Amine Abadi
email: amine.abadi@etu.univ-orleans.fr

Adnen El Amraoui
email: adnen4@gmail.com

Hassen Mekki
email: mekki.hassen@gmail.com

Nacim Ramdani
email: nacim.ramdani@univ-orleans.fr

Guaranteed Tracking Controller for Wheeled Mobile Robot Based on Flatness and Interval Observer

This paper proposes a guaranteed tracking controller for a Wheeled Mobile Robot (WMR) based on the differential flatness theory and the interval observer. Using the flatness property, it is possible to transform the non linear WMR model into a canonical Brunovsky form, for which it is easier to create a state feedback controller. Since, in most real applications, the WMR is subjected to uncertainties such as slip, disturbance and noise, control algorithms must be modified to take into account those uncertainties. Therefore, based on the information of the upper and lower limits of the initial condition and all the uncertainties, an interval observer that generates an envelope enclosing every feasible state trajectory is developed. After that, based on the center of the obtained interval observer, a new control law is proposed to guarantee the tracking performance of the WMR despite the existence of un-measurable states and bounded uncertainties. The closedloop stability of the system is proven analytically using the Lyapunov theorem. A lot of numerical simulation is realized in order to demonstrate the efficiency of the suggested guaranteed tracking control scheme.

I. INTRODUCTION

In the last decade, special attention has been paid to mobile robots in view of their particular structure, automatic programming and practical challenges. The application domain of these systems has become very varied, such as national defense, hospital tasks, logistics industry and other areas. In order to accurately realize these different practical tasks, a highly performant guidance scheme needs to be proposed for the mobile robot. Recently, the differential flatness property introduced by Fliess [1] has proven to be a good tool to ameliorate the trajectory planning and to create tracking controllers for linear and nonlinear systems. Thus, with flatness, all the state and control inputs of the system can be written as a function of the flat outputs and their derivatives. This property allows us to eliminate the utilization of the complex integration process. In addition, the flatness has specific advantages when used in nonlinear control systems. Indeed, by permitting an accurate linearization of the system's dynamical model, it can be possible to avoid utilizing the linear models with limited validity in the controller design. This characteristic makes

flatness a good tool for solving various problems in many areas [START_REF] Kim | Flatness-based nonlinear control for position tracking of electrohydraulic systems[END_REF][START_REF] Poutney | Robust Tracking Control of Quadrotors based on Differential Flatness: Simulations and Experiments[END_REF][START_REF] Sangwan | Effects of Viscous Damping on Differential Flatness-Based Control for a Class of Under-Actuated Planar Manipulators[END_REF]. In the last decade, the flatness property has been extensively used for the planning and tracking of mobile robots. In [START_REF] Abadi | Optimal trajectory generation and flatness tracking control for a mobile robot[END_REF], Abadi put forward an optimal trajectory generation algorithm for Wheeled Mobile Robot (WMR) based on the transcription method, flatness and the B-spline curve. In [START_REF] Luviano-Juárez | Trajectory Tracking Control of a Mobile Robot Through a Flatness-Based Exact Feedforward Linearization Scheme[END_REF], Luviano combined the flatness feedforward control and the generalized proportional integral to ensure good trajectory tracking for the WMR. In [START_REF] Nasr | A multi-scroll chaotic system for a higher coverage path planning of a mobile robot using flatness controller[END_REF], Nasr proposed higher coverage trajectory generation strategies of a WMR utilizing the flatness property. In the majority of studies on the tracking control of the WMR, nonholonomic assumption has been utilized. This leads to neglecting the existence of slip between the ground and the wheels. However, in real applications, slip is caused by many reasons such as high speed and uneven terrain. Generally, the slippage is not the only problem that can affect the tracking task of the WMR because most of the controllers and observers applied to this latter are based on the assumption that all the states of the system are available and that the disturbances and the measurement noise are negligible. Consequently, a new guidance law is necessary to be developed in order to guarantee the tracking result of the WMR subjected to unknown but bounded uncertainties (slippage, disturbance, measurement noise). In the recent years, interval observers have become a robust approach to dealing with the state estimation problems for a system affected by disturbances and/or uncertainties, which are supposed to be unknown but bounded. This theory was originally introduced in [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF][START_REF] Rapaport | Parallelotopic and practical observers for nonlinear uncertain systems[END_REF] and has been successfully applied in several applications [START_REF] Bernard | Closed loop observers bundle for uncertain biotechnological models[END_REF][START_REF] Lamouchi | Interval observer framework for fault-tolerant control of linear parametervarying systems[END_REF][START_REF] Ifiqir | Interval observer for LPV systems: Application to vehicle lateral dynamics[END_REF][START_REF] Kwassi | Interval Observers for Secure Estimation in Cyber-Physical Systems[END_REF]. Starting from the knowledge of the upper and lower limits of the initial conditions and the uncertainties, interval observers can be developed to produce upper and lower bounds of state variables of dynamical systems at every time instant. Accordingly, the bounds offer intervals where the estimated variables are sure to stay for transient periods during which the classical observers cannot ensure any guarantee. One of the important conditions to create interval observers treated the cooperativity of the estimation error dynamics, which was relaxed in [START_REF] Raïssi | Interval state estimation for a class of nonlinear systems[END_REF]. It was demonstrated that according to a constructive procedure, a Hurwitz matrix could be transformed to a Metzler and Hurwitz one (cooperative). The interval observer design for linear and nonlinear systems with model uncertainty was discussed in [START_REF] Wang | Interval observer design for LPV systems with parametric uncertainty[END_REF][START_REF] Efimov | Design of interval observers for uncertain dynamical systems[END_REF]. However, the existing results showed that the observation error would converge to an interval whose size depended on the value of uncertainty. In addition, any state belonging to this interval could be considered as a robust state estimation for the system. Hence, compared to a punctual observer, the estimation interval guarantees more robustness when dealing with unknown bounded uncertainties. This property encourages us to exploit the advantage of the interval observer in practical applications such as tracking the trajectory of the WMR. In this paper, our contribution consists in designing of guaranteed trajectory tracking control for the WMR despite the existence of unknown but bounded uncertainties, such as slippage, external environmental perturbation and measurement noise. Based on the differential flatness properties, it is possible to change the WMR equation model into a linear canonical (Brunovsky) form. For the obtained linearized system, it is simpler to develop a stabilizing feedback controller. To improve the tracking robustness of the WMR, an interval observer is developed to create an envelope containing all possible state estimations based on the upper and lower values of initial conditions and all uncertainties. Subsequently, the center of the interval observer is considered as a robust state estimation of the WMR. Finally, based on this robust estimation, flatness control, combined with an estimated feedback law, is developed in order to guarantee that the WMR tracks the reference trajectory in a precise interval. To the best of our knowledge, it is the first time in the literature that the advantages of interval estimation techniques are exploited in the design of tracking control for an uncertain WMR system. This article is organized as follows. In section II, the WMR model is presented. In section III, the tracking controller for the WMR without considering any uncertainties is defined. In section IV, the uncertain model of the WMR is defined as well. In section V, the interval observer design is described. Section VI is devoted to the creation of guaranteed tracking control. Section VII deals with the simulation results, and section VIII concludes the paper.

II. WHEELED-ROBOT MOBILE MODEL

The system utilized in this paper consists of a two-wheeled differential driven mobile robot. It is equipped with two independently driven wheels (right and left) and a front wheel to ensure the equilibrium of the robot movement. The generalized configuration of the driven robot mobile is given by q = [x, y, θ], where x and y are the center position coordinates of the mobile robot in the fixed frame (O, X, Y ) and θ represents the robot orientation angle with respect to the X axis. The kinematic model of the WMR without slip is defined as follows:

     ẋ = cos(θ)v ẏ = sin(θ)v θ = w (1)
where v and w are the transnational and rotational velocity of the robot, respectively. This latter can be written as a function of right and left angular speeds of the wheels (w r and w l ) as follows:

v = ( w r + w l 2 )r (2) w = ( w r -w l 2b )r (3)
where r is the radius of the wheel, and 2b is the distance between the wheels. According to the non slip condition, the non-holonomic constraint is defined as follows:

ẋsinθ -ẏcosθ = 0 (4)
Equation ( 4) signifies that accurate tracking will be realizable only if the desired trajectories are feasible for the physical platform.

