1 Tianjin Key Laboratory of Intelligent Robotics, College of Computer & Control Engineering, Nankai University, Tianjin 300071, China2 School of Mechanical Engineering, Liaoning Technical University, Fuxin, Liaoning 123000, China3Department of Electrical and Mining Engineering, University of South Africa, Florida 1710, South Africa
Correspondence should be addressed to Mingwei Sun; [email protected]
Received 12 March 2014; Accepted 29 April 2014; Published 14 May 2014
The trajectory optimization problem subject to terminal impact time and angle specifications can be reformulated as a nonlinearprogramming problem using the Gauss pseudospectral method. The cost function of the trajectory optimization problem ismodified to reduce the terminal control energy. A receding horizon optimization strategy is implemented to reject the errors causedby the motion of a surface target. Several simulations were performed to validate the proposed method via the C programminglanguage. The simulation results demonstrate the effectiveness of the proposed algorithm and that the real-time requirement canbe easily achieved if the C programming language is used to realize it.
1. Introduction
To enhance the penetration probability of a cruise missile,controlling both the impact angle and the impact time is ofspecial significance in practice. In the past three decades, sev-eral guidance laws with terminal impact angle specificationswere proposed within different frameworks [1–11]. It shouldbe noted that the impact time is a crucial factor in the salvoattack against a highly valued surface target equipped withadvanced air defense installations such as close-in weaponsystems (CIWS). However the impact time specification isdifficult to achieve and there were few reports in this field.Jeon et al. [7] proposed a 2-dimensional salvo attack strategywith terminal impact angle and time specifications, whichwas obtained by a linearized model based optimal control.An analytical guidance law was derived explicitly for thisstrategy. There are two control loops in this approach: theouter and inner loops are designed to regulate the time-to-goand the angle, respectively. When the time-to-go predictionis larger than the specified time, a trajectory curve will beplanned online to consume the excessive time; otherwise,a comparatively straight line will be generated to meetthe impact time requirement. Despite its simplicity, twosevere pitfalls accompany this method: (1) the small heading
angle assumption could cause large terminal errors whenthe practical angle is large and (2) the process constraintswith the inequality forms, for example, the familiar no-flyzone constraints, are unable to be incorporated. To copewith the no-fly zone constraints, many trajectory planningmethodologies based on A∗ algorithm [12–15] and intelligentoptimization algorithms, such as genetic algorithm (GA) [16–18], ant colony optimization (ACO) [19, 20], and particleswarm optimization (PSO) [21–23], have been extensivelyinvestigated. Nevertheless, these approaches can only gener-ate an open-loop optimal trajectory offline due to the com-putational complexity and it is not applicable to use them toimpact amoving surface target.Therefore, it is urgent to solvethe trajectory optimization problem with terminal impactangle and time specifications as well as process inequalityconstraints simultaneously in real-time.
Recently, the Gauss pseudospectral method (GPM) hasattracted a wide attention within aerospace industry, espe-cially in the fields of guidance and trajectory design [24–28].GPM is a direct optimal control solving approach for the gen-eral nonlinear systems with various constraints. Nowadays, amature GPM solving softwareGPOPS (Gauss PseudospectralOPtimization Software) is available, which makes GPM atechnology to be more than an art. In this paper, the GPM
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 604705, 8 pageshttp://dx.doi.org/10.1155/2014/604705
2 Mathematical Problems in Engineering
Y
OX
A
VMissile(Xm, Ym)
Target (Xt, Yt)
𝛾m
𝛾t
Vt
Figure 1: Missile-target relative kinematics.
is employed in a receding horizon way to obtain an adaptivetrajectory to impact a moving surface target, which satisfiesthe terminal and process constraints and can be realizedonline. In the design, a time-to-go dependent penalty func-tion is utilized to ensure a smooth trajectory at the terminalend and reduce the miss distance.
The remaining parts of this paper are organized as follows.The problem is formulated in Section 2. Section 3 gives a briefintroduction about GPM. The receding horizon strategy toimplement GPM is presented in Section 4. In Section 5, thesimulation results are provided and analyzed.The concludingremarks are summarized in Section 6.
