Dynamic modelling and control of flexible link manipulators: methods and scope- Part-1

Objectives: This paper addresses two key issues in the area of flexible robotics. The issues are dynamic modelling and control of flexible link robots. A brief, yet, significant review is provided that addresses these issues. Methods: The various approaches used by researchers for dynamic modelling and control of flexible robots are presented. Besides that,methods used for achieving optimal control are also discussed. Findings: After a review of 153 research papers from the year 1975 to 2021, it has been found that a good dynamic model of flexible manipulator helps in reducing the control and computational efforts. Recent trends in research in the area of flexible manipulators are towards the use of sliding mode control and vision-based control techniques. Novelty: Inclusion of the effect of torsional vibrations besides lateral vibrations on the positional accuracy of flexible manipulators makes the current research work


Introduction
Nowadays, robots are used in various fields like marine applications, medicine, the agricultural sector, space explorations, industry, and many other diverse fields. Many of these applications of robots require them to be lightweight, fast, and at the same time accurate and precise. The potential advantages offered by elastic link manipulators are higher operation speed and greater payload-to-manipulator weight ratio than the rigid robots. But, the flexibility of arms leads to deformation and vibrations at the tips of the links during the motion. As a consequence, dynamics modelling becomes extremely complicated. Also, the appearance of oscillations makes the control of such manipulators an arduous task. Thus, consideration of structural flexibility in robotic arms is a real challenge. The study in the area of flexible manipulators can be broadly classified as shown in Figure 1 below. https://www.indjst.org/ There can be two types of flexibilities present within a flexible manipulator: joint flexibility and link flexibility. Only link flexibility has been considered in this paper. The present literature survey gives a brief description of work done by various authors regarding various approaches adopted by them during the mathematical modelling of flexible manipulators. It also provides information about the application of control schemes used by researchers for accurate tip position control of flexible manipulators. The techniques of vibration suppression and trajectory control are reviewed in Part-2 of the study. For the establishment of an accurate mathematical model, firstly the frame of reference is chosen. Three different types of frames of reference are available in the literature. These are: floating frame of reference, corotational frame of reference, and inertial frame of reference. The flexible links can be modelled using Euler-Bernoulli beam theory, Timoshenko beam theory, and nonlinear beam theory available in the literature. If the length to thickness ratio of the beam is more than 10 then Euler-Bernoulli beam theory is suitable otherwise Timoshenko beam theory gives accurate results. It is advisable to incorporate the effect of geometrical stiffening during mathematical modelling for accurate results. Dwivedy and Eberhard [2006] (1) , Benosman and Vey [2004] (2) ,   (3) , and Jing et al. [2019] (4) provided a comprehensive review on various techniques used by different researchers for modelling and control of flexible multi-body systems. A deep and significant insight is provided by Tokhi and Azad [2017] (5) with regards to the modelling, simulation and control of flexible-link manipulators that serve as a lighthouse for the researchers working in the area of flexible robotics. Many industrial applications require the use of moving platforms comprised of flexible mechanisms. Precise control of these platforms is necessary as it directly impacts the industrial production and hence the economy of any country (6) .
The present survey is divided into the following sub-headings: 1. Dynamic modelling using assumed modes method (AMM) 2. Dynamic modelling using finite element method (FEM) 3. Other approaches for dynamic modelling 4. Control strategies for accurate tip positionin 5. Optimization techniques https://www.indjst.org/

Dynamic modelling using assumed modes method
The initial work on flexible robots started with the study of a single flexible link. Euler-Bernoulli (EB) beam theory was used for modelling the lateral vibrations. The solution to the elastic deformation of the flexible link was obtained using the assumed modes method (AMM). From the literature, it was found that the most preferred approach for developing the mathematical model of flexible manipulators is: Lagrangian dynamics. Book et al. [1975] (7) pioneered the research in this area. They developed the frequency domain model and the time domain model and used the linear feedback schemes for vibration control. The correct input torques for the flexible manipulator were found out by Luh et al. [1980] (8) using the resolved-motion-ratecontrols (feedback) method. Book and Majette [1983] (9) determined the feedback gains of the controller using frequencydomain and state-space representations. Judd and Falkenburg [1985] (10) studied the control problem from the point of view of generating the desired trajectory and avoiding singularities simultaneously. Kanoh et al. [1986] (11) developed the technique of quasilinearization for implementing a non-linear PID controller. Bakr and Shabana [1986] (12) developed a method for the dynamic analysis of geometrically nonlinear and inertia-variant flexible systems using Lagrange's multipliers. The issues involved in utilizing end-point sensing for Two-Link Flexible manipulators were raised by Oakley and Cannon [1989] (13) .
