Fei Liu

Angel Ricardo Licona

Arnaud Leleve
email: arnaud.leleve@insa-lyon.fr

Damien Eberard

Minh Tu Pham

Tanneguy Redarce

An Energy-Based Approach for n-d.o.f. Passive Dual-user Haptic Training Systems

Keywords: Computer Simulation, Dual-user System, Hands-On Training, Haptic interfaces, Port-Hamiltonian Framework, Shared Control

This paper introduces a dual-user training system whose design is based on an energetic approach. This kind of system is useful for supervised hands-on training where a trainer interacts with a trainee through two haptic devices, in order to practice on a manual task performed on a virtual or teleoperated robot (for example for an MIS task in a surgical context). This paper details the proof of stability of an Energy Shared Control (ESC) architecture we previously introduced for one degree of freedom (d.o.f.) devices. An extension to multiple degrees of freedom is proposed, along with an enhanced version of the Adaptive Authority Adjustment (AAA) function. Experiments are carried out with 3 d.o.f. haptic devices in free motion as well as in contact contexts in order to show the relevance of this architecture.

Introduction

As in every profession where dexterous manipulation is necessary, medical staff require initial and continuing hands-on training on ever evolving medical methods. Medical simulators such as cadavers and animals have been a convenient way to learn by trial for decades in medical universities. However, due to the growing cost of providing them, noncontinuous availability problems and ethical issues, phantoms are increasingly used. Yet, phantoms provide a limited set of common cases to practice on. Providing a larger set of typical medical cases requires ownership of a great variety of available phantoms, which becomes rapidly costly. Nowadays, the over-crowded medical student populations are still offered too few opportunities to perform hands-on training during their curriculum, due to a limited access to hands-on training resources. For instance, epidural anesthesia requires much practice to be mastered enough to practice it on a real patient. A study [START_REF] Vaughan | A review of epidural simulators: Where are we today?[END_REF] by Vaughan et al. reported that it requires at least 80 anesthesia to reach a 90% success rate. Not so many trials are performed during students' studies. Therefore, it is necessary to provide cost-efficient solutions they could use during supervised sessions but also autonomously in order to sufficiently train themselves during their curriculum.

Over the last decade, Virtual Reality (VR) simulators (see [START_REF] Yiannakopoulou | Virtual reality simulators and training in laparoscopic surgery[END_REF] for a recent review on this topic) have been designed to overcome the drawbacks of aforementioned models. With virtual patient models which can be parameterized on-line, it becomes possible to provide an infinite set of medical cases with various difficulty levels.VR simulators have been progressively improved to provide to trainees a more realistic environment in 2D and more recently in 3D (with one example [START_REF] Delorme | Neurotouch: A physics-based virtual simulator for cranial microneurosurgery training[END_REF] ). With haptic training simulators, the additional force feedback provides a realistic tool behavior, which leads to an efficient training for advanced tasks. [START_REF] Panait | The role of haptic feedback in laparoscopic simulation training[END_REF] More recently, the interest of haptic feedback in roboticassisted surgery has been proven by Talasaz et al. in the case of suturing application. [START_REF] Talasaz | The role of direct and visual force feedback in suturing using a 7-dof dual-arm teleoperated system[END_REF] It is demonstrated that force feedback plays a key role in helping users to manipulate the instruments better and to reduce tissue stress, tissue damage and accidental hits. These systems feature a haptic interface (i.e. a device by which tool-environment interaction forces are transmitted back to a human operator based on their hand motion) which acts as a master and a software architecture which connects the master to the slave. The slave is generally a software simulating a virtual tool inside a virtual environment.

Even if some simulators usually provide solutions to autonomously train oneself, for some difficult cases, it remains useful that a trainer guides the trainees' motions for a more accurate and efficient training. A trainer can directly guide the hands of a trainee to perform a correct motion but this "four hand fellowship" does not permit for the trainee to feel and dose the correct level of force to apply on their device in case of interaction of their tool with its environment. Dual-user systems are a practicable solution to this problem, as they can reproduce this important force information to both users, each one interacting with their own haptic interface. These systems extend the aforementioned master-slave architecture by adding a second master, as shown in Figure 1. The common concept is that the users share the slave control according to a dominance factor (α ∈ [0, 1]). When α = 1 (respectively 0), the trainer (respectively trainee) has full authority on the slave and on the trainee's (respectively trainer's) device. When 0 < α < 1, both users share the slave control with a dominance (over the other user) which is function of α. According to the architectures found in the literature, the effect of α on the force feedback provided to the users differs. These differences are highlighted in section 2.

Fig. 1. Architecture of a dual-user system

The objective of the current project is to design a dual-user hands-on haptic training system. In this system, trainer and trainee manipulate their own haptic interface. They share the same "slave" which can be either a real surgical robot dived into a real environment, located in a near black box for instance, or a virtual robot in a virtual environment, provided by a software simulation. Such an application, providing haptic mentoring, has been introduced by Chebbi et al. [START_REF] Chebbi | A collaborative virtual haptic environment for surgical training and tele-mentoring[END_REF] We focus on applications with two masters and one slave, all located in the same room. The models proposed hereafter will be extended in the future to cope with transport delay consequences on stability and transparency in order to permit a distant use.

The following typical use case helps determine the main requirements of the system. Suppose, at first, that the trainer (an experienced surgeon) aims at demonstrating the right trajectories of their surgical tool to perform a task featuring free motions and some tool-environment contacts. This implies that they require a realistic force feedback to dose their force, as in a bilateral teleoperation context. The trainer manually sets α = 1 to become leader (the trainee becomes follower): it is a mentoring mode. They then get a full force feedback from the slave in order to perform their task as if they were handling the real instruments. Meanwhile, the trainee's device follows the trajectory of the leader one. If the trainee deviates from this reference trajectory when in free motion, the compliance of the device brings them back to the right position. In case of interaction between the tool and its environment, the trainee can also feel in their hands the right level of effort to provide to the tool, by means of a display which guides them to set their device at the right position with the right applied force. Afterwards, the functioning can be inverted by reversing α so that the trainee manipulates and the trainer follows and evaluates trainee's motions and applied forces: it is an evaluation mode.

The analysis of the scientific literature showed that the existing solutions did not fit all the requirements of such a use case. Reasons are detailed in section 2. Thus, a dual-user control architecture had been previously introduced: the Energy based Shared Control (ESC). [START_REF] Liu | A dual-user teleoperation system with online authority adjustment for haptic training[END_REF] It reuses the robust control approach introduced by Stramigioli, [START_REF] Stramigioli | Modeling and IPC control of interactive mechanical systems: A coordinate-free approach[END_REF] which ensures passivity even in presence of robot model uncertainties, limited bandwidth, or non-linearities (saturations). It thus provides robust passive and compliant interfaces. In Liu et al., [START_REF] Liu | A dual-user teleoperation system with online authority adjustment for haptic training[END_REF] ESC was developed for single degree of freedom (d.o.f.) mechanisms, with experiments conducted with only two real haptic devices and a virtual one. We used an energetic approach which guarantees the passivity of the system whatever evolution of α, which is not always proven in existing frameworks (in LTI models, α is supposed to be constant or sufficiently slowly evolving to consider it as a constant in the models). We then proposed to embed an Adaptive Authority Adjustment (AAA) function to automatically revert, in evaluation mode, the authority from the trainee to the trainer in case of bad gestures performed by the trainee. [START_REF] Liu | A dual-user teleoperation system with adaptive authority adjustement for haptic training[END_REF] However, this solution required to tune two task-dependent parameters which were hard to be optimized by any trainer. ESC has been compared with two other dual-user architectures from Khademian and Hashtrudi-Zaad 7 (the "Complementary Linear Combination" (CLC) and the "Masters Correspondence with Environment Transfer" (MCET)), by means of simulations. [START_REF] Liu | An energy based approach for passive dual-user haptic training systems[END_REF] This comparative study concludes, from simulation results, that the ESC has position tracking performance equivalent to the one of CLC and MCET architectures. Yet, ESC force tracking performance is intrinsically better for hands on training purpose as the former architectures cannot perform mentoring and evaluation involving simultaneous force and position tracking for both users. We also enhanced the ESC (where the environment and users were assumed as passive) by introducing a Time Domain Passivity Controller (TDPC) to keep the system passive whatever user or environment and whatever value of α and every parameter tuning the system compliance. [START_REF] Liu | An energy based approach for passive dual-user haptic training systems[END_REF] More recently, we reported some preliminary experiments testing the opportunity to extend this approach to several degrees of freedom. [START_REF] Licona | Collaborative hands-on training on haptic simulators[END_REF] As this provided encouraging results, we introduce here this novelty.

