Optimum Design of a New Craniotomy Robotic Manipulator ... · Optimum Design of a New Craniotomy...

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The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.OS8.003 Optimum Design of a New Craniotomy Robotic Manipulator Based on Experimental Motion and Force Analyses in Real Craniotomy Environment Terence Essomba Medical Augmented Reality Research Center, Chang Gung Memorial Hospital, Linkou, Taiwan Chieh-Tsai Wu Medical Augmented Reality Research Center, Chang Gung Memorial Hospital, Linkou, Taiwan Shih-Tseng Lee Department of Neurosurgery, Chang Gung Memorial Hospital, Linkou, Taiwan Chin-Hsing Kuo Department of Mechanical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan e-mail: [email protected] Abstract: The paper presents an experimental study on the workspace and force requirements for general craniotomy tools and introduces a new manipulator for a craniotomy robot that is optimized to meet these requirements. The craniotomy is a surgical technique that allows the surgeon to access organ (brain) inside the skull. Surgical drill and cutter are used to open the skull prior to the neurosurgery. The technical requirements of this application are studied through discussion with medical personnel, observation of the real operation, and experimentations. The surgical tool has been instrumented and used on animal bones to collect clinical data including cutting force and motion range. A serial spherical arm is adopted for the present application, and its kinematic and force transmission indices are defined to identify an optimal mechanism through two different optimization methods. Last, a computer-aided design model of the optimized robotic manipulator for craniotomy application is presented. Keywords: Craniotomy, Surgical robot, Spherical linkage, Kinematics, Force transmission, Workspace, Optimization. 1 Introduction In traditional neurosurgery, the surgeon needs to operate directly the patient’s brain. However, the brain is protected by the solidest bone of the human body, i.e., the skull. In order to provide a physical access the brain, the craniotomy is a surgical operation for temporarily removing a section of the skull. The surgeon must first use a semi-automatic surgical tool to drill several holes called “burr hole” on the skull [1], he/she then use another one to cut and remove an entire section by linking one burr hole to another until a complete piece can be removed, namely “bone flap,” from the skull. Once the bone flap is removed, the surgeon can access the brain to perform the brain surgery. Today, surgical tools dedicated to craniotomy are all semi-automatic. The intervention using semi-automatic tools requires an experienced physicians who can provide a high degree of dexterity while manipulating his/her tools. But at the same time, the surgeon can rely on robotic technology that provide better stability for the process to suppress the surgeon’s physiological factor such as vibration, fatigue or errors [2], [3]. The high risk associated to the craniotomy has been subject to a lot of research around the world. Research teams have developed robotic systems to assist surgeon while performing brain surgery. However, most of them has begun with semi-automatic surgical tools for skull milling purposes [4], [5]. Cases of full automatic robotic systems remain unusual. As result, only a small number of craniotomy robots were reported [6], [7], [8], [9]. Traditionally, when performing the bone flap removal procedure, surgeon will use a hand-held semi-automatic tool. Since 1998, the traditional semi-automatic craniotomy procedure can be assisted by a surgical robotic system [3]. As introduced in [2], robotic systems dedicated to skull surgery can be divided in three categories regarding their function: burr hole drilling, skull milling and full craniotomy. These system are briefly summarized as follow. 1.1 Burr hole drilling robots Burr hole drilling robots are designed to drill only a hole in the skull to allow the surgeon to access to a specific intracranial area. No bone flap needs to be removed during the drilling process. The Dalhousie University in Canada, for example, has developed such a system using a PA-10 industrial robot [10]. NeuroBot from Nanyang Technological University [11], the headset surgical device of University of Hannover [12] and the Stewart parallel platform of Yuan Ze University in Taiwan [13], can all be considered in this category. 1.2 Milling robots The second category concerns neurosurgical interventions that requires the use of the milling cutter to drill grooves in limited area of the skull surface. If the lesion is located on the skull, it can be milled by a milling instrument to remove the patient’s skull lesions as a surgery procedure. Several robotic systems respond to this characteristic. A system using an RX-130 industrial robot has been developed by the German Saarland University [14] in order to mill the skull off for cochlear implants. The Johns Hopkins University in USA has developed the

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The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.OS8.003

Optimum Design of a New Craniotomy Robotic Manipulator Based on Experimental Motion and Force Analyses in Real Craniotomy Environment

Terence Essomba Medical Augmented Reality Research Center, Chang Gung Memorial Hospital, Linkou, Taiwan

Chieh-Tsai Wu Medical Augmented Reality Research Center, Chang Gung Memorial Hospital, Linkou, Taiwan

Shih-Tseng Lee Department of Neurosurgery, Chang Gung Memorial Hospital, Linkou, Taiwan

Chin-Hsing Kuo Department of Mechanical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan

e-mail: [email protected]

Abstract: The paper presents an experimental study on the workspace and force requirements for general craniotomy tools and introduces a new manipulator for a craniotomy robot that is optimized to meet these requirements. The craniotomy is a surgical technique that allows the surgeon to access organ (brain) inside the skull. Surgical drill and cutter are used to open the skull prior to the neurosurgery. The technical requirements of this application are studied through discussion with medical personnel, observation of the real operation, and experimentations. The surgical tool has been instrumented and used on animal bones to collect clinical data including cutting force and motion range. A serial spherical arm is adopted for the present application, and its kinematic and force transmission indices are defined to identify an optimal mechanism through two different optimization methods. Last, a computer-aided design model of the optimized robotic manipulator for craniotomy application is presented.

