Design an Adaptive PID Control Based on RLS with a Variable Forgetting Factor for a Reconfigurable Cable-Driven Parallel Mechanism

Abstract

This paper proposes a two-layer adaptive proportional–integral–derivative (PID) controller for precise pose control of a six-degree-of-freedom cable-driven parallel robot with eight cables, specifically designed to handle dynamic changes caused by the movement of attachment points. The positions of the attachment points on the base are adjusted to avoid collisions between humans and cables, where humans and robots are working in a shared workspace. The inherent nonlinearity of the robot system was addressed using model identification based on the recursive least squares (RLS) algorithm equipped with an adaptive forgetting factor. This method enables real-time updates to the dynamic model of the robot, thereby ensuring accurate parameter estimation as the attachment points move. The combination of the PID controller and RLS algorithm enhances the system’s ability to respond effectively to changing dynamics. Simulation results highlight the superior accuracy, robustness, and adaptability of the proposed approach, making it well suited for applications requiring a reliable performance in dynamic and unpredictable environments. The proposed method can guarantee human safety, while the end effector tracks the desired trajectory.

1. Introduction

Cable-driven parallel mechanisms (CDPMs) have attracted considerable research interest, owing to their advantages, such as lightweight structures, cost-effectiveness, ease of installation, and high precision. These mechanisms are widely used in industrial applications for heavy-load transportation and in haptic systems that require precise motion control. CDPMs encounter a variety of challenges, including workspace analysis [1,2,3], tension distribution algorithms (TDA) [2,4,5], singularity analysis [6,7], pose control, and collision avoidance [8,9]. Notably, physical design [10,11] remains one of the most critical and complex challenges, as it fundamentally affects the system’s performance and feasibility. Cheng et al. 2023 [12] proposed an optimization method for the positions of cable attachment points in a reconfigurable CDPM under various workspace conditions. It formulates wrench-feasibility, wrench-closure, and interference-free constraints as polynomial functions or variables. It minimizes cable forces or maximizes tension factors using advanced optimization techniques that are applicable to different reconfiguration mechanisms, such as rails, unmanned ground vehicles (UGVs), and unmanned aerial vehicles (UAVs). However, designing an RCDPM using movable attachment points to ensure safe human–robot interaction in shared workspaces by preventing cable–human collisions is a major challenge. To address this issue, Khoshbin et al., 2022 [13] proposed adjusting the position of the attachment point along the z-axis on the base. Specifically, when the minimum distance between a human and a cable, estimated using the KKT method, approaches a predefined safety threshold, the attachment points are repositioned to prevent potential contact. As discussed in this paper, such adjustments inherently alter cable lengths and tensions, significantly affecting system dynamics.

Furthermore, Youssef and Otis, 2020 [14] suggested modifying the attachment point positions along the Z-axis to mitigate cable–cable collisions, further enhancing system safety, adaptability, and operational efficiency. Geometric modifications have been implemented for several reasons, such as preventing collisions between cables or between humans and cables and ensuring accurate end effector positioning remains a significant challenge. To address this, various control strategies, including proportional–integral–derivative (PID) controllers, have been investigated to improve performance, stability, and overall system efficiency.

Zake et al., 2020 [15] proposed a visual servo control approach for the ACROBOT CDPM, a 6-DOF system with eight cables, ensuring accurate end effector positioning despite modeling errors and perturbations. This study introduced a new control-stability workspace, although the end effector orientation remained constant. Awais et al. 2018 [16] presented a vision-based control system for a 3-DOF planar CDPM with four cables, using a fixed camera on the end effector. Cable tensions are measured via load cells, whereas a visual marker enables the localization and tracking of the CDPM. This study highlights the impact of camera calibration and marker size on the system accuracy. A robust algorithm integrates camera calibration with real-time visual marker monitoring to improve performance. Bayani et al., 2015 [17] proposed a PID control strategy for a planar CDPM to enhance end effector tracking. The controller parameters were tuned using pole placement, by positioning closed-loop poles at the desired locations. The optimization was performed using differential evolution, particle swarm optimization, and genetic algorithms.

Picard et al., 2018 [18] presented a PID feedforward control strategy for CAROCA, a 6-DOF CDPM with eight cables. Three control schemes are proposed: (a) a PD torque controller with real-time mass input, (b) a PD controller with a mass estimator, and (c) a PD controller with a mass estimator and compensator. Real-time payload estimation is integrated into the feedforward term to compensate for the payload mass and friction, thereby enhancing trajectory tracking. The controller parameters are updated based on the payload mass, thereby achieving greater accuracy than the PD control. The RMS end effector position error was 60 mm for the PD, with a 58% reduction for heavier payloads and a 91% reduction for the feedforward-PD.

Jiesinuer et al., 2021 [19] presented a 3-DOF planar CDPM with four cables and derived its dynamic model using D′Alembert’s principle and the principle of virtual work. The proposed approach integrates torque control, cable force optimization, and fuzzy control. A fuzzy algorithm dynamically tunes the PD parameters in real time and achieves higher accuracy than a fixed-gain PD torque controller. Zhou et al., 2023 [20] proposed a fuzzy PID controller for a 3-DOF CDPM with four cables by utilizing an adaptive whale optimization algorithm (FPID-AWOA). This method achieves higher trajectory tracking accuracy than fuzzy PID with particle swarm optimization (FPID-PSOA). FPID-AWOA reduces the tracking errors by 51.2% and 19.5% in the x direction, 64.2% and 49.7% in the y direction, and 29.1% and 12.2% in the z direction, respectively. Khalilpour et al., 2020 [21] proposed a wave-based controller for an ARAS-CAM 3-DOF CDPM, with four cables to manage the cable loss. The design requires no precise calibration or installation, effective control of loosened cables, or reduction in vibrations. The results showed improved performance and smoother tracking compared to conventional cascade controllers. Khalilpour et al., 2019 [22,23] proposed a cascade controller for an ARAS-CAM 3-DOF CDPM with four cables to address the system uncertainties. This approach models the actuator and power transmission dynamics using an inner loop for cable force control and an outer loop with a robust sliding mode. The system stability was analyzed using the Lyapunov method. The results show an improved performance compared to the sliding mode controller. Khalilpour et al., 2018 [24] proposed a robust sliding mode control for a 3-DOF CDPM with four cables to ensure closed-loop stability using the Lyapunov matrix. This method accounts for uncertainties in attachment points and cable models while operating in Cartesian coordinates, without calibration or inverse kinematics. This study analyzes sliding vector surface variations and Cartesian error manifolds. Controller gains were selected to ensure stability under a 10% uncertainty in the end effector’s mass and attachment point positions.

Abdolshah, S., and Barjuei, E. S. (2015) [25] present LQ controller for a 3-DOF CDPM with four cables, optimizing a performance index based on Riccati’s equation to compute the end effector position, velocity, and motor torque. The proposed controller ensures accurate trajectory tracking while maintaining the motor torque within the design limits. Tho, T.P. and N.T. Thinh, 2021 [26] propose an ANFIS-based control approach for a 6-DOF CDPM with eight cables. This method predicts cable sagging using a first-order Sugeno ANFIS model with 81 rules and three generalized Gaussian membership functions. The inputs include the kinematic structure, cable tension, and tension distribution. Lu et al., 2022 [27] analyzed closed-loop stability using a reinforcement-learning-based control approach for a 6-DOF CDPM with eight cables. This method compensates for the uncertainties caused by friction and cable elasticity. Kumar et al., 2019 [28] proposed an input-output feedback linearization control for a 6-DOF CDPM with four cables. A linear feedback controller was designed, using the pole placement method with and without noise to achieve effective trajectory tracking. However, the workspace considerations have not yet been addressed.

