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Abstract

The inverted pendulum robot is a classical problem in controls. The inherit instabilities in the setup make it a natural target for a control system. Inverted pendulum robot is suitable to use for investigation and verification of various control methods for dynamic systems. Maintaining an equilibrium position of the pendulum pointing up is a challenge as this equilibrium position is unstable. As the inverted pendulum robot system is nonlinear it is well-suited to be controlled by fuzzy logic. In this paper, Lagrange method has been applied to develop the mathematical model of the system. The objective of the simulation to be shown using the fuzzy control method can stabilize the nonlinear system of inverted pendulum robot.

Keywords

Inverted Pendulum Robot Disturbances Servo Control Fuzzy Control

1. Introduction

In recent years, the researchers wish to simulate human stance on the machine. In this paper, the model is constructed based on purely inverted pendulum dynamics and on a movable supportive base. This work was based on the assumption that the act of maintaining an erect posture could be viewed. However, the problems often are a complicated nonlinear system (Figure 1) [1].

Inverted pendulum robot is the abstract model of many control problems which the gravity center is upper and the fulcrum is lower and it is unstable object. In the control process it can reflect a number of key issues effectively such as the system stability, nonlinear problem, controllability, robustness and so on. As a controlled object it is a higher order, nonlinear, multi-variable, strong coupling and unstable rapid control system. Walking robot joint control, the vertical degree of control in rocket launch, satellite attitude control, those all related to the stability of the control problem that of upside-down objects. Therefore, the inverted pendulum control strategies can be applied to aerospace, military, robotics, industry and others areas, to solve the balance problems.

Recently, some authors proposed several control methods to control the nonlinear systems by using fuzzy system models [1 –9]. The affine fuzzy system means the fuzzy system of which consequent part is affined and which has a constant bias term. It is well known that such models can describe or approximate a wide class of nonlinear systems. Hence, it is important to study their stability and the design of stabilizing controllers. Besides, robust stability also has been considered in literatures which have presented robust stability analysis and methods for designing robust fuzzy controllers to stabilize a class of uncertain fuzzy systems. In general, stability analysis and synthesis can be extended to the time-delay systems. Time delays often appear in industrial systems and information networks. Thus, it is also important to analyze time-delay effects for the affine fuzzy systems.

Figure 1.

Inverted pendulum robot system to simulate human stance

Nowadays, fuzzy logic is one of the most used methods of computational intelligence robot [10 –14] and with the best future. Oscar Castillo et al. [15] describe the application of Ant Colony Optimization (ACO) and Particle Swarm Optimization (PSO) on the optimization of the membership functions parameters of a fuzzy logic controller (FLC) in order to find the optimal intelligent controller for an autonomous wheeled mobile robot. The results obtained by the simulations performed are statistically compared among them and the previous work results obtained with GAs in order to find which is the best optimization technique for this particular robotics problem. And [16] presented a hybrid architecture, which combines Type-1 or Type-2 fuzzy logic system (FLS) and genetic algorithms (GAs) for the optimization of the membership function (MF) parameters of FLS, in order to solve to the output regulation problem of a servomechanism with nonlinear backlash. In this approach, the fuzzy rule base is predesigned by experts of this problem. The proposed method is far from trivial because of nonminimum phase properties of the system.

Intelligent control of robotic systems is a difficult problem because the dynamics of these systems is highly nonlinear [17–18]. Oscar Castillo et al. [19] develop a tracking controller for the dynamic model of unicycle mobile robot by integrating a kinematic controller and a torque controller based on fuzzy logic theory. Both the kinematics and dynamics models are used currently to design, simulate, and control robot manipulators [20 –25]. The kinematics model is a prerequisite for the dynamics model and fundamental for practical aspects like motion planning, manufacturing cell graphical simulation and fuzzy servo control. For example, the inverted pendulum robot is a classical problem in controls [26–27]. The inherit instabilities in the setup make it a natural target for a control system. Inverted pendulum robot is suitable to use for investigation and verification of various control methods for dynamic systems. Maintaining an equilibrium position of the pendulum pointing up is a challenge as this equilibrium position is unstable. As the inverted pendulum robot system is nonlinear it is well-suited to be controlled by fuzzy logic. In this paper, Lagrange method has been applied to develop the mathematical model of the system. The objective of the simulation to be shown using the fuzzy control method can stabilize the nonlinear system of inverted pendulum robot.

