Abstract- Applying Trilateration concept for target localizing demands measuring distance between target and at least three recognized reference points. The errors of measured radiuses due to the accuracy of applied measurement technique resulted large divergence in Trilateration localizing.
The presented paper utilized standard Trilateration calculations with a view to likely measurement error, and then a new algorithm is presented to improve calculated target's position. The new Error Correction Algorithm is a good alternative for standard Trilateration calculations. It is easy to implement, isolated from applied measurement technique.
Since ECA is isolated from the applied distance measurement technique, it examined the practical result of distance measurement based on received signal strength (RSS) technique. Although practical distance measurement had great offset from actual values but ECA could lessen the offset error by more than 50%.
Keywords-component; Error Correction; Trilateration
Position estimation is implemented in various types of application such as mapping by civil engineers, animal tracking, fleet tracking, mobile tracking, navigation systems, etc. Regardless of how the implementations, the similarity of all applications is that they all supposed to calculate the relative coordinate of their target.
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Determining a point's coordinate in the specific coordinate system requires identifying different components. For example, a point in Cartesian coordinate system is recognized by at least two values of horizontal and vertical axis. Polar system uses as a two-Dimensional (2D) coordinate system and determines the point with two parameters, distance from a point to a center coordinates (r) and the angle between the line drawn from the centre point and axis length. Cartesian coordinate system is preferred for target localizing applications because of its simplicity.
Several studies have been done to increase Trilateration quality [2-7] but a limitation through them.
In this paper, firstly, Trilateration and its mathematical-computations are reviewed. Secondly, in part IV error possibility is discussed for practical applications and offset error is analyzed in Trilateration. After that, a new algorithm is investigated for positioning target in Trilateration. It is evaluated for practical data of experiments in part VI. Finally, results from the new algorithm are compared with traditional Trilateration computation and indicate the proposed error correction algorithm (ECA).
In Trilateration, positioning a point in Cartesian system requires measuring its distance in different alignments respect to the base point. For 2-D positioning, point's distance is measured apart from two axes (x, y) .
Having three distances and three references' coordinate, then target's position is calculated by the following equation system:
(x1, y1), (x2, y2), (x3, y3) are the references' coordinate and (xt, yt) is target position. r1, r2, r3 are three measured distance between target and the respective reference point. As positioning is relative therefore one of the reference points is deemed as coordinate origin. If the first reference node is taken as the coordinate origin then (1) is simplified to (2):
The above equation system is written from three circles with the radius of the measured distance and center of the reference points. Intersection between these three circles is calculated as target position. A solution for this equation system is matrix method. Equation (1) should be written in matrix form like:
T is the unknown values of target position in (2). A and B are define as the following:
Simply and are calculated by (4):
Calculated position with (4) is exactly the circles' intersection if three radiuses were accurately measured; otherwise calculated position with (4) contains some offset error. Offset error in horizontal or vertical alignments is dependent to that in which measurement error has occurred.
If δ1, δ2, δ3 suppose as the offset error of r1, r2, r3 respectively then:
ra1, ra2 and ra3 are three actual radiuses and rm1, rm2, rm3 are the relative measured radiuses.
(x't , y't) is the calculated point which it might have (Dx, Dy) offset error.
Assuming maximum of offset error (δE) as the expected error then positioning offset error is calculated by the following equation:
Equation (6) shows that positioning offset error is associated with references' coordinates, distance from target to references (radiuses) and occurred offset error (δE).
Practically, occurred error for three times of distance measurement has rarely same value, perhaps the occurred error is too small and negligible. Besides, variation of offset error during several measurements is associated with the measurement technique and its accuracy. Here is an example of target placement within three references, and different states of error incidence are discussed.
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New Error Correction Algorithm
Error Correction Algorithm (ECA) solution is drawing three circles with the measured radiuses and then determines target location based on circles overlaps. There might be no cross among circles or no intersection shared between three circles therefore different states of circle's overlap are explained in the next section and the best answer is decided for each case.
Considering the inaccuracy of distance measurement, circles might not intersect each others at a unique point or may never intersect others.
The most common error state through Trilateration is when three circles have intersected but it's not focused on one point. It might happen if distance measurement technique is accurate enough and stable to have similar offset error in three times. In this state each circle has two intersections with other circles and 6 points are indicated as the circles intersection. Three of these points are placed closer to each other. Suggested location for target by ECA is the centre of these three points. Fig. 1 describes state 1 if target is in between three nodes but Fig. 2 shows this state when target is placed outside the invented triangle of three nodes.
Fig. 1- Trilateration Error State 1, if target is surrounded by three nodes
Fig. 2- Error state when target is placed outside the invented triangle of three nodes
In this state one of the circles has two intersections with others but the other two circles don't meet each other. In this case 4 points are as the intersections of circles. These four intersects are all stated on one of the circles, two of which are shared with the last circle. Centre of two closer intersections is the ECA suggestion for this error state (see Fig. 2). One of the Selected Intersection Points (SIP) fits in two of the circles and second SIP must belong to the third circle.
