Tip Trajectory Characteristics and Nonlinear Stability Analysis of Robotic Manipulator With Flexible Links-joints


 An efficient and new dynamic model of two-link flexible manipulator connected and controlled by flexible joints i.e., prismatic and revolute pairs has been developed to explore the modal analysis to study tip-trajectory characteristics and subsequently investigate the nonlinear steady-state responses under harmonic motion at flexible joints. The governing equations including joint dynamics have been derived using extended Hamilton’s principle. Modal parameters have been graphically presented to highlight the influences of various system parameters on the determination of eigenfrequencies and eigenspectrums. Obtained reduced order equations have then used to study the trajectory characteristics of tip displacements, angular and actuator positions by imparting the appropriate actuator force and joint torque. Further, nonlinear studies have been carried out to compute the steady state responses and their stability and local bifurcation by using 2nd order method of multiple scales. Investigation of the influences of various design parameters on the nonlinear stability and local bifurcation of steady state responses have been demonstrated and those results have been found to be in good agreement with numerically obtained findings. The obtained results find very useful in the applications of long-reach robot manipulators performing complex operations assigning with translating and rotary motion together.


Introduction
The modeling and subsequent control problems in flexible robotic system has been a major center of attention in recent years on the grounds that robots are being widely used in modern industrial operations, medical utilities, agricultural production, defense tasks and space exploration. In order to perform hazardous operations, where human reach is not possible, for example in nuclear industry, long reach robotic manipulators are being used effectively. In medical industry, robots are being employed to accomplish tedious and intricate surgical tasks where the accuracy is of main concern. There has been a constant pursuit to have low power consumption, high speed, and light-weight manipulator in industrial and space applications in order to boost the productivity or reduce the payload of spacecraft. In pick and drop operation, the manipulator is supposed to move on a planar surface to reach the required location and then perform the requisite tasks. Flexible robot manipulator with a prismatic joint are functioning in many such engineering applications like industrial robots, telescopic members attached to loading vehicles, spacecraft antenna, magnetic tape-drivers, printers, band saws, and weaving mechanisms. Hence, developing at least a two-link flexible manipulator with harmonically driven prismatic and revolute joints incorporating required payload can have useful engagement in various industrial operations, especially space applications where a long-reach and lightweight robotic manipulator is highly necessitate. The flexibilities in the links, actuator as well as the joints have been introduced in resemblance with the actual manipulator models. Numerous studies have been carried out with regard to the investigation of modeling, and dynamic control of single-link manipulator having revolute joint, with and without payload. The reported studies have been divided into two categories based on the type of joints used to drive the manipulator i.e. revolute [1][2][3][4][5] and prismatic joints [6][7][8][9][10][11]. On many occasions, beam element is often used to model the flexible link and integrating it with rotating hub makes a robotic manipulator with revolute joint [12]. Fundamental beam element has been often used to model one-dimensional flexible link in vibration analysis of flexible robots [13]. The linear and nonlinear dynamic characteristics of these one-dimensional flexible links have been investigated using either classical or refined beam theories (Timoshenko or higher order shear beam). However, Euler-Bernoulli beam hypothesis [14], has been one of the most elementary beam element theory widely used to estimate the transverse deflection and eigenfrequencies of flexible robots arms. It is very evident from the existing literature that considering appropriate beam theory to suitably model a flexible robotic arm always leads to accuracy in evaluating the desired dynamic responses when subjected to external influences. Appropriate identification of tip mass attached at tip of manipulator link [15] that can often assumed as payload or end-effector of the robotic manipulator not only affects vibration frequencies but also changes the joint requirements as torques or forces for a specific maneuver in a pick and place operation. The dynamics of longreach manipulator link with end-effector has mostly been modeled using Euler-Bernoulli's beam with attached tip mass. The dynamics of beam deformation, link flexibility results coupled nonlinear partial differential