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A computationally efficient method for modeling flexible robots based on the assumed modes method
Applied Mathematics and Computation 218 (2011) 4483–4493
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A computationally efficient method for modeling flexible robots based on the assumed modes method Laura Celentano ⇑, Alessandro Coppola Università degli Studi di Napoli, Federico II, Dipartimento di Informatica e Sistemistica, Via Claudio 21, I-80125 Napoli, Italy
a r t i c l e
i n f o
Keywords: Modeling Simulation Flexible robots Assumed modes method Computational efficiency
a b s t r a c t In this paper a simple method to obtain the analytical model of a flexible robot is presented, which turns out to be more efficient, from a computational point of view, than the classic assumed modes method. The presented method consists of using appropriate linear combinations of the modes of each link as basis functions to evaluate the deflection, in such a way as to minimize the dependency of the position of the generic link on the Lagrangian variables of the previous links. Hence, the number of terms of the inertia matrix and of the Coriolis and centrifugal vectors is significantly reduced. First, the model is derived, provided that the links are homogeneous and with constant cross-section, by analytically or otherwise by numerically calculating the parameters of the closed-form expression of the Lagrangian function of the generic link supposed free; afterwards, the analytical dynamic model of the whole robot is obtained by using an iterative interconnection algorithm, which can be easily implemented by using a symbolic manipulation language. The simplicity and efficiency of the proposed method is illustrated by considering the analytic expression of the kinetic energy of the end-effector in different cases and with significant comparison examples. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The modeling of robots with flexible links is a historic topic of robotics research [1–10] and nowadays it remains very interesting for the scientific community [16,18,20–22,26,27]. Since flexible robots must be operated and/or solicited at low frequencies in order to avoid their breaking and/or annoying noises, the model must be precise especially at low frequencies. On the other hand, it is well known that if the endeffector of a flexible robot is controlled by a closed-loop controller, the spillover phenomenon easily occurs. Therefore, when the assumed modes method is used to model the robot, it is necessary to consider a high number of modes both in order to have a good precision at very low frequencies and in order to avoid the spillover phenomenon [7,11–13,17,21,25,27]. Unfortunately, the above considerations lead to models which are difficult to obtain and very burdensome both in terms of memory usage and of simulation times. In this paper a method is presented, which allows to easily obtain, by using a symbolic manipulation language, the analytical model of a robot with any number of links and with any number and any type of basis functions per link. Moreover, it will be shown that, by choosing as basis functions a suitable linear combination of the assumed modes, the obtained model
⇑ Corresponding author. E-mail addresses: [email protected] (L. Celentano), [email protected] (A. Coppola). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.10.029
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Fig. 1. Schematic representation of a flexible robot.
turns out to be more efficient with respect to the ones which can be obtained with the classic modal methods proposed in literature [4–8,14–16,20,22,26]. In details, in Section 2 the hypotheses about the typology of the robots considered in this paper are presented. In Section 3 the proposed modeling technique is illustrated: first, the kinetic and potential energies of a generic flexible link, supposed in free-free boundary conditions, are calculated, then the kinetic and potential energies of the whole flexible robot are derived by using an interconnection algorithm based on the congruence technique, and finally the dynamic model of the flexible robot is derived by using the Lagrangian approach. In Section 4 a method to reduce the computational cost of the classic assumed modes method is illustrated and in Section 5 some examples are reported, from which it emerges that the proposed method allows obtaining a clear improvement of the computational efficiency with respect to the classic assumed modes one. Finally, in Section 6 conclusions and future developments are reported. 2. Hypotheses, notations and preliminaries In this paper it is considered, for brevity, the case of planar robots with fixed base, constituted by m flexible links having a straight line as unstressed configuration, both rigid ends of negligible dimensions with respect to its length, and rotation axes orthogonal to the vertical plane. In Fig. 1 a schematic representation of a robot with three flexible links is reported, whereas in Fig. 2 a detailed representation of the i-th link in the stressed and unstressed configuration is shown. For the sake of clarity, the following preliminary notations are introduced for the generic i-th link: Li is the length of the link; Ei is the Young’s modulus of the link’s material; Ii is the area moment of inertia for the cross-section of the link; mi is the þ þ mass per unit length of the flexible part of the link; M i ; M i are the masses of both rigid ends of the link; J i ; J i are the inertia moments of both rigid ends of the link with respect to rotation axes; xoi, yoi, ai are the absolute motion coordinates of the link supposed rigid; di(z, t) is the deflection of the link in relative coordinates (with reference to Fig. 2 it is d = di(Li, t) and c ¼ @di@zðz;tÞ . z¼Li
3. Robot model 3.1. Kinematic model of a flexible link In this subsection the kinematic model of a flexible link is derived. Let xoi, yoi, ai be the Lagrangian coordinates of absolute motion of the generic link in the unstressed configuration with respect to an inertial frame, and let di(z, t) be the transversal displacement of a generic point of the i-th link at abscissa z and at time t due to the distributed flexibility (see Fig. 2). Then the coordinates (xi, yi) of a generic point of the i-th deformed link can be expressed as:
xi ¼ xoi þ z cos ai di sin ai ; yi ¼ yoi þ z sin ai þ di cos ai :
ð1Þ
A flexible link is a distributed parameter system having an infinite number of freedom degrees due to its flexibility, which is represented in the model by di(z, t). In order to obtain a finite dimension model which is oriented to analysis and control it is necessary to approximate the deflection di(z, t). In literature different methods have been proposed in order to approximate the deflection of a flexible link
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Fig. 2. Kinematic schematic representation of a flexible link.
having constant cross-section. These methods consist in finding a complete set of functions {wk(z)}, through which the deflection of the i-th link is approximated as
di ðz; tÞ
ni X
gk ðtÞwk ðzÞ;
ð2Þ
k¼1
where gk(t), k = 1, . . . , ni, are the Lagrangian deformation variables. Different choices of the set {wk(z)} have been proposed in literature. The assumed modes method uses the modes obtained from the solution of the Euler–Bernoulli beam equation with distributed inertial load
! @2d @ 2 dðz; tÞ @ 2 dðz; tÞ þm EI ¼ 0: 2 2 @z @z @t 2
ð3Þ
The above modes can be easily analytically computed if EI = const, otherwise they can be numerically computed by using standard software packages or by using the method proposed in [27]. Remark 1. If the generic link is subjected to large deformations (see e.g. [11]), it is still possible to apply the proposed methodology, by fictitiously subdividing the link into sublinks as shown in Fig. 3. 3.2. Lagrangian function of a flexible link In this subsection the kinetic and potential energies of a generic flexible link are obtained. By taking the derivatives of (1) and by using (2), the velocity vector components of a generic point along the deformed link are obtained
Fig. 3. Fictitious subdivision of the generic link into sublinks.
