EntranceBasic research. Basic research With a fixed fulcrum
1The work is devoted to the problem of stabilizing an inverted pendulum, which receives much attention in control theory, since algorithms for maintaining the vertical position of anthropomorphic technical devices are constructed using the example of this unstable system. This article describes a strategy for bringing the inverse pendulum into a vertical unstable position; an opto-mechanical system for stabilizing the inverse pendulum has been developed, consisting of a laboratory stand TR-802 from Festo and a displacement control device. It is shown that after the pendulum is brought to its highest position, the stabilization system holds the pendulum in this position by moving the carriage a certain number of steps depending on the angle of inclination of the pendulum. Algorithms have been developed for bringing the pendulum to an unstable equilibrium position and then maintaining it in this position, as well as corresponding software.
reverse pendulum
equilibrium
stabilization
feedback
algorithm
photo emitter
microprocessor
software
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The problem of controlling pendulum-type objects is fundamental for a number of fields of science, since its solution is reflected in the theory of automatic control, robotics and is used in modeling aircraft, in solving problems of stabilizing the position of objects on a moving platform, in the development of special means of transportation - Segways, etc.
A physical pendulum is one of the simplest and most common physical models, representing a load oscillating on an inextensible thread or a rigid rod. A special case of such a system is a reverse pendulum, which is an unstable physical object that has two equilibrium positions: at the lower and upper points. In this case, any disturbance, no matter how small, can bring the pendulum out of the upper equilibrium position with its subsequent tendency to move to the lower equilibrium position. To stabilize the pendulum at the top point, the system can be supplemented with various elements that provide feedback - a necessary component of the control system.
Works are devoted to solving the problem of stabilizing the upper position for an inverted pendulum. The system model is expressed by the following equation:
where m is the mass of the pendulum; l is the length of the pendulum suspension; J is the moment of inertia of the pendulum; θ is the angle of inclination of the pendulum from the vertical; a - acceleration of movement of the pendulum suspension point (carriage); g - acceleration free fall. After carrying out the transformations, we get
Consequently, the movement of the system is influenced by the following parameters: the mass and length of the pendulum suspension and the acceleration of movement of its suspension point - the carriage.
Description of the system operation
In this work, the task was set to simulate the process of bringing the pendulum to the uppermost position with subsequent stabilization of this position using the laboratory stand TR-802 from Festo (Germany) as a setter for the uppermost position of the pendulum, as well as other components used for the created stabilization system.
1.Strategy for bringing the pendulum to the upper equilibrium position
Obviously, the possibilities of bringing the pendulum to the uppermost equilibrium position are limited by the parameters (in particular, the length of the drive and the maximum possible acceleration value of the carriage) of the laboratory bench TR-802 from Festo, on the basis of which the problem is solved. So, the maximum acceleration of the carriage is a=4m/s2.
Through mathematical calculations, it was established that the threshold value of the acceleration of the pendulum, which determines the required amount of change by the carriage in the direction of its movement, is a0 = 13.1 m/s2. Since, when using the Festo TP-802 laboratory bench, this value is much higher than the maximum possible value of the acceleration of the carriage, in this work we used a reverse pendulum output strategy in which the direction of movement of the carriage is repeatedly changed and the displacement of the carriage from the current position is increased.
2. Mathematical description of bringing the pendulum to its uppermost position
It is known that for a pendulum to reach its upper equilibrium position, its potential energy must reach the value Ep=2mpgl, where mp is the mass of the pendulum; l is the length of the pendulum; g is the acceleration of free fall. It is taken into account that mp=0.06kg, l=0.25m, g=10m/s2. Thus, to solve the problem, the potential energy of the pendulum must become equal to Ep=0.3J.
It was decided that the pendulum would be swung as follows: an electromechanical drive moves the carriage relative to its original position by a fixed number of steps, first in a negative direction, then in a positive direction. The amount of displacement from the original position increases each time the carriage moves in either direction. To bring the pendulum to the uppermost equilibrium position, an algorithm was developed, presented in Fig. 1. In this case, it is assumed: (1) the carriage moves along the Ox axis between points X=0mm and X=300mm; (2) initial position of the carriage - coordinate X=150mm; (3) N is the value (in mm) of the carriage displacement from the initial position, (4) K is the specified increment (in mm) of the carriage displacement from this position.
