Open AccessArticle

Map Representation and Navigation Planning for Legged Climbing UGVs in 3D Environments

Ao Xiang

Chenzhang Gong

¹ and

Li Fan

^1,2,*

College of Control Science and Engineering, Zhejiang University, Hangzhou 310058, China

Huzhou Institute of Zhejiang University, Huzhou 313000, China

Author to whom correspondence should be addressed.

Drones 2024, 8(12), 768; https://doi.org/10.3390/drones8120768

Submission received: 20 November 2024 / Revised: 13 December 2024 / Accepted: 17 December 2024 / Published: 19 December 2024

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Figure 1
The overall framework of the proposed method. The input information consists of RGB images and depth information captured by a depth camera. The RGB images are used to estimate the relative pose, while the depth images are converted into point clouds. The point clouds are first used to extract plane and boundary information, which are then used to construct the proposed MLEM. The numbers 1, 2 indicate the code of the different planes. Based on the MLEM, a universal hierarchical planning framework is used for path planning and motion generation in challenging 3D environments. The outputs of the planning framework are all joint angles trajectories. We use a position controller to control the joints. "> Figure 2
The multi-layer elevation map construction pipeline. Based on the plane and boundary information obtained, the point clouds of each plane are used to construct the MLEM. Each plane and its corresponding elevation map are labeled, and the boundary information stores the connection information for different planes. "> Figure 3
The transformation process of two adjacent planes. "> Figure 4
Cross-plane path planning and map unfolding process. After the MLEM is constructed and each plane is numbered, a search is conducted in the constructed 3D space map for possible paths from the start plane to the target plane. Based on priority, the planes involved in different paths are unfolded and a path search is conducted until a viable path is found. "> Figure 5
The different unfolding results with the same planes in a different order. "> Figure 6
The two forms of collision checking. For soft-collision checks, the surrounding grids validation of the foot are checked. A foot is considered supportive when the amount of valid grids around the foothold exceeds a specified threshold. A state is deemed valid only when all six feet are supportive. "> Figure 7
The sampling space <math display="inline"><semantics> <mrow> <mo>[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>θ</mi> <mo>]</mo> </mrow> </semantics></math> represents the coordinates on the map and rotation about the z-axis. "> Figure 8
Boundary motion state. (a). The simplified model and a hexapod prototype. (b) The three independent basic motion patterns. The transition between two adjacent states is realized through a single basic motion. "> Figure 9
The convex hull of the foot−valid workspace. "> Figure 10
The region transition process between the plane region and boundary region. "> Figure 11
The transition between two states is considered as either a translation or a rotation around a certain point. "> Figure 12
(a). The world coordinate system and the body coordinate system. (b). The change of the foot end within the body coordinate system when body posture adjusts. "> Figure 13
(a). The obstacle−free planned path and its simulation scene. (b). The soft−collision check planned path and its simulation scene. Numbers 1–4 represent the sequence of movements. "> Figure 14
In the simulation environment, the UGV first walks along the wall boundary and passes through a narrow gap. Numbers 1−16 represent the sequence of movements. It then executes a plane-to-plane transition to move from the ground to the wall. After performing obstacle avoidance maneuvers on the wall, it returns to the ground. This process fully demonstrates the locomotion capabilities of a legged climbing UGV in a complex 3D environment. "> Figure 15
(a). The red line represents the planned trajectory, while the blue line connecting the scatter points represents the measured trajectory. (b). The absolute motion error over time. "> Figure 16
(1–4) shows that the physical UGV moves along plane boundaries to traverse narrow gaps. (a–d) shows that the UGV conducts the plane-to-plane transition. More experiment and simulation details can be found in the video provided. "> Figure 17
The plane transition process under different angles. ">

Versions Notes

Abstract

Legged climbing unmanned ground vehicles (LC-UGVs) possess obstacle avoidance and wall transition capabilities, allowing them to move in 3D environments. Existing navigation methods for legged UGVs are only suitable for ground locomotion rather than 3D space. Although some wall transition methods have been proposed, they are specific to certain legged structures and have not been integrated into the navigation framework in full 3D environments. The planning of collision-free and accessible paths for legged climbing UGVs with any configuration in a 3D environment remains an open problem. This paper proposes a map representation suitable for the navigation planning of LC-UGVs in 3D space, named the Multi-Level Elevation Map (MLEM). Based on this map representation, we propose a universal hierarchical planning architecture. A global planner is applied to rapidly find cross-plane topological paths, and then a local planner and a motion generator based on motion primitives produces accessible paths and continuous motion trajectories. The hierarchical planning architecture equips the LC-UGVs with the ability to transition between different walls, thereby allowing them to navigate through challenging 3D environments.

Keywords:

legged climbing UGVs; map representation; cross-plane planing

1. Introduction

Legged climbing unmanned ground vehicles (LC-UGVs) are a type of UGV that possesses a legged structure (such as quadruped or hexapod) and moves by adhering to surfaces like the ground and walls. LC-UGVs have promising potential for applications such as wind turbine blade inspection [1], high-altitude glass cleaning [2], on-orbit services [3,4], etc.

Many LC-UGVs have been reported in recent years. Although these LC-UGVs have shown advanced climbing abilities, researchers have mainly focused on the adhesion mechanism and the design of the system. While these legged climbing UGVs only climb on vertical surfaces in tests, it is essential in reality that they are equipped to move in challenging 3D environments (e.g., stepping over obstacles and crossing walls). Only a few works have been conducted on the navigation planning of legged climbing UGVs in 3D environments [5,6]. However, their approaches are only suitable for specific configurations and for pre-constructed environments.

