Optimal Path Finding for Direction, Location and Time Dependent Costs, with Application to Vessel Routing

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1 1 Optimal Path Finding for Direction, Location and Time Dependent Costs, with Application to Vessel Routing Irina S. Dolinskaya Department of Industrial Engineering and Management Sciences Northwestern University, Evanston, IL Robert L. Smith Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI Transportation Center, Northwestern University, Evanston, IL March 4, 2010

2 Outline 1 Motivation Optimum Vessel Performance in Evolving Nonlinear Wave Fields Project 2 3 4

3 Outline Optimum Vessel Performance in Evolving Nonlinear Wave Fields 1 Motivation Optimum Vessel Performance in Evolving Nonlinear Wave Fields Project 2 3 4

4 4 Motivation Optimum Vessel Performance in Evolving Nonlinear Wave Fields Optimum Vessel Performance in Evolving Nonlinear Wave Fields Project Overview A five-year project funded by the Office of Naval Research (ONR) Multidisciplinary University Research Initiative (MURI) Grant In collaboration with: The Department of Naval Architecture and Marine Engineering at the University of Michigan The Department of Electrical and Computer Engineering at the Ohio State University The Applied Physics Laboratory at the University of Washington

5 5 Motivation Optimum Vessel Performance in Evolving Nonlinear Wave Fields Optimum Vessel Performance in Evolving Nonlinear Wave Fields Project Overview The goal of this project is to develop a system that can control the behavior of a vessel in real-time, based on real-time measurements and forecasts of the wave field surrounding the vessel. Three objectives: Minimize travel time, thus decreasing operational costs Minimize fuel consumption, consequently reducing environmental effects Minimize vessel motions, thus increasing passengers comfort and crew s efficiency

6 Optimum Vessel Performance in Evolving Nonlinear Wave Fields Optimum Vessel Performance in Evolving Nonlinear Wave Fields Project Overview Real-Time Measurement of Ocean Wave Fields Short-Term Forecasts of Evolving Nonlinear Wave Fields Time-Domain Computation of Nonlinear Ship Motions Dynamic Real-Time Path Optimization and Vessel Control

7 Optimum Vessel Performance in Evolving Nonlinear Wave Fields Problem Characteristics The cost function (vessel speed) and operability constraints are not available in closed form, and an optimal path cannot be found analytically. Information about the surrounding environment is available up to the radar visibility horizon. A minimum turning radius function constrains the curvature of feasible paths and problem controllability. Small run-time of an optimal path finding algorithm is essential for real-time implementation.

8 Outline 1 Motivation Optimum Vessel Performance in Evolving Nonlinear Wave Fields Project 2 3 4

9 9 Motivation Problem Statement Given: (s, θ s ) - starting position, (t, θ t ) - target position, R H - radar visibility horizon, V (a, θ a, t a ) - maximum attainable speed for s a R H, R(a, θ a, t a ) - minimum turning radius for s a R H. Find: Fastest path from (s, θ s ) to (t, θ t ).

10 10 Motivation Optimal-Path Finding Dynamic Programming (DP) Model Our objective is to develop computationally efficient and numerically robust algorithms to solve path optimization problems in a time-varying environment. Since the cost function (e.g., vessel speed) is evaluated by simulation models, it is not available in closed form, and an optimal path cannot be found analytically.

11 11 Motivation Optimal-Path Finding Dynamic Programming (DP) Model We discretize the path space into a set of waypoints and construct a dynamic programming model that finds an optimal ordered set of waypoints to traverse from start s to a target point t.

12 12 Motivation Dynamic Programming Aspects to be Addressed 1 Complete information about the environment (i.e., cost function) is not available. 2 Traditional optimal-path finding dynamic programming (DP) approaches optimize over a set of piecewise-linear paths, which are control-infeasible. 3 Optimal-path finding in the network with time-dependent costs traditionally calls for adding a time variable to the DP state space, resulting in significant increase in number of states to be considered.

