A new method to compute the probability of collision for short-term - PowerPoint PPT Presentation

Introduction Computing the probability of collision Conclusion A new method to compute the probability of collision for short-term space encounters R. Serra, D. Arzelier, M. Joldes, J-B. Lasserre, A. Rondepierre and B. Salvy Journ´ ees Nationales de Calcul Formel CIRM November, 2014

Introduction Computing the probability of collision Conclusion Fiction... Credit Gravity (2013)

Introduction General Model Computing the probability of collision Short-term encounter probability of collision Conclusion On-orbit collision Figure: Cerise hit by a debris in 1996 (source : CNES/D. Ducros) 2 / 23

Introduction General Model Computing the probability of collision Short-term encounter probability of collision Conclusion On-orbit collision Figure: Space debris population model (source : ESA) 3 / 23

Introduction General Model Computing the probability of collision Short-term encounter probability of collision Conclusion On-orbit collision Context Two objects: primary P (operational satellite) and secondary S (space debris) Information about their geometry, position, velocity at a given time � Affected by uncertainty Needs: Risk assessment Design of a collision avoidance strategy � compute the probability of collision 4 / 23

Introduction General Model Computing the probability of collision Short-term encounter probability of collision Conclusion On-orbit collision Context Two objects: primary P (operational satellite) and secondary S (space debris) Information about their geometry, position, velocity at a given time � Affected by uncertainty Needs: Risk assessment Design of a collision avoidance strategy � compute the probability of collision P Classical assumptions R p Spherical objects R c Gaussian probability density functions S Independent probability distribution laws R s Figure: Combined spherical object 4 / 23

Introduction General Model Computing the probability of collision Short-term encounter probability of collision Conclusion On-orbit collision Context Two objects: primary P (operational satellite) and secondary S (space debris) Information about their geometry, position, velocity at a given time � Affected by uncertainty Needs: Risk assessment Design of a collision avoidance strategy � compute the probability of collision P Classical assumptions R p Spherical objects R c Gaussian probability density functions S Independent probability distribution laws R s Figure: Combined spherical object Probability of collision Generally: 12-dimensional, Gaussian integrand, Complex integration domain Computation: Monte-Carlo trials and/or simplified models 4 / 23

Introduction General Model Computing the probability of collision Short-term encounter probability of collision Conclusion Short-term encounter model and probability of collision Framework: High relative velocity Assumptions: Rectilinear relative motion No velocity uncertainty Infinite encounter time horizon � Probability of collision: 2-D integral over a disk. 5 / 23

Introduction General Model Computing the probability of collision Short-term encounter probability of collision Conclusion Short-term encounter model and probability of collision Framework: High relative velocity Assumptions: Rectilinear relative motion No velocity uncertainty Infinite encounter time horizon � Probability of collision: 2-D integral over a disk. Formula − ( x − x m ) 2 − ( y − y m ) 2 1 � � � P = exp d x d y, 2 σ x 2 2 σ y 2 2 πσ x σ y B ((0 , 0) ,R ) 5 / 23

Introduction General Model Computing the probability of collision Short-term encounter probability of collision Conclusion Short-term encounter model and probability of collision y level sets Framework: High relative velocity Assumptions: y m Rectilinear relative motion No velocity uncertainty Infinite encounter time horizon � Probability of collision: R x m x 2-D integral over a disk. combined object Figure: 2-D Gaussian integral over a disk Formula − ( x − x m ) 2 − ( y − y m ) 2 1 � � � P = exp d x d y, 2 σ x 2 2 σ y 2 2 πσ x σ y B ((0 , 0) ,R ) where R : radius of combined object x m , y m : mean relative coordinates σ x , σ y : standard deviations of relative coordinates 5 / 23

Introduction Existing methods Computing the probability of collision Our method Conclusion Examples Existing methods Methods based on numerical integration schemes: Foster ’92, Patera ’01, Alfano ’05. Analytic methods: Chan ’97 � uses some simplifying assumptions ( σ x = σ y ) Pro’s and Con’s 6 / 23