The navigation problem for LC-UGVs can be described as planning non-collision accessible discrete sequences of contacts in three-dimensional space from a start point to a destination. Although navigation planning for legged UGVs in complex terrains has been extensively studied, the studied methods are limited to ground environments. The major difference from legged UGVs is that the start and target points for legged climbing UGVs may lie on different planes (e.g., the ground and the ceiling). The legged climbing UGVs must not only move on a single plane but also transition across different planes. Therefore, the algorithm for the legged UGVs is hard to apply to solve the navigation planning problem for legged climbing UGVs directly.

A number of wall transition methods have been developed; however, they are usually specific to particular configurations of LC-UGVs, such as biped robots [5], quadruped robots [6], and hexapod robots [7]. There is a lack of a wall transition method that is applicable to any configuration of legged climbing UGVs. Additionally, the significant differences between wall transition motions and movements within a single plane make it challenging to integrate them into a universal navigation planning framework.

Therefore, this paper proposes a novel navigation planning architecture for LC-UGVs in 3D environments. Considering that LC-UGVs can not only cross obstacles on the ground but also perform plane transitions, it is necessary to redefine the map representation of the 3D environment. Traditional 2D maps, including the occupancy grid map [8] and the elevation map [9], are unable to represent information across different planes. Full 3D map representations such as Octomap [10] are redundant because the UGVs cannot fly in 3D space like a UAV. We propose a novel map representation, which we refer to as the multi-layer elevation map (MLEM).

Based on this map representation, a universal hierarchical planning framework is proposed. The planning framework includes three modules: a global planner, a local path planner, and a motion generator. The framework solves challenges brought on by the differences between wall transition and single-plane movement.

The hierarchical framework divides the complex planning problem into three linked and progressive levels. It enables the LC-UGVs to transition between different wall surfaces and even move along the boundary between two walls.

Our contributions are summarized as follows:

A novel map representation suitable for LC-UGVs, known as the multi-layer elevation map, is proposed.
We introduce a universal hierarchical planning framework for LC-UGVs in 3D environments. This architecture integrates wall transition and single-plane movements into the unified path planning method and is applicable to various configurations of LC-UGVs. Based on the framework, LC-UGVs can not only traverse obstacles on the ground but also perform wall transitions and even move along gaps between planes.
To validate the effectiveness of the proposed framework, extensive simulations alongside physical tests are performed.

In the following, we survey related works in Section 2. In Section 3, the proposed map representation and navigation planning framework are introduced in detail. Section 4 presents experimental and simulation testing to validate the effectiveness of our proposed methods. Finally, we provide a conclusion.

2. Related Work

Navigation for LC-UGVs in challenging terrain is a hot issue. In the process of planning, two key aspects need to be addressed. The first is the representation of the surrounding environment, as safe and efficient navigation requires knowledge about the shape and its traversability. The second concerns methods for navigation planning. We provide a review of these two aspects in this section.

2.1. Map Representation for Navigation

UGVs typically perceive their environment through on-board sensors such as depth cameras and LIDAR. The three-dimensional environmental information obtained by these sensors is often represented as point clouds. However, point clouds are not directly applicable to navigation planning; thus, many map representations for UGVs have been proposed.

Common map representations include Occupancy Grid Maps [8], Octomap [9], 2.5D Elevation Maps [10], and Euclidean Signed Distance Filed (ESDF). Occupancy Grid Maps store the probabilities of obstacles occupying a two-dimensional space and are suitable for UGVs that need continuous contact with the ground, such as wheeled UGVs. For robots operating in three-dimensional space, such as UAVs, Octomap and ESDF are commonly used. Octomap utilizes a hierarchical octree structure to store occupancy probabilities for a 3D environment. Each voxel in the ESDF contains the Euclidean distance to the nearest obstacle, making it suitable for obstacle avoidance.

For legged UGVs, 3D spatial information is redundant, and it requires high-resolution maps to obtain more accurate terrain information, which imposes a significant memory burden on 3D representations. 2.5D elevation maps store terrain heights in a two-dimensional grid of fixed resolution. It is efficient for legged UGVs to assess the traversability of terrains [10]. Therefore, 2.5D elevation maps are used for navigation planning for legged UGVs.

However, the elevation map is inadequate for planning in 3D space for LC-UGVs. Therefore, we propose a map representation method known as multi-layer elevation maps (MLEMs). It enables the LC-UGVs to plan in 3D environments without the need to represent the entire 3D environment.

2.2. Navigation Planning for Legged Climbing UGVs

Navigation planning for legged UGVs has been well studied. Most planning approaches firstly compute a single geometric traversability value per terrain patch [10]. Then, they perform state sampling and evaluation for foothold and body posture [11]. And finally, a graph-based path planning method is applied to find the optimal path [12]. Then, the robot can perform autonomous movements along the planned path by applying appropriate trajectory planning and motion control [13,14].

However, compared to legged UGVs, LC-UGVs possess more diverse motion capabilities. A typical distinction is that LC-UGVs not only need to move within a single plane but also need to transition between different planes.

A number of wall transition methods have been developed. Chen et al. [15] propose a single-step motion planning method suitable for biped pole-climbing robots to generate collision-free motion on Spatial Trusses. In their other work [5], they propose a planning method that moves across multiple wall surfaces. Kim et al. [6] perform a quadruped gait planning method for cross-wall movement, which can achieve smaller angle plane conversion. Gao et al. [7] propose a method for a hexapod UGV to perform vertical wall-surface conversion.

