The control strength quantification analysis of outer pendulum rod for double inverted pendulum

Wencong Fu1 , Chan Wang2 , Peng Xu3 , Yuanhong Dan4

1, 2, 3School of Electrical and Electronic Engineering, Chongqing University of Technology, Chongqing, China

4School of Computer Science and Engineering, Chongqing University of Technology, Chongqing, China

1Corresponding author

Vibroengineering PROCEDIA, Vol. 28, 2019, p. 99-104. https://doi.org/10.21595/vp.2019.21040
Received 19 September 2019; accepted 28 September 2019; published 19 October 2019

Copyright © 2019 Wencong Fu, et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Abstract.

Due to the complexity of the dynamics characteristics of an inverted pendulum, and the problem that the linearization analyze method cannot satisfy the controlling requirement, a nonlinear dynamics analyze method was proposed. Through decoupling the dynamics model of a double inverted pendulum, the outer pendulum rod motion equation was derived. And then, aiming at the control strength function of outer pendulum rod, the qualitative and quantitative relationship between spatial position of pendulum rod and the control strength of outer rod, and the quantification relationship between dynamics parameters and the control strength of outer rod were separately analyzed. And the simulation verified the correctness of the analysis.

The control strength quantification analysis of outer pendulum rod for double inverted pendulum

Highlights
  • The structure of rotary double pendulum and its coordinate system
  • Nonlinear dynamics analysis method
  • simulation verified

Keywords: double inverted pendulum, quantification analysis, control strength of outer rod, dynamics parameter, spatial position of pendulum.

1. Introduction

The inverted pendulum is a classical underactuated mechanical system with nonholonomic constraints. Such kind of non-completely controllable system is important for the dynamics characteristic analysis. In article [1], the method based on Lyapunov stability theorem to conduct the stability control of rotating inverted pendulum was adopted. Jiang built up the mathematical model for rotary inverted pendulum with Lagrange function [2]. The parameter identification for the physical model of two-link and three-link was done through the least square method [3, 4]. The stability control of triple inverted pendulum with cloud control method was realized in [5]. And the stability control of quadruple inverted pendulum with variable universe fuzzy control algorithm was firstly realized in [6]. The controllability of double inverted pendulum and triple inverted pendulum close to equilibrium points with linearization method was analyzed in [4, 8]. But it is not suitable for the area except equilibrium point. Liu applied the Lyapunov exponent formed by state equation to express the motion stability of the system [9]. It is an efficient method to evaluate the quality of closed-loop system after the fact. But with little valueness for designing the controlling system beforehand. The angular acceleration expression of passive joint was derived through dynamics decoupling for a class of rootless underactuated systems in [10]. And simulation and analysis of the controllability of passive joint were carried out. But the quantification relationship between control input and passive joint did not get further discuss. Also, the quantification relationship between physical parameters and system controlling characteristics did not get further study in most relative researches.

The article mainly discussed the control strength quantification analysis of outer pendulum rod for double inverted pendulum. In Section 2, through decoupling the dynamics model, the outer pendulum rod action strength function was derived. In Section 3, the strength analysis, including maximum positive control and maximum negative control, of the outer pendulum rod and the quantification analysis of dynamics parameters and control strength of the outer rod were mainly discussed. Section 4 includes the discussion and analysis of simulation results. Section 5 presents the concluding remarks.

2. Mathematical model and dynamics decoupling

2.1. Mathematical model

The coordinate system was established shown in Fig. 1.

From [11], the mathematical model of the double inverted pendulum can be derived as:

(1)
M Θ ¨ + C Θ ˙ + G = F ,

where:

