COLLISION DETECTION AND RESPONSE AND

HAPTIC INTERACTION IN VIRTUAL ENVIRONMENT

SIMULATIONS WITH DEFORMABLE OBJECTS

by

PAUL D. JACOBS

Submitted in partial fulfillment of the requirements

for the degree of Master of Science

Thesis Advisor:

Dr. M. Cenk Çavuşoğlu

Department of Electrical Engineering and Computer Science

CASE WESTERN RESERVE UNIVERSITY

May, 2006

Contents

List of Tables iii

List of Figures v

Abbreviations vi

Abstract vii

1 Introduction 1

1.1 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Background and Related Literature 3

2.1 Haptic Devices and Surgical Simulation . . . . . . . . . . . . . . . . . 3

2.2 Deformable Surface Models . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Multi-Rate Simulation in Haptic Interaction . . . . . . . . . . . . . . 8

3 Methods 10

3.1 Multi-Rate Simulation Algorithm Outline . . . . . . . . . . . . . . . . 10

3.2 Local Model Construction . . . . . . . . . . . . . . . . . . . . . . . . 13

i

3.2.1 Two Dimensional Case - Sample Construction of the Local Lin-

ear Approximation Model . . . . . . . . . . . . . . . . . . . . 17

3.3 Collision Detection and Response in Multi-Rate Simulation . . . . . . 21

3.3.1 1D Example of Hybrid Collision Resolution . . . . . . . . . . 22

3.3.2 Hybrid Collision Resolution in 3D MSD models . . . . . . . . 25

3.4 Mesh Deformation in Collision Response . . . . . . . . . . . . . . . . 28

3.5 Determination of Constraint Forces . . . . . . . . . . . . . . . . . . . 28

3.5.1 Sticking Constraint . . . . . . . . . . . . . . . . . . . . . . . . 30

3.5.2 Sliding Constraint . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.5.3 Frictional Sliding Contact Constraint . . . . . . . . . . . . . . 34

3.6 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Results 38

4.1 Comparison of Local and Global Model Simulations . . . . . . . . . . 40

4.2 Pure Slipping Constraint Results . . . . . . . . . . . . . . . . . . . . 42

4.3 Pure Sticking Constraint Results . . . . . . . . . . . . . . . . . . . . 45

4.4 Stick-Slip Frictional Sliding Friction Results . . . . . . . . . . . . . . 47

5 Discussion 49

5.1 Extensions to Literature . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Bibliography 52

ii

List of Tables

4.1 Parameters used in MSD model simulation . . . . . . . . . . . . . . . 39

iii

List of Figures

2.1 The PHANTOM Premium 1.5 Haptic Interface [18]. . . . . . 4

2.2 The da Vinci robotic surgical system [17]. . . . . . . . . . . . . 5

2.3 2D Mass-Spring-Damper Model . . . . . . . . . . . . . . . . . . 6

3.1 Haptic Simulation Block Diagram . . . . . . . . . . . . . . . . . 11

3.2 Haptic Simulation Block Diagram, Previous Work . . . . . . . 12

3.3 Local Model Order Reduction . . . . . . . . . . . . . . . . . . . 16

3.4 Four node fully connected 2-D mass-spring-damper model . . 17

3.5 1-D Example of hybrid collision resolution model . . . . . . . 23

3.6 One dimensional hybrid collision resolution model flowchart 24

3.7 Three dimensional hybrid collision resolution model flowchart 27

3.8 Triangle displacement in sticking and sliding mode. Diagram

a. shows the position of the instrument at the beginning

and end of a time step. Diagram b. illustrates the sliding

displacement. Diagram c. illustrates the sticking displacement. 29

3.9 Baryocentric Co-ordinates . . . . . . . . . . . . . . . . . . . . . . 29

4.1 Haptic simulation screenshot . . . . . . . . . . . . . . . . . . . . 38

4.2 Screenshot showing contact between the instrument and the

deformable object . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

iv

4.3 Multi-rate simulation results. Dotted line is global model

running at 1000 Hz. Dashed line is global model running at

30 Hz. Solid line is local model running at 1000 Hz. . . . . . 41

4.4 Multi-rate simulation results, y-position of test node vs time.

Solid line is global model running at 30 Hz, dashed line is

local model running at 1000 Hz. . . . . . . . . . . . . . . . . . . 43

4.5 Pure Slipping constant contact results. . . . . . . . . . . . . . . 44

4.6 Pure Slipping dragging contact results. . . . . . . . . . . . . . . 45

4.7 Pure Sticking contact results. Solid line is magnitude of nor-

mal force, dotted line is magnitude of tangential force. . . . . 46

4.8 Hybrid stick-slip dragging results, co-efficient of friction 1.0.

Solid line is magnitude of force normal, dotted line is magni-

tude of tangential force. . . . . . . . . . . . . . . . . . . . . . . . 47

v

Abbreviations

MSD - Mass-Spring-Damper

FEM - Finite Element Model

HCI - Human-Computer Interface

DOF - Degree of Freedom

vi

Collision Detection and Response and Haptic Interaction in Virtual

Environment Simulations With Deformable Objects

Abstract

by

PAUL D. JACOBS

An increasingly common new modality in human-computer interaction is haptic in-

terfacing, especially in the field of medical simulation. The order-of-magnitude dif-

ference in update rates between graphical deformable object simulations and haptic

interfaces can be bridged using local low-order approximations. However, providing

force feedback using local models complicates collision detection and response with

the virtual tool, since the user interacts with lower-order proxies rather than the full

simulated objects, and instability can result. A novel approach focusing on rolling

contact with stick-slip friction is presented where all collision detection and response

with the virtual tool is performed at the local level at the haptic time-scale, utilizing

linearized low-order local models that approximate the behavior of the full model for

the short time steps and small deformations involved.

vii

Chapter 1

Introduction

The value of haptic interaction in surgical simulation has led to a great deal of research

interest into the challenges involved in providing haptic force-feedback in virtual en-

vironment simulations with deformable surfaces. The key obstacle to overcome in

haptic interaction is the difference in update rates of the virtual environment simula-

tion, which is typically linked to the graphical update rate of between 10 and 60 Hz,

and the update rate of the haptic interface, which must be on the order of 1 KHz in

order to be convincing to the operator. One method of bridging this gulf is through

multi-rate simulation, where the virtual environment in its full complexity is simula-

tion at the visual update rate, while a simpler simulation, often encompassing only

parts of the environment local to the virtual instrument, is run in parallel at the hap-

tic update rate and periodically re-synchronized with the full model. The purpose

of this project is to explore multi-rate simulation techniques involving performing

collision response using a stick-slip friction model in the haptic-rate simulation.

