Integral Backstepping Approach for Mobile Robot Control

This paper presents the trajectory tracking problem of a unicycle-type mobile robots. A robust output tracking controller for nonlinear systems in the presence of disturbances is proposed, the approach is based on the combination of integral action and Backstepping technique to compensate for the dynamic disturbances. For desired trajectory, the values of the linear and angular velocities of the robot are assured by the kinematic controller. The control law guarantees stability of the robot by using the lyapunov theorem. The simulation and experimental results are presented to verify the designed trajectory tracking control.


Introduction
In the recent years, the mobile robots have been used to performing tasks that are too dangerous or tedious for humans. They can also be found in: industry, science, education, medical, domestic machines, entertainment and military applications [1].They are used in a range of applications [2,3].
Many research articles have been proposed for the trajectory tracking problem in the literature. Studies and approaches have been developed in this field [4,5]. Most controllers designed in control systems are based on kinematic models of the mobile robots and control theory to achieve and accomplish the designing control laws.
For the path-following controller, in [6] the control algorithm of nonlinear PID-based adaptive control for the wheeled mobile robot (WMR) is developed. In [7], a controller based on distance measures using the sonar sensors is proposed to follow a wall, and to guide a mobile robot along a corridor while avoiding obstacles this algorithm is tested in the real implementation cases.
In the last-cited paper, the designed controllers do not include the dynamic model of mobile robots. However, it is necessary to using the tools from control theory and dynamic systems in order to perform at high speeds and heavy work. Thus, some controllers based on dynamic modeling have been proposed. As an example in [8], A fuzzy Logic Controller is proposed to track a desired trajectory of a unicycle mobile robot using Mamdani model and backstepping technique, the dynamics parameters for tracking control of mobile robot are included using the Euler-Lagrange method (masses, inertias, damping, etc.). Moreover, no experimental results were reported.
In [9], an adaptive fuzzy logic based controller is designed to control mobile robot with unknown dynamics parameters for tracking control of mobile robots, the proposed algorithm is implemented on mobile robot system with PIC 16f877 microcontroller, but the real implementation required high-performance computer architecture based on a multiprocessor system. An adaptive trajectory-tracking controller based on the model dynamic of a unicycle-like mobile is designed in [10], a stability analysis using Lyapunov theory is analyzed to ensure the robot stability, the parameters are updated on-line where the parameters of the system are not known or change ISSN: 1693-6930 the task, this method of trajectory tracking control is tested on a Pioneer 3-DX mobile robot. Fuzzy adaptive trajectory tracking control of nonholonomic wheeled mobile robot is proposed in [11], to compensate for the kinematic and dynamic disturbances relying on the state variables, the proposed controller is tested only with simulation results and no experimental results were presented.
The robust trajectory tracking of mobile robots is proposed in [12] to compensate the unknown nonlinear dynamics of the robot. The control structure is based on the combination of the neural adaptation technique with sliding mode control with the use real implementation. In [13], the Backstepping approach is introduced to solve the tracking control problem, the commands of linear and angular velocities delivered to the robot servos are studied and tested with real implementation by giving the robot forward and angular reference velocities, collecting the actual velocities, the stability analysis of the system is proved via Lyapunov theory.
Some of the parameters of the systems are not modeled or are prone to change in time, with is often impossible to control. In this paper, a robust control of PI/Backstepping is proposed to compensate for the dynamic disturbances, and its stability is analyzed using the Lyapunov theory, the approach is based on the combination of nonlinear PI controller and a Backstepping technique. Integral action control is added to reduce the error of the trajectory tracking. A comparative study between the proposed approach and the backstepping control is presented by simulations in term of accuracy and stability under different load conditions. The rest of the paper is organized as follows. Section 2 gives the dynamic unicycle-like robot model. Section 3, presents and explains the proposed PI/backstepping controller design. Respectively, in section 4, we presented simulations and experimental results to verify the designed trajectory tracking control method. Finally, in section 5 we present the conclusion of this paper.

Robot model
In this section a unicycle-like mobile robot proposed by De La Cruz and Carelli in [14] is considered, the mobile robot is illustrated in Figure1 , where G is the center of mass, h is the point of required to track a trajectory, u and ω are, respectively, the linear and angular velocities, ψ is the robot orientation, a, b, c and d are distances. The force and moment equations for the robot are written as [15]: Where F rrx and F rry are the longitudinal and lateral tire forces of the right wheel, F rlx and F rly are the longitudinal lateral tire forces of the left wheel, F cx and F cy are the longitudinal and lateral force exerted on C by the castor, I z is the robot moment of inertia at G. The kinematics of point h is written in [14] as : Where u andū are the longitudinal and lateral velocities of the center of mass. The complete model of the unicycle-like mobile is given in [14] byDe La Cruz and Carelli as: Where the parameters of the dynamic model are defined in [14] as: where m is the robot mass, I e and B e are the moment of inertia and the viscous friction coefficient, r is the right and left wheel radius, and R t is the nominal radius of the tire, k a is the constant of torque,k b is electromotive constant of motors, R a is the electrical resistance of the motors. PD controllers are implemented with proportional gains k P T and k P R , and derivative gains k DT and k DR to control the velocities of the right and left motor.

