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Locomotion inside water
Locomotion inside water includes swimming or non-swimming locomotion. The latter includes specialized actions such as flying and gliding, as well as jet propulsion. In this section, we will describe only the swimming locomotion because it is the most used in the field of miniature robot and it is divided into fish swimming mechanisms (at high and moderate Re) and micro-organisms swimming mechanisms (at low Re).
Fish swimming mechanisms
To aid in the description of fish swimming mechanisms; Figure 1.16 identifies the elements of fish. The fish swims either by body and/or caudal fin (BCF) movements or using median and/or paired fin (MPF) propulsion. The BCF propulsion has been classified into five swimming modes: Anguilliform, subcarangiform, carangiform, thurnniform and ostraciform. The latter is the only oscillatory BCF mode; it is characterized by the pendulum oscillation of the caudal fin, while the body remains essentially rigid. The others are undulatory BCF modes. In anguilliform mode, ondulations of large amplitude are obtained by whole body.
Similarly for subcarangiform mode, but the amplitude of undulations is limited in front and increased in the latter half of the body. For carangiform mode, the amplitude of the undulations is limited to the last third of the body. In thunniform mode, the thrust is generated by the caudal fin only.
The MBF propulsion has been classified into seven swimming modes: Rajiform, diodontiform, amiiform, gymnotiform, balistiform, labriform and tetraodontiform. The two latter modes are classified into undulatory fin motions, while the five remaining are classified into oscillatory fin motions. In rajiform mode, the most of the body length undulate vertically along the pectorals that are flexible and very long. Similarly, in diodontiform mode, thrust is generated by the pectoral fins but they are not in the same level and the same shape as that in mode Rajiform. In amiiform mode, propulsion is obtained by undulations of a long-based dorsal fin. Contrary, in gymnotiform mode, propulsion is obtained by undulations of a long-based anal fin. In balistiform mode, both the anal and dorsal fins undulate to generate thrust. In labriform mode, thrust is generated by oscillatory movements of the pectoral fins. In tetraodontiform mode, the dorsal and anal fins beat together, either in phase or alternating phase to generate thrust. Figure 1.17 describes fish swimming modes.
Some piezoelectric miniature robots like swimming fishes are developed in [(Fukuda, et al., 1994),(Tzeranis, et al., 2003), (Borgen, et al., 2003),(Deng, et al., ICRA 2005), (Kodati, et al., 2007), (Wiguna, et al., 2006),(Hu, et al., 2006)], where piezoelectric actuators are used for producing the movements of BCF and MPF for miniature robots. The choice between swimming modes in liquid depends on the application expected. For example the design of thunniform swimmers miniature robots is optimized for high speed swimming in calm waters and is not well suited to other actions such as slow swimming, turning maneuvers and rapid acceleration from stationary and turbulent water [(Sfakiotakis, et al., 1999)]. Example is given in Figure 1.18.
Micro-organisms swimming mechanisms
All the micro swimming mechanics such as flagella, spermatozoa, cilia, and amoeba crate in one way or another traveling wave, advanced in the opposite direction of the micro organisms locomotion. The swimming mechanics of micro-organisms are divided into flagellar and ciliary swimming.
Flagellar swimming is the simplest swimming method for micro system and is produced by a sinusoidal or helical wave in an elastic tail. In contrast of flagella, cilia beat in asymmetrical fashion i.e. by orienting the cilia parallel to the flow during the recovery stroke much lower drag is produced than when they beat in a more perpendicular orientation during the propulsive stroke [(BIEWENER, 2003)].
Piezoelectric actuators are used in [(Kosa, et al., 2007), (Kosa, et al., 2008)] for creating the travelling wave needed for the movements of swimming micro organisms (Figure 1. 19).
Locomotion at the water surface
Locomotion at the water surface is divided into two different locomotion principles: striding on the water surface like water striders and running on the water surface like a basilisk lizard. Water striders take advantage of their size by using the surface tension of water to generate forces in order to step over the surface of water. These forces increase with the depth of the unwetted limb of the water striders [(BIEWENER, 2003)]. Basilisk lizard have a weight which is larger than the surface tension can support, it takes advantage of the mass density of the water, which exerts a reactive force when running rapidly with its webbed feet [(BIEWENER, 2003)]. As example we take the water strider miniature robot walking on water describes in [(Suhr, et al., 2005)] (Figure 1.20).
Validation process
We begin the process of model validation by comparing the resonant frequencies of the model (1D and 2D) with the experimental ones, then we apply a sinusoidal electrical voltage to one patch, we measure the transverse displacement of the beam/plate and the obtained electrical voltage for the other patch in the case of open circuit and resistance shunt circuit and we compare it with the model. All measures are done using a high resolution laser interferometer LK-G3001PV Keyence France.
