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Dynamic Analysis of a Two-link Flexible Manipulator Subject to Different Sets of Conditions
Available online at www.sciencedirect.com
Procedia Engineering 41 (2012) 1253 – 1260
International Symposium on Robotics and Intelligent Sensors 2012 (IRIS 2012)
Dynamic analysis of a two-link flexible manipulator subject to different sets of conditions Atef A. Ataa, Waleed F. Faresb, Mohamed Y. Sa'adehc a
Department of Engineering Mathematics and Physics, Faculty of Engineering, Alexandria University, Egypt. b Department of Mechanical Engineering, Faculty of Engineering, IIUM, Kuala Lumpur, Malaysia. c Department of Mechanical Engineering, Faculty of Engineering, Southeastern Louisiana University, USA.
Abstract Dynamic analysis for flexible manipulator is very important for selecting the actuator size and designing the proper control strategy. The effect of different sets of initial and boundary conditions on the joints torques is investigated in this paper. The elastic deflection for each link is computed using the assumed modes methods for four modes of vibration. A third order polynomial trajectory is designed in the joint space and the required torques are obtained through the solution of the inverse dynamics problem. The obtained results showed
that the boundary conditions have a considerable effect on the elastic deflection and the corresponding actuators’ torques as well. © 2012 The Authors. Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the Centre of Humanoid Robots and Bio-Sensor (HuRoBs), Faculty of Mechanical Engineering, Universiti Teknologi MARA. Keywords: Flexible manipulator, torque, assumed modes, trajectory, initial conditions.
Nomenclature ݍ ݍ ܯ ܯ ܯ ݄ ݄ ݄ ݄ ܥ ܥ K ߬ w( x, t )
generalized joint variable generalized flexible deformation variable manipulator configuration dependent generalized mass matrix of rigid part manipulator configuration dependent generalized mass matrix of flexible part representing the interaction of rigid and flexible variables generalized matrix of Coriolis and Centrifugal terms for rigid part generalized matrix of Coriolis and Centrifugal terms for flexible part representing the interaction of rigid and flexible variables representing the interaction of flexible and rigid variables vector of gravitational terms for rigid part vector of gravitational terms for rigid part generalized structural stiffness matrix vector of input torque applied at the joints elastic displacement
W (x)
spatial function
T (t ) time function Greek symbols ωn natural frequency _________
a
Corresponding author. Tel.: +2-03-592-550 ; fax: +2-03-592-1853 . E-mail address: [email protected]
1877-7058 © 2012 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.07.308
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1. Introduction Flexible link manipulators have many advantages over rigid manipulators such as higher payload to mass ratio, large reachability due to long and light members and precise high speed operation. The performance of these manipulators are severely limited by the oscillatory response which persists for a period of time after the move is completed, in other words, large and simultaneous motion of the links induce oscillations that persist beyond the nominal final completion time [1], thus delaying any subsequent operation. Modelling and simulation of flexible manipulators in a simplified manner requires many approaches and assumptions. Many researchers applied the computed torque method which was originally developed for rigid manipulators on flexible link systems [2]. However, the complexity of the inverse dynamics makes a straightforward application of computed torque method impossible [3]. Alternatively, some approximate schemes were proposed for open and closed loop control ([4], [5], [6]). A two link rigid-flexible manipulator moves in the horizontal plane was proposed [3], thus the gravity was not included. The manipulator was modelled using Bernoulli-Euler beam theory to describe the flexural motion of the second link. The kinematics analysis was described using frames moving with the rigid links. The problem of modelling and control of flexible manipulators has been also investigated by several authors . In most of these works, the integro-partial differential Eq. (IPDE) describing the systems dynamics are reduced to ordinary differential equations using approximate methods such as modal expansion [7], or finite element techniques [8]. Modelling using Singular Perturbation approach has been used by many researchers ([9], [10], [11]). They assumed a single flexible link manipulator and used perturbation methods to model and control the proposed manipulators. Two link flexible manipulators has been also investigated in terms of modelling and control [12]. They assumed that both manipulators’ links are flexible and that the tip mass is in contact with a constraint surface. They applied Lagrange multiplier method and Hamilton's principle; they derived the dynamic relations of joint angles, vibrations of the flexible links, and the constraint force. Ata et al.,[13] investigated the effect of boundary conditions including tip mass and tip inertia on the mode shapes and the corresponding natural frequencies for a single link flexible manipulator and they concluded that the boundary conditions has a considerable effect on the mode shape and the natural frequency as well.In this work, a mathematical modeling of a two-link flexible manipulator is derived using Bernoulli-Euler beam theory where the elastic deformation is described using Tangential Coordinate System. The elastic displacement is measured between the beam and the axis describing the beam when no flexibility exists [13] where the maximum deformation occurs at the end point of each link. The equations describing the torques of the first and the second joints are then derived from the extended Hamilton principle. Numerical simulations of the required torques using rest-to-rest third order polynomial trajectories are introduced and compared.
