Hierarchical collision-free motion planning method for biped truss-climbing robots

Abstract

Biped truss climbing robots (BTCRs) are employed to perform high-rise tasks within truss environments, benefiting from their superior transition capabilities and flexible mobility. However, the intricate geometry of these structures poses challenges for robot navigation and operation. To tackle this issue, this paper proposes a novel BTCRs climbing path planning framework based on a progressive multi-layer architecture. The robot’s transition regions between adjacent members are determined efficiently by unfolding three-dimensional truss members onto two-dimensional planes and discretizing them. Initially, leveraging transition analysis, a global truss member route using graph search methods is generated. Subsequently, a mathematical optimization model is introduced to determine transition grips along the global route, minimizing the total number of grips. Finally, the single-step path planner employs an improved rapidly-exploring random tree (RRT)-connect algorithm, guaranteeing collision-free motion between adjacent grips. By integrating these three layers, the framework demonstrates the feasibility and effectiveness of the proposed analysis and algorithms for climbing path planning in simulation tests, using a self-developed BTCR, Climbot.

Keywords

Biped truss-climbing robots transition performance analysis path planning collision avoidance

Introduction

Numerous truss environments are present in human production and living environments, involving a large amount of high-altitude operations. These operations include the assembly, disassembly, and maintenance of truss structures such as electric towers, fruit picking in agriculture and forestry, inspection of truss structures in the construction industry, and the erection and dismantling of scaffolding. These operations are characterized by their high intensity, high difficulty, and high risk, and are currently mainly dependent on manual labor. The utilization of robots to replace or assist humans in carrying out truss-related high-altitude operations has practical implications and application prospects (Fang and Cheng, 2023). Biped truss climbing robots (BTCRs) exhibit various configurations, including legged (Kim et al., 2018), wheeled (Liu et al., 2020), crawler (Su et al., 2023), and hybrid (Longo and Muscato, 2006) types, as well as encircling, hook, and grasping variants (Chu et al., 2010). Inspired by inchworms and apes, biped climbing robots feature a multi-joint body and operable grippers at both ends, enabling them to achieve alternate climbing (Muscolo et al., 2014). BTCRs adeptly surmount obstacles and seamlessly transition between truss members within the intricate 3D environment.

BiCRs have made significant progress in structural analysis, system development, and research. However, current research challenges primarily focus on autonomous climbing, including environmental perception and path planning. Environmental perception relies on onboard multi-sensor systems to achieve simultaneous localization and mapping (SLAM) within truss structures (Chen et al., 2023; Zhu et al., 2024). Path planning is a fundamental and critical technology for both autonomous perception in truss environments and robot control. Therefore, the study centers on climbing path planning. Ground mobile robots employ path planning methods that fall into three distinct categories: complete planning algorithms (Heo et al., 2022; Zhaoying et al., 2022), probabilistic complete algorithms (Lau et al., 2022; Novosad et al., 2023), and intelligent algorithms (Li et al., 2024; Xu et al., 2022). However, the climbing trajectory of BTCRs is characterized by a discrete sequence of gripping points within the truss environment. The path planning process for BTCRs necessitates the incorporation of position and direction information, along with ensuring collision-free motion between consecutive grips. Consequently, directly applying the path planning techniques used for ground mobile robots to the BTCR context presents challenges.

Researchers have conducted valuable investigations concerning the path planning challenges for BTCRs within a three-dimensional (3D) truss environment. SM² has demonstrated the capability to execute operational and maintenance tasks on the space station using a proposed gait planning algorithm that leverages fixed attachment points within the environment (Xu et al., 1994). ROMA, an energy-efficient solution designed to minimize power consumption, relies solely on the endpoints and intermediate nodes of each member, restricting its effectiveness for task-oriented traversal (Balaguer et al., 2002, 2000). Frambot computes the energy consumption associated with its fundamental gait and subsequently determines an energy-optimal climbing path using a genetic algorithm (Chung and Xu, 2011). Shady3D employs the Dijkstra shortest path algorithm to determine an optimal sequence of clamping points, starting from a given point and reaching the target (Yoon and Rus, 2007). To derive climbing paths on rugged surfaces, such as tree branches, Treebot initially discretized the trunks and subsequently determined an optimal path to the target using a dynamic programming algorithm (Lam and Xu, 2013). This approach considers constraints related to action cost, gravity direction, robot state, and accessibility. Addressing the path planning challenges for cellular robots within truss-based space stations, Dai et al. (2020) introduced an optimized ant colony algorithm that integrates gravitational search principles. The modular biped climbing robot named Climbot was designed, consisting of five joint modules and two gripper modules (Gu et al., 2022). Climbot functioned as a research platform for comprehensive investigations into non-collision climbing path planning for BTCRs within the 3D truss environment (Gu et al., 2018; Zhu et al., 2018). However, these approaches are specifically suited for BTCRs possessing a unique planar configuration and are less suitable for more flexible BTCRs, such as Climbot-6D (Figure 1).

