Surrogate-Assisted Coordinated Optimization of Mechanism Parameters and Motion Trajectories for a Variable-Link-Length Robotic Manipulator

Abstract

Fixed-link manipulators have limited adaptability to changes in task locations and obstacle layouts, while sequential mechanism design and trajectory planning restrict their coordinated performance. This study proposes a surrogate-assisted bilevel optimization method for a four-degree-of-freedom PRRR variable-link-length manipulator. The three link lengths are treated as outer-layer mechanism variables, whereas B-spline control points and trajectory duration are optimized in the inner layer subject to joint, motion, endpoint, and collision constraints. An objective-decoupled surrogate predicts trajectory duration, path length, jerk cost, and minimum clearance, and is embedded in an adaptive reference vector-guided multi-operator multi-objective beluga whale optimization algorithm. The framework combines inverse-kinematics prescreening, surrogate evaluation, high-fidelity trajectory re-optimization, dense constraint verification, and preference-based decision-making. Blind-test, ablation, and high-fidelity verification results show that the method efficiently identifies high-quality, physically feasible mechanism–trajectory candidates. Factorial analysis of an obstacle-constrained handling task indicates that trajectory optimization primarily improves smoothness and clearance, whereas mechanism adaptation redistributes joint motion and further enhances overall trajectory quality. Physical experiments demonstrate the executability of the selected mechanism–trajectory solutions without observed cylinder collision or joint-limit activation in the tested trials. These results demonstrate that the proposed framework provides an effective approach to task-adaptive mechanism–trajectory co-optimization in constrained environments.

1. Introduction

As manufacturing systems move toward small-batch, high-mix, and flexible production, industrial robots are increasingly required to operate under changing target locations, workspace requirements, and obstacle layouts. Conventional manipulators are generally designed with fixed link dimensions and joint configurations, which restrict subsequent motion planning to the kinematic capability and feasible workspace imposed by the predetermined mechanism. When a task approaches a workspace boundary or is performed in a cluttered environment, an unfavorable fixed configuration may result in insufficient reachability, excessive joint excursion, path detours, or reduced collision-clearance margins. Reconfigurable robots and morphology–control co-design have therefore attracted increasing attention as means of adapting robot geometry or morphology to changing task requirements [1,2,3].

Recent studies have further shifted from general morphological adaptation toward task-oriented manipulator configuration and mechanism–motion co-design. Zhao et al. investigated a reconfigurable space manipulator with lockable passive telescopic links and showed how link-length variation changes inverse-kinematics solutions and workspace characteristics [4]. Sung and Joe developed a wire-driven variable-link manipulator whose workspace can be actively adjusted according to operational requirements [5]. Beyond variable-link mechanisms themselves, Vaish and Brock used task-relevant environmental constraints to structure the hardware–control co-design space for robotic manipulation [6], while Stein et al. jointly optimized actuator design and manipulator motion for application-specific task sets [7]. More recently, Zhu et al. integrated continuous structural parameter selection and motion planning for a serial manipulator within a reinforcement-learning-based co-design framework [8]. These studies demonstrate the growing importance of considering mechanism parameters together with task-level motion requirements. However, the computational treatment of continuously adjustable link lengths remains challenging when every candidate mechanism must be evaluated through a newly solved constrained trajectory-optimization problem.

Motion planning itself must simultaneously account for execution efficiency, smoothness, joint limits, velocity and acceleration constraints, endpoint accuracy, and collision avoidance. Classical studies established time-optimal manipulator motion [9] and minimum-jerk movement generation [10], while optimization-based planners such as CHOMP [11], STOMP [12], and sequential convex trajectory optimization [13] provided practical frameworks for generating smooth collision-free trajectories. Most trajectory-planning methods, however, assume that the manipulator geometry has already been fixed. For a variable-link-length manipulator, changing the link lengths alters the workspace boundary, inverse-kinematics solution distribution, joint-motion allocation, and the feasible set of the trajectory-optimization problem. Conversely, the trajectory quality achievable under a particular mechanism configuration determines whether that configuration is suitable for the task. Mechanism-parameter selection and trajectory optimization are therefore coupled through their shared kinematic and constraint environment rather than being two independent design stages.

A strict nested solution of this coupled problem requires inverse kinematics, constrained trajectory optimization, collision checking, and dense constraint verification to be repeated for a large number of candidate mechanism configurations. Surrogate-based optimization provides a means of reducing the computational burden associated with such expensive evaluations [14,15,16]. Recent research has further extended surrogate-assisted evolutionary optimization toward more systematic model-management and multi-objective frameworks. Liang et al. reviewed current surrogate-assisted evolutionary algorithms for expensive optimization and highlighted surrogate construction, model management, and expensive-evaluation allocation as central issues [17]. Yan et al. proposed a multiple-surrogate-assisted evolutionary algorithm for high-dimensional expensive multi-objective optimization, illustrating the continued development of surrogate management for complex objective spaces [18]. In robotics, Liu et al. combined surrogate modeling and multi-objective evolutionary optimization for simultaneous packing and robot motion-planning decisions, demonstrating the potential of surrogate evaluation to reduce repeated motion-planning cost [19]. These developments indicate that surrogate models are increasingly used not merely as static regressors, but as computational tools for screening and prioritizing candidates in expensive optimization workflows.

Comparison with Related Work and Research Gap

To clarify the position of the present study with respect to recent task-oriented robot design and surrogate-assisted optimization, representative related studies are compared in

Table 1. Comparison of representative mechanism–motion co-design and surrogate-assisted optimization studies.

Study	Design/Mechanism Variables	Motion Variables	Solution Method	Surrogate/Model Update	Hardware Validation
Zhao et al. [4]	Lockable passive telescopic links/manipulator configuration	IK configuration; workspace characteristics	IK and workspace analysis with optimal configuration search	None	Not reported as the main validation
Sung and Joe [5]	Wire-driven variable-link mechanism	Joint position/current control and manipulation motion	Variable-link mechanism design with hybrid control	None	Yes; object-grasping experiments
Vaish and Brock [6]	Manipulator, gripper, and hand hardware parameters	Control/motion variables	Gradient-based co-design using task-relevant constraints and a co-design Jacobian	None	Not the primary focus
Stein et al. [7]	Motor geometry/design parameters	6-DOF manipulator motion	Application-oriented motor–motion co-design with differentiable simulation/direct collocation	None	No; simulation-based validation
Zhu et al. [8]	Continuous manipulator link lengths	Continuous joint-level motion/control policy	SAC-based structure–motion co-design	None; end-to-end RL rather than surrogate assistance	Custom PRRR platform; main quantitative evaluation includes simulation/high-fidelity physics
Yan et al. [18]	Generic high-dimensional decision variables	Generic multi-objective decision space	Multiple-surrogate-assisted evolutionary optimization with adaptive diffusion mapping	Multiple surrogates with model-management mechanisms	No robotic hardware
Liu et al. [19]	Packing/layout decision variables	Robot motion-planning decisions	Surrogate-assisted multi-objective optimization	Surrogate-assisted expensive evaluation	No mechanism reconfiguration validation
This study	Task-level adjustable link lengths $L_1,{ L}_2,{ L}_3$	B-spline control points and trajectory duration T	Surrogate-assisted bilevel optimization + mechanism-conditioned high-fidelity trajectory re-optimization	Offline objective-decoupled surrogate; frozen during outer search; no online update	Yes; laboratory PRRR prototype execution

from five aspects: mechanism/design variables, motion variables, solution strategy, surrogate/model-management scheme, and hardware validation. The selected studies cover reconfigurable and variable-link manipulators, mechanism–motion co-design, and recent surrogate-assisted expensive multi-objective optimization. The comparison is intended to identify the methodological combination addressed in the present study, rather than to imply that any individual component, such as B-spline trajectory parameterization, surrogate regression, or multi-objective evolutionary optimization, is itself new.

As shown in Table 1, the representative studies address different parts of the overall problem. Reconfigurable and variable-link manipulator studies mainly focus on workspace adaptation, inverse kinematics, or hardware/control realization [4,5], whereas recent robot co-design approaches jointly optimize hardware or structural parameters with control or motion variables using gradient-based optimization, differentiable simulation, or reinforcement learning [6,7,8]. Surrogate-assisted evolutionary methods, in contrast, primarily address the computational cost of expensive optimization through surrogate construction and model-management strategies [17,18,19]. Among the representative studies summarized here, comparatively limited attention has been given to the specific combination considered in this work: task-level continuous link-length selection, mechanism-conditioned constrained trajectory re-optimization, surrogate-based outer candidate screening, and subsequent high-fidelity feasibility restoration on a physical variable-link manipulator.

Accordingly, the novelty of the present study does not lie in the individual use of B-spline trajectory parameterization, HGB regression, TOPSIS, Pareto-based multi-objective optimization, or the general concept of adjustable robot morphology. Rather, the principal contribution lies in integrating these established tools into a mechanism-conditioned bilevel workflow in which each outer candidate represents a different physical link-length configuration, its endpoint IK solutions and inner constrained trajectory are recomputed accordingly, and the optimized trajectory performance is fed back to the mechanism-level search. The objective-decoupled surrogate is used to reduce the cost of large-scale outer candidate evaluation, but it does not replace the high-fidelity inner problem: representative candidates are subsequently subjected to renewed B-spline–SLSQP trajectory optimization and dense constraint verification before final decision-making and physical execution.

Within this workflow, ARV-MOBWO constitutes a problem-oriented BWO-based outer search architecture rather than the sole source of novelty of the overall framework. Its methodological contribution is the integration of multi-operator candidate generation, adaptive operator selection, archive-reference recombination, and reference vector-based environmental/archive selection for the three-dimensional link-length search. The ablation study is used to evaluate these modifications relative to the underlying MOBWO framework. The comparison with NSGA-II and MOPSO is interpreted as an assessment of competitive search behavior on the same surrogate landscape rather than evidence of universal superiority over established multi-objective or surrogate-assisted optimizers.

To address this problem, this study develops a surrogate-assisted bilevel coordinated optimization framework for a laboratory-developed four-degree-of-freedom PRRR variable-link-length manipulator. The three adjustable link lengths are treated as task-level outer decision variables, whereas the B-spline control points and trajectory duration constitute the inner trajectory variables. Based on the research gap identified above, the main contributions are as follows. First, a bilevel formulation establishes an explicit mechanism–trajectory relationship in which the inverse-kinematics configurations and constrained trajectory are recomputed for each candidate link-length vector. Second, an objective-decoupled trajectory-performance surrogate rapidly predicts the optimized trajectory duration, Cartesian path length, jerk cost, and minimum clearance associated with candidate mechanisms, thereby reducing the number of repeated high-fidelity trajectory evaluations required during large-scale screening. Third, an adaptive reference vector-guided multi-operator multi-objective beluga whale optimization algorithm (ARV-MOBWO) is developed as a BWO-based search architecture integrating IK reachability prescreening, surrogate evaluation, nondominated archive management, and representative-candidate selection. Finally, surrogate candidates are subjected to renewed high-fidelity B-spline–SLSQP trajectory optimization, dense constraint verification, preference-based decision-making, mechanism–trajectory factor analysis, and physical prototype execution, so that surrogate screening does not replace the final feasibility assessment.

