Feasibility Study of a Fully Compliant Statically ...

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Proceedings of DETC’04 of DETC 2004 ASME 2004 Design EngineeringProceedings Technical Conferences and ASME Design Engineering Technical Conferences Computers and Information in Engineering Conference and Computers and Information in Engineering Conference September 28-October 2, 2004, Salt Lake City, Utah, USA Salt Lake City, Utah, September 28 – October 2, 2004

DETC2004-57242 DETC2004-57242 FEASIBILITY STUDY OF A FULLY COMPLIANT STATICALLY BALANCED LAPAROSCOPIC GRASPER 1

Aäron Stapel , Just L. Herder

2

Delft University of Technology, Department of Design, Engineering and Production 1: Section of Production Technology and Industrial Organisation (PTO) 2: Section of HuMan-Machine Systems (MMS) Mekelweg 2, 2628 CD Delft, The Netherlands V: +31-15-2784713, F: +31-15-2784717, E: [email protected]

ABSTRACT Compliant mechanisms have many advantages over their rigidbody counterparts. One disadvantage however is the fact that motion of the mechanism is associated with elastic energy storage in the compliant parts. This is a problem especially in cases where accurate force transmission is of primary concern, such as in medical graspers. A solution to this problem is to statically balance the elastic forces by the addition of a spring force compensation mechanism, such that the effect of the compliance is neutralized. The complete resulting mechanisms resulting from this concept are called statically balanced compliant mechanisms (SBCMs). This paper presents a feasibility study into the design of a grasper for medical purposes and demonstrates that the concept is possible and practically viable. It is shown that the compliant gripper of a laparoscopic forceps can be statically balanced with a singlepiece compliant compensation mechanism, with a balancing error of only 0.03N while dimensions are such that the compensation part of the mechanism can be stored inside the hand grip of the instrument. Keywords: Compliant mechanism, static balancing, minimally invasive surgery, laparoscopic instrument.

play and highly reduced force transmission due to friction [1]. This reduces the perception of forces by the surgeons, and therefore their sense of touch [2]. Alternative designs have been proposed, based on rolling joints, with significantly improved mechanical efficiency, however at the cost of a fairly complex mechanism (Fig. 1b; [3,4]). As an alternative for rolling link mechanisms, compliant mechanisms are excellent candidates. Advantages of compliant mechanisms are well recognized and include the absence of friction and backlash, and the reduction of parts. A disadvantage though, in certain applications, is the energy storage in the flexible members distorting the input-output relationship (e.g. [5,6]). Due to the energy storage in the elastic members of the mechanism, energy is not conserved between input and output ports, which is a serious drawback in some cases [6]. Particularly in manually operated instruments, such as surgical forceps, the compliance requires operating effort even when no external work is done. More importantly, the force feedback quality is reduced because the force transmission from the gripper to the hand grip is disturbed by the elastic forces. In principle, because the elastic forces are conservative, the undesirable effect of compliance can be eliminated by static balancing [7]. Perfect compensation would cancel the need for operating force and energy during free movement of the gripper: the elastic influence of the compliant members on the operation op the gripper would be fully eliminated, as if there were no compliance present. Compliant mechanisms in which

INTRODUCTION Conventional endoscopic forceps typically are based on rigid body linkages connected by pin-in-hole joints (Fig. 1a) that cannot be lubricated adequately as they need to operate inside the human body. Consequently, they suffer from considerable

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Copyright © 2004 by ASME

(a)

(a)

(b) Figure 1 Gripper mechanisms: (a) currently used linkages with pinin-hole joints, (b) experimental prototypes with rolling contact joints [3,4]. (b) Figure 2 Statically balanced compliant gripper according to Type I: (a) overview of the mechanism, (b) close-up of the gripper [5].

the elastic forces are thus eliminated were called statically balanced compliant mechanisms (SBCMs; [5]). Several categories of SBCMs can be distinguished [5]. A first category consists of a compliant part and a conventional compensation mechanism, consisting of links, joints and springs. In a second category, the compensation mechanism is also fashioned as a compliant mechanism. This category comprises mechanisms that are fully compliant, yet incorporate one or more separate springs as compensation energy container. In a third category, the compensation energy is stored in a compliant part of the mechanism, rather than in a separate spring element. A fourth category of statically balanced compliant mechanisms would be adaptive, accommodating for the phenomenon that under loaded conditions, compliant mechanisms generally behave different from the unloaded situation. Previously, a statically balanced compliant grasper according to type I has been designed, consisting of a compliant gripper part and a separate compensation mechanism (Fig. 2; [5]). The neutral position of the grasper is half open. The grasper can be opened further by pushing the central part, which is connected by two compliant sections to the jaws, whereas pulling the central part will close the grasper. The forces introduced by the bending of the compliant parts were eliminated by compensation mechanism, consisting of a rolling link mechanism and two helical springs. Although the principle works well and several physiological quantities of tissue can be perceived, the design is rather complex and difficult to clean.

