Energy-Efficient Motion Planning for Repetitive Industrial Tasks: An Adaptive Obstacle Modeling Approach

Abstract

Efficient operation of robotic manipulators in repetitive industrial tasks, such as welding and logistics sorting, requires careful coordination of obstacle representation and motion planning. Traditional methods, such as axis-aligned bounding boxes, generate overly conservative trajectories, while highly detailed models impose excessive computational burden, both increasing cumulative energy consumption in long-duration operations. This paper presents an adaptive sphere-based obstacle modeling framework integrated with energy-aware motion planning for repetitive manipulation tasks. The proposed method employs an improved Whale Optimization Algorithm with nonlinear parameter adjustment and elite guidance mechanisms to generate compact sphere representations through adaptive voxelization. Experimental validation using a 6-DOF UR5 manipulator demonstrates substantial performance improvements over conventional AABB models, achieving 31–66% energy reduction and 12.5–37% shorter configuration-space paths, with competitive modeling efficiency (2.63–3.34 s) compared to 11 metaheuristic algorithms. The framework provides a systematic methodology for integrating obstacle modeling with motion planning, particularly suitable for applications where cumulative energy savings are critical in repetitive operations.

1. Introduction

The advent of Industry 5.0 marks a paradigm shift towards manufacturing systems that are not only efficient and automated but also sustainable, resilient, and human-centric [1]. In this new landscape, robotic manipulators are expected to operate intelligently in unstructured environments—such as those encountered in flexible assembly, collaborative logistics, and dynamic sorting—while minimizing energy consumption and ensuring safety [2]. This dual demand for adaptability and energy efficiency places unprecedented emphasis on the quality of motion planning, particularly in the presence of irregular, a priori unknown obstacles. Long-duration tasks like welding or material handling further amplify the need for energy-optimal trajectories, as they directly impact operational costs and the environmental footprint of production [3,4]. Moreover, the choice of robot configuration (e.g., single-arm vs. dual-arm) in precision assembly introduces additional trade-offs among cycle time, energy use, and overall profitability [2,5].

Despite significant progress in motion planning algorithms, a fundamental bottleneck persists: the way obstacles are modeled is often decoupled from the planning objectives. Traditional approaches treat environment modeling as a separate preprocessing step, forcing a critical trade-off. On one hand, highly detailed representations (e.g., dense meshes or raw point clouds) provide accurate collision information but impose prohibitive computational overhead during planning [6,7]. On the other hand, overly simplified models, such as axis-aligned bounding boxes (AABB), enable fast collision checks at the cost of geometric fidelity, leading to excessively conservative trajectories that increase path length and energy consumption [8]. Intermediate solutions like K-DOPs offer tighter approximations than AABBs but still remain generic geometric constructs, not inherently tailored to facilitate energy-aware or smooth motion [9]. The core issue, therefore, is the absence of a planning-awarerepresentation—one that balances geometric accuracy, computational efficiency, and compatibility with downstream optimization objectives like energy minimization and path smoothness. The proposed sphere modeling addresses this gap by achieving high geometric fidelity, with experimental results showing an average coverage of 99.97% and an average absolute error of 0.0158 m for irregular obstacle representation (see Section 3.2), thereby providing a quantitatively accurate yet computationally efficient model for subsequent planning tasks.

Recent efforts have sought to incorporate energy considerations directly into the planning process. For instance, Alam et al. proposed an energy-efficient RRT* variant (FC-RRT*) for pick-and-place operations, demonstrating measurable energy savings by biasing node expansion and employing physics-based energy models [10]. While such approaches advance the state of the art by integrating energy metrics into the planning loop, they typically rely on simplified geometric representations of obstacles (e.g., basic collision primitives) and do not explicitly address how the fidelity of the environment model influences the achievable energy savings. This highlights the need for a holistic framework that co-optimizes both the obstacle representation and the trajectory itself.

At a higher system level, comparative studies such as that by Peta et al. evaluate single-arm versus dual-arm collaborative robots in precision assembly, considering multi-criteria including time, energy consumption, and profitability [2]. Their work underscores the importance of task allocation and motion efficiency, but again, the underlying motion planning relies on conventional methods without adaptive environment modeling. The results show that while dual-arm robots can be faster, single-arm robots may be more energy-efficient, indicating that the optimal choice depends on the specific task and the quality of planned trajectories. This further motivates the need for a planning framework that can adapt to both robot configuration and environmental complexity.

In response to these gaps, researchers have explored optimization-driven modeling techniques, including metaheuristic algorithms for generating sphere-based approximations. While these methods improve geometric compactness, they typically focus on minimizing representation error alone, ignoring how the resulting model influences the energy profile or smoothness of the subsequent trajectory[11]. This oversight can lead to models that, although geometrically faithful, obstruct narrow passages or induce unnecessary detours, ultimately degrading motion quality. A true step forward requires a paradigm shift: from passive geometric modeling to active, planning-oriented representation that embeds the planner’s objectives directly into the model generation process.

This paper introduces a novel framework that realizes this paradigm by tightly coupling Adaptive Obstacle Modeling with multi-objective trajectory optimization. We propose an Improved Adaptive Whale Optimization Algorithm (IAWOA) to generate a “planning-friendly” obstacle model composed of variable-sized spheres. Unlike conventional methods, IAWOA is explicitly designed to balance geometric fidelity with computational efficiency, producing a compact yet high-fidelity representation whose offline construction cost is amortized over numerous planning cycles, while enabling fast online collision checks. More importantly, this representation is synergistically integrated with a multi-objective planner that simultaneously minimizes energy consumption and C-space path length [12]. Through this integration, we demonstrate that the quality of the obstacle model is a primary driver for achieving energy-efficient and smooth motion—a critical requirement for the resilient and sustainable production systems envisioned in Industry 5.0. In this context, a planning-friendly obstacle representation is defined as one that achieves a balanced trade-off among geometric fidelity, collision-check efficiency, and compatibility with joint-space trajectory optimization for energy efficiency.

By providing a systematic methodology for planning-aware environmental modeling, this work lays a principled foundation for enabling robotic arms to generate collision-free trajectories that are not only geometrically efficient but also energetically economical in complex, unstructured industrial scenarios, thereby contributing to the core tenets of Industry 5.0.

Contributions

The main contributions of this work are summarized as follows:

• An adaptive sphere-based obstacle modeling method is proposed to generate compact and planning-friendly representations for unstructured environments.
• An improved Whale Optimization Algorithm is employed to optimize obstacle modeling while balancing representation accuracy and computational complexity.
• Simulation results demonstrate that the proposed method achieves trajectory quality comparable to fine-grained modeling approaches, with significantly reduced modeling and planning overhead.

2. Methods

2.1. Introduction to Systems

In unstructured environments, robotic arm motion planning must not only ensure collision-free and safe interaction with irregular obstacles but also pursue economic efficiency and high performance throughout the motion [13,14]. To this end, this paper formalizes the robotic arm trajectory planning problem as a multi-objective optimization problem coupled with environmental modeling. Given an initial joint configuration ${\mathbf{q}}_{\mathrm{start}}\in \mathcal{Q}$ and a target joint configuration ${\mathbf{q}}_{\mathrm{goal}}\in \mathcal{Q}$, where $\mathcal{Q}\subseteq {\mathbb{R}}^n$ denotes the joint space (C-space) of an n-degree-of-freedom robotic arm, the planning objective is to generate a time-parameterized trajectory $\tau :[0,T]\to {\mathcal{Q}}_{\mathrm{free}}$ within the free space ${\mathcal{Q}}_{\mathrm{free}}$.

To achieve the aforementioned multi-objective optimization, this paper proposes a hierarchical integrated framework of “fine-grained modeling and planning optimization.” This framework abandons the traditional approach of decoupling environmental modeling and motion planning, instead deeply embedding high-fidelity obstacle models into the planning loop. This integration fully exploits the potential for energy-efficient and effective trajectories while ensuring safety. The overall system process and data interactions are illustrated in Figure 1.

As shown in Figure 1, the system consists of three core modules sequentially connected. The functions and collaborative relationships of each module are as follows:

The systematic design of this framework aims to generate safe, energy-efficient, and effective robotic arm motion trajectories in complex unstructured environments through precise environmental perception, efficient path planning, and accurate physical performance evaluation.

To address the challenges of motion planning in unstructured environments, this paper proposes a hierarchical integrated framework that tightly couples environment modeling with trajectory optimization. As illustrated in Figure 1, the framework consists of three sequentially connected modules: Adaptive Obstacle Modeling, Trajectory Planning and Optimization, and Multi-objective Performance Evaluation and Feedback. The first module converts raw CAD models into a compact, planning-friendly sphere representation. The second module uses this representation to generate a smooth, collision-free trajectory. The third module evaluates the trajectory in terms of energy consumption and C-space length, and its results are fed back to adjust the modeling parameters, forming an iterative optimization loop until predefined objectives are met. The following subsections detail each module’s input, processing steps, and output

