An Overview of Piezoelectric Self-Sensing Actuation for Nanopositioning Applications: Electrical Circuits, Displacement, and Force Estimation <s> C. Optimal Design of Piezoelectric-Based Nanopositioning Systems for Charge-Based SSA <s> A method for the design of displacement amplifying compliant mechanisms for piezoelectric actuators is developed using a topology optimization approach. The overall stroke amplification or geometric advantage of the mechanism, and the overall mechanical efficiency of the mechanism are considered as objective functions. The maximization of these objective functions is accomplished using two different solution methods, Sequential Linear Programming and an Optimality Criteria method. The focus of this paper is on the underlying topology optimization problem formulations and the solution methods. Design examples are presented which illustrate the method, with comparisons of the computation time and mechanism performance for the two formulations and solution methods. A procedure has also been developed to automatically convert the topology optimization results to a solid CAD model. A prototype design has been fabricated to serve as proof of concept. <s> BIB001 </s> An Overview of Piezoelectric Self-Sensing Actuation for Nanopositioning Applications: Electrical Circuits, Displacement, and Force Estimation <s> C. Optimal Design of Piezoelectric-Based Nanopositioning Systems for Charge-Based SSA <s> Analyses of the performance of piezo-microactuators for various thicknesses of piezoelectric layers are presented. If the value of the Young's modulus of the piezoelectric layer is not more than five time less than that of nonpiezoelectric layers, to realize maximum displacement of the actuator it is better to use a piezoelectric layer as thin as possible (thickness 2–5 µm) with a high breakdown electrical field property. A unimorph lead zirconate titanate (PZT)/stainless steel actuator was successfully fabricated using the aerosol deposition method (ADM) and its performance was investigated for various thicknesses of piezoelectric layers. The electrical energy consumption of actuators when they deflect with the same displacement did not strongly depend on the thickness of the piezoelectric layer. A breakdown electrical filed of more than 500 kV/cm for a lead zirconate titanate [Pb(Zr0.52,Ti0.48)O3] film with thickness ranging from 2 to 80 µm was reported as well. <s> BIB002 </s> An Overview of Piezoelectric Self-Sensing Actuation for Nanopositioning Applications: Electrical Circuits, Displacement, and Force Estimation <s> C. Optimal Design of Piezoelectric-Based Nanopositioning Systems for Charge-Based SSA <s> In this paper, a novel approach to the design of piezoelectric plate and shell actuators using topology optimization is described. A new piezoelectric material model PEMAP-P (piezoelectric material with penalization and polarization) is proposed, which is an extension of the SIMP (solid isotropic material with penalization) model used for elastic materials. In addition to the pseudo-density ρ1, which describes the 'amount' of piezoelectric material in each finite element, a new design variable ρ2 is introduced for the polarization of the piezoelectric material. The optimization problem consists in distributing the piezoelectric actuators in such a way as to achieve a maximum output displacement in a given direction at a given point of the structure, while simultaneously minimizing the structural compliance. Sequential linear programming (SLP) is used to solve the optimization problem. Examples are given demonstrating the potential of the proposed approach for the optimal design of piezoelectric actuators for multi-layer plate and shell structures. <s> BIB003 </s> An Overview of Piezoelectric Self-Sensing Actuation for Nanopositioning Applications: Electrical Circuits, Displacement, and Force Estimation <s> C. Optimal Design of Piezoelectric-Based Nanopositioning Systems for Charge-Based SSA <s> Piezoelectric tubes with quartered external electrodes have been widely used as scanners in modern scanning probe microscopes. Various feedback control techniques have been developed to improve bandwidth and accuracy of these scanners. Non-contact displacement sensors, e.g. capacitive and inductive sensors, have been used for position feedback. However, their measurements contain a significant noise component if operated over large bandwidths. As an alternative measure of displacement, the piezoelectric voltage induced in a tube nanopositioner has been proposed recently. This technique provides a much improved noise figure, up to three orders of magnitude better than capacitive sensors. In this arrangement, an electrode is used to actuate the tube, while