Figure 1.

BTCR climbing in a truss environment: (a) an illustration of robot climbing scene and (b) the kinematic diagram of Climbot, one self-developed BTCR with six-DOF.

Previous research has primarily concentrated on identifying gripping points on individual truss members or devising single-step motions, with insufficient emphasis on obstacle avoidance. Consequently, the establishment of a comprehensive climbing path planning framework for BTCRs remains an open challenge. In this study, a hierarchical framework for planning the 3D climbing path of BTCRs is presented. Specifically, a quadtree grip map for the transitional region between adjacent truss members is constructed by simplifying the robot’s workspace and conducting geometric intersection tests with the truss members. Based on transition analysis, the global member sequence is determined using a graph search method. To plan gripper placements along the global truss member route, the transitional grips for each route member are mathematically optimized, thereby minimizing the number of required grippers. In addition, a single-step planner based on the RRT-connect algorithm is introduced to ensure collision-free movements between adjacent gripper pairs. The feasibility and effectiveness of the proposed analysis and algorithms for climbing path planning are validated through simulations conducted with Climbot, a self-developed biped truss-climbing robot. BTCRs and biped wall-climbing robots share many commonalities. The theories presented in this paper can also be extended and applied to the path planning problems of biped wall-climbing robots in spatial wall environments.

The rest of this paper is organized as follows. Section Method Overview introduces the climbing path planning problem and proposes the hierarchical framework. Section Transition Performance Analysis presents a novel method to analyze the transition performance of BTCRs. Based on the transition analysis, Section Climbing Path Planning details each component of the climbing path planning framework. Section Simulations conduct simulations to verify the proposed analysis and algorithms. Section Conclusion presents concluding statements and future research directions.

Method overview

The problem of 3D climbing path planning

Truss environments are common in large buildings or bridges, which usually consist of straight cylindrical members to form large 3D multi-member structures. The following conventions are used in this paper to aid with comprehension and expression:

The truss environmental map is known, which can be retrieved from BIM (building information modeling) or collected by a reasonable sensing system;

The truss is exclusively made up of cylindrical elements, regardless of how they are connected;

Other than truss members, no other forms of obstacles.

Without loss of generality, this paper represents a truss member using the coordinates of the two ends of the member $p_{0}, p_{1}$ in the world coordinate frame ${W}$ and the radius of the member r,

M = (p_{0}, p_{1}, r) .

(1)

The spatial truss environment $T_{t}$ for BTCRs to perform climbing operations can be described as follows:

T_{t} = [M_{1}, M_{2}, \dots, M_{n_{t}}]^{T},

(2)

where M represents one of the truss members, and $n_{t}$ denotes the total number of members in the structure.

BTCR climbing in a truss environment while avoiding obstacles can be seen in Figure 1(a). Given the environmental map of the truss member, the objective is to determine a feasible climbing path for the robot. This involves setting the starting point, destination, and relevant robot configuration. Along this path, sequentially positioned grips are distributed across various truss elements. The robot’s body connects adjacent grips, and the end gripper follows a single-step motion trajectory to link alternate grips. Importantly, the path must remain collision-free within its surroundings. Due to the robot’s slow speed, dynamic constraints are not considered in this study.

Proposed planning method

This research proposes a hierarchical framework with three components to efficiently handle the climbing path planning problem of BTCR in 3D truss environment.

Judgment of transition between members and solving transitional region

Transitioning between truss members is a fundamental and essential capability for a BTCR; without it, it can only be moved on a single member. As a result, to assess the BTCRs’ transitional performance, the rapid transition ability judgment method must be discussed and the transitional region between any two members must be solved.

