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# Active vibration control of cable-driven parallel robots

T Garcia, Didier Rémond, Simon Chesné

► To cite this version:

T Garcia, Didier Rémond, Simon Chesné. Active vibration control of cable-driven parallel robots. 31st International Conference on Noise and Vibration Engineering (ISMA2024 - USD2024), KU Leuven, Sep 2024, Louvain, Belgium. hal-05128171

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https://hal.science/hal-05128171v1

Submitted on 24 Jun 2025

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# Active vibration control of cable-driven parallel robots

T.Garcia, D. Rémond, S. Chesné

INSA LYON, CNRS, LAMCOS, UMR5259
69621 Villeurbanne, France

e-mail: thibault.garcia@insa-lyon.fr

## Abstract

A cable-driven parallel robot (CDPR) is a manipulator characterized by its end-effector (EE) being connected to the base by cables. An electric motor/winch system enables the adjustment of each cable length and tension. The primary drawback of cables lies in their low stiffness, resulting in end-effector vibrations along the trajectory. This study aims to manage EE vibrational response, not by directly controlling its degrees of freedom, but through active control of cable tensions. The contribution involves the collocated utilization of two types of actuator-sensor pairs for active control: piezoelectric transducers embedded on cables and electric motors with embedded measurements of their torques. The limitations and consequences of this integration are explored here. Motors are usually considered as low-frequency actuators. The challenge here is to optimize the dynamics of the motor/winch systems in order to achieve control. An analysis, both analytical and numerical, of integrating these control devices and their respective laws is conducted.

## 1 Introduction

Cable-Driven Parallel Robots (CDPRs) represent a type of parallel robot where the mobile platform called end-effector (EE) is operated by cables controlled by winding motor systems. Such a system is well appreciated for camera visualization or manipulation with big CDPRs for instance [1]. Nevertheless, the primary drawback of cables lies in their low stiffness and internal damping, resulting in EE vibrations, preventing high precision processes. In order to solve this problem, Lacaze explores the possibility of integrating collocated piezoelectric sensor-actuator (SA) pairs on the cables and to realize active control by a simple control law and some improvements [2], [3], but at one pose (a given position and orientation of the EE). The complex and evolving vibration architecture of CDPRs fundamentally prevents optimizing a control law for all possible poses, the active cables selection for embedding these pairs being also critical. Thus to remedy this problem, Garcia proposes in [4] both a proper formulation of the cable collocated active control architecture and an active cables selection method along a trajectory on fully constrained (unsuspended EE and more cables than DOFs) medium-sized CDPRs in order to perform high precision trajectories.

The low cost of CDPRs, their high speeds due to the low weight of the cables and their large workspace lead to a growing interest in using CDPRs for additive manufacturing, a process that has seen spectacular growth in recent years. This paper is part of the CABTIVE project (continuation of the CABFAB project [2],[3],[5],[6],[7]) which focuses on the development and study of fully constrained medium-sized CDPRs (typically $1mx1mx1m$) for additive manufacturing via the polymer extrusion process. This paper consists in an application case of the article of Garcia [4] on a 8-cables 3D cable printer. An originality of this contribution consists in the integration of motor-winders in the 3D printer model, in order to realize collocated active control by cables. A selection method of modes to control and of active cables for a given plane is proposed, and then it is illustrated with both motors and piezoelectric transducers for performing a high-quality 3D printing. A discussion on the advantages and drawbacks of each actuator is finally proposed.
## 2 3D cable printer

This section focuses on a presentation of the 3D cable printer studied in this paper. The modeling is detailed in [4]. Here, only the basic equations are reminded in order to help the reader to understand the subject of this paper. The studied CDPR (without the representation of piezoelectric transducers) can be seen in Fig. 1.

