2.1 Map accuracy

At the beginning of the 20th century, paper maps began to be used for automobile navigation. The first automobile map was published by the American Automobile Association (AAA) as early as 1905 (Ristow, Reference Ristow1946; Bauer, Reference Bauer2009). There was almost no requirement for accuracy (Liu et al., Reference Liu, Wang and Zhang2020). The advent of digital maps brought requirements for accuracy. Digital maps date to the 1960s and they are associated with the first Geographic Information Systems, especially in North America. The best known systems included SYMAP, ODYSSEY, GRID, MAP and MIDAS. In the 1970s, Esri's ARC/INFO appeared on the market as one of the first vector programs. Digital maps have been common since the 1990s and their accuracy ranged from 5 m to 10 m. In 2000, deliberate degradation of the Global Positioning System (GPS) ceased, allowing civilian users to achieve up to 10 times more accurate positioning. This expanded support for digital maps included support for advanced driver assistance systems. In 2007, Toyota introduced Map on Demand, the world's first-of-its-kind technology for distributing map updates to car navigation systems. The following year, in 2008, Toyota was the first to combine the functions of a brake assist system and a navigation system that was connected to adaptive variable suspension (i.e. NAVI/AI-AVS) in its Toyota Crown model. This extension affected the required accuracy for the maps, which was 50 cm. In 2010, HD maps appeared, which, unlike the previous iterations, are processed in 3D and require an accuracy of 10–20 cm (for details, see Liu et al., Reference Liu, Wang and Zhang2020).

High demands are placed on HD maps. As a result, their creation and maintenance are demanding and expensive. Maintenance (i.e. updating) is perhaps even more complicated than the initial creation because the HD map requires quality data that are supplied quickly and cheaply.

We can achieve these conflicting requirements with the Medium map (‘MD’ map), a term introduced by Carmera (Sorrelgreen, Reference Sorrelgreen2021). Whereas the HD map is characterised by its high accuracy for traffic, the surrounding elements and spatial accuracy, the MD map provides high accuracy for elements, but with lower spatial accuracy. The MD map, thus, divides the dense and complex information of HD maps into more manageable blocks. Each of these blocks has a different degree of accuracy due to the complexity of the traffic. For example, a block that includes an intersection can be constructed and maintained with HD map accuracy, while a block that contains a motorway can be created with lower accuracy because transport through this block is easier. Thus, these logical units hold different levels of trust. The MD map records all of the elements in the block (i.e. traffic lights, pedestrian crossings, cyclists), but not their exact location in the real world (for more information, see Sorrelgreen, Reference Sorrelgreen2021).

It is possible that an MD map, which is considered theoretical, would fill the gap between the extended digital map and the HD map, as shown in Figure 1, where a digital map (i.e. SD map) focuses on the discovery of the topology of maps, such as the road network, and does not usually contain lane level information, see Ort et al. (Reference Ort, Walls, Parkison, Gilitschenski and Rus2022). They only require a relatively low-resolution image with roughly metre-level accuracy.

Figure 1. Types of maps arranged according to their accuracy

2.2 Factors that influence the location of the vehicle in an HD map

Today, the accurate and reliable localisation of an autonomous vehicle is perceived to be fundamental. Highly accurate localisation, in combination with an HD map, will cause a less demanding perception and an understanding of the traffic environment for autonomous vehicles (for more information, see Liu et al., Reference Liu, Li, Tang, Wu and Gaaudiot2008; Javanmardi et al., Reference Javanmardi, Javanmardi, Gu and Kamijo2018; Shin et al., Reference Shin, Park and Park2020; Endo et al., Reference Endo, Javanmardi, Gu and Kamijo2021). As mentioned above, the standard approach is to use GNSS, which is relatively affordable and reliable in open spaces. The disadvantage is in the urban environment or when the surroundings include vegetation because the accuracy decreases due to the indirect view and the reflectivity of the signals (for details, see Alam et al., Reference Alam, Kealy and Dempster2013; Iqbal et al., Reference Iqbal, Georgy, Abdelfatah, Korenberg and Noureldin2013; Yozevitch et al., Reference Yozevitch, Ben-Moshe and Dvir2014; Heng et al., Reference Heng, Walter, Enge and Gao2015; Javanmardi et al., Reference Javanmardi, Javanmardi, Gu and Kamijo2018; Arman and Tampere, Reference Arman and Tampere2021).

