Experimental Results - TOF Estimation - Experiment Result and Comparison

Chapter 5 Experiment Result and Comparison

5.2 TOF Estimation

5.2.1 Experimental Results

In this section we compare the standard deviation of TOF estimation in Case1 and Case2, where the targets are all stationary. The two maximums algorithm we use here is applied to the same fitting window as in Newton’s method. In Case 1 and Case 2, there are 700 testing frames, and 300 testing frames in Case 3. Note that here we compare the standard deviation of TOF in centimeter unit for analyzing convenience.

Case1:

Figure 5.2-1 : Marker pen, comparison of std of different methods among 5 channels.

Table 5.2-1 : Marker pen, comparison of TOF standard deviation (unit: cm)

TOF

Case 2:

Figure 5.2-2 : Human’s index finger, comparison of std of different methods among 5 channels

Table 5.2-2 : Human’s index finger, comparison of TOF standard deviation (unit: cm)

TOF

5.2.2 Comparison and Discussion

From Figure 5.2-1 and Figure 5.2-2, we observe that the threshold method can have less variation in TOF estimation when target is stationary and with smooth surface. Echo signal under this case is usually stable. However since the threshold value is determined arbitrary, the result of estimation could be seriously affected by disturbance wave with amplitude larger than the threshold value whether the target is marker pen or human’s hand, and cause several outliers.

The estimation result of two maximums algorithm is relative stable and accurate than threshold method since it applies linear interpolation based on envelope model in (2.3-1) on the rising edge to estimate TOF, and the rising edge is also located in desired echo region derived from the method in section 3.2. However the way of linear interpolation is very sensitive to the variation of slope on the rising edge, causing the corresponding interference to measured TOF.

Comparing Newton’s method with the two models (Figure 5.2-3), the general echo model characterizes the actual envelope better than the double exponential model, so that the residual of curve fitting using (2.2-1) would be smaller than (2.2-2). However, general echo model would be more sensitive to the variation of the measured echo signal since it fits the onset of echo well, yielding the interference to TOF. On the other hand, although there would be a little bias of TOF estimation from the double exponential model, the rising edge of the model is less sensitive to the variation of echo, which is consistent to the experiment result in [8]. Hence we conclude that the method of Newton’s optimization with double exponential model provides much more stable TOF estimation than other methods, and the derived TOF data is used by all cases in object localization and tracking in the section 5.3.

*From Figure 5.2-3 it is observed that since the general model fits the rising edge better than the double exponential model, so that the estimated TOF would be much more sensitive to the variation of slope on rising edge although the residual of curve fitting is smaller than double exponential model.

Figure 5.2-3 : Comparison between 2 model (a) curve fitting with general model (b) curve fitting with double exp. model

(a)

(b)

tˆ

5.3 Objection Localization and Tracking

In this section three methods of deriving target’s coordinate are tested, including Least Square, Newton Ralphson’s method, and EKF. At first all the 5 channels are used among the three methods, then the outlier rejection issue would then be discussed in

Table 5.3-2 : Marker pen, mean of target coordinate (x,y) among different methods (unit: cm)

Target Localization

Method

Least Square Newton’s

method EKF

Target

Coordinate

axis

(14, 37) x

13.44 13.48 13.49

y

36.56 38.51 38.49

(18, 37) x

17.74 17.80 17.79

y

35.84 38.58 38.55

(14, 41) x

14.09 14.07 14.10

y

41.72 41.56 41.28

(18, 41) x

18.24 18.23 18.22

y

41.05 41.60 41.35

Figure 5.3-1 shows the XY-plot of the three methods with coordinate of marker pen

at (18, 41).

(a) (b)

(c)

Figure 5.3-1 : Object localization comparison a) Least Square

b) Newton’s Method c) EKF

Note that the ground truth is measured manually.

R2 R3

Case2:

Table 5.3-3 and Table 5.3-4 show the standard deviation and mean of each target coordinate, we can see that the standard deviation of target coordinate is the smallest using EKF.