With the onset of the nineties, the researchers forayed into the research of multi-link flexible manipulators. Luca and Siciliano [1991] (14) derived a closed-form finite-dimensional dynamic model for planar multilink lightweight robots. Li and Sankar [1993] (15) (19) showed the importance of inverse dynamic analysis for flexible manipulators. They also showed the effect of boundary conditions used during the mathematical modelling of flexible manipulators on actuator torques. Furthermore, various nonlinear phenomena like friction and gravity greatly affect the modal vibrations of a flexible manipulator (20) . Thus, it is desirable to have an accurate dynamic model of the flexible manipulator from the point of view of control effectiveness. A mathematical model of a Two-Link Flexible manipulator having two revolute joints undergoing both small bending and small torsional deformations is described by Mishra and Singh [2019] (21) . An investigation into the effect of payload on the modal response of the flexible manipulator is carried out by Kumar and Pratiher [2020] (22) .  Figure 2, it can be seen that the dynamics of a flexible manipulator can be obtained by superimposing the dynamics of the rigid manipulator and the dynamics of a vibrating structure. The vibrating structure is usually taken as a beam which is a distributed parameter system. Distributed parameter systems exhibit infinite modes and hence it is necessary to use AMM. AMM helps in incorporating a sufficient number of modes that can help describe the vibration behaviour of the flexible system satisfactorily. But as the structure of the flexible manipulator becomes complex, it becomes difficult to use the AMM. The solution obtained using AMM is prone to the number of modes used and the type of boundary conditions chosen. From the literature, it is found that the mathematical model of a flexible manipulator depends upon the type of modelling technique used (3) . In the following section, a survey on dynamic modelling using FEM is provided. https://www.indjst.org/ 3 Dynamic modelling using finite element method While many researchers were using AMM for modelling flexible manipulators, Sunada and Dubowsky [1981] (23) developed a method of dynamic analysis based upon finite elements. To reduce the size of dynamic equations obtained by FEM, component mode synthesis (CMS) was used by them. After 1986, most of the authors started to use the finite element method along with Lagrangian dynamics for dynamic modelling. Dado and Soni [1986] (24) carried out the forward and the inverse dynamics analyses of flexible manipulators. Naganathan and Soni [1986] (25) used the Newton-Euler approach for dynamic modelling of flexible links modelled as Timoshenko beam. The response of a Two-Link Flexible manipulator under gravity was observed by Usoro et al. [1986] (26) . Bayo [1987] (27) found the joint torque of a single flexible link for producing the desired tip motion. He also performed the frequency domain analysis of the flexible manipulator system. Simo and Vu-Quoc [1987] (28) highlighted the role of nonlinear beam theory in transient dynamic analysis of flexible structures. Tzou and Wan [1990] (29) implemented Rayleigh and viscoelastic damping for preparing the mathematical model of flexible manipulators. Chedmail et al.
