Rot orcraft Pursuit-Evasion in Nap-of4he-Earth Flight Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3455 NASA Ames Research Center Moffett Field, CA 94035 Abstract Two approaches for studying the pursuitevasion problem between rotorcraft executing napof-the-earth flight are presented. The first of these employs a constant speed kinematic helicopter model, while the second approach uses a three degree of freedom point-mass model. The candidate solutions to the first differential game are generated by integrating the state-costate equations backward in time. The second problem employs feedback linearization to obtain guidance laws in nonlinear feedback form. Both approaches explicitly use the terrain profile data. Sample extremals are presented. Introduction Recently, there has been significant research activity in developing approaches for automating the nap-of-the-earth flight regime for high performance helicopters1". This research is driven by a need to decrease the pilot work load to acceptable levels in such an operational mode. Technology requirements for achieving this objective include not only the synthesis of guidance and control laws, but also the development of various sensors for gathering information about the terrain and obstacles. Several of these issues have been addressed in Reference .I. Specifically, the development of image-based ranging systems for obstacle and terrain detection have been discussed in References 2 - *Member AIAA, School of Aerospace Engineering, Georgia Institute of technology, Atlanta. Mailing Address : FSN Branch, MS 210-9 'Graduate Student, School of Aerospace Engineering, Georgia institute of Technology, Atlanta $Research Scientist, FSN Branch, MS 210-9 Copyright @ 1990 American Institute of Aeronautics and Astronautics, Inc. All rights reserved. 490-y77~7 and 3. A heuristic guidance law for trajectory selection in the presence of obstacles was outlined in Reference 4, while the ~ r o b l e mof optimal trajectory planning using macroscopic terrain features was examined in detail in References 5 and 6. The research in References 5 and 6 have focussed on planning methods useful for generating trajectories that optimize a linear combination of flight time and terrain masking connecting initial conditions with specified terminal conditions. Present research is an outgrowth of the work reported in Reference 5. This paper examines the trajectory planning process wherein two vehicles with conflicting objectives are involved. The first participant in this process, designated as the evader, is a rotorcraft attempting to reach a specified terminal condition while masking its trajectory using the terrain. The second participant, designated as the pursuer, has the objective of intercepting the evader before it completes its mission, while masking its trajectory using the available terrain. The motivation for maximizing terrain masking is to escape detection from each other and also from other ground based or airborne detection devices. It is assumed here that the evading rotorcraft has no offensive capabilities. Both vehicles may attempt to accomplish their objectives in a time optimal fashion. This problem is suitable for analysis via the theory of differential games7's. Two differential games incorporating models with increasing complexity are considered in this paper. The first of these uses a kinematic model with local heading angle as the control variable while the second employs a point-mass model with the rotor thrust, pitch and yaw attitudes of vehicle as the three control variables. The performance index for the f i s t Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3455 game is a Linear combination of flight time and terrain masking, while the second pursuit-evasion game considers a quadratic form in the terminal miss distance together with the square of the rotorcraft acceleration magnitude as the performance payoff. The candidate trajectories for the differential game formulation using the kinematic vehicle model are obtained by backward integration of the state-costate system. This approach was motivated by the research discussed in Reference 9. The second differential game formulation is patterned after the procedure discussed in Reference 10. In this problem, the solution is obtained by first transforming the nonlinear vehicle models into a linear, time invariant form using the theory of feedback linearizati~n'~*".This process is followed by the solution of the differential game in transformed coordinates. TEe resulting guidance laws are transformed back to the original coordinates to obtain nonlinear feedback laws. Each of these differential game solutions will be discussed in the ensuing. Rotorcraft Pursuit-Evasion with a Kinematic Model In Reference 5, a trajectory planning scheme useful for nap-of-the-earth flight of a rotorcraft using optimal control theory12was discussed. In that work, the terrain profile was used to perform coordinate transformations to reduce the dimension of the problem from three to two dimensions. Secondly, by invoking a constant of motion in the