Aug 23, 2010 - Morris and Madani (1996). They have shown that the ... Vol. 29 No. 3 2010. M. Khairudin, Z. Mohamed, A. R. Husain and M. A. Ahmad. 209 ...
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Dynamic modelling and characterisation of a two-link flexible robot manipulator M. Khairudinl, Z. Mohamedl, A. R. Husain1 and M. A. Ahmad2 1Faculty of Electrical Engineering, Universiti Teknologi Malaysia, 81310 UTM Skudai, Johor, Malaysia 2Faculty of Electrical and Electronics Engineering, Universiti Malaysia Pahang, Malaysia Received 23rd August 2010
ABSTRACT This paper presents an investigation into the dynamic modelling and characterisation of a two-link flexible robot manipulator. A planar two-link flexible manipulator incorporating structural damping, hub inertia and payload that moves in the horizontal plane is considered. A dynamic model of the system is developed using a combined Euler-Lagrange and assumed mode method. Simulation is performed to assess the dynamic model and system responses at the hub and end-point of both links are presented and analysed in time and frequency domains. Moreover, effects of payload on the dynamic characteristics of the flexible manipulator are studied and discussed. Keywords: Assumed mode, modelling, two-link flexible manipulator.
1. INTRODUCTION Flexible manipulators have several advantages over rigid robots: they require less material, are lighter in weight, consume less power, require smaller actuators, are more manoeuvrable and transportable, have less overall cost and higher payload to robot weight ratio. These types of robots are used in a wide spectrum of applications starting from simple pick and place operations of an industrial robot to microsurgery, maintenance of nuclear plants and space robotics (Dwivedy and Eberhard, 2006). However, control of flexible manipulators to maintain accurate positioning is extremely challenging. Due to the flexible nature of the system, the dynamics are highly non-linear and complex. Problems arise due to lack of sensing, vibration due to system flexibility, imprecise positional accuracy and the difficulty in obtaining an accurate model for the system (Martins et al., 2003). The complexity of the problem increases dramatically for a two-link flexible manipulator where several other factors such as coupling between both links have to be considered. Moreover, the complexity of this problem increases when the flexible manipulator carries a payload. Practically, a robot is required to perform a single or sequential task such as to pick up a payload, move to a specified location or along a preplanned trajectory and place the payload. Previous investigations have shown that the dynamic behaviour of the manipulator is significantly affected by payload variations (Tokhi et al., 2001). If the advantages associated with lightness are not to be sacrificed, accurate models and efficient controllers have to be developed. The main goal on the modelling of a two-link flexible manipulator is to achieve an accurate model representing the actual system behaviour. It is important to recognise the flexible nature and dynamic characteristics of the system and construct a suitable mathematical framework. Modelling of a single-link flexible manipulator has been widely established. Various approaches have been developed which can mainly be divided into two categories: the numerical analysis approach and the assumed mode method (AMM). The numerical analysis methods that are Vol. 29 No. 3 2010
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Dynamic modelling and characterisation of a two-link flexible robot manipulator utilised include finite difference and finite element methods. Both approaches have been used in obtaining the dynamic characterisation of single-link flexible manipulator systems incorporating damping, hub inertia and payload (Aoustin et al, 1994) Performance investigations have shown that the finite element method can be used to obtain a good representation of the system (Tokhi et al, 2001). AMM looks at obtaining approximate modes by solving the partial differential equation characterising the dynamic behaviour of the system. An ordinary differential equation can be obtained by representing the deflection of the manipulator as a summation of modes. Each mode is assumed to be a product of two functions, one as a function of the distance along the length of the manipulator and the other, as a generalized co-ordinate dependent upon time. Previous study utilising this approach for modelling of a single-link flexible manipulator has shown that the first two modes are sufficient to identify the dynamic of flexible manipulators A good agreement between theory and experiments has also been achieved (Marlins