Lucas Quesada

Dorian Verdel

Olivier Bruneau

Bastien Berret

Michel-Ange Amorim

Nicolas Vignais

EMG-to-torque models for exoskeleton assistance: a framework for the evaluation of in situ calibration

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

EMG-to-torque models for exoskeleton assistance: a framework for the evaluation of in situ calibration Lucas Quesada , Dorian Verdel , Olivier Bruneau , Bastien Berret , Michel-Ange Amorim , Nicolas Vignais

Abstract-In the field of robotic exoskeleton control, it is critical to accurately predict the intention of the user. While surface electromyography (EMG) holds the potential for such precision, current limitations arise from the absence of robust EMG-to-torque model calibration procedures and a universally accepted model. This paper introduces a practical framework for calibrating and evaluating EMG-to-torque models, accompanied by a novel nonlinear model. The framework includes an in situ procedure that involves generating calibration trajectories and subsequently evaluating them using standardized criteria. A comprehensive assessment on a dataset with 17 participants, encompassing single-joint and multi-joint conditions, suggests that the novel model outperforms the others in terms of accuracy while conserving computational efficiency. This contribution introduces an efficient model and establishes a versatile framework for EMG-to-torque model calibration and evaluation, complemented by a dataset made available. This further lays the groundwork for future advancements in EMG-based exoskeleton control and human intent detection.

Index Terms-Exoskeleton, Electromyography, EMG-to-torque models, Intention detection, Human-machine interaction

I. INTRODUCTION

A CTIVE exoskeletons are a major potential tool to support neurorehabilitation protocols and prevent the appearance of musculoskeletal disorders (MSDs) at work. Concerning rehabilitation, they can be used to improve gait and upper limb control during daily activities for those who have suffered a stroke or a spinal cord injury [START_REF] Federici | The effectiveness of powered, active lower limb exoskeletons in neurorehabilitation: A systematic review[END_REF]- [START_REF] Proietti | Upperlimb robotic exoskeletons for neurorehabilitation: A review on control strategies[END_REF]. Exoskeletons can also be used to decrease the biomechanical load on the body [START_REF] Moeller | Effects of upper-limb exoskeletons designed for use in the working environment-a literature review[END_REF], thus contributing to preventing the onset of MSDs when the workspace cannot be adapted to improve ergonomics [START_REF] Moore | Practical demonstrations of ergonomic principles[END_REF], although more evidence is needed [START_REF] Theurel | Occupational exoskeletons: Overview of their benefits and limitations in preventing work-related musculoskeletal disorders[END_REF], [START_REF] Mcfarland | Considerations for industrial use: A systematic review of the impact of active and passive upper limb exoskeletons on physical exposures[END_REF]. In such situations, integrating the human intention into the exoskeleton's control loop is necessary to ensure safe, appropriate, and intuitive assistance [START_REF] Bi | A review on emg-based motor intention prediction of continuous human upper limb motion for humanrobot collaboration[END_REF].

Human intention detection (HID) involves continuously estimating the motion of one or more human joints. There are several approaches to this aim. Kinematics can be used to anticipate the trajectory of a movement based on prior Manuscript received xxx; revised xxx. L. Quesada, D. Verdel, B. Berret, M-A. Amorim and N. Vignais were with the CIAMS, Université Paris-Saclay, 91405 Orsay, France and with the CIAMS, Université d'Orléans, Orléans, France. Corresponding e-mail: lucas.quesada@ens-paris-saclay.fr.

L. Quesada, D. Verdel and O. Bruneau were with the LURPA, ENS Paris-Saclay, 91190 Gif-sur-Yvette, France. D. Verdel was with Human Robotics Group, Department of Bioengineering, Imperial College London, W12 0BZ London, United-Kingdom.

This work is supported by the French National Agency for Research (grant ANR-19-CE33-0009) and the École Normale Supérieure Paris-Saclay. knowledge of typical trajectories. These techniques, often based on probabilistic models [START_REF] Mainprice | Human-robot collaborative manipulation planning using early prediction of human motion[END_REF], [START_REF] Mainprice | Predicting human reaching motion in collaborative tasks using inverse optimal control and iterative re-planning[END_REF], have already been used to assist with an active exoskeleton [START_REF] Jamsek | Predictive exoskeleton control for arm-motion augmentation based on probabilistic movement primitives combined with a flow controller[END_REF]. Another method is to employ a bio-electric signal and a model to estimate the movement. Data sources such as mechanomyography (MMG), force myography (FMG) [START_REF] Xiao | A review of force myography research and development[END_REF], electromyography (EMG), or a fusion of signals [START_REF] Zhang | Evaluation of a fused sonomyography and electromyography-based control on a cable-driven ankle exoskeleton[END_REF], [START_REF] Hu | Development of sensory-motor fusion-based manipulation and grasping control for a robotic hand-eye system[END_REF] may be processed to control an exoskeleton. Thanks to the electromechanical delay, i.e., the latency between EMG signal and muscle force production [START_REF] Cavanagh | Electromechanical delay in human skeletal muscle under concentric and eccentric contractions[END_REF], human intention detection can be predictive: the movement estimation can occur before the onset of the movement itself. This potential predictive capability can be advantageously used in real-time assistive systems, making EMG signals an appealing choice for human/machine interactions [START_REF] Kiguchi | An EMG-based control for an upper-limb power-assist exoskeleton robot[END_REF]- [START_REF] Huang | Development and continuous control of an intelligent upper-limb neuroprosthesis for reach and grasp motions using biological signals[END_REF].

Several approaches can be employed to predict intent based on EMG signals, but two general categories can be defined: i) categorical classification and ii) continuous regression. Movement classification tends to use convolutional neural networks [START_REF] Jiang | Shoulder muscle activation pattern recognition based on semg and machine learning algorithms[END_REF], [START_REF] Treussart | Controlling an exoskeleton with EMG signal to assist load carrying: A personalized calibration[END_REF], support vector machine [START_REF] Tsai | A comparison of upper-limb motion pattern recognition using emg signals during dynamic and isometric muscle contractions[END_REF], [START_REF] Phinyomark | Feature extraction and selection for myoelectric control based on wearable EMG sensors[END_REF], gaussian mixture models [START_REF] Trigili | Detection of movement onset using emg signals for upper-limb exoskeletons in reaching tasks[END_REF] and other various techniques [START_REF] Jaramillo-Yánez | Realtime hand gesture recognition using surface electromyography and machine learning: A systematic literature review[END_REF] linked to pattern recognition. This type of HID allows to predict which type of movement is performed (e.g., elbow flexion/extension) but has to be trained on a predetermined set, preventing its generalization. On the other hand, continuous movement regression aims to predict a continuous feature of the movement, such as joint angle or torque. This is generally achieved through a mathematical model linking EMG signals to the predicted feature. For example, neuromusculoskeletal models [START_REF] Lotti | Adaptive model-based myoelectric control for a soft wearable arm exosuit: A new generation of wearable robot control[END_REF]- [START_REF] Pan | Comparing EMG-based humanmachine interfaces for estimating continuous, coordinated movements[END_REF], simple linear regressions [START_REF] Kiguchi | An EMG-based control for an upper-limb power-assist exoskeleton robot[END_REF], [START_REF] Pan | Comparing EMG-based humanmachine interfaces for estimating continuous, coordinated movements[END_REF]- [START_REF] Li | An enhanced EMG-driven musculoskeletal model based on non-negative matrix factorization[END_REF], various synergybased [START_REF] Camardella | Towards online myoelectric control based on muscle synergiesto-force mapping for robotic applications[END_REF], [START_REF] Li | An enhanced EMG-driven musculoskeletal model based on non-negative matrix factorization[END_REF]- [START_REF] Chen | Muscle-synergies-based neuromuscular control for motion learning and generalization of a musculoskeletal system[END_REF] and other custom models [START_REF] Li | semg-based joint force control for an upper-limb power-assist exoskeleton robot[END_REF]- [START_REF] Ameri | Support vector regression for improved real-time, simultaneous myoelectric control[END_REF] currently coexist. However, comparing models in the literature is difficult due to methodological differences in evaluation. To provide accurate predictions, continuous regression typically requires first calibrating the model to fit its internal parameters to an individual [START_REF] Buongiorno | A linear approach to optimize an EMG-driven neuromusculoskeletal model for movement intention detection in myo-control: A case study on shoulder and elbow joints[END_REF]. Some studies propose to estimate joint torques from EMG signals during isometric tasks [START_REF] Buongiorno | A linear approach to optimize an EMG-driven neuromusculoskeletal model for movement intention detection in myo-control: A case study on shoulder and elbow joints[END_REF], [START_REF] Camardella | Towards online myoelectric control based on muscle synergiesto-force mapping for robotic applications[END_REF], [START_REF] Ziai | Comparison of regression models for estimation of isometric wrist joint torques using surface electromyography[END_REF], [START_REF] Dai | Comparison of constantposture force-varying EMG-force dynamic models about the elbow[END_REF], [START_REF] Huang | Divide-and-conquer muscle synergies: A new feature space decomposition approach for simultaneous multifunction myoelectric control[END_REF], and specific or periodic dynamic tasks [START_REF] Grimmelsmann | sEMG-based prediction of human forearm movements utilizing a biomechanical model based on individual anatomical/ physiological measures and a reduced set of optimization parameters[END_REF], [START_REF] Pan | Comparing EMG-based humanmachine interfaces for estimating continuous, coordinated movements[END_REF], [START_REF] Li | An enhanced EMG-driven musculoskeletal model based on non-negative matrix factorization[END_REF], [START_REF] Wu | Adaptive cooperative control of a soft elbow rehabilitation exoskeleton based on improved joint torque estimation[END_REF], [START_REF] Peternel | Adaptive control of exoskeleton robots for periodic assistive behaviours based on EMG feedback minimisation[END_REF]. However, similarly to categorical classification, a realistic general assistive system cannot be restricted to isometric or predetermined tasks. Individualizing a calibration procedure might also involve a long computation time or specialized equipment, reducing its practicality for future commercial or clinical applications. Therefore, developing a model that allows short in situ calibration procedures capable of performing HID for arbitrary movements is crucial. To validate and compare models, the data on which the model is evaluated must be distinct from the data used for individual- ized calibration (e.g., different movements, efforts, etc.). Apart from very few studies using this type of cross-validation [START_REF] Ziai | Comparison of regression models for estimation of isometric wrist joint torques using surface electromyography[END_REF], most do not exploit this aspect, thus increasing the risk of overfitting models and misinterpreting their performance.

In the present paper, we address these concerns by: i) introducing an in situ calibration framework enabling a generic calibration of upper limb EMG-to-torque models, ii) applying this framework to elbow and shoulder flexion/extension concurrently and separately, iii) evaluating and comparing the performances obtained by four models, one of which is introduced. This paper is structured as follows: first, we provide an overview of the experimental protocol that produced our dataset, including the exoskeleton control law, arbitrary task generation, and EMG placement; next, we explain the data processing and model evaluation procedures; then, we describe the four EMG-to-torque models we evaluated; and finally, we present our results, along with their limitations and potential future directions.

