# Active Cables Selection for Collocated Vibration Control of Small-Sized Overconstrained Cable-Driven Parallel Robots

Thibault Garcia, Didier Rémond, Simon Chesné

► To cite this version:

Thibault Garcia, Didier Rémond, Simon Chesné. Active Cables Selection for Collocated Vibration Control of Small-Sized Overconstrained Cable-Driven Parallel Robots. Journal of Mechanisms and Robotics, 2025, 17 (7), pp.071001. 10.1115/1.4067622. hal-04971787

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# Active Cables Selection for Collocated Vibration Control of Small-Sized Overconstrained Cable-Driven Parallel Robots

### Thibault Garcia

INSA Lyon, CNRS, LaMCoS, UMR5259
69621, Villeurbanne, France

Email: thibault.garcia@insa-lyon.fr

### Didier Rémond

INSA Lyon, CNRS, LaMCoS, UMR5259
69621, Villeurbanne, France

Email: didier.remond@insa-lyon.fr

### Simon Chesné

INSA Lyon, CNRS, LaMCoS, UMR5259
69621, Villeurbanne, France

Email: simon.chesne@insa-lyon.fr

Cable-driven parallel robots are well appreciated for high dynamics applications, due to their lightweights moving parts. Nevertheless, due to the low stiffness of cables, vibrations can occur and can degrade performances if high precision is required, such as in additive manufacturing for instance. Previous works have studied techniques to counteract vibrations, like using motor command or embedded devices. Based on a previous first exploration of using piezoelectric transducers on cables for this type of robot, this paper presents a proper formulation of the collocated active vibration control to damp the end-effector oscillations of small-sized overconstrained CDPRs by the measure of the variation in cable tensions. This goes through a modeling of such a robot with embedded piezoelectric transducers under appropriate assumptions. From this control formulation, it is shown that the collocated nature of these transducers are fundamental. It is thus possible to highlight an energetic index of active cables selection, regardless of the used control law. The proposed technique is developed theoretically and analyzed through simulations on an 8-cables robot.

## INTRODUCTION

### Context

A Cable-Driven Parallel Robot (CDPR) is a manipulator characterized by its end-effector being

linked to the frame by flexible cables instead of rigid links. An electric motor/winch system enables the adjustment of each cable length and tension. This type of robot can be used for material handling in order to move heavy loads in a large space like a room, e.g. the CDPR CoGiRo, which has dimensions of 15mx11mx6m [1]. Other applications of CDPRs have emerged last years, like motion simulator or to move a camera in a stadium. CDPRs can be classified into two types : partially constrained and overconstrained. To be overconstrained, a CDPR has to be redundant (more cables than degrees of freedom) and unsuspended [2], [3]. This type of CDPR typically demonstrates better stiffness by applying tension to the platform with all cables [4].

The low mass of cables allows the end-effector (EE) to achieve high speeds. However, their limited stiffness and internal damping induce oscillations along the EE trajectory. These vibrations can severely degrade CDPR performance, especially in high-precision tasks. Effective damping of EE oscillations during motion is therefore essential.

The context of this paper focuses on small-sized (typically $1m^3$) overconstrained CDPRs with thin and lightweight cables, for applications requiring speed, static and dynamic accuracy, like for additive manufacturing process. The EE has a very low mass, and its instrumentation may be limited by the pro-
cess. This type of CDPR, termed "high-frequency" (HF), exhibits natural EE mode frequencies typically above 20Hz, due to its higher stiffness and lower EE mass compared to CDPRs commonly studied in the literature.

## Literature survey

Many research have focused on methods for damping EE vibrations. Firstly, some researches consider the command of the motors with some model prediction. In [5], Khosravi assumes axial vibrations to be predominant and applies classic control using a rigid cable model, adding corrective terms to account for the cable elasticity. In [6], Jamshidifar develops a dynamic model that decouples the equation of motion into two parts, one for the desired trajectory and another for undesired vibrations. The control scheme calculates the nominal cable tensions for trajectory tracking and generates auxiliary signals to adjust these tensions. In [7], Kozak linearizes the nonlinear equation of motion of a parallel manipulator to identify modal frequencies and damping ratios. Using it and the trajectory, he can adjust the control to compensate for vibrations using input shaping.

Increasing cable pretension boosts stiffness and reduces oscillations [4], but it is limited by motor torque and mechanical constraints.

Lesellier tests passive on-board devices, such as stabilizers, on suspended CDPRs [8]. He models and studies devices like solid inertial wheels or gyroscopic actuators attached to the platform.

A last known way is to control actuators using measurement signals. For instance the motors. Begey [9] proposes controlling motors with an inner position loop instead of torque control. Motors adjust the unstretched cable length based on its proportionality to rotations. An Inertial Measurement Unit (IMU) measures EE speed, and a camera tracks pose. This data recalculates the trajectory using gain matrices, and a torque command is applied to the inner loop. In [10], [11], Weber, Cuvillon, and Gangloff use an inner motor control loop for active control. They linearize the dynamic equation to derive the vibration equation and, using an IMU to measure EE velocity, develop six modal-space controllers. The control adjusts motor rotations to dampen vibrations, with gain scheduling ensuring performance and stability under minimal parameter variation. In [12], the author considers a CDPR with non negligible mass cables, and proposes a motor command scheme using fuzzy logic and PI controller. Cold-gas thrusters can also be added, as shown in [13], [14]. Similarly, [15] uses active pendulums to compensate for out-of-plane vibrations.

Traditionally, robot vibrational frequencies are low, within the bandwidth of the actuators or techniques used. This limits their application to HF CDPRs in this paper.

Active control using collocated piezoelectric actuators with sensors is studied in papers like [16], [17] or [18]. The resonance frequency of these actuators is typically in the $kHz$ range, with displacements on the order of micrometers. They are usually integrated into rigid, lightly damped structures like trusses, beams, and plates, where the force exerted by a piezoelectric actuator—stiffness multiplied by displacement—is sufficient to counteract vibrations. In [16], Preumont explains collocation, where an actuator is driven by a same location sensor. This sensor-actuator (SA) pair provides beneficial properties for control, such as improved phase margin. Lacaze, inspired by works on cable structures from Preumont [16], [17], first proposes and experimentally explores embedding collocated piezoelectric SA pairs on cables [19], [20] [21], achieving good performances. The high bandwidth of piezoelectric actuators is here valuable in HF CDPRs.

## Motivations and contributions of the paper

Although piezoelectric SA pairs could be used with motor-based techniques, as in [9], [10], and [11], these methods require complex instrumentation like cameras and IMUs. In contrast, Lacaze simplifies the approach by using only a force sensor on the cable. He applies Preumont's formalism [17] to a simplified CDPR model for 3D printing, focusing on one control law with some improvements. However, his model does not address SA pair collocation or variations in CDPR parameters, limiting control law design. The placement of SA pairs is as crucial as the control law itself, and the variability in CDPR parameters complicates finding an optimal control.

A solution would be to embed and activate a SA pair on each cable. If vibration damping is needed in a specific area, it is more cost-effective to selectively embed SA pairs in chosen cables. However, if damping is required throughout the entire workspace, all cables may need activation at certain EE positions. To save energy and reduce the risk of instabilities due to interactions between the individual control loops, limiting simultaneous activation of
SA pairs is preferable. Lacaze illustrates in [19] [21] performances does not improve with all cables activated.

Consequently, the question of the selection of cables to embed and/or to activate arises, which is named "active cables selection" problem. The cable selection is similarly treated in [22], but focuses on selecting a set of cables for force control and another for length control, within the context of overall tension distribution control.

