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An inverted pendulum is a
pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the ...
that has its
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
above its
pivot Pivot may refer to: *Pivot, the point of rotation in a lever system *More generally, the center point of any rotational system *Pivot joint, a kind of joint between bones in the body *Pivot turn, a dance move Companies *Incitec Pivot, an Austra ...
point. It is
unstable In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stable are unstable; systems can also be mar ...
and without additional help will fall over. It can be suspended stably in this inverted position by using a
control system A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial c ...
to monitor the angle of the pole and move the pivot point horizontally back under the center of mass when it starts to fall over, keeping it balanced. The inverted pendulum is a classic problem in dynamics and
control theory Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
and is used as a benchmark for testing control strategies. It is often implemented with the pivot point mounted on a cart that can move horizontally under control of an electronic servo system as shown in the photo; this is called a cart and pole apparatus. Most applications limit the pendulum to 1
degree of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
by affixing the pole to an
axis of rotation Rotation around a fixed axis is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rota ...
. Whereas a normal pendulum is stable when hanging downwards, an inverted pendulum is inherently unstable, and must be actively balanced in order to remain upright; this can be done either by applying a
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
at the pivot point, by moving the pivot point horizontally as part of a
feedback Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled ...
system, changing the rate of rotation of a mass mounted on the pendulum on an axis parallel to the pivot axis and thereby generating a net torque on the pendulum, or by oscillating the pivot point vertically. A simple demonstration of moving the pivot point in a feedback system is achieved by balancing an upturned broomstick on the end of one's finger. A second type of inverted pendulum is a
tiltmeter A tiltmeter is a sensitive inclinometer designed to measure very small changes from the vertical level, either on the ground or in structures. Tiltmeters are used extensively for monitoring volcanoes, the response of dams to filling, the small ...
for tall structures, which consists of a wire anchored to the bottom of the foundation and attached to a float in a pool of oil at the top of the structure that has devices for measuring movement of the neutral position of the float away from its original position.


Overview

A pendulum with its bob hanging directly below the support
pivot Pivot may refer to: *Pivot, the point of rotation in a lever system *More generally, the center point of any rotational system *Pivot joint, a kind of joint between bones in the body *Pivot turn, a dance move Companies *Incitec Pivot, an Austra ...
is at a stable equilibrium point; there is no torque on the pendulum so it will remain motionless, and if displaced from this position will experience a restoring torque which returns it toward the equilibrium position. A pendulum with its bob in an inverted position, supported on a rigid rod directly above the pivot, 180° from its stable equilibrium position, is at an
unstable equilibrium In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the equation at this solution has the form ...
point. At this point again there is no torque on the pendulum, but the slightest displacement away from this position will cause a gravitation torque on the pendulum which will accelerate it away from equilibrium, and it will fall over. In order to stabilize a pendulum in this inverted position, a feedback control system can be used, which monitors the pendulum's angle and moves the position of the pivot point sideways when the pendulum starts to fall over, to keep it balanced. The inverted pendulum is a classic problem in dynamics and
control theory Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
and is widely used as a benchmark for testing control algorithms (
PID controller A proportional–integral–derivative controller (PID controller or three-term controller) is a control loop mechanism employing feedback that is widely used in industrial control systems and a variety of other applications requiring continuou ...
s,
state space representation In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations or difference equations. State variables are variables wh ...
,
neural networks A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...
,
fuzzy control A fuzzy control system is a control system based on fuzzy logic—a mathematical system that analyzes analog input values in terms of logical variables that take on continuous values between 0 and 1, in contrast to classical or digital logic, ...
,
genetic algorithm In computer science and operations research, a genetic algorithm (GA) is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms (EA). Genetic algorithms are commonly used to gene ...
s, etc.). Variations on this problem include multiple links, allowing the motion of the cart to be commanded while maintaining the pendulum, and balancing the cart-pendulum system on a see-saw. The inverted pendulum is related to rocket or missile guidance, where the center of gravity is located behind the center of drag causing aerodynamic instability. The understanding of a similar problem can be shown by simple robotics in the form of a balancing cart. Balancing an upturned broomstick on the end of one's finger is a simple demonstration, and the problem is solved by self-balancing
personal transporter A personal transporter (also powered transporter, electric rideable, personal light electric vehicle, personal mobility device, etc.) is any of a class of compact, mostly recent (21st century), motorised micromobility vehicle for transporting an ...
s such as the
Segway PT The Segway is a two-wheeled, self-balancing personal transporter invented by Dean Kamen and brought to market in 2001 as the Segway HT, subsequently as the Segway PT, and manufactured by Segway Inc. ''HT'' is an initialism for "human trans ...
, the self-balancing hoverboard and the
self-balancing unicycle An electric unicycle (often initialized as EUC or acronymized yuke or Uni) is a self-balancing personal transporter with a single wheel. The rider controls speed by leaning forwards or backwards, and steers by twisting or tilting the unit side to ...
. Another way that an inverted pendulum may be stabilized, without any feedback or control mechanism, is by oscillating the pivot rapidly up and down. This is called
Kapitza's pendulum Kapitza's pendulum or Kapitza pendulum is a rigid pendulum in which the pivot point vibrates in a vertical direction, up and down. It is named after Russian Nobel prize, Nobel laureate physicist Pyotr Kapitza, who in 1951 developed a theory which ...
. If the
oscillation Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
is sufficiently strong (in terms of its acceleration and amplitude) then the inverted pendulum can recover from perturbations in a strikingly counterintuitive manner. If the driving point moves in
simple harmonic motion In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the ...
, the pendulum's motion is described by the
Mathieu equation In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation : \frac + (a - 2q\cos(2x))y = 0, where a and q are parameters. They were first introduced by Émile Léonard Mathieu, ...
.


