Dynamic modeling, optimized design, and fabrication of a 2DOF piezo-actuated stick-slip mobile microrobot

Mechanism and Machine Theory 133 (2019) 514–530

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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory

Research paper

Dynamic modeling, optimized design, and fabrication of a 2DOF piezo-actuated stick-slip mobile microrobot Navid Asmari Saadabad, Hamed Moradi∗, Gholamreza Vossoughi Centre of Excellence in Design, Robotics & Automation, Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 18 July 2018 Revised 30 November 2018 Accepted 30 November 2018

Keywords: Mobile microrobot Stick-slip locomotion Non-dominated sorting genetic algorithm Piezo-actuated microrobot

a b s t r a c t This paper proposes an optimized design for a 2DOF linear-motion mobile microrobot. The forward motion of the microrobot is achieved by simultaneous excitation of the vertical and horizontal oscillators that create the stick-slip locomotion. Dynamic equations of motion for the microrobot are derived and simplified based on Coulomb friction. The horizontal and vertical elements are connected to the main mass by piezoelectric actuators and are set in oscillatory motion by applying a harmonic voltage to the actuators. The design parameters, including the perpendicularly-mounted masses, frequency, and phase difference of the excitations are tuned in order to achieve high locomotion velocity and low energy consumption. The parameters are set such that the motion is confined to the stick-slip mode. Because of the nonlinear dynamic equations and abundance of objective functions, the non-dominated sorting genetic algorithm, NSGA ii, is implemented to optimize the parameters. The selected solution is capable of achieving an average velocity of 0.08 mm/s while keeping the delivered mechanical power as low as 0.07 mW. The optimized design is further investigated by analyzing the effect of excitation parameters on its motion characteristics. Finally, in order to verify the modeling and simulations, a prototype is fabricated and the experimental and theory results are shown to be in agreement. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction The need for manipulating small-sized devices with a relatively high accuracy has increased the demand for inventing micro-scale and nano-scale tools. Emergence of new applications for such tools and machines, and simultaneous achievements in the methods of creating these tools, has given a rise to the interest of research in the field of miniature, micro-, and nano- machines and robots [1–4]. The research and the recent breakthroughs in electronics is probably going to cause imperative achievements in the near future [5–7]. Microrobotics is one of the fields that has been popularized not long ago. A microrobot is defined as a micro-machine that is capable of interacting with and adapting to the outer environment. In other words, a microrobot can be defined as a macrorobot in which the dimensions are relatively small. The potential applications of microrobots include micro-assembly, micro-surgery, and micro-manipulation [8–11]. Microrobots might be categorized based on their structure, dimensions, or applications. One specific categorization divides microrobots into two groups of mobile and stationary. In this grouping, a mobile microrobot is defined as a robot which takes up several cubic micrometers of volume and it is capable of providing a motion range of at least several times ∗

Corresponding author. E-mail address: [email protected] (H. Moradi).

https://doi.org/10.1016/j.mechmachtheory.2018.11.025 0094-114X/© 2018 Elsevier Ltd. All rights reserved.

