September 21, Modeling of Gradient-Based Controllers IPresentation Transcript
Multi-Robot Systems CSCI 7000-006 Monday, September 21, 2009 NikolausCorrell
So far Reactive Algorithms Local interactions Threshold-based dynamics Gradient-based We know it works, but how to prove it? Deliberative Algorithms Local planning Collaborative planning Tight coordination
Today Part II of the course: Modeling of Multi-Robot Systems Gradient-based models for reactive control Cost-function over position of the robots Basic behaviors, nomode switching M. Schwager, A Gradient Optimization Approach to Adaptive Multi-Robot Control, Ph.D. Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, September, 2009.
Gradient-based approaches “Gradient” estimate Use sensor and position information Compare value and positions with local neighbors Requires local range and bearing for collaboration Gradient can describe Environmental sensing Neighborhood relations
Example: Maximizing Visual Coverage Flying robots are equipped with downward facing cameras Cost function of robot positions encodes information gain Robots move to locally optimize information gain Course question: come up with a reactive controller to do this M. Schwager, B. Julian, and D. Rus, Optimal coverage for multiple hovering robots with downward-facing cameras, In Proc. of the International Conference of Robotics and Automation (ICRA 09), Kobe, Japan, May, 2009.[
M. Schwager, B. Julian, and D. Rus, Optimal coverage for multiple hovering robots with downward-facing cameras, In Proc. of the International Conference of Robotics and Automation (ICRA 09), Kobe, Japan, May, 2009.[ Example: Maximizing Visual Coverage
Example 2: Optimally sample an environmental distribution Goal: deploy more robots into regions with high information density Information density unknown at first Learn parameterized model while moving
Example 2: Optimally sample a environmental distribution M. Schwager, J. McLurkin, J.-J. E. Slotine, and D. Rus, From theory to practice: Distributed coverage control experiments with groups of robots, In Proc. of the International Symposium on Experimental Robotics (ISER 08), Athens, Greece, July, 2008.
Multi-Robot System Model State-space of each robot is its position in P Vectoris a single point indescribing the system Cost function Control input (speed) pi
Closed-Loop Control Speed for robot i calculated such that each robot moves towards a local minima of H Stability depends on properties of H! M. Schwager, A Gradient Optimization Approach to Adaptive Multi-Robot Control, Ph.D. Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, September, 2009.
Stability of Dynamical Systems Entire classes on this topic! Straightforward for linear systems Non-Linear systems: Find function that bounds the system Prove properties of this function Keyword: Lyapunov stability and its variations Convexity and Continuity
Lyapunov Stability System Find function V(x) >= 0 and V(x)=0 only if x=0 V’(x(t)) < 0 (negative definite) System is Lyapunov stable if V(x) with the above properties exist V(x) is called the Lyapunov candidate Common analogy, spring-damper, energy of the system the Lyapunov candidate, energy only decays, system stable
Other important concepts: Convexity A set is convex when all points on the line between any two points is also on the set Non-convex sets are concave Local minima of convex functions are global minima Convex Set Concave Set Convex function
Other important concepts: Continuity A continuous function f has no abrupt changes A function is Lipschitz continuous if there is a positive b so that Limits the maximum slope A locally Lipschitz first-order differential equation has a unique solution! Lipschitz: Locally Lipschitz:
Voronoi Cost Function Introduce sensory function fover P Example: Oil spill Goal: more robots where fis high Cost functionwith the cost of measuring a value at q from pi Optimal solution:minimize H(P) Q pi
Voronoi Cost Function Goal: mini has only one solution (closest robot to q) Voronoi tesselation of q with cells Vi Result: V(s) consisting of all points closer tosthan to any other site J. Cortes, S. Martınez, T. Karatas, and F. Bullo. Coverage control for mobile sensing networks. IEEE Transactions on Robotics and Automation, 20(2):243– 255, April 2004.
Optimizing the Voronoi Cost Function Solve Calculate 1st derivative Let , e.g. light sensor Define Move pi to the centroid of its Voronoi cell! Mass of Vi First moment of Vi Centroid of Vi S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, 28(2):129–137, 1982.
Course question The gradient is given by What is the control law for robot i ui=
Proof of Convergence (Sketch) Theorem: converges assymptotically to the set of centroidalVoronoi Configurations Tool: LaSalle’s invariance principle Requirements:
Q, P are bounded
P invariant under control law -> trajectories are bounded
Control law locally Lipschitz
From theory to practice Voronoi neighbors are not communication neighbors Approximate decomposition (noise on position information) Time discrete execution vs. continuous dynamics Voronoi Neighbors are not necessarily communication neighbors
Summary Convergence can be proven for a subset of reactive control laws Tools: gradient descent on cost function Key: encoding of the problem into an analytically tractable cost function Tricks: Voronoi decomposition Develop applications that can be broken down into known systems
This week Other important classes of reactive multi-robot problems with provable properties Friday: Prairie-Dog, hand in results from last week to Till Sunday: Project proposal, 1 page (max), 12pt
Project Proposal Everybody should think about a project What is the objective? I want to test hypothesis A I want to apply method B to problem C I want to prove conjecture D What is the method? This is about coming up with a scientific research project in the domain of this course You don’t have to do this project! We will put together teams next week to work on feasible projects