Sahra Amouri

Julien Averseng

Jérôme Quirant

Jean-François Dube

Structural design and active control of modular tensegrity systems

Keywords: tensegrity, structural design, optimization, vibration control

Tensegrity systems are self stressed reticulate structures composed of a set of compressed struts assembled inside a continuum of tendons. This principle is at the origin of lightweight and transparent structures that can cover large spaces and be erected, in particular cases, by deployment. In this paper, we propose a general design and optimization procedure adapted to modular structures following this principle. An application is presented on the case of a curved deployable footbridge.

Besides, as lightweight frames, these systems are subject to deformation and vibration issues when faced to varying actions such as climatic, human, or seismic loads. Active control is as solution that allows, using actuators integrated into the structure, to attenuate these effects. On the case of a real plane modular tensegrity grid, we present a specific methodology for the active damping of the first two modes and its experimental validation.

Introduction

Tensegrity systems are selfstress space reticulate structures, composed of a set of compressed struts in stable equilibrium inside a continuum of tendons [START_REF] Motro | Tensegrity: Structural systems for the future[END_REF].

Following this principle developed during the fifties [START_REF] Fuller | The dymaxion world of Buckminster Fuller[END_REF] [START_REF] Snelson | Tensegrity masts[END_REF] can emerge lightweight, large spans and transparent structures (Figure 1) that, in some cases, can be erected by deployment [START_REF] Smaili | Foldable/unfoldable curved tensegrity systems by finite mechanism activation[END_REF] [START_REF] Quirant | Conceptual Design and Analysis of a Deployable Structure with Flexible Joints[END_REF]. Although providing many benefits, very few systems of this kind are present in structural applications, mainly because they require strong design and computation prerequisites. [START_REF] Averseng | Design, Analysis and Self Stress Setting of a Lightweight Deployable Tensegrity Modular Structure[END_REF] [START_REF] Tibert | Deployable tensegrity mast[END_REF] Indeed, their stability and rigidity imply being in a state of selfstress, which imposes a specific design approach and using not so widespread non-linear analysis techniques. Besides, as any lightweight system, their damping properties are low, which makes them sensible to dynamic actions, climatic or human, that may induce resonance and then a loss of comfort of damages in secondary fragile elements like glass panels. So, it is necessary to control the dynamic behaviour in order to attenuate these effects.

In this paper, we present the synthesis of two studies focused on the structural design and control of tensegrity systems on the case of deployable modular systems. We propose a new design methodology for determining the form, selfstress level and crosssections characteristics of a whole structure so as to optimize the weight and flexural rigidity. In a second part, we develop a general process for controlling the first vibration modes, using integrated actuators, and we present an application on the case of a plane tensegrity grid.

Optimal design

The design of tensegrity systems is the result of a form-finding process, which consists in optimizing the balance between form and internal forces. In addition, as for any structure, the engineer has to consider realistic project situations and justify the elements according to design rules provisions, while optimizing material cost. With these lightweight systems, self weight is not negligible, which imposes an iterative process of form-finding and design. This apparent complexity and the fact that a few structural design studies (Rhode-Barbarigos et al.,2010) [START_REF] Safaei | Study of Various Tensegrity Modules as Building Blocks for Slender Booms[END_REF] exists at the time certainly explains the low development of this kind of solution. Another more fundamental aspect is related to the intrinsic low structural stiffness of these systems [START_REF] Hanaor | Debunking" Tensegrity"-A Personal Perspective[END_REF], partly of second order geometric nature. That is why a real application is possible only with structures that limit finite mechanisms, similarly to conventional space truss. In that case, optimum design involves all of its parameters: the form, selfstress level, materials and cross-section characteristics of elements. This problem can be carried out in an exploratory manner in three parts: establishing a set of different configurations with varying geometry and material, designing the selfstress level and the appropriate cross-sections characteristics for each one and evaluating its structural performances, and finally, looking for the optimum solution.

Modular tensegrity system

We demonstrate this methodology on the case of a footbridge composed of two tensegrity beams generated by replication of 4 bars tensegrity modules. Each beam is deployable in a short time and stiffened by transverse cables added in the upper layer in order to block mechanisms and then limit vertical deflection [START_REF] Averseng | Design, Analysis and Self Stress Setting of a Lightweight Deployable Tensegrity Modular Structure[END_REF].

A simply supported rigid deck joints the two beams. Curvature is induced by mapping the system on a horizontal axis cylindrical surface (Figure 2). The whole structure is modelled as a space reticulate system in which the deck is represented by a series of transverses bars. So as to avoid torsion in the supporting beams, they are jointed at their ends to a specific reticulate sub-system that distribute the vertical forces to the four lower nodes of each module. In every selfstress system, the internal forces can be represented as a state vector built by combination of fundamental states. Those vectors actually constitute a base of the kernel of the equilibrium matrix, established from the static global equilibrium equation [START_REF] Quirant | Designing tensegrity systems: the case of a double layer grid[END_REF] [START_REF] Pellegrino | Matrix analysis of statically and kinematically indeterminate frameworks[END_REF]. In modular systems, those basis states are generally localized in every module [START_REF] Sanchez | Selfstress states identification and localization in modular tensegrity grids[END_REF]. To simplify the problem and to set a global uniform selfstress state realistically, we practice by similitude with the tensioning process proposed by [START_REF] Averseng | Prestressing tensegrity systems-application to multiple selfstress state structures[END_REF] by simulating the controlled shrinking of some elements qualified as "actives". In the presented system, those elements are the diagonal cables shared by two successive modules.