Introduction Existing methods Computing the probability of collision Our method Conclusion Examples Existing methods Methods based on numerical integration schemes: Foster ’92, Patera ’01, Alfano ’05. Analytic methods: Chan ’97 � uses some simplifying assumptions ( σ x = σ y ) Pro’s and Con’s use truncated power series, but no rigorous proof about convergence rate 6 / 23

Introduction Existing methods Computing the probability of collision Our method Conclusion Examples Existing methods Methods based on numerical integration schemes: Foster ’92, Patera ’01, Alfano ’05. Analytic methods: Chan ’97 � uses some simplifying assumptions ( σ x = σ y ) Pro’s and Con’s use truncated power series, but no rigorous proof about convergence rate truncation orders fixed by trial and error and by comparing with other existing software: Approximately 60,000 test cases were used to evaluate the numerical expression [...] The reference (”truth”) probability was computed with MATHCAD 11 [...] - Alfano’05 6 / 23

Introduction Existing methods Computing the probability of collision Our method Conclusion Examples Existing methods Methods based on numerical integration schemes: Foster ’92, Patera ’01, Alfano ’05. Analytic methods: Chan ’97 � uses some simplifying assumptions ( σ x = σ y ) Pro’s and Con’s use truncated power series, but no rigorous proof about convergence rate truncation orders fixed by trial and error and by comparing with other existing software: Approximately 60,000 test cases were used to evaluate the numerical expression [...] The reference (”truth”) probability was computed with MATHCAD 11 [...] - Alfano’05 Fast and already used in practice 6 / 23

Introduction Existing methods Computing the probability of collision Our method Conclusion Examples Existing methods Methods based on numerical integration schemes: Foster ’92, Patera ’01, Alfano ’05. Analytic methods: Chan ’97 � uses some simplifying assumptions ( σ x = σ y ) � provides truncation error bounds Pro’s and Con’s use truncated power series, but no rigorous proof about convergence rate truncation orders fixed by trial and error and by comparing with other existing software: Approximately 60,000 test cases were used to evaluate the numerical expression [...] The reference (”truth”) probability was computed with MATHCAD 11 [...] - Alfano’05 Fast and already used in practice Our purpose Give a ”simple”, ”analytic” formula, suitable for double-precision evaluation and effective error bounds. 6 / 23

Introduction Existing methods Computing the probability of collision Our method Conclusion Examples Our method - Underlying techniques Laplace transform: 1 � Lasserre and Zeron, Solving a Class of Multivariate Integration Problems via Laplace Techniques, Applicationes Mathematicae, 2001. 7 / 23

Introduction Existing methods Computing the probability of collision Our method Conclusion Examples Our method - Underlying techniques Laplace transform: 1 � Lasserre and Zeron, Solving a Class of Multivariate Integration Problems via Laplace Techniques, Applicationes Mathematicae, 2001. D-finite functions 2 � solution of linear differential equation with polynomial coefficients � power series coefficients satisfy a linear recurrence relation with polynomial coefficients Example: { f ′ − f = 0 , f (0) = 1 } ↔ { ( n + 1) f n +1 = f n , f 0 = 1 } f ( x ) = exp( x ) ↔ 7 / 23

Introduction Existing methods Computing the probability of collision Our method Conclusion Examples Our method - Underlying techniques Laplace transform: 1 � Lasserre and Zeron, Solving a Class of Multivariate Integration Problems via Laplace Techniques, Applicationes Mathematicae, 2001. D-finite functions 2 � solution of linear differential equation with polynomial coefficients � power series coefficients satisfy a linear recurrence relation with polynomial coefficients Example: { f ′ − f = 0 , f (0) = 1 } ↔ { ( n + 1) f n +1 = f n , f 0 = 1 } f ( x ) = exp( x ) ↔ cos , arccos , Airy functions, Bessel functions, ... 7 / 23