<figure><img src="image_1.png" /></figure>

# Performance of a 2D lidar device mounted on a mobile platform for 3D scanning of an urban environment

ASTEK Group

**Abstract-** Nowadays, research in the field of autonomous cars is progressing rapidly. This is true both for software, with ever more advanced recognition algorithms, and for hardware, with ever more powerful computers and ever more efficient sensory equipment. All of this is linked to important safety, societal and economic issues. Indeed, autonomous cars must above all guarantee the safety of users and outsiders. The improvement of this safety requires an improvement of the data processing algorithms, using artificial intelligence, but also an improvement of the data collected by all the sensors scattered outside and inside the vehicle. Although there are solutions for 3D mapping, these are expensive in terms of both cost and computing power. A conventional lidar, on the other hand, rotates around a single axis of revolution, which does not provide 3-dimensional information about the scene. However, it is because of its good performance in mapping the environment around the vehicle in 3 dimensions that lidar is considered an indispensable asset for the autonomous car, being able to detect very small obstacles with great precision. The objective of the current work is to measure the performance of a rotating 2D lidar device mounted on two platforms with different architectures, enabling it to scan a scene for 3D mapping purposes. The work allowed the theoretical characterization of the scene scanning geometries across the two architectures by quantifying the average point density and its standard deviation, the solid angle scanned. In a second step, the work allowed the quantification of the obstacle detection rate as a function of its distance and the measurement error by simulation and experimentation.

**Index Terms**- Autonomous vehicle, 2D Lidar, Rotating platform, 3D Mapping, Scanned solid angle, Point cloud density, Measurement error

## I. INTRODUCTION

The main function of lidars is to measure the relative distance between them and a potential obstacle. They are active sensors consisting of a laser transmitter, a photosensitive receiver, and an on-board electronic circuit. Lidar telemetry works on the principle of "time-of-flight" technology, i.e. the distance measurement is obtained by measuring the time between emission and reception of a light wave. Other sensors such as Sonar and Radar, with acoustic and radio waves respectively, work on the same principle. However, compared to these sensors, lidar has a better resolution, allowing it to perform well in 3D mapping tasks of the environment. Despite this, lidar remains an expensive solution and often scans the scene in a single plane, which prevents accurate 3D mapping. They are therefore referred to as 2D lidar. In the current work, it is proposed to integrate the same 2D lidar in two different architectures of mobile platforms allowing a 3D scan of the scene and to compare their 3D scanning performances. The interest of such a device is to be able to benefit from the telemetric accuracy of a 2D lidar while performing a 3D mapping of the scene and to make the obstacle detection robust for autonomous car navigation needs. More precisely, this paper proposes to model the scanning trajectory of the laser beam as a function of the type of platform architecture on which the 2D lidar could be mounted, to quantify the resolution of the measurement points, i.e. the number of points scanned per unit of solid angle, the total solid angle scanned, the detection rate of a given obstacle as a function of its distance and the error in the telemetric measurement. After a state of the art of lidar technologies, the work will first model the theoretical performances concerning the kinematic and geometric aspects of laser scanning for the two architectures. Secondly, the work will quantify some real performances for the best-chosen architecture such as the detection rate of the points, the number of points scanned on an obstacle of standard dimension and this as a function of its distance and angular position with respect to the chosen architecture.
## II. STATE OF THE ART

In this section, an analysis of the existing literature, where the articles most related to the problem of 3D lidar mapping will be made. More specifically, these articles study 2D or 3D lidar sensors or are interested in the autonomous navigation of any system and ideally in the navigation of cars equipped with lidar. The paper [1] presents the work of researchers who are trying to set up an autonomous 2D room mapping system based on a robot that is able to find a path without collision and move around it. The robot in question is equipped with a 2D lidar, communicating the data to a Raspberry which in turn communicates with an Arduino controlling the wheel motors according to the data returned by the sensor. The accuracy of the device's distance measurement observed in the experiment is 20 millimeters. The navigation is done in two stages: in the first stage, the environment is scanned, and the data is processed, and only in the second stage does the vehicle start moving, after having chosen a path. However, this work aims to carry out a flat mapping, in 2D and not in 3D. The influence of different obstacles (color, shape, material) on the quality and reliability of the mapping is not addressed. The article [2] presents work on the mapping of a part in 3D, using only a 2D lidar. To do this, the lidar is mounted on a gimbal driven by a servo motor. Rotating vertically, the scanning is done according to the axis of rotation of the Lidar but also that of the servomotor, allowing to recover the data of a 3D environment. All the actuators and sensors are programmed via ROS and a Raspberry PI serves as the on-board computer. The robot manages to generate a map of the laboratory experiment, but some dark surfaces are not detected by the device. This is because the robot only scans once it is stationary and then calculates its trajectory to the next position. In this study, the environment is mapped after several shots, but as in the previous paper, navigation and scanning are done in two steps, which makes the system poorly adapted to a dynamic environment. The paper mentions poor performance in the presence of black surfaces but does not really quantify the influences of color, material or obstacle size. Finally, the paper does not quantify the performance in terms of spatial accuracy of the mapping or the time to complete the task.