2. Problem Formulation
2.1. Kinematics Equations. The missile-target relative kine-matics is shown in Figure 1. Without loss of generality, itis assumed that the velocity of the missile, 𝑉, is a constant.The earth coordinate system is chosen as 𝑋𝑂𝑌. The currentstates of themissile are the position (𝑋
𝑚, 𝑌𝑚) and the heading
angle 𝛾𝑚. The control variable for the missile is the normal
acceleration, 𝐴, being perpendicular to 𝑉, which can changethe velocity direction. The corresponding states of the targetare the position (𝑋
𝑡, 𝑌𝑡) and the heading angle 𝛾
𝑡.
According to Figure 1, the mass-point dynamics of themissile and the target are [9, 29]
𝑑𝑋𝑚
𝑑𝑡= 𝑉 cos (𝛾
𝑚) ,
𝑑𝑌𝑚
𝑑𝑡= 𝑉 sin (𝛾
𝑚) ,
𝑑𝛾𝑚
𝑑𝑡= 𝑢𝑚=
𝐴
𝑉,
𝑑𝑋𝑡
𝑑𝑡= 𝑉𝑡cos (𝛾
𝑡) ,
𝑑𝑌𝑡
𝑑𝑡= 𝑉𝑡sin (𝛾𝑡) .
(1)
2.2. Optimal Trajectory Requirement
2.2.1. Terminal Constraints. The terminal impact conditionsare
𝑋𝑚(𝑡𝑓) = 𝑋
𝑡, 𝑌
𝑚(𝑡𝑓) = 𝑌𝑡, 𝛾
𝑚= 𝛾𝑓, (2)
where 𝛾𝑓and 𝑡𝑓are the specified impact angle and time,
respectively. When multiple missiles are synchronized bytheir onboard clocks separately, a salvo attack can be achievedautomatically without any communication. Therefore, thereis less risk when encountering sophisticated electronic jam-ming and deception environment.
2.2.2. Control Constraint. The acceleration is bounded as
|𝐴| ≤ 𝐴max, 𝐴max > 0, (3)
which are determined by the airframe and the engine operat-ing conditions of specific missiles.
2.2.3. No-Fly Zone Constraints. The cruise missile often fliesover a wide range of area. There exist several types of no-fly zones, such as dangerous terrains and adverse air defenseareas, which should be evaded completely. Without loss ofgenerality, these no-fly zones can be represented by disks as
(𝑋𝑚(𝑡) − 𝑥
𝑖)2
+ (𝑌𝑚(𝑡) − 𝑦
𝑖)2
≤ 𝑅2
𝑖, 𝑖 = 1, 2, . . . 𝑛,
(4)
where (𝑥𝑖, 𝑦𝑖) and 𝑅
𝑖are the center and the radius, respec-
tively, and 𝑛 is the number of no-fly zones.
2.2.4. Cost Function. Theaccumulative control energy duringthe guidance course
𝐽 (𝑡0) =
1
2min∫
𝑡𝑓
𝑡0
𝑢2
𝑚𝑑𝑡 (5)
is often employed as the cost function to produce a more orless straight trajectory, where 𝑡
0is the initial time instant.
Formost guidance laws, themaximal control energy oftenoccurs at the terminal end, which could cause large missdistance. To solve this problem, a time-weight function isintroduced in the cost objective as
𝐽 (𝑡0) =
1
2min∫
𝑡𝑓
𝑡0
(𝑓𝑝(𝑡) ⋅ 𝑢
2
𝑚) 𝑑𝑡. (6)
As the impact time is fixed, 𝑓𝑝(𝑡) can be designed as a
monotonically increasing function according to the terminalrequirements.
3. A Brief Introduction to GPM
Consider a dynamic equation
�̇� = 𝑓 (𝑥 (𝑡) , 𝑢 (𝑡) , 𝑡) , (7)
where 𝑥(𝑡) ∈ 𝑅𝑛 and 𝑢(𝑡) ∈ 𝑅𝑚 are state and control vari-ables, respectively, and the function 𝑓: 𝑅𝑛 × 𝑅𝑚 × 𝑅 → 𝑅𝑛.
