Loïc Cuvillon
email: l.cuvillon@unistra.fr

Xavier Weber
email: .xavier.weber@gmail.com

Jacques Gangloff
email: jacques.gangloff@unistra.fr

Modal Control for Active Vibration Damping of Cable-Driven Parallel Robots

Cable-driven parallel robots are well suited for applications that require a very large workspace. Thanks to their lightweight moving parts, they can achieve high dynamics while remaining pretty safe for nearby human workers. Furthermore, their size only depends on the length of the cables, thus their scale is almost totally decoupled from their cost. However, due to the cables, the stiffness is very low with respect to rigid link robots, inducing slowly damped oscillations of the end effector. Previous works have shown those vibrations can be effectively damped by the winch actuators thanks to active vibration damping techniques. In this paper, a gain-scheduling approach is proposed based on a linearized model of the robot dynamics. This model is projected in modal space yielding 6 decoupled transfer functions for 6 degrees of freedom of a cable-driven parallel robot using thin cables. The stability of the proposed control law is analyzed for a static and a moving end effector. The proposed control algorithm is validated experimentally on an 8-cables suspended robot prototype.

Introduction

Cable-Driven Parallel Robots (CDPR) are parallel robots using flexible cables instead of rigid sets of articulated links. This specificity allows for lightweight robots with large payloads and very large workspace. A robot with a very large workspace opens a lot of new opportunities, e.g. moving a camera over a stadium [1], moving a stretcher for rescue operations [2] or moving dexterously a heavy payload in a large warehouse [START_REF] Gouttefarde | Geometry Selection of a Redundantly Actuated Cable-Suspended Parallel Robot[END_REF], but several problems arise when long cables are used.

With thin cables, the CDPR stiffness is extremely low compared to a rigid link robot. With thick cables, sagging occurs. The sagging is a deflection of the cable under its own weight, thus altering the geometrical model and reducing stiffness [START_REF] Gouttefarde | Simplified static analysis of large-dimension parallel cable-driven robots[END_REF]. Low stiffness is responsible for low frequency natural vibrations of the CDPR end effector. Low frequency vibrations are especially annoying because of the long settling time of the end effector. Several approaches have been studied to tackle CDPR vibrations issues. One of them consists simply in increasing the stiffness by increasing cable tensions in order to reduce the settling time. This approach is proposed in [START_REF] Kawamura | Development of an ultrahigh speed robot FAL-CON using wire drive system[END_REF] but it is limited by the admissible tension in the cables. Furthermore, a lot of power is lost for maintaining this tension. It is also unsuitable for underconstrained or suspended CDPR where some antagonist cables are lacking. The other approach is active damping of these vibrations with either embedded actuators (reactionbased mass inertia [START_REF] Weber | Active vibration canceling of a cable-driven parallel robot using reaction wheels[END_REF][START_REF] De Rijk | Out-of-plane vibration control of a planar cable-driven parallel robot[END_REF] or thrusters [START_REF] Sellet | Active damping of parallel robots driven by flexible cables using cold-gas thrusters[END_REF]) or with cable actuation and exteroceptive feedback [START_REF] Yanai | Antisway control for wire-suspended mechanism based on dynamics compensation[END_REF][START_REF] Caverly | Dynamic modeling and noncollocated control of a flexible planar cable-driven manipulator[END_REF][START_REF] Khosravi | Dynamic Mod-Cuvillon, L., et al. 12 eling and Control of Parallel Robots With Elastic Cables: Singular Perturbation Approach[END_REF][START_REF] Khosravi | Stability analysis and robust pid control of cable-driven robots considering elasticity in cables[END_REF][START_REF] Begey | Dynamic control of parallel robots driven by flexible cables and actuated by position-controlled winches[END_REF][START_REF] Meunier | Control of an overactuated cable-driven parallel mechanism for a radio telescope application[END_REF][START_REF] Jamshidifar | Vibration decoupled modeling and robust control of redundant cable-driven parallel robots[END_REF][START_REF] Baklouti | Vibration reduction of cable-driven parallel robots through elasto-dynamic model-based control[END_REF].

Inspired by earlier studies on robot manipulator with flexible joints [START_REF] Ghorbel | Adaptive control of flexible-joint manipulators[END_REF], Cavarly [START_REF] Caverly | Dynamic modeling and noncollocated control of a flexible planar cable-driven manipulator[END_REF] and Khosravi [START_REF] Khosravi | Dynamic Mod-Cuvillon, L., et al. 12 eling and Control of Parallel Robots With Elastic Cables: Singular Perturbation Approach[END_REF] use a torque control signal of the winch actuators to ensure a stable trajectory of the CDPR. In [START_REF] Caverly | Dynamic modeling and noncollocated control of a flexible planar cable-driven manipulator[END_REF], a passivity-based control is proposed based on a dynamic model that includes the end effector, the inertia of the winches and a lumped-mass model of the cables. A proportional feedback on the stretching velocity of the massless cables combined respectively with a computed torque control or a robust PID control is studied respectively in [START_REF] Khosravi | Dynamic Mod-Cuvillon, L., et al. 12 eling and Control of Parallel Robots With Elastic Cables: Singular Perturbation Approach[END_REF] and [START_REF] Khosravi | Stability analysis and robust pid control of cable-driven robots considering elasticity in cables[END_REF]. Simulations show the effectiveness of the method to track a known trajectory, while damping vibrations. Assuming high cable stiffness, i.e. that cable vibrations are fast, a proof of stability is derived using the singular perturbation theory [START_REF] Khalil | Nonlinear Systems[END_REF]. More recently, a control law combining a PID feedback and a feed-forward term was proposed in [START_REF] Baklouti | Vibration reduction of cable-driven parallel robots through elasto-dynamic model-based control[END_REF]. The feed-forward aims at rejecting the end effector vibrations by inverting the elasto-dynamic model.

Instead of using torque control of the winches actuators, Meunier [START_REF] Meunier | Control of an overactuated cable-driven parallel mechanism for a radio telescope application[END_REF] and Beguey [START_REF] Begey | Dynamic control of parallel robots driven by flexible cables and actuated by position-controlled winches[END_REF] propose a cascade controller based on an inner loop controlling the winch angular positions. By controlling the unstretched cable length through the winding of the cable around the pulley, the elongation and thereby tension of the elastic cable can be effectively regulated. In [START_REF] Begey | Dynamic control of parallel robots driven by flexible cables and actuated by position-controlled winches[END_REF], it is established how the inner position loop can efficiently cope with nonlinearites such as dry friction, usually associated with actuators with high reduction ratio gearboxes. To track a desired trajectory and reject the disturbances, an outer loop regulates the end effector Cartesian coordinates with a computed torque control using the inversion of the CDPR dynamic model and a PD controller. Pose and velocity feedback of the end effector are respectively measured with a high-speed camera and an Inertial Measurement Unit (IMU). Stability of the cascade control scheme is proved using the singular perturbation theory where the assumption of a high cable stiffness in [START_REF] Khosravi | Dynamic Mod-Cuvillon, L., et al. 12 eling and Control of Parallel Robots With Elastic Cables: Singular Perturbation Approach[END_REF] is replaced by the assumption of a high gain of the inner position loop controller. In [START_REF] Meunier | Control of an overactuated cable-driven parallel mechanism for a radio telescope application[END_REF], the outer loop is composed of a PID controller that rejects the joint-space position errors in order to achieve the desired end effector pose. Using an inversedynamics controller, the output of the PID controller is then converted into tension references for the inner position loop of the winches.

Lastly, Jamshidifar [START_REF] Jamshidifar | Vibration decoupled modeling and robust control of redundant cable-driven parallel robots[END_REF] et al. focuses on vibration rejection of a planar robot. A decoupled model of the vibration dynamics is derived from the first order linearization of the CDPR dynamics. The decoupled model is time varying, since its stiffness matrix depends on the current CDPR pose and cable tension. Thus, a linear parameter variant (LPV) controller is designed to cope with vibrations over the whole workspace. The output of the LPV controller is a tension variation that is added to the nominal tension computed using the full inverse dynamics model of the CDPR in order to track a desired trajectory. The control law is tested on a planar robot with active vibration damping along one Degree of Freedom (DoF).

In this paper, active vibration damping is considered, based on a linearized model of a CDPR along its end effector trajectory. The dynamics of the CDPR are projected in the end effector vibrations modal space. A decoupled modal controller is designed in modal space, whereas it is designed in Cartesian space in other works. The projection in the modal space converts complex coupled end effector dynamics into fully decoupled dynamics, called modes. Applying this technique to CDPRs is the main contribution of this work. It allows mainly for an easier tuning strategy: since all modes are decoupled, simple SISO controllers can be used for each mode.

The design of an active damping control algorithm usually requires a good knowledge of the CDPR vibrations characteristics. Natural frequencies and vibration modes of a CDPR are calculated from its stiffness model. Several methods have been proposed to estimate the stiffness of a CDPR as in [START_REF] Behzadipour | Stiffness of Cable-based Parallel Manipulators With Application to Stability Analysis[END_REF][START_REF] Diao | Vibration analysis of cable-driven parallel manipulators[END_REF][START_REF] Yuan | Static and dynamic stiffness analyses of cable-driven parallel robots with non-negligible cable mass and elasticity[END_REF]. A simple model proposed in [START_REF] Diao | Vibration analysis of cable-driven parallel manipulators[END_REF] only relies on the knowledge of cable stiffness and CDPR geometry.

But, in some specific configurations, this model may be inaccurate and antagonistic stiffness should be considered [START_REF] Behzadipour | Stiffness of Cable-based Parallel Manipulators With Application to Stability Analysis[END_REF].

Generic developments on modal control are presented in [START_REF] Gould | On the modal control of distributed systems with distributed feedback[END_REF]. Modal control has been widely used for vibration damping of flexible mechanical structures. Some control algorithms like Direct Velocity, Positive Position, Integral Force Feedback and more advanced optimal control algorithms are reviewed in chapters 7 and 11 of [START_REF] Preumont | Vibration Control of Active Structures[END_REF]. Position and velocity feedback gains can be chosen to optimize the modal control energy [START_REF] Singh | Efficient modal control strategies for active control of vibrations[END_REF]. Linear Quadratic Regulator (LQR) control algorithm applied to modal control is named Independent Modal Space Control (IMSC). However, so far, modal control seems to have raised relatively few interests in robotics. In [START_REF] Wang | Experimental Identification and Active Control of Configuration Dependent Linkage Vibration in a Planar Parallel Robot[END_REF], IMSC has been applied to a parallel robot with one flexible segment. In [START_REF] Kozak | Locally Linearized Dynamic Analysis of Parallel Manipulators and Application of Input Shaping to Reduce Vibrations[END_REF], the dynamic model of a flexible parallel robot is linearized along a path and modal analysis is used for input shaping.

