IEEE TRANSACTIONS ON ROBOTICS, VOL. 27, NO. 4, AUGUST 2011

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Active Stabilization for Robotized Beating Heart Surgery Wael Bachta, Pierre Renaud, Edouard Laroche, Antonello Forgione, and Jacques Gangloff

Abstract—In this paper, control strategies for an active stabilizer dedicated to beating heart coronary artery bypass grafting are investigated. The active stabilizer, which consists of a piezoactuated compliant mechanism, has to be controlled to compensate for the displacements induced by the beating heart in order to provide the surgeon with a locally motionless myocardium surface. Three controllers, including different levels of prior knowledge about the heart motion, are presented. Their performance with respect to modeling uncertainties, arising unknown interactions of the stabilizer with its positioning mechanism, and the heart, is studied through simulations, as well as laboratory and in vivo experiments. Finally, the selection of the most adequate control scheme and the performance of the device from a clinical point of view are discussed. Index Terms—Beating heart surgery, high-speed visual servoing, medical robotics.

I. INTRODUCTION

C

ORONARY artery bypass grafting, which is one of the most common surgical interventions in the field of heart surgery, requires the suturing of grafts on coronary arteries that have a diameter of no more than 2–3 mm. Such a delicate task requires a submillimeter accuracy, i.e., typically 0.1 mm. Cardiopulmonary bypass is, as a consequence, traditionally used to stop the heart, and the gesture is performed on a motionless heart. Its deleterious effects [1] have lead the surgeons to try to operate on a beating heart using cardiac stabilizers. These passive mechanical devices are designed to locally immobilize the heart surface. Their performances are nevertheless reported to be unsatisfying [2], [3], especially in a minimally invasive surgery (MIS) context [4], [5], because of their inherent flexibility: Heart-surface displacements of several millimeters are observed.

Manuscript received October 4, 2010; revised March 8, 2011; accepted March 26, 2011. Date of publication May 16, 2011; date of current version August 10, 2011. This paper was recommended for publication by Associate Editor R. S. Dahiya and Editor B. J. Nelson upon evaluation of the reviewers’ comments. W. Bachta is with the Institut des Syst`emes Intelligents et de Robotique, Universit´e Pierre et Marie Curie–Paris 6, Paris 75005, France (e-mail: [email protected]). P. Renaud, E. Laroche, and J. Gangloff are with Laboratoire des Sciences de l’Image, de l’Informatique et de la T´el´ed´etection, University of Strasbourg, Strasbourg 67084, France (e-mail: [email protected]; [email protected]; [email protected]). A. Forgione is with the Niguarda C Granda Hospital, Milan 20162, Italy (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TRO.2011.2137770

As the manual tracking of the heart motion is too complex [6], many contributions have been proposed to overcome the difficulty that arises from this disturbance that is characterized by large amplitudes and a high bandwidth because of significant high-order harmonics [7]. In several approaches [8]–[10], a teleoperation scheme is considered, where a slave robot is synchronized with the heart motion, similarly to breathing motion compensation approaches [11], [12]. The performance in terms of accuracy of different control schemes has been evaluated [7], [13]–[15]. Predictive control [16] is seen as the most efficient approach, since it is possible to get a tracking bandwidth larger than the bandwidth of the robotic system itself (e.g., by compensating for the delays). In such a case, the tracking accuracy is closely related to the ability to predict the heart motion. As a consequence, prediction algorithms, giving the necessary prior information to predictive control schemes, have been also widely studied [5], [17], [18]. Because of the heart motion complexity, repetitive [19], [20] or iterative learning control [21] approaches that are usually used for respiratory motion compensation [22] are not sufficient. To be clinically efficient and MIS compatible, the synchronization approach requires the design of a complex slave robot that exhibits high dynamics to be compatible with the heart motion, as well as intracavity mobilities, to provide enough dexterity to the surgeon. We have proposed in [23] an alternate way to compensate for heart motion, with the introduction of the active cardiac stabilization principle [24], [25]: The active stabilizer is a compliant actuated mechanism that compensates in real time for the residual heart motion because of the stabilizer flexibility thanks to an exteroceptive measurement. Vision is used since it is already available to give a feedback to the surgeon. The use of a compliant structure for disturbance canceling can be related to other work on hand tremor compensation for microsurgery [26]. We proposed two versions of active cardiac stabilizers [23], [27] compatible with MIS and presenting, respectively, one (see Fig. 1) and two degrees of freedom (DOF). A schematic of the considered active stabilization setup is given in Fig. 2: The active stabilizer is fixed to a positioning system and controlled using a high-speed camera mounted on an endoscope. An independent surgical robot can then be used to perform the surgical task. Surgical and compensation tasks are totally decoupled, which allows us not only to simplify the mechanical design but to improve the safety of the surgical setup as well. It is important to notice that the geometry of the stabilizer is constrained by the access to the heart: The device is characterized by a long shaft that obviously introduces a significant flexibility. This yields an upper bound on the control bandwidth, which is a first characteristic specific to our approach. A second one is

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Fig. 1.

IEEE TRANSACTIONS ON ROBOTICS, VOL. 27, NO. 4, AUGUST 2011

CAD view of Cardiolock 1, an active stabilizer with 1 DOF.

Fig. 3. Principle of active compensation by superimposition of initial and compensation situations in the case of a single deformation of the shaft (deformations are amplified for sake of clarity).

developed. Simulation and experimental results are presented, and robustness issues with respect to the modeling uncertainties are discussed. In vivo results are presented in Section IV-B with a discussion on the performance of the proposed control schemes. Conclusions are then given on the most adequate control scheme, and the compatibility of active stabilization with the medical requirements is discussed. II. SYSTEM MODELING Fig. 2.