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x_ i ¼ x_ oi z sinðai Þa_ i y_ i ¼ y_ oi þ z cosðai Þa_ i þ
g_ k wk sinðai Þ
k¼1 ni X
ni X
gk wk cosðai Þa_ i ;
k¼1 ni X
g_ k wk cosðai Þ
k¼1
ð4Þ
gk wk sinðai Þa_ i ;
k¼1
where the dependency on z and t have been omitted for simplicity and ni is the number of Lagrangian deformation variables of the i-th link. The kinetic energy of the i-th link can be computed as follows
1 T i ¼ mi 2
Z
Li
0
x_ 2i
þ
y_ 2i
1 i X 1 1 1 g_ k w0k jz¼Li dz þ Mi x_ 2oi þ y_ 2oi þ Ji a_ 2i þ Mþi x_ 2i jz¼Li þ y_ 2i jz¼Li þ J þi a_ i þ 2 2 2 2 k¼1 n
!2 :
ð5Þ
By substituting (4) into (5), after some tedious manipulations and by omitting, for simplicity of notations, the subscript i, it is
2 2 2 1 _ þ J þ qTf Bf qf þ Mþ L2 þ mTf ðLÞqf a_ 2 þ q_ Tf Bf q_ f M x_ 2o þ y_ 2o þ Mþ mTf ðLÞq_ f þ J þ m0T f ðLÞqf 2 T T þ T þ T _ _ _ _ _ _ þ2 k q_ f þ J þ m0T f ðLÞqf þ M Lmf ðLÞqf a 2 M mf ðLÞqf þ h qf aðxo cos a þ yo sin aÞ o T þ T þ2 Na_ þ h q_ f þ M mf ðLÞq_ f ðy_ o cos a x_ o sin aÞ ;
T¼
ð6Þ
where: M ¼ M þ mL þ M þ ;
3
J ¼ J þ m L3 þ J þ ;
N ¼ M þ L þ 12 mL2 ;
the matrix Bf 2 Rnn is derived by using the relationship
1 m 2
Z
L 0
2 1 mTf q_ f dz ¼ q_ Tf Bf q_ f ; 2
ð7Þ
the vector h 2 Rn is derived by using the relationship
m
Z
L
0
T mTf q_ f dz ¼ h q_ f ;
ð8Þ
the vector k 2 Rn is derived by using the relationship
m
Z 0
L
T mTf q_ f zdz ¼ k q_ f ;
ð9Þ
qf 2 Rn represents the vector of the Lagrangian deformation variables
qTf ¼ ½g1 g2 gn ;
ð10Þ
mf(z) 2 Rn represents the vector of the basis functions
mTf ¼ ½w1 w2 wn :
ð11Þ
Remark 2. It is worth noting that the matrix Bf and the vectors h and k can be computed off-line, just one time, for each link and the kinetic energy of the i-th link (5) can be conveniently rewritten in a compact matricial form as
Ti ¼
1 _T e _ ~ Biq ~i ; q 2 i
ð12Þ
where:
" ~Ti ¼ ½q ~Tli qTli ; q
~Tli ¼ ½xoi yoi ; q
qTli
T i qfi ;
¼ ½a
ei ¼ B
e 11i B eT B 12i
e 11i ¼ B
Mi
0
0
Mi
;
e 12i B
# e 12i B ; e 22i B
2 T M þi mTfi ðLi Þqfi þ hi qfi cos ai Ni sin ai 6 ¼4 T M þi mTfi ðLi Þqfi þ hi qfi sin ai þ Ni cos ai
~T sin a h i i ~T cos a h i i
3 7 5;
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2
2 þ 2 T T J þ q B q þ M L þ m ðL Þq i fi i fi fi i i fi fi 6 ¼4 ~ ki
e 22i B
3 ~T k i 7 5; e fi B
~T ¼ hT þ M þ mT ðL Þ; h i i i fi i ~T ¼ kT þ J þ m0T ðL Þ þ M þ L mT ðL Þ; k i i i i i fi i i fi e fi ¼ Bfi þ M þ mfi ðLi ÞmT ðLi Þ þ J þ m0 ðLi Þm0T ðLi Þ: B i i fi fi fi
Once the kinetic energy has been derived, it is necessary to calculate the elastic potential energy Uei and the gravitational potential energy Ugi of the i-th link. The elastic potential energy due to the deformation of the i-th link is
U ei ¼
Z
1 Ei Ii 2
Li
0
m00T fi qfi
2
dz ¼
1 T q K i qfi : 2 fi
ð13Þ
The gravitational potential energy turns out to be
U gi ¼ mg
Z
Li
0
yi dz þ M i gyoi þ Mþi gyi jz¼Li ;
ð14Þ
where g is the gravity acceleration. By substituting the second of (1) into (14) and by using (2) it is
T U gi ¼ M i gyoi þ Ni g sin ai þ M þi mTfi ðLi Þqfi þ hi qfi g cos ai :
ð15Þ
Remark 3. Note that, by virtue of the choice made for the Lagrangian deformation variables, the contributions to the kinetic energy and to the gravitational potential energy of the ending inertias M+ and J+ depend generally on all the Lagrangian deformation variables of the vector qf. 3.3. Interconnection algorithm and Lagrangian function of the robot In this subsection an algorithm for the calculation of the Lagrangian function of a robot constituted by several interconnected flexible links is presented, starting from the results, valid for a single link, stated above. This algorithm allows calculating the kinetic and potential energies of a robot constituted by v flexible links having kinetic energy given by (12), elastic potential energy given by (13), and gravitational potential energy given by (15). It is useful to observe that, being the 1-st link hinged to the base, the variables xo1 and yo1 do not appear in the kinetic energy expression, hence
T1 ¼
1 T q_ B1 q_ l1 ; 2 l1
ð16Þ
e 221 . Moreover, the rigid translation variables xoi and yoi, 2 6 i 6 m are redundant, since they depend on ak, qf k, where B1 ¼ B k = 1, . . . , i 1; in fact from Eq. (4) the following recursive relationship can be derived
x_ oiþ1 y_ oiþ1
¼
a_ i x_ oi þ Ai ðai ; qfi Þ ; q_ fi y_ oi
ð17Þ
where Ai 2 R2ðni þ1Þ has the following expression
" Ai ¼
Li sin ai mTfi ðLi Þqfi cos ai
sin ai mTfi ðLi Þ
Li cos ai mTfi ðLi Þqfi sin ai
cos ai mTfi ðLi Þ
# ð18Þ
:
Therefore, the expression (12), for i P 2, can be rewritten as a function of the sole Lagrangian variables as follows
Ti ¼
1 T T e i A1...i1 q_ 1...i ¼ 1 q_ T Bi q_ 1...i ; B q_ A 2 1...i 1...i1 2 1...i
qT1...i ¼ qTl1 qTli ;
A1...i1 ¼
A1 . . . Ai1
O
O
Ini þ1
;
ð19Þ
in which Ip denotes the identity matrix of order p and O is a zero matrix of suitable dimensions. Finally, the kinetic energy of the robot constituted by m flexible links turns out to be
T¼
1 T _ q_ Bq; 2
ð20Þ
where q = q1. . .m and the inertia matrix B is obtained ‘‘by adding’’ the matrices Bi according to the recursive scheme reported in Fig. 4.
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Fig. 4. Composition scheme of the matrix B.