the opposite electrode is used as a sensor. Operating the tube in this mode has two drawbacks: i) the operating range of the tube is reduced to half, and ii) the tube is driven asymmetrically, thus the opposite sides of the tube are no longer stressed antisymmetrically, i.e. the tube cannot be considered as a perfectly collocated system. In this paper, a new electrode pattern for piezoelectric tube scanners is presented. This new approach addresses the aforementioned problems and allows simultaneous sensing and actuation of the tube in an efficient way. <s> BIB004 </s> An Overview of Piezoelectric Self-Sensing Actuation for Nanopositioning Applications: Electrical Circuits, Displacement, and Force Estimation <s> C. Optimal Design of Piezoelectric-Based Nanopositioning Systems for Charge-Based SSA <s> This article presents in a unified way, the various optimization criteria used by researchers for optimal placement of piezoelectric sensors and actuators on a smart structure. The article discusses optimal placement of piezoelectric sensors and actuators based upon six criteria: (i) maximizing modal forces/moments applied by piezoelectric actuators, (ii) maximizing deflection of the host structure, (iii) minimizing control effort/maximizing energy dissipated, (iv) maximizing degree of controllability, (v) maximizing degree of observability, and (vi) minimizing spill-over effects. Optimal piezoelectric sensor and actuator locations on beam and plate structures for each criterion and modes of interest are presented in a tabular form. This technical review has two objectives: (i) practicing engineers can pick the most suitable philosophy for their end application and (ii) researchers can come to know about potential gaps in this area. <s> BIB005 </s> An Overview of Piezoelectric Self-Sensing Actuation for Nanopositioning Applications: Electrical Circuits, Displacement, and Force Estimation <s> C. Optimal Design of Piezoelectric-Based Nanopositioning Systems for Charge-Based SSA <s> Soft piezoelectric actuators are extensively used in micro- and nanopositioning applications. Traditional sensorless position control approaches use a hysteresis mapping between voltage and position in a voltage feedforward control scheme in order to compensate for hysteresis errors. However, several factors such as frequency, temperature, and aging influence this mapping. Recently, charge control for positioning is also attracting interest among researchers due to the linear relationship between position and charge. However, a sophisticated hardware design is required to minimize charge drift. In this study, a new sensorless position control technique is proposed which requires neither an accurate inverse mapping nor a sophisticated charge amplifier in order to compensate for hysteresis. A constitutive relationship is employed to infer position from charge measurement through current integration. To eliminate drift from the charge-based measurement, an observer is presented in this study which uses a second self-sensing position estimate that is based on the variation of effective piezoelectric capacitance with stroke. The proposed observer is tested as a position feedback sensor inside a simple integrating control loop and applied to step profiles of variable stroke and sinusoidal profiles of constant and mixed frequencies. The responses are compared with a laser interferometer and the maximum observer error between the laser and the observer is 0.89 μm ( 3% of the maximum stroke) over a frequency range of 10 to 100 Hz. <s> BIB006 </s> An Overview of Piezoelectric Self-Sensing Actuation for Nanopositioning Applications: Electrical Circuits, Displacement, and Force Estimation <s> C. Optimal Design of Piezoelectric-Based Nanopositioning Systems for Charge-Based SSA <s> This paper describes a self-sensing technique for a piezoelectrically driven magnetic resonance imaging (MRI)-compatible tweezer style end effector, suitable for robot assisted MRI guided surgery. Nested strain amplification mechanisms are used to amplify the displacement of the piezo actuators to practical levels for robotics. By using a hysteretic piezoelectric model and a two port network model for the compliant nested strain amplifiers, it is shown that force and displacement at the tweezer tip can be estimated if the input voltage and charge are measured. One piezo unit is used simultaneously as a sensor and an actuator, preserving the full actuation capability of the device. An on-site calibration procedure is proposed that calibrates the combined electromechanical model without requiring specific loading conditions on the inner piezoelectric actuators. Experimental validation shows an average of 12% error between the