2. Dynamical equation and the assumed modes method The equations of motion for a two-link flexible manipulator can be written in a closed form as rigid and elastic parts [ 14]. The rigid part is given by : ࡹ࢘࢘ ሷ ࢘ ࡹ࢘ࢌ ሷ ࢌ ࢎ࢘࢘ ሶ ࢘ ࢎ࢘ࢌ ሶ ࢌ ࢘ ሺሻ ൌ ࣎ (1) And the elastic part as: ࡹࢀ ࢘ࢌ ሷ ࢘ ࡹࢌࢌ ሷ ࢌ ࢎࢌ࢘ ሶ ࢘ ࢎࢌࢌ ሶ ࢌ ࢌ ሺሻ ࡷࢌ ൌ (2) Where: ൌ ሺࢀ࢘ ǡ ࢀࢌ ሻࢀ For the details of the elements of the abovementioned matrices, the reader is referred to [15]. In order to obtain the joints torques, a trajectory is assumed for each joint and the elastic deflection for each link should be calculated by solving Equation (2) and substituting the results in Equation (1). Getting a closed form solution for Equation (2) for each link is very difficult since it is highly nonlinear, so an alternative approach is solving the general lateral vibration of the flexible beam subject to different combinations of initial and boundary conditions. There are three main methods used to model flexible manipulators; namely, the assumed modes, the finite element, and the lumped parameters methods and each model has its own advantages and disadvantages [15]. In their finite element model, fewer mathematical operations are required for inertia matrix computation in comparison with the assumed modes formulation. On the other hand, since the number of state space equations is more in the finite element models, the computation time may be greater. They have also shown that the use of the finite element model to approximate flexibility usually gives rise to overestimated stiffness matrix, and analytically concluded that overestimation of structure stiffness may lead to unstable closed-loop response of the original manipulator system. In this study, and based on the aforesaid fact, the assumed modes model is considered here. On the other hand, finding and exact solution for Eqs. (2) for the elastic deflection is very difficult if not impossible. An alternative approach is to find a solution for the general equation of flexural stiffness subject to any set of boundary conditions and substitute the solution into Eqs. (1) to find the required actuator torque [16].
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The general equation of motion for the forced lateral vibration of a non-uniform beam is: º ∂2 ª ∂2w ∂2w EI ( x) 2 ( x, t )» + ρ ( x) 2 ( x, t ) = f ( x, t ) 2 « ∂x ¬ ∂x ∂t ¼ For a uniform beam, Eq.(3) reduces to [16]: ∂4w ∂2w EI 4 ( x, t ) + ρ ( x, t ) = f ( x, t ) ∂x ∂t 2 For free vibration, f ( x, t ) = 0 and so the equation of motion becomes:
(3)
(4)
∂4w ∂2w EI ( x, t ) + 2 ( x, t ) = 0 , (5) c= 4 ρ ∂x ∂t There exist three common end conditions; these are clamped, pinned, and free (the mathematical expressions for these conditions are explained in section (5)). For the clamped (fixed) end, the displacement and the slope of the displacement curve are zero. So, the resulting boundary conditions are both geometric. For the pinned end, the displacement and the bending moment are zero. So the first boundary condition is geometric and the second is natural. For the free end, both the bending moment and shearing force are zero. The resulting boundary conditions are both natural [14]. Many works considered the boundary condition of a flexible link as fixed-free ([3], [12], [17], [18]). The rule that has been followed by most of these researchers’ states that if the frame x i − y i rotates such that it is always tangent to the beam at xi = 0 , then the pseudo-clamped boundary conditions can be applied [19]. Then, the transverse displacement is found using the method of separation of variables which can be expressed as: c2
n
w( x, t ) = ¦ Wi ( x) Ti (t ) i =1
(7)
Where W (x ) is a function of the spatial coordinate (x ) , T (t ) is the time function, and i denotes the number of the mode shape. For any beam, there will be an infinite number of normal modes with one natural frequency associated with each normal mode [20]; the first four modes are considered here, so n = 4 . Substituting Eq.(7) into Eq.(5) and rearranging leads to: 1 d 2T (t ) c 2 d 4W ( x) 2 == ωn (8) 4 W ( x) dx T (t ) dt 2 Where is the natural frequencies
ωn
ωn = β n 2
of the beam are computed as follows [18]:
EI
ρ
= (β n l ) 2
EI ρ l4
Eq.(18) can be written as two equations: d 4W ( x) - β 4W ( x) = 0 dx 4 d 2T (t ) 2 + ω n T (t ) = 0 dt 2 Where
β4 =
ωn 2 2
=
ρω n 2
EI c The solution of Eq.(11) can be expressed as T (t ) = A cos ω n t + B sin ω n t
(9)
(10) (11)
(12) (13)
Where A and B are constants to be determined from the initial conditions. A is the initial displacement x 0 , and B is the initial velocity of the beam. Since the joints will undergo a rest-to-rest motion, B = 0. The solution for Eq.(10) can be expressed in the following form [20]: W ( x) = C1 (cos βx + cosh βx) + C 2 (cos βx − cosh βx) + C 3 (sin βx + sinh βx) + C 4 (sin βx − sinh βx)
(14)
The function W (x ) is known as the normal mode or characteristic function of the beam. The unknown constants C1 to C 4 in Eq. (21) can be determined based on the different combinations of the boundary conditions of the beam.
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3. Trajectory selection In order to obtain the actuator torque for each joint, a trajectory should be assigned either in joint space or Cartesian space. This trajectory will be substituted into the equations of motion to find the actuator torque for each flexible link. There are many trajectories for the joint space among them polynomials of different orders are the simplest and easiest to design once the trajectory constraints are identified. In our study, and just for comparison, we will select polynomial of third order to express the joint space trajectory for both joints as follows: The manipulator joints are required to move from start point to goal point according to fourth order polynomial trajectory given by [21]: θ (t ) = A0 + A1 t + A2 t 2 + A3 t 3 (15) Where
A0 through A3 are constants to be determined from initial conditions. For a rest-to-rest manoeuvring, where the
joint starts from rest, increases its speed gradually, and comes to rest at the end of the task, one can get A0 = θ 0i and A1 = 0 , where θ 0i is the initial angle value of joint i.
