Dynamical analysis and control of rotatory manipulators with time varying mass loads

Abstract

Manipulators are widely used in industrial fields as automation equipment and robotic structures. Besides moving objects, manipulators often work with variable payloads, and the residual vibration significantly affects the position accuracy of manipulators. This paper mainly investigates the dynamic characteristics of flexible manipulators with time varying mass payloads and active control of the residual vibration is carried out. Finite element method is utilized to construct the dynamical model, and responses of the system are calculated by Bathe’s method. The influence of the time varying mass is analyzed in details, including an increasing mass case and a decreasing mass case. Then active control of the residual vibration is given based on a PID controller. To improve the control effect, genetic algorithm (GA) is applied to tune the parameters of the PID controller. Simulations demonstrate that the time varying mass affects the residual vibration of the system remarkably. For a decreasing mass system, a nonstructural negative damping is induced and the residual vibration may be strengthened, while for an increasing mass system, a nonstructural positive damping is caused and the residual vibration may be attenuated. The comparison of the control methods demonstrates that the GA-PID controller results in a significant improvement in the vibration reduction.

Keywords

Rotatory manipulator active vibration control time varying payload GA-PID controller non-structural damping

1. Introduction

Robotic manipulators play critical roles in industrial manufacturing and advanced devices [1–3], and higher motion resolution and accuracy are required to satisfy the demand of sophisticated equipment, however the flexibility of robotic manipulators leads to residual vibrations, which are greatly influenced by material, the dimension, the payload and the motion speed. Residual vibration is not expected in structural designs, and smart materials have attracted significant attentions in vibration control, such as shunt damping layers [4], piezoelectric actuators, electromagnetic actuators [5] and shape memory alloy [6]. Based on linear matrix inequality, Mohamed et al. [7] utilized a robust PD controller to control of the end-point deflection of a planar two-link flexible manipulator, and the robustness of the controller was verified by experimental measurements and numerical simulations. Xu and Yuan [8] conducted a two-degrees of freedom model to analyze the precision control of a flexible-link manipulator, and a feedback control scheme was successfully implemented to enhance the tracking performance. Gao et al. [9] carried out a neural network controller to control the vibration of a two-link flexible manipulator, and demonstrated the feasibility using the Quanser platform. Wu et al. [10] modeled a two-link manipulator by simplifying the motor using a virtual spring and suppressed the residual vibration of the manipulator by a non-collocated controller. Qiu et al. [11] introduced a fuzzy neural network control scheme to control the vibration of a two-link flexible manipulator, showing that the control scheme was effective for the small amplitude residual vibration.

From the research reviews, the influence of the mass payload has been investigated in some studies, however only the case of constant mass payload was taken into consideration. Therefore, in this study, we mainly focus on the dynamic characteristics and active control of rotatory flexible manipulators with variable mass payloads. In Section 2, the governing equation of a manipulator with a variable mass on the free end is modeled using the finite element theory. Dynamical analysis of the time varying system is performed in Section 3, and the influence of the time varying payload is discussed. Active vibration control is carried out in Section 4. Finally, the study is briefly summarized in Section 5.

2. Finite element Model

Figure 1(a) shows a flexible manipulator with a payload connected to the free end, while the mass of the payload is a function of time, denoted by m (t). A revolute joint is attached at the left endpoint, and a servo motor is used to rotate the manipulator. The servo motor drives the flexible link with an angular velocity ω, and the flexible link rotates an angle of θ in the xoy coordinate. Then residual vibration continues for a

period until the motor stops. The finite element model is displayed in Fig. 1(b): the flexible link is discretized by beam elements, and the servo motor is simplified to a rotation spring with a defined displacement condition, in addition the payload is modeled as a point mass with an inertia. The flexible link has a total length of L, and is uniformly meshed with an elemental length l.

Fig. 1.

Rotating manipulator with variable mass load and the finite element model.