III. FLATNESS-BASED TRACKING CONTROL

The differential flatness concept represents a fundamental property that characterizes some nonlinear systems. It can be shown that the WMR is a differently flat system, whose flat outputs are given by σ = [σ 11 , σ 21 ] = [x, y]. Therefore, all the states and the control of the WMR system can be written in terms of flat output and their derivatives as follows:

θ = arctan( σ21 σ11 ) (5)
v = σ2 11 + σ2 21 (6)
w = σ11 σ21 -σ11 σ21 σ2 11 + σ2 21 (7)
The relationship between the control input vector, w and v, and the flat output's highest derivatives is not invertible. This problem obviously exposes an obstacle to realize static feedback linearization. To overcome this fact, the control input v is considered as an additional state for the kinematics model (1), consequently, the new extended system is defined as follows:

         ẋ = cos(θ)v ẏ = sin(θ)v v = u 1 θ = u 2 (8)
where the state system of the mobile robot is X = [x, y, v, θ] T and the new control input is defined by u 1 = v and u 2 = w. The invertible relation between the inputs u 1 and u 2 and the higher derivatives of the flat outputs σ 11 = x and σ 21 = y is defined as follows:

σ11 σ21 = B u 1 u 2 (9)
with B = cos(θ) -vsin(θ) sin(θ) vcos(θ) The matrix B is not singular if v = 0. Under this assumption, the control can be defined as follows:

u 1 u 2 = B -1 σ11 σ21 (10)
Thus, the model of WMR can be written in the two-linear canonical (Brunovsky) form as follows:

BF 1      σ11 = σ 12 σ12 = v x Y 1 = σ 11 = x BF 2      σ21 = σ 22 σ22 = v y Y 2 = σ 21 = y (11)
where v x and v y are appropriate feedback controllers that permit the flat outputs σ 11 and σ 21 to track the desirable reference trajectories σ xd and σ yd , respectively. The feedback controllers are defined as follows:

v x = σxd -K x2 ėrx1 -K x1 e rx1 (12)
v y = σyd -K y2 ėry1 -K y1 e ry1 (13)
where

e rx1 = σ 11 -σ xd , e ry1 = σ 21 -σ yd , K x1 , K x2 , K y1
and K y2 are a controller gain that can be chosen so that the characteristic polynomial associated to each flat output tracking error is Hurwitz. The characteristic polynomials of the Burnovsky systems [START_REF] Lamouchi | Interval observer framework for fault-tolerant control of linear parametervarying systems[END_REF] are defined as follows:

s 2 + K x2 s + K x1 = s 2 + 2ξ x Γ xc s + Γ 2 xc ( 14
)
s 2 + K y2 s + K y1 = s 2 + 2ξ y Γ yc s + Γ 2 yc ( 15
)
where the parameters ξ x and ξ y are the damping coefficient, and Γ xc and Γ yc are the bandwidths of the controller. Based on equations ( 14) and ( 15), the controllers gain can be calculated as follows:

K x1 = Γ 2 xc , K x2 = 2ξ x Γ xc , K y1 = Γ 2 yc , K y2 = 2ξ y Γ yc (16)
The Flatness-Based Tracking Control (FBTC) applied to the WMR can be obtained when replacing σ11 and σ12 by the feedback controllers v x and v y in the control input [START_REF] Bernard | Closed loop observers bundle for uncertain biotechnological models[END_REF] as follows:

u F BT C1 u F BT C2 = B -1 σxd -K x2 ėrx1 -K x1 e rx1 σyd -K y2 ėry1 -K y1 e ry1 (17)
The dynamics of the closed-loop tracking error of x and y are defined as follows:

ėrx = H rx e rx (18)
ėry = H ry e ry [START_REF] Zheng | Interval observer for a class of uncertain nonlinear singular systems[END_REF] with:

e rx = [e rx1 , ėrx1 ] T , e ry = [e ry1 , ėry1 ] T , H rx = 0 1 -K x1 -K x2 , H ry = 0 1 -K y1 -K y2
The closed-loop stability of the error tracking systems can be ensured by appropriately choosing the controller poles.

IV. UNCERTAIN KINEMATIC MODEL

We consider that the WMR is subjected to three sources of uncertainties, namely as slippage, external environmental disturbance, and measurement noise. In this case, the Kinematic model of the WMR will be considered as follows:

     ẋ = cos(θ)v + v t cos(θ) + v s sin(θ) + d x ẏ = sin(θ)v + v t sin(θ) -v s cos(θ) + d y θ = w + w s + d θ (20)
where d x , d y and d θ represent the external disturbances, v t and v s represent the slip velocities in the forward direction and normal to the forward direction, respectively, and w s represents the angular slip velocity component. Based on [START_REF] Ryu | Differential flatness-based robust control of mobile robots in the presence of slip[END_REF], we assume that the slip component are defined as follows:

v t (t) = v s (t) = w s (t) = γ 1 v(t) (21)
where γ 1 is a positive constant. We assume that the component velocity and the external disturbance and their derivatives are bounded as follows:

||v t || ≤ γ 1 ||v||, ||v s || ≤ γ 2 ||v||, ||w s || ≤ γ 3 (22) || vt || ≤ γ 4 , || vs || ≤ γ 5 , || ẇs || ≤ γ 6 ( 23
)
d 1 ≤ d x ≤ d 1 , d 2 ≤ d y ≤ d 2 , d 3 ≤ d θ ≤ d 3 ( 24
)
d 4 ≤ ḋx ≤ d 4 , d 5 ≤ ḋy ≤ d 5 , d 6 ≤ ḋθ ≤ d 6 ( 25
)
where γ i , d i and d i i = 1, .., 6 are known values. When choosing the control input defined by equation [START_REF] Bernard | Closed loop observers bundle for uncertain biotechnological models[END_REF] and considering that WMR is subjected to slippage, disturbance and noise, the two integral chains of the Burnovsky form [START_REF] Lamouchi | Interval observer framework for fault-tolerant control of linear parametervarying systems[END_REF] are modified as follows:

M BF 1      σ11 = σ 12 σ12 = v x + ∆ x Y 1 = σ 11 + η x M BF 2      σ21 = σ 22 σ22 = v y + ∆ y Y 2 = σ 21 + η y (26)
where ∆ x and ∆ y are two lumped uncertainties that collect the slippage and external disturbances affecting the x and y channels, and η x and η y are an additive noise affecting the x and y measurement. The existence of ∆ x , ∆ y , η x and η y in a linearized obtained form represents an obstacle to ensure the convergence of the tracking error to zero. To deal with this problem, many researchers have proposed the Active Disturbance Rejection Control (ADRC) [START_REF] Ramírez-Neria | Trajectory Tracking for an Inverted Pendulum on a Cart: An Active Disturbance Rejection Control Approach[END_REF] as a solution. The principle of the ADRC approach is to estimate the effect of lumped uncertainties to the system by the extended state observer strategies and then compensate them. This concept is very complicated and difficult when considering important values of uncertainties. To overcome this problem, a new control algorithm based on the interval observer is proposed to guarantee the tracking results of the uncertain WMR.

V. INTERVAL OBSERVER DESIGN

In this section, interval observers are applied for the Burnovsky system defined by equation (26). In a matrix form, this latter is written as follows:

σ1 = A 1 σ 1 + B 1 v x + ∆ 1 Y 1 = C 1 σ 1 + η x , σ2 = A 2 σ 2 + B 2 v y + ∆ 2 Y 2 = C 2 σ 2 + η y (27) with σ 1 = [σ 11 , σ 12 ] T , σ 2 = [σ 21 , σ 22 ] T , A 1 = A 2 = 0 1 0 0 , B 1 = B 2 = 0 1 T , ∆ 1 = 0 ∆ x T , ∆ 2 = 0 ∆ y T , C 1 = C 2 = 1 0 .
The main idea of designing interval observers of the uncertain system (27) is to create lower and upper bounds of the real states σ 1 (t) and σ 2 (t), which allows us to guarantee that both latter belong to a specific interval. The observer interval design of system (27) requires the following assumption: A1. Pairs (A

     σ1 = A 1 σ 1 + B 1 v x + L 1 C 1 (σ 1 -σ 1 ) + ∆ 1 + L 1 η x σ1 = A 1 σ 1 + B 1 v x + L 1 C 1 (σ 1 -σ 1 ) + ∆ 1 + L 1 η x σ 1 (0) ≤ σ 1 (0) ≤ σ 1 (0) (28)      σ2 = A 2 σ 2 + B 2 v y + L 2 C 2 (σ 2 -σ 2 ) + ∆ 2 + L 2 η y σ2 = A 2 σ 2 + B 2 v y + L 2 C 2 (σ 2 -σ 2 ) + ∆ 2 + L 2 η y σ 2 (0) ≤ σ 2 (0) ≤ σ 2 (0) (29) For system (27), it is impossible to compute gains L 1 and L 2 such that (A 1 -L 1 C 1 ) and (A 2 -L 2 C
2 ) are Metzler and Hurwitz, which presents an obstacle for the construction of the interval observers. This drawback is surmounted via a change in coordinates

Z 1 = G 1 σ 1 and Z 2 = G 2 σ 2 such that E 1 = G 1 H ob1 G -1 1 and E 2 = G 2 H ob2 G -1
2 are Metzler and Hurwitz. The detail of calculating the transformation matrices G 1 and G 2 can be found in [START_REF] Raïssi | Interval state estimation for a class of nonlinear systems[END_REF]. Actually, when introducing the new variables Z 1 = G 1 σ 1 and Z 2 = G 2 σ 2 , system (27) can be presented as follows:

Ż1 = G 1 A 1 G -1 1 Z 1 + G 1 B 1 v x + G 1 ∆ 1 Y 1 = C 1 G -1 1 Z 1 + η x (30) Ż2 = G 2 A 2 G -1 2 Z 2 + G 2 B 2 v y + G 2 ∆ 2 Y 2 = C 2 G -1 2 Z 2 + η y (31)
The interval observers of the new systems (30) and (31) are defined as follows:

                   Ż1 = E 1 Z 1 + G 1 B 1 v x + G + 1 ∆ 1 -G - 1 ∆ 1 + G 1 L 1 Y 1 + G 1 L 1 η x Ż1 = E 1 Z 1 + G 1 B 1 v x + G + 1 ∆ 1 -G - 1 ∆ 1 + G 1 L 1 Y 1 + G 1 L 1 η x Z 1 (0) = G + 1 σ 1 (0) -G - 1 σ 1 (0) Z 1 (0) = G + 1 σ 1 (0) -G - 1 σ 1 (0) (32)                      Ż2 = E 2 Z 2 + G 2 B 2 v y + G + 2 ∆ 2 -G - 2 ∆ 2 + G 2 L 2 Y 2 + G 2 L 2 η y Ż2 = E 2 Z 2 + G 2 B 2 v y + G + 2 ∆ 2 -G - 2 ∆ 2 + G 2 L 2 Y 2 + G 2 L 2 η y Z 2 (0) = G + 2 σ 2 (0) -G - 2 σ 2 (0) Z 2 (0) = G + 2 σ 2 (0) -G - 2 σ 2 (0) (33)
with