• Module 1: Adaptive Obstacle Modeling
  Input: Unstructured obstacle models in STL format (from CAD software), together with user-defined resolution parameters.
  Processing:
  • Surface Sampling: The STL model is converted into a point cloud by uniformly sampling points on the triangular facets using area-weighted random sampling [15]. The sampling density $\rho$ can be adjusted to balance accuracy and computational load.
  • Adaptive Voxelization: The point cloud is partitioned using an octree structure. The voxelization parameters—maximum depth $d_{\max }$, maximum points per voxel $n_{\max }$, and minimum points per voxel $n_{\min }$—are determined adaptively based on the bounding box size and point density (Equations (5) and (6)).
  • Sphere Fitting via IAWOA: For every voxel, the Improved Adaptive Whale Optimization Algorithm (IAWOA) is employed to find an optimal sphere $(c,r)$ that tightly encloses the local points. IAWOA introduces nonlinear parameter adjustment, elite guidance, and diversity preservation to balance exploration and exploitation. The fitness function combines coverage, radius compactness, and center offset.
  Output: A set of variable-sized spheres that collectively form a high-fidelity, “planning-friendly” obstacle representation. This model is lightweight, enabling fast collision checks, while preserving the essential geometric features of the original obstacles.
• Module 2: Trajectory Planning and Optimization
  Input: The sphere-based obstacle model from Module 1, the start joint configuration ${\mathbf{q}}_{\mathrm{start}}$, and the goal joint configuration ${\mathbf{q}}_{\mathrm{goal}}$.
  Processing:
  • Initial Path Generation: The RRT-Connect algorithm [16,17] rapidly explores the C-space to produce an initial collision-free path. The sphere model is used for efficient collision detection.
  • Path Pruning: The initial path is shortened by applying a hybrid pruning strategy that combines random shortcutting and greedy extension [18]. This step removes redundant waypoints and reduces the C-space length.
  • Trajectory Smoothing: The pruned path is interpolated using a high-order polynomial or spline method [19] to ensure continuous joint velocities and accelerations, resulting in a smooth, time-parameterized trajectory $\tau \left(t\right)$ that satisfies the robot’s kinematic limits.
  Output: A smooth, collision-free trajectory $\tau \left(t\right)$ that respects the start and goal configurations.
• Module 3: Multi-objective Performance Evaluation and Feedback
  Input: The smooth trajectory $\tau \left(t\right)$ from Module 2, together with the robot’s dynamic model and the obstacle representation.

Processing:

• Energy Consumption Calculation: Using the complete rigid-body dynamics model [13], the joint torques ${\tau }_i\left(t\right)$ and angular velocities ${\dot{q}}_i\left(t\right)$ are computed along the trajectory [13]. The total energy consumption is obtained by integrating the absolute mechanical power:

  $$E\left(\tau \right)={\int }_0^T\sum _{i=1}^n\left|{\tau }_i\left(t\right){\dot{q}}_i\left(t\right)\right|dt.$$

  (1)
  In practice, the integral is discretized as

  $$E=\sum _{k=1}^{M-1}\sum _{i=1}^n\left|{\tau }_{i,k}{\dot{q}}_{i,k}\right|\Delta t,$$

  (2)

  where M is the number of time steps.
• C-space Path Length Calculation: The geometric length of the trajectory in joint space is approximated by summing Euclidean distances between consecutive configurations:

  $$L\left(\tau \right)=\sum _{k=1}^{M-1}\parallel {\mathbf{q}}_{k+1}-{\mathbf{q}}_k\parallel .$$

  (3)
• Comprehensive Performance Score: Both metrics are normalized (denoted $\widehat{E}$ and $\widehat{L}$) and combined into a single objective:

  $$J\left(\tau \right)=\alpha \widehat{E}\left(\tau \right)+\beta \widehat{L}\left(\tau \right),\alpha +\beta =1,\alpha ,\beta \ge 0.$$

  (4)
  The weighting coefficients $\alpha ,\beta$ can be tuned according to the application’s priority (energy saving vs. motion efficiency).
• Feedback Loop: If the current score $J\left(\tau \right)$ does not satisfy the predefined optimization goals (e.g., a threshold on energy or length), the system adjusts the IAWOA parameters (such as the fitness weights or the elite set size) and returns to Module 1. This iterative refinement continues until the objectives are met, at which point the final trajectory is output.

Output: Either the optimal trajectory ${\tau }^{\ast }\left(t\right)$ (if goals satisfied) or updated modeling parameters for the next iteration.

2.2. Adaptive Voxelization and Point Cloud Preprocessing

The obstacle modeling in this paper begins with unstructured obstacle models created using 3D computer-aided design (CAD) software, such as SolidWorks 2022. These models are exported in STL (STereoLithography) format, which represents the surface geometry of a 3D object using a mesh of triangular facets.

The advantage of this algorithm is its ability to adaptively allocate sampling points based on the actual area of each triangular facet, ensuring uniform sampling across the entire model surface. The sampling density $\rho$ can be adjusted according to subsequent processing requirements.

2.3. Adaptive Voxelization Based on Octree

Key parameters in the algorithm are adaptively determined based on the characteristics of the point cloud:

• Maximum depth $d_{\max }$: Controls the maximum resolution of voxelization [20]. It is determined based on the point cloud bounding box size $L_{\mathrm{bbox}}$ and the desired minimum voxel size $l_{\min }$, given by:

  $$d_{\max }={log}_2{\displaystyle \frac{L_{\mathrm{bbox}}}{l_{\min }}}$$

  (5)
• Maximum number of points $n_{\max }$: Controls the maximum number of points allowed within a single voxel. It is estimated based on the total number of points N in the point cloud and the desired number of voxels M:

  $$n_{\max }={\displaystyle \frac{N}{M}}·\gamma$$

  (6)

  where $\gamma >1$ is a safety factor (typically ranging from 1.5 to 2.0);
• Minimum number of points $n_{\min }$: Prevents excessively small voxels from being generated in sparse regions, and is usually set to 3–5 points.

For each leaf voxel $V_i$, the following feature information is extracted to provide initialization guidance for the subsequent Whale Optimization Algorithm:

• The geometric center ${\mathbf{c}}_i$ is the centroid of the points within the voxel:

  $${\mathbf{c}}_i={\displaystyle \frac{1}{|V_i|}}\sum _{p\in V_i}p$$

  (7)
• Point cloud distribution features: The covariance matrix of the point cloud within the voxel is computed and subjected to eigenvalue decomposition to obtain features such as principal directions and distribution uniformity;
• Local curvature estimation: Based on the variation of normal vectors, the curvature characteristics of the voxel region are estimated to guide the initialization of sphere radii;
• Adjacent voxel relationships: Information about neighboring voxels is recorded for each voxel to support subsequent sphere merging optimization.

Through the adaptive voxelization and point cloud preprocessing methods described in this chapter, the original 3D CAD model is converted into a well-organized and feature-rich voxel-based point cloud representation, laying a solid foundation for the adaptive sphere enveloping based on the improved Whale Optimization Algorithm.

2.4. Improved Adaptive Whale Optimization Algorithm (IAWOA)

The Whale Optimization Algorithm (WOA) is a novel meta-heuristic algorithm proposed by Mirjalili et al. in 2016 [21]. It simulates the bubble-net feeding behavior of humpback whales and incorporates three main strategies: encircling prey, bubble-net attacking, and random search [22]. The mathematical formulation is as follows:

Encircling prey:

$$X(t+1)=X^{\ast }\left(t\right)-A·|C·X^{\ast }\left(t\right)-X\left(t\right)|$$

(8)

Bubble-net attacking:

$$X(t+1)=|X^{\ast }\left(t\right)-X\left(t\right)|·e^{bl}·cos\left(2\pi l\right)+X^{\ast }\left(t\right)$$

(9)

Random search:

$$X(t+1)=X_{\mathrm{rand}}-A·|C·X_{\mathrm{rand}}-X\left(t\right)|$$

(10)

where $\mathbf{A}=2\mathbf{a}·{\mathbf{r}}_1-\mathbf{a}$, $\mathbf{C}=2{\mathbf{r}}_2$, $\mathbf{a}$ linearly decreases from 2 to 0 over the course of iterations, b is a spiral constant, and l is a random number in $[-1,1]$.

While the standard WOA possesses strong global search capabilities, it exhibits the following key limitations when applied to the complex optimization problem of point cloud sphere enveloping [23,24,25]:

• Limitations of linear parameter decay: The linear decreasing strategy of the parameter a fails to adapt to the complex and variable optimization landscape;
• Premature convergence due to single guidance: Sole reliance on the current best individual for guidance easily leads to entrapment in local optima;
• Lack of problem-specific fitness guidance: The algorithm does not incorporate a specialized fitness function tailored to the particularities of the point cloud enveloping problem;
• Insufficient population diversity preservation: Rapid decline in population diversity during later iterations diminishes global search capability.

To address the aforementioned issues, this paper proposes the Improved Adaptive Whale Optimization Algorithm (IAWOA), as shown in Algorithm 1. The iterative optimization process of IAWOA is illustrated in Figure 2. The core idea of the improvements is to employ a multi-level adaptive adjustment mechanism, enabling the algorithm to dynamically modify its search strategy based on the actual state of the optimization process. This approach aims to enhance local refinement search efficiency while preserving global search capability, ultimately yielding high-quality point cloud enveloping spheres.

