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Closed form model of manipulators with highly flexible links
8th Vienna International Conference on Mathematical Modelling 8th Vienna18International Conference on Mathematical Modelling February - 20, 2015. Vienna University of Technology, Vienna, 8th Vienna Vienna International International Conference on Mathematical Mathematical Modelling 8th Conference on Modelling February - 20, 2015. Vienna University of Technology, Vienna, 8th Vienna18International Conference on Mathematical Modelling Austria Available online at www.sciencedirect.com February 18 20, 2015. Vienna University of Technology, February 18 18 -- 20, 20, 2015. 2015. Vienna Vienna University University of of Technology, Technology, Vienna, Vienna, Austria February Vienna, Austria Austria Austria
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IFAC-PapersOnLine 48-1 (2015) 653–654
Closed Closed Closed Closed
form model of manipulators form model of manipulators form model of form model of manipulators manipulators highly flexible links highly flexible links highly flexible links highly flexible links ∗ ∗
with with with with
Bruno Scaglioni ∗ Gianni Ferretti ∗ Bruno Scaglioni ∗∗ Gianni Ferretti ∗∗ Bruno Bruno Scaglioni Scaglioni ∗ Gianni Gianni Ferretti Ferretti ∗ ∗ Politecnico di Milano, Dipartimento di Elettronica, Informazione e ∗ Politecnico di Milano, Dipartimento Elettronica, Informazione ∗ Bioingegneria, 20133 Milan, Piazza di Leonardo da Vinci 32, Italy ee ∗ Politecnico di Dipartimento di Elettronica, Informazione ∗ Politecnico di Milano, Milano, Dipartimento di Elettronica, Informazione Bioingegneria, 20133 Milan, Piazza Leonardo da Vinci 32, Italy e (e-mail: 20133 [bruno.scaglioni,gianni.ferretti]polimi.it). Bioingegneria, Milan, Bioingegneria, Milan, Piazza Piazza Leonardo Leonardo da da Vinci Vinci 32, 32, Italy Italy (e-mail: 20133 [bruno.scaglioni,gianni.ferretti]polimi.it). (e-mail: (e-mail: [bruno.scaglioni,gianni.ferretti]polimi.it). [bruno.scaglioni,gianni.ferretti]polimi.it). © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. A closed form dynamic model of manipulators with highly A closedlinks formisdynamic model of manipulators with highly flexible presented in this abstract. The model is A closed closed form form dynamic dynamic model of manipulators manipulators with highly A model of with highly flexible links is presented in this abstract. The model is based on a Newton-Euler formulation of motion equations flexible links is presented in this abstract. The model is flexible links is presentedformulation in this abstract. The equations model is based on a Newton-Euler of motion and a substructuring approach is used to account for based on a Newton-Euler formulation of motion equations based on a Newton-Euler formulation of motion equations and substructuring is used toin account for large a deformations. Theapproach model, formulated closed form and substructuring approach is for and aadeformations. substructuring approach is used used to toin account account for large The model, formulated closed form with respect to joints and elastic coordinates, accounts large deformations. deformations. The The model, model, formulated formulated in closed form large in closed form with respect to joints andterms. elasticThe coordinates, accounts also for quadratic velocity formulation of the with respect to joints joints and elastic elastic coordinates, coordinates, accounts with respect to and accounts also for quadratic velocity terms. The formulation of the motion equations starts from a data set which can be also for quadratic velocity terms. The formulation of the the also for equations quadratic velocity terms. The formulation of motion starts from a data set which be either analytically numerically computed by FEcan codes. motion equations or starts from aa data data set which which can be motion equations starts from set can be either analytically numerically computed by simulation FE codes. Validation has beenor out bycomputed comparing either analytically analytically orcarried numerically by FE codes. either or numerically computed by FE codes. Validation carried out by comparing results withhas twobeen different multibody softwares. simulation Validation has been carried out by by comparing comparing simulation Validation carried out results withhas twobeen different multibody softwares. simulation results with two different multibody softwares. results with twofloating different multibody softwares. In the classical frame of reference (FFR) approach In the classical floating frame of referenceof(FFR) approach (Shabana (1998)), where superposition large body moIn the classical floating frame of reference (FFR) approach In the classical floating frame of referenceof(FFR) approach (Shabana (1998)), where superposition large body motion and small linear deformations expressed in local refer(Shabana (1998)), where superposition of large body mo(Shabana (1998)), where superposition of large bodyrefermotion and small linear deformations in local ence frame is considered, every linkexpressed i is characterized by a tion and and small linear deformations deformations expressed in local local referrefertion small linear expressed in ence is considered, linkin i isFig.1, characterized by a local frame FFR {O , xi , yi , zi } every as shown while another ence frame is iconsidered, considered, every link ii is is characterized characterized by aa ence is every link by local frame FFR {O , xi , yi , zi } as shown in Fig.1, while another i frame {O , x , y , z } placed at the link tip, is here local FFR {O , x , y , z } as shown in Fig.1, while another