Claire Paya
email: claire.paya@st.com

Euriell Le Corronc
email: euriell.le.corronc@laas.fr

Yannick Pencolé
email: yannick.pencolé@laas.fr

Philippe Vialletelle
email: philippe.vialletelle@st.com

Observer-based detection of time shift failures in (max,+)-linear systems

In this paper, we address the problem of failure detection in production lines modeled as Timed Event Graphs (TEG). The proposed method represents TEGs as (max,+)-linear systems with disturbance and aims at detecting time shift failures in the underlying production lines. To do so, we will reconstruct the state of the observed system and define an indicator relying on the residuation theory on (max,+)-linear systems.

Introduction

In industry, Discrete Event Systems (DES) can be used to model and solve fault diagnosis problems in automated production lines. The objective is to detect, localize and ultimately identify failures as soon as possible to avoid further equipment unavailability. In systems like production lines, failures can be not only caused by complete equipment breakdowns but also by the occurrence of time shifts so the production line can dramatically slow down and not be able to comply with the specified production objectives. This paper addresses the problem of detecting and localizing the source of such time shifts. One of the first methods used to diagnose failures in DES is proposed in [SSL + 95] on untimed automata. Then, [START_REF] Tripakis | Fault Diagnosis for Timed Automata[END_REF] defines the diagnosis problem on timed automata so that diagnostic decisions are refined by using timed observations. In [START_REF] Ghazel | State observer for des under partial observation with time petri nets[END_REF], the diagnosis is based on time Petri Nets that is well-suited to model competition and parallelism within the system. One subclass of time Petri Nets, called Timed Event Graph (TEG), can also be used to specifically represent systems like production lines. In TEGs, places are associated with a punctual duration and they can be modeled by (max,+) algebra as introduced in [BCOQ92, Max91]. [KLBvdB18] presents the history of DES with the use of (max,+) algebra. For example, [START_REF] Kim | Feedback control of cluster tools for regulating wafer delays[END_REF] uses (max,+) algebra to control wafer delays in cluster tools for semiconductor production. The problem of failure diagnosis by the use of (max,+) algebra has been introduced in [START_REF] Sahuguède | Design of indicators for the detection of time shift failures in (max, +)-linear systems[END_REF] where the proposed detection method relies on the residuation theory and compares observable outputs with expected ones to detect output time shifts. Failure localisation is then performed by an adhoc structural analysis of the underlying TEG that does not use (max,+) algebra. The objective of this paper is to design a new set of time shift failure indicators that are not based on the observable outputs of the system only but on the estimation of the internal state of the system so that the fail-ure localisation problem is also solved in an algebraic way. To do so, the proposed failure indicator will rely on an observer that is proposed in [START_REF] Hardouin | Observer design for (max,+) linear systems[END_REF] and aim at rebuilding system'states based on the observations. The paper is organized as follows. Section 2 presents a motivation example inspired from the semiconductor industry. Section 3 summarizes the necessary mathematical background about (max,+)-linear systems. Section 4 describes how the time shift failure problem in system can be turned into the problem of state estimation in a system with input disturbance and gives the construction of an observer of such a system. Section 5 finally defines the proposed time shift failure indicators.

Motivation example

The problem that we address is motivated by a real production line that is at STMicroelectronics Crolles300 plant. STMicroelectronics is among the world's largest semiconductor companies, serving all electronics segments. Semiconductor manufacturing is complex and one of its most important challenges is to succeed in detecting production drifts before they have real impact on production plan. STMicroelectronics has complex production lines of wafer batches with many pieces of equipment running in parallel. One of the objectives is to detect as soon as possible that an equipment is late to ensure that products (wafer batches) are delivered on time or at least with minimal delays. If the line is not faulty, the wafer batch that arrives at t=1 on Eq2 (resp. Eq1) is processed in 4 hours (resp. 3 hours) and then takes 1 hour (2 hours) to arrive in front of Eq3. Synchronization is done at t=7 and the batches are processed by Eq3 in 4 hours. So, the fully processed batch should come out at t=11. However, the first real output comes out one hour later: a time drift has happened. Following the same idea, considering output y 3 , the batch that arrives at t=1 on Eq2, should be processed in 4 hours and process information should come at t=6, hence another time drift.

In production lines, such time drifts are considered as time shift failures. This paper aims at designing a (max,+)algebraic indicator that detects them based on a model as the one presented in Figure 1. TEGs can be formally defined as (max,+)-linear systems that are introduced in the next section. It relies on the dioid M ax in [[γ, δ]] where the successive fires of a transition x of the TEG are characterized by a series representing the events γ and the fire timings δ.

Mathematical background

This section recalls the mathematical background used in this paper for describing (max,+)-linear systems [START_REF] Baccelli | Synchronization and linearity: an algebra for discrete event systems[END_REF]Max91].

Dioid theory

The dioid theory is used to describe the inputs and the behavior of the studied system. In particular, series of a specific dioid are defined to obtain the trajectories of inputs and states flows of timed events. Definition 1. A dioid D is a set composed of two internal operations ⊕ and ⊗. The addition ⊕ is associative, commutative, idempotent (i.e. ∀a ∈ D, a⊕a = a) and has a neutral element ε. The multiplication ⊗ is associative, distributive on the right and the left over the addition ⊕ and has a neutral element e. Element ε is absorbing by ⊗. When there is no ambiguity, the symbol ⊗ is omitted. Definition 2. A dioid is complete if it is closed for infinite sums and if ⊗ is distributive over infinite sums. Example 1. The dioid Z max = (Z ∪ -∞) endowed with the max operation as addition ⊕ and the addition as multiplication ⊗ with neutral element denoted ε = -∞ and e = 0. The dioid Z max is not complete because +∞ does not belong to the set Z max so the infinite sum is not set to +∞. By adding +∞ to the dioid Z max , we get the complete dioid Z max where Graphically, a series of B[[γ, δ]] is described by a collection of point of coordinates (n, t) in Z 2 with γ as horizontal axis and δ as vertical axis. For instance, Figure 2 shows a series u 1 = u 2 = γ 0 δ 1 ⊕ γ 1 δ 2 ⊕ γ 2 δ 3 ⊕ γ 3 δ 4 ⊕ γ 4 δ 5 ⊕ γ 5 δ 6 ⊕ γ 6 δ 7 ⊕ γ 7 δ +∞ (monomials with e as Boolean coefficient). In the following, we will consider the dioid

(-∞) + (+∞) = (-∞). Theorem 1 ( [BCOQ92]). Let D be a complete dioid, x = a * b is the solution of x = ax ⊕ b,
M ax in [[γ, δ]]. It 1 2 3 4 5 6 1 2 3 4 5 6 7 u 1 = u 2 γ δ Figure 2: Representation of inputs u 1 = u 2 is the quotient of the dioid B[[γ, δ]] by the modulo γ * (δ -1 ) * . The dioid M ax in [[γ, δ]] is a complete dioid with ∀a, b ∈ M ax in [[γ, δ]]: a = b ⇔ aγ * (δ -1 ) * = bγ * (δ -1 ) * .
(n) from Z → Z such that s = n∈Z γ n δ Ds(n) .
Considering the TEG of Figure 1 and the associated scenario defined in Section 2, inputs u 1 and u 2 are represented by the series

u 1 = u 2 = γ 0 δ 1 ⊕γ 1 δ 2 ⊕γ 2 δ 3 ⊕γ 3 δ 4 ⊕γ 4 δ 5 ⊕ γ 5 δ 6 ⊕ γ 6 δ 7 ⊕ γ 7 δ +∞ from the M ax in [[γ, δ]] dioid.
The absence of an 8 th wafer batch in the scenario is indicated by +∞ in monomial γ 7 δ +∞ . Figure 2 is the graphical representation of u 1 = u 2 . This series is composed of monomials γδ, where γ represents the events of the series, and δ represents the dates of the series. Series u 1 has for dater function

D u1 (0) = 1, D u1 (1) = 2, D u1 (2) = 3, D u1 (3) = 4, D u1 (4) = 5, D u1
(5) = 6 and D u1 (6) = 7. This dater function lists all the dates of the event occurrences. u 2 has obviously the same dater function (as

u 1 = u 2 ). Definition 5. Let s ∈ M ax in [[γ, δ]] be a series, its canonical form is s = K k=0 γ n k δ t k with K ∈ N∪{+∞} and n 0 < n 1 < . . . t 0 < t 1 < . . .
∀n ∈ Z, D b• /a (0) ≤ T a,b (n) ≤ -D a• /b (0), where D b• /a (0) is obtained from monomial γ 0 δ D b• /a (0) of se- ries b• /a and D a• /b (0) is obtained from γ 0 δ D a• /b (0) of series a• /b. Definition 9. Let a, b ∈ M ax in [[γ, δ]],
the time shift between series a and b is

∆(a, b) = [D b• /a (0); -D a• /b (0)],
(2)

where γ 0 δ D b• /a (0) ∈ b• /a and γ 0 δ D a• /b (0) ∈ a• /b.
In this interval, the series from which the time offset is measured is the series a. It is called the reference series of the interval.