Preliminary results in [START_REF] Weber | Active vibration canceling of a cable-driven parallel robot in modal space[END_REF] show experimentally that a modal controller yields an effective CDPR vibration damping around a linearization point. To reject the effects of actuators nonlinearities, the winch motors are position-controlled as in [START_REF] Begey | Dynamic control of parallel robots driven by flexible cables and actuated by position-controlled winches[END_REF][START_REF] Meunier | Control of an overactuated cable-driven parallel mechanism for a radio telescope application[END_REF] and the higher level modal vibration damping loop sends set points to these inner position loops. However, the efficiency of the active damping decreases when the robot moves away from the state where the controller has been tuned. Thus, a gain-scheduling approach is considered here to generalize the active vibration damping method proposed in [START_REF] Weber | Active vibration canceling of a cable-driven parallel robot in modal space[END_REF] for the whole CDPR workspace. Furthermore, a thorough analysis underlines that the stability of the proposed gain-scheduling control law is guaranteed for a moving end effector if the trajectory velocity is slow enough. Finally, the approach is validated experimentally on a new 6 DoF, 8-cables suspended robot, that shares the same cable geometry as the CoGiRo robot described in [START_REF] Gouttefarde | Geometry Selection of a Redundantly Actuated Cable-Suspended Parallel Robot[END_REF].

This paper is divided into 4 main sections. Section 2 describes the system, explains the assumptions that are made and defines the notations. Section 3 derives the equations of the dynamics and the modal control law. Section 4 presents a stability analysis of the proposed gain scheduling control law. Section 5 presents the experimental setup and discusses the experimental results.

Context

The approach proposed in this paper is suited for CD-PRs whose cable sagging can be neglected. Sagging is the bending of a cable due to its own weight. It increases with cable linear mass and length and decreases with cable tension. There are several models for this phenomenon. A complex model (elastic catenary) is needed when the sagging is large, and a simplified (parabolic) is sufficient when it is smaller [START_REF] Gouttefarde | Simplified static analysis of large-dimension parallel cable-driven robots[END_REF]. It has been shown experimentally in [START_REF] Yuan | Static and dynamic stiffness analyses of cable-driven parallel robots with non-negligible cable mass and elasticity[END_REF] that sagging has a significant impact on the CDPR natural frequencies and, in some configurations, a simplified model may yield very inaccurate results. Nevertheless, when sagging is negligible, the cable can be effectively modeled by a linear spring. This model may be inaccurate for some types of cables Cuvillon, L., et al. used to build CDPRs. Studies on cable characteristics show that cables usually have a non-linear relationship between their tension and elongation [START_REF] Riehl | On the determination of cable characteristics for large dimension cable-driven parallel mechanisms[END_REF]. In what follows, it is assumed that this relationship can be linearized to provide a quite accurate linear spring model within nominal tension boundaries [START_REF] Gouttefarde | Simplified static analysis of large-dimension parallel cable-driven robots[END_REF]. Thus, a cable is modeled as a linear spring with a specific stiffness k s = es (N) related to its Young modulus e (N m -2 ) and its cross-section s (m 2 ). Then, the axial stiffness k of the cable is given by Eq. (1).

k = k s l 2 = es l 2 (1)
In Eq. (1), l 2 represents the free length of the cable, i.e. its length when its tension equal to zero. Winding cable around its pulley reduces l 2 , and therefore increases the cable axial stiffness. Vibrations occurring on CDPRs are mainly due to the axial stiffness of the cables.

Modeling and Control

With modal analysis theory, the dynamics of a multiple DoF system can be split in decoupled modes, whose characteristics (frequency, direction) depend on the equilibrium point. The dynamic model of a CDPR linearized in the vicinity of its equilibrium point and projected in modal space is derived in this section. It is then used to tune a vibration rejection modal controller.

CDPR Modeling

Let x = [p T φ φ φ T ] T be the pose vector of the CDPR end effector with respect to the base frame R g (Fig. 1). Vector φ φ φ contains angular coordinates (e.g. Euler angles) of the rotation R gr (φ φ φ) from the inertial frame R g to the end effector frame R r .

Assuming massless cables, the dynamics of a CDPR end effector relatively to the inertial frame R g can be described by Newton-Euler equations. This model has been used and derived in many works [START_REF] Begey | Dynamic control of parallel robots driven by flexible cables and actuated by position-controlled winches[END_REF][START_REF] Alp | Cable suspended robots: feedback controllers with positive inputs[END_REF] and yields the following equation:

M(x)ẍ + C(ẋ, x)ẋ + g(x) = -JT (x) τ τ τ + ψ ψ ψ (2)
where M(x) is the mass and inertia matrix, C(ẋ, x) is the Coriolis and centrifugal matrix and g(x) is the vector of the gravitational forces. The vector τ τ τ denotes the cable tensions, while the vector ψ ψ ψ is an external wrench disturbance (wind, impact, . . . ) applied to the end effector. The Jacobian matrice J maps the time derivative of the pose ẋ to joint velocities l1 = [ • • • l1 i • • • ] T , i.e. the time derivative of distance A i B i between cables ends points:

J = ∂l 1 ∂x = J S = J I 3×3 0 0 S(φ φ φ) (3)
where S(φ φ φ) links the time derivative of φ φ φ to the rotational velocity vector ω ω ω, ω ω ω = S(φ φ φ) φ φ φ, and J is the inverse Jacobian of the robot linking the velocity screw of the end effector to the joint velocities l1 . The inverse Jacobian transpose of a CDPR is given by:

J T (x) = • • • g u i • • • • • • g b i × g u i • • • (4)
with u i the unit vector giving the direction of the i th cable from its end point on the base frame to its end point on the end effector and b i = PB i the vector from the end effector center of gravity P to the end point B i of cable i on the end effector, expressed in the base frame (see Fig. 1).

With the assumption of elastic massless cables, forces applied by the cables to the end effector can be computed using the distances l 1 between cable endpoints and the cable free lengths l 2 (the length of the cable with no tension). The free length can be calculated using the winch angular positions θ θ θ: l 2 = rθ θ θ where r is a scalar representing the pulley radius. Note that l 1 depends only on x since it is obtained by the inverse kinematics of the robot. The diagonal matrix of cables axial stiffness K = diag(k i ) links cable tensions τ τ τ to cable lengths:

τ τ τ = K(l 2 )[l 1 (x) -l 2 ] (5)
with k i the stiffness of the i th cable depending on its free length l 2 i (Eq. ( 1)). It is assumed that cable tension remains positive thanks to a dedicated tension management algorithm [START_REF] Oh | Cable suspended planar robots with redundant cables: controllers with positive tensions[END_REF][START_REF] Gouttefarde | A versatile tension distribution algorithm for n -dof parallel robots driven by n + 2 cables[END_REF]. Finally, from Eq. ( 2), the model of a CDPR dynamics with massless elastic cables becomes:

M(x)ẍ + C(ẋ, x)ẋ + g(x) = JT K(l 2 )[l 2 -l 1 (x)] + ψ ψ ψ (6)

Modal Analysis

Modal analysis is a method for decomposing the oscillatory response of a mechanical structure around an equilibrium point into elementary decoupled sinusoid components called "modes".

A linearized model of the CDPR dynamics (Eq. ( 6)) in the vicinity of a feasible or equilibrium trajectory Cuvillon, L., et al.

( ẍ(t), ẋ(t), x(t), l 2 (t), ψ ψ ψ(t)) is obtained by a first-order Taylor expansion (see appendix A for a detailed derivation) :

M(x)∆ẍ + C( ẋ, x) + ∂C(ẋ, x) ∂ẋ ẋ,x ẋ ∆ẋ -JT (x)diag -1 (l 2 )diag(τ τ τ) + JT (x)K(l 2 ) ∆l 2 + ∂M(x) ∂x x ẍ + ∂C(ẋ, x) ∂x ẋ,x ẋ + ∂ JT (x) ∂x x τ τ τ + JT (x)K(l 2 ) J(x) ∆x -∆ψ ψ ψ = 0 (7)
with ∆x = xx, the difference between the current pose and the equilibrium one, and τ τ τ = K(l 2 )[l 1 (x)l 2 ] (Eq. ( 5)).

Considering a static equilibrium ( ẍ = 0, ẋ = 0) of the CDPR end effector, the linearized model of the CDPR dynamics is reduced to :

M(x)∆ẍ + K x (x, l 2 )∆x = T(x, l 2 )∆l 2 + ∆ψ ψ ψ (8)
with

K x (x, l 2 ) = ∂ JT (x) ∂x x τ τ τ + JT (x)K(l 2 ) J(x) (9)
T(x, l 2 ) = JT (x)K(l 2 )diag(l 1 (x))diag -1 (l 2 ) (10)
The matrix K x is the Cartesian stiffness of the robot, i.e. the relationship between elementary Cartesian displacements of the end effector and corresponding variations of the wrench applied to the end effector by the cables. It is symmetric if no external moment (ψ ψ ψ = 0) is applied to the end effector [START_REF] Azadi | Antagonistic variable stiffness elements[END_REF]. Furthermore, K x is positive definite except in a few singular situations detailed in [START_REF] Behzadipour | Stiffness of Cable-based Parallel Manipulators With Application to Stability Analysis[END_REF].

Based on the linear model (Eq. ( 8)), a classic modal analysis is applied to convert the coupled system into six single DOF linear systems. Let η η η be the six dimensional vector of modal coordinates. Those coordinates are related to the pose variation ∆x by the 6 × 6 modal transformation matrix E:

∆x = Eη η η (11)
The modal matrix E has to verify the following conditions:

E T ME = I 6×6 E T K x E = Λ Λ Λ (12)
where I 6×6 is the 6 × 6 identity matrix and Λ Λ Λ is the 6 × 6 diagonal matrix of the system eigenvalues.

The matrix E is obtained by decomposition in generalized eigenvalues Λ Λ Λ of the system verifying the following equation:

K x E = MEΛ Λ Λ (13)
As K x and M are symmetric, E T K x E and E T ME are both diagonal matrices [START_REF] Preumont | Vibration Control of Active Structures[END_REF]. Each column eigenvector e i of E = [e 1 , e 2 , . . . , e 6 ] can be normalized such that e i T Me i = 1 in order to verify the first condition of Eq. ( 12). This normalization is possible since if one e i is solution of the generalized eigenvalues problem, i.e. K x e i = Me i λ i , then the vector α i e i is also a solution with α i an arbitrary constant.

