1. Introduction

Robotic manipulators are widely used in environments that are hazardous to humans and in situations where there are substantial amounts of repetitive work to reduce the burden on humans and to assist in accomplishing special tasks. To ensure sufficient positioning accuracy at the end of a manipulator, the links of the manipulator can generally be stiffened by a bulky structure to prevent the links from vibrating and to avoid elastic deflection during operation [1–4]. However, heavy links limit the speed of the manipulator’s movement and consume excessive amounts of energy. In some cases, thickening the structure of the links to ensure the positioning accuracy of the end of the manipulator may not be practical. For example, the weight of a manipulator operating on the space station must be as light as possible because of launch costs [5]. In addition, to ensure that the manipulator can perform in a greater workspace, the boom of a concrete pump truck structure should be more slender [6]. Robotic manipulators are also used in nuclear maintenance [7, 8], for spraying large aircraft [9], etc. Under these circumstances, these robotic manipulators have to be designed as flexible-link manipulators (FLMs) with a slender structure.

FLMs have been found in extensive applications because of their advantages in terms of being lightweight and having high load ratios (the ratio of load mass to manipulator mass), high movement speeds, and low energy consumption [10–12]. Moreover, in human-robot interactions, a lightweight FLM is safer to humans and has smaller workload requirements on joint drivers, sensors [13, 14], etc. However, such an FLM experiences vibration and elastic deformation during movement, which will greatly affect the stability of the entire system and the positioning accuracy of the end [15, 16]. Therefore, it is important to study the effects of the dynamic characteristics of FLMs, in terms of the vibration suppression and elastic deformation compensation of the flexible links. For example, the dynamic model of a single-joint flexible manipulator has been developed to suppress the vibration of the flexible manipulator [3, 13].

Because multijoint FLMs are highly nonlinear, strongly coupled, multi-degree-of-freedom (DOF) systems, the dynamic modeling is complex. In addition, considering the payload mass at the end of the manipulator and the lumped mass (for example, the mass of the actuators) on each link, multi-DOF rigid-flexible coupling dynamic modeling is more complicated. These challenging topics have attracted a large number of excellent researchers, who have conducted a considerable amount of research on the dynamic modeling of FLMs. In Giorgio and Del Vescovo’s work [17], the lumped parameter method is used to model the dynamics of flexible links, and although this modeling method is simple, the number of elements must be as large as possible to obtain an adequate model accuracy. The modeling and control of a special class of single-link, an arbitrary number of lumped masses, flexible arms is presented in [18], in which the arm consists of massless flexible structures. Subudhi and Morris adopted the Lagrange equation and the assumed mode method (AMM) to establish the dynamic model of a manipulator with flexible links and joints [19]. A dynamic model of 3-PRR parallel manipulator with flexible links based on AMM was proposed in [20]. Wei et al. provided a global mode method for dynamic modeling of a flexible arm flexible joint manipulator with a tip mass [21]. Some other modeling methods, such as the finite element method (FEM), are also commonly used. For example, the dynamic modeling of the flexible links using FEM is presented in [22]. The FEM is used to derive the dynamic equations for finding the maximum allowable dynamic load of flexible-link mobile manipulators [23]. In addition, the Newton–Euler method is also used to dynamically model the flexible arms [24]. Nonlinear dynamics of motion are developed for flexible manipulator links consisting of rotary joints that connect pairs of flexible links [25]. Bascetta et al. developed a closed-form dynamic model of flexible manipulators based on the Newton–Euler equations [26]. However, the dynamic modeling in the above studies only considers flexible arms with a certain number of DOFs or flexible links without payload mass, and the expressions of the dynamic model are still very computationally expensive in practice.

