Physics-Informed Neural Network-Based Input Shaping for Vibration Suppression of Flexible Single-Link Robots

Abstract

The vibration suppression of flexible robotic arms is challenging due to their nonlinear spatiotemporal dynamics. This paper presents a novel physics-informed neural network (PINN)-based input-shaping method for the vibration suppression problem. Through a two-phase training process of a neural network based on a loss function that follows both the physical model constraints and the vibration modal conditions, we identify optimal input-shaping parameters to minimize residual vibration. With the use of powerful computational resources to handle multimode information about the vibration, the PINN-based approach outperforms traditional input-shaping methods in terms of computational efficiency and performance. Extensive simulations are carried out to validate the effectiveness of the method and highlight its potential for complex control tasks in flexible robotic systems.

1. Introduction

The application of robotic arms is becoming more prevalent as a consequence of technological advancements and the increasing demands of industrial production [1]. Nevertheless, conventional rigid robotic arms encounter considerable difficulties when attempting to perform intricate operations in complex environments due to the inherent rigidity of their physical structure. These difficulties include restricted flexibility and adaptability, which can result in suboptimal performance or even failure in tasks that require high precision and adaptability. Flexible robotic arms have been proposed as a solution, offering increased adaptability to various complex environments while ensuring operational accuracy [2]. They can deform themselves to navigate around obstacles and conform to different shapes, making them suitable for applications where conventional rigid arms would struggle. These devices are utilized in areas such as flexible spacecraft [3,4], minimally invasive surgery [5,6], and automated manufacturing processes where precision and flexibility are paramount [7]. Despite their advantages, the nonlinear spatiotemporal dynamics of flexible robotic arms introduce significant control challenges, particularly for objectives such as point-to-point motion regulation, endpoint trajectory planning and tracking, and vibration suppression [8]. These challenges necessitate the development of advanced motion-planning and control strategies to fully harness the potential of flexible robotic arms.

Among these challenges, vibration suppression for flexible robotic arms has been extensively studied using various motion-planning and control strategies [8,9]. These methods are crucial as vibrations can significantly impair the performance and precision of flexible robotic systems. Successful feedback strategies such as backstepping control [7], sliding-mode control [10], and filtered feedback linearization [11] allow for real-time adjustments to mitigate unwanted oscillations of the flexible arms. Intelligent control methods such as fuzzy neural networks [12] and reinforcement learning [13] can be leveraged for their ability to learn and adapt to new conditions, offering a robust solution to the vibration suppression problem in flexible robotic arms. Another promising approach to vibration suppression is input shaping, which preemptively modifies control inputs to counteract anticipated vibrations [14,15]. This can be used as a trajectory planner for controllers, making it compatible with other vibration suppression control methods. This method has been successfully applied for controlling the swing of an underactuated tower crane, effectively controlling vibrations under varying payload conditions [16]. Maghsoudi et al. [17,18] presented improved input-shaping techniques using particle swarm optimization for nonlinear 3D crane models, which led to significant reductions in payload sway. The situation becomes more complex when this method is applied to a multimode vibration system. Traditional methods use multiple filters chained in series to address these issues. However, the computational effort required would increase significantly with the number of modes considered. For effective implementation, Masoud and Alhazza [19] proposed a frequency modulation technique to tune the second-mode frequency to an odd multiple of the closed-loop first-mode frequency and solved the problem with the primary input shaper. Thomsen et al. [20] introduced a modified cascaded structure to account for two vibration modes in industrial robot arms.

Recent advances in machine learning, particularly deep neural networks, have provided new insights for solving complex computational problems in physics and engineering. In particular, machine learning methods have demonstrated their versatility and effectiveness in optimizing mechanical processes, predicting system behaviors, and enhancing control strategies [21,22,23,24]. Among these methods, physics-informed neural networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs), which can be used to describe complex spatiotemporal dynamics. By embedding physical laws directly into the neural network training process, PINNs combine data-driven insights with theoretical models. Raissi et al. [25] demonstrated the capability of PINNs in solving both forward and inverse problems by minimizing data and physical residuals. Subsequent developments, such as the fractional PINN developed by Pang et al. [26] and the physics-informed neural operators developed by Li et al. [27], have expanded the applicability of PINNs to more complex domains, including fractional differential equations and high-dimensional PDEs. Continued innovation in this area has enabled PINNs to address a wide range of complex phenomena and problems that could not be efficiently solved by traditional methods [28,29]. In particular, PINNs’ application to mechanical problems has achieved superior performance compared to conventional methods. The work in [30] has shown that PINNs are effective, robust, and accurate in predicting both displacement and stress fields in 2D and 3D solid mechanics. PINNs also demonstrate capabilities in the parameter identification and modeling of various mechanical systems [31,32,33]. Based on the PINN model, model predictive control was proposed in [34] to solve an optimal control problem for multilink manipulators. There have been few attempts to apply the PINN method to the control of flexible mechanical systems. Nevertheless, PINNs’ inherent ability to handle spatiotemporal dynamics and their powerful computational resources for solving PDEs make them a promising method for vibration suppression in flexible robotic systems.

In this paper, we use a PINN as a solver for the input-shaping problem of a flexible single-link robot to achieve vibration suppression. The optimal impulse sequence in the input-shaping scheme is trained using a two-phase network learning approach based on a loss function that incorporates both physical model knowledge and vibration modal information. The key benefits of the proposed method are as follows:

1.

Network learning for input shaping: The proposed method leverages efficient neural network techniques and its powerful computational resources to search for optimal input shapers for vibration suppression, with results that outperform conventional methods in complex motion scenarios.

2.