Given the numerous embedding and configuration possibilities in a CDPR, this paper proposes a simple and robust method for active cables selection. Using the collocation principle, a selection index based on an energetic approach is developed. Inspired by Lacaze's work, the goal is to properly formulate the collocated active control through variation in cable tensions. To build this architecture, a small-sized overconstrained CDPR with integrated SA pairs is rigorously modeled. The relationship between cable tension variations and EE modal shapes, underexplored in the literature, is then explored, enabling the creation of the selection index. Simulations on an 8-cables and 6-DOFs CDPR demonstrate the benefits and potential of the proposed approach. This paper focuses on proposing dynamic analytical tools using the collocation property to design controllers for collocated active control with piezoelectric devices for HF small-sized CDPRs, rather than optimizing damping performance.

## Paper organization

This paper is divided into 4 main sections. Section 1 focuses on assumptions and a modeling of CDPR with embedded SA pairs. Section 2 focuses on the collocated control architecture. Section 3 focuses on the establishment of the energetic active cables selection index. Section 4 presents simulation results on a 8-cables CDPR and 6 DOFs.

# 1 CDPR MODELING

## 1.1 Assumptions on cables

The cable model determines the CDPR dynamics, so a balance of realism and simplicity is needed, guided by relevant literature assumptions.

Cable dynamics, including weight, sagging, and cable modes, can be considered or neglected. Sagging, due to cable weight, complicates the geometric model, as cables cannot be treated as geometric vectors [23], [24]. It also affects vibration

behaviour. Yuan [25], [26] considers sagging and builds a frequency-dependent dynamic stiffness matrix, coupling EE vibration response with cable modes. Ferravante [27] models CDPR in 3D, capturing local resonances of cables using finite elements but neglects sagging. Sagging should also be considered when cable tension varies significantly [28]. Alternatively, cables can be modeled as straight rods with negligible diameter, ignoring their weight and dynamics. This choice is relevant for a small-sized overconstrained CDPR with thin and lightweight cables.

Cables can be modeled as inextensible for trajectory generation [2], but an elastic model is required for vibration behaviour. A choice exists between linear and non-linear models. Baklouti [29] proposes a non-linear model for large strains, while hysteresis is considered in [30], [31]. The linear model, where cables act as springs, is typically used, with vibration modes obtained by linearizing the dynamic equation [11], [13]. The stiffness matrix from [32] is widely adopted. Diao [33] shows transverse vibrations of fully constrained CDPRs contribute minimally compared to axial flexibility. Gueners [34] concludes the linear model is valid only for thin steel cables in 3D cable printing. Here, the linear elastic model suffices, as the considered cable, like steel, exhibits linear properties. Internal damping is neglected due to the cable low damping in practice.

## 1.2 Other assumptions

The EE is of any form and is considered rigid. The EE point of interest (C) coincides with the center of mass (G) (Fig. 1). The cable unstretched length $l_{0i}$ controls the trajectory of the CDPR, as in [9], assuming a perfect inner-loop with respect to trajectory bandwidth and EE modes, and neglecting motor-winder dynamics. The piezoelectric material is linear with applied voltage. Due to the cable low stiffness and the considered tension distribution, the piezoelectric transducers are treated as rigid. The SA pair mass is considered embedded on the EE, and the actuator side controls cable length and tension (Fig. 1).

## 1.3 Modeling

This part is dedicated to establish the dynamic equation of the EE.
proximating the global equation. These trajectories involve large translations and zero rotation commands, focusing on low rotations (vibrations). This leads to S = 1, leading to a simplification of Eq. (5), which becomes:

$$ 
\begin{aligned}
f(\mathbf{X}, \dot{\mathbf{X}}, \ddot{\mathbf{X}}, l_0, \delta, W_{nc}) &= \\
&\mathbf{M}(\mathbf{X})\ddot{\mathbf{X}} + \mathbf{C}(\mathbf{X}, \dot{\mathbf{X}})\dot{\mathbf{X}} - \mathbf{J}^T(\mathbf{X})t(\mathbf{X}, \delta, l_0) - W_g - W_{nc} \\
&= 0 \quad (6)
\end{aligned}
$$

Where $\mathbf{M}(\mathbf{X}) = \begin{pmatrix} m_e l_3 & 0_3 \\ 0_3 & l_c \end{pmatrix}$ and $\mathbf{C}(\mathbf{X}, \dot{\mathbf{X}}) = \begin{pmatrix} 0_3 & 0_3 \\ 0_3 & [\dot{\alpha} \times] l_c \end{pmatrix}$. The nominal trajectory is considered as an equilibrium state, then Eq. (6) is linearized by development of Taylor in the 1st order around the nominal trajectory $\mathbf{P}^* = [\mathbf{X}^*, \dot{\mathbf{X}}^*, \ddot{\mathbf{X}}^*, l_0^*, \delta^*, W_{nc}^*]$. Rotation commands are always zero thus $\dot{\alpha}^* = 0$ and $\ddot{\alpha}^* = 0$, leading to $\frac{\partial \mathbf{M}}{\partial \mathbf{X}} \ddot{\mathbf{X}}^* = 0$, $\frac{\partial \mathbf{C}}{\partial \mathbf{X}} \dot{\mathbf{X}}^* = 0$, $\frac{\partial \mathbf{C}}{\partial \dot{\mathbf{X}}} \dot{\mathbf{X}}^* = 0$ and $\mathbf{C}(\mathbf{X}^*, \dot{\mathbf{X}}^*) = 0$ (see Appendix A). Finally:

$$ 
\begin{aligned}
\mathbf{M}(\mathbf{X}^*)\delta\ddot{\mathbf{X}} + \mathbf{K}(\mathbf{X}^*, l_0^*, \delta^*)\delta\mathbf{X} &= \\
&-\mathbf{J}^T(\mathbf{X}^*)\mathrm{diag}(k_i(l_{0i}^*)\frac{l_i^*}{l_{0i}^*})\delta l_0 + \mathbf{J}^T(\mathbf{X}^*)\mathrm{diag}(k_i(l_{0i}^*))\delta\delta \\
&+ \delta W_{nc}
\end{aligned}
\quad (7)
$$

Where $\mathbf{M}$ is the mass matrix and $\mathbf{K} = \mathbf{K}_{axial} + \mathbf{K}_t$ is the stiffness matrix given by [32]:

$$ \mathbf{K}_{axial}(\mathbf{X}^*, l_0^*) = \mathbf{J}^T(\mathbf{X}^*)\mathrm{diag}(k_i)(l_0^*)\mathbf{J}(\mathbf{X}^*) \quad (8) $$

$$ \mathbf{K}_t(\mathbf{X}^*, l_0^*, \delta^*) = \sum_{i=1}^{m} \mathbf{K}_{t,1}^i - \sum_{i=1}^{m} \mathbf{K}_{t,2}^i \quad (9) $$

With:

$$ \mathbf{K}_{t,1}^i = \frac{t_i}{d_i} \begin{pmatrix} 1_3 - u_i u_i^T & (1_3 - u_i u_i^T)[\mathbf{CB}_i \times]^T \\ [\mathbf{CB}_i \times](1_3 - u_i u_i^T) [\mathbf{CB}_i \times] & (1_3 - u_i u_i^T)[\mathbf{CB}_i \times]^T \end{pmatrix} \quad (10) $$

$$ \mathbf{K}_{t,2}^i = t_i \begin{pmatrix} 0_3 & 0_3 \\ 0_3 & [u_i \times][\mathbf{CB}_i \times] \end{pmatrix} \quad (11) $$

$-\mathbf{J}^T(\mathbf{X}^*)\mathrm{diag}(k_i \frac{l_i}{l_{0i}})\delta l_0$ represents the forces applied by the motor on the EE, whereas $\mathbf{J}^T(\mathbf{X}^*)\mathrm{diag}(k_i)(l_0^*)\delta\delta$ is for the piezoelectric actuators. The stiffness matrix $\mathbf{K}$ is symmetric when no external moments are applied on the EE [35]. In the following, $W_{nc}^* = 0$, which means $\delta W_{nc}$ represents variations in external forces from equilibrium, such as a local shock. Equation (7) (which cannot directly formulate the collocated control architecture based on variations in cable tensions) parallels the static equilibrium case, treating the trajectory as successive static poses governed by mass-spring dynamics from modal analysis. This static equilibrium case is considered below.