E 1 = G 1 (A 1 -L 1 C 1 )G -1 1 , G + 1 = max(0, G 1 ), G - 1 = G + 1 -G 1 , E 2 = G 2 (A 2 -L 2 C 2 )G -1 2 , G + 2 = max(0, G 2 ), G - 2 = G + 2 -G 2 .
After designing the interval observers of the new systems (30) and (31), we can deduce the upper and lower states of the original systems (27) in the following way:

     σ1 = R + 1 Z 1 -R - 1 Z 1 σ1 = R + 1 Z 1 -R - 1 Z 1 σ 1 (0) ≤ σ 1 (0) ≤ σ 1 (0) ,      σ2 = R + 2 Z 2 -R - 2 Z 2 σ2 = R + 2 Z 2 -R - 2 Z 2 σ 2 (0) ≤ σ 2 (0) ≤ σ 2 (0) (34) with R 1 = G -1 1 , R + 1 = max(0, R 1 ), R - 1 = R + 1 -R 1 , R 2 = G -1 2 , R + 2 = max(0, R 2 ), R - 2 = R + 2 -R 2 . Consider the error equations e z1 = Z 1 -Z 1 , e z1 = Z 1 -Z 1 , e z2 = Z 2 -Z 2 and e z2 = Z 2 -Z 2 .
The dynamics of upper and lower observer errors are defined as follows:

E Obx ėz1 = E 1 e z1 + Π z1 ėz1 = E 1 e z1 + Π z1 E Oby ėz2 = E 2 e z2 + Π z2 ėz2 = E 2 e z2 + Π z2 (35) with Π z1 = G + 1 ∆ 1 -G - 1 ∆ 1 -G 1 ∆ 1 + G 1 L 1 (η x -η x ), Π z1 = G 1 ∆ 1 + G - 1 ∆ 1 -G + 1 ∆ 1 + G 1 L 1 (η x -η x ), Π z2 = G + 2 ∆ 2 -G - 2 ∆ 2 -G 2 ∆ 2 + G 2 L 2 (η y -η y ) and Π z2 = G 2 ∆ 2 + G - 2 ∆ 2 -G + 2 ∆ 2 + G 2 L 2 (η y -η y )
. Since E 1 and E 2 are assumed to be non-negative, Π z1 , Π z1 , Π z2 and Π z2 are non-negative. As a result, for any initial condition Z 1 (0), Z 1 (0), Z 2 (0) and Z 2 (0) chosen such that e z1 (0), e z1 (0), e z2 (0) and e z2 (0) are non-negative, the dynamics of the interval estimation errors e z1 , e z1 , e z2 and e z2 stay always non-negative for all time t, so the state is bounded as follows:

Z 1 (t) ≤ Z 1 (t) ≤ Z 1 (t), Z 2 (t) ≤ Z 2 (t) ≤ Z 2 (t) (36)
Moreover, functions Π z1 , Π z1 , Π z2 and Π z2 are globally Lipschitz, and consequently for Z 1 ≤ Z 1 ≤ Z 1 and Z 2 ≤ Z 2 ≤ Z 2 and for a chosen submultiplicative norm, there exist positive constants π z1i and π z2i i = 1..6 such that:

||Π z1 || ≤ π z11 ||Z 1 -Z 1 || + π z12 ||Z 1 -Z 1 || + π z13 (37) ||Π z1 || ≤ π z14 ||Z 1 -Z 1 || + π z15 ||Z 1 -Z 1 || + π z16 (38) ||Π z2 || ≤ π z21 ||Z 2 -Z 2 || + π z12 ||Z 2 -Z 2 || + π z23 (39) ||Π z2 || ≤ π z24 ||Z 2 -Z 2 || + π z25 ||Z 2 -Z 2 || + π z26 (40)
According to [START_REF] Zheng | Interval observer for a class of uncertain nonlinear singular systems[END_REF], if there exist positive definite and symmetric matrices P x2 , P y2 , Q x2 , Q y2 , δ x and δ y such that the following Linear Matrix Inequality (LMI) are satisfied:

E T z1 P x2 + P x2 E z1 + δ x P 2 x2 + γz1 δx I 1 + Q x2 P x2 P x2 -I1 δx ≤ 0, (41) E T z2 P y2 + P y2 E z2 + δ y P 2 y2 + γz2 δy I 2 + Q y2 P y2 P y2 -I2 δy ≤ 0, (42) with E z1 = diag(E 1 , E 1 ), E z2 = diag(E 2 , E 2 ), γ z1 = 2 max(π 2 z11 , π 2 z12 , π 2 z13 , π 2 z14 , π 2 z15 , π 2 z16 ) and γ z2 = 2 max(π 2 z21 , π 2 z22 , π 2 z23 , π 2 z24 , π 2 z25 , π 2 z26 ). Then, variables Z 1 (t), Z 1 (t), Z 2 (t)
and Z 2 (t) are bounded for all the time. The interval observer design enables obtaining deterministic dynamic intervals containing a real state vector. Accordingly, the centre of the interval can be chosen as a robust state estimation for the uncertain WMR as follows:

σ1 = σ 1 + σ 1 2 , σ2 = σ 2 + σ 2 2 , θ = θ + θ 2 , v = v + v 2 (43)
Based on equations ( 5) and ( 6), the lower and upper values of θ and v can be deduced as follows:

θ = arctan( σ 22 σ 12 ), θ = arctan( σ 22 σ 12 ) (44) v = σ 2 12 + σ 2 22 , v = σ 2 12 + σ 2 22 ( 45
)
Choosing the center of the interval as a robust state estimation has no influence on the trajectory tracking results because any σ ∈ [σ, σ] can be considered as a guaranteed state estimation for the WMR. Generally, the center of the interval is a classic choice used in many researchers [START_REF] Lamouchi | Interval observer framework for fault-tolerant control of linear parametervarying systems[END_REF][START_REF] Efimov | Application of interval observers to estimation and control of air-fuel ratio in a direct injection engine[END_REF] studying the control based on the interval observer.

VI. GUARANTEED TRACKING CONTROL

In this section, based on the interval observer result, a guaranteed tracking control is developed for the uncertain WMR. Replacing the real state by a robust state estimation defined by equation ( 43) in the feedback controllers ( 12) and ( 13), the estimated feedback law can be obtained as follows:

vx = σxd -K x2 ėrx1 -K x1 êrx1 (46) vy = σyd -K y2 ėry1 -K y1 êry1 (47)
with êrx1 = σ11 -σ xd and êry1 = σ21 -σ yd . When replacing state σ by the robust state estimation σ in the u F BT C control [START_REF] Ryu | Differential flatness-based robust control of mobile robots in the presence of slip[END_REF], the Guaranteed Flatness-Based Tracking Control (GFBTC) applied to the uncertain WMR system can be obtained as follows:

u GF BT C1 u GF BT C2 = B-1 σxd -K x2 ėrx1 -K x1 êrx1 σyd -K y2 ėry1 -K y1 êry1 (48)
u GF BT C differs from u F BT C in the way that in u GF BT C the robust state estimation σ is utilized, but in u F BT C state σ is applied. To analyze the stability of the WMR system, the stability of the tracking error dynamics of positions x and y must be studied. To prove the stability of the tracking error dynamics of position x, let us define the Lyapunov function candidate as follows:

V r1 = e T rx P x1 e rx (49)
where e rx = [e rx1 , ėrx1 ] T , and P x1 is a symmetric positive defined matrix. The derivative of the Lyapunov function V r1 is defined as follows:

Vr1 = ėT rx P x1 e rx + e T rx P x1 ėrx (50)
When substituting ėrx by equation [START_REF] Ramírez-Neria | Trajectory Tracking for an Inverted Pendulum on a Cart: An Active Disturbance Rejection Control Approach[END_REF] in equation ( 50), the derivate of the Lyapunov function V r1 is defined as follows:

Vr1 = e T rx (H T rx P x1 + P x1 H rx )e rx (51)
Gains k x1 and k x2 are chosen such that matrix H rx is Hurwitz. Thereby, for any symmetric positive definite matrix Q x1 , there exists a symmetric positive definite matrix P x1 satisfying the Lyapunov equation:

H T rx P x1 + P x1 H rx = -Q x1 (52)
According to (52), the derivative of the Lyapunov function V r1 can be written as follows:

Vr1 = -e T rx Q x1 e rx (53)
Thus, based on the Lyapunov method, the asymptotic stability for trajectory tracking is deduced.

