# Kinematic Modelling and Motion Analysis of a Humanoid Torso Mechanism
## 3 Results
### Experimental Validation  
To validate the proposed kinematic model, the workspace result was compared to the reachable workspace of the prototype as per Figure 6, which was evaluated in the configuration space of the robot by measuring the orientation of the upper torso. The experimental setup is illustrated in Figure 7 with its main components, which include the torso prototype with an embedded IMU on the upper torso on the last Functional Spinal Unit (FSU) to extract the orientation data, an Arduino microcontroller with the kinematic model, a power supply, and a joystick (spring-loaded to the centre) to teleoperate the system. All the motors are embedded in the waist of the robot not to hinder spine motion. While a wired control was used in this experimental validation, wireless communication can enable autonomous navigation with the humanoid robot. The motion of the torso was mapped to the joystick motion according to the proposed kinematic model, with the direction of the joystick controlling the direction of bending \(\varphi\) and the position of the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Backbone Length \(\ell\)** & **Bending Angle \(\theta\)** & **Bending Direction \(\varphi\)** \\ \hline
129.0 mm & [0; 15] deg & [0; 360] deg \\ \hline
**Backbone compression \(\Delta\ell\)** & **Tendon length \(l_{i}\)** & **Tendon radius \(r\)** \\ \hline
0.0 mm & [120.0; 138.0] mm & 34.0 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Size and motion parameters of the torso mechanism prototype in Figure 2.  
Figure 5: Workspace of the LARMbot’s torso mechanism evaluated through the proposed PCCK model with elongation and torsion. joystick linearly mapped to the bending angle \(\theta\). The extreme positions of this mapping are illustrated in Figure 6 with the corresponding upper torso pose. Figure 6: Motion control of the LARMbot’s torso mechanism through a joystick, with motion mapping to angle of direction and bending angle. F: forward; FL: forward-left; L: left; BL: backward-left; B: backward; BR: backward-right; R: right; FR: forward-right. In the reported experiment, the robot was driven from a central position (straight backbone) to a point on the outer border of its workspace and then was moved around the entire border once before going back to the central position (see Figure 6). This motion was repeated four times, and the motion data of the upper torso were reported in Figures 8 and 9 as acquired by the IMU. Figure 8 illustrates the acquired motion as bending around the \(x\)- and \(y\)-axes of the torso mechanism, whereas Figure 9 reports the overall bending angle and axial torsion of the torso. The proposed torso mechanism can reach and move around the limits on its workspace, as illustrated in Figure 10, which shows the acquired IMU data points during a manually operated motion around the reachable workspace that was computed with the new analytical model. The experiments also show a negligible backbone elongation and torsion in absence of any external wrench, as motion is here obtained by tendon action only. The proposed model achieves a smooth motion, with acceleration values always below 36 rad/s\({}^{2}\), as reported in Figure 11. The power consumption is also fairly low, with a maximum value below 15 W, as reported in Figure 12. The experimental results are summarised in Table 3.  
Figure 8: Raw orientation data acquired by the IMU on the upper torso expressed as bending around the \(x\)- and \(y\)-axes of the upper torso of the robot in Figure 6.  
Figure 7: Experimental setup for the validation of the proposed kinematic model, including the torso mechanism, an Arduino microcontroller, a joystick for teleoperation, and a power supply. Figure 2.0 Acceleration data acquired by the IMU on the upper torso mapped as angular acceleration around the instantaneous bending axis \(\alpha_{\theta}\) and around the backbone axis \(\alpha_{z}\).