Mechanism and Machine Theory 39 (2004) 75–95
www.elsevier.com/locate/mechmt
Application of line geometry and linear
complex approximation to singularity analysis of the 3-DOF
CaPaMan parallel manipulator
Alon Wolf
a,*
, Erika Ottaviano b, Moshe Shoham a, Marco Ceccarelli
b
a
Department of Mechanical Engineering, Technion-Israel Institute of Technology, Technion City,
Haifa 32000, Israel
b
Laboratory of Robotics and Mechatronics, DiMSAT––University of Cassino, Via Di Biasio 43,
03043 Cassino (Fr), Italy
Received 3 June 2002; received in revised form 6 May 2003; accepted 8 July 2003
Abstract
This paper analyzes the singularities of a three degree of freedom (DOF) spatial parallel manipulators
utilizing line geometry and linear complex approximation. First, the 6 · 6 matrix, mapping external
wrenches acting on the moving platform to internal forces/moments at the manipulatorÕs joints, is derived
as a set of six governing lines. By analyzing the dependencies of these lines, all singular configurations of the
manipulator are obtained. Then, the closest linear complex to these six governing lines is derived. The
closest linear complexÕs axis and pitch provides information and understanding of the robotÕs self-motion
when in a singular configuration. Moreover, in the neighborhood of a singular configuration the instantaneous motions arising from manufacturing tolerances and low rigidity are determined. The proposed
analysis has been applied to the 3-DOF parallel robots CaPaMan (Cassino Parallel Manipulator) and
CaPaMan2.
2003 Elsevier Ltd. All rights reserved.
Keywords: Parallel robot; Singularity; Line geometry; CaPaMan
*
Corresponding author. Address: Mechanical Engineering and the Robotics Institute, Carnegie Mellon University,
The Robotics Institute, Newell-Simon Hall 5000 Forbes AV, Pittsburgh, PA 15213, USA. Tel.: +1-412-268-8871.
E-mail addresses: [email protected] (A. Wolf), [email protected] (E. Ottaviano), [email protected] (M. Shoham), [email protected] (M. Ceccarelli).
0094-114X/$ - see front matter 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/S0094-114X(03)00105-8
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A. Wolf et al. / Mechanism and Machine Theory 39 (2004) 75–95
1. Introduction
The 6-DOF (degrees of freedom) Gough–Stewart platform was first introduced as a tyre testing
mechanism by Gough [1] and later as a flight-simulator by Stewart [2]. It took almost fifteen years
from StewartÕs flight simulator to see a parallel mechanism being first combined in an assembly cell
[3]. Yet, only recently machine industry has discovered the potential of integrating parallel manipulators with reduced number of DOFs. These mechanisms have simpler mechanical structure,
simpler control system, high speed performance, and low manufacturing and operations cost. Hence
they can be found in production lines, performing tasks where full mobility (6-DOF) is not required
due to the nature of the task. For instance, three-DOF spatial parallel mechanisms have been
presented for telescope applications [4], flight simulation [5], and beam aiming applications [6].
Numerous works investigating parallel mechanisms stress out the various advantages of these
mechanisms, yet one of their major drawbacks is their performance while in or close to singular
configurations. In these configurations, the mechanism tends to lose its stiffness while gaining
extra degrees of freedom [7,8]. Physically, when the mechanism is in a singular configuration, the
structure cannot resist or balance an external wrench applied at the moving platform. This situation can cause a general failure or permanent damage to the manipulator and surrounding
equipment. The characteristics of singular configurations in parallel robots, as described above,
make the singular analysis phase be one of the most important and earliest steps in their design
procedure. In this phase, the designer identifies the singular configurations of the designed
mechanism and alters its design in order to achieve continuous singularity-free paths within the
workspace [9].
The phenomena of singularity in parallel robots have been extensively studied. Gosselin and
Angeles [10] have suggested a classification of singularities for parallel manipulators, based on the
rank deficiency associated with the Jacobian matrices, into three main groups. The first type of
singularity occurs when the manipulator reaches internal or external boundaries of its workspace
and the output link loses one or more DOF. The second type of singularity is related to those
configurations in which the output link is locally movable even if all the actuated joints are locked.
