Thesis_Jos_Lassooij.pdf

Master of Science Thesis

Design of a Bi-Stable Compliant Grasper Using Bi-Stable Beams and Disc springs

J. Lassooij August 22, 2012

Title: Design of a Bi-Stable Compliant Grasper Using Bi-Stable Beams and Disc Springs

Author: J. Lassooij

Student nr: 1263900

Date: August 22, 2012

Institute: Delft University of Technology

Faculty of Mechanical, Maritime and Material Science Engineering

Department Bio Mechanical Engineering

Board of examiners:

Prof. Dr. F.C.T. van der Helm, Biomechanical Engineering, 3mE, TU Delft

Dr. Ir. J.L.Herder, Biomechanical Engineering, 3mE, TU Delft

Dr. Ir. N. Tolou, Biomechanical Engineering, 3mE, TU Delft

Content

Preface ..................................................................................................................................................... 2

Acknowledgements ................................................................................................................................. 2

General Introduction ............................................................................................................................... 4

Paper 1: ................................................................................................................................................... 8

Design of a Bi-Stable Compliant Grasper with Bi-Stable Beams ......................................................... 9

Abstract ........................................................................................................................................... 9

Introduction ..................................................................................................................................... 9

Method .......................................................................................................................................... 10

Conceptual Design ......................................................................................................................... 10

Dimensional Design ....................................................................................................................... 16

Evaluation ...................................................................................................................................... 18

Discussion ...................................................................................................................................... 22

Conclusion ..................................................................................................................................... 22

References ..................................................................................................................................... 23

Paper 2: ................................................................................................................................................. 24

Feasibility Study: Using Disc Springs to Create Bi-Stable Compliant Grasper ................................... 25

Abstract ......................................................................................................................................... 25

Introduction ................................................................................................................................... 25

Method .......................................................................................................................................... 27

Results ........................................................................................................................................... 30

Discussion ...................................................................................................................................... 36

Conclusion ..................................................................................................................................... 37

References ..................................................................................................................................... 38

General Discussion ................................................................................................................................ 39

General Conclusion ............................................................................................................................... 40

References ............................................................................................................................................. 41

Appendices ............................................................................................................................................ 42

Appendix A: ANSYS modeling and sensitivity study .......................................................................... 42

Appendix B: ANSYS coding disc spring .............................................................................................. 56

Appendix C: Results 1st bi-stable element ........................................................................................ 60

Appendix D: Standard disc springs in holder ..................................................................................... 66

Appendix E: Alternative design disc spring ....................................................................................... 70

2

Preface The research I have performed on the design of a bi-stable compliant grasper is presented in this

thesis. During the course of my graduation, I investigated two different bi-stable elements that could

be used in the design of a bi-stable compliant grasper. This has divided the research (and thesis) in

three parts.

In the first part I used a bi-stable beam to create the negative stiffness needed to make the

compliant grasper bi-stable. In the second part I investigated the use of disc springs to make the

compliant grasper bi-stable. Finally, in the third part both designs are compared to each other.

In the appendices A and B steps taken in the research are explained in more detail. In Appendices C-E

some other experiments are presented. In Appendix C the results of a different bi-stable beam

combined with a compliant grasper are given. In Appendix D experiments are given in which the disc

spring is constrained in all translational degrees of freedom instead of only the z-translation. In

Appendix E a different fabrication method for disc springs is described. This method is using a

preload displacement to obtain the desired shape of the disc spring, instead using of plastic

deformation.

Acknowledgements During the (long) process of my graduation I have had help from many fellow office members: Gerard

Dunning, Pieter Pluimers, Toon Lamers, Lodewijk Kluit, Sergio de Paula Pellegrini and Juan Gallego. I

would like to thank them for all answering all my questions, their new ideas, their different view on

problems and the many coffee breaks we had. In particular I would like to thank my daily supervisors

Doctor (!) Nima Tolou and Just Herder for their patience, guidance and critical reviews during my

graduation project. Furthermore I would like to thank Prof. Frans van der Helm for reviewing my

concept thesis. Also thanks to Hans Drop, Gerwin Smit, Aad Beeloo and Rien Waaijer, for producing

the disc springs, letting me use the measurement setup, and helping me countless times in the

workshop.

4

General Introduction In the past decades the field of surgery has undergone much technical advancement. One of the

largest changes is the introduction of Minimally Invasive Surgery (MIS). This is a type of surgery in

which surgeons use small incisions and long slender instruments to operate on people, instead of

making one large incision and using their hands. Due to the smaller incisions the recovery time and

chance of infection are reduced.

Despite the benefits, this type of surgery also has some disadvantages. With MIS, the surgeon has

lost direct contact with the patients they are operating on. Instead of their hands, they now have to

use long slender instruments. As a result, all tactile information is lost. Also, the number of degrees

of freedom inside the patient is reduced (Gallagher, McClure et al. 1998). The surgeons also lose

direct (3D) vision on the operating area. Instead, they now have to look at a (2D) tv-screen

(Breedveld, Stassen et al. 1999). Finally, the surgeon often has to stand in uncomfortable positions

for longer periods of time (Van Veelen and Meijer 1999). These disadvantages make the surgery

more complex, and increase the chance for errors (Joice, Hanna et al. 1998).

In order to help the surgeon Intuitive Surgical Inc. made the DaVinci Surgical System. This is a master-

slave system that allows the surgeon to operate on a patient through multiple robotic arms. The

instruments of the DaVinci have more degrees of freedom (DoF) than the conventional MIS

instruments due to an extra wrist joint near at the tip. The surgeon moves the handles in the master

console, and this movement is copied electronically to the slave arms. The system uses a 3D camera

that allows the surgeon to have a better view of the operating area. Disadvantages of the system are

that the DaVinci lacks force feedback, is very expensive to buy and maintain.

In order to overcome the problems of the conventional MIS instruments and the DaVinci system, the

Minimally Invasive Manipulator (MIM) was designed (Jaspers, Shehata et al. 2004). The MIM fills the

gap between the conventional (simpler) MIS instruments on the one hand, and the very complex

DaVinci system on the other hand. The MIM is a purely mechanical system. It uses cables and pulleys

to copy the movement of the hands of the surgeon to the tip of the instruments inside the patients.

Similar to the DaVinci system, the MIM also has the extra wrist joint and can be used with 3D vision.

Unlike the DaVinci system the MIM has some force feedback due to the mechanical coupling

between the surgeon and the instrument.

In Figure 1 the MIM is shown. In total the MIM copies seven DoF from the handle of the surgeon to

the tip of the instrument: six DoF for moving the tip and one to actuate the grasper. The first two DoF

are copied using a configuration of parallelograms. The other five DoF are copied using many cables

and pulleys.

The force feedback inside the MIM is influenced by the friction inside the system. This friction is

affected by the reaction forces on the cables and pulleys (De Vries, 2012). When the surgeon is

moving the MIM the reaction forces and friction are low. This changes when the surgeon starts to

grasp and hold objects like a common suturing needle. Due to the required force to hold the objects,

the reaction forces on the cables and pulleys of the MIM increase. This can be illustrated by an

example.

5

(a)

(b)

Figure 1: Schematic view of the Minimally Invasive Manipulator (MIM). The MIM copies seven DoF from the handle to the tip. The seven DoF are: rotations round insertion (1&2), rotation of instrument around its axis (3), translation of instrument along its axis (4), wrist rotations (5&6) and opening/closing of instrument (7).

In Figure 2 a close up of the tip of the instrument is shown. The tip is opened via a torsion spring and

is closed by pulling the two cables. The friction between the object and tip and in the pulley and

cables is not shown. In the figure, the tip is applying a pinch force to hold an object. This force is

generated by the cable force. The cable force (Fcable) multiplied by its arm (rcable) is equal to the pinch

force (Fpinch) multiplied by its arm (rpinch). In the current design for the tip the moment arm of the

pinch force is almost six times larger than the moment arm of the cable force. As a result, the

required cable force is six times larger than the desired pinch force. The DaVinci is able to generate a

pinch force of around 17 N (De Vries, 2012). For such a pinch force, the resulting cable and pulley

forces in the MIM are 85 N and 170 N respectively

The results of these high cable and pulley forces are: (1) high wear, (2) cables break as a result of too

much cable force and (3) a significant increase in friction in the MIM. In experiments, users

experienced that moving the MIM became harder when an object was held. As a result, their hands

became fatigued (De Vries, 2012).