Keywords: Craniotomy, Surgical robot, Spherical linkage, Kinematics, Force transmission, Workspace, Optimization.

1 Introduction In traditional neurosurgery, the surgeon needs to

operate directly the patient’s brain. However, the brain is protected by the solidest bone of the human body, i.e., the skull. In order to provide a physical access the brain, the craniotomy is a surgical operation for temporarily removing a section of the skull. The surgeon must first use a semi-automatic surgical tool to drill several holes called “burr hole” on the skull [1], he/she then use another one to cut and remove an entire section by linking one burr hole to another until a complete piece can be removed, namely “bone flap,” from the skull. Once the bone flap is removed, the surgeon can access the brain to perform the brain surgery. Today, surgical tools dedicated to craniotomy are all semi-automatic. The intervention using semi-automatic tools requires an experienced physicians who can provide a high degree of dexterity while manipulating his/her tools. But at the same time, the surgeon can rely on robotic technology that provide better stability for the process to suppress the surgeon’s physiological factor such as vibration, fatigue or errors [2], [3].

The high risk associated to the craniotomy has been subject to a lot of research around the world. Research teams have developed robotic systems to assist surgeon while performing brain surgery. However, most of them has begun with semi-automatic surgical tools for skull milling purposes [4], [5]. Cases of full automatic robotic systems remain unusual. As result, only a small number of craniotomy robots were reported [6], [7], [8], [9].

Traditionally, when performing the bone flap removal procedure, surgeon will use a hand-held semi-automatic tool. Since 1998, the traditional semi-automatic craniotomy procedure can be assisted by a surgical robotic system [3]. As introduced in [2], robotic systems dedicated to skull surgery can be divided in three categories regarding their function: burr hole drilling, skull milling and full craniotomy. These system are briefly summarized as follow.

1.1 Burr hole drilling robots Burr hole drilling robots are designed to drill only a

hole in the skull to allow the surgeon to access to a specific intracranial area. No bone flap needs to be removed during the drilling process. The Dalhousie University in Canada, for example, has developed such a system using a PA-10 industrial robot [10]. NeuroBot from Nanyang Technological University [11], the headset surgical device of University of Hannover [12] and the Stewart parallel platform of Yuan Ze University in Taiwan [13], can all be considered in this category.

1.2 Milling robots The second category concerns neurosurgical

interventions that requires the use of the milling cutter to drill grooves in limited area of the skull surface. If the lesion is located on the skull, it can be milled by a milling instrument to remove the patient’s skull lesions as a surgery procedure. Several robotic systems respond to this characteristic. A system using an RX-130 industrial robot has been developed by the German Saarland University [14] in order to mill the skull off for cochlear implants. The Johns Hopkins University in USA has developed the

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Steady Hand robot [15] and improved the NeuroMate robot [13] to mill at the bottom of the skull. These machines are programmed to operate in tandem. The parallel skull milling robot “CRANIO system” has been created by the Aachen University in Germany [15]. The robot was mainly used to remove the tumor from the skull. 1.3 Full craniotomy robots

In this category, the robotic systems are able to provide the whole process of craniotomy, i.e., drilling burr hole and cutting a part of the skull until a complete bone flap can be removed. Currently, only a handful of research teams can propose robotic systems to help surgeons perform a craniotomy. In 1998, the University of Karlsruhe in Germany, used a PUMA 260 industrial robot to perform a craniotomy procedure [6]. Later in 2009, they introduced a KUKA LWR Industrial robot [7] for craniotomy application. The same research team in collaboration with the University of Heidelberg in 2003 has also raised the RobaCka robot, and successfully conducted human experiments [8] in order to assist physicians for premature malformations healing. However, for security reasons, RobaCka robot cannot complete the entire craniotomy process. After the removal of half of the skull thickness, the surgeon has to continue the cutting process using a hand-held tool to cut the rest of the skull thickness and to remove the skull deformity. After that, the University of Karlsruhe used another industrial robot (KUKA LWR) and discarded the traditional craniotomy tool by using the method based on CO2 laser craniotomy research [7]. In 2009, Lunghwa University of Science and Technology in Taiwan [9] proposed a method to guide robotic arm using a laser scanning system and to detect the thickness of the skull under the robot trajectory with computer tomography. This method allows guiding of an industrial six-axis robotic arm (Fanuc LR Mate 200iB) for craniotomy purposes. 1.4 Literature review conclusion