The third type occurs when both first and second types of singularities are involved. Ma and
Angeles [11] have introduced a comprehensive classification for singularities, namely configuration singularities, architecture singularities and formulation singularities. The first type of singularity is an inherent manipulator property that occurs at some points within its workspace.
Architecture singularities are caused by the manipulator architecture and can prevail over the
entire workspace. Formulation singularities are caused due to the adopted analysis and can be
avoided simply by changing the formulation method. Another method of mapping singular
configurations consists of writing closed-loop velocity equations for the manipulator as a function
of the linear and angular velocities of the joints. This method was used by Tsai [12], and Wang
and Gosselin [13]. The method was applied to several parallel manipulators, see for example: Shi
and Fenton [14], Wang and Gosselin [13], Zanganeh and Angeles [15], and Ebert-Uphoff and
Gosselin [16]. Algebraic formulation of the velocity equations can also be obtained by direct
differentiation of the position loop. This method is compared with the vector analysis for a 3-DOF
spatial parallel manipulator in [17].
In the present investigation, the singularity analysis problem is addressed from line geometry
point of views. This approach was used by several researchers since the Jacobian matrix of the
A. Wolf et al. / Mechanism and Machine Theory 39 (2004) 75–95
77
Gough–Stewart platform is actually the Plucker line coordinates of the robotÕs actuators [18].
Hence, linear dependency of these lines means a rank deficit Jacobian matrix and singular configuration of the robot. Collins and Long [19] showed that the six columns of the Jacobian matrix
are zero-pitch wrenches (lines) acting on the top platform by which all singular configurations can
be obtained. Based on Dandurand stability analysis of spatial varieties of lines [21], Ben-Horin
[20], determined the singularity configurations of 17 platforms. This research also, showed how a
redundant robot can be designed to circumvent all singularities within its workspace. Simaan and
Shoham [22] investigated the singularity of a class of non-fully parallel manipulators that share
the same direct instantaneous kinematic matrix. Hao and McCarthy [23] introduced screw theory,
and based on geometry of the screws, singular conditions were presented and their equations were
developed. Using line geometry, Fichter [24] found the singular configurations of a Gough–
Stewart platform obtained by 90 deg rotation about a normal to the base platform.
The present investigation extends the application of line geometry to parallel manipulators and
determines not only their singular configurations but also the manipulator behavior in these
configurations or in the neighboring ones. For this purpose the 6 · 6 matrix mapping external
wrenches acting on the moving platform to the internal forces/moments of the moving platformÕs
joints, is derived as a set of six lines. Then, the closest linear complex to these six governing lines is
obtained using the Linear Complex Approximation Algorithm (LCAA) as was presented by
Pottmann et al. [26] and later applied to kinematics analysis of parallel robots by Wolf and
Shoham [27]. The closest linear complexÕs axis and pitch provides additional information and
understanding of the type of singularity and the nature of instantaneous motions occurring in or
at the vicinity of singular points. As an example, this method is applied in this investigation to the
CaPaMan robot described in Ottaviano et al. [25].
2. Mechanisms architecture
The 3-DOF CaPaMan (Cassino Parallel Manipulator) robot is composed of a fixed platform
FP and a moving platform MP that is driven by 3 limbs through spherical joints located at
H1 ; H2 ; H3 (Fig. 1). Each limb is composed of a parallelogram AP, a prismatic joint SJ and a
connecting bar CB. The connecting bar CB is able to translate along the passive prismatic joint SJ,
keeping its vertical posture. Each AP plane is rotated by p=3 radians with respect to the neighbor
one. The input kinematic variables of the robot are hi (Fig. 4), the input crank angles. Points H
and O denote the center points of MP and FP, respectively.
CaPaMan2 (see Fig. 2) is a modified version of CaPaMan in which a passive revolute joint RJ,
replaces the passive prismatic joint SJ, in each limb.