self-sensed and true values. <s> BIB007 </s> An Overview of Piezoelectric Self-Sensing Actuation for Nanopositioning Applications: Electrical Circuits, Displacement, and Force Estimation <s> C. Optimal Design of Piezoelectric-Based Nanopositioning Systems for Charge-Based SSA <s> Piezoelectric materials are well recognized for the development of systems and actuators working at the micro/nano-scale such as microsystems. This recognition is thanks to the high resolution, high bandwidth, and high force density that they can offer. However, piezoelectric actuators are typified by low range of displacement relative to other actuators like magnetic or thermal actuators. To obtain the sufficient range of displacement with a piezoelectric actuator, either we use high input voltages or we redesign the actuator to have larger dimensions. The former solution may lead to the destruction of the actuators and the latter is not congruent with the objectives of microsystems where the dimensions should be miniaturized. Furthermore, increasing the dimensions of the actuators reduces their rapidity and bandwidth. This paper proposes an approach based on interval analysis to design piezoelectric actuators with cantilever structures. The aim consists in reducing their dimensions while still satisfying some specified performances in term of output range and in term of resonant frequency (and thus bandwidth). The problem of the design is formulated as a set-inversion problem which can be solved using interval techniques. The obtained results, validated with prototype fabrication and experimental characterization, demonstrate the efficiency and the interests of the proposed method for designing systems and actuators working at the micro/nano-scale in general. <s> BIB008 </s> An Overview of Piezoelectric Self-Sensing Actuation for Nanopositioning Applications: Electrical Circuits, Displacement, and Force Estimation <s> C. Optimal Design of Piezoelectric-Based Nanopositioning Systems for Charge-Based SSA <s> Piezo modal transducers in 2d can be designed theoretically by tailoring polarity of the surface electrodes. However, it is also necessary to include null-polarity phases of known width separating areas of opposite polarity in the manufacturing process in order to avoid short-circuiting. Otherwise the performance of such devices could be spoiled. In this work, we propose an appropriate topology optimization interpolation function for the electrode profile such that the effect of this new phase (hereafter gap-phase) is included in the formulation of the design problem. The approach is density-based, where the interface is controlled by including the gradient norm in the electrode profile interpolation. Through a detailed case study in 1d, conclusions on how to control the width of this gap-phase are extracted, and subsequently extended to the 2d case. <s> BIB009 </s> An Overview of Piezoelectric Self-Sensing Actuation for Nanopositioning Applications: Electrical Circuits, Displacement, and Force Estimation <s> C. Optimal Design of Piezoelectric-Based Nanopositioning Systems for Charge-Based SSA <s> Modal transducers can be designed by optimizing the polarity of the electrode which covers the piezoelectric layers bonded to the host structure. This paper is intended as a continuation of our previous work (Donoso and Bellido J Appl Mech 57:434---441, 2009a) to make better the performance of such piezoelectric devices by simultaneously optimizing the structure layout and the electrode profile. As the host structure is not longer fixed, the typical drawbacks in eigenproblem optimization such as spurious modes, mode tracking and switching or repeated eigenvalues soon appear. Further, our model has the novel issue that both cost and constraints explicitly depend on mode shapes. Moreover, due to the physics of the problem, the appearance of large gray areas is another pitfall to be solved. Our proposed approach overcomes all these difficulties with success and let obtain nearly 0-1 designs that improve the existing optimal electrode profiles over a homogeneous plate. <s> BIB010 </s> An Overview of Piezoelectric Self-Sensing Actuation for Nanopositioning Applications: Electrical Circuits, Displacement, and Force Estimation <s> C. Optimal Design of Piezoelectric-Based Nanopositioning Systems for Charge-Based SSA <s> A new piezoelectric actuator with collocated sensor is designed, modeled and simulated. The structure has three piezoelectric layers where the two external layers serve for the actuation by a convenient application of electrical potentials, and the middle layer serves as the sensor. After presenting the principle of the structure, a model is developed for the actuator and as well as for the