Performance study of propulsion of N-link artificial Eukaryotic flagellum swimming microrobot within a fractional order approach: From simulations to hardware-in-the-loop experiments

Abstract

The understanding of how bioinspired artificial microrobots propel themselves by propagating a planar wave along their flagellum is crucial to improve their mechanical design, as well as their performance. Likewise, the implementation of such a planar wave motion in N-link swimming microrobots involves several challenges, among with the motion control of actuators can be highlighted, whose load (viscous drag forces) does not only depend on their own link and motion, but also on their position along the flagellum. This paper proposes an improved locomotion for an N-link artificial Eukaryotic flagellum (AEF) swimming microrobot taking into account a fractional order approach for both the waveform design for propulsion and the control of the flagellum distributed dynamics. On the one hand, the novel way of swimming, based on a fractional order power law for the amplitude modulation, allows to preserve the motion properties obtained applying classical traveling waveforms, but presenting some benefits in terms of propulsion. On the other, a robust fractional order proportional-derivative (PD^μ) controller is designed for the motion control of the microrobot. To demonstrate the advantages and validate both the waveform and the control strategy proposed, a hardware-in-the-loop (HIL) testbed for a 4-link robot is built. It consists of a microrobot simulator developed with the physical modeling tools in the MATLAB/Simulink environment and the embedded microcontroller Atmel ATmega32u4, where the control of the robot is programmed. The testbed, thanks to the simulator, allows to select different modes of swimming and geometry for the microrobot, as well as evaluating the performance of the locomotion in terms of propulsion, power efficiency or tracking. Experimental and simulation results are given to show that the best efficiency, with regard to both the way of swimming and the energy consumption with the control applied, is achieved with the proposed fractional approach.

Introduction

The study of bioinspired artificial microrobots reproducing the functionality of biological systems at micro and nanoscales has attracted a great attention in recent years. However, working at such scales involves, among other challenges, exploring new and efficient ways to propel such a kind of swimmers [9], [31], [36], which can lead to many applications, such as making existing therapeutic and diagnostic procedures less invasive [29] and robots that manipulate and/or interact with tiny entities of the environment [5], [8]. Likewise, conventional actuation principles can not be applied in those applications, due to the fact that microrobots have to navigate within environments characterized by a low Reynolds number (Re, which is a dimensionless parameter that quantifies the ratio between inertial and viscous forces in a fluid), i.e., within environments dominated by viscous forces. A review on the theoretical framework for locomotion at low Re can be found in [21], [35].

Although the analysis of the ways of swimming at low Re has been extensively addressed in the literature and from different perspectives, in this paper we are especially focused on those related to the well-known N-link Purcell’s swimmers. For example, approaches based on primitives of motion and resistive force were reported in [2], or on optimal-coordinate in [14], or from a qualitative point of view in [4], or for the particular case of 3-link Purcell’s swimmers in [17], [18], [43]. And in what generating planar waves by means of an Eukaryotic flagellum is concerned, the main tends are approached with several methods: (1) by distributed actuation [1], [19], [28], [39], (2) by two-point actuation [13], [32], and (3) by single-point actuation with absorption of the reflected wave [13], [32], or (4) with a non-uniform distribution of mass [25]. However, in the literature there is a lack of studies that analyze and compare between different propulsion strategies for microswimmers that navigate within low Re environments.

In robotics, hardware-in-the-loop (HIL) testbeds are recognized to be effective tools to validate controls when prototyping, rather than only using simulations, since a more realistic environment can be achieved before building the robot. This naturally allows robot designers to find out possible bugs on digital implementation of controllers, as well as limitations of the system hardware, avoiding or reducing the main drawbacks that would involve the use of a real prototype, i.e., high costs, time-consuming implementation, and certain inflexibility [38].

In the last decades, fractional calculus (FC), a branch of mathematical analysis that deals with non-integer order derivatives and integrals, has emerged as an efficient and powerful mathematical tool not only for accurate modeling many complex phenomena that can be found in several fields of science and engineering, but also for controlling complex systems by improving and generalizing well-established control methods and strategies. As a result, hundreds of research papers and monographs have been published for both kinds of applications (refer to, e.g., the recent book series on FC with applications in [26]). However, in the field where this work is framed, to the best of authors’ knowledge, the only contributions reported were those related to a fractional model of Navier–Stokes equation (see e.g. [15], [20]), expecting to find other relevant results as discussed in [42].

Given this motivation, this paper proposes an improved locomotion for an N-link artificial Eukaryotic flagellum (AEF) swimming microrobot within the FC framework for both the waveform for propulsion and its motion control. Firstly, a novel mode of swimming is presented based on a fractional order power law for the modulation of the wave amplitude. The study of the proposed waveform is carried out in terms of forward thrust and velocity by means of the implementation of numerical algorithm that allows to solve the mathematical problem that it involves. Secondly, a robust fractional order proportional-derivative (PD^μ) controller is designed for the control of the microrobot to reproduce a non-reciprocal motion using three different tuning methods. To guarantee this kind of motion, the robot actuators have to describe the same dynamics in closed-loop with independence of its position along the flagellum (notice that the load of each actuator depends on its position at the flagellum).

To validate both the waveform and the control strategy proposed, a HIL testbed for a 4-link AEF swimming microrobot is built. It consists of a microrobot simulator developed with the physical modeling tools in the MATLAB/Simulink environment and the embedded microcontroller Atmel ATmega32u4, where the controllers are programmed. The testbed, thanks to the simulator, allows to select the mode of swimming among four types and the robot geometry (namely, size of the swimmer and number of links in which its flagellum is divided to), as well as evaluating the performance of the locomotion with regard to propulsion, power efficiency or tracking. For comparison purposes, classical waveforms and controllers are also applied. Experimental and simulation results are given to show that the best power efficiency of the microrobot is achieved with the proposed fractional approach in terms of both the way of swimming and the energy consumption. Preliminary results of this study can be found in [39], [40], [41].

In summary, the main contributions of this paper are two: (1) a new waveform for propulsion of swimming microrobots based on a fractional order power law for amplitude modulation; and (2) a study of the propulsion performance for different traveling waves and controllers. Hence, this article provides a significantly different perspective from those presented in the aforementioned works by looking at propulsion optimization: firstly, before the control design, i.e., maximizing the propulsion thrust and velocity by searching for both the optimal traveling wave and the number of links of the swimmer, and secondly taking into account control, improving the energy efficiency with the consideration of different types of control strategies and tuning methods. These points of view could provide important guidelines for future research, overcoming the problem of controlling swimming microrobots in such microscales.

The contents of the paper are organized as follows. Section 2 describes the mathematical background concerning the hydrodynamics of swimming robots at low Re environments and the classical waveforms for propulsion. Section 3 presents the new waveform for propulsion based on a fractional order power law for amplitude growth. Section 4 describes the prototype of the swimming microrobot developed to test propulsion and locomotion with the different waveforms and control strategies. The control of the swimming microrobot required to emulate a non-reciprocal motion within low Re environments is addressed in Section 5. The analysis of propulsion performance based on both simulated and experimental results is given in Section 6. Finally, Section 7 draws the concluding remarks and perspectives on the future work.