Most of these studies are limited to specific configurations (such as bipedal, hexapod, etc.), and the environment is assumed to be known. At present, there is a lack of a universal LC-UGVs navigation planning framework in 3D environments. Therefore, we propose a hierarchical planning framework designed for legged climbing UGVs.

3. Method

3.1. Overall Framework

In this section, we provide an overview of the proposed framework, which consists of several modules. The functions of the modules and the information flow are illustrated in Figure 1.

UGVs can use RGB image data and depth data obtained from the depth camera to estimate their position and construct 3D maps of the environment. This has been implemented in many SLAM frameworks [16,17]. We obtain 3D point clouds of the surrounding environment through RTABMAP [17]. Then, the 3D point clouds are represented as multi-layer elevation maps via plane segmentation and boundary detection.

Based on the proposed multi-layer elevation map, we apply a universal hierarchical planning framework to facilitate the movement of the LC-UGVs in the 3D environment.

The global planner quickly searches for possible transition paths from the starting plane to the target. It is worth noting that the global path is only the topological path. Then, the multi-layer elevation maps are unfolded onto a single elevation map based on the topological path, which is the core of the proposed MLEM. The advantage of global planning lies in reducing the search space to a finite number of planes.

In the local path planning stage, a discrete state sequence from the starting point to the destination is planned. We use a strategy based on motion primitives, dividing the LC-UGV’s motion into plane motion and boundary motion. The former involves the feet moving on the same plane and the latter includes movement on different planes. This strategy solves the challenges posed by the differences between the two movement modes, thereby allowing the planning of a complete path in 3D space.

Lastly, the motion generator produces the footholds and the continuous trajectory between neighbor path points.

In the following sections, we present detailed information on map representation, path planning, and motion generation methods.

3.2. Multi-Layer Elevation Map

A multi-layer elevation map (MLEM) is an extension of a conventional elevation map. A conventional elevation map stores terrain information on the ground. In contrast, an MLEM preserves the terrain information of all traversable planes for LC-UGVs.

The construction process of the multi-layer elevation map is illustrated in Figure 2. To obtain the multi-layer elevation map, we first segment the acquired point cloud data into planes. The RANSAC [18] algorithm is applied to detect planes. Considering that LC-UGVs can only traverse relatively large planes, recognition of small planes is disregarded in the plane segmentation algorithm.

After extracting all passable planes, the boundaries of these planes are detected. The boundary information is crucial for the multi-layer elevation map. The boundary between two planes is extracted by calculating the distance from points to the plane equation; points that are at a distance less than the threshold from both planes are considered boundaries.

However, this can lead to fake boundaries where two planes do not touch. Thus we calculate the minimum distance between the boundary point cloud and the two plane point clouds to eliminate such boundaries. The multi-plane segmentation and boundary detection algorithms are shown in Algorithm 1.

Algorithm 1: Plane segmentation and boundary detection.

Input:: point clouds $C l o u d_{a l l}$
Output:: plane equations coefficients S, plane point clouds $C l o u d_{p l a n e s}$ , boundary point clouds $C l o u d_{b o u n d a r i e s}$
1:: $C l o u d_{w o r k}$ = $C l o u d_{a l l}$ .copy()
2:: Initial S, $C l o u d_{p l a n e s}$ , $C l o u d_{b o u n d a r i e s}$
3:: while $C l o u d_{w o r k}$ .size() > $C l o u d_{a l l}$ .size()/5 do
4:: $C l o u d_{p}, [A, B, C, D] \leftarrow R A N S A C (C l o u d_{w o r k})$
5:: S.append( $[A, B, C, D]$ ), $C l o u d_{p l a n e s}$ .append( $C l o u d_{p}$ )
6:: $C l o u d_{w o r k}$ .delete( $C l o u d_{p}$ )
7:: end while
8:: $n u m \leftarrow l e n g t h (C l o u d_{p l a n e s})$
9:: for i in range( $n u m$ ) do
10:: for j in range(i, $n u m$ ) do
11:: for p in $C l o u d_{a l l}$ do
12:: $d \leftarrow D i s t a n c e (p, S [i], S [j])$
13:: if $d < t h r e s h o l d$ then
14:: $C l o u d$ .append(p)
15:: end if
16:: $v a l i d \leftarrow D i s t a n c e (C l o u d, C l o u d_{p l a n e s}, i, j)$
17:: if $v a l i d$ then
18:: $C l o u d_{b o u n d a r i e s}$ .append( $C l o u d$ )
19:: end if
20:: end for
21:: end for
22:: end for

The conversion from point cloud to elevation map is typically accomplished using the grid-map library [19]. It uses the ground as a reference, dividing the two-dimensional space (

X, Y

dimensional) into grids of fixed resolution. The corresponding 3D point cloud for each grid cell is used to calculate the max Z value. Then, the value is stored in the 2D grid. The map requires only a two-dimensional array, resulting in high efficiency.

The core point of the proposed MLEM is that we use many planes as references to calculate the elevation maps. The process is realized through the homogeneous transformation of point clouds. Specifically, a homogeneous transformation matrix is computed based on aligning the plane with the ground; then, the matrix is applied to the transformation.

In addition to the elevation maps of all traversable planes, the MLEM also stores the equations of all planes and their boundary point clouds.

3.3. Global Planner and Map Unfold

The construction of multi-layer elevation maps is intended to identify the passable regions, effectively representing a reconfiguration of 3D space. Since a single boundary only represents the connection between two planes, we consider the plane as a node and the boundary as an edge. Thus, the 3D space can be represented as a topological map.