The above review reflects that several research teams have developed robotic systems to assist surgeon while performing brain surgery. However, these studies are mostly focused on image guided robots and safety devices. Robotic systems proposed so far are based on the use of anthropomorphic industrial robots equipped with adapted surgical tools. While the craniotomy process (like other kinds of surgical fields) requires a technological response to very specific needs, a special-purpose robotic for craniotomy robots is not presented yet. Consequently, there is a large development potential in the research of mechanical architecture dedicated to craniotomy robots. In 2014, the National Taiwan University of Science and Technology (NTUST) proposed an approach based on a mechanical architecture especially dedicated to the craniotomy requirement where a novel Remote Center of Motion (RCM) mechanism has been proposed [16]. Their robotic manipulator, based on a decoupled two rotational Degrees of Freedom (DoF), simplifies the control and the trajectory planning programing and the kinematic aspect is optimized. However, there is no guaranty that the architecture has enough force to drill and cut the patient’s skull.

The present paper introduces the design of a new robotic manipulator for craniotomy application. The

Section 2 is an analysis of the craniotomy requirement including data measurement experimentations. A dedicated mechanical architecture is proposed and analyzed in the Section 3. This mechanism is optimized to reach the user requirements in Section 4. The Section 5 provides conclusions and perspective. 2 Craniotomy Requirement Analysis

As for many type of surgical intervention, the craniotomy has very specific requirements. Discussions with involved neurosurgeons and experts in craniotomy have highlighted several specifications for the robotic system. These have been confirmed by visual observation of real craniotomy in surgical room and described by numerical data through experimentation. 2.1 Motion requirement analysis

The process of craniotomy performed by a surgeon or by a robotic manipulator requires a very high level of stability in order to guarantee the accordance with pre-planned trajectories. Traditional craniotomy uses first Computed Tomography (CT) to obtain the location of the patient’s brain lesion and for path surgery planning. Then, the craniotomy is performed by the surgeon using semi-automatic hand tools. According to surgeons, the craniotomy task includes two different phases: (1) drilling several burr holes around the skull zone expected to be removed; and (2) cutting the skull by linking these burr holes until the bone flap can be removed. The first prototype of our craniotomy robot has been designed to respect the following specification [16]: - The craniotomy task requires 2 rotational DoF and 1 linear DoF aligned with the end effector longitudinal axis; - Rotational and linear DoF are better to be decoupled to reduce the complexity of system control when executing the drilling and cutting motions; - The drill axis has to be perpendicular to the skull surface which is reasonably assumed a perfect sphere, but a limited angle is still acceptable; - The workspace has been assessed to a cone of 25° apex angle based on the experience of clinic need. This paper introduces the second generation of our design where the rotational workspace size is increased to a cone of 45° apex angle. Also, the decoupled DoF characteristic is not requested in order to have more choice in terms of mechanical architecture candidates. The center axis of this spherical wrist (zW) aligned with the patient’s skull sagittal plan (xz) and oriented from 45° from the longitudinal axis (x) as illustrated on Fig. 1.

Fig. 1. Rotational workspace of the craniotomy process.

2.2 Force and motion capture experiments According to medical experts, the force aspect of a

craniotomy robot shall be the priority. The skull it the solidest bone of the human body it has been noticed that

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during craniotomy intervention, the surgeon seems to push the surgical cutter very hard (something using both hands). So it is important to measure the force necessary for drilling the bone and used as a specification for the robotic manipulator.

Such experimentations have been performed to measure the force and acceleration and the speed of the cutting instrument for craniotomy. The hospital involved in the project is using a GA520R surgical cutter from AESCULAP, Fig. 2 (a). The same surgical cutter has been instrumented with a force sensor an optical tracker, Fig. 2 (b). The motion capture system Polaris Vicra has been used to track the motion of the tracker. These data (force and motion) have been recorded while the cutter was operated, Fig. 2 (c). Pork bones were used for these experimentation because this animal’s tissues is closest to human’s ones. The investigated data are: - The bone reaction force, necessary to cut the bone; - The 3D position of the cutting point.

These data are synchronized and sampled in time allowing computing the time-dependent velocity and acceleration.

Fig. 2. Surgical cutter (a). Instrumentation of the tool (b).

Force and motion experimentation protocol.

As result, a total of 8 cutting experiments have been performed. For each of them, the above data have been collected. Fig. 3 shows the evolution of the bone reaction force recorded in all experiments. It can be interpreted that the force in the cutting direction does not exceed 60 N. In order to keep a safety margin, this skull reaction force to be inserted as an input data is increased to 80 N. For the 3D position, it is used to determine the velocity and the acceleration of the surgical tool. Their evolutions are respectively illustrated in Fig. 4 and Fig. 5.

