A Review on Design, Modeling and Control Technology of Cable-Driven Parallel Robots

Abstract

In view of the limitations of traditional rigid joint robots, cable-driven parallel robots (CDPRs) have shown significant advantages such as wide working space and high payload-to-weight ratio by replacing rigid connectors with flexible cables. Therefore, CDPRs have received widespread attention in the academic community in recent years and been applied to many fields. This review systematically reviews and categorizes the research progress in related fields in the past decade, focusing on mechanical structure design, mainstream mathematical models, and typical planning and control algorithms. In terms of mechanical structure, the advantages and disadvantages of three types of mainstream configurations and their application scenarios are summarized in detail. As for mathematical models, the dynamic modeling methods and various disturbance compensation models are mainly sorted out, and their action mechanisms and inherent limitations are explained. In terms of planning and control, four main research directions are discussed in detail, and their core ideas, evolution context, and development prospects are deeply analyzed. Although significant results have been achieved in the field of CDPR research, it is still necessary to continue to explore the direction of configuration diversification and intelligent autonomy in the future.

1. Introduction

Due to the rapid development of robotics technology, various types of robotic systems have been deeply integrated into many fields such as precision manufacturing, medical care, aerospace, disaster relief, etc., becoming a key force in promoting industrial upgrading and scientific and technological progress. However, the rigid connection and serial joint configuration of traditional robots have limitations, which restrict technological breakthroughs and application expansion [1]. Traditional robots are composed of rigid links connected in series through joints. Although this design achieves precise control, it has disadvantages: large mass and volume, increased cost, and limited space; large joint inertia, poor dynamic response, and difficulty in meeting high-speed tasks; limited load capacity, heavy weight, and reduced payload, especially in large-load scenarios [2].

In order to break through the limitations of traditional robots, cable-driven parallel robots (CDPRs) came into being and showed significant advantages. CDPRs adopt an innovative design concept. The design fixes the drive to the base and replaces the traditional rigid link with flexible cables, which brings many performance improvements [3]. Firstly, integrating the drive into the base effectively reduces the weight of the end-effector and reduces the overall inertia of the system, thereby significantly improving the dynamic response performance [4]. Secondly, the use of flexible cables not only simplifies the mechanical structure, but also makes the robot system lighter and further improves the flexibility of movement. The advantages of CDPRs are also reflected in the expansion of their workspace. Due to the scalability and retractable nature of the cables, the end-effector (EE) can reach a large workspace and even cover some spaces that are difficult for traditional robots to reach [5]. This large workspace feature means that CDPRs have unique potential applications in the fields of construction, space exploration, and large component assembly [2].

At present, CDPRs have been widely used in application scenarios with strict spatial constraints requiring high speed, high precision, or large working range. Typical fields include medical rehabilitation, construction manufacturing, and large precision equipment [4]. In the field of medical rehabilitation, researchers [6,7,8] have successfully developed an efficient cable-driven parallel-exoskeleton rehabilitation training system by coupling driving cables with elastic and rigid components (such as springs and rigid connecting rods) to accurately simulate the complex biomechanical properties of the human musculoskeletal system, including its flexibility and support mechanism. The system is designed with full consideration of rehabilitation needs and can provide patients with a stable and controllable training environment, significantly improving rehabilitation effects and usability. In the field of construction manufacturing, the application of CDPR technology is becoming increasingly widespread. Reference [5] introduced CDPR into the building exterior wall image-drawing process, and realized the automatic drawing of exterior wall patterns through a high-precision robot arm, thereby improving construction efficiency and design flexibility. In the field of large-scale precision equipment, the most representative application example is the high-precision attitude adjustment mechanism of the radio telescope feed cabin, FAST project [9]. The cable transmission system adopts a multi-cable coordinated drive mechanism to achieve millimeter-level precision and attitude control, effectively supporting the stable operation of the feed cabin in complex environments.

The existing literature has systematically reviewed various research works in the field of CDPRs, covering mechanical design, modeling, trajectory planning and control algorithms, and provided theoretical support for applications such as industrial automation and rehabilitation medicine. In [2], the development history and application scenarios of CDPRs are systematically reviewed, focusing on the theoretical progress of their design modeling, performance optimization and control planning, including lightweight structure innovation and robustness improvement, and looking forward to future research directions such as intelligent control and composite material integration, such as adaptive control strategies based on deep learning and the application potential using carbon-fiber materials. In [4], the latest research results of CDPRs in design modeling, control strategies, and trajectory planning are summarized, such as real-time feedback mechanisms and multi-objective optimization algorithms. At the same time, technical challenges and future research directions in dynamic environments are proposed, such as external disturbance compensation and uncertain load processing, and the development of efficient self-calibration systems is required. In [10], a comprehensive review of the dynamic model and various control strategies of parallel robots were conducted. The stability and response speed of each method were compared in detail, and its performance optimization in high-speed and high-precision scenarios was analyzed, such as trajectory-tracking error minimization and vibration suppression technology, which is suitable for high-demand fields such as semiconductor manufacturing. In [11], the configuration design, kinematic/dynamic analysis, workspace optimization, trajectory planning, and control strategy of the CDPR are systematically summarized, emphasizing the importance of parameter calibration and simulation verification, and an in-depth discussion of cable mass and elastic modeling issues and future research directions, such as nonlinear dynamic effect analysis and multi-scale material modeling, to improve system reliability. In [6], the mechanism design, drive mechanism, control strategy and clinical trial progress of the shoulder rehabilitation robot are reviewed, covering patient adaptability assessment and safety protocols. At the same time, future research directions are discussed, such as enhanced human–machine interaction and integration of personalized treatment plans to promote rehabilitation efficiency. Reference [12] reviews the theoretical research and application progress of CDPR, covering core areas such as configuration design, cable force distribution, workspace and stiffness analysis, performance evaluation, optimization and motion control, and explores its broad application potential in industry and technology.

