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A Review of Topological Perspectives in Robotic Manipulation

P E N G Q I

Master of Science Thesis Stockholm, Sweden 2012

A Review of Topological Perspectives in Robotic Manipulation

P E N G Q I

DD221X, Master’s Thesis in Computer Science (30 ECTS credits) Master Programme in Software Engineering of Distributed Systems 120 cr Royal Institute of Technology year 2012 Supervisor at CSC was Danica Kragic Examiner was Danica Kragic TRITA-CSC-E 2012:018 ISRN-KTH/CSC/E--12/018--SE ISSN-1653-5715 Royal Institute of Technology School of Computer Science and Communication KTH CSC SE-100 44 Stockholm, Sweden URL: www.kth.se/csc

A Review of Topological Perspectives in

Robotic Manipulation

Abstract

Topology is an important area of study in mathematics, which is mainly concerned with

qualitative properties of objects or spaces. It is broad and well developed. Current research

interests are to explore the possibilities to integrate topological perspectives in various

practical engineering problems. Robotics contains many challenging tasks and provides an

ideal playground for topological tools. In this thesis, we select several topics of

interdisciplinary research from topology and robotic manipulation and summarize some

interesting topological insights and approaches.

In the introductory chapter, several selected aspects of topological perspectives in robotics

are briefly reviewed. In the second chapter, we give a detailed account of widely used

topological modeling methods that are used for robotic manipulation, revealing the common

ground to involve topology. In the third chapter, topological considerations about

configuration spaces and topological reasoning on the problems such as finding a global

inverse kinematic solution or dealing with singularities are presented. In the fourth chapter,

we concentrate on topological properties of kinematic maps of redundant manipulators and

discuss methods that potentially enable an efficient exploration of redundancy resolution.

Finally, we mention some open problems for future research.

Acknowledgements This thesis would not have been possible without the generous support I received all the

time.

First and foremost, I would like to give the sincerest gratitude to my supervisor, Prof.

Danica Kragic, for offering me the opportunity to pursue my thesis research under her

supervision and guidance. Her insight and farsightedness has inspired me in the research and

helped me to overcome all the challenges on the way. I am deeply grateful of her help in the

completion of this thesis.

I would also like to thank Dr. Patric Jensfelt, my Master program director, who shows me a

colorful robotics world and makes me dived in the field, having fun. Many many thanks to Dr.

Florian Pokorny, you have been truly friendly in every way and have continuously supported

me in my reference reading – always prepared to clarify the intriguing details in topological

concept and theorem. I am indebted to Dr. Christian Smith, Dr. Yiannis Karayiannidis, Dr.

Lazaros Nalpantidis and Dr. Dimos V. Dimarogonas in DAM research group. I have enjoyed

the friendly and stimulating research atmosphere – thanks to everyone! It is a great learning

experience in my life and will be unforgettable. You all have undoubtedly contributed to my

thesis.

Special thanks go to my parents and other family members for their love and

encouragement. Your endless support will make me unceasing progress in my life.

Last but not least, great thanks to all the friends I met here in Sweden, you make my life

rich, leaving us friendship and fragrant recalling to last a lifetime.

Stockholm, January, 2012

Peng QI

Contents

Standard Notations .................................................................................................................. vii

1 Introduction ......................................................................................................................... 1

1.1 Understanding topology .............................................................................................. 1

1.2 Topology in robotics: a glance ..................................................................................... 2

1.3 Structure and literature sum-up ................................................................................... 4

1.4 Contributions ............................................................................................................... 4

1.5 Outline ......................................................................................................................... 4

2 Topological Modeling of Robotic Manipulation ................................................................. 7

2.1 Closed manifolds in robotic manipulation ................................................................... 7

2.1.1 Operational space of end-effector ..................................................................... 7

2.1.2 Configuration space of articulated robot arm ................................................... 8

2.3 Smooth mapping of robotic manipulation ................................................................. 10

2.4 Understanding singularity and redundancy ................................................................ 11

2.4.1 Singularity ....................................................................................................... 11

2.4.2 Redundancy .................................................................................................... 13

3 Aspects of Topological Reasoning .................................................................................... 15

3.1 On the topological structure of configuration spaces ................................................ 15

3.1.1 The matrix representation of topological changes .......................................... 15

3.1.2 Topological invariants .................................................................................... 17

3.2 On the global solution to an inverse kinematics problem .......................................... 18

3.2.1 A general formulation of path tracking .......................................................... 18

3.2.2 Different path lifting algorithms ..................................................................... 19

3.2.3 Arguments on inverse kinematic and identity mapping ................................. 21

3.3 On the singularity existence problem ........................................................................ 22

3.3.1 Rotation axes parallel to a plane ..................................................................... 22

3.3.2 Arguments on the singularity existence problem ........................................... 23

3.3.3 The seventh DOF in redundant robot arms .................................................... 23

4 Redundant Manipulator Self-Motion Topology ................................................................ 25

4.1 Overview on seven DOFs anthropomorphic robot arm ............................................ 25

4.1.1 Manipulator construction and model .............................................................. 25

4.1.2 Flexibility and redundancy parameterization ................................................. 27

4.2 Inverse kinematics solutions for seven DOFs anthropomorphic robot arms............. 28

4.2.1 Velocity based IK solutions at the local level ................................................ 28

4.2.2 Global approaches to the IK problem at the position level ............................ 29

4.2.3 Graphic illustration of self-motion manifolds ................................................ 30

4.3 Characterization of self-motion manifolds ................................................................ 33

4.3.1 The number of self-motion manifolds ............................................................ 33

4.3.2 Bundle connectivity graph .............................................................................. 33

4.4 Follow-up processing with the characterization of self-motion manifolds ............... 36

4.4.1 A two-phase approach of off-line analysis and on-line retrieving.................. 36

4.4.2 Global path planning optimization ................................................................. 36

5 Conclusion and Future Work ............................................................................................. 37

Bibliography ............................................................................................................................ 39

Standard Notations

the real numbers

the natural numbers

the integers

n-dimensional Euclidean space

M operational space

C configuration space

J Jacobian matrix

the unit sphere in

m-dimensional torus

SO(3) special orthogonal group of 3-by-3 matrices

I the unit interval [0, 1]

product of sets, groups, or spaces

the fundamental group functor

o function composition

isomorphic

the induced homomorphism

CHAPTER 1 INTRODUCTION

1

1 Introduction

This chapter aims at making clear the essential thoughts in topology, i.e. the study of

connectivity and continuity, and serves as a brief and general introduction to the

thesis work. We answer the question of what topology is and then take a glance at the

different topological perspectives in robotics. Next, we classify the main cited

references along the structure of the report. It is concluded with contributions and the

outline of the subsequent chapters.

1.1 Understanding topology

Topology is an area of mathematics that studies the properties of topological spaces and in

particular properties that are invariant under certain types of deformations. The idea of

topology originated from the investigation of some geometric problems. Leonhard Euler’s

demonstration in 1736 on the historically notable problem, the Seven Bridges of Königsberg,

is generally considered as the beginnings of topology. In Fig. 1.1 (a), there are seven bridges

separately crossing two branches of the river and two islands (A and B) are connected to each

other and the mainland (C and D) by those bridges. The problem is to check if it is possible to

plan a walk that would cross each of these bridges once but not more than once. Euler’s

analysis reformulated the problem in an abstract term (See Fig. 1.1 (b)), where he ignored

geometric properties such as length and angle but was interested in connectivity properties.

By this way, a collection of points and lines build a sufficient model containing all the vital

information of the problem. This mode of thinking laid a foundation for graph theory and

prefigured the idea of topology.

Figure 1.1. (a) shows the diagram of the Seven Bridges of Königsberg; (b) shows the

graphic demonstration of the Seven Bridges of Königsberg.


2

Topology is sometimes called ―rubber sheet geometry‖ [1], since many properties that are

of interest to topological studies stay invariant if the space is stretched and contracted like a

rubber sheet. In this sense, squares are considered the same as circles. You might also have

heard ―a topologist is a person who cannot tell a coffee cup from a doughnut…‖ (See Fig. 1.2)

Both a doughnut and a coffee mug with a handle have just one hole, and they can be

continuously deformed into each other.

Figure 1.2. A coffee mug and a doughnut. They are regarded as the same from

topological perspective.

To sum up the two motivating examples above, topology, in some aspects, can be defined

as the study of connectivity and continuity. Modern topology is a broader field including

many branches. Under basic divisions they are point-set topology, algebraic topology and

differential topology [2]. Point-set topology or general topology is the low level language of

topology and studies fundamental properties inherent to topological spaces such as

compactness and connectedness, while the topological spaces do not have to be similar to

manifolds. Algebraic topology focuses on algebraic invariants that classify topological spaces

(e.g. up to homotopy). Differential topology studies manifolds and maps between them. In

this thesis, a study on various topological methods or topological perspectives applied to

robotic manipulation such as topological analysis of configuration space, motion planning, etc,

will be proposed later on.

