A Pareto-Optimal-Based Fractional-Order Admittance Control Method for Robot Precision Polishing

Abstract

Traditional integer-order admittance control is widely used in industrial scenarios requiring force control, but integer-order models often struggle to accurately depict fractional-order-controlled objects, leading to precision bottlenecks in the field of precision machining. For robotic precision polishing scenarios, to enhance the stability of the control process, we propose a more physically accurate five-parameter fractional-order admittance control model. To reduce contact impact, we introduce a method combining the rear fastest tracking differential with fractional-order admittance control. The optimal parameter identification for the fractional-order system is completed through Pareto optimality and a time–frequency domain fusion analysis of the control system. We completed the optimal parameter identification in a simulation, which is applied to the robotic precision polishing scenario. This method significantly enhanced the force control precision, reducing the error margin from 15% to 5%.

1. Introduction

Robotic blade polishing [1] represents a particularly challenging task within the precision machining sector. Industrial robots, known for their high processing efficiency, cost-effectiveness, and expansive operational range, are increasingly poised to replace traditional CNC machine tools [2]. However, limitations in robot stiffness and precision complicate their application in precise control tasks. Techniques aimed at optimising stiffness [3] and speed [4] have proven effective in enhancing the stability of robotic movements. Given that contact force directly influences machining quality, this article focuses extensively on the exploration and enhancement of force control algorithms.

Impedance control [5], admittance control [6], and force–position hybrid control [7] are the most commonly used force control algorithms, delivering effective results across many domains requiring force control. However, due to inherent limitations in these control methods, they often struggle to maintain high precision when dealing with nonlinear control subjects and unstructured environments. Consequently, numerous studies have emerged that integrate these traditional methods with adaptive control [8], robust control [9], and artificial intelligence algorithms [10] to enhance performance and accuracy. With advancements in fractional calculus [11], fractional-order control [12,13] has emerged as an enhanced methodology, notable for its precision in depicting the actual physical properties of controlled objects.

Recent studies on fractional-order control have demonstrated its significant potential across various fields. For instance, in sensor technology [14], fractional-order filters and sensors show great promise, particularly in modern industrial sensing systems. In robotics, fractional-order control is primarily employed for modelling and control [15], with substantial prospects in specialised scenarios such as underwater exploration and radioactive environments. Additionally, it has applications in biomedicine [16], enhanced intelligent systems [17], and other domains. Furthermore, the inherent memory characteristic of fractional-order models significantly improves accuracy in control systems that involve friction. In recent years, the control field has witnessed significant research contributions. Notably, Aydin et al. [18] have employed fractional-order admittance controllers in physical human–machine interaction systems, with experiments demonstrating their superior robustness and transparency. Chen et al. [19] developed a two-degree-of-freedom fractional-order proportional-derivative (FOPD) controller, aligned with the separation principle, and demonstrated superior velocity tracking compared to traditional integer-order PID controllers. Li et al. [20] utilised fractional calculus to refine the admittance model, thereby more precisely characterising robot–environment interactions. Li et al. [21] developed a methodology to determine the Lyapunov exponent spectrum of fractional-order systems, validated to surpass existing techniques in accuracy and correctness. Zheng et al. [22] devised a fractional-order extended state observer (FOESO) to estimate multi-source disturbances and uncertainties, utilising a fractional-order-switching law to enable the rapid convergence of system states. Saif et al. [23] introduced a fractional-order quadrotor system, where the associated fractional-order sliding mode control (FOSMC) demonstrated superior control performance, robustness, and precision compared to traditional integer-order models. Ma et al. [24] developed a fractional-order finite-time disturbance observer to approximate and compensate for model uncertainties and unknown external disturbances, with its effectiveness confirmed through simulation experiments on unmanned surface vehicles.

It is apparent that fractional-order control significantly enhances control systems, yet it also presents considerable challenges. The complexity due to an abundance of control parameters, which complicates the process of parameter identification, is a primary concern. As a result, there is extensive research dedicated to optimising the identification of fractional-order control parameters. Chen et al. [25] modified the inertia, damping, and stiffness elements in the impedance model to fractional orders, providing a stepwise tuning criterion for fourteen parameters. Ma et al. [26] developed a fourth-order fractional sliding mode control rate, utilising fuzzy approximation to optimally tune the control parameters. Aydin et al. [27] developed a multi-objective optimisation framework for fractional-order admittance controllers, employing Pareto optimality for comprehensive joint optimisation. This framework facilitates fair comparisons across various interaction controllers. Chen et al. [28] introduced an innovative frequency domain analysis method tailored to their disturbance-rejection fractional-order controller, completing the parameter tuning process and elucidating the effects of disturbances and noise on controller performance. Ding et al. [29,30] utilised fractional-order systems to model damping forces, and proposed a frequency-based parameter-tuning method that demonstrated optimal time domain indices for step responses. To tackle the complexities of tuning fractional-order PID parameters, Chen et al. [31] initially proposed a holistic analytical design method based on frequency-specification-defined loop shaping, applying it effectively to motor speed control.

Given the inherent limitations of traditional control models, accurately characterising the dynamic behaviour of the controlled object is challenging. As a result, these models fail to meet the stringent requirements of precision force control and are insufficient for practical application in precision polishing scenarios. To address this issue and enhance polishing quality, this study employed the fractional-order admittance control (FOAC) method to achieve precise control on an independently developed polishing machine. To minimise the impact during tool entry and enhance machining quality, tracking differentiators from disturbance rejection control technology are amalgamated with fractional-order admittance control. The resultant control model, which incorporates five hyperparameters, meticulously evaluates both the time domain and frequency domain characteristics of the system. A tailored objective equation was developed, and optimal fractional-order control parameters were identified through Pareto optimality. Simulation and experimental outcomes affirmed the method’s superior control stability.

The rest of this paper is organised as follows: Section 2 introduces the self-developed robotic polishing system and its integration with the admittance control strategy. Section 3 outlines the methodology, beginning with the design and analysis of the fractional-order admittance controller. It then addresses the issue of zero overshoot by integrating the rear fastest tracking differential (Rear-FTD) with the fractional-order admittance controller (FOAC), concluding with the identification of optimal controller parameters. Section 4 presents the simulation analysis and experimental results. Section 5 concludes the article.

2. Problem Formulation and Preliminaries

2.1. Description of the Polishing System

This paper addresses high-precision robotic polishing tasks, focusing on a robotic polishing system as the controlled subject. The system consists of a six-degree-of-freedom industrial robot and a self-developed floating polisher, as shown in Figure 1.