Table 5.3-3 : Human’s index finger, standard deviation comparison of target coordinate (x, y) (unit: cm) Target

Localization Method

Least Square Newton’s

method EKF

Target

Coordinate

axis

(14, 37) x 1.191 1.336 0.400

y 3.514 0.498 0.143

(18, 37) x 0.995 0.968 0.216

y 3.521 0.430 0.222

(14, 41) x 1.734 1.036 0.574

y 5.203 0.448 0.292

(18, 41) x 1.449 1.178 0.718

y 4.146 0.530 0.246

Table 5.3-4 : Human’s index finger, mean of target coordinate (x, y) among different methods (unit: cm)

Target Localization

Method

Least Square Newton’s

method EKF

Target

Coordinate

axis

(14, 37) x 13.72 13.64 13.72

y 35.69 37.97 37.89

(18, 37) x 17.70 17.70 17.70

y 36.65 37.94 37.75

(14, 41) x 13.42 13.41 13.56

y 42.11 41.60 41.23

(18, 41) x 17.87 17.83 17.86

y 40.72 41.39 41.00

Figure 5.3-2 shows the XY-plot of the three methods with coordinate of human’s finger at (18, 41) .

(a) (b)

(c)

Figure 5.3-2 : Object localization comparison a) Least Square

b) Newton’s Method c) EKF

Note that the ground truth is measured manually.

R2 R3

Case3: target as human’s index finger

(*Note that blue dots is the estimated coordinate, red line is the reference target

trajectory. Finger stopped at the starting point and end point for a while before and after moving.) (1) Target moves from (8, 40) to (20, 40), velocity of x direction is about 1cm/sec.

Figure 5.3-3 : Object tracking comparison a) Least Square

b) Newton’s Method c) EKF

(a) (b)

(c)

R2 R3

Figure 5.3-4 : The EKF states change of x and y direction from Figure 5.3-3 (c)

(2) Target moves from (8, 36) to (20, 48), velocity of target is about 2 cm/sec.

Figure 5.3-5 : Object tracking comparison a) Least Square

b) Newton’s Method c) EKF

(a) (b)

(c)

R2 R3

Figure 5.3-6 : The EKF states change of x and y direction from Figure 5.3-5 (c)

(3) Target moves from (20, 36) to (20, 48), velocity of y direction is about 1cm/sec.

Figure 5.3-7 : Object tracking comparison a) Least Square

b) Newton’s Method c) EKF

(a) (b)

(c)

R2 R3

Figure 5.3-8 : The EKF states change of x and y direction from Figure 5.3-7 (c)

5.3.2 Comparison and Discussion

We first observe the least square problem from (4.1-3), the system matrix A that contains the range estimate would increase uncertainty of estimation. Newton’s method is better than least square method since the round trip model doesn’t contain any range estimation. However the Newton’s method only estimates the coordinate based on the current TOF data so that the estimation result would be directly affected by the measurement noise. Although the performance of Newton’s method and EKF are similar when target is stationary, the EKF can still provide more stable and smoother target localization and tracking since it estimates the target coordinate based on the previous state and inherently considers the interference of measurement.

Note that from Figure 5.3-4 and Figure 5.3-6 , the estimated velocity from EKF is close to the target moving velocity, but comparing the (x, y) estimation in Figure 5.3-5(c), tracking on x direction is delayed more than on y direction. It is because that the geometry of ultrasonic platform structure (Appendix and [13]) makes the variance of x direction bigger than y direction (we can also observe this phenomenon in Table 5.3-1 and Table 5.3-3 ). To solve this problem we simply set the process noise covariance of x direction smaller than that of y direction, which means that it would reduce the variance of the estimated x coordinate but the tradeoff is that the delay would increase when tracking on x direction. Finally from Table 5.3-2 and Table 5.3-4, it is observed that the estimated means of target coordinate are similar among three methods.

We then consider the outlier issue in case3, when target moves from (20, 36) to (20, 48), we observe that sometimes echo would be distorted seriously hence several outliers appear in measurement (Figure 5.3-9). The outliers cause a great effect on the tracking performance of EKF. In this section we use the outlier detection method mentioned in section 4.3 to improve the performance of tracking on target:

Figure 5.3-9 : Measurement data of Case3 (3)

Case3: target as human’s index finger

(3) Target moves from (20, 36) to (20, 48), velocity of y direction is about 1cm/sec

Figure 5.3-10 : Object tracking comparison a) single EKF

b) EKF with outlier rejection

(a) (b)

Figure 5.3-11 : The EKF states change of x and y direction from Figure 5.3-10 (b)

(1) Target moves from (8, 40) to (20, 40), velocity of x direction is about 1cm/sec

Figure 5.3-12 : Object tracking comparison a) single EKF

b) EKF with outlier rejection

(a) (b)

Figure 5.3-13 : The EKF states change of x and y direction from Figure 5.3-12 (b)

(2) target moves from (8,36) to (20, 48), velocity of target is about 2 cm/sec.

