Re Real Time Crowd Navigation From Fi First P Princi ncipl ples o of P Proba babi bility The Theory Pete Trautman and Karankumar Patel Honda Research Institute San Jose, CA, USA
What is Crowd Unstructured: no flow rules or static obstacles • Navigation? Collect time stamped x,y data (trajectories) •
Decoupling leads to Freezing Robot Problem Goal Executed Path Desired Path Even with perfect future True Pedestrian Movement: knowledge, “decoupling” Humans leverage cooperation prediction and navigation fails Optimality Theorems of Autonomous Crowd Navigation. CDC, 2017. [43] P. Trautman and A. Krause. Unfreezing the robot: Navigation in dense interacting crowds. In IROS , 2010.
Decoupling: Suboptimal at any Density Large safety decrement at all densities 1 % mgGP unsafe: 0.63492, total runs: 126 % mgIGP unsafe: 0.19444, total runs: 108 0.9 0.8 mgGP # unsafe/# runs: 5/19 12/22 12/16 17/21 19/21 7/8 8/8 0.7 mgIGP # unsafe/# runs: 2/30 6/28 2/13 2/7 4/9 3/3 2/2 Fraction of Unsafe Runs 0.6 0.5 0.4 0.3 No unsafe runs 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 ) Average Crowd Density Over Duration of Run (people/m Trautman, P.; Ma, J.; Krause, A.; and Murray, R. M. 2013. Robot navigation in dense crowds: the case for cooperation. In ICRA . Trautman, P. 2013. Robot Navigation in Dense Crowds: Statistical
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where p ( h , f R | z h T ) = ψ ( h , f R , γ ) p ( f R | z R T , z R T ) p ( h | z h T ) • How can ψ ( h , f R , γ ) model it? p ( f R , h 1 , h 2 | z ) [ f R , h 1 , h 2 ] ∗ <latexit sha1_base64="SXb5lu0WUYqriImUPUPtCG2L8g=">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</latexit> u ( t + 1)
<latexit sha1_base64="ruFEBLr3N4b4SI1g1F5oSOqM6VU=">ADnXicbVLbtNAELUbLiVcmsIjD6xIkHiAXCqlSd8qLhIPqAoSaSvVUbRej51V92J212mCZf6ED+MP+AzGdqhIy0grzZ65npkJU8Gt6/d/+TuNO3fv3d90Hz46PGTvdb+01OrM8NgyrTQ5jykFgRXMHXcCThPDVAZCjgL9+X9rMlGMu1+urWKcwkTRSPOaMOoXnrdxBCwlXOQDkwRTNwsHJV2txAVOTNjZ07kPw7oIMBVdMS0lVlAeChli7NBZ5JzA8WThqjL7qoGcJk4nRSx4BsQ4rWoeFhViTJRU8IrysSVnZCYkzVSudIFXa8STt3MhACRITxHEJROg6j07xi6DVAlk2A8Ce/rZa1N8Ns3mr3e/2KyG3lcFGaXsbmcz3/Z9BpFkmMZ4Jau3FoJ+6WU4NkhDlJDILKWXNIELVBWVYGd5NbuCvEIkIrE2+JQjFfpvRE6ltWsZoqekbmFv2krwf7aLzMXjWc5VmjlQrC4UZzgWTcr9kogbYA5HE3HKDMdeCVvQcsp4BVtVZBQb7DnaZmKXScXkzQKzGQPxLF/VnLaCQaYL+LYdihexnuX19YR6tR3Qm1psoJcC9tF1hmYO3XsfjE7Rtce0igEvi4HtJXhghrO3FceRBbaFs1bUi30aDgcjsbX68OFHo0PD0eDa+T0oDsYdvtfDtrH7zar3fWey+9197AG3nH3idv4k095p/4zi/8H40XjY+Nz42T2nXH38Q87akcfYHB1QwPQ=</latexit> The How and What of Interaction <latexit sha1_base64="Sw/fRlM8S848N4kMBftPTRavk=">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</latexit> Theorem: Crowd navigation cost is statistically valid ⇐ ⇒ cost only a function of the full set of mixture statistics ⇐ ⇒ ⇣ ⌘ ` ( t ) , µ f i µ R C k ( t ) , w ( t ) , Σ ( t ) | {z } |{z} |{z} intents at t preference flexibility → Provide statistically valid interaction function P IGP ¬ κ → Provide a real time locally optimal solver <latexit sha1_base64="thn/57XmCLriaYEJyA4S2ezTm3Y=">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</latexit> Corollary: Mismodeling flexibility leads to a) overaggressive or b) overcautious (FRP) robot
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