# A Self-Contained and Self-Checking LPS with High Accuracy

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. LOSNUS

#### 3.1. Design Rationale of LOSNUS

#### 3.2. Basic Principle of LOSNUS

#### 3.3. Signal Processing

**Figure 3.**(

**a**) 1-bit quantized linear frequency modulated chirp with Δ = 25 kHz, f

_{0}= 52.5 kHz and T = 256 μs; (

**b**) Binary autocorrelation function (Theoretical D = 6.4).

**Figure 4.**(

**a**) Directivity diagram without attached cone of Senscomp 600 electrostatic transducer; (

**b**) Directivity diagram after application of cone.

**Figure 5.**(

**a**) Received (analog) waveforms for burst with f

_{0}= 52.5 kHz and chirp with settings from Figure 3; (

**b**) Binary autocorrelation function showing pulse compression properties of the chirp compared to a burst.

#### 3.4. Transmitter Identification

**Figure 6.**(

**a**) Transmitter identification for n coding frequencies. Filter output values are only evaluated at discrete times as the starting time of the frame is known. (

**b**) Example transmitter decoding on binary input data using f

_{i}= 40 kHz + 4.5 kHz · i, 0 ≤ i ≤ 5. Identified transmitter is (40 kHz, 53.5 kHz). Red vertical lines mark sampling points.

#### 3.5. Localization

_{i}and T

_{j}, a receiver R and ToAs t

_{i}and t

_{j}the basic TDoA equation is given in Equation (3) where c is the speed of sound.

**Figure 7.**(

**a**) Calculation of the receiver positions by all fifteen combinations in case of no error. All positions are close and spread by the DOP. (

**b**) In case of a single outlier only five combinations are still valid and all others deliver different results. The assumed ToA error for this example was 1 cm.

## 4. Calibration

_{ij}be the ToF between transmitter i and receiver j. The defining equations are then given by

_{i}′, R

_{i}′ and a cone length l′. As the speed of sound was only relatively correct but not absolutely the output is a scaled version of the system. By calculation of the factor a according to Equation (7) the final calibrated transmitter and receiver positions as well as the cone length are given by

_{i}(k) are performed where the scaling factor c(k) is defined by the sum of the 24 ToFs where the first measurement is arbitrarily used as reference.

## 5. Uncertainty

#### 5.1. General Considerations

#### 5.2. Uncertainty in ToF and TDoA Measurements

_{0}described the unknown transmission time. In the case of ToF, the term t

_{0}contains any electrical and acoustic delays.

#### 5.3. Uncertainty in Calibration

_{i}is the coordinate of the transmitter, u

_{r}is the uncertainty of the reference path and r is the length of the reference path. The same applies for the receivers R

_{i}. All other systematic or random components are reduced either by the algorithms or averaging. A receiver-position used within the calibration is called reference position. Having two such positions available and a linear belt allows us to create arbitrary reference positions with a known uncertainty.

#### 5.4. Uncertainty in Localization

**Figure 8.**Graphical example for the calculation of uncertainty showing unknown value at the bottom, a systematic difference b(k) at each position, the uncertainty in determining b(k) as interval u

^{2}[b(k)] on the line tracking the value and the uncertainty of the individual measurements u

^{2}[P(k)].

_{c}is defined as the positive square root of Equation (15). This complicated expression is explained in Figure 8. The index k identifies the position on the belt (23 positions in our case) whereas P(k) defines a sufficient set of measurements at this position (20 in our case).

^{2}[P(k)] is simply the variance for the measurements mainly resulting from the uncertainty of the TDoA measurements and the DOP. At each position, a systematic correction factor b(k) can be determined being the mean distance of the measurements to the interpolated reference position. As the value is not known the uncertainty for determining this factor is expressed by u

^{2}[b(k)] resulting from the inaccurate reference positions and the variance of the mean value of the measurements. As only a single systematic value b is determined u

^{2}[b(k)] describes the variance of the factor itself. An important aspect of applying this formalism is that results can be transferred to different room positions as well. For example u

^{2}(P(k)) which is closely related to the DOP can be estimated by calculating the DOP and reevaluating Equation (15) again.