6

The increased friction distorts the force feedback in all degrees of freedom of the MIM. When

holding an object, the surgeon does not need force feedback in the grasping motion, but for the

other degrees of freedom force feedback is desired. For example, when holding a needle, the

surgeon would like to feel if he is puncturing tissue or bone.

The goal of this report is to present two designs for a bi-stable compliant grasper. The grasper will

have two stable positions: open and closed. In those positions there is no need for an actuation force

from the surgeon. When the grasper is closed, it will produce enough pinch force to hold an object.

This report is organized as follows. First, the two designs are described in two papers. The first design

is a compliant grasper combined with bi-stable beam. Both the grasper and bi-stable element will be

a planer design. A large scale prototype is presented. The second design uses a disc spring as bi-

stable element. This design is a 3D design, and is on the scale of the MIM. After these papers there

will be a general discussion. In this discussion the two designs will be compared to each other to see

which design should be used to design a bi-stable compliant grasper for the MIM. Finally some

conclusions are given about the entire project.

Figure 2: Actuation of the grasper. The grasper consists out of two jaws (Jaw 1 & 2). Each jaw is actuated by a cable (Cable 1 & 2). Each cable runs to the jaw over a separate pulley. The grasper is closed by pulling both cables. The grasper is opened by the torsion spring. Steering is done by pulling one of the two cables. The large reaction forces are a result of the large moment arm of the pinch force with respect to that of the cable force. The friction forces between the grasper and object, and the cables and pulleys are not shown.

8

Paper 1: Design of a Bi-Stable Compliant Grasper with Bi-Stable Beams

9

Design of a Bi-Stable Compliant Grasper with Bi-Stable Beams

Abstract The Minimally Invasive Manipulator (MIM) copies the movements of the hands of the surgeon to the

tip of the instrument inside the operating area. The movement is mechanically copied using cables

and pulleys. When the surgeon grasps and holds an object, the friction inside the MIM increases and

distorts the force feedback in the remaining degrees of freedom. The increased friction is the result

of the reaction forces on the cables and pulleys due to the required pinch force to hold the object.

The goal of this paper is to present a design of a bi-stable grasper. This grasper will have two stable

positions, open and closed. In these positions no actuation force from the surgeon is needed. Due to

the lack of actuation force, the grasper can hold an object, without high friction forces as a result.

The concept of a compliant grasper combined with bi-stable beams is presented. A large scale

prototype was designed fabricated. The prototype showed bi-stable behavior over a range of 8 mm.

Using finite element modeling, dimensions for a bi-stable beam suitable for the MIM are found.

Introduction In the past decades the field of surgery has undergone much technical advancement. One of the

largest changes is the introduction of Minimally Invasive Surgery (MIS). This is a type of surgery in

which surgeons use small incisions and long slender instruments to operate on people, instead of

making one large incision and using their hands. Due to the smaller incisions the recovery time and

chance of infection are reduced.

Despite the benefits, this type of surgery also has some disadvantages: (1) All tactile feedback is lost,

(2) the number of degrees of freedom inside the patient is reduced (Gallagher, McClure et al. 1998)

and (3) the surgeon often has to stand in uncomfortable positions for longer periods of time (Van

Veelen and Meijer 1999). These disadvantages make the surgery more complex, and increase the

chance for errors (Joice, Hanna et al. 1998).

The Minimally Invasive Manipulator (MIM) is a device that is designed for use in the surgical field

(Jaspers, Shehata et al. 2004). The MIM has three major advantages compared to the standard

minimally invasive surgery (MIS): (1) the MIM allows the surgeon to have more freedom of

movement inside the operating area. (2) It allows the surgeon to sit comfortably during the

procedure. (3) The MIM uses cables and pulleys to mechanically copy the movement of the hands of

the surgeon to the tip of the instrument. Due to the mechanical coupling there is some force

feedback.

The force feedback inside the MIM is influenced by the friction inside the system. This friction is

affected by the reaction forces on the cables and pulleys (De Vries 2012). When the surgeon is

moving the MIM the reaction forces and friction are low. This changes when the surgeon starts to

grasp and hold objects like a common suturing needle. Due to the required force to hold the objects,

the reaction forces on the cables and pulleys of the MIM increase.


much cable force and (3) a significant increase in friction in the MIM. In experiments, users

experienced that moving the MIM became harder when an object was held. As a result, their hands

became fatigued (De Vries 2012). Also, due to the increased friction, the force feedback in all DoF is

distorted. When holding an object, the surgeon does not need force feedback in the grasping motion,

10

but for the other degrees of freedom force feedback is desired. For example, when holding a needle,

the surgeon would like to feel if he is puncturing tissue or bone. These three results increase the risk

of complications and errors during surgery.

This paper presents the first design of a bi-stable compliant grasper. This grasper will have two stable

positions, open and closed. In these positions no actuation force from the surgeon is needed. When

closed, the grasper will produce enough pinch force to hold an object. Due to the lack of actuation

force from the surgeon when the grasper is holding an object, the friction remains low. As a result,

the surgeon has some force feedback when moving the instrument.

The rest of the paper is organized as follows. First the design approach and design requirements will

be given in the method. Next, the concept will be presented followed by the dimensional design.

Then the experimental validation will be presented, followed by a discussion of the results and

conclusions.

Method In order to remove all reaction forces, the bi-stable part of the grasper needs to be as close to the

grasper as possible. The instruments of the MIM are designed to fit through an 8 mm trocar. To the

authors knowledge, a combination of a bi-stable element and a grasper on this scale does not exist.

This small scale introduces difficulties in assembly and evaluation of the design. For these reasons,

the prototype of the bi-stable grasper will be on a larger scale than the scale of the MIM. The

prototype will serve as a proof of principle for the design concept presented in this paper.

The prototype should have two stable positions (i.e. positions in which no actuation force from the

surgeon is required). These positions should be when the prototype is open and closed. In the closed

position, the bi-stable element should provide an actuation force that is actively closing the grasper.

This force results into a pinch force.

First, the concept is presented. Then the dimensions of this concept are determined with the help of

finite element modeling (FEM). The finite element model used for the bi-stable element has been

validated by (Dunning, Tolou et al. 2012). Dimensions for two scales are determined. The first

dimensions are for the bi-stable beam on the required scale for the MIM. The behavior of the bi-

stable beam is combined with the behavior of the compliant grasper of Herder and Berg. The second

dimensions are for the large scale prototype. The prototype was validated by measuring the force-

displacement and comparing them to the FEM results.

After fabrication the dimensions of the prototype are checked to determine the fabrication error.

Using the FEM an estimate is made on the change in behavior based on these fabrication errors. This

is also compared to the measurement and FEM results.

Conceptual Design The design presented in this paper is adapted from the design of (Tolou and Herder 2009), shown in

Figure 1. It combines a compliant grasper with bi-stable elements.

Compliant grasper

Compliant mechanisms are mechanisms that obtain some or all of their movement through the

deformation of flexible members (Howell 1996). They typically have a low stiffness one direction of

motion, and high stiffness for the other directions of motion. Compliant mechanisms have no sliding

11

parts, and therefore no friction, wear, or the need for lubrication. Also, it is possible to make

compliant mechanisms out of one part (monolithic), eliminating the need for assembly and

simplifying the design. One disadvantage of compliant mechanisms is that they have (positive)

stiffness. Due the lack of friction and need for assembly, compliant mechanisms are chosen to be

used in the design.

In Figure 2, a conventional (Figure 2a) and compliant (Figure 2b) grasper are shown. In the compliant

grasper the joint is replaced by flexible members. The initial (i.e. relaxed) configuration of the

grasper is half open to reduce the stresses inside the grasper. To close the grasper a pulling force is

needed. To further open the grasper a pushing force is needed. The further the grasper opens or

closes, the more the force will increase, as shown in Figure 3a. In order to make the grasper bi-stable,

a higher negative stiffness needs to be added to the positive stiffness of the grasper.

Figure 1: Concept for a compliant grasper combined with bi-stable elements, presented in (Tolou and Herder 2009). The bi-stable elements have length l, in plane thickness t, out of plane thickness w and rise h.

(a) (b)

Figure 2: (a) Conventional grasper with 2 rigid parts connected by a joint. (b) Compliant version of figure 2a, grasper opens and closes by deformation of the flexible members.

l t h

12

(a) (b) Figure 3: (a) Force displacement characteristic of a compliant grasper when the relaxed position of the grasper is half opened. The grasper has a constant positive stiffness. (b) Force displacement characteristic of a typical bi-stable mechanism. The graph has two stable equilibriums. The behavior shows negative stiffness for -1.5

13

When the compliant grasper and bi-stable mechanism are combined, their two curves can be added

together. The new curve has the same stroke as the grasper, and shows bi-stable behavior. Normally

bi-stable elements have a zero force in their stable positions. When the grasper and bi-stable beam

are combined, the goal is to have a non-zero force. This non-zero force will then generate the pinch

force needed to hold an object. This can be achieved when the stroke and magnitude are chosen

large enough. The resulting curve is shown in Figure 4. In this situation, the surgeon only has to apply

force to switch between an open or closed grasper.

The bi-stable beams used in the concept presented by Tolou have the highest stroke and force with

respect to their volume compared to other elements used to create negative stiffness (Dunning,

Tolou et al. 2011). For this reason, they will be used as a basis for the design presented in this paper.

Before the bi-stable beams could be used in the design, some changes were made. The pin-pin joints

were removed to remove friction, and reduce the number of parts. Also, to further reduce the

complexity, the number of beams was reduced. This was done by using one bi-stable beam that

spans the full width of the grasper instead of two that each span half.

Tuning

The behavior of the combined system depends on the intersection of the characteristics of the

compliant grasper and bi-stable beam. In order to compensate for fabrication errors, assembly and

other unforeseen circumstances the intersection point needs to be tunable. This can be done by

adding nuts in the middle of the bi-stable beam as shown in Figure 5a. By rotating the nuts, the

relative distance between the bi-stable beam and the grasper is changed. This will lead to a change in

the force displacement characteristic, as shown in Figure 5b. The final concept for the bi-stable

compliant grasper (BSCG) can be seen in Figure 6.

(a) (b)

Figure 5: (a) Concept for preloading and tuning the bi-stable element using nuts. (b) By rotating the nuts, the curve of the bi-stable element shifts (blue dashed and blue dashed plusses lines). Due to this shift, the point of intersection of the grasper and bi-stable element changes. This leads to a change in behavior of the combined system (green dash dotted and green dash dotted plusses lines)

-3 -2 -1 0 1 2 3 4 5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Position [-]

Forc

e [

-]

Shift bi-stable element

Positive stiffness

Bi-Stable element

Combined system

Shifted Bi-stable element

Changed combined system

Bi-stable element

Nuts

Actuation

14

Figure 6: The final concept, the bi-stable element combined with the compliant grasper.

Symmetrical behavior

The bi-stable behavior of a bi-stable beam can be achieved in two ways. The first way is to preload an

initially straight beam from the sides, creating a curved preloaded beam. The second way is to

plastically deform a straight beam and shape it into a curved beam. The advantage of the second

method is that it does not need any preloading from the side. This means that the design would be

simpler to assemble.

The difference between these methods lies in the force-displacement behavior of the beams. Bi-

stable behavior has a so called symmetry. This is the ratio between the maximum positive and

negative forces. The symmetry is an indication for how much one of the stable points is preferred. If

both points are preferred equally, the behavior is symmetrical. This means that the maximum

positive and negative forces have the same magnitude.

A bi-stable beam made using the first method is has symmetrical force-displacement behavior, and

does not have a preference for one of the two stable positions. A bi-stable beam made using the

second method will have a preference for the first stable position, namely its position it was shaped

into.

Figure 7a illustrates the difference between the behaviors. In the figure, two bi-stable behaviors can

be seen, a symmetrical behavior (blue dashed line) and a non-symmetrical behavior (blue dashed

plusses line). Both behaviors have a stroke (2 units) and the same difference between maximum

positive and negative force (4 units). The difference in symmetry can be seen by the ratio between

the maximum positive and negative forces. This ratio is 1 for the blue dashed line, and 3 for the blue

dotted line.

Both bi-stable behaviors can be used to make a compliant grasper (CG) bi-stable. The symmetrical bi-

stable behavior is combined with a compliant grasper with symmetrical behavior (i.e. the relaxed

position of the grasper is halfway its stroke). The non-symmetrical bi-stable behavior is combined

with a compliant grasper that has a non-symmetrical behavior (i.e. the relaxed position of the

grasper is not in the middle of the stroke). This is illustrated in Figure 7b. As can be seen, both

grasper are made bi-stable, and have the same stroke. The symmetrical compliant grasper has a

symmetrical bi-stable behavior, and the non-symmetrical grasper has a non-symmetrical bi-stable

behavior.

Bi-stable beam Tuning nut

Flexible element Grasper tip

15

In theory, both bi-stable behaviors can be used to design a bi-stable compliant grasper. Which

system is going to be used will depend on the fabrication method and required dimensions of the bi-

stable beam. This will be discussed in the next section.

(a)

(b)

Figure 7: (a) Symmetry in bi-stable behavior. A symmetrical behavior (blue dashed line) and a non-symmetrical behavior (blue dashed plusses line). Both behaviors have a stroke (2 units) and the same difference between maximum positive and negative force (4 units). The difference in symmetry can be seen by the ratio between the maximum positive and negative forces. This ratio is 1 for the blue dashed line, and 3 for the blue dashed plusses line. (b) Influence of symmetry on behavior of combined system. The non-symmetrical bi-stable beam (blue dotted plusses line) is combined with the non-symmetrical compliant grasper (CG) (black solid plusses line). The combined system (green dash-dot plusses line) is also non-symmetrical.

-4 -3 -2 -1 0 1 2 3 4-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Displacement [-]

Forc

e [

-]

Symmetry in bi-stable behavior

Symmetrical behavior

Non-symmetrical behavior

-3 -2 -1 0 1 2 3 4 5-3

-2

-1

0

1

2

3

Displacement [-]

Forc

e [

-]

Non-Symmetrical and Symmetrical Bi-Stable Compliant Grasper

Symmetrical CG

Non-Symmetrical CG

Symmetrical BS-beam

Non-symmetrical BS-beam

Symmetrical BSCG

Non-symmetrical BSCG

16

(a)

(b)

Figure 8: (a) Expected behavior of bi-stable beam on MIM scale (b) Expected behavior of prototype based on ANSYS modeling. With the behavior of the Compliant Grasper (black solid line, starred), Bi-Stable Beam (blue dashed line, starred) and Bi-Stable Compliant Grasper (BSCG) (green dash-dotted line, starred)

Dimensional Design In order to provide a basis for the dimensions of the concept, a finite element modeling (FEM) was

performed using a commercially FEM package, ANSYSTM 11.0, and the forces-displacements

behaviors were analyzed [ANSYS Inc. Version 11.0 Manual]. Because of large deflections, a non-linear

-0.2 0 0.2 0.4 0.6 0.8 1-40

-30

-20

-10

0

10

20

30

40

Displacement [mm]

Forc

e [

N]

Bi-stable beam on MIM scale

Compliant Grasper

Bi-Stable Beam

BSCG

-4 -3 -2 -1 0 1 2 3 4-80

-60

-40

-20

0

20

40

60

80

Displacement [mm]

Forc

e [

N]

FEM Results for Bi-Stable Compliant Grasper

Compliant Grasper

Bi-Stable Beam

BSCG

17

static analysis has been performed. PLANE82 and BEAM3 elements were used to mesh the grasper

and the bi-stable element respectively. The first type is a 2-D eight nodes element, with two

translational degrees of freedom per node. It provides more accurate results for mixed

(quadrilateral-triangular) automatic meshes and can tolerate irregular shapes without as much loss

of accuracy as well. The uni-axial BEAM3 element gives the shortest computation time while the

actual out of plane properties can also be provided using the real constant capability of ANSYSTM. The

BEAM3 element has three degrees of freedom at each node with tension, compression, and bending

capabilities. The material is assumed to follow linear elastic stress-strain behavior and to be isotropic.

The model was used to predict the dimensions of the bi-stable beam and compliant grasper for the

prototype and the dimensions of the bi-stable beam for the small scale compliant grasper. The

dimensions were made such that the maximum stresses inside the materials were not exceeded.

More details about this model and simulation can be found in Appendix A.

Small scale

The dimensions of the bi-stable beam were chosen such that the bi-stable element would fit inside a

tube with an inner diameter of 7 mm (i.e. the same size as the current MIM instruments). The bi-

stable beam for the small scale compliant grasper has an effective length of 5.5 mm, an out of plane

thickness of 4.33 mm, a material thickness of 0.17 mm and a preload displacement of 0.022 mm. The

result is a bi-stable beam with a total stroke of 0.8 mm and maximum positive and negative forces of

32 N and -32 N respectively. The part of the graph with negative stiffness has a stroke of 0.46 mm.

This is shorter than the stroke of the compliant grasper of (Herder and Berg, 2000). The behavior of

the small scale bi-stable beam, compliant grasper from Herder and Berg and their combined behavior

can be seen in Figure 8a.

Large scale

The large scale bi-stable beam has a total length of the bi-stable beam is 45.50 mm. Due to the

tuning screws, and clamping, the effective length is lower: 30 mm. The preload displacement is 0.58

mm. The in plane thickness is chosen to be 0.15 mm. The out of plane thickness is 20 mm. The length

of the flexible elements of the grasper is 76 mm. The in plane thickness is 1.5 mm and the out of

plane thickness is 20 mm. The expected behavior of the large scale bi-stable beam, compliant grasper

and their combined behavior are shown in Figure 8b

Prototype

The bi-stable beam was cut out of a high quality stainless steel (ST 301) sheet. Material of this

thickness was easily found, and it is predicted to have sufficient negative stiffness to make the

grasper bi-stable. Relaxation in this material is low, and it can cope with high stresses. This material is

hard to machine. Due to limited fabrication methods, it was not possible to create the curvature in

the bi-stable beam by plastic deformation. For this reason, the desired curvature is created by

preloading from the side.

The grasper was made using a 3D printer (Solido SD300 Pro). The printer uses PVC sheets (0.168 mm

thick, E = 3.37E9) to build up the model. This method was chosen for its short production times and

capability to produces complex designs. The PVC used will have some relaxation, but this is expected

to be low since the stresses in the grasper are expected to be relatively low. The screw thread was

also used for actuating the bi-stable grasper. The sides of the bi-stable beam were fixed to the

grasper via two other screw threads and nuts. The prototype is shown in Figure 9.

18

Evaluation Validation of the prototype was done by measuring the force-displacement behavior of the

compliant grasper and the BSCG. The prototype was mounted on a setup shown in Figure 10. The

compliant grasper was attached to the force sensor by a pulling rod. For each experiment, the

compliant grasper was first brought from its relaxed to its closed position. Then the compliant

grasper was brought to its fully opened position and back to its closed position. During this

movement, the pushing/pulling force (FETE RIS components - B3G-C3-50kg-6B, resolution = 0.06 N,

range = [0,50] kg) and displacement (Positek P101.200CL100, resolution: 0.045 mm, range = [0,200]

mm, namely LVDT) were measured. An amplifier (Scaime CPJ 25) and DaQ-mx Data Acquisition

system (NI USB6008) were used to read the data. Each measurement was repeated ten times. The

measurements were done for the compliant grasper with and without the bi-stable beam attached.

The bi-stable beam alone was not measured. The force-displacement behavior of the bi-stable beam

alone was obtained from the data of the other measurements.

The experimental results were compared to the FEM results. An estimate of the fabrication errors

was made for the compliant grasper and bi-stable beam. The influence of these errors on the

behavior is compared to the measurement and FEM results.

Figure 9: Prototype of bi-stable compliant grasper.

Figure 10: The top view of test set-up: The stiffness characteristic and force displacement of Bi-Stable Compliant Grasper was determined by measuring actuation force (Fact) and displacement (Xact) during opening and closing of the Bi-Stable Compliant Grasper.

Actuation (x)

Bi-stable beam Flexible elements Grasper tip

Nut for preloading

19

(a)

(b)

Figure 11: (a) Force-displacement results from FEM (starred lines) and measurements of the compliant grasper (black solid lines) and bi-stable beam (blue dashed lines). (b) Force-displacement results from FEM (green dash dotted starred lines) and measurements (green dash-dotted lines) of Bi-stable Compliant Grasper.

-4 -2 0 2 4 6-80

-60

-40

-20

0

20

40

60

80

Displacement [mm]

Forc

e [

N]

Compliant Grasper (FEM)

Compliant Grasper (Exp)

Bi-Stable Beam (FEM)

Bi-Stable Beam (Exp)

-4 -3 -2 -1 0 1 2 3 4-80

-60

-40

-20

0

20

40

60

80

Displacement [mm]

Forc

e [

N]

BSCG (FEM)

BSCG (Exp)

20

Results

In Figure 11a, the results of the FEM models and measurements of the grasper and bi-stable beam

are shown. For the compliant grasper, the model predicted an almost linear behavior with a stiffness

of 2.6 N/mm. The measured stiffness was lower: 0.89 N/mm. When pulling, the stroke of the grasper

is only slightly shorter than predicted: -3.5 mm (exp) vs. -3.7 mm (FEM). The measurements show a

maximum hysteresis of around 0.665 N (25.8%).

For the bi-stable beam the model predicted a much higher negative stiffness (24 N/mm) compared to

the measurement results (10.55 N/mm). The measured stroke of the bi-stable element is slightly

larger as the stroke of the FEM, namely 8 mm and 7.6 mm respectively. The measurements show a

maximum hysteresis of 4.6 N (18.3%).

The force-displacement behavior of the combined bi-stable beam and compliant grasper is shown in

Figure 11b. As can be seen, when the grasper is closed (x = -3.5) there is a positive actuation force of

20 N, actively closing the grasper.

The following parameters have an influence on the behavior of the grasper: length, out of plane

thickness, in plane thickness and Youngs modulus. For the first three parameters the errors were

determined by measuring the actual dimensions. The real Youngs modulus could not be measured

and was estimated. An overview of the parameters and their values can be found in Table 1. The

influence of each parameter individually can be found in Appendix A1.

For the bi-stable beam the length, out of plane thickness, in plane thickness and preload

displacement determine the force-displacement behavior. The length, out of plane thickness and

preload displacement were slightly changed due to fabrication errors. The in plane thickness was

measured to be exactly 0.15 mm. An overview of the parameters and their values can be found in

Table 2. The influence of each parameter individually can be found in Appendix A1.

In Figure 12, the influence of all the fabrication errors combined can be seen for both the compliant

grasper and the bi-stable beam. A large influence on the stiffness of the compliant grasper due to the

fabrication errors can be seen (0.9 N/mm 5.1 N/mm). There is only a small influence on the

behavior of the bi-stable beam. Only the stroke of the bi-stable beam is affected, there is very little

change in the negative stiffness.

Parameter

Length [mm]

Out of plane thickness [mm]

In plane thickness [mm]

Youngs mod. [GPa]

Modeled 76 20 1.5 3.37

Measured errors 74-78 19-21 1.3-1.7 2.37-4.37 Table 1: Overview of parameters that influence the force-displacement behavior of the compliant grasper and their fabrication errors.

Parameter

Length [mm]

Out of plane thickness [mm]

In plane thickness [mm]

Preload [mm]

Modeled 30 20 0.15 1.16

Measured errors 29.9-30.7 19.6-20 0.15 0.76-1.56 Table 2: Overview of parameters that influence the force-displacement behavior of the bi-stable beam and their fabrication errors.

21

(a)

(b)

Figure 12: (a) Force-displacement characteristic of the FEM (black solid starred line) and measurements (black solid line) of the compliant grasper. The influence of fabrication errors is shown (red solid circled lines). (b) Force-displacement characteristic of the finite element modeling (blue dashed line) and measurements (blue dashed starred line) of the bi-stable element. The influence of fabrication errors in the dimensions is shown (red dashed circled lines).

-4 -2 0 2 4 6

-20

-15

-10

-5

0

5

10

15

20

Displacement [mm]

Forc

e [

N]

Compliant Grasper

FEM

FEM: Fabrication Errors

Measurements

-5 0 5-100

-80

-60

-40

-20

0

20

40

60

80

100

Displacement [mm]

Forc

e [

N]

Bi-Stable Beam

FEM

Measurements

FEM: Fabrication Errors

22

Discussion The small scale bi-stable beam is capable of making the compliant grasper of Herder and Berg bi-

stable. The negative stiffness stroke of the bi-stable element is smaller than the stroke of the

compliant grasper. When closed, there is a positive actuation force that creates a pinch force.

The measurement results show that the large scale bi-stable compliant grasper was successfully

made. The grasper has two clear equilibrium points. When the grasper is closed (x = -3.5) a positive

force of 20 N is seen. This force results into a pinching force that can be used to hold objects.

In Figure 11b the displacement of the bi-stable grasper continues when the grasper is closed. This

was possible because the flexible members of the grasper would still deform when the grasper was

closed. The measurement was continued after the grasper was closed, to see when the force would

become zero.

For both the compliant grasper and bi-stable compliant grasper the FEM results do not match the

measurement results (see Figure 11a). As can be seen in Figure 12, for the compliant grasper, this

difference can partially be explained by the large influence of fabrication errors and variation in

Youngs modulus. The given Youngs modulus for PVC is 3.37 GPa. The 3D printer uses multiple layers

of PVC and glues these layers together. The Youngs modulus of the compliant grasper will be

different due to the influence of the layers and the glue.

For the bi-stable beam, the fabrication errors do not explain the difference between the FEM and

measurement results. The difference can be explained by the clamping of the bi-stable beam with

nuts. The total length of the bi-stable beam is 45.5 mm. The bi-stable beam is clamped with three

nuts, one on each side and on in the middle. This reduces the effective length by 3 times the size of a

nut (5.5 mm). As a result, the bi-stable beam was modeled to have an effective length of 30 mm. In

reality there is still material around nuts that contributes to force-displacement behavior of bi-stable

beam. As a result, the actual effective length is longer than 30 mm, but it is not certain how long

exactly. The increase in effective length leads to a decrease in negative stiffness.

There is some hysteresis found in the measurements of the compliant grasper. This is relatively low

and can partially be explained by the relaxation of the grasper. Another reason could be some

backlash in the design due to the use of thread to actuate the prototype and connect the bi-stable

elements to the compliant grasper. This also partially explains the hysteresis in the measurements of

the bi-stable compliant grasper. There was also some friction inside the part of the prototype that

contains the bi-stable element.

Conclusion The MIM has important advantages over conventional MIS instruments. It allows the surgeon to have

more freedom of movement and sit comfortably while performing surgery. During surgery grasping

and holding objects for longer periods of time is a common need. Currently there are high reaction

forces on cables and pulleys when objects are grasped and held. Due to these forces there is a

significant increase in friction, high wear and breaking of cables. This increases the risk of

complications and errors during surgery.

We proposed to resolve this problem by combining a bi-stable element with a compliant grasper.

This grasper has two stable positions, open and closed. When closed the grasper is able to hold an

23

object without the need of an actuation force from the surgeon. The bi-stable element is located

close to the grasper to allow for compact design and remove the reaction forces in the cables and

pulleys.

The concept of the first bi-stable compliant grasper is presented. A large scale prototype was

designed, optimized and built based on finite element modeling. The prototype was evaluated by

measuring the force-displacement behavior. The prototype successfully shows bi-stable behavior and

generates a pinch force. A small scale design was presented that successfully made the compliant

grasper of Herder and Berg bi-stable. When this design implemented, the chance for errors and

complications during surgery will be reduced.

References De Vries, K. (2012). Master Thesis: Testing and modeling of the Minimally Invasive Manipulator. Dunning, A. G., N. Tolou, et al. (2011). "Review Article: Inventory of platforms towards the design of a

statically balanced six degrees of freedom compliant precision stage." Mech. Sci. 2(2): 157-168.

Dunning, A. G., N. Tolou, et al. (2012). "Bistable compliant mechanisms: Corrected finite element modeling for stiffness tuning and preloading incorporation." Journal of Mechanical Design, Transactions of the ASME 134(8).

Gallagher, A. G., N. McClure, et al. (1998). "An ergonomic analysis of the fulcrum effect in the acquisition of endoscopic skills." Endoscopy 30(7): 617-620.

Howell, L. L., Midha, A., Norton, T.W. (1996). "Evaluation of equivalent spring stiffness for use in a pseudo-rigid-bodymodel of large-deflection compliant mechanisms." Journal of Mechanical Design 118: 126-131.

Jaspers, J. E. N., M. Shehata, et al. (2004). Mechanical manipulator for intuitive control of endoscopic instruments with seven degrees of freedom, Salt Lake City, UT.

Joice, P., G. B. Hanna, et al. (1998). "Errors enacted during endoscopic surgery - A human reliability analysis." Applied Ergonomics 29(6): 409-414.

Tolou, N. and J. L. Herder (2009). Concept and Modeling of a Statically Balanced Compliant Laparoscopic Grasper. Proceedings of the ASME Design Engineering Technical Conference, San Diego, California, USA.

Van Veelen, M. A. and D. W. Meijer (1999). "Ergonomics and design of laparoscopic instruments: Results of a survey among laparoscopic surgeons." Journal of Laparoendoscopic and Advanced Surgical Techniques - Part A 9(6): 481-489.

24

Paper 2: Feasibility Study: Using Disc Springs to Create Bi-Stable Compliant

Grasper

25

Feasibility Study: Using Disc Springs to Create Bi-Stable Compliant

Grasper

Abstract The Minimally Invasive Manipulator (MIM) copies the movements of the hands of the surgeon to the

tip of the instrument inside the operating area. The movement is mechanically copied using cables

and pulleys. When the surgeon grasps and holds an object, the friction inside the MIM increases and

distorts the force feedback in the remaining degrees of freedom. This is the result of the reaction

forces on the cables and pulleys due to the required pinch force to hold the object. One solution to

this problem is to use a bi-stable compliant grasper. The bi-stable element used in paper 1 is a planar

design, whereas the instruments used by the MIM are circular. A bi-stable element that is circular

might be more effective in the circular environment of the MIM. Disc springs are commonly used to

apply a preload or flexible quality to a bolted joint or bearing. These springs are known to have the

potential to show bi-stable behavior. The goal of this paper is to present the first design of a bi-stable

compliant grasper which uses a disc spring as bi-stable element. The design will be the first design on

the required scale for the MIM. This was done by validating an ANSYS model and a database. Using

the model and database disc spring with suitable dimensions for the MIM scale was found. Two disc

springs in parallel were needed to make a bi-stable compliant grasper. Further investigation is

needed to see if the non-symmetric behavior of the disc spring will be a problem for a future design.

Introduction Minimally Invasive Surgery (MIS) is a type of surgery in which surgeons use small incisions and long

slender instruments to operate on people, instead of making one large incision and using their hands.

Due to the smaller incisions the recovery time and chance of infection are reduced.

This type of surgery has some disadvantages: (1) All tactile feedback is lost, (2) the number of

degrees of freedom inside the patient is reduced (Gallagher, McClure et al. 1998) and (3) the surgeon

often has to stand in uncomfortable positions for longer periods of time (Van Veelen and Meijer

1999). These disadvantages increase the chance for errors (Joice, Hanna et al. 1998).

The Minimally Invasive Manipulator (MIM) is a device that is designed for use in the surgical field

(Jaspers, Shehata et al. 2004). The MIM has three major advantages compared to the standard

minimally invasive surgery (MIS): (1) more freedom of movement inside the operating area. (2) The

surgeon can sit comfortably during the procedure. (3) Some force feedback due to the use of cables

and pulleys to copy the movement of the hands of the surgeon to the tip of the instrument.

The reaction forces on the cables and pulleys influence the friction inside the system. When the

surgeon is moving the MIM the reaction forces and friction are low. When the surgeon starts to grasp

and hold objects like a common suturing needle, the reaction forces on the cables and pulleys of the

MIM increase.


much cable force and (3) a significant increase in friction in the MIM (De Vries 2012). Also, due to the

increased friction, the force feedback in all DoF is distorted. When holding an object, the surgeon

does not need force feedback in the grasping motion, but for the other degrees of freedom force

feedback is desired. For example, when holding a needle, the surgeon would like to feel if he is

26

puncturing tissue or bone. Due to the three reasons mentioned above the risk of complications and

errors during surgery is increased.

In paper 1 a bi-stable compliant grasper was proposed to solve this problem. This grasper has two

stable positions (open and closed). When closed, the grasper is able to produce a pinching force that

is able to hold an object. The compliant grasper is made bi-stable by adding a bi-stable beam to the

compliant grasper. A large scale prototype was made which proved the concept. A design was

presented for a bi-stable compliant grasper with the right dimensions for the MIM.

The bi-stable beam used in the design of paper 1 is a planar design. The bi-stable beam is a leaf

spring that is preloaded via a displacement from the sides. The orientation of the beam will be

orthogonal to the centerline of the instrument. When this design is applied in a circular environment

a problem arises. Due to the circular environment, the effective length l of the beam is depending

on the out of plane thickness w and vice versa (i.e. the longer the beam is, the lower the out of

plane thickness can be and vice versa). As a result, the available space might not be used effectively.

As a result the space used to generate a force-displacement behavior might be more than necessary.

(a) (b) Figure 1: (a) a single disc spring leaning against a stack of disc springs. (b) dimensions of disc spring that determine the behavior. Figure showing: material thickness t, cone height h, outer radius ro and inner radius ri. All dimensions are in meters.

Figure 2: Normalized force deflection characteristic of disc spring. The three curves represent a cone height/material thickness ratio of 1, 2 and 3. As can be seen, the force becomes negative when the ratio is 3, meaning that the disc spring is bi-stable.

Forc

e /F

orc

e to

fla

t (

F/Fc

) [-

]

Deflection/Cone Height [-]

3.0

2.0

1.0

27

The goal of this paper is to present the first design of a bi-stable compliant grasper which uses a disc

spring as bi-stable element. By using the circular disc spring, the space is used more effectively.

The rest of this paper is organized as follows: In the method section the theory behind disc springs is

given, followed by two simulation models, and validation method. In the results section the results of

the validation are shown. The results are then discussed followed by the conclusions.

Method The feasibility of the disc springs was investigated in three steps. First a database was obtained and

finite element model was made in ANSYS to predict the behavior of the disc springs based on their

dimensions. Then the database and model were validated using a disc spring bought from a

manufacturer. This was done because disc springs with the required cone height/material thickness

ratios were not available in stores.

Second, a sensitivity analysis was performed after the validation. The sensitivity analysis was done in

order to gain more insight into the behavior of the spring, and how this behavior can be influenced.

In the third step, the database and model were used to find a disc spring with bi-stable behavior

dimensions of a disc spring with sufficient bi-stable behavior to make a compliant grasper bi-stable.

This is done by combining the behavior of the disc spring to that of a compliant grasper. As an

example for the laparoscopic application the dimensions of the MIM are used. The MIM uses

instruments with a diameter of 8 mm. For the compliant grasper the design of Herder and Berg was

used (Herder and Berg 2000).

Disc springs

Disc springs are commonly used to apply a preload or flexible quality to a bolted joint or bearing.

These springs are known to have the potential to show negative stiffness in part of their force-

displacement characteristic (Belleville Springs 2012). In Figure 1a a typical disc spring is shown. The

behavior of a disc spring depends on the dimensions of the disc spring shown in Figure 1b.

The shape of the force deflection curve of a disc spring depends on the ratio between the height of

the spring, and the thickness of the material. This is illustrated in Figure 2. When the ratio is larger

than 1.5, the behavior shows negative stiffness in the region of interest (ROI) (25% - 75% of the

stroke). In order for the disc spring to become bi-stable, the force has to become negative, this

occurs when the ratio is larger than 3.

Modeling

The database was obtained from a spring manufacturer. It shows the result of a disc spring based on

the input dimensions. It also gives the stresses at the edges of the disc spring. The finite element

model was made in ANSYS using SHELL93 elements. These elements are particularly well suited to

model curved shells. Each element has six nodes, where each node has six degrees of freedom:

translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z-axes. The

element has plasticity, stress stiffening, large deflection, and large strain capabilities.

28

(a)

(b)

Figure 3: (a) Disc spring used to validate database and ANSYS model. (b) Schematic overview of measurement setup used to determine behavior of disc springs. The disc spring is pushed against a fixed plate, with a pushing rod that has a slightly larger radius than the inner radius of the disc spring. The force and displacement are measured during the deformation of the spring.

Validation

The validation was done with two identical disc springs were ordered from a company. The disc

springs were chosen based on their high cone height/thickness ratio of 1.5. This is the highest ratio

that was available at the company. The ratio was chosen as high as possible so that the stiffness in

the ROI was as close to the desired negative stiffness as possible. The disc spring is shown in Figure

3a, and has the following dimensions: outer radius of 9 mm, inner radius of 3.1 mm, material

thickness of 0.4 mm and a cone height of 0.6 mm.

The measurements were done using a measurement setup. A schematic drawing of this setup is

shown in Figure 3b. The pushing/pulling force (FETE RIS components - B3G-C3-50kg-6B, resolution =

0.06 N, range = [0,50] kg) and displacement (LCIT 2000, range = [0,100] mm) were measured. An

amplifier (Scaime CPJ 25) and DaQ-mx Data Acquisition system (NI USB6008) were used to record the

data. Matlab was used to process the data. The actuation force was generated using an air pressured

29

cylinder. Each measurement was repeated 10 times. The disc spring was placed against a fixed plate,

positioned over a hole. During the measurements the disc spring was compressed using a pushing

rod with a slightly larger radius than the inner radius of the disc spring. The hole allowed the disc

spring to be compressed further than the flat position, making it possible to measure the entire

stroke of the disc spring.

The database and model were compared to each other for: force, stroke and stresses. The stresses at

the inner edge and outer edge were compared when the spring was at maximum deformation. The

database and model were compared to the measurement results for only force and stroke, since it

was impossible to measure the stresses inside the disc spring during the experiments.

Sensitivity analysis

The sensitivity analysis was performed using the ANSYS model and the data processed with Matlab.

The following parameters were changed: inner radius r1, outer radius r2, cone height h and

material thickness t. The effects of changing the parameters on the stroke, maximum force and

stiffness in the ROI were calculated.

Conceptual design

The design of the compliant grasper was based on the design presented in (Herder and Berg 2000),

shown in Figure 4a. The grasper was designed to fit in a 5 mm trocar. It has a stiffness of 43 N/mm

and a stroke of 0.6 mm. In Figure 4b the concept of the bi-stable compliant grasper is shown, in this

concept the disc spring is combined with the compliant grasper.

(a)

(b)

Figure 4: (a) design of Herder and Berg, 2000. (b) Concept of bi-stable compliant grasper. The disc spring is combined with the compliant grasper.

30

(a)

(b)

Figure 5: ANSYS model of disc spring with (a) top view and (b) front view. The outer diameter is constrained in z-translation and rotation. The top and left side of the disc spring are constrained in x and y-translation respectively. This to ensure the disc spring is not shifting, whilst allowing it to expand when moving.

Results

FEM

In Figure 5, the model used in ANSYS can be seen. The outer diameter is constrained in z-translation

and rotation. The top and left side of the disc spring are constrained in x and y-translation

respectively. This to ensure the disc spring is not shifting, whilst allowing it to expand when moving.

During the simulation, the inner diameter is pushed down until over a distance that is twice the cone

height. The coding used can be found in Appendix B.

Validation

The stresses given by both models when the disc springs are in maximally deformed are shown in

Table 1. As can be seen, the stresses are in the same order of magnitude. The difference between the

models is 320.9 MPa and 187.8 MPa for the inner and outer edge respectively. Note that for the

inner edge the ANSYS model is predicting higher stresses, whereas for the outer edge the database

predicts a higher stress.

31

Model Stress Inner edge [MPa] Stress outer edge [MPa]

ANSYS 3250.4 912.0

Database 2929.5 1099.8 Table 1: Comparison of stresses predicted by both models.

The force-displacement characteristics of the simulation models and the two springs are shown in

Figure 6a and b. The differences between the simulation models are very small. The measurements

show some differences with respect to the simulation models. The measured stroke of 1.4 mm of the

disc springs is larger than the predicted 1.2 mm. Also the models predict a small negative stiffness in

the ROI, whereas the experiments show a zero stiffness in this region. Hysteresis of 9.3 N (6.3%) and

11.2 N (7.0%) can be seen in the first and second disc spring respectively.

For both disc springs, the general shape of the force-displacement behavior is the same. In case of

the first disc spring the force are close to the predicted values. The measured constant force in the

ROI is slightly higher than the predicted value, 140 N and 135 N respectively. For the second disc

spring, the measured constant force in the ROI is much higher than predicted by the models, 160 N

and 135 N respectively. In both figures a clear step behavior can be seen.

Sensitivity analysis

In Figure 7a the effect of changing the outer diameter can be seen. When the outer diameter is

increased, the maximum force is decreasing while the stroke remains the same. The stiffness in the

ROI (0.3 0.9 mm) is almost unchanged when the outer diameter is increased.

In Figure 7b the effect of changing the inner diameter is shown. The effects on the maximum force

are small and the stroke does not change. The stiffness in the ROI is changing, but no clear pattern

can be seen.

Figure 7c shows the effect of changing the material thickness. By changing the material thickness, the

cone-height/material-thickness ratio is also changed. When the material thickness is increased, the

maximum force also increases. The stroke is not influenced by the change in material thickness. The

stiffness in the ROI increases and becomes positive.

Finally, in Figure 7d the effect of changing the cone height can be seen. The stroke and maximum

force increase when the cone height is increased. The stiffness in the ROI decreases and becomes

negative.

32

(a)

(b)

Figure 6: Force-displacement behavior of both disc springs. FEM (blue dashed and dotted lines) and measurement (solid lines) results are shown. (a) Disc spring 1. (b) Disc spring 2.

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-50

0

50

100

150

200

250

300

Displacement [mm]

Forc

e [

N]

Standard Spring 1

Ansys model

Database

Measurements

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-50

0

50

100

150

200

250

300

Displacement [mm]

Forc

e [

N]

Standard Spring 2

Ansys model

Database

Measurements

33

(a)

(b)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

50

100

150

200

250

300

350

400

450

500

550

Displacement [mm]

Forc

e [

N]

Sensitivity: Outer diameter

rout

= 8 mm

rout

= 8.5 mm

rout

= 9 mm

rout

= 9.5 mm

rout

= 10 mm

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

50

100

150

200

250

300

350

400

450

500

550

Displacement [mm]

Forc

e [

N]

Sensitivity: Inner diameter

rin

= 2.1 mm

rin

= 2.6 mm

rin

= 3.1 mm

rin

= 3.6 mm

rin

= 4.1 mm

34

(c)

(d)

Figure 7: Force-displacement behaviors of the disc spring. The influence of outer radius, inner radius, material thickness and cone height on the behavior is shown. (a) Influence of outer radius rout. (b) Influence of inner radius rin. (c) Influence of the material thickness t. (d ) Influence of the cone height h.

Suitable spring for laparoscopic application

The force-displacement characteristics of a suitable disc spring for the MIM, the compliant grasper

and the disc spring combined with compliant grasper are shown in Figure 8a. The dimensions of the

disc spring are: outer radius: 3.5 mm, inner radius: 1 mm, material thickness: 0.08 mm and cone

height: 0.32 mm. As can be seen, the negative stiffness is 75 N/mm. The maximum force is around

12.4 N, the minimum force around -4.8 N. The stroke that shows negative stiffness is 0.32 mm. The

stroke of the disc spring is smaller than that of the compliant grasper of (Herder and Berg 2000) (0.32

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

50

100

150

200

250

300

350

400

450

500

550

Displacement [mm]

Forc

e [

N]

Sensitivity: Material thickness

t = 0.30 mm

t = 0.35 mm

t = 0.40 mm

t = 0.45 mm

t = 0.50 mm

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

50

100

150

200

250

300

350

400

450

500

550

Displacement [mm]

Forc

e [

N]

Sensitivity: Cone height

h = 0.4 mm

h = 0.5 mm

h = 0.6 mm

h = 0.7 mm

h = 0.8 mm

35

mm vs. 0.6 mm). As a result, the stroke of the disc spring combined with the compliant grasper will

be 0.32 mm. As can be seen, one disc spring is not sufficient to make the compliant grasper bi-stable.

When closed, the bi-stable compliant grasper has a positive actuation force which results into a pinch

force. When opened a small actuation force (1 N) is needed to keep the grasper open. This problem

can be solved by using two disc springs in parallel. The behavior of the resulting bi-stable compliant

grasper is shown in Figure 8b. Now the pinch force has increased, and bi-stable behavior is achieved.

(a)

(b)

Figure 8: (a) Behavior of one disc spring combined with compliant grasper. Dimensions: Outer radius = 3.5 mm, inner radius = 1 mm, material thickness = 0.08 mm and cone height = 0.32 mm. (b) Behavior of two disc springs in parallel and compliant grasper combined.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-10

-5

0

5

10

15

Displacement [mm]

Forc

e [

N]

Bi-Stable Compliant Grasper

Compliant Grasper

Disc Spring: Ansys model

Disc Spring: Database


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-15

-10

-5

0

5

10

15

20

25

30

Displacement [mm]

Forc

e [

N]


Compliant Grasper

Disc Spring: Ansys model

Disc Spring: Database


36

Discussion The results show that it is possible to use disc springs as a bi-stable element for use in a circular

environment. When two disc springs are used in parallel the force-displacement behavior is clearly

bi-stable, and produces a high negative stiffness and maximum forces. When closed, the bi-stable

compliant grasper produces a pinch force.

The behavior is not symmetrical (i.e. the maximum force is higher in magnitude than the minimum

force. As a result, the second stable position of the bi-stable compliant grasper is less stable than the

first. This is no problem for the application of surgery.

The force-displacement behavior of the two simulation models used is virtually the same. There are

some differences in the predicted stresses. The difference between the models is 320.9 MPa and

187.8 MPa for the inner and outer edge respectively. For the inner edge the ANSYS model is

predicting higher stresses, whereas for the outer edge the database predicts a higher stress. It is

unclear as to why this is the case, but the assumption that it is due to the difference in complexity of

the models seems legit. The ANSYS model is far more complex, and accounts for non-linearity. The

database is a data base which interpolates between empirically obtained data.

The measurements show some differences with respect to the models. Firstly, the measured disc

springs show a 0.2 mm longer stroke. This can partially be explained by the deformation of the force

sensor. This was measured to be 0.13 mm when loaded with 250 N. The reason for the remaining

0.07 mm (5.8%) difference between the measurement results and the FEM is unknown.

The step behavior that is seen in the measurement results is due to the resolution of the

displacement sensor. Unfortunately a more accurate sensor was not available. For future research a

more accurate sensor needs to be used. Due to this limited accuracy, no conclusions can be made

about the exact forces and stroke of the disc spring. Nonetheless, the general force-displacement

behavior of the measurement results is clearly similar to that of the FEM.

The hysteresis found in the measurements can partially be explained by the friction between the disc

spring and the plate it is pushed against. During its deformation, the disc spring expands and slides

over the plate. Another explanation is backlash due to the screw-thread connection between the

sensor and the pushing rod.

The sensitivity study shows that only the maximum force is sensitive to changes in the outer radius.

This can be explained by looking at the cross section of the disc spring, the beams become longer

when the outer radius is increased, decreasing their stiffness. As a result, the force over the entire

stroke is lower.

The stroke, maximum force or the stiffness in the ROI are not sensitive to changes in the inner radius.

This is unexpected since when the inner diameter is decreasing, the difference between inner and

outer diameter is increasing. As a result the cross section of the disc spring will show longer beams.

It is expected that the stiffness of those beams would decrease. It seems that due to the circular

shape, these effects are cancelled.

The cone height is the only parameter that influences the entire behavior (stroke, maximum force

and stiffness in ROI) of a disc spring. This was expected since it was known that the stroke was only

depending on the cone height. When the cone height is changed, the cone height/material

37

thickness ratio is also changed. This explains why the stiffness in the ROI is decreasing when the

cone height (and with it the ratio) is increased.

The decrease in cone height/material thickness ratio also explains why the stiffness in the ROI

increases when the material thickness increased. Since the entire thickness is increased, the stiffness

outside the ROI also increases, resulting in a higher maximum force.

For the future, a prototype has to be made which successfully applies a disc spring in a design for a

bi-stable compliant grasper. The non-symmetrical behavior needs to be further investigated. It might

also be interesting to investigate if the circular disc spring is actually more efficient in a circular

environment as a bi-stable beam.

An advantage of conventional disc springs is the possibility to stack them in series to increase their

stroke. This could also be very interesting for the bi-stable disc springs, but due to the negative

stiffness, more study is required to see if this is feasible.

Conclusion The MIM has important advantages over conventional MIS instruments. It allows the surgeon to have






Using a bi-stable compliant grasper reduces the reaction forces in the cables and pulleys, thereby

reducing the risk of complications and errors. The bi-stable element used to make this grasper bi-

stable is a planar design. Due to the circular environment of the surgical instruments of the MIM, the

planar design might not use the available space efficiently.

We proposed the use of disc springs as bi-stable element to make a compliant grasper bi-stable. In

order to investigate if this is feasible an ANSYS model and a database were validated. The model and

database are accurately predicting the behavior of the disc spring and can be considered valid.

A sensitivity analysis was done to determine the influence of parameters of the disc spring on its

behavior. The analysis shows that the stroke of a disc spring is only influenced by the cone height.

The dependency of the stiffness in the ROI by the cone height and material thickness is confirmed.

The maximum force is depending on the cone height, material thickness and outer radius.

