# Influence of Joint Stiffness and Motion Time on the Trajectories of Underactuated Robots
## 2 Mathematical Model
It is worth noticing that \(f_{n3}\) is always finite, since \(I_{2}^{*}>I_{3}^{*}\), independently from the sizes of the two links. From the equations of motion, the motor torques \(\tau_{1}\) and \(\tau_{2}\) can be calculated as functions of the desired trajectory. First, by multiplying the second row by \(I_{3}^{*}\) and the third row by \(I_{2}^{*}\) and subtracting the latter to the former, it is possible to obtain motor torque \(\tau_{2}\):  
\[\tau_{2}=(I_{3}^{*}-I_{2}^{*})\tilde{q}_{3}-\frac{I_{2}^{*}}{I_{3}^{*}}(c_{3}q_ {3}+k_{3}q_{3}) \tag{5}\]  
It is worth noting that motor torque \(\tau_{2}\) depends only on the joint variable \(q_{3}\). Then, the motor torque \(\tau_{1}\) can be expressed as a function of motor torque \(\tau_{2}\) and of angular acceleration of the first joint \(\tilde{q}_{1}\). By subtracting the second row of Equations (1) from the first, this expression of the first motor torque (\(\tau_{1}\)) holds:  
\[\tau_{1}=\tau_{2}+\tilde{q}_{1}(I_{1}^{*}-I_{2}^{*}) \tag{6}\]  
The Laplace transform method is used to define the flat variables, since the equations are linear [38]. The equations of motion in the Laplace domain can be written as follows:  
\[ \begin{bmatrix}I_{1}^{*}&I_{2}^{*}&I_{3}^{*}\\ I_{2}^{*}&I_{2}^{*}&I_{3}^{*}\\ I_{3}^{*}&I_{3}^{*}&I_{3}^{*}\end{bmatrix}\begin{bmatrix}s^{2}Q_{1}(s)\\ s^{2}Q_{2}(s)\\ s^{2}Q_{3}(s)\end{bmatrix} + \begin{bmatrix}0&0&0\\ 0&0&0\\ 0&0&c_{3}\end{bmatrix}\begin{bmatrix}sQ_{1}(s)\\ sQ_{2}(s)\\ sQ_{3}(s)\end{bmatrix} + \begin{bmatrix}0&0&0\\ 0&0&0\\ 0&0&k_{3}\end{bmatrix}\begin{bmatrix}Q_{1}(s)\\ Q_{2}(s)\\ Q_{3}(s)\end{bmatrix} = \begin{bmatrix}T_{1}(s)\\ T_{2}(s)\\ 0\end{bmatrix}  \tag{7}\]where \(Q_{1}(s)\),\(Q_{2}(s)\) and \(Q_{3}(s)\) are the Laplace transforms of joint variables, whereas \(T_{1}(s)\) and \(T_{2}(s)\) are the Laplace transform of motor torques. From the last row, the first flat variable in the Laplace domain \(Y_{1}(s)\) can be defined:  
\[Q_{3}(s)=-\frac{I_{3}^{*}s^{2}(Q_{1}(s)+Q_{2}(s)+Q_{3}(s))}{k_{3}+c_{3}s} \tag{8}\]  
The term \(Y_{1}(s)=Q_{1}(s)+Q_{2}(s)+Q_{3}(s)\) is the first flat variable in the Laplace domain, and \(Q_{3}(s)\) can be expressed as a function of only flat variable \(Y_{1}(s)\) as follows:  
\[Q_{3}(s)=-\frac{I_{3}^{*}}{k_{3}+c_{3}s}s^{2}Y_{1}(s) \tag{9}\]  
The second flat variable coincides with the joint variable \(Q_{1}(s)\):  
\[Y_{2}(s)=Q_{1}(s) \tag{10}\]  
The Equation (5) of the motor torque \(\tau_{2}\) in the Laplace domain becomes  
\[T_{2}(s)=(I_{3}^{*}-I_{2}^{*})s^{2}Q_{3}(s)-\frac{I_{2}^{*}}{I_{3}^{*}}(c_{3} s+k_{3}q_{3})Q_{3}(s) \tag{11}\]  
and introducing Equation (9) into Equation (11), it is possible to obtain the torque \(T_{2}(s)\) as a function of only flat variable \(Y_{1}(s)\). The following equation holds:  
\[T_{2}(s)=I_{2}^{*}s^{2}Y_{1}(s)+\frac{I_{3}^{*}(I_{2}^{*}-I_{3}^{*})}{k_{3}+c_ {3}s}s^{4}Y_{1}(s) \tag{12}\]  
Collecting \(k_{3}\) from the denominator of the second term of Equation (12), the expression becomes  
\[T_{2}(s)=I_{2}^{*}s^{2}Y_{1}(s)+\frac{I_{3}^{*}(I_{2}^{*}-I_{3}^{*})}{k_{3}} \bigg{(}1+\frac{c_{3}}{k_{3}}s\bigg{)}^{-1}s^{4}Y_{1}(s) \tag{13}\]  
In underactuated robots equipped with steel springs and low-friction ball bearings, the equivalent damping coefficient \(c_{3}\) is rather small with respect to joint stiffness \(k_{3}\); hence, the second term of Equation (13) can be approximated.