sensor. Then simulation is carried out to evaluate their performances. The novel structure is very promising for applications that require control and automation, especially in situations where the use of sensors is unfeasible or difficult. <s> BIB011 </s> An Overview of Piezoelectric Self-Sensing Actuation for Nanopositioning Applications: Electrical Circuits, Displacement, and Force Estimation <s> C. Optimal Design of Piezoelectric-Based Nanopositioning Systems for Charge-Based SSA <s> This paper proposes a self-sensing measurement technique to perform the precise estimation of the displacements along two axes in a two-degrees-of-freedom (2-DOF) piezoelectric actuator. For that, a new electrical circuit scheme that permits charge amplification is first proposed to match the different electrodes combination of the 2-DOF actuator. Then, a new bivariable observer that precisely estimates the displacements is calculated and implemented experimentally in a cascade with the electrical circuit to complete the self-sensing. The experimental tests and results verification with external sensors revealed that the measured displacements given by the developed self-sensing measurement technique derive better than a micron of precision which well fits to micromanipulation applications. Discussion about the features and the performances improvement perspectives of the suggested approach are presented at the end of the paper. <s> BIB012 </s> An Overview of Piezoelectric Self-Sensing Actuation for Nanopositioning Applications: Electrical Circuits, Displacement, and Force Estimation <s> C. Optimal Design of Piezoelectric-Based Nanopositioning Systems for Charge-Based SSA <s> Abstract This paper presents the mechanical structure optimization of a piezoelectric energy harvesters devoted to supply embedded animal tracking devices. The harvester mechanical structure being a bilayer unimorph piezoelectric cantilever is composed of two elements: a piezoelectric layer and a non-piezoelectric layer (i.e, active and passive layers). The suggestion of this paper is to find the optimal ratio between their two thicknesses in order to maximize the output recuperated voltage at the electrodes. For that, from the model that links the mechanical excitation moment and the output electrical charge, a gradient-based optimization is carried-out. Comparison with existing mechanical piezoelectric harvester structure is made and which clearly demonstrates that the proposed structure permits to gain up to five-times in terms of the output charge and a significant gain in terms of output electrical power for the same condition. <s> BIB013 </s> An Overview of Piezoelectric Self-Sensing Actuation for Nanopositioning Applications: Electrical Circuits, Displacement, and Force Estimation <s> C. Optimal Design of Piezoelectric-Based Nanopositioning Systems for Charge-Based SSA <s> The technological advancement in today's world has made the world more dependent onto the technology thus increasing the demand for more power and energy. This directly caused the world to desperately find another way to produce energy as the world current main energy source has depleted. This paper focused on piezoelectric disk especially piezoelectric transducers as it is said to be the most effective in converting mechanical vibrations to into electrical energy. Pressure was applied onto the disk and it will then convert into an electrical output (voltage). The material that behaves in such a way is also known as piezoelectric sensor. The voltage produced by the transducer can easily be assessed by voltage measuring instruments that can be used to measure stresses or forces. The physical quantity like mechanical stress or force cannot be measured directly. Therefore, piezoelectric transducer can be used. First step in doing this research is to find the placement with the highest pressure point on the insole. After thorough research, two places with the highest pressure point were found which is at the front and back of the insole. After that, a small circuit was developed to get the data for the research. The circuit which includes a rectifier as an ACDC converter that converts AC to DC to be charged to the battery. From the output, data were taken and converted into joules which are then converted again into steps. <s> BIB014 </s> An Overview of Piezoelectric Self-Sensing Actuation for Nanopositioning Applications: Electrical Circuits, Displacement, and Force Estimation <s> C. Optimal Design of Piezoelectric-Based Nanopositioning Systems for Charge-Based SSA <s> This letter presents a displacement self-sensing method for piezoelectric actuators that takes into account the effect of the deformation of these latter on their electrical