1

1.1 Thesis Contributions

First, a novel method of constructing local approximation models of mass-spring-

damper networks was developed for use in multi-rate simulation. Second, a collision

response algorithm capable of providing stick-slip frictional interaction was developed.

Finally, these methods were used to implement a virtual environment simulation using

the PHANTOM(TM) 1.5 Haptic Interface.

1.2 Thesis Outline

Background information and existing research on surgical simulation, haptic inter-

faces, and deformable surface models are presented in the chapter on Background

and Related Literature (ch. 2). The Methods chapter (ch. 3) describes the algo-

rithms developed in this work and the implementation of the simulation. Findings

are reported in the Results chapter (ch. 4) and extensions to literature, limitations

of the study, and future research directions are described in the Discussion chapter

(ch. 5).

2

Chapter 2

Background and Related Literature

This section will discuss information regarding haptic devices, multi-rate simulation

techniques, and commonly used deformable surface models.

2.1 Haptic Devices and Surgical Simulation

Although the term ‘haptics’ refers to any human-computer interface (HCI) device that

provides force feedback, including “rumble-packs” common to consumer video game

controllers and actuated yokes and steering wheels for driving and flying simulations,

the type of haptic interface focused on here are multi-link rigid armatures capable

of three-dimensional force feedback and possessing at least three degrees of freedom.

This type of haptic armature significantly predates surgical simulation; the first haptic

devices were electrically actuated teleoperators built in the 1950s at the Argonne

National Laboratory to handle radioactive materials. Haptic interaction was added

to the teleoperators in order to handle glassware without crushing it, which proved

to be an extremely challenging task without force feedback.

The specific haptic device considered here is the PHANTOM Premium 1.5 Haptic

3

Figure 2.1: The PHANTOM Premium 1.5 Haptic Interface [18].

Interface developed by Sensable Technologies shown in Figure 2.1[18]. The PHAN-

TOM interface is commonly used in virtual environments haptic simulations due to

it’s relatively large workspace and low inertia and friction. The frequency response

of the PHANTOM has a large resonance at 125 Hz which denotes the highest fre-

quency signal that can accurately be represented by the PHANTOM due to mechan-

ical damping and limitations of the electronics. This response is considered sufficient

for surgical simulation applications [3].

The advent of minimally invasive surgery ushered in a new role for haptics both

in surgical simulation and surgical robotics. While surgical simulation of traditional

surgery is still an open research problem, training simulation systems have been devel-

oped that take advantage of the more limited types of interaction present in minimally

invasive surgeries, where the surgeon interacts with the patient only through a series

4

Figure 2.2: The da Vinci robotic surgical system [17].

of long instruments inserted through small incisions. This often makes laparoscopic

surgeries more difficult to perform, especially since they require a skill set distinct

from that required by open surgical methods. At the same time, the fact that the sur-

geon interacts with the patient through a highly constrained and specialized interface

with limited degrees of freedom is a boon for surgical simulation, since these factors

greatly simplify the task of providing haptic interaction. The difficulty, unfamiliarity,

and relative rarity of some of these surgeries also make them attractive prospects for

a training simulator. For these reasons, existing surgical haptic simulations focus on

endoscopic surgical techniques using interfaces that closely mimic endoscopic-specific

surgical tools. Numerous research groups have constructed simulators for tasks such

as laparoscopic cholecystectomy, colon endoscopy, arthroscopy, bronchoscopy, spinal

biopsy, hysteroscopy, and radiosurgery, among others. [19] [13] [15] [2]. Haptic teleop-

erators have also been developed as part of robotic surgical systems. In systems like

the da Vinci Surgical System [17], complex robotic instruments capable of dexterous

5

manipulation are inserted through narrow incisions and operated remotely with a

haptic device. The use of a force-reflecting teleoperator in the actual surgery makes

this type of procedure an excellent candidate for future work in surgical simulation.

Relatively few researchers have investigated the extremely challenging open re-

search problem of simulating more traditional types of surgery in which the surgeon

interacts directly with the patient through hand-held as opposed to fixed surgical in-

struments. Morris describes a temporal bone surgery simulator, but the interaction is

focused on bone drilling, which utilizes rigid-body interaction and avoids deformable

surfaces [14]. A simulator for curvilinear capsulorhexis cataract surgery has been built

by Webster et. al., but uses a very specific model not suitable for modeling general

soft body structures [20]. Work has also been done on developing haptic techniques

for simulating needle insertion procedures [7] [8].

2.2 Deformable Surface Models

Figure 2.3: 2D Mass-Spring-Damper ModelSpring-damper pairs are represented by damped springs.

6

Several different types of deformable surface model have been suggested for use in

haptic interaction. Lumped mass-spring-damper (MSD) models represent deformable

objects with layers of masses connected by spring-damper pairs or damped springs.

MSD models are a common choice for surgical simulation for their ease and simplicity

in modeling and for their low computational requirements [10]. A significant drawback

to the mass-spring-damper approach is the difficulty in modeling a volumetric object

using mass-spring elements. Although the appearance of volume can be obtained

through the use of cross-springs binding the mass elements into a three-dimensional

grid, there is no clear procedure to match the properties of a real substance to the

spring and damper parameters in the model. The main alternative to MSD systems

are finite-element models (FEM), which provide a continuum model that is strongly

physically based. However, FEM models incur larger computational costs [11]. These

concerns lead many FEM implementations to include off-line computation, which

limits their use in interactive haptic simulations.

Other volumetric methods have been suggested to capture the realism of FEM

techniques with lower computational requirements. Kim and Srinivasan have devel-

oped a method involving dynamically subdividing a volumetric element into a col-

lection of randomly ‘sprinkled’ spherical volume elements referred to as the method

of finite spheres [10]. Mendoza, Sundaraj, and Laughier have suggested the “Long

Element Model” in which a body is broken down into columns of incompressible fluid

governed by Pascal’s Principle [11]. Although these models have been found to be

useful in haptic simulation, the discussion here is limited to the more widely used

mass-spring-damper and FEM models.

7

2.3 Multi-Rate Simulation in Haptic Interaction

A major difficulty in providing high-fidelity force feedback is the extremely demanding

update rate requirements of haptic feedback. The human sense of touch is remarkably

sensitive, and can distinguish between changes in force into the range of hundreds of

Hz. It is generally accepted that the update rate of the haptic interface must be

five to seven times higher than the highest frequency event that is to be simulated.

Therefore, in order to render events in the 100 to 200 Hz range matching the capa-

bilities of the PHANTOM and similar haptic interfaces, 1 KHz is widely considered

the minimum update rate for realistic haptic feedback [6]. Real-time virtual envi-

ronment simulations, on the other hand, are tailored towards visual update rates,

and typically run between 10 - 60Hz, and it is infeasible to significantly increase the

update rate of the physical simulation due to computational limitations. Overcoming

this orders-of-magnitude difference in update rate has been the focus of the bulk of

haptic research. As of the late 1990’s, the typical practice was to low-pass filter the

generated force to the bandwidth of the model update rate [19]. This was found to

not be an adequate solution, as this effectively reduced the haptic update rate to the

visual update rate.

