TRAJECTORY TRACKING CONTROL FOR FOUR-WHEELED OMNI ROBOT USING MULTIPLE SLIDING SURFACE CONTROL ALGORITHM

ABSTRACT

This paper discusses the development of a nonlinear controller design methodology and its application to an Omni robot control problem. The method is called the "Multiple Sliding Surface" method and is closely related to sliding mode control, input/output linearization and integrator backstepping. The method was developed for a class of systems, typical of robot control systems, where the uncertainties are "mismatched" and where many of the equations contain sparse, experimentally obtained maps. The error bounds on these maps are often unknown and their sparseness makes them difficult to differentiate. The developed method does not require any derivatives and has guaranteed semi-global stability.

Results of paper presents the synthesis of a trajectory tracking control system for Omni Robot by building the Multiple Sliding Surface Control (MSSC) algorithm including: modeling the four-wheel Omni Robot, building the MSSC algorithm for the Ommi robot, and simulating it through Simulink Matlab.

АННОТАЦИЯ

В этой статье обсуждается разработка методологии проектирования нелинейного контроллера и ее применение к проблеме управления роботом Omni. Метод называется методом «Multiple Sliding Surface» и тесно связан с управлением скользящим режимом, линеаризацией ввода/вывода и интегратором backstepping. Метод был разработан для класса систем, типичных для систем управления роботами, где неопределенности «несоответствуют» и где многие уравнения содержат разреженные, экспериментально полученные карты. Границы погрешности на этих картах часто неизвестны, а их разреженность затрудняет их дифференциацию. Разработанный метод не требует никаких производных и имеет гарантированную полуглобальную устойчивость.

Результаты статьи представляют синтез системы управления отслеживанием траектории для робота Omni путем построения алгоритма управления множественной скользящей поверхностью (MSSC), включая: моделирование четырехколесного робота Omni, построение алгоритма MSSC для робота Ommi и его моделирование с помощью Simulink Matlab.

Keywords: MSSC, Omni Robot; Simulnk Matlab, Control, Trajectory, Simulating, Algorithm.

Ключевые слова: MSSC, Omni Robot; Simulink Matlab, Управление, Траектория, Моделирование, Алгоритм.

Introduction

The problem here is to control the robot position to follow the preset trajectory. The joint variable vector is usually chosen as q = [x y θ]^T  where  are the coordinates of the robot in the system. global coordinates,  is the robot's direction.

Depending on the dynamic equation system of the robot is a constrained system (nonholonomic) or unconstrained (holonomic) we also divide autonomous robots into constrained robots and unconstrained robots [4].

An example of an unconstrained robot is the Omni Robot [1–4; 6; 7]. The joint variables of the robot Now it will be freely controlled, the robot will move completely flexibly. In conventional Mobile Robots, the wheels are placed along the axis of the robot. But in Omni robot the wheels are placed on the perimeter of the robot.

Omni Robot Modeling

Figure 1. Omni Robot 4 wheels

In figure 1 we see that the Omni robot has 4 wheels offset  , the distance from the wheel to the center of the robot is D, Oxy is the global coordinate axis, v is the linear velocity of the robot, vn is the normal velocity of the robot and w is the brain velocity of the robot;

 (with )

 is the speed of wheel .

q = [x y θ]^T  is the robot's position and orientation vector in the global coordinate system.

Dynamic model

Robot kinematic equation:

Projecting vectors  onto the global coordinate system  we have:

                                       (1)

- is calculated based on the wheel speed as follows:

with constant 

The Omni robot's dynamic equation is based on the Euler-Lagrange formula:

                        (2)

Where:

- is the joint variable vector chosen in the above section.

- is the bounded uncertain noise vector (small and ignored in the calculation )

- is the input signal vector (here we choose m torque applied to each wheel) 

is the matrix of inertia coefficients and moments of inertia; is the matrix of viscosity and radial coefficients; is the gravity matrix; is the input transformation matrix.

; ;

We have the dynamic equation of Omni robot:

                 (3)

Multiple Sliding Surface Control (MSSC) method

MSSC is a control method with steps following the Backstepping design method [5].

- Consider nonlinear system:

                                                        (4)

with  and being a continuous function

First we construct the first sliding surface:  . The derivative of  is based on (4)

                                                (5)

- Next we consider the second sliding surface:

- Where  is called the virtual input designed to drive . Derivative  :

                                                          (6)

- u will be designed so that , i.e.  is chosen as follows:

                                                       (7)

- The positive constant  will be determined later, u is chosen as:

                                                        (8)

- From (5) - (8) we have the following system:

                                                       (9)

- Choose the Lyapunov function . From (9) we can calculate the derivative of :

                            (10)

- Using the inequality we have:

                                                (11)

- Choosing with  we get:

                                                                   (12)

According to Luapunov's theorem, the system is stable.