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of its body length [12]. The mobilizing mechanism that is implemented in a microrobot is of foremost importance, because it can affect the overall size of a microrobot, dramatically [13]. The utilized locomotion mechanism should not only be relatively small in size, but it should also provide high-precision motion for the mobile microrobot. Driesen classifies a microrobot into various modules: powering, communication, electronics, locomotion, and tool modules; in which the mobilization principle and the actuators of the microrobot constitute the locomotion module [12]. Breguet et al. elaborated on the various locomotion modules of microrobots and claimed that in general, the combination of two energy conversions provide the means for motion in a mobile microrobot: electrical-to-mechanical and mechanicalto-mechanical energy transformations [14]. Piezoelectric, polymer, and electro-static actuators are among the most common actuators that function as electrical into mechanical energy convertors [15]. Comparing various mechanisms that transform the mechanical energy into the motion, the friction inertial drive is very well-known for two main advantages: long-range motion and high-resolution motion [16]. This motion mechanism is mostly utilized in conjunction with piezoelectric actuators which operate at relatively high bandwidths. The combination of friction inertial drive and piezoelectric actuators have been frequently utilized in creation of mobile microrobots. Driesen had a thorough review on this principle and designed a microrobot with the same mechanism [12]. He coined the term modulated friction inertial drive (MFID) and tried to simplify the motion by separating two principles of slip generation and slip variation. Edeler et al. presented a comprehensive modeling for the stick-slip motion and friction characteristics [17]. Zhou et al. studied the microrobots that are propelled by stick-slip locomotion concepts (SSL) and concluded that these mechanisms are energy-efficient but they are not the fastest available mechanisms for locomotion [18]. Hariri et al. proposed a high-velocity stepping-motion microrobot by employing a traveling wave in a piezo-actuated beam structure and discussed the effects of design parameters on motion [19]. They also identified the important factors of SSL for achieving forward motion with minimum energy dissipation in the form of friction. Oldham et al. studied the legged locomotion of microrobots and proposed control methods for corresponding piezoelectric actuators [20–23]. Rios, Flemming, et al. designed a legged robot driven by piezoelectric actuators and analyzed the effects of frequency and phase on the motion [24,25]. Adibnazari et al. developed a tripedal stick-slip microrobotic stage with sub-micrometer precision [26]. Kamali et al. proposed a novel A-shaped microrobot based on stick-slip principle and validated the design experimentally [27,28]. Jalili et al. employed the SSL principle and analyzed the hybrid motion of a microrobot in three separate modes: stick, slip, and jump [29,30]. Following the endeavors on development of a high-precision mobile microrobot, various designs are proposed for linear and planar microrobots. The microrobots that used stick-slip as the locomotion mechanism and were actuated using piezoelectric actuators, have successfully achieved the objective of providing precise motion. However, the scope of most of the previous attempts were confined to the analysis of the motion regardless of the design parameters that might affect the motion characteristics. In this research, a novel design that relies on stick-slip principle is introduced and analyzed. The basic idea behind this design is the oscillation of vertically and horizontally mounted masses which can create slip generations and slip variations, respectively. The presented 2DOF microrobot is capable of transporting on a linear path in small-sized steps. This microrobot consists of a main body connected to two auxiliary parts. These parts are set in oscillatory motion using piezoelectric actuators which are driven by the reverse piezoelectric effect. These vibrating external masses cause a variation of the friction force that is applied to the main body; resulting in a forward or backward motion. By manipulating the excitation command applied to the piezoelectric actuators, one can control the motion characteristics of the microrobot. The physical structure of the proposed design is presented and the governing equations of motion are derived. Afterwards, the equations are numerically simulated with a set of initial values that are set on the mathematical model. Since the motion performance is dependent on the mass ratios and actuation features, the search for an optimum design is noteworthy. In order to search for the optimized designs, several objective functions are defined and optimized in a non-dominated sorting genetic algorithm (NSGA) by manipulating the values of physical parameters. The optimization algorithm looks for the design parameters that minimize the defined objective functions. Afterwards, the selected design is simulated for investigating the effect of each parameter on its velocity and the mechanical power delivered to the microrobot. Finally, the dynamic modeling principles are examined and verified by experimental implementation of a 2DOF microrobot which is designed and fabricated based on the proposed design idea. 2. Mathematical modeling of 2DOF piezo-actuated stick-slip microrobot In this section, dynamics of the 2DOF linear-motion microrobot attached to the vertical and horizontal oscillators is analyzed. The general arrangement of the oscillators in the 2DOF microrobot is depicted in Fig. 1. The main body of the microrobot is attached to the horizontal and vertical auxiliary oscillators; each oscillator consisting of a mass connected to a piezoelectric actuator. The harmonic excitation of the actuators results in a vibratory motion of the masses that provides the means for stepping motion of the main body. Manipulation of the oscillators’ vibration, influences the characteristics of the forward or backward motion. Fig. 2 visualizes the general pattern of the motion. This plot represents a simulation of the dynamic model that will be discussed shortly. At each instant, the vertical oscillator is either increasing or decreasing the value of the normal force applied to the main body. In the absence of vertical oscillation, the microrobot goes through a back and forth motion with zero net displacement. However, simultaneous vibration of perpendicular-mounted oscillators, prevents the microrobot’s displacement in one direction; leading to a net forward or backward oscillatory motion in the opposite direction.

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Fig. 1. Arrangement of the oscillators in the 2DOF microrobot.

Fig. 2. Microrobot’s general pattern of motion under the effect of stick-slip locomotion principle.

Fig. 3. Schematic of the 2DOF microrobot and corresponding elements.

Fig. 3 represents the detailed model of the 2DOF microrobot. The main mass, M, is attached to two auxiliary masses, mh , mv , using horizontal and vertical piezoelectric actuators. The piezoelectric actuators are assumed to be bolt-clamped Langevin [31]. Hence, in order to simplify the modeling, each actuator is substituted with a set of spring-dampers and a voltage coefficient. The voltage coefficient transforms the applied voltage into the mechanical stress. Fig. 4 shows the free body diagram of the system depicted in Fig. 3.

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Fig. 4. Free body diagram of the 2DOF microrobot.