13 Complete information about the environment (i.e., cost function) is not available. There is complete information within the radar (or sensor) visibility horizon, R H. The environment (e.g., wave field) beyond the radar visibility horizon is assumed to be a stationary random process characterized by a global parameter (e.g., sea state).

14 For each instance of the problem, sea state is a fixed parameter describing a stochastic system (wave field). Consequently, the distribution of waves does not change, and the medium beyond R H is assumed to be time and space homogeneous.

15 Vessel Speed Prediction Model for Homogeneous Media Sea is time and space homogeneous characterized by a fixed parameter, sea state. For each sea state, the maximum attainable speed is computed as a function of the heading relative to the dominant wave direction. Figure: Speed Polar Plot

16 16 Motivation Fastest Path in Homogeneous Media 1,2 Optimal path is piecewise linear with at most one waypoint (e.g., st 1 and stz 2 ) 1 Dolinskaya, I. S., Kotinis, M., Parsons, M. G., and Smith, R. L., Optimal Short-Range Routing of Vessels in a Seaway, Journal of Ship Research, Vol. 53, No. 3, September 2009, pp Dolinskaya, Irina and Robert L. Smith, Fastest Path Planning for Direction Dependent Speed Function, Technical Report 08-02, July 20, 2008, University of Michigan, (submitted to Transportation Science).

17 17 Motivation Radar Visibility Horizon in The dynamic programming model evaluates the fastest paths to each point on the border of the radar visible region, then the homogeneous wave field result finds the best path to continue to the target point.

18 18 Motivation As the mobile agent moves along a path, the information about surrounding environment is updated, and an optimal path is continuously reevaluated.

19 19 Motivation System Dynamic Restrictions Traditionally, optimal-path finding and path-following are the separate stages of the problem. Classical dynamic programming model finds the optimal piecewise-linear path. However, in many applications a mobile agent cannot instantaneously change its heading, making piecewise-linear paths infeasible. We integrated the systems operability and dynamics constraints (R(a, θ a, t a )) in the optimization model, which results in a control-feasible solution.

20 20 Motivation Traditional DP applies a straight line path between a pair of waypoints. We find a control-feasible path by integrating heading angle in the state space.

21 Assume local homogeneity at each waypoint and time instance. Find fastest path between a pair of waypoints for a given maximum attainable speed function V max (θ) and minimum turning radius R min (θ).

22 Fastest Path with Bounded Curvature Given a starting location and heading (a, θ a ), and the target location and heading (b, θ b ), find the fastest path from a to b such that the path curvature is limited by the minimum turning radius R(θ). L.E. Dubins 1 found a fastest path with bounded curvature for a constant (direction-independent) speed and minimum turning radius. We extend his results to direction-dependent speed and curvature bounded functions. 1 L. E. Dubins, On Curves of Minimal Length with a Constraint on Average Curvature, and with Prescribed Initial and Terminal Positions and Tangents, American Journal of Mathematics, Vol. 79, No. 3. (Jul., 1957), pp

23 23 Motivation Control Model of the Problem Such that, min J u = T with the boundary conditions: 0 ẋ(t) 2 + ẏ(t) 2 dt = V (α(t)) ẋ(t) = V (α(t)) cos(α(t)), ẏ(t) = V (α(t)) sin(α(t)), α(t) = V (α(t)) R(α(t)) u(t), T (x(0), y(0), α(0)) = (x a, y a, θ a ), (x(t ), y(t ), α(t )) = (x b, y b, θ b ). 0 dt = T

24 Optimal Path with Bounded Curvature 1 System Controllability The problem is reduced to Dubins car problem Existence of an Optimal Path Filippov s theorem for minimum-time problems Necessary Conditions for Optimality Pontryagin s Minimum Principle Theorem 1 Any optimal path is the concatenation of the arcs with minimum turning radius R(θ) and the straight line segments. 1 Dolinskaya, I. S., Optimal Path with Bounded Curvature in a Direction-Dependent Medium, submitted to IEEE Transactions on Robotics.

