# Regressed Terrain Traversability Cost for Autonomous Navigation Based on Image Textures
## 2 Problem Definition  
Terrains considered in this study are not always uniform due to irregular obstacles, such as large stones and roots of trees, and are assumed to be traversable even at a higher cost than homogeneous terrains. Typical obstacle size varies between 40 to 80 mm in height, which is a significant mobility challenge for the mobile robot as the wheels radius is 105 mm. Terrains are assumed to be rigid. Deformable terrains [11] that can lead the mobile robot to lose balance, or cause slippage/sinkage [26], are not covered in this paper. In addition, untraversable terrains are also beyond the scope of this paper. Samples of the terrains investigated in this paper are introduced in Figures 1 and 2.  
This study was conducted with a widely used mobile platform (Pioneer 3 AT) for navigation and traversability projects in outdoor environments [27; 28; 29; 30]. The platform is driven by four wheels and is capable of traversing bumps up to 100 mm, gaps up to 150 mm, and slopes up to 19\(\circ\). The mobile platform is equipped with a vision sensor and an acceleration sensor, as shown in Figure 3. The camera gathers terrain images, and the IMU registers the acceleration signal generated during a sequence of a run. The goal is to estimate vibrations as a measure of traversability only from terrain images. The predicted traversability cost will be used to anticipate the motion nature of subsequent terrains and is expected to enable a safe run. The focus is not on which robot offers the best handling of terrain  
Figure 1: Natural terrains. (**a**) Gravel; (**b**) Stones; (**c**) Non-uniform; (**d**) Grass; (**e**) Woodchip; (**f**) Non-uniform.  
unevenness but on enabling any type of robot, regardless of its construction/configuration, to traverse non-uniform terrains.  
In the bird's-eye view given by Figure 4, the gray circle represents the mobile robot and the black rectangles are the wheels. The wheel base and the wheels radius are denoted by \(L\) and \(r\) respectively. The system accepts as input \(v_{1}\) and \(v_{\text{r}}\) as input, respectively, which are the angular velocities of the left and right wheels. The mobile robot motion is described as follows:  
\[\begin{cases}z(t)=\frac{r}{2}\times(v_{1}(t)+v_{\text{r}}(t))\times t\times \cos(\phi(t)),\\ x(t)=\frac{r}{2}\times(v_{1}(t)+v_{\text{r}}(t))\times t\times\sin(\phi(t)),\\ \phi(t)=\frac{r}{L}(v_{1}(t)-v_{\text{r}}(t)),\end{cases} \tag{1}\]  
where \(x\) and \(z\) are the robot position and \(\phi\) is the orientation of the platform. The angular velocities for both the left and right wheels are similar; therefore, the robot will run straight according to the world \(Z_{w}\)-axis. Moreover, the effects of steering and acceleration/deceleration are nonexistent. The mobile robot has no suspension system, that is, it is a rigid body. Hence, effects of interaction of the mobile platform on the right and left wheels will appear on the sensor values. Operating in outdoor environments comes with a very important lightning conditions challenge. To simplify the problem setting, experiments were conducted under fair light conditions with neither shadows nor strong backlight.  
Figure 3: Pioneer 3 AT hardware configuration.  
Figure 2: Artificial terrains. **(a)** Slick asphalt; **(b)** Granulated asphalt; **(c)** Tiles; **(d)** Wood.