2. Problem Formulation for Coordinated Optimization

2.1. Manipulator Architecture and Constraints

The experimental platform is a laboratory-developed four-degree-of-freedom PRRR variable-link-length manipulator consisting of one vertical prismatic joint and three planar revolute joints, as shown in Figure 1. Each link between adjacent revolute joints incorporates a sliding-slot and bolt-locking mechanism, allowing continuous adjustment and locking before task execution, as illustrated in Figure 2. The link lengths remain fixed during execution and are therefore treated as task-level offline mechanism parameters rather than real-time control degrees of freedom.

It should be emphasized that the continuous outer-level search is performed entirely in the computational design space and does not imply repeated physical adjustment of the links during optimization iterations. The sliding-slot mechanism allows each link length to be set continuously within its prescribed mechanical interval. After the offline optimization is completed, only the selected link-length vector is physically adjusted and locked before task execution, and the link lengths remain unchanged throughout the motion. Therefore, the proposed framework addresses task-level offline reconfiguration rather than online morphological adaptation.

The joint vector is denoted by $q={\left[d_1,{ \theta }_2,{ \theta }_3,{ \theta }_4\right]}^{\mathrm{T}}$, and the mechanism-parameter vector by $L={\left[L_1,{ L}_2,{ L}_3\right]}^{\mathrm{T}}$. The main constraints are summarized in

Table 2. Main mechanism and motion constraints.

Parameter Category	Variable	Range	Description
Link length	$L_2$	0.20–0.25 m	Continuously adjusted and locked before a task
	$L_3$	0.20–0.25 m	Continuously adjusted and locked before a task
	$L_4$	0.25–0.30 m	Continuously adjusted and locked before a task
Joint position	$d_1$	0–0.80 m	Vertical prismatic joint
	${ \theta }_2$	−90 to 90 deg	Revolute joint
	${ \theta }_3,{ \theta }_4$	−150 to 150 deg	Revolute joints
Velocity limit	$d_1; {\theta }_2{, \theta }_3,{\theta }_4$	0.25 m/s; 1.50 rad/s	Used in planning and verification
Acceleration limit	$d_1; {\theta }_2{, \theta }_3,{\theta }_4$	0.80 m/s2; 4.00 rad/s2	Trajectory constraint
Required clearance	$d_{safe}$	0.020 m	Hard constraint for obstacle tasks

.

2.2. Outer-Level Mechanism Variables and Inner-Level Trajectory Variables

For a given motion-planning task, the task conditions are represented as

$$\tau =\left\{p_s, p_g, c, O\right\}$$

(1)

where $p_s$ and $p_g$ denote the initial and goal end-effector poses, respectively; c is the task category; and O contains the obstacle geometry and position. For obstacle-free tasks, O is an empty set. The outer-level mechanism variables are the three task-level adjustable link lengths:

$$L={\left[L_2,L_3,L_4\right]}^{\mathrm{T}} \in {\mathsf{\Omega }}_L$$

(2)

Here, L denotes a continuous task-configuration variable in the offline optimization model rather than an online actuated state.

For a task $\tau$ and a candidate link-length vector L, the initial and goal IK configurations are recomputed. If either endpoint has no IK solution satisfying the joint limits, the candidate is declared infeasible at the outer level. After the mechanism and endpoint configurations are determined, the four-dimensional joint trajectory is represented by a cubic B-spline, and the inner-level decision variables are defined as

$$\xi ={\left[P, T\right]}^{\mathrm{T}}$$

(3)

where P is the set of B-spline control points, and T is the total trajectory duration. The endpoint constraints $q\left(0\right)=q_s$ and $q\left(T\right)=q_g$ fix the boundary configurations. The internal control points determine the geometric shape of the trajectory, whereas T scales the velocity, acceleration, and jerk profiles. The inner feasible set depends on both the task and the link lengths and is constrained by joint positions, velocities, accelerations, endpoint accuracy, and collision clearance:

$${\Omega }_{\xi }\left(\mathrm{L};\mathsf{\tau }\right)=\left\{\mathsf{\xi }\hspace{0.17em}|\hspace{0.17em}\begin{array}{l}{\mathrm{q}}_{\mathrm{m}\mathrm{i}\mathrm{n}}\le \mathrm{q}\left(t\right)\le {\mathrm{q}}_{\mathrm{m}\mathrm{a}\mathrm{x}},\\ \left|\dot{\mathrm{q}}\left(t\right)\right|\le {\dot{\mathrm{q}}}_{\mathrm{m}\mathrm{a}\mathrm{x}}, \left|\ddot{\mathrm{q}}\left(t\right)\right|\le {\ddot{\mathrm{q}}}_{\mathrm{m}\mathrm{a}\mathrm{x}},\\ {\mathrm{e}}_{\mathrm{e}\mathrm{n}\mathrm{d}}\le {\mathsf{\epsilon }}_{\mathrm{e}\mathrm{n}\mathrm{d}}, d_{\mathrm{m}\mathrm{i}\mathrm{n}}\left(\mathrm{L},\mathsf{\xi };\mathsf{\tau }\right)\ge d_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}\end{array}\right\}$$

(4)

Because the cubic B-spline trajectory optimization is nonconvex, a multistart strategy is combined with sequential least-squares quadratic programming (SLSQP) [20,21]. A baseline set of control points is first generated by joint-space interpolation between the initial and goal configurations. Several perturbed initial solutions are then constructed around the baseline and optimized independently using SLSQP. The feasible trajectory with the minimum composite objective value is retained. The main optimizer settings are listed in

Table 3. Main parameters of the trajectory optimizer.

Parameter	Value
B-spline degree	3
Number of control points	8
Optimization samples	81
Final verification samples	301
Number of initializations	5
Maximum SLSQP iterations	140
Minimum trajectory duration	0.30 s
Maximum trajectory duration	8.00 s
Trajectory duration safety factor	1.08
SLSQP convergence tolerance	$1\times {10}^{-7}$
Constraint tolerance	$2\times {10}^{-5}$
Endpoint tolerance	$1\times {10}^{-7}$
Singularity regularization ${\epsilon }_{\kappa }$	$1\times {10}^{-6}$
Preferred clearance $d_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{f}}$	0.045 m

.

2.3. Bilevel Multi-Objective Optimization Model

The inner trajectory optimization is formulated as a constrained scalar optimization problem. For a given task τ and link-length vector L, the decision vector ξ contains the internal cubic B-spline control points and the total trajectory duration T, as defined in Section 2.2. The trajectory objective combines execution efficiency, Cartesian travel, joint-motion burden, dynamic smoothness, singularity avoidance, and obstacle clearance using established trajectory-performance concepts [9,10,11,12,13]:

$$J_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{j}}={\omega }_T\hat{T}+{\omega }_S\hat{S}+{\omega }_Q\hat{Q}+{\omega }_A\hat{A}+{\omega }_J\hat{J}+{\omega }_{\kappa }\hat{\kappa }+{\omega }_D\hat{D}$$

(5)

where $\hat{T}, \hat{S},\hat{Q}, \hat{A},\hat{J}, \hat{\kappa }$ and $\hat{D}$ denote the normalized trajectory duration, Cartesian path-length, joint-motion, acceleration, jerk, singularity-risk, and safety-distance cost terms, respectively. The fixed weight vector used throughout this study is

$$\left[{\omega }_T,{\omega }_S,{\omega }_Q,{\omega }_A,{\omega }_J,{\omega }_{\kappa },{\omega }_D\right]=\left[0.26, 0.18, 0.16, 0.10, 0.10, 0.12, 0.08\right]$$

(6)

The weights are nonnegative and sum to unity. They represent a fixed engineering trade-off rather than a task-dependent or theoretically unique optimal weighting scheme, and they were kept unchanged during both dataset-label generation and subsequent high-fidelity trajectory re-optimization. Therefore, differences in optimized trajectory performance arise from changes in the task conditions and mechanism parameters rather than from task-specific objective reweighting.

The individual objective terms are calculated consistently with the trajectory optimizer used for dataset generation. The Cartesian path term is the accumulated three-dimensional end-effector displacement over the sampled trajectory. The joint-motion term accumulates the Euclidean norm of successive joint increments after each joint component is normalized by its admissible joint range. The acceleration cost is obtained by trapezoidal integration of the squared joint accelerations normalized by their corresponding acceleration limits, whereas the jerk cost is obtained by trapezoidal integration of the squared joint jerks normalized by the corresponding joint ranges. The singularity-risk term is the mean reciprocal planar manipulability, where the manipulability at each trajectory sample is defined as the product of the two singular values of the 2 × 3 planar Jacobian of the three revolute joints, with a regularization constant of ${10}^{-6}$.

For the first six terms, normalization is performed relative to a baseline trajectory generated for the same task and link-length configuration. To prevent unstable division by very small baseline values, the normalization scales for duration, path length, joint motion, acceleration, jerk, and singularity risk are lower-bounded by 0.25, 0.05, 0.05, ${10}^{-4}, {10}^{-3}$, and 1.0, respectively. The baseline metrics are evaluated on the dense 301-point trajectory grid. This task- and mechanism-specific normalization places heterogeneous trajectory-performance quantities on comparable scales without using a global dataset-wide min–max transformation.

For obstacle-constrained tasks, the clearance quantity $d_{\mathrm{m}\mathrm{i}\mathrm{n}}$ is the minimum signed surface distance, over the complete sampled trajectory, between all obstacles and the composite manipulator envelope consisting of the L2–L4 link capsules, O2–O4 joint disks, and the gripper oriented bounding box. The soft clearance cost is defined by the shortfall from a preferred clearance $d_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{f}}$ = 0.045 m,

$$\hat{D}={\left[\mathrm{m}\mathrm{a}\mathrm{x}\left(0, {\displaystyle \frac{d_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{f}}-d_{\mathrm{m}\mathrm{i}\mathrm{n}}}{d_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{f}}}}\right)\right]}^2,$$

and $\hat{D}=0$ for obstacle-free tasks. This soft preference is distinct from the hard feasibility requirement $d_{\mathrm{m}\mathrm{i}\mathrm{n}}\ge d_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}=0.020 \mathrm{m}$. Consequently, a trajectory with $0.020\le d_{\mathrm{m}\mathrm{i}\mathrm{n}}<0.045 \mathrm{m}$ remains feasible but receives a nonzero clearance penalty, whereas no additional clearance cost is imposed when $d_{\mathrm{m}\mathrm{i}\mathrm{n}}\ge 0.045 \mathrm{m}$.

For numerical robustness during SLSQP search, the performance objective is augmented by a quadratic penalty on temporary violations of velocity, acceleration, endpoint-accuracy, and clearance constraints. The penalty coefficient is 200, and the normalized violations are defined relative to the corresponding velocity and acceleration limits, endpoint tolerance, and required clearance. This penalty only guides the numerical search when intermediate trial points enter infeasible regions; it does not replace the explicit constraints or the subsequent dense feasibility verification. During optimization, each candidate trajectory is evaluated at 81 samples. Five initializations are used for each inner solve, with a maximum of 140 SLSQP iterations and a convergence tolerance of $1\times {10}^{-7}$. The general constraint tolerance is $2\times {10}^{-5}$, and the endpoint tolerance is $1\times {10}^{-7}$. After optimization, the retained candidate is recomputed and verified on a 301-point dense trajectory grid.