Structural design

For any given geometric and material configuration, the parameters to optimize are the selfstress level, which we define as the highest compression force among struts, and the cross-sections of every element. This dimensioning is processed iteratively with three steps (Figure 3): calculation of cross-sections given a selfstress level, structural analysis and serviceability check (no slackening of tendons under SLS load) then Ultimate Limit State checks (resistance in tension and buckling). First natural frequencies are not considered because they mainly depend of the geometry that, given a configuration, is fixed. In ULS situation, slackening is admitted, which involves carrying a non-linear structural analysis, through dynamic relaxation in our case [START_REF] Barnes | Applications of dynamic relaxation to the design and analysis of cable, membrane and pneumatic structures[END_REF]) [START_REF] Averseng | Interactive design and dynamic analysis of tensegrity systems[END_REF]. The global stability and resistance are checked according to the Eurocode standards. For the buckling of composite struts, provisions are derived from those applicable to steel, assuming higher imperfection and security factors. The process iterates, increasing either the selfstress level or cross-sections until all criteria are validated.

Finding an optimum

Each optimized solution of the panel of potential configurations is evaluated through the performance index 𝐼 !"#$ defined in equation ( 1).

𝐼 !"#$ = ! ! !"# •! (1)
In this expression, 𝑓 !"# is the ELS deflection (in meters) and 𝑚 the total mass of the structure (in 10 3 kg). As we can see, a high index signifies low deflection and low mass, which means supposedly high structural performances. To identify an optimal set of parameters, the performance indexes are used to built an explicit meta-model, actually a quadratic response surface as shown in equation ( 2), which is a continuous and derivable representation of the performance index in function of the geometric parameters.

𝐼 !"#$,!"# 𝑥 = 𝑎 + 𝑏 ! 𝑥 ! ! + 𝑐 !! 𝑥 ! ! ! + 𝑐 !" 𝑥 ! !!! ! = 𝑎 + ⋯ 𝑏 ! ⋯ 𝑥 + ! ! 𝑥 ! 𝐻𝑥 (2)
In this expression, 𝑥 is the column vector of the parameters defining the structure: the initial deflection 𝑓 ! , the width 𝑏, the height ℎ and the number 𝑛 of modules. The coefficients 𝑎, 𝑏 ! and 𝑐 !" are obtained by least square fitting on the set of performance indexes of the designed configurations. We present in Table 2 the response surfaces obtained for the three materials considered, with correlation indices of 0.987, 0.989 and 0.938 respectively for steel, timber and composite. These functions can be written under a matrix form, in which appear the Hessian matrix 𝐻 of the problem, from which an extremum can be deduced as in equation (3).

∇ 𝐼 !"#$,!"# 𝑥 = 0 𝑏 + 𝐻 𝑥 = 0 (3)

𝑥

The set of parameters of each optimum, one per considered material, are presented in Table 3.

𝐴 !".!"#$% cm 2 1 1 1 𝐴 !"#.!"#$% cm 2 1 1 1 𝐴 !"#$ cm 2 2.1 1.6 1.6
As can be seen, the optimized timber and steel structures share a similar weight (110 kg/m 2 ) that is lower than conventional solutions. Although the three solutions require a same module width, we observe that the timber footbridge needs significantly higher cross-section areas and module height, to improve the flexural inertia. Among the three solutions (Figure 4), the one based on composite differentiates clearly in lightness, due to a lower number of modules, despite longer struts. In all cases, the initial selfstress level needs to be around 115 kN, which is rather important. flexural rigidity. This suggests also improving the formulation of the performance index in order to better account for this criterion.

Active control

Active vibration control aims to lower the amplitude of vibrations induced in a structure by external actions that may excite its resonances. It consists in integrating actuators in the structure and commanding them so as to cancel vibrations. Several studies were raised on the control of tensegrity systems, using instantaneous optimal control [START_REF] Djouadi | Active control of tensegrity systems[END_REF], establishing control law from exact kinematics [START_REF] Skelton | Dynamics and control of tensegrity systems[END_REF], shifting soft modes through selfstress (Ali & Smith, 2010), or using robust synthesis [START_REF] Averseng | Active control of a tensegrity plane grid[END_REF] [START_REF] Tuanjie | Robust vibration control of flexible tensegrity structure via µ synthesis[END_REF]. In the continuity to these studies, we propose to control the first vibration modes of a plane tensegrity system with a new approach that consists in decomposing the actuation mode and control law by frequency domains, around each Eigen mode. Using 𝐻 ! synthesis technique, this method allows optimizing the efficiency of actuators while ensuring the robustness and stability of control law, by keeping them simple.

Experimental model

The methodology is applied to a plane modular tensegrity grid derived from the Tensarch project [START_REF] Motro | Tensarch : A tensegrity double layer grid prototype[END_REF] showed in Figure 1. It is formed by discontinuous sets of weaved struts in equilibrium inside a continuous network of tense cables. Two hydraulic actuators are integrated in this structure, in positions inducing tension locally in the lower layer, inducing vertical deflection (Figure 5). This solution is also effective in inducing vibration up to 30 Hz, which covers the major part of the dynamic behaviour of the structure. Vertical vibration levels are measured on the upper layer, above one of the actuators (A1), allowing observing all flexion modes of even order. An electrodynamic shaker is used to introduce an external force in (I1), at a certain offset from the symmetry axis to induce flexion and also torsion.