In order to arrive at a less complex design, a conceptional design according to type III was considered, because then the complete mechanism could be made out of essentially one piece. The objective of this paper is to present a feasibility study of a laparoscopic grasper according to SBCM Type III, the main question being whether or not such a design would fit inside the slender contours of these instruments. CONCEPTIONAL DESIGN The design presented in this paper is based on an existing gripper mechanism [5]. The neutral position of this gripper is half opened. The stiffness is known to be almost linear and equal to a nominal value of 50 N/mm. Based on these data, it was aimed to design a compensation mechanism according to Type III, i.e. a compliant compensation mechanism, directly connected to the gripper. Design criteria The compliant compensation mechanism should have the following properties. First, it should have a negative stiffness being the opposite of the gripper stiffness. The force needed for closing the gripper without force compensation amounts to 15N. As a reference for the balancing quality, a maximum balancing error of 0.07N was assumed, being the performance value of the existing balanced design [5]. The stiffness should be adjustable after production to make up for production errors.

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The stiffness of the grasper varies, due to fabrication tolerances, by about 20%. The nominal stiffness, without fabrication errors, of the grasper is 50N/mm. According to the dimensions of the balancer, the nominal stiffness of the balancer will vary as well. The nominal stiffness of the balancing mechanism therefore, should be equal to the negative of the grasper and be adjustable with at least 20% as well. The compensation mechanism should also have a straight-line output motion for direct connection with the gripper. The input motion of the gripper is a path of 0.3mm along a straight line. In order to be able to produce a single compliant architecture which comprises both the grasper and the balancer, the output motion of the balancer should be equal to the input motion of the grasper. Preferably the whole mechanism should be made out of one piece. It should fit in the ergonomic dimensions of the instrument. The surgeon should be able to manipulate the device freely and easily. As a maximum volume for the compensation mechanism a cylinder with a height and diameter of 40mm has been chosen.

l

O

(a)

(b)

Approach Bistable mechanisms have a negative stiffness, but in addition to the demand of a negative stiffness, the present application demands a correct force-deflection characteristic as well. From all possible mechanisms there have been selected configurations featuring a symmetric energy characteristic since the gripper has a symmetric energy characteristic as well. Considering the boundary conditions stated above, the following groups of mechanisms were found. (1) Buckling mechanism [8,9,10] and snap-through buckled beams [12]. Within the application at hand, a relatively long path is desired with a prescribed stiffness. The required dimensions for the compensation mechanisms for the prescribed stiffness would lead to unpractical dimensions of the compensation mechanism. Therefore these mechanisms were not considered further. (2) Compliant versions of bistable four-link mechanism, cam mechanisms, slider-crank or slider-rocker mechanisms, and double-slider mechanisms. The adjustability of the four-link mechanisms in the application on hand is poor in combination with a desired straight output motion. Therefore the four-link mechanism was not developed further. The desire of a gripper and a compensation mechanism out of one piece is the reason for not choosing a bistable cam mechanism. In this analysis two configurations and different possibilities for stiffness adjustment will be discussed. The dimensions of the best two configurations will be briefly dealt with. Finally, the mechanism with the better properties will be further dimensioned.

(c)

(d)

Figure 3 Conceptional design based on a slider-rocker mechanism: (a) Statically balanced rigid-body mechanism [11], (b) one slider replaced by compliant rocker, (c) proposed stiffness adjustment, (d) double embodiment to reduce friction in slider.

Slider-rocker A first conceptional design was based on statically balanced rigid-body mechanism as shown in Fig. 3a [11]. By assuming one of the springs as the gripper to be balanced and changing the other springs into a compliant segment, a bistable sliderrocker mechanism [12] was found with a straight-line output motion (Fig. 3b). The stiffness can be adjusted by changing the effective length of the leaf spring as shown in Fig. 3c. The stiffness of the leaf spring can be found with:

c=

3

F 3EI = 3 d L

(1)


By changing the effective length by ∆L, the stiffness becomes:

c=

3EI ( L + DL)3

(2)

where the factor

( L + DL)3 L3

(a)

(3)

is a measure of the influence of ∆L on the stiffness of the leaf spring. It can be seen that the larger the length of the leaf spring, the smaller the influence of ∆L. To achieve a stiffness variation of 20% with a leaf spring length of 30mm the required ∆L is 1.78mm. Because a slider would bring friction in the mechanism and with that a loss of sensation for the surgeon, two mechanisms can be used as suggested in Fig. 3d.

