CC BY
0Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 2, pp. 265-279. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd230301
NONLINEAR ENGINEERING AND ROBOTICS
MSC 2010: 70E60, 70Q05
The Control of an Aquatic Robot by a Periodic Rotation of the Internal Flywheel
A. V. Klekovkin, Yu. L. Karavaev, I. S. Mamaev
This paper presents the design of an aquatic robot actuated by one internal rotor. The robot body has a cylindrical form with a base in the form of a symmetric airfoil with a sharp edge. For this object, equations of motion are presented in the form of Kirchhoff equations for rigid body motion in an ideal fluid, which are supplemented with viscous resistance terms. A prototype of the aquatic robot with an internal rotor is developed. Using this prototype, experimental investigations of motion in a fluid are carried out.
Keywords: mobile robot, aquatic robot, motion simulation
1. Introduction
This paper is devoted to the development and experimental investigations of a prototype of an aquatic robot actuated by rotating a rotor installed inside a shell with a special form. The development of the prototype of an aquatic robot is based on the results of studies of simpler
Received January 16, 2023 Accepted February 10, 2023
The work of A. V. Klekovkin (Section 4) was supported by the Russian Science Foundation under grant No. 21-71-30011, the work of Yu. L. Karavaev (Sections 2) was carried out within the framework of the state assignment of the Ministry of Education and Science of Russia FZZN-2020-0011.
Anton V. Klekovkin klanvlad@mail.ru
P. G. Demidov Yaroslavl State University ul. Sovetskaya 14, Yaroslavl, 150003 Russia
Yury. L. Karavaev karavaev_yury@istu.ru
Kalashnikov Izhevsk State Technical University ul. Studencheskaya 7, Izhevsk, 426069 Russia
Ivan. S. Mamaev mamaev@rcd.ru
Institute of Mathematics and Mechanics of the Ural Branch of RAS ul. S. Kovalevskoi 16, Ekaterinburg, 620990 Russia
models of robots moving on the surface of a fluid [1-4]. The research results obtained in these studies have not only confirmed that bodies with asymmetric form are capable of moving on the surface of a fluid by changing the internal angular momentum or changing the position of the center of mass, but also demonstrated agreement between theoretical and experimental results.
It should be noted that this concept of actuating aquatic robots is under intense development at present, and that there exist both theoretical studies and experimental investigations [5-14]. The use of only internal actuation mechanisms allows one to ensure complete sealing of the shell and to exclude external moving elements, thus enhancing the maneuverability and significantly simplifying the design of the robot.
The first attempt to equip the above-water robot with an internal rotor with a module that ensures changes in buoyancy is presented in [15], but experiments with the developed prototype turned out to be unsuccessful. Based on the experience gained, the design of the aquatic robot was significantly modified. But, as before, the rotor, which produces the variable angular momentum, is the main mechanism actuating the robot. The motion of the aquatic robot is described using the equations of motion of a flat nondeformable body which are supplemented with fluid resistance forces whose parameters have been determined by numerical and experimental methods.
2. The design of an aquatic robot actuated by rotating the internal rotor
Consider the design of the prototype of an aquatic robot. The shell of the robot has in its cross-section the form of a symmetric airfoil NACA 0040 with overall dimensions 430 x 160 mm. The aquatic robot has been designed using the modular approach, which facilitates the assembly and adjustment of each module (see Fig. 1). The following parts are fastened on a special platform (base): a rotor with a driving motor, a buoyancy adjustment module, a module for changing the position of the center of mass, which can be used to change the angle of pitch, a control system, and a power supply accumulator.
The overall dimensions of this special platform are smaller than the internal dimensions of the shell, which allows one to move the platform inside the shell during installation. This makes it possible to change the position of the center of mass of the whole robot.
The rotor is made of steel, is 90 mm in diameter and 10 mm in thickness. A brushed direct current motor with an encoder is used as the drive of the rotor. The shaft of the motor is connected to a worm reduction gearbox. The characteristics of the reduction gearmotor are presented in Table 1. The output shaft of the reduction unit is installed at an angle of 90° to the shaft of the motor, which makes a more convenient and compact assembly possible. The output shaft of the reduction unit is connected to the rotor through a cylindric transmission consisting of a pair of pinions with reduction ratio 1:1. This drive enables the rotor to be sped up to an angular velocity of 50 rad/s while ensuring acceleration of order 130 rad/s2. The encoder makes it possible to keep track of the angular displacement of the motor shaft and thus the angular displacement of the rotor.
The module for changing the position of the center of mass consists of two linear rails which serve to align the robot in the fluid in the presence of a pitch and are symmetric relative to the longitudinal axis of the foil. Each linear rail has its own drive from the brushed direct current motor with a reduction unit and is equipped with a screw-type gear for transferring a load with
Fig. 1. 3D-model of an aquatic robot (without lid), where 1 is the shell, 2 is the platform with internal elements of the robot, 3 is the reduction gearmotor which actuates the rotor, 4 denotes additional mechanisms with mass, 5 is the buoyancy adjustment module which includes a cylinder for water intake, and 6 is the controlling board
Table 1. Characteristics of the reduction gearmotor actuating the rotor
Characteristic Value
Rated supply voltage 12 V
Reduction ratio of the reduction unit 18.75:1
Shaft torque 0.847 Nm
Maximal rotational velocity 530 RPM
No-load current 0.2 A
Stall current 5.5 A
Table 2. Characteristics of the reduction gearmotor of the additional drives
Characteristic Value
Rated supply voltage 12 V
Reduction ratio of the reduction unit 51.45:1
Shaft torque 0.066 Nm
Maximal rotational velocity 650 RPM
No-load current 0.08 A
Stall current 0.75 A
a weight of 115 g over a distance of up to 60 mm. The characteristics of the motor are presented in Table 2. The extreme positions of the moving load are limited by end contact probes.
To submerge the aquatic robot, two embodiments of the buoyancy adjustment module have been developed. One of them is a cylinder with a shaft whose linear displacement ensures the injection or pumping-out of fluid from the cavity conjugate to the surrounding fluid through
a special hole in the housing. The piston is moved using 2 motors which operate in parallel and are similar to the motors actuating movable loads. As in modules with movable loads, the rotational motion is transformed into translational motion by means of a screw-nut gear. When the piston moves out of the cylinder, a low-pressure region forms inside the cylinder, which allows the fluid from the external medium to fill it. The cylinder can hold about 50 ml of fluid. When the piston moves backwards, the fluid from the cylinder is pushed out into the external medium. The extreme positions of the piston rod are limited by end contact probes.
The other embodiment is implemented by a hydraulic peristaltic pump which injects fluid into an elastic cavity capable of changing its volume. The basic feature of the chosen pump is that the structural elements of the pump do not contact the fluid. It involves using three rollers which press the hose with fluid to the pump housing and, while rotating, pump the fluid from the external medium into the cavity located inside the robot body. When the pump rollers rotate backwards, the fluid is pumped back from the cavity into the external medium. A reduction gear motor is used as the pump drive (its characteristics are presented in Table 3).
Table 3. Characteristics of the reduction gearmotor of the pump drive
Characteristic Value
Rated supply voltage 12 V
Reduction ratio of the reduction unit 34.014:1
Shaft torque 1.079 Nm
Maximal rotational velocity 300 RPM
No-load current 0.25 A
Peak current 5.0 A
Each of the mechanisms is connected to the external medium (fluid) by means of a flexible hose 6 mm in diameter. Both mechanisms are shown in Fig. 2.
(a) (b)
Fig. 2. 3D model of a buoyancy adjustment module: in the form of a cylinder (a) and in the form of a pump injecting the fluid into an elastic cavity (b)
Among the advantages of the buoyancy adjustment module in the form of a cylinder we mention fast response. Its disadvantage is that only a small amount of fluid gets into the cylinder (50 ml). By virtue of its structural design, the disadvantage of the buoyancy adjustment module with a peristaltic pump is a small velocity of fluid intake, and the advantage is the volume of fluid intake (150-200 ml), which is 3 to 4 times larger than the volume of fluid in a cylinder. The
elasticity of the shell of the cavity into which the fluid is injected allows it to assume complex form and to completely fill the internal space.
Both embodiments of the buoyancy adjustment module make it possible to change the mass of the robot without changing its external volume, which is equivalent to changing the buoyancy and allows submersion and emergence of the robot.
Here it should be noted that, both when the piston is removed from the cylinder and when fluid is injected into the elastic cavity, the air pressure inside the shell of the robot increases, and it is necessary to ensure that all detachable joints of the robot body withstand high pressure. For this prototype, two circular lids with two sealing rings on each (see Fig. 3) are used as detachable joints.
Fig. 3. Arrangement of lids on the 3D-model of an aquatic robot, where 1 denotes the lids, 2 is the first sealing ring, and 3 is the second sealing ring
The structural arrangement of the aquatic robot is shown in Fig. 4.
Fig. 4. Structural arrangement of the aquatic robot
To control the motion of the robot, a printed-circuit board has been developed. The primary element of the board is an STM32F103CBT6 microcontroller. The board also has: the driver VNH3SP30 of the drive motor of a rotor, four motor drivers DRV8835 for additional actuators, a Bluetooth module, a data transmission radio module which operates at a frequency of 433 MHz (and can be replaced with a module operating at a frequency of 27 MHz), a DC-DC converter (LD1117) to ensure a stable voltage of 3.3 V necessary for power supply of the microcontroller, adapter modules and digital circuits of the motor drivers. The primary power supply element is an Li-Po accumulator with a rated voltage of 11.1 V and a capacity of 550 mAh.
Control commands were transmitted using radio modules operating at a frequency of 433 MHz. This frequency of the radio transmitter allowed a reliable signal reception for the robot submerged into a 1.5 x 1.5 x 3 m pool. To increase the radio coverage, for example, in larger pools or natural water reservoirs, a radio signal receiver operating at a frequency of 27 MHz can be installed on the control board or a removable floating receiver can be used.
3. Mathematical model
3.1. Equations of motion
To describe the motion, we define two coordinate systems: an inertial (fixed) coordinate system OXYZ and a moving coordinate system rigidly attached to the body, Cxyz, with origin at the center of mass of the robot and with the axis Cx directed to the frontal part. The position of point C is defined by the radius vector r = (x, y, z)T. Let the vector d denote the distance from the center of the moving coordinate system to the axis of rotation of the rotor. A diagram of the robot with the coordinate systems is shown in Fig. 5.
For the developed prototype of the robot, the motion in a fluid can be broken up into 2 cases: motion along the vertical axis OZ (rx = const; ry = const) and plane-parallel motion in a plane parallel to OXY (rz = const).
To describe the plane-parallel motion, we introduce the vector v = (v1, v2, 0)T which denotes the linear velocity of the robot's center of mass referred to the moving axes, the vector u = = (0, 0, w)T, which is the angular velocity vector of the body, and denote the angle between the axis OX and Cx by a. Then we can write the following kinematic relations:
X = v1 cos a — v2 sin a, y = v1 sin a + v2 cos a, a = w. (3.1)
The motion of a rigid body in a fluid in the absence of external forces can be described by the Kirchhoff equations supplemented with viscous resistance terms:
d dT _ dT_ d_dT_ _ dT_
dtdi^ dv2 17 dtdv2 dvx 2'
d dT dT dT (3'2)
dt du) ' ^ dvx "idv2
where T is the kinetic energy of the system, f1 and f2 are the components of the force describing viscous resistance, and g is the moment of viscous resistance force.
For this object the kinetic energy consists of three components:
T = Tf + Tb + Tr, (3.3)
where Tf is the kinetic energy of the fluid, Tb is the kinetic energy of the shell, and Tr is the kinetic energy of the rotor.
Writing the kinetic energy and substituting into Eqs. (3.2), we obtain equations of motion, and supplementing them with kinematic relations, we obtain a complete system of equations describing the motion of the object considered:
(m + An )v1 — d2mr w = (m + X22 )v2w + (X23 + d1mr )w2 — f1, (m + X22)v2 + (X23 + d1 mr)w = —(m + X11)v1w + d2mr w2 — f2,
X23 V2 + (I + X33 )w + mr (d1 V2 — d2 V1) = (\n — X22 )V1V2— (3'4)
—A23 V]_w — mr w(d1 v1 + d2v2) — g — k(t), x = v1 cos a — v2 sin a, y = v1 sin a + v2 cos a, a = w,
where m is the mass of the entire system, mr is the mass of the rotor, I is the moment of inertia of the system relative to the vertical axis passing through point C, and X^ are the added-mass coefficients. The role of control action is played here by the gyrostatic momentum k(t) = Ir&(t), where Q is the angular velocity of the rotor, and Ir is the moment of inertia of the rotor relative to the axis of rotation. A more detailed derivation of the equations of motion can be found in [1, 2].
Remark 1. We note that the question of the choice of a model of resistance to the motion of a rigid body in a fluid is a controversial issue. The impossibility of an accurate quantitative description of motion in a fluid within the framework of different resistance models is pointed out in [16]. The choice of a model often depends on the form of the body and on the regimes of its motion. For example, in recent experimental investigations of resistance forces for a flat shovel, a linear dependence on the logarithm of the linear velocity of motion has been obtained, and the moment of resistance depends linearly on the angular velocity [17], although the velocities of motion were low. For bodies having nonplanar form (the form of an ellipse or airfoil form), the motion was described using the quadratic dependences of the resistance forces on the velocity of motion [1, 3, 4].
Consider two models of viscous resistance: the linear model — f1 = clvl, f2 = c2v2, g = c3w, and the quadratic model — f1 — Ci ViVi I, f2 — c2v2\v21, g — C3w|w|, where ci, c2 and C3 are the coefficients of viscous resistance. Figure 6 shows the results of simulation of the system (3.4) for the linear and quadratic resistance models for an aquatic robot moving along a straight line. The graph showing the time dependence of the linear velocity components for the linear model of resistance is shown in red and that for the quadratic model of resistance is shown in dark blue. The experimental results corresponding to the controls considered are shown in black.
0.02
0.01
0.00
(a)
cc
0.00
-0.01-
mil WW ilii mL m
\ If if Iff Vf 4 1 ff ft Vf v\
(b)
Fig. 6. Graphs showing the time dependences of the linear velocity components v1 (a) and v2 (b) for the equations of motion with the linear friction model (red curve), with the quadratic friction model (dark blue curve) and similar dependences obtained from the experiments (black curve)
As can be seen from the figures, the graphs showing the time dependence of the velocity components v1 and v2 and obtained using the model with quadratic resistance are in qualitative agreement with the experimental ones, in contrast to the graphs of velocities obtained by using the model with linear resistance. The model of the linear resistance of motion for the form of the body and the regimes of motion considered above provides neither a qualitative nor a quantitative description of its motion, and the further procedure of calculating the resistance coefficients for this model will not lead to a more exact agreement with the experiment. The quadratic model for the object under consideration is more suitable since the form of the velocity curve agrees with the experimental values, and a more exact quantitative agreement with the experimental data can be achieved by a choice (a definition based on the experimental data) of the viscous resistance coefficients c1, c2 and c3. Therefore, in what follows we will use equations of motion with the quadratic resistance model.
3.2. Control action
Since the rotor is caused to rotate by the direct current motor with an encoder, it is more convenient to control its velocity. The equations of motion contain the acceleration of the rotor. Since the maximal velocity of rotation of the rotor is limited, variable acceleration needs to be used in the experiments. Therefore, for the law of changing the angular velocity of the rotor we choose a function that is periodic in nature. In practice it will be more convenient to implement a piecewise-linear periodic function consisting of four intervals in a period. The time intervals ti and t3 specify the time of rotation of the rotor with constant velocity, and the time intervals t2 and t4 specify the time intervals of rotation of the rotor with acceleration. In [1, 14] it is found that the rotation of the rotor with the largest angular acceleration will give the largest effect. Then the angular velocity of the rotor in the intervals t1 and t3 must be maximal and the time intervals t2 and t4 must be made as small as possible, and then they will define the time of transition from the maximal rotational velocity of the rotor in one direction to the maximal rotational velocity in the opposite direction, with t2 = t4 = tacc. A graph of the control law is presented in Fig. 7, and is written in analytical form as follows:
f nmaX, t e [nT; nT + ti ], Ti - T31, t e [nT + ti; nT + ti + tacc], — ^max, t e [nT + t1 + tacc; nT + t1 + t3 + tacc], T3t - T2, t e [nT + ti + t3 + tacc\ T(n + 1)], where ^ = nmaA2h+2nT+tacc) ^ ^ = 2m (ra+l)) Ta = 20^ n = Q) 2) 3) . . .
Q(t) =
acc
u u.
t
'max
T
Fig. 7. Time dependence of the angular velocity of the rotor
4. Experimental investigations
4.1. Methods of conducting the experiments
The experiments were conducted in a 1.5 x 1.5 x 3 m pool filled with water. The trajectory of motion of the robot was followed up using a system consisting of 4 cameras operating in an optical range and intended for underwater filming. This motion capture system has been developed by the company Contemplas and includes, in addition to the cameras, a calibration object, a workstation and specialized software Templo and Vicon Motus, which is intended for video records and subsequent processing and generation of data on the objects in the video, respectively. Two cameras were mounted on each of the opposite sides of the pool. Each of the cameras ensures filming with a recording rate of 50 frames per second.
Prior to conducting the basic experiments, it is necessary to calibrate the system. This is done using a calibration object consisting of 24 markers with known spatial coordinates of each of the markers. The marker is a sphere 20 mm in diameter, covered with retro-reflecting material (see Fig. 8). During calibration, a fixed calibration object is placed in the visibility range of the cameras, then the position of the markers of the calibration object is recorded in the frames obtained from each of the cameras. This calibration object determines the origin of coordinates relative to which the motion of the bodies will be determined.
When conducting basic experiments, active LED markers are placed on the moving body and the motion is recorded using Templo software. To restore the trajectory using Vicon Motus software, it is necessary to record in the initial frame the position of each of the markers, its dimensions in the frame and the region of search for the marker in the following frames. Thus, we obtain the position of all tracked markers in the frames of the cameras. Next, using calibration information, it is possible to calculate the spatial coordinates of the markers, their linear velocities and accelerations. In the presence of two and more markers on the object, the angular displacement, angular velocity and angular acceleration can be determined.
When experiments were conducted with the developed prototype of the aquatic robot with an internal rotor, 5 markers were fastened on it: two markers on the lateral sides and one marker on the bottom of the robot.
Fig. 8. Calibration object in a pool
4.2. Parameters of the prototype
Based on the developed design of a screwless aquatic robot, a prototype with characteristics presented in Table 4 was developed (see Figs. 9, 10).
Table 4. Characteristics of an aquatic robot
Characteristic Value
Dimensions The mass of the shell The mass of the rotor The moment of inertia of the shell The moment of inertia of the rotor 430 x 160 x 80 mm 2.72 kg 0.48 kg 0.01091 kg • m2 0.00035 kg • m2
Next, it is necessary to determine the added mass coefficients and the coefficients of viscous resistance, which are incorporated in the model of motion. The added mass coefficients describe the resistance of fluid for accelerated motion and are determined by the form of the body. The coefficients of viscous resistance describe the influence of the viscosity of fluid and the pressure distribution on the surface of the body and, in addition to the form of the body, also depend on the regime of motion. For bodies with simple form, such as a sphere, a cylinder or an ellipsoid, other added mass coefficients can be calculated analytically [18]. For bodies of more complex form, the approach described in [19, 20] is the most correct method to determine the added mass coefficients and the coefficients of viscous resistance using the data obtained from experiments. The added mass and viscous resistance coefficients for this prototype were determined using the
Fig. 9. Photo of the internal platform of an aquatic robot
Fig. 10. Photo of the prototype of an aquatic robot with an internal rotor in the assembled condition
methods described in [1]. During simulation they were taken to be equal to:
An = 0.451, A22 = 1.314, A33 = 0.013, A23 = 0.119, c1 = 1.808, c2 = 323.120, c3 = 0.028.
Next, we consider the results of experimental investigations and numerical simulation for different values of the parameters involved in the law of control. As basic maneuvers we consider the motion along a straight line and the motion in a circle.
