ILIAN ALEXANDROV BONEV
1. Kinematic Geometry of Mechanisms
GEOMETRIC ANALYSIS OF PARALLEL MECHANISMS
You can download the entire thesis:
http://www.gpa.etsmtl.ca/prof/ibonev/Publications/ThesisBonev.pdf
The publication of Lagrange’s masterwork M'ecanique Analytique in 1788 marked a new
analytic era in dynamics. Indeed, his revolutionary achievement was to transform the
study of rigid body movement into a branch of calculus. Prior to that, due to the
rudimentary development of analytic tools, mechanics was, perforce, a geometric art.
The reader will find no figures in this work. The methods which I set forth do not require either
constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a
regular and uniform rule of procedure. (Excerpt from M'ecanique Analytique by J.L. Lagrange, 1788).
Basic geometry was developed by the ancient Greeks and Euclid’s Elements was
written as early as 300 BC. The foundations of algebra as we know it, on the other
hand, were laid down much later—in the third century AD—and it was only after the
development of calculus in the 17th century that the analytical study of mechanics
became possible. The arguable preference of the algebraic over the geometric approach
is not an issue of the past. The recent advent of the computer brought a revolution in
mechanical design. While certainly the computer proved to be of great assistance to the
engineer, it has also had negative effects on the readiness to seek deeper understanding
of the principles of mechanical motion. This trend was quickly noticed and eloquently
described by the two most famous advocates of kinematic geometry:
With a computer at his elbow an engineer is often tempted to pay little if
any attention to principles, but rather plunge into a particular problem of
synthesis without considering either the fundamental theory or the criteria
that limit the performance of the devices he aims to produce. [...] But
more importantly [...] the geometric principles reveal a map of a terrain,
regions within which can then be explored in greater detail by analytical or
graphical methods... If the map shows that there are inaccessible regions
on the terrain, if it warns of hazards and dangerous frontiers, and if it can
guide the explorer along safe paths by which he can reach his goal quickly
with simple transport, then it should have some value. (Hunt, 1978)
The digital computer demands on the part of its machine-designing users
a ruthless competence in the algebraic processes needed for the manipulation
of mechanical information and its numerical analysis. It is accordingly
fashionable just now in the field of the theory of machines not so much to
denigrate as simply to ignore the main bases in actual mechanical motion
from which these algebraic processes grow. The main bases are essentially
pictorial, geometrical. They arise from natural philosophy. Students in
the mechanical sciences are becoming increasingly unable to contemplate
a piece of ordinary reality in machinery accordingly, and to extract from
that reality the geometric essence of it. It is of course true that without
algebra there can be no programme, no numerical data, and no numerical
result; but without an underlying geometry of the reality there can be no
applicable algebra. Without a diagram we cannot write an equation. But
without geometry we cannot even begin to draw. (Phillips, 1984)
So well have Profs. Kenneth Hunt and Jack Phillips warned against the treacherous
trend of over-dependence on computer-based solutions. However, it was only recently
that the advances in computer algebra systems such as MapleTM and MathematicaTM
has critically worsened the situation. Students and even researchers are relying entirely
on such mathematical packages without even mastering these tricky systems. While
the powerful programs for symbolic computations are undoubtedly helpful in design,
they should be used only with complete understanding of their limitations (e.g., when
dealing with trigonometric expressions).
Paradoxically, it is exactly the development of the computer that has made geometry
important again. As computers and automatic control algorithms have become
more powerful, designs of increasingly complicated mechanisms have become practical.
If prior to that, analytic methods were su_cient for the study of mechanisms, this
was because these mechanisms were of outstanding simplicity. However, the complex
spatial machines of nowadays can no longer be completely analysed by purely analytic
or numerical methods. While most researchers were occupied developing or using
computer-aided engineering tools, the two Australian professors, Kenneth Hunt and
Jack Phillips, were among the few who realised the need for a revival of the geometric
methods.
Kinematic geometry is the first and simplest segment of kinematics that deals exclusively
with displacements (Hunt, 1978). Time, as a variable, is usually not required
to be brought into account. Indeed, the use of screw theory eliminates that need completely.
