A Novel Bidirectional Beetle-Informed RRT* Connect Path Planning Algorithm for Angle-Steel Tower Operation Robots

Abstract

This paper proposes a novel Bidirectional Beetle-Informed RRT* (BBI-RRT*) Connect algorithm to enhance the safety and path planning efficiency of 6-DOF robotic manipulators operating in the complex high-altitude environment of angle-steel towers. By digitally reconstructing the tower environment through model registration, the algorithm establishes an accurate foundation for subsequent path planning. A bidirectional beetle antennae search mechanism is employed to guide node sampling, effectively accelerating the convergence rate of the algorithm. To ensure the generation of feasible path, a multi-constraint objective function is designed to balance path length, smoothness, and operability. Additionally, an Informed RRT* process is integrated to refine the path within an adaptive 3D ellipsoid, achieving global path optimization. Both simulation tests on the Unity platform and real-world experiments are conducted to validate the effectiveness and superiority of the proposed algorithm.

1. Introduction

Angle-steel towers are critical infrastructures in power transmission systems, as their structural integrity directly influences the reliability and safety of the power grid [1,2]. Conventional maintenance operations are still largely dependent on manual labor performed at high altitudes, which brings significant safety risks to the workers. For this reason, tower-climbing robots equipped with multi-axis manipulators have emerged as a research priority for automating hazardous tasks such as bolt tightening, insulator replacement, and foreign object removal [3,4].

For reliable autonomous operation, the manipulator end-effector-equipped with specific tools and cameras must move safely to target positions within the tower workspace. This requirement gives rise to a Cartesian-space path planning problem, whose objective is to generate collision-free paths from initial to target poses. This capability is critical for the robot [5,6]. However, the complex spatial lattice structure of angle-steel towers introduces unique challenges that probably render traditional planning methods inadequate [7,8,9].

Two major issues limit the performance of existing methods. First, overly simplified geometric primitives (e.g., cuboids or cylinders) lead to inaccurate collision detection when modeling obstacles [10,11,12]. For example, Liu et al. [13] used cylindrical approximations to represent substation equipment. Their study demonstrated that such simplifications cannot closely match the true spatial complexity of the structures, resulting in excessive obstacle-avoidance redundancy. Angle-steel towers are openwork truss structures, with multiple operation targets (such as bolts) distributed inside. If simplified geometric primitives are used for envelope modeling, internal operation points will be unreachable. Second, conventional path planning algorithms have inherent performance limitations. The A* algorithm guarantees the shortest path but has exponential computational complexity in high-dimensional spaces. This makes it impractical for real-time applications [14]. In contrast, the rapidly exploring random tree (RRT) algorithm reduces the dimensionality issue through random sampling. However, it requires long iterations and often generates suboptimal paths, which have excessive length and discontinuous turning angles [15,16]. Recent advances include dynamic obstacle avoidance with path reshaping on probabilistic roadmaps (PRM-DNNCC) for high-DOF robots [17,18], and estimated informed anytime search with adaptive sampling (MIT*) to further accelerate convergence in complex environments [19]. These state-of-the-art planners achieve strong performance in general robotic scenarios but have not been validated in the narrow, lattice-structured angle-steel tower environment with strict kinematic feasibility requirements.

To overcome these challenges, this paper proposes a BBI-RRT* Connect algorithm integrated with digital models of angle-steel towers. The main contributions are summarized as follows:

• A high-precision digital twin of the tower is constructed through model registration to enhance collision detection accuracy.
• A bidirectional beetle antennae search (BAS) mechanism is introduced to guide early-stage sampling, improving convergence speed and search efficiency.
• A multi-constraint objective function combining location, optimization, and operability costs is introduced to ensure feasible and smooth motion of the 6-DOF manipulator under tower environment constraints.

2. Problem Description and Preliminaries

2.1. Problem Description

To achieve climbing and autonomous operations on angle-steel towers, an aerial work robot is designed, as shown in Figure 1. The aerial work robot consists of a climbing mobile platform and a 6-degree-of-freedom (6-DOF) manipulator. The climbing mobile platform is composed of two mechanical grippers and a lead screw mechanism. The lower gripper is fixedly connected to the robot body, and the upper gripper can move along the linear guides under the drive of the lead screw. The climbing platform can realize step-by-step climbing along the outer edge line of the angle-steel tower. When the robot moves to a centain position, both mechanical grippers close to achieve stable attachment. Then, the 6-DOF manipulator can perform various operations on the tower.