equation with complex boundary conditions. Linear analysis [16][17][18] is sometimes enough to describe the small vibration oscillation of flexible robotic link. Mostly, linear analysis has been restricted to calculate the eigenfrequencies and corresponding eigenfunctions. However, nonlinear analysis [18][19][20] is very imperative to compute the vibration behaviors of a flexible robotic manipulator under external and/or parametric influences as the system exhibits inadmissible vibrations when the forcing frequency becomes equal or nearly equal to its natural frequency. Various types of nonlinearities exist in a manipulator system such as inertial, geometric, damping, material and physical nonlinearities. While inertial nonlinearities [5,18] generally occur due the nonlinear terms in the velocities of the links, geometric nonlinearity [6,19,20] arises in the potential/strain energy terms of manipulator for inextensible, large deformation or nonlinear curvature conditions of the links that result into cubic nonlinear terms in the equations of the motion. Often the nonlinear stress-strain relationship [6] due to inelastic deformation in the link enables substantial effect on the structural responses under forcing conditions. The dynamics of the long-reach [21] or flexible manipulator are significantly complex by the virtue of flexible nature of the system. As a result, these flexibilities in the manipulator links and joints possess a major challenge to achieve and maintain accurate positioning and desired trajectory tracking. As the robot manipulator in various capacities and configurations are being used in different industrial operations for different applications where the end point control is of major concern, it becomes inevitable to study the system performance under parametric variations. However, the rich and diverse dynamics of flexible robots have been discretized using various solution techniques such as assumed mode method (AMM) [22][23][24], finite element method (FEM) [25][26][27], and lumped parameter method (LPM) [28][29] to study its behavior under different working conditions. The suitable understanding of the eigenfrequencies [12,16,18] and vibration characteristics of flexible manipulator enhances the ability of a design engineer to detect, locate and quantify the extent of damages within the robotic structure. The fundamental frequency may vary significantly with the design parameters which in turn possesses a challenge to the robustness of a designed controller. Identifying the eigenfrequencies and corresponding mode shapes may offer necessary information about the deflected shape of structural vibration to design engineer when one of the natural frequencies comes closer to the external frequency called as resonance.
The effect of design parameters on the system's natural frequencies and subsequent mode shapes before moving into the investigation of the dynamic performance of flexible manipulator is substantial requirements. Investigation of parametric variations on modal analysis can offer the ways of improving and optimizing the dynamic aspects of flexible manipulators. Proper knowledge about modal parameters always guides the operator to appropriately select the desired operating conditions in order to avoid any resonance condition that causes the unwanted excessive vibration and noise.
Furthermore, flexible robots possess higher elastic compliance that leads to residual vibrations especially after a grasping maneuver. These light and flexible robot arms combined with fast motions or interacting with surrounding atmosphere or by stopping abruptly may cause considerable amount of vibration of the end point. This residual vibration may lead to positioning inaccuracy [33]. However, dynamics of robotic manipulator becomes more complex while considering both revolute and prismatic joints in addition to elastic compliance of flexible links. Flexible manipulator with both revolute and prismatic pairs raises the substantial difficulties in developing the mathematical model over the robotic system only with revolute joints [5,14]. The earlier works lacked in the nonlinear behavior of the two-link flexible manipulator with prismatic and revolute pairs and thus major focus here is to investigate the nonlinear dynamic responses along with parametric study on the determination of its modal parameters. Determining the nonlinear responses [6,[18][19][20] enables a deep insight and better understanding of dynamic performances of a long-reach two-link manipulator to identify the ranges of operation safety and thus avoiding undesirable behavior because of structural instabilities [18][19][20]. Simultaneously, catastrophic failure under various configurations can be avoided by the sufficient knowledge of the system's behavior operating at a certain speed.
Hence, investigation of nonlinear responses of flexible long-reach manipulator is quite necessary for accurately predicting the state of dynamic stability to avoid catastrophic failure under different working circumstances.