From this definition, if the time shift interval needs to be defined with series b as the reference series, the interval will be

∆(b, a) = [D a• /b (0); -D b• /a (0)].
Example 4. Generally speaking, let us consider two different series a = γ 0 δ 12 ⊕ γ 1 δ 15 ⊕ γ 2 δ 18 ⊕ γ 3 δ 21 ⊕ γ 4 δ +∞ and b = γ 0 δ 12 ⊕ γ 1 δ 15 ⊕ γ 2 δ 19 ⊕ γ 3 δ 23 ⊕ γ 4 δ +∞ . The minimal time shift between a and b is D b• /a (0) = 0 and is found in the monomial where the degree of γ is 0 in b• /a = γ 0 δ 0 ⊕ γ 1 δ 3 ⊕ γ 2 δ 7 ⊕ γ 3 δ 11 ⊕ γ 4 δ +∞ . The maximal time shift is -D a• /b (0) = 2 and is found in γ 0 δ -2 from a• /b = γ 0 δ -2 ⊕γ 1 δ 2 ⊕γ 2 δ 6 ⊕γ 3 δ 9 ⊕γ 4 δ +∞ . The time shift interval is ∆(a, b) = [0; 2]. The distance between a and b is a minimum of 0 and a maximum of 2 hours meaning that series a is faster than b.

Models of (max,+)-linear systems

The elements of the TEG are represented by equations in M ax in [[γ, δ]]. The equations can be grouped into a set of matrices A, B and C that contain information about the structure of the TEG. The state representation defines relations between any set of input event flows u and the state x, and the relations between the state x and the output event flows y. Let u ∈ M ax in [[γ, δ]] p×1 be the input vector of size p, x ∈ M ax in [[γ, δ]] n×1 be the state vector of size n and y ∈ M ax in [[γ, δ]] q×1 be the output vector of size q. The state representation is:

x = Ax ⊕ Bu, y = Cx, where A ∈ M ax in [[γ, δ]] n×n , B ∈ M ax in [[γ, δ]] n×p and C ∈ M ax in [[γ, δ]
] q×n . Equality x = Ax ⊕ Bu can be transformed to x = A * Bu thanks to Theorem 1 so we have y = CA * Bu.

Matrix H = CA * B represents the transfer function of the TEG, that is the dynamic of the system between the inputs and the outputs.

For the system of Figure 1 the matrices

A ∈ M ax in [[γ, δ]] 6×6 , B ∈ M ax in [[γ, δ]] 6×2 and C ∈ M ax in [[γ, δ]] 3×6
of the state representation are:

A =        . γ 1 δ 0 . . . . γ 0 δ 3 . . . . . . . . γ 1 δ 0 . . . . γ 0 δ 4 . . . . γ 0 δ 2 . γ 0 δ 1 . γ 1 δ 0 . . . . γ 0 δ 4 .        , B =        γ 0 δ 1 . . . . γ 0 δ 1 . . . . . .        , C =   . . . . . γ 0 δ 0 . γ 0 δ 0 . . . . . . . γ 0 δ 0 . .   .
The exponent n of γ represents the backward event shift between transitions (the n + 1 th firing of x 1 depends on the n th firing of x 2 ) and the exponent of δ represents the backward time shift between transition (the firing date of x 2 depends on the firing date of x 1 and time between 2 and 5). time shift failures?

The objective of the paper is to propose a method that detects time shift failures as proposed in Section 2 and that uses an observer as introduced in [HMCL10] and [HM-CSM10]. As later detailed in Section 4.2, this observer aims at computing a reconstructed state from the observation of the inputs and outputs of the system that is sensitive to a specific type of disturbance. These disturbances are characterized as new inputs w that slow down the system. Based on this characterisation, the system will then be assumed to behave with respect to the following state representation.

x = Ax ⊕ Bu ⊕ Rw, y = Cx.
(3) Section 4.1 describes how time shift failures can be characterized as such disturbances. Section 4.2 then introduces the observer that will be used in the proposed detection method.

Time shift failures as input disturbances

Throughout this paper, we consider that time shift failures are permanent phenomena that can occur at any step of the production. Formally speaking, a time shift failure is characterized by an unexpected and unknown delay d > 0 that is added to the normal duration time t of a place p.

As shown on Figure 3, this place is characterized by a transition upstream x i-1 , a duration t, a number of tokens o and a transition downstream x i . Let x i-1 = K n=0 γ sn δ hn (see Definition 5), where s n is the transition firing number, h n is the firing date and K the number of firing events. The normal downstream transition is

x i = K n=0 γ sn+o δ hn+t .
When a time shift failure d > 0 holds in a place, the downstream transition then becomes:

x i = K n=0 γ sn+o δ hn+t+d .
To characterize the same time shift failure over the place p by a disturbance, we will first modify the TEG. We add to the downstream transition x i an input w i , as shown in Figure 4, which slows down this transition. This new input w i is not observed because it is related to a failure in an equipment. To get the same effect of an offset d > 0 in the downstream transition, input w i has to be defined as

w i = k n=0 γ sn+o δ hn+t+d . ( 4
)
Back to Figure 1 where the place p 5 has a duration of t = 4. To characterize an offset, meaning a time shift failure, of d = 1, we add a disturbance w 4 to the transition x 4 after the place p 5 in the same configuration as Figure 4. Suppose that x 3 = γ 0 δ 2 ⊕ γ 1 δ 6 ⊕ γ 2 δ 10 ⊕ γ 3 δ 14 ⊕ γ 4 δ 18 ⊕ γ 5 δ 22 ⊕ γ 6 δ 26 ⊕ γ 7 δ +∞ . Since an offset of 1 time unit is present on

p 5 , x 4 = γ 0 δ 2+4+1 ⊕γ 1 δ 6+4+1 ⊕γ 2 δ 10+4+1 ⊕γ 3 δ 14+4+1 ⊕ γ 4 δ 18+4+1 ⊕ γ 5 δ 22+4+1 ⊕ γ 6 δ 26+4+1 ⊕ γ 7 δ +∞ .
By setting the disturbance w 4 = x 4 = γ 0 δ 7 ⊕γ 1 δ 12 ⊕γ 2 δ 17 ⊕γ 3 δ 22 ⊕ γ 4 δ 27 ⊕ γ 5 δ 32 ⊕ γ 6 δ 37 ⊕ γ 7 δ +∞ , the firing of transition x 4 is slowed down.

Based on this characterization, the faulty system that we consider will behave based on Equation (3) and input disturbances as defined by Equation (4). Let

w ∈ M ax in [[γ, δ]] l×1 t p x i-1 x i o Figure 3: Representation of a place t p x i-1 x i w i o
Figure 4: Representation of a place with disturbance be the input vector of disturbances of size l. The input w corresponds to the transition that will be disturbed. Matrix R ∈ M ax in [[γ, δ]] n×l is filled with γ 0 δ 0 monomials that represent the connections between disturbances and internal disturbed transitions. All the other entries are set to ε. Equality x = Ax ⊕ Bu ⊕ Rw can be transformed to x = A * Bu ⊕ A * Rw thanks to Theorem 1 so we have

y = CA * Bu ⊕ CA * Rw.
In the example of Section 2, all the internal transitions in Figure 1 will be disturbed so R is the matrix

R ∈ M ax in [[γ, δ]] 6×6 : R =        γ 0 δ 0 . . . . . . γ 0 δ 0 . . . . . . γ 0 δ 0 . . . . . . γ 0 δ 0 . . . . . . γ 0 δ 0 . . . . . . γ 0 δ 0       

Observer synthesis

In this paper we use the definition of an observer from the articles [START_REF] Hardouin | Max-plus linear observer: application to manufacturing systems[END_REF], [START_REF] Hardouin | Observer design for (max,+) linear systems[END_REF]. Figure 5 shows the system with disturbances w and from which we can observe the outputs y o . The observer is a new model obtained from the fault-free model and that will estimate the states of the system x r in the presence of such disturbances. (5) To obtain the estimated vector x r as close as possible to real state x, the observer relies on the largest matrix L ∈ M ax in [[γ, δ]] n×q such that:

x r x o (A ⊕ LC) * Bu ⊕ (A ⊕ LC) * LCA * Rw A * Bu ⊕ A * Rw
which is given by: L = (A * B• /CA * B) ∧ (A * R• /CA * R). The observer matrix L of the TEG of Figure 1 is