Considering the complexity of the dynamic modeling of multi-DOF FLMs with a lumped mass on the links, a general-purpose dynamic modeling method for N-DOF FLMs based on the Lagrange equation is presented in this paper, and the symbolic expression computation software for the dynamic equations of N-DOF flexible manipulators is developed. In a more general situation, where the lumped mass is located not only on the end of the links but also at other positions of the links (for example, the flexible arms composed of functionally graded materials (FGM) in the fields of spacecraft, nuclear industries, robotics, etc., the flexibility of the arms can be increased or changed, or changing the vibration frequency of the flexible arm structure by adding concentrated mass to the intermediate position of flexible arms, see [27]) is also considered in this paper. The modeling method is as follows. First, the flexible arms are divided into imaginary rigid and flexible parts and modeled separately. Then, the rigid and flexible parts are coupled, and the dynamic model is further simplified. Finally, the simplest general expression of the dynamic model of the flexible arm with N-DOFs is obtained by the induction method, and “symbolic expression computation software for dynamic equations of N-DOF flexible manipulators” is developed using the symbolic calculation software Mathematica. The test results indicate that compared with the traditional modeling method, the proposed modeling method can reduce the computation time by more than 90% and the software can quickly calculate and generate the dynamic model of the flexible manipulator, thereby greatly reducing the difficulty of the dynamic modeling of the flexible manipulator. Finally, the position tracking control experiment of a 2-DOF flexible manipulator is taken as an example and the dynamic model is generated automatically by the program. Meanwhile, a physical model is also constructed. In an experiment on the end effector of the manipulator tracking a trajectory of a circle, the numerical simulation results of the dynamic model are compared with the experimental results. The dynamic model and the dynamic modeling method are both verified to be correct.

The remainder of this paper is organized as follows. In Section 2, the motion of N-DOF FLM systems is described. Then, in Section 3, the dynamic equations of N-DOF FLMs are derived and established. “The symbolic expression computation software for dynamic equations of N-DOF flexible manipulators” is developed in Section 4, in which the dynamic equations of two-DOF FLMs are automatically generated by the program, and the experiments are conducted. Finally, conclusions are drawn in Section 5.

2. Motion of Flexible Links

To study the general situation of dynamic modeling for serial multijoint FLMs and to simplify the derivation process, fixed-base and small-deformation flexible manipulators moving in the vertical plane are the focus of this paper. Before we derive and establish the dynamic model of the multi-DOF FLM, the appropriate float frame for each flexible link should be assigned, as shown in Figure 1.

In general, the elements in matrix ${\mathbf{A}}_i^0$ are a function of (θ₁  θ₂   ⋯   θ_i) and the elements in vector ${\mathbf{r}}_i^i$ are a function of (f_i1  f_i2).

Equation (11) can be interpreted as the interaction effects of the motions of joint j and joint k on all points on virtual link i.

3. Derivation of Dynamic Equations for Flexible Links with Lumped Mass

3.1. Calculation of Kinetic Energy

After obtaining the velocity of points on link i, we can calculate the kinetic energy of link i. Assuming that the mass of flexible link i is m_i, let $T_i^m$ be the kinetic energy of link i and let $d{T}_i^m$ be the kinetic energy of a particle with differential mass dm on link i; then, we can obtain that

$$\begin{array}{c}d{T}_i^m=\frac{1}{2}\text{Tr}\left[{\dot{\mathbf{r}}}_i^0{\left({\dot{\mathbf{r}}}_i^0\right)}^T\right]dm=\frac{1}{2}\text{Tr}\left\{\left[\left(\sum _{j=1}^i{\mathbf{U}}_{ij}{\dot{\theta }}_j\right){\mathbf{r}}_i^i+{\mathbf{A}}_i^0{\dot{\mathbf{r}}}_i^i\right]\times {\left[\left(\sum _{j=1}^i{\mathbf{U}}_{ij}{\dot{\theta }}_j\right){\mathbf{r}}_i^i+{\mathbf{A}}_i^0{\dot{\mathbf{r}}}_i^i\right]}^T\right\}dm\\ =\frac{1}{2}\sum _{p=1}^i\sum _{r=1}^i\text{Tr}\left\{{\mathbf{U}}_{ip}\left[{\mathbf{r}}_i^i{\left({\mathbf{r}}_i^i\right)}^{T}dm\right]{\mathbf{U}}_{ir}^T\right\}{\dot{\theta }}_p{\dot{\theta }}_r+\sum _{j=1}^i\text{Tr}\left\{{\mathbf{U}}_{ij}\left[{\mathbf{r}}_i^i{\left({\dot{\mathbf{r}}}_i^i\right)}^{T}dm\right]{\left({\mathbf{A}}_i^0\right)}^T\right\}{\dot{\theta }}_j+\frac{1}{2}\text{Tr}\left\{{\mathbf{A}}_i^0\left[{\dot{\mathbf{r}}}_i^i{\left({\dot{\mathbf{r}}}_i^i\right)}^{T}dm\right]{\left({\mathbf{A}}_i^0\right)}^T\right\},\end{array}$$ (12)