[1991] (30) used modified Denavit-Hartenberg representation for kinematic analysis of a Two-Link flexible manipulator. Gaultier and Cleghorn [1992] (31) developed and applied a spatially translating and rotating-beam finite element to model flexible manipulators. Alberts et al. [1992] (32) evaluated the effectiveness of viscoelastic damping methods using finite element analysis. Hu and Ulsoy [1994] (33) used Galerkin's approach for preparing the mathematical model of a rigid-flexible spherical-coordinate robot from the point of view of the design of a robust controller. Stylianou and Tabarrok [1994] (34) developed governing equations of motion for an axially moving flexible beam and investigated its dynamic stability characteristics under the effects of physical damping, tip mass, tip support, and wall flexibility. They performed discretization using 'variable domain finite beam elements' . Theodore and Ghosal [2003] (35) modelled the flexible links using finite element approach and discussed important issues like robustness and stability for the controller used for trajectory control of a flexible manipulator. Fotouhi [2007] (36) studied the vibration analysis of a rotating flexible robotic arm undergoing large deformations. Before him all the authors considered only the small elastic deformation case. Grazioso et al. [2016] (37) made use of FEM based upon geometrically exact beam theory for obtaining the most accurate mathematical model for the flexible manipulators. Bamdad and Feyzollahzadeh [2020] (38) used discrete time transfer matrix method to analyze the large deformations of a flexible manipulator. They tried to reduce the computational complexity associated with such manipulators. The links of the flexible manipulator possess distributed parameters. Hence, to describe the dynamics of these links, discretization can be done using the finite element method. This involves the division of flexible links into some finite number of elements and finding the inertia and stiffness matrices that govern the dynamics of the system under consideration. Discretization of flexible links (21) can be done using Space-Frame Elements [Chandrupatla and Belegundu, 2012] (39) . A 'space-frame element' has two nodes with each node having six degrees of freedom: three translational and three rotational. Figure 3 shows the procedure of obtaining a finite element model for the flexible links. The derivation of the finite element model depends upon the type of finite element chosen for discretization, the number of finite elements used for discretization, and the type of boundary conditions chosen. The use of FEM facilitates the easy handling of boundary conditions. The boundary conditions exhibited by the flexible manipulator are time-dependent. This is due to timevarying boundary conditions (40,41) for the flexible links. Besides that, the change in configuration of the manipulator due to its motion causes the eigenvalues to change. https://www.indjst.org/

Other approaches for dynamic modelling
Few authors used some different approaches for the dynamic modelling and vibration control of flexible robots. These include the ERLS (Equivalent Rigid Link System) approach, the redundant manipulator or controllable local degrees of freedom approach, the wave-based approach, and Kane's method. Marino and Spong [1986] (42) studied the nonlinear problem for a rigid single link manipulator having joint elasticity. They used feedback linearization, singular perturbation, integral manifold, and composite control for designing the control system. Matsuno et al. [1994] (43) developed the mathematical model of the decoupled bending and torsion vibrations of a flexible beam. This was one of the newest researches in the area of flexible robotics as it considered the torsional modes of vibration also along with the flexural modes of vibration. Until now, only the flexural vibrations were studied by various authors. The vibration suppression was achieved by two rotation motors. The authors also developed a feedback system using a dynamic compensator and implemented the Voigt type internal damping in their system. Elmaraghy et al. [1994] (44) discussed the joints and links flexibility of robots and control of robots by dynamic hybrid force and position control and a model reference adaptive control schemes. Ubertini [2000] (45) performed the numerical analysis of flexible link systems based upon an iterative scheme called Grobner basis. He used a new approach for developing the mathematical model of flexible links. This approach was called Equivalent Rigid Link System, ERLS. Yushu et al. [2008] (46) tried to suppress the vibration in flexible manipulators using controllable local degrees of freedom. Connor [2007] (47) applied wave-based control in lumped flexible robotic systems through position control and active vibration damping. Gao and Yun [2008] (48) improved the dynamic performance of a flexible redundant manipulator through its kinematics redundancy feature. Bian et al. [2009] (49) also suggested improving the motion accuracy of a flexible manipulator by suppressing the vibration using the kinematic redundancy feature. Zimmert and Sawodny [2010] (50) developed a spring-damper-based mathematical model and used a two-degree of freedom control approach for the active damping of the bending oscillation of a forklift's mast using the flatness technique. They used a feed-forward control strategy. Kermani [2010] (51)