The resulting inner-level optimization problem is therefore written as

$${\mathit{\xi }}^{\mathrm{*}}\left(\mathit{L};\mathsf{\tau }\right)\in \mathrm{a}\mathrm{r}\mathrm{g}\underset{\mathsf{\xi }\in {\Omega }_{\xi }\left(\mathrm{L};\mathsf{\tau }\right)}{\mathrm{m}\mathrm{i}\mathrm{n}}{J}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{j}}\left(\mathit{L},\mathit{\xi };\mathsf{\tau }\right)$$

(7)

where the constraint set ${\Omega }_{\xi }$ is defined in Equation (4). The superscript ^∗ indicates that the trajectory is newly optimized for the current task and candidate link-length configuration rather than inherited from another mechanism or obtained by simple interpolation.

The outer level does not directly optimize the scalarized inner objective in Equation (5). Instead, it extracts physically interpretable performance metrics from the optimized inner trajectory. Following established criteria for execution time, motion smoothness, path planning, and collision avoidance [9,10,11,12,13], together with standard Pareto-based multi-objective optimization [22,23], the outer objective vector for obstacle-free tasks is

$$\underset{\mathit{L}\in {\mathsf{\Omega }}_L}{\mathrm{m}\mathrm{i}\mathrm{n}} \mathit{F}\left(\mathit{L}\right)=\left[T^{\mathrm{*}}\left(\mathit{L}\right),S^{\mathrm{*}}\left(\mathit{L}\right),J^{\mathrm{*}}\left(\mathit{L}\right)\right]$$

(8)

whereas for obstacle-constrained tasks the minimum clearance is additionally included and expressed in minimization form:

$$\underset{\mathit{L}\in {\mathsf{\Omega }}_L}{\mathrm{m}\mathrm{i}\mathrm{n}} F\left(L\right)=\left[T^{*}\left(\mathit{L}\right),S^{*}\left(\mathit{L}\right),J^{*}\left(\mathit{L}\right),-{d_{\mathrm{m}\mathrm{i}\mathrm{n}}}^{*}\left(\mathit{L}\right)\right]$$

(9)

The individual trajectory-performance criteria and their Pareto-vector representation are established concepts and are not claimed here as new mathematical objectives. The contribution of the present formulation lies instead in evaluating these criteria from a constrained inner trajectory that is re-optimized for every candidate link-length configuration and subsequently embedding the resulting mechanism-dependent performance into the surrogate-assisted bilevel workflow. The outer nondominated solutions are therefore coordinated pairs of mechanism parameters and their corresponding optimized trajectories.

2.4. Surrogate Evaluation and High-Fidelity Trajectory Verification

A strict nested solution would invoke the B-spline-SLSQP inner optimizer for every newly generated link-length candidate. To reduce this cost, the objective-decoupled trajectory-performance surrogate model (TD-TPSM) approximates the mapping from the task conditions and candidate link lengths to the optimized inner-level trajectory performance:

$${\widehat{\mathrm{F}}}_{\theta }\left(\mathsf{\tau },\mathrm{L}\right)\approx \mathrm{F} \left(\mathrm{L},{\mathsf{\xi }}^{\mathrm{*}}\left(\mathrm{L},\mathsf{\tau }\right);\mathsf{\tau }\right)$$

(10)

The IK prescreening used before surrogate evaluation is deliberately limited to endpoint reachability. For a candidate link-length vector, it verifies whether the prescribed initial and goal end-effector poses admit inverse-kinematics solutions satisfying the joint-position limits. Passing this prescreening does not imply full trajectory feasibility: intermediate collision, velocity-limit, acceleration-limit, endpoint-accuracy, or dense trajectory constraints are not certified by IK alone. These conditions are evaluated only when representative candidates undergo renewed high-fidelity trajectory optimization and dense verification.

The overall procedure, shown in Figure 3, comprises two stages. In the first stage, ARV-MOBWO generates link-length candidates. After endpoint-IK reachability prescreening, TD-TPSM predicts their objective vectors, and surrogate Pareto candidates are obtained through nondominated sorting, reference vector environmental selection, and external archive maintenance. In the second stage, representative candidates are selected from the surrogate archive and subjected to renewed IK, multistart B-spline-SLSQP optimization, and dense verification at 301 trajectory samples. Only high-fidelity feasible link-length-trajectory-performance solutions are retained.

3. Objective-Decoupled Trajectory-Performance Surrogate Model

3.1. Dataset and Fused Input Features

The surrogate models were trained using 5000 feasible trajectory samples generated through the task–mechanism–trajectory pipeline: 1500 regular tasks, 1250 workspace-boundary tasks, 1250 cylindrical obstacle tasks, and 1000 box obstacle tasks. Representative task scenarios are shown in Figure 4. For every task and link-length configuration, the IK solutions were recomputed, and trajectory labels were generated using cubic B-splines, multistart SLSQP optimization, and composite geometric-envelope collision checking. Every retained sample satisfied the joint-position, velocity, acceleration, endpoint-accuracy, and collision-clearance constraints.

The 5000 retained samples therefore constitute a feasible response regression dataset conditioned on successful high-fidelity trajectory generation. Failed trajectory-generation attempts are not used as regression targets in the present TD-TPSM, and the surrogate is consequently not trained to classify feasible and infeasible mechanism–trajectory combinations. Instead, its outputs estimate the expected performance of an optimized feasible trajectory when such a trajectory exists within the learned data distribution. Feasibility determination remains a separate stage of the proposed workflow.

The surrogate input combines mechanism parameters, initial and goal poses, relative-motion features, periodic orientation encodings, task categories, and obstacle-environment descriptors. The mechanism features are L2, L3, and L4. Relative-motion features include Δx, Δy, Δz, planar distance, and three-dimensional distance. End-effector orientations are encoded using sine and cosine functions to avoid discontinuity near -π and π. Obstacle features include obstacle type, center position, dimensions, and distances relative to the initial and goal poses.

3.2. Objective-Decoupled Modeling and Cross-Validation

Trajectory duration, Cartesian path length, jerk cost, and minimum clearance differ substantially in scale, governing factors, and valid sample ranges. To avoid interference among heterogeneous targets in a unified multi-output regressor, TD-TPSM models the four trajectory-performance quantities using separate regression submodels. Candidate regressors include a dummy mean baseline, ridge regression, random forest, extremely randomized trees, histogram-based gradient boosting (HGB), and a multilayer perceptron [24,25,26].

To select the target-specific regressors and evaluate sample-level predictive stability, five-fold cross-validation stratified by task class was repeated using five random seeds, {11, 23, 37, 51, 73}, resulting in 25 training-validation runs for each model-target pair. The splitting unit in this procedure was an individual trajectory sample rather than a unique task identity. Therefore, task-disjoint folds were not enforced, and different link-length configurations associated with identical or closely related task descriptors could occur in both the training and validation folds. Accordingly, the cross-validation results reported in this section are interpreted as sample-level model selection and predictive stability evidence, rather than as a strict assessment of generalization to entirely unseen tasks. A separate downstream evaluation is conducted using the 20 blind-test tasks and subsequent high-fidelity trajectory re-optimization in Section 5.

Model accuracy is evaluated using the coefficient of determination ${\mathrm{R}}^2$, mean absolute error (MAE), and normalized mean absolute error (NMAE). For n ground-truth values $y_i$ and corresponding predictions $\widehat{y_i}$,

$$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}\left|y_i-\widehat{y_i}\right|,$$

and

$$\mathrm{N}\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{\mathrm{M}\mathrm{A}\mathrm{E}}{y_{\mathrm{m}\mathrm{a}\mathrm{x}}-y_{\mathrm{m}\mathrm{i}\mathrm{n}}}}$$

Here, $y_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $y_{\mathrm{m}\mathrm{i}\mathrm{n}}$ denote the maximum and minimum ground-truth values over the complete valid modeling sample pool for the corresponding prediction target. The same target-specific range is used to normalize the MAE across all validation folds. For minimum clearance, the normalization range is calculated using valid obstacle-task samples only because minimum clearance is not used as a regression target for obstacle-free tasks. The target-specific cross-validation results are summarized in

Table 4. Sample-level cross-validation results for the target-specific TD-TPSM submodels.

Target	Selected Model	Mean ${\mathbf{R}}^2$	Std. of ${\mathbf{R}}^2$	MAE	NMAE
Trajectory duration	HGB	0.7452	0.0130	0.4203 s	0.0635
Cartesian path length	HGB	0.9212	0.0144	0.0806 m	0.0238
Jerk cost	HGB	0.4432	0.0305	1.1030	0.0470
Minimum clearance	HGB	0.3784	0.0439	0.0053 m	0.0438

, and the corresponding ${\mathrm{R}}^2$ distributions are compared in Figure 5.

HGB achieved the best aggregate performance among the tested regressors under the adopted sample-level cross-validation protocol. Cartesian path length was the most predictable target, with a mean ${\mathrm{R}}^2$ of 0.9212, indicating that relative endpoint geometry, task category, obstacle geometry, and link lengths explain most of the variation in path scale. The trajectory duration model achieved a mean ${\mathrm{R}}^2$ of 0.7452 and captured a substantial portion of the nonlinear relationship between task conditions, mechanism parameters, and optimized execution time.

Jerk cost and minimum clearance were substantially more difficult to predict, with mean ${\mathrm{R}}^2$ values of 0.4432 and 0.3784, respectively. These quantities are more sensitive to local B-spline control-point adjustment, optimizer convergence, active constraints, and the instantaneous spatial relationship between the composite manipulator envelope and the obstacles. Their lower predictive accuracy is therefore explicitly regarded as a limitation of the surrogate model rather than interpreted as high-precision prediction.

In particular, the cross-validation MAE of 0.0053 m for minimum clearance is non-negligible relative to the hard collision-clearance requirement of 0.020 m. Consequently, surrogate-predicted clearance is not used as a safety certificate or as the final collision-feasibility criterion. It serves only as an approximate objective for candidate screening, ranking, and risk indication during the outer search. For every representative candidate retained for final evaluation, the endpoint IK configurations and the B-spline trajectory are recomputed using the high-fidelity optimizer, followed by dense 301-point motion-constraint and geometric collision-clearance verification. Final feasibility and safety conclusions are therefore based on the high-fidelity trajectory verification rather than on the surrogate-predicted clearance.

For consistent deployment and a uniform inference interface, HGB was selected for all four target-specific TD-TPSM submodels because it achieved the best aggregate performance among the candidate regressors under the adopted sample-level cross-validation protocol. The trained submodels are subsequently used as fixed offline surrogates during the outer mechanism search; they accelerate candidate evaluation but do not replace the high-fidelity inner trajectory optimization or final feasibility verification.

4. ARV-MOBWO for Coordinated Optimization

4.1. Algorithmic Framework

The standard beluga whale optimization (BWO) algorithm has a simple structure and effective global exploration [27], but it was originally developed for single-objective optimization and does not maintain a nondominated archive. In the three-dimensional continuous link-length space, a single candidate generation mechanism may cause local clustering, while conventional crowding-distance selection may not preserve representative trade-off directions consistently across three- and four-objective tasks. To address these limitations, an adaptive reference vector-guided multi-operator multi-objective BWO algorithm, termed ARV-MOBWO, is proposed.