1.2 Topology in robotics: a glance

The interdisciplinary research of robotics and topology is exploring an ―inchoate world‖ [3]

in the field of robotics. In the late 1980s and 1990s, some topological considerations in

robotics were started being reported in publications such as [4-10], while a booming era of

learning and enabling topological perspectives in robotics or engineering issues was triggered

by a group of topologists and roboticists in the Topology & Robotics conferences [11], [12] in

2003 and 2006 both at ETH, Zürich.

In [13] the pilot studies have been concentrated on extracting a topology-based

representation for character motion synthesis that involves close contacts in the area of

computer animation. A unique state space called topology space is proposed where three

attributes of topology coordinates are used to represent how the segments of a single character

or multiple characters are tangled. Here topology space is homotopy equivalent to generalized

motion space, thus the complex movement planning can be achieved efficiently by processing


3

only few parameters. A principal superiority of topology representation is that we do not need

to adhere to complex numerical description which is computationally expensive.

A more general way of topology-based formalism, thinking and philosophy with regard to

robotic manipulation has been proposed similarly by [6], [9], [10], [14], which is widely

accepted by topologists and roboticists. In Chapter 2 we will reveal a comprehensive

interpretation.

Moreover, a novel approach to model topological changes of metamorphic mechanisms is

introduced in [15], where the topological configuration is decided by link connectivity and the

configuration of mechanisms is represented in the form of topological graph and adjacency

matrix. Here the information about joint rotation or translation is disregarded, while

topology-based abstractions in another aspect clearly lay out vital information about the

mechanism. This kind of topological perspective is well adapted to revealing the intrinsic

relationship between distinct topological configurations of metamorphic mechanisms. In this

sense, the topological configuration of the fixed robot manipulator will always remain the

same during manipulation tasks.

On the general solution to an inverse kinematics problem in the field of robotic

manipulation, researchers have proposed some insights and analysis from topological

perspectives, such as Baker in [9], Gottlieb in [4-6], Baker and Wampler II in [8], Pfalzgraf in

[10], etc. The theorems concerning inverse kinematic solutions in those papers are proved

based on methods from topology and we can see that the applications of topology can give

insight into engineering problems.

In [16] Balkcom and Mason come up with a graphical method to describe the topology of

origami configuration space. Authors first consider origami-paper as a mechanism (creases as

revolute joints, and facets as mechanism links) and then give a summary of seven distinct

topological classes of configuration space. The topological analysis to determine the

connectedness of origami configuration space is presented, but needs more rigorous

treatment.

Furthermore, Erdmann in [17] gives a broad exploration on a topological perspective of

planning discrete strategies with uncertainty. This is a graph-level planning and has been

receiving significant interests in recent years. With a topology-based representation, the

graph-level method can be applied in a number of different domains within robotics such as

spatial navigation, graph searching and planning, control level schematic, etc.

Regarding the self-motion manifolds of redundant robot manipulators, Lück in [18]

proposes a detailed exploration and introduces a unique graphic representation (bundle

connectivity graph) to delineate the relationship between the finite sets of kinematic map. The

technique has also been successfully applied to a spatial eight DOFs manipulator (AAI Arm)

[19], confirming the applicable potential of the proposed concepts.


4

1.3 Structure and literature sum-up

1.4 Contributions

The primary contribution of this thesis is a review of topological perspectives in the field of

robotic manipulation. There exist several long and thorough articles (e.g. [23]) with regard to

topology-based investigations in general robotics, but the research targeted specially to

robotic manipulation distributes separately. Although we can extract some scattered

discussions in this aspect from literatures, the same essence usually appears in different kinds

of formulations. In this report, we narrow down the scope of topological problems inspired by

robotics and summarize some selected topics, methods and results from topological

perspectives in robotic manipulation. Most parts of results in the text have been mentioned

separately by other literature. We also integrate some new considerations and demonstrate the

possibilities to apply the previously proposed methods to our thesis field.

A second contribution is that we associate different approaches and propose the feasibility

based on a plenty of relevant readings. Parts of them are verified while others might imply the

research directions of the future.

1.5 Outline

The remainder of this report is organized as follows: in Chapter 2, we establish a general

topological model of robotic manipulation and emphasize on topological thinking of

configuration space and kinematic map. This modeling approach provides the possibility to

apply methods and theorems from topology in practical manipulation problems.


5

In Chapter 3, we give an overview on some aspects of topological reasoning in robotic

manipulation and summarize the discussions from a number of different research papers.

Studies are classified into three categories: the topological structure of configuration spaces,

the global solution to an inverse kinematic problem and singularity existence problem.

Benefiting from various methods and results from algebraic topology, we get a greater

understanding of some aspects of robotics problems

After a general exploration on the topological considerations of configuration space,

kinematic mapping and singularities in Chapter 3, Chapter 4 gives an in-depth investigation

on redundant anthropomorphic robot arms and topological properties of self-motion

manifolds. As a result, the characterization and classification of inverse kinematics solutions

at a global and position level is derived. In Chapter 5, some open problems are discussed.

CHAPTER 2 TOPOLOGICAL MODELING OF ROBOTIC MANIPULATION

7

2 Topological Modeling of Robotic

Manipulation

The following chapter introduces the basic terminology of kinematic chains and

proposes a topology-based formalism for representing the manipulation operations.

As we will see later on, the mentioned spaces generally are smooth closed manifolds

and it is on this premise that methods from the abstract world of topology can be

applied to the concrete subject of robotic manipulation. The classical metrical notions

(Denavit-Hartenberg notation [26], for example) and frame transformations of a

manipulation system are rather remote from the topology-based abstraction, thus very

few or none of them will be mentioned here. The essence of the chapter is centered on

the topological perspective.

2.1 Closed manifolds in robotic manipulation

In the study of manipulation systems, we commonly consider two kinds of spaces—the

operational space of end-effector and the configuration space of robot arm. End-effector

refers to the tool attached on the free end of robot arm such as magnet, parallel gripper, hand,

or other devices. Robot arm here consists of multiple rigid links connected by joints. The

following mathematical interpretations of these two spaces act as a bridge between topology

and robotics, and lay the foundation of topological reasoning in robotics. Similar way of

thinking is accepted by topologists and roboticists and they have established or mentioned

similar mathematical models. Our review starts with a summary of the topological model for

robotic manipulation in this chapter. Throughout the thesis, mechanical manipulation related

notations are following Craig’s textbook [27].

2.1.1 Operational space of end-effector

When considering an object in three-dimensional space, there are two rough attributes,

position and orientation, to be specified. Position of a point is defined by the vector

in three-dimensional Euclidean space. Three-dimensional space orientation of

a body is uniquely determined by special orthogonal group SO(3), which is a rotation group

of orthogonal 3-by-3 matrices with determinant equal to +1. Hence the Cartesian product of dimension six totally specifies the operational or target space of the

end-effector. In topology, it is a smooth and closed differential manifold. In general

conditions the operational space can be explicitly identified with

(2.1)


8

where refers to the orientation in three-dimensional space. We do not think of its explicit

mathematical composition, because it is not necessary to know what the formula is in the

following topology analysis.

We also have to notice that the operational space above is a general description and the

specific one should correspond to the concrete manipulation tasks. Position can be in

two-dimensional plane space, i.e. , and sometimes we do not need the full space

of orientation, for example in line welding, we do not need the welding tool to rotate around

the pointing direction. Similarly, in two-dimensional Euclidean space, orientation of a body is

uniquely determined by another special orthogonal group SO(2).

2.1.2 Configuration space of articulated robot arm

As previously mentioned, joints in kinematic chains connect rigid links and allow relative

motion of neighboring links. We usually consider two kinds of joints: rotational revolute joint

and translational prismatic joint. Both of them have one degree of freedom (DOF)1.

Figure 2.1. (a1), (a2) revolute joint, which constitutes purely rotational motion along the

joint axis; (b1) prismatic joint, which constitutes purely translational motion along the

joint axis.

As shown in Fig. 2.1(a), the relative motion of revolute joint is specified by joint angles.

From the topological point of view, these joint angles correspond to points on the unit

circle . In the Euclidean plane, a point identifies a complex

number on a unit circle and it is only specified by angle . From the

relation , we have the algebraic form as .

Here the angle interval is theoretically defined, while in practice there are usually other limits.

The configuration space of a kinematic chain with all revolute joints will be a Cartesian

product

, which forms an m-dimensional torus with .

Torus is a topologist’s favorite surface which is n-dimensional compact manifold and

compact abelian Lie group. Two-dimensional torus is easy to depict by a closed surface of

1 Degree of freedom: any of the minimum number of independent parameters required to completely specify the

motion of a mechanical system, commonly abbreviated to DOF.


9

―donut‖ (See Fig. 2.2), while the m-dimensional torus is its analytical generalization to higher

dimensions.