Robust Controller design
In this work, two controllers are used: the first one is a kinematics controller for external loop and the second one is a PI/backstepping for an internal loop as see in Figure2.

Kinematic controller
The kinematic controller is developed with the inverse kinematics equations of the robot mobile and it is designed in order to generate the desired values of the linear and angular velocities. From (3), the kinematic model can be written as: Where u ref and ω ref are the desired values of the linear and angular velocities given by the kinematic controller. The matrix inverse is The control law can be chosen as: Where ε x = x d − x and ε y = y d − y the current are position errors, h(x, y) and h d (x d , y d ) are the current and the desired coordinates. The development and stability analysis of the kinematic controller is detailed in [13].

Nonlinear PI-Based Backstepping controller design
The combination of integral action and backstepping technique is used to design a control law that improves the robustness when the dynamics parameters of the robot are not well-known. The dynamic part of equation (3) is: For the first step, we consider the output errors between the reference and actual controls are given by: And the following Lyapunov function defined as The time derivative of the Lyapunov candidate functions is calculated aṡ The stabilization of the dynamics errors system can be obtained by introducing a virtual controls input: Where the integral actions are: With K 1u , K 1ω , α u and α ω are design parameters. At the second step the new errors given by:

ISSN: 1693-6930 1177
And From (11) and (14), it follows that: And Where K 2u , K 2ω are positive constant of stability. So that is result, the integral backstepping control laws of elevation, linear and angular velocities are: And The stability of the equilibrium at the origin of the error system can be obtained by Lyapunov theory Thus, choosing the Lyapunov candidate functions as follows: Whose first time derivative are negative defined, which means that, the tracking error is asymptotically stable.

Simulation and experiment results
In this section, the proposed control law of the robot was implemented on a Arduino Robot Mobile (Radius of 185 mm, height of 85 mm and weight of 1.50 kg), which has two processors based on the ATmega32u4, see Figure 3, The wheels are driven by DC motors having rated torque 30 mNm at 15000 rpm, each motor is equipped with an encoder output of 500 ticks/revolution, the sample time of the robot is 0.1 s. Arduino Yun with highly integrated and cost effective IEEE 802.11n 1x1 2.4 GHz System is mounted on the robot to transmit data from matlab to the robot using the Wireless network protocol. The odometric sensors are used to sensing the point h of robot position. The controller is implemented under the Windows operating system using Matlab. In order to test the proposed PI/Backstepping control law of this paper, several experiments were carried out with circular track reference trajectory of 2 m. The initial position for the mobile robot is as P 0 = (x, y, ψ) = (0, 0, 0). The gains of the implemented controller are selected as:   The robot mobile tracks the circular trajectory with a good performance (smaller tracking error) as see in Figure 4a, the evolutions of the desired and actual linear and angular velocities are plotted in Figure 4b, from these figures, it is clear that the tracking of the reference trajectory is very accurate and the tracking errors tend to zero. Figure 5a illustrates the control the robot's wheels speed signal generated by the Motor board processor. The control law of the linear and angular velocities are presented in Figure 5b, these velocities are sent to the robot through wireless communication using the Yun Arduino card.

Comparative study
The second experiment is designed to compare between the Backstepping approach proposed in [13] and PI/backstepping, we carried out the experiment with same circular track reference trajectory of 2 meters of radius and same initial posture P 0 =(x,y,θ) =(0,0,0). Figure 6 shows the evolution distance errors for experiments using the proposed approach and backstepping controller, it can be see that the PI/backstepping is the most stable and accurate method comparing the backstepping approach. Now in table 1, we have the numerical results: the mean error, variance and standard deviation of trajectory using backstepping and PI/backstepping approaches, we notice that the PI/backstepping presents the best performance in term of accuracy, the numerical results proved

Conclusion
This paper addressed the problem of tracking control of unicycle-like robot. The proposed approach is based on the backstepping technique combined with integral action; the stability of the system is analyzed using the Lyapunov theory, and the design controller is implemented on an Arduino mobile robot. The integral part of algorithm is added to eliminate the tracking errors and ISSN: 1693-6930 the disturbances and dynamics not modeled as wheel and tire diameters, mass, inertia, etc, the comparative results demonstrate the effectiveness the proposed controller, in term of accuracy in term of accuracy, stability and convergence. The force estimator design can be added to guarantee the robustness of the controller. This work can be applied to remote control using the virtual reality [16], and augmented reality [17]. We intend also to improve the system in an environment with obstacles.