A DIRECT KINEMATIC COMPUTATION ALGORITHM FOR …digitool.library.mcgill.ca/thesisfile33962.pdf · A DIRECT KINEMATIC COMPUTATION ALGORITHM FOR ALL PLANAR 3-LEGGEDPLATFORMS CHEN,CHAü

A DIRECT KINEMATIC

COMPUTATION ALGORITHM FOR

ALL PLANAR 3-LEGGED PLATFORMS

CHEN,CHAü

REng., Shanghai Jiaotong University, 1996

Department of Mechanical Engineering and

Centre for Intelligent Machines

McGill University

Montréal, Québec, Canada

November, 2001

A Thesis submitted to the Faculty of Graduate Studies and Research

in partial fulfilment of the requirements for the degree of

Master of Engineering

© C. CHAO, 2001

1+1 National Libraryof Canada

Bibliothèque nationaledu Canada

Acquisitions and Acquisitions etBibliographie Services services bibliographiques

385 Wellington Street 395. rue w"ngtonOttawa ON K1A 0N4 Ottawa ON K1A 0N4Canada can.cta

'-"~--

The author bas granted a nonexclusive licence alloWÎDg theNational Libnuy ofCanada toreproduce, loan, distnbute or seUcopies of tbis thesis in microform,paper or electronic formats.

The author retains ownershîp of thecopyright in this thesis. Neither thethesis nor substantial extracts nom itmay he printed or otherwisereproduced without the author'sperm1SS1on.

L'auteur a accordé une licence nonexclusive permettant à laBibliothèque nationale du Canada dereproduire, jriter, distribuer ouvendre des copies de cette thèse sousla forme de microfiche/film, dereproduction sur papier ou sur formatélectronique.

L'auteur conserve la propriété dudroit d'autem qui protège cette thèse.Ni la thèse ni des extraits substantielsde celle-ci ne doivent être imprimésou autrement reproduits sans sonautorisation.

0-612-79065-7

Canad~

AB8TRACT

ABSTRACT

An analysis and comprehensive solution to the direct kinematics problem(DK) of

aIl planar 3-legged platforms with lower pairs, called general planar Stewart-Gough

platform(PSGP), is presented. There are 10 types of PSGP DK problem formulation

including those with mixed leg architecture.

Planar kinematic mapping expresses pole position and rotation angle of a pla

nar displacement as a point in 3-dimensional projective space represented by 4 ho

mogeneous coordinates. This provides a universal tool for kinematic analysis. Its

application will be demonstrated in the derivation of a general algorithm for pla

nar DK. For each type of PSGP, the problem is reduced to a 6th order univariate

polynomial whose roots reveal aIl solutions. An example of a PSGP with 6 real as

sembly configurations is presented. Furthermore, this algorithm was implemented

and tested exhaustively. A complete self-contained version, coded in C, is available

at http:j jwww.cim.mcgil1.caj"'paulj. It should be easy to customize and adapt to

any given real time micro-controller application.

RÉSUMÉ

RÉSUMÉ

Cette thèse présente l'analyse et la solution détaillée du modèle géonétrique direct

(MGD) s'appliquant à toutes les plates-formes planeaires dont la base est un triplet

de pattes articulées par couples rotoïdes ou prismatiques. Ce type de plate-forme

est généralement appelée plate-forme Stewart-Gough planaire (PSGP). Il existe dix

formes de MGD concernant les PSGP, y compris les plates-formes possédant les deux

types de pattes (rotoïde et prismatique).

Les applications de cinématique planaire expriment la position des pôles et l'langle

de rotation d'un déplacement planaire comme point dans l'espace projectifreprésenté

par quatre coordonnées homogènes. Cette approche est un outil universel pour

l'anàlyse cinématique. L'efficacité de cette dernière avec les MGD des PSGP sera

démontrée et par conséquent il en résultera la construction d'un algorithme adapté

au MGD. Pour chaque type de PSGP, le problème se réduit à un polynôme univariable

du 6è degré dont les racines donnent lieu à toutes les solutions en question. Un exem

ple de PSGP avec six configurations réelles est présenté. En plus, l'algorithme utilisé

fut mis en application et testé de façon approfondie. Une version exhaustive, codée

en C, est disponible au site http:j jwww.cim.mcgill.caj",paul/. Selon toute vraisem

blance, il devrait être raisonnablement facile d'adapter l'algorithme à n'importe quelle

application en temps réel moyennant des microcontrôleurs.

ii

ACKNOWLEDGEMENTS

ACKNOWLEDGEMENTS

Here, 1 must thank Professor Paul Zsombor-Murray. It is he who introduced me to

robot kinematics, which perfectly suits my interests. He taught me about kinematic

mapping and helped me at each step of my investigation. What should be mentioned

is that Professor Zsombor-Murray's humorous approach to research made mine an

enjoyable game.

1 also acknowledge the help of Professor M. John D. Hayes. He provided me

with his thesis files, which helped me greatly in my thesis preparation. His excellent

example was an inspiration to me.

Xiaohui Song and Johnathan Shum are two fellow students who helped me ac

c1imatize to the environment at the Centre for Intelligent Machines at McGill and to

master the daily tasks of using Linux and Maple and "burning" a CD.

Finally, 1 am most grateful to my parents, Guangxin Chen, Qinghui Yang, and

my wife, Xiaoling Wu. They always encouraged me to focus on my research and took

care of my life.

iii

TABLE OF CONTENTS

TABLE OF CONTENTS

ABSTRACT .

RÉSUMÉ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ii

ACKNOWLEDGEMENTS ili

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. VI

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. viii

CHAPTER 1. INTRODUCTION........................ 1

1.1. THESIS SUBJECT DEVELOPMENT 1

1.2. THESIS OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . . . .. 2

CHAPTER 2. PLANAR KINEMATIC MAPPING 3

2.1. HOMOGENEOUS COORDINATES . . . . . . . . . . . . . . . . . .. 3

2.2. DUALITY................................. 4

2.3. COORDINATES OF POINT AND LINE ON A PLANE . . . . . .. 5

2.4. COORDINATES OF POINT AND PLANE IN 3-D SPACE . . . . .. 6

2.5. KINEMATIC MAPPING OF PLANAR DISPLACEMENT . . . . .. 7

CHAPTER 3. PLANAR 3-LEGGED PLATFORM . . . . . . . . . . . . .. 9

3.1. TYPES OF JOINTS AND LEGS . . . . . . . . . . . . . . . . . . .. 9

3.2. TYPES OF PLANAR PLATFORM . . . . . . . . . . . . . . . . . .. 11

CHAPTER 4. DIRECT KINEMATICS . . . . . . . . . . . . . . . . . . . .. 13

4.1. RR LEG EQUATION. . . . . . . . . . . . . . . . . . . . . . . . . .. 13

4.2. PR LEG EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . .. 15

4.3. RP LEG EQUATION. . . . . . . . . . . . . . . . . . . . . . . . . .. 16

IV

TABLE OF CONTENTS

4.4. pp LEG EQUATION. . . . . . . . . . . . . . . . . . . . . . . . . .. 18

4.5. ALGORITHM FOR EACH TYPE OF PLATFORM . . . . . . . . .. 18

4.5.1. ALGORITHM FOR A C-C-C PLATFORM . . . . . . . . . . .. 18

4.5.2. ALGORITHM FOR A L-L-L PLATFORM . . . . . . . . . . . .. 22

4.5.3. ALGORITHM FOR A V-V-V PLATFORM . . . . . . . . . . .. 23

4.5.4. ALGORITHM FOR A C-C-L PLATFORM 26

4.5.5. ALGORITHM FOR A C-C-V PLATFORM . . . . . . . . . . .. 29

4.5.6. ALGORITHM FOR A L-L-C PLATFORM. . . . . . . . . . . .. 31

4.5.7. ALGORITHM FOR A L-L-V PLATFORM. . . . . . . . . . .. 33

4.5.8. ALGORITHM FOR A V-V-C PLATFORM . . . . . . . . . . .. 36

4.5.9. ALGORITHM FOR A V-V-L PLATFORM 38

4.5.10. ALGORITHM FOR A C-L-V PLATFORM . . . . . . . . . . .. 41

4.5.11. ALGORITHM FOR AT-TYPE PLATFORM . . . . . . . . .. 43

4.6. OBTAINING PLANAR DISPLACEMENT . . . . . . . . . . . . . .. 51

CHAPTER 5. APPLICATION OF THE ALGORITHM 53

5.1. EXAMPLE TO ILLUSTRATE ALGORITHM USE. . . . . . . . .. 53

5.1.1. DECIDING THE TYPE OF THE PLATFORM . . . . . . . . .. 53

5.1.2. CHOOSING THE REFERENCE FRAME . . . . . . . . . . . .. 54

5.1.3. COMPUTATION........................... 55

5.1.4. POSITION OF MOVING PLATFORM. . . . . . . . . . . . . .. 55

5.2. EXAMPLE TO SHOW HOW THE CODE WORKS . . . . . . . . .. 56

CHAPTER 6. CONCLUDING REMARKS. . . . . . . . . . . . . . . . . 58

6.1. CONCLUSION . . . . . 58

6.2. A SUGGESTION TO PROMOTE POPULARITY. . . . . . . . . 59

REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

v

LIST OF FIGURES

LIST OF FIGURES

3.1 Types of legs 10

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

4.14

4.15

4.16

4.17

4.18

4.19

MA moves on the circle centered on FA 13

MA moves on the fixed Hne determined by FA and FA1 15

The moving Hne determined by MA and MAI moving on the fixed

point FA . . . . . . . . . 17

The translating platform 18

An ideal reference frame for C-C-C type platforms 19

An ideal reference frame for L-L-L type platforms 21

An ideal reference frame for V-V-V type platforms 23

An ideal reference frame for C-C-L type platforms 26

An ideal reference frame for C-C-V type platforms 28

An ideal reference frame for L-L-C type platforms 31

An ideal reference frame for L-L-V type platforms 34

An ideal reference frame for V-V-C type platforms 36

An ideal reference frame for V-V-L type platforms 39

An ideal reference frame for C-L-V type platforms 41

An ideal reference frame for T-C-C type platforms 44

An ideal reference frame for T-L-L type platforms 45

An ideal reference frame for T-V-V type platforms 46

An ideal reference frame for T-C-L type platforms 48

An ideal reference frame for T-C-V type platforms 49

Vi

LIST OF FIGURES

4.20

5.1

5.2

5.3

An ideal reference frame for T-L-V type platforms

A platform moves on a fixed base

Reference frames of this platform .

