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Analysis of technology. Reverse pendulum

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 theoretical mechanics, Whitney’s problem was originally formulated as follows: suppose that an inverted material pendulum is installed on a cart, the cart moves in a straight line, 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 mechanical system, for example a man 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 a narrow, practical problem: to simplify experiments in the study of skiing equipment. Usually, to study the movements of skiers, it is necessary to continuously record its position, and many forces acting on the skis and the skier himself require complex equipment and long preparation 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. Now, for some areas of the USA and Europe, there are 3D satellite maps of similar accuracy in the public domain, their coverage area is rapidly expanding.

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 the skier is described by the model of a generalized inverse pendulum, with a variable length, then from the 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 the simple angle formulas 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. An amendment was also received for precise definition position of the CM, but it turned out: it is not very large, and fits into the accuracy of 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 the following works. the main problem: 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!

Material from Wikipedia - the free encyclopedia

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:

$\ddot \theta - (g \over \ell) \sin \theta = 0$

When translated, it will have the same sign of angular acceleration:

$\ddot \theta = (g \over \ell) \sin \theta$

Thus, the reverse pendulum will accelerate from the vertical unstable equilibrium in the opposite direction, 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 $\theta(t)$ pendulum angle length $l$ in relation to the vertical and the acting force of gravity and external forces $F$ in the direction $x$ . Let's define $x(t)$ trolley position. Lagrangian $L = T - V$ systems:

L = \frac(1)(2) M v_1^2 + \frac(1)(2) m v_2^2 - m g \ell\cos\theta Where $v_1$ is the speed of the cart, and $v_2$ - velocity of a material point $m$ . $v_1$ And $v_2$ can be expressed through $x$ And $\theta$ by writing velocity as the first derivative of position.

v_1^2=\dot x^2

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 $v_2$ leads to:

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:

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:

\frac(\mathrm(d))(\mathrm(d)t)(\partial(L)\over \partial(\dot x)) - (\partial(L)\over \partial x) = F

\frac(\mathrm(d))(\mathrm(d)t)(\partial(L)\over \partial(\dot \theta)) - (\partial(L)\over \partial \theta) = 0 Substitution $L$ into these expressions with subsequent simplification leads to equations describing the motion of a reverse pendulum:

\left (M + m \right) \ddot x - m \ell \ddot \theta \cos \theta + m \ell \dot \theta^2 \sin \theta = F

\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 $\theta\approx 0$ .

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:

$\left(-\ell \sin \theta , y + \ell \cos \theta \right)$

and the speed is found through the first derivative of the position:

$v^2=\dot y^2-2 \ell \dot y \dot \theta \sin \theta + \ell^2\dot \theta ^2.$ $\ddot \theta - (g \over \ell) \sin \theta = -(A \over \ell) \omega^2 \sin \omega t \sin \theta.$

This equation does not have an elementary solution in 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 $y$ , the pendulum falls quickly after leaving a stable vertical position.
If $y$ 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 ( $\theta = 0$ ).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.

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An excerpt characterizing the Reverse Pendulum

