Intent-aware control in kinematically redundant systems: Towards collaborative wearable robots Mahdi Khoramshahi, Guillaume Morel, and Nathanael Jarrassé

Abstract-Many human-robot collaboration scenarios can be seen as a redundant leader-follower setup where the human (i.e., the leader) can potentially perform the task without the assistance of the robot (i.e., the follower). Thus, the goal of the collaboration, beside stable execution of the task, is to reduce the human cost; e.g., ergonomic, or cognitive cost. Such system redundancies (where the same task be achieved in different manner) can also be exploited as a communication channel for the human to convey his/her intention to the robot; since it is essential for the overall performance (both execution and assistance) that the follower recognizes the intended task in an online fashion. Having an estimation for the intended task, the robot can assist the human by reducing the human cost over the task null-space; i.e., the null-space which arises from the overall system redundancies with respect to the intended task. With the prospective of supernumerary and prosthetic robots, in this work, we primarily focus on serial manipulation in which the proximal/distal part of the kinematic chain is controlled by the leader/follower respectively. By exploiting kinematic redundancies for intention-recognition and costminimization, our proposed control strategy (for the follower) ensures assistance under stable execution of the task. Our results (simulations and preliminary experimentation) show the efficacy of our method in providing a seamless robotic assistance (i.e., improving human posture) toward human intended tasks (i.e., reaching motions) for wearable robotics.

I. INTRODUCTION

Collaborative robotics has become the cynosure of research on pHRI (physical human-robot interaction) mainly due to its numerous applications: from industrial robotics (where mobile robots, manipulators, and humanoids aim to assist humans with cumbersome tasks [START_REF] Busch | Evaluation of an industrial robotic assistant in an ecological environment[END_REF][START_REF] Wansoo | Towards ergonomic control of collaborative effort in multi-human mobile-robot teams[END_REF][START_REF] Agravante | Human-humanoid collaborative carrying[END_REF]) to wearable robotics which aims to restore or augment human motor capacities (e.g., exoskeletons [START_REF] Goehlich | Magic triangle-human, exoskeleton, and collaborative robot scenario[END_REF], prosthetic and supernumerary arms [START_REF] Leigh | Robotic symbionts: Interweaving human and machine actions[END_REF][START_REF] Parietti | Supernumerary robotic limbs for human body support[END_REF]). The ubiquitous properties of such applications are (1) the presence of heterogeneous agents, (2) decentralized control, and (3) redundancy: On one side of the interaction, we have an agent with higher cognitive capacities (often a human) who has the intention of performing a specific task, and on the other side, an agent with higher actuation capacity (often a robot) who aims to provide assistance. Decentralization occurs since the intention of the leader is latent to the follower and the communication channel is often limited to kinesthetics, myoelectric signals, etc. Furthermore, redundancy often arises from the fact that the human can potentially perform the task alone; robots are employed to reduce human metabolic cost or improve human ergonomics, manipulability, etc. As an example, consider the case of a human-robot serial manipulation (illustrated in Fig. 1) where the human can perform his/her desired reaching motion even if the robot provides poor collaboration.

The major challenge for collaborative robots is to provide assistance toward an unknown task; i.e. the human-intended task. Methods relying on input devices to communicate the intentions (e.g., control panel, joystick, etc.) proved to be limited in their performance due to lack of realtime communication, necessity for extra hardware, or increase in human cognitive load [START_REF] Steinfeld | Common metrics for human-robot interaction[END_REF][START_REF] Villani | Survey on human-robot collaboration in industrial settings: Safety, intuitive interfaces and applications[END_REF][START_REF] Losey | A review of intent detection, arbitration, and communication aspects of shared control for physical human-robot interaction[END_REF]. Therefore, recent approaches aim at real-time integration of human intention (as discussed in [START_REF] Ajoudani | Progress and prospects of the human-robot collaboration[END_REF]) based on motion [START_REF] Aarno | Motion intention recognition in robot assisted applications[END_REF][START_REF] Huang | Control of upper-limb power-assist exoskeleton using a human-robot interface based on motion intention recognition[END_REF], forces [START_REF] Kucukyilmaz | Intention recognition for dynamic role exchange in haptic collaboration[END_REF][START_REF] Olivares-Alarcos | On inferring intentions in shared tasks for industrial collaborative robots[END_REF], impedance [START_REF] Li | Force, impedance, and trajectory learning for contact tooling and haptic identification[END_REF][START_REF] Liu | Intention recognition in physical human-robot interaction based on radial basis function neural network[END_REF], gesture [START_REF] Chen | Wristcam: A wearable sensor for hand trajectory gesture recognition and intelligent human-robot interaction[END_REF][START_REF] Zhu | Wearable sensor-based hand gesture and daily activity recognition for robot-assisted living[END_REF], etc. These approaches range from relying on delayed observation to prediction of human behavior which aims to increase robot's proactivity [START_REF] Aarno | Motion intention recognition in robot assisted applications[END_REF][START_REF] Mainprice | Human-robot collaborative manipulation planning using early prediction of human motion[END_REF][START_REF] Maeda | Learning interaction for collaborative tasks with probabilistic movement primitives[END_REF][START_REF] Mainprice | Goal set inverse optimal control and iterative replanning for predicting human reaching motions in shared workspaces[END_REF][START_REF] Liu | Human motion prediction for human-robot collaboration[END_REF][START_REF] Khoramshahi | From human physical interaction to online motion adaptation using parameterized dynamical systems[END_REF][START_REF] Khoramshahi | A dynamical system approach to task-adaptation in physical human-robot interaction[END_REF]. Beside human-intention recognition, stability (and subsequently safety) is another challenge; i.e., to guarantee that the human-intended task is executed stably [START_REF] Khoramshahi | A dynamical system approach to task-adaptation in physical human-robot interaction[END_REF][START_REF] Duchaine | Safe, stable and intuitive control for physical human-robot interaction[END_REF]. Finally, the robotic input needs to be perceived as assistive by the human; e.g., consider an undesirable scenario where the human rejects robot's input as a disturbance to his/her intended task. In summary, any seamless collaborative control design requires: (1) robust human-intention recognition, (2) stable task execution, and (3) human cost minimization.

Numerous types of robotic assistance (i.e., cost function) are considered in the literature: reducing human effort in exerting a force [START_REF] Landi | Admittance control parameter adaptation for physical human-robot interaction[END_REF], human tremor and oscillations [START_REF] Duchaine | Safe, stable and intuitive control for physical human-robot interaction[END_REF], muscle fatigue [START_REF] Peternel | A selective muscle fatigue management approach to ergonomic human-robot co-manipulation[END_REF], metabolic cost [START_REF] Nasiri | Reducing the energy cost of human running using an unpowered exoskeleton[END_REF], etc. In this work, we primarily focus on assisting human to adopt an ergonomic posture; see [START_REF] Wansoo | Towards ergonomic control of collaborative effort in multi-human mobile-robot teams[END_REF][START_REF] Thomas | Intuitive work assistance by reciprocal humanrobot interaction in the subject area of direct human-robot collaboration[END_REF][START_REF] Busch | Postural optimization for an ergonomic human-robot interaction[END_REF] for postural optimization in human-robot co-manipulation tasks. Posture is also of particular interest to supernumerary and prosthetic robots since human users compromise their posture to compensate for their lack of direct control over non-natural joints [START_REF] Metzger | Characterization of compensatory trunk movements during prosthetic upper limb reaching tasks[END_REF]. Several conventional metrics exist for measuring ergonomic cost of a human posture such as REBA and RULA; see [START_REF] Dwyer | Ergonomic assessment of robotic general surgeons: a pilot study[END_REF] as an application of such metrics. In this work, to focus on control aspect of our method, we assume that a leader preferred/ergonomic posture is given; e.g., having an upright torso. We show that stable convergence to both intended target position (of the reaching motion) and ergonomic posture can be achieved under our proposed control strategy.

