EVOLUTIONARY PATH PLANNING FOR AUTONOMOUS AIR VEHICLES USING MULTIRESOLUTION PATH REPRESENTATION

Transcription

1 IEEE International Conerence on Intelligent Robots and Systems (IROS), Wailea, HI, 2002 EVOLUTIONARY PATH PLANNING FOR AUTONOMOUS AIR VEHICLES USING MULTIRESOLUTION PATH REPRESENTATION Ravi Vaidyanathan 1,3*, Cem Hocaoglu 2,Troy S. Prince 1, Roger D. Quinn 3 1 Orbital Research Inc., Cleveland, OH, USA 2 State University o New York at Stony Brook, Stony Brook, NY, USA 3 Case Western Reserve University, Cleveland OH, USA Abstract There is a recognized need or automated path planning or unmanned air vehicles (UAVs) and guided munitions. Evolutionary programming approaches provide an alternative to classical unctional optimization methods with the capability o incorporating a variety o optimization goals, while tolerating vehicle constraints. In this work, we introduce an evolutionary light path planning algorithm capable o mapping paths or reelying vehicles unctioning under several aerodynamic constraints. An air-to-ground targeting scenario was selected to demonstrate the algorithm. The task o the path planner was to generate inputs lying a munition to a point where it could ire a projectile to eliminate a ground target. Vehicle light constraints, path destination, and inal orientation were optimized through itness evaluation and iterative improvement o generations o candidate light paths. Evolutionary operators comprised o one crossover operation and six mutation operators. Several cases or air-to-ground vehicle targeting have been successully executed by the evolutionary light path planning algorithm under challenging initial conditions. A easible path is typically ound rapidly (<100 generations), with urther optimization (3000 generations) insuring a strike very near target center. These results clearly demonstrate that evolutionary optimization using can achieve light objectives or air vehicles without violating limits o the aircrat. I. Introduction Autonomous ree-lying entities such as unmanned aircrat, cruise missiles and other precision guided air vehicles are increasingly dependent on automatic control. One highly relevant scenario is the need or automated path planning within critical vehicle constraints or unmanned air vehicles (UAVs) and guided munitions or target-seeking and/or goal orientation operations. Necessary eatures o such a system include: rapid autorouting capability, incorporation o vehicular and environmental constraints into path generation, and the ability to eiciently perorm path planning unctions within strict time constraints [1]. Current path planning technologies do not ully achieve uture demands or guided air vehicles [1,2]. Speciic deiciencies include: Many autorouters only address higher-level path planning behavior; vehicular dynamics and constraints are not ully considered [1,2] Many existing path planning methods are o little use in situations involving vehicle constraints, resulting in bottle-necks with many local minima [3], The need or autorouting or 6 degree o reedom (do) platorms under strict vehicle constraints has not been ully addressed [4], Conventional path planners typically use a ixed path representation; such methods cannot sel-adjust to problem complexity [5], Accounting or time-varying or non-holonomic constraints in dynamic environments oten hampers planner perormance [5], Optimization criterion oten are complex and subjective; many traditional planners optimize only with respect to the shortest path [5] without ull consideration o critical constraints. Global planning approaches [6,7] are in general complete in that i a path exists, it will be ound, yet the computational expense grows with vehicle complexity. Although local methods such as potential ield procedures [8,9] are more eicient, they can become trapped in local minima. In general, unctional optimization methods have tendencies to become trapped within local minima, and may be diicult to implement on ree-lying platorms since complexity grows exponentially with vehicle degrees o reedom [10]. Furthermore, enumerative techniques are susceptible to ineiciencies when dealing with complex vehicles due to size o search space. A class o heuristic searching methods based upon simulated evolution known broadly as Genetic Algorithms (GA) has become very popular lately or discrete optimization problems characterized by many local minima in nondierentiable, discontinuous or constrained problem spaces. These evolutionary techniques are population-oriented: successive populations o easible solutions are generated in a stochastic manner ollowing laws similar to that o natural selection. This contrasts standard programming techniques that normally ollow a single trajectory repeatedly until a satisactory solution is reached. Many previous path planners cannot accommodate a variety o optimization criteria or allow changes in these standards without changing the characteristics o the planner or the search map. Evolutionary approaches, on the other hand, can handle a variety o optimization goals and are very tolerant to the orm o the evaluation unction. Functions to be optimized need not be *corresponding author: raviv@orbitalresearch.com