Mathematical Problems in Engineering 3
The traditional optimal control problem can be formulated ina unified form as
𝐽 = Φ (𝑥 (𝑡𝑓) , 𝑡𝑓) + ∫𝑡𝑓
𝑡0
𝑔 (𝑥 (𝑡) , 𝑢 (𝑡) , 𝑡) 𝑑𝑡, (8)
subject to the boundary equality constraints
𝜙 (𝑥 (𝑡0) , 𝑡0, 𝑥 (𝑡𝑓) , 𝑡𝑓) = 0, (9)
and the process inequality constraints
𝐶 (𝑥 (𝑡) , 𝑢 (𝑡) , 𝑡0, 𝑡𝑓) ≤ 0, (10)
where 𝜙 : 𝑅𝑛 × 𝑅 × 𝑅𝑛 × 𝑅 → 𝑅𝑞 and 𝐶 : 𝑅𝑛 → 𝑅𝑟.A brief review of pseudospectral method (PSM), espe-
cially Gauss PM (GPM), is presented here. The detaileddescription of GPM can be found in [24].
Firstly, the original time interval [𝑡0, 𝑡𝑓] is transformed to
the interval [−1, 1] as
𝜏 =2𝑡
𝑡𝑓− 𝑡0
−𝑡𝑓+ 𝑡0
𝑡𝑓− 𝑡0
. (11)
Using (11), (8) can be reformulated in terms of 𝜏 as
𝐽 = Φ (𝑥 (1) , 𝑡𝑓) +
𝑡𝑓− 𝑡0
2∫1
−1
𝑔 (𝑥 (𝜏) , 𝑢 (𝜏) , 𝜏) 𝑑𝜏. (12)
Similarly, the dynamic (7) and the boundary constraints(9) are replaced by
2
𝑡𝑓− 𝑡0
𝑑𝑥
𝑑𝜏= 𝑓 (𝑥 (𝜏) , 𝑢 (𝜏) , 𝜏) , (13)
𝜙 (𝑥 (−1) , 𝑡0, 𝑥 (1) , 𝑡
𝑓) = 0, (14)
respectively.In the GPM, the Legendre-Gauss (LG) points, 𝜏
𝑘(1 ≤ 𝑘 ≤
𝑁 − 1), distributed on the interval [−1, 1], are defined as theroots of
𝑃𝑁(𝜏) =
1
2𝑁𝑁!
𝑑𝑁
𝑑𝜏𝑁[(𝜏2
− 1)𝑁
] , (15)
while 𝜏0= −1, 𝜏
𝑁= 1. The continuous state and control
variables are approximated by the𝑁th polynomials as
𝑥 (𝜏) ≈ 𝑋 (𝜏) =
𝑁
∑𝑖=0
𝐿𝑖(𝜏)𝑋 (𝜏
𝑖) , (16)
𝑢 (𝜏) ≈ 𝑈 (𝜏) =
𝑁
∑𝑖=1
𝐿𝑖(𝜏) 𝑈 (𝜏
𝑖) , (17)
where the Lagrange interpolation polynomial is
𝐿𝑖(𝜏) =
𝑁
∏𝑗=0,𝑗 ̸= 𝑖
𝜏 − 𝜏𝑖
𝜏𝑖− 𝜏𝑗
. (18)
Secondly, (16) is differentiated at the node points as [28]
�̇� (𝜏) ≈ �̇� (𝜏) =
𝑁
∑𝑖=0
�̇�𝑖(𝜏)𝑋 (𝜏
𝑖) , (19)
where
𝐷𝑘𝑖= �̇�𝑖(𝜏𝑘)
=
{{{{{{
{{{{{{
{
(1 + 𝜏𝑘) �̇�𝐾(𝜏𝑘) + 𝑃𝐾(𝜏𝑘)
(𝜏𝑘− 𝜏𝑖) [(1 + 𝜏
𝑖) �̇�𝐾(𝜏𝑖) + 𝑃𝐾(𝜏𝑖)], 𝑖 ̸= 𝑘,
(1 + 𝜏𝑖) �̈�𝐾(𝜏𝑖) + 2�̇�
𝐾(𝜏𝑖)
2 [(1 + 𝜏𝑖) �̇�𝐾(𝜏𝑖) + 𝑃𝐾(𝜏𝑖)], 𝑖 = 𝑘.