The aim of the paper [3] was to find a way to navigate a robot indoors with reduced sensory equipment to avoid as many potential collisions as possible. To achieve this, the researchers used a 2D RP lidar with a rotating mirror to redirect the transmitted and received beams to other points in the environment to obtain a 3D scan. The architecture of the robot allows real-time monitoring of the data, thanks to Bluetooth communication between the device and an external processor. The relative difference between the actual

measurement and that obtained by the lidar varies between 0.5% and 4%. The system takes about 11ms to process the acquired data for one scan of the environment. However, the robot remained static during the experiment. Furthermore, the obstacle scanned is a ceiling which represents a static and simple obstacle. Finally, we have no information on the performance of the telemetry according to the characteristics of the obstacle (size, color, shape...). The paper [4] proposes an architecture composed of several low-cost 2D lidars capable of mapping a 3D environment, the three lidars are mounted on the same platform with different inclinations allowing to scan different areas. The information is processed by an FPGA (Field Programmable Gate Array) which transforms the 2D coordinates into 3D. The environment obtained is composed of 6000 points, which corresponds to the simultaneous operation of the 3 Lidars for 0.5 seconds. The Lidars used scan a 360° horizontal plane and the inclination of the Lidars allows a 90° open vertical field of view. However, the accuracy of the infrared sensors is not discussed. In addition, three lidars are used in this paper, whereas we propose to use only one. The article [5] deals with the exploration of an outdoor environment using a UAV equipped with a 2D lidar, which is cheaper and less cumbersome than a 3D lidar. To achieve a 3D mapping of the environment, the drone follows a circular path around a point of interest during which it stops N times to perform a scan where the value of N can be chosen by the user and allows to modulate the accuracy of the resulting representation. To explore an entire area, several points of interest are chosen. This being the case, it should be noted that a drone has more freedom of movement than a terrestrial vehicle, the 3^e^ dimension is obtained thanks to this freedom of movement, which is not the case for the car. Furthermore, the scan is carried out over a period of 30 minutes, which is too long. Finally, we have no information on the accuracy of the perceived environment. The article [6] shows a method for mapping an urban environment using a 2D lidar with a SLAM (simultaneous localization and mapping) algorithm. The lidar is positioned on a vehicle that moves on a university campus. However, the data collected in this study is 2D, for exploration and place recognition purposes, not for safety during navigation. The only performance indicators we have are the superposition between satellite vision and experimental data, which does not give us any indication of the effective accuracy of infrared sensors on a smaller scale. Finally, the paper [7] proposes to study an autonomous vehicle equipped with a 3D Lidar and a camera on its ability to recognize a pedestrian and other obstacle in different situations under different weather conditions (clear weather and rain). The study compared the recognition results

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with lidar and with camera separately. The camera, which performs very well in these recognition tasks, is taken as a reference in the tests. It was shown that Lidar performs better in clear weather than in rainy weather. The failure rate increases from 33% to 66% respectively. Other parameters include the orientation of the pedestrian (front or side), the position of the pedestrian (in front of, to the side of, or behind the car), and the presence of other nearby obstacles. The article focuses on the recognition of an obstacle type and not only on obstacle detection. It does not allow for distance measurement by lidar between the obstacle and the vehicle.

In conclusion, there are no studies in the scientific literature that address our problem. The articles studied are all related to lidar technology. Only four articles are interested in mapping a 3D environment from a 2D lidar, but not for dynamic navigation purposes, which is a key point for autonomous car needs. The topic of dynamic navigation is covered by only one paper. There are only two papers for which precise computation and environment scanning times have been given, which can give us indications on the dynamic performance of the system. A full mapping time is given in one paper, but the estimated time is too long for our real time need.