It is necessary to plan a short, smooth, and safe path for the manipulator. However, the difficulty lies in the inability to accurately describe the spatial position distribution of the angle steels. To avoid complex mathematical descriptions of the spatial geometry of the angle steels, this paper uses the Unity platform to build a high-precision digital twin model of the robot and the angle-steel tower. It relies on the collision detection function of the digital engine to guide the search for the optimal path in space.

2.2. Kinematics Model of the Manipulator

To perform end-effector path planning for the manipulator, its kinematic model must first be established. The link coordinate frames of the robot are defined as shown in Figure 2. From these coordinate frames, it can be clearly observed that the climbing mobile platform contains only one prismatic joint; in contrast, the 6-DOF manipulator consists of six rigid revolute joints connected in series. These two parts share a common base coordinate frame. The Modified Denavit–Hartenberg (MDH) convention is adopted to establish the kinematic model of the manipulator: the link coordinate frames $\left\{i\right\}$ are assigned at each joint, the Z-axis aligns with the i-th joint axis pointing toward joint $i+1$; the X-axis defines the common normal between adjacent Z-axes; and the Y-axis aligns to complete the right-handed coordinate system.

The specific MDH parameter table of the manipulator is given in

Table 1. MDH parameters of the manipulator.

i	${\mathit{\alpha }}_{\mathit{i}-1}$ (deg)	${\mathit{a}}_{\mathit{i}-1}$ (mm)	${\mathit{d}}_{\mathit{i}}$ (mm)	${\mathit{\theta }}_{\mathit{i}}$ (deg)
1	0	0	240.5	${\theta }_1$
2	90	0	0	${\theta }_2+90$
3	0	256	0	${\theta }_3+90$
4	90	0	210	${\theta }_4$
5	−90	0	0	${\theta }_5$
6	90	0	172.5	${\theta }_6$

. The detailed solution process of forward and inverse kinematics is included in Appendix A.

2.3. Traditional RRT* Algorithm

As one of the most well-known path planning algorithms, the RRT* algorithm explores the space of interest incrementally by expanding a tree structure. In this tree, nodes represent feasible states, and edges denote valid transitions [20]. Compared with the standard RRT, the RRT* algorithm introduces parent reselection and rewiring operations. These operations iteratively minimize the path cost, thus ensuring asymptotic optimality. However, in cluttered or high-dimensional workspaces, the algorithm usually needs thousands of iterations to generate a satisfactory path. This significantly limits its practical efficiency. To solve this problem, a novel BBI RRT* Connect algorithm is proposed.

3. Bidirectional Beetle-Informed RRT* Connect Algorithm

3.1. Digital Model Construction

Angle-steel towers have complex three-dimensional structures. Their components, such as beams, columns and bolts, are densely distributed and interconnected. This brings great challenges to robotic path planning and collision avoidance [21,22]. Therefore, accurate modeling of these components is crucial. The fidelity of the digital model directly influences the safety and reliability of robotic path planning [23]. In this study, the solution is to use Unity3D (version 2021.3.26f1c1) as the physics engine for reconstructing and describing high-altitude operations. Direct modeling in Unity3D is not practical because of the complex and numerous components. Therefore, some auxiliary modeling tools are used to build accurate and compatible models.

The tower geometry is reconstructed from high-precision lofting models using TMA software. The robot model, which includes the climbing mobile platform and the manipulator, is built in professional 3D mechanical design software such as SolidWorks 2021. 3ds Max software is used as an intermediate platform for the format conversion and optimization of the tower geometry and the robot model, as shown in Figure 3. Several preprocessing steps are applied before integration into Unity3D:

• Model simplification: Redundant meshes and hidden components are removed to reduce polygon count.
• Coordinate alignment: Model axes are rotated and adjusted to match the left-handed coordinate system in Unity3D.
• Scale correction: Units are standardized in 3ds Max to ensure a 1:1 scale correspondence after import.

