#
On the Calibration of GNSS-Based Vehicle Speed Meters^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- (a)
- A National Metrology Institute (NMI) whose service is suitable for the intended need and is covered by the CIPM MRA (International Committee for Weights and Measures—Mutual Recognition Arrangement). CIPM MRA is the framework through which National Metrology Institutes demonstrate the international equivalence of their measurement standards and the calibration and measurement certificates they issue.
- (b)
- An accredited calibration laboratory whose service is suitable for the intended need and the Accreditation Body is covered by the ILAC Arrangement or by Regional Arrangements recognized by ILAC.
- (c)
- An NMI whose service is suitable for the intended need but not covered by the CIPM MRA.
- (d)
- A calibration laboratory whose service is suitable for the intended need but not covered by the ILAC Arrangement or by Regional Arrangements recognized by ILAC.

## 2. The Considered Measurement Setup

_{ref}) and by the instrument under calibration (V

_{UUC}).

_{UUC}is obtained as the average of the N speed (${V}_{UU{C}_{i}}$) measured by the GNSS device during the time interval T.

_{UUC}is made on the vehicle while the measure of V

_{ref}is made on the ground, both a suitable choice of the distance between the photocells and a suitable synchronization method must also be considered for the comparison of these two speeds (to be sure that both the reference system and the instrument under calibration measure the vehicle speed in the same time and in the same place).

_{i}” from 1 to 0.01 s. Considering a vehicle travelling at speed “v

_{i}”, the speed measured by a GNSS receiver (installed on board) is made at a distance interval ${d}_{i}={v}_{i}\xb7{t}_{i}$. As it is possible to see in Table 1, the distance interval between two consecutive measurements made by the GNSS receiver changes from 3 mm to about 83 m. This means that the distance between the photocells cannot be fixed and its choices must be made appropriately to be sure that at least one point (${V}_{UU{C}_{i}}$) of the speed measured by the GNSS device is made during the time interval T (in other words, inside the space between the two photocells).

- (i).
- The first one is based on the use of a device to be able to communicate to the GNSS receiver (UUC) the start and stop instant times. This can be made, for example [12,13], by means of two flashlight emitters, placed on the ground, that are triggered correspondingly with the start and stop instant times and two flashlight receivers, positioned on the vehicle, that sense the start and stop instant times allowing for the creation of start and stop flags in the speed values acquired by UUC. The application of this solution is conditioned by the availability of a GNSS device equipped with a synchronization data input port or the availability of an additional central unit to be able to acquire, simultaneously, both the speed values coming from the UUC and the synchronization data coming from flashlight receivers.
- (ii).
- The second one is based on the use of UTC time (Coordinated Universal Time). Since any GNSS system always furnish the measured data (e.g., the speed) accompanied by corresponding UTC time, the idea is to use a GNSS receiver with a high update rate (e.g., 100 Hz), and equipped with a synchronization data input port. It is positioned on the ground and connected with the photocells. In this way, it is possible to create flags in the UTC time registered by the GNSS receiver triggered with the start and stop instant times. The flagged UTC times, stored by the GNSS system on the ground, are then used for the synchronization with the speed values measured by the UUC.

## 3. Analysis of the Uncertainty Contributions

_{UUC}), others are connected to the considered calibration method ($\u2206{V}_{ref}$ related to the measure of V

_{ref}, and $\u2206{V}_{SYNC}$ related to the synchronization capability between V

_{ref}and V

_{UUC}).