## 2 COLLOCATED CONTROL ARCHITECTURE

This section presents the collocated control architecture, aiming to minimize $\delta X$ responses to perturbations by controlling a set of cable tension variations $\delta t_k$ by collocated actuators $k_k \delta \delta_k$, as explored by Lacaze [19] [21]. This Low Authority Control (LAC) method [16] uses limited measurements at specific points.

### 2.1 Control at the cable level

Let establish the transfer matrix between actuator forces and variations in cable tensions $\delta t$. Noting that:

$$ \delta t = -\mathrm{diag}(k_i)\mathbf{J}\delta\mathbf{X} - \mathrm{diag}(k_i \frac{l_i}{l_{0i}})\delta l_0 + \mathrm{diag}(k_i)\delta\delta \quad (12) $$

The vibration equation (7) can be rewritten into Laplace domain:
$$[s^2\mathbf{M} + \mathbf{K}_t]\delta\mathbf{X} = \mathbf{G}^{-1}\delta\mathbf{X} = \mathbf{J}^T\delta\mathbf{t} + \delta\mathbf{W}_{nc} \quad (13)$$

With Eq. (12), it becomes:

$$\delta\mathbf{t} = \mathbf{G}_o(s)[\delta\mathbf{F} + \mathrm{diag}(k_i)\delta\delta] \quad (14)$$

Where $\delta\mathbf{F} = -\mathrm{diag}(k_i \frac{l_i}{l_{0i}})\delta\mathbf{l}_{0i}$ - $\mathrm{diag}(k_i)\mathbf{J}\mathbf{G}\delta\mathbf{W}_{nc}$ represents the disturbance effort acting at the cable level. Finally, with $\mathbf{1}_m$ the $mxm$ identity matrix:

$$\mathbf{G}_o = [\mathbf{1}_m + \mathrm{diag}(k_i)\mathbf{J}\mathbf{G}\mathbf{J}^T]^{-1} \quad (15)$$

It is the $mxm$ transfer matrix linking cable tension variations to forces applied on the EE at the cable level. Non-diagonal terms $G_{okj}(s)$ represent coupling between cables, while diagonal terms $G_{okk}(s)$ are the collocated transfer functions for each cable, currently determined numerically for a specific pose.

The collocated control architecture scheme is represented in Fig. 2.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Collocated control architecture block diagram</figcaption></figure>

The measured variation in cable tensions $\delta t$ is compared to a tension variation "reference" $\delta t_{ref}$, signal injected into a control law $H(s)$. The output drives the piezoelectric force applied on the plant in order to decrease $\delta t$ and reject the disturbance $\delta F$. The control being decentralized, $H(s)$ is a diagonal matrix of controllers, defined by its diagonal elements $H_k(s) = \frac{k_k \delta \delta_k}{\delta t_{ref,k} - \delta t_k}$. The force of the $k^{th}$ piezoelectric actuator is thus driven by the measure

of the variation in the $k^{th}$ cable tension, which is an image of the EE vibrations. In vibration control $\delta t_{ref} = 0$ thus:

$$\delta\mathbf{t} = [1 + \mathbf{G}_o\mathbf{H}]^{-1}\mathbf{G}_o\delta\mathbf{F} \quad (16)$$

The variation in tension of a controlled cable $k$ can be expressed as (see Appendix B):

$$\delta t_k = \frac{G_{okk}\delta F_k + \sum_{j \neq k} G_{okj}[\delta F_j - H_j\delta t_j]}{1 + G_{okk}H_k} \quad (17)$$

To increase the modal damping ratio $\xi_i$, the oscillating poles of each mode must be controlled. In case of a single active cable $k$ ($H_j = 0, j \neq k$) the poles can be designed by solving $1 + G_{okk}H_k = 0$. The rootlocus visualizes this by plotting closed-loop poles as the control gain varies, aiding in performance tuning and stability, ensuring poles stay in the left half-plane. However, rootlocus analysis requires a defined control law. Since the CDPR structure changes with pose, selecting the active cable is not straightforward. Optimal controller design works for a single pose [19] [21], but cable authority can change with pose. Therefore active cable selection must precede controller design. In case of multiple active cables, loop interactions complicate pole design and increase instability risks due to the presence of $-G_{okj}H_j\delta t_j, j \neq k$.

## 2.2 Control at the end-effector level

Computation of EE global poles when several active cables are used is led by state space representation. Using Eq. (12) and the control law:

$$\mathrm{diag}(k_i)\delta\delta = -\mathbf{H}(s)\delta t \quad (18)$$

$$\delta t = -\mathrm{diag}(\frac{k_i}{1+H_i(s)})\mathbf{J}\delta\mathbf{X} - \mathrm{diag}(\frac{k_i}{1+H_i(s)}\frac{l_i}{l_{0i}})\delta\mathbf{l}_{0i} \quad (19)$$

Then using Eq. (13):
$$[s^2\mathbf{M} + \mathbf{Q}_{\text{control}}(s)]\delta\mathbf{X} = -\mathbf{J}^T\text{diag}\left(\frac{k_i}{1+H_i(s)}\frac{l_i}{l_{0i}}\right)\delta\mathbf{l}_0 + \delta\mathbf{W}_{\text{nc}} \quad (20)$$

With $\mathbf{Q}_{\text{control}}(s) = \mathbf{J}^T\text{diag}(\frac{k_i}{1+H_i(s)})\mathbf{J} + \mathbf{K}_t$ the dynamic matrix depending on $s$ and containing both stiffness and damping terms. Considering perturbations to 0, and with state representation $\delta\mathbf{X} = (\delta\mathbf{X}^T s\delta\mathbf{X}^T)^T$:

$$s\delta\mathbf{X} = \begin{pmatrix} 0_6 & 1_6 \\ -\mathbf{M}^{-1}\mathbf{Q}_{\text{control}}(s) & 0_6 \end{pmatrix} \delta\mathbf{X} = \mathbf{A}_{\text{cl}}(s)\delta\mathbf{X} \quad (21)$$

The closed-loop poles of the EE DOFs can be computed by solving $\det(s\mathbf{1}_{12} - \mathbf{A}_{\text{cl}}(s)) = 0$. This representation checks stability and performances but does not allow proper pole design due to MIMO complexity, making rootlocus design unsuitable.

## 2.3 Interest of an active cables selection index

Firstly, when only one cable is active, the correct cable must be selected based on the desired controlled modes, and the control law designed using SISO techniques like rootlocus. Secondly, control performance should be understood and managed, even with interactions between loops when multiple cables are active.

# 3 ACTIVE CABLES SELECTION INDEX

## 3.1 Single active cable

### 3.1.1 Collocated transfer function

This part presents the analytical formulation of the collocated transfer function $G_{okk}(s)$ for a cable $k$. It highlights the collocated properties and provides a robust index for selecting active cables.

