This extensive review paper, which involves 204 papers,
discusses comprehensively a number of performance indices that are
instrumental in the design of parallel kinematics manipulators. These
indices measure the workspace as well as its quality including the distance
to singularity, dexterity, manipulability, force transmission, accuracy,
stiffness, and dynamic performance. After being classified, the indices are
discussed in terms of some important aspects including definition, physical
meaning soundness, dependency, consistency, scope of applicability, and
computation cost. For the sake of completeness, some key mathematical
expressions of the indices are provided.
Introduction
Parallel kinematics manipulators (PKMs) have been extensively studied in the
last two decades with a few of them deployed into commercial market
particularly for demanding tasks such as high-speed pick-and-place and
aircraft component machining as they offer high dynamics, high agility, high
payload-to-weight ratio, distributed joint errors, and simple inverse
kinematics. Nevertheless, they are also attributed with some drawbacks such
as smaller workspace, singularity issue, and complicated forward kinematics.
To optimize the performance of a PKM, some performance measures or
performance indices are usually evaluated and optimized during the design
phase.
Patel and Sobh (2015) categorized the performance measures into either local
or global, kinematic or dynamic, and intrinsic or extrinsic. The local
measures depend on the manipulator posture whereas the global measures
evaluate the manipulator performance across the whole workspace. The global performance index GPI can be
obtained by analytically averaging the local performance indexP over the workspace W as given by Eq. (1). If the local performance index cannot be expressed analytically in
terms of the workspace, Eq. (2) can be used to perform discrete averaging.
In the latter case, the number of points N being evaluated may affect the
accuracy of the averaging. The evaluation of a large number of points can be
computationally expensive. In such a case, less representative sampling
points can be used.
1GPI=∫WP⋅dW∫WdW2GPI=1N∑i=1NPi
To evaluate the performance index P in a single cross-section of the
manipulator workspace, section performance index SPI (Wang et al., 2014) has been proposed and is given
by:
SPI=∫SP⋅dS∫SdS
where S denotes the cross-section area of the workspace.
Furthermore, Zhang et al. (2014) proposed the use of a global performance
index which evaluates the distribution characteristics of a performance
index, including central tendency, dispersion, and shape. This global index
integrates four performance indices: global average index (GAI), global volatility index (GVI), global skewness index (GSI), and global kurtosis index (GKI).
The kinematic measures evaluate the kinematic performance of the manipulator
whereas the dynamic measures indicate the dynamic properties of the
manipulator. The intrinsic measures indicate the inherent characteristics of
the manipulator regardless of its task whereas the extrinsic measures are
related to the manipulator task. From the viewpoint of workspace, the
performance measures can be classified into two categories (Lou et al.,
2014): (i) measures on workspace geometric properties, i.e. area/volume and
shape, and (ii) measures on workspace quality such as distance to
singularity, kinematic dexterity, manipulability, transmissibility,
accuracy, repeatability, stiffness, dynamic dexterity, etc.
A discussion on well-known classical performance indices can be found in
Merlet (2006c). To the authors' knowledge, the most recent and
comprehensive review on the manipulator performance measures is Patel and
Sobh (2015). Although the paper discusses a good number of indices, it does
not include some important indices for parallel kinematics manipulators such
as stiffness indices. In this paper, a comprehensive review of a number of
performance indices for PKMs is presented. More focus is given to indices
commonly considered in the design of PKMs. In fact, some of the indices are
borrowed or extended from those for serial kinematics manipulators (SKMs)
while some others have been developed from the beginning for PKMs. The
development of the indices to the current state of the art along with the
description of their advantages as well as limitations and disadvantages is
presented. Some key mathematical expressions are included to give a
self-contained, clearer description on the indices. However, a reader
interested in more details should refer to corresponding references.
This paper is a major extension of the authors' shorter review previously
published (Rosyid et al., 2017) with the following differences. First, while
the previous paper only discusses the kinematic performance measures, this
paper also includes the dynamic performance measures. Furthermore, more
kinematic performance measures are introduced in this paper for more
completeness. In particular, force transmission performance measures are
very little discussed in the previous review but they are extensively
discussed in this paper. Second, this paper includes key mathematical
expressions of most of the reviewed performance indices along with their
description while the previous paper only provides a concise description of
the indices and does not include the mathematical expressions of the
indices. Third, a new classification of the performance measures is used in
this paper to improve the taxonomy of the performance measures. Fourth, more
discussions and insights on the performance measures are provided in this
paper. Furthermore, some more recommendations are added in this paper for
future research direction. Finally, as a result of the more completeness of
this paper, the number of relevant references are significantly increased.
Workspace
The workspace is the set of points (locations) the end effector can reach.
It is determined by the manipulator's mobility which is represented by the
number and types of its degrees of freedom (DOF), and constrained by the
link lengths, the joint limits, and the interference between the components.
The workspace is usually evaluated in terms of its size (area/volume) and
shape. The latter is commonly quantified by the aspect ratio of the regular
workspace. In general, it is preferred to have larger and better-shaped
workspace. The workspace can also be characterized by using workspace volume index (Merlet,
2006a), space utilization index (Lee et al., 2010), or footprint ratio (Shijun, 2015) which are all defined as the
ratio between the workspace volume and the volume of the machine. Recently,
a normalized index bounded between 0 and 1 called global workspace conditioning index (GWCI) (Enferadi and
Nikrooz, 2017) has been introduced and applied in a spherical PKM. This
index is given by the workspace volume divided by the product of motion
ranges in each degree of freedom.
Prior to optimizing the geometrical parameters in order to achieve better
workspace, selecting an appropriate topology is crucial. Many manipulators,
such as Hexaglide, Linapod, Pentaglide, sliding H4, and Triaglide, have
sliders (gliders) to provide a larger workspace.
The manipulator workspace is commonly classified into several types (Merlet,
2006a) which include constant-orientation workspace (translation workspace), orientation workspace, reachable workspace (maximal workspace), inclusive workspace, dexterous workspace, and total orientation workspace. Useful workspace, sometimes also called used workspace, which
indicates part of the workspace to be used for a specific application is
usually a regular workspace such as a rectangle, circle, cuboid, sphere, or cylinder. A
designer is often much interested in the useful workspace as it indicates
part of the workspace which can be really utilized for the application. Baek
et al. (2001) presented a method to find maximally inscribed rectangle in
the workspace of SKMs and PKMs. Furthermore, Bonev and Ryu (2001) introduced
a two-dimensional subspace of orientation workspace called projected orientation workspace and defined it
as the set of possible directions of the approach vector of the moving
platform.
The workspace can be represented completely in six-dimensional space.
However, a graphical representation is only possible up to three-dimensional
space. Therefore, the position workspace is usually represented separately
from the orientation workspace. A two-dimensional plot can be used to show
the workspace of a planar manipulator, whereas a three-dimensional plot can
be utilized to represent the workspace of a spherical or spatial
manipulator. The two-dimensional workspace plot can be presented in either
Cartesian or polar coordinate system whereas the three-dimensional workspace
plot can be presented in Cartesian, cylindrical, or spherical coordinate
system. In general, the workspace of a manipulator with more than 3 DOF can
be graphically illustrated by fixing (n-3) DOF. Depending on which DOF to be
fixed, the workspace will be different (Merlet, 2006a). Furthermore, while
plotting the workspace of SKMs is straight forward, that is typically not
the case for PKMs.
Generally, three main approaches namely geometrical approach, discretization numerical approach, and non-discretization numerical approach can be used to determine
and plot the workspace. The geometrical approach has been extensively used
in the early works on the workspace determination of PKMs. Using the
geometrical approach, Bajpai and Roth (1986) studied the workspace of PKMs
and the effects of legs' length to the workspace. Three years later,
Gosselin (1989) presented his work on the determination of
constant-orientation workspace of PKMs. Subsequently, Merlet (1995)
presented the determination of orientation workspace of PKMs. Afterwards,
Kim et al. (1997) proposed an algorithm to determine the reachable and
dexterous workspace of PKMs. As planar PKMs need different treatments,
Gosselin and Jean (1996) followed by Merlet et al. (1998) investigated the
workspace determination of planar PKMs.
In the geometrical approach, the true boundaries of a PKM workspace can be
quickly and accurately obtained from the intersection of the boundaries of
every open-loop chains composing the PKM. The use of computer-aided design
(CAD) software can make this process easier and faster such as the work
proposed by Arrouk et al. (2010). Nevertheless, this approach lacks general
applicability as it usually should be tailored to the considered
manipulator. In addition, it is difficult to include all the constraints by
using this approach.
In the discretization numerical approach, a discretized bounding space
covering all possible points in the workspace is created. Subsequently, the
inverse kinematics along with the constraints is used to check whether or
not a possible point belongs to the workspace. This approach is sometimes
called binary representation since a reachable point is usually indicated by
“1” and plotted whereas an unreachable point is indicated by “0” and not
plotted. This approach is intuitive, applicable to all types of PKMs, and
able to include all the constraints. However, the size of the discretization
steps affects the accuracy of this approach. In addition, small voids inside
the workspace cannot be detected unless the discretization steps are small
enough to capture them. Monte Carlo method (Alciatore and Ng, 1994; Rastegar
and Perel, 1990) is similar to this approach as a large number of random,
discrete active joint points within the joint range are input to the forward
kinematics and accordingly the end effector position points are plotted.
However, forward kinematics for PKMs with the exception of the very simple
ones is typically complicated. Therefore, Monte Carlo approach is also
conducted in the other direction where a large number of random, discrete
possible workspace points in a bounding space are transformed to the joint
positions by using the inverse kinematics. If a joint position falls within
the joint range and meets the constraints, the corresponding workspace point
is reachable.
Some recent works using the discretization numerical approach include a new
approach proposed by Bonev and Ryu (2001) to determine the three-dimensional
orientation workspace of 6-DOF PKMs. An algorithm based on the
discretization approach was proposed by Castelli et al. (2008) to determine
the workspace, the workspace boundaries, the workspace volume, and the
workspace shape index of SKMs and PKMs. A discretization method to determine
the reachable workspace and detect available voids of PKM was presented by
Dash et al. (2005).
