Piezoelectric inertial robot for operating in small pipelines based on stick-slip mechanism: modeling and experiment

Abstract

Small pipes exist in industrial and biomedical fields, and require microrobots with high operational precision and large load capacity to inspect or perform functional tasks. A piezoelectric inertial pipeline robot using a “stick-slip” mechanism was proposed to address this requirement. In this study, the driving principle of the proposed robot was analyzed, and the strategy of the design scheme was presented. A dynamics model of the stick-slip system was established by combining the dynamics model of the driving foot system and the LuGre friction model, and the simulation analysis of the effect of system parameters on the operating trajectory was performed. An experimental system was established to examine the output characteristics of the proposed robot. Experimental results show that the proposed pipeline robot with inertial stick-slip mechanism has a great load capacity of carrying 4.6 times (70 g) its own mass and high positioning accuracy. The speed of the pipeline robot can reach up to 3.5 mm/s (3 mm/s) in the forward (backward) direction, with a minimum step distance of 4 μm. Its potential application for fine operation in the pipe is exhibited by a demonstration of contactless transport.

Key words： pipeline robot; piezoelectric; inertial drive; stick-slip; large load capacity; dynamics model; small pipeline

1 Introduction

A wide range of small pipes, such as gas pipes, condensation tubes, test tubes, and human intestines, exists in the industrial and biomedical fields. These tubular structures have long lengths, relatively small inner diameters, and most of them have complex external geometric shapes [1–3]. Such pipelines create difficulty when performing inspection and maintenance operations. Thus, the demand for miniature pipeline robots that can achieve stable and accurate motion and carry actuators, such as probing instruments and work tools, is increasing [4,5].

Pipeline robots can be classified in accordance with the drive source as fluid driven, electric motor driven, hydraulic and pneumatic driven, electromagnetic driven, and functional material driven [6,7]. Depending on their characteristics, pipeline robots with different working principles play a crucial role in applications [8,9]. Fluid-driven pipeline robots, one of the first pipeline robots, are driven directly from the fluid without any additional power elements, and can only be driven effectively in large pipes with sufficient pressure [10,11]. The pipeline robot driven by electric motors is the most commonly used drive method and has the maximum drive speed and drive force in applications with larger pipe diameters [12–15]. However, it cannot enter the pipe or the driving speed and driving force are reduced when the pipe diameter is small due to the size of the motor and transmission device [16,17]. Hydraulic and pneumatic drives are mostly identical in terms of drive principle, and they can be applied to miniature-sized pipe robots. However, the essential hose for media transfer and control limits the use of pipe robots for long-distance inspection [18–22]. Electromagnetic field-driven pipeline robots are easy to miniaturize, especially the field-driven capsule robots using Helmholtz and Maxwell coils that can travel in the human intestine; however, their control system is more complicated, and the positioning accuracy is low [23–25]. The technology of using functional materials as the actuators of micro pipeline robots has received extensive attention in recent years, in which piezoelectric materials, shape memory alloys, and magnetostrictive materials present the characteristics of easy intelligence, integration, and miniaturization. Although the pipeline robots with memory alloy and magnetostrictive materials as actuators can realize locomotion in small pipes, their response speed and locomotion accuracy are relatively low and sensitive to ambient temperature due to the limitation of the characteristics of the actuators [26–29]. Piezoelectric materials have the advantages of small size, light mass, high displacement resolution, high energy density, and fast response [30,31]. The response speed, driving force, and control accuracy of miniature pipeline robots with piezoelectric materials are improved, which are better choices as the driving element [32–34].

Idogaki et al. [35] of Denso (Japan) first developed a pipeline robot using a piezoelectric stack that can perform inspection tasks in pipes with a diameter of 8 mm; this robot functions on the principle of “main body + mass block” inertial impact type. Tsuruta et al. [36] improved the previous structure by replacing the stacked piezoelectric actuator with a four-layered lead zirconate titanate bimorph actuator. Gong et al. [37,38] proposed two types of inertial impact pipeline robots with piezoelectric stack and piezoelectric bimorph as driving elements for vertical and horizontal locomotions, respectively. Liu et al. [1] and Sun [39] also developed an inertial impact type of pipeline robot with piezoelectric bimorph. Liu et al. [40] proposed different configurations to fabricate pipeline robots with piezoelectric materials. The abovementioned miniature pipe robots exhibit mainly inertial impact type motion of the “main body + mass block” pattern. Some researchers have explored alternative drive principles with piezoelectric materials. Yang et al. [41] proposed an approach of elliptical motion trajectory using ultrasonic vibrations of piezoelectric materials to excite the driving foot. Liu et al. [42] presented a pipeline robot in the form of bending and telescoping composite vibration using a piezoelectric stack. The new drive mechanism will enrich the advantages of piezoelectric material-driven pipeline robots and address the requirements of pipeline robots in different working conditions.

