Abstract
The stabilization problem of nonholonomic mobile robots with unknown system parameters and environmental disturbances is investigated in this paper. Considering the dynamic model and the kinematic model of mobile robots, the transverse function approach is adopted to construct an additional control parameter, so that the closed-loop system is not underactuated. Then the adaptive backstepping method and the parameter projection technique are applied to design the controller to stabilize the system. At last, simulation results demonstrate the effectiveness of our proposed controller schemes.
1. Introduction
The motion control of nonholonomic mobile robots has been an active research field in the past few decades and remains a challenging control problem. For nonholonomic mobile robots, it cannot be stabilized by any static continuous state feedback [1], due to the Brockett’s condition [2]. To handle the stability of nonholonomic systems, it has to resort to either a time-varying [3, 4], a discontinuous [5, 6], or a dynamic state feedback controller [7, 8].
The early work in [9] studies the tracking problem based on backstepping method for both a kinematic model and a dynamic model of mobile robot. In [10], a neural network adaptive controller based on input-output linearization design is presented to guide a mobile robot during trajectory tracking. However, these methods are only applicable when the system dynamic parameters are known. In order to handle system disturbances that are inevitable in real applications, an adaptive tracking controller has been proposed for a class of mobile robots with uncertainties [11]. Based on Lyapunov’s direct method and backstepping technique [12], a time-varying global adaptive controller at the torque level that simultaneously solves both tracking and stabilization problems in the case of unknown dynamic parameters has been developed. Similar work can be found in [13]. Moreover, a robust adaptive controller is designed for a mobile robot with bounded unknown disturbances in [14]. The works in [15, 16] extend the control law design for more general uncertain nonholonomic systems. Subsequent related works on the stabilization and tracking control of nonholonomic mobile robots include, but are not limited to, [17–21] and many references therein. However, all the stabilization schemes mentioned above need the assistance of the sinusoidal function; thus, the convergence rate of stabilization is slow.
To overcome this problem, in this paper, we consider the stabilization control of the nonholonomic mobile robot with unknown parameters and bounded external disturbances. In contrast to the aforementioned results, all the system parameters as well as the disturbance are assumed to be unknown. Based on the transverse function approach [22] and backstepping techniques, a robust adaptive controller will be developed to guarantee the asymptotical stability of the nonholonomic mobile robot. By employing the traverse function approach, an “auxiliary manipulated variable” is introduced, with which the difficulty encountered in controlling an underactuated system can be overcome. The rest of this paper is arranged as follows. In Section 2, we present the model of nonholonomic mobile robots and problem formulation. Then we propose an adaptive control scheme to achieve stabilization in Section 3, where the stability of the overall system is discussed. Simulation results are provided in Section 4. Finally, some concluding remarks are drawn in Section 5.
2. Problem Formulation
A unicycle mobile robot is considered, which consists of two driving wheels located at the same axis and a passive self-adjusted supporting wheel. The actuated two wheels are driven by two DC servomotors independently. As shown in Figure 1, the geometric center and center of mass of the mobile robot do not coincide. The origin of Po- frame is the geometric center Po, the center of mass Pc is on axis, and the distance to the origin Po is . The position of Po in global coordinate frame O-XY is and is the orientation of the local frame Po-. For the sake of simplicity, it is assumed that the robot does not slip and there is no sliding between the tire and the road; that is, there is no Coulomb-like friction. Then the system can be described by the following dynamic model and kinematic model [23], where the parameters are shown in Table 1: where denotes the position and orientation of the robot, denotes the velocities of the left and right wheels, represents the control torques applied to the wheels, is a symmetric, positive definite inertia matrix, is the centripetal and Coriolis matrix, denotes the surface friction, and is the bounded unknown external disturbance. Matrices , , and are the same as those in [12], which are given as The upper bound of the external disturbance is assumed to satisfy where is an unknown positive constant.
Two control vectors are introduced as and , and then the kinematic model (1) can be written as
Assumption 1. The parameters and fall in known compact sets; that is, there exist some known positive constants , , , and , such that and .
Remark 2. The degrees of freedom of the nonholonomic mobile robot are three, but there are only two independent control inputs, so the system (5) is underactuated.