impedance. A self-updating digital electrical impedance equivalent to that of the piezoelement is suggested in order to overcome the mismatch problem met when considering a static analog electrical impedance as classically used. Then, the design of an observer that performs a thorough separation of the electric current due to the direct piezoelectric effect from the measured electric current is proposed in order to estimate the piezoelectric actuator displacement. The experimental results validate the proposed approach and demonstrate the efficiency of the developed self-sensing circuit and its observer to estimate the displacement of the piezoceramics actuator for long duration measurement of constant step signal. <s> BIB015 </s> An Overview of Piezoelectric Self-Sensing Actuation for Nanopositioning Applications: Electrical Circuits, Displacement, and Force Estimation <s> C. Optimal Design of Piezoelectric-Based Nanopositioning Systems for Charge-Based SSA <s> Abstract A novel technique for the multi-parameter optimization of distributed piezoelectric actuators is presented in this paper. The proposed method is designed to improve the performance of multi-mode vibration control in cylindrical shells. The optimization parameters of actuator patch configuration include position, size, and tilt angle. The modal control force of tilted orthotropic piezoelectric actuators is derived and the multi-parameter cylindrical shell optimization model is established. The linear quadratic energy index is employed as the optimization criterion. A geometric constraint is proposed to prevent overlap between tilted actuators, which is plugged into a genetic algorithm to search the optimal configuration parameters. A simply-supported closed cylindrical shell with two actuators serves as a case study. The vibration control efficiencies of various parameter sets are evaluated via frequency response and transient response simulations. The results show that the linear quadratic energy indexes of position and size optimization decreased by 14.0% compared to position optimization; those of position and tilt angle optimization decreased by 16.8%; and those of position, size, and tilt angle optimization decreased by 25.9%. It indicates that, adding configuration optimization parameters is an efficient approach to improving the vibration control performance of piezoelectric actuators on shells. <s> BIB016 </s> An Overview of Piezoelectric Self-Sensing Actuation for Nanopositioning Applications: Electrical Circuits, Displacement, and Force Estimation <s> C. Optimal Design of Piezoelectric-Based Nanopositioning Systems for Charge-Based SSA <s> A method is presented to design an optimal unimorph beam that flexes and generates voltage in response to a distributed load acting over its length using an analytical model. The objective is to design a beam that resonates at a specified frequency at which there is a peak in the excitation. The associated open-circuit voltage sensitivity and gain bandwidth product are functions to be maximized. For specific boundary conditions and pairs of materials, the dimensional ratio of the thickness of the piezoelectric layer to the square of the length of the beam is chosen such that the beam resonates at the specified frequency. The two optimum values of the dimensionless ratio of the thicknesses of the elastic and piezoelectric layers associated with the maximum values of the two functions of interest are determined through a simple one-dimensional search. Figures of merit and numerical results are used to show the effect of boundary conditions and properties of materials on the functions of interest. All beams with the optimum ratio of thicknesses are equally good and the dimensions of a beam for a specific application are to be chosen by considering principles of design for manufacture and the external electronic circuit. <s> BIB017 </s> An Overview of Piezoelectric Self-Sensing Actuation for Nanopositioning Applications: Electrical Circuits, Displacement, and Force Estimation <s> C. Optimal Design of Piezoelectric-Based Nanopositioning Systems for Charge-Based SSA <s> The objective of the research is to design a broadband energy harvester device through the multi-beam approach and non-linear trapezoidal geometry approach. The performance of the composite piezoelectric PZT-PZN polycrystalline ceramic material is simulated using COMSOL Multiphysics, and results are compared using series configuration of a composite bimorph energy harvester which vibrates at its 1st fundamental frequency. We chose a five cantilever multibeam harvester to demonstrate that individual fundamental modes of the beams can achieve a broader frequency band and generate power. Authors also