Compared with existing research, this paper is dedicated to systematically reviewing the latest technological progress in the field of CDPR, focusing on analyzing its research progress and core challenges in key areas such as innovative mechanical structure design, dynamics modeling and solution, high-precision trajectory planning algorithms, and robust/intelligent control strategies. Related research not only addresses fundamental theoretical issues such as redundant drive optimization, workspace analysis, flexible cable dynamics, and time-varying tension distribution, but also lays a solid theoretical foundation for CDPR’s diversified configuration optimization, high-fidelity mathematical model construction, and high-performance real-time control system design, and clearly points out its future development direction and potential technological breakthroughs in application scenarios such as large-scale space operations, rehabilitation medical equipment, and precision manufacturing equipment.

2. Mechanical Structure Design of CDPR

The structures of CDPRs are diverse and can be divided into three categories according to their design features and functional requirements: traditional structures, multi-agent frameworks, and special structures. Most CDPRs use a traditional structure, which consists of core components such as a fixed base, multiple high-strength cables, and a movable platform. The base is usually rigid and cannot be deformed or moved independently to provide a stable support foundation; the cables are precisely controlled in length by servo motors to drive the platform to achieve precise positioning and movement in three-dimensional space. The reconfigurable framework is designed to allow CDPR to adapt to more complex and changeable environments, such as narrow indoor spaces or areas full of obstacles. This framework currently uses drone swarms or unmanned vehicle swarms as dynamic bases to replace fixed structures, allowing the system to freely adjust its shape and disperse its layout, thereby effectively avoiding obstacles or entering restricted areas, significantly improving environmental adaptability [13]. Special structures are highly customized designs developed specifically for a specific application task, such as cable-driven exoskeleton systems used in the field of medical rehabilitation to assist patients in limb rehabilitation training [14], or cable-driven bionic manipulators used to perform fine grasping operations [2]; in addition, this type of structure may also include special configurations in industrial inspection or special rescue scenarios to meet specific performance requirements. In short, these three types of structures have their own advantages and together support the wide applications of CDPRs in different fields. Figure 1 shows the different types of CDPRs.

2.1. Traditional Structure

CDPRs with traditional structures can be roughly divided into two categories: underconstrained CDPRs and fully constrained CDPRs. The number of drive cables m in an underconstrained CDPR satisfies the inequality $m<n+1$, where n represents the number of degrees of freedom (DOF) of the end-effector (EE). This structural characteristic makes it impossible for the system to fully constrain the posture of the EE through kinematics. In this configuration, the position and orientation of EE will spontaneously tend to the equilibrium position where the system’s gravitational potential energy is minimum, and its overall state is extremely susceptible to external disturbances such as wind loads, inertial forces, or accidental contact. A typical application case is a fixed platform consisting of a large base and three cable suspensions: the load is constrained only by three cables, and its posture stability is highly dependent on gravity, lacking an active constraint mechanism [15]. Due to the lack of cables below to provide counter-tension, the working-space range of this type of underconstrained system is significantly larger than that of fully constrained CDPRs, making it particularly suitable for operating scenarios that require EE to move flexibly in a wide space, such as inspections of large outdoor facilities or material transportation at construction sites [2]. However, this design has some defects: the position and direction of the EE of the underconstrained CDPR partially rely on the balance of gravitational potential energy, which makes it only suitable for low-speed scenarios, and the working state is easily affected by unexpected external forces, and the stability is significantly lower than that of the full/redundant constraint system [16]; due to the strong coupling characteristics between kinematics and static equilibrium, the underconstrained CDPR requires simultaneous geometric–static joint analysis; this type of system cannot actively control all DOF through cable tension, resulting in the positioning accuracy of EE being affected by both the gravitational field distribution and the elastic deformation of the cable.

The fully constrained CDPR adopts a redundant drive scheme in which the number of cables is greater than or equal to the DOF of EE, and ensures that each DOF is always in an active control state through geometric constraints. This topology eliminates the underconstrained degrees of freedom of EE and avoids the passive degrees-of-freedom problem of suspended CDPRs [4]. Its control system uses redundant drive and tension optimization algorithms to ensure that all cables maintain positive tension in any positions, effectively eliminating vibration and energy dissipation during movement [17]. Compared with the underconstrained CDPR, the fully constrained CDPR adopts a redundant drive scheme with at least 6 DOF, and ensures active control of all DOF through geometric constraint mechanisms, eliminating the underconstrained DOF of the EE and avoiding the problem of passive DOF. In addition, its advantages include the position of the EE is determined by the geometric length of the cable, and the kinematic analysis only requires geometric parameters and tension distribution, without considering gravity or interference, simplifying trajectory planning and control complexity; the cable is always tensioned to form a rigid closed loop, improving anti-disturbance performance, and suitable for high-precision positioning; all DOF are independently controlled, and high-precision tracking of complex spatial trajectories is achieved. However, this design also has significant defects: the redundant cable configuration increases the risk of cable interference, especially when operating in a three-dimensional space, and the probability of cable entanglement increases significantly; in order to maintain the continuous tension of all cables, the system needs to apply a higher base preload, which not only leads to increased energy consumption, but also accelerates the wear of drive components, making preload management a key technical bottleneck affecting the system’s lifespan [9]; compared with the theoretical maximum working space, the effective working space of the fully constrained system is reduced by about $25\%$ to $40\%$; the optimal tension distribution of the redundant driving force needs to be solved in real time, which not only significantly increases the complexity of the control algorithm, but also significantly increases the complexity of system design and development. The following table summarizes some CDPRs with traditional structures and lists their structural characteristics, accuracy indicators, etc. in detail. The features and applications of these CDPRs are shown in

Table 1. Traditional CDPRs.