Identification

Two aspects of the dynamic behaviour are identified: the passive part, which results from external actions, and the active part, which is the transfer between the command to active 𝐺 ! .

To optimize the impact of actuators on each vibration resonance, they are coupled: they are commanded in phase for inducing flexion, and in opposition of phase to generate torsion. This means that the active part of the behaviour has to be defined for each regime. From the experimental results, identification is made by analogy with a simple rheological model (Figure 7). Pour caractériser la réponse passive de la structure, on réalise un essai de type balayage sinus à l'aide d'un pot vibrant (Fig. 2) situé en un point décalé par rapport à l'axe longitudinal de la structure afin d'en exciter les modes de flexion et torsion. Les signaux mesurés sont la force introduite par le pot en entrée et l'accélération verticale en nappe supérieure de la structure. Le comportement est ensuite déterminé sous la forme de la fonction de transfert accélération sur force (Fig. 3a). Sur ce graphe, on voit apparaitre les pics de résonnance des modes de torsion (13,4 Hz) et de flexion (17,4 Hz). L'objectif du contrôle actif sera de les attenuer.

Le comportement actif

Afin d'optimiser l'impact des activateurs sur chacun des régimes de vibration de la structure, leur commande est couplée : il sont pilotés en phase lorsqu'il s'agit d'agir sur le mode de flexion et en opposition de phase pour le mode de torsion. Le comportement actif est cette fois déterminé par l'intermédiare de la réponse fréquentielle (Fig. 3) entre la commande en entrée et l'accélération mesurée en sortie.

Identification du comportement

Le comportement dynamique de la structure étudiée est décomposé et identifié par similitude avec un modèle rhéologique simple. Trois paramètres M, C, K représentent

M K C actuator x0(t) u(t) x(t)

F(t)

element of variable length put in series. From the equilibrium equation, develop equations ( 4)-( 5):

𝑀𝑥 = 𝐹 -𝐾 𝑥 -𝑢 -𝐶 𝑥 -𝑢 (4)
𝑀𝑥 + 𝐶𝑥 + = 𝐹 + 𝐾𝑢 + 𝐶𝑢 (5)
The transfer function is obtained from the Laplace transform of equation ( 5) which leads to equations ( 6)-( 7).

𝐿[𝑀𝑥 + 𝐶𝑥 + 𝐾𝑥] = 𝐿[𝐹 + 𝐾𝑢 + 𝐶𝑢] (6)
𝑀𝑠 ! 𝑥(𝑠) + 𝐶𝑠𝑥(𝑠) + 𝐾𝑥(𝑠) = 𝐹(𝑠) + 𝐾𝑢(𝑠) + 𝐶𝑠𝑢(𝑠) (7)
We can then represent the behaviour of the structure by the two frequency response functions defined in equations ( 8)-( 9), for the passive part 𝐺 ! and the active part 𝐺 ! .

𝐺 ! = ! ! (! !!!) !! ! !! !!! (8) 𝐺 ! = ! ! !! ! !! !!! (9)
Around the resonance peak of each mode, identified experimentally, we can first evaluate the parameter 𝐾 by fixing 𝑀. We can then adjust the damping coefficient and the global level using a factor 𝛼, which represents the gain of the acquisition chain. The results are presented in Table 4.

Robust active control

Among the existing modern control methods, we are interested in the robust synthesis algorithms (LQG, PRLQG, 𝐻 ! , µ) because they consider the uncertainties and external disturbances that affect the model or the signals. These uncertainties vary a lot and can involve the real behaviour of joints, materials, and loading (snow for instance).

Figure 8. Typical closed-loop

In the 𝐻 ! approach (Fezans & el., 2008), the problem consists in minimizing the norm of the transfer between disturbances and criteria outputs. In a typical closed-loop (Figure 8.a), we note the system to control as 𝐺, the controller as 𝐾, the measured output 𝑦 and the command 𝑢. The uncertainties are modelized as external signals 𝑤 ! or 𝑤 ! added to the in the loop. The relationship between the output y and the other signals is developed in equation ( 10).

y = (1 + GK) !! w ! + (1 + GK) !! Gw ! + (1 + GK) !! GK(r -n) (10)
We introduce the following notations: 𝑆 = (1 + 𝐺𝐾) !! the output sensitivity, and 𝑇 = 𝐺𝐾(1 + 𝐺𝐾) !! the complementary output sensitivity. These functions are used to build specifications for the controller. In our case, the loop takes the form presented in Figure 9.a. The command u is in input of the active part 𝐺 ! of the system. T must be high in the bandwidth of the system, and low beyond to eliminate measurement noise. The external actions excite the Eigen modes of 𝐺 ! and can be assimilated to an external noise 𝑤 ! , so we need a low sensitivity 𝑆. Finally, the uncertainties of the system may be viewed as perturbations 𝑤 ! on the input, so robustness requires a low 𝐾𝑆 function. In the 𝐻 ! method, these conditions writes under a "standard" form (Figure 9.b), centered on the controller K, where P, presented in a matrix form in equation ( 11), describes the other part of the closed loop, the connections between external signals (𝑤 !"# and 𝑧) and internal ones (𝑦 and 𝑢).