Multimode vibration suppression: The proposed method enhances the effectiveness of vibration suppression by utilizing a larger number of modes, while its computational burden remains manageable as the number of modes increases.

3.

Direct feed-forward control design: The optimal impulse sequence and network output can be used directly in feed-forward control algorithms, eliminating the need to solve the system.

The remainder of this paper is organized as follows: In Section 2, we describe the distributed parameter system model of the flexible single-link robot and discuss its motion-planning problem. In Section 3, we detail the proposed PINN-based input command-shaping method. Section 4 outlines how the proposed PINN solutions are applied to design feed-forward control algorithms. Section 5 provides experimental validation, where we test and evaluate the effectiveness of the proposed method through extensive simulations. Finally, we conclude with a summary of our findings in Section 6.

2. Problem Formulation

In this paper, we consider a flexible link clamped to a rigid cart moving along the $x^0$-axis of the inertial reference coordinate system $C{S}^0$, which is the same model as in [7]. The aim is to drive the cart from point to point while suppressing the vibration of the flexible link as much as possible. The schematic diagram in Figure 1 represents the flexible system considered in this paper. In particular, the following idealizations and simplifications are considered. First, the flexible link is assumed to be uniform and symmetric, with a constant mass density $\rho$, a uniform cross-sectional area A, and a constant flexural rigidity $EI$. Second, since the cross-sectional area, A, of the flexible link is small relative to its length, l, we only consider the deflection of the link, $w(x,t)$, along the $x^1$-axis of the body coordinate system, $C{S}^1$. The origin of the body coordinate system, $C{S}^1$, is the attachment point between the flexible link and the cart, and the $x^1$-axis of $C{S}^1$ is the neutral axis of the undeflected beam. Third, the tip load fixed at the outermost end of the link is simplified as a point with mass $m_t$ and a moment of inertia $J_t$. The tip load undergoes vibration along with the link, with a displacement $w_t\left(t\right)$ and corresponding angle $w_t^{\prime }\left(t\right)$. The model of the flexible system can be described by the following PDEs:

$$\left\{\begin{array}{cc}\hfill \ddot{w}(x,t)+\beta {\displaystyle \frac{EI}{\rho A}}{w}^{″″}(x,t)+{\displaystyle \frac{EI}{\rho A}}{w}^{″″}(x,t)& =a\left(t\right)\hfill \\ \hfill {\ddot{w}}_t\left(t\right)-\beta {\displaystyle \frac{EI}{m_t}}{\dot{w}}_t^{‴}\left(t\right)-{\displaystyle \frac{EI}{m_t}}{w}_t^{‴}\left(t\right)& =a\left(t\right)\hfill \\ \hfill {\ddot{w}}_t^{\prime }\left(t\right)+\beta {\displaystyle \frac{EI}{J_t}}{\dot{w}}_t^{″}\left(t\right)+{\displaystyle \frac{EI}{J_t}}{w}_t^{″}\left(t\right)& =0\hfill \end{array}\right.$$

(1)

with the following boundary conditions:

$$\left\{\begin{array}{c}{w}_c\left(t\right)=0\hfill \\ w_c^{\prime }\left(t\right)=0\hfill \end{array}\right.$$

(2)

where $\beta$ is the damping factor and $a\left(t\right)$ is the acceleration of the cart. In this formulation, we define $\dot{(·)}={\displaystyle \frac{\partial }{\partial t}}$ as the derivative with respect to time t and ${(·)}^{\prime }={\displaystyle \frac{\partial }{\partial x}}$ as the derivative with respect to position x. The variable with the subscript c denotes $x=0$ (e.g., $w_c\left(t\right)=w(x=0,t)$), and the variable with the subscript t denotes $x=l$ (e.g., $w_t\left(t\right)=w(x=l,t)$). For model (1), we can define the linear stiffness operator ${\mathcal{K}}_f={\left[\begin{array}{ccc}{\displaystyle \frac{EI}{\rho A}}& -{\displaystyle \frac{EI}{m_t}}& {\displaystyle \frac{EI}{J_t}}\end{array}\right]}^T$ and the linear damping operator ${\mathcal{D}}_f=\beta {\mathcal{K}}_f$, which represent the effects of stiffness and damping on the system’s deflection, respectively. These operators ensure that the dynamic behavior of the flexible system is accurately captured.

Specifically, the motion-planning problem is to set the acceleration, $a\left(t\right)$, of the cart for a given translational motion task while suppressing the deflection of the link, $w(x,t)$, as much as possible. We will present a PINN-based input-shaping method to solve this problem in the next section.

3. PINN-Based Input-Shaping Method

The framework for the PINN-based input-shaping technique is illustrated in Figure 2. A neural network is constructed to approximate the vibration deflection, $w(x,t)$. An input shaper is used to generate the motion curve of acceleration, $a\left(t\right)$, through a set of trainable impulses. The network parameters and the impulses are trained according to a loss function based on the physical model of the system, along with the modal coefficients of the vibration. In this manner, optimal network parameters and impulses are sought to suppress the vibration while satisfying the physical constraints. For effective learning, a two-phase training strategy is utilized. The details of the method are introduced as follows.