Fig. 3. Evolution of the measured bone reaction force.

Fig. 4. Evolution of the linear velocity of the surgical cutter.

Fig. 5. Evolution of the linear acceleration of the surgical

cutter.

For the velocity and the acceleration, the peaks observed at the beginning and at the end of each curve are respectively due to the motion while placing the instrumented cutter on the bone and shake forward when the bone is completely cut. The determination of their respective maximum values shall be focused on the part between these two peaks. The linear velocity is limited to 0.02 m/s and the linear acceleration to 0.0001 m/s2. Regarding its low value, the linear acceleration will not be considered in following research. The linear velocity will be used for upcoming tasks such as actuator selection. These data tend to confirm visual observation of the craniotomy cutting phase, which motion appears to be very slow.

The results listed above are used as input data for the definition and the optimization of a mechanical architecture for the robotic manipulator.

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3 Mechanical Architecture Definition and Analysis The mechanical architecture of the robotic system has

to provide very accurate response to the requirements stated in the previous section. The literature study reveals that the novelty of these researches is usually based on a very accurate control to provide the safety of the patient’s brain. The last prototype of this project constitutes a rupture with traditional industrial anthropomorphic manipulator used for full craniotomy. As result, this mechanism provides a high level of dexterity and a very simple control of decoupled DoF. However, the mechanism was not optimized in term of force transfer at this point of the project. 3.1 Mechanical architecture definition

One important specification for the new prototype is to propose a simpler and smaller mechanism. As respecting the motion requirements described in Section 2.1 several compact RCM architectures have been assessed for the present application and a RRP serial spherical mechanism has been selected. This architecture is famous and widely used for medical application that requires RCM [17], [18]. It is a lot more compact than anthropomorphic robots, and its simplicity reduced the number of design variables, which makes its kinematic analysis and optimization very simple. Fig. 6 illustrates such a mechanism adapted for craniotomy application.

Fig. 6. RRP serial spherical architecture for craniotomy.

The base of the robot is located to the right side of the

patient’s skull for practical reasons: the robot is supposed to be attached to a rail on the side of the clinical bed. Its axis z1 is aligned with the y axis of the workspace to simplify the kinematic analysis. The Circular Segment 1 (CS1) of the RRP arm is described by the angle α and is limited by axes z1 and z2. The Circular Segment 2 (CS2) is described by the angle β between axes z2 and zE. All the above axes are aligned with the RCM point. The angle between the axis x1 and the direction of CS1 is the first actuator variable θ1. And the second one θ2 is located between x2 and the direction of CS2. Angles α and β are the design variables. The linear DoF is directed by the axis zE and is actuated by the actuator variable d. The global size of the mechanism is also given by the radius R of the CS1 and CS2. This variable is fix and its value has been selected 300mm as a compromise between compactness and free space between the patient and the manipulator.

3.2 Forward and inverse kinematic model

All design and actuator variable are defined and allow the analysis of the mechanism kinematic model. The forward kinematic model is first resolved. To determine the Euler variables (ψ,θ) reached by the end effector for a given set of actuator variables (θ1,θ2), the end effector axis using Euler rotation zEI is expressed in the workspace reference frame RW(xW,yW,zW) and written as follows:

zEI = R zW ,ψ( ).R xW ,θ( ).zW =−sinψ sinθ−cosψ sinθcosθ

"

#

$$$$

%

&

''''

, (1)

where R(u,ϕ) represents the rotation matrix of angle ϕ about the axis u. The same axis is expressed using the mechanism variables zED as follows:

zED = R xW ,π 2( ).R z1,θ1( ).R y1,α( ).R z2,θ2( ).R y2,β( ).zW , (2)

zED =

Sθ1SαCβ + Sβ Cθ1Sθ2 + Sθ1Cθ2Cα( )Cθ2SαSβ −CαCβ

Cθ1SαCβ + Sβ Sθ1Sθ2 −Cθ1Cθ2Cα( )

"

#

$$$$

%

&

''''

, (3)

where Cα = cos α and Sα = sin α. Since zED and zEI vectors are equal, it is possible to identify their respective terms to isolate Euler variables. The angle θ is first determined:

θ = ±cos−1 Cθ1SαCβ + Sβ Sθ1Sθ2 −Cθ1Cθ2Cα( )( ) . (4)

Then ψ is computed as follows:

ψ = ATAN2 −zED 1( ) Sθ , zED 2( ) Sθ( ) . (5)

For the inverse kinematic model, the actuator variables (θ1,θ2) allowing the mechanism to reach the end effector rotational coordinates (ψ,θ) are determined using a similar method. This time, the actuator variables are isolated. The second one, θ2 is computed first:

θ2 = ±cos−1 CαCβ −CψSθ( ) CSαSβ( ) . (6)

The first actuator variable θ1 is then computed written as follows:

θ1 = ±cos−1 C A2 +B2( )( )+ ATAN2 B,A( ) , (7)

where A = Sθ2Sβ , B = SαCβ +CαSβCθ2 ,C = SψSθ 3.3 Force transmission analysis

Among all quantitative criteria of the new manipulator, medical experts involved in the project have given the priority to the force transfer. Ended, it has been observed that the surgical cutter is subject to a certain amount of force while operating. So it is mandatory to guarantee that the robot will provide enough force to counter balance the skull reaction force. To study the force transfer in the mechanism, each segment of the RRP serial arm is studied separately. All external forces and moments applied on each of them are shown in Fig. 7.

