## 4 Fractional-Order PID Adaptive Impedance Control  
### Control Method Design  
In traditional impedance controllers, the end effector usually exhibits significant overshoot upon contact with the environment, and it has a slow response to the error in contact force [28]. In order to improve the performance of the manipulator when interacting with the environment, this paper combines the advantages of fractional-order PID and proposes a novel fractional-order PID adaptive impedance control method. The fractional-order integration and differentiation help prevent integral saturation, reduce contact force overshoot, enhance system robustness, and further improve control performance. Recently, fractional calculus has been applied in many fields, such as artificial neural networks [29], computer vision [30], and the extension of PID controllers [31]. Due to the differentiation and integration orders no longer being limited to integers but extended to any real number, it can provide additional flexibility to PID controllers, allowing them to be extended to fractional-order PID (FOPID) controllers [32]. This enables it to better accommodate complex systems with time-varying and nonlinear characteristics. The equation for fractional calculus is as follows:  
\[{}_{a}D_{t}^{q}=\left\{ \begin{array}{c}\frac{d^{q}}{dt^{q}}:q>0\\ 1:q=0\\ \int_{d}^{t}(dt)^{-q}\colon q<0\end{array} \right. \tag{27}\]where \(D\) is the fundamental operator of calculus, \(a\) and \(t\) are the boundaries of this operator, and \(q\) denotes that this operation is not restricted to integer orders. Currently, various mathematical representations of fractional calculus can be found in the literature, with the most common ones being Caputo, Riemann-Liouville (RL), and Grunwald-Letnikov (GL) [33; 34; 35]. Equation (28) provides the definition of RL:  
\[{}_{a}D_{t}^{q}f(t)=\frac{1}{\Gamma(n-1)}\frac{d^{n}}{dt^{n}}\int_{a}^{t}\frac {f(\tau)}{f(t-\tau)^{q-n+1}}d\tau \tag{28}\]  
Equation (29) provides the definition of Caputo:  
\[{}_{a}D_{t}^{q}f(t)=\frac{1}{\Gamma(q-n)}\int_{a}^{t}\frac{f^{(n)}(\tau)}{f(t- \tau)^{q-n+1}}d\tau, \tag{29}\]  
where \((n-1<q<n)\), and \(\Gamma(q)\) is represented by the following equation. \[\Gamma(q)=\int_{0}^{\infty}e^{-u}u^{q-1}du, \tag{30}\]  
Similarly, Equation (31) provides the definition of GL:  
\[{}_{a}D_{t}^{q}f(t)=\lim_{h\to 0^{+}}\frac{1}{h^{q}}\sum_{j=0}^{(t-a)/h}(-1)^{j} \binom{q}{j}f(t-jh) \tag{31}\]  
Unlike integer-order functions, fractional-order functions cannot obtain their exact solutions, and approximate methods are usually used for calculations. Among them, the pole distribution-based approach proposed by [PERSON] has shown good approximation effectiveness in achieving the desired accuracy [36]. Therefore, this provides an implementation method for fractional-order PID. The approximate fractional filter within the frequency range \((w_{b},w_{h})\) can be represented as  
\[s^{\alpha}\approx K\prod_{k=1}^{N}\frac{s+\omega_{k}^{\prime}}{s+\omega_{k}} \tag{32}\]  
where \(\alpha\) represents the order of the differentiator, \(w_{k}\), \(w_{k}\prime\), and \(K\), respectively, denote its zeros, poles, and gain. \[\left\{ \begin{array}{l}K=\omega_{h}^{\alpha}\\ \\ \omega_{k}^{\prime}=\omega_{b}\omega_{u}^{\frac{2k-1-a}{N}}\\ \\ \omega_{k}=\omega_{b}\omega_{u}^{\frac{2k-1+a}{N}}\end{array} \right. \tag{33}\]  
where \(\omega_{u}=\sqrt{\omega_{h}/\omega_{b}}\) and the control effort of the PID controller is expressed as  
\[u(t)=K_{p}e(t)+K_{i}\int_{0}^{t}e(t)dt+K_{d}\frac{de(t)}{dt} \tag{34}\]  
Its transfer function \(C(s)\) can be represented as  
\[C(s)=k_{p}+\frac{k_{i}}{s}+k_{d}s \tag{35}\]When introducing fractional order into PID control, the transfer function of fractional order PID can be represented as  
\[C(s)=k_{p}+\frac{k_{i}}{s^{\lambda}}+k_{d}s^{\mu} \tag{36}\]  
where, \(\mu\) and \(\lambda\) represent the differentiation and integration orders of the controller, respectively.