Type	Features	References
Underconstrained CDPRs	The number of cables is less than or equal to DOF of EE	[15,18,19,20]
Fully-Constrained CDPRs	The number of cables is more than DOF of EE	[17,21,22,23,24]

.

2.2. Reconfigurable Structure

The reconfigurable CDPR achieves real-time and flexible reconfiguration of its own topological structure by dynamically adjusting the position of the cable anchor points on the fixed platform or EE, or deeply integrating it into the collaborative framework of the Multi-Agent System (MAS). This type of innovative design organically combines the dexterity of the under-constrained CDPR with the stability advantages of the fully constrained CDPR, and exhibits the following core characteristics: firstly, by increasing or decreasing the number of cable branches in real time and optimizing the spatial configuration of the terminal towing points, the reconfigurable CDPR can actively change its spatial topological configuration, effectively avoiding collisions with complex static or dynamic obstacles in the workspace, making it significantly adaptable in unstructured complex terrain (such as ruins search and rescue, rugged planetary surfaces) or working environments that need to deal with dynamic interference (such as aircraft assembly lines); secondly, its reconstruction mechanism supports the dynamic optimization of the overall stiffness, load-bearing capacity, and accuracy of the system by the online adjustment of key parameters such as the number of branches, cable spatial layout, and tension distribution, so as to meet the differentiated performance requirements of different mission stages; more importantly, with the help of the collaborative innovation of structural reconstruction strategies, global path-planning optimization and intelligent driving algorithms, the reconfigurable CDPR has significantly broken through the inherent bottlenecks of traditional CDPR such as the small working space and limited shape due to the unilateral tension constraint of the cable (which can only withstand tension but not pressure) and the rigid boundaries of the fixed structure.

However, this design also has several key challenges and shortcomings: firstly, the topology reconstruction process requires frequent dynamic adjustment of the number of physical connection points or branches of the cables, which means that kinematic modeling must consider the switching and transition problems between multiple sets of discrete topological parameter states. The real-time control algorithm needs to efficiently integrate and switch multiple sets of heterogeneous dynamic models, and the transient during structural switching will introduce a significant time-varying nonlinear dynamics (such as impact and friction mutations), which greatly increases the difficulty of high-precision parameter identification and model convergence of the system; secondly, structural reconstruction (such as adding or removing cables or moving anchor points) will change the overall mass distribution, center-of-gravity position, and inertia tensor parameters of the system in real time, directly and significantly affecting its dynamic response characteristics (such as acceleration and vibration decay time), forcing the control system to have stronger robustness and adaptability; thirdly, when adjusting the number of cable branches, the vibration mode and natural frequency of EE may undergo discontinuous mutations, which may easily induce the risk of residual vibration aggravation or instability. It is urgent to redesign the adaptive damping compensation algorithm based on online modal identification to suppress transient oscillations. In order to systematically sort out the current research progress, the following table summarizes some typical CDPR prototype systems that use a reconfigurable framework, and lists in detail key information such as their structural topological characteristics, drive configuration, reconstruction method, accessible workspace, motion accuracy indicators, and main application scenarios.

Table 2. Reconfigurable CDPRs.

Base Type	Features	References
Autonomous Cars	Ground Mobility, Flexibility, Great Load Capacity and Intelligence	[13,25,26]
Drones	Flexibility, Rapid Deployment, Scalability, Adaptability and Intelligence	[27]

has shown the features and applications of these CDPRs.

2.3. Special Structure

Medical rehabilitation exoskeleton systems equipped with CDPRs are usually based on lightweight flexible frame designs, which not only reduce the overall weight of the equipment, but also improve the flexibility of wearing, especially for long-term rehabilitation scenarios.

This type of system uses bionic principles to optimize the drive path, such as imitating the coordination mechanism of human skeletal muscles to make the layout of the drive cables more consistent with the natural motion trajectory, thereby reducing energy loss [28]. By adopting lightweight and highly elastic materials such as carbon-fiber composite materials and a modular layout, the system can achieve coordinated movement of multiple degrees of freedom of joints. The modular design can customize joint components according to different rehabilitation needs, and support independent or linkage control of parts such as shoulders, elbows, and wrists. At the same time, the system integrates sensor feedback mechanisms such as electromyography sensors and position encoders to monitor the user’s motion status in real-time, and reduces system energy consumption and motion inertia through safety constraint mechanisms (such as torque limitation, emergency stop function), while ensuring accurate trajectory tracking and natural human–computer interaction [29]. These features ultimately form a comprehensive solution that integrates motion flexibility, wearable adaptability, and functional scalability, supporting seamless upgrades from basic assistance to advanced rehabilitation training [6].