<figure><img src="image_3.png" /><figcaption>Figure 1: 3D cable printer plot</figcaption></figure>

The CDPR is 1mx1mx1m dimension. The $A_i$ points are the assumed fixed exit points on the frame. The extrusion head (EE) is a $m_e = 500g$ cylindrical mass of 30mm of radius and 50mm of height. The cables are thin steel cables of diameter of 0.54mm with a Young modulus $E = 6.9 * 10^9$ Pa. The orange plot represents the polymer cord for the Fused Deposition Modeling (FDM) 3D printing process. It is supposed here that this process is mastered and does not affect the dynamic behaviour of the robot. The center of mass $G$ is supposed at the center of the EE. Using the Euler X-Y-Z angle representation, the pose of the EE is described using:

$$X = \begin{pmatrix} \mathbf{OG} \\ \alpha \end{pmatrix} = (x \ y \ z \ \phi \ \theta \ \psi)^T \quad (1)$$

The point of interest, the tip of the 3D printing nozzle (which is 30mm long), is represented by the point $C$. Its position can be deducted from the EE pose by the formula:

$$OC = OG + R_{OE}(\alpha)GC^E \quad (2)$$

With $GC^E = (0 \ 0 \ -h')^T$, $h' = h/2 + 30mm$ the distance between $G$ and $C$ and $R_{OE}$ the rotational matrix which depends on the orientation of the frame attached to the EE $E$ with respect to the fixed inertial frame $O$ attached to the cubic frame. A representation can be seen in Fig. 7.

The 3D printing process demanding quasi-zero efforts, only the weight can be taken into account. The EE dynamic equation (considering that all the mass is concentrated in $G$) is:
$$ \bar{M}\ddot{X} + \bar{C}\dot{X} = \begin{pmatrix} \mathbf{1}_3 & \mathbf{0}_3 \\ \mathbf{0}_3 & \mathbf{S}^T \end{pmatrix} [\mathbf{J}^T \mathbf{t} + \mathbf{W}_g] \quad (3) $$

With $\bar{M}$ the mass matrix, $\bar{C}$ the coriolis and centrifugal matrix, $\mathbf{W}_g = (0 \ 0 \ -m_{eg} \ 0 \ 0 \ 0)^T$ the weight of the EE, and $\mathbf{J}^T = (\cdots \ \mathbf{u}_i \ \cdots \ \mathbf{G} \mathbf{B}_i \times \mathbf{u}_i \ \cdots)$ the classical jacobian matrix projecting the cable tensions into the cartesian space. The vector $\mathbf{u}_i = \frac{\mathbf{B}_i \mathbf{A}_i}{\|\mathbf{B}_i \mathbf{A}_i\|}$ is defined as the unit cable vector, with $\mathbf{B}_i$ the hang point of the cable on the EE. The cable tension is defined by the elastic model $t_i = k_i(l_i - l_{0i})$, with $k_i = \frac{EA}{l_{0i}}$ its stiffness. $l_i$ is the stretch length of the cable and $l_{0i}$ is the unstretched length, which is the variable for controlling the trajectory of the EE.

## 2.1 Piezoelectric actuators

The piezoelectric actuator is a APA60SM from CEDRAT TECHNOLOGIES, is embedded on the EE on the $\mathbf{B}_i$ points and is driven in voltage by $\delta_i = \alpha V_i$, with $\delta_i$ the free displacement in $m$, $\alpha = 4.5 * 10^{-7}$ a constant in $m/V$, and $V_i$ the applied voltage in $V$. The cable length then the cable tension is changed by the actuator displacement:

$$ l_i = ||\mathbf{B}_i \mathbf{A}_i|| - e_{i,\text{rest}} + \delta_i \quad (4) $$

With $e_{i,rest}$ the sensor-actuator (SA) pair length at rest. In the following, all the cables have an embedded SA pair.