3.1. PINN Structure

We use a PINN as a network solver for the motion-planning problem. The neural network used in this PINN solver is a fully connected neural network, $w_{\mathrm{NN}}(x,t;{\theta }_w)$, which takes the position x and the time t as inputs and returns $\widehat{w}(x,t)$ as an approximation for $w(x,t)$ as follows:

$$\begin{array}{cc}\hfill \mathrm{input}\mathrm{layer}:& {\mathcal{N}}^0=(x,t)\in {\Re }^2\hfill \\ \hfill \mathrm{hidden}\mathrm{layers}:& {\mathcal{N}}^m=\sigma (W^m{\mathcal{N}}^{m-1}+b^m)\in {\Re }^{N_m},\hfill \\ \hfill & \mathrm{for}1\le m\le M-1\hfill \\ \hfill \mathrm{output}\mathrm{layer}:& w_{\mathrm{NN}}=W^M{\mathcal{N}}^{M-1}+b^M\in \Re \hfill \end{array}$$

(3)

where ${\mathcal{N}}^0$ is the input to the network, ${\mathcal{N}}^m$ is the output of the m-th layer, and $w_{\mathrm{NN}}$ is the output of the network. $\sigma$ is a given nonlinear activation function. $W^m$ and $b^m$ are denoted as the weight matrix and bias vector in the m-th layer. These parameters are rearranged into a parameter vector ${\theta }_w$. The network input in this design is the concatenation of the spatial coordinate x and the temporal coordinate t, forming a two-dimensional vector. This vector is processed through the hidden layers and passed to the output layer to produce the final scalar output, $w_{\mathrm{NN}}$, which represents an estimation of the deflection $w(x,t)$.

The loss function, $\mathcal{L}$, of the neural network should account for the performance of vibration suppression while satisfying the physical model constraints. It is a weighted sum of two indices:

$$\mathcal{L}={\mathcal{L}}_{\mathrm{PINN}}+\lambda {\mathcal{L}}_{\mathrm{vs}}$$

(4)

where $\lambda$ is the weight term. The index ${\mathcal{L}}_{\mathrm{vs}}$ evaluates the performance of vibrational suppression. It is a functional of the deflection $w(x,t)$, which will be formulated in part C of this section. The index ${\mathcal{L}}_{\mathrm{PINN}}$ primarily incorporates the PDE model constraints. Specifically, it consists of the internal loss, ${\mathcal{L}}_{\mathrm{PDE}}$, and the boundary condition loss, ${\mathcal{L}}_{\mathrm{BC}}$.

$${\mathcal{L}}_{\mathrm{PINN}}={\mathcal{L}}_{\mathrm{PDE}}+{\mathcal{L}}_{\mathrm{BC}}.$$

(5)

${\mathcal{L}}_{\mathrm{PDE}}$ represents the sum of the residuals of the PDEs given in Equation (1), calculated using the approximated state $\widehat{w}(x,t)$ at the collocation points.

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{PDE}}& ={\displaystyle \frac{1}{N_r}}\sum _{i=1}^{N_r}\left|\right|R_1(x_i,t_i){\left|\right|}^2+{\displaystyle \frac{1}{N_t}}\sum _{i=1}^{N_t}\left(\left|\right|R_2(x_i,t_i){\left|\right|}^2+\left|\right|R_3(x_i,t_i){\left|\right|}^2\right),\hfill \end{array}$$

(6)

where

$$\left\{\begin{array}{c}{R}_1(x,t)=\ddot{\widehat{w}}(x,t)+\beta {\displaystyle \frac{EI}{\rho A}}{\dot{\widehat{w}}}^{″″}(x,t)+{\displaystyle \frac{EI}{\rho A}}{\widehat{w}}^{″″}(x,t)-a\left(t\right),\hfill \\ R_2(x,t)={\ddot{\widehat{w}}}_t\left(t\right)-\beta {\displaystyle \frac{EI}{m_t}}{\dot{\widehat{w}}}_t^{‴}\left(t\right)-{\displaystyle \frac{EI}{m_t}}{\widehat{w}}_t^{‴}\left(t\right)-a\left(t\right),\hfill \\ R_3(x,t)={\ddot{\widehat{w}}}_t^{\prime }\left(t\right)+\beta {\displaystyle \frac{EI}{J_t}}{\dot{\widehat{w}}}_t^{″}\left(t\right)+{\displaystyle \frac{EI}{J_t}}{\widehat{w}}_t^{″}\left(t\right).\hfill \end{array}\right.$$

(7)

The boundary condition loss also consists of two parts: the residuals from the two boundary restrictions given in Equation (2) at the collocation points.

$${\mathcal{L}}_{\mathrm{BC}}={\displaystyle \frac{1}{N_b}}\sum _{j=1}^{N_b}\left(B_1{(x_j,t_j)}^2+B_2{(x_j,t_j)}^2\right).$$

(8)

where

$$\left\{\begin{array}{c}{B}_1(x,t)={\widehat{w}}_c\left(t\right),\hfill \\ B_2(x,t)={\widehat{w}}_c^{\prime }\left(t\right).\hfill \end{array}\right.$$

(9)

The collocation points are uniformly distributed across the spatiotemporal domain $[0,l]\times [0,T]$, where T denotes the terminal time of the operation. During the training phase, $N_r$ and $N_t$ represent the number of collocation points used to evaluate the residuals of the PDEs, while $N_b$ denotes the number of collocation points applied to enforce the boundary conditions. The sets ${\{x_i,t_i\}}_{i=1}^{N_r}$, ${\{x_i,t_i\}}_{i=1}^{N_t}$, and ${\{x_j,t_j\}}_{i=1}^{N_b}$ represent the training points for the PDEs given in Equation (1) and the boundary conditions given in Equation (2), respectively.