Yet for convenience, velocity may sometimes be introduced in the study of
the special, so-called singular, configurations of mechanisms. The main subject of this
thesis is the displacements of parallel mechanisms or the geometry of two relatively
moving bodies connected by a multitude of kinematic chains.
2. Preliminaries
The geometric approach used in this thesis as well as many of the results in it have a
wide application. Apart from parallel mechanisms, some results may also be applied
to the study of serial robots or biomechanical systems, to computer animation, and
to many other fields. However, we are addressing chiefly a specialised audience which
is already acquainted with the basic principles of parallel mechanisms. Therefore, one
will not find here any of the often seen pro/con comparisons between serial and parallel
robots, nor a vague attempt for extensive literature review on parallel robots in general,
nor the commonly seen photos of flight simulators. The field of parallel robots, despite
the scarcity of specialised textbooks, is already too advanced to allow us to review it
on a couple of pages.
What we are obliged to do is to present a brief overview of the terminology and
nomenclature used in this thesis together with the corresponding definitions. This
is because, unfortunately, there exist no well-established terminology in the field of
parallel mechanisms. Or, should we say parallel robots?
The most controversial term relates to the very focus of our thesis—the parallel
mechanism. A plethora of loose synonyms may be found in the literature such as Stewart
platform, Gough platform, hexapod, parallel robot, parallel manipulator, or closed-loop
kinematic chain. Sometimes, these terms are properly used based on their connotations,
but most frequently, they are not. For example, according to the terminology for
the Theory of Machines and Mechanisms defined by IFToMM, a robot is a mechanical
system under automatic control that performs operations such as handling and automation,
while a kinematic chain is simply an assemblage of links and joints. This thesis
studies parallel kinematic chains which are sometimes actuated, sometimes not (as in Chapter 3). However, we are always exclusively interested in
the relative motion between the mobile platform and the base. Hence, the use of the
term parallel mechanism throughout this thesis.
An n-DOF (n-degree-of-freedom) fully-parallel mechanism is composed of n independent
legs connecting the mobile platform to the base. Each of these legs is a serial
kinematic chain that hosts one and only one motor which actuates, directly or indirectly,
one of the joints. The variables that describe the actuated joints will be referred
to as the input variables or also as the active joint variables. Other authors refer to the
same variables as articular coordinates. On the other hand, the variables that describe
fully the pose of the mobile platform (the end-effectors) will be referred to as output
variables. In other works, the same variables are referred to as generalised coordinates.
In our thesis, we will deal mainly with fully-parallel mechanisms, each having identical
legs. Our investigation will cover 3-DOF planar fully-parallel mechanisms, 3-DOF
parallel mechanisms, and 6-DOF spatial fully-parallel mechanisms. Most of the results
may be extended to other parallel mechanisms as well. For example, in Chapter 3, we
do not even mention actuators—all results remain valid even if two actuators are used
per leg. In fact, a mechanism which is not covered by the above definition of fullyparallel
mechanism is sometimes called a hybrid mechanism. Anyway, we will loosely
use the term parallel mechanism to refer mostly to fully-parallel mechanisms but, in
some cases, to hybrid mechanisms as well.
The configuration of an n-DOF parallel mechanism is not simply defined by the
pose of its mobile platform. In general, for a given pose, i.e., for a given set of output
variables, there are several valid sets of input variables. The task of computing the
input variables out of the output variables will be referred to as the inverse kinematic
problem (IKP). The (typically) two solutions to the inverse kinematics of a single chain
will be identified by a branch index. The solutions to the inverse kinematics of the whole
parallel mechanism will be called the working modes (Chablat and Wenger, 1998) or
branch sets. When the inverse kinematic problem of a chain degenerates, we will talk
about a Type 1 singularity (Gosselin and Angeles, 1990). These singularities are also
referred to as Redundant Input (RI) singularities (Zlatanov et al., 1994b).