## 2.2 Motor-winders

<figure><img src="image_4.png" /><figcaption>Figure 2: Motor-winder scheme</figcaption></figure>

Figure 2 depicts the motor-winder scheme. The motor-winder system is chosen without reducer to limit friction and inaccuracies, and by a threaded winder system. In this way, the cable does not wrap around itself, which prevents a change in the radius of the winder by superimposing the cables [8]. In addition, overlapping cables leads to vibrations. The winder frame is defined by $(C_i, x_i, y_i, z_i)$. The motor axis is defined by $\vec{u}_{\Delta_i}$ and is opposite to the $x_i$ axis. The motor rotation $q_i$ is positive if it is directed according the trigonometric convention thus here along the motor axis. The motor shaft (rotor) is assumed rigid thus the motor torque is perfectly transmitted to the point $C_i$. Moreover, the link between the winder and the rotor is considered perfect, thus the forces are perfectly transmitted in the point $C_i$. The length $A'_i A_i$ ($A'_i$ the exit
point on the winder in green in Fig. 2) is assumed small and negligible regarding the length $l_i$ in the CDPR's cube, thus this portion of cable is small enough to consider it inextensible, and therefore perfectly transmits the cable tension $t_i$. The thread pitch is defined by $p$. The motor $i$ controls the unstretched length $l_{0i}$ by the equation:

$$l_{0i} = l_{0i}^{\mathbf{X}_{\text{init}}} - Rq_i \quad (5)$$

With $l_{0i}^{\mathbf{X}_{\text{init}}}$ is the initial value when rotations are zeros (here $\mathbf{X}_{\text{init}}$ is the pose when the point $C$ is at the cube center) and $R = \sqrt{r_{enr}^2 + (\frac{p}{2\pi})^2}$ is the winch ratio. The chosen motor is a permanent magnet synchronous machine from BECKHOFF AUTOMATION AM8053 of shaft inertia $J_m = 593*10^{-6}kg*m^2$. The threaded winder is in aluminium with a radius $r_{enr} = 50mm$ and an inertia equal to $J_{enr} = 253 * 10^{-6}kg * m^2$. Its pitch is equal to $0.8mm$ because the cables are thin. Its length is equal to $11mm$ thus the winder can be considered as a disc with a negligible deviation torque (Fig. 2). Moreover, this means that in practice $R \cong r_{enr}$.

The proposed placement of motor-winder systems allows to describe all the motors by Fig. 2. The cable torque in $C_i$ is equal to $\vec{M}_{C_i}(\vec{t}_i) = -r_{enr}t_i u_{\vec{\Delta}_i}$. The driven torque $C_{em_i}$ transmitted in $C_i$, obtained by the electromechanical interaction between the stator and the rotor is considered positive along the motor axis, thus $C_{em_i} = C_{em_i} u_{\vec{\Delta}_i}$. The proposed motor-winder placement unifies the dynamic equation and therefore the motor control scheme, regardless of the placement in the CDPR. The theorem of the dynamic moment expressed in $C_i$, projected on the motor axis $u_{\vec{\Delta}_i}$ and in a matricial form is:

$$J_{tm}\ddot{q} + f_v\dot{q} = C_{em} - r_{enr}t \quad (6)$$

With $J_{tm} = J_m + J_{enr}$ the total inertia and $f_v = 0.01Nm/(rad/s)$ the estimated motor shaft viscous friction coefficient.

The motor control algorithm is inspired by [9] and Fig. 3 presents its inner loop scheme:

<figure><img src="image_5.png" /><figcaption>Figure 3: Inner loop scheme</figcaption></figure>

The frequency bandwidth of the electrical dynamics of the motor (establishment of the current) is assumed very high enough to be neglected, and thus the associated transfer function is considered as equal to 1.

