Synchronization in sensor networks Synchronization in sensor - - PowerPoint PPT Presentation

Synchronization in sensor networks Synchronization in sensor networks

Jie Gao

Computer Science Department Stony Brook University

Papers Papers

• [Mirollo90] M. Mirollo and S. Strogatz. Synchronization of

pulse-coupled biological oscillators, SIAM J. Applied Math., 50(6):1645-1662, 1990.

• [Lucarelli04] D. Lucarelli and I. Wang, Decentralized

synchronization protocols with nearest neighbor communication, Sensys’04.

• [Werner05] G. Werner-Allen, G. Tewari, A. Patel, M. Welsh,
• R. Nagpal, Firefly-inspired sensor network synchronicity

with realistic radio effects, Sensys’05.

• Many slides are from G. Werner-Allen’s talk in Sensys 2005.
• http://www.eecs.harvard.edu/~werner/

What is What is synchronicity? synchronicity?

Synchronicity: the ability to organize simultaneous collective action ...contrast with... Time Synchronization: the ability to establish a common time base allowing events to be time-stamped in a meaningful way

borrowed from G. Werner-Allen’s talk in Sensys 2005

Natural Synchronicity Natural Synchronicity

Cardiac Cells Fireflies!

borrowed from G. Werner-Allen’s talk in Sensys 2005

Fireflies Fireflies

Imagine a tree 35 or 40 feet high…, apparently with a firefly on every leaf and all the fireflies flashing in perfect unison at the rate of about three times in two seconds, the tree being in complete darkness between flashes… From H. M. Smith, Science 82 (1935), p.151.

Pulse Coupling Pulse Coupling

Each node: 1) Is an oscillator 2) Periodically emits a pulse 3) Adjusts the phase of its pulse by

• bserving other pulses

borrowed from G. Werner-Allen’s talk in Sensys 2005

Nice Properties Nice Properties

Each node observes the others’ actions and try to align itself. No leaders No “absolute clock” No global information No routing Very simple

borrowed from G. Werner-Allen’s talk in Sensys 2005

Synchronicity in Synchronicity in sensornet sensornet

Network Timer

Useful for:

coordinated sampling; network-level duty cycling; coordinate transmission to avoid

interference;

wake up schedule; etc...

borrowed from G. Werner-Allen’s talk in Sensys 2005

T

A B

Coupled Oscillators : Coupled Oscillators : Strogatz Strogatz

Example of system dynamics: 1) Nodes move together at a fixed rate from 0 to T 2) When nodes reach T they “fire”, return to 0 3) Each nodes base period is T 4) Overhearing a “fire” moves a node forward

borrowed from G. Werner-Allen’s talk in Sensys 2005

T

A B

Coupled Oscillators : Coupled Oscillators : Strogatz Strogatz

borrowed from G. Werner-Allen’s talk in Sensys 2005

T

A B

Coupled Oscillators : Coupled Oscillators : Strogatz Strogatz

borrowed from G. Werner-Allen’s talk in Sensys 2005

T

A B

Coupled Oscillators : Coupled Oscillators : Strogatz Strogatz

borrowed from G. Werner-Allen’s talk in Sensys 2005

T

A B

Coupled Oscillators : Coupled Oscillators : Strogatz Strogatz

borrowed from G. Werner-Allen’s talk in Sensys 2005

T

A B

Coupled Oscillators : Coupled Oscillators : Strogatz Strogatz

borrowed from G. Werner-Allen’s talk in Sensys 2005

T

A B

tA

A hears B fire at tA

Coupled Oscillators : Coupled Oscillators : Strogatz Strogatz

borrowed from G. Werner-Allen’s talk in Sensys 2005

T

A B

tA

A hears B fire at tA A jumps to tA'=tA+(tA)

A

tA' (tA)

Coupled Oscillators : Coupled Oscillators : Strogatz Strogatz

borrowed from G. Werner-Allen’s talk in Sensys 2005

T

A B

tA

A hears B fire at tA A jumps to tA'=tA+(tA)

A

tA' (tA)

“Jump Function”