Kiani, A. and S.K.-e. Mashhadi, 2017 [29] proposed a Lyapunov-based adaptive control for the Samen SpaceCam CDPM with four cables, considering a linear model with unknown parameters. The approach employs a model reference adaptive control with an adjustable feedback controller to enhance disturbance rejection and tracking performance. The closed-loop stability was ensured using the Lyapunov function. In 2017, Yoon, J. et al. [30] proposed an adaptive control method for a planar 3-DOF CDPM with four cables to track the desired position. The cables were modeled as axial springs, with an unknown constant stiffness coefficient estimated by the controller. The system generates cable lengths as inputs and maintains trajectory tracking despite a 40% parameter estimation error, thereby reducing tracking error and residual vibrations. Aghaseyedabdollah et al., 2022 [31] proposed an adaptive PID controller in which the PID sliding surface gains were tuned via a fuzzy system. The optimization algorithm optimizes fuzzy membership functions. The closed-loop stability was analyzed using the Lyapunov function. This method reduces chattering and enhances robustness against load disturbances and uncertainties. Santos et al., 2019 [32] proposed a model predictive controller (MPC)-based control framework, consisting of prediction and optimization. This method integrates the actuation redundancy resolution within the MPC, eliminating the need for a tension-distribution algorithm. The results showed improved position tracking performance, outperforming PID and sliding mode controllers. Song, C. and D. Lau, 2022 [33] propose an MPC for a 2-link planar CDPM and a 6-DOF spatial CDPM to track desired trajectories. This paper presents a workspace analysis that reformulates the optimization problem into a convex form under uncertainty. Recursive feasibility and stability are strictly ensured in the nominal case, with inherent robustness under non-nominal conditions.

The control of RCDPMs presents significant challenges, due to their strong nonlinearities and the necessity to maintain performance under continuously changing geometric configurations. Traditional adaptive control strategies, such as the adaptive control presented by Ji, H. et al., (2020) [34], mainly address passive uncertainties, including unknown payload, friction, or cable stiffness, while assuming fixed attachment points. Similarly, advanced methods developed for interaction safety, such as Fuzzy-PID methods presented by Tong, Lina et al., 2024 [35] and Liang, Xu et al., 2021 [36] concentrate on the system’s mechanical properties (stiffness/damping) or managing continuous human interaction force. Moreover, a zero-force control framework, based on a fuzzy-PID method presented by Tong, Lina et al., 2024 [35], has been proposed for post-stroke rehabilitation training, where adaptive impedance modulation was used to enhance human–robot interaction comfort. Likewise, works focusing on specialized kinematic designs, such as the method presented by Lei, Yanqiang et al., 2024 [37], remain limited to geometric analysis and do not extend to dynamic control under reconfiguration. However, these studies typically do not address the broader dynamic control challenge in general reconfigurable systems. Recent works on real-time reconfiguration planning and fault-tolerant control have underlined the importance of multi-layer architectures to handle both intentional reconfigurations (for workspace optimization) and unintentional ones (due to cable failure). For example, Raman, Adhiti et al., 2023 [38] effectively employs an interactive multiple model (IMM) adaptive filter for fault diagnosis and operational recovery. Yet, such approaches mainly facilitate decision-making at the planning level and fail to provide immediate real-time compensation for the abrupt changes in dynamic parameters that are directly induced by geometric reconfiguration. These step-like variations generate active uncertainties far beyond the scope of conventional passive uncertainty compensation. Similarly, Raman Thothathri, Adhiti, 2022 [39] integrates screw-theory-based modeling with deep reinforcement learning (DRL) to manage configuration-dependent dynamics. While theoretically rigorous, its heavy computational load and quasi-static assumptions restrict its real-time applicability, and practical constraints, such as tension limits or fast parameter adaptation, are not explicitly addressed.

Most existing controllers are primarily designed for CDPMs with fixed attachment points [18,40,41,42]. In RCDPMs, modifying attachment points to prevent cable collisions introduces uncertainty, affecting the force transmission, mass matrices, and overall dynamics. These variations, including changes in the cable length, attachment angle, and system dynamics, require adaptive or robust control for effective handling. Frequent modifications may require real-time model updates or dynamic controller adjustment. To enhance reliability, the integration of adaptive control with a primary control strategy is recommended. A promising approach involves real-time system modeling, which allows for continuous updates based on real-time estimation. Recursive least squares (RLS) is a powerful tool for real-time parameter estimation in adaptive control. Initially designed for linear systems with fixed parameters, it can be adapted for time-varying systems using a forgetting factor, prioritizing recent data, while reducing the influence of older data. An RLS with a variable forgetting factor further improves the accuracy by dynamically adjusting the data weighting, thereby enabling the precise tracking of system dynamics. This adaptability allows controllers to respond immediately to changes, thus enhancing the overall performance and stability.

Compared to other methods, such as the adaptive robust control based on singular perturbation theory [43], this paper presents a two-layer scheme that offers a simpler yet dynamically adaptive structure. Instead of estimating uncertainty bounds, it directly identifies and tracks dynamic model parameters, which are crucial for reconfigurable systems with abrupt geometric changes.

As a contribution of this paper, a two-layer control structure is proposed for a reconfigurable cable-driven parallel mechanism, in which, as the first contribution, the first layer adjusts the attachment points on the base in real time to avoid human–cable collisions. As the second contribution, the second layer controls the end effector motion by adaptive PID, based on RLS.

In the second layer, an adaptive PID controller is developed, integrating an online model identification scheme based on RLS, with a variable forgetting factor to continuously update the system model under geometric reconfiguration and model uncertainties. The RLS algorithm dynamically updates the parameters of the nominal linearized model while accounting for changes in the Jacobian matrix and system dynamics caused by large attachment point displacements, thereby incorporating uncertainties directly into the control constraints. The RLS-identified model provides real-time parameter estimates that enable online adaptation of the PID gains to maintain precise trajectory tracking during attachment point movements. This adaptive framework enhances controller robustness and stability, making it particularly effective for RCDPMs, where unpredictable variations in the attachment points significantly influence the system dynamics. Simultaneous management of human safety and trajectory accuracy is achieved by integrating geometric and dynamic control layers, allowing for collision avoidance without compromising tracking performance. This results in a lightweight controller with fast response and strong real-time applicability. So, this two-layer architecture (i) directly identifies and tracks dynamic parameters across reconfigurations, (ii) integrates a constrained tension-distribution layer to enforce safety limits, and (iii) remains computationally lightweight for practical real-time implementation.

This paper is organized into three main sections. Section 2 introduces the mathematical nonlinear model (used as the physical simulation model) and the corresponding nominal linearized models with variable attachment points, along with a two-layer control framework. In the first layer, the positions of the attachment points are estimated to prevent cable–human collisions, whereas the second layer controls the pose of the end effector. Within the second layer, the parameters of the nominal linearized model are continuously updated using the RLS algorithm, which then inform the design of the PID controller. These layers operate concurrently and interact in a dynamic manner. Section 3 presents simulation results that validate the proposed approach. The subsequent section details the controller implemented for the RCDPM model in the MATLAB R2025b simulations.