The displacement of neighboring links with prismatic joint is mathematically interpreted by

a real interval [a, b] of , or more generally by the whole real line . Compared with a

circle above, real line is homeomorphic to the open interval with two ends, which is a

one-dimensional topological manifold and only locally compact. Due to these minor

differences, a general description for the configuration space of a robot arm with both

revolute and prismatic joints is reasonably with two parts:

(2.2)

where m and n are the number of revolute joint and prismatic joint, respectively. All the joints

are independent. The dimension of configuration space manifold defines the DOF of the robot

arm, i.e. .

Given the collection of the link lengths , the configuration of

robot arm is fully determined by a set of rotational angles and translational

displacements . This set can be seen as a point on a (m+n)-dimensional

manifold—the configuration space . We also should realize that in practical cases

there are inevitable restrictions on joint motions and thus the configuration space will be some

subset of .

Explicitly, the configuration space can be identified as

(2.3)

An intuitive illustration is given by Fig. 2.2: there is a basic robot arm consisting of two

general revolute joints as elbow and wrist, and thus its corresponding configuration space is

the surface of a two-dimensional torus. Any pair of elbow angle and wrist angle defines a

specific point on the donut, and vice versa. If we think of latitudes and longitudes (both

are ) on the surface, the latitude variable and longitude variable are exactly

consistent with elbow angles and wrist angles. Any arm movements can be illustrated by a

continuous path from one point to another on the torus surface.


10

Figure 2.2. Configuration space from a topological point of view [3]. Robot arm

configurations A and B respectively corresponds to points A and B positioned by the

―latitude‖ and ―longitude‖ on the torus surface.

Note that we only consider the two kinds of joints—revolute joint and prismatic joint,

which most manipulators have. Another popular joint is the spherical joint, or called ball joint.

It is of three degrees of freedom and we can find it used in some cases such as humanoid

robot arms and closed kinematic chains. However, its mechanical complexity is much higher.

In mathematics, we use the sphere to interpret that joint.

2.3 Smooth mapping of robotic manipulation

We have given the mathematical interpretations for operational space of the end-effector

and configuration space of the robot arm. Both of them are nice differentiable manifolds. In

topology, we thus think of a smooth map from a topological space to another. Now consider

the mapping for the two spaces, clearly a function can be defined as , which is a

so-called forward kinematics map. Specifically, the mapping between Eq. (2.1) and Eq. (2.3)

can be described as

(2.4)

A general operational space is of six DOFs. If a specific robot arm satisfies m+n 6 and

m 3, every desired position and orientation of end-effector can be achieved by a certain

configuration of this robot arm. It partially explains why the six DOFs robot arm is quite

popular in industrial settings. We take the Unimation PUMA 560 for instance. It is a classical

and popular six DOFs robot manipulator, whose end-effector can reach any point within its

workspace with any orientation (See Fig. 2.3).


11

Figure 2.3. PUMA 560 manipulator. The first three rotations specify the position of

end-effector and another three in the wrist orient the end-effector.

The established forward kinematics map f is onto function but not one-to-one. Any one

robot arm configuration corresponds to one position and orientation of end-effector, while

there may be more than one configuration solutions for a fixed end-effector. A simple

example could be ―elbow up‖ and ―elbow down‖ when we fix the free end of the arm. Here

comes another mapping known as inverse kinematics map. This function usually

receives more researchers’ recognition, as the details and requirements in operational space

are known. Besides, the control is derived based on information in operational space, so a

definite description and analysis on the inverse kinematics solution is highly valuable.

2.4 Understanding singularity and redundancy

The two concepts mentioned in this section may not be relevant to topological modeling,

but in order to achieve a complete understanding of robotic manipulation, we include them.

These two concepts are ―singularity‖ and ―redundancy‖ and will be frequently mentioned

later on.

2.4.1 Singularity

Let us define as a robot arm configuration and as an end-effector position

within operational space. Given there is the smooth mapping according to Eq. (2.4), we

have . By taking its partial derivatives, we derive another function— the Jacobian J,

which maps joint velocities to end-effector velocities, i.e.

(2.5)


12

A detailed description of the differentials of Jacobian function is beyond the scope of this

thesis. For more information, see the related chapters in [27].

Having the Jacobian matrix in Eq. (2.5), we say that a point is called a singular

configuration if the Jacobian matrix J at this point does not have maximal rank. In [10],

Pfalzgraf interprets this singular condition from a linear approximation point of view. It is

mentioned that for a small neighborhood of a singular configuration q, the tangent map

(derivative) cannot guarantee a full linear approximation of a neighborhood of its image f (q)

due to losing rank. Meanwhile, Gottlieb in [4] admits this interpretation as a good intuitive

picture, but it is not mathematically strict. Moreover, the author adds that the singular

condition originates from the fact that mapping f is not onto. Locally not every instantaneous

motion of the end-effector can be tracked by instantaneous robot arm joint motion.

Thus we could say the singularities do not ―prevent the robot arm position anywhere‖

within its workspace, but rather ―cause problems with motion of the arm in their

neighborhood‖ [27].

The Jacobian matrix J at the singular point p loses rank, i.e. the corresponding determinant

vanishes, |J (p)|=0. On the other hand, by a transformation of Eq. (2.5) under normal

conditions (|J (p)| 0 and ) we get

(2.6)

Regarding a path tracking problem with bounded end-effector velocity , Eq. (2.6) hints

that the joint velocities tend to infinity when the end-effector trajectory approaches the

singularity. However, as mentioned in [9], when the trajectory goes directly through the

singularity, the velocities problem surprisingly does not arise (See Fig. 2.4).

Figure 2.4. Path tracking by a two-link planar robot arm. Two links are equal length and

connected by revolute joint. One end of robot arm is fixed in the center point with one

rotational DOF ( ). The disc represents two-dimensional operational space. In

(a) the robot arm tracks a straight line going near the singularity and we can see that

from point B to point D the ―elbow‖ is forced to toward the opposite direction by high


13

joint angle velocities. In (b) the straight line passing through the singularity but the

singularity configuration as a transitive state makes the tracking more smooth and

harmonious than that in (a).

Singularities cause many technical problems and should be taken care. Although they

cannot be eliminated, they can be efficiently avoided. Based on the location of the

end-effector, singularities are classified into two main categories [27]:

1. Workspace-boundary singularities: the robot arm is fully stretched or folded back;

2. Workspace-interior singularities: two or more joint axes become collinear.

2.4.2 Redundancy

First, we give a description of the redundancy: if the DOF of configuration space is more

than needed, i.e. dim (C)>dim (M), there will be ―extra‖ free joints and we call such an arm as

redundant arm. It offers extra freedoms for robot arm, which intuitively enlarges the

flexibility of certain manipulations. However, an infinite number of possible joint motions

also complicate the control strategies [28].

Redundancy has very close relation with singularity avoidance. In next chapter, we will

apply topological viewpoints on this issue which results in some insights. Human arm is

seven DOFs and thus seven DOFs redundant robot arms are receiving more and more

recognition nowadays. In this thesis, there are some analysis related to the Schunk LWA3

robot arm (See Fig. 2.5).

Figure 2.5. Schunk LWA3 seven DOFs robot arm.

CHAPTER 3 ASPECTS OF TOPOLOGICAL REASONING

15

3 Aspects of Topological Reasoning

The following chapter describes some topological considerations of configuration

space, kinematic mapping and singularities. It becomes clear that the concepts and

theories from algebraic topology fit quite naturally into robotics problems and result

in some interesting and broadening insights [5]. Although a topological viewpoint

does not provide precise numerical or optimal solutions, it enriches our

understanding on robotics. In this chapter, we give an overview of some aspects of the

use of topology focusing on a general open kinematic chain and summarize the

discussions from a number of different research papers. Although significant research

has been carried out in this direction, the “topological robotics” is none the less an

“inchoate world” [3].

3.1 On the topological structure of configuration spaces

3.1.1 The matrix representation of topological changes

Topology is a branch of mathematics and initially related to geometry. When we talk about

topology, we commonly refer to the geometrical connectivity or the relative positions of

points, excluding the intricacies of distance and curvature. Thus what we consider is whether

two points are path-connected or not. In our daily life, the subway map is a typical example

derived from topology, where the actual layout of the underground system is simplified to

only display its connectivity properties.

In this subsection, our discussion is focused on ―topological structure‖ and ―configuration

spaces‖. If we did not mention the term ―algebraic topology‖ and did not present the

mathematical description for ―configurations‖ in the previous chapter, one may naturally

suppose that exploring the topological structure of configuration spaces is about the

mechanisms structure analysis of robot manipulators. This kind of topological study is

reasonable and does exist, thus in the following paragraphs we will first present this point of

view on robot arms.