An example of the motion of the robot in a pool is shown in Fig. 11.
4.3. Motion along a straight line
For the motion along the trajectory which describes on average a straight line, experimental investigations were conducted with the following parameters involved in the law of control: t1 = = t3, tacc = 0.3 seconds (the value of time tacc depends on the specific model of the motor, the design of the transmission mechanisms and on supply voltage, and was determined experimentally). The maximal angular velocity of the rotor is Qmax = 50 rad/s. The experiments
Fig. 11. Photo of an aquatic robot in a pool
were conducted for three different values of the period of control action: T = 2, 3, 4 seconds. Figure 12 shows experimental and theoretical trajectories of motion. Since the robot moves in limited space, the time of motion for all constructed trajectories was taken to be 75 seconds. This is the minimal time in which the robot has overcome the largest possible distance for this pool.
0.10-o-
y, m
0
0.25
0.50 0.75 T — 2 sec
(a)
y, m
y, m
x, m
0 0.25 0.50 0.75 T = 3 sec
(b)
0 0.25 0.50 0.75 1 T = 4 sec
(c)
1.25
-x, m
x, m
1.25
Fig. 12. The theoretical trajectory (solid line) and the experimental trajectory (dashed line) of the aquatic robot with an internal rotor along a straight line for different periods of control action: (a) T = 2 sec; (b) T = 3 sec; (c) T = 4 sec
The results of comparison of the average velocity of motion along a straight line for experiment and simulation are presented in Table 5.
The experiments show that the trajectory of motion has a small curvature radius. This can be explained by the displacement of the center of mass of the entire robot from the symmetry axis of the shell.
Table 5. Average velocity of the motion of the robot along a straight line
Period, s Velocity, m/s
Simulation Experiment
T = 2 0.013 0.015
T = 3 0.017 0.017
T = 4 0.019 0.018
4.4. Motion along a circle
As was shown in the investigation of the law of control [2], the most efficient method of motion along a circle is a control such that tacc ^ min and t1 = t3. We introduce the coefficient k defining the inequality of the time intervals ^ and i3: k = j-. For motion in a circle, experiments were conducted with the fixed period T = 2 seconds and with the values k = 2, 3, 5. Figure 13 shows the results of experiments on the motion in a circle versus the theoretical trajectories. All trajectories are plotted for a duration of motion equal to 100 seconds.
The results of comparison of the average velocity of motion along the trajectory in the form of a circle and the curvature radius of the trajectory for experiment and simulation are presented in Table 6.
Table 6. The average velocity of motion and the curvature radius of the trajectory for a robot moving in a circle
k Velocity, m/s Radius, m
Simulation Experiment Simulation Experiment
k = 2 0.014 0.011 0.40 0.35
k = 3 0.012 0.010 0.24 0.18
k = 5 0.009 0.007 0.16 0.14
5. Conclusion
In this paper we have presented the design of an aquatic robot actuated by one internal rotor, which has in its cross-section the form of a symmetric airfoil. We have presented a real prototype of the robot for which we have proposed a theoretical model of motion in a fluid and a law of control, and presented the results of our experimental research. The investigation of the theoretical model has allowed us to implement in practice the motion along rectilinear trajectories and the motion along trajectories having a certain curvature radius. Experiments have shown that this prototype of the aquatic robot, which is controlled by rotating one rotor, can move in the fluid in a plane parallel to the fluid surface, and that, by combining the motion by rotating the rotor with submersion/emersion by means of buoyancy adjustment modules (intake and discharge of fluid into the robot body) it is possible to implement motion in the volume of the fluid. By comparing the simulation of motion with the experiment, we have shown that the model provides an adequate description of the motion for the parameter values considered.
In our future research we plan to investigate the control by combining different mechanisms of actuation, in particular, to enhance the performance of the aquatic robot.
(c)
Fig. 13. The theoretical trajectory (solid line) and the experimental trajectory (dashed line) of the aquatic robot with an internal rotor moving in circles for the period of control action T = 2 seconds: (a) k = 2; (b) k = 3; (c) k = 5
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