“This is Bezukhova’s brother, Anatol Kuragin,” she said, pointing to the handsome cavalry guard who walked past them, looking somewhere from the height of his raised head across the ladies. - How good! is not it? They say they will marry him to this rich woman. And your sauce, Drubetskoy, is also very confusing. They say millions. “Why, it’s the French envoy himself,” she answered about Caulaincourt when the countess asked who it was. - Look like some kind of king. But still, the French are nice, very nice. No miles for society. And here she is! No, our Marya Antonovna is the best! And how simply dressed. Lovely! “And this fat one, with glasses, is a world-class pharmacist,” said Peronskaya, pointing to Bezukhov. “Put him next to your wife: he’s a fool!”
Pierre walked, waddling his fat body, parting the crowd, nodding right and left as casually and good-naturedly as if he were walking through the crowd of a bazaar. He moved through the crowd, obviously looking for someone.
Natasha looked with joy at the familiar face of Pierre, this pea jester, as Peronskaya called him, and knew that Pierre was looking for them, and especially her, in the crowd. Pierre promised her to be at the ball and introduce her to the gentlemen.
But, before reaching them, Bezukhoy stopped next to a short, very handsome brunette in a white uniform, who, standing at the window, was talking with some tall man in stars and a ribbon. Natasha immediately recognized the short man young man in a white uniform: it was Bolkonsky, who seemed to her very rejuvenated, cheerful and prettier.
– Here’s another friend, Bolkonsky, do you see, mom? - Natasha said, pointing to Prince Andrei. – Remember, he spent the night with us in Otradnoye.
- Oh, do you know him? - said Peronskaya. - Hate. Il fait a present la pluie et le beau temps. [The rainy or good weather. (French proverb meaning that he is successful.)] And such pride that there are no boundaries! I followed my daddy's lead. And I contacted Speransky, they are writing some projects. Look how the ladies are treated! “She’s talking to him, but he’s turned away,” she said, pointing at him. “I would have beaten him if he had treated me the way he treated these ladies.”

Suddenly everything began to move, the crowd began to speak, moved, moved apart again, and between the two parted rows, at the sound of music playing, the sovereign entered. The master and hostess followed him. The Emperor walked quickly, bowing to the right and left, as if trying to quickly get rid of this first minute of the meeting. The musicians played Polskoy, known then by the words composed on it. These words began: “Alexander, Elizabeth, you delight us...” The Emperor walked into the living room, the crowd poured to the doors; several faces with changed expressions hurriedly walked back and forth. The crowd again fled from the doors of the living room, in which the sovereign appeared, talking with the hostess. Some young man with a confused look stepped on the ladies, asking them to move aside. Some ladies with faces expressing complete obliviousness to all conditions of the world, spoiling their toilets, pressed forward. The men began to approach the ladies and form Polish pairs.
Everything parted, and the sovereign, smiling and leading the mistress of the house by the hand, walked out of the living room door. Behind him came the owner with M.A. Naryshkina, then envoys, ministers, various generals, whom Peronskaya kept calling. More than half of the ladies had gentlemen and were going or preparing to go to Polskaya. Natasha felt that she remained with her mother and Sonya among the minority of ladies who were pushed to the wall and not taken in Polskaya. She stood with her slender arms hanging down, and with her slightly defined chest rising steadily, holding her breath, her shining, frightened eyes looked ahead of her, with an expression of readiness for the greatest joy and the greatest sorrow. She was not interested in either the sovereign or all the important persons to whom Peronskaya pointed out - she had one thought: “is it really possible that no one will come up to me, will I really not dance among the first, will all these men who are now not notice me?” It seems that they don’t even see me, and if they look at me, they look with such an expression as if they were saying: Ah! it's not her, there's nothing to watch. No, this cannot be! - she thought. “They should know how much I want to dance, how great I am at dancing, and how much fun it will be for them to dance with me.”
The sounds of the Polish, which continued for quite a long time, were already beginning to sound sad - a memory in Natasha’s ears. She wanted to cry. Peronskaya moved away from them. The Count was at the other end of the hall, the Countess, Sonya and she stood alone as if in a forest in this alien crowd, uninteresting and unnecessary to anyone. Prince Andrey walked past them with some lady, obviously not recognizing them. Handsome Anatole, smiling, said something to the lady he was leading, and looked at Natasha’s face with the same look as one looks at the walls. Boris walked past them twice and turned away each time. Berg and his wife, who were not dancing, approached them.
Natasha found this family bonding here at the ball offensive, as if there was no other place for family conversations except at the ball. She did not listen or look at Vera, who was telling her something about her green dress.
Finally, the sovereign stopped next to his last lady (he was dancing with three), the music stopped; the preoccupied adjutant ran towards the Rostovs, asking them to step aside somewhere else, although they were standing against the wall, and the distinct, cautious and fascinatingly measured sounds of a waltz were heard from the choir. The Emperor looked at the audience with a smile. A minute passed and no one had started yet. The adjutant manager approached Countess Bezukhova and invited her. She raised her hand, smiling, and placed it, without looking at him, on the adjutant’s shoulder. The adjutant manager, a master of his craft, confidently, slowly and measuredly, hugging his lady tightly, first set off with her on a glide path, along the edge of the circle, and picked her up at the corner of the hall left hand, turned it, and because of the ever-accelerating sounds of the music, only the measured clicks of the spurs of the adjutant’s fast and dexterous legs could be heard, and every three beats at the turn, the fluttering velvet dress of his lady seemed to flare up. Natasha looked at them and was ready to cry that it was not she who was dancing this first round of the waltz.
Prince Andrei, in his colonel's white (cavalry) uniform, in stockings and shoes, lively and cheerful, stood in the front rows of the circle, not far from the Rostovs. Baron Firgof spoke with him about tomorrow's supposed first meeting state council. Prince Andrei, as a person close to Speransky and participating in the work of the legislative commission, could give correct information about the meeting tomorrow, about which there were various rumors. But he did not listen to what Firgof told him, and looked first at the sovereign, then at the gentlemen who were getting ready to dance, who did not dare to join the circle.