Inspired by the method proposed in [5], we use a tree graph to construct the topological path graph. If a boundary exists between two planes, then these two planes are considered neighbors. In the tree graph, the starting plane is the root node, and the plane’s neighbors are its child nodes.

Only the target plane can be the leaf (node with no child). All nodes appear only once in the branch from the root to the leaf, except when the root node and the leaf are same plane. Hence, every branch of the tree from the root to the leaf represents a route for plane transitions. Unlike the work of [5], we do not assess the feasibility of plane-to-plane conversion in the global planning phase; this step takes place in the local path planning phase.

The resulting paths are prioritized according to the number of crossing planes; then, the planes are unfolded based on this priority. The term “unfold” may cause confusion. In fact, it introduces another crucial component in the global planning process: map unfolding.

Although elevation maps corresponding to different planes have been constructed, the problem for applying these multiple elevation maps to the path planning of LC-UGVs remains unresolved. Therefore, we introduce the concept of “map unfolding”, which involves projecting the elevation maps of multiple planes onto a single 2D plane. The unfolding of the planes along the boundaries is based on a naive idea. We conduct point cloud transformation to make the direction of the normal vectors of the two planes consistent, and the position of the boundaries remains unchanged after the transformation.

In the model illustrated in Figure 3,

n_{1}

and

n_{2}

denote the normal vectors of the two planes,

v = [v_{x}, v_{y}, v_{z}]

represents the axis of rotation, and

θ

signifies the angle required to rotate the planes until they are parallel. The axis of rotation

v

and the angle

θ

can be derived from the cross product and dot product of the two normal vectors, respectively. The rotation matrix R can then be constructed using Rodrigues’ rotation formula as shown in (1).

R = I + sin (θ) K + (1 - cos (θ)) K^{2}

(1)

I is the identity matrix, and K is the skew-symmetric matrix of the rotation axis.

K = (\begin{matrix} 0 & - v_{z} & v_{y} \\ v_{z} & 0 & - v_{x} \\ - v_{y} & v_{x} & 0 \end{matrix})

(2)

Let

p

be the point of the boundary line. Then, the translation transformation

t

can be expressed as (3).

t = p - R p

(3)

Through iteration, multiple planes can be unfolded onto the same one. Map unfolding utilizes elevation maps with boundary information to represent the 3D space. This method allows us to employ well-established 2D path planning algorithms for the path planning of LC-UGVs in 3D environments, resulting in higher efficiency compared to direct path searching in the 3D space.

A typical example is illustrated in Figure 4. In the example, the goal of the UGV is to move from the point start to the point target. First, within the limited moving space, the UGV performs a three-dimensional reconstruction of the surrounding environment and generates a multi-layer elevation map.

Then, the global planner searches possible plane transition paths based on the topological map. The multiple elevation layers are unfolded onto the same plane along their boundaries for further local path planning.

Notably, the unfolding results with the same planes in a different order are different. An example is shown in Figure 5; the unfolding elevation map restricts the LC-UGVs to move according to the plane transition sequence planned by the global planner.

3.4. Local Path Planner

The local path planner requires data inputs including the unfolded elevation map, the UGV model, the start point, and the target point. Its task is to search a collision-free and accessible path from start to destination on the elevation map.

Since legged climbing UGVs can establish stable contact with different planes via the adhesive force, their mobility exhibits significant differences across various regions on the map. Traditional path planning methods cannot be applied directly.

Inspired by the work of [20], we propose a path planning method based on motion primitives. The UGV’s movement is divided into plane motion and boundary motion. The two types of motion primitives correspond to different path planning methods and motion generation strategies.

The local path planning stage consists of two parts: map pre-processing and path planning. The former determines the region of plane motion and boundary motion, and calculates an valid foothold map. Then, path planning is carried out separately in the two regions.

3.4.1. Map Pre-Processing

In the pre-processing stage, we first establish the scope of the two types of motion primitives. The interval

[- r, + r]

on both sides of the boundary is considered the boundary motion area (r is the UGV radius), and the rest is the plane motion area. Then, the elevation map is used to assess the validity of the footholds.

Typically, researchers evaluate footfall scores based on terrain roughness and slope. However, the conditions for foot placement are relatively more stringent for climbing UGVs because uneven terrain carries a significant risk of detachment. Therefore, we have calculated a foothold map based on flatness, which assesses whether each grid cell can accommodate the foot end. The flatness over grid coordinate

(i, j)

can be calculated through (4).

m (i, j)

denotes the height value for grid coordinate

(i, j)

a (i, j) = \sum_{x = i - t}^{i + t} \sum_{y = j - t}^{j + t} {(m (x, y) - \frac{\sum_{x = i - t}^{i + t} \sum_{y = j - t}^{j + t} (m (x, y))}{{(2 t + 1)}^{2}})}^{2}

(4)

t controls the range of surrounding grids to be traversed; it can be calculated via (5) with the diameter of the foot end d and the resolution of the elevation map s.

t = ⌈\frac{d - s}{2 s}⌉

(5)

The foothold map can be determined as (6); it is used for state validation checks.

f o o t h o l d (i, j) = \{\begin{matrix} 0, i f a (i, j) < t h r e s h o l d \\ 1, e l s e \end{matrix}

(6)

3.4.2. Path Planning in Plane Motion Region

We utilize the Probabilistic Roadmap (PRM) algorithm for path planning. The method initially samples valid states in the configuration space and then use the A* algorithm to search for the shortest path in the sampled graph. It is a well-established path planning algorithm, which has also been used in [7,12].

