Adaptive H-Infinity MPC-Fuzzy Control Integrated With Sliding-Mode Disturbance Observer for Trajectory Tracking of a 5-DOF Robotic Manipulator With Gripper

Abstract

Existing control schemes for 5-DOF robotic manipulators suffer from fundamental limitations: conventional PID controllers cannot handle nonlinear dynamics and coupling effects, adaptive methods exhibit slow response to sudden parameter changes, robust H∞ controllers sacrifice performance for worst-case robustness, and intelligent methods lack formal stability guarantees. These limitations become critical in high-precision applications requiring simultaneous handling of model uncertainties, external disturbances, nonlinearities, and physical constraints. This paper addresses these challenges by proposing a novel integrated control architecture that systematically combines four complementary control methodologies within a unified framework: (1) adaptive H-infinity control providing formal robustness guarantees, (2) Model Predictive Control enabling constraint handling and optimization, (3) Fuzzy Logic control managing complex nonlinearities, and (4) Sliding Mode Control with enhanced disturbance observer for high-frequency noise rejection. The primary contribution is a novel adaptive weighting mechanism that dynamically adjusts each controller's contribution based on real-time performance metrics using exponential functions of tracking error magnitude, ensuring smooth transitions and optimal resource allocation without hard-switching limitations. The integrated framework employs a comprehensive mathematical analysis including composite Lyapunov stability proofs to guarantee asymptotic convergence of tracking error and boundedness of control signals. Rigorous theoretical analysis provides formal convergence guarantees and robustness bounds for parameter variations up to ±30%. The proposed approach is validated through extensive simulations conducted on MATLAB/Simulink using a detailed 5-DOF robotic manipulator model with clearly defined Denavit-Hartenberg parameters, mass distribution, and inertial properties. Simulation results demonstrate significant performance enhancement compared to conventional control methods. The proposed controller achieves superior tracking accuracy with RMS tracking error of 1.35 mm compared to 2.50 mm for H∞ control, 2.87 mm for adaptive control, and 5.58 mm for PD control, representing up to 46% improvement over the best existing robust control method. The system exhibits exceptional robustness with only 11.5% performance degradation under ±30% parameter variations compared to 31.2% degradation for H∞ control and 85.3% for PD control. Disturbance rejection capability is demonstrated with maximum tracking error of 3.21 mm and recovery time of 0.43 s under 5N external force disturbance, significantly outperforming conventional methods.

Keywords

hybrid control trajectory tracking 5-DOF robotic manipulator adaptive weighting MATLAB/Simulink

1. Introduction

Robotic manipulators are increasingly deployed in complex environments, demanding precise trajectory tracking despite uncertainties, external disturbances, and parameter variations. Traditional control methods often struggle to maintain high performance under these challenging conditions, particularly for multi-degree-of-freedom (DOF) systems with complex kinematics and dynamics. The control challenges are further amplified when considering the robot's full operational range, including high-speed movements, variable loads, and interactions with uncertain environments.

Conventional control strategies typically address specific aspects of the control problem individually. PID controllers offer simplicity but often fail to effectively handle nonlinearities and coupling effects. Model-based methods, such as computed torque control, provide enhanced performance but heavily rely on accurate system modeling. Adaptive controllers manage parametric uncertainties but may react slowly to sudden changes. Robust control methods effectively handle disturbances but typically at the expense of reduced nominal performance. Recent intelligent control approaches using neural networks or fuzzy systems have demonstrated potential for handling nonlinearities, though often lacking formal stability guarantees.

This paper addresses the above limitations by proposing a novel integrated control framework that combines the strengths of multiple control methodologies while mitigating their individual weaknesses. Key innovations in our approach include:

–
An adaptive H-Infinity control component ensuring robust performance against uncertainties and disturbances with formal stability guarantees.
–
A Model Predictive Control (MPC) component optimizing future control actions based on predicted system behavior while respecting physical constraints.
–
A Fuzzy Logic component managing nonlinearities using linguistic rules derived from expert knowledge.
–
A Sliding Mode Control (SMC) component featuring an innovative disturbance monitoring and compensation mechanism to effectively mitigate high-frequency noise.
–
An adaptive weighting mechanism dynamically adjusting each control component's contribution based on real-time performance metrics.

The robotic platform used for validation through simulation is a 5-DOF robotic manipulator with an attached gripper (effectively controlled through six servo-driven joints), constructed of aluminum with clearly defined Denavit-Hartenberg parameters, mass distribution, and inertial properties. We present comprehensive mathematical modeling, controller design, stability analysis, and simulation validation to demonstrate the effectiveness of the proposed approach.

This paper presents four unique contributions that distinguish it from existing robotic control approaches. The first contribution is a comprehensive four-component integration that surpasses existing methods, which typically combine at most 2–3 controllers with limited theoretical analysis. In contrast, this work introduces the first systematic integration of four complementary control methodologies, namely adaptive H∞, Model Predictive Control (MPC), Fuzzy Logic Control (FLC), and Sliding Mode Control (SMC) with a disturbance observer. This integration comes with complete formal stability guarantees, addressing the fundamental limitation of single-method approaches that cannot simultaneously handle model uncertainties, external disturbances, nonlinear dynamics, physical constraints, and optimization requirements.

The second contribution involves a novel, mathematically rigorous dynamic adaptive weighting algorithm that uses exponential functions of tracking error magnitude. Specifically, the algorithm is expressed as $α_{i} = \frac{exp (- β_{i} ‖ e ‖^{2})}{\sum_{j} exp (- β_{j} ‖ e ‖^{2})}$ , ensuring smooth transitions and automatic optimization without the need for manual tuning or hard-switching discontinuities. This mechanism eliminates performance jumps during mode transitions and provides optimal resource allocation based on real-time system performance.

The third contribution introduces a new nonlinear disturbance observer, designed to achieve exponential convergence. The observer is expressed as ${\hat{τ}}_{a} = z + p (q, \dot{q})$ with $\dot{z} = - L (z + p + M^{- 1} (τ - C \dot{q} - G))$ , and is systematically integrated across all control components rather than operating independently. This integration provides superior disturbance rejection capabilities, with an average convergence time of 123 ms, compared to standalone observer implementations.

The fourth contribution presents a complete theoretical framework. This is the first rigorous stability analysis of a four-component integrated controller, using composite Lyapunov functions defined as $V_{total\;} = V_{H \infty} + V_{parameter\;} + V_{observer\;} + V_{MPC}$ . This analysis provides formal convergence rate bounds, robustness margins, and performance specifications for the entire integrated system, thereby addressing the critical theoretical gap in multi-component control system analysis.

Together, these contributions significantly advance the state-of-the-art by providing a comprehensive solution that addresses the full spectrum of robotic control challenges within a unified and theoretically sound framework.

The remainder of this paper is structured as follows: Section 2 reviews related research on robotic manipulator control. Section 3 details the 5-DOF robotic manipulator model. Section 4 formulates the control problem with specific objectives and constraints. Section 5 elaborates on the proposed control methodology and mathematical analysis. Section 6 provides rigorous stability analysis. Section 7 presents extensive simulation results. Section 8 discusses the strengths and limitations of the proposed method, and Section 9 concludes the paper.
2. Literature Review

2.1 Classical Control Methods

Robotic manipulator control has developed significantly over decades. Early approaches relied on independent joint control using PID controllers (Astrom & Hagglund, 2006), which remain prevalent in industrial applications due to their simplicity and reliability. However, PID controllers encounter difficulties with nonlinearities and coupling effects inherent in multi-degree-of-freedom robot dynamics.

Computed torque control (Spong et al., 2006) addresses these limitations by utilizing dynamic models to compensate for nonlinear effects. This method performs well when the model is accurate but degrades significantly under model uncertainties. Spong et al. (2006) demonstrated that computed torque control could achieve asymptotic tracking for robotic manipulators under perfect model knowledge conditions but requires enhancements to handle uncertainties.

Fossen et al. (2009) developed similar model-based techniques for underwater robotic systems, emphasizing the complexity of control in environments with multiple disturbances and nonlinearities. Similarly, Kreutz (1989) proposed exact linearization methods for complex robotic systems, but still required precise model knowledge.

2.2 Robust and Adaptive Control Strategies

Robust control methods aim to maintain performance despite bounded uncertainties. H-infinity control, as studied by Zhou and Doyle (1998), provides formal guarantees for worst-case disturbance attenuation but may sacrifice nominal performance to achieve robustness. Chen and Gao (2019) applied H-infinity control to a 2-DOF robotic manipulator, achieving disturbance rejection with guaranteed attenuation levels, but reported conservative control gains due to worst-case design considerations.

Adaptive control addresses parameter uncertainties through online parameter estimation. Slotine and Li (1991) developed adaptive control schemes for robotic manipulators ensuring global stability despite uncertain inertial parameters. These methods demonstrate excellent convergence properties but may respond slowly to abrupt parameter changes. Wang et al. (2020) proposed a composite adaptive controller combining tracking error and prediction error for faster parameter convergence, achieving a 40% improvement in convergence speed compared to conventional adaptive control.

Zhang et al. (2018) developed hybrid adaptive methods combining neural networks with classical adaptive controllers to improve responsiveness to parameter variations. Similarly, Cheah et al. (2006) proposed an integrated adaptive framework for force and position control tasks, emphasizing the importance of multi-objective solutions for complex robotic systems.

2.3 Predictive Control Approaches

Model Predictive Control (MPC) optimizes future control actions by predicting system behavior over a finite horizon. Camacho and Bordons (2007) provided a comprehensive framework for MPC implementation. For robot control, MPC offers the advantage of explicitly handling state and input constraints. Tang et al. (2020) implemented MPC for a 6-DOF industrial robot, reporting improved trajectory tracking while satisfying joint and actuator constraints. However, they noted computational challenges for real-time implementation at high sampling rates.

Kayacan et al. (2016) extended classical MPC concepts with fuzzy predictive models, allowing better application to nonlinear systems. Similarly, Avanzini et al. (2018) developed computationally efficient MPC methods specifically for robotic manipulators, addressing the critical challenge of computation time.

2.4 Intelligent Control Methods

Intelligent control paradigms employ techniques from artificial intelligence to handle complex nonlinear systems. Fuzzy control, as described by Passino and Yurkovich (1998), translates expert knowledge into linguistic rules for control decisions. Neural networks provide learning capabilities to approximate complex nonlinear functions. Rahmani and Rahman (2022) developed a neural network controller for a 7-DOF robotic manipulator learning inverse dynamics online, demonstrating improved performance over model-based methods when dynamics are uncertain.

Chao et al. (2019) focused on optimization techniques for fuzzy parameters to enhance overall control performance. Zhang and Mo (2021) explored the integration of deep learning techniques into robotic control frameworks, showing significant improvements in handling complex nonlinearities.

2.5 Hybrid and Integrated Control Approaches

Recent research has focused on approaches combining multiple control methodologies. Jung Seul (2018) integrated sliding mode control with neural networks to combine robust performance with learning capabilities. Their method showed a 35% reduction in tracking error compared to conventional sliding mode control but required significant computational resources.

Ferrentino et al. (2023) combined MPC with adaptive mechanisms for robotic manipulators, achieving constraint satisfaction while adapting to parameter variations. Their method demonstrated superior performance in constrained environments but required careful tuning of multiple parameters.

Yang and Chen (2015) investigated disturbance observer-based control for robotic systems, emphasizing the effectiveness of explicitly estimating and compensating for disturbances. Their analysis showed that disturbance observers significantly improve disturbance rejection, particularly for low-frequency disturbances, but may amplify measurement noise if not properly designed.

Kharabian and Mirinejad (2020) proposed an adaptive fuzzy sliding mode controller with H-infinity performance for robotic manipulators. Their method addresses both parameter uncertainties and external disturbances but does not incorporate predictive capabilities or consider physical constraints.

Additionally, Chettibi Taha (2024) explored multi-objective optimization methods for trajectory tracking problems using hybrid evolutionary algorithms, providing complementary perspectives on control structure optimization. Yin et al. (2019) developed machine learning-based energy-efficient control techniques for robotic systems, expanding the range of control objectives considered.

2.6 Research Gaps and Contributions

Current robotic control methodologies suffer from inherent limitations that hinder optimal performance under diverse operating conditions. Traditional PID and PD controllers, while widely used in industrial applications due to their simplicity, struggle to effectively handle the nonlinearities and coupling effects present in multi-degree-of-freedom robotic systems (Astrom and Hagglund, 2006). These linear controllers are based on the assumption of system linearity and fail to compensate for complex joint interactions, leading to degraded performance during dynamic movements. Adaptive control methods, such as those developed by Slotine and Li (1991) and further enhanced by Wang et al. (2020), offer the ability to handle parameter uncertainties through online estimation. However, these methods exhibit characteristically slow responses to sudden parameter changes and can become unstable when there are rapid system variations.

Robust control approaches, particularly H∞ control as described by Zhou and Doyle (1998) and applied to robotics by Chen and Gao (2019), provide formal robustness guarantees against bounded uncertainties. However, they often result in conservative performance due to their worst-case design philosophy. Model Predictive Control (MPC), as comprehensively covered by Camacho and Bordons (2007) and applied to robotics by Tang et al. (2020), offers optimal control capabilities with explicit constraint handling. Despite this, MPC suffers from high computational costs and limited disturbance rejection for high-frequency disturbances.

Recent attempts at dual-method combinations have sought to overcome the limitations of individual approaches but still face fundamental constraints. Zhang et al. (2020) developed a combination of SMC and Fuzzy Logic in 2022, which lacks predictive capabilities and formal H∞ performance guarantees. Ferrentino et al. (2023) proposed a combination of MPC and Adaptive control in 2021 but without the integration of a disturbance observer and with limited robustness analysis. Wang et al. (2020) combined Adaptive and Neural Network control in 2020, introducing computational complexity without incorporating the robustness benefits of sliding mode control.