Schematic representation of the active stabilizer.

related to the device behavior and is influenced by its interaction with the heart surface and with the positioning system on which it is mounted. The dynamic model that describes the relationship between the actuators and the stabilizer tip position is altered by these interactions. Furthermore, they are subject to modeling uncertainties: Heart tissues have mechanical properties that are variable in time and among patients, and the configuration of the positioning system is dependent on the surgery location. As a consequence, we propose in this paper to study the control of an active stabilizer, considering Cardiolock 1, which is the active stabilizer with 1 DOF. This system has been designed to compensate for the heart residual motion in the main direction [24]. Its control can be considered as a generic problem for other active stabilizers. Thus, the implementation issues related to the surgical context are highlighted in this paper, as is the novelty of the device. Three aspects of the control of an active stabilizer are indeed discussed in this paper. We focus first on the influence of the modeling uncertainties on the performance of the active stabilizer. In particular, we show that H∞ control methodology that has been selected [28]–[30] allows us to cope efficiently with the model uncertainties and is well suited to our context. Second, an analysis of the most relevant prior information on heart motion, which can be included in the control law, is performed. A discussion is conducted, using results obtained with laboratory experiments, as well as in vivo tests. Finally, conclusions on the performance of an active stabilizer for this medical application are presented. The sequel of this paper is organized as follows. In Section II, the modeling of the active stabilizer is described with the analysis of its interaction with the heart and the positioning system. In Section III, the design of three H∞ controllers, including different degrees of knowledge of the heart motion, is

After the introduction of the active stabilizer and its modeling, the influence of the flexibilities of the positioning system and the interaction with the heart are evaluated experimentally and discussed. A. Description of Cardiolock 1 Cardiolock 1 is designed to compensate for residual displacement in a single direction, corresponding to the maximum cardiac force and displacement. The device consists of two parts (see Fig. 1). A first active part is composed of a 1-DOF closedloop mechanism remaining outside the patient’s body. The other part is a passive shaft of 10-mm diameter and 300-mm length. The closed-loop mechanism is composed of a piezoelectric actuator (Cedrat Technologies APA120ML) and a compliant slidercrank system, enabling the transformation of the piezoactuator linear displacement into a rotation motion. The compensation principle is schematically represented in Fig. 3. The deflection of the stabilizer shaft because of the heart motion produces a distal displacement uc , which is illustrated by the upper shaft in Fig. 3. The distal displacement is compensated thanks to the displacement ua of the actuator. The lower shaft of Fig. 3 represents the shaft after compensation for the residual motion. The piezoelectric actuator is connected to a driver including a power amplifier and a local position control loop using a strain gauge measurement. Using this driver, it is possible to accurately control the actuator displacement with a bandwidth that exceeds 500 Hz. On the current prototype, the position of the distal end is obtained by a 333-Hz camera with a resolution of 128 pixels per mm (see Fig. 4). B. Device Modeling 1) Dynamic Modeling: The mechanism is represented in Fig. 5 using a pseudo-rigid-body model (PRBM) [31]. In this modeling, flexure hinges are approximated by revolute joints associated with torsion springs describing the material elasticity.

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system, the angle α, the output of the system, and the endeffector position. 2) Identification: Classical parametric identification is performed using a step response. Experiments show that the numerator of G(s) can be approximated by a constant term λ, which means that G(s) is equivalent to a second-order function: G(s) =

Fig. 4. Cardiolock 1 prototype during laboratory (left) and in vivo (right) experiments.

M22

s2

K λ = ξ + f2 s + K2 1 + 2 ωn s +

1 ωn

s2

(4)

with a static gain K = 0.321 m, a damping coefficient ξ = 0.014, and a natural frequency ωn = 369.1 rad/s. In the following, we evaluate the influence of the flexibilities introduced by the positioning system on the function G(s), as well as the interaction between the device and the heart.

C. Influence of the Holder Elasticity on Cardiolock Dynamics

Fig. 5. PRBM modeling of the Cardiolock 1 device. The modeling of the interaction with the heart is in the lower right square.

The deflection of the beam can also be modeled by a rotating rigid beam associated with a torsion spring and damper. The first DOF α is controlled by the piezoelectric actuator, through the closed-loop mechanism, and the second DOF β corresponds to the nonactuated rotation of the stabilizer shaft. The relationship between α and the actuator displacement can be derived easily. Since the latter is controlled with a high bandwidth, the angle α can be considered equal to its reference value and is, thus, an input of the system. Small displacements are involved, and therefore, we can express the dynamic model of the system around the nominal position [α, β] = [0, 0] after linearization ¨ M22 β¨ + f2 β˙ + K2 β = Lf − M21 α

(1)

y = (b + L)α + Lβ.

(2)

with M21 = 12 m2 Lb + 13 m2 L2 and M22 = 13 m2 L2 . Parameter m2 describes the weight of the second arm, representing the shaft of length L located after the crank. K2 and f2 are, respectively, the stiffness and the damping modeling the shaft behavior. f describes the applied cardiac force. For small displacements, the stabilizer tip position can be expressed as

Using (1), (2), and the Laplace variable s, we can finally write in the frequency domain Y (s) = G(s)α(s) + P (s)F (s) with

!

G(s) =

(3)

2

((b+L )M 2 2 −L M 2 1 )s +(b+L )(f 2 s+K 2 ) M 2 2 s 2 +f 2 s+K 2 L2 M 2 2 s 2 +f 2 s+K 2 .

P (s) = The transfer function G(s) characterizes the Cardiolock dynamics: It establishes the relationship between the input of the

The active stabilizer is attached to a positioning system, that can be either a passive or an active device. This system aims to give to the base of the stabilizer a static position during the surgery, as well as at the tuning of the position of the stabilizer to access any suturing site on the heart. The workspace of the positioning system is thus quite large. Furthermore, this system is supposed to remain compact and relatively lightweight to avoid being cumbersome on the operation table. As a consequence, the system will probably present flexibilities that will alter the dynamic model of the stabilizer. We propose hereafter to quantify their influence. The positioning system used for our active stabilizer is a custom medical robot having a selective compliant assembly robot arm like kinematics with a wrist at its end (see Fig. 6). In order to study the influence of its flexibility on the stabilizer dynamics, two experiments are performed. First, the stiffness of the system is assessed in static conditions for one position, which will later be called the nominal configuration of the robotic arm (see Fig. 6). In that position, the main source of flexibility is due to the wrist on which the stabilizer is mounted. Thus, we simply model the flexibility of the positioning system in that position by a torsion spring located at the base of the stabilizer. Its stiffness is evaluated to be equal to Kp = 390 N.m/rad. Second, step responses of the stabilizer are analyzed when the stabilizer is installed on the robot. The results allow us to conclude that the holder does not influence the static gain. The damping coefficient is, however, significantly altered: It reaches 0.06 in the nominal configuration. The variation of this coefficient is 30% for different holder configurations. The natural frequency is 19 Hz lower and reaches 40 Hz in the nominal position. It is important to note that the natural frequency varies with respect to the robot arm configuration, and this variation can reach 20%. The value of the natural frequency in the nominal configuration corresponds to the eigenfrequency that can be computed from the stiffnesses K2 , Kp , and the inertia of the stabilizer around its base. This validates the modeling of the flexibility by a simple torsion spring, as described previously.

