ADVANCES IN MANUFACTURING SCIENCE AND TECHNOLOGY
Vol. 34, No. 4, 2010
OPTIMIZATION PROBLEM FOR PLANNING A TRAJECTORY OF A MANIPULATOR WITH A FLEXIBLE LINK Iwona Adamiec-Wójcik
S u mm ary
The paper presents an application of the rigid finite element method to modelling of a flexible link of a manipulator. The modification of the classical rigid finite element method which takes into account only bending and torsion enables us to consider large deflections of the links. In the case when the deflections are small a simple form of the equations can be used. Applicability of the models is discussed on the basis of simulation results. One of the problems connected with realisation of desired motion of a mechanism is a proper choice of drive functions. The flexibility influences a desired trajectory of the end-effector. In order to compensate the flexibility effects an optimization task is formulated and it consist in such a choice of the courses of joint angles that the desired trajectory of the end-effector is realized in spite of the vibrations caused by flexible links. The required drive functions are calculated and the results are presented. Keywords: dynamics of manipulators, flexibility of links, rigid finite element method, trajectory, compensation for flexibility Zadanie optymalizacji w planowaniu trajektorii manipulatora z podatnym członem S t r e s zc z e n i e
W artykule omówiono zastosowanie metody sztywnych elementów skończonych do modelowania podatnego członu manipulatora. Modyfikacja klasycznej metody sztywnych elementów skończonych, która uwzględnia tylko podatność gietną i skrętną, umożliwia analizę dużych ugięć. Dla przypadku małych ugięć – prosta liniowa postać równań ruchu może być stosowana. Przedstawione wyniki symulacji są podstawą do analizy zastosowań modeli. Jednym z problemów związanych z realizacją zadanej trajektorii ruchu jest właściwy dobór funkcji napędowych. Podatność członów wpływa na realizacje zadanej trajektorii. Sformułowano zadanie optymalizacji, które kompensuje wpływ podatności poprzez taki dobór funkcji napędowych, aby zadana trajektoria była realizowana niezależnie od drgań wywołanych przez podatne człony. Uzyskane wyniki przedstawiono i sformułowano wnioski końcowe. Słowa kluczowe: dynamika manipulatorów, podatność członów, metoda sztywnych elementów skończonych, trajektoria, kompensacja podatności
Address: Iwona ADAMIEC-WÓJCIK, D.Sc. Eng., University of Bielsko-Biala, 2 Willowa St., 43-300 Bielsko-Biala, e-mail:
[email protected]
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1. Introduction Although research in the field of flexible multibody systems has been carried out for decades, the large number of recent publications demonstrates the continuing interest in the subject [1-4] especially when flexible links undergo large deflections [5-8]. Flexible links are normally discretised using the finite element method and thus the modal method is usually employed to reduce the number of generalized coordinates describing the motion of the system. Much research, mainly in Poland, employs the rigid finite element method first formulated by Kruszewski [8]. In this paper a modification of the rigid finite element method introduced by Wojciech [9] will be used. The new formulation of the method has been developed by means of homogenous transformations and joint coordinates [10, 11]. This method has been successfully used in dynamic analysis of both planar and spatial systems such as cranes, textile machines, and manipulators. Flexibility effects in dynamics of machines and mechanisms is especially important when the system contains long slender links (satellite antennae, rotor blades) or if there is a rapid movement. It has been proved that flexibility effects cannot be omitted and the modification of the rigid finite element method is an efficient way of modeling mechanisms with slender, whippy links [12-15]. Flexibility effects arise not only when the links are flexible but also when the support of the machine is flexible. The influence of the flexible support on the motion has been investigated (among others) in [16]. There the optimization method has been used to compensate vibrations caused by the flexible support and the results have been presented. In this paper the manipulator with four links (Fig. 1) will be modeled and analysed. The third link of the manipulator can be flexible. Dynamics of the manipulator will be discussed when in order to model the flexible link linear and nonlinear models are used. Then an optimization problem will be solved and drive functions compensating the flexibility of the link will be calculated.
2. Model of the manipulator The manipulator considered is presented in Fig. 1. It is assumed that the motion of the manipulator can be realized by means of the rotation angles shown. The third link of the manipulator can be flexible and in such a case it is discretised using the modification of the rigid finite element method (RFE). The discretisation of the flexible link with constant cross-section is carried out in two stages (Fig. 2): first, the link is divided into n equal segments and the spring-damping element reflecting spring features of the segment is placed in the
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center of each segment. Then the final discretisation takes place and a system of n+1 rigid finite elements connected by n spring damping elements is obtained.
