Innovations in Intelligent Machines 1 - Javaan Singh Chahl et al (Eds) part 8 pdf

132 A. Pongpunwattana and R. Rysdyk 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Time (sec) Score α F max α min F Fig. 11. Normal task score profile Table 1. Planner parameters Parameter Description Value α F min minimum task score weighting factor 3500 α F max maximum task score weighting factor 7000 α V vehicle cost weighting factor 2500 α Q fuel cost weighting factor 300 α L loss function scaling factor 250 Table 2. Simulation parameters Parameter Description Value Unit η O obstacle payload effectiveness 0.5– R O obstacle payload range 30 km σ O obstacle uncertainty radius 20 km σ G target uncertainty radius 20 km η V vehicle payload effectiveness 0.7– R V vehicle payload range 20 km V a vehicle speed 150 m/s Fuel init vehicle initial fuel level 6.0 liters Fuel max vehicle max fuel level 7.0 liters It simulates effects caused by actions or events occurring during the simu- lation. Simulation entities are customizable and may be used to represent a variety of air and ground vehicles, targets, and buildings. Unless otherwise specified, the values of parameters listed in Table 1 and 2 are used in all of the simulations presented here. Evolution-based Dynamic Path Planning for Autonomous Vehicles 133 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 1 Longitude (deg) Latitude (deg) step = 0 time = 0 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 1 Longitude (deg) Latitude (deg) 1 1 1 step = 15 time = 0 (a) (b) 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 1 Longitude (deg) Latitude (deg) 1 1 1 step = 30 time = 0 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 1 Longitude (deg) Latitude (deg) 1 1 1 step = 40 time = 0 (c) (d) Fig. 12. Snapshots of off-line path planning with multiple targets Inthisexampleproblem,therearethreetargetsandthreeobstacles.Figure 12 shows snapshots of the planning results for different generations in the evolution process. The vehicle is represented by a triangle with vehicle number on it. The dashed circlearound thevehicle represents the range of the payload on-board the vehicle. This payload can be a sensor or offensive payload. The square markers represent actual locations of sites. Each of these square markers will have a vehicle number on it if the site is a target and assigned to that vehicle. A solid circle located near each square marker represents an area which covers all of the possible locations of the site represented by the square marker. Each filled square marker with a dashed circle around it represents a site with defensive capabilities which can destroy or change the health states of vehicles if they are within the area marked by the dashed circle. The goal location where the vehicle is required to be at the end of the mission is represented by a hexagram in the plots. This representation of the scenario in the plots is also used in all other planning examples. The results show the ability of the planner to generate an effective path to visit all the assigned targets and avoiding collision with the 134 A. Pongpunwattana and R. Rysdyk 0 20 40 60 80 100 120 140 160 180 200 100 120 140 160 180 200 220 240 260 280 300 Generation Expected value of loss function Fig. 13. Evolution of the loss function of the candidate path in each generation Current spawn point Best path Next spawn point Follow this trajectory Current spawn point Fig. 14. Concept of dynamic path planning algorithm which retains the knowledge gained from the previous planning cycle obstacles. Figure 13 shows that the expected value of the loss function decreases dramatically in the early generations. The path planner then fine tunes the resultant path in later generations. 4.2 Algorithm for Dynamic Planning Dynamic path planning is a continuous process. A diagram describing the concept of the dynamic path planning is shown in Figure 14. The planning problem in each cycle is a similar problem to that in the previous cycle. This approach attempts to preserve some information of the past solutions and Evolution-based Dynamic Path Planning for Autonomous Vehicles 135 uses it as the basis to compute new solutions even though the new problem is slightly different from the previous problem. This takes an advantage of evolutionary algorithms that several candidate solutions are available at any time during the optimization process. The planner of the vehicle continually updates its path while the vehicle is moving in the field of operation. The planning process starts with the static path planning process to generate an initial population P 0 and find the first best candidate path Q(0) ∈ P 0 depicted as the black path in Figure 14. The location of the first spawn point is at the desired vehicle position ¯z V (s 1 )at time t s 1 specified by the path Q(0). The following steps in the dynamic path planning algorithm are described below 1. Generate a new population P i+1 from the current population P i by updat- ing all the paths in the current population to begin at the location of the next spawn point. The paths are modified by removing a small number of segments from the start of the paths and adding other segments to join the paths to the spawn point. 