problem, the task of finding the optimal trajectory was reduced to that of determining the initial value of the control variable. The computational effort involved in this trajectory planning process was also estimated. In this section, the same frame work will be used to formulate the two sided trajectory planning problem. As in Reference 5, let the composite profile F(z,y) be given by with h, being the desired clearance above the terrain profile, f (2, y) being the actual terrain profile and P(x, y) the specified threat overlay. The kinematic models for the pursuer and evader moving over this composite profile are given by5 . Ze Ve C O S X ~ = VeFx, F,, sin X, JZ + J(1+ F.,Z)(~ + Fxe2 + Ffi2) here, the symbol V denotes rotorcraft airspeeds and x denotes the heading angle defined on the local tangent plane. The subscript p and e denote the pursuer and the evader respectively. The vehicle altitude rates being given by the equations he = V'siny,, hp = vPsinyp (6) The pursuer-evader flight path angles yp, 7, are given by the equations It is assumed that the rotorcraft airspeeds Vp, V, either remain constant throughout the game duration or is specified at each terrain location. Note that as in Reference 5, it is also possible to consider a case in which the rotorcraft airspeeds are considered to be control variables. The variables F,, ,F,, ,FaP, F,, are the composite terrain profile gradients at the instantaneous evader-pursuer locations. The heading angles xp and xe are the control variables in this . Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3455 problem. These equations are obtained by resolving the rotorcraft velocity vector in the local tangent plane using the terrain gradients to define a transformation between the local tangent plane coordinate system with an inertial frame. These coordinate systems are illustrated in Figure 1. For further details on this coordinate system, the reader is directed to References 5 and 6. The rotorcraft pursuit-evasion begins at a certain set of initial conditions when the participants first become aware of one another. One of the participants in the game will be identified as the pursuer, while the other participant is assumed to be the evader. Further, it is assumed here that this designation continues unchanged throughout the duration of the encounter. In the present game, the objective of the evader is to reach a specified set of final conditions without being captured by the pursuer. The pursuer on the other hand, is attempting to capture the evader before it gets to the desired final conditions. Further, the participants in the pursuit-evasion encounter attempt to accomplish their objectives in a time-optimal fashion while providing for terrain masking. As in reference 5 and 6, it is assumed here that the terrain masking will be accomplished if the participants minimize their altitude above the specified datum. Thus, the performance index employed here is Here, Fp and Fe are the altitudes at the current pursuer-evader locations defined by the composite profile given in equation (1). Wp, We are weighting factors for the pursuer and evader altitudes. These reflect the participant's concern about degree of desired terrain masking. The negative sign in the second term explicitly recognizes the fact that the evader is attempting to maximize the performance index. At this stage, the definition of this differential game is incomplete because no criterion has been laid down for the termination of the game. In this paper, the pwsuit-evasion game is assumed to terminate the first instant the pursuer succeeds in approaching the evader within the firing range of its weapon system. The pursuer's weapon envelope is assumed to be spherical. Note that alternate weapon envelope geometries can also be employed in the present formulation. It needs to be emphasized that the present formulation does not allow for any offensive capabilities for the evader. Additionally, the game is declared a draw if the pursuer-evader trajectories leave a predefined region in the terrain at any inst ant during the encounter For the case of a circular weapon envelope centered at the pursuer, the capture condition can be expressed by requiring that . The quantity d is a specified constant. The condition (11) ensures that sufficient opportunity exists for weapon usage. The time for capture is then determined as the first time instant when the inequality (10) is met as an equality. The game termination constraint may be satisfied by augmenting the performance index with equation (10) as an equality, i.e., The quantity v is an undetermined multiplier. Given the performance index (12) together with the differential constraints (2) - (5), one may derive necessary conditions that must be satisfied by the optimal trajectories. To this end, define variational Hamiltonian12 as +Mp + A42ip (13) The Euler-Lagrange equations for the evader can then be obtained as - B1 sin ~e A2 + (B2 sin xe + B3 cos Xe)X1 K A13A23 (14) Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3455 - B4 sin xe -k (B5 sin xe 4- B6 cos xe)X1 A13 K (15) where