Bilateral control architectures

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Université libre de Bruxelles (U.L.B.) Faculté des Sciences appliquées

Contribution to the design of control laws for bilateral teleoperation with a view to applications in minimally invasive surgery

Thèse présentée en vue de l’obtention du grade de Docteur en Sciences de l’Ingénieur par

Thomas Delwiche

Promoteur : Prof. Michel Kinnaert Octobre 2009

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Université libre de Bruxelles (U.L.B.) Faculté des Sciences appliquées

Contribution to the design of control laws for bilateral teleoperation with a view to applications in minimally invasive surgery

Thèse présentée en vue de l’obtention du grade de Docteur en Sciences de l’Ingénieur par

Thomas Delwiche

Promoteur : Prof. Michel Kinnaert Octobre 2009

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Acknowledgements

The financial support of the following institutions is gratefully acknowledged. This work has been supported by a grant of the FRIA and by a grant of the Van Buuren foundation. The experimental test bench has been financed by the FNRS. Travel costs have been partially supported by the FNRS and the by UAE New-York.

I would like to thank my supervisor, Prof. Michel Kinnaert, who allowed me to explore this interesting research direction and who introduced me to many people in the "automation" community. The quality of his supervision greatly helped me overcome the hurdles inherent in the completion of a PhD thesis.

My great thanks to Laurent Catoire and to Serge Torfs who built the experimental setup used in this work.

Many thanks are due to Prof. Johan Schoukens and to all members of the ELEC department of the Vrije Universiteit Brussel, who welcomed me in their department. I have been impressed by the quality of their research and also really enjoyed the very pleasant working atmosphere.

I would like to express my great thanks to Samir Aberkane, Joseph Yamé and their colleagues of the CRAN laboratory in Nancy. Beside our fruitful collabora- tion, I particularly enjoyed their friendly welcome during my stay there.

Moreover, I would like to acknowledge the members of the Dutch-Belgian Haptics Network and especially Pierre Letier, Bert Willaert, Emmanuel Vander Poorten and Dennis Van Raaij, for interesting discussions.

Thanks to all people who indirectly contributed to this work by their comments and suggestions.

I would like to express my gratitude to Prof. Raymond Hanus who allowed me to work in the department of Control Engineering and Systems Analysis and to my colleagues for the their contribution to the good atmosphere of the lab, especially during the famous coffee breaks.

Special thanks to my friends, my parents and my sisters, Julie and Céline, who supported me during these four years and especially during the delicate writing process.

Finally, thanks to Aline,ma moitié.

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Abstract

Teleoperation systems have been used in the operating rooms for more than a decade. However, the lack of force feedback in commercially available systems still raises safety issues and forbids surgical gestures like palpation. Although force feedback has already been implemented in experimental setups, a systematic methodology is still lacking to design the control laws.

The approach developed in this thesis is a contribution towards such a systematic methodology: it combines the use of disturbance observers with the use of a structured fixed-order controller. This approach is validated by experiments performed on a one degree of freedom teleoperation system. A physical model of this system is proposed and validated experimentally.

Disturbance observers allow to compensate friction, which is responsible for performance degradation in teleoperation. Contrary to alternative approaches, they are based on a model of the frictionless mechanical system. This allows to compensate friction with a time varying behavior, which occurs in laparoscopy.

Parametric uncertainties in this model may lead to an unstable closed-loop. A kind of "separation principle" is provided to decouple the design of the closed-loop system from the design of the observer. It relies on a modified problem statement and on the use of available robust design and analysis tools.

A new metric is proposed to evaluate the performance of friction compensation systems experimentally. This metric evaluates the ability of a compensation system to linearize a motion system, irrespective of the task and as a function of frequency. The observer-based friction compensation is evaluated with respect to this new metric and to a task-based metric. It correctly attenuates the friction in the bandwidth of interest and significantly improves position and force tracking during a palpation task.

Structured fixed-order controllers are optimized numerically to achieve robust closed-loop performance despite modeling uncertainty. The structure is chosen among classical teleoperation structures. An efficient algorithm is selected and implemented to design such a controller, which is evaluated for a palpation task.

It is compared to a full-order unstructured controller, representative of the design approach that has been used in the teleoperation literature up to now. The comparison highlights the advantages of our new approach: order-reduction steps and counter-intuitive behaviors are avoided.