In the PINN solver, the network parameter ${\theta }_w$ and the acceleration $a\left(t\right)$ are the unknown variables that must be learned during the training phase by minimizing the loss function $\mathcal{L}$.

$$({\theta }_w^{*},a^{*}\left(t\right))=\underset{{\theta }_w,a\left(t\right)}{argmin}\mathcal{L}.$$

(10)

However, this is not easily solved because the unknown acceleration, $a\left(t\right)$, is a continuous function of time. The conventional gradient descent method cannot be directly applied to find the minimum in this infinite-dimensional setting. To enable feasible optimization, we introduce the input-shaping technique in the next part to simplify the problem.

3.2. Input-Shaping Technique

The main idea behind the input-shaping technique is to plan an object’s motion by fine-tuning a given motion using a sequence of command impulses. In the input-shaping technique, the motion typically does not need to be pre-specified. The input impulse can be designed independently of a specific motion. However, this approach is only effective for simple motion tasks. When the motion function becomes complex, especially with high nonlinearity, the system dynamics can be significantly affected. In such cases, pre-specifying the motion and incorporating it into the design process is a better strategy to achieve optimal performance. Thus, in our design, we pre-specified the motion. In our flexible system, the cart’s acceleration, $a\left(t\right)$, is determined by convolving a predefined acceleration profile, $a_0\left(t\right)$, with a command signal, $u\left(t\right)$.

$$a\left(t\right)=(u\ast a_0)\left(t\right)={\int }_{-\infty }^{\infty }u\left(\tau \right)a_0(t-\tau )d\tau ,$$

(11)

The command signal, $u\left(t\right)$, is characterized by a sequence of impulses, $\mathit{u}=\left[\mathit{A},\mathit{\tau }\right]$, where $\mathit{A}=(A_1,A_2,\dots ,A_n)$ represents the vector of specific impulse amplitudes, and $\mathit{\tau }=({\tau }_1,{\tau }_2,\dots ,{\tau }_n)$ represents the vector of the delay intervals corresponding to these impulses. The sequence of impulses, $\mathit{u}$, is chosen to ensure that the cumulative residual vibrations from each impulse cancel each other out, resulting in a reference motion that deviates slightly from the initial motion but does not generate residual vibrations in the system.

In conventional input command-shaping techniques, calculating the command sequence, $\mathit{u}$, requires knowledge of the system’s natural frequency, ${\omega }_n$, and damping ratio, $\zeta$. For instance, common input shapers, such as the zero vibration (ZV) shaper and the zero vibration derivative (ZVD) shaper, may use the frequency of the system’s first mode as ${\omega }_n$ to calculate the impulse sequence as follows:

3.2.1. ZV Shaper

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{A}_1\hfill & ={\displaystyle \frac{1}{1+K}},{\tau }_1=0,\hfill \\ A_2\hfill & ={\displaystyle \frac{K}{1+K}},{\tau }_2=T\hfill \end{array}\right.\end{array}$$

(12)

3.2.2. ZVD Shaper

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{A}_1\hfill & ={\displaystyle \frac{1}{1+2K+K^2}},{\tau }_1=0,\hfill \\ A_2\hfill & ={\displaystyle \frac{2K}{1+2K+K^2}},{\tau }_2=T,\hfill \\ A_3\hfill & ={\displaystyle \frac{K^2}{1+2K+K^2}},{\tau }_3=2T\hfill \end{array}\right.\end{array}$$

(13)

where $K=e^{{\displaystyle \frac{\pi }{\sqrt{1-{\zeta }^2}}}}$ and $T={\displaystyle \frac{\pi }{{\omega }_n\sqrt{1-{\zeta }^2}}}$.

However, in the case of multimode systems, deriving a general analytical solution becomes challenging due to the need to satisfy multiple constraint conditions [14].

The computational burden can be transferred to the PINN framework proposed above. The shaped acceleration, $a\left(t\right)$, defined by an unknown impulse command, $\mathit{u}$, is fed into the network. The unknown impulse command, $\mathit{u}$, is modeled as a set of learnable parameters within the PINN framework. Since it is a finite-dimensional vector, it is possible to search for its optimum using the conventional gradient descent method.

3.3. Modal-Based Loss Function

The design of the loss functional, $\mathcal{L}$, based on the deflection, $w(x,t)$, can significantly affect both the speed and the outcome of network training. Especially for the vibration deflection, $w(x,t)$, designing the loss functional becomes more challenging as it is a continuous function defined over the spatiotemporal domain. To design a functional that directly extracts the information from a continuous function, the absolute or quadratic regulation area of the function can be used. Also, the maximal amplitude of the function could also be used. However, to design a functional that better supports network learning, we utilize modal information as the foundation for its construction. The complexity is reduced by using modal analysis, a method commonly employed to study the dynamic behavior of vibrations. The modal analysis method decomposes the vibration response into a series of modes, leveraging their orthogonality properties to simplify complex vibration problems into manageable components. Then, by considering a sufficient number of these components, we can extract the primary dynamical characteristics of the vibration. Since this modal information represents the dominant dynamics of the system, it guides the network to learn within the system’s dominant subspace.

For the flexible system under consideration, the deflection state, $w(x,t)$, can be expressed as a linear combination of mode shapes using a modal basis.

$$w(x,t)=\sum _{i=1}^{\infty }{W}_i\left(x\right)q_i\left(t\right)$$

(14)

where $W_i\left(x\right)$ is the spatial basis function and $q_i\left(t\right)$ is the projection of the state $w(x,t)$ onto the corresponding basis function.

