Title: Quantization in Robotic Devices and Resource Allocation
1Quantization 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
2Important 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
3Outline
- 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
4Quantization 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)
5Behavior 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
6Motion 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)
7Important 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
8Sequential 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.
9Finite-State (?) Machine
Gain-scheduling or other
Trim surface 1
Immelman
Trim Surface 2
Upright
Roll
Flip
Split S
Upside down
10Aerospace example
11Composition 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
12From constructive quantization to quantization
Engineering
Consider the unstable system S For a given
quadratic CLF Find a quantizer such that
Elia/Frazzoli/Liberzon
13PERTURBATION APPROACH
- Design ignoring constraint
- View as approximation
- Prove that this still solves the problem
14Result Logarithmic Quantization
(Single Input)
X
Elia Mitter TAC 01
15Design Criterion
Optimal sampling minimizes R(T)
System independent constants
Elia Mitter TAC 01
16Quantization for Two-Input Systems.
Elia / Frazzoli
17Example of Joint Log Partition
18WEIGHTED MULTICENTER PROBLEM
.
Logarithmic partition
Minimize H(Q,W)
19Resource 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
20Some 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?
21Distributed 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
22Lloyds 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
23Connection 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
24Connection 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)
25Deterministic 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
-
26Deterministic 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
27DA 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
-
28Example 20 targets, 15 resources
29Conclusions
- Quantization effects ubiquitous in multi-resource
allocation and planning problems - Significant potential for cross-fertilization
between seemingly unrelated basic research and
applications
30Variations
- 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
31Other 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
32Other 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
33animations
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37conclusions