# Nonlinear Control of Multibody Flexible Mechanisms: A Model-Free Approach
###### Abstract  
In this paper a novel nonlinear controller for position and vibration control of flexible-link mechanisms is introduced. The proposed control strategy is model-free and does not require the measurement of the elastic deformation of the mechanism, since the control relies only on the knowledge of the angular position of the actuator and on its time derivative, which can be measured simply with a quadrature encoder. The conditions for the closed-loop stability are evaluated using Lyapunov theory. The performance of the proposed technique is evaluated on a four-bar flexible-link mechanism. Superior vibration damping and more accurate trajectory tracking is obtained in comparison with a PD controller and a fractional order controller, which relies on the same set of measurement as the proposed nonlinear controller.  
nonlinear control; flexible mechanism; dynamic modeling; fractional control 20212021110821108211  
multibody mechanisms. It should also be considered that control practitioners often rely on simple models as the basis for model-based controls [1], such as modal systems with a limited number of state variable. In this case, the reduced accuracy in the modeling should be compensated by increasing the robustness properties [23; 24] of the controller. This is necessary to avoid the crucial problem of spillover effect, i.e., the neglecting of higher-order modes, which can be a source of instability if combined with linear time-varying parameter perturbations [25; 26]. While the research community has proven to be able to cope with these difficulties by including robustness in the control formulation [24; 27; 28], the solution to the problem is still not within the availability of most industrial applications.  
The aim of this paper is to introduce a simple nonlinear control strategy that is shown to be very effective for the simultaneous position control and vibration damping of flexible multibody mechanisms. The proposed approach is of straightforward implementation, has limited computational requirements and, therefore, could be potentially made available to standard industrial control systems. Moreover, the control strategy introduced in this paper has two important characteristics:  
1. the control is model-free;
2. the control relies on one measured variable only.  
The first characteristic is fundamental in all industrial applications, where the required knowledge to operate with complex nonlinear models is not often found. The field of application of the proposed solution is further enlarged, since the control strategy relies only on the measurement of the angular position of the mechanism, which can be easily obtained with quadrature encoders.  
The paper is outlined as follows: Section 2 briefly recalls the dynamic modeling of planar flexible mechanisms used for the numerical evaluation and the proof of stability of the proposed control strategy. Control design and stability conditions are explained in Section 3, whereas Section 4 reports some numerical evaluation of the closed-loop performance and a comparison with PD control and fractional order PD\({}^{\mu}\) controllers. The conclusions are given in Section 5.