Machine Learning Applications in Parallel Robots: A Brief Review

Abstract

Parallel robots, including cable-driven parallel robots (CDPRs), are widely used due to their high stiffness, precision, and high dynamic performance. However, their multi-chain closed-loop architecture brings nonlinear, multi-degree-of-freedom coupled motion and sensitivity to geometric errors, which result in significant challenges in their modeling, error compensation, and control. The rise in machine learning technology has provided a promising approach to address these issues by learning complex relationships from data, enabling real-time prediction, compensation, and adaptation. This paper reviews the progress of typical applications of machine learning methods in parallel robots, covering four main areas: kinematic modeling, error compensation, trajectory tracking control, as well as other emerging applications such as design synthesis, motion planning, and CDPR fault diagnosis. The key technologies used, their implementation architecture, technical difficulties solved, performance advantages and applicable scope are summarized. Finally, the review outlines current challenges and future directions. It is proposed that hybrid learning physics modeling, transfer learning, lightweight deployment, and interdisciplinary collaboration will be the key directions for advancing the integration of machine learning and parallel robotic systems.

1. Introduction

Parallel robots represent a significant class of robotic mechanisms, characterized by multiple independent kinematic chains connecting a movable platform to a fixed base in a parallel closed-loop structure [1,2]. Since the successful application of the Gough–Stewart platform [3,4], parallel robots have attracted growing attention from both academia and industry and have undergone rapid development. Parallel robots exhibit outstanding features such as high stiffness [5], high payload-to-weight ratio [6], excellent dynamic performance [7], and superior motion accuracy [8]. Their outstanding features have enabled their widespread application in fields such as motion simulators [9], high-speed pick-and-place devices [10], machine tools [11], etc. Representative systems include the Gough–Stewart motion platform [12], the Delta robot [13], and the Sprint Z3 tool head [14]. To meet the increasing demands for high dynamic performance and large operational workspaces, cable-driven parallel robots (CDPRs) have emerged as a significant subclass of parallel robots and a focal point of research in recent years [15,16]. By replacing rigid links with flexible cables, CDPRs achieve substantial weight reduction and workspace expansion [17,18], making them especially suited for large-scale [19,20] or highly dynamic tasks [21]. Both rigidly actuated parallel robots and CDPRs continue to play an irreplaceable role in industrial and technological domains, offering broad application potential and economic impact.

Despite their advantages, parallel robots—particularly CDPRs—face significant challenges in modeling and control. Their multi-chain parallel structure leads to complex kinematics and non-unique inverse solutions [22], especially in systems with high degrees of freedom (DOFs) or redundant configurations [23]. Moreover, parallel robots are sensitive to structural errors, and the strong coupling among multiple-DOF motion and kinematic chains complicates motion accuracy and error compensation [24,25]. Their complex dynamic behavior and pose-dependent inertia distribution impose additional difficulties in controller design [26]. In the case of CDPRs, the unidirectional nature of cable tension imposes strict constraints to maintain structural stability and prevent slack or degraded performance [27]. The presence of nonlinearities such as cable elasticity, friction, pulley transmission effects, and environmental disturbances further increases the difficulty of accurate modeling and robust control for CDPRs [28]. Traditional model-based methods are increasingly limited in their ability to address these challenges with sufficient precision and real-time capability, motivating the need for alternative approaches.

Machine learning (ML), a major domain within artificial intelligence (AI), has emerged in recent years as a transformative technology with powerful capabilities in modeling and optimizing complex robotic systems [29]. Among various ML techniques, artificial neural networks (ANNs) serve as a fundamental and widely adopted approach due to their powerful capability in approximating complex nonlinear functions [30]. Common structures of ANNs include multi-layer perceptrons (MLPs), convolutional neural networks (CNNs), and recurrent neural networks (RNNs) such as long short-term memory (LSTM) networks [31]. MLPs serve as universal approximators for static nonlinear mappings [32], CNNs are effective for spatial feature extraction, and LSTMs are well-suited for capturing temporal dependencies [33,34]. A representative feedforward ANN architecture is illustrated in Figure 1, showing the layered structure of input, hidden, and output nodes with nonlinear activation functions enabling complex function approximation. These architectures are typically trained using gradient-based optimization algorithms such as backpropagation combined with stochastic gradient descent (SGD) or its variants. When ANNs are composed of multiple hidden layers, they are often referred to as deep learning (DL) models, which have gained increasing attention for their ability to represent complex, high-dimensional patterns. Unlike traditional methods that rely on well-defined physical and mathematical models, ML algorithms like ANNs can extract complex, nonlinear relationships from large volumes of historical or real-time data [35]. This data-driven paradigm is particularly beneficial for complex parallel robot systems, whose analytical solutions to kinematics, dynamics, or control problems are difficult to solve analytically or computationally expensive.