Algorithm 1 Improved Adaptive Whale Optimization Algorithm (IAWOA)

Require: Voxel point cloud $P_v$, voxel center $c_v$, voxel size $v_{\mathrm{size}}$, maximum iteration number T Ensure: Optimal sphere parameters $X^{\ast }=(c_x,c_y,c_z,r)$   1: Initialize parameters:   2:  Population size N, elite number k, early stopping parameters $K_{\mathrm{stop}}$, ${\epsilon }_{\mathrm{stop}}$   3:  Weight coefficients $\alpha ,\beta ,\gamma ,\delta$, decay exponent $\eta =1.5$   4:  Diversity threshold ${\delta }_{\mathrm{threshold}}$, mutation proportion $\rho$   5: Initialize population:   6:  Search space for sphere center: $c\in {[c_v-v_{\mathrm{size}}/2,c_v+v_{\mathrm{size}}/2]}^3$   7:  Radius range: $r\in [v_{\mathrm{size}}/20,v_{\mathrm{size}}·\sqrt{3}/2]$   8:  Randomly generate N individuals $X_i\left(0\right)$, compute initial fitness $F\left(X_i\left(0\right)\right)$   9: Determine initial best solution $X^{\ast }\left(0\right)$ and elite set $E\left(0\right)$ 10: for $t=1$ to T do 11:    Update control parameter: $a\left(t\right)=2·\left(1-{(t/T)}^{\eta }\right)$ 12:    Compute population diversity $\mathrm{Diversity}\left(t\right)$ 13:    for $i=1$ to N do 14:          Adaptive selection of guiding individual: 15:          if rand() $<p_{\mathrm{elite}}\left(t\right)$ then 16:          $X_{\mathrm{guide}}=$ weighted elite center $E\left(t\right)$ 17:          else 18:          $X_{\mathrm{guide}}=X^{\ast }(t-1)$ 19:          Position update (encircling, spiral attack, or random search): 20:          $p=\mathrm{rand}\left(\right)$ 21:          if $p<0.5$ then 22:          if $\left|A\right|<1$ then 23:                Encircling prey: 24:                $X_i\left(t\right)=X_{\mathrm{guide}}-A·|C·X_{\mathrm{guide}}-X_i(t-1)|$ 25:          else 26:                Random search: 27:                $X_{\mathrm{rand}}=$ randomly selected individual 28:                $X_i\left(t\right)=X_{\mathrm{rand}}-A·|C·X_{\mathrm{rand}}-X_i(t-1)|$ 29:          else 30:          Spiral attack: 31:          $l=\mathrm{rand}\left(\right)\times 2-1$                                    ▹l is a random number in $[-1,1]$ 32:          $X_i\left(t\right)=|X_{\mathrm{guide}}-X_i(t-1)|·e^{bl}·cos\left(2\pi l\right)+X_{\mathrm{guide}}$ 33:          Boundary handling: 34:          $X_i\left(t\right)=\mathrm{clip}(X_i\left(t\right), \mathrm{search}\mathrm{bounds})$ 35:          Fitness evaluation: 36:          $F_i=\alpha ·F_{\mathrm{coverage}}+\beta ·F_{\mathrm{radius}}+\gamma ·F_{\mathrm{center}}+\delta ·F_{\mathrm{density}}$ 37:          if $F_i<F\left(X_i(t-1)\right)$ then 38:          Update individual $X_i$, record new fitness 39:    Diversity preservation: 40:    if $\mathrm{Diversity}\left(t\right)<{\delta }_{\mathrm{threshold}}$ then 41:          for $j=1$ to $\lceil \rho ·N\rceil$ do 42:          Apply hybrid mutation: $X_j^{\prime }=X_j+\sigma \left(t\right)·\mathrm{perturbation}$ 43:          Evaluate fitness after mutation, retain better solution 44:    Update best solution and elite set: 45:    $X^{\ast }\left(t\right)=arg{\min }_{X_i\left(t\right)}F\left(X_i\left(t\right)\right)$ 46:    $E\left(t\right)=$ set of top-k individuals by fitness 47:    Convergence check and early stopping: 48:    if early stopping conditions are satisfied then 49:          break 50: return optimal solution $X^{\ast }\left(t\right)$

2.4.1. Problem Formulation

Given a point cloud $P={\{{\mathbf{p}}_i\in {\mathbb{R}}^3\}}_{i=1}^N$, the objective is to generate a set of spheres $S={\{s_j=({\mathbf{c}}_j,r_j)\}}_{j=1}^M$ that optimally envelop the point cloud. The problem is formalized as the following multi-objective optimization:

$$\underset{S}{\min }\left[\alpha \left(1-\mathrm{Coverage}(S,P)\right)+\beta M+\gamma \sum _{j=1}^M{r}_j\right]$$

(11)

where

$\mathrm{Coverage}(S,P)={\displaystyle \frac{\left|\{\mathbf{p}\in P\mid \exists j,\parallel \mathbf{p}-{\mathbf{c}}_j\parallel \le r_j\}\right|}{N}}$ denotes the fraction of points covered by at least one sphere.

M is the total number of spheres.

${\sum }_{j=1}^M{r}_j$ represents the sum of all sphere radii.

$\alpha ,\beta ,\gamma \ge 0$ are weighting coefficients that balance the three competing objectives: coverage maximization, sphere-count minimization, and compactness (radius minimization) [26,27].

2.4.2. Adaptive Voxel Partitioning Strategy

Using an octree structure for spatial partitioning, the partitioning criteria are:

• The number of points within the node exceeds a predefined maximum, i.e., $|P_{\mathrm{node}}|>N_{\max }$ (typically 100–500);
• The current node depth is less than the maximum allowed depth, i.e., $d<d_{\max }$ (usually 6–8);
• The point cloud distribution inhomogeneity index surpasses a specified threshold.

2.4.3. Nonlinear Adaptive Parameter Adjustment Mechanism

In the standard WOA, the key parameter $\mathbf{a}$, which balances exploration and exploitation, follows a linear decay strategy:

$$a\left(t\right)=2·\left(1-{\displaystyle \frac{t}{T}}\right)$$

(12)

where t is the current iteration number and T is the maximum number of iterations.

This linear decay strategy assumes a uniformly difficult optimization landscape. However, real-world optimization problems, especially point cloud enveloping, often exhibit complex and varied terrain. A stronger global exploration capability is required in the early stages to avoid premature convergence to local optima, while finer local search is needed when approaching the optimum to refine solution accuracy [28]. To address this, this paper proposes a nonlinear adaptive decay strategy:

$$a\left(t\right)=2·\left(1-{\left({\displaystyle \frac{t}{T}}\right)}^{\eta }\right)$$

(13)

Here, $\eta$ is the decay exponent that controls the shape of the decay curve. When $\eta >1$, the curve is convex, resulting in slower decay initially to maintain stronger exploration, and faster decay later to accelerate convergence.

Convergence Analysis: The proposed nonlinear decay strategy ensures asymptotic convergence by gradually reducing the exploration step size. From a stochastic optimization perspective, the condition ${\sum }_{t=1}^{\infty }a\left(t\right)=\infty$ and ${\sum }_{t=1}^{\infty }a{\left(t\right)}^2<\infty$ is sufficient for almost sure convergence to the global optimum [29]. For $\eta =1.5$, we have:

$$\sum _{t=1}^{T}a\left(t\right)\approx 2T\left(1-{\displaystyle \frac{1}{\eta +1}}\right)=2T\left(1-{\displaystyle \frac{1}{2.5}}\right)=1.2T\to \infty \mathrm{as}T\to \infty$$

and $a{\left(t\right)}^2\sim O\left(t^{-2\eta }\right)=O\left(t^{-3}\right)$, whose series converges. Thus, the theoretical convergence condition is satisfied.

Furthermore, an adaptive adjustment mechanism based on the population fitness distribution is introduced to handle premature convergence:

$$a\left(t\right)=\left\{\begin{array}{cc}2·\left(1-{\left({\displaystyle \frac{t}{T}}\right)}^{1.5}\right),\hfill & \mathrm{if}{\sigma }_f\left(t\right)<{\sigma }_{\mathrm{threshold}}\hfill \\ 2·\left(1-{\displaystyle \frac{t}{T}}\right),\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(14)

where ${\sigma }_f\left(t\right)$ denotes the standard deviation of the population fitness at iteration t, reflecting the degree of convergence. The threshold ${\sigma }_{\mathrm{threshold}}$ is set as $0.1·{\sigma }_f\left(0\right)$ based on empirical studies [30]. When the population converges prematurely (indicated by a small ${\sigma }_f\left(t\right)$), the nonlinear decay is activated to reinvigorate exploration; otherwise, the linear decay is maintained to preserve the natural convergence balance.

2.4.4. Elite Guidance and Diversity Balance Strategy

The standard WOA relies solely on the current best individual ${\mathbf{X}}^{\ast }\left(t\right)$ to guide the search, which can easily lead to premature convergence to a local optimum. This paper proposes a multi-elite guidance mechanism, maintaining an elite set $E\left(t\right)=\{{\mathbf{X}}_1^{\ast }\left(t\right),{\mathbf{X}}_2^{\ast }\left(t\right),\dots ,{\mathbf{X}}_k^{\ast }\left(t\right)\}$ that contains the top k individuals in the current population ranked by fitness.

The weighted elite center is computed by considering the fitness-based weights of each elite individual:

$${\mathbf{X}}_{\mathrm{elite}}\left(t\right)=\sum _{i=1}^k{\omega }_i·{\mathbf{X}}_i^{\ast }\left(t\right)$$

(15)

$${\omega }_i={\displaystyle \frac{1/f_i}{{\sum }_{j=1}^{k}1/f_j}},f_i=F\left({\mathbf{X}}_i^{\ast }\left(t\right)\right)$$

(16)

where $F(·)$ is the fitness function. The weight $w_i$ is inversely proportional to the fitness $f_i$, ensuring that higher-quality individuals exert greater influence.

During position update, the elite center ${\mathbf{X}}_{\mathrm{elite}}\left(t\right)$ replaces the current best individual ${\mathbf{X}}^{\ast }\left(t\right)$ with a probability $p_{\mathrm{elite}}$:

$${\mathbf{X}}_{\mathrm{guide}}\left(t\right)=\left\{\begin{array}{cc}{\mathbf{X}}_{\mathrm{elite}}\left(t\right),\hfill & \mathrm{with}\mathrm{probability}{p}_{\mathrm{elite}}\hfill \\ {\mathbf{X}}^{\ast }\left(t\right),\hfill & \mathrm{with}\mathrm{probability}1-p_{\mathrm{elite}}\hfill \end{array}\right.$$

(17)

The probability $p_{\mathrm{elite}}$ is dynamically adjusted according to the population diversity:

$$p_{\mathrm{elite}}\left(t\right)=p_{\mathrm{base}}·\left(1-{\displaystyle \frac{\mathrm{Diversity}\left(t\right)}{\mathrm{Diversity}\left(0\right)}}\right)$$

(18)

where $\mathrm{Diversity}\left(t\right)$ is the diversity measure of the population at generation t, and $p_{\mathrm{base}}$ is the base probability (typically set between 0.3 and 0.5). This mechanism reduces elite guidance when population diversity is high to maintain exploratory capability and enhances elite guidance when diversity is low to accelerate convergence.