2. Run the static planning algorithm continuously to update the population and to find the best candidate path. 3. Send the updated candidate path to the vehicle navigator once the vehicle reaches the current spawn point. 4. Update the estimates of the locations of sites in the environment. 5. Return to step 1 To demonstrate dynamic path planning, we revisit the scenario presented in the last planning example. Starting from the off-line planning result shown in frame (d) of Figure 12, the results of dynamic planning are shown in Figure 15. Frames in the figure show snapshots of the simulation at various simulation time steps. In this simulation, the vehicle is assumed to have an on-board radar which can improve the estimates of nearby sites’ locations. The radar can detect a site within the range of 60 kilometers with 40 meters standard deviation. During the simulation, the planning algorithm updates the candidate path every 10 seconds of the simulation time. The size of the execution time horizon, which is the time difference between two consecu- tive spawn points, is 100 seconds. Since the scenario does not change during the simulation, the dynamically updated path is little different to the off-line planned path. The vehicle follows the path to successfully observe the first two targets, but misses the last target in its first attempt. Frame (c) of Figure 15 shows that the planner is able to quickly update the path to guide the vehicle back to the target and eventually observe the target as shown in Frame (d). 5 Planning with Timing Constraints To incorporate time-of-execution specifications into the path planning prob- lem, the task score weighting factor α F i used in the objective function is defined as a time-dependent function. This time-dependent score weighting function 136 A. Pongpunwattana and R. Rysdyk 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 1 Longitude (deg) Latitude (deg) 1 1 1 step = 9 time = 900 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 Longitude (deg) Latitude (deg) 1 1 step = 22 time = 2200 (a) (b) 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 Longitude (deg) Latitude (deg) 1 step = 46 time = 4600 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 1 Longitude (deg) Latitude (deg) step = 56 time = 5600 (c) (d) 1 1 1 1 1 1 1 1 Fig. 15. Snapshots of dynamic path planning at different time steps. Each observed target is marked with a cross symbol α F i (q) can be used to define a time window for the vehicles to execute each task. This function gives a high positive value during the time period in which we want the vehicle to perform the task. The function gives a small positive value or zero value during the time period in which executing the task does not meet the mission objectives. In this section, we present an example showing the ability of the planner to generate paths which satisfy the imposed timing constraints. The mission objective is to observe a target site which is protected by a nearby defensive site. There are two vehicles each of which has its own path planner. The task of Vehicle 1, which has an offensive payload, is to destroy the defensive site before the beginning of the execution time window of the target site. Vehicle 2, which is equipped with a sensor payload, has to observe the target after the beginning of the the execution time window. That is at 2000 seconds after the mission starts. The duration of the execution time window is 500 seconds. Observing the target site after the expiration time of the execution time window yields Evolution-based Dynamic Path Planning for Autonomous Vehicles 137 a smaller task score. The profiles of score weighting functions of both tasks are given in Figure 16. Figure 17 and 18 show the results of an off-line path planning problem with timing constraints. In this simulation, each vehicle is equipped with a planner which has identi- cal knowledge of the environment and planning parameters. The static off-line planning result in Figure 17 shows that the planner of Vehicle 1 decides to go directly to the defensive site while Vehicle 2 takes a longer path to wait for the expiration of the the execution time period for reaching the target site. 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Time (sec) Score 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 Time (sec) Score 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 (a) (b) Fig. 16. Frame (a) shows the profile of the task score weighting function of the defensive site. Frame (b) shows the profile of the task score weighting function of the target 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 Longitude (deg) Latitude (deg) 1 step = 30 time = 0 1 2 2 Fig. 17. Off-line path planning with execution time window 138 A. Pongpunwattana and R. Rysdyk 0 50 100 150 70 80 90 100 110 120 130 140 150 Generation Expected value of loss function vehicle1 vehicle2 Fig. 18. Evolution of off-line path planning loss function with execution time window To verify that the path planners are capable of generating plans with timing constraints, we ran a simulation starting with