Introduction

Computational techniques are now regularly used to investigate the locomotion of extinct species. For example finite element analysis is now frequently used to investigate the strength of skeletal elements under load (Rayfield et al. 2001; Manning et al. 2006; Sereno et al. 2007), and imaging techniques such as LIDAR are powerful tools in the analysis of fossil trackways of vertebrates (Bates et al. 2008). However the most directly applicable technique is locomotor modelling. Models vary from the highly theoretical (e.g., Alexander 1992; McGeer 1992; Minetti and Alexander 1997; Srinivasan and Ruina 2006) to more realistic simulations (e.g., Yamazaki et al. 1996; Sellers et al. 2003; Nagano et al. 2005), and these approaches have been used both to understand the fundamental mechanics of terrestrial gait and also to predict gait parameters: either those internal values that are difficult to measure directly or for fossil vertebrates where experimentation is impossible.

Simple models have the great advantage of being straightforward to understand and unequivocal in their predictions. More complex models depend on a much greater number of modelling parameters and are therefore more difficult to interpret. However, because they are based more closely on real organisms they can be directly tested through comparison of their predictions to those obtained experimentally. For example Srinivasan and Ruina's recent model (Srinivasan and Ruina 2006) predicts that there are three types of stable, efficient bipedal locomotion that they describe as walking, pendular running and running. The model itself is highly simplified with mass-less, linear sprung limbs and a point mass body, and so its predictions cannot be tested experimentally. More realistic models are similarly able to produce these gaits spontaneously (Sellers et al. 2004; Sellers et al. 2005) but because they are more closely modelled on the morphology of experimental subjects, their predictions are more accurate and can be directly compared with experimental data.

The earliest musculoskeletal models for use in reconstructing gait in vertebrate fossils date back to the pioneering work of Yamazaki et al. (Yamazaki et al. 1996) who produced a highly sophisticated neuromusculoskeletal simulation to investigate the evolution of bipedality in humans and other primates. Indeed it is early human fossils that have received the most attention with a number of models of extinct bipeds having been produced (Crompton et al. 1998; Kramer 1999; Sellers et al. 2004; Nagano et al. 2005; Sellers et al. 2005; Ogihara and Yamazaki 2006). However, the study of terrestrial vertebrate locomotion has also involved computer simulations: terror birds (Blanco and Jones 2005) and dinosaurs (Gatesy et al. 1999; Stevens 2002; Hutchinson et al. 2005; Sellers and Manning 2007). Simulations require a musculoskeletal model and are either kinematically based where a movement pattern is provided for the animal based on either trackway data or motion-capture information from extant species, or kinetically based where the simulation is driven by muscular forces usually with a global optimisation criterion to produce efficient models of high speed locomotion. The latter approach is particularly valuable in situations where there are no suitable modern analogues (usually because a particular morphological form is no longer found) or where there is little trackway data (for example most forms of high speed locomotion).

Simulation technology is now relatively mature with a range of both commercial [e.g., SC.ADAMS, SDFast, MADYMO] and open-source [e.g., Dynamechs, ODE, PhysSym, and Tokamac] along with front ends that allow simplified construction of biomechanical models [e.g., SIMM, OpenSIMM, Marilou Robotics Studio, and LifeMod]. Gait production also requires gait controllers, and for complex models it is impractical to explore exhaustively the whole solution space so a selective search is needed to find suitable controller parameters. Popular techniques are to use finite steps within the search space (Srinivasan and Ruina 2006); to constrain parts of the model using functional linkage and pre-designed neural networks (Yamazaki et al. 1996); and to use genetic algorithms to explore selectively profitable areas of the search space (Sellers et al. 2003). These approaches are frequently combined with genetic algorithms and this combined approach is particular popular in robotics leading to the term evolutionary robotics (Nolfi and Floreano 2000).

Hadrosaurian dinosaurs are an ideal study animal for gait reconstruction given they were particularly diverse, and their fossil remains relatively abundant (Horner et al. 2004). Trackway interpretations have suggested that they were gregarious (Carpenter 1992; Lockley and Matsukawa 1999). Their gait is particularly interesting because there has been a debate on the function of their well-developed ossified tendons arrayed along their spinal columns (Organ 2006), and also as to whether they were quadrupedal, bipedal or indeed facultatively able to switch between the two gaits (Galton 1970; Maryańska and Osmólska 1984; Lockley 1992; Meyer and Thuring 2003). Added to that a number of extraordinarily well-preserved specimens have been found with probable soft tissue preservation (for review see Manning 2008). One particular aspect of gait that has received considerable attention is the prediction of maximal running speed. Chasing down prey is a vital factor in the lives of extant predators, as is the avoidance of being captured for prey animals. It is, therefore, of little surprise that speed estimation is of such interest to vertebrate palaeobiologists. Recent work estimating dinosaur maximum running speeds (Sellers and Manning 2007) provided a good match between predicted and simulated top speeds of extant bipeds. However, the results for bipedal dinosaurs were surprising in that there was a strong inverse relationship between body mass and top speed. Theoretical considerations have suggested that maximal running speed should be independent of body mass (Hill 1950) or should increase with body mass (Blanco and Gambini 2007). However, experimental and observational data suggest that there is an optimum body size for running speed at about 100 kg with both smaller and larger animals running slower (Garland 1983). These data suggest that simple theoretical models are unable to adequately represent the diversity of physical processes involved in maximal speed running and in particularly the differences associated with varying gaits (bipedal, quadrupedal, symmetrical and asymmetrical), and it is this aspect that is the focus of this paper.