INNOCENTI AIAA2014 FINAL

Guidance Augmentation for improved Target Visibility

L.Cancemi∗ , M.Innocenti † and L.Pollini‡

Dipartimento di Ingegneria dell’Informazione, University of Pisa, Pisa, 56100, Italy

The paper describes the synthesis of a new guidance law, which aims at keeping highvisibility of a target during the engagement. The scenario under consideration is a urbanenvironment with a known obstacle distribution. A probabilistic target visibility mapis defined, and a guidance law is developed, which moves the vehicle in the directionof increasing visibility while chasing the target. The approach is verified with severalsimulation examples with moving targets, showing that the performance of a standardguidance such as Proportional Navigation can be obtained even in the presence of obstaclesalong the line of sight. In addition, simulations are presented in the case of evasive actionby the target.

I. Introduction

The problem of target visibility in a structured environment, characterized by stationary or mobile obsta-cles, has received a lot of attention in the literature, especially for applications involving target observation,and surveillance. In reference1 an algorithm for calculating the motion of a vehicle is presented, whichsatisfies visibility constraints of a predictable or partially predictable moving target, minimizing the totaldistance traveled by the vehicle. In reference2 an observation task is considered, in which the vehicle isconstrained to maintaining a fixed distance from the target and conditions are given for the existence of asurveillance strategy. The problem of the observation of a target is also treated in3 and in,4where the targetsto be observed may be mobile and multiple. Their motion is considered unpredictable, within a known,stationary environment. The resulting path planner computes a motion strategy that maintains visibility,by maximizing a ”minimum distance of escape”, which is the minimum distance that the target needs inorder to escape from the visibility region of the observer. In4 in particular, the algorithm is applied to areal robot in a scenario one vs. one. A dual interpretation of the problem is given in,5 where an observationtask is still considered, with the target equipped with an optimal guidance law to escape from the visibilityregion.

The problem of navigation between obstacles towards a goal (or target) in structured environments hasbeen often approached using path planning techniques like in most mobile robotics applications (especiallythose related to stationary targets)6,7,8 . In this work, however, we propose a guidance law that does notrequire a path planning step but appears as an acceleration augmentation to an otherwise classical guidancelaw (like PNG). This visibility-based augmentation term may be incorporated into typical pursuer-evaderscenarios, also found in traditional and recent literature, related to missile guidance9,10,11 and to manageguided and/or autonomous maneuvering targets.

The definition of visibility usually found in the literature has a Boolean characterization, based onthe presence or absence of a line-of-sight (LOS) between the target and the vehicle. Here we extend thevisibility concept by associating a probabilistic knowledge of the position of the target; this generalizesearlier approaches, and incorporates uncertainties about the target location in the analytical characterizationof the visibility region. The probabilistic knowledge of the target position, can also be interpreted as aprediction, within some time interval, of its future position. The methodology is presented at the simulationlevel, by comparing it to standard guidance laws such as proportional navigation (PNG)12,13,14 . Theproposed methodology appears suited for path planning applications, as well as more traditional intercept-type scenarios.

∗Graduate Student.†Professor, Associate Fellow AIAA.‡Assistant Professor, Senior Member AIAA.

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American Institute of Aeronautics and Astronautics

The paper is organized as follow. Section II describes the general framework that is used to model targetvisibility. Section III shows the application to an intercept task, and results are given by comparing theproposed guidance law with classical PNG guidance. Finally, conclusions are drawn in Section IV.

II. Definition of a Target Visibility Map

The objective of this section is to analyze the visibility level of a generic vehicle also (defined as stationaryor moving target) in a structured environment, which coincides with a typical urban scenario, characterizedby the presence of fixed obstacles. The problem is to identify areas of the environment, where the target isvisible, and also to compute those with higher visibility as the scenario changes. For this purpose, we seeka spatial function that computes the visibility level of the target, for each point of the environment. In thefollowing, we consider 2D scenario, however the problem can be extended to a three dimensional space withno substantial variations. We also limit ourselves to a 1 vs.1 situation.