These equations were derived using the symbol manipulation program MACSYMA13. The Euler-Lagrange equations and the optimality condition for the pursuer has exactly the same form as expressions (15) - (23), and can be written down by inspection. In the interests of conserving space, these will not be given here. It may be verified that the form of the necessary conditions remain the same regardless of the order of the minimization and maximization. Next, the terminal conditions on the costates can be obtained as : Since the final time is open and the variational Hamiltonian does not explicitly depend on time, this problem has a constant of motion, Viz., H(t) = 0, 0 5 t 5 tf (26) Using this fact , together with the terminal conditions on the various costates, the undetermined multiplier may be computed as: where B6 = -A23~xeFxeye and the optimality condition is: (22) The denominator of this equation is simply the negative of the product of range and range rate at the final time, i. e., the closing rate. It Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3455 is possible to verify that as in the planar pursuitevasion game, the value of this parameter will turn out to be positive if the terrain masking weights were zero. Extremals for the pursuit-evasion problem can be constructed if the initial conditions on the costates or the terminal conditions on the states were known. Since the final condition to be reached by the evader is normally known, in this research, the extremals are obtained by integrating the state-cost ate system backward in time until the trajectories reach the boundary of the admissible region. Since the last opportunity for the pursuer to catch the evader is at its final position, this represents the upper bound on the duration of the game. The detailed procedure for constructing the trajectories is as follows. The pursuer's capture set is first superimposed at the evader's final position. The terminal condition of the pursuer is selected from this set and the equations are integrated backwards till the trajectories reach the boundary of the admissible region. The relative distance between the two vehicles are continuously monitored along the trajectories. Trajectories are constructed for several conditions in the capture set. From the final time, if the relative distance between the vehicles increase as the trajectories evolve in retrogressive time, it indicates that the pursuer can catch the evader only at the final time. On the other hand, if the relative distance decreases or stays constant while the trajectories are evolving in retro-time, it indicates that capture is possible anywhere along this segment of the trajectories. The results of this exercise can be used to determine the set of initial conditions for the pursuer and the evader that will not result in capture at the specified final conditions. These computations are also useful to delineate the set of initial conditions from which capture is assured. Construction of these sets will require extensive numerical computations and will be a future research item. Some sample trajectories will be illustrated in the ensuing. Figure 2 shows the pursuer and evader trajectories in the down-range cross-range plane. The dotted lines and the lines with numerals attached are the constant altitude contours on the terrain. A fixed terrain clearance of 50 feet together with unity terrain masking weight were used in this computation. The value of the parameter v corresponding to this set of final conditions turned out to be 5,793-03.The trajectory duration was approximately 95 seconds. The intricate behavior of the trajectories are apparent from this figure. The corresponding altitude evolution is illustrated in Figure 3. Note that this trajectory is given in retrogressive time and the tgo = 0 corresponds to the terminal instant. During the course of the present investigation, several such trajectories were generated. It may be verified that if the terrain gradients were zero, the resulting pursuit-evasion game produces what is called a simple motion7>*. In this case, the trajectories will turn out to be straight lines joining the initial conditions and the intercept point. Rotorcraft Pursuit-Evasion with a Point-mass Model From the previous discussions, it is clear that the differential game solution is complex even in the case of simple vehicle models. The usefulness of the foregoing analysis is limited to addressing the question of capturability since generating large number of trajectories in real-time is not a viable proposition. In practical situations, it is desirable to obtain feedback solutions. It has been shown in Reference 10 that the theory of feedback linearizationll can be used to solve a class of differential games with high order nonlinear dynamics and a quadratic payoff. The approach here is to transform a nonlinear vehicle model into a linear, time invariant form using feedback. This approach has been subsequently extended to study spacecraft pursuit-evasion problems14. In this section, this approach will be illustrated for rotorcraft