After an initial population of link-length candidates is generated, endpoint-IK reachability prescreening is performed. Candidates whose prescribed start or goal pose has no joint-limit-satisfying IK solution are rejected, whereas the remaining candidates proceed to surrogate performance evaluation. This prescreening does not certify full trajectory feasibility. Candidates that pass the prescreening are evaluated by TD-TPSM for trajectory duration, path length, jerk cost, and minimum clearance. In each generation, offspring are produced by the baseline BWO operator, a current-to-pbest differential evolution operator, and an archive-reference recombination operator. Reflection-based boundary handling, IK prescreening, and surrogate evaluation are then applied. The parent and offspring populations are merged, and the next generation is formed using Pareto nondominated sorting and reference vector niching, while the external archive is updated. After the termination criterion is reached, representative candidates are selected for high-fidelity trajectory re-optimization and TOPSIS-based decision-making.

4.2. Multi-Operator Candidate Generation and Adaptive Selection

The baseline BWO operator retains migration-foraging-style global search and local exploitation, with a decreasing exploration probability to shift gradually from exploration to exploitation. The current-to-pbest differential evolution operator combines the current individual, a high-quality individual, and random difference vectors to improve convergence toward promising regions [28]. The archive-reference recombination operator selects two nondominated archive members representing different trade-off directions and generates new candidates using simulated binary crossover, a perturbation along the second archive direction, and local mutation. The external archive therefore serves not only as a repository of solutions but also as reusable search knowledge.

The three candidate generation operators are not used at fixed proportions. Their selection probabilities are updated according to whether their offspring survive environmental selection, dominate their parents, and enter the external archive. If offspring i is generated by operator k, its reward is defined as

$$R_i=I_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{v}\mathrm{i}\mathrm{v}\mathrm{e}}+0.75I_{\mathrm{d}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{e}}+1.00I_{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{v}\mathrm{e}}$$

(11)

Operator quality is updated using an exponential moving average, and the selection probabilities for the next generation are normalized subject to a minimum-probability constraint. The mechanism adaptively allocates surrogate evaluation resources according to the task type and search stage.

4.3. Reference Vector Selection, Archive Maintenance, and Representative Solution Decision

After the parent and offspring populations are merged, complete nondominated fronts are retained according to Pareto rank. If the critical front exceeds the remaining population capacity, Das–Dennis reference vector association and niche-occupancy control are used for environmental selection [29]. When the external archive exceeds its capacity, truncation is likewise based on directional representativeness and front-approximation quality. This strategy prevents the population from remaining concentrated in a few local trade-off regions and retains representative candidates emphasizing duration, path length, smoothness, and safety.

Multi-objective optimization produces a Pareto candidate set rather than a single optimum. For candidates that pass high-fidelity trajectory verification, comprehensive compromise, duration-priority, path-priority, smoothness-priority, and safety-priority decision profiles are defined. TOPSIS is used to compute the relative closeness of each candidate to the positive and negative ideal solutions [30], thereby selecting a task-adaptive link-length configuration and its one-to-one matched optimized trajectory. The preference weights used for the different decision profiles are summarized in

Table 5. TOPSIS preference weights for representative solution selection.

Task Type	Preference Profile	Duration	Path Length	Jerk Cost	Minimum Clearance
Obstacle-free	Comprehensive compromise	0.4118	0.2941	0.2941	—
	Duration priority	0.7778	0.1111	0.1111	—
	Path priority	0.1667	0.6667	0.1667	—
	Smoothness priority	0.1667	0.1667	0.6667	—
Obstacle-constrained	Comprehensive compromise	0.35	0.25	0.25	0.15
	Duration priority	0.70	0.10	0.10	0.10
	Path priority	0.15	0.60	0.15	0.10
	Smoothness priority	0.15	0.15	0.60	0.10
	Safety priority	0.10	0.10	0.10	0.70

.

These weight vectors are predefined engineering preference profiles rather than parameters learned from the optimization results. For each priority-oriented profile, a dominant weight is assigned to the preferred criterion, while nonzero weights are retained for the remaining criteria to avoid degenerating the decision stage into a single-objective selection. The comprehensive compromise profile distributes the weights more evenly to represent a balanced trade-off among the available objectives.

For obstacle-constrained tasks, the safety-priority profile assigns a weight of 0.70 to minimum clearance because collision avoidance is given dominant importance when selecting among candidates that have already satisfied all hard feasibility constraints. This preference weight should not be confused with the collision-clearance constraint itself. A minimum clearance of 0.020 m is independently enforced as a hard constraint during high-fidelity trajectory optimization and dense verification. Therefore, the TOPSIS weights only affect the ranking of already feasible high-fidelity candidates and do not modify the surrogate training, ARV-MOBWO search process, Pareto archive, or feasibility constraints.

5. Experimental Design and Results

5.1. Experimental Setup

The algorithmic experiments used 20 independent blind-test tasks that were not involved in method development or parameter tuning. The tasks covered regular, workspace-boundary, cylindrical obstacle, and box obstacle scenarios. ARV-MOBWO, standard MOBWO, NSGA-II, and MOPSO were assigned the same evaluation budget. Each run used a population of 36, 60 iterations, and an external archive capacity of 60, and each algorithm was evaluated with five independent random seeds. Solution-set quality in the surrogate objective space was measured using hypervolume (HV), inverted generational distance (IGD), generational distance (GD), and spacing. Higher HV is preferred, whereas lower IGD, GD, and spacing are preferred.

Because the same tasks and random seeds were evaluated by all compared algorithms, nonparametric paired statistical tests were used to account for the matched experimental design. The Friedman test was used as an omnibus test to assess whether multiple algorithms differed systematically on the same set of tasks. When pairwise comparison between ARV-MOBWO and an individual benchmark algorithm was required, the paired Wilcoxon signed-rank test was applied. To control the family-wise Type-I error associated with multiple comparisons, the resulting pairwise p-values were adjusted using the Holm procedure. A difference was described as statistically significant only when the Holm-adjusted value satisfied $p_{\mathrm{H}\mathrm{o}\mathrm{l}\mathrm{m}}<0.05$.

For the surrogate-space comparison, the statistical analysis uses the matched task–seed results from the 20 blind-test tasks and five random seeds. The mean values reported in the summary tables are therefore used only as descriptive measures of central tendency and are interpreted together with the paired statistical tests rather than as independent evidence of significance. For the high-fidelity stage, one median-HV surrogate run is selected for each task–algorithm combination before candidate verification; statistical comparisons of high-fidelity performance are consequently conducted across the matched blind tasks rather than by treating individual verified candidates as independent samples.

5.2. Ablation Study of ARV-MOBWO

ARV-MOBWO augments MOBWO with multi-operator candidate generation and reference vector-based selection mechanisms. To quantify the contribution of each component, variants were created by removing one mechanism at a time under identical experimental settings. HV, IGD, GD, and spacing were compared, as summarized in

Table 6. Ablation results for the main ARV-MOBWO mechanisms.

Variant	HV	IGD	GD	Spacing	Main Observation
Full ARV-MOBWO	0.393534	0.002512	0.001150	0.002857	Baseline
Without AOS	0.392917	0.002729	0.001667	0.002891	GD worsened by 44.96%
Without pDE	0.394024	0.002530	0.001221	0.002934	Weaker local approximation and uniformity
Without ARR	0.392329	0.003177	0.002120	0.002961	IGD and GD deteriorated substantially
Without RVAT	0.393773	0.002572	0.001611	0.002449	Spacing improved, but GD worsened by 40.09%
Without RVES	0.393767	0.002560	0.001464	0.002768	Weaker front approximation

. AOS denotes adaptive operator selection, pDE denotes current-to-pbest differential evolution, ARR denotes archive-reference recombination, RVAT denotes reference vector archive truncation, and RVES denotes reference vector environmental selection.

Removing ARR decreased HV by 0.306% and worsened IGD and GD by 26.473% and 84.35%, respectively, with degradation across all four metrics. ARR is therefore the key mechanism for improving front approximation. Removing AOS increased GD by 44.96%, indicating that fixed operator proportions cannot adapt effectively to different task categories and search stages. Removing reference vector archive truncation or environmental selection improved spacing but worsened GD by 40.09% and 27.30%, respectively, revealing a trade-off between geometric uniformity and front-approximation accuracy.

The mechanisms make distinct contributions to ARV-MOBWO. ARR has the strongest effect on overall performance because its removal degrades HV, IGD, GD, and spacing simultaneously. AOS mainly improves adaptability and search stability. Although pDE does not increase HV in isolation, it improves IGD, GD, and spacing, thereby supporting local exploitation and solution-set regularity. RVAT and RVES expose a trade-off between distribution uniformity and front proximity: their removal yields smaller spacing but substantially poorer GD. Thus, uniform distribution alone does not guarantee accurate approximation of the Pareto front. Because all ablation variants use the same surrogate models, population size, evaluation budget, and experimental tasks, this analysis isolates the incremental effects of the mechanisms added to the underlying MOBWO framework. The results indicate that ARR, AOS, pDE, RVAT, and RVES contribute differently to front approximation, convergence, and distribution quality under the present fixed-surrogate setting. These ablation results, rather than the cross-algorithm comparison alone, provide the primary evidence for the algorithmic contribution of ARV-MOBWO relative to standard MOBWO.

5.3. Comparison with Multi-Objective Optimizers

ARV-MOBWO, standard MOBWO [27], NSGA-II [22], and MOPSO [23] were compared on the 20 blind-test tasks under identical surrogate models and evaluation budgets. Each algorithm was independently executed using five random seeds, giving 100 matched task–seed observations per algorithm. The mean performance values are summarized in

Table 7. Comparison of multi-objective algorithms in the surrogate objective space.

Algorithm	HV	IGD	GD	Spacing	Mean Overall Rank
MOPSO	0.434073	0.003784	0.002227	0.002630	3.124
NSGA-II	0.436677	0.002724	0.001102	0.002803	1.821
Standard MOBWO	0.432724	0.004087	0.002543	0.003331	3.974
ARV-MOBWO	0.435418	0.002914	0.001074	0.003157	2.359

, while Figure 6 presents the corresponding distributions using boxplots. As described in Section 5.1, statistical conclusions are based on paired Wilcoxon signed-rank tests with Holm correction rather than on the mean values alone.

Under the common fixed-surrogate setting, ARV-MOBWO obtained better mean HV, IGD, and GD values than MOPSO. Relative to MOPSO, its mean HV increased by 0.310%, while its mean IGD and GD decreased by 22.99% and 51.78%, respectively. The paired Wilcoxon signed-rank tests with Holm correction confirmed statistically significant differences for HV ($p_{\mathrm{H}\mathrm{o}\mathrm{l}\mathrm{m}}=7.54\times {10}^{-7}$), IGD ($p_{\mathrm{H}\mathrm{o}\mathrm{l}\mathrm{m}}=3.55\times {10}^{-13}$), and GD ($p_{\mathrm{H}\mathrm{o}\mathrm{l}\mathrm{m}}=2.83\times {10}^{-17}$). In contrast, MOPSO achieved a smaller spacing value, and the corresponding difference also favored MOPSO significantly ($p_{\mathrm{H}\mathrm{o}\mathrm{l}\mathrm{m}}=5.74\times {10}^{-7}$). The comparison therefore indicates different strengths in convergence and distribution quality rather than uniform superiority of ARV-MOBWO across all indicators.