### III. STRATEGY FOR CHOOSING ARCHITECTURES

Two simple platform architectures, on which the 2D lidar will be mounted, will be proposed. A first architecture, inclined and rotating at constant speed, will allow the device to scan a 3D scene.

<figure><img src="image_3.png" /><figcaption>Figure 1: Architecture 1 - Rotating platform (blue) with constant speed in time and integrated 2D lidar (orange)</figcaption></figure>

In this first architecture, the rotating platform is controlled by a stepper motor which allows the rotation of the assembly around the axis (Mz) with the speed of rotation $\vec{\Omega}$ constant in time. The 2D lidar rotates around the axis (Lz') with a rotation speed $\vec{\omega}$ of constant norm in time but the vector is variable in time if the axes (Mz) and (Lz') are not merged. For this purpose, we note by $\theta$ the tilt angle of the lidar platform corresponding to the angle between the axes (Mz) and (Lz'). The second architecture is also

composed of a platform integrating the 2D lidar whose movement is oscillating.

<figure><img src="image_4.png" /><figcaption>Figure 2: Architecture 2 - Oscillating platform (blue) with time-varying speed and integrated 2D lidar (orange)</figcaption></figure>

The platform allows the whole 2D lidar device to oscillate around the axis (My) between two extreme angles, the angular position of the lidar with respect to the "horizontal" plane and which we note $-\beta_{max}$ and $\beta_{max}$. In a general framework, we call by $\beta$ the angle between the axes (Lx') and (Mx) such that $\dot{\beta} = \frac{d\beta}{dt} = \Omega(t)$ represents the instantaneous speed of rotation of the rotating platform. The sign convention of $\beta$ is in the trigonometric direction and therefore in the figure above, $\beta$ is considered as negative. The 2D lidar rotates with a constant rotation speed $\vec{\omega}$ around the axis (Lz'). We note, with reference to the figure, that $M = L$.

### IV. KINEMATIC PERFORMANCE MODELLING OF ARCHITECTURE

#### 1

We note by $\vec{u}$ a unit vector carried by the 2D lidar laser emission. It will be assumed that $\vec{u}$ is orthogonal to $\vec{\omega}$. We can also consider that $\vec{u}$ is collinear to the axis (Ly'). The kinematic equations (in the permanent rotation regime) governing the time evolution of $\vec{\omega}$ and of $\vec{u}$ are:

$$ \frac{d\vec{\omega}}{dt} = \vec{\Omega} \times \vec{\omega} $$

And:

$$ \frac{d\vec{u}}{dt} = (\vec{\omega} + \vec{\Omega}) \times \vec{u} $$

With, as a reminder, $\vec{\Omega}$ is the speed of rotation of the platform and is constant over time.

We can then write:

$$ \frac{d^2\vec{\omega}}{dt^2} = \cos\theta \omega \Omega \vec{\Omega} - \Omega^2 \vec{\omega} $$

And:

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$$ \frac{d^2 \vec{u}}{dt^2} = \vec{u} \cdot \vec{\Omega} (2\vec{\omega} + \vec{\Omega}) - (\omega^2 + 2\omega\Omega \cos\theta + \Omega^2)\vec{u} $$

For the solutions to these two differential equations, it is easier to reason in projection. By noting by $(\vec{i}, \vec{j}, \vec{k})$ the unit vectors carried respectively by the axes $(Mx), (My)$ and $(Mz)$ we can write:

$$ \vec{\omega}(t) = \omega(\sin\theta \cos(\Omega t + \psi)\vec{i} + \sin\theta \sin(\Omega t + \psi)\vec{j} + \cos\theta \cos(\Omega t)\vec{k}) $$

With $\psi$ a time constant. In the following, we set the values of all the quantities linked to the initial conditions to 0 without detracting from the generality. Here, we therefore set $\psi = 0$. We can then write:

$$ \vec{u}(t) = (\cos(\omega t) \cos(\Omega t) - \cos\theta \sin(\omega t) \sin(\Omega t))\vec{i} + (\cos(\omega t) \sin(\Omega t) + \cos\theta \sin(\omega t) \cos(\Omega t))\vec{j} - \sin\theta \sin(\omega t) \vec{k} $$

Let $F$ be the sampling frequency of the 2D lidar. Two consecutive 2D lidar emissions will, at time $t$ have as carrier vector $\vec{u}(t)$ and $\vec{u}(t + \frac{1}{F})$ such that if $F \gg \omega$, $\Omega$ at order 1:

$$ \vec{u}\left(t + \frac{1}{F}\right) \approx \vec{u}(t) + \frac{1}{F} \frac{\partial \vec{u}(t)}{\partial t} $$

