# Experimental validation of the analytic-centre based tension distribution algorithm for cable driven parallel robots

Daria Berretta, Vincenzo Di Paola, Stéphane Caro, Matteo Zoppi

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Daria Berretta, Vincenzo Di Paola, Stéphane Caro, Matteo Zoppi. Experimental validation of the analytic-centre based tension distribution algorithm for cable driven parallel robots. Meccanica, 2025, 60 (9), pp.3069-3081. 10.1007/s11012-025-02044-9. hal-05390730

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# Experimental validation of the analytic-centre based tension distribution algorithm for cable driven parallel robots

Daria Berretta · Vincenzo Di Paola ·

Stéphane Caro · Matteo Zoppi

Received: 6 November 2024 / Accepted: 19 August 2025

© The Author(s) 2025

**Abstract** Cable-Driven Parallel Robots (CDPRs) offer several advantages over traditional parallel robots. Guiding them accurately and safely requires fast Tension Distribution Algorithms (TDAs) able to work in real-time. This paper proposes the experimental validation of our previous TDA namely, the Analytic Centre. The contribution consists of comparing the simulated results with the measured tension profiles to infer the validity of the theoretical outcome. In other words, the idea is to find evidence of the predictions made in simulations. Experiments confirm simulated results supporting the theoretical advantages of the presented Analytic Centre, proving its usefulness.

**Keywords** Cable-driven robots · Tension distribution algorithms · Analytic centre · Experimental validation

## 1 Introduction

A Cable-Driven Parallel Robot (CDPR) uses cables to control the pose of a rigid platform. Compared to conventional parallel robots with rigid links, CDPRs offer greater configuration flexibility and a larger workspace

Recently, the industrialization of robotics has increased the demand for human-robot collaboration, which in turn requires higher precision and greater operational safety [12]. Although cable flexibility makes them compliant, they constitute a risk when a failure occurs as underlined in Di Paola et al. [4].

In this context, the choice of cable tensions play a key role when controlling the platform's motion. Indeed, tensions must be carefully chosen to ensure accurate movements and safety. However, their choice is not always unique. To solve this problem, several TDAs have been developed to determine the cable tensions while considering practical constraints to fulfil a wide range of tasks. Pott et al. [18] presented an algorithm determining feasible tension distributions for CDPRs in Closed-Form (CF), subsequently

<footnote>D. Berretta · V. D. Paola · M. Zoppi
Dipartimento di Ingegneria Meccanica, Enegergetica, Gestionale e dei Trasporti, Università degli Studi di Genova, 16143 Genoa, Italy</footnote>

<footnote>D. Berretta · V. D. Paola (✉) · S. Caro
Nantes Université, École Centrale Nantes, CNRS, LS2N, UMR 6004, 1, Rue de la Noe, 44321 Nantes, France
e-mail: vincenzo.dipaola@edu.unige.it</footnote>
improved in Pott [17] to enlarge the coverage of the Wrench Feasible Workspace (WFW).<fnref n="1" /> Borgstrom et al. [1] used Linear Programming (LP), guaranteeing “Optimally Safe” (OS) tension distributions. In Oh and Agrawal [15] Quadratic Programming (QP) is used. This latter was taken up by Bruckmann et al. [3] for CDPR with maximum Degree of Redundancy (DoR)<fnref n="2" /> equal to 2. Many other methods exist as shown in Pott [17] and Gouttefarde et al. [11] however we focus on the commonly used one in the literature.

This paper aims to implement and compare several aspects of various TDAs using a real prototype. The objective is to establish the soundness of the theoretical results presented in our previous paper, Di Paola et al. [7].

This work is organised as follows, Sect. 2 focuses on the system modelling and recalls the main equations for the dynamics of a generic Cable-Driven Parallel Robot (CDPR). Section 3 provides an overview of existing tension distribution algorithms. Section 4 presents the experimental framework, the prototype, and the methodology used to compare the performance of different TDAs. Finally, Sect. 5 summarises the main results and suggests potential future works.

## 2 System modeling

This section recalls the main equations to model the dynamics of a generic CDPR.

Typically, the inverse geometric model (Sect. 2.1) describes the relationship between the pose of the Moving Platform (MP) and the cable lengths. The inverse kinematic model (Sect. 2.2) relates the desired pose and velocities of the MP to the required cable lengths and their derivatives. Together with the dynamic model (Sect. 2.3), these relationships describe the interaction between the wrenches acting on the load and the cable lengths.

