Parametric Study of Double Link Flexible Manipulator

In this work, finite element method based on Lagrangian formulation is used for obtaining the equations of motion of the double link flexible revolute-jointed robotic manipulator. Both the links are considered as Euler-Bernoulli beams. A parametric study is carried out for the double link flexible robotic manipulator through linear modeling technique. A comparative study for dynamic response is carried out for the uniform beam manipulators under various types of excitations.


Introduction
In conventional robotic system, links are rigid giving small static deflection and hence it is possible to obtain high positional accuracy. However, most of the power is used to overcome the inertia of the system. On other hand, lightweight and large dimension robotic manipulators have become popular due to their higher manipulation speed, less weight/overall cost, transportability, better energy consumption, enhanced payload capacity etc in comparison to conventional robotic system. There is a wide range of applications of such new generation robotic systems in many areas nowadays. It is desired to design lighter robots to carry out heavier payloads as well as to operate it at higher speeds. Thus, flexible link manipulators are a subject of intensive research. Dwivedy and Eberhard (2006) presented a wide review on dynamic analysis of flexible manipulator done by various researchers. Book et al. (1975) linearized the equations of motion about a nominal configuration for a two-link flexible manipulator. Chang and Hamilton (1991), and Usoro et al. (1986) presented a Lagrangian finite element approach for the mathematical modeling of the manipulators with flexible links. Yigit (1994) modeled a two-link rigid-flexible manipulator and derived the equations of motion by applying the Hamilton's principle. Ankarah and Diken (1997) used the Euler-Bernoulli beam theory and solved the transient vibration theorem with the mode summation method to control the residual vibration of a single flexible link.
The dynamics of a flexible arm and flexible joint manipulator carrying a payload with rotary inertia was studied by Bedoor and Almusallam (2000). Meghdari and Fahimi (2001) derived the improved elastic generalized coordinates. Kane's equation of motion for arbitrary number of rigid and elastic bodies is presented. Also, equations of motion are de-coupled in first order terms. Zhang and Bai (2012) established Lagrangian dynamic equations of two-link flexible manipulator through integrated model and multi body dynamics method. Dynamic response reliability is analyzed by using Monte Carlo and extremum surface method.
Most of the published work focuses on modelling and pays less attention for its optimal design. Asada et al. (1991) presented optimum structure along with control aspect of flexible robot arms. Coordinates used by finite element model are treated as design variables, which are optimized for obtaining the optimal shape and structure of the arm mechanism. Wang (1994) addressed optimum design of a single link manipulator to maximize its fundamental frequency. He formulated the design problem as a nonlinear eigenvalue problem and used variational method. He demonstrated the increase of fundamental frequency as a result of optimization by considering a few numerical examples.
In the present work a linearized model for small rigid body motion and small flexural deflection is used. Based on this model, complete parametric study is done to predict the dynamic behaviour of the system due to the variation of various design parameters. In addition, shape optimization is done to increase the fundamental frequency and dynamic response of the optimized links is studied.

Obtaining Elemental Equation of Manipulator
Rotating flexible beams have significant transverse deflections. They behave as a nonlinear elastic beams and exhibit vibratory motions in both chord wise and flap wise directions. However, Robotic manipulators usually work at moderate peak speed. Induced transverse force in the chord wise direction due to the applied excitation torque is much higher compared to the gravity force in flap wise direction and vibrations are predominantly in chord wise directions. In this work, model of Usoro at el. (1986) is adopted. However, formulations are consistently linearized for small angular/transverse deflections under linear beam theory to reduce the complexity of the system modeling. Fig.1(a) shows single link flexible manipulator in which XOY and X 1 OY 1 represents the stationary and moving co-ordinate frames respectively. Motion of the link is represented by fixed XOY co-ordinate frame. The link is considered slender. Hence, transverse shear and rotary inertia effects are neglected allowing it to be treated as an Euler-Bernoulli beam. Beam is assumed to vibrate predominantly in horizontal plane (XOY), neglecting gravity effects.

Modeling of First Link
In the FEM formulation the manipulator is divided into finite elements, each element having five degrees of freedom. Detail of th i element of the first link is shown in Fig. 1 P with respect to inertial system XOY for smaller angular displacement and small flexural deflection is given by In finite element method, variables are converted to nodal variables.
2.1.1. Kinetic energy computation of the i th element of the 1 st link: Kinetic energy of th i element of the first link is given by We have Substituting Equation 5 in Equation 4, we get r r (6) Thus elemental mass matrix is given by where, Thus, elemental stiffness matrix is given by

Modeling of Second Link
Hermitian shape functions are expressed byS i . In the FEM formulation the manipulator is divided into 10 elements, each element having eight degrees of freedom. Detail of th j element of the second link is shown in Figure   2(b). In the Figure 2b and

Kinetic Energy Computation of the j th element of the 2 nd link:
Kinetic energy of the second link of the th j element is Thus, mass matrix of the element become All the constants of the above matrix may be obtained by integrating 2 j M for different vector elements of Z 2 .