#### 3.1. Uncertainty Sources on the Measure of ${V}_{UUC}$

#### 3.2. Uncertainty Sources on the Measure of ${V}_{ref}$

#### 3.2.1. Uncertainty Sources on the Distance “d”

- Uncertainty of the measure of the distance between the two photocells (${u}_{\u2206{d}_{ACC}}$). It is connected to the accuracy ($AC{C}_{d}$) of the instrument used to measure the distance d. Considering a uniform distribution, this uncertainty contribution can be written as:$${u}_{\u2206{d}_{ACC}}=\frac{AC{C}_{d}}{\sqrt{3}}$$
- Uncertainty in the calibration of the instrument used to measure the distance (${u}_{\u2206{d}_{CAL}}$). It can be taken from the calibration certificate of the instrument.
- Uncertainty due to the difference in the plane formed by the two couples of photocells respect to the track plane. As is shown in Figure 3, if the installation heights of the two photocells from the road surface (${h}_{1}$ and ${h}_{2}$) are not the same, the distance ${d}_{dp}^{\prime}$, passed through by the vehicle, is different with respect to the known distance “d” between the photocells. The magnitude of this effect is connected to the influence quantities that depend on the choice made for the installation of the photocells couples: On the rod or tripods. Generally, in addition to the accuracy of the instrument used to measure ${h}_{1}$ and ${h}_{2}$, possible influence quantities are the asphalt roughness and/or subsidence, track planarity (especially for high values of “d”), etc.The error, $\u2206{d}_{dp}$, in the knowledge of “d” can be evaluated as:$$\u2206{d}_{dp}=d-{d}_{dp}^{\prime}=d-\sqrt{{d}^{2}-{\left({h}_{1}-{h}_{2}\right)}^{2}}.$$This effect can be considered with a zero mean value and with uncertainty ${u}_{\u2206{d}_{dp}}$ that, considering a uniform distribution, can be written as:$${u}_{\u2206{d}_{dp}}=\frac{\u2206{d}_{dp}}{\sqrt{12}}$$
- Uncertainty due to the difference between the vehicle trajectory and the longitudinal axis of the track.Considering the schemes shown in Figure 4a, the vehicle could travel along a trajectory that is not perfectly aligned to the track axis (to have a clear draw, the cars positions, and consequently the effects of car trajectory, were amplified with respect to the reality). In this condition, the distance ${d}_{ct}^{\prime}$ passed through by the vehicle and is different with respect to the known distance “d” between the photocells. The error, $\u2206{d}_{ct}$, in the knowledge of “d” can be evaluated as:$$\u2206{d}_{ct}=|d-{d}_{ct}^{\prime}|=|d-\frac{d}{\mathrm{cos}\alpha}|=\left|d\xb7\left(1-\frac{1}{\mathrm{cos}\alpha}\right)\right|.$$The amplitude of this effect is strictly connected to the skills of the driver, the value of the distance d, and the speed of the test vehicle. Fixing the driver skills, the greater the distance d and the lower the speed value, the greater $\u2206{d}_{ct}$ can be. For instance, in Figure 4b, a case in which for longer distance d the driver correcting the trajectory around the track axis can have an amplification of the effect on $\u2206{d}_{ct}$ is shown.This effect can be considered with a zero mean value and with an uncertainty ${u}_{\u2206{d}_{ct}}$ that, considering a uniform distribution, can be written as:$${u}_{\u2206{d}_{ct}}=\frac{\u2206{d}_{ct}}{\sqrt{12}}$$
- Uncertainty due to the error into the installation of the photocells transmitter-receiver in a direction not perpendicular to the longitudinal axis of the track.As is drawn in Figure 5, the laser beams emitted by the photocells can form an angle (β) respect to the direction orthogonal to the track axis. This can be caused by the not ideal methodology used to install of the photocells couples on the track. As a consequence of this effect, the distance ${d}_{al}^{\prime}$, passed through by the vehicle, ranges from a minimum of ${d}_{a{l}_{min}}^{\prime}=d-2\text{}L\text{}tan\beta $ (condition shown in Figure 5a) to a maximum of ${d}_{a{l}_{max}}^{\prime}=d+2\text{}L\text{}tan\beta $ (condition shown in Figure 5b), where L is the distance from the photocell laser emitters at which the vehicle passes through on the track.Correspondingly, the maximum error, $\u2206{d}_{al}$, in the knowledge of “d” can be evaluated as:$$\u2206{d}_{al}={d}_{a{l}_{max}}^{\prime}-{d}_{a{l}_{min}}^{\prime}=4\xb7L\xb7\mathrm{tan}\beta $$This effect can be considered with a zero mean value and with uncertainty ${u}_{\u2206{d}_{al}}$ that, considering a uniform distribution, can be written as:$${u}_{\u2206{d}_{al}}=\frac{\u2206{d}_{al}}{\sqrt{12}}$$
- Uncertainty due to the not collimated laser beam.As shown in Figure 6, the laser beams emitted by the photocells could not be perfectly collimated and the presence of the vehicle (start and stop conditions) can be detected in a point that lies inside a cone around the photocells emitter-receiver line. As a consequence, the start and stop conditions can occur in a point upstream or downstream this line, resulting in a variation of the distance “d”. The effect increases with the distance of the vehicle from the photocell emitters and the maximum value (g) can be taken in the points the photocells receivers are positioned.As a consequence of this effect, the distance ${d}_{co}^{\prime}$, passed through by the vehicle, ranges from a minimum of ${d}_{c{o}_{min}}^{\prime}=d-2\text{}g$ to a maximum of ${d}_{a{l}_{max}}^{\prime}=d+2\text{}g$.Correspondingly, the maximum error, $\u2206{d}_{co}$, in the knowledge of “d” can be evaluated as:$$\u2206{d}_{co}={d}_{c{o}_{max}}^{\prime}-{d}_{c{o}_{min}}^{\prime}=4\xb7g$$This effect can be considered with a zero mean value and with uncertainty ${u}_{\u2206{d}_{co}}$ that, considering a uniform distribution, can be written as:$${u}_{\u2206{d}_{co}}=\frac{\u2206{d}_{co}}{\sqrt{12}}$$
- Uncertainty due to the thermal expansions.Since the measurement collection can take several hours and in this time interval the ambient temperature can obviously change, a thermal expansion of the setup components could occur. In particular, there may be a thermal expansion of the rigid bar on which the photocells are mounted (if the tripod solution is chosen to install the photocells on the track, this uncertainty contribution can be neglected). Obviously, the rod should be made of a material with a low coefficient of thermal expansion.Indicating with the λ the linear coefficient of thermal expansion of the material taken into account, and with ΔK the maximum temperature difference during the calibration, the error $\u2206{d}_{te}$ in the knowledge of “d” can be evaluated as:$$\u2206{d}_{te}=\lambda \xb7d\xb7\u2206\mathrm{K}$$This effect can be considered with a zero mean value and with uncertainty ${u}_{\u2206{d}_{te}}$ that, considering a uniform distribution, can be written as:$${u}_{\u2206{d}_{te}}=\frac{\u2206{d}_{te}}{\sqrt{12}}$$In conclusion, the distance “d” can be expressed by the following equation:$$d=d+\u2206{d}_{Acc}+\u2206{d}_{dp}+\u2206{d}_{ct}+\u2206{d}_{al}+\u2206{d}_{co}+\u2206{d}_{te}$$$${u}_{d}=\sqrt{{u}_{\u2206{d}_{Acc}}^{2}+{u}_{\u2206{d}_{CAL}}^{2}+{u}_{\u2206{d}_{dp}}^{2}+{u}_{\u2206{d}_{ct}}^{2}+{u}_{\u2206{d}_{al}}^{2}+{u}_{\u2206{d}_{co}}^{2}+{u}_{\u2206{d}_{te}}^{2}}$$

