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

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10-09-2025
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Authors: Daria Berretta; Vincenzo Di Paola; Stéphane Caro; Matteo Zoppi
Published in: Meccanica | Issue 9/2025

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Abstract

This article delves into the experimental validation of various tension distribution algorithms (TDAs) for cable-driven parallel robots (CDPRs), with a special focus on the analytic-centre based approach. The study systematically compares different TDAs, including closed-form, linear programming, quadratic programming, and the analytic centre method, using a real prototype called CRAFT. The key topics covered include system modeling, existing tension distribution algorithms, and the experimental framework used for validation. The article highlights the importance of choosing the right cable tensions to ensure accurate movements and safety in CDPRs. Through extensive experiments, it is shown that the analytic-centre based TDA outperforms classical methods by generating continuous and smooth tension profiles, covering the entire wrench feasible workspace, and being capable of including non-linear constraints. The study also discusses the real-time capabilities of the analytic-centre approach, making it particularly suited for broader deployment scenarios. The conclusion emphasizes the flexibility and applicability of the analytic-centre method in real-world, large-scale systems, paving the way for future applications in various industries.

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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 improved in Pott [17] to enlarge the coverage of the Wrench Feasible Workspace (WFW).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)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.

2.1 Inverse geometric model

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

Fig. 1

Generic loop describing the geometric parameters of a CDPR

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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 $\textbf{A}_{i}$ of the cables are located on pulleys fixed to the rigid frame, while the anchor points $\textbf{B}_{i}$ are located on the moving platform. The unit vector $\textbf{u}_{i}$ describes the direction of the cable between points $\textbf{A}_{i}$ and points $\textbf{B}_{i}$.

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

$$\begin{aligned} ^{b}\textbf{l}_{i} =\ l_{i} \mathbf {^{b}u}_{i} =\ ^{b}\overrightarrow{\textbf{A}_{i} \textbf{B}_{i}} =\ ^{b}\textbf{R}_{p}\ ^{p}\textbf{b}_{i} + ^{b}\textbf{p} - ^{b}\textbf{a}_{i} =\ ^{b}\textbf{b}_{i} + ^{b}\textbf{p} - ^{b}\textbf{a}_{i} , \end{aligned}$$

(1)

where $i \in [1,..., m]$, m is the number of cables, $^{b}\textbf{l}_{i}$ is the vector describing the $i-th$ cable, $^{b}\textbf{a}_{i}$ is the coordinate vector of the output point $\textbf{A}_{i}$ expressed in the $F_{b}$ frame, $^{p}\textbf{b}_{i}$ is the coordinate vector of the anchor point $\textbf{B}_{i}$ expressed in frame $F_{p}$, $^{b}\textbf{p}$ is the vector going from $O_{b}$ to $O_{p}$ expressed in frame $F_{b}$ and $^{b}\textbf{R}_{p}$ is the rotation matrix from frames $F_{b}$ and $F_{p}$.

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

$$\begin{aligned} ^{b}\textbf{u}_{i} = \frac{^{b}\textbf{l}_{i}}{l_{i}} , \end{aligned}$$

(2)

where $l_{i}$ is the length of the $i-th$ cable defined as $l_{i} = || \textbf{l}_{i}||_2$.

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 cable length between the pulley entry point, $\textbf{A}_{i}$, and the MP anchor point, $\textbf{B}_{i}$, consider also the portion of the cable that is wrapped on the pulley.

The corresponding joint position is obtained from the total cable length $\textbf{l}_{f}$ and the winch radius $r_{w}$ as

$$\begin{aligned} \textbf{q} = \frac{\textbf{l}_{f}}{r_{w}}. \end{aligned}$$

(3)

2.2 Inverse kinematic model

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

$$\begin{aligned} \textbf{At}_{d} = \dot{\textbf{l}}_{d}, \end{aligned}$$

(4)

where $\textbf{t}_{d} = [\textbf{v}_{d}\ \varvec{\omega }_{d}]$ is the platform twist, with $\textbf{v}_{d}$ and $\varvec{\omega }_{d}$ representing the desired translational and rotationl velocity, respectively. The matrix $\textbf{A}$ is

$$\begin{aligned} \textbf{A} = \begin{bmatrix} ^{b}\textbf{u}'_{1} & ^{b}\textbf{b}_{1} \times ^{b}\textbf{u}'_{1} \\ \vdots & \vdots \\ ^{b}\textbf{u}'_{i} & ^{b}\textbf{b}_{i} \times ^{b}\textbf{u}'_{i} \\ \vdots & \vdots \\ ^{b}\textbf{u}'_{m} & ^{b}\textbf{b}_{m} \times ^{b}\textbf{u}'_{m} \end{bmatrix}_{m \times 6} . \end{aligned}$$

(5)

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

$$\begin{aligned} \dot{\textbf{q}}_{d} = \frac{\dot{\textbf{l}}_{d}}{r_{w}}. \end{aligned}$$