Several methods have been proposed to solve this problem through multi-rate

simulation. Multi-rate simulation techniques aim to divide a virtual environment

simulation into two parallel simulations, one running at the visual update rate, the

other at the haptic update rate. The visual update rate simulation models the vir-

tual environment in its full complexity, and provides visual feedback to the user, while

the haptic update rate simulation uses a simpler and more computationally tractable

model to provide only force feedback. Commonly, the visual update rate simulation

models the entire virtual environment, while the haptic rate simulation models only

8

an area local to the point of haptic interaction. Cavusoglu and Tendick used a local

linearization of mass-spring-damper models, and showed that the spacial dependence

of the state variables of each node of the linearized system is primarily on nearby

nodes, and therefore a local model consisting of only a few layers surounding the con-

tact point can be used to provide satisfactory haptic interaction [5]. Multi-resolution

methods with coarser meshes used for the entire model and more detailed meshes

for areas of local interest have also been used successfully in multi-rate simulation.

[1][21][9].

9

Chapter 3

Methods

In order to test novel approaches to multi-rate simulation and collision response in

haptic virtual environments, a haptic implementation was built using a 3 DOF manip-

ulator using mass-spring-damper as the modeling technique for deformable surfaces.

The twin focuses of this chapter are a low-order linearization technique for building

local approximation models for MSD deformable surfaces, and a collision response

system that incorporates ideas from traditional constraint and penalty-based contact

resolution algorithms.

3.1 Multi-Rate Simulation Algorithm Outline

The core of the approach to multi-rate simulation in this work is to divide the neces-

sary computational tasks into those that must be performed at the servo-loop update

rate of the haptic interface and those that can be performed at the same rate as the

overall simulation. The methods presented here were initially developed in [5] and

[2]. Like other multi-rate methods, the algorithm is divided into two basic blocks,

as shown in Figure 3.1. The ‘global’ simulation incorporates the entire virtual envi-

10

ronment and runs at the visual update rate in the order of magnitude of 10 Hz. A

‘local’ simulation runs at or near the haptic server loop update rate, and simulates

the behavior of a subset of the global model.

GlobalSimulation

LocalSimulation

Haptic Device

10 Hz 1,000 Hz

geometry

force model

geometry

force

instrumentposition

Figure 3.1: Haptic Simulation Block Diagram

After each global update is generated at 100 ms intervals corresponding to the

10 Hz update rate, a local linear approximation model is generated and passed to a

second simulation, running either in a separate process or thread in single-computer

operation, or running on a second computer in networked operation. This second

simulation uses the local linear approximation model to provide force output to the

user and then sends the state of the local model and haptic instrument back to the

global model, which incorporates this information back into the global model, and

then re-computes a new linear approximation for the next cycle.

This approach differs significantly from previous work in that the flow of informa-

tion back from the local to the global model includes geometric information. Typi-

cally, haptic simulations follow the outline of Figure 3.2. The flow of information from

the local model to the global model is very simple and consists only of position and

velocity information from the haptic instrument. Either contact-resolving collision

response is done only on the global model, or else collision response is performed in

11

GlobalSimulation

LocalSimulation

Haptic Device

10 Hz 1,000 Hz

force model

instrumentposition

force

instrumentposition

Figure 3.2: Haptic Simulation Block Diagram, Previous Work

parallel on both global and local simulations but any deformations made to the local

model are discarded every time a new local model is constructed.

This type of algorithm has several drawbacks. First, it restricts the rate that input

from the haptic instrument to the virtual environment to the global simulation update

rate. At first glance, this may not seem to be an important constraint; voluntary

motion of humans in tasks such as handwriting tops out at a motion frequency of

about 6 Hz, and a study of hand motions of eye surgeons has indicated that a motion

frequency of 2 Hz accounts for most voluntary motion [16]. However, in the case of

stick-slip frictional contact, an instrument skipping across a rough surface would be

restricted from changing mode of contact faster than the visual update rate. Second,

it introduces latency between haptic input and changes in the global model that may

result in a haptically apparent discontinuity. Since the altered local model that results

from interaction with the haptic instrument is discarded at the end of the haptic time

step and replaced with a new local model that is computed based solely on the new

position of the haptic instrument, the new model may have a large discontinuity in

force output. While it is possible to interpolate between force values generated at the

end of one global time-step with the initial forces generated by the next local model,

12

if the discontinuity is large and the global time step is long, then the force output

may seem to be ‘muddy’ or delayed.

The alternative method presented here is built on the hypothesis that these draw-

backs can be avoided by including model geometry in the feedback from the local to

the global model. Therefore, any changes to the local model that have been made

at the haptic update rate are incorporated back into the global model, under the

assumption that if collision detection and resolution techniques are performed at the

haptic rate on a model local to the point of haptic interaction, then a higher-fidelity

haptic experience can be provided. The challenging case of stick-slip frictional contact

is used as a testbed for this algorithm. This approach has several requirements; first,

the local model must accurately simulate the behavior of the global model, at least for

the short time intervals and small deformations involved. Second, the collision and

detection response algorithm needs to operate effectively with the highly demanding

1 ms time-step.

3.2 Local Model Construction

At each global time-step, a local linearized approximation model of the deformable

surface is constructed and sent to the local model simulation for simulating the dy-

namic behavior at the haptic update rate. In this work, mass-spring-damper network

are used as the global model, but a similar treatment is possible using finite element

models but not presented here. This method is an extension of the method developed

by Cavusoglu in work originally presented in [2].

The model used for deformable objects under consideration here is a network of n

masses connected by damped springs which is being deformed by a virtual instrument

at a single contact point on the outer surface of the mesh. The outer surface of the

13

mesh is composed of an array of triangular polygons which are constructed using

the mass nodes as vertices. The behavior of the system is governed by a non-linear

differential equation of the form

d

dt

XẊ

= Ẋ

M−1f(X, Ẋ)

(3.1)where X and Ẋ are respectively the state vectors containing the positions and veloc-

ities of each node, f(X, Ẋ) is a function that maps the state vectors to a vector of

forces on each node, and M−1 is a 3n× 3n matrix of the form

M−1 =

1m1

1m1

1m1

0

. . .