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TABLE I ˜ MODAL PARAMETERS OF THE TRANSFER FUNCTIONS G AND G

Fig. 6. Active stabilizer mounted on the medical robot in its nominal configuration.

D. Influence of the Heart on Cardiolock Dynamics The influence of the heart on the active stabilizer can be determined through in vivo experimentation on pigs. Because of the animal anatomy, an auxiliary end-effector is added in order to let the stabilizer fingers reach the myocardium without interaction between the stabilizer and the ribcage (see Fig. 4). Of course, its weight affects the dynamics of the system. The evaluation of the heart influence is thus achieved in three steps. First, a simple mechanical model of the heart is introduced as well as an analytical modeling of the interaction. Heart-model identification is then performed with the auxiliary end-effector. Finally, the influence of the heart on the active stabilizer itself is evaluated. 1) Modeling: The mechanical behavior of the heart depends on the cardiac muscle properties, the blood pressure in the heart cavities, and the forces exerted by the surroundings organs. This behavior varies also during the heart cycle [32]. To obtain an accurate model is thus not simple and would lead to a complex model [33] not suitable for the device dynamics analysis. We, therefore, prefer to derive a simple model, corresponding to a mass-spring-damper system (see Fig. 5), which captures the macroscopic behavior of the heart mechanical properties. The input of this model represents the physiological signals that are responsible for the heart-surface motion, while its output is the force applied on the stabilizer tip. With this model, the force f developed by the heart on the stabilizer is equal to " " # # ˙ − d˙c (t) − mc y¨(t) (5) f = −Kc y(t) − dc (t) − fc y(t)

where Kc , fc , and mc are, respectively, the equivalent stiffness, damping, and mass of the stabilized area on the heart. y, y, ˙ and y¨ are the position, velocity, and acceleration of the heart surface, whereas dc and d˙c represent a nonmeasurable displacement and velocity due to the heart activity. Using the Laplace transform, we then have, from (3) and (5) λ 2 c α(s) Y (s) = M +L 2 m K 2 +L fK+L 2f 2 c 22 c 2 s + K 2 +L 2 K c K 2 +L 2 K c s + 1 %& ' $ (6) ˜ G

+

L 2 (K c + fc s ) K2+L2Kc 2 M 22 + L m c 2 f +L 2 fc s + 2 2 K2+L Kc K2+L2Kc

s+1

Dc (s).

The transfer function G between α and y is transformed into a ˜ whose modal parameters are described in Table I. function G,

The heart stiffness Kc increases the value of the resonance frequency while decreasing its static gain. This result is logical, since the heart stiffness can be considered as a spring added in parallel between the actuator and the stabilizer tip. In the same way, it can be concluded that the overall system damping is increased by the cardiac tissue damping fc but decreased by the additional heart mass mc . Furthermore, the heart mass decreases the overall natural frequency. We now need to identify Kc , fc , and mc to evaluate quantitatively the influence of the heart on the stabilizer. 2) Identification: A sternotomy is achieved to access to the heart of a pig under general anesthesia. The experimental setup (see Fig. 4) consists of the Cardiolock 1 prototype mounted on the medical robot in its nominal configuration (see Fig. 6), equipped with an end-effector that hosts a 6-DOF force sensor (ATI Nano 17) with a resolution of 0.0125 N and a visual marker to evaluate the stabilizer tip position. In (5), one can notice that dc is a nonmeasurable displacement due to the artificial ventilation of the pig and the cardiac activity. The identification is, therefore, processed on the heart of the pig just after its death. A pseudorandom binary sequence is generated with the piezoelectric actuator for the identification. The dynamic response depends on the heart, the stabilizer with its end-effector, and the positioning system behaviors. A model for each element is available, except for the parameters of the heart model, which are therefore identified. The heart mass mc is identified to be equal to 0.025 kg, which is approximately 8% of the total heart mass, the damping coefficient fc is equal to 4.8 N.s/m, and the stiffness Kc is equal to 380 N/m. 3) Influence on the Active Stabilizer Dynamic Model: The identified values of mc , fc , and Kc allow us to compute, thanks to Table I, new values for the modal parameters of G(s). The natural frequency and the static gain of the active stabilizer decrease, respectively, by 13% and 4%, to reach 319 rad/s and 0.303 m. The damping coefficient is significantly increased by a factor 5, to reach 0.1. Therefore, the heart can be almost seen as a pure damper for the stabilizer. E. Discussion The interaction with the heart most significantly influences the device behavior by introducing an additional damping. This influence is favorable for the control of the stabilizer. It seems, however, rather delicate to take it into account for the design of the stabilizer control for two reasons. First, the identification of the mechanical properties of the heart are experimentally very delicate to conduct. In our experiments, we had to stop the heart to eliminate the influence of the nonmeasurable input dc . The behavior of the positioning system has furthermore to be

BACHTA et al.: ACTIVE STABILIZATION FOR ROBOTIZED BEATING HEART SURGERY

taken into account during the identification, since its behavior influences the experimental data. Second, a modeling of the heart behavior would require experiments on several areas on the heart, corresponding to each possible surgery location on the myocardium. We, therefore, decide to design the stabilizer control with the nominal model and, thus, get a conservative design, which is indeed desirable for safety reasons for a medical device. The positioning system considered during the experiments dramatically reduces the bandwidth of the active stabilizer. Indeed, the robot used as a holder is a lightweight arm of relatively low stiffness, since it was designed for a medical application [19]. We thus obtain here a worst case scenario of the influence of the positioning system. The flexibilities of this latter could be taken into account by identifying the stabilizer behavior when mounted on the positioning system and designing a controller according to the system dynamics. Such an approach would be very limitative for a clinical use, since the identification process and the control synthesis should be performed each time the positioning system configuration is modified during the surgery. A solution to overcome this issue could be to derive an analytical model of the flexible holder and include it in the design of the controller. This solution seems tedious to implement, if we keep in mind the complexity of the dynamic models of flexible manipulators. Moreover, it would remain holder dependent, i.e., modeling should be performed for any new passive or active system. Finally, we again consider that the design of the stabilizer control has to be performed with the nominal model of the device. The proposed approach is to include in the design process robustness margins to ensure that we can handle the model uncertainties. The laboratory and the in vivo experiments will demonstrate the validity of this choice. The H∞ control design method is chosen in order to ensure such robustness margins. This method is presented and developed in the Section III, after deriving the model of the whole system to be controlled.