Let consider only one active cable $k$. Moreover, let $\delta\mathbf{W}_{\text{nc}} = 0$ and $\delta\mathbf{l}_0 = 0$. Let $\mathbf{\Phi} = (\mathbf{\Phi}_1 ... \mathbf{\Phi}_6)$ be the modal shapes matrix and $\delta\mathbf{\eta}$ the modal amplitudes vector, thus $\delta\mathbf{X} = \mathbf{\Phi}\delta\mathbf{\eta}$. The modal mass matrix is $\mu = \text{diag}(\mu_i) = \mathbf{\Phi}^T\mathbf{M}\mathbf{\Phi}$ and the modal stiffness matrix is $k' = \text{diag}(k'_i) = \mathbf{\Phi}^T\mathbf{K}\mathbf{\Phi}$.

Starting from Eq. (7) and moving into modal space:

$$\delta\mathbf{X} = \sum_{i=1}^{6} \frac{\mathbf{\Phi}_i \mathbf{\Phi}_i^T}{k'_i + s^2 \mu_i} \left( \mathbf{u}_k \times \mathbf{C}\mathbf{B}_k \times \mathbf{u}_k \right) k_i \delta\delta_k = \sum_{i=1}^{6} \frac{\mathbf{\Phi}_i \mathbf{\Phi}_i^T}{k'_i + s^2 \mu_i} \mathbf{J}_k^T k_i \delta\delta_k \quad (22)$$

By computing $\mathbf{J}\mathbf{\Phi}_i$, Eq. (23) is obtained, and using Eq. (12), Eq. (24) is derived.

$$\text{diag}(k_i)\mathbf{J}\delta\mathbf{X} = \sum_{i=1}^{6} \frac{1}{k'_i + s^2 \mu_i} \begin{pmatrix} \mathbf{J}_1 \mathbf{\Phi}_i \mathbf{\Phi}_i^T \mathbf{J}_k^T k_1 \\ \vdots \\ \mathbf{J}_m \mathbf{\Phi}_i \mathbf{\Phi}_i^T \mathbf{J}_k^T k_m \end{pmatrix} k_i \delta\delta_k \quad (23)$$

$$\delta t = -\sum_{i=1}^{6} \frac{1}{k'_{i} + s^2 \mu_{i}} \begin{pmatrix} \mathbf{J}_1 \mathbf{\Phi}_i \mathbf{\Phi}_i^T \mathbf{J}_k^T k_1 \\ \vdots \\ \mathbf{J}_m \mathbf{\Phi}_i \mathbf{\Phi}_i^T \mathbf{J}_k^T k_m \end{pmatrix} k_i \delta\delta_k + \begin{pmatrix} 0 \\ \vdots \\ 0 \\ k_i \delta\delta_k \\ 0 \\ \vdots \\ 0 \end{pmatrix} \quad (24)$$

Considering the line $k$ in $\delta t$, noticing that $\mathbf{J}_k \mathbf{\Phi}_i \mathbf{\Phi}_i^T \mathbf{J}_k^T = (\mathbf{J}_k \mathbf{\Phi}_i)^2$, and defining $\omega_i = \sqrt{\frac{k'_i}{\mu_i}}$ the modal frequency of mode $i$, the following transfer function is:

$$G_{okk}(s) = \frac{\delta t_k}{k_k \delta \delta_k} = 1 - \sum_{i=1}^{6} \frac{\frac{k_i}{k'_i} (\mathbf{J}_k \mathbf{\Phi}_i)^2}{1 + \frac{s^2}{\omega_i^2}} = 1 - \sum_{i=1}^{6} \frac{\nu_i^k}{1 + \frac{s^2}{\omega_i^2}} \quad (25)$$

This result aligns with Preumont's papers for truss structures in [16], [18], and [36], where an alternation of zeros and poles arises due to $\nu_i^k > 0$, with $\nu_i^k$ being the energetic selection index detailed below. This alternation, valued in control design, remains true despite changes in structural parameters, as SA pairs are always collocated. For CD-PRs, inherently evolutive structures, non-collocated SISO systems may encounter instabilities along a trajectory. In contrast, collocated systems with a law designed to avoid energy injection at one pose, remain stable during the trajectory due to this alternation. Here, the primary variation is in control au-
thority, which can exhibit peaks or total losses without instability. Additionally, the alternation ensures a phase shift between 0° and 180°, improving phase margin. In this study, the equal number of zeros and poles prevents roll-off at high frequencies. Lastly, the pole-zero (PZ) distance, a key criterion for SA pair placement (e.g., [37]), and Eq. (25) provide a robust energetic index for active cable selection.

### 3.1.2 Energetic index for active cable selection

This part presents the meaning of the energetic index $\nu_i^k$, how to exploit it and its practical interest. Let $U_i^{tot}$ be the total deformation energy when the EE vibrates according to the mode $i$. Then $\delta\mathbf{X} = \mathbf{\Phi}_i\delta\eta_i$ and:

$$U_i^{tot} = \frac{1}{2} \delta\mathbf{X}^T \mathbf{K} \delta\mathbf{X} = \frac{1}{2} \mathbf{\Phi}_i^T \mathbf{K} \mathbf{\Phi}_i \delta\eta_i^2 = \frac{1}{2} k_i' \delta\eta_i^2 \quad (26)$$

Let $U_i^k = \frac{1}{2}k_k\delta l_k^2$ be the deformation energy of the cable $k$, when the EE vibrates according to mode $i$ (without actuation of the SA pair). By noticing that $\delta l_k = \delta d_k + \delta\delta_k = \delta d_k$ and that $\delta\mathbf{d} = -\mathbf{J}\delta\mathbf{X}$, meaning $\delta d_k = -\mathbf{J}_k\delta\mathbf{X} = -\mathbf{J}_k\mathbf{\Phi}_i\delta\eta_i$:

$$U_i^k = \frac{1}{2} k_k (\mathbf{J}_k \mathbf{\Phi}_i)^2 \delta\eta_i^2 \quad (27)$$

Finally:

$$\nu_i^k = \frac{\frac{1}{2} k_k (\mathbf{J}_k \mathbf{\Phi}_i)^2 \delta\eta_i^2}{\frac{1}{2} k_i' \delta\eta_i^2} = \frac{U_i^k}{U_i^{tot}} \quad (28)$$

$\nu_i^k$ represents the fraction of modal strain energy of the cable $k$, associated with the mode $i$, meaning the part of the total deformation energy of the mode $i$ in the structure stored in the cable $k$. The larger this term is, the higher the energetic contribution of cable $k$ to mode $i$. Consequently, $\nu_i^k$ links the modal shapes of the EE and the cables from an energetic perspective. A similar index for truss structures, used for active truss selection, is presented in [36].