The remainder of this paper is organized as follows: In Section 2, the expression of problems is presented. In Section 3, the fuzzy servo control method is proposed. In Section 4, linearization and stability analysis and the fuzzy rule is shown. Section 5 is the simulation results. Section 6 is the conclusion.

2. The expression of problems

Design of the robot naturally centered on the basic idea of an inverted pendulum. There would be a weight at the top of a long shaft and there would be some sort of mechanism attached to the bottom of this shaft that would be used to balance the weight at the top. Unlike most inverted pendulum experiments that consist of a cart on a track that limits mobility, this project was constructed much like a Segway to enable the robot to have a greater range of mobility. Kinematic and dynamic models for wheeled inverted pendulum robot as shown in Figure 2.

Figure 2.

Kinematic and dynamic models for wheeled inverted pendulum robot

Lagrange method is adopt to establish the mathematical model because the number of equations equal to the degree of freedom system and modeling process can be simplified. The dynamics of inverted pendulum robot can be described in general form as follows [3].

\frac{d}{d t} (\frac{\partial ℓ}{\partial {\dot{q}}_{i}}) - \frac{\partial ℓ}{\partial q_{i}} + \frac{\partial F}{\partial {\dot{q}}_{i}} = Q_{i}

(1)

H (q) \ddot{q} + C (q, \dot{q}) \dot{q} + G (q) = Q u

(2)

where $H (q)$ is an $n \times n$ inertia matrix dependent on position vector $q, \dot{q}$ and $\ddot{q}$ are the velocity and acceleration vectors, respectively, $C (q, \dot{q}) \dot{q}$ is the coriolis and centripetal force vector, and $G (q)$ is the gravity force vector.

3. Fuzzy servo control method

Fuzzy method has been used as alternative method to develop control rule for complex system. The method has been successfully implemented to the real-world application like engine control system of subway train. The idea behind of control method in this paper is to divide the operating region of nonlinear system into small area, and treated as a collection of local linear systems. After that, in order to develop the control law, fuzzy method has been apply to each local linear system and combines it as new control law as shown in Figure 3 ( $D_{i}$ : small domain Ω: all domain).

Figure 3.

Driving domain

Fuzzy method has been apply to each local linear system and combines it and can be define as

R_{i} \begin{matrix} ​ \end{matrix} IF \begin{matrix} ​ \end{matrix} z_{n} \in D_{i} \begin{matrix} ​ \end{matrix} THEN \begin{matrix} ​ \end{matrix} S \begin{matrix} ​ \end{matrix} is \begin{matrix} ​ \end{matrix} S_{i}

where $D_{i}$ is the local driving area, S is the nonlinear system, $S_{i}$ is the local linear system and the fuzzy rules are $i = - N, \dots, N$ . $z_{w} \in R^{n}$ is a vector of nonlinear elements, and $z_{w} = C_{w} {[\begin{matrix} x & u \end{matrix}]}^{T}$ , which are included in controlled system in the equation. $C_{w} \in R^{n_{w} \times (n + m)}$ is a matrix of which its elements are 1 or 0.

Example

Let $x^{T} = (\begin{matrix} x_{1} & x_{2} & x_{3} \end{matrix}),$ $u^{T} = (\begin{matrix} u_{1} & u_{2} \end{matrix})$ , where $x_{1}$ , $x_{2}$ , $u_{1}$ are nonlinear, and $C_{w}$ can be defined as

z_{w} = [\begin{matrix} x_{1} \\ x_{2} \\ u_{1} \end{matrix}], \begin{matrix} ​ \end{matrix} C_{w} = [\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{matrix}]

Let $ω_{i}$ define as Gaussian type fuzzy membership function as follows.

\begin{array}{c} ω_{i} = e^{- {(z_{w} - z_{w i})}^{T}} Q_{w} (z_{w} - z_{w i}) \\ Q_{w} = Q_{w} ​^{T} > 0 \end{array}

$ρ_{i}$ can be define as

ρ_{i} = w_{i} / \sum_{i = - N}^{+ N} ω_{i} .

