There was another, unusual approach to describing skiing technique, which is also NOT associated with movements in the hinge system corresponding to the parts of the skier’s body. It is based on the model of an inverted pendulum, also called an inverted pendulum or a Whitney pendulum.
This is a very interesting object of theoretical mechanics; Whitney’s problem was originally formulated as follows: suppose that an inverted material pendulum is installed on a cart, the cart moves rectilinearly, but NOT uniformly. It is required to find the initial position of the pendulum such that it will NOT fall on the cart if the dependence of the speed on time is known in advance, and its 2nd derivative is continuous.
The Whitney problem is still of interest to mathematicians, but the inverse problem is much more important: dynamic control of the movement of the cart, such that the pendulum maintains a given initial position, or oscillates around it. This task is important for robotics, navigation, production automation, orientation spacecraft, it is also realized during normal walking.
But the problem can be generalized: to a pendulum with 2 degrees of freedom, the support of which moves along an arbitrary, curvilinear trajectory, with variable speed, but also under the condition of continuity of 2 derivatives. The simplest example of a generalized inverse pendulum: put a long rod on the palm of your hand and hold it in an unstable position, moving your hand along an arbitrary trajectory.
If we generalize further, we can make a pendulum with a variable length: in this case, its natural frequency will change, the task becomes much more difficult. This is already a general model of unstable equilibrium of a mechanical system, for example a person on a rope. But this task can also be formulated differently: to ensure the balance of the pendulum, with uneven movement of the support along a given curved path, by actively changing the inclination and length of the pendulum. We see: in this formulation, the task fully corresponds to the movement of the skier along the slope!
It turned out that back in 1973, Polish mathematician Janusz Morawski described the mechanics of a skier using a reverse pendulum, but this work was forgotten for 40 years.
J. Morawski's model was not perfect: he did not take into account the lateral slip of the pendulum support, which was necessary in ski equipment in the early 1970s. But modern athletes high level, the technique is no longer related to slippage, and the model more closely matches reality.
New research on the inverse pendulum began with the solution of the narrow, practical problem: to simplify experiments to study skiing techniques. Usually, to study the movements of skiers, it is necessary to continuously record his position, and the many forces acting on the skis and the skier himself, require complex equipment and long preparation of experiments.
In 2013, Matthias Gilgien, a well-known specialist in ski mechanics, proved that if the trajectory of the center of mass relative to the snow surface is known, then using the generalized inverse pendulum model one can calculate the trajectory of the skis, as well as all the acting forces during the descent. As a result, all complex measuring equipment can be replaced with a conventional GPS navigator!
The experiment was carried out with a geodetic navigator operating using the differential navigation method, with an accuracy of determining coordinates: 1 cm in the horizontal plane and 2 cm in the vertical. A detailed 3D terrain model obtained using a geodetic scanner was also used. Currently, for some areas of the US and Europe, open access, there are satellite 3D maps of similar accuracy, their coverage area is rapidly increasing.
Taking into account the micro-relief, which continuously changes on the slope, the accuracy of the heights is 10-20 cm, those. an order of magnitude lower than navigation accuracy. The navigator antenna was located on the skier's helmet, the position of the COM was calculated based on the previous results of Robert Reid, who found out that among athletes at the national level, the COM does not deviate far from the straight line passing through the middle of the neck and the middle of the distance between the skis. And the skier, when turning, tries to keep his head vertical, the middle of the neck is approximately under the antenna. The “surface-CM” distance is always approximately 0.45-0.5 of the “surface-head” distance, sometimes the CM may deviate from this position, but taking into account the accuracy of the surface representation, errors in calculating the position of the CM are not significant, strong deviations occur only with rough mistakes with loss of balance.
If a skier is described by a model of a generalized inverse pendulum, with a variable length, then from a known trajectory and the speed of the CM relative to the surface, it is possible to calculate the angles of its deviation from the vertical position, such that the pendulum does not fall. You can also get the support trajectory: points in the middle of the distance between the ski mounts. And from the position of the CM relative to the support, it is possible to obtain the centering of the skier in the longitudinal direction, and the inclination to the center of rotation, although it is impossible to calculate the position of body parts and the relative loading of the skis.
