Gang Zheng, Senior Member, IEEE Lingxiao Xun
email: lingxiao.xun@inria.fr

Sofiane Ghenna
email: sofi-ane.ghenna@uphf.fr

Alexandre Kruszewski
email: alexandre.kruszewski@centralelille.fr

Éric Cattan
email: eric.cattan@uphf.fr

Christian Duriez
email: christian.duriez@inria.fr

Sébastien Grondel
email: sebastien.grondel@uphf.fr

Gang Zheng
email: gang.zheng@inria.fr

M E Degrees

Modeling and Control of Conducting Polymer Actuator

Keywords: Conducting polymer actuator, IEAP, parameter identification, optimal control. I

Conducting polymer actuator has nonlinear dynamic characteristics during its charge process. In this study, we proposed an electromechanic model and an optimal controller for a type of ionic electroactive polymer (IEPA) actuator with sub-millimeter scale, which can produce large deformation under low actuation voltage. The electronic model is to describe the evolution of charge state in time domain. The mechanic model is to calculate the deformation of conducting polymer actuator under the actuation force and external force. Based on the electromechanic coupling model, a parameter identification method is proposed to estimate the nonlinear parameter of conducting polymer actuator. The experiments show that our electromechanic model successfully predicts the deformation of actuator under different input voltages with the identified parameters. In the last step, an optimal controller is designed to control the orientation of IEAP actuator, which achieves at a high control performance in our experiments. The success of the modeling and control lays the foundation work for the subsequent biomedical applications.

of applications in robotics [START_REF] Kim | Electroactive polymers for robotic applications[END_REF] or biomedical applications [START_REF] Smela | Conjugated polymer actuators for biomedical applications[END_REF]. The work of this paper focuses on a typical actuator based on IEAP as the conducting polymer, which contains its nonlinear modeling, parameter estimation and control.

In this paper the IEAP based trilayer micro-actuator was micro-fabricated according to the method described in [START_REF] Seurre | Demonstrating full integration process for electroactive polymer microtransducers to realize soft microchips[END_REF]. The upper and lower layers are made of electronically conducting polymers (CP) material basically poly (3,4ethylenedioxythiophene) polystyrene sulfonate as reported in [START_REF] Plesse | Synthesis and characterization of conducting interpenetrating polymer networks for new actuators[END_REF]. The middle layer is considered as an ion reservoir providing the system with ionic conductivity and mechanical properties [START_REF] Plesse | Synthesis and characterization of conducting interpenetrating polymer networks for new actuators[END_REF]. This middle layer is formed with a semiinterpenetrated polymer network (IPN) composed of two polymers: poly ethylene oxide (PEO) and linear nitrile butadiene rubber (NBR) network. When a potential difference is applied to these two CP layers, one of the layers will undergo an oxidation reaction and the other one has a reduction reaction. It results in a corresponding increase or decrease in the charge density of the two layers. The change in ionic charge density will cause the two electrodes to produce opposite strains, which will cause one layer to expand and one layer to shrink. The expansion and contraction effect can be regarded as the bending of the actuator caused by the equivalent moment of force exerted on it. It should be noted that, the bending direction is occurring toward the anode [START_REF] Rohtlaid | Pedot: Pss-based micromuscles and microsensors fully integrated in flexible chips[END_REF].

Usually, an RC circuit model to describe the charge change process in the two electrode layers is used, and then the internal stress and strain are analyzed to obtain the curvature of the actuator. At present, some equivalent RC circuit models have been proposed [START_REF] Nguyen | An advanced mathematical model and its experimental verification for trilayer conjugated polymer actuators[END_REF], [START_REF] Mazzoldi | Conducting polymer actuators: Properties and modeling[END_REF]. These models usually use isolated RC combinations or limited RC grid circuits to equivalent charge change processes. Although these models are efficient, their disadvantage is that they do not take into account the nonlinear changes of physical parameters in the actuator. The research of [START_REF] Farajollahi | Characterization and dynamic charge dependent modeling of conducting polymer trilayer bending[END_REF] shows that the resistivity and volumetric capacitance will both have greater nonlinearity during the charging process. The change interval affects the corresponding time of the actuator, so the nonlinear characteristics of the material are a consideration that can not be ignored in the modeling. [START_REF] Farajollahi | Nonlinear two-dimensional transmission line models for electrochemically driven conducting polymer actuators[END_REF] has proposed a finite nonlinear 2D-RC model, where this model takes into account the nonlinear characteristics of material properties following changes in charge density. In the work of Nishida [START_REF] Nishida | Multi-scale distributed parameter modeling of ionic polymer-metal composite soft actuator[END_REF], the infinite RC circuit was used to approximate the real system and finite difference method was adopted to get the numerical solution. However, the methods mentioned above struggle to achieve a balance between the dimensionality and accuracy of the model.

The second part of modeling for conducting polymer actuator is calculating its deformation. It is noticed that if the charging time is enough long to reach the equilibrium charge state, the deformation will be uniform along the polymer, leading to the uniform curvature, which has been calculated by several studies [START_REF] Nguyen | An advanced mathematical model and its experimental verification for trilayer conjugated polymer actuators[END_REF], [START_REF] Moghadam | Development of a novel soft parallel robot equipped with polymeric artificial muscles[END_REF]. In fact, it usually takes a long time to reach the equilibrium charge state, thus the local deformation is necessary to study. [START_REF] Nishida | Multi-scale distributed parameter modeling of ionic polymer-metal composite soft actuator[END_REF] and [START_REF] Liu | Energy-based modeling of ionic polymer-metal composite actuators dedicated to the control of flexible structures[END_REF] proposed the Timoshenko model to predict the deformation of conducting polymer actuator with the assumption of small deformation of 2 degrees of freedom. Meanwhile, in most cases the conducting polymer actuator does not work alone but interacts with the environment, such as external forces applying on it or multi conducting polymer actuators being connected in series.

Another difficulty related to the nonlinear model lies in the measurement of electronic model parameters. The nonlinear changing of various physical parameters of materials requires a large amount of experimental data to construct. At the same time, it is difficult to realize the control of a single variable during the experiment due to the excessive change of material parameters, which affects the accuracy of the experimental results. This process is often cumbersome and is limited by measurement accuracy, resulting in error stacking.

In terms of drive control, [START_REF] Khawwaf | Adaptive microtracking control for an underwater ipmc actuator using new hyperplane-based sliding mode[END_REF] designed PID and adaptive robust controller for controlling the displacement of conducting polymer actuator. However, the above-mentioned controller design is only based on the system model after the end point displacement is identified, which does not include the actuator curvature and bending moment information, thus this simplified model makes it difficult to further meet the control requirements of the actuator, such as actuator rotation angle control and multi-drive coupling control.

B. Contributions

Facing to the limitations of the current research, the contributions of our proposed model and controller are as follows:

1) This work aim to find a more general format for both the electronic and mechanic model in order to reduce the computational complexity without losing accuracy and facilitate the control design. In this paper, we put forward a parameterization of both charge field and strain field to model the real system, which greatly reduces its dimension. The Cosserat rod theory is adopted to deal with the exact nonlinearity in the large deformations of conducting polymer in six degrees of freedom under the interaction with the environment. Comparing with the current modeling methods, our proposed coupling model, with lower dimension and a balance between accuracy, robustness and computational complexity, intent to lay the foundation for real-time simulation and control works.

2) To overcome the problems related to the unknown or imprecise physical parameters, this paper proposes a parameter identification method based on a nonlinear electro-mechanical coupling model. A visual identification system is designed to measure the curvature of actuator which is used for parameter identification.

3) An optimal controller based on the proposed electromechanic model is designed in this paper, which lays the foundation work for the subsequent multi-drive coupling trajectory tracking control in the future. Experiments are carried out to validate the controller design and its robustness.

In summary, this paper is organized as follows: A nonlinear modeling of the electronic and mechanical part is introduced in Section II. Then parameter identification is addressed in Section III. Controller design is presented in Section IV. Numerical simulations and experimental validation are presented in Section V. Finally, Section VI concludes this article.

II. NONLINEAR MODEL OF POLYMER

The modeling task of conducting polymer can be mainly divided into two parts: the electronic model and the mechanic model. Noting that the working state of IEAP actuator depends on the actuation frequency [START_REF] Nguyen | Nonlinear dynamic modeling of ultrathin conducting polymer actuators including inertial effects[END_REF] , it can be classified into two cases: (i). When the actuation frequency is below the mechanical resonance frequency of the actuator, the deformation is quasi-static. (ii). When the actuation frequency is greater than the mechanical resonance frequency, the deformation varies mainly due to inertial mass effects. Thus the dynamics in mechanic model is necessary.

A. Electronic model

In this part we will first build the continuous electronic model which describes the dynamic evolution of charge along actuator. Then the discrete model is built by discretization of the continuous model.

1) Strong form of electronic dynamics: Denote the total length of polymer as L where two electrode layers outside wrap the middle ion reservoir membrane. When the polymer is charged, the research in [START_REF] Nguyen | Nonlinear dynamic modeling of ultrathin conducting polymer actuators including inertial effects[END_REF] shows that the process of charge can be regarded as an infinity RC circuit grid (see Fig. 1(a)). Let us consider an infinitesimal element dx, and denote ∆R e1 and ∆R e2 as the electric resistance of outer layers and ∆R i as the ionic resistance of middle layer, ∆C 1 and ∆C 2 as the corresponding capacitances of outer layers, which satisfy the following equations according to its definition:

∆R e1 = dx ω e1 bh e , ∆R e2 = dx ω e2 bh e , ∆R i = h i ω i bdx ∆C 1 = C v1 bh e dx , ∆C 2 = C v2 bh e dx,
where b represents the width of polymer, h e is the thickness of the CP layer, C v1 and C v2 are the volumetric capacitance of the top CP layer and the bottom CP layer , ω e1 and ω e2 are the volumetric conductivity of the top CP layer and the under CP layer, ω i is the volumetric conductivity of the separator layer. For each RC grid element depicted in Fig. 1(a), its scheme can be simplified as Fig. 1(b), where

   ∆R i = hi ωibdx = ri dx ∆R e = ∆R e1 + ∆R e2 = ( dx ωe1bhe + dx ωe2bhe )dx = r e dx ∆C = ∆C 1 + ∆C 2 = (C v1 bh e + C v2 bh e )dx = c e dx
(1) Without loss of generality, it is assumed that the electric resistances and ionic resistance are not constant. In fact, this electronic nonlinear characteristic has been observed in [START_REF] Farajollahi | Nonlinear two-dimensional transmission line models for electrochemically driven conducting polymer actuators[END_REF], which implies that the values of those resistances depend on the local charge.

As shown in Fig. 1(b), for the RC grid element located at x with dx length, denote v as the voltage of capacity ∆C, ∆q as the charge of ∆C, i 1 and i 2 as the currents of the two branches, λ as the linear charge density at x, then v, i 1 and i 2 can be represented by λ(x, t) via the following equations:

v(x, t) = λ c e , i 1 (x, t) = ∂λ ∂t dx, i 2 (x, t) = l x ∂λ ∂t dr. (2)
For each grid as Fig. 1(b), according to Kirchhoff laws (i.e., U = 0), we have

i 2 (x, t)∆Re + λ(x+dx,t) ce + i 1 (x + dx, t)∆R i -i 1 (x, t)∆R i -λ(x,t) ce = 0,
With (1) and (2), we can then get the strong form of electronic dynamics from the above equation:

r e (λ) L x ∂λ ∂t dr + 1 c e ∂λ ∂x + r i (λ) ∂ 2 λ ∂x∂t = 0, (3)
which is a partial differential equation (PDE) describing the charge density along the polymer, satisfying the following boundary conditions:

λ(x, 0) = 0, λ(0, t) c e + r i (λ(0, t)) ∂λ(0, t) ∂t -V in = 0, (4)
where V in is the tension input applied on the two CP layers. Remark 1: Due to the balance of charge in the charging process, the charge along actuator will tend to become stable when time tends to infinity, which means ∂λ ∂t = 0, thus ∂λ ∂x = 0 according to (3). Consequently, the distribution of charge along actuator at the equilibrium point will become geometrically uniform.

Up to now, we have built the continuous equation of charge density as PDE (3), and with (4) they compose a BVP (Boundary Value Problem). However, analytically solving such a highly nonlinear BVP is quite complicated. In the following, we derive its weak form by applying Galerkin method to approximate the solution via spatially parameterizing the charge field.

2) Weak form of electronic dynamics: In order to simplify the writing of ( 3) and ( 4), we use prime in place of ∂ ∂x and dot in place of ∂ ∂t . Thus ( 3) and ( 4) can be written as follows:

r e (λ) L x λdr + c e -1 (λ)λ + r i (λ) λ = 0, (5)
with the boundary condition:

c e -1 (λ(0, t))λ(0, t) + r i (λ(0, t)) λ(0, t) -V in (t) = 0. (6)
Now we can define the "virtual charge displacement" δλ of each micro RC grid, then the energy balance equation of conducting polymer might be deduced, which corresponds to the weak form of PDE (5) as follows:

∀x ∈ [0, L] → δλ(x) ∈ R, L 0 δλ T (x) r e L x λdr + c e -1 λ + r i λ dx = 0. (7)
3) Parameterization charge field: Note that the solution of ( 7) is defined in an infinite set. To discretize the system and find the numerical solution, we consider that the field of charge can be approximated by a set with limited degrees of freedom. To generically handle this kind of approximation, the field of charge λ is defined by the product of basis function and coefficients. i.e.,

λ(x, t) := Φ(x)λ(t), s ∈ [0, L], (8)
where

Φ(x) = [Φ 0 , Φ 1 , . . . , Φ m ] ∈ R 1×m is the basis function and λ = [λ 0 , λ 1 , . . . , λ m ] T ∈ R m×1 is the coefficients.
In order to globally parameterize the charge field without losing physical meaning, the basis function Φ(x) is chosen by these two assumptions below: Assumption 1: The conducting polymer can be divided into several sections and for each section the charge density is distributed linearly. Assumption 2: All the physical parameters are homogeneous in each section.

Based on the piece-wise linear assumption 1, basis function Φ(x) holds following structure:

Φ =                    l-x l , x l , 0, . . . , 0 , 0 x < l . . . . . . 0, . . . , 0, kl-x l , x-(k-1)l l , 0, . . . , 0 , (k -1)l x < kl . . . . . . 0, . . . , 0, nl-x l , x-(n-1)l l , (n -1)l x < nl
As a result, (8) leads to two relations:

δλ(x) = Φδλ, λ(x) = Φ λ. (9)
4) Discrete electronic model: Note that the weak form ( 7) is equivalent to (3). This weak form holds the advantage of realizing the approximation from the continuous dynamics of electronic model to discrete dynamics. Concretely, by substituting ( 8)-( 9) into [START_REF] Seurre | Demonstrating full integration process for electroactive polymer microtransducers to realize soft microchips[END_REF], one can get:

δλ T L 0 Φ T r e L x Φ λdr + c e -1 Φ λ + r i Φ λ dx = 0.
The equation above should hold for any δλ, thus after removing δλ T and adding the boundary condition [START_REF] Ghaffari | Retracted: Aligned nano-porous microwave exfoliated graphite oxide ionic actuators with high strain and elastic energy density[END_REF], we obtain the following succinct ODE representation of the electronic model:

(Re 0 P + Ri W Q ) λ + Ce -1 W Q λ = Vin 0 , (10)
with

W = 1 0 . . . 0 ∈ R m+1 , P = L 0 Φ T (x) L x Φ(r)drdx, Q = L 0 Φ T (x)Φ (x)dx, Re = diag{re(λ0), re( λ0 + λ1 2 ), . . . , re( λm-1 + λm 2 )}, Ri = diag{ri(λ0), ri( λ0 + λ1 2 ), . . . , ri( λm-1 + λm 2 )}, Ce = diag{ce(λ0), ce( λ0 + λ1 2 ), . . . , ce( λm-1 + λm 2 )}.
After having established the discrete dynamic equation [START_REF] Plesse | Synthesis and characterization of conducting interpenetrating polymer networks for new actuators[END_REF] of electronic model to describe the evolution of charge density along actuator, the next step is to show how the charge density of each section can create the deformation of polymer.

B. From electric charge to actuation moment

Consider now one element of 3-layer polymer depicted in Fig. 2 with E e and E i being the Young's modulus of the CP layer and the separator layer respectively. Denote α as the strain-to-charge ratio of the CP layers, and λ as the linear charge density of CP layers. Obviously, the values of charge density for these two CP layers should be opposite due to the balance of charge. Thus we define the linear charge density of the upper CP layer as λ, so that of the bottom CP layer should be -λ. As shown in Fig. 2, for a micro element of the actuator, the stress on its cross section S is comprised with two parts (see Fig. 2): the elastic stress σ e and the redox stress σ r . Since the redox strain is proportional with charge density [START_REF] Farajollahi | Characterization and dynamic charge dependent modeling of conducting polymer trilayer bending[END_REF], the equivalent redox stress can be calculated as follows:

σ r (y) =    -E e αλ bhe , hi 2 < y hi 2 + h e 0, -hi 2 y hi 2 E e αλ bhe , -hi 2 -h e y < -hi 2
Consequently, for any cross section we can compute its actuated force defining in the local frame of cross section. It can be deduced directly that the total contribution of the electric stress is a moment while the forces of the two CP layers counteract with each other:

F a = S σ r dS = t 1 S σ r dS = t 1 h i 2 +he - h i 2 -he σ r dydz,
where vector t 1 is the identical direction vector of σ r (i.e., t 1 = 1 0 0 T ). The integral part

h i 2 +he - h i 2 -he σ r dy = 0, which yields F a = 0 0 0 T , Ma = S y × σrdS = S t2 × t1σrydS = b 0 t h i 2 +he - h i 2 -he σrydydz = t b 0 Eeα(he + hi)λ b dz = tEeα(he + hi)λ = tβλ = 0 0 βλ T , (11)
where t 2 is the identical direction vector of y (i.e., t 2 = 0 1 0

T ), t = t 2 × t 1 = 0 0 -1 T , β = E e α(h e + h i )λ.
For ease of expression, we use an actuation wrench Λ a ∈ R 6 to represent F a and M a : [START_REF] Mazzoldi | Conducting polymer actuators: Properties and modeling[END_REF] where Γ = 0 0 1 0 0 0 T .

Λa = Ma T Fa T T = 0 0 βλ 0 0 0 = Γβλ
Once we deduced the actuation moment of polymer actuator, the dynamic state of polymer actuator can be then analyzed.

C. Mechanic model

In the aspect of mechanic analysis, it is natural that the polymer actuator can be regarded as a slender beam. The local charge accumulated inside the polymer will induce the internal stress which causes the deformation of the polymer actuator. In this section we will first build the relation between the internal force and local charge. Then the dynamics of the polymer actuator will be analyzed in the following subsections, where we consider the deformation of polymer actuator in 6 DOF (bending, shear, torsion, extension) under actuation force and external force.

1) Continuous geometries & kinematics of Cosserat rod: In cosserat rod theory, the configuration of rod is described by the rotation matrix R ∈ SO(3) and position vector u ∈ R 3 , which represent the orientation and position of each cross section of the rod with respect to global frame. Thus, the configuration space of rod is defined as follows:

C = {g : x ∈ [0, L] → g(s) ∈ SE(3)}, g = R u 0 T 1
The space-time variation of the configuration can be defined with two vector fields ξ and η which stand for the strain and velocity of rod respectively in local frame: ∀x ∈ [0, L] → ξ(x) ∈ se(3):

ξ = g -1 g = κ 0 T 0 ∈ se(3), ξ = ξ ∨ = κ T T T ∈ R 6
where κ stands for the angular strain and represents the linear strain. ∀x ∈ [0, L] → η(s) ∈ se(3):

η = g -1 ġ = w v 0 T 0 ∈ se(3), η = η ∨ = w T v T T ∈ R 6
where w stands for the angular velocity and v represents the linear velocity with respect of local frame.

Based on the definitions of ξ and η, one can derive the geometries and kinematics through the following two equations respectively:

g = g ξ , η = ξ -ad ξ η (13)
which connect the velocity and acceleration with strain.

2) Strong form of dynamics: To analyze the mechanic behavior of beam such as polymer actuator, the strong form of dynamic model can be demonstrated as [START_REF] Farajollahi | Characterization and dynamic charge dependent modeling of conducting polymer trilayer bending[END_REF], which is detailed in [START_REF] Black | Parallel continuum robots: Modeling, analysis, and actuation-based force sensing[END_REF] and [START_REF] Boyer | Dynamics of continuum and soft robots: A strain parameterization based approach[END_REF]:

M η -ad T η Mη = Λ i -ad T ξ Λ i -Λ a + ad T ξ Λ a + Λ e , (14)
satisfying the following boundary conditions:

Λ i (L) -Λ a (L) = Λ e (L), (15)
where M represents the screw inertia matrix, M = diag{ρI x , ρI y , ρI z , ρA, ρA, ρA} ∈ R 6×6 . ρ is the equivalent mass density of the total actuator. A is the area of crosssection. I x , I y and I z are the second moment of the area of the three axis respectively. Λ i ∈ R 6 represents the internal force wrench of polymer actuator which can be calculated by the constitutive equation based on Cosserat model hypotheses:

Λ i = Kξ with K = diag{GI x , EI y , EI z , EA, GA, GA} ∈ R 6×6
, the equivalent stiffness matrix of the entire electrodes plus polymer layers, where G represents the shear modulus and E represents the young's modulus. Λ a ∈ R 6 represents the actuation force wrench of polymer actuator. Λ e ∈ R 6 represents the distributed load applied on the polymer actuator.

3) Weak form of dynamics: According to the Lagrange-d'Alembert principle, if we define the virtual displacement of each micro solid along polymer actuator δr(x), one can write the virtual work of polymer actuator which leads to the weak form of equation ( 14) as follows:

∀x ∈ [0, L] → δr(x) ∈ R 6 , L 0 δr T (x)(M η -ad T η Mη -Λ i + ad T ξ Λ i + Λ a -ad T ξ Λ a -Λ e )ds = 0 (16
) 4) Parameterization strain field: To find the solution of equation ( 16), the same parameterization idea with electric model is introduced here in order to get the discretize the mechanic model. [START_REF] Rohtlaid | Pedot: Pss-based micromuscles and microsensors fully integrated in flexible chips[END_REF] represents the geometries of the rod. i.e., its configuration space can be reconstructed by knowing the strain ξ and the initial condition g(0). As a result, this yields to the second definition of the configuration space: C = SE(3) × S, where S = {ξ : x ∈ [0, L] → ξ ∈ R 6 }. Note that space S is an infinite dimensional space. For the sake of approximating the solution of ( 16), we consider that a finite dimensional space S can be adopted by parameterizing the strain field ξ, i.e., ξ(s)

:= ξ 0 + Ψ(x)q, x ∈ [0, L], (17)
where

Ψ(x) = [Ψ 0 , Ψ 1 , . . . , Ψ n ] ∈ R 6×n
is the basis function and q = [ξ 0 , ξ 1 , . . . , ξ n ] T ∈ R n×1 is the coefficients. ξ 0 stands the constant part of ξ which we do not want to chose to be variable.

5) Discrete mechanic model: As a result of parameterization of strain field, the geometries and kinematics can be analytically reconstructed by q from (13). For instance, if we chose Ψ(x) as a piecewise constant basis function, the analytical solutions of configuration g and η are as follows:

for x ∈ [l i-1 , l i ], g(x, t) = g(l i-1 , t)e ξ i (x-li-1) , (18)
η(x, t) =e -(x-li-1) ad ξ i η(l i-1 , t) +

x li-1 e (s-li-1) ad ξ i ds ξi .

Thanks to the work of [START_REF] Renda | Discrete cosserat approach for multisection soft manipulator dynamics[END_REF], these mapping between ξ and η can be represented by the Jacobian matrix J (x, q): η(x, t) = J (x, q) q(t).

Furthermore, the relationship of their time derivative can be deduced by the derivation of (20):

η(x, t) = J (x, q, q) q(t) + J (x, q)q(t).

From ( 20) one can deduce δr(x) = J (x)δq. Taking it into equation ( 16) yields:

δq T L 0 J T (M η -ad T η Mη -Λ i + ad T ξ Λi + Λ a -ad T ξ Λa -Λe)ds = 0. (22)
After removing δq and taking (20) ( 21) into ( 22), we can finally get the discrete dynamic equation: M (q)q+C(q, q) q = F i (q)+F a (q)+F e (q)+N (q)G [START_REF] Renda | Discrete cosserat approach for multisection soft manipulator dynamics[END_REF] where Fe(q) = L 0 J T Λeds, N (q) = L 0 J T MAd -1 g ds, G = [0 1×4 9.81 0] T is the twist of gravitational acceleration,

Fi = L 0 J T (Λ i -ad T ξ Λi)ds, Fa = - L 0 J T (Λ a -ad T ξ Λa)ds. (24)
Form kinematics in (13) one can get η = -ad ξ i η + ξi for x ∈ [l i-1 , l i ]. Taking (20) into the above equation yields:

J q = -ad ξ i J q + ξi = -ad ξ i J q + Ψ q.

Thus, one can get J = -ad ξ i J + Ψ. Then, by substituting this relationship into F i in (24) and using the integral by parts one can obtain

Fi = L 0 (J T Λi) -(-ad ξ J + Ψ) T Λi -J T ad T ξ Λids = L 0 (J T Λi) -Ψ T Λids = J T Λi| L 0 - L 0 Ψ T Λids.
The two boundary conditions J (0) = 0 and

Λ i (L) = Λ a (L) lead to J T Λ i | L 0 = J T (L)Λ a (L).
The piecewise constant strain assumption leads to Λ i = Kξ = KΨq. As a result, F i can be simplified by substituting these two relations into the equation above:

F i = J T (L)Λ a (L) - L 0 Ψ T KΨdsq.
Thanks to the same mathematical form of F a and F i , the same structure holds:

F a = -J T (L)Λ a (L) + L 0 Ψ T ΓΦdsβλ.
After deducing F i and F a , the final mechanic model can be rewritten as following:

M q + C q + Kq = Hz(t) + P , (25)
with the following definitions of all matrices:

• z(t) = [z 0 , z 1 , . . . , z m ] T = βλ(t) ∈ R m , the actuation input; • M (q) = L 0 J T MJ ds ∈ R n×n , the mass matrix; • C(q, q) = L 0 J T (M J -ad T J q MJ )ds ∈ R n×n , the Coriolis matrix; • K = L 0 Ψ T KΨds ∈ R n×n , the stiffness matrix; • H = L 0 Ψ T ΓΦds ∈ R n×m , the actuation matrix; • P (q, t) = Fe(q, t) + N (q)G ∈ R n , the contribution of external force and gravity.

D. Whole coupling model

After having deduced the electric model and the mechanical model of polymer, we can then gather these two models together to describe the whole dynamics of input tension and the corresponding deformation by the combination of the deduced electric system:

(Q 1 Σ 1 + Q 2 Σ 2 ) ż + Q 3 Σ 2 z = U V in , (26)
with

Q1 = 1 β Re, Q2 = 1 β Ri, Q3 = 1 β Ce -1 , Σ1 = 0 P , Σ2 = W Q , U = 1 0 ,
and the obtained mechanic system:

M q + C q + Kq = Hz(t) + P . ( 27
)
If the actuator works in quasi-static state, one can obtain its static model by just deleting the first two items M q and C q of (27).

III. PARAMETERS IDENTIFICATION

In the case of polymer actuator, the parameters of electric system are normally hard to measure directly, thus an identification work is necessary via which we can estimate the parameters from the data that are easily to measure. The electric model ( 26) deduced in the above section is nonlinear and contains unknown matrix Q 1 , Q 2 , Q 3 which are nonlinear function of z. Therefore, a parameter identification procedure is presented in this section to reveal the nonlinearlity of those parameters.

From the definition of R e , R i and C e , we define:

Q1 = diag{q1(z0), q1( z0 + z1 2 ), q1( z1 + z2 2 ), . . . , q1( zm-1 + zm 2 )}, Q2 = diag{q2(z0), q2( z0 + z1 2 ), q2( z1 + z2 2 ), . . . , q2( zm-1 + zm 2 )}, Q3 = diag{q3(z0), q3( z0 + z1 2 ), q3( z1 + z2 2 ), . . . , q3( zm-1 + zm 2 )}.
It is assumed that these nonlinearlities of function q can be approximated by polynomials with certain order k, i.e., System ( 26) can be written into the following algebraic form:

q 1 (s) = k i=0 a i s i , q 2 (s) = k i=0 b i s i , q 3 (s) = k i=0 c i s i . (28) By noting a = a0, . . . , a k , b = b0, . . . , b k , c = c0, . . . , c k and Π = I diag(z) diag 2 (z) . . . diag k (z) , the matrix Q 1 , Q 2 and Q 3 can be represented as: Q 1 = Πa T , Q 2 = Πb T , Q 3 = Πc T . Set θ = a b c T and
Ξθ = Ψ, where Ψ = U T Vin(t1), • • • , U T Vin(tn p ) T .
It is clear that, if all the z can be measurable and Ξ is full row rank, then θ can be identified by using classical least square method.

IV. CONTROLLER DESIGN This section is devoted to designing controller to control the orientation of polymer at free state, as depicted in Fig. 3.

A. State-space representation

In our study, the actuation frequency is below the mechanical resonance frequency of the actuator, thus only quasistatic model is considered in the experimental tests. From [START_REF] Zhao | Preprocessing and postprocessing for skeletonbased fingerprint minutiae extraction[END_REF] one can then get the mechanic system of quasi-static model ignoring the gravity and external force:

Kq = -Hz(t). ( 29
)
It is clear that in this scene there is only one variable degree of freedom which is the angular strain κ along axis z. Thus here we firstly reduce the order of (29) by a matrix

B = [0 1×2 1 0 1×3 ]: κ = -BK -1 Hz. ( 30
)
From κ one can calculate the geometric matrix g by ( 18) and extract the position of end-effector and its orientation from g:

p = Dg(κ)T , (31)
with D = I 3×3 0 , T = 0 0 0 1 T . As a result, the bending θ L can be deduced by p:

θ L = arctan(p). ( 32
)
Consider the deduced system (26) (30) (31) and (32), it can be written into the following state-space form:

ż = f (z) + g(z)u(t) θ L = h(z) (33)
where

f (z) = -(Q 1 Σ 1 + Q 2 Σ 2 ) -1 Q 3 Σ 2 z, g(z) = (Q 1 Σ 1 + Q 2 Σ 2 ) -1 U , u(t) = V in (t)
and function h(z) is implicitly defined by (30) (31) and (32).

B. Observer design

In the aspect of control, considering the different application scenarios of polymer actuator, we suppose that one can only measure the position of end effector, thus in order to design state-feedback controller to control θ L , we need to design firstly an observer to estimate z. Considering that the system is nonlinear and observable, extended Kalman filter [START_REF] Ribeiro | Kalman and extended kalman filters: Concept, derivation and properties[END_REF] has been adopted in this paper, whose form is written as

ż = f (ẑ) + g(ẑ)u + P H T R -1 [θ L -h(z)], (34)
with P is the solution of the following equation:

Ṗ = AP + P A T + Q -P H T R -1 HP , where A = ∂f ∂z | z=ẑ , H = ∂h ∂z | z=ẑ
. matrix Q and R are respectively the power spectral density of process noise and that of the observation noise, which represent the errors of system and measurement respectively.

C. Interpolation and linearization

As shown in Fig. 3(a), for the end-effector of polymer, we firstly divide the workspace between θ Lmin and θ Lmax by m+

1 points θ Lp i to form m zones U i = [θ Lp i-1 , θ Lp i ] with θ 0 = θ Lmin and θ Lp m = θ Lmax , i.e., [θ Lmin , θ Lmax ] = ∪ m i=1 U i . Then the reference path θ L ref (t)
is interpolated by θ Lp i , and nonlinear system (33) whose output is located into the ith zone U i is linearized at the equilibrium point θ Lp i , based on which linear controller will be designed to drive θ L moving toward reference θ L ref .

According to Remark 1 in Section II, the equilibrium charge density λ for a given orientation θ Lp i is geometrically uniform. Thus, for any

θ Lp i ∈ [θ Lmin , θ Lmax ] we have z 0 (θ Lp i , ∞) = • • • = z n (θ Lp i , ∞).
As shown in Fig. 3(b), the geometry leads to the following relation:

κ(θ Lp i , ∞) = β Lp i L = 2θ Lp i L , ∀θ Lp i ∈ [θ Lmin , θ Lmax ] . (35)
Thus one can deduce z(θ Lp i , ∞) from (30):

z 0 (θ Lp i , ∞) = • • • = z n (θ Lp i , ∞) = EI z κ(θ Lp i , ∞).
For simplicity, at each θL p i ∈ [θL min , θL max ], note the equilibrium z as zeq(θL

p i ) = z0(θL p i , ∞), • • • , zn(θL p i , ∞) T .
By taking z eq (θ Lp i ) in (33), the input voltage u eq of equilibrium point can be solved.

Such a linearization of nonlinear system (33) at θ Lp i yields

     ż = A zeq(θ Lp i ) (z -z eq (θ Lp i )) +B zeq(θ Lp i ) (u -u eq (θ Lp i )) θ L = H zeq(θ Lp i ) (z -z eq (θ Lp i )) + θ Lp i , ∀θ Lp i ∈ U i (36) where    A zeq (θ Lp i ) = ∂f ∂z | z=zeq (θ Lp i ) , B zeq (θ Lp i ) = ∂g ∂z | z=zeq (θ Lp i ) H zeq (θ Lp i ) = ∂h ∂z | z=zeq (θ Lp i ) (37)

D. Controller design

The above linearized system can be represented by tracking error with extended state below:

x = e z T e iθ T ,
where e z = z eq (θ Lp i ) -z, e θ = θ Lp i -θ L and e iθ = e θ dt.

With the above state, we can obtain the following observation error dynamics:

ėz ėiθ = A zeq (θ Lp i ) 0 H zeq (θ Lp i ) 0 Ā ez e iθ + -B zeq (θ Lp i ) 0 B (u -ueq(θL p i )) ū . (38)
Remark 2: Integrator e iθ in ( 38) is added on purpose to get a precise steady state. By doing so, we ensure that at steady state (when ėz = 0) we get the output at the right value. Note that this integrator will produce a pole at location zero. The location of this pole in closed loop (when u = Kx , K computed by our proposed LQR method) will be in the left hand plane, i.e. the system becomes stable [START_REF] Hespanha | Linear systems theory[END_REF].

With the estimated curvature ẑ via extended Kalman filter (34), then the iterative LQR controller [START_REF] Hespanha | Linear systems theory[END_REF] is used to stabilize the linearized system at each time step, which guarantees the convergence speed and robustness of system by its advantage as the optimal controller, thus we replace z by ẑ in implementation. Concretely, by noting e z = z eq (θ Lp i ) -ẑ in the new state x of system, in order to minimize the following cost function:

J = ∞ ti (x T Qx + ūT Rū)dt,
where t i is the initial time when end-effector enters the current linearized zone of θ Lp i , R and Q are the weight matrices for ū and x, the following analytical solution of LQR controller is obtained:

ū = u LQR = R-1 B P x, (39)
with P is the solution of the following algebraic Riccati equation:

Ā P + P ĀT + Q -P BT R-1 B P = 0.
Thus, the input voltage u = u LQR + u eq (θ Lp i ).

V. NUMERICAL SIMULATION & EXPERIMENT In this section, experimental tests will be presented to valid the deduced nonlinear model, and the proposed LQR controller.

A. Experimental setup

As shown in Fig. 4, our platform contains the polymer actuator which is connected to Arduino card and a camera (Microscope USB Dino-Lite) which enables us to catch the deformation of actuator with 20 fps. Specific image processing program has been developed by us to track the shape of polymer actuator in real time and compute the corresponding curvatures and its time derivatives for each divided segment, which will be detailed in the next subsection. Tab.I shows the measured parameters of the tested actuator. B. Nonlinear model validation 1) Parameter identification: In order to identify the unknown parameters (q 1 , q 2 , q 3 ), we use a ramp signal V in = at with a = 0.01 V /s as the excitation input signal to deform the polymer. In the work of Section III, the variable z is calculated from the curvature of actuator, which can be measured directly by our designed visual system. Since the actuation speed is very low, the actuator can be regarded as quasi-static state. Since there is no external force applied on the actuator, the internal force equals to the actuation one, i.e., for s ∈ [0, L], Λ i (s) = Λ a (s), which means:

κ(s, t) = βλ(s, t) EI z = z(s, t) EI z , ( 40
)
where κ is the curvature along axis z of polymer actuator. Thus z can be calculated by observing κ:

z(s, t) = EI z κ(s, t). (41)
Based on the image captured by the rapid camera, the shape of actuator can be extracted, from which we can compute the curvature along actuator and its derivative of time. An algorithm of image binarization, inter-frame difference [START_REF] Cheng | A motion image detection method based on the inter-frame difference method[END_REF] and skeleton extraction [START_REF] Zhao | Preprocessing and postprocessing for skeletonbased fingerprint minutiae extraction[END_REF] is applied via Matlab for obtaining the center skeleton of actuator. Then the curvature could be calculated by the following equation:

κ = y (1 + y 2 ) 3/2 , κ = ∂κ ∂t . (42)
However, it is inconvenient to compute the curvature and its derivative directly by the point data set of skeleton, due to the repeated calculation and discontinuity of data. To overcome this problem a polynomial curve fitting method is used here to get the curve function of actuator in real time which is of the following form:

y = w i=0 p i x i . ( 43
)
The analytical expression of the curvature and its derivative can be obtained directly by taking (43) into (42):

κ = w i=0 i(i -1)p i x i-2 (1 + ( w i=0 ip i x i-1 ) 2 ) 3/2 , κ = ∂κ ∂t . ( 44
)
Based on the methods above, we got the curvature and its derivative of all sections as shown in Fig. 5. From the figure we can observe that the overall derivative of curvature of the actuator shows a trend of increasing first and then becoming steady. This is because the charge inside actuator gradually reaches saturation from the near power end (section 1) to the far power end (section 10). From the physical analysis about the micro structure of conducting polymer actuator [START_REF] Farajollahi | Nonlinear two-dimensional transmission line models for electrochemically driven conducting polymer actuators[END_REF], the resistance of the inter-layer is little varied by the charge density. Thus it is reasonable to suppose that q 2 in our model is constant, which mostly corresponds to the ionic mass transport. On the contrary, there exists a variation of conductivity and volumetric capacitance of the electrode which can not be ignored during the procedure of charge [START_REF] Farajollahi | Nonlinear two-dimensional transmission line models for electrochemically driven conducting polymer actuators[END_REF]. Indeed the electrical conductivity of the conducting polymer is a nonlinear function of redox level and the volumetric capacitance changes with the oxidation state. Hence, in the identification process, we finally adopt the following 5-order polynomial to represent the nonlinearity of q 1 , and q 3 :

q 1 (s) = 5 i=0 a i s i , q 2 (s) = b 0 , q 3 (s) = 5 i=0 c i s i .
The identified values are given in Tab. II.

2) Comparison and validation: In order to show the deduced model is reliable, we simulate our model with the identified nonlinearity of q 1 , q 2 and q 3 , and compare the deformation result with experimental measurements. Fig. 6 shows the evolution of the curvature for the 1st section, the 5th section and the 10th section, indicating the fitting level between the experiment and the deduced model with the parameter fitting function of 4 order and 5 order. The error percentage of average curvature of the latter is less than 5% which is acceptable in our study. Fig. 7 shows the coincidence experiment with external load is also carried out. As shown in Fig. 8, a payload mass of 63mg is fixed on the tip of actuator. In this case, by changing the boundary condition of (15) to:

Λ i (L) -Λ a (L) + Ad -1 g L M p G = M p ηL -ad T η L M p η L , one can deduce the mechanical dynamics model of actuator in case of adding payload, where M p is the inertial matrix of payload and Ad g L is the transformation matrix (see appendix). The comparison between simulation and experiment is shown in Fig. 8 under the step input voltage of amplitude 1.5V and 0.1Hz.

C. Orientation control

Two control tests were implemented respectively for moving to fixed orientation and tracking a variable reference. 1) Moving to fixed orientation: It is expected to control polymer's end-effector orientation θ L to a given desired orientation θ L f in from its initial orientation θ Lini . A linear trajectory is planned to avoid input saturation (i.e., V in ≤ 3V ) and ensures that the actuator might arrive at desired orientation in a stable speed. Therefore, we have the following output reference with k = 2.5:

θ L ref = kt + θ Lini , t < θ L f in -θ L ini k θ L f in , t θ L f in -θ L ini k
where θ Lini = π/12, θ L f in = π/4 and k = 0.045.

2) Tracking variable reference: A sinusoidal reference is tested in the experiment, which is as follows:

θ L ref = θ Lini + θ a sin(ωt),
where the initial orientation θ Lini = π/6, amplitude θ a = π/25 and frequency ω = 0.02 rad/s.

3) Parameter setting: By following the proposed procedure in Section V, we first linearize the deduced nonlinear model in each zone, and then an Extended Kalman filter of form (34) has been constructed with Q = 10I and R = 0.01. After that, the proposed LQR controller of form (39) has been realized with Q = diag{1, ..., 1, 10} ∈ R 11×11 and R = 100.

4) Result analysis: The corresponding experimental results have been depicted in Fig. 9 and Fig. 10. Clearly, we can observe that the trajectory of polymer's end-effector can track the reference trajectory accurately with an error within 1%. The system can maintain stable when an external vibration disturbance was applied on it. Additionally, the input signal is within 3V , which is in the ideal input region.

VI. CONCLUSION

In this paper, a nonlinear modeling and control for conducting polymer actuator are investigated. The study focuses on electric and mechanic model, allowing to consider nonlinearity which occurs in the micro-actuator. Electro-mechanical model is proposed, based on which the parameter identification approach has been addressed where measured curvatures are well predicted by the deduced model. Then controller design using Linear-Quadratic Regulator is proposed, where an Extended Kalman filter has been introduced to estimate polymer's curvatures. With the proposed LQR controller, actuator bending angle is then controlled where measurements are in good agreement with the imposed bending angle. The results of this study is illustrated with simulation and experimental validations. Future work would focus on the design and control for multi-body (in series/parallel) micro-robot based on IEAP actuators, in order to validate the final application (such as robots for navigation in vessel and micro gripper).

Fig. 1 .

 1 Fig. 1. 2D-RC infinite grid for electronic model. Based on the above two different cases, this paper investigates different types of models: a nonlinear electronic dynamic model in Section II-A, a mechanic quasi-static model for case (i) and a mechanic dynamic model for case (ii) in Section II-C. The final coupling model is then deduced in Section II-D.

Fig. 2 .

 2 Fig. 2. The equivalent redox stress distribution on the cross section S of polymer based actuator.

Fig. 3 .

 3 Fig. 3. (a). Geometry of polymer based actuator and its linearization in workspace. αs is the tangent angle of point A; θ L is the included angle between vector of end point and axis y; θ L ref is the reference angle; the figure on right side shows the linearization of system in workspace. (b). Geometry at θ Lp i when the curvature of actuator is uniform.

Fig. 4 .

 4 Fig. 4. Experiment device. The actuator is fixed on the workbench and charged by micro-controller (MakeBlock MegaPi). A Dino-Lite micro camera on the right side is installed for observing the deformation of actuator.

Fig. 5 .

 5 Fig. 5. Curvature and its derivative measured under input u = 0.01t.

Fig. 6 .

 6 Fig. 6. The evolution of the curvature for each segment of the actuator. level of the deformation of actuator with time between the experiment and the simulation via quasi-static model and dynamic model, under a ramp input voltage u = 0.0075t and a step input voltage of amplitude 1.5V respectively. An

Fig. 7 .

 7 Fig. 7. (a). Deformation of the polymer actuator under the ramp input comparing with mechanical static model; (b). Deformation of the polymer actuator under the step input comparing with mechanical dynamic model.

Fig. 8 .

 8 Fig. 8. Displacement of tip of the polymer actuator with external load under the step input voltage of amplitude 1.5V and 0.1Hz comparing with mechanic dynamic model.

Fig. 9 .

 9 Fig. 9. Control performance of tracking ramp reference.

Fig. 10 .

 10 Fig. 10. Control performance of tracking sinusoidal reference.

  concatenate the following values at different time from t 1 to t np with n p > m where m is the dimension of z: Σ1Π(tn p ) ż(tn p ) Σ2Π(tn p ) ż(tn p ) Σ2Π(tn p )z(tn p )

			Σ1Π(t1) ż(t1)	Σ2Π(t1) ż(t1)	Σ2Π(t1)z(t1)	
	Ξ =	 	. . .	. . .	. . .	 

TABLE II IDENTIFIED

 II RESULT OF q 1 , q 2 , q 3 .814 0.111 -6.9e -4 1.4e -6 b 0.0275 c i 0.0110 -5.9e -5 -2.4e -7 6.1e -9 2.8e -11 4.2e-14

	i	0	1	2	3	4	5
	a i 526	249	-6

ACKNOWLEDGMENT

This work was supported by the French Government through the National Research Agency (ANR) : ROBOCOP (ANR-19-CE19), PIA EQUIPEX LEAF (ANR-11-EQPX-0025) projects, H2020 project TWINNIMS (Grant agreement 857263), and the French RENATECH network.

APPENDIX

A. Lie group notations

The adjoint representation of the Lie algebra is given by

where the operator (•) represents a conversion from a 3dimensional vector to its skew-symmetric matrix.

B. Transformation matrix

The matrix transforming the velocity or acceleration twist from body frame to inertial frame is given by


Shutao Wang

Kostiantyn Kravchyk

Stéphanie Pigeot-Remy

Weiqiang Tang

Frank Krumeich

Michael Wörle

Maryna I Bodnarchuk

Sophie Cassaignon

Olivier Durupthy

Shuangliang Zhao

# Kostiantyn

V Kravchyk

Stephanie Pigeot-Rémy

Clément Sanchez

Maksym V Kovalenko
email: mvkovalenko@ethz.ch

Anatase TiO 2 Nanorods as Cathode Materials for Aluminum-Ion Batteries

Keywords: Al-ion battery, aluminum, titanium (IV) oxide, energy storage, nanorod, synthesis

INTRODUCTION

Al-ion batteries (AIBs) are appealing as an inexpensive electrochemical energy storage technology and have attracted significant attention in the last few years. [1][2] Aluminum has numerous advantages such as high natural abundance, high volumetric/gravimetric (8046 mAh cm -3 /2980 mAh g -1 ) capacities and low redox potential. [3][4][5][6][7][8] Contrary to metallic Li, Na, and K, Al could be utilized in the form of metallic foil on the anode part of the battery due to smooth and fast electrodeposition/stripping. Thus far, the main barrier for the progress of AIBs has been the scarcity of cathodes supporting Al 3+ ion intercalation/deintercalation. In this arena, an important difference exists in comparison to univalent Li + /Na + cations.

Intercalation of Al 3+ cations, being much smaller than Li + ions, was revealed to be difficult as a result of strong Coulombic interaction with the lattice of the host cathode material. 9 In the last few years, various materials have been explored with respect to Al 3+ ion storage, such as transition metal oxides, [10][11][12][13][START_REF] Wei | Molybdenum Oxide as Cathode for High Voltage Rechargeable Aluminum Ion Battery[END_REF][START_REF] Zhang | Porous CuO Microsphere Architectures as High-Performance Cathode Materials for Aluminum-Ion Batteries[END_REF][START_REF] Liu | Aluminum Storage Behavior of Anatase TiO 2 Nanotube Arrays in Aqueous Solution for Aluminum Ion Batteries[END_REF][START_REF] Liu | The Electrochemical Behavior of Cl -Assisted Al 3+ Insertion into Titanium Dioxide Nanotube Arrays in Aqueous Solution for Aluminum Ion Batteries[END_REF][START_REF] Zhong | Electrochemically Conductive Treatment of TiO 2 Nanotube Arrays in AlCl 3 Aqueous Solution for Supercapacitors[END_REF][START_REF] Koketsu | Reversible Magnesium and Aluminium Ions Insertion in Cation-Deficient Anatase TiO 2[END_REF][START_REF] Tang | First-principles Investigation of Aluminum Intercalation and Diffusion in TiO 2 Materials: Anatase versus Rutile[END_REF][START_REF] Sang | The electrochemical behavior of TiO 2 -NTAs electrode in H + and Al 3+ coexistent aqueous solution[END_REF][START_REF] Lahan | Anatase TiO 2 as an Anode Material for Rechargeable Aqueous Aluminum-Ion Batteries: Remarkable Graphene Induced Aluminum Ion Storage Phenomenon[END_REF] sulfides, [START_REF] Geng | Reversible Electrochemical Intercalation of Aluminum in Mo 6 S 8[END_REF][START_REF] Hu | A Binder-Free and Free-Standing Cobalt Sulfide@Carbon Nanotube Cathode Material for Aluminum-Ion Batteries[END_REF][START_REF] Li | A Highly Reversible Co 3 S 4 Microsphere Cathode Material for Aluminum-Ion Batteries[END_REF][START_REF] Wu | A Rechargeable Aluminum-Ion Battery Based on a VS 2 Nanosheet Cathode[END_REF][START_REF] Zhang | Flowerlike Vanadium Suflide/Reduced Graphene Oxide Composite: An Energy Storage Material for Aluminum-Ion Batteries[END_REF][START_REF] Hu | An Innovative Freeze-Dried Reduced Graphene Oxide Supported SnS 2 Cathode Active Material for Aluminum-Ion Batteries[END_REF][START_REF] Liang | Self-Supported Tin Sulfide Porous Films for Flexible Aluminum-Ion Batteries[END_REF][START_REF] Li | Rechargeable Aluminum-Ion Battery Based on MoS 2 Microsphere Cathode[END_REF][START_REF] Takami | Studies on the Electrochemical Behaviour of the FeS 2 Electrode in Molten AlCl 3 •NaCl by the AC Impedance Method[END_REF][START_REF] Wang | A Novel Aluminum-Ion Battery: Al/AlCl 3 -EMImCl/Ni 3 S 2 @Graphene[END_REF][START_REF] Yu | Hexagonal NiS Nanobelts as Advanced Cathode Materials for Rechargeable Al-Ion Batteries[END_REF][START_REF] Geng | Titanium Sulfides as Intercalation-Type Cathode Materials for Rechargeable Aluminum Batteries[END_REF][START_REF] Kulish | Aluminium and Magnesium Insertion in Sulfur-Based Spinels: a First-Principles Study[END_REF][START_REF] Wang | High-Performance Aluminum-Ion Battery with CuS@C Microsphere Composite Cathode[END_REF] selenides, [START_REF] Cai | Stable CoSe 2 /Carbon Nanodice@reduced Graphene Oxide Composites for High-Performance Rechargeable Aluminum-Ion Batteries[END_REF][START_REF] Jiang | One-Dimensional Cu 2-x Se Nanorods as the Cathode Material for High-Performance Aluminum-Ion Battery[END_REF] phosphite, [START_REF] Tu | Facile Synthesis of Ni 11 (HPO 3 ) 8 (OH) 6 /rGO Nanorods with Enhanced Electrochemical Performance for Aluminum-Ion Batteries[END_REF] molybdite, [START_REF] Nacimiento | Exploring an Aluminum Ion Battery Based on Molybdite as Working Electrode and Ionic Liquid as Electrolyte[END_REF] vanadate, [START_REF] Jiang | Investigation of the Reversible Intercalation/Deintercalation of Al into the Novel Li 3 VO 4 @C Microsphere Composite Cathode Material for Aluminum-Ion Batteries[END_REF] carbide, [START_REF] Vahidmohammadi | Two-Dimensional Vanadium Carbide (MXene) as a High-Capacity Cathode Material for Rechargeable Aluminum Batteries[END_REF] Prussian blue analogs, [START_REF] Reed | A Rechargeable Aluminum-Ion Battery Utilizing a Copper Hexacyanoferrate Cathode in an Organic Electrolyte[END_REF][START_REF] Liu | Copper Hexacyanoferrate Nanoparticles as Cathode Material for Aqueous Al-Ion Batteries[END_REF] iodine, [START_REF] Zhang | Naturally Abundant High-Performance Rechargeable Aluminum/Iodine Batteries Based on Conversion Reaction Chemistry[END_REF] selenium, [START_REF] Huang | Rechargeable Aluminum-Selenium Batteries With High Capacity[END_REF] sulfur [START_REF] Yang | An Aluminum-Sulfur Battery with a Fast Kinetic Response[END_REF][START_REF] Bian | Using an AlCl 3 /Urea Ionic Liquid Analog Electrolyte for Improving the Lifetime of Aluminum-Sulfur Batteries[END_REF][START_REF] Cohn | A Novel Non-Aqueous Aluminum Sulfur Battery[END_REF] and oxygen. [START_REF] Wang | A High-Capacity Dual-Electrolyte Aluminum/air Electrochemical Cell[END_REF] Overall, the reversibility of the Al 3+ ion intercalation/deintercalation redox processes has been demonstrated, although nearly all known cathodes possess poor rate capabilities and limited cycle life. Of all possible cathode materials for AIBs, titanium oxide (TiO 2 ) is probably the most appealing candidate due to high natural reserves of the constituent elements, its nontoxicity, and high oxidation stability. Reversible intercalation of Al 3+ ions into TiO 2 was first demonstrated in 2012 by Liu et al. [START_REF] Liu | Aluminum Storage Behavior of Anatase TiO 2 Nanotube Arrays in Aqueous Solution for Aluminum Ion Batteries[END_REF] on anatase TiO 2 nanotube arrays in an aqueous AlCl 3 electrolyte. The reduction of Ti 4+ to Ti 3+ /Ti 2+ during Al 3+ ion intercalation has been confirmed by XPS studies. Later, reversible Al 3+ ion storage was shown on anatase TiO 2 nanospheres. [START_REF] Kazazi | High Surface Area TiO 2 Nanospheres as a High-Rate Anode Material for Aqueous Aluminium-Ion Batteries[END_REF] Despite these achievements, Al 3+ ion intercalation in TiO 2 has been reported up to recently only in aqueous electrolytes, which are, however, susceptible to parasitic reactions such as chemical oxidation of the aluminum anode and hydrogen evolution during aluminum deposition. In the context of minimization of these side reactions, Koketsu et al. [START_REF] Koketsu | Reversible Magnesium and Aluminium Ions Insertion in Cation-Deficient Anatase TiO 2[END_REF] reported the electrochemical performance of cation-deficient anatase TiO 2 in a chloroaluminate ionic liquid electrolyte which demonstrated a high capacity of 120 mAh g -1 in comparison with pure bulk TiO 2 (30 mAh g -1 ). However, the electrochemical performance of cation-deficient anatase TiO 2 was only reported for three cycles.

Encouraged by both the initial reports on TiO 2 in an aqueous electrolyte [START_REF] Liu | Aluminum Storage Behavior of Anatase TiO 2 Nanotube Arrays in Aqueous Solution for Aluminum Ion Batteries[END_REF][START_REF] Kazazi | High Surface Area TiO 2 Nanospheres as a High-Rate Anode Material for Aqueous Aluminium-Ion Batteries[END_REF][START_REF] He | Black Mesoporous Anatase TiO 2 Nanoleaves: a High Capacity and High Rate Anode for Aqueous Al-Ion Batteries[END_REF] and the recent publications of Koketsu et al., [START_REF] Koketsu | Reversible Magnesium and Aluminium Ions Insertion in Cation-Deficient Anatase TiO 2[END_REF] and others [START_REF] Das | Electrochemical Study on the Rechargeability of TiO 2 as Electrode Material for Al-ion Batteries with Chloroaluminate Ionic Liquid Electrolyte[END_REF] we sought to thoroughly evaluate the Al 3+ ion storage performance of nanosized anatase TiO 2 in chloroaluminate ionic liquid. Herein, we demonstrate that anatase TiO 2 nanorods (NRs) deliver a high capacity of 112 mAh g -1 and 165 mAh g -1 at a current density of 50 mA g -1 in mild and highly acidic formulations (AlCl 3 /1-Ethyl-3-methylimidazolium chloride (EMIMCl) with molar ratios of 1.1:1 and 2.0:1), respectively. The mechanism of Al 3+ ion intercalation into TiO 2 NRs in both formulations has been assessed using ex situ X-ray photoelectron (XPS) and energydispersive X-ray (EDX) spectroscopies. Density functional theory (DFT) calculations were performed to predict the volume change of the TiO 2 structure during Al 3+ ion intercalation.

RESULTS AND DISCUSSION

In this work, monodispersed TiO 2 NRs were synthesized using a previously reported method; [START_REF] Dufour | Morphological Control of TiO 2 Anatase Nanoparticles: What Is the Good Surface Property to Obtain Efficient Photocatalysts?[END_REF][START_REF] Sugimoto | Synthesis of Uniform Anatase TiO 2 Nanoparticles by Gel-Sol Method: 3. Formation Process and Size Control[END_REF] briefly, the synthesis of NRs was performed by hydrothermal treatment of titanium isopropoxide in the presence of triethanolamine and ethylenediamine (see Figure S1 and Experimental Section for details). As follows from transmission electron microscopy (TEM) measurements, TiO 2 NRs were crystalline with length and width on the average order of 50-200 nm and 7-14 nm, respectively (Figure 1a). As illustrated in Figure 1b, the powder X-ray diffraction (XRD) and selected area electron diffraction patterns of TiO 2 NRs show a tetragonal anatase structure with the space group I4 1 /amd. The lattice parameters, as obtained from a Rietveld refinement, are: a = b = 3.80002(10) Å, c = 9.5010(2) Å. Notably, in 5 On anode:

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𝑥𝐴𝑙 + 7𝑥𝐴𝑙𝐶𝑙 - 4 ↔4𝑥𝐴𝑙 2 𝐶𝑙 - 7 + 3𝑥𝑒 - On cathode: (2) 4𝑥𝐴𝑙 2 𝐶𝑙 - 7 +𝑇𝑖𝑂 2 +3𝑥𝑒 -↔𝐴𝑙 𝑥 𝑇𝑖𝑂 2 +7𝑥𝐴𝑙𝐶𝑙 - 4
As follows from Figures 1c and1d, TiO 2 NRs showed superior electrochemical performance over bulk TiO 2 . Bulk TiO 2 shows only a small capacity of 35 mAh g -1 , which is also in agreement with results of Koketsu et al. [START_REF] Koketsu | Reversible Magnesium and Aluminium Ions Insertion in Cation-Deficient Anatase TiO 2[END_REF] Apparently, decreasing the crystal grain size of the TiO 2 shortens the ionic diffusion length and increases the active material/electrolyte interactions. Consequently, TiO 2 NRs possess significantly higher electrochemical activity as compared to micron-sized TiO 2 particles (Figure S4).

accordance with Rietveld refinement, about 6.3% of brookite phase was present in TiO 2 NRs (see Figure S2, Table S1 and Experimental Section for details). As follows from highresolution transmission electron microscopy (HRTEM) imaging (see inset of Figure 1b), the growth direction of TiO 2 NRs is [001] and the top facet of TiO 2 NRs is a plane (010).

For the electrochemical tests, TiO 2 NRs were mixed with polyvinylidene fluoride (PVDF) binder, carbon black (CB), titanium nitride (TiN) and N-methylpyrrolidone (NMP). The obtained slurry was cast onto tungsten plates. A homemade plastic cell design was used for the electrochemical measurements (Figure S3). Aluminum ionic liquid electrolyte consisting of AlCl 3 :EMIMCl with molar ratios of 1.1 or 2.0 was employed as an electrolyte.

Figure 1c shows the voltage profiles of the AIB at a current density of 50 mA g -1 using TiO 2 NRs as a cathode material. During the first discharge, the intercalation of approximately 0.12 mol of Al 3+ ions per mole of TiO 2 was observed, which corresponds to a total initial discharge capacity of 121 mAh g -1 . During the first charge cycle, all inserted Al 3+ ions could be extracted from the TiO 2 structure. The discharging half-reactions can be written as follows: 2p, and Ti 2p XPS peaks for pristine, discharged and charged samples. For discharged electrode, at both 2.0 and 1.1 molar ratios, the Ti 2p peak displaced to higher binding energy, as a result of Ti 4+ reduction to Ti 3+ . Upon the following charge, Ti 3+ is oxidized back to Ti 4+ . 6

As evident from the Al 2p XPS spectra for both the 2.0 and 1.1 ratios, the Al peak appears after discharge and fully vanishes after the following charge. However, in the case of the 2.0 ratio, the Cl 2p signal was also observed for the discharged electrode, which disappeared after the following charge. Appearance of Cl was also observed on the EDX maps for the 2.0 ratio, in contrast to the 1.1 ratio (see Figures S10 for details). The quantitative XPS analysis of the discharged TiO 2 electrode using AlCl 3 /EMIMCl ionic liquid with 2.0 molar ratio revealed the following atomic ratios for Al and Cl: 2.5(±0.1) and 2(±0.1).

We assume that the appearance of the Cl in the discharged TiO 2 electrode can be attributed to the presence of AlCl 2 + species in the highly acidic AlCl 3 :EMIMCl ionic liquid electrolyte, and this has been confirmed in a series of previous reports [START_REF] Tsuda | Review-Electrochemical Surface Finishing and Energy Storage Technology with Room-Temperature Haloaluminate Ionic Liquids and Mixtures[END_REF][START_REF] Abood | Do All Ionic Liquids Need Organic Cations? Characterisation of [AlCl 2 •nAmide] + AlCl 4and Comparison with Imidazolium Based Systems[END_REF][START_REF] Angell | High Coulombic Efficiency Aluminum-Ion Battery Using an AlCl 3 -Urea Ionic Liquid Analog Electrolyte[END_REF][START_REF] Noth | An Investigation of AlCl 3 Solutions in Ethers by 27 Al NMR Spectroscopy[END_REF][START_REF] Kim | Rechargeable aluminium organic batteries[END_REF] as well as in the present work by solution state [START_REF] Zhang | Flowerlike Vanadium Suflide/Reduced Graphene Oxide Composite: An Energy Storage Material for Aluminum-Ion Batteries[END_REF] Al NMR measurements (Figure S11). It is possible that the AlCl 2 + species is too large for intercalation (r(AlCl 2 + ) = 3.76 Å vs. r(Al 3+ ) = 0.53 Å) and can be attracted by Coulombic forces to the reduced Ti 3+ ion sites on the surface of the TiO 2 NRs, contributing to the aluminum charge storage mechanism. These assumptions might explain why the capacity of TiO 2 in the highly acidic electrolyte is higher than that of the mildly acidic electrolyte. To better understand the structural changes during Al 3+ ion intercalation into the anatase TiO 2 structure, DFT calculations were performed. [START_REF] Tang | First-principles Investigation of Aluminum Intercalation and Diffusion in TiO 2 Materials: Anatase versus Rutile[END_REF] Specifically, we gradually increased the number of Al 3+ ions in the intercalation sites of an anatase supercell and fully relaxed the supercell volume and the atomic positions of the host. We note that at low Al 3+ ion content (the molar fraction of aluminum x < 0.167, and the capacity of 163 mAh g -1 ), the repulsive force between the Al-Ti is the major factor affecting the stability of the Al 3+ ion intercalation site. As follows from the DFT study during Al 3+ ion intercalation into TiO 2 , the anatase TiO 2 mechanisms that are discussed in detail above. Lower coulombic efficiency being observed in chloroaluminate ionic liquid electrolyte with 2.0 : 1 molar ratio (AlCl 3 :EMIMCl) as compared to that of slightly acidic liquid (1.1 : 1 molar ratio) can be explained by the enhanced corrosivity of the highly acidic melt with respect to the W current collector. Having established that the higher acidity of the electrolyte solution offers higher capacity, we studied the electrochemical performance of TiO 2 NRs in greater detail using a higher acidity electrolyte (Figure 4c,d). Namely, we revealed that the constant current-constant voltage (CCCV) charge at a constant voltage step of 1.2 V, improves the discharge capacity of TiO 2

NRs to 91 mAh g -1 . Notably, irreversibility between charge and discharge curves within first cycles (Figure 4d) might be associated with partial trapping of Al 3+ ions in TiO 2 lattice. The cycling stability tests of half-cells employing TiO 2 NRs with a CCCV protocol at a current density of 0.5 A g -1 showed high capacity retention for 150 cycles with a Coulombic efficiency of ~99.9% (Figure 4b). TEM measurements of TiO 2 NRs after 150 cycles revealed that their structure and morphology was fully preserved (Figure S13). Importantly, TiO 2 NRs possess one of the most stable Al-ion cathodic performances with high capacity published to date for AIBs (for details see Table S2).

CONCLUSIONS

In summary, we report the assessment of TiO 2 NRs as a cathode material for Al-ion storage. TiO 2 NRs exhibit high capacities of 112 mAh g -1 and 165 mAh g -1 at a current density of 50 mA g -1 in mild and highly acidic (AlCl 3 /EMIMCl molar ratios of 1.1:1 and 2.0:1) chloroaluminate ionic liquid electrolytes, respectively. Moreover, TiO 2 NRs can be cycled at high current densities of 0.5 A g -1 (2.0:1 molar ratio), and they deliver a high capacity of approximately 90 mA g -1 without capacity fading over 150 cycles. The mechanism of Al 3+ ion intercalation/deintercalation was probed by ex situ XPS and EDX measurements. The result evidence the electrochemical reduction/oxidation of Ti 4+ /Ti 3+ ions and insertion of Al 3+ ions for both the 1.1:1 and 2.0:1 ratios during the discharging/charging of TiO 2 NRs.

12 Structural changes during Al 3+ ion intercalation into an anatase TiO 2 structure were assessed by using DFT calculations which indicated the structural stability of the Al x TiO 2 structure at a low Al 3+ ion content of x < 0.167.

EXPERIMENTAL SECTION

Synthesis of TiO 2 NRs. Titanium oxide NRs were synthesized using a previously reported method. A volume of 6.9 mL of titanium isopropoxide (TIPO) was added to 6.7 mL of triethanolamine under stirring. Then, 16.4 mL of deionized water was added into the reaction mixture following the addition of 1.4 mL of ethylenediamine (in 15 min) under vigorous stirring. Afterward, the solution was brought to a total volume of 100 mL with water. The closed flask was heated to 100 °C for 24 h. The synthesized gel was transferred into autoclave for a second heat-treatment at 140 °C for 72 h. Lastly, as-synthesized TiO 2 NRs were collected from the autoclave and washed with water and HNO 3 (3 mol L -1 ), following their centrifugation and drying under a dry air flow for 12h. %) using a Fritsch Pulverisette 7 classic planetary mill for 1 h at 500 rpm. The obtained homogeneous suspension was painted onto W current collectors following the drying step at 80 °C for 12 hours under vacuum (loading ~1 measurements of TiO 2 brookite NPs (Figure S18). It has been revealed that bookite TiO 2 structure has lower electrochemical performance in comparison with anatase structure. TiO 2 mg cm -2 ). It is worth noting that, the contribution from empty W current collector to the capacity of TiO 2 electrodes prepared on W plates is negligible (see Figure S15).

Electrochemical

Electrochemical measurements were conducted in homemade plastic cells (Figure S3), in an Ar-filled glove box (O 2 < 0.1 ppm, H 2 O < 0.1 ppm) using glass fiber filter as a separator, and AlCl 3 :EMIMCl ionic liquid as an electrolyte. Aluminum foil was used as both reference and counter electrodes. AlCl 3 :EMIMCl ionic liquid (1.1:1 and 2:1 molar ratios) was used as the electrolyte. We note that highest molar ratio of AlCl 3 :EMIMCl is ≈ 2 because above this ratio AlCl 3 can not be dissolved anymore in the choloroaluminate ionic liquid electrolyte. 8 Notably, in case of the use of chloroaluminate ionic liquid with molar ratio of 2:1, small corrosion of Al anode was observed as visible from SEM images of cycled Al foil (Figure S16). Electrochemical measurements were performed on an MPG2 multi-channel workstation (BioLogic) using a voltage range of 0.2-2.2 V vs. Al 3+ /Al. For constant current-constant voltage (CCCV) measurements, constant voltage step was applied during charge at 1.2 V vs.

Al 3+ /Al. CCCV protocol at 1.2 V had been chosen mainly because it allowed to boost the discharge capacity of TiO 2 while having high coulombic efficiency during cycling. CV step at voltages of > 1.2 V significantly decreased the Coulombic efficiency. On the contrary, CV step at voltages of < 1.2 V had a limited contribution to the capacity. The charge storage capacity was normalized by the mass of TiO 2 NRs. Importantly, although TiO 2 electrodes were composed of TiN, pVdF and CB additives, the latter have a negligible contribution to the charge storage capacity of TiO 2 as indicated in Figure S17. TiN was used, as an additive, which helps to improve the electronic conductivity of TiO 2 electrodes. [START_REF] Wang | Aluminum Chloride-Graphite Batteries with Flexible Current Collectors Prepared from Earth-Abundant Elements[END_REF] In order to understand whether brookite impurity has any impact on the electrochemical performance of anatase TiO 2 NRs, we have performed separately electrochemical brookite NPs have been synthesized following the procedure reported by Pottier et al. [START_REF] Pottier | Synthesis of Brookite TiO 2 Nanoparticles by Thermolysis of TiCl 4 in Strongly Acidic Aqueous Media[END_REF]

Rietveld Refinement of the Pristine TiO 2 (Anatase).

A Rietveld refinement with GSAS II [START_REF] Toby | GSAS-II: the Genesis of a Modern Open-source All Purpose Crystallography Software Package[END_REF] was performed using the structural parameters of TiO 2 in the anatase structure. [START_REF] Ballirano | Rietveld Refinements on Laboratory Energy Dispersive X-ray Diffraction (EDXD) data[END_REF] A very small peak 30.72° (broad) is assigned to TiO 2 in the brookite structure (about 6.3 wt.%, crystallite size of 20 nm was assumed). The values for the lattice constants of TiO 2 (anatase) are a = 3.80002 (10) Å, c = 9.5010(2) Å, the cell volume is V = 137.20(1) Å 3 . The values for the crystallite size were described with an "uniaxial" model, (anisotropic axis [0, 0, 1]) and refined to 0.012 and 0.029 µm, respectively.The figures of merit are Rp = 0.024, Rwp = 0.034, R(F)anatase = 0.028, R(F2)anatase = 0.045, χ2 = 8.679, GOOF = 2.946.

Computational Details. Calculations were carried out using periodic density functional theory (DFT) within the Vienna Ab initio Simulation Package (VASP). [START_REF] Kresse | Ab Initio Molecular Dynamics for Liquid Metals[END_REF][START_REF] Kresse | Efficient Iterative Schemes for ab Initio Total-energy Calculations Using a Plane-Wave Basis Set[END_REF] The projected augmented wave (PAW) method was used to describe the inner cores and electron-ion interactions. [START_REF] Blöchl | Projector Augmented-wave Method[END_REF][START_REF] Kresse | From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method[END_REF] The exchange-correlation effects were treated by the Perdew-Burke-Ernzehof (PBE) functional with the generalized gradient approximation (GGA). [START_REF] Zhang | Generalized Gradient Approximation Made Simple[END_REF] A supercell containing 108 atoms with 3×3×1 unit cells was modeled for TiO 2 anatase. [START_REF] Tang | First-principles Investigation of Aluminum Intercalation and Diffusion in TiO 2 Materials: Anatase versus Rutile[END_REF][START_REF] Dawson | Improved Calculation of Li and Na Intercalation Properties in Anatase, Rutile, and TiO 2 (B)[END_REF] To maximize computational efficiency without affecting the accuracy of the calculation, a plane wave cut-

Figure 1 .

 1 Figure 1. (a) Transmission electron microscopy (TEM) image and (b) X-ray diffraction (XRD) pattern of TiO 2 NRs [inset: high-resolution TEM image and selected area electron diffraction (SAED) image of TiO 2 NRs]. First three galvanostatic charge-discharge curves of TiO 2 NRs (c) and microcrystalline bulk TiO 2 (d) using AlCl 3 :EMIMCl electrolytes (1.1:1 molar ratio).Next, we compared the electrochemical behavior of TiO 2 NRs in mild and highly acidic

Figure 2 .

 2 Figure 2. (a) Typical galvanostatic charge-discharge curves (3 rd cycle) of TiO 2 NRs measured at a current density of 50 mA g -1 in mild (1.1:1 molar ratio) and acidic (2.0:0 molar ratio) AlCl 3 /EMIMCl ionic liquid electrolytes. See Figure S6 for the electrochemical performance of TiO 2 NRs in chloroaluminate ionic liquid with 1.1, 1.3, 1.5, and 2.0 AlCl 3 /EMIMCl molar ratios measured at a current density of 0.5 A g -1 . (b) Ex situ XPS measurements of electrodes composed of TiO 2 NRs before and after discharge and charge in AlCl 3 /EMIMCl ionic liquid electrolytes (1.1:1 and 2.0:0 molar ratios). See Figures S7 and S8 for the detailed O 1s and Ti 2p XPS spectra. Full XPS survey spectra are shown in Figure S9.

Figure 3 .

 3 Figure 3. (a) Representation of the crystal structures of TiO 2 with different A 3+ ion content ( x = 0; x = 0.028; x = 0.167). Gray, blue, and red spheres represent O, Ti, and Al atoms, respectively. (b) The dependence of computed cell volume change of anatase TiO 2 structure on the Al 3+ ion content ( x ).We then measured the electrochemical performance of TiO 2 NRs at a high current density of 0.5 A g -1 . Figure4ashows the typical voltage profiles of the TiO 2 NRs. After the 20th

Figure 4 .

 4 Figure 4. (a) Galvanostatic charge-discharge curves of TiO 2 NRs during the 20 th cycle and (b) their cyclic stability measured at a current density of 0.5 A g -1 using AlCl 3 :EMIMCl electrolytes (1.1:1 and 2.0:1 molar ratios). (c) Charge and discharge voltage profiles and (d) cyclic performance of TiO 2 NRs tested at current density of 0.5 A g -1 using CCCV mode ( 2:1 (AlCl 3 /EMIMCl molar ratio). See also Figure S14 for cyclic stability measurements of TiO 2 NRs at a current density of 50 mA g -1 .

  chemicals were used: carbon black (Super C65, TIMCAL), poly(vinylidene fluoride) (PVdF, Sigma-Aldrich), titanium nitride (TiN, 99.8%), Nmethyl-2-pyrrolidone (NMP, 99%, Sigma-Aldrich), anhydrous AlCl 3 (99.99%, ABCR), 1-ethyl-3-methylimidazolium chloride (EMIMCl, 99%, Iolitec), W plate (Bocheng Molybdenum Co., Ltd), titanium (IV) oxide (anatase, 99.9%, Aldrich), Al foil (MTI Corporation), and glass microfiber separator (GF/D, Whatman). TiO 2 electrodes were prepared by mixing of TiO 2 NRs (45 wt. %), carbon black (CB, 22.5 wt. %), titanium nitride (TiN, 22.5 wt, %), and poly(vinylidene) fluoride binder (pVdF, 10 wt.

(seeFigure

  Figure S19for XRD and TEM images).

  14 off energy of 500 eV and the Monkhorst Pack grid of 5×5×2 were used in all calculations. The tetrahedron method with Blöchl corrections with a smearing width of σ = 0.05 eV was used to minimize the errors in the Hellmann-Feynman forces. Full relaxation of the supercell volume and atom positions was performed until the forces converged to less than 0.1 meV Å -1 . All structures were optimized with a convergence criterion of 0.01 meV for the energy calculation. Polypyrenes as High-Performance Cathode Materials for Aluminum Batteries. Adv. Mater. 2018, 30 (15), 1705644. (7) Kravchyk, K. V.; Wang, S.; Piveteau, L.; Kovalenko, M. V. Efficient Aluminum Chloride Natural Graphite Battery. Chem. Mater. 2017, 29 (10), 4484-4492. (8) Kravchyk, K. V.; Kovalenko, M. V. Rechargeable Dual-Ion Batteries with Graphite as a Cathode: Key Challenges and Opportunities. Adv. Energy Mater. 2019, 1901749. (9) Levi, E.; Levi, M. D.; Chasid, O.; Aurbach, D. A Review on the Problems of the Solid State Ions Diffusion in Cathodes for Rechargeable Mg Batteries. J. Electroceram. 2009, 22 (1), 13-19. (10) Gu, S.; Wang, H.; Wu, C.; Bai, Y.; Li, H.; Wu, F. Confirming Reversible Al 3+ Storage Mechanism Through Intercalation of Al 3+ Into V 2 O 5 Nanowires in a Rechargeable Aluminum Battery. Energy Storage Mater. 2017, 6, 9-17. (11) Chiku, M.; Takeda, H.; Matsumura, S.; Higuchi, E.; Inoue, H. Amorphous Vanadium Oxide/Carbon Composite Positive Electrode for Rechargeable Aluminum Battery. ACS Appl. Mater. Interfaces 2015, 7 (44), 24385-24389. (12) Wang, H.; Bai, Y.; Chen, S.; Luo, X.; Wu, C.; Wu, F.; Lu, J.; Amine, K. Binder-Free V 2 O 5 Cathode for Greener Rechargeable Aluminum Battery. ACS Appl. Mater. Interfaces 2015, 7 (1), 80-84. ANR within the Photonorm project (ANR-11-ECOT-0008). The authors are grateful to the research facilities of ETH Zürich (ETH Electron Microscopy Center, Department of Chemistry and Applied Biosciences) and Empa (Empa Electron Microscopy Center and Laboratory for Mechanics of Materials & Nanostructures) for access to the instruments and for technical assistance.

	ASSOCIATED CONTENT
	REFERENCES
	Supporting Information. Supporting Figures S1-S19 and Tables S1-S2. This material is
	available free of charge via the Internet at http://pubs.acs.org.

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ACKNOWLEDGMENTS

This research is part of the activities of SCCER HaE, which is financially supported by Innosuisse -Swiss Innovation Agency This work was also financially supported by the National Natural Science Foundation of China (Nos. 91834301 and 21878078). SPR, SC and OD greatly acknowledge the financial support of the synthesis work by the French

AUTHOR INFORMATION

Corresponding Authors:

*E-mail: mvkovalenko@ethz.ch

Author Contributions

The manuscript was written through the contributions of all authors. All authors have given approval to the final version of the manuscript.