The system consists of an actuation system and a feed system. The polishing head is used to complete the machining process in contact with the workpiece. The transducer is a six-dimensional force sensor, and the data obtained are the input of the admittance control, which acts on the actuation system. Traditional force control algorithms require the real-time adjustment of six-degree-of-freedom parameters [x,y,z,a,b,c], and the nonlinearity and high coupling of the robot’s six joints pose significant control challenges and reduce precision. This paper decouples the control of the robot and the floating polisher. The robot manages offline motion, ensuring alignment between the machining surface coordinate system and the grinding point coordinate system. The floating polisher handles online force control, using the admittance control strategy to adjust the contact force in real time. Given that the feed system is the controlled object and the interaction environment involves a six-degree-of-freedom robot holding the workpiece, the environmental dynamics are first established using the Lagrangian method, resulting in the following expression. The Lagrangian method is an energy-based approach to dynamic calculations. Unlike the Newton–Euler method, which relies on force balance and requires detailed mechanical analysis, the Lagrangian method simplifies the process by focusing on energy. It calculates the kinetic and potential energy of the manipulator and uses these to establish the Lagrangian equation L.

$$\left\{\begin{array}{l}{M}_{ij}={\displaystyle \sum _{m=max(j,k)}^6}Trace({\displaystyle \frac{\partial {}_0{}^{m}L}{\partial q_j}}·I_m·{({\displaystyle \frac{\partial {}_0{}^{m}L}{\partial q_k}})}^T)\\ C_{ijk}={\displaystyle \frac{1}{2}}({\displaystyle \frac{{\partial M}_{ij}}{\partial q_k}}+{\displaystyle \frac{{\partial M}_{ik}}{\partial q_j}}+{\displaystyle \frac{{\partial M}_{jk}}{\partial q_i}})\\ G_i=-{\displaystyle \sum _{j=i}^6}{m}_j·g·{\displaystyle \frac{\partial {}_0{}^{j}L}{\partial q_i}}·{}^{j}r_{cj}\end{array}\right.$$

(1)

Here, the MCG matrix represents the robot’s dynamic parameters in the joint space. L is the Lagrangian function, $I_m$ is the inertia matrix of the robot model which can be derived from the model, q represents the joint positions of the robot, $m_j$ denotes the mass of the j link, and ${}^{j}r_{cj}$ represents the vector position of the centre of the $j$ link. The Jacobian matrix $J\left(q\right)$ is introduced to map the dynamic equations from the joint space to the Cartesian space.

$$F=M_{rob}·\ddot{\chi }+C_{rob}·\dot{\chi }+G_{rob}·\chi$$

(2)

The $[M_{rob},C_{rob},G_{rob}]$ matrix comprises the robot’s n × n mass matrix, n × n centrifugal force matrix, and n × 1 gravity vector. F denotes the n × 1 force–torque vector acting on the robot’s end effector. $\chi$ is an n × 1 Cartesian vector that accurately represents the position and orientation of the end effector. n stands for the number of joints.

Based on the transformation between the joint space and the Cartesian space, and the relationship between joint space velocity $\dot{q}$ and Cartesian space velocity $\dot{\chi }$, the expressions for the robot’s dynamic parameters $M_x$, $C_x$, and $G_x$ matrices in the Cartesian space dynamics equation are derived as follows:

$$\left\{\begin{array}{l}{M}_{rob}\left(q\right)=J^{-T}\left(q\right)M\left(q\right)J^{-1}\left(q\right)\\ C_{rob}\left(q\right)=J^{-T}\left(q\right)\left(C\left(q\left(t\right)\right)-M\left(q\right)J^{-1}\left(q\right)\dot{J}\left(\dot{q}\right)\right)J^{-1}\left(q\right)\\ G_{rob}\left(q\right)=J^{-T}\left(q\right)G\left(q\right)\end{array}\right.$$

(3)

The dynamics of the controlled object are simplified to a single-axis feed system, considering factors such as ball screw transmission efficiency, model inertia, friction, and elastic deformation. The resulting dynamic model is presented as follows:

$$J_{grd}·{\ddot{x}}_d+D_{grd}·{\dot{x}}_d+L_{grd}·x_d=U_{grd}·\mu +F_f+F_n$$

(4)

$[J_{grd},D_{grd},L_{grd}]$ represents the equivalent inertia matrix, equivalent friction coefficient, and equivalent stiffness of the floating grinding head. $U_{grd}$ is a constant matrix associated with transmission efficiency and related factors. $[x_d,{\dot{x}}_d,{\ddot{x}}_d]$ denotes the position, velocity, and acceleration of the grinding head’s feed axis. $[J_{grd},D_{grd},L_{grd}]$ includes the control torque, feed direction friction force, and normal contact force during the grinding process.

2.2. Analysis of Control Strategy

The control objective is to maintain a constant normal contact force on the workpiece. A six-axis force sensor is mounted at the robot’s end effector. After gravity compensation, the sensor reading $F_e$ must be converted to the normal contact force ${\stackrel{\rightharpoonup }{F}}_n$. A force analysis is performed on the blade and the fixture at the sensor’s end in Figure 2.

$$\left\{\begin{array}{l}{\stackrel{\rightharpoonup }{F}}_{sensor}+{\stackrel{\rightharpoonup }{F}}_g+{\stackrel{\rightharpoonup }{F}}_{contact}=m·{\stackrel{\rightharpoonup }{\ddot{x}}}_{ext}\\ {\stackrel{\rightharpoonup }{F}}_e={\stackrel{\rightharpoonup }{F}}_{sensor}+{\stackrel{\rightharpoonup }{F}}_g\end{array}\right.$$

(5)

${\stackrel{\rightharpoonup }{F}}_{sensor}$ represents the external force applied by the sensor to the fixture and is also the force signal detected by the sensor. ${\stackrel{\rightharpoonup }{F}}_g$ denotes the gravitational force of the end fixture and the blade, numerically equal to ${}^{B}G_{workpiece}$. ${\stackrel{\rightharpoonup }{F}}_{contact}$ indicates the contact force at the grinding point in the current state, composed of several components.

The sensor signal, ${}^H\stackrel{\rightharpoonup }{F}_e$, is measured in the sensor coordinate system {H} and needs to be converted to the base coordinate system {B}.

$${}^B\stackrel{\rightharpoonup }{F}_{contact}={}_H{}^{S}R\times {}_S{}^{B}R\times {}^H\stackrel{\rightharpoonup }{F}_e$$

(6)

Here, ${}_H{}^{S}R$ and ${}_S{}^{B}R$ are rotation matrices. The normal direction at the grinding point is assumed to be the feed direction of the grinding head, ${}^B\stackrel{\rightharpoonup }{n}=(\mathrm{0,1},0)$. The component of the contact force ${}^B\stackrel{\rightharpoonup }{F}_{contact}$ in the base coordinate system along the normal direction ${}^B\stackrel{\rightharpoonup }{n}$ is the normal force ${\stackrel{\rightharpoonup }{F}}_n$. The final expression for the normal contact force is then derived.

$${\stackrel{\rightharpoonup }{F}}_n={\displaystyle \frac{({}_H{}^{S}R\times {}_S{}^{B}R\times {}^H\stackrel{\rightharpoonup }{F}_e)·{}^B\stackrel{\rightharpoonup }{n}}{\left|{}^B\stackrel{\rightharpoonup }{n}\right|}}$$

(7)

The normal contact force ${\stackrel{\rightharpoonup }{F}}_n$ serves as the input for the outer loop of the force control. Using the admittance control strategy illustrated in Figure 3, the force control algorithm is implemented. The output variable of the admittance control is the location of the feed device $x\left(t\right)$.