Writing the equations of motion for each mass and in each direction based on Fig. 4, and simplifying the surface friction as the Coulomb friction, the governing equations for the 2DOF model are derived as:

(M + mv )x¨ = kh (u − x ) + bh (u˙ − x˙ ) − kVh Vh − Fhf mh u¨ = −kh (u − x ) − bh (u˙ − x˙ ) + kVh Vh − fhf

My¨ = kv (v − y ) + bv (v˙ − y˙ ) − kVv Vv + N − Mg

mv v¨ = −kv (v − y ) − bv (v˙ − y˙ ) + kVv Vv − mv g

(1)

where, kVh , Vh and kVv , Vv are the simplified voltage coefficients and the excitation voltages of the horizontal and vertical f

f

actuators, respectively. Fh and fh are the friction forces that act on M and mh , and N is the normal surface force applied to the main mass, M. The first two equations, denote the dynamics of motion in the horizontal direction for the main mass and the horizontal auxiliary mass connected to it, while the last two equations identify the vertical direction dynamics of the main mass and its vertically attached component, mv . f f In the absence of horizontal friction forces, Fh and fh , the components oscillate in their initial position and no net motion occurs. However, the friction terms act in opposition to motion in specific directions and contribute to the net displacement of the system. The value that each friction term can attain is dependent on the other terms of force that act on the same part. The friction forces cannot surpass a specific value that is dependent on the surface normal force. If the sum of forces applied to each body, does not surpass this critical value, the body stays stationary since the friction force prevents motion. However, if the net forces acting on the body surpass that critical value, the friction force will not increase anymore and the body goes under forward or backward motion. Eqs. (2) and (3) summarize the value ascribed to each friction term. Since the Coulomb friction law is implemented in the modeling, the surface roughness is modeled using the coefficient of friction, μ.



| F | < μN → Fhf =  F | F | ≥ μN → Fhf = μNsgn(x˙ )  | f | < μmh g → fhf =  f  f = −kh (u − x ) − bh (u˙ − x˙ ) + kVh Vh → | f | ≥ μmh g → fhf = μmh gsgn(u˙ )  F = kh (u − x ) + bh (u˙ − x˙ ) − kVh Vh →

(2)

(3)

The analysis of the motion is confined to the stick-slip mode and the excitation of the actuators is controlled such that the masses and the substrate do not separate. Thus, y = 0 and the last two equations are rewritten as:



y = y˙ = y¨ = 0 ⇒

0 = kv v + bv v˙ − kVv Vv + N − Mg ⇒ mv v¨ = −kv v − bv v˙ + kVv Vv − mv g



N = mv v¨ + mv g + Mg mv v¨ = −kv v − bv v˙ + kVv Vv − mv g

(4)

The second part in Eq. (4), provides the vertical position of the vertical auxiliary mass, v, as a function of the excitation voltage and the first part of Eq. (4), presents the normal force applied to the main mass as a function of v.

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Table 1 Initial physical properties of the 2DOF microrobot. Physical property

Piezoactuator equivalent stiffness

Piezoactuator voltage coefficient

Main body mass

Horizontal/vertical auxiliary masses

Value

k = 1.22 × 107 kN/m

kV = 2.28N/V

M = 50 g

mh = 30 g mv = 5 g

Excitation voltage amplitude

Excitation voltage frequency

V¯ = 100 V

 = 7 kHz

Excitation voltages phase difference

φ h − φv = π / 2

Fig. 5. Linear motion of the initial design of the 2DOF microrobot.

As mentioned earlier, the excitation voltages applied to the actuators are harmonic as in Eq. (5), in which V and  are the amplitude and frequency of these harmonic excitations:

Vh = V¯ sin (t + φh ) Vv = V¯ sin (t + φv )

(5)

Having the differential form of the equations of motion and the excitation commands fully defined, the obtained dynamic model of the microrobot is numerically simulated so as to study its motion characteristics. The set of initial physical properties of the microrobot are represented in Table 1. In order to analyze the motion, the set of equations of motion in the form of Eq. (6), is numerically simulated in Simulink toolbox of MATLAB for a given time interval, [T0 TF ]. In this formulation x0 , u0 , x˙ 0 , u˙ 0 represent the initial position and velocity of the horizontal coordinates.

  x u

    = f˜

x0 , u0

x˙ 0 u˙ 0

 ,  , V¯ ,

φh , φv

(6)

Fig. 5 represents the position of the microrobot and its horizontal and vertical auxiliary masses for the first 3 ms of its progress on the surface. This plot is generated by simulating the derived dynamic model in Simulink. The parameters used in this model are adopted from Table 1. As Fig. 5 shows, the main body and the horizontal auxiliary mass follow an undulatory motion with a nonzero net displacement over time, while the vertical mass oscillates up and down around its initial position. Fig. 6 shows the fluctuations of the surface normal force acting on the main body of the microrobot in the initial 1 ms of motion. The minimum value of the surface normal force is always greater than zero, meaning that the surface and the edge do not separate and the stick-slip condition is not violated in the course of robot locomotion. Figs. 5 and 6 show the simulation results over a 3 ms time span. The presented mathematical model and its simulation, provide a straightforward means for analyzing the effect of physical parameters and excitation properties on the motion features of the 2DOF microrobot. The prepared model acts like a function: the inputs to the function are the physical parameters and excitation characteristics and the output of the function is the motion behavior of various components of the microrobot.

