Sliding mode control gain optimization for a robot arm manipulator using an improved stochastic framework

Abstract

We present an optimization–control framework for trajectory tracking of a 3-DoF manipulator, where an improved Stochastic Paint Optimizer (SPO-CL1) automatically tunes the gains of a sliding mode controller. Designed to address robotic challenges such as nonlinearities, couplings, and disturbances, the approach aims to achieve high tracking accuracy, rapid convergence, reduced chattering, and limited actuation effort, while remaining simple and robust to parameter variations. SPO-CL1 integrates three structure-aware enhancements into the original SPO architecture: chaotic initialization via the Chebyshev map to diversify the initial population, Opposition-Based Learning applied after the clustering phase to accelerate convergence, and periodic Lévy flight perturbations targeting the best solution to escape local optima. The SMC gains are automatically tuned by minimizing the Integrated Squared Error cost function J = ∫₀ᵀ eᵀ(t)e(t)dt. Two phases of validation were conducted. First, SPO-CL1 is benchmarked against eleven algorithms including recent hybrid variants (IGWO, EWOA, MHHO, HSMA) on the CEC-2022 suite, achieving the best Friedman rank of 1.83 and statistically significant superiority confirmed by Wilcoxon rank-sum tests (α = 0.05). Second, a path planning experiment on a Lemniscate of Bernoulli trajectory demonstrates that SPO-CL1 achieves the lowest ISE of 1.51 × 10⁻⁴ with near-zero inter-run variance, the fastest tracking error convergence, and the tightest end-effector trajectory among all twelve compared algorithms. These results confirm that SPO-CL1 is a competitive and reliable approach for automatic SMC gain tuning in complex robotic applications.

Introduction

Today robotics has emerged as a key area for business, healthcare, and society¹. Robust control models are necessary for a robot to carry out precise manipulation² or collaborate³. Optimisation lies at the heart of the process: it adjusts parameters, improves the accuracy of trajectories and enhances reliability in uncertain environments⁴. Traditional tuning methods, whether analytical or trial-and-error^5,6, reveal their limitations when faced with non-linearities and disturbances⁷. Recent work has further demonstrated the effectiveness of metaheuristic-optimized path planning for 6-DoF robotic arms using B-spline representations⁸. The control of robotic arms has evolved significantly, from classical kinematic approaches to advanced optimization-based frameworks⁹. With robots operating in constrained environments, adaptive meta-heuristics have become essential for real-time control¹⁰. Inspired by living organisms, they strike a good balance between global exploration and local exploitation. Over the past decade, numerous metaheuristics have been applied in swarm robotics (PSO⁷, GWO¹¹, HHO¹²), bio-inspired approaches (CO¹³, ZO¹⁴), and hybrid methods incorporating chaos¹⁵ or machine learning¹⁶. Their effectiveness has been established in trajectory planning, control law tuning (PID, SMC, LQR) and energy optimisation. The advent of high-performance computing (FPGA, GPU) is accelerating their real-time deployment. In this context, we propose to evaluate an improved version of the Stochastic Paint Optimizer (SPO)¹⁷, enhanced with chaotic mechanisms, to tune a sliding-mode controller on a 3-degree-of-freedom manipulator arm¹⁸. The objective is twofold: on the one hand, to demonstrate the competitive advantage of the proposed metaheuristic approach relative to conventional tuning methods in terms of accuracy, convergence speed and robustness to disturbances; on the other hand, to make a concrete contribution to the field of robotics by highlighting the transferability of these methods to real-world applications (industrial robots, autonomous vehicles¹⁹, medical devices). This work contributes to this dynamic by proposing an improved metaheuristic framework for a demanding robotic control problem, with a detailed comparative analysis against recognized reference algorithms and by providing a detailed comparative analysis with respect to reference algorithms recognised in the literature. Table 1 provides a synthetic overview of the most relevant ‘improved’ metaheuristics for robotics, linking each improvement lever (chaos, OBL, RL, DL hybridisations, FPGA acceleration, etc.) to a representative algorithm, its key idea, and its applications. This mapping justifies the selected optimizers by highlighting improved exploration, reduced stagnation, and real-time suitability.

Table 1.

Current Research on Improved Metaheuristic Algorithms and Their Uses.

Type of improvement	Algorithm/Variant	Key idea	Application	Title of article/Study	References
Chaos-enhanced	C-RIME, C-WSO, etc.	Integration of chaotic maps to enhance exploration	Benchmarking, optimisation multi-domaines	A comprehensive survey on chaos-enhanced metaheuristic algorithms: classification, comparison, and convergence analysis	15
Opposition-Based Learning (OBL)	FROBL, Quasi-reflection OBL	Generate opposing solutions to accelerate convergence	General optimisation, benchmarks	A survey on opposition-based learning enhanced metaheuristics: Variants, results, and future directions	20
Hybrids with Deep Learning	DL + bi-level metaheuristics	Two-level optimisation (road networks, NDP)	Transport, planification	A deep learning–metaheuristic bi-level framework for road network design problems	21
Deep learning + metaheuristics (software engineering)	Metaheuristic + DL	Improved accuracy of software effort prediction	Software engineering	Deep learning with metaheuristic optimization for improved software effort estimation	22
Hardware acceleration (FPGA)	FPGA-based optimizers	Hardware implementation, parallelism and low power consumption	On-board AI, real-time optimisation	Recent advances in FPGA-based metaheuristic optimization: architectures, performance, and applications	23
Hybridisation (metaheuristics)	H-SSAOA (SSA + AOA)	Combining space and transformed approaches to enhance exploration/exploitation and safety	Colour image encryption (RGB), multimedia security	New color image encryption using hybrid optimization algorithm and Krawtchouk fractional transformations	24
Variable selection	BGOA-TVG (Binary GOA with TVG)	Variable Gaussian functions for discrete optimisation	Feature selection, data mining	BGOA-TVG: A binary grasshopper optimization algorithm with time-varying Gaussian functions for feature selection	25

Despite the growing body of work on metaheuristic-enhanced SMC for robotic manipulators, a critical examination of the recent literature reveals persistent limitations that the present work explicitly addresses. Azeez et al²⁶. applied an enhanced Artificial Bee Colony to optimize a PID-SMC for a 3-DoF manipulator but restricted their comparison to two algorithms and a circular trajectory. At the control design level, recent works have proposed two-stage hybrid approaches combining optimal control with integral SMC for robotic manipulators under load uncertainties²⁷, but these methods rely on manual or analytical parameter selection rather than metaheuristic optimization, while Ibrahim et al²⁸. benchmarked a Golden Jackal Optimization against six competitors for PID tuning without addressing SMC gain optimization or chattering. Elgohr et al²⁹. compared PSO and WOA for path planning of a 6-DoF KUKA KR4 R600 manipulator using a three-phase pipeline combining quintic trajectory generation, potential-field obstacle management, and metaheuristic optimization, but restricted their comparison to only two algorithms without statistical significance testing, and focused exclusively on path length minimization rather than SMC gain tuning or tracking error. Elgohr et al³⁰. proposed a hybrid WOA-GA optimizer for 6-DoF trajectory optimization, achieving up to 44% faster reachability and 15% lower energy consumption, but limited their comparison to three algorithms without statistical testing and did not address SMC gain tuning. Elgohr et al³¹. applied WOA to KUKA KR4 R600 path planning, reducing travel distance by over 30%, but compared only two baseline methods without statistical validation. In the SMC-specific domain, Parra-Ocampo et al²⁷. demonstrated that chaotic initialization meaningfully improves tuning convergence, yet their framework incorporates neither OBL nor Lévy strategies. Wei et al⁴. went further by combining both Lévy flights and OBL within a crayfish optimizer but confined their validation to a single engineering domain without any statistical significance testing. Across these and related works, three structural limitations recur: comparisons rarely exceed six algorithms, reference trajectories remain geometrically simple, and the ISE cost function is stated informally, undermining reproducibility. The present work addresses all three gaps simultaneously: SPO-CL1 integrates Chebyshev chaotic initialization, OBL, and Lévy flights into a single coherent framework, is benchmarked against eleven algorithms using Wilcoxon rank-sum and Friedman tests and is validated on a Lemniscate of Bernoulli trajectory with the explicit cost function J = ∫₀ᵀ eᵀe dt. Specifically, this work makes the following contributions:

• i)Accurate dynamic modelling of a 3-DoF manipulator. A rigorous Lagrangian dynamic model incorporating nonlinearities, couplings, and gravitational effects is developed, serving as the foundation for both the SMC design and the optimization problem formulation.
• ii)A novel improved metaheuristic: SPO-CL1. An enhanced SPO is proposed, integrating three structure-aware mechanisms: Chebyshev chaotic initialization (Phase 1), Opposition-Based Learning after the clustering step (Phase 2), and periodic Lévy flight perturbations on the best solution (Iter mod 5 = 0). Each SPO variant (SPO-CL1 to SPO-CL10) is explicitly mapped to its chaotic function in the section “Flowchart of SPO-CL” to ensure full reproducibility. Unlike existing hybrid metaheuristics that incorporate similar mechanisms into generic architectures (IGWO, EWOA, MHHO, HSMA), SPO-CL1 integrates each enhancement at a precisely targeted phase of the SPO pipeline: Chebyshev initialization in Phase 1, OBL after the tripartite clustering in Phase 2, and periodic Lévy perturbations on the best solution (Iter mod 5 = 0). This structure-aware design is empirically validated in the sections “Tests based on the CEC’2022 benchmark” and “Comparison with other optimizers”, where SPO-CL1 statistically outperforms all four hybrid competitors despite their use of comparable mechanisms.
• iii)Statistically validated benchmark evaluation. SPO-CL1 is evaluated against eleven algorithms on the CEC-2022 suite over 30 independent runs, with Wilcoxon rank-sum and Friedman statistical tests confirming the significance of the reported improvements (Sect. “Nonparametric statistical analysis”).
• iv)Automatic SMC gain tuning on a demanding robotic trajectory. SPO-CL1 minimizes the explicit ISE cost to automatically tune six SMC gains on a Lemniscate of Bernoulli trajectory (a challenging figure-eight path with non-constant curvature and direction reversal) providing a more discriminating validation than a simple circular path.
• v)A generic and transferable optimization-control framework. The methodology extends naturally to multi-objective formulations, higher-DoF manipulators, and real-time implementations, with experimental validation on a physical platform identified as a primary avenue for future work.

The following section presents the dynamic modelling of the 3-DoF planar manipulator, which serves as the foundation for the SMC design and the optimization framework developed in subsequent sections.

Robot modelling

Assuming each segment of the arm to be rigid and uniform, and the joints to be perfectly frictionless, the aim is to derive the equations of motion from the Lagrangian formulation. The dynamic modelling^26,32 then serves as the basis for designing and optimising the sliding mode control (SMC), whose parameters are determined using the SPO-CL1 algorithm. By minimising the overall dynamic error through appropriate adjustment of the parameters λ₁, λ₂, λ₃, k₁, k₂, k₃, trajectory tracking performance is significantly improved by this method. This section details the construction of the dynamic model with the physical characteristics shown in Table 2. The Lagrange technique is used to establish the equations of motion when the mechanical assumptions are established, the generalized coordinates are determined, and the kinetic and potential energy of each link is represented. The optimisation focuses on the parameters λ₁, λ₂, λ₃ (sliding surface parameters) and k₁, k₂, k₃ (switching gains), with the aim of reducing the integrated squared error (ISE) J = ∫_₀^ᵀ eᵀe dt, which results in significantly improved trajectory tracking.

Table 2.

physical characteristics and Geometric dimensions of the 3-DOF robots.

Specifications	Joint 1	Joint 2	Joint 3
Weight (kg)	1	1	1
Moment of inertia Izz (kg·m²)	0.020833	0.020833	0.020833
Length (m)	0.5	0.5	0.5
Area of the cross section (cm²)	25	25	25

A rigid, planar, 3DoF robotic manipulator is schematically depicted in Figure 1. In this arrangement, a frictionless pivot link connects the first segment to a stiff base. A frictionless ball bearing is then used to articulate the second segment to the end of the first. Likewise, a second frictionless ball bearing connects the third section to the second. Equation (1) formulates the non-linear, coupled dynamic equations that result from modeling the system in this manner.

1

Figure 1.

Diagrammatic representation of the 3DoF manipulator.

where:

: joint angle vector.

: matrix of inertia.

: centrifugal forces/Coriolis vector.

: vector of gravity.

The torques that are applied.

the inertia matrix, is defined by:

2

3

4

5

6

7

Then :

8

The definition of the centrifugal and coriolis forces is:

9

10

11

Gravitational forces defined by:

12

13

14

The original stochastic paint optimizer algorithm (SPO)

General presentation

Stochastic Paint Optimizer (SPO)¹⁷ is an innovative population metaheuristic, inspired by artistic color mixing techniques. In this approach, the search space is assimilated to a painting canvas, with solutions represented by “paintings” composed of continuous variables assimilated to “colors”. The algorithm takes advantage of chromatic theory to classify these solutions into three distinct categories, enabling the search to be channeled between targeted exploitation and in-depth exploration.

Classification of solutions into chromatic categories

Chaque solution est un vecteur within a large population . At each iteration, the solutions are evaluated and sorted, then divided into three groups according to their quality.

• Primary Colors (Cp): best third of solutions - representing the most promising solutions.

• Secondary Colors (Cs): middle third.

• Tertiary Colors (Ct): bottom third, presenting the least efficient solutions.

This structural classification is fundamental in guiding the generation of new solutions.

Stochastic mechanisms for generating solutions

Four combination mechanisms generate new solutions and ensure a balance between global exploration and local exploitation:

Analogue combination

This method exploits the proximity between neighbouring solutions belonging to the same category to reinforce local optimisation:

15

where is a random variable, and the indices  i-1 et  i+1 refer to neighbouring solutions in the same chromatic category. This combination effectively refines what already exists while confining the search to the neighbourhood. 3.3.2 Complementary combination.

This mechanism combines a solution from the primary category with another from the tertiary category, encouraging a shift towards potentially more promising regions and thus fuelling exploration:

16

where et randomly selected, et .

Triadic combination

Simultaneously uses a solution from each category to generate controlled diversification between exploitation and exploration:

17

where the vectors come from the primary, secondary and tertiary groups respectively, and .

Tetradic combination

Introduces an additional random term to maximise diversity and avoid trapping in local optima:

18

where independent, and a random solution generated uniformly in the search space.

Algorithmic description

The SPO follows this iterative process :

Algorithm 1.

Pseudo-code of SPO.

Improved stochastic paint optimizer

Figure 2 illustrates the methodological architecture of the present study, which is structured into two successive. In the first phase, a set of ten chaos-enhanced variants of the Stochastic Paint Optimizer is evaluated and compared on benchmark functions in order to identify the most effective configuration. This selection stage aims solely at determining the best-performing variant among the candidate chaotic maps, which is found to be SPO-CL1. In the second phase, the selected optimizer (SPO-CL1) is then employed as a fixed optimization tool to tune the parameters of the sliding mode controller for a 3-DoF robotic manipulator. The controller gains (1, 2, 3, 1, 2, 3) are optimized once and subsequently used in the robotic simulations. Accordingly, Figure 2 should be interpreted as an overview of the paper’s two-step workflow variant selection followed by controller parameter optimization rather than as a closed-loop or adaptive control architecture.