Equations of motion

The
equations of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
of inverted pendulums are dependent on what constraints are placed on the motion of the pendulum. Inverted pendulums can be created in various configurations resulting in a number of Equations of Motion describing the behavior of the pendulum.


Stationary pivot point

In a configuration where the pivot point of the pendulum is fixed in space, the equation of motion is similar to that for an uninverted pendulum. The equation of motion below assumes no friction or any other resistance to movement, a rigid massless rod, and the restriction to
2-dimensional In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...
movement. :\ddot \theta - \sin \theta = 0 Where \ddot \theta is the
angular acceleration In physics, angular acceleration refers to the time rate of change of angular velocity. As there are two types of angular velocity, namely spin angular velocity and orbital angular velocity, there are naturally also two types of angular acceler ...
of the pendulum, g is the
standard gravity The standard acceleration due to gravity (or standard acceleration of free fall), sometimes abbreviated as standard gravity, usually denoted by or , is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. ...
on the surface of the Earth, \ell is the length of the pendulum, and \theta is the angular displacement measured from the equilibrium position. When \ddot \theta added to both sides, it will have the same sign as the angular acceleration term: :\ddot \theta = \sin \theta Thus, the inverted pendulum will accelerate away from the vertical unstable equilibrium in the direction initially displaced, and the acceleration is inversely proportional to the length. Tall pendulums fall more slowly than short ones. Derivation using torque and moment of inertia: The pendulum is assumed to consist of a point mass, of mass m , affixed to the end of a massless rigid rod, of length \ell, attached to a pivot point at the end opposite the point mass. The net
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
of the system must equal the
moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
times the angular acceleration: :\boldsymbol_=I \ddot \theta The torque due to gravity providing the net torque: :\boldsymbol_= m g \ell \sin \theta\,\! Where \theta\ is the angle measured from the inverted equilibrium position. The resulting equation: : I \ddot \theta= m g \ell \sin \theta\,\! The moment of inertia for a point mass: :I = m R^2 In the case of the inverted pendulum the radius is the length of the rod, \ell . Substituting in I = m \ell ^2 : m \ell ^2 \ddot \theta= m g \ell \sin \theta\,\! Mass and \ell^2 is divided from each side resulting in: :\ddot \theta = \sin \theta


Inverted pendulum on a cart

An inverted pendulum on a cart consists of a mass m at the top of a pole of length \ell pivoted on a horizontally moving base as shown in the adjacent image. The cart is restricted to
linear motion Linear motion, also called rectilinear motion, is one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion, with co ...
and is subject to forces resulting in or hindering motion.