Most of the current inertia-driven pipeline robots use the “main body + mass block” inertia impact drive principle, which is good for miniaturization, large stroke, and high precision. However, most of the existing research focuses on the realization of inertia impact motion and ignores the problems of carrying actuators and operating in small pipes after miniaturization. The micro- and nano-scale fine manipulation in microelectromechanical systems and biomedical fields, such as manipulation of viruses or cells in test tubes and fine manipulation of biological tissues in outer space or in endoscopic tubes deep into the human body, has rapidly developed, especially in high-risk or inaccessible environments to humans. Therefore, small-sized pipeline robots that can perform precision operations with the capacity of large loads are urgently needed. For the development of a miniature pipeline robot with a large load and high position accuracy, a piezoelectric pipeline robot driven by the inertial stick-slip mechanism is proposed. This driving method is used for pipeline robots, which has not been reported before. The benefits of the proposed pipeline robot are its capability of bidirectional operation in small-sized pipes and carrying precision actuators of large mass, such as operating devices and endoscopes. The remaining parts of this paper are structured as follows. Section 2 illustrates the working principle and designed structure of the proposed pipeline robot. Section 3 demonstrates the modeling of the dynamic contact between the driving foot and pipe wall to analyze the drive process. Section 4 presents the fabricated prototype and discusses the experimental results and dynamic output characteristics. Section 5 provides the conclusions and perspectives of this work.

2 Working principle and structure

2.1 Working principle

An inertial actuator with the functional principle of the “stick-slip” mechanism usually employs a sawtooth wave signal (Fig.1(a)) to supply the piezoelectric elements. In the stick phase, the piezoelectric actuator slowly expands along with the rising edge (slow positive ramp) of the drive signal, and then the slider moves with the driving object by a distance Δx due to the friction. In the slip phase, the piezoelectric actuator rapidly contracts with the falling edge (steep decay) of the drive signal, and the slider keeps the position due to the inertia, whereas the driving object moves to the original position. Thus, the slider moves a distance Δx with respect to the driving object (Fig.1(b)). Inspired by this drive mechanism, the process is transformed into a drive method suitable for pipeline robots. In Fig.1(c), the slider is transformed into a pipeline, and the driving object is transformed into a driving foot. The driving process of the proposed pipeline robot is explained in detail below.

Fig.1 Driving principle of the pipeline robot: (a) drive signal, (b) driving principle of “stick-slip,” (c) driving principle of the proposed pipeline robot.

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1) Initial phase: When time t = 0, the drive signal is in the low-level stage, and the piezoelectric stack is not excited. Thus, the driving foot mechanism is not deformed, and the robot is in a rest state.

2) Stick phase: When the signal of the excitation voltage slowly rises during the time 0−t₁ phase, the piezoelectric stack slowly elongates and pushes the driving feet to produce deformation. The friction force at the contact points between the driving feet and the inner wall of the pipe is increased, and the contact points of the driving feet are relatively stationary with the pipe wall. The main body moves to the left by a distance Δx to comply with the deformation of the driving foot. Here, the frictional force between the Δx support feet and the pipe wall is set to be smaller than the thrust of the main body.

3) Slip phase: When the drive signal is in the phase of time t₁−t₂, the voltage of the drive signal declines rapidly from the maximum value to zero, and the piezoelectric stack contracts rapidly. The driving feet restore deformation due to the flexure hinge recovery force. At this time, the friction force between the contact points of the driving feet and the pipe wall is reduced. The position of the body relative to the pipe remains unchanged, and the relative slip between the driving feet and the pipe wall occurs because of the inertial force of the main body. Therefore, the pipeline robot as a whole will be displaced to the left by ∆x. However, in practice, the process still has a backward phenomenon, so the actual displacement will be slightly smaller than ∆x.

The above described process can be repeated to achieve the continuous stepping motion of the proposed pipeline robot. The step length, speed, and motion direction of the robot can be controlled by adjusting the drive voltage amplitude, frequency, and duty cycle of the sawtooth wave.

2.2 Structure design

The devised pipeline robot is presented in Fig.2 in accordance with the analysis of the operating principle. The main components of the proposed pipeline robot include two piezoelectric stacks, a main body, four support feet, and two driving feet. The main body is composed of two blocks, which use a hollow structure to minimize the mass of the main body. Four support feet are evenly distributed around the body and are bolted to the body. The bending structure of the support foot can provide the pre-pressure to the pipe wall and adapt to the various diameters of the pipeline by adjusting the screws. Two piezoelectric stacks are symmetrically embedded into the body with a preload provided by screws. Two flexure hinges with restoring force are mounted on one block of the main body, which amplifies the output displacement of the piezoelectric stack and supports the rotation of the driving feet. The driving foot mounted on the flexure hinge is an L-shaped plate spring with a half-sphere contact point that is set at the cantilever end to interact with the pipe wall. For the other block of the main body, a mounting hole is devised to assemble the camera module, injectors, and noncontact gripper. The proposed pipeline robot limited by the size of selected piezoelectric stacks (20 mm × 5 mm × 5 mm) has the capacity to adapt the range of the pipeline diameter of 25–35 mm. Therefore, it can further be miniaturized.

Fig.2 Configuration of the devised pipeline robot and portable devices for the application.

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3 Modeling and analysis of the driving process

3.1 Modeling of the driving process

A dynamic model considering the dynamic frictions needs to be established to describe the driving process and investigate the dynamic behavior theoretically. The physical model must be simplified to a dynamics model. Half of the structure of the pipeline robot is taken as a research object that includes a piezoelectric stack, a flexure hinge, a driving foot, and the pipe wall due to the symmetrical structure of the proposed pipeline robot. In actual operation, the robot is in motion, and the pipe is stationary. For ease of analysis, the robot is assumed to be motionless and the pipe moves relative to the robot to build the dynamic model, as shown in Fig.3.

Fig.3 Dynamic model of the proposed pipeline robot.