AH the modes of this example ..

50

53

54

57

vii

LIST OF TABLES

LIST OF TABLES

3.1 Practical leg types . 10

3.2 The types of PSGP 11

3.3 The possible architectures of T-type PSGP 12

4.1 The coordinates of each point(C-C-C) · . 19

4.2 The coefficients of each equation(C-C-C) 19

4.3 The coordinates of each point(L-L-L) 22

4.4 The coefficients of each equation(L-L-L) 22

4.5 The coordinates of each point(V-V-V) . 24

4.6 The coefficients of the equations(V-V-V) 24

4.7 The coordinates of each point (C-C- L) · . 26

4.8 The coefficients of each equation(C-C-L) . 27

4.9 The coordinates of each point (C-C-V) · . 29

4.10 The coefficients of each equation(C-C-V) 29

4.11 The coordinates of each point(L-L-C) · . 31

4.12 The coefficients of each equation(L-L-C) . 32

4.13 The coordinates of each point (L-L-V) · . 34

4.14 The coefficients of each equation(L-L-V) . 34

4.15 Thecoordinates of each point(V-V-C) .. 36

4.16 The coefficients of each equation(V-V-C) 37

4.17 The coordinates of each point (V-V-L) · . 39

4.18 The coefficients of each equation(V-V-L) 39

viii

LIST OF TABLES

4.19 The coordinates of each point(C-L-V) · . 41

4.20 The coefficients of each equation(C-L-V) 42

4.21 The coordinates of each point(T-C-C) . . 44

4.22 The coefficients of each equation(T-C-C) 44

4.23 The coordinates of each point(T-L-L) · . 45

4.24 The coefficients of each equation(T-L-L) . 46

4.25 The coordinates of each point(T-V-V) .. 47

4.26 The coefficients of each equation(T-V-V) 47

4.27 The coordinates of each point(T-C-L) · . 48

4.28 The coefficients of each equation(T-C-L) . 48

4.29 The coordinates of each point(T-C-V) .. 49

4.30 The coefficients of each equation(T-C-V) 50

4.31 The coordinates of each point(T-L-V) · . 51

4.32 The coefficients of each equation(T-L-V) 51

5.1 The parameters of this platform ..... 56

IX

CHAPTER 1

INTRODUCTION

This thesis is an investigation of the direct kinematics (DK) of planar three-legged

fully-parallel platform manipulators, called planar Stewart-Gough platforms (PSGP).

The kinematic analysis presented here turns out to be general enough to handle all

possible architectures with lower pairs.

1.1. THESIS SUBJECT DEVELOPMENT

In 1965 D. Stewart [19] first suggested that fight simulators could be built as fully

parallel platform type manipulators with six degrees of freedom (DOF). In subsequent

years such manipulators came to be known as Stewart platforms. However, V. E.

Gough [8] had made a design for a tire test-stand nine years earlier, which more

closely resembles the architecture of a modern fiight simulator platform than does

Stewart 's. Therefore, the term Stewart-Gough platform (SGP) is used to refer to this

kind of architecture.

This thesis will focus on a universal approach to the direct kinematics of much

simpler planar Stewart-Gough platforms (PSGP). Considerable effort has been ex

pended on this topic. Much of the earlier research [16, 17, 18] concentrated on

1.2. THE818 OVERVIEW

numerical solutions while yielding insufficient insight into relating practical imple

mentation to theoretical issues. Recent papers [10, 12, 13] on this topic report

algorithms for the topological symmetric PSGP. However, the derivation of a general

and neat univariate polynomial to solve the DK problem of any lower pair jointed

PSGP, including those with mixed legs, was unsolved. That is precisely our goal.

Kinematic mapping is a uniquely powerful tool in the analysis of DK problems.

To give a full explanation of this tool is also a purpose of this thesis. This tool will

be used to derive the general univariate polynomial for the DK problem of any lower

pair jointed PSGP. Furthermore, our results were implemented in an algorithm coded

in C which computes all assembly poses for any PSGP of specified design parameters

and actuated joint variable values.

1.2. THESIS OVERVIEW

In Chapter 2 the projective plane and planar kinematic mapping are introduced.

These concepts will be applied in Chapter 4.

Chapter 3 gives the classification of leg and PSGP architectures with only lower

pair joints. Also the elementary projective planar concepts are put to good use to

deal with situations involving unactuated P joints.

Chapter 4 represents the main contribution of this thesis. AlI algorithms and

their derivation are detailed there. The final univariate polynomials are given with

explicit expression of their coefficients.

Examples are provided in Chapter 5 to show how the algorithms discussed in

Chapter 4 are used.

Finally, Chapter 6 summarizes novel contributions and anticipates future, wider

application of kinematic mapping.

2

CHAPTER 2

PLANAR KINEMATIC MAPPING

In this chapter the geometric and algebraic tools used in the kinematic analysis of

PSGP will be discussed. We focus on mapping planar Euc1idean displacements, a

special case of 3-D Euc1idean displacements. Planar mapping represents the pole po-

sition and rotation angle, associated with any given finite displacement of a planar

rigid body, as a point in a 3-D projective space given by four homogeneous coordi

nates. We start with some simpler, more familiar concepts, preliminary to introducing

coordinates in this kinematic "image space" .

2.1. HOMOGENEOUS COORDINATES

A point in the planar Cartesian coordinate frame (E2 ) is located by two param

eters, (x, y). However, the ratio of three parameters, (Xl: X2 : X3) may also represent

the position of this point. These two kinds of coordinates have relationship as follows.

Xl X2x=-,y=-

X3 X3(2.1)

Therefore, any triplet of the form (ÀXI : ÀX2 : ÀX3) describes exactly the same point

(x, y). The (Xl : X2 : X3) coordinates are called homogeneous coordinates. When

2.2. DUALITY

X3 = 1, the magnitude scaled Cartesian coordinate pair (x, y) is recovered. X3 = adenotes a point on the line at infinity which closes the projective plane.

Similarly, any point in 3-D space also can be represented by both the Cartesian

coordinates (x, y, z) and the homogeneous coordinates (Xl: X2 : X3 : X4)' The relation

between them are given below.

Xl X2 X3x=-,y=-,z=-

X4 X4 X4(2.2)

AIso, X4 = 1 recovers the Cartesian coordinates and X4 = a denotes a point on the

plane at infinity which closes the 3-D projective space.

2.2. DUALITY

In the Euclidean plane, a line is described by a linear equation relating the coor

dinates of any point on it

Ax+Ey+C = a (2.3)

A, E and C are the given homogeneous coordinates of the line. This equation may

also be represented with homogeneous point coordinates

(2.4)

ÀA, ÀE, ÀC, À 1= a while (Xl: X2 : X3) are the homogeneous

point coordinates. (Xl: X 2 : X 3 ) represent a given line. Any point (Xl: X2 : X3)

which satisfies Eq. 2.4 is on the line (Xl: X 2 : X 3 ); similarly, any line (Xl: X 2 : X 3 )

satisfying Eq. 2.4 is on the point (Xl: X2 : X3)' Therefore, points and lines are dual

elements in the projective plane.

In 3-D projective space we also have the equation

(2.5)

4

2.3. COORDINATES OF POINT AND LINE ON A PLANE

where (Xl: X 2 : X 3 : X 4) are the homogeneous coordinates of a plane, while (Xl:

X2 : X3 : X4) are the homogeneous coordinates of a point. So, points and planes are

dual elements in the projective space.

2.3. COORDINATES OF POINT AND LINE ON A PLANE

Any two distinct points determine a unique line on a plane. Assume two distinct

points (Y1 : Y2 : Y3) and (Zl : Z2 : Z3)' (Xl: X2 : X3) is any other point on the line

determined by these two points. Obviously, these three points are collinear. Hence,

we can write the singular relation [1, 3, 5]

det Y1 Y2 Y3 = 0

Then, using Grassmannian expansion [9, 14] gives

[Y2 Y3] [Y3 Y1] [Y1det Xl + det X2 + det~ ~ ~ ~ ~

Therefore, the line coordinates are given by

(2.6)

Another proposition is that any two distinct lines determine one and only one

point. Similarly, we may have two distinct lines (Y1 : Y2 : Y3) and (Zl : Z2 : Z3), then

(Xl: X 2 : X 3) is any line on the intersection point determined by the previous two

5

2.4. COORDINATES OF POINT AND PLANE IN 3-D SPACE

lines. Obviously, these three lines are concurrent. Renee, we have

Xl X2 X3

det YI Y2 Y3 = 0

Zl Z2 Z3

Then, using Grassmannian expansion gives

Therefore, the point coordinates are given by

Again, the duality of points and lines on a plane is evident.

(2.7)

2.4. COORDINATES OF POINT AND PLANE IN 3-D SPACE

Three distinct points determine a unique plane. Assume that these three points

are (Xl: X2 : X3 : X4), (YI: Y2 : Y3 : Y4) and (Zl : Z2 : Z3 : Z4) while (WI : W2 : W3 : W4)

is any point on the plane determined by these three points. Renee, we have [21]

WI W2 W3 W4

Xl X2 X3 X4det =0

YI Y2 Y3 Y4

Zl Z2 Z3 Z4

The coordinates of the plane can be obtained without difficulty.

6

2.5. KINEMATIC MAPPING OF PLANAR DISPLACEMENT

(Xl: X 2 : X 3 : X 4 ) =

X2 X3 X4 Xl X3 X4 Xl X2 X4 Xl X2 X3

det Y2 Y3 Y4 : -det Yi Y3 Y4 : det Yi Y2 Y4 : -det Yi Y2 Y3

Z2 Z3 Z4 Zl Z3 Z4 Zl Z2 Z4 Zl Z2 Z3

(2.8)

Similarly,if we are given three distinct planes, (Xl: X 2 : X 3 : X 4 ), (Yi: Y2 : Y3 : Y4)

and (Zl : Z2 : Z3 : Z4), the coordinates of the point determined by them may also be

given by [21]

(Xl: X2 : X3 : X4) =

X 2 X 3 X4 Xl X3 X 4 Xl X 2 X 4 Xl X 2 X 3

det Y2 Y3 Y4 : -det Yi Y3 Y4 : det Yi Y2 Y4 : -det Yi Y2 Y3

Z2 Z3 Z4 Zl Z3 Z4 Zl Z2 Z4 Zl Z2 Z3

(2.9)


Three parameters, (a, b, cP), define a planar displacement. The Cartesian coordi

nates of the origin of the moving platform frame, E, measured in the fixed reference

frame, ~, are (a,b) and cP is the rotation angle measured from the X-axis of ~ to the

x-axis of E, the positive sense being counter-clockwise. A point in E relative to ~ can

be given by the homogeneous linear transformation

X

Y

Z

cos(cP)

sin( cP)

o

-sin(cP) a

cos (cP) b

o 1

X

Y

Z

(2.10)

where (x : Y : z) are the coordinates of a point in E while (X : Y : Z) are those of

the same point in ~.