4. Numerical simulation and discussion In this work, different combinations of boundary conditions have been chosen to be applied and the resulting output torque performances have been compared. For consistency purposes, and since the first and second links are connected at the second hub, we will assume that the same boundary condition at this point apply for both links. Following the aforementioned rule, the boundary conditions used at this work are limited to one of the following cases: Table 1. initial and boundary conditions sets First link
Second link
Case 1
Pinned-pinned
Pinned-free
Case 2
Fixed-pinned
Pinned-free
Case 3
Fixed-fixed
Fixed-free
The parameters of the two-link flexible manipulator are shown in table (2) Table 2. Parameters of the mobile manipulator [15]
Parameter Length of link 1 Length of link 2 Hub mass Tip mass
Notation
Hub inertia Flexural Rigidity Moment of Inertia of link 1 Moment of Inertia of link 2 Mass density per unit length Initial displacement of link 1 Initial displacement of link 2 The arbitrary parameter First joint initial and final angle values
Ih EI I1 I2 ρ x01 x02 Cn
Second joint initial and final angle values
l1 l2 mh mp
θ 01 , θ f1 θ 02 , θ f2
Numerical value 1m 0.8 m 0.1 kg 0.05 kg 0.0015 kg. m2 0.227 N.m2 0.0174 kg. m2 0.0140 kg. m2 0.25 kg/m 4 mm 5 mm 0.5 0, 86o 0, 57o
Atef A. Ata et al. / Procedia Engineering 41 (2012) 1253 – 1260
First, considering case 1, so that Pinned-Pinned for the first link and Pinned-Free for the second link, thus the boundary conditions for the first link are:
wi (0, t ) = 0;
wi′′(0, t ) = 0
wi (li , t ) = 0;
wi′′(li , t ) = 0
and for the second link are:
wi (0, t ) = 0;
wi′′(0, t ) = 0
wi′′(li , t ) = 0;
wi′′′(li , t ) = 0
For the solution of Eq.(20), it is assumed that the solution of the first normal mode has the following form:
Wn ( x) = C n [sin β n x] with the frequency equation as:
sin β n l = 0
β1l = π , β 2 l = 2π , β 3l = 3π , β 4 l = 4π while the solution of the second one has the form:
Wn ( x) = C n [sin β n x + η n sinh β n x] the associated frequency equation is:
tan β n l - tan hβ n l = 0
η n = (sin β n l / sinh β n l ) β1l = 3.926602 , β 2 l = 7.068583 , β 3l = 10.210176 , β 4 l = 13.351768 Considering case 2, thus Fixed-Pinned for the first link and Pinned-Free for the second link, then the boundary conditions for the first link are:
wi (0, t ) = 0;
wi′ (0, t ) = 0
wi (li , t ) = 0;
wi′′(li , t ) = 0
wi′′(li , t ) = 0;
wi′′′(li , t ) = 0
and for the second link are:
wi (0, t ) = 0;
wi′′(0, t ) = 0
For the solution of Eq.(20), it is assumed that the solution of the first normal mode has the following form:
Wn ( x) = C n [(sin β n x − sinh β n x) + α n (cosh β n x − cos β n x)] with the frequency equation written as:
tan β n l - tanhβ n l = 0
α n = (sin β n l − sinh β n l ) /(cos β n l − cosh β n l ) β1l = 3.926602 , β 2 l = 7.068583 , β 3l = 10.210176 , β 4 l = 13.351768 and the solution of the second one has the form:
Wn ( x) = C n [sin β n x + η n sinh β n x]
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and the frequency equation is:
tan β n l - tan hβ n l = 0
η n = (sin β n l / sinh β n l ) β1l = 3.926602 , β 2 l = 7.068583 , β 3l = 10.210176 , β 4 l = 13.351768 Finally, assuming that the boundary conditions are Fixed-Fixed for the first link and Fixed-Free for the second link (case 3), thus the boundary conditions for the first link are:
wi′ (0, t ) = 0
wi (0, t ) = 0;
wi (li , t ) = 0;
wi′ (li , t ) = 0
and for the second link are:
wi′ (0, t ) = 0
wi (0, t ) = 0;
wi′′(li , t ) = 0;
wi′′′(li , t ) = 0
For the solution of Eq.(20), it is assumed that the solution of the first normal mode has the following form:
Wn ( x) = C n [(sinh β n x − sin β n x) + α n (cosh β n x − cos β n x)] with the frequency equation having the following form:
cos β n l . cos hβ n l = 1
α n = (sinh β n l − sinh β n l ) /(cos β n l − cosh β n l ) β1l = 4.730041, β 2 l = 7.853205 , β 3l = 10.995608 , β 4 l = 14.1371651 and the solution of the second one has the form:
Wn ( x) = C n [(sin β n x − sinh β n x) − η n (cos β n x − cosh β n x)] and the frequency equation is:
cos β n l . cos hβ n l = −1 § sin β n l + sinh β n l · ¸ η n = ¨¨ cos β l + cosh β l ¸ ©
n
n
¹
β1l = 1.875104 , β 2 l = 4.694091, β 3l = 7.854757 , β 4 l = 10.995541 In order to analyze the effect of the boundary conditions on the required hub torque for both links, simulation has been carried out for the three above mentioned cases using MATLAB. The parameters of the flexible manipulator are summarized in the following table. The simulation algorithm can be conducted as follows 1. Solving Eq. (12) using the assumed modes method and calculating the elastic deflection for the two links w( x, t ) for the three cases using different combinations of mode shapes as illustrated in Eq. (21) and Table (1). ( x, t ) . 2. Differentiating the resulting elastic deflection to find w ( x, t ) and w 3. Design the joint space trajectory as a third order polynomial (Eq. 24) and apply the trajectory constraints (Table 2) to find the coefficients A0 , A1 , A2 , A3 Substituting the resulting data from steps (2) and (3) into Eq. (1) to find the actuator torque for the two joints for the three cases and representing the obtained torque history by graphs. The required hub torque in the four modes for each
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link in the first case is shown in Fig. (2.1 to 2.4). For the second and third cases, the results are shown in Fig. (3.1-3.4) and (4.1-4.4) respectively. First mode
0.2
0.2
0.2
0.1
0.1
0.1
0 -0.1
T o rqu e N .m
0.3
0 -0.1
0 -0.1
-0.2
-0.2
-0.2
-0.3
-0.3
-0.3
-0.4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-0.4
5
0
0.5
1
1.5
2
time (sec)
2.5
3
3.5
4
4.5
-0.4
5
Fig 2.1: case 2, first mode
Second mode
1
0.5
0.5
0.5
0 -0.5
T o rqu e N .m
1.5
1
T o rqu e N .m
1.5
1
0 -0.5 -1
-1
-1.5
-1.5
2.5
3
3.5
4
4.5
-2
5
0
0.5
1
1.5
2
time (sec)
3.5
4
4.5
-2
5
1.5
4
1
2
2
0.5
T o rqu e N .m
6
4
T o rqu e N .m
6
0 -2 -4
-1
-6
-1.5
1
1.5
2
2.5
3
3.5
4
4.5
-8
5
0
0.5
1
1.5
2
time (sec)
Fig 1.3: case 1, third mode
2.5
3
3.5
4
4.5
-2
5
4
5
5
2
T o rq ue N .m
6
10
T o rq ue N .m
15
10
-5
0 -5 -10
-4
-15
-6
-20 0
-20
3
3.5
4
4.5
Fig 1.4: case 1, fourth mode
5
0
2
4
6
8
10
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