A structured fixed-order controller combined with a disturbance observer is implemented during a needle insertion experiment and allowed to obtain excellent performance.

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Introduction

What is minimally invasive surgery?

Minimally-invasive surgery (MIS) appeared in the mid eighties as an alternative to open surgery. The surgeon starts by performing small incisions (less than10mm) through which trocars are inserted. A typical trocar used in MIS is depicted in figure 1(a). It is generally made of a hollow tube closed by a rubber valve. During some procedures like abdominal surgery, the belly of the patient is inflated with CO2gas to increase the operating space. The trocars prevent gas leakages through the incisions and maintain them open. Once the trocars are in position, dedicated long instruments, like the one depicted in figure 1(b), and an endoscope are inserted trough them. The surgeon manipulates the instruments and relies on the visual feedback provided by the endoscope to perform the operation (see figure 2).

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Figure 1: Trocar (left) and instrument (right) used in MIS.

This technique presents aesthetic advantages and reduced trauma for the patient, due to smaller cuts in comparison with classical surgery. This result in shorter hospital stays, lower costs and less pain [28]. Depending on the field of

1

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Figure 2: MIS: long instruments passing through trocars are manipulated by the surgeon who relies on images, obtained via an endoscope and displayed on a monitor, to perform surgical gestures.

application, MIS is named laparoscopy (abdominal cavity), thoracoscopy (chest cavity), arthroscopy (joints), pelviscopy (pelvis) or angioscopy (blood vessels).

The first laparoscopy, a cholecystectomy¹, was performed in 1985. In less than a decade, the amount of laparoscopies in the case of relatively simple operations like cholecystectomies increased significantly. For instance, in 1993, 67% of the cholecystectomies performed in the USA were realized with the minimally invasive technique (see [28]).

Unfortunately, if MIS offers many advantages to the patient, it complicates the task of the surgeon. A trocar allows rotational motions around the insertion point in the body of the patient and a translation motion along the shaft of the instrument only. The four degrees of freedom of the instrument are depicted in figure 3. According to Michel Degueldre, surgeon at Saint-Pierre Hospital in Brussels, operating through a trocar is like manipulating an object with a plastered wrist.

Moreover, all motions are inverted. If the surgeon wants to move the tip of the instrument to the left, he must move his hand to the right. Beside this reduction of the dexterity, the surgeon is most of the time in an uncomfortable position, manipulating the instruments above the patient and simultaneously looking at the monitor displaying the image taken by the endoscope. Finally, the perception of the task is reduced because the image displayed by the monitor is two-dimensional and because the forces felt by the surgeon are polluted by the friction between the rubber membrane of the trocar and the instrument. A summary of the different drawbacks of manual MIS can be found in the review paper [94].

1Removal of the gall bladder.

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Introduction 3 A few works present experimental measurements of the friction force, in the case of various types of trocars. In [91], 6 trocars, presenting various types of sealing mechanisms, are investigated on a test bench. Friction levels ranging from 0.25 N up to 3 N have been measured for the different trocars and variations between inward and outward motions have been observed. Lubricating the shaft of the instrument with water allows a significant reduction of the friction level, between 15% and 45% of the initial friction level. The authors recommend the use of lubricants in clinical practice, even if the trocar is already lubricated by the manufacturer. Another set of 3 trocars, commonly used in laparoscopy, is investigated in [80]. A surgical instrument is equipped with force and velocity sensors.

Inward and outward movements are performed with the lubricated instrument. As for the previous study, friction levels depend on the direction of motion. This is pointed out by the authors as an additional source of disturbances for the surgeon.

The friction levels were found to vary between 0.5 N and 3.5 N. A third study is reported in [90]. This time, friction is not measured on a test bench but directly during an operation. Mean friction levels measured during laparoscopy procedures are equal to 1.48 N for the outward motion and equal to 1.32 N for the inward motion. These values were found to be lower in the case of thoracoscopy because valve-less trocars are used. Indeed, in thoracoscopy, the chest does not need to be inflated. Mean friction levels in this case are equal to 0.42 N for the outward motion and equal to 0.34 N for the inward motion.

The friction force heavily depend on the technology of the sealing mechanism.