The finite element modeling theory is given in many textbooks [12]. For a planar link, the deformation can be decomposed to an axial deformation and a bending deformation. Hence each beam element has three degrees, including two nodal displacements (u, w) and a flexural rotation of the cross section φ: $\begin{eqnarray}\displaystyle \boldsymbol{u}_{i}^{\text{T}}=\left[\begin{array}{@{}cc@{}}u_{i} & u_{i+1}\end{array}\right],\quad \boldsymbol{w}_{i}^{\text{T}}=\left[\begin{array}{@{}cccc@{}}w_{i} & {\varphi}_{i} & w_{i+1} & {\varphi}_{i+1}\end{array}\right] & & \displaystyle\end{eqnarray}$ (1) where symbol T represents the matrix transpose. Here the mass and inertia of the payload are denoted by m and I_m, respectively, and the rotational spring constant of the motor is expressed by K_o. Then the kinetic energy K and the potential energy V of the system are derived as: $\begin{eqnarray}\displaystyle \text{} & \text{}\displaystyle K={\displaystyle \frac{1}{2}}\int _{0}^{l}{\rho}A(w_{t})^{2}dx+{\displaystyle \frac{1}{2}}\int _{0}^{l}{\rho}A(u_{t})^{2}dx+\left[{\displaystyle \frac{1}{2}}m(u_{t})^{2}+{\displaystyle \frac{1}{2}}m(w_{t})^{2}+{\displaystyle \frac{1}{2}}I_{m}(w_{x})_{t}^{2}\right]_{x=L} & \displaystyle\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \text{} & \text{}\displaystyle V={\displaystyle \frac{1}{2}}\int _{0}^{l}EI(w_{xx})^{2}dx+{\displaystyle \frac{1}{2}}\int _{0}^{l}EA(u_{x})^{2}dx+\left[{\displaystyle \frac{1}{2}}K_{o}(w_{x})^{2}\right]_{x=0} & \displaystyle\end{eqnarray}$ (3) where 𝜌, E, A and I are density, modulus of elasticity, area of cross section and moment of cross section of the flexible link. Subscript t and x represent the derivatives with respect to the time and the spatial coordinate, respectively. Based on the finite element method, the kinetic energy and the potential energy is rewritten in matrix forms: $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}K={\displaystyle \frac{1}{2}}\dot{\boldsymbol{w}}_{e}^{\text{T}}\boldsymbol{m}_{w}^{b}\dot{\boldsymbol{w}}_{e}+{\displaystyle \frac{1}{2}}\dot{\boldsymbol{u}}_{e}^{\text{T}}\boldsymbol{m}_{u}^{b}\dot{\boldsymbol{u}}_{e}+\left[{\displaystyle \frac{1}{2}}\dot{\boldsymbol{w}}_{e}^{\text{T}}\boldsymbol{m}_{w}^{m}\dot{\boldsymbol{w}}_{e}+\frac{1}{2}\dot{\boldsymbol{u}}_{e}^{\text{T}}\boldsymbol{m}_{u}^{m}\dot{\boldsymbol{u}}_{e}\right]\\[12.0pt] V={\displaystyle \frac{1}{2}}\boldsymbol{w}_{e}^{\text{T}}\boldsymbol{k}_{w}\boldsymbol{w}_{e}+{\displaystyle \frac{1}{2}}\boldsymbol{u}_{e}^{\text{T}}\boldsymbol{k}_{u}\boldsymbol{u}_{e}+{\displaystyle \frac{1}{2}}\boldsymbol{w}_{e}^{\text{T}}\boldsymbol{k}_{o}\boldsymbol{w}_{e}.\end{array} & & \displaystyle\end{eqnarray}$ (4)

Superscripts b and m represent the parameters of the link and the payload, respectively, while subscripts u and w represent the variables related to axial motion and transversal motion. Using variational principle, the governing equations for the axial vibration and transversal vibration can be obtained [12]: $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\boldsymbol{m}_{u}^{b}\ddot{\boldsymbol{u}}_{e}+\boldsymbol{m}_{u}^{m}\ddot{\boldsymbol{u}}_{e}+\dot{\boldsymbol{m}}_{u}^{m}\dot{\boldsymbol{u}}_{e}+\boldsymbol{k}_{u}\boldsymbol{u}_{e}=\boldsymbol{f}_{e}^{u}\\[12.0pt] \boldsymbol{m}_{w}^{b}\ddot{\boldsymbol{w}}_{e}+\boldsymbol{m}_{w}^{m}\ddot{\boldsymbol{w}}_{e}+\boldsymbol{I}_{m}\ddot{\boldsymbol{w}}_{e}+\dot{\boldsymbol{m}}_{w}^{m}\dot{\boldsymbol{w}}_{e}+\boldsymbol{k}_{w}\boldsymbol{w}_{e}=\boldsymbol{f}_{e}^{w}.\end{array} & & \displaystyle\end{eqnarray}$ (5)