(20)
The static form of (7) can be obtained by equating the rightside of both (19) and (7) at the discrete node points.
Thirdly, the continuous-time cost function (8) is approx-imated by using Gauss quadrature formula as
𝐽 = Φ (𝑋0, 𝑡0, 𝑋𝑓, 𝑡𝑓) +
𝑡𝑓− 𝑡0
2
𝑁
∑𝑘=1
𝑤𝑘𝑔 (𝑋𝑘, 𝑈𝑘, 𝜏𝑘; 𝑡0, 𝑡𝑓) ,
(21)
where 𝑤𝑘are the Gauss weights.
Similarly, the terminal and process constraints can also bereformulated.
According to the procedures described above, the originaloptimal control problem can be approximated as a staticnonlinear programing (NLP) one. The essence of GPM isto replace the original infinite-dimension dynamic optimalcontrol problem with a finite-dimension static NLP by elim-inating differential and integral equations. There are manyeffective methods to solve NLP and among them the sequen-tial quadratic programing (SQP) is a famous one which hasbeen widely used because of its maturity. Nowadays, a kindof reliable software, SNOPT (Sparse Nonlinear OPTimizer),is available for solving SQP problems in a unified framework.
4. Receding Horizon GPM
The GPM provides us with an effective way to deal withnonlinear optimal control problem subject to various con-straints directly, avoiding the shortcomings existing in thelinearization approximation approach [7]. Moreover, a freeGPOPS is available to seek GPM solution in real-time andreduce the huge computational complexity existing in thetraditional trajectory planning [14, 19, 21]. However, theoptimal solution obtained by using GPOPS is an open-loopone which is unable to cope with moving surface targets suchas warships.Therefore, a closed-loop optimal solution shouldbe generated instead. Enlightened by the mechanism ofmodel predictive control [30], this obstacle can be eliminatedby using receding horizon scheme which can be realizedonline.
The receding horizon GPM can be implemented asfollows.
Step 1. Input the number of LG points, 𝑁, the specifiedimpact time 𝑡
𝑓, and terminal heading angle 𝛾
𝑓. Set 𝑡
0as the
initial time instant and 𝑖 = 1.
Step 2. According to the guidancemeasurement information,plan an optimal trajectory from the current position to thetarget and obtain a control law 𝑢
∗(𝑡𝑖) using GPM. Assume
that this computational period is Δ𝑇𝑖. During the time
interval [𝑡𝑖, 𝑡𝑖+ Δ𝑇𝑖], or when the new optimization results
are not available, the corresponding values of 𝑢∗(𝑡𝑖−1
) on thisinterval are implemented. And then 𝑢∗(𝑡
𝑖) is used after the
time instant of 𝑡𝑖+ Δ𝑇𝑖.
Step 3. Let 𝑖 = 𝑖 + 1 and repeat Step 2 until 𝑡 = 𝑡𝑓.
This method can be illustrated in Figure 2.
5. Numerical Simulations
In this section, the effectiveness of the proposed methodis illustrated by several numerical simulations. The firstsimulation is to illustrate the effectiveness of the time-weight function𝑓
𝑝(𝑡), wherein a stationary target is employed
without using receding horizon optimization. The secondand the third simulations combined are designed to test thecapacity of the proposed method to deal with no-fly zoneconstraints. Here a moving target is used to evaluate thereceding horizon optimization strategy. In all the simulations,the velocities of missiles keep a constant of 250m/s and theinitial numbers of LG points, 𝑁, are universally selected as20.
