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Mechanism and Machine Theory xxx (2003) xxx–xxx
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www.elsevier.com/locate/mechmt
Dynamics of parallel manipulators by means of screw theory
J. Gallardo
, J.M. Rico a, A. Frisoli b, D. Checcacci b, M. Bergamasco
b
Departamento de Ingenierıa Mec
anica, Instituto Tecnol
ogico de Celaya, Av. Tecnol
ogico y Garcıa Cubas,
38010 Celaya, Gto., Mexico
b
PERCRO-Scuola Superiore S. Anna, Viale Rinaldo Piaggio 34, 56025 Pontedera, Pisa, Italy
PR
a
a,*
Received 4 March 2002; received in revised form 22 January 2003; accepted 29 January 2003
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An approach to the dynamic analysis of parallel manipulators is presented. The proposed method, based
on the theory of screws and on the principle of virtual work, allows a straightforward calculation of the
actuator forces as a function of the external applied forces and the imposed trajectory. In order to show the
generality of such a methodology, two case studies are developed, a 2-DOF parallel spherical mechanism
and a Gough–Stewart platform.
Ó 2003 Published by Elsevier Science Ltd.
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Abstract
16 1. Introduction
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Over the last two decades parallel manipulators have received increasing attention by kinematicians as demonstrated by the relevant amount of published works on this subject, see for
instance [1–9] among many others.
As it is well known a general parallel manipulator is a mechanism composed of a mobile
platform connected to the ground by several independent kinematic chains, called serial connector
chains. Each serial connector chain can be regarded as a serial manipulator with both actuated
and passive joints, the former providing the actuation to the mobile platform. Despite of a reduced workspace and a more complex solution of the direct kinematic problem than serial manipulators, the higher stiffness, accuracy and payload/weight ratio which can be achieved by
parallel manipulators make them attractive systems for applications ranging from motion simulators to positioning robotic systems. The first suggestion of this class of mechanisms is his-
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15 Keywords: Parallel manipulators; Virtual work; Klein form; Kinematics; Dynamics
*
Corresponding author.
E-mail address: [email protected] (J. Gallardo).
0094-114X/03/$ - see front matter Ó 2003 Published by Elsevier Science Ltd.
doi:10.1016/S0094-114X(03)00054-5
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torically attributed to Gough, who constructed a prototype of a parallel mechanism for measuring
the tire wear and tear under different conditions [2]. Afterwards a similar mechanism was proposed by Stewart, in 1965, as a flight simulator; and nowadays mechanisms that employ the same
architecture of the Gough mechanism are without doubt the most studied type of parallel manipulators, universally known in literature as Gough–Stewart platforms.
The dynamic analysis of parallel manipulators has been traditionally carried out through
several different methods, i.e. the Newton–Euler method, the Lagrange formulation and the
principle of virtual work, which nevertheless present some drawbacks. The Newton–Euler method
usually requires large computation time, since it needs the exact calculation of all the internal
reactions of constraint of the system, even if they are not employed in the control law of the
manipulator. On the other hand, both the Lagrangian and the principle of virtual work formulations are based on the computation of the energy of the whole system with the adoption of a
generalized coordinate framework, whereby the system dynamics equations are expressed. Such
an energy approach to the analysis of parallel manipulator can be further simplified and standardized by means of the theory of screws.
In this work the analysis of parallel manipulators is developed through a novel methodology
based on the theory of screws. The kinematics is approached by extending results previously
obtained by the authors [10,12,13], in the analysis of open serial and closed chains to the kinematics of parallel manipulators. Then the dynamics is approached by an harmonious combination
of screw theory with the principle of virtual work. Finally, in order to show the effectiveness of the
method two applications are given, the first one dealing with a 2-DOF spherical parallel mechanism, and the second one dealing with a Gough–Stewart platform.
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50 2. Preliminary concepts: kinematics of open serial and closed chains
$
¼ ði ^siþ1 ; i~
sOiþ1 Þ;
CO
i iþ1
RR
51
This section summarizes a few results, obtained via screw theory, dealing with the kinematics of
52 open serial and closed chains.
53
Consider two rigid bodies i and i þ 1, that are connected to each other by means of a helicoidal
54 joint described by the normalized screw i $iþ1 . The screw i $iþ1 can be expressed in terms of Pl€
ucker
55 coordinates as:
ð1Þ
i hiþ1 i ^siþ1 þ i ^siþ1 ~
rO=P ;
ð2Þ
UN
57 where i ^
siþ1 is a unit vector defined on the instantaneous rotation axis and i~
sOiþ1 is usually given as a
58 function of the screw pitch i hiþ1 and the vector ~
rO=P , directing from an arbitrary point P of the
59 instantaneous screw axis to the point O:
i iþ1
~
sO
61 where is the usual cross-product of three-dimensional vector algebra.
62
At the beginning of the last century, Ball [14] 1 defined the velocity state of a rigid body, V~O , as a
!
~Þ ¼ x
~, and a dual component, Dð VO Þ ¼ ~
63 screw with a primary component, PðV
vO , so that
1
For a historical overview of early analyses on the screw axis, see Ceccarelli [15].
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~
x
;
V~O ¼
~
vO
3
~ is the angular velocity of the considered rigid body and ~
where x
vO is the velocity of the body
point that is coincident with point O. Note that while the primary component does not depend on
the point O and it can be regarded as a property of the particular motion, the dual part depends
on the choice of the point O. Hereafter the representation point O will not be indicated as a
subscript of the velocity state. Further, the velocity state of a rigid body can be considered as a
twist on a screw, and so the expression (3) can be rewritten as:
OO
F
65
66
67
68
69
70
ð3Þ
V~O ¼ x$:
ð4Þ
PR
~O changes
72 When the representation point is changed from O to P , only the dual part of the twist V
73 as:
vP ¼ DðV~Þ þ PðV~Þ ~
rP =O :
DðV~P Þ ¼ ~
0 ~m
V
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The direct velocity analysis of a serial manipulator consists of determining the velocity of the
end effector when the rates i xiþ1 of the actuated joints are known. It is well known that in a serial
manipulator, see Fig. 1, the velocity state of the end effector, body m, with respect to the base link,
body 0, can be expressed as a linear combination of the screws associated with the serial chain
joints:
¼ 0 x1 0 $1 þ 1 x2 1 $2 þ þ m1 xm m1 $m ;
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75 2.1. Kinematics of an open chain manipulator
ð5Þ
ð6Þ
2
82 where i xiþ1 is the joint rate between two consecutive links.