Compared to these former publications, we introduce in this new paper the following contributions: r The Port Hamiltonian modeling of this architecture, only evoked in previous papers, is detailed. r The extension to multiple degrees of freedom for haptic devices with same kinematics is discussed, which makes this architecture usable in practice in many cases, as the both users should use same device types. r The proof of passivity is extended to n-d.o.f. systems, with the help of an updated TDPC. r The Adaptive Authority Adjustment (AAA) is extended to n-d.o.f. and also enhanced:

it henceforth requires only one threshold parameter which can be tuned on-line by the trainer, without specific skill.

Real-time experiments are performed with a 3-d.o.f. architecture and three real haptic devices in order to validate these enhancements. They are shown in user (cartesian) space for the sake of clarity.

This paper is organized as follows. Section 2 describes the evolution of the dual-user concept in the literature. Section 3 details the ESC based dual-user system model. Section 4 proves the passivity of the architecture with several d.o.f. along with an

Existing Dual-user Architectures

Various architectures have been proposed for the control of dual-user haptic systems.

The "shared control" concept has been introduced by Nudehi et al. in [START_REF] Nudehi | A shared-control approach to haptic interface design for minimally invasive telesurgical training[END_REF] for Minimal Invasive Telesurgical Training purpose, i.e. to "allow experienced surgeons mentor trainee surgeons through shared control of a surgical robot". The control of the slave is shared between both users, and each one experiences feedback forces proportional to the difference of position of their own device, and to α. In any case, an interaction between the robot and its environment would not feed back interaction forces to any user. Therefore, this kind of training system can only be used to train users on motions, not on efforts. Khademian and Hashtrudi-Zaad overcame this limitation by providing a three-port multilateral control architecture. [START_REF] Khademian | Shared control architectures for haptic training: performance and coupled stability analysis[END_REF] Two distinct architectures are proposed by the authors. The first one is the Complementary Linear Combination (CLC) architecture which provides feedback forces combining the environment and the other user forces (typically F e and F h2 , for user 1). The desired position and force commands for each device are a complementary weighted sum of positions and forces of the other two devices. When α = 1, master 1 and the slave form a four-channel bilateral teleoperation system. The position of the device 1 and the forces the user 1 applies on it, also serve as input for the master 2 controller (the behavior is symmetric when α = 0). When 0 < α < 1 both users can perform a task collaboratively in a shared environment. On initial examination, this architecture is the solution which best fits this project's requirements. Unfortunately, when trying to perform a mentoring (respectively evaluation) session, the trainee (respectively trainer) has no direct force feedback from the environment, only from the other user. This is why the authors advise to use this system with values of α = 0.2, 0.5, 0.8 to get the best transparency. Yet, in these cases, both users influence the slave trajectory, which may lead from inaccurate to trembling motions, not compliant with mentoring and evaluation modes. The second architecture Khademian and Hashtrudi-Zaad introduced is called "Masters Correspondence with Environment Transfer" (MCET), [START_REF] Khademian | Shared control architectures for haptic training: performance and coupled stability analysis[END_REF] where both user devices follow the motion of each other and the effort fed back to both users is F e /2. The command to the slave is weighted accordingly by α and (1 -α). Kinesthetic performance analysis and experimental user perception studies performed by these authors showed that the MCET architecture provides a better kinesthetic feedback to both users with the least sensitivity to the dominance factor, compared with CLC. Yet, still according to the authors, this architecture transmits a distorted environment force to both users (which was equal to F e /2 in addition with a force expressed as a function of the difference of the positions of the devices of both users), which does not comply with the aforementioned use-case.

Ghorbanian et al. defined two dominance factors α and β. [START_REF] Ghorbanian | A novel control framework for nonlinear time-delayed dual-master/single-slave teleoperation[END_REF] α determines the balance of authority between the trainer and the trainee, while β indicates the supremacy of both trainer and trainee over the slave robot. This provides an additional degree of freedom in the authority mechanism which is used to set a nonlinear relation between α and β in order to adjust the authority of the leader (the user for which α > 0.5) over the slave with respect to their authority over the follower. It also supports uncertain communication channels in presence of delays, packet loss, data duplication and packet swapping, which corresponds to practical issues to overcome as soon as masters and slave are away. However, when α = 1, the desired Master 2 force (F h2 d ) is only linked to the position of Master 1, not to the tool-environment interaction force, which does not fit our training requirements either. Also, as this work is not compared with CLC or MCET, it cannot be determined whether the transparency of this architecture is better.

The architecture introduced by Razi et al. [START_REF] Razi | Analysis of coupled stability in multilateral dual-user teleoperation systems[END_REF] transmits to the follower a force F h2 d = (F e -αF h1 ), where F h1 is the force applied by user 1 on their device, which also does not correspond to this project requirements. An interesting work has been introduced by Shamaei et al. [START_REF] Shamaei | Design and evaluation of a trilateral shared-control architecture for teleoperated training robots[END_REF] where two dominance factors are used. Several architectures are provided. The one which tracks the positions of all three devices is experimentally evaluated. This architecture's performance is analyzed in terms of impedance matching between trainer and trainee during soft and hard interactions. This provides interesting but partial information during interaction: no information about position nor force absolute tracking error is provided, only a comparison versus the two dominance factors. Also, the authors specify that this approach is limited in terms of control freedom as derivative operators may destabilize it.

In, [START_REF] Shahbazi | A new set of desired objectives for dual-user systems in the presence of unknown communication delay[END_REF] the users' haptic feedback is a weighted (with α) sum of the virtual toolenvironment interaction force F e and the other user interaction (with their master device) force F h . More precisely, the desired haptic feedback applied to user 1 is F h1d = α.F e + (1 -α)F h2 . Thus, when α = 1, user 1 only feels F e and user 2 only feels F h1 . Experiments show that when α = 0.5 and both users follow very close trajectories, they feel very close force feedback, in presence of time delays between each master and the slave. This is an interesting result. Yet, both users must kind of cooperate to guide the slave tool with the same dominance. In our point of view, it may disturb the mentoring/evaluation as each user are implied in the motions and the interactions without distinction. Also the experiments have been performed with constant environment interaction characteristics. More realistic experiments should mix free motion and interaction phases in the same session.

Furthermore, in a dual-user hands-on training system, the authority factor α should be switched anytime by the trainer. The architectures should guarantee that fast changes of α may not destabilize the system. Yet CLC, MCET and Razi et al. architectures have been modeled as LTI systems. Therefore, the robustness of these architectures should be evaluated according to the varying parameter α. This has not been ensured yet as far as we know. As far as we could observe, the only architectures for which robustness versus α variations is proven are the ones in 6 and, [START_REF] Shahbazi | A new set of desired objectives for dual-user systems in the presence of unknown communication delay[END_REF] through a Lyapunov function, and the one from Shamaei et al., [START_REF] Shamaei | Design and evaluation of a trilateral shared-control architecture for teleoperated training robots[END_REF] through an unconditional stability approach based on the Llewellyn's criterion.

In 2011, a survey 15 has been conducted concerning passivity-based controllers for nonlinear bilateral teleoperation. In our point of view, an energy approach seems promising as it already provided interesting results in more classical teleoperation studies (with two examples 8 9 ). Since this work has been realized, Lu et al. proposed to use three dominance factors to provide more flexibility to the users. [START_REF] Lu | Enhanced transparency dual-user shared control teleoperation architecture with multiple adaptive dominance factors[END_REF] These factors are dynamically adapted by extending the Online Authority Adjustment (OAA) mechanism (introduced in [START_REF] Liu | A dual-user teleoperation system with online authority adjustment for haptic training[END_REF] ), in the presence of communication delays. They use a linear modeling to prove the stability and study the transparency of the system. Also, Zakerimanesh et al. proposed to consider the actuator saturations [START_REF] Zakerimanesh | Dual-user nonlinear teleoperation subjected to varying time delay and bounded inputs[END_REF] (they only validated this proposal in simulation so far).

This review shows that despite of advanced works in this topic, none fully complies with the aforementioned use-case. The equivalent performance of the ESC compared to CLC and MCET architectures, established in simulation [START_REF] Liu | An energy based approach for passive dual-user haptic training systems[END_REF] motivates to further work on this approach.

Energy Shared Control Based Dual-User Architecture

This section details the design of a passive dual-user architecture, using an energetic approach. The scalability of this approach is a clear advantage: adding passive parts to a passive network, with lossless (and therefore passive) interconnections preserves the passivity. Thus, extending a teleoperation system with a second or more users is practicable with a moderated increase of model complexity. The design of this architecture started from a dual-user intrinsically passive system detailed in subsections 3.1 and 3.2. Yet, as this architecture was not able to provide full slave feedback to both users, it has been enhanced (see subsection 3.3).