L =        . γ 1 δ 0 (γ 1 δ 3 ) * . . γ 0 δ 0 (γ 1 δ 3 ) * . . . γ 1 δ 0 (γ 1 δ 4 ) * . . γ 0 δ 0 (γ 1 δ 4 ) * γ 1 δ 0 (γ 1 δ 4 ) * γ 0 δ 2 (γ 1 δ 4 ) * γ 0 δ 1 (γ 1 δ 4 ) * γ 0 δ 0 (γ 1 δ 4 ) * γ 0 δ 6 (γ 1 δ 4 ) * γ 0 δ 5 (γ 1 δ 4 ) *       
Based on the previous observer, suppose that the system behaves with respect to the inputs u 1 and u 2 defined in Section 3 but transition x 4 disturbed with w 4 = γ 0 δ 7 ⊕ γ1 δ 12 ⊕ γ 2 δ 17 ⊕ γ 3 δ 22 ⊕ γ 4 δ 27 ⊕ γ 5 δ 32 ⊕ γ 6 δ 37 ⊕ γ 7 δ +∞ then the reconstructed state is

x r = (A ⊕ LC) * Bu ⊕ (A ⊕ LC) * LCA * Rw which is the vector x r = [x r1 , . . . , x r6 ] T =        γ 0 δ 2 ⊕ γ 1 δ 5 ⊕ γ 2 δ 8 ⊕ γ 3 δ 11 ⊕ γ 4 δ 14 ⊕ γ 5 δ 17 ⊕ γ 6 δ 20 ⊕ γ 7 δ +∞ γ 0 δ 5 ⊕ γ 1 δ 8 ⊕ γ 2 δ 11 ⊕ γ 3 δ 14 ⊕ γ 4 δ 17 ⊕ γ 5 δ 20 ⊕ γ 6 δ 23 ⊕ γ 7 δ +∞ γ 0 δ 2 ⊕ γ 1 δ 7 ⊕ γ 2 δ 12 ⊕ γ 3 δ 17 ⊕ γ 4 δ 22 ⊕ γ 5 δ 27 ⊕ γ 6 δ 32 ⊕ γ 7 δ +∞ γ 0 δ 7 ⊕ γ 1 δ 12 ⊕ γ 2 δ 17 ⊕ γ 3 δ 22 ⊕ γ 4 δ 27 ⊕ γ 5 δ 32 ⊕ γ 6 δ 37 ⊕ γ 7 δ +∞ γ 0 δ 8 ⊕ γ 1 δ 13 ⊕ γ 2 δ 18 ⊕ γ 3 δ 23 ⊕ γ 4 δ 28 ⊕ γ 5 δ 33 ⊕ γ 6 δ 38 ⊕ γ 7 δ +∞ γ 0 δ 12 ⊕ γ 1 δ 17 ⊕ γ 2 δ 22 ⊕ γ 3 δ 27 ⊕ γ 4 δ 32 ⊕ γ 5 δ 37 ⊕ γ 6 δ 42 ⊕ γ 7 δ +∞       
The state x r estimated by the observer takes into account the disturbance w 4 . If the disturbance w 4 were not present, the estimated state would be x r4 = γ 0 δ 6 ⊕ γ 1 δ 10 ⊕ γ 2 δ 14 ⊕ γ 3 δ 18 ⊕ γ 4 δ 22 ⊕ γ 5 δ 26 ⊕ γ 6 δ 30 ⊕ γ 7 δ +∞ (no time shift: monomial γ 0 δ 6 instead of γ 0 δ 7 ).

5 Time shift failure detection in (max,+)-linear systems with observer

I xi (u, y o ) = false if ∆(x ri , x si ) = [0; 0], true otherwise, with x s = [x s1 . . . x sn ] T = A * Bu x r = [x r1 . . . x rn ] T = Ax r ⊕ Bu ⊕ LCx r ⊕ Ly o ∆(x ri , x si ) = [D xri• /xsi (0); -D xsi• /xri (0)].
Theorem 4. The indicator I xi (u, y o ) returns true only if a time shift failure involving x i with ∆(x ri , x si ) = [0, 0] has occurred in the system. A time shift failure involves a transition x i if the time shift failure occurs in a place of the TEG that is in the upstream 1 of transition x i .

To prove the result, we show that if the system has no failure in the places in the upstream of x i then the indicator I xi (u, y o ) necessary returns false. Suppose the system does not have such a time shift failure, it means by definition of the observer that the estimated state x ri is the same as the fault-free model state x si as no place in the upstream of x i is disturbed. If In the example of Section 2, based on the previous observer, suppose that the system behaves with respect to the inputs u 1 and u 2 defined in Section 3. Suppose that in reality there was an incident on Equipment 2: the operation lasts longer with a processing time of 5 hours in p 5 instead of 4 hours (see Figure 1). The real system is then characterized by Equation (3) with the disturbance w 4 that is defined in Section 4.1. The estimated state is the same as given at the end of Section 4.2. In particular, x r3 is represented with plain line in Figure 7. The expected state x s is the vector [x s1 , . . . , x s6 ] T =        γ 0 δ 2 ⊕ γ 1 δ 5 ⊕ γ 2 δ 8 ⊕ γ 3 δ 11 ⊕ γ 4 δ 14 ⊕ γ 5 δ 17 ⊕ γ 6 δ 20 ⊕ γ 7 δ +∞ γ 0 δ 5 ⊕ γ 1 δ 8 ⊕ γ 2 δ 11 ⊕ γ 3 δ 14 ⊕ γ 4 δ 17 ⊕ γ 5 δ 20 ⊕ γ 6 δ 23 ⊕ γ 7 δ +∞ γ 0 δ 2 ⊕ γ 1 δ 6 ⊕ γ 2 δ 10 ⊕ γ 3 δ 14 ⊕ γ 4 δ 18 ⊕ γ 5 δ 22 ⊕ γ 6 δ 26 ⊕ γ 7 δ +∞ γ 0 δ 6 ⊕ γ 1 δ 10 ⊕ γ 2 δ 14 ⊕ γ 3 δ 18 ⊕ γ 4 δ 22 ⊕ γ 5 δ 26 ⊕ γ 6 δ 30 ⊕ γ 7 δ +∞ γ 0 δ 7 ⊕ γ 1 δ 11 ⊕ γ 2 δ 15 ⊕ γ 3 δ 19 ⊕ γ 4 δ 23 ⊕ γ 5 δ 27 ⊕ γ 6 δ 31 ⊕ γ 7 δ +∞ γ 0 δ 11 ⊕ γ 1 δ 15 ⊕ γ 2 δ 19 ⊕ γ 3 δ 23 ⊕ γ 4 δ 27 ⊕ γ 5 δ 31 ⊕ γ 6 δ 35 ⊕ γ 7 δ +∞

Figure 1 :

 1 Figure 1: Fault free model Figure 1 presents a fault-free behavioural model of such a

  where x = a * b, and a * = i≥0 a i is the Kleene star operator with a 0 = e and a i+1 = a ⊗ a i . Definition 3. For a dioid D, denotes the order relation such that ∀a, b ∈ D, a b ⇔ a ⊕ b = b. Example 2. The complete dioid B[[γ, δ]] is the set of formal series with two commutative variables γ and δ with Boolean coefficients in {ε, e} and exponents in Z. A series s ∈ B[[γ, δ]] is written s = n,t∈Z s(n, t)γ n δ t where s(n, t) = e or ε (respectively representing the presence or the absence of the monomial). The neutral elements are ε = n,t∈Z εγ n δ t and e = γ 0 δ 0 .

  The internal operations are the same as in B[[γ, δ]] and neutral elements ε and e are identical to those of B[[γ, δ]]. Definition 4. Let s ∈ M ax in [[γ, δ]] be a series, the dater function of s is the non-decreasing function D s

Definition 6 .

 6 Let Π : D → C an application defined on ordered sets. We say: Π isotone ≡ ∀s, s ∈ D s s ⇒ Π(s) Π(s ). Definition 7. Let Π : D → C be an isotone mapping, where D and C are complete dioids. The largest solution of Π(x) = b, if it exists, is called the residual of Π and is denoted Π . When Π is residuated, Π is the unique isotone mapping such that Π • Π I dC and Π • Π I dD where I dC and I dD are respectively the identity mappings on C and D. Example 3. The mappings L a : x → a ⊗ x and R a : x → x ⊗ a defined over a complete dioid D are both residuated. Their residuals are denoted by L a (x) = a • \x and R a (x) = x• /a. Theorem 2 ( [Max91]). Let D be a complete dioid and A ∈ D n×m be a matrix. Then, A • \A = (A • \A) * (1) Time comparison between series can then be defined based on residuals. Definition 8. Let a, b ∈ M ax in [[γ, δ]] and their respective dater functions D a and D b . The time shift function representing the time shift between a and b for each n ∈ Z is defined by T a,b (n) = D a -D b . Theorem 3 ( [Max91]). Let a, b ∈ M ax in [[γ, δ]], the time shift function T a,b (n) can be bounded by:

Figure 5 :

 5 Figure 5: Observer structure with disturbance From articles [HMCSM10], [HMCL10] we get the following observer's equations:    x r = Ax r ⊕ Bu ⊕ L(y r ⊕ y o ) = (A ⊕ LC) * Bu ⊕ (A ⊕ LC) * LCA * Rw, y r = Cx r .

Figure 6

 6 Figure6shows how the proposed set of indicators is designed: the system is ruled by the observable inputs u, the unobservable disturbances w and produces the observable outputs y o ; the observer estimates the states x r based on the observation of u and y o . States x s result from the simulation of the fault-free model (as in Figure1) based on u, the proposed indicator then relies on a series comparison denoted ∆(x ri , x si ) (see Definition 9) for every transition x i .

Figure 6 :

 6 Figure 6: Detection method structure Definition 10. The indicator for state x i is I xi (u, y o ) defined as the Boolean function:

  x si = x ri , then we havex si • /x ri = x ri • /x si = x ri • /x ri but x ri • /x ri = (x ri • /x ri )* according to Theorem 2 and with Definition 1 of the Kleene star:(x ri • /x ri ) * = e ⊕ • • • = γ 0 δ 0 ⊕ . . . . So if x ri = x si , one has D xri• /xsi (0) = -D xsi• /xri (0) = 0.