0

1mn

1mn

1mn

(3.2)

where mi is the mass of the node with index i. The computational requirements

of this relatively simple deformable model prevent it from being simulated at the

haptic update rate. For the relatively small 15x4x15 node networks used for testing,

update rates of about 120 Hz were achievable. Therefore, at each global time step, a

14

linearized discrete model is constructed by taking the tangent behavior of the system:

d

dt

XẊ

≈

Ẋ

M−1

f0 + F ∆X

∆Ẋ

(3.3)

where

f0 = f(X0, Ẋ0)

∆X = X −X0

∆Ẋ = Ẋ − Ẋ0

F =

[∂f

∂X

∂f

∂Ẋ

]∣∣∣∣X=X0

Ẋ=Ẋ0

(3.4)

The linearization process does not reduce the order of the model, and so a re-

duction of the model to components spacially close to the virtual instrument contact

point is needed to reduce the computational complexity so as to arrive at a model

that can be simulated at the haptic update rate. The method used here to construct

a low order approximation follows the method of Cavusoglu and Tendick that in-

cludes in the low order approximation the closest node on the surface of the mesh

to the contact point and the immediately adjacent nodes, as well as all the springs

and dampers connected to any of these nodes [5]. The springs and dampers connect-

ing these nodes to the rest of the deformable model are replaced with springs and

dampers with the same parameters connected to a stationary and immovable ‘wall’

with the justification that the high frequency response of the model depends mostly

15

on contributions from the nodes that are immediate neighbors, and only secondarily

on the rest of the mass-spring-damper network, and so for the high-frequency de-

pendent haptic interaction, the remainder of the nodes can be considered stationary

during each haptic time step.

Figure 3.3: Local Model Order Reduction

At each time step of the virtual environment simulation, a local linearization is

constructed from the global deformable object simulation and used to provide haptic

feedback in between the updates of the global simulation. The agreement of the local

simulation with the results provided by the global simulation are naturally imperfect.

However, the haptic feedback is qualitatively similar enough to be convincing to the

16

user, and can both be computed from the local model rapidly at every global update

cycle and also simulated at the haptic update rate.

3.2.1 Two Dimensional Case - Sample Construction of the

Local Linear Approximation Model

x x

x x

1 2

3 4

Figure 3.4: Four node fully connected 2-D mass-spring-damper model

Consider the two dimensional model shown in Figure 3.4 consisting of four nodes

fully connected by damped springs represented by spring and damper pairs. We wish

to linearize each spring and damper. Let p = [p1, p2] and q = [q1, q2] be the location

of a pair of nodes np and nq in the network. For a spring with natural length L0 and

spring constant k connecting nodes ns and nt, the force on node np is given by

fs = k

[(‖q − p‖ − L0)

(q − p)‖q − p‖

]= k

[(q − p)− L0

q − p‖q − p‖

](3.5)

17

We wish to find a linearized equation for fs in the form

fs ≈ fs0 +[

∂fs∂p

∂fs∂q

] ∆p∆q

(3.6)Define

Apqi = −∂fsi∂pi

= k

[1− L0

‖q − p‖‖q − p‖2 − (qi − pi)2

‖q − p‖2

]

Bpqi,j = −∂fsi∂pj

= k

[L0

‖q − p‖(qi − pi)(qj − pj)

‖q − p‖2

], i 6= j

(3.7)

Kp,q =

Apq1 Bpq1,2Bpq2,1 A

pq2

(3.8)Then, define

K =

[Kp,q −Kp,q

](3.9)

so that the action of the spring is linearized by

fs ≈ fs0 + K

∆p∆q

(3.10)The construction of the linearization of each damper is similar. Let w = [w1, w2]

and v = [v1, v2] be the velocities of the nodes np and nq. For a damper with damping

constant b, the force fd on node np due to the damper is given by

fd = b

[(w − v)T (q − p)

‖q − p‖

](q − p)‖q − p‖

(3.11)

We wish to find a linearized equation for fd in the form

18

fd ≈ fd0 +[

∂fd∂p

∂fd∂q

] ∆p∆q

+ [ ∂fd∂w

∂fd∂v

] ∆w∆v

(3.12)Define

Cpqi = b

[(wi − vi)

qi − pi‖q − p‖2

+ ((w − v)T (q − p))‖q − p‖2 + 2(qi − pi)2

‖q − p‖4

]Dpqi,j = b

[(wj − vj)

qi − pi‖q − p‖2

+ ((w − v)T (q − p))2(qi − pi)(qj − pj)‖q − p‖4

]Epqi = b

(qi − pi)2

‖q − p‖2

F pqi,j = b(qi − pi)2

‖q − p‖2(3.13)

V p,q =

Cpq1 Dpq1,2Dpq2,1 C

pq2

Bp,q =

Epq1 F pq1,2F pq2,1 E

pq2

(3.14)Then, define

V =

[V p,q −V p,q

](3.15)

B =

[Bp,q −Bp,q

](3.16)

so that the action of the spring-damper pair is linearized by

19

ftotal ≈ fs0 + fd0 + (K + V )

∆p∆q

+ B ∆w

∆v

(3.17)The linearization of all the springs and dampers in the network of figure 3.4 would be

f ≈ f0 + P∆X + B∆Ẋ = fo + F

∆X∆Ẋ

(3.18)with

F =

[P B

](3.19)

with P and B being n× n matrices of the forms

P =

−K1,2−K1,3−K1,4

−V 1,2−V 1,3−V 1,4K1,2+V 1,2 K1,3+V 1,3 K1,4+V 1,4

K2,1+V 2,1−K2,1−K2,3−K2,4

−V 2,1−V 2,3−V 2,4K2,3+V 2,3 K2,4+V 2,4

K3,1+V 3,1 K3,2+V 3,2−K3,1−K3,2−K3,4

−V 3,1−V 3,2−V 3,4K3,4+V 3,4

K4,1+V 4,1 K4,2+V 4,2 K4,3+V 4,3−K4,1−K4,2−K4,3

−V 4,1−V 4,2−V 4,3

(3.20)

20

B =

−B1,2−B1,3−B1,4 B1,2 B1,3 B1,4

B2,1 −B2,1−B2,3−B2,4 B2,3 B2,4

B3,1 B3,2 −B3,1−B3,2−B3,4 B3,4

B4,1 B4,2 B4,3 −B4,1−B4,2−B4,3

(3.21)

3.3 Collision Detection and Response in Multi-

Rate Simulation

The method explored here incorporates a novel local collision detection algorithm

in which the local simulation performs collision detection and response ‘in-between’

frames of the global model. The rationale behind performing collision detection and

response at the local level is two-fold. First, it allows for a more realistic high-

frequency interaction. In particular, intermittent contact, like dragging an instrument

across a rough surface, may be negatively affected by latency from the global model.

In a networked environment in particular, the local side must continue to provide

high fidelity interaction during unexpected delays. Maintaining geometric information

sufficient to perform collision response can alleviate these issues. Secondly, for simple

models, the response of the local model may be superior to the computed response

of the global model, as the local model has more detailed information about the

trajectory of the instrument during the global model time period.

