FOG – Fractional Order Central Pattern Generators for Legged Robots Gait Generation

A significant part of the Earth is inaccessible to wheeled mechanisms. Natural obstacles like rocks, loose soil, deep ravines, and steep slopes conspire to render rolling locomotion ineffective. Hills, mountains, seashores, seabeds, as well as the moon and other planets present similar terrain challenges. In many of these natural terrains legs are well-suited. They can avoid obstacles by making discrete contacts and passing up undesirable footholds. Legged robots can climb over obstacles and step across ditches, surmounting terrain discontinuities of body-scale while staying level and stable.

Although under study since several years, the existing artificial locomotion systems show limited mobility in controlled conditions and none shows capabilities near those revealed by biological systems. The reasons are, among others, due to the complexity of the legs control and coordination and the limited knowledge regarding locomotion gaits (Holmes et al., 2006).

In most cases legged robots joint trajectories are computed based in geometric considerations. But this method of generating the joint trajectories for robots to implement a specific gait if different from the one used in nature, in biological systems, although their kinematic structure often imitate the structure of animals skeletons (Holmes et al., 2006; Ijspeert, 2008).

Biologist have found physiological evidences that the rhythmic movements in humans and animals, such as walking, swimming, crawling, flying and breathing, are implemented through synchronized rhythmic movements, generated by neuronal elements, called Central Pattern Generators (CPG) (Grillner, 1985; Cohen et al., 1988). A large number of degrees of freedom (dof) are involved in legged locomotion and it is essential that the movements of all these dof are coordinated. This coordination is mainly implemented in the central nervous system that generates signals according to the desired gait. This process can be modeled by the CPG, that can be implemented through a group on non-linear neural oscillators, arranged into a network and mutually coupled in a way that allow the generation of stable limit cycles with an appropriate frequency and phase.

Each gait depends on the oscillator parameters and of their coupling. The generated signals can then be used as reference trajectories for the feedback control system of an artificial legged locomotion system (Ijspeert, 2008). This way locomotion gaits can be changed by changing a small number of parameters, being needed to implement a supervisory controller, in order to compute and supply the parameters for the CPG to implement a desired gait.

CPG architectures have been extensively studied for the locomotion of a variety of animals and their mathematical models have been developed and validated comparing the results obtained though simulation with experimental observations (Ijspeert, 2008).

Recently was shown that human and animal locomotion present periodic fractional order (FO) characteristics (West and Latka, 2005). In spite of this fact, all existent CPG models only consider integer order differential equations - never were developed approaches based in FO differential equations.

FO systems are a generalization of the classical differential equation description, in which the integrals and derivatives can have an arbitrary order (usually called fractional-order by historical reasons) (Samko et al., 1993; Podlubny, 1999). FO models are adequate to model the constitutive behavior of materials and physical systems exhibiting hereditary and memory properties (Oldham and Spanier, 1974; Hilfer, 2000). This fact constitutes a major advantage of fractional derivatives in comparison with classical integer models, where these effects are simply neglected.

Taking advantage of this research team previous experience in the areas of robotic locomotion systems modeling and control (Silva et al., 2005; Silva and Machado, 2006), FO systems (Barbosa et al., 2007), system optimization using genetic algorithms and particle swarm optimization (Reis et al., 2007) and biological inspired flying robots (Silva and Machado, 2008b), it is our idea to extend some of the CPG models, already developed and frequently adopted for generating the reference trajectories for the joints of legged robots, in such a way that the differential equations are generalized to FO.

After developing these models, they will be tested on a multi-legged robot locomotion simulator (Silva and Machado, 2005), adapted to quadruped robots, and it will be verified if the joint reference trajectories, generated this way, make the locomotion more efficient than the one achieved with “conventional” CPG models, from the viewpoint of some pre-established indices (Silva and Machado, 2008). We will also verify if the developed CPG models can be extended to the generation of the wing beat of biological inspired flying robots (Silva et al., 2008b).