# Influence of Joint Stiffness and Motion Time on the Trajectories of Underactuated Robots
## 3 Effect of Joint Stiffness on Trajectories  
In order to evaluate the effect of joint stiffness of the last nonactuated joint, different simulations were carried out, considering the same motion time (\(t_{f}=0.6\,\mathrm{s}\)) and varying the stiffness of the nonactuated joint \(k_{3}\) in the range \((1.5\div 5.0)\times 10^{-3}\,\mathrm{Nm}/\mathrm{rad}\). The considered stiffness values are rather small but are compatible with the inertial parameters of the small robot, which are summarized in Table 1.  
The values of the natural frequency of the robot corroborate this statement. They can be calculated by means of Equation (4) and are reported in Table 2. With the smallest value of stiffness considered in the present parametric analysis, the natural frequency of the robot is \(1.1\,\mathrm{Hz}\), whereas with the largest value of torsional stiffness, the natural frequency increases to \(2.0\,\mathrm{Hz}\). These values are compatible with a small prototype made of light materials. Since the model includes a viscous damper in parallel with the torsional spring to take into account friction losses, the value of the damping coefficient \(c_{3}\) has to be defined as well. In this parametric analysis, a constant damping ratio \(\zeta_{3}\) is assumed [38]; hence, \(c_{3}\) is given by the following equation:  
\[c_{3}=\frac{\zeta_{3}k_{3}}{\pi f_{n3}} \tag{23}\]  
The value \(\zeta_{3}=0.03\) is adopted, which is compatible with a joint equipped with a ball bearing and a steel spring. Figure 2: Outline of the proposed approach. It is worth noting that the end-effector reaches the final position with a negligible positioning error. This result is achieved because the friction of the nonactuated joint is taken into account in the model. The end-effector trajectories obtained considering large values of stiffness are close to an arc of circumference, which is the trajectory of an equivalent robot with 2-DOF without the nonactuated joint and with the last link length \(a_{2}^{\prime}=a_{2}+a_{3}\). When torsional stiffness decreases, there is a modification in the central part of the trajectory with a reduction in the swept area and the formation of a protuberance. For larger values of stiffness, the protuberance tends to a cusp. Finally, for very low values of torsional stiffness, the protuberance degenerates into a loop, which is connected with the initial and final part of the trajectory by means of two small loops. Joint variables corresponding to the trajectories of Figure 3 are shown in Figure 4. Joint variable \(q_{1}\) does not depend on joint stiffness \(k_{3}\) because \(q_{1}\) coincides with flat variable \(y_{2}\)  
The trajectories of the end-effector depicted in Figure 3 are interesting, but the area of the workplace swept by the links of the robot is a more useful parameter, if the robot has to be implemented in a work cell. In fact, the swept area makes it possible to evaluate the capability of the robot to move in the presence of obstacles in the work cell. To calculate the area swept by the robot during the motion, a specific algorithm was developed. This algorithm operates as follows: every point of a link in configuration \(i\) is connected to the adjacent points in the same configuration and to the same points in configuration \(i+1\); some quadrilaterals are defined, which are represented in Figure 6a using different colors. Such quadrilaterals are joined to define a more complex polygon that describes the area swept by the robot during the motion from configuration \(i\) to configuration \(i+1\), which is represented in Figure 6b. The overlapping areas are counted once in the united area. This process is iterated between the initial and the final configuration. This algorithm is suitable both for links represented as segments, which is the case depicted in Figure 6, and for links described by more complex geometries (e.g., CAD files). Figure 4: Joint variables calculated considering different stiffness values and the same motion time (0.6 s).