This leads to:

$$q_i = \frac{1}{1 + \frac{f_v + K_v}{K_v K_p} s + \frac{J_{tm}}{K_v K_p} s^2} q_i^* + \frac{-\frac{r_{enr}}{K_v K_p}}{1 + \frac{f_v + K_v}{K_v K_p} s + \frac{J_{tm}}{K_v K_p} s^2} t_i \quad (7)$$

The gains $K_v$ and $K_p$ have to be high enough in order to decouple the dynamics of the EE from that of motors ($f_0 > f_6$ with $f_0$ the inner loop natural frequency and $f_6$ the 6th mode frequency). Here, the gains
are settled so as to have an equivalent damping equal to $\frac{1}{\sqrt{2}}$. $K_v$ is set to 2 and this leads to $K_p = 1.2 * 10^3$ and $f_0 \approx 267Hz$. This leads also to $\frac{r_{enr}}{K_v K_p} \approx 2 * 10^{-5}$, thus the perturbation $t_i$ is considered rejected by this loop. Thus here $q_i \approx q_i^*$ and $l_{0i} \approx l_{0i}^*$. However, this model is limited by electrical, digital, thermal and other mechanical aspects not being taken into account, this paper focusing on a concept illustration.

# 3 Active control scheme

## 3.1 EE vibrations

The CDPR can be locally seen as an evolutive structure with mass and stiffness matrices depending on the EE pose [4], [10]:

$$ M\delta\ddot{X} + K\delta X = -J^T\text{diag}(k_i \frac{l_i}{l_{0i}})\delta l_0 + J^T\text{diag}(k_i)\delta\delta + \delta F \quad (8) $$

With $\delta F$ some disturbances, like cable friction or the friction of the printing nozzle tip C on the part. $\delta X$ represents the infinitesimal displacement of the EE with respect to the equilibrium $X^*$.

## 3.2 Collocated active control with motors or piezoelectric actuators

In [4], the formulation of the collocated active control using piezoelectric transducers is formulated. The gains $K_v, K_p$ being large enough, thus $\delta l_0 \approx \delta l_0^*$, $\delta l_0 = -R\delta q$ and thus the same principle can be applied with motors, which is illustrated in Fig. 4.

<figure><img src="image_6.png" /><figcaption>Figure 4: Collocated control scheme at the cable level using a motor or a piezoelectric actuator</figcaption></figure>

This figure depicts the collocated vibration control scheme at a cable level k, using either a SA pair on the cable or the motor associated to this cable, illustrated by the switch. The case where there are both actuators to perform vibration control on a cable is not described here. The transfer function $G_{okk}$ is the collocated transfer function of the cable k. The transfer function $G_{okj}$ represents the coupling between cables. The control law is described by $H_k$. Vibration control on an active cable (either a SA pair or a motor) produces a force that will be seen by an other active cable as a disturbance, which is depicted by the term $-H_j\delta t_j$ [4]. Nevertheless, although the variation in cable tension $\delta t_k$ is the variable to control, it is in the reality impossible to measure perfectly this component. The piezoelectric sensor can not measure the continuous component of a cable tension. Moreover, a conditioner is generally added. For modeling this, a 1Hz high-pass filter $\frac{s}{s+\omega_c}$ with $\omega_c = 2\pi rad/s$ is considered. For the motors, the measurement of the cable tension is indirectly determined by the measurement of both the shaft rotation (simple and double derivation) and the currents:
$$t_{k, \text{estimated}} = \frac{1}{r_{\text{enr}}} (C_{em_k} - J_{tm} \ddot{q}_k - f_v \dot{q}_k) \quad (9)$$

To be consistent with the piezoelectric SA pairs, a virtual high-pass filter $\omega_c$ of 1Hz for conditioning is added to the motor's vibration control scheme. Moreover, Lacaze proposes a bandpass filter $\omega_{PH}$ and $\omega_{PB}$ between 20 and 300Hz in [3] in order to counteract excessive variations in cable tensions during the trajectory. To be consistent with Fig. 4, these filters have to be taken into account in the setting of the control law which is done in the static case, thus in $H_k(s)$.

