Quantization in Robotic Devices and Resource Allocation

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Quantization in Robotic Devices and Resource
Allocation

• Eric Feron, MIT, introducing
• F. Bullo (UIUC)
• N. Elia (U. Iowa/MIT)
• E. Feron / S. Salapaka (MIT)
• E. Frazzoli (UIUC)
• D. Liberzon (UIUC)
• MURI 6-month review
• 1/24/2003

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Important message

• Quantization is a need and an opportunity
• in cooperative networked control of dynamical
  peer-to-peer systems

• Traditional View Quantization issue is an
  afterthought
• New View Quantization can be made part of
  design process

• Application to
• System guidance and control under limited
  communication
• capabilities
• System guidance and control under limited
  computational power
• Coarse resource allocation

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Outline

• Quantization in control robotic systems
• Vehicle Guidance with Finite Automata
• Control Systems with quantized inputs
• Quantization for sensor networks and resource
  allocation
• Problem statement
• Geometric Solution
• Technical issues
• Technology Transitions Transition Opportunities

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Quantization in control systems

• A fundamental question
• What is the minimal information about/description
  of the current state of the system I need in
  order to achieve a given goal, within a given
  class?
• Focus on Quantization
• Quantization of inputs
• Quantization of behaviors (outputs)

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Behavior Management

• Optimize Performance(time, fuel, e.m.
  signature, safety)
• Subject to Vehicle Dynamics (including under
  failed mode)
• Structural/Aerodynamic Constraints
• Uncertain current position, final position,
  intermediate points
• Obstacle and vehicle avoidance requ'ts
• Report Progress toward goal

• Need to solve in real time an optimal control
  problem for a nonlinear, high-dimensional,
  high-bandwidth system

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Motion Description Languages

• Main purpose Reduction in Computation/Communicati
  on /Implementation of control strategies
• Motion Description Languages (Brockett et al.)
• Control Quanta (Bicchi et al.)
• Maneuver Automata (Frazzoli et al.)
• Kinematic Decoupling (Bullo Lynch)
• Motion Graphs (Kovar et al.)
• Experimental Data (Feron, Frazzoli, Gavrilets,
  Mettler)

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Important considerations

• System Symmetries
• Continuous Car driving in Boston vs. Cambridge,
  headed south vs. north.
• Discrete Identical UAVs are insensitive to
  position swaps.
• Automatic state complexity reduction
  trajectories and maneuvers modulo symmetries
• Classes of objective functions ?Important motion
  classes

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Sequential Combination of Primitives

• Two main classes of motion primitives
• Trim primitives prefix- and postfix-closed
  repeatable primitives
• Maneuvers Finite time transitions from one state
  to another From trim to trim or otherwise.
• Recent evolutions Replace /Complement trims with
  LTI modes . Match flight reality better. May
  simplify planning tasks (under investigation) via
  Mixed LP planning/ polynomial optimization.

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Finite-State (?) Machine
Gain-scheduling or other
Trim surface 1
Immelman
Trim Surface 2
Upright
Roll
Flip
Split S
Upside down
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Aerospace example
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Composition of automata Operating several
vehicles together

• Composition of Maneuver Automata is not a
  Maneuver Automaton
• Recompute new multi-vehicle automaton, with new
  trims, etc ? Scalability issues.
• Use single-vehicle maneuvers trim states to
  generate scalable maneuvers and trim states for
  vehicle group.

or
More on this with Frazzoli
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From constructive quantization to quantization
Engineering
Consider the unstable system S For a given
quadratic CLF Find a quantizer such that
Elia/Frazzoli/Liberzon
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PERTURBATION APPROACH

• Design ignoring constraint
• View as approximation
• Prove that this still solves the problem

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Result Logarithmic Quantization
(Single Input)
X
Elia Mitter TAC 01
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Design Criterion
Optimal sampling minimizes R(T)
System independent constants
Elia Mitter TAC 01
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Quantization for Two-Input Systems.
Elia / Frazzoli
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Example of Joint Log Partition
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WEIGHTED MULTICENTER PROBLEM
.
Logarithmic partition
Minimize H(Q,W)
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Resource Allocation Problems