In the plane motion region, the legged UGVs likely have a high degree of freedom; thus, direct sampling of joint angles would result in low computational efficiency. We simplify the sampling space of the UGV’s state into

[x, y, θ]

[x, y]

represents the coordinates on the map, and

θ

represents the yaw angle, which is the rotation about the z-axis. Although this simplification limits the height variation and body posture inclination, it makes our method applicable to various configurations of legged UGVs (bipedal, quadrupedal, and hexapodal). If other research is focused on a specific LC-UGVs, more states can be added to the sampling space to improve the UGV mobility.

There are two core steps in the sampling process of the PRM algorithm: the validity check of nodes and the neighboring nodes check. The local path planner provides two types of validity check. For obstacle-free planning, we require all grids within the active range to be valid. Therefore, a sub-map of the current position on the foothold map is extracted to verify that its values are all 0. The size of the sub-map is determined by the UGV’s size. For soft-collision planning, we only consider whether the surrounding grid of the foothold is valid under the current state. The state is safe when the number of valid grids around each foot is greater than a given threshold. The two forms of collision checking are provided as illustrated in Figure 6.

The check of neighboring nodes firstly calculates the distance between two nodes, then verifies the feasibility of the transition between the nodes. A hexapod UGV example is shown in Figure 7. The difference between neighboring states is limited to a small value, allowing the UGV to transition states via a single step motion. Therefore, the feasibility check of the transition only needs to assess whether the height of the obstacles crossed from start to end footholds is lower than the maximum leg swing height.

3.4.3. Path Planning in Boundary Motion Region

In the boundary motion region, we also apply the PRM algorithm for path planning.

However, the positioning of the feet across diverse planes introduces substantial complexity to the path planning. Concurrently, the absence of a standardized plan is underscored by the variance in leg number across different configurations.

To propose a more universal and efficient planning method, we utilize a simplified model, as illustrated in Figure 8a. Considering that LC-UGVs typically have an even number of legs and are axisymmetric, we model these UGVs as comprising a body and two sets of legs on each side. The same set of legs reside on the same plane, with equal distance to another plane, and the body posture is assumed to only rotate along the x-axis.

Thus, the configuration space is reduced to the

[x, y, z, α, d_{1}, d_{2}]

space.

d_{1}

and

d_{2}

represent the distance from the foot to the boundary, respectively.

[x, y, z, α]

is the position and roll angle of the body.

Due to the mutual constraints between the foothold and posture, it remains challenging to plan a continuous trajectory between two neighbor states in the current configuration space.

The advantage of this simplified model is that we can decompose the complex boundary motion into more basic motion patterns. Three basic motions are utilized: the adjustment of the foothold (

d_{1}

d_{2}

), the alteration of the body posture (

y, z, α

), and the progression along the boundary (x).

As shown in Figure 8b, the conduct of these three motions is independent. Therefore, when performing state sampling in the PRM algorithm, we sample intermediate states between two adjacent states based on the three basic motions to ensure that the transition between each state can be achieved using only one type of basic motion.

For the kinematic feasibility check in the sampling process, we calculate the foot-end workspace of the two sets of the feet based on the body position and posture

x, y, z, α

. The foot-end workspace is divided into multiple convex hulls as shown in Figure 9. The convex hulls are used for foothold kinematic feasibility checks, which avoids frequent inverse kinematics calculations, thereby enhancing efficiency. Then, a foothold collision check is applied on the foothold map based on the value of

[d_{1}, d_{2}]

3.4.4. Inter-Regional Key-State

Our method can search for collision-free feasible discrete paths in both plane regions and boundary regions. However, how to integrate the planning of the two regions to form a complete path from the starting point to the end point is unclear.

The key to the problem lies on the boundary line between the two regions. Note that the boundary line here does not refer to the planar boundary, but rather to the delineating boundary that defines the ranges of the two regions in the map processing.

When the planning algorithm of one region samples the boundary location

(x, y)

, it should consider whether to enter the other region for further planning. Due to the different sampling spaces used by the plane region and boundary region, it is essential to adjust the current state to meet the constraint of the other region’s sampling space. This state is termed the inter-regional key-state, and this process is known as the region transition process.

The transition from the plane region to the boundary region is simple, requiring only the sampling of appropriate

d_{1}

and

d_{2}

to adjust the foothold. Conversely, the transition from the boundary region to the plane region requires adjusting

α = 0

and ensuring the foothold lies on the same plane. Figure 10 depicts the two types of region transition.

In the A* algorithm, additional costs are incorporated for region transitions to prevent frequent crossings between different regions.

3.5. Motion Generator

Following local path planning, the paths within both plane motion and boundary motion regions are composed of a series of discrete states, thereby necessitating single-step motion generation between states to foster a continuous motion trajectory. Given that motion generation implicates the configuration of the UGV, we take the case of a hexapod climbing UGV, which presents a particularly complex configuration.

3.5.1. Plane Motion Single-Step Generation

The motion generation method for conducting state transformations between neighbor path points depends on the type of collision check. For planning that requires obstacle-free, periodic gaits such as straight gaits and rotation gaits, these gaits can be employed directly because there are no obstacles within the range of motion.

However, for paths considering soft collisions, it may be difficult to achieve a single transition with conventional gaits due to the potential differences in position and orientation between two path points.

