Method of Evolving Junctions: Finding the Shortest Paths in - PowerPoint PPT Presentation

Method of Evolving Junctions: Finding the Shortest Paths in Cluttered Environment Jun Lu collaboration with Shui-Nee Chow, Magnus Egerstedt Yancy Diaz-Mercado, Haomin Zhou Georgia Institute of Technology SIAM Gators, 2014

Introduction Shortest Path Problem Given a finite number of obstacles in a region M in R 2 or R 3 , what is the shortest path connecting two given points X , Y in M while avoiding the obstacles. X Y 1 / 25

Existing Methods • Polygonal obstacles: optimal alogrithm with complexity O ( n log n ). (Hershberger and Suri) X Y 2 / 25

Existing Methods • Polygonal obstacles: optimal alogrithm with complexity O ( n log n ). (Hershberger and Suri) X Y • Polyhedral obstacles, NP-hard in the framework configuration space . (Canny and Reif) Y X 2 / 25

• Approximation Methods. (Papadimitriou) 3 / 25

• Approximation Methods. (Papadimitriou) • Continuous Settings: Penalty Method X Y 3 / 25

• Approximation Methods. (Papadimitriou) • Continuous Settings: Penalty Method X Y • Front Propagation (Dijkstra, A ∗ , HJB). |∇ d ( x ) | = 1 , d ( Y ) = 0 . 3 / 25

Shortest Path Problem • Starting point X , ending point Y in domain M ⊂ R n . 4 / 25

Shortest Path Problem • Starting point X , ending point Y in domain M ⊂ R n . • Obstacles { P k } k ≤ 1 . The boundary ∂ P k of P k are piecewise C 2 and has level set representation φ k ( x ) = 0. 4 / 25

Shortest Path Problem • Starting point X , ending point Y in domain M ⊂ R n . • Obstacles { P k } k ≤ 1 . The boundary ∂ P k of P k are piecewise C 2 and has level set representation φ k ( x ) = 0. • Admissible paths connecting X and Y is A ( X , Y , M ) = { γ : [0 , 1] → M | γ (0) = X , γ (1) = Y , φ k ( γ ( θ )) ≥ 0 , 1 ≤ k ≤ N } . 4 / 25

Shortest Path Problem • Starting point X , ending point Y in domain M ⊂ R n . • Obstacles { P k } k ≤ 1 . The boundary ∂ P k of P k are piecewise C 2 and has level set representation φ k ( x ) = 0. • Admissible paths connecting X and Y is A ( X , Y , M ) = { γ : [0 , 1] → M | γ (0) = X , γ (1) = Y , φ k ( γ ( θ )) ≥ 0 , 1 ≤ k ≤ N } . • The shortest path is γ opt = argmin γ ∈A J ( γ ) . where J ( γ ) is the Euclidean length of γ . 4 / 25

Separable Path • For any two points x , y ∈ M , define γ 0 ( x , y ) = argmin γ ∈A ( x , y , M ) J ( γ ) , = { z ( λ ) = (1 − λ ) x + λ y , λ ∈ [0 , 1] } , J 0 ( x , y ) = J ( γ 0 ( x , y )) = � x − y � . 5 / 25

Separable Path • For any two points x , y ∈ M , define γ 0 ( x , y ) = argmin γ ∈A ( x , y , M ) J ( γ ) , = { z ( λ ) = (1 − λ ) x + λ y , λ ∈ [0 , 1] } , J 0 ( x , y ) = J ( γ 0 ( x , y )) = � x − y � . • For any two points x , y ∈ ∂ P k , k ≥ 1, define γ k ( x , y ) = argmin γ ∈A ( x , y ,∂ P k ) J ( γ ) , J k ( x , y ) = J ( γ k ( x , y )) . 5 / 25

Separable Path A path γ ∈ A ( X , Y , M ) is SEPARABLE if there are finite number of points, called junctions ( x 0 , x 1 , · · · , x n +1 ) such that γ = γ 0 ( x 0 , x 1 ) · γ k 1 ( x 1 , x 2 ) · · · · γ 0 ( x n , x n +1 ) . Here γ 1 · γ 2 is the concatenation defined by � θ ∈ [0 , 1 γ 1 (2 θ ) , 2 ]; ( γ 1 · γ 2 )( θ ) = θ ∈ [ 1 γ 2 (2 θ − 1) , 2 , 1] . x i +5 x i +1 x i +2 x i x i +3 x i +4 Figure : γ 0 ( x i , x i +1 ) · γ k 1 ( x i +1 , x i +2 ) · γ 0 ( x i +2 , x i +3 ) · 6 / 25

Separable Path Theorem γ opt of the shortest path problem is separable. 7 / 25

Separable Path Theorem γ opt of the shortest path problem is separable. • γ opt is completely determined by finitely many junctions. 7 / 25

Separable Path Theorem γ opt of the shortest path problem is separable. • γ opt is completely determined by finitely many junctions. • Dimension Reduction Finding γ opt = ⇒ Finding junctions ( x 0 , · · · , x n +1 ) Infinite Dimensional = ⇒ Finite Dimensional 7 / 25

Gradient Flow • The length of the path connecting junction x i and x i +1 is � J 0 ( x i , x i +1 ) , i even ; J ( x i , x i +1 ) = J k i ( x i , x i +1 ) , i odd . 8 / 25

Gradient Flow • The length of the path connecting junction x i and x i +1 is � J 0 ( x i , x i +1 ) , i even ; J ( x i , x i +1 ) = J k i ( x i , x i +1 ) , i odd . • The length of the shortest path is n � J ( x 0 , · · · , x n +1 ) = J ( x i , x i +1 ) , x i ∈ ∂ P k i i =0 8 / 25