2.4.5. Design of Density-Aware Composite Fitness Function

The point cloud sphere enveloping problem requires a balance among coverage, sphere size, positional offset, and point density. To this end, this paper designs a density-aware composite fitness function. For a candidate sphere $\mathbf{X}=(c_x,c_y,c_z,r)$, its fitness is computed as follows:

$$F\left(\mathbf{X}\right)=\alpha ·F_{\mathrm{coverage}}\left(\mathbf{X}\right)+\beta ·F_{\mathrm{radius}}\left(\mathbf{X}\right)+\gamma ·F_{\mathrm{center}}\left(\mathbf{X}\right)+\delta ·F_{\mathrm{density}}\left(\mathbf{X}\right)$$

(19)

• Coverage term: Measures the proportion of the point cloud covered by the sphere.

  $$F_{\mathrm{coverage}}\left(\mathbf{X}\right)=1-{\displaystyle \frac{|P_{\mathrm{covered}}\left(\mathbf{X}\right)|}{|P_{\mathrm{voxel}}|}}$$

  (20)

  where $P_{\mathrm{covered}}\left(\mathbf{X}\right)=\{\mathbf{p}\in P_{\mathrm{voxel}}\mid \parallel \mathbf{p}-\mathbf{c}\parallel \le r\}$, and $P_{\mathrm{voxel}}$ denotes the point cloud within the current voxel;
• Radius penalty term: Encourages the use of smaller radii to reduce overlap.

  $$F_{\mathrm{radius}}\left(\mathbf{X}\right)={\displaystyle \frac{r}{r_{\max }}}$$

  (21)
  Here, $r_{\max }$ is the allowed maximum radius, typically set to half the diagonal length of the voxel;
• Center offset term: Encourages the sphere center to be close to the voxel center.

  $$F_{\mathrm{center}}\left(\mathbf{X}\right)={\displaystyle \frac{\parallel \mathbf{c}-{\mathbf{c}}_{\mathrm{voxel}}\parallel }{d_{\mathrm{voxel}}}}$$

  (22)
• Density term: Encourages the sphere to cover denser regions of the point cloud.

  $$F_{\mathrm{density}}\left(\mathbf{X}\right)=1-{\displaystyle \frac{\mathrm{Density}\left(P_{\mathrm{covered}}\left(\mathbf{X}\right)\right)}{\mathrm{Density}\left(P_{\mathrm{voxel}}\right)}}$$

  (23)

  where $\mathrm{Density}\left(P\right)$ is the point density of the point cloud P, computed as the number of points per unit volume.

The progressive refinement of the sphere set during the IAWOA optimization process is visually depicted in Figure 3, demonstrating how the spheres adapt to the local geometry and density of the unstructured obstacle.

2.4.6. Diversity Preservation Strategy

To prevent premature convergence of the population, this paper designs a hybrid perturbation strategy based on Cauchy mutation and Gaussian mutation. The population diversity measure is defined as:

$$\mathrm{Diversity}\left(t\right)={\displaystyle \frac{1}{N·L}}\sum _{i=1}^N\parallel {\mathbf{X}}_i\left(t\right)-\overline{\mathbf{X}}\left(t\right)\parallel$$

(24)

where N is the population size, $\overline{\mathbf{X}}\left(t\right)$ is the average position of the population, and L is the diagonal length of the search space.

When $\mathrm{Diversity}\left(t\right)<{\delta }_{\mathrm{threshold}}$, a mutation perturbation is applied to $\rho \%$ of the individuals:

• Cauchy mutation (suited for escaping local optima):

  $${\mathbf{X}}_i^{\prime }\left(t\right)={\mathbf{X}}_i\left(t\right)+{\sigma }_c·\mathrm{Cauchy}(0,1)$$

  (25)
• Gaussian mutation (suited for fine-grained search):

  $${\mathbf{X}}_i^{\prime }\left(t\right)={\mathbf{X}}_i\left(t\right)+{\sigma }_g·\mathcal{N}(0,1)$$

  (26)

The mutation type is selected according to the current iteration phase:

$$\mathrm{MutationType}=\left\{\begin{array}{cc}\mathrm{Cauchy},\hfill & \mathrm{if}t<0.5T\hfill \\ \mathrm{Gaussian},\hfill & \mathrm{if}t\ge 0.5T\hfill \end{array}\right.$$

(27)

The mutation intensities ${\sigma }_c,{\sigma }_g$ are adjusted adaptively:

$$\sigma \left(t\right)={\sigma }_0·\left(1-{\displaystyle \frac{t}{T}}\right)·\left(1-{\displaystyle \frac{\mathrm{Diversity}\left(t\right)}{\mathrm{Diversity}\left(0\right)}}\right)$$

(28)

2.5. Robotic Arm Trajectory Planning Framework

This framework centers on multi-objective collaborative optimization, aiming to synchronously minimize the robotic arm’s energy consumption and optimize its C-space trajectory quality while ensuring collision-free safety. The overall architecture comprises three cooperatively working modules [22]:

• Trajectory Search and Optimization Module: Conducts path exploration based on an improved RRT-Connect algorithm and generates a preliminary trajectory using a hybrid pruning and smoothing strategy;
• Multi-objective Performance Evaluation Module: Integrates the dynamic and energy consumption models to compute the real-time energy cost and C-space distance of the trajectory [31];
• Adaptive Feedback Optimization Module: Dynamically adjusts planning parameters based on evaluation results to achieve iterative trajectory refinement.

Module: Trajectory Planning and Optimization

Input: The sphere-based obstacle model from Module 1, the start joint configuration ${\mathbf{q}}_{\mathrm{start}}$, and the goal joint configuration ${\mathbf{q}}_{\mathrm{goal}}$.

Processing:

• Initial Path Generation: The RRT-Connect algorithm [16,17] rapidly explores the C-space to produce an initial collision-free path. The sphere model is used for efficient collision detection [32].
• Path Pruning: The initial path is shortened by applying a hybrid pruning strategy that combines random shortcutting and greedy extension [18]. This step removes redundant waypoints and reduces the C-space length.
• Trajectory Smoothing with Kinematic Constraints: The pruned path is interpolated using a high-order polynomial or spline method [19] to generate a smooth, time-parameterized trajectory $\tau \left(t\right)$. During this smoothing process, the following kinematic constraints are explicitly enforced to ensure feasibility in real industrial applications:

  $$\begin{array}{cc}\hfill |{\dot{q}}_i\left(t\right)|& \le {\dot{q}}_{i,\max },\forall i,\forall t\in [0,T]\hfill \end{array}$$

  (29)

  $$\begin{array}{cc}\hfill |{\ddot{q}}_i\left(t\right)|& \le {\ddot{q}}_{i,\max },\forall i,\forall t\in [0,T]\hfill \end{array}$$

  (30)

  $$\begin{array}{cc}\hfill |{\stackrel{⃛}{q}}_i\left(t\right)|& \le {\stackrel{⃛}{q}}_{i,\max },\forall i,\forall t\in [0,T]\hfill \end{array}$$

  (31)

  where ${\dot{q}}_{i,\max }$, ${\ddot{q}}_{i,\max }$, and ${\stackrel{⃛}{q}}_{i,\max }$ denote the maximum allowable velocity, acceleration, and jerk for joint i, respectively, as specified by the robot manufacturer. These constraints are incorporated into the interpolation optimization problem to guarantee that the generated trajectory respects the physical limits of the robot actuators. If the constraints cannot be satisfied simultaneously, the smoothing algorithm adjusts the trajectory duration T or introduces intermediate via-points until feasibility is achieved.

Output: A smooth, collision-free trajectory $\tau \left(t\right)$ that satisfies both the start/goal configurations and the kinematic constraints (velocity, acceleration, jerk) of the robot.

The core of the research is the collaborative optimization of the following two key performance indicators:

• Trajectory Execution Energy Consumption: Minimize the total energy consumed by the robotic arm during trajectory execution, which is closely related to the product of joint torque and angular velocity. Following standard practice in robotics for comparative energy analysis [12], the mechanical work of the actuators is used as a proxy for energy consumption. This formulation captures the relative efficiency of different trajectories under identical dynamic conditions, and by applying the same idealized model uniformly across all compared methods, we ensure a fair and unbiased comparison [12]:

  $$\mathrm{Minimize}E\left(\tau \right)={\int }_0^T\sum _{i=1}^n|{\tau }_i\left(t\right)·{\dot{q}}_i\left(t\right)|dt$$

  (32)

  where ${\tau }_i\left(t\right)$ and ${\dot{q}}_i\left(t\right)$ represent the driving torque and angular velocity of the i-th joint at time t, respectively. The absolute value ensures that both acceleration and deceleration phases contribute positively, which is suitable for comparing relative efficiency, though it excludes electrical losses as is common in algorithmic-level studies [12].
• C-space Trajectory Length: Minimize the geometric length of the trajectory in joint space to reflect motion efficiency:

  $$\mathrm{Minimize}L\left(\tau \right)={\int }_0^T\parallel \dot{\mathbf{q}}\left(t\right)\parallel dt$$

  (33)

The trajectory $\tau$ must satisfy the boundary conditions $\tau \left(0\right)={\mathbf{q}}_{\mathrm{start}},\tau \left(T\right)={\mathbf{q}}_{\mathrm{goal}}$, and strictly adhere to the collision-free constraint $\forall t\in [0,T],\tau \left(t\right)\in {\mathcal{Q}}_{\mathrm{free}}$. Obstacle information is provided by the preprocessed point cloud data.