In recent years, an increasing number of studies have explored the applications of ML in parallel robots and achieved promising results. Studies have been conducted to utilize neural networks (NNs) for inverse and forward kinematic calculation, improving the speed and accuracy of the computations [36,37]. Data-driven models have been developed to model and compensate for terminal errors caused by geometric and nongeometric factors, significantly improving the positioning accuracy of parallel robots [38,39]. Moreover, DL and reinforcement learning (RL) have been employed to optimize trajectory tracking and control strategies in complex systems and environments, enhancing the robustness and adaptability of parallel robots [40], especially for CDPRs [41]. These studies show that ML provides a promising alternative for modeling, error compensation, trajectory tracking, and control strategy development for parallel robots, including CDPRs.

This paper briefly reviews the application of ML in parallel robots, focusing on the research progress of ML methods used in kinematic modeling, error compensation, trajectory tracking, and control strategies. The review summarizes representative ML methods, implementation frameworks, technical challenges addressed, and performance outcomes. Finally, it discusses current limitations and directions for future research.

2. Machine Learning for Kinematics

Solving the kinematics of parallel robots, especially the forward kinematics (FK) problem, remains a fundamental challenge due to the complexity, strong coupling, and nonlinearity in these systems. ML techniques provide a promising alternative by learning the complex mapping between joint variables and terminal poses directly from data.

Although the inverse kinematics (IK) problem for parallel robots can usually be solved analytically, there remains room for improvement in computational efficiency. ML methods have been introduced to accelerate the solution process, enabling faster prediction. For instance, Thomas et al. [42] tested several ML algorithms—including linear regression, support vector machine (SVM), decision tree (DT), random forest (RF), and MLP network— to calculate the IK solution for a 3-PPSS parallel robot to overcome the computational difficulties and approximation problems involved in analytical methods. Similarly, Ghasemi et al. [43] applied MLP and radial basis function (RBF) network to approximate the IK solution for the Tricept parallel machine, which has a faster calculation speed to meet the requirements of real-time control. Figure 2a shows the framework diagram of the implementation scheme of applying ML methods to IK. The desired pose s_d is used as the input of the ML model, and the error ε_q between the predicted result q_p and the numerical calculation result q_d is fed back to the model for training, to minimize ε_q. Compared with traditional methods, the ML-based IK calculation method can shorten the calculation time while ensuring accuracy, which is beneficial for real-time control.

The most important application of ML in kinematics is to solve the FK problem, as analytical FK solutions are generally difficult to derive for parallel robots. Traditional FK calculations are generally based on numerical iterative methods, such as the Newton–Raphson method [44], which is usually time-consuming and requires considerable effort to ensure the stability and accuracy of the calculation. To improve the efficiency of the numerical solution, Parikh and Lam [45] proposed combining MLPs with numerical methods, using the MLP algorithm to provide an improved initial guess for the FK solver, thereby reducing the number of iterations and improving the convergence efficiency. The implementation architecture is shown in Figure 2c. More commonly, ML models are trained as universal approximators to directly learn the mapping from joint variables to end-effector poses. Dehghani et al. [46] proposed an MLP-based approach for solving the FK of the HEXA parallel manipulator, effectively learning the nonlinear mapping to improve pose accuracy in the absence of a closed-form solution. Morell et al. [47] introduced an SVM-based method with offline preprocessing and fast online evaluation to model the FK of parallel robots across various configurations, enabling accurate, geometry-independent approximations within specific pose regions. Zubizarreta et al. [48] employed MLPs to approximate the FK of a 3PRS parallel robot, achieving a 7.5-fold speedup over the Newton–Raphson method while maintaining high accuracy. Prado et al. [49] applied ANNs to solve the FK of wearable parallel robots, achieving real-time and robust performance despite model uncertainties such as link deflections and joint misalignments. Khalapyan et al. [50] proposed an MLP-based method that incorporated the previous platform position as an additional input to enhance solution accuracy and speed. Zhang and Lei [51] demonstrated that ν-SVM with a linear kernel outperforms MLP and RBF networks in estimating the FK of a redundant parallel robot in terms of accuracy, generalization, and convergence speed. In addition, several studies have addressed both FK and IK problems simultaneously [52,53].