Rewriting Eq. ( 8) with modal coordinates (Eq. ( 11)) yields:

ME η η η + K x Eη η η = T∆l 2 + ∆ψ ψ ψ (14)
Left product by E T gives:

I 6×6 η η η + Λ Λ Λ(x, l 2 )η η η = E(x, l 2 ) T [T(x, l 2 )∆l 2 + ∆ψ ψ ψ] (15)
Left side of equation ( 15) is fully decoupled and so can be decomposed into 6 independent equations, one for each mode. Matrix E contains 6 vectors of vibration modes and diagonal matrix Λ Λ Λ contains 6 squared natural frequencies corresponding to these modes. Therefore in this particular space, the response of the CDPR to a joint position variation ∆l 2 or an external wrench disturbance ∆ψ ψ ψ is a vector of six independent sinusoids.

Control Architecture and Decoupled SISO Model of the System

Let us consider an equilibrium state of the CDPR defined by (x, l 2 ): the CDPR pose x and the associated free cable lengths l 2 that unequivocally define the cable tensions (Eq. ( 5)). The goal of the vibration damping algorithm is to regulate towards zero the end effector velocity ẋ as fast as possible. To achieve this rejection, the controller acts on the cable tensions through a variation of the unconstrained cable length ∆l 2 around l 2 . The general architecture of the disturbance rejection control loop is presented in Fig. 2.

Using pulley actuators to wind or unwind unconstrained cable length, the control signal is equivalent to a pulley angular position variation ∆θ θ θ. An inner winch position loop can efficiently cope with nonlinearites such as dry friction, usually associated with actuators with high reduction ratio gearboxes. In [START_REF] Begey | Dynamic control of parallel robots driven by flexible cables and actuated by position-controlled winches[END_REF], a stability analysis using singular perturbation theory shows that, if the gain of the inner position loop is high enough, the stability of the outer CDPR pose loop is not compromised by the inner loop dynamics. Assuming a high-gain inner position loop, its high-bandwidth dynamics are negligible with respect to the low frequency CDPR oscillatory modes and thus are omitted thereafter. Regulating ∆ẋ toward zero, or equally ẋ (since at equilibrium ẋ = 0), is equivalent to regulate η η η towards zero in modal space, since ∆ẋ = E η η η. As the vibration problem can be decoupled in the modal space (Eq. ( 15)), simple Single-Input Single-Output (SISO) control strategies can be used for active vibration damping. Setting ∆ψ ψ ψ = 0, accordingly to the superposition theorem, and applying the Laplace transform to Eq. (15) yields:

Y m = L η U m - + C m (s) 0 s[s 2 I 6×6 + Λ] -1 Gm(s)
s 2 I 6×6 L η η η +Λ Λ Λ L η η η = E T T L ∆l 2 (16) L η η η = s s 2 I 6×6 +Λ Λ Λ -1 E T T L ∆l 2 L u m (17)
where L {.} stands for the Laplace transform and s is the Laplace variable. Finally, let Y m = L η η η and U m = L u m be the modal control signal for the regulation defined by U m = E T T L ∆l 2 . The decoupled dynamics of the CDPR in modal space (Eq. ( 17)) are modeled by a diagonal transfer function matrix G m :

Y m (s) = s s 2 I 6×6 +Λ Λ Λ -1 U m (s) (18)
G m U m (s)
The transfer function G m (s) = s s 2 I 6×6 +Λ Λ Λ -1 is a 6 × 6 diagonal matrix of transfer functions with undamped second order dynamics, consistent with the fact that frictions are neglected. The regulation loop based on this model is described in Fig. 3. The controller C m (also a diagonal matrix of transfer functions) can be designed and tuned to efficiently regulate towards zero the modal velocity Y m . So, the 6 × 6 Multiple-Input Multiple-Output (MIMO) coupled model in Cartesian space can be easily controlled using 6 independent SISO controllers in modal space. The implementation details of the modal control loop are given in Fig. 4. Based on the definition of the modal control signal u m , the output of the modal controller is back projected into the joint space using T + , the right pseudoinverse of T, yielding the control signal ∆l 2 = T + (E T ) -1 u m . The modal velocity, evaluated by projecting the end effector velocity ẋ in the modal space with the matrix E -1 , is fed back to the controller: η

η η = E -1 ∆ x = E -1
x. The estimate of ẋ, denoted x, is measured thanks to an IMU (Inertial Measurement Unit, a 3-axis accelerometer combined with a 3axis gyrometer). Since TT + = I 6×6 by definition, the block diagram in Fig. 4 is equivalent to the modal loop in Fig. 3.

Modal analysis assumes that x stays constant. So the proposed modal stabilization control law should be restricted to the rejection of external disturbances (transient external forces applied to the end effector by the environment, like wind blowing or contact) around a static pose. In the upcoming sections, it will be shown that the proposed control law remains stable even with a slowly varying x. Thus, this approach allows for rejection of oscillations even during a slow trajectory.

In order to track a varying pose x, a corresponding cable tension τ τ τ can be computed using the CDPR inverse dynamics (Eq. ( 2)) and the desired trajectory parameters (x, ẋ, ẍ): [START_REF] Behzadipour | Stiffness of Cable-based Parallel Manipulators With Application to Stability Analysis[END_REF] = τ τ τ +τ τ τ Null where ( JT ) + stands for the Moore-Penrose pseudoinverse of the wrench matrix, τ τ τ is the tension solution of Eq. (2) with minimal norm, and τ τ τ Null ∈ Null( JT ) such that JT τ τ τ Null = 0. In case of a redundant CDPR (more cables than degrees of freedom), the vector τ τ τ Null can be selected in the null space of the wrench matrix to ensure positive tension of the cables (τ τ τ > 0). Various algorithms [START_REF] Oh | Cable suspended planar robots with redundant cables: controllers with positive tensions[END_REF][START_REF] Gouttefarde | A versatile tension distribution algorithm for n -dof parallel robots driven by n + 2 cables[END_REF] are proposed in the literature to select an adequate τ τ τ Null vector.

τ τ τ = ( JT (x)) + M(x) ẍ + C( ẋ, x) ẋ + G +τ τ τ Null
Since the wrench applied on the CDPR end effector results from modifications of the cable elongation, the cable tensions are converted into unconstrained cable length l 2 . Based on the linear elastic cable model (Eq. ( 5)), l 2 is given by:

l 2 =   I + 1 k s diag(τ τ τ +τ τ τ Null τ τ τ )   -1 l 1 (x) (20)
Note that the positive tension condition (τ τ τ > 0) implies l 2 < l 1 , i.e. the cable is stretched. The complete trajectory generation for a varying x is described in Fig. 5.

space projection cartesian → modal modal → joint space projection controller modal L η L η Ẋ ∆L 2 U m Y m X - + C m (s) T + (E T ) -1 E E -1 E T T 0 IMU s[s 2 I 6×6 + Λ] -1
CDPR with position-controlled winches

Gain Scheduling Control Law

In modal space, the open-loop transfer function G m depends only on the current Λ Λ Λ(x, l 2 ) (Eq. ( 18)). This transfer function is simple enough in modal space to find an algebraic pole placement law for the controller that guarantees constant disturbance rejection performances over the whole workspace using gain scheduling. So, a basic pole placement approach can be used to tune 6 SISO controllers

C m i in C m = diag(C m 1 , ...,C m 6 ) according to the 6 SISO open-loop transfer functions G m i in G m = diag(G m 1 , ..., G m 6 ).
Model G m i has an invariant structure but has varying parameters since it has two pure imaginary conjugate poles that depends on the frequency of the corresponding mode in Λ Λ Λ. So, for given tuning specifications, the structure of the controller should also be constant.

A classical strategy when doing pole placement is to cancel stable system poles/zeros with controller zeros/poles. However, applied to the modal conjugate poles, this approach would yield a controller with a 0 dB gain at the system modal frequency (notch filter). It means that such a controller would be unable to reject modal oscillations due to external disturbances. Obviously, this is not the kind of behavior that is expected. Instead, for each mode, a controller is tuned for fast disturbance rejection and measurement noise resilience.

Two complex conjugate poles and one zero are added to the controller to place the four closed loop poles like in the root locus of Fig. 6. The controller gain k c i is tuned in order to obtain a closed loop with two identical pairs of complex conjugate poles. The complex conjugate poles and the zero of the controller have the same negative real part -a and Fig. 6. Root locus design of a modal controller the imaginary part of the controller complex poles is taken equal to the imaginary part of the process complex poles, i.e. √ Λ i where Λ i is the i th eigenvalue in Λ Λ Λ. With this design, measurement noise is well filtered and the controller is robust to small modeling errors. At the end, the modal controller for the i th modal velocity has the form:

C m i = k c i (s + a) s 2 + 2as + (a 2 + Λ i ) (21)
Cuvillon, L., et al.

with

k c i = 2( Λ i (a 2 + Λ i ) -Λ i ).
The negative real part -a of the controller poles and zero is the only design parameter. Increasing a yields a faster closed loop, but at the cost of a higher control signal energy. Thus, a trade-off must be found to avoid actuator saturation. With a ∈]0, 2Λ i 2 + √ 5[, the two pairs of closed loop complex conjugate poles are:

- a 2 ± j Λ i (a 2 + Λ i ) - a 2 4 ( 22
)
Since the real part -a/2 of the closed loop poles is negative, the system is stable. So, with a reference equal to zero, the modal velocity η converges towards zero. An implementation of this approach on the CoMiRo robot is presented in section 5.

Stability Analysis 4.1 Near the Equilibrium Point

A static equilibrium pose of the CDPR is fully defined by (x, l 2 ). While x is the pose of the end effector, l 2 is one solution of the inverse kinetostatic (IKS) problem which fully defines the cable tensions and verifies G(x) -JT K(l 2 ) l 2l 1 (x) = 0 in Eq. [START_REF] Weber | Active vibration canceling of a cable-driven parallel robot using reaction wheels[END_REF]. Around this equilibrium point, the CDPR dynamic model is parameterized by Λ Λ Λ(x, l 2 ) in the modal space with:

η η η +Λ Λ Λη η η = u m (23)
and a generic state space representation of the previous decoupled modal controller C m is:

ẋc = F c (Λ) x c + G c (Λ) η η η u m = H c (Λ) x c (24)
with x c , the controller state vector, u m the modal control signal, η η η, the measured modal velocities, F c , G c , H c , respectively the state, input and output matrices. Since the modes are decoupled, these matrices are all block diagonal. The closed loop system composed of Eq. ( 23)-( 24) is stable, since the design of the controller ensures that the poles or state matrix eigenvalues of the closed loop system have all strictly negative real parts.

