Adaptive Integral Sliding Mode Stabilization of Nonholonomic Drift-Free Systems

This article presents adaptive integral sliding mode control algorithm for the stabilization of nonholonomic drift-free systems. First the system is transformed, by using input transform, into a special structure containing a nominal part and some unknown terms which are computed adaptively. The transformed system is then stabilized using adaptive integral sliding mode control. The stabilizing controller for the transformed system is constructed that consists of the nominal control plus a compensator control.The compensator control and the adaptive laws are derived on the basis of Lyapunov stability theory. The proposed control algorithm is applied to three different nonholonomic drift-free systems: the unicycle model, the front wheel car model, and the mobile robot with trailer model. The controllability Lie algebra of the unicycle model contains Lie brackets of depth one, the model of a front wheel car contains Lie brackets of depths one and two, and the model of a mobile robot with trailer contains Lie brackets of depths one, two, and three. The effectiveness of the proposed control algorithm is verified through numerical simulations.


Introduction
Designing feedback control laws for the stabilization of mechanical control systems has been an interesting subject for researchers in the field of control theory.These systems have attracted intensive attention from the control community because of their wide practical applications in robotics, industry, and automobiles.Due to mechanical design and configuration, these systems are classified into two categories: holonomic and nonholonomic.In holonomic systems, the control input degrees are equal to total degrees of freedom, whereas, nonholonomic systems have less controllable degrees of freedom as compared to total degrees of freedom and have restricted mobility due to the presence of nonholonomic constraints.Roger Brockett showed that the nonholonomic systems cannot be stabilized by continuous static state feedback laws [1].Later on Murray et al. showed that the dependence of the stabilizing control on time is essential [2].
To solve this problem, different control approaches have been presented in the literature.A detailed survey of stabilization of nonholonomic systems can be found in [3] and a survey of underactuated mechanical systems is given in [4].In the literature, several control techniques have been developed for stabilization of nonholonomic systems.Some of these include discontinuous time-invariant techniques [5][6][7][8], time-varying techniques [9][10][11][12], adaptive techniques [13,14], and sliding mode control technique [15][16][17][18][19][20].Sliding mode control (SMC) is a special nonlinear control technique.The objective of the SMC technique is to force the system states to a certain surface, known as the sliding manifold.Once the surface is reached, the system is forced to remain on it thereafter.
The main disadvantage of the SMC is the requirement of discontinuous control law across the sliding manifold.In practical systems, this leads to an undesirable phenomenon called chattering.The closed loop dynamics of the system in SMC depends only on the design parameters of the switching sliding manifold.Sliding mode control also offers several advantages such as simplicity, fast response, and robustness to external disturbance and parameter variation.
Our objective in this article is to propose a scheme for the construction of stabilizing control for nonholonomic mechanical systems.The suggested sliding mode controller can stabilize systems, which do not fulfill Brockett's necessary conditions, as the sliding mode control is inherently discontinuous.Since sliding mode control is insensitive towards model errors, parametric uncertainties, and other disturbances; therefore, it is extensively used.Sliding surface will show system's behavior when the system reaches the sliding manifold [21][22][23][24].
The integral sliding mode control guarantees the robustness of the motion in the whole state space [25,26] because of eliminating the reaching phase.Since the reaching phase is eliminated, therefore the robustness of the system can be guaranteed throughout the system response, starting from the initial time instance.The integral sliding mode control combines the nominal control that stabilizes the nominal system and a discontinuous control that rejects the uncertainty.
The control algorithm presented in this paper is general and applicable to a large class of nonholonomic control systems without drift.The proposed algorithm is applied to three different nonholonomic drift-free systems: the unicycle model, the front wheel car model, and the mobile robot with trailer model.The effectiveness of the proposed algorithm is verified through numerical simulations.
The rest of the article is organized as follows.Section 2 presents problem formulation.Section 3 presents the proposed control methodology in its general form.Section 4 presents application examples of the unicycle model, the front wheel car model, and the car with trailer model.Section 5 presents simulation results for the application examples, and finally Section 6 concludes the paper.

Mathematical Model of Nonholonomic System.
The kinematic model for a drift-free nonholonomic system is given as where   () are linearly independent vector fields on R  ,   are locally bounded in , and piece-wise continuous control functions are defined on the interval [0, ∞).These systems are difficult to control as revealed by the fact that linearization of system (1) is uncontrollable.The most difficult issue from a theoretical viewpoint is the design of feedback laws that can stabilize these systems about an equilibrium position.

Problem Statement.
Given a desired set point  des ∈ R  , construct a feedback strategy in presence of the control   : R  → R,  = 1, 2, . . ., , so that the desired set point  des is an attractive set for (1), such that (; 0,  0 ) →  des , as  → ∞ for any initial condition.Generally, by appropriate translation of coordinate system,  des = 0 can be achieved.