Searching and optimizing the climbing path

Path planning is more difficult since the BTCRs, particularly the ones with 6 degree of freedom (DoF), can travel around the truss member to change the direction in which they grasp it. This part, based on the truss map and the transition analysis, focuses on solving the grip sequence that connects the starting point and destination by utilizing the truss member route search and optimizing the grips that are distributed on these members.

Single-step collision-free path planning

The single-step path between the alternate grips must be solved after the discrete grip sequence has been generated. The configuration is always changing as the robot moves in single steps. Consequently, it is essential to identify potential collisions between the robot and the truss member as well as with itself when planning to create a safe and dependable path.

Transition performance analysis

Transition between truss members

The transition analysis model between two truss members is depicted in Figure 2(a). While ${B}$ and ${E}$ represent the robot coordinate systems of the base and end gripper, respectively, ${W}$ denotes the world coordinate system. The reference coordinate systems of the two members are ${U}$ and ${V}$ . When the two members have transitional regions, one robot gripper can grasp a grip inside one member’s transitional region while the other can reach another member in a specific pose. As such, the robot’s reachable space and pose limits should be taken into account while evaluating transition ability. Currently, there exists an inverse kinematics solution that permits the robot’s base and end grippers to simultaneously grasp both transition members.

Figure 2.

Transition performance analysis method: (a) transition analysis model; (b) discretization and expansion of truss member; and (c) diagram of the proposed transitional region solving method.

The workspace of the robot is drawn using the Monte Carlo approach with different grippers serving as the base to assess the reachability of the robot. Because of the discrete spherical contours of the robot workspace on different bases, a sphere is employed to envelop the robot’s workspace,

x^{2} + y^{2} + {(z - {{}^{r}l}_{0} - {{}^{r}l}_{1})}^{2} = {(\sum_{i = 2}^{n_{dof}} {{}^{r}l}_{i})}^{2}

(3)

where $n_{dof}$ represents the robot’s number of degrees of freedom and ${{}^{r}l}_{i}$ denotes the robot link’s length.

As a result, by determining if the member is within the robot’s spherical workspace, the BTCR transition between truss members can be roughly assessed. In particular, the length between the target member’s axis and the center of rotation of the first T-shaped module close to the base gripper is computed. If the minimum distance between the target truss member and the sphere’s center exceeds the sphere’s radius, the robot cannot reach that member. However, if it is less than or equal to the sphere’s radius, additional testing is indispensable to determine the robot’s transitional feasibility. This is described in detail below.

Truss member pretreatment

As seen in Figure 2, the grasping orientation of a grip ${}_{W}^{E}R = [n, o, a]$ , where $n$ lines up with the member axis when robot grasps a cylindrical truss member, $a$ denotes the approaching direction of the robot, which can be established by the angle α around the member axis, then $o = a \times n$ . Furthermore, since the gripping point is on the member’s axis, the grasping location may be obtained by calculating the length l from the member’s reference point (Gu et al., 2022). Consequently, two values of rotation angle and length $(α, l)$ can be used to represent the six-dimensional grasping pose of the BTCR.

Preprocessing the truss members using the discrete operation can help with transition region solving and path planning, as shown in Figure 2(b). The truss member is divided into multiple rings with a certain length along the axis direction, and each ring represents the grasping point location. Then the ring is discretized into a certain number of points to represent the angle on the circumference of the member. Therefore, the grasping pose of the robot can be expressed by the discrete points.

Without losing generality, by expanding along the axis in the opposite direction of gravity, the two-dimensional plane of the discrete cylindrical surface of the truss member can be obtained. For the unfolded plane, the abscissa represents the rotation angle of the gripper around the axis of the member, which is within $[0, 2 π]$ . The ordinate represents the length of the member, ranging from 0 to l, where l is the total length of the member.

Solving transitional region

The 3D truss member can be transformed into a 2D plane through discretization and unfolding. Using a specific pose on one member as the base, the robot’s reachable pose set on the other member represents its transitional region $R_{t}$ . These poses must meet the reachable conditions for both position and the orientation.