25 25 Motivation Optimal Path with Bounded Curvature for Convex Speed Polar Plot Theorem 2 In the case of a convex speed polar plot, an optimal path from (s, θ s ) to (t, θ t ) is of the form {C, C, C}, or {C, S, C}, where C denotes a sharpest turn curve and S denotes the straight line segment. It is implied that a path of the form {C, C, C} alternatively switches between clockwise and counterclockwise sharpest turn curves.

26 26 Motivation The state of the system is the location of a mobile agent as well as the heading at which it arrived. For a given waypoint and current heading, the DP selects the waypoint to travel to next and the direction to arrive at. The fastest path with bounded curvature (Theorem 2) gives the minimum time to travel to each of the next waypoints to be considered.

27 27 Motivation Due to the time dependency of the environment and cost function, one might not always want to arrive to each intermediate waypoint as soon as possible. To account for this fact, a time variable is traditionally added to the DP state in order to keep track of all possible times at which we might arrive at, and consequently leave, point a. This additional variable significantly increases the number of DP states to be considered. We present an alternative formulation of the DP functional equation, which allows us to eliminate the time variable from the state space.

28 28 Motivation Time-Dependent DP in Literature Dreyfus 1 was able to eliminate the time variable from the state space for problems where unrestricted waiting is allowed at each waypoint (node). Kaufman and Smith 2 eliminated the time variable from DP state when the cost function satisfies the consistency conditions. Stopping at waypoints is not practical or feasible for many applications, and the consistency condition is too restrictive. We adapt Dreyfus approach by allowing the agent to slow down as needed. 1 Dreyfus, Stuart E. [1969], An Appraisal of Some Shortest-Path Algorithms, Operations Research, Vol. 17, No. 3, pp Kaufman, D. E. and Smith, R. L. [1993], Fastest Paths in Time-Dependent Networks for Intelligent Vehicle-Highway Systems Application, Intelligent Transportation Systems Journal, Vol. 1(1) pp

29 29 Motivation Dynamic Programming Formulation Let τ(a, θ a, b, θ b, t a ) denote the travel time along the fastest path with bounded curvature from point a to point b starting at heading angle θ a at time t a and arriving at heading angle θ b. Let T (a, θ a, b, θ b, t a ) denote the smallest elapse of time from the moment of arriving to point a at time t a to arriving to the position (b, θ b ). Think about what is the earliest time a mobile agent can arrive from position (a, θ a ) to position (b, θ b ) starting the travel no earlier than time t a (i.e., we can choose to leave a any time after arriving there at t a ). T (a, θ a, b, θ b, t a ) = min t 0 { t + τ(a, θ a, b, θ b, t a + t )}

30 30 Motivation Dynamic Programming Formulation After we find the optimal path and the optimal times to leave each waypoint, we can choose to slow down along the link prior to arriving to each waypoint such that we arrive there at the optimal time to continue the travel.

31 31 Motivation Dynamic Programming Functional Equation Define g(a, θ a ) to be the minimum travel time from the starting position (x s, θ s ) to point a, arriving at a with the heading angle θ a. Without loss of generality, assume starting travel time to be zero. min {a,θa: b a =l}[g(a, θ a ) + T (a, θ a, b, θ b, g(a, θ a ))] g(b, θ b ) = 0 for (b, θ b ) = (s, θ s )

32 32 Motivation Optimal-Path Planning 1 Apply fastest path with bounded curvature to find values τ(a, θ a, b, θ b, t a ). 2 Compute smallest elapse of time function T = min t { t + τ}. 3 Use DP recursive equation to find fastest path to all the boundary points of the radar visibility horizon g(b, θ b ) = min {a,θa}[g(a, θ a ) + T (a, θ a, b, θ b, g(a, θ a ))]. 4 Apply fastest path in homogenous media to compute the cost to reach target point from the boundary R H.