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Fig. 6. Fluctuations of the surface normal force acting on initial design of the 2DOF microrobot.

3. Optimization of the proposed design process The proposed arrangement of the masses and actuators in the 2DOF microrobot is a general solution for achieving the objective of precise and micro-scale motion. The motion dynamics are highly dependent on the excitation voltages and physical attributes of the microrobot. In order to search for the optimum values of these parameters, we employ an optimization method and search for the parameters that satisfy the design objectives. In this regard, the objectives, the parameters, and the optimization algorithm should be determined. In this section, all the figures are adopted from the simulations of motion in a specific time-interval.

3.1. Objective functions and constraints The objectives of the design process are specified in the form of objective functions each of which relies on the motion of the microrobot in a specific time interval. Average locomotion velocity and average delivered power are the main motion features of microrobot that are taken into account throughout the design process. Moreover, since the dynamic modeling is contingent on the basic assumption of stick-slip motion, this constraint is also defined as an objective function and acts on the optimization process as a penalty term. Since the instantaneous speed is constantly oscillating within positive and negative bounds, the average velocity is defined over a time interval and measures the motion proficiency. In this definition, as Eq. (7) represents, T0 and TF depict the boundaries of the analysis time interval. In order to maximize the average velocity, its inverse value forms a function that should be minimized: f1 = 1/v¯ .

   x − xT0  1 v¯ =  TF = TF − T0

(7)

f1

In order to analyze the power consumption of the design, the mechanical power that is delivered by the microrobot is calculated. This value depends on the forces that act on the contacting edges of the microrobot. Eq. (8) defines the general method for calculating the average mechanical power delivered to the microrobot in a given time interval. In this definition, k denotes the contact point while vk and Fsk represent the instantaneous speed and friction force at that point, respectively. The second objective function (f2 ) is established upon the mechanical power delivered to the microrobot. It should be noted that the mechanical power delivered to the microrobot is not the same as the electrical power consumed by the mechanism. In fact, it represents the portion of the power that is converted to mechanical power. In this equation, n represents the total number of contact points between the microrobot and the substrate. Here we have n = 2.

n

TF 

k=1

P¯ =

 vk .Fsk dt

T0

TF − T0

= f2

(8)

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Table 2 Optimization parameters and their bounds. Parameter

Horizontal auxiliary mass mh

Vertical auxiliary mass mv

Excitation frequency 

Phase difference of excitations φh − φv

Lower bound Upper bound

10 g 100 g

2g 20 g

5 kHz 20 kHz

π rad

0 rad

Table 3 Tuned parameters for the utilized genetic algorithm. Parameter

Number of optimization parameters

Number of objective functions

Population size

Mutation rate

Sharing parameter

Value

4

3

400

0.02

0.05

One of the major assumptions of the proposed modeling is that the microrobot’s vibration is confined to the stick-slip mode and no separation occurs at the contact points. In correspondence to this assumption, an external constraint is set on the optimization process in the form of the third objective function. In each simulation of a specific design’s motion, the normal force acting on the main mass, defined in Eq. (1) as N, is ˜ . The microrobot does not go through the separation phase, unless the minimum value of the recorded as time-series N ˜ , takes negative values. Otherwise, this negative value is imposed on the optimization process as recorded time-series, N the third nonnegative objective function. Nonzero values of this last objective function correlate to designs that violate the stick-slip assumption.



˜ f3 = f3 N

(9)

3.2. Design parameters The next step would be determining the design parameters. It is evident that the piezoelectric actuators are mounted on the microrobot and their physical characteristics are fixed. On the other hand, the auxiliary masses connected to the actuators can be adjusted and then mounted on the microrobot. Hence, in addition to the excitation parameters, the horizontal and vertical oscillating masses can be adjusted in the course of optimization. Table 2 summarizes the design parameters and their lower and upper bounds. Other physical parameters are fixed with the values given in Table 1. 3.3. Optimization algorithm The optimization process should combine the dynamics of the 2DOF microrobot and the objective functions to search for the set of desired parameters. The set of desired parameters should result in the aimed motion features. Due to the nonlinear dynamics and the abundance of design parameters and design objectives, a multi objective evolutionary algorithm (MOEA) can guarantee an efficient search for the optimum solutions. Among the various evolutionary algorithms, the non-dominated sorting genetic algorithm ii (NSGA ii) is a constrained multi-objective algorithm that provides relatively less computational complexity, emphasizes elitism, and preserves diversity throughout the optimization process [32]. The main framework of this algorithm is similar to the genetic algorithm; however, sorting the population is executed based on two factors: the domination rank and the crowding distance of each solution. The most important advantage of this sorting is that the several objective functions are not required to get mixed in a single objective. As a result, each objective can have its own mean value and variance, hence, defining weighting coefficients will be useless. After generating each solution, the physical parameters corresponding to that solution are set on the dynamic model and the model is simulated in a specific time-interval. The simulation provides the motion characteristic which are employed in computing the objective functions. Once the objective functions are determined for each member of the population, the solutions are divided into various fronts based on the domination rank. The first front includes the members which are not dominated by any other population member. The second front is composed of the members that are only dominated by the members of the first front, and so on. Fig. 7 shows the arrangement of the solutions and how they are compared for a case of two objective functions. Several configurations are formed and plotted based on two objective functions f1 and f2 . As seen in Fig. 7, these points have been classified into three separtae groups. The points in group F3 are inferior than the points in groups F1 and F2 because they have higher values for both objective functions. The same comparison can be applied to members of groups F1 and F2 . But the members of each group are not superior than one another. This comparison is the basis of non-dominated sorting. F3 is dominated by F2 , and F2 is dominated by F1 . The approximate values of the objective functions is not important because they do not have to get mixed in a single objective function. Properties of the implemented genetic algorithm are represented in Table 3. In the implemented genetic algorithm, the population goes through a binary tournament selection while the sorting process preserves the diversity using a crowded comparison operator. The crowding distance of each solution represents how compact the solutions are distributed in the solution space. The computational complexity of this algorithm is in the form: O(Mf Npop 2 ), in which Mf is the number of objectives and Npop is the population size. The optimization process is carried out until the successive generations reach a convergent set of solutions.

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Fig. 7. Basic structure for comparing solutions and finding Pareto-optimal configurations.

Fig. 8. The change in the first front of population throughout the optimization algorithm.

4. Optimized designs for the proposed stick-slip microrobot Carrying out the optimization algorithm on the dynamic model leads to a set of solutions that exhibit lower values of objective functions compared to the initial randomly generated designs. The elements of the first front of population, F1 , go through a monotonic increase in numbers throughout the process. Fig. 8 shows this increase until the first front of population saturates at its maximum value; this value is equal to the population size. The results of this section correspond to the simulations executed in Simulink. 4.1. Selecting the optimized design In order to select one design as the final result of the optimization algorithm, we need to take a look at the distribution of population in the last step of the algorithm. The first front which includes all of the population members in the last step of the optimization process, comprises the final solutions that are not superior to any other solution. The first step is to sort out the solutions that violate the basic stick-slip assumption. The value of the third objective function determines whether a configuration is acceptable or not. If f3 = 0 for a specific configuration, then surface separation occurs for that design. Fig. 9 displays the distribution of optimized designs with respect to the values of the first two objective functions, f1 and f2 . In

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Fig. 9. Distribution of solutions with respect to first and second objective functions.

Fig. 10. Final acceptable solutions with respect to average velocity and delivered mechanical power.

order to distinguish the arrangements that violate the fundamental stick-slip assumption from the ones that do not, the designs that present a nonzero value of third function, f3 , are plotted in another color in Fig. 9. These designs represent the cases where the microrobot separates from the substrate. As it is observed in Fig. 9, the solutions do not surpass a specific area of the two-dimensional space spanned by the first two functions. In other words, there is a trade-off between the average mechanical power delivered to microrobot and the average velocity of the microrobot. A design cannot achieve high values of velocity while keeping the delivered mechanical energy low at the same time. Based on the definition of non-dominated sorting, there are points in Fig. 9 that are evidently dominated by the crowded area of the solutions. However, these points are not dominated by any other solution because of the value that they assign to the third objective function. The 3-dimensional view of the distribution is relatively more informative about the shape of the first front of solutions. The main goal of the optimization process is to come up with a design that minimizes the two distinct objective functions. In this regard, one solution should be selected out of the 47 acceptable solutions in the last generation of genetic algorithm’s population. Fig. 10 shows them in the two-dimensional space that is spanned by average velocity, v¯ , and delivered mechanical power, P¯ . A third criterion defined as deliverance of maximum velocity to mechanical power ratio is defined and imposed on the solutions. This criterion can be defined mathematically as: max(v¯ /P¯ ) ≡ min(P¯ /v¯ ). The red line

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Table 4 Physical parameters of the selected design. Parameter

Main body mass

Horizontal auxiliary mass

Vertical auxiliary mass

Excitation frequency

Phase difference of excitations

Value

M = 50 g

mh = 14.7 g

mv = 3.6 g

 = 9.306 kHz

φh − φv = 0.35π rad

Fig. 11. Time response of the selected optimized design.