2 IEEE International Conerence on Intelligent Robots and Systems (IROS), Wailea, HI, 2002 dierentiable or continuos. Evolutionary path planning approaches are also lexible to changes in environment and are robust to uncertainties. Evolutionary approaches provide an alternative orm o path generation capable o incorporating a variety o optimization goals, while tolerating vehicle constraints. These beneits have lead to the development o several evolutionary path planners (a brie summary is presented in 5 ). In this work, our past research [5,11,12] in multidimensional path planning is expanded to create an evolutionary light-path planning algorithm capable o mapping paths or ree-lying vehicles unctioning under aerodynamic constraints. The generation o light trajectories or air-to-ground targeting or autonomous munitions is selected as a benchmark situation to demonstrate the utility o the path planning genetic algorithm. The organization o this paper is as ollows: in Section II the dynamics o a candidate autonomous air vehicle are modeled, Section III gives a description o the representation used or encoding paths, Section IV enumerates work perormed within evolutionary evaluation unction to incorporate aerodynamic constraints and light objectives, Section V delineates the genetic operators o the path planning algorithm, Section VI summarizes the operation o the path planner or airto-ground target seeking, Section VII presents results o the algorithm when implemented on a candidate aircrat, and Section VIII briely discusses conclusions, ongoing, and uture work. II. Air Vehicle Model Development The general equations o motion o a 6 degree o reedom (do) rigid airrame may be described through Newton s Laws in terms o the nomenclature enumerated in Table 1: s& ω F = m + s& t (1 a, b) ( I ω ) M = + ω ( I ω ) t Aerodynamic orces acting on an air vehicle, are oten expressed in the orm 13 : F ( A = [ C z )][ Q ] [ ( (2 a, b) M A = Cm z )][ Qm ] Where both [C] matrices are dimensionless coeicients which are unctions primarily o aircrat state z = (V, α, β, p, q, r), and each [Q] is a product o light dynamic pressure, and aircrat reerence area or characteristic length, respectively. The system inputs, u(t), include aerodynamic orces developed by actuator delections and propulsive orces, and environmental eects, whose impact on the air vehicle may be relected in state space orm as: Z X Y Figure 1 z & = Az + B u (3) The essential mode o operation or air vehicle autopilot systems is to move control suraces (δp, δq, δr) in response to desired roll rate (p), pitch rate (q), and yaw rate (r) commands based on a linearized airrame model [13]. The evolutionary algorithm also plans unctions or the aircrat based upon a linear airrame response. The actual control inputs to the system (i.e. control surace delections) are generated and modiied to achieve desired targeting objectives. Additionally, since the algorithm also changes the time o each control input, the minimum time step is constrained by the requency response characteristics o the actuators themselves. These inputs are integrated over time to produce the time history o the state variables, including aircrat roll, pitch and yaw rates. During operation, these rates orm inputs to the autopilot, which directs the aircrat to ollow the developed trajectory As a case study or evolutionary light path generation, an air-to-ground targeting scenario was chosen. In this scenario, a small airborne autonomous munition ires a z Projectile Trajectory Variable Parameter Description F Total orce vector acting on airrame M Total moment vector acting on airrame s Position vector o mass center o airrame ω Angular velocity o airrame (body-ixed) m Air vehicle mass I Inertia matrix o air vehicle δp, δq, δr Elevator, aileron, and rudder delections V Absolute vehicle airspeed (global) u Forward velocity (body centered) v Side velocity (body centered) w Downward velocity (body centered) α Angle o attack=tan -1 (w/u) β Sideslip angle=tan -1 (v/u) p Angular roll rate q Angular pitch rate r Angular yaw rate Table 1 Nomenclature x y

