Abstract
Vibrationdriven locomotion systems are widely used in various industries, particularly, in the form of capsuletype robots, wheeled platforms, wormlike units, etc. Because of the changeable operating conditions, such systems require continuous control of their kinematic and dynamic characteristics. The main purpose of the present paper is to define the optimal excitation conditions (forced frequencies and phase shifts) of a wheeled twomodule vibrationdriven robot equipped with two unbalanced rotors. The research methodology contains four stages: developing the robot’s dynamic diagram and mathematical model describing its motion; designing the robot’s simulation model in the MapleSim software; numerical modeling of the system locomotion conditions in the Mathematica software; simulating the system dynamic behavior in the MapleSim software. The obtained results show the time dependencies of the system’s kinematic characteristics at different phase shift angles of the unbalanced rotors. The major scientific novelty of this paper consists in substantiating the possibilities of adjusting the system’s operational parameters in accordance with the changeable technological requirements by means of changing the phase shift angles of the unbalanced rotors. The proposed ideas and obtained results can be used while developing new designs of robots based on the twomodule vibrationdriven systems and while improving the control systems for adjusting their performance in accordance with the changeable operational conditions.
Highlights
 The dynamic diagram of the wheeled twomodule vibrationdriven robot equipped with two unbalanced rotors is developed and the mathematical model describing its motion is derived.
 In order to provide the system’s nearresonance operational conditions, the mathematical dependencies for determining the corresponding forced frequency and spring stiffness are deduced.
 The numerical modeling of the system locomotion conditions is carried out in the Mathematica software, and the virtual experiments (computer simulation) are conducted in the MapleSim software.
 The obtained results may be used while developing new designs of robots based on twomodule vibrationdriven locomotion systems with two unbalanced rotors and while improving their control systems.
1. Introduction
Vibrationdriven systems are widely used for actuating various mobile machines in different industries, particularly, in the medicine and healthcare sector [1, 2], mining, road and housebuilding, oil and gas industries, etc. In a vast majority of vibrationdriven machines, the stickslip locomotion conditions are implemented. The dynamic behavior of a vibroimpact capsuletype system equipped with a delayed feedback controller is thoroughly studied in [1]. The peculiarities of implementing this system while performing the colonoscopy are theoretically and experimentally investigated in [2]. The paper [3] considers an improved doublecrank vibration exciter based on a twin crankslider mechanism and used for actuating the sliding locomotion of a platform. In [4], the authors studied the motion characteristics of a vibroimpact system under different operational conditions (friction levels, excitation force magnitudes, angles of inclination of the supporting surfaces, etc.). The paper [5] is focused on modeling the dynamics of a sliding platform actuated by a vibroimpact exciter equipped with doublesided elastic constraints and disturbed by a halfsine periodic force.
In distinction to the sliding locomotion systems mentioned above, wheeled vibrationdriven robots are also widely studied. The interesting designs of the selfactuated locomotion systems equipped with synchronized centrifugal exciters and unidirectionally rotating wheels are proposed and investigated in [6, 7]. Similar research on the wheeled vibroimpact platforms with cranktype inertial exciters was performed in [8, 9, 10], where the authors carried out theoretical and experimental studies on the platform’s dynamic behavior under different design parameters (impact gap values, disturbing body and platform masses, springs stiffnesses, etc.) and operational conditions (forced frequencies, friction levels, etc.).
Another prospective design of mobile vibrationdriven robots is based on multimodule locomotion systems, whose members are connected by springdamper elements. The problems of maximizing the robot’s translational speed with simultaneous minimization of the power consumption are currently comprehensively studied, particularly, in [11, 12]. The papers [13, 14] are dedicated to analyzing the dynamic behavior and sliding bifurcations of a multimodule vibrationdriven robot under different dryfriction conditions.
The present paper is based on the authors’ previous research published in [15, 16]. The peculiarities of implementing the centrifugal exciter for actuating the wheeled robot are studied in [15], while the sliding twomodule system dynamics is analyzed in [16]. The main purpose of the present paper is to define the optimal excitation conditions (forced frequencies and phase shifts) of a wheeled twomodule vibrationdriven robot equipped with two unbalanced rotors.
2. Research methodology
2.1. Dynamic diagram and mathematical model of the vibrationdriven locomotion system
Let us consider the wheeled twomodule vibrationdriven locomotion system, whose dynamic diagram is shown in Fig. 1. The system consists of two movable platforms, which can roll along a smooth horizontal surface. The platforms are connected by a springdamper element characterized by stiffness $k$ and damping coefficient $c$. To actuate the system, two inertial vibration exciters (unbalanced rotors) are mounted on each platform. The rotary motions of the rotors are described by the corresponding laws ${\phi}_{1}\left(t\right)$ and ${\phi}_{2}\left(t\right)$, while the positions of the movable platforms relative to the inertial reference system $xOy$ are defined by the generalized coordinates ${x}_{1}$ and ${x}_{2}$. The masses of the front and rear platforms are denoted as ${m}_{1}$, ${m}_{2}$, respectively, and the masses of the corresponding unbalanced rotors are ${m}_{3}$, ${m}_{4}$. In addition, the rotors are characterized by the eccentricities ${r}_{1}={A}_{1}{B}_{1}$, ${r}_{2}={A}_{2}{B}_{2}$.