The fuzzy servo system can be rewrite as.

\dot{z} = \sum_{i = - N}^{+ N} ρ_{i} (A_{z i} x + B_{z i} u + d_{z i})

(3)

In order to develop new control rule, we calculate each local linear system control rule by using

u = - K_{i} z

(4)

$K_{i}$ is the feedback coefficient matrix of the input. This matrix calculated using Hikita method [28]. Let $f_{i j} = - {(λ_{j} I - A_{z i})}^{- 1} B_{z i} g_{j}$ , $j = 1, 2, 3, …, n + l)$ , $g_{j} \in R^{m}$ , $f_{i j} \in R^{n + l}$ . $λ_{j}$ be the eigenvalues and $f_{i j}$ be eigenvector of the system. The feedback co-efficient matrix can be represented as

\begin{array}{c} K_{i} = [\begin{matrix} g_{i 1} & \dots & g_{i (n + l)} \end{matrix}] {[\begin{matrix} f_{i 1} & \dots & f_{i (n + l)} \end{matrix}]}^{- 1} \\ K_{i} = [\begin{matrix} k_{i 1} & \dots & k_{i (n + l)} \end{matrix}] \end{array}

Applied fuzzy method to each local linear system and combines it as new control law.

The control input u is

u = - \sum_{i = - N}^{+ N} ρ_{i} K_{i} z

(5)

The input u in (5) is applied to system in (1) and (2).

4. Linearization and stability analysis and the fuzzy rule

The inverted pendulum robot used in this paper is shown in Figure 2. It consists of cart and a pendulum. The cart is free to move the horizontal direction when the force F applies to it. We assume that the mass of the pendulum and cart are homogenously distributed and concentrated in their center of the gravity and the friction of the cart is proportional only to the cart velocity and friction generating by the pivot axis is proportional to the angular velocity of the pendulum. The parameters used for simulation are shown in Table 1. The mathematical model of inverted pendulum system can be described as follows.

(M + m) \ddot{x} + m l \cos θ \ddot{θ} + D \dot{x} - m l \sin θ {\dot{θ}}^{2} = F

(6)

m l \cos θ \ddot{x} + \frac{4}{3} m l^{2} \ddot{θ} - m g l \sin θ + C \dot{θ} = 0

(7)

With output as $y = x_{1}$ . Define an error of the system as $\dot{v} = y - r$ . Rearranged the equation in term of $x_{1} = x$ , $x_{2} = \dot{x}$ , $x_{3} = θ$ , $x_{4} = \dot{θ}$ and $F = u$ . The new equation for inverted pendulum are shown in below.

\begin{array}{l} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = - 3 \cos x_{3} (m g l \sin x_{3} - C x_{4}) \\ \begin{matrix} ​ & ​ \end{matrix} - \frac{4 l (u - D x_{2} + m l \sin x_{3} x_{4}^{2})}{4 l (M + m) + 3 m l \cos x_{3}^{2}} \\ {\dot{x}}_{3} = x_{4} \\ {\dot{x}}_{4} = \frac{- 3 ((M + m) (m g l \sin x_{3} - C x_{4})}{m l^{2} (- 4 (M + m) + 3 m \cos x_{3}^{2})} \\ \begin{matrix} ​ & ​ \end{matrix} + \frac{m l \cos x_{3} (u - D x_{2} + m l \sin x_{3} x_{4}^{2})}{m l^{2} (- 4 (M + m) + 3 m \cos x_{3}^{2})} \end{array}

Rom the equation, we know that the nonlinear coefficient for the system is $x_{3}$ and $x_{4}$ . The linearization has been done to mathematical model of the inverted pendulum robot system using Taylor expansion as shown in (5) and (6) around of the operating points for nonlinear variables $x_{3}$ and $x_{4}$ .

Table 1.

