It can also be inclined (which is more meaningful for the study of walking stability control of actual robots). Real-time stability of double inverted pendulum system
The stability research of inverted pendulum system is a challenge faced by modern control theory, and the experiment of stability research of inverted pendulum system is of great significance to control theory.
Experience. In this paper, the modeling of double inverted pendulum is studied from two angles, namely, from the angle of easy-to-understand motion synthesis and convenience.
The angle of Lagrange equation is deduced and compared, so that readers with basic mechanical knowledge can understand the mode of double inverted pendulum system.
Type a is easier to understand.
1 system description
The double inverted pendulum system in the experiment consists of the following parts: there are
A smooth guide rail with an effective length of 90 cm can move back and forth on the guide rail.
The mobile trolley is hinged with a swinging rod made of aluminum, and the auxiliary pendulum
The lever is connected with the first-stage swing rod in the same way, and their hinge mode.
It determines their motion in the vertical plane, the first-order pendulum and the second-order pendulum.
The specifications are the same, and the effective length is 525 cm ... The driving system of the car includes
DC torque servo motor and synchronous belt drive system, trolley
The relative displacement of the reference point (that is, the center position of the guide rail) is determined by the following formula
Potentiometer 0 is obtained by measuring the transmission belt, and the first swing rod is perpendicular to the vertical direction.
The included angle consists of a potentiometer fixed at the hinge of the first-stage swing lever and the trolley.
Measured by 1, the included angle between the secondary pendulum and the vertical direction is determined by the potential.
By measuring the angle difference between two pendulums. And get it indirectly. DC。
The servo motor generates the driving force F, which makes the car change according to the swing angle.
Move on the guide rail, thus realizing the balance of the double inverted pendulum system.
The establishment of mathematical model of double inverted pendulum system and its significance 49
2 Mathematical modeling
■ The mathematical model of the inverted pendulum system is based on the following assumptions:
Each stage of the 1) pendulum is a rigid body.
2) The length of the synchronous belt remains unchanged during the experiment.
3) The driving force is directly proportional to the input of the amplifier and directly acts on the trolley without delay.
4) All I friction forces, such as Coulomb friction force and dynamic friction force, are very small and can be ignored in the modeling process.
2, 1 Write the mathematical model of the double inverted pendulum according to Newtonian mechanics and rigid body dynamics.
Using the principle of motion synthesis: absolute motion, relative motion and implicated motion,
Firstly, the kinematics of the system is analyzed. Because the dynamic coordinate system is based on the pendulum 1,
The center of mass of pendulum 2 is well understood, and the analysis process is based on it. Using Newton
The dynamics of the system is analyzed by mechanics, and the mathematical model of the double inverted pendulum is obtained.
Using the isolation method in mechanics, the double inverted pendulum system is divided into trolley and simple pendulum.
The blue part of lever 1 and swing lever 2 First, the analysis of the trolley is shown in Figure 2.
Swing bar 1 The force exerted on the trolley is divided into vertical component and horizontal component.
Direction component. The equation in the horizontal direction is 1 = mo2.
The force analysis of pendulum 1 and pendulum 2 is shown in fig. 3 and fig. 4.
● Swing bar
/ l
\ ^.
L/- one
ⅲ-g
Fig. 3 Stress of pendulum 1
Fig. 2 Force analysis of trolley
J
0/ 1
Each basket: /f
Fig. 4 Force Analysis of Swing Bar 2
Using Newton's second law and moment of momentum theorem, the kinematics and dynamics equations of simple pendulum are obtained.
2-2=ml +ml, l sprout cos0l-m, l sprout sin0 L.
M g-l+F2 =...sin0l+m 1fl ~ eos0l。
Sin. S- 1 (L. )COS
According to Newton's second law and the theorem of moment of momentum, the kinematics and dynamics equations of a simple pendulum are obtained:
2= Pa+M: l 1o ~ cos0l+ pang /~ 2cosoz- pang sin0- pang sin.
M2g-FZ = M2L SIN0L+M2L0 ~ SIN0: Ten m L P~eos0l+m2 cos02.