#### 3.2.2. Uncertainty Sources on the Time Interval “T”

- $\u2206{T}_{Acc}$ is connected to the accuracy ($AC{C}_{T}$) of the instrument (e.g., electronic counter) used to measure the time interval T. This effect can be considered with a zero mean value and with uncertainty ${u}_{\u2206{T}_{ACC}}$ that, considering an uniform distribution, can be written as:$${u}_{\u2206{T}_{ACC}}=\frac{AC{C}_{T}}{\sqrt{3}}$$
- $\u2206{T}_{RES}$ is connected to the resolution ($RE{S}_{T}$) of the instrument (e.g., electronic counter) used to measure the time interval T. This effect can be considered with a zero mean value and with uncertainty ${u}_{\u2206{T}_{RES}}$ that, considering an uniform distribution, can be written as:$${u}_{\u2206{T}_{RES}}=\frac{RE{S}_{T}}{\sqrt{12}}$$
- $\u2206{T}_{CAL}$ is connected to the calibration of the instrument used to measure the time interval T. The uncertainty value (${u}_{\u2206{T}_{CAL}}$) can be taken from the calibration certificate of the instrument.
- $\u2206{T}_{rd}$ is connected to the response delay of the photocells, and more generally, of the whole electronics used to transform the vehicle passage through the photocells to electronic signals that drive the start and stop condition in the measure of the time interval T. Knowledge about this effect can be taken from the manual of the photocells and the used electronics as well as by a metrological characterization of these components. This effect can be considered with a zero mean value and with uncertainty ${u}_{\u2206{T}_{rd}}$ that, considering a uniform distribution, can be written as:$${u}_{\u2206{T}_{rd}}=\frac{\u2206{T}_{rd}}{\sqrt{12}}$$In conclusion, the uncertainty of the time interval T can be expressed by the following equation:$${\mathrm{u}}_{\mathrm{T}}=\sqrt{{u}_{\u2206{T}_{Acc}}^{2}+{u}_{\u2206{T}_{RES}}^{2}+{u}_{\u2206{T}_{CAL}}^{2}+{u}_{\u2206{T}_{rd}}^{2}}$$

#### 3.3. Uncertainty Sources Due to the Synchronization

_{UUC}is made on the vehicle while the measure of V

_{ref}is made on the ground, a suitable synchronization method must be considered for the comparison of these two speeds (to be sure that both the reference system and the instrument under calibration measure the vehicle speed in the same time and in the same place). The GNSS systems measure and store the speed ${V}_{UUC}^{i}$ with a specific update rate. Figure 7 shows a schematic representation of the speeds measured by the UUC installed on the vehicle while it goes through the reference system based on the photocells. Two update rates (${F}_{1}$ and ${F}_{2}$, with ${F}_{1}=2\xb7{F}_{2}$) of the GNSS system is represented.

## 4. Variability Range of the Considered Uncertainty Contributions

- Speed—While the GNSS systems are usually able to measure speeds above 1000 km/h, the variability range of the speeds to be investigated using the calibration method under investigation has to be limited to a value allowed by a vehicle that travel on a track with a suitable speed stability. For this reason, a maximum speed value equal to 300 km/h has been chosen.
- d—The choice of the maximum value of the distance between the two photocells couples was made, taking into account the considerations made in Section 2 regarding the distance interval between two consecutive measurements made by the GNSS receiver related to the update rate of the GNSS systems and the speed testing point (see Table 1). Therefore, in order to guarantee that at least one sample of the speed measured by GNSS system is inside the two pairs of photocells for all possible GNSS update rates and a speed up to 300 km/h, the maximum value of d was fixed at 85 m. As far as the minimum value of d is concerned, it has been chosen, taking into account that, typically, reducing d increases the uncertainty. A value equal to 1 m seemed to be sufficiently low to have an analysis with a wide view.
- $AC{C}_{d}$—The values reported in Table 2 have been obtained analyzing the typical accuracies of the instruments present on the market for the measure of distance in field. The minimum value has been chosen, taking into account the typical values of their resolution.
- (h
_{1}− h_{2})—As far as the knowledge of h_{1}and h_{2}, in addition to the typical accuracies of the instruments present on the market for the in field measure of distance, the effect due to the roughness and the planarity of the asphalt of the track ground surface should be taken into account. The value of $\u2206{d}_{dp}$ is obtained by means of Equation (9), starting from the chosen values of h_{1}− h_{2}and d inside the ranges shown in Table 2. - α—As described in Section 3, the car trajectory effect and, consequently, the variability range of α is strictly connected to the skills of the driver, to the value of the distance d and to the speed of the test vehicle. The lower the vehicle speed, the greater the possibility to have a slight deviation of the car trajectory with respect to the track axis. At the same time, the possibility to have a car trajectory deviation increases versus the distance d. Obviously, fixing the deviation dv (see Figure 4a), increasing the distance d decreases the angle α, but at the same time, increasing the distance d will increase the possibility of having bigger deviations (also due to the multiple deviation, as represented in Figure 4b). Finally, as shown in Table 2, a variability range from 0 ° to 2 ° has been chosen for the parameter α. In order to take into account the dependency of α with distance d, the following sub-range has been considered:
- -
- 0° < α < 2° for 1 m < d < 5 m (as consequence, considering α = 2°, dv ranges from 3.5 cm to 17 cm changing d from 1 m to 5 m respectively);
- -
- 0° < α < 1° for 6 m < d < 20 m (as consequence, considering α = 1°, dv ranges from 11 cm to 34 cm changing d from 6 m to 20 m respectively);
- -
- 0° < α < 0.5° for 20 m < d < 85 m (as consequence, considering α = 0.5°, dv ranges from 22 cm to 75 cm changing d from 25 m to 85 m respectively)

- β—The value of this parameter is strictly connected to both the methodology and the instrumentations used to grant the orthogonality among the photocell lasers and the road axis. The maximum value of β is obtained, assuming a basic methodology based on the measure of the distance among the two photocells couples (emitter and receivers) and considering a maximum accuracy in the measure of the distance equal to the maximum value of $AC{C}_{d}$.
- $\u2206{d}_{co}$—The maximum value of $\u2206{d}_{co}$ is obtained by means of Equation (15), considering typical values achieved by the sizes of the laser cone area of commercial photocells for g.
- $\u2206{d}_{te}$—As described in Section 3, the effect due to the thermal expansion is not negligible only in the case of the photocells installed by means of a rigid bar. The maximum value of $\u2206{d}_{te}$ is obtained by means of Equation (17), considering the maximum values of $\lambda $, d, and ΔK. As far as $\lambda $ is concerned, the value of the thermal coefficient of the aluminum has been chosen. Considering that the use of a rigid bar is both manageable and feasible for distances d that are not too long, the maximum value of d was set to 10 m. Since the measurements could be made in all year days, a maximum temperature variation ΔK = 40 K has been considered.
- $AC{C}_{T}$—There is a wide choice of instruments to be able to measure the time. The corresponding accuracy can vary in a wide range that depends on the cost of the instrumentation. The values of $AC{C}_{T}$ reported in Table 2 are representative of the instrumentation on the market.
- $RE{S}_{T}$—Consideration similar to the parameter $AC{C}_{T}$ can be made.
- $\u2206{T}_{rd}$—The values reported in Table 2 for this parameter are derived by the typical values of the response delay of the photocells available on the market.

## 5. Results and Discussion

#### 5.1. Case Study 1

- -
- ${u}_{\u2206{V}_{m}}$ is strongly influenced by the testing speed, varying from 0 to about 0.36 km/h, with the speed range increasing from 1 to 300 km/h, respectively (Figure 8a);
- -
- As shown in Figure 8b, the distance d also has a strong effect on ${u}_{\u2206{V}_{m}}$, especially for lower values of the distance (from 1 to 20 m). The influence is always dependent on the selected speed (100, 200, and 300 km/h) with an increase of ${u}_{\u2206{V}_{m}}$, increasing the speed value. This effect is related on the greater weight of the uncertainty contributions on both the distance d and the time T when d and T are smaller (in this case d is at its minimum value equal to 1 m and T is small with low values of d and high values of speed).
- -
- The influence on ${u}_{\u2206{V}_{m}}$ due to the parameter $AC{C}_{d}$ is slightly lower with a variation of ${u}_{\u2206{V}_{m}}$ from 0.36 to 0.5 km/h for the case related to 300 km/h and maximum values of about 0.12 km/h and 0.28 km/h for 100 and 200 km/h, respectively (see Figure 8c).
- -
- The effects due to $\u2206{d}_{dp}$ (${h}_{1}-{h}_{2}$), $\u2206{d}_{ct}$ ($\alpha $), $\u2206{d}_{te}$, and $RE{S}_{T}$ (see Figure 8d,e,h,l, respectively) are all dependent on the speed value (with greater values of ${u}_{\u2206{V}_{m}}$, up to about 0.35 km/h, for higher values of speed), but they give negligible variation to ${u}_{\u2206{V}_{m}}$.
- -
- The effects due to $\u2206{d}_{al}$ ($\beta $), $\u2206{d}_{co}$, $AC{C}_{T}$, $\u2206{T}_{rd}$, and $\u2206{V}_{SYNC}$ (see Figure 8f,g,i,m,n, respectively) give significant variation to ${u}_{\u2206{V}_{m}}$ that increases with the increasing of the speed value. Looking at Figure 8f it is possible to note that the effect of $\u2206{d}_{al}$ gives rise to a ${u}_{\u2206{V}_{m}}$ greater than 1 km/h already at $\beta $ values of about 0.1 °, reaching more than 15 km/h for the higher values of speed and $\beta $ (this effect is connected to the methodology and instrumentation to be used for the calibration setup and it is so high because of the weight of $\u2206{d}_{al}$ on the distance d that is fixed to its minimum value).
- -

#### 5.2. Case Study 2

- -
- The first general consideration is that, thanks to the high value of d, all ${u}_{\u2206{V}_{m}}$ values are heavily lower than the case study 1.
- -
- -
- The effects due to speed, $AC{C}_{d}$, $\u2206{d}_{dp}$ (${h}_{1}-{h}_{2}$), $AC{C}_{T}$, and $RE{S}_{T}$ (see Figure 9a,c,d,i,l, respectively) give rise to negligible values of ${u}_{\u2206{V}_{m}}$ that are always lower than 0.006 km/h.
- -
- All the other effects, $\u2206{d}_{ct}$ ($\alpha $), $\u2206{d}_{al}$ ($\beta $), $\u2206{d}_{co}$, $\u2206{d}_{te}$, $\u2206{T}_{rd}$, and $\u2206{V}_{SYNC}$ (see Figure 9e,f,g,h,m,n, respectively) give significant variation to ${u}_{\u2206{V}_{m}}$ that increase with the increasing of the speed value. Greater amplitude of ${u}_{\u2206{V}_{m}}$ are connected to $\u2206{d}_{al}$ and $\u2206{V}_{SYNC}$ with ${u}_{\u2206{V}_{m}}$ values that can reach more than 0.1 km/h even in this particularly favorable condition. This result is very important in order to understand both the limitations and the caution to be used in the applicability of this methodology in the calibration of high accuracy GNSS instrumentation.

#### 5.3. Case Study 3

#### 5.4. Case Study 4

#### 5.5. Case Study 5

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Representation of the vehicle that is seen by the first (

**a**) and second (

**b**) photocells couple.

**Figure 3.**Uncertainty due to the difference in the plane formed by the two couples of photocells with respect to the track plane.

**Figure 4.**Uncertainty contribution due to the vehicle trajectory along the track: short distances (

**a**) and long distances (

**b**).

**Figure 5.**Uncertainty contributions due to the error into the installation of the photocells transmitter-receiver in a direction not perpendicular to the longitudinal axis of the track. Underestimation (

**a**) and overestimation (

**b**) of the distance between the two photocells laser beams passed through by the vehicle.

**Figure 8.**Uncertainty of the calibration method (${u}_{\u2206{V}_{m}}$) versus the uncertainty contribution parameters. In each plot from (

**a**–

**n**) the independent variable is one parameter. The plots are relative to the case study 1 (d = 1 m, speed = 100, 200, and 300 km/h, all the other parameters are set to their minimum value, as reported in Table 2).

**Figure 9.**Uncertainty of the calibration method (${u}_{\u2206{V}_{m}}$) versus the uncertainty contribution parameters. In each plot from (

**a**–

**n**) the independent variable is one parameter. The plots are relative to case study 2 (d = 85 m, speed = 100, 200, and 300 km/h, all the other parameters set to their minimum value, as reported in Table 2).

**Figure 10.**Uncertainty of the calibration method (${u}_{\u2206{V}_{m}}$) versus the uncertainty contribution parameters. In each plot from (

**a**–

**n**) the independent variable is one parameter. The plots are relative to the case study 3 (d = 5 m, speed = 100, 200, and 300 km/h, all the other parameters set to their minimum value, as reported in Table 2).

**Figure 11.**Uncertainty of the calibration method (${u}_{\u2206{V}_{m}}$) versus the uncertainty contribution parameters. In each plot from (

**a**–

**n**) the independent variable is one parameter. The plots are relative to the case study 4 (all the parameters are randomly selected inside their own range of variability reported in Table 2—except the case (a) where the speed is fixed at 300 km/h). The black bars near the mean values represent the standard deviation of the ${u}_{\u2206{V}_{m}}$ calculated on the 1000 random repetitions.

**Table 1.**Distance interval between two consecutive measurements made by the Global Navigation Satellite System (GNSS) receiver as function of the GNSS update rate and the vehicle speed.

Distance Interval d_{i} [m] | |||||||
---|---|---|---|---|---|---|---|

Update Rate [Hz] | 1 | 2 | 5 | 10 | 20 | 100 | |

Vehicle Speed v_{i} [km/h] | 30 | 0.278 | 0.139 | 0.056 | 0.028 | 0.014 | 0.003 |

100 | 27.778 | 13.889 | 5.556 | 2.778 | 1.389 | 0.278 | |

200 | 55.556 | 27.778 | 11.111 | 5.556 | 2.778 | 0.556 | |

300 | 83.333 | 41.667 | 16.667 | 8.333 | 4.167 | 0.833 |

Parameter | Range | Units | |
---|---|---|---|

Min | Max | ||

Speed | 1 | 300 | km/h |

d | 1 | 85 | m |

$AC{C}_{d}$ | 0.1 | 2 | mm |

(h_{1} − h_{2}) | 0.5 | 2 | cm |

α | 0 | 2 | ° |

β | 0 | 0.5 | ° |

$\u2206{d}_{co}$ | 0 | 0.02 | m |

$\u2206{d}_{te}$ | 0 | 9 | mm |

$AC{C}_{T}$ | 10^{−10} | 10^{−5} | s |

$RE{S}_{T}$ | 10^{−10} | 10^{−5} | s |

$\u2206{T}_{rd}$ | 5∙10^{−5} | 6∙10^{−4} | s |

$\u2206{V}_{SYNC}$ | 0 | 0.5 | km/h |

**Table 3.**Summary results for case study 4 considering 1000 random repetitions: Maximum (max), minimum (min), and mean (mean) of the obtained uncertainties ${u}_{\u2206{V}_{m}}$ versus the different uncertainty parameters.