The stability properties and eigenvalues of the state matrix are conserved for a linear time-invariant (LTI) system through a similarity or coordinate transformation. Thus, the stability of the closed loop in the modal space yields the stability of the closed loop in the Cartesian space due to the coordinate transformation η η η = E -1 ∆x. Using M = (E T ) -1 E -1 and K x = (E T ) -1 Λ Λ ΛE -1 (Eq. ( 12)) and the definition of u m (Eq. ( 17)), the coordinate transformation of the closed loop (Eq. ( 23)-( 24)) from the modal space to the Cartesian space yields the stable system: M(x)∆ẍ + K x (x, l 2 )∆x = T(x, l 2 )∆l 2 ( 25)

∆l 2 = T + (E T ) -1 H c (x, l 2 ) x c ẋc = F c (x, l 2 ) x c + G c E -1 (x, l 2 ) ∆ẋ (26)
which is the interconnection of the linearized dynamics model of the CDPR (Eq. ( 8)) around the constant equilibrium (x, l 2 ) and a LTI controller. Finally, as the linearized system is stable, the Lyapunov indirect method theorem [START_REF] Khalil | Nonlinear Systems[END_REF] points out that the interconnection of the CDPR nonlinear dynamics (Eq. ( 6)) and the linear controller (Eq. ( 26)) is asymptotically stable in a neighborhood of the constant equilibrium (x, l 2 ).

Along a Trajectory

A continuously differentiable trajectory of the CDPR end effector pose, defined and scheduled by the exogenous input signal x(t), is considered. For each point on this trajectory, it is assumed that a corresponding continuous cable length l 2 (t) = l 2 (x(t)), solution of the IKS model, is known or assessed with the trajectory generation scheme in Fig. 5 when the value of ẍ and ẋ reaches zero.

In this case, the plant control input that ensures tracking of the desired trajectory and vibration damping (Fig. 5) is l 2 (t) = l 2 (x(t)) + ∆l 2 with ∆l 2 , the output of the vibration damping controller (Eq. ( 26)). Thus, the closed loop system composed of the CDPR dynamics and the gain scheduled controller is defined by the state equations:

M(x)ẍ + C(ẋ, x)ẋ + G(x) = JT (x)K(l 2 ) [l 2 -l 1 (x)] (27)
l 2 = l 2 + T + (E T ) -1 H c (x, l 2 ) x c ẋc = F c (x, l 2 ) x c + G c E -1 (x, l 2 ) ẋ ( 28
)
The system has now time varying parameters, due to the dependence of the controller on the scheduling signal x(t) that defines the trajectory. Some stability properties for such a gain scheduled closed loop can be assessed using theorems dealing with the stability of nonlinear system with a "slowly" varying input [START_REF] Rugh | Research on gain scheduling[END_REF][START_REF] Lawrence | On a stability theorem for nonlinear systems with slowly varying inputs[END_REF]. Let x(t) be the varying input. The system of Eq. ( 27)-( 28) can be rewritten using the standard state-space representation:

ẋcl (t) = f (x cl (t), x(t)) (29)
with

x cl = [ẋ T x T x c T ]
T , the closed loop state vector, concatenation of the CDPR and controller states, and f , an appropriate nonlinear function.

This closed-loop system has a manifold of exponentially stable equilibrium parametrized by the input signal x(t), since it verifies the following properties: (P1) the nonlinear function f is twice continuously differentiable along a trajectory included in the wrench feasible workspace of the robot;

(P2) there is a continuously differentiable equilibrium state xcl (x(t)) = [0 T x(t) T 0 T ] T such that for each constant input value of x, f (x cl , x) = 0. This property can be verified by substituting xcl for x cl in system of Eq. ( 27)-( 28) and keep in mind that l 2 verifies G(x) -JT K(l 2 ) l 2l 1 (x) = 0 as a solution of the IKS model. (P3) for each constant input value x, the linearized closed loop of the system (Eq. ( 27)-( 28)) around the equilibrium (x cl (x), x) has eigenvalues with a negative real part. Actually, the linearization yields the system:

     M(x)∆ẍ + K x (x, l 2 )∆x = T(x, l 2 )∆l 2 ∆l 2 = ∂l 2 ∂x (x cl ,x) ∆x + T + (E T ) -1 H c (x, l 2 ) x c ẋc = F c (x, l 2 ) x c + G c E -1 (x, l 2 ) ∆ẋ (30)
which has the same state matrix and thus eigenvalues than the linearized system of Eq. ( 25)-( 26) around a constant equilibrium. Compared to Eq. ( 25)-( 26), the system exhibits an additional input term in ∆x. This term acts like an exogenous input disturbance (called "hidden coupling terms" in [START_REF] Rugh | Research on gain scheduling[END_REF]). It is due to the dependence of the plant (CDPR) input signal on the time-varying signal l 2 (x(t)) in order to track the trajectory.

The theorem [START_REF] Rugh | Research on gain scheduling[END_REF], detailed in [START_REF] Lawrence | On a stability theorem for nonlinear systems with slowly varying inputs[END_REF], thus states: for a given ρ > 0, there are constants δ 1 > 0 and δ 2 > 0 such that if the input signal satisfies for t > 0, || ẋ(t)|| < δ 2 and if

x cl (0) -xcl (0) =   ẋ(0) x(0) x c (0)   -   0 x(0) 0   < δ 1 ( 31
)
then the solution of the system (Eq. ( 29)) verifies for t > t 0

x cl (t) -xcl (t) =   ẋ(t) x(t) x c (t)   -   0 x(t) 0   < ρ ( 32
)
This result points out that, if the nonlinear closed loop system -composed of the CDPR (Eq. ( 27)) and gain scheduled controller (Eq. ( 28)) -is close enough to the equilibrium at time t = 0 and if the input signal x(t) does not vary too fast, then the system will remain in the neighborhood of the equilibrium trajectory defined by xcl (x(t)). This neighborhood is defined by the open ball of radius ρ and center xcl (x(t)) as shown in Fig. 7. Extensive simulations can be carried out to estimate the numerical values of the bounds on the transient disturbance and trajectory velocity ẋ in Fig. 5 that guarantee the stability property.

Experiments 5.1 Experimental setup: CoMiRo

A suspended 8-cables and 6-Degrees of Freedom (DoF) CDPR called CoMiRo has been built (Fig. 8). The geometry of the CoMiRo is similar to the geometry of the CoGiRo, a heavy payload CDPR developed in Montpellier (France) at the LIRMM by [START_REF] Gouttefarde | Geometry Selection of a Redundantly Actuated Cable-Suspended Parallel Robot[END_REF]. The CoMiRo cables anchor points on the room walls form a cuboid of 6.94 per 3.78 per 3.00 meters, along -→ x , -→ y and -→ z directions (see Fig. 9). The pulley output points at the end effector form another cuboid of 0.155 per 0.185 per 0.07 meters, along -→ x , -→ y and -→ z directions. The end effector weights 2 kg. It is built with Lego Mindstorms parts except for the cables, pulleys and some electronic components. Lego Mindstorms are an interesting solution to experiment with lightweight CDPRs for the following reasons:

• a rapid prototyping due to fast assembly and easily reconfigurable structure, • a low cost, easily duplicable solution, with lightweight and compact components made of high-quality plastics embedding functions like motor drivers, sensors conditioners, power and logic, • a worst case benchmark for control algorithms: due to the gearboxes and bearing sleeves made of plastic, high Cuvillon, L., et al. Two cables (Shimano Blue Wing TM , 3.2 kg, 0.2 mm, k s = 140.18 N) per actuator, instead of one, make the constraints on the pulley shaft more even, minimizing the frictions on the plastic sleeve bearing of the actuators. Each pair of cables is modeled as a single cable of specific stiffness k s = 280.36 N.

The robot controller, like the structure, is designed with a rapid prototyping approach for highly modular robotics applications. RPIt 1 is a Simulink toolbox, developed in our laboratory, that uses the code generation feature of Simulink Coder TM on a Raspberry Pi target. This toolbox provides all Simulink blocks needed for the experiment: multiple EV3 blocks, various IMU blocks, ... A custom firmware has been developed for the EV3 brick called EV3.14 2 . It improves encoder resolution from 1 to 0.5 • and reduces USB communication delays between the brick and the Raspberry Pi from 5 ms to 1 ms. Simulink external mode is used during the experiments to send motion commands, modify block parameters on the fly and monitor signals with scopes. A sampling interval of 7 ms is used in all experiments.

Each motor of the CoMiRo Lego is controlled by an independent position loop using encoder measurement feedback. Since this robot has 8 cables, the cable tension management algorithm should deal with a two-dimensional null space. Tensions can be obtained with the optimization method proposed in [START_REF] Gouttefarde | A versatile tension distribution algorithm for n -dof parallel robots driven by n + 2 cables[END_REF].

1 http://icube-avr.unistra.fr/en/index.php/RPIt 2 http://icube-avr.unistra.fr/en/index.php/EV3.14 Furthermore, a NDI Polaris localizer was used to learn the current Cartesian pose x(t) associated with the current cable free length l 2 (t). Indeed, the pose of the CDPR end effector cannot be estimated uniquely using only the sensors embedded into the end effector (motor encoders, IMU). However, after this learning phase, the proposed damping control law needs only the IMU feedback and does not require the localizer feedback anymore.

The proposed control law in Fig. 4 is implemented on the CoMiRo. Note that the translational velocity is obtained by integrating linear accelerations from IMU over time with a high-pass filter in order to avoid drifts. This velocity screw yields finally an estimate of ẋ using the definition of Eq. ( 3): ẋ = S -1 [ ṗT ω T ] T .

Modal Control Law Validation

In this section, all experiments are done around a unique static equilibrium point x 1 . It is located on the vertical symmetry axis of the robot workspace at a height z = 0.95 m. Angles are set to zero, the base frame and robot frame are aligned. Euler angles xyz are used to represent the rotational part of x. At this pose, the matrix K x has the following value:

K x =        
316.3 0 0 -0.24 11.18 0 0 101.2 0 -4.10 0.24 0 0 0 110.0 0 0 0 -0.24 -4.10 0 1.81 -0.32 0 11.18 0.24 0 -0.32 2.09 0 0 0 0 0 0 5.57

        (33)
To compute K x , the tension at equilibrium τ is needed (Eq. ( 9)). This tension is directly related to the cable elongation obtained from the pose x 1 and the cable free length l 2 measured by the winch encoders (Eq. ( 5)).

With the inertia tensor I r expressed in the robot frame, the mass matrix M of the end effector takes the following value:

M = 2.0 I 3×3 0 0 I r with I r =   0.0138 0 0 0 0.121 0 0 0 0.0049   (34
) Knowing K x and M, modal matrices E and Λ Λ Λ are computed using the method described by Eq. ( 13).