Control strategies for accurate tip positioning
Many researchers adopted the approach of 'model-based control' for accurate tip positioning of the flexible manipulator. Morgul [1992] (55) performed the dynamic boundary force and torque control of an EB beam. He successfully showed exponential decay of vibrations using Lyapunov control. Hillsley and Yurkovich [1993] (56) achieved the endpoint position control of a Two-Link Flexible mechanism using input shaping control and acceleration feedback. Luca and Siciliano [1993] (57) designed a Lyapunov controller based on proportional-derivative (PD) feedback and feedforward control schemes for the flexible robots under the influence of gravity. Khorrami et al. [1995] (58) controlled the tip vibrations of flexible manipulators under varying payload conditions using adaptive control and input preshaping. Lu et al. [1996] (59) designed and implemented the controller and sensors based on a reduced-order model obtained through assumed mode analysis of flexible arm. Tso et al. [2003] (60) minimized tip deflection using a non-linear Lyapunov-type controller designed for trajectory control. Sun et al. [2004] (61) performed the active vibration control of a flexible link through the placement of segmented piezoelectric actuators. They used a PD-based feedback control scheme and command shaping together. The researchers also performed the dynamic modelling using Lagrangian dynamics and performed the stability analysis by virtual joint model. Monje et al. [2007] (62) studied the control of single link, lightweight flexible manipulators in the presence of payload using PD and fractional-order derivative controller. Pereira et al. [2008] (63) implemented integral resonant control (IRC) scheme for controlling the vibrations of a cantilever beam. Many authors used new control approaches that included integral resonant control, sliding mode control, adaptive control schemes, fuzzy-based control methods, etc. for vibration suppression and optimal trajectory control. Chaoui et al. [2009] (64) proposed a control strategy based on artificial neural networks (ANNs) while Korayem et al. [2009] (65) developed an algorithm to improve the maximum load-carrying capacity (MLCC) of flexible robots. They used a sliding mode controller and non-linear observer in their control system. Pereira et al. [2009] (66) also proposed a new control approach for a singlelink flexible manipulator, based on Integral Resonant Control (IRC) scheme and thus addressed the two main problems which complicate the design of a control system viz. the high order of the system and the non-minimum phase dynamics that exist between the tip position and the input. Mamani et al. [2009] (67) applied a sliding mode control (SMC) design methodology based upon a unit vector approach and a state estimator based on polynomial approximations. Liu et al. [2010] (68) presented an FBFN-based (Fuzzy Basis Function Network) adaptive control algorithm for flexible manipulators. Diaz et al. [2010] (69) https://www.indjst.org/ dealt with active vibration cancellation in flexible manipulators using a robust input shaper. Takahashi and Sasaki [2010] (70) investigated a feedback controller for tip angular position control of a single-link flexible robot arm. They considered the passivity of the flexible robot arm based on a distributed parameter system while designing the controller. Macchelli and Melchiorri [2011] (71) developed a procedure for improving the transient response of a boundary-controlled non-linear flexible elastic beam through a feed-forward action. The boundary conditions were time-varying. Bossi et al. [2011] (72) proposed the use of model predictive control (MPC) to control a fast mechanical system having a flexible arm with limited oscillations during the manoeuver. Chaoui and Sicard [2012] (73) proposed a control structure based upon ANN that compensated the nonlinear friction terms and flexibility and accurately tracked the desired trajectory. It could be used to improve the static and dynamic performance of electromechanical systems. Kulakov et al. [2015] (74) provided a new approach of force-torque control for service robots that involve contact with the environment. In their approach, they successfully transformed the joint control forces into artificial potential forces.

Optimization techniques
Various optimization schemes were used by the researchers to achieve the optimal control of flexible manipulators. Command shaping control strategy for vibration reduction was used by Alam et al. [2007] (75) . They used particle swarm optimization (PSO) for achieving the optimal control. Zebin and Alam [2010] (76) (80) discussed the precise control of the end-point position of a planar single-link flexible manipulator using PD-type Mamdani FLC and GA optimized FLC. He modelled the flexible link using Timoshenko beam theory and considered Kelvin-Voigt damping in the system. The efficiency of the flexible manipulator is greatly influenced by the presence of residual vibrations. In one research, these vibrations are controlled using internal frictional dampers (81) . The optimization of frictional dampers was performed using PSO and some other techniques.