${\mathit{u}}_{\u2206{\mathit{V}}_{\mathit{m}}}\text{}[\mathbf{km}/\mathbf{h}]\text{}@300\text{}\mathbf{km}/\mathbf{h}$ | ${\mathit{u}}_{\u2206{\mathit{V}}_{\mathit{m}}}\text{}[\mathbf{km}/\mathbf{h}]\text{}@200\text{}\mathbf{km}/\mathbf{h}$ | ${\mathit{u}}_{\u2206{\mathit{V}}_{\mathit{m}}}\text{}[\mathbf{km}/\mathbf{h}]\text{}@100\text{}\mathbf{km}/\mathbf{h}$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Max | Min | Mean | Max | Min | Mean | Max | Min | Mean | ||

Speed | min | 0.14 | 0.00048 | 0.074 | 0.14 | 0.00048 | 0.074 | 0.14 | 0.00048 | 0.074 |

max | 9.59 | 0.053 | 0.57 | 4.72 | 0.026 | 0.31 | 3.09 | 0.017 | 0.21 | |

d | min | 0.26 | 0.028 | 0.15 | 0.20 | 0.020 | 0.12 | 0.16 | 0.011 | 0.088 |

max | 19.25 | 1.341 | 10.52 | 12.83 | 0.764 | 6.87 | 6.42 | 0.315 | 3.39 | |

$AC{C}_{d}$ | 9.60 | 0.053 | 0.57 | 6.26 | 0.034 | 0.39 | 3.09 | 0.017 | 0.21 | |

h_{1} − h_{2} | 9.59 | 0.053 | 0.57 | 6.25 | 0.033 | 0.39 | 3.08 | 0.017 | 0.21 | |

$\alpha $ | 9.59 | 0.075 | 0.58 | 6.25 | 0.048 | 0.39 | 3.08 | 0.025 | 0.21 | |

$\beta $ | min | 3.79 | 0.021 | 0.26 | 2.45 | 0.013 | 0.18 | 1.20 | 0.007 | 0.11 |

max | 18.29 | 0.221 | 1.02 | 12.15 | 0.148 | 0.69 | 6.06 | 0.074 | 0.36 | |

$\u2206{d}_{co}$ | min | 9.45 | 0.020 | 0.52 | 6.15 | 0.011 | 0.35 | 3.03 | 0.008 | 0.20 |

max | 10.71 | 0.097 | 0.67 | 7.01 | 0.067 | 0.45 | 3.47 | 0.034 | 0.24 | |

$\u2206{d}_{te}$ | 9.59 | 0.053 | 0.57 | 6.25 | 0.033 | 0.39 | 3.08 | 0.017 | 0.21 | |

$AC{C}_{T}$ | 9.60 | 0.053 | 0.57 | 6.25 | 0.033 | 0.39 | 3.08 | 0.017 | 0.21 | |

$RE{S}_{T}$ | 9.59 | 0.053 | 0.57 | 6.25 | 0.033 | 0.39 | 3.08 | 0.017 | 0.21 | |

$\u2206{T}_{rd}$ | 9.73 | 0.072 | 0.61 | 6.30 | 0.039 | 0.40 | 3.09 | 0.018 | 0.21 | |

$\u2206{V}_{SYNC}$ | min | 9.59 | 0.024 | 0.56 | 6.25 | 0.013 | 0.36 | 3.08 | 0.004 | 0.18 |

max | 9.59 | 0.146 | 0.60 | 6.26 | 0.145 | 0.42 | 3.09 | 0.144 | 0.26 |

**Table 4.**Values of the set of random parameters associated to the maximum and minimum ${u}_{\u2206{V}_{m}}$ for case studies 4 and 5.

Speed | d | $\mathit{A}\mathit{C}{\mathit{C}}_{\mathit{d}}$ | h_{1} − h_{2} | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\u2206{\mathit{d}}_{\mathit{c}\mathit{o}}$ | $\u2206\mathit{K}$ | $\mathit{A}\mathit{C}{\mathit{C}}_{\mathit{T}}$ | $\mathit{R}\mathit{E}{\mathit{S}}_{\mathit{T}}$ | $\u2206{\mathit{T}}_{\mathit{r}\mathit{d}}$ | $\u2206{\mathit{V}}_{\mathit{S}\mathit{Y}\mathit{N}\mathit{C}}$ | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

[Km/h] | [m] | [m] | [m] | [°] | [°] | [mm] | [K] | [μs] | [μs] | [μs] | [km/h] | ||

Case Study 4 | Min | * | 70.6 | 0.0014 | 0.010 | 0.0048 | 0.0036 | 4.8 | 25.4 | 7.73 | 5.05 | 99.9 | 0.00031 |

Max | * | 1.3 | 0.00037 | 0.014 | 0.033 | 0.0060 | 6.6 | 24.6 | 6.18 | 7.98 | 512 | 0.074 | |

Case Study 5 | Min | * | 5 | 0.0014 | 0.011 | 0.0033 | 0.00046 | 6.5 | 5.8 | 0.23 | 8.57 | 87.2 | 0.0015 |

Max | * | 5 | 0.0011 | 0.010 | 0.031 | 0.0087 | 19 | 9.3 | 4.41 | 2.66 | 397 | 0.12 |

**Table 5.**Summary results for the case study 5 considering 1000 random repetitions (for all parameters except d that is fixed to 5 m): Maximum (max), minimum (min), and mean (mean) of the obtained uncertainties ${u}_{\u2206{V}_{m}}$ versus the different uncertainty parameters.

