Abstract
The idea of fractional calculus is not new. Fractional derivatives are almost as old as integer-order definition. In 1695 Leibniz discussed this problem with L'Hospital, but many other contributions are due to investigators such as Liouville, Abel, Heaviside and Riemann, that formalized the theory of the non-integer order systems. The area of fractional calculus has primarily been the domain of mathematicians, and only had the theoretical foundation. Nowadays, this concept is employed in physics, engineering, biology, economy and other scientific fields. In our work, we apply the concepts of fractional calculus and the theory of electrical impedance to botanical elements. The fractional order behaviour of these type of systems are studied and the relation with the electrical impedance is formulated.

Abstract
In this paper, we incorporate fuzzy reasoning in fractional-order PD controllers. The resulting fuzzy fractional PD controller is investigated in terms of its digital implementation and robustness. The combined features of both algorithms will result in a better control system performance. Also, the effectiveness and robustness of the new methodology is illustrated through an application example.

Abstract
This paper compares the performance of classical position PD algorithm with a cascade controller involving position and force feedback loops, for multi-legged locomotion systems and variable ground characteristics. For that objective the robot prescribed motion is characterized in terms of several locomotion variables. Moreover, we formulate several performance measures of the walking robot based on the robot and terrain dynamical properties and on the robot hip and foot trajectory errors. Several experiments reveal the performance of the different control architectures in the proposed indices.

Abstract
This paper analyzes the performance of a hexapod robot under the action of a cascade controller involving position and force feedback loops. In order to evaluate the system properties we formulate several measures of the walking system based on the system dynamics and the trajectory errors. Simulation experiments reveal the influence of the different control algorithms upon the proposed indices.

Abstract
This paper studies quasi-periodic gaits of multi-legged robot locomotion systems based on the analysis of the dynamic model. The purpose is to determine the system performance during walking and the best strategy to overcome an obstacle. For that objective the robot prescribed motion is characterized in terms of several locomotion and obstacle variables. In this perspective, we formulate three performance measures of the walking robot namely, the mean absolute power, the mean power lost in the joint actuators and the mean force of the interface body-legs per walking distance. A set of model-based experiments reveals the influence of the obstacle position and dimensions in the proposed indices.

Abstract
This paper presents the energy analysis of periodic gaits for multi-legged locomotion systems. The main purpose is to determine the system performance during walking and the best set of locomotion variables that minimizes a cost function related to energy. For that objective, the prescribed motion of the robot is completely characterized in terms of several locomotion variables such as gait, duty factor, body height, step length, stroke pitch, maximum foot clearance, link lengths, body and legs mass and cycle time. In this work, we formulate three indices to quantitatively measure the performance of the walking robot namely the mean absolute power, the mean power dispersion and the mean power lost in the joint actuators. A set of experiments reveals the influence of the locomotion variables in the proposed indices.