Taking into account that when a carriage with a pendulum attached to it moves along a horizontal axis, the kinetic energy of movement of the carriage Ek is converted into potential energy of movement of the pendulum Ep, the increase in the energy of the pendulum can be calculated. Let's say that the displacement value of the carriage from the initial position is N=50mm, the value of the specified increment of the carriage displacement from the initial position is K=50mm. Then the magnitude of the potential energy of the pendulum after the first displacement of the carriage
after the second -
Thus, after three movements of the carriage, the potential energy of the pendulum must exceed the value required to bring it to the upper equilibrium position.
3. Algorithm for bringing the pendulum to the uppermost position
In practice, it turned out that the conclusions drawn in the previous paragraph, taking into account the transformation kinetic energy carriages into the potential energy of the pendulum do not correspond to experimental data. Most of the energy is wasted into the environment due to imperfect design, carriage friction and pendulum suspension.
Thus, the physical control object is a reverse pendulum, brought to the uppermost unstable position for a finite number of movements of the electromechanical drive carriage driven by an MTR-ST stepper motor, which is controlled by a PC computer through a coordinate positioning controller SPC-200. The start of operation of the system for stabilizing the position of the reverse pendulum follows the withdrawal of the pendulum to its uppermost position. To solve this problem, taking into account , the algorithm presented in Fig. 1 and the corresponding application program for positioning the carriage were developed. It is assumed that N is the displacement of the carriage from the center of the drive axis, and K is the specified increase in the displacement of the carriage from the center of the drive axis.
Rice. 1. Algorithm for the subroutine for bringing the pendulum to the upper position
A listing of the program for bringing the pendulum to its highest position, developed during an experiment on the “carriage-pendulum” system using the Festo WinPisa 4.41 software application, is presented below. Comments explaining the program code are given opposite the corresponding lines after the “;” sign.
At the start of the program, the carriage moves to the center of the drive axis. The next 9 lines of the program correspond to increasing oscillations of the pendulum, at the end of which the carriage makes 2 more movements in order to briefly stabilize the pendulum at the top point.
Immediately at the moment the pendulum reaches the upper equilibrium position, the motion control of the “carriage - pendulum” system passes to the developed stabilization system.
4.Operation of the stabilization system
One of the important components of this system is the optical device for motion control, described in the work. The structure of the system is shown in Fig. 2.
On a fixed base1 there is a carriage3 moving along the X axis, on which a pendulum7 with a weight8 containing a radiator9 is attached. The carriage is rigidly connected to the stepper motor4 via a linear electromechanical drive2. Stepper motor4 is controlled via motor controller5 using position controller6. The computer14 controls the operation of the emitter9 and the decoder13, the inputs of which receive signals from photodetectors 10, 11, 12 containing a device for converting into current values, and their outputs are connected to the computer14. In this case, the photodetector 10 is central and generates a signal at its output only when the reverse pendulum is in a vertical position (the highest point).
The system works as follows: the pendulum7 is brought to the uppermost unstable equilibrium position for a finite number of movements of the carriage of an electromechanical drive controlled by a stepper motor5, and the maximum travel distance of the carriage is 300 mm. The light emitter 9 attached to the weight of the pendulum 8 is turned on from the moment the pendulum 7 begins to move upward, and at the photodetector 10 at the moment the pendulum 7 is in a vertical position, a signal is generated, which is sent through the decoder 13 to the computer 14 and is programmed to be fixed, which corresponds to the extreme upper position of the pendulum. While under the influence physical strength, the pendulum cannot remain in this position for long and begins to deviate. When the pendulum deviates from the vertical, the direction of light from the photoemitter changes, which is recorded by photodetectors. Based on which photodetector closest to photodetector 10 was the first to register the emitter signal (Lk or Pk), it is possible to establish the coordinates of the pendulum (the angle of deflection of the pendulum from the vertical) and the direction of deflection. The number of photodetectors and the step of their alternation directly depend on the required measurement accuracy.