K G

P = -𝑊 ! 𝐺 ! -𝐺 ! 𝑊 ! 𝑊 ! 𝐺 ! 𝐺 ! 𝑊 ! -𝐺 ! -𝐺 ! (11)
The 𝑧 ! signals are criteria output, images of 𝑆 and 𝑇 shaped by 𝑊 ! the functions that express performance requirements. The synthesis problem consists then in finding K respecting the closed loop conditions in equation ( 12) that summarize all the performance requirements.

𝑊 ! 𝑆 ! < 1 𝑊 ! 𝑇 ! < 1 (12)

Experimental results

Applying this method, we synthetized two controllers, one for each resonance mode in torsion and flexion. Each controller is implemented as a z filter in a control program under LabView™. The dominant frequency of the acceleration signal is used to switch, between the two modes at 15 Hz, the controller and the actuation mode from one regime to the other. We present in figure 10 a comparison of the behaviour acceleration over external force between the passive and the controlled system, measured by performing swept sine analysis. While the attenuation is modest in the domain corresponding to the torsion mode, we observe a more interesting impact on the flexion mode. In figure 11, we present the comparison of the spectrums of the response under random force, with and without control. Under this more realistic loading, obtained for each controller in torsion and flexion, we confirm a good attenuation of the peak in flexion and an encouraging result on torsion.

Conclusion

Tensegrity systems are little represented in construction, mainly because of a lack of specific analysis and design methods. In this paper, we presented a design methodology associated to the search of an optimum solution. The case is a whole footbridge composed of two curved modular tensegrity beams that can be erected by deployment.

Several materials are proposed and a large set of geometric parameters is explored.

Using the surface response technique on a set of designed configurations, a solution can be found that optimizes the structural performances, minimizing the mass and maximizing the flexural rigidity. Among three materials, similarities appear on the value of the initial deflection and the shape of the modules, confirming the relevance and equilibrium of the proposed solutions. Although they may not appear as competitive compared to conventional solutions, it is mainly the erection process, by deployment, that gives a strong advantage to these systems.

Figure 1 .

 1 Figure 1. Tensegrity systems : (a) Needle Tower and (b) Tensarch project

Figure 2 .

 2 Figure 2. Geometry composition of the footbridge : (a) module, (b) beam assembly, (c) initial deflection, (d) final structure with deck

Figure 3 .Figure 1 :

 31 Figure 3. Structural configuration design procedure

Figure 4 .

 4 Figure 4. Rendering of the final solutions.

Figure 5 .

 5 Figure 5. The active tensegrity grid : geometry, actuators (A1 & A2) and force exciter (I1)

  Figure Frequency response function of the active grid: (a) passive part 𝐺 ! and (b)

Figure 7 .

 7 Figure 7. Equivalent rheological model

Figure 3 .

 3 Figure 3. a) Réponse fréquentielle de la structure passive, b) Réponse fréquentielle de la structure active

Figure 9 .

 9 Figure 9. (a) Closed loop including 𝐺 ! and 𝐺 ! parts of the system and (b) standard form.

Figure 10 .

 10 Figure 10. Comparison of the passive and controlled response (a) in torsion and (b) in flexion.

Figure 11 .

 11 Figure 11.Comparison of the passive and controlled response for each controller, identified under random excitation.

  In addition to structural design, we developed a control strategy for attenuating first vibrations modes. The principle is to synthetize a closed loop controller using robust 𝐻 ! synthesis technique for each vibration mode. To manage this, the behaviour of the structure is identified in several frequency domains by similitude with a springmass system. The implemented controller is then adapted for each control regime, in torsion or flexion. This original methodology gives encouraging results on the attenuation of the first resonances. It is being extended to other modes and other structures, using simulations, in order to propose new active tensegrity structures.

Table 1 ,

 1 are the unknowns of this problem. Elements are classified in groups: lower and upper layer cables, struts, diagonal cables. All sections are considered full and circular, expect for steel and composite struts, made of circular hollow sections with a fixed diameter over thickness ratio.

	Table 1. Material and geometrical range of parameters
	number of modules	𝑛	6 to 14
	width	𝑏	40 cm to1.6 m
	height	ℎ	80 cm to 1.8 m
	initial deflection f0	𝑓 !	-0.5 to 1 m
	material		timber (C24, k mod = 0.6)
			steel (S235)
			carbon/epoxy composite (f c = 1200 MPa, E c = 125 GPa)

Table 2 .

 2 Structural performance index response surfaces for steel, timber and composite

	tensegrity footbridges
	Material						𝐼 !"#$,!"#
			-35.1				26.8	12.4 -6.57 1.76
	S235	31.3 +	-40.8 71.5	𝑥 +	1 2	𝑥 !	12.4 -6.57 -25.8 9.28 -5.07 19.1 -25.8 4.75	𝑥
			-8.35				1.76	4.75 -5.07 0.865
			-43.5				16.8	11.7 0.941 2.07
	C14	17.4 +	-65.4 101	𝑥 +	1 2	𝑥 !	11.7 0.941 -29.6 -5.92 -6.33 32.2 -29.6 7.99	𝑥
			-9.19				2.07	7.99 -6.33 1.01
			-102				117	23.3 -2.51 1.23
	composite	35.3 +	-63.6 107	𝑥 +	1 2	𝑥 !	23.3 -2.51 -97.2 75.9 -11.5 75.5 -97.2 11.8
			-9.87				1.23	11.8 -11.5 1.25

Table 3 .