The projection of $\vec{u}$ onto the plane $(xMy)$ is a vector of norm $\sqrt{\sin^2(\omega t) \cos^2\theta + \cos^2(\omega t)}$ which is independent of $\Omega$. The azimuth angle in cylindrical coordinates of the scene scan by the vector $\vec{u}$, which we note $\varphi$ is such that:

$$ \cos \varphi = \frac{\cos(\omega t) \cos(\Omega t) - \cos \theta \sin(\omega t) \sin(\Omega t)}{\sqrt{\sin^2(\omega t) \cos^2 \theta + \cos^2(\omega t)}} $$

$$ \sin \varphi = \frac{\cos(\omega t) \sin(\Omega t) + \cos \theta \sin(\omega t) \cos(\Omega t)}{\sqrt{\sin^2(\omega t) \cos^2 \theta + \cos^2(\omega t)}} $$

For a given angle $\varphi$ angle, looking for the set of times $t$ verifying the 2 equations above amounts to measuring the times for which the 3D lidar device scans the same direction of the scene, i.e. the same value of $\varphi$. By simplifying the 2 equations into:

$$ \tan \varphi = \frac{\cos(\omega t) \sin(\Omega t) + \cos \theta \sin(\omega t) \cos(\Omega t)}{\cos(\omega t) \cos(\Omega t) - \cos \theta \sin(\omega t) \sin(\Omega t)} $$

Thus, at $\varphi$ given, searching for the set of times verifying this equation is equivalent to searching for the set of times for which the device in question scans the same or the opposite direction of the scene (A $\varphi$ or to $\varphi + \pi$). By exploiting the differential equation in $\vec{u}$ we can observe that the characteristic period for the laser beam to scan the same direction in space (the same angle $\varphi$) is literally $\Delta t$ such that:

$$ \Delta t = \frac{2\pi}{\sqrt{\omega^2 + 2\omega\Omega \cos\theta + \Omega^2}} $$

There is therefore a third scanned point closest to the other two defined points, as a reminder for $\vec{u}(t)$ and $\vec{u}(t + \frac{1}{F})$ for the vector carried for a later time $\Delta t$ characterised for $\vec{u}(t + \Delta t)$ such that the projection along the axis $(Mz)$ would be:

$$ \vec{u}(t + \Delta t) - \vec{u}(t)|_z \approx 2 \sin \theta \sin\left(\frac{\omega \Delta t}{2}\right) \cos\left(\omega\left(t + \frac{\Delta t}{2}\right)\right) \vec{k} $$

The area of the triangle, formed by the 3 lidar shots in $t, t + \frac{1}{F}$ and $t + \Delta t$ on a spherical obstacle of radius $r$ whose centre is the device, is then:

$$ A(t) \approx \frac{1}{2} r^2 \left\| \frac{1}{F} \frac{\partial \vec{u}(t)}{\partial t} \times (\vec{u}(t + \Delta t) - \vec{u}(t)|_z) \right\| $$

Then we have:

$$ A(t) = \left| \frac{r^2}{F} \sin\theta \sin\left(\frac{\omega\Delta t}{2}\right) \cos\left(\omega\left(t + \frac{\Delta t}{2}\right)\right) \sqrt{\cos^2(\omega t)(\omega\cos\theta + \Omega)^2 + \sin^2(\omega t)(\omega + \Omega\cos\theta)^2} \right| $$

Since the obstacle is a sphere with the center of the device, it faces the laser shot. Then the number of points scanned per unit solid angle is $N$ such that:

$$ N(t) = \frac{r^2}{A(t)} $$

In the very special but representative case where $t = 0$ we would have:

$$ N(t=0) = \left| \frac{2F}{\sin\theta \sin\left(\frac{2\pi\omega}{\sqrt{\omega^2 + 2\omega\Omega\cos\theta + \Omega^2}}\right)(\omega\cos\theta + \Omega)} \right| $$

The total solid angle swept is:

$$ \gamma = 4\pi \sin\theta $$

A compromise must therefore be found between the value of $\gamma$ and the value of $N$ by choosing the value of $\theta$.