4.5
5
0
-15 2.5
3.5
-2
-10
time (sec)
3
Fourth mode
15
0
2.5
Fig 3.3: case 3, third mode 8
2
2
Fourth mode 20
1.5
1.5
time (sec)
Fig 2.3: case 2, third mode
Fourth mode
1
1
time (sec)
20
0.5
0.5
0
-6 0.5
5
-0.5
-4
0
4.5
Third mode 2
0
4
Fig 3.2: case 3, second mode
8
-8
0
Third mode
8
-2
3.5
time (sec)
Fig 2.2: case 2, second mode
Third mode
T o rqu e N .m
3
time (sec)
Fig 1.2: case 1, second mode
T o rq ue N .m
2.5
3
0
-1
2
2.5
-0.5
-1.5 1.5
2
Second mode
1.5
1
1.5
Fig 3.1: case 3, first mode 2
0.5
1
Second mode 2
0
0.5
time (sec)
2
-2
0
time (sec)
Fig 1.1: case 1, first mode
T o rqu e N .m
First mode
0.3
T o rqu e N .m
T o rqu e N .m
First mode 0.3
12
14
16
18
time (sec)
Fig 2.4: case 2, fourth mode
20
-8
0
0.5
1
1.5
2
2.5
3
3.5
4
time (sec)
Fig 3.4: case 3, fourth mode
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5. Conclusions The simulation results presented show the first four modes of the output torques for the three used boundary conditions. The fluctuations in the generated torques about their nominal values become tougher and higher from a mode to another. This phenomenon has been highlighted by in ([14] , [23]). It can be observed clearly from the simulated results of the torque history for the three cases that the boundary conditions have a considerable effect on the mode shapes of vibration and the corresponding torques as well. The characteristics of the first and second cases were almost the same in performance and magnitude. This is due to the fact that β nl values are very close in both cases for the first link’s boundary conditions. In addition, the second link has the same boundary condition in both cases. The difference occurs in the third case, where the first link was assumed to have fixed-fixed boundary condition. Although this assumption is valid from a theoretical point of view, it remains unjustified from a practical point of view. This is because the second end of the first link is attached to the second link by a hub, but at the same time the hub and the second link are attached to the movable frame x2 − y2 , thus it is inconsistent to regard it as fixed. Since the first mode's frequency is very small, no oscillations in the torques values in the first mode are detected. The oscillations start to appear in the torques of the second mode onward due to the high frequency content.
References [1] De Luca, A., Di Giovani, G., 2001, Rest-to-Rest motion of a two-link robot with a flexible forearm, Proceedings of IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Italy, pp. 929-935. [2]. Luh, J.Y.S, Walker, W.P., Paul, R.P., 1982, Resolved Motion Force Control of Robot Manipulators, ASME Journal of Dynamic Systems, Measurement and Control, 102, pp. 126. [3] Yigit, A.S., 1992, A Robust Controller for a Two-Link Rigid-Flexible Manipulator, ASME Dynamic Systems and Control, Anaheim, CA, 42, pp.229. [4] Pfeiffer, F., 1989, A Feedforward Decoupling Concept for the Control of Elastic Robots, Journal of Robotics Systems, 6 (4), pp. 407. [5] Bayo, E., Papadopoulos, P., and Stubbe, J., 1989, Inverse Dynamics and Kinematics of Multi-Link Elastic Robots: An iterative frequency domain Approach," The International Journal of Robotics Research 8 (6), pp. 49 [6] Asada, H. Ma, Z.D., Tokumaru , H., 1990, Inverse Dynamics of Flexible Robot Arms: Modeling and Computation for Trajectory Control, ASME Journal of Dynamic Systems, Measurement and Control 112, pp. 177. [7] Krishnan, H. Vidyasagar, M., 1988, Control of a single link flexible beam using a Hankel-norm-based reduced order model manipulators, Proceedings of IEEE International Conference on Robotics and Automation, Philadelphia, CA, Vol.1, 9-14. [8] Usoro, P.B., Mahil, S.S. , Nadira, R.,1984, A Finite Element/Lagrange Approach to Modeling Lightweight Flexible Manipulators, part one: One-link System," Sensors and Controls for Automated Manufacturing and Robotics, pp. 215. [9] Bouaziz, Dochain, B. D., Piedboeu, J.C. F, Hurteau R.,1993, Singular Perturbation Approach to modelling and control of flexible manipulators, Second IEEE Conference on Control Applications, Vancouver, B.C., pp. 391-396. [10] Khorrami, F., Ozguner, U., 1988, Perturbation Methods in control of flexible link manipulators, Proceedings of the 1988 International Conference on Robotics and Automation, San Fransisco, CA, Vol.1, pp. 365-373. [11] Sicilano, B., Book, W.J., De Maria, G., 1986, Invariant Manifold Approach to control of one flexible arm, Proceedings of the 25th IEEE Conference in Decision and Control, Athens, Greece, Vol. 2, pp. 1131-1134. [12] Matsuno, F., Asano, T., Sakawa, Y., 1994, Modeling and Quasi-Static Hybrid Position/Force Control of Constrained Planar Two-Link Flexible Manipulators, IEEE Transactions on Robotics and Automation, 10 (3), pp. 287. [13] Ata, A. A., Elkhoga, S. M., Shalaby, M. A.,1996, Inverse Dynamics of a Flexible Hub- Arm System through Liapunov Second Method, Inverse Problems in Engineering 2, pp. 301. [14] Mohri, A., Sarkar, P.K., Yamamoto, M., 1998, An Efficient motion planning of flexible manipulator along specified path, Proceeding of the IEEE International Conference on Robotics and Automation, Leuven, Belgium, (1998) 1104-1109. [15] Faris, W. F., Ata, A. A. ,Sa’adeh, M. Y. ,2009, Energy Minimization Approach for flexible Manipulators Using Genetic Algorithm, Journal of Vibration & Control 15(4), pp. 497. [16] Meirovitch, L., 1967. Analytical Methods in Vibrations. Macmillan Publishing Inc. [17] Theodore, R.J., Ghosal, A., 1995, Comparison of the Assumed Modes and Finite Element Models for Flexible Multilink Manipulators, The International Journal of Robotics Research 14 (2), pp. 91. [18] Ulsoy, G., Hu, F.L., 1994, Dynamic Modelling of Constrained Flexible Robot Arms for Controller Design, Transactions of the ASME, Journal of Dynamic Systems Measurement, and Control 116 (1), pp. 56. [19] Theodore, R.J., Ghosal A., 1997, Modeling of Flexible-Link Manipulators with Prismatic Joints. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 27 (2), pp. 296. [20] J. Fei, Active vibration control of flexible steel cantilever beam using Piezoelectric actuators, Proceedings of the 37th Southeastern Symposium on System Theory SSST'05, (2005), 35-39. [21] Rao, S.S.,1995, Mechanical Vibrations (3rd ed.). Addison-Wesley Publishing Company, Inc. [22] Spong, M.W. and Vidyasagar, M., 1989. Robot Dynamics and Control. John Wiley & Sons, Inc. [23] Sarkar, P. K., Yamamoto, M., Mohri, A., 1997, Significance of spline curve in path planning of flexible manipulator," Proceedings of the IEEE International Conference on Robotics and Automation, Albuquerque, New Mexico, pp. 2535-2540.