et al, 2003). Besides, several other methods have also been studied. These include modelling of a single-link flexible manipulator system using the particle swarm optimisation algorithm (Alam and Tokhi, 2007) and extended Hamilton’s principle and generalised Galerkin’s method (Pratiher and Dwivedy, 2007) Similar to the case of a single-link manipulator, the finite element method and AMM have also been investigated for modelling of a two-link flexible robot manipulator. However, the complexity of the modelling process increases dramatically as compared to the case of a single-link flexible manipulator Yang and Sadler (1990) have developed the finite element model to describe the deflection of a planar two link flexible robot manipulator. A dynamic model has been developed using the finite element methods utilising a generalised inertia matrix (Usoro et al, 1986) De Luca and Siciliano (1991) have utilised the AMM to derive a dynamic model of multilink flexible robot arms limiting to the case of planar manipulators with no torsional effects. A single-link flexible manipulator model which is suitable for development of a two-link flexible manipulator model has been described by Morris and Madani (1996). They have shown that the accuracy of a single-link model can be improved by inclusion of a shear deformation term. Subudhi and Morris (2002) have also presented a systematic approach for deriving the dynamic equations for an n-link manipulator. In this work, two-homogenous transformation matrices are used to describe the rigid and flexible motions respectively. A modelling technique for a two-link flexible manipulator based on the absolute nodal coordinate method has also been investigated (Tian et al, 2009). The simulation results have indicated that the absolute nodal coordinate formulation is effective in modelling such a system. This paper presents an investigation into the dynamic modelling and characterisation of a two-link flexible robot manipulator incorporating structural damping, hub inertia and payload. For the flexible manipulator system, the effects of other physical parameters such as payload on the dynamic characteristics of the system should be further explored. The payloads are attached at the end-point of the second-link whereas hub inertias are considered at the actuator joints. A combined Euler-Lagrange and assume mode discretisation technique are used to derive the dynamic model of the system. Simulation of the dynamic model is performed in Matlab and Simulink. System responses, namely angular position, modal displacement, end-point acceleration and the power spectral density (PSD) of the end-point acceleration are evaluated in both time and frequency domains. Moreover, the works presents the effects of varying payload on the dynamic characteristics of the system. The work presented forms the basis of the design and development of suitable control strategies for two-link flexible manipulator systems. The rest of the paper is structured as follows: Section 2 provides a brief description of the two-link flexible manipulator system considered in this study. Section 3 describes the modelling process of the system using AMM. Simulation results are presented in Section 4 and the paper is concluded in Section 5
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M. Khairudin, Z. Mohamed, A. R. Husain and M. A. Ahmad 2. THE TWO-LINK FLEXIBLE MANIPULATOR SYSTEM Figure 1 shows a structure of a two-link flexible manipulator system considered in these investigations. The links are cascaded in a serial fashion and are actuated by rotors and hubs with individual motors. The ith link has length li with uniform mass density per unit length ρi. The first link is clamped at the rotor of the first motor whereas the second motor is attached at the tip of the first link. E and I represent Young modulus and area moment of inertia of both links respectively. A payload is attached at the end-point of link-2. X0Y0 is the inertial co-ordinate frame, XiYi the ^ ^ rigid body coordinate frame associated with the ith link and XiYi is the moving coordinate frame θ1, and θ2 are the angular positions and vi (xi, t) is the transverse component of the displacement vector. Mp is an inertial payload mass with inertia Jp at the end-point of link-2 The physical parameters of the two-link flexible manipulator system considered in this study are shown in Table I. Mh2 is the mass considered at the second motor which is located in between both links, Jhi is the inertia of the ith motor and hub. The input torque, τi(t) is applied at each motor and Gi is the gear ratio for the ith motor. Both links and motors are considered to have the same dimensions.
Figure 1.
Structure of a two-link flexible manipulator.