From a comprehensive analysis of the literature, four critical research gaps emerge. First, no existing work systematically integrates more than three control methods within a unified theoretical framework while providing formal stability analysis for the complete integrated system. Second, current integration methods often rely on hard-switching mechanisms or fixed weighting schemes, which lead to performance discontinuities and suboptimal resource allocation during mode transitions. Third, existing disturbance observers tend to operate independently, without systematic integration with multiple control strategies, limiting their effectiveness in comprehensive frameworks. Fourth, no existing research offers a complete stability analysis for multi-component integrated systems, with most studies focusing on individual component analysis and overlooking the interaction effects between them.

This research addresses these identified gaps through four distinct contributions. The first contribution is the comprehensive four-component integration, which represents the first systematic combination of adaptive H∞, MPC, Fuzzy Logic, and SMC with a disturbance observer within a single unified framework. The second contribution is the dynamic adaptive weighting mechanism, which uses exponential functions to ensure smooth transitions and automatic optimization without discontinuities. The third contribution is the enhanced disturbance observer design, which achieves exponential convergence while being systematically integrated across all control components. The fourth contribution is the complete theoretical framework, providing the first rigorous stability analysis for a four-component integrated controller using composite Lyapunov functions.

Recent developments in robotic control continue to explore specialized approaches to address specific challenges, although they lack comprehensive integration. He et al. (2018) recently developed learning-based model predictive control with neural network disturbance compensation, demonstrating improved performance but remaining limited to MPC-based frameworks. Fang et al. (2022) proposed adaptive neural sliding mode control with finite-time convergence, successfully combining neural networks with sliding mode but lacking predictive capabilities. Zhao and Wang (2022) investigated event-triggered adaptive control with a focus on communication efficiency rather than comprehensive performance optimization.

Wang et al. (2024) addressed distributed cooperative control with an emphasis on multi-robot coordination, but not on individual manipulator performance enhancement. Luo et al. (2023) developed reinforcement learning-enhanced sliding mode control, combining learning with robustness but requiring extensive training data. Recent hybrid approaches include Chakraborty and Raghuvanshi (2025), who combined fuzzy logic with deep reinforcement learning, and Chen et al. (2020), who developed adaptive barrier function control for collision avoidance. While these contributions advance specific aspects of robotic control, none provides the systematic four-component integration with comprehensive formal stability analysis presented in this work.

Additionally, recent studies have explored advanced control strategies for robotic and manipulator systems, focusing on optimizing performance through innovative controller designs. In the study “Hybrid spiral-bacterial foraging algorithm” by Kasruddin Nasir et al. (2022), a hybrid spiral-bacterial foraging algorithm (HSBFA) was employed to optimize fuzzy logic control parameters for a flexible manipulator system. This controller enhances the hub angle tracking precision by improving convergence speed and fitness accuracy, effectively balancing exploration and exploitation to avoid local optima. The research “Optimal fully actuated system approach” by Tian et al. (2024) utilized an optimal control framework based on a fully actuated system approach for trajectory tracking in robot manipulators. This controller ensures precise trajectory tracking by leveraging a tensorial state-space formulation, enhancing dynamic interaction modeling and operational efficiency. In “Application of improved sliding mode” by Pham et al. (2024), an improved sliding mode control integrated with artificial neural networks was proposed for robot control. This controller improves robustness and adaptability, enabling precise motion control under varying conditions and external disturbances. Lastly, “Adaptive learning control” by Xu and Wu (2024) introduced an incremental hybrid neural network-based controller for robot manipulators. This approach enhances adaptive learning capabilities, allowing the manipulator to handle complex, dynamic tasks with improved accuracy and stability through continuous learning.

2.7 Justification for Multi-Controller Integration

The justification for combining MPC, H-infinity, and fuzzy logic controllers stems from their complementary strengths and individual limitations when applied independently to robotic manipulator control.

Standalone MPC Limitations: While MPC excels in constraint handling and optimization, it suffers from computational complexity for real-time implementation and limited robustness to model uncertainties. Tang et al. (2020) reported that standalone MPC for 6-DOF robots requires simplifications that compromise performance under disturbances.

Standalone H-infinity Limitations: H-infinity control provides excellent robustness guarantees but often results in conservative performance due to worst-case design philosophy. Chen and Gao (2019) demonstrated that pure H-infinity control for robotic systems achieves only 60% of optimal tracking performance to maintain robustness margins.

Standalone Fuzzy Logic Limitations: Fuzzy controllers handle nonlinearities effectively but lack formal stability guarantees and systematic disturbance rejection capabilities. Rahmani and Rahman (2022) showed that standalone fuzzy control exhibits 40% performance degradation under external disturbances compared to model-based methods.

Integration Benefits: The proposed integration addresses these limitations systematically:

- MPC provides optimization and constraint handling during nominal conditions

- H-infinity ensures robustness guarantees throughout operation

- Fuzzy logic manages complex nonlinearities that linear methods cannot handle

- SMC with disturbance observer provides high-frequency disturbance rejection

This integration is superior to sequential switching approaches (Jung Seul, 2018) because it maintains all capabilities simultaneously through adaptive weighting, avoiding performance degradation during mode transitions.

Recent advancements in nonlinear control strategies have introduced notable improvements, yet they fall short of the comprehensive integration achieved in the proposed work. Sigmoid PID controllers, as discussed in (Ates et al., 2020), leverage sigmoid activation functions to enhance nonlinear compensation, achieving approximately 25% improved tracking performance compared to conventional PID controllers. However, these controllers exhibit limited disturbance rejection and lack robustness guarantees under significant parameter variations, restricting their applicability in dynamic environments. Brain Emotional Learning-Based Intelligent Controllers (BELBIC) PID variants, described in (Saeed et al., 2025), incorporate emotional learning mechanisms inspired by neurological processes, demonstrating up to 30% improvement in adaptive behavior for specific applications (Saeed et al., 2025). Nevertheless, their reliance on extensive training datasets and the absence of formal stability analysis for multi-degree-of-freedom (multi-DOF) systems limit their scalability. Neuroendocrine PID controllers, explored in (Ding et al., 2017), employ hormone-inspired adaptation mechanisms, offering superior responsiveness to changing conditions (Ding et al., 2017). However, their computational complexity and lack of integration with predictive control strategies hinder real-time implementation. Tilt Integral Derivative (TID) controllers, as presented in (Singh et al., 2020), replace traditional integral action with tilt compensation, providing enhanced disturbance rejection compared to standard PID controllers (Singh et al., 2020). Yet, they lack a systematic framework to address model uncertainties and physical constraints effectively. In contrast, the proposed integrated controller unifies the strengths of these methodologies, incorporating formal stability guarantees, robust disturbance rejection, and adaptive weighting mechanisms. This holistic approach ensures superior performance across diverse operating conditions, surpassing the limitations of these standalone variants.

3. Robotic Manipulator Modeling

3.1 Kinematic Model

The 5DOF + 1Gripper robotic manipulator is modeled using the standard Denavit-Hartenberg (D-H) convention. The D-H parameters define the relationship between consecutive joint frames and are provided in Table 1.

Table 1.
D-H Parameters of 5DOF Manipulator.

$J o i n t i$ $a_{i} (m m)$ $α_{i} (\circ)$ $d_{i} (m m)$ $θ_{i} (\circ)$

1 0 90 28 $θ_{1}$

2 150 0 0 $θ_{2}$

3 85 0 0 $θ_{3}$

4 50 90 0 $θ_{4}$

5 0 0 40 $θ_{5}$

$J o i n t i$	$a_{i} (m m)$	$α_{i} (\circ)$	$d_{i} (m m)$	$θ_{i} (\circ)$
1	0	90	28	$θ_{1}$
2	150	0	0	$θ_{2}$
3	85	0	0	$θ_{3}$
4	50	90	0	$θ_{4}$
5	0	0	40	$θ_{5}$

The gripper is modeled as an additional degree of freedom with variable opening width but fixed orientation relative to link 5. The homogeneous transformation matrix between consecutive frames is given by:

\begin{aligned} ^{i - 1} T_{i} = [\begin{array}{cccc} \cos θ_{i} & - \sin θ_{i} \cos α_{i} & \sin θ_{i} \sin α_{i} & a_{i} \cos θ_{i} \\ \sin θ_{i} & \cos θ_{i} \cos α_{i} & - \cos θ_{i} \sin α_{i} & a_{i} \sin θ_{i} \\ 0 & \sin α_{i} & \cos α_{i} & d_{i} \\ 0 & 0 & 0 & 1 \end{array}] \end{aligned}

(1)

For joint 1, substituting the D-H parameters ( $α_{1} = 0, α_{i} = 90 \circ, d_{i} = 28$ ):

\begin{aligned} ^{0} T_{1} = [\begin{array}{cccc} \cos θ_{1} & 0 & \sin θ_{1} & 0 \\ \sin θ_{1} & 0 & - \cos θ_{1} & 0 \\ 0 & 1 & 0 & 28 \\ 0 & 0 & 0 & 1 \end{array}] \end{aligned}

(2)

Similarly, for the remaining joints, we calculate:

\begin{aligned} ^{1} T_{2} = [\begin{array}{cccc} \cos θ_{2} & - \sin θ_{2} & 0 & 150 \cos θ_{2} \\ \sin θ_{2} & \cos θ_{2} & 0 & 150 \sin θ_{2} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] \end{aligned}

(3)

\begin{aligned} ^{2} T_{3} = [\begin{array}{cccc} \cos θ_{3} & - \sin θ_{3} & 0 & 85 \cos θ_{3} \\ \sin θ_{3} & \cos θ_{3} & 0 & 85 \sin θ_{3} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] \end{aligned}

(4)

\begin{aligned} ^{3} T_{4} = [\begin{array}{cccc} \cos θ_{4} & 0 & \sin θ_{4} & 50 \cos θ_{4} \\ \sin θ_{4} & 0 & - \cos θ_{4} & 50 \sin θ_{4} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}] \end{aligned}

(5)

\begin{aligned} ^{4} T_{5} = [\begin{array}{cccc} \cos θ_{5} & - \sin θ_{5} & 0 & 0 \\ \sin θ_{5} & \cos θ_{5} & 0 & 0 \\ 0 & 0 & 1 & 40 \\ 0 & 0 & 0 & 1 \end{array}] \end{aligned}

(6)

The complete forward kinematics from the base frame to the end-effector is:

\begin{aligned} ^{0} T_{5} =^{0} T_{1} \cdot^{1} T_{2} \cdot^{2} T_{3} \cdot^{3} T_{4} \cdot^{4} T_{5} \end{aligned}

(7)

This provides the position and orientation of the end-effector given the joint angles.

The Jacobian matrix maps joint velocities to end-effector velocities:

\begin{aligned} v = J (q) \dot{q} \end{aligned}

(8)

where

v = [v_{x}, v_{y}, v_{z}, ω_{x}, ω_{y}, ω_{z}]^{T}

represents the linear and angular velocities of the end-effector, and

J (q)

is the

6 \times 5

Jacobian matrix.

For revolute joints, the Jacobian columns are calculated as follows:

\begin{aligned} J_{i} = [\begin{matrix} z_{i - 1} \times (o_{5} - o_{i - 1}) \\ z_{i - 1} \end{matrix}] \end{aligned}

(9)

where

z_{i - 1}

is the rotation axis of joint i, and

o_{5} - o_{i - 1}

is the vector from the origin of frame

i - 1

to the end-effector.

3.2 Dynamic Model

The dynamic model of the robot is derived using the Euler-Lagrange formulation:

\begin{aligned} \frac{d}{d t} (\frac{\partial L}{\partial \dot{q}}) - \frac{\partial L}{\partial q} = τ + τ_{e} \end{aligned}

(10)

where

L = T - U

is the Lagrangian function, T is the kinetic energy, U is the potential energy,

τ

is the control input, and

τ_{e}

represents external forces/moments.

The resulting dynamic equation is:

\begin{aligned} M (q) \ddot{q} + C (q, \dot{q}) \dot{q} + G (q) + F (\dot{q}) + τ_{d} = τ \end{aligned}

(11)

where:

$q = [q_{1}, q_{2}, q_{3}, q_{4}, q_{5}]^{T} \in R^{5}$ is the joint position vecto r

$\dot{q} = [{\dot{q}}_{1}, {\dot{q}}_{2}, {\dot{q}}_{3}, {\dot{q}}_{4}, {\dot{q}}_{5}]^{T} \in R^{5}$ is the joint velocity vector (rad/s)

$\ddot{q} = {[{\ddot{q}}_{1}, {\ddot{q}}_{2}, {\ddot{q}}_{3}, {\ddot{q}}_{4}, {\ddot{q}}_{5}]}^{T} \in R^{5}$ is the joint acceleration vector $(rad / s^{2})$

$M (q) \in R^{5 \times 5}$ is the positive definite inertia matrix

$C (q, \dot{q}) \in R^{5 \times 5}$ contains Coriolis and centrifugal component s

$G (q) \in R^{5}$ is the gravity vector

$F (\dot{q}) \in R^{5}$ represents friction component s

$τ_{d} \in R^{5}$ represents external disturbances and unmodeled dynamics

$τ \in R^{5}$ is the control input vecto r

Note: Throughout this paper, we use the following notation conventions:

- Single dot ( $\dot{q}$ ) represents first time derivative (velocity)

- Double dot ( $\ddot{q}$ ) represents second time derivative (acceleration)

- Subscript i denotes the i-th joint component

- Bold symbols represent vectors and matrices

- Non-bold symbols represent scalars

The elements of the inertia matrix are constructed from the inertial properties of each link:

\begin{aligned} M_{ij} (q) \; = \; \sum_{k = max (i, j)}^{5} [m_{k} . {(J_{v_{k}}^{i})}^{T} . J_{v_{k}}^{j} + {(J_{w_{k}}^{i})}^{T} . I_{k} . J_{w_{k}}^{j}] \end{aligned}

(12)

where

J_{v_{k}}^{i}

and

J_{ω_{k}}^{i}

are the linear and angular Jacobian matrices, respectively, and

I_{k}

represents the inertia tensor of link k.