In this work, we take inspiration from the literature of movement-based control of wearable device (particularly prosthetic arms) for impaired users. This line of research emerged to overcome the limitations of Myoelectric control [START_REF] Geethanjali | Myoelectric control of prosthetic hands: state-of-theart review[END_REF]; mainly to decode human intention from surface EMG signal. Movement-based methods can be categorized into synergy-based and compensation cancellation control. The first category exploits the fact that there are kinematic synergies across joints in human reaching motion [START_REF] Bockemühl | Inter-joint coupling and joint angle synergies of human catching movements[END_REF]. Therefore, the motion of robotic joints are coupled (via a synergy model) to human joints motion; e.g., prosthetic elbow coupled to human shoulder [START_REF] Popovic | Cloning biological synergies improves control of elbow neuroprostheses[END_REF][START_REF] Garcia-Rosas | Personalized online adaptation of kinematic synergies for humanprosthesis interfaces[END_REF]. This approach provides a reliable interface through which the human can indirectly control the robotic joint. Compensation cancellation control [START_REF] Legrand | Movement-based control for upper-limb prosthetics: Is the regression technique the key to a robust and accurate control?[END_REF], however, leverages the fact that impaired users utilize their kinematic redundancies to create their desired motion at the end-effector; e.g., utilizing hip and shoulder to reach forward which compromises the ergonomics [START_REF] Montagnani | Exploiting arm posture synergies in activities of daily living to control the wrist rotation in upper limb prostheses: a feasibility study[END_REF][START_REF] Hussaini | Categorization of compensatory motions in transradial myoelectric prosthesis users[END_REF]. Our previous works [START_REF] Merad | Intuitive prosthetic control using upper limb inter-joint coordinations and imu-based shoulder angles measurement: a pilot study[END_REF][START_REF] Merad | Can we achieve intuitive prosthetic elbow control based on healthy upper limb motor strategies?[END_REF][START_REF] Merad | Assessment of an automatic prosthetic elbow control strategy using residual limb motion for transhumeral amputated individuals with socket or osseointegrated prostheses[END_REF] show the effectiveness of movement-based approach (over Myoelectric control) in providing an ergonomic posture to the human user. However, these current approaches only operate in the joint-space and do not consider the humanintended task. In this work, we extend this line of research to a case where the robot recognizes the human-intended motions, which allows to provides assistance for both task execution and ergonomic posture. More specifically, we assume that measurements for the end-effector velocity and human joint positions are available for intention-recognition purposes. Furthermore, we analyze the performance of our proposed control strategy rigorously in terms of stability, convergence, and optimally.

II. METHOD

In this section, we first formulate our problem and later propose our control approach.

A. Problem formulation

Let us consider a generic n-DoF redundant robot with its joint configuration q ∈ R n with its end-effector position x ∈ R m where m < n. We assume that the robot is controlled in a leader-follower fashion, where the task is only known to the leader. Furthermore, we assume that the first l proximal joints are controlled by the leader (q L = [q 1 , q 2 , ..., q l ]) while the rest is controlled by the follower; i.e., n-l distal joints q F = [q l+1 , q l+2 , ..., q n ]. To control the robot, the leader and the follower command their respective joint-velocities; i.e., qL and qF assuming perfect joint velocity tracking. Therefore, the velocity of the end-effector is obtained by using the task Jacobian:

ẋ = J L J F J qL qF (1)
The Jacobian matrix (J(q) ∈ R m×n ) is segmented into respective parts for the leader (J L ∈ R m×l ) and the follower (J F ∈ R m×(n-l) ). We consider the following reaching task for the leader.

v L = -γ(x -x d ) (2)
where x d is the target position for the end-effector, γ is a positive scalar, and v L is the leader desired task-velocity. Beside, we consider the following cost (C ∈ R + ):

C = 1 2 ( q -q) T W L 0 0 W F W ( q -q) (3)
where q = -K(q -q) represents joint velocities toward desired posture q ∈ R n with K ∈ R n×n a positive definite matrix. W L and W F (positive definite matrices) account for the leader and follower weights respectively. The unified/centralized solution ( q * ) to this problem (e.g., using method of Lagrange multipliers) is:

q * = J # v L + N q (4)
where the weighted pseudo-inverse of the Jacobian and its null-space projector can be segmented into the respective parts for the leader and the follower.

J # L J # F J # = W -1 J T (JW -1 J T ) -1 , and N L N F N = (I -J # J)
(5) However, in our leader-follower formulation, the follower is not aware of the task (x d nor v L ), only the state of the system (q and ẋ) and the leader desired posture (q) and the null-space gain (K) . In the next section, we propose our control strategy for solving this problem.

B. Proposed control strategy

We propose the following control strategy:

     qL = J # L v L + N L q qF = J # F v F + N F q vF = -α(v F -ẋ) (6)
The first equation represents the leader's strategy which attempts to create its desired velocities (v L and q). The second and third equations represent the follower's strategy which uses a lower-pass filtered version of ẋ as an approximation for the leader intended velocity; i.e., v F is the follower inferred velocity, and α is positive scalar. In this strategy, the leader creates its desired velocity (v L ) to reach the intended goal (x d ), while the follower assists the leader via executing an inferred velocity (v F ) based on previous observation of the end-effector motion ( ẋ).

Finally, let us note that the ODEs in Eq. 6 describe the desired dynamics where forward Euler method is used for numerical implementation.

In Appendix, we analyze the dynamics of the system under our proposed control strategy. In short, we show that when 1) the leader is not singular (i.e., capable of performing the task alone), and 2) the follower estimation rate is faster than the task dynamics (i.e., α > γ and α > K) then 1) the task is achieved in a stable manner, and 2) the follower stably tracks (and converges to) the centralized solution (i.e., when the follower knows v L )

III. ILLUSTRATIVE EXAMPLE

Let us consider the most simple example where we have two prismatic joints and one-dimensional task x = q L + q F . where q L and q F represent the leader and the follower joints respectively. Therefore, the Jacobian is J = [1 , 1], which means J L = J F = 1. For inverse kinematics (IK), we consider the following general parameterization:

W = 1 0 0 w F → J # = λ λ , N = λ - λ -λ λ (7)
with the follower IK weight w F > 0, λ = 1/(1 + w F ), and λ = 1 -λ. For null-space velocity, we consider q = [-kq L , 0] T , with k > 0, in order to perform the task with q L = 0. In this case, the dynamics (derived from Eq. 6) are:

     qL = (1 -λ)v L -λkq L qF = λv F + λkq L vF = -α(v F -qF -qL ) (8)
At this point, it is interesting to consider two extreme cases:

w F = ∞ → λ = 0 → qF = 0 w F = 0 → λ = 1 → qF = -qL (9)
It is intuitive to see that in the case of w F = ∞, zero velocities are expected from the follower as it has a high cost. On the other hand, when w F = 0, the leader is not performing any portion of the task since it is expected that the follower performs the task completely. Therefore, the follower recognizes v L = 0 as the intended motion. The follower moves only in order to cancel the leader's motion on the end-effector ( ẋ = 0) as the leader still moves in order to reach q L = 0. For other cases (where 0 < λ < 1), we can write (by taking the time-derivative of qF in Eq. 8):

qF = -α λ( qF -q * F -∆ qF ) (10)
where q * F = λv L + λkq L represent the centralized solution for the follower, and ∆ qF = kα -1 (λv L -λ 2 λ-1 kq L ) accounts for the error induced by filtering (i.e., intention estimation). This error is mitigated by higher values for α. Moreover, since -α λ < 0, the follower velocity converges to q * F (when ∆ qF can be neglected). These dynamics also show that small values of λ lead to faster convergence to the centralized solution. However, in centralized solutions with low λ (i.e., high w F ), the follower contributes less (i.e., q * F ∝ λ). Therefore, in the decentralized scenario, we face a trade-off between intention-recognition and assistance; where the leader is required to perform the task partially in order to communicate the intended goal to the follower and subsequently receive assistance.

Position

[rad] q 1 q 1 q 2 q 2 q 3 q 3

(b) We simulate the system behavior for different values of w F and compare against the centralized solution. For reaching task, we consider v L = -1.2(x -1) and k = 1 for the null-space. We initialized the system at q L = 0.5, q F = 0.1, and v F = 0, and all their derivatives at zero. The centralized solution is generated with w F = 0 and v F = v L . The results show the aforementioned trade-off. Higher values of w F lead to similar results to those of the centralized solution at the end-effector; see the first subplot. However, lower values of w F perform similarly in the null-space; see q L in the second subplot. Finally, the third subplot (q F ) shows that with w F = 0.1, q F behaves similar to q * F . The fourth subplot shows this trade-off in a quantitative manner where w F has opposite effect on task and null-space error.