This index relates to the poles (dependent on EE modes) and zeros (dependent on cables) of

the transfer function $G_{okk}$, serving as a criterion for evaluating a cable authority in active control. The PZ alternation enables the expression of $G_{okk}$ as

$$G_{okk}(s) = G_o^k \prod_{i=1}^{6} \frac{\frac{s^2}{z_i^{k^2}} + 1}{\frac{s^2}{\omega_i^2} + 1}, \quad \text{with } G_o^k \text{ the static gain, and } z_i^k < \omega_i < z_{i+1}^k, \text{ property due to alternation property of collocation and to the use of force transducers. The term } \pm j\omega_i \text{ is the pole of mode } i \text{ characterized by EE vibration modes, and } \pm jz_i^k \text{ is in this paper named the associated zero of mode } i \text{ for cable } k. Exciting harmonically the EE via cable } k \text{ at this frequency } z_i^k \text{ causes a variation in this cable tension equal to zero. The zero } \pm jz_{i+1}^k \text{ is here named the adjacent zero of mode } i.}$$

In the following, the PZ distance of mode $i$ represents the distance between the pole and the associated zero $|\omega_i - z_i^k|$. At the location of a SA pair, a greater distance indicates higher authority on mode $i$. Indeed, it is well known in structural control that the maximum damping ratio, when a Low Authority Controller (LAC) is applied to a SISO system with collocated SA transducers, is directly linked to the PZ distance ([16], [37], [38]). This can serve as an empirical rule when the modes are well separated. Let focus on a targeted mode $j$ to be controlled. Passing into harmonic regime $s = j\omega$, and by definition of anti-resonance: $G_{okk}(\omega = z_j^k) = 0$. Then using Eq. (25):

$$G_{okk}(\omega = z_j^k) = 0 = 1 - \sum_{i=1}^{6} \frac{\nu_i^k}{1 - \frac{z_j^{k^2}}{\omega_i^2}} \quad (29)$$

Isolating mode $j$ yields:

$$\nu_j^k = (\omega_j^2 - z_j^{k^2}) \frac{1 - \sum_{i \neq j} \frac{\nu_i^k}{1 - \frac{z_j^{k^2}}{\omega_i^2}}}{\omega_j^2} \quad (30)$$

Equation (30) defines a proportional relationship between the energetic index $\nu_j^k$ and the PZ distance $\omega_j^2 - z_j^{k^2}$ for mode $j$, serving as an authority criterion. However, interactions between modes can occur when they are close, as is typically the case for CDPRs. The energetic index $\nu_j^k$ also accounts for
this interaction. In the same way as Eq. (30), Eq. (31) establishes interaction between a mode j of interest and an other mode p:

$$ \nu_j^k = (\omega_j^2 - z_p^{k2}) \frac{1 - \sum_{i \neq j} \frac{\nu_i^k}{1 - \frac{z_p^{k2}}{\omega_i^2}}}{\omega_j^2} \quad (31) $$

Here, due to the collocation, let focus on the adjacent zero $z_{j+1}^k$ ($p = j + 1$). If the pole of mode j, $\omega_j$, is very close to or equal to the adjacent zero $z_{j+1}^k$, the energetic index will be low, even in case of a large PZ distance of mode j, which would typically indicate high authority. This phenomenon, known as *PZ cancellation*, results in a complete loss of authority on mode j when a LAC is applied. This effect is clearly shown in the root locus in Figure 4 of [37].

To conclude, the energetic index reflects both the authority (PZ distance) and interactions between modes, including PZ cancellation. It helps identify which cables to avoid (low PZ distance, PZ cancellation) and which to select (high PZ distance, no PZ cancellation). Intuitively, it quantifies how a mode is expressed on a cable, and in collocated control, the more a mode is expressed at the SA pair location, the better it can be controlled. Since the index depends on the EE pose, it allows tracking along a trajectory to guide active cable selection.

## 3.2 Several active cables

This section illustrates the control behaviour with multiple active cables and its relation to the energetic index. While it focuses on a specific case, the index relevance is clearly demonstrated in the context of CDPRs. Let consider the simple control law named Integral Force Feedback (IFF), well known and unconditionnaly stable [16]. Considering Eq. (20), the control law $IFF(H(s) = \frac{-g}{s}$, applying the same gain for all active cables and passing into $s = j\omega$:

$$ \begin{aligned} & [-\omega^2\mu + k' - \frac{g^2}{g^2 + \omega^2}\Phi^T J^T \mathrm{diag}(k_i) J\Phi + \\ & \qquad j\omega \frac{g}{g^2 + \omega^2}\Phi^T J^T \mathrm{diag}(k_i) J\Phi] \delta\eta \\ &= \Phi^T \delta W_{nc} = \delta \tilde{F} \quad (32) \end{aligned} $$

This equation holds under the assumption that the modal shapes are not altered by the control, which is typically the case. Let define $D = k^{-1}[\Phi^T J^T \mathrm{diag}(k_i) J\Phi] = \sum_k D_k$, with $D_k = (d_{ij}^k = \frac{k_i(J_k\Phi_i)(J_k\Phi_j)}{k_i^2})$. It can be noticed that the diagonal elements of $D_k$ are the energetic indexes for each mode $\nu_i^k = d_{ii}^k = \frac{k_i(J_k\Phi_i)^2}{k_i^2}$. Then the matrix $D$ contains in its diagonal elements the sum of the energetic indexes of all cables of each mode, then $d_{ii} = \sum_k \nu_i^k$. The matrix $D$ is fully populated, but is in general diagonal dominant. Neglecting the off-diagonal elements, the modal amplitude for each mode can be expressed as:

$$ \delta\eta_i \approx \frac{\frac{1}{k_i^2}}{1 - (\frac{\omega}{\omega_i})^2 - \frac{g^2}{g^2 + \omega^2} \sum_k \nu_i^k + j\omega \frac{g}{g^2 + \omega^2} \sum_k \nu_i^k} \delta\tilde{F}_i \quad (33) $$

If the excitation is at resonance, $\omega = \omega_i$, and assuming a small gain, $g \ll \omega_i$ (which is preferred to avoid actuator saturation and prevent altering the structure characteristics), the modal damping ratio is approximately:

$$ \xi_i \approx \frac{g \sum_k \nu_i^k}{2\omega_i} \quad (34) $$

The result is similar to that in [18] for truss structures. Thus the modal damping ratio is proportional to the sum of the energetic index contributions of active cables. Active cables selection depends on the user's goals: for a critical mode, a set of cables with dominant energetic indexes should be chosen. If, in a given set, one cable dominates a mode compared to the others, the modal damping at the cable level (using a SISO technique like rootlocus) will dictate the EE modal damping. Then, selecting active
cables with well-separated indexes helps decouple the modal damping design from control loop interactions.

While this result applies to a simple, stable law, it can extend to other integrative laws. More complex laws may increase mode coupling, emphasizing the need to carefully select active cables. This result highlights the trend of the authority on modal damping with the energetic index even with several active cables, with the final selection depending on user needs and desired performances.

# 4 SIMULATIONS

## 4.1 CDPR parameters

The objective of this simulation part is to illustrate the concepts described in this paper, particularly to demonstrate the value and utility of the energetic selection tool.

Figure 3 depicts the simulated CDPR when the EE is at the center of the cubic frame.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Simulated CDPR with global axis definition</figcaption></figure>

The CDPR parameters are described in Table 1. The EE is cylindrical. The used cables are steel from CarlSthal. The frame is cubic. The piezoelectric actuator is a APA60SM from CEDRAT TECHNOLOGIES, a type of amplified actuator also used in [39], and used by Lacaze [19] [21]. The maximal displacement is around 80µm. The typical cable stiffness ranges from 10⁴ N/m to 5 * 10⁴ N/m, resulting in a maximum control force between 1 and 4N,

which is comparable to the typical variation in cable tensions for this CDPR type during EE vibrations. Consequently, this type of actuator is well-suited. It is driven in voltage with the relation $δ_i = αV_i$. The mass of the SA pairs is negligibly compared to that of the EE. The angle formed by the axis $x_E$ and the $B_i$ points is 20°. The inertia matrix exprimed at point C in $R_E$ is equal to $I_C^E = \text{diag}(\frac{m_e r^2}{4} + \frac{m_e h^2}{12}, \frac{m_e r^2}{4} + \frac{m_e h^2}{12}, \frac{m_e r^2}{2})$. Unless stated otherwise, $x, y, z$ are expressed in $m$ and $φ, θ, ψ$ are expressed in $rad$. The tension distribution is computed by the closed-form algorithm [40] with $t_{min} = 0N, t_{max} = 120N$. Simulations are conducted in Matlab-Simulink. Simulink handles temporal simulations with the general dynamic model in Eq. (5) and the ode3 solver, while Matlab simulates the linearized model. Figure 4 shows the Simulink scheme. The motor inner loop is assumed perfect, with $l_0 = l_0^*$.