From Eq. (5), two additional damping appears in the form of $\dot{\boldsymbol{m}}_{u}^{m}$ and $\dot{\boldsymbol{m}}_{w}^{m}$ . It is an interesting phenomenon that a changing mass causes a damping-like term, which plays a significant role in the vibration of the flexible system. For the ith element, we rewrite the displacements and the rotation angle into one vector: $\begin{eqnarray}\displaystyle \boldsymbol{d}_{i}^{\text{T}}=[\begin{array}{@{}cccccc@{}}u_{i} & w_{i} & {\varphi}_{i} & u_{i+1} & w_{i+1} & {\varphi}_{i+1}\end{array}]. & & \displaystyle\end{eqnarray}$ (6) The elemental governing equation can be obtained as: $\begin{eqnarray}\displaystyle \boldsymbol{m}_{e}^{b}\ddot{\boldsymbol{d}}_{e}+\boldsymbol{m}_{e}^{m}\ddot{\boldsymbol{d}}_{e}+\dot{\boldsymbol{m}}_{e}^{m}\dot{\boldsymbol{d}}_{e}+\boldsymbol{k}_{e}^{b}\boldsymbol{d}_{e}=\boldsymbol{f}_{e} & & \displaystyle\end{eqnarray}$ (7) where m ^b and m ^m are the elemental mass matrix induced by the elastic link and induced by the payload, respectively. $\dot{\boldsymbol{m}}^{m}$ is the additional damping caused by the mass change of the payload. k ^e and f _e are the stiffness matrix and the force vector. Using Rayleigh damping scale coefficient 𝛼 and 𝛽, structural damping $\boldsymbol{c}_{e}={\alpha}\boldsymbol{m}_{e}^{b}+{\beta}\boldsymbol{k}_{e}^{b}$ is considered, and the actuation forces f _V are applied by piezoelectric patches, therefore the total elemental equation becomes: $\begin{eqnarray}\displaystyle \boldsymbol{m}_{e}^{b}\ddot{\boldsymbol{d}}_{e}+\boldsymbol{m}_{e}^{m}\ddot{\boldsymbol{d}}_{e}+\dot{\boldsymbol{m}}_{e}^{m}\dot{\boldsymbol{d}}_{e}+\boldsymbol{c}_{e}\dot{\boldsymbol{d}}_{e}+\boldsymbol{k}_{e}^{b}\boldsymbol{d}_{e}=\boldsymbol{f}_{e}+\boldsymbol{f}_{V}. & & \displaystyle\end{eqnarray}$ (8)

As the flexible link rotates, a transformation matrix is needed to describe the displacement, force, stiffness and mass in global coordinates, the transformation matrix T is of the following form [12]: $\begin{eqnarray}\displaystyle \boldsymbol{T}=\left[\begin{array}{@{}cc@{}}\boldsymbol{C}\boldsymbol{S} & \boldsymbol{0}\\ \boldsymbol{ 0 } & \boldsymbol{C }\boldsymbol{ S}\\ \end{array}\right],\quad \boldsymbol{C}\boldsymbol{S}=\left[\begin{array}{@{}ccc@{}}\cos {\theta}_{i} & \sin {\theta}_{i} & 0\\ -\sin {\theta}_{i} & \cos {\theta}_{i} & 0\\ 0 & 0 & 1\\ \end{array}\right],\quad \boldsymbol{0}=\left[\begin{array}{@{}ccc@{}}0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\\ \end{array}\right]. & & \displaystyle\end{eqnarray}$ (9)