ii , i xii , i yii , i zii }i as shown in Fig.1, while another local local FFR frame{O {O i i , ix i , iyi , izi } placed at the link tip, is here considered, with the of link the FFR inhere the local frame {O placed at the tip, is i , x i , ysame i , zi } orientation local frame with {Oii , xthe placed at the link FFR tip, isinhere ii , ysame ii , zii } orientation considered, the undeformed configuration. The angle θiof isthe theFFR coordinate considered, with the same orientation of the in the considered, with the same orientation of the FFR in the undeformed configuration. The angle θ is the coordinate i of the joint connecting link The i to link i −θθ1, while zˆi−1 is the undeformed configuration. angle the coordinate ii is undeformed configuration. The angle is the coordinate of theofjoint connecting link i to 1, zˆi−1 the i while axis rotation in the frame {Olink , iix− , yi−1 ,z }.is of connecting link link − 1, is i−1 ˆi−1 of the theofjoint joint connecting link ii to to 1,,while while zˆ is the the i−1 axis rotation in the frame {Olink , ix−i−1 y , }. i−1 i−1 i−1 i−1 axis ,, x ,, z }. i−1 , yi−1 i−1 axis of of rotation rotation in in the the frame frame {O {Oi−1 x , y z }. i−1 i−1 i−1 i−1 i−1 i−1 i−1 i−1
Fig. 1. Reference frames and substructures Fig. 1. Reference frames and substructures Fig. Fig. 1. 1. Reference Reference frames frames and and substructures substructures The motion equations for link i, expressed in the local The motion equations for link i, expressed in the local reference frame, can be then developed using the principle The motion equations for i, in local The motion equations for link link i, expressed expressed in the the local reference frame, can be then developed using the principle of virtual frame, work (Meirovitch (1967); Shabana (1998)) reference can developed using the reference can be be then then(1967); developed using (1998)) the principle principle of virtual frame, work (Meirovitch Shabana of (1967); Shabana ofvirtual virtual work work (Meirovitch (Meirovitch Shabana (1998)) (1998)) (1967); T ˜ T ¯ ¯ v ¯˙ i 03 C i U mi d ˜¯TTC,i T ¯t,i m ˙˙ i T ¯ 0 v m˜ C ˜ T 3 i U mi d T C,i t,i ¯ ¯ 0 = ω ¯ T v U m C d ¯ ¯ mm ¯ ¯ ˜ i T ˙ 3 i i i 033 t,i C ω ii¯ ¯˙ iii C,i miiJ iC,i v U d C,i t,i r,i 0 ˜ = T mmii d C,i C t,i ¯ ¯ ˙ J d −K q − D q q˙¨ = ˜ C,i e,i i03 e,i i ¯ T ω ¯ii M ¯ ¯d ¯J ˜¯ T mC C 3 De,i q˙i C ¯q˙¨iiiii = T ii¯d C,i −Ke,i qi0− m t,i e,i ω 3 ¯ ¯r,i r,i C C,i ¯J¯r,i r,i i t,i C,i ii −K q − D q M C C r,i ¨ q e,i i e,i r,i e,i i −K q − D q˙˙i ¨ ¯ ¯ q e,i ii e,i M C C e,i e,i ii ¯t,i ¯r,i hre,i ii hrω,i t,i r,i e,i Me,i C C t,i r,i e,i r r hrθe,i hrθω,i + h (1) h r r + hre,i hrω,i θ θ ω,i e,i (1) ω,i e,i h h + + hθfω,i θ f e,i θ θ θ θ + h ω,i he,i f + f (1) ω,i e,i + (1) +h ω,i e,i hω,i h f f e,i f f f h h e,i hfω,i h ω,i e,i ω,i e,iMi ×Mi Mi ×M i ×Mi i where Me,i ∈ RM Mi ×Mi , De,i ∈ RMi ×Mi , Ke,i ∈ RMi ×Mi where M ∈ R , D ∈ R , K ∈ RM e,i e,i e,i M ×M M ×M i mass, i i and i stiffness matrix i ×Mi are theM damping where ∈ RM ∈ RM Mii ×M ×Mii ,, D Mii ×M ×Mii ,, K Mii ×M ×Mii e,i e,i ∈ where Mstructural De,i ∈ R RM K e,i e,i are the structural mass, damping and stiffness e,i ∈ R e,i e,i ∈ 3Rmatrix r e,i θ respectively, regarding the damping flexible dofs, hstiffness ∈ R , h are the structural mass, and matrix ω,i r θ ∈ are the structural mass, andhstiffness respectively, regarding the damping flexible dofs, ∈ R33 , matrix hω,i ∈ r θ ω,i ω,i r ∈ R3 θ ∈ respectively, regarding the flexible dofs, h , h r 3 θ ω,i ∈ respectively, regarding the flexible dofs, hω,i ω,i ω,i ∈ R , hω,i ω,i
Table Table Table Table
1. 1. 1. 1.
Link data Link data LinkNumber data of modal coords. Link data
Mi 1 i 2 3 , I 4 , I5 , I6 , I7 , I8 , I9 IM M i1 i, Ii2 , Ij,i i j,i i i j,i 9 M , I3 , I44i , I55j,i , I66i , I77i , I88j,i , Ijk,i IM ii, I2 9 i1 jk,i 1e,i 3 6 , I7 , I8 9 ,, IIi2i2 ,, II3j,i IK 3 ,,, I 5 ,,, I 8 ,,, I 9 I4ii4 ,,, III5j,i j,i j,i j,i , Iˆii , Ij,i I IIii6i ,, IIii7i ,, IIj,i IIjk,i IIKiii1e,i jk,i j,i i j,i j,i jk,i Sie,i , Si K Kie,i ˆi e,i K S , S ¯ u ˆ 0i Sii ,, S S ˆ ii ˆ S ¯ˆ i0i, S S u i z ¯ u 0i ¯ˆi−1 u 0i ¯ u z 0i i−1 ˆ z zˆ ˆi−1 i−1 f i−1 3z M
Number of modal coords. Inertia invariants Number of modal coords. Number of Number of modal modal coords. coords. Inertia invariants Structural stiffness matrix Inertia invariants Inertia invariants Inertia invariants Structural stiffness matrix Shape functions matrices Structural stiffness matrix Structural stiffness matrix Structural stiffness matrix Shape functions matrices rel. position between FFRs Shape functions matrices Shape functions matrices Shape functions matrices rel. position between FFRs Joint rotation axis rel. position between FFRs rel. position position between FFRs rel. between Joint rotation axis FFRs Joint rotation axis Joint rotation axis Joint rotation axis