Table I Parameters of a two-link flexible manipulator
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Dynamic modelling and characterisation of a two-link flexible robot manipulator 3. DYNAMIC MODELLING This section focuses on the development of a combined Euler-Lagrange and AMM simulation algorithm characterising the dynamic behaviour of the two-link flexible manipulator system The description of kinematics is developed for a chain of n serially connected flexible links Considering revolute joints and motion of the manipulator on a two-dimensional plane, the rigid transformation matrix, Ai, from Xi-1Yi-1 to XiYi can be written as (1)
The elastic homogenous transformation matrix, Ei, due to the deflection of the link i can be obtained as
(2)
where νi(xi, t) is the bending deflection of the ith link at a spatial point xi (0 ≤ xi ≤ li) The global transformation matrix Ti transforming co-ordinates from X0Y0 to XiYi follows a recursion as (3)
Where I is an identity matrix. Let
be the position vector that
describes an arbitrary point along the ith deflected link with respect to its local coordinate frame (XiYi) and 0ri, be the same point referring to X0Y0. The position of the origin of Xi+1Yi+1 with respect to XiYi is given by (4) where 0pi is its absolute position with respect to X0Y0. Using the global transformation matrix, 0ri and 0pi, can be written as (5) To derive the dynamic equations of motion of a two-link flexible manipulator, the total energies associated with the manipulator system needs to be computed using the kinematic formulations. The total kinetic energy of the manipulator is given by (6) where TR, TL and TPL are the kinetic energies associated with the rotors, links and the hubs, respectively. The kinetic energy of the ith rotor can be obtained as (7)
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M. Khairudin, Z. Mohamed, A. R. Husain and M. A. Ahmad . where αi iS the angular velocity of the rotor about the ith principal axis. Moreover, the kinetic energy of a point ri(xi) on the ith link can be written as (8) . where 0ri (xi) is the velocity vector. The velocity vector can be computed by taking the time derivative of Equation (5): (9) . where 0pi can be deterrnined by using Equations (4) and (5) along with (10) . Furthermore, the time derivative of the global transformation matrix Ti can be recursively calculated from (De Luca and Siciliano, 1991; Subudhi and Morris, 2002) (11) . . Computation of A i, and E i, in Equation (ll) can be performed as follows: (12)
Evaluation of the transpose and derivative of transpose terms of the velocity vector in Equation (8) can be accomplished by utilising the following identities: (13) (14) where I is the identity matrix of appropriate dimensions. Once, the kinetic energy associated with the ith link is obtained, the kinetic energy of all the n links can thus be found as
(15)
As shown in Figure 1 and the kinematics formulation described previously, the kinetic energy associated with the payload can be written as
(16)
where ; n being the link number, prime and dot represent the first derivatives with respect to spatial variable x and time, respectively.
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Dynamic modelling and characterisation of a two-link flexible robot manipulator The total potential energy of the system due to the deformation of the link i by neglecting the effects of the gravity can be written as (17)
where EI is the flexural rigidity of the system. The dynamics of the link at an arbitrary spatial point xi along the link at an instant of time t can be written using Euler-Beam theory as (18)
On the other hand, bending deflections νi (xi, t) can be expressed as a superposition of mode-shapes and time dependent modal displacements as
(19)
where qij(t) and φij(xi) are the jth modal displacement and jth mode shape function for the ith link. The solution of equation (19) is in the form of (20) where mi is the mass of link i and γij, is given as
(21)
and βij, is the solution of the following equation:
(22)
Subsequently, the natural frequency for the jth mode and ith link, ωij, is determined from the following expression (Tao Fan, 2007): (23)
In this work, the dynamic model of the system incorporating payload is investigated. In this case, the effectiveness masses at the end of the individual links (ML1 for link-1 and ML2 for link-2) are set as
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M. Khairudin, Z. Mohamed, A. R. Husain and M. A. Ahmad
(24)
and the effectiveness inertia of the individual links (JL1 for link-1 and JL2 for link-2) are
(25)
where m2 is the mass of link-2 and Jo2 is the joint inertia of link-2 about joint-2 axis. The co-ordinate vector consists of link positions, (θ1, θ2) and modal displacements (q11, q12, q21, q22). The force vector is F = {τ1, τ2, 0, 0, 0, 0}T, where τ1 and τ2 are the torques applied at the hubs of link-1 and link-2, respectively. The dynamic equations of motion of a two-link flexible manipulator can be derived utilising the EulerLagrange’s equations with the Langrangian, L = T – U . With i = 1 and 2 and j = 1 and 2, the equation can be obtained as: (26)
and (27)
Considering the damping, the desired dynamic equations of motion of a two-link flexible manipulator can be obtained as (28)
where f1 and f2 are the vectors containing terms due to coriolis and centrifugal forces, M is the mass matrix and g1 and g2 are the vectors containing terms due to the interactions of the link angles and their rates with the modal displacements. K is the diagonal stiffness matrix which the values of ωij2mi and D is the passive structural damping as
(29)
(30)
4. RESULTS In this section, simulation results of the dynamic behaviour of the two-link flexible manipulator system are presented in the time and frequency domains. The system is considered with and without payload. Bang-bang signals with amplitude of 0.2 Nm, shown in Figure 2, are used as input torques applied at the hubs of the manipulator. A bang-bang torque has a positive (acceleration) and negative (deceleration) period
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Dynamic modelling and characterisation of a two-link flexible robot manipulator allowing the manipulator to, initially, accelerate and then decelerate and eventually stop at a target location System responses are monitored for a duration of 5 s, and the results are recorded with a sampling time of 1 ms. Three system responses namely the angular positions, modal displacement and end-point acceleration at both links with the PSD of the end-point acceleration are obtained and evaluated. Table I shows the physical parameters of the two-link flexible manipulator used in this simulation. The developed simulation algorithm based on combined Euler-Lagrange and AMM model was implemented within the Matlab and Simulink environment on Intel Celeron 1.73 GHz processor. Table II shows the parameters of βij for both links and modes with different payload profiles (0 kg, 0.1 kg and 0.5 kg). It is noted that the parameter of, Bij which effect the behaviour of modes shapes as in Equation (22) is gradually decreased with increasing value of payload
Figure 2.