The configuration of an n-DOF parallel mechanism is not defined by its input variables
either. The task of finding the valid set of output variables corresponding to a
set of input variables, referred to as the direct kinematic problem, has usually a multitude
of solutions, referred to as assembly modes. In fact, some mechanisms allow an
infinite number of solutions to their direct kinematics—a situation referred to as self
motion (Karger and Husty, 1996). More precisely, self motion means a finite mobility
from some points of the workspace, whereas the confusingly similar term architecture
singularity refers to a singularity in every point of the workspace (Ma and Angeles,
1992). When two, or more, of the assembly modes are coinciding, we say that there
is a Type 2 singularity (Gosselin and Angeles, 1990). These are also referred to as
Redundant Output (RO) singularities (Zlatanov et al., 1994b).
The configuration of an n-DOF parallel mechanism is not even defined by both the
input and output variables. Indeed, some mechanisms exist which will allow passive
motion even when the motors and the mobile platform are fixed. Such particular
singularities are called Redundant Passive Motion (RPM) singularities (Zlatanov et al.,
1994b). The configuration of a parallel mechanism is defined by all its joint variables.
The set of all feasible sets of joint variable values will be referred to as the configuration
space. When a singularity exists in the configuration space of a parallel mechanism, an
Increased Instantaneous Mobility (IIM) occurs (Zlatanov et al., 1994b). If the increased
mobility involves the platform, a constraint singularity appears (Zlatanov et al., 2002a).
Most frequently, however, the user and the designer of a parallel mechanism will
be interested only in the set of feasible output variables which we will refer to as
the complete workspace. The complete workspace of a 6-DOF parallel manipulator
is a six-dimensional highly coupled entity which is practically impossible to visualise.
Therefore, the complete workspace of such mechanisms is studied only through its
di_erent subsets. Most of these are also defined for parallel mechanisms with less than
six degrees of freedom.
The most common subset of the complete workspace is the constant-orientation
workspace (Merlet, 1994) which is the set of permissible positions for the centre of the
mobile platform while the platform is kept at a constant orientation. Conversely, the
orientation workspace is the set of permissible orientations of the mobile platform, while
the platform centre is held fixed.
We will use the standard character-based notation for the di_erent architectures
of n-DOF fully-parallel mechanisms with identical legs. We will use the letters R,
P, U, and S to denote respectively revolute, prismatic, universal, and spherical joints.
When a joint is actuated, its corresponding letter will be underlined. Thus, the chain of
letters (one of which is underlined) designating the joints in a leg, ordered consecutively
from the base to the mobile platform, will be used to denote the leg. The sequence
of characters, preceded by “n-,” will be used to denote the architecture of an n-DOF
fully-parallel mechanism with n such legs.
3. Objectives and Contributions of the Thesis
The principal goal of this thesis is to show how a geometric approach to the study
of the kinematics of well-known parallel mechanisms can reveal and correctly explain
numerous previously unknown properties. Our investigation was propelled by the desire
to foster the reliance on such geometric methods. As we go through the kinematic
analysis of parallel mechanisms, we show how geometry brings an in-depth insight into
the very principles of motion, much better than the study of a mystifying algebraic
equation. If, after reading this thesis, a researcher starts visualising circles, spheres,
and tori, where earlier he or she saw nothing but quadratic or quartic equations, our
first goal would have been achieved.
Geometry develops creativity. It is not through the use of a computer algebra system
that one will come up with a new design or architecture. It is unlikely that a genetic
algorithm will automatically generate an innovative optimal design. It is intuition and
a confident grasp of the principles of motion that will lead to ingenious mechanical
solutions.
In trying to convert the reader to a more geometry-aware approach, we also show
concern for the weary readers of lengthy scientific papers. Numerous are the examples of
papers with arrays of lengthy equations or obscure graphical results, struggling with the
width limits of the common double-column format. The ancient Chinese proverb that
“a picture is worth a thousand words” is very true in the field of mechanical design.