To prove the stability of the error observer of the x position, let us recall the proof defined in [START_REF] Zheng | Interval observer for a class of uncertain nonlinear singular systems[END_REF], which shows that variables Z 1 (t) and Z 1 (t) are bounded. Let Consider the positive definite quadratic Lyapunov function as follows:

V z1 = e T z1 P x2 e z1 (54)
where e z1 = (e T z1 , e T z1 ) T , and P x2 is a positive definite symmetric matrix. The observation error of system E obx can be rewritten as:

ėz1 = E z1 e z1 + Π z1 (55)
with

Π z1 = [Π T z1 , Π T z1 ]
T . Due that matrix E 1 is Hurwitz and Metzler, so is the matrix E z1 . The derivate of V z1 can be defined as follows:

Vz1 = e T z1 (E T z1 P x2 + P x2 E z1 )e z1 + 2e T z1 P x2 Π z1 (56)
≤ e T z1 (E T z1 P x2 + P x2 E z1 )e z1 + δ x e T z1 P 2 x2 e z1 + 1 δ x ||Π z1 || 2
(57) According to Corollary 1 defined in [START_REF] Zheng | Interval observer for a class of uncertain nonlinear singular systems[END_REF], there exists a positive constant γ z1 such that:

||Π z1 || 2 ≤ γ z1 (||e z1 || 2 + 1) (58)
with

γ z1 = 2 max(π 2 z11 , π 2 z21 , π 2 z31 , π 2 z41 , π 2 z51 , π 2 z61
). Thus, equation (57) can be written as:

Vz1 ≤ e T z1 (E T z1 P x2 + P x2 E z1 + δ x P 2 x2 + γ z1 δ x I 1 )e z1 + γ z1
(59) Therefore, if there exist positive definite symmetric matrices P x2 and Q x2 and a positive scalar δ x such that the LMI defined in (41) is satisfied, then equation (59) can be defined as follows:

Vz1 ≤ e T z1 (-Q x2 )e z1 + γ z1 (60)
This implies that variables Z 1 (t) and Z 1 (t) are bounded for all t 0 ≥ 0. To prove the global stability of the complete closed-loop system (system + controller + state observer), let us define the following Lyapunov function candidate:

V 1 = V r1 + V z1 (61)
It follows from the asymptotic stability of each subsystem that the global asymptotic stability of the error dynamics of position x is guaranteed. The stability of the error dynamics of position y can be studied in the same way as position x.

VII. SIMULATION AND RESULTS

In this section, two kinds of simulation are presented to validate the performance obtained by the guaranteed tracking control. The parameters of the considered WMR are given by: r = 0.1 m, b = 0.15 m. A reference trajectory is generated for the WMR permitting its movement from the initial state X(0) = [0, 0, √ 2, 45] T to the final state X(5) = [4, 7, √ 2, 45] T . In addition, the obtained trajectory must respect the following constraint:

0 m ≤ σ xd ≤ 5 m, 0 m ≤ σ yd ≤ 10 m, -90 • ≤ θ d ≤ 90 • , 0.1 m.s -1 ≤ v d ≤ 10 m.s -1 , v x (0) = v y (0) = v x (5) = v y (5) = 1 m.s -1 (62)
An 11-order Bezier curve for each flat output is considered to generate the path for the WMR, which can be formulated as follows:

σ xd (t) = 11 k=1 B k,11 (t)α xk , σ yd (t) = 11 k=1 B k,11 (t)α yk (63)
where α xk and α yk , k = 0, .., 11 are the control parameters of the Bezier curve, and B k,11 (t) is a Bernstein polynomial defined as follows:

B k,11 (t) = 11! k!(11 -k)! ( t f -t t f -t 0 ) 11-k ( t -t 0 t f -t 0 ) k (64)
A. Simulation 1

In this simulation, the lower and upper values of the initial state and all the uncertainties are taken as σ This choice of eigenvalues makes the observer dynamics faster than the system. According to [START_REF] Raïssi | Interval state estimation for a class of nonlinear systems[END_REF], matrices G 1 and G 2 , chosen such that matrices E 1 and E 2 are Metzler and Hurwitz, are defined as

G 1 = G 2 = -2 0.2 3 -0.2 .
Take π z1i = π z2i = 1, i = 1, .., 6, so γ z1 = γ z2 = 2. By solving the LMI defined in (41) and (42), we find that δ x = δ y = 1, P x2 = diag(P x21 , P x21 ), P y2 = diag(P y21 , P y21 ),

Q x2 = diag(Q x21 , Q x21 ) and Q y2 = diag(Q y21 , Q y21 ),
where Q x21 and Q y21 are the identity matrices 2 * 2, and P x21 and P y21 are defined as P x21 = P y21 = 0.1650 0 0 0.1076 .

In order to show the robustness of the proposed control, FBTC and GFBTC are applied to the uncertain WMR system under the same conditions. As a consequence, the WMR first starts from an uncertain initial condition X(0) ∈ [X(0), X(0)] defined as X(0) = [x(0), ŷ(0), v(0), θ(0)] T = [1, 1, 7.74, 60] T . Second, it is subjected to unknown uncertainties with known bounds defined previously. Figure 1 depicts the following results:

-The red curves represent the desired trajectory of the WMR.

-The blue and black curves illustrate the upper and lower state estimations of the uncertain WMR system.

-The pink and green curves show the tracking results when applying FBTC and GFBTC, respectively, to the uncertain WMR system.

From Figure 1, it can be firstly observed that choosing a properly gain enables the interval observer to provide a time-varying enclosure, which contains all possible real state vectors of the uncertain WMR. This type of observers allows guaranteeing the estimation result of a WMR despite the presence of bounded uncertainties. In addition, it can be demonstrated that exploiting the estimation result obtained by the interval observer in the design of GFBTC permits the WMR to move in a precise interval containing the desired reference trajectory. Whereas, when applying FBTC to the uncertain WMR system, this latter diverges strongly from the reference. As a result, the controllers that are not based on an uncertain model, even if they are feedback controllers, may not work correctly.

B. Simulation 2

A second simulation collection is carried out with the same references and the same bound values of the initial state considered in simulation 1 and with other important values of uncertainty conditions for checking robustness. In this simulation, we assume that the WMR is subjected to unknown uncertainties with known bounds defined as 2 where comparative simulation with FBTC and GFBTC is also given. From Figure 2, the effectiveness of GFBTC can be seen compared to FBTC. Furthermore, the width of the estimated interval increases compared the interval width obtained in simulation 1, and this is due to the augmentation of the value of uncertainties. In spite of this disadvantage, it can be observed that the WMR still moves in a precise interval containing the reference trajectory. Consequently, it can be deduced that the combination between the flatness control and the interval observer permits obtaining a guaranteed guidance law for the WMR despite the existence of unknown uncertainties with important bound values.

γ 1 = γ 2 = γ 3 = 0.8, γ 4 = γ 5 = γ 6 = 0.

VIII. CONCLUSION

In this paper, the problem of tracking trajectories for a WMR system subjected to unknown but bounded uncertain slip, external disturbances and measurement noise have been studied. To resolve this problem, a guaranteed tracking controller based on the flatness property and the interval observer has been designed. The simulation results have shown that the proposed approach guarantees the trajectory tracking for the WMR in a precise interval despite the presence of all uncertainties with important bound values. In future work, observer gains need to be further optimized in order to reduce the effect of uncertainties on the interval estimation accuracy of the WMR.

  1 (0) = [-2, 0.2] T , σ 2 (0) = [-2, 0.2] T , σ 1 (0) = [4, 4.2] T , σ 2 (0) = [4, 7.47] T , γ 1 = γ 2 = γ 3 = 0.4, γ 4 = γ 5 = γ 6 = 0.5, d 1 = d 2 = d 4 = d 5 = -0.3 , d 3 = d 6 = -0.25, d 1 = d 2 = d 4 = d 5 = 0.3 , d 3 = d 6 = 0.25, η x = -0.1, η y = -0.2, η x = 0.1, η y = 0.2. The controller design parameters are chosen as Γ xc = Γ yc = 5 rad/s and ξ x = ξ y = 1. When choosing the matrix gain L 1 = L 2 = 25 150 T , matrices (A 1 -L 1 C 1 ) and (A 2 -L 2 C 2 ) are Hurwitz and have the eigenvalues -10 and -15.

  7, d 1 = d 2 = d 4 = d 5 = -0.6 , d 3 = d 6 = -0.5, d 1 = d 2 = d 4 = d 5 = 0.6 , d 3 = d 6 = 0.5, η x = -0.5, η y = -0.4, η x = 0.5, η y = 0.4. The feedback gain of the controllers and the observer have been adjusted to obtain a smooth and fast tracking performance. The tracking performances related to simulation 2 are illustrated in Figure

Fig. 1 :Fig. 2 :

 12 Fig. 1: Tracking simulation results of WMR subjected to uncertainties with known bound values defined in simulation 1

  1 , C 1 ) and (A 2 , C 2 ) are observable. A2. There exist L 1 and L 2 gains such that matrices (A 1 -L 1 C 1 ) and (A 2 -L 2 C 2 ) are Hurwitz and Metzler (offdiagonal elements are positive). A3. The lumped uncertainties and the measurement noise are unknown but bounded with known bounds ∆ 1 , ∆ 2 , η x , η y , ∆ 1 , ∆ 2 , η x and η y . According to assumptions 1, 2 and 3, the interval observers of system (27) are defined as follows:

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 823887.


Stéphanie Sarda
email: stephanie.sarda@iut-tlse3.fr

Farid Errassifi

Olivier Marsan

Anne Geffre

Catherine Trumel

Christophe Drouet

Adsorption of tranexamic acid on hydroxyapatite: Toward the development of biomaterials with local hemostatic activity

Keywords:

This work proposes to combine tranexamic acid (TAX), a clinically used antifibrinolytic agent, and hydroxyapatite (HA), widely used in bone replacement, to produce a novel bioactive apatitic biomaterial with intrinsic hemostat ic properties. The aim of this study was to investigate adsorptive behavior of the TAX molecule onto HA and to point out its release in near physiological conditions. No other phase was observed by X ray diffraction or transmission electron microscopy, and no apparent change in crystal size was detected. The presence of TAX on the powders was lightly detected on Raman spectra after ad sorption. The adsorption data could be fitted with a Langmuir Freundlich equation, suggesting a strong interac tion between adsorbed molecules and the formation of multilayers. The concentration of calcium and phosphate ions in solution remained low and stable during the adsorption process, thus ion exchange during the adsorption process could be ruled out. The release of TAX was fast during the first hours and was governed by a complex pro cess that likely involved both diffusion and dissolution of HA. Preliminary aPTT (activated partial thromboplastin time) hemostasis tests offered promising results for the development of osteoconductive apatitic biomaterials with intrinsic hemostatic properties, whether for dental or orthopedic applications.