Authority Sharing Mechanism

Inspired by the CLC architecture, [START_REF] Khademian | Shared control architectures for haptic training: performance and coupled stability analysis[END_REF] we proposed an Authority Sharing Mechanism (ASM) by considering an energetic approach, as shown in Figure 2 (both "MFS" blocks can be ignored as of now; they will be introduced in the next subsection). The ASM, based on parameter α, is located in the inner dashed box. It is composed of three linear Multiple-Input-Multiple-Output (MIMO) subsystems D m1 , D m2 and D s which behavior is introduced in equations ( 1), ( 2) and (3)1 . They, all together, redirect the energy flows (composed of angular velocity and torque signals) exchanged between the trainer device ( θr1 , t r1 ), the trainee's one ( θr2 , t r2 ) and the slave ( θs , t s ). The two additional power ports ( θsf1 , t sf1 ), ( θsf2 , t sf2 ), and the effective role of D m1 and D m2 are introduced in next subsection. ( θsf1 , t sf1 ) and ( θsf2 , t sf2 ) can be considered null as of now. Three dominance factors α, β 1 , β 2 ∈ [0, 1] are introduced to achieve the goal of shared control.

D m1 :   θr1 t s1 t sf1   =   0 α (1 -α) -α 0 0 (α -1) 0 0     t r1 θs1 θsf1   (1)
D m2 :   θr2 t s2 t sf2   =   0 (1 -α) α (α -1) 0 0 -α 0 0     t r2 θs2 θsf2   (2)
D s :   θs1 θs2 t rs   =   0 0 β 1 0 0 (1 -β 2 ) β 1 (1 -β 2 ) 0     t s1 t s2 θrs   (3)
The relationships between α, β 1 and β 2 are defined as,

β 1 = 0, α = 0 1, α > 0 β 2 = 1, α = 1 0, α < 1 (4)
The introduction of β 1 and β 2 provides a nonlinear solution to avoid the repetition of the authority distribution in D s (which would provoke some port relations embedding α 2 terms, which was not desired), since the shared power flow is already formed at D m1 and D m2 . The role of D s is to sum the torques from the masters to the slave and deliver the velocities to both users. We obtain three modes by changing the dominance factor α: mentoring mode (α = 1), guidance mode (0 < α < 1) and evaluation mode (α = 0).

The three nodes (D m1 , D m2 and D s ) form a power-conserving interconnection. This means that the global authority sharing mechanism is passive and lossless. This is represented by the following power balance:

t T r1 θr1 + t T r2 θr2 + t T rs θrs + (t T sf1 θsf1 + t T sf2 θsf2 ) = 0 (5)
which indicates the ASM passivity.

Haptic Interface Control

ASM does only redirect signals between the three power ports, it does not permit an efficient control of the end interfaces (Master 1, Master 2 and Slave). For this purpose, we inserted three Intrinsically Passive Controllers (IPC) (respectively IPC M1, IPC M2 and IPC S) in between them (see Appendix). The IPC controllers, introduced by Stramigioli, [START_REF] Stramigioli | Modeling and IPC control of interactive mechanical systems: A coordinate-free approach[END_REF] have the property to ensure a robust passive interaction of the controlled system with an unknown environment, without having to model both ones. As the three interconnected devices (Master 1, 2 and Slave) are interacting respectively with users and any environment, IPC controllers are interesting solutions for the three of them. Therefore, by linking them to the ASM architecture through power-conserving interconnections, we obtain a new passive system, whatever controllers' parameters, devices', users' and environment's characteristics. This ensures a structural robustness and leaves freedom to tune every IPC parameter for performance and global transparency. Moreover, IPC controllers add some compliance which helps in manipulation and training tasks.

Providing Force and Position Feedback to Both Users

This ASM architecture has a defect when α = 1 (respectively 0): the trainer (respectively trainee) exchanges all their energy exclusively with the slave. The system acts this way as a classical teleoperation system and completely isolates the trainee (respectively trainer). The latter cannot control the slave, which is desired, but they also do not get any feedback from it, which is opposed to the aforementioned training requirements. In order to solve this problem, we added two identical Modulated Flow Sources (MFS) visible in Figure 2 which role is to "copy" θrs coming from the slave IPC without directly exchanging any energy. The aim of these two flow sources is to feed back the complementary motion information from the slave which is missing in the ASM architecture. Consequently, both users experience a full motion feedback regardless of the control authority (i.e. the value of α). This is checked by introducing θsfi = θrs (i ∈ {1; 2}) into Eq. ( 1) and ( 2):

θr1 = α θs1 + (1 -α) θsf1 = α θrs + (1 -α) θrs = θrs θr2 = (1 -α) θs2 + α θsf2 = (1 -α) θrs + α θrs = θrs (6)
It is worth mentioning that additional energy may be injected into the system by the modulated flow sources. This could violate the passivity of the whole architecture which may cause unstable behaviors. This aspect is discussed in section 4.

Remark: This way, thanks to the IPC controller properties (see section 3.2 and Appendix), for example, when α = 0, t h2 tends towards t e but also t h1 tends towards t e when the trainer moves their master interface such a way that θ m1 → θ m2 . One can see an illustration of it in the plots of the experimental results in Section 5.

Thus, all together, we obtain an intrinsically passive dual-user system, named "Energy Shared Control" (ESC), which corresponds to the outer dash-dot box in Figure 2.

Extension to Multiple Degrees of freedom

As the training applications with only one d.o.f. are limited, we applied the ESC architecture to systems with multiple degrees of freedom. As a first approach, we deal with training systems having master and slave devices with same kinematics. This is not a strong assumption as for a given training application, master devices should be identical. Using the same device model is also coherent. Concerning the slave robot, this assumption is a little more restricting as it should be more practical to use a dedicated robot to handle some tools. The aforementioned one d.o.f. architecture has been expanded to n d.o.f. by duplicating the previous one-d.o.f. architecture independently for each joint (see Figure 3). We assumed that the potential interactions between each d.o.f. would naturally balance each other as we use same device kinematics and that the robustness of the IPC controllers would absorb potential disturbances. In future works, we will deal with interfaces with different kinematics.

Adaptive Authority Adjustment (AAA) Enhancement

The Adaptive Authority Adjustment (AAA) function, [START_REF] Liu | A dual-user teleoperation system with adaptive authority adjustement for haptic training[END_REF] permits the automatic adjustment of α in real time during an evaluation. In evaluation mode, the trainee is leader by default but when the trainer voluntarily deviates from the trainee's trajectory (potentially because of an error performed by the trainee), the trainer automatically gets back the authority over the slave. When the trainee's device comes back close to the position of the trainers' one, the authority is automatically set back to the trainee.

We propose here to improve the previous solution which was introduced for a one d.o.f. system. This new algorithm applies for n d.o.f. systems and now requires only one parameter adjustable in real time by the trainer (versus two parameters which were hard to tune in the previous proposal): e max which can be tuned on-line by the trainer according to the desired strictness during the evaluation. It is based on a Bézier polynomial proposed by Beltran-Carbajal in another context, 2 which sets α (the same value for every joint) in real time, as a smooth function (see Figure 4 and Eq. ( 7)) of the deviation of position between the master devices e(t):

α(t) =    e r (t) 5 [r 1 -r 2 e r (t) + r 3 e r (t) 2 - r 4 e r (t) 3 + r 5 e r (t) 4 -r 6 e r (t) 5 ] e r (t) ≤ 1 1 e r (t) > 1 (7)
where e r (t) = e(t) emax is the relative error, e(t) = n 1 e i (t) 2 is the global magnitude of the instantaneous articular errors e i (t) = |θ m1 i (t) -θ m2 i (t)| with i ∈ {1..n}, and e max is the deviation which implies a return to full authority of the trainer over the slave. r 1 = 252, r 2 = 1050, r 3 = 1800, r 4 = 1575, r 5 = 700 and r 6 = 126. In practice, it has not been necessary to compute a mean error on a floating time frame. Remark: Section 5.4 shows how the system behaves in practice. This is a simple but efficient solution. A more evolved approach has been proposed by Shahbazi et al. [START_REF] Shahbazi | An expertise-oriented training framework for robotics-assisted surgery[END_REF] with the help of fuzzy logic (FL). This approach cannot be directly used with our architecture as the FL simultaneously tunes online two dominance factors as in Ghorbanian's architecture. [START_REF] Ghorbanian | A novel control framework for nonlinear time-delayed dual-master/single-slave teleoperation[END_REF] A modification of the ASM part would be necessary to ensure that it remains passive whatever dominance values of α.

