⎥
⎣
⋅ 14 ⎥⎦
x = [ x y z θ ]
For the H4 structure and design parameters described above, when the nacelle is in the
origin pose and the actuated joint accelerations are q = [10 −10 10
]
10 m/s2, the linear
acceleration of the nacelle can reach 23m/s2 and the angular acceleration can reach
2.5rad/s2.
3.6 Kinetostatic performance indices
3.6.1 Manipulability
The inverse kinematic jacobian matrix reflects a linear relation between the manipulator
accuracy x
Δ and the measurement errors q
Δ on q . If the measurement errors are bounded,
then the hyper-sphere in the joint error space can be mapped into an ellipsoid in the
generalized Cartesian error space with using the Euclidean norm. This ellipsoid is usually
called the manipulability ellipsoid. If using the infinity norm, the joint errors are restricted to
lie in a hyper-cube in the joint error space. The hyper-cube in the joint error space can be
mapped into the kinematic polyhedron in the positioning errors space (Merlet, 2006). The
shape and volume of the manipulability ellipsoid and the kinematic polyhedron
characterize the manipulator dexterity. It is necessary to set up some kinetostatic
performance indices that are used to quantify the dexterity of a robot. The famous
Yoshikawa’s manipularity index
T
JJ is used for serial robots for a long time. Another index
is the condition number that is often used for parallel robots.
κ
J 1
−
=
J
(42)
The condition number expresses how a relative error in joint space gets multiplied and leads
to a relative error in work space. The condition number is dependent on the choice of the
metric norm and the most used norms are the 2-norm and the Euclidean norm. The
condition number is mentioned as the main index for characterizing the accuracy of parallel
robots. But there is major drawback to the condition number for H4, since the elements of
the matrix corresponding to translations are dimensionless, whereas those corresponding to
the rotations are lengths. A direct consequence is that the condition number has no clear
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Parallel Manipulators, Towards New Applications
physical meaning, as the rotations are transformed arbitrarily into “equivalent” translations
(Merlet, 2006). There are various proposals that have been made to avoid this drawback
(Gosselin, 1990; Kim & Ryu, 2003). But all these proposals have their own special
drawbacks. How to design a general law for parallel robots’ manipulability is still an open
problem.
3.6.2 Isotropy
Poses with a condition number of 1 are called isotropic poses, and robots having only such
type of poses are called isotropic robots (Merlet, 2006). At the given pose as shown in Fig.11,
the condition number of H4 is about 8.56 and there is no any isotropic poses in the
workspace of H4 (Company et al., 2005). While Part4 robots which are the evolutional
structure of H4 are isotropic at their original poses.
There are some other manipulability and accuracy indices, such as global conditioning
index, uniformity of manipulability and maximal positioning error index et al. All the
above definitions of the kinetostatic indices do not take into account other factor affecting
the accuracy of parallel robots, such as manufacturing tolerances, clearance and friction in
the joints. In order to include these factors in accuracy analysis, Monte Carlo statistical
simulation technique can be used to evaluate the accuracy of parallel manipulators (Ryu &
Cha, 2003).
4. Singularity analysis
This section will identify all the singular configurations of the H4 and analyze the
manipulator’s self-motion when in or closed to a singular configuration. Generally, singular
configurations refer to particular poses of the end-effector, for which parallel robots lose
their inherent infinite rigidity, and in which the end-effector will have uncontrollable
degrees of freedom. But for this new family of parallem manipulator H4, singularities are
associated with either loss or gain of DOF. This section utilizes line geometry tools and
screw theory to deal with singularity analysis of H4. Firstly, the basic theory including in the
process of singularity analysis is introduced briefly. Then the static equilibrium condition of
the end-effector is derived to obtain the full inverse kinematic jacobian 6×6 matrix, which is set
of governing lines of the manipulator. Based on linear dependency of these lines, the
singular configurations of the manipulator can be identified. Moreover, in order to deal with
singularities associated with loss of DOFs (serial singularity), the static equilibrium of the
actuators is also defined. Secondly, architecture and constraint singularities associated with
gain of DOFs (parallel singularity) are defined and analyzed using linear complex
approximation algorithm (LCAA), which is employed to obtain the closest linear complex,
presented by its screw coordinates, to the set of governing lines. The linear complex axis and
pitch provide additional information and a better physical understanding of the
manipulator’s self-motion when in or closed to a singular configuration. Lastly, various
singularities of an example H4 manipulator are presented and analyzed using the proposed
methods (Wu et al., 2006).
