# Nonlinear Control of Multibody Flexible Mechanisms: A Model-Free Approach
## 2 Dynamic Model of Multibody Flexible-Link Mechanisms  
In this section, the analytic model of a generic planar multibody mechanism composed of flexible links is briefly explained. The complete development of the equations of motion can be found in [29] together with an experimental validation of the accuracy of the model. The same dynamic model has been validated and used in several works, such as in [30] for the development of predictive control strategies for position control and vibration damping, and in [31] for the development of state observers for flexible-link mechanisms. Recently, such model has also been extended to 3D mechanisms in [32; 33; 34]. The dynamic model of flexible-link planar mechanisms is based on an Equivalent Rigid-Link System (ERLS) formulation, and on the use of a discretization based on the finite element method (FEM). First of all, each link of the mechanism must be divided into finite elements. The kinematics of the mechanism, which can have either a closed-loop kinematic chain or an open-loop one, is defined by means of the following vectors, measured in the fixed global reference frame \(\{X,Y,Z\}\): \(\mathbf{r}_{i}\) and \(\mathbf{u}_{i}\) are the vectors of nodal position and nodal displacement in the \(i\)-th element of the ERLS and of their elastic displacement, respectively; \(\mathbf{p}_{i}\) is the position of a generic point inside the \(i\)-th element; \(\mathbf{q}\) is the vector of the generalized coordinates of the ERLS. The relationship between these vectors is shown in Figure 1.  
The application of the principle of virtual works, which states that the sum of the virtual works of inertial, elastic and external forces is equal to zero, leads to:  
\[\sum_{i}\int_{\mathbf{v}_{i}}\phi\mathbf{p}_{i}^{T}\mathbf{\dot{p}}_{i}\mathrm{d} \upsilon+\sum_{i}\int_{\mathbf{v}_{i}}\delta e_{i}^{T}\mathbf{D}_{i}e_{i} \mathrm{d}\upsilon=\sum_{i}\int_{\mathbf{v}_{i}}\delta\mathbf{p}_{i}^{T} \mathbf{g}\mathrm{d}\upsilon+(\delta\mathbf{u}^{T}+\delta\mathbf{r}^{T}) \mathbf{F} \tag{1}\]  
\(e_{i}\), \(\mathbf{D}_{i}\), \(\rho_{i}\) and \(\delta e_{i}\) are the strain vector, the stress-strain matrix, the mass density of the \(i\)-th link and the virtual strains, respectively. \(\mathbf{F}\) is the vector of the external forces, including gravity. Equation (1) shows the virtual works of inertial, elastic an external forces, respectively. From this equation, \(\mathbf{p}_{i}\) and \(\mathbf{\dot{p}}_{i}\) can be evaluated, for a generic point of the \(i\)-th link from:  
\[\delta\mathbf{p}_{i}=\mathbf{R}_{i}\mathbf{N}_{i}\mathbf{T}_{i}\mathbf{\dot{ \sigma}}_{i} \tag{2}\]  
\[\mathbf{\dot{p}}_{i}=\mathbf{R}_{i}\mathbf{N}_{i}\mathbf{T}_{i}+2(\mathbf{R}_ {i}\mathbf{N}_{i}\mathbf{T}_{i}+\mathbf{R}_{i}\mathbf{N}_{i}\mathbf{T}_{i}) \mathbf{\dot{u}}_{i} \tag{3}\]  
where \(\mathbf{T}_{i}\) is a matrix that describes the transformation from global to local reference frame of the \(i\)-th element, \(\mathbf{R}_{i}\) is the local-to-global rotation matrix, and \(\mathbf{N}_{i}\) is the shape function matrix, which is obtained using a proper FEM description.