Proof of Passivity

In this section, the passivity condition is evaluated. To do so, the system is modeled using the port-Hamiltonian approach [START_REF] Van Der Schaft | Port-hamiltonian systems theory: An introductory overview[END_REF] as it provides a compact model of the form of Eq. (8). A passivity controller is then proposed to ensure a stable behavior.

Global Model

In the following, uppercase variables signify vectors of n variables with the same name for each joint (typ. A = (a 1 a 2 ... a n ) T ). The energetic model of the whole system including the ESC and the haptic devices can be expressed as:

                 Ẋ = [J -R(X)] ∂H ∂X + GU + QV Y = G T ∂H ∂X S = Q T ∂H ∂X (8)
where the state variables are

X := (X T c,m1 X T c,m2 X T c,s X d,m1 X d,m2 X d,s ) T
with X c,i (for i ∈ {m 1 , m 2 , s}) being the state variables of each IPC controller duplicated for each d.o.f., and X d,i the state variables of each haptic device (see Appendix).

J is a skew-symmetric matrix which models the internal state variable interconnections. R(X) is a matrix which represents the internal dissipation. H(X) is the global system Hamiltonian.

The input U = (T h1 T h2 T e ) T (of size n × 3) is composed of the external torques and G is a matrix which describes how these torques are distributed into the system. Y = ( Θm1 Θm2 Θs ) T is the angular velocity output (of size n × 3). The input V = ( Θsf1 Θsf2 ) T (of size n × 2) represents the virtual velocities provided by both MSF and S = (T sf 1 T sf 2 ) T (of size n × 2) is composed of the virtual output torques sent to the MSF. This model is built by lossless interconnection of subsystems which can be modeled each one in a port hamiltonian way. The only light assumption is that the haptic devices can be modeled this way, as proposed in. [START_REF] Duindam | Modeling and Control of Complex Physical Systems: The Port-Hamiltonian Approach[END_REF] The energetic ports (angular velocity -torque) of the environment, users (trainer and trainee), master and slave devices, IPC master and slave controllers and the ESC are all interconnected following the Figure 2. See the Appendix concerning the definition of ports of devices and IPC controllers.

dH dt = - ∂ T H ∂X R ∂H ∂X + Y T U + S T V (9)
or, equivalently:

H(X(t)) -H(X(0)) = -D(t) + t 0 Y T (τ )U (τ )dτ + t 0 S T (τ )V (τ )dτ (10)
where D(t) is a non-negative function which represents the energy dissipated in the system.

Only the human/environment power port (U, Y ) is associated with the external world. Therefore, H(X(t)) -H(X(0)) is a storage function if the following inequality holds:

H(X(t)) -H(X(0)) ≤ t 0 Y T (τ )U (τ )dτ (11)
As D(t) ≥ 0 , a sufficient condition to respect this property is to impose that : From the leader user point of view (trainer in mentoring mode, trainee in evaluation mode), the system is composed of lossless interconnections of passive subsystems.

Otherwise, the two MFS could inject extra energy into the system and potentially destabilize it. By integrating Eq. ( 1) and ( 8) and knowing that θsf1 = θsf2 = θrs , the previous passivity condition can be rewritten as:

E p (t) = E p (0) + t 0 S T (τ )V (τ )dτ = 0 + t 0 T T sf1 (τ ) Θsf1 (τ ) + T T sf2 (τ ) Θsf2 (τ ) dτ = t 0 [(α -1)T r1 (τ ) -αT r2 (τ )] T Θrs (τ ) dτ (13)
So, the system will remain passive as long as:

E p (t) ≤ 0 ∀t ≥ 0 (14)
4.2.2. Passivity Controller. In practice, in order to respect the passivity condition of Eq. 14, we modulate the Modulated Flow Source outputs as indicated:

Θsf1 = Θsf2 = Θrs , if E p ≤ 0 0, otherwise (15)
As long as the condition in Eq. 14 is satisfied, both MFS are activated, providing full feedback to both users (mainly to the follower one). Otherwise, both MFS are deactivated, forcing the modulated flows Θsf1 and Θsf2 to 0. This passivity controller always guarantees the global passivity and only deteriorates the additional feedback in case of potential loss of passivity. This means that in critical situations, the quality of the feedback is downgraded to the one provided in section 3.1. The main drawback is that, in mentoring and evaluation modes, the follower user becomes isolated as long as the passivity controller triggers. Experiments did not permit us to encounter such a situation. This is why we did not look for a less conservative solution up to now. This could be a suggestion of improvement.

Real-time Experiments

Setup

The setup is composed of three Geomagic 3D Touch haptic devices (see Figure 6). The devices' kinematic and dynamic parameters are available in. [START_REF] Sansanayuth | Teleoperation with inverse dynamics control for phantom omni haptic device[END_REF] These devices are six d.o.f. systems but only three d.o.f. are actuated. Figure 5 shows the haptic device with the considered actuated degrees of freedom (θ 1 , θ 2 and θ 3 ). The control software was implemented in Matlab Simulink® (see Figure 7). Concerning the software connection with the devices, the Open Haptics® software library was used along with the Phantorque block introduced in 1 and extended to simultaneously work with three devices. During all the tests, the real time clock is provided by this block. The IPC controllers were tuned for compliant motion following the guideline provided in. [START_REF] Stramigioli | Creating artificial damping by means of damping injection[END_REF] Resulting parameters corresponding to the models introduced in the Appendix are visible in Table II.

Free Motion

In this first experiment, in order to show how mentoring and evaluation modes work, we set the dominance factor α = 1 at first and we suddenly changed it to 0 at t=25 s in order to determine how the system behaves on a sudden change of α Also, in order to show that the device of the following user effectively follows the one of the leader, each user performed a motion to take their device away from the trajectory prescribed by the leader, at their own turn. It it is visible in Figures 8,9, and 10, for the trainee during mentoring stage a and for the trainer during evaluation stage b .

Figure 8 depicts the global trajectory performed in the user/carterian space by the users in free motion. This plot shows that, out of the two attempts to take one of the devices away, they all haptic follow close trajectories. The root mean squared (RMS) value of the tracking error (computed from the positions expressed in the user/cartesian space) between the leading device (Master 1 when α = 1 and Master 2 when α = 0) and the slave was 1.8 mm, for trajectories in a range of 18 cm × 4 cm × 10 cm corresponding to a volume of 72.10 -5 m 3 . This level of precision corresponds to the one affordable with this type of off-the-shelf haptic device. The main limitation is the available maximum force around 3 N. Some proposals to enhance the user experience are discussed later on. The upper plots of Figures 9 and10 show the positions of the three devices along x and y axis (z is not shown for reasons of space and the results are close to the other axis) in the user frame shown in Figure 5. They were computed from the angular positions Θ m1 , Θ m2 , and Θ s of the three actuated joints, using the geometrical model provided in. [START_REF] Aldana | Operational space consensus of multiple heterogeneous robots without velocity measurements[END_REF] In practice, these signals are directly available from inside the IPC controllers which exchange angular velocities with each other, for the dual-user operation needs (see Figure 2).

We obtained the best position tracking for x (error less than 1 mm RMS between Master 1 and the slave when α=1) and the worst for y (3.3 mm RMS). As y is in the vertical direction, it highlights that the weight of the whole arm disturbs a little the IPC controllers. This controller was parameterized to provide the best transparency. Attempts to raise the gains of this joint controller led to oscillations and worst performance. A gravity compensation approach should be embedded to raise the precision in this direction.

Besides, the position tracking precision between the device of the following user and the slave is logically lower than between the leading user device, as the following user is guided by their device to follow the slave trajectory. According to the resistance they bring (the stiffness of their hand), the follower's position is shifted from the reference trajectory. The more the following users resist the motion of the device, the more the position tracking error raises (and so the force feedback) in order to bring them back to the leader's device position.

Lower plots of Figures 9 and 10 show the actuating forces of each device F m1 , F m2 , F s (in the user frame), computed from the torques T m1 , T m2 , T s applied on the three actuated joints, using the inverse dynamic model of the Geomagic 3D Touch from. [START_REF] Aldana | Operational space consensus of multiple heterogeneous robots without velocity measurements[END_REF] These torques, generated by the IPC controllers for each actuated joint of each device are the only source of information about haptic feedback, as there is no force sensor measuring the real interaction forces in our setup. We must assume that the internal torque controller of the devices accurately applies them on each joint. This also means that these torques (which are also transmitted to the other devices) embed the (environment or user) interaction torques, the gravity effect and the friction torques. Therefore, the global system control is performed with these disturbed torques. Let notice that these torques are limited so that the resulting force is limited to 3.3 N in any direction of the cartesian space, according to the manufacturer. We took care to work below this threshold. In order to enhance the precision of the force measure, we should embed a real force sensor on each effector of these devices. Nevertheless, as the three devices feature the same equipment which have very close motions, these disturbances are, in practice, nearly the same for the three of them. Actually, they offset one another and the user only feels the feedbacks from tool-environment interaction. Also, the spikes visible in these plots (with some of them greater than 3N) when the devices are quickly moved are are not felt by users. We assume that the internal torque controller of the devices filters them. One can observe that, as the motion is performed in free motion, these forces are close to zero. Indeed, in mentoring (respectively evaluation) mode, the master 1 (respectively master 2) and slave forces overlap. Moreover, one can observe the forces generated by the devices to bring back the following user to the reference trajectory at marks a and b . The sudden change of α has also no negative effect neither on forces nor on energy levels (see Figure 11) which remain negative.