Conclusions from literature survey
In this section, conclusions from the literature survey will be provided. Most of the papers deal with the planar single-link flexible robotic arms with small 3D motions. For such links, a linear model is sufficient to describe the dynamic characteristics. A lot of research is going on, on non-linear models of flexible arms. The simple case of non-linearity in flexible arms is that of a two-link case which was described in a few papers. Furthermore, links having revolute joints were studied a lot. The establishment of an accurate mathematical model for a flexible manipulator can be done in the floating frame of reference, corotational frame of reference, and inertial frame of reference (82) . Table 1 describes these three frames of reference.  (96) https://www.indjst.org/ Most of the authors focused upon the design of controllers according to the accurate dynamic model of the flexible arm. Researchers used both the AMM and FEM to model the flexible arm without compromising the accuracy. Some researchers tried to use intelligent control techniques like fuzzy logic, neural networks, and genetic algorithms for designing robust controllers as these do not require complicated mathematical modeling. There are two types of vibration control schemes: feedforward and feedback. It is also found that to describe the active control of vibrations, the state-space model can accurately represent the system without sacrificing its simplicity. Trajectory control using forward and inverse dynamics methods [Judd and Falkenburg, 1985] (7) , [Dado and Soni, 1986] (19) and effect of gravity [Luca and Siciliano, 1993] (134) on the motion of flexible links were studied in one or two papers. Few authors recommended the use of nonlinear beam theory ([Simo and Vu-Quoc, 1987] (23) , [Hu and Ulsoy, 1994] (28) ) to account for geometric nonlinearities and a nonlinear controller ( [Marino and Spong, 1986] (35) , [Khorrami et al., 1995] (135) ) for effentrol of flexible manipulators. To improve the dynamic performance of flexible manipulators, few optimization techniques were also proposed. But, the most attractive one is the use of the 3kinematic redundancy feature for minimizing joint torques and vibration suppression. To deal with the nonlinear behaviour associated with nonlinear elasticity and large deflection effects, wave-based control may be suitable. In a rare case, both the bending and torsional vibrations of a single flexible link were controlled by using two electric motors. Mishra and Singh [2019] (21) tried to achieve the vibration control of a Two-Link Flexible manipulator using both passive and active damping control methods. They considered both the bending and torsional vibrations of the flexible links. All these authors considered the small deformation of the flexible links. Banerjee and Lemak [1991] (97) used a different approach for modelling the flexible links. They used Kane's method in association with AMM for obtaining the equations of motion of a Flexible multibody system. They introduced the concept of dynamic stiffness into their dynamic model and proposed that to obtain correct simulation results it is necessary to incorporate this. Besides that, Hu and Zhang [2015] (98) also used this method for obtaining the equations of motion. The literature also gives a comparison between AMM and FEM ( [Meghdari and Ghassempouri, 1994] (99) , [Theodore and Ghosal, 1995] (100) and provides information about stability analysis of flexible manipulators. The proper placement of actuators made up of smart materials on flexible links plays a crucial role in damping the vibrations [Sun et al., 2004] (61) . This approach is known as active vibration control. A survey related to active vibration control is provided in Part-2 of this review. The effect of axial foreshortening on bending vibrations, the effect of the actuator on vibration mode shapes, and the question of using the number of modes for accurately representing the vibratory motion of a flexible link are highlighted. A summary of the work done by various researchers in the area of flexible robotics is given below in Table 2. https://www.indjst.org/ From the literature survey, it is found that different design methods were used by various researchers for preparing the mathematical model of the flexible manipulators. These are provided in Table 3.  (101) , [Wang and Gao, 2016] (102) It is a linear beam theory. Shear effects are not included. The theory holds good for long and slender beams.
Continued on next page https://www.indjst.org/ Table 3 continued  10 Newton-Euler-FEM [Naganathan and Soni, 1986] (25) , [Mohan, 2006] (115) , [Scaglioni et al., 2017] (116) The Newton-Euler approach is based upon Newton's second law of motion. FEM is used to find out the elastic deformation of the flexible links. This method can be easily implemented in computer programs. 11 Decoupled Natural Orthogonal Complement [Mohan, 2006] (115) ; [Mohan and Saha, 2007] (117) ; [Saha et al, 2013] (118) It helps in obtaining the generalized inertia matrix required for recursive forward dynamic analysis of manipulators. It decomposes a given matrix into an upper block triangular and a block diagonal matrix. This helps in achieving good computational efficiency and numerical stability. 12 Kane's Method Banerjee and Lemak [1991] (97) , Hu and Zhang [2015] (98) , [Jia et al, 2017] (119) This method is more convenient than the Lagrangian dynamics. It makes use of the concepts of generalized active forces and generalized inertial forces.