3 IEEE International Conerence on Intelligent Robots and Systems (IROS), Wailea, HI, 2002 kinetic energy projectile straight downward rom its center o gravity to strike a ground target. The iring action o the munition is depicted in Figure 1, with global and body ixed reerence rames delineated, along with the projectile trajectory. The task o the path planner was to generate a set o inputs which would ly the munition to a point where the projectile trajectory would strike as near as possible to the target center. Note that the angular orientation o the vehicle as well as its position is critical or proper target strike. Thus, unlike the majority o munition targeting where only global position at strike is relevant, this problem is a six do optimization. Also, or realistic depiction o targeting scenarios, the aircrat was orced into glide mode (orward thrust could not be generated), to provide a state analogous to endgame situations where engines delay is unreliable to reach a target at close range. To develop realistic light model path planning tests, the Flight Dynamics and Control Analysis (FDC) 1.2 toolbox or the MatLab/Simulink environment was used [14] The FDC toolbox includes a complete non-linear model o a DHC-2 Beaver; a light, single engine, high wing aircrat. This model was modiied to improve responsiveness, and more closely resemble the light characteristics o autonomous airborne munitions. The linearized model comprised o equations (1), (2), and (3) was implemented within the genetic algorithm to evaluate system light perormance during evolutionary optimization. The planner modiied control surace inputs, and time step between inputs (limited by actuator responsiveness) to produce an eective light path to the target. In light, roll, pitch, and yaw rates resulting rom the control inputs may be ed directly to an autopilot to guide the air vehicle along the path. III. Multiresolution Flight Trajectory Representation A undamental aspect o any heuristic optimization is the speciic structure o encoded inormation enabling iterative progression towards an ideal solution. For air vehicle light trajectory generation, a multiresolution binary tree representation is implemented such that complex paths o higher dimensionality to allow eicient evolutionary operation over generations o light trajectories. Multiresolution methods have been used in signal and image processing to provide eicient data representation adapted to the complexity o the signal content, as well as in our own past work [5,11,12] or path planning o autonomous robots with less stringent motion constraints. Furthermore, tree-like representations have seen extensive use in genetic programming [15], particularly in situations where mutation is relied on to play a signiicant role in evolution [16,17,18]. The algorithm implemented or automatic routing o autonomous air vehicles in this work utilizes a related approach to path representation. In addition to accurate path encryption, the representation can reduce expected search lengths or trajectory generation. I a successul path is ound early in the search hierarchy urther expansion o that portion is not necessary; the structure o the binary tree is thus internally optimized based upon problem complexity. This advantage is mapped into the search space and the string length is adjusted accordingly, enhancing precision and computational eiciency. Within the binary tree, air vehicle light paths are represented by hierarchically ordered nodes, each containing a speciic set o control surace delections, i n = (δp n, δq n, δr n ), accompanied by a time shit, t n, to orm a complete array o parameters, u n, at each node. t n represents the delay rom the last input i n-1, to the current input i n. The tracking o all such nodes maps the total set o control inputs and time intervals piloting the air vehicle over the light trajectory. These nodes are organized in a tree-like structure, as shown in the let section o Figure 2. The complete path rom start to inish is relected by this binary tree, with the n th node corresponding to the input u n, which is comprised o the control surace delection i n, occurring at the time t = j = n j= 0 t j. The in-order traversal o this binary tree provides a sequence o knot points representing all control inputs over the light trajectory rom start to inish, as described by each P i in the right section o Figure 2, shown with start (S) and end (G) points. Intermediate nodes in the igure are generated randomly, and may subsequently be perturbed, inserted, or removed to modiy and optimize the path. The sequence o nodes deined by the bottom o Figure 2 is:(s = start, G = goal):s P 4 P 3 P 5 P 2 P 6 P 1 P 8 P 9 P 7 G.) Although the number o nodes varied rom path to path, no particular constraints or limits based on size were implemented with regard to node selection or crossover or mutation. The trajectories represented through multiresolution binary trees do not correspond directly to actual aircrat light paths; rather they decode into a set o piloting commands, which may be passed directly to an autopilot or the air vehicle to maneuver itsel along the light trajectory. Large arrays o candidate light trajectories u u 2 2 u 4 u 5 6 u 6 u u 8 9 u 9 u 7 0 S P 4 1 Figure 2 P P 5 P P 6 P 1 6 P P 7 10 G 8 P 9