show that composite trapezoidal beam design leads to high power density broadband frequency response. The multibeam approach resulted in broader bandwidth of 18 Hz while generating a power density of 0.0913 mW/cm 3 whereas the trapezoidal shape generated 2.3 – 2.5 mW/cm 3 with a bandwidth of 4 to 6 Hz. Authors believe that these results could help design broadband energy harvesters to enhance power density as well as bandwidth. <s> BIB018 </s> An Overview of Piezoelectric Self-Sensing Actuation for Nanopositioning Applications: Electrical Circuits, Displacement, and Force Estimation <s> C. Optimal Design of Piezoelectric-Based Nanopositioning Systems for Charge-Based SSA <s> Quasi-static piezoelectric actuators are used in micro-robotics due to the small displacements they can achieve, and they have been proved suitable for Self-Sensing Actuation (SSA) applications. The fabrication parameters of an actuator (i.e. morphology, stiffness, capacitance) will have an impact on the performance of the SSA implementation, and this study intends to examine how to optimize those parameters for a piezoelectric cantilever. A model is established for this type of actuator, from which an objective function is defined. Boundary conditions are set on the optimization study by imposing constraints on the design. An analytical optimization approach is then adopted to study the evolution of the objective function across the design space. The study evidences the existence of a subset of solutions satisfying the constraint and maximizing the objective function, and the monotonicity of this subset is demonstrated. It is concluded that to obtain an optimal solution, one must maximize the length, minimize the number of layers and their thickness, and derive the appropriate width from these parameters applied to the constraint imposed on the objective function. <s> BIB019
Even though charge-based SSA has demonstrated a great potential BIB015 , BIB012 , BIB006 , BIB007 , its effectiveness depends on the morphology of the PEA and the choice of the mechanical properties of its constituent materials. Therefore, the first step to take full advantage of the potential of charge-based SSA is the PEA's design. 1) PEA Design Optimization: The distribution of material inside the piezoelectric layers influences the actuators' performance, and hence, the number, shape, size, and placement of the actuators have to be optimized. Among the optimization methods for PEAs, there are some methods that are given in the following. 1) Parametric optimization in which parameters of the PEA are varied in order to determine the dimensions and material properties that guarantee improved performances in terms of output range BIB002 , BIB016 (δ pre /v c ) and in terms of bandwidth BIB008 . δ pre is the bending due to the piezoelectric reverse effect under the application of v c . 2) Topology optimization that is based on the piezoelectric material with penalization (PEMAP) model, where the design variable is the pseudo density ρ 1 , which describes the amount of piezoelectric material in each finite element in the piezoelectric layer(s). Topology optimization is employed to find an optimal distribution of piezoelectric material in a multilayer plate or shell structure to provide the maximum displacement δ pre or generated forces in a given direction at a given point of the domain BIB001 - BIB009 . 3) Simultaneous topology and polarization optimization that uses the piezoelectric material with penalization and polarization (PEMAP-P) and work to find the optimum actuator layout and polarization profile simultaneously BIB003 - BIB010 . For this method, in addition to the pseudo density ρ 1 , a new design variable ρ 2 is introduced for the polarization of the piezoelectric material. The optimization problem consists in distributing the PEAs in such a way as to achieve a maximum output displacement δ pre in a given direction at a given point of the structure while simultaneously minimizing the structural compliance. All these methods aim to determine the geometric feature that will enhance the PEA's generated displacement due to an applied input voltage without regard to the sensor aspect of the piezoelectric material. 2) Piezoelectric Sensor Optimization: On the one hand, applying an input voltage will result in the piezoelectric material elongation/contraction, and on the other hand, a pressure applied onto a piezoelectric material will be converted into an electrical output (strain-induced charges). When a piezoelectric material is used to convert mechanical into electrical energy, it is called a piezoelectric sensor. Like for PEAs, works have been conducted for the optimization of piezoelectric sensors. These searches can be grouped into two categories as follows. 