Figure 5.3-14 : Object tracking comparison a) single EKF

b) EKF with outlier rejection

(a) (b)

In case3 (3),we can see that when the outliers are detected and rejected, the variance of x coordinate in Figure 5.3-11 is relative smaller than Figure 5.3-8, and the estimated target trajectory in Figure 5.3-10 (b) can follow up the desired path smoother and more consistent than in Figure 5.3-10 (a). Note that since there is no obvious outlier in measurement in case 3 (1) and (2), the results of Figure 5.3-12 (a)(b) and are similar, so do the results in Figure 5.3-14 (a)(b).

Figure 5.3-15 : The EKF states change of x and y direction from Figure 5.3-14 (b)

Finally we observe another trajectory using the EKF with outlier detection.

Case3: target as human’s index finger

(4) target moves along circle with center at (16, 42), both the starting point and the end point are at (16, 48).

From Figure 5.3-16 we see that the estimated target coordinate is around the reference trajectory. However there is delay of x direction near the end point. This is caused by the outliers that continuously appear when target is moving, and since the Kalman gain would be small during presence of outliers, it would make the estimated coordinate close to prediction trajectory.

Figure 5.3-16 : Target moves along circle

Chapter 6 Conclusions and Future Work

In this work we provide a strategy for TOF estimation and tracking on target coordinate. Several methods were compared and experimentally evaluated in different scenarios.

In the part of TOF estimation, the threshold method provides a simple way to find the TOF and can have stable estimation when the amplitude of desired echo is sufficiently large, but there would be always biases on each estimation. Moreover, threshold method would do the wrong estimation easily because of other disturbance waves. TOF measurements from Newton’s method and two maxima algorithm using general model would be more accurate than threshold method since the two method use the similar envelope model ((2.2-1) and (2.3-1)) to do the estimation. However, the Newton’s method using nonlinear curve fitting would provide more reasonable TOF than Two maximums which use linear interpolation. Although general model fit the rising edge better than double exponential model, it is found that TOF estimation from general model would be much more sensitive to the variation of rising edge. Therefore we conclude that Newton’s method with double exponential model is the best method with tolerable delay in TOF estimation in our application.

When deriving target coordinates from the TOF data, using EKF can provide more stable and smoother estimation than least square method and Newton’s method according to the experiment result. The outlier rejection strategy is also provided to EKF to improve the tracking performance.

There are several areas for improvement. For single target, both marker pen and human’s hands are kept vertically to the screen platform, even when the human’s hand is moving. The result of target inclining to screen is not presented since the echo we received would attenuate fast and would be too difficult to be detected, or is distorted very seriously causing too many outlier in TOF measurement (the relationship between amplitude decay of echo and the incline angle of target can be investigated in [14]).

Improving the hardware directly such as other sensor or the circuit module would be a straight solution. Secondly, the localization of multiple targets is another problem since it requires a strategy for identification of corresponding echoes. Finally, since there are wide beam angle of our sensors for both transmitter and receivers, it is possible to realize the 3D localization and tracking as future work.

Appendix

Geometry Analysis of Ellipse

Consider the sensor shelf as follows; the target coordinate can be viewed as the intersection point between ellipses formed by a transmitter and other receivers.

Let



_i: be the traveling time,

v

_s:Sound velocity,

d

_i: the traveling distance =



_i×

v

_s,

a  d

, distance between a focus and ellipse center 2

i i

c  r

and the semi-minor axis

b

 a

_i²

 c

_i² , i = 1~4.

So for any ellipse formed by

R

_iand T, the equation can be expressed as

) 1

From the simulation using EKF method mentioned in section 4.3, we can observe that the variance of x and y direction of target coordinate could be significantly affected by the intersection condition of ellipses.

Figure 1 : Target coordinates by ellipses from transmitter and two of all receivers.

Figure 2 : Target coordinates by ellipses from transmitter all receivers.

Simulation result shows that intersection condition could be vary from the relative position between targets and sensors, as we can see in Figure 1 when the y coordinate of target increase the variance of x direction would also increase.

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