It is crucial to investigate the case where the leader and the follower have different assumptions about each other's contribution. Note that in our formulation, we assumed that both agents utilize the same gain matrix W in their inverse kinematics. Last subplot in Fig. 2 shows the performance of our control approach (in terms of joint-level error) where the leader and the follower have different assumption for w F . Intuitively, the leader assuming small values or the follower assuming big values (i.e., relying heavily on the other partner) are counter-productive. When the follower assumes w F < 1 (i.e., to contribute more than the leader), the performance is not only satisfactory, but also less sensitive to the leader's assumption.

IV. THREE-LINKS PLANAR ROBOT

In this simulation, we consider a 3-DoF planar model illustrated in Fig. 3a where the first two joints are controlled by the leader and the last joint by the follower. This resembles a simplified model for human reaching in the sagittal plane utilizing hip, shoulder, and a prosthetic elbow. In this model, we have l 1 = 0.6, l 2 = 0.3, and l 3 = 0.25m. The task is to reach x d = [.5, .5] T with v L = -0.1(x -x d ). The preferred posture is q = [0, 0, 0] T with K = diag[2, 0, 0]; i.e., only keeping the first link upright. The robot is initialized at q = [0; -7π/8; π/6] and simulated with dt = 0.01. The centralized solution is generated using W = diag([1, .1, .05]), whereas in the leader-follower scenario W = diag([1, .1, .5]). We choose w F = 0.5 (lower expected contribution from the follower) to investigate the performance of the robust case which we expect following the insight from the previous results presented in Fig. 2.

The results presented in Fig. 3b compares the centralized and collaborative scenarios in terms of joint angles. As expected by the weight matrix W in the centralized scenario, the reaching motion is mostly achieved by moving the third and the second joint. However, in the collaborative scenario, the leader moves its joints (q 1 and q 2 ) more to convey the intended motion. Interestingly, the leader only uses q 1 on the onset of the motion and goes back to its preferred value. Compared to the centralized case, the follower motion has less amplitude (due to higher w F ) and is delayed (due to filtering). It can be seen that after 0.2s, the joint values converge to the centralized solution. Furthermore, Fig. 3c shows the joint-space error (compared to the centralized solution) when the leader and the follower have different assumption about w F . The graph is similar to the one obtained from the 1-DoF example; i.e. the results are less sensitive to the leader's assumption when the follower assumes w F = 0.4. However, a lower error can be achieved if both agree upon w F = 0.1.

Tracking markers

Reaching target

Fig. 4: The experimental setup where a human subject controls a kinematic chain on the screen with his/her respective joints in order to reach a designated target. We use this setup to compare different control strategies for the third joint.

V. EXPERIMENTAL RESULTS

In this section, we validate our proposed control method facing a leader with higher complexity. To do so, we asked 10 participants to control the 3-DoF manipulator (of the previous section) with their respective kinematic chain in the sagittal plane; hip, shoulder, and elbow to control q 1 , q 2 , and q 3 respectively. We use the Optitrack system to track four marker-clusters attached to the subject upper body (hip, shoulder, elbow and wrist) from which we obtains the values for q. The participants were asked to reach for a series of target positions in the frontal reachable workspace. These targets position, as well as the 3-DoF robot, was displayed to the participants on a monitor screen as illustrated in Fig. 4.

To better investigate our approach, we considered the following conditions. In HC, the participant controls all the tree joints; i.e., a centralized control. For FE, the third joint (elbow) is fixed at q 3 = π 6 , and the participant controls q 1 and q 2 . In SS, q 3 is controlled using a simple shoulder-elbow synergy; i.e., q3 = q2 where q 1 and q 2 is controlled by the subject. In PC, the proposed control method is used to control the third joint. For PC-S, only the synergy part of the propose method is used; i.e., q = 0 in Eq. 6. Finally in PS-N, only the null-space part of the proposed method is used; i.e., v F = 0 in Eq. 6. The sequence of the target position (consists of G = 28 goals) were kept fixed across the conditions. We designed this sequence to accommodate motions in different direction; since the previous reached-goal will be the initial position to reach another. Each target was presented for a period of 6s making the trial T = 168s long. Prior to each trial, the participants had a short training session to familiarize with the control strategy. For our controller, we used α = 40, W = diag([1, 1, .15]), K = diag([2, .2, 0]), and q = [0, -2.7, 0]rad. Finally, the control loop frequency, in average, was around 50Hz.

We consider the four following metric (listed in the table below) to evaluate the performance of participants. Hip utilization is computed as the average absolute value of the first joint (q 1 ). Distance error is obtained as the average of distance between the end-effector and the target e = x -x d where . denotes the vector norm. Time to reach is the average time spent to reach a target position with tolerance δ = 0.02m: Finally, Deviation from straight line is the average of the velocity components that are orthogonal to the error vector e. The participants' performances across conditions are summarized in Fig. 5. In this figure, the performance of each participant in each conditions is compared (in percentages) to their individual performances in the HC condition.

Metric Equation Average in HC

Hip utilization The obtained results show that our approach is effective in reducing the hip utilization in human participants; 47% compared to 250% when the elbow is locked, and 150% when the simple synergy method is used. However, it can be seen that the equality of reaching motion (in terms of the other three metrics) is worse compared to HC and FE when the human (even though the third joints is locked) controls the system in a centralized manner. Such lower performance is expected since in the decentralized control the follower has uncertainties (about the intended task and over control parameters) and suffers from delayed observations. Nevertheless, comparison between SS and PC reveals that more complex dynamics (which reduces the hip utilization by 102%) does not exacerbate the quality of reaching drastically; 10% slower reaching with 17% higher deviation from straight line. Moreover, the results shows that due to the faster dynamics of the synergy part of our method, the target is reached faster but with higher deviation from the straight line. On the other hand, slower dynamics of the null-space part lead to less deviation but slower reaching is achieved. These observations are useful in tuning parameters such as w F and K which adjust the reactivity of each respective part. Finally, it is important to note the variability among participants (as the distributions have high variance) which hints at 1) personalized parameter tuning, and 2) possibility for improvement after training.

VI. DISCUSSION AND CONCLUSION

In this work, we present an intent-aware control method for collaborative robots in kinematically redundant setups with applications towards prosthetic and supernumerary arms. Our proposed method is inspired by the movement-based control approach in which it was observed that impaired users utilize their kinematic redundancies (and subsequently, compromise the ergonomics of their posture) to perform their intended task. Our method exploits the fact that the intended task can be recognized through the observations of the end-effector velocity created by the leader. We formulate our method based on the segmentation of the Jacobian matrix which allowed us to investigate the stability and convergence behavior rigorously. The simulation results are in line with the theoretical aspects of the method and shed light on how certain parameters affect the final performance; e.g., w F as the follower IK gain which sets a trade-off between optimality and robustness.

Our experimental results show that our proposed method is effective in recognizing the intended task, and thus, reducing the ergonomic cost for human-leaders. We compare our method against a simple linear synergy model which often used in the literature. Most often, the synergy model is extracted from healthy subjects ( [START_REF] Vinjamuri | Candidates for synergies: linear discriminants versus principal components[END_REF][START_REF] Lambert-Shirzad | On identifying kinematic and muscle synergies: a comparison of matrix factorization methods using experimental data from the healthy population[END_REF]) to be used for individuals with amputations ( [START_REF] Garcia-Rosas | Personalized online adaptation of kinematic synergies for humanprosthesis interfaces[END_REF][START_REF] Alshammary | Efficacy of coordinating shoulder and elbow motion in a myoelectric transhumeral prosthesis in reaching tasks[END_REF]). However, such methods neglect the kinematic changes in individuals with amputations; e.g., higher hip and scapula utilization to compensate for the lack of control over the distal joints. Moreover, we put our method under further scrutiny by investigating the performance of the separate parts of the controller; namely the synergy and the null-space part in Eq. 6. The obtained results hint at how the respective gains of each part (w F and K) can be tuned effectively in order to provide satisfactory dynamics for the human-user; slow dynamics increase the reaching time and fast dynamics cause fluctuations and overshoots.