Fig. 12. Surface normal force applied to the main body of the optimized design.

in Fig. 10 is the least steep line which represents the configuration filtered by the last defined criterion. The physical parameters corresponding to this design are represented in Table 4. 4.2. Time response of the selected configuration The selected configuration and its excitation properties are set on the dynamic model and its simulation response is recorded. Fig. 11 depicts the position behavior of this specific design for the first 3 ms of simulation. As indicated before, the normal force acting on the main body should not pass the zero boundary. Fig. 12 represents the ˜ ) for the first 3 ms of simulation. This signal shows that after less than a millisecond, the surface normal force signal (N

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Fig. 13. Effect of excitation amplitude and frequency on average velocity of the optimized design.

normal force enters its steady-state oscillation and does not transfer the motion into separation mode. The analyses of the average velocity and delivered mechanical power are also conducted for the data-points in which the microrobot has entered its steady-state motion, meaning that the transient part which might go through separation is truncated. 5. Further analysis of the motion and the effect of various design parameters After selecting a specific configuration as the optimal solution, the selected model is further analyzed to investigate the effect of excitation parameters and physical parameters on its motion. The excitation parameters include the frequency, amplitude, and the phase difference of the excitation voltages shown in Eq. (5). In the beginning of the optimization, it was assumed that the amplitude and frequency of the horizontal and vertical oscillators are the same, while their phases are different. The changes applied to the dynamic model obey the same rule, meaning that changes are applied to both horizontal and vertical actuators. The performed analyses of this section is the result of several simulations with different physical and excitation parameters. 5.1. Effect of excitation’s amplitude and frequency on motion In order to inspect the role of excitation frequency and amplitude on the average velocity of the 2DOF microrobot, the physical parameters of the optimized design are set on the model and the frequency and amplitude are varied in specific intervals. As depicted in Fig. 13, the frequency is increased in the span of  = [5 20] kHz, while the voltage amplitude is held constant at 8 evenly distributed values of V¯ = {20, 40, 60, 80, 100, 120, 140, 160}. In some cases, the microrobot’s motion might shift into the separation phase due to the increase in the frequency, hence, crossing the zero limits in the surface normal force. The configurations that do not comply with the stick-slip assumption and pass this zero limit, are marked off in their respective plots. As expected, the higher the amplitude of excitation voltage, V¯ , is, the sooner the surface separation occurs, f3 = 0. Fig. 13 shows the change in average locomotion velocity in response to alterations in the excitation voltage. The approximately linear relation between the average locomotion velocity and excitation frequency can be explained in terms of the number of forward steps the microrobot takes in a specific time interval. Increasing the frequency results in a proportional increase in the number of steps in that time interval. Hence, the average locomotion velocity increases linearly with frequency. The corresponding values of mechanical power delivered to microrobot are plotted in Fig. 14. Similar to Fig. 13 the points at which separation occurs are represented differently. Comparison of Figs. 13 and 14 reveals the fact that higher locomotion velocity necessarily requires more power to be delivered to the microrobot. 5.2. Effect of excitation’s phase difference on motion In the next step, the phase difference of horizontal and vertical excitation commands is varied and its effect on average locomotion velocity and delivered mechanical power is plotted in Figs. 15 and 16, respectively. The phase difference is created by fixing the vertical phase at zero, φv = 0, and changing the horizontal one in the φh = [0 2π ] rad span.

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Fig. 14. Effect of excitation amplitude and frequency on delivered mechanical power of the optimized design.

Fig. 15. Effect of phase difference on average velocity of the optimized design.

The point annotating the optimum design is located at the peak of Fig. 15 which verifies the effectiveness of the utilized optimization procedure. It means that the phase difference value found by the optimization algorithm is in agreement with the value found by this single analysis. The data presented in Fig. 15 can be employed to drive the microrobot with a specific velocity in a desired direction. In this regard, the phase difference value corresponding to each velocity should be set on the excitation voltages. The effect of phase difference on the average delivered mechanical power is to some extent in correlation with the diagram representing the average velocity. For the phase values where the velocity, v¯ , is in the vicinity of its maximum value, the delivered power, P¯ , takes its minimum value. This correlation indicates the high efficiency of the motion in the vicinity of this configuration. In contrast, at the boundaries or in the middle of the interval ([0 2π ] rad), the velocity reaches approximate zero values, while the power diagram implies high rate of delivered power. In such cases, the various parts of the 2DOF microrobot are in constant oscillatory motion, while the net average displacement is zero; it denotes a stationary but power-consuming microrobot.

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Fig. 16. Effect of phase difference on delivered mechanical power of the optimized design.

Fig. 17. Effect of surface friction coefficient on average velocity of the optimized design.

5.3. Effect of surface roughness on motion In further attempts and in order to analyze the effect of surface friction on motion, the coefficients of static and kinetic friction of the surface are altered and the simulation response is recorded. Since the surface friction is the main driving principle of this mechanism, the results of this investigation are of considerable importance. The friction coefficient is varied in the range μ = [0 1] and the dynamic system is simulated for each value. Similar to previous analyses, the results are displayed for the average velocity and delivered mechanical power in Figs. 17 and 18. As demonstrated, the dependence of microrobot’s average locomotion velocity, v¯ , on the friction coefficient of the substrate is not uniform for various values of the coefficient. For the values less than approximately 0.5 (μ ≤ 0.5), the locomotion speed does not go through considerable change, while for the values greater than 0.5 (μ > 0.5) the surface roughness acts in opposition to motion. In this case, the average velocity that the microrobot can attain decreases with the constant increase in friction coefficient. The monotonic decline of the velocity continues up to the point where the velocity reduces to half

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Fig. 18. Effect of surface coefficient of friction on delivered mechanical power of the optimized design.

its initial value. The other important point about Fig. 17 is the sudden rise of velocity from zero to its maximum value for the near-zero values of surface roughness. The noteworthy point is that a smooth surface does not inevitably result in zero locomotion velocity, unless the coefficient of friction is approximately equal to zero, μ∼ =0. As illustrated in Fig. 18, surface roughness which is embodied in dynamic equations in the form of friction coefficient, μ, has a direct impact on the mechanical power delivered by the microrobot; the less smooth the substrate is, the harder it gets for the stick-slip mechanism to drive the microrobot. The results presented in this section, were derived by varying one or several parameters and observing the behavior of the microrobot in simulation in response to these modifications. These responses are of significant importance since they can be used to manipulate the microrobot at a desired velocity in a specific direction. The results presented in Figs. 13–18 can be used to construct look-up diagrams for driving a fabricated microrobot. 6. Experimental results of the proposed locomotion mechanism The dynamic modeling presented for the 2DOF microrobot is the basis for the optimization procedure. The results which showed the effect of physical parameters on motion characteristics are dependent on the dynamic modeling. Although the governing equations were derived by simplifying most of the dynamic attributes, the most prominent features of the microrobot’s dynamics were preserved. Hence, the most significant motion characteristics are inherited in the dynamic equations and their corresponding models. In order to verify this notion, an experimental setup is prepared and a sample microrobot with the proposed mechanism is implemented and tested in this setup. Fig. 19 shows the designed and fabricated prototype of the microrobot. The physical properties of the piezoelectric actuators are similar to the data presented in Table 1 and the masses are measured to be M = 30 g, mh = 35 g, and mv = 5 g. Assigning the actual values and properties of the masses and actuators to the previously-developed dynamic model, the motion of the microrobot can be simulated and compared with the experimental results. In order to obtain the experimental results, an initial harmonic voltage excitation with amplitude of V¯ = 100 V ,  = 6 kHz, and φh − φv = π /2, as described in Eq. (5) is applied to the stack piezoelectric actuators. Since the motion steps are too small to be detected, the displacement of the microrobot is recorded after considerable amount of time. This way, the average locomotion velocity is calculated and compared with the theoretical results. 6.1. Effect of voltage amplitude on the locomotion velocity: experiment and modeling comparison In the first step, fixing all other parameters, the amplitude of the excitation is changed and its effects on velocity are recorded. Fig. 20 displays a comparison of the experimental results and the simulation results for the effect of voltage amplitude on the average velocity. As seen in Fig. 20, the slope to both result series is positive, which means that the displacement in a given time interval is proportional to the voltage applied to the actuators.

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Fig. 19. The (a) designed and (b) fabricated prototype of the 2DOF microrobot.

Fig. 20. Comparison of the experimental and the simulation results for the effect of voltage amplitude on average locomotion velocity.

6.2. Effect of phase difference on the locomotion velocity: experimental and modeling comparison Similarly, the effects of phase difference on average velocity is recorded and compared in Fig. 21. The phase difference applied to the piezo-actuators is varied in the interval of φh − φv = [0 π ] rad. The experimental results show that for certain values of the phase difference, the microrobot does not move on the substrate. This fact can be explained by referring to the micro-scale structure of the surface which is, in contrast to the friction assumptions, not uniform. The bottom edge of the microrobot which is in contact with the substrate has a non-smooth micro-scale structure. This non-uniform edge might get stuck in the porous structure of the substrate at specific positions. In these cases, the inertial drive of the main body cannot initiate the motion progress of the stuck microrobot and the recorded value of velocity is zero. In modeling the system, it was assumed that the friction force applied to each mass is linearly proportional to the normal surface force regardless of the exact position of the microrobot. However, the probability of getting stuck in the porous micro-scale structure which is dependent on the geometry of interaction point, can largely affect the dynamics. The comparison of the motion for the fabricated microrobot tested in the experimental setup and the presented dynamic modeling, show that general characteristics of the dynamic modeling are in accordance with the experimental tests. Although the values of the locomotion velocity are not the same for the two sets of data in Figs. 20 and 21, but the dependence of the velocity on the varying parameter is verified for both models. The un-modeled dynamics, non-perfect experimental setup, simplified modeling of the piezo-actuators, and the uncertainties in the values assigned to physical parameters, are the main reasons for these inconsistencies. The presented dynamic model considers the stacked piezo-actuators as a linear spring-dashpot system, which is a simplification of the higher order true dynamic model. Also, the adopted friction model preserves much of the dynamics but it does not model the interaction

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Fig. 21. Comparison of the experimental and the simulation results for the effect of phase difference on average locomotion velocity.

completely. Moreover, the parameters assigned to the linear piezo-actuator are the nominal values that are adopted from the actuators’ catalog while the true values may deviate from these values. Much of this difference stems from the stacking process of the single piezo-actuators. The other important factor that affects the recorded average locomotion velocity is the experimental setup. The fabrication process and the prepared substrate are not ideally flat. A thick layer of glass is used in this regard, but it should be noted that even a small amount of dust can affect the contact point of the microrobot and substrate. 7. Conclusions The recently shaped demand in developing high-precision micro-scale tools has created a rise in development of mobile microrobots. Mobile microrobots are defined as robots capable of creating high-precision micro-scale motion. Modulated friction inertial drive (MFID) and stick-slip locomotion (SSL) are two main locomotion mechanisms that have been effectively utilized in various mobile microrobots to create the precise motion. In order to further investigate the SSL mechanism and to optimize the motion characteristics of this mechanism, a design for a 2DOF piezo-actuated microrobot is proposed. The idea is then modeled, optimized, and fabricated. The proposed design comprises three separate masses that are connected using piezoelectric actuators. Two harmonic excitation voltages applied to the actuators, create slip generation and slip variation in the contacting point of the microrobot and the substrate. The simultaneous vibration of the horizontal and vertical mounted parts creates an undulatory motion with non-zero displacement. The microrobot’s response goes through a transient and later a steady-state phase. During the transient state, the presented dynamic equations are not applicable as the main body may detach from surface. Instead, the steady-state response is recorded in order to analyze each configuration. The motion characteristics highly depend on the physical parameters and excitation features. As a result, in order to find the optimal designs, the parameters and the dynamic model of the general mechanism are fed into a multi-objective optimization algorithm so as to minimize/maximize several objective functions. High average locomotion velocity and low mechanical power consumption are the main objectives of the design process, while confinement to the stick-slip mode of motion acts on the optimization as a design constraint. Instead of weighting the objective functions and combining them into a single objective, a multi-objective optimization method is utilized to look for the optimal configurations. The employed non-dominated sorting genetic algorithm (NSGA ii) results in a front of solutions each of which is not superior to the others with respect to various objective functions. After selecting one of the optimized solutions, the physical parameters corresponding to that solution are set on the dynamic model for simulations and further analysis. The effect of various excitation parameters on the average velocity and delivered mechanical power of the design is achieved by varying each parameter and recording the microrobot’s response. The results of these simulations indicate that average locomotion velocity is proportional to the frequency and amplitude of excitation that is applied to the actuators. This relationship is also observed for the delivered mechanical power of the micro-robot, meaning that an increase in frequency and amplitude of excitation, results in higher mechanical power consumption. The results can be used as look-up diagrams to manipulate the device in a specific direction with a desired velocity, while holding the power at its lowest possible value. The phase difference of the horizontal and vertical actuators can also affect the average locomotion velocity. The results show that

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velocity reaches a peak value at specific phase differences. The voltage amplitude, frequency, and phase value, are the three parameters that can be utilized in driving the microrobot with given motion properties. Finally, a prototype of this design is fabricated and tested in an experimental setup so as to determine the inconsistencies and similarities of the modeling and experiment. It is observed that although the results are not completely in agreement, the general motion behavior is in accordance for the model and the physical prototype. The un-modeled dynamics, simplifications applied to the friction force, uncertainties in the physical parameters, and the non-perfect substrate are the main reasons for the difference of the two theoretical and experimental data sets. Declarations of interest None. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.mechmachtheory. 2018.11.025. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

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