Essentials of stabilization

The essentials of stabilizing the inverted pendulum can be summarized qualitatively in three steps. 1. If the tilt angle \theta is to the right, the cart must accelerate to the right and vice versa. 2. The position of the cart x relative to track center is stabilized by slightly modulating the null angle (the angle error that the control system tries to null) by the position of the cart, that is, null angle = \theta + k x where k is small. This makes the pole want to lean slightly toward track center and stabilize at track center where the tilt angle is exactly vertical. Any offset in the tilt sensor or track slope that would otherwise cause instability translates into a stable position offset. A further added offset gives position control. 3. A normal pendulum subject to a moving pivot point such as a load lifted by a crane, has a peaked response at the pendulum radian frequency of \omega_p = \sqrt . To prevent uncontrolled swinging, the frequency spectrum of the pivot motion should be suppressed near \omega_p . The inverted pendulum requires the same suppression filter to achieve stability. Note that, as a consequence of the null angle modulation strategy, the position feedback is positive, that is, a sudden command to move right will produce an initial cart motion to the left followed by a move right to rebalance the pendulum. The interaction of the pendulum instability and the positive position feedback instability to produce a stable system is a feature that makes the mathematical analysis an interesting and challenging problem.


From Lagrange's equations

The equations of motion can be derived using
Lagrange's equations In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lo ...
. We refer to the drawing to the right where \theta(t) is the angle of the pendulum of length l with respect to the vertical direction and the acting forces are gravity and an external force ''F'' in the x-direction. Define x(t) to be the position of the cart. The kinetic energy T of the system is: : T = \frac M v_1^2 + \frac m v_2^2, where v_1 is the velocity of the cart and v_2 is the velocity of the point mass m. v_1 and v_2 can be expressed in terms of x and \theta by writing the velocity as the first derivative of the position; : v_1^2=\dot x^2, : v_2^2=\left(\right)^2 + \left(\right)^2. Simplifying the expression for v_2 leads to: : v_2^2= \dot x^2 -2 \ell \dot x \dot \theta\cos \theta + \ell^2\dot \theta^2. The kinetic energy is now given by: : T = \frac \left(M+m \right ) \dot x^2 -m \ell \dot x \dot\theta\cos\theta + \frac m \ell^2 \dot \theta^2. The generalized coordinates of the system are \theta and x, each has a generalized force. On the x axis, the generalized force Q_x can be calculated through its virtual work : Q_x\delta x=F \delta x,\quad Q_x=F, on the \theta axis, the generalized force Q_\theta can be also calculated through its virtual work : Q_\theta\delta\theta=mgl\sin\theta \delta \theta,\quad Q_\theta=mgl\sin\theta. According to the
Lagrange's equations In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lo ...
, the equations of motion are: : \frac - = F, : \frac - = mgl\sin\theta, substituting T in these equations and simplifying leads to the equations that describe the motion of the inverted 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 nonlinear, but since the goal of a control system would be to keep the pendulum upright the equations can be linearized around \theta \approx 0.


From Euler-Lagrange equations

The generalized forces can be both written as potential energy V_x and V_\theta, According to the
D'Alembert's principle D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond d'Alembert. D'Alember ...
, generalized forces and potential energy are connected: :Q_j = \frac\frac - \frac \,, However, under certain circumstances, the potential energy is not accessible, only generalized forces are available. After getting the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
\mathcal=T-V, we can also use
Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
to solve for equations of motion: :\frac - \frac \left ( \frac \right ) = 0, :\frac - \frac \left ( \frac \right ) = 0. The only difference is whether to incorporate the generalized forces into the potential energy V_j or write them explicitly as Q_j on the right side, they all lead to the same equations in the final.