II. METHODS

A. Participants

B. Material 1) Exoskeleton:

The ABLE upper limb active exoskeleton [START_REF] Garrec | ABLE, an innovative transparent exoskeleton for the upper-limb[END_REF] was used to impose a viscous resistive force on the participant's wrist to induce torques in its elbow and shoulder. Its movement was limited to a parasagittal plane, only allowing flexion and extension of the elbow and the shoulder. Participants were attached to the exoskeleton through a previously developed mechanical interface [START_REF] Verdel | Influence of the physical interface on the quality of human-exoskeleton interaction[END_REF] composed of an orthosis fixed on a force/torque (FT) sensor (1010 Digital FT, ATI, Apex, USA) itself attached to a slider on the exoskeleton's forearm. This mechanical interface includes a passive rotation between the exoskeleton and the human limb, which reduces unwanted hyperstatic constraints and increases comfort while permitting the measurement of interaction forces [START_REF] Verdel | A trade-off between complexity and interaction quality for upper limb exoskeleton interfaces[END_REF].

2) Electromyography: Electromyographic signals were recorded using wireless MiniWave sensors with a WavePlus receiver (Cometa, Bareggio, Italy). The receiver was connected via USB to a computer, and the acquisition was performed through the Qualisys Track Manager (QTM) software (Qualisys, Göteborg, Sweden). Muscle head recording locations were identified following SENIAM recommendations [START_REF] Hermens | Development of recommendations for SEMG sensors and sensor placement procedures[END_REF]. The skin was locally shaved, cleaned with alcohol pads, and dried before placing the Ag/AgCl EMG electrodes (F3010, Fiab, Firenze, Italy). The muscle set included the long and short head of the biceps (BICShort, BICLong), long, lateral, and median heads of the triceps (TRILong, TRILat, TRIMed), brachioradialis (BRD), brachialis (BRA), posterior, medial, and anterior heads of the deltoids (DELTPost, DELTMed, DELTAnt), clavicular head of pectoralis major (PECT), and latissimus dorsi (LATI).

3) Motion capture: Kinematic data of the participant and the exoskeleton were captured using a total of 10 Oqus 500+ motion capture cameras (Qualisys, Göteborg, Sweden). Eight 10 mm markers were placed on anatomical landmarks, including the acromion, seventh cervical vertebra, sternal end of the clavicle, styloid processes of the ulna and radius, lateral and medial epicondyles and distal end of the first metacarpus. Additionally, two markers were placed on the anterior and posterior sides of the arm, specifically at the midpoint of the upper arm. Additional markers were placed on the exoskeleton and orthosis (see Fig. 1) to track their movement for later data synchronization and frame transformation purposes. Motion capture data was recorded in synchronization with EMG using the QTM software.

C. Procedure 1) MVC and anthropometric measurements: Participants first completed maximum voluntary contraction (MVC) tasks for flexion and extension of the elbow and shoulder. For the elbow, they placed their arm vertically against their body, resting their wrist on a horizontal bar set at the same height as their elbow joint. To perform flexion and extension MVCs, they adjusted the position of their wrist either above or below the bar and pushed against it. The same process was applied to the shoulder joint by moving the bar to the shoulder level and asking the participant to extend their elbow fully. Each task was repeated twice and lasted for 3 seconds. Following the MVC trials, after placing markers on specific anatomical landmarks as described in Section II-B3, participants were instructed to stand still within the motion capture area to capture a frame of a static pose.

2) Motor tasks: The experiments were based on a trajectory tracking task and divided into two sessions. During the first session, participants tracked a trajectory using only elbow Fig. 2: Illustration of the different experimental conditions. a) During the single-joint task, the movement was limited to elbow flexion/extension. To limit hyperstatic constraints in the human-exoskeleton kinematic chain and increase comfort, a passive translation along X s (slider) and a passive rotation around Y s were allowed by the physical interface [START_REF] Verdel | Influence of the physical interface on the quality of human-exoskeleton interaction[END_REF]. Both shoulder and elbow flexion/extension were involved during the multi-joint task. Participants were only attached to the exoskeleton at the wrist level, which implied that the slider needed to be fixed to transmit efforts from the human to the exoskeleton. The Z s axis represents the sensor frame, while the Z o axis represents the orthosis frame. It is necessary to consider this angular misalignment to compute accurate inverse dynamics. b) Participants were positioned in front of a projected screen during the tasks. The trajectory that needed to be tracked was visualized as white dots, while a yellow dot represented the participant's current position. As participants successfully touched the white dots, they would turn green, indicating a successful catch. The displayed score corresponded to the number of dots participants successfully caught. In the single-joint task, the height of the yellow dot representing the participant's position was determined by the angular position of their elbow. During the multi-joint task, participants were turned 90 degrees, and the Cartesian coordinates of their wrist determined the position of the yellow dot. Two screen dimensions were effectively utilized for spatial representation, where white dots represented the future trajectory, and red dots represented the set of accessible positions during the current period (500 ms). c) In both cases, B-splines are being used to produce the trajectories with control points (gray dots) generated into the light-gray area that contains the reachable workspace. Trajectory examples are shown as black lines. The section of the trajectory currently displayed on the screen of the b panel is dashed. flexion/extension (i.e., single-joint). During the second session, participants tracked a trajectory using elbow and shoulder flexion/extension (i.e., multi-joint). Two different kinematic arrangements of the exoskeleton were employed, depending on the condition (see Fig. 2.a). The first setup enabled only the elbow to rotate while the wrist was secured to a slider. The second configuration allowed both the elbow and the shoulder to rotate. As previously stated in the material section, these two configurations avoided hyperstaticity between the subject and the exoskeleton, reducing unwanted interaction efforts. The trajectory was projected on a screen positioned frontally for the single-joint task and sagittally for the multijoint task (see Fig. 2.b). Visual feedback included a yellow dot representing the participant's current position, synchronized with white dots depicting the desired trajectory scrolling across the screen. For multi-joint tasks, red dots indicated specific positions participants were expected to reach at a given time. Participants earned a point each time the yellow dot coincided with a target dot, displaying an interactive score to enhance engagement.

a) Exoskeleton Control: The exoskeleton was controlled to impose a viscous resistance at the elbow and shoulder joints, increasing the muscle effort induced by the task. This allowed to extend the range of measured muscle activities and, therefore, the range of validity of the resulting calibration. Furthermore, to reduce the effort asymmetry generated by the exoskeleton's weight between upward and downward movements, it compensated for its weight based on a former identification [START_REF] Verdel | An identification-based method improving the transparency of a robotic upper limb exoskeleton[END_REF]. For the single-joint condition, the resulting exoskeleton control torque τ e was computed as follows:

τ e = g e (q e ) -µ c qe (1)
with µ c representing the viscous coefficient, and qe denoting the angular velocity of the exoskeleton's elbow joint. For the multi-joint condition, the same gravity compensation and viscous resistance scheme was applied, yielding control torques computed from task space coordinates using the following equation: τ e = g e (q e ) -µ c J e (q e ) ṗh

where J e (q e ) and q e respectively represent the Jacobian matrix and the angular positions of the exoskeleton and ṗh the velocity vector of the hand expressed in the task space.

In both equations, g e or g e denotes the gravity model of the exoskeleton.

b) Trajectory generation: The target trajectories were generated using random b-splines (see Fig. 2.c). A set of random control points was generated in time and space to create a candidate trajectory. Among the generated trajectories, the one maximizing the mechanical work while covering the whole reachable workspace was chosen. This approach to random trajectory generation can produce a broad spectrum of trajectories, ranging from flat to very curved. To effectively scan the workspace, two criteria were established to select trajectories that displayed a wide range of movements. These criteria were the amount of mechanical work and the evenness of its distribution in the workspace. For each component i of the trajectory (e.g., i ∈ [Y, Z] for multi-joint condition), according to equations 1 and 2, a substitute to the mechanical work created by the viscous field can be computed as

W i = µ c T 0 ṡ2 i (t)dt (3)
where T was the duration of the trajectory, and s corresponds to q e and p h for single and multi-joint conditions, respectively. The distribution of mechanical work within the movement can be assessed by analyzing partial work, which quantifies the amount of work dedicated to movement in either the positive or negative direction:

w + i = µ c T 0 ⌊ṡ i (t)⌋ 2 + dt w - i = µ c T 0 ⌊ṡ i (t)⌋ 2 -dt (4)
where w + i and w - i respectively denote the work of the movement in the upward (i.e., positive and denoted ⌊ṡ i (t)⌋ + ) and downward (i.e., negative and denoted ⌊ṡ i (t)⌋ -) directions. The normalized work balance criterion is then expressed as the mean of relative partial work contributions. With the total work as the sum of individual components work W = D i=1 W i , the criterion is thus expressed as:

C WB = 1 2D D i=1 2 - |w + i -W/(2D)| W/(2D) - |w - i -W/(2D)| W/(2D) (5
) here, D represents the dimension of the task space (e.g., D = 1, 2 for single and multi-joint conditions, respectively). For example, in multi-joint condition, forward, backward, upward, and downward movements should all account for a quarter of the total work. With equation 5, trajectories with an evenly distributed work throughout space maximize C WB . Before executing the task, a hundred trajectories were generated, and their respective scores were computed as

C = W × C WB (6)
Finally, the trajectory with the highest score is selected and used for the task.

3) Trials: Before initiating the main trials, preliminary test tasks were conducted to iteratively set the viscous coefficient for the robot controller. In these initial trials, a baseline viscous coefficient was employed, and participants were requested to rate their perceived effort on a subjective Borg scale. Based on the participants' ratings, the viscous coefficient was then manually adjusted to achieve a moderate effort. Then, twenty trials were performed, each lasting thirty seconds and featuring a different random trajectory. Ten trials were conducted for each condition: single-joint and multi-joint. The trial procedure was as follows: a random trajectory was generated, and participants were instructed to stand ready. A countdown with an accompanying sound cue was initiated when the trial began. A score, displayed on the top right of the screen, kept track of the number of successfully caught dots, encouraging participants to track the trajectory accurately. A one-minute rest period was observed between each trial to prevent muscle fatigue. D. Data processing 1) Data synchronization: EMG and motion capture data were synchronized using the QTM software. However, the exoskeleton kinematic data obtained from internal encoders and force/torque measurements were recorded separately, necessitating temporal synchronization with the EMG and motion capture data. To address this, the exoskeleton's positions were measured using its internal encoders and the motion capture system. The alignment process consisted of determining the optimal temporal shift that minimized the root mean square error between the encoder-informed and motioncapture-informed positions of the exoskeleton arm. Once this optimal shift was determined, it was applied to the force/torque data to ensure proper alignment across all data sources.

2) EMG feature extraction: EMG signals were processed with successive filtering. First, a fourth-order Butterworth 20-450Hz bandpass filter was applied, and the signal was rectified. Second, the EMG envelope was extracted using a 3Hz fourthorder Butterworth low-pass filter [START_REF] Potvin | Less is more: high pass filtering, to remove up to 99% of the surface EMG signal power, improves EMG-based biceps brachii muscle force estimates[END_REF]. In this study, all filters were bidirectionally applied to mitigate latency effects. Finally, EMG signals were normalized using MVC data.