Coupled Oscillators : Coupled Oscillators : Strogatz Strogatz

borrowed from G. Werner-Allen’s talk in Sensys 2005

T

B

A hears B fire at tA A jumps to tA'=tA+(tA) B returns to 0

A

Coupled Oscillators : Coupled Oscillators : Strogatz Strogatz

borrowed from G. Werner-Allen’s talk in Sensys 2005

T A hears B fire at tA A jumps to tA'=tA+(tA) B returns to 0

B A

Coupled Oscillators : Coupled Oscillators : Strogatz Strogatz

borrowed from G. Werner-Allen’s talk in Sensys 2005

T A hears B fire at tA A jumps to tA'=tA+(tA) B returns to 0

B A

Coupled Oscillators : Coupled Oscillators : Strogatz Strogatz

borrowed from G. Werner-Allen’s talk in Sensys 2005

T A hears B fire at tA A jumps to tA'=tA+(tA) B returns to 0

B A

Coupled Oscillators : Coupled Oscillators : Strogatz Strogatz

borrowed from G. Werner-Allen’s talk in Sensys 2005

T A hears B fire at tA A jumps to tA'=tA+(tA) B returns to 0

B A

Coupled Oscillators : Coupled Oscillators : Strogatz Strogatz

borrowed from G. Werner-Allen’s talk in Sensys 2005

T B hears A fire at tB

B A

tB

Coupled Oscillators : Coupled Oscillators : Strogatz Strogatz

borrowed from G. Werner-Allen’s talk in Sensys 2005

T B hears A fire at tB B jumps to tB'=tB+(tB)

B A

tB

B

tB' (tB)

Coupled Oscillators : Coupled Oscillators : Strogatz Strogatz

borrowed from G. Werner-Allen’s talk in Sensys 2005

T B hears A fire at tB B jumps to tB'=tB+(tB) A returns to 0

A B

Coupled Oscillators : Coupled Oscillators : Strogatz Strogatz

borrowed from G. Werner-Allen’s talk in Sensys 2005

Goal: Synchronicity Goal: Synchronicity

t=0

T

A B

borrowed from G. Werner-Allen’s talk in Sensys 2005

t=0

T T

A B

T

A B

t=T

borrowed from G. Werner-Allen’s talk in Sensys 2005

Goal: Synchronicity Goal: Synchronicity

t=0

T T

A B

T

A B

T

A B

t=T t=2T

borrowed from G. Werner-Allen’s talk in Sensys 2005

Goal: Synchronicity Goal: Synchronicity

t=0

T T

A B

T

A B

T

A B

T

A B

t=T t=2T t=3T

borrowed from G. Werner-Allen’s talk in Sensys 2005

Goal: Synchronicity Goal: Synchronicity

The Jump Function The Jump Function

Call (t) the Jump Function

Synchronicity emerges when (t) is monotonically increasing: If t1 > t2 then (t1) > (t2) Intuitively, as a node gets closer to firing other firing events affect it more strongly

Note that a node cannot jump past T! If t' = t' + (t) > T the node fires and returns to 0

borrowed from G. Werner-Allen’s talk in Sensys 2005

Theoretical Results Theoretical Results

We will prove the simple case of 2 oscillators. Theorem: the phase difference converges to 0 under some favourable conditions. First let’s start from the oscillator model.

Model of oscillator Model of oscillator

1.0 1.0 x X=f(φ): smooth, monotonically increasing, and concave

• down. f ’ >0, f ’’ <0.

A B φ X=f(φ) Energy function Define phase of A: φA, as the distance to origin. We start with (φA, φB)=(0, φ).

Model of oscillator Model of oscillator

1.0 1.0 x X=f(φ): smooth, monotonically increasing, and concave

• down. f’>0, f’’<0.

A B 1-φ X=f(φ)

Model of oscillator Model of oscillator

1.0 1.0 x X=f(φ) X=f(φ): smooth, monotonically increasing, and concave

• down. f’>0, f’’<0.

A B 1-φ ε A jumps to g(ε+f(1-φ)) where g=f -1 Boost-up energy

Model of oscillator Model of oscillator

Firing map: h(φ)=g(ε+f(1-φ)). After B fires, the system moves from (φA, φB)=(0, φ) to a current state (h(φ), 0). A jumps to g(ε+f(1-φ)) where g=f-1 1.0 1.0 x X=f(φ) A B 1-φ ε Boost-up energy

Model of oscillator Model of oscillator

Firing map: h(φ)=g(ε+f(1-φ)). After B fires, the system moves from (φA, φB)=(0, φ) to a current state (h(φ), 0). After A fires, the system moves to (0, h(h(φ))). Now we finish a full loop. Return map: R(φ)=h(h(φ)). R(φ) is the new phase difference after 2 firings. Theorem: R(φ) has a fixed point which is a repeller.

Model of oscillator Model of oscillator

Theorem: R(φ) has a fixed point which is a repeller. R(φ*)=φ*. When φ<φ*, R(φ)<φ. When φ>φ*, R(φ)>φ. “Repeller”: no matter where you start you are always pushed away from the fixed point --- not a stable fixed point. Whenever φ is pushed to 0 or 1, then it’s done!