In the modal analysis method, the spatial basis function, $W_i\left(x\right)$, is called the mode shape and is chosen to be the eigenvector of the system’s stiffness operator. It can be obtained by solving the following eigenvalue problem associated with the system’s PDEs:

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{EI}{\rho A}}{W}_i^{″″}\left(x\right)& ={\lambda }_i{W}_i\left(x\right)\hfill \\ \hfill -{\displaystyle \frac{EI}{m_t}}{W}_i^{‴}\left(l\right)& ={\lambda }_i{W}_i\left(l\right)\hfill \\ \hfill {\displaystyle \frac{EI}{J_t}}{W}_i^{″}\left(l\right)& ={\lambda }_i{W}_i^{\prime }\left(l\right)\hfill \end{array}\right.$$

(15)

with the following boundary conditions:

$$\left\{\begin{array}{c}{W}_i\left(0\right)=0\hfill \\ W_i^{\prime }\left(0\right)=0\hfill \end{array}\right.$$

(16)

where ${\lambda }_i$ is the eigenvalue. The general solution for the mode shape, $W_i\left(x\right)$, is given by

$$W_i\left(x\right)=Acos{\kappa }_{i}x+Bsin{\kappa }_{i}x+Ccosh{\kappa }_{i}x+Dsinh{\kappa }_{i}x$$

(17)

where ${\kappa }_i^4={\displaystyle \frac{\rho A}{EI}}{\lambda }_i$, and A, B, C, and D are the constants to be determined.

To solve for the eigenvalues ${\lambda }_i$, the general solution is substituted into the eigenvalue problem given in Equations (15) and (16). This results in a nonlinear characteristic equation, the roots of which are the eigenvalues. The numerical solutions of the preceding eigenvalues, ${\lambda }_i$, for our system are given by

$$\begin{array}{cc}\hfill {\lambda }_i=& 0.9641,2.4175,4.0507,5.6771,\dots (i=1,2,3,4,\dots )\hfill \end{array}$$

(18)

The mode shape, $W_i\left(x\right)$, can be calculated accordingly, as illustrated in Figure 3.

Each mode’s contribution to the overall response is weighted by the corresponding projection coefficient, $q_i\left(t\right)$, whose dynamics are given by

$${\ddot{q}}_i\left(t\right)+\beta {\omega }_i^2{\dot{q}}_i\left(t\right)+{\omega }_i^2{q}_i\left(t\right)=0$$

(19)

where ${\omega }_i=\sqrt{{\lambda }_i}$ is the natural frequency of the i-th mode. The projection coefficient, $q_i\left(t\right)$, on the mode with a larger eigenvalue decays more quickly. After an extremely short time, only the $q_i\left(t\right)$ corresponding to modes with smaller eigenvalues will remain, representing the dominant dynamics of the system. Thus, we can truncate the expansion of the state, $w(x,t)$, into its first few modes.

$$w(x,t)=\sum _{i=1}^{N_{\mathrm{modal}}}{W}_i\left(x\right)q_i\left(t\right)$$

(20)

where $N_{\mathrm{modal}}$ is the number of modes chosen to be sufficient for the approximation. The main information about the system dynamics is now encapsulated in the $q_i\left(t\right)$ of the first few modes. The projection coefficient, $q_i\left(t\right)$, is the projection of the response, $w(x,t)$, onto the modal shape function, $W_i\left(x\right)$, which can be approximately calculated using a sufficiently large number of discrete points of the estimated state, $\widehat{w}(x,t)$.

$${\widehat{q}}_i\left(t\right)=\sum _{j=1}^Z\widehat{w}(x_j,t)·W_i\left(x_j\right).$$

(21)

We discretize the spatial coordinate x into discrete positions $x_j$ with an equally spaced partition in the above equation, where Z denotes the number of discrete points.

Now, we can design the loss function, ${\mathcal{L}}_{vs}$, based on the dominant vibration modal information, ${\widehat{q}}_i\left(t\right)$. In order to achieve maximal vibration suppression, we require that the modal coefficient, ${\widehat{q}}_i\left(t\right)$, be as close to zero as possible. If only one mode is considered, the loss function can be designed as follows:

$${\mathcal{L}}_{{\mathrm{modal}}_i}=\sum _{k=1}^{N_T}{\widehat{q}}_i{\left(t_k\right)}^2,$$

(22)

where we sample the modal coefficient error at discrete times, with $N_T$ representing the number of sample times. The loss function can easily be extended to consider more modes through a weighted sum of different modal losses.

$${\mathcal{L}}_{\mathrm{modal}}=\sum _{i=1}^{N_{\mathrm{modal}}}{\gamma }_i{\mathcal{L}}_{{\mathrm{modal}}_i},$$

(23)

where ${\gamma }_i$ is the scalar weight of the i-th mode.

This extension allows us to directly solve for the input-shaping command in a multimode scenario if we use it as a functional of the deflection state, $w(x,t)$, to evaluate the vibration level of the system, where ${\mathcal{L}}_{\mathrm{vs}}={\mathcal{L}}_{\mathrm{modal}}$. As it represents the dominant vibration modal information about the flexible dynamics, it can guide the network training toward a dominant coefficient space where the flexible dynamics can be accurately approximated and the motion can be appropriately determined.

In this multimode scenario with $N_{\mathrm{modal}}$ modes, the number of impulses in the input-shaping command, $\mathit{u}=\left[\mathit{A},\mathit{\tau }\right]$, can be selected as $2N_{\mathrm{modal}}+1$ [14].

3.4. Two-Phase Training

In the proposed PINN solver, there are two classes of trainable parameters: the network parameters, ${\theta }_w$, and the impulse command, $\mathit{u}$. The network parameters, ${\theta }_w$, are primarily trained to satisfy the governing PDE constraints, while the impulse command, $\mathit{u}$, is designed for vibration suppression. It is difficult for them to be trained simultaneously for a composition with two targets. For effective training, we propose a two-phase strategy to decompose the parameter learning process into a coarse-tuning phase and a fine-tuning phase for the composite target.

