Distributed Algorithms for Guiding Navigation across a Sensor Network

Qun Li, Michael DeRosa, and Daniela Rus Dartmouth College

MOBICOM 2003

Issue Find an optimal path to guide a user

through a region or to a goal, and avoid dangerous area through cooperation among sensors

Sensors in dangerous area

Overview of the proposed solution

Basic idea Model detected dangerous events as

obstacles in dynamic robot motion planning problem

Static robot motion planning problem Guiding a robot from a source to a destination

location while avoiding all encountered obstacles Dynamic robot motion planning :

Dynamic path planning is required when only partial a priori information is available about the obstacles, and the environment is unpredictable and time-varying

Overview of the proposed solution

Apply potential field theory to the navigation problem The goal has the lowest potential to attract moving object

(attractive force) Obstacles (area with detected danger) raise potential values to

repulse moving object (repulse force) Moving object follows the gradient of the artificial potential

field (from high potential level to low potential level) Design issue: local minima

Some specific types of potential function, e.g., harmonic function, are used to avoid local minima

Artificial potential fieldAreas of detected dangerous events in a (100,100) grid

attractive force

repulse force

Algorithm Algorithms to solve the navigation problem

Algorithm 1: establish potential field according to the detected dangerous events in the network

Algorithm 2; establish potential integration field toward the goal

Guarantee no local minimal Algorithm 3: guide moving object’s next step

movement based on gradient of the potential field established through Algorithm 2

Assumption Distance between sensors are measured

through hop-count The moving object is equipped with a device

that can talk to the field sensors The moving object queries the nearby sensors for

next moving direction periodically Sensors know their location All sensor has the same transmission range R Sensors detect information of dangerous event

in the area they cover E.g. a sensor detect high temperature caused by fire

in the nearby area

Algorithm 1: Potential Field

Algorithm 1: establish an artificial potential field based on detected dangerous events

Three steps of Algorithm 11. Broadcast of initial potential value (max) from

source sensors detecting danger2. Update of received potential values at the

neighbors based on hop distance to the source The potential value received at sensor i from a source j is inverse of

the square of the shortest hop distance from i to j Potential =

Potential values from all sources are added up at each sensor i

3. Flooding of received potential values with updated hop distance at each hop

2

1

hop

Algorithm 1: Potential Field

(a) Initial phase

Potential

(A , 0)

(b) one-fire event detected (c)

(d)

Source ID= B Hop count=0

Node ID

(B , 0) (C , 0)

(D , 0) (E , 0) (F , 0)

(G , 0) (H , 0) (I , 0)

(A , 0) (B , 0) (C , 0)

(D , 0) (E , 0) (F , 0)

(G , 0) (H , 0) (I , 0)

goalgoal

fire

(A , 1) (B , max) (C , 1)

(D , 0) (E , 1) (F , 0)

(G , 0) (H , 0) (I , 0)

goal

(1)

(2)

EX:

hopB=hop+1 = 0+1=1

potB=1/(hopB)2=1

potC= potC+ potB=0+1=1

Potential =2

1

hop

(A , 1) (B , max) (C , 1)

(D , 1/4) (E , 1) (F , 1/4)

(G , 0) (H , 1/4) (I , 0)

goal

From node C:

hopB(C)=hop(C)+1 = 1+1=2

potB(C)=1/(hopB(C))2=1/4From node E:hopB(E)=hop(E)+1 = 1+1=2potB(E)=1/(hopB(E))2=1/4

potC= potC+ min(potB(C), potB(C))= 0+1/4=1/4 (d) Final

(A , 1) (B , max) (C , 1)

(D , 1/4) (E , 1) (F , 1/4)

(G , 0) (H , 1/4) (I , 1/9)

goal

Example : Two Fire Event

(a)

(A , 1) (B , max) (C , 1)

(D , 1/4) (E , 1) (F , 1/4)

(G , 0) (H , 1/4) (I , 1/9)

goal

(b)

(A , 1) (B , max) (C , 1)

(D , 1/4) (E , 1) (F , 5/4)

(G , 0) (H , 5/4)(I , max)

goal

EX:

hopI=hop+1 = 0+1=1

potI=1/(hopI)2=1

potF= potF+ potI=1/4+1=5/4

(c)

(A , 1) (B , max) (C , 5/4)

(D , 1/4) (E , 5/4)

(F , 5/4)

(G , 0) (H , 5/4)(I , max)

goal

(d)

(A , 1) (B , max) (C , 5/4)

(D , 13/36) (E , 5/4)

(F , 5/4)