Through this workflow, a digital twin environment of the tower–robot system is established, providing a basis for path planning and collision detection. The accuracy of physical equipment models may degrade after multiple layers of software conversion. However, the digital environment for high-altitude operations has stringent accuracy requirements. Therefore, this study adopts a model registration method based on PL-ICP [24] method to ensure model fidelity.

Point cloud data of the digital twin model are read from Unity3D. Meanwhile, the point cloud data of the original physical model are collected separately. These two datasets are used to obtain the initial transformation matrix. Then, the nearest-neighbor points of each point in the original model are identified in the digital twin model by using the nearest-neighbor method, and points with large registration errors are removed. The objective function is constructed as follows:

$$\begin{array}{cc}\hfill mi{n}_{R_{k+1},t_{k+1}}J(R_{k+1},t_{k+1})& ={\displaystyle \sum _{i=1}^N}{\left(n_i^T\left[R_{k+1}{P}_i+t_{k+1}-P_{j_i}\right]\right)}^2,\hfill \end{array}$$

(1)

where J denotes the objective function, $R_{k+1}$ and $t_{k+1}$ represent the rotation matrix and translation vector at the (k + 1)-th iteration respectively; N is the number of corresponding point pairs, $n_i$ and $P_i$ represent the unit normal vector and the vector of the i-th point in the original physical model’s point cloud respectively, and $P_{j_i}$ is the vector of the nearest-neighbor point corresponding to $P_i$ in the digital twin model.

The registration accuracy is quantitatively evaluated using the root mean square error (RMSE) of the point-to-plane distance:

$$\begin{array}{c}\hfill \mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(n_i^T\left(R_{k+1}{P}_i+t_{k+1}-P_{j_i}\right)\right)}^2}.\end{array}$$

(2)

The RMSE of the raw tower model before registration is measured as 16.74 mm, which exceeds the 2 mm accuracy requirement. After PL-ICP-based registration, the RMSE is reduced to 0.62 mm, meeting the precision demand for high-altitude operation tasks.

The verified high-fidelity digital model is saved and displayed in Unity3D platform, as shown in Figure 4.

After importing the high-precision registered model into Unity3D, collision properties are assigned to the model to support the subsequent path planning. In Unity3D, available collider types include simple geometric primitives (Sphere Collider, Box Collider, Capsule Collider), Compound Collider, and Mesh Collider. In this work, Mesh Collider is adopted for collision detection of complex angle-steel components. This collider generates a tightly fitted, shape-adaptive collision envelope that conforms to the irregular 3D geometry of each structural part. Compared with simple geometric approximations, Mesh Collider achieves pixel-level accurate collision detection. When a collision is detected, a trigger signal is generated to instruct the planning algorithm to adjust its spatial expansion direction for obstacle avoidance.

3.2. Bidirectional Beetle Antennae Search Mechanism

In high-dimensional environments, conventional RRT* algorithms often suffer from slow convergence due to random sampling, which requires extensive computation and a large number of iterations. For the digital twin environment constructed in this study, the standard RRT* generally requires more than 4000 iterations to obtain a feasible path. To enhance convergence efficiency, a BAS mechanism is incorporated to guide the sampling process adaptively.

The BAS algorithm is inspired by the foraging behavior of longhorn beetles. A beetle perceives the intensity of environmental stimuli through two antennae and moves toward the side with a stronger signal. By iteratively adjusting its direction according to this sensory difference, the beetle converges toward the optimal position.

To extend BAS for high-dimensional optimization, a step attenuation coefficient is introduced. A larger initial step promotes global exploration, whereas a gradually decaying step enables local refinement near the optimum. As the step decreases, the antenna spacing $d_0$ correspondingly shrinks, allowing the search process to transition smoothly from global to local exploration. Because only a few beetles are used, the computational burden remains low while maintaining high convergence speed.

For the 3D optimization problem considered here, let $L_{\mathrm{step}}$ denote the beetle’s step length, $\eta$ denote the attenuation coefficient, and $d_0$ denote the antenna spacing. The search process can be described as follows.