Using the provided inertial properties:

Link 1: $m_{1} = 0.40 \; kg$ ,

I_{1} = diag (0.007, 0.007, 0.0005) kg \cdot m^{2}

Link 2: $m_{2} = 0.25 \; kg$ ,

I_{2} = diag (0.005, 0.005, 0.0003) kg \cdot m^{2}

Link 3: $m_{3} = 0.20 \; kg$ ,

I_{3} = diag (0.004, 0.004, 0.0002) kg \cdot m^{2}

Link 4: $m_{4} = 0.15 \; kg$ ,

I_{4} = diag (0.003, 0.003, 0.0002) kg \cdot m^{2}

Link 5: $m_{5} = 0.20 \; kg$ ,

I_{5} = diag (0.002, 0.002, 0.0001) kg \cdot m^{2}

The Coriolis and centrifugal matrix is constructed using Christoffel symbols:

\begin{aligned} C_{i j} (q, \dot{q}) = \sum_{k = 1}^{5} Γ_{i j k} (q) \cdot {\dot{q}}_{k} \end{aligned}

(13)

where:

\begin{aligned} Γ_{i j k} (q) = \frac{1}{2} \cdot (\frac{\partial M_{i j} (q)}{\partial q_{k}} + \frac{\partial M_{i k} (q)}{\partial q_{j}} - \frac{\partial M_{j k} (q)}{\partial q_{i}}) \end{aligned}

(14)

The gravity vector is derived from the potential energy:

\begin{aligned} G_{i} (q) = \frac{\partial U (q)}{\partial q_{i}} = \sum_{k = i}^{5} m_{k} \cdot g \cdot \frac{\partial h_{k} (q)}{\partial q_{i}} \end{aligned}

(15)

where

h_{k} (q)

is the height of the center of mass of link k, and

g = 9.81 \; m / s^{2}

is the gravitational acceleration.

The friction model combines both Coulomb and viscous friction:

\begin{aligned} F_{i} (\dot{q}) = F_{c, i} \cdot sgn ({\dot{q}}_{i}) + F_{v, i} \cdot {\dot{q}}_{i} \end{aligned}

(16)

where

F_{c_{i}} \approx 0.02 \; N \cdot \; m

is the Coulomb friction torque and

F_{v_{i}} \approx 0.001 \; N \cdot \; m \cdot \; s / deg

is the viscous friction coefficient.

Remark 1:

The friction parameters are identified from experimental measurements and represent average values over the operating range. Actual friction may vary with temperature, load, and joint position.

The inertia tensor $I_{k}$ for each link k is determined through a systematic approach that combines geometric analysis, material properties, and experimental validation.

In the geometric analysis, each link is modeled as a combination of basic geometric shapes based on the actual robot construction. For instance, Link 1 is modeled as a cylindrical base with a radius of 25 mm and a height of 28 mm. Link 2 is a rectangular beam with dimensions of $150 \; mm \times 30 \; mm \times 25 \; mm$ , while Link 3 is another rectangular beam with dimensions of $85 \; mm \times 25 \; mm \times 20 \; mm$ . Link 4 is modeled as a cylindrical joint with a radius of 15 mm and a height of 50 mm. Finally, Link 5 is modeled as a cylindrical end-effector with a radius of 12 mm and a height of 40 mm.

Regarding material properties, all links are constructed from aluminum alloy 6061-T6, which is uniformly distributed across each link. The relevant material properties include a density of $ρ = 2700 \; kg / m^{3}$ , Young's modulus $E = 68.9 GPa$ , and Poisson's ratio $ν = 0.33$ .

The inertia tensor for each link is calculated based on standard formulas. For a rectangular beam with length l, width w, height h, and mass m, the components of the inertia tensor are given by:

I_{x x} = \frac{m}{12} (w^{2} + h^{2}), \; I_{y y} = \frac{m}{12} (l^{2} + h^{2}), \; I_{z z} = \frac{m}{12} (l^{2} + w^{2})

For a cylindrical shape with radius r, height h, and mass m, the inertia tensor components are calculated as:

I_{x x} = I_{y y} = \frac{m}{12} (3 r^{2} + h^{2}), \; I_{z z} = \frac{m r^{2}}{2}

The calculated inertia values are then validated through several approaches to ensure accuracy. This includes verification using CAD software, specifically SolidWorks, which models the links with identical geometry and material properties. In addition, physical pendulum tests are conducted on critical links, with a maximum error of less than $3 %$ . The calculated values are also compared with manufacturer specifications where available, and cross-validation is performed using different geometric decomposition methods.

This systematic approach guarantees an accurate dynamic model, which is essential for the proposed control design. Validation confirms that the inertia parameter accuracy remains within acceptable engineering tolerances.

3.3 Workspace Analysis and Singularities

The workspace of the robot is defined by joint limits and link lengths. The reachable workspace is approximately a sphere with radius 285 mm (combined length of links 2–5) centered at the first joint axis.

Singularities occur when the Jacobian matrix loses rank, resulting in a loss of controllability in certain directions. Three types of singularities are identified:

- Shoulder singularity: When the wrist center lies on the axis of the first joint (arbitrary $θ_{1}, θ_{2} + θ_{3} = 0$ )

- Elbow singularity: When the arm is fully extended or folded back $(θ_{3} = 0$ or $θ_{3} = 180^{\circ})$

- Wrist singularity: When the axes of joints 4 and 5 are aligned $(θ_{4} = 0$ or $θ_{4} = 180^{\circ})$

These singularities are considered in the controller design to ensure robust performance throughout the workspace.

According to the analysis by Siciliano and Khatib (2016), singularities in the robot workspace represent inherent structural limitations, requiring special strategies such as singularity avoidance techniques based on manipulability measures. Yoshikawa (1985) proposed a manipulability index that can be integrated into trajectory planning algorithms to avoid regions near singularities.

3.4 Model Validation and Verification

The 5-DOF robotic manipulator model is developed based on a laboratory prototype constructed at Gyeongsang National University's Hydraulic-Pneumatic and Marine Systems Laboratory, following Yahboom DOFBOT 5DOF specifications with modifications for research purposes (Herath et al., 2023). The physical prototype specifications are as follows:

Physical System Specifications:

- Base: Aluminum 6061-T6 alloy construction

- Joint actuators: Maxon EC-4pole 30 servo motors with planetary gearboxes

- Encoders: Incremental encoders with 4096 pulses/revolution

- Workspace: Spherical with 285 mm radius

Model-to-Reality Validation Process:

Geometric Validation: The Denavit-Hartenberg parameters were measured directly from the physical system using a FARO Arm coordinate measuring machine (CMM) with ±0.1 mm accuracy. The forward kinematics equations were verified against CAD model analysis using SolidWorks Professional 2023, showing maximum position error < 0.1 mm across the entire workspace.

Dynamic Parameter Identification: Inertial parameters were identified through systematic experiments:

- Compound pendulum tests for each link's moment of inertia (accuracy within 3% of theoretical values)

- Static torque measurements at 25 different configurations for gravity compensation validation

- Friction parameter identification through slow-motion experiments (0.1–10 deg/s)

Cross-Validation with Multi-body Simulation: The MATLAB/Simulink model was validated against Adams multibody simulation software, showing:

- Maximum torque prediction error < 2.5% for standard trajectories

- Dynamic response correlation > 98% for frequency range 0–50 Hz

- Model accuracy suitable for control system design

Experimental Correlation: Basic PD control was implemented on the physical prototype to validate model accuracy:

- 95% correlation between simulation and experimental tracking error for sinusoidal trajectories (0–1 Hz)

- Model prediction error < 5% for step responses

- Successful validation of the complete dynamic model for control design purposes

All geometric and inertial parameters were measured in-house. The prototype construction followed standard industrial robot design principles with custom modifications for research flexibility.

4. Problem Formulation

4.1 Assumptions

Assumption 1:
The desired trajectory $q_{d} (t), {\dot{q}}_{d} (t)$ , and ${\ddot{q}}_{d} (t)$ are bounded and continuously differentiable.

This assumption ensures that the reference trajectory is physically feasible and does not require infinite accelerations or jerks. Assumption 2:
External disturbances $τ_{d}$ are bounded by an unknown finite constant:
$‖ τ_{d} ‖ \leq ρ, ‖ {\dot{τ}}_{d} ‖ \leq ρ_{d}$
This assumption allows the design of a disturbance observer with finite estimation error.
Assumption 3:
Parameter uncertainties in $M (q), C (q, \dot{q})$ , and $G (q)$ can be represented as:
$M (q) = M_{0} (q) + Δ M (q) C (q, \dot{q}) = C_{0} (q, \dot{q}) + Δ C (q, \dot{q}) G (q) = G_{0} (q) + Δ G (q)$
where $M_{0} (q), C_{0} (q, \dot{q})$ , and $G_{0} (q)$ are nominal values, and the uncertainties satisfy: $∥ Δ M (q) ∥\leq δ_{M}, ∥ Δ C (q, \dot{q}) ∥\leq δ_{C}, ∥ Δ G (q) ∥\leq δ_{G}$ , with $δ_{M}, δ_{C}$ , and $δ_{G}$ being unknown bounded positive constants.
Assumption 4:
The inertia matrix $M (q)$ is positive definite and uniformly bounded:
$λ_{min} \cdot I \leq M (q) \leq λ_{max} \cdot I$
where $λ_{min}$ and $λ_{max}$ are positive constants, and I is the identity matrix.

4.2 Control Objectives

Given a desired trajectory $q_{d} (t) \in R^{5}, {\dot{q}}_{d} (t) \in R^{5}$ , and ${\ddot{q}}_{d} (t) \in R^{5}$ , the control objectives are:

Tracking Error Convergence: Ensure that the tracking error $e (t) = q_{d} (t) - q (t)$ converges to zero asymptotically or to a small bounded region: $li m_{t \to \infty} ∥ e (t) ∥\leq ε$ , where $ε$ is a small positive constant.

Disturbance Rejection: Maintain tracking performance in the presence of external disturbances and unmodeled dynamics: $∥ e ∥_{L_{2}} \leq γ \cdot ‖ τ_{d} ‖_{L_{2}}$ , where $γ$ is the disturbance attenuation level.

Parameter Adaptation: Adapt to parameter uncertainties by estimating unknown parameters: $li m_{t \to \infty} ∥ \tilde{θ} (t) ∥\leq δ$ , where $\tilde{θ} (t) = θ - \hat{θ} (t)$ is the parameter estimation error, and $δ$ is a small positive constant.

Control Effort Minimization: Minimize the control effort while achieving the above objectives: $mi n_{τ} \int_{0}^{T} ∥ τ (t) ∥^{2} d t$ , subject to achieving objectives 1–3.

4.3 Control Constraints

The robotic system is subject to the following constraints:

Joint Position Limits: $q_{min} \leq q \leq q_{max}$ , where:

Joint 1: $q_{1, min} = - 180^{\circ}, q_{1, max} = 180^{\circ}$

Joint 2: $q_{2, min} = - 120^{\circ}, q_{2, max} = 120^{\circ}$

Joint 3: $q_{3, min} = - 120^{\circ}, q_{3, max} = 120^{\circ}$

Joint 4: $q_{4, min} = - 180^{\circ}, q_{4, max} = 180^{\circ}$

Joint 5: $q_{5, min} = - 120^{\circ}, q_{5, max} = 120^{\circ}$

Joint Velocity Limits: $| {\dot{q}}_{i} | \leq {\dot{q}}_{i, max}, i = 1, 2, 3, 4, 5$ , where ${\dot{q}}_{i, max}$ typically ranges from 120–180 deg/s depending on the joint.

Joint Acceleration Limits: $| {\ddot{q}}_{i} | \leq {\ddot{q}}_{i, max}, i = 1, 2, 3, 4, 5$ , where ${\ddot{q}}_{i, max}$ typically ranges from 400–800 $deg / s^{2}$ depending on the joint.

Torque Limits: $| τ_{i} | \leq τ_{i, max}, i = 1, 2, 3, 4, 5$ , where $τ_{i, max}$ is determined by servo motor specifications.

Singularity Avoidance: $σ_{min} (J (q)) \geq σ_{0}$ , where $σ_{min} (J (q))$ is the minimum singular value of the Jacobian matrix, and $σ_{0}$ is a small positive threshold.

4.4 Integrated Control System Architecture

The proposed control architecture integrates four complementary control methodologies as shown in Figure 1. The system architecture consists of:

Figure 1.

Integrated Control Architecture for 5-DOF Robotic Manipulator with Gripper. (a) Overall System Architecture. (b) H∞ Controller. (c) Model Predictive Controller. (d) Fuzzy Logic Controller. (e) SMC + Disturbance Observer. Adaptive Weighting and Final Integration.

5. Proposed Control Methodology

Our novel control architecture integrates four main components:

–
Adaptive H-infinity controller
–
Model Predictive controller
–
Fuzzy Logic controller
–
Sliding Mode controller with Disturbance Observer
Each component addresses specific aspects of the control problem, and their integration provides a comprehensive solution.
5.1 Adaptive H-Infinity Control Component

The H-infinity component aims to minimize the effect of disturbances on tracking performance while adapting to parameter uncertainties.

Definition 1:
The H-infinity performance criterion is defined as:
$γ = sup_{ω \in R, τ_{d} \neq 0} \frac{∥ e (j ω) ∥}{‖ τ_{d} (j ω) ‖}$
where $γ$ is the disturbance attenuation level.

The control objective is to ensure that $γ < γ_{0}$ , where $γ_{0}$ is a prescribed level.

We propose an adaptive H -infinity controller of the form:
$τ_{H \infty} = \hat{M} (q) {\ddot{q}}_{r} + \hat{C} (q, \dot{q}) {\dot{q}}_{r} + \hat{G} (q) - K_{e} . e - K_{r} . r$
where:
${\dot{q}}_{r} = {\dot{q}}_{d} + Λ \cdot e$ is a reference velocity vecto r

$r = \dot{q} - {\dot{q}}_{r} = \dot{e} + Λ \cdot e$ is a composite error measure

$\hat{M} (q), \hat{C} (q, \dot{q})$ , and $\hat{G} (q)$ are parameter estimate s

$K_{e}$ and $K_{r}$ are positive definite gain matrices

The reference acceleration is:
${\ddot{q}}_{r} = {\ddot{q}}_{d} + Λ \cdot \dot{e}$

This controller structure combines the benefits of model-based compensation through adaptive parameters and robust feedback through error components.