The diagram for solving the transitional region is displayed in Figure 2(c). First, by expanding and discretizing each truss member, the 3D truss environment is described with 2D data. The transitional judgment is then carried out. The points outside the simplified robot workspace are regarded as non-transition regions. A robot kinematics test is conducted for other candidate points. Points without kinematic inverse solutions are marked as non-transitional. At last, the transitional region is obtained, allowing the robot to transition to another member from a specific grasping point.

The transitional region is described using a quadtree grid map in this paper. There are three states for the quadtree nodes: transitional, non-transitional, and mixed. A node is considered to be in a mixed state if any of its four partitions include both transitional and non-transitional child nodes. All of the quadtree’s child nodes are generated once the minimum splitting scale (depth) has been determined. After that, divide them to the predetermined depth and solve the node’s transitional region to determine each node’s state. Transitional child nodes continue in their state while non-transitional child nodes are merged. By merging up from deep to shallow in this way, the corresponding quadtree transitional region can be obtained.

Climbing path planning

Truss member route planner

Given the truss map, starting point and destination, the robot lacks sufficient information to determine the next action in the complex truss environment, such as whether to move along the current member or perform a transition between members, which member to transition to, and where to grasp. The planning algorithm’s complexity will skyrocket if it is searched blindly. Therefore, a truss member route planner is proposed, which reduces the complexity of the climbing path planning algorithm and provides heuristic information for grip planning.

Member route planning can be divided into three steps: transitional judgment, adjacency matrix construction, and member sequence search. The transitional judgment model is shown in Figure 3. The corresponding member is first extracted using the specified starting and target points, which are used to define the starting and target members. Obtaining the assumed grasping pose of the robot on the two transition members beforehand is required before making a transition judgment. Within the truss environment, the shortest distance between any two members is solved, yielding the hypothetical grips that correspond to the nearest point pair ${P_{1}, P_{2}}$ . To find the matching rotation angle α, the line joining the two points is then projected onto the yOz plane of the member coordinate system. Two assumed grips are used as the robot base to judge the transition. The adjacency matrix is constructed to express the transition between the two members. Finally, the truss member route from the start member to the target member is generated by the depth-first search (DFS) method in graph theory.

Figure 3.

Transition judgment model.

Grip sequence planner

The truss member route planning algorithm determines the sequence of members to be traversed by the robot, but not the position and orientation of the grasping points on each member. Hence, it is imperative to plan the grip sequence that enables the robot to reach the target point incrementally. This paper adopts the strategy that first solves the transition grips in the transition regions between the route members using an optimization model and then searches for the grips on the grasping regions of each route member sequentially, as shown in Figure 4(a). The black rectangular area represents the expanded member, while the green region corresponds to the transition region determined through transition analysis. Within this region, blue dots represent transition grips obtained via mathematical optimization. These points can be used to establish both the initial and terminal grips on each route member. Starting from the initial grip, a candidate grip set (depicted by the gray area) can be computed. Then appropriate grasp points (indicated by yellow dots) are selected as subsequent bases. This iterative process continues until all grips on the given route member are identified, initiating the search for grips on the next route member.

Figure 4.

The method of grip sequence planning: (a) grip sequence planning model and (b) diagram of the search for grips.

Optimization model of transition grips

Moving distance and climbing gaits

The total number of grips of a BTCR is directly related to the robot’s gait. A gait with a larger movement distance requires fewer steps to climb the same displacement, while a gait with a smaller movement distance can fine-tune the position of the grips. BTCRs, such as Climbot, have three basic gaits, namely the inchworm-like gait, the swinging-around gait, and the flipping-over gait (Guan et al., 2016). The minimum step length of the inchworm-like gait is $d_{0} = 0$ , while the maximum length is $d_{1} = d_{3} - d_{2}$ . The swinging-around gait and the flipping-over gait have the same minimum and maximum step length, $d_{2} = 2 l_{2} \cos \frac{θ_{\max}}{2}$ and $d_{3} = 2 l_{2}$ , respectively.

A grip of BTCRs includes not only the moving distance l, but also the rotation angle α, which can be represented by the state vector $s = (α, l)$ . Let $s_{k}^{s}$ and $s_{k}^{e}$ represent the starting and ending grips of the robot on the k th member, then the moving distance and the rotation angle on this member are,

{\begin{matrix} d_{k} & = l_{k}^{e} - l_{k}^{s} \\ α_{k} & = α_{k}^{e} - α_{k}^{s} \end{matrix}

(4)

where $l_{k}^{e}$ and $l_{k}^{s}$ are the position components of the starting and ending grips, while $α_{k}^{e}$ and $α_{k}^{s}$ are the angle components.

