# Influence of Joint Stiffness and Motion Time on the Trajectories of Underactuated Robots
## 2 Mathematical Model
The factor \((1+(c_{3}/k_{3})s)^{-1}\) has the form \((1+x)^{p}\) with \(p=-1\) and \(x=(c_{3}/k_{3})s\), and can be approximated by means of the Taylor expansion arrested at the second term:  
\[(1+x)^{p}\approx 1+px \tag{14}\]  
Hence, Equation (13) can be written as  
\[T_{2}(s) =I_{2}^{*}s^{2}Y_{1}(s)+\frac{I_{3}^{*}(I_{2}^{*}-I_{3}^{*})}{k_ {3}}\bigg{(}1-\frac{c_{3}}{k_{3}}s\bigg{)}s^{4}Y_{1}(s) \tag{15}\] \[T_{2}(s) =I_{2}^{*}s^{2}Y_{1}(s)+\frac{I_{3}^{*}(I_{2}^{*}-I_{3}^{*})}{k_ {3}}s^{4}Y_{1}(s)-\frac{c_{3}I_{3}^{*}(I_{2}^{*}-I_{3}^{*})}{k_{3}^{2}}s^{5}Y_ {1}(s) \tag{16}\]  
Motor torque \(\tau_{2}\) in the time domain can be obtained by calculating the inverse Laplace transform of Equation (16):  
\[\tau_{2}(t)=I_{2}^{*}\dot{y}_{1}+\frac{I_{3}^{*}(I_{2}^{*}-I_{3}^{*})}{k_{3}} y_{1}^{(4)}-\frac{c_{3}I_{3}^{*}(I_{2}^{*}-I_{3}^{*})}{k_{3}^{2}}y_{1}^{(5)} \tag{17}\]The motor torque of joint 1 in the Laplace domain is  
\[Q_{3}(s)=-\frac{I_{3}^{*}}{k_{3}+c_{3}s}s^{2}Y_{1}(s) \tag{9}\]0  
and in the time domain, it becomes  
\[Q_{3}(s)=-\frac{I_{3}^{*}}{k_{3}+c_{3}s}s^{2}Y_{1}(s) \tag{9}\]1  
In the next sections, the robot is simulated considering a point-to-point motion, and the initial and final configurations of the robot for these simulations are  
\[\boldsymbol{q}_{i}= \begin{bmatrix}q_{1i},q_{2i},q_{3i}\end{bmatrix} ^{\mathsf{ T}}= \begin{bmatrix}0,-\pi/3,0\end{bmatrix} ^{\mathsf{T}},\boldsymbol{q}_{f}= \begin{bmatrix}q_{1i},q_{2i},q_{3i}\end{bmatrix} ^{\mathsf{T}}= \begin{bmatrix} \pi,\pi/4,0\end{bmatrix} ^{\mathsf{T}} \tag{20}\]  
In a point-to-point motion, the robot moves from one configuration to another, starting and stopping with null velocity. Hence, the derivatives of \(q_{1}\) and \(q_{2}\) are null. Coming to \(q_{3}\), the torsional spring on the passive joint tries to push \(q_{3}\) towards the rest position, hence at starting and end positions, the passive joint has both null angle and derivatives. Since Equation (17) requires the continuity of the fifth derivative of \(y_{1}\), this flat variable is calculated by means of an 11-degree polynomial, whereas since Equation (19) requires only the continuity of the second derivative of \(y_{2}\), this flat variable is calculated by means of a 5-degree polynomial. Hence, the initial and final conditions for the flat variables are  
\[\boldsymbol{y}(t_{i}) = \begin{bmatrix}y_{1},y_{1},y_{1},y_{1}^{(3)},y_{1}^{(4)},y_{1}^{ (5)},y_{2},\dot{y}_{2},\dot{y}_{2}\end{bmatrix} ^{\mathsf{T}}= \begin{bmatrix} q_{1i}+q_{2i}+q_{3i},0,0,0,0,q_{1i},0,0\end{bmatrix} ^{\mathsf{T}} \tag{21}\] \[\boldsymbol{y}(t_{f}) = \begin{bmatrix}y_{1},y_{1},\dot{y}_{1},y_{1}^{(3)},y_{1}^{(4)},y_ {1}^{(5)},y_{2},\dot{y}_{2},\dot{y}_{2}\end{bmatrix} ^{\mathsf{T}}= \begin{bmatrix} q_{1f}+q_{2f}+q_{3f},0,0,0,0,0,q_{1f},0,0\end{bmatrix} ^{\mathsf{T}} \tag{22}\]  
It is worth noting that the robot will not fall into singular configurations, since the robot is controlled in the joint space rather than in Cartesian space. In the framework of this research, a small prototype robot is simulated. The inertial parameters of the robot are summarized in Table 1, and will be used for the simulations. The operational scheme utilized to carry out the analysis is illustrated in Figure 2. Initially, the start point and the end point were selected, and subsequently, the joint variables were determined using inverse kinematics. Following that, the flat variables were computed. Motor torques \(\tau_{1}\) and \(\tau_{2}\) were calculated using the flat variables and their derivatives. All these calculations were performed analytically. On the one hand, the torques were employed to simulate the system in Matlab-Simulink solving the equations of motion using standard integration routines. On the other hand, the same torques were used as inputs for the motors in the experimental tests. Finally, the Cartesian trajectories were obtained. Since neither optimization processes nor stochastic methods were used, time and space complexity are not a concern.