3. Static analysis of CaPaMan manipulator
In order to obtain the 6 · 6 matrix that maps the external wrench applied at the moving
platform to the joints forces and moments, static equilibrium is analyzed. This matrixÕ rows describe the six governing lines, from which the manipulatorÕs singularities are obtained.
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A. Wolf et al. / Mechanism and Machine Theory 39 (2004) 75–95
Fig. 1. Kinematic architecture and design parameters of CaPaMan1 (Cassino Parallel Manipulator).
Fig. 2. Kinematic architecture and design parameters of CaPaMan2 (Cassino Parallel Manipulator).
In our example––CaPaMan manipulator––the static analysis can be divided into two parts.
One regards the parallel chain architecture which is composed of the three prismatic joints SJ, the
A. Wolf et al. / Mechanism and Machine Theory 39 (2004) 75–95
79
three connecting bars CB, the three spherical joints BJ and the moving platform MP. The second
deals with the static equilibrium conditions of the parallelograms. Hence, one can analyze the
singularity of the entire robotic structure by conducting separate singularity analysis for the
parallelogram and for the parallel part of the manipulator. Observing the structure of the parallel
part of the CaPaMan manipulator as described in Fig. 3, one can detect that due to the spherical
joint, connecting link CB to the platform MP, there are no moments exerted by the limbs to the
platform hence, the static equilibrium for the ith limb is given by (see Fig. 3):
3
X
f1;i ^s2;i þ
i¼1
3
X
3
X
f2;i ^s3;i Fe ¼ 0
i¼1
w
Rp r^i ^s2;i f1;i þ
X
ð1Þ
w
Rp ^
ri ^s3;i f2;i Me ¼ 0
i¼1
where w Rp is a rotation matrix, which expresses the orientation of the moving platform with respect to the fixed reference frame; r^i is a vector connecting H to the center Hi of the ith spherical
joint. ^sk;i and fk;i (k ¼ 1; 2; 3) are the unit direction vectors and the internal forces acting on the ith
link in the kth direction respectively. Writing Eq. (1) in a matrix form yields:
3
2
f1;1
7
6
6 f1;2 7
6
^s2;2
^s2;3
^s1;1 ^s2;1
^s1;2 ^s2;2
^s1;3 ^s2;3 6 f1;3 7
^s2;1
7
w
w
w
w
w
w
7
Rp ^
Rp r^2 ^s2;2
Rp r^3 ^s2;3
Rp r^1 ^s3;1
Rp ^r2 ^s3;2
Rp ^r3 ^s3;3 6
r1 ^s2;1
6 f2;1 7
4 f2;2 5
f2;3
Fe
¼
Me
ð2Þ
f1,i
Fe
Me
r̂i
H
f 2 ,i
Sˆ3,i
Sˆ2,i
Sˆ1,i
Fig. 3. Forces transmitted to the moving platform (MP) by link i.
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A. Wolf et al. / Mechanism and Machine Theory 39 (2004) 75–95
and the forces at the robots joints due to external load are given by:
3
2
f1;1
7
6
6 f1;2 7
7
6
f
F
1;3 7
JS1 e ¼ 6
7
6
Me
6 f2;1 7
4 f2;2 5
f2;3
ð3Þ
where ^s3;i ¼ ^s1;i ^s2;i .
Hence, for the analysis of the singularity of the parallel part, one can investigate the transformation matrix JS given in Eq. (3).
Observing JS , one can detect that its columns are lines lying on the Klein quadric M24 , as they
satisfy KleinÕs equation [28,29]:
p01 p23 þ p02 p31 þ p03 p12 ¼ 0
ð4Þ
where each column of Eq. (3) is represented by its Pl€
ucker line coordinates given by
ðp01 ; p02 ; p03 ; p23; p12 ; p31 ÞT .