First, the movement direction of the beetle is determined by

$$\begin{array}{c}\hfill \overrightarrow{\mathrm{DIR}}={\displaystyle \frac{\mathrm{rand}(n,1)}{\parallel \mathrm{rand}(n,1)\parallel }},\end{array}$$

(3)

where n is the dimensionality of the search space.

Then, the coordinates of the left antennae $X_{\mathrm{left}}$ and right antennae $X_{\mathrm{right}}$ are calculated as

$$\begin{array}{cc}\hfill X_{\mathrm{left}}& =X_{\mathrm{best}}+\overrightarrow{\mathrm{DIR}}{d}_0,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill X_{\mathrm{right}}& =X_{\mathrm{best}}-\overrightarrow{\mathrm{DIR}}{d}_0.\hfill \end{array}$$

(5)

where $X_{\mathrm{best}}$ represents the best-known position, and $d_0$ (the distance between the two antennae) is defined as $d_0=L_{\mathrm{step}}·e^{-\eta t}$.

Next, the objective function $F\left(X\right)$ is evaluated to measure the Euclidean distance between one’s position X and the target position $X_{\mathrm{target}}$:

$$\begin{array}{c}\hfill F\left(X\right)=\parallel X-X_{\mathrm{target}}{\parallel }_2.\end{array}$$

(6)

The corresponding evaluations at the two antenna positions are

$$\begin{array}{cc}\hfill F\left(X_{\mathrm{r}}\right)& =F\left(X_{\mathrm{right}}\right),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill F\left(X_{\mathrm{l}}\right)& =F\left(X_{\mathrm{left}}\right).\hfill \end{array}$$

(8)

Finally, using the sign function $\mathrm{sign}(·)$, the beetle updates its position according to

$$\begin{array}{c}\hfill X_{\mathrm{new}}=X_{\mathrm{best}}-d_0·\overrightarrow{\mathrm{DIR}}·\mathrm{sign}\left(F\left(X_l\right)-F\left(X_r\right)\right).\end{array}$$

(9)

If $X_{\mathrm{new}}$ falls on the angle-steel tower components, the collision detection function of the digital engine will be triggered. Then, the search direction is recalculated by (3). The iterative updates are performed by comparing $F\left(X_{\mathrm{new}}\right)$ and $F\left(X_{\mathrm{best}}\right)$. If $F\left(X_{\mathrm{new}}\right)<F\left(X_{\mathrm{best}}\right)$, the best position is updated by $X_{\mathrm{best}}=X_{\mathrm{new}}$; otherwise, the step size is gradually reduced according to the attenuation rule until convergence is achieved.

3.3. Multi-Constraint Objective Function

To obtain a high-quality feasible path for the six-degree-of-freedom (6-DOF) manipulator, multiple constraints must be imposed to address obstacle avoidance and kinematic limitations. Unlike the original RRT* algorithm [25], the proposed improved RRT* algorithm integrates an extended objective function evaluation into the nearest-neighbor search procedure. It traverses all nodes in the node list and selects the one with the minimum total cost for tree expansion. Specifically, the objective function $O\left(x\right)$ is augmented with three penalty terms to satisfy the requirements of tower inspection and maintenance tasks: the position cost, the optimization cost, and the operability cost.

The location cost $P\left(x\right)$ integrates both the current and cumulative distance costs to improve path efficiency. And the optimization cost $R\left(x\right)$ quantifies the variation in joint angles to maintain smooth motion during operations such as bolt tightening or component inspection. They are defined as:

$$\begin{array}{cc}\hfill P\left(x\right)& ={∥X_{\mathrm{new}}-X_{\mathrm{goal}}∥}_2+{\displaystyle \sum _{i=0}^{j-1}}{∥X_{\mathrm{new}}\left(i\right)-X_{\mathrm{goal}}∥}_2,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill R\left(x\right)& ={\displaystyle \sum _{i=0}^l}{∥{\theta }_i(x+1)-{\theta }_i\left(x\right)∥}_2,\hfill \end{array}$$

(11)

where j denotes the number of parent nodes in the tree, and l represents the rotational degrees of freedom of the manipulator.