θ3 θ2
θ 4( y ) θ 4( z )
θ 4( x) P
θ1 Z0 Y0 {0}
X
x0P , y 0P
Fig. 1. Manipulator considered L ∆
∆
∆
∆
∆
sde i
∆
rfei
∆
∆
∆
sde i +1
rfei −1
rfei +1
Fig. 2. Discretisation of the flexible link
The modification of the RFE method takes into account only bending and torsional flexibility, which means that each rigid finite element („rfe”) can have up to three degrees of freedom in the relative motion (Fig. 3). The motion of „rfe” is considered with respect to the preceding element and thus the vector of generalized coordinates of the flexible link takes the following form: T
T
T
q% f = [q% (0) ... q% (fi ) ... q% (fn) ]T f
(1)
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where q% (fi ) = [ϕ1( i) ϕ 2( i ) ϕ 3( i ) ]T , i = 1,..., n , q% (0) f = [θ 3 ]. Zˆ ( j )
ϕ 3( j) Yˆ ( j)
ϕ 2( j )
rfe j−1 Xˆ ( j −1) Zˆ ( j −1)
Yˆ ( j −1)
rfe j
ϕ1( j ) Xˆ ( j)
Fig. 3. Degrees of freedom of the j-th „rfe”
An advantage of the RFE method in comparison to the finite element method is that the equations of motion are derived in a general unified approach based on the algorithm derived for rigid systems. The equation of motion of a rigid link can be obtained as a special case of the model of a flexible link assuming n = 0. Having assigned the coordinate systems to each „rfe” in a way described by Craig [17] using the Denavit-Hartenberg notation the transformation matrix from the reference system {i} to {i-1} is defined by the following formula:
~ B (i )
c1( i ) c3(i ) (i) (i) s c = 3 (2i ) − s2 0
c3( i ) s2(i ) s1( i ) − s3(i ) c1( i ) s3( i ) s2( i ) s1(i ) + c3(i ) c1( i ) c2(i ) s1( i ) 0
c3( i ) s2( i ) c1( i ) + s3( i ) s1(i ) s3(i ) s2(i ) c1(i ) − c3(i ) s1( i ) c2(i ) c1( i ) 0
l (i −1) 0 0 1
(2)
(i) (i ) (i ) (i ) (i ) where: s k = sin ϕ k , ck = cos ϕ k , l is the length of the i-th element.
The transformation matrix presented enables us to take into account large deformations of the flexible link. When the displacements of rigid elements are small the transformation matrix takes the following form:
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~ B (i )
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− ϕ 3(i ) 1
1 (i ) ϕ = 3 (i ) − ϕ 2 0
ϕ1(i ) 0
ϕ 2(i ) − ϕ1(i ) 1 0
l ( i−1) 0 0 1
(3)
The vector of generalised coordinates describing the motion of the manipulator considered can be presented in the form: T
T
T
q = [ q% (1) q% (2) q% Tf q% (4) ]T
where: q% (1) = [θ1 ],
q% (2) = [θ 2 ],
(4)
q% (4) = [θ 4x θ 4y θ 4z ]T .
The equations of motion are derived from the Lagrange equations and to this end the kinetic and potential energies of all links together with the rigid and spring-damping elements have to be calculated. Finally, the equations of motion are written as: && + B( q, q& )q& + Cq = F( M) A( q ) q
(5)
where: q is the vector of generalised coordinates with m = 3n + 3 components, A, B, C – matrices m × m , with elements dependent on generalised coordinates and velocities, M = [ M (1) M (2) M (3) M (4) ]T . When a kinematic input is assumed moments M (1) ÷ M (4) are unknown and the following constraint equations have to be fulfilled:
θ i = α ( i ) for i = 1,...,4
(6)
where: θi – angles defined in Fig. 1, α (i ) – drive functions given for θi respectively. In order to carry out numerical simulations a computer programme has been worked out for the model presented. Dynamics of the manipulator has been considered assuming that the third link is flexible and the motion is realised by means of the change of angles θ1 and θ2 . Results presented below have been obtained for the following parameters of the links: the first link is a rotary
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column of length 0.58 m, with a circular cross-section of re = 0.04, ri = 0.03 m as external and internal radiuses respectively and mass 14.8 kg. All other links have a box cross-section and the following parameters are assumed: l2 = 1.0 m, l3 = 2.5 m, l4 = 0.6 m, external and internal dimensions of the cross-section for the second and third links are the same: a e(2) = a e(3) = 0.06 m, ai(2) = ai(3) = 0.0598 m, a e(4) = 0.1, ai(4) = 0.08 m, mass density: ρ = 7801 kgm–3, E = 2.1⋅1011 NM–2. Drive functions are calculated in such a way that the accelerations of the angles follow the course shown in Fig. 4.