E =         0
0.45 -0.25 -0.48 0 0.03 0 0.02 0.18 -0.11 0 -0.68 0 0 0 0 -0.71 0 0 -1.57 -7.66 2.36 0 -2.43 0 6.77 0.66 6.02 0 -0.58 14.35 0 0 0 0 0

        (35)
The modal frequencies F m of the robot are the square root of the eigenvalues Λ i in Λ Λ Λ. Furthermore, the mode corresponding to the i th frequency in F m (in rad s -1 ) has the direction of

        (36)
Experimental results of the modal vibration damping control loop in the frequency domain are shown in Fig. 10. An impulse disturbance is added to the signal u m (see Fig. 4) to mainly excite vibration mode 2. With the proposed modal control, the residual vibration magnitude on all modes is significantly reduced compared to the free response.

Gain Scheduling 5.3.1 Motivations

To validate the need for a gain scheduling approach like the one proposed in Eq. ( 28), the controller parameters calculated at the pose x 1 defined in the previous section are frozen and performance of the disturbance damping control law is assessed after moving the robot at a significantly different pose without updating the controller. This assessment is referred as "Active" or "Active damping" in what follows. Then, the controller is updated according to this new position and a new performance assessment is made. This is referred as "GS" or "Gain Scheduling" in what follows.

Thanks to its simple form presented in section 3.4, the control algorithm can be easily and quickly updated online accordingly to this model within the sampling time of 7 ms:

• the matrix T + (E T ) -1 (x(t), l 2 (t)) is updated in the control loop of Fig. 4 to convert modal control signals into actuators signals; • and each SISO controller is updated as detailed in section 3.4 according to the current value of the modal frequencies matrix Λ Λ Λ. On the CoMiRo robot, modal frequencies vary from 4.8 rad s -1 to 41.9 rad s -1 inside a Static Feasible Workspace defined by a 4.5×1.5×2 m 3 cuboid centered in x 1 where end-effector Euler angles vary from -15 • to +15 • . The modal frequencies are evaluated on discrete poses within the workspace with translational quantization steps of 0.1 m and angular steps of 5 • . To select an admissible equilibrium tension at each point, the tension distribution algorithm proposed in [START_REF] Gouttefarde | A versatile tension distribution algorithm for n -dof parallel robots driven by n + 2 cables[END_REF] with minimization of the L 2 tension norm is used. The lower admissible cable tension is 0.1 N and the upper one is 17.5 N.

Results

Table 1 and Fig. 11 show the added value of the gainscheduling approach for vibration damping at a static equilibrium pose x 2 , far from the pose x 1 where the active damping has been computed. The pose x 2 chosen for this experiment is translated by x 2x 1 = [-2, 0, 1] T meters from the workspace center pose x 1 defined in the previous section. Measured velocity data are projected into modal space with the modal matrix E computed at the current pose 2. The figures 11d and 11e show that performance degrades when the controller is not tuned ("active damping") to the mode frequency of the current pose: instead of being damped, vibrations on modes 4 and 5 are amplified. This degradation is also visible in the frequency domain as shown in Fig. 12. Thus, gain-scheduling control is required to guarantee optimal damping over the whole workspace.

The figures 14 and 15 compare the performance of the proposed control strategy without and with gain-scheduling when the suspended robot is moving at a constant velocity along x axis (Fig. 13). The maximum translational velocity is 10 cm s -1 due to limitations of the Lego actuators. To highlight the vibration rejection, a modal velocity impulse disturbance is applied simultaneously on the 6 modes. The result is a clear reduction of the vibrations (Fig. 15 compared to Fig. 14) for all modes when a gain scheduling control law is activated during the whole trajectory. This vibration damping is also visible in the frequency domain as shown in Fig. 16. The variations ∆θ of the winch angular positions yielding cable tension variations that damp the CDPR vibrations with the GS controller are given at the bottom of Fig. 15. This control signal issued by the modal controller is used as reference for the inner loop that controls the unstretched cable length ∆l 2 = r∆θ θ θ with r the winch radius.

The video of this experiment can also be viewed online at https://youtu.be/tbOvdrJk3Nk. At time 00:02:50, two superimposed videos are displayed: the same

Conclusions

Using modal analysis, a linear decoupled model of a CDPR with elastic cables is described. Projecting the robot linearized dynamics into the modal space yields 6 decoupled linear second order transfer functions which can be easily controlled with standard SISO techniques. This is the main benefit of the proposed approach. A control strategy, that cancels the vibrations of the end effector caused by disturbances is proposed. Furthermore, thanks to the decoupled nature of the model, the modal controller can be easily up- dated with a trivial, computational efficient algorithm. Since natural modes and frequencies are related to the end effector pose, a gain-scheduling approach is introduced to improve the efficiency of the vibration damping algorithm over the whole workspace of the CDPR. Finally, stability analysis shows that the control law remains stable along a slowly varying trajectory. The proposed approach is validated experimentally on a small suspended CDPR built with Lego Mindstorms, which shares the same architecture as its big Cuvillon, L., et al. ∆x -∆ψ ψ ψ = 0 (37) And using:

∂K(l 2 ) ∂l 2 l 2 [l 2 -l 1 (x))] = ∂diag -1 (l 2 ) ∂l 2 l 2 diag(es)[l 2 -l 1 (x)] = -diag -1 (l 2 ) ∂diag(l 2 ) ∂l 2 l 2 diag -1 (l 2 )diag(es)[l 2 -l 1 (x)] = -diag -1 (l 2 ) ∂diag(l 2 ) ∂l 2 l 2 K(l 2 )[l 2 -l 1 (x)]
-τ τ τ ∂K(l 2 ) ∂l 2 l 2 [l 2l 1 (x)] = diag -1 (l 2 )diag(τ τ τ) (38) (since for u scalar, ∂A -1 ∂u = -A -1 ∂A ∂u A -1 ) finally yields Eq. [START_REF] De Rijk | Out-of-plane vibration control of a planar cable-driven parallel robot[END_REF].

List of Figures

Fig. 1 .

 1 Fig. 1. CDPR notations

Fig. 2 .

 2 Fig. 2. Architecture of the vibration damping control loop

Fig. 3 .

 3 Fig. 3. Vibration damping control loop scheme in modal space

Fig. 4 .Fig. 5 .

 45 Fig. 4. Practical implementation of the modal vibration damping control on the CDPR

Fig. 7 .

 7 Fig. 7. Stability of the gain-scheduling modal damping along a trajectory: if x(t) varies slowly, the closed loop system state x cl (t) remains in the neighborhood of the equilibrium trajectory xcl (x(t))

Fig. 9 .

 9 Fig. 9. CoMiRo notations, dimensions and frame definitions

Fig. 10 .

 10 Fig. 10. Frequency-domain response of the CDPR to a pulse disturbance at pose x 1

6 Fig. 11 .Fig. 12 .

 61112 Fig. 11. Modal damping assessment with a velocity impulse disturbances on mode 1 and 3 at pose x 2

Fig. 13 .

 13 Fig. 13. Cartesian trajectory of CDPR end effector during a translation motion without modal damping (free) and with gain scheduling modal damping. A impulse modal disturbance is applied on all the vibration mode at one point of the trajectory to highlight the vibration rejection.

Fig. 14 .Fig. 15 .

 1415 Fig. 14. Vibration in the modal space during motion in the absence of active damping (on top) and associate position motor control signal ∆θ θ θ (on bottom) issued by the modal controller (inactive here)

Fig. 16 .

 16 Fig. 16. Frequency-domain response of the CDPR to a pulse disturbance on all modes applied at one point of the trajectory

Table 1 .

 1 RMS errors on modal velocities (without unit) at pose x 2 for a velocity impulse disturbances on mode 1 and 3

	Modes	1	2	3	4	5	6	∑ modes
	Free	9.31 1.47 6.10 0.98 0.40 2.99	21.25
	Active	4.66 2.40 3.79 3.25 0.59 2.86	17.55
	GS	4.25 0.83 2.57 1.04 0.24 1.70	10.63

  1 CDPR notations . . . . . . . . . . . . . . . . 3 2 Architecture of the vibration damping control loop . . . . . . . . . . . . . . . . . . . . 5 3 Vibration damping control loop scheme in modal space . . . . . . . . . . . . . . . . . . 5 4 Practical implementation of the modal vibration damping control on the CDPR . . . . . . 6 5 Block diagram of the CDPR nominal trajectory generation (left side) combined with the vibration damping control (right side) . . . . 6 6 Root locus design of a modal controller . . . 6 7 Stability of the gain-scheduling modal damping along a trajectory: if x(t) varies slowly, the closed loop system state x cl (t) remains in the neighborhood of the equilibrium trajectory xcl (x(t)) . . . . . . . . . . . 8 8 CoMiRo robot . . . . . . . . . . . . . . . . . 8 9 CoMiRo notations, dimensions and frame definitions . . . . . . . . . . . . . . . . . . . 9 10 Frequency-domain response of the CDPR to a pulse disturbance at pose x 1 . . . . . . . . . 10 11 Modal damping assessment with a velocity impulse disturbances on mode 1 and 3 at pose x 2 . . . . . . . . . . . . . . . . . . . . 11 12 Frequency-domain response of the CDRP to a pulse disturbance on mode 1 and 3 at pose x 2 11 13 Cartesian trajectory of CDPR end effector during a translation motion without modal damping (free) and with gain scheduling modal damping. A impulse modal disturbance is applied on all the vibration mode at one point of the trajectory to highlight the vibration rejection. . . . . . . . . . . . . . . 11 14 Vibration in the modal space during motion in the absence of active damping (on top) and associate position motor control signal ∆θ θ θ (on bottom) issued by the modal controller (inactive here) . . . . . . . . . . . . . . . . 12 15 Vibration in the modal space during motion with gain-scheduling modal damping (on top) and associate position motor control signal ∆θ θ θ issued by the modal controller (on the bottom) . . . . . . . . . . . . . . . . . . 12 16 Frequency-domain response of the CDPR to a pulse disturbance on all modes applied at one point of the trajectory . . . . . . . . . . . 12

	List of Tables
	1	RMS errors on modal velocities (without
		unit) at pose x

2 for a velocity impulse disturbances on mode 1 and 3 . . . . . . . . . . 10 Cuvillon, L., et al.

Cuvillon, L., et al.