M = 1 0 0 M 21 M 22 M 23 M 31 M 32 M 33 ,           G = 0 - m 1 g l 1 + m 2 g L 1 sin θ 1 - m b g L 1 sin θ 1 - m 2 g l 2 sin θ 2 ,
C = 0 0 0 C 21 C 22 C 23 C 31 C 32 C 33 ,           F = u 0 0 ,           M 21 = k 1 cos θ 1 ,           M 22 = k 2 ,         M 33 = k 5 ,
M 23 = M 32 = k 4 cos θ 1 - θ 2 ,           M 31 = k 3 cos θ 2 ,           Θ = θ 0 θ 1 θ 2 T ,          
C 21 = - 1 2 J 1 + m 1 l 1 2 + m 2 L 1 2 θ ˙ 0 sin 2 θ 1 - k 4 θ ˙ 0 cos θ 1 sin θ 2 - 1 2 m b L 1 2 θ ˙ 0 sin 2 θ 1 ,
C 22 = c 1 + c 2 ,           C 23 = k 4 θ ˙ 2 sin θ 1 - θ 2 - c 2 ,           C 33 = c 2 ,          
C 31 = - 1 2   k 5 θ ˙ 0 sin 2 θ 2 - k 4 θ ˙ 0 sin θ 1 cos θ 2 ,           C 32 = - k 4 θ ˙ 1 sin θ 1 - θ 2 - c 2 .

The state variables and physical parameters of the rotating secondary inverted pendulum were defined as follows: m1, m2, mb are the mass of inner rod, outer rod and encoder. l1, l2 are the centroid position of inner and outer rod. L0, L1 are the length of horizontal rotating rod and inner rod. c1, c2 are the rotational friction of cart-inner rod and inner rod-outer rod axis. J1, J2 represent the moment of inertia of inner and outer rod. θ0, θ1, θ2 represent the angle of the horizontal rotating rod, and the inner and outer rod. θ˙0, θ˙1, θ˙2 represent the angular speed of the horizontal rotating rod, and the inner and outer rod. And u is the control variable.

Fig. 1. The structure of rotary double pendulum and its coordinate system

The structure of rotary double pendulum and its coordinate system

2.2. Dynamics decoupling

The meaning of the physical parameters abbreviation in the mathematical model, and the derivation of the outer rod motion equation are shown as follows:

(2)
k 1 = m 1 l 1 + m 2 L 1 + m b L 1 L 0 ,           k 2 = J 1 + m 1 l 1 2 + m 2 L 1 2 + m b L 1 2 ,           k 3 = m 2 l 2 L 0 ,
k 4 = m 2 l 2 L 1 ,           k 5 = J 2 + m 2 l 2 2 ,             k 6 = m 1 l 1 + m 2 L 1 + m b L 1 ,           k 7 = m 2 l 2 ,
θ ¨ 2 = K K u 2 u + K g 2 g + K v 02 θ ˙ 0 2 + K v 12 θ ˙ 0 2 + K v 22 θ ˙ 0 2 + K e 12 θ ˙ 0 2 + K e 22 ( θ 1 ˙ - θ 2 ˙ ) .

Then, the expressions of the intensity function factor Ku2 in Eq. (2) is shown as follow:

(3)
K u 2 = k 1 k 4 cos θ 1 cos θ 1 - θ 2 - k 2 k 3 cos θ 2 .

3. Quantitative analysis of spatial position of rod and control strength of the outer rod

cos θ 1 cos θ 1 - θ 2 is a continuous derivable function. Therefore:

(4)
d cos θ 1 cos θ 1 - θ 2 d θ 1 = 0             θ 1 = n π + θ 2 2 ,     n Z .

Substituting Eq. (4) into cosθ1cosθ1-θ2:

(5)
max cos θ 1 cos θ 1 - θ 2 = cos θ 2 2 2 ,           n   i s   e v e n ,
(6)
m i n cos θ 1 cos θ 1 - θ 2 = - sin θ 2 2 2 ,           n   i s   o d d .

3.1. Analysis of maximum positive control strength for outer rod

(7)
m a x K u 2 = 1 2 ( k 1 k 4 - 2 k 2 k 3 ) c o s θ ^ 2 + 1 2 k 1 k 4 ,
(8)
d [ m a x ( K u 2 ) ] d θ ^ 2 = 0         2 k 2 k 3 - k 1 k 4 2 s i n θ ^ 2 = 0 .

In Eq. (7), the Ku2 is the relative maximum, when the outer rod angle is θ^2.

Considering that Ku2 is a continuous derivable function, the derivative is zero at the relative maximum value of Ku2. Then, θ^2 can be obtained in Eq. (8).

Substitute 2k2k3-k1k4=0 into Eq. (7): maxKu2=12k1k4=k2k3.