The visibility level of a vehicle in a complex environment is a quantity that depends on several factors,in order to formalize the problem we assume the following:

• Target position

• Obstacles position

• Uncertainty on target position

A first definition of the visibility level may coincide with the existence or absence of a line-of-sight (LOS)between the vehicle and the target. Absence or loss of a LOS can be caused for instance by the presence ofobstacles in the path between the two vehicles due to their relative motion, or jamming. In this case, thedefinition of visibility level is Boolean and coincided with the existence of a clear LOS or not. This approachdoes not take into considerations uncertainty of actual target position, and does not consider possible targetmotion within the time-frame of successive visibility map computations. The following section describes aprobabilistic approach to the definition of a visibility map.

II.A. The Probabilistic Visibility Map

In order to improve on the limitations highlighted above, we first describe the target position with a stochasticvariable T ; assuming motion in 2D, we describe target position with the probability density function fT (x, y),where (x, y) ∈ R2; as anticipated above, several approaches may be followed to define fT according to thespecific problem under consideration: the probability density function could be considered as a probabilisticuncertainty about the exact location, but also as a prediction of the target future position after some timeinterval, and centered at the last known target position. Clearly, the interpretation of the resulting visibilitymap associated to the target depends heavily on the meaning given to the probability density function. Wecan now define the Target Visibility Map as:

Definition 1 : Target Visibility Map The visibility map pTv(x∗, y∗) of target T , whose stochastic positionis described by the probability density function fT (x, y), is the probability of having a clear LOS between agiven point of the environment (x∗, y∗) and the target, and it is computed as:

pTv(x∗, y∗) =

∫ ∫R2

fT (x, y) · fLOS(x∗, y∗, x, y)dxdy (1)

where fLOS(x∗, y∗, x, y) = {0, 1} is the binary function (valued 1 or 0) that represents the existence ornot of a clear LOS between the points (x∗, y∗) ∈ R2 and (x, y) ∈ R2 of the environment.

Thus, the visibility level of a target from a point in 2D space is defined as the probability associatedwith a possible position of the target, multiplied by a Boolean factor that indicates the existence of a joining(LOS) between the probable position and the point itself.

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A possible choice for the target density distribution is the bi-variate Gaussian probability density function,centered at the ideal target position:

fT (x, y) =1

σXσY 2πe− (x−xT )2

2σ2X

− (y−yT )2

2σ2Y (2)

where, xT and yT represent the ”exact” target position (be that measured or provided), and σx and σyare the standard deviations along X and Y axes. This function, in the case of some normalized numericalvalues given by (xT , yT ) = (50, 50) and σX = σY = 10, shows the familiar shape in Figure 1.

Figure 1: Bi-variate Gaussian Density Function

The probability density function could be considered as a probabilistic uncertainty about the exactlocation, but also as a prediction of the target future position a some time intervals, and centered at thelast known target position. Clearly a visibility level associated to the target depends heavily on the meaninggiven to the function in Eq.(2).

This probability density function will be used for the simulations presented in the paper; the selection ofa specific probability density function is just for modeling purposes.

For what regards real-time implementation, the numerical computation of the visibility level can beperformed by discretizing the environment as follows:

pTv(x∗, y∗) =∑

(x,y)∈I

fT (x, y) · fLOS(x∗, y∗, x, y) (3)

wher I is a set of points centered at the center of the probability density function, or target positionrt = (xT , yT ), and spread around it to include the majority of high probability points (up top a radius of 3σfor instance).

Figure 2 shows an example of the LOS connecting the points belonging to an area near the vehicle. Thered points indicate the region around the target where the fT associates a probability, this probability willbe filtered by fLOS : in particular is multiplied by 1 if there is a connection between the point and the vehicle(LOS green), and by 0 otherwise (LOS black, or equivalently no LOS).