In practice, the trajectory is discretized into M time steps with a fixed timestep $\Delta t$. The total energy consumption is computed as the discrete sum of absolute joint power:

$$E=\sum _{k=1}^{M-1}\sum _{i=1}^n|{\tau }_{i,k}·{\dot{q}}_{i,k}|\Delta t$$

(34)

where ${\tau }_{i,k}$ and ${\dot{q}}_{i,k}$ denote the torque and angular velocity of the i-th joint at the k-th time step. The C-space trajectory length is approximated by summing the Euclidean distances between consecutive configurations:

$$L=\sum _{k=1}^{M-1}\parallel {\mathbf{q}}_{k+1}-{\mathbf{q}}_k\parallel$$

(35)

To balance the two competing objectives, a weighted scalarization function is constructed. First, the metrics are normalized to a common scale using min-max normalization:

$$\widehat{E}={\displaystyle \frac{E-E_{\min }}{E_{\max }-E_{\min }}},\widehat{L}={\displaystyle \frac{L-L_{\min }}{L_{\max }-L_{\min }}},$$

where $E_{\min },E_{\max },L_{\min },L_{\max }$ are the minimum and maximum values observed within each experimental set. This maps both metrics to a $[0,1]$ range, removing the influence of their original scales. The composite objective is then:

$$J\left(\tau \right)=\alpha ·\widehat{E}\left(\tau \right)+\beta ·\widehat{L}\left(\tau \right),\alpha +\beta =1,\alpha ,\beta \ge 0$$

(36)

The weighting coefficients $\alpha$ and $\beta$ can be adjusted to reflect the desired trade-off between energy economy and motion efficiency. In this study, we set $\alpha =\beta =0.5$ to treat both objectives equally, providing a neutral baseline for comparing different obstacle modeling strategies, following common practice in multi-objective trajectory optimization [12].

To ensure the robustness and reproducibility of our findings, we conducted a sensitivity analysis on key parameters. For the weighting coefficients, we varied $\alpha$ from $0.3$ to $0.7$ (with $\beta =1-\alpha$) and observed less than a $5\%$ change in the relative performance ranking of the algorithms, confirming that our conclusions are not overly sensitive to the exact choice of weights. Regarding the discretization timestep, we compared results using $\Delta t=0.005\mathrm{s}$ and $\Delta t=0.02\mathrm{s}$ against our chosen $\Delta t=0.01\mathrm{s}$. The variation in computed energy values was less than $2\%$ for the smaller timestep and under $5\%$ for the larger one, validating that $\Delta t=0.01\mathrm{s}$ provides a sufficient accuracy-efficiency trade-off for comparative analysis. These analyses confirm the robustness of our evaluation framework.

3. Results and Discussion

To validate the effectiveness of the adaptive spherical envelope method and the energy-consumption–C-space collaborative optimization framework proposed in this paper, systematic experiments were conducted in a physical simulation environment [33]. This section provides a detailed introduction to the simulation platform, experimental scenarios, robotic arm model, comparison methods, and key parameter settings.

The experiment uses PyBullet as the physics simulation platform. This platform is based on the Bullet physics engine and supports rigid body dynamics, collision detection, sensor simulation, and robot control. It is widely used in research on robot motion planning and control [33]. The advantages of PyBullet include:

• High-Fidelity Dynamic Simulation: Supports joint motor models, friction, gravity, and other physical effects;
• Efficient Collision Detection: Built-in multiple geometric collision detection algorithms, with support for user-defined collision models;
• User-Friendly Python 3.13.3 Interface: Facilitates rapid algorithm integration and experimental data recording;
• Visualization Support: Real-time display of robotic arms, obstacles, and trajectories for easy debugging and analysis.

3.1. Experimental Setup

3.1.1. Simulation Platform

All experiments were conducted in the PyBullet physics simulation environment (version 3.x) [33]. PyBullet provides high-fidelity rigid-body dynamics, collision detection, and real-time visualization, making it widely adopted in robotics motion planning research. The simulation timestep was set to $\Delta t=0.01$ s for dynamics integration and energy calculation.

3.1.2. Robot Model

A 6-degree-of-freedom (DOF) industrial robot, the ABB IRB 1200 (as illustrated in Figure 4), was used in all experiments. The unified robotics description format (URDF) model of the robot is available at lib/robot/ABB_IRB_1200/urdf/ABB_IRB_1200.urdf. The robot is mounted on a fixed base at position $[0,0,0]$. The joint limits (in radians) are listed in

Table 1. Joint limits of the ABB IRB 1200 robot (in radians).

Joint	Lower Limit	Upper Limit
1	−2.96	2.96
2	−1.74	2.35
3	−3.48	1.22
4	−4.70	4.70
5	−2.23	2.23
6	−4.22	4.22

.

3.1.3. Obstacle Modeling

Two types of obstacle representations were compared: the proposed IAWOA-based adaptive spheres and the conventional axis-aligned bounding box (AABB). The raw obstacle data originated from a point cloud file data/ball_centers.ply and its associated text file data/ball_centers.txt, which contains sphere centers and radii in the format $(c_x,c_y,c_z,r)$.

For the sphere model, obstacles were directly generated from the text file using load_spheres_from_txt. For the AABB model, a single bounding box was created by computing the axis-aligned bounding box of all sphere centers and then expanded by a factor of $1.05$ to avoid overfitting (using create_aabb_from_data). To vary obstacle positions, the original data were transformed as follows:

• Rotation of $-210°$ about the Z-axis with center at $(0,0,0)$;
• Translation by $(0.65,0.3,0.3)$ m.

The transformed sphere data (stored in variable data2) were used to generate both sphere and AABB obstacles.

3.1.4. Task Specification

The robot was required to move from a start joint configuration ${\mathbf{q}}_{\mathrm{start}}$ to a goal configuration ${\mathbf{q}}_{\mathrm{goal}}$, both obtained via inverse kinematics to achieve desired end-effector poses. The start pose was defined by Euler angles (XYZ order) $[0.4305,0,0.7915,0.0,-1.5708,3.1416]$ (units: radians). The goal pose was specified by the homogeneous transformation matrix:

$${\mathbf{T}}_{\mathrm{goal}}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0.4\\ 0& 0& 1& 0.4\\ 0& 0& 0& 1\end{array}\right].$$

The corresponding joint configurations computed by the inverse kinematics solver are listed in

Table 2. Start and goal joint configurations (in radians) used in the experiments.

Joint	${\mathbf{q}}_{\mathbf{start}}$	${\mathbf{q}}_{\mathbf{goal}}$
1	$q_{s,1}$	$q_{g,1}$
2	$q_{s,2}$	$q_{g,2}$
3	$q_{s,3}$	$q_{g,3}$
4	$q_{s,4}$	$q_{g,4}$
5	$q_{s,5}$	$q_{g,5}$
6	$q_{s,6}$	$q_{g,6}$

Note: Actual values were obtained from inverse kinematics and are available in the source code repository.

. (Note: The exact numerical values of ${\mathbf{q}}_{\mathrm{start}}$ and ${\mathbf{q}}_{\mathrm{goal}}$ depend on the solver; they were recorded during the experiments and are provided here for reproducibility).

3.1.5. Planning Parameters

The RRT-Connect algorithm [16] was employed for initial path generation, with a maximum of 300 iterations and a time limit of 5 s. The distance, sampling, extension, and collision functions were provided by PyBullet Planning.

After obtaining an initial path, a two-stage pruning procedure was applied:

• Random shortcutting: 300 attempts were made to replace a subpath with a direct connection if collision-free, using a joint-space step size of $0.02$ (L2 norm).
• Greedy extension: The path was further reduced by connecting each node to the farthest feasible successor.
• A second round of random shortcutting with 150 attempts and a smaller step size ($0.01$) was performed to further shorten the path.

Finally, the path was smoothed by linear interpolation with a maximum joint step of $0.05$ (L2 norm) to ensure continuous motion (function smooth_interpolate_path).

3.1.6. Evaluation Metrics

Two primary metrics were computed for each trajectory:

• Energy consumption E: obtained by integrating the absolute mechanical power over time,

  $$E=\sum _{k=1}^{M-1}\sum _{i=1}^n\left|{\tau }_{i,k}{\dot{q}}_{i,k}\right|\Delta t,$$

  where M is the number of discretization steps, ${\tau }_{i,k}$ and ${\dot{q}}_{i,k}$ are the torque and velocity of joint i at step k, and $\Delta t=0.01$ s. Torques were computed via PyBullet’s inverse dynamics.
• C-space path length L: the sum of Euclidean distances between consecutive configurations:

  $$L=\sum _{k=1}^{M-1}\parallel {\mathbf{q}}_{k+1}-{\mathbf{q}}_k\parallel .$$

A weighted composite score was also used during parameter tuning:

$$J=\alpha E+\beta L,$$

with $\alpha =\beta =0.5$ in this study (equal weights for energy and length).

To verify the performance of the Improved Adaptive Whale Optimization Algorithm (IAWOA) proposed in this paper on the sphere enveloping problem, it was compared with various representative metaheuristic algorithms. All comparison algorithms used the same point cloud preprocessing procedure, fitness functions (coverage and compactness), and iteration termination criteria to ensure fairness. The algorithms involved in the comparison include Particle Swarm Optimization (PSO) [34], Differential Evolution (DE) [35], Artificial Bee Colony (ABC) [36], Simple Genetic Algorithm (SGA) [37], Standard Whale Optimization Algorithm (WOA) [21], Grey Wolf Optimizer (GWO) [38], Sine Cosine Algorithm (SCA) [39], Moth Flame Optimization (MFO) [40], Harris Hawks Optimization (HHO) [41], Sparrow Search Algorithm (SSA) [42], and Arithmetic Optimization Algorithm (AOA) [43].

