Path Planning for Time-Optimal Information Collection

Transcription

1 Path Planning for Time-Optimal Information Collection Andy Klesh University of Michigan April 2, 2008

2 Motivation

3 Motivation Problem Many Intelligence, Surveillance and Reconnaissance (ISR) Missions require autonomous vehicles to collect information.

4 Motivation Problem Many Intelligence, Surveillance and Reconnaissance (ISR) Missions require autonomous vehicles to collect information. Time is limited

5 Motivation Problem Many Intelligence, Surveillance and Reconnaissance (ISR) Missions require autonomous vehicles to collect information. Time is limited Many targets exist

6 Motivation Problem Many Intelligence, Surveillance and Reconnaissance (ISR) Missions require autonomous vehicles to collect information. Time is limited Many targets exist Enhance autonomous information collection through:

7 Motivation Problem Many Intelligence, Surveillance and Reconnaissance (ISR) Missions require autonomous vehicles to collect information. Time is limited Many targets exist Enhance autonomous information collection through: 1 Model the coupling between information collection and vehicle kinematics

8 Motivation Problem Many Intelligence, Surveillance and Reconnaissance (ISR) Missions require autonomous vehicles to collect information. Time is limited Many targets exist Enhance autonomous information collection through: 1 Model the coupling between information collection and vehicle kinematics 2 Exploit this interaction to form optimal flight paths

9 Related Work A Mathematical Theory of Communication, Shannon, 1948 Optimal Path Planning for Unmanned Combat Aerial Vehicles to Defeat Radar Tracking, Kabamba, Meerkov and Zeitz, 2006

10 Related Work A Mathematical Theory of Communication, Shannon, 1948 Optimal Path Planning for Unmanned Combat Aerial Vehicles to Defeat Radar Tracking, Kabamba, Meerkov and Zeitz, 2006 Path Planning for Cooperative Time-Optimal Information Collection, Klesh, Girard and Kabamba, ACC 2008, Accepted Real Time Path Planning for Time-Optimal Exploration, Klesh, Girard and Kabamba, AIAA GNC 2008, Accepted A Path Planning Framework for Autonomous Exploration, Klesh, Girard and Kabamba, CDC 2008, Submitted Autonomous Exploration and Remote Sensing, Klesh, Girard and Kabamba, CDC 2008 Invited Session, Submitted

11 Unanswered Questions Problem How can we improve the rate of information return from ISR missions?

12 Original Contributions

13 Original Contributions Formulate the exploration problem as the communication of information over a noisy channel

14 Original Contributions Formulate the exploration problem as the communication of information over a noisy channel Identify characteristics for information optimal flight paths

15 Outline 1 Motivation 2 UAV Exploration Models Problem Formulation Optimal Path Planning 3 Generalized Exploration 4 Information Constraints Isolated Objects Clustered Objects Application to Real-Time Controllers 5 Conclusions Future Work

16 The Exploration Problem Exploration The collection of information about objects of interest with known locations, where information is understood in the classical sense of Shannon as selection from a set. We do not seek to classify the object based on our acquired knowledge, merely collect information in a time-optimal way.

17 Communication and Exploration Object of Interest

18 Communication and Exploration Object of Interest Sensing Process

19 Communication and Exploration Object of Interest Sensing Process Signal

20 Communication and Exploration Object of Interest Sensing Process Signal Sensor

21 Communication and Exploration Object of Interest Sensing Process Signal Sensor Transmitter Noisy Channel Information Receiver

22 Communication and Exploration Object of Interest Sensing Process Signal Sensor Transmitter Noisy Channel Information Receiver Exploration Exploration can be viewed as a communication process where each object of interest is a transmitter, the explorer is the receiver, the sensing processes are noisy communication channels, and the sensor signals carry information that allows the explorer to identify the objects of interest.

23 Information Collection Model The rate of information received by the sensor: İ = W log 2 (1 + SNR) (1)

24 Information Collection Model The rate of information received by the sensor: İ = W log 2 (1 + SNR) (1) The signal-to-noise ratio of a signal with an active sensor: SNR = K 4 ((X A) 2 +(Y B) 2 ) 2 (2)

25 Information Collection Model The rate of information received by the sensor: İ = W log 2 (1 + SNR) (1) The signal-to-noise ratio of a signal with an active sensor: SNR = K 4 ((X A) 2 +(Y B) 2 ) 2 (2) The rate of information received about a given object: İ = W log 2 (1 + K 4 ((X A) 2 + (Y B) 2 ) 2 ) (3) where (A, B) is the Cartesian location of the object of interest.