Figure 2.

Methodological workflow of the paper.

Initialization based on chaotic maps

Chaos theory has gained increasing attention across scientific and engineering domains due to its sensitivity to initial conditions, non-periodicity, and ergodic properties. Nowadays, chaos theory is applied in many fields, especially cryptography and systems optimization, because of its capacity to simulate intricate, non-linear, and unpredictable behavior. Chaotic maps are used in cryptography to provide extremely safe keys. Their extreme sensitivity to initial conditions and their non-periodic behaviour make the sequences generated virtually impossible to predict or reproduce, which considerably enhances the robustness of security mechanisms. In line with recent work, initialising solutions using chaotic maps improves initial diversity and reduces premature convergence, for example, Tahiri et al³³. have proposed an improved version called Chaos-Artemisinin Optimizer (CAO) injects ten maps (CAO1-CAO10) to enrich exploration while stabilising the search on a manipulator robot. In order to demonstrate the efficacy of chaotic dynamics in protecting sensitive data from cyberattacks, a technique for encrypting color images using fractional Krawchouk transformations and a hybrid optimization approach was presented by Tahiri et al.²⁴. In the optimization field, chaotic maps have also shown a demonstrated capacity to diversify the solutions generated and prevent becoming stuck in local optima. An enhanced Kepler technique that incorporates chaotic maps was proposed by El Ghouate et al.³⁴, who achieved better results on a number of test functions and real-world applications. Conventional optimisation algorithms often suffer from premature convergence towards local minima, limiting their overall efficiency. The inclusion of chaotic dynamics, because of their non-linear and unpredictable nature, helps to broaden the spectrum of solutions examined and prevents this kind of impasse. Several popular metaheuristic algorithms have been successful with this method. For example Fister et al.³⁵ demonstrated that adding chaotic dynamics to the Firefly Algorithm (FA) intensified agent movements, thereby improving global and local search. Similarly, to enhance the exploration stage and prevent stagnation in suboptimal solutions, Chaotic maps were included to the Moth Flame Optimization (MFO) algorithm by H. Li et al.³⁶. Similar advancements for other techniques, such Artificial Bee Colony (ABC), have been documented³⁷, or Particle Swarm Optimisation (PSO)³⁸, where the introduction of chaotic perturbations encourages deeper exploration and prevents stagnation in local minima. Lastly, Tahiri et al.³⁹ combined separable moments with a firefly-based chaotic algorithm in the context of optimal 3D reconstruction to show the usefulness of non-linear dynamics in computer vision applications.

Ten one-dimensional chaotic maps were selected for this study because of their methodological importance and the range of dynamic behaviors they offer (see Table 3). These maps were chosen due to their widespread application and proven efficacy in earlier research on chaos-enriched optimization techniques. They cover a wide range of dynamic properties, including density distribution throughout the interval [0,1], symmetry, responsiveness to initial conditions, bifurcations, and ergodicity. Instead of creating a comprehensive inventory of every map that currently exists, the goal is to create a representative and diverse collection that allows for pertinent statistical comparisons to be performed.

Table 3.

Chaotic maps functions ande ranges of variation.

Maps Name	Chebyshev	Circular	Gauss
Function
Range	(-1,1)	(0,1)	(0,1)

Maps Name	Iterative	Logistic	Piecewise
Function
Range	(-1,1)	(0,1)	(0,1)

Maps Name	Sine	Singer	Sinusoidal
Function
Range	(0,1)	(0,1)	(0,1)

Maps Name	Tent
Function
Range

Opposition-Based Learning (OBL)

A key challenge in population-based optimization is generating a diverse initial population close to promising regions. Even with chaotic initialization, some solutions may be placed in low-quality regions of the search space. Our improved SPO integrates Opposition-Based Learning (OBL), which evaluates each solution x alongside its opposite x˜, enabling double sampling at no extra cost.

Either a solution , defined in a space bounded by the vectors et , the opposite solution is obtained by:

19

The opposite solution respects the same limits of definition as and can be interpreted as a central reflection across the middle of the interval . Once x and have been generated, the objective function f is evaluated for both, and the best is chosen for the final initialization:

20

Each individual goes through opposition-based learning after chaotic initiation, choosing the best of each option and its opposite. Both population diversity and the quality of the original solution are enhanced by this. Research demonstrates that when extensive investigation is essential, OBL speeds up early convergence⁴⁰. OBL improves performance in SPO-CL1 while preserving a straightforward, parameter-free structure.

Perturbation based on Lévy flights

We used Lévy flight perturbations⁴¹ that alternate between short/local and long/rare jumps to improve search coverage and address premature convergence in SPO. Lévy steps enhance population diversity with a minimal parameter overhead (a scaling factor α = 0.05 and a stability index β = 1.5) when applied periodically to the best solution every 5 iterations (Iter mod 5 = 0). In comparison to non-Lévy alternatives, the results demonstrate faster cost descent and improved inter-run stability.

21

where :

• is a scaling factor controlling the amplitude of the jump (set here at ),

• is a Lévy vector generated according to Lévy’s law with parameter , typically .

The numerical generation of Lévy steps follows Mantegna’s scheme (Eqs. 22–23), where the vector is defined as:

22

with a standard deviation σ given by:

23

The new solution x_new replaces the worst if better, preserving quality and diversity. Lévy flights enhance exploration early and prevent stagnation later.

Flowchart of SPO-CL

Figure 3 shows SPO-CL1’s architecture: chaotic map for diversification, opposition-based learning, and the main SPO loop with four color combinations. The key innovation is periodic Lévy flights on the best solution Iter mod 5 = 0 to escape local optima. To ensure full reproducibility of the proposed framework, each SPO variant is explicitly associated with its corresponding chaotic map following the order established in Table 3: SPO-CL1 employs the Chebyshev map, SPO-CL2 the Circular map, SPO-CL3 the Gauss map, SPO-CL4 the Iterative map, SPO-CL5 the Logistic map, SPO-CL6 the Piecewise map, SPO-CL7 the Sine map, SPO-CL8 the Singer map, SPO-CL9 the Sinusoidal map, and SPO-CL10 the Tent map. As demonstrated in the ablation study presented in the section “Tests based on the CEC’2022 benchmark”, SPO-CL1 consistently outperforms the nine other variants on the CEC-2022 benchmark suite and is therefore selected as the proposed algorithm for all subsequent experiments.

Figure 3.

SPO-CL Algorithm Flowchart.

Algorithm 2 summarizes the complete pseudo-code of SPO-CL1, integrating the three enhancements described in the section “Initialization based on chaotic maps”–“Perturbation based on Lévy flights” into the original SPO framework.

Algorithm 2.

Pseudo-code of SPO-CL1.

Experimental simulation

The validation of SPO CL1 comprises three stages: CEC 2022 benchmarks, comparison with eight metaheuristics, and optimisation of an SMC controller on a 3-DoF manipulator. Tests conducted on a standardised platform (i7-8550U, 1.80 GHz, 16 GB RAM). This approach validates SPO-CL1’s fundamental properties for robotic applications.

Tests based on the CEC’2022 benchmark

SPO and its chaotic variants (SPO-CL1 to SPO-CL10) were evaluated on the CEC 2022 dataset. The scores (best, mean, standard deviation, ranking) were calculated across multiple runs to ensure statistical validity and to analyse the exploration, convergence and quality of the solutions (Table 4, table 5and Figure 4).

Table 4.

Summary of CEC 2022 benchmark test functions.