Dai and Jones in [15] represent a configuration of mechanisms in the form of topological

graph and adjacency matrix. The topological configuration is decided by link connectivity,

rather than by specifying rotational angles or translational displacements. It means that

configuration of the robot arm will remain the same during task manipulations if its structure

does not change. Below we briefly explain how the matrix representation of topological

changes is derived in [15].

First of all, in the initial configuration each link or virtual link is numbered from 1 to n.

And then both the rows and columns of a square matrix are labeled with these link numbers in

sequence. Finally, the element (i, j) in the square matrix is given as 1 if the i-th and j-th links


16

are connected, while 0 is assigned to the corresponding disconnected cases and all the

elements on the main diagonal are equal to 0.

Figure 3.1. Topological changes of one four-link revolute joint robot arm with an extra

virtual link.

Figure 3.1 illustrates two configurations of a four-link revolute joint robot arm. The dotted

line in subfigure (a) denotes the virtual link connecting the end-effector and the manipulated

object. In subfigure (b), the end-effector grips the object and the 5th link degenerates. Based

on the matrix representation rules described above, topological configurations in subfigure (a)

and subfigure (b) are represented by

(3.1)

(3.2)

As we can see from the matrix representation, the topological configuration variation

brings on the synchronous changing in matrix order and some element values. In order to

describe the configuration transformation, an E U-elementary matrix operation is introduced

in [15]. It applies an elimination Ej-elementary matrix together with a Ui,j-elementary matrix

to set up the relationship between two matrices of different topological-configurations in the

form of

B = (E U) A (E U)T (3.3)


17

Details about the formation of E U-elementary matrix and its structural evolution can be

found in [15], [21]. In addition, an example of topological configuration transformation of a

spatial metamorphic mechanism is presented in [15].

Although the proposed topological analysis of configurations is originally focused on

metamorphic mechanisms, we can still adopt this kind of formalism and topological thinking

to structure-fixed manipulators. In such cases, the manipulated objects are modeled together

with one arm or dual arms and the configuration of objects changes in the process of

manipulation. Besides, for dexterous grasp manipulations, the proposed topological modeling

becomes quite intuitive. It is because the topological configuration of a multi-finger hand with

the object changes frequently, not when the robot hands hold the object for a closer inspection,

but rather every time when the fingertips touch the object, i.e. connect and disconnect.

On the other hand, since the topological configuration of a structure-fixed robot arm

remains unchanged, we can set it as one of motion planning criterions, i.e. in the whole

moving process our control will try to keep configuration invariant. (See Fig. 3.2) In order to

hold the topological configuration unchanged, we are allowed to plan a kind of behavior as

the red character does. For the blue character, its left arm penetrates through the right arm,

which causes two wrists attached together. Consequently, the topological configuration is

changed. This situation is forbidden when planning the motion.

Figure 3.2. The snapshots of posture changing from (a) to (c), and the corresponding

topological configurations are represented by articulated line segment representations.

From (a) to (c) the topological configuration of red characters remains the same, while

that of blue characters changes due to two points becoming one in (b). (Character figures

originate from [13])

3.1.2 Topological invariants

Robotic manipulation research is mainly concerned with two closed manifolds

[14]—operational space and configuration space, and therefore investigating the topological

properties of both spaces, to a great extent, will assist us in understanding and characterizing

robot arm mechanisms and its behaviors or system capabilities. It is because topology is good


18

at extracting intrinsic structures. The term ―structure‖ now does not refer to the physical

skeleton but rather to the properties of sets of the space such as compactness, connectedness,

homology, etc. There are many meaningful applications by this kind of topological

investigation such as comparing and evaluating mechanisms [22], aiding robotics motion

planning [29], [30], etc.

Some particular topological invariants under discussion here are algebraic invariants such

as homology and homotopy groups. Besides, we are curious about more topological

properties. For example, when comparing two mechanisms, we compare their behaviors

rather than their structure [31], thus we expect to transform configurations and motions of one

mechanism to those of another [22]. This naturally raises our interest on the topological

invariants related to homeomorphism and homotopy equivalence.

Simply speaking, homeomorphism addresses the problem of whether two topological

spaces look similar. By definition, the transformation function between these two spaces

should be bijective, continuous, and its inverse function should be continuous. An interesting

illustration is that between a coffee mug and a donut there exists a continuous deformation.

By using homeomorphisms we could translate motions among spaces, or perhaps the problem

in one space can be put into another space where it can be seen from another perspective.

Homotopy equivalence describes a process in which one space is deformed continuously

into another. The fundamental group is an often used invariant of homotopy equivalence, on

which we will explain more details in the following sections.

Investigating topological invariants undoubtedly provides a more subtle underlying

interpretation than pure connectivity properties as mentioned in the preceding subsection,

however, regarding this pure mathematical considerations we still can’t give a confident

conclusion on whether it will sufficiently satisfy the practical needs. So to speak, topological

investigation from the qualitative viewpoint is not an adequate criterion, sometimes there are

requirements for extra information such as a metric.

3.2 On the global solution to an inverse kinematics problem

Manipulator inverse kinematics problems have been addressed by numerous experts.

Mathematicians and in particular topologists have also proposed lots of insights and analysis

into the corresponding engineering aspects. Here we present a summing-up of topological

reasoning on inverse kinematics solutions. I would like to point out that the solutions

subsequently mentioned are quite a personal selection, not a complete overview due to my

limited view and depth. Since most common robotic manipulators constructed today only

consist of revolute joints, in the analysis we consider all joints to be revolute, i.e. . The same results can be obtained by using similar methods for prismatic joints.

3.2.1 A general formulation of path tracking

Given an end-effector trajectory in the operational space , i.e. a path ,

where , and the forward kinematics map


19

, we want to find an acceptable continuous path in configuration space, i.e. ,

where , such that . From the topological point of

view, it can be formulated as a path lifting problem [10]: does there always exist such a map q

so that o The commutative diagram is shown as below:

It is desirable for the path to be a continuously differentiable curve in

configuration space [10]. On the other hand, we have to notice that the forward kinematics

map f inevitably contains singularities which should be avoided by the path lifting. Moreover,

the path lifting solutions also need to be conveniently produced by computers as mentioned in

[9].

3.2.2 Different path lifting algorithms

Baker in [9] explored several possible approaches to the path lifting problem, among which

the most natural way is to integrate the proposed differential equation:

(3.4)

where q represents a robot arm configuration and x is an end-effector placement.

In addition, to derive the function g( ), one can combine some optimal conditions with

regard to manipulator performances such as minimizing the joint angle rotations or speeds.

Although it is easy for computers to process those integrations, there are no considerations

that have been made for singularities. Meanwhile, in [32] the functional constraints method

has been considered to be conceptually simple, which resolves the redundancy and avoids the

singularities. In order to stay away from the singular configurations among , one can

propose another onto mapping to set a functional constraint. The dimension n is

equal to the effective dimension of the operational space, and the composition mapping

becomes:

where S is a subset that contains all singular and near singular configurations.

Given that the new domain is also a closed manifold, we desire the procedure o to

be invertible, i.e. to have no singularities. In fact it just works fine on the previously identified

avoidable singularities, but is not in full force and effect with regard to the constrained

M f

I M

C

q f

x


20

mechanism.

Furthermore, a special functional constraint known as the extended Jacobian technique is

proposed in [32]. Similar with the thought in Eq. (3.4), one first derives an objective

function based on a certain optimization criterion (commonly optimized locally) and

then induces a smooth mapping G through the proposed function and Jacobian matrix

information on the forward kinematics map f.

(3.5)

where m is the dimension of the configuration space, n is the dimension of the operational

space and P is a (m-n)-dimensional manifold. The function should also be a smooth

mapping between two topological spaces.

In order to track a prescribed end-effector trajectory x( ) with the initial configuration q(0)

and the forward kinematics map f, we establish the equation

(3.6)

where a denotes a constant and is determined by the initial configuration q(0).

By differentiating both sides of Eq. (3.6) with regard to time t, we derive the relationship

between joint angle velocities and end-effector velocities.

(3.7)

where we call the m-by-m square matrix of the form

(3.8)

to be the extended Jacobian .


21

Theoretically, the mapping G is designed to make Eq. (3.6) invertible. However, the

algorithm can occasionally generate non-singular robot arm configurations where

has its

maximal rank but

becomes singular.

Although effort has been made and some possibilities were explored on the inverse

kinematics problem, no global solution to inverse kinematics problem can be provided, in

general. With regard to redundant manipulators, some important remarks and analysis of the

closed-loop inverse kinematics algorithms are proposed in [7], [8], [33].

3.2.3 Arguments on inverse kinematic and identity mapping

Given the continuous forward kinematics map , the mathematical

argument on the global solution to the inverse kinematics problem can start with an

assumption that there exists such a cross-section to f, that is to say, a continuous

mapping so that the composition o is an identity function, i.e.

o

[4].

