# Nonlinear Control of Multibody Flexible Mechanisms: A Model-Free Approach
## 3 Nonlinear Control Design
The time derivative of (12) is:  
\[\dot{E}_{k}(t)+\dot{E}_{p}(t)=\tau(t)\dot{q}(t) \tag{13}\]  
which, substituted into (11), leads to:  
\[\dot{V}(t)=\tau(t)\dot{q}(t)-\dot{q}(t)k_{p}(q^{*}-q(t)) \tag{14}\]  
Using the formulation of the control torque \(\tau(t)\) as defined in (9), the last equation leads to:  
\[\dot{V}(t)=\dot{q}(t)\big{(}k_{p}(q^{*}-q(t))-k_{d}q(t)-k_{n}\dot{q}(q^{*}-q(t ))^{2}\big{)}-\dot{q}(t)k_{p}(q^{*}-q(t)) \tag{15}\]  
which can also be written as:  
\[\dot{V}(t)=-\dot{q}^{2}(t)\Big{(}k_{n}(q^{*}-q(t))^{2}+k_{d}\Big{)} \tag{16}\]  
The derivative of the Lyapunov function \(\dot{V}(t)\) is negative as long as the term \(k_{n}(q^{*}-q(t))^{2}+k_{d}\) is positive, i.e., if \(k_{n}(q^{*}-q(t))^{2}>-k_{d}\). Under this condition, the closed loop system is dissipative and global stability is guaranteed. It should be mentioned that this condition is quite conservative, since it does not take into account the dissipative action of the damping terms neglected in the Lyapunov candidate function in (12). The third condition imposed by [PERSON]'s lemma is met if \(\dot{V}(t)\) is uniformly continuous. According to [23], a sufficient condition for the uniform continuity of \(\dot{V}(t)\) is the boundedness of \(\dot{V}(t)\). The latter condition will be demonstrated here. First of all, \(V(t)\leq V(0)\) for every \(t\geq 0\), as a consequence of the results presented so far. On the other hand, \(V(t)\) is the sum of nonnegative terms (see Equation (10)). Therefore, the boundedness of \(V(t)\) implies the boundedness of all the three terms on the right-hand side of Equation (10). The boundedness of the kinetic energy \(E_{k}(t)\) also implies the boundedness of \(\dot{q}(t)\) and of \(\dot{u}(t)\). At the same time, the existence of bounds on the potential energy \(E_{p}(t)\) implies that \(u(t)\) is bounded [35]. Therefore, also \(\tau(t)\) is bounded, according to the control law in Equation (9). Now, from Equation (16), the second time derivative of the Lyapunov candidate function is:  
\[\ddot{V}(t)=-2\dot{q}(t)\ddot{q}(t)\Big{(}k_{n}(q^{*}-q(t))^{2}+k_{d}\Big{)}+ 2k_{n}\dot{q}^{3}(t)q(t) \tag{17}\]  
This equation clearly shows that if also \(\dot{q}(t)\) is bounded, then \(\ddot{V}(t)\) is bounded as well. On the other hand, the boundedness of \(\tau(t)\) implies the boundedness of the vector of external forces \(\mathbf{f}\) that appears in (7), both in the presence and the absence of gravity force. Now, Equation (8) can be rewritten as:  
\[\left[ \begin{array}{c}\ddot{\mathbf{u}}\\ \ddot{\mathbf{q}}\end{array} \right]=\left[ \begin{array}{cc}\mathbf{M}^{ \dagger}&(\mathbf{MS})^{\dagger}\\ (\mathbf{S}^{T}\mathbf{MS})^{\dagger}&\mathbf{S}^{T}\mathbf{MS}\end{array}  \right]^{-1}\!\!\left[ \begin{array}{c}\mathbf{f}^{T}\\ \mathbf{S}^{T}\mathbf{f}\end{array} \right] \tag{18}\]Up to now it has been shown that \(\mathbf{t}(\mathbf{u},\dot{\mathbf{u}},\mathbf{q},\dot{\mathbf{q}})\) is bounded, and as the consequence of the boundedness of \(\dot{\mathbf{q}}\) also the sensitivity matrix \(\mathbf{S}\) is bounded. Therefore, the left-hand side of Equation (18) is bounded as long as the matrix inversion can be performed. But the matrix inversion can be performed as long as the mechanism does not encounter, during the motion, any singular configuration, and this condition is met provided that a proper choice of the zeroed elastic displacement is made [29]. Now, if the left-hand side of (18) is bounded, also \(\ddot{\mathbf{q}}(t)\) and consequently \(\ddot{V}(t)\) are bounded as well, according to (17). Therefore, all the conditions imposed by [PERSON]'s lemma are met and the control is stable, provided that the control tuning parameters are set within the above prescribed limits.