3) Anthropometrics: Data collected during the static motion capture phase is used to scale an upper limb model [START_REF] Holzbaur | A model of the upper extremity for simulating musculoskeletal surgery and analyzing neuromuscular control[END_REF] using OpenSim's scaling tool [START_REF] Seth | OpenSim: Simulating musculoskeletal dynamics and neuromuscular control to study human and animal movement[END_REF]. This process involves adjusting the dimensions of the model's bones to match the marker spacing on anatomical landmarks. By adapting the bone dimensions, the model accurately represents the proportions of the individual's limb. Furthermore, the limb mass is computed by considering the ratio of the new bone dimensions to the original dimensions of Holzbaur's model. This ensures that the model incorporates the appropriate mass distribution (mass matrix) to simulate the upper limb dynamics accurately.

4) Kinematics:

Although human kinematics could be deduced from the exoskeleton's internal encoders into infield situations, here it was instead computed from motion capture data to improve accuracy and compensate for a known misalignment between the human and exoskeleton joint axes [START_REF] Verdel | Human weight compensation with a backdrivable upper-limb exoskeleton: Identification and control[END_REF]. Positions of the reflective markers were extracted and labeled using the automatic identification models (AIM) feature provided by the QTM software. The methods used for inverse kinematics varied depending on whether the trials involved single or multiple joints. In single-joint trials, which involved strictly only elbow flexion/extension, the angle of the elbow was directly computed using the marker coordinates. Conversely, in multi-joint trials, although the main focus was on shoulder and elbow flexions and extensions, it was essential to consider the potential movements of the entire upper body to take into account its full dynamics. Consequently, a comprehensive inverse kinematic analysis was conducted using OpenSim. Finally, kinematic data were filtered (4 th order 3Hz low-pass Butterworth) to align the harmonic content of kinematic data to the EMG envelope and ease the extraction of time derivatives using numerical differentiation. Joint angles, velocities and accelerations are referred to as q h , qh and qh .

5) Human dynamics: Human dynamics were automatically computed from kinematics and anthropometric data previously introduced using Opensim. Classically, the motion of the human upper limb is described by the following dynamics, for J joints:

τ τ τ h = M(q h )q h + c(q h , qh ) + g g g h (q h ) + τ τ τ i (7)
where, τ τ τ h ∈ R J represents the torque exerted at the human's joints, M ∈ R J×J denotes the mass matrix, c ∈ R J corresponds to the vector of Coriolis and centrifugal forces, g g g h ∈ R J signifies the vector of gravitational torques, and τ τ τ i represents the vector of human-exoskeleton interaction torques. By considering these factors, the equation provides a comprehensive representation of the complex dynamics involved in the motion of the human upper limb. Interaction torques are computed by first transforming the FT data from its original frame to the forearm's frame and using the upper limb's jacobian matrix J h (q h ) so that

τ τ τ i = J T h (q h )f s + τ τ τ s (8)
where f s and τ τ τ s are the forces and torques measured from the sensor underneath the orthosis. 6) Resampling: All signals were resampled at a rate of 30Hz to reduce the number of samples and speed up the calibration of the models. All signals being 3Hz low-pass filtered, no information was lost.

E. Analysis

As the present study aims to evaluate and compare EMGto-torque models, it is necessary to properly define the criteria for selecting one model over another. In particular, for future implementations, the model accuracy and its computational cost (i.e., the calibration and evaluation durations) are essential performance criteria. The specific criteria to quantify these aspects are detailed below.

1) Cross validation: It is more conservative to provide different calibration and evaluation datasets to mitigate the risk of evaluating overfitted models. To that end, a K-fold approach was employed. This partitioning consists of dividing the dataset in k folds and then performing k iterations of training/evaluation with k -1 fold in the training set and one fold in the evaluation set [START_REF] Refaeilzadeh | Cross-validation[END_REF]. In this research, a two-fold division approach was taken due to the limited data (10 trials per condition) and after testing various values of k. For each participant and condition (single and multi-joint), the ten 30 s trials were randomly divided into two folds, each containing five trials. This procedure resulted in two calibration and evaluation scores for each model, condition, and participant.

2) Performance metric: a) Joint torque estimation: The accuracy of the models was evaluated using two metrics: the normalized root mean square error (NRMSE) and Pearson's coefficient of correlation (R). These metrics provide insights into the performance of calibrated models when estimating a joint torque in terms of data distance and error trends. The NRMSE is defined as follows

NRMSE j = 1 N i=N i=0 (τ h,j (t i ) -τh,j (t i )) 2 max(τ h,j ) -min(τ h,j ) (9)
where N is the number of samples in the fold and t i is the i th time sample. The reference torque τ h,j for the j th joint is calculated from equation 7, while τh,j is an estimated torque obtained from the tested EMG-to-torque model. The NRMSE is normalized with respect to the amplitude of the reference torque, obtained using max(τ h,j ) and min(τ h,j ). Similarly, Pearson's coefficient of correlation (R j ) is computed as

R j = cov(τ h,j , τh,j ) σ τ h,j σ τh,j (10)
where cov(•, •) denotes the covariance, and σ τ h,j represents the standard deviation. The NRMSE and R values were compared during the cross-validation process to evaluate and compare the performance of the models. b) Computation efficiency: While EMG-to-torque model accuracy is undoubtedly the most critical feature for exoskeleton control, it is also crucial to be as computationally efficient as possible. Indeed, for future applications, it is necessary to induce short calibration times, which will influence the usability of the control on a daily basis with different users, and quick evaluation times, which will impact the controller's responsiveness. As such, during cross-validation, calibration and evaluation durations were measured. The calibration duration was measured for each k-fold iteration across conditions and participants. The estimation duration was measured similarly and then divided by the number of samples to evaluate a feasible control loop period in a real-time application.

F. Statistical analysis

The 2-fold partitioning and validation process resulted in two values for each criterion, which were then averaged, leaving one value per subject and model as data. R scores were normalized using the Fisher transform. The data was initially evaluated for its sphericity using Mauchly's test. Subsequently, a repeated measures analysis of variance (rm-ANOVA) was conducted with the models as the within-subjects factor, and a Greenhouse-Geisser correction was applied if the sphericity was violated. Model-to-model post-hoc comparisons were performed with a paired sample t-test and a Holm-Bonferroni correction. In the results, ± corresponds to the standard deviation (SD). Datapoints with a Z-score over 3 were considered to be outliers and not included in the statistical analysis.

III. EMG-TO-TORQUE MODELS

The following section describes models using K muscles and J joints. Data matrices contain N samples.

A. Linear mapping model

The multivariate linear regression (MVLR) model is a straightforward formulation used to compute joint torques by linearly combining muscle excitations [START_REF] Camardella | Towards online myoelectric control based on muscle synergiesto-force mapping for robotic applications[END_REF]. This is achieved with a mapping matrix, denoted as H ℓ ∈ R J×K , which establishes the link between the estimated torques τ τ τ h (t) ∈ R J and the muscle excitations m(t) ∈ R K + . This relationship can be expressed as

τ τ τ h (t) = H ℓ • m(t) (11)
The model is calibrated by first defining the data matrices T ∈ R J×N and M ∈ R K×N + , which contain torque and muscle excitation samples from the training dataset, respectively. The calibration process involves finding the mapping matrix H ℓ that satisfies the equation:

T = H ℓ • M (12)
To achieve this, the mvregress function from Matlab (Math-Works, Natick, MA, US) is employed on T and M to determine the optimal values for H ℓ .

B. Neuromusculoskeletal model

Neuromusculoskeletal (NMS) models are a combination of a neuromuscular Hill-type model [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF] and a musculoskeletal representation of the body. A previously developed [START_REF] Holzbaur | A model of the upper extremity for simulating musculoskeletal surgery and analyzing neuromuscular control[END_REF] and benchmarked [START_REF] Saul | Benchmarking of dynamic simulation predictions in two software platforms using an upper limb musculoskeletal model[END_REF] NMS model implemented on the OpenSim simulation software [START_REF] Seth | OpenSim: Simulating musculoskeletal dynamics and neuromuscular control to study human and animal movement[END_REF] was used in this study. Muscle activations are first derived from excitations via a nonlinear activation function [START_REF] Buchanan | Neuromusculoskeletal modeling: Estimation of muscle forces and joint moments and movements from measurements of neural command[END_REF] as follows

a k (t) = e A k m k (t) -1 e A k -1 (13)
where A k , a k (t), and m k (t) are, respectively, the shape factor, activation and excitation of the k th muscle. For a given muscle, the force can then be expressed as follows (see also [START_REF] Millard | Flexing computational muscle: Modeling and simulation of musculotendon dynamics[END_REF]),

F M = F M iso a(t)f L ( lM )f V (ṽ M ) + f P E ( lM ) cos α (14)
where F M iso denotes the maximal isometric force, while lM and ṽM represent the normalized fiber length and velocity, respectively. The function f L ( lM ) characterizes the forcelength relationship, describing how a muscle fiber's maximum active force varies with its length. The normalized fiber length is given by lM = l M /l M 0 , where l M 0 corresponds to the optimal fiber length at which f L ( lM ) reaches its maximum value (i.e., lM = 1). Similarly, f V (ṽ M ) describes the forcevelocity relationship, reflecting the changes in active force generation of the muscle with fiber velocity (i.e., f V (0) = 1 for isometric contractions), and f P E ( lM ) accounts for the passive elastic behavior of the fiber. Lastly, α denotes the pennation angle, which quantifies the angle between the axis of muscle force and the orientation of its fibers. This model incorporates a serial elastic element representing the tendon to ensure equilibrium between the muscle fiber and tendon forces and compute their lengths. This is captured by ensuring the following constraint during forward dynamics:

F M -F M iso f T ( lT ) = 0 (15)
where lT represents the normalized tendon length, and f T ( lT ) describes the force-length relationship of the tendon's passive elastic behavior. The normalized tendon length is given by lT = l T /l T 0 , where l T 0 corresponds to the tendon slack length, such that f T ( lT ) = 0 when lT < 1. Equation 15 is solved at each time step during forward estimation of muscle force. Once the muscle force is obtained, the resulting estimated joint torques can then be derived as

τh,j (t) = k=K k=1 F M k (t) × r k→j (t) (16)
where r k→j (t) is the moment arm of k th muscle on the j th joint. During estimation and calibration, both the muscle force and the moment arm [START_REF] Sherman | What is a moment arm? calculating muscle effectiveness in biomechanical models using generalized coordinates[END_REF] are computed via OpenSim. When a muscle is not acting on a joint (i.e., brachioradialis on shoulder flexion), r k→j is null.