#### 5.5. Computation of Uncertainty Parameters

_{1}and R

_{2}be two reference points. Then arbitrary reference points can be created in between if these two points are connected by a linear belt and the positions R

_{1}and R

_{2}have been part of the calibration (For accuracy reasons). An interpolated reference point at position k with a step size of Δ is given by

_{k}and the center of gravity of the measurement points.

_{1}and u

_{2}be the uncertainty of the reference points R

_{1}and R

_{2}. Assuming a simple linear interpolation the uncertainty for the point R

_{k}is given as

#### 5.6. Temperature Compensation during Operation

_{1}, T

_{2}, T

_{3}, T

_{4}, three time differences t

_{12}, t

_{13}, t

_{14}and the temperature T for calculating speed of sound.

#### 5.7. Impact of Motion and Doppler Effect

_{0}the Doppler shift can be approximated by [31] where is the direction and speed of motion of the receiver R. The direction of wave propagation between the transmitter and receiver is given by the unity vector .

_{0}. The error for a located position R can therefore be defined according to Equation (32).

**Figure 10.**Simulation Results: (

**a**) Effect of receiver motion with 0 ms interface spacing (only possible in CDMA system), (

**b**) Effect of receiver motion with 10 ms interframe spacing suitable for LPS LOSNUS with up to 3.4 m minimal spacing between consecutively firing transmitters.

## 6. Experimental Results

#### 6.1. System Configuration

**Figure 11.**(

**a**) Test system configuration consisting of six US transmitters. (

**b**) Ultrasonic transmitter adjustable in two direction including wiring.

#### 6.2. Calibration

**Figure 14.**(

**a**) Compensation factor c(k) calculated according to Equation (10); (

**b**) Example of two ToFs compensated with factor c(k). Data points marked with “x” are before compensation and data points marked with “o” are after compensation.

Tx1 | Tx2 | Tx3 | Tx4 | Tx5 | Tx6 | |
---|---|---|---|---|---|---|

x | 0 | 0 | 0.3221 | 0 | 0.1121 | 0.1762 |

y | 1.6126 | 0 | 0.0769 | 1.6271 | 2.903 | 3.8394 |

z | 0 | 0 | −0.9454 | −1.9408 | −1.9240 | −0.8845 |

#### 6.3. Localization

**Figure 16.**Correlation results of a complete LOSNUS sequence showing ToA for individual transmitters and an example for a Non Line of Sight (NLOS) signal for Tx1.

^{T}|| = 4.7 mm ± 1.3 mm. A graphical visualization of the uncertainty components is shown in Figure 19a. In this case b(k) calculated according to Equation (18) shows the systematic component of the uncertainty, i.e., the mean distance between the points and the reference position. The value of is the standard deviation of the measured points at position k. This value is closely related to the DOP by the standard deviation of the ToA measurements and the DOP for the transmitter/receiver combination used. Due to the large and accurate reference path used the component is rather small.

**Figure 18.**Measurement on linear belt at 23 positions with 20 measurements each. Temperature = 24.8 °C taken from Figure 17.

^{T}|| = 6 mm ± 2.2 mm. The deviations between the two results, although small, are due to the finite sample size. In addition modeling does not take into account all possible effects.

**Figure 20.**(

**a**) Repeated measurement at different date and temperature; (

**b**) Temperature obtained by using reference position was 29.1 °C.

#### 6.4. Practical Realization Considerations

**Figure 21.**(

**a**) Picture showing a test realization of a sensor node with a ZigBee communication interface. (

**b**) Suggested software architecture for implementation.

## 7. Conclusion

## Conflicts of Interest

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**MDPI and ACS Style**

Walter, C.; Syafrudin, M.; Schweinzer, H.
A Self-Contained and Self-Checking LPS with High Accuracy. *ISPRS Int. J. Geo-Inf.* **2013**, *2*, 908-934.
https://doi.org/10.3390/ijgi2040908

**AMA Style**

Walter C, Syafrudin M, Schweinzer H.
A Self-Contained and Self-Checking LPS with High Accuracy. *ISPRS International Journal of Geo-Information*. 2013; 2(4):908-934.
https://doi.org/10.3390/ijgi2040908

**Chicago/Turabian Style**

Walter, Christian, Mohammad Syafrudin, and Herbert Schweinzer.
2013. "A Self-Contained and Self-Checking LPS with High Accuracy" *ISPRS International Journal of Geo-Information* 2, no. 4: 908-934.
https://doi.org/10.3390/ijgi2040908