Antagonistic activity of one-joint muscles in three-dimensions using non-linear optimisation q A. Jinha a , R. Ait-Haddou a , P. Binding b , W. Herzog a, * a Human Performance Laboratory, The University of Calgary, Calgary, AB, Canada T2N 1N4 b Department of Mathematics and Statistics, The University of Calgary, Calgary, AB, Canada T2N 1N4 Received 19 February 2004; received in revised form 4 October 2005; accepted 16 March 2006 Available online 30 March 2006 Abstract Non-linear optimisation, such as the type presented by R.D. Crowninshield and R.A. Brand [The pre- diction of forces in joint structures: Distribution of intersegmental resultants, Exercise Sports Sci. Rev. 9 (1981) 159], has been frequently used to obtain a unique set of muscle forces during human or animal move- ments. In the past, analytical solutions of this optimisation problem have been presented for single degree- of-freedom models, and planar models with a specific number of muscles and a defined musculoskeletal geometry. Results of these studies have been generalised to three-dimensional problems and for general for- mulations of the musculoskeletal geometry without corresponding proofs. Here, we extend the general solu- tion of the above non-linear, constrained, planar optimisation problem to three-dimensional systems of arbitrary geometry. We show that there always exists a set of intersegmental moments for which the given static optimisation formulation will predict co-contraction of a pair of antagonistic muscles unless they are exact antagonists. Furthermore, we provide, for a given three-dimensional system consisting of single joint muscles, a method that describes all the possible joint moments that give co-contraction for a given pair of antagonistic muscles. Ó 2006 Elsevier Inc. All rights reserved. 0025-5564/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.mbs.2006.03.018 q Research supported by I.W. Killam Foundation, NSERC of Canada, and the Canada Research Chair Programme. * Corresponding author. Tel.: +1 403 220 3438; fax: +1 403 284 3553. E-mail address: walter@kin.ucalgary.ca (W. Herzog). www.elsevier.com/locate/mbs Mathematical Biosciences 202 (2006) 57–70