R3 , hω,i ∈ RMii are the vectors of gyroscopic and cenR33 , hffω,i ∈ RM are rthe vectors of gyroscopic and ceni f M θ gyroscopic Mi f R , h ∈ R vectors ceni 3 Mand i arehthe ∈ R33 , hof ∈ R33 , hffe,iand ∈ R tripetal ω,i R , h ∈ R are the vectors of and cene,i e,i ω,i terms r θ gyroscopic Mi ω,i ∈ R , h ∈ R , h ∈ R tripetal terms and hre,i f 3 θ 3 M e,i e,i f r 3 , hθ θ applied 3 , at Miii f are the vectors of external forces, the∈ body ∈ R ∈ R h R tripetal terms and h r 3 3 M e,i e,i e,i ∈ R , h ∈ R , h ∈ R tripetal terms and h e,i e,i e,i are the vectors of external forces, e,i e,i applied ate,ithe body connectors. The ofterms of the generalized mass matrix, are external forces, applied at body are the the vectors vectors ofterms external forces, applied mass at the thematrix, body connectors. Thegyroscopic of and the generalized as well as the centripetal terms, can be connectors. The terms of the generalized mass matrix, connectors. The terms of the generalized mass matrix, as well as the gyroscopic and centripetal terms, can be computed from data Table 1 asterms, described in as well the and centripetal can as well as as starting the gyroscopic gyroscopic and of centripetal terms, can be be computed starting from data of Table 1 as described in Ferretti et starting al. (2014). computed from data of Table 1 as described computed from data of Table 1 as described in in Ferretti et starting al. (2014). Ferretti (2014). Ferretti et et aal. al.spatial (2014).vector notation (Fijany and FeatherAdopting Adopting a spatial vector notation (Fijany and Featherstone (2013)), thus defining: Adopting aa spatial vector notation and FeatherAdopting spatial vector notation (Fijany (Fijany defining: and Featherstone (2013)), thus stone (2013)), thus defining: i defining: stone (2013)), ω ∈ R6 , F = nii ∈ R6 Vi =thus (2) ω fii ∈ R66 viii ∈ R66 , Fii = n n Vi = ω (2) ω n 6 , 6 f v V = F = ∈ R ∈ R ii ii 6 6 i i i i Vii =vi v , Fii =andf (2) ∈ R angular ∈ Rvelocities(2) ii ii vare f the linear of where ωi and i i and v are the angular and linear velocities where ω i i each FFR expressed inthe theangular global and reference frame and nof and v are linear velocities where ω i i vii are inthe angular and linearframe velocities nof ofii where ωii and eachfFFR expressed the global reference and the force and torque applied fromframe link i and to link each FFR in the global reference and n i are expressed each FFR expressed in the global reference frame and n areOthe force and torque fromand link dynamic i to linkiii iat iand − 1f all theapplied kinematic and f the force and torque applied from link ii to i respectively, ii are and f are the force and torque applied from link to link link i − 1 iat Orelative all the kinematic dynamic i respectively, equations to the each single link canand be collected ii − 11 at O all the kinematic and dynamic ii respectively, − at O respectively, all the kinematic and dynamic equations to the each single linkform canmodel be collected i into globalrelative equations, defining a closed of the equations relative to each link can be equations relative to the the each single single linkform canmodel be collected collected into global equations, defining a closed of the flexible manipulator: into global equations, defining aa closed form model of into global equations, defining closed form model of the the flexible manipulator: flexible manipulator: ¨ ˙ flexible manipulator: ˙ = τ (3) Mθθ (θ, q)θ + Mθq (θ, q)q¨ + C θ (θ, q, θ, ˙ q) ˙ = τ (3) Mθθ (θ, q)訨 + Mθq (θ, q)q¨ + C θ (θ, q, θ, ˙˙ q) ¨ ˙˙ = (θ, q) θ + M (θ, q) q + C (θ, q, θ, q) M θq θ ¨ ¨ (θ, q) θ + M (θ, q) q + C (θ, q, θ, q) = τ τ (3) (3) MTθθ θθ θq θ θθ θq θ ¨ + Mqq (θ, q)q¨ + D e q˙ + Ke q+ Mθq T (θ, q)θ ¨ (θ, q)θ¨ + Mqq (θ, q)q¨ + D e q˙ + Ke q+ MTθq T ˙˙ + (θ, q)q¨¨ + q+ K M T (θ, q)θ ˙ q) ¨ + Mqq ˙ = 0 (4) θ, qq (θ, q)q +D D+eee q q+ qC + Keeq, Mθq θq q (θ, θq (θ, q)θ + Mqq ˙ q) ˙ = 0 (4) + C q (θ, eq, θ, ˙˙ q) ˙˙the C (θ, q, θ, = (4) Where θ = col(θi ), q = col(q+ ), q being q i Cq i q, θ, q) q (θ, + =0 0vector (4) Where θ = col(θ ), q = col(q ), q being the vector i i i of elastic coordinates of link i. It must be recalled that Where θ = col(θ ), q = col(q ), q being the vector ii ), q = col(qii ), qii being the vector Where θ = col(θ of elastic coordinates of link matrices i. Iti must recalled that, i equations (3,4), in iparticular Mbe Mθq , M of elastic of i. be that θθ , recalled qq of elastic coordinates coordinates of link link matrices i. It It must must that equations (3,4), in particular Mbe Mθq , closed M θθ , recalled qq , D , K and vectors C and C , can be computed in equations (3,4), in particular M , M , M , e e θ qmatrices θθ θq qq equations (3,4), in particular matrices M , M , M θθ θq qq , D , K and vectors C and C , can be computed in closed θθ θq qq e starting e θ links data q form from the summarized in Table 1. D , K and vectors C and C , can be computed in closed e e θ q D , K and vectors C and C , can be computed in closed e starting e q form from theθθ links data summarized in Table 1. e e q form the summarized in 1. form starting starting from the links links data summarized in Table Tablenot 1. However, the from standard FFRdata approach is generally However, the standard FFR approach is generally not applicable in the case of large deflections, thus when the However, the standard FFR approach is generally not However, the standard approach is generally not applicable in the case are of FFR large deflections, the elastic displacements not “small”. In thus this when case, the applicable in the case of large deflections, thus when applicable in the case of large deflections, thus when the elastic displacements are not “small”. In this case, absolutedisplacements nodal coordinates formulation (ANCF) has been elastic are “small”. In the elastic displacements are not not “small”. (ANCF) In this this case, case, the absolute nodal coordinates formulation has been proposed (Yakoub and Shabana (1999)), requiring the absolute nodal coordinates formulation (ANCF) has absolute nodal coordinates formulation (ANCF) has been been proposed (Yakoub and Shabana (1999)), requiring the definition of global shape functions for every element with proposed (Yakoub and Shabana (1999)), requiring the proposed (Yakoub and Shabana (1999)), requiring the definition of global shape functions for every element with respect to of theglobal absolute reference frame. On theelement other hand, definition shape functions for every