Master 1 Master 2 Slave Fig. 11. Energy Ep generated by both MFS for each joint in free motion

In Contact Experiments

In this second experiment, the dominance factor α is also changed from 1 to 0, at t =45 s. The trainer leads their device to keep the slave effector on a surface which had the shape of a vertical circular guide slot (diameter of 133 mm) and was rapid printed with ABS material (visible in Figure 6). This shape was chosen to illustrate a motion involving every joint. The surface is located in the plane x = 3 mm in the cartesian space. It was chosen so that the users push in the x+ direction to simplify the reading of the plots.

To evaluate the force feedback precision of the system, the leading user was assigned to keep their effector in contact while traveling in the circular slot. At two distinct instants, they had to stop their motion and to apply a higher push force, in order to ask the follower user to apply the same interaction force. This can be very useful in a training context to learn how to dose forces to apply. As any teleoperation system, bound to a balance between transparency and stability, the positions of the masters are virtually beyond the surface (around 18 mm) while the slave effector stopped in contact, which corresponds to a stiffness of 130 N/m. We tried to increase the gains of the IPC controllers but it resulted in some annoying chattering. This stiffness could be raised by using more powerful off-the-shelf haptic devices (these ones are limited to ≈ 3 N in each direction).

Figure 12 shows the trajectory performed by the effectors. The spikes labeled with numbers from 1 to 4 correspond to the periods of time (5< t <14 s, 20< t <40 s, 45< t <60 s, and 65< t <85 s) when the leading users stopped their motion and pushed on the shape. It is clearly visible that the slave does not follow the leader master in the x+ direction as it is in contact with the rigid surface. Only the masters go beyond this limit because of the aforementioned lack of stiffness. Figure 14 shows the actuating forces of each device F m1 , F m2 , F s (in the user frame), as in previous free motion experiment. One can observe a good force tracking in the two dimensions x and y (the same for z). The more the leaders push their own device "inside the piece", the more the force feedback increases, as visible in Figure 14. In these plots, one can observe that the slave forces overlap or, at least, are very close to the leading master feedback interaction forces, which attests a good force tracking, and which is confirmed by the negligible level of the global force tracking error (computed from the magnitude of the difference of the forces in the three cartesian dimensions) between the slave and the leaders' forces: 0.2 N RMS, corresponding to 5 % of the magnitude of the torques.

When the leaders decide this corresponds to the level of force they want the followers to replicate, they ask the followers to join them at the same (master) location. Its is done at t = 32 s and t = 75 s. The followers align then their own device at the same location as the trainer's only by way of a video feedback and real-time plots displayed on their monitor, as visible in Figure 6. For instance, at t = 38.5 s, the trainee manages to have their own haptic device located at a distance of less than 1 mm in the x direction. The interaction force in this direction is then at 2.4 N, the trainer feels 2.5 N and the trainee 2.3 N, which globally corresponds to an error of 4 % compared to the slave force value. At t = 78 s, the trainee and the slave are located at x = 17.2 mm while the trainer located their device at x = 16 mm (error of 1.2 mm: 7%). The trainee feels the slave interaction force (2.6 N with a precision better than 1 %) while the trainer feels 2.4 N, which corresponds to an error of 8 %.

This clearly shows that the better the following users position their device close to the same position as the leading user, the better the force feedback. On the one hand, a better alignment graphic tool should help minimize this error, and on the other hand, one has to take into account the Just Noticeable Difference (JND) of force that users can detect. In, 18 a JND of 10% has been experimentally determined for forces detected by the index finger with forces from 2 to 5 N. This corresponds to a JND force magnitude of 0.5 N. Assuming it applies in our case, the force feedback errors measured in the experiments are below this level. They, therefore, should not be tangible by a human. More detailed experiments should be performed to confirm this assumption in our case.

Figure 15 clearly shows that MSF energy levels are not influenced by the contacts between the slave effector and its environment. Fig. 15. Global Energy Ep generated by both MFS for all the actuated joints, in contact case

AAA Validation

Using the aforementioned AAA algorithm, an experimental validation has been performed to evaluate the behavior of the system. Some previous experiments took place only on a one d.o.f. system with the previous AAA algorithm. [START_REF] Liu | A dual-user teleoperation system with adaptive authority adjustement for haptic training[END_REF] In Figure 18, the trajectories, in free motion, of the devices are shown in the user space with the corresponding force feedback. At first, α = 0. The trainee, by manipulating master 2, guides the slave and the trainer (on master 1). At t=5.6 s, the trainer voluntarily progressively pulls away their device from the trainee's trajectory. As soon as the distance between the two master devices raises, α automatically grows and the trainer gets more control on the slave which progressively approaches the trainer's trajectory and then, when sufficiently far from the trainee's trajectory (at t=10.6 s), it fully follows them instead of the trainee: α grew up to 1 so that the trainer was full leader when 11 < t < 15 s. It can be observed that they get the force feedback from the slave during this period of time, in force plots of Figures 16 and17.

When 0 < α < 1, the slave is in between both trajectories. It is sufficient to avoid, for instance, a mistake in the gesture of the trainee which could have damaged the tool or the environment. As soon as the trainee draws near the position of the trainer, α decreases so that the trainee can continue their exercise. In this experiment, the distance which raised α to 1 was 19 mm. In practice, it should be adapted in real time by the trainer to the level of confidence in the skills of the trainee and according to the potential harmfulness of bad gestures.

Conclusions about these experimental results

These experimental results demonstrate that the ESC architecture complies with the requirements evoked in the introduction. For a three d.o.f. system, the follower user is able to get the same force feedback as the leader when both users position their own device at the same location. Also, the Adaptive Authority Adjustment mechanism proved it was efficient; it does not disturb the force feedback as long as the trainer follows (and, by the way, acknowledges) the trainee's motion. It permits the trainer to alternatively and transparently follow the trainee in evaluation mode, and to instantaneously take back the control of the slave to avoid any insecure situation. Experiments showed a global position tracking precision around 3 mm with a very good force tracking precision (less than 0.2 N). We observed a larger tracking error in every situation in the z axis which may be explained by the fact that, during the experiments, the arm is the most extended in this direction from the central vertical axis rotation. There may be a greater accumulation of sensor measure errors and model inaccuracy in this direction.

Conclusion

In order to provide a supervised hands-on training system, dual-user architectures suitable for such purpose have been surveyed. We proposed a typical use case which permitted to extract the requirements for such an activity and we observed which existing solutions could fit. As we found no solution responding to all these requirements, and considering the advantages of energy based modeling approaches, we extended in this paper the Energy Shared Control (ESC) dual-user architecture, from one to n d.o.f., assuming each device is identical. . We detailed its modeling and motivated its architecture using the port-Hamiltonian framework. We also reported the extension to n d.o.f. and the enhancement of the Automatic Authority Adjustment mechanism previously introduced in 11 for one d.o.f. The ESC performance has been validated experimentally: the results demonstrate a good slave position tracking performance (with a precision about 3 mm) in both mentoring and evaluation mode and a full force feedback provided to both users in case of slave contact with environment with a precision around 0.2 N. Experiments demonstrated that the AAA not only did not decrease this performance but it also facilitates the setting in real time of the authority, notably in emergency situations. Nevertheless, the assumption that all devices share same kinematics may limit the applications. We will work on carrying this control architecture into cartesian space to open it to any combination of devices with different kinematics. We will also study a way to compensate for the gravity to enhance the positioning precision. Also, in this paper, we do not study the effect of delays in the loop as every device is supposed to be in the same room. The situation occurs when users are distant will be studied further, as in practice mentor surgeons work in hospitals while trainees study in their universities. A remote usage would be appreciated to gain travel time. Actually, it would be interesting to apply this architecture to a real supervised laboratory work and perform a comparative psychometric study to conclude definitely about the advantages of such an approach. and X d := [Θ T d P T d ] T is the state vector for each interface (X d ∈ R 2m , m ≥ n). Θ d ∈ R n is a vector of n joint positions and P i ∈ R n a vector of n joint momenta. B d (Θ d , P d ) ∈ R n×n is a symmetric positive-definite matrix, representing the dissipation elements.

