# Comparative Study of Advanced Control Strategies of Cable-Driven Parallel Robots

Ahmed Nassim Bellila, Hélène Chanal, Olivier Company, Marc Gouttefarde, Chedli Belhassen Bouzgarrou

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Ahmed Nassim Bellila, Hélène Chanal, Olivier Company, Marc Gouttefarde, Chedli Belhassen Bouzgarrou. Comparative Study of Advanced Control Strategies of Cable-Driven Parallel Robots. CFM 2025 - 26e Congrès Français de Mécanique, Laboratoire d'Etude des Microstructures et de Mécanique des Matériaux (LEM3 - UMR CNRS 7239), Aug 2025, Metz, France. hal-05312918

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# Comparative Study of Advanced Control Strategies for Cable-Driven Parallel Robots

Ahmed Nassim Bellila<sup>a, b</sup>, Hélène Chanal<sup>a</sup>, Olivier Company<sup>b</sup>, Marc Gouttefarde<sup>b</sup>, Chedli Bouzgarrou<sup>a</sup>

a. Université Clermont Auvergne, Clermont Auvergne INP, CNRS, Institut Pascal, F-63000, France,

b. LIRMM, Univ Montpellier, CNRS, Montpellier, France

ahmed.bellila@lirmm.fr

## Abstract :

Cable-Driven Parallel Robots (CDPRs) offer large workspaces and high payload capacities, but their control remains challenging due to cable elasticity, unidirectional actuation, and model uncertainties. This paper presents a comparative study of three control strategies—Computed Torque Control (CTC), linear Model Predictive Control (MPC), and Sliding Mode Control (SMC)—applied to a 6-DoF CDPR with elastic cable. Each method is evaluated in simulation on the HRPcable CDPR, considering tracking accuracy, robustness, computational cost, and implementation complexity. Particular attention is given to the integration of tension distribution and the controllers ability to handle singularities and tension feasibility constraints. Results highlight trade-offs between precision, robustness, and real-time feasibility, offering guidance for controller selection in practical CDPR applications.

**Key words:** Cable-driven parallel robot, Control strategies, Trajectory Tracking.

## 1 Introduction

In recent years, the control of CDPRs has evolved to address challenges such as cable elasticity, dynamic uncertainties, and actuator constraints. The work in [1] proposes a control strategy based on singular perturbation theory to explicitly address cable elasticity in CDPRs. It combines a slow dynamics controller with a fast corrective term to ensure overall stability and improved tracking accuracy. Simulations on a suspended cable robot highlight the importance of modeling elasticity, showing that neglecting it may lead to instability and large positioning errors.

SMC has also been widely applied in CDPRs for its robustness to uncertainties and finite-time convergence. Its relative simplicity compared to other nonlinear control methods has made it a popular choice for both simulation and experimental studies. El Ghazaly et al in [5] developed an adaptive version of SMC. Oh and Agrawal [3] applied SMC to estimate feasible workspaces, using the control theory introduced in [4]. Further improvements led to advanced formulations such as terminal SMC [5] and the super-twisting algorithm [6], which enhance robustness and reduce steady-state errors. However, chattering remains a known limitation of SMC, and while various smoothing and saturation techniques have been proposed, it still appears in recent results [6].

Regarding MPC, only a few studies have applied it to CDPRs [7, 8, 9, 10]. These works primarily focus on tension-based control to ensure positive and feasible cable tensions during motion planning
and guaranteeing that the Mobile Platform (MP) remains within the robot feasible wrench workspace. MPC has also been used to generate optimal dynamic trajectories [11].

Building on these contributions, the present work provides a comparative study of three control strategies—CTC, MPC and SMC—applied to a 6-DoF CDPR with elastic cable. Their performance is evaluated in simulation with respect to tracking accuracy, robustness, constraint handling, and implementation feasibility.

The proposed strategies are evaluated in simulation on the HRPcable robot, a 6-DoF CDPR actuated by eight cables. This paper is organised as follows. Section 2 provides an overview of the CDPR modeling. Sections 3, 4, and 5 detail the CTC, MPC, and SMC strategies, respectively. Section 6 presents the simulation results and discussion, while Section 7 concludes the paper.