with definition of global shape functions for every element with respect to the absoluteisreference On the other hand, the ANCF approach actuallyframe. only suitable for beams respect to absolute frame. On hand, respect to the the absoluteisreference reference frame. On the the other other hand, the ANCF approach actually only suitable forcomplex beams and shells elements, but it is hardly applicable to the ANCF approach is actually only suitable for beams the ANCF approach is actually only suitable for beams and shells elements, but it is hardly applicable to complex shapes andelements, cannot benefit of hardly the results of FEM and shells but applicable to complex and shells but it it is is applicable to analysis. complex shapes andelements, of hardly the results of FEM Accuracy ofcannot resultsbenefit for large deformation fieldsanalysis. can be shapes and cannot benefit of the results of FEM analysis. shapes and cannot benefit of the results of FEM analysis. Accuracy of results for large deformation fields can be Accuracy of results for large deformation fields Accuracy of results for large deformation fields can can be be
Copyright © 2015, 2015,IFAC IFAC (International Federation of Automatic Control) 653Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © Copyright © 2015, IFAC 653 Peer review under responsibility of International Federation of Automatic Copyright © 2015, IFAC 653 Copyright © © 2015, 2015, IFAC IFAC 653Control. Copyright 653 10.1016/j.ifacol.2015.05.013
MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria 654 Bruno Scaglioni et al. / IFAC-PapersOnLine 48-1 (2015) 653–654
Fig. 3. Angle of the second joint
Fig. 2. Angle of the first joint anyway achieved in the context of the FFR approach by means of the substructuring technique (Shabana (1985)) as demonstrated in the context of this work. Highly flexible bodies can thus be substructured in several elements, applying on each element the theory of linear elasticity. To this aim, link i can be subdivided into more rigidly connected flexible elements or substructures, as shown in Fig. 1, each one described through a FFR approach introducing Mk modal coordinates. Define with Ne the total number of elements of the whole manipulator N e and with M = k=1 Mk the total number of modal coordinates. Let θi = θk¯e , i = 1, . . . , N denote the “real” joint angle between element k¯ − 1 and k¯ and consider a “dummy” joint angle θke if a rigid connection exists between element k − 1 and k. Then, the model remains formally identical by removing the columns corresponding to the “dummy” joint angles in the assembly matrices. The developed model has been validated by comparison of simulation results obtained with a Matlab/Simulink implementation of the closed form model with multibody simulations, obtained with Modelica/Dymola and with MSC/Adams. The benchmark has been taken from Yakoub and Shabana (1999), where the ANCF is used to describe links compliance in the case of a flexible double pendulum. The structure consists of two flexible bodies connected together and to the ground by revolute joints, initially in horizontal position and free to fall under the effect of gravity. The physical parameters, summarized in Table 2, have been chosen in order to allow large deformation on the second pendulum. The models, developed in the different environments, implements substructuring by subdividing the first link in two elements with length 0.1 m and the second link in 12 elements with lentgh 0.075 m. All figures show comparisons among Matlab/Simulink simulation (solid line), Modelica/Dymola simulations (dashed line) and MSC/Adams simulations (dotted line). It must be pointed out that the Modelica model accounts for damping in exactly the same way as the closed form model, while Adams adopts a different approach, hence, small differences in the simulation results are appreciable. Figures 2 and 3 show the relative angle of the first and second joint, while figure 4 shows the transverse deflection of the tip point of the second link. As it is apparent, results are in good accordance, in particular, the results obtained from Matlab/Simulink and Modelica/Dymola simulations are undistinguishable, while the results provided by MSC/Adams shows some small differences. A sequence of 3D snapshots of the simulation at intermediate instants is shown in Fig. 5. 654
Fig. 4. Transverse deflection of second pendulum tip
Fig. 5. 3D representation of the simulation Table 2. Double Pendulum data Property
Body 1
body 2
Mass (Kg) Length(m) Cross sectional Area(m2 ) Second moment of area (m4 ) Mass moment of inertia (Kg × m2 ) Modulus of elasticity (N/m2 )
0.6810972 0.2 1.26E − 03 1.26E − 07 2.27E − 03 8.00E + 07
2.4740052 0.9 1.26E − 03 1.26E − 07 1.67E − 01 5.00E + 05
REFERENCES Ferretti, G., Leva, A., and Scaglioni, B. (2014). Objectoriented modelling of general flexible multibody systems. Mathematical and Computer Modelling of Dynamical Systems, 20(1), 1–22. Fijany, A. and Featherstone, R. (2013). A new factorization of the mass matrix for optimal serial and parallel calculation of multibody dynamics. Multibody System Dynamics, 29, 169–187. Meirovitch, L. (1967). Analytical Methods in Vibration. Macmillan Publishing, New York. Shabana, A.A. (1998). Dynamics of Multibody Systems. Cambridge University Press, New York. Shabana, A. (1985). Substructure synthesis methods for dynamic analysis of multi-body systems. Computers and Structures, 20(4), 737 – 744. Yakoub, R. and Shabana, A. (1999). Use of Cholesky coordinates and the absolute nodal coordinate formulation in the computer simulation of flexible multibody systems. Nonlinear Dynamics, 20(3), 267–282.
ScienceDirect
IFAC-PapersOnLine 48-1 (2015) 653–654
Closed Closed Closed Closed
form model of manipulators form model of manipulators form model of form model of manipulators manipulators highly flexible links highly flexible links highly flexible links highly flexible links ∗ ∗
with with with with
Bruno Scaglioni ∗ Gianni Ferretti ∗ Bruno Scaglioni ∗∗ Gianni Ferretti ∗∗ Bruno Bruno Scaglioni Scaglioni ∗ Gianni Gianni Ferretti Ferretti ∗ ∗ Politecnico di Milano, Dipartimento di Elettronica, Informazione e ∗ Politecnico di Milano, Dipartimento Elettronica, Informazione ∗ Bioingegneria, 20133 Milan, Piazza di Leonardo da Vinci 32, Italy ee ∗ Politecnico di Dipartimento di Elettronica, Informazione ∗ Politecnico di Milano, Milano, Dipartimento di Elettronica, Informazione Bioingegneria, 20133 Milan, Piazza Leonardo da Vinci 32, Italy e (e-mail: 20133 [bruno.scaglioni,gianni.ferretti]polimi.it). Bioingegneria, Milan, Bioingegneria, Milan, Piazza Piazza Leonardo Leonardo da da Vinci Vinci 32, 32, Italy Italy (e-mail: 20133 [bruno.scaglioni,gianni.ferretti]polimi.it). (e-mail: (e-mail: [bruno.scaglioni,gianni.ferretti]polimi.it). [bruno.scaglioni,gianni.ferretti]polimi.it). © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. A closed form dynamic model of manipulators with highly A closedlinks formisdynamic model of manipulators with highly flexible presented in this abstract. The model is A closed closed form form dynamic dynamic model of manipulators manipulators with highly A model of with highly flexible links is presented in this abstract. The model is based on a Newton-Euler formulation of motion equations flexible links is presented in this abstract. The model is flexible links is presentedformulation in this abstract. The equations model is based on a Newton-Euler of motion and a substructuring approach is used to account for based on a Newton-Euler formulation of motion equations based on a Newton-Euler formulation of motion equations and substructuring is used toin account for large a deformations. Theapproach model, formulated closed form and substructuring approach is for and aadeformations. substructuring approach is used used to toin account account for large The model, formulated closed form with respect to joints and elastic coordinates, accounts large deformations. deformations. The The model, model, formulated formulated in closed form large in closed form with respect to joints andterms. elasticThe coordinates, accounts also for quadratic velocity formulation of the with respect to joints joints and elastic elastic coordinates, coordinates, accounts with respect to and accounts also for quadratic velocity terms. The formulation of the motion equations starts from a data set which can be also for quadratic velocity terms. The formulation of the the also for equations quadratic velocity terms. The formulation of motion starts from a data set which be either analytically numerically computed by FEcan codes. motion equations or starts from aa data data set which which can be motion equations starts from set can be either analytically numerically computed by simulation FE codes. Validation has beenor out bycomputed comparing either analytically analytically orcarried numerically by FE codes. either or numerically computed by FE codes. Validation carried out by comparing results withhas twobeen different multibody softwares. simulation Validation has been carried out by by comparing comparing simulation Validation carried out results withhas twobeen different multibody softwares. simulation results with two different multibody softwares. results with twofloating different multibody softwares. In the classical frame of reference (FFR) approach In the classical floating frame of referenceof(FFR) approach (Shabana (1998)), where superposition large body moIn the classical floating frame of reference (FFR) approach In the classical floating frame of referenceof(FFR) approach (Shabana (1998)), where superposition large body motion and small linear deformations expressed in local refer(Shabana (1998)), where superposition of large body mo(Shabana (1998)), where superposition of large bodyrefermotion and small linear deformations in local ence frame is considered, every linkexpressed i is characterized by a tion and and small linear deformations deformations expressed in local local referrefertion small linear expressed in ence is considered, linkin i isFig.1, characterized by a local frame FFR {O , xi , yi , zi } every as shown while another ence frame is iconsidered, considered, every link ii is is characterized characterized by aa ence is every link by local frame FFR {O , xi , yi , zi } as shown in Fig.1, while another i frame {O , x , y , z } placed at the link tip, is here local FFR {O , x , y , z } as shown in Fig.1, while another ii , i xii , i yii , i zii }i as shown in Fig.1, while another local local FFR frame{O {O i i , ix i , iyi , izi } placed at the link tip, is here considered, with the of link the FFR inhere the local frame {O placed at the tip, is i , x i , ysame i , zi } orientation local frame with {Oii , xthe placed at the link FFR tip, isinhere ii , ysame ii , zii } orientation considered, the undeformed configuration. The angle θiof isthe theFFR coordinate considered, with the same orientation of the in the considered, with the same orientation of the FFR in the undeformed configuration. The angle θ is the coordinate i of the joint connecting link The i to link i −θθ1, while zˆi−1 is the undeformed configuration. angle the coordinate ii is undeformed configuration. The angle is the coordinate of theofjoint connecting link i to 1, zˆi−1 the i while axis rotation in the frame {Olink , iix− , yi−1 ,z }.is of connecting link link − 1, is i−1 ˆi−1 of the theofjoint joint connecting link ii to to 1,,while while zˆ is the the i−1 axis rotation in the frame {Olink , ix−i−1 y , }. i−1 i−1 i−1 i−1 axis ,, x ,, z }. i−1 , yi−1 i−1 axis of of rotation rotation in in the the frame frame {O {Oi−1 x , y z }. i−1 i−1 i−1 i−1 i−1 i−1 i−1 i−1