pursuit-evasion in nap-of-the-earth flight. More recently, the analysis has been extended to include a realistic weapon envelope and the time-of-flight in the performance index15. The point-mass model for a rotorcraft is given by the following six, fist-order nonlinear differential equations : . T sin6 V = --- - g sin 7 m 2 = T cos 0sinq5 mV cos 7 (31) g T cos6 cos q5 - cos 7) mg (32) Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3455 4=v( Here, z is the down range, y the cross range, h altitude, V airspeed, y the flight path angle, x the heading angle, T the main rotor thrust, m the vehicle mass and g is the acceleration due to gravity. The coordinate system for this model is illustrated in Figure 4. The control variables in this model are the pitch attitude of the helicopter 8, the roll attitude q5 and the main rotor thrust T. Note that the rotor-fuselage drag is not included in the foregoing set of equations. It is sometimes convenient to decompose the rotor thrust into two componentsthe fist one being fixed, while the second one is allowed to vary. To ensure that the rotorcraft executes a terrain following flight, it is necessary that the vehicle altitude follow the terrain profile. This requirement may be enforced in the present problem by by differentiating the terrain profile equation (1) twice with respect to time and substituting for various quantities from the vehicle model. This results in a state-control constraint of the form : T cos 8 cos 4 = mV2 cos8 7 (FEEcos x [ +Fwsin X)cos X+ (F,~cos X+ Fuusin x) sin X] +(F, cos X- F, sinx)T COST cos 0 sin 4+mg cos 7 (36) This constraint specifies the relationship between various state and control variables while the vehicle is in terrain following flight. The equations of motion and the terrain constraint for both vehicles have the same form. However, the model parameters may be different. If one attempts to formulate a pursuit-evasion problem using the nonlinear model described in the foregoing, the solution can only be obtained using numerical methods. The objective here, however, is to obtain a feedback law. To meet this objective, the nonlinear equations will be f i s t transformed to a linear time invariant form via a coordinate transformation. Differentiating the down-range cross-range equations once with respect to time, it is possible to write the pursuer-evader equations of motion in the following form. Ze = ael, ye = ae2 (38) For example, the right hand sides of these equations are given by -jpVp sin 7, cos xp -j; (39) sin ye sin xe (40) The variables V ,j,i for both the pursuer and the evader may be eliminated from equations (37), (38) using expressions (30) - (32). If the variables apl, ap2, ael, ae2 are known, the actual control variables can be computed as 4, = tan" ap2 cos xp - apl sin xp rp 0, = tan-' [{sin q5,(aPl cos X, + ap2 sin ;yp apl cos X, - apl sin xP TP= cos 0, sin 4, (43) Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3455 with model and the quadratic performance index, this problem has a feedback solution. The control laws emerging from such a differential game formulation can be shown1' to be of the form With the feedback gains Kl, K2,K3 being computed using the formulae Similar inverse transformations can be written for the evading rotorcraft also. Once the vehicle models are transformed into linear time invariant form, the differential game between the two rotorcraft may be formulated. If the performance index is defined in terms of transformed quantities, a solution may be obtained in a straightforward manner. If mini-maximization of the relative distance at the final time is taken to be the performance index, then the differential game may be solved as in Reference 10. In order to ensure that the emerging strategies are implementable, an acceleration constraint will be imposed on the problem. Thus the objective of the pursuit-evasion game may be defhed as : where : The variables a, a,P are three scalar weighting factors between relative distance at the h a 1 time, and square of the pursuer-evader acceleration magnitudes. The superscript T denotes the transpose operation. The final time t j is assumed to be given. The necessary conditions for optimizing the performance index (45) subject to the differential constraints (37) and (38) can be obtained by proceeding formally by defining the variational Hamiltonian12. Due to the linear nature of the Note that the time-to-go should be available in order to complete the solution. Although it was assumed to be specified, due to inaccuracies in modeling, this parameter has to be estimated while implementing the pursuit-evasion guidance law. It is possible to obtain one such estimate by invoking a constant of motion H(t) = constant in the problem as in Reference 14. Once the control variables apl, ap2, ael, ac2 in the transformed problem are computed, the control variable transformations (41)-(43) may be used to obtain nonlinear pursuit-evasion guidance law. A typical pursuit-evasion scenario using this pursuit-evasion strategy is illustrated in Figure 5. The vehicle trajectories are denoted with