Bipedal Locomotion: A Fractional CPG for Generating Patterns

Proceedings of the 10th Conference on Dynamical Systems Theory and Applications

Complex order biped rhythms

Animal locomotion is a complex process, involving the central pattern generators (neural networks, located in the spinal cord, that produce rhythmic patterns), the brainstem command systems, the steering and posture control systems and the top layer structures that decide which motor primitive is activated at a given time. Pinto and Golubitsky studied an integer CPG model for legs rhythms in bipeds. It is a four-coupled identical oscillators' network with dihedral symmetry. This paper considers a new complex order central pattern generator (CPG) model for locomotion in bipeds. A complex derivative Dα±jβ, with α, β ∈ ℜ+, j = √-1, is a generalization of the concept of an integer derivative, where α = 1, β = 0. Parameter regions where periodic solutions, identified with legs' rhythms in bipeds, occur, are analyzed. Also observed is the variation of the amplitude and period of periodic solutions with the complex order derivative.

Fractional central pattern generators for bipedal locomotion

Locomotion has been a major research issue in the last few years. Many models for the locomotion rhythms of quadrupeds, hexapods, bipeds and other animals have been proposed. This study has also been extended to the control of rhythmic movements of adaptive legged robots. In this paper, we consider a fractional version of a central pattern generator (CPG) model for locomotion in bipeds. A fractional derivative D α f(x), with α non-integer, is a generalization of the concept of an integer derivative, where α=1. The integer CPG model has been proposed by Golubitsky, Stewart, Buono and Collins, and studied later by Pinto and Golubitsky. It is a network of four coupled identical oscillators which has dihedral symmetry. We study parameter regions where periodic solutions, identified with legs’ rhythms in bipeds, occur, for 0<α≤1. We find that the amplitude and the period of the periodic solutions, identified with biped rhythms, increase as α varies from near 0 to values close to unity.

Complex-order dynamics in hexapod locomotion

This paper studies the dynamics of foot–ground interaction in hexapod locomotion systems. For that objective the robot motion is characterized in terms of several locomotion variables and the ground is modelled through a non-linear spring-dashpot system, with parameters based on the studies of soil mechanics. Moreover, it is adopted an algorithm with foot-force feedback to control the robot locomotion. A set of model-based experiments reveals the influence of the locomotion velocity on the foot–ground transfer function, which presents complex-order dynamics.

Properties and Sustainability of Biodiesel from Animal Fats and Fish Oil

This work presents and analyses the fat and fuel properties and the methyl ester profile of biodiesel from animal fats and fish oil (beef tallow, pork lard, chicken fat and sardine oil). Also, their sustainability is evaluated in comparison with rapeseed biodiesel and fossil diesel, currently the dominant liquid fuels for transportation in Europe. Results show that from a technological point of view it is possible to use animal fats and fish oil as feedstock for biodiesel production. From the sustainability perspective, beef tallow biodiesel seems to be the most sustainable one, as its contribution to global warming has the same value of fossil diesel and in terms of energy efficiency it has the best value of the biodiesels under consideration. Although biodiesel is not so energy efficient as fossil diesel there is room to improve it, for example, by replacing the fossil energy used in the process with renewable energy generated using co-products (e.g. straw, biomass cake, glycerine).