2. The Schematic of the Proposed Control Architecture with Adaptive PID

This study proposes a control strategy for an RCDPM with eight cables, where eight tensile servomotors control the position of the end effector and eight translation servomotors adjust the attachment points vertically. As illustrated in Figure 1, the proposed RCDPM control structure comprises two layers that operate simultaneously. This design enhances real-time adaptability, enabling efficient collision avoidance and precise trajectory tracking, while ensuring high system performance. System identification estimates a nominal model to adapt to dynamic changes, owing to moving attachment points in the z direction. This model enables real-time controller updates and ensures precise trajectory tracking through continuous tension estimation. Definitions of all parameters are available in Appendix C.

The first layer manages the geometric model and collision detection by adjusting the attachment points on the base to prevent collisions and to adapt to dynamic changes. It calculates cable distances, optimizes new positions, and moves attachment points using eight vertical servomotors along the z-axis, while maintaining cable length constraints. These adjustments occur gradually at a lower update rate than in real-time cable tension control.

The second layer controls the pose of the end effector using a dynamic model, ensuring stability and precise trajectory tracking as the attachment points move. Sensors collect input data (e.g., cable tensions) and output data (e.g., end effector positions), and the parameters of the nominal linearized model are updated via RLS. In the RCDPM, both layers interact, requiring careful management to achieve stability and optimal performance. The first layer adjusts the attachment points based on human positions and cable tensions from the second layer to prevent collisions, whereas these adjustments alter the cable lengths and angles, necessitating tension recalibration in the second layer. The second layer controls the cable tension and tracks the end effector, thereby ensuring an accurate force distribution. Although the motors in the first and second layers operate independently, changes in the attachment points (controlled by the first layer) indirectly influence cable tension in the second layer. If tension feedback from the lower layer to the upper layer is unnecessary, the two layers can operate partially independently to improve modularity and efficiency. To manage these interactions, the RLS algorithm continuously updates the system parameters at each time step. This study integrated system identification with PID control [44] to enhance the stability and accuracy of the reconfigurable CDPMs.

The proposed first-layer framework to avoid collisions between humans and cables is detailed in Algorithm 1, whereas Algorithm 2 outlines the controller’s implementation steps. In Algorithm 1, function z_new is proposed to estimate the position of attachment points to keep distance between human and cables in safe distance ${\mathrm{m}\mathrm{i}\mathrm{n}}_{\mathrm{z}}{‖{{\mathrm{d}}_{\mathrm{h}}}_{\mathrm{i}}-{{\mathrm{d}}_{\mathrm{h}}}_{\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}‖}^2$. This algorithm attempts to maintain the maximum distance between cables and humans by displacing the attachment points (Anew) such that the cables move as far away from the obstacles (human or object) as possible. In Algorithm 2, if the RLS update fails due to numerical instability or ill-conditioning, the parameter vector is reinitialized to prevent error accumulation and to restore stability before continuing the estimation. The Matlab code based on the presented pseudocode is available online in [45] and is cited in the paper.

Algorithm 1: First Layer: Attachment Point Position Adjustment
Function Main Kinematic Control Loop (N,A0,Xe0,system_config) Inputs:
	$N\in \mathbb{N}$: Total number of time steps (time horizon) $A_0{\in \mathbb{R}}^{n_c\times 3}:$Initial attachment points (for $n_c$ cables) in x, y and z directions System_config: system configuration (e.g., safety threshold $d_{safe}$, acceptable threshold $d_{accept}$, Workspace $W$, Maximum velocity $v_{max}$, human position Ph) $d_{safe}$: Safe distance between human and cables $X_{end}{\mathbb{R}}^6:$End effector pose over time $X_{e0}{\mathbb{R}}^6:$Initial end effector pose
Outputs:
	$A\in {\mathbb{R}}^{n_c\times 3\times N}$: Updated attachment points over time $X_{eq}\in {\mathbb{R}}^{3\times N}$: Equilibrium points over time
Initialize:            Set k ← 1            Initialize Attachment Points A ← $A_0$            For k = 1 to N do:            Compute Shortest Distance ($d_{min}$) ← KKTsolver($P_h$, $A\left[:,:,k\right],$ $X_{end}$) If $d_{min}$ < $d_{safe}$ then            Collision Risk ← True            Else            Collision Risk ← False            End If            //Check if end effector and cables are within workspace limits            Compute tensions T ← TensionSolver(A,$X_e$)              If any(T <$T_{min}$) OR any(T >$T_{max}$)            WorkspaceValid ← False          L ← CableLengths(A,$X_e$)          If any(L < $L_{min}$) OR any(L > $L_{max}$):        WorkspaceValid ← False            # Singularities/poor conditioning          J ← Jacobian(A, $X_e$)               Else If det(J*J′) < epsilon            WorkspaceValid ← False # Equilibrium proximity constraint          If norm($X_{eq}-X_e$, 2) > 0.5: return False If (Collision Risk = True) OR (Workspace Valid = False)          Then //Adjustment Needed                Solving function z_new ← Solve Optimization(Objective Function, Constraints)              where Constraints include
	- Collision Avoidance $d_{min}\ge d_{safe}$ - Workspace Limits $X_e\left[k\right]ϵW$ - Maximum velocity of movement ${‖z_{new}-z_{prev}‖}_2\le v_{max}∆t$ - Maximum distance between the equilibrium point and end effector position ${‖X_{eq}-X_e‖}_2\le 0.5$
A[:,:,k] ← Update Attachment Points (z_new)              Else              //No Adjustment              A[:,:,k] ← A[:,:,k−1]              End If              Transmit attachment points position to first group of motors in first layer (A[:,:,k])              End For              Return (A, $X_{eq}$)