Beside the aforementioned two approaches, several non-discretization
numerical methods have been proposed to determine the workspace of PKMs.
Some of the methods are as follows (Merlet, 2006a; Wang et al., 2001):
Jacobian rank deficiency method (Jo and Haug, 1989b) which is only practical
to determine constant orientation workspace.
Numerical continuation method (Haug et al., 1995; Jo and Haug, 1989a) which
can avoid singularity points but is only practical to determine constant
orientation workspace.
Constrained optimization method (Snyman et al., 2000) which is modified from
the numerical continuation method.
Boundary search method (Wang et al., 2001) which is based on constrained
non-linear programming.
The principle that the velocity vector of the moving platform cannot have a
component along the normal of the boundary (Kumar, 1992), but this method
cannot be applied to manipulators with prismatic joints and cannot easily
include the mechanical limits and interference between links.
Interval analysis method (Kaloorazi et al., 2014) which can handle almost
any constraints and any number of DOF.
Recently, Bohigas et al. (2012) proposed branch-and-prune technique to
determine all the workspace boundary points of general lower-mobility (3-DOF
or lower) SKMs and PKMs. This technique overcomes the limitation of
numerical continuation method. Gao and Zhang (2011) presented Simplified
Boundary Searching (SBS) method which integrates geometrical approach,
discretization method, and inverse kinematics model of a PKM. Saputra et al. (2014) proposed swarm optimization approach to determine the workspace of
PKM. Finally, Zhou et al. (2017) proposed a novel numerical approach to
determine simultaneously both position workspace and orientation workspace
for both lower-mobility and six-DOF PKMs.
Jacobian matrix and dependency issues
The Jacobian matrix relates the velocities of the moving platform to the
velocities of the active joints. Furthermore, it also relates the active
joint wrench to the task wrench. It is discussed here because it serves as
the base of many performance measures.
The velocity kinematics of the manipulator can be expressed by:
Aq˙+Bx˙=0
where x˙ is the end effector velocity and q˙ is the
actuator velocity. The matrices A and B are called the forward and inverse
Jacobian matrices, respectively. The total Jacobian matrix J can be defined
as:
J=-B-1A
The properties of the Jacobian matrix, unfortunately, is dependent on the
scale and frame. Dependency on the scale means that the value is
significantly affected by the choice of unit. Changing the unit can
dramatically change the values of its properties (Yu et al., 2008).
Furthermore, the Jacobian matrix is not invariant with respect to the
reference frame used (Doty et al., 1995).
Furthermore, when a manipulator mixes translational and rotational DOF, the
Jacobian matrix becomes unit-inconsistent (non-homogeneous) and accordingly
lacks sound physical meanings (Lipkin and Duffy, 1988). To deal with this
issue, several techniques have been proposed to normalize inconsistent
(non-homogeneous) Jacobian matrix including the following:
Using characteristic length or natural length (Angeles, 2002; Ma and
Angeles, 1991b; Ranjbaran et al., 1995; Fattah and Hasan Ghasemi, 2002;
Angeles, 2006; Tandirci et al., 1992; Chablat et al., 2002): All Jacobian
matrix components having units of length are divided by a characteristic
length. The characteristic length giving the best performance is called the
natural length by Ma and Angeles (1991b). The characteristic length can be
derived based on the evaluation of manipulator isotropy (Chablat et al.,
2002; Chablat and Angeles, 2002). In case it is not derivable, a common
practice is dividing the components having units of length by the average
platform radius or the platform radius of the corresponding limb. Despite
the wide use of the characteristic length, its geometric interpretation is
not straightforward. To deal with this issue, the use of homogeneous space
(Khan and Angeles, 2006) reveals a more direct geometric interpretation of
the characteristic length.
Using scaling matrix: Stocco et al. (1998, 1999) proposed the concept of
scaling matrix which includes task-space scaling matrix, which can be
considered a generalized extension of the characteristic length, and
joint-space scaling matrix. Another concept of scaling matrix has been
proposed in Lou et al. (2004) where the distances between the center of the
moving platform to the joints connecting the limbs to the moving platform
are used in the scaling matrix.
Using weighting factor (Hosseini et al., 2011; Hosseini and Daniali, 2011):
The Jacobian matrix components having units of length are divided by a
length (as in characteristic length method) and at the same time the
corresponding velocity coordinates are multiplied by the same length.
Moreover, different weights can be used for different coordinates. Both the
length and the weights are constant across the workspace.
Using velocities of three points on the moving platform (Pond and Carretero,
2006, 2007; Kim and Ryu, 2003; Kong et al., 2007): As a continuation of
Gosselin's work (Gosselin, 1990a) which proposed the use of two points on
the end effector to get homogeneous Jacobian matrix, the use of distinct and
non-collinear three points on the moving platform leads to differential
kinematic expressions giving dimensionless Jacobian matrix.
Point-based method (Altuzarra et al., 2006; Gosselin, 1992; Altuzarra et
al., 2008): This method is based on computational velocity formulation which
can easily incorporate all kinematic variables in the manipulator and is
suitable for general-purpose computational kinematics software. The Jacobian
matrix obtained is dimensionless.
General and systematic method (Liu et al., 2011): Giving f×f homogeneous
Jacobian matrix, this method is implemented by firstly formulating the
linear map between the joint rates and velocity twist using the generalized
Jacobian and subsequently generating a linear map between the velocity twist
and f linear velocities at a set of selected points on the end-effector.
However, this method works only for manipulators with one type of actuators.
Homogeneous extended Jacobian matrix (Nurahmi and Caro, 2016): This method
has been formulated for non-redundant manipulators by considering a set of
linear independent axes at the points of a tetrahedron, which represents the
permitted motions and the restricted motions of the moving platform.
Singularity
Singularity inside the workspace is a common drawback of PKMs, and therefore
should be evaluated at the design stage. It has a very important role since
most of the performance measures such as dexterity, manipulability,
transmissibility, and stiffness are related to the singularity. Therefore,
it has attracted much attention and accordingly various works have been
conducted to characterize different types of singularities. A broad
classification of singularities has been made by Ma and Angeles (1991a) who
introduced three types of singularities, namely architectural singularity,
configuration singularity, and formulation singularity. The architectural
(or structural) singularity is a permanent singularity occurring inside the
whole or a part of a manipulator workspace, caused by a particular
architecture of a manipulator. This is the worst type of singularity. This
type of singularity leads to so-called self-motion or sometimes called the
Borel–Bricard motion. Karger and Husty (1998) and Karger (2008) studied all
self-motions in a Stewart-Gough PKM. Wohlhart (2003, 2010) termed this type
of singularity as architectural shakiness and pointed out that it means more
than mere singularity; it means “architectural mobility”. Moreover, he
introduced degree of shakiness (Wohlhart, 2010) and pointed out that a PKM is architecturally
mobile if it is shaky to the fourth degree, i.e. it is mobile if we start
with locked active joints from any architecturally possible position and
hence it is uncontrollable.
The configuration singularity is caused by a particular configuration
(posture) of a manipulator. Most of the discussions on singularity are on
this type of singularity. The formulation singularity is caused by the
failure of a kinematic model at a particular posture of a manipulator, such
as gimbal-lock when using a particular Euler angle representation. This type
of singularity can be overcome by changing the kinematic model. It is worth
mentioning that the architecture singularity overrides the configuration
singularity whereas the configuration singularity overrides the formulation
singularity.
In general, singular configurations can be determined either analytically or
geometrically. A well-known analytical method was introduced by Gosselin and
Angeles (1990) who evaluate singularity by observing the Jacobian matrices.
They classified the singularity in PKMs into the following three types:
Type 1 singularity (also called: direct kinematic singularity, forward
kinematic singularity, serial singularity, or sub-mobility). Mathematically,
this is indicated by the singularity of the forward Jacobian matrix A. In
this singularity, some velocities of the end-effector cannot be generated.
In other words, very small changes in the joint space do not affect the
end-effector pose. Hence, the manipulator loses one or more degrees of
freedom.
Type 2 singularity (also called: inverse kinematic singularity, parallel
singularity, or over-mobility). Mathematically, this is indicated by the
singularity of the inverse Jacobian matrix B. In this singularity, the
end-effector can move although the joints are locked. Hence, the manipulator
gains one or more uncontrollable DOF. Accordingly, the stiffness of the PKM
is locally lost.
Type 3 singularity (also called: combined singularity). Mathematically, this
is indicated by the singularity of both the forward Jacobian matrix A and the
inverse Jacobian matrix B. In this singularity, the end-effector can move
when the joints are locked, and at the same time, the end-effector pose does
not change due to very small changes in the joints. Ma and Angeles (1991a)
pointed out that this type of singularity is a case of architecture
singularity because of its design-dependence feature. However, the
architecture singularity is not only this type of singularity.
A more general discussion on singularities was delivered by Zlatanov et al. (1995) who included the passive joints in the singularity evaluation. Based
on differential kinematics equations which involve the passive joints, they
defined a configuration as singular when the kinematics of the mechanism is
indeterminate with respect to either the input or output velocities. In
other words, a configuration is singular if either the forward or inverse
kinematics problem does not have a general solution. Moreover, they
classified singularities into six types based on the physical phenomena
which occur in singular configurations. The six types are: (i) redundant
input (RI) singularity which corresponds to serial singularity, (ii) redundant output (RO) singularity which covers parallel singularity, (iii) impossible input (II) singularity, (iv) impossible output (IO) singularity,
(v) increased instantaneous mobility (IIM) singularity, and (vi) redundant
passive motion (RPM) singularity. In addition, there are 21 possible
combinations of the six singularity types. Although the method is a general
framework, it was developed only for non-redundant PKMs. Later, Zlatanov et
al. (2002) introduced constraint singularity in lower mobility PKMs, which
occurs when the screw system, formed by the constraint wrenches in all legs,
loses rank. They pointed out that the constraint singularity is an IIM
singularity, but not every IIM configuration is a constraint singularity.