In addition, f1;i and f2;i are forces generated on the parallelogram and are functions of the
torques exerted by the motors and the input joint variable angles hi , see Fig. 4, and are given by:
X
Fx ¼ 0 f2;i þ R1;i cos hi þ R2;i cos hi fi sin hi ¼ 0
ð5Þ
X
ð6Þ
Fz ¼ 0 f1;i þ R1;i sin hi þ R2;i sin hi þ fi cos hi ¼ 0
X
l2
l2
l2
ð7Þ
MQ ¼ 0
R1;i sin hi þ fi cos hi R2;i sin hi ¼ 0
2
2
2
f1,i Sˆ2,i
f 2,i Sˆ3,i
l2
Q
f i*
R1,i
τi
R2,i
Z
l1
θi
X
Fig. 4. Parallelogram static analysis.
A. Wolf et al. / Mechanism and Machine Theory 39 (2004) 75–95
81
where R1;i , R2;i are two forces, exerted by the two limbs of the ith parallelogram at the input and
output cranks, respectively. si is the torque exerted by motor i; fi is a virtual force perpendicular
to R1;i so that si ¼ fi l1 .
Observing Eqs. (5)–(7), the three unknown parameters are: fi , R1;i , R2;i . Solving (5)–(7) for fi
yields:
fi ¼ f1;i cos hi þ f2;i sin hi
ð8Þ
Writing si ¼ fi l1 using Eq. (8) for i ¼ 1; 2; 3, si can be written as:
3
2
f1;1
6 f1;2 7 2 3
7
6
s1
6 f1;3 7
7 ¼ 4 s2 5
6
JP 6
7
6 f2;1 7
s3
4 f2;2 5
f2;3
ð9Þ
where
2
l1 cos h1
JP ¼ 4
0
0
l1 sin h1
0
0
0
l1 cos h2
0
0
l1 sin h2
0
0
0
l1 cos h3
3
0
0 5
l1 sin h3
Hence the singular analysis of the CaPaMan should focus both on JS (Eq. (3)) and JP (Eq. (9)).
The analysis of JS provides the singularity of the parallel part of CaPaMan robot known as the
‘‘parallel singularity’’ and JP provides the singularity due to the parallelogram structure, known as
the ‘‘serial singularity’’.
3.1. Singularity analysis of the CaPaMan manipulator
Examining JP one can see that singularity occurs whenever:
1. hi ¼ 0 þ pkðk ¼ 1; . . . ; n; i ¼ 1; . . . ; 3Þ. In this case one or more columns of JP are zero hence it
can not move the platform in the horizontal direction (denoted x in Fig. 4). These singularities
are demonstrated in Fig. 5A and B (per one limb).
2. hi ¼ p2 þ pkðk ¼ 1; . . . ; n; i ¼ 1; . . . ; 3Þ. In this case one or more columns of JP are zero hence it
can not move the platform in the normal direction (denoted z in Fig. 4). This singularity is demonstrated in Fig. 5C (per one limb).
Singular configurations of the parallel part of CaPaMan are identified by investigating matrix
JS . The first three columns of JS define the three forces of actuation which are along the three
connecting bars CB of the robot (see Figs. 3 and 6). When the three forces are coplanar and
intersect in a common point they belong to a flat pencil defined by any two of them, hence, are
linearly dependent [21]. Observing the robotsÕ structure and JS , one can detect that the three lines
along CBs are always parallel, hence the three are linearly dependent if and only if they are coplanar. This singularity of flat pencil type occurs only when the platform is rotated by p=2 around
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A. Wolf et al. / Mechanism and Machine Theory 39 (2004) 75–95
Fig. 5. Singularity of the parallelogram.
H3
Ŝ 2, 2
Ŝ 3,3
r3
Platform (MP)
H
Ŝ 2,1
r1
Ŝ 3, 2
r2
Ŝ 2, 2
H1
H2
Ŝ 3,1
Fig. 6. Lines of actions while platform parallel to base platform.
the y-axis, in this case ^s2;1 ¼ ^s2;2 ¼ ^s2;3 ¼ ½0; 0; 1, are coplanar. Examining JS in this configuration,
its rank is of order 5 and the robot is in a singular configuration.