The next figure shows the above construction extended to a larger scenario with multiple obstacles.Figure 3 (a) shows the LOS that connects the target to the various points of the environment, thus indicatingwhich areas have visibility of the target (green areas) and which have none (areas in red). In this examplethe range is up to 30 normalized spatial units.

In this case, the visibility information is only Boolean (target in view or not), and no distinction existswithin the areas of the same color. When we introduce the uncertainty region around the target, described bythe Gaussian density function fT (x, y), we obtain a visibility information between zero for each point of theenvironment, if none of the points in the target uncertainty region is in sight, and 1, if the entire uncertaintyregion is in view. This is shown in Figure 3 (b), where level curves of visibility are described, obtained usinga fT (x, y) with σX = σY = 3. The areas bounded by darker colors represent a higher visibility.

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(a) Without Obstacle (b) With Obstacle

Figure 2: Sample computation of the stochastic Visibility Map

(a) Line-of-Sight rays (b) Stochastic Visibility Map

Figure 3: Binary and stochastic Visibility Levels for a Stationary Target

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II.B. Use of the Visibility Map

The knowledge of the visibility level of a target with respect to all points in environment can be exploited,and a guidance structure can be devised, that provides a desired trajectory in the direction of improving thecurrent visibility level. A simple algorithm that takes into account the spatial variation of the probability, andproduces a direction of motion, in terms of velocity vector characterized by orientation and magnitude canbe used to evaluate preliminarily the applicability of the proposed Visibility Map to the goal of maximizingvisibility while approaching the target.

Assuming a purely kinematic problem, the computation of the desired velocity vector that brings thevehicle toward areas of higher target visibility can be performed by following the direction of maximumgradient of the Target Visibility Map pTv. The ”optimal” motion direction velocity vector v̄ is given by:

v̄ =∂pTv

∂r̄|r̄=r̄v (4)

where r̄v represents the current position vector of the vehicle pursuing the target. Following the directionsin which the gradient is positive and increasing, we obtain an increase of target visibility. Therefore, forthe calculation of the vehicle motion direction it is not necessary to compute the visibility level relative tothe entire environment, but only for the points in its vicinity; this might lead to a local optimization onlyand could bring the vehicle into local maxima of the target visibility; this issue is not considered here andanalysis of the convexity of the Visibility map will be subject of future study.

In order to compute in real-time the direction of maximum increase of the visibility map, a quick searchin the neighborhood of the vehicle position is performed by evaluating the value of pTv in a ring of pointsaround vehicle position, and taking the point r∗ that maximizes the increase:

r∗ = arg maxr

[pTv(r)− pTv(rv)], r ∈ B(rv, d) (5)

where B(rv, d) is an area centered at the vehicle position with radius r: B(rv, d) = {r ∈ R2 : dist(r, rv) ≤d}.

In the planar case then, which is the object of this paper, the optimal direction is computed in polarcoordinates as:

γvis = arctan(rv − r∗)Y(rv − r∗)X

, |v̄| = pTv(r∗)− pTv(rv)

d∼=∣∣∣∣∂pTv

∂r̄|r̄=r̄v

∣∣∣∣ · (rv − r∗) (6)

The effectiveness of the proposed algorithm is described in Figure 4 in which the algorithm is appliediteratively. The vehicle is initially at position (70, 70) and the target is in (70, 58). Figure 4 shows thetrajectory suggested by the algorithm, also shows the same trajectory on the visibility level curves, in whichit is evident that the vehicle goes in a position in which the visibility is maximum.

III. Visibility Aware Proportional Navigation Guidance

This section describes the application of the visibility approach to an intercept problem with a movingtarget. The baseline guidance used is Proportional Navigation, and it is modified to take into account targetvisibility. PNG guidance is a well-established guidance law in many areas, and it is based on the fact thattwo vehicles are on a collision course when their Line-of-Sight does not change direction.