The transfer function of the IOAC controller is as follows. $\left[M_a,B_a,K_a\right]$ denote the desired parameters of the control system: inertia, damping, and stiffness, respectively.

$$Y\left(s\right)={\displaystyle \frac{X\left(s\right)}{F\left(s\right)}}={\displaystyle \frac{1}{M_a{s}^2+B_{a}s+K_a}}$$

(8)

Stability analysis of the control system is performed using Lyapunov’s second method. This is a typical second-order control system. In theory, the characteristic roots ${\lambda }_1$ and ${\lambda }_2$ of the non-homogeneous linear differential equation can be determined, leading to the general solution shown below.

$$\left\{\begin{array}{l}x\left(t\right)=C_1{e}^{{\lambda }_{1}t}+C_2{e}^{{\lambda }_{2}t}+{\displaystyle \frac{F_d}{K}}\\ v\left(t\right)={\displaystyle \frac{\partial x\left(t\right)}{\partial t}}\end{array}\right.$$

(9)

The initial conditions define the system’s initial state: $x\left(t\right)=0$ and $v\left(t\right)=0$ when $t=0$. The particular solution of the non-homogeneous linear differential equation is then calculated.

$$C_1={\displaystyle \frac{F_e}{K}}\left({\displaystyle \frac{{\lambda }_2}{{\lambda }_1-{\lambda }_2}}\right), C_2={\displaystyle \frac{F_e}{K}}\left({\displaystyle \frac{{\lambda }_1}{{\lambda }_2-{\lambda }_1}}\right)$$

(10)

According to Lyapunov’s second method, the energy function is defined as $V\left(t\right)=0.5 m{v}^2\left(t\right)$.

$$\dot{V}\left(t\right)=mv\dot{v}=\left(C_1{\lambda }_1{e}^{{\lambda }_{1}t}+C_2{\lambda }_2{e}^{{\lambda }_{2}t}\right)\times \left(C_1{{\lambda }_1}^2{e}^{{\lambda }_{1}t}+C_2{{\lambda }_2}^2{e}^{{\lambda }_{2}t}\right)$$

(11)

$V\left(t\right)$ is a positive definite function. Since ${\lambda }_1$ and ${\lambda }_2$ are less than 0, $\dot{V}\left(t\right)\le 0$ holds for all $t\ne 0$, making $\dot{V}\left(t\right)$ a negative definite. According to Lyapunov’s second method, this control system is globally asymptotically stable.

3. Methodology

To fulfil the high-precision machining requirements of the bespoke robotic precision polishing system, enhancements in control accuracy and disturbance resistance are necessary. The methodology outlined in the article comprises three principal components: (1) the adoption of a fractional-order admittance controller to more accurately approximate the actual dynamic model, accompanied by analyses of its rationality and stability; (2) the integration of the nonlinear differential tracker (Rear-FTD) with the fractional-order admittance controller (FOAC) to resolve issues of pulse overshoot and enhance the system’s resistance to disturbances; (3) the identification of optimal controller parameters.

3.1. Fractional-Order Admittance Controller

According to the transfer function of the integer-order admittance controller shown in Equation (8), the $s^2$ term denotes the controller’s inertial energy storage characteristics, the $s^1$ term signifies energy dissipation properties, and the $s^{\alpha }$ term and the $s^{\beta }$ term (where $\alpha /\beta$ is fractional) represent a compromise between different control features, thus enabling a more accurate depiction of the actual dynamics of the physical environment. Building on this foundation, the FOAC is developed as detailed below [30].

$$Y_{FOAC}\left(s\right)={\displaystyle \frac{1}{M_a{s}^{\alpha }+B_a{s}^{\beta }+K_a}} (1<\alpha <2, 0<\beta <1)$$

(12)

With variations in order, the physical significance of the coefficients $[M_a,B_a,K_a]$ also shifts. Utilising definitions from fractional calculus, frequency domain analysis provides values for equivalent inertia $M_{af}$, equivalent damping $B_{af}$, and equivalent stiffness $K_{af}$, as detailed subsequently.

$$\left\{\begin{array}{l}{M}_{af}=M_a{\omega }^{\alpha }·\mathrm{s}\mathrm{i}\mathrm{n}\left({\displaystyle \frac{\alpha \pi }{2}}\right)\\ B_{af}=M_a{\omega }^{\alpha }·\cos \left({\displaystyle \frac{\alpha \pi }{2}}\right)+B_a{\omega }^{\beta }·\mathrm{s}\mathrm{i}\mathrm{n}({\displaystyle \frac{\beta \pi }{2}})\\ K_{af}=B_a{\omega }^{\beta }·\cos \left({\displaystyle \frac{\beta \pi }{2}}\right)+K_a\end{array}\right.,(1<\alpha <2, 0<\beta <1)$$

(13)

In this framework, $\omega$ denotes frequency. The equivalent control parameters of the designed FOAC are influenced by its frequency. By modulating the orders [α, β], there is a theoretical potential to facilitate more adaptable interactions between the belt sander and the robot, consequently enhancing both the response speed and stability of the system. The robotic polishing control diagram based on the FOAC is illustrated in Figure 4.

$Y_{FOAC}\left(s\right)$, $F_{cs}\left(s\right)$, $Z_e\left(s\right)$, and $H\left(s\right)$ denote the transfer functions of the FOAC, the controlled system, the interaction dynamics, and the filter, respectively. $x_{exp}$ and $x_{cmd}$ donate the initial position and the position control instruction. According to the description, the open-loop transfer function $G_0\left(s\right)$ and the closed-loop transfer function $G\left(s\right)$ are defined as follows.

$$\left\{\begin{array}{l}{G}_0\left(s\right)={\displaystyle \frac{F_{cs}\left(s\right)·Z_e\left(s\right)·H\left(s\right)}{M_a{s}^{\alpha }+B_a{s}^{\beta }+K_a}}\\ G\left(s\right)={\displaystyle \frac{F_{cs}\left(s\right)}{M_a{s}^{\alpha }+B_a{s}^{\beta }+K_a+F_{cs}\left(s\right)Z_e\left(s\right)H\left(s\right)}}\end{array}\right.$$

(14)

The values for $\beta$ and $\alpha$ range between $0\le \beta \le 1$ and $1\le \alpha \le 2$, respectively. Utilising the definition of fractional-order operators and applying the inverse Laplace transform, the time domain differential equation for the fractional-order system is derived as follows:

$${\mathcal{L}}^{-1}\left(F_{cs}\left(s\right)\right)·F\left(t\right)=M_a·D^{\alpha }\left(x\right(t\left)\right)+B_a·D^{\beta }\left(x\right(t\left)\right)+K_a+{\mathcal{L}}^{-1}\left(F_{cs}\left(s\right)Z_e\left(s\right)H\left(s\right)\right)$$

(15)