7


We may map the three displacement parameters(a, b, cP) to an image point in a

projective kinematic image space with homogeneous coordinates,

(Xl: X 2 : X 3 : X 4 ) =

((a sin(cP/2) - bcos(cP/2)) : (a cos(cP/2) + bsin(cP/2)) : 2 sin(cP/2) : 2 cos(cP/2))

(2.11)

as explained in detail by Bottema and Roth[2].

By expressing the displaeement in terms of image coordinates, Eq. 2.11 becomes

X

Y

Z o

-2X3X 4

(Xl-X?)

o

2(XI X3 + X 2X 4 )

2(X2X 3 - X I X 4 )

(Xl + Xl)

x

y

z

(2.12)

8inee any real displaeement described by (a, b, cP) has a corresponding unique image

point, the inverse mapping can be obtained by

tan(cP/2)

a

b

X 3/X4

2(XI X3 + X 2X 4 )/(Xl + Xl)

2(X2X 3 - X I X 4 )/(Xl + Xl)

(2.13)

From Eq. 2.13 we notiee that X 3 = X 4 = 0 cannot be mapped to a displaeement.

Renee, the mapping is injective, not bijective [4]. That is, we can map any planar

displaeement to a unique image point in 3-D projective spaee. On the other hand,

any image point that is not on the Hne containing image points (Xl : X 2 : 0 : 0)

in 3-D projective spaee can be mapped to one and only one planar displaeement, if

cP < 0 is excluded.

8

CHAPTER 3

PLANAR 3-LEGGED PLATFORM

A planar Stewart-Gough platform(PSGP) consists of a moving platform connected

to a fixed base by three articulated legs. Each leg connects 2: to E via two rigid links

and three one DüF joints, one of which is actuated. Thus, the moving platform will

have 3 DOF. If the value of three actuators' coordinates are specified, we can consider

these joints to be locked [10]. Then, the moving platform becomes a structure unless

the platform assumes a singular pose.

3.1. TYPES OF JOINTS AND LEGS

We deal here with only one-DOF lower kinematic pairs. Since displacements of

the platform are confined to the plane, only revolute (R) and prismatic (P) pairs are

relevant. The leg architecture is described by three letters indicating the succession

of joints beginning with the one on 2:. The possible. combinations are: RRR, RPR,

RRP, PRR, RPP, PRP, PPR, PPP [6, 15]. The last is excluded because three P-pairs

represent three translations in the plane, which cannot be independent. The seven

possible topologies are illustrated in Fig. 3.1.

3.1. TYPES OF JOINTS AND LEGS

RRR FiF?F' RPP

PRP PRF' PF'R

FIGURE 3.1. Types of legs

Onee the active joint is locked, there remain only two free joints on each leg.

These must be one of four types: either RR, PR, RP or pp [10]. Henee, there exist

only the four practical leg types shawn in Table 3.1.

TABLE 3.1. Practicalleg types

RR-type PR-type RP-type PP-typeRRR RPR RRP RPPRRR PRR RRP PRPRRR PRR RPR PPRPRR PPR PRPRPR PPR RPPRRP PRP RPP

Each type, RR, PR and RP, engenders six possible three-joint legs while PP defines

three possible legs. In Table 3.1 the actuated joints are underlined.

Since any RR leg implies that the R-joint represented by a point on E is bound

ta the circumference of a circ1e with a known radius eentered on the R-joint on

~, for convenienee, "C", standing for "çirc1e", will represent RR. PR denotes a leg

where the R-joint on E is bound to the Hne determined by the P-joint on~. Henee,

10

3.2. TYPES OF PLANAR PLATFORM

"L", standing for "line", will represent PR. Then, RP denotes a leg where the Hne

determined by the P-joint on E is bound to the R-joint on L:. So, "V", standing

for "moying Hne", will be used to represent RP. Finally, pp denotes a leg which will

unconditionally create a degenerate PSGP wherein E may only translate, without

rotation not withstanding the choice of the remaining legs. Therefore, "T", standing

for ",translation", will represent such a pp leg. These are included for the sake of

completeness since they do not represent any practical design.


Each leg must be one of the four types, C, L, V and T. However, we notice that

there can only be one T-Ieg on a PGSP because two or more T-Iegs will cause the

platform either to move uncontrollably or to be unassemblable. Hence, we divide the

PSGP into 11 types, including mixed leg architectures. The legs' sequence about E

is not important because one may imagine viewing the platform from above or below

to get the cycle right. The first ten types in Table 3.2 are the only practical ones.

TABLE 3.2. The types of PSGP

type of PSGP first leg type second leg type third leg typeCCC C C CLLL L L L

VVV V V VCCL C C L

CCV C C VLLC L L CLLV L L V

VVC V V CVVL V V LCLV C L V

T-type T C, L or V C, L or V

11


We notice that there are 6 possible architectures in T-type PSGP. They are shawn

in Table 3.3

TABLE 3.3. The possible architectures of T-type PSGP

type of PSGP first leg type second leg type third leg typeTCC T C CTLL T L LTVV T V VTCL T C LTCV T C VTLV T L V

12

CHAPTER 4

DIRECT KINEMATICS

The direct kinematic(DK) problem involves determining aIl possible poses of the

moving platform when the actuated joint variables are set to specifie values. There

are three types of legs, RR, PR and RP. Constraint equations for each leg type are

as follows.

4.1. RR LEG EQUATION

FIGURE 4.1. MA moves on the circle centered on FA

As Fig. 4.1 shows, a moving point(MA) is bound to the circumference of a circle

centred on fixed point(FA) with radius ra' The homogeneous coordinates of MA are

4.1. RR LEG EQUATION

(XMA : YMA : 1) in E while those of FA are (XFA : YFA : 1) in L;. The homogeneous

coordinates of MA in L; are related to those in E by

(4.1)

1

Where Td is the transformation matrix in Eq. 2.12.

The circle equation is

(4.2)

which, when expanded, becomes

It is convenient to express Eq. 4.3 in the following form:

Substituting Eq. 4.1 into Eq. 4.4 produces

(Xl +Xi) (-XlxitA -X1YitA +2X1K1xMA -X1K3 +2X1K2YMA -4X4 XMA X 2

4X4 K 1X 3YMA + 4X4 K 2X 3XMA + 4X4 K 1X 2 + 4X4yMA X 1 - 4X4 K 2X 1 - xitAXi +

4X3yMAX 2 - 4X? - 4Xi - K3Xi - XiY'iIA + 4K1X1X3 + 4XMA X 1X 3 + 4K2X 2X 3

2XiK1xMA - 2XiK2YMA) = 0

(4.5)

As we know, Xl + Xi = 0 cannot represent any planar displacement. Renee, we can

divide Eq. 4.5 by (Xl + Xl)/4 to obtain the constraint surface in the image space.

14

4.2. PR LEG EQUATION

Xr + X~ + ((-KI - XMA)X3 + (-YMA + K 2 )X4 )XI + ((-K2 - YMA)X3 + (-KI +

XMA)X4 )X2 + (1/2K2YMA + 1/2KIxMA + 1/4K3 + 1/4YXtA + 1/4x~A)Xl + (KIYMA

K 2xMA)X4 X 3 + (1/4x~A + 1/4Y~A - 1/2KIxMA + 1/4K3 - 1/2K2YMA)Xl = 0

(4.6)

Eq. 4.6 is the point equation of a quadric surface in the 3-D projective image space.

This general quadric is the geometric image of the kinematic constraint that a point

in E moves on a circle in ~.

4.2. PR LEG EQUATION

FIGURE 4.2. MA moves on the fixed line determined by FA and FA!

As Fig. 4.2 shows, a moving point(MA) is bound to a fixed line determined by

two points FA and FAL The homogeneous coordinates of MA are (XMA : YMA : 1) in

E while those of FA and FA1 are (XFA : YFA : 1) and (XFAI : YFAI : 1) in ~. The

homogeneous coordinates of MA in ~ are given by

(4.7)

1

15

4.3. RP LEG EQUATION

The line equation is

X MA YMA ZMA

det X FA YFA 1 = 0

X FA1 YFA1 1


(4.8)

Substituting Eq. 4.7 into Eq. 4.9 gives

((2YFA - 2YFAdX3 + (-2XFA1 + 2XFA )X4)X1 + ((2XFA1 - 2XFA )X3 + (2YFA

2YFAl)X4)X2+(XFAYMA+YFAlXMA-XFAlYMA+XFAYFAl-YFAXMA-YFAXFA1)Xj+

(-2YFAYMA + 2YFA1YMA + 2XFA1 XMA - 2XFAXMA)X4X 3 + (YFAxMA + XFAYFAI +

XFA1YMA - XFAYMA - YFAXFAI - YFA1xMA)Xl = 0

(4.10)


This special quadric is the geometric image of the kinematic constraint that a point

in E moves on a line in ~.


As Fig. 4.3 shows, a moving line determined by two points MA and MAI in E,

is bound to a fixed point(FA). The homogeneous coordinates of MA and MAI are

(XMA : YMA : 1) and (XMAI : YMAI : 1) in E while those of FA are (XFA : YFA : 1) in

~. The homogeneous coordinates of FA in E are given by

(4.11)

1

16


F

L--------_ ....._~

FIGURE 4.3. The moving line determined by MA and MAI moving on thefixed point FA

The equation of the line is

det YMA 1 =0 (4.12)


XMAI YMAI 1

(4.13)

Substituting Eq. 4.11 into Eq. 4.13 gives

((2YMA - 2YMAl)X3 + (2XMAI - 2XMA)X4)X1 + ((2XMAI - 2XMA)X3 + (-2YMA +

2YMAl)X4)X2+(XMAYFA+YMAIXFA-YMAXMAI-YMAXFA+XMAYMAl-xMAlYFA)Xj+

(-2XMAIXFA - 2YMAIYFA + 2XMA X FA + 2YMAYFA)X4X3 + (yMAXFA - YMAIXFA

YMAXMAI - xMAYFA + XMAYMAI + xMAIYFA)Xl = 0

(4.14)


This special quadric is the geometric image of the kinematic constraint that a hne in

E moves on a point in 1;.