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In Fig.3, the equivalent mass of the piezoelectric stack is denoted by m₁, the equivalent mass of the flexure hinge is denoted by m₂, the equivalent mass of the driving foot is denoted by m₃. Here, the pipeline is equivalent to a beam with the equivalent mass denoted by m₄. Considering the transition of the motion state of the robot and the pipe and the measurement of inertial forces in the equilibrium equation, the magnitude of the mass m₄ is equivalent to the overall mass of the robot. x₁ is the output displacement of the piezoelectric stack, that is, the input displacement of the system. x₂, y₃, and x₄ denote the displacements of mass blocks m₂, m₃, and m₄, respectively. k₁, k₂, and k₃ denote the equivalent stiffness of the piezoelectric stack, flexure hinge, and plate spring, respectively, and c₂ and c₃ denote the equivalent damping of the flexure hinge and plate spring, respectively. F₁₂ and F_{12}^{\prime } are the acting force and reacting force between the piezoelectric stack and the flexure hinge. F₁ and F_1^{\prime } are the acting force and reacting force between the driving foot and the pipe wall.

In accordance with Newton’s Second Law, the equations describing the dynamics behavior can be presented as

(1)\{\begin{array}{c}\left(m_2+m_3\right){\ddot{x}}_2=F_{\text{p}}-k_2{x}_2-c_2{\dot{x}}_2-f_1,\\ m_3{\ddot{y}}_3=F_1-k_3{y}_3-c_3{\dot{y}}_3,\\ m_4{\ddot{x}}_4=f_1-f_2,\end{array}

where F_p is the output force of the piezoelectric stack at an output displacement of x₁, F₁ is the pressure of the driving foot on the pipe wall, f₁ is the friction between the driving foot and the pipe wall, and f₂ is the friction between the support foot and the pipe wall. From Ref. [43], the output force F_p of the piezoelectric stack can be written as

(2)F_{\text{p}} =k_{1}n{d}_{33}U-k_1{x}_1,

where n is the number of the piezoelectric sheets in the piezoelectric stack, U is the voltage of the drive signal, and d₃₃ is the piezoelectric coefficient. The stiffness k₁ of the piezoelectric stack can be calculated as k_{\text{1}} =A/\phantom{A\left(s_{\text{33}}{L}_{\text{p}}\right)}\left(s_{\text{33}}{L}_{\text{p}}\right), where A is the cross sectional area of the piezoelectric stack, L_p is the length of the piezoelectric stack, and s₃₃ is the elastic compliance coefficient of the piezoelectric stack.

In Eq. (1), the pressure F₁ of the driving foot is varied with the position of the contact point. Thus, the friction f₁ determined by F₁ is a varying force in the operation process. The cause of the change in force F₁ can be illustrated in Fig.4. The generation of displacement x₂ is determined by the rotation angle θ₁ of the flexure hinge. If the driving foot has no constraint, then the contact point will have a displacement y₃ at a rotation angle θ₂. However, the pipe wall restricts the position of the driving foot and creates a deformation of the driving plate spring. The value of the pressure F₁ is determined by the unconstrained displacement y₃ of the contact point due to the angle of rotation θ₁. In Fig.4, l is the length from the rotation center to the top of the flexure hinge displacement amplification structure, l₁ is the distance from the center of rotation of the hinge to the point of action with the piezoelectric stack, l₂ is the vertical distance from the top of the flexure hinge to the center of rotation under rotation θ₁, l₃ is the vertical distance between the top of the hinge and the initial position after rotation θ₁, and l₄ is the length of the driving foot. Given that the elongation of the piezoelectric stack is in the order of microns, the rotation angle θ₁ of the flexure hinge will be extremely small and can be approximated as

Fig.4 Analysis of the force and geometric relationship of the driving foot: (a) deformation of the driving foot and (b) geometric relationship of the driving foot.

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(3){\theta }_1=\frac{x_1}{l_1}.

From Fig.4(b), the relationship between y₃ and x₁ can be derived from the geometric features as

(4)y_3\left(t\right)=l_4\sin \frac{x_1}{l_1}-l_3.

The relationship between the output displacement and the pressure F₁ can be obtained by substituting Eq. (4) into Eq. (1). The dynamic model of the friction f₁ is established by employing F₁.

The dynamic friction model is introduced to consider the hysteresis phenomenon, variable static friction, and the Stribeck effect, and to accurately describe the nonlinear stick-slip driving principle. In accordance with the LuGre model, the frictional contact between the contact point of the driving foot and the pipe wall is simulated as the contact of two surfaces with elastic bristles. When a tangential force is applied, the bristles deform similar to a leaf spring, generating a frictional force. The bending displacement of the bristles is random due to the irregularity of the friction surface, and the LuGre model can be expressed as [44]

(5)\frac{\text{d}\mathit{\text{z}}}{\text{d}t}=v -\frac{\left|v\right|}{g\left(v\right)}\mathit{\text{z}},

where v is the relative velocity of two friction surfaces, that is, v={\dot{x}}_4- {\dot{x}}_2, z is the average deformation of bristles, and g(v) is a function describing the Stribeck effect. In Ref. [45], the function g(v) is derived as

(6)g\left(v\right)=\frac{F_{\mathrm{C}}+\left(F_{\mathrm{S}}-F_{\mathrm{C}}\right)\exp \left(-{\left({\displaystyle \frac{v}{v_{\mathrm{s}}}}\right)}^2\right)}{{\sigma }_0},

where F_C is the Coulomb friction force, F_S is the static friction force, v_s is the velocity when the Stribeck effect occurs, and σ₀ is the stiffness of the bristles. The Coulomb friction force can be calculated as F_{\mathrm{C}}=\mu F_1, where μ is the friction coefficient between the driving foot and the pipe. The friction force generated by the deflection of bristles can be expressed as

(7)f_1={\sigma }_0\mathit{\text{z}}+{\sigma }_1\frac{\text{d}\mathit{\text{z}}}{\text{d}t}+{\sigma }_{2}v,

where σ₁ is the damping coefficient of the bristle, and σ₂ is the viscous damping coefficient. If the bristles are in the steady-state motion, then \mathrm{d}\mathit{\text{z}}/\phantom{\mathrm{d}\mathrm{z}\mathrm{d}t}\mathrm{d}t=0. Thus, the relation between velocity and friction force for the steady-state motion in accordance with Eqs. (5)−(7) is expressed as

(8)f_1=F_{\mathrm{C}}\mathrm{s}\mathrm{g}\mathrm{n}\left(v\right)+\left(F_{\mathrm{S}}-F_{\mathrm{C}}\right) \exp \left(-{\left(\frac{v}{v_{\mathrm{s}}}\right)}^2\right)\mathrm{s}\mathrm{g}\mathrm{n}\left(v\right)+{\sigma }_{2}v.

In accordance with the characteristic of the moving process, the “slip-stick” motion of the pipeline robot can be divided into four phases. The movement state of each phase of the system is analyzed and discussed by using the LuGre model of the dynamic friction force. In our hypothesis of the system model, the runner is the pipeline. Thus, the dynamic equations in each phase are presented separately with it as the research object.

(1) Initial “stick” phase: The state between the driving foot and pipe wall can be considered the preload phase. At this moment, the friction between the contact point of the driving foot and the pipe belongs to the category of static friction. However, as described by the LuGre model, the small deformation z of two contact surfaces can be regarded as the occurrence of relative displacement x₄, that is, \mathit{\text{z}}=x_4. Substituting Eq. (7) into the equation that describes the motion of runner m₄ in Eq. (1) and assuming right is the positive direction, the dynamics equation describing this phase can be obtained as

(9)-m_4\frac{{\text{d}}^2{x}_4}{\text{d}{t}^2}={\sigma }_0\mathit{\text{z}}+{\sigma }_1\frac{\text{d}\mathit{\text{z}}}{\text{d}t}+{\sigma }_{2}v-f_2.

With \mathit{\text{z}}=x_4 and v=\mathrm{d}{x}_4/\phantom{\mathrm{d}{x}_4\mathrm{d}t}\mathrm{d}t, Eq. (9) can be rearranged as

(10)m_4\frac{{\text{d}}^2{x}_4}{\text{d}{t}^2}+\left({\sigma }_1+{\sigma }_2\right)\frac{\text{d}{x}_4}{\text{d}t}+{\sigma }_0{x}_4-f_2=0.

(2) “Stick” phase: The runner is driven by the contact point of the driving foot and moves together. From the macroscopic point of view, no relative displacement occurs between the runner and the contact point of the driving foot. From the microscopic point of view, the two still maintain the relative displacement z in the initial “stick” phase, and the friction force is still static at this time based on the LuGre model. Since the displacement z is not increasing, the friction force of the LuGre model can be written as f_1= {\sigma }_0\mathit{\text{z}}. Thus, the equations that can describe the dynamic behavior of the runner in this phase are written as

(11)m_4\frac{{\text{d}}^{2}x}{\text{d}{t}^2}-{\sigma }_0\mathit{\text{z}}+f_2=0.

(3) “Slip” phase before velocity {\dot{x}}_4=0: In this phase, the voltage of the drive signal has a steep decay, and the driving foot restores to its original shape with the piezoelectric stack. A substantial relative sliding motion occurs between the contact point of the driving foot and the runner due to the inertia. The runner starts to decelerate under the action of friction forces f₁ and f₂ until the velocity is zero. This phase is named the preslip-zero phase. If the right side is the positive direction, then the dynamics equation describing this phase can be obtained as

(12)m_4\frac{{\text{d}}^2{x}_4}{\text{d}{t}^2}=-\left({\sigma }_0\mathit{\text{z}}+{\sigma }_1\frac{\text{d}\mathit{\text{z}}}{\text{d}t}+{\sigma }_{2}v\right)-f_2.

In this phase, the average tangential bending displacement of the bristles in the LuGre model can be regarded as reaching the steady state, that is, \mathrm{d}\mathit{\text{z}}/\phantom{\mathrm{d}\mathit{\text{z}}\mathrm{d}t}\mathrm{d}t=0. Thus, the dynamic equation of the preslip-zero phase can be rearranged as

(13)m_4\frac{{\text{d}}^2{x}_4}{\text{d}{t}^2}+{\sigma }_{2}v+{\sigma }_0\mathit{\text{z}}+f_2=0 .

(4) “Slip” phase after velocity {\dot{x}}_4=0: During the phase of restoring the deformation of the driving foot, the runner may follow the driving foot to move back a certain distance due to the existence of sliding friction between the driving foot and the runner after the velocity decreases to 0. The retreating distance is called the backlash. The runner is a reverse acceleration process before the arrival of the next periodic signal. The dynamic equation is rewritten as

(14)m_4\frac{{\text{d}}^2{x}_4}{\text{d}{t}^2}+{\sigma }_{2}v+{\sigma }_0\mathit{\text{z}}-f_2=0.

3.2 Simulation analysis

In the “slip” phase, the runner is in a process of deceleration and then reverse acceleration. The driving foot overcomes the friction before the next stick motion arrives to complete the deformation recovery, where it is ready to start the initial “stick” phase again. The motion process of the pipeline robot is simulated by using the equations of motion established above to investigate the motion state of the robot in one cycle and the effect of the system parameters on the drive performance. The system parameters of the pipeline robot can be obtained by testing and referring to Refs. [44,45]. The detailed values of these parameters are listed in Tab.1–Tab.3.

Tab.1 Parameters of the employed piezoelectric stack

Symbol	Quantity	Value
d33	Piezoelectric coefficient	635 pm/V
S33	Elastic compliance constant of the piezoelectric stack	18.1 × 10−12 m−2/N
Lp	Length of the piezoelectric stack	20 mm
A	Cross section area of the piezoelectric stack	25 mm2
n	Number of the piezoelectric sheets	181
Vmax	Peak voltage of the drive signal	120 V
x1max	Maximum output displacement of the piezoelectric stack	15 μm

Tab.2 Parameters of the LuGre model

Symbol	Quantity	Value
FS	Maximum static friction	0.025 N
μ	Friction coefficient	0.78
σ0	Bristle stiffness	105 N/m
σ1	Bristle damping	{10}^{5/\phantom{52}2} N∙s/m
σ2	Viscous damping coefficient	0.4 N∙s/m
vs	Stribeck velocity	0.001 m/s
f2	Friction between support foot and pipe wall	0.03 N

Tab.3 Parameters for the structure properties of the robot

Symbol	Quantity	Value
k2	Stiffness of the flexure hinge	3.9×106 N/m
c2	Damping of the flexure hinge	0.002 N∙s/m
k3	Stiffness of the driving foot	1.8×105 N/m
c3	Damping of the driving foot	0.008 N∙s/m
m2	Equivalent mass of the flexure hinge	0.66 g
m3	Equivalent mass of the driving foot	0.15 g
m4	Equivalent mass of the pipe	13.38 g

The system parameters in Tab.1–Tab.3 are substituted into the dynamic equations of the four phases. The motion trajectory of the runner in one cycle of the drive signal is presented in Fig.5(a). Given that the motion states of the pipe and the robot are transformed relatively to establish the dynamics equations, the motion state of the runner in Fig.5(a) reflects the motion state of the robot. The key parameters the equivalent mass m₄ of the pipe, the stiffness k₂ of the flexure hinge, and the stiffness of the driving foot k₃ are selected to investigate the effect on the motion state of the runner. The motion trajectory curves of the runner with different variables are presented in Fig.5(b)–Fig.5(d).

Fig.5 Simulation for the motion trajectory of the runner: (a) motion trajectory of the runner in four phases, (b) motion trajectories of the runner with different variables m₄, (c) motion trajectories of the runner with different variables k₂, and (d) motion trajectories of the runner with different variables k₃.

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In Fig.5(a), area a is the initial “stick” phase of the system, where the runner enters area b of the stick phase in a relatively short period of time, that is, the preload between the driving foot and the pipe wall is completed in a relatively short period of time. In area b, the contact point of the driving foot sticks to the pipe wall, and the deformation of the driving foot increases with the increase in the drive signal voltage. Therefore, a large relative displacement of the main body to the pipe wall occurs at this phase. When the drive voltage reaches its peak and the drive signal is about to turn to the falling edge, the pipeline robot still moves forward due to the inertial force. From the area c, this state is maintained for an extremely short time, and the relative distance produced is extremely small. When the drive signal is at the falling edge, the driving foot quickly recovers its deformation, and the driving foot contact point is in the “slip” phase with the pipe wall due to inertia. However, the runner may follow the driving foot to move backward a certain distance owing to the sliding friction between the driving foot and the runner, that is, the backlash, as shown in area d. In one cycle, the actual step distance of the pipeline robot is Δx.

From Fig.5(b), the output displacement decreases with the increase in the runner mass (i.e., the main body mass) during an operating cycle. Specifically, larger runner masses (i.e., body masses) produce smaller relative displacements for the proposed robot during the “stick” phase and smaller backlash during the “slip” phase. This condition is because larger masses of the main body have larger inertial forces. Therefore, if a higher step resolution and smaller backlash are required, then the robot body mass can be increased appropriately for the design. Similarly, a larger flexure hinge stiffness corresponds to a smaller displacement output. However, it has a limited effect on the reduction of the backlash, as shown in Fig.5(c). Fig.5(d) shows that the increase in the stiffness of the drive plate spring increases the relative displacement generated in the stick phase and the amount of backlash in the slip phase, but the overall output displacement is slightly larger. Therefore, the stiffness of the drive plate spring can be increased appropriately to improve the operating speed when designing the structure.

4 Experiment and discussion

A prototype of the proposed pipeline robot is fabricated to verify the correctness of the theoretical analysis and the numerical simulation of the dynamics model, as shown in Fig.6. Acrylic is used to make the main body. Aluminum alloy 7075 is used to create the flexure hinge. The material of the support foot and drive plate spring is brass. The contact point of the driving foot is made of epoxy resin. The piezoelectric stack model is PST (20 mm × 5 mm × 5 mm). The body of the prototype has a diameter of 25 mm and a length of 30 mm. Thus, it can work in pipes with the diameter range of 25–35 mm. A test system was constructed to drive the pipeline robot and to measure the output characteristics under different drive states. The established experimental system is presented in Fig.7. A sawtooth wave signal was generated by an arbitrary waveform generator (DG1022) and was supplied to a power amplifier (E000.B4). The power-amplified sawtooth wave signal was used to drive two piezoelectric stacks in pipeline robots. A laser Doppler vibrometer (Optomet Vector-Basis-Plus) was applied to measure the displacement response of the proposed pipeline robot. The relationship between the output speed, displacement of the proposed pipeline robot, and the voltage, frequency, duty cycle of the excitation signal, and the external load was investigated.

Fig.6 Prototype of the proposed robot.

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Fig.7 Experimental system for measuring the output characteristics.

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4.1 Characteristics of output displacement

The sawtooth wave drive signal was set to a frequency of 100 Hz with a rising edge duty cycle of 80% for forward motion and 20% for backward motion to investigate the relationship between drive voltage and output displacement. The diameter of the test pipe was selected as 35 mm. The displacement responses of the forward and backward motions of the pipeline robot are presented in Fig.8, which were measured by the laser Doppler vibrometer at drive signal voltages of 80, 100, and 120 V.

Fig.8 Displacement response of the pipeline robot: (a) forward motion and (b) backward motion.

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The observations in Fig.8 are as follows:

1) The forward displacement response curves show a sawtooth-like waveform with the same period and drive signal, which are similar to the simulation results from the dynamics analysis, thereby verifying the correctness of the theoretical analysis. Such a form indicates that the motion of the pipeline robot will run a large step forward accompanied by a small amount of backlash in one cycle, but the overall motion direction is forward, as shown in Fig.8(a).

2) Similarly, the pipeline robot obtains a larger inertial force on the rising edge (vertical rise) of the drive signal at a duty cycle of 20%, producing a slip phase between the driving foot and the pipe wall. A smaller inertial force on the falling edge (slow negative ramp) produces a stick phase. The entire pipeline robot moves backward, but traces of forward movement are still observed within one movement cycle, as shown in Fig.8(b).

(3) With the increase in the peak voltage of the drive signal, the step length of the pipeline robot running in one cycle increases, and the accumulated displacement increases at the same drive frequency. For the forward motion, a larger drive voltage causes the driving foot to have a greater inertial force during the falling edge phase of the drive signal. The slip effect between the driving foot and pipe wall is more remarkable, and the backward displacement of the pipeline robot decreases.

A motion course of the driving foot in a cycle is taken from the test data to compare with the simulation result from Eq. (1). The test data for one step with exciting voltage peaks of 100 and 120 V, and exciting frequencies of 120 Hz are shown in Fig.9. From Fig.9, the two curves of the motion course are highly similar, but the values of the test data are greater than the simulation results in the whole process. At the end of the “stick” phase, the deviation between the simulation and experiment is 7 μm (6 μm) under an exciting voltage of 120 V (100 V), whereas the deviation of the whole process is 2 μm (1.5 μm). Thus, the relative deviation for a step motion is calculated to be about 6%. The reasons for the deviation are caused by larger contact stresses of the drive feet adjustment in the experiment. The deviation is also evidenced by the time of the initial “stick” phase from the two curves, that is, t₁ < t₂. Higher preload forces in the drive reduce the predeformation time of the bristles.

Fig.9 Comparison of a motion course of the driving foot in a cycle: voltages of (a) 120 V and (b) 100 V.

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4.2 Characteristics of output speed

In the measurement of the output speed, the frequency of the drive signal frequency was set to 100 Hz, and the voltage peak of the drive signal was adjusted in the range of 20–120 V to record the speed of the pipeline robot operating forward and backward for a certain distance, as shown in Fig.10(a). The effect of the drive frequencies on the operating speed was investigated by fixing the voltage peak of the drive signal at 120 V, as presented in Fig.10(b). The key findings in Fig.10 are as follows:

Fig.10 Output speed of the pipeline robot under various voltages and frequencies of the drive signal: (a) changed voltages, (b) changed frequencies, (c) max step distance with drive signal of 120 V and 120 Hz, (d) min step distance with drive signal of 50 V and 50 Hz.

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1) When the drive voltage amplitude is gradually adjusted from 50 to 120 V, the linearity between the robot output speed and the drive voltage amplitude is considered to be good, and the relationship between them is approximately proportional. However, Fig.10(a) shows that a difference is observed in the speed of the forward and backward motions of the proposed robot under the same drive signal properties. The main reason for the speed difference can be explained by the fact that the friction force between the driving foot and the pipe wall is gradually decreasing during the recovery deformation of the flexure hinge and drive plate spring in the slip phase in the forward direction. The friction force between the driving foot and the pipe wall is gradually increasing during the recovery deformation of the flexure hinge and drive plate spring in the slip phase in the backward motion.

2) The output speed of the robot is positively correlated with the frequency of the drive signal. From Fig.10(b), the output speed of the robot increases linearly with the increase in the excitation voltage frequency when the driving voltage is 120 V at the driving frequency of 50–120 Hz. The maximum output speed is 3.5 mm/s with a step distance of 30 μm for forward motion and 3 mm/s with a step distance of 25 μm for backward motion at a frequency of 120 Hz.

3) From Fig.10(c) and 10(d), the speed of the pipeline robot is 2.98 mm/s in the forward direction, and the step distance is 30 μm when the drive voltage is 120 V. In the backward direction, the robot has a speed of 2.54 mm/s and a step distance of 25 μm. When the drive voltage is 50 V, the speed of the pipeline robot is 0.5 mm/s in the forward direction, and the step distance is 5 μm. The speed is 0.39 mm/s in the backward direction, and the step distance is 4 μm.

4) The change rate of speed difference between forward and backward motions is more obvious with the frequency and voltage of the drive signal at the lower drive frequency of 50–70 Hz and drive voltage of 50–0 V. The change rate of speed difference is insignificant at the drive signal amplitude over 70 V and drive frequency over 70 Hz. At the lower drive frequency or lower drive voltage, the robot is in the slip phase, and the retraction is larger because the driving foot obtains a smaller inertial force. Higher stepping accuracy can be obtained in this case, but the stability of the system’s operation is relatively weak from the experiment.

Theoretically, if the slope of the rising edge of the drive signal is small and the falling time is short to be infinitely close to zero, then the amount of robot backtracking will be small, and the vibration and backlash can be eliminated. However, the backlash phenomenon is inevitable in the actual drive due to the limitation of the charging and discharging times of the piezoelectric element and the mechanical response time of the system. At the same time, we can use the backlash phenomenon to improve the operating accuracy and resolution of the inertial pipeline robot. For example, if the pipeline robot moves a distance of x₁ in the stick phase, then the robot moves back a distance of x₂ in the slip phase due to the backlash. The overall displacement for a cycle time of the robot is (x₁ − x₂). Therefore, a reasonable backlash will improve the resolution of the pipeline robot. On the basis of the above investigation of the effect of drive voltage and frequency on output speed, the proposed robot exhibits the maximum speed at a drive voltage of 120 V and frequency of 120 Hz. In the investigation of the effect of the drive signal duty cycle on the output speed, the drive signal was selected to have a voltage of 120 V and a frequency of 120 Hz. The speed curves of the proposed robot changed with the duty cycle of the drive voltage, as presented in Fig.11, and the speed profile of the robot is V-shaped. When the duty cycle is 50%, the output speed of the robot is 0. If the drive signal duty cycle is greater than 50%, the robot produces a forward movement, and the speed increases linearly with the increase in the drive signal duty cycle. When the drive signal duty cycle is less than 50%, the robot moves in the reverse direction, and the speed increases linearly with the increase in the drive signal duty cycle. The robot presents a maximum speed of 3.5 mm/s in the forward direction at a duty cycle of 90% and a maximum reverse speed of 3 mm/s at a duty cycle of 10%.

Fig.11 Effect of drive signal duty cycle on output speed.

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4.3 Output characteristics of speed versus load

The pipeline robot was originally designed to have the ability to transport materials, carry inspection, and actuation equipment within the pipeline. Therefore, the maximum load capacity of the proposed robot needs to be investigated. Here, permanent magnets were gradually attached to the pipeline robot, and the relationship between the carrying mass of the proposed robot and the operating speed was recorded under the conditions of the drive signal voltage of 120 V, operating frequency of 120 Hz, and duty cycle of 90% for forward motion and 10% for backward motion, as shown in Fig.12. From Fig.12(a), the speed of the pipeline robot decreases with the increase in load. The load capacity of the robot is better when operating forward than operating backward, which is determined by the larger friction force available to the driving foot when moving forward. However, the speed of the robot is less affected by the load when the load mass is 0–20 g; when the load mass is 2040 g, the speed of the robot decreases more rapidly with the increase in load. When the load exceeds 40 g, the speed of the robot decreases with the increase in load in an approximately linear trend. When the mass of the carrying magnet is increased to 70 g (65 g), the proposed robot has no stable displacement output in forward (backward) motion. Therefore, the maximum load capacity of the robot is about 70 g for forward motion and 60 g for backward motion under the drive signal of 120 V and 120 Hz. At this point, the proposed robot carries a maximum mass of 4.6 times its own mass, where the mass of the robot is 15 g. For the climbing capacity, the proposed pipeline robot without load can climb to a pipe angle range of 0°–70°, as shown in Fig.12(b). When the climb angle increases, the running speed of the pipeline robot drops linearly. The running speed is 0.5 mm/s with a climb angle of 70°. Fig.12(c) presents the forward motion of the pipeline robot under no load and the backward motion state when the load is 50 g at different moments. From Fig.12(c), the speed of the pipeline robot with a load is slightly reduced compared with the unloaded state, but it still has a stable operating condition.

Fig.12 Load capacity of the proposed pipeline robot: (a) relationship between load and speed, (b) climbing capacity, and (c) operating state.

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An acoustic levitation griper was designed to be installed into the proposed pipeline robot. The pipeline robot was endowed with the ability to transport small particles, liquid droplets, and biological tissues without contact holding. Fig.13 exhibits the process where the proposed pipeline robot carried a small particle with a diameter of 1 mm to be transported by 75 mm within 27 s. The success of this demonstration illustrates the potential of the proposed pipeline robot for future research in high-precision positioning and transportation in miniature pipes (e.g., test tubes) for contact-free operation. The detailed demonstration can be seen in the Electronic Supplementary Materials.

Fig.13 Demonstration of the contactless transport of a microparticle.

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On the basis of the above experimental results, several reported typical piezoelectric inertial pipeline robots are selected for comparison to clearly show the advantages of the pipeline robot designed in this article. The results given in Tab.4 [6,18,35] allow us to infer that.

Tab.4 Performance comparison between the proposed pipeline robot and several typical pipeline robots [6,18,35]

Robot	Driving mode	Dimension/(mm × mm)	Weight/g	Adaptable pipe diameter/mm	Operating speed/(mm∙s−1)		Resolution/μm	Load capacity/g	Climbing ability/(° )
					Forward	Backward
This pipe-robot	Inertial stick-slip	30.0 × 35.0	15	20–30	3.50	3.00	4	70.0	0–70
Sun’s [35]	Inertial impact	9.8 × 22.0	5	10–20	2.19	2.48	6	2.5	0–40
Takaharu’s [18]	Inertial impact	10.0 × 70.0	10	10–20	5.00	−a)	10	−	−
Li’s [6]	Inertial impact	20.0 × 55.0	30	20–30	2.86	2.69	9	−	−

Note: Parts of data are estimated with the information offered by authors. “–” indicates that the information is not available.