The operability cost represents the ability of the manipulator to move its end-effector flexibly within the tower’s narrow and obstacle-rich workspace. It is expressed as:

$$\begin{array}{c}\hfill M={\displaystyle \frac{1}{j+1}}\left[C\left(\mathbf{q}\right)+{\displaystyle \sum _{i=0}^{j-1}}C\left(\mathbf{q}\right(i\left)\right)\right],\end{array}$$

(12)

where $C\left(\mathbf{q}\right)$ quantifies the manipulability of the configuration, and $\mathbf{q}$ represents the joint angles. A smaller value implies that the manipulator is close to a singular configuration, and a larger value corresponds to higher dexterity and motion flexibility. When its value is zero, the manipulator may suffer from structural stress or loss of controllability. Therefore, manipulability acts as an important indicator for judging whether the manipulator is in or near a singular pose.

The manipulability measure is defined as:

$$\begin{array}{c}\hfill C\left(\mathbf{q}\right)=\sqrt{det\left(J\left(\mathbf{q}\right)J^T\left(\mathbf{q}\right)\right)},\end{array}$$

(13)

where $J\left(\mathbf{q}\right)$ is the Jacobian matrix of the 6-DOF manipulator, derived from its kinematic model, and $det(·)$ represents the determinant operator.

Then, all cost terms are normalized to the range [0, 1] using min–max scaling to eliminate magnitude discrepancies. This ensures that no single term dominates the optimization due to unit or scale differences. The normalized cost is defined as:

$$\begin{array}{c}\hfill \overline{P}\left(x\right)={\displaystyle \frac{P\left(x\right)-P_{min}}{P_{max}-P_{min}}}\overline{R}\left(x\right)={\displaystyle \frac{R\left(x\right)-R_{min}}{R_{max}-R_{min}}}\overline{M}=1-{\displaystyle \frac{M}{C_{max}}}\end{array}$$

(14)

where $P_{min}={\sum }_{i=0}^{j-1}{∥X_{\mathrm{new}}\left(i\right)-X_{\mathrm{goal}}∥}_2$, $P_{max}=L_{\mathrm{step}}+{\sum }_{i=0}^{j-1}{∥X_{\mathrm{new}}\left(i\right)-X_{\mathrm{goal}}∥}_2$, $R_{min}=0,R_{max}=6\pi$, and $C_{max}$ is the maximum manipulability measure over all feasible robot configurations.

Finally, the overall objective function is formulated as:

$$\begin{array}{c}\hfill O\left(x\right)=min\left\{\overline{P}\left(x\right)+\alpha \overline{R}\left(x\right)+(1-\alpha )\overline{M}\right\}\end{array}$$

(15)

where $\alpha \in [0,1]$ is a weighting coefficient to balances the optimization and operability costs. This balance ensures smooth motion via $\overline{R}\left(x\right)$ while avoiding singularities via $\overline{M}\left(x\right)$ in the tower environment, and $\overline{P}\left(x\right)$ maintains path efficiency.

When $\alpha \to 0$, the optimization focuses on manipulability and singularity avoidance, which is suitable for obstacle-dense and constrained environments. When $\alpha \to 1$, the optimization emphasizes motion smoothness, generating more stable trajectories in open spaces. In practical implementation, one may first set $\alpha =0$ to generate a collision-free and singularity-free feasible trajectory, and then gradually increase $\alpha$ to improve the trajectory smoothness.

3.4. Informed RRT* Algorithm

The Informed RRT* algorithm introduces a heuristic sampling mechanism after an initial feasible path is obtained. Instead of sampling uniformly across the entire configuration space, the search is constrained within an ellipsoidal region centered between the start and goal. As the path cost decreases during iteration, the ellipsoid gradually contracts, focusing sampling on promising areas and speeding up convergence to the optimal solution.