Compared with standard MOBWO, ARV-MOBWO achieved better mean values for HV, IGD, GD, and spacing, and a substantially better mean overall rank (2.359 versus 3.974). These descriptive results are consistent with the ablation study in Section 5.2, which more directly isolates the incremental contributions of the mechanisms introduced into the underlying MOBWO framework. Accordingly, the improvement over standard MOBWO is interpreted primarily through the controlled ablation evidence rather than through the cross-algorithm mean comparison alone.

ARV-MOBWO did not outperform NSGA-II overall in the surrogate objective space. NSGA-II achieved a larger mean HV (0.436677 versus 0.435418), a smaller mean IGD (0.002724 versus 0.002914), a smaller mean spacing (0.002803 versus 0.003157), and the best mean overall rank of 1.821. The Holm-adjusted pairwise tests favored NSGA-II for HV ($p_{\mathrm{H}\mathrm{o}\mathrm{l}\mathrm{m}}=1.60\times {10}^{-10}$), IGD ($p_{\mathrm{H}\mathrm{o}\mathrm{l}\mathrm{m}}=7.34\times {10}^{-4}$), and spacing ($p_{\mathrm{H}\mathrm{o}\mathrm{l}\mathrm{m}}=1.35\times {10}^{-5}$). ARV-MOBWO obtained a slightly smaller mean GD than NSGA-II (0.001074 versus 0.001102), but this difference was not statistically significant ($p_{\mathrm{H}\mathrm{o}\mathrm{l}\mathrm{m}}=0.804$). Therefore, the small numerical difference in GD should not be interpreted as a practically or statistically meaningful superiority over NSGA-II.

The distributional information in Figure 6 further complements the mean values in Table 7. The boxes and whiskers show that the quality metrics vary across blind tasks and random seeds, making it inappropriate to infer algorithmic superiority from small differences between the reported means alone. The boxplots are therefore interpreted together with the paired statistical tests. In particular, the overlap between algorithm distributions reinforces the need to distinguish statistically supported differences from descriptive mean trends.

Taken together, NSGA-II provides the strongest overall surrogate-space performance among the four algorithms, particularly in front coverage and distribution quality, whereas ARV-MOBWO ranks second overall and shows competitive convergence behavior. The purpose of the comparison is therefore not to establish ARV-MOBWO as a universally superior multi-objective optimizer. Rather, it evaluates the behavior of different outer-search algorithms on the same offline-trained and frozen surrogate landscape. The incremental algorithmic contribution of ARV-MOBWO relative to standard MOBWO is supported mainly by the ablation study, while the ability of surrogate-generated candidates to transfer to the high-fidelity trajectory space is evaluated separately in Section 5.4.

Moreover, because the surrogate models remain fixed throughout the search for all four algorithms, the present experiment compares outer multi-objective optimizers under identical approximate objective information; it does not constitute a benchmark against complete adaptive surrogate-assisted evolutionary algorithms that iteratively acquire new high-fidelity samples and update their surrogate models during optimization.

5.4. High-Fidelity Trajectory Verification and Verification-Budget Analysis

To determine whether surrogate candidates can be converted into physically feasible coordinated mechanism–trajectory solutions, the median-HV run was selected for each task and algorithm. Five representative candidates were then sampled from each nondominated archive. The initial and goal IK configurations were recomputed, the internal B-spline control points and trajectory duration were re-optimized using SLSQP, and all constraints were verified. In total, 400 high-fidelity trajectory verifications were performed, of which 392 candidates were feasible, giving an overall feasibility rate of 98%. The feasibility rates of ARV-MOBWO, MOPSO, NSGA-II, and standard MOBWO were 98%, 100%, 96%, and 98%, respectively. Most failures occurred in box obstacle tasks.

Because the high-fidelity verification records retain both the surrogate-predicted objective values and the corresponding values obtained after renewed trajectory optimization, they also provide a direct assessment of surrogate-to-high-fidelity transfer error on the blind-task candidate set. The aggregate prediction errors obtained from the archived verification records are summarized in

Table 8. Surrogate-to-high-fidelity prediction errors for the verified blind-task candidates.

Performance Metric	MAE	Median Relative Absolute Error	Interpretation
Trajectory duration	0.3724 s	6.60%	Moderate prediction error
Cartesian path length	0.04349 m	3.86%	Most stable transferred objective
Jerk cost	0.9368	25.02%	Largest relative prediction error
Minimum clearance	0.004785 m	8.71%	Used for screening only; safety requires high-fidelity verification

. These errors are distinct from the sample-level cross-validation errors in Section 3.2 because they are evaluated on candidates generated during the downstream blind-task optimization experiments and subsequently recomputed using the full trajectory optimizer.

The blind-task verification errors follow the same qualitative pattern as the cross-validation results: Cartesian path length is the most stable objective, whereas jerk is more difficult to predict because it is highly sensitive to local trajectory shape and high-order derivatives. The minimum-clearance MAE is approximately 4.8 mm in the high-fidelity verification records. Although this is slightly smaller than the 5.3 mm sample-level cross-validation MAE, it remains substantial relative to the 20 mm hard clearance requirement. Consequently, neither value is interpreted as sufficient for direct safety certification. The surrogate provides an inexpensive approximate ranking of candidate mechanisms, while the renewed trajectory optimization and geometric collision verification determine final feasibility.

The high-fidelity front quality under different verification budgets is presented in

Table 9. High-fidelity front quality under different trajectory verification budgets.

Verification Budget	Algorithm	Real HV	Real IGD	Real GD
B = 1	MOPSO	0.195634	0.603499	0.146858
	NSGA-II	0.153965	0.661414	0.265738
	MOBWO	0.235808	0.692224	0.214075
	ARV-MOBWO	0.170436	0.681895	0.227529
B = 3	MOPSO	0.338256	0.306173	0.139769
	NSGA-II	0.300595	0.332645	0.186808
	MOBWO	0.385895	0.321454	0.084140
	ARV-MOBWO	0.388491	0.240235	0.077311
B = 5	MOPSO	0.396930	0.217256	0.125202
	NSGA-II	0.399856	0.232517	0.131277
	MOBWO	0.477403	0.177277	0.039518
	ARV-MOBWO	0.481448	0.139961	0.027108

and Figure 7.

To assess whether the differences observed in the high-fidelity objective space were statistically supported, the matched blind-task results were further analyzed using the nonparametric procedure defined in Section 5.1. The results are summarized in

Table 10. Nonparametric statistical tests for the high-fidelity comparison.

Statistical Comparison	Metric	p-Value/Adjusted Result	Interpretation
Friedman omnibus test	Real HV	0.5766	Not significant
Friedman omnibus test	Real IGD	0.8254	Not significant
Friedman omnibus test	Real GD	0.1422	Not significant
Friedman omnibus test	Spacing	0.9832	Not significant
ARV-MOBWO vs. MOPSO, Wilcoxon + Holm	HV	Raw p = 0.0441 $p_{\mathrm{H}\mathrm{o}\mathrm{l}\mathrm{m}}=0.1322$	Not significant after correction
ARV-MOBWO vs. MOPSO, Wilcoxon + Holm	IGD	$p_{\mathrm{H}\mathrm{o}\mathrm{l}\mathrm{m}}=0.4949$	Not significant
ARV-MOBWO vs. MOPSO, Wilcoxon + Holm	GD	$p_{\mathrm{H}\mathrm{o}\mathrm{l}\mathrm{m}}=0.4083$	Not significant
ARV-MOBWO vs. NSGA-II/standard MOBWO	Core metrics	$\mathrm{All} p_{\mathrm{H}\mathrm{o}\mathrm{l}\mathrm{m}}\ge 0.05$	No corrected significant advantage

.

The Friedman tests did not detect statistically significant overall differences among the four algorithms for any of the high-fidelity quality metrics. Likewise, none of the pairwise comparisons involving ARV-MOBWO remained significant after Holm correction. Notably, although the uncorrected ARV-MOBWO–MOPSO HV comparison yielded p = 0.0441, the Holm-adjusted value increased to $p_{\mathrm{H}\mathrm{o}\mathrm{l}\mathrm{m}}=0.1322$. The high-fidelity differences reported below are therefore interpreted as descriptive mean trends rather than statistically demonstrated superiority.

As the number of high-fidelity candidates verified per task and algorithm increased from B = 1 to B = 5, the real-objective HV generally increased, whereas the real-objective IGD and GD generally decreased for all four algorithms. Verifying more candidates expands the high-fidelity feasible nondominated set, covers a larger number of objective trade-off directions, and improves both proximity to and coverage of the empirical reference Pareto front.

At B = 1, only the top-ranked candidate from each task-algorithm combination was re-optimized and verified. The result therefore reflects the high-fidelity transferability of a single preferred surrogate candidate and is sensitive to ranking error and surrogate bias. Standard MOBWO achieved the largest real-objective HV, whereas MOPSO achieved the smallest real-objective IGD and GD. ARV-MOBWO was not the best on any of the three metrics, indicating that its archive-level advantage cannot be observed reliably when only one candidate is verified.

At B = 3, ARV-MOBWO obtained the largest mean real-objective HV and the smallest mean real-objective IGD and GD among the four algorithms under the present verification protocol. Its mean HV was close to that of standard MOBWO, whereas its mean IGD and GD were lower than those of MOPSO and NSGA-II. These results indicate a favorable mean high-fidelity transfer trend when three candidates per task are verified; they are not interpreted as evidence of statistically established superiority over the other algorithms.

At B = 5, ARV-MOBWO again obtained the largest mean real-objective HV and the smallest mean real-objective IGD and GD under the present protocol, with standard MOBWO showing the next closest mean performance. This result suggests that the ARV-MOBWO surrogate archive contains multiple candidates with favorable high-fidelity transfer characteristics. However, the observed differences are descriptive mean trends for the present test set and verification budget and should not be generalized as statistical or universal superiority.

Overall, ARV-MOBWO showed no clear advantage under the extremely small verification budget B = 1. At B = 3 and B = 5, it exhibited favorable mean HV, IGD, and GD values after high-fidelity re-optimization. These observations indicate that, for the present test set, the surrogate archive generated by ARV-MOBWO contains several candidates that transfer effectively to the high-fidelity trajectory space when a moderate verification budget is available. They should be interpreted as problem-specific transferability evidence rather than as proof that ARV-MOBWO is generally superior to NSGA-II, MOPSO, MOBWO, or established surrogate-assisted expensive optimization algorithms.

This high-fidelity result should be distinguished from the surrogate-space comparison in Section 5.3. It does not imply that ARV-MOBWO is generally superior to NSGA-II as a multi-objective optimizer; rather, it indicates that the candidate set generated by ARV-MOBWO exhibited favorable high-fidelity transferability under the specific verification protocol and budgets considered in this study.

5.5. Coordinated Optimization for a Cylindrical Obstacle Handling Task

A pick–transport–place task was designed to further investigate the coupling between mechanism parameters and trajectories. The key task poses are listed in

Table 11. Key poses of the pick–transport–place task.