17

4.5. ALGORITHM FOR EACH TYPE OF PLATFORM

4.4. pp LEG EQUATION

FIGURE 4.4. The translating platform

Sinee pp type legs prohibit rotation of the moving platform, we always can choose

the frames of E and ~ properly to make cP zero as shown in Fig. 4.4. That means

X 3 = 0 and a pp leg provides no bound on Xl and X 2 .


There are 10 types of PSGP DK problem formulations. In each formulation,

there is a group of three equations determined by the types of the three legs. It is

easy to check that X 4 = 0 will always lead to X 3 = 0 from Eqs. 4.6, 4.10 and 4.14.

As we know, the point on the (Xl: X 2 : 0 : 0) line in the 3-D projective image spaee

cannot represent any planar displaeement. Renee, X 4 cannot be zero in cases of RR,

PR and RP. And also in the case of PP, X 4 cannot be zero in order to ensure cP = O.

Sinee (Xl: X 2 : X 3 : X 4 ) are homogeneous coordinates, we can set X 4 = 1 to exclude

solutions at infinity.

4.5.1. ALGORITHM FOR A C-C-C PLATFORM. Without loss of gen-

erality, the special coordinate frames in Fig. 4.5 can be used. The homogeneous coor

dinates of each point are shown in Table 4.1. Note that C, which stands for "çircle",

18


Fe

MC

(FIGURE 4.5. An ideal reference frame for C-C-C type platforms

is an abbreviation for an RR leg as described in Section 4.1.

TABLE 4.1. The coordinates of each point(C-C-C)

in ~ in E lengthFA (0 : a: 1) MA (0:0:1) leg A raFB (XPB : a: 1) MB (XMB : a: 1) leg B rbFC (Xpc : Ypc : 1) MC (XMC : YMC : 1) leg C re

Substituting the parameters in TabA.1 into Eq. 4.6 gives three equation: Ha = 0,

Hb = a and He = 0, whose coefficients are shown in Table 4.2.

TABLE 4.2. The coefficients of each equation(C-C-C)

X 2 X 2 X IX 3(1) X 2X 3 (2) X I (3) X 2 (4) X 3 (5) Xl(6) (7)l 2

Ha: 1 1 a a a a a a6 a6

Hb: 1 1 bl a a b4 a b6 b7

He: 1 1 Cl C2 C3 C4 C5 C6 C7

Hba : a a bl a a b4 a b6 - a6 b7 - a6

Hea : 0 a Cl C2 C3 C4 C5 C6 - a6 C7 - a6

where

Ka = -r~

19


K b = X~B - rt

K e = X~C + Y;C - r~

b4 = -XFB + XMB

b6 = (2XFB XMB + Kb + x~B)/4

b7 = (X~B - 2XFB XMB + K b)/4

C2 = -YFC - YMC

c3 = -YMC + YFC

C4 = -XFC + XMC

C6 = (2YFCYMC + 2XFCXMC + Kc + Y~c + x~d/4

c7 = (X~c + y'Juc - 2XFC XMC + Kc - 2YFCYMC)/4

Solving Hba = 0 and Hea = 0 for Xl and X 2 gives

Xl = dIX: + d2Xi + d3X 3 + d4

d5Xj + d6X 3 + d7

X _ elX: + e2X i + e3X 3 + e42 - d5X;} + d6X 3 + d7

where

dl = -C2b6 + C2a6

d2 = c4 a 6 - C4 b6 + C6 b4 - a6 b4

d3 = C5 b4 - C2b7 + C2 a 6

d4 = -a6b4 - c4h + C7 b4 + C4 a 6

d5 = C2bl

d6 = -cl b4 + C4 b l

d7 = -C3b4

(4.15)

(4.16)

20


Eqs. 4.15 and 4.16 express Xl and X 2 in term of X 3 for a C-C-C type DK problem.

Substituting Eqs. 4.15 and 4.16 into Ha = 0 gives

where

A6 = ef + a6d~ + dfA5 = 2d l d2 + 2a6d5d6 + 2ele2

A4 = a6d~ + 2d l d3+ e~ + 2ele3 + d~ + 2a6d5d7 + a6d~

A3 = 2d l d4 + 2e2e3 + 2a6d6d7 + 2a6d5d6 + 2d2d3+ 2el e4

A2 = 2e2e4 + e~ + d~ + a6d? + 2d2d4 + 2a6d5d7 + a6d~

Al = 2e3e4 + 2d3d4 + 2a6d6d7

Ao = a6d? + e~ + d~

Eq. 4.17 is the univariate polynomial for C-C-C type DK problems.

----------------

(4.17)

FC FCl

MA FAI

HB

U FEI

FIGURE 4.6. An ideal reference frame for L-L-L type platforms

21


4.5.2. ALGORITHM FOR A L-L-L PLATFORM. Without loss of gen-

erality, the special coordinate frames in Fig. 4.6 can be used. The homogeneous

coordinates of each point are shown in Table 4.3. Note that L, which stands for

"line", is an abbreviation for a PR leg as described in Section 4.2.

TABLE 4.3. The coordinates of each point(L-L-L)

in ~ in ~ in EFA (0:0:1) FA1 (1:0:1) MA (0 : 0 : 1)FB (XFB : YFB : 1) FB1 (XFBI : YFBI : 1) MB (XMB : 0 : 1)FC (XFC : YFC : 1) FC1 (XFCI : YFCI : 1) MC (XMC : YMC : 1)

Substituting the parameters in Table 4.3 into Eq. 4.10 gives three equations: Ha = 0,

H b = 0 and He = 0, whose coefficients are shown in Table 4.4.

TABLE 4.4. The coefficients of each equation(L-L-L)

X I X 3 (1) X 2X 3(2) X I (3) X 2(4) X 3(5) Xj(6) (7)Ha: 0 1 -1 0 0 0 0H b: bl b2 -b2 bl bs b6 b7He: Cl C2 -C2 Cl Cs C6 C7

Fe: 0 0 0 0 Al A 2 A o

where

Fe = (Hb - H ab2)CI - (He - HaC2)bl

bl = YFB - YFBI

b2 = X FBI - X FB

b6 = (YFBlXMB + XFBYFBI - YFBXMB - YFB X FBl )/2

b7 = (YFBXMB + XFBYFBI - YFBXFBI - YFBl XMB)/2

Cs = -YFCYMC + YFClYMC + XFCIXMC - XFCXMC

C6 = (XFCYMC + YFCIXMC - XFCIYMC + XFCYFCI - YFCXMC - YFCX FCI)/2

22

(4.18)


C7'= (YFCxMC + XFCYFCl + XFClYMC - XFCYMC - YFCXFCl - YFCl XMC)/2

A 2 = cl b6 - C6bl

Al = clbs - cSbl

Ao = -C7bl + cl b7

Fe = °is the univariate polynomial for L-L-L type of DK problem.

Solving Ha = °and Hb = °for Xl and X2 gives

X __ X 3 (bsX 3 + b6Xl + b7 )

1- bl (Xt+ 1)

X - _ bSX 3 + b6Xl + b7 (4.19)2 - bl(Xl + 1)

Eqs. 4.18 and 4.19 express Xl and X2 in term of X3 for L-L-L type DK problems.

~,-'8-"-"--Mrr"",~-o- -,-' M[ 1 '~I "",

FC l',

MAt0/L 1

l /: ,~-,--J)FA FE

FIGURE 4.7. An ideal reference frame for V-V-V type platforms

4.5.3. ALGORITHM FOR A V-V-V PLATFORM. Without loss of gen-

erality, the special coordinate frames in Fig. 4.7 can be used. The homogeneous COOf

dinates of each point are shown in Table 4.5. Note that V, which stands for "moying

line", is an abbreviation for an RP leg as described in Section 4.3.

Substituting the parameters in Table 4.5 into Eq. 4.14 gives three equations: Ha = 0,

Hb = °and He = 0, whose coefficients are shown in Table 4.6.

where

23


TABLE 4.5. The coordinates of each point(V-V-V)

in ~ in E in EFA (0:0:1) MA (0 : 0 : 1) MAl (XMAl : YMAl : 1)FB (XFB : 0 : 1) MB (XMB : 0 : 1) MB1 (XMBl : YMBl : 1)FC (XFC : Y FC : 1) MC (XMC : YMC : 1) MC1 (XMCl : YMCl : 1)

TABLE 4.6. The coefficients of the equations(V-V-V)

X 1X3(1) X 2X 3(2) X l (3) X 2(4) X 3(5) Xl(6) (7)Ha: al a2 a2 -al 0 0 0Hb: bl b2 b2 -bl b5 b6 b7

He: Cl C2 C2 -Cl C5 C6 C7

Hba: 0 d2 d2 0 d5 d6 d7

Hea : 0 e2 e2 0 e5 e6 e7

Fe: 0 0 0 0 Al A2 Ao

a2 = XMAl

b6 = (YMBlXFB + XMBYMBl)/2

b7 = (-YMBlXFB + XMBYMBl)/2

Cl = YMC - YMCl

C2 = X MCl - XMC

C5 = -XMC1XFC - YMC1YFC + XMCXFC + YMCYFC

C6 = (XMCYFC + YMC1XFC - YMCXMCl - YMCXFC + XMCYMCl - XMC1YFC)/2

C7 = (YMCXFC - YMC1XFC - YMCXMCl - XMCYFC + XMCYMCl + XMC1YFC)/2

d2 = b2al - bl a 2

24


d5 = al b5

d6 = al b6

d7 = al b7

A 2 = -e2d6 + d2e6

Al = d2e5 - e2d5

A o = -e2d7 + d2e7

Fe = 0 is the univariate polynomial for a V-V-V type DK problems.

Solving Hb = 0 and He = 0 for Xl and X 2 gives

X _ dlXg + d2Xl + d3X 3 + d4

l - d5Xl + d7

X _ elXg + e2 X l + e3X 3+ e42 - d5Xj + d7

where

dl = -( -b6C2 + b2C6)

d2 = -(b6 CI + b2C5 - bl C6 - b5C2)

d3 = -(b5CI - blC5 - b7C2 + b2C7)

d4 = -b7CI + bl C7

d5 = (b2CI - b1C2)

d7 = b2CI - blC2

el = (b1C6 - b6CI)

e2 = (b1C5 + b2C6 - b6C2 - b5CI)

e3 = (b1C7 - b7CI + b2C5 - b5C2)

(4.20)

(4.21)

25


Eqs. 4.20 and 4.21 express Xl and X 2 in terms of X 3 for V-V-V type DK problems.