Meghdari and Ghassempouri [1994] (99) performed a comparison of Lagrangian AMM and FEM. Theodore and Ghosal [1995] (100) also did a comparison of the assumed modes method and finite element models but for flexible multi-link manipulators. The conclusion is that less number of mathematical operations is required for computation of inertia matrix in case of finite element model as compared to assumed modes model but with a large number of differential equations of motion. The use of FEM is recommended for manipulator links with complex geometries and having flexibilities and the use of AMM for manipulator links with uniform geometries and numerical simulation. The AMM has a faster convergence than FEM. Furthermore, the FEM overestimates the natural frequencies of the structure being considered.
Various control approaches were used by the researchers to achieve the tip position control of flexible manipulators. These approaches are tabulated in Table 4.  (7) , [Luh et al., 1980] (8) , [Book and Majette, 1983] (9) , [Takahashi and Sasaki, 2010] (70) , [Zimmert et al., 2012] (101) , [Chaoui and Sicard, 2012] (73) This type of control employs a comparison between the controlled variable and the reference variable. The error thus generated is minimized by using a suitable controller. Any type of control system that utilizes feedback can be considered as feedback control.  (120) In feedforward control, the adjustment of the controlled variable is not based on error. It is based upon the knowledge of the process being controlled gained through either the mathematical model or the measurements. Feedforward control is based on a command signal from an external source. 3 Integral resonant control [Pereira et al., 2008] (63) , [Pereira et al., 2009] (66) , [Bian et al, 2018] (121) This control scheme is based upon the fact that the positioning bandwidth of vibrating systems like beams can be improved significantly if the lower dominant modes of vibration are shifted to higher resonating frequencies. 4 Artificial neural network/ Adaptive control [Khorrami et al., 1995] (58) , [Chaoui, 2009] (64) , [Liu et al., 2010] (68) , [Abe, 2011] (122) , [Chaoui and Sicard, 2012] (73) , [Rahmani and Mohammed, 2019] (123) An adaptive control system can modify its mode of operation and takes care of uncertainties that may adversely affect the controlled variable. It suitably modifies the control gains such that the system performance remains within the specified limits. Most of the time, the artificial neural network is used to predict any kind of uncertainty. Continued on next page https://www.indjst.org/ Table 4 continued  5 Input shaping [Hillsley and Yurkovich, 1993] (56) , [Alam and Tokhi, 2007] (75) , [Diaz et al., 2010] (69) , [Chen and Huang, 2017] (124) It is an open-loop control technique that utilizes a command signal for the reduction of vibrations of digital control systems. The command signal chosen is such that the vibration induced by the previous part of the signal is cancelled by the vibration induced by the latter part. 6 Sliding mode control [Korayem et al., 2009] (65) , [Mamani et al., 2009] (67) , [Lochan et al., 2017] (125) , [Yanmin et al., 2015] (126) , [Yang and Tan, 2018] (127) , [Aghababa and Saif, 2021] (128) , [Homaeinezhad et al., 2021] (129) It is a nonlinear control technique that makes use of discontinuous control signals for controlling the output of a nonlinear system. It is a kind of variable structure control.

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Singular Perturbation   (140) , [Yang et al., 2017] (113) , [Hu and Zhang, 2015] (98) , [Liu and He, 2018] (141) It can be understood with the help of the following equation: εx 2 − 5x + 6 = 0 where ε → 0. If ε = 1 then the above equation has two roots, viz. x = 2 and x = 3, but when ε = 0 then the above equation has only one root and that is x = 6/5. When the parameter ε is very small, then the roots of the above quadratic equation are obtained using singular perturbation theory.