Cable-driven exoskeleton systems have the advantages of high power-to-weight ratio and high mobility. Their lightweight structure effectively reduces the inertia of the device and enhances dynamic interaction capabilities. By integrating elastic drive units, safe human–machine collaboration is achieved, avoiding the risk of injury caused by sudden movements. Such systems use parallel mechanisms and elastic elements to optimize the force transmission path, significantly reduce the load on the shoulder joint, and adapt to complex anatomical movements. For example, the CAREX system avoids axis-alignment problems through flexible joint design, accurately reproduces the coordinated movement of the scapula and humerus, and its cable-driven layout simulates the natural arc of human shoulder rotation [28].

Recently, inspired by the structure of biological tendons, researchers have used cables to simulate biological tendons and invented a new CDPR. Y. Sun proposed a fishtail-inspired method to optimize the stress distribution of serial flexible joints in a tendon-driven continuum robot (TDCR) to solve the fatigue and fracture problems caused by uneven stress [30]. Q. Peyron conducted a comprehensive elastic stability analysis of tendon-driven continuum robots for the first time, revealing their buckling phenomenon and multi-configuration characteristics, and proposed an open-loop control strategy that used the buckling phenomenon to achieve active softening [31].

3. CDPR Modeling Research

The mathematical modeling research of CDPRs focuses on the establishment and optimization of kinematic and dynamic models, covering rigid body kinematic analysis, construction of static equilibrium equations, and modeling of the dynamic response of elastic cables.

3.1. Kinematic and Dynamic Modeling

At present, most researchers use Lagrange equations to model the dynamics of CDPR. In order to maintain generality, this section explains in detail how to establish the dynamic model of CDPR with traditional structure, which can be found in paper [32]. Consider a 6-DOF CDPR with a mobile platform controlled by n cables.

As shown in Figure 2, it is a spatial schematic diagram of a 6-DOF CDPR controlled by n cables, where the blue object represents n winches connected by n cables, and the black object represents EE, which can be of any shape, meaning that the specific shape is not given.

We define the world coordinate system $O-O_x{O}_y{O}_z$ and the body coordinate system $P-O_x{O}_y{O}_z$ on EE as shown in Figure 2, where $\mathit{p}$ represents the coordinate vector from the origin of $P-O_x{O}_y{O}_z$ to the origin of $O-O_x{O}_y{O}_z$, $p_i$ stands for the connection point of the i-th cable on EE, ${\mathit{a}}_i$ is the coordinate vector of the connection point in $P-O_x{O}_y{O}_z$, ${\mathit{c}}_i$ is the coordinate vector of the connection point of the i-th pulley in $O-O_x{O}_y{O}_z$, and ${\mathit{l}}_i$ is the coordinate vector of the i-th cable in $O-O_x{O}_y{O}_z$. The displacement vector of EE in $O-O_x{O}_y{O}_z$ is $\mathit{p}={\left[x_p{y}_p{z}_p\right]}^{\mathsf{T}}$ and the rotation vector is $\mathit{\varphi }={\left[{\alpha }_p{\beta }_p{\gamma }_p\right]}^{\mathsf{T}}$, while the motion vector is $\mathit{x}=\left[{\mathit{p}}^{\mathsf{T}}{\mathit{\varphi }}^{\mathsf{T}}\right]$. Many papers have adopted Lagrangian functions to build the dynamic model of CDPRs. Here is a general dynamic model of CDPRs:

$$\left(\mathit{M}\left(\mathit{x}\right)+{\mathit{J}}^{\mathsf{T}}{\mathit{R}}_T^{-1}{\mathit{I}}_m{\mathit{R}}_T^{-1}\mathit{J}\right)\ddot{\mathit{x}}+\left({\mathit{J}}^{\mathsf{T}}{\mathit{R}}_T^{-1}{\mathit{I}}_m{\mathit{R}}_T^{-1}\dot{\mathit{J}}+\mathit{C}(\mathit{x},\dot{\mathit{x}})\right)\dot{\mathit{x}}+\mathit{G}={\mathit{J}}^{\mathsf{T}}{\mathit{R}}_T^{-1}\mathit{u}.$$

(1)

in which $\mathit{M}\left(\mathit{x}\right)$ stands for the positive-definite symmetric inertia matrix of EE, and $\mathit{C}(\mathit{x},\dot{\mathit{x}})$ means the Coriolis centrifugal matrix, while $\mathit{G}$ represents the gravity vector. ${\mathit{I}}_m$ and ${\mathit{R}}_T$ are the inertia matrix and drive ratio of winches, respectively. $\mathit{J}$ stands for the Jacobian matrix.

This standard dynamic model can effectively characterize the dynamic characteristics of CDPRs and is therefore widely applied. V. Nguyen established an extended Kalman filter and a multiplicative extended Kalman filter based on this model combined with a data-driven approach, which significantly improved the accuracy of position estimation of the EE in a high-noise environment [33]. Y. Zou built a neural network model based on this model and successfully reduced the tracking error to less than $2^{\circ }$ through a data-driven approach [34]. H. Sun combined this model with the finite-element method, using the shape function matrix and the curvature matrix to describe the dynamic behavior of the plate structure, and verified the accuracy and effectiveness of this method through physical experiments [35]. M. Joyo proposed a reinforcement learning algorithm adopting the model, and verified the applicability of this method in a complex robot structure environment through physical experiments [36]. Based on this, Y. Peng proposed a new anchor-point model to solve the static equilibrium reachable workspace of underconstrained CDPRs [37]. This method reduces nonlinearity by simplifying the model, making the problem more suitable for reformulated linearization technique. To improve the accuracy of the model, researchers have improved the standard model by coupling the tension constraint model, friction model, elastic cable model, catenary model, and cable sag effect.