For any given θ2, if the inner rod angle θ1=2nπ+θ22, Ku2 achieves the highest controllability.

Substitute 2k2k3-k1k40 into Eq. (8): θ^2=nπ, nZ and:

(9)
max K u 2 = k 1 k 4 - k 2 k 3 ,         n   i s   e v e n ,
(10)
max K u 2 = k 2 k 3 ,           n   i s   o d d .

When 2k2k3-k1k4>0, Eq. (10) is the global maximum. Then, θ2=(2n+1)π, and θ1=(n+12)π(nZ). The spatial position of inner and outer rods are shown in Fig. 2.

For the same reason, Eq. (9) is the global maximum when 2k2k3-k1k4<0. At this time, the outer rod angle θ2=2nπ and the inner rod angle θ1=nπ shown in Fig. 3.

Fig. 2. The angle of both rods on max Ku2 when 2k2k3-k1k4>0

The angle of both rods on max Ku2  when 2k2k3-k1k4>0

Fig. 3. The angle of both rods on max Ku2 when 2k2k3-k1k4<0

 The angle of both rods on max Ku2  when 2k2k3-k1k4<0

3.2. Analysis of maximum negative control strength for outer rod

(11)
min K u 2 = - k 1 k 4 sin θ ˇ 2 2 2 - k 2 k 3 cos θ ˇ 2 = - k 1 k 4 2 - 2 k 2 k 3 - k 1 k 4 2 cos θ ˇ 2 .

For the same reason, the minimum value min(Ku2) can be obtained in Eq. (11):

(12)
min K u 2 = - k 1 k 4 / 2 = - k 2 k 3 ,             2 k 2 k 3 - k 1 k 4 = 0 ,
(13)
min K u 2 = - k 2 k 3 ,               2 k 2 k 3 - k 1 k 4 > 0 ,
(14)
min K u 2 = k 2 k 3 - k 1 k 4 ,           2 k 2 k 3 - k 1 k 4 < 0 .

In Eq. (12), for any given θ2, the outer rod obtains the reverse maximum controllability when θ1=nπ+θ22. In Eq. (13), θ2=2nπ, θ1=2n+1π+θ22=nπ+π2(nZ) shown in Fig. 4. In Eq. (14), θ2=(2n+1)π, the inner rod angle θ1=2n+1π+θ22=nπ(nZ) shown in Fig. 5.

Fig. 4. The angle of both rods on min Ku2 when 2k2k3-k1k4>0

 The angle of both rods on min Ku2  when 2k2k3-k1k4>0

Fig. 5. The angle of both rods on min Ku2 when 2k2k3-k1k4<0

 The angle of both rods on min Ku2  when 2k2k3-k1k4<0

3.3. Quantification analysis of dynamics parameters and control strength of outer rod

(15)
K u 2 = 2 m 1 cos θ 1 cos θ 1 - θ 2 - 2 3 cos θ 2 + 4 m 2 + m b cos θ 1 cos θ 1 - θ 2 - cos θ 2 16 9 m 1 + 4 m 2 1 3 + sin θ 1 - θ 2 2 + 16 3 m b *   L 0 l 2 .

It can be inferred from the Eq. (15) that l1 has no relationship with the control strength of outer rod Ku2. Besides that, simulations of dynamics parameters including rotating arm length L0, the inner rod mass m1, the outer rod length and mass l2, m2 are shown in Section 4.

The relationship of inner rod mass m1 and control strength of outer rod can be simplified as:

(16)
K u 2 = a m 1 + b + c ,
c = L 0 l 2 2 cos θ 1 cos θ 1 - θ 2 - 4 3 cos θ 2 ,         b = 9 4 m 2 1 3 + sin θ 1 - θ 2 2 + 3 m b ,
a = 16 9 4 m 2 1 3 + sin θ 1 - θ 2 2 + 16 3 m b 2 cos θ 1 cos θ 1 - θ 2 - 4 3 cos θ 2
            + 16 9 4 m 2 cos θ 1 cos θ 1 - θ 2 + m b cos θ 1 .

