Abstract
Thanks to their advantages over rigid ones, interest for lightweight parallel manipulator was increased. Besides, structural flexibility effects at high operational speeds are more significant. Thus, developing an appropriate model for the assessment of the dynamic properties of flexible mechanisms and linkages to gain effective vibration control will raise high demand. Therefore, this paper represents the dynamic and kinematic modeling using the assumed mode method and first-type Lagrange equations of the 2-DOF planar parallel manipulator with two flexible links. To truly predict vibrations of the manipulator without any major simplifying assumptions, nonlinear dynamic modeling, which thoroughly attempts to represent the flexible behavior of the links, is considered. As a result, an active damping approach is being studied with PZT actuators. The results show that this approach is effective in damping the vibrations of the links that give accurate trajectory control.
1. Introduction
With regard to parallel manipulators with a lightweight structure, a planar parallel manipulator with lightweight linkages provides a high-speed alternative positioning mechanism for manipulators of serial architecture. These robots are used in a wide range of applications, from simple selection and location of robotic systems for industrial applications to microsurgical applications, maintenance of nuclear power plants, or space robotics [1]. The interest in research into flexible connection manipulators and mechanisms was significantly increased to make full use of the potential offered by flexible manipulators. It is, nevertheless, particularly challenging to control flexible manipulators so that precise positioning can be maintained. For a two-link flexible manipulator, the problem becomes more complex. The dynamics are highly nonlinear and complex due to the flexibility of the system [2, 3]. Although lightweight links are more likely to meet high-speed and high-acceleration requirements, the inertia and forces from the actuators are more likely than ever to deflect and vibrate [4]. The structural flexibility effects at high end-effector speeds are much more significant. Manipulators and mechanisms with flexible links are systems with a variety of degrees of freedom. They are described by coupled nonlinear partial differential equations of motion. The dynamic model formulation of manipulators with flexible links and mechanisms was based on different discretization ways of flexible links to devise and apply a real-time controller for joint movements and vibration removal. The most popular approaches are the finite element method (FEM) [5, 6] and the assumed mode method (AMM) [7, 8]. It has been commonly established to model a flexible single link manipulator. Various approaches were developed, mainly divisible into two categories: the approach to numerical analysis and the assumed method mode (AMM) [9, 10]. AMM examines approximate models by solving a partial differential equation that characterizes the system’s dynamic behavior.
Previous studies have been reported using this approach to model a flexible single-link manipulator [11, 12]. Zhou et al. recently developed dynamic equations for a flexible three-PRS manipulator with regard to vibrational analyses using the FEM method [13], taking into consideration link flexibility. Kang and Mills introduced a dynamic flexible-link 3-PRR planar parallel manipulator by employing AMM [14]. The existing parallel 2-DOF manipulator with solid links for an optimal design was studied [7]. In high-speed pick-and-place applications, this manipulator is also productive [15]. Nevertheless, no research has been conducted on the dynamic modeling of the mechanism, taking into account the flexibility in which industrial operations are inevitable. This paper considers a method of active damping using piezoelectric materials. Deformation of the flexible links produces shear stress that PZT materials can counteract due to the voltage control applied. PZT can achieve better performance in vibration damping than other transducer materials as PZT has higher stress constant [16–19].
Due to the promising results, the parallel kinematic machine (PKM) is the greatest increasing need of the machining and pick and place industry. Due to their high structural stiffness and rigidity, PKMs’ absolute positioning error is reduced. The 2-DOF PRRRP PKM machine tool proves to achieve competing accuracies in the end tool [15]. However, the heavy and bulky links used to give adequate stiffness and accuracy significantly increase equipment costs, motor torques (power), and energy consumption. To overcome this issue, as a real-world requirement, research on the use of lightweight robot links is carried out in this paper. To overcome the positioning error due to the flexibility of the links, an active vibration control system based on PZT actuators is implemented.