## 4 Trajectory generation scheme with active control

The temporal simulation of the generation of trajectory and of the active control loops is given by Fig. 5:

<figure><img src="image_7.png" /><figcaption>Figure 5: Temporal simulation scheme</figcaption></figure>

Knowing the trajectory of the nozzle tip $C$, it is possible to determine the trajectory of the EE to be generated. Then, by the inverse geometric model, a tension distribution algorithm (the system being overactuated), and the cable elastic model, the unstretched length and therefore the commands of the motor rotations are generated. The APA is driven in voltage $V_i$. The motor active control generates an instruction of small variation of rotation $\delta q_k^*$ which is added to the rotation trajectory (in green in Fig. 5), like in [11], [12]. When a cable is designated to be an active cable, thus the loop of Fig. 4 is activated which corresponds to either the motor-winder loop (in green) or the piezoelectric transducer loop (in purple). The real implementation gives:

$$\delta q_k^* = \frac{1}{R} \frac{1}{k_k \frac{l_k}{l_{0k}}} (-H_{k, \text{law}}^{\text{mot}}) \frac{s}{s + \omega_{PH}} \frac{\omega_{PB}}{s + \omega_{PB}} \frac{s}{s + \omega_c} t_{k, \text{estimate}} \quad (10)$$

$$\delta \delta_k = \alpha \frac{1}{k_k} (-H_{k, \text{law}}^{\text{piezo}}) \frac{s}{s + \omega_{PH}} \frac{\omega_{PB}}{s + \omega_{PB}} \frac{s}{s + \omega_c} t_k \quad (11)$$

Actuators are in fact driven in force. The driven force for motors is $k_k \frac{l_k}{l_{0k}} R \delta q_k$, and for piezoelectric actuators it is $k_k \delta \delta_k$. $H_{k, \text{law}}^{\text{mot}}$ and $H_{k, \text{law}}^{\text{piezo}}$ are the basic laws to be chosen. In this paper, the simple law Integral Force Feedback [3], [13], [14] $H_{k, \text{IFF}} = \frac{g_k}{s}$ is chosen, with $g_k$ the gain to set up. The gains are set to $4.4 * 10^8$ for piezoelectric actuators and to 200 for motor-winders in order to have the same force generated by the two actuators on a cable by the relation $g_k^{\text{mot}} = \alpha g_k^{\text{piezo}}$. In this way, the law is settled fairly for both actuators,
which can then be compared. These gains are the same for all the trajectory, and are chosen because they
bring a sufficient damping to be visible (when a cable has authority) without oversaturating the actuators.
The APAs are saturated between −20 and 150V due to the power amplificator (displacement between around
−10 and 70µm), and the motors are virtually saturated between ±0.08°, a choice which is equivalent to a
translation displacement between ±70µm. Moreover, the maximum torque of the motor is equal to 53Nm,
thus the motor torque is saturated at ±53Nm.

5 3D printing

The objective of this part is to apply active control in printing the word "ISMA" in the plane $z_c = -0.4m$, when $x_c, y_c$ are limited to $\pm0.25m$. The speed profil of the point C coordinates is depicted in Fig. 6, respecting the maximum printing speed of $0.1m/s$ [5]:

<figure><img src="image_8.png" /><figcaption>Figure 6: Speed profil of the trajectory "ISMA"</figcaption></figure>

This speed profile will cause sudden changes in the trajectory, which will excite structural modes via the
cable elasticity. The simulation is performed using ode3 from Simulink software. The following parts will
highlight the importance of both the mode choice and active cables selection.

**5.1 Targeted mode selection**

Firstly, it is necessary to select the modes to control. Indeed, it is not relevant to control all the six modes for
each pose, because modal shapes will not move the printing nozzle (point C) in the same way, like illustrated
in Fig. 7.

<figure><img src="image_9.png" /><figcaption>Figure 7: Mode selection illustration</figcaption></figure>
On the left, the modal shape of pure rotation along the EE axis does not cause a displacement of point C, thus this mode can be neglected. On the right, the modal shape of pure rotation on the plane causes a displacement δOC which has to be minimized by vibration control. Thus generally, to obtain a good quality 3D part, the nozzle displacement must be minimized according to modal shapes.