Merlet (2006a) pointed out that the architectural singularity can be seen as
a special case of constraint singularities in which the motions of the
end-effector are finite.
Using a geometric approach, Merlet (2016, 2006a) studied singularity of PKMs
based on Grassmann line geometry and applied the method to various classes
of PKMs. A generalized classification based on a geometric framework was
made by Park and Kim (1999) who classified singularities into three types.
The three types are: (i) configuration space singularity which occurs when
the joint configuration space manifold is singular, (ii) actuator
singularity which occurs at configurations where the mechanism loses one or
more DOF as a result of the choice of actuated joints, and (iii) end-effector singularity which occurs when the moving platform loses DOF of
an available motion. It is also possible that more than one of the above
three types holds. This classification is frame-independent and works also
for a redundant case, including actuation redundancy. Following Park's
classification, Liu et al. (2003) presented a thorough geometric treatment
for each of the three singularity types. They also discussed second order
singularities namely degenerate singularity and non-degenerate singularity
which hold for each of the three singularity types. The degenerate
singularity applies if it is continuous whereas the non-degenerate applies
if it is isolated (Liu et al., 2003). A degenerate actuator singularity
corresponds to a configuration where some of the links can move although all
the actuators are locked, whereas a non-degenerate actuator singularity
corresponds to configurations where certain actuator forces may cause the
mechanism to break apart (Park and Kim, 1999).
Huang et al. (2013) classified the singularities of a Stewart-Gough PKM into
seven types based on the physical kinematic status: (i) dead-point
singularity, (ii) extreme-displacement singularity, (iii) constraint-dependency singularity, (iv) full-cycle geometry singularity, (v) instantaneous mobility increase, (vi) mobility transfer to a local, and
(vii) variety-mobility-property singularity. Yuefa and Tsai (2016)
classified singularities for lower mobility PKMs into three types: (i) limb
singularity which occurs when the limb twist system degenerates and
accordingly the moving platform loses one or more DOF, (ii) platform
singularity which occurs when the overall wrench system of the moving
platform degenerates and accordingly the moving platform gains one or more
DOF, and (iii) actuation singularity which occurs when inappropriate joints
are chosen as the active joints and hence the moving platform still
possesses certain DOF although all the actuators are locked. Recently, Chen
et al. (2015b) classified singularities of lower mobility PKMs into four
types by considering the motion/force transmissibility and constrainability.
The four types are: (i) input constraint singularity, (ii) output constraint
singularity, (iii) input transmission singularity, and (iv) output
transmission singularity. Two advantages are offered by this new
classification. First, it can identify all possible singularities. Second,
it provides a significant physical meaning regarding the transmissibility
and constrainability performance of a PKM.
Beyond all the aforementioned types of singularities, there is also
so-called control singularity. The control singularity occurs when a single
motor actuates two joints (Merlet, 2006a). Finally, it can be seen that all
the introduced characterization and classification methods vary in some
aspects including the scope of discussion, the analysis method, the basis of
classification, the inclusion of passive joints, and the scope of
applicability.
Since the performance of a manipulator deteriorates when the manipulator is
coincident with or close to the singular configuration, many performance
measures have been defined to indicate the distance to singularity. In fact,
quite many performance measures are capable to indicate the distance but
only a few can specify the type of singularity being faced or identify a
more specific type of singularity. Furthermore, various techniques including
actuation redundancy and providing additional constraints have been
developed to deal with singularity problem.
Dexterity measuresJacobian condition number
The Jacobian condition number κJ is given by:
κJ=‖J‖‖J-1‖
where each double bar bracket indicates a norm.
The Jacobian condition number, often simply called the condition number for
short, has a value from 1 to infinity, where infinity indicates singularity.
Alternatively, it can also be expressed by its inverse value, called the
inverse Jacobian condition number, which has a value from 0 to 1 where 0
indicates singularity. Hence, the Jacobian condition number indicates the
distance to the singularity.
The Jacobian condition number indicates how large the error in the task
space will be if a small error occurs in the joint space. The more
ill-conditioned the Jacobian matrix, the larger the error in the task space
will be due to a small error in the joint space. In other words, the
Jacobian condition number also indicates the accuracy of a manipulator.
In addition, the Jacobian condition number is a measure of kinematic
dexterity (or simply called dexterity for short) and kinematic uniformity
(isotropy). The kinematic dexterity is defined as the capability of a
manipulator to move the end-effector in all directions with ease. The
Jacobian matrix is called isotropic when its condition number or inverse
condition number is 1. At this state, the velocity amplification is
identical in all directions, which means that the manipulator can move with
the same ease in all directions. However, the Jacobian condition number does
not provide a complete information about the dexterity as it only informs
how equal the ease in different directions, but not how easy. It is possible
that either the motion in all directions requires small effort or the motion
in all directions require a large effort. Manipulability which will be
reviewed soon provides a more complete information about the kinematic
dexterity.
The drawbacks of the Jacobian condition number inherit the dependency
problems possessed by the Jacobian matrix as discussed earlier. Furthermore,
the Jacobian condition number, as any other condition number of a matrix, is
dependent on the choice of norm definition used. The commonly used norms to
define the Jacobian condition number are as follows:
2-norm, which is defined as the ratio of the maximum and minimum singular
values of the Jacobian matrix.
Frobenius norm, which is given by an analytical function of the manipulator
parameters and hence suitable if its gradient is evaluated (Altuzarra et
al., 2011), as well as it is computationally cheaper since it does not
compute the singular values.
Weighted Frobenius norm, which can be rendered to specific context by
adjusting its weights (Khan and Angeles, 2006) in addition to all of the
mentioned advantages of the Frobenius norm.
Using the 2-norm definition, the Jacobian condition number can be written as
follows:
κJ=σmax(J)σmin(J)
where σmax(J) and σmin(J) denote the maximum and
minimum singular values of the Jacobian matrix, respectively.
Using the Frobenius norm definition, the components in Eq. (6) are defined
by the following:
8‖J‖=trace(JJT)9‖J-1‖=trace(J-1(J-1)T)
When weighted Frobenius norm definition is used, the expressions in Eqs. (8)
and (9) are modified to the following:
10‖J‖=trace(JWJT)11‖J-1‖=trace(J-1W(J-1)T)12W=1nI
where W is the weighting matrix, n is the number of degrees of freedom, and
I is the identity matrix.
The Jacobian condition number is a local property as it is dependent on the
manipulator posture. The global condition number (or global conditioning index, GCI) is defined to represent this property
globally. The GCI is obtained by averaging the local condition index (LCI), i.e. the local
Jacobian inverse condition number, over the workspace as indicated in Eqs. (1) and (2). Since the Jacobian condition number is commonly used to
indicate the dexterity, GCI is also commonly called the global dexterity index (GDI). A map
showing the values of LCI across the workspace is commonly called the
dexterity map. Different global indices of the Jacobian condition number
have also been proposed. Kurtz and Hayward (1992) introduced global gradient index (GGI) defined
as the maximum local flatness of the inverse Jacobian condition number in
the workspace:
GGI=max∇(1/κJ)
Zhixin et al. (2005) introduced global kinematic performance fluctuation indexσ given by:
σ=∫WLCI-GCI2dW∫WdW
which represents the kinematic performance volatility in the workspace.
Huang et al. (2003, 2004) introduced global comprehensive kinematic performance indexψ given by:
ψ=GCI2+fD2
where D is the ratio of the maximum Jacobian condition number to the minimum
one across the workspace whereas f denotes the weight factor of D relative to
GCI.
Other dexterity measures
The minimum singular value, another commonly used term for the velocity minimum, is a better
indicator of distance to singularity than the manipulability index or the
Jacobian condition number (Patel and Sobh, 2015). To make dimensionless the
minimum singular value, the relative minimum singular value was defined in Kim and Khosla (1991) as the
division of the minimum singular value over a non-dimensionalizing factor
having identical dimension to the singular values.
To more intuitively represent the physical performance specifications of
manipulators, which are typically worst-case, Olds (2015) introduced
worst-case velocity index and worst-case error index at a point in the workspace as follows:
16μmin=vwcvj=θ˙∞=1minx˙=θ˙∞=1minJθ˙217μmax=ewcej=θ˙∞=1maxx˙=θ˙∞=1maxJθ˙2
where vwc is the worst-case (slowest) end effector velocity,
ewc is the worst-case (largest) end effector positioning error,
vj is the joint velocity magnitude, ej is the joint error,
θ˙ is the joint velocity vector, and x˙ is the end
effector velocity.
The worst-case velocity index given by Eq. (16) indicates the minimum
possible magnitude of the end effector velocity given a unit infinity-norm
joint speed input. If the manipulator is not capable of moving in any
direction, the worst-case velocity index will be 0. Therefore, it is
expected to have as large worst-case velocity index as possible.
Intuitively, the worst-case velocity index can be visualized as the smallest
velocity in the manipulability polytope. In similar token, the worst-case
error index given by Eq. (17) implies the maximum possible error magnitude
of the end effector given a unit infinity-norm joint speed input. It is
expected to have as small worst-case error index as possible. Intuitively,
the worst-case error index can be visualized as the largest velocity in the
manipulability polytope.
Accordingly, Olds (2015) defined a new isotropy index given by:
l=μminμmax
The value of l is bounded between 0 and 1. When the worst-case velocity index
is 0, l will be zero. On the other hand, if the best-case and worst-case
velocities are equal, l will be 1 which indicates a full isotropy.
ManipulabilityManipulability as Jacobian-based dexterity and motion/force transmission measure
Manipulability indicates the quality of velocity and force transmission
(amplification). In fact, it provides more information about the velocity
and force amplification than the Jacobian matrix condition number. While the
latter only informs how isotropic the velocity and force amplification, the
former informs both the magnitude and the isotropy of the velocity and force
amplification. Therefore, the manipulability serves as both kinematic
dexterity and motion/force transmission measure.