5.1. Selection of Time-Weight Function 𝑓𝑝(𝑡). For most guid-
ance laws, the control energy almost reaches its peak at theterminal end due to the divergent line-of-sight (LOS) rate,which would lead to a large miss distance.Therefore, it wouldbe preferred to ensure the terminal trajectory to be straightfor a period of time. In order to find a better time-weightfunction, five types of time-dependent functions 𝑓
𝑝(𝑡) are
investigated and they are
(1) 𝑓𝑝(𝑡) = 1;
(2) 𝑓𝑝(𝑡) = 𝑡2;
−10000 −8000 −6000 −4000 −2000 0 2000−6000
−5000
−4000
−3000
−2000
−1000
0
1000
1
2
3
4
5
y(m
)
x (m)
Figure 3: Flight trajectories.
0 10 20 30 40 50−30
−20
−10
0
10
20
30
40
50
Con
trol
Time (s)
1
2
3
4
5
Figure 4: Acceleration curves.
(3) 𝑓𝑝(𝑡) = 1/(𝑡
𝑓− 𝑡 + 0.5);
(4) 𝑓𝑝(𝑡) = 1/(𝑡
𝑓− 𝑡 + 0.5)
2;
(5) 𝑓𝑝(𝑡) = 1/(𝑡
𝑓− 𝑡 + 0.5)
3.
Because of the terminal impact time specification, thesefunctions can be easily realized. Assume a scenario that themissile impacts a stationary target subject to 𝑡
𝑓= 50 s and
𝛾𝑓= 0∘. The simulation results are shown in Figures 3 and 4
when using a one-phase GPM. According to Figure 3, it canbe seen that all the terminal constraints are not violated foreach function. From Figure 4, it is clear that when 𝑓
𝑝(𝑡) = 1,
the terminal control amplitude is the maximal one amongall; when 𝑓
𝑝(𝑡) = 𝑡2, the initial acceleration is large, and the
control curve is similar to a bang-bang control. However, thisfunction is free of 𝑡
𝑓, which could be sensitive to the case
configuration; when 𝑓𝑝(𝑡) = 1/(𝑡
𝑓− 𝑡 + 0.5)
2, the terminalflight trajectory is straight. But with the increment of powerin the denominator, both the initial acceleration and the totalenergy consumption are increased. Therefore, a comprehen-sive tradeoff is made and 𝑓
𝑝(𝑡) = 1/(𝑡
𝑓− 𝑡 + 0.5)
2 is chosenand fixed for the following simulations.
Mathematical Problems in Engineering 5
−10000 −8000 −6000 −4000 −2000 0 2000−5000
0
5000
10000y
(m)
x (m)
Actual curvet = 0
t = 20
t = 30
(a) Flight trajectories
0 5 10 15 20 25 30 35 40 45 50−50
−40
−30
−20
−10
0
10
20
30
40
Actual curvet = 0
t = 20
t = 30
Time (s)
Acce
lera
tion(m
/s2)
(b) Acceleration curves of the first missile
0 5 10 15 20 25 30 35 40 45 50−20
0
20
40
60
80
100
Actual curvet = 0
t = 20
t = 30
Time (s)
Hea
ding
angl
e (de
g)
(c) Heading angle curves of the second missile
Figure 5: Salvo attack without inequality constraints.
Table 1: Case configurations without inequality constraints.
Missile (𝑋0, 𝑌0) (km) 𝛾
0(deg) 𝛾
𝑓(deg) 𝑡
𝑓(s)
1 (−3.5, 8.5) 0 −105 502 (−9, −4.5) 45 0 50
Table 2: Simulation results without inequality constraints.
5.2. Without Process Inequality Constraint. In this scenario,two antiship missiles attack a moving warship without anyno-fly zone constraint. The fixed sampling period is Δ𝑇 =
0.5 s. The velocity of the warship is 15m/s with 𝛾𝑡= −135∘.