If a good trocar is selected and if suitable lubrication is applied, the surgeon will barely notice the friction. However, according to [80], the friction level is not the only criterion to chose a trocar. Economical and technical arguments are also taken into account. This explains that some trocars used in laparoscopy present friction levels up to 3 N. This is disturbing for fine motions, like a suturing task for instance. According to the authors of [78], who investigated several types of suturing needles, the peak value of the interaction force during suturing remains below1.5N. In some cases, the friction force of the trocar may then be twice that of the interaction force.

Robotically-assisted surgery appeared in the late nineties as a solution to most of these drawbacks.

Robotically-assisted MIS

With robotically-assisted MIS, also called teleoperated MIS, the surgeon interacts with the patient through a teleoperation device, as depicted in figure 4. The system is made of two distinct robots: the master, manipulated by the surgeon and the slave, holding the surgical tools passing through trocars. The robots are connected by communication lines. The position of the master, imposed by the surgeon, is measured and sent as a command to the slave which tracks this position. The surgeon is not directly in contact with his patient anymore but is comfortably seated

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Figure 3: The four degrees of freedom (in red) of the surgical tool in MIS (figure taken from [89]). The tool is inserted through a trocar maintaining the incision open.

at the master console.

The teleoperation system is made of five parts interconnected as depicted in figure 5. When the position information is sent from the master to the slave but no information is sent back to the master, one refers to unilateral teleoperation.

Robotically-assisted MIS alleviates almost all drawbacks mentioned for manual MIS. The daVincir robot depicted in figure 4, whose first version was released in 1999, allows 3D vision, 6 degrees of freedom for each instrument, thanks to articulated wrists, and motion scaling, allowing microsurgery. As can be seen in the figure, the position of the hands of the surgeon is replicated by the robotic wrists in the body of the patient, making manipulation as intuitive as in open surgery. However, despite these impressive features, the surgeon has no feeling of touch anymore which is sometimes crucial in surgery².

To make the user feel the interaction force between the slave and the environment, the master must be actuated and suitable control laws must be established.

One then refers to bilateral teleoperation or to force feedback teleoperation.

Restoring the sense of touch in teleoperated MIS

According to Luc Soler, head of the R&D department at IRCAD (Institut de Recherche contre les Cancers de l’Appareil Digestif), located in Strasbourg (France), most of the surgeons complain about the lack of sense of touch when manipulating the daVincir for the first time but eventually appear to become used to it. In facts, many surgical interventions are performed daily with teleoperation systems

2According to the commercial website of Intuitive Surgical [4], the company that produces the daVincir, the latter is provided with force feedback capabilities. However according to [25]

and to our own trials with the daVincir robot at St-Pierre Hospital in Brussels, stiff surfaces can be felt but they appear much softer than they actually are. Soft tissues cannot be felt at all.

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Introduction 5

Figure 4: The daVincirsurgical teleoperation robot is depicted in the top figure. The bottom figures represent close-up views of the hands of the surgeon and of the tools inside the body of the patient. These figures are taken from [4].

Surgeon Master

Communication

Slave Patient

Visual feedback

Figure 5: Unilateral teleoperation principle: position information is sent from the master to the slave. A position controller allows the slave to follow the position of the master.

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that are not equipped with force feedback capabilities. Should force feedback then be considered as an unnecessary feature? Certainly not.

First of all, the fact that the surgeon is not able to feel the forces applied to the patient, but relies on visual clues to evaluate them, raises questions about the patient’s safety. For instance, if the surgical tools are moved outside the vision field of the endoscope, they may touch an organ and damage it without the surgeon noticing.

Next, some techniques relying specifically on the sense of touch are impossible.

For example, locating cancerous nodules hidden in a lung is performed by palpation in open surgery. In thoracoscopy, palpation is replaced by imaging techniques which are by far less intuitive. The same comment applies when looking for hidden arteries.

Finally, applications like percutaneous interventions may also benefit from force feedback. During percutaneous interventions, a radiologist inserts a needle through the skin of the patient to reach an inner organ and to deliver a therapy. He evaluates the position of the needle using CT-scan images. In order to avoid exposing the radiologist to harmful X-rays, a teleoperated needle insertion device with force feedback capabilities has been build in Strasbourg (see [79]). Between two images, the radiologist relies only on force feedback to drive the needle.