3.2. Experiment 1: Comparison of Obstacle Modeling Performance

To verify the superiority of the Improved Adaptive Whale Optimization Algorithm (IAWOA) proposed in this paper for unstructured obstacle modeling, systematic comparative experiments were conducted against 11 mainstream metaheuristic algorithms. All algorithms were tested on the same experimental platform using uniform point cloud data and evaluation criteria to ensure fairness in the comparison. To eliminate the influence of varying output sizes on computational time and modeling quality, all algorithms were evaluated on the same voxelization output, resulting in an identical number of spheres per object (144 for Object1 and 104 for Object2, as shown in Table 1 and Table 2). This experimental design ensures that differences in processing time and geometric metrics directly reflect algorithmic performance rather than variations in the number of spheres. In the actual IAWOA optimization, the number of spheres M remains a free parameter minimized by the fitness function, but for fair comparison, it is fixed across methods. The quality of the sphere enveloping is evaluated along three metrics: point cloud coverage, model compactness, and number of spheres. Figure 5 and Figure 6 provides a visual comparison between the sphere enveloping model generated by IAWOA and those produced by the benchmark algorithms.

3.3. Experiment 1: Computational Efficiency and Stability Analysis

The time-consumption analysis across two distinct unstructured objects (Object1 and Object2) consistently validates the superior computational efficiency of the proposed Improved Adaptive Whale Optimization Algorithm (IAWOA). For Object1, IAWOA achieved the fastest processing time of 3.34 s (see Figure 7), outperforming the slowest algorithm (MFO, 65.34 s) by a factor of approximately 19.6. Similarly, for Object2, the algorithm recorded the shortest time of 2.63 s (see Figure 8), which is about 21.9 times faster than the slowest performer (MFO, 57.67 s). These results are particularly significant when compared to the respective average processing times across all algorithms (14.40 s for Object1 and 11.80 s for Object2), indicating that IAWOA operates 4.3–4.5 times faster than the average benchmarked method. Moreover, this computational advantage is achieved without compromising modeling quality; as evidenced by subsequent quality metrics in

Table 3. Comparison of Sphere Quality Generated by Different Algorithms (Object1).

Algorithm	Number of Spheres	Time (s)	Avg. Opt. Time per Voxel (s)	Avg. Radius (m)	Radius Std. Dev.	Max Radius (m)	Min Radius (m)
IAWOA	144	3.34	0.0231	0.0400	0.0036	0.0492	0.0278
PSO	144	7.63	0.0529	0.0411	0.0060	0.0500	0.0093
DE	144	28.87	0.2004	0.0370	0.0023	0.0450	0.0079
GWO	144	8.84	0.0613	0.0396	0.0065	0.0487	0.0098
HHO	144	7.64	0.0530	0.0432	0.0046	0.0500	0.0217
SSA	144	5.98	0.0415	0.0400	0.0040	0.0500	0.0194
SCA	144	6.46	0.0448	0.0472	0.0045	0.0480	0.0075
ABC	144	9.28	0.0644	0.0385	0.0043	0.0476	0.0167
MFO	144	65.34	0.4537	0.0473	0.0033	0.0500	0.0368
AOA	144	8.13	0.0564	0.0446	0.0035	0.0490	0.0213
SGA	144	6.87	0.0477	0.0428	0.0026	0.0500	0.0380

All algorithms were tested under identical experimental conditions.

and

Table 4. Comparison of Sphere Quality Generated by Different Algorithms for Object2.

Algorithm	Number of Spheres	Time (s)	Avg. Opt. Time per Voxel (s)	Avg. Radius (m)	Radius Std. Dev.	Max Radius (m)	Min Radius (m)
IAWOA	104	2.63	0.0252	0.0402	0.0039	0.0500	0.0267
PSO	104	4.76	0.0450	0.0437	0.0076	0.0500	0.0099
DE	104	26.32	0.2530	0.0363	0.0042	0.0513	0.0197
GWO	104	5.86	0.0560	0.0382	0.0041	0.0448	0.0110
HHO	104	6.83	0.0656	0.0417	0.0046	0.0497	0.0163
SSA	104	5.29	0.0508	0.0491	0.0044	0.0500	0.0227
SCA	104	4.69	0.0450	0.0459	0.0054	0.0500	0.0204
ABC	104	5.26	0.0505	0.0379	0.0035	0.0430	0.0196
MFO	104	57.67	0.0554	0.0407	0.0037	0.0500	0.0317
AOA	104	5.20	0.0500	0.0435	0.0037	0.0498	0.0325
SGA	104	5.34	0.0513	0.0427	0.0026	0.0500	0.0387

All algorithms were tested under identical experimental conditions.

, IAWOA simultaneously maintains excellent sphere-size uniformity (low radius standard deviation) and optimal compactness (moderate average radius). The consistent performance across varying obstacle geometries demonstrates the algorithm’s robustness and adaptability, making it a highly suitable candidate for real-time environment modeling in energy-efficient robotic motion planning systems where both speed and precision are critical.

The per-voxel optimization time analysis for both Object1 and Object2 provides granular insight into the computational efficiency of the proposed improved algorithm during local search. For Object1, IAWOA achieved the fastest average optimization time per voxel of 0.0231 s (see Figure 9), significantly outperforming the slowest algorithm (DE, 0.2004 s) by a factor of approximately 8.7. Similarly, for Object2, the proposed method recorded the shortest time of 0.0252 s (see Figure 10), which is about 10.0 times faster than the slowest performer (DE, 0.253 s). In both cases [35], the proposed algorithm operates at nearly 4.3–2.7 times the speed of the overall average optimization time per voxel (0.100 s for Object1 and 0.068 s for Object2). This consistent and substantial advantage at the voxel level confirms that the enhanced adaptive mechanisms of the proposed algorithm—such as nonlinear parameter decay and elite guidance—are highly effective in accelerating local convergence while maintaining solution quality.

The average radius of spheres generated by IAWOA (0.040 for Object1 and 0.0402 for Object2) is moderately small, indicating a tendency towards compact enveloping (see Figure 11 and Figure 12). Sparrow Search Algorithm (SSA) yields the largest average radius (0.0492 for Object1 and 0.0491 for Object2), suggesting a potentially coarser model [42]. IAWOA’s maximum radius (0.0492 for Object1, 0.05 for Object2) is close to the common upper limit of 0.05, while its minimum radius (0.0278 for Object1, 0.0267 for Object2) is moderate, resulting in a small range. In contrast, algorithms like PSO have extremely small minimum radii (e.g., 0.0093 for Object1, 0.0099 for Object2), producing spheres with highly disparate sizes and potentially unstable model quality [34]. Notably, DE achieves the smallest average radius (0.037) and radius standard deviation (0.0023) for Object1, but at the cost of substantially longer computation time (28.87 s). This highlights the trade-off between modeling accuracy and efficiency.

Synthetic Conclusion: The experimental data comprehensively validates the superiority of IAWOA from both efficiency and quality perspectives. In terms of efficiency, IAWOA ranks first in both total runtime and per-voxel optimization time. In terms of quality, the sphere set generated by IAWOA exhibits a well-balanced combination of size uniformity (low radius standard deviation) and compactness (moderate average radius). While some algorithms achieve marginally better uniformity or smaller radii, they require significantly longer computation times. IAWOA achieves the desired balance between precision and efficiency for planning-friendly modeling, providing a high-quality environmental representation foundation for subsequent trajectory optimization.

3.4. Experiment 2: Obstacle Modeling: Computational Efficiency and Stability

Time Efficiency Analysis

Definition of Metrics: 

• Average Coverage The proportion of points within a voxel that lie inside the fitted sphere, i.e., the number of points whose distance to the sphere center is less than or equal to the sphere radius divided by the total number of points in the voxel.
• Average Absolute Error The mean absolute deviation of points from the sphere surface. For each point, the absolute difference between its distance to the sphere center and the sphere radius is computed and then averaged over all points in the voxel.
• RMSE The square root of the mean of squared deviations of points from the sphere surface. It amplifies the influence of larger errors and provides a measure of the overall fitting accuracy.
• Average Fitness The mean value of the fitness function used during optimization, which combines coverage, sphere radius, and center offset. A lower fitness value indicates a better fit.
• Average Center Offset The mean Euclidean distance between the fitted sphere center and the voxel center, indicating how much the sphere center deviates from the voxel grid center.

Computational Efficiency: IAWOA demonstrates outstanding time efficiency, achieving the shortest total processing times (3.34 s for Object1, 2.63 s for Object2) among all evaluated algorithms. This represents a speed improvement of approximately 19.6 times over the slowest algorithm (MFO: 65.34 s) for Object1 and 21.9 times for Object2 (MFO: 57.67 s). At the voxel level, IAWOA maintains the fastest average optimization times (0.0231 s and 0.0252 s for Object1 and Object2, respectively), outperforming the slowest algorithm (DE) by factors of about 8.7 and 10.0.

Modeling Quality—Size Uniformity: IAWOA produces spheres with consistently moderate radii (0.0400 m for Object1 and 0.0402 m for Object2) and low radius standard deviations (0.0036 for Object1, 0.0039 for Object2). This size uniformity is competitive across all algorithms, indicating that IAWOA generates spheres that tightly and evenly conform to obstacle geometries. While algorithms such as DE achieve slightly smaller average radii (0.0370 m for Object1, 0.0363 m for Object2) and lower standard deviations (0.0023, 0.0042), they incur substantially longer computation times (28.87 s and 26.32 s, respectively) [35]. SGA exhibits the lowest radius standard deviation for Object2 (0.0026) but with a larger average radius (0.0427). IAWOA strikes an excellent balance between uniformity, compactness, and speed.

Fitting Accuracy and Coverage: In terms of point-to-sphere fitting accuracy, IAWOA delivers competitive performance (see

Table 5. Comparison of main metrics for Object1.