Fig. 4. Standard course of the acceleration of a drive
It can be seen that the braking phase of the motion is a mirror reflection of that of accelerating. Dynamics of the manipulator has been calculated for t = 5 s when the angles changed as follows: θ1 from 0 to 90° and θ2 from 45 to 50°. Configuration of the manipulator is defined by other angles which are x y z constant during the motion: θ3 = 0, θ 4 = 0, θ 4 = 135, θ 4 = 0 . Figure 5 presents the deflections of the end of the flexible link calculated using both linear and nonlinear model. The realised motion of the manipulator causes large deflections of the flexible link and thus the equations of motion of the manipulator has to be in the nonlinear form. It can be seen that the linear model neglects deflections caused by centrifugal forces [10, 18]. The above results has been obtained for the flexible link divided into n = 10 rfes. The results obtained for different numbers of rigid elements are presented in Fig. 6.
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Fig. 5. Deflections of the end point of the flexible link
Fig. 6. Deflections of the end of the flexible link for a different number of rfes
The results obtained for n = 5,7 and 10 do not differ much and for further considerations the flexible link is divided into 5 elements.
3. Optimisation problem Flexibility of links, especially in the case of large deflections, can considerably disturb the motion of the end-effector of a manipulator and thus it can influence realisation of some technological processes which require significant precision. Let us consider rotary motion of the manipulator presented in Fig. 1. During rotation of the column point P projected onto plane X ( 0) Y ( 0) realises a trajectory, which is an arc of a circle with a radius:
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rP0 =
(x ) + ( y ) 0 2 P
0 2 P
(7)
if all the links are ideally rigid. However, the flexibility of links can cause vibrations which will disturb the trajectory. The optimisation task considered here consist in searching of function θ 2 = α ( 2) which will compensate the flexibility of link 3. So the drive 2 should move in such a way that the projection of a trajectory of point P onto the plane is as close as possible to the one for a rigid manipulator. The formulation of the problem is: Let us assume that the rotation of the column of the manipulator is performed during 5 seconds in which the angle θ1 changes for a given value. ( 2) which will ensure the minimum of the following Find such a course of α functional:
F (X ) =
1 tk
tk
∫ [rp − rp ] dt 0 2
(8)
t0
The drive function is calculated according to the formula:
α ( 2 ) = α ( 2 , 0 ) + ∆α ( 2 )
(9)
( 2 ,0 ) is defined as in Fig. 4, ∆α ( 2 ) is a spline function of the third order. where α
It is assumed that the spline function is defined by means of decisive values ∆α j = 1, ..., ni + 1 shown in Fig. 7. (i ) j ,
Fig. 7. Interpolation of ∆α ( i ) by means of spline functions
In order to find values of functional F defined in (8) the equations of motion (5) have to be integrated at each optimisation step.
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Two motions are considered. First, angle θ1 changes from 0 to 90°, and then from 0 to 180°. The number of points at which the decisive variables are calculated for the first task is n 3 = 5 , and for the second one n 3 = 7 . Figure 8. presents trajectories of the end-effector for both cases before and after optimisation.
Fig. 8. Trajectory of point P before (a) and after optimisation (b)
In order to compensate the flexibility effect the second drive has been set to motion and the courses of velocities are presented in Fig. 9. Those velocities have been achieved for ∆α ( 2) , calculated while solving the optimization task, shown in Fig. 10. The deflections of the flexible link before and after the optimization process are shown in Fig. 11.
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Fig. 9. Velocity of θ 2 : a) 90°, b) 180°
Fig. 10. Courses of ∆α(2): a) 90°, b) 180°
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Fig. 11. Deflections of the end of the flexible link
It can be seen that the deflections caused by the flexibility of the link have been considerably reduced. The value of the objective function has been reduced for 58% in the first case and 48% in the second case. The equations of motion have been integrated using the Newmark method and the Nelder-Meads method has been applied in order to solve the optimisation problem. Such an approach requires high numerical efficiency in solving the equations of motion and because of the calculation time it is impossible to use this method for controlling the motion of the manipulator in real time.
