Scaling contact force parameters across body size, limb count, and number of contact spheres

A popular way to model contact interactions in musculoskeletal simulations uses Hertz theory applied to contact spheres, with Hunt Crossley based dissipation. Suitable contact parameters for dynamic simulations will be highly dependent on the morphology, scale, materials, and movement in question. Inappropriate parameter choices can manifest in unpredictable ways during simulations, potentially resulting in misinterpretations or failed simulations. Here, I demonstrate that both the plane strain modulus and the dissipation parameters are not scale invariant. I derive equations to scale the contact parameters in dimensionless form, which allows accounting for differences in body size, number of legs, contact sphere radius, and number of spheres per foot. As a demonstration of this scaling approach, I scale the contact parameters of a 62 kg human to a 500 kg human, a mouse (0.02 kg), an emu (37.8 kg), a horse (545 kg), and a giraffe (1190 kg), and demonstrate that geometrically and dynamically similar contact behaviour is achieved in all cases. The scaling approach presented here can be used to scale parameters known to work for one model to a completely different model, which is particularly useful in studies that simulate the effects of allometric scaling. I also provide equations to estimate suitable contact parameters for a model directly, without using a different model as a starting point. The limitations of Hertz Hunt Crossley contact models in biomechanical simulations are discussed. Lastly, I derive dimensionless expressions and scaling guidelines for the smoothed contact force implementation "SmoothSphereHalfSpaceForce" in the popular biomechanical simulator OpenSim.