Fig. 1. Reference frames and substructures Fig. 1. Reference frames and substructures Fig. Fig. 1. 1. Reference Reference frames frames and and substructures substructures The motion equations for link i, expressed in the local The motion equations for link i, expressed in the local reference frame, can be then developed using the principle The motion equations for i, in local The motion equations for link link i, expressed expressed in the the local reference frame, can be then developed using the principle of virtual frame, work (Meirovitch (1967); Shabana (1998)) reference can developed using the reference can be be then then(1967); developed using (1998)) the principle principle of virtual frame, work (Meirovitch Shabana of (1967); Shabana ofvirtual virtual work work (Meirovitch (Meirovitch Shabana (1998)) (1998)) (1967); T ˜ T ¯ ¯ v ¯˙ i 03 C i U mi d ˜¯TTC,i T ¯t,i m ˙˙ i T ¯ 0 v m˜ C ˜ T 3 i U mi d T C,i t,i ¯ ¯ 0 = ω ¯ T v U m C d ¯ ¯ mm ¯ ¯ ˜ i T ˙ 3 i i i 033 t,i C ω ii¯ ¯˙ iii C,i miiJ iC,i v U d C,i t,i r,i 0 ˜ = T mmii d C,i C t,i ¯ ¯ ˙ J d −K q − D q q˙¨ = ˜ C,i e,i i03 e,i i ¯ T ω ¯ii M ¯ ¯d ¯J ˜¯ T mC C 3 De,i q˙i C ¯q˙¨iiiii = T ii¯d C,i −Ke,i qi0− m t,i e,i ω 3 ¯ ¯r,i r,i C C,i ¯J¯r,i r,i i t,i C,i ii −K q − D q M C C r,i ¨ q e,i i e,i r,i e,i i −K q − D q˙˙i ¨ ¯ ¯ q e,i ii e,i M C C e,i e,i ii ¯t,i ¯r,i hre,i ii hrω,i t,i r,i e,i Me,i C C t,i r,i e,i r r hrθe,i hrθω,i + h (1) h r r + hre,i hrω,i θ θ ω,i e,i (1) ω,i e,i h h + + hθfω,i θ f e,i θ θ θ θ + h ω,i he,i f + f (1) ω,i e,i + (1) +h ω,i e,i hω,i h f f e,i f f f h h e,i hfω,i h ω,i e,i ω,i e,iMi ×Mi Mi ×M i ×Mi i where Me,i ∈ RM Mi ×Mi , De,i ∈ RMi ×Mi , Ke,i ∈ RMi ×Mi where M ∈ R , D ∈ R , K ∈ RM e,i e,i e,i M ×M M ×M i mass, i i and i stiffness matrix i ×Mi are theM damping where ∈ RM ∈ RM Mii ×M ×Mii ,, D Mii ×M ×Mii ,, K Mii ×M ×Mii e,i e,i ∈ where Mstructural De,i ∈ R RM K e,i e,i are the structural mass, damping and stiffness e,i ∈ R e,i e,i ∈ 3Rmatrix r e,i θ respectively, regarding the damping flexible dofs, hstiffness ∈ R , h are the structural mass, and matrix ω,i r θ ∈ are the structural mass, andhstiffness respectively, regarding the damping flexible dofs, ∈ R33 , matrix hω,i ∈ r θ ω,i ω,i r ∈ R3 θ ∈ respectively, regarding the flexible dofs, h , h r 3 θ ω,i ∈ respectively, regarding the flexible dofs, hω,i ω,i ω,i ∈ R , hω,i ω,i
Table Table Table Table
1. 1. 1. 1.
Link data Link data LinkNumber data of modal coords. Link data
Mi 1 i 2 3 , I 4 , I5 , I6 , I7 , I8 , I9 IM M i1 i, Ii2 , Ij,i i j,i i i j,i 9 M , I3 , I44i , I55j,i , I66i , I77i , I88j,i , Ijk,i IM ii, I2 9 i1 jk,i 1e,i 3 6 , I7 , I8 9 ,, IIi2i2 ,, II3j,i IK 3 ,,, I 5 ,,, I 8 ,,, I 9 I4ii4 ,,, III5j,i j,i j,i j,i , Iˆii , Ij,i I IIii6i ,, IIii7i ,, IIj,i IIjk,i IIKiii1e,i jk,i j,i i j,i j,i jk,i Sie,i , Si K Kie,i ˆi e,i K S , S ¯ u ˆ 0i Sii ,, S S ˆ ii ˆ S ¯ˆ i0i, S S u i z ¯ u 0i ¯ˆi−1 u 0i ¯ u z 0i i−1 ˆ z zˆ ˆi−1 i−1 f i−1 3z M
Number of modal coords. Inertia invariants Number of modal coords. Number of Number of modal modal coords. coords. Inertia invariants Structural stiffness matrix Inertia invariants Inertia invariants Inertia invariants Structural stiffness matrix Shape functions matrices Structural stiffness matrix Structural stiffness matrix Structural stiffness matrix Shape functions matrices rel. position between FFRs Shape functions matrices Shape functions matrices Shape functions matrices rel. position between FFRs Joint rotation axis rel. position between FFRs rel. position position between FFRs rel. between Joint rotation axis FFRs Joint rotation axis Joint rotation axis Joint rotation axis
R3 , hω,i ∈ RMii are the vectors of gyroscopic and cenR33 , hffω,i ∈ RM are rthe vectors of gyroscopic and ceni f M θ gyroscopic Mi f R , h ∈ R vectors ceni 3 Mand i arehthe ∈ R33 , hof ∈ R33 , hffe,iand ∈ R tripetal ω,i R , h ∈ R are the vectors of and cene,i e,i ω,i terms r θ gyroscopic Mi ω,i ∈ R , h ∈ R , h ∈ R tripetal terms and hre,i f 3 θ 3 M e,i e,i f r 3 , hθ θ applied 3 , at Miii f are the vectors of external forces, the∈ body ∈ R ∈ R h R tripetal terms and h r 3 3 M e,i e,i e,i ∈ R , h ∈ R , h ∈ R tripetal terms and h e,i e,i e,i are the vectors of external forces, e,i e,i applied ate,ithe body connectors. The ofterms of the generalized mass matrix, are external forces, applied at body are the the vectors vectors ofterms external forces, applied mass at the thematrix, body connectors. Thegyroscopic of and the generalized as well as the centripetal terms, can be connectors. The terms of the generalized mass matrix, connectors. The terms of the generalized mass matrix, as well as the gyroscopic and centripetal terms, can be computed from data Table 1 asterms, described in as well the