Being under the influence of physical forces, the pendulum cannot remain in this position for long and begins to deviate. When the pendulum deviates from the vertical, the direction of light from the photoemitter changes, which is recorded by photodetectors. Based on which photodetector closest to the photodetector was the first to register the emitter signal (Lk or Pk), it is possible to establish the coordinates of the pendulum (the angle of deflection of the pendulum from the vertical) and the direction of deflection. The number of photodetectors and the step of their alternation directly depend on the required measurement accuracy. Information about the position of the pendulum7 is received from photodetectors into the computer14, processed according to a given program, on the basis of which a control action is generated for the positioning controller 6: move the carriage in the direction of the pendulum deflection by a certain number of steps, depending on the pendulum’s deviation from the vertical. Thus, this system is closed and allows you to stabilize the reverse pendulum in a vertical position. The system operation algorithm is presented in Fig. 3.
Rice. 2. System structure
Rice. 3. System operation algorithm
Conclusion
Thus, in this work, algorithms were developed for bringing the pendulum to the uppermost equilibrium position and then maintaining it in a vertical (unstable) equilibrium position. The imperfection of the pendulum design led to the need to perform more movements of the carriage to bring the pendulum to the top point. The principles of constructing an opto-mechanical system for stabilizing the position of the reverse pendulum at the top point were also developed, consisting of a laboratory electro-mechanical stand TR-802 from Festo and an optical displacement control device. As recommendations, it is proposed to use the results obtained for the development of monitoring systems for technological objects when moving controlled scanning bodies along three coordinates.
Bibliographic link
Kuzyakov O.N., Andreeva M.A. OPTO-MECHANICAL SYSTEM FOR STABILIZING THE POSITION OF THE REVERSE PENDULUM // Basic Research. – 2016. – No. 5-3. – P. 480-485;URL: http://fundamental-research.ru/ru/article/view?id=40326 (access date: 03/23/2020). We bring to your attention magazines published by the publishing house "Academy of Natural Sciences"
DOI: 10.14529/mmph170306
STABILIZATION OF A BACK PENDULUM ON A TWO WHEEL VEHICLE
V.I. Ryazhskikh1, M.E. Semenov2, A.G. Rukavitsyn3, O.I. Kanishcheva4, A.A. Demchuk4, P.A. Meleshenko3
1 Voronezh State technical university, Voronezh, Russian Federation
2 Voronezh State University of Architecture and Civil Engineering, Voronezh, Russian Federation
3 Voronezhsky state university, Voronezh, Russian Federation
4 Military Training and Research Center Air Force « Air Force Academy named after Professor N.E. Zhukovsky and Yu.A. Gagarin", Voronezh, Russian Federation
Email: [email protected]
We consider a mechanical system consisting of a two-wheeled cart, on the axis of which a reverse pendulum is located. The task is to form such a control action, formed according to the principle feedback, which, on the one hand, would ensure the given law of motion of the mechanical means, and on the other, would stabilize the unstable position of the pendulum.
Keywords: mechanical system; two-wheeler; reverse pendulum; backlash; stabilization; control.
Introduction
The possibility of controlling unstable technical systems has been theoretically considered for a long time, however practical significance such management was clearly manifested only in lately. It turned out that unstable control objects, with suitable control, have a number of “useful” qualities. Examples of such objects include spacecraft at the take-off stage, a thermonuclear reactor and many others. At the same time, in case of failure automatic system control, an unstable object can pose a significant threat, danger to both humans and environment. As a catastrophic example of the results of turning off automatic control, we can cite an accident at Chernobyl nuclear power plant. As control systems become more and more reliable, an increasingly wide range of technically unstable objects in the absence of control are used in practice. One of the simplest examples of unstable objects is the classical inverse pendulum. On the one hand, the task of stabilizing it is relatively simple and obvious; on the other hand, it can find practical application when creating models of bipedal creatures, as well as anthropomorphic devices (robots, cybers, etc.) moving on two supports. IN recent years Works have appeared devoted to the problems of stabilizing a reverse pendulum associated with a moving two-wheeled vehicle. These studies have potential applications in many fields such as transportation and reconnaissance due to the compact design, ease of operation, high maneuverability and low fuel consumption of such devices. However, the problem under consideration is still far from a final solution. It is known that many traditional technical devices have both stable and unstable states and operating modes. Typical example- Segway, invented by Dean Kamen, an electric self-balancing scooter with two wheels located on either side of the driver. The two wheels of the scooter are located coaxially. The Segway automatically balances when the driver's body position changes; For this purpose, an indicator stabilization system is used: signals from gyroscopic and liquid tilt sensors are sent to microprocessors, which generate electrical signals that act on the motors and control their movements. Each wheel of the Segway is driven by its own electric motor, which reacts to changes in the balance of the machine. When the rider's body tilts forward, the Segway begins to roll forward, and as the angle of the rider's body increases, the speed of the Segway increases. When the body is tilted back, it self-
The car slows down, stops, or rolls in reverse. Steering in the first model occurs using a rotary handle, in new models - by swinging the column left and right. Problems of control of oscillatory mechanical systems are of significant theoretical interest and great practical importance.