The last three rows of JS represent three moments of constraint, each perpendicular to the plane
spanned by ^s1;i and ^s2;i (see Fig. 6). Rotation of the platform by p=2 around the Z-axis while the
platform is parallel to the base causes ri to become parallel to ^s3;i hence r^i ^s3;i ¼ 0 (Fig. 6). This
means that the last three lines in JS (columns 4–6 of Eq. (2)) have a zero moment relative to the
origin hence they intersect at the origin (their last three coordinates are zero). Consequently, there
are three coplanar lines intersecting at the origin forming a flat pencil of lines with the origin as its
A. Wolf et al. / Mechanism and Machine Theory 39 (2004) 75–95
83
vertex. In this case JS Õs rank is of order 5 and the robot is in a singular configuration. This kind of
singularity is also known as FichterÕs singularity [24].
3.2. Applying the linear complex approximation for the CaPaMan manipulator
In this section, the Linear Complex Approximation Algorithm (LCAA) which was presented at
[26] and further used for robotic analysis in [27] is used to investigate the type and instantaneous
motion the robot tends to perform while in singular configurations. The LCAA is based on
minimizing the instantaneous work generated by the wrenches defining the rows of the robots
Jacobian matrix Li ðli ; li Þ while the robotsÕ platform undergoes an infinitesimal motion defined by
its linear complex given by C ¼ ðc; cÞ. The instantaneous work is given by the reciprocal product
of Li with C [28–30]:
jc l þ c lj
ð10Þ
mðL; CÞ ¼
kck
This minimization is equivalent to solving the positive semidefinite quadratic form given by [26]:
k
X
2
F ðX Þ ¼
ðx li þ x li Þ ¼ X T MX
ð11Þ
i¼1
Under the normalization condition 1 ¼ kxk2 ¼ X T DX , where D ¼ diagð1; 1; 1; 0; 0; 0Þ ¼ C, X
presents the set of all linear complexes given by X ¼ ðx; xÞ 2 R6 , and M is the Gramian matrix of
JS . The solution of Eq. (11) is an eigenvalue problem and the result of this algorithm provides the
closest linear complex to the given set of lines. Moreover, the algorithm also provides the direction
of the axis of the linear complex and its pitch. For further information regarding this method see
[26,27]
Two singular configurations of the parallel part of CaPaMan robot were identified in the
previous section. Figs. 7–11 demonstrate the result of the LCAA when rotating the robotsÕ
LCA
60
40
20
0
40
20
y
0
-20
-40
-40
-20
0
20
40
x
Fig. 7. Rotation of the platform in p=2 relative to platformÕs z-axis. Linear complex axis marked as LCA (3D view).
84
A. Wolf et al. / Mechanism and Machine Theory 39 (2004) 75–95
40
30
20
10
y
0
LCA
-10
-20
-30
-40
40
30
20
10
0
-10
-20
-30
-40
x
Fig. 8. Rotation of the platform in p=2 relative to platformÕs z-axis. Linear complex marked as LCA (top view).
70
60
50
40
z
30
20
10
0
-40
-30
-20
-10
0
10
20
30
40
y
Fig. 9. Rotation of the platform in p=2 relative to platformÕs z-axis. JacobianÕs lines in black (side view).
platform in p=2 degrees around the platforms z-axis while the platform is parallel to the base. The
closest linear complexÕs axis found by the algorithm is plotted in Fig. 7 and Fig. 8, as the line
perpendicular to and centered st the moving platform. This line coordinates are given by:
A ¼ ½0 0 1 0 0 0, and has a zero pitch. Hence, the instantaneous screw motion of the platform
while in this singular position is of a zero pitch motion, meaning a pure rotation with respect to A.
A. Wolf et al. / Mechanism and Machine Theory 39 (2004) 75–95
85
70
60
50
40
z 30
20
10
0
40
20
y
0
-20
-40 -40
-30
-20
-10
10
0
20
30
40
x
Fig. 10. Rotation of the platform in p=2 relative to platformÕs z-axis. JacobianÕs lines in black (3D view).
-40
-30
-20
-10
y
0
10
20
30
40
-40
-30
-20
-10
0
10
20
30
40
x
Fig. 11. Rotation of the platform in p=2 relative to platformÕs z-axis. JacobianÕs lines in black (top view).