The model is calibrated by first scaling the musculoskeletal component using anthropometric data from motion capture, adjusting the length and width of each limb. Then, the muscle parameters are optimized using a single objective genetic algorithm [START_REF] Buongiorno | A linear approach to optimize an EMG-driven neuromusculoskeletal model for movement intention detection in myo-control: A case study on shoulder and elbow joints[END_REF], [START_REF] Mohammadi | Openga, a c++ genetic algorithm library[END_REF] with a stalling criterion of 20 generations, 16 chromosomes per generation, and a maximum of 1000 generations. Crossover fraction and mutation rate were determined with a parametric study showing that a value of 0.7 for each provided the best calibration. The cost function is defined as the quadratic mean of the estimation mean square error of each joint:

C(τ h , τh ) = j=J j=1 1 N n=N n=0 (τ j h (t n ) -τ j h (t n )) 2 (17)
Five parameters are optimized for each muscle: maximal isometric force F M iso , optimal fiber length l M 0 , tendon slack length l T 0 , pennation angle at optimal fiber length α and shape factor A.

C. Synergy-based model

Synergies are described as a coherent activation, in space or time, of a group of muscles [START_REF] Avella | Combinations of muscle synergies in the construction of a natural motor behavior[END_REF]. A variety of synergy decompositions, including spatial, temporal, spatiotemporal, or space-by-time, can be extracted from a set of muscle excitations depending on the specific analysis or application [START_REF] Delis | A unifying model of concurrent spatial and temporal modularity in muscle activity[END_REF].

However, since our dataset consists of randomly generated trajectories without specific gait or periodic movements, we do not anticipate the emergence of temporal synergies. Therefore, we have decided to focus on a spatial synergy model for our analysis. Considering the number of spatial synergies as S, the muscle excitation data matrix M ∈ R K×N + can be factorized into two nonnegative matrices as follows

M = W • A + ε ε ε (18)
with W ∈ R K×S + the synergy weight matrix, mapping a synergy activation to a group of muscles, and A ∈ R S×N + the synergies activation matrix, containing the corresponding synergy activation for each sample of the data. ε ε ε ∈ R K×N is the muscle excitation residual matrix that contains the residual error resulting from the factorization. After factorization, the synergy-based EMG-to-torque models are formulated similarly to the MVLR,

τ τ τ h (t) = H s • α(t) (19)
where H s ∈ R J×S is the synergy mapping matrix, which maps a synergy to a joint, and α(t) ∈ R S + is the synergy activation vector (i.e., one of the columns of A). In this framework, α(t) is not measured directly and must be calculated from the previously factorized synergy weight matrix:

α(t) = W + • m(t) (20)
with • + the pseudo-inverse operator. Therefore, the final formulation of the synergy-based model is

τ τ τ h (t) = H s • W + • m(t) ( 21
)
The model is calibrated in several steps. First, the muscle excitation matrix M from the training data (with a corresponding torque matrix T) is factorized via Matlab's NNMF function to obtain the synergy weights and activation matrices W and A. Then, the calibration problem corresponds to finding H s such that

T = H s • A (22)
Similarly to the MVLR model, this is done by using Matlab's mvregress function on T and A. In this paper, synergy-based models are abbreviated SYN; if the number of synergies is relevant in the context, the model is abbreviated SYNX, with X replaced by the number of synergies. To ensure at least one synergy per degree of freedom and direction, a minimum of 2 and 4 synergies were used for single and multi-joint conditions, respectively.

D. Nonlinear mapping model

While the MVLR model is easy to formulate and calibrate, its main limitation stems from its inability to consider nonlinear muscle activation effects. To address this, we incorporate the shape factor, as described for the neuromusculoskeletal model, to obtain a nonlinear mapping (NLMap). The model formulation thus becomes:

a k (t) = e A k m k (t) -1 e A k -1 τ τ τ h (t) = H nℓ • a(t) (23)
where A k , m k and a k are the corresponding shape factor, excitation, and nonlinear activation of the k th muscle, and H nℓ ∈ R J×K the nonlinear mapping matrix. a is the activation vector.

The model is calibrated iteratively using a single-objective genetic algorithm with multiple steps in each iteration. Initially, the genetic algorithm generates a set of shape factors as a chromosome. These parameters are then used to compute the nonlinear muscle activations of the training dataset, which are compiled into an activation matrix A. Next, the mapping matrix is computed using the torque data matrix T through the mvregress function, solving the matrix form of equation 23.

Since the mapping problem has a unique solution, a specific set of shape factors corresponds to a unique mapping, and hence, the matrix coefficients are not part of the chromosome. The final score of the chromosome is then computed with the cost function described in equation 17, and the best chromosome is determined through crossover and mutation operations in an iterative way as part of a typical genetic algorithm, with a 0.8 crossover rate and 200 chromosomes per generation.

IV. RESULTS

A. Trajectory Generation

The trajectory generation framework exhibited Gaussianlike distributions for elbow positions during single-joint and multi-joint conditions. The central tendency of elbow movements was observed around the 90 • position, with the majority falling within the 45 • to 135 • range (see Fig. 3). Shoulder positions were concentrated in the 0 • to 90 • range. During single-joint tasks, reference elbow torques were uniformly distributed within the -10 N m to 20 N.m range. A Gaussianlike distribution emerged during multi-joint tasks, mainly featuring positive elbow torques ranging from -5 N.m to 20 N.m and shoulder torques spanning -10 N.m to 30 N.m.

B. Torque estimation

1) Single-joint: For the estimation of single-joint torque, the repeated measures ANOVA exhibited a notable influence of the chosen model on the NRMS error and the correlation coefficient (p < 0.001). Figure 4 presents a comprehensive summary of the performance outcomes. Specifically, the NLMap and NMS models attained NRMS errors of 4.66 ± 0.54% and 5.93 ± 0.96%, respectively. In contrast, the MVLR had a score of 7.11 ± 1.17%, and SYN models ranged from 7.31 ± 1.28% (SYN5) to 8.46 ± 2.37% (SYN2). Similarly, the correlation coefficient for the NLMap and NMS models were 0.971±0.011 and 0.958±0.013, respectively, while the MVLR was 0.942±0.019 and SYN models exhibited a range spanning from 0.943±0.016 (SYN6) to 0.918±0.0418 (SYN2). Overall, the NLMap model proved to be significantly better in terms of NRMSE and R (p < 0.001) than any other model with a strong effect (Cohen's d > 1.65). The NMS model was also significantly better than MVLR and SYN models for both criteria (p < 0.05) with a strong effect (Cohen's d > 1). Fig. 3: Generated trajectory kinematics and reference torque distributions. For the single joint condition, each line represents a participant. For the multi-joint condition, lines and fills represent the mean and standard deviation of the data distribution across all participants. Solid and dashed lines represent the elbow and shoulder joints, respectively. A 0 • angle corresponds to a fully extended elbow and no shoulder elevation. Light red lines represent reference positions observed during daily living tasks for the elbow [START_REF] Haverstock | Elbow motion patterns during daily activity[END_REF] and the shoulder [START_REF] Chapman | Assessing shoulder biomechanics of healthy elderly individuals during activities of daily living using inertial measurement units: High maximum elevation is achievable but rarely used[END_REF]. 2) Multi-joints: The repeated measures ANOVA showed that model selection significantly affected the NRMS error and correlation coefficient for the shoulder and elbow joints (p < 0.01 and p < 0.001, respectively). The performance results are presented in Figure 6 for both joints. An illustration of the torque reconstruction capabilities of the NLMap mode is shown in Figure 5.

a) Elbow: NLMap and NMS models obtained NRMS errors of 6.58 ± 0.73% and 8.97 ± 1.51%. The MVLR model had an error of 8.54 ± 1.75%, while SYN models ranged from 9.07 ± 2.12% (SYN9) to 10.24 ± 2.72% (SYN4). Likewise, NLMap and NMS models, respectively, obtained a correlation score of 0.913 ± 0.019 and 0.844 ± 0.043, while MVLR was 0.863 ± 0.041 and SYN ranged from 0.857 ± 0.042 (SYN9) to 0.798 ± 0.082 (SYN4). Overall, the NLMap model proved to be significantly (p < 0.01) better in terms of NRMSE and R than any other model with a strong effect (Cohen's d > 1.45). On the other hand, the NMS model did not show significant differences from the MVLR model.

b) Shoulder: The NRMS error for the NLMap and NMS models were 6.87 ± 1.4% and 8.0 ± 2.1%, respectively. The MVLR model had an error of 7.79±1.4%, while SYN models exhibited a range from 8.08 ± 1.5% (SYN7) to 8.47 ± 1.9% (SYN10). Similarly, the correlation scores for the NLMap and NMS models were 0.885 ± 0.030 and 0.840 ± 0.052, while MVLR was 0.855 ± 0.033 and SYN models ranged from 0.849 ± 0.034 (SYN9) to 0.828 ± 0.075 (SYN10). In terms of NRMSE, the NLMap model demonstrated statistically significant superiority (p < 0.05) with a moderate to strong effect (Cohen's d > 0.62). Regarding the correlation coefficient, statistically significant differences (p < 0.01) were found between the NLMap and the other models. In particular, no significant differences were observed between the NMS and MVLR models.

C. Computation efficiency

Duration results are shown in Table I. For linear models such as MVLR and SYN, the calibration time ranged from 1s to 2.5s in both conditions, while the NLMap model took an average of 22s to 45s to calibrate, depending on the condition. Finally, NMS models took the most time on average, ranging from 20 minutes in the single-joint condition to one hour in the multi-joint condition. Similarly, linear models were the fastest to estimate in a real-time setup, with an estimation time of less than a microsecond. The NLMap model took longer, with an

V. DISCUSSION

The present paper introduced a general framework for the in-situ calibration and evaluation of EMG-to-torque models to enable predictive exoskeleton control to assist arbitrary human movements. We outlined a set of common models, their ontology, and the associated calibration technique. We also proposed a novel nonlinear mapping model designed to be simple in its formulation, with low calibration and estimation duration, yet able to consider nonlinear effects. We generated random, non-periodic trajectories, enabling us to test these models in various kinematic and dynamic situations. We investigated the difficulty of predicting torques with different degrees of freedom by testing single-joint and multi-joint strategies, which has yet to be addressed in the literature concerning arbitrary motions [START_REF] Bi | A review on emg-based motor intention prediction of continuous human upper limb motion for humanrobot collaboration[END_REF]. Finally, we evaluated the models using a cross-validation approach to prevent biases related to overfitting.

Creating a task to generate data with a wide range of kinematic and dynamic situations is essential to obtain a general calibration of EMG-to-torque models. While recent studies have used predefined or periodic movements [START_REF] Grimmelsmann | sEMG-based prediction of human forearm movements utilizing a biomechanical model based on individual anatomical/ physiological measures and a reduced set of optimization parameters[END_REF], [START_REF] Lotti | Myoelectric or force control? a comparative study on a soft arm exosuit[END_REF], the proposed framework was able to generate arbitrary random trajectories. Although the experimental setup prevented full coverage of the elbow positions, the distributions of angular positions were representative of daily tasks [START_REF] Haverstock | Elbow motion patterns during daily activity[END_REF], [START_REF] Chapman | Assessing shoulder biomechanics of healthy elderly individuals during activities of daily living using inertial measurement units: High maximum elevation is achievable but rarely used[END_REF].