4 IEEE International Conerence on Intelligent Robots and Systems (IROS), Wailea, HI, 2002 may be created rapidly using multiresolution path representation by generating and ordering random sets o control surace inputs ( i), along with random time intervals ( t) between inputs. Provided each i alls within acceptable control surace delections, and each t is greater than or equal to the system actuator delay, the light path will be achievable within vehicle actuator constraints. Vehicle light dynamics and constraints, as well as path destination and inal orientation, may be optimized through proper itness evaluation and iterative improvement o collections o multiresolution light paths. IV. Evolutionary Fitness Evaluation Given a generation o candidate multiresolution light trajectories or air vehicle targeting, some method o measuring positive and negative aspects o paths is critical or the selection o paths or genetic union (crossover) and mutation. This measure must relect both the validity o the light path in relation to violations o the vehicle s light envelope and maintenance o light stability, and or target strike accuracy. This problem cannot be solved by a single perormance measure. Such situations are also commonly encountered outside the ield o aircrat control; many real-world problems are composed o a set o variables that must be speciied, and a set o constraints which restrict the those variables [19]. The concept o multiobjective evolutionary algorithms have been described [20,21,22,23] as a modiication o the standard genetic algorithm at the selection level to attack such constraints, in particular or aircrat [24] i. The main dierence between a conventional evolutionary algorithm and a multiobjective genetic algorithm resides in itness assignment [22]. In a review o evolutionary approaches to multiobjective optimization, Fonseca and Fleming [25], categorize multicriteria evolutionary algorithms and compare itness assignment strategies. In particular, they distinguish plain aggregating approaches, population-based non-pareto based approaches, and Pareto-based approaches. Aggregation methods combine the objectives into a higher scalar unction that is used or itness calculation. Population aggregation methods include: the weightedsum approach, target vector optimization, and the method o goal attainment [25]. Population-based non-pareto approaches are able to evolve multiple nondominated solutions concurrently in a single simulation run. By changing the selection criterion during the reproduction phase, the search is guided in several directions simultaneously [25]. Pareto-based itness was proposed in [26]. All approaches o this type use Pareto dominance in order to determine the reproduction probability o each individual [23]. A major advantage o the Pareto-based itness approach is the lack o sensitivity to nonconvexity i Note that this work ocused on wing design as opposed to path planning in Pareto-optimal sets [25]. An aggregation method o assigning itness scoress was implemented in this work, primarily due to the act that only a single solution was desired. Three itness stages were summed or scalarization. Prior to itness assignment actuator commands represented by the binary tree must irst be decoded into a mapping o the light motion o the aircrat. The decoding process begins rom the air vehicle initial state (z 0 ), to the time ( t 1 ) o the irst actuator input ( i 1 ) by: 1) solving and integrating equation (3) rom t=0 t 1 with input u= i 0 to obtain all aircrat states (z) over the interval, 2) solving equations (2 a & b) using the derived states (z) to determine all applied orces (F) and moments (M) rom t=0 t 1, and 3) solving and integrating equation (1 a & b) to obtain a global map o all aircrat translational positions (s) and angular orientations (ω) rom t=0 t 1. The end result o this decoding is the set o all states, global positions, and angular orientations o the aircrat ollowing the light trajectory rom starting position to the irst node on the multiresolution path. This process is repeated rom node 1 to node 2 or t= t 1 t 1 + t 2, with the actuator input u= i 1, and continues until the interval t = j = n 1 j = 1 t j j = n j = 1 t j with input u= i n-1 is reached covering node (n-1) to node n. In the inal decoding step, the actuator delay, t min is added to the time at node n to provide integration limits rom node n to end node G or the inal input i n, hence providing a map o aircrat transnational positions, angular orientations, and light states over the entire multiresolution light path. Based upon