**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 closedform 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:


• *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.

**Author Contributions:** 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.; writing—original draft preparation, M.R.; writing—review and editing, M.R., M.C., and D.C.; visualization, M.R. and D.C.; supervision, M.C. and D.C.; project administration, M.C. and D.C.; funding acquisition, D.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded by a gran<sup>t</sup> from Ministero della Salute (Ricerca Corrente 2021).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
