# Nonlinear Control of Multibody Flexible Mechanisms: A Model-Free Approach
## 3 Nonlinear Control Design  
The proposed control algorithm is based on the following control action:  
\[\tau(t)=k_{p}(q^{*}-q(t))-k_{d}q(t)-k_{n}q(t)(q^{*}-q(t))^{2} \tag{9}\]  
being \(q^{*}\) the reference for the angular position of the mechanism \(q(t)\). \(k_{p}\), \(k_{d}\) and \(k_{n}\) are scalar quantities that represent the tuning parameters of the controller. The control action is nonlinear, given the presence of the term \(k_{n}q(t)(q^{*}-q(t))^{2}\). In the following, it will be shown that the nonlinear control action, without which the control acts as a pure PD controller, can be effectively used to improve the damping of high frequency vibration induced on the mechanism during the motion. The PD controller is often used for the control of this class of mechanisms, since the derivative action acts as a damping factor for the elastic displacement. A notable work on the subject [35] also reports that any flexible-link mechanism, even in the presence of gravity, can be asymptotically stabilized by a simple joint PD controller, therefore without the need to measure elastic displacements. The proposed controller belongs to the wide class of nonlinear PID control, according to the definition of [36], which lists in this class any PID control with time-varying proportional, derivative and integral gains. The use of nonlinear PID controllers is recognized by the capability of improving the performance when used to control nonlinear plants, or to improve damping, tracking accuracy or to reduce rise time for rapid inputs for linear plant. The control law (9) can be written as a PD control with a nonlinear derivative gain described by the equivalent gain \(k_{d}^{\prime}(t)=k_{d}+k_{n}(q^{*}-q(t))^{2}\), and, therefore, as a derivative gain modulation based on the magnitude of the state of the plant, according to [36]. The numerical results provided in this work will show that the modulation of the derivative gain can improve both the position tracking and the vibration damping for fast-changing reference signals. The conditions for the stability of the closed loop system can be derived by taking into account the following Lyapunov function:  
\[V(t)=E_{k}(t)+E_{p}(t)+\frac{k_{p}}{2}(q^{*}-q(t))^{2} \tag{10}\]  
\(E_{k}(t)\) and \(E_{p}(t)\) are the kinetic and the potential energy of the mechanism, that cannot assume negative values. If \(k_{p}\) is a positive scalar, also \(V(t)\geq 0\). According to [PERSON]'s lemma [37], the asymptotic stability can be proved if \(V(t)\) is lower bounded, \(\dot{V}(t)\) is negative definite, and \(\dot{V}(t)\) is uniformly continuous in time. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline crank length & 0.373 m \\ coupler length & 0.525 m \\ follower length & 0.632 m \\ ground link length & 0.36 m \\ beam width & 6 mm \\ beam thickness & 6 mm \\ [PERSON]’s modulus & \(2\times 10^{11}\) N/m\({}^{2}\) \\ flexural inertia moment & \(11.102\times 10^{-2}\) m\({}^{4}\) \\ link mass/length & 0.272 kg/m \\ Rayleigh damping \(\alpha\) & \(8.7\times 10^{-2}\) \\ Rayleigh damping \(\beta\) & \(2.1\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical parameters of the flexible mechanism. Let us verify also the second condition imposed by [PERSON]'s lemma. The time derivative of \(V(t)\) is:  
\[\dot{V}(t)=\dot{E}_{k}(t)+\dot{E}_{p}(t)-q(t)k_{p}(q^{*}-q(t)) \tag{11}\]  
If damping is neglected from the dynamic model of the mechanism, under conservative conditions, the variation of the total energy of the system over a time \(t\) is only due to the work done by the actuator over the same time interval:  
\[\Delta E_{k}(t)+\Delta E_{p}(t)=\int_{0}^{t}\tau(s)\dot{q}(s)\mathrm{d}s \tag{12}\]  
being \(\tau\) the torque provided by the actuator.