All these control approaches require the application of proper control and sensor system (142) . The current trends show the use of sliding mode control and command shaping in vibration control of flexible manipulators. The performances of the control systems used for flexible manipulators were optimized using various optimization techniques like Genetic Algorithm, Particle Swarm Optimization, Vector Evaluated Particle Swarm Optimization (143) , Nelder-Mead Simplex Algorithm (144) , and Fuzzy Basis Function Network adaptive control algorithm (145) .
The present survey on flexible link manipulators identified few research gaps for manipulators having revolute joints. More research is required on flexible manipulators with prismatic joints. This may involve modelling the phenomenon of friction present within the system. Besides that, the effect of torsional vibrations (105,146) on positional accuracy must be studied. It is also found that the flexible manipulators exhibit nonlinear frequency that depends upon vibration amplitudes and mass ratios   (147) . This further increases the positional inaccuracy. Thus, it is necessary to damp the vibrations of links of a flexible manipulator so that positional accuracy can be maintained. This becomes significant for space robots used for performing on-orbit assembly missions (148) . Various other open-problems (149) related to the area of flexible multibody systems include the choice of a suitable reference frame (Table 2), definition of exact boundary conditions at the location where the https://www.indjst.org/ two successive flexible links are joined together, inclusion of a sufficient number of vibration modes into the dynamic model, convergence issue during finite element analysis of flexible links, the effect of centrifugal stiffening, and various non-linear terms present in the dynamics equation of the flexible manipulator that involves coupling between rigid and elastic motions. Besides that, certain modelling issues need to be addressed. These are explained in the following paragraphs.
The governing equations of motion of the flexible link manipulators can be described using the following matrix equation (105,150,151) .
In Similarly, the equations of motion of the flexible links can be obtained as follows. ( Equation-3b is an integro-partial differentiation equation that describes the vibratory motion of the flexible links. It is difficult to solve this equation. The solution of this equation is rarely found in the literature. Most of the researchers have solved the vibratory motion of the flexible links by solving the partial-differentiation equation using the assumed-modes method. In the case of lateral vibrations of the flexible links, the vibration of the flexible links has been described by the Euler-Bernoulli beam equation. Thus, the accuracy of the solution depends upon the number of 'assumed modes' incorporated into the equation. To describe the behavior of the flexible manipulator exactly, it is required to solve equation-3 as it is. Literature also describes the use of the finite element method for obtaining the oscillatory behavior of flexible links. While using FEM, the order of the mass matrix increases proportionally to the number of nodes (151) . If a flexible link is divided into n finite elements having two nodes each then the maximum number of elastic degrees of freedom will be 6 x (n+1). Thus, the order of the inertia matrix described in equation-3 will be N + 6 x (n+1). Now equation-3b will not be an integro-partial differentiation equation. It will then consist of a summation of various terms. The accuracy of the solution will now depend upon the number of finite elements (n) used. If n→∞, then equation-3b will become an integral equation as per the definition of 'integration' described using the ab-initio method. Thus, it is very cumbersome to solve equation-3 using FEM. Furthermore, results will not be accurate also. To improve the accuracy of results, a new error function must be defined as a minimization problem.
In equation-3, coupling terms M r f and M f r are present within the inertia matrix. These terms represent the coupling between rigid and elastic motions. It is required to study the effect of these terms (105) on simulation.
It is crucial to find out the exact boundary conditions of a flexible manipulator. Besides satisfying the geometric constraints, the boundary conditions must also satisfy the compatibility equations at the location where the two successive links are joined together. While using the assumed modes method, it is essential to correctly determine the boundary conditions. Various https://www.indjst.org/ researchers have described the types of boundary conditions to be used. In the present work, a new method for finding out the boundary conditions is proposed by using equation-3. The boundary equation can be represented as follows. ( In equation-3c, the subscript '*' represents the motion of the point where the previous and the successive flexible links are joined together; q f * represents the elastic motion at the tip of the previous link and the starting point of the successive link. Such a boundary condition is called the dynamic boundary condition. The exact solution of this equation will give information about the exact mode shapes of the flexible links. Recent trends show that sliding mode control, variable structure control, model-predictive control, vision-based control (152) and, reinforcement learning control (153) offer some promising results in control of flexible manipulators. Through this review paper, the authors feel that awareness and interest will be created among the readers, and they will get motivation for doing further research in the area of flexible robotics.