Fig. 7.

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Modeling of the control loop.

Fig. 8. Bode diagram of the identified transfer function in red and the experimentally measured frequency response in green.

the acquisition and the processing of the current image. After identification using an output-error method [35], the open-loop model can be written as being equal to Gd (z −1 ) = z −2

8.207 + 9.608z −1 + 5.594z −2 . 1 − 0.8827z −1 + 0.9747z −2

(7)

Fig. 8 shows the good superimposition of the experimental and identified Bode diagrams of the system. Fig. 8 also validates the simplification of the numerator of G(s) to a simple gain introduced in Section II-B1. B. Control Design Approaches

III. CONTROL DESIGN A. System Modeling We now consider the whole system, i.e., the active mechanism with its actuator and the high-speed camera used as a feedback for the control. Fig. 7 shows the block diagram of the visual loop. The computed control signal u is converted into an analog voltage with a digital-to-analog conversion modeled by a zeroorder holder. This voltage is the reference of the piezo low-level position control loop, converted to an angle α by a factor µ that corresponds to the amplification provided by the closedloop mechanism. The output y is measured using the position of a visual marker v in the image given by the camera with a sampling period Ts = 3 ms. The camera is positioned so that the image plane is parallel to the visual marker displacement plane. A simple gain Kv thus describes the transformation between v and y. The dynamic behavior of the camera can be modeled as an averaging filter representing the exposure time effect [34]: z −1 ) . A time delay of one period is required for Cm (z) = K v (1+ 2

From a control point of view, the stabilization task consists in designing an appropriate feedback control law that rejects the disturbance caused by the heart force f , while guaranteeing sufficient stability margins to ensure robustness with respect to the modeling uncertainties discussed earlier. Close problems have been investigated for the control of flexible structures, for which robustness to the reduction of the model order has to be ensured [36]. The H∞ control framework [37] is indeed suitable in such a context: This control methodology allows us to define explicitly the stability margins and has been proven to provide good results when dealing with flexible systems [28], [29]. Moreover, the H∞ controller design is achieved via a shaping of the closed-loop system frequency responses, which is adequate with our knowledge of the system parameters and the disturbance properties. We have pointed out in Section I the interest demonstrated in the literature of taking into account a priori information for physiological motion compensation. For investigation purposes, we thus consider three different control schemes, corresponding

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Fig. 9.

IEEE TRANSACTIONS ON ROBOTICS, VOL. 27, NO. 4, AUGUST 2011

Four blocks augmented plant.

to three different levels of prior knowledge on the disturbance. First, no information is considered: In other words, a simple feedback controller is introduced. Then, an a priori value of the fundamental cardiac frequency is integrated by using a modified feedback controller. These two controllers will be introduced together. Finally, a prediction of the perturbation is explicitly taken into account, by introducing an H∞ controller with a preview. The three controllers are evaluated in simulation to investigate the level of performances that can be expected and how the modeling uncertainties due to the influence of the positioning system and the heart are handled by the controllers. C. Feedback Controllers 1) H∞ Problem Formulation: The augmented plant that we consider is shown in Fig. 9. The controller synthesis is performed in the continuous time domain for ease of interpretation. The conversion between continuous and discrete time domains is performed using the bilinear transform and, for implementation, a model reduction to an order of 10 is performed using the Hankel singular values. Gc (s) is the bilinear transform of the identified transfer function Gd (z). Pc (s) represents the bilinear transform of the transfer function Pd (z) between the cardiac force and the visual measurement v. Pd (z) is determined by multiplying the bilinear transform of P (s) by Cm (z)z −1 (see Fig. 7). P (s) has the same denominator than G(s) (see Section II-B1) and a numerator which is experimentally identified. We use standard notations in this section: The reference is denoted r, the control signal u, the error signal e, and the perturbation p. Wi , i ∈ [1, 3] are the weighting transfer functions that are used to shape the closed-loop system behavior. The controller K is designed to get a stable closed-loop system and minimize the H∞ norm of the performance channel, i.e., the transfer between w = [r, w2 ] and z = [z1 , z3 ] [37]. Denoting γ the resulting H∞ norm, the following inequalities will be satisfied for any angular frequency ω:  |Ter (jω)| < γ|W1 −1 (jω)|   |Tu r (jω)| < γ|W3 −1 (jω)| (8) −1 −1  2 (jω)|  |Tep (jω)| < γ|W1−1 (jω)W−1 |Tu p (jω)| < γ|W3 (jω)W2 (jω)| where Tij designates the transfer from j to i. The respective roles of the weighting functions W1 , W2 , and W3 appear by

looking at the behavior in low and high frequencies. Indeed, |K(jω)Gc (jω)| $ 1 in low frequencies. Therefore, the sensitivity function S are equal to Ter , and the functions Tep and Tu r can be approximated, respectively, by K −1 Gc −1 , Pc K −1 Gc −1 , and Gc −1 . On the contrary, |K(jω)Gc (jω)| % 1 in high frequencies so that Ter , Tep , and Tu r can be, respectively, approximated by 1, Pc , and K. One can observe that the controller K impacts only Ter and Tep in low frequencies, while it influences only Tu r in high frequencies. In the sequel, W1 and W2 will, therefore, be used to tune the system modulus margin and frequency behavior in low frequencies, while W3 will be used to shape the controller frequency response in high frequencies. The choice of these weighting functions as well as the way they define the control task requirements is presented in the following. 2) Simple Feedback Controller: W1−1 shapes the transfer Ter and is thus an upper bound of the sensitivity function S. Since the modulus margin is equal to 1/&S&∞ , W1 is chosen in the sequel equal to a constant gain δ, that will constrain the modulus margin of the controlled system to be greater or equal to that gain. This way, we can set the robustness of the closed-loop system with respect to the modeling uncertainties. W1−1 W2−1 shapes the transfer Tep which expresses the performance of the disturbance rejection. W1 is now known, and W2 is chosen equal to the following general equation: W2 =