The control action is provided by the magnitude of the lateral acceleration APNG that is taken nor-mal to the velocity vector, resulting in a True Proportional Navigation implementation. The commandedacceleration is thus:

APNG = NVcσ̇ (7)

where Vc is the closing velocity, σ̇ is the LOS rate and N is the navigation ratio. PNG performance interms of miss distance and generated acceleration profiles over moving targets have been extensively studiedin the past, even though the presence of obstacles has been rarely taken into account; if this is the case,PNG could not be used since the generated trajectory would likely intersect obstacles or target motion couldbring the LOS to intersect an obstacle.

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(a) Pursuer Trajectory (b) Pursuer Trajectory and level curves of the Visibility Map

Figure 4: Pursuer Trajectory along the direction of maximum gradient of the Visibility Map for a stationaryTarget

The aim of this PNG augmentation law is to provide PNG with the capability to maneuver betweenobstacles until near enough to the target to be able to complete the final part of the flight using LOS rateinformation only. The PNG law is now modified with an augmentation term in order to take into accounttarget visibility: the main idea is to translate the direction of motion used in the previous section, generatedas a desired velocity vector from the Visibility Map, into a commanded acceleration suitable for augmentationof Eq. (7). This is achieved by defining the angular error between the current velocity vector angle of thepursuer (γ) and the optimal angle (γvis) computed using the Visibility Map, and by using this error toaccelerate the pursuer:

Avis = K · kvis · sin(γvis − γ) (8)

where K is the guidance gain, and kvis is a gain proportional to the improvement of visibility that thevehicle would achieve if it was to move in the direction established by γvis. It is in fact computed as thedifference between the visibility in the new direction and that in the current one; thus, using the syntax ofprevious section:

kvis = pTv(r)− pTv(rv) ∝∣∣∣∣∂pTv

∂r̄|r̄=r̄v

∣∣∣∣ (9)

Given the fact that pTv ∈ [0, 1], then also the gain kvis ∈ [0, 1]; values of kvis close to 1 indicate a maximumimprovement of visibility, while values near 0 indicate little change of visibility. The term sin(γvis − γ)manages full 360 degrees of possible directions of γvis around γ; if the error is 0 < (γvis − γ) < π, then theacceleration Avis > 0; if π < (γvis − γ) < 2π then Avis < 0.

Now, the acceleration contribution Avis, computed according to the directions of visibility improvement,can be added to the standard PNG guidance law, yielding the following definition:

Definition 2 : Visibility Aware Proportional Navigation Guidance (VA-PNG) The VisibilityAware Proportional Navigation Guidance law is defined as :

AV A−PNG = APNG +Avis = NVcσ̇ +K · kvis · sin(γvis − γ) (10)

It should be noted that, since kvis·sin(·) ∈ [0, 1], then the gain K actually represents the maximum level ofacceleration that the augmentation part of the VA-PNG can generate; selecting its value as trade-off betweenthe control authority of LOS rate guidance (APNG), and the visibility aware part is then straightforward.

At this point, we must mention that we neglected actuator saturation, other types of commanded accel-eration limitations, and physical constraints on the vehicle’s trajectory (maximum turn rate, etc.). Generalmodeling complexity issues and uncertainties are not part of the present work.

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III.A. Examples and Simulations

This section presents some numerical results to validate the beneficial contribution of visibility levels withinga standard proportional navigation structure. Initially a moving but non maneuvering target is considered,later some escape capabilities are added. Scenarios were designed so that PNG would achieve target interceptin the absence of obstacles, and do not include agility issues, nor off-boresight maneuvers,14 which will beconsidered in a different study. Simulations were performed with both standard PNG and the VA-PNG. Bothvehicle and target were modeled as pure kinematic objects, as standard in guidance analysis. In order tosimulate a realistic implementation of the visibility augmentation, the term Avis is computed at the lower ratethen APNG; this results into non-smooth variations of the commanded accelerations due to target visibility,but does not compromise actual performance. A throughout study of the effect of visibility map computationrate versus intercept performance was not performed, and will be subject of future investigations.