The dynamic model can be linearly parameterized as:
$M (q) {\ddot{q}}_{r} + C (q, \dot{q}) {\dot{q}}_{r} + G (q) = Y (q, \dot{q}, {\dot{q}}_{r}, {\ddot{q}}_{r}) \cdot θ$
where $Y (q, \dot{q}, {\dot{q}}_{r}, {\ddot{q}}_{r})$ is the regression matrix and $θ$ is the parameter vector.

The parameter adaptation law is defined as:
$\dot{\hat{θ}} = - Γ \cdot Y^{T} (q, \dot{q}, {\dot{q}}_{r}, {\ddot{q}}_{r}) \cdot r$
where:
- $\hat{θ}$ is the vector of estimated parameter s

- $Γ$ is a positive definite adaptation gain matrix
For the 5DOF robotic manipulator, the parameter vector $θ$ includes inertial parameters, link masses, center of mass positions, and friction coefficients. The regression matrix $Y (q, \dot{q}, {\dot{q}}_{r}, {\ddot{q}}_{r})$ is constructed from the dynamic model by extracting the coefficients of these parameters.
Theorem 1:
Under Assumptions 1–4, the adaptive H-infinity controller ensures that:
- All signals in the closed-loop system remain bounded

- The tracking error converges asymptotically to zero in the absence of disturbances

- For bounded disturbances, the H-infinity performance criterion is satisfied with $γ < γ_{0}$ , where $γ_{0}$ is a prescribed constant

Proof:
Consider the Lyapunov function:
$V_{H \infty} = \frac{1}{2} \cdot r^{T} \cdot M (q) \cdot r + \frac{1}{2} \cdot {\tilde{θ}}^{T} \cdot Γ^{- 1} \cdot \tilde{θ}$
where $\tilde{θ} = θ - \hat{θ}$ is the parameter estimation error.

Taking the time derivative:
${\dot{V}}_{H \infty} = r^{T} M (q) \dot{r} + \frac{1}{2} r^{T} \dot{M} (q) r - {\tilde{θ}}^{T} Γ^{- 1} \dot{\hat{θ}} = r^{T} M (q) \dot{r} + \frac{1}{2} r^{T} \dot{M} (q) r + {\tilde{θ}}^{T} Y^{T} r$

From the error dynamics:
$M (q) \cdot \dot{r} + C (q, \dot{q}) \cdot r = Y \cdot \tilde{θ} - K_{e} \cdot e - K_{r} \cdot r + τ_{d}$

Substituting into the Lyapunov derivative:
$\begin{aligned} {\dot{V}}_{H \infty} & = r^{T} [Y \tilde{θ} - K_{e} e - K_{r} r + τ_{d} - C (q, \dot{q}) r] + \frac{1}{2} r^{T} \dot{M} (q) r + {\tilde{θ}}^{T} Y^{T} r \\ = r^{T} [- K_{e} e - K_{r} r + τ_{d}] + r^{T} Y \tilde{θ} - r^{T} C (q, \dot{q}) r + \frac{1}{2} r^{T} \dot{M} (q) r + {\tilde{θ}}^{T} Y^{T} r \end{aligned}$

Using the property that $M (q) - 2 \cdot C (q, \dot{q})$ is skew-symmetric:
${\dot{V}}_{H \infty} = - r^{T} \cdot K_{e} \cdot e - r^{T} \cdot K_{r} \cdot r + r^{T} \cdot τ_{d} \leq - λ_{min} (K_{e}) \cdot ∥ r ∥ \cdot ∥ e ∥ - λ_{min} (K_{r}) \cdot ∥ r ∥^{2} + ∥ r ∥ \cdot ‖ τ_{d} ‖$

To satisfy the H -infinity criterion, we need:
${\dot{V}}_{H \infty} + e^{T} \cdot Q \cdot e - γ_{0}^{2} \cdot τ_{d}^{T} \cdot τ_{d} \leq 0$
where Q is a positive definite weighting matrix.

We can establish that:
${\dot{V}}_{H \infty} + e^{T} Q e - γ_{0}^{2} τ_{d}^{T} τ_{d} \leq - λ_{m i n} (K_{e}) | r | | e | - λ_{m i n} (K_{r}) | r |^{2} + | r | | τ_{d} | + λ_{m a x} (Q) | e |^{2} - γ_{0}^{2} | τ_{d} |^{2}$

By completing the square for the term $∥ r ∥ \cdot ‖ τ_{d} ‖$ :
$∥ r ∥ \cdot ‖ τ_{d} ‖ \leq \frac{1}{2 α} \cdot ∥ r ∥^{2} + \frac{α}{2} \cdot ‖ {τ_{d}}^{2} ‖$
where $α > 0$ is a design parameter.

Substituting:
$\begin{aligned} {\dot{V}}_{H \infty} + e^{T} Q e - γ_{0}^{2} τ_{d}^{T} τ_{d} & \leq - λ_{m i n} (K_{e}) | r | e | - λ_{m i n} (K_{r}) | r |^{2} + \frac{1}{2 α} | r |^{2} + \frac{α}{2} | τ_{d} |^{2} + λ_{m a x} (Q) | e |^{2} - γ_{0}^{2} | τ_{d} |^{2} \\ = - λ_{m i n} (K_{e}) | r | e | - (λ_{m i n} (K_{r}) - \frac{1}{2 α}) | r |^{2} + λ_{m a x} (Q) | e |^{2} + (\frac{α}{2} - γ_{0}^{2}) | τ_{d} |^{2} \end{aligned}$

This expression is negative if:
$λ_{min} (K_{r}) > \frac{1}{2 α}$

$λ_{min} (K_{e})$ is large enough to ensure $- λ_{min} (K_{e}) \cdot ∥ r ∥ \cdot ∥ e ∥ + ∥ e ∥^{2} \cdot λ_{max} (Q) \leq 0$
$γ_{0}^{2} > \frac{α}{2}$

By choosing $K_{r} = \frac{1}{2} \cdot Λ^{T} \cdot Q \cdot Λ + \frac{1}{4 γ_{0}^{2}} \cdot I$ and $K_{e}$ such that $λ_{min} (K_{e}) \geq \frac{λ_{max} (Q)}{λ_{min} (Λ)}$ , and choosing $α = \frac{1}{γ_{0}^{2}}$ , we ensure that all conditions are satisfied.

Therefore, the H-infinity performance criterion is met with disturbance attenuation level $γ < γ_{0}$ . In the absence of disturbances, ${\dot{V}}_{H \infty} \leq - λ_{min} (K_{r}) \cdot ∥ r ∥^{2}$ , which ensures asymptotic convergence of r and thus e to zero.

The disturbance attenuation level $γ_{0}$ is systematically determined through performance requirements and design trade-off analysis.

For performance requirements, based on application specifications for high-precision manipulation, the maximum allowable tracking error is set to $e_{max} = 2.5 \; mm$ (or 0.0025 rad), and the expected maximum disturbance is $τ_{d, max} = 5 N \cdot m$ . From these, the required attenuation ratio is calculated as:
$γ_{0} = \frac{e_{max}}{τ_{d, max}} = 0.4$

In the trade-off analysis, the selection of $γ_{0}$ involves fundamental design trade-offs. A smaller $γ_{0}$ provides superior disturbance rejection but may result in higher control gains, leading to increased actuator effort and the potential for saturation. On the other hand, a larger $γ_{0}$ allows for smoother control actions but reduces the disturbance rejection capability.

The design procedure follows a systematic approach. The first step is to define the $H \infty$ performance criterion as:
$\frac{‖ e ‖_{L_{2}}}{‖ τ_{d} ‖_{L_{2}}} < γ_{0}$

Next, the relationship to control gains is established through Linear Matrix Inequality (LMI) conditions, given by:
$[\begin{array}{cc} - Q & C^{T} \\ C & - γ_{0}^{2} I \end{array}] < 0$

Finally, an optimization problem is solved to determine the minimum $γ_{0}$ subject to specific constraints, including:
Stability: $λ_{min} (K_{r}) > \frac{1}{2 γ_{0}^{2}}$

Actuator limits: $‖ τ ‖ \leq τ_{max}$

Bandwidth requirements: $ω_{c} \geq 10 rad / s$

For the 5-DOF manipulator system, the selected value for $γ_{0}$ is 0.35. This value ensures the following:
$35 %$ disturbance attenuation, meaning the tracking error is no more than 0.35 times the disturbance magnitude

Sufficient stability margins with reasonable control gains

Avoidance of actuator saturation under normal operating conditions

Adequate closed-loop bandwidth for effective tracking performance

The selected $γ_{0}$ provides an optimal balance between disturbance rejection capability and practical implementation constraints.
5.2 Model Predictive Control Component

For MPC implementation, we linearize the nonlinear dynamics around the operating point at each sampling instant. Let the state vector be $x = [q^{T}, {\dot{q}}^{T}]^{T} \in R^{10}$ and the control input be $u = τ \in R^{5}$ .

The linearized state-space model at time step k is:

\begin{matrix} x_{k + 1} = A_{k} \cdot x_{k} + B_{k} \cdot u_{k} + w_{k} \\ y_{k} = C_{k} \cdot x_{k} \end{matrix}

where:

$A_{k} = {\frac{\partial f}{\partial x} |}_{x_{k}, u_{k}}$ is the state transition matri x

$B_{k} = {\frac{\partial f}{\partial u} |}_{x_{k}, u_{k}}$ is the input matrix

$C_{k}$ is the output matrix (identity for full state feedback)

$w_{k}$ represents linearization error and external disturbances

For robot dynamics, the linearization matrices are constructed as:

\begin{matrix} A_{k} = [\begin{array}{cc} I_{5 \times 5} & T_{s} \cdot I_{5 \times 5} \\ - {M {(q_{k})}^{- 1} \cdot \frac{\partial G (q)}{\partial q} |}_{q_{k}} & I_{5 \times 5} - T_{s} \cdot M {(q_{k})}^{- 1} \cdot C (q_{k}, {\dot{q}}_{k}) \end{array}] \\ B_{k} = [\begin{matrix} 0_{5 \times 5} \\ T_{s} \cdot M {(q_{k})}^{- 1} \end{matrix}] \end{matrix}

where

T_{s}

is the sampling time.

The MPC optimization problem is formulated as:

min_{u_{k ∣ k}, \dots, u_{k + N p} - 1 ∣ k} \sum_{i = 0}^{N_{p} - 1} ‖ y_{k + i ∣ k} - y_{d, k + i} ‖_{Q}^{2} + ‖ u_{k + i ∣ k} ‖_{R}^{2} + ‖ Δ u_{k + i ∣ k} ‖_{S}^{2}

subject to:

System dynamics: $x_{k + i + 1 ∣ k} = A_{k} \cdot x_{k + i ∣ k} + B_{k} \cdot u_{k + i ∣ k}$

Input constraints: $u_{min} \leq u_{k + i ∣ k} \leq u_{max}$

State constraints: $x_{min} \leq x_{k + i ∣ k} \leq x_{max}$

Rate constraints: $Δ u_{min} \leq Δ u_{k + i ∣ k} \leq Δ u_{max}$

where:

$N_{p}$ is the prediction horizon

$Q, R$ , and S are positive definite weighting matrices

$Δ u_{k + i ∣ k} = u_{k + i ∣ k} - u_{k + i - 1 ∣ k}$ is the control incremen t

$y_{d, k + i} = [q_{d, k + i}^{T}, {\dot{q}}_{d, k + i}^{T}]^{T}$ is the desired state

For the unconstrained case, the optimal control sequence can be computed analytically:

U_{k}^{*} = {(H^{T} \cdot Q_{p} \cdot H + R_{p} + S_{p})}^{- 1} \cdot (H^{T} \cdot Q_{p} \cdot (Y_{d} - F \cdot x_{k}) - S_{p} \cdot Δ U_{k - 1})

where:

\begin{aligned} U_{k} & = {[u_{k ∣ k}^{T}, u_{k + 1 ∣ k}^{T}, \dots, u_{k + N_{p} - 1 ∣ k}^{T}]}^{T} \end{aligned}

\begin{aligned} Y_{d} & = {[y_{d, k + 1}^{T}, y_{d, k + 2}^{T}, \dots, y_{d, k + N_{p}}^{T}]}^{T} \end{aligned}

\begin{aligned} Q_{p} & = diag (Q, Q, \dots, Q) \end{aligned}

\begin{aligned} R_{p} & = diag (R, R, \dots, R) \end{aligned}

\begin{aligned} S_{p} & = diag (S, S, \dots, S) \end{aligned}

$F$ and H are the free and forced response matrices.

This is based on canonical models developed by Mayne et al. (2000) and extended for multivariable systems with constraints. Rawlings and Mayne (2009) proved that this analytical solution guarantees global optimality for the unconstrained case.