1.

Coarse-Tuning Phase: Network Parameter Selection

In the coarse-tuning phase, we focus on determining the network parameters for the flexible system represented by PDEs. During this phase, only the physical model constraints are considered.

$${\theta }_w^{*\prime }=\underset{{\theta }_w}{argmin}{\mathcal{L}}_{\mathrm{PINN}}$$

(24)

In this phase, the impulse command, $\mathit{u}$, is given and fixed. Starting from the randomly initialized network parameters, ${\theta }_w^0$, we employ gradient-based optimization to find an optimal set of values, ${\theta }_w^{*\prime }$, that minimize the system model loss function, ${\mathcal{L}}_{\mathrm{PINN}}$. During each iteration k, the parameters are updated as follows:

$${\theta }_w^{k+1}={\theta }_w^k-{\eta }^{\left(k\right)}{\nabla }_{{\theta }_w}{\mathcal{L}}_{\mathrm{PINN}}\left({\theta }_w^k\right),$$

(25)

To enhance training efficiency and convergence, a StepLR learning rate scheduler is employed. This scheduler decays the learning rate by a factor of $\gamma$ every s epochs, following the rule

$${\eta }^{\left(k\right)}={\eta }_0·{\gamma }^{\lfloor {\displaystyle \frac{k}{s}}\rfloor },$$

(26)

where ${\eta }_0$ is the initial learning rate and s denotes the number of epochs after which the learning rate is decayed by a factor of $\gamma$.

These learned network parameters are crucial for effectively addressing the dynamics of the PDEs and serve as a foundation for subsequent vibration suppression optimization.

2.

Fine-Tuning Phase: Learning Optimal Impulse Command

In the second phase, we utilize the established network structure, $w_{\mathrm{NN}}$, with the learned network parameters ${\theta }_w^{*\prime }$ and enhance it by introducing input-shaping commands, $\mathit{u}=[\mathit{A},\mathit{\tau }]$, as trainable parameters within the neural network. This phase involves minimizing a composite loss function, $\mathcal{L}$, which consists of both the system model residuals and additional vibration modal residuals.

$$({\theta }_w^{*},{\mathit{u}}^{*})=\underset{{\theta }_w,\mathit{u}}{argmin}\mathcal{L}$$

(27)

The weights, $\lambda$, in the compound loss function, $\mathcal{L}$, are adjusted to prioritize the vibration suppression performance. The exact values of these weights are chosen based on observations from initial training runs, aiming to minimize residual vibrations as effectively as possible while maintaining model accuracy (i.e., ensuring the model constraints are satisfied). In this fine-tuning phase, the network parameters, ${\theta }_w$, are initialized to the optimal values, ${\theta }_w^{*\prime }$, obtained in the first phase, and the impulse command, $\mathit{u}$, is initialized to the values used in the first phase. They are updated simultaneously according to the following rule:

$$\begin{array}{cc}\hfill {\theta }_w^{k+1}& ={\theta }_w^k-{\eta }^{\left(k\right)}{\nabla }_{{\theta }_w}\mathcal{L}({\theta }_w^k,{\mathit{u}}^k),\hfill \\ \hfill {\mathit{u}}^{k+1}& ={\mathit{u}}^k-{\eta }^{\left(k\right)}{\nabla }_{\mathit{u}}\mathcal{L}({\theta }_w^k,{\mathit{u}}^k)\hfill \end{array}$$

(28)

The learning rate, ${\eta }^{\left(k\right)}$, decays according to the same rule as described in Equation (26).

Ultimately, the training yields an optimized control solution, ${\mathit{u}}^{*}$, and a corresponding improved system response, $w_{\mathrm{NN}}$, with network parameters ${\theta }_w^{*}$, thus achieving an effective balance between accurately modeling system dynamics and mitigating undesirable vibrations.

4. Feed-Forward Control

The proposed PINN solver provides a planned motion $a\left(t\right)$, which can be used as a reference trajectory for the controller. It also offers an estimated state to facilitate control design.

In this flexible link system, the cart with the drive unit is modeled as a rigid body with mass $m_c$, capable of translational motion along the $x^0$-axis of the inertial reference coordinate system $C{S}^0$. The force, $F_c\left(t\right)$, acting on the cart in the direction of the $x^0$-axis of $C{S}^0$ serves as the control input. Müller et al. [7] proposed a feed-forward control algorithm to calculate the force, $F_c\left(t\right)$, acting on the cart as follows:

$$F_c\left(t\right)=m_{c}a\left(t\right)-EI{w}_c^{‴}\left(t\right)-\beta EI{\dot{w}}_c^{‴}\left(t\right)$$

(29)

This feed-forward control algorithm requires knowledge of the cart’s acceleration, $a\left(t\right)$, the beam deflection, $\widehat{w}(x,t)$, and its derivatives. In conventional methods, the acceleration is a pre-prescribed reference trajectory, while the deflection and its derivatives are estimated using the flexible system model. Finite-difference or finite-element methods are commonly employed for these estimations.

An advantage of the proposed PINN method is that the knowledge required for the controller can be directly obtained from the network’s learning results. By leveraging the input-shaping commands, $\mathit{u}$, learned during the second phase of neural network training, the cart’s acceleration, $a\left(t\right)$, can be computed using convolution methods, as described in Equation (11). Additionally, the output of the neural network, $w_{\mathrm{NN}}$, is an appropriate estimation of the deflection. Consequently, ${\dot{w}}_c^{‴}\left(t\right)$ and $w_c^{‴}\left(t\right)$ can be directly computed from the network’s output, $w_{\mathrm{NN}}$. Compared to conventional methods, our approach uses the PINN results to directly compute the control signal. This simplifies the process by leveraging the neural network’s ability to learn the underlying physics and dynamics, thereby reducing the reliance on state estimation through complex system discretization methods.