The bang-ban input torque
Table II Relation between payload and parameter of Bij for link-1 and link-2
4.1. System without Payload Figures 3, 4 and 5 show the angular positions, modal displacement and end-point acceleration responses of the two-link flexible manipulator without payload for both links. It is noted that a steady-state angular position levels of 54.56 degrees and -122.33 degrees were achieved within 1.8 s for link-1 and link-2 respectively. The time responses specifications of angular position for link-1 were achieved within the settling time and overshoot of 2.00 s and 1.35 % respectively. On the other hand, the settling time and overshoot for link-2 were obtained as 1.99 s and 0.35 % respectively. The modal displacement responses of link-1 and link-2 were found to oscillate between -6.1x10-3 m to 9.3x10-3 m and -11.2x10-3 m to 17.3x10-3 m 214
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M. Khairudin, Z. Mohamed, A. R. Husain and M. A. Ahmad respectively. On the other hand, the end-point acceleration responses were found to oscillate between -0.89 to 0.61 m/s2 for link-1 and -2.98 to 3.96 m/s2 for link-2. Figure 6 shows the PSDs of the end-point acceleration representing the system response in the frequency domain. It is noted for the first two modes of vibrations, the resonance frequencies for link-1 were obtained at 4 Hz and 55 Hz and for link2 as 4 Hz and 52 Hz respectively
Figure 3.
Angular position of the system without load.
Figure 4.
Modal displacement of the system without load.
Figure 5.
End-point acceleration of the system without load. Vol. 29 No. 3 2010
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Figure 6.
PSD of end-point acceleration of the system without load.
Figure 7.
Angular position of the system with incorporating payload.
4.2. System with Payload To demonstrate the effects of payload on the dynamic characteristics of the system, various payloads of up to 0.5 kg were examined. Figure 7 shows the system responses of the flexible manipulator with payloads of 0.1 kg and 0.5 kg for link-1 and link-2 respectively. It is noted that the angular positions for link-1 decrease towards the positive direction whereas the angular positions of link-2 decrease in the negative direction with increasing payloads. For payloads of 0.1 kg and 0.5 kg, the steady-state angular position levels for link-1 were obtained as 35.97 degrees and 17.40 degrees respectively whereas for link-2, these were obtained as -71.87 degrees and -34.32 degrees. The time response specifications of angular positions have shown significant changes with the variations of payloads. With increasing payload, the system exhibits higher settling times and overshoots for both links. For link-1, the response exhibits an overshoot of 1.99 % with the settling time of 2.53 s and an overshoot of 11.97 % with the settling time of 4.80 s for 0.1 kg and 0.5 kg respectively. For link2, the response exhibits a percentage overshoot of 1.83 % with the settling time of 2.39 s for 0.1 kg payload and overshoot of 5.7 % with the settling time of 3.93 s for 0.5 kg payload. 216
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M. Khairudin, Z. Mohamed, A. R. Husain and M. A. Ahmad Figures 8 and 9 show modal displacements and end-point acceleration responses of the system with payloads of 0.1 and 0.5 kg. It is noted with increasing payloads, the magnitudes of vibration of modal displacement increase for both links. In this case, the modal displacement responses of link-2 were found to oscillate between -29.6x10-3 m to 46.2x10-3 m and -56.4x10-3 to 83.3x10-3 m for payloads of 0.1 kg and 0.5 kg respectively. However, the magnitudes of the acceleration responses decrease with increasing payloads. For link-2 the responses were found to oscillate between -2.60 to 3.25 m/s2 and -1.14 to 1.30 m/s2 for payloads of 0.1 kg and 0.5 kg respectively. The result shows that with a payload attached at the end-point, the effective mass of link-2 increases and thus the system moves with less acceleration. In this work, PSD of the end-point acceleration response is utilised to investigate the effects of payload on the dynamic behaviour of the system in the frequency domain. Figure 10 shows the PSD of the end-point acceleration response with payloads of 0.1 kg and 0.5 kg for both links. It is noted that the resonant modes of vibration of the system shift to lower frequencies with increasing payloads. For a payload between 0.1 to 0.5 kg the resonant frequencies for link-l shifted from 3 Hz and 17 Hz to 1 Hz and 7 Hz for the first two modes of vibration respectively. On the other hand, the resonant frequencies of link-2 shifted from 2 Hz and 17 Hz to I Hz and 7 Hz. This implies that thc manipulator oscillates at lower frequency rates than those without payload.