FTl

MB

----+-+-~--_.-.- )

fB

FIGURE 4.8. An ideal reference frame for C-C-L type platforms

4.5.4. ALGORITHM FOR A C-C-L PLATFORM. Without loss of gen-


coordinates of each point are shown in Table 4.7.

TABLE 4.7. The coordinates of each point(C-C-L)

in ~ in E lengthFA (0:0:1) MA (0 : 0 : 1) leg A Ta

FB (XPB : 0 : 1) MB (XMB : 0 : 1) leg B Tb

FC (XPC : YpC : 1) MC (XMC : YMC : 1)FC1 (XPCI : YPCI : 1)

Substituting the parameters in Table 4.7 into Eqs. 4.6 and 4.10 gives three equations:

Ha = 0, Hb = 0 and He = 0, whose coefficients are shown in Table 4.8.

where

26


TABLE 4.8. The coefficients of each equation(C-C-L)

X2 X2 XI X3 (1) X 2X 3(2) XI (3) X 2(4) X 3(5) Xj(6) (7)l 2

Ha: 1 1 0 0 0 0 0 a6 a6

Hb: 1 1 bl 0 0 b4 0 b6 b7

He: 0 0 Cl C2 -C2 Cl Cs C6 C7

Hba : 0 0 bl 0 0 b4 0 b6 - a6 b7 - a6

bl = (-XFB - XMB)

b4 = -XFB + XMB

b6 = (2XFB XMB + Kb + Xi1B)/4

b7 = (XL-B - 2XFB XMB + K b)/4

Cs = -YFCYMC + YFCIYMC + XFCIXMC - XFCXMC

C6 = (XFCYMC + YFCIXMC - XFCIYMC + XFCYFCI - YFCXMC - YFCXFCI)/2

C7 = (YFCXMC + XFCYFCI - XFCYMC - YFCIXMC - YFCXFCI + X FCIYMc)/2

Solving Hba = 0 and He = 0 for Xl and X2 gives

Xl = dIX: + d2Xj + d3X 3 + d4

dsXj + d6X 3 + d7

X ~ e l X5 + e2Xj + e3X 3 + e42-

dsXj + d6X 3 + d7

where

dl = -a6C2 + C2 b6

d2 = cl b6 - Cla6 - C6 b4

ds = -C2bl

d6 = cl b4 - clbl

d7 = -C2b4

(4.22)

(4.23)

27


e2 = bl C5 + C2b6 - a6c2

e3 = cla6 + bl C7 - c l b7

Eqs. 4.22 and 4.23 express Xl and X2 in terms of X3 for V-V-V type DK problems.


where

A6 = dî + a6d~ + eîA5 = 2d l d2+ 2ele2 + 2a6d5d6

A4 = 2el e3 + a6d~ + 2dl d3+ a6d~ + 2a6d5d7 + d§ + e§

A3 = 2d2d3+ 2e2e3 + 2ele4 + 2dl d4 + 2a6d6d7 + 2a6d5d6

A2 = 2e2e4 + 2d2d4 + 2a6d5d7 + d~ + a6d? + a6d~ + e~

Al = 2d3d4 + 2a6d6d7 + 2e3e4

Ao = d~ + a6d?+ e42

Eq. 4.24 is the univariate polynomial for C-C-L type DK problems.

(4.24)

FB

t 1

,Ld---+----+---------------(,)FA

MB

FIGURE 4.9. An ideal reference frame for C-C-V type platforms

28


4.5.5. ALGORITHM FOR A C-C-V PLATFORM. Without loss of gen-



TABLE 4.9. The coordinates of each point(C-C-V)

in ~ in E lengthFA (0 : 0 : 1) MA (0 : 0 : 1) leg A Ta

FB (XFB : 0 : 1) MB (XMB : 0 : 1) leg B Tb

FC (XFC : YFC : 1) MC (XMC : YMC : 1)MC1 (XMCI : YMCI : 1)

Substituting the parameters in Table 4.9 into Eqs. 4.6 and 4.14 gives three equations:

Ha = 0, H b = 0 and He = 0, whose coefficient are shown in Table 4.10.

TABLE 4.10. The coefficients of each equation(C-C-V)

X 2 Xi X I X 3 (1) X 2X 3 (2) X I (3) X 2 (4) X 3 (5) Xi(6) (7)l

Ha: 1 1 0 0 0 0 0 a6 a6

H b: 1 1 bl 0 0 b4 0 b6 b7

He: 0 0 Cl C2 C2 -Cl C5 C6 C7

H ba : 0 0 bl 0 0 b4 0 b6 - a6 b7 - a6

where

K -X2 2b - FB - Tb

b4 = -XFB + XMB

b6 = (2XFB XMB + Kb + xitB)/4

b7 = (X~B - 2XFB XMB + K b)/4

Cl = YMC - YMCI

C2 = X MCI - XMC

29

4.5. ALGORITHM FOR EACH TYPE OF· PLATFORM

C5 = -XMCIXFC - YMCIYFC + XMCXFC + YMCYFC

c6 = (XMCYFC + YMCIXFC - YMCXMCI - YMCXFC + XMCYMCI - XMCIYFC)/2

C7 = (YMCXFC - YMCIXFC - YMCXMCI - xMCYFC + xMCYMCI + XMCIYFC)/2

Solving Hba = 0 and He = 0 for Xl and X 2 gives

where

dl = (a6 c 2 - C2 b6)

d2 = (c l b6 - Cla6 + C6 b4)

d3 = (C5b4 + a6C2 - b7C2)

d4 = c l b7 - Cla6 + C7 b4

d5 = C2 bl

d6 = (-cl b4 - clbl )

d7 = -C2b4

el = -(bIC6 - c l b6 + cla6)

e2 = -(bl C5 - C2 b6 + a6 c2)

e3 = -(cla6 + bl C7 - c l b7 )

e4 = b7C2 - a6c2

Xl = dlX~ + d 2Xj + d 3X 3 + d 4

d5X~ + d 6X 3 + d 7(4.25)

(4.26)

Eqs. 4.25 and 4.26 express Xl and X2 in terms of X 3 for C-C-V type DK problems.


where

(4.27)

30


As = 2ele2 + 2d l d2+ 2a6dSd6

A4 = d~ + e~ + 2a6dSd7 + a6d~ + 2dl d3+ 2ele3 + a6d~

A3 = 2ele4 + 2e2e3 + 2d2d3 + 2a6dSd6 + 2a6d6d7 + 2dl d4

A2 = a6d~ + 2a6dSd7 + 2d2d4 + d~ + e~ + a6d? + 2e2e4

Al = 2e3e4 + 2d3d4 + 2a6d6d7

Ao = e42 + d~ + a6d?

Eq. 4.27 is the univariate polynomial for C-C-V type DK problems.

FC

"

LLM.;/./

,//i:(

FA

FIGURE 4.10. An ideal reference frame for L-L-C type platforms

4.5.6. ALGORITHM FOR A L-L-C PLATFORM. Without loss of gen-



TABLE 4.11. The coordinates of each point(L-L-C)

in:E frame in E lengthFA (0 : 0 : 1) MA (0 : 0 : 1) leg A Ta

FB (XFB : YFB : 1) MB (XMB : 0 : 1)FB1 (XFBl : YFBl : 1) MC (XMC : YMC : 1)FC (XFC : YFc : 1)

FC1 (XMGl : YMCl : 1)

Substituting the parameters in Table 4.11 into Eqs. 4.6 and 4.10 gives three equa

tions: Ha = 0, Hb = 0 and He = 0, whose coefficients are shown in Table 4.12.

31


TABLE 4.12. The coefficients of each equation(L-L-C)

X2 X 2 X I X 3(1) X 2X 3 (2) X I (3) X 2 (4) X 3 (5) Xj(6) (7)l 2

Ha: 1 1 0 0 0 0 0 a6 a6

Hb: 0 0 bl b2 -b2 bl b5 b6 b7

He: 0 0 Cl C2 -C2 Cl C5 C6 C7

where

K - 2a - -ra

bl = Y FB - Y FBI

b2 = X FBI - X FB

b5 = XFBlXMB - XFBXMB

b6 = (YFBlXMB + XFBYFBI - YFBXMB - Y FB X FBl )/2

b7 = (YFBXMB + XFBYFBI - YFBXFBI - Y FBl XMB)/2

C2 = XFCI - X FC

C5 = -YFCYMC + YFCIYMC + XFCIXMC - XFCXMC

C6 = (XFCYMC + YFCIXMC - XFCIYMC + XFCYFCI - YFCXMC - Y FCX FCd/2

C7 = (YFCXMC + XFCYFCI + XFCIYMC - XFCYMC - YFCXFCI - Y FCI XMC)/2


X _ dIX: + d 2Xj + d 3 X 3 + d 4

l - d5X§ + d7

X_ elX: + e2X §+ e3X 3 + e4

2-d5X§ + d7

where

dl = b2 C6 - b6 C2

d2 = -b6CI - b5C2 + bl C6 + b2C5

d3 = -b5CI + b2C7 + bl C5 - b7 C2

(4.28)

(4.29)

32


d4 = -b7 Cl + bl C7

ds = -b2Cl + blC2

d7 = -b2Cl + bl C2

el = b6 Cl - bl C6

e2 = -bICS + b2C6 - b6 C2 + bSCl

e3 = b7Cl - blC7 + b2Cs - bSC2

e4 = b2C7 - b7C2

Eqs. 4.28 and 4.29 express Xl and X 2 in terms of X 3 for L-L-C type DK problems.

Substituting Eqs. 4.28 and 4.29 into Ha = Ogives

(4.30)

where

A6 = dî + eî + a6dg

As = 2dld2+ 2ele2

A 4 = d~ + e~ + 2a6dSd7 + a6dg + 2el e3 + 2dld3

A3 = 2ele4 + 2dld4 + 2e2e3 + 2d2d3

A 2 = 2a6dSd7 + d~ + e~ + 2d2d4 + a6d~ + 2e2e4

Al = 2e3e4 + 2d3d4

Ao = d~ + a6d~ + e42

Eq. 4.30 is the univariate polynomial for L-L-C type DK problems.