5. Simulation Results

In this simulation, we evaluate the vibration suppression performance of the proposed PINN method when the flexible system is in motion. We first compare the proposed PINN method, considering a single mode, with the traditional ZV and ZVD methods to demonstrate its efficiency. Next, the proposed PINN method is extended to incorporate multiple modes, and the effects of the multimode approach are analyzed. Finally, the PINN-based feed-forward control design is tested through experiments.

In our simulation experiments, the parameters for the flexible beam are given as follows: $l=1.9\mathrm{m}$, $\rho A=0.86\mathrm{kg}/\mathrm{m}$, $EI=10\mathrm{Nm}$, $m_t=0.04\mathrm{kg}$, $J_t=1.5·{10}^{-5}\mathrm{kg}·{\mathrm{m}}^2$, $\beta =5.38·{10}^{-4}$, and $m_c=8.64\mathrm{kg}$. The simulation model and these system parameters are based on real-world applications and experimental data [7]. The cart moves along the $x^0$-axis of the inertial reference coordinate system, $C{S}^0$, with a pre-planned acceleration, $a_0$, given as follows:

$$a_0\left(t\right)=\left\{\begin{array}{cc}0{\mathrm{m}/\mathrm{s}}^2,\hfill & t\in \left\{\right[0,2)\cup [4,5)\cup [7,10\left]\right\}\mathrm{s},\hfill \\ 2{\mathrm{m}/\mathrm{s}}^2,\hfill & t\in \left\{\right[2,3)\cup [6,7\left)\right\}\mathrm{s},\hfill \\ -2{\mathrm{m}/\mathrm{s}}^2,\hfill & t\in \left\{\right[3,4)\cup [5,6\left)\right\}\mathrm{s}\hfill \end{array}\right.$$

(30)

The cart moves forward to the terminal with an acceleration phase of $2{\mathrm{m}/\mathrm{s}}^2$ and a deceleration phase of $-2{\mathrm{m}/\mathrm{s}}^2$ and then returns to the origin in a similar manner. According to this setting, the cart returns to the origin at time $t=7\mathrm{s}$.

We propose two indices to evaluate the vibration suppression performance of the flexible system. The first is the maximum residual vibration, defined as the maximum absolute value of the vibrations at the top of the flexible link after the cart returns to the origin.

$${\mathcal{R}}_{max}=\underset{t\in \left(t_{\mathrm{end}},T\right),x\in l}{max}\left|w(x,t)\right|,$$

(31)

where $t_{\mathrm{end}}$ denotes the time at which the cart returns to the origin, and $T=10\mathrm{s}$ is the final time considered for the evaluation. The second index is the total residual vibration, calculated as the sum of the absolute values of the vibrations along the entire beam after the cart returns to the origin.

$${\mathcal{R}}_{\mathrm{total}}={\int }_{t_{\mathrm{end}}}^T{\int }_0^l\left|w(x,t)\right|dxdt.$$

(32)

The PINN design in the experiment employs a network with five hidden layers, each containing 50 neurons, and an output layer with a single neuron. The hyperbolic tangent (tanh) activation function is used in the hidden layers. Network parameters are updated using the two-phase training scheme, following the proposed update rule, with an initial learning rate of ${\eta }_0={10}^{-3}$.

5.1. Input Shaping on Single Mode

In the first part of the simulation, only the first vibration mode is considered for the input-shaping design. A comparison is conducted between the conventional ZV and ZVD methods and the proposed PINN method.

In the design of the proposed PINN method, the modal loss function with the first mode, ${\mathcal{L}}_{{\mathrm{modal}}_1}$, is constructed as shown in Equation (22) using the network output. A specific number of discrete spatiotemporal points, $(x_j,t_k)$, are sampled for its estimation. The x-sampling points are equidistant, with $Z=100$ points distributed over the range $x\in [0,1.9]\mathrm{m}$. The t-sampling points are randomly selected, with $N_T=20$ points within the interval $t\in [7,10]\mathrm{s}$, to capture the characteristics of the first mode when the cart returns to the origin. The input-shaping impulses for the first mode, using the conventional ZV and ZVD shapers (involving two and three pulses, respectively), are calculated using Equations (12) and (13). The solutions of different input-shaping methods are illustrated in Figure 4, where the impulses, the shaped acceleration, $a\left(t\right)$, and the corresponding displacement of the cart for the ZV shaper, ZVD shaper, and proposed PINN method are presented, respectively. The spatiotemporal vibration results, $w(x,t)$, for different motions are shown in Figure 5. For a better comparison, the tip vibrations of the flexible robotic arm over time for different motions are plotted on the same graph.

Compared to the unshaped motion, all three input shapers demonstrate the ability to suppress vibrations to some extent. The overall performance is evaluated using the maximum residual vibration index, ${\mathcal{R}}_{\max }$, and the total residual vibration index, ${\mathcal{R}}_{\mathrm{total}}$, with their values presented in

Table 1. Comparison of maximum and total residual vibration for ZV, ZVD, and PINN solutions.

Solution Type	Max Residual Vibration (m)	Total Residual Vibration (${\mathbf{m}}^2·\mathbf{s}$)
No Input Shaping	0.1116	1140.9643
ZV	0.0948	669.7387
ZVD	0.0823	537.7970
PINN Solution (First Mode)	0.0027	17.9913

. Although all the shapers demonstrated vibration reduction effects, the proposed PINN shaper achieved the best performance in both indices.