The input U d and output Y d matrices are:

U d = U d E U d C = T ext d T des d Y d = Y d E Y d C = Θd Θd ( A3
H d (X d ) = 1 2 P T d M -1 d (Θ d )P d + V d (Θ d ) (A4)

IPC Controller

This section recalls the controller model used for each master, for a single d.o.f. It is an Intrinsically Passive Controller introduced by Stramigioli. [START_REF] Stramigioli | Modeling and IPC control of interactive mechanical systems: A coordinate-free approach[END_REF] Each master's IPC controller (c, j), j ∈ {m 1 , m 2 }3 (see Figure 19) consists of a virtual mass, two springs and a damper The angular displacements of the two springs are represented by q rm ∈ R and q cm ∈ R, respectively, and the momentum of the virtual mass by p cm ∈ R. Each controller state variable is x m := [q rm p cm q cm ] T and its Hamiltonian is

H m (x m ) = 1 2 k rm q 2 rm + 1 2 k cm q 2 cm + p 2 cm 2m m (A5)
where k rm ∈ R and k cm ∈ R are the elasticity factors, and m m ∈ R represents the inertial element.The port-Hamiltonian model of Master IPC is:

         ẋm = [J m (x m ) -R m (x m )] ∂H m ∂x m + G m u m y m = G T m ∂H m ∂x m (A6)
where

J m =   0 -1 0 1 0 -1 0 1 0   R m =   0 0 0 0 b cm 0 0 0 0   G m =   0 1 0 0 1 0   u m = θm R θm N y m = t m R t m N (A7)
where subscripts R and N respectively mean Robot and Network controller sides and b cm is the dissipation element. Correspondingly, for the slave IPC controller, one spring has been removed (in comparison with the master IPC controller), to ensure a causal interconnection, as shown in Figure 20. The slave IPC controller, with a state vector x s := [p T s q T s ] T has the following Hamiltonian:

H s (x s ) = 1 2 k is q 2 s + p s 2m s (A8)
It is modeled in the following port-hamiltonian form:

         ẋs = [J s (x s ) -R s (x s )] ∂H s ∂x s + G s u s y s = G T s ∂H s ∂x s (A9)
where As the IPC controller parameters can be set arbitrarily without affecting its own (and the global) system passivity, one can tune them to get the best transparency from this system. The stiffness and damping parameters (in other words, the compliance) of each interface should be set so that the slave robot perfectly tracks the motion of the leader user and the users' interfaces feed back realistic environment force applied the on slave tool. Also, their damping should be set in order to avoid overshooting behaviors and undesirable oscillatory effects. A guideline for the IPC controllers is available. [START_REF] Stramigioli | Creating artificial damping by means of damping injection[END_REF]

J s = 0 -1 1 0 R s = b s 0 0 0 G s = 0 1 1 0 u s = θs R t s N y s = t s R θs N (

Fig. 3 .

 3 Fig. 3. Multiple d.o.f. articular ESC architecture, here for three joints

Fig. 4 .

 4 Fig. 4. Bézier based function to adapt α according to the position deviation between master devices

4. 2 .

 2 System Passivity 4.2.1. ESC Passivity Analysis. The global system features both the human/environment power port (U, Y ) and the modulated source power port (V, S). The energy balance is given by

An 0 S

 0 Energy-Based Approach for n-d.o.f. Passive Dual-user Haptic Training Systems 11 t T (τ )V (τ )dτ ≤ 0 (12)

Fig. 5 .Fig. 7 .

 57 Fig. 5. The 3 controlled joints of the Geomagic 3D Touch and the location of the user/cartesian frame

Fig. 8 .

 8 Fig. 8. Projection in space of the trajectory performed in free motion

Fig. 9 .Fig. 10 .

 910 Fig. 9. Cartesian positions (upper plot) and forces (lower plot) for masters and slave in x direction in the user frame in free motion

Fig. 12 .

 12 Fig. 12. Projection in space of the trajectory performed in contact case

Fig. 13 .Fig. 14 .

 1314 Fig. 13. Position for the trajectory performed in contact case in the cartesian space

Fig. 16 .

 16 Fig. 16. Trajectories and feedback forces in x direction in the user frame, with AAA function.

Fig. 17 .Fig. 18 .

 1718 Fig. 17. AAA validation: trajectories and feedback forces in y direction in the user frame, with AAA function.

  ) where E and C subscripts respectively indicate Environment and Controller sides. For the inputs, T des d ∈ R n is the desired torque while T ext d ∈ R n is an external torque exerted on the considered device. The total energy of the device H d (Hamiltonian) is the sum of the global kinetic energy due to the inertial elements M d (Θ d ) ∈ R m×m and of a global potential energy V d (θ d ) due to the gravity:

Fig. 19 .

 19 Fig. 19. Master IPC controller (in rotation context)

Fig. 20 .

 20 Fig. 20. Slave IPC controller

Table I .

 I Nomenclature.

	Notation	Meaning (i ∈ {1, 2})	Unit
	α	Main dominance factor
	β 1 , β 2	ASM internal energy distribution factors
	( θmi , t hi )	User i -Master i power port	(rad/s, N.m)
	( θmi , tm i )	Master i -IPC Mi power port	(rad/s, N.m)
	( θri , tr i )	IPC Mi -ASM power port	(rad/s, N.m)
	( θrs , tr s )	ASM power port -IPC S power port	(rad/s, N.m)
	( θs, ts)	IPC S -Slave power port	(rad/s, N.m)
	( θs, te)	Slave -Environment power port	(rad/s, N.m)
	improved version of the Adaptive Authority Adjustment (AAA) mechanism. Section
	5 provides experimental results to demonstrate the relevance of the aforementioned
	developments.

  Fig. 2. Energy Shared Control (ESC) Dual-user system connected to haptic devices in attendance of both users and an environment (the authority sharing mechanism is located inside the inner dashed box).

	Authority Sharing Mechanism (ASM)			Energy Shared Control (ESC)
			t
	t	t	t	t
				t	t	t
	t	t	t	t
			t

The notations of time-varying variables with * (t) are abridged for the sake of clarity.

all the same, every subscript is written here m for "master controller instead of (c, j).

Acknowledgements

The authors acknowledge the financial support of the China Scholarship Council (CSC) and the Consejo Nacional de Ciencia y Tecnologia (CONACyT) in Mexico.

Appendix

Modeling the Master/Slave Devices The dynamics of each n-d.o.f. haptic device (d, i) 2 with i ∈ {m 1 , m 2 , s} (each master and the slave) can be modeled in the following port-Hamiltonian form with:

where

2 for reasons of clarity, every subscript is written d for "device" instead of (d, i). Nevertheless, signal values are different between masters and slave


Iris Naudin

Minh Tu Pham

Richard Moreau

Arnaud Lelevé

A Robotic Platform For Endovascular Aneurysm Repair

An EndoVascular Aneurysm Repair (EVAR) is a procedure used to fix an aneurysm of the aorta. In this procedure, a guide is inserted by the femoral artery. This guide goes through to the height of the aneurysm and then a catheter follows the guide. Next, a stent graft is deployed in order to repair the aortic aneurysm. The objectives of our work is to develop a low-cost robotic system and implement a program that helps the trajectory planning during an endovascular operation. More precisely, this program can predict if the aorta will break or not depending on the guide used. Such a robotic platform could serve as a teaching instrument by creating an environment for young surgeons in which they will be able to practice their skills to perform an EVAR. This paper describes the different components of this platform and provides some experimental results.