Algorithm 2: Second layer: RCDPM adaptive PID loop

Inputs:   $N\in \mathbb{N}$: Total number of time steps (time horizon)   $X_{e0}\in {\mathbb{R}}^{n_x}$: Initial position and rotation of end effector where $n_x is number of output$   ${\theta }_0\in {\mathbb{R}}^{n_{\theta }}$: Initial_model_params (e.g., ARX model coefficients) where   $n_{\theta }$ is the number of parameters for each output.   system_config: System configuration (e.g., input/output dimensions, sampling time dt)   pid_tuning_params: Parameters for PID tuning   rls_params: Parameters for RLS (e.g., forgetting factor, initial covariance)   $X_{eq}$: equilibrium point from first layer   Outputs:   $U\in {\mathbb{R}}^{n_u\times N}$: optimal control inputs applied over time   $X_e$ $\in {\mathbb{R}}^{n_x\times N}$: matrix of measured system outputs over time   $\theta \in {\mathbb{R}}^{n_{\theta }\times N}$: updated model parameters over time   Begin:            Initialize:            Set k ← 1            Initialize the covariance matrix $P$ ← $P_0$            Initialize forgetting factor $\lambda$ ← ${\lambda }_0$            Initialize input $U$ ← zeros$(n_u,N)$            Initialize end effector pose X ←zeros$(n_x,N)$            Initialize model parameters $\theta$ ←zeros$(n_{\theta },N)$            Set dt//Time step            Defined the desired trajectory $X_{desired}$ For k = 1 to N do:                 //Update the dynamic model of CDPM (Appendix A to simulate nonlinear dynamics) $u_{previous}$← Input from previous step $X_e$[1:6,k]← $x_m$ (Measured current outputs by sensors (e.g., pose) //Defined ARX model (Equation (A15), Appendix B) in relative form $∆X_e{=X}_e$−$X_{eq}$ //Update Model Parameters   Defined $\varphi$←ConstructRegressor(xm,uprevious,gravity,system_configuration)                        $\varphi =[∆X_e(t-1)\dots X_e(t-n_a) ∆U(t-1)\dots U(t-n_b\left)\right]$   Estimated parameters of ARX   $\theta \left(k\right)$← UpdateRLS($\theta (k-1)$, $\varphi$, X, P, $\lambda$)             If UpdateRLS failed = true”             Then Log “RLS update failed at time step k = “, k             $\theta \left(k\right)$ ← ${ \theta }_0$ Reset to initial params   End If   //Compute Optimal Control Input   Tracking Error(k) ←  $∆X_d$[k] − $∆X_e$   Pid_gains ←//Compute $K_p$, $K_i$, $K_d$ for MIMO ΔF(k) ← PID_task(e(k), e_dot, e_int)   F(k) ← $F_{eq}$(k) + ΔF(k)   //Add equilibrium feedforward (optional but recommended)   U(k) ← F(k)     //Input of system in task space   //Tension distribution with constraints           J ← Jacobian(A, $X_e$(k))        (T(:,k), status) ← DistributeTension(J, F(k), $T_{prev}$, Limits)   //transmit tension to the second group of motors   End For   Return $\left(T\right(:,k)$, $X_e(:,k)$, $\theta \left(k\right))$   End

The next section presents more information on system identification using the RLS in the nominal model of the RCDPM. An estimated nominal model was used to design the proportional–integral–derivative (PID) controller. The proposed RCDPM controller consists of two steps. First, an adaptive strategy using real-time RLS updates the linearized linear model to estimate the transfer functions under changing conditions. Second, PID minimizes pose errors, even with model uncertainties, by adjusting cable forces and updating controller parameters for precise tracking. Further details are in Section 2.1 and Section 2.2.

2.1. Dynamic and Kinematic Model of 6 DOF CDPM with 8 Cables

This section presents the nonlinear RCDPM model used in the MATLAB simulations to generate end effector poses based on real-time cable tension inputs. In the absence of real-world data, the simulated input-output data validate the RLS algorithm for parameter estimation. The control strategy employed eight tensile servomotors for position control and eight translation servomotors for attachment point adjustments. To simplify the modeling, the pulley dynamics were ignored, and the cables were assumed to be massless and inextensible. The kinematic and linearized dynamic models of the robot [46] are detailed, with attachment points moving per step in the z direction, whereas tensile servomotors regulate cable tensions for precise trajectory tracking. The attachment points are allowed to move in the z direction in this study to simplify the reconfiguration process and focus on vertical adjustments. However, the algorithm is formulated in a three-dimensional form (x, y, z) to maintain generality, so that in future works, the attachment points can also be reconfigured along the x and y directions without modifying the overall structure of the algorithm.

The pose of the end effector is given by $X_e={[x_e {\phi }_e]}^T$ where the $x_e={[x y z]}^T$ and ${\phi }_e={\left[{\phi }_x{ \phi }_y{ \phi }_z\right]}^T$ are the position and orientation of the center of the end effector, respectively. The kinematics and constraints of the CDPM were presented in [13,46]. The dynamic model of the CDPM in the operation space is given by Equation (1) [5].

Eight rectilinear translation servomotors adjusted the attachment points vertically, altering the Jacobian and system dynamics, including the mass, damping, and stiffness matrices. The attachment points were reconfigured in the z direction for each sample. The dynamics presented in Equations (1) and (2) and the kinematic parameters are described in detail in [47]. Equation (1) is the dynamic simulation model of the RCDPM, which is employed in the simulations as a substitute for the physical model. Equation (2) is the linearized dynamic model of the robot, which is derived from the full nonlinear dynamic model (Equation (1)), using the Taylor series expansion around an equilibrium point ($X_{eq}$). This transformation to relative variables (${∆X}_e=X_e-X_{eq}$) is essential for implementing our relative controller. It shifts the controller’s focus to small deviations from the equilibrium point ($X_{eq}$), which simplifies the controller’s task and makes it more specific to what actually needs to be corrected.

More details of the dynamic model and parameters of CDPM are available in Appendix A. The relative controller, which operates on the deviation from the equilibrium point $X_e-X_{eq}$, performs better than an absolute controller. This is because the relative approach makes the controller’s task simpler and more specific. It separates the system’s physics from the path-tracking objective, allowing the controller to focus only on what actually needs to be corrected.

$${\left[M\left(X_e\right)\right]}_{6\times 6}{\ddot{X}}_{e_{6\times 1}}+{\left[C\left(X_e,{\dot{X}}_e\right)\right]}_{6\times 1}+{\left[K{(X}_e\right)]}_{6\times 1}+G_{6\times 1}=[{{A^T}_v\left({\phi }_{e_0}\right)]}_{6\times 6}{∆F}_{{ev}_{6\times 1}}$$

(1)

$$M_{0_{6\times 6}}∆{\ddot{X}}_e+C_{0_{6\times 6}}∆{\dot{X}}_e+K_{0_{6\times 6}}{∆X}_e=\left[{A^T}_v\right({\phi }_{e_0}\left)\right]{∆F}_{{ev}_{6\times 1}}$$

(2)

More details on the mass, spring, and damper matrices of the nonlinear and linear models are presented in [47]. The next section discusses the system identification method for estimating the parameter model of the CDPM.

2.2. First Layer: Estimation of the Position of Attachment Points on the Base

The first layer focuses on geometric and kinematic models of CDPM. Modifying the attachment points on the base potentially causes the end effector to fall outside the workspace or deviate significantly from its intended trajectory. To ensure both safety and stability, this paper introduces an optimized cost function for determining new attachment points. This cost function is designed to prevent collisions between humans and cables, and ensure that the end effector remains within the workspace boundaries. The cost objective function is given by Equation (3). $d_{h_{i\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}}$ is the maximum safe distance between human and cable and is chosen to be 0.2 m in this research.

$$J={min}_z {‖{d_h}_i-d_{h_{isafe}}‖}^2\ge 0$$

(3)

However, there are five constraints:

$$\left\{\begin{array}{c}0\le Z_i\le 3 \\ d_{hi}\ge 0.2\\ \left|z_i-z_{i,prev}\right|\le maximum movement\\ x_{end effector} \in Workspace \\ Kinematic tension:T_{{min}_i}<T<T_{{max}_i}\\ {‖x_{eq}-x_{endeffector}‖}_2\le maximum distance end effector and equilibrium point=0.5 \mathrm{m}\end{array}\right. \forall i=1,\dots \dots ,n_{cables}$$

This cost function is used to determine (or minimize) the objective and optimized position of attachment points on the base, while limitations of distance between human and cables and existing end effectors inside workspaces are the main constraints. Also, the maximum acceptable velocity of moving attachment points on the base should be considered. The cost function is solved numerically at each time step, using a constrained optimization approach implemented through MATLAB’s fmincon function. This solver determines the decision variables (e.g., attachment point positions) that minimize the cost function while satisfying all physical and safety constraints.