Based on the existing literature, it is quite apparent that the single-link and two-link manipulators with revolute joints have been modeled and studied thoroughly. Subsequent various control strategies [30][31][32] have been developed for vibration suppression in such manipulators. In addition, researchers [6,18,19] have extensively investigated the resonant responses of singlelink flexible manipulator with prismatic joint under different working excitations. However, twolink manipulators considering prismatic joint has been constantly eluded from the research. As a result, there have been lacks of studies on dynamic responses of flexible manipulator with harmonically excited prismatic and revolute joints. Consequently, in this work, the dynamic model of two-link manipulator incorporating a payload driven by prismatic and revolute motions has been attempted. The governing equations of motion and boundary conditions assimilating geometric nonlinearities have been derived using extended Hamilton's principle with subsequent modal analysis to determine the eigenfrequencies and eigenfunctions. Due to behavioral uncertainty at payload end, accurate dynamic modeling of flexible manipulator with prismatic and revolute pairs possesses a challenging task and hence, investigation of influences of critical design parameters on the system responses can highlight its parametric instability under certain working situations. In nonlinear analysis, perturbation technique has been used to determine resonant responses under different conditions. Effect of design parameters such as payload mass, joint variables, mass density and flexural rigidity of the links onto the modal parameters, nonlinear behaviors and stability of the system has been exhibited graphically. The conferred results shall be noteworthy contribution in development of effective control strategies for vibration suppression of long reach manipulators.

Flexible Two-link manipulator system: Kinematic model
A schematic diagram of two-link planar flexible manipulator incorporating a payload is shown in an Euler-Bernoulli beam element; 4) prismatic and revolute joints are modeled as linear springmass system, torsional spring-inertia system, respectively are taken into consideration. The expressions for the total kinetic energy ( T ) and potential energy ( U ) of the system, respectively are given by  (2) Here, T constitutes the kinetic energy associated with masses of the links ( 1,2  ), masses at the link terminals ( 1,2 m ) and joint inertia ( j ) while U represents the bending strain energy of links ( 1,2 EI  ), energy due to axial stretching ( 1,2 EA  ), strain energy of the prismatic and revolute joints ( , a kk  ). Subscripts 1, 2 designate the first and second link respectively.
δδ tt  equal to zero between two times stages. Now, by substituting equations (1)(2) in extended Hamilton's principle, fourth-order coupled partial differential equations for the link motions and complex boundary conditions along with the joint dynamics are expressed as: The governing equation of first link motion is expressed as: The associated boundary conditions for the first link are: Similarly, the governing equation for second link motion can be expressed as: The associated boundary conditions for the second link are: m s + + v + L s + L θ + θ θ θ s θv θ θv s ms θ s s s v θ s v s -s θ -s s -2s θs - Equation of motion of joint dynamics is expressed as:

Free Vibration: eigenfunction and eigenfrequencies
In this section, the natural frequencies and related mode shapes are sought for the dynamic model using free vibration analysis. The effect of coupled nonlinear terms is then neglected and the transverse deflections of first and second links are expressed in terms of new functions in space and time as which on substitution in Eqs. (3-7) results into following equations given by: Now, the deflection functions   Here 1 3 1 4 R R , S S LL are the integration constants which are obtained by substituting the Eqs. (13)(14) in the boundary conditions. After substitution, a set of seven algebraic equations in terms of nondimensional system parameters and nondimensional eigenfrequency parameter (  ) is obtained as: The transcendental eigenfrequency equation in terms of known system parameters is obtained for the existence of non-trivial solution for the Eq. nondimensional parameters are given in appendix.

2.3: Nonlinear Analysis: problem formulation
The nonlinear behaviors and stability characteristics of the two-link flexible manipulator with harmonically driven prismatic and revolute motion of frequency 1,2  have been analyzed. The nonlinear coupled terms and the cubic nonlinearity arising due to geometric stretching have been retained in the governing equations of motions of the links expressed in Eqs. (3)(4)(5) and viscous damping   1,2 c is included in both the links.
