1 Hybrid Estimation (Filt ering) Problem Conditional Dependencies - PDF document

Motivation method that uses efficient Gaussian representation Multi-Modal Particle Filtering Life support systems BIO-Plex courtesy NASA JPL for Hybrid S ystems NASA robotic missions MSL Embedded systems: with Autonomous Mode Transitions • Continuous and discrete behavior • Highly complex artifacts S tanislav Funiak, Brian Williams • Need for autonomous, robust operation transitions between modes that Hybrid model, estimate state from observations depend on continuous state MIT Space Systems and Artificial Intelligence Laboratories Extract diagnosis from subtle symptoms Cambridge MA, USA Legged bipeds, intelligent assistants courtesy MIT LEG Laboratory DX-2003 / SafeProcess 2003 Bridge Presentation 1 2 Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions DX-2003 / SafeProcess 2003 Bridge Presentation Outline S imple Hybrid S yst em: Acrobatic Robot � Hybrid modeling, Hybrid estimation problem � Prior work: multi-modal methods, particle filtering � Multi-modal particle filtering � Derivation using Rao-Blackwellised particle filtering � Unification with prior multi-modal methods � Discussion � Experimental results � Approximations in the filter 3 4 Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions DX-2003 / SafeProcess 2003 Bridge Presentation Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions DX-2003 / SafeProcess 2003 Bridge Presentation Hybrid Discrete/ Continuous Model Hybrid Discrete/ Continuous Model Probabilistic Hybrid Automata (Hofbaur, Williams) Probabilistic Hybrid Automaton for acrobatic robot state x : x d discrete state (mode) with finite domain state x : x d x c continuous state x c input/output: u d discrete command input/output: u d u c continuous command u c T 1 y c continuous output (observation) y c + noise T 1 > 0: p= 0.01 T 1. probabilistic transitions between modes dynamics: dynamics: m 0, ok m 0, f set of transitions & guard conditions over θ 1 > 0.7: p= 0.02 θ 1 > 0.7: p= 0.02 and θ 1 < 0.7: p= 0.01 θ 1 < 0.7: p= 0.01 p= 0.01 p= 0.01 m 1, ok m 1, f F 2. discrete-time dynamics for each mode T 1 > 0: p= 0.01 mode-dependent non-linear functions white Gaussian noise 5 6 Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions DX-2003 / SafeProcess 2003 Bridge Presentation Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions DX-2003 / SafeProcess 2003 Bridge Presentation 1

Hybrid Estimation (Filt ering) Problem Conditional Dependencies in PHA PHA PHA T 1 > 0: p= 0.01 T 1 > 0: p= 0.01 continuous input m 0, ok m 0, f m 0, ok m 0, f continuous output u c ,0: t θ 1 > 0.7: p= 0.02 θ 1 > 0.7: p= 0.02 θ 1 > 0.7: p= 0.02 θ 1 > 0.7: p= 0.02 (observation) θ 1 < 0.7: p= 0.01 θ 1 < 0.7: p= 0.01 θ 1 < 0.7: p= 0.01 θ 1 < 0.7: p= 0.01 y c ,1: t discrete input p= 0.01 p= 0.01 u d ,0: t p= 0.01 p= 0.01 m 1, ok m 1, f m 1, ok m 1, f Hybrid DBN of T 1 > 0: p= 0.01 T 1 > 0: p= 0.01 PHA models autonomous transition Task: autonomous mode transitions: Given a PHA model, initial distribution , and sequence conditioned on continuous state of command inputs and observations coupling between discrete and continuous state � estimate x d , t and x c , t the process is not Markov � approximate posterior distribution 7 8 Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions DX-2003 / SafeProcess 2003 Bridge Presentation Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions DX-2003 / SafeProcess 2003 Bridge Presentation Background: Multi-Modal Estimation Background: Particle Filt ering Verma et al., 2001 Dearden and Clancy 2002, Koutsoukos et al. 2002 e.g. IMM, HME, GPB � � Represent posterior distribution by samples evolved probabilistically � represent efficiently as mixture of Gaussians � Sample the whole state space ) suffer from state space explosion � (except Freitas, 2002) Beam filter (HME), (Hofbaur, Williams): • Tracks leading set of likely trajectories • For each, estimates continuous state Initialization step • Interleaves mode trajectory tracking (1) sample the initial distribution and continuous state estimation Importance sampling step evolve each particle according to (2) proposal distribution q evaluate importance weights Selection step (3) resample particle according to their nonlinearities and merging ) bias importance weights 9 10 Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions DX-2003 / SafeProcess 2003 Bridge Presentation Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions DX-2003 / SafeProcess 2003 Bridge Presentation Outline Background: Rao-Blackwellised Particle Filtering � Reduce size of sampled space by factoring out � Hybrid modeling, Hybrid estimation problem subspace with analytical solution (Murphy, Russell) � Prior work: multi-modal methods, particle filtering � Divide state variables x into two sets, r and s: � Multi-modal particle filtering sample r, analytically solve for s � Derivation using Rao-Blackwellised particle filtering � Unification with prior multi-modal methods � Discussion � Experimental results � Approximations in the filter analytical solution particle filter I f able to compute analytically, only need to sample r 11 12 Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions DX-2003 / SafeProcess 2003 Bridge Presentation Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions DX-2003 / SafeProcess 2003 Bridge Presentation 2