In control tasks, open-loop system responses offer direct insights into system stability, while closed-loop responses are typically used for a time domain analysis to assess the suitability of controller designs. The stability analysis for the control system shown in Figure 4 involves defining the base order of the fractional-order system as $\sigma =\alpha /m_1=\beta /m_2$ (with $m_1$ and $m_2$ being integers). Consequently, the open-loop transfer function of the fractional-order system is transformed into an integer-order form in terms of $\lambda$ as follows:

$$\left\{\begin{array}{l}\lambda =s^{\alpha /m_1}=s^{\beta /m_2}\\ G_0\left(\lambda \right)={\displaystyle \frac{K_1{\lambda }^{n_1}+K_2{\lambda }^{n_2}+\cdots +K_i}{M_a{\lambda }^{m_1}+B_a{\lambda }^{m_2}+K_a}}\end{array}\right.$$

(16)

The poles of $G_0\left(\lambda \right)$, denoted as $p_i$, can be calculated. As the base order $\sigma$ varies, the corresponding stability regions of the system are depicted in Figure 5.

$$\left|arg\left(p_i\right)\right|>\sigma ·{\displaystyle \frac{\pi }{2}}$$

(17)

In summary, the system reaches local asymptotic stability at the equilibrium point when all $p_i$ satisfy Equation (17).

3.2. System Disturbance-Based Rear-FTD

Zero-position overshoot is a significant issue in robotic precision polishing systems, especially in reducing the impact during tool entry. Theoretically, when the controller’s response speed is high, a noticeable overshoot in the force signal at initial contact is inevitable. Traditional control systems often struggle to simultaneously achieve rapid responsiveness and minimise overshoot. The active disturbance rejection control theory has proven particularly effective in this area. The paper proposes a strategy that combines a tracking differentiator with FOAC, introducing a transition process $x_{ftd}\left(t\right)$ to smooth the entry phase. The implementation proceeds as follows:

$$\left\{\begin{array}{l}{x}_{ftd}\left(k+1\right)=x_{ftd}\left(k\right)+h·v\left(k\right)\\ v\left(k+1\right)=v\left(k\right)+h·a_{st}\left(k\right)\end{array}\right.$$

(18)

Since the actual control model is a discrete control system, discrete time modelling is used to ensure consistency between the model and the real system. In this setup, $x_{ftd}\left(k\right)$ and $v\left(k\right)$ represent the output signals of the transition process. Here, $x_{ftd}\left(k\right)$ provides a restoration tracking of the controlled signal $x_{ftd0}\left(k\right)$, and $v\left(k\right)$ acts as a differential signal of the tracking signal $x_{ftd}\left(k\right)$. The acceleration signal $a_{st}\left(k\right)$, which determines the tracking speed, is derived from the fastest control synthesis function $fst\left(x_{ftd}\left(k\right)-x_{ftd0}\left(k\right),v\left(k\right),\delta ,h\right)$. The parameter $\delta$ specifies the tracking speed, termed the velocity factor, while $h$, the integration step, corresponds to the sampling period. The fastest control synthesis function $\mathrm{f}\mathrm{s}\mathrm{t}$ is designed as follows:

$$fst=\left\{\begin{array}{l}-\delta sign\left(a\right), \left|a\right|>d\\ -\delta {\displaystyle \frac{a}{d}} ,\left|a\right|\le d\end{array}\right.$$

(19)

$$a=\left\{\begin{array}{l}v+{\displaystyle \frac{a_0-d}{2}}sign\left(y\right),\left|y\right|>d_0\\ v+{\displaystyle \frac{y}{h}} ,\left|y\right|\le d_0\end{array}\right.$$

(20)

Within this framework: $d=\delta h$; $d_0=hd$; $y=x_{ftd}+hv$; and $a_0=\sqrt{d^2+8\delta \left|y\right|}$. In FTD, the fastest control synthesis function is crucial for determining signal-tracking performance. This article employs a nonlinear velocity function, which has been validated through practical applications in various fields and demonstrates effective noise suppression and robustness. The performance of this function is largely dependent on parameter settings. The fastest differential tracker (FTD) fundamentally smooths the control signal. As depicted in Figure 6, within the FOAC-based robotic polishing control setup, the placement of the FTD relative to the FOAC within the control loop markedly affects the results.

Figure 6 illustrates the functions of the FTD and FOAC within the algorithm. The lower part of the figure depicts the block diagram of the fractional-order admittance control. Using Rear-FTD as an example, the output $x_{cmd}$ from the admittance control block serves as the input to Rear-FTD, and the output of Rear-FTD becomes the input command for the control system. In other words, the command intended for the controlled system is first filtered by FTD before being transmitted. This integrated approach enhances the system’s anti-interference capability. The Front-FTD is tasked with smoothing the input pulse signals to the FOAC, while the Rear-FTD smooths the output control signals from the FOAC. Theoretically, the Rear-FTD, which directly manages the input signals of the controlled system, has a more significant impact on system response than the Front-FTD. Figure 7 shows the force control performance of both the Front-FTD and Rear-FTD under actual sensor signal conditions.

The experimental data largely align with the theoretical analysis. The Rear-FTD surpasses the Front-FTD method in terms of pulse overshoot suppression and disturbance rejection capabilities. Consequently, this study adopts a combined approach using Rear-FTD and FOAC for control.

From Equations (18)–(20), it is evident that the Rear-FTD has two hyperparameters: the integration step h and the velocity factor δ. To achieve optimal performance, h should be consistent with the control cycle of the control system. The value of δ represents a trade-off between response speed and filtering effectiveness; a smaller δ enhances the filtering effect but slows down the response speed. It is essential to find an appropriate balance during experimentation. Since the coupling between these two parameters is minimal, their optimisation is relatively straightforward and can be adjusted and analysed independently in the experiment.

3.3. Optimal Control Parameter Identification

Before adjusting the parameters $[\delta ,h]$ of the Rear-FTD, it is crucial to identify the optimal control parameters for the FOAC, encompassing the admittance control parameters and the orders $[M_a,B_a,K_a,\alpha ,\beta ]$. Among these parameters, $M_a$ primarily influences the system’s inertia, while $K_a$ predominantly affects the system’s stiffness. These two parameters must be adjusted to align with the actual inertia and stiffness of the controlled object. The damping parameter $B_a$ has the most significant impact on the system because the control system in this study employs a speed control method, where $B_a$ serves as the speed gain, directly determining the control response speed. Parameters $\alpha$ and $\beta$ are exponents, and different fractional-order values exhibit entirely distinct system characteristics, enabling a more accurate representation of the fractional friction properties inherent in the physical world. Given this, a strategy is developed for tuning the fractional-order control parameters, which is based on frequency domain responses and Pareto multi-objective optimisation.