83
By rearranging Eq. (6) in matrix form, the 6 m Jacobian J of the manipulator is found as:
V
85 where
6
6
¼ J6
4
J ¼ ½ 0 $1
..
.
m1 xm
1 2
$
7
7
7;
5
...
ð7Þ
m1 m
$ :
Conversely, the inverse velocity analysis requires the determination of the joint rates i xiþ1 when
the motion of the end effector with respect to the base link is given. If the manipulator is not
kinematically redundant and it is not at a singular configuration, the Jacobian is represented by an
invertible square matrix, thus:
UN
87
88
89
90
1 x2
3
CO
0 ~m
0 x1
RR
2
2
If the screw i $iþ1 is
associated with a prismatic joint, then the joint rate and its screw are
~
0
i iþ1
;
¼ i viþ1
i xiþ1 $
siþ1
i^
where i ^siþ1 is a unit vector along the direction of the prismatic joint.
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Fig. 1. An open chain serial manipulator.
7
7
.. 7 ¼ J 10 V~m :
. 5
m1 xm
1 x2
ð8Þ
D
6
6
6
4
3
0 x1
PR
2
OO
F
4
ARTICLE IN PRESS
~
AO TE
92 The acceleration analysis can be conducted following Brand [16], who defined the acceleration
93 motor ~
AO of a rigid body, also known as reduced acceleration state or briefly accelerator, in terms
~ ~
~_ , and a dual component, Dð~
94 of a primary component, Pð~
AO Þ ¼ x
aO x
vO :
AO Þ ¼ ~
~_
x
;
~ ~
~
vO
aO x
ð9Þ
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~_ and ~
96 where x
aO are, respectively, the angular acceleration of the rigid body and the linear ac97 celeration of a point O fixed to the rigid body used as a representation point. Further, the ac98 celeration of any other point P of the rigid body, can be easily computed as:
ð10Þ
where ~
rP =O is directed from O to P . Note that while the primary component of the reduced acceleration state is clearly the angular acceleration, the interpretation of the dual part is not so
straightforward. 3 By using concepts of elementary kinematics, it can be easily demonstrated how
the dual part of an accelerator is transformed as a screw for a change of the representation point:
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100
101
102
103
RR
~ ð~
~_ ~
~
aP ¼ ~
aO þ x
x ~
rP =O Þ;
rP =O þ x
~ ~
~ ð~
~ ~
~_ ~
aP x
vP ¼ ~
aO þ x
x ~
rP =O Þ x
vP
Dð~
AP Þ ¼ ~
rP =O þ x
~_ ~
~ ð~
~ ~
~ ~
~_ ~
¼~
aO þ x
rP =O x
vP x
rP =O Þ ¼ ~
aO x
vO þ x
rP =O
UN
¼ Dð~
AO Þ þ Pð~
AO Þ ~
rP =O :
ð11Þ
105 Due to the difficulty of representing the accelerator in screw form, previous works limited the
106 application of screw theory to velocity analysis only, usually called first order analysis [18].
3
The velocity and reduced acceleration states can be treated in a unified form by means of the concept of helicoidal
vector field [17]. In that way, the velocity analysis can be extended not only to the acceleration analysis but also to the
so-called higher order analyses.
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107
Rico et al. [10] showed that the reduced acceleration state of the end effector, body m, of a serial
108 chain can be expressed, in terms of the screws associated to the kinematic pairs of the serial chain,
109 with respect to an inertial reference frame fixed to body 0, as
0~m
m
A ¼ 0 x_ 1 0 $1 þ 1 x_ 2 1 $2 þ þ m1 x_ m1
m $ þ $L ;
ð12Þ
$L ¼ ½ 0 x1 0 $1
1 x2
OO
F
111 where the Lie screw term $L , is
1 2
m
m2 m1
$ þ þ m1 xm1
$
m $ þ þ ½ m2 xm1
m1 m
m1 xm $
ð13Þ
113 and the brackets [ $1 $2 ] represent the Lie product between the elements $1 and $2 of the Lie
114 algebra eð3Þ. By reordering expression (12) in matrix form the direct acceleration analysis can be
115 solved through
6
6
A ¼ J6
4
0~m
3
_1
0x
_2
1x
..
.
_m
m1 x
PR
2
7
7
7 þ $L ;
5
ð14Þ
117 while, under the same conditions of Eq. (8), the inverse acceleration analysis is solved by
_2
1x
..
.
_m
m1 x
3
7
7
Am $L Þ:
7 ¼ J 1 ð0~
5
D
6
6
6
4
_1
0x
TE
2
ð15Þ
EC
119 In what follows the results obtained for an open serial chain are extended to closed chains. It is
120 important to note that Eq. (12), and thus Eqs. (14) and (15) are valid under any value of the
121 angular velocity of the end effector, for details the lector is referred to [8,12,17].
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122 2.2. Kinematics of a single closed chain manipulator
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123
A single closed chain manipulator, see Fig. 2, can be obtained by fixing the end effector, body
~m[0 ¼ 0~
Am[0 ¼ ~
124 m, of a serial manipulator to the base link, body 0, so that 0 V
0. Thus, according
125 with Eqs. (6) and (12), the velocity and acceleration analysis of single closed chains must satisfy:
Fig. 2. A single closed chain parallel manipulator.
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m
~
$ þ 1 x2 1 $2 þ þ m1 xm1
m $ ¼ 0;
_ 1 0 $1 þ 1 x_ 2 1 $2 þ þ m1 x_ m m1 $m þ $L ¼ ~
0:
0x
0 x1
0 1
ð16Þ
i xiþ1
i ¼ 0; . . . ; m 1:
ð17Þ
PR
The generic influence coefficient Gji xiþ1 , expresses the velocity i xiþ1 that results at a passive joint i
for a unitary rate of the jth actuated joint. The computation of the influence coefficients does not
present particular difficulty, since for a parallel manipulator it is always possible to write one or
more closure equations.
The resulting system of the closure equations is linear in the joint velocities and can be rearranged to express the passive joints rates in terms of the active ones. The influence coefficients for
the ith actuated joint can then be computed by solving the system of equations for q_ i ¼ 1 and
q_ j ¼ 0 with j 6¼ i. An example of computation is reported in the examples and for further details
the reader is referred to Gallardo and Rico [11].