• Quantization
• How to partition state space and assign control
  values?
• Objective Obtain coarsest quantization
• Coverage control and resource allocation
• How to partition a domain
• Assign a resource to each cell
• Objective minimize
• resources,
  
• partition

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Some applications

• Mobile Sensing Networks in surveillance and
  exploration
• Ad hoc sensor networks
• Multi vehicle systems
• Coverage Control optimal placement and tuning of
  sensors, and optimal space partitioning via
  decentralized/scalable control protocols
• Resource Allocation Problems
• Facility location where to place mailboxes in a
  city/ cache servers on internet?
• Data compression how to assign codebook vectors
  to input data to minimize distortion?
• Clustering analysis how to cluster, i.e.
  optimally partition a set of events?
• Weapon pre-positioning how to preposition
  Weapons wrt target distribution?

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Distributed Coverage Problem

• Objective Given sensors
  moving in an environment , to
  achieve optimal coverage in closed loop
• Assumptions
• Identical isotropic sensors
• the coverage or performance at point q taken
  from i th sensor at
  degrades with distance
• Cost function
• is a
  distribution/information/prob. density function

is a partition of
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Lloyds Algorithm sensor platform dynamics (eg
UAV platform)

• Each agent i performs
• determine own Voronoi cell of the partition
• determine the centroid of cell
• compute the advance strategy
• using Lyapunov function
• is the mass of , is the
  area moment about
• Convergence to local minimum

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Connection with MICA program Facility Location
(resource allocation)

• (Problem suggested by Jeff Shamma / Jerry
  Wohletz)
• Objective to optimally place the facilities
  in a domain
• i.e. find optimal partition and
  resource locations
• for a given distribution function
• Facility/domains
• mailboxes/cities
• water towers/forest fires
• weapons/targets
• cache servers/internet
• Cost function is the same

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Connection with MICA program

• Cost function is called distortion
• non convex and computationally complex
• e.g. 30 targets/ 20 weapons implies checking over
  30 million partitions
• emphasis on global minimum
• Related areas
• signal compression (minimum-distorsion quantizer
  design)
• statistical pattern recognition (learning vector
  quantization)

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Deterministic Annealing Algorithm

• Deterministic Annealing algorithm
• solves an approximate problem
• solution is an upper bound on actual problem
• The approximate problem includes
• a modified distortion term
• a new parameter called associated
  probability
• is convex w.r.t. for given
  
• an entropy term
• measures randomness of association

-
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Deterministic annealing

• Cost function at th iteration
• is called temperature,
• analogous to free energy FE-TS in statistical
  physics
• Where E is energy, T is temperature and S is
  entropy
• is determined by using Free Energy
  Principle
• minimum of free energy determines the
  distribution at thermal equilibrium

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DA Algorithm

• Free energy principle gives
• is the gibbs function,
• Substituting this into the cost function and
  solving for optimal resource location gives
• centroid with a posteriori weights

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Example 20 targets, 15 resources
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Conclusions

• Quantization effects ubiquitous in multi-resource
  allocation and planning problems
• Significant potential for cross-fertilization
  between seemingly unrelated basic research and
  applications

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Variations

• New problems
• In the algorithm all resources were assumed
  identical
• Variations by having constraints on resources
• Associated problems are solved by modifying the
  free energy term
• (A) Total mass constraints
• Scenario (Water tower/Forest Fire)
• Water towers can be of different sizes
• Total amount of water W is given
• Modified free energy term
• is the weighting parameter on the i th
  resource

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Other constraints

• (B) capacity constraint
• each water-truck has a respective capacity
• Modified free energy term
• (C) unit constraints
• water has to be transported in buckets of
  different sizes,
• Limit the number of th type of bucket by
• Modified free energy term

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Other constraints (contd.)

• (D) multi capacity constraint
• There are types of targets on the ground
• th type of target can be destroyed by th type
  weapon
• relative capacities of weapons in each UAV is
  given
• Modified free energy
• is the weighting function of
  the target of type at location

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animations
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conclusions