For link 1: Nondimensionalization of Eqs. (16)(17) has been established using the dimensionless terms: For link 2: In further text, prime and over dot shall be read as the differentiation with respect to space parameter  and time τ , respectively. Now, Galerkin's method is used to discretize spatiotemporal Eqs. (18)(19) into temporal equations by using the expressions,  are the eigenfunction of the first and second links for first mode of vibrations, respectively given in Eqs. (13)(14) and expressed here as: The coefficient matrix of Eq. (15) is numerically solved for the given nondimensional system parameters to determine the nondimensional eigen frequency parameter,  and the expressions for 13 RG  , and 14 SS  are given in the appendix. Now, first using the orthogonal property of the mode shapes and then using a small book keeping parameter (ε) for ordering the Eqs. (20)(21), the resultant non-dimensional nonlinear second order ordinary differential equations of motion of the links are obtained as: The approximate solutions of Eqs. (22)(23) are determined by using second order method and 1,2 q are expressed in terms of fast   0 T=τ and slow time ( 1 T=ετ , 2 2 T=ετ) scales as: Using chain rule, time derivatives in terms of 0 T , 1 T and 2 T become: The equations for second link are obtained in a similar manner as: The general solution of differential equations Eqs. (27 & 30) can be expressed as:          . These small divisor terms lead to unbounded solutions. In order to have bounded solutions of the respective equations these secular or small divisor terms should be removed. Also, it has been numerically found that the normalized link frequencies   The equilibrium points 10 r , and 10  satisfy the steady state equations for first link and the stability of the steady state solutions depends on the nature of the eigenvalues of the coefficient matrix of Eq. (42). The steady state solutions will be stable if and only if the real parts of the eigenvalues are negative. The steady state solutions stability for the second link can also be determined in a similar manner.  2 2  2  3 2  1  3  3  1 3 2  3  1  1  3  2  3  1  3  1   2 2  2 2 2  2  2  3  1 2  2  3  1  9  1 1  3  3  7  1 1

Internal resonance in second link
Here, 1

Modal analysis: Modal Parameters
The eigenspectrums of a system are the essential parameters whose understanding is crucial for its safe operation and acceptable performance as well. The system vibrates at inadmissible amplitude leading to its failure or catastrophic injuries to the operator involved when the system is operated at a frequency equal or nearly equal to one of its natural frequencies. The variation of the eigenfrequencies with the system parameters provides a better understanding of the operational territory of the system in order to avoid such occurrences. The variation of system parameters has been accomplished by alteration of nondimensional parameters and thus apprehending its effect on the eigenspectrums of the two-link flexible manipulator with prismatic and revolute joints. The variation of payload mass parameter    decreasing the stiffness of actuator joint. It is also evident from the figure that the system experiences a jump into higher eigenfrequencies at the unit magnitude of the actuator frequency.
As a result, manipulator may undergo higher mode of vibration. It has been noticed that adding stiffness to the system either by increasing the flexural rigidity    or by increasing the stiffness of the rotary joint   J  increases the overall eigenfrequency of the system. In addition, the unit magnitude of the joint frequency   J  represents the condition when the natural frequency of the manipulator becomes equal to that of natural frequency of revolute joint comprising a lumpparameter model of torsional spring-inertia system depicting the flexible joint. At unit magnitude, joint dynamics gets decouple from the system and behaves as point mass. It has also been observed that the system eigenfrequency experience a sudden jump at unit magnitude of the joint frequency and hence, beyond this value the system starts oscillating at a higher mode of vibration. It is also observed that a lower eigenfrequency generates approximately at 2 1.5  and due to which the system again tends to vibrate at a lower eigenfrequency. These significant effects of the system dynamic characteristics on the fundamental eigenfrequency of the system may considerably influence the modal deflections and control parameters. In order to develop efficient control strategies and have a better understanding of actual dynamic responses of flexible robots, the development of accurate eigenspectrums is a perquisite requirement. The effect of system parameters on the dynamic responses is investigated through the eigenfunction which has been used to discretize the infinite dynamic model of the manipulator governed by the partial differential equations. Therefore, a brief investigation into the influence of system parameters on the eigenfunctions of the two-link flexible manipulator has been presented in Figs 3-6. The arrows in the figures indicate the decreases or increase in modal amplitude of the manipulator.   that also leads to a shift in modes shapes as shown in Fig. 6. In addition, it has also been noticed that, as we increase the joint frequency   J  beyond 1.5, system again tends to vibrate at a lower mode of vibration. A similar phenomenon has also been observed previously [18] in the case of flexible manipulator driven only by revolute joints. The presented results furnish the vital information regarding modal parameters which shall enable the identification of resonance conditions when operating frequency becomes equal or nearly equal to the system natural frequency and hence, facilitate the proper attenuation of system to avoid the inadmissible vibrations.