Remark 4. It is important to note that, by virtue of the choice which has been made of the Lagrangian deformation variables, the transformation matrix Ai depends generally on all the Lagrangian deformation variables of the vector qf. For the above consideration and for Remarks 2 and 3, the inertia matrix of the robot B and the vector of Coriolis and centrifugal terms c are very burdensome to compute because of the very large number of their terms, which notably increases when the number of the Lagrangian deformation variables of each link increases. Concerning the elastic potential energy of the whole robot, it is easy to verify that it turns out to be
Ue ¼
1 T q Kq; 2
ð21Þ
where the matrix K is the block diagonal matrix K = diag(0, K1, 0, K2, . . . , 0, Km). Finally, the gravitational potential energy of the whole robot Ug is obtained as the sum of
U gi ¼ M i g
i1 X T Li sin ai þ mTfi ðLi Þqfi cos ai þ N i g sin ai þ M þi mTfi ðLi Þqfi þ hi qfi g cos ai ;
ð22Þ
k¼1
where Ugi is the gravitational potential energy of the i-th link, which has been obtained from (15) by expressing yoi as a function of the Lagrangian deformation variables. Remark 5. It is useful to note that, once the kinetic and potential energies of the generic flexible link have been calculated in an hybrid symbolical-numerical form, the ones of the whole robot can be obtained by implementing the proposed interconnection algorithm with a symbolic manipulation software language. 3.4. Dynamic model of the robot In this subsection the dynamic model of the whole robot is considered in the more suitable form presented in [19,23,24] by using the Euler–Lagrange method. It is easy to show that this model, under the assumptions that control actions uc and disturbances ud are the ones reported in Fig. 5, is
Fig. 5. Control actions and disturbances acting on the robot.
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d 1 @ T @ _ ðBðqÞqÞ q_ BðqÞq_ þ Kq þ U g ¼ Hc uc þ Hd ud ; dt 2 @q @q
ð23Þ
where:
uTc ¼ ½C 1 C 2 C m ; 2
1
1
uTd ¼ ½C d F d ; 0
6 O m0 ðLÞ O 6 f1 6 60 1 1 6 6O O m0f 2 ðLÞ 6 6 60 0 1 6 Hc ¼ 6 O O 6O 6 60 0 0 6 6 O O O 6 6 60 0 0 4 .. .. .. . . .
ð24Þ 3
7 7 7 7 7 7 7 7 7 7 ; 7 7 7 7 7 7 7 7 7 5 .. .
2
hd11
6h 6 d21 Hd ¼ 6 6 .. 4 . hdm1
hd12
3
hd22 7 7 7; ... 7 5
ð25Þ
hdm2
in which: T
T
hdi1 ¼ ½0 O; i ¼ 1; . . . ; m 1; hdm1 ¼ ½1 m0f m ðLm Þ; h i T hdi2 ¼ Li sinðai ad Þ mTfi ðLi Þqfi cosðai ad Þ mTfi ðLi Þ sinðai ad Þ ;
ð26Þ
i ¼ 1; 2; . . . ; m;
being O a zero vector of suitable dimension. 4. A method to reduce the computational cost 4.1. Linear transformation of the deformation variables In order to reduce the computational cost of model (23), note that if the classic basis assumed modes is chosen, functions
wk(z) = mk(z), k = 1, . . . , ni, for z = L are all not null with their first derivatives (see Fig. 6). Therefore, by pointing with lk(t) the corresponding Lagrangian deformation variables, the deformation variables d(t) and c(t) (as depicted in Fig. 2), given by dðtÞ ¼ m1L l1 ðtÞ þ m2L l2 ðtÞ þ þ mni L lni ðtÞ;
cðtÞ ¼ m01L l1 ðtÞ þ m02L l2 ðtÞ þ þ m0ni L lni ðtÞ;
mkL ¼ mk ðLÞ – 0; @mk ðz; tÞ 0 mkL ¼ – 0; @z
ð27Þ
z¼L
depend on all the Lagrangian variables lk(t). ^ k ðzÞ (pseudo-modes) are chosen an opportune linear combination of the mode shape funcIf as basis functions wk ðzÞ ¼ m tions mk(z) as follows
3 2 1 0
mode 1 mode 2 mode 3 mode 4
-1 -2 -3 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
z/L Fig. 6. The first ni = 4 modes of a generic link with constant section.
0.9
1
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1 0.8 0.6
pseudomode 1 pseudomode 2
0.4 0.2 0 -0.2 -0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.8
0.9
1
0.8
0.9
1
z/L Fig. 7. Pseudo-modes of a generic link for ni = 2.
1.5
pseudomode 1 pseudomode 2 pseudomode 3
1
0.5
0
-0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
z/L Fig. 8. Pseudo-modes of a generic link for ni = 3.
1.2 1
pseudomode 1 pseudomode 2 pseudomode 3 pseudomode 4
0.8 0.6 0.4 0.2 0 -0.2 -0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
z/L Fig. 9. Pseudo-modes of a generic link for ni = 4.
1 1 0 ^ 1 ðzÞ m1 ðzÞ m C C Bm B B ^ 2 ðzÞ C B m2 ðzÞ C C ¼ TB C; B @ A @ A ^ ni ðzÞ mni ðzÞ m 0
ð28Þ
with T such that
^ k ðLÞ ¼ m
1; k ¼ 1; 0; k – 1;
^ k ðz; tÞ @m ¼ 1; k ¼ 2; @z 0; k – 2; z¼L
ð29Þ
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^ k ðtÞ, it is easy to verify that (see Figs. 7–9), denoted the new Lagrangian variables by l
dðtÞ ¼ g1 ðtÞ;
cðtÞ ¼ g2 ðtÞ:
ð30Þ
Hence d(t) and c(t) depend only on a single Lagrangian deformation variable (i.e. themselves are Lagrangian variables) and not on all the deformation Lagrangian variables as in the classic modal methods. Remark 6. If the proposed mode shape functions are used, according to (29), the matrices Ai given by (18) are very sparse and they always depend only on the deformation variable dini , whatever the number of the mode shape functions is; this fact leads to strongly reduce the dependency of the inertia matrix and of the vector of Coriolis and centrifugal terms on the Lagrangian deformation variables; in this way, significant improvements in terms of computational cost and memory usage are obtained. Remark 7. Moreover, the proposed pseudo-modes allow an easier solution of the inverse kinematic problem and an easier implementation of closed-loop control laws.