3.2. Friction Model

In the process of establishing the CDPR model, friction is one of the main disturbance factors to be considered. M. Kim clearly pointed out that the friction between the cable and the pulley significantly increased the system error of the CDPR [38]. This part will establish the CDPR dynamic equation considering a general friction model. The Coulomb viscous friction model currently widely used is a simplified model that can accurately characterize the friction characteristics:

$${\mathit{F}}_f={\mathit{F}}_v+{\mathit{F}}_c={\mathit{K}}_v\dot{\theta }+{\mathit{K}}_c\mathrm{sign}\left(\dot{\theta }\right),$$

(2)

in which ${\mathit{F}}_f$, ${\mathit{F}}_v$ and ${\mathit{F}}_c$ stand for total friction, viscous friction, and Coulomb friction, respectively, while ${\mathit{K}}_v$ and ${\mathit{K}}_c$ are the viscous friction coefficient and Coulomb friction coefficient, respectively. According to (1), we are able to obtain the improved dynamic model of CDPR incorporating friction:

$$\begin{array}{cc}\hfill & \left(\mathit{M}\left(\mathit{x}\right)+{\mathit{J}}^{\mathsf{T}}{\mathit{R}}_T^{-1}{\mathit{I}}_m{\mathit{R}}_T^{-1}\mathit{J}\right)\ddot{\mathit{x}}+\left({\mathit{J}}^{\mathsf{T}}{\mathit{R}}_T^{-1}{\mathit{I}}_m{\mathit{R}}_T^{-1}\dot{\mathit{J}}+\mathit{C}(\mathit{x},\dot{\mathit{x}})+{\mathit{J}}^{\mathsf{T}}{\mathit{R}}_T^{-1}{\mathit{K}}_v{\mathit{R}}_T^{-1}\mathit{J}\right)\dot{\mathit{x}}\hfill \\ \hfill & +{\mathit{J}}^{\mathsf{T}}{\mathit{R}}_T^{-1}{\mathit{K}}_c\mathrm{sign}\left(\dot{\mathit{l}}\right)+\mathit{G}={\mathit{J}}^{\mathsf{T}}{\mathit{R}}_T^{-1}\mathit{u}.\hfill \end{array}$$

(3)

There are a large number of existing studies that adopt this improved model. M. Bajelani adopted this model and used a delay-time learning algorithm for compensation, reducing the error rate to 1–2%, effectively improving the accuracy and robustness of the system [22]. Reference [39] also adopted this model and used adaptive sliding mode control to suppress the influence of friction and improve system accuracy. References [15,19,40] used this model to design an adaptive sliding-mode controller, which improved the accuracy of CDPRs by more than $50\%$. Y. Lu combined this model with the Lyapunov function and proposed a reinforcement learning method to reduce the root mean square error index of the system by $30\%$ [41]. R. Wang combined the tension constraint model with this model and suppressed the influence of friction through feedforward control, reducing the cable-length error to 1.23 mm [42]. M. Harandi assumed that the robot was always in motion, so he used a simplified linear viscous friction model and used adaptive control to reduce the error by 40–50% [43]. P.A. Voglewede and I. Ebert-Uphoff first introduced the planar friction cone model and the spatial soft-finger contact model, breaking through the limitations of the traditional friction model and significantly improving the intuitiveness and practicality of force-closure analysis [44].

3.3. Tension Constraint Model

In the CDPR system design, the tension constraint model is a mathematical model that describes the physically feasible conditions that the cable tension must meet. This model constitutes the core foundation for the study of system dynamics and control. B. Kin has shown through a large number of experimental studies that the introduction of tension constraints in the CDPR control system design can significantly improve the system posture accuracy and tension distribution accuracy, reducing the system posture error and tension distribution error by more than $40\%$ [25]. Therefore, tension constraints play an irreplaceable core role in the control of CDPRs. The tension constraints of CDPR are usually expressed as the following set of inequalities [17]:

$$T_{i min}\le T_i\le T_{i max}(i=1,2,\dots ,n),$$

(4)

where $T_i$, $T_{imin}$, and $T_{imax}$ mean the current, the minimum, and the maximum tension of the i-th cable, respectively. Based on this model, R. Wang constructed a cable-length constraint containing high-order derivatives, which not only improved the positioning accuracy of the CDPR by $15\%$, but also significantly suppressed the vibration amplitude and acceleration peak of the EE [17,42]. M. Bajelani [22] used tension constraints to construct the pseudo-inverse of the Jacobian matrix, ensuring all cable tensions remain positive, thereby enhancing the robustness of the system. P.A. Voglewede and I. Ebert-Uphoff [44] proposed the “plane dual cable theorem” and “space dual cable theorem” based on tension constraints, transforming the force-closure conditions in grasping theory into geometric criteria suitable for cable-driven robots. M. Harandi [43] introduced free vectors in the controller and used the redundant characteristics of the right null space of the Jacobian matrix to satisfy the tension constraints, thereby achieving the optimal distribution of the cable drive force. S.W. Hwang [18], S. Khoshkam [20], Q. Kui [15], M.C. Kim [38], E. Idà [45], and D. Song [46], respectively, achieved high-precision motion control based on tension constraints by applying different control algorithms including sliding-mode control (SMC) and model predictive control (MPC).

3.4. Sagging-Cable Model

The sagging-cable model is one of the core tools for dynamic modeling of CDPRs. The model takes into account both the mass and elastic properties of the cable, and can accurately describe the cable deformation and tension distribution caused by its own weight during movement, thereby supporting the development of high-precision kinematic models and analyzing force singularities [47]. Under high-speed and high-acceleration conditions, the model optimizes the cable configuration through the static equilibrium equation, effectively enhances the stiffness and stability of the system, and suppresses positioning errors and vibrations caused by cable sagging [11]. The basic principles of the model are briefly described as follows.