So the velocity state of an intermediate body n can be computed as
D
131
132
133
134
135
136
137
138
139
140
¼ G1i xiþ1 q_ 1 þ G2i xiþ1 q_ 2 þ þ GFi xiþ1 q_ F ;
OO
F
127 Further, if the resulting closed chain has F degrees of freedom, with F P 1, then the passive joint
128 rates i xiþ1 can be expressed in terms of first order influence coefficients Gji xiþ1 , j ¼ 1; . . . ; F , and of
129 generalized velocities q_ i , as
0 ~n
TE
V ¼ 0 x1 0 $1 þ þ n1 xn n1 $n
ð18Þ
142 or substituting Eq. (17), in terms of the first order influence coefficients it follows that
V ¼ ðG10 x1 q_ 1 þ þ GF0 x1 q_ F Þ0 $1 þ þ ðG1n1 xn q_ 1 þ þ GFn1 xn q_ F Þn1 $n
n ¼ 1; . . . ; m 1:
EC
0 ~n
ð19Þ
144 If partial screws $in , expressing the contribution to the motion of body n due to the actuated joint i,
145 are defined as:
i ¼ 1; . . . ; F
RR
$in ¼ Gi0 x1 0 $1 þ þ Gin1 xn n1 $n
n ¼ 1; . . . ; m 1;
147 thus expression (19), collecting the generalized velocities q_ i , can be arranged in the following form
0 ~n
V ¼ $1n q_ 1 þ þ $Fn q_ F
154
155
156
157
158
ð20Þ
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Of course, for numerical computations all the screws must be expressed with respect to the same
coordinate system and the same representation point. When the representation point is changed,
e.g. from point O to the center of mass Cm of body n, the expression of partial screws $ij in (20) is
modified according to (5):
Pð$in Þ
i
:
ð21Þ
$Cm n ¼
rCm =O
Pð$ij Þ ~
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149
150
151
152
n ¼ 1; . . . ; m 1:
An , the
Further, after having expressed the state of motion of a generic body n through 0 V~n and 0~
0 n
0 n
aP of any arbitrary point of body n, like its center of mass Cm ,
velocity ~
vP and the acceleration ~
can be computed respectively by (5) and (10).
Finally, taking into proper account the enumeration of the closed chains, and their links, the
results obtained for a single closed chains can be extended to the analysis of parallel manipulators.
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159 3. Kinematics of parallel manipulators
"
0 ~pl
VO
¼
OO
F
160
The results obtained in Section 2 for open serial and closed chains can be applied to solve
161 kinematics of parallel manipulators.
162
Consider a parallel manipulator composed of k serial connector chains, as shown in Fig. 3, and
~0pl with respect to an inertial
163 suppose that the mobile platform has an instantaneous motion 0 V
164 reference frame OXYZ , fixed to the base link, body 0, given by
#
~pl
x
0 pl :
~
vO
0
ð22Þ
..
.
ðiÞ
m1 xm
3
7
7
1
7 ¼ ðJ ðiÞ Þ 0 V~pl
5
i ¼ 1; . . . ; k;
169 where
1 2ðiÞ
$
m1 mðiÞ
...
$
:
EC
J ðiÞ ¼ ½ 0 $1ðiÞ
D
6
6
6
4
ðiÞ
0 x1
ðiÞ
1 x2
ð23Þ
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2
PR
166 After expressing all the screws with respect to the point O, the inverse velocity analysis can be
167 individually solved for each serial chain through expression (8):
UN
CO
RR
171 Here the superscript ðiÞ denotes the ith connector chain.
Apl with respect to the
172
Suppose now that the mobile platform has a reduced acceleration state 0~
173 inertial reference frame OXYZ . Then, the inverse acceleration analysis is solved by simple appli174 cation of expression (15) to each connector chain of the parallel manipulator:
Fig. 3. A parallel manipulator.
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6
6
6
6
4
_ 1ðiÞ
0x
_ 2ðiÞ
1x
..
.
_ ðiÞ
m1 x
m
3
7
7
ðiÞ
7 ¼ ðJ ðiÞ Þ1 ð0~
Apl $L Þ i ¼ 1; . . . ; k:
7
5
ð24Þ
OO
F
2
176 The general expressions (23) and (24) are the basis of inverse velocity and acceleration analysis of
177 parallel manipulators. Given a velocity and reduced acceleration state of the moving platform, all
178 the serial connector chains must satisfy Eqs. (23) and (24).
PR
179 4. Dynamics of parallel mechanisms
D
180
Recently the inverse dynamics of a Gough–Stewart platform has been significantly simplified
181 by means of the principle of virtual work [7,9]. In this section it is shown how this approach can be
182 further simplified by applying the Klein form, a bilinear symmetric form of the Lie algebra.
183
Assume that $1 ¼ ð^s1 ;~
sO1 Þ and $2 ¼ ð^s2 ;~
sO2 Þ are two elements of the Lie algebra, eð3Þ. The Klein
184 form KLð$1 ; $2 Þ, a non-degenerate symmetric bilinear form, is defined as
TE
KL : eð3Þ eð3Þ ! R KLð$1 ; $2 Þ ^s1 ~
sO2 þ ^s2 ~
sO1 ;
ð25Þ
186 where denotes the usual dot product of three-dimensional real vector algebra.
187
The wrench ~
FO acting on a rigid body is defined as a screw with a primary component,
~
~
188 PðFO Þ ¼ f , and a dual component, Dð~
FO Þ ¼ ~
sO , as follows
~
f ;
~
sO
ð26Þ
RR
where ~
f and ~
sO respectively are the force vector and the torque vector expressed by using O as
representation point.
AnCm , describe the motion of
Suppose that the velocity and reduced acceleration state, 0 V~Cnm and 0~
a rigid body n of mass m, adopting its mass center Cm as representation point.
n
By assuming the same representation point Cm , the inertial wrench, ~
FI;C
, acting on body n
m
according with DÕAlembertÕs principle is given by:
m~
aCm
n
~
;
ð27Þ
¼
FI;C
m
~ In x
In x_ x
CO
190
191
192
193
194
195
EC
~
FO ¼
UN
197 where In ¼ In0 is the centroidal body inertia matrix expressed in the reference frame OXYZ fixed to
198 the base link. This matrix can be computed by transforming the local inertial matrix In through
199 the rotation matrix 0 Rn , as
In ¼ 0 Rn Inn ð0 Rn ÞT :
ð28Þ
201 Suppose that body n is subject to the effect of a gravity field, then the resulting gravity wrench,
n
FG;C
202 ~
, is given by
m
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"
n
~
FG;C
¼
m
9
#
m~
g
;
~
0
ð29Þ
OO
F
204 where ~
g is the gravity acceleration vector. Moreover if an external force, ~
fEn , and an external
n
!
n
205 torque, sE , are applied to the mass center of the considered rigid body, the external wrench ~
FE;C
m
206 is given by:
"
n
~
FE;C
m
ð30Þ
In the computation of the overall system dynamics, however, the same coordinate system OXYZ
and the same representation point Cm , must be used for the screw and wrench of every link. So Eq.