The basic concepts of bilateral teleoperation are presented in the next section.

Bilateral teleoperation

The principle of bilateral teleoperation is depicted in figure 6. To allow force feedback, position and/or force measurements must be made at the slave and sent back to the master. The master is provided with actuators to be able to exert forces on the user. The actuated master is often called a haptic interface, where the term haptic refers to the sense of touch³.

At this point, it is important to give some explanations about the force sensing mechanism of a human being. The sense of touch, also called the haptic sensation, is basically made of two components: the tactile and the kinesthetic components (see [18]).

Kinesthetic information regroup data such as the position, the velocity of the limbs and the forces acting on them. These information are gathered by sensors located on the muscles, the joints and the tendons.

3Haptic interfaces are also used to interact with virtual environments.

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Introduction 7

Surgeon Master

Communication

Slave Patient

Visual feedback

Figure 6: Bilateral teleoperation principle: information are exchanged between the master and the slave. Information like position and/or force are measured at the slave and sent to the master to allow force feedback.

Tactile information concern the pressure and indentation distributions, in space but also in time, on the cutaneous surface and it is provided by mechanoreceptors situated in the derma and the epidermis of the fingerpads [18]. Examples of tactile interfaces can be found in the literature: [77] describes a tactile interface which is able to transmit information in Braille language, [52] describes a teleoperation setup with tactile feedback which can be applied in the context of surgery and [18]

describes in a detailed way another system with sensors and actuators especially dedicated to softness discrimination.

Although a complete haptic feedback implies both information (tactile and kinesthetic), it is difficult, due to technological limitations, to implement the tactile part. Indeed, many receptors located at different places in the fingers must be excited and researchers who are designing tactile devices are dealing with problems of miniaturization, robustness and simplicity. So, in this work, haptic feedback is limited to its kinesthetic component. From this point, force feedback or haptic feedback must be understood as kinesthetic feedback, if there is no explicit men- tion of the contrary.

A challenge among others consists in reflecting the interaction forces without reflecting the parasitic forces like those corresponding to trocar friction. It will be seen in this work that a good placement of the sensors, combined with adequate control laws, allows to tackle this problem.

Remark 1. As it will be seen along this work, the bilateral coupling between the master and the slave raises delicate stability questions. Although force feedback remains obviously the most intuitive way to display a force, several authors investigate sensory substitution to suppress the coupling between master and slave.

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Visual and/or audio feedback have been investigated in the litertaure. Interested readers may refer to [60] for more information.♦

Remark 2. Vibratory feedback has been suggested as an additional information to reinforce kinesthetic feedback. Vibrations are measured at the slave side by an accelerometer and displayed to the user by small loudspeakers, attached to the master manipulator. This seems to enhance the perception of contact or puncture.

Interested readers may refer to [63], [75], [74], [97] and [62] for more information about vibratory feedback.♦

The contributions of this thesis are presented in the next section.

Contributions of the thesis:

This work proposes several contributions to the design of control laws for bilateral teleoperation, with a view to applications in minimally invasive surgery. Exper- imental validations of the proposed concepts are performed on a one degree of freedom teleoperation setup, designed and build at the department of Control En- gineering and systems Analysis. A physical model of the system is proposed and validated using frequency domain identification tools.

1. Friction is highlighted as a major source of performance degradation for most teleoperation control schemes. It must therefore be compensated. MIS dif- fers from other teleoperation applications by the fact that the slave enters the body of the patient through a trocar. The latter opposes significant friction forces to the motion of the slave. Although numerous techniques have been proposed in the literature to compensate friction in robot joints (see [19]) and recently in MIS robot joints (see [70]), compensation of the friction of the trocar is addressed here for the first time. According to [19], most friction compensation techniques are based on a fixed model of the friction.

However, trocar friction presents a time-varying behavior which cannot be captured by a fixed model. The compensation is therefore based on a disturbance observer relying on a frictionless model of the mechanical system rather than on a fixed model of the friction. Discrepancies between the model and the real dynamics may lead to instabilities of the compensated system.