It is known that during the operation of mechanical systems, due to aging and wear of parts, backlashes and stops inevitably arise, therefore, to describe the dynamics of such systems, it is necessary to take into account the influence of hysteresis effects. Mathematical models In accordance with classical concepts, such nonlinearities are reduced to operators that are considered as transformers on the corresponding function spaces. The dynamics of such converters are described by the “input-state” and “state-output” relations.
Statement of the problem
In this paper, we consider a mechanical system consisting of a two-wheeled cart, on the axis of which a reverse pendulum is located. The task is to form a control action that, on the one hand, would ensure the given law of motion of the mechanical device, and on the other, would stabilize the unstable position of the pendulum. In this case, the hysteresis properties in the control circuit of the system under study are taken into account. Below are graphically presented the elements of the mechanical system being studied - a two-wheeled vehicle with a reverse pendulum attached to it.
Rice. 1. Basic structural elements mechanical device in question
here / 1 / I feili / Fr I
" 1 " \ 1 \ 1 i R J
Hr! / / / / /1 / / /
Rice. 2. Left and right wheels of a mechanical device with control torque
Parameters and variables that describe the system under consideration: j - angle of rotation of the vehicle; D is the distance between two wheels along the center of the axle; R - wheel radius; Jj - moment of inertia; Tw is the difference in torque between the left and right wheels; v-
longitudinal speed of the vehicle; c is the angle of deflection of the pendulum from the vertical position; m is the mass of the inverted pendulum; l is the distance between the center of gravity of the body and
wheel axle; Ti - the sum of the torques of the left and right wheels; x - movement of the vehicle in the direction of longitudinal velocity; M - chassis mass; M* - wheel mass; And - backlash solution.
System dynamics
The dynamics of the system are described by the following equations:
n = - + - Tn, W in á WR n
in = - - ml C0S in Tn,
where T* = Tb - TJ; Тп = Ть + ТЯ; Mx =M + m + 2(M* + ^*); 1в = t/2 + 1С; 0.=Мх1в-т2/2 съ2 в;
<Р* = Рл С)Л = ^ С № = ^ О. (4)
A model that describes the dynamics of changes in system parameters can be represented in the form of two independent subsystems. The first subsystem consists of one equation - the p-subsystem,
determining the angular movements of the vehicle:
Equation (5) can be rewritten as a system of two equations:
where e1 = P-Pd, e2 = (P-(Pa.
The second subsystem, which describes the radial movements of the vehicle, as well as the oscillations of the pendulum mounted on it, consists of two equations - (y,v) -subsystem:
U =-[ Jqml in2 sin in- m2l2 g sin in cos in] + Jq Tu W in S J WR u
in =- - ml С°*в Tv W WR
System (7) can be conveniently represented as a system of first-order equations:
¿4 = TG" [ Jqml(qd + e6)2 sin(e5 +qd) - m¿l2g sin(e5 + qd) cos(e5 +qd)] + TShT v- Xd,
¿6 =~^- ^^^ +c)
where W0 = MxJq- П121 2cos2(qd + e5), e3 = X - Xd, ¿4 = v - vd, ¿5 =q-qd, ¿6 =q-qd
Let's consider subsystem (6), which we will control using the feedback principle. To do this, we introduce a new variable and define the switching surface in the phase space of the system as ^ = 0.
5 = in! + с1е1, (9)
where c is a positive parameter. It follows directly from the definition:
■I = e+c1 e1 -sry + c1 e1. (10)
To stabilize the rotational motion, we define the control torque as follows:
Т№ Р - ^ в1 - -М§П(51) - к2 (11)
where, are positive parameters.
We will similarly construct the control of the second subsystem (8), which will also be controlled using the feedback principle. To do this, we introduce a new variable and define the switching surface in the phase space of the system as ■2 = 0.