A key issue when calibrating and using EMG-to-torque models is their accuracy. We found that the NLMap model was the most accurate for both single-joint and multi-joint conditions. Given its simplicity, the MVLR model also demonstrated remarkable accuracy and should not be ignored for potential applications. Moreover, the NLMap and MVLR models do not require kinematic data as input features, making their implementation easier and necessitating fewer sensor types. The NMS model was surprisingly underperforming: while it had a satisfactory accuracy for single-joint estimation, it did not perform better than the MVLR in the multi-joint condition. This is due to the wide variety and the number of internal parameters that must be fitted, making it difficult to find a global minimum during calibration. Thus, while the NMS model should theoretically have the best accuracy due to its knowledge-based nature, this same nature mitigates the practicability of its calibration for infield applications. The SYN models did not offer any notable advantages compared to their counterparts. Although the decrease in dimensionality may provide insight into how the central nervous system utilizes muscle redundancy during different movements, this does not appear to enhance the accuracy of movement prediction. It is difficult to compare the accuracy of other models in the literature because the evaluation method differs. Generally, models are evaluated on a different limb, with a different metric, feature, and task, and more often than not, without any cross-validation procedure. However, Table II provides an overview of some model scores reported in the literature using the NMRSE metric. The NRMSE scores for single-joint and multi-joint conditions indicated that the accuracy of the estimation for a single joint (i.e. the elbow) may be reduced when a more complex movement (e.g., shoulder rotations) is included. This effect could be caused by various factors, such as bi-articular muscles and movement artifacts due to the EMG sensor cables.

While we tested the models on a powerful computer, exoskeleton controllers are typically embedded systems with limited computing power. Thus, it is important to assess the computational complexity of the models. However, to our knowledge, this kind of analysis has not been done in the literature. We found that simpler models, such as MVLR and SYN, were quicker to calibrate and provided faster real-time torque estimation. The calibration process took less than three seconds, and the duration of the estimation was so short that it could not be accurately measured (< 10 -3 ms). The NLMap model calibration process took less than one minute, a suitable wait time for most applications. The estimation duration was less than one millisecond, allowing for a 1kHz exoskeleton control frequency. On the contrary, the NMS model did not demonstrate the same level of performance. Calibration took anywhere from 20 minutes to an hour, depending on the condition, and the estimation duration was more than ten milliseconds. This is due to the number of parameters that need to be adjusted during calibration (five parameters per muscle, sixty in total for multi-joint condition). Moreover, two steps are necessary for the computation of this model: muscle equilibrium and moment arm computation. The muscle equilibrium step is an iterative process that must be repeated at each time step, and for each muscle to balance the lengths of the fibers and tendons, it therefore influences computation duration both during calibration and estimation. Because muscle paths and insertions are not adjusted, the time taken to calculate the moment arm does not significantly impact the calibration duration since it is only necessary to do it once. However, during real-time estimation, each muscle's moment arm must be computed for each actuated degree of freedom, impacting its duration. Consequently, numerical methods have been developed to accelerate this process [START_REF] Sartori | Estimation of musculotendon kinematics in large musculoskeletal models using multidimensional b-splines[END_REF]. This study offers new insights into the estimation of joint torques from EMG signals. However, certain limitations restrict the scope of these findings. To begin with, this research did not include all existing models as it would not have been feasible: we compared models commonly encountered in the literature and developed our model with practicality in mind. However, we proposed an evaluation protocol that enables anyone to test their model against others. To that end, we provided a standardized dataset that can be used to test new models and compare them with our results [START_REF] Quesada | Dataset: Will be made available upon publication[END_REF]. In addition, even though sagittal movements encompass many potential applications of exoskeleton assistance, these results should be extended to include all the degrees of freedom of the upper limbs. Furthermore, research has shown that the EMG signal can be degraded due to changes in the electrode/skin impedance over time, affecting the accuracy of the model estimation [START_REF] Ziai | Comparison of regression models for estimation of isometric wrist joint torques using surface electromyography[END_REF]. Consequently, sweat and skin degradation over time could impede exoskeleton assistance in a workplace environment, potentially making it unsafe. Although current EMG sensors allow the measurement of the electrode/skin impedance, no model has yet been developed to consider it in the torque estimation. Finally, muscle fatigue may emerge as a factor impacting torque estimation accuracy over time. Fatigued muscles typically exhibit a lower mean EMG frequency [START_REF] Chandra | Muscle fatigue induced hand tremor clustering in dynamic laparoscopic manipulation[END_REF] that current models fail to consider.

We anticipate conducting further research based on the results of this study. First, we will explore EMG feature extraction and reducing the number of measured muscles to enhance the robustness and widespread use of EMG-to-torque models. Comparing various feature extraction methods and muscle sets will be essential for improving the information embedded in the signal. Several studies have examined feature selection for EMG-based movement classification [START_REF] Tsai | A comparison of upper-limb motion pattern recognition using emg signals during dynamic and isometric muscle contractions[END_REF], [START_REF] Phinyomark | Feature extraction and selection for myoelectric control based on wearable EMG sensors[END_REF], [START_REF] Phinyomark | Feature reduction and selection for emg signal classification[END_REF], but to our knowledge, no comprehensive comparison of EMG features has been conducted for continuous movement regression. Also, minimizing the number of measured muscles can simplify the placement of EMG sensors, which is a timeconsuming and specialized task. Second, using our dataset and evaluation protocol, we plan to investigate artificial neural network-based (ANN) models, including Long Short-Term Memory (LSTM) networks. ANN models have the potential to offer more accurate estimations while maintaining a short calibration [START_REF] Treussart | Controlling an exoskeleton with EMG signal to assist load carrying: A personalized calibration[END_REF], [START_REF] Li | An enhanced EMG-driven musculoskeletal model based on non-negative matrix factorization[END_REF]. They can be adapted to incorporate multiple EMG features and other relevant factors, such as kinematics. Finally, we aim to assess the impact of the estimation error on exoskeleton assistance and user feedback, considering potential safety concerns related to unexpected feedback loops and system instabilities resulting from variable human behavior and signal degradation over time. Despite these challenges, real-time estimation of human torques holds significant promises for the development of versatile assistive control strategies.

Fig. 1 :

 1 Fig. 1: Participant with EMG sensors and reflective markers in the exoskeleton Visible reflective markers and EMG sensors are highlighted with blue and pink lines, respectively. The setup corresponds to the multi-joint condition.

  17 healthy subjects (11 males, age 28.2 ± 7 years, height 175.4 ± 7 cm, weight 70 ± 11 kg) took part in the experiment. The experimental protocol was approved by the ethics committee of Université Paris-Saclay (CER-PS-2021-048/A1). Participants signed a written informed consent form before starting the experiment.

Figure 7

 7 visually compares correlations between the NLMap and MVLR models. The S-shaped correlation curve of the MVLR model indicates the presence of nonlinear residual

Fig. 4 :

 4 Fig. 4: Single-joint model accuracy. a) Pearson's correlation coefficient b) Normalized root mean square error

Fig. 5 :

 5 Fig. 5: Multi-joint torque estimation with the NLMap model for a representative participant. a) Processed EMG signals. b) Angular position and velocity of the Shoulder and Elbow. c) Shoulder (τ sho ) and Elbow (τ elb ) reference and estimated torques.

Fig. 6 :Fig. 7 :

 67 Fig. 6: Multi-joint model accuracy. In both panels, data for the elbow joint is represented in gray, and data for the shoulder joint in white. a) Pearson's correlation coefficient b) Normalized root mean square error

TABLE I :

 I Calibration and estimation duration. Calibration duration is the time necessary to perform the complete calibration of a model. Estimation duration is the time necessary to compute one estimation time step. The results with a value of < 10 -3 indicate that the estimation duration was less than the resolution of the measurement. < 10 -3 < 10 -3 average of a tenth of a millisecond. The NMS model was the longest, with an estimation time of 10.8ms for the single-joint condition and 15.6ms for the multi-joint condition.

	Single-joint	NLMap NMS	MVLR SYN6	SYN2
	Calibration [s]	22.1	1216	1.096	2.02	1.37
	SD [s] 0.27	538	0.33	0.32	0.5
	Estimation [ms] 0.08	10.8	< 10 -3 < 10 -3 < 10 -3
	SD [ms] 0.003	0.84 < 10 -3 < 10 -3 < 10 -3
	Multi joints NLMap NMS	MVLR SYN10 SYN4
	Calibration [s]	44.7 3662.14	1.49	2.45	1.89
	SD [s]	2.6	1200	0.13	0.22	0.22
	Estimation [ms]	0.1	15.6	< 10 -3 < 10 -3 < 10 -3
	SD [ms] 0.005	5.7	< 10 -3

TABLE II :

 II Overview of NMRSE model evaluations. Results are reported "as is", taking the best score showcased in each paper. These results must be compared cautiously as they are reported using various evaluation methods, limbs, movements, and features. KF, LSTM, CNN, and NMF, respectively, stand for Kalman Filter, Long-Short Term Memory, Convolutional Neural Network, and Non-negative Matrix Factorization.

	Ref	Type	Feature	Joint	NRMSE [%]
	-	NLMap	Torque	Elbow	6.50
	-	NLMap	Torque Shoulder (Flex/Ext)	6.87
	[32]	NMS	Torque	Wrist (Flex/Ext)	2.02
	[65]	LSTM	Torque	Hip (Flex/Ext)	3.6
	[38]	KF	Torque	Elbow	5.81
	[66] CNN-LSTM Torque Shoulder (Flex/Ext)	12.44
	[66] CNN-LSTM Torque	Elbow	12.76
	[30]	NMS	Angle	Wrist & Elbow	13
	[33] NMF-NMS Angle	Wrist (Pro/Sup)	17


Dr Susan E Matthews
email: susan.matthews@uea.ac.uk

Dr Samy Cecioni

Dr John E O'brien

Colin J Macdonald

Dr David L Hughes

Dr Garth A Jones

Dr Stephen H Ashworth

Dr Sébastien Vidal

Dr C J Mcdonald

Fixing the conformation of calix[4]arenes: When are three carbons not enough?

Keywords: Calix[4]arene, Conformational Analysis, Molecular Switching, Kinetics, 1 H NMR

arenes are unique macrocycles that through judicious functionalisation at the lower-rim can be either fixed in one of four conformations or remain conformationally flexible. Introduction of propynyl or propenyl groups unexpectedly provides a new possibility; a unidirectional conformational switch, with the 1,3alternate and 1,2-alternate conformers switching to the partial cone conformation, whilst the cone conformation is unchanged, under standard experimental conditions. Using 1 H NMR kinetic studies, rates of switching have been shown to be dependent on the starting conformation, upper-rim substituent, where reduction in bulk enables faster switching, solvent and temperature with 1,2-alternate conformations switching fastest. Ab initio calculations (DFT) confirmed the relative stabilities of the conformations and point towards the partial cone conformer being the most stable of the four. The potential impact on synthesis through the 'click' reaction has been investigated and found not to be significant.