  Series x s3 is represented with dotted line in Figure7. The

Figure 7 :

 7 Figure 7: Graphical representation of x r3 and x s3 intervals computed by the set of indicators are: ∆(x r1 , x s1 ) = [D xr1• /xs1(0) , -D xs1• /xr1 (0)] = [0, 0], ∆(x r2 , x s2 ) = [D xr2• /xs2(0) , -D xs2• /xr2 (0)] = [0, 0],

  This production line corresponds to three pieces of equipment (namely Eq1, Eq2, Eq3). Eq1 is modeled with a couple of places p 3 , p 4 : it is available (i.e. no current processing) if a token is in place p 3 while it is processing its input if a token is in place p 4 .

	The process of Eq1 is carried out in 3 hours. Similarly Eq2
	and Eq3 are respectively modeled by the couple of places
	p 6 , p 5 (processing time: 4 hours) and p 9 , p 10 (processing
	time 4 hours). Places p 7 , p 8 model wafer batch transporta-
	tion between Eq1, Eq2 and Eq3. From Eq1 to Eq3 it takes
	2 hours while from Eq2 to Eq3 it takes only 1 hour. Eq3
	requires processed wafer batches from both Eq1 and Eq2 to
	operate. It is modeled by a transition x 5 that can be fired
	only when Eq3 is available and wafer batches from Eq1
	and Eq2 are finally in front of Eq3. For i = {1, 2}, a trig-
	ger of an input transition u i represents the occurrence of an
	event from sensors on the production line that indicates the
	arrival of unprocessed wafer batches in front of Eqi. The
	output to the production line is a stream of fully processed
	wafer batches modeled by firing transition y 1 . Outputs y 2
	and y 3 provide observable information about the end of the
	process of Eq1 and Eq2. Suppose now that the underly-
	ing production line is faulty, i.e. it does not comply with
	the behaviour defined by Figure 1. Consider that a stream
	of 7 wafer batches arrive at Eq1 (input u 1 ) respectively at
	time t ∈ {1, 2, 3, 4, 5, 6, 7} and a similar stream with the
	same dates arrive at Eq2 (input u 2 ). Then, suppose that
	processed wafer batches are successively available at time
	{12, 17, 22, 27, 32, 37, 42} (output y 1 ), and process infor-
	mation is available at time {5, 8, 11, 14, 17, 20, 23} for out-
	put y 2 and at time {7, 12, 17, 22, 27, 32, 37} for output y 3 .
	Then, the question is: based on the fault-free model of Fig-
	ure 1, can we detect and localize a time shift failure in the
	underlying production line?

A place p is in the upstream of a transition x in a TEG if there is a path of arcs from p to x.

The set of indicators that return true are associated with transitions x 3 , x 4 , x 5 , x 6 . Indicators for transitions x 1 , x 2 return false. Now, if we assume that there is only one type of time shift failure in the system (as it is the case in the proposed scenario where there is only one incident on Equipement 2), Proposition 4 ensures that the time shift failure occurs in a place that is in the upstream of every transition x 3 , x 4 , x 5 , x 6 . It follows that the time shift failure occurs in Eq2, either in place p 2 (transportation delay before the arrival in front of Eq2), or in place p 6 (processing start of Eq2 is delayed), or in place p 5 (process of Eq2 longer than expected, which is the real case).

Conclusion

In this paper, we define a method for detecting time shift failures in systems modeled as Timed-Event Graphs using an observer that estimates the real states of the system based on the observations. Our work is motivated by the monitoring and the detection of time shift in production lines like in semiconductor manufacturing industry. The method defines a formal (max,+) algebraic indicator on the residuation theory. The proposed indicator is able to detect the presence of time shift failures as soon as it returns true and provides first localisation results. As a perspective, we aim at improving the accuracy of this indicator to better exploit the quantitative information contained in the interval ∆(x ri , x si ). We expect that a further analysis about the bounds of the intervals may actually provide more information about failure localization and identification. Indeed, observer matrix L actually represents the connections between the observed output y o and the internal transitions x r of the observer. In the matrix L of Section 4.2, we can notice that all the rows are filled which means that L is able to provide an estimate of any of the states x i 's (one row per x i ). Looking at the first monomial γ j δ l of a series γ j δ l (...) * it should be possible to know when a state x i starts to be effectively reconstructed by L. A monomial like γ j δ l asserts that the estimation of the corresponding state only starts after the j th +1 event. By using this property of L, our perspective would be to make the indicators more accurate and take benefit of the quantitative information from the computed intervals to estimate the delay of the time failures.


MSc Davergne Thomas Pt

Pallot Adrien Pt

MD, PhD Dechartres Agnès

MD Fautrel Bruno

Laure Gossec

PhD Md

Thomas Davergne
email: thomas.davergne@gmail.com

Adrien Pallot

Agnès Dechartres

Use of wearable activity trackers to improve physical activity behavior in rheumatic and musculoskeletal diseases: A systematic review and meta-analysis

Keywords: Wearable activity tracker, physical activity, effectiveness, low back pain, osteoarthritis, chronic inflammatory rheumatic disease

Objective: Wearable activity trackers (WATs) could be a promising strategy to improve physical activity (PA) in patients with rheumatic and musculoskeletal diseases (RMDs). The aim was to assess the adherence and effectiveness of WATs to increase PA levels in RMDs.

Methods

: A systematic review was performed to identify all cohorts and controlled trials evaluating WATs in RMDs, published between 2000 and 2018, by searching MEDLINE, EMBASE, PsycINFO and Cochrane. Data collected pertained to (a) adherence, (b) effectiveness on PA or (c) effectiveness on symptoms (pain, function, quality of life or fatigue). Meta-analyses were performed with a random effect model. Results: Of 2378 references, 17 studies were included with a total of 1588 patients; 8 studies (47%) in osteoarthritis, 5 (29%) in low back pain and 3 (18%) in inflammatory arthritis. Adherence assessed in 4 studies was high (weighted mean time worn: 92.7% (standard deviation 4.6%). A significant increase in PA was noted (mean difference 1520 steps, 95% confidence interval [580 -2460], I²=77% or 16 minutes [2 -29] of moderate to vigorous PA, I²=0%). A significant increase in pain was found for long interventions (>8 weeks) (standardized mean difference 0.25 [0.07, 0.43], I²= 0%).

Conclusion:

WATs in RMDs had a high short-term adherence with a significant increase in number of steps and time spent in moderate to vigorous PA though pain should be monitored. WATs may be an effective option to increase PA in this at risk population. Registered in PROSPERO: CRD42018083532.

Significance and Innovations:

• Short-term adherence to wearable activity trackers was high in people with rheumatic and musculoskeletal diseases.

• Interventions using wearable activity trackers were effective to increase physical activity levels in rheumatic and musculoskeletal diseases, with a mean difference of 1520 steps per day and 16 daily minutes spent in moderate to vigorous physical activity.

• Symptoms did not worsen with short term use of wearable activity trackers though pain increased in long-term interventions.

Physical inactivity has been identified as the fourth leading risk factor for global mortality around the world (1). The positive effects of physical activity on health, wellness and reduced mortality are widely established and documented for all ages [START_REF] Almeida | 150 minutes of vigorous physical activity per week predicts survival and successful ageing: a population-based 11-year longitudinal study of 12 201 older Australian men[END_REF](3)[START_REF] Kahn | The effectiveness of interventions to increase physical activity. A systematic review[END_REF][START_REF] Orrow | Effectiveness of physical activity promotion based in primary care: systematic review and meta-analysis of randomised controlled trials[END_REF].

Patients with rheumatic and musculoskeletal diseases (RMDs), such as low back pain, lower limb osteoarthritis and rheumatic inflammatory diseases are more prone to physical inactivity [START_REF] O'dwyer | Physical activity guidelines: is the message getting through to adults with rheumatic conditions?[END_REF]. However, patients with RMDs derive specific benefits from regular physical activity. Physical activity and exercises are a key component of clinical practice guidelines for the management of several rheumatic conditions [START_REF] Van Der Heijde | update of the ASAS-EULAR management recommendations for axial spondyloarthritis[END_REF][START_REF] Hochberg | American College of Rheumatology 2012 recommendations for the use of nonpharmacologic and pharmacologic therapies in osteoarthritis of the hand, hip, and knee[END_REF][START_REF] Qaseem | for the Clinical Guidelines Committee of the American College of Physicians. Noninvasive Treatments for Acute, Subacute, and Chronic Low Back Pain: A Clinical Practice Guideline From the American College of Physicians[END_REF].

Self-monitoring of physical activity, e.g. with wearable activity trackers (WATs) is one of the most used strategies to increase physical activity for adults with disability [START_REF] Castro | A scoping review on interventions to promote physical activity among adults with disabilities[END_REF].

Via sensors, these devices help users track their daily movement and provide feedback on activity, e.g. with monitor displays or companion smartphone tools [START_REF] Mercer | Behavior Change Techniques Present in Wearable Activity Trackers: A Critical Analysis[END_REF]. This technology aims to educate and motivate users toward better physical activity habits and better health behavior [START_REF] Patel | Wearable devices as facilitators, not drivers, of health behavior change[END_REF]. An issue with WATs is medium term adherence. A survey on 6223 consumers revealed that more than half of individuals who purchased an activity tracker stopped using it; and, of these, one third did within 6 months [START_REF] Ledger | How the Science of Human Behavior Change Offers the Secret to Long-Term Engagement[END_REF]. It is possible that adherence to WATs may be different in patients with RMDs, potentially given physical limitations [START_REF] Jacquemin | Physical Activity Assessment Using an Activity Tracker in Patients with Rheumatoid Arthritis and Axial Spondyloarthritis: Prospective Observational Study[END_REF].