(5) has to be applied in order to express the above written wrenches with respect to a congruent
representation pole Cm . Finally, the overall wrench ~
F n acting on body n is then
PR
208
209
210
211
#
~
fEn
¼ n :
~
sE
n
n
n
~
Fn ¼ ~
FI;C
þ~
FG;C
þ~
FE;C
:
m
m
m
ð31Þ
D
213 With these conventions the power wn , produced by the resulting wrench ~
F n acting on body n on a
0 ~n
214 generic motion VCm , can be determined with the aid of the Klein form, Eq. (25), as follows
wn ¼ KLð~
F n ; 0 V~Cnm Þ:
ð32Þ
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216 If the body n belongs to a closed chain, with m 1 (intermediate) bodies, by substituting Eq. (20)
217 into Eq. (32), the total power w performed on the mechanism by the external forces and actuated
218 joints acting on all links is found as:
EC
w ¼ KLð~
F 1 ; $1Cm 1 q_ 1 þ þ $FCm 1 q_ F Þ þ KLð~
F 2 ; $1Cm 2 q_ 1 þ þ $FCm 2 q_ F Þ þ þ KLð~
F m1 ; $1Cm m1 q_ 1 þ þ $FCm m1 q_ F Þ þ s1 q_ 1 þ þ sF q_ F
ð33Þ
RR
220 where si is the generalized force actuating on the ith joint with the generalized velocity q_ i . So
221 collecting the generalized velocities q_ i :
F 2 ; $1Cm 2 Þ þ þ KLð~
F m1 ; $1Cm m1 Þ þ s1 q_ 1 þ w ¼ ½KLð~
F 1 ; $1Cm 1 Þ þ KLð~
þ ½KLð~
F 1 ; $FCm 1 Þ þ KLð~
F 2 ; $FCm 2 Þ þ þ KLð~
F m1 ; $FCm m1 Þ þ sF q_ F :
CO
The principle of virtual work states that if a multi-body system is in equilibrium under the effect of
external actions, then the global work produced by the external forces with any virtual velocity
must be zero, for details the reader is referred to Tsai [8]. Introducing the generalized virtual
velocities dq_ i , and taking into account that the work due to the internal reactions of constraint is
zero, the global virtual work dw, from (34), is given by:
UN
223
224
225
226
227
ð34Þ
F 2 ; $1Cm 2 Þ þ þ KLð~
F m1 ; $1Cm m1 Þ þ s1 dq_ 1 þ dw ¼ ½KLð~
F 1 ; $1Cm 1 Þ þ KLð~
þ ½KLð~
F 1 ; $FCm 1 Þ þ KLð~
F 2 ; $FCm 2 Þ þ þ KLð~
F m1 ; $FCm m1 Þ þ sF dq_ F :
ð35Þ
229 Further, since the generalized virtual velocities dq_ i are arbitrary, dw ¼ 0 if and only if:
F 2 ; $iCm 2 Þ þ þ KLð~
F m1 ; $iCm m1 Þ þ si ¼ 0
KLð~
F 1 ; $iCm 1 Þ þ KLð~
i ¼ 1; . . . ; F
ð36Þ
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231 and so
f 1 ~
siCm 1 þ~
s1Cm ~
si1 þ ~
f 2 ~
siCm 2 þ~
s2Cm ~
si2 þ ~
f m1 ~
siCm m1 þ~
sm1
sim1 si ¼ ½~
Cm ~
i ¼ 1; . . . ; F :
ð37Þ
Finally, from expression (37) the generalized forces si can be computed directly. It is important to
note that from the above solution, si are independent from the virtual velocities dq_ i , but they
obviously still depend on the real generalized velocities q_ i , contained in the inertial terms of (37),
through the Lie bracket expression contained in the computation of the links accelerations.
OO
F
233
234
235
236
CO
RR
EC
TE
D
This section shows a first application of the method to a 2-DOF spatial parallel mechanism
used as a haptic interface to simulate a virtual car gearshift. The calculation of the system dynamics will be approached by solving the direct kinematic equation of the system. The kinematic
representation of the spherical mechanism is shown in Fig. 4 together with the ground coordinate
system OXYZ and the joint axes.
The differential kinematic of the mechanism is easily derived using screw algebra. Since the
system is spherical, the fixed point O is conveniently chosen as the representation point.
Taking into account that all the axes of the screws are concurrent and noting that the axes of
screws 1 $2ð2Þ and 1 $2ð1Þ are always orthogonal to the knob OP, then in Pl€
ucker coordinates the
infinitesimal screws are given by
(
0 1ð1Þ
$ ¼ ð1; 0; 0; ~
0Þ 1 $2ð1Þ ¼ ð0; cosðq1 Þ; sinðq1 Þ; ~
0Þ 2 $3ð1Þ ¼ ð2 ^s3ð1Þ ; ~
0Þ;
0 1ð2Þ
1 2ð2Þ
~
~
$ ¼ ð0; 1; 0; 0Þ $ ¼ ðcosðq2 Þ; 0; sinðq2 Þ; 0Þ;
UN
238
239
240
241
242
243
244
245
246
247
PR
237 5. Example 1, 2-DOF spherical mechanism
Fig. 4. The gearshift mechanism and its scheme.
