Abstract—In this paper, a planar waypoint guidance synthesis for missiles or unmanned aerial vehicles using the optimal impact-angle-control laws is proposed. The energy optimal trajectory optimization problem with waypoint constraints can be converted to an unconstrained optimization problem of finding the optimal waypoint pass angles for the energy impact- angle-control laws. These optimal waypoint pass angles can be simply determined from a set of linear algebraic equations. Our approach does not require time-consuming numerical optimizations so that the energy optimal trajectory passing through all the waypoints can be generated in real time.

I. INTRODUCTION

OST missions of Unmanned Aerial Vehicle (UAV) including cruise missiles can be accomplished by

passing through several waypoints of a given order. Waypoints are defined as spatially fixed points that should be passed by the vehicle. In most cases, the selection of waypoints is done in the stage of the path planning before flight. Waypoint guidance to deliver a vehicle from one waypoint to the next waypoint is the basic requirement for autonomous flight of vehicles.

By regarding the next waypoint as the fixed target, typical missile guidance laws can be directly used for waypoint guidance. PNG [1] is a candidate for waypoint guidance, but it produces large guidance commands when the next waypoint is suddenly assigned. Also, PNG does not have any design parameters that can be used for trajectory shaping. In [2], the LOS waypoint guidance has been proposed for autonomous marine vehicles. In this method, the LOS connecting the current waypoint and the next waypoint is imbedded in the guidance synthesis for smooth command transition. In general, mere application of the guidance law to each fight region between the waypoints, even though a guidance law has any optimality, does not provide the optimality of the entire trajectory including all waypoints. In [3], a waypoint guidance algorithm to follow the straight-line segment between two waypoints has been proposed. The basic idea of this method is to regulate the flight path along the line segment by a linear quadratic regulator (LQR) and to transfer the vehicle to the next line-segment with the

This work was supported by Flight Vehicle Research Center (FVRC), Korea.

C. K. Ryoo is a senior researcher of the Agency for Defense Development (ADD), Korea (Corresponding author, phone: +82-42-869-3758; fax: +82-42-869-3710; e-mail: ckryoo@fdcl.kaist.ac.kr).

H. S. Shin is a MS student of the Dept. of Aerospace Eng., Korea Advanced Institute of Science and Technology (KAIST) (e-mail: hsshin@fdcl.kaist.ac.kr).

M. J. Tahk is a professor of the Dept. of Aerospace Eng., KAIST, (e-mail: mjtahk@fdcl.kaist.ac.kr).

minimum acceleration turn. The application of this method is restricted to the line-following mission.

In this paper, a waypoint guidance synthesis based on the Optimal Guidance Law with the impact angle constraint (OGL-T) for lag-free systems [4,5] is proposed. OGL-T are obtained as a solution of the LQ optimal control problem to find a control input which minimizes the control energy subject to the terminal impact angle constraint. OGL-T is useful when anti-tank or anti-ship missiles need to attack vulnerable sides of the target. For waypoint guidance applications, unlike other guidance laws, OGL-T provides a great flexibility for trajectory shaping since the trajectory can be easily modified by the pass angle at each waypoint.

Using the analytic time solution of OGL-T, we can approximate the energy cost as a quadratic function of the waypoint pass angles. By differentiating the energy cost by the waypoint pass angles, simple linear algebraic equations to calculate the sub-optimal waypoint pass angles are derived. The energy optimal waypoint guidance is synthesized in two steps: the sub-optimal waypoint pass angles are calculated first by using the linear algebraic equations, and then, OGL-T is applied to guide the vehicle between two adjacent waypoints. Since the proposed method does not require any numerical optimization processes, the optimal trajectory can be generated in real time even under sudden changes of the waypoint set, which may happen to avoid pop-up threats or to meet mission changes during the flight. If the vehicle has a response lag, the optimal guidance law with terminal constraints on impact angle and acceleration [6] can be used as waypoint guidance. In this case, the proposed synthesis should be slightly modified.

II. EQUIVALENCE OF OPTIMAL CONTROL PROBLEMS

Consider N waypoints that will be visited by the vehicle in a given order as shown in Fig. 1. We define the flight region between the ( 1)thi - and the thi -waypoint as the

thi -segment. The position and the flight path angle of the vehicle are denoted as ( )x t , ( )z t and ( )t , respectively. It is assumed that the control input ( )u t is realized without any lag and is applied normal to the velocity vector. We also assume that the speed of the vehicle, V , is maintained constant during the flight by engine control. The position and the pass angle of the thi -waypoint are represented as ix , izand i , respectively.