Two common paradigms are used for collision response algorithms; constraint-

based and penalty-based collision response. Constraint-based methods, sometimes

referred to as geometrically-based methods, resolve the collision by enforcing con-

straints that prevent the model from entering an invalid state. For example, in a

21

simple simulation of ball rolling across a surface, a constraint-based method might

apply the constraint that the ball’s position in the y direction must remain above the

level of the surface. Constraint-based methods have the disadvantage that they do

not directly calculate contact forces, which is necessary for haptic feedback. Penalty

force based methods operate by connecting virtual springs to interpenetrating objects

that pull them apart. Penalty force methods require little computation and compute

a contact force suitable for haptic interaction, but have several important limitations.

Penalty force methods have difficulty taking friction into account, have trouble with

very rigid objects, and are typically hard to generalize to multi-point contact between

objects [12].

For this simulation, a hybrid approach is used for haptic instrument-deformable

surface collisions that applies a geometric deformation to the deformable object di-

rectly, as in constraint-based methods, and then computes a ‘constraint force’ neces-

sary to overcome the internal forces of the deformable surface model that oppose the

deformation. This constraint force is applied to the nodes of the mesh, and is also

used to compute a contact force that is applied to the haptic interface, much as a

‘penalty’ force would be.

The development of the hybrid method is presented in two stages. First, a one

dimensional example algorithm is presented to demonstrate the rationale behind the

hybrid technique, and then in the next section, the full hybrid algorithm is described.

3.3.1 1D Example of Hybrid Collision Resolution

The collision response algorithm is a hybrid approach that applies a geometric con-

straint, and then computes constraint forces necessary to maintain that constraint.

First, the simulation is advanced by one time step, and the system is checked for

22

instrument

object surface internal spring

fixed `wall’

X∆ X∆f

f

instrument

object

A. B.

C. D.

Figure 3.5: 1-D Example of hybrid collision resolution model

collisions. If a collision is occurring between the instrument and the deformable sur-

face, the simulation is reversed in time to the collision time, and a deformation is

computed from this state that will result in no collision occurring. Then, constraint

forces are determined that will prevent the system from entering an illegal state when

the remainder of the time-step is re-incremented. The constraint forces are then used

to generate a contact force to apply to the haptic instrument. To illustrate this pro-

cess on a simplified one dimensional model, consider figure 3.5-A. A single mass is

connected to a fixed ‘wall’ by a linear spring with constant k, and can be depressed

by an instrument in one dimension. If the instrument is pushed into the surface,

the situation in figure 3.5-B. occurs. To resolve this collision, first a geometric defor-

mation is applied to the model to resolve the collision, as in figure 3.5-C. Then, the

constraint force is determined that will prevent the collision from re-occurring when

the time step is computed; in this model, the constraint force is equal to the spring

force

23

fc = k∆X (3.22)

In the one-dimensional model, this constraint force is also used as the contact

force, so finstrument = fc is applied to the haptic instrument as feedback to the user,

as in figure 3.5-D. The time step is then advanced again. Figure 3.6 shows a flowchart

diagram of the logic behind this collision response algorithm.

Computecontact force and

apply to instrument

Check for collisionbetween mesh

and instrumentduring last time step

no collisionfound

collision found

Advance model to time t+1

Read positionof Instrument

from haptic device

Reverse state of model to time

of collision

Apply Deformation

to Mesh

Computeconstraint force

on Mesh

Advance model to time t+1, applying

constraint force

Go to next time step

Figure 3.6: One dimensional hybrid collision resolution model flowchart

It is assumed that the system is modeling the behavior of a soft deformable surface

with mass evenly distributed throughout the region between the node ‘surface’ on the

‘wall.’ Therefore, inertial effects resulting from directly changing the position of the

node is considered to be of negligible magnitude compared to the force generated by

the spring, and are ignored.

24

3.3.2 Hybrid Collision Resolution in 3D MSD models

The collision response algorithm uses the same basic strategy as the 1-D example

case presented in figure 3.6, with the addition of a more complex stick-slip friction

model. In this frictional model, there are three possible modes of interaction between

the haptic instrument and the deformable surface model:

1. Pure Sticking - the tip of the instrument drags the contact point of the mesh

along with it, as if the mesh has a very high coefficient of friction,

2. Pure Sliding - the tip of the instrument pushes down on the mesh, but exerts

no force tangential to the mesh surface, as if the mesh were frictionless,

3. Frictional Sliding - the tip of the instrument applies both a normal and a tan-

gential force on the surface, but the initial contact point does not remain un-

derneath the instrument tip throughout the contact.

In the hybrid collision response algorithm, the collision is resolved geometrically

before constraint forces are applied. However, unlike the 1-D algorithm, in which

there is only one strategy to deform the model and resolve the collision, each one of

these modes of interaction has associated with it a different geometric deformation

and method of determining constraint forces. For objects with extremely low surface

friction, the Pure Sliding mode is used, while for objects with extremely high surface

friction or very ‘sticky’ surfaces, Pure Sticking is used.

Objects that have intermediate values of friction are handled by trying the Pure

Sticking mode first, and evaluating how consistent the resulting constraint forces are

with the stick-slip frictional model. If the Pure Sticking behavior is rejected by the

heuristic, a hybrid mode referred to as Frictional Sliding is used to resolve the collision.

As an example, when the Pure Sticking constraint is determined, a large contact force

25

would be required to counteract the action of the springs and drag the mesh along

with the instrument at the point of contact. If the frictional force caused by moving

the instrument along the mesh surface could result in this contact force, it is deemed

to be realistic by the heuristic, and the results of this mode are retained and other

modes are not considered. Otherwise, Frictional Sliding is evaluated instead.

In combined Pure Sticking/Frictional Sliding collision response, Pure Sticking is

evaluated first using this basic procedure; first, the mesh is deformed so as to resolve

a collision between a node and a triangle of the deformable surface mesh. Then,

constraint forces for each node of this triangle are determined that will prevent the

system from re-entering an interpenetrating state during this time step are computed.

These constraint forces are the forces necessary to counteract the springs of the mesh

and maintain the mode of interaction. The contact force is then set to be the sum of

the three node constraint forces. This contact force is then broken down into forces

normal (N) and tangential (T ) to the mesh surface. The maximal frictional force

consistent with the friction model is calculated as

Ffr = µN (3.23)

If the force T is greater than Ffr, then the evaluation heuristic rejects the case as

inconsistent with the friction model, and therefore unrealistic to use to resolve the

interaction, and the given mode of interaction is rejected. The Frictional Sliding case

is then computed using a similar procedure and is used instead.