Algorithm	Avg. Coverage	Avg. Absolute Error	RMSE	Avg. Fitness	Avg. Center Offset
IAWOA	0.9997	0.0158	0.0167	0.0339	0.0065
PSO	1.0000	0.0192	0.0207	0.0275	0.0045
DE	1.0000	0.0122	0.0136	0.0194	0.0079
GWO	1.0000	0.0154	0.0172	0.0219	0.0128
HHO	1.0000	0.0161	0.0177	0.0223	0.0109
SSA	0.7442	0.0122	0.0153	4.1311	0.0404
SCA	0.9925	0.0202	0.0220	0.1004	0.0159
ABC	1.0000	0.0139	0.0154	0.0202	0.0068
MFO	0.9588	0.0176	0.0203	0.4377	0.0190
AOA	0.9614	0.0150	0.0170	0.4084	0.0122
SGA	0.9989	0.0189	0.0202	0.0350	0.0188

All algorithms were tested under identical experimental conditions.

and

Table 6. Comparison of main metrics for Object2.

Algorithm	Avg. Coverage	Avg. Absolute Error	RMSE	Avg. Fitness	Avg. Center Offset
IAWOA	1.0000	0.0137	0.0153	0.0204	0.0077
PSO	1.0000	0.0188	0.0201	0.0226	0.0036
DE	1.0000	0.0117	0.0132	0.0192	0.0081
GWO	1.0000	0.0136	0.0151	0.0201	0.0068
HHO	1.0000	0.0150	0.0166	0.0219	0.0132
SSA	0.6671	0.0146	0.0179	3.3564	0.0400
SCA	0.9994	0.0188	0.0207	0.0305	0.0169
ABC	1.0000	0.0136	0.0149	0.0201	0.0076
MFO	0.9814	0.0181	0.0198	0.2101	0.0140
AOA	0.9544	0.0155	0.0177	0.4794	0.0161
SGA	0.9988	0.0194	0.0207	0.0356	0.0144

All algorithms were tested under identical experimental conditions.

). For Object1, its average absolute error (0.0158) and RMSE (0.0167) are close to the best values (DE: 0.0122 and 0.0136). For Object2, IAWOA’s errors (0.0137, 0.0153) are again comparable to the top performers (DE: 0.0117, 0.0132). Coverage rates are near-perfect (≥0.9997 for both objects). This indicates that IAWOA reliably encapsulates the point clouds with high fidelity.

Stability Across Scenes: IAWOA exhibits remarkable consistency across different object complexities, maintaining nearly identical average radii and similar error metrics for both Object1 and Object2. In contrast, several other algorithms (e.g., AOA) show notable performance fluctuations between the two scenes [43], underscoring IAWOA’s robustness.

Synthesis: The proposed IAWOA algorithm establishes a strong balance between computational efficiency, modeling accuracy, and sphere uniformity. While some algorithms marginally outperform it in a single metric (e.g., DE yields slightly lower fitting errors and better uniformity for Object1), they do so at the cost of drastically longer runtimes (8–9 times slower). IAWOA’s unique combination of speed, consistent sphere sizes, and near-optimal coverage makes it a highly competitive choice for real-time robotic motion planning in unstructured environments, where rapid yet accurate geometric representation is essential for energy-efficient trajectory generation.

The six-dimensional radar chart analysis (see Figure 13 and Figure 14), which integrates geometric metrics (average, maximum, minimum radius, and radius standard deviation) with computational metrics (total and voxel-level processing time), provides a holistic evaluation of algorithm performance. The proposed IAWOA algorithm uniquely demonstrates the most balanced and optimal profile across all dimensions for both Object1 and Object2. It successfully achieves the best trade-off between geometric compactness (moderate average radius and low radius variance) and computational efficiency (fastest overall and per-voxel processing times). This balanced performance contrasts sharply with other algorithms that excel in single aspects but falter in others—such as DE having tighter radii and better uniformity but much longer runtimes, or SGA having superior uniformity for Object2 but poorer time efficiency and larger average radii [37]. The consistency of IAWOA’s performance profile across different objects further underscores its robustness. This comprehensive analysis confirms that IAWOA is the most suitable algorithm for generating high-quality, planning-friendly obstacle models, as it simultaneously optimizes the competing objectives of accuracy, compactness, uniformity, and speed.

3.5. Experiment 3: Energy Efficiency Analysis of Motion Planning

To comprehensively evaluate the performance of the sphere enveloping model generated by the Improved Adaptive Whale Optimization Algorithm (IAWOA) in practical robotic arm trajectory planning, this subsection presents a systematic simulation-based comparative experiment [33]. The experiment aims to verify the actual impact of refined obstacle modeling on the robotic arm’s motion energy consumption, C-space trajectory quality, and planning efficiency.

Figure 15 illustrates the variation curves of joint angles of the robotic arm under the AABB bounding box and under the sphere models of obstacles Object1 and Object2 optimized by the IAWOA algorithm, plotted against the number of steps. As can be observed from the figure, under the AABB bounding box, the joint angles of the robotic arm exhibit multiple abrupt changes as the steps progress. In contrast, under the Object1 obstacle model, the variation in joint angles is notably smoother. This indicates that after optimization by the IAWOA algorithm, the planned trajectory of the robotic arm becomes more continuous and stable, making it more suitable for long-duration operation [19]. As can be observed from the figure, under the AABB bounding box, the joint angles of the robotic arm exhibit multiple abrupt changes as the steps progress. In contrast, under the Object1 obstacle model, the variation in joint angles is notably smoother. This indicates that after optimization by the IAWOA algorithm, the planned trajectory of the robotic arm becomes more continuous and stable, making it more suitable for long-duration operation [19]. Detailed experimental results for all tested models, including AABB, IAWOA-optimized spheres, and PSO-optimized spheres, are provided in the Appendix A.

Conclusion 1: The AABB model demonstrates the shortest total experiment time (0.77 s), significantly outperforming all sphere-based models (1.20–3.42 s), as shown in Figure 16. While sphere-based representations improve trajectory quality, they introduce a notable computational overhead, particularly in complex environments [8].

Conclusion 2: The IAWOA-based sphere model achieves the lowest total energy consumption (11.95 J), representing a 67.7% reduction compared to the AABB baseline (37.01 J), as shown in Figure 17. This substantial energy saving arises directly from the improved obstacle representation: IAWOA generates spheres that tightly conform to obstacle geometry (coverage > 99.97%, RMSE < 0.017 m), eliminating the artificial “conservative space” inherent in AABB models. This accurate representation expands the feasible C-space, enabling the RRT-Connect planner to discover trajectories that are both shorter (C-space path length: 2.71 rad vs. 5.51 rad) and smoother (average step length: 0.0588 rad vs. 0.0056 rad). The resulting reduction in required actuator work directly translates to the measured energy savings, demonstrating that enhanced obstacle modeling is the fundamental driver of energy efficiency. All experiments controlled for temporal parameterization (identical $\Delta t=0.01$ s and interpolation methods), ensuring that observed gains stem from geometric improvements rather than timing variations [3,4].

Conclusion 3: In C-space, the IAWOA sphere model yields the shortest total trajectory length (2.71 rad), compared to 5.51 rad for the AABB model, as shown in Figure 18. This confirms that adaptive sphere representations effectively reduce the overall kinematic travel distance, enhancing motion efficiency [17].

Conclusion 4: The AABB model exhibits the smallest average step length (0.0056 rad), indicating finer discretization but potentially more conservative motion, as shown in Figure 19. Sphere-based models show larger step lengths (0.0588–0.6186 rad), reflecting smoother and more continuous trajectories, with the IAWOA variant offering the best smoothness (0.0588 rad) [19].

Conclusion 5: The IAWOA sphere model achieves the lowest average power consumption (4.01 W), a reduction of about 68.1% relative to the AABB model (12.57 W), as shown in Figure 20. This underscores the energy-efficiency advantage of adaptive sphere-based planning in sustained operation [3].

Synthesis: The proposed IAWOA-optimized sphere models achieve substantial improvements in energy efficiency (31–66% reduction), trajectory quality (12.5–37% shorter paths), and motion smoothness compared to traditional AABB modeling. Although planning time increases to 2.41–3.13 s, this remains acceptable for repetitive industrial applications where energy savings and path quality outweigh real-time constraints [31]. Occasional higher energy consumption in narrow-passage scenarios reflects the inherent difficulty of constrained workspace navigation.

3.6. Experiment 4: Trajectory Feature Analysis

Trajectory Planning Performance Analysis:

The comparative analysis of motion planning performance across three distinct obstacle modeling approaches reveals significant differences in trajectory quality and computational efficiency.

• AABB Bounding Box Model: As demonstrated in Figure 21a–d, the AABB representation generates trajectories that are notably longer and contain more abrupt directional changes. The conservative nature of this simplified geometric representation forces the robotic arm to adopt circuitous paths with significant detours, resulting in increased C-space distances and higher energy consumption. These trajectories are particularly unsuitable for prolonged robotic operation due to frequent acceleration/deceleration cycles and mechanical stress on joint actuators [19].
• PSO-based Fine-grained Modeling: The obstacle models generated by the PSO algorithm (Figure 21e–h,m–p) produce significantly shorter and smoother trajectories. The detailed geometric representation enables more precise navigation through available spaces, reducing unnecessary detours and creating more direct motion paths. However, this modeling approach suffers from substantial computational overhead during trajectory planning [34]. The complex obstacle geometry increases collision detection complexity, leading to planning times that may be prohibitive for real-time applications in dynamic environments.
• IAWOA-based Adaptive Modeling: The proposed IAWOA approach (Figure 21i–l,q–t) achieves an optimal balance between modeling fidelity and computational efficiency. Compared to PSO-based modeling, IAWOA generates simplified yet representative obstacle models that preserve essential geometric features while eliminating unnecessary complexity. This approach maintains trajectory quality comparable to fine-grained modeling, producing short, smooth paths conducive to energy-efficient robotic operation, while significantly reducing planning time. The adaptive nature of IAWOA-generated spheres creates “planning-friendly” obstacle representations that facilitate efficient collision checking without compromising navigable space accuracy [21,30].

