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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.
CHAPTER 1 INTRODUCTION
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
CHAPTER 1 INTRODUCTION
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.
CHAPTER 1 INTRODUCTION
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.
CHAPTER 1 INTRODUCTION
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)
CHAPTER 2 TOPOLOGICAL MODELING OF ROBOTIC MANIPULATION
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.
CHAPTER 2 TOPOLOGICAL MODELING OF ROBOTIC MANIPULATION
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.
CHAPTER 2 TOPOLOGICAL MODELING OF ROBOTIC MANIPULATION
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).
CHAPTER 2 TOPOLOGICAL MODELING OF ROBOTIC MANIPULATION
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)
CHAPTER 2 TOPOLOGICAL MODELING OF ROBOTIC MANIPULATION
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
CHAPTER 2 TOPOLOGICAL MODELING OF ROBOTIC MANIPULATION
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
CHAPTER 3 ASPECTS OF TOPOLOGICAL REASONING
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)
CHAPTER 3 ASPECTS OF TOPOLOGICAL REASONING
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
CHAPTER 3 ASPECTS OF TOPOLOGICAL REASONING
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
CHAPTER 3 ASPECTS OF TOPOLOGICAL REASONING
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
CHAPTER 3 ASPECTS OF TOPOLOGICAL REASONING
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 .
CHAPTER 3 ASPECTS OF TOPOLOGICAL REASONING
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].
CHAPTER 3 ASPECTS OF TOPOLOGICAL REASONING
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.
CHAPTER 3 ASPECTS OF TOPOLOGICAL REASONING
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
CHAPTER 3 ASPECTS OF TOPOLOGICAL REASONING
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
CHAPTER 4 REDUNDANT MANIPULATOR SELF-MOTION TOPOLOGY
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
CHAPTER 4 REDUNDANT MANIPULATOR SELF-MOTION TOPOLOGY
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
CHAPTER 4 REDUNDANT MANIPULATOR SELF-MOTION TOPOLOGY
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.
CHAPTER 4 REDUNDANT MANIPULATOR SELF-MOTION TOPOLOGY
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:
CHAPTER 4 REDUNDANT MANIPULATOR SELF-MOTION TOPOLOGY
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 }
}
CHAPTER 4 REDUNDANT MANIPULATOR SELF-MOTION TOPOLOGY
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
CHAPTER 4 REDUNDANT MANIPULATOR SELF-MOTION TOPOLOGY
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.
CHAPTER 4 REDUNDANT MANIPULATOR SELF-MOTION TOPOLOGY
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
CHAPTER 4 REDUNDANT MANIPULATOR SELF-MOTION TOPOLOGY
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.
CHAPTER 4 REDUNDANT MANIPULATOR SELF-MOTION TOPOLOGY
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.
CHAPTER 4 REDUNDANT MANIPULATOR SELF-MOTION TOPOLOGY
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.
CHAPTER 5 CONCLUSION AND FUTURE WORK
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.
CHAPTER 5 CONCLUSION AND FUTURE WORK
39
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