Parameters of inverted pendulum system

Parameter	Description	Value
Mass of the cart	M	0.18kg
Mass of the pendulum	m	0.10kg
Gravitational constant	g	9.8m/s²
Length of pendulum's center of gravity to its rotation axis	l	0.20m
Coefficient of friction between cart and system	D	5.0kg/s
Coefficient of friction between pendulum and rotation axis	C	0.02kgm/s

where $δ x_{3} = x_{3} - x_{3 i}$ and $δ x_{4} = x_{4} - x_{4 i}$ . The linearization has been done and the linear servo system can be represent as.

\dot{z} = A_{z i} z + B_{z i} u + d_{z i}

(8)

where

\begin{array}{l} A_{z i} = [\begin{matrix} 0 & 1 & 0 & 0 & 0 \\ 0 & a_{22} & a_{23} & a_{24} & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & a_{42} & a_{43} & a_{44} & 0 \\ 1 & 0 & 0 & 0 & 0 \end{matrix}] \\ B_{z i} = {[\begin{matrix} 0 & b_{21} & 0 & b_{41} & 0 \end{matrix}]}^{T} \\ z = {[\begin{matrix} x_{1} & x_{2} & x_{3} & x_{4} & v \end{matrix}]}^{T} \\ d_{z i} = {[\begin{matrix} 0 & d_{21} & 0 & d_{41} & 0 \end{matrix}]}^{T} \end{array}

$d_{z i}$ is disturbance vector. Each coefficient can be described as follows.

\begin{array}{l} a_{22} = (- 2 D (5 m + 8 m - 3 m \cos (2 x_{3 i})) \\ \begin{matrix} ​ \end{matrix} + 6 m x_{3 i} \sin (2 x_{3 i})) / {(4 (M + m) - 3 m \cos^{2} (x_{3 i}))}^{2} \\ a_{23} = (8 M - m + 3 m \cos (2 x_{3 i})) (4 m l^{2} x_{4 i}^{2} \cos (x_{3 i}) \\ \begin{matrix} ​ \end{matrix} \begin{matrix} + 3 m g l (3 m - (8 M + 5 m) \cos (2 x_{3 i})) \end{matrix} \\ \begin{matrix} ​ \end{matrix} \begin{matrix} - \end{matrix} 3 C x_{4 i} (8 M + 11 m + 3 m \cos (2 x_{3 i})) \sin (x_{3 i}) \\ \begin{matrix} ​ \end{matrix} / {(2 l (M + m) - 3 m \cos^{2} (x_{3 i}))}^{2} \\ a_{43} = ((3 m g (M + m) \cos (x_{3 i})) (8 M - m \\ \begin{matrix} ​ \end{matrix} + 3 m \cos (2 x_{3 i})) + 3 x_{4 i} ​^{2} m l^{2} (3 m \\ \begin{matrix} ​ \end{matrix} - 6 ((8 M + 5 m) \cos (2 x_{3 i})) \\ \begin{matrix} ​ \end{matrix} \cdot ((M + m) \sin (2 x_{3 i}))) \\ \begin{matrix} ​ \end{matrix} / {(2 l (4 l (M + m) - 3 m l \cos^{2} (x_{3 i})))}^{2} \\ a_{44} = 3 (C (M + m) + m^{2} l^{2} x_{4 i} \sin (2 x_{3 i})) \\ \begin{matrix} ​ \end{matrix} / m l^{2} (- 4 (M + m) + 3 m \cos^{2} (x_{3 i})) \\ b_{21} = 2 (8 M + 5 m - 3 m \cos (2 x_{3 i}) \\ \begin{matrix} ​ \end{matrix} + 6 m x_{3 i} \sin (2 x_{3 i})) \\ \begin{matrix} ​ \end{matrix} / {(4 (M + m) - 3 m \cos^{2} (x_{3 i}))}^{2} \\ b_{41} = ((- 6 \cos (x_{3 i})) (8 M + 5 m - 3 m \cos (2 x_{3 i}) \\ \begin{matrix} ​ \end{matrix} + 2 (8 M + 11 m + 3 m \cos (2 x_{3 i})) \\ \begin{matrix} ​ \end{matrix} / {(4 l (4 (M + m) - 3 m \cos^{2} (x_{3 i})))}^{2} \end{array}

Base on the Lyapunov law: the nonlinear control system can be analyzed by the linearization system. The controllability matrix of the control system

S = [\begin{matrix} B & A B & A^{2} B & \dots & A^{n - 1} B \end{matrix}]

(9)

is full-rank. Rank (S) = 4. So the inverted pendulum robot system is controllable.