4.1 Basic theory
In the context of designing a parallel manipulator, understanding the intrinsic nature of
singularities and their relations with the kinematic parameters and the configuration spaces
A Novel 4-DOF Parallel Manipulator H4
427
is of great importance. The phenomena of singularity in parallel manipulators have been
approached from different points of view. One way to introduce singular configurations is
to examine the relations obtained for inverse kinematics. Singularities correspond to the
roots of the parallel manipulator Jacobian’s determinant. Based on the rank deficiency
associated with the Jacobian matrices, singularities of parallel manipulators have been
explained (Gosselin &Angeles, 1990; Zlatanov et al., 1994). Using the linear decomposition
method and co-factor expansion, St-Onge and Gosselin (St-Onge & Gosselin, 2000) studied
the singularity loci of the Gough-Steward platform, and obtained a graphical representation
of these loci in the manipulator workspace. Due to the complexity of the kinematic model,
several authors proposed numerical methods to analyze the singularities (Feng-Cheng
&Haug, 1994; Funabashi & Takeda, 1995). However, even with symbolic computation
software, the calculation of the determinant of Jacobian matrix is a complicated task.
Moreover, identifying kinematic singularity through matrices does not provide physical
insight into the nature of singular configuration of parallel manipulators.
A parallel manipulator is naturally associated with a set of constraint functions defined by
its closure constraints (geometric relations of the closed-chain mechanism). The differential
forms arising from these constraint functions completely characterize the geometric
properties of the manipulator. So Liu et al. (Liu et al., 2003) used differential geometric tools
to study singularities of parallel mechanisms and provided a finer classification of
singularities. In their works, they classified singularity into configuration space singularities
and parametrization singularities which include actuator and end-effector singularities as
their special cases. But the method is too abstract and complicated when dealing with
singularities of spatial parallel manipulators.
Another approach to analyze the parallel manipulators’ singularity is based on line
geometry and screw theory. Merlet (Merlet, 1992) considered the Jacobian matrix of the
Gough-Steward platform as the Plücker line coordinates of the robot’s actuators and
identified the singular configurations of the robot. In (Romdhane et al., 2002), a mixed
geometric and vector formulation is used to investigate the singularities of a 3-translational-
DOF parallel manipulator. In (Collins, 1995), singularities of an in-parallel hand controller
for force-reflected teleoperation were analyzed, the six columns of the Jacobian matrix were
viewed as zero-pitch wrenches (lines) acting on the top platform, then line geometry and
rank determining geometric constructions were used to obtain all configuration
singularities. Basu and Ghosal (Basu & Ghosal, 1997) presented a geometric condition to
deal with the singularity analysis associated with gain of DOF in a class of platform-type,
multi-loop spatial manipulators. Joshi and Tsai (Joshi & Tsai, 2002) developed a
methodology for the Jacobian analysis of limited-DOF parallel manipulators by making use
of the theory of reciprocal screws. A 6×6 Jacobian matrix so derived provided information
about both architecture and constraint singularities.
The present investigation identifies all the singular configurations of the H4 robot and
provides physical insight into the nature of these singular configurations from the view
point of geometry. Moreover, the behavior in these singular configurations or in the
neighboring ones is also determined using linear complex approximation. All these analyses
need some basic theories, include Grassmann geometry, Line geometry and Screw theory
etc. The following gives a brief presentation of these theories.
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Parallel Manipulators, Towards New Applications
4.1.1 Line geometry and Plücker coordinates
Line geometry investigates the set of lines in three-space. The ambient space can be a real
projective, affine, Euclidean or a non-Euclidean space. Line geometry possesses a close
relation to spatial kinematics (Bottema & Roth, 1990) and has therefore found applications in
mechanism design and robot kinematics. A parallel manipulator has a singular position if
the axes of its hydraulic cylindrical legs lie in a linear complex, a special three-parameter set
of lines. In practice, several sources for errors (manufacturing, material properties,
computing, etc.) can hardly be avoided. Thus, the question is whether the lines on the
objects near their realization are close – within some tolerance – to the lines of a linear
complex. This is an approximation or regression problem in line space (Helmut et al., 1999).