■2 = inc + C2inc, (12)
where c2 is a positive parameter, then
1 . 2 2 2
■2 = e3 + c2 e3 = (b + b6) ^5 + bе) - m 1 § ^5 + c1)C08(e5 + bе)] +
7^T - + c2 ez
To stabilize the radial motion, we determine the control torque:
mt"2/2 ^k T = -Kt/ (vj+eb)r^t(eb + bj)+jn^ + bj)e08(e5 + bj)--0- \сr ez - +^n^) +kA ^],(14)
where k3, k4 are positively specified parameters.
In order to simultaneously control both subsystems of the system, we introduce an additional control action:
= § Hapv--[va + c3(v-vy) - k588p(^3) - kb 53], (15)
where § is the acceleration of free
falls; c3, k5, kb - positive parameters; 53 - switching surface determined by the ratio:
53 = e6 + c3e5.
Let us formulate the main results of the work, which consist in the fundamental possibility of stabilizing both subsystems, in the assumptions made regarding the control actions, in the vicinity of the zero equilibrium position.
Theorem 1. System (6) with control action (11) is absolutely asymptotically stable:
Nsh || e11|® 0,
Nsh || e2 ||® 0. t®¥u 2
Proof: we define the Lyapunov function as
where a = Dj 2 RJр.
Obviously, the function V > 0, then
V = Ш1 Si = Si. (18)
Substituting (14) into V, we get
V = -(£ Sgn(S1) + k2(S1))S1. (19)
Obviously, V1 Theorem 2. Consider subsystem (8) with control action (14). Under the assumptions made, this system is absolutely asymptotically stable, i.e., under any initial conditions, the following relations are satisfied: lim ||e3 ||® 0, t®¥ (20) lim 11 e41|® o. Proof: we define the Lyapunov function for system (8) by means of the relation where b =Wo R!Je. Obviously, the function V2 > 0, and V2 = M S2 = S2, since zones of insensitivity to the control action arise. Let's give brief description the hysteresis converter used in the future is backlash, based on the operator interpretation. The converter output - backlash at monotonic inputs is described by the relation: x(t0) for those t for which x(t0) - h< u(t) < x(t0), x(t) = \u(t) при тех t, при которых u(t) >x(t0), (24) u(t) + h for those t for which u(t)< x(t0) - h, which is illustrated in Fig. 3. Using the semigroup identity, the action of the operator extends to all piecewise monotonic inputs: Г x(t) = Г [ Г x(t1), h]x(t) (25) and with the help of a special limit construction on all continuous ones. Since the output of this operator is not differentiable, the backlash approximation by the Bouk-Wen model is used in what follows. This well-known semiphysical model is widely used for the phenomenological description of hysteresis effects. The popularity of the Bouka-Wen shoe model famous for its ability to cover analytical form various shapes hysteresis cycles. The formal description of the model is reduced to the system of the following equations: Fbw (x, ^ = akh() + (1 -a)Dkz(t), = D"1(AX -р\х \\z \п-1 z -ухе | z |п). (26) Fbw(x,t) is treated as the output of the hysteresis converter, and x(t) as the input. Here n > 1, D > 0 k > 0 and 0<а< 1. Rice. 3. Dynamics of input-output backlash correspondences Let us consider a generalization of systems (6) and (8), in which the control action is supplied to the input of the hysteresis converter, and the output is the control action on the system: Fbw (x, t) = akx(t) + (1 - a)Dkz(t), z = D_1(Ax-b\x || z \n-1 z - gx | z\n). ¿4 = W-J mlQd + eb)2 sin(e5 + q) - m2l2g sin(e5 + ed) cos(e5 + 0d)] + ¿b = W -Fbw (x, t) = akx(t) + (1 - a)Dkz(t), ^ z = D_1(A x- b\x\\z\n-1 z-gx\ z\n). As before, in the system under consideration, the main issue was stabilization, i.e., the asymptotic behavior of its phase variables. Below are graphs for the same physical parameters of the system with and without backlash. This system was studied through numerical experiments. This problem was solved in the Wolfram Mathematica programming environment. The values of the constants and initial conditions are given below: m = 3; M = 5; Mw = 1; D = 1.5; R = 0.25; l = 0.2; Jw = 1.5; Jc = 5; Jv = 1.5; j(0) = 0;x(0) = 0; Q(0) = 0.2; y(0) = [ j(0) x(0) Q(0)f = )