<figure><img src="image_2.png" /><figcaption>**Fig. 1** Generic loop describing the geometric parameters of a CDPR</figcaption></figure>

### 2.1 Inverse geometric model

Figure 1 shows a generic loop describing the geometric parameters involved when describing a CDPR.

Two frames are typically used, a base frame $F_b = (O_b, x_b, y_b, z_b)$ and a platform frame denoted as $F_p = (O_p, x_p, y_p, z_p)$. The output points $\mathbf{A}_i$ of the cables are located on pulleys fixed to the rigid frame, while the anchor points $\mathbf{B}_i$ are located on the moving platform. The unit vector $\mathbf{u}_i$ describes the direction of the cable between points $\mathbf{A}_i$ and points $\mathbf{B}_i$.

The equation describing the loop associated with each cable is expressed by

$$^{b}\mathbf{I}_{i} = l_{i}^{b}\mathbf{u}_{i} = {}^{b}\overline{\mathbf{A}_{i}\mathbf{B}_{i}} = {}^{b}\mathbf{R}_{p}{}^{p}\mathbf{b}_{i} + {}^{b}\mathbf{p} - {}^{b}\mathbf{a}_{i} = {}^{b}\mathbf{b}_{i} + {}^{b}\mathbf{p} - {}^{b}\mathbf{a}_{i}, \quad (1)$$

where $i \in [1, ..., m]$, $m$ is the number of cables, ${}^b\mathbf{I}_i$ is the vector describing the $i$-th cable, ${}^b\mathbf{a}_i$ is the coordinate vector of the output point $\mathbf{A}_i$ expressed in the $F_b$ frame, ${}^p\mathbf{b}_i$ is the coordinate vector of the anchor point $\mathbf{B}_i$ expressed in frame $F_p$, ${}^b\mathbf{p}$ is the vector going from $O_b$ to $O_p$ expressed in frame $F_b$ and ${}^b\mathbf{R}_p$ is the rotation matrix from frames $F_b$ and $F_p$.

Hence, the unit vector of the $i$-th cable is given by

$$^{b}\mathbf{u}_{i} = \frac{{}^{b}\mathbf{I}_{i}}{l_{i}}, \quad (2)$$

where $l_i$ is the length of the $i$-th cable defined as $l_i = ||\mathbf{I}_i||_2$.

In this paper, the effect of pulley geometry is taken into account within the model, with a detailed explanation provided in Metillon [14]. In other words, the

<footnote n="1">¹ The WFW for a CDPR refers to the set of all external wrenches (forces and moments) the robot can exert on its end-effector while maintaining a stable and feasible configuration.</footnote>

<footnote n="2">² Degree of Redundancy is defined as $DoR = m - n$ where $m$ is the number of cables and $n$ the number of Degrees of Freedom (DoF).</footnote>
cable length between the pulley entry point, $\mathbf{A}_i$, and the MP anchor point, $\mathbf{B}_i$, consider also the portion of the cable that is wrapped on the pulley.

The corresponding joint position is obtained from the total cable length $\mathbf{l}_f$ and the winch radius $r_w$ as

$$ \mathbf{q} = \frac{\mathbf{l}_f}{r_w}. \qquad (3) $$

## 2.2 Inverse kinematic model

The desired cable velocity vector $\dot{\mathbf{l}}_d$ is obtained using the forward Jacobian matrix $\mathbf{A}$ as

$$ \mathbf{A}\mathbf{t}_d = \dot{\mathbf{l}}_d, \qquad (4) $$

where $\mathbf{t}_d = [\mathbf{v}_d \omega_d]$ is the platform twist, with $\mathbf{v}_d$ and $\omega_d$ representing the desired translational and rotational velocity, respectively. The matrix $\mathbf{A}$ is

$$ \mathbf{A} = \begin{bmatrix} {}^b\mathbf{u}'_1 & {}^b\mathbf{b}'_1 \times {}^b\mathbf{u}'_1 \\ \vdots & \vdots \\ {}^b\mathbf{u}'_i & {}^b\mathbf{b}'_i \times {}^b\mathbf{u}'_i \\ \vdots & \vdots \\ {}^b\mathbf{u}'_m & {}^b\mathbf{b}'_m \times {}^b\mathbf{u}'_m \end{bmatrix}_{m \times 6}. \qquad (5) $$

