Multifractal Terrain Generation for Evaluating Autonomous Off-Road Ground Vehicles

TL;DR

The paper presents a multifractal terrain generation framework based on the 3D Weierstrass-Mandelbrot function to control roughness via the fractal dimension $D$. By combining three monofractal DEMs and varying $D$ across $2.3$, $2.45$, and $2.6$, it creates $60$ unique terrains and categorizes their roughness with gradient maps. These terrains are rendered in Unreal Engine for AGV testing, where 20 randomized straight-line paths are evaluated per map, revealing that higher $D$ reduces low-roughness areas and increases semi- and high-roughness regions, lowering success rates and increasing RMS accelerations and angular rates. The approach provides a practical, tunable tool for stress-testing terrain-aware path planning and navigation, adaptable to different simulators and vehicle scales.

Abstract

We present a multifractal artificial terrain generation method that uses the 3D Weierstrass-Mandelbrot function to control roughness. By varying the fractal dimension used in terrain generation across three different values, we generate 60 unique off-road terrains. We use gradient maps to categorize the roughness of each terrain, consisting of low-, semi-, and high-roughness areas. To test how the fractal dimension affects the difficulty of vehicle traversals, we measure the success rates, vertical accelerations, pitch and roll rates, and traversal times of an autonomous ground vehicle traversing 20 randomized straight-line paths in each terrain. As we increase the fractal dimension from 2.3 to 2.45 and from 2.45 to 2.6, we find that the median area of low-roughness terrain decreases 13.8% and 7.16%, the median area of semi-rough terrain increases 11.7% and 5.63%, and the median area of high-roughness terrain increases 1.54% and 3.33%, all respectively. We find that the median success rate of the vehicle decreases 22.5% and 25% as the fractal dimension increases from 2.3 to 2.45 and from 2.45 to 2.6, respectively. Successful traversal results show that the median root-mean-squared vertical accelerations, median root-mean-squared pitch and roll rates, and median traversal times all increase with the fractal dimension.

Figures (7)

• Figure 1: Top and perspective views of three example monofractal terrains (a, c), (b, f), (c, g) generated using the 3D Weierstrass-Mandelbrot function using different parameters. An example multifractal terrain (d,h) using our approach which combines the three example terrains to create more varied roughness for autonomous ground vehicle testing.
• Figure 2: Example low- (a), mid- (b), and high-frequency (c) monofractal digital elevation maps (DEMs), combined via the pixel-wise product to create a fourth multifractal DEM (d) which is imported into Unreal Engine for autonomous ground vehicle testing. Note that (a-c) have Weierstrass-Mandelbrot ridges ($M$) equal to 16, 32, and 64, fractal dimensions ($D$) equal to 2.2, 2.45, and 2.45, and elevation scaling coefficients ($G$) equal to 1e-6, 8e-8, and 1e-8, all respectively.
• Figure 3: An example multifractal digital elevation map (DEM) with a top-down view of its grayscale image (a). The DEM in (a) is imported, scaled, and rendered in Unreal Engine (b). A perspective view of the Unreal Engine-rendered terrain is shown in (c) with a simulated Clearpath Husky autonomous ground vehicle in view, with its corresponding map location shown in (b) as a white rectangle.
• Figure 4: An example Moore-neighborhood-gradient map with pixels categorized into one of three groups: low (blue), semi (orange), or high roughness (yellow) (a). The gradient map from (a) is then filtered with a morphological closing filter for random mission selection, where high-roughness areas cannot contain a start/goal location, semi-rough areas cannot contain a start location, and low-roughness areas are free to contain start/goal locations (b). The filtered gradient map from (b) is shown with 20 randomly selected missions with straight lines connecting respective start (green) and goal (red) locations (c).
• Figure 5: A bar graph of the multifractal digital elevation maps' (DEM) median areas of low (a), semi (b), and high roughness (c), all by the fractal dimension used in the high-frequency DEM. Twenty terrain maps are generated for each of the three fractal dimension values of 2.3, 2.45, and 2.6, for a total of 60 maps.