From Newton's second law

Oftentimes it is beneficial to use
Newton's second law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
instead of
Lagrange's equations In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lo ...
because Newton's equations give the reaction forces at the joint between the pendulum and the cart. These equations give rise to two equations for each body; one in the x-direction and the other in the y-direction. The equations of motion of the cart are shown below where the LHS is the sum of the forces on the body and the RHS is the acceleration. : F-R_x = M \ddot x : F_N - R_y - M g = 0 In the equations above R_x and R_y are reaction forces at the joint. F_N is the normal force applied to the cart. This second equation only depends on the vertical reaction force thus the equation can be used to solve for the normal force. The first equation can be used to solve for the horizontal reaction force. In order to complete the equations of motion, the acceleration of the point mass attached to the pendulum must be computed. The position of the point mass can be given in inertial coordinates as : \vec r_P = (x-\ell \sin \theta) \hat x_I + \ell \cos \theta \hat y_I Taking two derivatives yields the acceleration vector in the inertial reference frame. : \vec a_ = (\ddot x + \ell \dot \theta^2 \sin \theta - \ell \ddot \theta \cos \theta ) \hat x_I + (-\ell \dot \theta^2 \cos \theta - \ell \ddot \theta \sin \theta) \hat y_I Then, using Newton's second law, two equations can be written in the x-direction and the y-direction. Note that the reaction forces are positive as applied to the pendulum and negative when applied to the cart. This is due to Newton's Third Law. : R_x = m(\ddot x + \ell \dot \theta^2 \sin \theta - \ell \ddot \theta \cos \theta ) : R_y - m g = m (-\ell \dot \theta^2 \cos \theta - \ell \ddot \theta \sin \theta) The first equation allows yet another way to compute the horizontal reaction force in the event the applied force F is not known. The second equation can be used to solve for the vertical reaction force. The first equation of motion is derived by substituting F-R_x = M \ddot x into R_x = m(\ddot x + \ell \dot \theta^2 \sin \theta - \ell \ddot \theta \cos \theta ) which yields : \left (M+m \right) \ddot x - m \ell \ddot \theta \cos \theta + m \ell \dot \theta^2 \sin \theta = F By inspection this equation is identical to the result from Lagrange's Method. In order to obtain the second equation the pendulum equation of motion must be dotted with a unit vector which runs perpendicular to the pendulum at all times and is typically noted as the x-coordinate of the body frame. In inertial coordinates this vector can be written using a simple 2-D coordinate transformation : \hat x_B = \cos \theta \hat x_I + \sin \theta \hat y_I The pendulum equation of motion written in vector form is \sum \vec F = m \vec a_. Dotting \hat x_B with both sides yields the following on the LHS (note that a transpose is the same as a dot product) : (\hat x_B)^T\sum \vec F = (\hat x_B)^T (R_x \hat x_I + R_y \hat y_I - m g \hat y_I) = (\hat x_B)^T(R_p \hat y_B - m g \hat y_I) = -m g \sin \theta In the above equation the relationship between body frame components of the reaction forces and inertial frame components of reaction forces is used. The assumption that the bar connecting the point mass to the cart is massless implies that this bar cannot transfer any load perpendicular to the bar. Thus, the inertial frame components of the reaction forces can be written simply as R_p \hat y_B which signifies that the bar can only transfer loads along the axis of the bar itself. This gives rise to another equation which can be used to solve for the tension in the rod itself : R_p = \sqrt The RHS of the equation is computed similarly by dotting \hat x_B with the acceleration of the pendulum. The result (after some simplification) is shown below. : m(\hat x_B)^T(\vec a_) = m(\ddot x \cos \theta - \ell \ddot \theta) Combining the LHS with the RHS and dividing through by m yields : \ell \ddot \theta - g \sin \theta = \ddot x \cos \theta which again is identical to the result of Lagrange's method. The benefit of using Newton's method is that all reaction forces are revealed to ensure that nothing will be damaged.


Variants

Achieving stability of an inverted pendulum has become a common engineering challenge for researchers. There are different variations of the inverted pendulum on a cart ranging from a rod on a cart to a multiple segmented inverted pendulum on a cart. Another variation places the inverted pendulum's rod or segmented rod on the end of a rotating assembly. In both, (the cart and rotating system) the inverted pendulum can only fall in a plane. The inverted pendulums in these projects can either be required to only maintain balance after an equilibrium position is achieved or be able to achieve equilibrium by itself. Another platform is a two-wheeled balancing inverted pendulum. The two wheeled platform has the ability to spin on the spot offering a great deal of maneuverability. Yet another variation balances on a single point. A
spinning top A spinning top, or simply a top, is a toy with a squat body and a sharp point at the bottom, designed to be spun on its vertical axis, balancing on the tip due to the gyroscopic effect. Once set in motion, a top will usually wobble for a few se ...
, a
unicycle A unicycle is a vehicle that touches the ground with only one wheel. The most common variation has a bicycle frame, frame with a bicycle saddle, saddle, and has a human-powered vehicle, pedal-driven direct-drive mechanism, direct-drive. A two spee ...
, or an inverted pendulum atop a spherical ball all balance on a single point.


