## 2 Methods and Results
### Moment Equations and Muscle Torques
In three dimensions, the mathematic formalism relative to the closed-form dynamic equations of the system becomes extremely complex [4]. An example of such 3D analysis has been recently presented for a six-link chain, with sagittal plane symmetry, to model the biomechanics of the barbell bench press exercise [26]. Another limitation stems from the assumption that only one contact external resistance acts on link \(i\). In general, multiple contact external resistances can simultaneously act on an individual link of the chain. In two dimensions, the system of contact external resistances acting on a link constitutes a co-planar system of forces acting on a rigid body. This system can always be reduced to the resultant force \(\overset{\rightarrow}{R}\) applied at a point of the central axis (if \(\overset{\rightarrow}{R}\
eq 0\)), or to a force couple (if \(\overset{\rightarrow}{R}=0\)). The first case (\(\overset{\rightarrow}{R}\
eq 0\)) was considered in this study. In the latter case (\(\overset{\rightarrow}{R}=0\)), the dynamic force equations and the joint reaction forces (Equations (9) and (10)) are not affected by the contact external resistances, while the torque of the equivalent force couple should simply be added to the moment Equations (20) and (21). The 2D reaction forces exerted on link \(i\) by link \(i-1\) constitute a co-planar system of forces acting on a rigid body as well. In general, the resultant \(\overset{\rightarrow}{\Phi}_{i,i-1}\) of these forces is non-zero in musculoskeletal joints. Therefore, this system of forces was reduced to the vector \(\overrightarrow{\Phi}_{i,i-1}\) applied at a point of the central axis. It was assumed that the line of action of this applied vector passes through \(\mathsf{J}_{i}\). This implies that the _moment_ of \(\overrightarrow{\Phi}_{i,i-1}\) about \(\mathsf{J}_{i}\) is zero. In biomechanics, this assumption is often considered acceptable as long as the joint is not close to the extremes of its range of motion [27]. Furthermore, this assumption does not affect the force dynamic equations and the closed-form expression of the joint reaction forces (Equations (9) and (10)). With these limitations, the analytical results derived in this paper can be applied to the study of the biomechanics of multilink musculoskeletal chains with deep/superficial and segmental/global muscles. One possible use is for modeling spinal dynamics during spine flexion, extension, and lateral flexion movements, where each vertebra is represented as a link. Another possible application that concerns chains with more than three links and deep/superficial multi-joint muscles is related to the flexion/extension movements of the fingers. The extrinsic flexor muscles of the fingers, the flexor digitorum superficialis and flexor digitorum profundus, cross more than three joints, from the elbow to the proximal interphalangeal joint, and from the wrist to the distal interphalangeal joint, respectively. Many natural lower limb movements, such as those occurring during walking, running, and jumping, involve sagittal-plane motion of the hip, knee, ankle, and metatarsophalangeal joints. The dynamics of these movements can be analyzed by Equations (1)-(24), as long as the toes (the basis of the chain) remain in contact with the ground. It is important to point out that, in contrast to numerical computational models of human motion, the closed-form analytical model established in the present study clearly highlights the functional contribution to the joint forces and torques of each relevant biomechanical parameter (force of each single-joint or multi-joint muscle spanning or non-spanning the joint; weight and contact external resistances acting on each individual link of the chain; position, angular velocity, and angular acceleration of each individual link of the chain). Therefore, the effect of a change in any of these parameters can be readily assessed and understood.