Presented at NACOB 98: North American Congress on Biomechanics Canadian Society for Biomechanics - American Society of Biomechanics University of Waterloo Waterloo, Ontario, Canada August 14-18, 1998

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PROCEDURES AND EXAMPLES OF MATRIX BASED FORWARD SOLUTION
DYNAMICS APPLIED TO SYMMETRIC HUMAN MOVEMENT SKILLS

T. Duck
Kinesiology and Health Science, York University
North York, Ontario, M3J 1P3

INTRODUCTION

One approach to forward solution dynamics of human movement skills is based on applying Newtonian mechanics to an N-link system. The segment equations can be developed using either forces and joint moments as driving functions, or alternately, using hypothesized joint movement as the input control. The latter approach assumes a specific associated force and joint moment pattern. If segment free body diagrams are appropriately constructed, the matrix for equation and subsequent differential solution can be formulated using specific sequences which are applicable from one to 'N' links. The current task, then, is to develop a matrix based forward dynamics solution for an N- link model of symmetric human movement skills. Specifically, a six-link model of a symmetric human body is assumed, with a single fixed support point. Example applications of such a model would be symmetric jumps (vertical, standing long, back somersault etc.), non-twisting platform dives, swim starts, and symmetric non- twisting horizontal bar skills.

THEORY AND PROCEDURES

Figure 1 illustrates a free body diagram for a jump and horizontal bar skill. For a jump the support force is assumed to be applied at the ball of the foot. For a bar skill, the bar is a fixed point, with bar torque set to zero or to an assumed or known function. Figure 2 identifies segment parameters and relevant angles for a three-segment system, which can be extended to "N" segments by adding similarly constructed links.

Figure 1: Six-Segment Symmetric Models

Figure 2: Segment Parameter and Angle Identification

Figure 3 illustrates a universal single segment free body diagram that can be used for all segments (i=1 to N). For the link used in support contact, moment T₁ is zero or prescribed, and for the last open link the distal force and moment variables are set to zero.

Figure 3: Universal Free Body Diagram

Planar equations using Newtonian mechanics, are established for each segment using the absolute angles defined as theta. The absolute angle of the first segment (theta 1) is the variable to be determined by the differential equation solution, when subject controlled angles (phi) are given. The controlled angles (phi) are replaced with equivalent theta (theta) based substitutions. The support contact for all models is a fixed point, which also permits the substitution of linear segment accelerations with their equivalent angular components. The equations are also refined based on relational information known about linked segments.

The result is 3 x N equations for the system with 3 x N unknowns, specifically joint linear forces (A), moments (T) and the orientation of the immediate support segment, (theta 1). The system of equations is solved using Gaussian elimination to determine the acceleration (theta 1"), followed with a differential solution for (theta 1), initially using Runge-Kutta procedures for the first four values, followed by the Adams- Moulton procedure for the remaining values.

When the planar equations are determined for the first three segments using the free body diagram and procedures described, then sequences are identifiable and can be used to construct the required N by N matrix. Figure 4 provides an example of a 9 by 9 matrix for a 3-segment, 9-equation system. The '#' symbol identifies constants that require defining, with similar group definitions identified in the table by the letters B to H.

Figure 4: Matrix for 3-Segment System

Using these procedures it is possible to solve forward simulation equations for any N-link symmetric system.

RESULTS

Figure 5 shows the results of a forward solution for a six-segment platform dive and horizontal bar skill.

Figure 5: Six-Segment Symmetric Forward Solution Examples

The present paper is limited to the support phase. Although the results show a clear intuitive validity, established validity boundaries are now required. This approach also provides an entry point for future joint moment controlled simulation.