It is clear that b>0. When a>0, c>0, Ku2c,ab+c, the Ku2 decrease with m1. When a>0, c<0, Ku2ϵc,2b+c, Ku2 increases with m1, if ab+c>0, Ku2 decreases before m1 passing zero point. When a<0, c<0, Ku2ϵ ab+c, c, Ku2 decreases with m1. When a<0, c>0, Ku2 increases with m1, if ab+c<0, Ku2 decreases before m1 passing zero point.

Also, the relationship between m1 and Ku2 can be simplified as:

(17)
K u 2 = a m 2 + b + c .

The analyzing process is similar to m1. With different dynamics parameters, three situations such as increasing, decreasing, decreasing first and then increasing exist for the control strength of outer rod. The simulation under different spatial position is given in Section 4.

4. The simulation verification of control strength for outer rod

In order to verify the analysis process and conclusion mentioned above, the equivalent dynamics parameters [11] were put into Eq. (3). Then, calculate the Ku2 within the range of pendulum angles θ1,θ2-2π,  2π. Finally, the 3D surface graphs were drawn as follow.

Fig. 6 and Fig. 7 show the relationship between Ku2 and rod angles from different perspectives. In these figures, the red area indicates the part in which Ku2 is greater (the deep red shows the maximum of Ku2). And the blue area is the part which Ku2 is smaller. Also, when Ku2 reaches the maximum, the outer rod angles could be ±π, and the inner rod angles could be ±π2, ±3π2. And when Ku2 reaches the minimum, the outer rod angles could be 0, ±2π, and the inner rod angles could be ±π2, ±3π2. Finally, the rod angles with the maximum and the minimum of Ku2 from Fig. 7 fit the rod angles derived from Eq. (10) and Eq. (13) separately which verified the correctness of analysis about the extremum of Ku2 mentioned above.

The simulation result of control strength of outer rod and dynamics parameters including L0, l2, m1, and m2, were separately shown in Fig. 8, Fig. 9, Fig. 10 and Fig. 11.

Fig. 6. The perspective view between Ku2 and θ1, θ2

The perspective view  between Ku2 and θ1, θ2

Fig. 7. The left and right view between Ku2 and θ1, θ2

The left and right view between Ku2 and θ1, θ2

Fig. 8. The relationship between L0 and Ku2

 The relationship between L0 and Ku2

a) Inverted point

 The relationship between L0 and Ku2

b) Hanging point

 The relationship between L0 and Ku2

c) Horizontal point

Fig. 9. The relationship between l2 and Ku2

The relationship between l2 and Ku2

a) Inverted point

The relationship between l2 and Ku2

b) Hanging point

The relationship between l2 and Ku2

c) Horizontal point

Fig. 10. The relationship between the mass of inner rod m1 and Ku2

 The relationship between the mass of inner rod m1 and Ku2

a) Down point, Ku2 increases with m1

 The relationship between the mass of inner rod m1 and Ku2

b) Spatial position of rods (80°, 110°), Ku2 decreases with m1

It can be inferred from those figures that the dynamics parameters are positively or negatively correlated with Ku2. For the parameters L0, l2, the influence for Ku2 is much greater than m1, m2. And the Ku2 hardly changes with m1, m2. Hence, adjusting l2 could be the main method to satisfy Ku2. However, in the actual system, the improvement of the control action intensity does not mean that the control difficulty is low. And it is also necessary to comprehensively consider that the dynamic and static characteristics such as the servo precision and response speed of the drive mechanism (motor, controller, transmission mechanism).

Fig. 11. The relationship between the mass of inner rod m2 and Ku2

 The relationship between the mass of inner rod m2 and Ku2

a) Down point, Ku2 decreases with m2

 The relationship between the mass of inner rod m2 and Ku2

b) Spatial position of bars (80°,110°), Ku2 increases with m2

5. Conclusions

In this article, the control strength quantification analysis of the outer pendulum rod for double inverted pendulum was mainly discussed from two perspectives. One is the qualitative and quantitative relationship between the control strength of the outer rod and the spatial position of the pendulum rod. And the other is the quantification relationship between the control strength of the outer rod and the dynamics parameters (including the length of pendulum rod L0, l1, l2, and the mass of pendulum rod m1, m2). Finally, the correctness of the relationship was illustrated by simulation results.

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