In parallel with the GPS measurements, conventional equipment was installed at the control site, which is used in studies of ski equipment using MOCAP methods, based on a model of the hinge system, with the calculation of the dynamics of body parts using long-proven methods. The collected data on the movement of the CM were then compared: they turned out to be very close, there are strong discrepancies only in the areas between turns, in which the length of the pendulum changes sharply during unloading.
But the task was not limited to constructing a new model of the movement of the CM, independent of the position of the skier: no one needs this! Practical goal: based on the inverse pendulum model, obtain external forces acting on the skier and skis: surface reaction, snow resistance, and aerodynamic drag. Dr. M. Gilgien and his collaborators obtained the equations for all the forces and compared them with the values that were calculated from the dynamics of the body parts. In the surface reaction graph taken as an example: the blue curve shows the force calculated from the inverse pendulum model, the red curve from the hinge system model as a reference.
Swiss scientist, Rolf Adelsberger, conducted a similar experiment, but also measured the deformation of skis during descent, using sensors glued to the skis. The measurement results corresponded to the forces, which were also calculated on the basis of GPS data, according to the method of M. Gilgien, this proves the correctness of the method.
Slovenian mathematician Boyan Nemec also studied the inverse pendulum model with athletes from the Slovenian national team, but installed an antenna on the skier's neck to better approximate the position of the CM. He obtained an equation for the spatial angle of inclination: depending on the effective accelerations and the length of the pendulum.
We see: the equation is much more complex than simple formulas angles that are constantly discussed on ski sites! But this equation was obtained on the basis of experimental data, and more accurately corresponds to the real processes that occur during descent. A correction was also obtained to accurately determine the position of the CM, but it turned out that it is not very large and fits within the accuracy of the surface measurements, as M. Gilgien had previously suggested.
Professor B. Nemets also noticed strong discrepancies in the unloading areas, and suggested: the error is associated with the linear law of change in the length of the pendulum. If you introduce longitudinal elasticity, the length will change nonlinearly, and the errors will sharply decrease. But at the same time, the pendulum will receive a new degree of freedom: the length will tend to harmonic oscillations, this requires a complete reworking of the model, B. Nemets plans to do this in next works. The main problem: the introduction of the elasticity coefficient, on which the natural frequency of longitudinal vibrations depends, because it is possible that the value of the coefficient is also not constant.
In this case, it is possible to obtain a new effect: if the pendulum support vibrates in the vertical direction, with a high frequency and small amplitude, then an additional force arises that keeps the pendulum in vertical balance: this phenomenon was discovered by P. Kapitsa, and he determined the minimum frequency of oscillations and their limit amplitude. In response to a single blow on an elastic surface, damped oscillations arise; therefore, a reverse pendulum mounted on an elastic support will also be in equilibrium, but very a short time after the impact: before the oscillations die out. A similar phenomenon is possible with a sharp change in the load on the skis, but their longitudinal elasticity depends on the amount of bending, the task becomes even more complicated.
But calculating the forces was also not the final goal: Dr. M. Gilgien received loads on the skier's knees, which can lead to joint injuries. His method makes it possible to obtain an assessment of the route, from a safety point of view, only based on GPS data during control passes.
Another direction is, as always, the creation of a tool for coaches that continuously displays the dynamics of the skier, which are hidden from direct observation: equilibrium conditions, effective accelerations and forces. This method does not require complex, expensive equipment, because even a very expensive GPS receiver is several times cheaper than MOCAP systems, or inertial sensors, and is much easier to use.
We see: old idea, to describe ski equipment without connection with the movements of the skier, is still not forgotten, despite the emergence of new technologies. It is possible that we said goodbye to the cute spherical horses early.
Good luck and balance!
An inverted pendulum is a pendulum that has a center of mass above its fulcrum, attached to the end of a rigid rod. Often the fulcrum is fixed to a trolley, which can be moved horizontally. While a normal pendulum hangs steadily downwards, a reverse pendulum is inherently unstable and must be constantly balanced to remain upright, either by applying a torque to the fulcrum or by moving the fulcrum horizontally as part of the system's feedback. A simple demonstration would be balancing a pencil on the end of your finger.
Review
The inverted pendulum is a classic problem in dynamics and control theory and is widely used as a benchmark for testing control algorithms (PID controllers, neural networks, fuzzy control, etc.).
The inverse pendulum problem is related to missile guidance, as the missile motor is located below the center of gravity, causing instability. The same problem is solved, for example, in the Segway, a self-balancing transportation device.