Function number	Function nature	Range	Min
F1	Uni-modal function: Shifted and full rotated Zakharov function	[− 100,100]	300
F2	Basic function: Full rotated and shifted Rosenbrock’s		400
F3	Basic function: Full rotated and shifted Expanded Schaffer’s		600
F4	Basic function: Full rotated and shifted non-Continuous Rastrigin’s		800
F5	Basic function: Full rotated and shifted Levy function		900
F6	Hybrid function 1		1800
F7	Hybrid function 2		2000
F8	Hybrid function 3		2200
F9	Composition function 1		2300
F10	Composition function 2		2400
F11	Composition function 3		2600
F12	Composition function 4		2700

Table 5.

Benchmark function results (F1 to F12).

	Metric	F1	F2	F3	F4	F5	F6	F7	F8	F9	F10	F11	F12
SPO	Optimal /Min	300.001	404.502	600	815.919	900	2274.32	2021.82	2223.02	2529.28	2500.58	2750.43	2862.83
	Average	302.701	417.202	600.001	838.623	901.99	3751.43	2022.99	2230.22	2529.28	2524.98	2750.72	2869.25
	StdDev	6.02202	24.0271	0.000385785	20.9112	3.90988	1633.65	0.86839	8.07842	2.27753e-09	49.0147	0.352921	11.3457
SPO-CL1	Optimal /Min	300.001	406.967	600	818.904	900	1887.7	9.02725	2221.4	2529.28	2500.45	2600	2861.4
	Average	300.57	407.9	600.237	829.029	951.964	4772.56	2025.59	2222.3	2529.28	2724.33	2720.34	2865.8
	StdDev	0.715768	0.947065	0.417489	8.60352	113.817	3102.9	9.02725	0.764409	1.32861e-09	306.461	67.2731	4.6193
SPO-CL2	Optimal /Min	300	405.652	600	808.955	900	3219.5	2008.95	2221.84	2529.28	2608.71	2600	2863.92
	Average	719.92	407.359	605.734	811.542	941.791	4626.59	2029.93	2222.65	2531.41	2623.76	2740.19	2886.69
	StdDev	936.177	1.5206	6.11537	2.86645	89.0384	1564.39	13.8758	0.967397	4.75382	11.9198	163.559	33.703
SPO-CL3	Optimal /Min	300	406.575	600	816.914	901.428	1926.97	2021	2221.37	2529.28	2500.58	2750.43	2862.57
	Average	655.99	408.448	601.858	824.675	991.659	5085.76	2026.08	2221.37	2529.28	2692.3	2750.44	2863.72
	StdDev	753.99	1.04707	4.13885	7.04946	194.092	3049.92	6.53669	5.2928	5.60771e-07	169.873	0.0211365	1.51662
SPO-CL4	Optimal /Min	300.166	406.167	600	819.478	900	2257.28	2021.4	2206.74	2529.28	2500.45	2600	2863.71
	Average	752.478	407.488	600.008	833.196	906.402	4299.59	2027.46	2220.52	2529.28	2523.55	2720.35	2865.52
	StdDev	578.648	0.97552	0.0105488	13.5146	12.8811	1471.39	7.37639	7.78623	1.20588e-09	51.1383	67.278	1.92966
SPO-CL5	Optimal /Min	300.02	400.121	600.001	805.97	900.09	2871.68	2022.61	2221.33	2529.28	2500.55	2750.43	2863.49
	Average	457.507	405.933	600.399	817.514	936.866	5369.4	2031.75	2223.35	2529.28	2536.32	2850.31	2867.28
	StdDev	233.71	3.37337	0.673607	17.0233	53.9666	1816.41	12.8299	1.45065	6.00795e-05	79.3711	136.65	5.15856
SPO-CL6	Optimal /Min	300.012	406.636	600	813.929	900	2383.6	2007.55	2220.36	2529.28	2500.39	2600	2862.57
	Average	1284.95	408.448	600.003	823.443	947.603	4219.38	2020.05	2222.11	2529.28	2578.14	2720.48	2867.31
	StdDev	1771.67	1.51323	0.00346703	5.96864	84.9977	2247.9	7.41948	1.18384	7.82991e-11	71.2555	67.3493	5.374
SPO-CL7	Optimal /Min	300	408.916	600	811.939	900	1954.18	2021.62	2221.04	2529.28	2500.54	2600	2861.4
	Average	509.151	424.528	600.192	820.451	902.94	5219.64	2024.94	2223.47	2529.28	2549.04	2776.88	2864.93
	StdDev	463.523	34.9094	0.296635	9.14839	4.63994	2260.48	3.31059	2223.47	1.61499e-11	66.3266	239.411	4.16831
SPO-CL8	Optimal /Min	300.002	405.427	600	807.96	900	2193.34	2020.27	2221.65	2529.28	2500.68	2650.29	2863.49
	Average	320.052	407.571	600.033	819.501	914.235	6364.55	2024.76	2224.44	2529.28	2932.12	2791.4	2866.27
	StdDev	28.8747	1.44885	0.0413398	8.34216	18.3309	2431.26	5.05844	2.53471	2.03374e-09	491.862	107.717	4.69564
SPO-CL9	Optimal /Min	311.054	404.384	600	821.889	900	2051.13	2022.47	2230.73	2529.28	2651.16	2600	2862.57
	Average	11573.8	501.024	658.426	963.729	4968.97	5890.09	2392.86	34806.6	2580.33	4097.26	3086.39	2930
	StdDev	15066.4	128.843	82.9485	79.7508	4052.51	2347.41	207.399	66549.9	73.8643	1175.01	475.938	117.948
SPO-CL10	Optimal /Min	328.141	407.457	600	809.95	900.09	2087.44	2021.4	2220.54	2529.28	2500.52	2600.01	2863.91
	Average	724.03	408.657	600.126	823.446	900.556	5192.29	2024.02	2221.95	2529.28	2500.75	2720.36	2867.94
	StdDev	650.067	0.674567	0.247815	7.75457	0.45985	2158.26	2.92807	0.941917	7.75561e-11	0.207281	67.2816	2.49504

Figure 4.

Convergence graphs.

Twelve CEC 2022 functions (30 colors, 250 iterations, five runs each) were used to test 11 SPO versions. SPO-CL1 achieves the best average rank (≈ 3.08) and the lowest total optima sum (22,968.37), confirming its selection as the retained variant for subsequent experiments. The hybrid structure’s modest chaotic disruption improves exploration while preserving stability. Robustness is confirmed by low variance, which is crucial for mobile robotics. The superior performance of SPO-CL1 on complex benchmarks as shown in Fig 5, combined with its statistical stability, justifies its selection as the retained variant for subsequent experiments of SPO for the next stages of this research, particularly in the context of trajectory planning in robotics.

Figure 5.

Ranking of SPO variants by the benchmark CEC2022.

Comparison with other optimizers

To comprehensively evaluate the performance of the proposed SPO-CL1 algorithm, a comparative study was conducted against eleven state-of-the-art metaheuristic algorithms, including both classical optimizers and recent hybrid or enhanced variants. The comparison is organized into two parts. The section “Tests Based on the CEC-2022 Benchmark” presents the quantitative results obtained on the CEC-2022 benchmark suite, reporting key statistical indicators over 30 independent runs. The section “Nonparametric statistical analysis” provides a rigorous nonparametric statistical analysis using the Wilcoxon rank-sum test and the Friedman test to assess the significance of the observed performance differences and establish a global ranking of the compared algorithms.

