## 3 Mathematical Modeling of the System  
To simplify, let us start with the derivation of the dynamics of an \(n\)-link under actuated manipulator. Thus, let \(\mathbf{q}=[q_{1},q_{2},...,q_{n}]^{T}\in\mathbb{R}^{n}\) denote the vector of generalized coordinates for the joint angles, and \(\mathbf{U}=[\Gamma_{1},\Gamma_{2},...,\Gamma_{n}]^{T}\in\mathbb{R}^{n}\) the vector of inputs and generalized torques. Hence the Lagrange dynamic equations read  
\[\mathbf{M}(\mathbf{q})\ddot{\mathbf{q}}+\mathbf{C}(\mathbf{q},\dot{\mathbf{q} })\dot{\mathbf{q}}+\mathbf{G}(\mathbf{q})=\mathbf{U}, \tag{1}\]  
where \(\mathbf{M}(\mathbf{q})\in\mathbb{R}^{n\times n}\) is a symmetric positive definite inertia matrix; \(\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\in\mathbb{R}^{n\times n}\) is the gyroscopic and Coriolis matrix; \(\mathbf{G}(\mathbf{q})\in\mathbb{R}^{n}\) is the vector of gravitational terms. To complete the dynamics we add the nonlinear external torque exerted by the claw at the base, i.e., \(q_{1}\). To this end, let us split the dynamics (1) into passive (claw) and active (control) subsystems as follows  
\[\begin{bmatrix}M_{1}&\mathbf{M_{c}}\\ \mathbf{M_{c}^{T}}&\mathbf{M_{2n}}\end{bmatrix}\begin{bmatrix}\ddot{q}_{1}\\ \ddot{\mathbf{q}}_{2n}\end{bmatrix}+\begin{bmatrix}C_{1}&\mathbf{C_{c1}}\\ C_{c2}&\mathbf{C_{2n}}\end{bmatrix}\begin{bmatrix}\dot{q}_{1}\\ \dot{\mathbf{q}}_{2n}\end{bmatrix}+\begin{bmatrix}g_{1}\\ \mathbf{g_{2n}}\end{bmatrix}=\begin{bmatrix}\tau_{c}\\ \boldsymbol{\tau}\end{bmatrix} \tag{2}\]  
where we have defined \(\mathbf{q_{2n}}:=[q_{2},q_{3},...,q_{n}]^{T}\), \(\mathbf{g_{2n}}\) in a similar way, \(\boldsymbol{\tau}:=[\Gamma_{2},\Gamma_{3},...,\Gamma_{n}]^{T}\) and \(\Gamma_{1}:=\tau_{c}\) stands for the external friction torque modeling the claw of a bird (see Figure 2c). Now, from the first equation of (2), let us denote the resulting dynamic torque at the base \(q_{1}\) whenever \(\dot{q}_{1}=0\) as \(\tau_{1}:=\mathbf{M_{c}}\ddot{\mathbf{q}}_{2n}+\mathbf{C_{c1}}\dot{\mathbf{q} }_{2n}+g_{1}\). Thus, as described in Section 2.3, the grasping torque provided by the claw, namely \(\tau_{c}\in\mathbb{R}\), can be modeled as a static friction--so called stictian--as follows  
\[\tau_{c}:=\begin{cases}\tau_{c,l}\cdot sgn[\dot{q}_{1}]&\quad\quad\quad\quad \quad if\ \dot{q}_{1}\
eq 0;\\ \tau_{c,l}\cdot sgn[\tau_{1}]&\quad\quad if\ \dot{q}_{1}=0\ and\ |\tau_{1}|>\tau_{c,l};\\ \tau_{1}&\quad\quad\quad\quad if\ \dot{q}_{1}=0\ and\ |\tau_{1}|<\tau_{c,l}; \end{cases} \tag{3}\]  
where \(\tau_{c,l}\) is its maximum value of the grasping torque. Notice that, if \(\tau_{1}\) does not surpass \(\tau_{c,l}\), the angular coordinate of the first joint remains unchanged, i.e., \(\dot{q}_{1}(t)\equiv 0\), \(t\geq 0\). More information about friction models see [23].  
**Remark 1**.: _Notice that the grasping torque of the claw \(\tau_{c}\) in (3) strongly modifies the dynamics when compared with a frictionless underactuated manipulator with a passive joint at its base, as it can be seen e.g., [15; 16; 24]. The static friction, so-called stiction, induces a highly nonlinear behavior around \(\dot{q}_{1}=0\), \(q_{1}=q_{10}\) constant._  
Finally, the Cartesian position (\(x_{t}\), \(y_{t}\)) of the end effector and joint coordinates are related through the geometric constraint, with respect to the reference frame (\(X,Y\)) defined in Figure 2c, and it reads  
\[\begin{bmatrix}x_{t}\\ y_{t}\end{bmatrix}=\begin{bmatrix}\sum_{i=1}^{n}l_{i}\cos\left(\sum_{i=1}^{n}q_{ i}\right)\\ \sum_{i=1}^{n}l_{i}\sin\left(\sum_{i=1}^{n}q_{i}\right)\end{bmatrix}. \tag{4}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\Delta x\ (\text{mm})\) & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 \\ \hline \(\tau_{c,l}(\text{Nm})\) & 0 & 0.015 & 0.036 & 0.060 & 0.081 & 0.1 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Mechanism characteristics.