Another way to stabilize a reverse pendulum is to rapidly oscillate the base in a vertical plane. In this case, you can do without feedback. If the oscillations are strong enough (in terms of acceleration and amplitude), then the reverse pendulum can stabilize. If a moving point oscillates according to simple harmonic oscillations, then the motion of the pendulum is described by the Mathieu function.
Equations of motion
With a fixed fulcrum
The equation of motion is similar to a straight pendulum except that the sign of the angular position is measured from the vertical position of the unstable equilibrium:
texvc not found; See math/README for setup help.): \ddot \theta - (g \over \ell) \sin \theta = 0
When translated, it will have the same sign of angular acceleration:
Unable to parse expression (Executable filetexvc not found; See math/README for setup help.): \ddot \theta = (g \over \ell) \sin \theta
Thus, the reverse pendulum will accelerate from the vertical unstable equilibrium in the opposite side, and the acceleration will be inversely proportional to the length. A tall pendulum falls more slowly than a short pendulum.
Pendulum on a trolley
The equations of motion can be obtained using Lagrange's equations. We are talking about the above figure, where Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \theta(t) pendulum angle length Unable to parse expression (Executable file texvc not found; See math/README for setup help.): l in relation to the vertical and acting force gravity and external forces Unable to parse expression (Executable file texvc not found; See math/README for setup help.): F in the direction Unable to parse expression (Executable file texvc . Let's define Unable to parse expression (Executable file texvc not found; See math/README for setup help.): x(t) trolley position. Lagrangian Unable to parse expression (Executable file texvc not found; See math/README for setup help.): L = T - V systems:
texvc not found; See math/README for help with setup.): L = \frac(1)(2) M v_1^2 + \frac(1)(2) m v_2^2 - m g \ell\cos\theta
Where Unable to parse expression (Executable file texvc is the speed of the cart, and Unable to parse expression (Executable file texvc - velocity of a material point Unable to parse expression (Executable file texvc not found; See math/README for setup help.): m
.
Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): v_1 And Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): v_2 can be expressed through Unable to parse expression (Executable file texvc not found; See math/README for setup help.): x And Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \theta by writing velocity as the first derivative of position.
texvc not found; See math/README - help with setup.): v_1^2=\dot x^2
Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): v_2^2=\left((\frac(d)(dt))(\left(x- \ell\sin\theta\right))\right)^2 + \left((\frac(d)(dt))(\left(\ell\cos\theta \right))\right)^2
Simplifying an Expression Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): v_2 leads to:
texvc not found; See math/README - help with setup.): v_2^2= \dot x^2 -2 \ell \dot x \dot \theta\cos \theta + \ell^2\dot \theta^2
The Lagrangian is now determined by the formula:
Unable to parse expression (Executable filetexvc not found; See math/README for help with setup.): L = \frac(1)(2) \left(M+m \right) \dot x^2 -m \ell \dot x \dot\theta\cos\ theta + \frac(1)(2) m \ell^2 \dot \theta^2-m g \ell\cos \theta
and equations of motion:
Unable to parse expression (Executable filetexvc not found; See math/README for setup help.): \frac(\mathrm(d))(\mathrm(d)t)(\partial(L)\over \partial(\dot x)) - (\partial( L)\over\partial x) = F
Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \frac(\mathrm(d))(\mathrm(d)t)(\partial(L)\over \partial(\dot \theta)) - (\partial (L)\over\partial\theta) = 0
Substitution Unable to parse expression (Executable file texvc not found; See math/README for setup help.): L into these expressions with subsequent simplification leads to equations describing the motion of a reverse pendulum:
texvc not found; See math/README for setup help.): \left (M + m \right) \ddot x - m \ell \ddot \theta \cos \theta + m \ell \dot \theta^2 \sin \theta =F
Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \ell \ddot \theta - g \sin \theta = \ddot x \cos \theta
These equations are non-linear, but since the goal of the control system is to keep the pendulum vertical, the equations can be linearized by taking Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \theta \approx 0
.