## 2 CDPR Modeling Overview

This section provides a brief overview of the kinematic and dynamic models used for the control of the CDPR MP. The pose of the MP is described by position vector **p** and orientation vector **ψ**, leading to the full pose $\mathbf{x} = [\mathbf{p}^T \ \mathbf{\psi}^T]^T$. The Figure 1 shows the main components of CDPR and kinematic notations.

<figure><img src="image_2.png" /><figcaption>Figure 1: CDPR components and kinematic notations.</figcaption></figure>

The Inverse Kinematics (IK) relates the MP pose to the cable lengths, while the cable longitudinal elasticity in each cable is modeled as a linear spring, proportional to the elongation from its rest length. Cable elasticity is taken into account, but cable mass is neglected. The MP dynamics are governed by the following equation:

$$M\ddot{x} + C(x, \dot{x})\dot{x} = G + Wt \quad (1)$$

where **M** is the mass matrix of the MP, $C(x, \dot{x})\dot{x}$ represents Coriolis and centrifugal effects, **G** is the gravitational vector and **Wt** represents the wrenches exerted by the cables on the MP, with **t** being the vector of cable tensions. A simplified model of the winch dynamics is also included, in which the motor inertia is explicitly taken into account. The model relates the motor torque to the cable force through the dynamics of the motor and winch drum, enabling the closed-loop computation of cable tensions based on motor commands and MP motion. The detailed dynamic model serves as the basis for the control
design presented in the following sections.

## 3 Computed Torque Control

This section introduces a model-based feed-forward control strategy, often referred to as Computed Torque Control (CTC). The objective is to generate a desired wrench $\mathbf{f}_{d,ff}$ that drives the MP along a predefined trajectory using the dynamic model of the CDPR. Assuming perfect model knowledge and negligible external disturbances, the required wrench is computed from the motion equation of the MP:

$$ \mathbf{f}_{d,ff} = \mathbf{M}(\mathbf{x})\ddot{\mathbf{x}}_d + \mathbf{C}(\mathbf{x}, \dot{\mathbf{x}})\dot{\mathbf{x}}_d - \mathbf{G}(\mathbf{x}) \quad (2) $$

This feed-forward term compensates for inertia, Coriolis, and gravitational effects based on the desired motion $\mathbf{x}_d$. The resulting desired wrench $\mathbf{f}_{d,ff}$ is mapped into a set of cable tensions by solving a Tension Distribution Algorithm (TDA). This distribution ensures equilibrium and respects tension limits. Figure 2 illustrates the overall architecture of the CTC control scheme implemented in simulation. The reference trajectory provides the desired pose, velocity, and acceleration of the MP, denoted as $\mathbf{x}_d, \dot{\mathbf{x}}_d, \ddot{\mathbf{x}}_d$. The CTC block computes the desired wrench $\mathbf{f}_{d,ff}$ based on the CDPR dynamic model. This wrench is then mapped into a set of feasible cable tensions $t$ using the TDA. In parallel, the desired pose $\mathbf{x}_d$ is converted into desired cable lengths $l_d$ through IK, and then into desired winch angles $\mathbf{q}_d$, considering cable elasticity effects. A PID controller is used to generate the motor torques $\tau_m$ required to track $\mathbf{q}_d$, based on the simulated winch positions $\mathbf{q}$. The resulting motor actions are applied to the CDPR physical model, which outputs the actual pose $\mathbf{x}$ of the moving platform.

<figure><img src="image_3.png" /><figcaption>Figure 2: Block diagram of the CTC strategy.</figcaption></figure>

## 4 Model Predictive Control

MPC is a control strategy that is particularly well-suited for systems with constraints. In the context of CDPRs, MPC is advantageous due to its ability to integrate the dynamics of the system with physical limitations such as unidirectional cable tensions and bounds on actuator torques. It offers a predictive framework that enables anticipatory control actions and ensures constraint satisfaction at all times. The control architecture corresponding to the MPC strategy is summarized in the block diagram shown in Figure 3.

### 4.1 Prediction Model and State Propagation

The linear MPC strategy as introduced by [12] relies on a discrete-time dynamic model of the CDPR obtained through Euler integration. The state vector is defined as $\mathbf{y}_k = [\mathbf{x}_k^T \ \dot{\mathbf{x}}_k^T]^T$, and the control
<figure><img src="image_4.png" /><figcaption>Figure 3: Block diagram of the MPC strategy.</figcaption></figure>

input is the vector of cable tensions $t_k \in \mathbb{R}^m$, where $m$ is the number of cables. The evolution of the system is expressed as:

$$y_{k+1} = Ay_k + Bt_k + v \quad (3)$$

This model is applied recursively over a prediction horizon $h_p$ to estimate the future trajectory of the robot based on candidate tension inputs.