Using the transformation matrix, the displacement vector d , stiffness matrix k , damping matrix c , mass matrix m and force vector f in global coordinates are transformed to: $\begin{eqnarray}\displaystyle \boldsymbol{d}_{i}=\boldsymbol{T}_{i}\boldsymbol{d}_{i},\quad \boldsymbol{k}_{i}=\boldsymbol{T}_{i}^{\text{T}}\boldsymbol{k}_{i}\boldsymbol{T}_{i},\quad \boldsymbol{m}_{i}=\boldsymbol{T}_{i}^{\text{T}}\boldsymbol{m}_{i}\boldsymbol{T}_{i},\quad \boldsymbol{c}_{i}=\boldsymbol{T}_{i}^{\text{T}}\boldsymbol{c}_{i}\boldsymbol{T}_{i},\boldsymbol{f}_{i}=\boldsymbol{T}_{i}\boldsymbol{f}_{i}. & & \displaystyle\end{eqnarray}$ (10)

Finally, the matrix equation for the motion of the manipulators including added damping is obtained as: $\begin{eqnarray}\displaystyle \boldsymbol{M}(t)\ddot{\boldsymbol{d}}+\left[\boldsymbol{C}\text{(t)}+\dot{\boldsymbol{M}}\text{(t)}\right]\dot{\boldsymbol{d}}+\boldsymbol{K}(t)\boldsymbol{d}=\boldsymbol{F}_{e}+\boldsymbol{F}_{V} & & \displaystyle\end{eqnarray}$ (11) where M , C and K are the mass matrix, damping matrix and stiffness matrix in the global coordinates, and d , F _e and F _V are the displacement vector, excitation force vector and control force vector, respectively. From Eq. (11), a non-structural damping $\dot{\boldsymbol{M}}\text{(t)}$ appears in the equation of motion. This damping is the time derivative of the mass. For a decreasing mass system with $\dot{\boldsymbol{M}}\text{(t)}<0$ , a negative damping is induced while for an increasing mass system, an additional positive damping is induced as $\dot{\boldsymbol{M}}\text{(t)}>0$ . To quantitatively analyze the influence of the additional damping, the case neglecting the additional nonstructural damping is also investigated: $\begin{eqnarray}\displaystyle \boldsymbol{M}(t)\ddot{\boldsymbol{d}}+\boldsymbol{C}(t)\dot{\boldsymbol{d}}+\boldsymbol{K}(t)\boldsymbol{d}=\boldsymbol{F}_{e}+\boldsymbol{F}_{V}. & & \displaystyle\end{eqnarray}$ (12)

3. Dynamical analysis

The motor rotates the flexible link from θ =0° to θ =90°, following a trapezoidal velocity profile [1]. The position and the angular velocity profile of the motor are shown in Fig. 2. The motor stops at 3s, then residual vibration appears resulting from the inertial force. The total simulation time is 7s. The Bathe’s method is applied for the transient analysis, which is a powerful direct implicit time integration scheme in dynamic response calculations [12]. The values for the simulation are given in Table 1 [2].

Fig. 2.

The position and the angular velocity profile of the motor.

In the robotic application, variable mass load is frequently encountered, however the influence of the mass change on the vibration frequency is mainly investigated. Furthermore in this study, the additional damping is also focused on by the transient analysis. The displacement responses of the free end are displayed in Fig. 3. The case for a decreasing mass load from 2.5 kg to 0.5 kg in 4s and the case for an increasing mass load from 0.5 kg to 2.5 kg are computed, meaning the rate of mass change is −0.5 kg/s or 0.5 kg/s. The Rayleigh damping coefficients 𝛼 =0.001 and 𝛽 =0.0002 are applied. The solid lines denote the real responses by solving Eq. (11), while the dash lines denote the responses solved by Eq. (12) neglecting the added damping.

Table 1

Properties of experimental system

Flexible Link		Motor
Properties	Values	Properties	Values
Geometry size/m³	0.4635 ∗ 0.08 ∗ 0.006	Weight of the motor/kg	3.26
Elasticity modulus/GPa	71	Inertia of cross section/(kg ⋅ m²)	0.0134
Density/kg ∗ m⁻³	2700	Rotational spring constant/(N ⋅ m/rad)	16000
Poisson’s ratio	0.3