where Tr is a trace operator and $\text{Tr}\left(\mathbf{X}\right)\triangleq {\sum }_{i=1}^n{x}_{ii}$. Now, let us compute the kinetic energy of lumped mass $T_i^M$ on flexible link i. We assume that the lumped mass can locate at any position on flexible link i. Similarly, we use the same method as equations (10) and (12) and obtain

$$\begin{array}{c}{T}_i^M=\frac{1}{2}\text{Tr}\left[{\dot{\mathbf{R}}}_i^0{\left({\dot{\mathbf{R}}}_i^0\right)}^T\right]M_i=\frac{1}{2}\sum _{p=1}^i\sum _{r=1}^i\text{Tr}\left\{{\mathbf{U}}_{ip}\left[{\mathbf{R}}_i^i{\left({\mathbf{R}}_i^i\right)}^T{M}_i\right]{\mathbf{U}}_{ir}^T\right\}{\dot{\theta }}_p{\dot{\theta }}_r+\sum _{j=1}^i\text{Tr}\left\{{\mathbf{U}}_{ij}\left[{\mathbf{R}}_i^i{\left({\dot{\mathbf{R}}}_i^i\right)}^T{M}_i\right]{\left({\mathbf{A}}_i^0\right)}^T\right\}{\dot{\theta }}_j+\frac{1}{2}\text{Tr}\left\{{\mathbf{A}}_i^0\left[{\dot{\mathbf{R}}}_i^i{\left({\dot{\mathbf{R}}}_i^i\right)}^T{M}_i\right]{\left({\mathbf{A}}_i^0\right)}^T\right\},\end{array}$$ (13)

where M_i is the lumped mass and ${\mathbf{R}}_i^0$ and ${\dot{\mathbf{R}}}_i^0$ are the position vector and velocity vector of the lumped mass M_i in the global inertial reference frame {0}, respectively. ${\mathbf{R}}_i^i$ and ${\dot{\mathbf{R}}}_i^i$ are the position vector and velocity vector of lumped mass M_i in float frame {i}, respectively. Thus, we let ${\mathbf{R}}_i^i={\left[\begin{array}{ccc}{X}_i& Y_i& 1\end{array}\right]}^T$, as shown in Figure 2.

Because we ignore the deformation of the flexible link in the longitudinal direction, the length of flexible link i is always L_i. To simplify the derivation, we define X_i = μ_iL_i, ${\mu }_i\in \left[\begin{array}{cc}-1,& 0\end{array}\right]$. When μ_i = 0, lumped mass M_i locates on the end of flexible link i if μ_i = −(1/2) represents M_i at the middle of flexible link i. Thus, we can obtain ${\mathbf{R}}_i^i={\left[\begin{array}{ccc}{\mu }_i{L}_i& Y_i& 1\end{array}\right]}^T$, and according to equation (4), we can obtain

$$\begin{array}{}{Y}_i=\text{sin}\left(\pi {\mu }_i\right)f_{i1}+\text{sin}\left(2\pi {\mu }_i\right)f_{i2}=\sum _{k=1}^2{f}_{ik}{S}_{ki}.\end{array}$$ (14)

To simplify the notation, we define S_ki = sin(kπμ_i), k = 1,2. The total kinetic energy of flexible link i and lumped mass M_i can be computed as

$$\begin{array}{c}{T}_i={\int }_{-L_i}^{0}d{T}_i^m+T_i^M=\frac{1}{2}\sum _{p=1}^i\sum _{r=1}^i\text{Tr}\left[{\mathbf{U}}_{ip}{\mathbf{J}}_{ui}{\mathbf{U}}_{ir}^T\right]{\dot{\theta }}_p{\dot{\theta }}_r+\sum _{j=1}^i\text{Tr}\left[{\mathbf{U}}_{ij}{\mathbf{J}}_{vi}{\left({\mathbf{A}}_i^0\right)}^T\right]{\dot{\theta }}_j+\frac{1}{2}\text{Tr}\left[{\mathbf{A}}_i^0{\mathbf{J}}_{wi}{\left({\mathbf{A}}_i^0\right)}^T\right],\end{array}$$ (15)