arrows in this figure. The lines with numerals appended are the terrain altitude contours. The pursuer is initially behind the evader with its velocity vector in a direction perpendicular to that of the evader. The weighting factors used in this study were a = 1.0, a = 0.0007, P = 0.00001. The engagement time corresponding to this initial condition was 100 seconds. The vehicle altitude histories corresponding to this encounter are given in Figure 6. At the final time, the pursuer altitude was same as that of the evader, although the flight path angle and the heading angles were different. The maximum load factor corresponding to this maneuver was about 2 and the maximum'vehicle roll-pitch attitudes were about 30 degrees. This solution is currently being studied in further detail with a view t o testing on a manned simulation of rotorcraft. Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3455 Conclusions This paper presented a preliminary study of the rotorcraft pursuit-evasion problem in nap-ofthe-earth flight. Two games of increasing complexity were discussed. The first of these uses a kinematic vehicle model, while the second approach employs a full point-mass model. In the first case, the solutions were obtained by integrating the state- costate system backwards in time from the h a l position of the evader. The solution for the second differential game was obtained via feedback linearization. in this case, the vehicle point-mass models were f i s t transformed into a linear, time-invariant form and the differential game was solved in transformed coordinates. The solution was transformed back t o the original space to yield nonlinear pursuit-evasion guidance laws. Sample extremals for both these cases were presented. The solution t o the first pursuit-evasion problem is useful for developing pilot decision aids for rotorcraft pursuit-evasion, while the solution to the second differential game is useful for onboard implement a t ion. Acknowledgement Partial research support for the first two authors under NASA Grant NAG 2 - 463 from Ames Research Center is gratefully acknowledged. References [I] Cheng, V. H. L., and Sridhar, B., "Considerations for Automated Nap-of-The-Earth Rotorcraft Flight", Proceedings of the 1988 American Control Conference, Atlanta, GA, June 15-17,1988. [2] Sridhar, B., and Phatak, A. V., "Simulation and Analysis of Image-Based Navigation System for Rotorcraft Low-Altitude Flight", AHS National Specialist's Meeting on Automation Applications of Rotorcraft, Atlanta, GA., April 4-6, 1988. Menon, P. K. A., and Sridhar, B., "Passive Navigation Using Image Irradiance Tracking", AIAA Guidance, Navigation, and Control Conference, August 14-16, 1989, Boston, MA. Cheng, V. H. L., "Concept Development of Automatic Guidance for Rotorcraft Obstacle Avoidance", IEEE Transactions on Robotics and Automation, Vol. 6, No. 2, pp. 252-257, April 1990. Menon, P. K. A., Kim, E. and Cheng, V. H. L., "Helicopter Trajectory Planning Using Optimal Control Theory", Proceedings of the 1988 American Control Conference, Vol. 2, Atlanta, GA., June 15-17, 1988, pp. 14401447. Kim, E., "Optimal Helicopter Trajectory Planning for Terrain Flight", Ph. D. Dissertation, Georgia Institute of Technology, Atlanta, GA., December 1989. Isaacs, R., Differential Krieger, New York, 1975. Games, Robert Friedman, A., Differential Games, WileyInterscience, New York, 1971. Rajan, N., and Ardema, M. D., "Barriers and Dispersal Surfaces in Minimum Time Interception", Journal of Op tirnization Theory and Applications, Vol. 42, No. 2, February 1984, pp. 201-228. [lo] Menon, P. K. A., "Short Range Nonlinear Feedback Strategies for Aircraft PursuitEvasion", Journal of Guidance, Control, and Dynamics, Vol. 12, No. 1, Jan.-Feb. 1989, pp. 27-32. [ll]Hunt, L. R., Su, R. and Meyer, G., "Global Transformations of Nonlinear Systems", IEEE 'Transactions on Automatic Control, Vol. AC-28, No. 1, January 1983, pp. 24-30. [12] Bryson, A. E. and Ho, Y. C., Applied Optimal Control, Hemisphere, New York, 1975. Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3455 [13] Bogen, R., et al., MACSYMA Reference Manual, The Mathlab Group Laboratory for Computer Science, M.I.T., Cambridge, MA, Version 10, 1983. [14] Menon, P. K. A., Calise, A. J., and Leung, S. K. M., "Guidance Laws for Spacecraft Pursuit -Evasionw, AlAA Guidance, Navigation, and Control Conference, Minneapolis, Minnesota, August 15-17, 1988. (151 Menon, P. K. A., and Duke, E. L., "TimeOptimal Aircraft Pursuit-Evasion with A weapon Envelope Constraint", Proceedings of the 1990 American Control conference, May 23-25, San Diego, CA, pp. 2337-2342. - tH \ r TANGENT PLANE A Fig.1. The Coordinate System for Kinematic Vehicle Models Fig. 2. Pursuer-Evader Trajectories Purrua ....-..-.. Evdcr Fig.3. Altitude Hbtorier for the Pursuer and the Evader Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3455 X Fig.4. The Coordinate System for Point-Mass Vehicle Models n '= (=) Fig.0. Altitude Histories for the Pursuer and the Evader Fig.6. Pursuer-Evader Trajectories

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