Pendulum with oscillating base
The equation of motion for such a pendulum is related to a massless oscillating base and is obtained in the same way as for a pendulum on a cart. The position of a material point is determined by the formula:
Unable to parse expression (Executable filetexvc not found; See math/README for setup help.): \left(-\ell \sin \theta , y + \ell \cos \theta \right)
and the speed is found through the first derivative of the position:
Unable to parse expression (Executable filetexvc not found; See math/README for setup help.): v^2=\dot y^2-2 \ell \dot y \dot \theta \sin \theta + \ell^2\dot \theta ^2.
Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \ddot \theta - (g \over \ell) \sin \theta = -(A \over \ell) \omega^2 \sin \omega t \sin \theta .
This equation does not have elementary solution in a closed form, but can be studied in many directions. It is close to the Mathieu equation, for example, when the amplitude of oscillations is small. Analysis shows that the pendulum remains vertical during rapid oscillations. The first graph shows that with slowly fluctuating Unable to parse expression (Executable file texvc , the pendulum falls quickly after leaving a stable vertical position.
If Unable to parse expression (Executable file texvc not found; See math/README for setup help.): y oscillates rapidly, the pendulum can be stable near a vertical position. The second graph shows that, after leaving a stable vertical position, the pendulum now begins to oscillate around the vertical position ( Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \theta = 0).The deviation from the vertical position remains small, and the pendulum does not fall.
Application
An example is balancing people and objects, such as in acrobatics or unicycle riding. And also a Segway - an electric self-balancing scooter with two wheels.
The inverted pendulum was a central component in the development of several early seismographs.
see also
Links
- D. Liberzon Switching in Systems and Control(2003 Springer) pp. 89ff
Further reading
- Franklin; et al. (2005). Feedback control of dynamic systems, 5, Prentice Hall. ISBN 0-13-149930-0
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An excerpt characterizing the Reverse Pendulum
Grandfather’s sister Alexandra Obolensky (later Alexis Obolensky) and Vasily and Anna Seryogin, who voluntarily went, were also exiled with them, who followed their grandfather along own choice, since Vasily Nikandrovich long years was my grandfather's attorney in all his affairs and one of his closest friends.Alexandra (Alexis) Obolenskaya Vasily and Anna Seryogin
Probably, you had to be truly a FRIEND in order to find the strength to make such a choice and go along at will where they were going, as if they were going only to their own death. And this “death”, unfortunately, was then called Siberia...
I have always been very sad and painful for our beautiful Siberia, so proud, but so mercilessly trampled by the Bolshevik boots! ... And no words can tell how much suffering, pain, lives and tears this proud, but tormented land has absorbed... Is it because it was once the heart of our ancestral home that the “far-sighted revolutionaries” decided to denigrate and destroy this land, choosing it for their own devilish purposes?... After all, for many people, even many years later, Siberia still remained a “cursed” land, where someone’s father, someone’s brother, someone’s died. then a son... or maybe even someone's entire family.
My grandmother, whom I, to my great chagrin, never knew, was pregnant with my father at that time and had a very difficult time with the journey. But, of course, there was no need to wait for help from anywhere... So the young Princess Elena, instead of the quiet rustling of books in the family library or the usual sounds of the piano when she played her favorite works, this time she listened only to the ominous sound of wheels, which seemed to menacingly They were counting down the remaining hours of her life, so fragile and which had become a real nightmare... She sat on some bags by the dirty carriage window and incessantly looked at the last pathetic traces of the “civilization” that was so familiar and familiar to her, going further and further away...
Grandfather's sister, Alexandra, with the help of friends, managed to escape at one of the stops. By general agreement, she was supposed to get (if she was lucky) to France, where this moment her whole family lived there. True, none of those present had any idea how she could do this, but since this was their only, albeit small, but certainly last hope, giving it up was too great a luxury for their completely hopeless situation. Alexandra’s husband, Dmitry, was also in France at that moment, with the help of whom they hoped, from there, to try to help her grandfather’s family get out of the nightmare into which life had so mercilessly thrown them, at the vile hands of brutal people...
Upon arrival in Kurgan, they were settled in cold basement without explaining anything or answering any questions. Two days later, some people came for my grandfather and said that they allegedly came to “escort” him to another “destination”... They took him away like a criminal, without allowing him to take any things with him, and without deigning to explain, where and for how long he is being taken. No one ever saw grandfather again. After some time, an unknown military man brought his grandfather’s personal belongings to the grandmother in a dirty coal sack... without explaining anything and leaving no hope of seeing him alive. At this point, any information about my grandfather’s fate ceased, as if he had disappeared from the face of the earth without any traces or evidence...