Introduction

Orthopedic, dental, and trauma surgeries are accompanied, among others, by a high risk of bleeding, and uncontrolled hemorrhage can be the leading cause of blood transfusions, morbidity, and mortality. Blood transfusions may be a life saving measure in such cases, but it re mains an expensive resource and can result in a variety of problems for patients like the risk of infection transmission.

Hemostasis is defined as "the stoppage of bleeding." The three major steps of hemostasis are 1.) vasoconstriction, which diminishes blood loss; 2.) exposure of collagen, which initiates a series of reactions called the coagulation cascade on the wall of blood vessels, leading to the for mation of a primary hemostatic plug; and finally, 3.) adhesion of platelets onto the blood vessel wall, leading to the conversion of fibrinogen into fibrin to create the final clot [1]. Effective and rapid hemostasis is critical to optimize surgical outcomes. In this context, hemostatic agents are ad juvants used in surgery to improve hemostasis when conventional tech niques (clips, staples, sutures, ligatures, and electrocoagulation) are insufficient [2]. Various hemostatic drugs have thus been used and show some efficacy in many types of surgery [START_REF] Royston | Seminars in anesthesia[END_REF].

Tranexamic acid [trans 4 (aminomethy1) cyclohexanecarboxylic acid, TAX] is a clinically used antifibrinolytic agent (Fig. 1). It is a synthetic lysine analog and is used to manage of different bleeding disorders as well as to reduce postoperative blood loss in patients undergoing high bleeding risk surgeries. The hemostatic property of TAX mainly relates to its ability to inhibit the conversion of plasminogen to plasmin, thereby preventing excessive blood loss in hyperfibrinolytic conditions. Plasmin breaks down fibrinogen and a series of other proteins involved in coagu lation [START_REF] Royston | Seminars in anesthesia[END_REF]4]. TAX however presents a low bioavailability of only 30 50% by oral administration [5]. In orthopedic surgery such as knee or hip arthroplasty, TAX appears to be more efficacious when a dose of N30 mg/kg is used [4]. Moreover, it has been found to cause gastrointes tinal side effects when administered orally, whereas this has not been re ported with intravenous administration. However, rapid intravenous boluses may cause hypotension, so it is recommended that it is given ei ther over 10 min or via infusion, although its dose needs to be reduced in cases of renal impairment. Thus, especially for surgical procedures known to present a high risk for bleeding, the search for improved hemostasis approaches or methodologies is still undergoing.

One appealing approach when dealing with bone related surgeries is to confer hemostatic properties to the implanted biomaterial itself. Calcium phosphates with an apatitic structure (apatites) constitute the mineral part of calcified tissues such as bone and teeth [6] and are for this reason widely used for bone related biomedical applications, in various forms including powders, scaffolds, self setting cements, or coatings [7,8]. Surface functionalization of apatite crystals with biomol ecules of biological interest (e.g., growth factors, drugs, enzymes, etc.) also increases its potential for medical uses: for example, in the field of bone repair, for activating bone tissue regeneration, for limiting oste oclastic resorption, or preventing bone infections. Due to their excep tional surface reactivity, apatitic compounds have been widely studied with drug delivery objectives [START_REF] Errassifi | Adsorption on apatitic calcium phosphates: applications to drug delivery[END_REF]. Adsorption on apatitic surfaces (as op posed to simple impregnation) leads to a stable association and control of the amount of bioactive molecules associated in the solid implant, and thus of the dose released, that increase its potential for the develop ment of osteoconductive biomaterials with drug delivery properties [START_REF] Gómez-Morales | [END_REF]. Also, the understanding of biomineralization events involves a bet ter comprehension of biomolecules adsorption phenomena. For in stance, several bone proteins such as osteocalcin, osteopontin, as well as biomolecules including albumin, or else phospholipids, among others, have been shown to adsorb strongly onto apatite [11,12]. In the biomaterials field, some bone adsorbing molecules used in the treatment of bone diseases, like bisphosphonates, have been used to control bone turnover and their biological activity has been shown to correlate with their adsorption properties [START_REF] Errassifi | Adsorption on apatitic calcium phosphates: applications to drug delivery[END_REF]. The surface characteris tics of apatites are also largely used in separation chromatography of proteins, and in fact, preparatory and development studies for these ap plications have initiated research on adsorption on apatite surface [13].

Among the calcium phosphate compounds widely used nowadays in bone replacement, stoichiometric hydroxyapatite (HA, Ca 10 (PO 4 ) 6 (OH) 2 ) is the most commonly found. The aim of this work was to combine TAX and hydroxyapatite to produce a novel bioactive apatitic biomaterial with intrinsic hemostatic properties. Such biomaterials may be used in orthopedic or dental surgery to control blood loss while avoiding the ad verse effects of systemic medication. The adsorption of TAX on HA could allow controlling the release of the hemostatic drug. Thus, the aim of this study was in priority to investigate the adsorptive behavior of TAX onto HA and to point out its release in near physiological conditions. Full char acterizations of the powders before and after adsorption were also per formed to better understand the mechanism of adsorption.

Materials and methods

Materials

Stoichiometric hydroxyapatite (HA) was synthesized by double de composition by adding dropwise a phosphate salt aqueous solution (1.54 M (NH 4 ) 2 HPO 4 , 433.3 mL) into a calcium salt aqueous solution (1 M Ca(NO 3 ) 2 .4H 2 O, 1100 mL) at 90 °C under continuous stirring [14]. The pH of the suspension during the synthesis was approximately 10 by adding NH 4 OH. After filtration, the precipitate was washed and freeze dried. The powder obtained was sieved (b125 μm) and stored in a freezer.

The phosphate content in the HA sample was determined by spectrophotometry using the phosphovanadomolybdic acid complex method and the calcium content was measured by complexometry with EDTA (relative error of 0.5%) [START_REF] Charlot | Chimie Analytique Quantitative[END_REF]. The determination of the atomic Ca/P ratio of the precipitate was reached with a relative error of 1%. The specific surface area of the powder was determined by nitrogen adsorp tion according to the Brunauer Emmett Teller (BET) method with a Quantachrome Instruments Monosorb NOVA 1000.

Powder of tranexamic acid (trans 4 (aminomethy1)cyclohexanecar boxylic acid, TAX) was obtained from Sigma Aldrich (purity 97%). Cis 4 (aminomethyl)cyclohexanecarboxylic acid was used as the internal standard (IS) for TAX titration and was purchased from Sigma Aldrich (purity 95%).

Adsorption experiments

Adsorption experiments were performed in 1 mM potassium chlo ride aqueous solution of TAX (initial concentration varied from 0 to 6 g/L or 3.8 × 10 -2 M). 50 mg of the HA sample was dispersed in 5 mL of the adsorption medium in a polypropylene tube and sonicated for a few minutes (initial pH about 7.2). Preliminary tests demonstrated that the kinetics of adsorption of TAX on HA was slow; therefore, the suspensions obtained were incubated in the presence of TAX for 24 h (without stirring and at ambient temperature). After incubation, the ad sorption tubes were centrifuged for 20 min at 5000 rpm. The resulting supernatants were filtered through Millipore filters (pore size 0.2 μm) and analyzed for calcium and phosphate titrations. The solids (HA TAX) retrieved after adsorption were freeze dried and stored in a freez er before characterization. Blanks containing only the adsorption solu tion incubated without solid were used as controls.

Solution analysis

Phosphate and calcium levels in the supernatants were determined by Inductively Coupled Plasma Atomic Emission Spectroscopy (ICP OES, Horiba Jobin Horibo).

There is no chromophore in the TAX molecule enabling direct anal ysis by UV visible spectrophotometry. Therefore, TAX was titrated in solution using a liquid chromatography/mass spectrometry (LC/MS) method previously adapted from Grassin et al. [START_REF] Grassin Delyle | [END_REF]. Chromatographic separation was performed on a 6140 Quadrupole 1200 Series LC system using an Uptisphere® Strategy™ C18 2 Column (5 μm, 4.6 × 150 mm, Agilent Technologies). The mobile phase was prepared by adjusting formic acid solution (0.1%) and methanol in the ratio 48:52 (v/v). The flow rate was 5 μL/s, the run time was equal to 4 min and the injection volume was 10 μL. A peak was observed at 2.29 min for TAX and 2.30 for IS. A mass spectrometer, equipped with an electrospray ionization (ESI) source was used for detection and quantification.

Stock solutions of TAX (1 g/L) and of the internal standard (IS, 0.4 g/L) were prepared separately in ultra pure water. TAX (from 0 to 12 ppm) and IS (5 ppm) calibration samples were prepared in triplicate by dilution of their stock solutions in ultra pure water. The calibration curve (Fig. 2) was obtained by plotting the peak area ratio of TAX to IS (TAX/IS) as a function of TAX concentration. The calibration equation was obtained by linear regression analysis (r 2 = 0.9997).

Release kinetics

TAX release profiles were obtained with a flow through cell system (USP Apparatus 4, Sotax AG, Switzerland) with a ø = 12 mm cell (6 mL) and a piston pump (Sotax CY, Sotax AG, Switzerland). Release tests were performed at 37.0 ± 0.5 °C under sink conditions according to the European Pharmacopoeia guidelines [17]. The dissolution medium consisted of ultra pure water pumped through the column at a flow rate of 15 mL/min and the closed system consisted of 50 mL of recycled dissolution medium. In all experiments, laminar flow was used with a bed of 1 g of glass beads covered with 100 mg of HA powder after TAX adsorption (175 μmol TAX/g HA).

