DX-05 MSL Entry Decent & Landing Sequence Diagnosis as Approximate Belief State Enumeration for Probabilistic Concurrent Constraint Automata DX-05 Oliver B. Martin Michel D. Ingham Brian C. Williams michel.ingham@jpl.nasa.gov Jet Propulsion Laboratory {omartin, williams}@mit.edu Massachusetts Institute of Technology June 2, 2005 June 2, 2005 DX-05 DX-05 MSL Entry Decent & Landing Sequence Paper Contribution � Best-First Belief State Enumeration � Increased PCCA estimator accuracy by computing the Optimal Constraint Satisfaction Problem (OCSP) utility function directly from the HMM propagation equation. � Improved PCCA estimator performance by framing estimation as a single OCSP. June 2, 2005 June 2, 2005 DX-05 DX-05 IMU Constraint Automaton Simplified Decent Stage IMU/PS System � Two Components � Inertial Measurement Unit (IMU) � Power Switch (PS) June 2, 2005 June 2, 2005 1
DX-05 DX-05 PS Constraint Automaton Estimation of PCCA Compute the belief state for each estimation cycle � Belief State Evolution visualized with a Trellis Diagram 0.7 0.2 0.1 � Complete history knowledge is captured in a single belief state by exploiting the Markov property � Belief states are computed using the HMM belief state update equations June 2, 2005 June 2, 2005 DX-05 DX-05 HMM Belief State Update Equations � Propagation Equation t t+1 � Update Equation June 2, 2005 June 2, 2005 DX-05 DX-05 Belief State Representation Approximations to PCCA Estimation � Best-First Trajectory � Best-First Belief State The belief state can be accurately 1. Enumeration (BFTE) Enumeration (BFBSE) approximated by maintain the k most likely estimates The probability of each state can be 2. accurately approximated by the most likely trajectory to that state The observation probability can be 3. t+1 reduced to 1.0 for observations 1.0 or 0.0 consistent with the state, and 0.0 for observations inconsistent with the state June 2, 2005 June 2, 2005 2
DX-05 DX-05 Simple IMU/PS Scenario PCCA Estimation as an OCSP � Best-First Trajectory � Best-First Belief State Enumeration (BFTE) Enumeration (BFBSE) � For PCCA Estimation: � x is the set of reachable target modes � C ( y ) requires that the observations, modal constraints, and interconnections must be consistent � f ( x ) is the estimate probability for state x June 2, 2005 June 2, 2005 DX-05 DX-05 PS Automaton Conflict-directed A* heuristic � HMM propagation equation: � Split into admissible heuristic for partial assignments: June 2, 2005 June 2, 2005 DX-05 DX-05 Accuracy Results Complexity Analysis � EO-1 Model (12 components) � 30 estimation cycles (nominal operations) Recall A* best case: n·b , worst case: b n � Best-First Trajectory � Best-First Belief State Enumeration (BFTE) Enumeration (BFBSE) � n arithmetic computation � n·k arithmetic computations � k OCSPs � 1 OCSP June 2, 2005 June 2, 2005 3
DX-05 DX-05 Performance Results Performance Results � Heap Memory Usage � Run-time (1.7 GHz Pentium M, 512MB RAM) June 2, 2005 June 2, 2005 DX-05 DX-05 Current Work � Extend BFBSE to use both HMM belief state update equations Backup Slides � Use observation probabilities within the conflict- directed search to avoid unlikely candidates � Done efficiently using a conditional probability table (Published in S.M. Thesis and i-SAIRAS ’05) June 2, 2005 June 2, 2005 DX-05 DX-05 Simple Two Switch Approximate Belief State 0.7 0.343 Sw1=on Scenario Enumeration (k=2) (0.7)(0.7) Sw2=on Sw1=on Sw2=on (0.7)(0.3) 0.7 0.343 Switch on t+1 bkn t+1 Switch on t+1 bkn t+1 Sw1=on 0.147 Sw1=on (0.7)(0.7) Sw1=on Sw2=bkn Sw2=on Sw2=on on t 0.7 0.3 (0.3)(0.7) on t 0.7 0.3 0.3 (0.7)(0.3) bkn t 0 1 Sw1=bkn bkn t 0 1 0.147 Sw2=bkn (0.3)(0.3) Sw1=bkn 0.147 0.3 Sw1=on (0.3)(0.7) Sw2=on Sw2=bkn Sw1=bkn Sw2=bkn � Most likely trajectories (k=2) (1)(1) � Complete probability (0.3)(0.3) 0.147 � Very reactive � Assuming approximate belief Sw1=bkn 0.063 Sw1=bkn � Gross lower-bound state is the true belief state Sw2=bkn Sw2=on (1)(1) � Extraneous computation � Tighter lower bound � Multiple OCSP instances 0.363 � Single OCSP 0.3 � Estimates generated and thrown away Sw1=bkn Sw1=bkn � Best-first order using A* Sw2=bkn Sw2=bkn � Unfortunately, Not MPI June 2, 2005 June 2, 2005 4
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