The comparison between simulation and experimental results

show that better performance is achieved when the leader and the follower have complementary dynamics; i.e., the leader and the follower solving the IK problem by using the respective parts of a unified Jacobian. However, in the experimental scenario, the unmodeled/unexpected dynamics seen by each agents deteriorates the tracking performance. In our future work, we will try to improve our approach by accommodating human-like dynamics; e.g., through online learning and personalization. Moreover, in future, we will implement our method on a prosthetic arm where we need to tackle technical challenges such as embedded and noisy measurements, imperfect physical coupling and robot dynamics. APPENDIX 1) Proof of stability: The dynamics of this system can be studied by plugging Eq. 6 into Eq. 1, which results in

ẋ = -γH L (x -x d ) + H F v F (11)
where H L = J L J # L and H F = J F J # F with property that H L + H F = JJ # = I; I being the identity matrix. The dynamics of the error can also be investigated with respect to the target position; i.e., e = x-x d and its time-derivative:

ė = -γH L e + H F v F (12)
The second time-derivative yields the dynamics as

ë + (γ + α)H L ė + αγH L e = η x (13)
where the η x = ḢL (λe -v F ). Since γ and α are positive scalars, the homogeneous system is stable if all eigenvalues of H L are positive. To show this, we expand H L as follows.

H L = J L W -1 L J T L (J L W -1 L J T L + J F W -1 F J T F ) -1 (14)
Here, we assume that J L W -1 L J T L is invertible; i.e., the leader is not singular. Therefore, H L can be simplified to:

H L = (I + A) -1 (15)
where A = (J F W -1 F J T F )(J L W -1 L J T L ) -1 has non-negative eigenvalues since it is the multiplication of a semi-positive and a positive definite matrix. Therefore, if λ ≥ 0 is eigenvalue for A, 1 1+λ > 0 is an eigenvalue for H L .

The previous analysis shows that the task can be achieve in a stable manner. However, we also need to show that internal dynamics are stable; e.g., the undesirable behavior when the leader compensates for the follower's unstable dynamics. Thus, we study the follower's dynamics which are qF = -α(I -J # F J F )( qF -q * F ) + η q [START_REF] Liu | Intention recognition in physical human-robot interaction based on radial basis function neural network[END_REF] where q * F = J # F v L + N F q represent the centralized solution of the follower; where v F is replaced by v L . Moreover, η q = N F q + ṄF q + J# F v F represents the disturbances due to higher order dynamics. The matrix J # F J F has the same non-zero eigenvalues of H F . Since H L + H F = I, it follows that (I -J # F J F ) has eigenvalues of form 1 1+λ > 0 as well. This guarantees a stable convergence of qF to its centralized solution when η q can be neglected.

Finally, in our analysis, we assume a perfect velocitycontroller and we ignore the second-order dynamics of the system; e.g., inertia, centrifugal forces and gravity. However, such neglected discrepancies can be accommodated into η x and η q as disturbances that can be tolerated by the velocity controller. Our analysis (at the velocity level) and proposed method (on motion planning for the follower) still hold even if one imagines other types of controllers such as torquelevel control (e.g., inverse dynamics control [START_REF] Chung | Motion control[END_REF]) as long as the control objective is to track qF proposed by Eq. 6.

Fig. 1 :

 1 Fig.1: Humans utilizing their kinematic redundancies to compensate for the lack of collaboration in wearable robots. In (a), the human leans backward to create an upward motion at the endeffector of the prosthetic arm, and in (b), leans forward to create forward motion at the end-effector of the supernumerary arm.

Fig. 2 :

 2 Fig. 2: The simulation results for the illustrative example of n = 2 and m = 1. The first plot shows the position of the end-effector under various values of wF . the second plot show the joint position. The third plot shows how the performance is compared to a centralized solution for different values of wF . The fourth plot shows the system performance when the leader and the follower have different assumption about wF .

Fig. 3 :

 3 Fig. 3: The simulation results for the 3-DoF planar manipulator. (a) The model. (b) The joint values. (c) The joint-level error (compared to the centralized solution) when the agents have different assumptions about each other's contribution (i.e., wF ).

Fig 2

 2 Fig 2 shows the results of a simulation for this 1-D case. We simulate the system behavior for different values of w F and compare against the centralized solution. For reaching task, we consider v L = -1.2(x -1) and k = 1 for the null-space. We initialized the system at q L = 0.5, q F = 0.1, and v F = 0, and all their derivatives at zero. The centralized solution is generated with w F = 0 and v F = v L . The results show the aforementioned trade-off. Higher values of w F lead to similar results to those of the centralized solution at the end-effector; see the first subplot. However, lower values of w F perform similarly in the null-space; see q L in the second subplot. Finally, the third subplot (q F ) shows that with w F = 0.1, q F behaves similar to q * F . The fourth subplot shows this trade-off in a quantitative manner where w F has opposite effect on task and null-space error.

Fig. 5 :

 5 Fig.5:The experimental result where humans controls the first 2 joints and different strategies used for the third joint. The results are presented in percentages in comparison to the HC condition where the human controls all the three joints. Each distribution is presented in terms of min, 25th percentile, median, 75th percentile, and max. Outliers are indicated by '+' symbol.

  0.05m/s


Marylène Vayer
email: marylene.vayer@univ-orleans.fr

Aynur Guliyeva

Fabienne Warmont

Atsushi Takano

Yushu Matsushita

Christophe Sinturel
email: christophe.sinturel@univ-orleans.fr

Transition between tetragonal and hexagonal pattern in binary

Keywords: Block Copolymer, Order-Order-Transition, Double hexagonal morphology, Tetragonal morphology, Thin film effect

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Microphase separation of linear ABC terpolymer has been the object of numerous works [START_REF] Hadjichristidis | Linear and non-linear triblock terpolymers. Synthesis, self-assembly in selective solvents and in bulk[END_REF][START_REF] Meuler | Ordered Network Mesostructures in Block[END_REF][START_REF] Bates | 50th Anniversary Perspective: Block Polymers Pure Potential Macromolecules[END_REF][START_REF] Tyler | SCFT Study of Nonfrustrated ABC Triblock Copolymer Melts[END_REF][START_REF] Nagpal | Morphologies of Linear Triblock Copolymers from Monte Carlo Simulations[END_REF][START_REF] Qin | Phase Behavior of Nonfrustrated ABC Triblock Copolymers: Weak and Intermediate Segregation[END_REF][START_REF] Phan | Morphology of Symmetric ABC Triblock Copolymers in the Strong Segregation Limit[END_REF][START_REF] Zheng | Morphology of ABC Triblock Copolymers[END_REF][START_REF] Chatterjee | Comprehensive Phase Behavior of Poly(isoprene-bstyrene-b-ethylene oxide) Triblock Copolymers[END_REF][START_REF] Tureau | Inducing Order from Disordered Copolymers: On Demand Generation of Triblock Morphologies Including Networks[END_REF][START_REF] Tureau | Phase behavior of neat triblock copolymers and copolymer/homopolymer blends near network phase windows[END_REF][START_REF] Mogi | Superlattice Structures in Morphologies of the ABC Triblock Copolymers[END_REF][START_REF] Brinkmann | New Structural Motif in Hexagonally Ordered Cylindrical Ternary (ABC) Block Copolymer Microdomains[END_REF][START_REF] Asai | Asymmetric double tetragonal domain packing from ABC triblock terpolymer blends with chain length difference[END_REF][START_REF] Izumi | A New Periodic Pattern with Five-Neighbored Domain Packing from ABC Triblock Terpolymer/B Homopolymer Blend[END_REF][START_REF] Asai | Formation of Tetragonally-Packed Rectangular Cylinders from ABC Block Terpolymer Blends[END_REF]. The formed structures and their domains of stabilization depend strongly on the blocks volume fractions and the incompatibility of the blocks with each other ( [START_REF] Tyler | SCFT Study of Nonfrustrated ABC Triblock Copolymer Melts[END_REF][START_REF] Nagpal | Morphologies of Linear Triblock Copolymers from Monte Carlo Simulations[END_REF][START_REF] Qin | Phase Behavior of Nonfrustrated ABC Triblock Copolymers: Weak and Intermediate Segregation[END_REF][START_REF] Phan | Morphology of Symmetric ABC Triblock Copolymers in the Strong Segregation Limit[END_REF][START_REF] Zheng | Morphology of ABC Triblock Copolymers[END_REF]). In the case of non-frustrated (i.e. the incompatibility between the end-blocks is higher than the incompatibility between the middle and end blocks) linear polymers, these materials selfassemble into simple ordered structures with spherical, cylindrical, gyroid and lamellar domains. Whereas cylindrical domains are arranged exclusively into a hexagonally packed structure for diblock copolymers, three types of cylindrical morphologies are predicted for triblock terpolymers [START_REF] Tyler | SCFT Study of Nonfrustrated ABC Triblock Copolymer Melts[END_REF][START_REF] Nagpal | Morphologies of Linear Triblock Copolymers from Monte Carlo Simulations[END_REF].