5. Validation and comparison results In this section some validation and comparison results are presented, which demonstrate the computational improvement obtained with the proposed method. As the computational complexity of the modeling methods depend, in general, also on the symbolic manipulation languages, on the programming languages and on the computers used, in order to clearly show the advantages of the proposed method, besides reporting tables with complexity indices (by comparing the obtained results with the ones obtained with the classic methods in different realistic cases, whose models are quite complex), the analytic expressions (suitably simplified) of the kinetic energy due to the tip mass in the case of a single and two-link robot is provided. (1) In the case of a single-link robot, the kinetic energy due to the tip mass M turns out to be: (a) Proposed pseudo-modal method The kinetic energy T is
T¼
i 1 h 2 M L1 þ d21 a_ 21 þ d_ 21 þ 2L1 a_ 1 d_ 1 ; 2
ð31Þ
(b) Classic modal method The kinetic energy T is obtained by posing in (31)
d1 ðtÞ ¼ m1L1 l1 ðtÞ þ m2L1 l2 ðtÞ þ þ mn1 L1 ln1 ðtÞ;
d_ 1 ðtÞ ¼ m1L1 l_ 1 ðtÞ þ m2L1 l_ 2 ðtÞ þ þ mn1 L1 l_ n1 ðtÞ:
ð32Þ
(2) In the case of a two-link robot, the kinetic energy due to the tip mass M turns out to be: (a) Proposed pseudo-modal method The kinetic energy T is
1 n 2 M L1 þ d21 a_ 21 þ d_ 21 þ 2L1 a_ 1 d_ 1 þ L22 þ d22 a_ 22 þ d_ 22 þ 2L2 a_ 2 d_ 2 þ 2½L1 d2 sinða1 a2 Þ 2L2 d1 sinða1 a2 Þ 2 þ2d1 d2 cosða1 a2 Þ þ L1 L2 cosða1 a2 Þa_ 1 a_ 2 þ 2½L1 cosða1 a2 Þ d1 sinða1 a2 Þa_ 1 d_ 2 þ 2½L2 cosða1 a2 Þ o þd2 sinða1 a2 Þd_ 1 a_ 2 þ 2 cosða1 a2 Þd_ 1 d_ 2 ;
T¼
ð33Þ
(b) Classic modal method The kinetic energy T is obtained by posing in (33)
d1 ðtÞ ¼ m11L1 l11 ðtÞ þ m12L1 l12 ðtÞ þ þ m1n1 L1 l1n1 ðtÞ; d2 ðtÞ ¼ m21L2 l21 ðtÞ þ m22L2 l22 ðtÞ þ þ m2n2 L1 l2n2 ðtÞ;
d_ 1 ðtÞ ¼ m11L1 l_ 11 ðtÞ þ m12L1 l_ 12 ðtÞ þ þ m1n1 L1 l_ 1n1 ðtÞ; d_ 2 ðtÞ ¼ m21L l_ 21 ðtÞ þ m22L l_ 22 ðtÞ þ þ m2n L l_ 2n ðtÞ: 2
2
2 2
ð34Þ
2
The reader or everybody wants to apply the proposed results, by using the above expressions of T, can easily evaluate for different values of n1 and n2 the advantages of the proposed method with respect to the classic modal one. Now consider a robot constituted by flexible links made of aluminium having square hollow constant cross-section (see Fig. 10) with Li = 2 m, li = 20 mm, si = 1 mm, Ei = 6.4 1010 N/m2, qi = 2.7 103 kg/m3, where qi denotes the density of the ith flexible link. In Table 1 (Table 2) the number of multiplies to evaluate the inertia matrix B (the gradient of the kinetic energy c) required by the classic assumed modes method and by the proposed one are compared, under the assumption that the number of modes per link is 4 and by increasing the number of links.
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Fig. 10. i-th flexible link having square hollow constant cross-section.
Table 1 Number of multiplies to evaluate the inertia matrix B with 4 modes per link.
2 links 3 links 4 links
Proposed modal method
Classic modal method
Ratio
138 362 683
407 1178 2308
2.95 3.25 3.38
Table 2 Number of multiplies to evaluate the gradient of the kinetic energy c with 4 modes per link.
2 link 3 link 4 link
Proposed modal method
Classic modal method
Ratio
336 896 1770
1052 2887 5624
3.13 3.22 3.18
Table 3 Number of multiplies to evaluate the inertia matrix B with 3 links.
3 modes 4 modes 5 modes
Proposed modal method
Classic modal method
Ratio
268 356 460
734 1178 1849
2.74 3.31 4.02
Table 4 Number of multiplies to evaluate the gradient of the kinetic energy c with 3 links.
3 modes 4 modes 5 modes
Proposed modal method
Classic modal method
Ratio
691 898 1150
1786 2887 4391
2.58 3.21 3.82
Note that the computational advantage of the proposed method is really notable (over 300%). In Table 3 (Table 4) the number of multiplies to evaluate the inertia matrix B (the gradient of the kinetic energy c) required by the classic assumed modes method and by the proposed one are compared, under the assumption that the number of links is 3 and by increasing the number of modes per link. Note that the computational advantage increases when the number of modes per link increases. The obtained computational improvement allows reducing the time required by the dynamic simulation of the robot and allows improving the implementation of the robot dynamic model in terms of memory usage. Remark 8. The proposed method can be extended also to spatial flexible robots. To easily convince yourself of this fact, it is important to note that if the section of a generic link has two axes of symmetry n, g orthogonal to axis f of the link, it is wellknown by the experts of building science that the cutting center coincides with the torsional center and the modes of the link
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are the union of the flexion modes in the plane n, f, of the flexion modes in the plane g, f and of the torsion modes around axis f. By linearly combining the modes in the plane n, f and the modes in the plane g, f as it has been made in the case of planar robots and the torsion modes such that the terminal torsion angle of each link coincides with a Lagrangian variable, it is easy to realize how it is possible to obtain, also in the spatial case, a more computationally efficient method with respect to the classic modal method.
6. Conclusions In this paper a method to obtain a closed-form model of a planar flexible robot with any number of links has been presented. Moreover, a method to improve the computational efficiency of the assumed modes method has been illustrated. The examples reported show that the computational efficiency of the presented method is really notable and it increases with the number of modes per link. Related results can be found in [27]; moreover, authors are successfully working in order to introduce the internal friction into the model and in order to design efficient control laws for flexible robots having small damping. References [1] W.J. Book, O. Maizza-Neto, D.E. Whitney, Feedback control of two-beam two-joint systems with distributed flexibility, ASME J. Dyn. Syst. Meas. Control (1975) 424–431. [2] M.J. Balas, Feedback control of flexible systems, IEEE Trans. Autom. Control 23 (1978) 673–679. [3] A. Truckenbrodt, Control of elastic mechanical systems, Regielungstechnick 30 (1982) 277–285. [4] P.C. Hughes, Space structure vibration modes: How many exist? Which are important, IEEE Control Syst. Mag. 7 (1987) 22–28. [5] G.G. Hasting, W.J. Book, A linear dynamic model for flexible robot manipulators, IEEE Control Syst. Mag. 7 (1987) 61–64. [6] Y. Chait, M. Miklavcic, C.R. Maccluer, C.J. Radcliffe, A natural modal expansion for the flexible robot arm problem via a self-adjoint formulation, IEEE Trans. Robot. Autom. 6 (5) (1990) 601–603. [7] T. Yoshikawa, H. Murakami, K. Hosoda, Modeling and control of a three degree of freedom manipulator with two flexible links, in: Proceedings of the 29nd Conference on Decision and Control, Honolulu, Hawaii, 1990, pp. 2532–2537. [8] A. De Luca, B. Siciliano, Closed form dynamical model of planar multilink lightweight robots, IEEE Trans. Syst. Man Cybern. 21 (4) (1991) 826–839. [9] D. Wang, M. Vidyasagar, Modeling a class of multilink manipulators with the last link flexible, IEEE Trans. Robot. Autom. 8 (1) (1992) 33–41. [10] C.J. Li, T.S. Sankar, Systematic methods for efficient modeling and dynamics computation of flexible robot manipulators, IEEE Trans. Syst. Man Cybern. 23 (1) (1993). [11] F. Boyer, W. Khalil, Kinematic model of a multi-beam structure undergoing large elastic displacement and rotations. Part one: model of an isolated beam, J. Mech. Mach. Theory 34 (1999) 205–222. [12] G. Celentano, R. Setola, The modelling of a flexible beam with piezoelectric plates for active vibration control, J. Sound Vibrat. 223 (3) (1999) 483–492. [13] A. De Luca, Feedforward/feedback laws for the control of flexible robots, in: Proceedings of the 2000 IEEE International Conference on Robotics and Automation, San Francisco, CA, 2000, pp. 233–240. [14] G. Wang, Y.F. Li, W.L. Xu, Regularization-based recovery scheme for inverse dynamics of high-speed flexible beams, Appl. Math. Comput. 115 (2-3) (2000) 161–175. [15] R.D. Robinett, C. Dohrmann, G.R. Eisler, J. Feddema, G.G. Parker, D.G. Wilson, D. Stokes, Flexible Robot Dynamics and Controls, Kluver Academic/Plenum Publishers, New York, 2002. [16] S.E. Khadem, A.A. Pirmohammadi, Analytical development of dynamic equations of motion for a three-dimensional flexible link manipulator with revolute and prismatic joints, IEEE Trans. Syst. Man Cybern. 33 (2) (2003) 237–249. [17] X. Hou, S.