Then, a piecewise function can be used to express the mathematical relationship between the minimum number of climbing steps $N_{k}$ and the corresponding moving state $s_{k}$ on a member,

N_{k} = {\begin{matrix} 0 & d_{k} = 0, Δ α_{k} = 0 \\ 2 & d_{k} = 0, Δ α_{k} \neq 0 \\ 2 & d_{k} \in (0, d_{1}] \\ 2 & d_{k} \in (d_{1}, d_{2}], Δ α_{k}^{e} \notin R_{k} \\ 1 & d_{k} \in (d_{1}, d_{2}], Δ α_{k}^{e} \in R_{k} \\ 2 & d_{k} \in (d_{2}, d_{3}), Δ α_{k}^{e} \notin R_{k} \\ ⌈ d_{k} / d_{3} ⌉ & others \end{matrix}

(5)

When both the moving distance and the rotation angle are 0, the step number on the member is 0. The robot can directly transition to the next route member without any adjustment. If the moving distance is 0, but the rotation angle is not, then the robot needs to climb two steps with inchworm-like gait on this route member. The moving distance is within $(0, d_{1}]$ , the robot needs to execute the inchworm-like gait with two steps. When the moving distance is within $(d_{1}, d_{2}]$ , and the rotation angle is not in the grasping region $R_{k}$ (the set of all grips of one gripper on one member when another one grasping on the same member), two climbing steps are also needed. However, if the rotation angle is in the grasping region with the same moving distance, the robot only needs to climb one step. If the robot’s movement distance is within $(d_{2}, d_{3})$ , and the rotation angle is not in the grasping region, two climbing steps are needed. For other situations, the robot needs to execute the swinging-around gait or flipping-over gait, and the minimum number of climbing steps is $⌈ d_{k} / d_{3} ⌉$ , where the symbol $⌈ ⌉$ denotes the ceiling operation.

Collision detection strategy

BTCRs may collide with the environment during each climbing step. It is necessary to extract the potential collision members according to the robot configuration and conduct collision detection with the robot, so as to select a safe and non-collision configuration that meets the requirements of collision-free climbing.

If two adjacent grips are known, the climbing configuration of the robot can be obtained by solving the inverse kinematics. Therefore, based on the spatial position of the robot links, a minimum sphere that envelops the entire robot can be constructed (Gärtner, 1999). Hence, a three-layer collision detector is designed to further filter the obstacle members and judge the collision state between the robot and the members.

First layer: when the robot transits between members, the transitional analysis can extract the members inside the robot’s workspace that may collide with the robot, but due to the large judgment range, it may include a large number of safe members;

Second layer: the adjacency matrix constructed by transitional analysis and the current configuration of the robot are used as input, and the potential collision members obtained by the first layer of collision detection are screened by constructing the minimum enveloping sphere;

Third layer: calculating the minimum distance between the potential collision members and the robot links, which can precisely determine whether the robot collides with the member.

The optimization model

Assuming that the truss member route includes n members, the robot on each member needs to climb $s_{i}$ steps, and each transition step requires one step. Because the transition step is not needed after reaching the destination, the total number of steps is to be subtracted by one step. With the minimum total number of grips as the optimization goal, a nonlinear optimization mathematical model is established

\begin{matrix} \min f = \sum_{i = 1}^{n} s_{i} + n - 1 \\ s . t . {\begin{matrix} s_{i} s . t . Equation (4) \\ g_{i}^{s} \in R_{t} (g_{i - 1}^{e}, i) \\ R \overset{O}{\cap} = \emptyset \end{matrix} \end{matrix}

(6)

The model first needs to meet the constraints of the moving distance and rotation angle of the climbing gait. The grip constraints require that the transition grip $g_{i}^{s}$ should be located on the i th route member, i.e. within the transition region $R_{t} (g_{i - 1}^{e}, i)$ solved when taking $g_{i - 1}^{e}$ as the base. During transiting between the members, robot R should not collide with the obstacles O under the current grip. The grips other than the starting and ending grips on one route member are unknown, so the constraints of collision-free for those grips are not considered. The subsequent analysis will be carried out in the search and generation of the grips on each route member.