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Fig. 7. Force and moment interaction in the mechanism.

The objective is to determine forces Rix,y,z and

moments Mix,y,z applied in the joint (with i = 1,2). It can be noticed that the values of actuator variables θ1 and θ2 have an impact on the above force interactions. These are computed first for CS2 as shown in Appendix A. The same method is used for CS1 as shown in Appendix B. Theoretically, additional moments resulting from the rotational acceleration of each segment should be added. However, it depends on the linear acceleration that has been measured in Section 2.2 and is very closed to zero. Therefore, the rotational acceleration is assumed to be zero. 4 Kinematic and Force Optimization

The mechanical architecture is optimized with two criteria: the kinematic performance and the force performance. Also, it has to respect the constraint of minimum necessary workspace. 4.1 Optimization indexes and constraints

All criteria and constraints are listed in this section. They are all converted into numerical indexes to evaluate the performance of each candidate mechanisms. 4.1.1 Workspace constraint

The workspace is not considered as a criterion to be optimized but as a constraint in the optimization process. A minimum necessary workspace for the robotic manipulator has to be respected. The mechanism must be able to reach all points of the geometry representing this workspace. It is described in Section 2.1. A structure which is unable to respect this constraint is rejected. . If a structure is able to reach a point of the workspace, then the mechanism inverse kinematic expression has a solution for a given orientation (ψ,θ) of the end effector. The existence condition is given using the expression of θ1 (6) and θ2 (7). Ended, the use of trigonometric functions in both expressions indicates that there are conditions where an orientation cannot be reached. A given orientation (ψ,θ) is reach by the mechanism if the following condition is respected.

CαCβ −CψSθ( ) SαSβ <1

SψSθ ≤1

CβSα + SβCαCθ2( )2 + SβSθ2( )2 =1

#

$

%%

&

%%

'

(

%%

)

%%

(8)

For the first expression, the value of 1 allows the existence of a solution. However, this particular condition indicates that the RR arm is completely extended, which results in a direct singularity. Consequently, this particular condition is rejected. These conditions must be respected for each point of the workspace. A constraint index is created to indicate if a mechanism respect the minimum necessary workspace. For a given mechanism m, this index is computed as follows:

Wm = P ψ,θ( )θ=0

j

∑ψ=0

i

∑ , (9)

where i = 2π/Δψ, j = π/4Δθ and P(ψ,θ) = 1 if condition stated by Eq. (8) is respected. As result, Wm gives the number of unreachable points of the minimum necessary workspace by a given mechanism m. Such a mechanism will be accepted only when Wm = 0. 4.1.2 Kinematic performance index

The ability of the mechanism to move its end effector from a position to any direction is also an important criterion. In the present application, the craniotomy robot has to be able to accurately follow a pre-planned trajectory as part of the user requirements. Several indices that measure the kinematic performance can be found in the literature. The kinematic conditioning has been selected to optimize the project first prototype [16]. For the present RR serial arm, the same index is used to evaluate the local kinematic performance. For a given orientation of the end effector, the local kinematic conditioning is computed as follows:

C ψ,θ( ) Jm( ) = Jm . Jm−1 (10)

where Jm the Jacobian matrix of a mechanism m. The Jacobian matrix of the mechanism is obtained by differentiating the 3D position of the end effector. This allows us to determine the velocity model that links the linear velocity of the end effector vE with the actuator variables velocities q! . It is then written in a matrix form as follows:

vE =!x!y!z

!

"

###

$

%

&&&= Jm !q = Jm

!θ1!θ2!d

!