Consider the following optimal control problem. OCP-1: Find ( )u t which minimizes

1

2 2

01

[ ( )] [ ( )]f i

i

Nt t

ti

J u t dt u t dt (1)

Optimal Waypoint Guidance Synthesis Chang-Kyung Ryoo, Hyo-Sang Shin, and Min-Jea Tahk, Member, IEEE

M

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

WA6.3

0-7803-9354-6/05/$20.00 ©2005 IEEE 1349

subject to cos , sin ,x V z V u V (2) with the constraints of

( )i ix t x and ( )i iz t z , for 0,1, ,i N . (3) where it is defined as the time when the vechicle reaches the

thi -waypoint. By definition, 0 0t and N ft t .In general, the solution to OCP-1 is obtained by applying

typical numerical optimization techniques. This approach includes a number of control parameters and constraints so that it takes a long time to converge.

Let the cost of each segment in (1) be defined by

1

2[ ( )]i

i

t

i tJ u t dt for 1, 2, ,i N (4)

and * ( )u t be the solution to OCP-1. Then, the minimum cost *J can be represented by

* *

1

N

ii

J J (5)

where 1

* * 2[ ( )]i

i

t

i tJ u t dt for 1, 2, ,i N . (6)

We also define the optimal waypoint pass angles, * ( )i itfor 0,1, ,i N , which are directly calculated from the trajectory solution to OCP-1. We define the set of the optimal waypoint pass angles as

* * , 0,1, ,i i N (7)

Consider another optimal control problem with the terminal constraints on a zero miss distance and a desired pass angle for the thi -segment. OCP-2: Find ( )u t which minimizes

1

2[ ( )]i

i

t

i tJ u t dt (8)

subject to (2) with the initial conditions,

1 1( )i ix t x , 1 1( )i iz t z , 1 1( )i it (9)and the terminal constraints,

( )i ix t x , ( )i iz t z , and ( )i it . (10)Let * ( )u t be the solution to OCP-2, then the minimum cost

*iJ is obtained as

1

* * 2[ ( )]i

i

t

i tJ u t dt (11)

Note that OCP-2 is associated with the trajectory optimization for the single sector between the ( 1)thi - and the thi -waypoint while OCP-1 is associated with the optimization of the entire trajectory including all the waypoints. OCP-2 is a typical guidance problem with a terminal impact angle constraint.

The following theorem provides the relationship between

OCP-1 and OCP-2. Theorem 1: If *

i i for 0,1, ,i N , then * *( ) ( )u t u tfor 1[ , ]i it t t and * *

i iJ J for 1, ,i N .Proof: For the thN -segment, *

NJ obtained by using * ( )u tfor 1[ , ]N Nt t t is the minimum cost for OCP-2. Therefore, we have

1 1

* * 2 * 2[ ( )] [ ( )]N N

N N

t t

N t tJ u t dt u t dt . (12)

On the other hand, from the principle of optimality [7], *NJ

with * ( )u t for 1[ , ]N Nt t t is the minimum cost for OCP-1. Therefore,

1 1

* * 2 * 2[ ( )] [ ( )]N N

N N

t t

N t tJ u t dt u t dt . (13)

From the assumption, *N N and *

1 1N N . The equations of motion given in (2) for OCP-1 and OCP-2 have the same boundary conditions so that the minimum energy cost is unique. Hence, we have * *

N NJ J . This is true only when * *( ) ( )u t u t for 1[ , ]N Nt t t .

By repeating the procedure up to the first segment, we can prove the theorem.

Theorem 1 implies that the optimal control history passing through all the way points can be obtained as the combination of the independent solutions of each segment if the set of optimal waypoint pass angles * is given. We did not consider the case where some waypoint pass angles are prescribed. In this case, the prescribed waypoint pass angles can be treated as the optimal waypoint pass angles and we can easily show that the theorem is still hold.