Two kinds of manipulability namely velocity (twist) manipulability and force (wrench) manipulability are commonly used. They are
commonly represented by velocity manipulability ellipse/ellipsoid and force manipulability ellipse/ellipsoid, respectively. In the velocity manipulability
ellipsoid, the axes lengths indicate the velocity minimum, velocity maximum,
and velocity isotropy (Wu et al., 2010; Zhang et al., 2014), whereas the
ellipsoid volume indicates the manipulability value. Although the ellipsoid
volume provides an information on the magnitude, it does not contain any
information on the directionality.
The flattening of the ellipse/ellipsoid or equivalently the ratio of the
minimum velocity to the maximum velocity indicates the velocity isotropy.
This isotropy represents the kinematic transmission ability in different
directions. When the ellipsoid is a sphere or equivalently the ratio is
unity, the kinematic transmission ability is uniform in all directions.
Furthermore, the velocity and force manipulability are linked by the duality
relation between differential kinematics and statics. The major axis of the
manipulability ellipsoid indicates the direction along which the manipulator
can move easily with the minimum effort, while the minor axis indicates the
direction along which the manipulator has the highest stiffness, i.e., the
manipulator's actuators can resist forces with minimum effort along this
direction.
Beside manipulability ellipse/ellipsoid, manipulability polytope (Lee,
1997), which can include the exact joint constraints, can also be used to
represent the manipulability. For six-dimensional manipulator, two separate
polytopes can be used to handle the translational and rotational degrees of
freedom independently and hence avoid calculation complexity.
The manipulability index w was first introduced by Yoshikawa (1985a) as
follows:
w=det(JJT)=σ1σ2…σm
where σ1,σ2,…,σm are the singular values
of the Jacobian J.
In the case of square Jacobian, Eq. (19) can be written as:
w=det(J)
Alternatively, the manipulability can also be defined by using scaled
Jacobian matrix J̃ as follows (Lee, 1997):
21we=det(J̃J̃T)=σ̃1σ̃2…σ̃m22J̃=JR
where R is a diagonal matrix whose diagonal elements are the maximum joint
velocity components.
Tanev and Stoyanov (2000) introduced the use of normalized manipulability index which is dimensionless and
bounded between 0 and 1. The normalized manipulability index is given by:
wn=wimax(w1,w2,…,wn)
where wi denotes the manipulability index at a given point and
max(w1,w2,…,wn) denotes the maximum
manipulability index in the entire workspace.
The most widely used measure to evaluate the force transmission quality is
the force manipulability or sometimes called the payload index (Wu et al., 2010). Similar to the velocity
manipulability, it is defined to represent how much unit-norm actuator
wrench is amplified to the task space. The force manipulability wf is
defined as follows:
wf=det(J-1J-1T)
where J-1 denotes the inverse Jacobian of the manipulator.
Comparing Eq. (24) to the expression of the velocity manipulability in Eq. (19), it can be intuitively understood that the force manipulability is
shown to be the best at a posture when the velocity manipulability is the
worst, and vice versa. Analogous to the velocity manipulability, the force
manipulability can be represented by an ellipse/ellipsoid called force
manipulability ellipse/ellipsoid.
The manipulability as given by Eqs. (19)–(24) is a local measure
(local manipulability index, LMI) which means that it is dependent on the manipulator posture since it
is based on the Jacobian matrix. It can be evaluated globally by using
global manipulability index (GMI) which is the local manipulability measure averaged over the workspace
as indicated in Eqs. (1) and (2). Another measure is uniformity of manipulability which indicates how
uniform the manipulability across the workspace (Pham and Chen, 2004).
Improved manipulability indices
The manipulability indices described earlier, unfortunately, are dependent
on order, scale, and dimension. Dependency on the order means that the order
is dictated by the number of DOF whereas dependency on the scale means that
the value is significantly affected by the choice of unit. The
earlier-mentioned manipulability indices are dependent on the dimension
since they are based on the Jacobian matrix, and hence face unit
inconsistency issue when translation and rotation are mixed.
To overcome the order dependency, Kim and Khosla (1991) introduced
order-independent manipulability indexwo given by:
wo=wn=det(JJT)n
where n is the number of DOF possessed by a manipulator.
Furthermore, they introduced relative manipulability indexwr which is independent from both the
order and the scale:
wr=wofM=wol2
where fM is a function having the dimension of [length]2 and l is
the total length of the manipulator.
Doty et al. (1995) proposed weighted twist-manipulability and weighted wrench-manipulability where each of them is defined by
incorporating a weighting matrix such that all degrees of freedom (either
translational or rotational) are treated uniformly.
To overcome the consistency problem which appears in manipulator mixing
translational and rotational DOF, Hong and Kim (2000) introduced a new
manipulability measure for PKMs which distinguishes between the
translational and rotational components. In this case, they offered four
different manipulability measures: translational velocity manipulability, rotational velocity manipulability, force manipulability, and moment manipulability.
Finally, Mansouri and Ouali (2009, 2011) introduced homogeneous power manipulabilitywp to
overcome the unit inconsistency problem. The expression of wp is not
included here since it is lengthy. Unlike manipulability index based on the
common Jacobian where the mix of translation and rotation results in unit
inconsistency, the utilization of a common quantity between translation and
rotation, i.e. power, naturally leads to homogeneous manipulability index.
This new measure, similar to the conventional manipulability, is also
represented by an ellipse/ellipsoid called power manipulability ellipse/ellipsoid. Furthermore, they introduced
three indices to evaluate the manipulability: power manipulability volumeGvol, power manipulability isotropyGiso, and
minimum power transmissibilityGmin:
27Gvol=1∑k=1nλk28Giso=λminλmax29Gmin=1λmax
where λmin, λmax and λk
denote the minimum, maximum, and kth eigenvalues of wp, whereas n is the
number of the degrees of freedom of the manipulator. These three indices are
local measures and can be extended to their corresponding global measures by
utilizing Eqs. (1) and (2).
Force transmission measures
The force transmission quality includes two kinds of analysis: forward force
transmission analysis and inverse force transmission analysis. The earlier
attempts to evaluate the bounds of the end effector wrench when the actuator
wrench is known, whereas the latter is to determine the bounds of actuator
wrench for given end effector wrench. In general, the force transmission
measures introduced in the literature can be classified into five broad
categories. The first four categories are Jacobian-based, angle-based, screw
theory-based, and matrix orthogonal degree-based indices. The first category
is represented by the well-known force manipulability, whereas the screw
theory-based category is based on the concept of virtual coefficient or
power coefficient. The last category is indices recently introduced to
consider external load in the transmission evaluation. Since the
Jacobian-based transmission measures have been discussed earlier, the
following will only discuss the remaining categories.
Angle-based force transmission indices
Pressure angleα and transmission angleμ, first introduced by Alt (1932) and developed by Hain (1967), are formerly defined for gears, cam-follower mechanisms, and planar
linkages and later extended for spatial linkages. The pressure angle is
defined as the angle between the direction of the driving force and the
direction of the velocity of the contact point, as pertaining to the driven
element. The transmission angle is the complementary angle to the pressure
angle. In general, the pressure angle is expected to be as small as possible
whereas, accordingly, the transmission angle is expected to be as large as
possible. These angles can also be used to indicate the distance to
singularity. Besides, a good transmission angle indicates a good force
transmission quality and less sensitivity to mechanical error. A survey on
the use of transmission angles in 4, 5, 6 and 7 bar linkages can be found
in Balli and Chand (2002). Furthermore, some works have attempted to extend
the use of pressure angle or transmission angle to spatial PKMs (Zhang et
al., 2011; Zhao et al., 2016). Although these angles have clear physical
meaning when used in planar linkages, their definitions become difficult and
complex with vague physical meaning when used in complex spatial PKMs with a
large number of DOF (Shao et al., 2017).
Screw theory-based force transmission indices
Through the screw theory, Ball (1900) introduced the concept of virtual coefficient.
Subsequently, Yuan et al. (1971) utilized the virtual coefficient between
the transmission wrench screw (TWS) and the output twist screws (OTS) to
propose the use of transmission factor having a range between -∞ to +∞ to
measure the transmission quality in spatial PKMs. The transmission factor
represents the virtual power delivered by the TWS on the OTS. Later,
Sutherland and Roth (1973) introduced the transmission index (TI) which is a normalized
version of the mentioned transmission factor:
TI=vc/vcmax
where |vc| denotes the virtual coefficient between the TWS and
OTS whereas and |vc|max denotes the maximum of the
virtual coefficient. Notice that Eq. (30) is a general expression of TI
whereas the specific expression of the Sutherland and Roth's TI can be found
in the mentioned reference. Shimojima and Ogawa (1979) also introduced a
normalized TI by proposing a unique definition of TWS, which depends on the
output link's load condition.
Later, Tsai and Lee (1994) proposed the total transmission index (TTI) composed of both
transmissivity (defined as the ability of a mechanism to generate output)
and manipulability (defined as the ability of an input motion to be
transmitted into the TWS):
TTI=TI×MI
where TI and MI denote the transmissivity and manipulability indices.
Finally, a generalization was made by Chen and Angeles (2007) who proposed
the generalized transmission index (GTI) which serves as a generalized form of not only the Sutherland and
Roth's TI but also the pressure and transmission angles in their feasible
range. Furthermore, they also proposed the transmission quality (TQ) based on the GTI to measure
globally the transmission of a mechanism.
Furthermore, Takeda and Funabashi (1995) proposed a definition of TI based
on virtual power transmitted from input to output link. In their approach,
all input links except one are fixed and the resulting pressure angles are
analysed at the connection between the input and output links. They defined
the TI of a manipulator with n DOF as:
TI=min(cos∝1cos∝2,…,cos∝n)
where αi denotes the pressure angle corresponding to a limb
where its input link is driven. The Takeda and Funabashi's TI is
dimensionless and bounded between 0 and 1 where 0 indicates singularity.