Gradient Flow • The length of the path connecting junction x i and x i +1 is � J 0 ( x i , x i +1 ) , i even ; J ( x i , x i +1 ) = J k i ( x i , x i +1 ) , i odd . • The length of the shortest path is n � J ( x 0 , · · · , x n +1 ) = J ( x i , x i +1 ) , x i ∈ ∂ P k i i =0 • The gradient flow on the manifold ∂ P k i is dx i � � d θ = −∇ J ( x 0 , · · · , x n +1 ) = −∇ x i J ( x i − 1 , x i ) + J ( x i , x i +1 ) . � �� � J ( x i ) 8 / 25

Gradient Flow • For i odd, J ( x i ) = J 0 ( x i − 1 , x i ) + J k i ( x i , x i +1 ). 9 / 25

Gradient Flow • For i odd, J ( x i ) = J 0 ( x i − 1 , x i ) + J k i ( x i , x i +1 ). • T ( x i , x i +1 ) ∈ T x i ∂ P k i - unit tangent of γ k i ( x i , x i +1 ) at x i . γ k i ( x i , x i +1 ) x i +1 x i T ( x i , x i +1 ) 9 / 25

Gradient Flow • For i odd, J ( x i ) = J 0 ( x i − 1 , x i ) + J k i ( x i , x i +1 ). • T ( x i , x i +1 ) ∈ T x i ∂ P k i - unit tangent of γ k i ( x i , x i +1 ) at x i . γ k i ( x i , x i +1 ) x i +1 x i T ( x i , x i +1 ) • Gradient of J k i is ∇ x i J k i ( x i , x i +1 ) = − T ( x i , x i +1 ) . 9 / 25

Gradient Flow • For i odd, J ( x i ) = J 0 ( x i − 1 , x i ) + J k i ( x i , x i +1 ). • T ( x i , x i +1 ) ∈ T x i ∂ P k i - unit tangent of γ k i ( x i , x i +1 ) at x i . γ k i ( x i , x i +1 ) x i +1 x i T ( x i , x i +1 ) • Gradient of J k i is ∇ x i J k i ( x i , x i +1 ) = − T ( x i , x i +1 ) . • Gradient of J 0 is � x i − x i − 1 � ∇ x i J 0 ( x i − 1 , x i ) = � x i − x i − 1 � · T T 9 / 25

Intermittent Diffusion • The gradient flow on the manifold ∂ P k i is � x i − x i − 1 � dx i d θ = − � x i − x i − 1 � · T − 1 · T . 10 / 25

Intermittent Diffusion • The gradient flow on the manifold ∂ P k i is � x i − x i − 1 � dx i d θ = − � x i − x i − 1 � · T − 1 · T . • Gradient descent only finds local minimizeers. 10 / 25

Intermittent Diffusion • The gradient flow on the manifold ∂ P k i is � x i − x i − 1 � dx i d θ = − � x i − x i − 1 � · T − 1 · T . • Gradient descent only finds local minimizeers. • There could be 2 N locally shortest paths in an environment with N obstacles. 10 / 25

Intermittent Diffusion • Idea: add noise to gradient flow intermittently, dx i = −∇ x i J ( x i ) d θ + σ ( θ ) d W ( θ ) , 11 / 25

Intermittent Diffusion • Idea: add noise to gradient flow intermittently, dx i = −∇ x i J ( x i ) d θ + σ ( θ ) d W ( θ ) , where • W ( θ ) is a standard Brownian motion in tangent space T x i ∂ P k i . 11 / 25

Intermittent Diffusion • Idea: add noise to gradient flow intermittently, dx i = −∇ x i J ( x i ) d θ + σ ( θ ) d W ( θ ) , where • W ( θ ) is a standard Brownian motion in tangent space T x i ∂ P k i . • For 0 = S 1 < T 1 < · · · < S m , m � σ ( θ ) = σ i χ [ S i , T i ] ( θ ) . i =1 y σ 2 σ 3 σ 1 x S 1 T 1 S 2 T 2 S 3 T 3 S 4 11 / 25

Intermittent Diffusion • Diffusion on, get out of local trap. 12 / 25

Intermittent Diffusion • Diffusion on, get out of local trap. • Diffusion off, converges to local minimizers. 12 / 25

Intermittent Diffusion • Diffusion on, get out of local trap. • Diffusion off, converges to local minimizers. • After m diffusions, obtain m local minimzers. 12 / 25

Intermittent Diffusion • Diffusion on, get out of local trap. • Diffusion off, converges to local minimizers. • After m diffusions, obtain m local minimzers. • Set γ opt to be the best minimizer. 12 / 25

Intermittent Diffusion • Diffusion on, get out of local trap. • Diffusion off, converges to local minimizers. • After m diffusions, obtain m local minimzers. • Set γ opt to be the best minimizer. Theorem For any given δ > 0 , the probability that γ opt is the globally shortest path is larger that 1 − δ for appropriate selected σ . 12 / 25

Solving SDEs • The full SDE is � x i − x i − 1 � dx i = − � x i − x i − 1 � · T − 1 d θ · T + σ ( θ ) d W ( θ ) = f ( x i ) d θ · T + σ ( θ ) d W ( θ ) · (1) 13 / 25

Solving SDEs • The full SDE is � x i − x i − 1 � dx i = − � x i − x i − 1 � · T − 1 d θ · T + σ ( θ ) d W ( θ ) = f ( x i ) d θ · T + σ ( θ ) d W ( θ ) · (1) • SDE (1) can be discretized by √ x ∗ i − x n i = f ( x n i )∆ θ T + σ n ∆ θξ, ξ ∼ N ( 0 , I ) . 13 / 25