Now suppose that * ( )u t , the optimal control of OCP-2, is given by a state-feedback closed-form guidance law

* ( ) : ( , ( ), ( ), ( );P)u t t x t z t t for 1[ , ]i it t t (14) where P denotes the parameter set which consists of the boundary conditions

P , , , for 0,1, ,i i i it x z i N (15) Let all elements except i be given. Define a set of waypoint pass angles as

, 0,1, ,i i N (16) Under the assumption that such a state-feedback law exists, we can consider the following nonlinear programming problem to find the optimal pass angles: NLP-1 : Find i for 0,1,2,...,i N , which minimizes

1

2

1( ) ( , ( ), ( ), ( ); )i

i

N t

ti

J t x t z t t dt (17)

subject to (2).NLP-1 is another trajectory optimization problem with waypoint constraints. Since the guidance law given by (14) satisfies the constraints on the positions and the pass angles of the waypoints, NLP-1 does not include any boundary conditions. The following theorem indicates the relationship between NLP-1 and OCP-1. Theorem 2: Let * * , 0,1, ,i i N be the solution to NLP-1 and the minimum cost *J be given by

1

* * 2

1[ ( , ( ), ( ), ( ); )]i

i

N t

ti

J t x t z t t dt . (18)

( )it

iz

0 ( )NtNz

( )u t

( )t( )z t

V-w.p.thi

-w.p.thN

( 1) -w.p.thi

1iz

UAV

1ix ( )x t ix Nx

segmentthi

0x0z

Fig. 1. Planar waypoint guidance geometry.

1350

Then, * * where * is obtained from OCP-1. Proof: Suppose that * * , then

1

* * 2

1[ ( , ( ), ( ), ( ); )]i

i

N t

ti

J t x t z t t dt . (19)

Recall that * ( )u t for 1[ , ]i it t t given in (14) provides the optimal control for OCP-2. From Theorem 1 if * , then

1

* 2 * *

1 1[ ( , ( ), ( ), ( ); )]i

i

N Nt

iti i

t x t z t t dt J J (20)

Therefore, (19) can be satisfied only by violating the fact that *J is the minimum cost for OCP-1. Hence, by contradiction,

the theorem is true. Theorem 2 implies that * * if * ( )u t is applied. Since

*J = *J , we know that * *( ) ( )u t u t for 1[ , ]i it t t .Therefore, Theorem 2 says that the converse of Theorem 1 is also true. From the theorems, the optimal trajectory obtained from OCP-1 can also be achieved by using the optimal impact-angle-control guidance law given in (14) if * is specified. Compared with OCP-1, NLP-1 can be solved with significantly less numerical effort. The dynamic constraints of (2), which should be numerically integrated to evaluate the cost of (17), are the only time-consuming part in solving NLP-1.

Up to now, we have showed that the energy optimal trajectory passing through all the waypoints can be obtained in a simple way of solving NLP-1. All of these are possible only when the closed-form state-feedback optimal guidance law given in (14) exists. The closed-form state feedback solution of OCP-2 with the linearized kinematics, so called the optimal-impact-angle control guidance law for lag-free system (OGL-T), has been already known [4,5].

III. OPTIMAL IMPACT-ANGLE-CONTROL GUIDANCE LAW

Fig. 2 shows a guidance geometry in which the vehicle is to reach WP2 from WP1 with a given impact angle f . Here, all state variables are defined with respect to the reference line connecting WP1 and WP2. Under the assumption that the vehicle is a lag-free system, we have u a . Assuming that

is small, we linearize (2) as

0

, (0) 0

, (0)

z V z

u V (21)

Note that the x equation of (2) is removed since x is not affected by u in the linearized equations of motion.

Now consider the following linear quadratic (LQ) optimal

control problem. OCP-2(LQ): Find u that minimizes

2

0

1 ( )2

ftJ u t dt (22)

subject to (21) and the terminal constraints,( ) 0fz t and ( )f ft . (23)

OCP-2(LQ) is the linearized version of OCP-2, which is the same optimal control problem as treated in [5]. The optimal control is obtained by using the sweep method [8]:

* ( )

6 ( ) 4 ( ) 2R go S

go f

u t C t C

V t t t (24)

where got represents the time-to-go defined by ft t and RC and SC are constants although they are expressed in

terms of the state variables 26 2 ( ) ( )R go fC V t t t (25)

2 3 ( ) ( ) 2S go fC V t t t (26)

Equation (24) is called the optimal-impact-angle control guidance law (OGL-T). The time to go can be estimated by the range over averaged speed along the LOS as

(1 )go

Rt k

V (27)

where k is a factor to compensate for the time increment taken along a curved trajectory [4,5]. Note that (24) with (27) corresponds to in (14) for solving NLP-1.