In the rest of this section, for the sake of completeness, we present the criteria for the evaluation of HD maps, which are described in detail by Javanmardi et al. (Reference Javanmardi, Javanmardi, Gu and Kamijo2018). Each HD map can be evaluated according to the following criteria:

• • functional adequacy of the map;

• • layout of objects on the map;

• • local similarity;

• • representative quality.

2.2.2 Layout of objects on the map

In addition to the number and quality of objects, an important factor is their location in space. In fact, the concept of function distribution comes from GNSS. If a GPS-based location does not have an even distribution of satellites (i.e. the satellites are grouped), then the calculated position will be incorrect (see details in Jwo, Reference Jwo2021). Prompted by this possibility, Javanmardi et al. (Reference Javanmardi, Javanmardi, Gu and Kamijo2018) defined the factor for the formulation of the distribution of elements in the foundation of the map. The basic idea is illustrated in Figure 2, (initially introduced by Javanmardi et al., Reference Javanmardi, Javanmardi, Gu and Kamijo2018), where, in Situations 2(a) and 2(b), the vehicle is erroneously located in the longitudinal direction, while in Situation 2(c), it is possible to achieve localisation in the longitudinal direction.

Figure 2. Basic idea for the formulation of the distribution of elements in the foundation of the map. (a) Erroneously located. (b) Erroneously located. (c) Correctly located

2.2.3 Local similarity

The environment, like the abstraction shown on the map, can cause local similarity and, if the abstraction ratio is too large, the map loses detail and the local similarity increases. In other words, some points on the map have similar properties in different environments and, with increased abstraction, it is no longer possible to distinguish objects, see Figure 3.

Figure 3. Map detail

3.1 Margin of error during creation

An accurate map is essential for orienting and determining the position of the vehicle. It is important to accept that the actual data collection and mapping includes a margin of error. The accuracy of maps is affected by several factors:

• • global accuracy error ($E_{glob}$);

• • local accuracy error ($E_{locacc}$);

• • local sampling error ($E_{locamp}$).

Each of these factors is affected by additional aspects: the global accuracy error depends on the recording of the car location and the GPS delay (i.e. reflectivity); the local accuracy error is affected by the sensors on the AV itself, the AV speed and the ability to accurately target the object; and the local sampling error depends on the accuracy of the processing and the model creation. As mentioned above, HD map makers create maps with varying accuracy. It can be seen that the company Atlatec provides specific data for individual factors (Dahlström, Reference Dahlström2020). Specifically, the global accuracy error is less than 3 cm (i.e. $E_{glob} < 3$ cm), the local accuracy error is less than 2 cm (i.e. $E_{locacc} <2$ cm) and the local sampling error is less than 4 cm (i.e. $E_{locamp} < 4$ cm).

Thus, it can generally be stated that for the creation of an HD map, the parameters for $E_{glob}, E_{locacc}$ and $E_{locamp}$ are given in centimetres. This is in line with the overall data provided by the other companies mentioned above. The total error for HD mapping is given as the sum of the maximum deviations of these factors:

(3.1)\begin{equation} E_{making}=E_{glob} + E_{locacc} + E_{locamp} \end{equation}

Figure 4 illustrates the factors that affect the accuracy of an HD map as it is created by a recording vehicle. It is clear that the total deviation is the sum of the deviations of the factors that affect the accuracy because the position of a landmark depends on the location from which the landmark was observed by the vehicle. Other inaccuracies increase with the creation of the HD map. Note that there is a possibility for error cancellation (i.e. if the landmark is located with an error, like 5 cm to the right of the actual location) and when, conversely, the processing margin of error is from the other direction such that the two cancel each other out. Unless otherwise specified, we will continue to work with the worst possible inaccuracy.