Yaokang Lv

Xing Yang

Weishi Du

Peihua Ma

Antoine Bonnefont

Dominic S Wright

Laurent Ruhlmann

Cheng Zhang

An efficient electrochromic supercapacitor based on solutionprocessable nanoporous poly(tris[4-(3,4-ethylenedioxythiophene) phenyl]amine)

Keywords: Tris[4-(3,4-ethylenedioxythiophene)phenyl]amine, Molecular structure, Electrochromic supercapacitor, nanoporous, microemulsion

A new green-synthetic route to the tris [4-(3,4ethylenedioxythiophene)phenyl]amine (TEPA) monomer has been developed and the molecular structure of TEPA determined by a single-crystal X-ray study for the first time. Solution-processable nanoporous poly (tris[4-(3,4-ethylenedioxythiophene)phenyl]amine) (PTEPA) was prepared via chemical-oxidative polymerization in a microemulsion. Based on the distorted structure of TEPA found in the solid-state, we speculate that dendritic PTEPA has a distorted threedimensional conformation with multiple twisted channels and pores which are narrowed and blocked by the bifurcation and distortion of PTEPA, which is consistent with the observed hierarchical pore structure. As a cathode material, PTEPA exhibits a discharge capacity of 89.5 mAh/g in the initial cycle with a very sloping two-stage discharge curve and relatively stable cycling performance. Beyond excellent energy storage properties, PTEPA also shows relatively good electrochromic performance. On this basis, an efficient all-solidstate electrochromic supercapacitor (ECSC) with good electrochromic performance and high energy storage capacity (13.3 mF cm -2 ) was assembled based on PTEPA and nanoporous graphene films. During the charging and discharging processes, the color of the ECSC changes between yellow-green and steel-blue. Thus, the energy storage level of the ECSC can be monitored by the corresponding color changes. The fabricated ECSC may have practical applications, for example, in self-powered electrochromic smart windows.

Introduction

The Electrochromic devices (ECDs) such as electrochromic smart windows, which can efficiently adjust light transmission through reversible color and transparency changes under an external voltage, [START_REF] Beaujuge | [END_REF][2][3][4] and supercapacitors (SCs), which are advanced electrochemical energy storage devices characterised for their high power density and short charging times, [4][5][6][7] have similar working mechanisms and device structures. As a result, there is the interesting prospect of the integrating energy storage and electrochromic functions into one device. As a promising smart device, such electrochromic supercapacitors (ECSCs) have attracted considerable interest over the past decade. [8][9][10][11][12][13][14] In comparison with traditional ECDs, ECSCs not only have the function of energy storage, but also can act simultaneously as a smart energy monitor, indicating the state of energy storage capacity through color changes. In addition, ECSCs are also expected to overcome problems concerning external power sources and the limited range of application of conventional ECDs.

Conjugated polymers have been widely viewed as an attractive option for electrode materials due to their adjustable band gaps, large theoretical specific capacitances, good electrical conductivity, high elemental abundance, light weight, and tunable mechanical flexibility. [START_REF] Beaujuge | [END_REF]15] Among these, many triphenylamine(TPA)-based polymers, which possess ultrafast electron-transfer rate constants and good hole-transporting ability, have been intensively studied as electrode materials for organic batteries and SCs. [16][17][18][19][20][21][22][23] In addition, a large number of TPA-based polymers also exhibit electrochromic behavior during redox processes that have potential applications in ECDs. [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36] In order to improve the charge carrier transport properties and structural stabilities of polymer chains, some thiophene derivatives, such as 2,2'-bithiophene and 3,4-ethylenedioxythiophene (EDOT), have been incorporated within the polymer backbone as "bridge" linkages. A series of monomers consisting of a TPA core substituted by thiophene derivatives were synthesized recently, and the resulting polymers possess good electrochemical stability and various electrochromic properties. [31][32][33][34][35][36] Tris [4-(3,4ethylenedioxythiophene)phenyl]amine (TEPA) is one of reported monomers, which has been synthesized via the Stille crosscoupling reaction. [31,37] M. Chahma et al. [37] and K. Idzik et al. [38] have prepared poly(tris [4-(3,4ethylenedioxythiophene)phenyl]amine) (PTEPA) films by electrochemical polymerization and found that they have significant electroactivity and reasonable redox stability, although these films are insoluble in all common solvents. C. Xu et al. 31 firstly studied the electrochromic properties of PTEPA films, which exhibit four different colors (yellow green, green, blue and steelblue) upon applied stepwise oxidations from -0.4 to 0.95 V. This property indicates that PTEPA is a potential candidate as an electrode material for ECSC. Nevertheless, this material still has several drawbacks that hinder its practical application. Firstly, because of the toxicity of organotin reagents, conventional Stille coupling reactions are not suitable for environmentally-friendly production, [START_REF] Albéniz | Polymers for green C-C couplings[END_REF] alternative "greener" synthesis procedures for the TEPA monomer should be investigated.

Secondly, electrochemically polymerized PTEPA films can not be dissolved in common organic solvents, their molecular weights were not determined and the processability was limited. [40][41][42][43] Last but not least, there has been no report on the energy storage properties of PTEPA, not to mention the application of this material in ECSCs.

In view of the issues mentioned above, here we have developed a novel synthetic route to the TEPA monomer and determined its molecular structure in the solid state for the first time. We also report the preparation of PTEPA via a chemical oxidative polymerization in a microemulsion. These nanoporous polymers exhibit excellent energy storage properties and are soluble in chloroform with a number-average molecular weight of 13843 g/mol. Thin electrochromic films were prepared by drop casting, and an ECSC with good electrochromic performance and high energy storage capacity was assembled based on PTEPA. The smart visual monitoring of the energy storage state of the ECSC is provided by means of the observed color during the charging-discharging process. phenylamine and 2-iodo-3,4-(ethylenedioxy)thiophene, the overall yield of TEPA through this four step synthetic route can be reached 55%. Compared with the previous route, our new route is greener and uses environmentally benign reagents. It will therefore be more suitable for industrial production.

Results and Discussion

The purity of TEPA produced by our method was confirmed by elemental analysis, NMR, MS and FT-IR spectra (Figs. S1-S3). In addition, the simulated powder XRD pattern is identical to that of authentic samples of TEPA (Fig. S4). The molecular structure of TEPA was determined via sXRD analysis. This compound crystallizes in the monoclinic centrosymmetric space group P21/c, with three (chemically identical) crystallographically independent molecules within the unit cell. The carbon atoms at the 2-position of the thiophene rings of the EDOT groups are bonded to the para position of the triphenylamine groups of the TPA core. As shown in Fig. 2, C-H•••O intermolecular hydrogen bonds connect adjacent molecules and form two-dimensional layers within the lattice (Table S2), which further extend into a 3D supramolecular structure through Van der Waals interactions (Fig. 3). As shown in Fig. 4 and Table S3, molecules of TEPA have a highly distorted arrangement. The three phenyl groups of the TPA core within individual molecules are not in the same plane, with the dihedral angles between the neighboring phenyl groups of TPA core being in the range of 59.9-74.5 o (Fig. 4a). In addition, the phenyl and thiophene rings are also not co-planar, with the dihedral angles between these rings being in the range of 18.6-23.3 o (Fig. 4b). These features are important because, in general, planarity and the resulting conjugation are strongly related to the band gap and conductivity of polymers, while the planarity of monomer units determines the planarity of the conjugated polymer to a large extent. [46][47][48] Small conformational differences may produce large band-gap effects in conjugated polymers. [49] The structure of the TEPA monomer is therefore not only related to the conformation of PTEPA but also to its physical properties. In some previous reports, due to the lack of accurate structural informations, monomeric units have been approximated to completely planar, 'star-shaped' building blocks. [19,23] As a result, it has been assumed that the highly porous polymers observed result from the aggregation of monomers into cyclic units which build together into planar, honey-comb arrangements e.g., in poly[1,3,5-tris(4-diphenylamino-phenyl)benzene] which has been shown to possess a highly porous structure. As represented in Fig. 5a, the TEPA monomer is in fact not a completely planar building block but is slightly pyramidalised at the central N1 atom with respect to the terminal C atoms (C1A, C1B, C1C), which will form C-C bonds to other monomer units upon polymerisation. As shown in Table S4, the distances from the central N atoms to the terminal C atoms over all three independent molecules in the unit cell are in the range of 0.808-0.817 nm, while the associated C-N-C angles are in the range of 117.4-122.9 o . Fig. 5b shows two possible assemblies of TEPA, although it is possible to assemble a nanoring through three connected TEPA molecules (shown on the left of Fig. 5b) which would form the previously proposed honeycomb sheet arrangement, the inherent distortion of the building blocks of TEPA are, we believe, more likely to result in the formation polymer chains (Fig. 5b, right). In this case polymerization of TEPA would result in irregular spiral channels (Fig. 5c) instead of the ordered columnar channels which would occur in a honeycomb structure. As shown in Fig. 5d, cross-linking of this arrangement will give a three-dimensional porous dendritic structure. As the degree of polymerization increases, the affect of the bifurcation and distortion of PTEPA will be accumulation throughout the structure of the polymer, while the pores and channels in PTEPA will be also gradually twisted and blocked by the external branches. This less porous structure is consistent with porosity measurements of PTEPA (section 3.2). It was reported that polytriphenylamine(PTA) displays negligible microporous structure with a very small surface area, [19,20] which, it was proposed, may be due to the difficulty in building a porous honeycomb structure using a monomer with shorter connecting groups. [19] Based to our hypothysis, we speculate that the structure of PTA contains similar twisted pores and channels to that proposed for PTEPA.

Synthesis and Structure of PTEPA

Dendritic homopolymer and copolymer films based on TEPA have been synthesized via electropolymerization in previous reports. [21,37] But all of these polymers exhibit insolubility in conventional organic solvents, which introduces difficulties for large-scale production. Microemulsion polymerization has been regarded as a simple and reliable process for preparation of nanostructured conjugated ploymers and oligomers. [50][51][52][53] In this work, a PTEPA powder was prepared via micelle-induced selfassembly and in situ polymerization of TEPA monomers. As a surfactant, DTAB forms micelles between the stirred aqueous and organic biphasic system during the microemulsion polymerization, TEPA diffuses into the micelles, where the oxidizing agent oxidizes it and subsequent polymerization occurs. As shown in Fig. 6, the obtained PTEPA powder is composed of irregular nanoparticles, most of which loosely agglomerated together. Nitrogen adsorption-desorption experiments were conducted to further explore the microstructure of the PTEPA. As presented in Fig. 6d, the PTEPA exhibits a type IV isotherm with hierarchical pore structure: an obvious hysteresis loop between the sorption curve and the desorption curve suggests the presence of mesopores. [54,55] A significant rise in the sorption curve in the high pressure region (0.80 < P/P0 < 0.95) indicates the presence of macropores, and a sharp uptake at low pressures (P/P0 < 0.01) is a characteristic feature for microporous materials. [56,57] The Brunauer-Emmett-Teller (BET) surface area of PTEPA calculated in the low pressure region (0.009 < P/P0 < 0.05) of the isotherm is 67 m²/g, and the total pore volume was calculated as 0.25 cm³/g (P/P0 = 0.992) for PTEPA. Pore size distributions calculated from desorption branches by the Barrett-Joyner-Halenda (BJH) model are given in Fig. 6e. PTEPA display typical hierarchical pore size distribution with a highest peak centering at ca. 1.94 nm, a lower peak centering at ca. 7.98 nm and micropores smaller than 1.53 nm. The average pore radius is 6.8 nm. In addition, micropore distribution of PTEPA analyzed by the Horvath-Kawazoe method (Fig. 6f) reveal that PTEPA has a narrow micropore size distribution with median pore width of 1.07 nm. These results prove that PTEPA is nanoporous, which is in accord with the schematic polymer structure of PTEPA (Fig. 5). The macropores and mesopores in PTEPA are mainly derived from the removed solvent and surfactant after polymerization, while the micropores stem from the porous framework of PTEPA. Inherent nanopores make the numerous active sites of PTEPA accessible by electrolyte ions during electrochemical processes. This may benefit the performance in electrical energy storage systems.