We propose a simple and effective method for state transition. The motion between states can be viewed as a translation or a rotation around a point on the plane, as shown in Figure 11. Then, a free gait can be applied to achieve the state transition using a single swing motion.

The center of rotation is located on the perpendicular bisector of the positions in neighbor states, and the angle between the line connecting the rotation point and the position of the neighbor states and the orientation of that state remains constant (

θ_{1} = θ_{2}

). The rotation center position

[x^{'}, y^{'}]

can be obtained by solving the binary linear equations as shown in (7).

\begin{matrix} \{\begin{matrix} (Δ x) x^{'} + (Δ y) y^{'} = (x_{1} + \frac{Δ x}{2}) Δ x + (y_{1} + \frac{Δ y}{2}) Δ y \\ a x^{'} + b y^{'} = d \end{matrix} \end{matrix}

(7)

d, a,

and b are intermediate variables, which can be calculated through (8).

\begin{matrix} d = a x_{1} + Δ x cos (θ_{1} + Δ θ) + b y_{1} + Δ y sin (θ_{1} + Δ θ) \\ a = (cos (θ_{1} + Δ θ) - cos θ_{1}) \\ b = (sin (θ_{1} + Δ θ) - sin θ_{1}) \end{matrix}

(8)

The rotation angle is

Δ θ

. Based on this method, the trajectory of the body can be calculated, which determines the stance leg motion.

3.5.2. Boundary Motion Single-Step Generation

Owing to the proposed simplified model, we decouple the boundary motion into three fundamental modules: the adjustment of foothold, displacements along boundaries and body posture transformation. For each basic module, the variation in the joints follows fixed patterns, which reduces planning complexity. This simplification makes our method suitable for various configurations of LC-UGVs.

The movement of the basic motion modules can be achieved through the following steps:

Step 1. Establish the body coordinate system. Construct the kinematics and inverse kinematics of the foot end to the center of mass within this body coordinate system using the D-H method.

Step 2. Transform the swinging trajectory of the foot end from the world coordinate system to the body coordinate system. For body posture adjustment where the foothold position remains unchanged, it also can be viewed as the change of the foot end relative to the body’s center of mass within the body coordinate system.

Step 3. Solve for the joint angle trajectories through inverse kinematics.

For body posture transformation, which excludes leg swing, we generate continuous motion trajectories simply by interpolating the center of mass states. However, for the adjustment of foothold, and displacements along boundaries, we employ sixth-order polynomial curves to generate continuous motion trajectories for the foot ends. The implementation follows the methodology outlined in our previous work [3].

The inverse kinematics for legged UGVs has been well established in previous research such as [1,3]. Therefore, the primary problem is the conversion of movement from the world coordinate system to the body coordinate system.

Considering the coordinate system shown in Figure 12, let

α

represent the body’s roll angle, and

c = [c_{x}, c_{y}, c_{z}]

denote the displacement of the body relative to the origin of the world coordinate system. Then, the homogeneous transformation matrix T is defined as (9):

T = (\begin{matrix} 1 & 0 & 0 & c_{x} \\ 0 & cos α & - sin α & c_{y} \\ 0 & sin α & cos α & c_{z} \\ 0 & 0 & 0 & 1 \end{matrix})

(9)

The transformation of the foot end coordinates from the world coordinate system

p_{o}

to the body coordinate system

p_{b}

is represented as (10):

p_{b} = T^{- 1} p_{o}

(10)

As shown in Figure 12, when the body posture changes to

[Δ α, Δ c_{y}, Δ c_{z}]

, the foothold remains unchanged in the world coordinate system. The foot end position change

Δ p

in the body coordinate system is defined as (11):

Δ p = T^{' - 1} p_{o} - T^{- 1} p_{o}

(11)

T^{'}

is the homogeneous transformation matrix between the world coordinate system and the body coordinate system after the body posture change, which is defined as (12).

T = (\begin{matrix} 1 & 0 & 0 & c_{x} \\ 0 & cos α + Δ α & - sin α + Δ α & c_{y} + Δ c_{y} \\ 0 & sin α + Δ α & cos α + Δ α & c_{z} + Δ c_{z} \\ 0 & 0 & 0 & 1 \end{matrix})

(12)

4. Experiment Results

4.1. Hexapod Climbing UGV Platform

The proposed framework was verified with a hexapod climbing UGV in both simulation and physical experiments. Detailed information about this hexapod climbing UGV can be referred to in our previous work [1]. Its foot end is equipped with vacuum suckers; each leg has four degrees of freedom, allowing for control of the foot end’s position and the rotation of the suckers. Moreover, each foot end is equipped with a passive ball joint, enabling the foot-end suction cups to adapt to planar surfaces within a certain angular range. This four-degree-of-freedom structural design and passive joint endow the hexapod climbing UGV with the hardware capability to adhere and climb in complex environments.

Since the UGV is capable of overcoming its own weight with a single suction cup, we did not consider potential sliding or tipping issues during wall-to-wall transition. For universal LC-UGVs, it is necessary to further consider their static constraints within the navigation framework proposed in this paper. For some extreme scenarios, it is even necessary to consider how the robot can regain its original mobility when some of its legs are damaged [21,22,23], thus maintaining the resilience and robustness of the legged UGVs during the navigation process.