I. INTRODUCTION

The need of advanced technology in the endovascular field has attracted many research studies. As a result we can find an extensive number of improvements and new ideas with the goals of reducing time, facilitate surgery and improve the outcome of the operation [START_REF] Qureshi | no turn back approach" to reduce treatment time for endovascular treatment of acute ischemic stroke[END_REF]. [START_REF] Wang | Remote-controlled vascular interventional surgery robot[END_REF] is an impressive work which consists of the develpment of a robot for a vascular interventional surgery (VIS) that has a masterslave structure and involves a fluoroscopic guidance. When one compares conventional surgery to robotic surgery it is common to find out that it takes a little bit more time for the robotic one to perform the surgery [START_REF] Kingdom | Endovascular versus open repair of abdominal aortic aneurysm[END_REF]. This can be due to the lack of experience in the surgeon to use the robot itself. Furthermore, it minimizes surgeon-based error. However, VIS can affect the surgeon with radiation sickness and injuries to skin and eyes. This is one of the reasons why medical robots were designed. [START_REF] Kingdom | Endovascular versus open repair of abdominal aortic aneurysm[END_REF] presents several robots with different assets: Firstly, it deals with the NaviCath system which includes a bedside unit and an operator control unit. It is controlled by a touch screen-controlled console and a joystick. Secondly, it mentions the Sensei robotic catheter control system (CCS) which facilitates not only the control but also the accurate positioning of a catheter within the cardiovascular system. Furthermore, it shows a catheter operating robot system (CORS) for vascular neurosurgery which the surgeon manipulates the robot away from the site where the surgery is made. Finally a remotecontrol VIS robot dedicated to cardiovascular interventions is developed. It has three outstanding improvements to the systems mentioned above: Its mechanism has a supporting manipulator and a catheter navigator, the image navigation systems takes the model from the 2D projection image into a All the authors are with Laboratoire Ampère, UMR CNRS 5005, Universiteé de Lyon, INSA Lyon, F69621, Villeurbanne, France name.surname@insa-lyon.fr 3D vascular model, a novel piezoelectric sensor to measure the contact force between the catheter and the arterial wall is presented. The advantages of such a system is i) to avoid Xray radiations of the surgeon by the provision of a convenient working environment, ii) to reduce a surgical risk due to artificial reasons by quantifying the catheter motion, iii) to improve surgical safety by the measurement of the contact force between the arterial wall and the catheter tip. In [START_REF] Wang | Intravascular catheter navigation using path planning and virtual visual feedback for oral cancer treatment[END_REF] the authors mention the use of position electromagnetic tracking system to localized a sensor attached to the catheter in combination of CT angiography. This method can produce high quality 3D images with blood vessels clearly presented, which will serve as the anatomical atlas during the surgical navigation. The imagery is created by two virtual cameras that are placed in the same direction of the two sensors and provide feedback to the surgeon in order to know what he/she is doing at the moment of placing the catheter in the targeted vessel.

The present study aims to develop a low cost robot which will help the planning of an endovascular operation and could also be implemented in education using the same instruments that a normal EVAR operation. This robot will act as a simulator of an endovascular surgery to repair an abdominal aortic aneurysm, converting it into a great asset for education. It is important to understand that for education purposes it is not necessary to have an exact replica of the operation, however there are important elements that are crucial for the knowledge we want surgeons to acquire. One of them is a haptic feedback. Haptics refers to a set of technological devices that interact with humans through the sense of touch, in other words, it is any form of nonverbal communication involving touch [START_REF] Maclean | Haptic interaction design for everyday interfaces[END_REF]. This feedback will help the surgeons to recognize the forces applied on the aorta. The advantages of this method in comparison with other methods such as optical based, acoustic based and x-ray based are that it is not time consuming; it can be done completely by the robot; it does not involve any kind of radiation exposure for the surgeon; with the knowledge of the characteristics of the stent grafts and the radiology we can chose the appropriate equipment and predict the results without being affected by the shape changing characteristics of the organs; its precision is better than that of a human; it is a low cost robot.

The paper is divided into several parts. Section 2 deals with abdominal aortic aneurysm. Section 3 is dedicated to a presentation of the a new prototype of robotic platform for computer-aided surgery. The new platform is composed of different components: a robotic system, an aorta phantom, a haptic device, and a real-time numerical simulation of an operation. Section 4 shows the experimental results of the whole system. Finally, a conclusion and some perspectives are provided.

II. ENDOVASCULAR ANEURYSM REPAIR

An Abdominal Aortic Aneurysm (AAA) (Figure 1) happens when the aorta gets dilated, to exceed more than fifty percent its initial diameter. This can lead to an explosion of the aorta causing the patient's death. In France, 5000 ruptures are experienced each year and the AAA is the 12 th leading cause of death. There are two ways to operate this disease: either by an open surgery or by an endovascular procedure. The difference of the procedures is that the open surgery takes from four to six hours and the recovery time is from five to ten days meanwhile the endovascular surgery takes from two to three hours and the recovery time is from two to three days. In other words, Endovascular Aneurysm Repair is not only minimally invasive but it is also faster and needs less recovery time than open surgery. The endovascular procedure is as follows: First the femoral artery has to be identifying, then, a catheter is inserted in that artery. Second, the catheter is guided to the aneurysm carrying a stent graft. Finally, the surgeon deploys it into place, once the stent graft is in position he/she removes the catheter (Figure 2 ) [START_REF] Amiot | and Association Universitaire de Recherche en Chirurgie Vasculaire. Fenestrated endovascular grafting: The french multicentre experience[END_REF].

However, it also presents complications such as endoleak and stent-graft migration [START_REF] Maleux | Complications after endovascular aneurysm repair[END_REF]. Endoleak happens when the stent graft does not seal well with the vessel wall; as a result, a bad outcome is obtained. In a study made by the division of vascular surgery of the Stanford University it is found that low initial deployment of the device, below the renal arteries, and short proximal fixation length are the principal causes of stent-graft migration. It is important to know that as we insert the guide and the stent-graft the shape of the aorta changes. The next section presents a new robotic platform dedicated for EVAR.

III. MATERIAL AND METHODS

A. Prototype

The proposed robot has to ensure two motions: The displacement of the guide wire along the z axis and the rotation around the same axis. A kinematic scheme is provided on Figure 3. Fig. 3. Kinematic scheme of our robot S 2 and S 3 are two rollers which let the guide wire to pass between them. Each roller is driven by a DC motor to ensure their rotations around the x 1 axis. These motions will allow the pinched guide wire to translate along the z 0,1 axis. As the link between S 1 and S 2 is a cylindrical joint, a translation of S 2 over S 3 will lead to rotate the pinched guide wire around the z 0,1 axis. Finally, the link between S 1 and S 0 is a prismatic joint allowing the translation along the y 0,1 axis. Thus the distance between S 2 and S 3 can be modified in order to use guide wires with different diameters. It is also possible to drive a catheter by this system instead of a guide wire.

A photo of our robot is shown on Figure 4. It consists of two conveyors placed one over the other which correspond to S 2 and S 3 on the kinematic scheme (Figure 3). The velocity of the conveyors is modified in order to control the position of the guide wire. The robotic platform also has a X-Y table which will be able to control the displacement of one conveyor over the other one, providing a rotation degree of freedom to the guide (Figure 5). This rotation will let the guide pass through some curvatures of the aorta. 6). This component is held by springs, modeling the environment surrounding the aortic tree in the human body. This phantom is placed in a tank filled with water to allow the displacement of a hydrophilic flexible guide inside, by two entrance spots sealingly attached to the wall of the tank. This makes it possible to compare the actual insertion of the guide in the replica with the numerical model described in Section III.B.

Finally, in order to manipulate the robot it is necessary to have the control that best fits our requirements. We Fig. 6. Aorta phantom immersed in a tank of water have to take in to account the price and specifications of different models and controls. A haptic device developed by Geomagic (Geomagic, Morrisville, North Carolina USA, 2013) was chosen (Figure 7). This device was chosen since we can achieve a good precision and it allows to perform the same movements as those performed by the surgeon during a conventional procedure. This device has six degrees of freedom of which three are motorized. It is also possible to collect the values of the six angles of rotation of the system including three gimbal angles. We can therefore describe different shapes in the defined workspace by holding the stylus of the haptic device. The latter is also equipped with two buttons that can be used for programming. Connected in USB to the computer, this hapic device allows the sending of a position reference to the conveyors and the X-Y table.

The phantom gives the sensation of contact. This is done since it is a motorized device that applies a force feedback on the users hand, allowing him/her to feel virtual objects and producing touch sensations. As a result, a realistic sense of touch is produced.

B. Numerical Simulation Of An EVAR

Another aspect of our work consists in combining the physical prototype described in section III.A and a numercial simulation of the movement of the guide in the aorta [START_REF] Mouktadiri | Angiovision -Pose d'endoprothese aortique par angionavigation augmentée[END_REF]. This work required the creation of a 3D digital model of the aorta and iliac arteries of a patient from medical imaging. The simplified whole structure of the aorta was reconstructed with the software ABAQUS and modeled with elastic, hyperelastic and linear composite materials. The mesh finite elements of the structure were created and several simulations were conducted. The model could be verified and optimized by superimposing medical images with the results of the simulation. The guide is inserted at a constant speed and angle and the model calculates the dynamic deformation of the aorta at the passage of the guide as well as the stress distribution in the structure. More details are available in [START_REF] Mouktadiri | Aortic endovascular repair modeling using the finite element method[END_REF]. Figure 8 shows the deformations caused by the insertion of different guides as well as the stress distribution in the aorta. The importance of the deformations makes it possible to understand the difficulty of the operation and the need for visual feedback during the procedure. Fig. 8. Deformation and distribution of stress in the blood vessels under the action of the guides [START_REF] Mouktadiri | Aortic endovascular repair modeling using the finite element method[END_REF] For our project, this simulation makes it possible to obtain information on the contacts between the guide and the aorta, their respective deformations and the trajectory of the guide. It also allows the trajectory planning of the operation and provides visual support for the practitioner. These numerical simulation will be essential in order to reproduce a realistic haptic feedback. The overall system includes all of the components described above. Figure 9 shows a picture of the overall system.