(b)

w Double-slider A very practical compliant bistable mechanism was found in [12] and is shown in Fig. 4. This mechanism can be used with several elements in series. The stiffness can be adjusted by changing the number of elements used in the compensation mechanism. This leads to a certain step size that is undesirable. The stiffness can also be adjusted by adjusting the stiffness per element as will be derived below. The force on a spring, changing by moving the slider as shown in Fig. 4c can be described as follows. The spring shown in Fig. 4c is without tension. The length of the tensionless spring is equal to l. When the slider connected to the spring is moved by a distance w, a force F(w) is needed. The angle before the movement w is equal to θ. The horizontal distance between the two ends of the spring is called a. In the tensionless situation as shown the movement w is equal to zero. An equal situation has been described in [13] for a link. The resulting force F(w) equals

EA æ 3 w2 1 w3 ö 2 w q + q + sin sin ç ÷ = F (w) l è 2 l 2 l2 ø

a F(w) e

(c)

l

Figure 4 Conceptional design based on a double-slider mechanism: (a) the systematic representation (b) a compliant fore bent element (c) illustration of the terms used in the stiffness adjustment discussion.

With

v=

w F ,s = sin q , f = , this equation becomes: l Cb l

3 1 u + u 2 + u3 = h 2 2

(4)

In the double-slider mechanism, instead of a link, a spring is used. The constants EA/l can therefore be changed into Cb, the stiffness of the spring consisting of the fore bend element. To get more insight in the above written equation, the equation is parameterized to the following equation:

3 1 v s 2 + v 2 s+ v3 = f 2 2

θ

(6)

where u=v/s and h=f/s3. In Fig. 5, a graph of this equation is shown. What becomes visible agrees with the bistable behavior expected from the mechanism. The stiffness is found by differentiation with respect to the displacement w:

æ w 3 w2 ö dF Cb ç sin 2 q + 3 sin q + ÷= l 2 l 2 ø dw è

(5)

4

(7)


Substituting w = -a = -lsinq yields the stiffness in the unstable equilibrium position. The stiffness for w = -a is:

3 æ ö Cb ç sin 2 q - 3sin 2 q + sin 2 q ÷ = Ccomp 2 è ø

0,25

0,15

(8)

0,05

So for one element, the compensation stiffness is

u

C comp = - C b sin q 1 2

2

(9)

-2,5

-2

-1,5

The stiffness for the total compensation mechanism therefore will be:

C comp = - 12 nC b sin 2 q = - 12 nC b

a

2

l

2

h

-1

-0,5

-0,05

0

0,5

-0,15

æe ö = - 12 nC b ç 2 - 1÷ çl ÷ è ø (10) 2

-0,25

where n is the number of elements and where the equality a2 = e2 - l2 is used. Changing the value for e has a stronger effect than changing ∆L in the slider-rocker mechanism. The slider-rocker mechanism is better adjustable in comparison with the double slider, due to the different influence of changing a characteristic length, e or ∆L, the slider-rocker mechanism will be developed further.

Figure 5 Parameterized force-deflection graph where h is the dimensionless force and u the dimensionless displacement.

spring, as can be derived from the bending an stiffness equations. The deflection of a fixed beam can be derived with Eq. 11. The stiffness of one leaf spring can be derived with Eq. 12, and when using perfect balancing amounts half the gripper stiffness, 25N/mm. The maximal stress, σmax, due to deflection of the leaf spring can be derived with Eq. 13. The maximal moment on the fixed beam is the deflection force multiplied with the length of the beam. With Eq. 13 and z being half the thickness, with a deflection of 10mm, a σmax equal to 600MPa, the maximal thickness becomes 0,018mm. This leads, with Eq. 12 to a width for the leaf spring of 2315m.