ρs + ωs s + (ωs

(9)

where ρ, (, and ωs are three constants. With such a function, the high- and low-frequency gain of Tep are upper bounded by, respectively, 1/ρδ and (/δ. The product δωs also constitutes a lower bound of the bandwidth of Tep . The disturbance rejection bandwidth and the maximum allowed steady error can thus be tuned with ωs and (. W3−1 shapes the transfer Tu r , and is therefore used to create a roll-off effect in high frequencies, i.e., to constrain the controller to have a low gain at high frequencies in order to ensure robustness with respect to the nonmodeled plant dynamics. W3 is chosen equal to W3 =

1 k1 s + ωs' k1 k2 s + ωs'

(10)

with k1 and k2 representing the upper bound of Tu r gain, respectively, in low and high frequencies. A small value of k2 is needed to ensure robustness with respect to nonmodeled dynamics. ωs' is the crossover frequency of W3 , which value indicates the frequency beyond which the controller gain has to be attenuated. The parameters that define the weighting functions are finally chosen equal to δ = 0.5, (ρ, (, ω) = (0.05, 5e−3 , 82), and (k1 , k2 , ωs' ) = (9, 1e−3 , 82). The modulus margin is then imposed to be greater than 0.5, the controller gain is attenuated beyond 20 Hz, and the system bandwidth is larger than 6.5 Hz. The obtained performance index is γ = 1.04. The resulting closedloop transfers and their respective templates are represented in Fig. 10. To choose a roll-off frequency equal to 20 Hz allows us to take into account the modeling uncertainties: When the stabilizer

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Fig. 10. Controller synthesis (templates are shown as the dotted red line, obtained transfers are shown as the blue line). (Left) Feedback controller. (Middle) Feedback controller with notch filters. (Right) Predictive controller.

is in interaction with the positioning system and the heart, it has been demonstrated experimentally that the system natural frequency can decrease down to 37 Hz when the positioning system is in its nominal configuration and even 30 Hz with other configurations. The obtained bandwidth of the system allows us to attenuate at least four cardiac disturbance harmonics, which are the most significant. To obtain a modulus margin greater than 0.5 improves the damping of the controlled plant compared with the open-loop system and allows a relative robustness with respect to modeling uncertainties. 3) Feedback Controller With Notch Filters: As a first step to integrate information on the perturbation, we consider the fact that the cardiac signal is characterized by a fundamental frequency and several harmonics. The same weighting functions are considered, except for W2 , whose gain is amplified by bandpass filters of narrow bandwidths centered on the cardiac harmonics. Let T be the transfer function of such a filter T =

s2 + 2ζnum ωc s + ωc2 s2 + 2ζden ωc s + ωc2

(11)

where ωc is the fundamental cardiac angular frequency, and ζnum > ζden ∈ [0, 1]. The ratio between ζden and ζnum defines the amplification around wc . Higher values of ζden decreases the sensitivity to the uncertainty on ωc . The transfer Tep is shaped by the inverse of W2 . As a consequence, the filters introduced in W2 can be seen as notch filters that attenuate the perturbation effect around the cardiac frequencies. This control strategy is thus designated in the following as the use of a feedback controller with notch filters. Hereafter, ζnum = 0.45 and ζden = 0.01, allow us to obtain an attenua-

tion of 33 dB of Tep around the considered frequencies. Four filters centered around the first four heart-frequency harmonics are considered. W3 is divided by the filters so that the controller gain is not constrained. The obtained performance index is γ = 1.03. Fig. 10 shows the resulting closed-loop transfers and their respective templates. D. Predictive Controller H∞ controllers with preview have been already proposed in the literature [38], [39], but the design was not achieved using the standard H∞ design tools. In the sequel, a standard H∞ controller with a preview component is introduced, after building a prediction of the disturbance acting on the system. 1) Disturbance Prediction: The disturbance d caused by the heart on the stabilizer cannot be measured. A simple estimator, based on the system model, the input, and the output measurement, is therefore introduced: d := v − u × Gd (z).

(12)

This disturbance is related to the heart motion, which has been extensively studied in the literature. This motion is indeed composed of respiratory and cardiac components. A coupling between both components exists [5], [40] so that we express d as d(k) = Mr (k) + Cc (k) + Cc1 (k)Cr (k)

(13)

where d(k) is the disturbance at sample time k and Mr the respiratory component. Cc contains all the significant heartbeat harmonics, whereas Cc1 contains only the first low-frequency

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a bandwidth of approximately 3 Hz, which is half of the previous feedback controllers bandwidth, in order to highlight the prediction capabilities. Finally, W3 is a second-order transfer function to increase the roll-off slope and allow us to limit the feedback controller gain in low frequencies. The delay has been set equal to four times the sampling period, in accordance with the plant order and the number of delays in the system transfer function. The obtained performance index is γ = 0.69. Fig. 10 shows the resulting closed-loop transfers and their respective templates. The gain of Tep remains below the 0-dB line, which indicates that any frequency contained in the disturbance signal can be attenuated.

A 2-DOF controller with a preview term.

E. Simulations

Fig. 12. view.

Augmented plant corresponding to the 2-DOF controller with pre-

harmonics (a truncated part of Cc ). Cr is the modulating respiratory component. It has the same fundamental frequency than Mr but is different in amplitude and has a more limited number of harmonics. If we replace Mr (k), Cc (k), Cc1 (k), and Cr (k) in (13) by truncated Fourier series and consider the products of the Fourier coefficients as new terms [5], we can write d(k) = ΦT (k) x(k)

(14)

with x(k) the parameter vector to be estimated, and Φ(k) a regression vector known at sample k. Using (14), the parameter vector is updated at each sample by a recursive least-squares algorithm with a forgetting factor [5]. As the regression vector can be computed several samples ahead, it is possible to predict the heart motion from this model: d(k + n) = ΦT (k + n) x(k).