Periodically, 5 mL of dissolution medium was collected and filtrated through Millipore filters (pore size 0.2 μm) and the TAX content was de termined by LP MS (25 samples from 5 to 2100 min). The same volume of dissolution medium was replaced back after each sampling in order to maintain a constant volume and remain in sink conditions. Each re lease test was duplicated. The results are presented as kinetics of cumu lative TAX release (in %) as a function of time:

TAX released% Q r cumul =Q ads ð Þ Ã 100 ð1Þ
where Q r cumul is the cumulative amount of TAX released and Q ads is the initial quantity of TAX adsorbed.

Characterization of solids after adsorption

The solid phases before and after TAX adsorption were characterized by X ray diffraction (XRD), using an Equinox 1000 curved counter INEL diffractometer with a cobalt anticathode (λ Co = 1.78892 Å). Raman microspectroscopy analysis was performed using a confocal microspectrometer (Horiba Jobin Yvon Labram HR800) with a helium neon laser (λ = 632.82 nm) over a wavenumber range of 350 1750 cm -1 . Transmission electron microscopy (TEM) micrographs were obtained on a JEOL JEM 2100F microscope set at an acceleration tension of 200 kV.

Hemostasis tests

Preliminary hemostatic tests were carried in this work by measuring the activated partial thromboplastin time (aPTT), which is a global screening assay to evaluate the coagulation cascade. The aPTT is the clotting time obtained when "partial thromboplastin" is added to plas ma. Partial thromboplastin is the phospholipid fraction of a tissue ex tract and differs from a complete tissue extract (i.e., "thromboplastin") by the lack of tissue factor [18,19].

Blood specimens from 10 healthy adult dogs of various breeds were obtained by jugular venipuncture, using minimal restraint and hand compression. Blood specimens were drawn using a 0.8 × 25 mm needle (Neolus; Terumo Europe N.V., Leuven, Belgium) connected to a vacuum system (VenojectLuer adapter; Terumo Europe, N.V.) into a citrated tube (Venosafe; Terumo France, Guyancourt, France). Trisodium citrate (0.129 M) is a good calcium ligand, and in the ratio 1:9 (v/v), the decal cification prevents clot formation.

To examine the effect of two solid samples on the aPTT, 50 mg HA with and without adsorbed TAX (175 μmol TAX/g HA) was added to 1.2 mL of citrated whole blood, and the results were compared with the results obtained when only TAX was added (100 μL TAX in aqueous solution, 1 g/L). Citrated plasmas were then obtained after centrifuga tion of the blood specimens for 5 min at 2900 g (Hettich Rotofix 32 A, Tuttlingen, Germany).

aPTT was automatically measured using a STA Compact analyzer (Diagnostica Stago, Asnières sur Seine, France) after quality control using STA Coag Control N + P (Diagnostica Stago, Asnières sur Seine, France). Briefly, citrated plasma was incubated at 37 °C with an activator and phospholipid (partial thromboplastin). CaCl 2 was then added to start the coagulation cascade. Results are expressed in seconds, as the time taken for a fibrin clot to form. Each test was performed in duplicate.

Results and discussion

Characterization of powders

The synthesized HA sample had a Ca/P molar ratio of 1.66 ± 0.02 corresponding to stoichiometric hydroxyapatite (Ca 10 (PO 4 ) 6 (OH) 2 ). The specific surface area of HA was 58 (±3) m 2 /g. The solids were char acterized after adsorption. Its X ray diffraction pattern displayed char acteristic diffraction lines of well crystallized hydroxyapatite and no other crystalline phase was detected after TAX adsorption (Fig. 3a); these results were further confirmed by TEM. Furthermore, there was no apparent change in crystal size, as all crystals remained approxi mately 100 nm in length (Fig. 3b andc).

Raman spectra of the powders before (HA) and after (HA TAX) ad sorption are presented in Fig. 4 and compared with the Raman spectrum of pure TAX. Raman spectra of the solids showed characteristic lines of apatitic phosphate groups: ν 1 (960 cm -1 ), ν 2 (430 450 cm -1 ), ν 3 (1040 1074 cm -1 ), and ν 4 (580 608 cm -1 ) modes of PO 4

3- [20]. More over OH -apatitic vibration was also detected around 3560 cm -1 . Char acteristic vibration bands corresponding to TAX around 1500 cm -1 (ν C\ \C , ν C O , ν CH2 , ν NH2 ), and between 2800 and 3000 cm -1 (ν CH2 , ν CH ), were slightly detected beside the HA spectrum after adsorption, probably because the amount of adsorbed TAX was limited and also be cause TAX molecule exhibits Raman vibration bands of low intensity.

Adsorption of tranexamic acid

The evolution of the amount of TAX adsorbed, Q ads (μmol/m 2 ), on HA from dilute aqueous solutions (0 40 mM) as a function of its remaining concentration in solution, C eq (mmol/L), is plotted in Fig. 5. As shown in the graph, no adsorption plateau was observed as the general tendency was a continuous increase of the adsorbed amount versus C eq .

The adsorption of (bio)molecules with a strong affinity for apatite surfaces can often be described by the Langmuir isotherm [START_REF] Errassifi | Adsorption on apatitic calcium phosphates: applications to drug delivery[END_REF]. This is the case for example for some biomolecules exhibiting phosphate or phosphonate endgroups as in phosphoserine or bisphosphonates. In this model, the solid surface is considered to be composed of energeti cally equivalent adsorption sites and no interaction exists between ad sorbate molecules [21,22] that form a monolayer on the surface of the substrate. Also, the adsorption of molecules exhibiting a high affinity for apatitic supports is often accompanied by a simultaneous ion ex change process, involving ions such as phosphate anions from the surface of apatite crystals. When the amount adsorbed on apatitic sur faces is small, the adsorption is generally more accurately described by the Freundlich model. This is the case for molecules such as glycine, acetic acid, or else serine [START_REF] Errassifi | Adsorption on apatitic calcium phosphates: applications to drug delivery[END_REF]22]. Such molecules weakly interact with apatite surface due to the presence of endgroups that have only a low affinity for apatite surface, such as carboxylate. However, the present data give graphical evidence that TAX adsorption on HA cannot be de scribed by the Langmuir isotherm model or the Freundlich model. In deed, rather than a tendency toward stabilization, an increasingly greater adsorption versus C eq is observed here. This evolution reminds a power law. Indeed, a mathematical analysis of the adsorption data showed that the curve can be satisfactorily (r 2 = 0.994) described by an equation of the form:

Q ads K Â C eq n ð2Þ
with a value of "n" N 1, namely n = 1.99 ± 0.11, and K = 2.64 ± 0.41 (with Q ads in μmol/m 2 and C eq in mmol/L). Note that this equation should not be confounded with the one describing the typical Freundlich model which has an equivalent form but with an exponent "n" lower than 1.

At this point, it was also interesting to check how the adsorption data could mathematically be fitted with a Langmuir Freundlich equa tion, also known as Sips, which also affects an exponent to the equilib rium concentration, as in the equation:

Q ads N m  K  C eq À Á n 1 þ K  C eq À Á n ð3Þ
(Note that there exists also an equivalent form of this equation that replaces K n with the constant K S ). The mathematical treatment of the data showed that such an equation could indeed describe the evolution of Q ads versus C eq in the case of TAX adsorption on hydroxyapatite, with a correlation coefficient of r 2 = 0.999, and the parameters n = 2.98 ± 0.16, K = 0.032 ± 0.002 L/mmol, and N m = 11.85 ± 1.14 μmol/m 2 . Ac cording to the Langmuir Freundlich theory, departure from an ideal ad sorption situation as depicted by Langmuir arises from the existence of surface heterogeneities with non equivalent sites and/or of non negligible interactions between adsorbed species. As previously men tioned in the literature, an exponent n N 1 suggests the existence of pos itive cooperativity between adsorbed molecules. Taking into account the shape of the isotherm observed here, the formation of multilayer of adsorbed molecules seems to be highly probable. The adsorbed amounts reached on the adsorption isotherm (Fig. 5) and by N m value were both particularly high for molecules that do not expose high affinity groups for the surface of apatite surfaces. Indeed, only one car boxyl and one amino group are present on the TAX molecule, which present relatively low affinity for the surface of the apatite, as has been previously reported for example for the amino acid serine [22]. This was particularly true for the present apatite sample, which is stoi chiometric and well crystallized, thus affected by a limited surface reac tivity. In contrast, the adsorption of molecules such as bisphosphonates, for example, which exhibit high affinity phosphonate groups, led to maximum amounts adsorbed of approximately 3 μmol/m 2 on apatitic substrates [21]. The high adsorbed amounts observed here can only be explained by the formation of TAX multilayers, with molecules probably interacting with each other through respectively their \ \COO -and \ \NH 3 + terminal groups. This is consistent with the exponent value found with the Langmuir Freundlich model, suggesting a strong impact of intermolecular interactions rather than interactions with the surface of the nanocrystals. This is also consistent with the equilibrium dissociation constants of TAX from the literature (the pKa for carboxylic acid and amine functional groups in TAX were estimated to be 4.9 and 10.6, respectively [START_REF] Grassin Delyle | [END_REF]). The mechanism of interaction of TAX with HA could tentatively be then proposed in two steps: (1) TAX carboxyl groups interact with calcium ions on the surface of HA to form a mono layer on the apatitic support, and simultaneously (2) interaction be tween protonated amino groups and carboxyl species between adsorbate molecules of TAX leads to the formation of multilayers (Fig. 6). Note that the data were also tentatively fitted to other isotherm models often used to describe multilayer adsorption, including on solid samples, like the BET isotherm [23]. Although a moderately good fit was reached (r 2 = 0.983), the Langmuir Freundlich still remained the best model to fit the experimental data.