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11
249 where
(
2 3ð1Þ
^s
¼ 1a ð1 ^s2ð2Þ 1 ^s2ð1Þ Þ ¼ 1a ðcosðq1 Þ sinðq2 Þ; sinðq1 Þ cosðq2 Þ; cosðq1 Þ cosðq2 ÞÞ;
a ¼ k1 ^s2ð2Þ 1 ^s2ð1Þ k;
ð38Þ
q1 ¼ arctanðPY =PZ Þ
OO
F
251 where, given a position of the free end of the knob by ~
P ¼ ðPX ; PY ; PZ Þ, the generalized coordinates
252 q1 and q2 can be obtained as
q2 ¼ arctanðPX =PZ Þ:
ð39Þ
~3ð1Þ ¼ ð0 x
~3ð1Þ ; ~
254 Assume that a velocity state of the knob with respect to the base link, 0 V
0Þ, is re255 quired. Then, following the two connector chains, it is possible to write
V
¼ q_ 2 0 $1ð2Þ þ 1 x2ð2Þ 1 $2ð2Þ ¼ q_ 1 0 $1ð1Þ þ 1 x2ð1Þ 1 $2ð1Þ þ 2 x3ð1Þ 2 $3ð1Þ :
PR
0 ~3ð1Þ
ð40Þ
257 Thus, after cancelling the dual parts of the infinitesimal screws, the joint rates can be computed.
258 Indeed
D
2
3
8
q_ 2
>
>
>
>
~3ð1Þ ;
½ Pð0 $1ð2Þ Þ Pð1 $2ð2Þ Þ ~
0 4 1 x2ð2Þ 5 ¼ 0 x
>
>
<
0
2
3
_
q
>
1
>
>
> 4 1 x2ð1Þ 5 ¼ ½ Pð0 $1ð1Þ Þ Pð1 $2ð1Þ Þ Pð2 $3ð1Þ Þ 1
>
>
:
2 x3ð1Þ
~3ð1Þ :
x
TE
0
ð41Þ
260 Further, from expression (40) it follows that
½ 1 $2ð1Þ
2 3ð1Þ
$
3
EC
2
1 x2ð1Þ
1 $2ð2Þ 4 2 x3ð1Þ 5 ¼ q_ 1 0 $1ð1Þ þ q_ 2 0 $1ð2Þ :
1 x2ð2Þ
ð42Þ
266 Thus,
¼
j q_ 1 0 ^s1ð1Þ þ q_ 2 0 ^s1ð2Þ
j1 ^s2ð1Þ 2 ^s3ð1Þ
CO
1 x2ð1Þ
RR
262 Afterwards, the influence coefficients are calculated by simple application of CramerÕs rule. Only
263 the influence coefficients associated to the joint rate 1 x2ð1Þ will be presented here.
264
From expression (42) immediately emerges that:
^s
1 2ð2Þ
^s j
1 2ð2Þ
^s
j
:
¼ G11 x2ð1Þ q_ 1 þ G21 x2ð1Þ q_ 2 ;
UN
1 x2ð1Þ
2 3ð1Þ
ð43Þ
ð44Þ
268 where the first order coefficients are given by
j0 ^s1ð1Þ 2 ^s3ð1Þ 1 ^s2ð2Þ j sinðq1 Þ sinðq2 Þ cosðq2 Þ
¼
;
a2
j1 ^s2ð1Þ 2 ^s3ð1Þ 1 ^s2ð2Þ j
j0 ^s1ð2Þ 2 ^s3ð1Þ 1 ^s2ð2Þ j cosðq1 Þ
¼ 1 2ð1Þ 2 3ð1Þ 1 2ð2Þ ¼
:
a2
^s
^s j
j ^s
G11 x2ð1Þ ¼ G21 x2ð1Þ
ð45Þ
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270 Similarly by application of CramerÕs rule it is found:
j1 ^s2ð2Þ 0 ^s1ð1Þ 1 ^s2ð2Þ j cosðq1 Þ sinðq2 Þ
;
¼
a
j1 ^s2ð1Þ 2 ^s3ð1Þ 1 ^s2ð2Þ j
j1 ^s2ð2Þ 0 ^s1ð2Þ 1 ^s2ð2Þ j sinðq1 Þ cosðq2 Þ
¼ 1 2ð1Þ 2 3ð1Þ 1 2ð2Þ ¼
:
a
^s
^s j
j ^s
G12 x3ð1Þ ¼ ð46Þ
OO
F
G22 x3ð1Þ
272 Thus, the partial screws associated to the knob (body 3 of chain 1), in terms of influence coeffi273 cients are given by
$13ð1Þ ¼ 0 $1ð1Þ þ G11 x2ð1Þ 1 $2ð1Þ þ G12 x3ð1Þ 2 $3ð1Þ ;
PR
$23ð1Þ ¼ G21 x2ð1Þ 1 $2ð1Þ þ G22 x3ð1Þ 2 $3ð1Þ :
275 Finally, the corresponding partial screws of the knob relative to its mass center, Cm3ð1Þ , expressed
276 in the global reference frame OXYZ , result in
#
i ¼ 1; 2;
ð47Þ
TE
here ~
rCm3ð1Þ =O is the vector pointed from the common intersection of all the revolute axes, point O,
to the mass center of the knob.
It is straightforward to demonstrate that the partial screws $iCmjðkÞ of the remaining bodies are
obtained in a similar way.
Assume that a reduced acceleration state of the knob with respect to the base link,
0~3ð1Þ
~_ 3ð1Þ ; ~
A ¼ ð0 x
0Þ, is required. Then, according with the two connector chains, it is possible to
write
0~3ð1Þ
A
EC
278
279
280
281
282
283
284
¼
Pð$i3ð1Þ Þ
Pð$i3ð1Þ Þ ~
rCm3ð1Þ =O
D
"
$iCm3ð1Þ
¼€
q2 0 $1ð2Þ þ 1 x_ 2ð2Þ 1 $2ð2Þ þ $L1 ¼ €
q1 0 $1ð1Þ þ 1 x_ 2ð1Þ 1 $2ð1Þ þ 2 x_ 3ð1Þ 2 $3ð1Þ þ $L2 ;
$L1 ¼ ½ q_ 2 0 $1ð2Þ
$L2 ¼ ½ q_ 1 0 $1ð1Þ
1 2ð2Þ
$ ;
1 2ð1Þ
x
þ 2 x3ð1Þ 2 $3ð1Þ þ ½ 1 x2ð1Þ 1 $2ð1Þ
1 2ð1Þ $
1 x2ð2Þ
2 x3ð1Þ
2 3ð1Þ
$
:
ð49Þ
CO
(
RR
286 where
ð48Þ
288 After cancelling the dual part of all the six dimensional vectors, the joint acceleration rates can be
289 computed from expression (48). Indeed
UN
8
2
3
>
€
q2
>
>
>
>
~_ 3ð1Þ Pð$L1 ÞÞ;
½ Pð0 $1ð2Þ Þ Pð1 $2ð2Þ Þ ~
0 4 1 x_ 2ð2Þ 5 ¼ ð0 x
>
>
<
0
2
3
€
q
>
>
>4 1 5
>
_
>
x
~_ 3ð1Þ Pð$L2 ÞÞ:
¼ ½ Pð0 $1ð1Þ Þ Pð1 $2ð1Þ Þ Pð2 $3ð1Þ Þ 1 ð0 x
1 2ð1Þ
>
>
: 2 x_ 3ð1Þ
ð50Þ
291 Once the inverse analyses are completed, the velocity and the acceleration of the mass center of the
292 knob are computed by application of elementary kinematics.
