# Two-Dimensional Mechanical Model of Human Stability in External Force-Caused Fall
## 3 Human Body Biomechanical Model  
The two-dimensional biomechanical model proposed is based on the Dempster model, for which the inertial properties are defined in [14]. The model consists of 13 parts corresponding to the main human body segments. The upper limb divides into the arm, forearm, and hand. The foot, shank, and thigh are elements of the lower limb. One element called further the torso represents the trunk, head, and neck. All the parts of the model are assumed to be rigid bodies. The segments are numbered from one to thirteen. The numbering approach, shown in Figure 3, is important due to the system of denotations that we have introduced to distinguish mechanical quantities corresponding to particular body segments. The characteristic dimensions of the segments, including the lengths of the limb parts denoted by \(l_{i}\), where \(i=1,2,\ldots 13\), are calculated according to the position of markers attached to bony prominences on the body. The mass of each segment, \(m_{i}\), is defined as the percentage of total body mass accordingly to the anthropometric Dempster data. Due to the symmetry of the body segments assumed, their centres of mass \(C_{i}\) lie on the individual axes of symmetry passing through the joints. Only the case of the torso's centre of mass is  
Figure 3: Front view (in the plane XY) on the biomechanical model. Figure 2: Placement of passive markers on the subject’s body. different--there the centre lies on the line of symmetry. In the case of limb segments, the location of the centre of mass is defined by the distance \(d_{i}\) measured from the proximal joint. The distances between the centre of mass of the torso and the greater trochanter and the acromion are denoted by \(d_{s}\) and \(d_{a}\), respectively. All the distances mentioned are specified relative to the segment length accordingly to [PERSON]'s data. The exception to this rule is the method of determining the position of the centre of mass of the foot. Taking into account the shape of the foot, a more adequate model seems to be a triangle whose vertices correspond to the markers of the side ankle, toe and heel. Thus, the centre of mass of the foot was taken at the centre of mass of this triangle. The mass moments \(I_{i}\) of inertia of particular segments about the axes that pass through the centres of mass and are perpendicular to the sagittal plane are needed. The radii of gyration, given by [PERSON] in the relative form, are used to calculate these quantities. The values of mass and mass moments of inertia of individual segments of the person doing exercises are specified in Table 1.  
The body segments are assumed to be connected by revolute joints (also called hinge or pin joints). These model joints correspond to the main joints of the human skeletal system, although due to the assumption of the planar motion of the biomechanical model, they do not always reflect the actual mobility of a given joint. There are 12 revolute joints in the model. Musculoskeletal interactions between two segments connected by a joint are modelled by a force and a couple at a given joint. Passing through the joints on the right side from the right foot to the right hand, we encounter a situation in which the indexes of the body segments are arranged in ascending order. Then, the internal force and couple at the joint linking the _ith_ body segment with the (_i_+1)_th_ segment receive the lower of the two indexes. The internal forces and couples at the joints of the left upper limb are subscripted from seven to nine, and the numbering starts from the shoulder joint. In turn, the subscripts assigned to internal forces and couples at the joints of the left lower limb change from 10 to 12, starting from the ankle joint. The force \(\overset{\rightarrow}{F}_{i}\), where \(i=1,2,\ldots 12\), is resolved into two rectangular components parallel to \(Y\) and \(Z\) axes. The couples accompanying them are represented by the moments \(\overset{\rightarrow}{M}_{i}\) that are perpendicular to the \(YZ\)-plane.