33 Outline 1 Motivation Optimum Vessel Performance in Evolving Nonlinear Wave Fields Project 2 3 4

34 Efficient Implementation 1 Apply fastest path with bounded curvature to find values τ(a, θ a, b, θ b, t a ). 2 Compute smallest elapse of time function T = min t { t + τ}. 3 Use DP recursive equation to find fastest path to all the boundary points of the radar visibility horizon g(b, θ b ) = min {a,θa}[g(a, θ a ) + T (a, θ a, b, θ b, g(a, θ a ))]. 4 Apply fastest path in homogenous media to compute the cost to reach target point from the boundary R H. 1 Precomputed off-line 2 Precomputing τ() provides tighter bounds on t 3 The only on-line calculations: DP recursive equation 4 Precomputed off-line and scaled as needed

35 35 Optimum Vessel Performance Project S-175 Containership R H = 2500 meters Nominal ship speed = 11.4 m/s Global sea state No. 6.5 (Sig. wave height 7 m) Maximum attainable speed range is [8.4, 10.3] m/s l = 250 metes

36 36 Optimum Vessel Performance: Minimum Turning Radius

37 Algorithm Run Time Used MATLAB to stay consistent with other project teams. C++ is known to be faster and is expected to be used for real-life implementation. Test runs we performed on Microsoft Windows Vista operating system with a single 2.9 GHz processor. Each run explored between 57,000 and 132,000 DP states. Corresponding run time is between 105 and 413 seconds. Faster processors are available (3.5 GHz with 15% improvement). C++ code delivers at least 65% decrease in run-time.

38 Optimal Path Results Compare optimal travel time to travel time for paths p 1, p 2 and p 3. For paths p 2 and p 3 we compare the travel times to points x 2 and x 3, respectively. The remain parts of the paths are the same. 38

39 39 Optimal Path Results Run No. θ st (deg) t /t(p 1 ) t /t(p 2 ) t /t(p 3 )

40 Optimal Path Results Up to 9.5% improvement On average between 4% and 6% improvement over implementing the results of Theorem 2 (neglecting radar collected data). This analysis neglects bounded curvature for paths p 1, p 2 and p 3, therefore these estimates are very conservative. The DP model not only improves the travel time, but also finds a control-feasible path.

41 Limitations of the Data Available for Numerical Analysis The analysis is conducted on a large (175 m) and slow ( m/s) vessel with large turning radius ( m) Average wave length is 90 m. For comparison, we scale turning radius by 0.5 for test run 3 Min Turning Radius p 1 p 2 p 3 R 8.8% 5.3% 5.3% 0.5R 14.6% 9.5% 15.4% Simulated wave-field is stationary and the speed prediction model has limited accuracy.

42 42 Test Run Number 5: θ st = 80 degrees

43 43 Test Run Number 5: θ st = 80 degrees

44 44 Test Run Number 7: θ st = 120 degrees

45 45 Test Run Number 7: θ st = 120 degrees

46 Outline 1 Motivation Optimum Vessel Performance in Evolving Nonlinear Wave Fields Project 2 3 4

47 47 Applications Various vessels, airplane, unmanned aerial and surface vehicles.

48 ONR Center for Innovation in Ship Design: Autonomous Navigation of an Amphibious Vehicle A Sea Base cargo ship is located off shore. Unmanned amphibious vehicles are responsible for transporting the cargo from the ship to the target location inland. The selected path has to minimize travel time while avoiding dangerous regions, and must dynamically adjust as new information about the surrounding environment is obtained. 48

49 49 Integrated limited visibility horizon Created a DP model to find a smooth fastest path by integrating the system dynamics in the optimization process Integrated vessel controller (speed) into the decision space of the algorithm, resulting in improved efficiency

50 50 Questions?

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