"

####

$

%

&&&&

(11)

The Jacobian matrix Jm is shown in Appendix C. This local kinematic index is computed for each point of the workspace to create a global index. The global kinematic conditioning is written as a sum-squared deviation of successive local kinematic conditioning as follows:

( ) ( )( )∑∑= =

=i j

mm JCNK0 0

,/1ψ θ

θψ (12)

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where i = 2π/Δψ, j = π/4Δθ and N the number of points of the workspace. For each candidate mechanism m, this index will be computed. 4.1.3 Torque index

A certain amount of force is required to cut the patient’s skull. The force transfer analysis of the selected mechanical architecture has been provided in Section 3.3. The objective is to guarantee that the robotic manipulator has enough force to provide drilling and cutting task. The actuators will have to provide enough torque to supply the cutting force that will counter balance the skull reaction force. Also, the value of this minimum necessary torque has to be as low as possible to reduce the size and weight of mechanical equipment. According to the Fig. 7, two couple moments correspond to the actuators torque of the mechanism, M1z and M2z. Ended their axes are precisely located along actuators that respectively manage the control of variables θ1 and θ2. Preliminary study has revealed that the moment M1z is always very higher than M2z. Consequently, only M1z will be part of the index to evaluate the force performance of a candidate mechanism. The moment M1z is computed for each point of the workspace, however, its value depends on the direction of the skull reaction force. That direction is given by the two components Fx and Fy. Ended, the other component Fz is used only in the case of drilling, which apply a force perpendicular to the skull surface. For a given point of the workspace, the moment M1z(φ) is computed for every possible directions indicated by the angle φ. The highest moment is selected as the local torque index M1z(ψ,θ).

M1z ψ,θ( ) =maxφ M1z φ( )( ) , (13)

where φ = ATAN2 Fy F Fz( ),Fx F Fz( )( ) . Here, the component Fz is used only to obtain a non-complex solution through the function ATAN2. The local index is computed for every points of the workspace. The maximum value of all of them is selected as the global torque index for a candidate mechanism, Tm.

Tm =max ψ,θ( ) M1z ψ,θ( )( ) . (14)

4.2 Force transmission optimization For the second generation of our craniotomy robot, the

design priority is given to the force transfer performance. A Genetic Algorithm (GA) method is used to identify an optimal mechanism. Such an algorithm is integrated in the software Matlab and used for the present application. An objective function is programmed to collect the candidate mechanism variables as input data and to compute the corresponding global force transfer index as stated in Eq. (14) and as an output data. They are collected and placed in a vector. The highest of them is then selected. The minimum necessary workspace constraint is also integrated in this algorithm. Technically, the value of Wm is not really computed. The existence of the solution to the inverse kinematic model is tested. If there is not solution, then the program exits the fitness function with a strong penalty value. If there is a solution, the actuator variables are computed and used to determine the local torque index. So the optimum mechanism is identified using the following formulation:

Minimize Tm( ) (15)

subject to Wm = 0 with m = [ α β ], α ϵ [ αmin ; αmax ], β ϵ [ βmin ; βmax ].

The GA has been run and identified the mechanism with the variables α = 67.5° and β = 67.5°. The corresponding global torque index is 12.61 N-m. This mechanism is the optimal solution for the present application. However, this mechanism presents, a singularity in one point of its workspace. It is located at the border of the workspace. The configuration that allows the mechanism reaching this point requires the serial arm being completely extended. This corresponds to a common singularity to every serial spherical RR arms. 4.3 Multi-objective optimization method

The previous optimization method has resulted a mechanism optimized in terms of force transfer. But this same mechanism generates a singularity that locates at the border of the workspace. The specific point that presents a singularity could be removed from the programed workspace. However, the presence of a singularity indicates a poor local kinematic performance in the same zone. In order to avoid this issue, the kinematic performance index is added to the present optimization method. The software Matlab can also provide such an algorithm. A second fitness function is programed to compute the global kinematic performance index of a given mechanism. The algorithm is very similar to the one mentioned in Section 4.1.1. The difference is the method to compute the global index. Instead of selecting the maximum value, the average of all local indices values is computed. Also, the research field for the design variables has been modified to avoid any mechanism in which design variables would generate a “border singularity”. A series of optimal candidate mechanisms are identified using a multi-objective method. They are distributed in a Pareto graphic using kinematic and torque global indexes as shown in Fig. 8.

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30

Global  kinematic  index

Global  torque  index  (Nm)

Optimal  candidate  mechanismsOptimal  selected  mechanism

Fig. 8. Series of optimal identified candidate mechanisms. It can be noticed by the resulted curve that these two

indices have antagonistic requirements in terms of mechanism dimension. One of these optimal candidate mechanisms has to be selected as a compromise between these two antagonistic indices. 4.3 Optimization results and CAD prototype

The result of the multi-objective optimization process described above is a series of optimal candidate

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mechanisms. The serial spherical RR arm with α = 90° and β = 50° has been selected. This choice is motivated by the compromise generated by its design variables in terms of kinematic performance and force transfer. Also, it has the ability to reach every point of the minimum necessary workspace described in Section 2.1. The global torque index computed for this mechanism is 23.76 N-m. However, the weight of the mechanism is not considered at this step the design. Ended, it is still unknown. The next step is to use all design variables to generate a CAD model of the robotic manipulator. The software Solidworks allows providing a very accurate estimation of the structure weight. The CAD model is shown on Fig. 9.

Fig. 9. CAD model of the craniotomy robotic manipulator. The weights and the centers of mass of the

manipulators segments and the end effector are respectively estimated and listed in Table I.