Firstly, we assume that the cable used is made of uniform material with a line density of $\mu$. The static configuration is formed by the balance of deadweight and tension, and the dynamic vibration is a small perturbation of the static position. The widely used Irvine model introduces bending stiffness based on the above model, which means that the cable bending moment $M=EI\left(d^{2}y\right)/\left(d{x}^2\right)$ participates in the equilibrium. Therefore, the static equilibrium equation is expanded to:

$${\displaystyle \frac{d^2}{d{x}^2}}\left(EI{\displaystyle \frac{d^{2}y}{d{x}^2}}\right)-{\displaystyle \frac{d}{dx}}\left(T{\displaystyle \frac{dy}{dx}}\right)=q.$$

(5)

When $EI$ is a constant, it simplifies to the fourth-order differential equation:

$$EI{\displaystyle \frac{d^{4}y}{d{x}^4}}-T{\displaystyle \frac{d^{2}y}{d{x}^2}}=q.$$

(6)

Based on the Irvine model, J-P. Merlet proposed a new algorithm for calculating the horizontal cross-sectional boundary of the suspended CDPR workspace taking into account the cable’s own weight, focusing on solving the workspace solution problem under given cable tension limits, platform height, and attitude constraints [48]. In subsequent research, he systematically analyzed the singular configurations of CDPR using the Irvine model for the first time, explored the classification of singular points in inverse kinematics and forward kinematics [49], revealed the local extreme points of the potential energy functional corresponding to its geometric statics equations, and verified the system stability conditions through theoretical derivation and numerical simulation [50]. These pioneering works have promoted this direction to become an important research branch in the CDPR field. A. Baskar [51] used this model to propose a numerical extension method based on random single-valued cycles, successfully solving the global kinematic problem of large CDPR systems.

4. Planning and Control

Compared with traditional rigid-link parallel mechanisms, CDPRs introduce many challenges due to the use of flexible cables instead of rigid links [2]. As mentioned in the previous section, we have systematically reviewed the current status of modeling research, in which the friction effect between the cable and the pulley, the tension constraint, and the cable suspension characteristics are all key constraints for high-precision control. In order to achieve high-precision trajectory-tracking capabilities of CDPRs, the research path has gradually evolved from early traditional methods to intelligent algorithms integrated with deep learning and reinforcement learning. This Section will focus on reviewing the latest research progresses in this field in motion planning and control.

Table 3. Different control methods.

Type		Method	Reference
Traditional Planing Method	Dynamic Programming	Specific Frequency Selection Algorithm & Parameter Recursive Optimization	[16,52,53,54]
		Fifth-order polynomial & three-dimensional trigonometric function analysis trajectory	[54,55,56]
	Improved RRT* Method	Mixed Potential Field Functions & Manifold Tangent Space Theory & Adaptive Sampling	[57,58,59,60]
		Collaborative Optimization of Tree Topology & Dynamic Maintenance	[2,13,26]
	Energy Optimization Planning	Continuous Space Optimization & Competitive Particle Swarm Optimization	[61]
	Online Zoom Trajectory	Path Speed Breakdown & Forward-Looking Adjustments	[17]
	Equilibrium Planning	Balanced Configuration Analysis & Input Shaping Technology	[18]
Traditional Control Method	Improved PID Control	High Gain Parameter & Nonlinear Feedforward Compensation	[42,62,63,64,65]
	Sliding-Mode Control	Mixed Integral SMC	[15,19]
		Adaptive Integral SMC & Ring Topology	[19]
		Multiple Input and Multiple Output SMC	[66]
	Dynamic Inverse Control	Combined with Time Delay Learning Method	[22]
	Compound Control	Admittance Control & Dual-Mode Motion Strategy	[38]
Model Predictive Control	Standard MPC	Time Domain Iterative Optimization	[17,67,68]
	Robust MPC	Probabilistic Constraints& Unscented Kalman Filter	[20]
		Super-Helical Sliding-Mode Observer& Recursive Integral Terminal SMC	[69]
		Lyapunov Function	[70]
Intelligent Control	Neural Networks	Learning Algorithms	[34]
		Long Short-Term Memory Neural Network	[71]
	Fuzzy Control	Interval Type-2 Fuzzy Neural Network& Non-singular Terminal SMC	[72]
	Deep Reinforcement Learning	DRL & PID control	[36]
		DRL & Lyapunov Function	[35,41]
		Multi-agent DRL	[73]

shows the latest research results in this area, and the specific details of these methods are going to be described in the following four subsections.

4.1. Traditional Planning Method

Planning methods can be divided into two categories: path-planning algorithms and trajectory-planning algorithms. Trajectory planning needs to consider both the velocity and acceleration of the robot during motion, which means that its planning process contains dynamic information, while path planning only considers the robot’s position, which means that it only involves kinematic information.