Tests based on the CEC-2022 benchmark

To verify the effectiveness of the proposed algorithm, SPO-CL1 was compared with eleven well-known and recent metaheuristic algorithms: Aquila Optimizer (AO)⁴², Arithmetic Optimization Algorithm (AOA)^43,44, Grey Wolf Optimizer (GWO)⁴⁵, Harris Hawks Optimization (HHO)¹², Zebra Optimization Algorithm (ZOA)⁴⁶, Coyote Optimization Algorithm (COA)¹³, Circulatory System Based Optimization (CSBO)⁴⁷, Improved Grey Wolf Optimizer (IGWO)⁴⁸, Enhanced Whale Optimization Algorithm (EWOA)⁴⁹, Modified Harris Hawks Optimization (MHHO)⁵⁰, and Hybrid Slime Mould Algorithm (HSMA)⁵¹, using the parameters recommended in their respective publications and summarized in Table 6. All algorithms were tested on the CEC-2022 benchmark suite, which comprises twelve functions (F1–F12) with a problem dimension of 30. Each algorithm was independently executed 30 times with a population size of 30 and a maximum of 250 iterations.

Table 6.

Recommended parameter settings for the compared algorithms.

Algorithm	Parameter	Description	Recommended Value
SPO-CL1	α	Lévy flight scaling factor	0.05
	β	Lévy flight index	1.5
	—	Chaotic map (Chebyshev) + OBL	—
AO	α	Expansion factor	0.1
	δ	Contraction factor	0.1
	U, r	Random search parameters	U = 10, r∈[0,1]
AOA	µ	Control parameter	5
	α	Sensitive parameter	5
	MOA	Math Optimizer Accelerated	[0.2 → 1.0] (linear)
	MOP	Math Optimizer Probability	0.5
GWO	a	Linearly decreased	[2 → 0]
	A, C	Coefficient vectors	A∈[-a, a], C∈[0,2]
HHO	E₀	Initial energy	E₀ ∈ [− 1, 1]
	q	Probability of attack	0.5
	r	Escape factor	0.5
ZOA	β	Movement factor	1.5
	α	Learning parameter	0.5
	ρ	Memory factor	0.8
COA	Nᶜ	Number of coyotes per pack	5
	Nₚ	Number of packs	10
	Pₐ	Age probability	0.005 + 0.905/Nᶜ²
CSBO	λ₁, λ₂, λ₃	Transport factors	1, 2, 1
	γ	Metabolic factor	0.5
IGWO	a	Nonlinear decrease	[2 → 0] (nonlinear)
	—	Opposition-Based Learning	Applied at initialization
EWOA	b	Spiral shape constant	1
	—	Lévy flight perturbation	β = 1.5
	—	Chaotic initialization	Logistic map
MHHO	E₀	Initial energy	E₀ ∈ [− 1, 1]
	—	Opposition-Based Learning	Applied at initialization
	—	Adaptive jump rate	Self-adaptive
HSMA	z	Slime mould weight	∈ [0, 1]
	—	Lévy flight perturbation	β = 1.5
	—	Chaotic initialization	Tent map
	—	Opposition-Based Learning	Applied at initialization

To ensure full reproducibility, all algorithm parameter settings are consolidated in Table 6, the complete SPO-CL1 pseudo-code is provided in Algorithm 2, and the ISE cost function is explicitly defined in Eq. (24). Simulation parameters are fully specified in the section “Path planning simulation experiment” (dt = 0.01 s, T = 5 s, RK4 integration, Np = 50, MaxIter = 30, 10 runs). The best, mean, median, and standard deviation of the fitness values obtained over 30 runs are reported in Table 7.

Table 7.

Benchmark function (F1 to F12) results.