III.A.1. Simulation 1

In this simulation, the pursuer starts from position (65, 60) with a 180 degree orientation, while the targetstarts from position (54, 49), with an orientation of 210 degrees. The intercept is guaranteed by the vehiclehaving a speed twice as larger as the one of the target. The simulation results are shown in Figure 5. Theleft column shows the performance of a baseline PNG law, whereas the trajectory followed by the vehicleusing the VA-PNG are shown in the right column.

(a) PNG (b) VA-PNG

Figure 5: Sim 1 : Intercept Comparison for a Non Maneuvering Target

The vehicle’s trajectory is shown in blue, while the target trajectory is shown in red. Figure 5 (b) showshow the vehicle, starting from a situation of low visibility, moves towards a point where the visibility of thetarget reaches almost a maximum, and then continues toward the intercept.


A similar result is obtained in the simulation shown in Figure 6 (a)(b). The pursuer starts from a positionlocated at point (70, 70) with a 270 degree orientation, while the target starts from position (70, 58), with anorientation of 280 degrees. In this case the visibility contribution to the total acceleration moves the vehicleaway from the obstacle, and only later, when the target is in full visibility, the proportional navigationguidance concentrates on the intercept.

In order to better understand the behavior of the proposed guidance, it is necessary to analyze howthe two different terms APNG and Avis contribute to the total pursuer acceleration. Figure 7 shows theacceleration time histories of the whole engagement.

It is interesting to notice, how the visibility component prevails on total acceleration in the initial phaseof the engagement, since in this phase the situation of poor visibility of the target is critical. In fact, theimprovement made by the visibility algorithm in the direction suggested is considerable, since the gain kvis

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(a) PNG (b) VA-PNG

Figure 6: Sim 2 : Intercept Comparison for a Non Maneuvering Target

Figure 7: Sim 2 : Commanded Acceleration Contributions

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in the equation (10) is close to 1. After about 3 time units the acceleration component due to visibility isbasically brought to zero.

(a) PNG Contribution (b) Visibility Contribution

Figure 8: Percent Contributions of each Commanded Acceleration

Figure 8 shows the two acceleration components of VA-PNG as a percentage of total acceleration, thisconfirms the fact that initially the majority of the control comes from the visibility component, whereas lateron the PNG guidance takes over. The target visibility level changes of course during the engagement, andis clearly shown in Figure 9.

Figure 9: Visibility Level

As expected, the target visibility level, the value of pTv along the pursuer trajectory, reaches the maximumin the final stages of the engagement, with the target fully visible and proportional navigation guidance fullyoperational.

A heuristic analysis of the effect of the gain K, that is of the maximum acceleration level due to thevisibility component of VA-PNG, is carried out in Simulation 2 with K varying from 4 to 12. Figure 10shows the different trajectories obtained: higher gain naturally yields a trajectory more focused on obstacleavoidance, while lower gains produce trajectories that passes nearer to the obstacle.

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Figure 10: Trajectory Variation with Gain K


Let us consider now an intercept scenario with a target with ”‘smart” maneuvering capabilities: the targettries to escape and hide from the pursuer exploiting the presence of the obstacle in the scenario. A simple,and perhaps academic, way to achieve this is to provide the target with a guidance component dual tothe one used by the pursuer: the target’s commanded acceleration is generated essentially by the visibilitycomponent, seeking a maneuver that goes into the direction of decreasing its visibility with respect tothe pursuer, thus looking for ”hiding” areas behind the obstacles (note that this approach has analyticalsimilarities with a mini-max non cooperative game).

The computation of the target’s trajectory is done using the gradient of pTv, with respect to the vehicle’sposition. The target’s velocity vector is thus computed as:

v̄T ∝∂pTv

∂r̄t(11)

where r̄t is the position vector of the target. If we consider only the directions in which the gradient isnegative and decreasing, the target will move in the direction of decreasing its visibility. Similarly to whatwas done before for the vehicle, the gradient is calculated only in a neighborhood of target position. Theresult of this extension will produce a tendency by the target to hide behind obstacles.

