Golden gait: An optimization theory perspective on human and humanoid walking

Abstract

Human walking is a complex task which includes hundreds of muscles, bones and joints working together to deliver harmonic movements with the need of finding equilibrium between moving forward and maintaining stability. Many different computational approaches have been used to explain human walking mechanisms, from pendular model to fractal approaches. A new perspective can be gained from using the principles developed in the field of Optimization theory and in particularly the branch of Game Theory. In particular we provide a new insight into human walking showing as the trade-off between advancement and equilibrium managed during walking has the same solution of the Ultimatum game, one of the most famous paradigms of game theory, and this solution is the golden ratio. The golden ratio is an irrational number that was found in many biological and natural systems self-organized in a harmonic, asymmetric, and fractal structure. Recently, the golden ratio has also been found as the equilibrium point between two players involved into the Ultimatum Game. It has been suggested that this result can be due to the fact that the golden ratio is perceived as the fairest asymmetric solution by the two players. The golden ratio is also the most common proportion between stance and swing phase of human walking. This approach may explain the importance of harmony in human walking, and provide new perspectives for developing quantitative assessment of human walking, efficient humanoid robotic walkers, and effective neurorobots for rehabilitation.Optimization theory is a branch of mathematics aiming at identifying the best choice, from some set of available alternatives, that optimizes (maximizes or minimizes) a specific target function (Asghar Bhatti, 2000). A well-known application of optimization theory to human locomotion refers to the fact that the comfortable walking speed is, for healthy individuals, that minimizing the energy consumption (Ralston, 1958; Miller et al., 2012; Oh et al., 2012; Long and Srinivasan, 2013; Seethapathi and Srinivasan, 2015). Game Theory is a branch of Optimization Theory in which there is not just one function to optimize, but there is the need to identify the best compromise among some entities involved into the problem (Kolokoltsov and Malafeyev, 2010). Game Theory has become a large and powerful theoretical framework providing mathematical models for predicting the choices of rational entities (usually called players) in conflict or in cooperation tasks (Rapoport, 1974; Sanfey, 2007). Mainly used in psychology, economy, political science, logic, computer science, Game Theory has also been enlarged to biology (Maynard Smith, 1982). Following this approach, game theoretical methods have been used in biochemistry and biophysics (Schuster et al., 2008), with some studies considering cells (Gatenby and Vincent, 2003) and even molecules (Bohl et al., 2004) as ‘‘players’’ working together or being in competition for the same objective. The idea of applying game theory to human walking proposed in this article originates by the observation that current advances in these so different research fields reported the same solution for two apparently different problems. In fact, the same equilibrium point was found in human walking and Ultimatum game: this point coincides with the so-called golden ratio. The golden ratio (φ) is the solution of the problem already reported by Euclid in III century B.C. to cut a given straight line so that the proportion between the shorter part to the longer one is the same as the longer part to the whole. It is an irrational number already found in many physical, biological fractal structures that are self-organized so that the larger-scale structure resembles the subunit structure (King et al., 2004; Yamagishi and Shimabukuro, 2008). In fact, it was found in structures of animal bodies (Livio, 2003) and plants’ leaves (Okabe, 2011), in the solar systems (Lombardi and Lombardi, 1984), replicated in architecture (Hemenway, 2005) and in certain musical rhythms (Garland, 1995), as well as in financial market patterns (Agaian and Gill, 2017). In humans, harmonic proportions have been found in the physiological activity of the heart (Yetkin et al., 2013) and in anthropometry (Davis and Altevogt, 1979), as depicted in figurative arts (Hemenway, 2005). In general, the golden ratio has been found as the best choice for many biological processes (Bartl et al., 2010; Yetkin et al., 2013; Schuster et al., 2017).