Interestingly, this PTEPA powder can be dissolved in chloroform, the number-average molecular weight (Mn) of PTEPA is 13843 g/mol, with a PDI of 2.24, determined using GPC. The solubility of PTEPA facilitates its processability for coating or printing, thin films of PTEPA on ITO glass substrates were prepared by drop casting from a 1 g/L chloroform solution of PTEPA and dried in air at room temperature. As observed in Fig. 6c, these films exhibit a rough surface containing widely distributed nanopores.

Electrochemical Energy Storage Performance

The electrochemical energy storage performances of a PTEPA electrode as the cathode was investigated in a 2032 coin-type half-cell. As shown in Fig. 7a, CV profiles of the electrode exhibit pairs of broad redox peaks located at 3.69 V vs. Li/Li + (oxidation) and 3.37 V vs. Li/Li + (reduction). The separation of the redox waves in PTEPA is approximately 0.32 V, which is a slightly narrower than that previously reported for the polytriphenylamine (PTPA) electrode (0.34 V). [19] The symmetric peak shape of the oxidation and reduction peaks of PTEPA demonstrates its capacitance characteristic. The charge-discharge behavior of the PTEPA electrode was explored at 20 mA/g between 2.5 V and 4.3 V. As shown in Fig. 7b, the PTEPA electrode exhibits a discharge capacity of 89.5 mAh/g in the initial cycle with two sloping voltage stages. The high voltage stage between the voltage of 4.1 V and 3.3 V is attributed to the redox reaction of TPA moieties of the PTEPA, which provides discharge capacity of about 62 mAh/g, while the other lower stage between 2.8 V and 2.6 V is probably related to the EDOT moieties, which provides a discharge capacity of about 27.5 mAh/g. Compared with other reported active materials, PTEPA exhibits a more sloping discharge curve without a stable discharge voltage plateau, which indicates that it is more suitable for applications in SCs than in batteries.

The cycling performance of the PTEPA electrode was examined at a constant current of 20 mA/g during a 50 cycle test and the results are shown in Fig. 7c. It is found that the discharge capacity of PTEPA decreases from its initial 89.5 mAh/g to 84.9 mAh/g after 50 cycles, with about a 5% loss of capacity. This result is better than that of a PTPA electrode ( > 7% loss ). The relatively stable cycling performance of PTEPA can be attributed to its nanoporous structure which reduced potential electrochemical decomposition and morphological change during the repeated charge-discharge processes. As shown in Fig. 7d, PTEPA exhibits relatively low coulombic efficiency in the initial charge-discharge process that is related to formation of an solid electrolyte interface (SEI) membrane on the lithium anode. With increase in cycling, the coulombic efficiency increased, reaching 98.3% at the 50th cycle.

Electrochemical measurements show that the PTEPA film on ITO exhibits mainly a Faradaic pseudocapacitive behavior in 1M H2SO4 aqueous solution. As shown in Fig. 7e. CV profiles of the PTEPA electrode exhibit a pair of broad redox waves centered at 0.67 V vs. Ag/AgCl (oxidation) and 0.39 V vs. Ag/AgCl (reduction), with good reversibility. As shown in Fig. 7f., the galvanostatic charge-discharge profiles show a significant IR drop at the beginning of the discharge process, followed by slow discharge process (0.8 V to 0.2 V), the potential sharply reverts to the original value afterward. The coulombic efficiency of the PTEPA film is about 43%. However, since the thin PTEPA film is easily detached from the ITO substrate during the charge and discharge process, it is difficult to determine the film mass and to calculate specific capacity accurately.

Spectroelectrochemsitry Studies

Spectroelectrochemical analysis was used to study the electrochromic properties of the PTEPA films on ITO. Visible and NIR transmission spectra of the PTEPA film at different potentials (-0.5 to 1.0 V) are shown in Fig. 8a. When a -0.5 V voltage was applied, the PTEPA film in its reduced state exhibited a yellowgreen color with two absorption bands. The absorption band centered at 440 nm was attributed to the π-π* transition and the broad absorption band between 560 nm and 1050 nm was assigned to the charge transfer band from the TPA moiety to the EDOT moiety. [31] Upon increasing the applied potential, the intensity of absorption peak around 440 nm was decreased, while the broad absorption peak at higher wavelength increases, leading to a color change to steel-blue at the fully oxidized state (1.0 V).

In order to characterize the optical contrast (ΔT%) and switching properties of the PTEPA film, dynamic electrochromic experiments were studied at 440 nm and 660 nm. The square wave potential method was used between -0.4 V and 1.0 V at regular intervals of 5 s. As shown in Fig. 8b, the ΔT% was found to be 23.97% at 440 nm and 14.29% at 660 nm, which were are slightly lower than that of electroploymerized PTEPA films (26.84% at 448 nm and 17.03% at 660 nm). [31] In addition, the switching time of the PTEPA film was found to be 0.96 s for coloring and 1.89 s for bleaching at 440 nm (Fig. 8c), and 0.67 s for coloring and 2.89 s for bleaching at 660 nm (Fig. 8d). These times are longer compared to electroploymerized PTEPA film. These phenomena can be attributed to the relatively low polymerization degree of the drop-casted film. As summarized in Table S5, solution-processable PTEPA still exhibits relatively short coloring times, which may in part be due to the nanoporosity of PTEPA, resulting in relatively high interface area and small diffusion length. The reasonable optical contrast and the fast switching properties make PTEPA a promising electrochromic material and will be beneficial to its application in ECSCs.

Performance of the Solid-state ECSC

As shown in Fig. 9a, an all-solid-state ECSC prototype was constructed based on PTEPA and nanoporous graphene films (Fig. S6). Nanoporous graphene films and PVA/H2SO4 gel electrolyte were prepared according to reported methods. 6 The detailed fabrication process of the ECSC is described in the ESI. The CV curves of the ECSC in the voltage window of -0.8 V~0.2 V are shown in Fig. 9b at different scan rates, which exhibit a quasi-rectangular shape at sweep rates from 0.01 to 0.10 V/s, indicating ideal capacitance behavior. However, this rectangular behavior changes at sweep rates of 0.20 V/s, suggesting resistance-like electrochemical behavior. The GCD curves of the ECSC were almost linear (Fig. 9d) and exhibit good symmetry. The IR drop of the ECSC increased from 7.9 mV at 0.015 mA/cm 2 to 76 mV at 0.12 mA/cm 2 , such a low IR drop indicates that the device has high electron transfer efficiency and low equivalent series resistance. The later is estimated to be around 175 Ω cm 2 by using electrochemical impedance spectroscopy (see ESI). The area specific capacitance of the device can reach a maximum of 13.3 mF/cm 2 with a current density of 0.015 mA/ cm 2 , but the specific capacitance of the ECSC significantly decays with increase in the current density (Fig. 9c). When the current density is increased to 0.03 mA/cm 2 , the area specific capacity decreased to 10.9 mF/cm 2 , when the current density is increased to 0.12 mA/cm 2 , the area specific capacity dropped to 7.2 mF/cm 2 , while when the current density increased to 0.30 mA/cm 2 , the area specific capacity dropped to 5.4 mF/cm 2 . These capacitance values are in good agreement with the one determined of 9.2 mF cm -2 determined by electrochemical impedance spectroscopy.

Since energy storage and electrochromism rely on the same electrochemical processes, the color of the ECSC changed between yellow-green and steel-blue reversibly during the charging and discharging processes. When the device was charged to 0.2 V, the color of the ECSC was steel-blue (see inset to Fig. 9d). In the opposite direction, when the device was discharged to -0.8 V, the color of this ECSC changed to yellow green (inset to Fig. 9d). Therefore, the energy storage level of ECSC can be monitored by the corresponding color changes. As summarized in Table 1, the maximum specific capacitance of the ECSC is higher than that of other reported electrochromic supercapacitors. It should be noted that the CV and GCD curves of our ECSC were very different to that of PTEPA. Nanoporous graphene films have excellent conductivity and high electric double-layer capacitance. In the case of the ECSC fabricated with a PTEPA and nanoporous graphene films, both the oxidation and reduction peaks of PTEPA disappear from the CV curves and the corresponding GCD curves become triangular. In this simple ECSC prototype, a considerable amount of capacitance comes from nanoporous graphene film, and the PTEPA film also provides electrochromic properties. Various ECSCs with different capacitance and electrochromic properties can be assembled through adjusting the dimensions and area of PTEPA and graphene films. For instance, a self-powered electrochromic smart window could be assembled using a PTEPA coated window glass combined with nanoporous graphene film as a window frame.

Conclusions

In summary, the TEPA monomer was obtained through a new practical synthetic route using a safer and greener approach, and its molecular structure determined by sXRD for the first time. Its solution processable polymer PTEPA was further prepared via microemulsion polymerization. The distorted molecular structure of the TEPA monomer affects the planarity and three-dimensional conformation of the dendritic polymer PTEPA. We speculate that PTEPA has twisted channels and pores, which are narrowed and blocked by the bifurcation and distortion of PTEPA. Investigations of the microstructure of PTEPA indicate that PTEPA has a hierarchical pore structure with an average pore size of 6.8 nm. PTEPA exhibits excellent energy storage properties, as a cathode material, exhibiting a discharge capacity of 89.5 mAh/g in the initial cycle with a very sloping two-stage discharge curve and relatively stable cycling performance. Beyond excellent energy storage properties, PTEPA also shows relatively good electrochromic performance. Hence, an efficient all-solid-state ECSC prototype with good electrochromic performance and high energy storage capacity (13.3 mF cm -2 ) was assembled based on PTEPA and nanoporous graphene films. During the charging and discharging process, the color of the ECSC changed between yellow-green and steel-blue reversibly. Thus, the energy storage level of the ECSC can be monitored by the corresponding color changes. Through adjusting the dimensions and area of the PTEPA and graphene film, various ECSCs (including selfpowered electrochromic smart windows) with different capacitance and electrochromic properties might also be possible using a similar scheme. This work not only provides a new approach for the preparation of electrode materials for efficient ECSCs, but also provides an important direction for the fabrication of new elecchromic energy storage devices.