A new condition for evaluating stability in the presence of parametric uncertainty is derived. Finally, the disturbance observer is implemented on a one degree of freedom teleoperation test bench. The validation of the method is twofold. First, a teleoperation experiment is conducted showing significant performance improvement in terms of position and force tracking. Next, a new performance index is proposed to evaluate the ability of the compensation to attenuate friction with respect to frequency and irrespective of a given task.

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Introduction 9 2. The environment and the operator are not time-invariant systems. One the one hand, different organs may be encountered with different mechanical behaviors and on the other hand the dynamics of the operator evolve with the position of the limbs, with the contraction of the muscles and as a function of the operator himself. Systematic controller design methods based on optimization theory, like H∞ optimal control and µ-synthesis, have been applied to teleoperation systems. They rely on a precise problem setting, which can include modeling uncertainty and specific objectives, like nominal performance, robust stability and robust performance.

To the best of our knowledge, robust controllers designed in the teleoperation literature are unstructured and have the same order as the model of the plant (full-order controllers). Indeed, mature numerical tools exist since the nineties to solve the design problem in this case. However, due to the high order of typical models used in teleoperation, an order reduction step may be required before implementing the controller practically.

In this work, the design of structured fixed-order controllers is suggested.

Although the design problem is then much more complex than in the unstructured full-order case, efficient algorithms have been proposed recently to solve this non-smooth and non-convex problem, like the HIFOO algorithm.

The benefits of our new approach are highlighted by an experimental study, conducted on the one degree of freedom teleoperation system. Two robust controllers, an unstructured full-order controller and a structured fixed-order controller, are designed and compared for a palpation task. Beside avoiding order reduction steps, it is shown that counter-intuitive behaviors, observed with unstructured controllers, are avoided by imposing a suitable structure.

In some particular cases, the design of a structured controller can be formu- lated as a static output feedback design problem. More algorithms, like the cone complementary linearization algorithm (CCLA), are then available. In this work, the HIFOO algorithm and the CCLA algorithm, which are most of the time evaluated with respect to academic examples, are compared on a practical case.

Finally, a structured controller is implemented, together with a DOB, during a needle insertion task and the performance of the resulting control architecture are discussed.

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Structure of the work

In chapter 1, the basic notions of teleoperation are presented, namely the standard bilateral control architectures, the hybrid matrix representation of a teleoperation system and the notion of transparency. The impact of friction on performance is highlighted and the tradeoff between robust stability and performance is illus- trated by an example.

Robustness analysis tools are introduced in chapter 2. They allow to draw conclusions about stability and performance in the presence of model uncertainty, as it is the case in teleoperation. The fundamental tools introduced in this chapter are used throughout this work.

A frequency identification tool is presented next in chapter 3. It will be used for model identification in chapter 5 and to quantify the ability of a friction compensation system to attenuate friction in chapter 6.

A one degree of freedom teleoperation test bench is used to validate the different developments presented in this work. Its design is presented in chapter 4.

A global model of the teleoperation setup is presented in chapter 5. The electro- mechanical system designed in chapter 4 is represented by a physical model which is validated using the tools presented in chapter 3. The human and the environment are described using models available in the literature.

The properties of a disturbance observer used to attenuate the friction are analyzed in chapter 6. A criterion is provided to evaluate the stability of the compensated system in the presence of parametric uncertainties. The performance of the disturbance observer are assessed experimentally on the test bench using a newly proposed performance index and a more classical task-based index.

Chapter 7 builds on the notions of chapter 2 to present algorithms for robust controller design of unstructured full-order controllers and structured fixed-order controllers.

In chapter 8, a new approach for controller design in teleoperation is presented, namely the design of structured fixed-order controllers. A case study is presented to highlight the benefits of this new systematic approach. The chapter is concluded by a needle insertion experiment implementing a robust controller, designed according to this new approach.

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CHAPTER 1 Bilateral control architectures

In this chapter, the reader is introduced to the different bilateral teleoperation architectures, classified according to the type and amount of sensors. The performance of the different control architectures, in terms of position and force tracking, are studied and the negative impact of the friction of the trocar on these performance is highlighted. An example then illustrates how performance and robust stability trade off in teleoperation. Finally, a friction compensation system, based on a disturbance observer, is suggested to compensate the friction of the trocar and improve performance in terms of position and force tracking.

