Spherical one-degree-of-freedom models of the knee and ankle joints for lower limb modelling

In virtually unloaded conditions, the human tibiofemoral (knee) and tibiotalar (ankle) joints behave as single degree-of-freedom (1DOF) systems [1]. In these conditions, fibers within the ligaments remain nearly isometric throughout the flexion arc and articular surfaces nearly rigid. Relevant theoretical models are showing that the ligaments and the articular surfaces act together as mechanisms to control the passive joint kinematics [2-4]. In the knee joint, isometric fibers were identified within the anterior cruciate, posterior cruciate and medial collateral ligaments, whereas rigid contacts were associated to the two condylar articular surfaces [2,3]. In the ankle, isometric fibers were identified within the calcaneal-fibular and tibio-calcaneal ligaments, rigid contacts were associated to the articular surfaces between the tibio-fibular mortise and the talus [4]. Kinematic measurements and corresponding model predictions also revealed that the instantaneous screw axes of both the knee and ankle motion all pass near to a single point, hereinafter called the pivot point (Figure 1). This observation proves that a nearly-spherical motion is experienced by these human joints. Based on this experimental evidence, a first group of mechanisms was proposed by these authors for the kinematic modeling of the knee and ankle joints [3,4]. These models feature two members (i.e. the rigid bones including cartilages) in mutual contact at the articular surfaces and interconnected by rigid links (i.e. the ligaments’ isometric fibers). Different models were defined by increasing the quality of articular surface representation; in particular, the model that features a spherical approximation for each of the anatomical articulating surfaces (hereinafter called ESM2, for consistency with previous studies [3]) revealed to be very accurate. These mechanisms replicated well the natural kinematics of these joints by direct representations of the anatomical structures that guide the passive motion. However, these mechanisms are limited by high computational costs and overall mechanical complexity; they show also numerical instability of the relevant mathematical models, that is source of singularity problems, of high sensitivity to geometrical parameter variations and of oscillations of the simulated motion. All these aspects can be a problem when simpler analyses are necessary as in musculo-skeletal modeling of the lower limb, or when simple and stable prostheses and orthoses have to be designed. Thus, still novel mechanisms are necessary to drastically simplify the geometry of the models and to improve their numerical stability. Spherical Parallel Mechanisms (SPM) are being analyzed [5-7], based both on anatomy and kinematics of human joints: two isometric ligaments (the cruciates for the knee and the calcaneal-fibular and tibio-calcaneal ligaments for the ankle) are modeled as binary links of constant length, and the bones are also connected by a spherical pair centered at the pivot point, in order to approximate the nearly spherical motion observed for the knee and ankle joints (Figure 2). The model is still a 1DOF equivalent spatial mechanism. The present study is aimed at defining and testing original SPM models for the kinematics analysis of the human knee and ankle joints. For this purpose, anatomical and kinematics measurements were taken from in-vitro experiments, from which the models were defined. Model predictions were compared with original measurements, and with those from the previous equivalent mechanism ESM2. Advantages and disadvantages of these new models are discussed.