For the open-loop transfer function, as depicted in Equation (14), the controlled system $F_{cs}\left(s\right)$ is modelled as a single-axis control system, optimised to balance response speed and overshoot, thereby simplifying it to an ideal input–output system. The environmental dynamics are reduced to a pure stiffness system, represented by $Z_e\left(s\right)=K_s$. By applying the relationship between the Laplace and Fourier transforms, substituting $s=j\omega$ derives the frequency response function.

$$G_0\left(j\omega \right)={\displaystyle \frac{K_s}{M_a{\left(j\omega \right)}^{\alpha }+B_a{\left(j\omega \right)}^{\beta }+K_a}}$$

(21)

Utilising Euler’s formula and the definition of complex numbers, the frequency response function $G_0\left(j\omega \right)$ is simplified.

$$\left\{\begin{array}{l}{G}_0\left(j\omega \right)={\displaystyle \frac{K_s}{\left(M_a{p}_1+B_a{q}_1+K_a\right)+\left(M_a{p}_2+B_a{q}_2\right)j}}\\ \begin{array}{l}{p}_1(\omega ,\alpha )={\omega }^{\alpha }cos{\displaystyle \frac{\pi \alpha }{2}},p_2(\omega ,\alpha )={\omega }^{\alpha }sin{\displaystyle \frac{\pi \alpha }{2}}\\ q_1(\omega ,\beta )={\omega }^{\beta }cos{\displaystyle \frac{\pi \beta }{2}},q_2(\omega ,\beta )={\omega }^{\beta }sin{\displaystyle \frac{\pi \beta }{2}}\end{array}\end{array}\right.$$

(22)

Consequently, this enables the analysis of the amplitude–frequency characteristics $\left|G_0\left(j\omega \right)\right|$ and phase frequency characteristics $\phi (\omega )$ of the control system.

$$\left\{\begin{array}{l}{G}_0\left(j\omega \right)=\left|G_0\left(j\omega \right)\right|e^{j·\phi (\omega )}\\ \begin{array}{l}\left|G_0\left(j\omega \right)\right|={\displaystyle \frac{K_s}{\sqrt{{\left(M_a{p}_1+B_a{q}_1+K_a\right)}^2+{\left(M_a{p}_2+B_a{q}_2\right)}^2}}}\\ \phi \left(\omega \right)=-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}({\displaystyle \frac{M_a{p}_2+B_a{q}_2}{M_a{p}_1+B_a{q}_1+K_a}})\end{array}\end{array}\right.$$

(23)

In the ideal scenario of a five-parameter optimisation problem, if four equality constraints and one objective equation are provided, the optimisation problem can be effectively solved using the Lagrange multiplier method. The system’s cutoff frequency is established at ${\omega }_c$, with the phase margin set at $\gamma$. Based on the frequency response characteristics, the design includes the following four equation constraints:

(1) The control system should aim for a steady-state error that approaches zero, denoted as $e_{ss}=0$. The steady-state error for a unit step input is accordingly represented.

$$e_{ss}=\underset{s\to 0}{\lim }{\displaystyle \frac{s·R\left(s\right)}{1+G_0\left(s\right)}}=K_a=0$$

(24)

(2) As defined, at the cutoff frequency ${\omega }_c$, the system’s logarithmic magnitude gain equals zero. This is formulated as $20lg\left|G_0\left(j{\omega }_c\right)\right|=0$ and is subsequently simplified.

$${\left(M_a{{\omega }_c}^{\alpha }\right)}^2+{\left(B_a{{\omega }_c}^{\beta }\right)}^2+2M_a{B}_a{{\omega }_c}^{\alpha +\beta }cos{\displaystyle \frac{\pi (\alpha -\beta )}{2}}=2K_s$$

(25)

(3) The phase margin $\gamma$ is calculated based on the phase characteristics at ${\omega }_c$.

$$\mathrm{t}\mathrm{a}\mathrm{n}(\pi -\gamma )={\left.{\displaystyle \frac{M_a{p}_2+B_a{q}_2}{M_a{p}_1+B_a{q}_1+K_a}}\right|}_{{\omega }_c}$$

(26)

(4) The phase derivative is an essential indicator of system stability; a phase derivative nearing zero suggests a gradual phase angle change. This trait enhances the system’s stability against external disturbances and parameter changes, helping avert oscillatory or unstable behaviour.

$$\left\{\begin{array}{l}{\left.{\displaystyle \frac{d\phi \left(\omega \right)}{d\omega }}\right|}_{{\omega }_c}={\displaystyle \frac{1}{1+{\left(\psi \left(\omega \right)\right)}^2}}·{\left.{\displaystyle \frac{d\psi \left(\omega \right)}{d\omega }}\right|}_{{\omega }_c}\to 0\\ \psi \left(\omega \right)={\displaystyle \frac{M_a{p}_2+B_a{q}_2}{M_a{p}_1+B_a{q}_1+K_a}}\end{array}\right.$$

(27)

The above constraints pertain to frequency domain response analysis indicators. By comprehensively managing and optimising the system’s steady-state error, gain, and phase, both response speed and robustness requirements can be effectively met. The parameter optimisation process is recast as an optimisation problem-solving exercise. Once the equation constraints outlined in Equations (24)–(27) are established, theoretically, defining the objective function allows for the calibration of the five parameters of the FOAC. Traditional single-objective methods often fail to accurately depict the controller’s performance across both time and frequency domains. This paper adopts Pareto optimality for multi-objective optimisation of the controller, focusing on the following two aspects:

(1) Time Domain Performance Indicator: Utilising the integrated time absolute error (ITAE) framework, consideration is given to the high response speed requirements characteristic of adaptive polishing applications. The settling time $t_s$ is integrated as a weighting factor in the ITAE to provide a comprehensive evaluation of the control system’s response speed and damping effectiveness. The cost function for the time-domain performance indicator is outlined below.

$$C_t=e^{t_s}·{\int }_0^{\infty }t\left|x_o\left(t\right)-x_i\left(t\right)\right|dt$$

(28)

(2) Frequency Domain Performance Indicator: Chosen based on principles of robust control, this involves the introduction of the sensitivity function $S\left(j\omega \right)={(I+G_0\left(j\omega \right))}^{-1}$, which gauges the stability of the control system. Theoretically, a smaller $S\left(j\omega \right)$ suggests a superior capacity of the system to mitigate disturbance signals and enhance stability. Consequently, the cost function for the frequency domain performance indicator is defined as follows.

$$C_f={\displaystyle \frac{-1}{max\left|{\left(I+G_0\left(j\omega \right)\right)}^{-1}\right|}}=-min\left(\sqrt{{\displaystyle \frac{{\left(M_a{p}_1+B_a{q}_1+K_s\right)}^2+{\left(M_a{p}_2+B_a{q}_2\right)}^2}{{\left(M_a{p}_1+B_a{q}_1\right)}^2+{\left(M_a{p}_2+B_a{q}_2\right)}^2}}}\right)$$

(29)

The two indicators represent the time-domain and frequency-domain characteristics of the system, respectively. $C_t$ focuses on the system’s response speed, while $C_f$ represents the system’s robustness. Both indicators are essential for precision control systems. The ultimate objective is to minimise both $C_t$ and $C_f$, however, optimising multiple targets simultaneously with a single set of parameters is challenging. Therefore, a Pareto optimal frontier is constructed to enable a balanced compromise between them. This method is commonly employed to address multi-objective optimisation problems. It involves combining two physical quantities with different dimensions to determine the optimal weight factor. The specific implementation process will be detailed in the experimental section. The final objective function, outlined below, involves the normalisation of $C_t$ and $C_f$ to provide a consistent standard for evaluation.