Experimental Section

Chemicals and Instrumentation

All reagent grade chemicals were purchased from The Aldrich or Aladdin Reagent (Shanghai) companies. Ltd., and were used without further purification. Indium tin oxide (ITO)-coated glass (sheet resistance Rs ≤ 10 Ω) was purchased from Guluo Glass Technology Co. Ltd. It was ultrasonically washed in deionized water, ethanol, acetone, toluene and (finally) ethanol solutions.

Proton ( 1 H) nuclear magnetic resonance (NMR) spectra were recorded with a Bruker Avance III instrument (Bruker, Switzerland) at 500 MHz. Fourier transform infrared (FT-IR) spectra were obtained using a Nicolet 6700 spectrometer (Thermo Fisher Nicolet, USA). Extracted ion chromatogram and mass spectra (MS) were explored using a ABSCIEX HPLC-TripleTOF 5600 Mass Spectrometer. Elemental analysis was obtained using a Perkin-Elmer 4200 elemental analyzer. Powder X-ray diffraction (pXRD) experiments were performed using a X'Pert Pro diffractometer with Cu-Kα radiation (λ= 1.5418 Å) operating at 40 kV and 40 mA and a scanning angle ranging from 5° to 50° of 2θ. Average molecular weight and polydispersity indices (PDIs) were measured through gel permeation chromatography (GPC) using a Waters Breeze 2 HPLC system. Scanning electron microscopy (SEM) images were collected using a field emission scanning electron microscope (Zeiss Gemini 300). Measurements of surface area and pore size were carried out using an ASAP 2460 surface area and porosimetry auto analyzer (Micromeritics, USA). Ultraviolet-visible (UV-vis) and near infrared (NIR) spectra were obtained using a UV-1800 spectrophotometer (Shimadzu, Japan).

Synthesis of the TEPA Monomer

As shown in Scheme 1, the TEPA monomer was obtained via a fourstep synthetic route using EDOT and TPA as the starting materials in an overall yield of 55%, a detailed description of the synthetic method is provided in the ESI. Samples obtained after chromatography were shown to be analytically pure by elemental analysis (calcd for TEPA (C36H27NO6S3): C: H: N: S= 65.0: 4.1: 2.1: 14.4; found: C: H: N: S = 65.0: 4.1: 2.1: 14.4). [START_REF] Beaujuge | [END_REF] H NMR and FT-IR spectra of the obtained TEPA samples are similar to the reported spectra 31 and are shown in in Fig. S1 and Fig. S2. The +ve ion electrospray MS (Fig. S3) shows a peak at m/z = 666.1085 for [M+H] + , which is very close to theoretical calculated value (666.1073).

Scheme 1. The new synthetic route for the TEPA monomer.

Single-Crystal X-ray Crystal Structure Determination

Yellow sheet-like crystals of TEPA suitable for single-crystal X-ray diffraction (sXRD) measurements were grown by slow evaporation from a CH2Cl2/CH3CN (1:1, by volume) solution at room temperature. sXRD analysis was carried out at T= 293(2) K on an Oxford Diffraction Gemini four-circle diffractometer, equipped with a Ruby CCD detector and using graphite-monochromatized Mo-Kα radiation. Data reduction included intensity integration, background and Lorentz-polarization correction performed with the CrysAlisPrO Version 1.171.37.35 software package (Agilent Technologies, 2014). The structures were solved by direct methods and then refined with full-matrix least-square methods using SHELXL-2014 program, [START_REF] Sheldrick | SHELXT -Integrated space-group and crystal-structure determination[END_REF] with all non-hydrogen atoms being refined anisotropically. A summary of the crystal data, collection parameters and refinement is given in Table 2 and selected bond lengths (Å) and angles ( o ) as well as hydrogen-bonding geometries are listed in Table S1 and Table S2. The dihedral angles between the two aromatic rings of TEPA molecule were calculated using the Diamond software. [START_REF]Diamond -Crystal and Molecular Structure Visualization Crystal Impact[END_REF] The calculated dihedral angles are summarized in Table S3. Polymerization of PTEPA 0.2 g dodecyltrimethylammonium bromide (DTAB) and 12 mL of 0.4 mol/L FeCl3 aqueous solution were mixed and stirred for 1 h at room temperature in a two-necked round bottom flask equipped with a dropping funnel. Subsequently, the TEPA monomer (1 mmol) was dissolved in 13 mL CH2Cl2, which was added dropwise to the previous solution with vigorous stirring. After polymerization for 18 h, the organic phase was decanted and washed several times with distilled water to remove the organic solvent. The powder of PTEPA produced was finally washed with methanol to wash out unreacted monomer and dried under vacuum at room temperature.

Preparation and Electrochemical Measurements of PTEPA Electrodes

The PTEPA electrodes were prepared by coating a mixture containing 50% of the as-prepared PTEPA polymers, 40% acetylene black, and 10% polyvinylidene fluoride binder on circular Al current collector foils, followed by drying at 60 o C for 12 hours The load mass of the composite on the current collector was about 4 mg. After this, CR2032 coin-type cells were assembled in an argon-filled glovebox using the prepared PTEPA electrode as the cathode and lithium foil as the anode. A polypropylene micro-porous film (Celgard 2300) was used as a separator and 1 mol/L LiPF6 dissolved in ethylene carbonate (EC) and dimethyl carbonate (DMC) (VEC/VDMC= 1:1) as the electrolyte.

Galvanostatic charge-discharge (GCD) measurements of PTEPA electrodes were carried out on a LAND CT2001A in the voltage range of 2.5-4.3 V versus Li/Li + , using a constant current density at room temperature. Cyclic voltammetry (CV) tests were carried out in the twoelectrode electrochemical cell assembled above with the scanning potential ranging from 2.5 V to 4.3 V and a scanning rate of 1 mV s -1 using a CHI 660E.

Preparation and Electrochemical Measurements of PTEPA Films

PTEPA powder was dissolved in chloroform to give a 1 g/L solution which was filtered through syringe filters before use, a digital photograph of this deep-blue PTEPA solution is shown in Fig. S5. PTEPA films were prepared via drop casting this solution onto ITO-coated glass substrates and drying in air at room temperature.

CV and GCD tests for energy storage performance of PTEPA films were carried out on a CHI660E electrochemistry workstation in a three-electrode system with the reference electrode being Ag/AgCl (silver wire coated with AgCl in saturated KCl solution) and the counter electrode being a platinum sheet. The electrolyte was a 1M H2SO4 aqueous solution. Spectroelectrochemsitry data was measured using Shimadzu UV-1800 UV-Vis spectrophotometer (Shimadzu, Japan) and Shanghai Chenhua CHI 660E electrochemical workstation in a three-electrode system with the reference electrode being Ag/AgCl, the counter electrode being a platinum wire and the electrolyte being 0.1M LiClO4/CH3CN.

  Synthesis and Structure of TEPA In 2007, M. Chahma et al. reported the synthesis of TEPA through a Stille coupling reaction between tris[4bromophenyl]amine and 2-tributylstannyl-3,4ethylenedioxythiophene. However, because of the toxicity of organotin reagents and the difficulties in separating organotin byproducts from target products, this synthetic pathway his problematic in large-scale synthesis. As shown in Scheme 1, we obtained a light yellow powder of TEPA via a Suzuki-Miyaura cross-coupling reaction between tris[4-(4,4,5,5-tetramethyl-1,3,2dioxaborolan-2-yl)

Figure 1 .

 1 Figure 1. Molecular structure of TEPA, H-atoms have been omitted for clarity (red = O, blue = N, gray = C, yellow = S), and only one of the three independent molecules in the unit cell is shown.

Figure 2 .

 2 Figure 2. Two-dimensional layers of TEPA.

Figure 3 .

 3 Figure 3. Molecular packing of TEPA.

Figure 4 .

 4 Figure 4. Key dimensions of the TEPA monomer units, indicating the distortion of the polymer building blocks.

Figure 5 .

 5 Figure 5. (a) Schematic representation of the TEPA monomer as a building block, (b) two possible assemblies of PTEPA, (c) a possible spiral channel of PTEPA, (d) a possible porous polymer structure of PTEPA.

Figure 6 .

 6 Figure 6. SEM images of PTEPA powder taken at a magnification of (a) 20k and (b) 60k; (c) SEM image of a PTEPA film; (d) Nitrogen adsorption-desorption isotherms of PTEPA powder; (e) BJH pore size distribution from the desorption isotherms of PTEPA; (f) microporous pore size distribution in PTEPA calculated from the adsorption isotherms.

Figure 7 .

 7 Figure 7. (a) CV curve of the PTEPA electrode at the scan rate of 1 mV/s; (b) Initial charge-discharge curves of the PTEPA electrode at a current density of 20 mA/g; (c) Cycling stability of the PTEPA electrode at a current density of 20 mA/g; (d) Coulombic Efficiency of the PTEPA electrode; (e) CV curve of the PTEPA film in 1 mol/L H2SO4 aqueous solution, at a scan rate of 100mV/s (f) GCD curves of the PTEPA film in 1 mol/L H2SO4 aqueous solution at approximately 10 A/g.

Figure 8 .

 8 Figure 8. (a) Spectroelectrochemical spectra of PTEPA films with different applied potentials between -0.4 V and 1.0 V; (b) Transmittance-time profiles of PTEPA films at 440 nm and 660 nm between -0.4 and 1.0 V with a residence of 5 s; (c) Switching response for PTEPA films monitored at 440 nm; (d) Switching response for PTEPA films monitored at 660 nm.

Figure 9 .

 9 Figure 9. (a) The assembly structure of the all-solid-state ECSC; (b) CV curves of the ECSC in the voltage window of -0.8 V~0.2 V at different scan rates; (c) Specific capacitance of the ECSC at different current densities; (d) Chargedischarge curves of the ECSC at different current densities and the corresponding digital photograph of the device.

Table 1

 1 Current density and area specific capacitance of reported electrochromic supercapatiors.

Table 2

 2 Details of the structure solution and refinement of TEPA

	Compound	TEPA
	Chemical formula	C36H27NO6S3
	FW	665.76
	Crystal system	Monoclinic
	Space group	P21/c
	a (Å)	10.6825 (4)
	b (Å)	15.4602 (9)

Acknowledgements

We sincerely thank the support from Zhejiang Provincial Natural Science Foundation of China (No. LY19B010003) and National Natural Science Foundation of China (NSFC, No. 21501148). LR and AB also thank the Université de Strasbourg as well as the Labex CSC (Chemistry of Complex Systems) which has also supported one part of this research.

Entry for the Table of Contents

Insert graphic for Table of Contents here.

Tris [4-(3,4-ethylenedioxythiophene)phenyl]amine with disorded structure were obtained via green-synthetic route, its solution-processable nanoporous polymer are applied for efficient all-solid-state electrochromic supercapacitor.