It has been discussed that the added damping plays a significant role in the transient responses. As shown in Fig. 3, for a decreasing mass, the vibration decays slower because of the added negative damping. Correspondingly, for an increasing mass system, a positive damping is induced, and the vibration decays faster. The action of the induced damping is directly related to the rate of mass change. Hence the action becomes stronger when a larger rate of mass change happens. Here a larger rate of mass change is considered, and the mass load changes from 4.5 kg to 0.5 kg in 4 s, i.e. dm∕dt = −1 kg/s. The responses shown in Fig. 4(a) illustrate that the vibration decays very slowly even there is a large structural damping, the main reason is that the influence of the structural damping is counterbalanced by the induced negative damping. For an increasing mass system with a rate of 1 kg/s, the influence of the additional damping also becomes more significant, as shown in Fig. 4(b). From the simulations, it can be concluded that in the dynamical analysis of systems with variable payloads, the influence of the induced damping must be considered, especially for a decreasing mass system with a negative damping, which is an unstable factor.

Fig. 3.

Displacement responses for the manipulator (|dm∕dt| =0.5 kg/s).

Fig. 4.

Displacement responses for the manipulator (|dm∕dt| =1 kg/s).

4. Feedback control

To overcome the residual vibration, a feedback controller based on piezoelectric materials is designed as shown in Fig. 5(a). The acceleration signals are acquired and processed by an integrator, then a feedback controller with velocity feedback is designed, and the piezoelectric patches are driven by an amplifier to supply control forces. Proportional integral derivative (PID) controller is the most widely used algorithm in engineering. However, to control the vibration of a time varying system, a PID controller with constant coefficients is not satisfactory. To improve the control effect, genetic algorithm (GA) is applied to tune the parameters of the PID controller, as shown in Fig. 5(b). The reference input signal and the measured process variable are r (t) and y (t), respectively; u (t) is the input signal applied to the plant model and e (t) = y (t) − r (t) is the error signal. In the PID controller, K_p, K_i and K_d denote the coefficients for the proportional, integral, and derivative terms, respectively. The three parameters are optimized in the genetic algorithm, according to crossover and mutation [13].

Fig. 5.

Feedback control model and GA-PID controller.

In the simulation the rates of mass change of the increasing mass system and the decreasing mass system are 1 kg/s and −1 kg/s, respectively. The displacement responses for the manipulator without and with control are calculated using Bathe’s method, and the responses of the free end are displayed in Fig. 6. It is visualized that the vibration decays much faster under control for the two cases, showing that both the PID controller and GA-PID controller have obvious suppressing effect on the flexible manipulators with variable payload. Comparing the classical PID controller and the GA-PID controller, it is also found that the GA-PID controller proposes a significant improvement in the vibration reduction.

Fig. 6.

Displacement responses for the manipulator with and without control.

For the manipulator with a constant mass of 0.5 kg and 4.5 kg, the natural frequency of are 10.78 Hz and 4.02 Hz, respectively. From frequency responses shown in Fig. 7, it is found that the frequency range is determined in consistent with the mass change while the frequency range shown in the two figures are also approximately from 4 Hz to 11 Hz. However, the disadvantage of fast Fourier transformation is that the variation of the frequency versus time cannot be obtained, and frequency changing is also a difficulty for active vibration control of variable mass systems.

Fig. 7.

Frequency responses for the system with variable mass load.

5. Conclusions

This paper investigated the vibration suppression of flexible manipulators with variable mass payloads. The influence of the variable mass was mainly analyzed and a GA-PID control scheme was carried out using piezoelectric actuators. Finite element method was used to model the time varying mass system and Bathe’s method was used to solve the dynamical equations in numerical simulations. The following conclusions could be summarized from the numerical simulations:

(1) The variable mass has remarkable effect on the residual vibration of the manipulator. Besides changing the vibration frequency of the system, an additional damping is induced by the variable payload, which is proportional to the rate of mass change. For an increasing payload, a positive damping is caused and the residual vibration is attenuated, while for a decreasing payload, a negative damping is caused and the residual vibration is strengthened.

(2) With active control, the residual vibration could be suppressed effectively by piezoelectric actuators in a wide range of vibration frequency, and the position precision of the manipulator can be improved. Comparing the control results between the PID controller and the GA-PID controller, it was found that the latter one resulted in a significant improvement in the vibration reduction.

Footnotes

Acknowledgements

This research received financial support from National Natural Science Foundation of China (Grant No. 11702162) and Shandong Key R&D Program (Grant No. 2019GGX104070).

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