 3 Material and geometrical characteristics of optimized solutions

		unit	S235	C24	composite
	𝑣 !	cm	40	60	60
	𝑏	cm	120	130	140
	ℎ	cm	130	170	120
	𝑛	-	10	8	6
	mass	kg	2474	2588	1807
	𝑓 !	Hz	3.26	4.22	3.7
	selfstress	kN	116.5	115	110

Table 4 .

 4 Mechanical parameters of equivalent models, for each mode

	frequency	mode	M (kg)	K (N/m)	C (Ns/m)	𝛼
	13.42	torsion	100 kg	280.63 10 3	300	4 10 -7
	17.42	flexion	100 kg	1198 10 6	350	3.4 10 -7


Florian Bailly
email: florian.bailly@psl.aphp.fr

Bruno Fautrel

Laure Gossec

Pain Assessment in Rheumatology -How Can We Do Better? A Literature Review

Keywords: Pain, Patient-reported outcomes, Questionnaire, Score

Patients' symptoms have a place of prominence in the rheumatology landscape, and among them pain is the most conspicuous. Several pain assessment tools have been validated. Onedimensional pain scales such as visual analog scales (VASs) and numeric rating scales (NRSs) are fast to administer but have limitations that must be acknowledged. Some clinical situations require the use of multidimensional scales such as the McGill Pain Questionnaire or the Multidimensional Pain Inventory (MPI). These tools allow the assessment and management of the diverse components of pain. Here, we review the main patient-reported outcomes that can serve to evaluate pain and the psychometric properties of pain assessment tools. We also discuss the selection of the tool most appropriate for each situation (e.g., everyday practice and research).

Introduction

Reliable pain assessments are crucial to the optimal management of patients with painful conditions. Most rheumatic diseases cause pain. Pain is both a common inaugural symptom and a major quality-of-life issue in chronic rheumatic diseases. In rheumatoid arthritis (RA), for instance, pain is the main determinant of the impact of the disease on the patient's life. During the follow-up of patients with painful diseases, the results of pain assessments play a major role in guiding treatment decisions.

When evaluating pain, in addition to the intensity of the symptom, specific characteristics such as location and triggers must be determined. Giving careful attention to this distressing symptom both benefits the patient-physician relationship and increases the chances of achieving optimal pain relief [START_REF] Machado | Efficacy and safety of paracetamol for spinal pain and osteoarthritis: systematic review and meta-analysis of randomised placebo controlled trials[END_REF].

In addition to open discussion with the patient, questionnaires designed to assess patient-reported outcomes can help in everyday practice. Familiarity with these questionnaires is also necessary to interpret the results of scientific research.

Here, we review the tools used to assess pain in patients with rheumatic diseases, their psychometric properties, and the interpretation of their results. We also discuss the selection of the tool best suited to each situation.

Psychometric properties of pain assessment tools

Assessment tools must be validated by studies of their metrological properties. The filter developed by methodologists working for the Outcome Measures in Rheumatology (OMERACT) group distinguishes three metrological properties: truth, feasibility, and discriminant capacity [START_REF] Boers | The OMERACT filter for Outcome Measures in Rheumatology[END_REF][START_REF] Kirwan | Updating the OMERACT filter: implications for patient-reported outcomes[END_REF]. The meaning of these words deserves discussion.

Truth

Truth (or validity) is the ability of the tool to accurately measure what it is designed to measure [START_REF] Boers | The OMERACT filter for Outcome Measures in Rheumatology[END_REF]. Truth is the most important metrological property.

Face validity reflects the subjective perception by the experts and/or patients of overall tool quality (wording and comprehension of the items, response modalities, and domains covered). Content validity refers to the relevance of the items and the ability of the tool to cover all the aspects of what is being measured. Construct validity evaluates the relevance of the various dimensions that make up the tool, i.e., the internal validity of the tool, as assessed by Cronbach's alpha (which can range from 0 to 1). Concurrent validity or criterion validity is assessed by comparing the new tool to previously validated tools. Criterion validity reflects the correlation between results provided by the new tool and those obtained using the external reference standard.

Reliability

Reliability reflects the extent of measurement error, i.

Sensitivity to change

Sensitivity to change is the ability of a measurement tool to detect a change in clinical status between two situations or two time points. A tool that is sensitive to change reflects variations that occur between two different time points, for instance under the effect of a treatment. Discriminant capacity, in contrast, is the ability to distinguish between two different states (e.g., healthy versus ill). Discriminant capacity and sensitivity to change are particularly important for assessing treatment responses. They can be measured by computing the effect size, which is considered relevant when greater than 0.80.

Feasibility

Feasibility, or simplicity, is the objective acceptability of the tool, which depends on factors such as administration time, inconvenience to the patient, cost and/or availability of the measurement devices. Feasibility stands apart from other metrological properties, as it cannot be computed statistically, except regarding the administration time and, in some cases, the missing data rate.

Pain assessment questionnaires, scores, and scales

Many pain assessment questionnaires can be used in rheumatology. However, we will confine our discussion to questionnaires whose metrological properties have been validated.

At least nine such questionnaires are available (Table 1). We will review their main features.

Generic one-dimensional scales

One-dimensional scales are simple to complete and can easily be used on multiple occasions. Therefore, they can serve to measure changes in pain intensity occurring spontaneously or in response to analgesic treatment. An absolute or relative cutoff can be readily defined. However, these scales fail to distinguish among the various components of pain.

