Scalable Nonlinear Dynamical Systems

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Scalable Nonlinear Dynamical Systems for Agent
Steering and Crowd Simulation
Siome Goldensteina, Menelaos Karavelasb, Dimitris
Metaxasa, Leonidas Guibasb, Eric Aarona, Ambarish
Goswamic a Computer and Information Science
Dept. Univ. of Pennsylvania b Computer Science
Dept. Stanford Univ. c Discreet
2004.4.21 Min Je Park
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Outline

• Introduction
• Overview
• Background
• Local System
• Global Environment Control
• Global Planning Layer
• Experiments
• Conclusions and Future Work

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Introduction
- Modeling low-level behavior (scalable and
efficient)
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Overview
- A novel 3-layer framework
Environmental
Local
Global
- local basic reactive actions - environmental
knowledge about agents immediate environment -
global general knowledge about the environment
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Overview
- How to model ?
kinetic data structure (KDS)
nonlinear dynamical system
Environmental
Local
Global
potential field hybrid systems
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Background
Dynamical system
Linear dynamical system (superposition principle)
Nonlinear dynamical system (hard to analyze,
chaotic?)
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Background
Critical points (or fixed points)
zero-crossings of f - a critical
point is a stationary point for the system -
local analysis about the critical points
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Background
An Example
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Background
Kinetic data structure (KDS)
a system with objects moving at vastly varying
speed
- Evaluating at each time step ?? - Exploiting
the continuity of motion and temporal coherence
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Background
Kinetic data structure (KDS)
Proof near points Certificates simple atomic
relationships (such as
distance comparisons)
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Background
Kinetic data structure (KDS)
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Overview
kinetic data structure (KDS)
nonlinear dynamical system
Environmental
Local
Global
potential field hybrid systems
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Local System
Modeling the low-level movement of every
agent What is the agent ? - heading angle -
forward velocity - some personality attributes
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Local System
Agent heading angle
f a nonlinear function env the state of the
environment
Design objective - avoiding an obstacle
(repeller) - pursuing a target (attractor)
Observation - repellers and attractors are
located at the critical points
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Local System
Agent heading angle
Positive any slight change will deviate farther
away with time
(repeller) Negative inverse situation
(attractor)
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Local System
Configuration revisited
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Local System
Attractor function
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Local System
Repeller function
collide !!!
Spatial extent, distance
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Local System
Repeller function
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Local System
Repeller function
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Local System
How to combine?
? artifacts !!!
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Local System
How to combine?
Nonlinear constraint competition dynamical system
4 critical points
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Local System
Designing choices Goldenstein2001 Stability
analysis Perko1991
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Local System
Forward velocity
rmin the distance to the closet obstacle d1
Safety distance t2c time to contact
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Local System
Personality - angular acceleration (moment of
inertia) - forward acceleration (mass) -
aggressiveness (subjective quantity)
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Local System
Extension to three dimensions
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Overview
kinetic data structure (KDS)
nonlinear dynamical system
Environmental
Local
Global
potential field hybrid systems
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Global Environment Control
How to keep environmental information up-to-data
as the agents move and the environment changes
Proof points inside a certain circle
Cx Certificates Delaunay triangularion
- containing a lot of proximity information -
easy to maintain as the points move continuously
(edge-flip operations). A
certain set of edges crossed by Cx
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Global Environment Control
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Global Environment Control
Advantages - conservative estimates for the
failure of the KDS certificates can be computed
using motion estimates provided by the local
system - O(nn) ? O(n)
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Overview
kinetic data structure (KDS)
nonlinear dynamical system
Environmental
Local
Global
potential field hybrid systems
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Global Planning Layer
Global minima vs. Local minima
Pacman is trapped !!!
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Global Planning Layer
Objective giving each agent the general
direction How? Precomputed velocity fields
- can be given by the designer
- can be generated by harmonic functions

(potential fields)
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Global Planning Layer
Harmonic functions (solutions to the Laplace
equation)
Ideal fluids (Incompressible and non-viscous)
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Global Planning Layer
Boundary conditions - targets potential -
obstacles potential (including the
configuration space boundary) Properties -
no local minima (no curl, no chaos) - global
minima at the targets potential
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Global Planning Layer
Implementation
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Global Planning Layer
Hybrid Dynamical Systems continuous
discrete dynamics Go in the direction of
the target Ignore the target for now Just
escaping this apparent local minima
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Experimental Result
A simple crowd simulation
A moving obstacle
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Experimental Result
A friendly target
A hostile obstacle
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Experimental Result
A hybrid system
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Conclusions
A novel three-layered system - agent steering -
customizable behavior - real-time environment
changes Technical novelty - nonlinear dynamical
systems - KDS - hybrid systems (global planning
layer)