and centripetal can as well as as starting the gyroscopic gyroscopic and of centripetal terms, can be be computed starting from data of Table 1 as described in Ferretti et starting al. (2014). computed from data of Table 1 as described computed from data of Table 1 as described in in Ferretti et starting al. (2014). Ferretti (2014). Ferretti et et aal. al.spatial (2014).vector notation (Fijany and FeatherAdopting Adopting a spatial vector notation (Fijany and Featherstone (2013)), thus defining: Adopting aa spatial vector notation and FeatherAdopting spatial vector notation (Fijany (Fijany defining: and Featherstone (2013)), thus stone (2013)), thus defining: i defining: stone (2013)), ω ∈ R6 , F = nii ∈ R6 Vi =thus (2) ω fii ∈ R66 viii ∈ R66 , Fii = n n Vi = ω (2) ω n 6 , 6 f v V = F = ∈ R ∈ R ii ii 6 6 i i i i Vii =vi v , Fii =andf (2) ∈ R angular ∈ Rvelocities(2) ii ii vare f the linear of where ωi and i i and v are the angular and linear velocities where ω i i each FFR expressed inthe theangular global and reference frame and nof and v are linear velocities where ω i i vii are inthe angular and linearframe velocities nof ofii where ωii and eachfFFR expressed the global reference and the force and torque applied fromframe link i and to link each FFR in the global reference and n i are expressed each FFR expressed in the global reference frame and n areOthe force and torque fromand link dynamic i to linkiii iat iand − 1f all theapplied kinematic and f the force and torque applied from link ii to i respectively, ii are and f are the force and torque applied from link to link link i − 1 iat Orelative all the kinematic dynamic i respectively, equations to the each single link canand be collected ii − 11 at O all the kinematic and dynamic ii respectively, − at O respectively, all the kinematic and dynamic equations to the each single linkform canmodel be collected i into globalrelative equations, defining a closed of the equations relative to each link can be equations relative to the the each single single linkform canmodel be collected collected into global equations, defining a closed of the flexible manipulator: into global equations, defining aa closed form model of into global equations, defining closed form model of the the flexible manipulator: flexible manipulator: ¨ ˙ flexible manipulator: ˙ = τ (3) Mθθ (θ, q)θ + Mθq (θ, q)q¨ + C θ (θ, q, θ, ˙ q) ˙ = τ (3) Mθθ (θ, q)訨 + Mθq (θ, q)q¨ + C θ (θ, q, θ, ˙˙ q) ¨ ˙˙ = (θ, q) θ + M (θ, q) q + C (θ, q, θ, q) M θq θ ¨ ¨ (θ, q) θ + M (θ, q) q + C (θ, q, θ, q) = τ τ (3) (3) MTθθ θθ θq θ θθ θq θ ¨ + Mqq (θ, q)q¨ + D e q˙ + Ke q+ Mθq T (θ, q)θ ¨ (θ, q)θ¨ + Mqq (θ, q)q¨ + D e q˙ + Ke q+ MTθq T ˙˙ + (θ, q)q¨¨ + q+ K M T (θ, q)θ ˙ q) ¨ + Mqq ˙ = 0 (4) θ, qq (θ, q)q +D D+eee q q+ qC + Keeq, Mθq θq q (θ, θq (θ, q)θ + Mqq ˙ q) ˙ = 0 (4) + C q (θ, eq, θ, ˙˙ q) ˙˙the C (θ, q, θ, = (4) Where θ = col(θi ), q = col(q+ ), q being q i Cq i q, θ, q) q (θ, + =0 0vector (4) Where θ = col(θ ), q = col(q ), q being the vector i i i of elastic coordinates of link i. It must be recalled that Where θ = col(θ ), q = col(q ), q being the vector ii ), q = col(qii ), qii being the vector Where θ = col(θ of elastic coordinates of link matrices i. Iti must recalled that, i equations (3,4), in iparticular Mbe Mθq , M of elastic of i. be that θθ , recalled qq of elastic coordinates coordinates of link link matrices i. It It must must that equations (3,4), in particular Mbe Mθq , closed M θθ , recalled qq , D , K and vectors C and C , can be computed in equations (3,4), in particular M , M , M , e e θ qmatrices θθ θq qq equations (3,4), in particular matrices M , M , M θθ θq qq , D , K and vectors C and C , can be computed in closed θθ θq qq e starting e θ links data q form from the summarized in Table 1. D , K and vectors C and C , can be computed in closed e e θ q D , K and vectors C and C , can be computed in closed e starting e q form from theθθ links data summarized in Table 1. e e q form the summarized in 1. form starting starting from the links links data summarized in Table Tablenot 1. However, the from standard FFRdata approach is generally However, the standard FFR approach is generally not applicable in the case of large deflections, thus when the However, the standard FFR approach is generally not However, the standard approach is generally not applicable in the case are of FFR large deflections, the elastic displacements not “small”. In thus this when case, the applicable in the case of large deflections, thus when applicable in the case of large deflections, thus when the elastic displacements are not “small”. In this case, absolutedisplacements nodal coordinates formulation (ANCF) has been elastic are “small”. In the elastic displacements are not not “small”. (ANCF) In this this case, case, the absolute nodal coordinates formulation has been proposed (Yakoub and Shabana (1999)), requiring the absolute nodal coordinates formulation (ANCF) has absolute nodal coordinates formulation (ANCF) has been been proposed (Yakoub and Shabana (1999)), requiring the definition of global shape functions for every element with proposed (Yakoub and Shabana (1999)), requiring the proposed (Yakoub and Shabana (1999)), requiring the definition of global shape functions for every element with respect to of theglobal absolute reference frame. On theelement other hand, definition shape functions