the off-line planning results. The dynamic planning simulation results are shown in Figure 19. Frame (b) of the figure shows that Vehicle 1 reaches the defensive site well before the simulation time 2000 seconds and successfully destroys the obstacle, although the vehicle is also destroyed. Frame (c) shows that Vehicle 2 reaches the target site at time 2200 seconds and successfully observes the target. If it is impor- tant for Vehicle 1 to survive, this can be insured by adjustment of the task score weighting function. However, the example illustrates the use of a vehicle in a sacrificial role. 6 Planning in Changing Environment In a changing environment, obstacles and targets may move unexpectedly during the operation. Dynamic planning is essential in this situation. The planner must be capable of replanning during the mission and predicting future states of the sites in the environment. In ECoPS, the site locations and their uncertainties are predicted using Equation 10 and 11. One advantage in using the approximation to the probability of intersec- tion described in Equation 30 or 31 is the ease with which it can be extended to include moving sites. It is the form of the solution which is a summation over a defined function that allows for the simple inclusion of time into the equations. This approach accommodates the integration of uncertainties and dynamics of the environment into the model and the objective function. This section provides two examples of planning in dynamic uncertain envi- ronments. The first example is a scenario with one moving target which is Evolution-based Dynamic Path Planning for Autonomous Vehicles 139 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 Longitude (deg) Latitude (deg) step = 5 time = 500 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 1 2 Longitude (deg) Latitude (deg) 1 2 step = 14 time = 1400 (a) (b) 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 1 2 Longitude (deg) Latitude (deg) 1 2 step = 22 time = 2200 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 1 2 Longitude (deg) Latitude (deg) 1 2 step = 34 time = 3400 (c) (d) 1 2 1 2 Fig. 19. Dynamic path planning with execution time window initially located at position (14.0, 2.5) and later heads west at the speed of 300 kilometers/hour after the vehicle has been moving for 100 seconds. In this example, the radius of the uncertainty circle of each obstacle and target is 10 kilometers. During the off-line planning period, the planner does not have the knowledge that the target will move in the future. The off-line planning result is shown in Figure 20. During the mission, the planner will need to dynam- ically adapt its path to intersect with the predicted location of the target. Frame (a) and (b) of Figure 21 show that the planner is adapting the path to intersect the target at a predicted location. In this simulation, the planner has knowledge of the velocity of the target site. The planner decides to wait until the target moves past the area covered by the top-right defensive site, and the vehicle successfully observes the target as shown in frame (c). The expected value of the loss function during the simulation is shown in Figure 22. The spike in the plot is due to the unexpected movement of the target which causes the planner to temporarily lose track of the target. The value of the loss function drops near zero when the vehicle intersects and successfully observes the target. 140 A. Pongpunwattana and R. Rysdyk 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 Longitude (deg) Latitude (deg) 1 step = 0 time = 0 1 Fig. 20. Off-line path planning result. The planner have no knowledge that the target will move in the future 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 Longitude (deg) Latitude (deg) 1 step = 2 time = 200 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 Longitude (deg) Latitude (deg) step = 11 time = 1100 (a) (b) 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 150 0.5 1 1.5 2 2.5 3 3.5 4 1 Longitude (deg) Latitude (deg) 1 step = 23 time = 2300 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 1 Longitude (deg) Latitude (deg) 1 step = 42 time = 4200 (c) (d) 1 1 1 Fig. 21. Snapshots of dynamic path planning with a moving target Evolution-based Dynamic Path Planning for Autonomous Vehicles 141 0 5 10 15 20 25 30 35 40 45 0 50 100 150 200 250 Time sample Expected value of loss function Fig. 22. Evolution of dynamic path planning with a moving target 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 1 Longitude (deg) Latitude (deg) 1 step = 0 time = 0 Fig. 23. Off-line path planning result. The planner does not have the knowledge that the obstacles will move in the future The second example is a dynamic planning problem with moving obstacles. During the off-line planning, the planner does not have the knowledge that the obstacles will move in the future. The off-line planning result shown in Figure 23 illustrates that the planner is able to find a near-optimal path to go almost directly to the target and the goal location. Almost immediately after the vehicle starts moving from its initial location, the obstacles begin moving north and south to block the vehicle from getting in and out of the area where the target is located. Figure 24 shows snapshots of the dynamically generated path during the simulation. These snapshots show that the vehicle is able [...].. .14 2 A Pongpunwattana and R Rysdyk 4 4 step = 6 time = 600 3.5 Latitude (deg) Latitude (deg) 3 2.5 2 1 1 1. 