Gottieb in [4-6] and Pfalzgraf in [10] have concentrated on the map and

argued that such a cross-section cannot exist, in general. It is a qualitative conclusion that is

obtained from a topological perspective. They first get the induced homomorphism and

and send both of topological spaces (i.e. and ) to their fundamental

groups . This thinking is derived from the fact that fundamental group could explain

essential differences between different kinds of topological spaces. Then, the identity function

further yields

. At this time the argument is transformed to

verify whether this induced homomorphism is one-to-one. However the corollary about the

induced homomorphism being one-to-one cannot be true. It finally leads to a conclusion

that no cross-section to exists.

Here we further extend the map into , while the same result can also

be obtained based on the following two facts.

1. It is commonly known that the fundamental group of is trivial fundamental group, i.e.

.

2. Proposition [20]: is isomorphic to if X and Y are

path-connected.

We find

, and thus the rest of

topological arguments will comply with the methodology that is applied for the map

in [4-6], [10].


22

3.3 On the singularity existence problem

Singularities are undesired in engineering, which makes engineers have to consider parts of

the configuration space as forbidden areas. Meanwhile, this treatment increases the

operational difficulty due to robot arms’ ―shrinking‖ movements and even causes some DOFs

to be lost. However, singularities cannot be eliminated in general, which can be seen from a

topological perspective below.

3.3.1 Rotation axes parallel to a plane

First we mention a practical engineering remark which provides us a direct impression on

singular configurations. We will quote one theorem appearing in [6], which states that a robot

arm configuration is singular for the map if and only if all the

rotation axes of the robot arm are parallel to a plane. Note that the above mentioned

singularities refer to the rotational singularities2 in the interior workspace.

Let denote all the rotation axes of the robot arm. Suppose that the rotation

axis and are parallel to a plane, then an instantaneous rotational velocity about the

axis will impart an instantaneous rotational velocity about the axis , which is

perpendicular to the plane. Furthermore, if all the rotation axes of the robot arm

are parallel to the plane, the corresponding velocity about the axis will take the

composition of all the instantaneous rotational joint velocities and be fixed only

perpendicularly to the plane. Consequently, the robot arm loses some DOFs. Figure 3.3 shows

an illustrational configuration of robot arm under the singular condition in which all the

rotation axes are parallel to a plane.

Figure 3.3. A robot arm configuration with all the rotation axes parallel to a plane.

Understanding this theorem may bring us a good practical habit of operating the robot arm,

since it is better not to settle or initialize a robot arm configuration with all its rotation axes

parallel to a plane. This is an easy mistake to make, because such configurations always make

the platform look formal.

2 A rotational singularity here refers to a robot arm configuration in which an instantaneous rotational velocity of

the end-effector can be arbitrarily specified.


23

3.3.2 Arguments on the singularity existence problem

Consider the six DOFs robot arm with full six-dimensional operational space, the

corresponding mathematical description of kinematic map then can be presented as . Among the six joint parameters that specify the

configuration of the robot arm, commonly the first three joint parameters are to define the

position of the end-effector in three-dimensional Euclidean space and the rest

three parameters specify the orientation of the end-effector.

The existence of workspace-boundary singularities is clearly apparent to the understanding,

and arguments on singularity existence problem are mainly focused on the persistence of

internal singularities. With a known clear-cut division of six joints in position and orientation,

research is carried out in the direction of increasing the DOF for the orientation to eliminate

the rotational singularities.

However, if one more DOF cannot completely eliminate the rotational singularities, what

about adding two, three and more DOFs ? This question is not simple as it seems to be. The

topological reasoning could help us clarify this practical engineering problem. First, a

mathematical formulation of the orientation mechanism part is presented as

(3.9)

Suppose that there are no singularities between the two closed manifolds and ,

then the smooth map f is a submersion which must be the projection map of a fibre bundle

based on the Ehresmann’s fibration theorem [34]. However, a topological argument based on

results from homology indicates that unfortunately the map cannot be a fibre bundle [5]. In

fact, no matter how many redundant DOFs are added, singularities will not be completely

eliminated. They must necessarily exist.

We do not give a detailed topological proof of the persistence of singularities. More strict

mathematical proof can be found in [5], [10].

3.3.3 The seventh DOF in redundant robot arms

Now that the singularities cannot be completely eliminated by adding extra redundant

joints, the problem is changed to avoid singularities. There are usually two main categories of

singularities—boundary and interior singularities—based on the location of the end-effector.

On the other hand, when dealing with anthropomorphic manipulators of six or seven DOFs,

we commonly classify singularities into another three types—shoulder, elbow and wrist

singularities—based on the locations where they occur [35]. (See Fig. 3.4) If the end-effector

lies along the rotation axis line of the first shoulder joint, the end-effector then cannot move in

the direction perpendicular to the arm plane. This is a type of shoulder singularity and occurs

inside the operational space; if the elbow of robot arm is fully stretched, the end-effector then

reaches the workspace boundary. This is a type of elbow singularity and can also be

considered as a boundary singularity; if two rotation axes of wrist joints are lying on the same


24

straight line, then the wrist inevitably loses one DOF. This is a type of wrist singularity and it

also occurs inside the operational space.

Figure 3.4. Singularities occur at the shoulder, elbow, and wrist joint of the six DOFs

anthropomorphic manipulator. (a): shoulder singularities; (b): elbow singularities; (c):

wrist singularities.

By adding the seventh revolute joint, the two types of interior singularities—shoulder and

wrist singularities—can be naturally avoided, and we also notice that both of them are

rotation singularities. In the case of elbow singularities they are inherent to the design of the

robot arm and only can be kept apart.

In [36] it is questioned why we just add the seventh DOF rather than an eighth or more

DOFs, and why a revolute joint rather than a prismatic joint. Firstly, in [36] Hollerbach set

forth four kinematic criteria for redundant arms: internal singularities elimination, workspace

optimization, kinematic simplicity and mechanical constructability. A seventh extra DOF can

adequately match all the four criteria, and there always exist the multiple degeneracies that

lead to internal singularities no matter how many extra DOF will be added.

In addition, shoulder and wrist singularities can be naturally avoided by adding the revolute

joint, while a prismatic joint cannot avoid the wrist singularities. So in this sense a revolute

joint is more adequate.

CHAPTER 4 REDUNDANT MANIPULATOR SELF-MOTION TOPOLOGY

25

4 Redundant Manipulator Self-Motion

Topology

Most of topological discussions in robotics publications arose in the late 1980s and

1990s. With the advent of the 21st century, researchers began to explore the

topological properties of data sets. They not only focused on theoretical aspects but

also perform experimental evaluation, such as inverse kinematics solutions, motion

planning solutions, etc.

In the previous chapter, we generally explored the topological considerations of

configuration space, kinematic mapping and singularities. Besides, we mentioned the

seven DOF of redundancy. In this chapter we concentrate on inverse kinematics of the

seven DOFs anthropomorphic robot arm by exploring its self-motion manifolds. The

characterization and classification of inverse kinematics solutions using a global

approach at the position level will be derived.

4.1 Overview on seven DOFs anthropomorphic robot arm

Human arm consists of shoulder, elbow and wrist joints and has has seven DOFs in total.

However, only six DOFs are required to position and orientate the hand in three-dimensional

space. Thus Human arm has the nature of redundancy. The seventh DOF offers considerable

flexibility and dexterity to execute more complex tasks, and human can grasp things from

different angles with the redundant DOF. It has been widely recognized that seven DOFs

anthropomorphic robot arm enlarges the operational possibility of tasks in a human-like

manner. Recent successful examples of anthropomorphic arm include Motoman SDA10D

assembly robot [37], ARMAR humanoid robot [38], WABIAN robot [39] and FRIDA

concept robot [40].

4.1.1 Manipulator construction and model

To some extent, robotics research takes inspiration from biological systems and creates the

corresponding artificial counterparts. They are not only similar in morphology but also

analogous or even more professional in function.

Commonly, robot arms are categorized by the number of DOFs, and with the increasing of

DOF, the dexterity and versatility are gradually enhanced. Meanwhile, if we view human arm

as an articulated three-link kinematic chain, the three DOFs planar manipulator (See Fig. 4.1)

can be regard as a basic model of anthropomorphic robot arm to be analyzed.

In [41] Gao et al. proposed criteria to design mechanisms with three moving links and

revealed the relationships between evaluation criteria and the link lengths. Three DOFs planar


26

manipulator belongs to three-link manipulators, while three-link manipulators also include

other mechanisms whose number of DOFs is more than three. In this sense, the evaluation

criteria from [41] can be adopted to analyze anthropomorphic robot arms.

Figure 4.1. Three DOFs planar manipulator diagram. It is fundamental to

anthropomorphic robot arm and three links correspond to upper arm, forearm and hand.