Visual analog scale (VAS) for pain

The pain VAS is probably the most widely used self-assessment tool in everyday practice. Pain intensity is measured by placing a mark on a 10-cm long horizontal or vertical line segment that goes from "no pain" to "worst imaginable pain" [START_REF] Huskisson | Measurement of pain[END_REF][START_REF] Scott | Graphic representation of pain[END_REF]. The scale can be used in print form or as a clear ruler with a slider that the patient places at the appropriate place on the line segment, allowing the observer to read the score in millimeters on the reverse. The score is a continuous variable that can range from 0 to 100 (corresponding to the distance in millimeters between the no-pain end of the line segment and the patient's mark). The VAS may also be graduated in centimeters and scored from 0 to 10 or colored toward increasingly bright red at the maximum-pain end of the line segment. To avoid the use by the patient of a preferred or habitual value, no words should be placed on the line segment between the two ends, and the presentation should be neutral [START_REF] Scott | Graphic representation of pain[END_REF].

Metrology:

The correlation coefficients between VAS pain scores and verbal rating scale (VRS) scores range from 0.70 to 0.75 [START_REF] Downie | Studies with pain rating scales[END_REF]. The VAS pain score is sensitive to change [START_REF] Joyce | Comparison of fixed interval and visual analogue scales for rating chronic pain[END_REF]. The minimal clinically important difference is 11 to 20 mm or a 30% score decrease, depending on the health condition [START_REF] Tashjian | Minimal clinically important differences (MCID) and patient acceptable symptomatic state (PASS) for visual analog scales (VAS) measuring pain in patients treated for rotator cuff disease[END_REF][START_REF] Wolfe | Assessment of pain in rheumatoid arthritis: Minimal clinically significant difference, predictors, and the effect of anti-tumor necrosis factor therapy[END_REF][START_REF] Farrar | Clinical importance of changes in chronic pain intensity measured on an 11-point numerical pain rating scale[END_REF].

Numerical rating scale (NRS)

The NRS is a one-dimensional pain assessment tool [START_REF] Childs | Responsiveness of the numeric pain rating scale in patients with low back pain[END_REF][START_REF] Jensen | Increasing the reliability and validity of pain intensity measurement in chronic pain patients[END_REF] available as several versions. The most widely used version, which is derived from the VAS, is a graduated tool marked with integers from 0 to 10, with higher numbers indicating greater pain intensity. As with the VAS, one end is marked "no pain" and the other end "worst imaginable pain" [START_REF] Downie | Studies with pain rating scales[END_REF][START_REF] Childs | Responsiveness of the numeric pain rating scale in patients with low back pain[END_REF][START_REF] Jensen | Increasing the reliability and validity of pain intensity measurement in chronic pain patients[END_REF]. The NRS can be administered orally or as a horizontal line bearing the numbers [START_REF] Jensen | The Measurement of Clinical Pain Intensity: a Comparison of Six Methods[END_REF]. Some patient with chronic pain prefer the NRS over the VAS [START_REF] Hush | Do numerical rating scales and the Roland-Morris Disability Questionnaire capture changes that are meaningful to patients with persistent back pain?[END_REF][START_REF] Williams | Simple pain rating scales hide complex idiosyncratic meanings[END_REF].

Metrology: NRS scores correlate strongly (0.86 to 0.95) with VAS pain scores [START_REF] Downie | Studies with pain rating scales[END_REF][START_REF] Ferraz | Reliability of pain scales in the assessment of literate and illiterate patients with rheumatoid arthritis[END_REF].

The minimal clinically important difference is 2 points or a 30% change [START_REF] Farrar | Clinical importance of changes in chronic pain intensity measured on an 11-point numerical pain rating scale[END_REF].

Verbal rating scale (VRS)

The VRS is a Likert-type scale consisting in a ranked series of verbal descriptors such as no pain, mild pain, moderate pain, severe pain, and very severe pain [START_REF] Keele | The Pain Chart[END_REF]. However, no validated French version is available, and the variability in the number of descriptors complicates the interpretation of the results. The VRS is used chiefly in patients who are unable to use a VAS or NRS (due to poor comprehension or difficulty with abstract concepts).

Metrology:

The results of the VAS, NRS, and VRS show correlations ranging from 0.70 to 0.75 [START_REF] Reading | A comparison of pain rating scales[END_REF][START_REF] Atkinson | Measurement of pain: patient preference does not confound pain measurement[END_REF]. However, the correlations are not linear and the three scales are not interchangeable. Furthermore, variations in the placebo effect across scale types have not been evaluated.

Multidimensional scales

Multidimensional scales provide information on the various aspects of pain, including its sensory and emotional components. However, they have longer administration and scoring times compared to one-dimensional scales.

The McGill Pain Questionnaire (MPQ)

The McGill Pain Questionnaire (MPQ) [START_REF] Melzack | The McGill pain questionnaire: major properties and scoring methods[END_REF] is an English-language list of 78 verbal descriptors grouped into 20 subclasses (each of which covers one aspect of the pain). These 20 subclasses are distributed among four classes: sensory or sensory-discriminative, affective or affective-emotional, evaluative, and miscellaneous (sensory-affective). The MPQ not only evaluates pain intensity, but also provides information on many other characteristics, of which knowledge is useful in patients with chronic pain. Two other sections evaluate pain intensity and changes in the pain over time. The score can range from 0 to 78.