$$J=\zeta {\displaystyle \frac{C_t-C_{tmin}}{C_{tmax}-C_{tmin}}}+(1-\zeta ){\displaystyle \frac{C_f-C_{fmin}}{C_{fmax}-C_{fmin}}}$$

(30)

In this context, $\zeta$ denotes the weighting factor, which varies between 0 and 1, employed to mediate the trade-off between response speed and stability. The maximum and minimum values of the parameters are obtained by comparing the array of $C_t$ and $C_f$ in discrete form. The formulation of the optimal controller parameter tuning problem is as follows:

$$\begin{gathered} \underset{[M_a,B_a,K_a,\alpha ,\beta ]}{\mathit{min}}J=\zeta {\displaystyle \frac{C_t-C_{tmin}}{C_{tmax}-C_{tmin}}}+(1-\zeta ){\displaystyle \frac{C_f-C_{fmin}}{C_{fmax}-C_{fmin}}} \\ s.t.\left\{\begin{array}{l}\begin{array}{l}\begin{array}{l}\underset{s\to 0}{\lim }{\displaystyle \frac{s·R(s)}{1+G_0\left(s\right)}}=K_a=0\\ {(M_a{{\omega }_c}^{\alpha })}^2+{(B_a{{\omega }_c}^{\beta })}^2+2M_a{B}_a{{\omega }_c}^{\alpha +\beta }cos{\displaystyle \frac{\pi (\alpha -\beta )}{2}}=2K_s\end{array}\\ \mathrm{t}\mathrm{a}\mathrm{n}(\pi -\gamma )={\left.{\displaystyle \frac{M_a{p}_2+B_a{q}_2}{M_a{p}_1+B_a{q}_1+K_a}}\right|}_{{\omega }_c}\end{array}\\ \begin{array}{l}{\displaystyle \frac{1}{1+{(\psi \left(\omega \right))}^2}}·{\left.{\displaystyle \frac{d\psi \left(\omega \right)}{d\omega }}\right|}_{{\omega }_c}\to 0\\ p_1={\omega }^{\alpha }cos{\displaystyle \frac{\pi \alpha }{2}},p_2={\omega }^{\alpha }sin{\displaystyle \frac{\pi \alpha }{2}}\\ q_1={\omega }^{\beta }cos{\displaystyle \frac{\pi \beta }{2}},q_2={\omega }^{\beta }sin{\displaystyle \frac{\pi \beta }{2}}\\ \psi \left(\omega \right)={\displaystyle \frac{M_a{p}_2+B_a{q}_2}{M_a{p}_1+B_a{q}_1+K_a}}\\ 0\le \beta \le 1\le \alpha \le 2, 0\le \lambda \le 1\end{array}\end{array}\right. \end{gathered}$$

(31)

Given that the aforementioned optimisation problem essentially involves a discretisation process, simplifying the calculation can be achieved if certain parameters are discretised. Since the variation range of the fractional exponents is limited ($0\le \beta \le 1$, $1\le \alpha \le 2$) and the required precision is to just one decimal place (where a change of 0.01 in the exponents has a negligible impact on the system), $\alpha$ and $\beta$ are initially discretised at equal intervals. This method effectively transforms Equations (24), (26) and (27) into linear constraints, while Equation (25) remains a second-order equality constraint. This transformation significantly accelerates the convergence speed of the optimisation problem and enhances the efficiency of the algorithm. The specific algorithmic approach for parameter tuning is outlined as Algorithm 1.

Algorithm 1: Optimal FOAC Parameter Identification
Input: Control system’s cutoff frequency ${\omega }_c$, phase margin $\gamma$, and environmental stiffness $K_s$
Output: Controller parameters ${[M_a,B_a,K_a,\alpha ,\beta ]}_{opt}$
1	for ${\alpha }_i\leftarrow 1$ to $2({\alpha }_i={\alpha }_i+0.1)$ do
2	for ${\beta }_i\leftarrow 0$ to $1({\beta }_i={\beta }_i+0.1)$ do
3	Substitute $\mathsf{\alpha }$ and $\mathsf{\beta }$ into the equality constraints in Equation (31)
4	Solve the numerical solution of $[M_{ai},B_{ai},K_{ai}=0]$
5	Compute $C_{ti}$ and $C_{fi}$ based on Equations (28) and (29)
6	Compute $i=i+1$
7	Calculate the normalised parameters $[C_{tmin},C_{tmax}]$ and $[C_{fmin},C_{fmax}]$
8	Based on the discrete rules of ${\alpha }_i$ and ${\beta }_i$, there are $j=121$ sets of controller parameters.
9	for $j\leftarrow 1$ to $121(j=j+1)$ do
10	for ${\zeta }_k\leftarrow 0$ to $1 ({\zeta }_k={\zeta }_k+0.01)$ do
11	Compute $J_k$ based on Equations (28) and (29)
12	Obtain the Pareto front line based on $[M_{aj},B_{aj},K_{aj},{\alpha }_j,{\beta }_j]$
13	Select the optimal weight factor ${\zeta }_j$ refer to $J_j$
14	Construct an exponential distribution map based on $[{\alpha }_j,{\beta }_j,J_j]$
15	Determine the parameter set ${[M_a,B_a,K_a,\alpha ,\beta ]}_{opt}$ at the minimum of $J_{min}$

Following the optimal controller parameter tuning algorithm outlined previously, a theoretically optimal set of controller parameters $[M_a,B_a,K_a,\alpha ,\beta ]$ can be derived.

4. Simulation and Experiment Studies

To implement the optimal fractional-order admittance controller parameter tuning algorithm proposed in Section 3, this Section employs a joint simulation based on MATLAB and Simulink. The simulation validation comprises three parts: (1) verification of the optimal parameter setting rules for FOAC, analysis of the specific impact of each parameter on the control system, and determination of the optimal controller parameters through algorithmic identification; (2) comparison of the control effects between the optimal FOAC and traditional integer-order admittance control, demonstrating the superiority of the optimal FOAC; (3) comparison of the effects with and without the addition of Rear-FTD, to verify the effectiveness of Rear-FTD in suppressing contact impact forces. The algorithm proposed in this paper is applied to an actual robotic polishing system, experimentally validating the practical force control effects of the algorithm.

4.1. The FOAC Optimal Parameter Identification

The simulation model is developed based on the content in Section 3, encompassing the admittance control module, Rear-FTD module, controlled system module, and robot module. The modelling relationships in Simulink are illustrated in Figure 8 below, which depicts the input and output of each component and the logical connections between them.

