# Kinematic Modelling and Motion Analysis of a Humanoid Torso Mechanism
###### Abstract  
This paper introduces a novel kinematic model for a tendon-driven compliant torso mechanism for humanoid robots, which describes the complex behaviour of a system characterised by the interaction of a complex compliant element with rigid bodies and actuation tendons. Inspired by a human spine, the proposed mechanism is based on a flexible backbone whose shape is controlled by two pairs of antagonistic tendons. First, the structure is analysed to identify the main modes of motion. Then, a constant curvature kinematic model is extended to describe the behaviour of the torso mechanism under examination, which includes axial elongation/compression and torsion in addition to the main bending motion. A linearised stiffness model is also formulated to estimate the static response of the backbone. The novel model is used to evaluate the workspace of an example mechanical design, and then it is mapped onto a controller to validate the results with an experimental test on a prototype. By replacing a previous approximated model calibrated on experimental data, this kinematic model improves the accuracy and efficiency of the torso mechanism and enables the performance evaluation of the robot over the reachable workspace, to ensure that the tendon-driven architecture operates within its wrench-closure workspace. humanoid robotics; assistive robotics; service robotics; mechanism design; kinematics; cable-driven robots; compliant mechanisms; underactuated mechanisms; motion analysis; workspace +
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humanoid robot designs have started introducing torso motion control [10] or one or more degrees of mobility in the torso to enhance performance. These designs range from simple serial mechanisms [11; 12; 13] to parallel architectures [14; 15; 16], which have demonstrated how much torso mobility can improve robot performance. Among these robots, the LARMbot humanoid [17; 18; 19; 20] is equipped with a cable-driven compliant torso mechanism with a complex behaviour inspired by the human spine: a central compliant element represents the backbone; actuated cables act as muscle tendons to define the shape of the central element, and rigid disk-like vertebrae route the cables in parallel to the backbone [15]. In previous studies, the behaviour of this mechanism has been numerically determined by linearly interpolating motion calibration data experimentally [16]. However, while a satisfying performance can be obtained with this procedure, the result is limited to a specific design and geometry and high precision can be ensured only in a limited motion range. Conversely, a model-based control would ensure correct behaviour of the mechanism over its whole range of motion and for any given geometry. While an analytical kinematic model would significantly improve torso performance, modelling the complex cable-driven compliant mechanism poses several challenges. While hybrid mechanisms with a rigid joint in parallel to two or more actuation cables have been kinematically modelled in the past [21; 22], previous results can only be partially applied to the proposed mechanism due to the increased mobility and compliance of the compliant backbone, which is capable of three main motion modes: bending, compression, and axial torsion. A solution to this can be found in continuum and snake-like robot literature. These systems are characterised by a compliant backbone with bending capabilities [23; 24] that are described through constant [25] or variable [26; 27] curvature kinematics. These models use two variables to define the bending mode, namely bending angle and direction of the bending plane, and they are also capable of describing backbone elongation [28]. In this paper, an extension of the constant curvature kinematic model which includes torsion and elongation is proposed as a novel solution to define a kinematic model that accurately describes the complex behaviour of the LARMbot's torso mechanism. First, the mechanical design of the torso is introduced with its main constructive details.