3. Controller Design
In this section, the transverse function approach is adopted firstly to perform suitable change of coordinates. With the change of coordinates, an additional controller will be created and thus the kinematic model (5) is no longer underactuated. In the second step, the adaptive controller and parameter estimator are designed such that the system is stabilized.
3.1. Control Objective
Design the control inputs to stabilize the nonholonomic mobile robot, as modeled in (2) and (5).
3.1.1. Coordinates Transformation
The new coordinates and additional controller are introduced, and the kinematic model (5) is transformed as follows: where and , for , are functions of designed as where and are arbitrarily small positive constants and satisfies . Then the following properties can be shown: Taking the derivatives of , , and yields where is ensured to be invertible [17]. Different from , the transformed coordinates can be controlled separately by , , and , where is an auxiliary manipulated variable to be designed; thus, (10) are not underactuated.
3.1.2. Controller Design
By replacing (5) with (10), the closed-loop system composed by (2) and (10) is of strict feedback form. Therefore, the backstepping technique [24, 25] method is applied to design the control inputs . Obviously, the design procedure can be divided into two steps. In the first step, the virtual controls and together with the “auxiliary manipulated variable” will be constructed to stabilize the system. In the second step, the actual control signals for will be delivered such that and in (10) can approach the virtual controls and , respectively. Apart from these, the adaptive laws for the unknown system parameters will also be provided.
Step 1. Define the parameter estimation errors , , where and are the estimated values of and , respectively.
Let , and choose the Lyapunov function candidate in this step as and then the derivative of is .
We then introduce two new error variables: where and are the virtual controls for and , respectively. , , and are chosen as where and are positive constants. The above design delivers the following results: The parameter estimators for and are designed as where . Note that denotes a Lipschitz continuous projection operator about which the design details and properties can be found in [26] and the following results are then obtained.
Lemma 3. If , then the projection satisfies , where .
Choose the Lyapunov function candidate in this step as We obtain that
Step 2. We are at the position to derive the actual control torque .
Define , where . Let ; thus, we have Multiplying the derivatives of both sides of (19) by and combining it with (14), we obtain that where
Introduce the estimate for unknown parameter vector . Then the local control torque and adaptive law are designed as where is a given positive matrix and is the estimate of the upper bound of , with The update law for is chosen as Choose the Lyapunov function for the overall system as where is a symmetric and positive definite matrix and . We obtain that
The main results in this section are formally presented in the following theorem.
Theorem 4. Consider the nonholonomic mobile robot system (1) and (2), with the controller (22) and parameter update laws (23) and (25) under Assumption 1. Then the closed-loop system is stable and satisfies
Proof. Considering the projection operation, are bounded. Thus from (27), all signals in are bounded. Hence, , , and are bounded. From (14), it is easy to check that , , and are bounded. Thus and are bounded. From (26), the boundedness of is concluded.
From (6) and (8), we obtain that
It then follows that
Since , , and will converge to zero asymptotically, (28) hold.
Remark 5. Since and are arbitrarily small positive constants; thus from (28) we know the stabilization errors are also arbitrarily small. The convergence rate of the system is actually close related to the design parameters , , and ; thus, compared to the results mentioned in the Introduction, the convergence rate can be much faster.
4. Simulation Results
In this section, we illustrate the design procedure and how to compute the bound of the control torque based on design parameters in Table 2 using Matlab.
The initial estimates for the unknown parameters are set to be 75% of their true values. The bounds of the external disturbances are assumed to be 10. The design parameters are chosen as follows: . The initial conditions are set as . The results on the mobile robot position and tracking errors evolving with time are shown in Figures 2 and 3, respectively. It can be seen that the system stabilized in 3 seconds with the torque in Figure 4. To show that our proposed control scheme renders a faster convergence rate, we make a comparison between our control scheme and the scheme proposed in [12] in terms of with the same control design parameters, as shown in Figure 5. We can clearly see that the convergence rate of our proposed scheme is much faster than that of [12].
5. Conclusions
In this paper, stabilization problem of nonholonomic mobile robot with unknown system parameters and external disturbances is investigated. By considering the kinematic model and dynamic model of the system, the traverse function approach and the backstepping method are used to stabilize the mobile robot.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.