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DYNAMIC MODELLING OF A FLEXIBLE LINK MANIPULATOR ROBOT USING AMM Moh. Khairudin Electrical Eng. Dept. Universitas Negeri Yogyakarta, Yogyakarta 55281, Indonesia e-mail:
[email protected] Abstrak Paper ini menyajikan pemodelan dari sebuah manipulator link fleksibel menggunakan teknik Lagrangian dalam hubungannya dengan metode modus diasumsikan (AMM). Link-link dimodelkan sebagai Euler-Bernoulli beams yang memenuhi kondisi batas massa. Sebuah beban yang terhubung (payload) ditambahkan ke ujung luar link, sedangkan hub inertias disertakan pada aktuator sendi. Pendekatan Lagrangian digunakan untuk mendapatkan model dinamis dari struktur. Model dinamis dari sebuah manipulator link fleksibel pada penlitian ini diverifikasi menggunakan simulasi Matlab/Simulink. Perumusan model yang diusulkan telah lengkap dengan turut mempertimbankan pengaruh beban yang terhubung (payload) dan redaman yang berada dalam kerangka kopling (structural link). Penekanan dari perumusan model ini telah di diatur untuk mendapatkan persamaan gerakan yang akurat yang menunjukkan aspek-aspek yang paling berpengaruh dalam kopling, khususnya untuk kasus kopling untuk tenaga gerak yang kaku (rigid) maupun yang selalu berubah serta lentur. Kata kunci: assumed method, dynamic model, Lagrangian.
Abstract This paper presents modeling of a flexible link manipulator using Lagrangian technique in conjunction with the assumed mode method (AMM). The links are modeled as Euler-Bernoulli beams satisfying proper mass boundary conditions. A payload is added to the tip of the outer link, while hub inertias are included at the actuator joints. The Lagrangian approach is used to derive the dynamic model of the structure. In this research, the dynamic model of a flexible link manipulator verified using Matlab/Simulink simulation. The model formulation proposed in this work is complete in the sense that it considers the effects of payload and damping structural of the link. The emphasis has been set on obtaining accurate equations of motion that display the most relevant aspects of the coupling between rigid and flexible dynamics. Keywords: assumed method, dynamic model, Lagrangian.
1. INTRODUCTION The first step of design procedure is to acknowledge the information of constructing the dynamic model of flexible manipulators using the combination of Euler-Lagrange and Assumed mode method (AMM). In order to have a successful modeling design, prior knowledge of AMM and Euler Lagrange equation are needed by integrating with Simulink. Simulation results are analyzed in both the time and frequency domains to assess the accuracy of the model in representing the actual system. Partial differential equations (PDE) and boundary equations of a flexible link manipulator system are obtained by matching the shear force and bending moment at the elbow joint, allowing the eigenvalues to be computed without recourse to dynamic formulations [1]. On the other hand, the vibration modes of a generic flexible link manipulator are studied as a function of the link, rotor and tip mass distribution. Necessary and sufficient conditions are developed for all vibration modes to exhibit a node at the manipulator. Various approaches have been developed which can mainly be divided into two categories: the numerical analysis approach and the AMM. The numerical analysis methods that are utilized include finite difference (FD) and finite element (FE) methods. The FD and FE approaches have been used in
Dynamic Modelling of A Flexible Link Manipulator Robot Using …… (Moh. Khoirudin)
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obtaining the dynamic characterization of a single-link flexible manipulator system incorporating damping, hub inertia and payload [2,3]. Subudhi and Morries [4] have used a combined Euler-Lagrange formulation and AMM approach to model the planar motion of a manipulator consisting of flexible links and joints. The conventional Lagrangian modeling of flexible link robots does not fully incorporate the bending mechanism of flexible link as it allows free link elongation in addition to link deflection. The equations of motion which can be arranged in a computationally efficient closed form that is also linear with respect to a suitable set of constant mechanical parameters have been obtained [5]. This paper presents modeling of a flexible link manipulator using Lagrangian technique in conjunction with the AMM. The links are modeled as Euler-Bernoulli beams satisfying proper mass boundary conditions. A payload is added to the tip of the outer link, while hub inertias are included at the actuator joints.
2. A FLEXIBLE LINK SYSTEM In this section, the flexible link kinematics is described. The kinematics description is developed for a chain of n serially connected flexible links as shown in Figure 1. The co-ordinate systems of the link are assigned referring to the Denavit–Hartenberg (D–H) description. X0Y 0 is ) ) the inertial co-ordinate frame (CF), XiYi the rigid body CF associated with the ith link and X i Yi is the flexible moving CF.