Since the RC and SC are constant, the boundary conditions for 0t give

206R f fC V t , 02 2S f fC V t (28)

From (24) and (28), we obtain 2

* 2 20 0

2f f

f

VJ

t (29)

By extending (29) to the multiple waypoint case, we can obtain analytical expression of the cost of NLP-2.

IV. SUB-OPTIMAL WAYPOINT PASS ANGLES

In OCP-2(LQ), and are defined with respect to the reference line connecting from WP1 to WP2 as shown in Fig. 3. Let i be the LOS angle of the line connecting the ( 1)thi -waypoint and the thi waypoint as depicted in Fig. 3. Let 1i i it t t , then it is approximated as

/i it R V (30) where iR is the distance between the ( 1)thi - and the

thi -waypoint. By using (29), the cost of OCP-1 or NLP-1 is

( )a t V( )t

( )t

( )R t

( )tf

f0

( )z t

Fig. 2. Impact-angle-control guidance geometry.

2z

Initial point

0

N

Nz1z

-w.p.thN1

1

21

1 -w.p.th

2 - w.p.nd

1R

2R

2

N

Fig. 3. Definition of i and iR .

1351

2 231 1

1

14N

i i i i i i i ii i

J VR

.(31)

Here, i and iR are fixed and easily calculated from the waypoint positions. Note that the energy cost is approximated as a quadratic function of the waypoint pass angles . Thus, the necessary condition to minimize (31) yields a simple linear algebraic equation to calculate * . If some waypoint pass angles are prescribed, the optimal trajectory can be independently obtained for each leg defined by a pair of waypoints. Cluster of waypoints belongs to one of the following four cases:

Case 1 : Fixed 0 and free N

In this case, the unknown parameter vector is defined by * * * * *

1 2 1

T

N N (32)

and * should satisfy the following necessary condition.

*

0, 1,2,...,i i

i

Ji N (33)

or*

* 2 0 1 21

1 2 2 1 1 2* *

*1 3 2 32

2 2 3 3 2 3

* **1 1

11 1 1

* *1

1 1 3 32

1 1 3 32

01 1 3 32

2 3

N N N NN

N N N N N N

N N N

N N N

R R R R R R

R R R R R R

R R R R R R

R R R

(34)

From (34), we have * 1

A RR (35) where

1 2 2

2 2 3 3

1 1

1 1 12 0 0...0 0

1 1 1 12 0...0 0

1 1 1 10 0 2

1 20 0 0

A

N N N N

N N

R R R

R R R RR

R R R R

R R

(36)

and

0 1 2 2 3 1

1 1 2 2 3 1

33

T

N N NR

N N NR R R R R R R R(37)

Case 2 : Free 0 and fixed N

Parameter vector * * * * *0 1 2 1

T

N N (38)

In a way similar to Case 1, we have

1 1

1 1 2 2

2 2 1 1

1 1

2 1 0 0...0 0

1 1 1 12 0...0 0

1 1 1 10 0 2

1 1 10 0 0 2

A

N N N N

N N N

R R

R R R RR

R R R R

R R R

(39)

1 1 2 2 1 1

1 1 2 2 1 1

33

T

N N N N NR

N N N N NR R R R R R R R

(40)Case 3 : Fixed 0 and fixed N

Parameter vector * * * * *1 2 2 1

T

N N (41)

1 2 2

2 2 3 3

2 2 1 1

1 1

1 1 12 0 0...0 0

1 1 1 12 0...0 0

1 1 1 10 0 2

1 1 10 0 0 2

A

N N N N

N N N

R

R R R

R R R R

R R R R

R R R

(42)

0 1 2 2 3

1 1 2 2 3

2 1 1

2 1 1

33

3

R

T

N N N N N

N N N N N

R R R R R

R R R R R

(43)

Case 4 : Free 0 and free N

Parameter vector * * * * *0 1 1

T

N N (44)

1 1

1 1 2 2

1 1

2 1 0 0...0 0

1 1 1 12 0...0 0

1 1 1 10 0 2

1 20 0 0

A

N N N N

N N

R R

R R R R

R

R R R R

R R

(45)

11 1 2

1 1 2 1

3T

N N NR

N N NR R R R R R (46)

If the prescribed waypoints require a sharp corner, the approximation of the flight time given by (30) may not be appropriate and the error in the optimal solution calculated from the linear algebraic equations can be large. Adding new waypoints around the corner can alleviate this difficulty.

1352

The optimal waypoint guidance synthesis is as follows: First, the algorithm checks whether the waypoints in the remaining flight region have been changed. If there are some waypoint changes, the optimal waypoint pass angles are calculated again by using the linear algebraic equations. Then, OGL-T is applied to each segment. The proposed guidance scheme produces the energy optimal trajectory without any in-flight numerical optimization.

V. NUMERICAL EXAMPLES

In this section, we investigate the performance of the proposed method via nonlinear simulations for two different mission scenarios. As shown in Fig. 4(a) and 5(a), nine waypoints on a plane are considered. The last waypoint (WP9) coincides with the initial point so the entire flight path is divided by 9 segments. Scenario 1 does not have constraints on the waypoint pass angles as shown in Fig 4(a). Scenario 1 corresponds to Case 4 defined in the previous section. In Scenario 2, two waypoint pass angles are prescribed as shown in Fig 5(a). The waypoint pass angle constraints are 45 degrees on WP3 and –180 degrees on WP7. Therefore, the entire flight path is divided into three distinct legs: Case 2 from WP1 to WP3, Case 3 from WP3 to WP7, and Case 1 from WP7 to WP9. The speed of the missile is 100m/s and remained constant during the flight.

We first compare three different trajectory solutions for a lag-free vehicle: 1) the solution to OCP-1, 2) the solution to NLP-1, and 3) the proposed optimal waypoint guidance (OWG) scheme illustrated in Fig. 6. OGL-T is used for NLP-1 and OWG.

In the optimization of OCP-1, for each segment ten discretized control inputs and the flight time are considered as the parameters to be optimized [9]. That is, the parameter set is given by

1

N

ii

X X (47)

where 9N and

1 2 10, ,..., ,i i i i iX u u u t for the thi -segment. (48) For OCP-1, therefore, the total number of the parameters to be optimized is 99. For NLP-1, the parameter set contains 10 waypoint pass angles for Scenario 1 and 8 for Scenario 2:

0 1 9

0 1 9 3 7

, ,..., for Scenario 1

, ,..., , for Scenario 2X (49)

The Runge-Kutta 4th-order method is used for the integration of (2) to calculate the cost and the violation of the constraints for OCP-1. In NLP-1, numerical integration is required only for the evaluation of the cost. A SQP (Sequential Quadratic Programming) method in [10] is adopted.

For Scenario 1, the three methods produce almost the same optimal waypoint pass angles as shown in Table 1. The cost obtained by OWG is slightly greater than the optimization result of OCP-1 or NLP-1. The minimum cost is obtained by OCP-1. The cost difference between OWG and OCP-1 is

within 0.6%. From Fig. 4, we observe that command profiles and trajectories obtained from three optimization methods are almost same. Calculation for OCP-1 and NLP-1 takes 278 seconds and 4 seconds, respectively. However, OWG requires only several milliseconds. The PC used for optimization has a 3.0GHz Intel CPU with 512MB RAM. All programs are coded by C++.

For Scenario 2, different linear algebraic equations should be applied to each case to obtain the sub-optimal waypoint pass angles. In this example, it takes 615 seconds and 6 seconds to obtain the optimal solutions of OCP-1 and NLP-1, respectively. More time is required for OCP-1 due to more constraints. Again, OWG requires few milliseconds to get the sub-optimal results as shown in Table 2. In this example, the cost difference between OWG and OCP-1 is only 0.3%. Fig. 5 shows that command profiles and trajectories obtained from three optimization methods are almost same.

VI. CONCLUDING REMARKS

Under the assumptions of a lag-free and unlimited vehicle, the proposed waypoint guidance scheme can generate the energy optimal trajectory passing through all the waypoints in real time.

TABLE 1 MINIMUM COST AND PASS ANGLES OF SCENARIO 1

Methods OCP-1 NLP-1 OWG

*J 4907.62 4916.91 4937.00 *0 84.35 84.17 84.17 *1 101.17 101.20 101.67 *2 50.96 50.59 49.15 *3 -14.65 -14.62 -14.28 *4 14.85 14.94 14.84 *5 -48.31 -48.50 -48.92 *6 -123.21 -123.40 -122.36 *7 -143.92 -143.79 -144.00 *8 -155.42 -155.80 -155.92 *9 167.70 167.66 167.96

TABLE 2 MINIMUM COST AND PASS ANGLES OF SCENARIO 2

Methods OCP-1 NLP-1 OWG

*J 9635.03 9636.80 9664.06 *0 81.46 81.13 82.08 *1 106.91 104.44 105.85 *2 30.78 33.85 34.54 *3 45(given)*4 -2.20 -2.78 -1.92 *5 -46.90 -45.08 -47.62 *6 -116.79 -117.79 -116.47 *7 -180(given)*8 -144.32 -144.40 -145.53 *9 162.04 161.71 162.77

1353

Command saturation is another important factor that leads guidance performance degradation. In most cases, command saturation comes from the guidance geometry. By choosing proper waypoints or by assigning proper waypoint pass angles, the magnitude of the guidance command can be reduced to avoid command saturation.