26 Time-Optimal Exploration Problem Statement Find a flight path that minimizes the total flight time for a UAV to collect a specified amount of information about each object of interest in a given area. min t f, subject to I (t f ) 1 (4) ψ( )

27 Necessary Conditions States: X Rate: Y Rate: Information Rate: I j = i=1 x i = v cos(ψ i ), 1 i n y i = v sin(ψ i ), 1 i n n kj 4 w log 2 (1 + ((x i a j ) 2 + (y i b j ) 2 ) 2 ), 1 j m Optimality Condition: 0 = vλ yi cos(ψ i ) vλ xi sin(ψ i ), 1 i n

28 Necessary Conditions Costates: X Costate: λ xi = Y Costate: λ yi = m j=1 m j=1 Information Costate: where j = (1 + 4k 4 j w(x i a j )λ Ij ((x i a j ) 2 + (y i b j ) 2 ) 3 j, 1 i n 4k 4 j w(y i b j )λ Ij ((x i a j ) 2 + (y i b j ) 2 ) 3 j, 1 i n λ Ij = 0, 1 j m k 4 j ((x i a j ) 2 +(y i b j ) 2 ) 2 ), 1 j m

29 Single UAV 4.5 Time-Optimal Flight Path Y/l c (m) X/l c (m) Constant Velocity Constant Altitude Assumptions: Instantaneous Turns Non-redundant additive information

30 Optimal Path Planning Proposition 1 If the objects of interest are isolated, then the optimal flight paths consist of sequences of straight lines (far from the objects of interest) connected by short turns (near the objects of interest) Corollary 1.1 If in addition to being isolated, the objects of interest are poorly visible, then the problem becomes a multi-vehicle TSP (MTSP). Proposition 2 When the visibility of all the objects of interest approaches infinity, t f 0 and the lengths of paths traveled by the UAVs approach zero.

31 Generalized Exploration Kinematics: χ i = f i (u i ), 1 i n Informatics: I j = ρ j (χ 1,..., χ n, w 1,..., w n ), 1 j m where the functions ρ j, 1 j m satisfy: ρ j (χ 1,..., χ n, w 1,..., w n ) = 0, χ 1,..., χ n / D j, > 0, otherwise

32 Generalized Exploration States: X Rate: Information Rate: Optimality Condition: χ i = f (u i ), 1 i n I j = ρ j (χ 1,..., χ n, w 1,..., w n ), 1 j m f (u 0 = λ i ) χi u i, 1 i n

33 Generalized Exploration Costates: Position Costate: λ xi = m j=1 Information Costate: ρ(χ 1,..., χ n, w 1,..., w n ) λ Ij, 1 i n χ i λ Ij = 0, 1 j m

34 Generalized Exploration Control Rate Magnitude: u i = n ρ j (χ 1,..., χ n, w 1,..., w n ) f (u i) u i n ( ) d f (ui ) λ χi dt u i j=1 i=1

35 Generalized Exploration UAV Control Rate Magnitude: ψ i = 4k4 j λ I j w i (cos(ψ i )(b j y i )+sin(ψ i )(a j x i )) cos(ψ i ), 1 i n λ xi r 6

36 Information Constraints Two constraints are active

37 Information Constraints Two constraints are active These are critical objects

38 Information Constraints One active constraint

39 Information Constraints One active constraint One critical object

40 Information Constraints - Clustering One active constraint

41 Information Constraints - Clustering One active constraint One critical object

42 Information Constraints - Clustering Two active constraints

43 Information Constraints - Clustering Two active constraints Two critical objects

44 Clustered Objects Proposed Procedure: 1 Connect the spheres of influence with a greedy path

45 Clustered Objects Proposed Procedure: 1 Connect the spheres of influence with a greedy path 2 Evaluate information gain

46 Clustered Objects Proposed Procedure: 1 Connect the spheres of influence with a greedy path 2 Evaluate information gain 3 Those items with an information gain equal to 1 bit are critical objects

47 Clustered Objects Proposed Procedure: 1 Connect the spheres of influence with a greedy path 2 Evaluate information gain 3 Those items with an information gain equal to 1 bit are critical objects 4 Optimize path for critical objects

48 Parametric Study Parametric Study: Vary only K 1, the visibility of object 1 Examine flight paths

49 Parametric Study Radius of Closest Approach f (K) K 3 Parametric Study:

50 Conclusions 1 ISR missions seek information

51 Conclusions 1 ISR missions seek information 2 Autonomous exploration can be aided improved information return

52 Conclusions 1 ISR missions seek information 2 Autonomous exploration can be aided improved information return 3 The exploration problem can be expressed as the communication of information over a noisy channel

53 Conclusions 1 ISR missions seek information 2 Autonomous exploration can be aided improved information return 3 The exploration problem can be expressed as the communication of information over a noisy channel 4 Path-planning can be optimized to minimize the flight time required to obtain a specified amount of information

54 Conclusions 1 ISR missions seek information 2 Autonomous exploration can be aided improved information return 3 The exploration problem can be expressed as the communication of information over a noisy channel 4 Path-planning can be optimized to minimize the flight time required to obtain a specified amount of information Conclusion Autonomous exploration can be enhanced through optimal path planning

55 Future Work Investigate Exploration vs Exploitation Expand exploration formulation for cooperative vehicles

56 Thank you