To simulate the equipment vibrations caused by the high-speed operation of the sanding belt in an actual processing environment, high-frequency white noise was introduced into the simulation. The intensity of the high-frequency band was set relatively high to rigorously test the system’s anti-interference capabilities. The desired parameters are initialised for the control system and the environmental stiffness: angular frequency ${\omega }_c=10 \mathrm{H}\mathrm{z}$, phase angle $\gamma ={45}^{°}$, and stiffness $K_s=1000$. Subsequently, the fractional-order coefficients $\alpha$ and $\beta$ are discretised within intervals [1, 2] and [0, 1], respectively, with a discretisation step of 0.1, resulting in a total of 121 parameter sets. Each set is subjected to the equation constraints outlined in Section 3.3, where the system’s phase derivative is used as a performance metric, thus producing 121 parameter sets. For each set of parameters, the five-parameter optimisation problem is simplified into a three-parameter optimisation problem, with the three constraint equations reduced to linear constraints. This simplification significantly streamlines the optimisation process, reduces the algorithm’s complexity, and enhances the feasibility of the algorithm’s practical application.

An initial screening of these sets is then conducted, since neither the mass $M_a$ nor the damping $B_a$ can be negative, unsuitable parameter sets are eliminated, leaving 43 sets, as shown in

Table 1. The effective parameter set.

Node	Parameter Set
	$\mathsf{\alpha }$	$\mathsf{\beta }$	${\mathit{M}}_{\mathit{a}}$	${\mathit{B}}_{\mathit{a}}$	${\mathit{K}}_{\mathit{a}}$
1	1.6	0	1.351	11.902	0
2	1.6	0.1	1.350	8.252	0
3	1.6	0.2	1.346	5.935	0
4	1.6	0.3	1.341	4.400	0
5	1.6	0.4	1.335	3.351	0
6	1.6	0.5	1.328	2.616	0
7	1.6	0.6	1.320	2.091	0
…	…	…	…	…	…
37	2	0.4	0.752	21.418	0
38	2	0.5	0.633	14.142	0
39	2	0.6	0.546	9.818	0
40	2	0.7	0.477	7.081	0
41	2	0.8	0.419	5.270	0
42	2	0.9	0.366	4.031	0
43	2	1	0.316	3.162	0

.

Based on the controller performance indices described in Equations (27)–(30), the time domain index $C_t$ and the frequency domain index $C_f$ are calculated, respectively. This indicator can be used to accurately assess the quality of the control system. Additionally, the magnitude of this value provides an intuitive measure of the impact of each parameter on the system. To achieve multi-objective optimisation, the optimal parameter set is identified by searching for Pareto-optimal solutions. The Pareto front of the optimal FOAC is illustrated in Figure 9.

The circles in the figure represent the values of $\zeta$. As the weighting factor $\zeta$ increases, it can be observed that $C_t$ decreases while $C_f$ increases. A compromise must be made between these indices. Based on Figure 8, the inflection point at $\zeta =0.64$ is selected. At this point, both $C_t$ and $C_f$ are relatively low, which effectively enhances the response speed and stability of the control system.

By substituting the solution at $\zeta =0.64$ for the original choice and facilitating a lateral comparison of system performance under different parameter sets, contour plots, profile views, and three-dimensional surface plots are generated, as seen in Figure 10. Additionally, the phase derivative (dphase) is also displayed as one of the evaluation indices.

Figure 10a–c represent the values of $C_t$, while Figure 10d–f represent the values of $C_f$, and Figure 10g–i depict the values of dphase. The contour lines clearly illustrate the magnitude of the evaluation indices corresponding to different parameter sets. It can be observed that as $\alpha$ increases, the time domain index of the system rises sharply before stabilising, while the frequency domain index, which reflects the system’s robustness, increases almost proportionally. Conversely, when $\beta$ increases, the impact on the control system is the opposite of that seen with $\alpha$. This indicates that an integer-order control model is not the optimal choice for balancing the system’s robustness and response.

Table 2. The evaluation indices under various parameter sets.

$\mathbf{N}\mathbf{o}\mathbf{d}\mathbf{e}$	${\mathit{C}}_{\mathit{t}}$	${\mathit{C}}_{\mathit{f}}$	$\mathit{d}\mathit{p}\mathit{h}\mathit{a}\mathit{s}\mathit{e}$	$\mathbf{N}\mathbf{o}\mathbf{d}\mathbf{e}$	${\mathit{C}}_{\mathit{t}}$	${\mathit{C}}_{\mathit{f}}$	$\mathit{d}\mathit{p}\mathit{h}\mathit{a}\mathit{s}\mathit{e}$
1	0.626	0.030	0	…	…	…	…
2	0.341	0.027	0.120	38	0.210	0.150	1 × 10−15
3	0.188	0.024	0.210	39	0.086	0.121	1 × 10−15
4	0.105	0.022	0.283	40	0.039	0.098	7 × 10−16
5	0.058	0.020	0.345	41	0.016	0.080	3 × 10−16
6	0.031	0.017	0.402	42	0.005	0.064	7 × 10−16
7	0.016	0.016	0.456	43	0	0.050	7 × 10−16

displays the values of these indices under various parameter sets, each corresponding to the sequence numbers listed in Table 1.

Based on the Figure 10 and Table 2, it is possible to visually discern the quality of the three performance indices [$C_t,C_f,dphase$] across different parameter set regions. Ultimately, by evaluating the Pareto objective function $\mathrm{J}$, the parameter set corresponding to the minimum value $J_{min}$ is identified. The evaluation indices for this optimal set are detailed in

Table 3. The evaluation indices under the optimal set.

Node	$\mathsf{\alpha }$	$\mathsf{\beta }$	${\mathit{M}}_{\mathit{a}}$	${\mathit{B}}_{\mathit{a}}$	${\mathit{K}}_{\mathit{a}}$	${\mathit{C}}_{\mathit{t}}$	${\mathit{C}}_{\mathit{f}}$	$\mathit{d}\mathit{p}\mathit{h}\mathit{a}\mathit{s}\mathit{e}$	$\mathsf{\zeta }$	${\mathit{J}}_{\mathit{m}\mathit{i}\mathit{n}}$
10	1.6	0.9	1.2853	1.2456	0	0.0009	0.0099	0.6253	0.64	0.0041

.

The parameter set listed in Table 3 represents the optimal control parameters for the control system established in the simulation. With this, the identification of the optimal parameters for the FOAC is now complete.

4.2. Fractional and Integer Admittance Control Comparison

To validate the effectiveness of the optimal fractional-order parameters, this section compares them with traditional integer-order parameters within the force control system built in Simulink. By observing the simulation results in both the time and frequency domains, we verify the superiority of the control system modifications. Considering that robotic polishing involves contact machining, which can cause significant vibrations, appropriate disturbances were added to the control system. The step response analysis is shown in Figure 11.

The objective is to maintain constant force control with a set force of 2 N. Although both control models stabilise the system at this force level, they differ in their ability to suppress overshoot and vibrations. The fractional-order model markedly outperforms the traditional integer-order control in mitigating both overshoot and vibrations.

Table 4. Settling times $t_s\left(s\right)$ for varying error band values.

$${\mathit{t}}_{\mathit{s}}$$	$$\pm 1\mathit{\%}$$	$$\pm 2\mathit{\%}$$	$$\pm 5\mathit{\%}$$	$$\pm 10\mathit{\%}$$
FOAC	+∞	0.15 s	0.05 s	0.05 s
Traditional	+∞	+∞	0.10 s	0.05 s

displays the settling times $t_s$ for varying error band values.