C. Gosselin et al. first proposed the concept of CDPRs dynamic trajectory planning by selecting a specific frequency to meet the cable tension requirements [52]. In subsequent studies, the team conducted an in-depth exploration of this method and optimized it through various strategies: by designing a parameter recursive relationship, combining frequency ratio and recursive accumulator, parameter transfer across trajectory segments and memory of historical trajectory parameters were achieved, thereby ensuring the spatial continuity of the trajectory [16,53]; based on the system’s inherent frequency characteristics, analytical trajectory expressions in the form of fifth-order polynomials and three-dimensional trigonometric functions were derived to achieve flexible control of trajectory morphology [54,55,56]; a trajectory-planning method based on oscillating spherical linear interpolation was developed, combined with a variable amplitude trochoid trajectory, to achieve high-precision point-to-point dynamic control [54]. E. Idà further validated the efficacy of the aforementioned methodology [45,74]. Y. Li applied this method to a cable-driven parallel waist rehabilitation robot to achieve smooth trajectory planning [33].

Afterwards, many traditional methods were applied to the path planning of CDPRs. References [57,58,59,60], respectively, integrated the classic RRT* algorithm with the hybrid potential field function, manifold tangent space theory, and adaptive sampling strategy, significantly reducing the number of algorithm iterations, greatly improving the success rate of path generation, and achieving fast and high-precision trajectory tracking control. References [2,13,26] introduced a tree topology structure based on existing research, and further improved the real-time computing performance and global optimization indicators of the algorithm through the coordinated optimization of the dynamic maintenance mechanism and collaborative search strategy. Other researchers have also proposed several trajectory planning methods based on traditional ideas. X. Wang combined continuous space optimization with sparse control and used the competitive particle swarm algorithm to achieve a minimum energy reconstruction strategy under physical constraints [61]. R. Wang proposed a forward-looking online scaling method based on the path velocity decomposition scheme to generate feasible trajectories that meet cable velocity, acceleration, jerk, and tension constraints [17]. S. Hwang proposed a balance configuration analysis method for under-constrained CDPRs and a balance-based trajectory generation method, which solved the problem of difficulty in determining stable balance configurations and effectively suppressed the vibration of the EE through input-shaping technology [18].

4.2. Traditional Control Method

Traditional control algorithms are the core elements to ensure efficient and reliable operation of CDPR systems. Such algorithms have been widely used in many industrial fields and have successfully proven their ability to effectively cope with challenges such as cable elasticity, friction, nonlinearity, and strong coupling, providing good stability and robustness for CDPRs. B. Kim proposed a simple model-based cable-length control algorithm, which significantly reduced position error and tension error [25] compared to the tension distribution algorithm and trajectory-planning algorithm, fully demonstrating the key role of the control system in CDPRs. This part will systematically review the traditional control algorithms that have been applied to CDPRs in recent years.

References [42,62,63,64,65] have shown that the adverse effects of nonlinear factors such as cable elasticity, gear clearance, and friction disturbance on the dynamic performance of the system can be effectively suppressed through the synergistic effect of high-gain PID parameter configuration and nonlinear feedforward compensation. This result also proves that the improved PID algorithm still has significant performance advantages when applied to this system. X. Li and Q. Kui proposed a hybrid integral sliding-mode control strategy to improve the trajectory tracking accuracy of CDPRs and suppress chattering. Based on the previous works, they further designed a parallel cooperative control scheme for CDPRs based on adaptive integral sliding mode, and achieved more robust and smooth cooperative control through adaptive weight matrix and ring topology structure [19]. R. de Rijk successfully suppressed out-of-plane vibrations using a multi-input multi-output sliding-mode controller [66]. S. Briot derived the stability criterion of CDPRs based on the Irvine cable model, combined with optimal control theory and variational principle [50]. S.K. Cheah introduced virtual filtering-rate trajectory and attitude error reconstruction technology, integrating adaptive feedforward control with a strict passive feedback controller [75]; and M.R.J. Harandi used a dual-parameter separation adaptive mechanism to construct a trajectory-tracking controller, effectively overcoming the negative effects caused by system model uncertainty [43]. M. Bajelani combined dynamic inverse control with a time-delay learning algorithm to successfully compensate for nonlinear factors such as cable elastic deformation and friction disturbance in the system [22]. M. Kim proposed a composite control method that integrates admittance control and dual-mode motion strategy, which improved key indicators by more than $50\%$ and ensured that the primary–subordinate CDPRs successfully completed the plug-in-hole task [38].

4.3. Model Predictive Control

Model predictive control is a hot topic in the current control field. This control method can adjust trajectory parameters (such as speed, acceleration, and jerk) in real time and satisfy multiple constraints (such as tension constraints, velocity constraints, and acceleration constraints) by performing rolling horizon optimization in the prediction domain [17]. Therefore, model predictive control has been widely used in various automation systems, for example, autonomous vehicles [76]. Currently, lots of studies are devoted to applying model predictive control to CDPRs to solve the optimal trajectories under multiple constraints.

In [67,68], MPC was applied to CDPRs, and the linear time-varying model and tension constraints were combined to effectively solve the problem of the system transitioning from a static state to a periodic motion or between different periodic motion modes. In order to improve the robustness of traditional nonlinear model predictive control, it was combined with probabilistic constraints and unscented Kalman filters to significantly suppress system vibration [20]. Qin Y. innovatively combined the super spiral observer with model predictive control to plan the motion trajectory of CDPR, and designed a recursive integral terminal sliding-mode controller based on adaptive disturbance observation, realizing the dynamic obstacle avoidance function of CDPR in complex environments [69]. Cao Y. successfully solved the input–output saturation problem in complex environments by combining the Lyapunov function with model predictive control [70].

4.4. Intelligent Control Method

In recent years, the development of intelligent control methods has been driven by algorithm innovation, deeply integrating emerging technologies such as edge computing, digital twins, and machine learning, and its application scope has expanded to a wide range of fields such as industrial automation, autonomous driving, and robotics. Studies have shown that intelligent control algorithms can effectively handle nonlinear characteristics and model uncertainties in CDPR systems. At the same time, for system characteristics that are difficult to accurately model, effective estimation and compensation can also be achieved by relying on machine learning algorithms.