## 4.2 Cost Function Design

The controller computes an optimal sequence of future cable tensions over a control horizon $h_c \le h_p$ by minimizing a cost function that captures:

* **Trajectory tracking error:** penalizing deviations from the desired trajectory.

* **Control effort:** penalizing large cable tension magnitudes.

* **Tension variation:** penalizing rapid changes in cable tensions.

The overall quadratic cost function is written as:

$$J = \|\gamma_d - \gamma(y, u)\|_{\mathbf{K}_{\gamma}}^2 + \|u\|_{\mathbf{K}_{u}}^2 + \|\Delta u\|_{\mathbf{K}_{D}}^2 \quad (4)$$

$$\left\{ \begin{array}{l} \mathbf{u}^* = \underset{\mathbf{u}}{\operatorname{argmin}} \quad J(\mathbf{u}) \\ \text{subject to} \quad \mathbf{u}_{\min} \le \mathbf{u} \le \mathbf{u}_{\max} \end{array} \right. \quad (5)$$

where $\gamma_d$ is the desired trajectory over $h_p$, $\gamma(y, u)$ the predictions over $h_p$, $\mathbf{u}$ is the stacked vector of future tensions over the horizon, and the weighting matrices $\mathbf{K}_\gamma$, $\mathbf{K}_u$, and $\mathbf{K}_D$ allow tuning of the controller behavior. At each sampling time, the MPC controller computes the optimal sequence of future tensions $\mathbf{u}^*$ by minimizing the cost function $J$ under system dynamics and tension constraints, as shown by problem (5). Only the first element of this optimal sequence is applied to the system (receding horizon principle). The computational cost of MPC is influenced by the prediction horizon $h_p$, the control horizon $h_c$ and the sampling time $\Delta t$. Longer horizons typically improve performance but increase the size of the optimization problem. In this work, the values of $h_p$ and $h_c$ are carefully tuned to achieve a trade-off between accuracy and real-time computation.
## 4.3 Constraints Handling and Tension Distribution

A key strength of MPC is its intrinsic ability to handle system constraints. The computed cable tensions are optimized under constraints such as:

$$ t_{\min} \le t_{k+i} \le t_{\max}, \quad \forall i \in \{0, \dots, h_c - 1\} \qquad (6) $$

These constraints ensure the physical feasibility of the cable tensions, including their unilateral nature (tensions must remain positive $t_{\min} \ge 0$). Unlike other control strategies, MPC does not require a separate tension distribution algorithm. The cable tensions are computed directly as part of the optimization process, guaranteeing: feasible tensions throughout the motion, satisfaction of static and dynamic equilibrium and consistent behavior even near workspace boundaries or singularities.

## 5 Sliding Mode Control

SMC is a nonlinear and robust control strategy designed to ensure convergence of tracking errors despite model uncertainties and external disturbances. Its principle relies on driving the system state to a predefined surface in the error space, called the sliding surface, and enforcing motion along this surface through a discontinuous control law. Following standard SMC formulations [13], the sliding surface is chosen to represent a weighted combination of the position and velocity errors, ensuring that once the system reaches the surface, the errors decay exponentially to zero. Thus, the sliding surface is defined as:

$$ s = e_2 + \Lambda e_1 \qquad (7) $$

where $e_1$ and $e_2$ are the position and velocity tracking errors, and $\Lambda$ is a positive definite gain matrix. The control law is then designed to stabilize the dynamics of $s$, ensuring that the system remains on the sliding surface once reached. To improve robustness, a switching term $K_s$ sat($s$) is introduced:

$$ f_d = M(\ddot{x}_d + \Lambda e_2 + K_s + K_s \text{sat}(s)) + C(x, \dot{x})\dot{x} - G \qquad (8) $$