An infinite stabilisation time indicates that the model fails to meet the accuracy requirements, with the actual force exceeding the error band’s range due to pulsations. If the error band for the settling time $t_s$ is set at ±1%, neither model achieves convergence. With an error band of ±2%, the FOAC achieves a settling time of 0.15 s, whereas the traditional integer-order control is unable to converge within this range. When the error band is increased to ±5%, the FOAC continues to surpass the performance of traditional integer-order control.

The frequency domain analysis of the control system is presented in Figure 12a illustrates the performance of the FOAC, and Figure 12b depicts the traditional integer-order controller. According to the Bode diagram, significant gain fluctuations occur in the integer-order controller in the mid-frequency range (frequency varies from ${10}^1$ rad/s to ${10}^2$ rad/s), leading to unstable transient responses, in contrast to the stability exhibited by the FOAC. In the high-frequency range (frequency varies from ${10}^2$ rad/s to ${10}^3$ rad/s), the FOAC’s gain decreases at about −20 dB/decade, compared to −40 dB/decade for the integer-order controller, indicating superior suppression of high-frequency noise by the fractional-order system. The Nichols diagram reveals a phase margin of 76.38 deg for the FOAC, significantly greater than the 14.45 deg for the integer-order controller, reflecting enhanced stability. Furthermore, the Nyquist diagram shows that the FOAC’s open-loop Nyquist trajectory is positioned further from (−1, j0), indicating a higher level of relative stability.

4.3. Contact Impact Suppression Analysis Based on Rear-FTD

In Section 3.2, ablation studies on the proposed rear fastest tracking differential (Rear-FTD) were conducted to assess its effectiveness in mitigating contact impacts within real robotic scenarios. The Rear-FTD algorithm incorporates two hyperparameters: the integration step h, equal to the sampling period, set at 0.01 s; and the velocity factor δ. The influence of different values of δ on the algorithm is illustrated in Figure 13. In alignment with the requirements of the actual polishing scenario, we set the target force to 1.5 N in the simulation.

As illustrated in

Table 5. An analysis of the time domain index for a controller with varying parameter $\delta$.

Items	${\mathit{t}}_{\mathit{p}}$	${\mathit{M}}_{\mathit{p}}$	${\mathit{t}}_{\mathit{r}}$	${\mathit{t}}_{\mathit{s}}$
Origin	1.29	1.78	1.28	1.38
$\delta =400$	1.35	1.73	1.32	1.90
$\delta =100$	1.44	1.68	1.38	1.48
$\delta =50$	1.53	1.57	1.48	1.44
$\delta =10$	1.93	1.58	1.80	1.73

, the peak time $t_p$, overshoot $M_p$, rise time $t_r$, and settling time $t_s$ for both the original and optimised signals were quantified. It was observed that a smaller $\delta$ results in a reduced overshoot $M_p$, demonstrating the algorithm’s enhanced capability to suppress zero overshoot. A comparison between the overshoot $M_p$ and settling time $t_s$ values show that at $\delta =50$, the system exhibits strong overshoot suppression capabilities while maintaining a rapid response speed.

4.4. Experimental Verification Based on Polishing Force Analysis

The fractional-order admittance control system, with its optimally identified parameters, was deployed in a robotic precision polishing scenario for comparison with a traditional integer-order control model. The experimental setup is illustrated in Figure 14. Given the minimal material removal characteristic of polishing tasks, a target control force of approximately 2 N was considered appropriate. The normal polishing forces throughout the process were recorded using a six-axis force sensor, as depicted in Figure 15. The time interval between each successive acquisition point is 10 ms. A comprehensive time-frequency domain analysis of the normal contact force on the polished surface was conducted, encompassing key parameters such as peak time $t_p$, overshoot $M_p$, rise time $t_r$, and settling times $t_s$ for different error band values.

Table 6. (1) The time domain index analysis of normal contact force (N). (2) The settling times ${\mathrm{t}}_{\mathrm{s}}\left(s\right)$ for varying error band values.

(1)
Item		${\mathit{t}}_{\mathit{p}}$	${\mathit{M}}_{\mathit{p}}$	${\mathit{t}}_{\mathit{r}}$
IOAC		1.03	2.433	0.98
FOAC		1.12	2.021	1.08
(2)
${\mathit{t}}_{\mathit{s}}$	$2\pm 0.1 \mathbf{N}$	$2\pm 0.3 \mathbf{N}$	$2\pm 0.5 \mathbf{N}$	$2\pm 0.7 \mathbf{N}$
IOAC	$+\infty$	6.62 s	2.80 s	0.88 s
FOAC	$1.02 \mathrm{s}$	$1$.00 s	0.99 s	0.98 s

demonstrates that the traditional control method is significantly inferior to the FOAC control method in terms of overshoot and control stability. Additionally, Figure 16 reveals that the traditional control method exhibits a noticeable amount of low-frequency noise. In contrast, the FOAC control model more effectively suppresses low-frequency noise, contributing to its superior control accuracy. From Figure 15, it is evident that under identical polishing conditions, the force control strategy using the FOAC model achieves a precision of 2 N ± 0.1 N, compared to 2 N ± 0.3 N with traditional integer-order admittance control. This represents an improvement in force control precision from 15% to 5%. Therefore, in robotic precision polishing tasks, the FOAC model markedly reduces contact impacts and enhances force control stability.

Optimising the control model enhances control accuracy, which in turn fundamentally improves blade processing quality and surface roughness, thereby meeting the stringent requirements of high-precision blade processing. For instance, when applied to the third-stage blade of a specific aircraft engine, a comparison of the blade’s back processing path and surface roughness is presented in Figure 17 and Figure 18. These figures illustrate that, compared to the traditional IOAC control model, the FOAC model proposed in this study significantly improves the surface quality of robotic blade polishing, effectively meeting the demands of precision polishing.

5. Conclusions

This paper introduces a fractional-order admittance control (FOAC) approach integrated with a rear fastest tracking differentiator (Rear-FTD) for robotic precision polishing applications, accompanied by an optimal parameter identification strategy. The principal innovations include the following: (1) fully leveraging the benefits of FOAC within robotic precision polishing to enhance force control stability; (2) combining Rear-FTD with FOAC to balance pulse overshoot and tracking speed, thus minimising contact impacts during the polishing tool entry phase; (3) utilising Pareto optimality and a combined time–frequency domain fusion analysis to address the complexities of identifying optimal parameters for fractional-order admittance control. Simulation and experimental results confirm that the proposed optimal FOAC strategy significantly enhances control precision and stability, while the integration of Rear-FTD with FOAC effectively mitigates pulse overshoot. This methodology provides substantial guidance for robotic precision polishing.

However, a challenge with this method is that the dynamics of the controlled object in the simulation differ from those in the actual environment. Consequently, the optimal parameters identified through simulation may not be optimal during actual implementation. In the future, integrating transfer learning approaches could help align the simulation with real-world conditions.