Introduction

Calix [4]arenes [1][START_REF] Gutsche | Calixarenes Revisited[END_REF][START_REF] Gutsche | Monographs in Supramolecular Chemistry[END_REF] have proved to be one of the most versatile macrocyclic platforms available and have been exploited for a myriad of different applications in the fields of materials and biological chemistry including sensing, catalysis, medicinal chemistry and drug delivery. [START_REF]For extensive reviews of applications see[END_REF] A particular advantage of the calix [4]arene skeleton, unlike larger analogues, is the ability to fix the macrocycle into one of four distinct conformations; cone, partial cone, 1,3-alternate or 1,2-alternate, by alkylation at the phenolic positions. 1 Whilst this property has been extensively exploited in the design of rigid systems for directed topological interactions, [1][START_REF] Gutsche | Calixarenes Revisited[END_REF][START_REF] Gutsche | Monographs in Supramolecular Chemistry[END_REF][START_REF]For extensive reviews of applications see[END_REF] less attention has been paid to harnessing conformationally flexible derivatives which offer potential in the design of switchable and stimulus responsive materials.

Two general categories of switches have been observed in the calixarene family; those which alter the macrocycle state from one extreme variant of a conformation to another extreme variant of the same conformation and, more limitedly, a true switch between two conformational isomers of the macrocycle. Examples of the first include: vase to kite switching [START_REF] Azov | [END_REF] of extended resorcin [4]arene cavitands where a variety of stimuli such as reduction in temperature 6 or pH, 7 redox 8 and metal cation 9 complexation facilitate the switch or calix [4]tubes 10 where complexation of potassium in the tube results in a change from flattened (or pinched) cone to cone conformations of the gating calix [4]arenes.

The most established examples of the second type are found with the calix [4]pyrroles 11 where complexation of Lewis basic anions or variation in pH result in a reversible change from the stable 1,3-alternate to the cone conformation driven by hydrogen bond formation. This phenomenon has been exploited in the design of self-assembled cages and capsules 12 and responsive polymers, 13 indicator displacement assays 14 and logic gates. 15 It is well established that introduction of propyl groups at the phenolic positions of calix [4]arenes is sufficient to lock 16,17 the individual conformations in place by blocking interconversion via ring inversion through the annulus. Accordingly heating of any of the conformations at 147 °C in 1,1,2,2-tetrachlorethane for 12 hours does not result in interconversion (Figure 1).

In contrast, calix [4]arenes bearing methyl groups are fully flexible [16][17][18][19][20][21] and freely rotate between the four conformations with the equilibrium being solvent and p-substituent dependent (Figure 1). 22,23 Both experimental NMR studies and MM3 calculations gave the relative stabilities for t-butyl calix [4]arene as partial cone > cone > 1,2-alternate > 1,3-alternate 21 at room temperature with the majority product at -30 °C being assignable to the partial cone 19,20 For calix [4]arenes functionalised with ethyl groups (Figure 1), a more complex situation results in which interconversion only occurs at elevated temperatures 17,19,20 Heating of a mixture of cone, partial cone and 1,3-alternate conformers for 2 hours in 1,1,2,2-tetrachloroethane results in a mixture of all four conformers, with the 1,2-alternate conformation being the most abundant, 16.17 whereas heating at lower temperatures in refluxing THF does not allow access to other conformers. 20 With acetoxy substituents, interconversion of the cone conformer is observed on heating at 150 °C to give, after 384 hours, predominantly the 1,2-alternate and 1,3-alternate conformations with a relative stability of 1,3-alternate > 1,2alternate > partial cone > cone (Figure 1). In this case, steric repulsion between the larger acetoxy groups was proposed to direct the conformer distribution away from the cone. 24 More recent studies on cyanomethoxy derivatives of both t-butyl calix [4]arene 25 and calix [4]arene 26 demonstrated that the partial cone and cone conformations were conformationally mobile, and that a thermodynamic equilibrium, with a biased distribution to the partial cone, could be achieved by heating either conformer. Interestingly, there was no evidence for rotation to either of the two alternate conformations. Stabilisation of a single conformation in equilibrium mixtures is possible through cation complexation, thus flexible tetramethoxycalix [4]arenes can be 'fixed' in the cone conformation by complexation with small cations such as Li + and Na + . 27 Whereas conformationally flexible calix [4]arenes bridged at the lower rim by a crown ether moiety 28 can switch from a mixture in which the majority conformation is the cone to a 1,3-alternate conformation through binding of a cesium cation within the crown. Recently, Neri 29 demonstrated that switching to a single conformation can be achieved through guest inclusion by functionalisation at the upper-rim. Thus for a methoxycalix [4]arene derivatised with two carboxylate groups an equilibrium mixture favouring the partial cone was switched entirely to the cone by inclusion of a dicationic paraquat guest and reversed by changing the pH.

Here, we investigated the conformational stability of tetrapropynyl calix [4]arenes and we report the first series of stimuli-responsive calixarene switches based on simple functionalisation of the lower rim. Through 1 H NMR, kinetic studies and ab initio calculations we demonstrate their tunability, response to two stimuli; temperature and solvent, and their unexpected ability to switch conformation in the solid state.

Results and Discussion

Whilst undertaking research on the widely used 1,3-alternate tetrapropynyl calix [4]arene 11,3 we were surprised to observe that upon heating in toluene overnight the material was entirely converted into the partial cone conformer 1p. This prompted us to investigate the relative stabilities of the four different conformations (Figure 2) of tetrapropynyl calix [4]arene 1 and the ease of switching between conformations.

Synthesis:

The cone conformation 1c was prepared following published procedures 30 and 1p and 11,3 were prepared in a single reaction, from the 1,3-disubstituted calix [4]arene 2, as previously reported by us (Scheme 1). 31 Single crystals of X-ray quality of 1p were grown by slow evaporation from dichloromethane/ methanol (Figure S40). The crystal structure analysis confirmed that the molecule exhibited a classic partial cone conformation with a similar core conformation to that found previously, 32 but with some of the propynyl groups in quite different orientations. Similar synthetic routes were employed for the synthesis of propynyl partial cone 5p, 6p and 1,3-alternate 51,3, 61,3 33 conformations of toctyl calix [4]arene and calix [4]arene respectively. Additionally the partial cone 7p 34 and the 1,3-alternate 71,3 35 conformations of the tetrapropenyl calix [4]arene, for comparison, were prepared as previously reported.

Scheme 1. Synthesis of 1,3-alternate and partial cone conformations.

The 1,2-alternate conformation was accessed through the 1,2dialkyne 8 which was prepared, in acceptable yield, by the treatment of t-butyl calix [4]arene with a stoichiometric amount of propargyl bromide in the presence of sodium hydride (Scheme 2). 36 Following the route of Lhoták et al., 37 the 1,2-alternate conformer 11,2 was synthesized using potassium trimethylsilanolate as base in THF. Unlike the published procedure the reaction was terminated after 48 hours to maximise the yield of the desired conformation and minimise conversion to the partial cone conformer 1p (vide infra). Formation of the 1,2alternate conformation was confirmed through 1 H NMR where the characteristic three signals (doublet, singlet, doublet) [START_REF] Gutsche | Monographs in Supramolecular Chemistry[END_REF] for the methylene bridge protons associated with C2h symmetry were observed (Figure 3d). The analogous calix [4]arene 61,2 was prepared in a similar two-step process, albeit in low yield. X-ray structure of 11,2: Slow evaporation from a dichloromethane / hexane solution yielded single crystals of 11,2 suitable for X-ray analysis (Figures 4 andS41-43). There are two independent calix [4]arene molecules in the crystal, one with 2/m symmetry, centred at (½, ½, ½); the second is centrosymmetric, about (1, ½, ½) and is disordered, lying over its mirror-image. The calix [4]arene rings in the two molecules have very similar shapes with the phenyl rings tilted similarly (ca 70 °) from the calix [4]arene ring mean-plane. The major differences between the two molecules are in the directions of the O-CH2 bonds from the phenolic oxygen atoms. Cone-shaped cavities are not formed in 1,2-alternate calix [4]arenes. Kinetic Studies: Initial studies focused on the stability of the four conformations of p-t-butyl calix [4]arene tetraalkyne in toluene-d8. Heating of the conformers at 80 °C indicated that in the case of the 1,3-alternate conformer 11,3 and the 1,2-alternate conformer 11,2 almost complete conversion to the partial cone conformer 2 was achieved within four hours with no evidence of any other conformers present. In contrast, both the cone and the partial cone conformers 1c and 1p showed no conformational change under the same conditions (Figures S11,S12).

A full 1 H NMR kinetic study was then undertaken on 11,2 and 11,3. Samples were heated for 3 hours at 5 °C intervals between 80 and 100 °C in toluene-d8, to determine both the first order rate coefficients for conformational switching and the activation energy. Example data showing the conversion for 11,3 are given in Figure 5 and Table 1. Full data for all compounds and all temperatures are provided in Supporting Information (S15-S18).

For both 11,2 and 11,3, rather than an equilibrium being established, a simple unidirectional switching mechanism is apparent and there is no evidence for back conversion of the partial cone or further conversion to another conformer from the partial cone.

The results clearly show that the rate of conversion of the 1,2-alternate conformer 11,2 to 1p is significantly faster (k = 3.275 × 10 -4 s -1 ) than that for the 1,3-alternate conformer 11,3 (k = 1.415 × 10 -4 s -1 ). However, it should be noted that the activation energy for both processes is comparable, pointing towards entropic factors leading to the differences in rates (Table 1). These data indicate that in the case of the propynyl functionalised calix [4]arenes, the conformational mobility is intermediate between calix [4]arenes functionalised with ethyl groups and those with propyl groups. Whilst the propyl functionalized calix [4]arenes retain fixed conformations, in the case of the ethyl derivatives, more flexibility is shown and it is possible to flip the cone conformation to the partial cone conformation in refluxing THF, an event not seen with 11,2 or 11,3. Interestingly, conformational flexibility of a monopropargylated derivative of t-butyl calix [4]arene has also been reported recently. 39 Here, the presence of three larger N,Ndimethylcarbamoylmethoxy groups restrict the rotation to a single aromatic moiety. In this case, a very slow interconversion and development of an equilibrium between a partial cone and a 1,2alternate conformation were observed at room temperature over many months. 11,3 [b] 0.492 (30) 105.4(72) [a] Uncertainty given in brackets quoted in units of the last decimal place. [b] Measurements made in DMSO-d6

Upper-rim substituent: In order to understand this phenomenon further, the effect of the upper-rim substituent was also considered. Although the 'oxygen through the annulus' mechanism 1-3 is well established for conformational interconversion of calix [4]arenes, the substituent at the upper-rim has previously been shown to affect conformer distribution significantly. For example, in calix [4]arenes exhibiting dynamic equilibrium the alternate conformations are not observed for ethyl substituted calix [4]arenes in contrast to the t-butyl calix [4]arene derivatives. 21 For our system, the effect of upper-rim substitution on the rate of conformational switching was investigated through an evaluation of both the 1,3-alternate conformations of calix [4]arene 61,3 and t-octyl calix [4]arene 51,3. Introducing the bulkier group, as in 51,3, results in slower conversion to 5p which can be attributed to increased 'drag' as the rings rotate, although the rate is not substantially different than that for 11,3 (Table 1) suggesting that the system will still switch with extensive upper rim substitution. However, replacing the t-butyl group with a hydrogen, as in 61,3, results in a much faster conversion (k = 2.421 × 10 -4 s -1 ) to the partial cone conformation 6p. Importantly, whilst switching was rapid for 61,3, again no conversion of the partial cone conformations 5p or 6p to other conformations was observed on heating for over 45 hours at 80 °C in toluene-d8 (Figure S13,S14).