Here, sat($\cdot$) is a smooth approximation of the sign function used to reduce chattering, and the matrices $K$ and $K_s$ are positive definite gain matrices. The stability of the closed-loop system is established using a Lyapunov function based on the sliding surface. The resulting control law guarantees convergence of the trajectory tracking error, with robustness to bounded uncertainties. Although SMC does not explicitly handle input constraints such as tension limits, its simplicity, low computational cost, and disturbance rejection make it an attractive candidate for CDPR control in scenarios where robustness is prioritized. The structure of the implemented SMC strategy, including the computation of the desired wrench and its integration into the control loop, is summarized in the block diagram shown in Figure 4.
<figure><img src="image_5.png" /><figcaption>Figure 4: Block diagram of the SMC strategy.</figcaption></figure>

# 6 Results and discussion

The study is conducted in simulation using MATLAB/Simulink, where the HRPcable system is accurately modeled. HRPcable is an eight-cable 6 degree-of-freedom CDPR. Its base dimensions are 8 m × 4 m × 3 m. Its MP is modeled as a cube of side length 0.5 m with a mass of $m_p = 25$ kg. The control of cable tensions is implemented in simulation. The simulation integrates realistic parameters, including cable elasticity, CDPR dynamics and external disturbances, to ensure a fair assessment of each control method. The Figure 5 illustrates the developed simulator in MATLAB/Simulink.

<figure><img src="image_6.png" /><figcaption>Figure 5: HRPcable prototype [9] and illustration of the developed MATLAB/Simulink simulator that reproduces its behavior.</figcaption></figure>

The simulations are conducted with tuned parameters, using error and trial, for each control strategy. For MPC, the weighting matrices used in the cost function are: $K_γ$ as a block-diagonal matrix with high weights on position errors ($\sim 10^9$) and moderate weights on velocity errors ($\sim 10^{-2}$), $K_u = 10^{-6} I$ to penalize control effort, and $K_D = 10^{-9}$ to avoid penalizing tension variations in this configuration. The prediction and control horizons are respectively set to $h_p = h_c = 10$ ms. For SMC, the sliding surface is defined using $\Lambda = 0.1 I$, and the control law includes gains $K = I$ for proportional correction and $K_s = 0.01 I$ for the switching term. These values were selected to balance tracking performance and robustness while mitigating chattering.

To evaluate and compare the tracking performance of the proposed control strategies, two metrics are considered: the Root Mean Square Error (RMSE) of the trajectory tracking error norm $\|e(t)\|_2$, and the maximum absolute tracking error on each individual component of the moving platform position vector.
These metrics are presented in Figure 6, where the first subplot displays the total RMSE norm for each control method, and the second subplot highlights the maximum tracking error observed across the six components of the MP pose. The simulation scenario consists of tracking a 6-DoF reference trajectory generated using a fifth-degree polynomial interpolation, ensuring smooth transitions with continuous velocity and acceleration profiles. No singularity or feasibility issues are encountered during the motion. However, to evaluate the robustness of the controllers, realistic modeling mismatches are deliberately introduced. The mass of the MP is underestimated in the control model with $\hat{m} = 25$ kg, whereas the true value used in simulation is $m = 25.71$ kg. Additionally, a 15% error is introduced in the identification of cable stiffness parameters to simulate inaccurate elasticity estimation.

<figure><img src="image_7.png" /><figcaption>Figure 6: Comparison of tracking performance across control strategies. The left plot shows the RMSE of the trajectory norm $||e(t)||_2$, while the right plot shows the maximum absolute error per position component.</figcaption></figure>

Another important distinction between the evaluated strategies lies in how they handle cable tensions. In the case of MPC, the desired cable tensions are directly computed as control inputs by solving a constrained optimization problem, thereby inherently ensuring feasibility and equilibrium. In contrast, SMC and CTC Control both generate desired wrenches to be applied on the MP. In these cases, a separate tension distribution algorithm from the literature [14] is employed to compute a feasible set of cable tensions that satisfies the desired wrench while respecting tension constraints.

The results in Figure 6 show that all three controllers successfully maintain trajectory tracking despite the presence of these modeling uncertainties. MPC achieves the highest RMSE norm $(2.69 \times 10^{-3}\text{m})$, mainly due to model mismatches and the limitations of a finite prediction horizon, but its peak error remains acceptable, particularly in the z-axis. SMC, while simpler and more robust to uncertainties, yields the lowest overall RMSE norm $(3.3 \times 10^{-4}\text{m})$ and maintains relatively low maximum errors across all axes. CTC Control sits between the two in terms of performance, offering better RMSE than MPC, but with slightly higher maximum deviation than SMC along the vertical axis.