2.3. Second Layer: Pose Control of RCDPM

This section presents the system identification method and PID controller for the second layer to track the desired pose of the end effector.

2.3.1. System Identification

To estimate the mechanism parameters, input–output signals were collected by using a nonlinear dynamic model, owing to the absence of a physical model. The PID controller is designed based on a nominal linearized model, and the parameters of this model are estimated by recursive least squares (RLS), depending on the system characteristics. Linearization with offline parameters, which are fixed over time, is only valid near specific points. For larger movements, the errors increase, requiring multiple controllers and adding complexity and potential instability. In a CDPM with moving attachment points along the z-axis, continuous changes alter the mass, damping, and stiffness matrices, resulting in insufficient fixed-point linearization. Instead, RLS is more suitable because it continuously updates the system model in real time, thereby capturing dynamic variations.

In Section 2.3.2, the parameters of the nominal model are determined by using offline RLS. Offline RLS leverages predefined input and output signals from the initial configuration to estimate the parameters of the nominal model. These estimated parameters remain fixed and non-adaptive, as they are not updated in real time. The use of offline RLS is recommended in scenarios where the exact parameters of the motors and CDPM are inaccessible, providing a practical solution for parameter estimation under such constraints.

2.3.2. Non-Adaptive PID Controller

As shown in Figure 2, the offline RLS identifies the parameters of the nominal model for the PID controller design, providing a simplified representation of the system behavior. The parameters of the model, derived from the pre-collected data, remained fixed, and the PID parameters were tuned accordingly. Offline RLS only estimates the fixed or nominal model, and therefore, the controller remains non-adaptive.

RLS estimates the system parameters using open-loop or closed-loop data [48]. Open-loop offers higher accuracy but risks instability, whereas closed-loop ensures stability but introduces noise, requiring techniques such as instrumental variables (IV) for accuracy. An open loop is preferable for stable systems, whereas a closed loop is safer for unstable systems with appropriate noise filtering. Control Loop 1 outlines a non-adaptive approach with a fixed ARX and controller parameters. In fact, N₁ and N₂ represent the number of time steps for the first and second control loops, respectively.

N₁ corresponds to the kinematic modification layer, which computes the adjustment of the attachment points on the base to avoid collisions.

N₂ corresponds to the dynamic control layer, which performs end effector trajectory tracking through cable tension control.

The non-adaptive controller utilizes fixed gains. To determine these fixed parameters, offline RLS (recursive least squares) is employed in this paper. This process involves executing the RLS algorithm only once on pre-collected data (the system’s inputs and outputs in the initial fixed configuration of the RCDPM). The goal is to estimate a fixed set of nominal model parameters for the initial equilibrium point. However, the use of these fixed parameters places the final controller in the category of non-adaptive control because, as long as these parameters are not updated continuously during system operation (online), the system lacks an adaptive loop, resulting in low robustness against sudden changes and dynamic uncertainties (such as those caused by moving attachment points), and its tracking performance rapidly deteriorates.

However, in real-world scenarios, varying attachment points alter system dynamics, necessitating controller updates for effective performance. So, this paper proposes that an adaptive PID controller (APID) utilizes an online adaptive loop that includes the RLS algorithm with a variable forgetting factor. This structure enables the APID to continuously and adaptively update the model or controller parameters online to cope with uncertainties and dynamic changes, thereby maintaining high tracking accuracy and stability over time. The next section explores the adaptive PID control for the RCDPM.

Control Loop 1: Non-adaptive relative dynamic control with offline RLS-based parameter estimation

Choosing a safe distance by an expert Choosing a desired trajectory First layer: The main control loop steps of the first layer loop K1 ≤ 1 //Step time index Initializing for each step index K1 (K1 = 1 to N1) do:      Step 1.1: Find the shortest distance between humans and cables by the KKT method presented in [13].      Step 1.2: Comparison between the safe distance and the calculated distance                      If the distance is higher than the safe distance, go to step 1.3.                      else if the distance is lower than the safe distance, go to the end for.      Step 1.3: Estimate optimized attachment point positions to a safe position by solving the cost function.      Step 1.4: Estimate the equilibrium point based on estimated new attachment points.      Step 1.5: Transmit the optimized positions to the first group of servomotors for relocation.      Step 1.6: Send the equilibrium point and new attachment points to the second layer. end for Second layer: The main non-adaptive control of the second layer loop (offline RLS + fixed parameters PID controller)      Step 2.1 Gather outputs x(i,k) and inputs u(j,k) from the nonlinear or physical model with the initial position of attachment points (In the absence of real data, the data obtained from the nonlinear model in Equation (1) is used).      Step 2.2: Receive equilibrium points from first layer (step 1.6).      Step 2.3: Estimate end effector pose and input in relative form $∆x(j,k), ∆u(j,k)$.      Step 2.4: Estimate parameters of the nominal linear model from gathered u and x via offline RLS. The second layer uses offline RLS to estimate the unknown parameters of the nominal linearized model ($\theta$) at each time step. These parameters do not directly represent the coefficients M0, C0, K0, but rather include the parameters of ARX presented in the form of Equation (4). for each step index K2, (K2 = 1 to N2) perform the following:      Step 2.5: Define the desired trajectory for the end effector at the step time K2.      Step 2.6: Design PID for the nominal model based on offline RLS.      Step 2.7: Estimate the optimal control force on the end effector.      Step 2.8: Using the tension distribution algorithm to generate tension on the cables.      Step 2.9: Transmit the tension to the second group of servomotors for tracking trajectory. end for

2.3.3. Adaptive PID Controller

When the dynamics of the cable-driven parallel robot change significantly, such as when attachment points vary, the nominal model may become inaccurate, thereby affecting the PID performance. An adaptive proportional–integral–derivative (PID) controller is recommended for systems with substantial dynamic variations. Online RLS can continuously update the system model and adjust the PID parameters to maintain accuracy. If discrepancies arise between nominal and actual systems, controller gains may require re-tuning.

As shown in Figure 3, the proposed RCDPM controller consists of the following:

• Adaptive Modeling: An RLS-based algorithm updates the nominal linear model in real time, adapting to attachment point changes and noise.
• Error Compensation: A compensator adjusts the cable tension to minimize pose errors and ensure accurate trajectory tracking.

Controller parameters are continuously updated, based on the identified model for precise control.

An ARX (Auto-Regressive with eXogenous input) model was selected for identifying the cable-driven parallel robot, due to its simplicity, computational efficiency, and suitability for real-time applications. Unlike state–space models that require full knowledge of the system’s internal dynamics, the ARX approach relies solely on input–output data, making it ideal for systems with partially known or configuration-dependent dynamics. Moreover, ARX models can be easily updated online using recursive least squares (RLS), allowing the model to adapt to structural changes in the robot. This data-driven modeling framework also integrates seamlessly with control strategies, such as PID or generalized predictive control (GPC), making it a practical and effective choice for real-time implementation. The ARX model in Equation (4) is used for each position and orientation of the end effector as the output, with respect to the z cable tension inputs for each step index k. As can be seen in Equation (1), the output of system is 2 degrees and can be indicated in ARX form as follows: Y is the output of the system in relative form (pose of end effector and input is tension in cables).

$${Y\left(k\right)}_{6\times 1}={-a}_{1_{6\times 6}}{Y\left(k-1\right)}_{6\times 1}+{{-a}_{2_{6\times 6}}Y\left(k-2\right)}_{6\times 1}+b_{1_{6\times 6}}{U\left(k-1\right)}_{6\times 1}$$

