Hybrid control of piezoelectric flexible manipulator based on Volterra filtered-xLMS algorithm

Abstract

To suppress the nonlinear vibration of the flexible manipulator during motion, this article presents a hybrid control strategy based on a servo motor and a piezoelectric actuator. The dynamic model of the piezoelectric flexible manipulator is established first. To realize the trajectory tracking, a proportional derivative control method is used to schedule the control torque. Because the Volterra filter can approximate the nonlinear system model, a Volterra filtered-xLMS algorithm based on a second-order Volterra filter structure is proposed, by which the active nonlinear vibration control of flexible link is realized. Simulation results show that the proposed Volterra filtered-xLMS algorithm can not only make use of the advantages of the classical filtered-xLMS algorithm but also solve the problem of effective modeling of nonlinear secondary path. The proposed hybrid control strategy based on Volterra filtered-xLMS algorithm and proportional derivative control algorithm can improve the position accuracy of joint and effectively suppress the vibration response of the nonlinear flexible link. A piezoelectric flexible manipulator with PZT (lead zirconate titanate) sensor and actuator is designed to demonstrate the validity and efficiency of the proposed method by experiments. Experiment results demonstrate that the attenuation time of vibration response is reduced from 5 s to 1.5 s, the vibration response at the first-order frequency is reduced by 60%, and the proposed methodology has an important advantage in application of active vibration control of piezoelectric flexible manipulator.

Keywords

active vibration control piezoelectric flexible manipulator piezoelectric smart structure Volterra filter Volterra filtered-xLMS

1. Introduction

Flexible manipulator has the advantages of light structure, high load ratio, and low energy consumption and is widely used in various fields such as industry, military, aerospace, and so on (Lochan et al., 2016). However, due to its low stiffness and low damping, low frequency elastic vibration is easy to be produced in the process of motion, which seriously affects the positioning accuracy, motion stability, and service life of the manipulator (Fareh et al., 2020; Huang and Ji, 2020; Shafei and Korayem, 2017).

As a basic part of the flexible manipulator, the joint servo motor has been used to suppress vibration of the flexible manipulator during motion. Lots of literatures improved the trajectory tracking accuracy and decreased vibration level of flexible manipulator by adjusting the joint driving torque (Alandoli et al., 2021; Zhao et al., 2021). The existing vibration control techniques based on the joint servo motor can be broadly classified into two categories, namely, feedback control and feed-forward control (Dong et al., 2019; Habibi et al., 2019). Input command shaping is a typical feed-forward control strategy in which the expected input is convolved with a series of impulse sequence, so as to avoid activating the vibration modes and eliminate the residual vibration of the flexible manipulator (Cole and Wongratanaphisan, 2013). Another alternative feed-forward control strategy was used to minimize the elastic vibration of the flexible arm by planning and optimizing the motion trajectory (Heidari et al., 2013; 2017; Korayem et al., 2011). The advantage of feed-forward control strategy is that it does not require any additional sensors. However, it belongs to the open-loop control technique, which has limitations in dealing with various disturbances and parameters variations. Feedback control strategy based on measuring the motion state of the joint and the flexible link is the most widely used, which has strong ability to resist external interference and can realize stable control. The control strategies, which only use joint servo motor as the driving mechanism, are easy to be implemented. However, the control strategy based on joint drive motor is essentially non-minimum phase system, which makes it very challenging to achieve high precision motion control and ensure the stability of the system. Therefore, it is very necessary to introduce other effective control methods.

In recent years, with the development of intelligent materials, intelligent actuators have provided a feasible solution to suppress elastic vibration of flexible manipulator by active control method (Garcia-Perez et al., 2019; Qiu et al., 2019). The intelligent actuator represented by piezoelectric actuator has the advantages of small size, flexible configuration, and simple arrangement, which is very suitable for vibration active control of flexible structures. So, more and more attention has been paid to the integrated control of flexible manipulator driven by joint servo motor and piezoelectric actuator. Shao (Shao et al., 2020) presented a finite-element model of a manipulator with a flexible link with flexible joint and embedded PZT actuators. Mirzaee (Mirzaee et al., 2010) used a hybrid structure control composed of a servo motor and an embedded piezoelectric actuator to achieve the trajectory tracking and vibration suppression of series and parallel flexible manipulators, respectively.

Traditional model-based active vibration method can suppress the elastic vibration in the trajectory tracking process on the basis of obtaining the precise dynamic characteristic parameters of the flexible manipulator. The main disadvantage of model-based methods is poor adaptability to the dynamic change of the controlled system. However, the flexible manipulator is uncertain and nonlinear, due to the change of terminal load, the hysteresis of driving joint, the flexibility of link and other reasons, it is difficult to establish an accurate real-time model of the flexible manipulator. In order to overcome this problem, sliding mode control (Ullah et al., 2021), robust control (Yang et al., 2019), neural networks control (Gao et al., 2019), fuzzy control (Ethem et al., 2019; Ozguney and Burkan, 2021), and other intelligent control methods (Rahimi and Nazemizadeh, 2014) had achieved certain control effect in theory and application, but there are still many problems, such as large computation, difficult real-time guarantee, parameter drift, and chattering phenomenon. Adaptive control adjusts the control parameters online according to the system state feedback information, which is widely used in the position and vibration control of flexible manipulator (Pradhan and Subudhi, 2014; Schnelle and Eberhard, 2017; Zhang and Liu, 2013). In the field of the active vibration control, the adaptive control also plays an important role by adjusting the output control signal through the adaptive control law to achieve the purpose of the vibration control. When the structure parameters of the controlled system are time-varying or seriously uncertain, the adaptive active vibration control can avoid the calculation of structural parameters and improve the control efficiency.

In the field of adaptive active vibration control, the classical adaptive control model is a feed-forward control based on Filtered-x Least Mean Square (FXLMS) algorithm, which has good control effect and adaptive performance, and is widely used in active vibration control of linear structures (Pu et al., 2019). This kind of algorithm is simple in structure and easy to implement. Its application to active vibration suppression of flexible link has important advantages, which can avoid complex nonlinear control algorithm and reduce the cost of control system. However, the characteristics of secondary path (the vibration path between secondary source and error sensor) have important influence on the convergence and control effect of the FXLMS algorithm (Pu et al., 2014). Due to the inherent hysteresis and creep characteristics of piezoelectric materials, complex hysteresis and nonlinearities exist between the input voltage and the output torque of the piezoelectric actuator. As a result, the transfer function of the secondary path containing the piezoelectric actuator is also nonlinear. For the FXLMS algorithm, this nonlinear characteristic of secondary path will not only reduce the control accuracy and limit the control performance of the system but also produce phase and harmonic distortion related to the input signal, which will weaken the feedback effect, and even cause the oscillation or instability of the control system. There are two main solutions to overcome the adverse effects caused by the nonlinear characteristics of the piezoelectric actuator in the active control system: on the one hand, consider the piezoelectric actuator itself, by establishing a suitable nonlinear hysteresis model, creep model, using inverse model to compensate the actuator nonlinearity, improve the actuator control performance (Juhasz et al., 2011; Shao et al., 2016). On the other hand, the nonlinear characteristics of a piezoelectric actuator are considered as part of the nonlinearity of the controlled structural system, and the vibration control is realized through the design of more complex nonlinear control algorithms (Rahimi and Nazemizadeh, 2014). However, these put forward higher hardware requirements for the implementation of active vibration control of flexible manipulator. Most of the existing secondary path modeling for FXLMS algorithm are based on FIR filtering structure, which cannot effectively model nonlinear secondary path. Since Volterra filter itself has nonlinear characteristics, it can approximate any continuous nonlinear system model, so it is very suitable for building nonlinear system model (Zhang et al., 2010). Combining FXLMS algorithm and Volterra filter, constructing a Volterra filtered-xLMS (VFXLMS) control algorithm is a novel and meaningful idea, in which the secondary path modeling is established based on Volterra filter structure. It can not only make use of the advantages of the classical FXLMS algorithm but also solve the problem of effective modeling of nonlinear secondary path.

The rest of the article is organized as follows. In the Section 2, a dynamic model of the flexible manipulator driven by a servo motor and a piezoelectric actuator is presented. Employing singular perturbation method, the dynamic model of the piezoelectric flexible manipulator is decomposed into a slow variable subsystem to characterize rigid motion and a fast variable subsystem to characterize elastic vibration. In the Section 3, a composite controller consisting of the proportional derivative (PD) control algorithm of the slow-varying subsystem and the VFXLMS algorithm of the fast-varying subsystem is proposed. In VFXLMS algorithm, the Volterra filter structure is used to model the nonlinear secondary path containing piezoelectric actuators online. In the Section 4, numerical analyses and simulations are performed. The effect of elastic vibration control in a piezoelectric flexible manipulator is verified. In the Section 5, the experimental device of flexible manipulator system is introduced, and the experimental results are analyzed. Finally, the full text is summarized in the section Conclusions.

2. Dynamics of piezoelectric flexible manipulator and singular perturbation decomposition

The flexible manipulator is a complex dynamic system with strong coupling and nonlinear characteristics. Nonlinear coupled partial differential equations are usually used to describe such continuous nonlinear dynamical systems with infinite degrees of freedom. However, in the implementation of vibration control, the dynamical equations with finite dimensions are necessary which capture the nonlinear flexible dynamics of the system, to facilitate the implementation of the control scheme. Assumed modes methods were widely used to model robot dynamic in order to capture the interaction between flexural vibrations and nonlinear dynamics (Rahimi and Nazemizadeh, 2014).

Different assumptions were made to simplify the problem as much as possible. The assumptions are made depending on the accuracy required. If the effects of shear deformation and rotational inertia are considered, the Timoshenko beam theory can be used in flexible manipulator modeling (Korayem et al., 2014; 2016). In this study, because the length is much larger than the size of the cross section, the flexible link is assumed to be a Euler–Bemoulli beam, considering only its bending deformation, ignoring the shear deformation.

A single-link piezoelectric flexible manipulator driven by servo motor and piezoelectric actuator is studied. The schematic diagram of the flexible piezoelectric manipulator system is shown in Figure 1. The flexible link is connected with the output shaft of the servo drive joint through a hub at its root. The rotating motion driven by the servo motor and the elastic vibration of the flexible link occurs in the plane perpendicular to the axis of rotation. $O x_{0} y_{0}$ is set as the fixed coordinate system, and $O x y$ is the reference coordinate system, which is always tangent to the root of the flexible manipulator. $w (x, t)$ is elastic deformation deflection of flexible manipulator. $τ (t)$ is the joint driving torque, and $θ$ is the rotation angle.

Figure 1.

Diagram of piezoelectric flexible manipulator.

2.1. Dynamic modeling of the piezoelectric flexible manipulator

In general, the first several order vibration modes of flexible manipulator are dominant, and the contribution of higher order vibration modes can be ignored. The elastic deformation deflection of flexible manipulator can be expressed by mode superposition

w (x, t) = \sum_{i = 1}^{\infty} φ_{i} (x) q_{i} (t) = \sum_{i = 1}^{n} φ_{i} (x) q_{i} (t)

(1)

where,

φ_{i} (x)

is the ith order mode shape vector,

q_{i} (t)

is the corresponding modal coordinate index, and n is the order of mode truncation.

The generalized mass $M_{i}$ and generalized stiffness $K_{i}$ of the ith mode of the flexible link are expressed by equations (2) and (3),respectively

M_{i} = \int_{0}^{l} ρ (x) A (x) ϕ_{i}^{2} (x) d x

(2)

K_{i} = \int_{0}^{l} E (x) I (x) {[ϕ_{i}^{″} (x)]}^{2} d x

(3)

where

l, A (x), ρ (x), E (x), I (x)

are the mass, length, cross section area, density, elastic modulus, and moment of inertia to the neutral axis of cross section, respectively.

The coordinate position of point p on the flexible link can be expressed as

{\begin{matrix} p_{x} = x \cos θ - w (x, t) \sin θ \\ p_{y} = x \sin θ + w (x, t) \cos θ \end{matrix}

(4)

The total kinetic energy of the single-link flexible manipulator system is composed of two parts: the rotational kinetic energy of the servo-driven joint (including the motor and the hub) and the elastic vibration kinetic energy of the flexible link (including the embedded piezoelectric plate)

T = \frac{1}{2} J_{m} {\dot{θ}}^{2} + \frac{1}{2} \int_{0}^{l} ρ (x) A (x) (p_{x}^{2} + p_{y}^{2}) d x

(5)

where

J_{m}

is the equivalent rotational inertia of the servo-driven joint.

ρ (x) A (x)

is the mass function of the flexible link. The total kinetic energy of the flexible manipulator can be further expressed as

T = \frac{1}{2} J {\dot{θ}}^{2} + \frac{1}{2} \sum_{i = 1}^{n} M_{i} q_{i}^{2} (t) {\dot{θ}}^{2} + \frac{1}{2} \sum_{i = 1}^{n} M_{i} {\dot{q}}_{i}^{2} (t) + \dot{θ} \sum_{i = 1}^{n} [\int_{0}^{l} ρ (x) A (x) x ϕ_{i} (x) d x] {\dot{q}}_{i} (t)

(6)

where

J = J_{m} + \int_{0}^{l} ρ (x) A (x) x^{2} d x

The rotating motion and the elastic vibration all occur in the plane perpendicular to the axis of rotation. It can be considered that there is no change of gravitational potential energy when the flexible manipulator moves, and only the potential energy caused by elastic deformation is considered. Therefore, the potential energy of the flexible manipulator can be expressed as

V = \frac{1}{2} \int_{0}^{l} E (x) I (x) {w^{″} (x, t)}^{2} d x = \frac{1}{2} \int_{0}^{l} E (x) I (x) {\sum_{i = 1}^{n} φ_{i}^{″} (x) q_{i} (t)}^{2} d x = \frac{1}{2} \sum_{i = 1}^{n} K_{i} q_{i}^{2} (t)

(7)

According to the inverse piezoelectric effect, the governing equation of a piezoelectric actuator (Jalili, 2010) is

M_{a} = \frac{1}{2} E_{a} d_{31} b_{a} (t_{a} + t_{f}) V = k_{p a} V

(8)

where

E_{a}, d_{31}

are the Young’s modulus and Piezoelectric strain coefficient of the piezoelectric actuator, respectively, and

b_{a}, t_{a}

are the width and thickness of the piezoelectric actuator, respectively.

t_{f}

is the thickness of flexible link on which the piezoelectric actuator is located.

V

is the input voltage of the piezoelectric actuator.

k_{p a}

is the equivalent piezoelectric moment coefficient. From equation (8),

k_{p a}

is generally regarded as a constant value, so the output bending torque

M_{a}

is proportional to the output control voltage

V

. However, due to the nonlinear properties of piezoelectric materials, such as hysteresis and creep, the input voltage and output torque of the piezoelectric actuator are nonlinear too.

The total virtual work done by the driving moment of the servo motor and the bending moments of L piezoelectric actuators (Sun et al., 2004) is

δ W = τ (t) δ θ + \sum_{j = 1}^{L} \int_{x_{s j}}^{x_{e j}} τ_{j} (x, t) b_{f} t_{f} (δ θ + Φ' (x) δ \dot{q}) d x

(9)

where

τ_{j} (x, t)

is the distributed shear stress applied by PZT,

b_{f}

and

t_{f}

are the width and the thickness of the flexible link, respectively.

x_{s j}

and

x_{e j}

represent the starting and ending positions of the jth piezoelectric actuator, respectively.

Φ

is the mode shape matrix of the flexible link. If the thickness of the paste layer is not considered, and the piezoelectric actuator and the flexible arm are considered to be perfectly paste, the virtual work expression is further simplified to obtain the virtual work of the system for perfectly bonded piezoelectric actuators can be evaluated as

δ W = τ (t) δ θ + \sum_{j = 1}^{L} M_{a j} (t) (δ θ + [Φ' (x_{e j}) - Φ' (x_{s j})] δ \dot{q})

(10)

where

M_{a j}

is the bending moment generated by the jth piezoelectric actuator, the term

[Φ' (x_{e j}) - Φ' (x_{s j})]

denotes the slope difference between two ends of the jth actuator.

The dynamic equation of the flexible manipulator obtained through the Lagrange function is

{\begin{matrix} J \ddot{θ} + \ddot{θ} \sum_{i = 1}^{n} M_{i} q_{i}^{2} (t) + 2 \dot{θ} \sum_{i = 1}^{n} M_{i} q_{i} {\dot{q}}_{i} (t) + \sum_{i = 1}^{n} {\int_{0}^{l} ρ (x) A (x) x ϕ_{i} (x) d x} {\ddot{q}}_{i} (t) = τ (t) + \sum_{j = 1}^{L} k_{p a} V_{j} (t) \\ M_{i} {\ddot{q}}_{i} (t) + \ddot{θ} {\int_{0}^{l} ρ (x) A (x) x ϕ_{i} (x) d x} - {\dot{θ}}^{2} M_{i} q_{i} (t) + K_{i} q_{i} (t) = \sum_{j = 1}^{L} k_{p a} V_{j} (t) [Φ' (x_{e j}) - Φ' (x_{s j})] \end{matrix}

(11)

2.2. Singular perturbation decomposition of dynamic equation

The dynamic equation can be expressed in matrix form

\begin{array}{l} [\begin{matrix} J + \sum_{i = 1}^{n} M_{i} q_{i}^{2} (t) & \int_{0}^{l} ρ (x) A (x) x ϕ_{1} (x) d x & \dots & \int_{0}^{l} ρ (x) A (x) x φ_{N} (x) d x \\ \int_{0}^{l} ρ (x) A (x) x φ_{1} (x) d x & M_{1} & \dots & 0 \\ \dots & \dots & \dots & \dots \\ \int_{0}^{l} ρ (x) A (x) x φ_{N} (x) d x & 0 & \dots & M_{N} \end{matrix}] [\begin{matrix} \ddot{θ} \\ {\ddot{q}}_{1} \\ \dots \\ {\ddot{q}}_{N} \end{matrix}] \\ + [\begin{matrix} \sum_{i = 1}^{n} M_{i} q_{i} {\dot{q}}_{i} (t) & \dot{θ} M_{1} q_{1} (t) & \dots & \dot{θ} M_{n} q_{N} (t) \\ - \dot{θ} M_{1} q_{1} (t) & 0 & \dots & 0 \\ \dots & \dots & \dots & \dots \\ - \dot{θ} M_{N} q_{N} (t) & 0 & \dots & 0 \end{matrix}] [\begin{matrix} \dot{θ} \\ {\dot{q}}_{1} \\ \dots \\ {\dot{q}}_{N} \end{matrix}] + [\begin{matrix} 0 & 0 & \dots & 0 \\ 0 & K_{1} & \dots & 0 \\ \dots & \dots & \dots & \dots \\ 0 & 0 & \dots & K_{N} \end{matrix}] [\begin{matrix} θ \\ q_{1} \\ \dots \\ q_{N} \end{matrix}] = [\begin{matrix} τ (t) + \sum_{j = 1}^{L} k_{p a}^{j} V_{j} (t) \\ \sum_{j = 1}^{L} k_{p a}^{j} V_{j} (t) [φ_{1}' (x_{e}) - φ_{1}' (x_{s})] \\ \dots \\ \sum_{j = 1}^{L} k_{p a}^{j} V_{j} (t) [φ_{N}' (x_{e}) - φ_{N}' (x_{s})] \end{matrix}] \\ = [\begin{matrix} 1 & k_{p a}^{1} & \dots & k_{p a}^{L} \\ 0 & k_{p a}^{1} [φ_{1}' (x_{e_{1}}) - φ_{1}' (x_{s_{1}})] & \dots & k_{p a}^{L} [φ_{1}' (x_{e_{1}}) - φ_{1}' (x_{s_{1}})] \\ \dots & \dots & \dots & \dots \\ 0 & k_{p a}^{1} [φ_{N}' (x_{e_{1}}) - φ_{N}' (x_{s_{1}})] & \dots & k_{p a}^{L} [φ_{N}' (x_{e_{1}}) - φ_{N}' (x_{s_{1}})] \end{matrix}] [\begin{matrix} τ (t) \\ V_{1} \\ ⋮ \\ V_{L} \end{matrix}] \end{array}

(12)

and

[\begin{matrix} M_{r r} & M_{r f} \\ M_{f r} & M_{f f} \end{matrix}] [\begin{matrix} \ddot{θ} \\ \ddot{q} \end{matrix}] + [\begin{matrix} H_{r r} & H_{r f} \\ H_{f r} & 0 \end{matrix}] [\begin{matrix} \dot{θ} \\ \dot{q} \end{matrix}] + [\begin{matrix} 0 & 0 \\ 0 & K_{f f} \end{matrix}] [\begin{matrix} θ \\ q \end{matrix}] = [\begin{matrix} τ (t) + g_{r} V \\ g_{f} V \end{matrix}]

(13)

where

g_{r} = [k_{p a}^{1}, k_{p a}^{2}, \dots k_{p a}^{L}], g_{f} = [\begin{matrix} k_{p a}^{1} [φ_{1}' (x_{e_{1}}) - φ_{1}' (x_{s_{1}})] & \dots & k_{p a}^{L} [φ_{1}' (x_{e_{1}}) - φ_{1}' (x_{s_{1}})] \\ ⋮ & ⋮ & ⋮ \\ k_{p a}^{1} [φ_{N}' (x_{e_{1}}) - φ_{N}' (x_{s_{1}})] & \dots & k_{p a}^{L} [φ_{N}' (x_{e_{1}}) - φ_{N}' (x_{s_{1}})] \end{matrix}]

Based on the difference of the state variables of the dynamic equation on the time scale, the flexible manipulator dynamic system is decomposed into two lower-order systems by singular perturbation method. First, the fast change phenomenon is ignored to obtain a simplified solution. The steady-state result represents the slow change phenomenon which plays a leading role in the system. Then, the boundary layer correction term is calculated on the “stretching” time scale to make error correction and compensation.

From the system dynamic characteristics, its mass matrix is positive definite symmetric, so there is an inverse matrix, and the inverse matrix of the mass matrix is N

N = [\begin{matrix} N_{r r} & N_{r f} \\ N_{f r} & N_{f f} \end{matrix}] = M^{- 1} = {[\begin{matrix} M_{r r} & M_{r f} \\ M_{f r} & M_{f f} \end{matrix}]}^{- 1}

(14)

Left-multiply both sides of the equation (14) by the matrix N , the dynamic equation becomes

[\begin{matrix} \ddot{θ} \\ \ddot{q} \end{matrix}] = - [\begin{matrix} N_{r r} & N_{r f} \\ N_{f r} & N_{f f} \end{matrix}] [\begin{matrix} H_{r r} & H_{r f} \\ H_{f r} & 0 \end{matrix}] [\begin{matrix} \dot{θ} \\ \dot{q} \end{matrix}] - [\begin{matrix} N_{r r} & N_{r f} \\ N_{f r} & N_{f f} \end{matrix}] [\begin{matrix} 0 & 0 \\ 0 & K_{f f} \end{matrix}] [\begin{matrix} θ \\ q \end{matrix}] + [\begin{matrix} N_{r r} & N_{r f} \\ N_{f r} & N_{f f} \end{matrix}] [\begin{matrix} τ (t) + g_{r} V \\ g_{f} V \end{matrix}]

(15)

A singular perturbation scale factor is introduced as follows

μ = \sqrt{1 / \min {K_{i}}} 1 \leq i \leq N

(16)

ε = \frac{1}{μ^{2}} q

(17)

{\tilde{K}}_{f f} = μ^{2} K_{f f}

(18)

By setting $μ = 0$ , the dynamic equation of the slow subsystem of the piezoelectric flexible manipulator can be obtained

M_{r r} \ddot{θ} + H_{r r} \dot{θ} = τ (t) + g_{r} V

(19)

For fast subsystem, introducing the boundary layer correction variable

z_{1} = ε - ε_{r}

(20)

where

ε_{r}

is the value of

ε

in slow subsystem, which is treated as constant in fast time scale.

A fast time scale is introduced on the boundary layer $t_{q} = t / μ$ , the dynamic equation of the fast subsystem can be written

\frac{d^{2} z_{1}}{d t_{q}^{2}} = - {\tilde{N}}_{f f} {\tilde{K}}_{f f} z_{1} + {\tilde{N}}_{r f} (τ + g_{r} V) + {\tilde{N}}_{r f} g_{f} V

(21)

Based on the singular perturbation method, the dynamic equation of the flexible manipulator is decomposed into the dynamic equations of the slow and fast subsystems as shown in equations (19) and (21), respectively. It can be seen that the slow subsystem represents rigid motion in a large-time range, while the fast subsystem represents the elastic vibration in the small-time range.

3. Hybrid control strategy of flexible manipulator

During the motion of the flexible manipulator, the dynamic posture, varying driving torque, and the nonlinear factors of joint will all cause the time-varying and uncertain dynamic characteristics of the flexible link, which makes it difficult to obtain accurate model parameters in real time. In order to overcome this problem, the proposed control strategy is based on the dynamic equations (17) and (19) obtained in the section 2, but without the precise model parameters.

In the following research, a hybrid control strategy of flexible manipulator is designed. The schematic diagram of hybrid control is shown in Figure 2. The trajectory tracking control is realized by using the servo drive joint torque for the slow subsystem, and the elastic vibration control is realized by using the piezoelectric actuator for the fast subsystem.

Figure 2.

Schematic diagram of hybrid control for piezoelectric flexible manipulator.

The control strategy proposed in this study consists of two parts. The first part is the PD feedback controller in joint space for trajectory tracking, using the feedback of the servo motor’s angular position and velocity. The second part is to design VFXLMS algorithm to suppress the vibration of the flexible manipulator, and realize the identification of nonlinear secondary path containing piezoelectric actuators through the Volterra filter structure.

3.1. Control algorithm for the slow subsystem-proportional derivative feedback control

Proportional derivative feedback control is the common control method for robot joint position control. It does not need to obtain the mathematical model of the control system, which reduces the difficulty of controller design and is easy to implement. The trajectory tracking control of flexible manipulator for the slow subsystem adopts PD feedback method as follows

τ (t) = K_{p} (θ_{d} - θ) + K_{d} ({\dot{θ}}_{d} - \dot{θ}) + \frac{g V}{N}

(22)

where

θ_{d}

and

{\dot{θ}}_{d}

are the designed angle position and designed angular velocity, and

θ

and

\dot{θ}

are the actual angle position and actual angular velocity of the servo motor obtained by the encoder, respectively.

K_{p}

and

K_{d}

represent the proportional gain and differential gain of the PD control law, respectively. The last term of equation (22) is the dynamic compensation of the effect of the piezoelectric actuator’s bending torque, which reflects the coupling effect of the elastic vibration control on the trajectory tracking control.

3.2. Control algorithm for the slow subsystem-VFXLMS

As mentioned in the Introduction section, although the FXLMS algorithm has important advantages in structural vibration active control, there are still some limitations when applied to active suppression of nonlinear vibration flexible link. Since the Volterra series can approximate any continuous nonlinear system, for the slow subsystem, the VFXLMS algorithm is proposed, in which the identification model of nonlinear secondary path is established by Volterra filter structure instead of FIR transverse filter structure in FXLMS algorithm.

3.2.1. Volterra series

The expression of Volterra series is as follows

\begin{array}{l} y (n) = \sum_{m_{1} = 0}^{M - 1} h_{m_{1}} (n) x (n - m_{1}) + \sum_{m_{1} = 0}^{M - 1} \sum_{m_{1} = m_{2}}^{M - 1} h_{m_{1}, m_{2}} (n) x (n - m_{1}) x (n - m_{2}) + \dots \\ + \sum_{m_{1} = 0}^{M - 1} \sum_{m_{1} = m_{2}}^{M - 1} \dots \sum_{m_{s} = m_{s - 1}}^{M - 1} h_{m_{1}, m_{2}, \dots m_{s}} (n) x (n - m_{1}) x (n - m_{2}) \dots x (n - m_{s}) + \dots \end{array}

(23)

where

h_{m_{1}, m_{2}, \dots m_{s}} (n)

is the S-order Volterra nucleus,

x (n)

is the input signal, and

y (n)

is the output signal. The Volterra series contains linear terms and nonlinear terms, which can completely model a nonlinear dynamic system. A second-order Volterra filter is usually used in practical applications, considering the computational complexity

y_{h} (t) = \sum_{m_{1} = 0}^{M - 1} h_{m_{1}} (n) x (n - m_{1}) + \sum_{m_{1} = 0}^{M - 1} \sum_{m_{2} = 0}^{M - 1} \sum h_{m_{1}, m_{2}} (n) x (n - m_{1}) x (n - m_{2})

(24)

where

h_{m_{1}} (n)

and

h_{m_{1}, m_{2}} (n)

are linear term weight coefficients and nonlinear term weight coefficients of Volterra filter, respectively.

y (n) = A^{T} X (n) + X^{T} (n) BX (n)

(25)

where

X (n) = {[x (n), x (n - 1), \dots x (n - M + 1)]}^{T}

However, when the order of the nonlinear term increases, the number of filter coefficients increased by power series, which greatly increases the difficulty of engineering application. An effective method is to transform the nonlinear kernel of the second-order Volterra filter into a diagonal matrix by using the discrete cosine transform (DCT), so that the number of nonlinear filter coefficients is greatly reduced and the computational complexity is reduced.

According to DCT matrix transformation principle, the second-order Volterra filter can be expressed as follows

y (n) = A^{T} X (n) + X^{T} (n) BX (n) = A^{T} X (n) + X^{T} (n) Q^{T} ΛQX (n)

(26)

where

Λ

is a diagonal matrix whose main diagonal element are eigenvalues of the matrix B.

Define

X_{1} (n) = QX (n) = {[x_{q} (n), x_{q} (n - 1), \dots x_{q} (n - M + 1)]}^{T}

(27)

y (n) = A^{T} X (n) + \sum_{i = 0}^{M} λ_{i} x_{q}^{2} (n - i) = H_{1}^{T} X (n) + H_{2}^{T} X_{q} (n)

(28)

where

H_{2} (n) = {[λ_{1} (n), λ_{2} (n), \dots λ_{M} (n)]}^{T}, X_{q} (n) = {[x_{q}^{2} (n), x_{q}^{2} (n - 1), \dots x_{q}^{2} (n - M + 1)]}^{T}

3.2.2. Identification of nonlinear secondary path based on Volterra filter

The accuracy of secondary path has an important effect on the control effect of elastic vibration. Next, the Volterra filter structure is used to replace the FIR transverse filter structure in the traditional FXLMS algorithm to establish the nonlinear secondary path model, and the offline identification of the nonlinear secondary path is carried out.

The block diagram of the proposed secondary path identification algorithm based on the second-order Volterra Filter is shown in Figure 3. The excitation signal $x (n)$ , amplified by power amplifier, acts on the piezoelectric actuator and causes the structural vibration response $y (n)$ at the measurement point. The same excitation signal $x (n)$ is used as the input of the second-order Volterra filter based secondary path model, and its output is $y_{h} (n)$

y_{h} (n) = {\hat{y}}_{1} (n) + {\hat{y}}_{2} (n) = {\hat{H}}_{1}^{T} X (n) + {\hat{H}}_{2}^{T} X_{q} (n)

(29)

where

{\hat{H}}_{1} (n)

and

{\hat{H}}_{2} (n)

are the linear and nonlinear term coefficients of the second-order Volterra filter, respectively, which is the estimate of the nonlinear secondary path.

Figure 3.

Block diagram of Identification of nonlinear secondary path.

The difference between the real structural response and the model output is defined as the secondary path identification error

e (n) = y (n) - y_{h} (n)

(30)

Based on the LMS algorithm, the coefficient vectors the second-order Volterra filter are updated as follows

{\hat{H}}_{1} (n + 1) = {\hat{H}}_{1} (n) + μ \frac{\partial e^{2} (n)}{\partial {\hat{H}}_{1} (n)} = {\hat{H}}_{1} (n) + 2 μ_{h 1} e (n) X (n)

(31)

{\hat{H}}_{2} (n + 1) = {\hat{H}}_{2} (n) + μ \frac{\partial e^{2} (n)}{\partial {\hat{H}}_{2} (n)} = {\hat{H}}_{2} (n) + 2 μ_{h 2} e (n) X_{p} (n)

(32)

where

μ_{h 1}

and

μ_{h 2}

is the corresponding convergent step size.

3.2.3. The VFXLMS algorithm

The proposed novel nonlinear active vibration control algorithm based on the second-order Volterra filter structure which named VFXLMS algorithm is given as follows. The block diagram of the VFXLMS algorithm is shown in Figure 4.

Figure 4.

Block diagram of Identification of nonlinear secondary path.

The filter reference signal based on the second-order Volterra filter structure is calculated by equations (33) and (34)

\hat{x} (n) = {\hat{H}}_{1}^{T} X (n)

(33)

{\hat{x}}_{q} (n) = {\hat{H}}_{2}^{T} X_{q} (n)

(34)

where

X (n)

and

X_{q} (n)

are the reference signal vectors.

X (n) = {[x (n), x (n - 1), \dots x (n - M + 1)]}^{T}, X_{q} (n) = {[x_{q}^{2} (n), x_{q}^{2} (n - 1), \dots x_{q}^{2} (n - M + 1)]}^{T}

The control output is calculated as follows

y (n) = W_{1} {(n)}^{T} X (n) + W_{2} {(n)}^{T} X_{q} (n)

(35)

According to the steepest descent principle, the updating formulas of linear term coefficients and nonlinear term coefficients of second-order Volterra filter are as follows

W_{1} (n + 1) = W_{1} (n) + 2 μ_{w 1} e (n) \hat{X} (n)

(36)

W_{2} (n + 1) = W_{2} (n) + 2 μ_{w 2} e (n) {\hat{X}}_{q} (n)

(37)

where

μ_{w 1}

and

μ_{w 2}

is the corresponding convergent step size,

\hat{X} (n) = {[\hat{x} (n), \hat{x} (n - 1), \dots \hat{x} (n - N + 1)]}^{T}, {\hat{X}}_{q} (n) = {[{\hat{x}}_{q}^{2} (n), {\hat{x}}_{q}^{2} (n - 1), \dots {\hat{x}}_{q}^{2} (n - M + 1)]}^{T}

.Thus, we propose a complete derivation process of hybrid control algorithm based on PD algorithm and VFXLMS algorithm. The flowchart procedure of the proposed method is provided in Table 1.

Table 1.

The appropriate flowchart procedure of the proposed method.

Step 1. Before the controller works, identify the nonlinear secondary path based on the second-order Volterra filter structure by equations (30) and (31).

Step 2. Calculate the joint driving torque in accordance with formula (21) based on the joint angular error and joint angular velocity error measured by the encoder, and the input voltage of the piezoelectric actuator (see step 5).

Step 3. Duplicate the identification result of step 1 to the nonlinear secondary path to the fast sub-controller.

Step 4. Take the vibration response signal measured by the piezoelectric sensor as the input reference signal of the VFXLMS algorithm, and calculate the control voltage of the piezoelectric actuator by using equation (34).

Step 5. Input the control voltage of the piezoelectric actuator into the slow subsystem controller, that is, in equation (21), to participate in the calculation of joint driving torque.

VFXLMS: volterra filtered-xLMS.

4. Simulation results and analysis

In order to verify the correctness of the dynamic model of the flexible manipulator and the effectiveness of the proposed control strategy, in this section, the dynamic characteristics of a flexible manipulator system are studied by numerical simulation. Then, the combined control strategy based on the servo motor and piezoelectric actuator is verified by numerical simulation. A series of simulation experiments on hybrid control of piezoelectric flexible manipulator are provided, comparing the performance of the proposed hybrid control algorithm based on VFXLMS algorithm with the classical FXLMS algorithm. The parameters of the piezoelectric flexible manipulator are shown in Table 2.

Table 2.

The parameters of the piezoelectric flexible manipulator in simulation.

	Lengthwidththickness (mm)	Young’s modulus (GPa)	Piezoelectric constant d31 (C/N)	Density (kg/m³)	Rotational inertia (kg*m²)
Flexible link	9851002	25	—	2400	5.4 × 10-4
Piezoelectric actuator	40401	62.9	187 × 10⁻¹²	7500	—

Based on the parameters of the piezoelectric flexible manipulator in Table 2, the first four natural frequencies of the flexible link can be calculated by numerical solution, as shown in Table 3. As low-order bending modes have a dominant effect on vibration energy, only the first two orders of bending vibration mode are of interest.

Table 3.

The natural frequency of piezoelectric flexible manipulator in simulation.

	Mode1 (Hz)	Mode2 (Hz)	Mode3 (Hz)	Mode4 (Hz)
Natural frequency	3.40	21.30	59.64	116.87

Next, the dynamic characteristics of the flexible manipulator system in the process of motion are studied. It is assumed that the flexible manipulator rotates according to the desired angular displacement trajectory as shown in equation (38), which is a quintic polynomial.

θ (t) = [6 {(\frac{t - t_{0}}{t_{e}})}^{5} - 15 {(\frac{t - t_{0}}{t_{e}})}^{4} + 10 {(\frac{t - t_{0}}{t_{e}})}^{3}] (θ_{e} - θ_{0}) + θ_{0}

(38)

where

t_{0}

and

t_{e}

are the starting and ending time of the flexible manipulator motion respectively, and

θ_{0}

and

θ_{e}

drive the initial and final angular positions of the joint, respectively.

The driving motor of the flexible manipulator starts to move after receiving the motion instruction. Because of the flexibility, the vibration is obviously in the process of motion, and after the driving joint reaches the designed position, the vibration of the link lasts for a long time. To suppress the vibration, the secondary bending moment is excited by the piezoelectric actuator to realize active vibration control.

The input torque of servo drive joint of servo drive motor calculated by (22) and the control voltage of piezoelectric actuator calculated by (35) are shown in Figure 5(a) and (b), respectively. The first two orders modal displacements of the flexible link with or without active vibration control by piezoelectric actuators are shown in Figure 6(a) and (b), respectively. After the piezoelectric actuator is activated, the first-order mode displacement is decreased obviously. Although the second-order modal displacement increases to a certain extent, the increase is smaller than decrease of the first-order modal displacement, so the elastic vibration of the flexible manipulator in the process of motion is generally reduced. Comparing the performance of the proposed hybrid control algorithm based on VFXLMS algorithm, the convergence step parameter must be set to a smaller value to ensure that the control system does not diverge in the FXLMS algorithm, which greatly deteriorates the convergence performance of the control system. Moreover, the residual vibration response at the end of the motion is not only greatly reduced but also reduced to zero in a shorter time, as shown in Figure 7(a). The frequency response curve of the vibration response at the end point is shown in Figure 7(b). The simulation result demonstrated that compared with the FXLMS algorithm, the proposed VFXLMS algorithm has a good adaptability to the nonlinear structural vibration, and can ensure a better control effect.

Figure 5.

(a) Input torque of servo drive joint. (b) The control voltage of PZT actuator.

Figure 6.

(a) The first-order modal displacement. (b) The second-order modal displacement.

Figure 7.

Vibration response at the end point: (a) time response curve; (b) frequency response curve.

5. Experimental results and analysis

In order to investigate the actual control effect of the active vibration control method based on the proposed VFXLMS method, experimental verification is necessary. In this section, we will experimentally evaluate the effectiveness of the proposed control method on a piezoelectric flexible manipulator.

5.1. The piezoelectric flexible manipulator control system

A piezoelectric flexible manipulator with PZT actuator and sensor is designed. The flexible manipulator rotates about the fixed shaft driven by a servo motor, and the elastic vibration is suppressed by a PZT actuator. The schematic diagram of the flexible manipulator experimental system is shown in Figure 8. The controller calculates the input torque and sends it to the servo driver, which drives the motor to drive the flexible manipulator to rotate about a fixed axis. The PZT sensor attached to the flexible link measures and converts the strain of the flexible link into an electrical signal, which is amplified by a charge amplifier and transmitted to the controller through the acquisition module. The controller calculates the control signal through the active control algorithm, and after the control signal is amplified by the power amplifier, the PZT actuator is driven to generate bending moment to suppress the vibration of the flexible link. As low-order bending modes have a dominant effect on vibration energy, only the first-order bending vibration mode is of interest. In the high-strain area of the smart cantilever plate, the PZT patches have good performance when used as sensors and actuators. The root region is the high-strain areas for the first-order mode. Two piezoelectric patches bonded on the top and bottom at the same location from the left edge are classified into one group as one actuator. The PZT sensor used to detect the vibration of the smart cantilever plate is placed next to the secondary actuator.

Figure 8.

Schematic diagram of the piezoelectric flexible manipulator experimental system.

The main components of the flexible manipulator are as follows:

The servo drive joint is composed of a servo drive motor, a coupling and a transmission component. (Servo drive motor is from China Qsino Dynatron Company, Model Numbers are COOLDRIVE RC3-A5550-T4-S1, TDA060-01330FA0-11B00).

The main structure of manipulator is smart flexible plate consisting of epoxy resin material which has good strength and high flexibility during the operation. The dimension of the smart plate is 680 × 100 × 2.5 mm.

A piezoelectric actuator is attached on the flexible link to suppress the elastic vibration, and the piezoelectric sensor is attached to measure the vibration response of the manipulator link. The dimension of the piezoelectric patches used as actuator in our experiments is 40 × 40 × 1 mm, and used as sensor is 10 × 10 × 1 mm. Considering the PZT actuator used in our experiments are strain type actuators, high-strain region of every order vibration mode should be identified to place the PZT actuator. The schematic diagram of the implemented real-time hybrid control system on the piezoelectric flexible manipulator based on NI CompactRIO controller is shown in Figure 9. Charge Amplifier Power Amplifier

Figure 9.

View of the experimental set-up.

The natural frequencies of the first fourth-order bending modes obtained by finite-element model (FEM) simulation and experiment are shown in Table 4. As the low-order bending mode plays a dominant role in the vibration energy, only the first four-order bending vibration mode is concerned.

Table 4.

The natural frequency of piezoelectric flexible manipulator in experiment.

	Mode1 (Hz)	Mode2 (Hz)	Mode3 (Hz)	Mode4 (Hz)
FEM	2.29	14.35	40.25	79.15
Experiment	2.46	14.22	40.19	79.21

FEM: finite-element model.

5.2. Hybrid control experiment of flexible manipulator

The trajectory and velocity of joint motion are designed based on the quintic polynomial, and the joint rotates 30° in 1 s. The control parameters of PD algorithm are $K_{p} = 0.004$ and $K_{d} = 0.0005$ . The step sizes for the second-order Volterra filter are $μ_{w 1} = 0.08$ and $μ_{w 2} = 0.002$ , and the lengths of filters W₁ and W₂ are set as 1000. The vibration response was measured at a sampling rate of 1652 Hz. The servo drive motor communicates with the upper computer through EtherCAT bus, and the communication cycle is 1 ms.

The input torque of servo motor and the control voltage of PZT actuator are shown in Figure 10(a) and (b), respectively. It can be seen that the torque of the servo motor is reduced both before and after the joint is moved to the designed position when combined with the piezoelectric actuator. The vibration response of the flexible link is shown in Figure 11. Compared with the control method that only uses servo to drive joint torque, the combined active vibration control by using PZT actuator is more effective. As can be seen from Figure 11(a), the attenuation time of vibration response is reduced from 5 s to less than 1.5 s. From the frequency response curve in Figure 11(b), the vibration response at the first frequency is reduced by 60%. The results show that the proposed hybrid control strategy based on VFXLMS algorithm and PD control algorithm is effective for vibration suppression of flexible manipulator.

Figure 10.

(a) The input torque of the servo motor; (b) Control voltage of the PZT actuator.

Figure 11.

Vibration response of the flexible link: (a) time response curve; (b) frequency response curve.

6. Conclusions

The proposed hybrid control strategy of track tracking and active vibration control of flexible manipulator system is effective. The advantages of the proposed method are in the following three aspects: (1) Compared with other competitive algorithms, such as fuzzy algorithms and neural network algorithms, the proposed VFXLMS algorithm has a relatively simple algorithm structure and a small computational complexity. (2) Compared with the control method based on precise model parameters, the proposed method is more suitable for the application of hybrid control because the proposed algorithm is a non-model control method and it does not require high accuracy of model parameters. (3) Compared to the classical FXLMS algorithm, the proposed VFXLMS algorithm has a good vibration control effect on the nonlinear flexible manipulator, which greatly enhances the adaptability of the control system. From what has been discussed above, the proposed hybrid control strategy not only ensures the high trajectory motion accuracy but also reduces the elastic vibration of the flexible link quickly, which significantly reduces the vibration attenuation time of the flexible link, and improves the operating efficiency of the flexible manipulator.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and astronautics) (Grant no. MCMS-E-0121G01) and the Fundamental Research Funds for the Central Universities (Grant no. PA2020GDSK0093)

ORCID iD

Yuxue Pu

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