The first one corresponds to a core-shell version, arranged in hexagonal packing (Figure 1a). In this case, one end-block (e.g. A) is the major component and constitutes the matrix in which core (end-block C)/shell (middle-block B) cylinders are formed. This concerns highly asymmetric non-frustrated linear triblock copolymers [START_REF] Tyler | SCFT Study of Nonfrustrated ABC Triblock Copolymer Melts[END_REF][START_REF] Nagpal | Morphologies of Linear Triblock Copolymers from Monte Carlo Simulations[END_REF][START_REF] Chatterjee | Comprehensive Phase Behavior of Poly(isoprene-bstyrene-b-ethylene oxide) Triblock Copolymers[END_REF].

The second type of morphology corresponds to cylinders consisting in one end-block (e.g. A) arranged in hexagonal packing in a matrix of B, the other end-block (C) being miscible with the middle-block (B) or forming interstitial domains in the matrix (Figure 1b). This concerns triblocks with B as major component, distinct proportions of A and C and intermediate segregation conditions [START_REF] Tyler | SCFT Study of Nonfrustrated ABC Triblock Copolymer Melts[END_REF][START_REF] Nagpal | Morphologies of Linear Triblock Copolymers from Monte Carlo Simulations[END_REF]. For example, Tureau et al. [START_REF] Tureau | Inducing Order from Disordered Copolymers: On Demand Generation of Triblock Morphologies Including Networks[END_REF][START_REF] Tureau | Phase behavior of neat triblock copolymers and copolymer/homopolymer blends near network phase windows[END_REF] observed for PI-PS-PMMA (ISM) cylinders of I arranged in hexagonal pattern in a matrix of SM.

The last one is the tetragonal structure, with alternating A and C cylinders within B matrix, observed for triblock with middle-block B as the major component and ratio of volume fractions of end-blocks is near unity (Figure 1c) [START_REF] Nagpal | Morphologies of Linear Triblock Copolymers from Monte Carlo Simulations[END_REF][START_REF] Qin | Phase Behavior of Nonfrustrated ABC Triblock Copolymers: Weak and Intermediate Segregation[END_REF]. For example, Mogi et al. [START_REF] Mogi | Superlattice Structures in Morphologies of the ABC Triblock Copolymers[END_REF] explored this structure with two kinds of cylinders formed by the two end-blocks for PI-PS-P2VP (ISP) system.

Beside the structures predicted above, the formation of a double hexagonal structure was repeatedly experimentally reported in the literature, whereas up to date, no arrangement with distinct A and B cylinders in a double regular hexagonal structure was predicted to our knowledge (Figure 1d). Brinkmann et al. [START_REF] Brinkmann | New Structural Motif in Hexagonally Ordered Cylindrical Ternary (ABC) Block Copolymer Microdomains[END_REF] observed in PS-PB-PMMA (SBM) triblocks having a (0.21:0.57:0.16) (S:B:M) composition, the formation of a new hexagonal morphology with S and M cylinders in a matrix of B (7 M cylinders forming centered hexagons and 6 S cylinders forming hexagon around each M cylinder). Such pattern was also described by Asai et al. [START_REF] Asai | Asymmetric double tetragonal domain packing from ABC triblock terpolymer blends with chain length difference[END_REF] for ISP system with a composition (I:S:P) (0.12:0.67:0.21), in which P cylindrical domains were arranged in centered hexagons and each P domain was surrounded by 6 I cylindrical domains. For ISP system, Izumi and al. [START_REF] Izumi | A New Periodic Pattern with Five-Neighbored Domain Packing from ABC Triblock Terpolymer/B Homopolymer Blend[END_REF] studied bulk system of binary ISP/S blends, the parent triblocks being all gyroid with composition between (0.13: 0.55: 0.32) and (0.34: 0.55: 0.11). Adding S homopolymer led to blend composition (0.25-0.10: 0.65-0.67: 0.08-0.24) and cylindrical structures. I and P were co-arrayed in a tetragonal symmetry up to a critical ratio of P/I that depended on the molecular mass of the homopolymer S (e. g. a ratio of 1 was observed for Mn=12 kg.mol -1 ). Above this ratio, cylinders of I and P were organized within a double hexagonal structure, where P cylinders form centered hexagons and I block form 6 cylinders surrounding the previous ones. Asai et al. [START_REF] Asai | Asymmetric double tetragonal domain packing from ABC triblock terpolymer blends with chain length difference[END_REF] demonstrated that blends of two terpolymers with a selected symmetric composition (volume fraction of S between 0.54 to 0.69) lead to alternating rods in a tetragonal structure. Moreover, if end-blocks are composed of a blend of short and long chains, the rods are no longer cylindrical domains but have non-constant mean curvature. [START_REF] Asai | Formation of Tetragonally-Packed Rectangular Cylinders from ABC Block Terpolymer Blends[END_REF][START_REF] Asai | Creation of Cylindrical Morphologies with Extremely Large Oblong Unit Lattices from ABC Block Terpolymer Blends[END_REF] In thin films, the cases are even more complex, since the limited amount of polymer and the enhanced effects of the interfaces lead to stabilization, destabilization and also to orientation of the structures. In addition, solvent swelling, thermodynamics, kinetics make the situation even more complex. [START_REF] Sinturel | Solvent Vapor Annealing of Block Polymer Thin films[END_REF] In PI-PS-PFS (ISFS) system, Chuang et al. [START_REF] Chuang | Templated Self-Assembly of Square Symmetry Arrays from an ABC Triblock Terpolymer[END_REF] demonstrated that neat copolymer with a (I:S:FS) (0.25:0.65:0.10) exhibits in thin films either square-packed patterns of I and FS cylinders with a composition (0.165:0.67:0.165) or FS-S core-shell cylinders in a hexagonal symmetry in a matrix of I with a composition (0.35:0.51:0.14). Addition of S leads to a (0.16:0.78:0.06) composition in volume and a stabilization of square pattern. In ISP thin films, Guliyeva et al. [START_REF] Guliyeva | Thin Films with Perpendicular Tetragonally Packed Rectangular Rods Obtained from Blends of Linear ABC Block Terpolymers[END_REF] demonstrated that, for the blend of two ISP triblocks studied by Asai et al. [START_REF] Asai | Formation of Tetragonally-Packed Rectangular Cylinders from ABC Block Terpolymer Blends[END_REF] with a symmetric composition (0.20:0.60:0.20), the square pattern is preserved in the inner part of the film under THF vapor annealing, producing rods perpendicular to the surface for short annealing time and rods parallel to the surface for long annealing time. It was shown in this paper that similarly to the bulk, the end-blocks domains are in this case no longer cylinders but rods with flat interface.

The problematic we address here is the structuration of block polymer blends with near symmetric composition in thin films, with a focus on the possible evolutions from tetragonal to double hexagonal morphology, due to the thin film effect. We also examine the effect of block chains length difference within the blends. For that purpose, we blended two triblocks of ISP with S as the major block, one with a very long chain of I and the other with a long chain of P.

We selected two blends compositions already studied in bulk in previous work by Asai et al. [START_REF] Asai | Asymmetric double tetragonal domain packing from ABC triblock terpolymer blends with chain length difference[END_REF] showing tetragonal structures in the bulk.