Figure 8.
Model displacement of the system incorporating payload.
Figure 9.
End-point acceleration of the system incorporating payload.
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Figure 10.
PSD of end-point acceleration of the system incorporating payload. 5. CONCLUSION The development of a dynamic model of a two-link flexible manipulator incorporating structural damping, hub inertia and payload has been presented. The model has been developed using a combined AMM and Euler-Lagrange approach. Simulations of the dynamic model have been carried out in the time and frequency domains where the system responses including angular positions, modal displacements and end-point acceleration are studied. Simulation results have shown that significant vibration occurs during movement of the system. It is found that the payload significantly affected the system behaviour both in time and frequency domains. These results are very helpful and important in the development of effective control algorithms for a two-link flexible robot manipulator incorporating payload Future work will consider validation of the dynamic model by comparing with an actual two-link flexible manipulator system. REFERENCES Alam, M. S. and Tokhi, M. (2007). “Dynamic modelling of a single-link flexible manipulator system: A particle swarm optimisation approach, Journal of Low Frequency Noise, Vibration and Active Control, Vol 26, 57-72 Aoustin Y, Chevallereau C, Glumineau A and Moog C. H. (1994). “Experimental results for the end-effector control of a single flexible robotic arm”, IEEE Transactions on Control Systems Technology, Vol 2, 371 381 De Luca A. and Siciliano B. (1991) ‘’Closed-form dynamic model of planar multilink lightweighl robots, “ IEEE Transactions on Systems, Man, and Cybernetics, Vol 21, 826-839 Dwivedy, S. K. and Eberhard, P. (2006). “Dynamic analysis of flexible manipulators, a literature review”, Mechanism and Machine Theory, Vol. 41, 749-777. Martins J. M., Mohamed Z., Tokhi M. O., Sa da Costa J. and Botto M. A. (2003). “Approaches for dynamic modelling of flexible manipulator systems”, IEE Proc-Control Theory and Application, Vol. 150, 401-411. Morris A. S. and Madani A. (1996). “Inclusion of shear deformation term to improve accuracy inflexible-link robot modelling”, Mechatronics, Vol. 6,631647.
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M. Khairudin, Z. Mohamed, A. R. Husain and M. A. Ahmad Pratiher B. and Dwivedy S. K. (2007). “Non-linear dynamics of a flexible single link Cartesian manipulator”, International Journal of Non-Linear Mechanics, Vol. 42, 1062-1073. Subudhi B. and Morris A. S. (2002). “Dynamic modelling, simulation and control of a manipulator With flexible links and joints “, Robotics and Autonomous System, Vol. 41, 257-270. Tao Fan. (2007). “Intelligent model predictive control of flexible link robotic manipulators”, PhD Thesis, University of British Columbia, May 2007. Tian, Q., Zhang, Y. Q., Chen, L. P. and Yang, J. (2009). “Two-linkflexible manipulator modeling and tip trajectory tracking based on the absolute nodal coordinate method’, International Journal of Robotics and Automation, Vol. 24, 103 - 114. Tokhi M. O., Mohamed Z. and Shaheed M. H. (2001). “Dynamic characterisation of a flexible manipulator system “, Robotica, Vol. 19, 571-580. Usoro, P. B., Nadira, R. and Mahil, S. S. (1986). “A finite element/Lagrangian approach to modelling light weight flexible manipulators”, ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 108,198-205. Yang Z. and Sadler J. P. (1990). “Large-displacement finite element analysis of flexible linkage”, ASME Journal of Mechanical Design, Vol. 112,175-182.
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