# Kinematic Modelling and Motion Analysis of a Humanoid Torso Mechanism
## 2 Materials and Methods
### Kinematic Modelling of a Compliant Tendon-Driven Torso Mechanism  
When compared to other similar tendon-driven mechanisms with four cables driving a joint, such as the rehabilitation device in [22], the LARMbot's torso mechanism poses additional challenges due to the compliance of the backbone. First, the backbone is not an idle joint, and, because of its stiffness, it exerts a wrench on the upper platform in any non-straight configuration. This wrench results in a higher actuation force required to move the torso mechanism but improves the dynamic behaviour of the mechanism by restoring a stable position when force closure is not achieved [22]. Furthermore, the torso mechanism, which is illustrated in the kinematic scheme in Figure 3, is also characterised by a degree of underactuation. The backbone has four DoFs, which can be characterised by the following parameters with reference to the kinematic schemes in Figure 3:  
* Backbone bending angle \(\theta\), which is defined as the angle between the \(x_{0}y_{0}\) plane of the lower torso frame \(\{S_{0}\}\) and the \(xy\) plane of the upper torso frame \(\{S\}\). * Direction of bending \(\varphi\), which is defined as the angle between the plane of bending and the \(x_{0}y_{0}\) plane of the lower torso frame \(\{S_{0}\}\). * Axial torsion \(\Delta\varphi\), which is defined as the angle between the \(x_{0}z_{0}\) plane of the lower torso frame \(\{S_{0}\}\) and the \(xz\) plane of the upper torso frame \(\{S\}\). * Axial elongation/compression \(\Delta\ell\), which is defined as the variation of the neutral backbone length \(\ell\) caused by the wrench acting on the component. However, only three of these DoFs can be actively controlled by the four actuators, as the tendons (routed as per Figure 3b,c) can only pull, but not push [22]. Thus, the configuration of the mechanism cannot be determined with motor position only, as its torsional mobility is underactuated and depends on the passive spring-like elements in the  
Figure 3: A kinematic scheme of the LARMbot’s torso mechanism: (**a**) Main geometrical and motion parameters of the compliant backbone; (**b**) Cross-section view of a vertebra with actuation tendon routing geometry; (**c**) Side representation of the entire system in a straight configuration with tendons routing points on the lower and upper torso. system, that is, the torso's backbone. Thus, kinematics must be integrated with a stiffness model of the backbone for complete characterization. As previously mentioned, a piecewise constant curvature kinematic (PCCK) model is used to describe the main bending of the backbone. PCCK models have been conceived as a convenient tool to analyse and control continuum robots, as explained in [25], even though they are recently being replaced by more complex models with variable curvature [26; 27] to accurately describe the hyper-redundant architecture of those systems, with complex motion coupling between independently bending sections. However, as the proposed torso mechanism can be represented by a single bending section, a constant curvature model can appropriately model the compliant backbone with limited approximation errors. To describe the pose of the lower and upper torso, the reference frames \(\{S_{0}\}\) and \(\{S\}\) can be defined at the base and at the end of the backbone, respectively. By assuming constant curvature along the backbone, the translation along its central curve \(\mathbf{t}\in\mathbb{R}^{3}\) can be expressed by an arc of a circle as:  
\[\mathbf{t}(\theta,\varphi,\Delta\ell)=\frac{\ell+\Delta\ell}{\theta}\left[ \begin{array} []{c}\cos\varphi(1-\cos\theta)\\ \sin\varphi(1-\cos\theta)\\ \sin\theta\end{array}