Hence, their TI is also an indicator of distance to singularity. However,
the approach only works for TWS with a zero pitch (i.e. a transmission force
line) and thus represents a special case of the GTI. In other words, in
order to define the pressure angle in a simple definition, the TWS can be
represented at a (spherical) joint where no moment is applied as constraint
(Brinker et al., 2018). This approach was later extended to spherical PKM
(Takeda et al., 1996), spatial 6-DOF PKM (Takeda et al., 1997), and
cable driven PKM (Takeda and Funabashi, 2001). Furthermore, Liu et al. (2014) proposed a novel approach to deal with a case where the TWS and OTS
are parallel by exploiting the dual property of the virtual coefficient and
accordingly enabling the use two, instead of one, characteristic points. It
was shown that the approach provides a more sensitive measure to the
platform configuration.
Based on the concept of the virtual coefficient and following Takeda's
approach of fixing all inputs except one, Wang et al. (2010, 2017) proposed
a general procedure for non-redundant spatial parallel manipulators where
the force transmission quality is expressed by input and output transmission
indices. The input transmission indexηinput is given by:
ηinput=min(ηi,i)
where ηi,i is defined as the ratio of actual power to potential
maximal power of the ith input member. In similar token, the output transmission index is given by:
ηoutput=min(ηo,i)
where ηo,i is defined as the ratio of actual power to potential
maximal power of the ith output member. Moreover, the force transmission index can be written as:
η=min(ηinputηoutput)
and is bounded between 0 and 1 where a larger value indicates better force
transmission. It also can be used to indicate the distance to singularity.
The indices given by Eqs. (33)–(35) are local performance measures. The
corresponding global index can be obtained by averaging the indices across
the workspace as indicated in Eqs. (1) and (2).
Finally, it should be admitted that the screw theory is a powerful and
systematic mathematical tool. However, definition and calculation of the
screw theory-based indices are quite complicated, and not intuitive (Shao et
al., 2017).
Matrix orthogonal degree-based force transmission indices
Shao et al. (2017) introduced the volume of matrix X as:
volX=detXTX
where matrix X is composed of n real column vectors, i.e. X=x1x2…xn.
Accordingly, they defined the orthogonal degree of matrix X as:
ortX=0ifminxi=0,i=1,2,…,nortX=volX∏i=1nxiifminxi≠0,i=1,2,…,n
They introduced three indices namely joint transmission index (JTI), branch transmission index (BTI), and end-effector transmission index (ETI) to indicate
the force transmission quality in a single joint, a single branch (limb),
and at the end effector, respectively. The JTI of the ith joint located in
the jth branch is given by:
JTIji=1-ortYji2
where Yji is a two-column matrix containing the afferent and efferent
force vectors at the ith joint in the jth branch.
The BTI of the jth branch is defined as the product of all JTIs:
BTIj=JTIj1×JTIj2×…×JTIjk
where k denotes the number of joints in the jth branch.
The ETI of a non-redundant manipulator having n DOF is defined as:
ETIj=ortEF
where E is a diagonal Boolean matrix representing the state, i.e. whether
free or constrained, of the manipulator DOF, whereas F=τ1τ2…τn is the branch force matrix each
column of which is composed of unit force vector and unit torque vector in
the jth branch.
Finally, the local total transmission index TIL of the manipulator is given by the product
between the minimum BTI and the ETI:
TIL=minBTI⋅ETI
All of the four indices are bounded between 0 and 1 where 0 indicates the
worst transmission whereas 1 represents the best transmission quality. Not
only these indices are able to indicate the distance to singularity but also
able to distinguish different types of singularities. When BTI is 0, it
means the manipulator is facing an inverse singularity. On the other hand,
the manipulator is facing a forward singularity if ETI is 0. Accordingly,
zero TIL indicates that the manipulator is in a singularity locus
whereas unit TIL indicates that the manipulator is at its best
transmission performance.
Force transmission indices considering external load
Lin and Chang (2002) found that the force transmission of a mechanism
depends not only on the posture of the mechanism but also on the selection
of the output link and the types of the loading. Therefore, the output link
should be specified and the external loads should be considered. They
proposed the force transmission index (FTI) defined as the effective force ratio (EFR) per unit
input torque/force, with a unit torque/force exerted on the output link i.
The EFR itself is defined as the total power transmitted to the output link
via the input-related joints versus the potential maximum power. The FTI for
a single-DOF manipulator is given by:
τFTI=Riη=RiToutTin=RiT′outT′infortorqueRiη=RiFoutFin=RiF′outF′inforforce
where Ri denotes the EFR of the output link i, η denotes the
mechanical advantage, Tin (Fin) is the magnitude of the input
torque (force) when an arbitrary amount of torque Tout (force
Fout) is applied to the output link, and T′in (F′in) is
the magnitude of input torque (force) when a unit torque T′out (a
unit force F′out) is applied to the output link. Notice that the FTI
is dimensionless. It is shown that the FTI provides a more accurate measure
of the force transmission quality than the Jacobian-based index.
Subsequently, Chang et al. (2003) extended the concept of FTI to n-DOF PKMs
with n limbs. In such as case, the total FTI is given by summing the FTI of
all of the n limbs:
τTFTI=∑p=1nτFTI(p)
To make it more intuitive, normalized force transmission index (NFTI) was introduced which bounds the index
values between zero and one. However, FTI as well as NFTI can only be used
for single-DOF manipulators or individual limbs of multi-DOF manipulators.
To extend the index to multi-DOF manipulators as a whole, mean force transmission index (MFTI) was
introduced by averaging the NFTI values of all limbs in the multi-DOF
manipulator. The values range from zero to one. Since MFTI depends on the
manipulator posture, a corresponding global index called global force transmission index (GFTI) is defined
to characterize the force transmission performance across the workspace.
Furthermore, a new force transmission index for PKMs was introduced by Chen
et al. (2015a), rooted in the concepts of pressure angle and transmission
angle in a single-DOF manipulator. It evaluates how well a unit-norm wrench
applied to the moving platform is transformed to the constraint forces of
manipulators, by evaluating the magnitude of force/torque of transmission
wrench based on given loads on the end-effector. It overcomes the unit
inconsistency and frame-dependency problems. The index Tn is normalized
and hence bounded between 0 to 1 as follows:
44Tn=Tmin/T45T=∑Fi2=∑1σi2
where T is the non-normalized force transmission index, Tmin is the
possible minimum value of T, Fi is the ith constraint force vector, and
σi is the ith singular value of the transmission matrix.
Furthermore, the worst scenario indicated by this index occurs when the
magnitude of the transmission wrenches reaches infinity, caused by parallel
singularity.
It should also be mentioned that, after realizing that all previously
proposed transmission indices only evaluate the relation between input and
output powers, Briot et al. (2013) investigated a complementary method to
evaluate the reactions at passive joints due to an external wrench, which
are typically high in the singularity neighbourhood and accordingly can
cause breakdown of a mechanism. They showed that the increase of reactions
at the passive joints depends not only on the transmission angle but also on
the position of the instantaneous center of rotation of the platform. To
avoid the breakdown, it should be ensured that the maximal reaction forces
at the passive joints are always below a certain threshold. However, they
developed the complementary method only for planar PKMs and admitted that an
extension to spatial PKMs would be challenging.
To this point, a large number of force transmission indices have been
proposed. Nevertheless, it is known that both the Jacobian-based force
manipulability and the other discussed force transmission measures are
intended to handle the static case. To measure dynamic force transmission
quality of mechanism, all of the discussed measures should be extended by
incorporating the force components appearing in a dynamic case, i.e.
inertial force, Coriolis and centrifugal forces, spring force, damping
force, and dynamic external force. This might involve defining or evaluating
an expected maximum speed and acceleration at which a manipulator will
operate. Moreover, the inclusion of friction force in both the static and
dynamic cases would give higher fidelity.
Accuracy measures
Accuracy indicates the ability of the manipulator to give the true position
as being commanded. It is very important for precision motion tasks such as
machining, measurement, and precision manipulation. It is affected by
manufacturing and assembly errors, joint clearance and compliance, link
compliance, transmission backlash, computational errors, actuator errors
(including finite resolution of the encoders, sensor errors, and control
errors), static and dynamic disturbances, thermal effect, etc. Among those
error sources, actuator errors are usually the most significant error source
(Merlet, 2006b).
Accuracy is often larger than the repeatability (Spong et al., 2004). The
accuracy can be predicted theoretically as well as checked by a test.
Increasing the accuracy can be done first at the design stage by
theoretically optimizing the theoretical accuracy indicators and
subsequently by calibration after the machine is built. Furthermore, the
inaccuracy which is represented by errors can be evaluated in two senses:
relative errors and absolute errors. For the latter one, the maximum
absolute errors are usually of utmost interest.
The theoretical accuracy indicators, commonly called accuracy indices, are
usually defined to quantify how much the source errors are amplified to the
end effector errors. This can be expressed in the following error model:
δX=Eδε
where δX and δε denote respectively the vector of
end effector errors and the vector of error source, whereas E transforms the
input errors to the end effector errors. It is quite common that the
Jacobian matrix is used to represent E. However, Ryu and Cha (2003) suggested
the use of total error transformation matrix E which includes not only the
Jacobian matrix, but also other transformation matrices which represent the
other error sources.
The theoretical accuracy indicators include the most commonly used Jacobian
condition number and worst-case error index as discussed earlier.
Furthermore, some more accuracy indices have been introduced.
In similar token to manipulability ellipsoid, error ellipsoid (Ropponen and Arai, 1995) is
used to quantify the amplification of source errors to the end-effector
errors. This ellipsoid is posture-dependent. Usually, unit-norm source
errors are used to define the error ellipsoid. However, Ryu and Cha (2003)
used normalized source errors instead of unit-norm source errors. Beside the
error ellipsoid, they also suggested the use of maximum singular value and condition number of the error transformation matrix as the accuracy indices.