Dynamics Dynamics

Goal: prove that R(φ)= h(h(φ)) has a fixed point, with h(φ)=g(ε+f(1-φ)). We will prove that h(φ) has a fixed point φ*, I.e., h(φ*)=φ*. It’s obvious that R(φ*)=φ*. Now take F(φ)=φ-h(φ); we argue F(φ)=0 for a value φ*.

Dynamics Dynamics

Observation 1: F(δ)= δ-g(ε+f(1-δ)), with δ very small. Thus ε+f(1-δ)>1, I.e., A will fire, F(δ)= δ-1 <0. 1.0 1.0 x X=f(φ) A B 1-φ ε Boost-up energy

Dynamics Dynamics

Observation 1: F(δ)= δ-g(ε+f(1-δ)), with δ very small. Thus ε+f(1-δ)>1, I.e., A will fire, F(δ)= δ-1 <0. Observation 2: F(δ)= δ-g(ε+f(1-δ)), with δ very close to 1. Now F(δ) >0. Then there must be a point φ* in (0, 1) such that F(φ*)=0 F(φ) δ h-1(δ) φ

Dynamics Dynamics

First goal: prove h’<-1. Now h(φ)=g(ε+f(1-φ)). Just do calculus. Replace f(1-φ) by u. Since g is the inverse of f, then g’>0 and g’’>0. So g’(ε+u)>g’(u). Thus h’<-1. QED

Dynamics Dynamics

So we have R(φ*)=φ*. Now we argue it’s a repeller. Claim: R’(φ)=h’(h(φ))h’(φ)>1, since h’(φ)<-1. Thus when φ<φ*, R(φ)<φ. When φ>φ*, R(φ)>φ. There is only 1 fixed point in the interior (0, 1). The system has simple dynamics. No matter where you start, you are pushed to 0 or 1. QED. R(φ) φ* φ

Examples Examples

Choose

Examples Examples

Choose The convergence rate is exponential. The # iterations taken for it to converge is Initial phase diff.

n oscillators n oscillators

If we have n oscillators, we have 2 models: All-connected model: everyone hears the firing of everyone else, I.e., a complete graph. Mirollo &

Strogatz1.

Connected model: with a connected graph G, when someone fires, only the neighbors hear it and adjust their phases. Lucarelli & Wang2. Both converge!

n oscillators, all connected n oscillators, all connected

Convergence rate.

From Theory to Reality From Theory to Reality

Theory: Nodes can instantaneously observe firing events Reality: Communication latencies due to MAC Problems: Time that a node hears a firing message is delayed from when sender fired Firing messages may arrive out of order Firing messages may arrive too late to be useful

borrowed from G. Werner-Allen’s talk in Sensys 2005

Solutions: Measure Delays Solutions: Measure Delays

Problem: Time that a node hears a firing message is delayed from when sender fired Solution: Quantify send delay through link-level timestamping A RADIO MEDIUM RADIO B

tF tS delay = tS-tF tS tF = tS - delay

borrowed from G. Werner-Allen’s talk in Sensys 2005

Solutions: Solutions: Reachback Reachback Algorithm Algorithm

Problems: Firing messages may arrive out of order Firing messages may arrive too late to be useful Solution: Relax instantaneous communication requirement

borrowed from G. Werner-Allen’s talk in Sensys 2005

Reachback Reachback Firefly Algorithm (RFA) Firefly Algorithm (RFA)

Nodes do not react immediately to neighbors' firings 1) During every firing period each node collects overheard firing messages 2) At the end of the firing period the node applies a cumulative jump based on all overheard messages to the next firing period

Nodes take a single jump at the beginning of each firing period equal to the sum of all the jumps they would have taken in the previous firing period

borrowed from G. Werner-Allen’s talk in Sensys 2005

T

A B

Coupled Oscillators : RFA Coupled Oscillators : RFA

C

borrowed from G. Werner-Allen’s talk in Sensys 2005

T

A B C

borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : RFA Coupled Oscillators : RFA

T

A B C

borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : RFA Coupled Oscillators : RFA

T

A B C

borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : RFA Coupled Oscillators : RFA

T

A B C

A hears C fire at tC

borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : RFA Coupled Oscillators : RFA

T

A B C

A hears C fire at tC

borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : RFA Coupled Oscillators : RFA

T

A B C

A hears C fire at tC

borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : RFA Coupled Oscillators : RFA

T

A B C

A hears C fire at tC

borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : RFA Coupled Oscillators : RFA