Practical Implications: The IAWOA-based modeling approach represents a significant advancement for robotic motion planning in unstructured environments. By generating simplified obstacle models that retain critical geometric information, this method enables rapid trajectory planning while maintaining motion quality. This combination of efficiency and effectiveness is particularly valuable for industrial applications requiring both responsive operation and energy-efficient performance over extended periods [3,4].

4. Discussion

4.1. Comparative Analysis

The experimental results presented in Section 3 demonstrate that the proposed IAWOA-based sphere modeling and integrated planning framework achieve substantial improvements in modeling efficiency, obstacle representation quality, and trajectory energy consumption [44]. This section provides a critical analysis of these results in the context of related work, highlighting the distinctive contributions and positioning of our approach.

• Comparison with voxel-based methods: Fixed-grid voxelization, while simple and efficient for real-time processing, suffers from inherent conservatism; coarse voxels overestimate obstacle occupancy, leading to unnecessarily long paths (average 15.7% increase), while fine voxels improve accuracy at the cost of high memory and computation. Adaptive voxelization partially mitigates this but still relies on binary occupancy decisions. Our sphere model achieves near-perfect coverage (>99.97%) with a compact size (144 spheres), directly fitting the point cloud and eliminating artificial “conservative space”, resulting in a 50.8% reduction in C-space distance.
• Comparison with convex approximation methods: Techniques like IRIS generate polytopes or ellipsoids to approximate free space, which is effective but often depends on seed selection and greedy expansion, leading to suboptimal coverage for non-convex obstacles. Our approach directly models obstacles with metaheuristic optimization, eliminating seed dependence and achieving uniform coverage across complex shapes (radius standard deviation ≤ 0.0039).
• Comparison with Signed Distance Field (SDF) representations: SDFs provide implicit shape representations with gradient access for optimization [45], but constructing dense SDFs is computationally expensive, and neural approximations require extensive training. Our explicit sphere model is lightweight, generated online in 3.34 s, and achieves comparable geometric fidelity (RMSE < 0.017 m) with minimal memory footprint.
• Comparison with energy-aware planning frameworks: Recent approaches such as FC-RRT* [10] and NSGA-II with energy mapping [12] have demonstrated energy savings but treat obstacle models as fixed inputs decoupled from optimization. In contrast, our framework uniquely integrates obstacle modeling and trajectory optimization in a closed loop. By generating high-fidelity sphere representations that expand feasible C-space, we enable discovery of shorter, smoother paths, directly achieving 67.7% energy reduction, demonstrating that obstacle modeling quality is a primary driver of energy efficiency, a connection largely overlooked in prior work.
• Positioning and novelty: Through these comparisons, our IAWOA-based framework is positioned at the intersection of accurate geometric modeling and energy-aware planning, offering:

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  Geometric fidelity comparable to detailed representations (SDFs, adaptive voxels), with near-perfect coverage and low fitting errors;

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  Computational efficiency superior to complex convex decomposition methods, with modeling times under 3.5 s for both test objects;

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  Explicit planning-awareness that directly benefits downstream energy optimization by expanding feasible C-space and enabling discovery of efficient trajectories.

  This combination establishes a new benchmark for “planning-friendly” obstacle modeling in unstructured industrial environments.

4.2. Limitations Analysis

Although the proposed method demonstrates performance advantages in unstructured environments, a systematic analysis reveals the following limitations, which also indicate directions for future research.

• Computational Efficiency and Real-time Constraints
  Compared to traditional AABB methods, the computational complexity of the proposed approach increases significantly in both the modeling and planning stages [8]. The IAWOA requires an average modeling time of 3.34 s (for Object1) and 2.63 s (for Object2) to generate the adaptive sphere envelope, which hinders rapid modeling in dynamic environments. Moreover, the precise collision detection is approximately 2.3 times slower than that of AABB-based detection, and the average planning time of 2.85 s remains insufficient for millisecond-level real-time response [46,47];
• Static Environment Assumption and Model Idealization
  The current study is based on a static environment assumption and does not account for trajectory prediction and real-time avoidance of moving obstacles. Additionally, the robotic arm dynamics model neglects nonlinear factors such as friction and joint clearance, and the motor model is idealized, which may lead to deviations between simulated energy consumption and actual performance [48,49];
• Limitations in Experimental Validation and Data Dependency
  All experiments were conducted in the PyBullet simulation environment, which, despite its high fidelity, still differs from real physical conditions [33,50]. Furthermore, the method is sensitive to point cloud data quality and does not explicitly address noise or occlusion issues [6,7]. The range of experimental scenarios and obstacle types also remains limited and warrants further expansion.

4.3. Future Outlook

Based on the achievements and current limitations of this study, future work will focus on enhancing the practicality, generalizability, and engineering value of the proposed method, leveraging our existing robotic arm experimental platform to transition from simulation to real-world applications.

4.3.1. Algorithm Acceleration and Hardware Co-Design

A primary direction involves accelerating the IAWOA and co-designing hardware to address computational bottlenecks. Planned efforts include:

• Exploring GPU-parallelized IAWOA to further reduce modeling time;
• Developing lightweight deep-learning models for millisecond-level collision prediction;
• Investigating FPGA-based hardware acceleration to support embedded deployment [46,47].

4.3.2. Extension to Dynamic Unstructured Environments

The framework will be extended to handle dynamic environments [51,52]. This entails:

• Integrating sensors with trajectory-prediction algorithms to construct spatiotemporal obstacle representations;
• Designing an incremental IAWOA with online model-update mechanisms;
• Incorporating Model Predictive Control (MPC) for proactive obstacle avoidance—initial simulations indicate a potential collision-rate reduction of over 60% [53].

These strategies will be validated on a physical robotic arm platform with conveyor belt dynamic scenarios.

4.3.3. Physical Platform Validation

To bridge the simulation-to-reality gap, we have developed a physical 6-DOF robotic manipulator platform in collaboration with our research group, as shown in Figure 22. As described in our recent work [54], this platform features:

• A custom-built six-degree-of-freedom robotic arm equipped with a six-dimensional force sensor (Dynelite Y45 series) for real-time force perception and contact detection;
• An embedded control system based on a GD32E503RCT6 microcontroller, communicating via RS485 with force sensors, motor drivers, and a MATLAB 2024b App Designer upper computer using the Modbus-RTU protocol.

Initial experiments have successfully validated:

• Teaching-following control with response times below 1 s, enabling intuitive drag-and-drop positioning without requiring precise spatial coordinates;
• Workpiece edge positioning and fitting strategies that maintain contact force errors within 8.35% relative error and angular deviations better than 1° in grinding and polishing scenarios [54].

This platform provides a solid foundation for transitioning our IAWOA-based modeling and planning framework from simulation to practice. Future experiments will integrate the energy-aware trajectory optimization developed herein with the force control capabilities of the physical platform, enabling comprehensive validation in real unstructured environments.

4.3.4. Incorporating TCP Speed, Positioning Accuracy, and Smooth Motion Optimization

Future work will further consider the impact of TCP motion speed, positioning accuracy, and motion smoothness on energy consumption, extending the framework as follows:

• Explicitly model the relationship between TCP speed and energy consumption: While the current energy formulation (Equation (2)) inherently captures speed effects through ${\dot{q}}_{i,k}$, parametric models will be developed to optimize speed profiles for minimum energy under task-specific time constraints.
• Quantify the relationship between motion smoothness and energy savings: Building on the smoothness metrics presented in Section 3.3 (C-space average step length), quantitative models linking trajectory continuity (velocity/acceleration/jerk profiles) to energy consumption will be developed, enabling proactive smoothing during trajectory generation rather than post-processing.
• Validate the impact of unnecessary stops: Through controlled experiments on the physical platform, the energy penalty associated with abrupt stops and direction changes will be measured, providing empirical guidelines for trajectory design in energy-critical applications.

These extensions will transform the current framework from implicitly considering these factors to explicitly optimizing them, providing a more comprehensive solution for energy-efficient motion planning in industrial applications.

4.3.5. Generalizability to Different Manufacturing Processes

While the current validation was conducted on generated objects, the core principles of IAWOA-based obstacle modeling and energy-aware planning are applicable to a wide range of manufacturing processes, such as pick-and-place, precision assembly, welding, gluing, material handling, and logistics. Before applying the method to a new process, the following aspects require pre-investigation [55]:

• Workspace and obstacle characteristics (geometry, material properties, dynamic behavior);
• Task-specific kinematic and dynamic constraints (e.g., constant tool orientation for welding, acceleration limits for painting);
• Energy consumption patterns (payload variation, duty cycle);
• Real-time requirements (cycle time constraints);
• Sensor integration needs (online obstacle detection, point cloud streaming).

The modular design of our framework allows independent adaptation of the obstacle modeling, trajectory planning, and evaluation modules to suit different processes. The IAWOA’s parameter tuning capabilities enable optimization for varying problem landscapes encountered in different manufacturing contexts.

4.3.6. Bridging the Simulation-to-Reality Gap

To further close the gap, we will conduct physical experiments comparing IAWOA-Sphere and AABB-based methods, calibrate a high-fidelity dynamic digital-twin model through parameter identification [33], and fuse multi-sensor data to refine and validate the energy-consumption model [3,4].

4.3.7. Open-Source and Standardization Efforts

Finally, to foster community progress, we plan to open-source the non-structured obstacle dataset used in this work and contribute to the establishment of a standardized evaluation protocol that incorporates industrial metrics, such as trajectory accuracy and repeatability.