Metrology:

A metaanalysis of 51 studies found that the MPQ score varied from 24% to 50% of the maximal value [START_REF] Wilkie | Use of the McGill Pain Questionnaire to measure pain: a meta-analysis[END_REF]. Reproducibility and sensitivity to change are good (Table 1) [START_REF] Burckhardt | Adult measures of pain: The McGill Pain Questionnaire (MPQ), Rheumatoid Arthritis Pain Scale (RAPS), Short-Form McGill Pain Questionnaire (SF-MPQ), Verbal Descriptive Scale (VDS), Visual Analog Scale (VAS), and West Haven-Yale Multidisciplinary Pain Inventory[END_REF][START_REF] Burckhardt | The use of the McGill Pain Questionnaire in assessing arthritis pain[END_REF]. Validity studies in patients with different types of joint pain showed that similar descriptors were selected by patients having the same health condition and that the number of selected descriptors correlated with the VAS score [START_REF] Papageorgiou | The quality of pain in arthritis: The words patients use to describe overall pain and pain in individual joints at rest and on movement[END_REF]. Nevertheless, the MPQ uses a rich vocabulary and is therefore not suitable for patients with a low level of literacy. No cutoff has been determined to assist in interpreting the results.

The Short-Form McGill Pain Questionnaire (SF-MPQ)

The Short-Form McGill Pain Questionnaire (SF-MPQ) uses only 15 descriptors from the full-length version [START_REF] Melzack | The short-form McGill Pain Questionnaire[END_REF] (11 in the sensory class and 4 in the affective class), one item on pain intensity, and a VAS for pain (Appendix 1). Each descriptor is rated on a 4-point intensity scale (0, absent; 1, mild; 2, moderate; and 3, severe). The maximum score is 60 points, and no cutoff has been determined. The administration time is shorter than for the full-length MPQ (2-5 minutes versus 5-15 minutes).

Metrology:

In two different populations, the correlation coefficient between the MPQ and SF-MPQ ranged from 0.67 to 0.87 [START_REF] Burckhardt | A Swedish version of the short-form McGill Pain Questionnaire[END_REF]. Internal consistency in patients with fibromyalgia or RA was 0.73 to 0.89 [START_REF] Burckhardt | A Swedish version of the short-form McGill Pain Questionnaire[END_REF]. Sensitivity to change and content validity varied across health conditions [START_REF] Boureau | Construction of a questionnaire for the selfevaluation of pain using a list of qualifiers. Comparison with Melzack's McGill Pain Questionnaire[END_REF].

The Saint-Antoine Pain Questionnaire

This tool is a modified French version of the MPQ [START_REF] Boureau | Construction of a questionnaire for the selfevaluation of pain using a list of qualifiers. Comparison with Melzack's McGill Pain Questionnaire[END_REF]. It is composed of 61 descriptors grouped into 17 subclasses including 9 sensory, 7 affective, and 1 evaluative. The results are heavily dependent on the sociocultural background and verbal expression skills of the patient.

Metrology: Content validity is similar to that of the full-length MPQ [START_REF] Boureau | Comparative study of the validity of four French McGill Pain Questionnaire (MPQ) versions[END_REF]. A short 15item version has been developed and is awaiting evaluation [START_REF] Boureau | Comparative study of the validity of four French McGill Pain Questionnaire (MPQ) versions[END_REF] (Appendix 2).

The West Haven-Yale Multidimensional Pain Inventory (WHYMPI)

A validated French version of the West Haven-Yale Multidimensional Pain Inventory (WHYMPI) is available. This 49-item tool evaluates several dimensions of the impact of chronic pain including patient experience with the pain, perceived support from the significant other (usually the spouse), and daily activities [START_REF] Kerns | The West Haven-Yale Multidimensional Pain Inventory (WHYMPI)[END_REF][START_REF] Laliberté | French translation of the Multidimensional Pain Inventory: L'inventaire multidimensionnel de la douleur[END_REF]. Three coping categories have been suggested: dysfunctional, interpersonally distressed, and minimizers/adaptive copers [START_REF] Turk | Toward an empirically derived taxonomy of chronic pain patients: integration of psychological assessment data[END_REF].

Metrology:

The metrological properties seem satisfactory (Table 1) [START_REF] Burckhardt | Adult measures of pain: The McGill Pain Questionnaire (MPQ), Rheumatoid Arthritis Pain Scale (RAPS), Short-Form McGill Pain Questionnaire (SF-MPQ), Verbal Descriptive Scale (VDS), Visual Analog Scale (VAS), and West Haven-Yale Multidisciplinary Pain Inventory[END_REF][START_REF] Flavell | Managing chronic back pain: impact of an interdisciplinary team approach[END_REF][START_REF] Kerns | Pain-relevant responses from significant others: Development of a significant-other version of the WHYMPI scales[END_REF].

The Short Form 36 Bodily Pain Scale (SF-36 BPS)

The Short Form 36 Bodily Pain Scale (SF-36 BPS) is one of the eight subscales of the SF-36, a generic quality-of-life scale of which a validated and copyrighted French version is available [START_REF] Leplège | The French SF-36 Health Survey: translation, cultural adaptation and preliminary psychometric evaluation[END_REF]. The two SF-36 BPS items evaluate pain intensity on a six-grade Likert scale and the impact of the pain on normal work on a five-grade Likert scale. The score is computed by summing the subscores on each item then converting the result to a value on a 0-100 scale.