We know that six DOFs are necessarily required to position and orientate the hand in

three-dimensional space. The six DOFs of an anthropomorphic robot arm are usually

distributed as following: two in shoulder, one at the elbow and three at the wrist. If we

continue to add the seventh DOF, there will be three possible kinematic designs—three DOFs

shoulder joint, two DOFs elbow or four DOFs wrist. In [36] Hollerbach argued that the

design of three DOFs shoulder joint is superior to others and it is kinematically equivalent to

human arm structure (See Fig. 4.2).

Figure 4.2. Diagram of human arm structure. Three DOFs are distributed in shoulder

joint, one DOF in elbow and three DOFs in wrist.

In many cases, three DOFs shoulder joint is regarded as a virtual spherical joint and wrist

joint is modeled in the same way. This is because their three joint axes intersect at one point.

We call this kind of anthropomorphic robot arm ―S-R-S‖ model [42]. It is worthwhile to note

that seven DOFs anthropomorphic robot arms are not restricted to ―S-R-S‖ model and there

are many alternative options [43]. Figure 4.3 depicts three kinds of shoulder joint designs. For


27

more information about mechanism designs of anthropomorphic robot arm, interested readers

are pointed to the related chapters in [44].

Figure 4.3. Diagrams of three different shoulder joint designs. Elbow joints and wrist

joints are omitted.

4.1.2 Flexibility and redundancy parameterization

The flexibility of anthropomorphic robot arm is provided by one DOF redundancy.

Redundant arm could accomplish any possible end-effector positions/orientations via an

infinite number of distinct configurations. The elbow of an arm could rotate about the

should-wrist line without any hand motions. This capability represents an important issue of

redundant arm and is named as self-motion (See Fig. 4.4). When seeking inverse kinematics

solutions or control strategies for redundant arms, we consider the self-motion as a critical

aspect.

Figure 4.4. The snapshots to show the self-motion of a seven DOFs anthropomorphic

robot arm. The elbow rotates about the should-wrist line without any hand motions.

To describe the arm redundancy, a parameter for representing the self-motion of redundant

arm is required. The selection of self-motion parameter is diversified. In [42], an overview of

several parameterization methods is given with the highlight in feasibility and adaptability. In

the following, we mention a commonly used parameterization method proposed in [45],

where self-motion is described by an arm angle (See Fig. 4.5). The arm angle is defined


28

as the angle between the arm plane and the reference plane. The arm plane is spanned by

should joint, elbow joint and wrist joint. Moreover, the trajectory of elbow joint is a circle or

an arc (with joint limits) and the arm angle defines the elbow position along the curve.

Figure 4.5. Self-motion of the spatial redundant arm. The arm angle is defined as the

angle between arm plane and reference plane.

4.2 Inverse kinematics solutions for seven DOFs anthropomorphic robot arms

Investigations of redundant manipulator inverse kinematics (IK) can be divided into two

main categories [24]: 1. using local methods to solve the IK problem at the velocity level; 2.

exploring IK solutions of global approaches at the position level. The first methodology is

mooted earlier and has achieved remarkable developments. It is the most used method but has

a number of disadvantages [25]. The second one aims at overcoming those difficulties of local

approaches and receives much interest recently. It requires topological tools to analyze the

structure of redundant manipulator kinematic map.

Table 4.1. Literature summarizing table on inverse kinematics solutions.

velocity based, local level position based, global level

[7], [28], [46-50] [18], [24], [25], [51]

4.2.1 Velocity based IK solutions at the local level

In Chapter 2, we derived the function (2.5) that maps joint velocities to end-effector

velocities by a linearized first order instantaneous kinematic relation. Assuming that the

manipulator has no redundant DOFs and is not positioned in a singular configuration, then the

Jacobian matrix J is invertible in Eq. (4.1) and the IK problem is solved.


29

(4.1)

Nonetheless, in the case of the seven DOFs anthropomorphic robot arm, the Jacobian

matrix J is of the size which is not full rank and thus there are an infinite number of

solutions to Eq. (2.5), i.e. redundancy problem. To address this issue, a number of methods at

a local level have been proposed in research, including pseudo-inverse techniques [7], [46-48],

Jacobian transpose methods [49] and augmentation methods [28]. Strategies are to use the

extra DOF to optimize robot arm performance or to propose an additional task function to

virtually eliminate the redundancy.

However, local approaches suffer from poor performances due to the convergence to local

minima [18], accumulated numerical errors [25] and being incomplete [25]. Thus global

approaches at the position level are explored to overcome those drawbacks.

4.2.2 Global approaches to the IK problem at the position level

Current global approaches are proposed by analyzing the topological properties of

manipulator kinematic map. Some results from topology are used to acquire the knowledge of

the entire system. In the following, we give a summary of the similar topology-based global

approaches in [18], [24], [25].

As discussed in the previous section, for a non-redundant manipulator, there are a finite

number of distinct configurations corresponding to a given end-effector position, while for a

redundant manipulator, an infinite number of configurations exist. The infinity of solutions

can be grouped into a finite set of discrete manifolds, termed as self-motion manifolds [24].

Each of manifolds is referred to as a fibre, and a rope consists of fibres [25]. Meanwhile, to

explore the structure of kinematic map at the global level, it is desired to update the previous

point-wise mapping to the manifolds mapping as:

(4.2)

where x is an end-effector position and {M} is a set of corresponding self-motion manifolds.

When a piece of operational space is mapped into configuration space, we can obtain part

of rope as a c-bundle and the corresponding piece of operational space is referred to as a

w-sheet [18]. Moreover, adjacent singularities [18] in configuration space form a singularity

manifold, which constitutes the boundary of c-bundles. The forward map of a singularity

manifold is referred to as a Jacobian surface in operational space, which serves as the

w-sheets partition [18]. Since the pre-image of a Jacobian surface consists of singular

configurations as well as regular configurations, it is termed as co-regular surface [18], and

singularity manifold as a set is contained within a co-regular surface. With all of the

categorizations above, the mapping relations can be shown as follows:


30

where q represents a robot arm configuration, x is an end-effector position and M is a

self-motion manifold. C, W, S, J and K denote c-bundle, w-sheet, singularity manifold,

Jacobian surface and co-regular surface, respectively.

After finishing the topology generation process of kinematic map, we can store this

information for further on-line computation. It results in a simplification of redundancy

resolution. Besides, it is worth noting that when kinematic constraints are considered in global

analysis, the topological structure of kinematic map undertakes a dramatic change [18].

4.2.3 Graphic illustration of self-motion manifolds

The two-dimensional torus (See Fig. 2.2) as a topologist’s favorite surface is commonly

used to capture the manifold structure of configuration space of 2R manipulator3. In a way,

we can construct a torus by gluing a rectangle along its edges (See Fig. 4.6). Thus a square

representation can be derived by cutting the torus along its two generators [24]. On this

analogy, the three-dimensional torus can be properly mapped into a cube. In the following

graphic illustrations, we apply this cube representation to reveal the structure of

three-dimensional configuration space of manipulators.

Figure 4.6. The snapshots to show the procedure of constructing a two-dimensional torus

from a rectangle.

3 One manipulator with n revolute joints is usually called nR manipulator for short, thus so does the 2R

manipulator mean.

{ S }

{ J }

f

{C }

{ W }

f

{M } f

{{C }}

}

{ K }

}


31

We take two kinds of examples to illustrate the topology-based methodology. The first one

is a planar 3R manipulator with link lengths L1 = 0.32, L2 = 0.28, L3 = 0.25. It has one DOF

redundancy in 2D space and the revolute joints have [ ] rotation without any kinematic

constraints. Based on the calculation equations in [24], self-motion manifolds of this planar

3R manipulator are visualized by MATLAB (See Fig. 4.7). As we can see, self-motion

manifolds are embedded in a cube and the pre-image of point A only contains one self-motion

manifold, which is a closed circle. It is imaginable that when the end-effector of the 3R

manipulator is fixed in point A, the physical transfer process from ―elbow up‖ configuration

to ―elbow down‖ configuration is smooth and continuous. However, the pre-image of point B

contains two distinct self-motion manifolds as shown by two non-closed arcs. In fact, all the

self-motion manifolds should be diffeomorphic to a circle [24]. The reason why the pre-image

of point B appears in non-closed form is the cube representation. Here, a 2 rotation of joint

angle exists due to the traversal of the self-motion manifold [51]. For more information

about the relation between the forms of expression of self-motion manifolds and the

end-effector locations, please refer to [51].

Figure 4.8 shows a composite graph of self-motion manifolds. As the end-effector moves

along a certain trajectory in operational space, circles and arcs that represent the self-motion

manifolds are gradually added from boundary to centre in the graph.

Figure 4.7. Illustrations of operational space, configuration space and self-motion

manifolds of the planar 3R manipulator. In the subfigures (a1) and (b1), concentric

circles represent the Jacobian surfaces. Configuration space is in the form of cube


32

representation. Self-motion manifold A in (a2) is the pre-image of point A in (a1) and

self-motion manifold B in (b2) is the pre-image of point B in (b1).