WATs have shown their effectiveness to increase physical activity in a variety of contexts. Systematic reviews of the literature showed that activity trackers led to an increase of physical activity in the general population [START_REF] Goode | The Impact of Interventions that Integrate Accelerometers on Physical Activity and Weight Loss: A Systematic Review[END_REF]16), in young populations [START_REF] Ridgers | Feasibility and Effectiveness of Using Wearable Activity Trackers in Youth: A Systematic Review[END_REF], in adults with overweight or obesity [START_REF] Vries Hj De | Do activity monitors increase physical activity in adults with overweight or obesity? A systematic review and meta-analysis[END_REF]. WATs could be a promising strategy to improve physical activity levels in patients with RMDs [START_REF] Mansi | A systematic review of studies using pedometers as an intervention for musculoskeletal diseases[END_REF]. However, WATs may pose specific challenges for this population due to pain in particular. Furthermore, for this population, being physically active could modify pain, fatigue, function and quality of life [START_REF] Van Der Heijde | update of the ASAS-EULAR management recommendations for axial spondyloarthritis[END_REF][START_REF] Hochberg | American College of Rheumatology 2012 recommendations for the use of nonpharmacologic and pharmacologic therapies in osteoarthritis of the hand, hip, and knee[END_REF][START_REF] Qaseem | for the Clinical Guidelines Committee of the American College of Physicians. Noninvasive Treatments for Acute, Subacute, and Chronic Low Back Pain: A Clinical Practice Guideline From the American College of Physicians[END_REF].

The aim of this systematic review with meta-analysis was to summarize the available evidence regarding adherence to WATs and effectiveness in increasing physical activity levels and improving symptoms for rheumatic patients and to identify factors that may have an impact on this effectiveness.

MATERIALS AND METHODS

This review was conducted in accordance with the Cochrane Handbook and the Preferred Reporting Items for Systematic Reviews and Meta-analysis (PRISMA) statement (see online supplement 1 for PRISMA checklist) [START_REF] Moher | Preferred reporting items for systematic reviews and meta-analyses: the PRISMA statement[END_REF]. The protocol was registered on PROSPERO CRD42018083532. More detailed information is available in online supplement 2.

Eligibility criteria

Participants

Patients of all ages with RMD were included, i.e. lower limb osteoarthritis, low back pain or chronic inflammatory rheumatic diseases (i.e. spondyloarthritis, rheumatoid arthritis, psoriatic arthritis or juvenile arthritis).

Type of intervention

We considered as a WAT, any device designed to be worn on the user's body; using accelerometers, with or without altimeters or other sensors to track the wearer's movements and/or biometric data; with or without the possibility to upload activity data to an online application that shows trends over time [START_REF] Hoy | Personal Activity Trackers and the Quantified Self[END_REF]. The announced objective of the WAT had to be an increase in physical activity (rather than gait or posture). Simple pedometers (e.g. Yamax) were distinguished from more 'advanced' WATs (e.g, Fitbit), which allow automatic transmission of information and could be linked with an App.

Study design and type of comparison

For adherence, all studies whatever the design were included. For effectiveness, randomised controlled trials were considered with the following comparison: WAT versus no intervention or waiting list; and for exploratory purposes WAT plus other adjunctive intervention (WAT+), e.g. educational programme, versus WAT only or with less important adjunctive intervention. When studies reported more than one intervention arm, the intervention with the less components associated with the WAT was chosen. Given our objective to analyze WAT versus no WAT, WAT+ was only analyzed when there was no "WAT alone" group.

Outcomes

The primary outcome was the amount of physical activity with (I) steps per day and / or (II) time spent in moderate-to-vigorous physical activity at the end of the intervention. Secondary outcomes were adherence to the device (e.g. duration of use, percentage of patients still in the study at completion (study completion rate); symptoms: pain, functional tests, disability scores, quality of life and fatigue (online supplement 2). Quality of life was reported using either Knee Injury and Osteoarthritis Outcome Score, or the Knee related Quality of Life subscale on healthrelated quality of life (EuroQol EQ-5D-3L Weighted Health Index). Measures were collected at intermediate time points (i.e. time points measured before end of intervention), end of the intervention and follow-up after end of intervention (if available).

Search strategy

A comprehensive search of articles available in English, with a publication date

Study selection

The online data manager Covidence (http://www.covidence.org) was used for the selection process. The study selection was performed by one reviewer (TD) firstly on title and abstract and secondly on full-length articles and verified by double reading (i.e., verified included studies) (AP). Authors were contacted when full-text articles were not available or complementary information was needed. For adherence, the percentage of days worn and retention rate over intervention duration was reported.

Data items and data extraction

Risk of bias assessment

Risk of bias was evaluated using the Risk of Bias tool developed by the Cochrane Collaboration [START_REF]Cochrane Handbook for Systematic Reviews of Interventions Version 5[END_REF] by two independent reviewers (TD and AP). Any discrepancies were resolved by discussion to reach a consensus.

Synthesis of results and heterogeneity

Adherence was pooled using weighted mean time worn and weighted mean study retention rate. The effects of the intervention were pooled using a meta-analysis if possible (i.e, standardized mean difference (SMD) using random effects models for physical activity and symptoms) in RevMan (version 5.3) [START_REF] Green | [END_REF]. For the meta-analysis, we adapted data (i.e. absolute values were used) in order to have a higher result meaning better outcome, thus by convention the right side of forest plots was always favoring WATs. To interpret the SMD, the following method was used: no effect (<0.2), small effect (0.2-0.5), moderate effect (0.5-0.8) or large effect (>0.8) [START_REF] Cohen | Statistical Power Analysis for the Behavioral Sciences[END_REF]. To illustrate the SMD for steps and time spent in moderate to vigorous physical activity, two methods were used. Firstly, the difference between groups in terms of step count increase was meta-analysed using mean difference in RevMan. Secondly, the step count in the intervention group (both at end of intervention and as a change from baseline) was calculated by weighted mean, to give indication of clinical relevance of results.

Heterogeneity was evaluated using forest plots to compare results across trials and the I 2 statistic measuring the proportion of variation (i.e., inconsistency) between studies that is due to heterogeneity rather than chance [START_REF] Higgins | Quantifying heterogeneity in a meta-analysis[END_REF]. An I 2 value of >40% indicates substantial heterogeneity, and an I² value >80% considerable heterogeneity, lower I² is considered better [START_REF]Cochrane Handbook for Systematic Reviews of Interventions Version 5[END_REF].

To identify factors that may have an impact on effectiveness, subgroup analyses were conducted for the primary outcome according to the type of WAT (pedometer versus more advanced WAT) and intervention duration and follow-up duration (short duration = 0-8 weeks; long duration >8 weeks). An analysis was performed on studies with lower limb osteoarthritis. For other outcomes, subgroup analyses regarding intervention duration and follow-up duration were performed.

RESULTS

Articles identified

The electronic search yielded 2,380 potentially relevant hits (Figure 1). In total, 17 studies were included in the review, of which 10 were pooled for meta-analysis [START_REF] Jacquemin | Physical Activity Assessment Using an Activity Tracker in Patients with Rheumatoid Arthritis and Axial Spondyloarthritis: Prospective Observational Study[END_REF][START_REF] Blitz | Do pedometers with or without education on exercise increase functional walking capacity and physical activity level in adolescents with juvenile idiopathic arthritis?[END_REF][START_REF] Dekker-Van Weering | A pilot study -the potential value of an activity-based feedback system for treatment of individuals with chronic lower back pain[END_REF][START_REF] Gordon | Influence of the Fitbit Charge HR on physical activity, aerobic fitness and disability in non-specific back pain participants[END_REF][START_REF] Hiyama | A four-week walking exercise programme in patients with knee osteoarthritis improves the ability of dualtask performance: a randomized controlled trial[END_REF][START_REF] Hurley | Supervised walking in comparison with fitness training for chronic back pain in physiotherapy: results of the SWIFT single-blinded randomized controlled trial (ISRCTN17592092)[END_REF][START_REF] Katz | Physical Activity to Reduce Fatigue in Rheumatoid Arthritis: A Randomized Controlled Trial[END_REF][START_REF] Krein | Pedometer-Based Internet-Mediated Intervention For Adults With Chronic Low Back Pain: Randomized Controlled Trial[END_REF][START_REF] Li | Efficacy of a Wearable-Enabled Physical Activity Counselling Program for People with Knee Osteoarthritis[END_REF][START_REF] Li | A Community-Based Physical Activity Counselling Program for People With Knee Osteoarthritis: Feasibility and Preliminary Efficacy of the Track-OA Study[END_REF][START_REF] Mcdonough | Pedometer-driven walking for chronic low back pain: a feasibility randomized controlled trial[END_REF][START_REF] Ng | Efficacy of a progressive walking program and glucosamine sulphate supplementation on osteoarthritic symptoms of the hip and knee: a feasibility trial[END_REF][START_REF] Paxton | A Feasibility Study for Improved Physical Activity After Total Knee Arthroplasty[END_REF][START_REF] Skrepnik | Assessing the Impact of a Novel Smartphone Application Compared With Standard Follow-Up on Mobility of Patients With Knee Osteoarthritis Following Treatment With Hylan G-F 20: A Randomized Controlled Trial[END_REF][START_REF] Talbot | A home-based pedometer-driven walking program to increase physical activity in older adults with osteoarthritis of the knee: a preliminary study[END_REF][START_REF] Brosseau | The implementation of a community-based aerobic walking program for mild to moderate knee osteoarthritis: a knowledge translation randomized controlled trial: part II: clinical outcomes[END_REF][START_REF] Christiansen | Preliminary findings of a novel physical therapist administered physical activity intervention after total knee replacement[END_REF]. One study was potentially relevant but full-text was not available [START_REF] Toda | Change in body fat, but not body weight or metabolic correlates of obesity, is related to symptomatic relief of obese patients with knee osteoarthritis after a weight control program[END_REF].