4.2. Map Representation Validation

Before conducting robot navigation experiments, we independently validated the proposed MLEM map representation. As mentioned earlier, the reason for proposing the MLEM is that traditional 2.5D elevation maps cannot provide the three-dimensional spatial information required for LC-UGV navigation. However, a complete 3D spatial representation is redundant for LC-UGVs, as they cannot fly in 3D space but rely on 3D surfaces instead. Therefore, we experimentally compared MLEM with commonly used complete 3D map representations [24,25], including the 3D occupancy map and Octomap. The 3D occupancy map represents environments by assigning occupancy probabilities to individual voxels, thus enabling detailed object shape and position representation. Octomap utilizes an octree structure for 3D voxel occupancy probability storage, and offers reduced memory consumption by pruning child nodes when they are consistently “occupied” or “unoccupied”.

We used point clouds as input to run three different mapping methods, calculating the time required to convert the point cloud to the corresponding map representation and the memory usage of the map. The input consisted of two point clouds of different sizes (3 MB and 60 MB, respectively). We tested the computation time and memory footprint of different map representations at resolutions of 0.025 m and 0.05 m. The results are shown in Table 1.

Experimental results highlight a substantial reduction in memory footprint with the MLEM compared to both 3D occupancy maps and Octomaps. The 3D occupancy map offers the fastest computation speed but consumes more memory compared to the other two methods, with its memory usage increasing significantly as the resolution rises. In contrast, the proposed MLEM map representation boasts the lowest memory footprint, even at high resolutions (0.025 m), requiring less memory than other methods at lower resolutions. However, its computation time is longer due to the need for plane extraction and boundary detection from point cloud input.

Therefore, a trade-off between processing speed and memory usage is necessary. LC-UGVs, typically requiring high-resolution maps for terrain assessment and moving at slower speeds. As the memory footprint of complete 3D maps dramatically increases with the map resolution, the MLEM offers a solution for significantly reducing the memory footprint of map representation. Therefore, the MLEM provides a greater advantage over other full 3D representations in LC-UGV navigation. Additionally, the MLEM incorporates boundary detection, which is crucial for LC-UGVs’ cross-plane motion.

4.3. Simulation

Owing to the hierarchical planning structure, the UGV exhibits movement capabilities across various challenge terrains. Simulations have been conducted on regular terrains with various obstacles and extremely challenging terrains featuring narrow gaps and insurmountable obstacles.

Figure 13 illustrates the routes and movements planned using two different collision check methods in the plane motion region. Based on soft-collision path planning, the UGV is able to traverse obstacles. The simulation results validate the robot’s ability to move and avoid obstacles by following the planned path in the plane.

Figure 14 presents a scenario where the LC-UGVs traverse extreme terrains that are difficult for regular UGVs to navigate. The specific layout of the simulation scenario is shown in Figure 1, where the robot is surrounded by complex obstacles. Based on the proposed hierarchical planning architecture, the UGV can adjust its posture to traverse narrow gaps along plane boundaries, execute plane-to-plane transitions, and move across multiple planes.

To further validate the effectiveness of our approach, we measured the center of mass (CoM) position change during the plane transition process and compared it with the planned trajectory. Figure 15a shows the planned and measured trajectories in 3D space, while Figure 15b illustrates the variation of the absolute motion error over time. Based on our observations, the error is primarily attributed to the robot’s swaying caused by leg swing and contact during the plane transition motion. The steady-state error after completing the motion is only 0.01 m, thus confirming the robot’s plane transition motion capability.

4.4. Physical Experiment

Physical experiments were performed with the hexapod climbing UGV platform as shown in Figure 16. Due to the limitations of the experimental site, we primarily verified the UGV’s capability for boundary motion.

The physical experiments showed that the proposed planning framework and motion generation method equipped the UGV with the ability to switch from plane to plane and move along boundaries. This extends beyond other work related to plane translation [6,7], as our method does not depend on specific motion trajectories and UGV configurations. The proposed general simplified model and the hierarchical planning method allow the UGV to explore more feasible ways of movement.

In contrast to existing wall transition work, the proposed hierarchical planning architecture and simplified model are not limited to motion trajectories designed for specific angles, such as the 90° transition from ground to wall. As illustrated in Figure 17, the robot successfully performs plane transitions under different angles, demonstrating the generalization capability of the simplified model and planning method presented in this paper.

Our physical experiments were primarily conducted in indoor environments. The current robotic platform and navigation methods face limitations in more complex outdoor or industrial settings. The robot relies on vacuum suction cups for planar adhesion, and the surface smoothness and roughness significantly affect the contact force. Moreover, our approach struggles to navigate non-planar terrains, such as cylinders. Future research will focus on developing more versatile navigation methods and robotic platforms to address these limitations.

5. Conclusions

In this paper, we propose a comprehensive framework enabling legged climbing UGVs to navigate challenging environments. We introduced the multi-layer elevation map (MLEM), which retains the terrain and boundary information of different planes. Based on the MLEM, a hierarchical planning framework was proposed. This framework first employs a global planner to discover the global path and unfold the elevation map. Then, in the local path planning stage, it adopts a strategy based on motion primitives, dividing the movement area into plane motion and boundary motion regions. Corresponding path planning and motion generation strategies are utilized in different areas. This method enables the legged UGV to step over obstacles, transition from one plane to another, and even move along plane boundaries.

Finally, the proposed methodology was tested and validated through simulations and physical experiments. Our study constitutes a significant endeavor toward enabling LC-UGVs to autonomously navigate and move in complex 3D environments. In our future work, we plan to develop more versatile navigation methods and robotic platforms that will work in more complex environments.

Supplementary Materials

The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/drones8120768/s1, Video S1: experiment.