(15)

2) H∞ Problem Formulation: A 2-DOF control scheme with preview capabilities is considered (see Fig. 11). The controller consists of two parts: a feedback part Kf and a preview one Kp . These two parts are designed simultaneously. Only causal transfer functions are permitted in classical H∞ controller design algorithms. Therefore, in the design scheme shown in Fig. 12, the preview term fed into the feedforward channel is transformed into a delay approximated with its Pade model D(s). 3) Controller Synthesis: For the controller design, the following weighting functions have been used:  0.5s + 20    W1 = s + 0.1 W2 = 1 (16)   0.0001s2 W = 3 1e − 12s2 + 2e − 06s + 1 where W2 is set to 1 to get a good tradeoff between the feedback and predictive components. W1 has been chosen to guarantee

1) Simulation Conditions: One can see from the model that is introduced in Section II-D1 that dc is a time-varying displacement which only depends on the heart activity. The force applied by the heart on the stabilizer is on the contrary a function of the stabilizer interaction with the heart and the positioning system. As a consequence, the disturbance d (see Fig. 9) in the image is also dependent on the stabilizer interactions and represents the stabilizer deflection due to the heart activity. For the simulations, we, therefore, use a perturbation in the image d based on in vivo data for which amplitude is set according to the simulation conditions, and the performances of the controllers are analyzed in three steps. First, the system modeling corresponds to the model that is considered during the controller synthesis. Then, the influence of the interaction of the stabilizer with the heart is introduced, and finally, the influence of the positioning system is introduced as well. 2) Results With the Nominal Model: The steady errors obtained with the different controllers are expressed through their standard deviation and peak-to-peak values in Table II, and the frequency analysis of the errors is represented in Fig. 13. The simple feedback controller allows an efficient stabilization of the device tip: The standard deviation and the peak-to-peak values are below 0.1 mm, which is a value that is considered as the targeted accuracy for such a device. Using notch filters to take into account the fundamental frequency of the heart and its harmonics appears as an efficient way to improve the device performance: Standard deviation and peak-to-peak values are divided by, respectively, 2.6 and 1.8 with respect to the results obtained with the simple feedback controller. Fig. 13 shows a significant decrease of the error around the first four cardiac harmonics, which was considered during the controller design. In order to evaluate the intrinsic capabilities of the predictive controller, an exact prediction of the future heart motion is fed into the controller. The standard deviation and the peak-to-peak value of the steady error then decrease to, respectively, 0.0024 and 0.014 mm. These results are approximately three times better than the other ones. 3) Influence of the Interaction With the Heart: To include the influence of the heart lowers slightly the performance of the two feedback controllers. On the contrary, the predictive controller performance is improved. This may look somehow astonishing,

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TABLE II SIMULATION RESULTS—VALUES REPORTED IN MICROMETERS

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TABLE III STANDARD DEVIATIONS (STD) AND PEAK TO PEAK VALUES (PTP) IN MICROMETERS FOR LABORATORY AND in vivo EXPERIMENTS

IV. LABORATORY AND IN VIVO EXPERIMENTS A. Laboratory Experiments

Fig. 13. Frequency analysis of the steady error (feedback control in blue, feedback control with notch filters in red, predictive control with exact preview in green).

since the controller has been designed using the nominal model, and the predictive component of the controller is an approximation of the model inverse. A comparison between the nominal model inverse and the inverse of the model, including the heart dynamics, shows that the gain of this latter is indeed closer to the gain of the controller in the frequency range of interest, which can explain the observed improvement. Last but not least, one can observe the very good robustness of the stabilizer control. Stability and efficiency are maintained with a system that no longer behaves like the nominal model used for the controller synthesis. 4) Influence of the Heart and the Positioning System: The influence of the positioning system on the system behavior is stronger. The standard deviation of the residual displacement is almost doubled. The level of performance remains, however, very interesting for the application: In this worst-case scenario, we can have peak-to-peak values of approximately 100 µm, which is the targeted accuracy, with the feedback controller with notch filters. The predictive controller is the most sensitive to the presence of the positioning system: The standard deviation is only three times better with respect to case of a feedback with notch filters, whereas it was six times better when only considering the heart interaction. Finally, once again, we can observe a very good robustness of the different controllers: The stiffness of the positioning system is low, which decreases sharply the natural frequency of the system, but stability and efficiency are still maintained.

The device is first evaluated in laboratory, with a heart simulator that introduces a simple interaction with the stabilizer and a mounting of the stabilizer that does not introduce significant additional flexibility. 1) Setup: A 2-DOF mechanism simulates the beating heart motion (see Fig. 4), using heart motion data acquired in vivo on a pig: A breathing motion of fundamental frequency equal to 0.25 Hz with six harmonics is superimposed to a beating heart motion characterized by a fundamental frequency equal to 1.5 Hz with eight harmonics. Following the observations reported in Section III-D1, the first four harmonics are modulated by the breathing motion. A spring reproduces the heart behavior. Contrary to a real heart, this simulator does not introduce a strong additional damping, which is less favorable in terms of stabilization performances. The other main difference with the simulations comes from the prediction that is now achieved online with the experimental data. 2) Results and Discussion: Standard deviation and peak-topeak values of the residual motion are evaluated over 17 s (see Table III). Temporal evolution is represented in Fig. 14. Simulation and experimental results cannot directly be compared, since the perturbations are not identical. Furthermore, they may change from one trial to another. This is why a column entitled “without control” has been added. It is, however, interesting to observe the relative efficiencies of the three controllers: The predictive controller is still the most efficient controller; however, its performance is now comparable with the one of the feedback controller with notch filters in terms of standard deviation. This decrease of the level of performance most likely comes from the use of the estimator that is introduced in Section III-D1 and the computation of the prediction. B. In Vivo Experiments Experiments on pigs have been performed to evaluate the performance of the active stabilizer in real conditions, since the pig heart has been proven to be an accurate model of the human heart.

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Fig. 15. Comparison of the simulation and in vivo results with the predictive controller (in vivo results in red and simulation results in blue).

Fig. 14. Steady residual motion using the different controllers. Laboratory experiments are on the left, and in vivo experiments are on the right. (a) No knowledge on the heart motion. (b) Prior knowledge on the cardiac frequency. (c) Use of a cardiac motion prediction.