In this matrix the vector $\mathbf{u}_i$ has been substituted with $\mathbf{u}'_i$ as we apply the correction for the pulley model mentioned before. Then the desired actuator speed vector $\dot{\mathbf{q}}$ is obtained as

$$ \dot{\mathbf{q}}_d = \frac{\dot{\mathbf{l}}_d}{r_w}. \qquad (6) $$

## 2.3 Static equilibrium and direct dynamic model

The platform is guided by applying a set of tensions along the cables. The equation describing its equilibrium is

$$ \mathbf{W}\boldsymbol{\tau} + \mathbf{w}_e = 0, \qquad (7) $$

where $\mathbf{W}$ is the wrench matrix

$$ \begin{bmatrix} \mathbf{u}'_1 & \mathbf{u}'_2 & \dots & \mathbf{u}'_m \\ \mathbf{b}_1 \times \mathbf{u}'_1 & \mathbf{b}_2 \times \mathbf{u}'_2 & \dots & \mathbf{b}_m \times \mathbf{u}'_m \end{bmatrix}_{6 \times m}, \qquad (8) $$

in which “X” defines the cross product, $\mathbf{w}_e$ the external wrench and $\boldsymbol{\tau}$ the cable tensions vector. This equation shows the relationship between $\boldsymbol{\tau}$ and the external

wrench $\mathbf{w}_e$ acting on the platform. For redundant CDPRs ($DoR \ge 1$) an infinite number of solutions for $\boldsymbol{\tau}$ exist. The set of solutions is defined as

$$ \boldsymbol{\Sigma} = \{ \boldsymbol{\tau} | \mathbf{W}\boldsymbol{\tau} + \mathbf{w}_e = \mathbf{0} \}. \qquad (9) $$

Furthermore, cable tension limits have to be considered hence, the following $m$-dimensional convex hypercube has to be included

$$ \boldsymbol{\Pi} = \{ \boldsymbol{\tau} | 0 < \underline{\boldsymbol{\tau}} \le \boldsymbol{\tau} \le \bar{\boldsymbol{\tau}} \}, \qquad (10) $$

where $\underline{\boldsymbol{\tau}}$ and $\bar{\boldsymbol{\tau}}$ are the positive tension limits.

Therefore, the set of feasible solutions $\boldsymbol{\Gamma}$ that comply with both Eqs. (9)–(10) is

$$ \boldsymbol{\Gamma} = \boldsymbol{\Sigma} \cap \boldsymbol{\Pi}. \qquad (11) $$

# 3 Existing tension distribution algorithms

Over the years, as highlighted in Sect. 1, several methods have been developed for computing cable tensions in CDPRs. Each method produces a unique tension distribution with distinct computational features. Broadly, TDAs can be classified into two groups: Analytical Methods (AM) which provide closed-form solutions to compute cable tensions, and Numerical Optimization Methods (NOP) which exploit optimization algorithms.

## 3.1 Comparison of various algorithms

This section provides both an overview of the existing methods and their peculiarities based on the work of Pott [17] with the addition of two relevant properties.

The first property is the Robustness Index (RI), which represents the distance of the tension solution w.r.t. the closest facet of $\boldsymbol{\Pi}$ (Eq. 10), and is defined as

$$ d(\boldsymbol{\tau}) = \min \{ \bar{\boldsymbol{\tau}} - \boldsymbol{\tau}_1, \boldsymbol{\tau}_1 - \underline{\boldsymbol{\tau}}, \dots, \bar{\boldsymbol{\tau}} - \boldsymbol{\tau}_m, \boldsymbol{\tau}_m - \underline{\boldsymbol{\tau}} \} \qquad (12) $$

where $\boldsymbol{\tau}_m$ is the average between the limits $\underline{\boldsymbol{\tau}}, \bar{\boldsymbol{\tau}}$.

A high RI value indicates that the selected tensions are distant from tension limits, providing a robust solution. Conversely, a low RI suggests that tensions are close to the limits which can lead to undesired vibrations and oscillations of the platform. The second property concerns the algorithm's ability to
<table><caption>**Table 1** Summary table of the implemented tension distribution algorithms and their properties</caption><thead><tr><th>Method</th><th>Real-time capable</th><th>Force niveau</th><th>Workspace coverage</th><th>Continuity</th><th>Max. redundancy</th><th>Computational speed</th><th>Robustness index</th></tr></thead><tbody><tr><td>Barycentric (AM)</td><td>Yes</td><td>Mi</td><td>Yes</td><td>Yes</td><td>r = 2</td><td>Fast</td><td>High</td></tr><tr><td>Improved closed-form (AM)</td><td>Yes</td><td>Any</td><td>No</td><td>Yes</td><td>Any</td><td>Fast</td><td>High</td></tr><tr><td>Linear programming (NOP)</td><td>No</td><td>Any</td><td>Yes</td><td>No</td><td>Any</td><td>Fast</td><td>High</td></tr><tr><td>Quadratic programming (NOP)</td><td>Yes</td><td>Hi, lo</td><td>Yes</td><td>Yes</td><td>Any</td><td>Medium</td><td>Low</td></tr><tr><td>Analytic center (NOP)</td><td>Yes</td><td>Mi</td><td>Yes</td><td>Yes</td><td>Any</td><td>Fast</td><td>Medium-high</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>**Fig. 2** Barycentric tension profiles</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Fig. 3** Closed form tension profiles</figcaption></figure>