Resonance frequencies validation
In short-circuit, the voltage is zero. In the case of an open circuit, the electric charge is constant. In this case, it is nil. The resonance frequencies and modes shapes of the system when the two electrodes of the piezoelectric patches are short circuited () are given by (referring to equation 4. 1):
([–[]){U}=0.
Table of contents :
Chapter 1: locomotion principle for piezoelectric miniature robots
1.1. Introduction
1.2. Piezoelectric miniature robot
1.3. Piezoelectric actuators
1.4. Locomotion on a solid substrate
1.4.1. Wheeled locomotion
1.4.2. Walking locomotion
1.4.3. Inchworm locomotion
1.4.4. Inertial drive
1.4.5. Resonant drive
1.4.6. Friction drive
1.4.7. Summary of miniature robots on a solid substrate
1.5. Locomotion in liquid
1.5.1. Locomotion inside water
1.5.1.1. Fish swimming mechanisms
1.4.1.2. Micro-organisms swimming mechanisms
1.5.2. Locomotion at the water surface
1.5.3. Summary of piezoelectric miniature robots inside and on liquid
1.6. Locomotion in air
1.6.1. Flapping wing MAV
1.7. Conclusion and discussion
1.8. References
Chapter 2: Introduction to the numerical modeling of thin structures with piezoelectric patches
2.1. Introduction
2.2. Mechanical equations
2.2.1 Strains
2.2.2 Stresses
2.2.3 Linear elasticity
2.3. Piezoelectricity
2.4. Unknowns to be determined
2.4.1. Bending vibrations of beams
2.4.2. Bending vibrations of plates
2.5. Static equation
2.6. Dynamic equation
2.7. Numerical modeling
2.7.1. Bending vibrations of beams
2.7.2. Bending vibrations of plates
2.7.3. Variational principle
2.7.4. Time discretization: Newmark method
2.8. Conclusion and discussion
2.9. References
Chapter 3: Modeling of non-collocated piezoelectric patches bonded on thin structures
3.1. Introduction
3.2. Constitutive equations
3.2.1 Case of piezoelectric patches bonded on a beam
3.2.1. 1 Mechanical constitutive equations
3.2.1. 2 Piezoelectric constitutive equations
3.2.2 Case of piezoelectric patches bonded on a plate
3.2.2.1 Mechanical constitutive equation
3.2.2.2 Piezoelectric constitutive equation
3.3. Displacement field
3.3.1 Neutral axis
3.3.2 Neutral plane
3.4. Variational formulation
3.4.1 Case of 1D formulation
3.4.2 Case of 2D formulation
3.5. 1D finite element formulation
3.6. 2D finite element formulation
3.7. Numerical equation
3.7.1 Case of beam
3.7.3 Beam-plate numerical equation
3.7.4 Actuator – sensor
3.7.5 Actuator – Actuator
3.8 Conclusion of the chapter
3.9 Appendix: Particular cases
3.10 References
Chapter 4: Experimental validation of models
4.1 Introduction
4.2 Experimented device
4.3 Rayleigh damping
4.4 Validation process
4.4.1 Resonance frequencies validation
4.4.2 Transverse displacement validation
4.4.2.1 Case of beam
4.4.2.2 Case of plate
4.4.2.3 Piezoelectric sensors Validation
4.4.2.4 Piezoelectric capacitance Validation
4.5 Conclusion and discussion
4.6 Appendix
4.7 References
Chapter 5: traveling wave piezoelectric beam robot
5.1 Introduction
5.2 Operation principle
5.2.1 Standing wave and traveling wave
5.2.2 Operation principle case of one mode excitation
5.2.3 Operation principle case of two modes excitation
5.3 Modeling of the piezoelectric beam robot
5.4 Optimal design
5.4.1 Thickness of piezoelectric patches and material used for the beam
5.4.2 Resonance frequency
5.4.3 Optimal operating frequency
5.4.3.1 Case of one mode excitation
5.4.3.1.1 Position 1
5.4.3.1.2 Position 2
5.4.3.1.3 Influence of positions to the performance of the traveling wave
5.4.3.1.4 Actuator-absorber & Absorber-actuator
5.4.3.2 Case of two modes excitation
5.4.3.2.1 Position 1
5.4.3.2.2 Position 2
5.4.3.2.3 Influence of positions to the performance of the traveling wave
5.4.3.2.4 Two modes excitation functionality
5.5 Conclusion and discussion
5.6 Appendix
5.7 References
Chapter 6: Robot manufacturing and experimental measurements
6.1 Introduction
6.2 Fabrication
6.3 Electronic and electric circuits design and realization
6.4 Experimental validation
6.5 Robot characterization
6.6 Significance and benefits
6.7 Conclusion of this chapter
6.8 Appendix
6.9 References
Chapter 7: Overview
7. 1 Introduction
7. 2 Piezoelectric transformers
7. 3 Damping vibration of thin beams and plates
7. 4 Active control of flexible structures
7. 6 Optimization topology
7. 7 Analytical model versus finite element model
7. 8 References
Conclusion and perspectives