Mineral ion composition of solutions after adsorption

The change in the mineral content of the solution is shown in Fig. 7. Note that there was no significant change in the pH of the solution dur ing the adsorption experiments (pH remained close to 7.4). The data point related to a Q ads equal to zero corresponded to a blank assay per formed under the same conditions as the adsorption experiments in the absence of TAX. The calcium and phosphate ion content in the solution remained low (corresponding to low solubility of the HA substrate) and most importantly stable during the adsorption process, at approximate ly 0.15 and 0.05 mM, respectively, for calcium and phosphate. The min eral ion content in the solution therefore did not seem to be related to the amount of TAX adsorbed and was only due to the congruent disso lution of the crystals in aqueous solution as the Ca/P ratio in solution was the same as in the solid (i.e. Ca/P ~1.66). Therefore, the possibility of ion exchange occurring during the adsorption process can be ruled out, contrary to the case of several (bio)molecules with a strong affinity for the surface of apatites, as mentioned above.

Release kinetics

The kinetics of release of TAX (in %) from HA are shown in Fig. 8. The release was initially fast, as 20% of TAX was released during the first 6 h. Then TAX release slowed down, impacted by the dissolution of HA but not only. By linearization using different models, some linear zones can be pointed out allowing us to conclude that the release mechanism was probably governed by a complex process involving both diffusion (as in Korsmeyer Peppas model for example) and dissolution (as in Hixson Crowell model), as suggested by Fig. 9 [24,25].

Preliminary hemostasis results via aPTT evaluations

aPTT results are presented in Fig. 10. With regards to the solid sam ples, the aPTT of HA powders decreased for each specimen with the presence of adsorbed TAX. This can be clearly visualized by the lower position of the HA/TAX curve (red circles) as compared with the HA sub strate alone (black squares). Note that the data points corresponding to free TAX (blue triangles) relate to a solution while apatitic samples (HA and HA/TAX) were solids; therefore, the dilution of TAX during aPTT experiments prevented a direct comparison with the powder samples.

Although preliminary, these results point out a reduction in aPTT; this is extremely promising for the development of apatitic biomaterials with intrinsic hemostatic properties.

Conclusion

The aim of this work was to combine tranexamic acid (TAX), a clini cally used antifibrinolytic agent, and hydroxyapatite (HA) widely used in bone replacement, to produce a novel bioactive apatitic biomaterial with intrinsic hemostatic properties. The study of the adsorptive behav ior of the TAX molecule onto HA was performed and its release in near physiological conditions was also followed. When the isotherm of ad sorption of TAX onto the HA surface was plotted, no adsorption plateau was observed as the general tendency was a continuous increase of the adsorbed amount. Despite the low affinity of TAX for the surface of HA crystals, likely due to the presence of only a carboxyl and an amino group known to have a relatively low affinity for the surface of apatite, significant adsorbed amounts could be reached: the adsorption data ex hibited an exponential tendency mathematically fitted to a Langmuir Freundlich equation with an exponent close to 3. This suggested a strong interaction between adsorbed molecules, resulting in the forma tion of multilayers. This tendency may be an advantage in practical he mostasis applications for potentially transporting large doses of TAX directly to the bleeding wound site for increased efficiency. Further more, it may be useful for adapting the dose of TAX to the condition of the patient (evaluation of bleeding risk, possible hemophilic condi tions…). The concentration of calcium and phosphate ions in solution remained low and stable during the adsorption process, thus ion ex change during the adsorption process can be ruled out. The release of TAX was fast during the first hours and was governed by a complex pro cess that likely involved both diffusion and dissolution of HA. The pre liminary hemostasis aPTT tests on the blood from ten healthy adult dogs showed promising results for the development of apatitic bioma terials with intrinsic hemostatic properties. Especially for surgical pro cedures known to present a high risk for bleeding, the search for improved hemostasis approaches or methodologies is still undergoing. One appealing approach when dealing with bone related surgeries is to confer hemostatic properties to the implanted biomaterial itself, and apatitic biomaterials appear well suited for this goal. This work con siders the possibility to develop apatitic based bone repair biomaterials with hemostatic activity and the preliminary results obtained are prom ising. This contribution can thus be seen as a first approach for the de velopment of osteoconductive and hemostatic biomaterials, which may be optimized in the future by the use of apatite nanoparticles allowing higher adsorbed amounts of drugs thanks to their larger spe cific surface areas and surface reactivity than HA, for further improving drug delivery properties.

Fig. 1 .

 1 Fig. 1. Tranexamic acid (TAX).

Fig. 2 .

 2 Fig. 2. Calibration curve of tranexamic acid by LC/MS method.

Fig. 3 .

 3 Fig. 3. (a) X-ray diffraction pattern of the powder sample after adsorption (HA-TAX) and indexation of the main diffraction lines in relation with JCPDS file ref. no.09-432. TEM observations of the powders (b) before (HA) and (c) after adsorption (HA-TAX), (175 μmol TAX/g HA).

Fig. 4 .

 4 Fig. 4. Raman spectra of the samples before (HA) and after (HA-TAX) adsorption of tranexamic acid (TAX), (175 μmol TAX/g HA).

Fig. 5 .

 5 Fig. 5. Adsorption isotherm of TAX on HA at 37 °C.

Fig. 6 .

 6 Fig. 6. Mechanism proposed for adsorption of TAX on the (001) surface plane of HA (using Crystalmaker and ChemDraw Softwares).

Fig. 7 .

 7 Fig. 7. Concentration of phosphate (blue squares) and calcium (red circles) ions in the solutions after adsorption upon Q ads of TAX on HA.

Fig. 9 . 3 p

 93 Fig. 9. Linear regression of (a) Korsmeyer-Peppas model (Release TAX% = K KP × t n , with K KP constant et n the diffusion parameter) and (b) Hixson-Crowell model ( Q 3 p

Fig. 8 .

 8 Fig. 8. Kinetics of release of TAX (%) from HA after adsorption. The amount of initial concentration of TAX was equal to Q ads (TAX) = 175 μmol/g HA.

Fig. 10 .

 10 Fig. 10. aPTT obtained for the powders of hydroxyapatite before (HA) and after adsorption (HA/TAX with Qads (TAX) = 175 μmol/g HA), and for TAX solution (1 g/L) performed on ten healthy adult dogs.


Magdalena Paul
email: magdalena.paul@iwb.mw.tum.de

Daria Leiber

Julian Pleli

Gunther Reinhart

Simulation of Reconfigurable Assembly Cells with Unity3D

Keywords: Simulation, Reconfigurability, Assembly, Unity3D

This paper introduces a Unity3D-based simulation of reconfigurable assembly cells. A systematic approach defining flexibility ranges and comparing product requirements and cell capabilities allows the automated proposal of reconfigurations in the assembly cell. With this approach, the suitability of an existing cell can be examined for different variants of products while taking reconfiguration aspects into account. The simulation simplifies the process of introducing a new product to an assembly line. Through the virtual approach, designing of the product and line planning can be executed simultaneously, thereby decreasing cost and time-to-market for new variants in an existing production system.

Introduction

With the dynamization of product life cycles and the increasing number of variants, manufacturing companies are facing new challenges. The trend towards individualized mass production leads to the need of frequent modifications in the assembly system. Manufacturers need to adjust to the increasing number of variants using flexible and reconfigurable systems. [1] Flexibility describes the ability of a production system to adapt quickly and cost-effectively to changing influencing factors. Reconfigurability focuses on special manufacturing facilities that combine the advantages of highly specialized and adaptable systems. [START_REF] Koren | Reconfigurable Manufacturing Systems[END_REF] Autonomous and standardized functional units allow for new machine configurations [START_REF] Wiendahl | Changeable Manufacturing -Classification, Design and Operation[END_REF].

To be economically successful, reconfigurable production systems alone are not sufficient. The necessary changes also need to be identified, planned and implemented with reasonable effort [START_REF] Koren | Reconfigurable Manufacturing Systems[END_REF]. In view of this, digital tools like simulation are of great importance. Through a simulative examination, the suitability of existing assembly cells for new products or variants can already be determined in early development phases. By detecting possible problems at an early stage, effort for redundant planning can be reduced [START_REF] Erdogan | Conti Temic microelectronic GmbH[END_REF]. The use of simulation tools thus contributes to exploiting the potential of reconfigurable assembly cells. In this paper, we present Unity3D as a software for the simulation of reconfigurable assembly cells.

2

State of the Art

Simulation of Reconfigurable Assembly Cells

Simulations can support the planning process of reconfigurable assembly cells in different ways. Gyulai et al. [START_REF] Gyulai | Simulation-based Production Planning and Execution Control for Reconfigurable Assembly Cells[END_REF] develop a two-level, simulation-based approach, making the model capable of evaluating production plans and analyzing the emulation of the cell control. Their model is able to support the management of a modular cell and is proposing a possibility to decrease the commissioning time of new cells. A framework for the modelling of reconfigurable manufacturing systems by using hybridized discrete event and agent-based simulation is provided by Khedri et al. [START_REF] Liraviasl | A Framework for Modelling Reconfigurable Manufacturing Systems Using Hybridized Discrete-Event and Agent-based Simulation[END_REF]. With the simulation, the authors provide insights on the emergent behavior of the agents of an intelligent manufacturing system. Further, Deif and ElMaraghy [START_REF] Deif | A multiple performance analysis of market-capacity integration policies[END_REF] use simulation augmented with Design of Experiment to analyze the performance of a reconfigurable manufacturing system with scalable capacity. Multiple performance measures lead them to propose a hybrid capacity scaling and marketing policy.