C. Methods

The essential aim is to reproduce as faithfully as possible the usual procedure, during which, the surgeon uses a guiding tool (Figure 10) that it serves around the guide to transmit a translation and / or a rotation through a force F and a moment M. For this, the Geomagic Touch haptic device is ideal with its stylus which has a geometry close to the original tool and which allows the same movements as the classical procedure. By pressing one of the button on the stylus of the haptic device as if it were pinching the guide, the surgeon will be able to transmit the prismatic movement or the rotation of the stylus to the robot. When the button is released, the stylus movements will no longer be recorded. In order to transmit the movements, the robot control can be separated into two parts: on one side the recovery of the data provided by the haptic device and their transformation into an usable signal, on the other hand, their use as a reference trajectory for the actuators of the robot.

The method used is therefore based on a master-slave system, where the haptic device control the robotic platform. For that, we used the results of the numerical simulation which include the efforts of contact between the end of the guide and the aortic wall as well as the length of guide inserted in the aorta at any moment. At the level of the robotic platform, this data is known thanks to measurement of the conveyors' speed and the insertion time. The approach of contact detection thus consists in comparing the length of guide actually inserted with the length of guide inserted in the simulation in order to find the corresponding contact in the simulation. This contact is finally returned to the haptic device in the form of a force whose the direction is opposite to the prismatic movement.

IV. RESULTS

A. Trajectory tracking

A program allowing the recovery of the reference position in translation from the haptic interface has been created. Figure 11 shows the juxtaposition of the stylus displacement curve and the switching curve of one of its buttons. The latter has been multiplied by 100 to allow a better reading. It has no unit for our application but corresponds to a square signal of amplitude 1 V.

When one of the buttons of the haptic device is pressed for the first time, the coordinate n 0 of the stylus on the z axis (Figure 7) is saved and is subtracted from all the values that will be recorded during the movement (see Figure 12). This is a correction of the signal to record the movement of the stylus from a zero value. When the button is released, meaning that the surgeon no longer wishes to transmit a movement to the guide, the last recorded value m 0 is maintained to allow the surgeon to start again from the point where it stopped as in the conventional procedure when the surgeon must move back the guide tool without changing the position of the guide. Once the button is pressed again, a new correction has be taken into account: it is therefore necessary both to cancel the first value n 1 and start from the last value m 0 . This is repeated for each movement of the surgeon performed with the stylus following the engagement of the button. Figure 12 shows the curve obtained when multiplying the curves of Figure 11between them. Only the movement made when the button is engaged is saved. This move will then be sent as a deposit to the robotic platform.

Finally, Figure 13 shows the position reference trajectory in its final form, obtained after data processing. The different movements of the stylus when the button 1 was engaged were accumulated to obtain this curve.

A real-time program using Simulink and ControlDesk softwares was developed. It uses the data collected through the haptic device and transmits it to the robotic platform through a proportional control law. It consists of three main parts: The first allows to send a voltage (0V or 5V) governing reference displacement (mm) time (s) Fig. 13. Displacement of the stylus along the z-axis as a function of time after data processing the direction of rotation of the conveyors through the sign of the derivative of the signal: the conveyors being coupled together, their directions of rotation will be opposed at any time. The second part sends a similar voltage to the two conveyors corresponding to the speed of the motors. This is between 0V and 10V. Finally, the third part makes it possible to collect the value of the speed of the motors in the form of tension and to apply a correction to the signal sent. The following graph (Figure 14) presents the results obtained via ControlDesk from the data of the previous curve. The error between the two curves has been multiplied by ten to allow better visibility. Similar to the prismatic movement of the guide, a program to collect the rotation of the stylus was written with Matlab-Simulink. It works on the same principle as the previous one and makes it possible to obtain a curve similar to those of Figure 13, with this time an angle value.

B. Haptic feedback

The numerical model programmed with Abaqus provides data for the haptic feedback. For the latter, the only reliable information we have at the robotic platform is the guide length inserted into the aorta. In the model, the speed of insertion of the guide being constant and equal to 20mm/s, the calculation of the guide length inserted is therefore to multiply the simulation time by this speed. Then, the Abaqus software allows to obtain the value of the contact forces exerted between the vascular wall and the end of the guide at each time step.

A Simulink program that combines rotation, translation movement and haptic feedback has been created. It retrieves data from previous curves and places them in a lookup table that returns the value of the corresponding contact force to the inserted guide length. Then, the inserted guide length is calculated in real time by multiplying the insertion speed by the time during which the motors of the conveyors operate thanks to the program managing the translation. This value is then sent to the look-up table so that finally the contact force is sent back to the haptic device.

V. CONCLUSIONS

Cardiovascular disease causes more deaths every year. Cases of abdominal aortic aneurysm rupture are no exception and physicians are increasingly resorting to endovascular surgery. This operation allows a faster recovery but remains risky: the slightest error can be fatal. To help surgeons during the procedure, a new technique has emerged: computerassisted surgery. In order to improve the existing techniques and secure the operation, we presented in this paper a remote control system of the procedure. The project is divided into two parts: a numerical study that involves modeling the insertion of the guide virtually and an automatic study that consists in managing the control of the insertion of the guide via a robotic platform. The current manuscript is focused on the second part of the study. Its main purpose is to present the different components of a new robotic platform and to implement a control law to a haptic device. This device allows both the control of insertion of the guide and the haptic feedback of the contact between the guide and the wall to allow the surgeon to perform the procedure remotely while keeping the same sensations as in a manual procedure. The work was divided into several phases. First of all, the design of a robotic platform, then the programming of the various functions allowing the retrieval of the data related to the movement of the haptic device. Then, these data were processed to allow the control of the robotic platform. Due to a compatibility problem between the interfaces (the MATLAB / Simulink software, the haptic interface and the dSPACE acquisition system), these two steps were performed separately. Despite the encouraging results, the real-time management of the platform was not possible so far. The haptic feedback has also been implemented in the system by importing the contact data from the numerical simulation. By means of a correspondence table, the appropriate force is returned to the haptic interface as a function of the length of the inserted guide. However, this force does not correspond to the force felt at the tool but to the force acting on the guide in contact with the wall. Finally, these different programs have been coupled to create a control interface. This one presents a haptic feedback without tremors, a management of the recoil and also an exit of the contact for an arbitrary rotation of 90 • .

In order to enable real-time management of the platform, a Microsoft Visual Studio application has been created. Based on the study of demonstration programs, it allowed to get rid of MATLAB / Simulink software, bypassing the problem of cross-compilation. The application, written in C language, interacts with the haptic device and provides access to the acquisition card and therefore with the robotic platform. We have been able to reprogram the control of the platform (translation, rotation) as well as the haptic feedback thanks to the data of the numerical simulation. In the continuation of the work, the most urgent will be to improve this application by optimizing these features. At the level of the haptic feedback, the measurement of the forces at the level of the guiding tool must be carried out in order to return suitable force values. The program should also be tested on several simulations to verify its robustness.

Fig. 1 .

 1 Fig. 1. Illustration of the Abdominal Aortic Aneurysm (National Institutes of Health, 2010)

Fig. 2 .

 2 Fig. 2. Illustration of a stent graft placement (National Institutes of Health, 2011)

Fig. 4 .

 4 Fig. 4. Two conveyors that provide the displacement of the guide wire

Fig. 5 .

 5 Fig. 5. X-Y table to ensure the rotation of the guide wire

Fig. 7 .

 7 Fig. 7. Haptic device (Geomagic, 2013)

Fig. 9 .

 9 Fig. 9. Overall system

Fig. 11 .

 11 Fig. 11. Stylus displacement curve during an acquisition trajectory (red) and switch-on curve of button of the haptic device (blue)

Fig. 12 .

 12 Fig. 12. Recorded stylus displacement curve for the creation of a reference trajectory

Fig. 14 .

 14 Fig. 14. Displacement of the guide as a function of time and tracking error curve

ACKNOWLEDGMENT

The authors would like to express my very great appreciation to Indira Gomez Osorio and Romain Rémy for their contributions to this work. The authors would also like to thank Dr. Vincenzo Brizzi and Pr. Patrick Lermusiaux for their valuable and constructive suggestions during the planning and development of this research project. Their willingness to give their time so generously have been very much appreciated.