(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

$$\begin{aligned} {\textbf {W}}\varvec{\tau }\mathbf { + w_{e}} = 0 , \end{aligned}$$

(7)

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

$$\begin{aligned} \begin{bmatrix} \mathbf {u'}_{1}& \mathbf {u'}_{2} & ... & \mathbf {u'}_{m} \\ \textbf{b}_{1} \times \mathbf {u'}_{1}& \textbf{ b}_{2} \times \mathbf {u'}_{2} & ... & \textbf{b}_{m} \times \mathbf {u'}_{m} \end{bmatrix}_{6 \times m} , \end{aligned}$$

(8)

in which “$\times$” defines the cross product, $\mathbf {w_{e}}$ the external wrench and $\varvec{\tau }$ the cable tensions vector. This equation shows the relationship between $\varvec{\tau }$ and the external wrench $\mathbf {w_{e}}$ acting on the platform. For redundant CDPRs ($DoR \ge 1$) an infinite number of solutions for $\varvec{\tau }$ exist. The set of solutions is defined as

$$\begin{aligned} \varvec{\Sigma } = \begin{Bmatrix} \varvec{\tau } |\ {\textbf {W}}\varvec{\tau }\mathbf { + w_{e} = 0} \end{Bmatrix} . \end{aligned}$$

(9)

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

$$\begin{aligned} \varvec{\Pi = \begin{Bmatrix} \tau \ |\ 0 < \underline{\tau } \le \tau \le \overline{\tau } \end{Bmatrix}}, \end{aligned}$$

(10)

where $\varvec{\underline{\tau }}$ and $\varvec{\overline{\tau }}$ are the positive tension limits.

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

$$\begin{aligned} \varvec{\Gamma = \Sigma \cap \Pi }. \end{aligned}$$

(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 $\varvec{\Pi }$ (Eq. 10), and is defined as

$$\begin{aligned} d(\varvec{\tau }) = \text {min} \begin{Bmatrix} \overline{\varvec{\tau }} - \varvec{\tau }_{1}, \varvec{\tau }_{1} - \underline{\varvec{\tau }}, ... , \overline{\varvec{\tau }} - \varvec{\tau }_{m}, \varvec{\tau }_{m} - \underline{\varvec{\tau }} \end{Bmatrix} \end{aligned}$$

(12)

where $\varvec{\tau }_{m}$ is the average between the limits $\underline{\varvec{\tau }}, \overline{\varvec{\tau }}$.

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 include nonlinear constraints while solving the optimisation problem.

Table 1

Summary table of the implemented tension distribution algorithms and their properties

Method	Real-time capable	Force niveau	Workspace coverage	Continuity	Max. redundancy	Computational speed	Robustness index
Barycentric (AM)	Yes	Mi	Yes	Yes	r = 2	Fast	High
Improved closed-form (AM)	Yes	Any	No	Yes	Any	Fast	High
Linear programming (NOP)	No	Any	Yes	No	Any	Fast	High
Quadratic programming (NOP)	Yes	Hi, lo	Yes	Yes	Any	Medium	Low
Analytic center (NOP)	Yes	Mi	Yes	Yes	Any	Fast	Medium-high

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 $\underline{\varvec{\tau }} = 20\ N$ while the maximum tensions $\overline{\varvec{\tau }} = 400\ N$.

Fig. 2

Barycentric tension profiles

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Fig. 3

Closed form tension profiles

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Fig. 4

Linear programming tension profiles

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Fig. 5

Quadratic programming tension profiles

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Fig. 6

Analtytic centre tension profiles

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Fig. 7

RIs trends for each TDAs

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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 Index if compared with the Analytic Centre. Additionally, as shown in the Table 1, it does not ensure full workspace coverage.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 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.

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)

Full size image

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.

Fig. 9

CRAFT robot frame and platform

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Key components of this prototype are: eight PARKER SME60⁴ motors with gearbox reducers (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 parallelepiped measuring $0.28 \times 0.28 \times 0.20$ m whose weight is 8.533 kg housing the sensor and many electronic components.

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)

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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.

Fig. 11

Tension sensors FUTEK FSH04097

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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 collect and process the data useful to command the electric motors (Fig. 12).

Fig. 12

Robot control architecture [14]

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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 circlular 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 those influencing cable tensions and sensor readings. 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 particular, the circular trajectory was traced with different 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 experiment 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 function “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 contemporarily prove what expected in Di Paola et al. [7]. The remaining methods and all the measured tension profiles fit the simulated ones allowing us to claim the soundness of the theoretical results (Figs. 16, 17, 18).