Kapitza's pendulum

An inverted pendulum in which the pivot is oscillated rapidly up and down can be stable in the inverted position. This is called
Kapitza's pendulum Kapitza's pendulum or Kapitza pendulum is a rigid pendulum in which the pivot point vibrates in a vertical direction, up and down. It is named after Russian Nobel prize, Nobel laureate physicist Pyotr Kapitza, who in 1951 developed a theory which ...
, after Russian physicist Pyotr Kapitza who first analysed it. The equation of motion for a pendulum connected to a massless, oscillating base is derived the same way as with the pendulum on the cart. The position of the point mass is now given by: :\left( -\ell \sin \theta , y + \ell \cos \theta \right) and the velocity is found by taking the first derivative of the position: :v^2=\dot y^2-2 \ell \dot y \dot \theta \sin \theta + \ell^2\dot \theta ^2. The
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
for this system can be written as: : L = \frac m \left ( \dot y^2-2 \ell \dot y \dot \theta \sin \theta + \ell^2\dot \theta ^2 \right) - m g \left( y + \ell \cos \theta \right ) and the equation of motion follows from: : - = 0 resulting in: : \ell \ddot \theta - \ddot y \sin \theta = g \sin \theta. If ''y'' represents a
simple harmonic motion In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the ...
, y = A \sin \omega t, the following
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
is: : \ddot \theta - \sin \theta = - \omega^2 \sin \omega t \sin \theta. This equation does not have elementary closed-form solutions, but can be explored in a variety of ways. It is closely approximated by the
Mathieu equation In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation : \frac + (a - 2q\cos(2x))y = 0, where a and q are parameters. They were first introduced by Émile Léonard Mathieu, ...
, for instance, when the amplitude of oscillations are small. Analyses show that the pendulum stays upright for fast oscillations. The first plot shows that when y is a slow oscillation, the pendulum quickly falls over when disturbed from the upright position. The angle \theta exceeds 90° after a short time, which means the pendulum has fallen on the ground. If y is a fast oscillation the pendulum can be kept stable around the vertical position. The second plot shows that when disturbed from the vertical position, the pendulum now starts an oscillation around the vertical position (\theta = 0). The deviation from the vertical position stays small, and the pendulum doesn't fall over.