the generated light map, the aircrat path planner perorms evolutionary itness evaluation in three stages or the inclusion o all vital optimization criteria: Stage 1 evaluates light paths purely in relation to maintaining light stability and avoiding light envelope violations, Stage 2 assesses path itness in terms o noncritical light parameters; i.e. constraints whose violation will not cause loss o aircrat control, yet whose limits ideally should not be exceeded, while Stage 3 analyzes the path with respect to light objectives by combining three actors: 1) proximity o path to desired goal, 2) orientation o aircrat, and 3) time o light. The total evolutionary itness or a given path is the weighted sum o scores rom each stage, each o which is scaled according to its importance to light goals. In its implementation or ground vehicle targeting, a light envelope consisting o limits on all light parameters (V, α, β, p, q, r, ψ, θ, φ, x e, y e, H) was developed based upon the aircrat s dynamics; lying within these limits maintains light control and stability. Stage 1 o itness evaluation analyzed the decoded light map or violations o these limits providing itness score S1:

5 IEEE International Conerence on Intelligent Robots and Systems (IROS), Wailea, HI, 2002 S i 1 = = k V i * ρ 1 i= 1 (4) where the summation represents the total number o states whose limits were exceeded, k is the amount o states, and ρ 1 is a scaling constant or normalization. For these experiments, the only relevant parameter in Stage 2 itness were altitude limits given by: [ V ] (5) S = wher 2 * ρ 2 e the quantity in brackets relects light path altitude violations, and ρ 2 provides normalization. The air vehicle ideally should not inringe these altitude limits (e.g. too low exposes the plane to ground ire, too high causes uncertainty in targeting), but in the event that no alternative options exist, these limits may be exceeded. Three actors comprised the Stage 3 itness: S = D p)* ρ + O( p)* ρ + T( p)* (6) D(p) 3 ( ρ33 measures the distance (Euclidean norm) rom the targeting point o the vehicle to the center o the target. D( p) = c (7) where c is a vector containing the target center coordinates, and is a vector o the current target o the aircrat given by: *(tan ˆ tan ˆ x cos ˆ sin ˆ ) e + H θ ψ + φ ψ = (8) *(tan ˆ cos ˆ tan ˆ sin ˆ ) ye + H φ ψ + θ ψ The subscript indicates states at the inal point o light and the ^ symbol represents the Euler angles transormed into angular orientations on the body ixed rame. Since a hit rom directly above the target is preerable to one at an angle, O(p) is included in Stage 3 as: 1 1 ( ) d O p = tan (9) H where d is the vector dierence between the target center and the point directly under the aircrat projected onto the plane o the target. O(p) thus approaches 0 as the angle or the target trajectory approaches 90. The inal itness measure in Stage 3 T(p) is simply the normalized time o light. Note that ρ 31, ρ 32, and ρ 33 are scaling constants. Thereore, or a path (p) represented by an ordered set o nodes rom beginning to end, the total evolutionary itness score, F(p) is deined as scalar the sum o each o the stages, or: F p ) = S + S + (10) ( 1 2 S 3 V. Evolutionary Operators The use o multiresolution representation with variable length encoding or air vehicle trajectories introduces many unique opportunities or evolutionary operation with respect to the binary tree structures. Given that each node u, represents a set o control inputs occurring at a speciied time with respect only to the preceding node, intermediate nodes may be reely interchanged with those rom binary trees o other candidate trajectories or perturbed within the same tree without any loss o generality in path representation. Evolutionary operators utilized by the light path planner expanded upon our past work 5 to generate one crossover operator, ive random mutation operators, and one intelligent mutation operator suitable or aircrat targeting trajectory optimization. Speciic operations consisted o: Swap-Subtree Crossover: combines two paths to reproduce two new ones by selecting a node at random rom each parent, and exchanging subtrees branching rom that node in their ospring. Figure 3 delineates crossover operation, (one parent in white; its mate in black), where the X marked nodes are selected.. Swap Subtrees Ospring - Genetic Union Parents Figure 3 Perturb 1 Mutator: randomly selects a node (n) in the binary tree and perturbs its contents ( u n ) a small amount. This operator allows or ine tuning o acceptable, but not ideal, paths. Perturb 2 Mutator: randomly selects a node (n) in the binary tree and perturbs its contents ( u n ) a large amount. This mutation makes signiicant alterations in light trajectory, ideally to change an ineasible path into a easible one, or to move an inaccurate path much closer to its target. Swap-Node Mutator: exchanges the contents ( u) o two randomly picked nodes in the binary tree, shown in Figure 4. Insert-Node Mutator: creates a new intermediate path input ( u), by inserting a node into the binary tree, shown in Figure 5. Flip Mutator: changes the sign o i within a randomly picked node Fix Mutator ( intelligent mutation): operates in a similar manner to the Perturb 2 Mutator, except that its Swap nodes Figure 4 Figure 5 Inserts new node