Studies were published between 2003 and 2018, with 9 studies (53%) published in 2017-2018. On top of the included publications, 10 ongoing studies were identified (online supplement 3).

General characteristics of included studies and population

The review included a total of 1,588 participants (Table 1 and online supplement 4).

The median sample size per study was 34 participants (range 17-246). The mean age of the population was 55 years (range 16-72 years) (Table 1). One study included a young population (<19 years with juvenile arthritis) ( 27) and 5 studies included elderly patients with osteoarthritis (the remaining 4 studies on osteoarthritis were not related specifically to elderly patients) [START_REF] Li | A Community-Based Physical Activity Counselling Program for People With Knee Osteoarthritis: Feasibility and Preliminary Efficacy of the Track-OA Study[END_REF][START_REF] Paxton | A Feasibility Study for Improved Physical Activity After Total Knee Arthroplasty[END_REF][START_REF] Talbot | A home-based pedometer-driven walking program to increase physical activity in older adults with osteoarthritis of the knee: a preliminary study[END_REF][START_REF] Brosseau | The implementation of a community-based aerobic walking program for mild to moderate knee osteoarthritis: a knowledge translation randomized controlled trial: part II: clinical outcomes[END_REF][START_REF] Christiansen | Preliminary findings of a novel physical therapist administered physical activity intervention after total knee replacement[END_REF]. Of the 17 studies, 15 were randomized controlled trials and 2 were cohort studies (used to assess adherence). Among the 17 studies, 9 (53%) were related to osteoarthritis, 5 (29%) to low back pain and only 3 (18%) to chronic inflammatory rheumatic diseases. When reported, mean disease duration was 8 to 14 years [START_REF] Jacquemin | Physical Activity Assessment Using an Activity Tracker in Patients with Rheumatoid Arthritis and Axial Spondyloarthritis: Prospective Observational Study[END_REF][START_REF] Hurley | Supervised walking in comparison with fitness training for chronic back pain in physiotherapy: results of the SWIFT single-blinded randomized controlled trial (ISRCTN17592092)[END_REF][START_REF] Katz | Physical Activity to Reduce Fatigue in Rheumatoid Arthritis: A Randomized Controlled Trial[END_REF][START_REF] Mcdonough | Pedometer-driven walking for chronic low back pain: a feasibility randomized controlled trial[END_REF][START_REF] Brosseau | The implementation of a community-based aerobic walking program for mild to moderate knee osteoarthritis: a knowledge translation randomized controlled trial: part II: clinical outcomes[END_REF]. Overall, 41.3% of the population was male.

All included studies were conducted in high-income countries: 10 (59%) in North America, 5 (29%) in Europe, 1 (4%) in Australia and 1 (4%) in Asia.

Risk of bias was low for 5 studies [START_REF] Hiyama | A four-week walking exercise programme in patients with knee osteoarthritis improves the ability of dualtask performance: a randomized controlled trial[END_REF][START_REF] Hurley | Supervised walking in comparison with fitness training for chronic back pain in physiotherapy: results of the SWIFT single-blinded randomized controlled trial (ISRCTN17592092)[END_REF][START_REF] Katz | Physical Activity to Reduce Fatigue in Rheumatoid Arthritis: A Randomized Controlled Trial[END_REF][START_REF] Krein | Pedometer-Based Internet-Mediated Intervention For Adults With Chronic Low Back Pain: Randomized Controlled Trial[END_REF][START_REF] Skrepnik | Assessing the Impact of a Novel Smartphone Application Compared With Standard Follow-Up on Mobility of Patients With Knee Osteoarthritis Following Treatment With Hylan G-F 20: A Randomized Controlled Trial[END_REF], unclear for 5 studies [START_REF] Blitz | Do pedometers with or without education on exercise increase functional walking capacity and physical activity level in adolescents with juvenile idiopathic arthritis?[END_REF][START_REF] Gordon | Influence of the Fitbit Charge HR on physical activity, aerobic fitness and disability in non-specific back pain participants[END_REF][START_REF] Mcdonough | Pedometer-driven walking for chronic low back pain: a feasibility randomized controlled trial[END_REF][START_REF] Paxton | A Feasibility Study for Improved Physical Activity After Total Knee Arthroplasty[END_REF][START_REF] Talbot | A home-based pedometer-driven walking program to increase physical activity in older adults with osteoarthritis of the knee: a preliminary study[END_REF] and high for 5 studies [START_REF] Li | Efficacy of a Wearable-Enabled Physical Activity Counselling Program for People with Knee Osteoarthritis[END_REF][START_REF] Li | A Community-Based Physical Activity Counselling Program for People With Knee Osteoarthritis: Feasibility and Preliminary Efficacy of the Track-OA Study[END_REF][START_REF] Ng | Efficacy of a progressive walking program and glucosamine sulphate supplementation on osteoarthritic symptoms of the hip and knee: a feasibility trial[END_REF][START_REF] Brosseau | The implementation of a community-based aerobic walking program for mild to moderate knee osteoarthritis: a knowledge translation randomized controlled trial: part II: clinical outcomes[END_REF][START_REF] Christiansen | Preliminary findings of a novel physical therapist administered physical activity intervention after total knee replacement[END_REF] (online supplement 5).

Characteristics of interventions and comparators

In all, 8 studies (47%) reported on advanced WATs and 9 (53%) on simple pedometers (Table 1). All studies but one used co-interventions in addition to the use of WATs. Among them, 12 (71%) studies used goal setting, 9 (53%) educational walking booklets, and 8 (47%) used counselling (Table 1). Weighted mean duration of intervention was 21.8 weeks (range: 2.0-52.0). Activity trackers were worn on wrist (6 studies (35%)) and waist (11 studies (65%)). Out of 13 studies measuring steps per day, the measurement was performed by the WAT itself in 9 studies (69%), a research device (e.g. Actigraph) in 3 studies (23%) and both in 1 study (8%).

In all, 10 studies (59%) used education, usual care or inactive comparator group, 5 studies (29%) used a WAT only or with less important adjunctive intervention in the comparator group and 2 studies (12%) (corresponding to cohorts used for adherence) had no comparator group.

Adherence to WATs

Adherence was reported in 4 studies (24%) including 2 RCTs and 2 cohort studies, in a total of 416 patients [START_REF] Jacquemin | Physical Activity Assessment Using an Activity Tracker in Patients with Rheumatoid Arthritis and Axial Spondyloarthritis: Prospective Observational Study[END_REF][START_REF] Dekker-Van Weering | A pilot study -the potential value of an activity-based feedback system for treatment of individuals with chronic lower back pain[END_REF][START_REF] Paxton | A Feasibility Study for Improved Physical Activity After Total Knee Arthroplasty[END_REF][START_REF] Skrepnik | Assessing the Impact of a Novel Smartphone Application Compared With Standard Follow-Up on Mobility of Patients With Knee Osteoarthritis Following Treatment With Hylan G-F 20: A Randomized Controlled Trial[END_REF]. The weighted mean time worn reported in 3 studies wearing WAT at the wrist was 92.7% (standard deviation (SD) 4.6%) for a weighted mean duration of 10.0 weeks (range 2.0-14.0). A fourth study wearing WAT at the hip reported that 63% of the participants wore the WAT more than 80% of study duration. Study retention rates were reported in all studies with a weighted mean retention rate of 90% (SD 11%) over a weighted mean duration of 24.3 weeks.

Effectiveness of WATs on physical activity

The effect of WATs of physical activity for rheumatic condition was investigated in 9 randomised controlled trials. Two trials were not included in the meta-analysis as one did not report results in the comparator group [START_REF] Mcdonough | Pedometer-driven walking for chronic low back pain: a feasibility randomized controlled trial[END_REF] and another did not report steps as outcome [START_REF] Li | A Community-Based Physical Activity Counselling Program for People With Knee Osteoarthritis: Feasibility and Preliminary Efficacy of the Track-OA Study[END_REF]. Meta-analysis based on 7 studies (i.e. 463 patients) showed a large effect on mean daily steps at end of intervention (mean study duration 13.9 weeks (SD 6.9)): the SMD was large: 0.83 [95% confidence interval (95%CI) 0.29 to 1.38], corresponding to a mean difference over the comparator group of 1,520 steps per day [95% CI 580 to 2460]) (Figure 2). The overall heterogeneity of study effects was large (I 2 =77%). A visual inspection of the forest plot according to risk of bias did not give indications of distortion of the results, which is concordant with the result of I 2 .

For indicative purposes, the calculation of weighted mean steps per day in the WAT groups at baseline and at end of intervention was 4,741 (SD 1,629) steps and 6,019 (SD 1297) steps respectively, corresponding to a weighted mean increase from baseline of 1,448 steps (SD 1098).