Modal decomposition leads to $\delta\mathbf{X} = (\delta\mathbf{O}\mathbf{G} \ \delta\boldsymbol{\alpha})^T = \sum_{i=1}^{6} \delta\eta_i \Phi_i$, with $\delta\eta_i$ the modal amplitude and $\Phi_i$ the modal shape. The associated displacement of point C is thus obtained by linearization of Eq. (2):

$$ \delta\mathbf{O}\mathbf{C} = \delta\mathbf{O}\mathbf{G} + [\delta\boldsymbol{\alpha} \times] \mathbf{R}_{\mathbf{O}\mathbf{E}}(\boldsymbol{\alpha}^*) \mathbf{G}\mathbf{C}^\mathbf{E} = \sum_{i=1}^{6} \delta\eta_i \left[ \begin{pmatrix} \Phi_i(x) \\ \Phi_i(y) \\ \Phi_i(z) \end{pmatrix} + \begin{pmatrix} \Phi_i(\phi) \\ \Phi_i(\theta) \\ \Phi_i(\psi) \end{pmatrix} \times \right] \mathbf{G}\mathbf{C}^* = \sum_{i=1}^{6} \delta\eta_i \delta\mathbf{O}\mathbf{C}_{\text{shape}}^i \quad (12) $$

The quantity $\delta\mathbf{O}\mathbf{C}_{\text{shape}}^i$ is the displacement vector of point C when the EE is vibrating according to mode i. The quantity to minimize is thus its norm $||\delta\mathbf{O}\mathbf{C}_{\text{shape}}^i||$ representing its modal displacement. To compare the modes between them, the following quantity is defined:

$$ \beta_i = \frac{||\delta\mathbf{O}\mathbf{C}_{\text{shape}}^i||}{\sum_{j=1}^{6} ||\delta\mathbf{O}\mathbf{C}_{\text{shape}}^j||} \quad (13) $$

$\beta_i$ represents the $i^{th}$ modal participation in the displacement of point C. If it is high compared to the other modes, thus this mode causes a high displacement, and thus this mode has to be damped.

Figure 8 depicts the map of modal participations for each mode, in case a layer of part to be printed is in the plane $z_c = -0.4m$:

<figure><img src="image_10.png" /><figcaption>Figure 8: Map of modal participations $\beta_i$ for plane $z_c = -0.4m$</figcaption></figure>

This figure 8 illustrates well the change in the dynamics of the EE, because according to the position of the point C the modes will not cause a displacement in the same proportions. For example it is clearly visible that modes 1 and 2 are not the modes to be particularly targeted ($\beta_i$ not above 10% in dark blue), unlike mode 3 with a large area where its modal participation is the maximum (from 82% at the center in red to
near 30% in cyan at the periphery). Modes 5 and 6 can also be targeted but not at the center because $\beta_5, \beta_6$ are in dark blue. If the part has delicate areas in the extremes, then it will be necessary to favor the modes 4,5 and 6 and no longer the mode 3 because in this case its participation decreases to near 10% in dark blue.

<figure><img src="image_11.png" /><figcaption>Figure 9: Map of modal participations $\beta_i$ for plane $z_c = 0m$</figcaption></figure>

Figure 9 depicts the same reasoning with the printing plane equal to $z_c = 0m$. Here, the dynamics is completely different and mode 6 has to be completely neglected. Moreover, modes 1 and 2 may be controlled because their $\beta_i$ can achieve 25% in some areas. Mode 3 is still dominant in red and cyan/green especially in the center.

## 5.2 Cable choice

Secondly, it is necessary to select active cables to control the selected modes. For this, one can use the energetic index method described in [4]. This index $\nu_i^k$ appears in the transfer function $G_{okk}$ of a cable $k$. The greater this term is, the greater the authority of a cable $k$ for controlling a mode $i$ is. After mode selection by $\beta_i$, the 6x8 matrix of energetic indexes can be computed:

$$ \nu = (\dots \quad \nu^k \quad \dots) \qquad (14) $$

With the 6x1 vector $\nu^k = (\dots \quad \nu_i^k \quad \dots)^T$ the vector of energetic indexes for the cable $k$. Then the weighting 6x1 vector $p$ can be defined, where $p_i = 0$ if $i$ is not one of the selected $\beta_i$, as the 8x1 choice vector:

$$ \mathbf{c} = (\dots \quad c_k = \sum_{i=1}^{6} p_i \nu_i^k \quad \dots)^T \qquad (15) $$

The chosen cables are those which have the highest $c_k$.