Fig. 1Dynamic diagram of the wheeled twomodule vibrationdriven locomotion system
While performing further investigations, let us assume zerofriction conditions when the wheels are rolling along a smooth horizontal surface. Let us study the case of the steadystate operational conditions of the inertial exciters when the rotors’ angular speeds are constant and equal to ${\omega}_{1}$ and ${\omega}_{2}$. Denoting the initial positions (phases of oscillations) of the front and rear unbalanced rotors as ${\phi}_{10}$ and ${\phi}_{20}\text{,}$ their motion laws can be described as follows: ${\phi}_{1}\left(t\right)={\phi}_{10}+{\omega}_{1}t$, ${\phi}_{2}\left(t\right)={\phi}_{20}+{\omega}_{2}t$. The mathematical model describing the locomotion conditions of the considered wheeled twomodule vibrationdriven system with two unbalanced rotors can be derived using the EulerLagrange equations and presented in the following form:
$\left({m}_{2}+{m}_{4}\right)\bullet {\ddot{x}}_{2}\left(t\right)+c\bullet \left({\dot{x}}_{2}\left(t\right){\dot{x}}_{1}\left(t\right)\right)+k\bullet \left({x}_{2}\left(t\right){x}_{1}\left(t\right)\right)={m}_{4}\bullet {r}_{2}\bullet {\omega}_{2}^{2}\bullet \mathrm{cos}\left({\phi}_{20}+{\omega}_{2}\bullet t\right),$
where the dots above ${x}_{1}\left(t\right)$ and ${x}_{2}\left(t\right)$ denote the first and secondorder derivatives of these displacements with respect to time, i.e., the corresponding velocities and accelerations.
Taking into account the fact that the considered locomotion system is semidefinite, its first natural frequency equals zero, which shows that the platforms move (oscillate) as a single body. The second natural frequency of the twomass vibratory system can be determined as follows:
Eq. (2) allows for determining the spring stiffness of the considered vibrationdriven locomotion system providing its energyefficient nearresonance operational conditions:
where $\xi $ is the correction coefficient providing the nearresonance operation.
Therefore, when the system’s inertial parameters (${m}_{1}$, ${m}_{2}$, ${m}_{3}$, ${m}_{4}$) are known, and the forced frequency ${\omega}_{f}$ is prescribed, the necessary natural frequency can be determined as follows: ${\omega}_{n}={\omega}_{f}/\xi $, where $\xi $ can take different values depending on a particular design and operational peculiarities of the system $(\xi =$ 0.93…0.99 [16]). Then, Eq. (3) can be used for calculating the necessary spring stiffness providing the system’s nearresonance locomotion.
Fig. 2Simulation model of the twomodule vibrationdriven system developed in MapleSim software
2.2. Simulation model of the twomodule vibrationdriven system
The simplified simulation model of the twomodule vibrationdriven system is developed in the MapleSim software (Fig. 2). Two prismatic sliders P_{1} and P_{2} with the connected masses RB_{1} and RB_{2} are used for simulating the movable platforms. The rods RBF_{1}, RBF_{2}, RBF_{3}, RBF_{4} with the revolute joints ${R}_{1}$, ${R}_{2}$ and joined masses RB_{3}, RB_{4} simulate the unbalanced rotors.
To actuate the unbalanced rotors, the corresponding constantspeed motion drivers (motors) CS_{1} and CS_{2} are used. The sliders P_{1} and P_{2} are connected by slider P_{3} equipped with the springdamper elements SD_{1} and S_{1}. The fixed frames FF_{1} and FF_{2} are used for prescribing the initial positions and motion directions of the sliders P_{1} and P_{2}. To define the kinematic characteristics of the platforms (masses RB_{1}, RB_{2}), the corresponding sensors Probe1 and Probe2 are applied.
3. Results and discussion
3.1. Numerical modeling of the system locomotion in the Mathematica software
Let us analyze the system’s locomotion conditions in the Mathematica software by numerical solving of the system of differential Eq. (1) using the RungeKutta methods. The following inertial and excitation parameters are considered: ${m}_{1}={m}_{2}=$ 0.25 kg, ${m}_{3}={m}_{4}=$ 0.025 kg, ${\omega}_{1}={\omega}_{2}=$ 157 s^{1}, $c=$2 (N∙s)/m, ${\phi}_{10}=$0. Using the Eq. (3), the spring stiffness can be calculated: $k\approx $3.7∙10^{3} N/m. To perform further numerical modeling, the following values of the phase shift angle ${\phi}_{20}$ of the rear unbalanced rotor are adopted: 0, 45° ($\pi $/4), 90° ($\pi $/2), 135° (3$\pi $/4), 180° ($\pi $), 225° (5$\pi $/4), 270° (3$\pi $/2), 315° (7$\pi $/4). The corresponding modeling results are shown in Fig. 3 using the black, blue, green, purple, red, gray, brown, and orange curves, respectively. Figs. 3(ac) show time dependencies of the front platform’s basic kinematic characteristics (displacement, velocity, acceleration) at different phase shift angles ${\phi}_{20}$. In Fig. 3(d), the dependence of the front platform’s average speed on ${\phi}_{20}$ is presented. Fig. 3(e) describes the relative positions (displacements) of the front and rear platforms in time. The initial distance between the platforms is set equal to 0.06 m.
The distance traveled by the front platform during the time period of 1 s, as well as the motion direction, significantly depends on the phase shift angle ${\phi}_{20}$ (see Fig. 3). The angles 45°, 90°, and 135° provide the backward motion, while at 225°, 270°, and 315° the platform moves forward. The largest average speeds of about 0.23 m/s are observed at ${\phi}_{20}=$90° and ${\phi}_{20}=$ 270°, while the lowest ones are equal to zero at ${\phi}_{20}=$ 0° and ${\phi}_{20}=$ 180°. The maximal instantaneous values of the platform’s speed and acceleration reach 3.7 m/s and 550 m/s^{2}, respectively, at ${\phi}_{20}=$ 135°. The maximal relative distance between the front and rear platforms exceeds 0.1 m, while the minimal one is about 0.015 m at ${\phi}_{20}=$ 180°. The obtained results will be used while designing the experimental prototype of the wheeled vibrationdriven robot.
4. Simulation of the system dynamic behavior in the MapleSim software
Let us carry out a computer simulation of the system locomotion in the MapleSim software in order to verify the correctness of the numerical modeling results obtained in the Mathematica software. While conducting virtual experiments, all inertial, stiffness, damping, and excitation parameters are similar to those mentioned above. Let us consider the case of the system’s best performance, when the phase shift angle ${\phi}_{20}=$ 90°. The corresponding simulation results are shown in Fig. 4. Considering Fig. 4(a), the front platform moves in a backward direction and during the time period of 1 s passed a distance of about 0.23 m. This allows for concluding that its average translational speed is 0.23 m/s. At ${\phi}_{20}=$90°, the maximal instantaneous values of the platform’s speed and acceleration exceed 2.8 m/s and 400 m/s^{2}, respectively. The maximal relative distance between the front and rear platforms is approximately 0.09 m, while the minimal one reaches 0.03 m. In general, all the obtained simulation results satisfactorily agree with the corresponding numerical modeling results shown by the green curves in Figs. 3(ac, e).
Fig. 3Numerically modeled kinematic characteristics of the vibrationdriven locomotion system
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b)
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d)
e)
Fig. 4Results of virtual experiments carried out in the MapleSim software
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b)
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d)
5. Conclusions
The research is focused on analyzing the dynamic behavior of the wheeled twomodule vibrationdriven locomotion system equipped with two unbalanced rotors under different excitation parameters. The system’s nearresonance operational conditions are substantiated by applying the corresponding forced frequency and spring stiffness. The phase shift angle ${\phi}_{20}$ of the rear unbalanced rotor is chosen as the controllable parameter providing the change in the system’s kinematic characteristics and motion direction. The corresponding dynamic diagram and simulation model of the wheeled vibrationdriven locomotion system are considered, and the differential equations describing its motion are derived. The numerical modeling of the system locomotion conditions is carried out in the Mathematica software, and the virtual experiments (computer simulation) are conducted in the MapleSim software. The results show that the angle ${\phi}_{20}$ of 45°, 90°, and 135° provides the system’s backward motion, while at 225°, 270°, and 315° the platform moves in forward direction. The largest average speeds of about 0.23 m/s are observed at ${\phi}_{20}=$ 90° and ${\phi}_{20}=$ 270°, while the lowest ones are equal to zero at ${\phi}_{20}=$ 0° and ${\phi}_{20}=$ 180°. The maximal relative distance between the front and rear platforms exceeds 0.1 m, while the minimal one is about 0.015 m at ${\phi}_{20}=$ 180°. The obtained results may be used while developing new designs of robots based on the twomodule vibrationdriven locomotion systems with two unbalanced rotors and while improving their control systems.
Acknowledgements
The authors have not disclosed any funding.
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