4.5.7. ALGORITHM FOR A L-L-V PLATFORM. Without loss of gen




tions: Ha = 0, Hb = aand He = 0, whose coefficient are shown in Table 4.14.

where

33


FIGURE 4.11. An ideal reference frame for L-L-V type platforms

TABLE 4.13. The coordinates of each point (L-L-V)

in ~ in EFA (0 : 0 : 1) MA (0: 0 : 1)FB (XFB : Y FB : 1) MAl (XMAI : YMAI : 1)

FB1 (XFBI : Y FBI : 1) MB (XMB : 0 : 1)FC (XFC : Y FC : 1) MC (XMC : YMC : 1)FC1 (XFCI : YFCI : 1)

TABLE 4.14. The coefficients of each equation(L-L-V)

X I X 3(1) X 2X 3(2) X I (3) X 2(4) X 3 (5) Xl(6) (7)Ha: al a2 a2 -al 0 0 0Hb: bl b2 -b2 bl b5 b6 b7

He: Cl C2 -C2 Cl C5 C6 C7

a2 = 2XMAI

bl = Y FB - Y FBI

b2 = X FBI - X FB

b5 = XFBlXMB - XFBXMB

b6 = (YFBlXMB + XFBYFBI - YFBXMB - Y FB X FBl )/2

b7 = (YFBXMB + XFBYFBI - YFBXFBI - Y FBl XMB)/2

Cl = Y FC - Y FCI

34



C6 = (XFCYMC + YFCIXMC - XFCIYMC + XFCYFCI - YFCXMC - Y FC X FC I)/2

C7 = (YFcxMC + XFCYFCI + XFCIYMC - XFCYMC - YFCXFCI - Y FCI XMC)/2

Solving Hb = a and He = a for Xl and X 2 gives

Xl = dlX~ + d 2Xl + d 3X 3 + d 4

dsXl + d 7

X _ elX~ + e2X l + e3 X 3 + e42 - dsXl + d 7

where

dl = b2C6 - b6C2

d 2 = -b6CI - bSC2 + bl C6 + b2Cs

d 3 = -bSCI + b2C7 + bICS - b7C2

d 4 = -b7CI + b l C7

e2 = -bl Cs + b2C6 - b6C2 + bSCI

e3 = b7CI - bl C7 + b2Cs - bSC2

(4.31)

(4.32)

Eqs. 4.31 and 4.32 express Xl and X 2 in terms of X3 for L-L-V type DK problems.

Substituting Eqs. 4.31 and 4.32 into Ha = Ogives

(4.33)

35


where

A2 = -ale2 + a2d2+ a2e3+ ald3

Al = -ale3 + a2d3 + ald4+ a2e4

Ao = -ale4 + a2d4

Eq. 4.33 is the univariate polynomial for L-L-V type DK problems.

F B

IL

~/f/F~

L 1('1----------+---~~--th

FA

FIGURE 4.12. An ideal reference frame for V-V-C type platforms

4.5.8. ALGORITHM FOR A V-V-C PLATFORM. Without loss of gen-



TABLE 4.15. The coordinates of each point(V-V-C)

in L: in E lengthFA (0:0:1) MA (0 : 0 : 1) leg A raFB (XFB : 0 : 1) MB (XMB : 0 : 1)FC (XFC : YFC : 1) MB1 (XMBI : YMBI : 1)

MC (XMC : YMC : 1)MC1 (XMCI : YMCI : 1)


tians: Ha = 0, Hb = 0 and He = 0, whase coefficients are shawn in Table 4.16.

36


TABLE 4.16. The coefficients of each equation(V-V-C)

X 2 X 2X l X 3(1) X 2X 3 (2) X l (3) X 2(4) X 3 (5) Xj(6) (7)1 2

Ha: 1 1 0 0 0 0 0 a6 a6

Hb: 0 0 bl b 2 b 2 -b l b5 b6 b7

He: 0 0 Cl C2 C2 -Cl C5 C6 C7

where

K - 2a - -ra

a6 = K a /4

bl = -YMBI

b5 = -XMBlXj + XMBXj

b6 = 1/2YMBlXj + 1/2xMBYMBl

b7 = -1/2YMBlXj + 1/2xMBYMBl

Cl = YMC - YMCl

C5 = -XMClXFC - YMClYFC + XMCXFC +YMCYFC

C6 = (XMCYFC + YMGlXFC - YMCXMCl - YMCXFC + XMCYMCl - XMClYFC)/2

C7 = (YMCXFC - YMClXFC - YMCXMCl - XMCYFC + XMCYMCl + XMClYFc)/2


X_ dIX: + d2Xj + d3X 3 + d4

1-d5Xj + d7

X_ elX: + e2X j + e3X 3 + e4

2-d5Xj + d7

where

dl = b6C2 - b 2C6

d2 = -b6Cl - b2C5 + bl C6 + b5C2

d3 = -b5Cl + b l C5 + b7C2 - b2C7

(4.34)

(4.35)

37


d4 = -b7CI + bl C7

d5 = b2CI - blC2

d7 = b2CI - bl C2

el = blC6 - b6CI

e2 = blC5 + b2C6 - b6 C2 - b5CI

e3 = bl C7 - b7CI + b2C5 - b5C2

e4 = b2C7 - b7C2

Eqs. 4.34 and 4.35 express Xl and X 2 in terms of X 3 for V-V-C type DK problems.


(4.36)

where

A6 = dî + eî + a6dg

A5 = 2d l d2+ 2ele2

A4 = d§ + 2a6d5d7 + 2el e3 + e§ + a6dg + 2dl d3

A3 = 2e2e3 + 2dl d4 + 2ele4 + 2d2d3

A2 = 2e2e4 + e~ + d~ + 2a6d5d7 + a6d? + 2d2d4

Al = 2d3d4 + 2e3e4

Ao = d~ + a6d?+ e42

Eq. 4.36 is the univariate polynomial for V-V-C type DK problems.

4.5.9. ALGORITHM FOR A V-V-L PLATFORM. Without loss of gen




tions: Ha = 0, Hb = 0 and He = 0, whose coefficients are shown in Table 4.18.

where

38


FT

FE

\ ~

l?f~~~ .()FA

FIGURE 4.13. An ideal reference frame for V-V-L type platforms

TABLE 4.17. The coordinates of each point(V-V-L)

in I; in EFA (0 : 0 : 1) MA (0 : 0 : 1)FB (XFB : 0: 1) MAI (XMAI : YMAI : 1)FC (XFC : YFC : 1) MB (XMB : 0 : 1)

FC1 (XFCI : YFCI : 1) MB1 (XMBI : YMBI : 1)MC (XMC : YMC : 1)

TABLE 4.18. The coefficients of each equation(V-V-L)

XI X3 (1) X 2X 3(2) XI (3) X 2 (4) X 3(5) Xi(6) (7)Ha: al a2 a2 -al 0 0 0Hb: bl b2 b2 -b l b5 b6 b7

He: Cl C2 -C2 Cl C5 C6 C7

a2 = 2XMAI

b2 = XMBI - XMB

b6 = (YMBlXFB + XMBYMBd/2


39




C7 = (YFCXMC + XFCYFCI + XFCIYMC - XFCYMC - YFCXFCI - YFCI XMC)/2

Solving Hb = 0 and He = 0 for Xl and X2 gives

Xl = dIX: + d2Xl + d3X 3 + d4

dsXl + d6X 3 + d7

X_ elX: + e2X l + e3X 3 + e4

2-dsX~ + d6X 3 + d7

where

dl = b2C6 - b6 C2

d2 = -b6CI - blC6 + b2Cs - bSC2

d3 = -bSCI - bICS + b2C7 - b7C2

d4 = -b7CI - bl C7

ds = bl C2 - b2CI

d6 = 2b2C2 + 2bl Cl

d7 = -b1C2 + b2CI

e2 = -bICS - b2C6 - b6C2 + bSCI

e3 = -bl c7 + b7CI - b2Cs - bSC2

(4.37)

(4.38)

Eqs. 4.37 and 4.38 express Xl and X2 in terms of X3 for V-V-L type DK problems.


(4.39)

40


where

A2 = -ale2 + a2d2 + a2e3+ ald3

Al = -ale3 + a2d3 + ald4 + a2e4

Ao = -ale4 + a2d4

Eq. 4.39 is the univariate polynomial for V-V-L type DK problems.

-------e-----~~~-~--~l~::~:~~-Fe·.'-,

'''-, ,. "" ..,

_-------- - /' MB-, <1/-

"""", ~--------/--.",--

FA FB

FIGURE 4.14. An ideal reference frame for C-L-V type platforms

4.5.10. ALGORITHM FOR A C-L-V PLATFORM. Without loss of

generality, the special coordinate frames in Fig. 4.14 can be used. The homogeneous


TABLE 4.19. The coordinates of each point(C-L-V)

in 'E in E lengthFA (0:0:1) MA (0 : 0 : 1) leg A raFB (XFB : 0 : 1) MB (XMB : 0 : 1)FC (Xpc : Ypc : 1) MB1 (XMBl : YMBl : 1)

FC1 (XFCl : YFCl : 1) MC (XMC : YMC : 1)

8ubstituting the parameters in Table 4.19 into Eqs. 4.6, 4.10 and 4.14 gives three

equations: Ha :- 0, Hb = 0 and He = 0, whose coefficients are shown in Table 4.20.

41


TABLE 4.20. The coefficients of each equation(C-L-V)

X2 X 2 X IX 3 (1) X 2X 3 (2) X I(3) X 2(4) X 3(5) Xj(6) (7)l 2Ha: 1 1 0 0 0 0 0 a6 a6

Hb: 0 0 b1 b2 b2 -bl bs b6 b7

He: 0 0 Cl C2 -C2 Cl Cs C6 C7

where

K - 2a - -ra

b6 = (YMBlXFB + XMBYMBl)/2


Cl = Y FC - Y FCI

C2 = X FCI - X FC

Cs = -YFCYMC + YFC1YMC + XFC1XMC - XFCXMC


C7 = (YFCXMC + XFCYFCI - XFCYMC - YFCIXMC - YFCXFCI + X FCIYMC)/2


Xl = dlX~ + d 2Xj + d 3X 3 + d 4

dsXj + d7

X _ elX~ + e2X j + e3X 3 + e42 - dsxj + d7

where

dl = C2b6 - C6 b2

d 2 = cl b6 - c Sb2 + C2 bS - bl C6

d 3 = -b2C7 + clbs - bICS + b7C2

(4.40)

(4.41)

42


d4 = clb7 - C7bl

d5 = b2CI - blC2

d7 = b2CI - blC2

el = -cl b6 + bl C6

e2 = blC5 - C6b2 + C2 b6 - clb5

e3 = C7bl - Cl b7 - C5 b2+ C2b5

e4 = -b2C7 + b7C2

Eqs. 4.40 and 4.41 express Xl and X 2 in terms of X 3 for C-L-V type DK problems.