include nonlinear constraints while solving the optimisation problem.

To practically grasp the differences between the various algorithms consider the following example; taken from Di Paola et al. [7]. This example examines a 4-cable CDPR with a point mass payload executing a circular trajectory of radius 0.5 m in 10 s. The initial and final velocities and accelerations are zero and the weighs of point mass is 20 kg. The minimum tensions are set to $\tau = 20$ N while the maximum tensions $\bar{\tau} = 400$ N.

The results show that all TDAs find solutions within the imposed tension limits.

Both the Barycentric Method (BARY) Fig. 2 and the Analytic Center (AC) are similar in tension profiles and RI. However, the use of the barycentric is limited to its DoR $\le 2$, making it unsuitable for more complex applications [5, 6].

The improved Closed Form (ICF) Fig. 3 provides a continuous solution coming with a lower Robustness

<figure><img src="image_5.png" /><figcaption>**Fig. 4** Linear programming tension profiles</figcaption></figure>

Index if compared with the Analytic Centre.
<figure><img src="image_6.png" /><figcaption>**Fig. 5** Quadratic programming tension profiles</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 6** Analytic centre tension profiles</figcaption></figure>

Additionally, as shown in the Table 1, it does not ensure full workspace coverage.<fnref n="3" />

Linear Programming Fig. 4 has the highest Robustness Index, but it may generate discontinuities in the tension profile, as shown in this case, thus potentially causing crashing or stopping scenarios in the robot.

The Quadratic Programming Fig. 5 minimizes the energy required by the actuators taking lower tension limits as reference. Moreover, this minimization results in the lowest RI and reduced stiffness of the

<figure><img src="image_8.png" /><figcaption>**Fig. 7** RIs trends for each TDAs</figcaption></figure>

robot potentially leading to vibrations and oscillations of the platform.

The Analytic Centre Fig. 6 is the only method that simultaneously preserves all mentioned qualities with the possibilities to include non-linear constraints as well (Fig. 7). A further example of its flexibility and applicability is shown in Dona et al. [8].

To provide a more compact and intuitive comparison among the tested TDAs, a radar plot is presented in Fig. 8. The plot summarises key performance metrics described in Table 1. Each axis represents a binary or normalised performance criterion, and each polygon corresponds to a TDA. This visualisation highlights the trade-off discussed above and illustrates that the Analytic Centre is the only method that consistently satisfies all desired criteria.

## 4 Experimental framework

Since the aim of this study is to propose a validation for the properties of the Analytic Centre, the CRAFT prototype Fig. 9, based at École Centrale de Nantes, is introduced. This CDPR has *m* = 8 cables and *n* = 6 degrees of freedom thus providing a 2-DoR. The robot's frame measures 3.77 m in width, 4.35 m in depth, and 2.75 m in height.

Key components of this prototype are: eight PARKER SME60⁴ motors with gearbox reducers

<footnote n="3">³ The workspace coverage or coverage of the wrench feasible workspace (CWFW) represents the ability of the specific TDA to find a tension distribution balancing an external wrench inside the robot workspace.</footnote>

<footnote n="4">⁴ Parker [16]: https://ph.parker.com/fr/en/product-list/sme-series-low-inertia-brushless-servo-motors-encoder-feedback.</footnote>
Barycentric

<figure><img src="image_9.png" /><figcaption>(a)</figcaption></figure>

Improved Closed Form

<figure><img src="image_10.png" /><figcaption>(b)</figcaption></figure>

Analytic Centre

<figure><img src="image_11.png" /><figcaption>(e)</figcaption></figure>

Linear Programming

<figure><img src="image_12.png" /><figcaption>(c)</figcaption></figure>

Quadratic Programming

<figure><img src="image_13.png" /><figcaption>(d)</figcaption></figure>

**Fig. 8** This figure presents radar charts evaluating the implemented TDAs: **a** Barycentric, **b** Improved Closed Form, **c** Linear Programming, **d** Quadratic Programming, and **e** Analytic Centre. Each method is assessed based on the previously mentioned criteria:

Robust Index (RI), continuity of the solution, Workspace Coverage, Real-time capability, and support for Any Degree of Redundancy (DoR)

<figure><img src="image_14.png" /></figure>
<figure><img src="image_15.png" /><figcaption>**Fig. 9** CRAFT robot frame and platform</figcaption></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /><figcaption>**Fig. 10** CoGiRo (Control of Giant Robots) is the acronym for the fully suspended configuration of the CDPR hereby used (https://anr.fr/Projet-ANR-09-SEGI-0018)</figcaption></figure>

(ratio 8), mounted at the base corners of the frame. Pulleys and anchor points are located at the top corners. The cables, made of VECTRAN (0.7 mm diameter), keep the mobile platform suspended, as shown in Fig. 10. The platform is an aluminium parallellepiped measuring 0.28 × 0.28 × 0.20 m whose weight is 8.533 kg housing the sensor and many electronic components.

Additionally, eight cable tension sensors FUTEK FSH04097⁵ (Fig. 11) are installed, positioned between the end of the cables and their anchor points on the platform.

<figure><img src="image_18.png" /><figcaption>**Fig. 11** Tension sensors FUTEK FSH04097</figcaption></figure>

These sensors have a maximum tension capacity of 150 N, but operational safety limits on the CRAFT prototype are set between 0 N and 111 N. The control system is managed by a supervisory PC using the RTIlib library [9] to ensure real-time operation. Moreover, dSPACE handles the robot's control enabling two-way communication between the algorithm and the sensors. The robot operates at a frequency of 1 kHz, while data are sampled every 0.001 s. Hence, the supervisory PC collects and processes the data useful to command the electric motors (Fig. 12).

The performances of the TDAs are assessed through experiments following the methodology presented in Di Paola et al. [7]. Therefore, the CRAFT is demanded to execute 5 circular trajectory of radius 0.5 m centred within the workspace. The trajectory is designed to start and stop with zero velocity and acceleration, achieving a steady-state speed during the intermediate phase of motion. This trajectory was selected to minimize transient effects, particularly

<footnote n="5">⁵ Futek [10]: https://www.futek.com/store/load-cells/s-beam-load-cells/miniature-s-beam-jr-LSB205/FSH04097.</footnote>
<figure><img src="image_19.png" /><figcaption>**Fig. 12 Robot control architecture [14]**</figcaption></figure>

those influencing cable tensions and sensor read-
ings. By focusing on the platform's behavior during
the third circular path, we ensure that it is free from
dynamical effects of the initial acceleration and final
deceleration phases.

**5 Analytic centre validation**

To confirm the results proposed in Di Paola et al. [7]
several tests were executed with all TDAs. In par-
ticular, the circular trajectory was traced with differ-
ent end-time constraints to see, for example, whether
the inertial effects of the load have influence on the
measured data. Because of no significant relevance
of external disturbance is detected during the experi-
ment here, only the case where the end-time is fixed
to 60 s is reported. During the experiments, it was
observed that both the input torques and the measured
forces were affected by significant noise. To address
this issue, the collected data were filtered using a
moving average implemented via the MATLAB func-
tion "movmean" [13]. The validation is carried out by
comparing the measured and simulated results for all

TDAs, considering both the tension profiles and the
Robustness Index (Figs. 13, 14, 15, 16, 17, 18).

**5.1 The experiments**

Expected and measured tension profiles are reported
in the following Figures where all but one TDAs find
a solution to the tension distribution problem. Indeed,
the Linear Programming Fig. 15 does not accomplish
the task. This behavior is due to the discontinuities
generated by the LP solver. These latter cause the
CRAFT to stop for safety reasons and contemporar-
ily prove what expected in Di Paola et al. [7]. The
remaining methods and all the measured tension pro-
files fit the simulated ones allowing us to claim the
soundness of the theoretical results (Figs. 16, 17, 18).