Table I. Weight and mass center of manipulator elements

Weight (N)

Coordinates (mm) x y z

Base 20 0 0 0 CS1 70 189 0 -78 CS2 55 139 0 -67 Tool 6 0 0 0

The center of mass coordinates of each element are

given in their respective reference frames. These new data are the input of the global torque index fitness function to compute the same index considering the manipulator weight and the direction of the gravity. The global torque index is finally increased to 40.36 N-m for the first actuator and 13.83 N-m for the second one. These data will allow the selected actuators and reducers to be integrated in the robotic manipulator.

5 Conclusions

The objective of our work is to design a robotic manipulator for craniotomy application. The similarity of most robotic manipulators (commercial anthropomorphic arms) has been highlighted. In this paper, we proposed an approach based on the mechanical optimization to design

a prototype with mechanical architecture that is specifically dedicated to the full craniotomy. The series of technical requirements for the manipulator were discussed with medical personal and measured using experimentations. The motion of the surgical cutter and the reaction force were investigated on animal bones. A simple and compact architecture was proposed for the robotic manipulator. Its kinematic model and force transfer were firstly defined. All these data were then integrated in GA to identify the mechanism optimal design variables that generates the best kinematic and force index values. Through two different optimization methods, an optimal mechanism was selected. A CAD model of the manipulator was provided. The resulting design provides an optimal multi-objective performance for the kinematic and force transmission ability of the craniotomy robot when executing the skull-cutting task. Acknowledgment

This work is financially supported by Chang Gung Memorial Hospital, Taiwan under grant No. CMRPD2C0051-53. References [1] Hsiao M.-H. , Kuo C.-H., “A Review to the Powered

Drilling Devices for Craniotomy,” Design of Medical Devices Conference, Minneapolis, MN, USA, 10-12 April, 2012.

[2] Kuo, C.-H., Dai, J.-S., Dasgupta, P., 2012, “Kinematic Design Considerations for Minimally Invasive Surgical Robots: An Overview,” International Journal of Medical Robotics and Computer Assisted Surgery, 8(2), pp. 127-145.

[3] Taylor, R. H., Stoianovici, D., 2003, “Medical Robotics in Computer-Integrated Surgery,” IEEE Transactions on Robotics and Automation, 19(5), pp. 765-781.

[4] Matinfar, M., Baird, C., Batouli, A., Clatterbuck, R., Kazanzides, P., 2007, “Robot-Assisted Skull Base Surgery,” Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, San Diego, CA, USA, 29 October-2 November, pp. 865-870.

[5] Guthart, G. S., Salisbury, J. K., Jr 2000, “The Intuitive™ Telesurgery System: Overview and Application,” Proceedings of IEEE International Conference on Robotics and Automation, San Francisco, California, USA, 24-28 April, pp. 618-621.

[6] Burghart, C., Raczkowsky, J., Rembold, U., Wörn, H., 1998, “Robot Cell for Craniofacial Surgery,” Proceedings of the 24th Annual Conference of the IEEE Industrial Electronics Society, Aachen, Germany, 31 August-4 September, pp. 2506-2511.

[7] Mönnich, H., Stein, D., Raczkowsky, J., Wörn, H., 2009, “System for Laser Osteotomy in Surgery with the Kuka Lightweight Robot - First Experimental Results,” World Congress on Medical Physics and Biomedical Engineering, Munich, Germany, 7-12 September, pp. 179-182.

[8] Korb, W., Engel, D., Boesecke, R., Eggers, G., Kotrikova, B., Marmulla, R., Raczkowsky, J., Wörn, H., Mühling, J., Hassfeld, S., 2003, “Development and First Patient Trial of a Surgical Robot for Complex Trajectory Milling,” Computer Aided Surgery, 8(5), pp. 247-256.

[9] Yuan, C.-C., The Study of Robot-Assisted Craniotomy (in Chinese), Master Thesis, Graduate School of Engineering and Technology, Lunghwa University of Science and Technology, Taiwan.

[10] Weimin, S., Jason, G., Yanjun, S., 2006, “Using Tele-Robotic Skull Drill for Neurosurgical Applications,” Proceedings of the IEEE International Conference on

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Mechatronics and Automation, Luoyang, China, 25-28 June, pp. 334-338.

[11] Sim, C., Ng, W. S., Teo, M. Y., Loh, Y. C., Yeo, T. T., 2001, “Image-Guided Manipulator Compliant Surgical Planning Methodology for Robotic Skull-Base Surgery,” International Workshop on Medical Imaging and Augmented Reality, Shatin, Hong Kong, China, 10-12 June, pp. 26-29.

[12] Kobler, J. P., Kotlarski, J., Öltjen, J., Baron, S., Ortmaier, T., 2012, “Design and Analysis of a Head-Mounted Parallel Kinematic Device for Skull Surgery,” International Journal of Computer Assisted Radiology Surgery, 7(1), pp. 137-149.