:l 12 sin02-L，cos02 d t。
2.2 Lagrange equation
In order to get the dynamic equation of the double inverted pendulum system, the Lagrange equation is applied. First, you can write
L = t-= ⊙ t-= ⊙ t ?+惉 +.+{ m. {[ sound (+in )] +[ strike (. s ])
+{:(strike (+Lt sin+sin )] +[ accuse (+0cos+] 2cos) r) a m.glc. s])
M2g(L，COS +t2 COS)
The expression of Lagrange equation is
Wait a minute: _l_2? Facing one "J one"?
Is the degree of freedom, that is, the generalized coordinate number. For the double inverted pendulum system, there are
S=3, today,
Because the value of mouth sum is very small in the experiment, the following approximation is used in the process of modeling simplification:
≈ ≈0; 1 ≈ 0; COS( 1)≈ 1； sin( 1)≈ 1； COS ≈COS ≈ 1:
Sin: sin
The linearization equation is arranged as follows
( 。 +m+:)+(。 ，+M2L。 ) Meng +:anti =F (1)
(.t+m)+(+m) Meng+m l CuO = (. +:L) Start
: quantity+:L. Meng +( +m CuO) is called winning g 12.
(2)
(3)
The meaning of each variable is as follows:
O is the quality of the car; It is the mass of pendulum 1; M is that mas of the swing rod 2; Is the length of the swing rod: f is the driving force of the car; for
The displacement of the trolley relative to the center position; Is the included angle between the swing rod 1 and the vertical direction; Is the included angle between the swing rod 2 and the vertical direction:,. It's just a swing.
1 distance from the center of mass to the hinge point: the distance from the center of mass of the swing rod 2 to the hinge point.
This procurement inspection, O = 2.328 7kg, -= 0.22kg,: = 0. 16kg, L =0.5m, = 0.32m, t2=0.26m ..
The motion of the double inverted pendulum system is absolutely unstable saddle point motion. According to the mathematical model and experimental results, the state feedback control is carried out.
Pole configuration should satisfy saddle point characteristics, so that the double inverted pendulum can never stand down.
3 application
In stability control, inverted pendulum is universal and typical. As a control device, the inverted pendulum system
Simple structure, low price, convenient for analog and digital realization of a variety of different control methods, as a controlled object, it is a kind of
High-order, unstable, multivariable, nonlinear and strongly coupled fast systems can only be realized by adopting effective control strategies.
Its stability. Inverted pendulum system can realize its stability control through PID, adaptive, state feedback and intelligence.
Ability control, fuzzy control and artificial neural network can all be realized in the control of inverted pendulum system, which is a new control method.
After the theory and method of control are put forward, when it cannot be strictly proved by theory, we can consider using inverted pendulum device to verify its correctness.
Sex and practicality.
In the research of control system, inverted pendulum system has been paid great attention, and "inverted pendulum system" has been recognized as a typical example of automatic control theory.
Experimental equipment is also a typical physical model of control theory in teaching and scientific research. By studying the inverted pendulum system,
Establishment and significance of mathematical model of double inverted pendulum system 5 1
It can not only solve the theoretical problems in control, but also integrate the three basic disciplines involved in control theory: mechanics, mathematics and electricity (including
Computer) are organically combined and comprehensively applied to the inverted pendulum system.
In modern mechanical control systems, such as helicopters, rocket launches, satellite operations and robots lifting weights, doing gymnastics and walking.
Walking robot, walking control, etc. , all have stability control problems similar to inverted pendulum. In the late 1960s, as a typical example,
The concept of inverted pendulum system was put forward, and people used to use it to test controllers.
This method has the ability to control unstable, nonlinear and fast systems. In practical teaching, as a means to verify the control strategy,
An inverted pendulum system is proposed. Because there is always a big difference between computer simulation results and actual experiments, the double inverted pendulum system
The development of this system provides students with the possibility of combining theory with practice.
4 conclusion
The double inverted pendulum system is an extremely complex nonlinear unstable control problem, which requires high control accuracy and rapidity.
However, the research results of a typical nonlinear unstable system are of great significance both in theory and methodology.
The establishment of the mathematical model of the second-order pendulum plays a guiding role in studying its stability. Experiments show that the state inversion is adopted on the basis of this modeling.
The feedback method is quite successful in the stability control of the double inverted pendulum system, and can be analyzed on this basis to provide a basis for computer control.
The foundation of theory and practice.
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