For $F = 5kHz$, $\omega = 12\pi \cdot s^{-1}$ and $\Omega = 3\pi \cdot s^{-1}$ (i.e. a period of $\frac{1}{3}$ of a second), for $\theta = 30^\circ$, we get $N = 261 \text{ points} \cdot \text{sr}^{-1}$. On the other hand, for $\theta = 15^\circ$, $N = 447 \text{ points} \cdot \text{sr}^{-1}$.

The average number of scans during a scan cycle period $\Delta t$ is $\bar{N}$ such that:

$$ \bar{N} = \frac{F\Delta t}{\gamma} $$

The time $t_0$ during which the entire solid angle $\gamma$ is scanned once, is $t_0 = n\Delta t$ such that:

$$ t_0 \approx \frac{N}{\bar{N}} \Delta t $$

For $\theta = 30^\circ$, $t_0 \approx 4,8\Delta t$ while for $\theta = 15^\circ$, $t_0 \approx 4,3\Delta t$. For different values of $\theta$ the graph below shows the evolution of the global density of the scatterplot for $F = 5kHz$, as a function of $\Omega$ in revolutions per second.
<figure><img src="image_7.png" /><figcaption>Figure 3: Evolution of the average density of the point cloud (in number of points per steradian), for different values of θ as a function of Ω (in revolutions per second) in one second</figcaption></figure>

Thus, the smaller the total solid angle, the θ is small and the greater the average density of scanned points per unit steradian.

## V. MODELLING OF THE KINEMATIC PERFORMANCE OF THE ARCHITECTURE 2

Similarly, the previous notation is maintained, i.e. we note by $\vec{u}$ a unit vector carried by the laser emission of the 2D lidar. Studying the same 2D lidar, we will admit that $\vec{u}$ is orthogonal to $\vec{\omega}$. We can also consider that $\vec{u}$ is collinear to the axis (Lx').

Ideally, the platform oscillates at constant speed between the extreme angular positions of $\beta$ which, as a reminder, we call $-\beta_{max}$ and $+\beta_{max}$ with a periodicity of $T$. Sometimes $\vec{\Omega} = \Omega \vec{j}$ over a period of $\frac{T}{2}$ and sometimes $\vec{\Omega} = -\Omega \vec{j}$ over the same period, with $\Omega = \frac{4\beta_{max}}{T}$. For modelling purposes, we will consider

that:

$$ \vec{\Omega}(t) = \Omega_0 \sin\left(\frac{2\pi t}{T}\right) \vec{j} $$

With $\Omega_0$ the oscillation amplitude such that:

$$ \Omega_0 = \frac{2\pi\beta_{max}}{T} $$

The kinematic equations (in the permanent rotation regime) governing the time evolution of $\vec{\omega}$ and of $\vec{u}$ are:

$$ \frac{d\vec{\omega}}{dt} = \vec{\Omega} \times \vec{\omega} $$

And:

$$ \frac{d\vec{v}}{dt} = (\vec{\omega} + \vec{\Omega}) \times \vec{u} $$

The differential equations are derived from this:

$$ \frac{d^2\vec{\omega}}{dt^2} = \frac{2\pi\Omega_0}{T} \cos\left(\frac{2\pi t}{T}\right) \vec{j} \times \vec{\omega} - \Omega_0^2 \sin^2\left(\frac{2\pi t}{T}\right) \vec{\omega} $$

And:

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$$ \frac{d^2\vec{u}}{dt^2} = \frac{2\pi\Omega_0}{T} \cos\left(\frac{2\pi t}{T}\right) \vec{j} \times \vec{u} + \vec{u} \cdot \vec{\Omega}(2\vec{\omega} + \vec{\Omega}) - (\omega^2 + \Omega^2)\vec{u} $$

Similarly, for the solutions to these two differential equations, it is easier to reason in projection. We then have:

$$ \vec{\omega} = \omega(-\sin\beta\vec{i} + \cos\beta\vec{k}) $$

And:

$$ \vec{u} = \cos\alpha \cos\beta \vec{i} - \sin\alpha \vec{j} + \cos\alpha \sin\beta \vec{k} $$

With $\alpha$ is the characteristic angle of rotation of the 2D Lidar. We can then write:

$$ \alpha(t) = \omega t \\ \beta(t) = -\frac{\Omega_0 T}{2\pi} \cos\left(\frac{2\pi t}{T}\right) $$