To complement the RMSE and maximum error analysis, Figure 7 illustrates the time evolution of the translational tracking error norm $e(t)$ for each control strategy over the full trajectory duration. This dynamic representation provides additional insight into the transient and steady-state behavior of the
<figure><img src="image_8.png" /><figcaption>Figure 7: Translational trajectory tracking errors over time for SMC, MPC, and CTC. Each subplot displays the position errors $(x, y, z)$ for a given control strategy.</figcaption></figure>

controllers. As shown in the figure, the SMC strategy exhibits a fast convergence and stable tracking with minimal error fluctuations along most of the trajectory. However, small initial oscillations are observed at the beginning of the motion, which can be attributed to the typical chattering effect associated with SMC, especially before the system fully reaches the sliding surface. The computed torque controller displays slightly higher peaks and more oscillatory behavior, especially during acceleration and deceleration phases, reflecting its sensitivity to modeling mismatches. The MPC controller, although showing larger errors during certain segments of the trajectory, maintains global stability and avoids abrupt deviations thanks to its anticipative optimization and constraint handling capabilities. These observations reinforce the quantitative results presented in the bar plots, confirming the robustness of SMC and the constraint compliance advantage of MPC.

A second simulation scenario is designed to assess how each control strategy handles workspace and tension feasibility limits. The same conditions as the previous test are reused, including model mismatches on mass and cable stiffness, but with a reduced maximum cable tension of $t_{max} = 400$ N, compared to 2000 N in the first scenario. Under these tighter constraints, the desired trajectory partially exceeds the robot feasible wrench workspace.

As expected, the MPC strategy proves effective at handling this situation, since it directly incorporates cable tension constraints into its optimization problem. Even when the desired wrench becomes unfeasible, MPC maintains stable behavior by computing admissible tensions that bring the MP as close as possible to the reference pose, while minimizing the tracking error. In contrast, both SMC and CTC rely on a pre-processing tension distribution algorithm to convert a desired wrench into feasible cable tensions. When the generated wrench exceeds the system capabilities (i.e unfeasible wrench), these controllers are unable to adjust, resulting in abrupt loss of control or tracking failure. In Figure 8, the tension distribution algorithm crashes and starts generating negative cable tensions, which are physically infeasible in CDPRs. While this phenomenon is observed in simulation, its implications in a real-world experimental context would be far more critical. CDPRs are often designed to operate at relatively high
<figure><img src="image_9.png" /><figcaption>Figure 8: Cable tension profiles for each control strategy under reduced tension limit ($t_{max}$ = 400 N).</figcaption></figure>

speeds and may handle significant payloads. An abrupt failure in tension computation, such as the gen-
eration of unfeasible or negative tensions, could lead to a sudden stop or uncontrolled motion of the
end-effector. This, in turn, poses a risk not only to the equipment itself but also to surrounding infras-
tructure and operator safety. The inability to anticipate or compensate for such situations in real time
underscores the importance of control strategies like MPC that inherently guarantee tension feasibil-
ity throughout the trajectory and ensure safe and continuous motion even near the limits of the robot
workspace.

7 Conclusion

This paper presents a comparative study of three advanced control strategies—MPC, CTC, and SMC—
applied to a 6-DoF CDPR. Each method is implemented in simulation on the HRPcable CDPR and
assessed based on tracking accuracy, robustness to model uncertainties, and ability to handle actuator
constraints. The results show that all three controllers succeed in maintaining stable motion and ac-
ceptable tracking despite the presence of modeling errors. SMC demonstrates excellent robustness to
modeling errors and yields the best RMSE, making it a strong candidate for scenarios with uncertain-
ties. CTC offers a good balance between simplicity and performance but shows limitations when the
required control actions approach the tension limits of the CDPR. MPC, while slightly less precise in
terms of RMSE, provides superior reliability in constrained conditions due to its predictive nature and
built-in constraint handling. Moreover, only MPC inherently guarantees tension feasibility throughout
the trajectory without requiring pre-processing. This is particularly valuable in safety-critical applica-
tions or near the limits of the robot workspace. The comparative analysis confirms that the selection
of a control strategy must consider the specific operational requirements, including robustness, imple-
mentation complexity, and proximity to physical constraints. Future work will aim to experimentally
validate these findings and investigate hybrid approaches that combine the robustness of SMC with the
constraint management capabilities of MPC.
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