In the present paper, an AMM modeling of the flexible links following Lagrangian method and a PD feedback control with linear velocity feedback (L-type) is used to correctly attenuate vibration due to trajectory tracking. This is followed by a proper PD trajectory control. The proposed active vibration damping approach was verified by simulations for flexible linkage manipulators.
2. Kinematic Modeling
Figure 1 illustrates the planar 2-DOF parallel manipulator with two flexible links. The manipulator architecture is 2-PRRRP, while R and represent revolute and prismatic joints, respectively. In a plane that works properly for pick and place tasks, the end-effector offers high precision 2-DOF translational motion. The end-effector position vector, two active prism joints, and two passive revolute joints are presented as follows, respectively, about the fixed X-y reference framework displayed in Figure 2:
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As a deformation assumption and design criteria of the flexible links, the influence of transverse, shear, and rotary inertia has not been taken into account since the beam is long and slender. Links only vibrate horizontally, and the torsion and vertical bending are not considered. Besides, the beam properties variation can be neglected across the whole body and cross section [20].
Consequently, Euler–Bernoulli beam theory can be employed to simulate the manipulator’s elastic behavior. The product of position-and time-dependent functions, i.e., AMM, expresses the deflection of the link, , as
In the equation above, , j and r denote the jth vibration mode and a finite number of assumed modes, respectively.
To select the boundary conditions, one can take many different approaches in the AMM. The optimum set is found closest to the system’s natural modes among the hypothesized modes in ideal situations. Thus, no assumption can be made about the employed set of hypothesized modes since several structural factors of the manipulator determine the natural modes [21]. In this study, pin-free modeling, implemented in a flexible PKM [16, 19], is considered for boundary conditions, which causes a significant deflection in the flexible links and remarkably investigates the effect of PZT actuators on damping the vibrations of flexible links. The alternative in the future needs an evaluation based on the robot’s actual structure to improve outcomes in the results.
Given the boundary conditions of the flexible links on and the end-effector, the selected normalized shape function that satisfies a pin-free boundary condition is as follows:where
The shape functions in the first three mode shapes of the flexible link are shown in Figure 3. The inverse kinematics problem is solvable by expanding the following restrictive equation, as shown in Figure 2:which yields
Equation (6) states that four solutions are available for the inverse kinematics of the mechanism. The four alternatives are consistent with four types of mechanism work modes. The deflection term, i.e., , should be drawn from dynamic modeling to approach the solutions.
3. Dynamic Modeling
In terms of link flexibility, the generalized coordinates are taken as follows:where
Figure 4 shows the manipulator’s closed kinematic chain with the flexible link deflection. Expanding equation (5), as mentioned, leads to four constraint equations, as follows:
As shown in Figure 5, the kinematic energy is taken into account in expecting the deformation of the links:wherewhere , , and are mass of the end-effector, mass of the sliders, and mass per length, respectively. Put the kinematic energy and the potential energy induced by link deformation for each coordinate in first-type Lagrangian equations to derive the dynamic modeling yielding the equations of motion for the flexible-link parallel manipulator:
Each of the components has been defined in the Appendix. Equation (12) is a differential-algebraic equation (DAE) that can be simplified by removing the Lagrangian constraint term as follows:
4. Vibration Control
Structural flexibility of the links transfers undesirable vibration to the end-effector, which leads to poor tracking efficiency. When the links are flexible for linear actuators, the vibration attenuation is difficult. Thus, an active damping approach with the help of PZT is implemented. Attaching the link surface, the PZT generates a shear force that suppresses the structural vibration of the links along the length. For flexible link manipulators, the aim of the above analysis is to make the rigid mode variable follow the required trajectory or converge to a certain point while suppressing the flexible link modes. The voltage of the PZT actuators will specify the L-type approach as explained in Appendix [22]. The voltage applied to the PZT actuator can be defined as follows:where represents the linear velocity feedback gain for PZT and are the start- and end-point positions of the PZT actuators from the linear actuators, , along the length. denotes the linear velocity of each link at . The virtual work conducted by the i-th PZT actuator can be measured as follows:where c denotes a positive constant implying the bending moment by applying voltage. Fulfilling the PZT actuator’s L-type layout is based on the position of the PZT actuator. The PZT actuator is to be installed in a region where the form function and the derivative has a similar variation trend within to achieve a stable control movement: .