(G , 0) (H , 5/4)(I , max)

goal

(d)

(A , 17/16) (B , max) (C , 5/4)

(D , 13/36) (E , 5/4)

(F , 5/4)

(G , 0) (H , 5/4)(I , max)

goal

Algorithm 2: Potential Integration Field

Algorithm 2: establish potential integration field according to the attraction potential values from the goal G Basic steps of Algorithm 2

1. Broadcast an initialization potential value from the goal sensor G

2. Update each neighbor’s potential value to G A sensor i’s potential value to G is the minimum sum of the p

otential value potk (established through Algorithm 1) of all nodes on a path from g to I

3. Flooding the potential value to G to the whole network with updated hop distance at each hop

pkkg potiP

node alli tog from ppath all

min

Algorithm 2: Potential Integration Field

(a) Potential field

(A , 1) (B , max) (C , 1)

(D , 1/4) (E , 1) (F , 1/4)

(G , 0) (H , 1/4) (I , 1/9)

goal (b)

goal

Goal ID Sender ID Hop count Potential

(G,G,0,0)

EX:

PG = received potential + potH =

0+1/4 = 1/4

1/4

1/4

Algorithm 2: Potential Integration Field

(b)

goal

1/4

1/4

(c)

goal

1/4

1/4EX:

PG = received potential + potI =

1/4+1/9 = 13/36

13/36

5/4

5/4

(d)

goal

1/4

1/4

13/36

5/4

5/4

EX:

From node I : 13/36 + 1/4 = 22/36

From node E: 5/4 + 1/4 = 6/4

PG = min( 22/36, 6/4) = 22/36

22/36

(e)

goal

1/4

1/4

13/36

5/4

5/4

22/36

58/36

Algorithm 2: Established potential field to target

location:

Artificial potential field established by Algorithm 1

Potential field to goal (80,20) established by Algorithm 2

goal

Algorithm 3: guide the moving object to the goal G The moving object queries the nearby sensors w

ith the goal G Sensors respond with their potential to G (PG), t

he predecessor priorG, and the hop distance to G (hopG)

The moving object chooses the location of priorG with minimum PG and hopG as the next step

First choose based on minimum Pg, then use minimum hopg to break ties

Improvements to the basic algorithms The basic algorithms assume bi-directional links (e.g.

the reverse link to priorg) Solution: each sensor purges unidirectional communication

links Sensors keep history of how frequently they receive messages f

rom a neighbor and only keep the links with high frequency Reducing the flooding message

Solution: each sensor wait for sometime before re-broadcasting a message out in algorithm 1 and 2

Since only the message with minimum hop distance in Algorithm 1 and the message with minimum potential value to g in Algorithm 2 are necessary to be re-broadcast, allow sensors to wait for the “minimum” message will reduce the message flooding

Validation of correctness There is no local minima that will stuck the obje

ct Since the potential Pg at sensor i is actually

Proof: for any node k other than g, suppose prior(k) is the sensor that is the predecessor on the minimum path from g to k, Pg = Pprior(k) + potk, where potk> 0. Therefore, for any node k, there at least exist a neighbor prior(k) that has a lower potential to g.

pkkg potiP

node alli tog from ppath all

min

Error in distance measurement Assumption used in the algorithm: hop distance =

physical distance

Analysis model Assume that the neighboring sensor that can

make the most progress toward the destination is chosen to be the next hop for routing.

Suppose the sensors are deployed as two-dimensional Poisson distribution with density

Goal : Find the average progress in each hop See the figure below, assume S is the sender and D

is the destination node. The area

Probability for a sensor at location A to be chosen as the next hop

(1) there is no node to the right of A P1 = probability that no sensors exist in S1 (2) There must be at least one node in that

square area P2 = probability that at least one sensor exist

at location A

S1

S1

1-e(-N) ~ N as N0

R=2

R=1

l-E[l’]

Experiment results Experiment configuration

Mote MOT300 series Atmel ATMEGA103 (4Mhz, 128KB memory, and 4K

RAM) processor 4Mbit flash memory (storage) RF Monolithic 916.50Mhz transceiver (TR1000) wit

h transmission range at 9 inch TinyOS operating system Sensing units:

A photo sensor, a power sensor, and a sound sensor

obstacle

goal

1. Time for a source to send the obstacle info to the whole network

2. Time for all the sensors to obtain the shortest distances to dangerous sources

3. Time to send the goal info to the whole network

4. Time for all sensors to find their safest paths to the goal

1 2 3 4 1 2 3 4

The response time : the period from the time when the topology change occurs to the time when the user finds the path to the goal.