Schematic illustration of an inverted pendulum on a cart. The rod has no mass. We denote the mass of the cart and the mass of the ball at the end of the rod by M And m. The rod has a length l.

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.).

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:

θ ¨ − g ℓ sin ⁡ θ = 0 (\displaystyle (\ddot (\theta))-(g \over \ell )\sin \theta =0)

When translated, it will have the same sign of angular acceleration:

θ ¨ = g ℓ sin ⁡ θ (\displaystyle (\ddot (\theta))=(g \over \ell )\sin \theta )

Pendulum on a trolley

The equations of motion can be obtained using Lagrange's equations. We are talking about the above figure, where θ (t) (\displaystyle \theta (t)) pendulum angle length l (\displaystyle l) in relation to the vertical and the acting force of gravity and external forces F (\displaystyle F) in the direction x (\displaystyle x). Let's define x (t) (\displaystyle x(t)) trolley position. Lagrangian L = T − V (\displaystyle L=T-V) systems:

L = 1 2 M v 1 2 + 1 2 m v 2 2 − m g ℓ cos ⁡ θ (\displaystyle L=(\frac (1)(2))Mv_(1)^(2)+(\frac (1) (2))mv_(2)^(2)-mg\ell \cos \theta )

where is the speed of the cart, and is the speed of the material point m (\displaystyle m). v 1 (\displaystyle v_(1)) And v 2 (\displaystyle v_(2)) can be expressed through x (\displaystyle x) And θ (\displaystyle \theta ) by writing velocity as the first derivative of position.

v 1 2 = x ˙ 2 (\displaystyle v_(1)^(2)=(\dot (x))^(2)) v 2 2 = (d d t (x − ℓ sin ⁡ θ)) 2 + (d d t (ℓ cos ⁡ θ)) 2 (\displaystyle 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 v 2 (\displaystyle v_(2)) leads to:

v 2 2 = x ˙ 2 − 2 ℓ x ˙ θ ˙ cos ⁡ θ + ℓ 2 θ ˙ 2 (\displaystyle 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:

L = 1 2 (M + m) x ˙ 2 − m ℓ x ˙ θ ˙ cos ⁡ θ + 1 2 m ℓ 2 θ ˙ 2 − m g ℓ cos ⁡ θ (\displaystyle 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)-mg\ell \cos \theta )

and equations of motion:

d d t ∂ L ∂ x ˙ − ∂ L ∂ x = F (\displaystyle (\frac (\mathrm (d) )(\mathrm (d) t))(\partial (L) \over \partial (\dot (x )))-(\partial (L) \over \partial x)=F) d d t ∂ L ∂ θ ˙ − ∂ L ∂ θ = 0 (\displaystyle (\frac (\mathrm (d) )(\mathrm (d) t))(\partial (L) \over \partial (\dot (\ theta)))-(\partial (L) \over \partial \theta )=0)