Our analysis leads to the conclusion that a force sensor must be placed at the tip of the slave, i.e. in the body of the patient, to guarantee acceptable performance in MIS. Issues related to the design of such sensors are also discussed.

1.1 Preliminaries

At the end of the eighties, Hannaford [47] and others proposed a representation of teleoperation systems inspired by network theory. The teleoperation system, including the master, the slave and the controller is represented by a two-port as depicted in figure 1.1, interconnecting the operator and the environment, in turn represented by one-ports . The different variables are respectively f_mh, the muscular force developed by the human, f_h, the force applied by the human to the master,f_ae, the active force developed by the environment,f_e, the force applied by the slave to the environment, x_m, the positions of the master andx_s, the position of the slave. With respect to the initial work of Hannaford, this representation uses positions instead of velocities. The reason is that position sensors are usually available in teleoperation systems rather than velocity sensors. The mechanical impedances of the human and the environment, Z_h and Z_e therefore relate forces

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to positions and not velocities.

Human operator Teleoperation Environment

system

fmh f^h

Zh Z_e

fae

fe

xm x_s

Figure 1.1: Teleoperation system modeled by a two-ports network. The human operator and the environment are modeled with one-ports. Figure adapted from [47].

The two-port is characterized by four variables: f_h, f_e, x_m and x_s related by the so-called hybrid matrix H:

f_h x_s

=

h₁₁ h₁₂ h₂₁ h₂₂

| {z }

:=H

x_m f_e

(1.1)

Although alternative representations are possible, the hybrid matrix is certainly one of the most popular because each of its components, which are transfer functions, can be interpreted physically. The different parameters are discussed in depth in [10].

• h₁₁ = _x^f^h

m|_f_e₌₀ describes the relation between the force applied by the user and its position during unconstrained motion (f_e = 0). In other words, this is the mechanical impedance felt by the user in unconstrained motion.

• h₁₂= ^f_f^h

e|_x_m₌₀is the relative force tracking when the user keeps a constant position. This parameter is related to force tracking but it does not correspond to a physically meaningful situation.

• h21= _x^x^s

m|fe=0 is the relative position tracking in unconstrained motion.

• h₂₂ = ^x_f^s

e|_x_m₌₀ is the contact admittance. It translates the sensitivity of position tracking to environment forces.

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1.1. Preliminaries 13 The parameters of the hybrid matrix completely define a teleoperation system. Ideal performance in teleoperation have been defined by Lawrence [65] and Yokokohji and Yoshikawa [96] as perfect position and force tracking. A teleoperation system with ideal performance is characterized by the following hybrid matrix, H_ideal:

H_ideal =

0 1 1 0

(1.2) Such system is said to be transparent. The term transparency can be best understood by considering figure 1.2. The dashed lines represent a transparent teleoperation system, i.e. a system where the user has the feeling of interacting directly with the remote environment. In practice, there exists a tradeoff between stability and transparency. Transparency should thus be defined with respect to the task to be performed and taking human sensing and actuation limitations into account. This way of thinking has been introduced in [27] and in [35].

Human operator Teleoperation Environment

system

fmh f^h

Zh Z_e

fae

fe

xm x_s

Figure 1.2: Transparent teleoperation system.

Analyzing performance using the hybrid matrix may lead to wrong conclusions since friction forces acting on the master and on the slave systems are not taken into account.

In a first time, only the friction opposed by the trocar to the motion of the slave, denoted f_{f ric} is taken into account. The results will be generalized later by taking the friction in the robot joints into account. To the best of our knowledge, a model describing the friction of the trocar has not been developed yet. However, f_{f ric} certainly depends nonlinearly on the velocity of the slave. Since the analysis and synthesis tools used in this work are based on linear models only,f_{f ric} will be

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treated as an independent disturbance. This introduces a new variable and leads to the definition of an extended hybrid matrix H¯:

fh

x_s

=

¯h11 ¯h12 ¯h13

¯h₂₁ ¯h₂₂ ¯h₃₂

| {z }

:= ¯H



 x_m

f_e f_{f ric}



 (1.3)

which takes the following value H¯ideal for a transparent system:

H¯_ideal =

0 1 0 1 0 0

(1.4) Classical bilateral control architecture are reviewed in the next section. Their performance are analyzed using the extended hybrid matrix.