Figure 1. Structure of a flexible link manipulator system Considering revolute joints and motion of the manipulator on a two-dimensional plane, the rigid transformation matrix, Ai , from Xi-1Yi-1 to XiYi is written as ⎡cos θ i Ai = ⎢ ⎣sin θ i
− sin θ i ⎤ cos θ i ⎥⎦
(1)
On using assumption (IV), the elastic homogenous transformation matrix, Ei, due to the deflection of the link i can be written as ⎡ 1 ⎢ Ei = ⎢ ⎢ ∂υi ( xi , t) ⎢ ∂x i ⎣
− xi =li
∂υ i ( xi , t ) ∂xi 1
xi =li
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
(2)
where υ i ( xi , t ) is the bending deflection of the ith link at a spatial point xi (0 ≤ xi ≤ li ) and li is the length of the ith link. The global transformation matrix T i transforming co-ordinates from X0Y 0 to XiYi follows a recursion as below
Ti = Ti −1 Ei −1 Ai
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(3)
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Let i
⎧ x i ⎫⎪ ri ( x i ) = ⎨ ⎬ ⎩υ i ( x i , t )⎪⎭
be the position vector that describes an arbitrary point along the ith deflected link with respect to 0
its local CF (XiYi ) and ri be the same point referring to X0 Y0. The position of the origin of Xi+1Yi+1 with respect to XiYi is given by i
and
0
p i +1 = i ri (l i )
(4)
pi is its absolute position with respect to X0Y0.
Using the global transformation matrix, 0
ri = 0 pi + Ti i ri ,
0
0
ri and 0 pi can be written as
pi +1 = 0 pi + Ti i pi +1
(5)
3. THE DYNAMIC MODELLING OF A FLEXIBLE LINK SYSTEM To derive the kinetic and potential energies associated with the manipulator, the procedure adopted in previous section is followed. By substituting for links (i = 1) and for two modes (j = 1, 2). The solution of the partial differential equation describing the flexible motion of the manipulator can be obtained following the general procedures. In this case, the effective masses at the end of the individual links are set as
M L1 = M p , J L1 = 0 , MD1 = 0
(6)
Here, the co-ordinate vector consists of link positions, θ1, and modal displacements (q11, q12). The force vector is F = {τ1, 0, 0} T, where τ is the torques applied by the rotor. Therefore, the following Euler–Lagrange’s equations result, with i = 1 and j = 1 and 2:
∂ ⎛⎜ ∂L ∂t ⎜⎝ ∂θ&i
⎞ ∂L ⎟− ⎟ ∂θ = τ i i ⎠
(7)
∂ ⎛⎜ ∂L ∂t ⎜⎝ ∂q& ij
⎞ ∂L ⎟− =0 ⎟ ∂q ij ⎠
(8)
Assembling the mass and stiffness matrices and utilising the Euler-Lagrange equation of motion, the dynamic equation of motion of the flexible manipulator system can be obtained as ..
.
M Q(t ) + D Q(t ) + KQ(t ) = F (t )
(9)
where M, D and K are global mass, damping and stiffness matrices of the manipulator respectively. The damping matrix is obtained by assuming the manipulator exhibit the characteristic of Rayleigh damping. The global damping normally determined through experimentation. The damping ratio typically ranges [6] from 0.007 to 0.01. For the flexible manipulator under consideration, the global mass matrix can be represented as
⎡M M = ⎢ θθ ⎣ Mθw
M θw ⎤ M ww ⎥⎦
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where M ww is associated with the elastic degrees of freedom (residual motion), M θw represents the coupling between these elastic degrees of freedom and the hub angle θ and M θθ is associated with the inertia of the system about the motor axis. Similarly, the global stiffness matrix can be written as
0 ⎤ ⎡0 K=⎢ ⎥ ⎣0 K ww ⎦ where K ww is associated with the elastic degrees of freedom (residual motion). It can be shown that the elastic degrees of freedom do not couple with the hub angle through the stiffness matrix. The global damping matrix
D in equation (9) can be represented as
0 ⎤ ⎡0 D=⎢ ⎥ 0 D ww ⎦ ⎣ where Dww denotes the sub-matrix associated with the structural damping. This proportional damping model has been assumed because it allows experimentally determined damping ratios of individual modes to be used directly in forming the global matrix. It also allows assignment of individual damping ratios to individual modes, such that the total manipulator damping is constituted with the sum of the damping associated with the modes. Using this assumption, the damping can be obtained as
Dww = α M ww + β K ww
(10)
where
α=
2 f 1 f 2 (ξ1 f 2 − ξ 2 f1 ) f 2 − f1 2
2
; β =
2 (ξ 2 f 2 − ξ 1 f1 ) f 2 2 − f12