Materials and methods

Poly(isoprene-block-styrene-block-(2-vinylpyridine)) triblock terpolymers (ISP) were synthesized by anionic polymerization following a protocol described elsewhere [START_REF] Mogi | Preparation and Morphology of Triblock Copolymers of the ABC Type[END_REF]. These ISP X and ISP Y terpolymers have been studied in details by Asai et al. [START_REF] Asai | Asymmetric double tetragonal domain packing from ABC triblock terpolymer blends with chain length difference[END_REF] d=95 nm LI= 48 nm LS=47 nm DP=12 nm Table1. Molecular characteristics of the used block copolymer and their bulk structure. [14] (1) Mn molecular weight, (2) φ volume fraction; N block length in number of Flory repeating units; (3) dcc center to center distance for the cylinders of the centered hexagon; D cylinder diameter. These characteristic lengths are deducted from TEM experiments (4) schematic representation with I in yellow, S is grey and P is blue; (5) representative TEM image, TEM scale bar =100 nm.

All the used solvents were purchased from Sigma Aldrich and Roth Sochiel and used as received. After the deposition, a solvent vapor annealing (SVA) post-treatment was performed. For that purpose, the films were exposed to toluene vapor in a closed glass vessel (1L) containing 3 mL of solvent at 30°C and the structures evolution depending on exposure time was detected.

AFM in the tapping mode was carried out in air with AFM Dimension Icon model (Bruker ICON) and silicon cantilevers RTESP300 from Bruker were used. All the displayed AFM images are height images processed with Gwyddion.

Transmission electron microscopy (TEM) was performed with a Philips CM20 operated at an accelerating voltage of 200 kV. For that purpose, the films were removed from the Si/SiO2

substrate by flotation at the surface of 0.5 M NaOH solution for more than 5 h. NaOH attacks the native silica layer and leads to the detachment of film fragments from the substrate. These fragments were washed by distilled water to remove NaOH traces and were transferred onto a TEM grid (for top view analysis) or on a rigid plastic substrate, 1 mm thick (for sectional view analysis). This plastic substrate was covered by another rigid plastic substrate and the resulting sandwich (with the fragment to be observed in the middle) was cut by ultramicrotome. The obtained sections were transferred onto grids. All the grids were dried at RT for 1 day and were stained. Two staining steps were performed for selectively revealing the domains, namely OsO4 staining for I and I2 exposure for P domains. The same staining protocol was carefully followed for all the samples in order to be able to compare the characteristic distances measured on different staining images. These distances were averaged on a minimum of 10 measures and the standard deviation estimated to at most ± 3 nm.

GISAXS experiments were performed on the "Xeuss" Xenocs X-ray scattering setup on thin films stained by I2 to increase the electronic density contrast. The two-dimensional GISAXS patterns were recorded using a high sensitivity low noise 2D detection (Pilatus 300K hybrid pixel) placed at a distance of 2329 mm from the sample, perpendicular to the X-ray beam (parallel beam at 8042eV) with a grazing incidence angle of 0.17° close to the critical angle of the films. The diffraction patterns were recorded, and in-plane GISAXS profiles were extracted.

Surface tensions of blocks and substrate were determined using the Owens Wendt approach.

Contact angles of 3 liquids (namely deionized water, diiodomethane and ethylene glycol) were performed using GBX Digidrop apparatus on I, S, P homopolymer films and substrate surfaces.

Results and discussion

The morphology evolution after toluene vapor exposure was first followed as a function of exposure time using AFM. To address the observed nanostructured patterns, GISAXS, TEM and AFM analyses were combined.

X/Y 85/15 blend

As a function of time, the morphology of the 85/15 X/Y film evolved from disorganized (as spun) to a hexagonal, then square pattern and then disorganized morphology as shown in Figure 2 and SI-1. This last morphology originates from a swelling of the film that allows passing through the order-disorder transition.

X/Y 85/15 blend hexagonal pattern

Figure 2a displays the AFM image of a surface exhibiting the hexagonal pattern. Despite numerous defects impairing the long-range order, the hexagonal arrangement is evident from this image, at least on a local length scale. The center-to-center distance between holes is evaluated to 72 nm. Figure 2b displays a TEM top-view image after successive selective staining by OsO4 and I2. This procedure stains I domains in black, P in dark grey and lets S unstained (light grey). The presence of black dots is due to gold nanoparticles used for other experimental purposes (including tomography, not shown here). On TEM image, round black domains of I adopt a centered hexagonal pattern. Each I domain is surrounded by 6 round dark grey domains of P. This morphology will be designed thereafter by "centered hexagons of I".

The center-to-center distance between two I domains is evaluated to 77 nm, the diameters of I and P domains are evaluated to 27 and 28 nm. The GISAXS pattern obtained on film after I2 staining presents peaks with an extension parallel to the qz axis (SI-2a). This is the signature of a homogeneous composition of the film perpendicularly to the substrate in agreement with the TEM-section view image (Figure 2c) which shows rods extending perpendicularly to the surface. The in-plane profile (Figure 2d) presents three main peaks at 0.090, 0.157 and 0.316 nm -1 corresponding to the (10) (11) and ( 22) diffraction peaks in ratio √3 and √12 to the first one of a hexagonal pattern in plane with a cell unit length of 70 nm in agreement with AFM and TEM results. distance between the features evaluated to 69 nm. This is consistent with TEM top-view image obtained after selective successive staining by OsO4 and I2, presenting round black domains of I adopting a square pattern (Figure 2f). Each I domain is surrounded by 4 round dark grey domains of P. I and P networks are interwoven. The characteristic distance is 69 nm and I and P domains sized respectively 32 and 30 nm. TEM section view exhibited rods perpendicular to the surface (Figure 2g). This is in agreement with the GISAXS pattern performed on film after staining by I2 presenting peaks with an extension parallel to the qz axis (see SI-2b). The in-plane profile (Figure 2h) presents two main peaks at 0.093 and 0.132 nm -1 corresponding to [START_REF] Tureau | Inducing Order from Disordered Copolymers: On Demand Generation of Triblock Morphologies Including Networks[END_REF] and [START_REF] Tureau | Phase behavior of neat triblock copolymers and copolymer/homopolymer blends near network phase windows[END_REF] in ratio √2 of a square pattern in plane with a cell unit length of 68 nm.

X/Y 60/40 blend

Under toluene exposure, thin film of X/Y 60/40 evolved through different types of morphologies in surface (displayed Figure 3 and in SI-3) starting from disorganized morphology to square pattern, then hexagonal pattern finally to disorganized film. domains exhibit marked difference of sizes, I domains diameter being higher than those of P (42 nm versus 33 nm). The GISAXS pattern obtained on this film after I2 staining presents continuous Bragg rods with an extension parallel to the qz axis (see SI-4a) revealing the axial extension of the structure perpendicularly to the surface. The in-plane profile (Figure 3c)

presents four main peaks at 0.078, 0.113, 0.156, 0.177 nm -1 corresponding to ( 10), ( 11), [START_REF] Guliyeva | Thin Films with Perpendicular Tetragonally Packed Rectangular Rods Obtained from Blends of Linear ABC Block Terpolymers[END_REF] and ( 21) diffraction peaks in ratio √2, 2 and √5 to the first one, in total agreement with a tetragonal pattern with the cylinders perpendicular to the surface plane within a unit cell length of 81 nm.

X/Y 60/40 blend hexagonal pattern

Figure 3d displays AFM image of a film exhibiting a centered hexagonal pattern of depressions, each depression being surrounded by 6 bumps hexagonally arranged. The center-to-center distance of the depression is evaluated to 95 nm. Figure 3e displays the TEM top-view image after successive staining by OsO4 and I2: round black domains of I adopt a centered hexagonal pattern. Each I domain is surrounded by 6 grey domains of P. This morphology will be designed thereafter by "centered hexagons of I". The distance between I domains is evaluated to 100 nm and reported in table 3. In this case, the I domains have a diameter (50 nm) at least twice the one of P domains (25 nm). The GISAXS pattern obtained on film after I2 staining presents peaks with an extension parallel to the q z axis (see SI-4b), signature of a homogeneity of the film perpendicularly to the substrate. The in-plane profile in Figure 3f presents two main peaks at 0.068 and 0.120 nm -1 corresponding to [START_REF] Tureau | Inducing Order from Disordered Copolymers: On Demand Generation of Triblock Morphologies Including Networks[END_REF] and [START_REF] Tureau | Phase behavior of neat triblock copolymers and copolymer/homopolymer blends near network phase windows[END_REF] in ratio √3 in agreement with a hexagonal pattern in plane with a cell unit length of 91 nm.