Grip search on a single member

By solving the optimization model of transition grips, the grip number and the starting and ending grips on each route member are obtained. The grip search algorithm mainly solves the problem of assigning discrete grips on the route members. Since there are countless grips for the robot to choose from in each movement process, the optimal grip should be selected.

Figure 4(b) illustrates the grip search diagram. Initially, the reference discrete grips are generated based on the transition grip optimization model results. When the grip count on the route member exceeds two, the reference discrete grips can be directly generated using uniform interpolation. If the grip count is zero or one, the robot either transitions directly to the next member or climbs only one step on the current member, eliminating the need to search for reference grips. When the grip count is two and the rotation angle satisfies the constraint in equation (5), the robot performs an inchworm-like gait, searching in both opposite directions along the member’s axis.

Then, the starting grip serves as the base for the robot, allowing it to generate the corresponding grasping region. Within this region, the robot selects the grip for the next climbing step using an appropriate heuristic function,

H (g) = \sqrt{{[g (l) - G_{r} (l)]}^{2} + {[g (α) - G_{r} (α)]}^{2}}

(7)

where $g$ denotes the searching grip, while $G_{r}$ denotes the reference discrete grip. The heuristic function, denoted as $H (g)$ , represents the projected distance between the two types of grips on the expanded member’s plane. A smaller heuristic value indicates a more favorable grip currently being searched.

If the next grip cannot be found, the algorithm switches to a backtracking search. The algorithm excludes the current grip, reverts to the previous step, and searches for an optimal grip as the new base before resuming the search process. To avoid excessive computational time, the backtracking search is limited to a single execution. Without successful backtracking, the robot cannot proceed with climbing along this member.

Given the distinct kinematics models for BTCRs with 6 DoF when grasping using different end grippers, a grasping flag is set to invoke the relevant robot kinematics algorithms. Furthermore, collision avoidance using the proposed three-layer collision detection strategy is recommended applying at each step.

Single-step motion planning

After obtaining the grip sequence of the BTCRs in the truss environment, it is necessary to plan the moving path between two grips at intervals. At this time, the robot takes the current grasping gripper as the base and moves the swinging gripper from the previous grip to the next one without collision. Figure 5 shows the single-step motion process of the BTCRs. The swinging gripper of the robot opens the fingers and exits the member to the takeoff intermediate point along the negative direction of the a-axis at a safe distance. Then, the swinging gripper moves to the landing intermediate point following the collision-free path, and finally, it approaches the target member along the positive direction of the a-axis of the swinging gripper and closes the gripper, thus completing a single-step climbing process.

Figure 5.

Single-step motion of BTCRs.

Therefore, the core of the single-step motion planning method is to solve a non-collision path connecting the takeoff intermediate point and landing intermediate point. In this paper, the bidirectional rapidly expanding random tree (RRT-connect) algorithm is used for path planning, and the proposed collision detection method is adopted to avoid obstacles. Since the obtained path may have a sudden change, a cubic uniform B-spline curve is used to smooth the path points.

Simulations

To verify the effectiveness of the proposed theory and algorithm of the climbing motion planning, a series of simulations were conducted with Climbot. The simulation environment was developed based on the RViz platform. The planning algorithms were run on a desktop with the configuration of 2.60GHz i5-3230M CPU, 8GB RAM and operating system Ubuntu 16.04.

The result of transition performance analysis

This part of simulations was designed to verify the feasibility and effectiveness of the transition performance analysis and the transition region solution algorithm. Figure 6(a) shows that the truss scene consisted of two members,

{}^{W}{Truss} = [\begin{matrix} 400 & 100 & - 300 & 400 & 100 & 2300 \\ 1500 & 200 & 300 & - 500 & 300 & 1800 \end{matrix}]

(8)

where each row represents a truss member with the coordinates of the member’s two endpoints. The coordinates are expressed in millimeters, and the truss member radius is fixed at 24 mm. The transition regions were determined when the base grippers of Climbot grasped Member I at different positions and rotation angles, and the swinging grippers moved to Member II.

Figure 6.

Simulations of transition performance analysis: (a) the scenario; (b) the results of transition region under same rotation angle with different grasping position; and (c) the results of transition region under same grasping position with different rotation angle.