Two consecutive 2D Lidar laser shots will form the two closest measurement points on the scene in $\vec{u}(t)$ and in $\vec{u}(t + \frac{1}{F})$ but also for $\vec{u}(t + \frac{2\pi}{\omega})$ corresponding to a rotation of the 2D Lidar. We always have the following approximation:

$$ \vec{u}\left(t + \frac{1}{F}\right) \approx \vec{u}(t) + \frac{1}{F} \frac{\partial \vec{u}(t)}{\partial t} $$

Where:

$$ \frac{\partial \vec{u}(t)}{\partial t} = -(\omega \sin \alpha \cos \beta + \Omega \cos \alpha \sin \beta)\vec{i} - \omega \cos \alpha \vec{j} + (-\omega \sin \alpha \sin \beta + \Omega \cos \alpha \cos \beta)\vec{k} $$

During the period $\frac{2\pi}{\omega}$ the size $\beta$ will change from $\Delta\beta = \beta(t + \frac{2\pi}{\omega}) - \beta(t)$ such that:

$$ \Delta\beta(t) = \frac{\Omega_0 T}{\pi} \sin\left(\frac{2\pi^2}{\omega T}\right) \sin\left(\frac{2\pi t}{T} + \frac{2\pi^2}{\omega T}\right) $$

And so:

$$ \vec{u}\left(t + \frac{2\pi}{\omega}\right) - \vec{u}(t) = -2 \cos \alpha \sin\left(\frac{\Delta\beta}{2}\right) \left(\sin\left(\beta + \frac{\Delta\beta}{2}\right)\vec{i} - \cos\left(\beta + \frac{\Delta\beta}{2}\right)\vec{k}\right) $$

Reminder (To be deleted after calculation)

$$ \vec{u}\left(t + \frac{1}{F}\right) - \vec{u}(t) = -\frac{1}{F}\left(\omega \sin \alpha \cos \beta + \Omega \cos \alpha \sin \beta\right)\vec{i} - \frac{1}{F}\cos \alpha \vec{j} + \frac{1}{F}\left(-\omega \sin \alpha \sin \beta + \Omega \cos \alpha \cos \beta\right)\vec{k} $$

The area of the triangle, formed by the 3 lidar shots in $t, t + \frac{1}{F}$ and $t + \frac{2\pi}{\omega}$ on a spherical obstacle of radius $r$ whose centre is the device, is then:

$$ A(t) \approx \frac{1}{2} r^2 \left\| \frac{1}{F} \frac{\partial \vec{u}(t)}{\partial t} \times \left( \vec{u}\left(t + \frac{2\pi}{\omega}\right) - \vec{u}(t) \right) \right\| $$

We have also:
<figure><img src="image_9.png" /></figure>

$$ 
\begin{aligned}
& \frac{1}{F} \frac{\partial \vec{u}(t)}{\partial t} \times \left( \vec{u}\left(t + \frac{2\pi}{\omega}\right) - \vec{u}(t) \right) \\
&= \frac{2 \cos \alpha \sin\left(\frac{\Delta\beta}{2}\right)}{F} \left[ -\cos\left(\beta + \frac{\Delta\beta}{2}\right) (\omega \sin \alpha \cos \beta + \Omega \cos \alpha \sin \beta) \right. \\
&\qquad \left. + \sin\left(\beta + \frac{\Delta\beta}{2}\right) (-\omega \sin \alpha \sin \beta + \Omega \cos \alpha \cos \beta) \right] \vec{j} \\
&\quad + 2 \frac{\omega}{F} \cos^2 \alpha \sin\left(\frac{\Delta\beta}{2}\right) \sin\left(\beta + \frac{\Delta\beta}{2}\right) \vec{k} \\
&\quad + 2 \frac{\omega}{F} \cos^2 \alpha \sin\left(\frac{\Delta\beta}{2}\right) \cos\left(\beta + \frac{\Delta\beta}{2}\right) \vec{l}
\end{aligned}
$$

This leads to:

$$ A(t) \approx \frac{\left| \cos \alpha \sin \left( \frac{\Delta\beta}{2} \right) \right|}{F} r^2 \sqrt{\left( \omega \sin \alpha \cos \left( \frac{\Delta\beta}{2} \right) - \Omega \cos \alpha \sin \left( \frac{\Delta\beta}{2} \right) \right)^2 + \omega^2 \cos^2 \alpha} $$

The number of points scanned per unit solid angle is $N$ such that:

$$ N(t) = \frac{r^2}{A(t)} $$

In the very special but representative case where $t = 0$ we would have:

$$ \Delta\beta(t = 0) = \frac{\Omega_0 T}{\pi} \sin^2\left(\frac{2\pi^2}{\omega T}\right) $$