Transition pathway from hexagonal towards tetragonal pattern and vice-versa

Before discussing the origin of the ordered phases, one has to note that both blends reach a disordered state for longer annealing time (>20h), with no evidence of short and long range order. This suggests that an order-disorder transition was reached during SVA, resulting from a major swelling degree (the fast evaporation freezes the system into a more or less disordered state). This suggests that the morphologies we will now discuss are not at equilibrium, but represent kinetically defined transitory structures.

For the two blends, Table 3 summarizes the characteristics of the morphologies in bulk (obtained from previous work [START_REF] Asai | Asymmetric double tetragonal domain packing from ABC triblock terpolymer blends with chain length difference[END_REF]) and in thin films (obtained in this work). In thin films, the characteristic distances measured in AFM, TEM and GISAXS are all in well agreement and allow to calculate the apparent composition of each phase (I, S and P) based on simple geometric considerations (see SI-5). The overall composition of the block copolymer in the film being constant, the variation of the apparent volume fraction determined by TEM can only be explained by a selective migration of one block towards the interface, forming a uniform wetting layer (non-observable on a top-view TEM micrograph) and depleting the inner portion of the film from this block. In turn, this will modify the morphology, as we will explain below. I:S:P apparent composition 0.17:0.66:0.17 0.11:0.65:0.24 0.17:0.68:0.15 0.26:0.62:0.12 0.21:0.66:0.13 0.23:0.66:0.11 Table 3: Characteristic distances (in nm) of the different observed patterns for X/Y 85/15, X/Y 60/40; dcc center to center distance for the cylinders of the centered hexagon or center to center distance of I (P) cylinders in the square pattern ; D cylinder diameter. For clarity, the shapes of all domains are represented by circular features, which may not reflect the actual geometry (see for example Fig. 5). (I:S:P) volume fraction of the bulk and observed patterns in thin film (I in yellow, S in grey and P in blue). *In bulk, the composition of the observed patterns is the same that the blend composition. In thin film, the given composition is determined from the TEM distances (apparent composition).

X/Y 85/15 blend

The X/Y 85/15 blend with a (I:S:P) native volume fraction of (0.17:0.66:0.17) was previously studied by Asai [START_REF] Asai | Asymmetric double tetragonal domain packing from ABC triblock terpolymer blends with chain length difference[END_REF]. In good agreement with predictions (volume fraction of the middle-block between 0.60 and 0.66 and equal volume fractions of both end-blocks), this blend exhibited in the bulk a tetragonal morphology. The cylinders of I and P have the same size (34 nm) as schematized in table 3. In the thin film version of this blend, the tetragonal morphology transitioned to an intermediate double hexagonal structure in the early stage of the toluene vapor annealing, with a modification of its apparent composition (0.11:0.65:0.24). This composition is depleted in I compared to the initial one (0.17:0.66:0.17). This could be explained by the tendency of I to segregate to the surface and to form a wetting layer at the interface film/air [START_REF] Guliyeva | Transition Pathway Between Gyroid and Cylindrical Morphology in Linear Triblock Terpolymer Thin Films[END_REF][START_REF] Son | Highly Ordered Square Arrays from a Templated ABC Triblock Terpolymer[END_REF] due its lowest surface energy. The surface energies of I, S, and P were herein determined on homopolymer films using contact angle measurements of different liquids and were estimated to 32, 40 and 47 mJ.m -2 for I, S and P in agreement with the literature [START_REF] Son | Highly Ordered Square Arrays from a Templated ABC Triblock Terpolymer[END_REF][START_REF] Zhang | Orientation Control in Thin Films of a High-χ Block Copolymer with a Surface Active Embedded Neutral Layer[END_REF]. The I segregation at the interface film/air consequently induces an inner composition of the films with a lack of I.

We observed centered hexagons of I cylinders (surrounded by P cylinders) in this work, whereas Asai obtained centered hexagons of P cylinders (surrounded by I cylinders) for the linear triblock terpolymer X (I:S:P) with a similar composition (0.12:0.67:0.21) [START_REF] Asai | Asymmetric double tetragonal domain packing from ABC triblock terpolymer blends with chain length difference[END_REF]. To explain these discrepancies, the stabilities of these two double hexagonal morphologies (centered hexagons of P versus centered hexagons of I) must be considered. As S-P is higher than S-I (S-I=0.06; S-P= 0.1; I-P=0.2 [START_REF] Ahn | Gyroid structures at highly asymmetric volume fractions by blending of ABC triblock terpolymer and AB diblock copolymer[END_REF]), the system tends to limit the amount of P/S interface compared to the I/S interface and thus to increase the ratio (Ir) between the I/S interface area and P/S interface area. From the interfacial energy standpoint, the most favorable system corresponds to centered hexagon of P (highest value of Ir in the figure 4a in good accordance with the morphologies reported literature. [START_REF] Asai | Asymmetric double tetragonal domain packing from ABC triblock terpolymer blends with chain length difference[END_REF][START_REF] Izumi | A New Periodic Pattern with Five-Neighbored Domain Packing from ABC Triblock Terpolymer/B Homopolymer Blend[END_REF] Figure 4: a) interface area ratio between I/S and P/S interfaces (Ir) versus I and P volume ratio (Vr) b) I and P diameters ratio (Dr) versus I and P volume ratio (Vr) for the three different proposed morphologies on the right of each graph (I in yellow, S in grey and P in blue). These results are obtained using only geometric considerations. The triangle and star symbols represent the positions of the X/Y 85/15 and 60/40 respectively, calculated from the TEM data. The circle represent the situation deduced from the native composition of the ISP X (it is not displayed in part b) since the conformational entropy (no segment with different chain length) does not concern this system).

In our case, we obtained centered hexagons of I, which corresponds to less favorable system from the interfacial energy standpoint. This can be seen in Figure 4a since the Ir values corresponding to the centered hexagons of I are below the values corresponding to the centered hexagons of P (the red triangle and star points represent the positions of the X/Y 85/15 and 60/40 respectively, calculated from the TEM data). However, chains with different sizes are present in each cylindrical domain since the studied system is a blend of two block copolymers with different compositions. In the X/Y studied blends, we mixed ISP X (I 357 and P 625 repeating units) with ISP Y (I 1649 and P 72 repeating units). In this X/Y 85/15 blend, I domains contain both chains of 357 units (85 %) and chains of 1649 units (15%). Consequently, we suggest that the system tends to adopt I domains as large as possible because it favors the location of the longest chains in the largest domains, to keep their conformational entropy loss as small possible. The double hexagonal morphology with centered hexagon of gives larger diameter for the I domains than for P domains (highest value of ratio between the I and P diameters (Dr) as in Figure 4b. It can be suggested that this "size effect" overcomes the P/S interface minimization, favoring the formation of a double hexagonal morphology with centered hexagon of I. Very interestingly, these results suggest that the presence of a moderate amount of long I chains (15%) induces an inversion of morphology, from centered hexagons of P to centered hexagons of I. For comparison purpose, in the ISP X case [START_REF] Asai | Asymmetric double tetragonal domain packing from ABC triblock terpolymer blends with chain length difference[END_REF] with a composition of (0.12:0.67:0.21) and centered hexagons of P, I has 357 units and P 625 units: the shortest chains locates in the smallest domains of I and the longest ones in the biggest ones (P domains).

The diameter of I domains is half the diameter of P domains. In this case, the interfacial stability is in good accordance with entropic considerations since the size of the domains are consistent with the size of the chains.