Procedia Engineering 41 (2012) 1253 – 1260
International Symposium on Robotics and Intelligent Sensors 2012 (IRIS 2012)
Dynamic analysis of a two-link flexible manipulator subject to different sets of conditions Atef A. Ataa, Waleed F. Faresb, Mohamed Y. Sa'adehc a
Department of Engineering Mathematics and Physics, Faculty of Engineering, Alexandria University, Egypt. b Department of Mechanical Engineering, Faculty of Engineering, IIUM, Kuala Lumpur, Malaysia. c Department of Mechanical Engineering, Faculty of Engineering, Southeastern Louisiana University, USA.
Abstract Dynamic analysis for flexible manipulator is very important for selecting the actuator size and designing the proper control strategy. The effect of different sets of initial and boundary conditions on the joints torques is investigated in this paper. The elastic deflection for each link is computed using the assumed modes methods for four modes of vibration. A third order polynomial trajectory is designed in the joint space and the required torques are obtained through the solution of the inverse dynamics problem. The obtained results showed
that the boundary conditions have a considerable effect on the elastic deflection and the corresponding actuators’ torques as well. © 2012 The Authors. Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the Centre of Humanoid Robots and Bio-Sensor (HuRoBs), Faculty of Mechanical Engineering, Universiti Teknologi MARA. Keywords: Flexible manipulator, torque, assumed modes, trajectory, initial conditions.
Nomenclature ݍ ݍ ܯ ܯ ܯ ݄ ݄ ݄ ݄ ܥ ܥ K ߬ w( x, t )
generalized joint variable generalized flexible deformation variable manipulator configuration dependent generalized mass matrix of rigid part manipulator configuration dependent generalized mass matrix of flexible part representing the interaction of rigid and flexible variables generalized matrix of Coriolis and Centrifugal terms for rigid part generalized matrix of Coriolis and Centrifugal terms for flexible part representing the interaction of rigid and flexible variables representing the interaction of flexible and rigid variables vector of gravitational terms for rigid part vector of gravitational terms for rigid part generalized structural stiffness matrix vector of input torque applied at the joints elastic displacement
W (x)
spatial function
T (t ) time function Greek symbols ωn natural frequency _________
a
Corresponding author. Tel.: +2-03-592-550 ; fax: +2-03-592-1853 . E-mail address: [email protected]
1877-7058 © 2012 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.07.308
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1. Introduction Flexible link manipulators have many advantages over rigid manipulators such as higher payload to mass ratio, large reachability due to long and light members and precise high speed operation. The performance of these manipulators are severely limited by the oscillatory response which persists for a period of time after the move is completed, in other words, large and simultaneous motion of the links induce oscillations that persist beyond the nominal final completion time [1], thus delaying any subsequent operation. Modelling and simulation of flexible manipulators in a simplified manner requires many approaches and assumptions. Many researchers applied the computed torque method which was originally developed for rigid manipulators on flexible link systems [2]. However, the complexity of the inverse dynamics makes a straightforward application of computed torque method impossible [3]. Alternatively, some approximate schemes were proposed for open and closed loop control ([4], [5], [6]). A two link rigid-flexible manipulator moves in the horizontal plane was proposed [3], thus the gravity was not included. The manipulator was modelled using Bernoulli-Euler beam theory to describe the flexural motion of the second link. The kinematics analysis was described using frames moving with the rigid links. The problem of modelling and control of flexible manipulators has been also investigated by several authors . In most of these works, the integro-partial differential Eq. (IPDE) describing the systems dynamics are reduced to ordinary differential equations using approximate methods such as modal expansion [7], or finite element techniques [8]. Modelling using Singular Perturbation approach has been used by many researchers ([9], [10], [11]). They assumed a single flexible link manipulator and used perturbation methods to model and control the proposed manipulators. Two link flexible manipulators has been also investigated in terms of modelling and control [12]. They assumed that both manipulators’ links are flexible and that the tip mass is in contact with a constraint surface. They applied Lagrange multiplier method and Hamilton's principle; they derived the dynamic relations of joint angles, vibrations of the flexible links, and the constraint force. Ata et al.,[13] investigated the effect of boundary conditions including tip mass and tip inertia on the mode shapes and the corresponding natural frequencies for a single link flexible manipulator and they concluded that the boundary conditions has a considerable effect on the mode shape and the natural frequency as well.In this work, a mathematical modeling of a two-link flexible manipulator is derived using Bernoulli-Euler beam theory where the elastic deformation is described using Tangential Coordinate System. The elastic displacement is measured between the beam and the axis describing the beam when no flexibility exists [13] where the maximum deformation occurs at the end point of each link. The equations describing the torques of the first and the second joints are then derived from the extended Hamilton principle. Numerical simulations of the required torques using rest-to-rest third order polynomial trajectories are introduced and compared.