(4.42)

where

A6 = eî + dî + a6dg

A5 = 2e2el + 2d2dl

A 4 = e~ + 2d3dl + 2e3el + 2a6d5d7 + d~ + a6dg

A3 = 2e3e2 + 2d3d2 + 2d4dl + 2e4el

A 2 = a6d? + e~ + 2d4d2+ 2e4e2 + d~ + 2a6d5d7

Al = 2e4e3 + 2d4d3

A o = d~ + a6d?+ e42

Eq. 4.42 is the univariate polynomial for C-L-V type DK problems.

4.5.11. ALGORITHM FOR AT-TYPE PLATFORM. There are 6 pos

sible architectures for aT-type platform. We will choose frames of E and ~ properly

to make sure X 3 = O. And we also set X 4 = 1 to exc1ude solutions at infinity. Then,

each architecture will be discussed in detail.

ALGORITHM FOR A T-C-C PLATFORM. Without loss of generality,

the special coordinate frames in Fig. 4.15 can be used. The homogeneous coordinates

43


FR

FA

FIGURE 4.15. An ideal reference frame for T-C-C type platforms

of each point are shown in Table 4.21.

TABLE 4.21. The coordinates of each point(T-C-C)

in ~ in E length

FA (0 : 0 : 1) MA (0 : 0 : 1) leg A raFB (XFB : YFB : 1) MB (XMB : 0 : 1) leg B rb

Substituting the parameters in Table 4.21 into Eq. 4.6 gives two equations: Ha = 0

and Hb = 0, whose coefficients are shown in Table 4.22.

TABLE 4.22. The coefficients of each equation(T-C-C)

X2 X2 X1(1) X 2(2) (3)1 2

Ha: 1 1 0 0 a3

Hb: 1 1 b1 b2 b3

Hba 0 0 b1 b2 b3 - a3

where

a3 = K a /4

K b = X~B + YiB - r~

b1 = YFB

44


b2 = -XFB + XMB

b3 = (X~B - 2XFB XMB + K b)/4

Solving H ba = 0 for Xl gives

Eq. 4.43 express Xl in terms of X 2 for T-C-C type DK problems.

Substituting Eq. 4.43 into Ha = 0 gives

Eq. 4.44 is the univariate polynomial for T-C-C type DK problems.

(4.43)

(4.44)

/\

L/~l ///.

/~~-~----/TA

F BJ

FIGURE 4.16. An ideal reference frame for T-L-L type platforms

ALGORITHM FOR A T-L-L PLATFORM. Without loss of generality,



TABLE 4.23. The coordinates of each point(T-L-L)

in L; in L; in EFA (0 : 0 : 1) FA1 (XFAI : YFAI : 1) MA (0 : 0 : 1)FB (XFB : YFB : 1) FB1 (XFBI : YFBI : 1) MB (XMB : 0 : 1)


45


and H b = 0, whose coefficients are shown in Table 4.24.

TABLE 4.24. The coefficients of each equation(T-L-L)

X I (l) X 2(2) (3)Ha: al a2 0Hb: bl b2 b3

where

bl = -XFBI + X FB

b2 = Y FB - Y FBI

b3 ...:... (YFBXMB + XFBYFBI - YFBXFBI - Y FBl XMB)/2

Solving Ha = 0 and Hb = 0 for Xl and X 2 gives

X_ a2b3

l --a2bl + b2a l

X2

= _ b3al

-a2bl + b2a l

Hence, Eq 4.45 and 4.46 give an explicit final solution.

FIGURE 4.17. An ideal reference frame for T-V-V type platforms

(4.45)

(4.46)

ALGORITHM FOR A T-V-V PLATFORM. Without loss of generality,


46



TABLE 4.25. The coordinates of each point(T-V-V)

in "E in E in EFA (0 : 0 : 1) MA (0 : 0 : 1) MAI (XMAI : Y MAI : 1)FB (XFB : Y FB : 1) MB (XMB : 0: 1) MB1 (XMBI : YMBI : 1)


and H b = 0, whose coefficients are shown in Table 4.26.

TABLE 4.26. The coefficients of each equation(T-V-V)

X l (l) X 2(2) (3)Ha: al a2 0

Hb: bl b2 b3

where

b2 = YMBI

b3 = (-YMBlXFB - xMBYFB + XMBYMBI + xMBlYFB )/2


X_ a2b3

1--a2bl + b2a l

X2

= _ b3al

-a2bl + b2a l

Henee, Eq 4.47 and 4.48 give an explicit final solution.

ALGORITHM FOR A T-C-L PLATFORM.

(4.47)

(4.48)

Without loss of generality,



Substituting the parameters in Table 4.27 into Eqs. 4.6 and 4.10 gives two equations:

47


FBl

FIGURE 4.18. An ideal reference frame for T-C-L type platforms

TABLE 4.27. The coordinates of each point(T-C-L)

in I; in E lengthFA (0 : a : 1) MA (0:0:1) leg A raFB (XFB : YFB : 1) MB (XMB : a: 1)

FB1 (XFBI : YFBI : 1)

Ha = a and Hb = 0, whose coefficients are shawn in Table 4.28.

TABLE 4.28. The coefficients of each equation(T-C-L)

X 2 X 2 X I (l) X 2 (2) (3)l 2

Ha: 1 1 a a a3

Hb: a a bl b2 b3

where

a3 = K a/4

bl = -XFBI + X FB

b3 = (YFBxMB + XFBYFBI - YFBXFBI - YFBl XMB)/2

Solving Hb = a for Xl gives

(4.49)

48


Eq. 4.49 expresses of Xl in terms of X 2 for T-C-L type DK problems.


(4.50)

Eq. 4.50 is the univariate polynomial for T-C-L type DK problems.

HBl

FIGURE 4.19. An ideal reference frame for T-C-V type platforms

ALGORITHM FOR A T-C-V PLATFORM. Without loss of generality,



TABLE 4.29. The coordinates of each paint(T-C-V)

in ~ in E lengthFA (0:0:1) MA (0:0:1) leg A raFB (XFB : YFB : 1) MB (XMB : 0 : 1)

MB1 (XMBI : YMBI : 1)

Substituting the parameters in Table 4.29 inta Eqs. 4.6 and 4.10 gives two equations:

Ha = 0 and Hb = 0, whose coefficients are shown in Table 4.30.

where

49


TABLE 4.30. The coefficients of each equation(T-C-V)

X 2 X 2 X I (l) X 2(2) (3)l 2

Ha: 1 1 0 0 a3

Hb: 0 0 bl b2 b3

b2 = YMBI

b3 = (-YMBlXFB - xMBYFB + XMBYMBI + xMBlYFB )/2

Solving Hb = 0 for Xl gives

Xl = _ b2 X 2 + b3

bl

Eq. 4.51 expresses of Xl in terms of X 2 for T-C-L type DK problems.


Eq. 4.52 is the univariate polynomial for T-C-V type DK problems.

//

H///FB

MAl./ /(/] //

// //

p>{/I:.() ..

FIGURE 4.20. An ideal reference frame for T-L-V type platforms

(4.51)

(4.52)

ALGORITHM FOR A T-L-V PLATFORM. Without loss of generality,



Substituting the parameters in Table 4.31 into Eqs. 4.10 and 4.14 gives two equations:

Ha = 0 and Hb = 0, whose coefficients are shown in Table 4.32.

50

4.6. OBTAINING PLANAR DISPLACEMENT

TABLE 4.31. The coordinates of each point(T-L-V)

in r; in E

FA (0:0:1) MA (0:0:1)FB (XFB : Y FB : 1) MAl (XMAI : Y MAI : 1)

FB1 (XFBI : Y FBI : 1) MB (XMB : 0 : 1)

TABLE 4.32. The coefficients of each equation(T-L-V)

X I (l) X 2(2) (3)Ha: al a2 0

Hb: bl b2 b3

where

bl = -XFBI + X FB

b2 = Y FB - Y FBI

b3 = (-YFBXFBI + YFBXMB + XFBYFBI - Y FBl XMB)/2


X_ a2b3

l --a2bl + b2al

X2

= _ b3al

-a2bl + b2al

Henee, Eq 4.53 and 4.54 give an explicit final solution.


(4.53)

(4.54)

The univariate polynomial in X 3 for each type, exeept for the T-type, can be

solved numerically, e.g., using Mueller's method [7]. Then, the values of Xl and X 2

can be obtained by substituting the value of X 3 into their expressions in term of X 3 .

51


Among T-type platforms, types T-L-L, T-V-Vand T-L-V already have algorithms

which yield closed form solutions. And T-C-C, T-C-L and T-C-V have algorithms

which yield quadratics in X 2 , which can also be solved in closed form. Then, their

expressions of Xl in terms of X 2 give the two corresponding values of Xl'

Finally, the displacement parameters are given by

tan(4)/2)

a

b

Xs!X4

2(X1X 3 + X 2X 4 )/(X§ + Xl)

2(X2X 3 - X 1X 4 )/(X§ + Xl)

(4.55)

and all real assemblies can be constructed.

52

CHAPTER 5

APPLICATION OF THE ALGORITHM

The purpose of this thesis being to provide a reliable algorithm for practical applica

tions, the algorithmic procedures of Chapter 4 will be implemented here. Examples

will help to accomplish this.

5.1. EXAMPLE TO ILLUSTRATE ALGORITHM USE

FIGURE 5.1. A platform moving on a fixed base

5.1.1. DECIDING THE TYPE OF THE PLATFORM. In Fig. 5.1 we

have two rigid bodies representing ~' and E'. Each leg in this PSGP is of PRP type.

Then, we assign the actuated joint in each leg by underlining it: the first leg, PRP,


the second leg, PRP and the third leg, PRP. Therefore, the types of these legs are

RP or V, pp or T, and PR or 1. Since the sequence of the legs is not important, the

type of this PSGP is T-L-V.