The maximal feasible acceleration is 𝐴max = 45m/s2. Thecase configurations are given in Table 1. The simulationresults are shown in Figure 5 and Table 2. In Figure 5(a),
the real lines are the actual trajectories and the point linesare the planned optimal trajectories at different samplingtime instants. It can be found that both missiles fly alongcircular arcs during their early stages in order to consumethe excessive time. According to Figures 5(b) and 5(c), theterminal trajectories are becoming straight gradually towardsspecified impact direction prior to impact instant; thereforetiny accelerations are needed in the final phase. In Table 2,(Δ𝑋, Δ𝑌) and Δ𝛾 are the terminal errors of position andimpact angle, respectively. It can be found that they arenegligible from Table 2. This fact suggests that the proposedmethod can effectively impact amoving surface target subjectto terminal constraints. It should also be noted that it is notnecessary to pay a lot of attention to the selection of theinitial values. In other words, this algorithm is insensitiveto the quality of initial solution. The setup of the computeris i5 processor and 2G memory. The GPOPS is running onMatlab platform. When the algorithm is implemented in Ccode, the convergent time can be reduced by a factor of
6 Mathematical Problems in Engineering
−10000 −8000 −6000 −4000 −2000 0 2000−5000
0
5000
10000
y(m
)
x (m)
Actual curvet = 0
t = 20
t = 30
Figure 6: Location of the no-fly zones.
−10000 −8000 −6000 −4000 −2000 0 2000−5000
0
5000
10000
y(m
)
x (m)
Actual curvet = 0
t = 20
t = 30
Figure 7: Salvo attack with no-fly zones.
up to 100. For this case, each trajectory can be obtainedwithin 10 s by using Matlab; therefore it meets real-timerequirement when using C programming language onboard.In fact, we rewrote the corresponding GPOPS functions usedin the abovementioned path planning by the C programminglanguage on the basis of the commercial SNOPT softwarefor practitioners. When implemented on the practical DOSplatform, the running time can be kept below 0.3 s for almostall situations.
5.3. With No-Fly Zone Constraints. In this case, we assumethat there are two no-fly zones along the trajectories of theprevious simulation and they are
(1) (𝑋𝑚+ 1300)
2
+ (𝑌𝑚− 8000)
2
≤ 122500;(2) (𝑋
𝑚+ 8000)
2
+ (𝑌𝑚− 0)2
≤ 122500.
They are shown as two disks in Figure 6, which can beconsidered as the process constraints. It is obvious that theno-fly zones are just located along the current trajectories;
hence the trajectories need to be replanned to evade the no-fly zones. The new trajectories are shown in Figure 7 whenadding the no-fly zone constraints. According to Figure 7,the two missiles evade the no-fly zones successfully in thesimulations. On the other hand, the running time is almostunchanged compared with the one in the above section; thatis, the process inequality constraints have little effect on theoptimization time for GPM.
5.4. Comparison with the Analytical Guidance Law. In [7], ananalytical guidance law with impact angle and time specifica-tions was derived based on the linearization approximationusing small heading angle assumption. In this section, thecomparative simulation is performed. The case configura-tions are shown in Table 3. The first and second missilesare guided by the method in [7] while the trajectory of thethird missile is obtained by using the proposed recedinghorizon GPM. The simulation results are shown in Figure 8and Table 4. It is evident that poor miss distance is causedfor the second missile due to the linearization approximationerrors. However, the third missile can still guarantee impactaccuracy.
Mathematical Problems in Engineering 7
6. Conclusions
This paper presented a trajectory optimization approachbased on the receding horizon GPM to impact a movingsurface target subject to the terminal impact angle and thetime specifications. A time-dependent weighting functionwas introduced to reduce the control energy at the terminalend. The receding horizon strategy could be applied to thetarget in motion. When using GPOPS, this comprehensivescheme can handle the process inequality constraints withoutcausing the increment of computational complexity and nolinearization needs to be done. The optimization algorithmcould be implemented real-time with an onboard computer.The effectiveness of the proposed method was illustrated byseveral numerical examples.
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper.
Acknowledgments
The authors would like to express their thanks to theanonymous reviewers of the paper. This work was supportedin part by the Natural Science Foundation of China (nos.61174094 and 61273138), the Tianjin Natural Science Foun-dation of China (nos. 14JCYBJC18700 and 13JCYBJC17400),South African National Research Foundation (no. 78673),and South African National Research Foundation IncentiveGrant (no. 81705).