6 IEEE International Conerence on Intelligent Robots and Systems (IROS), Wailea, HI, 2002 operation is not random. Within the selected binary tree, inputs u are changed speciically in relation to states violating light constraints. For example, a binary tree mapping a light path with too great a pitch angle, will be ixed by manipulating delections within i such that θ is reduced or that section o the light. The operation o this mutation is designed to ix input/output relationships o completely ineasible paths by altering the input and time o input, orcing the path into easible regions. VI. Evolutionary Target Seeking Implementation Process Figure 6 outlines the implementation process or the evolutionary generation o ground targeting trajectories or air vehicles. Since certain evolutionary operators (Section V) are best suited to correcting ineasible paths, while others optimize easible paths or greater accuracy, selection probabilities o each are adjusted once a target strike is achieved. The probability or each mutation was selected by trial and error. Two sets o probabilities were selected, one relecting the state o evolution beore a path striking the target was achieved, and one relecting the state ater a single successul path was achieved. A simple binary shit to the second set o probabilities was perormed when a single successul path was achieved. VII. Implementation Results Utilizing the ixed wing aircrat model, evolutionary itness scoring routines, and operators, the multiresolution aircrat path planning algorithm was implemented or optimal trajectory generation or ground target seeking. The movements o three control suraces (elevator, aileron, and rudder) were optimized to develop a trajectory or the vehicle to reach a position such that its iring trajectory would strike a circular land based target o 2.5m radius.. Elementary targeting scenarios were run or very simple targeting tasks in a LINUX C++ programming environment. Following these simple cases, the air vehicle was assigned to strike targets that the air vehicle could not reach without optimizing its angular orientation along with its global position (i.e. the target was located in a position that the air vehicle could not ly directly above). Delay was set at t min =0.15 sec, with input allowable at t=0. Sample results or initial populations varying rom 50 to 70 individuals, a mutation probability o 0.65, and a crossover probability o 0.35 are shown in Figures 7 to 10. Figure 7 shows rontal, lateral, and aerial views o critical munition positions along the path evolved to strike a target 30m directly in ront (X dimension) and 100 m below (Z dimension) the air vehicle s initial position. Dashed lines in the lateral and top views show the projectile trajectory. For this situation, the path planner created a trajectory piloting the vehicle to an orientation allowing accurate target strike (0.027m rom target center) in 0.59 sec o light time. The angular orientations o the aircrat through light (in radians) versus time are plotted in the lower right corner, while a logarithmic (base 10) plot o the best candidate in each generation versus the generation count is shown in the lower let (the lat line indicates the minimum itness score necessary to strike the target). Figures 8, 9, and 10 show successul targeting paths or varying goal positions lateral to the aircrat (Y dimension). The target is not shown in the aerial views in Figures 9 and 10 or better visualization, although the viewable portion o the projectile trajectory is still shown. Speciic inormation or each run is enumerated at the top o each igure. In each o these runs, the target position orced the angular orientations allowing accurate target strike; most notably in Figure 10 where roll and pitch angles are combined to achieve a iring trajectory 0.01m o a target center 10m lateral to plane starting position. VIII. Conclusions The results presented in Section VII demonstrate that evolutionary light path optimization using multiresolution representation can achieve light objectives without violating limits o the aircrat. Several cases or air-to-ground vehicle targeting have been successully executed under challenging initial conditions. In most situations, a easible path is ound rapidly (<100 generations), although