for every with definition of global shape functions for every element with respect to the absoluteisreference On the other hand, the ANCF approach actuallyframe. only suitable for beams respect to absolute frame. On hand, respect to the the absoluteisreference reference frame. On the the other other hand, the ANCF approach actually only suitable forcomplex beams and shells elements, but it is hardly applicable to the ANCF approach is actually only suitable for beams the ANCF approach is actually only suitable for beams and shells elements, but it is hardly applicable to complex shapes andelements, cannot benefit of hardly the results of FEM and shells but applicable to complex and shells but it it is is applicable to analysis. complex shapes andelements, of hardly the results of FEM Accuracy ofcannot resultsbenefit for large deformation fieldsanalysis. can be shapes and cannot benefit of the results of FEM analysis. shapes and cannot benefit of the results of FEM analysis. Accuracy of results for large deformation fields can be Accuracy of results for large deformation fields Accuracy of results for large deformation fields can can be be
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MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria 654 Bruno Scaglioni et al. / IFAC-PapersOnLine 48-1 (2015) 653–654
Fig. 3. Angle of the second joint
Fig. 2. Angle of the first joint anyway achieved in the context of the FFR approach by means of the substructuring technique (Shabana (1985)) as demonstrated in the context of this work. Highly flexible bodies can thus be substructured in several elements, applying on each element the theory of linear elasticity. To this aim, link i can be subdivided into more rigidly connected flexible elements or substructures, as shown in Fig. 1, each one described through a FFR approach introducing Mk modal coordinates. Define with Ne the total number of elements of the whole manipulator N e and with M = k=1 Mk the total number of modal coordinates. Let θi = θk¯e , i = 1, . . . , N denote the “real” joint angle between element k¯ − 1 and k¯ and consider a “dummy” joint angle θke if a rigid connection exists between element k − 1 and k. Then, the model remains formally identical by removing the columns corresponding to the “dummy” joint angles in the assembly matrices. The developed model has been validated by comparison of simulation results obtained with a Matlab/Simulink implementation of the closed form model with multibody simulations, obtained with Modelica/Dymola and with MSC/Adams. The benchmark has been taken from Yakoub and Shabana (1999), where the ANCF is used to describe links compliance in the case of a flexible double pendulum. The structure consists of two flexible bodies connected together and to the ground by revolute joints, initially in horizontal position and free to fall under the effect of gravity. The physical parameters, summarized in Table 2, have been chosen in order to allow large deformation on the second pendulum. The models, developed in the different environments, implements substructuring by subdividing the first link in two elements with length 0.1 m and the second link in 12 elements with lentgh 0.075 m. All figures show comparisons among Matlab/Simulink simulation (solid line), Modelica/Dymola simulations (dashed line) and MSC/Adams simulations (dotted line). It must be pointed out that the Modelica model accounts for damping in exactly the same way as the closed form model, while Adams adopts a different approach, hence, small differences in the simulation results are appreciable. Figures 2 and 3 show the relative angle of the first and second joint, while figure 4 shows the transverse deflection of the tip point of the second link. As it is apparent, results are in good accordance, in particular, the results obtained from Matlab/Simulink and Modelica/Dymola simulations are undistinguishable, while the results provided by MSC/Adams shows some small differences. A sequence of 3D snapshots of the simulation at intermediate instants is shown in Fig. 5. 654
Fig. 4. Transverse deflection of second pendulum tip
Fig. 5. 3D representation of the simulation Table 2. Double Pendulum data Property
Body 1
body 2
Mass (Kg) Length(m) Cross sectional Area(m2 ) Second moment of area (m4 ) Mass moment of inertia (Kg × m2 ) Modulus of elasticity (N/m2 )
0.6810972 0.2 1.26E − 03 1.26E − 07 2.27E − 03 8.00E + 07
2.4740052 0.9 1.26E − 03 1.26E − 07 1.67E − 01 5.00E + 05
REFERENCES Ferretti, G., Leva, A., and Scaglioni, B. (2014). Objectoriented modelling of general flexible multibody systems. Mathematical and Computer Modelling of Dynamical Systems, 20(1), 1–22. Fijany, A. and Featherstone, R. (2013). A new factorization of the mass matrix for optimal serial and parallel calculation of multibody dynamics. Multibody System Dynamics, 29, 169–187. Meirovitch, L. (1967). Analytical Methods in Vibration. Macmillan Publishing, New York. Shabana, A.A. (1998). Dynamics of Multibody Systems. Cambridge University Press, New York. Shabana, A. (1985). Substructure synthesis methods for dynamic analysis of multi-body systems. Computers and Structures, 20(4), 737 – 744. Yakoub, R. and Shabana, A. (1999). Use of Cholesky coordinates and the absolute nodal coordinate formulation in the computer simulation of flexible multibody systems. Nonlinear Dynamics, 20(3), 267–282.