DIMENSIONAL DESIGN The compensation mechanism consists out of several parts: two leaf springs, two cranks, a slider and the joints connecting these parts. The dimensions of these parts will be derived hereafter. Leaf spring The slider-rocker mechanism in combination with the gripper can be perfectly statically balanced according to Fig. 3a, where the horizontal spring represents the gripper. This configuration will be statically balanced if the end points of the springs will be located in point O when they are tensionless (zero free length). Furthermore, the two springs should have the same stiffness [11]. When the grasper and the compliant element in the slider-rocker mechanism are considered as springs the situation of Fig. 6 emerges. When using leaf springs as compliant members, the maximal deformation of the spring will equal the length of the crank. When the possible fabrication methods are taken into account though, it is preferable to maintain a value for the length of the crank of around 10mm. This way the compliant joints, that will be discussed later on, can be fabricated. This value however makes it difficult to design the leaf spring according to the first conditions of perfect balancing. A deflection of 10mm, with the proper stiffness, will lead to unworkable dimensions the height and the width of the leaf

F=

3EI d L3

(11)

C=

Ebh3 = 25 N / mm 4 L3

(12)

s max =

Mz I

(13)

This leaves the option of non-perfect balancing. In an iterative, computer assisted process using linear deflection equations, practical values for the dimensions have been found. A leaf spring length of 30mm, a width of 5.8mm, a thickness of 3.6mm and an eccentricity (e as shown in Fig. 6) of 9.5mm the gripper is approximately balanced. These dimensions will be verified later on with a PRB-Model analysis.

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θ

e

F

Figure 6

Influence of crank length. Figure 7

Compliant joints Other compliant parts of the mechanism are the compliant joints. Literature offers a solution in the use of tensile pivots (Fig. 7, [14]). These pivots were selected to avoid buckling of flexures. The dimensions can be derived according to the maximal allowable stresses combined with the minimal fabrication dimensions like the minimal thickness. The maximal possible rotation of a compliant joint can be investigated by comparing it with a fixed beam [15]. The maximum rotation of a fixed beam equals:

q =

ML EI

Crank and slider The total mechanism consists out of two leaf springs, two cranks, one slider and the tensile pivots between these parts. The dimensions of the cranks and the slider was chosen in such a way that the stiffness of these elements far exceeds the stiffness of the leaf springs. The length of the crank is chosen to be 10mm, out of practical considerations. In order to produce the mechanism out of one part, the width is preferably the same as the width of the leaf springs. The thickness is free to choose. The same holds for the slider but in case of the slider the length is free as well. Before choosing the further dimensions the behavior of the compensation mechanism will be verified using a Pseudo Rigid Body analysis.

(14)

The maximal bending moment in the beam equals

M =

2s max I h

(15)

PRB-MODEL In a PRB-Model the compliant elements are modeled as rigid elements connected with ideal joints with torsion springs as described in [12]. One half of the compensation mechanism is shown in Fig. 8a. The compliant members in this case are the leaf spring and the two tensile pivots. The leaf spring is modeled as a fixed beam with a length of 0,15L and a rigid element with a length of 0,85L. The stiffness K of the torsion spring is equal to [12]:

where h is the thickness of the beam and I the moment of inertia. Combining these two equations gives

q =

2s max Lt Eh

Tensile pivot [14] in three positions.

(16)

where Lt is the length of the tensile pivot. The total allowable stress is 600MPa. In the application at hand though, there are two stresses, due to the tensile and bending loads, respectively. The stress due to the tensile load is equal to 290MPa. Therefore the maximum bending stress of 310MPa is found. The maximum angle can be derived with the crank length of 10mm and the path of 0.3mm of the mechanism. This leads to a maximum angle θ of 0.03rad. Consequently, with a thickness of 0.15mm and a width of the tensile pivot equal to the width of the leaf springs (5.8mm), to a minimal length for Lt of 1.45mm.

K = 2.25

EI l

(17)

The length of the rigid elements in the tensile pivots is equal to half of the tensile pivot length. The stiffness of the torsion spring is equal to K=EI/Lt

(18)

To calculate the amount of stored energy the following equation is used:

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0,4 Residual force[N] 0,3

(a)

0,2 0,1 0 -0,3

-0,2

-0,1 -0,1 0

Displacement [mm] 0,1

0,2

0,3

-0,2 -0,3

(b)

-0,4

Figure 9

Residual force derived from PRB model.

0,03 Residual force[N]

(c)

0,02 0,01

Displacement [mm]

0 -0,3

Figure 8 Development of PRB model: (a) one half of the compensation mechanism, (b) compliant members replaced by rigidbody counterparts, (c) simplified representation.

Vp =

1 2

(K

2 2 2 lsq ls + K t1q t1 + K t 2q t 2

)

-0,2

-0,1 0 -0,01

0,1

0,2

0,3

-0,02

Figure 10 Residual force with spring width of 5.56mm.