Figure 4. Accuracy errors when creating an HD map

3.2 Margin of error for the static model

If we want to estimate the minimum necessary accuracy for an HD map such that the autonomous vehicle, according to its navigation, is able to move safely on the road network, we must work with the errors that occurred during its creation. The following estimate for the minimum required accuracy is based on HD map layers and additional data/limits. We determine the factors that will limit the accuracy of the HD map.

Below is an overview of the layers of the map, with a list of elements contained in each:

• • the Geometric map layer contains roads and a geometric representation of lines in space;

• • the Semantic map layer includes landmarks, lanes, curbs, intersections, traffic signs and traffic lights, the deceleration threshold, pedestrian crossings, properties of elements that affect traffic, road direction and two-way roads, speed limit, and lane departure restrictions;

• • the Map Priors layer includes the order of the traffic lights at an intersection, the average waiting time for traffic lights at a given time and the probability of parking space occupancy;

• • the Real-time layer includes the current traffic information and information sharing between traffic participants.

Autonomous vehicles will need to work with different levels of accuracy for the Map Priors layer and the Geometric layer, because it is clear that information about the traffic density in a given section and at a given time will not require as much accuracy as the information necessary for passing through intersections or narrow roadways (Figure 5).

Figure 5. Layers of an HD map, taken from Chellapilla (Reference Chellapilla2018)

The most restrictive conditions for accuracy are placed on the Geometric layer on which the vehicle moves. For an estimation based on the Geometric layer, it is necessary to consider the parameters of the design categories for roads, similar to that by Meng and Hsu (Reference Meng and Hsu2021) and Reid et al. (Reference Reid, Houts, Cammarata, Mills, Agarwal, Vora and Pandey2019). These parameters may be different for different countries, so we will maintain a general approach and only some situations will be demonstrated with specific examples. Figure 6 shows how the design parameters of roadways are defined.

Figure 6. Design categories of two line road, where $a$ is the width of the lane; $b$ is the categorical width of the road; $c$ is the width of the hard shoulder; $e$ is the width of the shoulder; $v$ is the edge of carriageway marking

Next, we will focus mainly on the parameter ‘$a$’ (i.e. the width of the lane). This width depends on the proposed speed. In the Czech Republic, widths from 2$\,{\cdot }\,$75–3$\,{\cdot }\,$50 m are established for speeds of 50–90 km/h, respectively.

This geometric view allows us to determine the deviation of the HD map, $\mu _m$, as follows:

(3.2)\begin{equation} \mu_m=\frac{a-s_v}{2} \end{equation}

where $s_v$ is considered the width of the vehicle (including the PL). Equation (3.2) corresponds to the formula of Meng and Hsu (Reference Meng and Hsu2021) and Jeong et al. (Reference Jeong, Cho and Kim2020), which determines the PL for driving in a directional curve. It follows that the accuracy of the HD map may depend on the type of road. Figure 7 shows the established estimate for the minimum required accuracy and other parameters considered for lanes.

Figure 7. New estimate based on the Geometric layer, where $a$ is the width of the lane, $s_v$ is the width of the autonomous vehicle, $\mu _m$ is defined by Equation (3.2) and $E_{making}$ is defined by Equation (3.1)

The Semantic layer of the HD map contains objects that perform the function of a landmark (i.e. points according to which the autonomous vehicle is located in the HD map, and thus also in space). It can be shown that the margin of error for the sensors and the margin of error for the computational algorithm have a significant effect on the accuracy of the localisation, while the margin of error for the location of the landmark is negligible. This is evident from the computational algorithm of triangulation for the distance from a certain object. Figure 8 shows the principle of the passive triangulation method.

Figure 8. Principle of passive triangulation

Camera A focuses on Object C at alpha and Camera B focuses on Object C at beta. Because the distance between the cameras is known, a formula for the distance of Object C can be derived according to the basic properties of a triangle:

(3.3)\begin{equation} x=\frac{c\,\cdot\,\sin\beta\,\cdot\,\sin\alpha}{\sin(180^\circ{-}\alpha-\beta)} \end{equation}

When locating an autonomous vehicle in an HD map, active methods are also used such that the distance is determined with a combination of LiDAR and a camera. The camera focuses on the subject and LiDAR measures the distance between the subject and the car, see e.g. Wen et al. (Reference Wen, Tang, Liu, Qian and Fan2023). This active method can be considered to be more accurate than calculating the distance with two cameras. However, as mentioned above, not every autonomous car is equipped with the same sensors. Therefore, we focus on a less accurate method for localisation. By generalising Equation (3.3) for the case where the distance of the object is not perpendicular to the line between the cameras, we have a new formula for the calculation:

(3.4)\begin{equation} l=\frac{\sqrt{2a^2+2b^2-d^2}}{a} \end{equation}

where $a={d\,\cdot\, \sin \alpha }/{\sin (180^\circ -\alpha -\beta )}$ and $b={d\,\cdot\, \sin \beta }/{\sin (180^\circ -\alpha -\beta )}$. Figure 9 shows the distance between the autonomous vehicle and the landmark. At least two landmarks are required for localisations. In the case of a higher number, the least-squares-approximation method is used to determine the position.

Figure 9. Distance from landmarks

The distance d between the cameras on the car is constant. The inaccuracy in the distance $l$ is mainly caused by the estimation of angles from the camera. This is one of the major problems for determining the position based on an image (i.e. distortion). The percentage of image distortion expresses the ratio between the real and the expected distance from the centre of the image in pixel coordinates. It is, therefore, necessary to calibrate the cameras (i.e. intrinsic matrix) to reduce this error. Table 1 shows the relationship between the degree of distortion and the margin of error for the estimated position at the camera's field of view at $90^\circ$. The distortion caused by the quality of the sensors (i.e. camera quality) acquires linear characteristics. At present, the quality requirements for the individual sensors with which the autonomous vehicle is equipped are not specified (for more information, see Meng and Hsu, Reference Meng and Hsu2021).

Table 1. Relationship between the degree of distortion and the error of the estimated position

We consider landmarks placed with the $E_{making}$ error (which can be neglected for insignificance) when locating the $E_{dis}$ distortion error, and we further assume that the distortion affects the position estimate, as shown in Figure 10. The magnitude of the localisation error depends on angle alpha (i.e. the angle between Landmark 1, the detection point and Landmark 2). If the angle alpha is small, the area in which the autonomous vehicle is detected increases. If the angle alpha approaches $90^\circ$, the detection area $m$ is small. This idea is from the Geometric Dilution of Precision (GDOP), where GDOP represents the geometric effect on the relationship between measurement error and positioning error. The GDOP value is calculated by an inversion matrix. The basis is the vector distance from the satellite to a located object. GDOP indicates how the distribution of satellites affects accuracy; in other words, a smaller value represents satellites that are in optimal positions and higher values indicate that the satellites are grouped. For example, a GDOP of less than $1$ is the highest possible confidence level to be used for applications that demand the highest possible precision at all times. Inspired by GDOP ideas, we propose determining $m$ in Equation (3.6) (i.e. the largest possible error for the location). Specific values are given in Table 2. Subsequently, we can, for the first time, determine the inaccuracy in the creation of the HD map as $\delta$, which satisfies the following inequality:

(3.5)\begin{equation} \delta\leq\mu_m-m \end{equation}

where

(3.6)\begin{equation} m=\sqrt{\left(\frac{E_{dis}(l_1)}{\cos(90^\circ{-}\alpha)}+E_{dis}(l_1)\cot\alpha\right)^2+E_{dis}^2(l_1)} \end{equation}

Figure 10. Angle-dependent localisation for angle alpha

Table 2. Value $m$ for distortion $1\%$ and landmarks from camera 20 and 5 m

Note that the value of $m$ is obtained from the linearised region of Figure 10. We demonstrate our considerations above for a specific situation in a static model: a lane width $a = 350$ cm and a vehicle with a width $s_v = 250$ cm (which includes the PL). From Equation (3.2), we can easily determine that $\mu _m = 50$ cm. The error caused by $1\%$ distortion at a distance from the landmark (L2) of 20 m and a distance from the landmark (L1) of 5 m is approximately 16 and 4 cm, respectively, when the angle alpha is $70^\circ$. Then it follows from Equation (5) that an HD map can be created for a given section with a minimum required accuracy of $32$ cm. Further, note that using a larger number of landmarks can provide more points to represent the vehicle. In other words, the area at the point of localisation in Figure 10 decreases more and the value of $m$ also decreases. Note that for an infinite number of points, the area in Figure 10 was represented by the circle and $m$ was the radius of the circle. Because we estimate the minimal required accuracy, we chose the worst possible situation for the estimation. In our case, therefore, we considered only two landmarks, and solved the error for the most important lateral direction, even though the other directions are also influenced. More specific values are provided in Table 4, where we consider $s_v=200$ cm for future autonomous vehicles.

Compare the values of $m$ and $m'$ from Tables 2 and 3, respectively. The distance of the landmark affects the value of $m$, but the trend of the size of the angle is the same for both tables. Note that some extreme positions of landmarks are not achievable in the practical world, e.g. $\alpha =20^\circ, L_1=20$ and $L_2=20$ m.

Table 3. Value $m'$ for distortion $1\%$ and landmarks from camera 20 and 20 m

Table 4 shows the minimum required lane-width accuracy for a vehicle width ($s_v = 200$ centimetres). It is clear that high inaccuracy cannot be considered for very narrow lanes because the HD map would not be able to make sense of the data. This depends on other design parameters, such as the number of passing cars. In the case of freight transport, the situation is further complicated in narrow sections because real traffic experience suggests the need for complex evasive manoeuvres. In these situations, even an absolutely accurate HD map would not be effective. The solution could provide car to car (C2C), or infrastructure to car (I2C). For example, Nordic countries specifically identify places to avoid on some roads and sometimes there is only one vehicle allowed in places on a two-way road. This avoidance manoeuvre is a matter for other algorithms. For standard lane widths from $300$ cm, the minimum required accuracy of the HD map is 31$\,{\cdot }\,$41 cm, which is shown in Table 4. Current technologies make it possible to create an HD map that is more accurate. A subject for further research is how to work with the accuracy of HD maps and the PLs of autonomous vehicles. Meng and Hsu (Reference Meng and Hsu2021) shows a technique for determining the PL, which is maximised to the overall width of the lane. A subject of interest may also be a study where the PL will be used more to avoid collisions and the creation of HD maps will depend only on the actual width of the vehicle.

Table 4. Minimum required accuracy for HD maps for $m=0\,{\cdot }\,1859$ cm

3.3 Margins of error for the dynamic model

Driving speed can significantly affect the location of a vehicle. Cameras capture 10–15 frames per second (FPS) in high resolution. At higher speeds, a car is able to travel up to 36 m in one second, so shooting takes place after approximately 2$\,{\cdot }\,$5 m. It is possible to reduce the resolution, but this is closely related to the quality of the HD maps. Another option is to reduce the speed or buy more expensive cameras with a higher frame rate. At higher speeds, the photo will be blurry or the shutter speed will need to be shortened, which will affect the quality in poor lighting conditions. A poor-quality photo reduces the success of landmark detection and classification. If we have more observed landmarks, then the probability for correct detection increases, while the method of least squares solves the calculation for the point that represents the vehicle. Figure 10 shows the number of shots per 100 m as a function of the speed when shooting at 15 FPS. Specific values are shown in Table 5.

Table 5. Number of frames according to vehicle speed

The number of frames on a given section decreases exponentially depending on the increasing speed, see Figure 11.

Figure 11. Number of frames per 100 m according to speed, where n$_{\mbox {F}}$ is the number of frames for 100 m

From a review of the available literature, if the source is cited, the tolerance for accuracy in the longitudinal direction is higher for HD map makers. However, the resulting localisation depends not only on the density of images captured by the camera but also on the speed of the computational algorithm. Therefore, a moving automated vehicle requires a high rate of position updating. For some systems, that means up to 200 Hz (Levinson et al., Reference Levinson, Montemerlo and Thrun2007), or 10 Hz with a speed limitation of 63 km/h (Levinson and Thrun, Reference Levinson and Thrun2010), or 20 Hz with a speed limitation of 60 km/h (Cui et al., Reference Cui, Xue, Du and Zheng2014). From these data, we are able to determine that the time of the computational algorithm that was designed to determine the position is from 5 to 100 ms. Then we are able to calculate that a car at a speed of 130 km/h and with a computational algorithm speed of 5 ms travels 18 cm during the calculation, and a car at a speed of 60 km/h with a computational algorithm speed of 0$\,{\cdot }\,$1 s travels 160 cm. If we add the distance that occurs between taking pictures at speeds of 130 and 60 km/h to both values, then we get that the vehicle is located at 268 and 261 cm, respectively. The error of estimating objects from the $E_{dis}$ camera was considered above in the lateral direction. Due to the calculated data in the longitudinal localisation, this error has a minimal effect. At a high frequency (i.e. 200 Hz and a speed of 50 km/h), the vehicle travels approximately 7 cm during the calculation with a shooting at 0$\,{\cdot }\,$92 m. This provides sufficient accuracy in an urban environment. From the point of view of the planar sections of the location at which this delay appears on motorways, it is not a major problem. This aspect will have a negative effect on directional arcs. The problem of delayed localisation is demonstrated in Figure 12, where the vehicle is actually at Point B while the evaluation of its position is at Point A.

Figure 12. Parameters of horizontal curve

If the vehicle was strictly driven according to the HD map, it would start to turn with a delay. The stated deviation $o$ can then be expressed as

(3.7)\begin{equation} o=\sqrt{u^2+v^2-2uv\,\cdot\,\cos\gamma} \end{equation}

where

(3.8)\begin{equation} \gamma= 90^\circ{-}\frac{180^\circ{-}\cos^{{-}1}\frac{u^2-2r^2}{-2r^2}}{2} \end{equation}

If we consider a deviation in the longitudinal direction of 268 cm at a speed of 130 km/h, which is allowed only on a highway, then the minimum radius of the directional curve is 1,025 m. Due to the sufficiently large $r$, we consider it to be a line instead of a circular arc, because the error is of the order of micrometres. From Equations (3.6) and (3.7) for the given values, we get $o= 24\,{\cdot }\,799$ mm.

Incorporating the results of the dynamic model into the requirements for the minimum necessary accuracy of HD maps from the static model leads to the modification of the original Equation (3.5) to

(3.9)\begin{equation} \delta'\leq \mu_m-m-o \end{equation}

Thus, we have determined a new formula for calculating the minimum necessary accuracy for HD maps, while including several factors, i.e. design parameters for roads, camera properties, computing power and location of landmarks. However, for large values for road-design parameters and the low computational time of the algorithm, the parameter $o$ is negligible in Equation (3.9).

Note that although the modification of the formula was caused by the considerations of the passage of an autonomous vehicle through a directional curve, it is possible to use this modification in general (i.e. in directional curves and on straight sections) because it follows from the formula that, on the straight section, $o = 0, \gamma = 0^\circ$ and $u = v$. Then we get the following with Equation (6):

(3.10)\begin{equation} o=\sqrt{u^2+u^2-2uu\,\cdot\,\cos(0^\circ)}=\sqrt{2u^2-2u^2\,\cdot\, 1}=\sqrt{0}=0 \end{equation}

To some extent, we have already shown that longitudinal inaccuracy affects the lateral in terms of vehicle location. In the dynamic model, the speed-dependent localisation delay affects the lateral accuracy mainly in the directional arcs, or their design parameters.