After the correct mode of interaction has been determined, the mesh is backed up

to the time of collision and the constraint force is applied to the nodes of the triangle.

When the simulation is advanced again to the end of the time step, the constraint

force enforces the mode of interaction.

26

Check for collisionbetween mesh

and instrumentduring last time step

no collisionfound

collision found

Advance model to time t+∆t

Read positionof Instrument

from haptic device

Reverse state of model to time

of collision

Advance model to time t+∆t, applying

constraint force

Go to next time step

if friction = 0

Apply Pure Sliding Deformation

to Mesh

Compute Pure Slidingconstraint force

on Mesh. Computecontact force and

apply toinstrument

if friction > 0Apply Pure Sticking

Deformationto Mesh

Compute Pure Stickingconstraint andcontact forces

Evaluate Pure StickingConstraint using

friction modelheuristic

Compute Frictional Sliding constraint force

Compute contactforce and applyto instrument

rejectPure

Sticking

acceptPure

Sticking Apply contact force to instrument

Figure 3.7: Three dimensional hybrid collision resolution model flowchart

27

3.4 Mesh Deformation in Collision Response

During collision detection, after the intersection point is determined and the model

is iterated to the collision time, a deformation must be applied to the mesh in order

to resolve the collision and prevent interpenetration from occurring. Each mode

of interaction in the collision response algorithm has a corresponding deformation,

illustrated in figure 3.8. In sliding contact, the triangle is pushed down in the normal

direction by an amount equal to the interpenetration distance, as shown in figure

3.8-b. In sticking contact, the triangle is instead pushed in the direction of the

instrument’s motion, so it is displaced in both the normal and tangential directions

relative to the triangle surface, as illustrated in figure 3.8-c. The frictional motion

case is handled in the same manner as sticking motion.

3.5 Determination of Constraint Forces

During collision resolution, after the deformation of the mesh, a constraint process

is used to determine the contact force. Separate constraints are used in the pure

sticking and pure sliding cases. In order to identify particular points on the surface of

a deformable triangle, baryocentric co-ordinates are used. Baryocentric co-ordinates

are a concept borrowed from FEM techniques. A given point P on a triangle is

identified by the areas of the sub-triangles formed by extending lines from the node

vertices to the interior point. Normalized baryocentric co-ordinates are used such

that the position of an interior point P on the triangle with vertices x1, x2, and x3

can be expressed as

p = αx1 + βx2 + γx3 (3.24)

28

a. b.

c.

Figure 3.8: Triangle displacement in sticking and sliding mode. Diagrama. shows the position of the instrument at the beginning and end of atime step. Diagram b. illustrates the sliding displacement. Diagram c.illustrates the sticking displacement.

x

x

x1

2

3

p

A3A2

A1

α =

A = A1 A2 A3+ +

A1A

β = A2A

γ = A3A

Figure 3.9: Baryocentric Co-ordinates

29

such that α + β + γ = 1.

3.5.1 Sticking Constraint

The sticking mode of interaction assumes that the instrument makes contact with the

mesh at a contact point with baryocentric co-ordinate p and that at the end of the

haptic time step, the instrument is still in contact with a point of the triangle with

the same baryocentric co-ordinate relative to the new positions of the nodes of the

triangle. This constraint can be expressed relative to p as

ṗ = p̈ = 0 (3.25)

Let the 9×9 matrix T11 be an orthogonal basis in baryocentric co-ordinates given

by

T11 =

αI3×3 βI3×3 γI3×3

......

...

......

...

(3.26)where I3×3 is the 3×3 identity matrix, and the empty entries of T11 are generated

using the Gram-Shmidt method. Define T to be a 3m × 3m transformation matrix

from the canonical object frame into baryocentric co-ordinates relative to the contact

triangle where m is the number of nodes in the model. We assume without loss of

generality that the contact triangle has nodes x1, x2, and x3. Then, T is given by

T =

T11 00 I

(3.27)

30

and

T

x1...

xn

=

p

...

...

(3.28)In order to enforce the constraint ṗ = p̈ = 0, define the P operator to be

P ≡ T−1

03×3 00 I

T (3.29)When computing the discrete equations of motion to advance the simulation to the

beginning of the next time step, the equations of motion

ẋnew = ẋold + M−1f(x, ẋ)∆t

xnew = xold + ẋnew∆t (3.30)

become

ẋnew = P (ẋold + M−1f(x, ẋ)∆t)

xnew = xold + ẋnew∆t (3.31)

The force on the instrument due to the collision is assumed to be the sum of the

force on each node of the triangle required to maintain the constraint. Define the P̃

operator to be

P̃ ≡ T T

I3×3 00 0

T−T (3.32)

31

The force on the instrument is then given by

fc1

fc2

fc3...

...

= P̃ f(x, ẋ)

finstrument = fc1 + fc2 + fc3 (3.33)

3.5.2 Sliding Constraint

The sliding mode of interaction assumes that the instrument makes contact with the

mesh as a contact point with baryocentric co-ordinate p, and at the end of the haptic

time step, the contact point has moved only in a plane tangential to the triangle. This

ensures that the mesh surface can move tangentially to the instrument, but cannot

inter-penetrate it at the end of the time step. This is a weaker constraint than the

sticking case. The constraint can be expressed relative to p and the triangle normal

n as

ṗ · n = 0

p̈ · n = 0 (3.34)

Define U0 to be a 3×3 orthogonal matrix of the form

32

U0 =

nT

· · · · · · · · ·

· · · · · · · · ·

(3.35)where the empty entries are generated using the Gram-Shmidt method. Define

U11 =

U0 0 0

0 I3×3 0

0 0 I3×3

(3.36)Let U be an 3m×3m transformation matrix from the canonical object frame into

baryocentric co-ordinates relative to the contact triangle, where m is the number of

nodes in the model. We assume without loss of generality that the contact triangle

has nodes x1, x2, and x3. Then, U is given by

U =

U11T11 00 I

(3.37)where T11 is as defined above in the description of sticking contact. In order to enforce

the constraint ṗ · n = p̈ · n = 0, define the Q operator to be

Q ≡ UT

01×1 00 I

U−T (3.38)When computing the discrete equations of motion to advance the simulation to

the beginning of the next time step, the equations of motion become

ẋnew = Q(ẋold + M−1f(x, ẋ)∆t)

33

xnew = xold + ẋnew∆t (3.39)

The force on the instrument due to the collision is assumed to be the sum of the

force on each node of the triangle required to maintain the constraint. Define the Q̃

operator to be

Q̃ ≡ U−1

11×1 00 0

U (3.40)The force on the instrument is then given by

fc1

fc2

fc3...

...

= Q̃f(x, ẋ)

finstrument = fc1 + fc2 + fc3 (3.41)

3.5.3 Frictional Sliding Contact Constraint

The frictional contact constraint is the same as the sticking contact constraint, except

that the constraint forces are divided into forces normal, FN , and tangential, FT , to

the triangle surface. The forces tangential to the node surface are bounded by µFN ,

the surface coefficient of friction, and then are applied as before. Since the sticking

mesh deformation is used for frictional sliding, during frictional sliding, the instrument

pushes the mesh surface, but the mesh can still slide around under the instrument

tip, since the frictional force is insufficient to cause pure sticking. Once again, we

34

assume without loss of generality that the contact triangle has nodes x1, x2, and x3.

The force on the nodes is then given by

f1

f2

f3...

...

= f(x, ẋ)

f̃1

f̃2

f̃3...

...

= P̃ f(x, ẋ) (3.42)

using the definitions presented above in the Pure Sticking section. For each force fi,

let fni and fti be the components of fi in the normal and tangential directions relative

to the triangle normal n. Then let the constraint forces f̂i be given by

f̂i = fni + µ‖fni‖fti‖fti‖

(3.43)

Then, the node forces for the frictional sliding case, F , is given by

F =

f1 − f̂1

f2 − f̂2

f3 − f̂3

f4

f5...

...

(3.44)

The equations of motion for the frictional sliding case are then

ẋnew = Q(ẋold + M−1F∆t)

35

xnew = xold + ẋnew∆t (3.45)

To determine the instrument force, first let finstrument be defined as

fc1

fc2

fc3...

...

= P̃ f(x, ẋ)

finstrument = fc1 + fc2 + fc3 (3.46)

as in the pure sticking case. Let finstrumentn and finstrumenttbe the normal and tan-

gential components of that force relative to the triangle normal. Then, the instrument

force for the frictional sliding case is

f̂instrument = finstrumentn + µ‖finstrumentn‖finstrumentt‖finstrumentt‖

(3.47)

3.6 Implementation

The collision response algorithm described above was implemented in a virtual en-

vironment simulation of a deformable object using multi-rate techniques. The test

objects were regular grids of lumped mass-spring-damper elements with sizes ranging

from 5×4×5 to 10×10×10, and the local-low-order model was generated from the

node on the object surface closest to the virtual instrument and the immediately

adjacent nodes in a 4×3×4 neighborhood. The full non-linear global simulation was

updated at 30 Hz, while the local simulation was run at 1 KHz.

36

The simulation was implemented in C++ using OpenGL as the graphics library.

The global simulation was run on a IBM PC running Windows XP, and the haptic

simulation was run on an IBM PC running the QNX real-time operating system.

The global simulation PC used dual Pentium Xeon(TM) processers running at 2.8

GHz. The local simulation PC used a Pentium 4(TM) processor running at 2.4 GHz.

A PHANTOM(TM) version 1.5 manipulator was used as the haptic interface. The

two computers were connected by an 100 Mbs ethernet network running a custom

network protocol implemented on top of UDP. The computers were connected through

a dedicated ethernet switch to eliminate the possibility of interference from other

traffic on the network.

37

Chapter 4

Results

Figure 4.1: Haptic simulation screenshot

A simulation implementing these methods was successfully implemented. Figure

4.1 is a screenshot of the implementation. The red cursor marking the location of

the haptic instrument is visible above the gray surface of the MSD model network.

The red box surrounds the area of the mesh that is used to generate the local model.

38

Figure 4.2: Screenshot showing contact between the instrument and thedeformable object

Figure 4.2 shows interaction between the network and the instrument. The first

section addresses the performance of the multi-rate simulation technique with the

mass-spring-damper model. The next two sections examines performance claims of

Pure Sticking and Pure Sliding modes of the algorithm, and the final section presents

results for full stick-slip friction algorithm.

Table 4.1: Parameters used in MSD model simulationK, spring constant 10.0 N/mm

Node spacing 1 cmNode mass 500 mg

Damping factor 1.5 N·s/mmGlobal time step 100 msHaptic time step 1 ms

39

4.1 Comparison of Local and Global Model Simu-

lations

The basic properties and performance of the multi-rate simulation technique were

tested using with instrument motion specified programmatically, rather than using

the haptic device for input, so that the input could be kept constant between test runs

in order to establish the difference in performance between the global and local model

simulations. Three test cases were constructed, each using the same pre-determined

instrument motion, in which a single node was pulled 2 cm above the surface of the

mesh, and then released. The force exerted on the instrument as the surfaces deforms

and presses against it was then computed.

The first test case uses the global model simulation only, updated at 30 Hz as

specified in the multi-rate algorithm, but does not run the local model simulation.

The second test case also uses the global model simulation only, but updates it at

1000 Hz. This is not possible in real time, but this test case provides a base line

for comparison of the behavior of the simulation. Finally, the last test case uses the

multi-rate simulation technique described above, in which the global model runs at

30 Hz, and the local model running at 1000 Hz is used to generate output ‘between’

global updates. The force output during the same part of the trial run is presented

in figure 4.3.

The global model running at 1000 Hz provides the most accurate output; the force

follows a smooth sinusoidal curve consistent with the mass-spring-damper model.

The global model running at 30 Hz exhibits the same approximate behavior, but the

force output shows a stair-step pattern due to the lower update rate. This method

results in a large discontinuity after every global update. The multi-rate simulation

follows the tangent behavior of the system between each global update. While there

40

2.4

2.45

2.5

2.55

2.6

1.2 1.3 1.4 1.5 1.6 1.7 1.8

Time, seconds

Forc

e, N

Figure 4.3: Multi-rate simulation results. Dotted line is global model run-ning at 1000 Hz. Dashed line is global model running at 30 Hz. Solid lineis local model running at 1000 Hz.

is a discontinuity as each global update is received and a new tangent behavior is

calculated, the magnitude of the jump is much lower than the height of the stair-step

discontinuity from the global model-only test case.

A second set of tests was performed to determine the effect of local model reduction

on the simulation. Since in the multi-rate simulation, part of the surface local to the

instrument tip is being simulated using the local model simulation while the rest is

being simulated using the global model simulation, tests were performed to establish

if this division resulted in inaccuracies in the model behavior. In these tests, the

41

instrument was held immobile over a section of the global model, so that a patch of

the surface under the instrument tip was simulated using the local model. The surface

was then deformed programmatically by pulling the node under the instrument tip

upwards and then releasing it, so that a wave-like motion spread through the model.

The instrument was held at a sufficient distance to ensure that it never touched the

surface during the trial. The test was then repeated with the instrument moved

away from the surface, so that no local model was used, and the entire simulation

was performed using the global model. The position of the deformed node along the

y-axis was then compared between the two tests, as presented in Figure 4.4.

The behavior of the multi-rate simulation was found to show a high degree of

agreement with the global model only simulation, with primarily a delay of one global

time step differentiating the two.

4.2 Pure Slipping Constraint Results

The Pure Slipping constraint was tested in several ways. First, in a single-axis contact

test, the instrument was constrained to move only in one dimension, perpendicular

to the mesh surface. The instrument was then pushed down 1.0 cm into the mesh

and held at that depth. The magnitude of the force output during the contact is

presented in Figure 4.5, and shows that the force response to the instrument initially

oscillates before settling to a steady-state value.

In a second test, the instrument was moved to one edge of the mesh, and dragged

over the surface. Figure 4.6 shows the magnitude of the force on the instrument

normal to the mesh surface. The magnitude of the tangential force response is zero

throughout the experiment. When the instrument is pushed into the surface, con-

tact forces are generated in the normal direction but not in the tangential direction,

42

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

0 0.5 1 1.5 2 2.5 3 3.5 4

time, seconds

y - p

ositi

on o

f nod

e ab

ove

orig

in, c

m

Comparison of local and global model behavior

Figure 4.4: Multi-rate simulation results, y-position of test node vs time.Solid line is global model running at 30 Hz, dashed line is local modelrunning at 1000 Hz.

43

0

1

2

3

4

5

6

4.5 9.5 14.5 19.5 24.5 29.5time, seconds

forc

e, N

Figure 4.5: Pure Slipping constant contact results.

44

consistent with a zero-friction model.

Sliding Contact

0

1

2

3

4

5

6

7

8

9

32 34 36 38 40 42time, seconds

forc

e, N

Figure 4.6: Pure Slipping dragging contact results.

4.3 Pure Sticking Constraint Results

The dragging contact experiment was also repeated to test pure sticking contact to

simulate a ‘sticky’ high co-efficient of friction surface. A tangential as well as a normal

force was observed, with a tangential force often larger than the normal, as the motion

of the instrument is mostly in the tangential direction.

45

Sticking Contact

0

2

4

6

8

10

12

14

10 10.5 11 11.5 12 12.5 13 13.5 14time, seconds

forc

e, N

Figure 4.7: Pure Sticking contact results. Solid line is magnitude of normalforce, dotted line is magnitude of tangential force.

46

4.4 Stick-Slip Frictional Sliding Friction Results

Stick-Slip Hybrid Algorithm

0

0.5

1

1.5

2

2.5

3

3.5

4

60 61 62 63 64 65

Forc

e, N

time, seconds

Figure 4.8: Hybrid stick-slip dragging results, co-efficient of friction 1.0.Solid line is magnitude of force normal, dotted line is magnitude of tan-gential force.

In this section, the full stick-slip frictional contact algorithm was tested. The

algorithm was allowed to switch between pure sticking and frictional sliding, and the

dragging contact experiment presented in the Pure Sticking constraint results above

was repeated. The test was performed with a co-efficient µ of friction of 1.0, with

results as shown in Figure 4.8. The tangential force is then bounded by the normal

force, since FT ≤ µFN . Therefore, when the force generated by the sticking constraint

rises above the normal force, the algorithm switches from pure sticking to frictional

47

sliding, leading to a decrease in force. A typical transition event can be seen at t =

61.8 Before the transition, the mesh is being dragged under the instrument tip by

the sticking constraint. As the mesh is deformed, the force required to maintain the

constraint increases, leading to an increase in both the normal and tangential force.

In the pure sticking results presented above, the tangential force increases above µ

times the normal force, since the dragging motion is mostly in the tangential direction.

However, in the hybrid model, instead the algorithm transitions and the tangential

force decreases.

48

Chapter 5

Discussion

5.1 Extensions to Literature

The main findings from this work extend the literature in two ways. First, the new

linearization method for Mass-Spring-Damper models is an improvement over existing

methods of building local approximations. Second, the stick-slip collision response

offer a new direction in contact resolution in haptic environments.

The linearization method was found to produce high quality force output at the

haptic update rate. In addition, it was established experimentally that the lineariza-

tion could be constructed and simulated both within the time constraints of the global

and haptic update rates and also with a high degree of accuracy.

Several important insights can be gained from the local hybrid collision resolution

algorithm. While the algorithm successfully generated reasonable contact forces, two

major caveats apply. First, the resolution of the mesh was relatively low, and so

triangulation was fairly coarse. As the instrument is dragged over the mesh, the facets

of the mesh are very haptically apparent. The periodic large discontinuities present

in both the pure sticking and hybrid algorithm dragging experiments were largely

49

the result of the instrument moving over a triangle boundary. Second, the method

requires that the high-frequency behavior of the model be dependent primarily on

local states. In the case of a non-homogenous material, this assumption could be

violated, resulting in discontinities and unrealistic force output.

5.2 Limitations

Several limitations qualify the results of this study. First, haptic virtual environments

with deformable surfaces remain rare, and detailed results from an alternative imple-

mentation from another research group with which to compare these results were

not available. This is compounded by the fact that MSD models are not physically

based, meaning that there is no straightforward method of determining MSD mesh

parameters from physical substances, like human tissue. Therefore, evaluation of the

algorithm remains largely qualitative. Second, the effects of highly non-homogenous

materials, which could potentially break the assumption that local effects are domi-

nant, have not yet been explored.

5.3 Conclusion

A novel method of generating linear approximations of MSD models was designed,

implemented, and successfully tested. A haptic update rate contact-resolution algo-

rithm was also test that provided a satisfactory haptic experience.

50

5.4 Future Work

The limitations found in testing of the implementation of this project suggest several

fruitful avenues for future work. The coarse granularity of the mesh was found to

cause discontinuities and loss of contact as the instrument rolls over triangle bound-

aries. Multi-resolution methods that dynamically re-triangulate the mesh are one

possibility to overcome these issues. In addition, it may also be possible to extend

the algorithm along lines inspired by the industry-standard Phong shading algorithm

by interpolating the triangle normals across the face of each triangle, so that a contin-

uous normal can be defined across multiple triangles. This may result in a smoother

force feedback when traveling over triangle boundaries, since the direction of the force

output would change continuously.

Additional work to establish a unified framework with which to compare results

from haptic simulations would strengthen the results of this and many other studies

by allowing quantitative comparisons of the haptic experience. Work has already

begun to adapt methods from this study to the GiPSi simulation framework, an

open source simulation framework for surgical simulation with haptic feedback [4]. It

would also extend the results of this study to make the deformable model strongly

physically based. The linearization method described for MSD could be adapted

for use with FEM models, which accurately model physical materials. In addition,

unpublished work by Cavusoglu and Natsupakpong aims to develop a method of

generating MSD mesh parameters from physical measurements, which would also

allow direct comparison of the haptic experience to actual manipulation.

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