Applies to Omni robot object

From the kinetic equation (1)

We have:

                                                   (13)

Infer

                                                     (14)

- From the dynamic equation (6), We can also calculate:

                                         (15)

Put: 

Hence:

From (14) and (15) we have the system:

                                                            (16)

We define the state variables as follows:

                                                (17)

From there we have the state model of the Omni Robot object as follows:

                                             (18)

-Consider sliding surface  :

                            (19)

- Derivative S ₁ combined with (18) we have:

Select        (20)

- Consider sliding surface  :

                                                    (21)

- Derivative  combined with (18) we get:  

- Select  

Omni Robot control diagram is shown in Figure 2.

Figure 2. Omni Robot control structure diagram

Simulation results

+ With the orbit set as a circle:

Initial coordinates of the robot : x(0)=0, y(0)=0, (0)=0

The selected controller parameters are:

Initial parameters:

The orbit is a circle:  

The simulation results of the Omni robot trajectory tracking problem using the MSSC algorithm are presented in Figure 3.

a) Omni robot motion trajectory with the orbit set as a circle

b) Omni robot motion trajectory error with the orbit set as a circle

Figure 3. Circular motion trajectory and trajectory error of Omni robot

from the results in figure 3 we can see that when using MSSC controller the quality of adhesion The convergence time is about 3s, the robot follows the trajectory throughout the simulation. When the parameters are set to The commonly defined value is . The trajectory deviation is small and there is no static deviation.

+ With the trajectory set as a straight line:

Initial coordinates of the robot :

The selected controller parameters are:

Initial parameters:

The trajectory is a straight line: 

The simulation results of the Omni robot trajectory tracking problem using the MSSC algorithm are presented in Figure 4.

a) Omni robot motion trajectory with the trajectory set as a line

b) Omni robot motion trajectory error with the trajectory set as a line

Figure 4. Straight motion trajectory and trajectory error of Omni robot

from the results in figure 4 we can see that, when the trajectory is set as a straight line, the MSSC controller resulting in better system adhesion quality, time The convergence time is about 3s, the robot follows the trajectory. During the simulation period, when changing parameter values, static errors and noise are very small.

Conclusion

Thus, the synthesis of the trajectory tracking control system for Omni Robot by building the Multiple Sliding Surface Control (MSSC) algorithm achieved good results, the Robot followed the initial trajectory with small errors. Surveying on different trajectory types, the simulation results showed that the robot achieved the trajectory tracking goal, the transition time was at the allowable level. However, the limitation of this method is the influence of noise, when the complexity of the system is higher, the noise is larger. Therefore, D. Swaroop and C. Gerdes proposed the DSC (Dynamic Surface Control) algorithm to solve this problem by using a first-order filter for each virtual controller synthesized at each step of the MSSC (Backstepping) design process.

References:

1. Ching-Chih Tsai, Li-Bin Jiang, Tai-Yu Wang, Tung-Sheng Wang, Kinematics Control of an Omnidirectional Mobile Robot // Proceedings of 2005 CACS Automatic Control Conference Tainan. – Taiwan, Nov 18-19, 2005.
2. Ehsan Hashemi, Maani Ghaffari Jadidi, Omid Bakhshandeh Babarsad, Trajectory Planning Optimization with Dynamic Modeling of Four Wheeled Omni-Directional Mobile Robots // CIRA. – Korea. – December 15-18, 2009.
3. Martynenko Yu.G., Formal'skij A.M. More often than not // RAN. Teoriya i sistemy upravleniya. – 2007. – № 6. – S. 142–149  [in English].
4.  Siahaan L. Motion control of an omnidirectional mobile robot // TA Baede DCT 2006.084. – Retrived from: https://www.academia.edu/11476769/Motion_control_of_an_omnidirectional_mobile_robot (accessed date: 05.06.2024).
5. Syu D., Mejer A. More often than not. – M.: Mashinostroenie, 1972. [in English].
6. Wang J., Chen J., Ouyang S., Yang Y. Trajectory tracking control based on adaptive neural dynamics for four-wheel drive omnidirectional mobile robots // Engineering Review. – 2014. – Vol. 34. – Iss. 3. – Pp. 235–243.
7. Wang Tai-Yu, Tsai Ching-Chih, Wang Der-An. Dynamic Control of An Omnidirectional Mobile Platform // Journal of Nan Kai. – 2010. – Vol. 7. – No. 1. – Pp. 9–18.