This paper being a first approach, here two modes are targeted, with two active cables to select. Arbitrarily, a weight of 5 is chosen for the mode with the highest $\beta_i$, and 1 for the second. Figure 10 depicts in the plane $z_c = -0.4m$ the poses when a cable is chosen (in green) or not (in red).
<figure><img src="image_12.png" /><figcaption>Figure 10: Map of active cables choice for plane $z_c = -0.4m$</figcaption></figure>

In Fig. 10, the green areas are the areas where the cable is chosen, contrary to the red areas. There are thus some cables which have to be preferred (top cables). It is clear that in practice it is not relevant to activate and desactivate the cables according to the selection areas, but this kind of map allows to choose the cable or cables to be preferred for the printing plane. For the following, the cable 4 which covers a relatively high green area of selection will be tested. The inverse reasoning of cable selection gives the worst cables like the cable 5 for instance, thus it will be tested in comparison with cable 4.

## 5.3 Active cable comparison: one active cable

Here the importance of the active cable choice is highlighted by testing two different active cables, using motor-winders for active control (in green in Fig. 5).

<figure><img src="image_13.png" /><figcaption>Figure 11: Comparison between cable 4 ON (blue) and cable 5 ON (red) with motor-winder</figcaption></figure>
The simulation is performed using the cable 4 in blue and the cable 5 in red. The result is depicted in Fig. 11. Here, the data OC = OC* + 10δOC is plotted in order to amplify the error. In this figure, the vibrations of z are more visible because of the large displacements of x and y which makes their associated vibrations less visible. The red plot corresponding to cable 5 is generally predominant in terms of oscillation amplitudes, which reflects a lower ability of cable 5 to reject disturbances in this plane. On the contrary, cable 4 which has a large green area in Fig. 10 is globally better. The authors wish to emphasize the global nature of this observation. Indeed, locally the behavior is complex, because a cable that has not been selected can still have good authority on a mode to control. In addition there are red areas for cable 4 that are green for the 5 in Fig. 10 (for instance in the extreme right), thus in these areas cable 5 is a little better. It can be illustrated in the extreme right of the plane where the blue curve (cable 4) seems to have a greater amplitude than the red curve (cable 5). Finally, depending on the excitement, the modes will be more or less present.

**5.4 Active cable comparison: several active cables**

A possibility for choosing the cable configuration is to cover the majority of the plane in green in Fig. 10 using several cables.

<figure><img src="image_14.png" /><figcaption>Figure 12: Comparison between cables 1 – 4 – 8 ON (blue) and cables 5 – 6 – 8 ON (red) with motor-winders</figcaption></figure>

Figure 12 depicts the printing of "ISMA" with a three cables configuration using motors. The blue curve corresponds to cables 1 – 4 – 8 ON, which is a good configuration according to Fig. 10, where these cables cover almost the whole plane. Cable 8 is chosen for the corners of the far right of the plane in Fig. 10. On the contrary, the 5 – 6 – 8 cable configuration in red can be seen as a bad configuration according to Fig. 10. It can be seen in Fig. 12 that the bad configuration in red is worse than the good configuration in blue, the amplitude of oscillations being poorly damped in red. Thus, for printing a good part in the plane $z_c = -0.4m$, using cables 1 – 4 – 8 is a good choice and permits to obtain a good quality 3D part.