Substitution L (\displaystyle L) into these expressions with subsequent simplification leads to equations describing the motion of a reverse pendulum:

(M + m) x ¨ − m ℓ θ ¨ cos ⁡ θ + m ℓ θ ˙ 2 sin ⁡ θ = F (\displaystyle \left(M+m\right)(\ddot (x))-m\ell ( \ddot (\theta))\cos \theta +m\ell (\dot (\theta))^(2)\sin \theta =F) ℓ θ ¨ − g sin ⁡ θ = x ¨ cos ⁡ θ (\displaystyle \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 θ ≈ 0 (\displaystyle \theta \approx 0).

Pendulum with oscillating base

(− ℓ sin ⁡ θ , y + ℓ cos ⁡ θ) (\displaystyle \left(-\ell \sin \theta ,y+\ell \cos \theta \right))

and the speed is found through the first derivative of the position:

v 2 = y ˙ 2 − 2 ℓ y ˙ θ ˙ sin ⁡ θ + ℓ 2 θ ˙ 2 . (\displaystyle v^(2)=(\dot (y))^(2)-2\ell (\dot (y))(\dot (\theta))\sin \theta +\ell ^(2) (\dot (\theta))^(2).)

The Lagrangian for this system can be written as:

L = 1 2 m (y ˙ 2 − 2 ℓ y ˙ θ ˙ sin ⁡ θ + ℓ 2 θ ˙ 2) − m g (y + ℓ cos ⁡ θ) (\displaystyle L=(\frac (1)(2) )m\left((\dot (y))^(2)-2\ell (\dot (y))(\dot (\theta))\sin \theta +\ell ^(2)(\dot ( \theta))^(2)\right)-mg\left(y+\ell \cos \theta \right))

the equations of motion follow from:

d d t ∂ L ∂ θ ˙ − ∂ L ∂ θ = 0 (\displaystyle (\mathrm (d) \over \mathrm (d) t)(\partial (L) \over \partial (\dot (\theta))) -(\partial (L) \over \partial \theta )=0)

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.).

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 file texvc not found; See math/README for setup help.): \ddot \theta = (g \over \ell) \sin \theta

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 the acting force of 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:

Unable to parse expression (Executable file 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.

Unable to parse expression (Executable file 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:

Unable to parse expression (Executable file texvc not found; See math/README for setup help.): 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 file texvc 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 file texvc 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:

Unable to parse expression (Executable file 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

Unable to parse expression (Executable file texvc 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 file texvc 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 an elementary solution in 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.

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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 by their own choice, since Vasily Nikandrovich for many years was grandfather’s attorney in all his affairs and one of the most his close 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 placed in a cold basement, without explaining anything and without 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 the real world - if someone started talking 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 family archives Alexandra 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 a relative of my grandfather along the Hesse line) 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?..

Diagnostics and diseases. Medicines from A to Z

Analysis of technology. Reverse pendulum

Review

Equations of motion

With a fixed fulcrum

Pendulum on a trolley

Pendulum with oscillating base

Application

see also

Links

Further reading

Write a review about the article "Reverse Pendulum"

Links

An excerpt characterizing the Reverse Pendulum

Review

Equations of motion

With a fixed fulcrum

Pendulum on a trolley

Pendulum with oscillating base

Review

Equations of motion

With a fixed fulcrum

Pendulum on a trolley

Pendulum with oscillating base

Application

see also

Links

Further reading

Write a review about the article "Reverse Pendulum"

Links

An excerpt characterizing the Reverse Pendulum