<table><caption>Table 1. Parameters of the CDPR</caption><thead><tr><td>Parameter</td><td>Description</td><td>Value</td></tr></thead><tbody><tr><td>$m_e$</td><td>EE mass</td><td>1 kg</td></tr><tr><td>$r$</td><td>EE radius</td><td>25 mm</td></tr><tr><td>$h$</td><td>EE height</td><td>100 mm</td></tr><tr><td>$a^3$</td><td>frame dimension</td><td>1 m³</td></tr><tr><td>$E$</td><td>cable Young modulus</td><td>6.9 * 10¹⁰ Pa</td></tr><tr><td>$d$</td><td>cable diameter</td><td>0.54 mm</td></tr><tr><td>$\alpha$</td><td>piezoelectric actuator coefficient</td><td>4.5 * 10⁻⁷ m/V</td></tr><tr><td>$e_{i,rest}$</td><td>SA pair length at rest</td><td>60 mm</td></tr><tr><td>$t_{min}$</td><td>minimal tension</td><td>0 N</td></tr><tr><td>$t_{max}$</td><td>maximal tension</td><td>120 N</td></tr></tbody></table>

<table><caption>Table 2. Parameters of the control law</caption><thead><tr><td>Parameter</td><td>Description</td><td>Value</td></tr></thead><tbody><tr><td>$g$</td><td>gain</td><td>10<sup>8</sup></td></tr><tr><td>$w_{PB}$</td><td>low-pass freq.</td><td>2π300 rad/s</td></tr><tr><td>$w_{PH}$</td><td>high-pass freq.</td><td>2π20 rad/s</td></tr><tr><td>$w_c$</td><td>conditioner freq.</td><td>2π rad/s</td></tr><tr><td>$H(s)$</td><td>filtered-IFF</td><td>$\frac{g}{s} \frac{\alpha}{s+\omega_{PB}} \frac{\omega_{PB}}{s+\omega_{PH}} \frac{s}{s+\omega_c}$</td></tr></tbody></table>
<figure><img src="image_5.png" /><figcaption>Fig. 4. Temporal simulation scheme</figcaption></figure>

The practical implementation of the active con-
trol is described in Fig. 4, and the parameters in
Table 2. The goal is to control the variation in ca-
ble tensions in the bandwith of EE modes. La-
caze proposes in [19] [21] to add a bandpass fil-
ter $\frac{s}{s+\omega_{PB}}$, in addition to the 1Hz cut-off fre-
quency high-pass filter corresponding to the signal
conditioner $\frac{s}{s+\omega_I}$ of the piezoelectric force sensor.
The piezoelectric voltage is driven by the control law,
thus the coefficient $\alpha$ is added. The IFF $\frac{g_k}{s}$ law is
used, as Lacaze in [19] [21]. This paper does not
aim to develop controllers or use an optimal gain;
therefore, this simple law suffices to demonstrate the
relevance of the energetic index. A gain of $g = 10^8$
is chosen for all active cables, as the equivalent gain
$g' = \alpha g$ satisfies $\alpha g = 45 \ll \omega_1 = 136$, providing
sufficient damping without significantly altering the
structure properties.

## 4.2 Evolution of the energetic selection index along a trajectory

This section illustrates the evolution of the
energetic index along a trajectory and its link to the
poles and zeros, showing how a cable authority on
modes changes.

Cable 5 is arbitrarily selected, and its transfer
function $G_{o55}$ is computed. The evolution of poles
(full lines) and zeros (broken lines) along the tra-
jectory $y = -x$ from $x = -0.25m, y = 0.25m$ to
$x = 0.25m, y = -0.25m$ on the $z = 0m$ plane is
shown in Fig. 5. Poles and zeros on the left repre-
sent the start, and those on the right, the end. The
trajectory and energetic indexes $\nu_i^5$ for the 6 modes
are mapped in Fig. 6 on the fixed $z = 0m$ plane with
varying $x$ and $y$.

These figures show that the CDPR vibration
behaviour evolves significantly. The map of Fig. 6
visualizes how mode shapes evolve and impact the
cables, providing a new perspective on the CDPR
dynamic behaviour. For example, the energetic
index of mode 6 is high in the lower right corner,
indicating significant deformation of cable 5, and
low in the higher left corner. The same reasoning
applies for other modes.

Discontinuities occur for modes 1 and 2. For
mode 1, around $x = -0.086m$, $y = 0.086m$, the en-
ergetic index of cable 5 drops from 25% to 5% (Fig.
6). At this point, modes 1 and 2 permute (Fig. 5, left
black vertical line). This happens when two modes
become very close in frequency and swap order.
These discontinuities are not present for modes 4,
5, and 6. Mode 3 shows similar discontinuities to
mode 2 in Fig. 6 (high and low center), but they are
not visible in Fig. 5 because the trajectory avoids
those zones. It highlights that a cable authority on a
mode can change abruptly along the trajectory, em-
phasizing the need for collocated control strategies
based on modal shapes, not frequencies.

The PZ distance and the energetic index evo-
lution are closely correlated. For mode 6 (black) in
Fig. 5, the PZ distance grows from small to large
values, matching the increase in the energetic index
in Fig. 6, making cable 5 effective for controlling
mode 6 at the end of the trajectory but not at the
start. For mode 3 (green) in Fig. 5, the PZ distance
decreases to cancellation, slightly increases, can-
cels again, and stabilizes at a low value, consistent
with the energetic index in Fig. 6, which starts at
13%, drops to 0%, rises to 13%, and stabilizes near
0%. It shows cable 5 is ineffective for mode 3 along
<figure><img src="image_6.png" /><figcaption>Fig. 5. Pole-zero evolution along the trajectory for cable 5 (pole: full line, zero: broken line)</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6. Map of energetic indexes for cable 5</figcaption></figure>

this trajectory. However, while the energetic index for mode 3 is similar at the beginning and center (Fig. 6), the PZ distance is greater at the start (Fig. 5). This difference arises from interactions with nearby modes in frequency. At the center in Fig. 5, mode 3 is close to modes 2 and 4, and these interactions are reflected in the right-hand terms of Eq. (30), where fractions like $\frac{z_j^k}{\omega_i^2}$ and the energetic

indexes $\nu_i^k$ of other modes come into play when modes are in proximity.

Finally, PZ cancellation is evident for mode 4 at the end of the trajectory. In Fig. 6, $\nu_4^5 = 0$ at the lower right corner, even though the PZ distance $\omega_4 - z_4^5$ (pink) is large in the right in Fig. 5. This is because the pole of mode 4 $\omega_4$ (full line in pink) aligns with the zero of the adjacent mode 5 $z_5^5$
(broken line in cyan), leading to $\omega_4 \approx z_5^5$ and thus $\nu_4^5 \approx 0$ per Eq. (31). The blue area in Fig. 6 confirms this, showing how the energetic index captures both PZ distance and cancellation.