A disc spring with suitable dimensions was found using the simulation models. Two of such disc

springs in parallel, combined with the compliant grasper, show bi-stable behavior. This is the first

concept for bi-stable compliant grasper on the required scale for the MIM using disc springs as bi-

stable elements.

38

References Belleville Springs, L. (2012). "http://www.bellevillesprings.com/disc-spring-characteristics.html."

Retrieved 21-08-2012, 2012. De Vries, K. (2012). Master Thesis: Testing and modeling of the Minimally Invasive Manipulator. Gallagher, A. G., N. McClure, et al. (1998). "An ergonomic analysis of the fulcrum effect in the

acquisition of endoscopic skills." Endoscopy 30(7): 617-620. Herder, J. L. and F. P. A. v. d. Berg (2000). Statically balanced compliant mechanisms (sbcm's) an

example and prospects. Proceedings of the ASME Design Engineering Technical Conference, Baltimore, Maryland, USA, ASME.



Van Veelen, M. A. and D. W. Meijer (1999). "Ergonomics and design of laparoscopic instruments: Results of a survey among laparoscopic surgeons." Journal of Laparoendoscopic and Advanced Surgical Techniques - Part A 9(6): 481-489.

http://www.bellevillesprings.com/disc-spring-characteristics.html.

39

General Discussion In both papers a different design was introduced and tested. The design in the second paper was

introduced because it was believed to have the potential to perform better in a circular environment.

Both designs have successfully shown bi-stable behavior, but they havent been compared to each

other. This will be done in this section.

The designs will be compared on the required scale for the Minimally Invasive Manipulator (MIM). In

order to make an objective comparison, two steps are taken. First, a cylindrical volume is defined

based on the scale of the MIM. Both designs have to fit within this volume.

The dimensions of both designs are chosen such that they fit inside the volume and both have

approximately the same behavior. Once these dimensions are found, the stresses inside both designs

are compared.

The volume was chosen based on the dimensions of the bi-stable disc spring found in paper 2. The

diameter of the cylindrical space was 7 mm. The height of the volume was 0.64 mm. The disc spring

had the following dimensions: outer radius = 3.5 mm, inner radius = 1 mm, material thickness = 0.1

mm, cone height = 0.4 mm. For the bi-stable buckling beam this led to the following dimensions:

length = 5.50 mm, out of plane thickness = 4.33 mm, in plane thickness = 0.17, preload displacement

= 0.016 mm.

Using the finite element models (FEM) of both designs, the force-displacement behavior and stresses

were computed. The results are shown in Figure 2.

Figure 2: Force-displacement behavior results from FEM of bi-stable buckling beam and disc spring.

The maximum stresses inside the disc spring are 2929.5 GPa for the excel model and 3.250 GPa for

the ansys model. The maximum stresses inside the bi-stable buckling beam are 1.140 GPa.

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-20

-15

-10

-5

0

5

10

15

20

Displacement [mm]

Forc

e [

N]

Disc spring: Excel

Disc spring: ANSYS

Bi-stable Beam

40

As can be seen, the maximum positive and negative forces of the bi-stable beam are higher in

magnitude than those of the disc spring (18 N vs. 12.4 N and -17.3 vs. -4.8). This results in a higher

negative stiffness for the bi-stable beam. The stroke of both designs is the same.

The stresses inside both designs are very different. Inside the bi-stable buckling beam, the stresses

are much lower than in the disc spring: 1.140 GPa vs. 2929.5 GPa and 3.250 GPa. This means that the

disc spring produces a lower negative stiffness with higher stresses compared to the bi-stable beam.

More study is needed to determine what the maximum force and stroke of the bi-stable buckling

beam are. In the introduction, an estimate of 20 N as a pinch force was given based on the generated

pinch force by the DaVinci. The forces generated by both designs seem to be too low to generate a

sufficient pinching force. This can be soled by using multiple disc springs or bi-stable buckling beams

and putting them in parallel. In theory the stroke and both designs can be increased by putting

multiple disc spring or bi-stable beams in series, but due to the unstable behavior, this might not be

possible in reality. More study is necessary to determine how this can be applied in practice.

General Conclusion The MIM has important advantages over conventional MIS instruments. It allows the surgeon to have






Using a bi-stable compliant grasper reduces the reaction forces in the cables and pulleys, thereby

reducing the risk of complications and errors. This grasper has two stable positions, open and closed.

When closed the grasper is able to hold an object without the need of an actuation force from the

surgeon. The bi-stable element is located close to the grasper to allow for compact design and

remove the reaction forces in the cables and pulleys. Two different designs are presented.

In the first paper the concept of the first bi-stable compliant grasper is presented. A large scale

prototype was designed, optimized and built based on finite element modeling. The prototype was

evaluated by measuring the force-displacement behavior. The prototype successfully shows bi-stable

behavior and generates a pinch force. In the future, this principle needs to be applied on the

required scale for the MIM. When implemented, the chance for errors and complications during

surgery will be reduced.

In the second paper we proposed the use of disc springs as bi-stable element to make a compliant

grasper bi-stable. In order to investigate if this is feasible an ANSYS model and a database were

validated. The model and database are accurately predicting the behavior of the disc spring and can

be considered valid. A sensitivity analysis was done to determine the influence of parameters of the

disc spring on its behavior. Using the FEM a disc spring with suitable dimensions was found. Two of

such disc springs in parallel, combined with the compliant grasper, results in a compliant grasper that

shows bi-stable behavior. This is the first concept for bi-stable compliant grasper on the required

scale for the MIM using disc springs as bi-stable elements.

41

Both the bi-stable beam and the disc spring can be used in the design for a bi-stable grasper. The bi-

stable beam is more effectively creating negative stiffness inside a circular space. As a result, it

should be able to produce more force and/or stroke in the same volume as a disc spring. This makes

the bi-stable beam more suitable for use in a bi-stable grasper than a disc spring.

References Belleville Springs, L. (2012). "http://www.bellevillesprings.com/disc-spring-characteristics.html."

Retrieved 21-08-2012, 2012. De Vries, K. (2012). Master Thesis: Testing and modeling of the Minimally Invasive Manipulator. Dunning, A. G., N. Tolou, et al. (2011). "Review Article: Inventory of platforms towards the design of a

statically balanced six degrees of freedom compliant precision stage." Mech. Sci. 2(2): 157-168.

Dunning, A. G., N. Tolou, et al. (2012). "Bistable compliant mechanisms: Corrected finite element modeling for stiffness tuning and preloading incorporation." Journal of Mechanical Design, Transactions of the ASME 134(8).

Gallagher, A. G., N. McClure, et al. (1998). "An ergonomic analysis of the fulcrum effect in the acquisition of endoscopic skills." Endoscopy 30(7): 617-620.

Howell, L. L., Midha, A., Norton, T.W. (1996). "Evaluation of equivalent spring stiffness for use in a pseudo-rigid-bodymodel of large-deflection compliant mechanisms." Journal of Mechanical Design 118: 126-131.



Tolou, N. and J. L. Herder (2009). Concept and Modeling of a Statically Balanced Compliant Laparoscopic Grasper. Proceedings of the ASME Design Engineering Technical Conference, San Diego, California, USA.

Van Veelen, M. A. and D. W. Meijer (1999). "Ergonomics and design of laparoscopic instruments:

Results of a survey among laparoscopic surgeons." Journal of Laparoendoscopic and Advanced

Surgical Techniques - Part A 9(6): 481-489.

http://www.bellevillesprings.com/disc-spring-characteristics.html.

42

Appendices

Appendix A: ANSYS modeling and sensitivity study In Figure A1, the ANSYS model of the compliant grasper is shown. The model is meshed with

PLANE82 elements. This type is a 2-D eight nodes element, with two translational degrees of

freedom per node (x and y). It provides more accurate results for mixed (quadrilateral-triangular)

automatic meshes and can tolerate irregular shapes without as much loss of accuracy as well. The

element does not have any rotations, or out of plane translation (z-direction) these degrees of

freedom were found to be not relevant in the design of the grasper. Initially a SHELL93 element was

used, which did have all six DoF, but the results from both models matched exactly. The PLANE82

elements were chosen decrease calculation speeds.

The outer legs of the grasper are constrained in all two translations. The inner leg of the grasper is

constrained in the y-translation and the x-translation is prescribed and used to actuate the grasper.

Figure A1: ANSYS model of compliant grasper, with the length of the flexible element l and the in plane thickness of the flexible element t. The out of plane t

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