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13
Consider now a frame Oi;j
xyz attached to each link, used to locally describe its inertial properties.
If we consider local frames whose origin Oi;j is coincident with that of the base coordinate system
O0 the transformation matrices reduce to rotation matrices. Defining the frame Oi;j
xyz of the link i; j
according to the Denavit–Hartenberg notation, we can quickly build the rotation matrix R0i that
transforms local properties in the ground frame O0 .
If rB ij =O is the vector, in local coordinates, directing from the origin O to the center of gravity Bi;j
of the link-i; j, in the ground frame it becomes rB0 ij =O ¼ R0i;j rB ij =O .
300
The inertia tensor in ground frame is expressed by:
OO
F
293
294
295
296
297
298
299
I0 ¼ R0i I0 ðR0i Þt :
ð51Þ
PR
302
Consider now a generic link, and express all the forces acting on it with respect to ground
303 frame. As the system is characterized by a spherical kinematics, only the momentum part of each
304 applied wrench must be considered for calculating the produced virtual work. The gravity force
305 acting on each link produces the momentum:
~g;iðjÞ ¼ R0 r
g:
M
i;j BiðjÞ =O miðjÞ~
ð52Þ
ð53Þ
TE
~IiðjÞ ¼ I0 x
~LiðjÞ I0 x
~LiðjÞ :
~_ LiðjÞ x
M
D
~Li;j and x
~_ Li;j are the i-link angular velocity and acceleration of serial connector chain j, then the
307 If x
308 momentum of the inertial forces is:
~e the external torque, applied to the handle by an external force ~
310 Indicating with M
Fe , we can now
311 compute all the torques acting on the mechanism. Therefore, the wrenches are given by
RR
EC
"
#
"
#
8
>
~
~
m
m
g
g
1ð1Þ
2ð1Þ
>
1ð1Þ
>
F 2ð1Þ ¼ ~
¼ ~
>
<F
~I1ð1Þ
~I2ð1Þ ;
Mg;1ð1Þ þ M
Mg;2ð1Þ þ M
"
#
"
#
>
~
~
m
g
~
g
þ
F
m
>
1ð2Þ
3ð1Þ
e
3ð1Þ
1ð2Þ
>F
>
F
¼
¼ ~
:
~I1ð2Þ :
~g;2ð1Þ þ M
~I2ð1Þ þ M
~e
Mg;1ð2Þ þ M
M
313 Finally, the application of expression (37) leads to
KLðF 1ð1Þ ; $1Cm 1ð1Þ Þ þ KLðF 2ð1Þ ; $1Cm 2ð1Þ Þ þ KLðF 3ð1Þ ; $1Cm 3ð1Þ Þ þ KLðF 1ð2Þ ; $1Cm 1ð2Þ Þ þ s1 ¼ 0;
KLðF 1ð1Þ ; $2Cm 1ð1Þ Þ þ KLðF 2ð1Þ ; $2Cm 2ð1Þ Þ þ KLðF 3ð1Þ ; $2Cm 3ð1Þ Þ þ KLðF 1ð2Þ ; $2Cm 1ð2Þ Þ þ s2 ¼ 0:
ð54Þ
CO
(
315 From this last expression, the generalized forces s1 and s2 can be computed directly:
UN
8
>
>
~g;1ð1Þ þ M
~I1ð1Þ Þ ~
>
s1 ¼ ½m1ð1Þ~
g ~
s1Cm 1ð1Þ þ ðM
g ~
s1Cm 2ð1Þ
s11ð1Þ þ m2ð1Þ~
>
>
>
>
~g;2ð1Þ þ M
~I2ð1Þ Þ ~
>
gþ~
Fe Þ ~
þðM
s12ð1Þ þ ðm3ð1Þ~
s1Cm 3ð1Þ
>
>
>
< þðM
~g;2ð1Þ þ M
~I2ð1Þ þ M
~e Þ ~
~g;1ð2Þ þ M
~I1ð2Þ Þ ~
g ~
s1Cm 1ð2Þ þ ðM
s13ð1Þ þ m1ð2Þ~
s11ð2Þ ~g;1ð1Þ þ M
~I1ð1Þ Þ ~
>
s2 ¼ ½m1ð1Þ~
g ~
s2Cm 1ð1Þ þ ðM
g ~
s2Cm 2ð1Þ
s21ð1Þ þ m2ð1Þ~
>
>
>
> þðM
~g;2ð1Þ þ M
~I2ð1Þ Þ ~
>
s22ð1Þ þ ðm3ð1Þ~
s2Cm 3ð1Þ
gþ~
Fe Þ ~
>
>
>
> þðM
~g;2ð1Þ þ M
~I2ð1Þ þ M
~e Þ ~
~g;1ð2Þ þ M
~I1ð2Þ Þ ~
>
g ~
s2
s2 þ m1ð2Þ~
þ ðM
s2 :
3ð1Þ
Cm 1ð2Þ
1ð2Þ
ð55Þ
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317 6. Example 2, Gough–Stewart manipulator
This section presents an application of the presented method to the analysis of a Gough–
Stewart manipulator Fig. 5.
The mobile platform is connected to the base link by means of six serial chains. Each serial
connector chain is composed of a lower spherical joint, a prismatic actuated joint and an upper
universal joint. The prismatic actuated joints provide six degrees of freedom to the mobile platform, so that it can assume an arbitrary pose. The actuators lengths are defined by 12 points,
namely U ðiÞ and S ðiÞ , respectively fixed to the mobile platform and to the base link. At once, each
serial connector chain is composed of a lower link, lðiÞ , and a upper link, uðiÞ .
OO
F
318
319
320
321
322
323
324
325
D
With respect to the direct kinematics problem, the inverse position analysis of parallel manipulators is quite simple and can be solved by using homogeneous transformation matrices, T44 .