The implementation framework of the FK solution of the above ML-based universal approximator is shown in Figure 2b. Given a set of desired poses s_d, the corresponding joint input q_d is calculated by inverse kinematics, and then q_d is input into the ML model to obtain the predicted pose s_p. The difference ε_s between desired end-effector pose s_d and predicted pose s_p is used as feedback to train the model, with the objective of minimizing ε_s through iterative learning. To improve the performance of ML models, intelligent optimization algorithms such as genetic algorithms (GAs) can be used to optimize NN parameters [54,55].

Compared to rigid parallel robots, CDPRs present even greater challenges for kinematic analysis due to nonlinearities introduced by cable mass and elasticity. The coupling between cable length and tension transforms their kinematics into a nonlinear statics problem. Chawla et al. [56,57] proposed a DL approach to solve the IK and FK of CDPRs considering cable mass and elasticity, achieving an 83% reduction in computation time compared to numerical methods. Mishra and Caro [58] proposed an unsupervised NN method for real-time forward geometric-static analysis of under-constrained CDPRs, efficiently handling cable elasticity and gravity-induced nonlinearity by approximating solutions along motion paths. Chen et al. [59] applied deep reinforcement learning (DRL) to enhance motion control and position estimation in CDPRs, introducing a delay strategy for synchronization and achieving improved accuracy and reduced control errors in high-payload scenarios. In addition, Zhang et al. [60] introduced CafkNet, a graph neural network (GNN)-based method for solving the forward kinematics of CDPRs, demonstrating promising performance improvements.

The summary of the ML methods used for the kinematic model in parallel robots are shown in

Table 1. Summary of ML methods for kinematic calculation in parallel robots.

Application Focus	Methods/Algorithms Used	Achievements	Limitations
IK	DT [42], RF [42], SVM [42]	Overcome the computational difficulties and approximation problems involved in analytical methods.	Inefficient for large-scale data.
	MLP [42,43], RBF [43]	Faster calculation speed meets the requirements of real-time control.	Training requires a large amount of data and computing resources.
IK for CDPRs	DNN [56,57]	Significant improvement in computational speed compared to numerical methods.	Requires large training time and a large dataset for initial training.
FK	SVM [47,51]	Fast online evaluation and better performance in convergence speed and generalization ability.	Model performance depends heavily on parameter selection.
	MLP [46,48,50,51,52,53,58]	Effectively learning the nonlinear mapping to improve pose accuracy; faster computation than numerical method.	Requires a large training dataset and fine-tuning.
	GA-MLP [52,54,55]	GA optimizes NN parameters and improves accuracy.	Accuracy is sensitive to training data distribution and GA optimization increases computation time.
	MLP and Newton–Raphson hybrid method [45]	Reducing iterations and improving convergence efficiency.	Still relies partially on numerical methods and the hybrid strategy introduces a slight increase in memory.
FK of CDPR	DL [56,57]	Significant improvement in computational speed.	A large training dataset is required due to complexity.
	DRL [59]	Improved position estimation accuracy of CDPR in high-load scenarios.	Complex trajectory applications require more training data and high computation GPUs.
	GNN [60]	Superior generality, high accuracy, and low time cost.	Graph-based methods are still emerging and complex, only considers straight cables now.

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ML has increasingly demonstrated its value in addressing the kinematic problems of parallel robots. Currently, a variety of ML methods are available; however, the effectiveness of models built using these methods varies [42,43,51]. While many ML approaches for solving IK and FK are broadly applicable to robotic systems, their deployment in parallel robots introduces unique complexities. Due to the closed-loop structure of parallel mechanisms, the FK problem often lacks a closed-form analytical solution and may involve multiple valid configurations, singularities, and ill-conditioned Jacobians. These characteristics make the learning task for ML models significantly more difficult than in serial manipulators. ML models must be able to approximate many-to-one or discontinuous mappings while preserving generalization capability, especially near singular regions. This necessitates careful dataset construction, model design, and sometimes hybrid approaches that combine ML with numerical solvers. For IK, although analytical solutions often exist in parallel robots, multiple solution branches may be present. ML models can be designed to learn feasible or task-optimal configurations, improving real-time applicability and robustness. To address these issues, researchers have explored strategies such as incorporating prior joint configurations, embedding constraint-aware loss functions, and using hybrid ML–numerical methods to enhance convergence and reliability.

Moreover, all the existing approaches primarily rely on simulation data or numerical values obtained from inverse kinematics calculations for the ML model training, rather than using real-world pose measurements. In the future, it is anticipated that the data measured from the real robot’s end-effector, as illustrated in the schematic diagrams in Figure 3, could be employed for error compensation, leading to higher accuracy. However, it may be difficult to collect a large amount of data for real robots. Therefore, how to efficiently collect data or reduce the data demand of ML models is the key.