And:

$$ A(t) = \frac{\left| \sin\left(\frac{\Delta\beta}{2}\right) \right| r^2 \omega}{F} $$

And so:

$$ N(t = 0) = \frac{F}{\left| \sin\left(\frac{\Delta\beta}{2}\right) \right| \omega} $$

The total solid angle is:

$$ \gamma = 8\beta_{max} $$

By choosing to equalize the solid angle of the two architectures for $\theta = 30^\circ$ for architecture 1, it is necessary to choose $\beta_{max} = 45^\circ$. For the period of platform 2, we will choose a period equivalent to that of platform 1, i.e. $T = \frac{1}{3}$ s. From this we can deduce $\Omega_0 = \frac{3\pi^2}{2} \text{rad.s}^{-1} \approx 14,8 \text{rad.s}^{-1}$. In the case where $\omega = 12 \text{tr/s}^{-1}$, we deduce $\Delta\beta = \frac{\pi}{4} \text{rad} = 45^\circ$ and therefore if $F = 5 \text{kHz}$ then $N(t = 0) = 173 \text{points.sr}^{-1}$. Similarly, by choosing to equalise the solid angle of the two architectures for $\theta = 15^\circ$ for architecture 1, it is necessary to choose $\beta_{max} = 23,3^\circ$. For $T = \frac{1}{3}$ s we have $\Omega_0 = 7,66 \text{rad.s}^{-1}$ and therefore $\Delta\beta = 0,40 \text{rad} = 23,3^\circ$ and therefore for $F = 5 \text{kHz}$, $N(t = 0) = 328 \text{points.sr}^{-1}$. For different values of $T$ corresponding to different values of scan frequency in Hz of $\frac{1}{T}$ the graph below shows the

evolution of the global density of the point cloud, also for different values of $\beta_{max}$.

<figure><img src="image_10.png" /><figcaption>Figure 4: Evolution of the average density of the point cloud (in number of points per steradian), for different values of $\beta_{max}$ as a function of the scan frequency $\frac{1}{T}$ (in Hz) in one second</figcaption></figure>

Similarly, the lower the $\beta_{max}$ is, the smaller the total solid angle scanned, but on the other hand, the greater the average density of points scanned per unit steradian.

## VI. COMPARISON OF THE TWO ARCHITECTURES

Comparing the performance of the two architectures, Architecture 1 has a significant advantage in terms of total solid angle scanned, but Architecture 2 outperforms Architecture 1 in terms of the density of points scanned.

<figure><img src="image_11.png" /><figcaption>Figure 5: Comparison of the total solid angles swept by the 2 architectures as a function of $\theta$ for architecture 1 and $\beta_{max}$ for architecture 2</figcaption></figure>

The graph in the figure below gives the ratio of the densities of the scanned point cloud per unit time as a function of $\theta$ for architecture 1 or $\beta_{max}$ for architecture 2.
<figure><img src="image_12.png" /></figure>

of a flat floor at a height of $z = 0$. We then add a cylinder whose center of gravity is at the distance $d$ from the device. We made this choice of shape in order to guarantee a scanning surface of 1 square meter: indeed, whatever the orientation of the cylinder with respect to the device, if its base remains on the ground, the device will see a square surface, contrary to a cube where the surface seen by the device would have been dependent on the orientation of the latter with respect to the device.

<figure><img src="image_13.png" /><figcaption>Figure 8: Geometry of the test environment</figcaption></figure>

We made 12 measurements, each time increasing the distance $d$ of 1 meter in 1 meter and 12 meters in distance. The detection rate is decreasing as a function of $d$:

<figure><img src="image_14.png" /><figcaption>Figure 9: Evolution of the obstacle detection rate as a function of $d$</figcaption></figure>

For an obstacle with a size equivalent to a cylinder of 1 meter in height and 1 meter in diameter, the device is 100% reliable up to 4 meters, beyond which the reliability of the device decreases linearly to reach 34% at 12 meters, the maximum range of the device. Such an obstacle will therefore certainly not be detected as soon as it enters the detection perimeter of the device, which is a sphere of radius 12 meters around the vehicle. Furthermore, the average number of points detected on the obstacle also decreases according to $d$.