1) Geometric optimization methods that aim for the optimization of geometric parameters, such as length, width, and thickness of the piezoelectric layer in order to maximize the output recuperated electrical charges at the electrodes Q δ . Pillai et al. BIB017 presented an analytical method to design an optimal unimorph beam that maximizes the strain-induced charges' sensitivity when acted upon by a uniform mechanical load p at a specified frequency ν s (Q δ / p(ν s )). Schlinquer et al. BIB013 suggested a unimorph and Chen and Bedekar BIB018 suggested a bimorph piezoelectric cantilever mechanical structure optimization for energy harvesting. The optimized design aimed to maximize the PEA's strain-induced charges due to external harmonic load (Q δ /F ext (ν)), where ν is the frequency of the harmonic load. 2) Localization optimization methods that aim to find the placement with the highest pressure point on a given structure and thus guarantee the highest possible output recuperated electrical charges at the electrodes BIB014 , BIB005 . 3) Simultaneous Piezoelectric's Actuation and Sensing Design Optimization: Some researchers have considered using both the direct and reverse effects of piezoelectric materials to ensure simultaneous good observability and controllability of a structure. Moheimani and Yong BIB004 proposed an electrode pattern on a piezoelectric tube actuator to simultaneous sensing and actuation. Moussa et al.  used a topological optimization method to design a compliant microactuator that optimally integrates actuating and sensing areas in a monolithic structure. Rougeot et al. BIB011 introduced a three-layered piezoelectric cantilever design for which the upper and lower layers were used for the PEA's actuation, while the middle layer served for the sensing of the PEA's displacement and perceived force. In BIB004 - BIB011 approaches, the actuation and sensing do not share the same electrodes. Therefore, the resulting structure cannot be considered to be an optimized PEA's design for charge-based SSA. Masson et al. BIB019 presented an analytical approach for the design of piezoelectric cantilever actuators, which aims to improve the SSA performance for external loads estimation while minimizing the dielectric effect (C pea ·v c ). The optimized piezoelectric cantilever mechanical structure they proposed was supposed to achieve a tip displacement of at least δ min = 25 μm, a minimum blocking force of F min bl = 100 mN, and withstand a maximum electric field of E max 3 = 3 V/μm (depolarizing field). The optimization problem was formulated as follows: where C pea is the PEA's capacitance in the absence of mechanical deformation and null electric field. The approach suggested an optimization for the external load estimation (Q δ /F ext ≡ Q δ /δ mec ) and did not account for the piezoelectric actuation-induced charges (Q δ /δ pre ). A simplified schematic of the superposition of bending under external load and piezoelectric bending due to the reverse piezoelectric effect is shown in Fig. 7 to help perceive the difference between them. Furthermore, the objective function F(l, w, h) just constraints the minimum displacement and blocking force and does not aim to maximize the actuation (δ pre /v c ). As highlighted in this section, earlier studies on the optimal design of piezoelectric structures aimed to optimize the sensor's sensitivity to mechanical loads and to increase the actuation. None intended to optimize both simultaneously in an SSA aspect, that is, using the same electrodes for both actuation and sensing. More recently, Liseli et al.  proposed an analysis of the charge-based SSA's performance dependence on the choice of the geometry and the properties of the constituent materials in case of a piezoelectric cantilever. They presented an analytical model for this type of PEA and defined a multiobjective function for simultaneous piezoelectric's actuation and sensing design optimization where actuation and sensing share the same electrodes. Experiments were conducted to corroborate the analytical modeling and analysis. Their studies and experiments results indicated a tradeoff between the PEA's actuation and sensor sensitivity, resulting from the choice of the mechanical properties of its constituent materials. Increasing the PEA's sensor sensitivity will be at the cost of its actuation and vice versa. In this case, the objective function is said to be conflicting, and there exists a (possibly infinite) number of Pareto optimal solutions. Without a piece of additional subjective preference information on either the PEA's actuation or sensor sensitivity, all Pareto optimal solutions are considered equally good.