Elastic potential energy of the j th element of 2 nd Link
The potential energy of the j th element of the 2 nd link due to elastic deformation is given as Thus, the elemental stiffness matrix is given by

Lagrange'S Equation of Motion in Discretized Form
The kinetic energy and the potential energy of the system are obtained by computing the kinetic energy and potential energy of the each element of the system and summing over all the elements.
where q F are the generalized forces. Being linear system, global mass and stiffness matrix is constant and equation of motion comes as Effect of hub mass and payload mass is incorporated in the global mass matrix and stiffness matrix using Dirac-delta function as described by Dixit et al. (2006). Hub mass and tip mass/payload is defined in terms of β (ratio of hub mass to beam mass), 2 µ (ratio of tip mass at link 2 to total beam mass), 1 µ (ratio of tip mass at link 1, mass of the motor at the link joint, to total beam mass) and α (ratio of length of link 2 to length of link 1).
Neglecting load vector, Eq. 23 becomes standard eignvalue problem, which is solved to obtain natural frequencies of the system. Numerical integration of Eq. 23 is carried out by using Newmark's integration scheme (Dixit, 2009) to obtain transverse deflections u & v, rigid body motions θ 1 & θ 2 and their derivatives.

Results and Discussion
In this section, a comparative analysis has been carried out for uniform as well as shape optimized double link flexible revolute manipulator. For the numerical study, a manipulator having uniform diameter 0.01 m, length 1.0 m, mass per unit length 217.3 gm/m, Young's modulus of elasticity 69 GPa is considered for both the links. Damping of the system is neglected. Most of the numerical simulations are done subjected to a sinusoidal torque given in Eq. 24 about the axis of rotation, The dynamic behaviors depend upon many parameters of double links flexible manipulator and also dynamic response consists of several desired objectives viz higher hub angle, less static deflection, less residual vibration, less response and settling time, etc. Improved dynamic response is a multi objective problem. Here some parametric study is done to analyze the dynamic behaviour of double link flexible manipulator.

Dynamic Response due to Different Payloads
Dynamic behaviour of the double link flexible manipulator changes with respect to the change of payloads at the tip of second link as shown in Fig. 4. It is observed that with the increase of payloads, the magnitude of hub angle and joint angle reduce. Residual vibration is considerably more at tip of first link or second link depending upon the input torque at hub joint or link joint respectively.
It is also observed that there is very small effect in dynamic response with the increase of hub inertia. Similar trend is observed in dynamic response due to the variation of motor mass (tip load at link1) and hub mass. For the sake of brevity, these results are not tabulated here.

Effect of Link Lengths on Dynamic Response
Dynamic response of the double link flexible manipulators also depends upon the links length ratios. The lesser the length of link 2 with respect to Link 1, the better the dynamic response i.e. more hub/joint angle and lesser residual vibration for the given set of torque. Therefore, designer should not prefer the longer second link with respect to the first link. τ is plotted in Fig. 6. It is observed that there is a decrease in hub angle and increase of joint angle with the increase of torque amplitude. As the magnitude of applied torque 2 τ increases, there is considerable increment in the residual vibration of the tip of second link. Similar trend is also observed due to the variation of input torque 1 τ at the hub joint (result not shown here). Thus, tip vibration increase for a particular link with the increase of torque amplitude acting in that particular link. Overall angular displacement depends upon the set of input torques at the joints.

Comparative Dynamic Response due to Different Torque Profile
Different torque profiles shown in Fig. 7 are considered for the comparison of dynamic response of double link flexible manipulator. All the torque profiles have same amplitude i.e. 0.5 N.m and same duration of excitation i.e. 4 sec. Dynamic response due to different torque profile is shown in Fig. 8. Triangular torque profile gives the lesser hub angles to the links. Bang-bang torque gives high input energy to the system giving high hub angle as well joint angle. However due to sudden change, bang-bang torque produces high residual vibration to the system. Sinusoidal torque may be preferred for smooth operations of the system.

Conclusions
Dynamics of double link flexible manipulator is highly complex and nonlinear in nature. Model is linearized to reduce the complexity of the model and tried to predict the behaviour of the system under low amplitude of vibration due to excitation. Parametric study suggests that dynamic response of the double link manipulator depends upon system parameters viz. payloads at tip and link joint, link lengths, input torque magnitude & profile and hub inertia.