(19)

where Vp is the total potential energy stored in the compensation mechanism, Kls is the stiffness of the PRB spring representing the leaf spring and Kti the stiffness of the i th tensile pivot. It is noted that the above equation is valid for one half of the compensation mechanism.

Potential energy [J] 62,342 62,341

RESULTS With the dimensions determined above, a residual force (unbalance force) was found exceeding the specified maximum value of 0,07N. The resulting residual force-deflection characteristic is shown in Fig. 9. By varying the width b of the leaf spring, an improved balanced system was found. The width used to obtain this result was 5.56mm. For this case the forcedeflection characteristic is shown Fig. 10. The total potential energy in the system is now almost constant, as is illustrated in Fig. 11. The non-symmetric behavior is probably due to the non-symmetric load around the unstable equilibrium position of the compensation mechanism. A CAD drawing of the compensation mechanism is shown in Fig. 12.

62,34 62,339 62,338 62,337

-0,3

-0,2

62,336 -0,1 0

0,1

0,2 0,3 Displacement [mm]

Figure 11 Potential energy of combined system. Note that the vertical axis has been cropped for clarity.

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CONCLUSION Compliant mechanisms can be statically balanced if their inherent stiffness or energy storage between input and output is undesired. Different kinds of these statically balanced compliant mechanisms (SBCMs) can be distinguished, of which SBCM Type III is fully compliant. This paper presented a feasibility study of such a design for a surgical instrument. If was found that a compliant gripper can be statically balanced by a fully compliant balancer, in such a way that the complete mechanism essentially consists of a single piece and has no preferred position as in rigid-link mechanisms. This is particularly relevant for the force feedback quality in surgical instruments like the one discussed. It was found possible to find dimensions that are expected to fit inside the contour of the instrument handgrip. Future investigations will be directed towards optimization of the design and implementation of a stiffness adjustment mechanism. Finally it is intended to manufacture a prototype to evaluate the instrument in practice.

Stiffness adjustment

Compensation Rod between Gripper mechanism mechanism and gripper

(a)

Stiffness adjustment

Compensation mechanism

Rod between mechanism and gripper

ACKNOWLEDGMENTS Special thanks are extended to the research group of Dr. Howell, who supplied me with of a lot of information about compliant bistable mechanisms. The first author wishes to thank the second author for initiating and guiding this project. NOMENCLATURE a distance A cross-sectional area b width c stiffness e distance E elasticity modulus (200Gpa for orthopedic steel) F force f dimensionless force h thickness I moment of inertia U unit vector L length of a leaf spring n number of elements s sine u dimensionless factor v dimensionless deflection w deflection z half the thickness h q angle maximal tension (600Mpa for orthopedic steel) σmax length of the tensile pivot Lt δ deflection of the leaf spring

(b)

(c) Figure 12 Impression of the resulting compliant static balancer: (a) diagram showing connection of gripper with compensation mechanism, (b) impression of fit in hand grip, (c) CAD drawing showing dimensions.

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[8] Dijksman, J.F., 1979, “A Study of Some Aspects of the Mechanical Behaviour of Cross-Springs Pivots and Plate Spring Mechanisms with Negative Stiffness”, Internal report, Department of Mechanical Engineering, Delft University of Technology. [9] Van Eijk J (1985) On the design of plate-spring mechanisms, Ph.D. Thesis Delft University of Technology. [10] Eijk, J., van, Dijksman, J.F., 1976, “Plate spring mechanisms with constant negative stiffness” Internal report, Department of Mechanical Engineering, Delft University of Technology. [11] Herder JL (2001) Energy-free Systems; Theory, conception and design of statically balanced spring mechanisms, Ph.D. Thesis Delft University of Technology, ISBN 90-37001920. [12] Howell LL (2001) Compliant Mechanisms, John Wiley & Sons, Inc., New York, ISBN 0-471-38478-X. [13] Keulen, van, F., 1998, “Stiffness and Strength 3B”, Internal report, Department of Mechanical Engineering, Delft University of Technology. [14] Masters, N.D., Howell, L.L., 2002, “A Three Degree of Freedom Pseudo-Rigid-Body Model for The Design of a Fully Compliant Bistable Micromechanism”, Proceedings ASME Design Engineering Technical Conferences, Sept 29 - Oct 2, Montreal, Quebec, Canada, paper number DETC2002/MECH-34202 [15] Pistecky, P.V., 1998, “Designing information transformers”, Internal report, Department of Mechanical Engineering, Delft University of Technology.

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