Introduction of Unity3D

Though Unity3D was primarily developed as a game engine, it offers a variety of advantages for the simulation of manufacturing systems. It is highly customizable and by using the integrated C#-code-base and physics engine various assembly line behaviors can be simulated. The simulation can consider mass, drag, springiness, bounciness and collision of objects. Furthermore, simulations generated with Unity3D can be easily used for demonstration purposes. They can be released on virtual reality, augmented reality and as program-and browser-exports [START_REF]Unity3D-Multiplatform[END_REF]. Not being bound to a specific medium, the display of the simulation can be ensured across different systems and locations.

Being used as the games engine for half of the games worldwide, Unity3D provides an extensive documentation and tool library for the user. [9]

Unity3D as a Simulation Tool

Through a systematic literature review, previous work concerning Unity3D as a simulation tool has been identified. On the level of factory and systems, Sun et al. [START_REF] Sun | International Conference on Information and Automation: Design and simulation of the production line system based on multi-channel controller[END_REF] propose a method to design a production line based in a multi-channel controller. Gaisbauer et al. [START_REF] Gaisbauer | CIRP: Presenting a Modular Framework for a Holistic Simulation of Manual Assembly Tasks[END_REF] introduce a concept to integrate arbitrary motion simulations in common frameworks with standardized motion model units. Um et al. [START_REF] Um | Procedia Manufacturing: Development a Modular Factory with Modular Software Components[END_REF] use a Unity3D-based simulation to develop a concept if a modular factory which makes it possible to reconfigure individual machine stations without extensive engineering effort. Furthermore, the possibility of Unity3D to be used in virtual reality applications is examined for designing a lean manufacturing environment [START_REF] Gamlin | International Conference on Interactive Technologies and Games: Immersive Virtual Reality Deployment in a Lean Manufacturing Environment[END_REF] and to develop interactive practice systems [START_REF] Ouyang | Computer Applications in Engineering Education, A Unity3D-based interactive three-dimensional virtual practice platform for chemical engineering[END_REF][START_REF] Radionova | International conference on Industrial Engineering: Interactive Educational System-Virtual Simulator "Sheet Rolling[END_REF]. On the level of cells and stations, Koechling et al. [16] present a procedure to enable production planners to virtually inspect the production including the material flow before the actual implementation. No publications have been found covering the topic of reconfigurability in manufacturing systems connected to Unity3D.

Approach for the Simulation of Reconfigurable Assembly Cells with Unity3D

The proposed approach aims to decrease time and effort when introducing a new product variant into an existing assembly line. It simplifies the production planning process for new products on existing assembly systems. By simulating the process with Unity3D, it is determined if a new variant can be produced in the existing assembly cell configuration. Through modelling the existing flexibility ranges in the reconfigurable assembly cells they can also be automatically considered. If the current line configuration is not suitable for the new product, but the line can be reconfigured to allow production, proposals on how to change the configuration are automatically made to the production planer. For this purpose an approach for modelling flexibility ranges of assembly cells is developed. The flexibility ranges are defined automatically via calculations based on the possible cell configurations. Basis for the suitability evaluation is a detailed model of the processes that are to be performed. The process requirements are then compared with the cell's capabilities.

The proposed approach consists of seven steps, as shown in Figure 1. As preparation for the simulation, relevant data is obtained, processed and entered. The input for the simulation consists of constraints, predetermined tasks, the process, available resources and products. The proposed approach contains five preparation steps: task definition, CAD-Model preparation, cell assembly and product import, definition of the flexibility ranges and identification of product requirements. After the complete data import, the simulation can be started, and the comparison between requirements and cell capabilities and the derivation of solutions can be executed in the application. For the detection of possible collision errors or reachability problems in the examined process, a high level of detail has to be chosen to describe process and product.

Simulation Preparation

First, the assembly task and the product requirements must be defined. Further relevant information are the characteristics of the current cell configuration. In order to achieve applicable results, the simulation model has to correspond to reality as closely as possible. Required data is therefore integrated by importing CAD-files of the cell into Unity3D. As Unity3D is not offering the possibility of importing those files directly, the models have to be converted to Unity3D format filmbox (.fbx). For this work, the CAD-program Autodesk 3DS Max was used. Further, the converted files are scaled to Unity3D standard, where one standard unit equals one meter. To be able to move the parts in the simulation relatively to each other the cell is rebuilt before the import in individual parts.

The product requirements must be identified before the start of the simulation. It is possible to load any product variant with differing requirements into the simulation. A program code automatically loads an object with default settings into a defined position in the simulation. First, it has to be determined how many and which process steps have to be carried out, what is required for the respective process step. Product requirements can be divided into direct and indirect. Direct requirements are defined by the dimensions of the product, number of machining steps, the resources required for the processes and the configuration of the needed equipment. Indirect requirements are independent of the product, like cycle times, quality standards and the use of certain components. Geometric dimensions are the only product feature that can be analyzed in Unity3D without additional information. Other parameters have to be inserted by the user and are stored in string variables to make them available for the comparison.

Depiction of Reconfigurability in the Simulation

To integrate flexibility, reconfigurability and modularity in the simulation model of the assembly cell the option in Unity3D to customize the program with C# scripts is used. For this the total flexibility of a cell is divided into two levels, as shown in Figure 2.

Fig. 2. Levels of flexibility in an assembly cell

The cell configuration provides the first flexibility range. This is done by determining whether equipment is built-in or modularly planned. For this purpose, the construction and processing areas, the process paths and the number of possible process steps in the cell are defined. At the equipment-level, flexibility is determined by the characteristics and settings of these. Together with the cell configuration, the equipment configuration determines the cell's solution space. The geometric representation of the equipment does not provide sufficient data for an informative simulation. The flexibility ranges are defined by specific queries and inputs. For individual process steps, it is defined which component is installed, if the concept is modular and retrofittable and if a new module has to be installed. This information is later used in the simulation process.

Comparison of Product Requirements and Cell Capabilities

In addition to geometric checks the parameters describing product requirements and cell skills are compared. The simulation can be used to examine if the components fit into the fixtures, if the machining areas are sufficient and if all processes can be executed without collisions. Entered parameters, information derived from the CAD-model and detected problems are stored automatically. In the next step it is checked whether the product requirements can be fulfilled within the defined flexibility ranges.

Automatic Derivation of Possible Solutions

The identified problems have to be converted in proposed solutions. Certain rules are defined for this purpose. The suggestions for solutions can be expressed product-or cell-related. In the present approach, changes can only be done to the cell, not to the product. The solution possibilities can be classified into three levels, as shown in Table 1. Category 1 includes modifications within the flexibility range [START_REF] Wiendahl | Changeable Manufacturing -Classification, Design and Operation[END_REF]. According to industry needs, the two further classifications divide the solutions into whether the user or the cell-manufacturer has to make adjustments. Each individual process step is compared, the result categorized and the user is shown the necessary steps for the reconfiguration of the cell. Finally the option to rebuild of the cell according to the proposals is offered.

Table 1. Categorization of solution possibilities

Category 1

The assembly cell can adapt itself within the flexibility range.

Category 2

The assembly cell user can implement new operating resources in modular provided areas or change them. Category 3

The assembly cell provider has to adjust the cell.

Validation of the Approach

The approach is evaluated based on an application case. The application example is the assembly of a surround view camera for vehicles at a large automotive supplier company. The selected process is dispensing, which is complex assembly process. In the process, glue is applied to the housing. The simulation is used to test which product requirements (e.g. connector geometries, dispensing-medium) necessitate a cell conversion. The simulation can run as an exported program without Unity3D. The program in Unity3D is structured in different scenes, representing different program windows.

At the beginning of the simulation, the machining areas of the processes and the possible, previously defined process areas are displayed, as shown in Figure 3. These behave according to the inputs of the process paths and their geometric restrictions and are automatically adapted in the visual representation.

Fig. 3. Machining and process areas

The structure of the cell corresponds to the previously defined configuration. If the product requirements of the new variant are entered and the CAD-model has been successfully imported, the simulation can be started. The comparison starts in the simulation with accessibility studies and collision monitoring. The product is moved to the processing position with a shuttle axis and the glue is dispended according to the requirements. If post-treatments and security methods are selected, these are performed via the transfer system. After the comparison, the cell can be rebuilt according to the product requirements. This is done either within the flexibility ranges or by automatically adding further operating resources. The detection of a collision is displayed in Figure 4. The information is saved and serves as a basis for the comparison, which is largely parameter-based. The comparison takes place for each process and all parameters declared process-relevant. An output window is generated, showing the results and the possible solutions.

Fig. 4. Collision detection

The implementation in the application case allows the comparison of the skills of the current assembly cell to the requirements of two new and different variants in just one simulation. The created process transparency helps the company to quickly recognize necessary changes in the reconfigurable cell.

Conclusion and Outlook

The simulation of reconfigurable assembly cells enables an accelerated reaction to changing economic conditions. The development and introduction of the presented method can reduce planning efforts and support simultaneous engineering. In addition, the simulation enables the user to virtually validate the modularity of the manufacturing cell. The next step is to further evaluate the usability and quality of the results. Therefore, further different applications have to be tested. The input, data collection and the attachment of the requirements to the product has to be automated and simplified in further research.

Fig. 1 .

 1 Fig. 1. Simulation procedure

Acknowledgments

The authors of this paper would like to thank the German Federal Ministry of Education and Research (BMBF) and the Project Management Agency Research Center Karlsruhe (PTKA) for funding this work, which is part of the research and development project ProMoA (funding number 02P15A100).