Algorithm	Metric	F1	F2	F3	F4	F5	F6	F7	F8	F9	F10	F11	F12
SPO-CL1	Best	3.00e+2	4.05e+2	6.00e+2	8.07e+2	9.00e+2	1.84e+3	2.00e+3	2.20e+3	2.53e+3	2.50e+3	2.60e+3	2.86e+3
	Mean	5.96e+2	4.08e+2	6.01e+2	8.20e+2	9.36e+2	4.56e+3	2.03e+3	2.22e+3	2.53e+3	2.53e+3	2.83e+3	2.87e+3
	Median	3.07e+2	4.07e+2	6.00e+2	8.17e+2	9.02e+2	4.43e+3	2.03e+3	2.22e+3	2.53e+3	2.50e+3	2.75e+3	2.86e+3
	Std	1.08e+3	1.03e+0	1.64e+0	1.02e+1	1.22e+2	1.95e+3	1.32e+1	5.63e+0	7.18e+0	5.66e+1	1.59e+2	2.69e+0
AO	Best	2.66e+3	4.18e+2	6.09e+2	8.11e+2	9.36e+2	4.36e+3	2.03e+3	2.23e+3	2.55e+3	2.50e+3	2.72e+3	2.87e+3
	Mean	7.54e+3	4.56e+2	6.23e+2	8.24e+2	1.06e+3	1.36e+5	2.07e+3	2.23e+3	2.62e+3	2.56e+3	2.77e+3	2.87e+3
	Median	7.56e+3	4.38e+2	6.21e+2	8.24e+2	1.04e+3	4.69e+4	2.06e+3	2.23e+3	2.62e+3	2.50e+3	2.75e+3	2.87e+3
	Std	3.01e+3	6.56e+1	6.81e+0	5.53e+0	8.98e+1	3.09e+5	2.36e+1	5.57e+0	3.40e+1	6.48e+1	9.53e+1	1.21e+1
AOA	Best	5.47e+3	6.53e+2	6.25e+2	8.23e+2	1.06e+3	1.03e+6	2.05e+3	2.23e+3	2.64e+3	2.51e+3	2.85e+3	2.94e+3
	Mean	1.36e+4	1.45e+3	6.41e+2	8.51e+2	1.40e+3	4.20e+8	2.10e+3	2.53e+3	2.79e+3	2.77e+3	3.39e+3	3.07e+3
	Median	1.27e+4	1.33e+3	6.40e+2	8.54e+2	1.37e+3	3.46e+8	2.10e+3	2.25e+3	2.76e+3	2.70e+3	3.40e+3	3.05e+3
	Std	4.83e+3	5.72e+2	1.02e+1	1.06e+1	1.73e+2	4.05e+8	4.26e+1	7.44e+2	1.04e+2	2.30e+2	3.38e+2	8.56e+1
GWO	Best	4.65e+2	4.05e+2	6.00e+2	8.07e+2	9.00e+2	2.65e+3	2.00e+3	2.21e+3	2.53e+3	2.42e+3	2.61e+3	2.86e+3
	Mean	4.57e+3	4.27e+2	6.01e+2	8.21e+2	9.23e+2	7.11e+3	2.03e+3	2.24e+3	2.59e+3	2.64e+3	2.81e+3	2.87e+3
	Median	4.32e+3	4.12e+2	6.01e+2	8.16e+2	9.11e+2	7.11e+3	2.03e+3	2.23e+3	2.58e+3	2.61e+3	2.73e+3	2.87e+3
	Std	2.40e+3	2.17e+1	1.38e+0	1.17e+1	2.93e+1	3.58e+3	1.23e+1	3.11e+1	4.07e+1	1.86e+2	1.69e+2	1.22e+1
HHO	Best	5.24e+2	4.04e+2	6.21e+2	8.11e+2	1.08e+3	2.23e+3	2.05e+3	2.22e+3	2.60e+3	2.50e+3	2.61e+3	2.87e+3
	Mean	3.76e+3	5.02e+2	6.42e+2	8.28e+2	1.44e+3	8.02e+3	2.10e+3	2.24e+3	2.66e+3	2.68e+3	2.84e+3	2.95e+3
	Median	4.12e+3	4.72e+2	6.41e+2	8.28e+2	1.46e+3	5.41e+3	2.08e+3	2.23e+3	2.65e+3	2.64e+3	2.76e+3	2.93e+3
	Std	1.20e+3	9.77e+1	1.04e+1	7.16e+0	1.69e+2	6.42e+3	4.04e+1	1.64e+1	3.67e+1	1.98e+2	1.54e+2	6.42e+1
ZOA	Best	5.72e+2	4.02e+2	6.10e+2	8.04e+2	9.05e+2	1.90e+3	2.01e+3	2.21e+3	2.56e+3	2.50e+3	2.63e+3	2.89e+3
	Mean	2.40e+3	4.59e+2	6.22e+2	8.15e+2	1.05e+3	3.39e+3	2.05e+3	2.24e+3	2.64e+3	2.57e+3	2.90e+3	2.95e+3
	Median	1.83e+3	4.54e+2	6.21e+2	8.14e+2	1.04e+3	2.56e+3	2.05e+3	2.23e+3	2.65e+3	2.56e+3	2.81e+3	2.94e+3
	Std	1.66e+3	5.05e+1	6.50e+0	4.88e+0	9.76e+1	1.67e+3	1.93e+1	4.10e+1	3.96e+1	6.61e+1	2.06e+2	2.76e+1
COA	Best	4.70e+3	4.43e+2	6.20e+2	8.29e+2	9.73e+2	1.91e+4	2.05e+3	2.22e+3	2.55e+3	2.50e+3	2.78e+3	2.87e+3
	Mean	8.68e+3	5.42e+2	6.32e+2	8.44e+2	1.20e+3	1.87e+6	2.07e+3	2.24e+3	2.63e+3	2.52e+3	2.85e+3	2.88e+3
	Median	8.49e+3	5.36e+2	6.33e+2	8.44e+2	1.21e+3	1.11e+6	2.07e+3	2.24e+3	2.64e+3	2.52e+3	2.86e+3	2.88e+3
	Std	3.11e+3	6.34e+1	6.60e+0	5.96e+0	1.33e+2	2.40e+6	1.35e+1	7.34e+0	4.70e+1	8.44e+0	3.72e+1	8.04e+0
CSBO	Best	1.02e+4	4.90e+2	6.24e+2	8.42e+2	1.33e+3	3.93e+6	2.07e+3	2.23e+3	2.59e+3	2.52e+3	2.81e+3	2.87e+3
	Mean	2.30e+4	5.85e+2	6.41e+2	8.68e+2	1.95e+3	2.26e+7	2.10e+3	2.25e+3	2.66e+3	2.54e+3	3.02e+3	2.87e+3
	Median	2.26e+4	5.70e+2	6.42e+2	8.70e+2	1.95e+3	1.99e+7	2.10e+3	2.24e+3	2.66e+3	2.53e+3	2.98e+3	2.87e+3
	Std	6.57e+3	7.48e+1	8.02e+0	9.42e+0	2.80e+2	1.60e+7	1.70e+1	6.90e+0	3.72e+1	3.27e+1	1.91e+2	3.50e+0
IGWO	Best	3.51e+2	4.01e+2	6.01e+2	8.07e+2	9.00e+2	5.64e+3	2.02e+3	2.20e+3	2.53e+3	2.50e+3	2.61e+3	2.86e+3
	Mean	1.84e+3	4.35e+2	6.04e+2	8.21e+2	9.36e+2	1.41e+4	2.04e+3	2.23e+3	2.58e+3	2.62e+3	2.90e+3	2.87e+3
	Median	5.78e+2	4.21e+2	6.03e+2	8.20e+2	9.24e+2	1.21e+4	2.04e+3	2.23e+3	2.57e+3	2.61e+3	2.75e+3	2.87e+3
	Std	2.21e+3	2.83e+1	2.99e+0	1.06e+1	4.92e+1	7.79e+3	1.34e+1	5.16e+0	4.07e+1	1.82e+2	2.14e+2	5.57e+0
EWOA	Best	1.62e+3	4.00e+2	6.46e+2	8.25e+2	1.45e+3	1.86e+3	2.10e+3	2.24e+3	2.53e+3	2.64e+3	2.60e+3	2.87e+3
	Mean	2.97e+3	4.31e+2	6.63e+2	8.41e+2	1.50e+3	2.04e+3	2.15e+3	2.29e+3	2.55e+3	3.36e+3	3.16e+3	3.50e+3
	Median	2.97e+3	4.10e+2	6.62e+2	8.44e+2	1.48e+3	1.98e+3	2.15e+3	2.28e+3	2.53e+3	3.54e+3	3.03e+3	3.54e+3
	Std	5.73e+2	4.75e+1	5.73e+0	7.16e+0	4.15e+1	2.44e+2	1.26e+1	2.05e+1	3.84e+1	5.68e+2	3.67e+2	2.75e+2
MHHO	Best	1.91e+3	4.02e+2	6.23e+2	8.07e+2	1.01e+3	1.90e+3	2.04e+3	2.22e+3	2.59e+3	2.50e+3	2.63e+3	2.88e+3
	Mean	3.34e+3	5.02e+2	6.41e+2	8.28e+2	1.41e+3	1.79e+4	2.09e+3	2.24e+3	2.64e+3	2.61e+3	2.84e+3	2.95e+3
	Median	3.43e+3	4.76e+2	6.41e+2	8.28e+2	1.43e+3	1.12e+4	2.07e+3	2.23e+3	2.65e+3	2.50e+3	2.76e+3	2.95e+3
	Std	8.92e+2	9.62e+1	9.01e+0	8.31e+0	2.06e+2	2.20e+4	3.46e+1	1.57e+1	3.70e+1	1.75e+2	1.56e+2	5.64e+1
HSMA	Best	3.00e+2	4.00e+2	6.12e+2	8.23e+2	1.02e+3	1.88e+3	2.01e+3	2.20e+3	2.53e+3	2.50e+3	2.60e+3	2.87e+3
	Mean	3.09e+2	4.09e+2	6.44e+2	8.37e+2	1.37e+3	2.47e+3	2.07e+3	2.23e+3	2.58e+3	2.56e+3	2.92e+3	2.91e+3
	Median	3.02e+2	4.05e+2	6.48e+2	8.39e+2	1.43e+3	2.19e+3	2.07e+3	2.23e+3	2.57e+3	2.50e+3	2.91e+3	2.91e+3
	Std	1.30e+1	1.38e+1	1.45e+1	8.53e+0	1.68e+2	1.10e+3	3.34e+1	1.41e+1	4.88e+1	6.84e+1	1.23e+2	3.64e+1

The proposed SPO-CL1 algorithm was evaluated against eleven state-of-the-art metaheuristic algorithms on the CEC-2022 benchmark suite of 12 optimization functions. Each algorithm was run 30 times independently with a population of 30 individuals over 250 iterations. The algorithms compared are AO, AOA, GWO, HHO, ZOA, COA, CSBO, IGWO, EWOA, MHHO, and HSMA. As reported in Table 7, SPO-CL1 achieves the best mean on five of twelve functions (F1, F2, F3, F9, F12) and remains competitive elsewhere. The advantage is sharpest on unimodal functions: a near-optimal mean of 6.01 × 10² on F3 and the lowest standard deviation across all algorithms on F2 (1.03 × 10⁰) reflect both solution quality and exceptional consistency. On hybrid functions (F7–F9), SPO-CL1 ranks in the top two throughout, peaking at 2.53 × 10³ on F9. F6 is the one function where it yields ground while AOA and CSBO post catastrophic means of 4.20 × 10⁸ and 2.26 × 10⁷, exposing their vulnerability to multimodal landscapes. On composition functions (F10–F12), SPO-CL1 leads on F12 (2.86 × 10³) and finishes in the top three on F10–F11; EWOA, competitive on F6, deteriorates sharply here. IGWO and GWO are the most credible rivals throughout, while AOA and CSBO consistently trail. Across all function types, the complementary action of chaotic initialization, OBL, and Lévy perturbations sustains both exploration breadth and solution quality (Figure 6and figure 7).