The major axis $c_{\max }$ is set to the length of the current best path. The minor axis $c_{\min }$ is set to the Euclidean distance between the start and goal configurations, defined as

$$\begin{array}{c}\hfill c_{\min }={∥X_{\mathrm{goal}}-X_{\mathrm{start}}∥}_2.\end{array}$$

(16)

To generate samples uniformly within the ellipsoid, random points are first generated inside a unit sphere, then transformed into the target ellipsoidal domain. The transformation is realized through a rotation matrix C and a scaling matrix L. The rotation matrix C is computed using the Kabsch [26] algorithm, which applies singular value decomposition (SVD) to the normalized vector connecting the start and goal points. The scaling matrix L is given by:

$$\begin{array}{c}\hfill L=\left[\begin{array}{ccc}{\displaystyle \frac{c_{\max }}{2}}& 0& 0\\ 0& {\displaystyle \frac{\sqrt{c_{\max }^2-c_{\min }^2}}{2}}& 0\\ 0& 0& {\displaystyle \frac{\sqrt{c_{\max }^2-c_{\min }^2}}{2}}\end{array}\right].\end{array}$$

(17)

As the path is continuously optimized, the ellipsoid shrinks dynamically to concentrate sampling around the optimal trajectories.

3.5. BBI-RRT* Connect Algorithm Summary

The BBI-RRT* Connect algorithm (integrating bidirectional beetle antennae search, a multi-constraint objective function, and Informed RRT*) provides a efficient path planning framework for aerial work manipulators in complex angle-steel tower environments. The full procedure in pseudocode is summarized in Algorithm 1.

Algorithm 1 BBI-RRT* Connect algorithm for high-altitude angle-steel tower operations.

1: Input: Digital twin model of angle-steel tower, start and goal positions, maximum iterations $ite{r}_{max}$, BAS step size, attenuation coefficient $\eta$   2: Output: Optimized collision-free path for the manipulator   3: Initialize iteration counter $iter=0$   4: Construct bidirectional BAS agents at start and goal positions   5: while  $iter<ite{r}_{max}$ do   6:     Update BAS agent positions to generate candidate sampling points   7:     Evaluate candidate nodes using the multi-constraint objective function   8:     Expand trees toward selected nodes, update parent nodes if necessary, and check collisions   9:     If collision occurs, discard node and resample; otherwise, update tree structure 10:     Attempt to connect the two trees to form an initial feasible path $L_1$ 11:     if initial path is found then 12:        Switch to Informed RRT* sampling within 3D ellipsoidal region defined by current best path 13:        Refine path, update parent nodes, and optimize path cost 14:     end if 15:     $iter\leftarrow iter+1$ 16: end while 17: Output the final optimized path

The theoretical analysis of computational overhead per iteration is supplemented as follow:

• The BAS module only computes left/right antenna coordinates in 3D space, evaluates the Euclidean distance to the target, and updates the beetle position with collision checking. All operations are fixed and constant-time, so the overhead is $O\left(1\right)$.
• The multi-constraint objective function includes position cost, smoothness cost, and manipulability cost. All calculations involve fixed dimension vectors and a 6 × 6 Jacobian matrix, with constant computation. The overhead is $O\left(1\right)$.
• The Informed RRT* ellipsoid sampling uses fixed 3 × 3 matrix transformation from a unit sphere. All operations are constant-time, so the overhead is $O\left(1\right)$.
• The nearest-neighbor search, parent reselection, and rewiring operations inherit from standard bidirectional RRT* Connect and remain $O(logN)$, where N is the number of nodes.

Thus, the per iteration time complexity of BBI RRT* Connect is identical to that of traditional bidirectional RRT* Connect, i.e., $O(logN)$. The extra overhead from BAS and the multi-constraint function is a small constant that does not change the asymptotic order.

4. Simulation and Experimental Results

4.1. Path Planning Simulation

Based on the digital environment built on the Unity3D platform in Section 3, path planning simulations were carried out to verify the effectiveness of the proposed BBI-RRT* Connect algorithm. Three representative regions of the angle-steel tower were chosen to generate efficient and collision-free paths for the 6-DOF manipulator. Region 1 contains multiple steel plates, densely distributed bolts, and several angle steels. Region 2 consists of one vertical wide angle steel, two horizontal thin angle steels, and four inclined thin angle steels. Region 3 includes one vertical wide angle steel and four inclined thin angle steels. To further validate the results, 6 benchmark algorithms—A*, Informed RRT*, RRT* Connect, BAS-RRT* Connect, PRM-DNNCC [17], and MIT* [19]—were tested under the same conditions. Each algorithm was run 50 times independently, and the average results were adopted for comparison.

