieves obstacle avoidance and a satisfactory simultaneous arrival. In this part, the effects of gain parameters Kn and Ka are discussed through some examples. First, we find that the convergence rate of the impact time is mainly influenced by the gain parameter Kn. Fig. 7 presents the numerical results of the time-to-go for a group of three interceptor missiles, in which the gain parameter Kn is set to be 20/(r0tgo0), 30/(r0tgo0), 40/(r0tgo0) and 50/(r0tgo0), respectively. It is quite clear that the time-to-go of the multi-missile network has the fastest convergence rate with gain Kn = 50/(r0tgo0). In contrast, it becomes slower as the gain gradually decreases to Kn = 20/(r0tgo0). The simulations demonstrate that a selection of gain parameter Kn between 20 and 50 generally result in better performance of the coordination of impact time. Further, we examine the effect of the gain parameter Ka by another example. Fig. 8 presents the numerical results of the
Obstacle avoidance for multi-missile network via distributed coordination algorithm
Fig. 7
Effect of gain parameter Kn on times-to-go.
Fig. 9
Fig. 8
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Effect of gain parameter Ka on control effort.
Effect of gain parameter Ka on ground tracks.
ground tracks for the multi-missile network, in which the gain Ka is set to 10 (dash-dot), 20 (dot), 30 (dash) and 40 (solid), respectively. It is found that the group of missiles move earlier and further rounds to avoid the obstacle as the gain increases from Ka = 10 to Ka = 40. As shown in Fig. 9, the control effort of the interceptor missiles (solid for M1, dash for M2 and dot for M3) is much lower when a larger gain Ka is chosen for component Aai . The simulations also suggest that the proper selection of the gain parameter Ka between 10 and 40 is able to meet the requirement for satisfactory obstacle avoidance. 5.2. Maneuvering target In this part, the scenario of cooperative engagement against a maneuvering target is used to evaluate the proposed coordination algorithm. The same initial conditions are selected as those in Section 4. Suppose that the velocity and acceleration of the target are set to constant values, i.e., VT = 100 m/s and AT = 2.0 9.81 m/s2, respectively. The initial heading angle of the target with respect to the inertial reference frame is set to be 60°. Figs. 10 and 11 illustrate the simulation results of the cooperative engagement against the maneuvering target. Cases
Fig. 10 Ground tracks of interceptor missiles against a maneuvering target.
3 and 4 demonstrate the coordination algorithm without component Aai and the complete coordination algorithm Eq. (9), respectively. It can be found that each member in the multi-missile network intercepts the maneuvering target with effective coordination of impact time. The obstacle avoidance is also achieved using the proposed guidance approach. Note that the final impact time is increasing since the head-pursuit guidance is performed in this scenario. 5.3. Size of obstacle Regarding the effect of the obstacle size in the coordination algorithm, another numerical simulation is performed in this part. Some different obstacle sizes (R = 2 km, 3 km, 4 km
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Fig. 12
Effect of obstacle size on ground tracks.
Fig. 11 Times-to-go of interceptor missiles against a maneuvering target.
and 5 km) are selected in the scenario of the cooperative engagement. The same initial conditions are selected as those in Section 4. Fig. 12 presents the ground tracks of the interceptor missiles. It is shown that the obstacle avoidance can be achieved using the proposed guidance law even though the size of the obstacle is up to 5 km. The coordination of the impact time is effective with satisfactory accuracy of target capture. Fig. 13 illustrates the effect of the obstacle size on the acceleration and the time-to-go. It can be found that a larger size of the obstacle typically results in the earlier saturation of the control effort. The time-to-go of the multi-missile network also increases with a larger size of obstacle. 6. Conclusions In this paper, a distributed coordination algorithm is proposed in which both the obstacle avoidance and the limited communication region among interceptor missiles are taken into account. (1) The proposed guidance law takes a simple form on the basis of the traditional PN algorithm. It enables a simultaneous arrival even if some interceptor missiles can only collect the information from nearest neighbors. (2) The simulation results show that the proper selection of gain parameter Ka can guarantee the obstacle avoidance with satisfactory accuracy of target capture. The convergence rate of impact time is determined by another gain Kn. (3) The distributed coordination algorithm is typically effective for the common size of obstacle. It also demonstrates that the cooperative interception against
Fig. 13
Effect of obstacle size on acceleration and time-to-go.
maneuvering target can also be achieved using the proposed algorithm.
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22. Jeon IS, Lee JI, Tahk MJ. Homing guidance law for cooperative attack of multiple missiles. J Guid Control Dyn 2010;33(1):275–80. 23. Zhang YA, Ma GX, Liu AL. Guidance law with impact time and impact angle constraints. Chin J Aeronaut 2013;26(4):960–6. 24. Lee JI, Jeon IS, Tahk MJ. Guidance law to control impact time and angle. IEEE Trans Aerosp Electron Syst 2007;43(1):301–10. 25. Zhao SY, Zhou R. Cooperative guidance for multimissile salvo attack. Chin J Aeronaut 2008;21(6):533–9. 26. Zhao J, Zhou R. Unified approach to cooperative guidance laws against stationary and maneuvering targets. Nonlinear Dyn 2015;81(4):1536–47. 27. Peng C, Liu X, Wu S, Li M. Consensus problems in distributed cooperative terminal guidance time of multi-missiles. Control Decision 2010;25(10):1557–66 Chinese. 28. Zhang YA, Ma GX, Wang XP. Time cooperative guidance for multi-missiles: A leader-follower strategy. Acta Aeronaut Astronaut Sin 2009;30(6):1109–18 Chinese. 29. Wang X, Hong X, Lin H. A method of controlling impact time and impact angle of multiple-missiles cooperative combat. J Ball 2012;24(2):1–6 Chinese. 30. Sun XJ, Zhou R, Hou DL, Wu J. Consensus of leader–followers system of multi-missile with time-delays and switching topologies. Optik 2014;125(3):1202–8. 31. Zou L, Ding Q, Zhou R. Distributed cooperative guidance for multiple heterogeneous networked missiles. J Beijing Uni Aeronaut Astronaut 2010;23(1):103–8 Chinese. 32. Zhao SY, Zhou R, Wei C, Ding QX. Design of time-constrained guidance laws via virtual leader approach. Chin J Aeronaut 2010;23(1):103–8. 33. Ha IJ, Hur JS, Ko MS, Song TL. Performance analysis of PNG laws for randomly maneuvering targets. IEEE Trans Aerosp Electron Syst 1990;26(5):713–21. 34. Zhou D, Qu P, Sun S. A guidance law with terminal impact angle constraint accounting for missile autopilot. J Dyn Syst Measur Control 2013;135(5):051009-1–051009-10. Zhao Jiang is currently a Ph.D. candidate in guidance, navigation and control from Beihang University, Beijing, China. His area of expertise includes autonomous flight control and guidance, constrained trajectory optimization, and flight mission planning. Zhou Rui is currently a professor in guidance, navigation, and control from Beihang University, Beijing, China. His area of expertise includes autonomous flight control and guidance, constrained trajectory optimization and flight mission planning.