To conclude, the energetic index visualizes the energetic link between cables and modal shapes. It quantifies a cable authority over a mode by connecting to the PZ distance, a widely used authority criterion. The simulation highlights the complex CDPR dynamics, where the assumption of well-separated modes does not hold. The energetic index also accounts for mode interactions, including PZ cancellation.

## 4.3 Simulation of a set of active cables

This section aims to demonstrate the use of the energetic index for selecting active cables. The goal is not to optimize performance but to show that a simple cable selection can achieve adequate damping.

The pose of interest is $X_s = (0.15 - 0.2 - 0.3 \mathbf{0}^T)^T$. Table 3 presents the energetic indexes for all the cables and modes at this pose.

The proposed set of active cables is [3 – 6 – 7]. Here, cable 3 is dominant for modes 1 – 2 – 3 (energetic index of 23.6%, 23.2%, 19.7%), cable 6 is dominant for modes 4 and 6 (energetic index of 20 and 22%) and cable 7 for mode 5 (energetic index of 44.5%).

### 4.3.1 Poles

This subsection aims to illustrate the behaviour of the MIMO system when the three cables are activated compared to the rootlocus of one cable.

<figure><img src="image_8.png" /><figcaption>Fig. 7. Rootlocus for cable 3 with filtered IFF control law</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8. Oscillating poles of the EE with filtered IFF control law for different activations of cables</figcaption></figure>

Figure 7 shows the rootlocus for cable 3 with poles for modes 1, 2, and 3 (red dots) at a gain of $10^8$, named $p_1^3 - p_2^3 - p_3^3$. The alternation of open-loop zeros (black circles) and open-loops poles (black crosses) is well illustrated. The well-separated loops reflect the modes separation. Large loops for modes 1, 2, and 3 (yellow, cyan, and blue in Fig. 7) are due to high energetic indexes of cable 3 in Table 3. Small energetic indexes for the last three modes (Table 3) result in nearly invisible loops at high frequencies in Fig. 7. The pole designed for mode 1 using cable 3 is $p_1^3 = -3 + j133$. Figure 8 compares EE oscillating poles without control (black crosses) and with cables 3, [3 – 6], and [3 – 6 – 7] activated (red, blue, green dots). This is consistent with Table 3 and Eq. (34), where poles combine the contributions of active cables. For mode 1, cable 3 dominates, and the pole for [3 – 6 – 7], $p_1^{3-6-7} = -3.4145 + j132.666$ (green), closely matches $p_1^3$ (red, designed with the rootlocus 7). Using the formula $\xi_i = \frac{|\text{real}(p_i)|}{\sqrt{\text{real}(p_i)^2+\text{imag}(p_i)^2}}$, the modal damping for mode 1 is $\xi_1^3 = 0.023$ and $\xi_1^{3-6-7} = 0.026$, indicating near-decoupled contributions from other cables of the set. For mode 2, however, cable 6 is significant (10% vs 23.6% for cable 3), thus the pole of mode 2 results from both cables 3 and 6 contributions, with cable 7 being negligible.

### 4.3.2 Temporal validation on the non-linear model

This section aims to validate the energetic index on temporal simulation with Simulink on the non-linear and complete Eq. (5).

Firstly, the EE is static and placed at $X_s$. A pulse disturbance at $4s$ is applied by the cable 5. The faster the transient response is attenuated, the greater is the modal damping. Simulink simulates the full non-linear and coupled Eq. (5). For small displacements $\delta X(t) = X(t) - X_s$, the modal shapes
<table><caption>Table 3. Table of fractions of modal strain energy for each mode and each cable (in %) at $X_s$</caption><thead><tr><th>Mode</th><th>Freq. (Hz)</th><th>Cable 1</th><th>Cable 2</th><th>Cable 3</th><th>Cable 4</th><th>Cable 5</th><th>Cable 6</th><th>Cable 7</th><th>Cable 8</th></tr></thead><tbody><tr><td>1</td><td>21.6</td><td>13.8</td><td>6.6</td><td>23.6</td><td>30.6</td><td>0.7</td><td>0.4</td><td>1.9</td><td>0.02</td></tr><tr><td>2</td><td>26.9</td><td>8.7</td><td>18.5</td><td>23.2</td><td>26.7</td><td>1.3</td><td>10</td><td>1.9</td><td>0.9</td></tr><tr><td>3</td><td>35.2</td><td>10</td><td>34.6</td><td>19.7</td><td>15</td><td>3</td><td>0.9</td><td>5.8</td><td>5.3</td></tr><tr><td>4</td><td>40.7</td><td>35.2</td><td>0.3</td><td>0.04</td><td>2</td><td>1.4</td><td>20</td><td>7.8</td><td>4.9</td></tr><tr><td>5</td><td>59.4</td><td>0.5</td><td>1</td><td>0.06</td><td>0.6</td><td>40</td><td>3.8</td><td>44.5</td><td>2.3</td></tr><tr><td>6</td><td>69.8</td><td>0.6</td><td>1.3</td><td>0.08</td><td>0.1</td><td>16.5</td><td>22</td><td>0.8</td><td>54.1</td></tr></tbody></table>

<figure><img src="image_10.png" /><figcaption>Fig. 9. Normalized DOF amplitudes for cable 3 activated</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 10. Normalized modal amplitudes for cable 3 activated</figcaption></figure>

matrix $\Phi$ is computed via modal analysis at pose $X_s$. Modal amplitudes are then obtained in Simulink from the signal $\delta X(t)$ (solution of the non-linear dynamic Eq. (5)) by $\delta\eta(t) = \Phi^{-1}\delta X(t)$. Figure 9 shows the six EE DOFs (numbered from 1 = x to 6 = $\psi$), and Fig. 10 presents the six corresponding modal amplitudes (normalized to peak-to-peak unit amplitude) during the time, just after the pulse perturbation, where cable 3 is activated. In Fig. 9, modes are expressed in the EE DOFs, with frequency components based on the modal shapes. The plot of modal amplitudes shows the effect of active cables choice on modes visually, because of the decoupling in modal space.

In Fig. 10, modes 1 to 3 are well damped, while modes 4 to 6 show nearly identical amplitudes for the blue and red curves. This aligns with the energetic index tool: cable 3 has high energetic indexes for modes 1, 2, and 3 (23.6%, 23.2%, 19.7%) and very low indexes for modes 4, 5, and 6 (near 0%). Although cable 3 does influence modes 4, 5, and 6, its effect is minimal. To damp the higher modes, additional cables are needed, as shown in Fig. 11 with cables 3, 6, and 7 activated.

<figure><img src="image_12.png" /><figcaption>Fig. 11. Normalized modal amplitudes for cables [3 – 6 – 7] activated</figcaption></figure>

All modes are effectively damped, consistent with Fig. 8 and the energetic indexes. It demonstrates near decoupling in modal dampings between the cables of the selected set. Mode 2 is better damped due to the combined contributions of cables 3 and 6.