In real Euclidean space E 3, a Cartesian coordinate system is used and the points are
represented by their coordinate vectors M. A straight line L can be determined by a point
p ∈ L and a normalized direction vector l of L, i.e. l = 1 . To obtain coordinates for L, one forms the moment vector
l =
: p × l
(43)
with respect to the origin. If p is substituted by any point q=p+λl on L, equation (43) implies
that l is independent of p on L. The six coordinates (l, l) with
l = ( l , l , l , and l = ( l , l , l
4
5
6 )
1
2
3 )
are called the normalized Plücker coordinates of L. The set of lines in E 3 is a four-dimensional
manifold and accordingly the six Plücker coordinates satisfy two relations. One is the
normalization and the other is, by equation (43), the so-called Plücker relation
l ⋅ l = 0
(44)
which expresses the orthogonality between l and l . Conversely, any six-tuple (l, l) with
l = 1 , which satisfies the Plücker relation l ⋅ l = 0 represents a line in E 3. As the orientation
are not concerned, (l, l) and (− l,−l) describe the same line L.
The topic about the Klein mapping and special sets of lines can refer to the paper written by
Pottmann et al. (Pottmann et al., 1999).
4.1.2 Grassmann geometry
As shown in the previous section, two lines with Plücker coordinates l = p , q , l = p , q
2
[ 2 2]
1
[ 1 1]
intersect if and only if p ⋅q + q ⋅p = 0 . Plücker vectors with p = 0 do not represent real
1
1
2
2
lines and are associated with a line at infinity. All lines at infinity belong to a plane, the
plane at infinity. A point may also be represented by the Plücker coordinates (α, r) so that its
coordinates are r/α. If α=0, then the point is at infinity, and a point (0, r) at infinity is on the
line at infinity (0, q) if and only if r·q=0. Consequently a point at infinity that belongs to the
two lines at infinity (0, s1), (0, s2) has coordinates (0, s1×s2).
The columns of the full inverse kinematic jacobian matrices of most parallel robots are
constructed from the Plücker vectors of lines associated with links of the manipulator. The
singularity of this matrix therefore means that there will be a linear dependence between
these vectors. Grassmann showed that linear dependence of Plücker vectors induced
A Novel 4-DOF Parallel Manipulator H4
429
geometric relations between the associated lines, so that a set of n Plücker vectors creates a
variety with dimension m< n. A thorough introduction to Grassmann geometry please refers
to (Pottmann et al., 1999). The applications of Grassmann geometry can be found in
references (Collins & Long, 1995; Hao & McCarthy, 1998; Wu et al., 2006).
Using the Grassmann’s geometrical conditions, an algorithm for finding the singular
configurations of any type of parallel robot whose full inverse kinematic jacobian consists of
Plücker vectors should be designed. Firstly consider all sets of n lines that are associated
with the Plücker vectors constructed from full inverse kinematic jacobian matrices, and then
the pose of the moving platform is determined so that the n lines satisfy one of the
geometrical conditions which ensure that they span a variety of dimension n-1, thereby
leading to a singularity of the robot. The following list the geometric conditions that ensure
that the dimension of the variety spanned by a set of n+1 Plücker vectors is n, for each
possible dimension n of the variety (Merlet, 2006). It should be noted that the notation of
intersecting lines also include parallel lines which intersect at infinity. To consider the
reading convenience, the following results is excerpted from the monograph written by
Merlet (Merlet, 2006).
dimension
a
b
c
d
3
2
1
Fig.12. Geometrical conditions that characterize the varieties of dimension 1, 2 and 3 (Merlet,
2006)
For the variety of dimension 1 (called a point) there is just one Plücker vector and one line.
A variety of dimension 2, called a line, may be constituted either by two Plücker vectors for
which the associated lines are skew, i.e. they do not intersect and they are not parallel, or be
spanned by more than two Plücker vectors if the lines that are associated with the vectors
form a planar pencil of lines, i.e. they are coplanar and possess a common point (possibly at
infinity, to cover the case of coplanar parallel lines). A variety of dimension 3, called a plane,
is the set of lines F that are dependent on 3 lines F 1, F 2, F 3. It is possible to show that all the points belonging to the lines F lie on a quadric surface Q. This quadric degenerates to a pair
of planes P 1, P 2 if any two of the three lines intersect.
–
condition 3d: all the lines are coplanar, but do not constitute a planar pencil of lines; F 1,
F 2, F 3 are coplanar and P 1, P 2 are coincident.
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Parallel Manipulators, Towards New Applications
–
condition 3c: all the lines possess a common point, but they are not coplanar; F 1, F 2, F 3
intersect at the same point, possibly at infinity (this covers the case of parallel lines).