2. Dynamical equation and the assumed modes method The equations of motion for a two-link flexible manipulator can be written in a closed form as rigid and elastic parts [ 14]. The rigid part is given by : ࡹ࢘࢘ ሷ ࢘ ࡹ࢘ࢌ ሷ ࢌ ࢎ࢘࢘ ሶ ࢘ ࢎ࢘ࢌ ሶ ࢌ ࢘ ሺሻ ൌ ࣎ (1) And the elastic part as: ࡹࢀ ࢘ࢌ ሷ ࢘ ࡹࢌࢌ ሷ ࢌ ࢎࢌ࢘ ሶ ࢘ ࢎࢌࢌ ሶ ࢌ ࢌ ሺሻ ࡷࢌ ൌ (2) Where: ൌ ሺࢀ࢘ ǡ ࢀࢌ ሻࢀ For the details of the elements of the abovementioned matrices, the reader is referred to [15]. In order to obtain the joints torques, a trajectory is assumed for each joint and the elastic deflection for each link should be calculated by solving Equation (2) and substituting the results in Equation (1). Getting a closed form solution for Equation (2) for each link is very difficult since it is highly nonlinear, so an alternative approach is solving the general lateral vibration of the flexible beam subject to different combinations of initial and boundary conditions. There are three main methods used to model flexible manipulators; namely, the assumed modes, the finite element, and the lumped parameters methods and each model has its own advantages and disadvantages [15]. In their finite element model, fewer mathematical operations are required for inertia matrix computation in comparison with the assumed modes formulation. On the other hand, since the number of state space equations is more in the finite element models, the computation time may be greater. They have also shown that the use of the finite element model to approximate flexibility usually gives rise to overestimated stiffness matrix, and analytically concluded that overestimation of structure stiffness may lead to unstable closed-loop response of the original manipulator system. In this study, and based on the aforesaid fact, the assumed modes model is considered here. On the other hand, finding and exact solution for Eqs. (2) for the elastic deflection is very difficult if not impossible. An alternative approach is to find a solution for the general equation of flexural stiffness subject to any set of boundary conditions and substitute the solution into Eqs. (1) to find the required actuator torque [16].
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The general equation of motion for the forced lateral vibration of a non-uniform beam is: º ∂2 ª ∂2w ∂2w EI ( x) 2 ( x, t )» + ρ ( x) 2 ( x, t ) = f ( x, t ) 2 « ∂x ¬ ∂x ∂t ¼ For a uniform beam, Eq.(3) reduces to [16]: ∂4w ∂2w EI 4 ( x, t ) + ρ ( x, t ) = f ( x, t ) ∂x ∂t 2 For free vibration, f ( x, t ) = 0 and so the equation of motion becomes:
(3)
(4)
∂4w ∂2w EI ( x, t ) + 2 ( x, t ) = 0 , (5) c= 4 ρ ∂x ∂t There exist three common end conditions; these are clamped, pinned, and free (the mathematical expressions for these conditions are explained in section (5)). For the clamped (fixed) end, the displacement and the slope of the displacement curve are zero. So, the resulting boundary conditions are both geometric. For the pinned end, the displacement and the bending moment are zero. So the first boundary condition is geometric and the second is natural. For the free end, both the bending moment and shearing force are zero. The resulting boundary conditions are both natural [14]. Many works considered the boundary condition of a flexible link as fixed-free ([3], [12], [17], [18]). The rule that has been followed by most of these researchers’ states that if the frame x i − y i rotates such that it is always tangent to the beam at xi = 0 , then the pseudo-clamped boundary conditions can be applied [19]. Then, the transverse displacement is found using the method of separation of variables which can be expressed as: c2
n
w( x, t ) = ¦ Wi ( x) Ti (t ) i =1
(7)
Where W (x ) is a function of the spatial coordinate (x ) , T (t ) is the time function, and i denotes the number of the mode shape. For any beam, there will be an infinite number of normal modes with one natural frequency associated with each normal mode [20]; the first four modes are considered here, so n = 4 . Substituting Eq.(7) into Eq.(5) and rearranging leads to: 1 d 2T (t ) c 2 d 4W ( x) 2 == ωn (8) 4 W ( x) dx T (t ) dt 2 Where is the natural frequencies
ωn
ωn = β n 2
of the beam are computed as follows [18]:
EI
ρ
= (β n l ) 2
EI ρ l4
Eq.(18) can be written as two equations: d 4W ( x) - β 4W ( x) = 0 dx 4 d 2T (t ) 2 + ω n T (t ) = 0 dt 2 Where
β4 =
ωn 2 2
=
ρω n 2
EI c The solution of Eq.(11) can be expressed as T (t ) = A cos ω n t + B sin ω n t
(9)
(10) (11)
(12) (13)
Where A and B are constants to be determined from the initial conditions. A is the initial displacement x 0 , and B is the initial velocity of the beam. Since the joints will undergo a rest-to-rest motion, B = 0. The solution for Eq.(10) can be expressed in the following form [20]: W ( x) = C1 (cos βx + cosh βx) + C 2 (cos βx − cosh βx) + C 3 (sin βx + sinh βx) + C 4 (sin βx − sinh βx)
(14)
The function W (x ) is known as the normal mode or characteristic function of the beam. The unknown constants C1 to C 4 in Eq. (21) can be determined based on the different combinations of the boundary conditions of the beam.