Figure 4.8. Integrated self-motion manifolds illustration. The ropes of fibres correspond

to the progress of end-effector moving along a certain trajectory in operational space

that may cross different w-sheets.

The second example is a regional 4R elbow-like manipulator. In this case, we consider part

of Schunk LWA3 robot arm—upper arm and forearm as our model (See Fig. 4.9) and analyze

its topological properties of self-motions. For ease of calculation, we take an approximate

treatment to let lengths of upper arm and fore arm be equal L = 0.32 and joint limits of

Schunk LWA3 robot arm are not taken into account. In simulations, the end-effector is fixed

at the point (0, 0.3, 0.2) of Cartesian coordinates and self-motion manifolds are projected to

the space and the space as shown in Fig. 4.10.

Figure 4.9. Schunk LWA3 seven DOFs robot arm. In this case, we only focus on the

upper arm and forearm to illustrate self-motion manifolds of the regional 4R

manipulator.


33

Figure 4.10. Illustrations of self-motion manifolds of the regional 4R manipulator. There

are four distinct self-motion manifolds marked by different colors in each cube. The

configuration space is a four-dimensional torus. (a) shows the projection in the

space, while (b) shows the projection in the space.

4.3 Characterization of self-motion manifolds

4.3.1 The number of self-motion manifolds

Regarding the number of self-motion manifolds, there are several discussions in literature.

From Fig. 4.7 we notice that the number of self-motion manifolds is relevant to end-effector

locations, and on this issue a detailed explanation in [51] is given in terms of planar 3R

manipulators. Besides, a theorem in [24] shows that the maximum number of distinct

self-motion manifolds is no more than the maximum number of inverse kinematics solutions

of the non-redundant manipulator in the same class, as listed in Tab.4.2.

Table 4.2. Maximum number of self-motion manifolds summarizing table for different

manipulator classes.

manipulator class max. number of self-motion manifolds

planar 2

regional 4

spatial 16

4.3.2 Bundle connectivity graph

We have studied the topological properties of generalized kinematic map and defined the

correspondence between the c-bundle in configuration space and the w-sheet in operational


34

space. In this subsection, we illustrate the various sets in graphics and make in-depth research

of their relationships. The proposed model is the previous mentioned planar 3R manipulator.

In Fig. 4.11, two figures are derived by MATLAB, which contain much topology information

of kinematic map. In addition, the corresponding relations of sets in operational space and

configurations are well represented. We can see that the c-bundles { }, { },

{ }, { } in subfigure (b) respectively correspond to the w-sheets , , , in

subfigure (a), and so do their boundaries.

Moreover, Lück in [18] introduced another unique graphic representation—bundle

connectivity graph, which is also focused on relationships among the finite sets of kinematic

map. In bundle connectivity graph, c-bundle is represented by node and co-regular surface

between neighboring c-bundles is represented by link connecting the corresponding nodes.

Meanwhile, c-bundles are clustered if their w-sheets are the same. Figure 4.12 shows such a

bundle connectivity graph, in which topology structure of the planar 3R manipulator

kinematic map is delineated.

Figure 4.11. Operational space (a) and configuration space (b) of the planar 3R

manipulator. Subfigure (b) shows the projection on the plane.


35

Figure 4.12. Bundle connectivity graph for the planar 3R manipulator. To a certain

extent, it is transited from the subfigure (b) of Fig. 4.11. The simple topology relation

responses to a trivial kinematic map due to no kinematic constraints considered in this

case.

Kinematic constraints cause disruptions in the topological structure of kinematic map.

Figure 4.13 shows a windowed topological structure of configuration space, where joint angle

limits are introduced. Compared with the subfigure (b) of Fig. 4.11, the changes of Fig. 4.13

are embodied in the additional c-bundles. Furthermore, Fig. 4.14 illustrates the topological

relationships by bundle connectivity graph. Obviously, the complexity of graph in Fig. 4.14 is

higher than that in Fig. 4.12.

Figure 4.13. A windowed topological structure of configuration space, cut from

subfigure (b) of Fig. 4.11. Joint angle limits and

are considered in this model, which cause the topological structure changes.


36

Figure 4.14. Bundle connectivity graph for the planar 3R manipulator with introduced

kinematic constraints. The graph is relative with the windowed topological structure of

configuration space.

4.4 Follow-up processing with the characterization of self-motion manifolds

So far we have discussed global approaches to the IK problem at the position level. On this

stage, we take a brief look at examples of follow-up processing with the characterization of

self-motion manifolds.

4.4.1 A two-phase approach of off-line analysis and on-line retrieving

Tarokh at al. in [25] proposed an extremely fast inverse kinematic computation for seven

DOFs anthropomorphic robot arms by utilizing the topological properties of global kinematic

map. The approach consists of two phases. The preceding phase is off-line analysis. We first

characterize the self-motion manifolds and classify the finite sets, then store the derived

topology information and generated parameters. The second phase is to retrieve the stored

information combining with the real-time hand position/orientation. The approach is complete

in the sense that it provides all the IK solutions for a given hand position/orientation.

4.4.2 Global path planning optimization

Except for the self-motion topology studies in terms of global kinematic map for redundant

manipulators, Lück in [18] also proposed some discretization strategies on the topological

structures of operational space and configuration space. All the discretization processing and

preparations are used to implement global optimum solutions of path planning, which can

avoid local minima and deadlocks.

CHAPTER 5 CONCLUSION AND FUTURE WORK

37

5 Conclusion and Future Work

Topology has been involved in a variety of robotics research as discussed in Chapter 1.

There exists a common ground between topology and robotics, which offers the potential to

apply a broader theory of topology to problems of practical robotics.

Herein, we have reviewed some robotic manipulation studies for which the topological

perspectives have been found to be applicable. Above all, it is shown that generally

operational space and configuration space of robotic manipulation are smooth closed

manifolds [14]. Secondly, an overview of some aspects of topological reasoning in robotic

manipulation is given. We clarified that no global solution to inverse kinematics problem can

be provided and singularities must necessarily exist no matter how many redundant DOFs are

added. Lastly, topological properties of kinematic map are exploited in order to develop a

global approach to the inverse kinematics problems at the position level. Furthermore,

self-motion manifolds of redundant manipulators were investigated by graphic illustrations.

The interdisciplinary research of topology and robotics is a new emerging area. On one

hand, it is inspiring and significant progress has been made roughly in a decade; on the other

hand, many new open problems ceaselessly surface and there still exist unsolved problems.

Below we list some of them related to robotic manipulation.

1. We have mentioned that topological invariants undoubtedly provide more subtle

underlying interpretation than pure connectivity properties. However, this pure

mathematical consideration cannot sufficiently satisfy the practical needs. Thus it is

desirable to exploit and study new invariants of topological spaces inspired by robotic

applications.

2. Topological perspective from the qualitative viewpoint is not an adequate criterion.

Sometimes there are requirements for additional information such as a metric. It may

achieve gratifying success to combine topological properties and metric information for

practical applications.

3. Studying topological spaces and configuration spaces of higher DOF manipulator or

multi coordinated manipulators is another research direction. On the basis of

topology-based formalism and its broader theory, various methods from topology can

be naturally applied in robotics such as motion planning of dual arm manipulation and

singularity problems of closed kinematic chains.


38

4. We used to separately model manipulation systems and manipulated objects, but if we

properly integrate them and build the corresponding topological model, it will make

our machine be more perceptual to the object interaction by analyzing the topological

properties of special topological spaces. Besides, this idea overcomes the shortcomings

of that topological analysis is not well suited for on-line approach [18] and of that

real-time collision avoidance and other dynamic constraints have to be omitted when

generating topology structure of manipulator kinematic map.


39

Bibliography

[1] D. A. Johnson and W. H. Glenn: Topology: The Rubber-Sheet Geometry (Exploring

Mathematics on Your Own), Webster Publishing Company, 1960.

[2] J. Munkres: Topology (2nd

Edition), Prentice Hall, 2000.

[3] D. Mackenzie, “Topologists and Roboticists Explore an „Inchoate World‟,” News focus,

Science, Vol.301, No. 5634, pp.756, 2003.

[4] D. H. Gottlieb, “Robots and Topology,” Proceedings of IEEE International Conference

on Robotics and Automation (ICRA), pp.1689-1691, 1986.

[5] D. H. Gottlieb, “Robot and Fibre Bundles,” Bulletin de la Societe Mathematique de

Belgique, Vol.38, pp.219-223, 1986.

[6] D. H. Gottlieb, “Topology and Robot Arm,” Acta Applicandae Mathematicae, Vol.11,

No.2, pp.117-121, 1988.

[7] C. Wampler, II, “Inverse Kinematic Functions for Redundant Manipulators,” Proceedings

of IEEE International Conference on Robotics and Automation (ICRA), pp. 610-617, 1987.