Meta-analysis based on 3 studies (i.e. 117 patients) showed a small effect for time spent in moderate to vigorous physical activity at end of intervention (mean study duration 12.0 weeks (SD 10.6)): the SMD was small: 0.41 [95%CI 0;04 to 0.77], (I 2 =0%) (Figure 3). This result corresponds to a mean difference over the comparator group of 16 minutes per day [95% CI 2 to 29].

No significant results were found for sedentary time (assessed in 2 studies). In studies with a prolonged follow-up after the end of the intervention period, no significant results were found for mean daily steps, time spent in moderate to vigorous physical activity and sedentary time (assessed in 2, 1 and 1 study respectively, data not shown).

Subgroup analysis suggested larger effects for advanced WATs than for simple pedometers (not significant, p= 0.42). Subgroups for study duration did not indicate a difference in effectiveness (Table 2). Regarding studies with lower limb osteoarthritis, meta-analysis based on 6 studies of patients with lower limb osteoarthritis (i.e. 406 patients) showed a large effect on mean daily steps at end of intervention: the SMD was large: 0.92 [95% confidence interval (95%CI) 0.33 to 1.52] with large heterogeneity (I²=84%).

Effectiveness of WATs on symptoms

Meta-analyses for pain, functional test, disability, quality of life and fatigue did not show any effect at end of intervention and end of follow-up (Figure 4). However, the analysis according to study duration indicated a small effect on pain in long interventions (>8 weeks); in the 5 randomised controlled trials (485 patients), the SMD was small: -0.25 [95%CI -0.43, -0.07], I²= 0%, indicating an increase in pain when using WATs for long term (Table 2).

Effectiveness of WATs assessed against a comparator comprising another WAT 5 studies assessed WATs plus other adjunctive interventions versus WAT only or

with less important adjunctive intervention, involving 531 participants for a weighted mean duration of 45.8 weeks (SD 22.8) (Table 1) [START_REF] Blitz | Do pedometers with or without education on exercise increase functional walking capacity and physical activity level in adolescents with juvenile idiopathic arthritis?[END_REF][START_REF] Gordon | Influence of the Fitbit Charge HR on physical activity, aerobic fitness and disability in non-specific back pain participants[END_REF][START_REF] Krein | Pedometer-Based Internet-Mediated Intervention For Adults With Chronic Low Back Pain: Randomized Controlled Trial[END_REF][START_REF] Ng | Efficacy of a progressive walking program and glucosamine sulphate supplementation on osteoarthritic symptoms of the hip and knee: a feasibility trial[END_REF][START_REF] Brosseau | The implementation of a community-based aerobic walking program for mild to moderate knee osteoarthritis: a knowledge translation randomized controlled trial: part II: clinical outcomes[END_REF]. No meta-analysis was performed because of lack of data reported and heterogeneity regarding components of interventions. Krein et al. reported a significant improvement in disability assessed with the Roland Morris Questionnaire at end of intervention and at end of follow-up compared with the control group (mean difference of 2.0 [95%CI 0.42, 3.55] and 1.70 [95%CI 0.10, 3.30] respectively) [START_REF] Krein | Pedometer-Based Internet-Mediated Intervention For Adults With Chronic Low Back Pain: Randomized Controlled Trial[END_REF]. In the other studies, no significant difference between groups was reported (data not shown).

DISCUSSION

This study has brought to light interesting results regarding the use of WATs for RMDs. Short term adherence to WATs was high. Interventions using wearable activity trackers were effective to increase physical activity levels in rheumatic and musculoskeletal diseases, with a mean difference of 1520 steps per day and 16 daily minutes spent in moderate to vigorous physical activity. Finally, the use of WATs did not change symptoms at short term though an increase in pain was noted for long study durations (>8 weeks).

This study has strengths and weaknesses. A relatively small number of studies were included in this review. However, a systematic search was completed in most major databases with complementary hand search of references to ensure that no relevant studies were missed. It is noteworthy that the majority of the references (10/17 studies) were only identified through the extensive hand search though they did not appear in the results of the electronic search, despite a sensitive search equation. This means that when checking back those references in the initial searches, they were not present (i.e. they were not missed in the selection phase). This could indicate a lack of clear identification of interventions using WATs in search engines, given the fact that Internet of Things is a relatively new topic. This is also indicated by the fact that half of studies included in this review were published in 2017-2018.

Only one reviewer performed the study selection process because this is a time consuming step which is not key for the quality of systematic review. Indeed alternatives to the conventional "double screening" approach exists and appear to be potentially more efficient approaches to identifying eligible studies for systematic reviews [START_REF] Shemilt | Use of cost-effectiveness analysis to compare the efficiency of study identification methods in systematic reviews[END_REF]. The key step is data extraction in double author rather than data selection as shown in Buscemi 2006 (45).

However, a second reviewer did verify the selection by screening included studies for inclusion criteria. This review was based on a protocol registered in PROSPERO.

Both SMD and mean difference were calculated for steps counts which is unusual but allows comparisons with other meta-analyses, whatever the technique used to report results.

Not all studies contributed to all outcomes (e.g. adherence = 4 studies, mean daily steps = 7 studies etc). This could come from a lack of standardisation in choice of outcomes during the conception of trials, to different objectives across trials or missing data in publications [START_REF] Li | A Community-Based Physical Activity Counselling Program for People With Knee Osteoarthritis: Feasibility and Preliminary Efficacy of the Track-OA Study[END_REF][START_REF] Mcdonough | Pedometer-driven walking for chronic low back pain: a feasibility randomized controlled trial[END_REF]. However, authors were contacted when outcomes of interest were missing.

Results showed a large heterogeneity. This heterogeneity is probably explained by differences in components of interventions and duration of intervention, type of comparator and type of RMD. Nevertheless, appropriate statistics such as SMD and random effects were used [START_REF]Cochrane Handbook for Systematic Reviews of Interventions Version 5[END_REF]. Moreover, subgroup analyses based on rheumatic condition indicated residual high heterogeneity in lower limb osteoarthritis.

Although using endpoint data can lead to misinterpretation in case of imbalance in baseline, change of measure or confidence interval of change were not available for 5 of the included studies [START_REF] Hiyama | A four-week walking exercise programme in patients with knee osteoarthritis improves the ability of dualtask performance: a randomized controlled trial[END_REF][START_REF] Li | Efficacy of a Wearable-Enabled Physical Activity Counselling Program for People with Knee Osteoarthritis[END_REF][START_REF] Li | A Community-Based Physical Activity Counselling Program for People With Knee Osteoarthritis: Feasibility and Preliminary Efficacy of the Track-OA Study[END_REF][START_REF] Paxton | A Feasibility Study for Improved Physical Activity After Total Knee Arthroplasty[END_REF][START_REF] Talbot | A home-based pedometer-driven walking program to increase physical activity in older adults with osteoarthritis of the knee: a preliminary study[END_REF].

In the present review, short term adherence to WATs was excellent. Among included studies, the percentage of wearing time was >90% over a mean duration of 10 weeks. Although scarce data were available for medium to long term use, this result is encouraging. A study assessing adherence to WATs in undergraduate students (range 20-24 years old) showed that more than half of the participants stopped using the FitBit activity tracker after two weeks (75% after four weeks) [START_REF] Shih | Use and adoption challenges of wearable activity trackers[END_REF]. This difference of results could come from extra efforts to maintain adherence in randomised clinical trials (e.g. weekly phone calls). Strategies to drive adoption and adherence have been identified, such as issues with remembering, issues with physical design and aesthetics, issues with data management, and should be further implemented in clinical research and clinical practice [START_REF] Ledger | How the Science of Human Behavior Change Offers the Secret to Long-Term Engagement[END_REF][START_REF] Shih | Use and adoption challenges of wearable activity trackers[END_REF].

The number of daily steps was significantly higher across studies at the end of intervention for interventions using WATs compared to interventions not using them, with a large SMD of 0.83 and mean difference of 1520 steps. Intervention groups using WAT moved from an average of 4741 steps to an average of 6019 steps, corresponding to a mean increase from baseline of 1448 steps. This physical activity represents at the group level a change of activity state from sedentary to low activity (45). As regards time in moderate to vigorous physical activity, results were quite difficult to interpret since, firstly the number was low, and secondly there was a discrepancy between the moderate SMD and the rather high mean difference (16 minutes per day). 16 minutes may be considered clinically relevant when time in moderate to vigorous physical activity is around 10 minutes per day in lower limb osteoarthritis [START_REF] Thoma | Are older adults with symptomatic knee osteoarthritis less active than the general population?: Analysis from the Osteoarthritis Initiative and NHANES[END_REF]. This activity is encouraging though smaller than the recommendation of 30 minutes of moderate intensity exercise [START_REF] Haskell | Physical activity and public health: updated recommendation for adults from the American College of Sports Medicine and the American Heart Association[END_REF]. The question of the effectiveness of WATs over the long term remains unsolved. Indeed in the present review, the two studies with follow-up after end of intervention did not evidence any increase in steps after stopping wearing the WATs.