Figure 3 - Trilateration Error State 2
State 3 is when only two circles have intersected and the other circle is so small that cannot reach the other circles or it is very big and two other circles are surrounded by it (Fig. 3). ECA solution for this state is to redraw circles with the new radius. Firstly, the single circle which has no intersection with two others will be recognized, then its radius will be added or subtracted by Δ as the new calculated radius for this circle.
Redrawing this circle, intersection will be checked again. After first correction, if circles' state might change to the State 1 then ECA continues the routine of state 1. After the first correction, state of circles' intersection doesn't change then the other two circles also will be redrawing. Redrawing of two crossed circles is the opposite of the single circle's radius correction. For example, if the single circle's radius is added with Δ and two crossed circles' radius will be subtract by Δ in order to acquire the corrected radius.
Δ is the correction factor which is correlated to maximum offset error (δ). When two radiuses are changed by Δ factor at the same time, then Δ is considered as δ/2 for this error state.
After the second correction, if circles' state doesn't change, which rarely happen in accurate enough practical applications, then ECA won't ensure the localization accuracy.
No intersection between circles is considered as an ambiguous situation. In this case ECA has a few choices to locate the target. Since three circles will definitely change therefore Δ factor is considered as δ/3 for this state.
ECA solution for this state is similar to state 3. The difference for this state is that all circles will be redrawing at the same stage.
Circles' radiuses are compared with distant between their centre points.
If R1+R2 < d1 then R1 and R2 will add with Δ, otherwise it subtract with Δ. The same is done for R1 and R3 but this time only R3 will be changed because R1 has changed once. Note that R3 is comparing with R1 after it has changed, therefore R3 will compare with the old value of R1.
Figure 4 - Trilateration Error State 3
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Figure 5 - Trilateration Error State 4
Result and Discussion
The following examples are comparing the standard Trilateration equations' with ECA.
Initially ECA is examined for the state of partial error occurrence. Three base nodes are considered at S1, S2, S3 points. Maximum offset error is δ and R1, R2, R3 are actual distance from target to S1, S2, S3 respectively. r1, r2, r3 are practically-measured radius and is the exact position of target. PECA is target's coordinate calculated by ECA and Pm is calculated by matrix solution in (2).
Examples are presented in table I and graphically presented in Fig. 6.
Calculated values of last collumns (Matrix Answer) are calculated with (2) and offset values are distance from actual position apart from calculated position.
Although in all above examples, ECA has improved the calculated position for target but to show that the examples are probable in practice, then practically distance measurement was done with the RSS technique implemention. A transmitter was mounted on a target and received signal strength was measured to calculate distance based of a practical propagation model.
Results of three practical Trilateration samples are compared in table II. Target position is calculated by matrix method with (2), positions also have computed with the ECA. Results prove that positioning offset is improved in ECA calculations. Fig. 7, Fig. 8 and Fig. 9 present the results graphically. Comparing the actual position for all samples, clearly ECA is improving the target estimation. For the cases with the larger error in distance measurements, ECA has reduced the offset error to less than 50% (sample 1 and 2).
In sample 1 maximum deviation was occurred for r3 and it was 4.49 meter, matrix answer has greater offset for Y axis. While the greatest offset with matrix computation is 6.82m, it was reduced to 3.68m by ECA and shows that error offset is about 47% of previous error. Similar improvement was observed while 1.13m offset in matrix answer was decreased to 0.73m by ECA and error is reduced to about 65% of matrix error.
Table I- Actual position, ECA and Matrix answer; offset is distance from actual position apart from the re;ated answer (ECA or Matrix)
Fig. 6- When three circles have intersection but more than one
Table II- Practical sampling result and localizing target with ECA and Matrix calculations
(1.68 , -6.82)
(2.0 , 0 )
(-0.32 , -6.82)
( 3.55 , 1.45)
(3.56 , 1.56 )
(3.5 , 2.8)
(0.05 , 1.35)
(0.06 , 1.24)
(-0.13 , 5.06)
(-0.5 , 6.21)
(-0.37 , 1.13)
Fig. 7- (a)-ECA estimation, (b)-Comparison between ECA and Matrix estimation for Sample 1
Fig. 8- (a)-ECA estimation, (b)-Comparison between ECA and Matrix estimation for Sample 2
Fig. 9- (a)-ECA estimation, (b)-Comparison between ECA and Matrix estimation for Sample 3
ECA advantages are that it is a general solution to improve target localizing via Trilateration. It is isolated from distance measurement technique but its quality is depends on the measurement accuracy. ECA can diagnose whether the error is happened or not and try to correct the coordinate only in incidence of error. Therefore, it is the superior algorithm for target estimation.
ECA seems to be more efficient if target is placed out of the area surrounded by nodes (Fig. 7 and Fig. 9). Nevertheless ECA doesn't claim to compute the exact position of target but at least it can solve the ambiguous states and is capable to reduce the offset error between actual and calculated points by about 50%. The weakness of ECA is that it doesn't support hybrid technique and considers the same error possibilities for all nodes.