<figure><img src="image_15.png" /><figcaption>Figure 6: Ratio of point densities between the 2 architectures as a function of $\theta$ for architecture 1 and $\beta_{max}$ for architecture 2</figcaption></figure>

The density ratio is less than or equal to 1, which means that the density of the point cloud generated by architecture 2 is higher. This ratio reaches 1 for an angle $\theta$ and $\beta_{max} = \frac{\pi}{2}$.

Finally, the figure below gives the values of the means and standard deviations of local point density for the two architectures.

### Moyennes et écart types des densités locales

<figure><img src="image_16.png" /><figcaption>Figure 7: Comparison of the means and standard deviations of the local densities of scanned points for architectures 1 and 2</figcaption></figure>

From the comparison results, architecture 1 has a better 360° scan of the scene. Indeed, architecture 2 provides a denser point cloud. Nevertheless, the latter is disparate: the local density of the point cloud is much more variable for architecture 2 than for architecture 1. Moreover, architecture 2 offers a very limited field of view on the sides, which could be problematic for the safety of the vehicle in several situations. In addition, the rotation of the platform in architecture 1 is continuous, which would limit jerking and therefore breakage. This is why Architecture 1 will be preferred.

## VII. TELEMETRY PERFORMANCE OF ARCHITECTURE 1

To obtain numerical performances of the architecture 1 we have chosen to test the detectability of a specific obstacle as a function of the distance between it and our device. To do this, we simulate our device on Gazebo with an "empty" world, i.e. simply composed
<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /><figcaption>Figure 10Evolution of the number of points scanned on the obstacle as a function of d</figcaption></figure>

The dispersion of the measurements does not appear to be a function of *d*. It can be observed that the detection rate is related to the number of points scanned: if the obstacle is too far away, then the number of points scanned on the obstacle decreases and the lack of regularity leads to detection failures. To measure the impact of the angular position of the obstacle $\varphi$ with respect to the architecture 1 device, we choose to move the centre of gravity of the obstacle on a circle of radius 6m as shown on the figure below.

<figure><img src="image_19.png" /><figcaption>Figure 11Diagram of the relative position of the obstacle to the device for simulation purposes</figcaption></figure>

The figure below shows the detection rate (out of 100 scans at each measurement) as a function of this angle $\varphi$ in degrees.

<figure><img src="image_20.png" /><figcaption>Figure 12Evolution of the detection rate as a function of $\varphi$</figcaption></figure>

The detection rate is not constant as a function of the angle $\varphi$. Nevertheless, no correlation pattern seems to emerge between the angle $\varphi$ and the detection rate. On this plot, we observe an average detection rate of 64% and a standard deviation of 4.3%. Finally, the figure below shows the average number of points detected per scan (100 measurements per $\varphi$).

<figure><img src="image_21.png" /><figcaption>Figure 13Evolution of the average number of points scanned on the obstacle as a function of $\varphi$</figcaption></figure>

Again, we see that the number of points scanned on the obstacle is not constant as a function of $\varphi$ and no pattern of evolution seems to emerge.

## VIII. CONCLUSION

The current work aims to develop a panel of laser rangefinders to map a 3D environment, in order to help a car, navigate in an urban environment. Based on a rotating 2D Lidar, the device must be externally operated to scan the third dimension. Several architectures have been proposed and studied to fulfil this task and to obtain the most suitable point cloud for autonomous vehicle navigation. The criteria considered were the density of the point cloud, its homogeneity, and its theoretical accuracy. In the scientific literature, we have shown that rotating 2D Lidar has already been used for 2D and 3D mapping, but only for recognition purposes, to identify the topography of a place and not to help a vehicle to navigate. The first stage of this study allowed us to theoretically quantify the performance of our two architectures and to choose which one would best meet the needs of autonomous land vehicle navigation. Thanks to simulation and numerical calculation tools such as Octave and Gazebo, we were able to obtain interesting numerical results showing, in the case of architecture 1, that the detection rate is constant up to 4m and then quasi-linear decreasing up to 12m for an obstacle of 1m² surface. We were also able to show that the number of points detected on the obstacle varies with the distance between the device and the obstacle of 1m² surface area and is perhaps represented by a power function. Current work has shown that it is possible to map an environment in 3 dimensions using a rotating 2D
<figure><img src="image_22.png" /></figure>

Lidar. However, the amount of data acquired by the Lidar and the number of full environment scans obtained per second is still too low to guarantee safe navigation of a car travelling at real speed. However, for vehicles travelling at lower speeds in simple environments, such as robots on an industrial site, this device could offer a workable solution.

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