# Nonlinear Control of Multibody Flexible Mechanisms: A Model-Free Approach
## 2 Dynamic Model of Multibody Flexible-Link Mechanisms
Taking \(\mathbf{B}_{i}(x_{i},y_{i},z_{i})\) as the strain-displacement matrix, the following relations holds:  
\[\epsilon_{i}=\mathbf{B}_{i}\mathbf{T}_{i}\mathbf{\dot{\sigma}}\mathbf{u}_{i} \tag{4}\]  
\[\delta\epsilon_{i}=\mathbf{B}_{i}\mathbf{\dot{\sigma}}\mathbf{N}_{i}\mathbf{T} _{i}\mathbf{u}_{i}+\mathbf{B}_{i}\mathbf{N}_{i}\mathbf{T}_{i}\mathbf{\dot{ \sigma}}\mathbf{u}_{i} \tag{5}\]  
Since nodal elastic virtual displacements \(\mathbf{\dot{\sigma}}\mathbf{u}\) and nodal virtual displacements of the ERLS \(\delta\mathbf{r}\) are independent from each other, the equations of motion of the system can be written as:  
\[\mathbf{M}\mathbf{\dot{u}}+\mathbf{M}\mathbf{S}\mathbf{\dot{q}}=\mathbf{f}( \mathbf{u},\mathbf{\dot{u}},\mathbf{q},\mathbf{\dot{u}}) \tag{6}\]  
\(\mathbf{S}^{T}\mathbf{M}\mathbf{\ddot{u}}+\mathbf{S}^{T}\mathbf{M}\mathbf{S} \mathbf{\dot{q}}=\mathbf{S}^{T}\mathbf{f}(\mathbf{u},\mathbf{\dot{u}},\mathbf{ q},\mathbf{\dot{u}}  
differential equations in (6) and (7) can be made solvable by forcing to zero as many elastic displacement as the generalized coordinates. In this way, the ERLS position is defined unequivocally. By removing the displacements forced to zero from Equations (6) and (7), an operation indicated by the operator \({}^{\dagger}\), the following equation is obtained:  
\[\left[ \begin{array}{cc}\mathbf{M}^{\dagger}&(\mathbf{MS})^{\dagger}\\ (\mathbf{S}^{T}\mathbf{MS})^{\dagger}&\mathbf{S}^{T}\mathbf{MS}\end{array}  \right]\left[ \begin{array}{c}\tilde{\mathbf{u}}^{\dagger}\\ \tilde{\mathbf{u}}\end{array} \right]=\left[ \begin{array}{c}\mathbf{f}^{ \dagger}\\ \mathbf{S}^{T}\mathbf{f}\end{array} \right] \tag{8}\]  
In this way, the values of the accelerations can be computed at each step by solving the system in Equation (8), while the values of velocities and displacements can be obtained by an appropriate integration scheme, provided that the integration step is set to a sufficiently small value. The structure of the mechanism taken into account for the numerical simulations is depicted in Figure 2. The mechanical parameters and the corresponding values are reported in Table 1. The finite element discretization of the mechanisms uses one finite element for the crank and the coupler links, while the follower, i.e., the longest link, is discretized with two finite elements. The ground link is considered to be perfectly rigid. The total number of elastic displacement for all the links taken as separated would be 21, but after assembling the mechanism and consequently taking into account the coupling between the links, the number of nonzero elastic displacement is reduced to 12. A visual representation of the elastic displacements is provided in Figure 3.  
Figure 3: Four-bar linkage: nodal elastic displacement \(u_{i}\) and rigid displacement \(q\). Figure 2: Four-bar linkage mechanism. The simulation results reported in Section 4 are provided, as regards the elastic motion, in terms of the link curvature instead of in terms of nodal elastic displacements, the latter being of less direct interpretation.