For the constrained case, the problem is formulated as a quadratic programming $(QP)$ problem:

min_{U_{k}} \frac{1}{2} \cdot U_{k}^{T} \cdot H_{q} \cdot U_{k} + f_{q}^{T} \cdot U_{k}

subject to:

A_{ineq\;} \cdot U_{k} \leq b_{ineq\;}

where:

\begin{aligned} H_{q} & = H^{T} \cdot Q_{p} \cdot H + R_{p} + S_{p} \end{aligned}

\begin{aligned} f_{q} & = - H^{T} \cdot Q_{p} \cdot (Y_{d} - F \cdot x_{k}) + S_{p} \cdot Δ U_{k - 1} \end{aligned}

$A_{ineq\;}$ and $b_{ineq\;}$ encode the input, state, and rate constraints

The MPC contribution to the control law is:

τ_{M P C} = u_{k ∣ k}^{*}

where

u_{k ∣ k}^{*}

is the first element of the optimal control sequence

U_{k}^{*}

5.3 Fuzzy Logic Control Component

The fuzzy controller manages nonlinearities and uncertainties by utilizing linguistic rules. We define five linguistic variables for each joint error $e_{i}$ and its rate of change ${\dot{e}}_{i}$ :

Negative Large (NL):

μ_{N L} (e_{i}) = {\begin{array}{ll} 1, & e_{i} \leq - 2 σ_{i} \\ \frac{- e_{i} - σ_{i}}{σ_{i}}, & - 2 σ_{i} < e_{i} < - σ_{i} \\ 0, & e_{i} \geq - σ_{i} \end{array}

Negative Small (NS):

μ_{N S} (e_{i}) = {\begin{array}{ll} \frac{e_{i} + 2 σ_{i}}{σ_{i}}, & - 2 σ_{i} < e_{i} < - σ_{i} \\ \frac{- e_{i}}{σ_{i}}, & - σ_{i} \leq e_{i} < 0 \\ 0, & \; otherwise\; \end{array}

Zero (ZE):

μ_{Z E} (e_{i}) = {\begin{array}{ll} \frac{e_{i} + σ_{i}}{σ_{i}}, & - σ_{i} < e_{i} < 0 \\ \frac{σ_{i} - e_{i}}{σ_{i}}, & 0 \leq e_{i} < σ_{i} \\ 0, & \; otherwise\; \end{array}

Positive Small (PS):

μ_{P S} (e_{i}) = {\begin{array}{ll} \frac{e_{i}}{σ_{i}}, & 0 < e_{i} < σ_{i} \\ \frac{2 σ_{i} - e_{i}}{σ_{i}}, & σ_{i} \leq e_{i} < 2 σ_{i} \\ 0, & \; otherwise\; \end{array}

Positive Large (PL):

μ_{P L} (e_{i}) = {\begin{array}{ll} 0, & e_{i} \leq σ_{i} \\ \frac{e_{i} - σ_{i}}{σ_{i}}, & σ_{i} < e_{i} < 2 σ_{i} \\ 1, & e_{i} \geq 2 σ_{i} \end{array}

Here, $σ_{i}$ represents the scaling factor for joint i.

Similar membership functions are established for the error rate ${\dot{e}}_{i}$ with the scaling factor $σ_{d, i}$ .

Definition 2:
A fuzzy rule base is defined as a set of IF-THEN rules: $R_{i}$ : IF e is $A_{i}$ AND $\dot{e}$ is $B_{i}$ THEN $τ$ is $C_{i}$ where $A_{i}, B_{i}$ , and $C_{i}$ are fuzzy sets.

For each joint, we define 25 rules (5 error levels $\times 5$ error rate levels). The rule base is structured as in Table 2:

Table 2.
Fuzzy Rule Base.

$e / \dot{e}$ NL NS ZE PS PL

NL PL PL PL PS ZE

NS PL PS PS ZE NS

ZE PL PS ZE NS NL

PS PS ZE NS NS NL

PL ZE NS NL NL NL

This rule base implements a nonlinear PD-type control strategy, with stronger control action when both error and error rate are large and of the same sign, and weaker when they are of opposite signs. The resulting control surface characteristics are illustrated in Figure 2.

Figure 2.
Fuzzy Control Surface Analysis.

According to Wang et al. (1992), this rule structure maximally leverages expert knowledge about the kinematics of the robotic system, helping achieve better response to various disturbances and uncertainties compared to conventional linear controllers.

The fuzzy inference process follows these steps:
Fuzzification: Convert crisp inputs $e_{i}$ and ${\dot{e}}_{i}$ into fuzzy values using membership functions.

*Rule evaluation: Calculate the firing strength of each rule using min operation for AND: $μ_{j} (e_{i}, {\dot{e}}_{i}) = min (μ_{A_{j}} (e_{i}), μ_{B_{j}} ({\dot{e}}_{i}))$

Aggregation: Combine the outputs of all rules.

Defuzzification: Convert the aggregated fuzzy output to a crisp control value using center of gravity method:
$τ_{F L C, i} = \frac{\sum_{j = 1}^{25} μ_{j} (e_{i}, {\dot{e}}_{i}) \cdot τ_{j, i}}{\sum_{j = 1}^{25} μ_{j} (e_{i}, {\dot{e}}_{i})}$

where $τ_{j, i}$ is the center of the output membership function for rule j and joint i.

The complete fuzzy control vector is:
$τ_{F L C} = [τ_{F L C, 1}, τ_{F L C, 2}, τ_{F L C, 3}, τ_{F L C, 4}, τ_{F L C, 5}]^{T}$

5.4 Sliding Mode Control With Disturbance Observer

$e / \dot{e}$	NL	NS	ZE	PS	PL
NL	PL	PL	PL	PS	ZE
NS	PL	PS	PS	ZE	NS
ZE	PL	PS	ZE	NS	NL
PS	PS	ZE	NS	NS	NL
PL	ZE	NS	NL	NL	NL

The sliding mode component provides robustness against disturbances through a nonlinear switching control law. The sliding surface is defined as:

s = \dot{e} + Λ e

where

Λ = diag (λ_{1}, λ_{2}, \dots, λ_{5})

is a positive definite matrix.

The sliding surface design ensures that once the system reaches the sliding manifold $s =$ 0, the error dynamics become:

\dot{e} = - Λ e

which ensures exponential convergence of the tracking error to zero.

To enhance robustness against external disturbances and unmodeled dynamics, we incorporate a nonlinear disturbance observer. The observer estimates the unknown disturbance $τ_{d}$ :

{\hat{τ}}_{d} = z + p (q, \dot{q})

where

z \in R^{5}

is the observer state and

p (q, \dot{q}) \in R^{5}

is an auxiliary function. The observer dynamics are designed as:

\dot{z} = - L (q, \dot{q}) z - L (q, \dot{q}) [p (q, \dot{q}) + M^{- 1} (q) (τ - C (q, \dot{q}) \dot{q} - G (q))]

where

L (q, \dot{q}) \in R^{5 \times 5}

is the observer gain matrix.

The auxiliary function $p (q, \dot{q})$ is designed to satisfy:

\frac{\partial p (q, \dot{q})}{\partial q} \cdot \dot{q} + \frac{\partial p (q, \dot{q})}{\partial \dot{q}} \cdot M^{- 1} (q) (τ - C (q, \dot{q}) \dot{q} - G (q)) = 0

A simple choice is:

p (q, \dot{q}) = - K_{p} \cdot q

where

K_{p}

is a positive definite matrix. This choice simplifies implementation while maintaining the necessary properties of the observer.

Theorem 2:
The disturbance estimation error ${\tilde{τ}}_{d} = τ_{d} - {\hat{τ}}_{d}$ converges exponentially to zero if $L (q, \dot{q})$ is properly designed and ${\dot{τ}}_{d}$ is bounded.
Proof:
Define the estimation error dynamics as:
${\dot{\tilde{τ}}}_{d} = {\dot{τ}}_{d} - {\dot{\hat{τ}}}_{d} = {\dot{τ}}_{d} - \dot{z} - \frac{\partial p (q, \dot{q})}{\partial q} \dot{q} - \frac{\partial p (q, \dot{q})}{\partial \dot{q}} \ddot{q}$

Substituting the observer dynamics and the expression for $\ddot{q}$ from the robot model:

With appropriate choice of $p (q, \dot{q})$ to satisfy the constraint equation, and using the relation ${\hat{τ}}_{d} = z + p (q, \dot{q})$ :

If we choose $L (q, \dot{q})$ such that:
$L (q, \dot{q}) τ_{d} + \frac{\partial p (q, \dot{q})}{\partial \dot{q}} \cdot M^{- 1} (q) τ_{d} \approx {\dot{τ}}_{d}$

Then the error dynamics simplify to:
${\dot{\tilde{τ}}}_{d} \approx L (q, \dot{q}) {\tilde{τ}}_{d}$

By choosing $L (q, \dot{q})$ to be negative definite (e.g., $L (q, \dot{q}) = - K_{L} \cdot I$ where $K_{L} > 0$ ), the estimation error converges exponentially to zero:
${\tilde{τ}}_{d} (t) \approx e^{- K_{L} \cdot t} \cdot {\tilde{τ}}_{d} (0)$

In practice, perfect cancellation is challenging, but with sufficiently high $K_{L}$ , the observer can achieve good disturbance estimation performance.

The sliding mode control law is designed as:
$τ_{S M C} = M (q) ({\ddot{q}}_{d} + Λ \dot{e}) + C (q, \dot{q}) \dot{q} + G (q) - {\hat{τ}}_{d} - K_{s} \cdot sgn (s)$
where $K_{s} = diag (k_{s, 1}, k_{s, 2}, \dots, k_{s, 5})$ is a positive definite gain matrix and $sgn (\cdot)$ is the sign function.

To mitigate chattering, we replace the discontinuous sign function with a continuous approximation:
$sgn (s) \approx \frac{s}{| s | + δ}$
where $δ$ is a small positive constant (typically 0.01–0.1).
Theorem 3:
The sliding mode controller with disturbance observer ensures finite-time convergence to the sliding manifold and subsequent exponential convergence of the tracking error if $K_{s}$ is properly chosen.
Proof:
Consider the Lyapunov function:
$V_{S M C} = \frac{1}{2} \cdot s^{T} \cdot M (q) \cdot s$

Taking the derivative:
${\dot{V}}_{S M C} = s^{T} M (q) \dot{s} + \frac{1}{2} s^{T} \dot{M} (q) s = s^{T} M (q) \dot{s} + \frac{1}{2} s^{T} \dot{M} (q) s$

Using the robot dynamics and sliding mode control law, we have:
$\begin{aligned} M (q) \ddot{q} & = τ - C (q, \dot{q}) \dot{q} - G (q) - τ_{d} = M (q) ({\ddot{q}}_{d} + Λ \dot{e}) + C (q, \dot{q}) \dot{e} + C (q, \dot{q}) {\dot{q}}_{d} + G (q) - {\hat{τ}}_{d} - K_{s} \cdot sgn (s) - C (q, \dot{q}) \dot{q} - G (q) \\ - τ_{d} = M (q) ({\ddot{q}}_{d} + Λ \dot{e}) - {\hat{τ}}_{d} - K_{s} sgn (s) - τ_{d} \end{aligned}$

Therefore, the error dynamics are:
$\begin{aligned} M (q) \ddot{e} & = - M (q) Λ \dot{e} - {\hat{τ}}_{d} - K_{s} sgn (s) - τ_{d} \end{aligned}$

Substituting this result into the Lyapunov derivative, we obtain:
$\begin{aligned} {\dot{V}}_{S M C} & = s^{T} (- M (q) Λ \dot{e} + {\hat{τ}}_{d} + K_{s} sgn (s) + τ_{d}) + \frac{1}{2} s^{T} \dot{M} (q) s = s^{T} ({\hat{τ}}_{d} + K_{s} sgn (s) + τ_{d}) + \frac{1}{2} s^{T} \dot{M} (q) s \\ = s^{T} ({\hat{τ}}_{d} + K_{s} sgn (s) + τ_{d}) + s^{T} C (q, \dot{q}) s = - s^{T} {\tilde{τ}}_{d} + s^{T} K_{s} sgn (s) + s^{T} τ_{d} + s^{T} C (q, \dot{q}) s \end{aligned}$
where ${\overset{↼}{τ}}_{d} = τ_{d} - {\hat{τ}}_{d}$ .

We have utilized the property that $\dot{M} (q) = C (q, \dot{q}) + C (q, \dot{q})^{T}$ .

Given that:
$s^{T} \cdot K_{s} \cdot sgn (s) = \sum_{i = 1}^{5} k_{s, i} \cdot | s_{i} |$
and assuming the following bounds (based on the disturbance observer): $‖ {\tilde{τ}}_{d} ‖ \leq ε_{d}, \; ‖ τ_{d} ‖ \leq ρ, \; ∥ C (q, \dot{q}) ∥\leq k_{c} \cdot ∥ \dot{q} ∥$

we obtain:
${\dot{V}}_{S M C} \leq | s | \cdot | {\tilde{τ}}_{d} | - \sum_{i = 1}^{5} k_{s, i} \cdot | s_{i} | + | s | \cdot | τ_{d} | + | s | \cdot | C (q, \dot{q}) | \cdot | s | \leq | s | \cdot ε_{d} - λ_{min} (K_{s}) \cdot ∥ s ∥_{1} + | s | \cdot ρ + k_{c} \cdot ∥ \dot{q} ∥ \cdot | s |^{2}$
where $∥ s ∥_{1} = \sum_{i = 1}^{5} | s_{i} |$ represents the L1 norm.

Given that: $∥ s ∥\leq∥ s ∥_{1} \leq \sqrt{5} \cdot ∥ s ∥$ we rewrite the above inequality as:
${\dot{V}}_{S M C} \leq | s | \cdot ε_{d} - \frac{λ_{min} (K_{s})}{\sqrt{5}} \cdot | s | + | s | \cdot ρ + k_{c} \cdot ∥ \dot{q} ∥ \cdot {| s |^{2} = | s | \cdot (ε_{d} - \frac{λ_{min} (K_{s})}{\sqrt{5}} + ρ) + k_{c} \cdot ∥ \dot{q} ∥ \cdot | s |}^{2}$

To guarantee ${\dot{V}}_{S M C} < 0$ for all $s \neq 0$ , the following condition must be satisfied:
$λ_{min} (K_{s}) > \sqrt{5} \cdot (ε_{d} + ρ + η)$
where $η > 0$ is a design parameter introduced to enhance robustness.

Selecting $K_{s}$ accordingly ensures that the system reaches the sliding manifold in finite time. Subsequently, the system follows the reduced-order dynamics: $\dot{e} = - Λ e$ which ensures exponential convergence of the tracking error.
5.5 Integrated Control Architecture

Our proposed integrated controller combines four control components using adaptive weighting:

τ = α_{1} \cdot τ_{H \infty} + α_{2} \cdot τ_{M P C} + α_{3} \cdot τ_{F L C} + α_{4} \cdot τ_{S M C}

where the adaptive weighting coefficients

α_{i}

are determined by:

α_{i} = \frac{exp (- β_{i} \cdot ∥ e ∥^{2})}{\sum_{j = 1}^{4} exp (- β_{j} \cdot ∥ e ∥^{2})}

Here, $β_{i}$ are positive constants determining each controller's contribution based on tracking performance.

The adaptive weighting mechanism automatically adjusts the contribution levels of each control element:

When tracking error is large, robustness-oriented components (SMC and $H_{\infty}$ ) receive higher weights.

When the error decreases, optimization-oriented components (MPC) become more dominant.

The fuzzy logic component provides nonlinear compensation consistently throughout the operation.

Park et al. (2021) demonstrated that a similar adaptive weighting approach yields superior performance compared to hard-switching mechanisms or using a single controller across diverse operating conditions.

Parameters $β_{i}$ are selected based on each controller's characteristics:

$β_{1} (H_{\infty})$ : Medium value to maintain stability over intermediate error ranges.

$β_{2}$ (MPC): Lowest value to emphasize MPC during small-error conditions.

$β_{3}$ (FLC): Intermediate value providing consistent nonlinear compensation.

$β_{4}$ (SMC): Highest value to prioritize SMC during large-error conditions.

Typical parameter values applied in simulation are:

\begin{aligned} β_{1} & = 10 \end{aligned}

\begin{aligned} β_{2} & = 5 \end{aligned}

\begin{aligned} β_{3} & = 15 \end{aligned}

\begin{aligned} β_{4} & = 20 \end{aligned}

These values were refined through extensive simulation trials to ensure smooth transitions between control modes. According to Wei Bin (2018), this parameter selection approach significantly reduces model dependence, particularly in the presence of substantial parametric variations.

Consider the composite Lyapunov function:

V_{t o t a l} = V_{H_{\infty}} + V_{M P C} + V_{F L C} + V_{S M C}

Taking its derivative with respect to time, we obtain:

{\dot{V}}_{t o t a l} = {\dot{V}}_{H_{\infty}} + {\dot{V}}_{M P C} + {\dot{V}}_{F L C} + {\dot{V}}_{S M C}

From the stability analyses of the individual controllers, it follows that:

\begin{aligned} {\dot{V}}_{H_{\infty}} & \; \leq - λ_{H_{\infty}} ∥ r ∥^{2} + γ_{H_{\infty}} ‖ τ_{d} ‖^{2} \\ {\dot{V}}_{M P C} & \; \leq - λ_{M P C} ∥ e ∥^{2} + γ_{M P C} ‖ τ_{d} ‖^{2} \\ {\dot{V}}_{F L C} & \; \leq - λ_{F L C} ∥ e ∥^{2} + γ_{F L C} ‖ τ_{d} ‖^{2} \\ {\dot{V}}_{S M C} & \; \leq - λ_{S M C} ∥ s ∥_{1} + γ_{S M C} ‖ \tilde{τ} ‖_{d} ∥ s ∥ + γ_{S M C}^{'} ‖ τ_{d} ‖ ∥ s ∥ \end{aligned}

where

λ_{*}

and

γ_{*}

are positive constants derived from each controller's stability analysis.

For the integrated control strategy, the closed-loop dynamics become:

M (q) \ddot{q} + C (q, \dot{q}) \dot{q} + G (q) + τ_{d} = α_{1} τ_{H_{\infty}} + α_{2} τ_{M P C} + α_{3} τ_{F L C} + α_{4} τ_{S M C}

The individual controllers are designed to drive the tracking error to zero. When combined using adaptive weights $α_{i}$ , which satisfy $\sum_{i = 1}^{4} α_{i} = 1$ , the integrated control maintains this property.

By substituting the integrated control law and utilizing the properties of the adaptive weighting functions, we have:

\begin{aligned} {\dot{V}}_{total\;} & \leq - min_{i} (α_{i}) [λ_{H_{\infty}} ∥ r ∥^{2} + λ_{M P C} ∥ e ∥^{2} + λ_{F L C} ∥ e ∥^{2} + λ_{S M C} ∥ s ∥_{1}] + max_{i} (α_{i}) [γ_{H_{\infty}} ‖ τ_{d} ‖^{2} + γ_{M P C} ‖ τ_{d} ‖^{2} \\ + γ_{F L C} ‖ τ_{d} ‖^{2} + γ_{S M C} ‖ {\tilde{τ}}_{d} ‖ ∥ s ∥ + γ_{S M C}^{'} τ_{d} ∥ s ∥] \end{aligned}

Since $mi n_{i} (α_{i}) > 0$ and $ma x_{i} (α_{i}) \leq 1$ , due to the exponential weighting functions’ properties, the above inequality can be simplified as:

{\dot{V}}_{total\;} \leq - η_{1} ∥ e ∥^{2} - η_{2} ∥ r ∥^{2} - η_{3} ∥ s ∥_{1} + γ_{total\;} ‖ τ_{d} ‖^{2} + γ_{obs} ‖ {\tilde{τ}}_{d} ‖ ∥ s ∥

where

η_{1}, η_{2}, η_{3}, γ_{total\;}

, and

γ_{obs\;}

are positive constants.

Considering the disturbance observer, it follows that $‖ {\tilde{τ}}_{d} ‖ \leq ε_{d} e^{- K_{L} t}$ ; thus, the term $γ_{obs} ‖ {\tilde{τ}}_{d} ‖ ∥ s ∥$ exponentially converges to zero.

For disturbances with bounded magnitudes, $‖ τ_{d} ‖ \leq ρ$ , we have:

{\dot{V}}_{total\;} \leq - η_{1} ∥ e ∥^{2} - η_{2} ∥ r ∥^{2} - η_{3} ∥ s ∥_{1} + γ_{total\;} ρ^{2} + γ_{obs\;} ε_{d} e^{- K_{L} t} ∥ s ∥

This implies that the tracking errors converge to a small neighborhood around zero, with the size of this neighborhood depending on the disturbance bound $ρ$ . As $t \to \infty$ , the observer-related error component vanishes, and we obtain:

lim_{t \to \infty} ∥ e (t) ∥\leq \sqrt{\frac{γ_{total\;} ρ^{2}}{η_{1}}}

When there is no disturbance present ( $ρ = 0$ ), the inequality simplifies to:

{\dot{V}}_{total\;} \leq - η_{1} ∥ e ∥^{2} - η_{2} ∥ r ∥^{2} - η_{3} ∥ s ∥_{1} + γ_{obs\;} ε_{d} e^{- K_{L} t} ∥ s ∥

ensuring asymptotic convergence of errors to zero as

t \to \infty

5.6 Controller Parameter Design Guidelines

To ensure the boundedness of tracking error and system stability, systematic parameter design procedures are established for each control component.

For the $H \infty$ controller, design constraints are set to ensure both stability and performance. The stability condition requires that $K_{e}$ and $K_{r}$ be positive definite. The performance requirement is defined by the condition $λ_{min} (K_{r}) > \frac{1}{2 γ_{0}^{2}}$ , where $γ_{0} = 0.35$ , resulting in $λ_{min} (K_{r}) > 4.08$ . The robustness condition is given by $λ_{min} (K_{e}) \geq \frac{λ_{max} (Q)}{λ_{min} (Λ)}$ .

The design procedure follows these steps:

Choose $Λ = diag (10, 10, 10, 10, 10)$ to set the desired convergence rate.

Set $Q = diag (100, 100, 100, 100, 100, 10, 10, 10, 10, 10)$ to prioritize tracking.

Calculate $K_{r} = 25 I$ , ensuring that it satisfies the performance requirement $K_{r} > 4.08 I$ .

Calculate $K_{e} = 15 I$ , which satisfies the robustness condition $K_{e} > \frac{100}{10} = 10 I$ .

For Model Predictive Control (MPC), the design constraints are focused on predictive capability and computational efficiency. The prediction horizon, $N_{p}$ , is selected to be at least $max (\frac{t_{rise\;}}{T_{s}}, 10) = max (\frac{0.5}{0.001}, 10) = 10$ , ensuring sufficient prediction capability. The control horizon $N_{c}$ is set to be $\leq \frac{N_{p}}{3}$ for computational efficiency, which gives $N_{c} = 3$ . The weight selection is also crucial, with $Q_{MPC} ≫ R_{MPC}$ to prioritize tracking.

The selected values are:

$N_{p} = 10$ , ensuring sufficient prediction capability .

$N_{c} = 3$ , optimizing computational efficiency.

$Q_{MPC} = 100 \times diag (1, 1, 1, 1, 1, 0.1, 0.1, 0.1, 0.1, 0.1)$ , prioritizing position .

$R_{MPC} = 0.1 \times diag (1, 1, 1, 1, 1)$ , applying moderate control effort penalization.

For Sliding Mode Control (SMC), the boundedness condition for finite-time convergence is given by:

K_{s} > \sqrt{5} (ϵ_{d} + ρ + η)

where

ϵ_{d} \leq 0.1

(observer error bound),

ρ \leq 5

(disturbance bound), and

η = 1

(safety margin). The design calculation results in:

K_{s} = diag (20, 20, 20, 20, 20) > \sqrt{5} (0.1 + 5 + 1) = 13.64

For adaptive weights, the convergence condition ensures all weights remain positive, represented as:

min_{i} (α_{i}) > 0 \; for\; all\; ‖ e ‖

The selected weighting parameters are:

$β_{1} = 10$ ( $H \infty$ : moderate response to error changes),

$β_{2} = 5$ (MPC: active during small error conditions),

$β_{3} = 15$ (Fuzzy: consistent contribution across the operating range),

$β_{4} = 20$ (SMC: dominant during large error transients).

The boundedness guarantee, given the above parameters, is expressed by:

‖ e (t) ‖ \leq ‖ e (0) ‖ e^{- λ t} + \frac{γ_{total\;} ρ^{2}}{η_{1}}

where:

λ = min (\frac{λ_{min} (K_{r})}{λ_{max} (M)}, λ_{min} (Λ)) = min (\frac{25}{2}, 10) = 10,

$γ_{total\;} = 0.35$ (proven from $H_{\infty}$ analysis),

η_{1} = min_{i} (α_{i}) \times λ_{min\;} (K_{e}) \geq 0.1 \times 15 = 1.5.

Thus, the tracking error bound is:

‖ e (\infty) ‖ \leq \sqrt{\frac{0.35 \times 25}{1.5}} = 2.42 \; mm \leq 2.5 \; mm

This ensures that the specification is satisfied. This systematic design approach guarantees the boundedness of tracking error while ensuring both stability and performance across all operating conditions.

6. Integrated System Stability Analysis

6.1 Composite Lyapunov Function Design

For the integrated control system, a composite Lyapunov function is used to account for the interaction between all four control components:

V_{integrated\;} = V_{tracking\;} + V_{parameter\;} + V_{observer\;} + V_{constraint\;}

where:

Tracking Component:

V_{tracking\;} = \frac{1}{2} r^{T} M (q) r + \frac{1}{2} e^{T} K_{e} e

Parameter Estimation Component:

V_{parameter\;} = \frac{1}{2} {\tilde{θ}}^{T} Γ^{- 1} \tilde{θ}

Observer Component:

V_{observer\;} = \frac{1}{2} {\tilde{τ}}_{d}^{T} P {\tilde{τ}}_{d}

MPC Constraint Component:

V_{constraint\;} = \frac{1}{2} x^{T} Q_{mpc} x

6.2 Integrated Stability Theorem

Theorem
(Integrated System Stability):

Under Assumptions 1–5 and with the proposed adaptive weighting mechanism $α_{i} = \frac{exp (- β_{i} ‖ e ‖^{2})}{\sum_{j} exp (- β_{j} ‖ e ‖^{2})}$ , the integrated control system guarantees:
Boundedness: All signals $q, \dot{q}, τ, \hat{θ}, {\hat{τ}}_{d} \in L^{\infty}$

Asymptotic tracking: $lim_{t \to \infty} ‖ e (t) ‖ \leq \sqrt{\frac{γ_{total\;} ρ^{2}}{η_{1}}}$

$H \infty$ performance: $\frac{‖ e ‖_{L_{2}}}{‖ τ_{d} ‖_{L_{2}}} < γ_{total\;} = 0.35$
Proof:
Taking the time derivative of the composite Lyapunov function:
${\dot{V}}_{integrated\;} = {\dot{V}}_{tracking\;} + {\dot{V}}_{parameter\;} + {\dot{V}}_{observer\;} + {\dot{V}}_{constraint\;}$

Step 1: Tracking Component Analysis
${\dot{V}}_{tracking\;} = r^{T} M (q) \dot{r} + \frac{1}{2} r^{T} \dot{M} (q) r + e^{T} K_{e} \dot{e}$
Substituting the integrated control law $τ = \sum \; α_{i} τ_{i}$ and using robot dynamics:
$M (q) \dot{r} = - C (q, \dot{q}) r + Y \tilde{θ} - \sum \; α_{i} K_{i, e} e - \sum \; α_{i} K_{i, r} r + τ_{d}$
This yields:
${\dot{V}}_{tracking\;} = r^{T} Y \tilde{θ} - r^{T} (\sum \; α_{i} K_{i, e}) e - r^{T} (\sum \; α_{i} K_{i, r}) r + r^{T} τ_{d} + e^{T} K_{e} \dot{e}$

Step 2: Parameter Component Analysis
${\dot{V}}_{parameter\;} = - {\tilde{θ}}^{T} Y^{T} r$

Step 3: Observer Component Analysis
${\dot{V}}_{observer\;} = {\tilde{τ}}_{d}^{T} P {\dot{\tilde{τ}}}_{d} = - {\tilde{τ}}_{d}^{T} P L {\tilde{τ}}_{d} + {\tilde{τ}}_{d}^{T} P {\dot{τ}}_{d}$

Step 4: Combined Analysis
${\dot{V}}_{integrated\;} = - r^{T} (\sum \; α_{i} K_{i, r}) r - e^{T} (\sum \; α_{i} K_{i, e} - K_{e}) e + r^{T} τ_{d} - {\tilde{τ}}_{d}^{T} P L {\tilde{τ}}_{d} + \; bounded\; terms\;$

Step 5: Adaptive Weight Properties
The exponential weighting functions satisfy:
Normalization: $\sum \; α_{i} = 1$

Positivity: $α_{i} \geq α_{min\;} = \frac{exp (- β_{max\;} ‖ e ‖^{2})}{4} > 0$

Smooth transitions: $| {\dot{α}}_{i} | \leq C_{α} ‖ \dot{e} ‖$ (bounded derivative)

Step 6: Negative Definiteness
Choose control gains such that:
$min_{i} (K_{i, r}) > \frac{1}{2 γ_{0}^{2} α_{min}}$ ensures $\sum \; α_{i} K_{i, r} > \frac{1}{2 γ_{0}^{2}} I$

$min_{i} (K_{i, e}) > \frac{λ_{max} (K_{e})}{α_{min}}$ ensures $\sum \; α_{i} K_{i, e} > K_{e}$

$λ_{min\;} (P L) > \frac{‖ P ‖^{2}}{λ_{min\;} (P)}$ ensures observer stability
This yields:
${\dot{V}}_{integrated\;} \leq - k_{r, min} ‖ r ‖^{2} - k_{e, min} ‖ e ‖^{2} - k_{obs} ‖ {\tilde{τ}}_{d} ‖^{2} + γ_{total\;} ‖ τ_{d} ‖^{2}$
where $k_{r, min}, k_{e, min}, k_{obs} > 0$ .
Step 7: Conclusions
Boundedness: $V_{integrated\;}$ bounded implies all signals are bounded.

Convergence: By Barbalat's lemma, $‖ e ‖, ‖ r ‖, ‖ {\tilde{τ}}_{d} ‖ \to 0$ as $t \to \infty$ .

$H \infty$ Performance: Integrating the inequality gives the desired attenuation .

Therefore, the integrated system is stable with guaranteed performance bounds.
6.3 Robustness Analysis

Theorem
(Parameter Variation Robustness):
The integrated system maintains stability under parameter variations $δ M, δ C, δ G \leq 0.5$ .

Proof:
Parameter uncertainties create additional disturbances:
$τ_{unc} = Δ M (q) \ddot{q} + Δ C (q, \dot{q}) \dot{q} + Δ G (q)$

With bounds $‖ τ_{unc\;} ‖ \leq ρ_{unc\;}$ , the sliding control component ensures:
$K_{s, eff} = α_{4} K_{s} > α_{min} K_{s} > ρ_{unc} + ρ + η$
guaranteeing finite-time convergence to the sliding manifold despite uncertainties.
6.4 Convergence Rate Analysis

Theorem
(Exponential Convergence):
The tracking error converges exponentially with rate:
$‖ e (t) ‖ \leq ‖ e (0) ‖ e^{- λ t} + \frac{γ_{total\;} ρ^{2}}{η_{1}}$

where $λ = min (\frac{λ_{min} (K_{r})}{λ_{max} (M)}, λ_{min} (Λ))$ and $η_{1} = α_{min} λ_{min} (K_{e})$ .

This comprehensive stability analysis validates the theoretical soundness of the proposed integrated control architecture with formal guarantees for stability, convergence, and robustness.
7. Simulation Results

7.1 Simulation Setup

The proposed controller was implemented in MATLAB/Simulink with the following parameters:

Sample time: 1 ms

Simulation time: 20 s

MPC prediction horizon: $N_{p} = 10$

MPC control horizon: $N_{c} = 3$

MPC weights: $Q = diag (100, 100, 100, 100, 100, 10, 10, 10, 10, 10), \; R = diag (0.1, 0.1, 0.1, 0.1, 0.1)$

Adaptation coefficients: $Γ = diag (5, 5, 5, 5, 5)$

$H_{\infty}$ coefficients: $K_{e} = diag (15, 15, 15, 15, 15), K_{r} = diag (25, 25, 25, 25, 25)$

Sliding mode parameters: $Λ = diag (10, 10, 10, 10, 10)$

SMC coefficients: $K_{s} = diag (20, 20, 20, 20, 20)$

Disturbance observer coefficients: $L = 50 I$

Fuzzy scaling factors: $σ = [0.1, 0.1, 0.1, 0.1, 0.1], σ_{d} = [0.05, 0.05, 0.05, 0.05, 0.05]$

Adaptation weights: $β_{1} = 10, β_{2} = 5, β_{3} = 15, β_{4} = 20$

The robot dynamics model was implemented using the full nonlinear equations with the provided Denavit-Hartenberg parameters and inertia properties. The model includes:

Gravitational effects

Coriolis and centrifugal forces

Joint friction (Coulomb and viscous)

Variable load simulation

External disturbances

Multiple reference trajectories were used to evaluate the controller's performance:

Circular trajectory in the $X Y$ plane:

x_{d} (t) = 150 + 50 \cdot \sin (0.5 t) mm y_{d} (t) = 150 + 50 \cdot \cos (0.5 t) mm z_{d} (t) = 100 + 30 \cdot \sin (0.2 t) mm

Square trajectory with smooth angles:

\begin{aligned} x_{d} (t) & = 150 + 50 \cdot sgn (\sin (0.25 t)) \cdot tanh (5 \cdot \sin (0.25 t)) mm y_{d} (t) \\ = 150 + 50 \cdot sgn (\cos (0.25 t)) \cdot \tanh (5 \cdot \cos (0.25 t)) mm z_{d} (t) = 100 \; mm \end{aligned}

Pick-and-place trajectory with multiple intermediate points:

\begin{aligned} [x_{d} (t), y_{d} (t), z_{d} (t)] \\ = PWL (t, {[0, 100, 150, 50], [5, 200, 100, 50], [10, 200, 100, 150], [15, 100, 150, 150], [20, 100, 150, 50]}) \end{aligned}

where PWL represents piecewise linear interpolation with smooth transitions between the intermediate points. The corresponding joint trajectories are computed using inverse kinematics.

7.2 Performance in Terms of Tracking Error

We evaluate the tracking performance using the circular trajectory under nominal conditions.

Example 1:
Consider the desired circular trajectory of the end-effector. Tracking error results using different control schemes:
Conventional PD Control

Model Predictive Control

Fuzzy Logic Control

Adaptive control

$H_{\infty}$ control

Proposed integrated control

The trajectory tracking comparison in Cartesian space is visualized in Figure 3. The RMS tracking errors for each controller are presented in Table 3.

Figure 3.
Comparison of Trajectory in Cartesian Space.

Table 3.
RMS Tracking Errors (mm).

Controller X-axis Y-axis Z-axis Average

PD Control 5.74 6.12 4.89 5.58

MPC 3.45 3.78 3.12 3.45

Fuzzy Logic 4.21 4.67 3.89 4.26

Adaptive 2.83 3.15 2.64 2.87

$H_{\infty}$ 2.41 2.73 2.36 2.50

Proposed 1.32 1.45 1.27 1.35

Comment 2: The proposed controller achieves a 43.2% reduction in tracking error compared to the next best method (H-infinity), demonstrating superior performance under nominal conditions.

The energy efficiency and control effort comparison is further illustrated in Figure 4. The control effort, measured by the integral of the squared torque, is compared in Table 4.

Figure 4.
Control Effort and Energy Efficiency Comparison.

Table 4.
Control Effort Comparison.

Controller Joint 1 Joint 2 Joint 3 Joint 4 Joint 5 Total

PD Control 105.32 12.81 73.55 6.22 4.75 72.5

Adaptive 84.71 78.41 45.94 5.12 0.54 74.6

$H_{\infty}$ 92.51 87.21 52.34 8.62 2.15 102.7

Proposed 88.11 65.61 39.24 2.31 9.84 55.0

Comment 3: The proposed controller not only achieves better tracking performance but also requires less control effort than conventional methods, proving its effectiveness.

These results are consistent with the analysis of Johansen and Fossen (2018), showing that integrated adaptive control methods can simultaneously improve both accuracy and energy efficiency.
Example 2:
During the trajectory tracking of the circular path, the evolution of the weights is as follows:
Initial stage (large error): $α_{4}$ (SMC) dominates with a value between 0.45 and 0.55.

Intermediate stage (moderate error): $α_{1}$ (H -infinity) increases to $0.35 - 0.40$ .

Steady state (small error): $α_{2}$ (MPC) becomes significant with a value between 0.30 and 0.35.

This adaptive weighting mechanism automatically adjusts the control strategy based on the current performance, which explains the superior results. The evolution of adaptive weights during different operating phases is depicted in Figure 5. These findings are consistent with those of Li et al. (2019), showing that smooth transitions between multiple control strategies can significantly improve overall performance under different operating conditions.

Figure 5.
Adaptive Weight Evolution During Different Operating Phases.
7.3 Noise Suppression Capability

Controller	X-axis	Y-axis	Z-axis	Average
PD Control	5.74	6.12	4.89	5.58
MPC	3.45	3.78	3.12	3.45
Fuzzy Logic	4.21	4.67	3.89	4.26
Adaptive	2.83	3.15	2.64	2.87
$H_{\infty}$	2.41	2.73	2.36	2.50
Proposed	1.32	1.45	1.27	1.35

Controller	Joint 1	Joint 2	Joint 3	Joint 4	Joint 5	Total
PD Control	105.32	12.81	73.55	6.22	4.75	72.5
Adaptive	84.71	78.41	45.94	5.12	0.54	74.6
$H_{\infty}$	92.51	87.21	52.34	8.62	2.15	102.7
Proposed	88.11	65.61	39.24	2.31	9.84	55.0

To evaluate the noise suppression capability, we applied an external force disturbance to the endeffector:

F_{e x t} (t) = {\begin{array}{ll} {[5, 5, 5]}^{T} \; N, & 5 \leq t \leq 7 \\ {[0, 0, 0]}^{T} \; N, & \; otherwise\; \end{array}

Example 3:
When subjected to a 5 N impulse disturbance at t = 0.5 s, the maximum tracking error and recovery time for each controller are compared. The time domain response characteristics under impulse disturbance are shown in Figure 6. The detailed performance metrics are summarized in Table 5.
Figure 6.
Time Response to Impulse Disturbance.

Table 5.
Noise Suppression Performance.

Controller Maximum Error (mm) Recovery Time (s)

PD Control 12.5 1.87

Adaptive 8.36 1.25

$H_{\infty}$ 7.45 0.98

Proposed 3.21 0.43

Comment 4: The proposed controller demonstrates superior noise suppression, with a 57% reduction in maximum error and a 56% reduction in recovery time compared to the next best method.

These results are consistent with findings by Chen et al. (2016), confirming that combining multiple noise suppression strategies can yield a greater cumulative effect than the sum of individual components.

The performance indices of the noise monitor are shown in Table 6.

Table 6.
Noise Monitoring Performance.

Controller Joint 1 Joint 2 Joint 3 Joint 4 Joint 5 Total

RMSE (N·m) 0.087 0.112 0.098 0.065 0.043 0.081

Maximum Error (N·m) 0.235 0.318 0.274 0.183 0.125 0.227

Convergence Time (ms) 127 135 121 118 114 123

Comment 5: The noise monitor effectively estimates external disturbances with an estimation error smaller than 5% after the initial transient. The fast convergence time (average 123 ms) allows for quick compensation, contributing to superior noise suppression.

This performance is significant and meets the high requirements of industrial and automation applications, as outlined by Chen and Patton (2012).
7.4 Robustness to Parameter Variations

Controller	Maximum Error (mm)	Recovery Time (s)
PD Control	12.5	1.87
Adaptive	8.36	1.25
$H_{\infty}$	7.45	0.98
Proposed	3.21	0.43

Controller	Joint 1	Joint 2	Joint 3	Joint 4	Joint 5	Total
RMSE (N·m)	0.087	0.112	0.098	0.065	0.043	0.081
Maximum Error (N·m)	0.235	0.318	0.274	0.183	0.125	0.227
Convergence Time (ms)	127	135	121	118	114	123

We tested robustness by changing the mass of each link by ±30% from its nominal value.

Example 4:
With a 30% increase in all link masses, the performance degradation in trajectory tracking error for each controller is evaluated. The comparative results are presented in Table 7.
Table 7.
Performance Degradation with Parameter Variations.

Controller Error Increase (%)

PD Controller 85.3

Adaptive 23.7

$H_{\infty}$ 31.2

Proposed 11.5

The adaptive component effectively compensates for parameter uncertainties, as parameter estimates converge to their true values. The performance degradation under various parameter variations is illustrated in Figure 7.

Figure 7.
Effect of Parameter Variations on Controller Performance.

Additional tests were conducted with variations in the friction coefficient (±50%) and inertia parameters (±40%).

Comment 6: The proposed controller maintains an error increase of less than 15% even with parameter variations of up to 50%, demonstrating exceptional robustness compared to conventional methods, which show error increases of 30–85%.

This result aligns with the findings reported by Peng et al. (2013), emphasizing the importance of combining multiple robust strategies to handle different uncertainties.
7.5 Detailed Simulation

Controller	Error Increase (%)
PD Controller	85.3
Adaptive	23.7
$H_{\infty}$	31.2
Proposed	11.5

To further test controller performance, we developed an advanced simulation environment in MATLAB/Simulink with the following features:

- High-frequency dynamics model (10 kHz) using the Runge-Kutta 4 method

- Advanced LuGre friction model including Stribeck effects and velocity dependence

- Joint elasticity model with varying stiffness and damping

- Sensor noise simulation with a realistic noise model

- Simulation of delays and bandwidth limitations of actuators

The simulation implementation considers several practical aspects:

- Computational optimization to obtain simulation results in a reasonable time

- Reduced MPC horizon (N_p = 6 instead of 10) for parameter testing

- Lookup table for fuzzy rules to accelerate computation

- Parallel processing for monitoring and control components

- Anti-windup mechanism for integral components

- State estimation and filtering

- Safety limits and monitoring

The comparative performance under different simulation configurations is presented in Table 8.

Table 8.
Results from Different Simulation Configurations.

Metric Basic Advanced Difference (%)

Average RMS Error (mm) 1.35 1.62 20.0

Maximum Error (mm) 2.73 3.45 26.4

Control Effort (normalized) 455.0 483.8 6.3

Computation Time (ms) 0.61 0.83 36.1

Metric	Basic	Advanced	Difference (%)
Average RMS Error (mm)	1.35	1.62	20.0
Maximum Error (mm)	2.73	3.45	26.4
Control Effort (normalized)	455.0	483.8	6.3
Computation Time (ms)	0.61	0.83	36.1

Comment 7: The advanced simulation results show a slight degradation in performance but still maintain an average tracking error of 1.62 mm under non-ideal conditions, compared to 4.83 mm for conventional PD control.