1) Setup: Experimentation is carried out on a 35-kg pig which underwent full sternotomy after receiving general anesthesia. The ventilation tidal volume was set with a frequency of 16 breaths/min. The recorded fundamental heart frequency was 1.16 beat/s. Cardiolock 1 is positioned on the myocardium surface using the previously presented medical robot. Because of the animal anatomy, a custom end-effector is added at the stabilizer tip (see Fig. 4), with fingers that have a suction capability to grip on the heart surface. A visual marker is located on the end-effector to evaluate its displacement with the highspeed camera. All the data acquisitions are synchronized on the camera frame rate with software running under the Xenomai real-time operating system. 2) Results: The three controllers with different a priori knowledge on the heart motion have been evaluated. The controllers are designed using a modified model of the stabilizer dynamics that captures the influence of the custom end-effector. Fig. 14 shows the residual motion for three in vivo active stabilization experiments. The standard deviation and the peakto-peak values of the residual motion, evaluated over 17 s, are reported in Table III. We notice that the controller with prior knowledge on the cardiac frequency now outperforms the predictive controller. This is most likely due to both the modeling uncertainties and the way we obtain the predictions of future motion. To further analyze

these results, simulations with the in vivo recorded disturbance data have been performed. The stabilizer dynamic model is now a model identified after the in vivo experiments that includes the dynamics of both the positioning system and the heart. Experimental and simulation results are in very good accordance with the first two controllers. In the case of the predictive controller (see Fig. 15), the black-line curve corresponds to a simulation performed with an exact prediction of the disturbance. According to this plot, the degradation of the performance of the predictive controller is clearly because of the lack of accuracy of the prediction. The prediction algorithm itself has been proven to be efficient [5]; however, it should be fed with a measurement of the current heart motion. In our setup, this measurement is not available and should be estimated. This is performed using (12), which gives poor results when the plant model is subject to uncertainties like during in vivo experiments, where the stabilizer is interacting with its holder and with the heart. Disturbance estimation is thus an important issue for predictive control in active stabilization context. Finally, it is important to notice that the stabilization performance is satisfactory with respect to the medical need: Using the feedback controller with notch filters, we are able to limit the standard deviation of the residual displacement to 50 µm. 3) Interspecimen Results: The in vivo results that are presented in this paper can be compared with the results that are presented in [23] for another experimentation on a pig. Standard deviation of the residual displacement was then limited to 30 µm, using a simple feedback control. This better result (30 versus 54 µm) can be explained by important differences in the spectrum of the disturbance. Indeed, in spite of similar experimental conditions (weight of the animal, area of interest, position of the holder, etc.), the disturbance is characterized by higher harmonics with significant higher amplitudes (see Fig. 16). V. DISCUSSION In vivo experiments validate the active stabilization approach, as well as its robustness with respect to modeling uncertainties and interspecimen variation. A deeper knowledge on the cardiac disturbance, if available, should improve the stabilization when used in the con-

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perturbation frequency content. The predictive controller may look like a promising approach, but the performances remain limited by the need for a correct perturbation estimation, which is an open problem in our context. Finally, the applicability of the active stabilizer, from a surgeon point of view, has been analyzed. The targeted accuracy of 0.1 mm has been obtained in vivo in pigs. REFERENCES Fig. 16. Frequency analysis of the steady error (first experiment in blue and the second in red).

trol scheme. This has been observed through simulation results, where the predictive controllers outperform the other ones when applied to the nominal model of the Cardiolock device, with an exact prediction. However, if the a priori information becomes erroneous, the performance of the active stabilization decreases. This is why the predictive controller performance decreases in vivo, where the prediction algorithm is fed with a poor estimation of the current value of the disturbance, based on the nominal model of the stabilizer dynamics. Nevertheless, the predictive control remains an attractive solution if the disturbance estimation issue can be handled correctly. Moreover, the prediction algorithm that is included in the prediction control scheme allows an online adaptation to changes of the disturbance. The feedback controller with notch filters has shown a better robustness, since its implementation is not directly dependent on the interaction dynamics. Its use in practice is also straightforward: A database of precomputed controllers covering the range of the possible cardiac frequency can be easily built so that a simple analysis of the disturbance frequency spectrum allows us to choose the adequate one. If a cardiac frequency change occurs during surgery, it is possible to adapt, in real time, the control law using gain-scheduling techniques. The simple feedback controller gives acceptable results and can be a good choice if the simplicity of implementation is a major requirement. VI. CONCLUSION In this paper, three aspects of the control of a novel active stabilizer for beating heart surgery have been discussed. First, the influence on the stabilizer behavior of the heart and the device holder has been investigated. The H∞ controller design methodology has been proposed to handle these sources of modeling uncertainties. The obtained robustness of the stabilization task has been demonstrated through simulations, in the laboratory, and by in vivo experiments. Second, the choice of the most adequate controller for such an active device has been discussed. Three different levels of prior information on the perturbation constituted by the beating heart have been considered. The experimental evaluation tends to show that the best choice is a simple feedback controller using notch filters to take into account the main harmonics of the