3. Machine Learning for Error Compensation

The positioning accuracy of parallel robots is affected by geometric and non-geometric errors arising from manufacturing defects, assembly misalignments, joint clearance, structural deformations, and environmental disturbances [61]. Traditionally, kinematic calibration techniques have been employed to improve accuracy by modifying the nominal kinematic model through parameter identification and optimization [62]. However, calibration typically involves the construction of complex error models and parameter identification processes, which can be especially burdensome for parallel robots due to their multiple kinematic chains and joints. In addition, model-based calibration methods are not flexible enough to address complex nonlinear error behaviors. In contrast, ML methods provide a powerful alternative that supports data-driven modeling and compensation strategies and bypasses explicit error formulas [63]. As illustrated in Figure 4a, ML models can learn the mapping between pose errors and corrective joint variables or geometric parameters, enabling direct compensation to improve positioning accuracy without requiring a detailed error model.

Various ML architectures have been proposed for this purpose. Yu [64] proposed a backpropagation (BP) trained MLP network to learn the kinematic model error of a parallel robot. The trained NN can be used to perform online pose accuracy compensation in the task. As an improvement, Yu [65] further proposed a hybrid ANN consisting of a BP network, an RBF network, and a control module to compensate for the pose error caused by the non-geometric parameter error of the Stewart parallel robot. Wang and Bai [66] divided the workspace of Stewart platform into subregions and used an MLP network to model the error mapping in each subregion. It improves the calibration accuracy, but the measurement and training tasks are greatly increased. The trained NN models can be embedded into real-time control systems for online compensation. It is important to note that the performance of ML models varies depending on architecture and training data, making the selection of appropriate models and hyperparameters essential to avoid overfitting and ensure generalization [67].

To address more complex and nonlinear error sources, DL methods have also been explored. Zhang et al. [68] proposed a DL method that combines attention-based mechanisms to capture both geometric and non-geometric error components, improving the accuracy and reliability of parallel robot operations. These attention modules dynamically adjust the importance of different input features, improving feature utilization without significantly increasing model complexity. In addition to attention mechanisms, RNN architectures such as LSTM models have also shown promise in learning time-dependent or history-aware pose error patterns. For example, Zhu et al. [38,69] proposed a CNN-LSTM hybrid network to predict the pose error of a Stewart platform, leveraging convolutional features for spatial abstraction and LSTM layers for temporal integration. Such architectures are particularly beneficial in dynamic environments or when systematic errors evolve over time due to thermal drift, wear, or deformation. These models complement feedforward networks by incorporating memory of prior states, thereby improving prediction stability and compensation accuracy. Given the high cost of acquiring real-world datasets for DL models, Akhmetzyanov et al. [70] introduced a transfer learning (TL) approach in which a model is first pre-trained on simulated data and then fine-tuned using a limited amount of experimental data. TL significantly shortens the data acquisition cost while maintaining high compensation accuracy, making it suitable for modular or reconfigurable parallel robots.

Hybrid strategies that combine model-based calibration and data-driven compensation have also been explored [71,72,73]. As shown in Figure 4b, hybrid methods typically involve a two-stage compensation framework. First, kinematic calibration based on numerical optimization techniques is performed to compensate for geometric errors. Subsequently, ML models—commonly MLP networks—are used to capture and correct residual non-geometric errors. This two-stage correction framework leverages the strengths of both physical modeling and data-driven learning, enabling more comprehensive error correction.

The summary of the ML methods used for error modeling and compensation in parallel robots are shown in

Table 2. Summary of ML-based methods for error modelling and compensation in parallel robots.

Application Focus	Methods/Algorithms Used	Achievements	Limitations
Universal approximators	MLP [64,66] MLP + RBF [65]	Achieved accurate pose error prediction and online compensation.	Limited generalization, sensitive to data quality; training complexity increases with hybrid model.
	DL with attention module [68]	Captures high-dimensional error features; uncertainty-aware prediction.	More complex networks and require large, labeled datasets and computing power.
	RNN (LSTM) [38,69]	Capturing time-dependent and sequential error patterns, beneficial in dynamic environments.	High data demand, computational burden, low interpretability, training instability.
	TL [70]	Shortens the data acquisition cost while maintaining compensation accuracy.	Pre-training may bias model, sim-to-real transfer gaps.
Hybrid modeling	Kinematic calibration + MLP [71,72,73]	Higher modeling accuracy by applying MLP to integrate non-geometric errors into the kinematic calibration model.	Calibration effort required, hybrid models harder to tune, require data collecting for both calibration and MLP.