Observing Figs. 9–11 one can visualize that as expected, the lines of the JS form a flat pencil of
lines with the origin at its vertex.
The second singularity occurs when the platform is rotated by p=2 degrees around its y-axis.
Figs. 12 and 13 present the axis of the closest linear complex as the vertical line, which is coplanar
with the moving platform. Moreover, the line passes at the platform center and its coordinates
are: A ¼ ½0 1 0 5 0 0 0 with a zero pitch. Hence, The instantaneous screw motion of the platform
while in this singular position is a zero pitch motion, meaning a pure rotation with respect to A.
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A. Wolf et al. / Mechanism and Machine Theory 39 (2004) 75–95
70
60
LCA
50
40
y
30
20
10
0
-40
-30
-20
0
-10
x
10
20
30
40
Fig. 12. Rotation of the platform in p=2 relative to platformÕs y-axis. JacobianÕs liner complex marked as LCA (front
view).
-40
-30
-20
-10
y
0
LCA
10
20
30
40
-40
-30
-20
-10
0
10
20
30
40
x
Fig. 13. Rotation of the platform in p=2 relative to platformÕs y-axis. Liner complex marked as LCA (top view).
Observing Figs. 12–16 one can see that as expected, the lines of the Jacobian form a flat pencil of
three parallel and coplanar lines with its vertex at infinity.
A. Wolf et al. / Mechanism and Machine Theory 39 (2004) 75–95
87
40
30
20
10
y
0
-10
-20
-30
-40
40
30
20
10
0
-10
-20
-30
-40
x
Fig. 14. Rotation of the platform in p=2 relative to platformÕs y-axis. JacobianÕs lines in Black (top view).
70
60
50
40
z
30
20
10
0
40
30
20
10
0
-10
-20
-30
-40
x
Fig. 15. Rotation of the platform in p:2 relative to platformÕs y-axis. JacobianÕs lines in black (front view).
4. Static analysis of the CaPaMan2 manipulator
The singularity analysis of CaPaMan2 is conducted similarly. Observing its structure presented
in Figs. 2 and 17, one can detect that the static equilibrium equations of the parallel part of the
robot are the same. This is due to the fact that both the prismatic joint in CaPaMan and the
88
A. Wolf et al. / Mechanism and Machine Theory 39 (2004) 75–95
70
60
50
40
z
30
20
10
0
40
30
20
10
0
-10
-20
-30
-40
y
Fig. 16. Rotation of the platform in p=2 relative to platformÕs y-axis. JacobianÕs lines in black (side view).
f1,i
Fe
Me
r̂i
f 2 ,i
sˆ3,i
Revolut Joint
Spherical Joint
H
sˆ 2,i
sˆ1,i
Fig. 17. Forces transmitted to the moving platform (MP) by link i.
revolute joint in CaPaMan2 can not resist a torque in the ^s1;i ^s2;i direction. Hence Eqs. (1) and
(3) stand for the CaPaMan2 with its interpretations as Pl€
ucker line coordinates. On the other
hand, the analysis of the parallelogram differs. Given f1;i and f2;i as the forces generated on the
A. Wolf et al. / Mechanism and Machine Theory 39 (2004) 75–95
89
Fig. 18. Parallelogram static analysis of CaPaMan2.
parallelogram, they are a function of the torques exerted by the motors and the angle hi , and are
given by: (Fig. 18).
X
f2;i þ R1;i cos hi þ R2;i cos hi fi sin hi ¼ 0
ð12Þ
Fx ¼ 0
X
X
Fz ¼ 0
MQ ¼ 0
f1;i
þ R1;i sin hi þ R2;i sin hi þ fi cos hi ¼ 0
ð13Þ
l2
l2
l2
R1;i sin hi þ fi cos hi R2;i sin hi ¼ 0
2
2
2
ð14Þ
where R1;i ; R2;i are two forces exerted by the two limbs of the ith parallelogram at input and output
cranks, respectively; si is the torque exerted by motor i; fi is a virtual force so that si ¼ fi l1 ;
¼ ðf1;i ^s2;i Þ^zi is the projection of the force along the ith limb to the ^zi direction of ith parallelf1;i
ogram. Hence
¼ ðf1;i ^s2;i Þ ^zi ¼ jf1;i ^s2;i j cosðai Þ
f1;i
ð15Þ
where
ai ¼ a cos
^s2;i ^zi
js2;i jj^zi j
!