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[20] L. Ott, P. Zanne, F. Nageotte, M. de Mathelin, and J. Gangloff, “Physiological motion rejection in flexible endoscopy using visual servoing,” in Proc. IEEE Int. Conf. Robot. Autom., 2008, pp. 2928–2933. [21] B. Cagneau, N. Zemiti, D. Bellot, and G. Morel, “Physiological motion compensation in robotized surgery using force feedback control,” in Proc. IEEE Int. Conf. Robot. Autom., 2007, pp. 1881–1886. [22] C. Riviere, J. Gangloff, and M. de Mathelin, “Robotic compensation of biological motion to enhance surgical accuracy,” Proc. IEEE, vol. 94, no. 9, pp. 1705–1716, Sep. 2006. [23] W. Bachta, P. Renaud, E. Laroche, J. Gangloff, and A. Forgione, “Cardiolock: An active cardiac stabilizer—first in vivo experiments using a new robotized device,” in Proc. Int. Conf. Med. Image Comput. Comput.-Aided Intervention, vol. LNCS-4791, 2007, pp. 78–85. [24] W. Bachta, P. Renaud, E. Laroche, A. Forgione, and J. Gangloff, “Design and control of a new active cardiac stabilizer,” in Proc. IEEE/RSJ Int. Conf. Intell. Robot. Syst., 2007, pp. 404–409. [25] J. Gagne, E. Laroche, O. Piccin, and J. Gangloff, “Active heart stabilization using adaptive noise cancelling techniques with gyroscopic actuation,” in Proc. 3rd IEEE/RAS-EMBS Int. Conf. Biomed. Robot., 2010, pp. 802–807. [26] C. Riviere, W. T. Ang, and P. Khosla, “Toward active tremor canceling in handheld microsurgical instruments,” IEEE Trans. Robot. Autom., vol. 19, no. 5, pp. 793–800, Oct. 2003. [27] W. Bachta, P. Renaud, E. Laroche, and J. Gangloff, “Cardiolock2: Parallel singularities for the design of an active heart stabilizer,” in Proc. IEEE Int. Conf. Robot. Autom., 2009, pp. 3839–3844. [28] R. Smith, C. Chu, and J. L. Fanson, “The design of H∞ controllers for an experimental non-collocated flexible structure problem,” IEEE Trans. Control Syst. Technol., vol. 2, no. 2, pp. 101–109, Jun. 1994. [29] R. Banavar and P. Dominic, “An LQG/H∞ controller for a flexible manipulator,” IEEE Trans. Control Syst. Technol., vol. 3, no. 4, pp. 409–416, Dec. 1995. [30] W. Bachta, E. Laroche, P. Renaud, and J. Gangloff, “H-infinity methodology for active cardiac stabilization,” in Proc. Int. Fed. Automat. Control World Congr., 2008, vol. 17, pp. 11642–11647. [31] L. Howell, Compliant Mechanisms. New York: Wiley-IEEE, 2001. [32] Y.-C. Fung, Biomechanics: Mechanical Properties of Living Tissues. New York: Springer-Verlag, 1993. [33] M. Sermesant, H. Delingette, and N. Ayache, “An electromechanical model of the heart for image analysis and simulation,” IEEE Trans. Med. Imag., vol. 25, no. 5, pp. 612–625, May 2006. [34] A. Ranftl, L. Cuvillon, J. Gangloff, and J. Sloten, “High speed visual servoing with ultrasonic motors,” in Proc. IEEE Int. Conf. Robot. Autom., 2007, pp. 4472–4477. [35] L. Ljung, System Identification: Theory for the User. Englewood Cliffs, NJ: Prentice-Hall, 1999. [36] I. Kar, T. Miyakura, and K. Seto, “Bending and torsional vibration control of a flexible plate structure using H-infinity based robust control law,” IEEE Trans. Control Syst. Technol., vol. 8, no. 3, pp. 545–553, May 2000. [37] J. Doyle, K. Glover, P. Khargonekar, and B. Francis, “State-space solutions to standard H2 and H∞ control problems,” IEEE Trans. Automat. Control, vol. 34, no. 8, pp. 831–847, Aug. 1989. [38] A. Cohen and U. Shaked, “Linear discrete-time H∞ -optimal tracking with preview,” IEEE Trans. Automat. Control, vol. 42, no. 2, pp. 270–276, Feb. 1997. [39] A. Kojima and I. Shintaro, “H∞ preview tracking in output feedback setting,” Int. J. Robust Nonlinear Control, vol. 14, no. 7, pp. 627–641, 2004. [40] L. Cuvillon, J. Gangloff, M. de Mathelin, and A. Forgione, “Toward robotized beating heart TECABG: Assessment of the heart dynamics using high-speed vision,” in Proc. Int. Conf. Med. Image Comput. Comput.Assist. Intervention, 2005, pp. 551–558. Wael Bachta received the Electr. Eng. degree from the Ecole Nationale Sup´erieure de Physique de Strasbourg, Strasbourg, France, in 2005 and the M.S. and Ph.D. degrees in robotics from the University Louis Pasteur, Strasbourg, in 2005 and 2008, respectively. He is currently an Associate Professor with Universit´e Pierre et Marie Curie–Paris 6, Paris, France, where he is a member of Institut des Syst`emes Intelligents et de Robotique. His research interests include medical robotics, visual servoing, and mechatronics. Dr. Bachta received the Young Scientist Award in “Computer Assisted Intervention Systems and Robotics” at the Medical Image Computing and Computer-Assisted Intervention Conference in 2007.

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Pierre Renaud received the M.Sc. degree in mechan´ ics and materials from the Ecole Normale Sup´erieure de Cachan, Cachan, France, in 2000 and the Ph.D. degree in robotics from the Clermont-Ferrand University, Clermont-Ferrand, France. Since 2004, he has been an Associate Professor with Institut National des Sciences Appliqu´ees, Strasbourg, France, where he is a member of the Control Vision and Robotics team, Laboratoire des Sciences de l’Image, de l’Informatique et de la T´el´ed´etection. His research interests include medical robotics, robot design, parallel robots, and mechatronics.

Edouard Laroche was born in 1971. He received the Eng. degree from the Ecole Nationale Sup´erieure d’Electricit´e et de M´ecanique, Nancy, France, in 1994 and the Agr´egation degree in electrical engineering and the Ph.D. degree from the Ecole Normale Sup´erieure de Cachan, Cachan, France, in 1995 and 2000, respectively. He is currently a Full Professor with the University of Strasbourg, Strasbourg, France, where he is with the Laboratoire des Sciences de l’Image de l’Informatique et de la T´el´edectection. His research interests include robustness issues in identification and control of electromechanical systems.

Antonello Forgione received the M.D. degree from the University of Naples, Naples, Italy, in 1996 and the Ph.D. degree in new technologies applied to general and oncologic surgery from the Federico II University, Naples, in 2006. In 2003, specialized in digestive and endoscopic surgery. From 2003 to 2007, he was a Research Fellow with the European Institute of Telesurgery. He is currently a Consultant Surgeon in general surgery with Niguarda C Granda Hospital, Milan, Italy, where he is the Scientific Director of the Advanced International Mini-Invasive Surgery Academy. His current research interests include mini-invasive surgery, robotic and microrobotic manipulators, and virtual and augmented reality. Dr. Forgione received a prize as one of the world’s leading researchers in the field of mini-invasive surgery during the World Congress of Surgery, Yokohama, Japan, in 2008.

Jacques Gangloff received the Agregation degree in electrical engineering from the Ecole Normale Sup´erieure de Cachan, Cachan, France, in 1995 and the M.S. and Ph.D. degrees in robotics from the University Louis Pasteur, Strasbourg, France, in 1996 and 1999, respectively. Between 1999 and 2005, he was an Associate Professor with the University of Strasbourg. In 2005, he was nominated as a Professor at the same University. He is currently a member of the Control Vision and Robotics team with the Laboratoire des Sciences de l’Image, de l’Informatique et de la T´el´ed´etection. His research interests include mainly visual servoing, predictive control, medical robotics, and, more recently, cable-driven parallel robotics. Dr. Gangloff received the Best Vision Paper Award at the IEEE International Conference on Robotics and Automation, the 2005 Best Paper Award of the IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, and the Best Conference Paper Award at the IEEE/RAS-EMBS International Conference on Biomedical Robotics and Biomechatronics.