Dynamic characteristics: system responses
A computationally efficient dynamic model is obtained by discretizing the governing equations of motion of links, joints and actuator given in Eqs. (3), (4), (5) and (7). The prismatic actuator has been provided a sinusoidal force of amplitude 0.1 N and having duty cycle of 0.5 sec. A similar torque profile of amplitude 0.5 N-m has been imparted to the joint for analyzing the system responses under parametric influences. In present analysis, the flexible links are considered with width, height and length as 1,2 b =0.03m , 1,2 h =0.003m , and 1,2 L =0.3m , respectively. The material of links are assumed to have mass density and Young's modulus of  The final position of the actuator remains unchanged at 0.32 m with the change in payload mass as evident in Fig. 7. It can also be noticed that the angular position of link-2 decreases from 2.4 radians to 1.45 radians with increase in payload from 0.1 Kg to 0.2 Kg, respectively. It can also be noticed that last link vibrates with reasonably larger amplitude as compared to the first link in response to the larger payload. In Fig. 8, we examine the influence of payload on the system responses when only prismatic joint is actuated while no input torque is provided at the revolute joint. No significant change in the actuator position is noticed but the response of the second link drastically varies if compared with the Fig. 7. If we compare the results in Figs. 7-8, it is apparent that the angular tip position of second link for payload mass of 0.1 kg is 2.4 rad and -0.79 rad for the manipulator with actuated and unactuated revolute joint, respectively. Here also, residual vibration amplitude increases with increase in payload mass. The influence of actuator mass is significant on the position of actuator as well as angular position of second link which is also shown in Fig. 9. It is visible that the actuator position decreases and the angular position of the second link increases, with increase in actuator mass. It is also noticeable in the figure that the tip of both the links vibrates at smaller amplitude for larger actuator mass. In Fig. 10, it can be observed that the settling time of actuator increases with increase in joint inertia without affecting the actuator position. However, the angular position of the second link significantly decreases and tip vibrations increase with increase in joint inertia.

Nonlinear analysis: primary and internal resonance
In the literature, the nonlinear behavior of single-link manipulator with prismatic joint has been extensively studied [12][13] and research has been confined to the development of control strategies of flexible two-link manipulator. When the manipulator is subjected to a prismatic motion, the system experiences a forced vibration and undergoes relatively large vibrations causing undesirable vibrations due to its low stiffness which is a major concern. The adequate familiarity with the manipulator behavior under parametric variation of system attributes enables the operator to avoid the catastrophic failure under such circumstances. For primary resonance condition in both the links, the internal resonance has been avoided by considering the length ratio    Fig. 11. The presence of geometric nonlinearities induces the multivalued solutions and jump phenomenon in the frequency response curves of both the links. It is observed that while the first link exhibits the spring hardening behavior with the bending of curve towards right, the second link illustrates the spring softening behavior with bending towards left. In case of first link the jump down phenomenon is observed at point D while increasing the motor speed and a sudden jump in amplitude of vibration is witnessed at point G as the motor speed is decreased. The numerical and the corresponding analytical time response, phase portrait and FFTs have been presented in Fig. 12 and results are found to be in good agreement.  From Fig. 14, it has been noticed that change in beam density may influence the overall end-effector position significantly. Maximum vibration amplitude has been substantially increased from 2 to 3 with increase in beam density from 0.5 to 1.0. The vibration amplitude can even increase upto 50% of initial value with a slight increase in beam density. With increase in beam density, jump length i.e., sudden change in amplitude due to jump phenomena, increases substantially as compared to actuator's mass effect. However, jump length in case of jump-down phenomena is comparatively higher than that of occurred during jump-up phenomena. Due to this sudden change into large vibration oscillation, system may undergo catastrophic structural damages. Thus appropriate selection of mass density and actuator mass can offer insignificant vibration oscillation and persuade a safe operating condition. The influence of flexural rigidity ratio on the frequency response characteristics is shown in Fig. 15 and it is visible that as the flexibility ratio    increases vibration amplitude of the first link decreases initially and then experiences a marginal change with further increase in flexural rigidity ratio    . However, vibration of second link is significantly influenced by the changes in flexural rigidity. With increase in flexural rigidity, vibration amplitude increases remarkably. For a slight increase in flexural rigidity, mechanical vibration may increase approximately double of the initial state. As a result, jump length increases in a similar order as amplitude of oscillation. Hence, catastrophic failure primarily may be observed in second link due to its severe vibration tendencies. It is thus advisable to select an appropriate value of flexural rigidity for individual link in order to avoid large oscillation which may later lead to catastrophic failure.  