Fig. 13

Desired ($\tau _{d}$) and measured ($\tau _{m_{mean}}$) tension for each cable using the BARY TDA; $\tau _{lmts_{rcd}}$ represents the minimum and maximum tension value obtained with the relative method

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Fig. 14

Desired ($\tau _{d}$) and measured ($\tau _{m_{mean}}$) tension for each cable using the CF TDA; $\tau _{lmts_{rcd}}$ represents the minimum and maximum tension value obtained with the relative method

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Fig. 15

Desired ($\tau _{d}$) and measured ($\tau _{m_{mean}}$) tension for each cable using the LP TDA; $\tau _{lmts_{rcd}}$ represents the minimum and maximum tension value obtained with the relative method

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Fig. 16

Desired ($\tau _{d}$) and measured ($\tau _{m_{mean}}$) tension for each cable using the QP TDA; $\tau _{lmts_{rcd}}$ represents the minimum and maximum tension value obtained with the relative method

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Fig. 17

Desired ($\tau _{d}$) and measured ($\tau _{m_{mean}}$) tension for each cable using the AC TDA; $\tau _{lmts_{rcd}}$ represents the minimum and maximum tension value obtained with the relative method

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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 tensions is reduced because the area of the feasible polygon $\Lambda$ 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.

Fig. 18

Desired and measured Robust Index for each TDA

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Fig. 19

Boxplot of the desired Robust Index

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Fig. 20

Boxplot of the measured Robust Index

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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.

Table 2

Desired and measured average Robust indices

	Desired RI	Measured RI
Barycentric	40.6016	42.2915
Improved closed form	41.2624	42.1171
Linear programming	–	–
Quadratic programming	41.1046	41.9785
Analytic centre	40.8050	41.7328

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

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.

Declarations

Conflict of interest

The authors declare no conflict of interest.

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Metadata
Footnotes
Literature

Title: Experimental validation of the analytic-centre based tension distribution algorithm for cable driven parallel robots
Authors: Daria Berretta
Vincenzo Di Paola
Stéphane Caro
Matteo Zoppi
Publication date: 10-09-2025
Publisher: Springer Netherlands
Published in: Meccanica / Issue 9/2025
Print ISSN: 0025-6455
Electronic ISSN: 1572-9648
DOI: https://doi.org/10.1007/s11012-025-02044-9

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.

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).

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.

Parker [16]: https://ph.parker.com/fr/en/product-list/sme-series-low-inertia-brushless-servo-motors-encoder-feedback.

Futek [10]: https://www.futek.com/store/load-cells/s-beam-load-cells/miniature-s-beam-jr-LSB205/FSH04097.

Borgstrom PH, Jordan BL, Sukhatme GS, Batalin MA, Kaiser WJ (2009) Rapid computation of optimally safe tension distributions for parallel cable-driven robots. IEEE Trans Rob. https://doi.org/10.1109/TRO.2009.2032957CrossRef

Bruckmann T, Pott A (2018) Cable-driven parallel robots. Springer tracts in advanced robotics. https://doi.org/10.1007/978-3-319-76138-1

Bruckmann T, Pott A, Hiller M (2006) Calculating force distributions for redundantly actuated tendon-based stewart platforms. In: Advances in robot kinematics: mechanisms and motion. Springer, pp 403–412

Di Paola V, Caro S, Zoppi M (2024) Design and performance investigation of a sliding-mode adaptive proportional-integral-derivative control for cable-breakage scenario. Meccanica 11:11. https://doi.org/10.1007/s11012-024-01875-2CrossRef

Di Paola V, Caro S, Zoppi M (2024) Design of a SM-APID control for rigid-body transportation under the effects of wind. New trends in mechanism and machine science proceedings of EuCoMeS 2024. https://doi.org/10.1007/978-3-031-67295-8-15

Di Paola V, Goldsztejn A, Zoppi M, Caro S (2023) Design of a sliding mode-adaptive pid control for aerial systems with a suspended load exposed to wind gusts. J Comput Nonlinear Dyn 10(1115/1):4062324

Di Paola V, Goldsztejn A, Zoppi M, Caro S (2024) Analytic center based tension distribution for cable-driven platforms (CDPs). J Mech Robot 10(1115/1):4065244

Dona D, Di Paola V, Zoppi M, Traversani A (2025) Stiffness-based analytic centre method for cable-driven parallel robots. In: Cable-driven parallel robots: proceedings of the 7th international conference on cable-driven parallel robots. https://doi.org/10.1007/978-3-031-94608-0-10

dSPACE (2023) Real-time interface (rti). https://www.dspace.com/en/ltd/home/products/sw/impsw/real-time-interface.cfm

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

Abstract

Abstract

Publisher's Note

1 Introduction

2 System modeling

2.1 Inverse geometric model

2.2 Inverse kinematic model

2.3 Static equilibrium and direct dynamic model

3 Existing tension distribution algorithms

3.1 Comparison of various algorithms

4 Experimental framework

5 Analytic centre validation

5.1 The experiments

6 Conclusion

Declarations

Conflict of interest

Publisher's Note

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