Multi-modal Particle Filt er – Concept Multi-modal Particle Filt er – Algorithm Combines particle filtering with multi-modal estimation � (1) Initialization step Derived with Rao-Blackwellised PF + approximations draw samples from the initial � m ok ,1 m f,0 m f,1 m ok ,0 distribution over the modes 1. Particle filter: initialize the corresponding m ok ,0 m ok ,1 continuous state estimates sample mode trajectories, m ok ,0 m ok ,1 m ok ,0 m ok ,1 not whole state space (2) Importance sampling step m ok ,0 m ok ,1 m ok ,0 m f ,1 - analogous to HME (H&W) evolve each sample trajectory m ok ,0 m ok ,0 m ok ,1 m ok ,1 according to the transition m ok ,0 m ok ,1 m ok ,1 m f ,1 model and previous m ok ,0 m ok ,1 m ok ,0 m ok ,1 continuous estimates Use continuous estimates from time t in particle filter at time t + 1 m ok ,0 m ok ,0 m ok ,1 (3) Selection (resampling) step m ok ,1 m ok ,0 m ok ,1 m ok ,1 m ok ,1 2. Analytical solution: For each sample trajectory, (4) Exact (Kalman Filtering) step determine transition & observ- estimate continuous state ation model for each sample with Kalman Filter update continuous estimates 13 14 Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions DX-2003 / SafeProcess 2003 Bridge Presentation Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions DX-2003 / SafeProcess 2003 Bridge Presentation Proposal Distribution (Import ance S Importance Weights (Importance S ampling S t ep) ampling S t ep) Conditioned on entire trajectory and prior observations PF: need to let m ok ,0 m ok ,1 � � Computed efficiently from previous continuous estimate y 1 m ok ,0 m ok ,1 m ok ,0 m f ,1 y 2 m ok ,0 m ok ,1 m ok ,0 m ok ,1 (expand in x c , t -1 ) Simplifies to � still hard to evaluate (non-standard) � expand in x c , t same for all x c , t -1 satisfying guard c probability of guard c conditional independences observation likelihood non-standard θ 1 0.7 mean of estimated θ 1 ignore autonomous transitions in second term guard boundary approximate weight using the prediction step of Kalman Filter 15 16 Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions DX-2003 / SafeProcess 2003 Bridge Presentation Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions DX-2003 / SafeProcess 2003 Bridge Presentation Unification with Prior Mult i-Modal Methods Outline Hybrid prediction & update equations (Hofbaur, Williams): � � Hybrid modeling, Hybrid estimation problem � Prior work: multi-modal methods, particle filtering � Multi-modal particle filtering � Derivation using Rao-Blackwellised particle filtering $ proposal distribution Hybrid transition function � � Unification with prior multi-modal methods $ weight function Hybrid observation function � Discussion Direct correspondence � Experimental results RBPF derives hybrid prediction & update equations � Approximations in the filter Joint improvements Meaningful merging of the two methods 17 18 Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions DX-2003 / SafeProcess 2003 Bridge Presentation Multi-modal Particle Filtering for Hybrid Systems with Autonomous Transitions DX-2003 / SafeProcess 2003 Bridge Presentation 3