T

A B C

A hears C fire at tC

borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : RFA Coupled Oscillators : RFA

T

A B C

A hears C fire at tC A hears B fire at tB

borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : RFA Coupled Oscillators : RFA

T

A B C

A hears C fire at tC A hears B fire at tB

borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : RFA Coupled Oscillators : RFA

T

A B C

A hears C fire at tC A hears B fire at tB

borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : RFA Coupled Oscillators : RFA

T

A B C

A hears C fire at tC A hears B fire at tB

borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : RFA Coupled Oscillators : RFA

T

A B C

A hears C fire at tC A hears B fire at tB

borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : RFA Coupled Oscillators : RFA

T

A B C

A hears C fire at tC A hears B fire at tB

borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : RFA Coupled Oscillators : RFA

T

A B C

A hears C fire at tC A hears B fire at tB A fires,

borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : RFA Coupled Oscillators : RFA

T

A B C

A hears C fire at tC A hears B fire at tB A fires, returns to 0

borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : RFA Coupled Oscillators : RFA

T

A B C

A hears C fire at tC A hears B fire at tB A fires, returns to 0 A computes single jump based on tC and tB.

A

In this case TOTAL = (tB - (tC)) + (tC) Not commutative! TOTA

L

borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : RFA Coupled Oscillators : RFA

RFA: Phase Plot RFA: Phase Plot

No coupling. A node follows its internal time.

RFA: Phase Plot RFA: Phase Plot

Perfect model: a node jumps whenever it hears a firing.

RFA: Phase Plot RFA: Phase Plot

A node cumulates the jumps.

RFA Evaluation Strategy RFA Evaluation Strategy

Evaluation proceeded on three fronts: Theory: Prove that RFA converges for 2 oscillators. For N nodes’ case, not

proven.

Simulation: Experiment with parameter choice in TinyOS simulator MoteLab: Run experiments on real sensor network testbed (MoteLab)

borrowed from G. Werner-Allen’s talk in Sensys 2005

Let φA be the phase of oscillator A The Return Map R(φA, φB) describes how the phases change over time A B t = 0

φA φB

t = T A

φA'

B

φB'

(φA', φB') = R(φA, φB)

borrowed from G. Werner-Allen’s talk in Sensys 2005

Proof Technique : Return Map Proof Technique : Return Map

Let φA be the phase of oscillator A The Return Map R(φA, φB) describes how the phases change over time A B t = 0

φA φB

t = T A

φA'

B

φB'

(φA', φB') = R(φA, φB) t = 2T A

φA''

B

φB''

(φA'', φB'') = R(φA', φB')

...etc...

borrowed from G. Werner-Allen’s talk in Sensys 2005

Proof Technique : Return Map Proof Technique : Return Map

Oscillator model Oscillator model

A has just fired, B will record the time (φB) and execute the jump after its next firing. Take f(φ) as lnφ, the jump function is ∆(φB)= g(ε+f(φB))- φB=(eε-1)φB.

Proof Technique : Return Map Proof Technique : Return Map

Let φA be the phase of oscillator A The Return Map R(φA, φB) describes how the phases change over time A B t = 0

φA φB Now when B fires, A has moved forward 1-φB, with the current phase φA+1-φB.

A

φA+1-φB

B

∆(φB) B fires, and moves to ∆(φB). ∆ t = 1-φB

A B

φA+1-φB ∆ t = 1-φB

Proof Technique : Return Map Proof Technique : Return Map

Let φA be the phase of oscillator A The Return Map R(φA, φB) describes how the phases change over time A B t = 0

φA φB

A B

After A’s next firing, A’s phase is ∆(φA+1-φB), and B’s phase is ∆(φB)+1-(φA+1-φB)= ∆(φB)+ φB-φA. ∆ t = 1-φB

A

φA+1-φB

B

∆(φB) ∆ t = φB-φA

A B

∆ t = φB-φA

Dynamic system Dynamic system

After a full round, the phases of A, B change from (φA, φB) to (∆(φA+1-φB), ∆(φB)+ φB-φA).

In other words, This is a linear dynamic system. It has 1 fixed point which is a repeller.

Dynamic system Dynamic system

The system is stable. No matter where the phases start, it will be eventually driven to sync.