where ${\mathbf{J}}_{ui}={\mathbf{J}}_{ui}^m+{\mathbf{J}}_{ui}^M$, ${\mathbf{J}}_{vi}={\mathbf{J}}_{vi}^m+{\mathbf{J}}_{vi}^M$, and ${\mathbf{J}}_{wi}={\mathbf{J}}_{wi}^m+{\mathbf{J}}_{wi}^M$. Additionally, ${\mathbf{J}}_{ui}^m$ is the pseudoinertia matrix of flexible link i, which, for a homogeneous Euler beam, can be calculated as

$$\begin{array}{c}{\mathbf{J}}_{ui}^m={\int }_{-L_i}^0{\mathbf{r}}_i^i{\left({\mathbf{r}}_i^i\right)}^{T}dm={\int }_{-L_i}^0\left[\begin{array}{c}{x}_i\\ y_i\\ 1\end{array}\right]\left[\begin{array}{ccc}{x}_i& y_i& 1\end{array}\right]dm={\int }_{-L_i}^0\left[\begin{array}{ccc}{x}_i^2& x_i{y}_i& x_i\\ x_i{y}_i& y_i^2& y_i\\ x_i& y_i& 1\end{array}\right]dm\\ =\left[\begin{array}{ccc}\frac{L_i^2}{3}& \frac{L_i\left(2f_{i1}-f_{i2}\right)}{2\pi }& -\left(\frac{L_i}{2}\right)\\ \frac{L_i\left(2f_{i1}-f_{i2}\right)}{2\pi }& \frac{1}{2}\left(f_{i1}^2+f_{i2}^2\right)& -\left(\frac{2f_{i1}}{\pi }\right)\\ -\left(\frac{L_i}{2}\right)& -\left(\frac{2f_{i1}}{\pi }\right)& 1\end{array}\right]m_i.\end{array}$$ (16)

As shown, the pseudoinertia matrix ${\mathbf{J}}_{ui}^m$ is a function of generalized coordinates f. The variation in the generalized coordinates f can reflect the deformation of the flexible link. Similarly, ${\mathbf{J}}_{ui}^M$ is calculated as

$$\begin{array}{c}{\mathbf{J}}_{ui}^M={\mathbf{R}}_i^i{\left({\mathbf{R}}_i^i\right)}^T{M}_i=\left[\begin{array}{c}{X}_i\\ Y_i\\ 1\end{array}\right]\left[\begin{array}{ccc}{X}_i& Y_i& 1\end{array}\right]M_i=\left[\begin{array}{ccc}{X}_i^2& X_i{Y}_i& X_i\\ X_i{Y}_i& {\left(Y_i\right)}^2& Y_i\\ X_i& Y_i& 1\end{array}\right]M_i\\ =\left[\begin{array}{ccc}{L}_i^2{\mu }_i^2& L_i{\mu }_i\sum _{k=1}^2{f}_{ik}{S}_{ki}& L_i{\mu }_i\\ L_i{\mu }_i\sum _{k=1}^2{f}_{ik}{S}_{ki}& {\left(\sum _{k=1}^2{f}_{ik}{S}_{ki}\right)}^2& \sum _{k=1}^2{f}_{ik}{S}_{ki}\\ L_i{\mu }_i& \sum _{k=1}^2{f}_{ik}{S}_{ki}& 1\end{array}\right]M_i.\end{array}$$ (17)

Note that ${\dot{x}}_i=0$, and we can easily obtain ${\mathbf{J}}_{vi}^m$ and ${\mathbf{J}}_{wi}^m$:

$$\begin{array}{c}{\mathbf{J}}_{vi}^m={\int }_{-L_i}^0{\mathbf{r}}_i^i{\left({\dot{\mathbf{r}}}_i^i\right)}^{T}dm=\left[\begin{array}{ccc}0& \frac{L_i\left(2{\dot{f}}_{i1}-{\dot{f}}_{i2}\right)}{2\pi }& 0\\ 0& \frac{1}{2}\left(\sum _{k=1}^2{f}_{ik}{\dot{f}}_{ik}\right)& 0\\ 0& -\frac{2{\dot{f}}_{i1}}{\pi }& 0\end{array}\right]m_i,{\mathbf{J}}_{wi}^m={\int }_{-L_i}^0{\dot{\mathbf{r}}}_i^i{\left({\dot{\mathbf{r}}}_i^i\right)}^{T}dm=\left[\begin{array}{ccc}0& 0& 0\\ 0& \frac{1}{2}\left({\dot{f}}_{i1}^2+{\dot{f}}_{i2}^2\right)& 0\\ 0& 0& 0\end{array}\right]m_i.\end{array}$$ (18)