(FBD

FIGURE 5.2. Reference frames of the triangle platform

5.1.2. CHOOSING THE REFERENCE FRAME. Fig. 5.2 shows the

referenee frames. The P-joint variable in the first leg is given as DFA , the R-joint

variable in the second leg is a, and that ofthe P-joint in the third leg is DMB . Henee,

the coordinates of FA, FB and FB1 in ~' and those of MA, MAI and MB in E' are

aH given.

The transformation matrix from ~ to ~' is given by

cosh) - sinh) D FA

TL, = sinh) cosh)

where '"'1 = Œ + {J - LOPQ

o oo1

(5.1 )

54


The transformation matrix from E to E' is given by

cos({3)

TE = sin({3)

o

- sin({3) 0

cos({3) 0

o l

(5.2)

Then, the homogeneous coordinates of FA, FB and FBI in ~ are obtained by

multiplying the inverse of TL, by their homogeneous coordinates in ~'. Similarly,

those of MA, MAI and MB in E are obtained by multiplying the inverse of TE by

theirs in E'.

5.1.3. COMPUTATION. Putting the parameters into the algorithm for T

L-V will give the displaeement parameters of E in ~, a, b and cP (here, of course,

cP = 0). Henee, the transformation matrix from E to ~ is given by

TD =

cos(cP)

sin(cP)

o

- sin(cP) a

cos(cP) b

o l

(5.3)

5.1.4. POSITION OF THE MOVING PLATFORM. Any point, whose

coordinates in E' are given, can be located in ~' by

xy

Z zE'

(5.4)

Therefore, the transformation matrix from E' to ~' is

(5.5)

55

5.2. EXAMPLE Ta SHOW HOW THE CODE WORKS

5.2. EXAMPLE TO SHOW HOW THE CODE WORKS

Assume we are dealing with a C-C-C platform. And the parameters for algorithm

of C-C-C type are shown in Table 5.1.

TABLE 5.1. The parameters of C-C-C type of platform

in ~ in E lengthFA (0:0:1) MA (0 : 0 : 1) leg A 3FB (6:0:1) MB (4:0:1) leg B 4FC (3:2:1) MC (1 : 5 : 1) leg C 5

Run "PSGP.exe" program as follows:

Please select the type of PSGP:1: CCC2: LLL3: VVV4: CCL5: CCV6: LLC7: LLV8: VVC9: VVL

10: CLV11: TCC12: TLL13: TVV14: TCL15: TCV16: TLV1

Please input length of link A:3

Please input length of link B:4

Please input length of link C:5

Please input x-coordinate of fixed point BO and moving point B:6 4

Please input coordinates of fixed point CO and moving point C:3 21 5

X3real[0]= 0.13902 X3image[0]= 0.00000 X1[1]= 1.16893X2[1]=-0.96286 X3[1]= 0.13902 aa[1]=-1.57035 bb[1]=-2.55617Phi [1] =15.82949

X3real[1]=-0.21204 X3image[1]= 0.00000 X1[2]=-1.12352X2[2]=-1.04349 X3[2]=-0.21204 aa[2]=-1.54121 bb[2]= 2.57384Phi [2]=-23.94356

56

5.2. EXAMPLE TO SHOW HOW THE CODE WORKS

X3real[2]= 0.32697 X3image[2]= 0.00000 X1[3]=-0.40256 X2[3]=1.52594 X3[3]= 0.32697 aa[3]= 2.51929 bb[3]= 1.62886Phi [3]=36. 21223

X3real[3]=-0.62155 X3image[3]= 0.00000 X1[4]=-0.82134 X2[4]=1.56353 X3[4]=-0.62155 aa[4]= 2.99214 bb[4]=-0.21709Phi [4]=-63.72635

X3real[4]= 0.85289 X3image[4]= 0.00000 X1[5]= 1.63762 X2[5]=1.09767 X3[5]= 0.85289 aa[5]= 2.88799 bb[5]=-0.81212Phi [5]=80.92075

X3real[5]=-0.90993 X3image[5]= 0.00000 X1[6]=-1.85105 X2[6]=0.82860 X3[6]=-0.90993 aa[6]= 2.74940 bb[6]= 1.20032Phi [6]=-84.60020

The position modes are shawn in Fig. 5.3

MUDE!

MUDE4

MUDE2

MODES

MODEJ

MODEr,

FIGURE 5.3. The six modes of this C-C-C type of PSGP

57

CHAPTER 6

CONCLUDING REMARKS

6.1. CONCLUSION

This thesis introduced a straightforward solutions to Direct Kinematic problems

of an types of PSGP including those with mixed legs. Kinematic mapping, used

to derive the final univariate polynomials, proved to be an invaluable tool for this

purpose.

A leg on a PSGP is classified to be one of four types: C, L, V and T, which

represent the free joint sequence RR, PR, RP and pp respectively. Furthermore,

PSGPs have been reduced to 11 types: CCC, LLL, YVY, CLL, CVV, LCC, LVV,

VCC, VLL, CLV and T. There are six possible T-type architectures: TCC, TLL,

TVV, TCL, TCV and TLV. This classification contains an possible architectures of

PSGP with three legs containing only lower pairs.

A truly compact solution to the DK problem for each type of PSGP has been

given herein for the first time. Engineering solutions to any of these problems can

now be obtained by simply applying the algorithms together with given design and

actuator parameters. One needs no familiarity with the algorithm nor with the

subtleties of kinematic mapping. Moreover, a program coded in C is available at

6.2. A SUGGESTION TO PROMOTE POPULARITY

http://www.cim.mcgill.ca/...paul;. It should be useful to design or test a PSGP and

to do animation and dynamic analysis.

6.2. A SUGGESTION TO PROMOTE POPULARITY

Kinematic mapping is a valuable tool to deal with difficult DK problems. It

should have successful application in other areas of kinematics as well. Rowever, it

has been ignored by most engineers. One reason may be that it is rooted in 19th

century projective geometry, while most engineers are familiar only with the older,

traditional Euclidean geometry. Rence, many reject kinematic mapping at the first

glance when confronted by its peculiar, sometimes intuitively unrealistic, possibly

confusing and overly complicated concepts. It is humbly suggested that we might

initially avoid the details of the basic geometry in order not to scare off potential users

when introducing them to kinematic mapping. The author found this tool to be easy

to use but difficult to understand. Therefore, it may obtain wider acceptance if we

focus first on actual applications before becoming enmeshed deeply in the theoretical

background.

59

REFERENCES

[1] AYRES, F" 1967, Projective Geometry, Schaum's Outline Series in Mathematics, McGraw-Hill.

[2] BOTTEMA, O., ROTH, B., 1990, Theoretical Kinematics, Dover.

[3] BUMCROT, R.J., 1969, Morden Projective Geometry, Holt, Rinehart and Winston.

[4] CLARK, A., 1971, Elements of Abstract Algebra, Dover.

[5] COXETER, H.S.M., 1954, The Real Projective Plane, second edition, Cambridge at the University Press.

[6] DANIALI, H.R.M., 1995, Contributions to the Kinematic Synthesis of ParaUelManipulators, Ph.D. Thesis, Dept. of Mech. Eng., McGill University.

[7] ENGELN-MüLLGES, G., 1996, Numerical algorithms with C, Springer.

[8] GOUGH, V.E., 1956, "Discussion in London: Automobile Stability, Control,and Tyre Performance", Proc. Automobile Division, Institution of Mech. Engrs.,pp.392-394.

[9] GRASSMANN, H., 1844, A New Branch of Mathematics, English translation1995 by Kannenberg, L.C., Open Court.

[10] HAYES, M.J.D., Kinematics of General Planar Stewart-Gough Platform,Ph.D. thesis, Dept. of Mech. Eng., McGill University.

[11] HAYES, M.J.D., HUSTY, M.L., ZSOMBOR-MURRAY, P.J., 1999, "Kine-matic Mapping of Planar Stewart-Gough Platforms", Proc. of 17th CanadianCongress of Applied Mechanics, Hamilton, 99-05, pp.319-320.

[12] HUSTY, M.L., 1996, "An Aigorithm for Solving the Direct Kinematics ofGeneral Stewart-Gough Platforms", Mechanism and Machine Theory, v.31, nA,pp.365-379

[13] HUSTY, M.L., 1994, "Kinematic Mapping of Planar Tree-Legged Platforms"Proc. of 15th Canadian Congress of Applied Mechanics, Hamilton, 95-05,pp.876-877.

[14] KLEIN, F., 1939, Elementary Mathematics from an advanced Standpoint:Geometry, Dover.

REFERENCES

[15] MERLET, J-P., 1996, "Direct Kinematics of Planar Parallel Manipulators",IEEE Int. Conf. on Robotics and Automation, Minneapolis, U.S.A., pp.37443749.

[16] ROONEY, J., EARLE, C.F., 1983, "Manipulator Postures and KinematicsAssembly Configurations", 6th World Congress on Theory of Machines andMechanisms, New Delhi, pp.1014-1020.

[17] ROTH, B., 1993, "Computations in Kinematics", in Computational Kinemat-ics, Angeles, J., Hommel, G., Koyacs, P., eds. Kluwer, pp.3-14.

[18] ROTH, B., 1994, "Computational Adyances in Robot Kinematies", in Ad-vances in Robot Kinematics and Computational Geometry, Lenarcic, J. andRayani, B., eds. Kluwer, pp.7-16.

[19] STEWART, D., 1965, "A Platform With Six Degrees of Freedom" , Proc. Instn.Mech. Engr., Vol. 180, Part 1, No. 15, pp.371-378.

[20] ZSOMBOR-MuRRAY, P.J., 1998, "Planar Kinematics and Geometrie Con-struction", Internal Report, Inst. f. Geom, TU-Graz.

[21] ZSOMBOR-MuRRAY, P.J., 1999, "Grassmannian Reduction of QuadraticForms" , website: http://www.cim.mcgill.ca/ rvpaulj.

61

Document Log:

Manuscript Version 1-August, 2001

Typeset by AMS-U\1EX-29 January 2002

CHEN,CHAü

CENTRE FOR INTELLIGENT MACHINES, MCGILL UNIVERSITY, 3480 UNIVERSITY ST., MONTRÉAL

(QUÉBEC) H3A 2A7, CANADA, Tel. : (514) 398-3938

E-mail address: [email protected]. ca

Typeset by AMS-U\'fEX

62

Documents

A DIRECT KINEMATIC COMPUTATION ALGORITHM FOR …digitool.library.mcgill.ca/thesisfile33962.pdf · A DIRECT KINEMATIC COMPUTATION ALGORITHM FOR ALL PLANAR 3-LEGGEDPLATFORMS CHEN,CHAü