${\mathit{u}}_{\u2206{\mathit{V}}_{\mathit{m}}}\text{}[\mathbf{km}/\mathbf{h}]\text{}@300\text{}\mathbf{km}/\mathbf{h}$ | ${\mathit{u}}_{\u2206{\mathit{V}}_{\mathit{m}}}\text{}[\mathbf{km}/\mathbf{h}]\text{}@200\text{}\mathbf{km}/\mathbf{h}$ | ${\mathit{u}}_{\u2206{\mathit{V}}_{\mathit{m}}}\text{}[\mathbf{km}/\mathbf{h}]\text{}@100\text{}\mathbf{km}/\mathbf{h}$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Max | Min | Mean | Max | Min | Mean | Max | Min | Mean | ||

Speed | min | 0.14 | 0.0023 | 0.074 | 0.14 | 0.0023 | 0.074 | 0.14 | 0.0023 | 0.074 |

max | 3.89 | 0.25 | 2.17 | 2.58 | 0.18 | 1.42 | 1.29 | 0.068 | 0.71 | |

d | 3.89 | 0.25 | 2.17 | 2.58 | 0.18 | 1.42 | 1.29 | 0.068 | 0.71 | |

$AC{C}_{d}$ | 3.89 | 0.26 | 2.17 | 2.58 | 0.19 | 1.42 | 1.29 | 0.071 | 0.71 | |

h_{1} − h_{2} | 3.89 | 0.25 | 2.17 | 2.58 | 0.18 | 1.42 | 1.29 | 0.068 | 0.71 | |

$\alpha $ | 3.89 | 0.25 | 2.17 | 2.58 | 0.18 | 1.42 | 1.29 | 0.069 | 0.71 | |

$\beta $ | min | 1.61 | 0.12 | 0.90 | 1.00 | 0.06 | 0.55 | 0.49 | 0.031 | 0.26 |

max | 3.97 | 3.63 | 3.75 | 2.62 | 2.42 | 2.49 | 1.30 | 1.210 | 1.24 | |

$\u2206{d}_{co}$ | min | 3.70 | 0.14 | 1.98 | 2.44 | 0.09 | 1.29 | 1.22 | 0.042 | 0.64 |

max | 3.95 | 1.39 | 2.49 | 2.61 | 0.93 | 1.64 | 1.30 | 0.464 | 0.82 | |

$\u2206{d}_{te}$ | 3.89 | 0.25 | 2.17 | 2.58 | 0.19 | 1.42 | 1.29 | 0.068 | 0.71 | |

$AC{C}_{T}$ | 3.89 | 0.25 | 2.17 | 2.58 | 0.18 | 1.42 | 1.29 | 0.068 | 0.71 | |

$RE{S}_{T}$ | 3.89 | 0.25 | 2.17 | 2.58 | 0.18 | 1.42 | 1.29 | 0.068 | 0.71 | |

$\u2206{T}_{rd}$ | min | 3.84 | 0.14 | 2.09 | 2.56 | 0.10 | 1.39 | 1.29 | 0.058 | 0.70 |

max | 3.94 | 0.87 | 2.30 | 2.59 | 0.40 | 1.46 | 1.29 | 0.112 | 0.71 | |

$\u2206{V}_{SYNC}$ | min | 3.88 | 0.21 | 2.17 | 2.57 | 0.12 | 1.42 | 1.28 | 0.058 | 0.70 |

max | 3.89 | 0.25 | 2.17 | 2.58 | 0.19 | 1.43 | 1.29 | 0.155 | 0.72 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Martucci, A.; Cerasuolo, G.; Petrella, O.; Laracca, M.
On the Calibration of GNSS-Based Vehicle Speed Meters. *Sensors* **2020**, *20*, 591.
https://doi.org/10.3390/s20030591

**AMA Style**

Martucci A, Cerasuolo G, Petrella O, Laracca M.
On the Calibration of GNSS-Based Vehicle Speed Meters. *Sensors*. 2020; 20(3):591.
https://doi.org/10.3390/s20030591

**Chicago/Turabian Style**

Martucci, Adolfo, Giovanni Cerasuolo, Orsola Petrella, and Marco Laracca.
2020. "On the Calibration of GNSS-Based Vehicle Speed Meters" *Sensors* 20, no. 3: 591.
https://doi.org/10.3390/s20030591