**5.5 Comparison between motors and piezoelectric actuators for vibration control**

The principle is to compare, for the 1 – 4 – 8 cable configuration, the motor and piezoelectric actuator actuations. The gains being equivalent in terms of produced forces, the only differences between the two actuators are the equilibrium state (centered or not) and the saturations.
<figure><img src="image_15.png" /><figcaption>Figure 13: Comparison between motor actuation (blue) and piezoelectric actuation (red) with cables 1–4–8 ON</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Figure 14: Piezoelectric actuator elongations</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Figure 15: Motor rotation instructions</figcaption></figure>

Figure 13 depicts the comparison between the two actuators, the blue curve being for the motor actuation and the red curve the piezoelectric one. One can remark that the blue curve is globally better than the red curve. This can be explained by the fact that the equilibrium elongation of a APA is $\delta_k^* = 0$, thus for active control the displacement $\delta\delta_k$ is oscillating between 70 and $-10\mu m$ but not centered, which is depicted in Fig. 14. On the contrary, the motor displacement is centered in Fig. 15, which is a choice. Moreover, the saturation of the APA is due to the power amplificator and can not be changed, whereas the motor saturation is here a choice and can be greater, although the motor is saturated in torque. Thus centering and saturations are impacting the performance of vibration control. Indeed, the bandpass filter of the control law has a response time, thus the variations in tensions due to the generation of the trajectory are felt by the vibration control. Here, the motor has more displacement amplitude than the APA, so the vibration control with the motors is better. For instance, it can be clairly illustrated in the figures 14 and 15 before 10s where the red curve for cable 8 is saturated for the APA but not for the motor. For solving the problem of centering for piezoelectric actuators, it is possible to define the displacement at equilibrium in a centered way between $-10$ and $70\mu m$, thus $\delta_k^* = 40\mu m$.

More generally, there are advantages and drawbacks for each actuators. Firstly, using SA pairs for the choice of active cables implies to embark a pair on the cable, but it is not desirable to have to embark all the cables because it can be expensive, and because the practical integration on cables can limit some desired poses. Thus it can limit the choice of active cables. Using the motors, which are anyway present for the generation of the trajectory, avoids this problem and allows to keep the possibility to use all the cables for all the printer poses, without additional cost. Nevertheless, the use of vibration control with the motors can in practice be limited by the experimental integration in terms of software, thermal, inertia but also in terms of friction that could distort the cable tension measurements. In addition, the positioning of the motors relatively to the mounting points $A_i$ must be considered in order to best control the elastic behavior of the cable, which leads to an additional design challenge. Thus SA pairs are preferable because the voltage control is much simpler and the use of a force sensor near the actuator allows to be perfectly collocated, which will not be really the case with the motors. Indeed, the collocation is formulated between the motor and the variation in the cable tension, and if the frequency bandwidth $f_0$ of the inner loop is too low, if the elastic behavior of the cable is not perfectly mastered, or if the current measurements are noisy, the collocation may be "lost" and some instabilities can occur.
# 6 Conclusion

In this contribution, a 3D cable printer with embedded piezoelectric actuators and motor-winders is studied, with a collocated active vibration control solution. The motor inner loop for rotation command is described and settled in order to be able to perform control on the modes of the end-effector corresponding to the printhead. The formalism is the same as piezoelectric sensor-actuator (SA) pairs previously described in an authors' article [4]. The real temporal integration of the trajectory generation and of the vibration control with motors or SA pairs is described. Based on the modal participation in the displacement of the tip of the printing nozzle for mode selection, and the energetic index previously highlighted by the authors [4], a method for choosing active cables on a given plane is proposed. Finally, temporal simulations to print a part in 3D allows to illustrate the importance of the modes to control and the choice of active cables. Moreover, the limitations and advantages of each actuator for performing vibration control are illustrated and discussed. This paper nevertheless remains a first approach, and some improvements will be led in future works.

# Acknowledgements

The authors would like to acknowledge the support of the Agence Nationale de la Recherche (ANR) under Grant ANR-22-CE10-0008 for funding this research. This work was conducted as part of the CABTIVE project.

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