发布于:2021-06-11 02:44:37

The design of a two-wheel balancing vehicle control system
Sensor Sensor

In this paper, the author analyzes the movement of the two-wheel balancing vehicle on the plane. Predigesting the system, and trying to setup a new mathematical model. And then do some research on optimal control system analyze. Finally, the simulation result shows that the scheme is feasible. KEY WORDS two-wheel vehicle、inverted pendulum、 optimal control、Riccati equation 1. Introduction The movement control about two-wheel balancing vehicle has the same principle with the rocket attitude control [1]. With the fast development of astronautics, the requirement about rocket attitude control is higher and higher. But the cost for this astronautic project is too high for testing. So the engineers attempt to find others ways to validate their design. Based on this concept the inverted pendulum comes into our vision. We can do kinds of experiment on the inverted pendulum platform to validate our design and to make some more improvement. In this paper, we analyze the system structure, and setup a mathematical model with some hypothesis, and then we do some research based on the model. Finally, we make a simulation to validate the system design. 2. Analyze of the system and control system design As we know that this kind of two-wheel balancing vehicle stands on the plane with two wheels, so that it cannot keep balance by itself. The external force is needed, if we need it to keep balance. This kind of force makes the vehicle in the condition of dynamic balance. While the vehicle is moving on the plane, the sensor on the body is working for detecting the horizontal acceleration, and giving some feedback to correct the movement in real time to keep the vehicle work well. Based on the concept above, the workflow we designed is shown in Fig.1

Data Processing

Data Processing

Computing, Policy analyze. ….

Date modulation

Date modulation

Terminal execute

Terminal execute

Fig.1 Work flow about the control system

For the vehicle is just working on the plane, we can design the vehicle is shown in Fig 2:

Fig. 2 Structure of the vehicle

In order to analyze, we treat the whole vehicle as a whole rigid-body, so we can predigest the whole vehicle as follow in Fig 3:

3. 4.

The two parts are both rigid-body; The one-class inverted pendulum can only move on a vertical plane, and the angle is less than 5° .

We carried out the theoretical analyze of this system based on the simplified model above (Fig 4) .The result is shown in the Fig 5:

Fig. 3 Predigest of the vehicle

To predigest future more, we can separate the vehicle into two parts: a car moving on the plane and a oneclass inverted pendulum. The inverted pendulum is putted on the above of the vehicle with link connection. The sketch map is shown in Fig 4.

Fig. 5 Theoretical analyze

Based on the Newton –Euler equation, we can get the horizontal and vertical loading process below [2]:
2 ? ?ml cos ? x ? ml ? ? gml sin ? ? 0 ? 2 ? ?? M ? m ? x ? lm cos ?? ? ml? sin ? ? f

In the functions above, the symbol stands for different meanings shown below: m: the mass of the inverted pendulum; M: the mass of the vehicle; l: the length of the inverted pendulum;

? : The move angle of the inverted pendulum

Fig. 4 Future predigest of the vehicle

As the vehicle is just working on the plane, we get the horizontal equation:

As we separate the vehicle into two parts, in order to analyze .We make some hypothesis: 1. 2. Ignore the air dynamic; Ignore all the other friction force ,the only friction force is between the wheel and the plane;

? M ? m? x ? lm cos ?? ? ml? 2 sin ? ?


This is a nonlinear function [3], so we should do some linear transformation: 1. Ignore the air dynamic;

2. 3. 4.

Ignore all the other friction force ,the only friction force is between the wheel and the plane; The two parts are both rigid-body; The one-class inverted pendulum can only move on a vertical plane, and the angle is less than 5° .

In the function above, the value P can be found in the Riccati equation:

P ? PA ? AT P ? PBR?1 BT P ? Q ? 0 .

As the angle is very small, generally can set that: sin ? ? ? ;cos ? ? 1 ;

? ?5

Future more, the Riccati equation above can be simplified as PA ? A

,so we

P ? PBR?1BT P ? Q ? 0 .

Based on the entire condition above (the hypothesis and the horizontal equation), we can get the state equation below:
? x? ? x? ? x? ? ? ? ? ? A ? x ? ? Bu ?? ? ?? ? ? ? ? ? ? ? ? ?? ? ? x ? ?1 y?? ??? ?? ? ? 0 ?0 ? ?0 ? A?? ?0 ? ?0 ? 0 0 ? x? ? 0? ? ? x ? ? ?0? u ? ? 0? ? ?? ? ? 0 ? ? ? ?? ? 0

Based on all the conclusion above, we get the best feedback gain is

K ? Q?1BT P .
3. Simulation In the whole design process, we ignore some facts. Based on the theory about optimal control of linear quadratic, we can get the simulation result. The details about the vehicle are below [4,5] : Terms Value 2kg 0.3kg Description mass of the car body mass of the one-class inverted pendulum length of the inverted pendulum the angle coefficient of sliding friction

0 1 1

?( I ? ml ) ? I ( M ? m) ? Mml 2 0 ? ml ? I ( M ? m) ? Mml 2

m gl I ( M ? m) ? Mml 2 0 mgl ( M ? m) I ( M ? m) ? Mml 2


0? ? 0? ? ? 1? ? 0? ?

M m

l φ μ

30cm 0°- 5° 0.73

0 ? ? ? ? 2 I ? ml ? ? ? I ( M ? m) ? Mml 2 ? B?? ? 0 ? ? ? ? ml ? 2 ? ? I ( M ? m) ? Mml ?

Table 1 Details for simulation

So we get the performance target:


1 tn T [ x Qx ? uT Ru ]dt 2 ?t1 1 x(t0 )T P(t0 ) x(t0 ) 2 ,
Fig. 6 Simulation result

And the minimum value of the target will be

J* ?

5. Reference: From the simulation result, we can find that the scheme is feasible. 4. Conclusion: From the simulation result above, we can infer that the vehicle working on the plane can be treated as a simply system. Because some hypothesis we make depends on the plane condition, it will work well on the plane. Future more, we can introduce some more accurate sensors; use some complex fitting algorithms, so that this system can work well in other condition, such as ramp or complex condition. We can make more improvement about the control law。 [1] An Introduction to the Mathematics and Methods of Astrodynamics , Revised Edition , AIAA Education Series 1999, Richard H. Battin,95-100 [2] Introduction to Robotics: Mechanics and Control (3rd edition) , China Machine Press , John J.Craig, 136-137 [3] Applied Nonlinear Control China Machine Press, Jean-Jacques E. Slotine , Weiping Li , 138-178 [4] Simulink-Dynamic System Simulation for MATLAB the MATHWORKS INC [5] STABILITY CONTROL OF INVERTED PENDULUM SYSTEM, Jian CHEN, Changchun University of Science and Technology, 2007