ð16Þ
Observing Eqs. (12)–(14), the three unknown parameters are: fi ; R1;i ; R2;i . Solving these equations
for fi yields:
fi ¼ f1;i
cos hi þ f2;i sin hi
Writing si ¼ fi l1 for i ¼ 1; 2; 3 yields:
ð17Þ
90
A. Wolf et al. / Mechanism and Machine Theory 39 (2004) 75–95
3
f1;1
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
2
3 6 f2;1 7
7
6
l1 cos h1 cos ai l1 sin h1
0
0
0
0
6 f1;2 7
7
6
4
5
0
0
0
0
l1 cos h2 cos a2 l1 sin h2
6 f2;2 7
7
6
0
0
0
0
l1 cos h3 cos a3 l1 sin h3 4
f1;3 5
f2;3
2 3
s1
¼ 4 s2 5
s3
JP
2
ð18Þ
Hence for singular analysis of the CaPaMan2 both JS of Eq. (3) and JP of Eq. (18) should be
analyzed. The analysis of JS provides the singularity of the parallel part of the CaPaMan2 known
as the ‘‘parallel singularity’’ and JP provides the singularity due to the parallelogram.
4.1. Singularity analysis of the CaPaMan2
Similar to the singular analysis of CaPaMan manipulator, two singular configurations of the
parallelogram occur when JS becomes singular. These configurations are given by:
1. hi ¼ 0 þ pkðk ¼ 1; . . . ; n; i ¼ 1; . . . ; 3Þ
2. hi ¼ p2 þ pkðk ¼ 1; . . . ; n; i ¼ 1; . . . ; 3Þ
For both cases see Fig. 5a–c.
The CaPaMan2 has an addition singular configuration of JS which does not appear at CaPaMan manipulator. This singularity occurs due to the replacement of the prismatic joints with
revolute ones. Observing JP one can see that whenever aiði¼1;...;3Þ ¼ p2 þ pk, columns 1, 3, 5 are zero
respectively. This means that the parallelogram canÕt transform applied force to the corresponding
motor torque. Fig. 19 depicts such a case.
Matrix JS is investigated next for the analysis of the parallel chain singularities. Observing JS ,
the first three columns define three forces of actuation, acting on the moving platform. These
forces are along the three connecting bars CB of the limbs of robot. In case the three forces are
coplanar and intersect in a common point then they belong to a flat pencil defined by any two of
them, hence, are linearly dependent. This type of singularity occurs whenever the moving platform
falls on top of the plane defined by the three revolt joints (Fig. 20).
Additional singularity occurs whenever all six lines represented by the columns of JS become
linearly dependent. This type of singularity occurs whenever all lines belong to a linear complex.
In this case, five of the six lines intersect one line joining H1 and H2 (denoted as LC in Fig. 21). The
sixth line represented by ½^s3;3 W Rp r3 ^s3;3 T intersects H3 and is parallel to LC hence intersecting it
at infinity. In this case all six reciprocal products of the lines of JS with respect to LC are zero, and
the six lines form a five system. In this singularity, the platform instantaneously gains a pure
rotation about the line joining H1 and H2 , which is the axis of the zero-pitch linear complex. This
kind of singularity is also known as HuntÕs singularity [8]. The last three rows of JS represent three
A. Wolf et al. / Mechanism and Machine Theory 39 (2004) 75–95
91
Fig. 19. Singularity of the parallelogram aiði¼1;...;3Þ ¼ p=2 þ pk.
r1
H1
l1
MP
H2
r2
l2
H2
r1
l3
l1
H1
MP
H2
l2
r2
r3
Fig. 20. Flat pencil singularity of the actuating forces.