References
[1] M. Kim and K. V. Grider, “Terminal guidance for impact atti-tude angle constrained flight trajectories,” IEEE Transactions onAerospace and Electronic Systems, vol. 9, no. 6, pp. 852–859, 1973.
[2] M. Idan, O. M. Golan, andM. Guelman, “Optimal planar inter-ceptionwith terminal constraints,” Journal of Guidance, Control,and Dynamics, vol. 18, no. 6, pp. 1273–1279, 1995.
[3] B. S. Kim, J. G. Lee, and H. S. Han, “Biased PNG law for impactwith angular constraint,” IEEE Transactions on Aerospace andElectronic Systems, vol. 34, no. 1, pp. 277–288, 1998.
[4] C. K. Ryoo, H. J. Cho, and M. J. Tahk, “Closed-form solutionsof optimal guidance with terminal impact angle constraint,” inProceedings of the IEEE Conference on Control Applications, vol.1, pp. 504–509, June 2003.
[5] C. K. Ryoo, H. J. Cho, and M. J. Tahk, “Optimal guidance lawswith terminal impact angle constraint,” Journal of Guidance,Control, and Dynamics, vol. 28, no. 4, pp. 724–732, 2005.
[6] B. Y. Jung and Y. D. Kim, “Guidance laws for anti-ship missilesusing impact angle and impact time,” in Proceeding of the AIAAGuidance, Navigation, and Control Conference, pp. 3048–3060,August 2006.
[7] I. S. Jeon, J. I. Lee, and M. J. Tahk, “Impact-time-control guid-ance law for anti-ship missiles,” IEEE Transactions on ControlSystems Technology, vol. 14, no. 2, pp. 260–266, 2006.
[8] M. G. Yoon, “Relative circular navigation guidance for theimpact angle control problem,” IEEE Transactions on Aerospaceand Electronic Systems, vol. 44, no. 4, pp. 1449–1463, 2008.
[9] P.Wu andM. Yang, “Integrated guidance and control design formissile with terminal impact angle constraint based on slidingmode control,” Journal of Systems Engineering and Electronics,vol. 21, no. 4, pp. 623–628, 2010.
[10] Y. Kim, J. Kim, and M. Park, “Guidance and control systemdesign for impact angle control of guided bombs,” in Proceed-ings of the International Conference on Control, Automation andSystems (ICCAS ’10), pp. 2138–2143, October 2010.
[11] H. B. Oza and R. Padhi, “Impact-angle-constrained suboptimalmodel predictive static programming guidance of air-to-groundmissiles,” Journal of Guidance, Control, and Dynamics, vol. 35,no. 1, pp. 153–164, 2012.
[12] R. J. Szczerba, P. Galkowski, I. S. Glickstein, and N. Ternullo,“Robust algorithm for real-time route planning,” IEEE Trans-actions on Aerospace and Electronic Systems, vol. 36, no. 3, pp.869–878, 2000.
[13] B. M. Sathyaraj, L. C. Jain, A. Finn, and S. Drake, “MultipleUAVs path planning algorithms: a comparative study,” FuzzyOptimization and Decision Making, vol. 7, no. 3, pp. 257–267,2008.
[14] D. Bodhale, N. Afzulpurkar, and N. T. Thanh, “Path planningfor a mobile robot in a dynamic environment,” in Proceedings ofthe IEEE International Conference on Robotics and Biomimetics(ROBIO ’09), pp. 2115–2120, February 2009.
[15] B. B. Meng and X. G. Gao, “UAV path planning based onbidirectional sparse A∗ search algorithm,” in Proceedings of theInternational Conference on Intelligent Computation Technologyand Automation (ICICTA ’10), vol. 3, pp. 1106–1109, Changsha,China, May 2010.
[16] Y. V. Pehlivanoglu, O. Baysal, and A. Hacioglu, “Path planningfor autonomousUAVvia vibrational genetic algorithm,”AircraftEngineering and Aerospace Technology, vol. 79, no. 4, pp. 352–359, 2007.