Key Pose	x (m)	y (m)	z (m)	End-Effector Orientation (deg)	Function
Home	0.725	0.000	0.428	0	Common start and end pose
Above pick	0.579	0.354	0.428	45	Safe transition before descent
Pick	0.579	0.354	0.064	45	Close gripper and grasp object
Above place	0.645	−0.195	0.428	−45	Safe transition before descent
Place	0.645	−0.195	0.338	−45	Open gripper and release object

. The median-link and optimized mechanisms used identical home, pick, place, and obstacle coordinates; only the three link lengths were changed, after which the IK configurations and trajectories were recomputed. The cylindrical obstacle was centered at (0.630, 0.100) m with a radius of 0.030 m. A required clearance of 0.020 m was imposed. The obstacle was placed near the direct path between home and the pick region to create a meaningful avoidance constraint, as shown in Figure 8.

After surrogate search, the top 15 candidates in the ARV-MOBWO archive were subjected to high-fidelity trajectory re-optimization and obstacle-clearance verification. The selected representative link lengths were L2 = 0.245657 m, L3 = 0.230277 m, and L4 = 0.275626 m. The results are listed in

Table 12. High-fidelity results for the representative cylindrical obstacle candidate.

Solution	Link Lengths (L2, L3, L4) (m)	Duration (s)	Path (m)	Jerk	Joint Motion	Acceleration Cost	Minimum Clearance (m)
Median-link mechanism—optimized trajectory	(0.225000, 0.225000, 0.275000)	28.2863	3.4880	30.5182	4.2299	2.9548	0.054647
ARV candidate—optimized trajectory	(0.245657, 0.230277, 0.275626)	29.4603	3.4633	24.7526	4.0574	2.5199	0.054406

. Relative to the median-link mechanism under the same optimized trajectory condition, the selected mechanism reduced path length by 0.71%, jerk cost by 18.89%, joint motion by 4.08%, and acceleration cost by 14.72%, while increasing duration by 4.15% and reducing minimum clearance by 0.44%. These changes represent a typical multi-objective trade-off.

A 2 × 2 mechanism–trajectory factor-combination experiment was constructed to examine the individual and interactive effects of mechanism adaptation and trajectory optimization. S0T0 denotes the median-link mechanism with the baseline trajectory, S0T1 the median-link mechanism with the optimized trajectory, S1T0 the optimized mechanism with the baseline trajectory, and S1T1 the optimized mechanism with the optimized trajectory. The inner trajectory optimization was repeated using random seeds 11, 23, 37, 51, and 73. All four combinations achieved 100% high-fidelity feasibility. The observed responses of the four factor combinations are summarized in

Table 13. Results of the 2 × 2 mechanism–trajectory factor combinations for the cylindrical obstacle task (mean ± standard deviation).

Solution	Duration (s)	Path (m)	Jerk	Joint Motion	Acceleration Cost	Minimum Clearance (m)	Joint-Limit Margin (deg)
S0T0	24.5066 ± 0.0000	3.6782 ± 0.0000	160.4096 ± 0.0000	4.2113 ± 0.0000	4.5778 ± 0.0000	0.035170 ± 0.000000	30.00 ± 0.00
S0T1	28.2863 ± 0.0003	3.4881 ± 0.0000	30.5149 ± 0.0011	4.2298 ± 0.0000	2.9550 ± 0.0001	0.054641 ± 0.000017	30.00 ± 0.00
S1T0	25.5472 ± 0.0000	3.6519 ± 0.0000	132.1096 ± 0.0000	4.0395 ± 0.0000	3.8828 ± 0.0000	0.035455 ± 0.000000	26.23 ± 0.00
S1T1	29.4555 ± 0.0071	3.4633 ± 0.0001	24.7911 ± 0.0561	4.0574 ± 0.0000	2.5215 ± 0.0027	0.054204 ± 0.000484	26.23 ± 0.00

.

To visualize the observed responses of the four cells in the 2 × 2 design, the mean metrics of S0T0, S0T1, S1T0, and S1T1 were normalized relative to S0T0, as shown in Figure 9. Trajectory duration, end-effector path length, jerk cost, normalized joint motion, and the maximum acceleration-to-limit ratio are cost-type metrics; lower normalized values indicate better performance. Minimum clearance is a benefit-type metric. To retain a common lower-is-better direction in the figure, it is normalized using the following reciprocal form:

$${\tilde{d}}_{\mathrm{m}\mathrm{i}\mathrm{n}}={\displaystyle \frac{1}{d_{\mathrm{m}\mathrm{i}\mathrm{n}}}}$$

(12)

A lower reciprocal clearance bar therefore corresponds to a larger physical clearance and a higher collision-avoidance margin. The dashed line (y = 1) represents the S0T0 baseline. Figure 9 is used as a descriptive comparison of the four observed factor combinations, whereas the baseline-referenced factor effects and their interaction are quantified separately below.

The four mechanism–trajectory combinations exhibit distinct performance patterns in Figure 9. For S0T1 and S1T1, both of which use optimized trajectories, normalized duration is greater than one, whereas path length, jerk cost, maximum acceleration-to-limit ratio, and reciprocal minimum clearance are all substantially lower than one. The inner optimizer therefore does more than time scaling: by jointly adjusting the B-spline control points and segment durations, it reshapes the end-effector avoidance path, suppresses abrupt motion and acceleration-limit utilization, and increases the actual separation between the composite manipulator envelope and the cylindrical obstacle.

At the baseline-trajectory level T0, comparing S0T0 with S1T0 shows that mechanism adaptation reduces path length, jerk cost, and joint motion, while requiring a longer trajectory duration. Even without inner-level trajectory optimization, changing the link lengths modifies the IK configurations and the allocation of motion among the joints during obstacle avoidance. However, the changes produced by mechanism adaptation alone are smaller than the smoothness and clearance improvements obtained when trajectory optimization is introduced. This result indicates that trajectory optimization is the dominant source of jerk reduction and collision-clearance enhancement in the present cylindrical obstacle task, whereas mechanism adaptation provides a complementary, metric-dependent contribution.

Both S0T1 and S1T1 maintain substantial obstacle clearance, but S1T1 further reduces end-effector path length, jerk cost, joint motion, and acceleration cost. Its trajectory duration is nevertheless longer than that of S0T1, and its actual minimum clearance is slightly smaller. Therefore, the S0T1–S1T1 comparison represents the conditional effect of mechanism adaptation when the trajectory factor is fixed at the optimized level T1. It should not be interpreted as the baseline-referenced mechanism effect used in the additive factor decomposition below. The resulting S1T1 solution reflects a multi-objective compromise among motion smoothness, joint-motion burden, safety margin, and execution time.

The error bars in Figure 9 are generally short, indicating concentrated results for S0T1 and S1T1 across the five random seeds. Although the minimum-clearance variability of S1T1 is slightly larger than that of the other metrics, its worst-case value remains well above the safety threshold. The dispersion is therefore attributable to small variations in local avoidance geometry rather than instability of trajectory feasibility.

To quantify the factor effects without mixing comparisons evaluated under different reference conditions, the four responses were further decomposed using S0T0 as a common reference. This formulation corresponds to reference-cell coding of a two-factor design, in which the trajectory effect, mechanism effect, and interaction contrast are all defined relative to the same baseline [31]. Let $M_{00}$, $M_{01}$, $M_{10}$, and $M_{11}$ denote the mean value of a given performance metric for S0T0, S0T1, S1T0, and S1T1, respectively.

For cost-type metrics, for which a smaller value indicates better performance, the baseline-referenced trajectory effect is defined as

$${\eta }_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{j}}={\displaystyle \frac{M_{00}-M_{01}}{M_{00}}}\times 100\%$$

(13)

the baseline-referenced mechanism effect as

$${\eta }_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}={\displaystyle \frac{M_{00}-M_{10}}{M_{00}}}\times 100\%$$

(14)

and the complete coordinated improvement as

$${\eta }_{\mathrm{c}\mathrm{o}}={\displaystyle \frac{M_{00}-M_{11}}{M_{00}}}\times 100\%$$

(15)

The interaction effect is then defined by the two-factor interaction contrast:

$${\eta }_{\mathrm{i}\mathrm{n}\mathrm{t}}={\displaystyle \frac{M_{01}+M_{10}-M_{00}-M_{11}}{M_{00}}}\times 100\%$$

(16)

For minimum clearance, which is a benefit-type metric, the improvement direction is reversed:

$${\eta }_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{j}}={\displaystyle \frac{M_{01}-M_{00}}{M_{00}}}\times 100\%$$

(17)

$${\eta }_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}={\displaystyle \frac{M_{10}-M_{00}}{M_{00}}}\times 100\%$$

(18)

$${\eta }_{\mathrm{c}\mathrm{o}}={\displaystyle \frac{M_{11}-M_{00}}{M_{00}}}\times 100\%$$

(19)

And

$${\eta }_{\mathrm{i}\mathrm{n}\mathrm{t}}={\displaystyle \frac{{M_{11}-M}_{01}-M_{10}+M_{00}}{M_{00}}}\times 100\%$$

(20)

With these definitions, the complete coordinated improvement is exactly decomposed as

$${\eta }_{\mathrm{c}\mathrm{o}}={\eta }_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{j}}+{\eta }_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}+{\eta }_{\mathrm{i}\mathrm{n}\mathrm{t}}$$

(21)

Thus, all three terms are evaluated using a consistent reference. A positive value denotes improvement, whereas a negative value denotes deterioration relative to S0T0. Under this sign convention, a negative interaction indicates that the combined improvement is smaller than the sum of the two baseline-referenced individual effects, whereas a positive interaction indicates an additional synergistic gain.

The resulting baseline-referenced factor effects are summarized in

Table 14. Baseline-referenced factor effects and interaction effects for the cylindrical obstacle task.

Metric	Baseline-Referenced Trajectory Effect (%)	Baseline-Referenced Mechanism Effect (%)	Complete Coordinated Improvement (%)	Interaction Effect (%)	Interpretation
Trajectory duration	−15.42	−4.25	−20.19	−0.52	Both factors increase duration; the interaction is weak
Path length	5.17	0.72	5.84	−0.04	Mainly improved by trajectory optimization; the two effects are nearly additive
Jerk cost	80.98	17.64	84.55	−14.07	Both factors improve smoothness, with substantial overlapping effects
Joint motion	−0.44	4.08	3.66	0.02	The improvement is mainly associated with mechanism adaptation
Acceleration cost	35.45	15.18	44.92	−5.71	Both factors reduce acceleration cost, with partial overlap
Minimum clearance	55.36	0.81	54.12	−2.05	Clearance improvement is dominated by trajectory optimization; the interaction offsets part of the gain

Note: The additive identity is satisfied using the unrounded values calculated from the four cell means. Minor apparent discrepancies may occur when the percentages displayed in the table are rounded to two decimal places.

.

Different performance metrics show distinct baseline-referenced factor effects, as summarized in Table 14. Relative to S0T0, introducing trajectory optimization while retaining the median-link mechanism produces improvements of 5.17%, 80.98%, 35.45%, and 55.36% in path length, jerk cost, acceleration cost, and minimum clearance, respectively. Inner-level trajectory optimization is therefore the dominant source of path reshaping, smoothness improvement, and collision-avoidance margin in the present cylindrical obstacle task. Its particularly strong effects on jerk and clearance result from changing the local trajectory geometry, B-spline control-point locations, and temporal allocation to avoid abrupt directional changes and excessive proximity to the obstacle.