Metrology:

The SF-36 has demonstrated good psychometric properties [START_REF] Quintana | Responsiveness and clinically important differences for the WOMAC and SF-36 after hip joint replacement[END_REF][START_REF] Escobar | Responsiveness and clinically important differences for the WOMAC and SF-36 after total knee replacement[END_REF].

However, the SF-36 BPS subscale has not been evaluated separately.

Measure of Intermittent and Constant Osteoarthritis Pain (ICOAP)

The measure of Intermittent and Constant Osteoarthritis Pain (ICOAP) is a multidimensional tool specifically designed to assess pain due to osteoarthritis [START_REF] Hawker G A | Development and preliminary psychometric testing of a new OA pain measure -an OARSI/OMERACT initiative[END_REF]. It evaluates pain intensity, frequency, and impact on mood, sleep, and quality of life, independently from disability. The French version has been validated [START_REF] Maillefert | Multi-language translation and crosscultural adaptation of the OARSI/OMERACT measure of intermittent and constant osteoarthritis pain (ICOAP)[END_REF]. Five items assess constant pain and six intermittent pain. The total score based on all 11 items is computed.

Metrology:

The validity [START_REF] Hawker | Development and preliminary psychometric testing of a new OA pain measure--an OARSI/OMERACT initiative[END_REF] and sensitivity to change of the ICOAP have been established. Experience with this recently introduced tool is still limited.

Discussion

An accurate and appropriate assessment of pain allows clinicians to develop an individually tailored management strategy. Pain assessment tools in French are available on the website of the French Society for Pain Assessment and Treatment (Société Française d'Etude et de Traitement de la Douleur, http://www.sfetd-douleur.org/adulte [40]).

One-dimensional tools such as the VAS, NRS, and VRS are fast and easy-to-use instruments for evaluating pain intensity. The validity of these three tools is similar in literate patients, but ease of administration and sensitivity to change are somewhat better with the NRS [START_REF] Jensen | The Measurement of Clinical Pain Intensity: a Comparison of Six Methods[END_REF]. In contrast, the VRS deserves preference in the very young, very elderly, and cognitively impaired [START_REF] Herr | Pain intensity assessment in older adults: use of experimental pain to compare psychometric properties and usability of selected pain scales with younger adults[END_REF]. For research purposes, the NRS is preferable because it can be administered either on paper or orally. None of these one-dimensional scales provides a comprehensive assessment of the pain component. However, their fast administration allows them to be used for repeated self-assessments. Completing a NRS at regular intervals (e.g., once a week) between two physician visits, particularly using electronic tools (e.g., websites), may improve the evaluation of the patient's pain experience. Nevertheless, in the medium term, many patients may be unwilling to perform regular self-assessments [START_REF] Koevoets | Autonomous online health assessment questionnaire registry in daily clinical practice[END_REF].

Also, repeated pain assessments might induce changes in the pain, particularly an increase in pain intensity. This increase may be promoted by the trivialization of repeated numerical ratings. The optimal frequency of pain assessments has not been determined and may constitute a relevant area for future research.

Furthermore, behavioral changes due to the knowledge that one is being observed or studied (Hawthorne effect) may lead to a decline in pain intensity [START_REF] Turner | The importance of placebo effects in pain treatment and research[END_REF]. One factor that may influence the Hawthorne effect is the personality of the evaluator.

When using a VAS or NRS, some patients report a pain intensity of 10/10 despite appearing to be free of distress. This puzzling behavior suggests an attempt to reflect the multidimensional components and consequences of pain in the one-dimensional assessment by indicating a higher pain intensity. For instance, the score assigned by the patient may be intended to reflect not only the intensity, but also the duration of the pain and its impact on psychological well-being or function. These apparently paradoxical score results indicate a need for using a multidimensional tool in order to capture the many aspects of the pain experience.

In studies that focus specifically on pain, multidimensional questionnaires are often given preference over one-dimensional tools. The MPQ and its French equivalent (Saint Antoine Pain Questionnaire) are widely used in research to measure not only the intensity, but also the type of pain. The SF-36 and SF-36 are useful for assessing pain within the overall setting of health and quality of life and for comparing different populations. In contrast to the Saint-Antoine Pain Questionnaire, the SF-36 BPS is sufficiently simple to be suitable for use in everyday clinical practice. Specific questionnaires such as the ICOAP provide detailed assessments in patients with specific diseases, both in clinical practice and research studies.

An important consideration is the appropriateness of the questionnaire for the cognitive and intellectual status of the patient. In patients who are unable to communicate verbally, options include the DOLOPLUS and ALGOPLUS questionnaires [START_REF] Rat | Validation of an acute pain-behavior scale for older persons with inability to communicate verbally: Algoplus®[END_REF] and the Faces Pain Scale. However, detecting difficulties with abstract concepts or subtle cognitive impairments may be difficult. This point is relevant to all patient-reported outcomes. A study from Great Britain of 10 patient-reported outcome questionnaires used in rheumatology found that 22% of the participants would be unable to complete any of the questionnaires [START_REF] Adams | Literacy levels required to complete routinely used patient-reported outcome measures in rheumatology[END_REF].

In conclusion, a formal pain assessment is an indispensable first step in the management of patients with pain. In this article, we have reviewed the main validated pain assessment tools and discussed the selection of the best tool for each specific situation. We hope this review will help clinicians gain familiarity with pain assessment issues in clinical practice and research.

Disclosure of interest:

None of the authors has any conflicts of interest to declare.

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