Consider two reference frames, the first one, oxyz , attached to the mobile platform, and the
second one, OXYZ , attached to the base link. The coordinates of a point fixed to the mobile
platform, expressed in the reference frame oxyz , can be expressed in the reference frame OXYZ as
follows
2 3
2 3
x
X
6 Y 7 0 pl 6 y 7
6 7 ¼ T 6 7;
ð56Þ
4z5
4Z5
1
1
EC
TE
327
328
329
330
331
332
PR
326 6.1. Position analysis
334 where the transformation matrix 0 T pl is given by the rotation matrix R and the translation vector
335 ~
ro=O , that points to the origin of the reference system oxyz from the origin of the reference system
336 OXYZ :
R
013
~
ro=O
:
1
RR
T pl ¼
UN
CO
0
Fig. 5. A Gough–Stewart manipulator.
ð57Þ
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ðiÞ
ðiÞ
15
ðiÞ
338 Thus the instantaneous position of point U ðiÞ ¼ ðUX ; UY ; UZ Þ, on the mobile platform, with
339 respect to the reference system OXYZ is given by
2 ðiÞ 3
ðiÞ 3
Ux
UX
6 U ðiÞ 7 0 pl 6 UyðiÞ 7
7
6 Y 7¼ T 6
4 U ðiÞ 5
4 U ðiÞ 5 i ¼ 1; . . . ; 6:
z
Z
1
1
2
OO
F
ð58Þ
341 Once the inverse position analysis is solved, the length of each actuator, connecting the mobile
342 platform to the base link can be determined as the distance between points U ðiÞ and S ðiÞ , where
343 i ¼ 1; . . . ; 6.
PR
344 6.2. Velocity analysis
V
¼ 0 V~mðiÞ ¼
3
7
7
7
7
7 i ¼ 1; 2; . . . ; 6;
q_ 7
6 4 ðiÞ 7
4 x5 5
5
ðiÞ
x6
J ðiÞ ¼ ½ 0 $1ðiÞ
1 2ðiÞ
$
2 3ðiÞ
$
3 4ðiÞ
$
4 5ðiÞ
$
5 6ðiÞ
$
:
CO
Note that the screw sets f0 $1ðiÞ ; 1 $2ðiÞ ; 2 $3ðiÞ g and f4 $5ðiÞ ; 5 $6ðiÞ g represent, respectively, the lower
spherical pair S ðiÞ and the upper universal joint U ðiÞ , while the screw 3 $4ðiÞ is associated to the
actuator joint.
ðiÞ
Introduce a unit line, $ðiÞ ¼ ð^sðiÞ ;~
sO Þ, whose direction is given by Pð3 $4ðiÞ Þ and passes through
ðiÞ
ðiÞ
U and S . It is evident that this line is reciprocal to all the screws representing the revolute pairs
of the ith serial connector chain. Thus, applying the Klein form to both sides of expression (59) it
follows that KLð$ðiÞ ; j $jþ1 Þ ¼ 0 for j ¼ 0; 1; 2; 4; 5 and therefore
UN
353
354
355
356
357
358
359
RR
351 where
ð59Þ
EC
0 ~pl
ðiÞ
0 x1
6 xðiÞ
61 2
6 2 ðiÞ
6
J ðiÞ 6 xðiÞ3
6
TE
2
D
345
The computation of the generalized velocities q_ i , a necessary step in the computation of the
346 influence coefficients, can be shortened by applying the properties of the Klein form, for details
347 the reader is referred to Rico and Duffy [5].
348
Assume that the platform has a velocity state, with respect to the base link, 0 V~pl , then according
349 with expression (7), it is possible to write
T
q_ i ¼ KLð$ðiÞ ; 0 V~pl Þ ¼ $ðiÞ D0 V~pl
i ¼ 1; 2; . . . ; 6;
361 where
0
D ¼ 33
I3
I3
033
¼ D1 :
363 Thus, it is possible to write in abbreviated form
ð60Þ
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2
3
q_ 1
6 q_ 2 7
6 7
ðJqT DÞ0 V~pl ¼ 6 . 7;
4 .. 5
ð61Þ
q_ 6
Jq ¼ ½ $ð1Þ
$ð2Þ
$ð3Þ
$ð4Þ
$ð5Þ
OO
F
365 where
$ð6Þ :
367 With expression (61) the generalized velocities can be quickly computed. Thus, the partial screws
j
368 of the mobile platform, $pl , are given by
PR
2 3
0
6 .. 7
6.7
1 6 7
$jpl ¼ ðJqT DÞ 6 1 7 j ¼ 1; 2; . . . ; 6:
6.7
4 .. 5
0
ð62Þ
3
3
q_ 1
7 6 7
7 6 q_ 2 7
7 6 7
7 6 q_ 3 7
7 ¼ 6 7 i ¼ 1; 2; . . . ; 6:
7 6 q_ 4 7
7 4 5
q_ 5
5
q_ 6
2
TE
ðiÞ
0 x1
6 xðiÞ
61 2
6 ðiÞ
T
ðiÞ 6 2 x3
Jq DJ 6
6 q_ i
6 ðiÞ
4 4 x5
ðiÞ
5 x6
EC
2
D
370 Further, the substitution of expression (59) into expression (61) leads to
ð63Þ
372 Expression (63) allows to determine, via CramerÕs rule, the influence coefficients, Gji xiþ1 for
373 j ¼ 1; . . . ; F , associated to each one of the joint rates. Thus, the joint rates can be computed as
ðjÞ
i xiþ1
i
iþ1
RR
¼ G1xðjÞ q_ 1 þ G2xðjÞ q_ 2 þ þ G6xðjÞ q_ 6
i
i
iþ1
i ¼ 0; . . . ; 5
j ¼ 1; 2; . . . ; 6:
ð64Þ
iþ1
CO
375 In what follows, the partial screws of the lower links will be computed. The velocity state of the
376 jth lower link, see expression (19), is given by
0 ~3ðjÞ
V
¼ ðG1xðjÞ q_ 1 þ þ G6xðjÞ q_ 6 Þ0 $1ðjÞ þ ðG1xðjÞ q_ 1 þ þ G6xðjÞ q_ 6 Þ1 $2ðjÞ
0
þ
ðG1xðjÞ q_ 1
2 3
þ þ
0
1
1
G6xðjÞ q_ 6 Þ2 $3ðjÞ
2 3
2
j ¼ 1; 2; . . . ; 6
UN
378 Thus
1
0 ~3ðjÞ
V
¼ $1lðjÞ q_ 1 þ $2lðjÞ q_ 2 þ $3lðjÞ q_ 3 þ $4lðjÞ q_ 4 þ $5lðjÞ q_ 5 þ $6lðjÞ q_ 6 ;
1
2
ð65Þ
ð66Þ
380 where the partial screws are given by
$ilðiÞ ¼ Gi xðiÞ 0 $1ðiÞ þ Gi xðiÞ 1 $2ðiÞ þ Gi xðiÞ 2 $3ðiÞ
0
1
1
2
2
3
i ¼ 1; 2; . . . ; 6:
ð67Þ
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17
382 Finally, the partial screws relative to the mass center of the ith lower link result in
"
$ilCm ðiÞ ¼
Pð$ilðiÞ Þ
i
rlðiÞ
Dð$lðiÞ Þ þ Pð$ilðiÞ Þ ~
#
i ¼ 1; 2; . . . ; 6;
ð68Þ
OO
F
384 where ~
rlðiÞ is a vector pointed from S ðiÞ to the mass center of the lower link. It is straightforward to
385 demonstrate that the partial screws of the upper links can be determined analogously.