with ξ 1 , ξ 2 , f 1 and f 2 represent the damping ratios and natural frequencies of modes 1 and 2 respectively. 4. RESULT AND DISCUSSION The dynamic model of a flexible link manipulator has been presented. The physical parameters of the manipulator are given in Table 1. In this study, the damping ratios were assumed as 0.0086 and 0.01 for vibration modes 1 and 2 respectively. The manipulator was excited with symmetric bang-bang torque inputs as shown in Figure 2. A bang-bang torque has a positive (acceleration) and negative (deceleration) period allowing the manipulator to, initially, accelerate and then decelerate and eventually stop at target location. The output responses of the manipulator are taken from both angle rotation and modal displacement. Table 1. Parameters of the flexible link manipulator Symbol ρ EI l Ir Ib
Parameter Mass density Flexural rigidity Length Rotor and hub inertia Beam inertia
Value
Unit
0.1648 3.73 0.9 5.8598x10 -4 0.04
kgm-1 Nm2 m kgm2 kgm2
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B a n g- b an g i np u t c o m m a n d 0.4 0.3
Torque (Nm)
0.2 0.1 0 -0 . 1 -0 . 2 -0 . 3 -0 . 4 0
0.5
1
1. 5 T im e ( s )
2
2 .5
3
Figure 2. The bang-bang input torque
Hub-angle
Hub-velocity
45
300
40
250
35 Hub v elocity (deg/sec)
200
Hub angle (deg)
30 25 20 15
50 0 -50
10
-100
5 0 0
150
0.5
1
1.5 Time (s )
2
2.5
-150
3
0
500
(a) Hub-angle.
End-point acceleration
2000
2500
3000
0
10
150
10
100
10
-1
-2
M agnitude (m /H z)
End point acceleration (m/sec/ sec)
1500 Time (s)
(b) Hub-velocity.
200
50
0
-3
10
-4
10
-5
-50
10
-100
-150 0
1000
-6
10
0.5
1
1.5 Time (s)
2
(c) End-point acceleration.
2.5
3
-7
10
0
10
20
30
40 50 60 Frequency (Hz)
70
80
90
100
(d) Spectral density of end-point acceleration.
Figure 3. Response of the simulation of one-link flexible manipulator
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However, a slightly different response was obtained in the frequency domain where only two modes of vibration can be obtained within simulation compared to three modes of vibration from experimental results. This is because only two modes of vibration are considered in deriving the mathematical model of the system. It is noted that the first two modes of vibration of the system converged to 13.94 Hz and 55.78 Hz for the simulation results. The experimental results, however, gave 11.72 Hz, 35.15 Hz and 65.60 Hz. The corresponding errors between the simulation and experimental results by taking the first two modes are accordingly 18.9% and 58.4 % respectively, which are considered a big difference. This is due to the limitation of number of modes (two modes) using in the simulation model. 5. CONCLUSSION A generalized modeling framework has been described to obtain the closed-form finite dimensional dynamic model for a flexible link manipulators by using the Euler-Lagrange approach combined with the AMM. The model formulation proposed in this work is complete in the sense that it considers the effects of payload and damping structural of the link. The emphasis has been set on obtaining accurate equations of motion that display the most relevant aspects of the coupling between rigid and flexible dynamics. REFERENCES [1]. A. S. Morris, A. Madani, “Inclusion of Shear Deformation Term to Improve Accuracy in Flexible-Link Robot Modeling”, Mechatronics, Vol. 6, No. 6., pp. 631-647, 1996. [2]. Tokhi MO, Mohamed Z, Azad AKM, “Finite difference and finite element approaches to dynamic modelling of a flexible manipulator”, Proceeding of IMechE-I: Journal of Systems and Control Engineering, 211:145-156, 1997. [3]. Tokhi MO, Mohamed Z, Shaheed MH, “Dynamic characterization of a flexible manipulator system”, Robotica, 19:571-580, 2001. [4]. Subudhi B, Morris AS., “Dynamic Modeling, Simulation and Control of a manipulator with flexible links and joints”, Robotics and Autonomous System, 41:257-270, 2002. [5]. M.Khairudin, Z. Mohamed, “A Technique For Modeling of a Two-Link Flexible Manipulator”, The 6rd IEEE Student Conference on Research and Development (SCOReD 2008), Johor, Malaysia, 26-27 Nov. 2008. [6]. Hasting, G. G. and Book, W. J., “A linear dynamic model for flexible robot manipulators”, IEEE Control Systems Magazine, Vol. 7, pp. 61-64, 1987.
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