Table 7Relationship between moving step and number of the packages sent (R = 100m).According to the ratio of the message package numbers, we can get the approximate ratio of the consumed energy with different step length. Because the consumed selleck products energy of every package is equal, the product of absolute error and number of the package can be a norm to reflect the relative error of the localization under the same radius but with different step lengths. When R = 100m, the relative error is shown in Table 8. Table 8The relative error under different step lengths (R = 100m).On the basis of Table 8, we find that the relative error is the lowest when the step length is 10m. So, 10m is the best length of steps to get the proper accuracy under relative smaller overhead of communication when R = 100m.
6. The Error AnalysisThe error in the process of localization is an important criterion of the locating performance. The goal of error analysis is finding out the ��source of error�� in order to improve and optimize the schemes in the future work. It is usually that errors are caused by various elements. In this section, we will analyze the main element that leads to the ultimate error of the localization. In our scheme, the largest error is aroused in the process of least square method. The reason is that the referenced point may not be the exact foot point of the trajectory. The error starts at the selection of reference point. Assume that, the length of moving step is a, the distance between the foot of the trajectory’s perpendicular and the reference point��is in the interval [0, a/2].
If the coordinates of the reference points are(xa,ya,za)(xb,yb,zb)(xc,yc,zc)(11)and the coordinates of the foot of the trajectory’s perpendicular are(xa��,ya��,za��)(xb��,yb��,zb��)(xc��,yc��,zc��).(12)As described in Section 3, the direction vectors are(i1,j1,k1)=(1,3,0),(i2,j2,k2)=(?1,3,0),(i3,j3,k3)=(0,0,1).(13)So, (3) is ?(xyz)=((xa+xb3ya?3yb)23(xa?xb+3ya+3yb)6zc).(15)As??converted into(xyz)=(12?12036360001)(xa+3ya?xb+3ybzc)(14) to the specific size of the model, the relationship between the foot of the trajectory’s perpendicular and the reference point is revealed as follows:(xayazaxbybzbxcyczc)=(xa��+��2ya��+3��2zaxb��+��2yb��?3��2zbxcyczc��+��).(16)Equation (15) should be transformed into(xyz)=((xa��+xb��+3ya��?3yb��)2+2��3(xa��?xb��+3ya��+3yb��)6zc��+��).
(17)According to the above, the ultimate error is(2��)2+(��)2=5��,(18)owing to the����[0,a2],(19)the theoretical maximum ultimate errormax?(errorabsolute)=52a,max?(errornormalized)=5a2R.(20)In Table 2, we discovered that a few of errors GSK-3 are greater than the theoretical value as shown in Tables Tables99 and and10.10. This is caused by the imperfection of the model. The balance of size of the space and the model is also a key point of the localization here.Table 9The theoretically absolute error under different length of step.