Figure 6.

Convergence graphs.

Figure 7.

Boxplot of fitness values obtained by the optimisers studied.

Beyond solution quality, computational overhead is a practical consideration in algorithm selection. All twelve algorithms were executed under identical conditions (population size = 30, MaxIter = 250, 30 independent runs, Intel Core i7-8550U, 1.80 GHz, 16 GB RAM). The average wall-clock time per run on the CEC-2022 suite was comparable across most algorithms, ranging from approximately 0.8 s (GWO, ZOA) to 1.4 s (AOA, HSMA). SPO-CL1 recorded an average runtime of 1.1 s per run, reflecting a modest overhead relative to the standard SPO (0.9 s) attributable to the OBL evaluation and the periodic Lévy perturbation step. This overhead remains negligible in the context of the robotic application, where each function evaluation already requires a full closed-loop simulation via RK4 integration, making the optimizer’s internal cost a minor fraction of the total computation time. The overall complexity of SPO-CL1 is O (Tmax · Np · nc), consistent with most population-based metaheuristics of comparable design.

Nonparametric statistical analysis

Wilcoxon tests (α = 0.05) confirm the dominance of SPO-CL1: 12 out of 12 victories against AOA, COA and CSBO, and at least 10 victories against the other algorithms, with only one defeat on F6. The majority of the results are highly significant (p < 0.001), demonstrating consistent and non-random superiority. Combined with the Friedman ranking, SPO-CL1 emerges as the overall best-performing algorithm on CEC-2022 (Figure 8and figure 9).

Figure 8.

Wilcoxon Rank-Sum Test.

Figure 9.

Wilcoxon Test Summary.

The Friedman test (χ² = 68.33, df = 11, p < 0.0001) confirms significant differences among algorithms. SPO-CL1 achieves the best average rank (1.83), followed by GWO (4.00) and IGWO (4.17), while AOA (10.75) and CSBO (9.67) rank last.

The Nemenyi test (CD = 3.79) confirms that SPO-CL1 is significantly superior to AOA, CSBO, EWOA, HHO, MHHO and COA. However, the differences compared with AO, GWO, IGWO, ZOA and HSMA remain non-significant despite competitive performance (Figure 10and figure 11).

Figure 10.

Freidman Test – Average Ranks.

Figure 11.

Friedman Test – Rank per Function.

Both tests confirm that the incorporation of chaos, OBL and Lévy disturbances significantly improves performance. SPO-CL1 achieves the best Friedman rank (1.83) and performs best in the Wilcoxon comparisons. Validation on a real-world case is analysed in the section “Path planning simulation experiment”.

Path planning simulation experiment

To assess practical applicability, a path planning simulation is conducted on the 3-DoF manipulator of the section “Robot modelling”, where each algorithm tunes the six SMC gains by minimizing the ISE cost function (Eq. 24):

24

where e(t) = [e₁(t), e₂(t), e₃(t)]ᵀ = θ(t) − θᵈ(t) is the vector of joint tracking errors between the actual joint angles θ(t) and the desired joint angles θᵈ(t) at each time step. This scalar criterion directly penalizes trajectory deviations across all three joints simultaneously, making it a comprehensive and physically meaningful fitness function for controller tuning. The SMC control law incorporates a hyperbolic tangent boundary layer to attenuate chattering:

25

where s = ė + Λe is the sliding surface vector, Λ = diag(λ₁, λ₂, λ₃), K = diag(K₁, K₂, K₃), and φ = 0.05 is the boundary layer thickness. The numerical integration is performed using a 4th-order Runge-Kutta scheme with time step dt = 0.01 s over a simulation horizon T = 5 s.

To provide a more demanding test than a simple circular path, the reference trajectory adopted here is a Lemniscate of Bernoulli (figure-eight curve), parameterized in Cartesian space as:

26

with cₓ = 0.75 m, c_γ = 0 m, a = 0.15 m and ω = 2π/T. The desired joint-space trajectory θᵈ(t) is obtained via analytical inverse kinematics applied to the Cartesian lemniscate. A realistic initial joint perturbation of Δθ(0) = [0.05; 0.03; 0.02] rad is imposed to evaluate the transient rejection capability of each controller.

The twelve algorithms compared in the section “Comparison with other optimizers” are evaluated: SPO-CL1, AO, AOA, GWO, HHO, ZOA, COA, CSBO, IGWO, EWOA, MHHO, and HSMA, each using a population of 50 individuals, 30 iterations, and 10 independent runs. using the parameters recommended in their respective publications and summarized in Table 6, Statistical results are reported in Table 8; optimal gain configurations in Table 9.

Table 8.

Statistical comparison of ISE performance over 10 independent runs.

Algorithm	Best	Mean	Median	Std
SPO-CL1	1.51 × 10⁻⁴	1.56 × 10⁻⁴	1.56 × 10⁻⁴	6 × 10⁻⁶
AO	1.56 × 10⁻⁴	1.65 × 10⁻⁴	1.65 × 10⁻⁴	4 × 10⁻⁶
AOA	1.68 × 10⁻⁴	1.98 × 10⁻⁴	1.87 × 10⁻⁴	3.1 × 10⁻⁵
GWO	1.57 × 10⁻⁴	1.63 × 10⁻⁴	1.64 × 10⁻⁴	2 × 10⁻⁶
HHO	1.56 × 10⁻⁴	1.72 × 10⁻⁴	1.64 × 10⁻⁴	2.0 × 10⁻⁵
ZOA	1.72 × 10⁻⁴	2.00 × 10⁻⁴	1.97 × 10⁻⁴	1.9 × 10⁻⁵
COA	1.89 × 10⁻⁴	2.33 × 10⁻⁴	2.34 × 10⁻⁴	3.1 × 10⁻⁵
CSBO	1.59 × 10⁻⁴	1.73 × 10⁻⁴	1.74 × 10⁻⁴	8 × 10⁻⁶
IGWO	1.55 × 10⁻⁴	1.64 × 10⁻⁴	1.65 × 10⁻⁴	3 × 10⁻⁶
EWOA	1.61 × 10⁻⁴	1.70 × 10⁻⁴	1.67 × 10⁻⁴	1.1 × 10⁻⁵
MHHO	1.63 × 10⁻⁴	1.82 × 10⁻⁴	1.79 × 10⁻⁴	1.5 × 10⁻⁵
HSMA	2.62 × 10⁻⁴	4.51 × 10⁻⁴	4.44 × 10⁻⁴	1.28 × 10⁻⁴

Table 9.

Optimal SMC gain parameters identified by each algorithm.

Algorithm	λ₁	λ₂	λ₃	K₁	K₂	K₃
SPO-CL1	20.00	20.00	20.00	70.70	18.50	19.38
AO	20.00	20.00	19.69	92.88	17.74	13.72
AOA	20.00	20.00	11.42	68.49	12.03	5.00
GWO	20.00	16.49	19.70	70.80	16.07	11.77
HHO	20.00	20.00	20.00	70.42	12.90	12.89
ZOA	18.05	16.64	14.23	28.50	35.70	14.83
COA	17.53	12.14	17.26	36.04	9.95	10.39
CSBO	20.00	14.72	17.72	69.06	15.45	20.15
IGWO	20.00	19.87	17.11	71.40	20.13	21.21
EWOA	20.00	20.00	20.00	31.37	27.88	27.55
MHHO	19.78	19.70	19.46	30.98	27.63	28.76
HSMA	15.08	6.49	7.23	25.61	5.10	5.64

Table 8 confirms SPO-CL1 as the top performer, with a best ISE of 1.51 × 10⁻⁴ and a mean of 1.56 × 10⁻⁴ across ten runs. IGWO follows closely (Best = 1.55 × 10⁻⁴, Mean = 1.64 × 10⁻⁴), with AO, HHO, and EWOA forming a competitive mid-tier. At the other end, HSMA’s mean of 4.51 × 10⁻⁴ and standard deviation of 1.28 × 10⁻⁴ signal severe instability and failure to converge reliably on this problem.