Figure 11 shows the trajectories for both vehicles. The target starts located at point (75, 67), andthe tracking vehicle starts from position (55, 72). A zero degree orientation is assumed for both. In thisparticular example, target evasion is not successful due to the energy properties assumed for the vehicles.Figure 12 (a) shows the visibility level of the target with respect to the vehicle. The visibility presents aminimum in the area ad about 7 sec., corresponding to the target moving behind the obstacle. Figure 12(b) shows the time history of the tracking vehicle’s acceleration contributions, the vehicle is commanded tomove to increase its visibility level in correspondence to the evading maneuver of the target.

IV. Conclusions

The paper presents a preliminary study for the augmentation of a standard guidance law with a term thattakes into account target visibility. The chosen scenario is a structured environment with known, stationaryobstacles. The guidance augmentation moves the vehicle in areas where maximum probability of visibilityis achieved, and where traditional intercept becomes possible. The concept of visibility levels associated tothe probability adds the flexibility of either considering the target position non deterministic, or avoidingBoolean modeling of the existence of a clear LOS. Results are promising for stationary as well maneuveringtargets in a 1 vs. 1 engagement.

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Figure 11: Engagement Trajectories with Maneuvering Target

(a) Target Visibility Level (b) Vehicle’s Commanded Acceleration Time Histories

Figure 12: Visibility and Guidance Law with Maneuvering Target

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References

1S.LaValle, H.Gonzales, C.Backer, Motion strategies for maintaining visibility of a moving target, IEEE InternationalConference on Robotics and Automation, 1997.2R.Murrieta, A.Sarmiento, S.Bhattacharya, Maintaining visibility of a moving target at a fixed distance: the case of observer

bounded speed, IEEE International Conference on Robotics and Automation, 2004.3R.Murrieta, H.Gonzales, B.Tovar, A reactive motion planner to maintain visibility of unpredictable targets, IEEE

International Conference on Robotics and Automation, 2002.4R.Murrieta, B.Tovar, S.Hutchinson, A simpling based motion planning approach to maintaing visibility of unpredictable

targets, Autonomous Robots, vol. 19, 2005, pp 285-300.5T.Muppirala, S.Hutchinson, R.Murrieta, Optimal motion strategies based on critical events to maintain visibility of a

moving target, IEEE International Conference on Robotics and Automation, 2005.6Antonios Tsourdos, Brian A.White and Madhavan Shanmugavel, Cooperative Path Planning of Unmanned Aerial Vehicles,

John Wiley and Sons,Llt, 2011.7L.J.Guibas, JC LaTombe, S.M.LaValle, D.Lin and R.Matwani, Visibility based Pursuer-Evasion in a polygonal

environment, Algorithms and Data Structures, 1997, pp. 17-30.8S.M.LaValle, D.Lin, L.J.Guibas, JC LaTombe, Finding an unpredictable target in a workspace with obstacles, IEEE

Robotics and Automation Proceedings, April 1997.9Bhattacharya S., Hutchinson S., Basar T., Game-theoretic analysis of a visibility based pursuit-evasion game in the

presence of obstacles, American Control Conference, 2009.10A.E.Bryson, S.Baron, Differential games and optimal Pursuit-Evasion strategies, IEEE Transactions on Automatic Control,vol. 10, 1965, pp. 385-389.11Arthur E. Bryson, Yu-Chi Ho, Applied Optimal Control, optimization, estimation and control, Hemisphere PublishingCorporation, 1975.12Paul Zarchan Tactical and Strategic Missile Guidance, AIAA 5th edition, January 1, 2007.13Yanushevsky, Rafael, Modern Missile Guidance, CRC Press, 2007.14M.Innocenti, Nonlinear guidance techniques for agile missiles, Control Engineering Practice, vol. 9, 2001, pp. 1131-1144.

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