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Compared to serial manipulators, parallel robots are particularly sensitive to both geometric and non-geometric errors due to the strong coupling between their multiple kinematic chains. Minor deviations in joint geometry, link lengths, or assembly conditions can result in amplified pose errors at the end-effector. Moreover, error characteristics are highly configuration-dependent, varying across the workspace in complex, nonlinear patterns. These factors pose significant challenges to ML-based compensation models, which must not only generalize across diverse operating conditions but also adapt to local sensitivities. Accurate pose labeling for training is especially demanding in parallel robots, often requiring high-precision external measurement systems or simulation-enhanced datasets. As a result, effective ML solutions for error compensation in parallel robots frequently adopt hybrid architectures or workspace segmentation strategies to ensure both accuracy and practical deployability.

While ML-based methods have demonstrated substantial potential in error modeling and compensation, several challenges remain. Key issues include the high cost of collecting large and diverse datasets, the lack of interpretability, and the difficulty of achieving real-time inference on computationally constrained systems. Future research should focus on addressing these limitations by developing more data-efficient learning algorithms (e.g., transfer learning, active learning), designing hybrid models that integrate physics-informed priors, enhancing model interpretability, and optimizing lightweight architectures for deployment on real-time embedded systems.

4. Machine Learning for Trajectory Tracking and Control

Achieving accurate and robust control of parallel robots presents persistent challenges due to their nonlinear, coupled dynamics, potential actuation redundancies, and sensitivity to disturbances. Traditional model-based controllers, such as proportional–integral–derivative (PID) and inverse dynamics controllers (IDCs), often face performance limitations in the presence of model uncertainties and unmodeled dynamics. In recent years, ML technologies have been increasingly applied to enhance trajectory tracking and improve control robustness in parallel robotic systems. These challenges are particularly pronounced in CDPRs, which introduce additional complexities such as maintaining positive cable tension to prevent slack, nonlinear couplings between actuators and end-effector pose, and configuration-dependent dynamics resulting from closed-loop constraints. In such systems, purely model-free learning approaches may struggle to ensure real-time feasibility, safety, and constraint satisfaction. Therefore, ML-based controllers for parallel robots must be carefully adapted to operate within strict physical and structural limits—often incorporating constraint-aware training objectives, hybrid model-based architectures, or real-time adaptation mechanisms. For CDPRs, unmodeled effects such as cable elasticity, pulley friction, and load dynamics further increase the need for data-efficient and physically interpretable control solutions. As a result, many of the most effective ML applications in parallel robot control combine learning modules with conventional control strategies, striking a balance between data-driven adaptability and model-based reliability.

4.1. Hybrid Machine Learning-Augmented Control

Hybrid control approaches embed NNs into classical control frameworks to create feedforward compensators or adaptive components that enhance the system’s ability to handle nonlinearities and uncertainties, as illustrated in Figure 5a. For instance, Li and Wang [74] integrated a diagonal recurrent neural network (DRNN) into a conventional PID controller, significantly reducing tracking error by accurately approximating nonlinear dynamics using the DRNN. Escorcia-Hernández et al. [75] employed a B-spline NN in parallel with a PD controller to compensate for dynamic errors and improve control precision. To address the problem of inaccurate dynamic models, Rahimi et al. [76] designed a self-adjusting IDC using an NN trained online with an arc length function, leading to reduced vibrations and improved trajectory smoothness. In CDPR control, Piao et al. [77] proposed an indirect end-effector force estimation method using an ANN to model and compensate for pulley friction and other unmodeled disturbances, thereby enhancing force control performance. These hybrid control strategies combine the structural stability of traditional controllers with the learning and compensation capabilities of NNs, making them well-suited for systems with incomplete dynamics or external perturbations.