[17] K. J. Obermeyer, “Path planning for a UAV performing recon-naissance of static ground targets in terrain,” in Proceedingsof the AIAA Modeling and Simulation Technologies Conference,AIAA-2009-5888, August 2009.
[18] M. Kapanoglu, M. Alikalfa, M. Ozkan et al., “A pattern-basedgenetic algorithm for multi-robot coverage path planning min-imizing completion time,” Journal of Intelligent Manufacturing,vol. 23, no. 4, pp. 1035–1045, 2012.
[19] A. Fridman, S. Weber, V. Kumary et al., “Distributed path plan-ning for connectivity under uncertainty by ant colony optimiza-tion,” in Proceedings of the American Control Conference (ACC’08), pp. 1952–1958, Seattle, Wash, USA, June 2008.
[20] M. A. P. Garcia, O. Montiel, O. Castillo et al., “Path planning forautonomous mobile robot navigation with ant colony optimi-zation and fuzzy cost function evaluation,” Applied Soft Com-puting, vol. 9, no. 3, pp. 1102–1110, 2009.
[21] J. L. Foo, J. S. Knutzon, J. H. Oliver, and E. H. Winer, “Three-dimensional multi-objective path planning of unmanned aerialvehicles using particle swarm optimization,” in Proceedingsof the 48th AIAA/ASME/ASCE/AHS/ASC Structures, StructuralDynamics, and Materials Conference, pp. 2170–2179, April 2007.
[22] J. L. Foo, J. Knutzon, V. Kalivarapu, J. Oliver, and E. Winer,“Path planning of unmanned aerial vehicles using B-splines andparticle swarm optimization,” Journal of Aerospace Computing,Information and Communication, vol. 6, no. 4, pp. 271–290,2009.
[23] M. P. Ran, H. B. Duan, X. G. Gao, and Z. Mao, “Improved parti-cle swarm optimization approach to path planning of amphibi-ous mouse robot,” in Proceedings of the 6th IEEE Conference on
8 Mathematical Problems in Engineering
Industrial Electronics andApplications (ICIEA ’11), pp. 1146–1149,Beijing, China, June 2011.
[24] D. Benson,AGauss pseudospectral transcription for optimal con-trol [Ph.D. dissertation], Department of Aeronautics and Astro-nautics, Massachusetts Institute of Technology, 2004.
[25] I. M. Ross and F. Fahroo, “Pseudospectral knotting methods forsolving optimal control problems,” Journal of Guidance, Control,and Dynamics, vol. 27, no. 3, pp. 397–405, 2004.
[26] G. T. Huntington and A. V. Rao, “Optimal configuration ofspacecraft formations via a gauss pseudospectral method,” inProceedings of the AAS/AIAA 15th Space Flight Mechanics Meet-ing, pp. 33–50, January 2005.
[27] G. T. Huntington and A. V. Rao, “Optimal reconfiguration ofspacecraft formations using the gauss pseudospectral method,”Journal of Guidance, Control, and Dynamics, vol. 31, no. 3, pp.689–698, 2008.
[28] R. Tawfiqur, H. Zhou, Y. Z. Sheng, Y. M. Younis, and K. N.Zhang, “Trajectory optimization of hypersonic vehicle usingGauss pseudospectral method,” in Proceedings of the 2nd Inter-national Conference on Mechanical and Aerospace Engineering,pp. 5232–5239, 2011.
[29] Y. H. Kim, C. K. Ryoo, and M. J. Tahk, “Guidance synthesis forevasive maneuver of anti-ship missiles against close-in weaponsystems,” IEEE Transactions on Aerospace and Electronic Sys-tems, vol. 46, no. 3, pp. 1376–1388, 2010.
[30] E. N. Hartley, P. A. Trodden, and A. G. Richards, “Model pre-dictive control systemdesign and implementation for spacecraftrendezvous,”Control Engineering Practice, vol. 20, no. 7, pp. 695–713, 2012.
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