1) Compared with the traditional inertial impact drive mechanism, the pipeline robot with an inertial stick-slip drive mechanism has the outstanding benefits of output performance, stepping accuracy, load capacity, and climbing ability. Some reported pipeline robots with inertial impact drive mechanisms do not have load and climbing capabilities.

2) The pipeline robot is in a moderate order of magnitude in terms of size, weight, and adaptable pipe diameter. However, if the structure is further optimized and the small size piezoelectric stacks are employed, then the size of the structure can be adapted to the pipe diameter below 20 mm, which is close to the diameter of the test tube.

3) This pipeline robot maintains a higher and stable operating speed with a slight increase in load. Although the robot employed the inertial impact drive mechanism, the variation of load can affect the matching relationship between the body mass and the inertia block, resulting in unstable output speed.

Overall, the inertial stick-slip driven pipeline robot outperforms the inertial impact driven pipeline robot in terms of accuracy and load capacity, which proves the importance and advantages of the research we have conducted.

5 Conclusions and perspectives

A small pipeline robot driven by the inertial stick-slip mechanism was proposed, and the feasibility of this drive scheme in pipeline robot applications was verified through theoretical and experimental analyses. The dynamic model of the drive system, including the LuGre friction model, was established, and the effect of the main parameters of the system on the operating state of the robot was investigated. An experimental investigation was conducted to test the relationships between output displacement, velocity, and drive signal voltage, frequency, duty cycle, and load capacity. The main conclusions of this study can be summarized as follows:

1) For the drive voltage range of 50–120 V and drive frequency of 50–120 Hz, the speed of the robot increases linearly with the increase in voltage and frequency. The maximum speed in the forward direction is 3.5 mm/s with a step of 30 μm, and the minimum speed is 0.5 mm/s with a step of 5 μm. The maximum speed in the backward direction is 3 mm/s with a step of 25 μm, and the minimum velocity is 0.39 mm/s with a step length of 4 μm.

2) The greater the rising (falling) edge duty cycle of the forward (backward) motion driven by the sawtooth wave excitation signal, the smaller the backlash of the robot motion and the greater the robot speed.

3) At a drive voltage of 120 V, a drive frequency of 120 Hz, and a duty cycle of 90%, the maximum load of the robot is 70 g, which is 4.6 times its own mass, and the maximum climb angle is 70°. This finding shows that the proposed robot has a good load capacity.

In future work, we will give the proposed pipeline robot more diverse executive operation functions, conduct research to improve its motion stability and control accuracy, and develop its potential for biomedical applications.

Nomenclature

A	Cross section area of the piezoelectric stack
c2, c3	Equivalent damping of the flexure hinge and the driving foot, respectively
d33	Piezoelectric coefficient
f1	Friction between the driving foot and pipe wall
f2	Friction between the support foot and pipe wall
F_1,F_1^{\prime }	Acting force and reacting force between the driving foot and the pipe wall, respectively
F_{12},F_{12}^{\prime }	Acting force and reacting force between the piezoelectric stack and the flexure hinge, respectively
FC	Coulomb friction force
Fp	Output force of the piezoelectric stack
FS	Static friction force
g(v)	Function describing the Stribeck effect
k1, k2, k3	Equivalent stiffness of the piezoelectric stack, the flexure hinge, and the driving foot, respectively
l	Length from the rotation center to the top of flexure hinge
l1	Distance from the center of rotation of the hinge to the point of action with the piezoelectric stack
l2	Vertical distance from the top of the flexure hinge to the center of rotation under the rotation θ1
l3	Vertical distance between the top of the hinge and the initial position after rotation θ1
l4	Length of the driving foot
Lp	Length of the piezoelectric stack
m1, m2, m3, m4	Equivalent mass of the piezoelectric stack, the flexure hinge, the driving foot, and the pipe, respectively
n	Number of the piezoelectric sheets in the piezoelectric stack
s33	Elastic compliance constant of the piezoelectric stack
T	Period of the drive signal
U	Output voltage of the drive signal
v	Relative velocity of two friction surfaces
vs	Stribeck velocity
Vmax	Peak voltage of the drive signal
x1	Output displacement of the piezoelectric stack
x_2, {\dot{x}}_{\text{2}}\text{,} {\ddot{x}}_{\text{2}}	Displacement, velocity, and the acceleration of the flexure hinge, respectively
x_4,{\dot{x}}_4\text{,}{\ddot{x}}_4	Displacement, velocity, and acceleration of the pipe, respectively
y_3,{\dot{y}}_3,{\ddot{y}}_3	Displacement, velocity, and acceleration of the driving foot, respectively
z	Average deformation of bristles
θ1, θ2	Rotation angle of flexure hinge and driving foot, respectively
Δx	Step distance of the pipeline robot and the slider
σ0	Bristle stiffness
σ1	Bristle damping of the bristle
σ2	Viscous damping coefficient
μ	Friction coefficient between the driving foot and the pipe

Acknowledgements

This work was supported by the State Key Laboratory of Robotics and System (HIT), China (Grant No. SKLRS-2022-KF-09). The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Electronic Supplementary Materials

The supplementary materials can be found in the online version of this article at https://doi.org/10.1007/s11465-022-0697-z and are accessible to authorized users.

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