M. Bajelani [22] proposed a data-driven method based on time-delay learning and successfully implemented high-precision trajectory tracking control using a black-box model. Y. Zou [34] used a neural network alternative learning algorithm to solve the kinematic modeling and control problem of CDPR with collision tolerance characteristics. E. Oghabi [72] constructed an adaptive interval type-II fuzzy neural network and integrated non-singular fast terminal sliding-mode control to effectively solve the high-precision finite-time trajectory tracking control problem under model uncertainty and external disturbances. X. Zhang [71] designed an adaptive controller based on long short-term memory neural network to handle complex nonlinear dynamic characteristics.

As an emerging learning algorithm, deep reinforcement learning has been widely used in various systems due to its powerful functions, especially in the field of CDPRs. M. Joyo [36] combined deep reinforcement learning with PID control to achieve accurate trajectory-tracking control under complex constraints. Y. Lu [35] proposed a control algorithm that combines deep reinforcement learning with Lyapunov function to solve the uncertainty problems caused by cable elasticity, mechanical friction, and other factors. S. Wu [73] applied this method to an underactuated dual-ship crane system and successfully coped with the control challenges of the dual-ship crane system in a complex marine environment. References [77,78] also proposed different DRL methods to effectively deal with control problems such as cable tension constraints and complex nonlinearities.

5. Conclusions and Outlook

The vigorous development of robotics has strongly promoted the transformation of various industries. The emergence of CDPR has effectively overcome the inherent limitations of traditional parallel mechanisms, such as limited workspace and insufficient load-bearing capacity. With the widespread applications of CDPRs in industry and scientific research, its related technical system has made significant progress. This paper systematically reviews the latest research results of CDPRs in the fields of mechanical configuration design, mathematical modeling methods, and planning and control algorithms. This Section will systematically summarize the work of this paper and look forward to the future development of the CDPR field:

• In the field of structural design, the traditional fixed-base structure has a high space occupancy rate and limits the working range of EE, which has prompted many studies to integrate the manipulator configuration with CDPR technology to develop special cable-driven robots mainly used in the field of medical rehabilitation. This type of design not only combines the advantages of traditional rigid robots and traditional CDPRs, but also significantly improves the robot’s environmental adaptability and movement flexibility. However, the inherent flexibility and nonlinear characteristics of the cable make its control system design more difficult and reduce positioning accuracy.
  The MAS that has emerged in recent years has become an innovative solution to replace the traditional fixed base. This design not only gives CDPR the ability to move autonomously, but also can adapt to diverse and complex environments through dynamic topological structure adjustment. However, the variability of the topological structure changes the tension-feasible domain constraints of the system, resulting in a significant increase in the complexity of control system design. At the same time, due to the lack of base constraints, the system will generate new structural vibration modes and tension distribution constraint problems, bringing new challenges to control system design. Currently, related research is still in its infancy, but this direction has broad research prospects and holds significant academic value and application potential.
• In the mathematical modeling research of CDPRs, dynamic and kinematic modeling has been developed to a relatively complete level, and a large number of papers have deeply explored its tension constraints and workspace characteristics. However, the research on the integration of friction model and catenary model is relatively scarce, and has failed to effectively combine the latest progress in these two fields. In the field of friction research, the Lugre model and the Stribeck model are widely used because of their accurate characterization of friction characteristics. Unfortunately, the current research on CDPRs still generally adopts the outdated Coulomb–viscous friction model, which ignores the nonlinear characteristics (including the stick–slip phenomenon) under low-speed conditions. There is also a lag in the research on the CDPR catenary model. It is worth noting that J. Merlet has filled the gap in this field through a series of studies, and systematically demonstrated the advanced nature of the Irvine model based on singular point analysis and workspace analysis. In summary, although the current research on mathematical modeling of CDPRs has made significant progress, it is still necessary to integrate multidisciplinary cutting-edge results to further improve the accuracy of the model.
• In terms of planning and control, there are currently two main development directions: one is model predictive control, and the other is intelligent control methods. Model predictive control has been a hot topic in the control field in recent years. It has the advantages of multi-variable complex constraint-processing capabilities and dynamic optimization, making it particularly suitable for systems with multiple constraints and multiple nonlinearities such as CDPR. However, the rolling iteration algorithm and multi-step prediction link increase the amount of calculations and are highly dependent on model accuracy. To achieve closed-loop stability, additional terminal constraints need to be designed. Intelligent control methods are emerging control methods that are sought after by researchers for their powerful fitting and learning capabilities. At present, intelligent control researches on CDPR have covered kinematic modeling, error compensation, trajectory-tracking control, etc. [79]. It has good adaptability, fault tolerance and robustness, can handle high-dimensional constraints well, and does not require precise mathematical models. However, the shortcomings of this algorithm are very obvious: lack of a unified stability proof framework, poor interpretability, strong data dependence, lack of real-time performance, and high computational cost. In general, MPC and intelligent control are currently the research focuses in this field and have extremely high research value. Overcoming the shortcomings of these algorithms has therefore become a top priority.

Although researchers have made significant progress in the field of CDPRs, research on this type of robot requires further exploration. With the continuous optimization of structural design and the iterative upgrade of control algorithms, CDPRs are expected to achieve application expansion in multiple industry scenarios and become a core component of the industrial robot field.