Given these results, combined with the faster conversion of 11,2 than 11,3, it is not unexpected that 61,2, the calix [4]arene tetraalkenyl compound in the 1,2-alternate conformation converts the most rapidly on heating in toluene-d8 (k = 8.38 × 10 -4 s -1 ) with extensive conversion occurring even after 15 minutes at 60 °C.

Solvent effects:

The role of solvent on the rate of conversion was also considered. Heating of 11,3 in DMSO-d6 resulted in conversion to 1p however the rate of conversion was considerably slower than in toluene (Figures S19-20) and slower than when a more bulky upper-rim substituent is in place. Thus the switch can be tailored both by structure and by solvent.

Ambient Temperature: The possibility of conversion also occurring at ambient temperature was investigated for 11,2, 11,3, 61,3 and 61,2. As expected for the 1,2-alternate conformation 61,2 rapid conversion was observed, with significant quantities of 6p produced within hours and full conversion taking place within 9 weeks in a CDCl3 solution (Figure 7 and Table 2). For 11,2 with the larger upper-rim substituent, conversion was slower although more than 50% conversion was observed over 9 weeks. These observations may help to explain why it is difficult to obtain high yields of 11,2 and 61,2 under the published synthetic conditions.

Interestingly, the order of rates of conversion in CDCl3 at ambient temperature is not consistent with those seen in toluene-d8 at elevated temperatures. The 1,3-alternate conformation 61,3, without an upper-rim substituent is converted more rapidly (k = 6.89 × 10 -7 s -1 ) than 11,2 (k = 3.03 × 10 -7 s -1 ) in the 1,2 alternate conformation. In contrast 11,3 converts most slowly under these conditions although the presence of 1p is apparent within 1 week of storage in CDCl3 (Figure S28).

Solvent related effects can also be observed at ambient temperature. Storage of 61,2 in either toluene-d8 or DMSO-d6 results in a slower conversion to 6p (Table 2, Figure S32). The results cannot be directly correlated with either dipole moment or viscosity of the solvents and it is likely that a combination of effects is occurring. 61,2 [b] 25.6 61,2 [c] 21.4

[a] Fractional uncertainty estimated to be 10%. [b] Measurements made in toluene-d8 [c] For 11,2, within the first hour a transformation from crystalline to an amorphous state was observed and an amorphous solid was formed that showed the presence of both the 1,2-alternate 11,2 and partial cone 1p conformations. Complete conversion to the partial cone 1p took place within 12 hours (k = 5.69 × 10 -5 s -1 ) (Figure S33). With 61,2 the material appeared to retain its crystaline apperance throughout the study and a slower transformation to 6p occurred over a period of 7 days (82% conversion, k = 1.40 × 10 -5 s -1 ) (Figure S34). Thus uni-directional switching is possible for the 1,2-alternate conformations even in the solid state.

Table 3. First order rate coefficients of the conversion at 100° C in bulk

Conformation

First order rate coefficient k × 10 5 / s -1 11,2 5.69 [a] 61,2 1.40

[a] Fractional uncertainty estimated to be 10%.

Ab initito calculations:

Experimentally, both of the alternate conformations of p-t-butyl calix [4]arene tetraalkyne 11,2 and 11,3 are prone to isomerisation, but the partial cone 1p and cone 1c conformations are not. It is not possible from the above experimental data to determine the thermodynamic global minimum and thus the most stable conformer, but it can be suggested this may be in line with results for the methyl and ethyl calixarene derivatives and be the partial cone conformation with the cone conformation inhabiting a thermodynamic local minimum. 21 This postulate was investigated using density functional theory (DFT) analysis. Optimization of the molecular geometries of the four conformations of 1 were performed on the Cartesian co-ordinates 41 using standard energy minimization techniques as implemented in Gaussian03, [START_REF] Frisch | Gaussian 03[END_REF] with the B3LYP functional in conjunction with the 6-31G(d) basis set. All of the optimized geometries were fully characterized and all the vibrational frequencies calculated. The energy calculations (Figure 8, Table 4) reveal that each of the conformations corresponds to a genuine minimum of the potential energy surface. The results confirm that the cone and partial cone conformations are both low energy with the partial cone being the more conformationally stable. In contrast both the 1,3-alternate and 1,2-alternate conformations are high energy and, in line with the experimental data, the least stable conformation is that of the 1,2-alternate. Lower-rim substitution: With these results in hand, we turned our attention to another situation; the conformational stability of the 1,3-alternate conformation of tetrapropenyl calix [4]arene 71,3. This material can be considered to be a useful comparitor between the switchable propynyl and the conformationally locked propyl materials. Heating of 71,3 in toluene-d8 at 80 °C resulted in a considerably slower conformational change (Figure S35) than that seen for 11,3, however, over a period of monitoring for 2 days at this elevated temperature, partial transformation into the partial cone conformation 7p was observed,. The results observed for both the propynyl and propenyl functionalised calixarenes indicate that neither can be considered to be conformationally fixed. However for 7, the interconversion is unlikely to be significant in any experimental applications.

Effect on synthesis:

Since the introduction of copper accelerated Huisgen [START_REF] Huisgen | [END_REF] azide-alkyne cycloadditions (CuAAC) by Meldal 44 and Sharpless, 45 there has been considerable interest in alkyne and azide functionalised calix [4]arenes for the formation of sensors 46 and biologically active calix [4]arenes. 47 Given the extensive use of these derivatives in synthesis and the previous reports that the partial cone conformation of hexahomotrioxacalix [3]arene 48 is fully converted to the cone conformation during a click reaction, it was interesting to determine whether the conformational switching could influence CuAAC click chemistry reactions. A series of model reactions between 11,3 and benzyl azide were undertaken and product formation monitored over time (Scheme 3).

In the case of standard Huisgen conditions 10 in the absence of a catalyst, no reaction was observed over the 4 hr period of heating and instead thermal interconversion occurred (Figure S39). However, in the presence of CuI, whether the reaction was undertaken with thermal or microwave heating, the click reaction was faster than the interconversion and only the 1,3-alternate product could be identified (Figure S38). This indicates that the observed conformational switching is unlikley to be problematic in synthetic applications.

Scheme 3. Effect of conformational interconversion on synthesis

Conclusions

Here we report the unexpected conformational flexibility of tetrapropynyl calix [4]arene which provides a unique switching system in which two conformations are stable (the cone and partial cone) and two flexible (the 1,2-alternate and 1,3-alternate) within normal operating conditions. Thus the molecule holds an intermediate position between fully flexible ethyl and fully fixed propyl derivatives (Figure 9). The switches are uni-directional from the flexible to stable conformations and the rates of conversion can be tuned by temperature and solvent. The system can be further tailored by alteration of upper-rim and lower-rim substituents and by choice of 1,2-alternate or 1,3-alternate conformations. Additionally, for the 1,2-alternate conformations, conformational conversion is observed in the bulk solid state both in amorphous and crystalline forms.

This new family of switches can be readily synthesized from commercial starting materials in short synthetic sequences and are amenable to further synthetic modification for the introduction of reporter systems and thus offer a new building block which can be exploited for the development of sensors and stimulus responsive materials.

Experimental Section

Compounds 1c, 30 1p, 31 11,3, 31 2, 31 4, 33 6p, 33 61,3, 33 7p, 34 71,3 35 and 9 49 were prepared according to literature procedures. Full experimental details for kinetics experiments and X-ray crystallography are given in Supporting Information. A suspension of p-t-octyl calix [4]arene 50 (3.00 g, 3.44 mmol) and K2CO3 (1.04 g, 7.54 mmol) was heated at reflux for 1 h in acetone (100 mL). Propargyl bromide (80% in toluene, 0.91 g, 7.65 mmol) was added and the mixture heated at reflux for 18 h. The reaction mixture was cooled and filtered. The filtrate was evaporated and the residue re-crystallised from dichloromethane/methanol to yield the title compound ( [4]arene 3 (1.896 g, 2 mmol) and caesium carbonate (1.06 g) was heated at reflux for 1 h in acetone (135 mL). Propargyl bromide (80% in toluene, 0.74 g, 6.22 mmol) was added and the mixture heated at reflux for 96 h. The reaction mixture was cooled and filtered. The filtrate was evaporated and the residue subjected to column chromatography (hexane/ dichloromethane 3:1) to yield the tetrasubstituted material as the 1,3alternate conformation (0.5 g) and the partial cone conformation (1.01 g) 51 (20.00 g, 30.9 mmol) in DMF (300 mL) and the mixture was stirred for 30 min. Propargyl bromide (80% in toluene, 9.68 g, 67.8 mmol) was added and the mixture stirred for a further 18 h. Water (50 mL) was added and the resulting solid collected and dried. Silica gel column chromatography (hexane/dicholoromethane 1:2) gave the desired compound as a white solid (5.29 g, 24% Potassium trimethylsilanolate (90%, 1.94 g, 15.11 mmol) was added to a stirred solution of 8 (2.19 g, 3.02 mmol) in THF (100 mL), the mixture stirred for 30 minutes before addition of propargyl bromide (80% in toluene, 4.32 g 30.2 mmol). The solution was stirred for 48 h and then poured into 1M HCl. The compound was extracted with dichloromethane (2 × 50 mL), washed with saturated Na2SO3 (2 × 50 mL), dried over MgSO4 and the solvent evaporated under reduced pressure. The residue was purified by silica gel column chromatography (hexane/dichloromethane 2:1) to give the title compound 11,2 as a white solid (1.05 g, 42%). Mp 107-109 °C Potassium trimethylsilanolate (90%, 2.36 g, 18.38 mmol) was added to a stirred solution of 9 (2.00 g, 4 mmol) in THF (100 mL), the mixture stirred for 30 minutes before addition of propargyl bromide (80% in toluene, 5.00 g, 35.00 mmol). The solution was stirred for 48 h and then poured into 1M HCl. The compound was extracted with dichloromethane (2 × 50 mL), washed with saturated Na2SO3 (2 ×50 mL), dried over MgSO4 and the solvent evaporated under reduced pressure. The residue was purified by column chromatography (hexane/dichloromethane 1:2) to give the title compound 61,2 as a white solid (584 mg, 25%). Mp 132-134 °C 1 H NMR δ 7.18 (4H, dd J = 7.5 Hz, 1.5 Hz, Ar), 7.08 (4H, dd J = 7.5 Hz, 1.5 Hz, Ar), 6.93 (4H, t J = 7.5 Hz, Ar), 4.44 (2H, d J = 13 Hz, ArCH2Ar), 4.03 (4H, s ArCH2Ar), 3.98 (4H, dd J = 15.7 Hz, 2.5 Hz, Alkyne CH2), 3.82 (4H, dd J = 15.7 Hz, 2.5 Hz, Alkyne CH2) 3.30 (2H, d J = 13 Hz, ArCH2Ar), 2.35 (2H, t J = 2.4 Hz, Alkyne CH)

Figure 1 .