Figure 6(b) displayed the results of the transition region where the rotation angle of the base gripper was fixed, but the grasping position increased gradually along the axis of Member I. While Figure 6(c) showed the results of the transition region with a fixed grasping position of the base gripper but different rotation angles (0°, 90°, 180°, and 360°, respectively). It was observed that there was no obvious linear relationship between the transition regions obtained by solving different grasping positions and rotation angles. When the grasping position was fixed, the rotation angles of 0° and 360° were the same grasping pose, so the corresponding transition regions were identical.

Specifically, the robot grasps Member I in the pose shown in Figure 6(a). The corresponding transitional region, as depicted in the second subfigure from the left in Figure 6(c), is represented with the vertical axis denoting the truss member length and the horizontal axis indicating the rotation angle of the gripper around the member axis. Notably, due to the upper-left portion of Member II extending beyond the robot’s reachable range, the upper half of the quadtree lacks a transitional region. In the lower-right portion of Member II, three relatively separate transitional regions emerge. Importantly, for different link lengths, there are generally corresponding transitional grips, with the robot having more viable transitional grips near the bottom end of the truss member, resulting in improved transition performance.

During the process of solving the transition region, the non-transitional grips were stored by a node, while the transitional nodes continued to be subdivided. In this paper, the quad-tree data structure was used to represent the transition region. To validate the validity of this data structure, the number of grids was compared when using the uniform method and the quadtree method to represent the transition region, and the results are presented in Table 1. The number of grids of the transition region stored by the quadtree was less than that of the uniform grid method, which effectively compressed the information of the transition region and reduced the complexity of subsequent motion planning. At the same time, the number of transitive grids using the quadtree was not lost, which could accurately describe the transition region.

Table 1.

Comparison of storage data structures in transition regions.

	Grid number
Methods	Transitional	Non-transitional	Total
Uniform	2015	14369	16384
Quadtree	2015	419	2434

Results of grip sequence planning

This part of the simulations was designed to verify the effectiveness and efficiency of the grip sequence planning algorithm. Two scenarios, consisting of 8 and 15 members, respectively, were set up for simulations.

In Scenario I, after randomly setting the starting and target poses of the robot, the transition between members in the environment was analyzed by the truss member route planner. An adjacency matrix was constructed to generate all feasible member sequences. Subsequently, the grip sequence planning algorithm planned two feasible routes, as shown in Figure 7(a) and (b). For Route I, the robot needed to climb nine steps on three members (the truss member sequence: Member 1, Member 3, and Member 7).

Figure 7.

The results of grip sequence planning in the truss environment with 8 and 15 members, respectively: (a) and (b) feasible Routes I and II in Scenario I; (c) to (f) feasible Routes I to IV in Scenario II.

However, in Route II, the robot traversed three different members (Member 1, Member 2, and Member 7), but required 11 steps to climb. It can be seen that even if the number of route members was the same, there was a significant difference in the grip number obtained by different truss member routes. To verify the reliability of the planning method, 50 tests were conducted by setting different starting and destination points in various truss environments. Out of 50, 49 successfully generated feasible grip sequences, yielding a success rate of 98%.

In Scenario II, after setting the starting point and destination, four feasible routes were solved through the truss member route planner and the grip sequence planner, as shown in Figure 7(c)–(f) and Table 2.

Table 2.

Comparison between feasible routes for scenario II.

Routes	Grip number	Optimize model solving time (s)	Grip searching time (s)
I	10	1.68	1.73
II	11	1.37	1.78
III	9	1.92	0.03
IV	10	1.86	0.03

In Route I, Climbot climbed 10 steps on 4 route members. It took 1.68 seconds to solve the optimization model of transition grips and 1.73 seconds to search for the grips on each member. While in Route II, the robot needed to climb 11 steps, and it took 1.37 seconds and 1.78 seconds to solve the optimization model and grip searching, respectively. In these two routes, the robot transitioned to Member 3 and Member 6, respectively, after adjusting the grips with different step numbers on Member 1, and finally to the same Member 8. The number of steps in the two routes was similar, and there was no significant difference in the time consumption of the path planning. Route III had the least grip number, which was 9 steps, and it took 1.92 seconds to solve the optimization model of transition grips, and 0.03 seconds to search for grips. While in Route IV, the robot climbed 10 steps, and it took 1.86 seconds and 0.03 seconds to solve the optimization model and grip searching, respectively. The grip number was relatively close in the two routes, and the planning time was reduced compared to the previous two routes. Upon comprehensive comparison, Route III was determined to be the optimal climbing path.