The controller was tested on a complex pick-and-place simulation task involving objects with various shapes and weights (50–200 g). The performance comparison across different controllers is detailed in Table 9.

Table 9.

Pick-and-Place Task Performance.

Controller	Success Rate (%)	Cycle Time (s)	Positioning Accuracy (mm)
PD Controller	78.3	12.5	4.27
Adaptive	89.5	10.8	2.64
$H_{\infty}$	91.2	10.3	2.35
Proposed	97.6	9.1	1.43

Comment 8: The proposed controller demonstrates superior performance in simulation tasks, with a success rate of 97.6% and a 33% improvement in positioning accuracy compared to conventional methods.

These results align with studies by Song et al. (2019), highlighting that control performance in trajectory tracking tasks does not always directly translate into corresponding task performance. However, in our case, the improvement in trajectory tracking performance has indeed resulted in a significant improvement in task performance.

8. Discussion

8.1 Performance Analysis

The proposed integrated control architecture offers several advantages:

Complementary Strengths: Each component addresses specific aspects of the control problem:

–
The adaptive H-infinity component provides fundamental robustness against uncertainties and disturbances with formal stability guarantees.
–
The MPC component optimizes control actions based on predicted future behavior while ensuring adherence to constraints.
–
The fuzzy component handles nonlinearities through linguistic rules.
–
The sliding mode component with the disturbance observer offers high-frequency disturbance rejection capabilities.

Adaptive Weighting Mechanism: The dynamic adjustment of control contributions results in: –
Robustness when needed (large errors or disturbances).
–
Optimality when possible (small errors, nominal conditions).
–
Smooth transitions between control strategies.
–
Automatic prioritization based on real-time performance.

Disturbance Observer: The estimation and compensation of disturbances provide: –
Outstanding external force rejection.
–
Compensation for unmodeled dynamics.
–
Reduced sensitivity to friction and other nonlinearities.
–
Faster recovery from disturbances.

Formal Stability Guarantees: Rigorous mathematical analysis ensures: –
Bounded tracking errors even under worst-case conditions.
–
Predictable convergence rates.
–
Known stability margins.
–
Guaranteed performance limits.

A comprehensive comparison of performance indices across all controllers is provided in Table 10.

Table 10.
Comprehensive Performance Indices.

Index PD Adaptive H-infinity Proposed

Nominal tracking error (mm) 5.58 2.87 2.50 1.35

Disturbance tracking error (mm) 8.72 4.59 3.91 1.86

Parameter variation error (mm) 10.34 3.55 3.28 1.51

Control effort (normalized) 1.00 0.83 0.88 0.79

Noise rejection (0–10) 3.2 6.5 7.3 9.1

Robustness to variations (0–10) 2.8 7.1 6.4 8.9

Computational load (normalized) 1.00 2.35 2.17 3.82

Comment 9: The proposed controller consistently outperforms conventional methods across all performance indices, except for computational load, where it requires approximately 3.8 times the computational resources of the standard PD controller.

This result aligns with previous research on complex integrated control architectures. For instance, Duchaine et al. (2007) also observed significant computational load for similar integrated controllers, but concluded that the performance benefits justify the additional computational requirements.
8.2 Limitations and Future Research Directions

Index	PD	Adaptive	H-infinity	Proposed
Nominal tracking error (mm)	5.58	2.87	2.50	1.35
Disturbance tracking error (mm)	8.72	4.59	3.91	1.86
Parameter variation error (mm)	10.34	3.55	3.28	1.51
Control effort (normalized)	1.00	0.83	0.88	0.79
Noise rejection (0–10)	3.2	6.5	7.3	9.1
Robustness to variations (0–10)	2.8	7.1	6.4	8.9
Computational load (normalized)	1.00	2.35	2.17	3.82

A significant limitation of the proposed method is its computational complexity, as the integrated controller necessitates solving the real-time Model Predictive Control (MPC) optimization problem, online parameter adaptation, fuzzy inference processing, and updating the disturbance observer. For intensive simulations, several compromises have been made to address this, such as reducing the MPC horizon, simplifying fuzzy rule evaluation, and adopting discrete parameter adaptation. Future research aims to overcome these limitations through the development of computationally efficient MPC algorithms like explicit MPC and move blocking, hardware acceleration using FPGA or GPU, optimized disturbance observer structures, and event-triggered control updates. These approaches align with the work of Zheng et al. (2022), who demonstrated that explicit MPC techniques could reduce computation time by up to an order of magnitude in similar applications. Another challenge is the tuning of multiple parameters across the controller's components, which in the current simulation setup requires manual adjustment of 12 coefficient matrices, 5 disturbance observer parameters, 8 fuzzy membership functions, and 4 adaptive weighting parameters. Future research will focus on automatic tuning methods, self-tuning capabilities, reduced parameterization, and sensitivity analysis to identify critical parameters. Additionally, the current disturbance observer assumes that disturbance derivatives are bounded, which may not hold in all practical scenarios, particularly for discontinuous impacts and friction forces. Future improvements will include advanced disturbance observer structures for impulse noise, higher-order observers for better estimation, machine learning-based disturbance prediction, and hybrid observer methods tailored for various disturbance types. Several promising research directions stem from this work, including expanding the control framework for human-robot interaction with force sensors and compliance control, extending the method for multi-robot coordination in simulated environments, integrating reinforcement learning-based control for improved disturbance prediction and parameter estimation, optimizing control objectives for energy efficiency in battery-powered robotic systems, and enhancing fault-tolerant control to maintain performance despite actuator or sensor faults in simulations.

9. Conclusion

This paper presents a novel integrated control architecture for the trajectory tracking problem of a 5DOF + 1Gripper robotic arm in a simulated environment, combining H-infinity adaptive control, MPC, fuzzy logic, and sliding mode control with disturbance supervision. Our comprehensive approach addresses key challenges in robot control: model uncertainty, external disturbances, nonlinearity, and physical constraints.

The main innovations of our control framework include:

A systematic integration of supplementary control methods, with each method addressing specific aspects of the control problem.

An adaptive weighting mechanism that automatically adjusts the controller's contributions based on real-time performance.

A nonlinear disturbance observer providing accurate estimation and compensation for external disturbances.

Rigorous stability analysis with formal guarantees for trajectory convergence and robustness.

The theoretical analysis provides formal guarantees on stability and performance, while extensive simulation validation confirms the effectiveness of the proposed method. Results show up to a 43% improvement in trajectory accuracy and significant disturbance rejection compared to conventional methods.

The adaptive weighting mechanism allows the controller to automatically adjust its behavior based on the simulation operating conditions, making it suitable for a wide range of robot control tasks that require high precision and robustness. Performance is demonstrated in simulated pick-and-place tasks, with a success rate of 97.6%, even under changing conditions.

Although computational complexity remains a challenge for real-time implementation, the significant performance improvements justify the additional computational requirements during the simulation phase, especially for applications demanding high precision. Future research will focus on optimizing computation, automatic parameter tuning, advanced disturbance supervision, and expanding to multi-robot systems and human-robot interaction in simulation environments.

The proposed control framework represents a significant step forward in robotic arm control simulation, providing a platform for developing high-precision, robust control capabilities for next-generation robotic systems. The methods presented in this simulation study lay a solid foundation for future experimental research on physical robot platforms.

This research presents four distinct contributions to robotic manipulator control that advance significantly beyond existing approaches. The primary contribution of this study is the first systematic integration of four complementary control methodologies, including adaptive H∞, Model Predictive Control (MPC), Fuzzy Logic Control (FLC), and Sliding Mode Control (SMC) with a disturbance observer, within a unified framework. This approach overcomes the fundamental limitations of existing single or dual-method strategies, which cannot simultaneously address model uncertainties, external disturbances, nonlinearities, and physical constraints.

A novel adaptive mechanism is introduced through a mathematically rigorous adaptive weighting algorithm that utilizes exponential functions, specifically $α_{i} = \frac{exp (- β_{i} ‖ e ‖^{2})}{\sum_{j} exp (- β_{j} ‖ e ‖^{2})}$ , dynamically optimizing controller contributions based on real-time tracking errors. This method eliminates hard-switching discontinuities and manual tuning requirements while ensuring smooth transitions between control modes.

The research also presents an enhanced disturbance rejection design with a new nonlinear disturbance observer. This observer achieves exponential convergence with an average time of 123 ms, supported by formal stability guarantees represented as $‖ τ_{a} ‖ \leq ϵ_{a} e^{- K_{L} t}$ . This integrated disturbance observer system provides superior noise rejection when compared to standalone observer implementations.

Furthermore, a comprehensive theoretical framework is provided through the first complete stability analysis of a four-component integrated controller, using composite Lyapunov functions defined as $V_{total\;} = V_{H \infty} + V_{MPC} + V_{FLC} + V_{SMC}$ . This analysis provides formal convergence rate bounds $‖ e (t) ‖ \leq ‖ e (0) ‖ e^{- λ t} + \frac{γ_{total\;} ρ^{2}}{η_{1}}$ and robustness guarantees for parameter variations of up to $\pm 50 %$ .

The performance validation demonstrates substantial improvements: a 60.9% enhancement in tracking accuracy compared to dynamic surface control, 65.4% better disturbance rejection (3.21 mm vs. 9.78 mm peak error), and exceptional parameter robustness, showing only 11.9% degradation in contrast to the 60–90% degradation observed with conventional methods. The computational overhead, which is 3.8 times the baseline, is justified by the significant performance gains, making this approach suitable for high-precision applications. Future research will focus on real-time optimization algorithms, experimental validation on physical platforms, and the extension of this framework to collaborative multi-robot systems.

Footnotes

Informed Consent Statement

Not applicable.

Author Contributions

D.-A.P.: formal analysis, software, resources, data curation, writing—original draft preparation, data collected and analyzed, visualization; S.-H.H.: supervision, project administration, writing—review and editing.

Funding

This work was supported by 2025 Ministry of Oceans and Fisheries (MOF) Marine Blue Tech Future Leader Training Project ‘Training Blue Tech Leaders for Eco-Friendly Ships (No. RS-2025-02220459)’ and the Glocal University 30 Project Fund of Gyeongsang National University in 2025.

Declaration of Conflicting Interests

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

Data Availability Statement

Dataset available on request from the authors.

ORCID iDs

Duc-Anh Pham

Seung-Hun Han

Institutional Review Board Statement

Not applicable.

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$e / \dot{e}$	NL	NS	ZE	PS	PL
NL	PL	PL	PL	PS	ZE
NS	PL	PS	PS	ZE	NS
ZE	PL	PS	ZE	NS	NL
PS	PS	ZE	NS	NS	NL
PL	ZE	NS	NL	NL	NL

$e / \dot{e}$	NL	NS	ZE	PS	PL
NL	PL	PL	PL	PS	ZE
NS	PL	PS	PS	ZE	NS
ZE	PL	PS	ZE	NS	NL
PS	PS	ZE	NS	NS	NL
PL	ZE	NS	NL	NL	NL

Adaptive H-Infinity MPC-Fuzzy Control Integrated With Sliding-Mode Disturbance Observer for Trajectory Tracking of a 5-DOF Robotic Manipulator With Gripper

Abstract

Keywords

1. Introduction

2.1 Classical Control Methods

2.2 Robust and Adaptive Control Strategies

2.3 Predictive Control Approaches

2.4 Intelligent Control Methods

2.5 Hybrid and Integrated Control Approaches

2.6 Research Gaps and Contributions

2.7 Justification for Multi-Controller Integration

3. Robotic Manipulator Modeling

3.1 Kinematic Model

Table 1. D-H Parameters of 5DOF Manipulator. J o i n t i a i ( m m ) α i ( ∘ ) d i ( m m ) θ i ( ∘ ) 1 0 90 28 θ 1 2 150 0 0 θ 2 3 85 0 0 θ 3 4 50 90 0 θ 4 5 0 0 40 θ 5

3.4 Model Validation and Verification

4. Problem Formulation

4.1 Assumptions

4.3 Control Constraints

4.4 Integrated Control System Architecture

5.3 Fuzzy Logic Control Component

5.6 Controller Parameter Design Guidelines

6. Integrated System Stability Analysis

6.1 Composite Lyapunov Function Design

6.2 Integrated Stability Theorem

7.1 Simulation Setup

7.2 Performance in Terms of Tracking Error

Table 8. Results from Different Simulation Configurations. Metric Basic Advanced Difference (%) Average RMS Error (mm) 1.35 1.62 20.0 Maximum Error (mm) 2.73 3.45 26.4 Control Effort (normalized) 455.0 483.8 6.3 Computation Time (ms) 0.61 0.83 36.1

8.1 Performance Analysis

9. Conclusion

Footnotes

Informed Consent Statement

Author Contributions

Funding

Declaration of Conflicting Interests

Data Availability Statement

ORCID iDs

Institutional Review Board Statement

References

Table 1.
D-H Parameters of 5DOF Manipulator.

$J o i n t i$ $a_{i} (m m)$ $α_{i} (\circ)$ $d_{i} (m m)$ $θ_{i} (\circ)$

1 0 90 28 $θ_{1}$

2 150 0 0 $θ_{2}$

3 85 0 0 $θ_{3}$

4 50 90 0 $θ_{4}$

5 0 0 40 $θ_{5}$

Table 8.
Results from Different Simulation Configurations.

Metric Basic Advanced Difference (%)

Average RMS Error (mm) 1.35 1.62 20.0

Maximum Error (mm) 2.73 3.45 26.4

Control Effort (normalized) 455.0 483.8 6.3

Computation Time (ms) 0.61 0.83 36.1

$e / \dot{e}$	NL	NS	ZE	PS	PL
NL	PL	PL	PL	PS	ZE
NS	PL	PS	PS	ZE	NS
ZE	PL	PS	ZE	NS	NL
PS	PS	ZE	NS	NS	NL
PL	ZE	NS	NL	NL	NL