4. Final remarks The paper presents a model of a manipulator with a flexible link. In order to discretise the flexible link the modification of the rigid finite element method is used. This modification allows to take into account large deflections of links. Yet, when the deflections are small simplified forms of equations can be applied. The problem of compensating flexible effects in realisation of the trajectory has been successfully solved by means of spline functions and the Nelder-Mead method. Since the calculation time of a single optimisation task can very up to 15 minutes on a PC according to the number of rigid elements and number of decisive points, this method cannot be applied to control in real time. Because the method is efficient in compensation of flexibility then it can be used for training a neural network which then could be used for control problems. This should be addressed in future work. It has to be underlined that the considerations presented can be easily generalised for manipulators with different structure and with more flexible links. Also the trajectory can be more complex and defined in space. References [1] A.A. SHABANA: Flexible multibody dynamics: review of past and recent developments. Multibody Systems Dynamics, (1997)1, 89-122.
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[2] B.O. AL-BEDOOR, M. HAMDAN: Geometrically non-linear dynamic model of a rotating flexible arm. Journal of Sound and Vibration, 240(2001)1, 59-72. [3] O.A. BAUCHAU, J. RODRIGUEZ: Formulation of modal based elements in nonlinear, flexible multibody dynamics. Journal of Multiscale Computational Engineering, 1(2003)2&3, 161-180. [4] A.L. SCHWAB, J.P. MEIJAARD: Comparison of three-dimensional flexible beam elements for dynamic analysis: classical finite element method and absolute nodal coordinate formulation. Journal of Computational and Nonlinear Dynamics, 5(2010)1, 110. [5] X. YUAN, Y. TADA: Nonlinear analysis of flexible beams undergoing large rotations via symbolic computations. Mathematical Problems in Engineering, 7(2001), 241-252. [6] L.G. MAQUEDA, A.A. SHABANA: Numerical investigation of the slope discontinuities in large deformations finite element formulations. Nonlinear Dynamics, 58(2009), 23-37. [7] C.-C. LAN, K.-M. LEE, J.-H. LIOU: Dynamics of highly elastic mechanisms using the generalized multiple shooting method: simulations and experiments. Mechanisms and Machine Theory, 44(2009), 2164-2178. [8] J. KRUSZEWSKI (red.): Metoda sztywnych elementów skończonych. Arkady, Warszawa 1975. [9] S. WOJCIECH: Dynamika płaskich mechanizmów dźwigniowych z uwzględnieniem podatności ogniw oraz tarcia i luzów w węzłach. Wydawnictwo Politechniki Łódzkiej, Zeszyty Naukowe nr 66, Łódź 1984. [10] E. WITTBRODT, I. ADAMIEC-WÓJCIK, S. WOJCIECH: Dynamics of flexible multibody systems. Rigid finite element method. Springer-Verlag, Berlin-Heidelberg 2006. [11] I. ADAMIEC-WÓJCIK: Modelling dynamics of multibody systems, Use of homogeneous transformations and joint coordinates. Lambert Academic Publishing, Köln 2009. [12] S. WOJCIECH, I. ADAMIEC-WÓJCIK: Nonlinear vibrations of spatial viscoelastic beams. Acta Mechanica, 98(1993), 15-25. [13] J. PŁOSA, S. WOJCIECH: Dynamics of systems with changing configuration and with flexible beam-like links. Mechanism and Machine Theory, 35(2000), 15151534. [14] I. ADAMIEC-WÓJCIK, K. AUGUSTYNEK: Modelling dynamics of flexible multibody systems by means of the rigid finite element method. Proc. Int. Conf. Multibody Dynamics, Madrid 2005. [15] I. ADAMIEC-WÓJCIK, E. WITTBRODT, S. WOJCIECH: Modelling of deflections of beam-like links by means of rigid finite element method. Proc. Int. Conf. Multibody Dynamics, Milano 2007. [16] A. URBAŚ, I. ADAMIEC-WÓJCIK, S. WOJCIECH: Dynamics of a manipulator fixed on a flexibly supported base. Proc. Int. Conf. Multibody Dynamics, Warszawa 2009. [17] J.J. CRAIG: Introduction to Robotics. Addison-Wesley, Massachusetts 1988. [18] E. WITTBRODT, S. WOJCIECH: Application of rigid finite element method to dynamic analysis of spatial systems. J. Guid. Control Dyn., 18(1995)4, 891-898. Received in September 2010