It is worth noting that if we calculate the modal loss function of the first mode for all the shapers, as shown in

Table 2. Modal loss, ${\mathcal{L}}_{\mathrm{modal}}$, of the first mode for ZV, ZVD, and PINN solutions.

Solution Type	Modal Loss of the First Mode
ZV	23.671
ZVD	23.374
PINN Solution (First Mode)	23.672

, their values are approximately equal. This indicates that all the shapers designed based on the first mode of vibration are equally effective in reducing this mode. However, while the conventional ZV and ZVD methods focus on the first mode and neglect other modes, the proposed PINN method is prone to finding an overall vibration suppression result with minimal first-mode vibration. Thus, in the overall performance evaluation, the proposed PINN method outperforms the conventional methods, even when all of them are based on the first mode.

5.2. Input Shaping on Multiple Modes

The overall performance of the proposed PINN method can be further enhanced by incorporating additional vibration modes into the loss function. We tested the method with multiple modes, ranging from $N_{\mathrm{modal}}=1$ to $N_{\mathrm{modal}}=10$, across a total of 10 sessions. The number of impulses was $3,5,7,\dots ,21$, respectively, for sessions with $N_{\mathrm{modal}}=1,2,3,\dots ,10$. For the modal loss function of multiple modes, the weight, ${\gamma }_i$, was set to 1, meaning no additional weight factor was applied. For each specific mode i, the modal loss function, ${\mathcal{L}}_{{\mathrm{modal}}_i}$, was sampled in the same manner as in the single-mode case. The trained impulses in the sessions with $N_{\mathrm{modal}}=1,2,5,9$, along with the corresponding shaped accelerations and displacements of the cart, are shown in Figure 6.

Figure 7 shows their spatiotemporal vibration results and, in particular, the vibrations at the tip. The performances are evaluated using the residual vibration indices presented in

Table 3. Maximum and total residual vibration for different numbers of modes.

Mode Number	Max Residual Vibration (m)	Total Residual Vibration (${\mathbf{m}}^2·\mathbf{s}$)
1	0.0027	17.9913
2	0.0024	13.1539
3	0.0021	12.2409
4	0.0019	11.2738
5	0.0017	10.5147
6	0.0016	10.1308
7	0.0012	7.9086
8	0.0008	3.9130
9	0.0008	4.8193
10	0.0008	4.4773

. As the number of modes considered increases, along with the corresponding quantity of impulses, the proposed PINN method is capable of planning smoother motions to decrease the overall residual vibration. It achieves a minimum–maximum residual vibration of approximately $0.0008\mathrm{m}$ after the eighth session, beyond which further reduction is not observed. The downward trends of both vibration indices are clearly visible in Figure 8 and Figure 9.

In the case of multiple modes, calculating the pulse using traditional methods becomes significantly more complex. As highlighted in [14], calculating the input-shaping parameters for multimode systems requires solving an increasing number of equations to satisfy the constraints for all modes. This complexity grows exponentially with the number of modes, making it impractical for implementation in high-order systems. Due to these computational challenges, we did not perform a direct comparison between the traditional methods and the proposed PINN-based input shaper under multimode conditions. However, these challenges with traditional methods highlight the advantage of the proposed PINN approach, which remains both effective and computationally feasible in multimode scenarios.

Overall, these results confirm the superior performance of the PINN-based input-shaping technique in reducing residual vibrations both at the tip and along the entire link. While the conventional ZV and ZVD designs are limited to using a single mode, the proposed PINN method can readily incorporate multiple modes to achieve substantial vibration suppression. This demonstrates the flexibility and efficiency of the PINN framework in addressing input-shaping problems for practical applications.

5.3. Feed-Forward Controller Design

The PINN results can be directly applied to the feed-forward controller design given in Equation (29). We used the optimal results of the PINN method with $N_{\mathrm{modal}}=10$ for the control algorithm in this experiment. The acceleration, $a\left(t\right)$, shaped by the trained impulses was used as the reference motion. The partial derivatives of the vibration, $w_c^{\prime \prime \prime }\left(t\right)$ and ${\dot{w}}_c^{\prime \prime \prime }\left(t\right)$, were calculated from the network output, $w_{\mathrm{NN}}$. Figure 10 illustrates the calculated $F_c\left(t\right)$, which shows the evolution of the control force applied to the cart.

The resulting vibration of the flexible system is shown in Figure 11. It is observed that both the spatiotemporal vibration and the tip vibration are significantly suppressed after the cart completes its movement and returns to the origin. The PINN-based input shaper can be used alongside other controllers, taking on the role of a reference trajectory planner and state estimator, which will be addressed in future work.

6. Conclusions

In this study, we applied a physics-informed neural network (PINN) to solve for the optimal impulses in the input shaper, achieving effective vibration suppression in a flexible single-link beam. By constructing the loss function based on the physical constraints of the system and the modal information of the vibration deflection, the neural network was able to search for the optimal impulse sequence, which was then convolved with the pre-set accelerator to produce planned motion for the cart. With the powerful computational resources of the neural network and the flexibility to extend the loss function to account for multiple modes in our proposed method, the PINN-based input-shaping approach demonstrates significant vibration suppression. In our simulation experiments of the flexible single-link robot system with a complex motion tasks, the proposed PINN method outperformed the conventional ZV and ZVD methods in both single-mode and multimode scenarios. We also discussed the potential of utilizing the proposed PINN-based input shaper as a motion planner and state estimator for control design. The efficiency and applicability of the proposed method make it a promising approach for vibration suppression in flexible systems with complex operational tasks.