Results of single-step motion planning

This part of simulations was to verify the efficiency and reliability of the single-step motion planning algorithm. Figure 8 presents a simple scenario consisting of three members. Climbot was tasked to climb two consecutive steps on Member I, while Members II and III served as obstacles to verify the effectiveness of the proposed algorithm.

Figure 8.

The results of single-step and multi-step motion planning: (a) Step I; (b) Step II; (c) multi-step motion for Route I in Scenario I; and (d) multi-step motion for Route III in Scenario II.

The blue curve in Figure 8(a) represented the moving path of the swinging gripper G1 in the first step. It started from the initial configuration, avoided the obstacle members and then approached and grasped the target member. The robot’s trajectory was smooth, with no sudden change point. The robot then exchanged the grasping base and initiated the second step of climbing. The motion path was depicted by the green curve in Figure 8(b). In this step, the obstacle members significantly influenced the robot’s movement, but it successfully avoided obstacles and reached the target grip smoothly. This simulation also validated the effectiveness of the set grasping flag, and the correct kinematics model could be called under different grasping bases to plan a collision-free path.

To further verify the reliability and stability of the planning method, 50 simulation tests were conducted in different truss environments by setting different target grips. All tests were able to solve the collision-free paths with a success rate of 100%. The average time for the robot to plan a single-step path was 0.76 seconds, and the maximum time was 2.30 seconds. Moreover, as the complexity of the environment increased, the path planning time also increased.

Multi-step collision-free motion planning was carried out for Route I in Scenario I and Route III in Scenario II, as shown in Figure 8(c) and (d). The blue and green curves in the figure correspond to the paths of the swinging gripper when the robot grasps the member with different base grippers. In the continuous multi-step motion generation process, the robot can effectively avoid the obstacle members, and the trajectories are smooth. Corresponding videos can be seen at the multimedia extension of the paper.

Conclusion

BTCRs possess significant potential to perform many high-rise tasks in the truss environment. To realize field application, BTCRs must be capable of efficient path planning to provide a collision-free path that guides the robot to climb flexibly.

A hierarchical framework for planning the 3D climbing path of BTCRs is presented in this paper. Following a transition performance analysis, the transition region between two transitive members is obtained and expressed using a quadtree structure. Based on the transitionability, the truss member route can be solved through the graph search method. To generate the grip sequence, the transition grips are first solved by a mathematical optimization model. Then, a heuristic search strategy is adopted to obtain all the grips along the truss member route. An RRT-connect-based single-step motion planning method is presented to plan the collision-free path. A series of simulations were conducted. The success rate of grip sequence planning in various truss environments was 98%. The single-step motion could be solved with a success rate of 100%. Multi-step collision-free paths were also planned in different scenarios. The results of the simulations verified the feasibility and effectiveness of the proposed theory and algorithms for climbing path planning in the truss environment.

However, the proposed path planning method in this paper is based on the ideal robot model. Due to the combined effects of the error between the truss map and the actual environment, the flexible and elastic deformation of the joints and links caused by self-weight, and the dynamic parameter error, the robot cannot accurately track the planned climbing path. Therefore, the interaction mechanism and modeling method of the multi-source motion error of BTCRs will be explored, and the strategy for improving the robot’s motion accuracy and the corresponding climbing path planning method will be studied accordingly. Moreover, the proficient regulation of BTCRs in complicated truss environments is identified as a crucial research focus. Hence, the employment of teaching by demonstration (TBD) (Su et al., 2021) and non-contact remote control technologies predicated on human gesture recognition (Qi et al., 2021) in the control of BTCRs heralds future research prospects.

Footnotes

Appendix

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the Basic and Applied Basic Research Foundation of Guangdong Province (grant no. 2024A1515011871, 2022A1515110750 and 2022A1515240013), China Postdoctoral Science Foundation (grant no. 2023M730736), and GDAS’ Project of Science and Technology Development (grant no. 2022GDASZH-2022010108).

ORCID iD

Shichao Gu

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