Figure 16 describes the effect of joint inertia on both the links and with increase in joint inertia, the amplitude of oscillation increases for the second link. Similarly, the variation in vibration amplitude has found to be insignificant for link-1 with increase in joint inertia. Thus, inappropriate selection of joint parameter can experience bifurcation and finally lead to structural damage under repetitive operations. The results presented here are in contrast with those determined in [18] for the robotic system driven by revolute joints, where the first link undergoes with an alternative vibration behavior from spring hardening to spring softening or vie-verse when parameter is subjected to alter. Thus, we can finally conclude by saying that the nonlinear system behavior is a critical function of the joint dynamics. As a result, any slight change in the configuration of joint dynamics may cause detrimental consequences on overall system dynamics. spring hardening to spring softening. It is interesting to note that this transformation primarily occurred when the actuator frequency is equal to 1.0 and at all other values; the response is restricted to spring hardening behavior with decreasing amplitude. Due to this sudden change in actuator frequency to unit magnitude from any other non-zero value, system may demonstrate fatigue characteristics and several such changes may lead to fatigue failure. Therefore, one must note that system should not operate with an actuator frequency equal to 1.0 to avoid any fatigue failure. However, responses of flexible second link are restricted to softening behavior for all values of actuator frequencies. But, it has been observed that with increase a  , amplitude of oscillation decreases except for a  equal to 1.0. When the system operates with a frequency equal to 1.0, it undergoes acute oscillation with large vibration amplitude. Thus the actuator frequency equal to 1.0 is found to be critical operating condition for both these links. This is because the sudden shift in natural frequency when actuator frequency becomes 1. A slight change in actuator frequency can raise the natural frequency to the higher value. Hence, one should avoid this situation to maintain safe and secure operating conditions. However, considering higher value of actuator frequency can enable minimum vibration characteristic. Similarly, the influence of joint frequency on the vibration characteristics has also been studied in Fig. 17. Similar trend is also observed here that with increase in joint frequency, amplitude of vibration of link-1 decreases except for J  equal to 1 while the nature of vibration characteristics always maintains to be of hardening behavior. Vibration amplitude increases to its maximum value when the system is subjected to J  equals to 1.0. In contrary, with increase in joint frequency, for J 1.0  , response amplitude of link-2 increases while it is exactly opposite for J 1.0 1.5    where vibration amplitude gets decreases significantly. It is interesting to note that for any value of J 1.0


, behavior changes to hardening from softening. Therefore, selection of joint frequency decides whether system exhibits hardening behavior or softening behavior or mix vibrations.  Figure 18 shows the behavioral differences between primary resonance condition and internal resonance condition in the second link. However, resonant behaviors found in these two resonance conditions are quite similar in nature and observed to be softening behavior. The response amplitude in internal resonance condition is found to be higher than that of primary resonance condition for the same values of design parameters. Primary resonance condition acts as a forced vibration phenomenon in the second link and synchronization of these two resonance conditions instigate the severity of vibration in the second link. It is interesting to note that, vibration amplitude of link-1 has a significant influence on the vibration of second link. It is observed that vibration amplitude of link-1 may offer a positive impact in order to reduce the vibration level of second link. As a result, with decrease in vibration amplitude 1 r , one can reduce the vibration level in second link and it behaves as a damping phenomenon in a same vibrating phase. Hence, appropriate selection of vibration parameters of link-1 may decline the possibility of vibration oscillation of link-2. Furthermore, the amplitude of the link-2 vibration increases and decreases, respectively with increase in payload mass parameter   m2  and joint inertia   J  parameter as illustrated in Fig. 19.