4.2. Adaptive Neural Network Controllers

Adaptive neural network controllers adapt to environmental changes and model uncertainties by updating control parameters in real time. Achili et al. [78] proposed an adaptive force/position hybrid controller based on an MLP network to compensate for the dynamic interaction disturbances between the robot and its environment, effectively suppress the oscillation introduced by the sliding mode term and significantly improve the trajectory tracking accuracy. Similarly, Zabihifar and Yuschenko [79] designed an adaptive force/position control scheme for a Delta robot using an RBF network with a weight adaptation law based on universal approximation properties. Asl and Janabi-Sharifi [80] introduced an MLP-based trajectory controller with bounded inputs and auxiliary dynamics to ensure stable cable tension and system stability. In more complex control tasks, NNs are often combined with sliding mode control (SMC) to further improve robustness. Nguyen et al. [81] proposed an adaptive chattering-free NN sliding mode controller that combines an RBF network and a sliding mode structure, which effectively solves the high-frequency chattering phenomenon in the traditional NN-SMC controller and achieves highly robust trajectory tracking. Zhang et al. [82] applied this strategy to a flexible spatial parallel robot, using an RBF network to compensate for the driving torque, thereby enhancing vibration suppression and tracking performance. Achili et al. [83] further demonstrated that combining MLP networks with SMC improved control robustness under varying dynamic conditions such as gravity and friction.

4.3. Fuzzy Neural Network Controllers

Fuzzy neural networks (FNNs) combine the interpretability and rule-based reasoning of fuzzy logic with the self-learning capability of NNs, making them highly effective in uncertain and nonlinear environments. Jalaeian et al. [84] proposed a dynamically growing FNN controller with self-organizing structure and adaptive capability, enabling a rapid response to parameter drift while maintaining system stability. Zhu and Fang [85] investigated an adaptive FNN control scheme that updates fuzzy rule weights online to maintain robustness under changing load conditions and disturbances. In dealing with friction and disturbance modeling errors, Hung and Na [86] introduced a PD-type fuzzy controller in a feedback linearization framework, thereby improving the stability and tracking accuracy under the conditions of modeling errors, disturbances, and noise. For perturbation compensation and load adaptation, Lu et al. [87] and Xu et al. [88] proposed adaptive fuzzy SMC controllers that integrated FNNs with SMC, achieving reliable trajectory tracking under dynamic uncertainties.

4.4. Deep Learning and Reinforcement Learning Approaches

DL and RL methods enable task planning and control decision-making through data-driven and policy learning without relying on a precise model of the system. The implementation framework is shown in Figure 5b. Zhang et al. [89] proposed a DL-based hybrid approach based on target detection and semantic segmentation, which was applied to the static and dynamic picking tasks of the Delta parallel robot, effectively improving the picking accuracy and response speed. Yadavari et al. [90] applied DRL to the PID controller tuning of the Stewart platform, demonstrating the feasibility of DRL in complex dynamic model control and offering a new path for intelligent parameter optimization.

RNNs—particularly the LSTM architecture—have demonstrated strong potential in parallel robot control due to their ability to model temporal dependencies and time-varying nonlinearities. Sabahi [91] introduced a novel convex BiLSTM-based controller for a 3-PSP spatial parallel robot, targeting frictional and inertial uncertainties. This architecture leveraged memory-based feedback and a convex combination of BiLSTM layers to adaptively compensate for dynamic variations, achieving notable improvements in tracking accuracy, robustness, and learning efficiency. Similarly, Zhang et al. [92] developed an LSTM-enhanced adaptive neurocontroller for a pneumatic artificial muscle-driven parallel robot system. Their approach used a continuous-time LSTM to approximate unknown time-varying dynamics, alongside a two-stage error transformation to enforce guaranteed transient and steady-state tracking performance. These LSTM-based controllers highlight the growing utility of deep temporal models in addressing time-dependent uncertainties and complex nonlinear behaviors that are difficult to capture using feedforward or model-based methods.

In CDPR systems, the DRL method shows great potential, especially under conditions of high system parameter uncertainty, complex cable force distribution, and significant external disturbances. Zhang and Guo [93] proposed a DRL-based control strategy for a reconfigurable CDPR with uncertain parameters after reconfiguration, verifying the application potential of DRL in reconstructing CDPR. Lu et al. [94] introduced DRL for compensating uncertainties caused by cable elasticity and mechanical friction, improving dynamic response. Lu et al. [95] further developed an RL-based disturbance observer and combined it with SMC to achieve high-precision control of CDPR under strong disturbance conditions. Addressing load instability, Grimshaw and Oyekan [96] proposed a cascaded control strategy combining NN reference control and nested PID, improving system state recognition and stability. LSTM-based RNNs also have demonstrated strong potential in CDPR control. Kang et al. [97] proposed a hybrid RNN combining RNN and LSTM to predict pose estimation errors in CDPRs caused by nonlinear cable behavior such as creep and hysteresis.

For dynamic trajectory control and tension optimization in CDPRs, the deep deterministic policy gradient (DDPG) algorithm has been widely adopted. Sancak et al. [98] proposed a DRL-based point-to-point and dynamic trajectory control strategy for planar CDPRs, emphasizing the optimization of cable tension distribution through the learning process without participation of the tension distribution algorithm. Bouaouda et al. [99] and Ma et al. [100] introduced the DRL algorithm based on DDPG into CDPR control, realizing the tension distribution optimization and high-precision control of CDPR without analytical modeling. Sun et al. [101] trained DRL agents with DDPG to actively suppress vibrations in CDPR-based flexible structures under low-tension conditions, reducing stabilization time. Raman et al. [102] and Xiong et al. [103] explored end-to-end and hybrid DRL architectures for trajectory tracking, concluding that hybrid strategies offer superior training efficiency and robustness. Nomanfar and Notash [104] highlighted the importance of reward function design in achieving high-performance RL-based control for CDPRs.

The summary of the ML methods used for error modeling and compensation in parallel robots are shown in

Table 3. Summary of ML-based control strategies for parallel robots and CDPRs.

Application Focus	Methods/Algorithms Used	Achievements	Limitations
Hybrid ML-Augmented Control	DRNN + PID [74] B-spline NN + PD [75] ANN + IDC [76] ANN + P [77]	NNs compensate the nonlinear terms of the system, reduce the tracking error and vibration, and improve trajectory smoothness.	The lack of modeling accuracy of the NN model can cause the control performance of the hybrid method to be inferior, needs careful design and model tuning.
Adaptive Neural Network Controllers	Adaptive MLP-based controller [78,80] Adaptive RBF-based controller [79]	Generalized approximation and adaptive law help to realize stable control of position and force without the need for a complete mathematical model of the system.	Slow training convergence, strong dependence on training data, poor interpretability.
	Adaptive RBF + SMC controller [81,82] Adaptive MLP + SMC controller [83]	The NNs extend the adaptability of SMC to the system model, enabling adaptive control of system dynamics and external environmental changes.	Difficulty in parameter adjustment, more controller design parameters, high design and debugging costs.
FNN	Adaptive FNN [84,85,86]	Ability to continuously update the fuzzy rules and network parameters according to the input data, adapting to the dynamic changes in the system.	High computational complexity, sensitive to parameters such as the shape of the initial affiliation function and the learning rate.
	FNN + SMC [87,88]	Enhanced suppression ability for system parameter uncertainty and external perturbation of SMC by combining with FNN.	More complex in design, including fuzzy rules, NN training, and sliding mold surface design, etc., with high debugging cost.
DL	DL [89]	Improves accuracy and response speed.	High data and resource dependency, overfitting and generalization risk.
	LSTM-based [91,92,97]	Captures time-varying error dynamics and improves robustness under uncertainty.	Training requires sequential labeled data, requires continuous adaptation, high resource use.
DRL	DRL [90,93,94] RL + SMC [95] RL + PID [96] DRL with DDPG [98,99,100,101,102,103]	Automatic feature extraction and dynamic environment adaptation capability.	High computational resource and time cost, sensitive reward function design and uncertain convergence stability.

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In summary, ML has shown significant promise in advancing intelligent control for parallel robots. The hybrid control and feedforward compensation methods improve the nonlinear modeling capabilities of the traditional control framework by introducing NNs. Adaptive control methods emphasize real-time compensation for model errors and external disturbances. FNN-based control offers strong robustness in scenarios with incomplete control rules by integrating rule logic and learning mechanisms. DRL-based strategies break through the dependence on system modeling and realize autonomous control based on policy learning.

5. Machine Learning for Other Applications

Beyond its wide application in kinematics, error compensation, and control of parallel robots, ML has demonstrated substantial potential in a variety of other applications for parallel robots, including design optimization, trajectory generation, and fault diagnosis.

In design and modeling, ML techniques have been applied to analyze robot structure–function relationships and predict system behavior. Huo et al. [105] proposed an ML-based approach for motion and constraint modeling. The robot topology is represented by symbols to establish mappings between structural configurations and mobility characteristics. The prediction model is further constructed using NNs to achieve robot mobility analysis. Elgammal et al. [106] explored the use of NNs in modeling inverse dynamics of 3-DOF translational parallel manipulators. By combining feedforward NNs with transfer learning strategies, their approach maintained high modeling accuracy while significantly reducing the need for large training datasets, thereby enhancing generalizability and portability.

