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226 M.A. Hurni et al.

Fig. 11.12 Obstacle intercept (with prediction)

Fig. 11.13 Obstacle

intercept (no prediction)

Fig. 11.12(b). For the ﬁrst 5 seconds, before obstacle motion begins, the vehicle

heads straight at the goal. Once the vehicle detects obstacle motion, it estimates the

obstacle’s course and speed, and autonomously makes a course correction (assum-

ing the obstacle’s course and speed stays the same) to avoid collision.

This obstacle intercept scenario was repeated without the use of course and speed

to predict obstacle position. The resulting trajectory collided with the obstacle, as

shown in Fig. 11.13.

11.3.5 Obstacle Intercept Window

The intercept window is deﬁned, in this work, as a ﬁnite period in an obstacle mo-

tion in which it is on a collision course with the vehicle, but the obstacle’s course

and speed is not constant and, therefore, the threat of collision only exists during a

11 An Info-Centric Trajectory Planner for Unmanned Ground Vehicles 227

Fig. 11.14 Obstacle intercept

window (no prediction)

Fig. 11.15 Obstacle intercept window (with prediction)

ﬁnite period of time. For example, two cars approaching an intersection have an in-

tercept window when after being on a collision course for a while one car turns onto

another road or slows down, thus, eliminating the chance of intercept. The obstacle

intercept problem was repeated, but the obstacle’s motion was modiﬁed to stop at

(x, y) = (20, 10) meters, eliminating any possibility of collision. In this scenario

the obstacle never obstructs the vehicle’s path to the goal. Figure 11.14 shows the

vehicle trajectory when prediction is not used. The small alteration in the vehicle’s

trajectory is due to the effect of the robustness term in the cost function [5]. The

overall maneuver time was 32.3 seconds.

The scenario was repeated using prediction, and the resulting trajectory is shown

in Fig. 11.15(a). The vehicle was baited off its original trajectory due to the fact that

it incorrectly predicted the obstacle’s collision course and autonomously replanned

to avoid the collision. The maneuver time for this scenario was 33.8 seconds, which

is longer than when prediction was not utilized (32.3 seconds). The evolution of the

trajectory is shown in Fig. 11.15(b). Frame 1 shows the trajectory before obstacle

motion commenced. Frames 2 and 3 show the predicted trajectory once the algo-

228 M.A. Hurni et al.

Fig. 11.16 Vehicle faster than target (no prediction)

rithm determined the obstacle’s course and speed. Frame 4 shows the new trajectory

after the obstacle stopped. Essentially, the obstacle’s motion faked out the vehicle

into executing an extra maneuver, resulting in it taking a longer time to reach the

goal. This intercept window scenario is the one instance when not using prediction

actually results in a less maneuver time than when using prediction. However, given

the fact that prediction still produced a valid trajectory (albeit a little longer) and

the fact that in all other cases prediction is the better way to go, this issue is an

acceptable disadvantage.

11.4 Target Motion Studies

11.4.1 Target Rendezvous: Vehicle Faster than Target

In this scenario, the vehicle must rendezvous with a moving target. Figure 11.16(a)

shows the vehicle trajectory without predicting the target position, and Fig. 11.16(b)

shows the evolution of that trajectory. Initially, the target is motionless, and after 15

seconds it commences its movement at 0.6 m/sec. Since the algorithm does not use

the target’s course and speed to predict a rendezvous point, each trajectory plan-

ning iteration simply aims the vehicle at the current target position. This results in

the vehicle falling in behind the target and catching up to it. The maneuver time

was 37.2 seconds. Using the target’s course and speed to predict a rendezvous point

resulted in the trajectory of Fig. 11.17(a) and a faster maneuver time of 33.3 sec-

onds. Figure 11.17(b) shows the evolution of that trajectory. Notice that the trajec-

tory follows an intercept course based on the prediction of the target future posi-

tions.

11 An Info-Centric Trajectory Planner for Unmanned Ground Vehicles 229

Fig. 11.17 Vehicle faster than target (with prediction)

Fig. 11.18 Ve hic le s low er

than target (no prediction)

11.4.2 Target Rendezvous: Vehicle Slower than Target

The same target rendezvous scenario was simulated, but this time the target speed

was faster than the vehicle’s maximum speed limit at 1.2 m/sec. The resulting

trajectory for the case of no prediction is shown in Fig. 11.18. The vehicle falls

in behind the target and can never catch up. Unless the target slows down, the

vehicle can never complete the mission. Figure 11.19 shows the optimal trajec-

tory using target course and speed to predict its future position. In this case, the

vehicle was able to rendezvous with the target by calculating an intercept ma-

neuver. It follows that if we want to maximize the vehicle’s chances of being

able to catch a moving target, then prediction is the way to achieve the best re-

sults.