They named these three indices error amplification factors (EAFs). It is a user preference to use any
of the three EAFs to evaluate the manipulator accuracy. Notice that they use
the total transformation matrix instead of only the Jacobian matrix to
define the EAFs. Xu and Li (2008) introduced a weighted mix between the
maximum singular value and the condition number and called it error amplification index (EAI).
Furthermore, Ryu and Cha (2003) defined the global error amplification factor (GEAF) by averaging the local
EAFs across the workspace as indicated in Eqs. (1) and (2).
Moreover, Li and Ye (2003) defined three error sensitivity measures:
comprehensive error sensitivity, directional error sensitivity, and absolute error sensitivity. The comprehensive error sensitivity W represents the ratio of the
volume of output error to that of input error and is given by:
W=(det(Jp))-1
where Jp is a square, invertible Jacobian matrix of the manipulator.
The directional error sensitivity C represents the isotropy of the error
transfer and is given by:
C=condJp
Absolute error sensitivity S represents the maximum number of time by which
the input error is amplified to form the output error, and is given by:
S=(σmin(Jp))-1
where σmin is the minimum coefficient by which the input errors
are amplified.
Furthermore, Li and Ye (2003) introduced a comprehensive error coefficiente given by:
e=f1W+f2C+f3S
where f1, f2, and f3 are the weights for W, C, and S, respectively.
However, W, C, S, and e represent the errors only for a specific posture. To measure
the errors for a group of postures, comprehensive error degreeE is used as follows:
E=∑i=1NaieiN
where N denotes the number of postures and ai is the weight of the
ith posture.
Unfortunately, the aforementioned accuracy measures face homogeneity issue
when dealing with mixed DOF. In response to this problem, beyond some
treatments to homogenize the Jacobian matrix, several authors proposed the
use of exact local maximum position error and local maximum orientation error for given actuator errors ε. Different from the
Jacobian condition number which represents the relative error of a
manipulator, these recently introduced accuracy measures indicate the
absolute error of the manipulator. Furthermore, these new indices represent
separately the position and orientation errors and therefore do not face an
issue with mixed DOF. To evaluate the indices, either geometric or numerical
approach may be conducted. However, the geometric approach is limited to
geometrically simple manipulators. For example, Yu et al. (2008) used a
geometric approach to measure the accuracy of 3-DOF planar PKMs. When a
manipulator has a complex kinematics, a numerical approach through either
the forward or inverse kinematics (Liu and Bonev, 2008) can be used to
evaluate the indices. Furthermore, to provide more physical insights, Briot
and Bonev (2008) introduced another method to evaluate the indices. This
method involves solving the direct kinematics for eight, or a maximum of
12n (n denoting the number of discretization steps) sets of inputs. However,
both Yu et al. (2008) and Briot and Bonev (2008) only work within the
workspace far from singularities and apply to only 3-DOF planar PKMs. A
similar approach to Briot and Bonev (2008) has been applied by Briot and
Bonev (2010) to 4-DOF 3T1R PKMs, which involves solving the direct
kinematics for 16, or a maximum of 16n (n denoting the number of
discretization steps) sets of inputs.
Although the local maximum position and orientation errors have clearer
physical meaning and do not have homogeneity problem, they require both
known input joint nominal values and errors. It means one should quantify
both of them prior to the determination of the indices. This, of course,
would be an additional effort for a designer. Using the condition number of
the error transformation matrix, it is not required to quantify any of them.
However, the condition number has homogeneity issue and is easily affected
by units and magnitude (Yu et al., 2008). To some extent, it is also
affected by the choice of norm definition. In authors' view, both the
condition number of the error transformation matrix and the maximum position
and orientation errors have their own advantages and disadvantages as
mentioned as well as their own specific meanings. The former index indicates
how much the input error, regardless of its source, will be amplified in the
task space, but not as absolute values, due to its distance to singularity.
On the other hand, the latter indices indicate how much exactly, as absolute
values, the known input error will be amplified in the task space. In fact,
which source of errors is taken into consideration affects the magnitude of
the input error. Accordingly, the choice of the index during the design
stage depends on the goal and situation. When the DOF are not mixed, the
condition number can be a valid yet cheap solution. In the case of mixed
DOF, it is a good practice to use both of the homogenized condition number
and the maximum position and orientation errors, and subsequently compare
the results. In case the results do not align well, the authors suggest to
use the latter indices due to its sound and straightforward physical
meaning.
Another thing to be considered is comprehensiveness when defining the input
error. As described earlier, several authors have introduced some
comprehensive accuracy measures which involve not only the Jacobian matrix
but also other sources of input joint errors. This comprehensiveness should
also be applied when using the maximum position and orientation errors. When
quantifying the input joint error, one should not only consider, for
example, the resolution of the input joint encoder as the source of error,
but also other significant sources of errors.
All of the aforementioned accuracy measures are defined mainly to be used at
the design stage, usually for dimensional synthesis. The real-time accuracy
observed in a fabricated manipulator is usually represented by tracking error of the end
effector. Accordingly, most of the control strategies are based on
minimizing the tracking error. In machines where the actuators need to work
synchronously to obtain an accurate end effector pose, another accuracy
indicator called synchronization error (sometimes also called contouring error) is also commonly used. The
tracking error is defined as the distance between the actual position and
the desired position, whereas the synchronization error is defined as the
shortest distance from the actual position to the desired trajectory
(contour) (Lou et al., 2011). The linear combination of the tracking error
and the synchronization error is called coupling error. Computing the synchronization
error is more involved, particularly in the PKMs, than computing the
tracking error. To simplify, the synchronization error can be approximated
by the distance of the actual position to the line tangential to the
trajectory at the desired position. All of these real-time accuracy measures
typically can be optimized through a control strategy following the
optimization of the manipulator design as well as kinematic calibration.
Stiffness measuresStiffness modeling
Stiffness or rigidity of a manipulator structure is important as it affects
the accuracy and repeatability of the manipulator. Stiffness is defined as
the ability of the manipulator structure to resist deformation caused by
wrench. A stiffness matrix defines the relation between deformation vector
and wrench vector. It can be derived as the Hessian of potential energy
associated to the wrench with respect to the coordinates (Ruggiu, 2012;
Quennouelle and Gosselin, 2012). Compliance (flexibility), defined as the
inverse of the stiffness, is also often used to indicate the stiffness.
Similarly, a compliance matrix is simply the inverse of the stiffness matrix
and vice versa. If a structure has high stiffness, it has low compliance.
The stiffness can be classified into static stiffness and dynamic stiffness.
The stiffness of a manipulator is dependent on its topology, geometry,
material, and control system. The overall stiffness is comprised of the
stiffness of the fixed base, the moving platform, the active and passive
joints, and the links. The influence of the passive joints on the
manipulator stiffness has been studied in some works (Zhang, 2005; Zhang et
al., 2009; Pashkevich et al., 2010b, 2011a, b; Klimchik et al., 2012; Sung Kim and Lipkin, 2014). Depending
on the significance, the stiffness of the links can be defined in the axial
direction (axial stiffness), transversal direction (bending stiffness), or
both of them. For simplicity, one or several components of the manipulator
are often assumed to be rigid. For example, joints can be considered elastic
while the links are assumed to be rigid, or vice versa. A more realistic
model can be provided by considering both of the joints and the links as
elastic. A trade-off between the simplicity and the fidelity should be made
by looking at how significant the compliance of a component contributes to
that of the whole manipulator. In a hybrid kinematics machine tool, a PKM is
often used for the spindle platform and SKM for the worktable, as the
spindle platform is usually the most flexible part of the machine and the
use of the PKM is expected to increase its stiffness. Furthermore, some PKMs
such as Tricept and Georg V have passive legs to increase their stiffness.
Stiffness is a local property as it is dependent on the manipulator posture.
Global stiffness measures are used to evaluate the stiffness globally.
Furthermore, stiffness varies with the direction in which it is evaluated as
well as the direction of the wrench. The stiffness can be identified in
either translational directions, hence called the translational stiffness,
or rotational directions, hence called the rotational stiffness. The
stiffness can also be evaluated by considering small deflections
(quasi-static state) or large deflections (loaded state) (Klimchik et al.,
2014).
Several expressions of stiffness have been used, including engineering
stiffness, generalized stiffness matrix, and Cartesian stiffness matrix. The
engineering stiffness is a one-dimensional stiffness expression obtained by
evaluating the displacement in the applied force direction (El-Khasawneh and
Ferreira, 1999). The generalized stiffness matrix, according to Quennouelle
and Gosselin (2012), is comprised of three components namely stiffness of
the unconstrained joints, stiffness due to dependent coordinates and
internal wrench, and stiffness due to external wrench. The Cartesian
stiffness matrix defines the relation between the variation of wrench
applied on the end-effector and the variation of the Cartesian displacements
(Ruggiu, 2012). It is the most widely used expression of stiffness in
manipulator field. Moreover, it is symmetric and either positive definite or
positive semi-definite. However, some researchers concluded that the
Cartesian stiffness matrix of the elastic structure coupling two rigid
bodies is asymmetric in general (Griffis and Duffy, 1993; Ciblak and Lipkin,
1994) and becomes symmetric if the connection is not subjected to any
preloading (Žefran and Kumar, 1996, 2002; Howard et al., 1998; Kövecses and Angeles, 2007). Quennouelle and Gosselin (2008) showed that the Cartesian stiffness matrix remains symmetric in the
presence of external loadings. Other expressions of Cartesian stiffness
matrix were presented by Klimchik (2011), Pashkevich et al. (2011b), and
Quennouelle and Gosselin (2012). The latter authors proposed a Cartesian
stiffness matrix which can take into account non-zero external loads,
non-constant Jacobian matrices, stiff passive joints and additional
compliances. Moreover, a direct relation between the Cartesian matrix and
the generalized stiffness matrix can be defined by utilizing the Jacobian
matrix (Quennouelle and Gosselin, 2008). New representations of Cartesian
stiffness matrix based on energetic perspective has been proposed by Metzger
et al. (2010).