5 step = 20 time = 2000 3.5 3 2.5 2 1 1 1. 5 1 1 0.5 0.5 0 10 10 .5 11 11 .5 12 12 .5 13 13 .5 14 14 .5 15 0 10 10 .5 11 11 .5 12 12 .5 13 13 .5 14 14 .5 15 Longitude (deg) Longitude (deg) (a) (b) 4 4 step = 27 time = 2700 3 1 2.5 1 2 1. 5 step = 39 time = 3900 3.5 Latitude... 2.5 2 1 1 1. 5 1 1 0.5 0.5 0 10 10 .5 11 11 .5 12 12 .5 13 13 .5 14 14 .5 15 0 10 10 .5 11 11 .5 12 12 .5 13 13 .5 14 14 .5 15 Longitude (deg) Longitude (deg) (c) (d) Fig 24 Snapshots of dynamic path planning with moving obstacles to avoid collision with the obstacles and successfully observe the target and finally reach the goal location The expected value of the loss function at each time step is given in Figure... 5(3) :16 9 19 1, June 20 01 11 J Holland Adaptation in Natural and Artificial Systems PhD thesis, University of Michigan, Ann Arbor, MI, 19 75 12 L E Kavraki et al Probabilistic roadmaps for path planning in highdimensional configuration spaces IEEE Transactions on Robotics and Automation, 12 (4):566– 580 , 19 96 13 S M LaValle and J J Kuffner Randomized kinodynamic planning In Proceedings of IEEE International... NJ, 19 96 G Song et al Customizing PRM roadmaps at query time In Proceedings of IEEE International Conference on Robotics and Automation, 20 01 S Thrun et al Map learning and high-speed navigation in rhino In Artificial Intelligence and Mobile Robots MIT Press, Cambridge, MA, 19 98 J Xiao et al Adaptive evolutionary planner/navigator for mobile robots IEEE Transactions on Evolutionary Computation, 1: 18 28, ... symmetric and satisfy triangle inequality Algorithms for Routing Problems Involving UAVs 15 1 UAV destination Fig 2 Find the minimum spanning tree UAV destination Fig 3 Double the edges in the minimum spanning tree to construct an Eulerian graph 1. 5-Approx Algorithm Christofides in [3] reduced the approximation factor from 2 to 1. 5 by coming up a better way of constructing the Eulerian graph from the minimum... References 1 R Rysdyk A Pongpunwattana Real-time planning for multiple autonomous vehicles in dynamic uncertain environments Journal of Aerospace Computing, Information, and Communication, 1( 12): 580 –604, 2004 2 J L Bander and C C White A heuristic search algorithm for path determination with learning IEEE Transactions of Systems, Man, and Cybernetics – Part A: Systems and Humans, 28 :13 1 13 4, January 19 98 3... Intelligence (SCI) 70, 14 7 17 2 (2007) c Springer-Verlag Berlin Heidelberg 2007 www.springerlink.com 14 8 S Rathinam and R Sengupta Minimum distance to travel from A to B ≠ minimum distance from B to A A B Fig 1 An example that illustrates the asymmetry in resource allocation problems involving UAVs by the vehicle is minimum is called the Travelling Salesman Problem (TSP) TSP is known to be NP-Hard [1] ,[2] That... and Automation, 19 99 14 A Mandow et al Multi-objective path planning for autonomous sensor-based navigation In Proceedings of the IFAC Workshop on Intelligent Components for Vehicles, pages 377– 382 , 19 98 15 C S Mata and J S Mitchell A new algorithm for computing shortest paths in weighted planar subdivisions In Symposium on Computational Geometry, pages 264–273, 19 97 16 R R Murphy Introduction to AI... Press, 2000 17 N J Nilsson Principles of Artificial Intelligence Tioga Publisher Company, Palo Alto, CA, 19 80 18 M H Overmars and P Svestka A probabilistic learning approach to motion planning In Goldberg, Halperin, Latombe, and Wilson, editors, Algorithmic Evolution-based Dynamic Path Planning for Autonomous Vehicles 19 20 21 22 23 24 25 26 14 5 Foundations of Robotics, The 19 94 Workshop on the Algorithmic... minimum spanning tree This gave rise to the following 1. 5-Approx algorithm for SVP: 1 Find the Minimum Spanning Tree (MST) of the graph G This step is same as the step 1 presented in the 2-Approx algorithm (Fig 2) 2 Find the minimum cost perfect matching (PM) on all the odd degree vertices in the MST (Fig 6) 3 Add all the edges in MST and PM to get an Eulerian graph (Fig 7) 4 Find an Eulerian walk on this . 0 (a) (b) 10 10 .5 11 11 .5 12 12 .5 13 13 .5 14 14 .5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 1 Longitude (deg) Latitude (deg) 1 1 1 step = 30 time = 0 10 10 .5 11 11 .5 12 12 .5 13 13 .5 14 14 .5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 1 Longitude. 500 10 10 .5 11 11 .5 12 12 .5 13 13 .5 14 14 .5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 1 2 Longitude (deg) Latitude (deg) 1 2 step = 14 time = 14 00 (a) (b) 10 10 .5 11 11 .5 12 12 .5 13 13 .5 14 14 .5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 1 2 Longitude. = 11 time = 11 00 (a) (b) 10 10 .5 11 11 .5 12 12 .5 13 13 .5 14 14 .5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 1 Longitude (deg) Latitude (deg) 1 step = 23 time = 2300 10 10 .5 11 11 .5 12 12 .5 13 13 .5 14 14 .5 15 0 0.5 1 1.5 2 2.5 3 3.5 4 1 Longitude