The application of this case to higher frequency modes is limited as the fulfillment of the equation for higher frequencies is only carried out in small areas on the link.
5. Simulation Results
Two linear actuators are equipped with a simple proportional-derivative-type (PD) feedback controller system as follows:where and represent PD feedback gains, respectively. and indicate desired values for linear actuators obtained from (5).
Tables 1 and 2, respectively, include dynamic parameters and feedback control gains. Using a fourth-order, Runge–Kutta method with MATLAB software was integrated into the normal differential equations at 1 msec integration intervals.
The desired trajectory, which accelerates and decelerates smoothly, has a sinusoidal function:
The objective is that the end-effector moves 2 mm within 10 msec .
Figure 6 illustrates the end-effector’s tracking error following the desired trajectory with and without PZT actuators in the X direction. The activated profile of the PZT actuator, known as “active damping,” decreases continuously as a result of the PZT actuator’s damping effect, while oscillation at the initial acceleration is significant. The tracking error in active damping mode is rapidly decreased, accordingly. The label “not damping,” shown in Figure 6, demonstrates that the PZT actuator is not activated. Thus, it refers to the typical features of the undamped system with flexible connections. The Y-direction movement of the end-effector, which should retain the Y position on 0 m, is also shown in Figure 7. The damping also damped the coordinate oscillations.
Also, to determine the effects of different acceleration values, the proposed path was tested using accelerations two and four times faster than the determined amount, which is exhibited in Figure 8 (i.e., the end-effector moves 4 mm and 8 mm in the same time frame in the second and third scenario). Figure 9 shows the tip deformation of every flexible link, , confirming the prominent role of the PZT actuator when vibrating the links structurally as the structural vibrations are damped thoroughly after 60 msec. As reflected, the increase of acceleration puts more vibration on the manipulator, resulting in more deflection of flexible links, which were fully dampened by the active damping method in all three scenarios.
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As discussed, the dynamic model for the manipulator is studied only by the first three modes. To address this, we performed a power spectral density (PSD) analysis of the manipulator based on frequency-domain for the proposed trajectories with and without active damping; the results plotted in Figures 10–12 determine that the first mode has the significant strength of the energy, while the variation at second and third modes are much weaker and had the same measures. Figures 10–12 also prove that the active damping method appropriately reduced the vibration in all given trajectories. This study’s results match the research conducted by Zhang et al., in which the vibrations of an experimentally moving platform with flexible links are damped using the PZT actuators as an active control method [23]. Furthermore, well-established researches in this area obtained comprehensive damping performance by considering less than three modes of the manipulator [19, 24] and validated by experimental setups [25], which indicates the accuracy of our results.
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6. Conclusions
2-DOF parallel manipulators containing lightweight and flexible materials are among the most discussed subjects by industrial scholars due to their extensive applications, such as pick and place tasks. In addition to the flexibility of robot links, working at high speeds makes it impossible to consider the deformation of the links as rigid. In this study, the deformation of the 2-PRRRP manipulators’ links during accelerated trajectories is addressed by approaching the assumed mode method. Using the first-type Lagrangian equations, the dynamic equations of the 2-DOF flexible-link planar parallel manipulator are represented. The active damping method with the PZT actuators is considered to attenuate the structural vibration of flexible links. The piezoelectric materials can achieve a suitable damping performance with an L-type control strategy to counter structural vibration of flexible connections, which substantially reduces the time of setting of the end-effector leading to precise tracking of the trajectory. Relevant results indicate no requirement to use solid heavy links in 2-DOF parallel manipulators and spend less energy by considering them flexible.
Appendix
First-type Lagrangian equations:
Potential energy induced by link deformation:
Equation of motion components:where .
Voltage generated by PZT sensor:
Using strain rate feedback,where is control gain applied to the PZT actuator.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflicts of interest.