For longer annealing time, the segregation of P at the interface substrate/film occurs [START_REF] Guliyeva | Transition Pathway Between Gyroid and Cylindrical Morphology in Linear Triblock Terpolymer Thin Films[END_REF], lowering the proportion of P in the inner film, which compensates for the depletion of I that previously occurred in the first stages of the annealing. P is also prone to migration (towards the substrate interface (the surface energy of the substrate being estimated to 46 mJ.m -2 [START_REF] Guliyeva | Transition Pathway Between Gyroid and Cylindrical Morphology in Linear Triblock Terpolymer Thin Films[END_REF]) but because I swells more, and faster than P2VP (as shown in SI-6), the formation of the wetting layer of I will occur first. The apparent composition value at that stage (extracted from the TEM) is now symmetric (0.17:0.68:0.15) and very close to the initial composition and drives the formation of the native tetragonal morphology. Note that the domains have similar sizes between hexagonal and tetragonal structures. This is in agreement with the fact that the size of the domains is imposed by the size of the blocks and not the volume fraction in the blend.

X/Y 60/40 blend

In the case of X/Y 60/40 blend, the same interpretation of the morphological evolution upon annealing can be given. The initial bulk volume (I:S:P) is (0.26:0.62:0.12). Despite a composition that would normally form a hexagonal pattern, a tetragonal morphology is formed in bulk, with interwoven tetragonal arrays of I and P cylinders, the domains of I being much larger (50 nm) than those of P (34 nm) [START_REF] Asai | Asymmetric double tetragonal domain packing from ABC triblock terpolymer blends with chain length difference[END_REF]. This is again due to the size of I and P chains. I domains contains 40% of chains of 1649 units whereas the maximal chain length in the P domains is only 625 units. In this case, the formation of a tetragonal pattern corresponds to a good balance between the limitation of the P/S interface (moderate value of Ir in During the first stage of annealing, I segregation at the surface occurs first, which in turn reduces the amount of I in the inner structure. However, this reduction lets the system with a composition (I:S:P) (0.21:0.66:0.13) depleted in I but still within the range of a tetragonal structure. Note that as previously, the I domains size (42 nm) is gouverned by the size of long chains of I, which in this case represents 40% of long chains.

Further solvent exposure led to P segregation at the interface film/substrate and enhanced proportion of I in the inner films projecting the system (0.23:0.66:0.11) outside the area of tetragonal structure, into the area of hexagonal one. Then again, the size of I domains is imposed by the chains length and this leads to centered hexagons of I surrounded by domains of P. Thin film stabilizes this structure although the composition is the same of the bulk which exhibits a tetragonal pattern. This phenomenon has been already observed for an ISP terpolymer which exhibits in bulk a gyroid pattern and in thin film evolves to a tetragonal morphology [START_REF] Ahn | Gyroid structures at highly asymmetric volume fractions by blending of ABC triblock terpolymer and AB diblock copolymer[END_REF].

In order to verify the central role of the length of the I chains in the formation of the morphology, we considered a blend of a ISP triblock and IS diblock with a (I:S:P) volume proportion of (0.20:0.59:0.21) in which both I and P chains are within the same range of repeating units number. This time, as show in SI-7, this blend led in thin film to a cylindrical morphology with centered hexagon (with a characteristic distance of 84 nm) of cylinders of P (diameter of 36 nm) surrounded by cylinders of I with a diameter of 27 nm leading to an (I:S:P) apparent composition (0.18:0.65:0.17). In this case, the interface area of P/S is less than the interface area of I/S (Interface ratio =1.5). The diameter ratio of the I and P domains is 0.75, compatible with the block lengths of I and P (59% with 96 units and 41% with 576 units for I chains; P domains contain chains of 513 units). This confirms the central role of the long chains of I in the inverted morphology observed previously. The presence of IS diblock allowing S to fill the space between the I domains is another favorable factor to observe this morphology.

Another point that has to be discussed is the shape of domains. It was demonstrated in previous works, that blending block copolymers exhibiting a bimodal distribution leads to non constant mean curvature (non-CMC) interface, e.g the domains are no more cylindrical but present flat interfaces [START_REF] Matsushita | Non-classical block-copolymer selfassembly resulting from a constrained location of chains and junctions[END_REF]. This is particularly visible for the square pattern of X/Y 60/40, I domains have clearly a square base as shown in figure 5a. The location of the blocks is proposed in figure 5b. This example is the most striking one in this study, but for all domains with a bimodal distribution of chains, the different locations of the blocks and the packing frustration impose the shape of the domains.

Conclusion

The morphology of blends of ISP linear block copolymer was investigated in thin film upon solvent vapor annealing. Unreported morphologies were obtained, with marked differences compared to the bulk morphology. A double hexagonal morphology was formed, with centered hexagons of I and surrounding hexagons of P, which is not predicted nor observed for neat block copolymer samples. In this thin film configuration, the formation of non-synchronized wetting layers of I and P leads to modification of the inner composition of the film, driving the observed transitions. Although the system must adopt the lowest P/S interface, our results

suggest that the presence of long chains (here the I chains) is playing a major role, favoring the formation of I domains with bigger size, in order to keep their conformation entropy loss as small as possible.

Figure 1 :

 1 Figure 1: schematic representations of the different ordered structures with cylindrical domains for linear, non frustrated ABC triblock terpolymer (A-block, B-block, C-block respectively in yellow, grey and blue). a) Hexagonal packing of core (C-block) shell (B-block) cylinders in a matrix of A; b) Hexagonal packing of (C-block) cylinders in a matrix of AB; c) Tetragonal packing of A-block and C-block cylinders in a matrix of B; d) Double hexagonal packing of A-block and C-block cylinders in a matrix of B. For a) b) and d), another version of this structure exists by replacing A by C and C by A. For c), four unit cells are shown for clarity.

Table 2 :

 2 Blending conditions, volume fraction and blocks length ratios Solutions of blends were prepared by dissolving polymers in solvent (1,2-dichloroethane) and were stirred for at least 2 hours. Si(100) substrates (kindly provided by STMicroelectronics Co) of 10*10 mm² were cleaned by sonication in tetrahydrofuran (THF), ethanol and acetone for 10 minutes each. Then the solutions were deposited by spin-coating on Si substrates at 3000 rpm for 30 seconds. The film thicknesses, measured by visible spectral reflectometry using F20 by Filmetrics, ranged from 100 to 120 nm. The accuracy of the determination of the thickness by visible spectral reflectometry was verified on polymer films using Atomic Force Microscopy (AFM) measurement of the depth of a scratch line.

Figure 2 :

 2 Figure 2: a), b), c), d) X/Y 85/15 blend thin films after SVA in toluene for 5h leading to a hexagonal pattern; e), f), g), h) for 10h leading to a square pattern; with a), e) AFM images and b) f) TEM top-view images after staining by OsO4 and I2. The schematic representations of the unit cells with I in black, P in dark grey and S in light grey are superimposed on the TEM images; c) g) section view TEM images; d), h) in-plane GISAXS profiles after film staining by I2

Figure 3 :

 3 Figure 3: a), b), c) X/Y 60/40 blend thin films after SVA in toluene for 10 h leading to a square pattern; d), e), f) for 15h leading to a hexagonal pattern; a), d) AFM images; b), e) TEM top-view images after staining of I and P (the schematic representations of the unit cells with I in black, P in dark grey and S in light grey are superimposed on the TEM images:for clarity, the shape of the domains are shown circular whereas evidence of non-CMC interfaces are seen on the TEM images.); c), f) in-plane GISAXS profiles after film staining by I2.

  Fig 4a) and formation of I domains large enough to accommodate the longer I chains (intermediate values of Dr in Fig 4b).

Figure 5 :

 5 Figure 5: a) 150*150 nm TEM image of square pattern for X/Y 60/40 thin film (I in black, S in grey and P in dark grey) and schematic representation

Acknowledgements

We gratefully acknowledge the financial support provided to the PIVOTS project by the Région Centre -Val de Loire (ARD 2020 program and CPER 2015 -2020) and the French Ministry of Higher Education and Research (CPER 2015 -2020 and public service subsidy to CNRS and Université d'Orléans). Frederic Mahut is deeply acknowledged for his experimental contribution. Pascal Andreazza and Marie-Pierre Faugère are thanked for fruitful discussions and contribution to the GISAXS experiments.

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time due to technical or time limitations.