 1 Figure 1. Summary of conformational flexibility of calixarenes (c = cone, p = partial cone, 1,2 = 1,2-alternate and 1,3 =1,3-alternate)

Figure 2 .

 2 Figure 2. The four possible conformations of tetrapropynyl tetra-t-butyl calix[4]arene

Scheme 2 .Figure 3 .

 23 Scheme 2. Synthesis of 1,2-alternate conformations

Figure 4 .

 4 Figure 4. Views of the fully ordered calix[4]arene 11,2. Hydrogen atoms have been omitted for clarity. Thermal ellipsoids are drawn at the 30% probability level

Figure 5 .

 5 Figure 5. Representative partial 1 H NMR spectra showing conversion of 11,3 to 1p in toluene-d8 at 85 ºC a) 15 minutes, b) 75 minutes, c) 135 minutes, d) 195 minutes (Red triangles 11,3, Blue circles 1p)

Figure 6 .

 6 Figure 6. Relative energies of the optimised ab initio structures and transition states to conformation 1p (not to scale)

Figure 7 .

 7 Figure 7. Conversion of 61,2 at ambient temperature in CDCl3. Partial 1 H MMR of methylene regions at a) 0 minutes, b) 24 hours, c) 7 days, d) 14 days, e) 6 weeks, f) 9 weeks. (Red triangles 61,2, Blue circles 6p)

Figure 8 .

 8 Figure 8. Optimised structures for a) cone conformation 1c, b) partial cone conformation 1p. c) 1,2-alternate conformation 11,2 d) 1,3-alternate conformation 11,3

Figure 9 .

 9 Figure 9. Summary of conformational flexibility of tetrafunctionalised calix[4]arenes

Table 1 .

 1 Experimental first order rate coefficients 40 and activation energies of the conversion at 85° C in toluene-d8 (unless otherwise indicated).

	Conformation	First	order	rate	Activation energy
		coefficient k × 10 4 / s -1	Ea / kJ mol -1
	11,2	3.275(53) [a]		103.2(20)
	11,3	1.415(58)		110.33(81)

Table 2 .

 2 First order rate coefficients of the conversion at ambient temperature in CDCl3 (unless otherwise indicated).

	Conformation	First order rate coefficient k × 10 7 / s -1
	11,2	3.03 [a]
	11,3	1.50
	61,3	6.89
	61,2	42.8

  Measurements made in DMSO-d6

	Solid State: Recently, Dehaen and Dobrzańska 38 reported an
	unprecedented conformational change of a homodithiacalixarene
	within a single crystal from a 1,3-alternate to a 1,2-alternate
	conformation in response to release of solvent on heating. The
	possibility of conversion in the solid state was also investigated
	for 11,2, 11,3, 61,3 and 61,2. Whilst both 11,3 and 61,3 showed no
	conversion over 7 days when heated as a solid at 100 ºC, the 1,2-
	alternate comformations 11,2 and 61,2 gave contrasting results.

Table 4 .

 4 Relative stability of the conformations of 1

		Energy / Ha	Erel / Ha	Erel /kJ mol -1
	1c	-2472.4620794	0.000332 0.87
	1p	-2472.4624118	0	0
	11.2	-2472.4484469	0.013965 36.67
	11,3	-2472.454857	0.007525 19.76

11,17,23-Tetra-p-t-Octyl-25,26,27,28-tetrapropargyloxycalix[4]- arene (1,3-alternate conformation 51,3), (partial cone conformation 5p)

  A suspension of 5,11,17,23-p-t-octyl-25,27-dipropargyloxy-26,28-dihydroxycalix

	2.50 g, 77%)
	Mp 131-133 °C 1 H NMR δ 7.03 (4H, s, Ar), 6.93 (2H, s, OH), 6.81 (4H, s,
	Ar), 4.74 (4H, d J = 2.4 Hz, Alkyne), 4.37 (4H, d J = 13 Hz, ArCH2Ar), 3.34
	(4H, d J = 13 Hz, ArCH2Ar), 2.50 (2H, t J = 2.4 Hz, Alkyne), 1.56 (4H, s,
	CH2), 1.36 (4H, s, CH2), 1.34 (12H, s, CH3), 1.06 (12H, s, CH3), 0.75 (18H,
	s, CH3), 0.27 (18H, s, CH3) 13 C NMR δ 31.1, 31.3, 31.9, 32.19, 32.23, 32.6,
	32.7, 37.9, 57.3, 57.7, 63.5, 76.4, 78.8, 126.2, 126.8, 127.5, 132.6, 140.0,
	140.6, 146.1, 149.2, 150.6. IR v = 3400, 3310, 2951, 2896, 2862, 1739,
	1600, 1476, 1363,1328, 1283, 1213, 1196 cm -1 ESI MS 966.7332 m/z
	[M+NH4] + calcd 966.7334.
	5,

1,3-alternate (51,3)

	Mp 25-127 °C 1 H NMR δ 7.05 (8H, s, Ar), 3.85 (8H, s, ArCH2Ar), 3.64 (8H,
	d J = 2.4 Hz, Alkyne), 2.24 (4H, t J = 2.4 Hz, Alkyne), 1.48 (8H, s, CH2),
	1.29 (24H, s, CH3), 0.68 (36H, s, CH3) 13 C NMR δ 31.4, 32.0, 32.6, 57.6,
	58.4, 74.4, 81.3, 127.9, 134.2, 144.5, 153.1 IR v = 3316, 3282, 2952, 2901,
	1594, 1473, 1362, 1197, 1152, 1112cm -1 ESI MS 1042.7645 m/z [M+NH4] +
	calcd 1042.7647.
	Partial cone (5p)
	Mp 160-162 °C 1 H NMR δ 7.30 (2H, s, Ar), 6.99 (2H, s, Ar), 6.93 (2H, d J
	= 2.3 Hz, Ar), 6.49 (2H, d J = 2.3 Hz, Ar), 4.41 (4H, d J = 2.4 Hz, Alkyne),
	4.30 (2H, d J = 14 Hz, ArCH2Ar), 4.16 (2H, d J = 2.4 Hz, Alkyne), 3.99 (2H,
	d J = 2.4 Hz, Alkyne), 3.81 (2H, d J = 14 Hz, ArCH2Ar), 3.70 (2H, d J = 14
	Hz, ArCH2Ar), 3.05 (2H, d J = 14 Hz, ArCH2Ar), 2.39 (2H, t J = 2.4 Hz,
	Alkyne), 2.33 (1H, t J = 2.4 Hz, Alkyne), 2.14 (1H, t J = 2.4 Hz, Alkyne)
	1.74 (4H, d J = 11 Hz, CH2), 1.48 (6H, s, CH3),1.44 (4H, d J = 11 Hz, CH2),
	1.32 (6H, s, CH3), 1.17 (6H, s, CH3), 0.97 (6H, s, CH3) 0.83 (9H, s, CH3),
	0.72 (9H, s, CH3), 0.62 (18H, s, CH3) 13 C NMR δ 31.0, 31.4, 31.9, 32.1,
	32.2, 32.3, 33.1, 38.0, 38.1, 38.3, 57.2, 57.3, 57.9, 58.7, 59.1, 60.9, 73.7,
	74.4, 74.64, 80.6, 81.3, 82.2, 126.3, 126.9, 127.1, 129.1, 132.0, 132.6,
	133.0, 136.1, 144.0, 144.4, 151.9, 153.4, 154.0 IR v = 3417, 3305, 2951,
	2896, 1737, 1594, 1474, 1363, 1198, 1152, 1111 cm -1 ESI MS 1042.7643
	m/z [M+NH4] + calcd 1042.7647.
	5,

11,17,23-Tetra-p-t-Butyl-25,26-dipropargyloxy-27,28-dihydroxy- calix[4]arene

	(8)
	Sodium hydride (3.42 g, 135.4 mmol) was added to a suspension of p-t-
	butyl calix[4]arene

5,11,17,23-Tetra-p-t-Butyl-25,26,27,28-tetrapropargyloxycalix[4]- arene (1,2-alternate conformation 11,2)

  ). 13 C NMR δ 31.4, 31.7, 32.5, 32.8, 32.9, 34.1, 34.3, 62.7, 76.2, 79.7, 125.3, 125.5, 126.2, 128.9, 132.8, 134.1, 142.8, 147.5, 149.1, 151.0. IR v = 3507, 3288, 2963, 2896, 2862, 1647, 1600, 1474, 1457, 1356, 1281, 1202, 1121, 990 cm -1 ESI MS 742.4832 m/z [M+NH4] + calcd 742.4830.

	Mp 171-173 °C 1 H NMR δ 8.27 (2H, s, OH), 6.95 (2H, d J = 2.5 Hz, Ar),
	6.92 (2H, d J = 2.5 Hz, Ar), 4.89 (2H dd J = 16, 2.4 Hz, Alkyne) 4.50 (1H,
	d J = 13 Hz, ArCH2Ar), 4.44 (2H, d J = 13 Hz, ArCH2Ar), 4.29 (1H, d J =
	13 Hz, ArCH2Ar), 3.32 (3H, d J = 13 Hz, ArCH2Ar), 3.30 (1H, d J = 13 Hz,
	ArCH2Ar), 2.60 (2H, t J = 2.4 Hz, Alkyne), 1.16 (18H, s, CH3), 1.09 (18H,
	s, CH3)

25,26,27,28-Tetrapropargyloxycalix[4]arene (1,2-alternate conformation 61,2)

  1 H NMR δ 7.13 (4H, d J = 2.5 Hz, Ar), 7.04 (4H, d J = 2.5 Hz, Ar), 4.36 (2H, d J = 13 Hz, ArCH2Ar), 3.97 (6H, br s ArCH2Ar + Alkyne CH2) 3.90 (2H, d J = 2.4 Hz, Alkyne CH2), 3.80 (4H, dd J = 15.6 Hz, 2.4 Hz, Alkyne CH2) 3.21 (2H, d J = 13 Hz, ArCH2Ar), 2.30 (2H, t J = 2.4 Hz, Alkyne CH), 1.28 (36H, s, CH3) 13 C NMR δ 31.7, 34.3, 38.2, 60.0, 74.6, 81.1, 125.7, 127.0, 132.9, 134.9, 145.3, 152.7 IR v = 3311, 3277, 2953, 2907, 2862, 1600, 1586, 1477, 1460, 1361, 1283, 1186, 1121, 1011 cm -1 ESI MS 818.5139 m/z [M+NH4] + calcd 818.5143

  13 C NMR δ 30.8, 37.1, 59.9, 74.6, 80.3, 123.4, 128.7, 129.9, 133.8, 135.9, 154.4 IR v = 3270, 2919, 1585, 1455, 1361, 1250, 1187, 1090, 1006 cm -1 ESI MS 594.2632 m/z [M+NH4] + calcd 594.2639.

Acknowledgements

The authors thank the University of East Anglia, the University Claude Bernard Lyon 1 and the CNRS for financial support. S. C. thanks the Région Rhône-Alpes (Cluster de Recherche Chimie) for additional funding. The authors would like to thank EPSRC UK National Mass Spectrometry Facility at Swansea University for provison of high resolution mass spectral analyses.