230 M.A. Hurni et al.

Fig. 11.19 Ve hic le s low er

than target (with prediction)

Fig. 11.20 Variable target

motion

11.4.3 Target Rendezvous: Variable Target Motion

In this section, we study the effect of changes in the target motion during the ve-

hicle’s maneuver. The scenario is the same as in Sect. 11.4.1, except instead of

the target moving indeﬁnitely, it stops after 10 seconds of motion. Figure 11.20

shows both cases with and without prediction. The case of no prediction pro-

duced a less tortuous maneuver as well as a faster one, posting a 33.9 second ma-

neuver time as opposed to the 36.8 second maneuver time when prediction was

used. This scenario shows that in a small number of cases where the obstacle’s

course and speed varies frequently and signiﬁcantly, not using prediction can ac-

tually produce better results than when using prediction. However, as stated pre-

viously, the price paid is acceptable when weighing the beneﬁts of using predic-

tion.

11 An Info-Centric Trajectory Planner for Unmanned Ground Vehicles 231

11.5 Conclusion

An information-oriented pseudospectral optimal control framework provides a nat-

ural approach for autonomous trajectory planning. Simulation results illustrate the

effects of varying levels of information and how more or less information inﬂuences

the performance of the planner. The simulations and discussions in this paper ex-

plain how the trajectory planning framework developed in [5] is an information cen-

tric planner that, with some exceptions, produces better results with more informa-

tion. Predicting obstacle and target positions using course and speed data produced

faster (with a few exceptions) and more reliable trajectories than using still snap-

shots of the environment. Having a priori knowledge of environmental conditions

will always produce the best results, however, providing such a level of information

is not always practical. In summary, when conducting trajectory planning, it is de-

sirable to include as much information in the optimal control problem formulation

as possible.

References

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hicles. In: AIAA Guidance, Navigation, and Control Conference and Exhibit, Keystone, CO,

AIAA-2006-6074 (2006)

3. Bollino, K.P., Lewis, L.R., Sekhavat, P., Ross, I.M.: Pseudospectral optimal control: a clear

road for autonomous intelligent path planning. In: AIAA Infotech@Aerospace 2007 Confer-

ence and Exhibit, Rohnert Park, CA, AIAA-2007-2831 (2007)

4. Choset, H., Lynch, K.M., Hutchinson, S., Kantor, G., Burgard, W., Kavraki, L.E., Thrun, S.:

Principles of Robot Motion; Theory, Algorithms, and Implementation. MIT Press, Cambridge

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and Exhibit, Honolulu, Hawaii, AIAA-2008-6305 (2008)

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cles. Master’s Thesis, Naval Postgraduate School, Monterey, CA (2006)

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Advanced Intelligent Mechatronics, Monterey, CA, pp. 843–850 (2005)

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bridge (2004)

Chapter 12

Orbital Evasive Target Tracking and Sensor

Management

Huimin Chen, Genshe Chen, Dan Shen,

Erik P. Blasch, and Khanh Pham

Summary In this chapter, we consider the sensor management problem for track-

ing space targets where the targets may apply evasive maneuvering strategy to avoid

being tracked by the space borne observers. We ﬁrst study the case of single target

tracking by a single observer and formulate the pursuit–evasion game with complete

information. Then we extend the tracking problem to a set of collaborative observers

and each observer has to decide when to sense which target in order to achieve the

desired estimation error covariance. A popularly used criterion for sensor manage-

ment is to maximize the total information gain in the observer-to-target assignment.

We compare the information based approach to the game theoretic criterion where

the observers are assigned according to the best response of the terminal result in

the pursuit–evasion game. Finally, we use realistic satellite orbits to simulate the

space resource management for situation awareness. We adopted NASA’s General

Mission Analysis Tool (GMAT) for space target tracking with multiple space borne

observers. The results indicate that the game theoretic approach is more effective

than the information based approach in handling intelligent target maneuvers.

12.1 Introduction

Over recent decades, the space environment has become more complex with a sig-

niﬁcant increase in space debris among densely populated satellites. Efﬁcient and

H. Chen

Department of Electrical Engineering, University of New Orleans, New Orleans, LA, USA

G. Chen (



) · D. Shen

DCM Research Resources, LLC, Germantown, MD, USA

e-mail: gchen@dcmresearchresources.com

E.P. Blasch

AFRL/RYAA, WPAFB, OH, USA

K. Pham

AFRL/RVSV, Kirtland AFB, NM, USA

M.J. Hirsch et al. (eds.), Dynamics of Information Systems,

Springer Optimization and Its Applications 40, DOI 10.1007/978-1-4419-5689-7_12,

233

234 H. Chen et al.

reliable space operations rely heavily on the space situation awareness where search

and tracking space targets and identifying their intent are crucial in creating a con-

sistent global picture of the space. Orbit determination with measurements pro-

vided by a constellation of satellites has been studied extensively [6, 7, 14, 16].