Examples

Arguably the most prevalent example of a stabilized inverted pendulum is a
human being Humans (''Homo sapiens'') are the most abundant and widespread species of primate, characterized by bipedality, bipedalism and exceptional cognitive skills due to a large and complex Human brain, brain. This has enabled the development of ad ...
. A person standing upright acts as an inverted pendulum with their feet as the pivot, and without constant small muscular adjustments would fall over. The human nervous system contains an unconscious
feedback Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled ...
control system A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial c ...
, the
sense of balance The sense of balance or equilibrioception is the perception of balance and spatial orientation. It helps prevent humans and nonhuman animals from falling over when standing or moving. Equilibrioception is the result of a number of sensory systems ...
or
righting reflex The righting reflex, also known as the labyrinthine righting reflex, is a reflex that corrects the orientation of the body when it is taken out of its normal upright position. It is initiated by the vestibular system, which detects that the body is ...
, that uses
proprioceptive Proprioception ( ), also referred to as kinaesthesia (or kinesthesia), is the sense of self-movement, force, and body position. It is sometimes described as the "sixth sense". Proprioception is mediated by proprioceptors, mechanosensory neurons ...
input from the eyes, muscles and joints, and orientation input from the
vestibular system The vestibular system, in vertebrates, is a sensory system that creates the sense of balance and spatial orientation for the purpose of coordinating movement with balance. Together with the cochlea, a part of the auditory system, it constitutes ...
consisting of the three
semicircular canals The semicircular canals or semicircular ducts are three semicircular, interconnected tubes located in the innermost part of each ear, the inner ear. The three canals are the horizontal, superior and posterior semicircular canals. Structure The ...
in the
inner ear The inner ear (internal ear, auris interna) is the innermost part of the vertebrate ear. In vertebrates, the inner ear is mainly responsible for sound detection and balance. In mammals, it consists of the bony labyrinth, a hollow cavity in the ...
, and two
otolith An otolith ( grc-gre, ὠτο-, ' ear + , ', a stone), also called statoconium or otoconium or statolith, is a calcium carbonate structure in the saccule or utricle of the inner ear, specifically in the vestibular system of vertebrates. The sa ...
organs, to make continual small adjustments to the skeletal muscles to keep us standing upright. Walking, running, or balancing on one leg puts additional demands on this system. Certain diseases and alcohol or drug intoxication can interfere with this reflex, causing
dizziness Dizziness is an imprecise term that can refer to a sense of disorientation in space, vertigo, or lightheadedness. It can also refer to disequilibrium or a non-specific feeling, such as giddiness or foolishness. Dizziness is a common medical c ...
and disequilibration, an inability to stand upright. A
field sobriety test Field sobriety tests (FSTs), also referred to as standardized field sobriety tests (SFSTs), are a battery of tests used by police officers to determine if a person suspected of impaired driving is intoxicated with alcohol or other drugs. ''FSTs ...
used by police to test drivers for the influence of alcohol or drugs, tests this reflex for impairment. Some simple examples include balancing brooms or meter sticks by hand. The inverted pendulum has been employed in various devices and trying to balance an inverted pendulum presents a unique engineering problem for researchers. The inverted pendulum was a central component in the design of several early
seismometer A seismometer is an instrument that responds to ground noises and shaking such as caused by earthquakes, volcanic eruptions, and explosions. They are usually combined with a timing device and a recording device to form a seismograph. The outpu ...
s due to its inherent instability resulting in a measurable response to any disturbance. The inverted pendulum model has been used in some recent
personal transporter A personal transporter (also powered transporter, electric rideable, personal light electric vehicle, personal mobility device, etc.) is any of a class of compact, mostly recent (21st century), motorised micromobility vehicle for transporting an ...
s, such as the two-wheeled
self-balancing scooter A self-balancing scooter (also hoverboard, self-balancing board, segway or electric scooter board) is a self-balancing personal transporter consisting of two motorized wheels connected to a pair of articulated pads on which the rider places their ...
s and single-wheeled
electric unicycle An electric unicycle (often initialized as EUC or acronymized yuke or Uni) is a self-balancing personal transporter with a single wheel. The rider controls speed by leaning forwards or backwards, and steers by twisting or tilting the unit side to ...
s. These devices are kinematically unstable and use an electronic feedback servo system to keep them upright. Swinging a pendulum on a cart into its inverted pendulum state is considered a traditional optimal control toy problem/benchmark.


See also

*
Double inverted pendulum A double inverted pendulum is the combination of the inverted pendulum and the double pendulum. The double inverted pendulum is unstable, meaning that it will fall down unless it is controlled in some way. The two main methods of controlling a dou ...
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Inertia wheel pendulum An inertia wheel pendulum is a pendulum with an inertia wheel attached. It can be used as a pedagogical problem in control theory. This type of pendulum is often confused with the gyroscopic effect, which has completely different physical nature. ...
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Furuta pendulum The Furuta pendulum, or rotational inverted pendulum, consists of a driven arm which rotates in the horizontal plane and a pendulum attached to that arm which is free to rotate in the vertical plane. It was invented in 1992 at Tokyo Institute of ...
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iBOT The iBOT is a powered wheelchair developed by Dean Kamen in a partnership between DEKA and Johnson & Johnson's Independence Technology division. History Development of the iBOT started in 1990 with the first working prototypes available in 199 ...
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Humanoid robot A humanoid robot is a robot resembling the human body in shape. The design may be for functional purposes, such as interacting with human tools and environments, for experimental purposes, such as the study of bipedal locomotion, or for other pur ...
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Ballbot A ball balancing robot also known as a ballbot is a dynamically-stable mobile robot designed to balance on a single spherical wheel (''i.e.'', a ball). Through its single contact point with the ground, a ballbot is omnidirectional and thus e ...


References

*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.


External links


YouTube - Inverted Pendulum - Demo #3 YouTube - inverted pendulumYouTube - Double Pendulum on a CartYouTube - Triple Pendulum on a Cart
* ttps://web.archive.org/web/20180619041024/http://www.engr.usask.ca/classes/EE/480/Inverted%20Pendulum.pdf Inverted Pendulum: Analysis, Design, and Implementationbr>Non-Linear Swing-Up and Stabilizing Control of an Inverted Pendulum SystemStabilization fuzzy control of inverted pendulum systems


{{DEFAULTSORT:Inverted Pendulum Pendulums Control engineering Machine learning task