The tormented, tormented heart of poor Princess Elena did not want to come to terms with such a terrible loss, and she literally bombarded the local staff officer with requests to clarify the circumstances of the death of her beloved Nicholas. But the “red” officers were blind and deaf to the requests of a lonely woman, as they called her, “of the nobles,” who was for them just one of thousands and thousands of nameless “license” units that meant nothing in their cold and cruel world ...It was a real inferno, from which there was no way out back into that familiar and kind world in which her home, her friends, and everything that she had been accustomed to from an early age remained, and that she loved so strongly and sincerely... And there was no one who could help or at least give the slightest hope of survival.
The Seryogins tried to maintain presence of mind for the three of them, and tried by any means to lift the mood of Princess Elena, but she went deeper and deeper into an almost complete stupor, and sometimes sat all day long in an indifferently frozen state, almost not reacting to her friends’ attempts to save her heart. and the mind from final depression. There were only two things that briefly brought her back to real world- if someone started a conversation about her unborn child or if any, even the slightest, new details came about the supposed death of her beloved Nikolai. She desperately wanted to know (while she was still alive) what really happened, and where her husband was, or at least where his body was buried (or dumped).
Unfortunately, there is almost no information left about the life of these two courageous and bright people, Elena and Nicholas de Rohan-Hesse-Obolensky, but even those few lines from Elena’s two remaining letters to her daughter-in-law, Alexandra, which were somehow preserved in Alexandra's family archives in France show how deeply and tenderly the princess loved her missing husband. Only a few handwritten sheets have survived, some of the lines of which, unfortunately, cannot be deciphered at all. But even what was successful screams with deep pain about a great human misfortune, which, without experiencing, is not easy to understand and impossible to accept.
April 12, 1927. From a letter from Princess Elena to Alexandra (Alix) Obolenskaya:
“I’m very tired today. I returned from Sinyachikha completely broken. The carriages are filled with people, it would be a shame to even carry livestock in them…………………………….. We stopped in the forest - there was such a delicious smell of mushrooms and strawberries... It’s hard to believe that it was there that these unfortunates were killed! Poor Ellochka (meaning Grand Duchess Elizaveta Fedorovna, who was related to my grandfather on the Hessian side) was killed nearby, in this terrible Staroselim mine... what a horror! My soul cannot accept this. Do you remember we said: “may the earth rest in peace”?.. Great God, how can such a land rest in peace?!..
Oh Alix, my dear Alix! How can one get used to such horror? ...................... ..................... I'm so tired of begging and humiliating myself... Everything will be completely useless if the Cheka does not agree to send a request to Alapaevsk...... I will never know where to look for him, and I will never know what they did to him. Not an hour goes by without me thinking about such a dear face to me... What a horror it is to imagine that he lies in some abandoned pit or at the bottom of a mine!.. How can one endure this everyday nightmare, knowing that he has already will I never see him?!.. Just like my poor Vasilek (the name that was given to my dad at birth) will never see him... Where is the limit of cruelty? And why do they call themselves people?..
DOI: 10.14529/mmph170306
STABILIZATION OF A BACK PENDULUM ON A TWO WHEEL VEHICLE
IN AND. 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 feedback principle, which, on the one hand, would ensure the given law of motion mechanical means, and on the other hand, it would stabilize the unstable position of the pendulum.
Keywords: mechanical system; two-wheeler; reverse pendulum; backlash; stabilization; control.
Introduction
Ability to control unstable technical systems has been theoretically considered for a long time, however practical significance Such management has clearly manifested itself only recently. It turned out that unstable control objects, with suitable control, have a number of “useful” qualities. Examples of such objects include spaceship 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 disastrous example of the results of a shutdown automatic control You can cite the accident at the 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 most simple examples 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 use when creating models of bipedal creatures, as well as anthropomorphic devices (robots, cybers, etc.) moving on two supports. IN last 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. A typical example is the Segway, an electric self-balancing scooter invented by Dean Kamen 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
the cat 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 such nonlinearities, in accordance with classical concepts, 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.
Formulation of the problem
IN this work a mechanical system is considered, 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. Main structural elements of the mechanical device under consideration
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 deviation 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/ (vy+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 short 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 special ultimate design for all continuous. 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 in an 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 = )