Therefore, one may choose appropriate values of payload and joint parameters to keep the operating conditions into a desire lever.
The present analysis can be satisfactorily applied to the long-reach robot manipulators which are involved in most of the pick and drop operations. The derived eigenspectrums can be adopted in developing the suitable control schemes to attain desire end-effector trajectories while nonlinear analysis of manipulators undergoing large deformations may predict the actual responses under any external influences. Nowadays, the manipulators are being used in different adverse conditions where the end-effector is subjected to various forcing conditions with different joint configurations and further, end-effector may sometimes move along the link to perform various operations. The present linear and nonlinear approach can offer insightful dynamics and certain design criteria of two-links flexible manipulator having both prismatic and revolute joint.

Conclusions
In current study, an attempt has been made to explore the modal analysis for calculating eigenfrequencies and eigenspectrums further to study the tip trajectory characteristics. In addition, nonlinear stability analysis has been carried put to inhibit the local bifurcations for long-term solutions when the flexible joints have been imparted with harmonic motion. Subsequently, the influence of parametric variation of design parameters on the eigenbehaviors, tip trajectories, stability and bifurcation phenomena has been graphically demonstrated. The critical and pin-point outcomes and conclusions thus obtained are bulleted as below.
 The addition of inertia through payload mass, joint inertia, actuator mass, and beam mass decreases the system natural frequencies. The eigenfrequencies tend to increase with the addition of stiffness of the system by increasing flexural rigidity and stiffness component of flexible joints. However, eigenfrequencies witness a sudden jump to a higher mode of vibration at unit magnitude of joint frequency.  The amplitude of roller-supported and terminal end decreases respectively with the increase of actuator and payload masses. The flexural rigidity of links significantly influences the amplitude of lower modes of vibration. At unit magnitude of the joint parameter represents the limiting condition when the joint dynamics gets decoupled from the manipulator system and system starts vibrating at a higher mode of vibration.  The actuator mass significantly affects the actuator position as well as the angular response of the joint. The increase in payload mass and joint inertia decreases the angular position of joint without affecting the position of actuator. The joint reverses its angular position when the joint torque is removed and only actuator force is imparted.  Second order method of multiple scales has been used to analytically investigate the system response under forcing conditions imparted though harmonically driven prismatic and revolute joints. The multi-valued solution and jump phenomenon due to static saddle-node bifurcations has been noticed. Numerically, it is established that internal resonance exists between both the links and hence, simple resonance in both the links and internal resonance in second link is investigated under the parametric variation of system attributes.  The influence of the payload mass, actuator mass and beam's mass density ratio has been found to be negligible on the nonlinear response of first-link of the manipulator while the amplitude at a specific frequency and hence the jump length of second link increases with the payload mass, actuator mass and beam's mass density ratio. U , U , U , U , U , U , U , U , U , U , U , U , U , U , U , and U 0.  (A.1) The non-dimensional parameters appearing in above equations are: