# Influence of Joint Stiffness and Motion Time on the Trajectories of Underactuated Robots
## 2 Mathematical Model  
The mathematical model presented in this section was developed according to the theory of differentially flat systems [24] for a robot moving in the horizontal plane. Figure 1 shows the scheme of the planar robot with 3-DOF with 2 actuated joints and 1 passive joint. In this paper, the actuated joints are considered infinitely rigid. In order to consider the friction in the nonactuated joint, a viscous damper (with viscous coefficient \(c_{3}\)) was placed in parallel with the torsional spring (with stiffness \(k_{3}\)). \(q_{1}\) and \(q_{2}\) are the angular positions of the actuated joints, whereas \(q_{3}\) is the angular position of the nonactuated joint. The mass of the \(i\)-th link prior to balancing is denoted by \(m_{i}\), while the counterbalancing mass of the \(i\)-th link is denoted by \(m_{ci}\). The barycentric moment of inertia of the \(i\)-th element prior to balancing is represented by \(I_{Gi}\); \(a_{Gi}\) is the distance from the center of mass of the \(i\)-th link to the \(i\)-th joint prior to balancing, and \(a_{i}\) is the total length of the \(i\)-th link. The distance of the counterbalancing mass from the axis of the \(i\)-th joint is denoted by \(a_{Ci}\). The robot is designed to work in the horizontal plane; hence, the counterbalancing mass of the first joint is not needed [25]. The mathematical model describing the dynamics of the system can be derived with the Lagrangian approach. The equations of motion in matrix form are:  
Figure 1: Scheme of the 3-DOF robot. \[ \begin{bmatrix}I_{1}^{*}&I_{2}^{*}&I_{3}^{*}\\ I_{2}^{*}&I_{2}^{*}&I_{3}^{*}\\ I_{3}^{*}&I_{3}^{*}&I_{3}^{*}\end{bmatrix}\begin{bmatrix}q_{1}\\ q_{2}\\ q_{3}\end{bmatrix} + \begin{bmatrix}0&0&0\\ 0&0&0\\ 0&0&c_{3}\end{bmatrix}\begin{bmatrix}q_{1}\\ q_{2}\\ q_{3}\end{bmatrix} + \begin{bmatrix}0&0&0\\ 0&0&0\\ 0&0&k_{3}\end{bmatrix}\begin{bmatrix}q_{1}\\ q_{2}\\ q_{3}\end{bmatrix} = \begin{bmatrix}\tau_{1}\\ \tau_{2}\\ 0\end{bmatrix}  \tag{1}\]  
where \(I_{i}^{*}\) is the moment of inertia of the system regarding joint \(i\) due to the following links (i.e., \(i,\dots,3\)), and \(\tau_{1}\) and \(\tau_{2}\) are the motor torques applied on the first and second joint, respectively. It is worth noting that if the conditions for differential flatness reported in Section 1 are met, the mass matrix is constant (thus, it is independent from the configuration), and the typical nonlinear terms of robot dynamics (Coriolis and centrifugal terms) disappear, as discussed by [PERSON] in [25]. Since the robot considered in this work moves in the horizontal plane, the nonlinear gravitational torque on joint 1, which is present in [25], disappears as well. The natural frequencies of the robot can be calculated by setting the motor torques and the damping coefficient to zero and solving the equations of free undamped vibrations assuming a harmonic solution \(\mathbf{q}=\mathbf{q}_{0}e^{i\omega t}\):  
\[ \begin{bmatrix}-I_{1}^{*}\omega^{2}&-I_{2}^{*}\omega^{2}&-I_{3}^{*}\omega^{2}\\ -I_{2}^{*}\omega^{2}&-I_{2}^{*}\omega^{2}&-I_{3}^{*}\omega^{2}\\ -I_{3}^{*}\omega^{2}&-I_{3}^{*}\omega^{2}&-I_{3}^{*}\omega^{2}+k_{3}\end{bmatrix} \begin{bmatrix}q_{01}\\ q_{02}\\ q_{03}\end{bmatrix} = \begin{bmatrix}0\\ 0\\ 0\end{bmatrix}  \tag{2}\]  
The natural frequencies are the zeros of the determinant of the matrix at the left hand side of Equation (2). The following equation holds:  
\[\omega^{4}[I_{3}^{*}(I_{2}^{*}-I_{3}^{*})(I_{2}^{*}-I_{1}^{*})\omega^{2}+k_{3}I _{2}^{*}(I_{1}^{*}-I_{2}^{*})]=0 \tag{3}\]  
Equation (3) clearly highlights the presence of two null frequencies, corresponding to rigid body motions; and one natural frequency, which is given by the following equation:  
\[f_{n3}=\frac{1}{2\pi}\sqrt{\frac{k_{3}I_{2}^{*}}{I_{3}^{*}(I_{2}^{*}-I_{3}^{*}) }} \tag{4}\]  
This natural frequency corresponds to a mode of vibration that chiefly involves the nonactuated link. It is noteworthy that the natural frequency of the robot depends only on the inertia of the second and the third link.