Also the analysis of the robust indices confirms
what was stated in Di Paola et al. [7]. Because of
the fully suspended configuration of the MP the
Quadratic Programming is no longer the TDA with
the lowest robustness and all algorithms assume
similar values among themselves, see Figs. 19 and
20. When the platform is suspended (i.e. no cable
below the platform) the ability to optimise ten-
sions is reduced because the area of the feasible
BARY Method

<figure><img src="image_20.png" /><figcaption>**Fig. 13** Desired ($\tau_d$) and measured ($\tau_{mmean}$) tension for each cable using the BARY TDA; $\tau_{lmts_{rd}}$ represents the minimum and maximum tension value obtained with the relative method</figcaption></figure>

CF Method

<figure><img src="image_21.png" /><figcaption>**Fig. 14** Desired ($\tau_d$) and measured ($\tau_{mmean}$) tension for each cable using the CF TDA; $\tau_{lmts_{rd}}$ represents the minimum and maximum tension value obtained with the relative method</figcaption></figure>

polygon A shrinks, reducing the ability to generate wrenches on the platform. Observe that the LP has not been considered in this analysis because the

discontinuous solution previously provided in the experiment.
LP Method

<figure><img src="image_22.png" /><figcaption>**Fig. 15** Desired ($\tau_d$) and measured ($\tau_{m,mean}$) tension for each cable using the LP TDA; $\tau_{lmts,rd}$ represents the minimum and maximum tension value obtained with the relative method</figcaption></figure>

QP Method

<figure><img src="image_23.png" /><figcaption>**Fig. 16** Desired ($\tau_d$) and measured ($\tau_{m,mean}$) tension for each cable using the QP TDA; $\tau_{lmts,rd}$ represents the minimum and maximum tension value obtained with the relative method</figcaption></figure>

To quantitatively summarise the data reported in Figs. 19 and 20, the average value, for each TDA, of the robustness index is reported in Table 2.

The small differences between measured and expected values can be attributed to the choice of the simplified model where, for example, the

<figure><img src="image_24.png" /></figure>
AC Method

<figure><img src="image_25.png" /><figcaption>**Fig. 17** Desired ($\tau_d$) and measured ($\tau_{m,mean}$) tension for each cable using the AC TDA; $\tau_{lmts,red}$ represents the minimum and maximum tension value obtained with the relative method</figcaption></figure>

Robust Index

<figure><img src="image_26.png" /><figcaption>**Fig. 18** Desired and measured Robust Index for each TDA</figcaption></figure>

elasticity of the cables is neglected. In other words, although the model can be improved to reduce the difference with the actual values, the increase in

accuracy does not justify the effort in complicating the model.
<figure><img src="image_27.png" /><figcaption>Fig. 19 Boxplot of the desired Robust Index</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>Fig. 20 Boxplot of the measured Robust Index</figcaption></figure>

<table><caption>Table 2 Desired and measured average Robust indices</caption><thead><tr><th></th><th>Desired RI</th><th>Measured RI</th></tr></thead><tbody><tr><td>Barycentric</td><td>40.6016</td><td>42.2915</td></tr><tr><td>Improved closed form</td><td>41.2624</td><td>42.1171</td></tr><tr><td>Linear programming</td><td>-</td><td>-</td></tr><tr><td>Quadratic programming</td><td>41.1046</td><td>41.9785</td></tr><tr><td>Analytic centre</td><td>40.8050</td><td>41.7328</td></tr></tbody></table>

## 6 Conclusion

The first aims of this work was implementing all the TDAs and test them on the CRAFT prototype. The idea was to compare them based on previous simulated tension-profile results. Therefore, looking at the reported graphs and tables, it can be concluded that the Analytic-Centre outperforms classical TDAs. Indeed, it applies to any DoR generating continuous and smooth tension profiles while covering the entire WFW with possible inclusion of non-linear

constraints. Real-time capabilities was another intrinsic objective of this work.

These features make the proposed method particularly suited for broader deployment scenarios. CDPRs are increasingly adopted in sectors such as manufacturing, inspection, construction, and logistics, where scalability, precision, and safety are critical, as explained in Bruckmann and Pott [2]. The flexibility offered by the Analytic Centre approach support integration in such domains, laying the groundwork for future applications in real-world, large-scale systems.

Many aspects of the tension distribution algorithms can still be developed. For example, a probable next study would focus on the performances of the algorithms using other configurations of the prototype to see whether the performances of the TDAs changes.

**Author Contributions** D.B. Experimental Validation and writing and V.D.P. Simulations, Writing and Revision S.C. and M.Z. Revision and Suggestion.

**Funding** Open access funding provided by Università degli Studi di Genova within the CRUI-CARE Agreement.

**Data availability** No datasets were generated or analysed during the current study.

## Declarations

**Conflict of interest** The authors declare no conflict of interest.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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