Furthermore, Zhang and Wei (2014) discussed and compared three stiffness
models for PKMs namely traditional stiffness model, kinetostatic compliance
model, and dexterous stiffness model. Klimchik et al. (2014) presented an
advanced stiffness modeling for PKMs with non-perfect chains under internal
and external loadings. The imperfectness occurs as the geometrical
parameters differ from the nominal ones and do not allow to assemble
manipulator without internal stresses that considerably affect the stiffness
properties and also change the end-effector location. Finally, Pashkevich et
al. (2010a) presented an enhanced stiffness modeling for parallelogram-based
PKMs while taking external loadings into consideration. The stiffness
modeling of parallelogram also can be found in Wu et al. (2016) and Wu and Zou (2016).
Furthermore, various models have been used to model the manipulator
stiffness. The use of the various models can be evaluated from several
aspects including the capability to handle complex geometry, ease of use,
computational cost, and effectiveness to achieve the goal. In general, the
stiffness models can be classified into three categories: (1) analytical,
continuous model, (2) lumped parameter model, and (3) distributed model. The
analytical, continuous stiffness model based on classical beam formulation
as well as geometrically exact beam formulation (Jafari and Mahjoob, 2010)
have been utilized but limited to simple manipulator geometry. The lumped
parameter model based on Jacobian matrix and commonly known as virtual joint
method (VJM) model has been widely used in the robotics field. The use of
VJM considering only the stiffness of rotational actuators was found in
(Salisbury, 1980). Furthermore, a one-dimensional VJM was applied to PKM by
Gosselin (1990b), followed by Pashkevich et al. (2009b) who introduced
multi-dimensional VJM. More recent works using VJM model include (Ceccarelli
and Carbone, 2002; Company et al., 2002; Arumugam et al., 2004; Zhang et
al., 2004; Majou et al., 2007; Vertechy and Parenti-Castelli, 2007). This
model is commonly used and preferred in robotics since it is analytical and
hence requires the same expression for all postures of the manipulator and
requires a lower computational cost. It usually still gives an acceptable
approximation of the stiffness despite its lower accuracy. For that reason,
this model is preferred for the initial estimation of the manipulator
stiffness as well as for design optimization purpose.
The most widely used distributed stiffness models are finite element
analysis (FEA) and matrix structural analysis (MSA) models, although some
other distributed models such as assumed mode method (AMM) and transfer
matrix method (TMM) models have also been utilized. As opposed to the lumped
parameter model, the FEA model discretizes the manipulator into a number of
elements and therefore is closer to the realistic, continuous model.
Structural mechanics commonly uses this model due to its high accuracy. It
is also able to handle a complex geometry by utilizing a suitable element
type. However, it is computationally intensive. Furthermore, it is not
practical as it requires new meshing at every different posture of the
manipulator. Due to its high accuracy, this model is often used to verify or
compare with another less accurate model such as VJM model (El-Khasawneh and
Ferreira, 1999; Long et al., 2003; Corradini et al., 2004; Rizk et al.,
2006). The MSA model, also known as direct stiffness method, can be
considered a special case of FEA model as it uses one-dimensional finite
elements instead of two- or three-dimensional ones. Therefore, it gives a
trade-off between accuracy and computational cost (Ceccarelli, 2008). Some
works using MSA model can be found in Li et al. (2002), Huang et al. (2002),
Deblaise et al. (2006) and Soares Junior et al. (2011).
Klimchik et al. (2014) quantitatively compared the computational complexity
of VJM, FEA, and MSA stiffness models based on their cost for matrix
inversion. They showed that the FEA and VJM models have the highest and
lowest complexity, respectively. As an extension of these three models,
modifications and improvements on the models have been conducted to
alleviate the drawbacks of each of the aforementioned models, such as
follows:
Online FEA by utilizing MSA using generalized springs (Taghvaeipour et al.,
2010)
VJM combined with FEA-based identification technique, which gives high
accuracy with low computational cost (Pashkevich et al., 2009a; Klimchik et
al., 2013)
Virtual spring approach which evaluates spring compliance based on FEA
concept and therefore gives high accuracy with low computational cost
(Pashkevich et al., 2009a, 2011b)
In authors' view, rather than obtaining a lumped parameter model through
FEA-based identification as in Pashkevich et al. (2009a) and Klimchik et al. (2013), one can use FEA followed by model order reduction to obtain an
accurate yet low-sized stiffness model of a manipulator with a complex
shape. This will allow an automated process in obtaining the low-sized
stiffness model. Various order reduction methods are available where each
offers different accuracy.
As mentioned earlier, the effectiveness to achieve the goal should also be
considered to determine a stiffness model to use. Unlike structural
mechanics which usually has an interest in the deformation of the whole
link, robotics typically only has an interest in the pose at the end
effector and possibly joints due to a deformation. In fact, different
stiffness models indeed may provide different pose values at those points of
interest, and therefore a more accurate model is preferred. Furthermore,
when a compliant behavior of manipulator links is considered, the rigid link
assumption used for the pose kinematics is no more valid. In such a case,
the pose of the end effector should be determined by solving constrained
flexible multibody dynamics. Discussion on this topic is beyond the scope of
this paper.
Finally, an experiment can be conducted to validate a stiffness model. In
this case, the stiffness is obtained from the relation between the measured
wrench and measured displacement. Based on a relation function, the
stiffness can be obtained by estimation using least squares algorithm or
other estimation algorithms.
Measures of stiffness magnitude
In the literature, the magnitude of manipulator stiffness has been presented
in the following different ways:
Graphical representations including stiffness maps, by which the stiffness
distribution can be plotted (Gosselin, 1990b; Mekaouche et al., 2015; Zhang
et al., 2017), and other graphical representations such as iso-stiffness
curves or surfaces (Merlet, 2006a)
Trace of the stiffness matrix (Carbone and Ceccarelli, 2007)
Weighted trace of the stiffness matrix (Gao et al., 2010; Zhang and
Gosselin, 2002)
Minimum, average, or maximum eigenvalues (and eigenvectors) of the stiffness
matrix (El-Khasawneh and Ferreira, 1999). For example, the evaluation of
minimum and maximum eigenvalues in Li and Xu (2008) and Wang et al. (2015) and
the average eigenvalue in Ruggiu (2017).
Mean value of the eigenvalues (Taghvaeipour et al., 2010)
Determinant of stiffness matrix (Carbone and Ceccarelli, 2007), which is the
product of the stiffness matrix eigenvalues, and indicates the area/volume
of a stiffness ellipse/ellipsoid. It also indicates the distance from
singularity.
Norm of the stiffness matrix, which can use Euclidian norm, Frobenius norm,
or Chebyshev norm (Carbone and Ceccarelli, 2007)
Center of stiffness or equivalently center of compliance (Patterson and
Lipkin, 1993)
Global compliance index which is given by mean value and deviation of
generalized compliance matrix (Xi et al., 2004)
Virtual work stiffness index which can avoid the problem caused by different
units of translation and orientation
Collinear stiffness value (CSV) (Shneor and Portman, 2010)
Trace of dexterous stiffness matrix (Zhang and Gao, 2012)
Carbone and Ceccarelli (2007) compared some of the abovementioned stiffness
representations from the numerical computation and effectiveness point of
view.
Measures of stiffness uniformity
Stiffness condition number is a local property as it is dependent on the
manipulator posture. In a similar fashion to the Jacobian condition number,
the stiffness condition number κs is defined by:
κs=KcKc-1
Similar to the Jacobian condition number, the stiffness condition number has
a value ranging from 1 to infinity. Alternatively, its inverse which has a
value ranging from 0 to 1 can also be used. The stiffness condition number
indicates the distance of the stiffness matrix Kc to singularity. It also
represents the isotropy or uniformity of the stiffness of any point in the
workspace, and therefore stiffness ellipses/ellipsoids are commonly used as
the graphical representation. The problem appears when the manipulator mixes
translational and rotational DOF and hence the Cartesian stiffness matrix is
nonhomogeneous. In this case, the Cartesian stiffness matrix is usually
normalized before it is used to compute the stiffness condition number as
given in Eq. (52).
Similar to Jacobian condition number, various norm definitions can be used
to evaluate the stiffness condition number. The 2-norm, Frobenius norm, and
weighted Frobenius norm are commonly used. The considerations in selecting
any of them are presented earlier when the Jacobian condition number is
discussed.
A global stiffness condition number commonly called the global stiffness index (GSI) is defined as
the inverse of the condition number of the stiffness matrix averaged over
the workspace as indicated in Eqs. (1) and (2). It indicates the
uniformity of stiffness within the whole workspace. Isotropic stiffness
workspace is defined as the workspace having stiffness isotropy (Shin et
al., 2011).
Consistency of stiffness measures
In manipulators with both translational and rotational DOF, the Cartesian
stiffness matrix does not have unit consistency. As a result, evaluation of
further stiffness indices such as eigenvalues and condition number of the
stiffness matrix becomes controversial. Attempting to overcome this issue,
several approaches have been proposed including the following:
Homogenizing the Jacobian matrix (such as using characteristic length or
another technique as discussed in Sect. 3) and subsequently using the
homogenized Jacobian matrix to calculate the stiffness matrix (Li and Xu,
2006, 2008).
Eigenscrew decomposition of the stiffness or compliance matrix (Ciblak and
Lipkin, 1999; Dai and Ding, 2006; Ding and Selig, 2004; Huang and Schimmels,
2000).