From the results shown in Figure 5, the proposed BBI-RRT* Connect algorithm generates smoother and more goal-oriented paths than other RRT-based variants, and effectively minimizes redundant branch growth even in dense obstacle environments.

For further quantitative comparison,

Table 2. Performance comparison in Region 1.

Algorithm	Time (s)	SD (s)	Cost	SD	SR (%)
A*	10.58	1.82	46.89	2.15	100
Informed RRT*	7.40	1.35	51.71	2.64	92
RRT* Connect	1.22	0.26	62.83	3.21	100
BAS-RRT* Connect	0.63	0.14	51.71	2.47	100
PRM-DNNCC	0.85	0.19	49.22	2.30	98
MIT*	0.71	0.16	48.05	2.11	99
BBI-RRT* Connect	0.35	0.08	47.38	1.86	100

,

Table 3. Performance comparison in Region 2.

Algorithm	Time (s)	SD (s)	Cost	SD	SR (%)
A*	1.74	0.31	33.52	1.68	100
Informed RRT*	0.96	0.22	35.51	1.92	98
RRT* Connect	0.61	0.15	41.54	2.35	100
BAS-RRT* Connect	0.49	0.11	40.36	2.16	100
PRM-DNNCC	0.56	0.13	36.88	2.01	97
MIT*	0.52	0.12	35.12	1.85	98
BBI-RRT* Connect	0.30	0.07	34.35	1.71	100

and

Table 4. Performance comparison in Region 3.

Algorithm	Time (s)	SD (s)	Cost	SD	SR (%)
A*	1.19	0.24	30.89	1.43	100
Informed RRT*	0.88	0.19	33.36	1.77	96
RRT* Connect	0.52	0.13	45.60	2.68	100
BAS-RRT* Connect	0.39	0.09	41.56	2.29	100
PRM-DNNCC	0.45	0.11	34.61	2.03	96
MIT*	0.41	0.10	33.69	1.88	97
BBI-RRT* Connect	0.27	0.06	32.22	1.54	100

present the key performance indicators including planning time, path cost, success rate (SR), and standard deviation (SD). Among all methods, the A* algorithm yields the shortest path cost but suffers from excessively long planning time, making it unsuitable for real-time tasks in high-dimensional and constrained environments. Informed RRT* improves optimality over conventional RRT* but still suffers from low sampling efficiency and unstable success rates. Although RRT* Connect achieves a 100% success rate, it generates significantly longer paths due to lack of heuristic guidance. By introducing beetle antennae search, BAS-RRT* Connect accelerates convergence, yet its path quality remains limited. The state-of-the-art planners, namely PRM-DNNCC and MIT*, deliver better path quality than BAS-RRT* Connect, but their planning time is noticeably longer. The proposed BBI-RRT* Connect algorithm achieves the shortest planning time, the lowest path cost (close to A*), and a 100% success rate in all three regions. Notably, BBI-RRT* Connect exhibits the smallest standard deviation in both time and cost, meaning its performance is highly consistent and insensitive to environmental variations.

4.2. Experimental Validation

To verify the practical performance of the proposed BBI-RRT* Connect algorithm, an angle-steel tower-climbing robot and a physical angle-steel tower were constructed in an indoor laboratory environment, as illustrated in Figure 6. A high-performance computer was utilized to accomplish high-fidelity environment modeling, dynamic 3D display, and high-precision collision detection in Unity3D, with detailed hardware specifications provided in

Table 5. Computer configuration parameters.

Item	Specification
Computer Model	HP 16-wf0032TX
CPU	13th Gen Intel Core i9-13900HX @ 2.20 GHz
GPU	NVIDIA GeForce RTX 4060 Laptop GPU
Operating System	Windows 11
Unity3D Version	2021.3.26f1c1

. In the experiment, only five algorithms with high performance—RRT* Connect, BAS-RRT* Connect, BBI-RRT* Connect, PRM-DNNCC and MIT*—were selected for experimental verification, since the A* and Informed RRT* algorithms showed significant gaps in performance. The offline pre-planned paths generated in the digital environment were converted into joint angles using the robotic arm’s kinematic model and subsequently executed on the physical robot to verify their feasibility and accuracy.