ML has also been leveraged for structural and performance optimization of parallel robots. Gao et al. [107] integrated BP NNs and the Levenberg–Marquardt algorithm to construct a system stiffness prediction model. This model served as the fitness function in multi-objective optimization, addressing the convergence challenges of traditional algorithms for complex performance objectives. Modungwa et al. [108] applied ensemble learning techniques to construct function approximation models for analytic cost functions, facilitating efficient search processes in the optimization of six-DOF parallel manipulators. For CDPRs, Umakarthikeyan and Ranganathan [109] developed a learning-assisted optimization strategy using random forest algorithms to determine optimal cable outlet positions and tension distributions for quadrotor-CDPR systems, significantly improving the system’s operable space and flexibility.

RL has increasingly been applied to trajectory planning and real-time obstacle avoidance. For instance, Liu et al. [110] proposed a dynamic path generation framework for CDPRs that integrates RL with real-time environmental awareness, enabling adaptive trajectory generation in dynamic environments with moving obstacles. Vu and Alsmadi [111] applied RL to pick-and-place trajectory generation for CDPRs and highlighted the critical role of reward function shaping in influencing policy behavior and task outcome.

Additionally, ML techniques have shown practical value in health monitoring and safety assurance for CDPR systems. Bettega et al. [112] developed a real-time fault detection framework for CDPRs using supervised classification algorithms. By analyzing variations in motor load torque, the system could effectively detect cable faults, demonstrating the feasibility of ML-based approaches for condition monitoring and system health assurance.

6. Conclusions and Outlook

The emergence and development of ML technologies have created transformative opportunities across key areas of parallel robot research, including kinematic modeling, error compensation, motion planning, control, and system design. Recent advances have demonstrated the successful application of ML techniques—particularly ANNs, DL, and RL—to both rigid parallel robots and CDPRs. A fundamental strength of ML lies in its ability to approximate complex system behaviors without relying on precise analytical models. This property makes ML especially valuable for highly nonlinear, high-DOF systems such as parallel robots, and positions it as a powerful solution to long-standing challenges related to system complexity, nonlinearity, and adaptability. The key achievements of using ML methods for parallel robots are as follows.

• Kinematic computation: ML has proven effective in improving the efficiency of IK computations and delivering reliable FK solutions. It also shows clear advantages in handling redundancy and singularities—areas traditionally difficult for analytical and numerical methods.
• Error compensation: data-driven models eliminate the need for explicit error modeling, enabling unified handling of both geometric and non-geometric errors.
• Control and trajectory tracking: ML-based controllers—especially hybrid and adaptive neural controllers— improve system adaptability and robustness under dynamic uncertainties through online learning, particularly in CDPR systems that are highly affected by nonlinearity and external disturbances.
• Emerging applications: In addition to the above, ML applications have also expanded into design synthesis, workspace analysis, trajectory generation, and fault diagnosis, accelerating the development of next-generation intelligent parallel robotic systems.

Nevertheless, several critical challenges remain. First, developing high-performance ML models often requires large volumes of high-quality, task-specific data, which can be costly and difficult to obtain in practical robotic systems. Second, a trade-off exists between model complexity and computational efficiency, which may limit the deployability of ML-based controllers on resource-constrained platforms such as embedded systems. Third, in certain complex subdomains—such as topology and type synthesis of parallel robots—the application of ML remains in its infancy and warrants further exploration.

It is also important to recognize that a purely ML-based approach—especially one relying exclusively on NNs—is unlikely to solve all challenges associated with parallel robotic systems. Given the structural complexity, real-time constraints, and safety-critical requirements of such systems—particularly in cable-driven variants—ML models often require large quantities of high-quality data and may struggle to generalize in the presence of singularities, unmodeled dynamics, or rare disturbances. In contrast, physics-based models and classical control theory offer structural interpretability, guaranteed stability, and proven robustness. Therefore, hybrid approaches that integrate machine learning with analytical models, control laws, or optimization techniques are emerging as more viable solutions. These frameworks combine the adaptability and learning capacity of ML with the predictability and efficiency of model-based methods. Future research should continue to explore and refine such hybrid strategies to fully leverage the strengths of both paradigms in parallel robot design, modeling, and control.

To address these limitations and fully realize the potential of ML in parallel robots, future research should focus on the following directions:

• Developing hybrid modeling frameworks that integrate physics-based models with data-driven techniques to improve generalization and robustness.
• Leveraging transfer learning and domain adaptation to reduce data requirements and enhance model transferability across different robot platforms.
• Advancing lightweight NN architectures and edge computing approaches to support efficient deployment on real-time embedded systems.
• Establishing open datasets and standardized benchmarking platforms, along with deeper interdisciplinary collaboration between the robotics and AI communities, to foster reproducibility and accelerate innovation in the field.