Principal axis decomposition through congruence transformation by making use
of the eigenvectors of the translational entry in the stiffness matrix (Chen
et al., 2015)
Decomposition of the dynamic inertia matrix by transforming variables into
dimensionless parameters (Kövecses and Ebrahimi, 2009),
which can be applied to the stiffness matrix (Taghvaeipour et al., 2012; Wu,
2014)
Decoupling of the stiffness matrix into translational and rotational parts
(Angeles, 2010; Wu et al., 2016; Wu and Zou, 2016)
It can be seen that the consistency problem in the stiffness measures have
been remedied by not only homogenizing the Jacobian matrix as the source of
the consistency problem, but also by decomposition and decoupling techniques
applied directly to the stiffness matrix.
Dynamic performance measures
Several dynamic performance measures have been used including mass, dynamic
dexterity, dynamic manipulability, dynamic stiffness, natural frequencies,
and some other dynamic performance measures.
Inertia – The inertia properties include the mass, the first mass moment of inertia, and the second mass moment of inertia (commonly called as
mass moment of inertia or simply moment of inertia). These inertial
properties are dictated by the material density and the geometry. In lumped
parameter model, the mass can be defined by the integration of the density
over the volume and the location of the center of mass. The mass and the
first moment of inertia are usually constant whereas the second moment of
inertia is usually constant with respect to the body frame but varying with
respect to the inertial frame. The generalized inertia matrix which
represents the inertia of the manipulator as a whole is posture-dependent as
it contains elements which vary with the manipulator posture.
Mass, or equivalently weight, is the most intuitive yet significant dynamic
performance measure. In fact, it is among the obvious advantages of PKMs
over SKMs since the PKMs commonly have their actuators attached to the base
and therefore the mobile mass is lower. However, larger reduction of the
mass usually results in lower stiffness. An optimization might target to
reduce the mass while keeping the stiffness above the required minimum
threshold. Further optimization of the mass can be achieved by obtaining
optimized detail geometry of the links and moving platform.
Dynamic dexterity – Analogous to the kinematic dexterity, the dynamic dexterityκM is defined by
the condition number of the mass matrix (inertia matrix) M:
κM=‖M‖‖M-1‖
where the mass matrix is the Hessian matrix of the kinetic energy expression
of the manipulator.
The norms in Eq. (53) are commonly defined by 2-norm definition, and hence
Eq. (53) can be rewritten as follows:
κM=σmax(M)σmin(M)
The dynamic dexterity is also a local property since every point at the
workspace has its own dynamic dexterity. The global dynamic dexterity index (GDDI) is measured by
averaging the local dynamic dexterity over the workspace as indicated in Eqs. (1) and (2).
Furthermore, Asada (1983) introduced generalized inertia ellipsoid which has principal axes along which
the inertia tensor is diagonal. The direction of these principal axes is
given by the eigenvectors of the generalized inertia tensor, whereas their
lengths are given by the reciprocal of the square root of the corresponding
eigenvalues. The major axis is corresponding to the smallest eigenvalue
which indicates the smallest generalized moment of inertia and consequently
the speed is the fastest in that direction. On the other hand, the minor
axis is corresponding to the largest eigenvalue which indicates the largest
generalized moment of inertia and consequently the speed is slowest in that
direction. If the lengths of the principal axes are identical, or
equivalently the generalized inertia ellipsoid is a sphere, the resultant
inertia is isotropic.
A normalized measure of inertia isotropyκ̃Λ bounded between 0 and 1 has been
introduced by Kilaru et al. (2015) as follows:
κ̃Λ=n2κ(Λ)
where n is the number of degrees of freedom of the manipulator, Λ is
a diagonal matrix with the eigenvalues of the inertia matrix as the diagonal
elements, and κ(Λ) is the condition number of Λ.
Furthermore, they also proposed the use of the maximum eigenvalue of
Λ as another dynamic performance measure of the manipulator.
Dynamic manipulability – Beyond the commonly used manipulability, dynamic manipulability which was
also introduced by Yoshikawa includes not only the Jacobian matrix but also
the mass matrix (Merlet, 1996; Yoshikawa, 1985b). The dynamic manipulability
wd is defined by:
wd=det(J(MTM)-1JT)
For non-redundant manipulators, the dynamic manipulability can be written
as:
wd=det(J)det(M)
Later, a new dynamic manipulability ellipsoid was proposed by Chiacchio (2000). Furthermore, Doty et al. (1995) proposed the use of so-called
dynamics frame-acceleration manipulability.
Dynamic stiffness – The dynamic stiffness is a function of the excitation force frequency
and depends on the manipulator mass, static stiffness, and damping (Yuan,
2015). Different from the static stiffness which indicates the resistance of
the manipulator structure against static loadings, the dynamic stiffness
indicates the resistance of the structure against dynamic loadings. The
dynamic stiffness can be evaluated by observing the frequency response under
excitation force with varying frequency. The lowest dynamic stiffness occurs
when the excitation force has the frequency equal to the natural frequency
of the manipulator (Alagheband et al., 2014). Azulay et al. (2014) evaluated
both of the static stiffness and the dynamic stiffness of a 3PPRS spatial
PKM after evaluating the tilting capability and the singularity of the
manipulator. They modeled the static stiffness of the manipulator by using
MSA, defined the dynamic stiffness of the manipulator by considering its
mass matrix, damping matrix, and static stiffness matrix, and modeled the
dynamic stiffness by using FEM. The dynamic stiffness Kd formulation
they used is given by:
Kd=(K-Mω2)2+(Bω)2
where M, B, and K are the mass matrix, the damping matrix, and the static
stiffness matrix of the manipulator, respectively.
Wang (2015) conducted an integrated stiffness analysis of redundant PKM
using FEM by considering static stiffness, dynamic stiffness, and moving
stiffness of the manipulator. They formulated the dynamic stiffness
Kd based on the following eigenvalue problem:
59det(Kd-λI)=060Kd=K-1M61λ=1/ω2
where K, M, ω, and I denote the manipulator stiffness matrix, mass
matrix, inherent frequency vector, and identity matrix, respectively.
Moreover, they defined the moving stiffness to represent the structure vibration when the
manipulator is moving.
Natural frequencies – Natural frequency (also called eigen-frequency) is one of the important
dynamic characteristics of a mechanical system, and a manipulator is not an
exception. The operation at or near the natural frequencies will result in
resonance. The natural frequencies depend on both the inertia and the
stiffness. They can be obtained theoretically or experimentally. The former
can be conducted either analytically or numerically. In fact, the analytical
method is only possible for simple geometries. For example, Germain et al. (2015) presented how the natural frequencies of PKMs are computed by using
Euler-Bernoulli beam model. For complex geometries, the theoretical modal
analysis should be conducted numerically such as by using FEA.
The natural frequencies are given by solving the following eigenvalue
problem:
62detωn2M-K=063ωn=2πfn
where M, K, ωn and fn are the mass matrix, the stiffness matrix,
the natural frequency in rad s-1, and the natural frequency in Hz,
respectively.
Other dynamic performance measures – In addition to the abovementioned dynamic performance measures, maximum driving (actuation) wrench (Yao et
al., 2017), power consumption (Han et al., 2017), maximum speed, and maximum acceleration are among the important dynamic
performance measures. In this case, it is always preferred to have a less
maximum driving wrench and power consumption, higher maximum speed, and
higher acceleration.
Recommendations
The workspace determination of PKMs is not straightforward as in SKMs.
Several approaches have been introduced. The geometrical approach can be
easily integrated with CAD system. However, it has a problem with general
applicability and constraint handling. Hence, a future work is required to
introduce a novel geometric technique which is able to overcome this
problem. On the other hand, although the numerical approach has wider
applicability and is able to handle the constraints, it is typically slow
and less accurate. Therefore, a novel numerical approach which is capable to
run faster with good accuracy is required.
Furthermore, a method to present the workspace as an analytical expression
is not available for most cases. As a consequence, although most of the
local performance measures can be expressed analytically, they cannot be
transformed to their global measures by analytical averaging across the
workspace indicated in Eq. (1) simply because the workspace cannot be
expressed analytically. In such a case, the only way is discrete averaging
as indicated in Eq. (2). Therefore, although finding a method to present the
workspace analytically with applicability to wider types of PKMs is very
challenging, it will be very useful once achieved.
For performance measures which represent the quality of the workspace, the
authors have provided some views and suggestions when discussing the
measures. In addition, it appears that mixed DOFs result in inconsistency of
the indices while many PKMs are required to have mixed DOFs to perform some
given tasks. Several attempts have been conducted to define homogeneous
indices to overcome the issue. Therefore, it is important to notice that any
new index introduced in the future should be able to avoid this issue or
overcome it in a more natural way so that the physical insights of the index
will be as sound and intuitive as possible. In addition, any new index in
the future should be bounded and independent from order, scale (choice of
unit), and frame. The formulation complexity, computation performance and
cost, physical meaning, information completeness, accurateness, and scope of
applicability should also be considered as other criteria. The computation
cost, in particular, is very significant when the indices are used in
optimization where a large number of iterations are involved. Some indices
which involve matrix inversion or eigenvalue computation typically require
high computation cost. Therefore, the introduction of cheaper performance
indices with sound interpretation as an alternative to the costly ones would
be advantageous. Furthermore, considering some special behaviors in a PKM
such as over-constraints, redundancy (including kinematic and actuation
redundancies), decoupling, joint clearance, component flexibility and
nonlinearity (including material and geometric nonlinearities), in the
performance measures should also be more discussed in the future.
Conclusion
This paper provided a comprehensive overview of performance indices of PKMs
which include both the workspace and its quality as well as both the
kinematic and dynamic performance indices. Not only this paper serves as a
comprehensive reference for designers of PKMs in optimizing the performance
of their design, but also provided the current state-of-the-art in the
research topics and recommended some future research work.
Data availability
No data sets were used in this article.
Author contributions
The manuscript was written through contributions of all authors. AR surveyed the literature and wrote the paper. BEK guided the work and reviewed the manuscript. AA reviewed the manuscript. All authors have given approval to the final version of the manuscript.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
This research was partially supported by the Khalifa University Internal Research Fund.
Review statement
This paper was edited by Marek Wojtyra and reviewed by Calin-Octavian Miclosina and two anonymous referees.
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