zi ðkÞ¼ xs ðkÞþ w¯ ðkÞ

where xs ðkÞ is the position of the odor source; w¯ ðkÞ denotes the measurement noise, which is a Gaussian process with zero mean and 1 Pk—1ðk — lÞr2  variance; and zi ðkÞ refers to the measurement for xs ðkÞ at time k。 On the basis of the dynamics model

k l¼0

of the position of the odor source (42), we use the Kalman filter to estimate the position of the odor source。

The Kalman filter algorithm consists of two major parts: prediction and estimation, which are described in the  following。

s   ðkÞ¼ ^xsðk — 1Þ

P—i ðkÞ¼ Pi ðk — 1Þ

Estimation:

8 1

ð43Þ

> Ki ðkÞ¼ P—i ðkÞðP—i ðkÞþ RðkÞÞ—

>

><

^xi ðkÞ¼ ^x—i ðkÞþ Ki ðkÞ

 PN     zj   k !

j¼1    ð Þ i

— ð Þ

ð44Þ

s s N s

>

>

>: Pi ðkÞ¼ ðI — Ki ðkÞÞP—i ðkÞ

where ^x—i ðkÞ is a priori position of the odor source at time k for the ith (i 2 lN ) robot given knowledge of the process prior to time k — 1; P—i ðkÞ is a priori estimate error covariance at time k for the ith robot while Pi ðk — 1Þ is a posteriori estimate error covariance at time k — 1 for the ith robot。 Ki ðkÞ is the Kalman gain; RðkÞ is a measurement noise covariance matrix; ^xi ðkÞ is   a

posteriori position estimate at time k for the ith robot; zj ðkÞ is the measurement for the jth robot。

By using the Kalman filter algorithm, ^xi ðkÞ is obtained。 Then, the probable position hi ðkÞ of the odor source can be calcu- lated by (41)。 In order to detect  more  odor  information  to  get  the  accurate  estimation  hi ðkÞ,  the  robot  should  track  the  plume and move along the plume until it  finds  the  source  of  odor。  This  idea  has  been  proposed  in  [23,24],  but  in  this  paper  we  de- rive the new equations that are accurate and different from ones given  by  [23,24]。  In  Fig。  11,  we  suppose  that  the  odor  re- leased by the odor source at the estimated position will move along the wind direction。 Hence, in order to detect the odor information, the robot at the current position  should  move  toward  the  goal  position  that  can  be  chosen  along  the  wind  direc- tion such that  the  goal  position  at  the  x  axis  are  located  in  the  center  between  the  estimated  position  and  the  current  position at  the  x  axis。

Moreover, let hxi ðkÞ and hyi ðkÞ denote the coordinates of the estimated position hi ðkÞ at the x axis and y axis, respectively。 xxiðkÞ and xyi ðkÞ denote the coordinates of the current position xi at the x axis and y axis, respectively。 Similarly, wx and wy are the coordinates of the wind velocity at the x and y axis, respectively。 Then, jxxi ðkÞ—hxi ðkÞj is the coordinate of the goal position at the   x   axis   if   the   estimated   position   is   regarded   as   the   origin。   Next,   according   to   the   rule   of   similar   triangles,