386 6.3. Acceleration analysis
387
Given a reduced acceleration state of the platform with respect to the base link, ~
Apl , by applying
388 the expression (24), the inverse acceleration analysis of each connector chain is determined by:
7
7
7
7
7
ðiÞ
1
Apl $L Þ
7 ¼ ðJ ðiÞ Þ ð~
7
6 €
q
6 iðiÞ 7
6 4 x_ 5 7
4 ðiÞ 5
_6
5x
i ¼ 1; 2; . . . ; 0;
390 where
ðiÞ
0 1ðiÞ
$L ¼ ½ 0 xðiÞ
1 $
ðiÞ 1 2ðiÞ
1 x2 $
PR
3
ð69Þ
D
ðiÞ
6 0 x_ 1
6 x_ ðiÞ
61 2
6 ðiÞ
6 2 x_ 3
6
ðiÞ
TE
2
ðiÞ
ðiÞ
þ 2 x3 2 $3ðiÞ þ q_ i 3 $4ðiÞ þ 4 x5 4 $5ðiÞ þ 5 x6 5 $6ðiÞ 1 2ðiÞ
þ ½ 1 xðiÞ
2 $
ðiÞ 2 3ðiÞ
2 x3 $
ðiÞ
ðiÞ
2 3ðiÞ
þ ½ 2 xðiÞ
3 $
q_ i 3 $4ðiÞ þ 4 x5 4 $5ðiÞ þ 5 x6 5 $6ðiÞ þ ½ q_ i 3 $4ðiÞ
4 5ðiÞ
þ ½ 4 xðiÞ
5 $
ðiÞ 5 6ðiÞ
5 x6 $
EC
þ q_ i 3 $4ðiÞ þ 4 x5 4 $5ðiÞ þ 5 x6 5 $6ðiÞ ðiÞ
ðiÞ
ðiÞ 4 5ðiÞ
4 x5 $
ðiÞ
þ 5 x6 5 $6ðiÞ ð70Þ
RR
;
392 that is a homogeneous linear system with six equations in the six unknowns,
ðiÞ
ðiÞ
ðiÞ
ðiÞ
ðiÞ
f0 x_ 1 ; 1 x_ 2 ; 2 x_ 3 ; €
qi ; 4 x_ 5 ; 5 x_ 6 g:
CO
394 When the velocity and acceleration analyses are completed, the velocity and the acceleration of
395 the mass center of each body are computed by application of elementary kinematics.
396 6.4. Computation of generalized forces
UN
397
Supposing that an external force ~
fE and torque ~
sE are applied to the mass center of the mobile
398 platform, then the wrench resulting, see expression (31), on the platform is given by
pl
#
pl
~
~
ð~
g
a
Þ
þ
f
m
Cm
E
~
FCplm ¼ pl plpl _
:
~pl I0pl x
~pl
~pl x
~
sE I0 x
pl
"
ð71Þ
400 Further, supposing, as a practical assumption, that there are no external forces applied to the
401 chain bodies, then the wrench resulting on the lower links of the manipulator are given by
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J. Gallardo et al. / Mechanism and Machine Theory xxx (2003) xxx–xxx
"
~
F lðiÞ ¼
ðiÞ
g ~
al Þ
mlðiÞ ð~
ðiÞ _ ðiÞ
ðiÞ
~3 I0pl 0 x
~ðiÞ
~3 0 x
I0 0 x
3
403 With regards for the upper links
"
~uðiÞ
F
¼
g ~
aðiÞ
muðiÞ ð~
u Þ
ðiÞ _ ðiÞ
ðiÞ
~
~ðiÞ
~
I0 0 x3 0 x3 I0pl 0 x
3
#
i ¼ 1; 2; . . . ; 6:
ð72Þ
i ¼ 1; 2; . . . ; 6;
ð73Þ
#
ðiÞ
OO
F
18
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SPS, Chennai
ARTICLE IN PRESS
405 where ~
al and ~
aðiÞ
u are the accelerations of the mass center of the lower and upper links respectively.
406
Finally, applying expression (37) it follows that
KLð~
F uð1Þ ; $iuCm ð1Þ Þ þ þ KLð~
F uð6Þ ; $iuCm ð6Þ Þ þ KLð~
F lð1Þ ; $ilCm ð1Þ Þ
i ¼ 1; . . . ; 6:
PR
FCplm ; $ipl Þ þ si ¼ 0
þ þ KLð~
F lð6Þ ; $ilCm ð6Þ Þ þ KLð~
ð74Þ
408 Thus, the generalized forces si can be computed directly from expression (74).
417 Acknowledgement
This work was supported by CONCyTEG, Mexico, and by MURST, Italy.
CO
418
EC
TE
This paper shows how the screw theory and the principle of virtual work can be used to systematically solve the dynamics of parallel manipulators. Unlike other procedures reported in the
literature, the proposed method does not require the computation of internal forces of constraint,
which is an unnecessary task when the goal of the analysis is neither the dimensioning of the
elements nor the computation of the kinetic energy of the whole system. Two examples were
analyzed, and the general expressions so obtained for the complete analysis show the versatility of
the presented method.
RR
410
411
412
413
414
415
416
D
409 7. Conclusions
419 References
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