-K. Tsui, A feedback control and a simulation of a torsional elastic robot arm, Appl. Math. Comput. 142 (2-3) (2003) 389–407. [18] X. Hou, S.-K. Tsui, Analysis and control of a two-link and three-joint elastic robot arm, Appl. Math. Comput. 152 (3) (2004) 759–777. [19] L. Celentano, R. Iervolino, A new method for robot modeling and simulation, ASME J. Dyn. Syst. Meas. Control 128 (2006) 811–819. [20] A. Macchelli, C. Melchiorri, S. Stramigioli, Port-based Modeling of a flexible link, IEEE Trans. Robot. 23 (2007) 650–660. [21] S. Moberg, S. Hanssen, A DAE approach to feedforward control of flexible manipulators, in: Proceedings of the 2007 IEEE International Conference on Robotics and Automation, Roma, Italy, 2007, pp. 3439–3444. [22] S. Liu, L. Wu, Z. Lu, Impact dynamics and control of a flexible dual-arm space robot capturing an object, Appl. Math. Comput. 185 (2) (2007) 1149–1159. [23] L. Celentano, R. Iervolino, A novel approach for spatial robots modeling and simulation, MMAR07, in: 13th IEEE International Conference on Methods and Models in Automation and Robotics, Szczecin, Poland, 27–30 Aug. 2007, pp. 1005–1010. [24] L. Celentano, An innovative method for robots modeling and simulation, in: Harald Aschemann, New Approaches in Automation and Robotics, I-Tech Education and Publishing, Vienna, 2008, pp. 173–196. [25] S. Moberg, J. Öhr, S. Gunnarsson, A benchmark problem for robust control of a multivariable nonlinear flexible manipulator, in: Proceedings of the 17th IFAC World Congress, Seoul, Korea, 2008. [26] S. Moberg, Modeling and Control of Flexible Manipulators. Ph.D. Thesis. Department of Electrical Engineering, Linköping University, 2010. [27] L. Celentano, A. Coppola, A wavelet based method to modeling realistic flexible robots, in: 18th IFAC World Congress, Milano, Italy, 28 Aug. – 2 Sep., 2011, pp. 929–937.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A computationally efficient method for modeling flexible robots based on the assumed modes method Laura Celentano ⇑, Alessandro Coppola Università degli Studi di Napoli, Federico II, Dipartimento di Informatica e Sistemistica, Via Claudio 21, I-80125 Napoli, Italy
a r t i c l e
i n f o
Keywords: Modeling Simulation Flexible robots Assumed modes method Computational efficiency
a b s t r a c t In this paper a simple method to obtain the analytical model of a flexible robot is presented, which turns out to be more efficient, from a computational point of view, than the classic assumed modes method. The presented method consists of using appropriate linear combinations of the modes of each link as basis functions to evaluate the deflection, in such a way as to minimize the dependency of the position of the generic link on the Lagrangian variables of the previous links. Hence, the number of terms of the inertia matrix and of the Coriolis and centrifugal vectors is significantly reduced. First, the model is derived, provided that the links are homogeneous and with constant cross-section, by analytically or otherwise by numerically calculating the parameters of the closed-form expression of the Lagrangian function of the generic link supposed free; afterwards, the analytical dynamic model of the whole robot is obtained by using an iterative interconnection algorithm, which can be easily implemented by using a symbolic manipulation language. The simplicity and efficiency of the proposed method is illustrated by considering the analytic expression of the kinetic energy of the end-effector in different cases and with significant comparison examples. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The modeling of robots with flexible links is a historic topic of robotics research [1–10] and nowadays it remains very interesting for the scientific community [16,18,20–22,26,27]. Since flexible robots must be operated and/or solicited at low frequencies in order to avoid their breaking and/or annoying noises, the model must be precise especially at low frequencies. On the other hand, it is well known that if the endeffector of a flexible robot is controlled by a closed-loop controller, the spillover phenomenon easily occurs. Therefore, when the assumed modes method is used to model the robot, it is necessary to consider a high number of modes both in order to have a good precision at very low frequencies and in order to avoid the spillover phenomenon [7,11–13,17,21,25,27]. Unfortunately, the above considerations lead to models which are difficult to obtain and very burdensome both in terms of memory usage and of simulation times. In this paper a method is presented, which allows to easily obtain, by using a symbolic manipulation language, the analytical model of a robot with any number of links and with any number and any type of basis functions per link. Moreover, it will be shown that, by choosing as basis functions a suitable linear combination of the assumed modes, the obtained model
⇑ Corresponding author. E-mail addresses: [email protected] (L. Celentano), [email protected] (A. Coppola). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.10.029
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Fig. 1. Schematic representation of a flexible robot.
turns out to be more efficient with respect to the ones which can be obtained with the classic modal methods proposed in literature [4–8,14–16,20,22,26]. In details, in Section 2 the hypotheses about the typology of the robots considered in this paper are presented. In Section 3 the proposed modeling technique is illustrated: first, the kinetic and potential energies of a generic flexible link, supposed in free-free boundary conditions, are calculated, then the kinetic and potential energies of the whole flexible robot are derived by using an interconnection algorithm based on the congruence technique, and finally the dynamic model of the flexible robot is derived by using the Lagrangian approach. In Section 4 a method to reduce the computational cost of the classic assumed modes method is illustrated and in Section 5 some examples are reported, from which it emerges that the proposed method allows obtaining a clear improvement of the computational efficiency with respect to the classic assumed modes one. Finally, in Section 6 conclusions and future developments are reported. 2. Hypotheses, notations and preliminaries In this paper it is considered, for brevity, the case of planar robots with fixed base, constituted by m flexible links having a straight line as unstressed configuration, both rigid ends of negligible dimensions with respect to its length, and rotation axes orthogonal to the vertical plane. In Fig. 1 a schematic representation of a robot with three flexible links is reported, whereas in Fig. 2 a detailed representation of the i-th link in the stressed and unstressed configuration is shown. For the sake of clarity, the following preliminary notations are introduced for the generic i-th link: Li is the length of the link; Ei is the Young’s modulus of the link’s material; Ii is the area moment of inertia for the cross-section of the link; mi is the þ þ mass per unit length of the flexible part of the link; M i ; M i are the masses of both rigid ends of the link; J i ; J i are the inertia moments of both rigid ends of the link with respect to rotation axes; xoi, yoi, ai are the absolute motion coordinates of the link supposed rigid; di(z, t) is the deflection of the link in relative coordinates (with reference to Fig. 2 it is d = di(Li, t) and c ¼ @di@zðz;tÞ . z¼Li
3. Robot model 3.1. Kinematic model of a flexible link In this subsection the kinematic model of a flexible link is derived. Let xoi, yoi, ai be the Lagrangian coordinates of absolute motion of the generic link in the unstressed configuration with respect to an inertial frame, and let di(z, t) be the transversal displacement of a generic point of the i-th link at abscissa z and at time t due to the distributed flexibility (see Fig. 2). Then the coordinates (xi, yi) of a generic point of the i-th deformed link can be expressed as:
xi ¼ xoi þ z cos ai di sin ai ; yi ¼ yoi þ z sin ai þ di cos ai :
ð1Þ
A flexible link is a distributed parameter system having an infinite number of freedom degrees due to its flexibility, which is represented in the model by di(z, t). In order to obtain a finite dimension model which is oriented to analysis and control it is necessary to approximate the deflection di(z, t). In literature different methods have been proposed in order to approximate the deflection of a flexible link
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Fig. 2. Kinematic schematic representation of a flexible link.