[8] D. Baker and C. Wampler, II, “Some Facts Concerning the Inverse Kinematics of

Redundant Manipulators,” Proceedings of IEEE International Conference on Robotics and

Automation (ICRA), pp. 610-617, 1987.

[9] D. R. Baker, “Some Topological Problems in Robitcs,” The Mathematical Intelligencer,

Vol.12, No.1, pp.66-76, 1990.

[10] J. Pfalzgraf, “On Geometric and Topological Reasoning in Robotics,” Annals of

Mathematics and Artificial Intelligence, Vol.19, No.3-4, pp.279-318,1997.

[11] http://www.fim.math.ethz.ch/conferences/2003/topology_robotics.pdf

[12] http://www.fim.math.ethz.ch/conferences/2006/topology_and_robotics1.pdf

[13] E. S. L. Ho and T. Komura, “Character Motion Synthesis by Topology Coordinates,”

Computer Graphics Forum, Vol.28, No.2, pp.299-308, 2009.

[14] M. Farber: Invitation to Topological Robotics, European Mathematical Society, Germany,

2008.

[15] J. S. Dai and J. R. Jones, “Matrix Representation of Topological Changes in

Metamorphic Mechanisms,” Journal of Mechanical Design, Vol. 127, No.4, pp.837-840,

2005.

[16] D. J. Balkcom and M. T. Masson, “Robotic Origami Folding,” The International Journal

of Robotics Research, Vol.27, No.5, pp.613-627, 2008.

[17] M. Erdmann, “On the Topology of Discrete Strategies,” The International Journal of

Robotics Research, Vol.29, No.7, pp.855-896, 2010.

[18] C. L. Lück, “Self-Motion Representation and Global Path Planning Optimization for

Redundant Manipulators through Topology-Based Discretization,” Journal of Intelligent and

Robotic Systems, Vol.19, No.1, pp.23-38, 1997.

http://www.fim.math.ethz.ch/conferences/2003/topology_robotics.pdf

http://www.fim.math.ethz.ch/conferences/2006/topology_and_robotics1.pdf


40

[19] C. L. Lück and S. Lee, ―Redundant Manipulator Self-motion Topology under Joint

Limits with An 8-DOF Case Study,‖ IEEE/RSJ International Conference on Intelligent

Robots and Systems (IROS), Vol.2, pp.848-855, 1993.

[20] A. Hatcher: Algebraic Topology, Cambridge University Press, 2002.

[21] G. Strang: Linear Algebra and Its Applications, Academic, New York, 1980.

[22] J. Sellen, “On the Topological Structure of Configuration Spaces,” Annals of

Mathematics and Artificial Intelligence, Vol.19, No.3-4, pp.335-354, 1997.

[23] R. Ghrist, ―Configuration Spaces, Braids, and Robotics,‖ Tutorial Talk Notes at the

National University of Singapore, 2007.

[24] J. W. Burdick, “On the Inverse Kinematics of Redundant Manipulators Characterization

of the Self-motion Manifolds,” Proceedings of IEEE International Conference on Robotics

and Automation (ICRA), Vol.1, pp.264-270, 1989.

[25] M. Tarokh, K. Keerthi and M. Lee, “Classification and Characterization of Inverse

Kinematics Solutions for Anthropomorphic Manipulators,” Robotics and Autonomous

Systems, Vol.58, No.1, pp.115-120, 2010.

[26] J. Denavit and R. S. Hartenberg, ―A Kinematic Notation for Lower-Pair Mechanisms

Based on Matrices,‖ Journal of Applied Mechanics, Vol.23, pp.215-221, 1955.

[27] J. J. Craig: Introduction to Robotics, Prentice Hall, America, 2005.

[28] H. Seraji, M. K. Long and T. S. Lee, “Motion Control of 7-DOF Arms: The

Configuration Control Approach,” IEEE Transactions on Robotics and Automation, Vol.9,

No.2, pp. 125-139, 1993.

[29] M. Farber, “Topology of Robot Motion Planning, Springer,” Morse Theoretic Methods

in Nolinear Analysis and in Symplectic Topology, pp. 185-230, 2006.

[30] M. Farber, “Collision Free Motion Planning on Graphs,” Springer Tracts in Advanced

Robotics, Vol.17, pp.123-138, 2005.

[31] L. Joskowicz, “Simplification and Abstraction of Kinematic Behaviours,” Proceedings of

the 11th International Joint Conference on Artificial intelligence (IJCAI), Vol.2,

pp.1337-1342, 1989.

[32] J. Baillieul, “Kinematic Programming Alternatives for Redundant Manipulators,”

Proceedings of IEEE International Conference on Robotics and Automation (ICRA), pp.

722-728, 1985.

[33] P. Falco and C. Natale, “On the Stability of Closed-Loop Inverse Kinematics Algorithms

for Redundant Robots,” IEEE Transactions on Robotics, Vol.27, No.4, pp.780-784, 2011.

[34] C. Ehresmann, “Les Connexions Infinitésimales Dans un Espace Fibré Différentiable,”

Colloque de Topologie, Bruxelles, pp.29-55, 1950.

[35] Y. Taki and K. Sugimoto, “Classification of Singular Configurations for 7-DOF

Manipulators with Kinematic Redundancy,” Proceedings of 6th Japan-France Congress on

Mechatronics and 4th Asia-Europe Congress on Mechatronics, pp.438-443, 2003.

[36] J. M. Hollerbach, “Optimum Kinematic Design for a Seven Degree of Freedom

Manipulator,” Robotics Research 2nd International Symposium Kyoto, pp.215-222. 1985.

[37] http://www.motoman.com/products/robots/models/sda10d.php

[38] http://his.anthropomatik.kit.edu/english/241.php

[39] http://www.takanishi.mech.waseda.ac.jp/top/research/index.htm

http://www.motoman.com/products/robots/models/sda10d.php

http://his.anthropomatik.kit.edu/english/241.php

http://www.takanishi.mech.waseda.ac.jp/top/research/index.htm


41

[40] http://www.abb.com/cawp/abbzh254/8657f5e05ede6ac5c1257861002c8ed2.aspx

[41] F. Gao, F. Guy and W. A. Gruver, “Criteria Based Analysis and Design of Three Degree

of Freedom Planar Robotic Manipulators,” Proceedings of IEEE International Conference on

Robotics and Automation, Vol.1, pp.468-473, 1997.

[42] M. Shimizu, H. Kakuya, “Analytical Inverse Kinematic Computation for 7-DOF

Redundant Manipulators with Joint Limits and Its Application to Redundancy Resolution,”

IEEE transactions on Robotics, Vol.24, No.5, pp.1131-1142, 2008.

[43] T. Zhao, J. Yuan, M. Zhao and D. Tan, “Research on the Kinematics and Dynamics of a

7-DOF Arm of Humanoid Robot,” IEEE International Conference on Robotics and

Biomimetics (ROBIO), pp.1553-1558, 2006

[44] Anthro Arm, The Design of a Seven Degree of Freedom Arm with Human Attributes.

Adam Paul Leeb, Thesis (S.B.), Dept. of Mechanical Engineering, Massachusetts Institute of

Technology, June 2007.

[45] K. Kreutz-Delgado, M. Long and H. Seraji, “Kinematic Analysis of 7-DOF

Manipulators,” International Journal of Robotics Research, Vol. 11, No. 5, pp.469-481, 1992.

[46] C. A. Klein and C. H. Huang, “Review of Pseudoinverse Control for Use with

Kinematically Redundant Manipulators,” IEEE transactions on. Systems Man Cybernet,

Vol13, No.3, pp.245-250, 1983.

[47] C. W. Wampler, “Manipulator Inverse Kinematic Solutions Based on Vector

Formulations and Damped Least-Squares Methods”., IEEE Transactions on Systems, Man

and Cybernetics, Vol.16, No.1, pp. 93-101, 1986.

[48] D. E. Whitney, “Resolved Motion Rate Control of Manipulators and Human Prostheses,”

IEEE Transactions on Man-Machine Systems, Vol.10, pp. 47-53, 1969.

[49] J. J. E. Slotine and D. R. Yoerger, “A Rule-based Inverse Kinematics Algorithm for

Redundant Manipulators,” International Journal of Robotics and Automation, Vol.2, No.2,

pp.86-89, 1987.

[50] S. R. Buss, “Introduction to Inverse Kinematics with Jacobian Transpose, Pseudoinverse

and Damped Least Squares methods,” IEEE Journal of Robotics and Automation, 2009.

[51] D. DeMers and K. Kreutz-Delgado, ―Canonically Parameterized Families of Inverse

Kinematic Functions for Redundant Manipulators,‖ Proceedings of IEEE International

Conference on Robotics and Automation (ICRA), Vol.3, pp. 1881-1886, 1994.

http://www.abb.com/cawp/abbzh254/8657f5e05ede6ac5c1257861002c8ed2.aspx

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