Due to the height limitation of the indoor space, only the foot part of the angle-steel tower was built. Therefore, experiments could only be conducted near Region 1, with two tests arranged at different start and end points. For Test A, the robot end-effector moved from the outer edge of the tower to the right area, as shown in Figure 7. The top row of Figure 7 compared the planned paths of five algorithms in the angle-steel tower environment: (a) RRT* Connect generated a relatively long path with redundant turns and detours; (b) BAS-RRT* Connect yielded a shorter and more efficient path under the guidance of the beetle antennae search; (c) PRM-DNNCC produced a feasible path with small fluctuations; (d) MIT* obtained a smoother path than the above three methods but still contains redundant sampling-induced turns; and (e) the proposed BBI-RRT* Connect generated the shortest, smoothest, and most direct path toward the target. The bottom row showed the physical execution process at some timestamps (the BBI-RRT* Connect was used for example). The manipulator followed the planned trajectory without collisions or singular configurations, successfully driving the end-effector to the target position. This verified that all five algorithms could generate feasible paths in the real-world tower scenario. From the quantitative data in

Table 6. Performance comparison in Test A.

Algorithm	Planning Time (s)	Path Cost	Success Rate
RRT* Connect	1.12	87.57	100% (40% execution)
BAS-RRT* Connect	0.93	81.96	100% (54% execution)
PRM-DNNCC	1.01	78.42	100% (54% execution)
MIT*	0.97	75.61	100% (64% execution)
BBI-RRT* Connect	0.66	71.89	100%

, the BBI-RRT* Connect algorithm outperformed the BAS-RRT* Connect and RRT* Connect algorithms in terms of planning time and path cost, which was consistent with the conclusion in simulation.

For Test B, the robot’s start point was set on the right side of the tower, and the end point was set on the adjacent left side of the tower. Figure 8 showed the planning results in Test B, where the end-effector needed to cross a thick angle-steel member. Although RRT* Connect, BAS-RRT* Connect, PRM-DNNCC and MIT* algorithms could generate geometrically collision-free paths in the digital environment, they did not consider singularity constraints of the 6-DOF manipulator. As marked in the red box, the robot fell into a singular configuration during actual movement, which resulted in failed execution and made the robot unable to reach the target stably. A manipulability measure curve shows the variation in the manipulability index along the path for all compared algorithms in Figure 8b. This curve clearly demonstrates that the proposed BBI-RRT* Connect maintains a high and stable manipulability value, while the baseline methods experience a sharp drop to near zero (indicating singular configurations). The quantitative results strongly validate the effectiveness of the proposed multi-constraint objective function in avoiding singularities.

In contrast, Figure 9 presented the successful execution achieved by the proposed BBI-RRT* Connect algorithm in Test B, with key moments recorded at t = 0.0 s, 5.0 s, 11.0 s, 18.0 s, and 24.0 s. The algorithm avoided both structural obstacles and singular configurations by using the multi-constraint objective function. The manipulator safely crossed the narrow region at about 11 s and steadily reached the target at 24 s. This demonstrated that the proposed method ensured safety, smoothness, and operational reliability in real-world angle-steel tower operations.

Overall, the above two tests prove the superiority of BBI-RRT* Connect algorithm in terms of convergence speed, path quality, and operational safety, particularly in complex angle-steel tower scenarios where conventional RRT* variants may encounter singularities or infeasible solutions.

5. Conclusions

This paper has discussed the path planning problem for 6-DOF robotic manipulators in the complex high-altitude environment of angle-steel towers via a novel BBI-RRT* Connect algorithm. It has demonstrated that the proposed algorithm can not only achieve fast obstacle-avoidance path planning but also prevent the robotic manipulator from entering singular poses. By model reconstruction in a digital engine, the proposed algorithm has integrated a bidirectional beetle antennae search mechanism, a multi-objective optimization function, and an informed sampling strategy, exhibiting superior performance compared with some other algorithms. Future work will focus on the trajectory tracking control of robotic manipulators.