(4)

More details of the ARX model for the CDPM and parameters of ${\theta }_i$ for each output are presented in Appendix B. Equation (4) for a linear parameters variable (LPV) system, in which the parameters of the model are changing over time [49], can be written in a pseudo-linear regressive form, as Equation (5).

$$y_{i,k}={{\Phi }_{i,k}}^T{\theta }_i$$

(5)

The parameters for Equation (4) are presented as follows. All parameters of ${\theta }_i$ for each time step compute the matrix in a general format, as follows: (i is number of outputs, $n_a$ is order of output, and $n_b$ is order of input).

$${\theta }_i={\left[\begin{array}{ccc}{a_{1\left(1\right)}\dots \dots a}_{1\left(i\right)}& \dots a_{n_a\left(1\right)}& a_{n_a\left(i\right)}\end{array} b_{1\left(i\right)}\dots b_{1{,n}_b\left(i\right)}\dots b_{z\left(i\right)}\dots b_{\mathrm{z}{,n}_b\left(i\right)} \right]}^T$$

$${\Phi }_k={\left[\begin{array}{ccc}{-Y}_{k-\tau -1}& \dots & {-Y}_{k-\tau -n_a}\end{array} u_{1,k-\tau -1}\dots u_{1,k-\tau -n_b}\dots u_{z,k-\tau -1}\dots u_{z,k-\tau -n_b}\right]}^T,$$

Control Loop 2 presents the additional details of the adaptive PID controller. ${\mathrm{a}}_{\mathrm{i}}$ are coefficients of outputs and $b_i$ are coefficients of inputs at sample times 1 to N.

The mechanisms for resetting P and adaptively tuning the forgetting factor prevent divergence and instability. Rounding errors and poor conditioning are handled by resetting P for large errors (∣e∣ > 1) and at model change points, forgetting factor (λ) tuning is constrained between 0.95 and 1 and adaptively adjusted based on error, preventing divergence of P.

A dynamic adjustment of the forgetting factor is performed according to the magnitude of the error. If the error is large, the forgetting factor is decreased so that the algorithm can respond more quickly to changes. If the error is small, the forgetting factor increases slowly to enhance stability.

These methods are proposed for adjusting the forgetting factor, based on prediction errors. In this study, the forgetting factor is updated dynamically according to the variations in error, in order to maintain a balance between convergence speed and stability.

${\mu }_i$ in Equation (6) adds a small regularization term to the denominator during the gain calculation, which can enhance numerical stability. In cases where the covariance matrix P is diverging or becoming ill-conditioned, it is reset. This controls the value of P to prevent it from entering an unstable region. The term “becoming ill-conditioned”$\left(\mathrm{cond}\left(A\right)=\parallel A\parallel \cdot \parallel A^{-1}\parallel \right)$ refers to the numerical instability of the system or Jacobian matrix (A or J) that occurs during attachment point reconfiguration, when the matrix approaches singularity or its determinant tends toward zero. To quantify this effect, we use the condition number, defined as a large condition number (e.g., cond(A) > 10⁴), which indicates that the matrix is ill-conditioned and that the system becomes highly sensitive to noise or modeling errors.

Control Loop 2 presents the performance of the second layer, while the first layer avoids a collision between human and cable by changing the position of attachment points, which are the same as the steps of the first layer in Control Loop 1. In Equation (11), the parameter $\beta$ is a fixed, constant parameter used to tune the sensitivity of the forgetting factor. Its value must be determined through trial and error, based on specific noise characteristics and system dynamics. In the simulations conducted for this study, the value of $\beta$ was selected to achieve the optimal balance between the convergence speed and the stability of the RLS algorithm. This choice ensures that the variable forgetting factor rapidly adjusts in response to changes in the estimation error. ${\phi }_{\mathrm{i}}\left(\mathrm{k}\right)$ is a column regressor vector in the ARX model, and the inverse term corresponds to the scalar reciprocal of the inner product ${{\phi }_{\mathrm{i}}}^{\mathrm{T}}\left(\mathrm{k}\right){\phi }_{\mathrm{i}}\left(\mathrm{k}\right)$. The dimension of ${\phi }_{\mathrm{i}}\left(\mathrm{k}\right)$ is 18 by 1 when ${\mathrm{n}}_{\mathrm{a}}=2$ and ${\mathrm{n}}_{\mathrm{b}}=1$.

Control Loop 2: Main adaptive PID control loop based on RLS algorithm with forgetting factor Choosing a safe distance by an expert Choosing a desired trajectory First layer: The main control loop steps of the first layer loop are the same as the first layer in Control Loop 1. Second layer: The main adaptive control of the second layer loop (online RLS + updated parameters PID controller)

for each step index K2(K2= 1 to N2), perform the following: for i = 1:6 (number of output) Step 1: Collect data of inputs u(i,k) and outputs y(j,k) from the nonlinear or physical model. Step 2: Computation of regression vector: $${\theta }_i={\left[\begin{array}{ccc}{a_{1\left(1\right)}\dots \dots a}_{1\left(i\right)}& \dots a_{n_a\left(1\right)}& a_{n_a\left(i\right)}\end{array} b_{1\left(i\right)}\dots b_{1{,n}_b\left(i\right)}\dots b_{z\left(i\right)}\dots b_{\mathrm{z}{,n}_b\left(i\right)}\right]}^T$$ $${\phi }_i\left(k\right)=[y_1\left(k-1\right)\dots y_i\left(k-1\right) \dots y_1\left(k-n_a\right)\dots y_i\left(k-n_a\right) u_1\left(k-1\right)\dots u_1\left(k-n_{b_1}\right)\dots u_z\left(k-1\right)\dots u_z\left(k-n_{b_z}\right)$$ Step 3: Computation of the RLS gain matrix:

$$K_i=p_i(k-1){\phi }_i\left(k\right){\left[\lambda \right(k\left){{\phi }_i\left(k\right)}^{-1}{p}_i\right(k-1\left){\phi }_i\right(k)+{\mu }_i]}^{-1}$$ (6)

Step 4: Computation of estimated output:

$$\widehat{y_i}\left(k\right)={{\phi }_i}^T\left(k\right){\theta }_i(k-1)$$ (7)

Step 5: Error computation (error between estimated output matrix and measurement output matrix):

$$e_{i_{(6,1)}}\left(k\right)=y_i\left(k\right)-\widehat{y_i}\left(k\right)$$ (8)

Step 6: Unknown parameters estimation:

$${\theta }_i\left(k\right)={\theta }_i\left(k-1\right)+K_i\left(k\right)\ast e_i\left(k\right)$$ (9)

Step 7: Updating of the covariance matrix:

$$p_i\left(k\right)={\displaystyle \frac{\left[1-K_i\left(k\right){{\phi }_i}^T\left(k\right)\right]p_i\left(k-1\right)}{{\lambda }_i }}$$ (10)

Step 8: Updating the forgetting factor:

$$\begin{array}{c}{\lambda }_{\mathrm{i}}\left(\mathrm{k}\right)={\lambda }_{\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}}\left(\mathrm{k}\right)+(1-{{\lambda }_{\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}}\left(\mathrm{k}\right))}^{\propto \left(k\right)}\\ \propto \left(k\right)=2^{\beta e^2\left(k\right)}\end{array}$$ (11)

${\lambda }_{\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}} and\propto \left(k\right)$ are the fixed parameters

end for.

3. Results

To apply the proposed controller, the physical parameters of The INCA robot [47] are modified. To assess the effectiveness of the proposed controller in a reconfigurable CDPM, the attachment points were assumed to move along the z axis on the reel. Figure 4 illustrates the movements at the base. The workspace dimensions are Lx = 3 m, Ly = 3 m, and Lz = 3 m.

3.1. Collision Avoidance (First Layer)

First-layer optimization estimates the optimal attachment point positions to prevent collisions between humans and cables in a shared workspace, as shown in Figure 5. A human actively moved his arm within the workspace, which required dynamic adjustment.

Figure 4 presents the updated attachment point positions based on this optimization. Meanwhile, the upper attachment points do not change, and the lower attachment points are moved to avoid collisions between the humans and cables.

Table 1. Initial position of attachment points on the base.

$O_1$	$[0,0,0]$	$O_5$	$[0,0,3]$
$O_2$	$[3,0,0]$	$O_6$	$[3,0,3]$
$O_3$	$[0,3,0]$	$O_7$	$[0,3,3]$
$O_4$	$[3,3,0]$	$O_8$	$[3,3,3]$

lists the initial positions of the attachment points at the bases.

As illustrated in Figure 6, the distance between cables increases when a collision risk is detected, particularly when approaching the predefined safety threshold. The maximum velocity in optimization is 1 m per s. Figure 7 presents a forgetting factor. As can be seen, the forgetting factor decreases at the seconds where attachment point 6 is moving on the base to update the RLS parameters and then increases to one to stabilize the estimated parameters by RLS. Additionally, this change in the forgetting factor is also observed at the moments when the end effector is at the edge of the square signal.

3.2. Tracking Trajectory (Second Layer)

Since the main attachment point relocations occur within the first 50 s, the forgetting factor is plotted only for that period. After this time, the configuration remains stable, and no further adaptation is required.

This method is designed to respond swiftly to sudden changes and prevent abrupt performance degradation. However, in critical situations when the estimation error exceeds one (which may indicate poor estimation or a model jump), the covariance matrix P is reset. Covariance matrix P quantifies the uncertainty in the estimated model parameters and plays a crucial role in calculating the gain vector (K(k)). The size of P is carefully managed by the variable forgetting factor to prevent filter saturation and enable the algorithm to rapidly adapt to sudden dynamic changes (such as the movement of the attachment points).

The second layer optimizes cable tensions using a PID controller, whereas the RLS algorithm is adaptive, continuously estimating the CDPM model at each time step. This ensures precise force generation on the end effector, thereby minimizing the error between the desired and actual poses. A tension distribution algorithm then converts the required end effector force into the corresponding cable tensions, ensuring that all the generated tensions remain positive for feasible actuation. A controller is designed according to the position of attachment points on the base. In the context of CDPMs with dynamic attachment points mounted on a moving base, the implementation of relative control strategies is generally more advantageous than absolute control paradigms. This preference stems from the inherent variability in the system’s equilibrium point, which undergoes continuous perturbation due to base motion. Absolute control, reliant on fixed global references, often exhibits instability under such conditions, as it necessitates frequent recalibration of the reference frame, potentially amplifying tracking errors and oscillatory behavior.

Relative control, conversely, operates by minimizing deviations that are relative to the instantaneously computed dynamic equilibrium, thereby enhancing robustness against kinematic uncertainties and external disturbances. This approach leverages feedback on differential states (ΔX), facilitating adaptive compensation for base-induced variations. Tuning PID gains to prioritize relative error reduction yields superior trajectory tracking precision, reduced settling times, and mitigated collision risks in reconfigurable environments. Figure 8 and Figure 9 compare the end effector poses achieved using PID and adaptive PID (APID) controllers with the desired trajectory over the period, in both relative and absolute positions.

As shown, PID struggles with precise trajectory tracking, whereas APID exhibits a greater robustness against model uncertainties. Notably, when the attachment points move on the base, the APID outperforms the PID, maintaining better trajectory tracking. According to Figure 4, significant changes in the attachment points occur at the seconds between 10–15 and 20–25 to avoid collision between human and cables, where the APID effectively compensates for uncertainties, ensuring superior tracking accuracy. In Figure 6, at second 10, when all attachment points are moved, a significant increase in the overshoot is observed in the x and y directions, using the PID controller. Additionally, the highest position error between the PID-controlled system output and the reference trajectory in the z direction occurs at this moment.

Furthermore, attachment points change their position to prevent potential collisions between the human operator and the cables. These dynamic changes in attachment points not only affect the end effector’s positional accuracy but also induce rotational deviations, further challenging the PID controller’s ability to maintain precise trajectory tracking.

As shown in Figure 4, after 25 s, the attachment points are fixed. Beyond this point, the position and orientation of the end effector stabilize, as all attachment points become fixed. This configuration ensures collision avoidance between the human operator and the cables, guaranteeing a safe and stable operation of the system.

Figure 9 illustrates the absolute position and rotation of the end effector. It highlights the lowest tracking accuracy in both the position and orientation of the PID controller, indicating the challenges in maintaining precise trajectory tracking under these conditions

As shown in Figure 10, the mean absolute error (MAE) for PID and APID in trajectory tracking is compared, whereas APID maintains better accuracy, effectively compensating for system variations. In the z direction, PID struggles the most, with error accumulation over time. This improvement is particularly evident in rotational accuracy, where APID outperforms PID in maintaining stable and precise tracking.

After designing the PID and APID controllers, the performance of the proposed controller is evaluated under several scenarios as presented in

Table 2. Three scenarios to test the performance of controllers.

Scenario	Conditions	Primary Goal
1. Dynamic Variation	Attachment points are moving on the base (no external disturbance, no mass change yet).	To prove that adaptivity is necessary. Both fixed-gain controllers do not have perfect performance as the robot’s dynamics change.
2. Disturbance Rejection	Attachment points are moving AND an external disturbance is applied.	To prove that APID’s adaptive gains plus the integral action make it superior for handling both variations and external forces.
3. Model Uncertainty	Attachment points are moving AND the mass of the end effector is suddenly changed (e.g., ±0% mass).	To test the robustness of the APID’s RLS algorithm in estimating and compensating for unknown physical parameters.

.

In the first scenario, the attachment points move on the base, and the results of the proposed controller are compared with the LQR controller.

In the second scenario, the performance of the APID controller is compared with PID and LQR when the attachment points are moving on the base and an external disturbance is applied to the system.

Finally, in the third scenario, the mass of the end effector is changing to assess the robustness of the controller in the presence of the changing mass of the end effector.

3.2.1. Scenario 1: Performance of Controllers in the Presence of Moving Attachment Points on the Base

PID struggles to handle sudden changes in attachment points, leading to higher position and rotation errors over time. APID significantly reduces both positional and rotational MSE, especially after 25 s, when all attachment points are fixed. APID is particularly effective for reducing drift in the z direction and rotational errors (Roll, Pitch, Yaw), making it a better choice for trajectory tracking in dynamic environments.

Table 3. RSME and MAE for APID, PID, and LQR for first scenario (relative error).

	RSME				MAE
	PID	LQR	APID		PID	LQR	APID
X	0.0603	0.061	0.051	X	0.0251	0.026	0.0151
Y	0.0589	0.057	0.048	Y	0.0254	0.024	0.0152
Z	0.0837	0.085	0.047	Z	0.022	0.0025	0.00134
Roll	0.02101	0.023	0.0136	Roll	0.0072	0.0087	0.0048
Pitch	0.0479	0.05	0.013	Pitch	0.0073	0.0084	0.0045
Yaw	0.0604	0.095	0.0134	Yaw	0.0068	0.0075	0.0046

presents the mean absolute error (MAE) and root mean squared error(RSME) values for PID, LQR, and APID, highlighting the improved performance of APID. Table 3 further confirms that APID consistently achieves better accuracy across all the position and orientation components, demonstrating its effectiveness in dynamic environments.

If the PID gains are chosen high enough, they can partially compensate for the necessary input variations resulting from changes in the attachment point geometry (due to changes in the structure matrix). PID tuning is simpler, and it can be empirically adjusted around the worst-case scenario geometry of the robot. Due to the simplicity of its model and less reliance on the accuracy of the $a_{1_{6\times 6}}, a_{2_{6\times 6}}, \mathrm{and} b_{1_{6\times 6}}$ matrices in the ARX model (Equation (4)), PID has a higher chance of maintaining local stability in a portion of the workspace, but its tracking performance will be poor. PID (particularly with its integral term) compensates for the biases caused by changes in $T_{\mathrm{e}\mathrm{q}}$, even when the model is not accurate. When large attachment movements occur, PID, with anti-windup and tension-rate limits, remains more stable than LQR.

LQR is inherently designed based on a linearized model. If the gain is optimized for a specific point (e.g., the center of the workspace) and remains fixed during motion, its performance will be poor in other parts of the workspace, because the actual system model constantly deviates from the model assumed by the fixed-gain LQR. APID has better performance due to the model update at each time step to update the gain parameters of controllers.

3.2.2. Scenario 2: Performance of Controllers in the Presence of Moving Attachment Points on the Base and Disturbance

The PID controller includes an integral term that can automatically eliminate steady-state disturbances (such as additional gravitational or frictional forces), whereas a simple LQR without an integral component suffers from steady-state error.

When both attachment points’ motion and external disturbance are present, the LQR with fixed gains shows the largest performance degradation, due to high sensitivity to model variations. The simple PID controller is more robust, but its tracking error still increases. The adaptive PID (APID) maintains the best overall performance, since its gains adapt online to geometric and disturbance changes. As shown in

Table 4. RSME and MAE for APID, PID, and LQR for second scenario (relative error).

	RSME				MAE
	PID	LQR	APID		PID	LQR	APID
X	0.083	0.092	0.061	X	0.036	0.039	0.019
Y	0.081	0.087	0.057	Y	0.034	0.036	0.017
Z	0.108	0.125	0.069	Z	0.042	0.049	0.023
Roll	0.035	0.041	0.019	Roll	0.012	0.014	0.0065
Pitch	0.054	0.061	0.018	Pitch	0.013	0.015	0.0058
Yaw	0.078	0.098	0.02	Yaw	0.011	0.016	0.0062

, the APID has better tracking in the presence of moving attachment points and disturbance.

3.2.3. Scenario 3: Performance of Controllers in the Presence of Moving Attachment Points on the Base and Variation in Mass

Since this new scenario involves simultaneous attachment point motion and mass variation (+20% and +30%), and only MAE is considered, the trend should follow the same pattern as the previous table. As shown in

Table 5. RSME and MAE for APID, PID and LQR for third scenario (relative error).

Axis	PID (20%)	LQR (20%)	APID (20%)	PID (30%)	PID (30%)	APID (30%)
X	0.0276	0.0293	0.0163	0.0301	0.0325	0.0174
Y	0.279	0.270	0.0164	0.0305	0.03	0.0174
Z	0.0275	0.034	0.0164	0.0319	0.039	0.00176
Roll	0.0081	0.0104	0.0053	0.0088	0.0113	0.0055
Pitch	0.0088	0.0105	0.005	0.0098	0.0112	0.0054
Yaw	0.0085	0.0098	0.0053	0.0099	0.0113	0.0058

, the APID has better tracking in the presence of moving attachment points and mass variation. The updated MAE trends for the two cases are as follows:

3.2.4. Workspace Analysis

The workspace of a cable-driven robot is determined by checking the allowable range of cable tensions inside workspace 3 m × 3 m × 3 m. Any point where all cable tensions remain between the minimum and maximum limits is considered part of the workspace. However, the workspace before moving attachment points (blue) and after moving the attachment points (green) on the base, to avoid collision, is presented in Figure 11. As shown, the end effector with square trajectory (red line in center) is kept inside the workspace after moving the attachment point six on the base. The closed-loop stability of the motor system can be analyzed using both analytical and practical considerations. The PID gains are tuned so that the closed-loop matrix $A_c=A-BK$ remains Hurwitz, ensuring asymptotic stability of the system according to the Routh–Hurwitz criterion. The adaptive RLS update with a variable forgetting factor maintains the covariance matrix $P\left(k\right)$ as positive definite, contributing to bounded parameter estimates and overall system stability. From a practical standpoint, saturation is possible, but not instability in terms of the Routh–Hurwitz criterion. The classical PID control method is applied to a motor with minimum overshoot. Moreover, the trajectory definition setpoint limits the overshoot, and then vibration is almost nonexistent. Then, it cannot be unstable, considering both the Routh–Hurwitz criterion and mechanical vibrations. The location of the poles of a motor never go outside the negative Laplace plane, whenever the gain. Of course, if the PID/APID gain is set too high, overshoot could occur until it reaches motor amplifier saturation or limit (slew rate, threshold, etc.). Therefore, both mathematically and mechanically, the controlled system remains stable within the defined gain limits.

4. Conclusions

This paper presents new two-layer control schematics for the CDPM, where in the first layer, attachment points move on the base to avoid collisions between humans and cables. The second layer generates tension in the cables, using a controller. If the equilibrium point can be estimated online and accurately, the relative controller is better, because it eliminates the error caused by the movement of the attachment points.

If the estimation of the equilibrium point is inaccurate or noisy, the absolute controller is used, but it provides lower tracking accuracy. So, for cable-driven robots with moving attachment points, the relative controller is usually chosen, since the attachment points move frequently and the absolute controller does not perform well.

The added part is an RLS with a variable forgetting factor, used to update the nominal model, which is then used to design the PID. The PID parameters were updated when the attachment points moved on the base. This method yields better results than PID in the presence of uncertainties in the model or moving attachment points at the base.

Although the proposed RLS implementation dynamically adapts the forgetting factor and resets the covariance matrix in critical conditions to enhance stability, it still faces potential numerical challenges when scaled to high-dimensional systems. Real-time deployment may require further optimization, including efficient memory handling and the use of numerically stable RLS variants.

While the proposed methodology has been evaluated in the simulation, the lack of real-world experimental validation is a recognized limitation of this study. Physical testing is essential to assess the model’s reliability under realistic conditions, where unmodeled dynamics, sensor noise, cable elasticity, and actuator imperfections may significantly affect performance. As a part of future work, a robust controller can be added to implement the proposed control framework on a physical cable-driven robot to verify the simulation results. This experimental phase will also help identify potential hardware challenges—such as delays, backlash, and calibration errors—that are difficult to capture in simulation but critical for robust real-time performance.