–
condition 3b: all the lines belong to the union of two planar pencils of non coplanar
lines that have a line L in common; two of the lines intersect at a point p, and L intersect
the last line at a. Two different cases may occur:
•
P 1, P 2 are distinct and intersect along the line L. The set of dependent lines are the
lines in P 1 that go through a, and the lines in P 2 that go through p
•
P 1, P 2 are distinct and parallel. This occurs if two of the lines Fi are parallel; L is a
line at infinity, and the set of dependent lines are two planes of parallel lines
–
condition 3a: all the lines belong to a regulus; F 1, F 2, F 3 are skew.
A variety if dimension 4, called a congruence, corresponds to a set of lines which satisfies
one of the following 4 conditions:
–
condition 4d: all the lines lie in a plane or meet a common point that lies within this
plane. This is a degenerate congruence.
–
condition 4c: all the lines belong to the union of three planar pencils of lines, in different
planes, but which have a common line. This is a parabolic congruence.
–
condition 4b: all the lines intersect two given skew lines. This is a hyperbolic
congruence.
–
condition 4a: the variety is spanned by 4 skew lines such that none of these lines
intersects the regulus that is generated by the other three. This is an elliptic congruence.
A variety of dimension 5, called a linear complex, is defined by two 3-dimensional vectors
(c, c) as the set of lines L with Plücker coordinates (l, l) such that c ⋅l + c⋅ l = 0. The complex may be:
–
condition 5b: all the lines of the complex intersect the line with Plücker coordinates
(c, c), namely, c ⋅ c = 0. This is a singular configuration.
–
condition 5a: c ⋅ c ≠ 0 . This is a general or non singular configuration.
The degree of freedom associated with a linear complex is a screw motion with axis defined
by the line with Plücker vector (c, c − pc) c , where p = c⋅ c c is the pitch of the motion.
4a
4b
4c
4d
5a
5b
Fig.13. Geometrical conditions that characterize the varieties of dimension 4 and 5 (Merlet,
2006)
4.1.3 Screw theory
According to screw theory, the instantaneous velocities of a rigid body can be described by a
6-dimensional vector of the form (ω, v), where ω is the angular velocity of the rigid body,
and v is its translational velocity. These elements are called velocity twists or screws. Forces
and torques exerted on the rigid body are important for motion and may be represented as a
A Novel 4-DOF Parallel Manipulator H4
431
couple of 3-dimensional vectors (F, M) called a wrench. A twist and a wrench will be said to
be reciprocal if
F ⋅ v + M ⋅ω = 0
(45)
When a kinematic chain is connected to a rigid body the key point is that the possible
instantaneous twists for the rigid body are reciprocal to the wrenches imposed by the
kinematic chains (called the constraint wrenches). In other words, the DOF of the rigid body
are determined by the constraint wrenches. So based on linear dependency of the constraint
wrenches, the singular configurations can be identified. Another function of the screw
theory is the mobility analysis (Li & Huang, 2003).
4.2 Static analysis of the H4
When deriving the Jacobian matrix of the H4 using the velocity-equation formulation
described in section 3.3, the result is a 4×4 Jacobian matrix because of the 4-DOF of the
manipulator. However, it is not clear as to this is the best way to express the Jacobian of this
type of limited-DOF parallel manipulator when singular analysis is processed. Because this
approach is valid for general-purpose planar or spatial parallel manipulators, for which the
connectivity of each serial chain (limb) is equal to the mobility of the end effector, it is not
necessary true for parallel manipulators with less than 6-DOF (Joshi & Tsai, 2002). For
example, this approach leads to a 3×3 Jacobian matrix for the 3- UPU parallel manipulator
assembled for pure translation (Tsai, 1996; Tsai & Joshi, 2000). However, such a 3×3 Jacobian
matrix cannot predict all possible singularities. Di Gregorio and Parenti-Castelli (Di
Gregorio & Parenti-Castelli, 1999) analyzed the singularities of the 3- UPU translational
platform. They derived the conditions for which the actuators cannot control the linear
velocity of the moving platform, generally known as architecture singularities. Bonev and
Zlatanov also found that under certain configurations the 3-UPU translational platform
would gain rotational DOFs, which was called constraint singularities (Bonev and Zlatanov,
2001). Constraint singularities occur when the limbs of a limited-DOF parallel manipulator
lose their ability to constrain the moving platform to the intended motion.
The 4×4 Jacobian matrix of