having constant cross-section. These methods consist in finding a complete set of functions {wk(z)}, through which the deflection of the i-th link is approximated as
di ðz; tÞ
ni X
gk ðtÞwk ðzÞ;
ð2Þ
k¼1
where gk(t), k = 1, . . . , ni, are the Lagrangian deformation variables. Different choices of the set {wk(z)} have been proposed in literature. The assumed modes method uses the modes obtained from the solution of the Euler–Bernoulli beam equation with distributed inertial load
! @2d @ 2 dðz; tÞ @ 2 dðz; tÞ þm EI ¼ 0: 2 2 @z @z @t 2
ð3Þ
The above modes can be easily analytically computed if EI = const, otherwise they can be numerically computed by using standard software packages or by using the method proposed in [27]. Remark 1. If the generic link is subjected to large deformations (see e.g. [11]), it is still possible to apply the proposed methodology, by fictitiously subdividing the link into sublinks as shown in Fig. 3. 3.2. Lagrangian function of a flexible link In this subsection the kinetic and potential energies of a generic flexible link are obtained. By taking the derivatives of (1) and by using (2), the velocity vector components of a generic point along the deformed link are obtained
Fig. 3. Fictitious subdivision of the generic link into sublinks.
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x_ i ¼ x_ oi z sinðai Þa_ i y_ i ¼ y_ oi þ z cosðai Þa_ i þ
g_ k wk sinðai Þ
k¼1 ni X
ni X
gk wk cosðai Þa_ i ;
k¼1 ni X
g_ k wk cosðai Þ
k¼1
ð4Þ
gk wk sinðai Þa_ i ;
k¼1
where the dependency on z and t have been omitted for simplicity and ni is the number of Lagrangian deformation variables of the i-th link. The kinetic energy of the i-th link can be computed as follows
1 T i ¼ mi 2
Z
Li
0
x_ 2i
þ
y_ 2i
1 i X 1 1 1 g_ k w0k jz¼Li dz þ Mi x_ 2oi þ y_ 2oi þ Ji a_ 2i þ Mþi x_ 2i jz¼Li þ y_ 2i jz¼Li þ J þi a_ i þ 2 2 2 2 k¼1 n
!2 :
ð5Þ
By substituting (4) into (5), after some tedious manipulations and by omitting, for simplicity of notations, the subscript i, it is
2 2 2 1 _ þ J þ qTf Bf qf þ Mþ L2 þ mTf ðLÞqf a_ 2 þ q_ Tf Bf q_ f M x_ 2o þ y_ 2o þ Mþ mTf ðLÞq_ f þ J þ m0T f ðLÞqf 2 T T þ T þ T _ _ _ _ _ _ þ2 k q_ f þ J þ m0T f ðLÞqf þ M Lmf ðLÞqf a 2 M mf ðLÞqf þ h qf aðxo cos a þ yo sin aÞ o T þ T þ2 Na_ þ h q_ f þ M mf ðLÞq_ f ðy_ o cos a x_ o sin aÞ ;
T¼
ð6Þ
where: M ¼ M þ mL þ M þ ;
3
J ¼ J þ m L3 þ J þ ;
N ¼ M þ L þ 12 mL2 ;
the matrix Bf 2 Rnn is derived by using the relationship
1 m 2
Z
L 0
2 1 mTf q_ f dz ¼ q_ Tf Bf q_ f ; 2
ð7Þ
the vector h 2 Rn is derived by using the relationship
m
Z
L
0
T mTf q_ f dz ¼ h q_ f ;
ð8Þ
the vector k 2 Rn is derived by using the relationship
m
Z 0
L
T mTf q_ f zdz ¼ k q_ f ;
ð9Þ
qf 2 Rn represents the vector of the Lagrangian deformation variables
qTf ¼ ½g1 g2 gn ;
ð10Þ
mf(z) 2 Rn represents the vector of the basis functions
mTf ¼ ½w1 w2 wn :
ð11Þ
Remark 2. It is worth noting that the matrix Bf and the vectors h and k can be computed off-line, just one time, for each link and the kinetic energy of the i-th link (5) can be conveniently rewritten in a compact matricial form as
Ti ¼
1 _T e _ ~ Biq ~i ; q 2 i
ð12Þ
where:
" ~Ti ¼ ½q ~Tli qTli ; q
~Tli ¼ ½xoi yoi ; q
qTli
T i qfi ;
¼ ½a
ei ¼ B
e 11i B eT B 12i
e 11i ¼ B
Mi
0
0
Mi
;
e 12i B
# e 12i B ; e 22i B
2 T M þi mTfi ðLi Þqfi þ hi qfi cos ai Ni sin ai 6 ¼4 T M þi mTfi ðLi Þqfi þ hi qfi sin ai þ Ni cos ai
~T sin a h i i ~T cos a h i i
3 7 5;
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2
2 þ 2 T T J þ q B q þ M L þ m ðL Þq i fi i fi fi i i fi fi 6 ¼4 ~ ki
e 22i B
3 ~T k i 7 5; e fi B
~T ¼ hT þ M þ mT ðL Þ; h i i i fi i ~T ¼ kT þ J þ m0T ðL Þ þ M þ L mT ðL Þ; k i i i i i fi i i fi e fi ¼ Bfi þ M þ mfi ðLi ÞmT ðLi Þ þ J þ m0 ðLi Þm0T ðLi Þ: B i i fi fi fi
Once the kinetic energy has been derived, it is necessary to calculate the elastic potential energy Uei and the gravitational potential energy Ugi of the i-th link. The elastic potential energy due to the deformation of the i-th link is
U ei ¼
Z
1 Ei Ii 2
Li
0
m00T fi qfi
2
dz ¼
1 T q K i qfi : 2 fi
ð13Þ
The gravitational potential energy turns out to be
U gi ¼ mg
Z
Li
0
yi dz þ M i gyoi þ Mþi gyi jz¼Li ;
ð14Þ
where g is the gravity acceleration. By substituting the second of (1) into (14) and by using (2) it is
T U gi ¼ M i gyoi þ Ni g sin ai þ M þi mTfi ðLi Þqfi þ hi qfi g cos ai :
ð15Þ
Remark 3. Note that, by virtue of the choice made for the Lagrangian deformation variables, the contributions to the kinetic energy and to the gravitational potential energy of the ending inertias M+ and J+ depend generally on all the Lagrangian deformation variables of the vector qf. 3.3. Interconnection algorithm and Lagrangian function of the robot In this subsection an algorithm for the calculation of the Lagrangian function of a robot constituted by several interconnected flexible links is presented, starting from the results, valid for a single link, stated above. This algorithm allows calculating the kinetic and potential energies of a robot constituted by v flexible links having kinetic energy given by (12), elastic potential energy given by (13), and gravitational potential energy given by (15). It is useful to observe that, being the 1-st link hinged to the base, the variables xo1 and yo1 do not appear in the kinetic energy expression, hence
T1 ¼
1 T q_ B1 q_ l1 ; 2 l1
ð16Þ
e 221 . Moreover, the rigid translation variables xoi and yoi, 2 6 i 6 m are redundant, since they depend on ak, qf k, where B1 ¼ B k = 1, . . . , i 1; in fact from Eq. (4) the following recursive relationship can be derived
x_ oiþ1 y_ oiþ1
¼
a_ i x_ oi þ Ai ðai ; qfi Þ ; q_ fi y_ oi
ð17Þ
where Ai 2 R2ðni þ1Þ has the following expression
" Ai ¼
Li sin ai mTfi ðLi Þqfi cos ai
sin ai mTfi ðLi Þ
Li cos ai mTfi ðLi Þqfi sin ai
cos ai mTfi ðLi Þ
# ð18Þ
:
Therefore, the expression (12), for i P 2, can be rewritten as a function of the sole Lagrangian variables as follows
Ti ¼
1 T T e i A1...i1 q_ 1...i ¼ 1 q_ T Bi q_ 1...i ; B q_ A 2 1...i 1...i1 2 1...i
qT1...i ¼ qTl1 qTli ;
A1...i1 ¼
A1 . . . Ai1
O
O
Ini þ1
;
ð19Þ
in which Ip denotes the identity matrix of order p and O is a zero matrix of suitable dimensions. Finally, the kinetic energy of the robot constituted by m flexible links turns out to be
T¼
1 T _ q_ Bq; 2
ð20Þ
where q = q1. . .m and the inertia matrix B is obtained ‘‘by adding’’ the matrices Bi according to the recursive scheme reported in Fig. 4.