The baseline-referenced mechanism effect is evaluated independently from S0T0 to S1T0. Mechanism adaptation reduces path length by 0.72%, jerk cost by 17.64%, joint motion by 4.08%, and acceleration cost by 15.18%, while slightly increasing minimum clearance by 0.81%. It also increases trajectory duration by 4.25%. These results indicate that changing the link lengths modifies the IK configurations and the allocation of motion among joints even before optimized trajectory reshaping is introduced. The mechanism factor therefore provides a meaningful but generally smaller contribution than the trajectory factor for the present obstacle-constrained task.

For completeness, the comparison between S0T1 and S1T1 can still be used to quantify the conditional effect of mechanism adaptation at the optimized trajectory level T1. This conditional comparison is not used as the mechanism term in the additive decomposition of Table 11. Under T1, mechanism adaptation further reduces path length by 0.71%, jerk cost by 18.76%, joint motion by 4.08%, and acceleration cost by 14.67%, while increasing trajectory duration by 4.13% and slightly reducing minimum clearance by 0.80%. These conditional changes show that the influence of link-length adaptation depends on the trajectory-optimization state and therefore provide an additional indication of mechanism–trajectory coupling.

For trajectory duration, the baseline-referenced trajectory effect is −15.42% and the baseline-referenced mechanism effect is −4.25%. Together with the −0.52% interaction effect, they yield the complete coordinated change of −20.19%. Neither factor minimizes execution time alone in this task. Instead, both accept a longer duration while improving other objectives, particularly jerk, acceleration, path length, and clearance. The increase in duration is therefore a multi-objective trade-off rather than evidence of optimization failure.

For joint motion, the baseline-referenced trajectory effect is −0.44%, whereas the mechanism effect is 4.08%. With an interaction effect of only 0.02%, the complete coordinated improvement is 3.66%. The near-zero interaction indicates that the effects are approximately additive for this metric. Trajectory optimization has little influence on cumulative joint excursion, whereas link-length adaptation changes the share of motion assigned to individual joints and is therefore the primary source of the reduction in joint motion.

The interaction effects for path length and joint motion are −0.04% and 0.02%, respectively, both close to zero. Thus, the effects of mechanism adaptation and trajectory optimization are approximately additive for these two metrics. The interaction effect for trajectory duration is also small (−0.52%), indicating that mechanism variation only weakly alters the time penalty associated with trajectory optimization.

In contrast, the interaction effects for jerk cost and acceleration cost are −14.07% and −5.71%, respectively. Under the present improvement-oriented sign convention, a negative interaction means that the complete coordinated gain is smaller than the sum of the two baseline-referenced individual gains. This result indicates overlapping rather than independent effects. Trajectory optimization already removes a large proportion of abrupt motion and high acceleration from the baseline trajectory, thereby reducing the remaining improvement available from mechanism adaptation. Nevertheless, the complete S1T1 solution still reduces jerk cost and acceleration cost by 84.55% and 44.92%, respectively, relative to S0T0.

Minimum clearance provides a particularly clear example of the interaction between the two factors. The baseline-referenced trajectory effect is 55.36%, and the baseline-referenced mechanism effect is 0.81%, whereas the interaction effect is −2.05%, yielding a complete coordinated improvement of 54.12%. Thus, changing the mechanism alone from S0T0 to S1T0 slightly increases the minimum clearance. However, after trajectory optimization has already established a substantially larger safety margin, the conditional S0T1–S1T1 comparison gives a −0.80% change, indicating that the optimized mechanism slightly reduces the already enlarged clearance. The change in the conditional mechanism response between T0 and T1 demonstrates that the effects of the mechanism and trajectory factors on obstacle clearance are not strictly independent.

For S1T1, the mean minimum clearance is 0.054204 m with a standard deviation of 0.000484 m. The worst value across the five seeds is 0.053338 m, leaving approximately 33.34 mm above the required 0.020 m clearance. Therefore, although the interaction redistributes part of the trajectory-derived safety margin, the final coordinated solution remains well within the prescribed collision-clearance constraint.

Taken together, Figure 9 and Table 14 show a clear but unequal division of roles between mechanism adaptation and trajectory optimization. In the present cylindrical obstacle task, trajectory optimization is the primary source of avoidance-path reshaping, jerk reduction, acceleration reduction, and clearance improvement. Mechanism adaptation provides a smaller and metric-dependent complementary effect by changing the IK configurations and joint-motion allocation; it further reduces path length, jerk cost, joint motion, and acceleration cost under the optimized trajectory condition. The nonzero interaction effects further indicate that the performance consequences of changing the mechanism depend on the trajectory-optimization state. Accordingly, the coordinated framework should not be interpreted as requiring comparable contributions from the two optimization levels. Rather, it allows trajectory optimization to operate on a task-adapted kinematic configuration and evaluates whether changing that configuration provides additional benefit for the specific task. For tasks in which the baseline mechanism already provides favorable reachability and joint configurations, the additional benefit of link-length adaptation may be modest. The mechanism layer is expected to become more relevant when task location, workspace-boundary effects, or obstacle geometry make the baseline configuration kinematically unfavorable.

Figure 10 shows the model-based geometric minimum clearance of S0T1 and S1T1 over normalized trajectory progress, calculated from the high-fidelity trajectory and the calibrated composite manipulator–obstacle geometry. These values are not direct physical clearance measurements obtained from an external sensor. Both high-fidelity clearance curves remain above the 0.020 m safety threshold over the entire trajectory. The low-clearance region occurs mainly during the avoidance segment between the home and pick region, whereas the pick descent, placement, and return motions have substantially larger calculated clearance. Although the local minimum of S1T1 is slightly lower than that of S0T1, the high-fidelity geometric model retains more than 33 mm of margin above the prescribed clearance threshold.

5.6. Physical Experiments

Figure 11 compares the absolute joint angles of S0T1 and S1T1 in the cylindrical obstacle handling task. For S1T1, θ2 is in [−63.77 degrees, −5.78 degrees], θ3 in [38.32 degrees, 69.97 degrees], and θ4 in [−64.96 degrees, 83.80 degrees]. All values satisfy the ±90 degrees limit of θ2 and the ±150 degrees limits of θ3 and θ4. The minimum joint-limit margin is 26.23 degrees. Although smaller than the 30.00 degrees margin of the median-link mechanism, it remains sufficient for execution.

The relative joint-angle curves in Figure 12 further show that both solutions return to their respective initial task configurations at the end of the motion. The optimized mechanism reduces several large joint excursions and redistributes the coordinated changes among θ2, θ3, and θ4, resulting in lower jerk and acceleration cost. Mechanism adaptation therefore changes not only the static workspace but also the division of motion among the joints for the same physical task.

The joint profiles in Figure 11 and Figure 12 are the high-fidelity reference trajectories used to construct the physical command sequences; they should not be interpreted as encoder-measured joint-tracking trajectories. Joint-level tracking errors were not independently logged during the present proof-of-concept experiments.

A fixed staged safety-validation protocol was used for the physical obstacle-constrained task. First, unloaded motion was executed without the cylinder and without grasping an object. Second, the cylinder was installed, and an unloaded obstacle-avoidance run was completed. Finally, the complete loaded pick–avoid–transport–place task was executed. This order was selected conservatively for laboratory safety and was kept fixed rather than randomized across conditions. Consequently, the physical experiment should be interpreted as a staged executability and safety demonstration rather than as a randomized comparative performance trial. The repeated loaded-task results are summarized in

Table 15. Repeated physical executability trials for the cylindrical obstacle task.

Solution	Trials	Grasp Success (%)	Placement Success (%)	Cylinder Collisions	Joint-Limit Triggers	Mean Placement Error (mm)	Total Physical Time (s)
Median-link optimized trajectory baseline (S0T1)	10	70	70	0	0	10.8	434.2
Optimized mechanism–trajectory solution (S1T1)	10	90	90	0	0	9.1	434.0

.

The two physical conditions correspond to the optimized trajectory cases used in the high-fidelity comparison: the median-link condition is S0T1, i.e., the median-link mechanism with its re-optimized trajectory, whereas the optimized condition is S1T1, i.e., the selected mechanism with its corresponding re-optimized trajectory. S0T0 was not used as the physical comparison baseline in Table 15. Both physical command sequences were derived from the corresponding high-fidelity collision-checked trajectories and converted into staged executable commands.

The total physical times of 434.2 and 434.0 s in Table 15 are wall-clock durations for the complete staged command sequences and are fundamentally different from the optimized trajectory durations reported in Table 12. The exported physical sequences contain explicit safety dwell periods, gripper opening and closing actions, vertical pick/place motions, command transmission, acknowledgement synchronization, and actuator execution time. In the archived command sequences, the prescribed dwell/wait periods alone total 388 s for each condition. Therefore, the approximately 434 s wall-clock values are not expected to reproduce the 28–29 s high-fidelity trajectory duration objective and are not used as evidence of trajectory-time improvement.

The observed grasping and placement success rates were 70% for S0T1 and 90% for S1T1. With only ten loaded trials per condition, this difference is treated only as a descriptive observation and is not used to claim a statistically significant improvement in physical task performance. The archived physical records distinguish success or failure at the grasping and placement outcome level, but they do not provide an instrumented post hoc classification of individual failure mechanisms, such as grasp misalignment, object slip, tracking-induced deviation, or other execution disturbances. Accordingly, no unsupported failure-mode percentages are introduced retrospectively.

The real-arm control program was configured with a nominal 20 Hz control-loop setting; however, execution of the present staged task was command- and acknowledgement-driven and included explicit dwell intervals, so this nominal value should not be interpreted as the effective constant trajectory-sampling rate of the physical experiment. A unified command-level speed-scaling factor, quantitative payload characterization, joint-tracking-error history, and directly instrumented physical clearance were not recorded as independent experimental variables for these proof-of-concept trials. The clearance values reported in Figure 10 are therefore high-fidelity geometric calculations rather than physical sensor measurements. For these reasons, the physical experiment is used primarily to demonstrate executable command conversion and the absence of observed cylinder collision or joint-limit activation in the tested trials, whereas quantitative trajectory-performance conclusions are based on the high-fidelity optimization and factor analysis.

Figure 13 presents representative stages of the physical cylindrical obstacle handling experiment and demonstrates that the optimized avoidance trajectory can be converted into executable manipulator commands.

6. Discussion

The factor analysis shows that mechanism adaptation and trajectory optimization play different and quantitatively unequal roles across the performance metrics. In the cylindrical obstacle task, trajectory optimization is clearly the dominant source of jerk reduction and clearance improvement. This result is reasonable because the inner optimizer directly modifies the B-spline control points and temporal parameters and therefore has direct influence on local path geometry, motion smoothness, and obstacle separation. Mechanism adaptation, by contrast, does not directly optimize the trajectory shape. Its role is to modify the endpoint IK configurations, joint-motion allocation, workspace geometry, and the mechanism-dependent feasible conditions under which the trajectory optimizer operates.