Fuzzy rule:

The state equation of inverted pendulum robot can be shown as.

\dot{x} = f (x, u) + d

(10)

y = g (x) + d_{0}

(11)

Set $z_{w} = C_{w} {[\begin{matrix} x & u \end{matrix}]}^{T} \in R^{n_{w}}$ , so $z_{w} \in D_{i}$ , the fuzzy rule is:

IF $z_{v} \in D_{i}$ THEN S is $S_{i}$ ( $- N \leq i \leq N$ )

And in this part, Sand $S_{i}$ are follows.

\begin{array}{l} S : \dot{x} = f (x, u) + d \\ \begin{matrix} ​ & y \end{matrix} = g (x) + d_{0} \end{array}

(12)

\begin{array}{l} S_{i} : \dot{x} = A_{i} x + B_{i} u + d_{x i} \\ \begin{matrix} ​ & y \end{matrix} = C_{i} x + d_{y i} \end{array}

(13)

5. Simulation results

Comprehensive analysis part B and C, feedback coefficient $K_{i}$ can be obtained from every small domain $D_{i}$ with pole assignment. Fuzzy input u is

u = - \frac{\sum_{i = - N}^{N} ω_{i} K_{i} x}{\sum_{i = - N}^{N} ω_{i}}

(14)

where $ω_{i} = e^{- q_{i} {(x - x_{i})}^{2}}$ , the initial angle of θ is $25^{\circ}$ and initial condition of simulation $x (0) = {[\begin{matrix} 0 & 0 & 0.44 & 0 \end{matrix}]}^{T}$ . The system simulate with $λ_{1} = λ_{2} = λ_{3} = λ_{4} = - 0.80$ , and $q = 5$ . The results of simulation are shown in Figure 4.

As the result of simulation, the system is stable with this fuzzy input, but it is stable which initial angle of θ is between $\pm 25^{\circ}$ . It is necessary for enlarging controllable angle to change fuzzy input. So set

ω_{i} = e^{- q_{1} {(x_{1} - x_{1 i})}^{2} - q (x_{2} - x_{2 i}) 2^{2}}

(15)

The initial angle of θ is $50^{\circ}$ and initial condition of simulation $x (0) = {[\begin{matrix} 0 & 0 & 0.87 & 0 \end{matrix}]}^{T}$ . The system simulate 3 times which $λ_{1} = λ_{2} = λ_{3} = λ_{4} = - 0.80$ were −0.80, −0.78 and −0.82 separately and $q_{1} = 5, q_{2} = 1$ . The results of simulation are shown in Figure 5.

Figure 4.

Result for simulation of θ

The simulation results show that with different poles, the system response different characteristics. With smaller pole, the overshoot of pendulum is smaller and settling time of pendulum is shorter. However, overshoot of cart is larger. Settling time of cart has the approximate same performance.

Figure 5.

Result for simulation of x from 0s to 15s

6. Conclusion

In this paper, mathematical model of inverted pendulum is established based on Lagrange method, and from the simulation results, the fuzzy method [29–30] can make nonlinear system stable. This fuzzy method adapt to most of nonlinear system. For different needs, the system can provide different response characteristics. In the future, servo system will be established and simulated to improve the system performance by changing the parameters.

Footnotes

7. Acknowledgments

This work was financially supported by the Innovation Program of Shanghai Municipal Education Commission (12YZ148), the Project-sponsored by SRF for ROCS, SEM (2011-1568) and the Scientific Research Foundation of SUES (2012-01). The authors would like to thank the editor and the reviewers for their constructive comments and suggestions which improved the quality of the paper.

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Study of Inverted Pendulum Robot Using Fuzzy Servo Control Method

Abstract

Keywords

1. Introduction

2. The expression of problems

3. Fuzzy servo control method

4. Linearization and stability analysis and the fuzzy rule

5. Simulation results

6. Conclusion

Footnotes

7. Acknowledgments

References