The tracking and data relay satellite system uses satellites in geostationary orbits

(GEO) to track the satellites in low-Earth orbit (LEO). The global positioning sys-

tem (GPS) uses a constellation of satellites with pseudo-range measurements, i.e.,

range measurements with clock differentials to determine the location of a user.

Unlike ground targets whose motion may contain frequent maneuvers, a satellite

usually follows its orbit so that long-term prediction of its orbital trajectory is pos-

sible once the orbital elements are known [6]. However, a space target can also

make an orbital change owing to its desired mission or intentionally hiding from the

space borne observers. Existing maneuvering target tracking literature mainly fo-

cuses on modeling target maneuver motion at random onset time (see, e.g., [1, 13]).

In space surveillance, the constellation of satellite observers is usually known to

the adversary and very unlikely to change frequently due to energy constraint. In

this case, evasive maneuvering motion can be intelligently designed to take advan-

tage of the sensing geometry, e.g., transferring to an orbit with maximum dura-

tion of the Earth blockage to an observer with known orbital trajectory. Accord-

ingly, a sensor management method has to optimally utilize the sensing resources

to acquire and track space targets with sparse measurements, i.e., with a typically

large sampling interval, in order to maintain a large number of tracks simultane-

ously.

Sensor management is concerned with the sensor-to-target assignment and a

schedule of sensing actions for each sensor in the near future given the currently

available information on the space targets. Sensor assignment and scheduling usu-

ally aim to optimize a certain criterion under energy, data processing and communi-

cation constraints. One popularly used criterion is the total information gain for all

the targets being tracked [11]. However, this criterion does not prioritize the targets

with respect to their types or identities. Alternatively, covariance control optimizes

the sensing resources to achieve the desired estimation error covariance for each

target [10]. It has the ﬂexibility to design the desired tracking accuracy according to

the importance of each target. We want to compare both informative based criterion

and the covariance control method in a game theoretic setting where the target can

intelligently choose its maneuvering motion and onset time based on the knowledge

of observers’ constellation.

The rest of the chapter is organized as follows. Section 12.2 presents the

space target motion and sensor measurement model. Section 12.3 provides a game

theoretic formulation of target maneuvering motion. Section 12.4 discusses the

information based performance metric for sensor management and covariance

control method with possibly evasive target maneuvering motion. Section 12.5

compares the performance of the proposed tracking and sensor management

scheme with the existing methods. Conclusions and future work are presented in

Sect. 12.6.

12 Orbital Evasive Target Tracking and Sensor Management 235

12.2 Fundamentals of Space Target Orbits

12.2.1 Time and Coordinate Systems

Several time systems are popularly used in the orbit determination problems. Satel-

lite laser ranging measurements are usually time-tagged in coordinated universal

time (UTC) while global positioning system (GPS) measurements are time tagged

in GPS system time (GPS-ST). Although both UTC and GPS-ST are based on

atomic time standards, UTC is loosely tied to the rotation of the Earth through the

application of “leap seconds” while GPS-ST is continuous with the relation GPS-

ST =UTC +n where n is the number of leap seconds since January 6, 1980. The

orbital equation describing near-Earth satellite motion is typically tagged with ter-

restrial dynamical time (TDT). It is an abstract, uniform time scale implicitly de-

ﬁned by the motion equation and can be converted to UTC or GPS-ST for any given

reference date.

The Earth centered inertial (ECI) coordinate system used to link GPS-ST with

UTC is a geocentric system deﬁned by the mean equator and vernal equinox at

Julian epoch 2000.0. Its XY -plane coincides with the equatorial plane of the Earth

and the X-axis points toward the vernal equinox direction. The Z-axis points toward

the north pole and the Y -axis completes the right hand coordinate systems.

The Earth centered Earth ﬁxed (ECEF) coordinate system has the same XY-plane

and the Z-axis as in the inertial coordinate system. However, its X-axis rotates with

the Earth and points to the prime meridian and the Y -axis completes the right hand

coordinate systems.

The local Cartesian system commonly referred to as east-north-up (ENU) coor-

dinate system has its origin at some point on the Earth surface or above (typically at

the location of an observer). Its Z-axis is normal to the Earth’s reference ellipsoid

deﬁned by the geodetic latitude. The X-axis points toward the east while the Y -axis

points toward the north. The conversion among these three coordinate systems is

provided in Appendix 1.

12.2.2 Orbital Equation and Orbital Parameter Estimation

Without any perturbing force, the position r of a space target relative to the center

of the Earth in ECI coordinate system should satisfy

r =−

r

r (12.1)

where μ is the Earth’s gravitational parameter. The target velocity is v



r and the

radial velocity is v



v·r

where r



=r is the distance from the target to the cen-

ter of the Earth. In order to determine the position and velocity of a satellite at any

time instance, six parameters are needed, typically, the three position components