# Kinematic Modelling and Motion Analysis of a Humanoid Torso Mechanism
## 4 Discussion  
In this paper, a kinematic model was proposed for a four-DoF tendon-driven compliant torso mechanism for humanoid robots. The proposed model is presented with a closed-form analytical formulation and it is used to compute the workspace of the mechanism under examination. The results are validated on a prototype, which is operated with a joystick on which the proposed model is mapped. The main findings of this paper can be summarised as:  
* _Mobility analysis:_ The mobility of a four-DoF tendon-driven compliant torso mechanism for humanoid robots is analysed to identify its main modes of motion. The degrees of mobility of the robot are further classified as active degrees of freedom, which can be controlled by the action of one or more actuators, and passive degrees of freedom, which depend on the intrinsic stiffness of the system only. * _Kinematic modelling:_ The proposed constant curvature kinematic model, which is usually used for continuum robots, is used to describe the main bending mode of motion \((\theta,\ \varphi)\) of the spine of the torso. The conventional constant curvature kinematic model is characterised for the analysed system, and an expanded model is here proposed to include backbone elongation \(\Delta\ell\) and torsion \(\Delta\varphi\). * _Stiffness modelling:_ A linearised stiffness model is introduced to create an efficient framework with lumped parameters to relate the deformation of the backbone to the wrench acting on the upper torso of the mechanism. This model also outlines how the axial torsion of the backbone cannot be controlled by the actuation tendons, as it can only be caused by external wrenches. * _Workspace analysis:_ The behaviour of the torso mechanism is characterised by evaluating its motion limit as a reachable workspace, bending and direction angles, and tendon displacement. * _Joystick mapping:_ A joystick mapping of the main motion parameters of the proposed kinematic model is proposed, with direction and angle of bending linearly mapped to joystick orientation and magnitude respectively. Figure 12: Power consumption acquired with a current sensor during the torso motion reported in Figures 8 and 9. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Maximum Angular Value** & **Maximum Acceleration** \\ \hline Backbone bending & \(\theta=17.95\) deg & \(\alpha_{\theta}=35.75\) rad/s\({}^{2}\) \\ \hline Backbone torsion & \(\Delta\varphi=1.54\) deg & \(\alpha_{z}=8.32\) rad/s\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Experimental results. * _Experimental validation:_ The workspace computed with the proposed model is confirmed by experiments with the prototype, whose motion is acquired by an onboard inertial measurement unit. The main advantages of the proposed model, which replaces a previous approximated model calibrated on experimental data, can be identified as higher accuracy and efficiency and a quick response thanks to the closed-form analytical formulation. Furthermore, this model can be also used to evaluate the operating performance of the robot over the reachable workspace and to ensure that the tendon-driven architecture operates within its wrench-closure workspace. Future works will focus on refining the current model by introducing a dynamic and stiffness model of the system, which will be validated with the addition of load cells on the tendons; by defining the performance of the torso through numerical indices that can be used to optimise motion planning; by analysing the compliant behaviour of the backbone with Finite Element analysis simulations in order to evaluate the error introduced by the lumped parameter model that is proposed in this manuscript; and finally by integrating this model with the kinematics of the whole humanoid, in order to implement a dynamic control of the system. Conceptualization, M.R., M.C., and D.C.; methodology, M.R. ; software, M.R. and D.C.; validation, D.C.; formal analysis, M.R. and M.C. ; investigation, M.R. ; resources, M.C. and D.C.; data curation, M.R.