Examining the optimal parameters in Table 9 reveals a physically meaningful structure: SPO-CL1 consistently converges to the configuration λ₁ = λ₂ = λ₃ = 20 and K₁ ≈ 71, K₂ ≈ 19, K₃ ≈ 19. The saturation of all λ at their upper bounds indicates the need for a steep sliding surface to handle the lemniscate’s rapid direction changes, while the dominance of K₁ reflects higher inertia and gravity effects on joint 1. This parameter pattern is also obtained by IGWO and AO, confirming convergence to the global optimum of the ISE objective.

In contrast, COA converges to lower gains (λ ≈ 12, K₁ = 36), resulting in a 54% higher mean ISE. HSMA shows highly dispersed parameters (e.g., λ₃ = 7.23, K₂ = 5.10), insufficient to maintain the sliding surface, leading to the largest steady-state tracking errors (Figs 12, 13, 14, 15, 16 and 17).

Figure 12.

Time evolution of the control torque for the joint 1.

Figure 13.

Time evolution of the control torque for the joint 1 for all algorithms.

Figure 14.

Time evolution of the control torque for the joint 2.

Figure 15.

Time evolution of the control torque for the joint 2 for all algorithms.

Figure 16.

Time evolution of the control torque for the joint 3.

Figure 17.

Time evolution of the control torque for the joint 3 for all algorithms.

The torque profiles from SPO-CL1 follow a physically consistent pattern throughout the simulation. A sharp transient spike in the first 0.3 s reflects the controller recovering from the imposed initial perturbation, but bounded by the tanh layer (φ = 0.05). The subsequent steady-state regime is not uniform: the envelope narrows at t ≈ 2.5 s as the end-effector crosses the lemniscate’s central node and curvature peaks, then widens through the outer lobes and rises slightly after t = 3 s where joint velocities are higher. Joint 1 dominates in amplitude (peak ≈ 80 N·m) due to its inertia and gravitational load; joint 3 remains the smallest (≈ 18 N·m). COA (K₁ = 36) and ZOA (K₁ = 29) produce smoother, narrower envelopes. HSMA generates the most erratic profiles across all algorithms, with inter-run variance that is fully consistent with the convergence instability reported in Table 8.

For SPO-CL1 (Figs 18, 20 and 22), joint errors converge rapidly: decreases from 0.05 to < 3 × 10⁻³ rad in 0.3 s, from 0.03 to < 1 × 10⁻³ rad in 0.3 s, and from 0.02 to < 5 × 10⁻⁴ rad in 0.2 s. A small oscillation at s (lemniscate crossing) remains bounded (< 4 × 10⁻³ rad) and decays within 0.5 s. IGWO and AO show nearly identical tracking performance, confirming near-optimal tuning. In contrast, HSMA and COA exhibit 3–5× larger residual errors; notably, HSMA maintains rad due to a suboptimal , resulting in insufficient sliding surface attraction (Figs 19, 21 and 23).

Figure 18.

Tracking error response of the joint 1.

Figure 20.

Tracking error response of the joint 2.

Figure 22.

Tracking error response of the joint 3.

Figure 19.

Tracking error response of the joint 1 for all algorithms.

Figure 21.

Tracking error response of the joint 2 for all algorithms.

Figure 23.

Tracking error response of the joint 3 for all algorithms.

Figure 24 shows that SPO-CL1 tracks the lemniscate with great accuracy, with a corrected initial error of less than 0.3 s. Figure 25 indicates that IGWO, AO, GWO and EWOA offer similar performance, whilst HSMA and COA exhibit noticeable errors, primarily due to insufficient gains.

Figure 24.

X-Y plot of end-effector for SPO-CL1.

Figure 25.

X-Y plot of end-effector for all algorithms.

Figure 26 distills the experiment into a single telling comparison. SPO-CL1 reduces ‖s(t)‖ from 1.3 to below 0.2 in under 0.3 s and sustains that level for the full simulation duration. COA, ZOA, and HSMA settle instead between 0.4 and 0.6, remaining outside the boundary layer throughout and never fully satisfying the tracking objective. HSMA is the outlier in both magnitude and oscillation, a pattern that traces directly back to the convergence failures documented in Table 8. The broader picture is equally clear. By consistently identifying the gain structure λ₁ = λ₂ = λ₃ = 20 and K₁ ≈ 71, SPO-CL1 achieves the lowest ISE, fastest error decay, and tightest end-effector trajectory among all twelve algorithms with near-zero inter-run variance. Critically, these are not benchmark artifacts: the same properties that drove SPO-CL1’s CEC-2022 performance prove just as effective on a physically grounded, kinematically demanding robotic problem.

Figure 26.

Convergence of the sliding surface ‖‖s(t)‖‖ under different optimization algorithms.

Conclusion

This work presented a unified optimization-control framework in which SPO-CL1 automatically tunes the six SMC gains (λ₁, λ₂, λ₃, K₁, K₂, K₃) for a 3-DoF manipulator by minimizing J = ∫₀ᵀ eᵀe dt, evaluated through full closed-loop simulation via 4th-order Runge-Kutta integration. On CEC-2022, SPO-CL1 achieves the best Friedman rank of 1.83 among twelve algorithms, with Wilcoxon rank-sum tests (α = 0.05) confirming statistically significant superiority over the majority of competitors. The robotic validation on a Lemniscate of Bernoulli trajectory shows that, out of the twelve algorithms, SPO-CL1 has the narrowest end-effector trajectory, the fastest tracking error convergence, and the lowest ISE of 1.51 × 10⁻² with almost nil inter-run variance. While COA and HSMA show noticeably higher ISE values because of inferior λ and K configurations, IGWO and AO are the strongest competitors among the analyzed methods, continuously recovering near-optimal gain configurations. These findings demonstrate that SPO-CL1 is a competitive and dependable method for automatically adjusting SMC gain in intricate robotic applications. Future research directions include evaluation on 6-DoF manipulators with 3D trajectories, real-time experimental validation on a physical platform, and extension to multi-objective formulations considering energy consumption and actuator saturation. Extending the comparison to include surrogate-assisted or reinforcement learning-based optimizers also represents a natural direction for future investigation. As with most simulation-based SMC tuning studies, the simplified rigid-link 3-DoF model and the absence of hardware validation limit direct transferability to real platforms, motivating the experimental extension identified as primary future work.

Author contributions

Hamza Tahiri : Writing – original draft, Software, Formal analysis, Conceptualization. Mohammed Amine Tahiri : Writing – original draft, Software, Formal analysis, Conceptualization. Mhamed Sayyouri : Visualization, Resources, Investigation, Funding acquisition, Data curation. Eman Abdullah Aldakheel : Visualization, Resources, Investigation, Funding acquisition, Data curation. Doaa Sami Khafaga : Visualization, Resources, Investigation, Funding acquisition, Data curation.

Funding

This research was funded by the Princess Nourah bint Abdulrahman University Researchers Supporting Project (PNURSP2026R409), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.