${\mathbf{J}}_{vi}^M$ and ${\mathbf{J}}_{wi}^M$ are computed in the same manner:

$$\begin{array}{c}{\mathbf{J}}_{vi}^M={\mathbf{R}}_i^i{\left({\dot{\mathbf{R}}}_i^i\right)}^T{M}_i=\left[\begin{array}{ccc}0& L_i{\mu }_i\sum _{k=1}^2{f}_{ik}{S}_{ki}& 0\\ 0& \left(\sum _{k=1}^2{f}_{ik}{S}_{ki}\right)\left(\sum _{k=1}^2{\dot{f}}_{ik}{S}_{ki}\right)& 0\\ 0& \sum _{k=1}^2{\dot{f}}_{ik}{S}_{ki}& 0\end{array}\right]M_i,{\mathbf{J}}_{wi}^M={\dot{\mathbf{R}}}_i^i{\left({\dot{\mathbf{R}}}_i^i\right)}^T{M}_i=\left[\begin{array}{ccc}0& 0& 0\\ 0& {\left(\sum _{k=1}^2{\dot{f}}_{ik}{S}_{ki}\right)}^2& 0\\ 0& 0& 0\end{array}\right]M_i.\end{array}$$ (19)

3.3. Relationship between the Generalized Forces and Actual Driving Forces

We continue to discuss the relationship between the generalized forces corresponding to the generalized coordinates and the actual forces applied to the manipulator’s joints. As shown in Figure 3, τ_i,i−1 is the actual force applied to joint i, where the first subscript i denotes the link that the force acts on and the second subscript i − 1 denotes the link that the force reacts on. According to the principle of virtual work, the virtual work acting on link i is

$$\begin{array}{}\delta W_i={\tau }_{i,i-1}\left(\delta {\theta }_i+\delta {\alpha }_i\right)+{\tau }_{i+1,i}\delta {\beta }_i,\end{array}$$ (29)

where ${\alpha }_i=\left(\partial y_i/\partial x_i\right)\left|{}_{x_i=0}\right.$ and ${\beta }_i=\left(\partial y_i/\partial x_i\right)\left|{}_{x_i=L_i}\right.$; thus, the total virtual work on all joints is

$$\begin{array}{}\delta W=\sum _{i=1}^n{\tau }_{i,i-1}\delta {\theta }_i+\sum _{\lambda =1}^2\left[{\tau }_{n,n-1}{\phi }_{i\lambda }^{\prime }\left(0\right)\right]\delta f_{n\lambda }+\sum _{i=1}^{n-1}\sum _{\lambda =1}^2\left[{\tau }_{i,i-1}{\phi }_{i\lambda }^{\prime }\left(0\right)+{\tau }_{i,i+1}{\phi }_{i\lambda }^{\prime }\left(L_i\right)\right]\delta f_{i\lambda },\end{array}$$ (30)

where ${\phi }_{i\lambda }^{\prime }\left(0\right)=\left(d{\phi }_{i\lambda }/d{x}_i\right)\left|{}_{x_i=0}\right.$ and ${\phi }_{i\lambda }^{\prime }\left(L_i\right)=\left(d{\phi }_{i\lambda }/d{x}_i\right)\left|{}_{x_i=L_i}\right.$, and the generalized forces corresponding to the generalized coordinates θ_i are

$$\begin{array}{}{F}_{\theta i}={\tau }_{i,i-1},i=\mathrm{1,2},\dots ,n.\end{array}$$ (31)

The generalized forces corresponding to the generalized coordinates f_iλ are

$$\begin{array}{c}{F}_{fi\lambda }={\tau }_{i,i-1}{\phi }_{i\lambda }^{\prime }\left(0\right)+{\tau }_{i+1,i}{\phi }_{i\lambda }^{\prime }\left(L_i\right),F_{fn\lambda }={\tau }_{n,n-1}{\phi }_{n\lambda }^{\prime }\left(0\right),\end{array}$$ (32)

where i = 1,2, …, n − 1,   λ = 1,2.