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3. Trajectory selection In order to obtain the actuator torque for each joint, a trajectory should be assigned either in joint space or Cartesian space. This trajectory will be substituted into the equations of motion to find the actuator torque for each flexible link. There are many trajectories for the joint space among them polynomials of different orders are the simplest and easiest to design once the trajectory constraints are identified. In our study, and just for comparison, we will select polynomial of third order to express the joint space trajectory for both joints as follows: The manipulator joints are required to move from start point to goal point according to fourth order polynomial trajectory given by [21]: θ (t ) = A0 + A1 t + A2 t 2 + A3 t 3 (15) Where
A0 through A3 are constants to be determined from initial conditions. For a rest-to-rest manoeuvring, where the
joint starts from rest, increases its speed gradually, and comes to rest at the end of the task, one can get A0 = θ 0i and A1 = 0 , where θ 0i is the initial angle value of joint i.
4. Numerical simulation and discussion In this work, different combinations of boundary conditions have been chosen to be applied and the resulting output torque performances have been compared. For consistency purposes, and since the first and second links are connected at the second hub, we will assume that the same boundary condition at this point apply for both links. Following the aforementioned rule, the boundary conditions used at this work are limited to one of the following cases: Table 1. initial and boundary conditions sets First link
Second link
Case 1
Pinned-pinned
Pinned-free
Case 2
Fixed-pinned
Pinned-free
Case 3
Fixed-fixed
Fixed-free
The parameters of the two-link flexible manipulator are shown in table (2) Table 2. Parameters of the mobile manipulator [15]
Parameter Length of link 1 Length of link 2 Hub mass Tip mass
Notation
Hub inertia Flexural Rigidity Moment of Inertia of link 1 Moment of Inertia of link 2 Mass density per unit length Initial displacement of link 1 Initial displacement of link 2 The arbitrary parameter First joint initial and final angle values
Ih EI I1 I2 ρ x01 x02 Cn
Second joint initial and final angle values
l1 l2 mh mp
θ 01 , θ f1 θ 02 , θ f2
Numerical value 1m 0.8 m 0.1 kg 0.05 kg 0.0015 kg. m2 0.227 N.m2 0.0174 kg. m2 0.0140 kg. m2 0.25 kg/m 4 mm 5 mm 0.5 0, 86o 0, 57o
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First, considering case 1, so that Pinned-Pinned for the first link and Pinned-Free for the second link, thus the boundary conditions for the first link are:
wi (0, t ) = 0;
wi′′(0, t ) = 0
wi (li , t ) = 0;
wi′′(li , t ) = 0
and for the second link are:
wi (0, t ) = 0;
wi′′(0, t ) = 0
wi′′(li , t ) = 0;
wi′′′(li , t ) = 0
For the solution of Eq.(20), it is assumed that the solution of the first normal mode has the following form:
Wn ( x) = C n [sin β n x] with the frequency equation as:
sin β n l = 0
β1l = π , β 2 l = 2π , β 3l = 3π , β 4 l = 4π while the solution of the second one has the form:
Wn ( x) = C n [sin β n x + η n sinh β n x] the associated frequency equation is:
tan β n l - tan hβ n l = 0
η n = (sin β n l / sinh β n l ) β1l = 3.926602 , β 2 l = 7.068583 , β 3l = 10.210176 , β 4 l = 13.351768 Considering case 2, thus Fixed-Pinned for the first link and Pinned-Free for the second link, then the boundary conditions for the first link are:
wi (0, t ) = 0;
wi′ (0, t ) = 0
wi (li , t ) = 0;
wi′′(li , t ) = 0
wi′′(li , t ) = 0;
wi′′′(li , t ) = 0
and for the second link are:
wi (0, t ) = 0;
wi′′(0, t ) = 0
For the solution of Eq.(20), it is assumed that the solution of the first normal mode has the following form:
Wn ( x) = C n [(sin β n x − sinh β n x) + α n (cosh β n x − cos β n x)] with the frequency equation written as:
tan β n l - tanhβ n l = 0
α n = (sin β n l − sinh β n l ) /(cos β n l − cosh β n l ) β1l = 3.926602 , β 2 l = 7.068583 , β 3l = 10.210176 , β 4 l = 13.351768 and the solution of the second one has the form:
Wn ( x) = C n [sin β n x + η n sinh β n x]
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and the frequency equation is:
tan β n l - tan hβ n l = 0
η n = (sin β n l / sinh β n l ) β1l = 3.926602 , β 2 l = 7.068583 , β 3l = 10.210176 , β 4 l = 13.351768 Finally, assuming that the boundary conditions are Fixed-Fixed for the first link and Fixed-Free for the second link (case 3), thus the boundary conditions for the first link are:
wi′ (0, t ) = 0
wi (0, t ) = 0;
wi (li , t ) = 0;
wi′ (li , t ) = 0
and for the second link are:
wi′ (0, t ) = 0
wi (0, t ) = 0;
wi′′(li , t ) = 0;
wi′′′(li , t ) = 0
For the solution of Eq.(20), it is assumed that the solution of the first normal mode has the following form:
Wn ( x) = C n [(sinh β n x − sin β n x) + α n (cosh β n x − cos β n x)] with the frequency equation having the following form:
cos β n l . cos hβ n l = 1
α n = (sinh β n l − sinh β n l ) /(cos β n l − cosh β n l ) β1l = 4.730041, β 2 l = 7.853205 , β 3l = 10.995608 , β 4 l = 14.1371651 and the solution of the second one has the form:
Wn ( x) = C n [(sin β n x − sinh β n x) − η n (cos β n x − cosh β n x)] and the frequency equation is:
cos β n l . cos hβ n l = −1 § sin β n l + sinh β n l · ¸ η n = ¨¨ cos β l + cosh β l ¸ ©
n
n
¹
β1l = 1.875104 , β 2 l = 4.694091, β 3l = 7.854757 , β 4 l = 10.995541 In order to analyze the effect of the boundary conditions on the required hub torque for both links, simulation has been carried out for the three above mentioned cases using MATLAB. The parameters of the flexible manipulator are summarized in the following table. The simulation algorithm can be conducted as follows 1. Solving Eq. (12) using the assumed modes method and calculating the elastic deflection for the two links w( x, t ) for the three cases using different combinations of mode shapes as illustrated in Eq. (21) and Table (1). ( x, t ) . 2. Differentiating the resulting elastic deflection to find w ( x, t ) and w 3. Design the joint space trajectory as a third order polynomial (Eq. 24) and apply the trajectory constraints (Table 2) to find the coefficients A0 , A1 , A2 , A3 Substituting the resulting data from steps (2) and (3) into Eq. (1) to find the actuator torque for the two joints for the three cases and representing the obtained torque history by graphs. The required hub torque in the four modes for each
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link in the first case is shown in Fig. (2.1 to 2.4). For the second and third cases, the results are shown in Fig. (3.1-3.4) and (4.1-4.4) respectively. First mode
0.2
0.2
0.2
0.1
0.1
0.1
0 -0.1
T o rqu e N .m
0.3
0 -0.1
0 -0.1
-0.2
-0.2
-0.2
-0.3
-0.3
-0.3
-0.4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-0.4
5
0
0.5
1
1.5
2
time (sec)
2.5
3
3.5
4
4.5
-0.4
5
Fig 2.1: case 2, first mode
Second mode
1
0.5
0.5
0.5
0 -0.5
T o rqu e N .m
1.5
1
T o rqu e N .m
1.5
1
0 -0.5 -1
-1
-1.5
-1.5
2.5
3
3.5
4
4.5
-2
5
0
0.5
1
1.5
2
time (sec)
3.5
4
4.5
-2
5
1.5
4
1
2
2
0.5
T o rqu e N .m
6
4
T o rqu e N .m
6
0 -2 -4
-1
-6
-1.5
1
1.5
2
2.5
3
3.5
4
4.5
-8
5
0
0.5
1
1.5
2
time (sec)
Fig 1.3: case 1, third mode
2.5
3
3.5
4
4.5
-2
5
4
5
5
2
T o rq ue N .m
6
10
T o rq ue N .m
15
10
-5
0 -5 -10
-4
-15
-6
-20 0
-20
3
3.5
4
4.5
Fig 1.4: case 1, fourth mode
5
0
2
4
6
8
10
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
4.5
5
0
-15 2.5
3.5
-2
-10
time (sec)
3
Fourth mode
15
0
2.5
Fig 3.3: case 3, third mode 8
2
2
Fourth mode 20
1.5
1.5
time (sec)
Fig 2.3: case 2, third mode
Fourth mode
1
1
time (sec)
20
0.5
0.5
0
-6 0.5
5
-0.5
-4
0
4.5
Third mode 2
0
4
Fig 3.2: case 3, second mode
8
-8
0
Third mode
8
-2
3.5
time (sec)
Fig 2.2: case 2, second mode
Third mode
T o rqu e N .m
3
time (sec)
Fig 1.2: case 1, second mode
T o rq ue N .m
2.5
3
0
-1
2
2.5
-0.5
-1.5 1.5
2
Second mode
1.5
1
1.5
Fig 3.1: case 3, first mode 2
0.5
1
Second mode 2
0
0.5
time (sec)
2
-2
0
time (sec)
Fig 1.1: case 1, first mode
T o rqu e N .m
First mode
0.3
T o rqu e N .m
T o rqu e N .m
First mode 0.3
12
14
16
18
time (sec)
Fig 2.4: case 2, fourth mode
20
-8
0
0.5
1
1.5
2
2.5
3
3.5
4
time (sec)
Fig 3.4: case 3, fourth mode
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5. Conclusions The simulation results presented show the first four modes of the output torques for the three used boundary conditions. The fluctuations in the generated torques about their nominal values become tougher and higher from a mode to another. This phenomenon has been highlighted by in ([14] , [23]). It can be observed clearly from the simulated results of the torque history for the three cases that the boundary conditions have a considerable effect on the mode shapes of vibration and the corresponding torques as well. The characteristics of the first and second cases were almost the same in performance and magnitude. This is due to the fact that β nl values are very close in both cases for the first link’s boundary conditions. In addition, the second link has the same boundary condition in both cases. The difference occurs in the third case, where the first link was assumed to have fixed-fixed boundary condition. Although this assumption is valid from a theoretical point of view, it remains unjustified from a practical point of view. This is because the second end of the first link is attached to the second link by a hub, but at the same time the hub and the second link are attached to the movable frame x2 − y2 , thus it is inconsistent to regard it as fixed. Since the first mode's frequency is very small, no oscillations in the torques values in the first mode are detected. The oscillations start to appear in the torques of the second mode onward due to the high frequency content.
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