Who would argue that a Bohemian dome is best understood through its geometric
definition and an intelligent cutaway drawing rather than via
studying its complicated algebraic equation? Who would dispute the advantage of a
nicely-shaped closed surface over a dispersed cloud of points for the
interpretation of the orientation workspace of a mechanism? It is geometry that allows
for the most compact description and interpretation of results.
The second objective of this thesis is to uncover the hidden properties of well-known
parallel mechanisms. Thus, our intention is to assist the numerous designers and users
of these popular architectures. It should be noted that the mechanisms that we analyse
were not specially chosen. We simply selected all 3-DOF planar parallel mechanisms,
several popular 3-DOF spatial parallel mechanisms with mixed degrees of freedom,
and the least-studied yet popular 6-DOF parallel mechanism with revolute actuators.
Dozens of prototypes exist for nearly all of the mechanisms that we study, many of
which are even commercially available.
While the revelations of our investigation would certainly not lead to product recalls,
they will help better understand and explain the properties of these mechanisms, and,
therefore, allow the optimal use of existing devices and the creation of new and improved
designs. For example, manufacturers can stop saying that their mechanisms have pitch
and roll capabilities but instead simply state that there is no torsion. They can improve
their control systems in order to make better use of the already small workspace of a
parallel mechanism. They can adopt new designs with few or no singularities at all.
Our last aim is to suggest topics and avenues of further research. Two particular
directions are identified in this thesis and detailed in the last chapter. While these
directions call for a sophisticated immersion in the theory of kinematic geometry, the
fruits of the research will not only be of theoretical but also of practical value.
4. Overview of the Results
Our work is presented in three main parts—Chapters 2, 3, and 4. While the progress
through the chapters leads us from the planar movement to three, and then six degrees
of freedom in space, the complexity does not necessarily follow the same progression.
In fact, it is probably Chapter 3 that is easiest to read and understand. On the other
hand, we go from the general study of all 3-DOF planar parallel mechanisms, to the
analysis of a class of 3-DOF spatial parallel mechanisms, to a single architecture of a
6-DOF parallel mechanism.
In Chapter 2, we analyse the singularities of all 3-DOF planar parallel mechanisms.
The velocity equations for all mechanisms are derived by using both screw theory
and the conventional approach of differentiating with respect to time the constraint
equations governing the motion of the mechanism. Therefore, a substantial part of the
chapter is dedicated to explaining the use of screw theory in the plane. Once these
velocity equations are set up, an exhaustive study on the various types of singularities
of these mechanisms is performed. Several new designs are identified that have few or
no singularities at all. Chapter 2 ends with a detailed discussion on one of the proposed
directions for research—the problem of workspace segmentation, working modes, and
assembly modes. Several examples are given to illustrate these intricate concepts.
In Chapter 3, we leave the plane and start with a discussion on the problem of
orientation representation. We present a concise treatise on the relatively unknown
Tilt & Torsion angles to demonstrate their numerous advantages. Then, using these
angles, we analyse several 3-DOF spatial parallel mechanisms with one translational
and two rotational degrees of freedom. We derive the relationships between the three
constrained and three feasible degrees of freedom and show clearly that these mechanisms
belong to a special class of mechanisms with zero torsion of the mobile platform.
In Chapter 4, we limit our investigation to the general 6-DOF 6-RUS parallel mechanism.
In the first part of the chapter, we set up a geometric method for the computation
of the edges of the constant-orientation workspace. Our method takes into account
the mechanical limits on the U joints. In the second part, we ignore those limits and
describe another geometric algorithm for the computation of the constant-orientation
workspace. This time, however, instead of computing only the edges, we also compute
the cross-sections of the workspace. In the final part of Chapter 4, we limit our study
to the special 6-RUS parallel mechanism with pair-wise coincident S joints and all six
centres of the U joints moving on the same circle. The Rotobot, as it is dubbed, allows
us to demonstrate once more the issue of workspace segmentation by working modes.
We propose geometric methods for the computation and representation of the horizontal
cross-sections of the singularity loci and constant-orientation workspace. One
interesting point relates to the astonishing geometric model of the constraint on the
circular order of the U joints.