DETERMINING LOADABILITY OF N-ARTICULAR CHAINS

Participants: R.E. Hughes, J. King

Keywords: finger, kinetics

Introduction

Understanding the mechanics of bi-articular chains is important for the conceptualization of finger mechanics. Leijnse (1996) has proposed a theoretical model for understanding the controllability of a planar model of a massless finger. The model explains the relationship between muscle moment arms at multiple joints and the ability of the finger to resist loads applied to the fingertip, which leijnse termed "loadability." Specific conditions (on moment arm magnitudes) were presented that guarantee the bi-articular chain is loadable. Leijnse concluded that the analysis cannot be extended to n-articular chains, e.g. chains with n articulations. However, each digit has three articulations (metacarpophalangeal, proximal interphalangeal, and distal interphalangeal), so fingers should be modeled as a 3-articular chains. The purpose of this project is to generalize the theory Leijnse to n-articular chains (n>2)

Materials and Methods

Leijnse considered a planar model of a bi-articular chain. Static equilibrium was formulated as

x₁R_E +x₂R_F + x₃R_I = -V (1)

where R_E=[r_E1 r_E2], R_F=[r_F1 r_F2], and R_I=[r_I1 r_I2] are vectors of muscle moment arms and V=[v_K1 v_K2] is a vector of moment arms of the force applied to the distal end of the chain (v_K1=d₁sinQ, v_K2=d₂sinQ, where d₁ and d₂ are the distances from the first and second joint to the point of application of external force). The subscripts F, E, and I denote the flexor, extensor, and interosseous muscles, respectively. The force generated by muscle i is x_i. The angle of load application is Q. The system is termed "loadable" by Leijnse (1996) if there exists finite x_i ³ 0, i=1, .., 3, satisfying (1) for all load directions 0<Q<360.

A more general set of conditions was obtained by applying Farkas’ Lemma (Rockafellar, 1970) to system (1), which gives the following result: either the system is loadable or there is a solution to

l ^. R_{E £} 0 (2)

l ^. R_{F £} 0 (3)

l ^. R_{I £} 0 (4)

l ^. V < 0 (5)

but not both. Therefore, the bi-articular chain is loadable if and only if there does not exist a solution to (2)-(5). Farkas’ Lemma is true for n-dimensional vector spaces. Therefore, this result applies if R_E, R_F, and R_I are n-dimensional. The theory applies to n-articular systems.

Progress

The theory has been applied to a model of the finger containing three articulations, and the results are in preparation for submission to ASME Transactions on Biomechanical Engineering.

References

1. Leijnse, J.N.A.L., "A graphic analysis of the biomechanics of the massless bi-articular chain. Application to the proximal bi-articular chain of the human finger," Journal of Biomechanics, Vol. 29, 1996, pp. 355-366.

2. Rockafellar, R.T., 1970, Convex analysis, Princeton University Press, Princeton, NJ, pp. 200-201.