Remark. Thanks to the analogy between a teleoperation system and an elec- trical network, passivity and stability criteria developed for network theory can be applied to teleoperation systems (see [9], [48] and [6] for instance). In this case however, a representation in terms of velocities is required. The hybrid matrix to be considered is denoted H^v and is given by:

f_h

−v_s

=

h^v₁₁ h^v₁₂ h^v₂₁ h^v₂₂

| {z }

:=H^∗

v_m f_e

(1.5)

The conversion between H and H^v is expressed as follows:

h^v₁₁ h^v₁₂ h^v₂₁ h^v₂₂

=

_h₁₁

s h₁₂

−h₂₁ −sh₂₂

♦ (1.6)

1.2 Fundamental bilateral control architectures

The main bilateral control architectures are introduced in this section. A teleoperation system is almost always equipped with position sensors, sometimes with one or two force sensors, less often with velocity sensors. In this work, position and force sensors will be considered. Not surprisingly, it will appear that increasing the number of sensors also increases the level of performance in terms of position and force tracking.

A general bilateral control architecture is depicted in figure 1.3. The lower box (green box) represents the block diagram of the master grasped by the user. Both are represented by their mechanical impedance, denotedZ_m andZ_h. The muscular force developed by the user is denoted f_mh. By analogy with a voltage source, the human can be seen as a non-ideal force source with an output impedance given by Z_h. The net force is therefore equal tof_mh−Z_hx_m and it is denotedf_h. Beside the force generated by the user, a second force is applied to the master, namely the

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1.2. Fundamental bilateral control architectures 15 actuator force. The latter is denoted fm,actu and it is computed by the controller (middle box).

Next, the upper box (red box) represents the block diagram of the slave interacting with its environment. The mechanical impedances describing their dynam- ical behavior are denoted Z_s and Z_e. The active force developed by the environment, which is equal to zero in many applications, is denoted fae. Three types of forces are applied to the slave: the net force applied by the environment (−f_e), the actuator force computed by the controller (f_s,actu) and the friction force of the trocar (ff ric), which is represented by a disturbance.

Finally, The middle box (blue box) represents the controller. The system to be controlled is fundamentally a multiple input multiple output (MIMO) system.

Even for a one degree of freedom setup, there are two inputs (the actuator inputs) and up to four outputs (in the most general case, position and force are measured at the master and slave sides). The controller is a transfer matrix with eight entries denoted C₁,..., C₆, C_m, C_s in figure 1.3.

In this section, fundamental control architectures are described and their performance are analyzed using the extended hybrid matrix. Each of them is a particular case of the general architecture of figure 1.3.

Remark. The name of the architectures refer to the number of communication channels and the type of information exchanged via these channels. For instance, one refers to a two-channel force-position scheme when x_m is sent as a command to the slave and when f_e is fed back to the master. Above two communication channels, the type of information exchanged between master and slave is usually omitted.♦

Remark 2. Notice the following sign convention: f_h is the net force applied by the human operator to the master while f_e is the net force applied by the slave to the environment. This is consistent with figures 1.1 and 1.2 and allows to define perfect force tracking as f_h =f_e.♦

Remark 3. Typical expressions used in the literature for the impedances of the master, the slave, the user and the environment are given below:

Z_m =m_ms²+b_ms (1.7)

Z_s =m_ss²+b_ss (1.8)

Z_h =m_hs²+b_hs+k_h (1.9)

Ze=ke (1.10)

These representations are used for the discussions of the present chapter.♦

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+

++

Z_e

_

Z_m^-1 Z_h

+

xs

C₆

C₅

fmh

fae

fe

Cm

C1 C⁴

Cs

fh x^m

+ ___

actu

fm_,

C3 C2

Z_s^-1

actu

fs_, _

_ +

_ + _

+

ffric ++

Figure 1.3: General bilateral control architecture involving force (fh,fe) and position (xm,xs) measurements at the slave and at the master sides. The master manipulated by the user is represented in the lower box, the slave interacting with its environment is represented in the upper box and the controller connecting these two systems is represented in the central box. Its outputs are the actuator torques (f_s,actu, f_m,actu).

Bilateral control architectures

Contents

Introduction

CHAPTER 1