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Fig. 4. Composition scheme of the matrix B.
Remark 4. It is important to note that, by virtue of the choice which has been made of the Lagrangian deformation variables, the transformation matrix Ai depends generally on all the Lagrangian deformation variables of the vector qf. For the above consideration and for Remarks 2 and 3, the inertia matrix of the robot B and the vector of Coriolis and centrifugal terms c are very burdensome to compute because of the very large number of their terms, which notably increases when the number of the Lagrangian deformation variables of each link increases. Concerning the elastic potential energy of the whole robot, it is easy to verify that it turns out to be
Ue ¼
1 T q Kq; 2
ð21Þ
where the matrix K is the block diagonal matrix K = diag(0, K1, 0, K2, . . . , 0, Km). Finally, the gravitational potential energy of the whole robot Ug is obtained as the sum of
U gi ¼ M i g
i1 X T Li sin ai þ mTfi ðLi Þqfi cos ai þ N i g sin ai þ M þi mTfi ðLi Þqfi þ hi qfi g cos ai ;
ð22Þ
k¼1
where Ugi is the gravitational potential energy of the i-th link, which has been obtained from (15) by expressing yoi as a function of the Lagrangian deformation variables. Remark 5. It is useful to note that, once the kinetic and potential energies of the generic flexible link have been calculated in an hybrid symbolical-numerical form, the ones of the whole robot can be obtained by implementing the proposed interconnection algorithm with a symbolic manipulation software language. 3.4. Dynamic model of the robot In this subsection the dynamic model of the whole robot is considered in the more suitable form presented in [19,23,24] by using the Euler–Lagrange method. It is easy to show that this model, under the assumptions that control actions uc and disturbances ud are the ones reported in Fig. 5, is
Fig. 5. Control actions and disturbances acting on the robot.
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d 1 @ T @ _ ðBðqÞqÞ q_ BðqÞq_ þ Kq þ U g ¼ Hc uc þ Hd ud ; dt 2 @q @q
ð23Þ
where:
uTc ¼ ½C 1 C 2 C m ; 2
1
1
uTd ¼ ½C d F d ; 0
6 O m0 ðLÞ O 6 f1 6 60 1 1 6 6O O m0f 2 ðLÞ 6 6 60 0 1 6 Hc ¼ 6 O O 6O 6 60 0 0 6 6 O O O 6 6 60 0 0 4 .. .. .. . . .
ð24Þ 3
7 7 7 7 7 7 7 7 7 7 ; 7 7 7 7 7 7 7 7 7 5 .. .
2
hd11
6h 6 d21 Hd ¼ 6 6 .. 4 . hdm1
hd12
3
hd22 7 7 7; ... 7 5
ð25Þ
hdm2
in which: T
T
hdi1 ¼ ½0 O; i ¼ 1; . . . ; m 1; hdm1 ¼ ½1 m0f m ðLm Þ; h i T hdi2 ¼ Li sinðai ad Þ mTfi ðLi Þqfi cosðai ad Þ mTfi ðLi Þ sinðai ad Þ ;
ð26Þ
i ¼ 1; 2; . . . ; m;
being O a zero vector of suitable dimension. 4. A method to reduce the computational cost 4.1. Linear transformation of the deformation variables In order to reduce the computational cost of model (23), note that if the classic basis assumed modes is chosen, functions
wk(z) = mk(z), k = 1, . . . , ni, for z = L are all not null with their first derivatives (see Fig. 6). Therefore, by pointing with lk(t) the corresponding Lagrangian deformation variables, the deformation variables d(t) and c(t) (as depicted in Fig. 2), given by dðtÞ ¼ m1L l1 ðtÞ þ m2L l2 ðtÞ þ þ mni L lni ðtÞ;
cðtÞ ¼ m01L l1 ðtÞ þ m02L l2 ðtÞ þ þ m0ni L lni ðtÞ;
mkL ¼ mk ðLÞ – 0; @mk ðz; tÞ 0 mkL ¼ – 0; @z
ð27Þ
z¼L
depend on all the Lagrangian variables lk(t). ^ k ðzÞ (pseudo-modes) are chosen an opportune linear combination of the mode shape funcIf as basis functions wk ðzÞ ¼ m tions mk(z) as follows
3 2 1 0
mode 1 mode 2 mode 3 mode 4
-1 -2 -3 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
z/L Fig. 6. The first ni = 4 modes of a generic link with constant section.
0.9
1
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1 0.8 0.6
pseudomode 1 pseudomode 2
0.4 0.2 0 -0.2 -0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.8
0.9
1
0.8
0.9
1
z/L Fig. 7. Pseudo-modes of a generic link for ni = 2.
1.5
pseudomode 1 pseudomode 2 pseudomode 3
1
0.5
0
-0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
z/L Fig. 8. Pseudo-modes of a generic link for ni = 3.
1.2 1
pseudomode 1 pseudomode 2 pseudomode 3 pseudomode 4
0.8 0.6 0.4 0.2 0 -0.2 -0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
z/L Fig. 9. Pseudo-modes of a generic link for ni = 4.
1 1 0 ^ 1 ðzÞ m1 ðzÞ m C C Bm B B ^ 2 ðzÞ C B m2 ðzÞ C C ¼ TB C; B @ A @ A ^ ni ðzÞ mni ðzÞ m 0
ð28Þ
with T such that
^ k ðLÞ ¼ m
1; k ¼ 1; 0; k – 1;
^ k ðz; tÞ @m ¼ 1; k ¼ 2; @z 0; k – 2; z¼L
ð29Þ
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^ k ðtÞ, it is easy to verify that (see Figs. 7–9), denoted the new Lagrangian variables by l
dðtÞ ¼ g1 ðtÞ;
cðtÞ ¼ g2 ðtÞ:
ð30Þ
Hence d(t) and c(t) depend only on a single Lagrangian deformation variable (i.e. themselves are Lagrangian variables) and not on all the deformation Lagrangian variables as in the classic modal methods. Remark 6. If the proposed mode shape functions are used, according to (29), the matrices Ai given by (18) are very sparse and they always depend only on the deformation variable dini , whatever the number of the mode shape functions is; this fact leads to strongly reduce the dependency of the inertia matrix and of the vector of Coriolis and centrifugal terms on the Lagrangian deformation variables; in this way, significant improvements in terms of computational cost and memory usage are obtained. Remark 7. Moreover, the proposed pseudo-modes allow an easier solution of the inverse kinematic problem and an easier implementation of closed-loop control laws.