moments of constraint, each perpendicular to the plane spanned by ^s1;i and ^s2;i . Rotation of the
platform by p=2 around the platformÕs z-axis while the platform is parallel to the base causes ri to
ri ^s3;i ¼ 0. Meaning that the last three lines in JS (columns 4–6) have
become parallel to ^s3;i hence ^
a zero moment relative to the three axis (their last three coordinates are zero) hence they intersect
at H , forming a flat pencil of lines, with H as its vertex. In this case, JS0 s rank is of order 5. This
kind of singularity is also known as FichterÕs singularity [24]. The instantaneous motion of the
robot while in FichterÕs singularity is a non-zero pitch screw motion with its axis at the platformÕs
z-direction.
4.2. Applying the linear complex approximation to the CaPaMan2
This section presents the results of the LCAA applied to the CaPaMan2. The method used in
this chapter is based on the analytical tools presented at Section 3.2.
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A. Wolf et al. / Mechanism and Machine Theory 39 (2004) 75–95
H1
LC
H1,2
LC
H2
MP
MP
l1
l2
l1,2
H3
r1
r2
H3
l1
l3
r3
r1,2
r3
Fig. 21. HuntÕs singularity of the CaPaMan2.
70
60
50
40
z
30
20
LCA
10
0
-40
40
20
-30
0
-20
-10
0
y
10
-20
20
30
40 -40
x
Fig. 22. HuntÕs singularity. JacobianÕs lines in black, linear complex marked as LCA.
Three singular configurations for the parallel part of CaPaMan2 were identified in the previous
section. The third singular configuration (FichterÕs singularity) was examined in chapter 3.2 (Figs.
7–11). Applying the LCAA, for the second singularity condition where the top platform falls on
top of the base platform (Fig. 20) one can observe that the lines describing JS form a line pencil
(Fig. 23). The linear complex found in this case is zero-pitched with having Pl€
uker coordinate
½0 1 0 0 0 10 joining H1 and H2 . The manipulator gains a DOF, which is an instantaneous pure
rotation about the axis representing the linear complex. In this configuration the rank of JS is of
order of 5.
In the second singularity condition (HuntÕs singularity, see Fig. 22), five of the lines describing JS
intersect the upper edge of the platform, connecting H1 and H2 while the sixth is parallel to it hence
intersecting it at infinity. Applying the LCAA at this configuration, the closest linear complex found
A. Wolf et al. / Mechanism and Machine Theory 39 (2004) 75–95
93
70
LCA
60
50
z
40
30
20
40
10
0
-40
20
0
-30
-20
-10
-20
0
y
10
20
30
40
x
-40
Fig. 23. Flat pencil singularity of the actuating forces. JacobianÕs lines in black, linear complex marked as LCA.
is a zero pitch screw with its axis coordinate are ½0 1 0 a 0 b, connecting H1 and H2 . This means that
the robot gains an instantaneous pure rotation around this axis and the rank of JS is of order 5.
5. Conclusions
Line geometry and linear complex approximation are utilized in this paper to investigate the
behavior of spatial three DOF robots in singular configurations. In order to use these tools the
6 · 6 matrix that maps the external wrenches acting on the moving platform to the forces/moments
at the joints is derived as a set of six lines. Then, by using known linear dependency of lines, the
singular configurations of the robot are identified, and the closest linear complex to these six
governing lines is obtained. The linear complexÕs axis and pitch not only provided information
and understanding on the type and location of the singularities, but also on the nature of the
instantaneous motions of the platform while in these configurations. This investigation reveals
extra singular configurations of the CaPaMan which were not discovered before. Also it was
found that at a p=2 rotation (FichterÕs singularity) both CaPaMan and CaPaMan2 obtain an
uncontrolled rotational motion about a line normal to the moving platform. In CaPaMan2 the
LCAA indicates that while in HuntÕs singularity the robot gains an uncontrolled motion about a
line connecting the two upper edges of the moving platform. This technique can be also applied
while the robot is in the vicinity of singular configuration to determine the uncontrolled direction
and nature of motions the robot possess due to manufacturing tolerances.
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