If the vehicle is assumed as a 1st-order lag system, acceleration is newly introduced in the equations of motion. In this case, the theorems can be generalized under the existence of the energy optimal guidance law with terminal constraints on impact angle and acceleration. This issue is currently being studied.

ACKNOWLEDGMENT

This study was supported by Flight Vehicle Research Center (FVRC) of the Agency for Defense Development (ADD) Korea.

REFERENCES

[1] P. Zarchan, Tactical and Strategic Missile Guidance, 4th ed., AIAA Inc., 2003, pp.11-15.

[2] V. Bakaric, Z. Vukic, and R. Antonic, "Improved basic planar algorithm of vehicle guidance through waypoints by the line of sight," Proceedings of IEEE International Symposium on Communications and Signal Processing, 2004, pp. 541-544.

[3] I. H. Whang and T. W. Whang, "Horizontal waypoint guidance design using optimal control," IEEE Transactions on Aerospace and Electronic Systems, Vol. 38. No. 3, July, 2002, pp. 1116-1120.

[4] C. K. Ryoo, H. Cho, and M. J. Tahk, “Closed-form solutions of optimal guidance with terminal impact angle constraint,” Proceedings of the 2003 IEEE Int'l Conference on Control Application, Istanbul, Turkey, June 2003, pp. 504-509.

[5] C. K. Ryoo, H. Cho, and M. J. Tahk, "Optimal guidance laws with terminal impact angle constraint," Journal of Guidance, Control, and Dynamics. Vol. 28, No. 4, Jul-Aug. 2005, pp. 724-732.

[6] Y. I. Lee, C. K. Ryoo, and E. Kim, "Optimal guidance with constraints on impact angle and terminal acceleration," Proceedings of the 2003 AIAA Guidance, Navigation, and Control Conference, AIAA 2003-5795, Austin, Texas, Aug. 2003.

[7] D. E. Kirk, Optimal Control Theory – An Introduction, Prentice-Hall, Englewood Cliffs, New Jersey, 1970, pp. 54-55.

[8] A. E. Bryson, Jr. and Y.-C. Ho, Applied Optimal Control, John Wiley & Sons, 1975.

[9] D. G. Hull, “Conversion of optimal control problems into parameter optimization problems,” Journal of Guidance, Control, and Dynamics,Vol. 20, No. 1, Jan.-Feb. 1997, pp. 57-60.

[10] C. Lawrence, K. L. Zhou, and A. L. Tits, User's Guide for CFSQP Version 2.5: A C Code for Solving (Large Scale) Constrained Nonlinear (Minmax) Optimization Problems, Generating Iterates Satisfying All Inequality Constraints, Institute for Systems Research, University of Maryland, 1997.

-500 0 500 1000 1500 2000 2500 3000 3500-500

0

500

1000

1500

2000

2500

WP9 OCP-1 NLP-1 OWG

Y p

ositi

on (m

)

X position (m)

WP1

WP2 WP3 WP4 WP5

WP6

WP7

WP8Launch point

Angle constraints

(a) Energy optimal trajectories

0 20 40 60 80 100-45

-30

-15

0

15

30

45

OCP-1 NLP-1 OWG

Gui

danc

e co

man

d (m

/s2 )

Time (sec)

(b) Profiles of the guidance command magnitude Fig. 5. Optimization results of Scenario 2.

-500 0 500 1000 1500 2000 2500 3000 3500-500

0

500

1000

1500

2000

2500

WP9 OCP-1 NLP-1 OWG

Y p

ositi

on (m

)

X position (m)

WP1

WP2 WP3 WP4 WP5

WP6

WP7

WP8Launch point

(a) Energy optimal trajectories

0 20 40 60 80 100-45

-30

-15

0

15

30

45

OCP-1 NLP-1 OWG

Gui

danc

e co

man

d (m

/s2 )

Time (sec)

(b) Profiles of the guidance command magnitude Fig. 4. Optimization results of Scenario 1.

1354