urther optimization is necessary to insure a strike very near target center. To our knowledge, this is the only work o its kind optimizing all rotational and translational degrees o reedom o an air-to-ground munition or target strike. Future work plans include Evolutionary Target Seeking Process or Autonomous Air Vehicles Target position T=(X,Y,R) Generate initial population o paths based upon inputs and time intervals: u=(δp,δq,δr, t) Calculate global translational positions (s) along each path rom aerodynamic orces, F = M & s& or s=(x,y,z) and angular orientation (ω) rom aerodynamic moments, T = I & ω & or ω=(ψ,θ,φ)*d Check each path or light envelope violations - record results Initial aircrat state z=(v,α,β,p,q,r) Calculate aircrat state (x) along each path in population rom z & = Az + Bu given initial x and time history or u Was a path achieving target kill within light envelope evolved? YES Alter mutation type and probabilities Calculate aerodynamic orces (F) rom F = C z and aerodynamic moments (T) romt = C m z Score each path based upon light envelope violations and kill probability Check each path or target kill NO Mutate and breed paths based on itness score to create new generation o paths Maintain initial mutation types and probabilities Genetic Algorithm Figure 6: Evolutionary Implementation Process

7 IEEE International Conerence on Intelligent Robots and Systems (IROS), Wailea, HI, 2002 IC: Initial α, β, θ, 2.5m rad. target 30m x, 0m y Evolutionary Evaluation: 500 Generations, Target Output: 0.59 sec light, target pt m o center IC: Initial α, β, θ, 2.5m rad. target 30m x, 5m y Evolutionary Evaluation: 3000 Generations, Target Output: 0.63 sec light, target pt. 2.35m o center Projectile Trajectory Projectile Trajectory Angular Orientation (rad vs. time) Pitch angle Fitness Score (LOG) vs. Generation Angular Orientation (rad vs.. time) Pitch angle Fitness Score (LOG) vs. Generation Roll angle Yaw angle Roll angle Yaw angle Figure 7 Figure 8 IC: Initial α, β, θ, 2.5m rad. target 30m x, -5m y Evolutionary Evaluation: 3000 Generations, Target Output: 0.47 sec light, target pt. 0.09m o center IC: Initial α, β, θ, 2.5m rad. target 30m x, 10m y Evolutionary Evaluation: 3000 Generations, Target Output: 0.45 sec light, target pt. 0.01m o center Projectile Trajectory Projectile Trajectory Angular Orientation (rad vs. time) Pitch angle Fitness Score (LOG) vs. Generation Angular Orientation (rad vs. time) Pitch angle Fitness Score (LOG) vs. Generation Yaw angle Yaw angle Roll angle Roll angle Figure 9 Figure 10 planning or target motion through optimization o intercept point, while current eorts involve the training Acknowledgements The authors grateully acknowledge the support o Mr. Johnny Evers o The Air Force Research Laboratory Munitions Directorate, Orbital Research Inc., and Dr. Arthur Sanderson or acilitating contact between our research groups. Reerences [1] Harman, W.L., Strike Mission: Real-Time Retargeting (RTR) Planning and Optimization, in Precision Strike Mission Area, presentation at the Naval Air and Surace Weaponry Technology BAA Industry Day, Lotus, T., 1998 [2] Helgason, R.V., Hayasuriya, A.C., Kennington, J.L., Lewis, K.H., Shortest Path Algorithms on Grid Graphs with Applications to Strike Planning, Technical Report to the Oice o Naval Research, 1997 [3] Koren, Y., Borenstein, J., Potential Field Methods and their Inherent Limitations or Mobile Robot Navigation, in Proc. IEEE Int. Con. on Robotics & Automation, 1991 [4] Henrich, H.,Wurll, C., Worn, H., 6 DOF Path Planning in Dynamic Environments A Parallel on-line o neural networks to reproduce generated light trajectories or instantaneous reaction. Approach, in Proc. IEEE Int. Con. on Robotics & Automation, 1998 [5] Hocaoglu, C., Sanderson, A.C., Planning Multiple Paths with Evolutionary Speciation, IEEE Tran. on Evolutionary Computation, due in press, 2001 [6] Sharir, M., Algorithmic Motion Planning in Robotics, in IEEE Symp. on Robotics & Automation, 1989 [7] Lozzano-Perez, T., A Simple Motion Planning Algorithm or General Robot Manipulators, IEEE Journal o Robotics & Automation, 3, 1987 [8] Khosla, P., Volpe, R., Superquadratic Artiicial Potentials or Obstacle Avoidance and Approach, in Proc. IEEE Int. Con. on Robotics & Automation, 1988 [9] Rimon, E., Doditschek, D.E., Exact Robot Navigation using Artiicial Potential Fields, IEEE Trans. on Robotics & Automation, 8, 1992 [10] Latombe, J., Robot Motion Planning, Kluwer Academic Publishers, 1991 [11] Hocaoglu, C., Sanderson, A.C., Evolutionary Path Planning using Multiresolution Path Representation, Proc. o IEEE Int. Con. on Robotics & Automation, 1998 [12] Hocaoglu, C., Sanderson, A.C., Multimodal Function Optimization using Minimal Representation

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