Secondly, the disturbance rejection is tested during a trajectory. A step5 trajectory $y = -\frac{4}{3}x$ is simulated in the plane $z = -0.3m$, from $X_1 = (0, 0, -0.3, 0^T)^T$ at 2s to $X_2 = (0.1875, -0.25, -0.3, 0^T)^T$ at 10s. The maximal ve-
<figure><img src="image_13.png" /><figcaption>Fig. 12. Trajectory of the point C for cables [3–6–7] activated</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 14. Frequency spectrum of accelerations for cables [3–6] activated</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 13. Frequency spectrum of accelerations for cable 3 activated</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 15. Frequency spectrum of accelerations for cables [3–6–7] activated</figcaption></figure>

locity of the trajectory for *x* and *y* is approximatively 60mm/s. At 7.38s, when $\mathbf{X} = \mathbf{X}_s$ (the pose of interest) with speeds of 34mm/s for *x* and 45mm/s for *y*, a pulse disturbance $\delta\mathbf{W}_{nc} = (10\ 0\ 0\ 10\ 10\ 0)^T N$ is applied to the EE to sufficiently excite all modes visually. Figure 12 shows the translational coordinates of point C, from before the disturbance at $\mathbf{X}_s$ (black cross) to the end $\mathbf{X}_2$, highlighting the oscillations.

Figures 13, 14, and 15 show the frequency spectrum of the acceleration of EE DOFs (numbered from 1 = *x* to 6 = *ψ*) during a period when the EE is near $\mathbf{X}_s$. These plots provide an equivalent view of the modal dampings, similar to the time-domain plots in Fig. 10 and 11.

The frequency spectra peaks correspond to the max of amplitude at natural frequencies (Table 3). Green circles indicate damped modes, while magenta circles (shown for DOF 6) indicate undamped modes. Cable 3 alone controls the first three modes (Fig. 13), but adding cable 6 does not fully damp mode 5 (Fig. 14). When cables 3, 6, and 7 are activated, all modes are damped (Fig. 15). The near decoupling in damping across active cables of the set is clearly illustrated, similar to the static case. In Fig. 12, with cables [3–6–7] activated (red curve), the disturbance is rejected,

and the EE follows the trajectory without oscillations.

In conclusion, the selection index is validated through temporal simulations with the non-linear coupled model for both static and trajectory cases. It demonstrates the effectiveness of the energetic selection index in easily choosing active cables to damp targeted modes, without the need for law optimization.

## 5 DISCUSSIONS

This model can be used to develop an algorithm for selecting active cables based on trajectory and process conditions, ideal for additive manufacturing of medium-sized parts for instance. It could also allow a CDPR with modular $A_i$ grip points to adapt modal shapes while keeping the same active cables configuration. Moreover, collocated control architecture and selection index for HF CDPRs could theoretically apply to motor winders instead of piezoelectric SA pairs, provided EE frequencies match motor position control bandwidth and friction is negligible. However, unlike the immediate feasibility of piezoelectric SA pairs, this practical implementation may
require low-level motor control in addition to me-
chanical limitations, making it less straightforward.
In terms of limitations, this model is limited to small
CDPRs with thin cables. For larger CDPRs, like the
CoGiRo robot [1], piezoelectric transducers are un-
suitable. If cable dynamics significantly impact the
EE, the active cables selection may be inaccurate.

6 CONCLUSIONS

This paper addresses the problem of damp-
ing end-effector (EE) vibrations in small-sized, over-
constrained CDPRs with thin and lightweight ca-
bles. This type of CDPR can be classified as high-
frequency (HF) CDPR, making the use of collocated
piezoelectric sensor-actuator (SA) pairs embedded
on the cables particularly suitable, as initially pro-
posed and explored in a previous work. This pa-
per proposes a model for such a CDPR based on
assumptions from the literature, aiming to use one
or several active cables. Each active cable con-
tains an embedded SA pair, with the actuator con-
trolled by a law and the collocated force sensor mea-
suring the variation in the cable tension. The first
contribution is, by linearizing the dynamic equation
around an equilibrium configuration, the reformula-
tion of the vibration equation of the EE, enabling a
collocated, decentralized active vibration control ar-
chitecture. The second contribution is the proposal
of an active cables selection method to minimize
costs, energy consumption, or active cables inter-
actions. This method uses the pole-zero alternation
property of collocated SA pairs to develop an ener-
getic index that links the EE modal shapes to vari-
ations in cable tensions. The decentralized but not
independent control loops of active cables interact
to define the EE poles. By selecting active cables
with the energetic index, modal poles can be more
easily controlled. Simulations on the complete non-
linear CDPR model validate the concepts, showing
that the control architecture and the energetic index,
derived from the linearized model, yield effective re-
sults when applied to the full non-linear model.
This paper does not focus on optimizing damping
performances, but provides analytical tools for future
implementation of complex collocated controllers,
with the energetic index potentially optimizing cable
control gains for gain scheduling along trajectories.

ACKNOWLEDGEMENTS

This work was supported by the French Na-
tional Research Agency (ANR) through the CAB-
TIVE project, Grant number ANR-22-CE10-0008.

REFERENCES

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## NOMENCLATURE

s Laplace variable

n number of EE DOFs

m number of cables ($m \ge n + 1$)

uᵢ unit vector of cable i

Aᵢ cable hang point on the frame

Bᵢ cable hang point on the EE

B'ᵢ cable hang point on the SA pair

R<sub>OE</sub> rotation matrix

X EE pose

X* EE equilibrium

δX EE oscillations with respect to the equilibrium

dᵢ distance between Aᵢ and Bᵢ

lᵢ cable length

δᵢ piezoelectric actuator displacement

α piezoelectric actuator coefficient between voltage and displacement

e<sub>i,rest</sub> SA pair length at rest

kᵢ cable stiffness

tᵢ cable tension

J<sup>T</sup> transposed Jacobian matrix

J<sub>k</sub><sup>T</sup> column of the transposed Jacobian matrix

M mass matrix

K stiffness matrix

K<sub>axial</sub> stiffness matrix due to the stiffness of cables

K<sub>t</sub> stiffness matrix due to the pretension of cables

G<sub>okk</sub> collocated dynamic transfer matrix for cable k

G<sub>okj</sub> dynamic transfer matrix between a force on cable j and the variation in cable tension k

Φᵢ modal shape of mode i

ηᵢ modal amplitude of mode i

ωᵢ natural frequency of mode i

k'ᵢ modal stiffness of mode i

μᵢ modal mass of mode i

νᵢ<sup>k</sup> energetic index of cable k for mode i

H<sub>k</sub>(s) = $\frac{k_k \delta \delta_k}{-\delta t_k}$ control law for cable k

## APPENDIX A: LINEARIZATION PROCESS DETAILS

$$ 
\begin{aligned}
& f(\mathbf{X}^* + \delta\mathbf{X}, \dot{\mathbf{X}}^* + \delta\dot{\mathbf{X}}, \ddot{\mathbf{X}}^* + \delta\ddot{\mathbf{X}}, \mathbf{l}_0^* + \delta\mathbf{l}_0, \delta^* + \delta\delta, \mathbf{W}_{\text{nc}}^* + \delta\mathbf{W}_{\text{nc}}) \\
& = f(\mathbf{P}^*) + \frac{\partial f}{\partial \mathbf{X}}(\mathbf{P}^*)\delta\mathbf{X} + \frac{\partial f}{\partial \dot{\mathbf{X}}}(\mathbf{P}^*)\delta\dot{\mathbf{X}} + \frac{\partial f}{\partial \ddot{\mathbf{X}}}(\mathbf{P}^*)\delta\ddot{\mathbf{X}} \\
& \quad + \frac{\partial f}{\partial \mathbf{l}_0}(\mathbf{P}^*)\delta\mathbf{l}_0 + \frac{\partial f}{\partial \delta}(\mathbf{P}^*)\delta\delta + \frac{\partial f}{\partial \mathbf{W}_{\text{nc}}}(\mathbf{P}^*)\delta\mathbf{W}_{\text{nc}} = 0
\end{aligned}
\quad (35) $$