5. Validation and comparison results In this section some validation and comparison results are presented, which demonstrate the computational improvement obtained with the proposed method. As the computational complexity of the modeling methods depend, in general, also on the symbolic manipulation languages, on the programming languages and on the computers used, in order to clearly show the advantages of the proposed method, besides reporting tables with complexity indices (by comparing the obtained results with the ones obtained with the classic methods in different realistic cases, whose models are quite complex), the analytic expressions (suitably simplified) of the kinetic energy due to the tip mass in the case of a single and two-link robot is provided. (1) In the case of a single-link robot, the kinetic energy due to the tip mass M turns out to be: (a) Proposed pseudo-modal method The kinetic energy T is
T¼
i 1 h 2 M L1 þ d21 a_ 21 þ d_ 21 þ 2L1 a_ 1 d_ 1 ; 2
ð31Þ
(b) Classic modal method The kinetic energy T is obtained by posing in (31)
d1 ðtÞ ¼ m1L1 l1 ðtÞ þ m2L1 l2 ðtÞ þ þ mn1 L1 ln1 ðtÞ;
d_ 1 ðtÞ ¼ m1L1 l_ 1 ðtÞ þ m2L1 l_ 2 ðtÞ þ þ mn1 L1 l_ n1 ðtÞ:
ð32Þ
(2) In the case of a two-link robot, the kinetic energy due to the tip mass M turns out to be: (a) Proposed pseudo-modal method The kinetic energy T is
1 n 2 M L1 þ d21 a_ 21 þ d_ 21 þ 2L1 a_ 1 d_ 1 þ L22 þ d22 a_ 22 þ d_ 22 þ 2L2 a_ 2 d_ 2 þ 2½L1 d2 sinða1 a2 Þ 2L2 d1 sinða1 a2 Þ 2 þ2d1 d2 cosða1 a2 Þ þ L1 L2 cosða1 a2 Þa_ 1 a_ 2 þ 2½L1 cosða1 a2 Þ d1 sinða1 a2 Þa_ 1 d_ 2 þ 2½L2 cosða1 a2 Þ o þd2 sinða1 a2 Þd_ 1 a_ 2 þ 2 cosða1 a2 Þd_ 1 d_ 2 ;
T¼
ð33Þ
(b) Classic modal method The kinetic energy T is obtained by posing in (33)
d1 ðtÞ ¼ m11L1 l11 ðtÞ þ m12L1 l12 ðtÞ þ þ m1n1 L1 l1n1 ðtÞ; d2 ðtÞ ¼ m21L2 l21 ðtÞ þ m22L2 l22 ðtÞ þ þ m2n2 L1 l2n2 ðtÞ;
d_ 1 ðtÞ ¼ m11L1 l_ 11 ðtÞ þ m12L1 l_ 12 ðtÞ þ þ m1n1 L1 l_ 1n1 ðtÞ; d_ 2 ðtÞ ¼ m21L l_ 21 ðtÞ þ m22L l_ 22 ðtÞ þ þ m2n L l_ 2n ðtÞ: 2
2
2 2
ð34Þ
2
The reader or everybody wants to apply the proposed results, by using the above expressions of T, can easily evaluate for different values of n1 and n2 the advantages of the proposed method with respect to the classic modal one. Now consider a robot constituted by flexible links made of aluminium having square hollow constant cross-section (see Fig. 10) with Li = 2 m, li = 20 mm, si = 1 mm, Ei = 6.4 1010 N/m2, qi = 2.7 103 kg/m3, where qi denotes the density of the ith flexible link. In Table 1 (Table 2) the number of multiplies to evaluate the inertia matrix B (the gradient of the kinetic energy c) required by the classic assumed modes method and by the proposed one are compared, under the assumption that the number of modes per link is 4 and by increasing the number of links.
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Fig. 10. i-th flexible link having square hollow constant cross-section.
Table 1 Number of multiplies to evaluate the inertia matrix B with 4 modes per link.
2 links 3 links 4 links
Proposed modal method
Classic modal method
Ratio
138 362 683
407 1178 2308
2.95 3.25 3.38
Table 2 Number of multiplies to evaluate the gradient of the kinetic energy c with 4 modes per link.
2 link 3 link 4 link
Proposed modal method
Classic modal method
Ratio
336 896 1770
1052 2887 5624
3.13 3.22 3.18
Table 3 Number of multiplies to evaluate the inertia matrix B with 3 links.
3 modes 4 modes 5 modes
Proposed modal method
Classic modal method
Ratio
268 356 460
734 1178 1849
2.74 3.31 4.02
Table 4 Number of multiplies to evaluate the gradient of the kinetic energy c with 3 links.
3 modes 4 modes 5 modes
Proposed modal method
Classic modal method
Ratio
691 898 1150
1786 2887 4391
2.58 3.21 3.82
Note that the computational advantage of the proposed method is really notable (over 300%). In Table 3 (Table 4) the number of multiplies to evaluate the inertia matrix B (the gradient of the kinetic energy c) required by the classic assumed modes method and by the proposed one are compared, under the assumption that the number of links is 3 and by increasing the number of modes per link. Note that the computational advantage increases when the number of modes per link increases. The obtained computational improvement allows reducing the time required by the dynamic simulation of the robot and allows improving the implementation of the robot dynamic model in terms of memory usage. Remark 8. The proposed method can be extended also to spatial flexible robots. To easily convince yourself of this fact, it is important to note that if the section of a generic link has two axes of symmetry n, g orthogonal to axis f of the link, it is wellknown by the experts of building science that the cutting center coincides with the torsional center and the modes of the link
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are the union of the flexion modes in the plane n, f, of the flexion modes in the plane g, f and of the torsion modes around axis f. By linearly combining the modes in the plane n, f and the modes in the plane g, f as it has been made in the case of planar robots and the torsion modes such that the terminal torsion angle of each link coincides with a Lagrangian variable, it is easy to realize how it is possible to obtain, also in the spatial case, a more computationally efficient method with respect to the classic modal method.
6. Conclusions In this paper a method to obtain a closed-form model of a planar flexible robot with any number of links has been presented. Moreover, a method to improve the computational efficiency of the assumed modes method has been illustrated. The examples reported show that the computational efficiency of the presented method is really notable and it increases with the number of modes per link. Related results can be found in [27]; moreover, authors are successfully working in order to introduce the internal friction into the model and in order to design efficient control laws for flexible robots having small damping. References [1] W.J. Book, O. Maizza-Neto, D.E. Whitney, Feedback control of two-beam two-joint systems with distributed flexibility, ASME J. Dyn. Syst. Meas. Control (1975) 424–431. [2] M.J. Balas, Feedback control of flexible systems, IEEE Trans. Autom. Control 23 (1978) 673–679. [3] A. 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