The smaller numerical contribution of the mechanism factor should therefore not be interpreted as implying that mechanism adaptation and trajectory optimization are redundant. The two levels act on different decision variables. Under the same optimized trajectory condition, changing the mechanism further reduces path length, jerk cost, cumulative joint motion, and acceleration cost, while the factor analysis shows that mechanism adaptation is the primary contributor to the reduction in joint-motion burden for the present task. At the same time, the results establish an important limitation: link-length adaptation is not expected to provide equally large improvements for every operating condition. If a fixed mechanism already provides favorable reachability and joint configurations, the additional benefit of changing the link lengths may be modest. The mechanism layer is therefore most relevant as a task-level reconfiguration option when task location, workspace-boundary effects, or obstacle geometry make the original configuration kinematically unfavorable.

The validation scope of the surrogate models also requires careful interpretation. The repeated cross-validation in Section 3.2 is stratified by task class at the individual-trajectory-sample level and does not enforce complete separation by task identity. Consequently, different link-length configurations associated with identical or highly similar task descriptors may appear in both the training and validation folds. This protocol is suitable for target-model selection and sample-level predictive stability assessment, but it may provide an optimistic estimate of generalization to completely unseen task geometries. The 20 blind-test tasks used in Section 5 constitute a separate downstream evaluation stage because they were not involved in method development or parameter tuning; however, this blind-task evaluation does not retroactively make the original cross-validation task-disjoint. The revised interpretation therefore distinguishes explicitly between sample-level cross-validation and downstream blind-task validation.

A stricter surrogate-validation protocol would group all link-length configurations belonging to the same task into the same fold, thereby preventing the same endpoint and obstacle descriptors from being shared across training and validation subsets. In addition, task-category-specific MAE and rank-correlation statistics were not included in the present validation protocol and are therefore not used to support claims of unseen-task ranking accuracy. Future validation should adopt task-grouped or leave-task-out cross-validation and report category-specific errors together with rank-correlation measures such as Spearman or Kendall coefficients. Such analyses would provide a more rigorous assessment of whether the surrogate preserves the relative ordering of candidate mechanisms for completely unseen task geometries.

The predictive accuracy of the four TD-TPSM outputs is also markedly different. Cartesian path length and trajectory duration are comparatively learnable and are therefore more suitable for large-scale candidate screening. Jerk and minimum clearance are more sensitive to local control-point adjustment, optimizer convergence, active constraints, and the instantaneous spatial relationship between the composite manipulator envelope and the obstacles. This is reflected both in the lower sample-level ${\mathrm{R}}^2$ values in Section 3.2 and in the surrogate-to-high-fidelity errors reported in Table 8. In particular, the 5.3 mm cross-validation MAE and approximately 4.8 mm blind-task high-fidelity MAE for minimum clearance are non-negligible relative to the 20 mm hard safety requirement. Consequently, surrogate-predicted clearance is not interpreted as a safety certificate. It is used only as an approximate screening, ranking, and risk-indication variable during the outer search.

The present TD-TPSM should also be distinguished from a feasibility classifier. Because the regression dataset contains only successfully generated feasible trajectories, the model learns mechanism-dependent trajectory-performance responses within the feasible response domain rather than the boundary between feasible and infeasible solutions. Final feasibility is instead established through renewed IK solution, multistart B-spline–SLSQP trajectory optimization, and dense 301-point constraint and geometric collision verification. The overall feasibility rate of 98% across 400 high-fidelity verifications indicates that the surrogate-assisted screening procedure is effective for identifying promising candidates, while the eight remaining infeasible candidates provide direct evidence that endpoint IK prescreening and surrogate regression cannot replace a complete trajectory-level feasibility assessment. Endpoint IK screening only establishes whether the prescribed start and goal poses admit joint-limit-satisfying configurations; it does not guarantee avoidance of intermediate collisions or satisfaction of velocity and acceleration constraints throughout the trajectory. A separate feasibility or constraint surrogate trained jointly on feasible and infeasible trajectory-generation samples would therefore be a useful extension and could reduce the number of infeasible candidates entering the expensive high-fidelity verification stage.

The present surrogate architecture is an offline, frozen model rather than an adaptive surrogate-assisted evolutionary optimization scheme. After the four target-specific regressors are trained, they remain unchanged during the ARV-MOBWO outer search, and new high-fidelity evaluations are used for candidate verification rather than for online model updating. This design simplifies deployment and separates inexpensive candidate screening from expensive physical-feasibility restoration, but it also limits the ability of the surrogate to adapt when the search encounters regions that are poorly represented in the original dataset. Uncertainty quantification, active sampling, online surrogate updating, or multiple-surrogate model management would therefore be relevant extensions, particularly for difficult workspace-boundary and box obstacle tasks.

The optimizer results should likewise be interpreted within the scope of the adopted comparison. ARV-MOBWO, standard MOBWO, NSGA-II, and MOPSO all search the same frozen surrogate landscape. The comparison therefore evaluates the behavior of the outer multi-objective search algorithms under identical surrogate information and evaluation budgets rather than comparing complete surrogate-assisted expensive optimization frameworks. The ablation study supports the contribution of the added mechanisms relative to the underlying MOBWO framework, whereas the comparison with NSGA-II shows that ARV-MOBWO does not dominate an established multi-objective optimizer in the surrogate objective space. The high-fidelity results at moderate verification budgets indicate favorable candidate transferability for ARV-MOBWO under the present protocol, but these mean-value differences should not be interpreted as evidence of universal optimizer superiority. The present study also does not benchmark complete adaptive surrogate-assisted evolutionary algorithms in which newly acquired high-fidelity samples are iteratively used to update the surrogate model. Accordingly, the contribution claimed here is limited to the ARV-MOBWO outer-search architecture and its integration into the proposed fixed-surrogate/high-fidelity-verification workflow, rather than superiority over the broader class of surrogate-assisted expensive optimization methods.

The physical experiments should be interpreted strictly within their proof-of-concept scope. The comparison used the S0T1 and S1T1 command sequences rather than S0T0, and only ten loaded trials were conducted for each condition. The observed 70% versus 90% grasping and placement success rates are therefore descriptive and do not establish a statistically significant physical performance improvement. Moreover, the trials followed a fixed staged safety protocol rather than a randomized comparative order, so possible order-related effects cannot be excluded. The approximately 434 s recorded times are complete wall-clock durations dominated by staged waits, gripper operations, communication/acknowledgement, and actuator execution and should not be compared directly with the 28–29 s high-fidelity trajectory duration objective. Failures were logged only at the grasp/placement outcome level rather than through an instrumented failure-mode taxonomy. Although the control program used a nominal 20 Hz loop configuration, the staged execution was acknowledgement-driven with explicit dwell intervals; a unified speed-scaling factor, quantitative payload characterization, joint-tracking-error history, and directly measured physical clearance were not independently recorded. Consequently, the physical results support executability and the absence of observed cylinder collision or joint-limit activation under the tested trials, while quantitative performance comparisons remain based primarily on the high-fidelity trajectory metrics and mechanism–trajectory factor analysis. Future physical validation should use randomized trial ordering, larger sample sizes, synchronized encoder and timing logs, payload characterization, and direct clearance measurement.

Recent nonlinear control approaches address complementary problems at the closed-loop execution level. State-filtered disturbance rejection control compensates matched and mismatched lumped disturbances through state filtering and disturbance-rejection mechanisms [32], whereas multilayer neuroadaptive reinforcement learning based on an actor–critic mechanism combines multilayer neural networks and disturbance estimations to improve tracking performance for uncertain nonlinear systems [33]. These methods mainly address closed-loop tracking robustness in the presence of modeling uncertainty and external disturbances. In contrast, the present study operates at the upstream task-planning and mechanism-configuration level by determining the task-specific link-length configuration and its matched reference trajectory offline while explicitly considering IK reachability, joint constraints, path geometry, motion smoothness, and obstacle clearance.

The proposed framework and these advanced control approaches should therefore be regarded as complementary rather than directly competing methods. The present approach explicitly incorporates mechanism geometry, task-dependent reachability, trajectory shape, and collision constraints into offline optimization, whereas disturbance-rejection and neuroadaptive controllers primarily improve the tracking of a prescribed reference trajectory during physical execution. Conversely, the current framework does not explicitly compensate for unmodeled dynamics, transmission backlash, actuator disturbances, or external perturbations. A natural extension is therefore to execute the optimized mechanism–trajectory solution using a disturbance-rejection or neuroadaptive lower-level controller to improve closed-loop robustness without changing the upstream coordinated-optimization formulation.

A fixed-link manipulator fabricated directly with the optimized dimensions would indeed be a more appropriate engineering solution for a single permanently repeated task because it could provide higher structural rigidity, lower mechanical complexity, and no reconfiguration overhead. The intended application scenario of the present variable-link platform is different. It targets small-batch and high-mix operation in which task positions, workspace requirements, or obstacle layouts vary between task batches. In this context, a single reconfigurable platform can realize different task-specific link-length configurations without manufacturing and maintaining multiple fixed-link manipulators.

Finally, the present sliding-slot and bolt-locking mechanism should be regarded as a laboratory proof-of-concept for task-level offline reconfiguration rather than an online morphing mechanism. Reconfiguration time, locking repeatability, stiffness variation, accumulated mechanical tolerance, and the economic trade-off relative to multiple fixed-link mechanisms are not included in the current optimization model. Together with task-grouped surrogate validation, category-specific error and ranking analysis, adaptive or uncertainty-aware surrogate management, a dedicated feasibility model, more statistically powered physical experiments, and more robust lower-level tracking control, these aspects constitute important directions for future work.

7. Conclusions

A surrogate-assisted bilevel framework was developed for the coordinated optimization of mechanism parameters and motion trajectories in a variable-link-length manipulator. The adjustable link lengths were treated as outer-level mechanism variables, and the B-spline control points and trajectory duration as inner-level trajectory variables. Candidate mechanisms were evaluated using the optimized trajectory performance obtained under the corresponding mechanism configuration, establishing a closed-loop relationship in which mechanism parameters alter the feasible trajectory set and trajectory performance feeds back to mechanism selection.

The objective-decoupled trajectory-evaluation surrogate models enable rapid prediction of trajectory performance for candidate link lengths. ARV-MOBWO combines multi-operator candidate generation, adaptive operator selection, archive-reference recombination, reference vector environmental selection, and external archive maintenance. Under the present fixed-surrogate setting, the ablation study shows that these mechanisms improve the underlying standard MOBWO framework in different aspects of convergence, front approximation, and solution distribution, while the comparison with NSGA-II indicates competitive rather than universally superior multi-objective search performance. Blind tests, ablation studies, and high-fidelity trajectory verification show that the framework reduces the number of expensive trajectory-optimizer calls while retaining coordinated mechanism–trajectory candidates with competitive front quality and physical feasibility.

Factor decomposition in the cylindrical obstacle handling task shows that trajectory optimization is mainly responsible for avoidance-path reshaping, smoothness improvement, and safety-margin enhancement, whereas mechanism adaptation provides more favorable kinematic configurations and joint-motion allocation. Their effects on smoothness and acceleration are nonlinearly coupled. Physical experiments further demonstrate that the selected link lengths and matched command sequences can be executed on the laboratory prototype for the prescribed pick–avoid–transport–place task without observed cylinder collision or joint-limit activation in the tested trials. The proposed framework therefore provides a practical approach for selecting task-adaptive mechanism parameters and generating high-quality trajectories for variable-structure manipulators.