[13] Tsai, T. C., Hsu, Y. L., 2007, “Development of a Parallel Surgical Robot with Automatic Bone Drilling Carriage for Stereotactic Neurosurgery,” Biomedical Engineering: Applications, Basis and Communications, 19(4), pp. 269-277.

[14] Federspil, P. A., Geisthoff, U. W., Henrich, D., Plinkert, P. K., 2003, “Development of the First Force-Controlled Robot for Otoneurosurgery,” Laryngoscope, 113(3), pp. 465-471.

[15] Matinfar, M., Baird, C., Batouli, A., Clatterbuck, R., Kazanzides, P., 2007, “Robot-Assisted Skull Base Surgery,” Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, San Diego, CA, USA, 29 October-2 November, pp. 865-870.

[16] Kuo, C.-H., Essomba, T., Li, G.-K., Wu, C.-T., Lee, S.-T., 2014, “Kinematic Optimization of a Novel Remote Center-of-Motion Mechanism for Robotic Craniotomy,” The 3rd IFToMM Asian Conference on Mechanism and Machine Science (Asian MMS), 9-10 July, Tianjin, China.

[17] Nouaille, L., Vieyres, P., Poisson, G., 2012, "Process of optimization for a 4 dof tele-echography robot", Robotica, vol. 30, pp. 1131-1135.

[18] Zhang, X., Nelson, C.A., 2010, “Multiple-Criteria Kinematic Optimization for the Design of Spherical Serial Mechanism Using Genetic Algorithm”, ASME In. Design Engineering Tech. Conf., Montreal.

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Appendix A Forces R2x,y,z and moments M2x,y,z are computed first for CS2:

R2 x = FxCβ −FzSβ( )Cθ2 −Wβ 1( )R2 y = FzSβ −FxCβ( )Sθ2 +FyCθ2 −Wβ 2( )R2z = FzCβ +FxSβ −Wβ 3( )

M2 x = aFz +Fx LβSβ − R+ S( )( )Sθ2 − bFyCθ2 −Wβ 2( )Cβ 3( )

M2 y = aFz −Fx LβSβ − R+ S( )( )Cθ2 − bFySθ2 −Wβ 3( ) Cβ 1( )2+Cβ 3( )

2

M2z = − a− c( )Fy −Wβ 2( )Cβ 1( )

With δ = β − tan−1 b a− c( )( ) , a = Lβ 1−Cβ( ) , b = R− S( )Cβ + LβSββ , c = R− S( )Sβ .

wβ = [ Wβ(1) Wβ(2) Wβ(3) ] is the weight force of CS2 directed by the gravity. cβ = [ Cβ(1) Cβ(2) Cβ(3) ] is the mass center of CS2. Appendix B Forces R1x,y,z and moments M1x,y,z are computed for CS1 using the same method as CS2: R1x = M2 x LαSα −M2z Lα 1−Cα( )( )Sθ1 + R2 xCα − R2zSα −M2 yCθ1 LαSα( )−Wα 1( )R1y = R2zSβ − R2 xCα −M2 y LαSα( )Sθ1 + R2 y −M2 xCθ1 LαSα +M2z Lα 1−Cα( )( )Cθ2 −Wα 2( )R1z = R2zCα + R2 xSα −M2 y Lα 1−Cα( )−Wα 3( )

M1x = M2 y − LαSαR2 x − Lα 1−Cα( )R2z( )Sθ1 + M2 xCα −M2zSα − SαR2 y( )−Wα 2( )Cα 3( )

M1y = M2zSα − LαSαR2 y( )Sθ1 + Lα 1−Cα( )R2z − LαSαR2 x −M2 xLαCα −M2 y( )1 −Wα 3( ) Cα 1( )2+Cα 3( )

2

M2z = −Lα 1−Cα( )R2 y +M2 xSα +M2zCα −Wα 2( )Cα 1( )

wα = [ Wα(1) Wα (2) Wα (3) ] is the weight force of CS1 directed by the gravity. cα = [ Cα(1) Cα(2) Cα(3) ] is the mass center of CS1. Appendix C The Jacobian matrix Jm is written as follows:

Jm =

Cθ1SαCβ − Sθ1Sθ2Sβ( ) R− d( ) Sβ Cθ1Cθ2 − Sθ2Cα( ) R− d( ) −Sθ1SαCβ − Sβ Cθ1Sθ2 +Cθ2Cα( )0 −Sθ2SαSβ R− d( ) CαCβ −Cθ2SαSβ

Sβ Cθ1Sθ2 + Sθ1Cθ2Cα( )− Sθ1SαCβ( ) R− d( ) Sβ Sθ1Cθ2 +Cθ1Sθ2Cα( ) R− d( ) Sβ Cθ1Cθ2Cα − Sθ1Sθ2( )−Cθ1SαCβ

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