Author Contributions

Conceptualization, A.X. and L.F.; methodology, A.X.; software, A.X. and C.G.; validation, A.X. and C.G.; formal analysis, A.X.; investigation, A.X. and C.G.; resources, A.X.; data curation, A.X.; writing—original draft preparation, A.X.; writing—review and editing, A.X. and L.F.; visualization, A.X. and C.G.; supervision, L.F.; project administration, L.F.; funding acquisition, L.F. All authors have read and agreed to the published version of the manuscript.

Funding

This paper was supported by the Intelligent Aerospace System Leading Innovation Team Program of Zhejiang (Grant No. 2022R01003).

Data Availability Statement

The original contributions presented in this study are included in the article/Supplementary Material. Further inquiries can be directed to the corresponding author(s).

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:

LC-UGVs	Legged Climbing Unmanned Ground Vehicles
MELM	Multi-Level Elevation Map
SLAM	Simultaneous Localization and Mapping
ESDF	Euclidean Signed Distance Filed

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Figure 1. The overall framework of the proposed method. The input information consists of RGB images and depth information captured by a depth camera. The RGB images are used to estimate the relative pose, while the depth images are converted into point clouds. The point clouds are first used to extract plane and boundary information, which are then used to construct the proposed MLEM. The numbers 1, 2 indicate the code of the different planes. Based on the MLEM, a universal hierarchical planning framework is used for path planning and motion generation in challenging 3D environments. The outputs of the planning framework are all joint angles trajectories. We use a position controller to control the joints.

Figure 2. The multi-layer elevation map construction pipeline. Based on the plane and boundary information obtained, the point clouds of each plane are used to construct the MLEM. Each plane and its corresponding elevation map are labeled, and the boundary information stores the connection information for different planes.

Figure 3. The transformation process of two adjacent planes.

Figure 4. Cross-plane path planning and map unfolding process. After the MLEM is constructed and each plane is numbered, a search is conducted in the constructed 3D space map for possible paths from the start plane to the target plane. Based on priority, the planes involved in different paths are unfolded and a path search is conducted until a viable path is found.

Figure 5. The different unfolding results with the same planes in a different order.

Figure 6. The two forms of collision checking. For soft-collision checks, the surrounding grids validation of the foot are checked. A foot is considered supportive when the amount of valid grids around the foothold exceeds a specified threshold. A state is deemed valid only when all six feet are supportive.

Figure 7. The sampling space

[x, y, θ]

represents the coordinates on the map and rotation about the z-axis.

Figure 7. The sampling space

[x, y, θ]

represents the coordinates on the map and rotation about the z-axis.

Figure 8. Boundary motion state. (a). The simplified model and a hexapod prototype. (b) The three independent basic motion patterns. The transition between two adjacent states is realized through a single basic motion.

Figure 9. The convex hull of the foot−valid workspace.

Figure 10. The region transition process between the plane region and boundary region.

Figure 11. The transition between two states is considered as either a translation or a rotation around a certain point.

Figure 12. (a). The world coordinate system and the body coordinate system. (b). The change of the foot end within the body coordinate system when body posture adjusts.

Figure 13. (a). The obstacle−free planned path and its simulation scene. (b). The soft−collision check planned path and its simulation scene. Numbers 1–4 represent the sequence of movements.

Figure 14. In the simulation environment, the UGV first walks along the wall boundary and passes through a narrow gap. Numbers 1−16 represent the sequence of movements. It then executes a plane-to-plane transition to move from the ground to the wall. After performing obstacle avoidance maneuvers on the wall, it returns to the ground. This process fully demonstrates the locomotion capabilities of a legged climbing UGV in a complex 3D environment.

Figure 15. (a). The red line represents the planned trajectory, while the blue line connecting the scatter points represents the measured trajectory. (b). The absolute motion error over time.

Figure 16. (1–4) shows that the physical UGV moves along plane boundaries to traverse narrow gaps. (a–d) shows that the UGV conducts the plane-to-plane transition. More experiment and simulation details can be found in the video provided.

Figure 17. The plane transition process under different angles.

Table 1. The validation of the three map representations.

	Octomap		3D Occupancy Map		MLEM
Input Size/Resolution	Time Cost (s)	Memory Footprint (MB)	Time Cost (s)	Memory Footprint (MB)	Time Cost (s)	Memory Footprint (MB)
3 MB/0.025 m	0.025	1.29	0.015	13	0.041	0.232
3 MB/0.05 m	0.012	0.368	0.004	1.64	0.019	0.058
60 MB/0.025 m	0.763	13.48	0.493	394	1.72	2.89
60 MB/0.05 m	0.32	3.15	0.146	49	1.35	0.732

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Xiang, A.; Gong, C.; Fan, L. Map Representation and Navigation Planning for Legged Climbing UGVs in 3D Environments. Drones 2024, 8, 768. https://doi.org/10.3390/drones8120768

AMA Style

Xiang A, Gong C, Fan L. Map Representation and Navigation Planning for Legged Climbing UGVs in 3D Environments. Drones. 2024; 8(12):768. https://doi.org/10.3390/drones8120768

Chicago/Turabian Style

Xiang, Ao, Chenzhang Gong, and Li Fan. 2024. "Map Representation and Navigation Planning for Legged Climbing UGVs in 3D Environments" Drones 8, no. 12: 768. https://doi.org/10.3390/drones8120768

APA Style

Xiang, A., Gong, C., & Fan, L. (2024). Map Representation and Navigation Planning for Legged Climbing UGVs in 3D Environments. Drones, 8(12), 768. https://doi.org/10.3390/drones8120768

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