Our result is consistent with other systematic reviews in the general population [START_REF] Goode | The Impact of Interventions that Integrate Accelerometers on Physical Activity and Weight Loss: A Systematic Review[END_REF][START_REF] Fanning | Increasing Physical Activity With Mobile Devices: A Meta-Analysis[END_REF]. Mansi et al reported in a systematic review for musculoskeletal conditions a significant effect in 2 studies out of 7 though no meta-analysis was performed [START_REF] Mansi | A systematic review of studies using pedometers as an intervention for musculoskeletal diseases[END_REF].

The present result demonstrates that interventions using WATs are an effective strategy to improve physical activity levels for rheumatic conditions. As shown in Table 1, there was a diversity of interventions and use of adjunctive components in the trials making it difficult to attribute the causality of the positive effects to the WAT itself rather than to the global intervention. However, analyses comparing WATs plus other adjunctive interventions versus WATs only or with less important adjunctive intervention did not show any effectiveness at increasing physical activity level, which could indicate that WATs may be a major component of effectiveness.

No effect was found (either increase or decrease) by use of WATs for pain, disability, functional tests, quality of life and fatigue. An improvement of symptoms may be expected since physical activity is encouraged by most of the clinical guidelines for RMDs [START_REF] Van Der Heijde | update of the ASAS-EULAR management recommendations for axial spondyloarthritis[END_REF][START_REF] Hochberg | American College of Rheumatology 2012 recommendations for the use of nonpharmacologic and pharmacologic therapies in osteoarthritis of the hand, hip, and knee[END_REF][START_REF] Qaseem | for the Clinical Guidelines Committee of the American College of Physicians. Noninvasive Treatments for Acute, Subacute, and Chronic Low Back Pain: A Clinical Practice Guideline From the American College of Physicians[END_REF]. It is possible that the moderate increase in physical activity was not sufficient to lead to clinical benefits. Surprisingly, a significant increase in pain was observed compared with the control groups in subgroup analysis for interventions longer than 8 weeks. Mansi et al reported a significant improvement of pain and function in 4/7 studies ( 20), but these results were found in intervention groups not considering the difference between groups. In the present review, in 5 trials (485 patients), the SMD was small (-0.25 [95%CI -0.43, -0.07]) which can be illustrated by a mean difference of 0.63 point in a 0-10 numeric pain rating scale [START_REF]Cochrane Handbook for Systematic Reviews of Interventions Version 5[END_REF]. This slight though significant increase in pain should be further confirmed and should not discourage activity given the potential benefice of physical activity. To deal with potential increase of pain, activity should be tailored to the individual characteristics such as type of disability and evolution of disability [START_REF] Jacquemin | Physical Activity Assessment Using an Activity Tracker in Patients with Rheumatoid Arthritis and Axial Spondyloarthritis: Prospective Observational Study[END_REF][START_REF] Hasenbring | Pain-related avoidance versus endurance in primary care patients with subacute back pain: psychological characteristics and outcome at a 6-month follow-up[END_REF].

In conclusion, WATs appear to be a promising strategy to improve physical activity for patients with RMDs. However, research may be needed to determine whether changes are maintained over time, ways to improve long-term use, and the most effective adjuncts.

5

METHODS

Protocol and registration

5 Indicate if a review protocol exists, if and where it can be accessed (e.g., Web address), and, if available, provide registration information including registration number.

6

Eligibility criteria 6 Specify study characteristics (e.g., PICOS, length of follow-up) and report characteristics (e.g., years considered, language, publication status) used as criteria for eligibility, giving rationale.

6

Information sources 7 Describe all information sources (e.g., databases with dates of coverage, contact with study authors to identify additional studies) in the search and date last searched.

8 Search 8 Present full electronic search strategy for at least one database, including any limits used, such that it could be repeated.

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Study selection 9 State the process for selecting studies (i.e., screening, eligibility, included in systematic review, and, if applicable, included in the meta-analysis).

8

Data collection process 10 Describe method of data extraction from reports (e.g., piloted forms, independently, in duplicate) and any processes for obtaining and confirming data from investigators.

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Data items 11 List and define all variables for which data were sought (e.g., PICOS, funding sources) and any assumptions and simplifications made.

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Risk of bias in individual studies 2 measures in means).

Synthesis of results

14 Describe the methods of handling data and combining results of studies, if done, including measures of consistency (e.g., I 2 ) for each meta-analysis.

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Risk of bias across studies 15 Specify any assessment of risk of bias that may affect the cumulative evidence (e.g., publication bias, selective reporting within studies).

NA due to small number of studies included

Risk of bias assessment

The risk of bias tool reviews the random sequence generation, allocation concealment, blinding of participants and personnel, blinding of outcome assessment, incomplete outcome data, selective reporting, and other biases. Since the blinding of participants is practically infeasible in a self-monitoring intervention, this was not assessed and blinding of personnel was scored separately.

We separately assessed the blinding of self-reported subjective outcomes (e.g., fatigue, quality of live) and the blinding of independent outcome assessors to objective outcomes (such as steps per day). Studies were classified as having: a low risk of bias if all key domains were assessed to be at low risk of bias and had no serious flaws; a high risk of bias if one or more categories was assessed at high risk of bias; and an unclear risk of bias if one or more key domains was assessed as having an unclear risk of bias.

Synthesis of result

For the meta-analysis, we adapted data in order to have a higher result meaning better outcome. By convention, the right side of forest plots was always favouring WATs. When an author used more than one scale for a same outcome, we selected a single one.

Subgroups analysis

Subgroup analysis based on the device localization was mentioned in the protocol but not completed because of enough lack of studies with location other than wrist.

  between 2000 and 2018 (May) were performed, in the following databases: MEDLINE via Ovid, EMBASE via EMBASE, PsycINFO via EBSCOhost and the Cochrane Central Register for Controlled Trials. A search equation involving both key-words and free-text words was initially developed for the MEDLINE search (online supplementary 2) and adapted for the other databases. We also conducted a search of ClinicalTrials.gov (www.ClinicalTrials.gov) for ongoing studies. Other databases were searched (online supplementary 2). The reference lists of included reports (backward citation searching) and references that refer to the included studies using Google Scholar and Web of Science (forward citation searching) were screened. Studies presented at the American College of Rheumatology et European League Against Rheumatism conferences of the two previous years were consulted to identify full-text published articles.

  Two review authors (TD and AP) independently extracted data from included studies and recorded information on a standardized extraction form accompanied by a codebook. Information were extracted regarding (I) general characteristics (i.e. study design, sample size, authors, year of publication, journal, title, country, objectives of the studies (as reported verbatim by the authors), and setting (e.g. healthcare center)), (II) characteristics of populations (i.e. number of patients included in each study, diseases and means duration of the diseases, physical activity levels at baseline, age, gender, medications), (III) interventions (i.e. intervention related to the activity tracker: availability of feedback, wearing position, duration, interface used for the feedback, nature of the feedback, characteristics of the device (brand, type of device, type of activity measured) and co-intervention: type, behavior targeted, duration) and (IV) outcomes of interest, tools to measure steps per day, and time point assessment. The main data collected in extraction forms were summarized in evidence tables. The results concerned adherence or effectiveness of intervention.

INTRODUCTION Rationale 3 5 Objectives 4

 354 Describe the rationale for the review in the context of what is already known. Provide an explicit statement of questions being addressed with reference to participants, interventions, comparisons, outcomes, and study design (PICOS).

1 Figure 1 :

 11 Figure 1: PRISMA flowchart: selection process of articles reporting WATs in rheumatic musculoskeletal diseases

2 Figure 2 :

 22 Figure 2: Forest plot of standardized mean differences between WAT and control in mean number of daily steps.

Figure 3 :

 3 Figure 3: Forest plot of standardized mean differences between WAT and control in moderate to vigorous physical activity (MVPA).

Figure 4 :

 4 Figure 4: Forest plot of standardized mean differences between WAT and control in pain, disability, functional tests, quality of life and fatigue.

  thype of study was included for results on adherence. OA= Osteoarthritis; CIRD= chronic inflammatory rheumatic disease; I/C= intervention / Comparator; WAT= wearable activity tracker; PA= physical activity

Table 2

 2 Risk of bias within studies19 Present data on risk of bias of each study and, if available, any outcome level assessment (see item 12). TD and AP) independently extracted data from included studies and record this information on a data on a standardized piloted extraction form accompanied by a codebook. The extraction form was first tested on 3 reports and then adjusted to ensure clarity and comprehensiveness of data collection. Extracted results were compared and any disagreements were resolved by consensus.

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Table 1 :

 1 Characteristics of interventions using WATs to increase physical activity levels for rheumatic musculoskeletal diseases.

				Rheumatic condition		WAT		Intervention / comparator Components associated			Outcome
	Author, publication year (ref)	N participants baseline	Study duration (weeks)	OA of lower limbs	Low back pain	CIRD	Advanced WAT	Pedometer	Mobile App or website	Activity goal setting	Written activity log	Counselling -education	PA Guideline or booklet	Exercise class	Usual care	Self-management	Adherence	Physical activity	symptoms
	WAT versus no WAT
	Paxton 2017 (38)	45 12 ✔				I			I		I	I		C		✔ ✔ ✔
	Skrepnik 2017 (39)	221 14 ✔			I		I				I/C				✔ ✔ ✔
	Katz 2018 (49)	96 21

Describe methods used for assessing risk of bias of individual studies (including specification of whether this was done at the study or outcome level), and how this information is to be used in any data synthesis.9 and online supplement Summary

Funding: this study was partly funded by an educational grant from the French society of rheumatology (patient education working group).