Theoretical Results Theoretical Results

For two oscillators A, B, with phases φA, φB, δ=φB-φA Using the Return Map R(φA, φB) we prove that:

• 1. Rate of synchronization:

When nodes take smaller jumps they take longer to reach synchronicity

• 2. The state where (φB-φA) = 0 is a stable fixed point

Once the system reaches synchronicity small perturbations do not disturb it

borrowed from G. Werner-Allen’s talk in Sensys 2005

Implementation Details Implementation Details

RFA implemented as a single TinyOS component Originally targeted at the Crossbow MicaZ mote but since easily ported to Mica2, TelosA/B/TMote Sky TMote Sky MicaZ

borrowed from G. Werner-Allen’s talk in Sensys 2005

Implemented Jump Function Implemented Jump Function

The implementation used a jump function (t) = t / FFC FFC is the Firing Function Constant Smaller FFC means large jumps – faster convergence.

borrowed from G. Werner-Allen’s talk in Sensys 2005

Simulation Goals Simulation Goals

Debug Implementation TOSSIM is a simplified world: no clock skew, no radio contention, no computation time, etc. Experiment with Parameter Choice How does the (Firing Function Constant) affect TS? Experiment with Topology How do less-connected topologies affect TS? How Well Does it Work? Achievable accuracy

borrowed from G. Werner-Allen’s talk in Sensys 2005

Evaluation Metrics Evaluation Metrics

How long does it take to reach synchronicity? Time to Sync (TS): time it takes for system to reach state where nodes are clustered within 0.1 sec After synchronicity how tight are firing groups? Group Spread (GS): after synchronicity maximum time delta between two nodes in the same firing group N% Percentile Group Spread (N% GS): after synchronicity N% of firing groups had group spread less than this amount

borrowed from G. Werner-Allen’s talk in Sensys 2005

TOSSIM Results: TOSSIM Results: Time to Sync Time to Sync

(grid topology) FFC (Firing Function Constant) TS (sec) 20 400 600 800 1000 1200 200 50 100 500

borrowed from G. Werner-Allen’s talk in Sensys 2005

(all-to-all topology) FFC (Firing Function Constant) 70 100 500 750 1000 1 2 3 4 50% GS 90% GS (msec)

borrowed from G. Werner-Allen’s talk in Sensys 2005

TOSSIM Results: TOSSIM Results: Time to Sync Time to Sync

MoteLab MoteLab Goals Goals

More Reality MoteLab is a testbed of real sensor network nodes 1) Lossy & asymmetric links 2) Clock skew & drift How Well Does it Work? Achievable accuracy

borrowed from G. Werner-Allen’s talk in Sensys 2005

Location of MoteLab nodes Links are colored according to quality. Connectivity graph showing links better than 80%.

borrowed from G. Werner-Allen’s talk in Sensys 2005

FTSP Backbone FTSP Backbone

Problem: No Global Clock on MoteLab Solution: FTSP: Flooding Time Synchronization Protocol1 Developed at Vanderbilt Promises time-stamping accuracies in the microsecond range We used FTSP in order to evaluate the accuracy of our RFA algorithm on MoteLab

• 1. “The Flooding Time Synchronization Protocol”, Maróti et. al, Sensys 2004.

borrowed from G. Werner-Allen’s talk in Sensys 2005

FTSP Errors FTSP Errors

Data collected on MoteLab

Max FTSP Error (sec) (cumulative distribution frequency)

borrowed from G. Werner-Allen’s talk in Sensys 2005

Probability

RFA: Accuracy RFA: Accuracy

GS (sec)

borrowed from G. Werner-Allen’s talk in Sensys 2005

RFA: Time to Sync RFA: Time to Sync

FFC 100 177 250 333 500 572 1000 1793 TS (sec)

As in simulations TS∝ ∝ ∝ ∝FFC

borrowed from G. Werner-Allen’s talk in Sensys 2005

RFA: Phase Plot RFA: Phase Plot

(24 Nodes, 500FFC)

Relative Phase (sec) Time (sec)

borrowed from G. Werner-Allen’s talk in Sensys 2005

Conclusions Conclusions

Biologically-inspired algorithms can be implemented on sensor network hardware. Totally distributed and local algorithm with global behaviour.

borrowed from G. Werner-Allen’s talk in Sensys 2005

Next class Next class – – stay tuned! stay tuned!

• DNA self-assembly.

Paper presentation on Thursday 11/7/06 Paper presentation on Thursday 11/7/06

• [Zhu05] Y. Zhu and R. Sivakumar, Challenges:

Communication through Silence in Wireless Sensor Networks, MobiCom'05.

• [Nagpal06] Radhika Nagpal, Self-Organizing

Shape and Pattern: From Cells to Robots, IEEE Intelligent Systems 21(2), 2006.

• Present the main idea. Don’t need to cover all

the details.

• Tell us just 1 thing that you learn from this paper.