The Proposed Control Algorithm
Step 1. Write system (1) in the following form: where   : R  × R  → R are nonlinear functions.
Step 2. Using the input transformation, transform system (2) into the following form: where ℎ  : R  × R  → R are nonlinear function and V is the new input.After some manipulation, system (3) can be rewritten as where   = − +1 + ℎ  ().
Step 3. Assume that   are uncertainties in the system.Let F ,  = 1, . . .,  be an estimate of   ,  = 1, . . .,  − 1, respectively.Apply the function approximation technique [27] to represent   and their estimates F as   =      () and F = ŵ is the function of basis vector and .Therefore, we can estimate   by estimating the weight vector   ; that is, F = ŵ    ().Define w =   − ŵ ; then system (4) can be written as (5) Step 4. Choose the nominal system for (5) as Step 5. Define the sliding surface for nominal system (6) as where   > 0 are chosen in such a way that  0 becomes Hurwitz polynomial.Then Choose We have ṡ 0 = − sign( 0 ).Therefore, nominal system ( 6) is asymptotically stable.
Step 6. Define the sliding surface for system (5) as where  is an integral term.To avoid the reaching phase, choose (0) such that (0) = 0. Choose V = V 0 + V  , where V 0 is the nominal input and V  is compensator term.Then where  1 = 1.
Step 7. Choose a Lyapunov function as Design the adaptive laws for w & ŵ ,  = 1, . . .,  and compute V  such that V < 0.

Theorem 1. Choose a Lyapunov function as
The following adaptive laws for w & ŵ and the value of V  will guarantee the time derivative of  in ( 13) to be strictly negative (i.e., V < 0) where  and   > 0,  = 1, . . .,  − 1. Proof.Since by using where  and   > 0,  = 1, . . .,  − 1, we have Choosing we have Using the chosen Lyapunov function we can write where  = 2  and  = 1.
In the following section we illustrate the above algorithm by applying it to three different nonholonomic drift-free systems.

The Unicycle Model.
A unicycle model or a two-wheel car model, shown in Figure 1, is basically a three-dimensional nonholonomic system having two inputs and three states with depth-one Lie bracket.A two-wheel car kinematic model is defined as [12] [ Introducing a new set of state variables 22) can be written as or where The kinematics model (22) satisfies the following assumptions: (P1) The vector fields  1 () and  2 () are linearly independent.(P2) System (24) satisfies the Lie algebra rank condition (LARC) for accessibility, where the Lie algebra, ( 1 ,  2 )(), spans R 3 at each point  ∈ R 3 .
Step 3. Assume  as an uncertainty and let F be an estimate of .The estimate of  by function approximating technique [27] is  =   .Then F = ŵ  and system (29) can be written as Step 4. Choose the nominal system for (30) as Step 5. Define the sliding surface for nominal system (31) as Then By choosing we have Therefore, nominal system (31) is asymptotically stable.
Step 6. Define the sliding surface for system (30) as Step 7. The adaptive laws for w, ŵ and the value of V  are as follows: where  and  1 > 0. Give where  = (1/2) 2 + (1/2)w  w. Choosing we have Using the chosen Lyapunov function we can write where  = 2 2 and  = 1.
Simulation results are shown in Figure 4.

The Front Wheel Car Model.
A front wheel car model, shown in Figure 2, is basically a four-dimensional nonholonomic system having two inputs and four states with depthtwo Lie bracket.A front wheel car kinematic model [6] can be defined as Assuming that  = 1 and introducing a new set of state variables  def = ( 1 ,  2 ,  3 ,  4 ) = (, , , ) the kinematics model (43) can be written as where  1 () = [ To verify property (P2), it is sufficient to calculate the following Lie brackets of  1 () &  2 (): which satisfy the LARC condition: span( 1 ,  2 ,  3 ,  4 )() = R 4 , ∀ ∈ R 4 .
Simulation results are shown in Figure 5.

The Mobile Robot with Trailer Model.
A car with trailer model, shown in Figure 3, is basically a five-dimensional nonholonomic system having two inputs and five states with depth-one, depth-two, and depth-three Lie brackets.A car with trailer kinematic model [8] can be defined as By assuming  =  = 1, system (65) can be written in the following standard form: where It can be verified that system (66) satisfies the following assumptions which are necessary for steering problem.
(P1) The vectors   (),  = 1, 2 are linearly independent and have no singular point for all  ∈  ⊆ R 5 , where  is some manifold in R 5 .

By choosing
Therefore, nominal system (73) is asymptotically stable.
Step 6. Define the sliding surface for system (72) as And choose V = V 0 + V  .Then Step 7. The following adaptive laws for w & ŵ and the value of with  and   > 0,  = 1, 2, 3, result in where  = 2 4 and  = 1.
From this we conclude that , w 2 , w 4 and w 5 → 0. Since  → 0, therefore  → 0. Simulation results are shown in Figures 4-6 for different initial conditions.

Conclusion
An adaptive integral sliding mode based control algorithm for the stabilization of nonholonomic drift-free control systems was presented.The objective was to steer the system from any arbitrary initial state to any desired state.The effectiveness of the method was tested on three different nonholonomic driftfree systems: the unicycle model, the front wheel car model, and the mobile robot with trailer model.The aim was to steer the systems to a desired value which was assumed to be zero.It is evident from the simulation results that the objective has been achieved.This method is general and can be employed to steer a variety of mechanical systems with nonholonomic constraints.

Figure 2 :
Figure 2: The front wheel car model.

Figure 3 :
Figure 3: The mobile robot with trailer model.

Figures 4 (
Figures 4(a) and 4(b) show simulation results of the unicycle model and represent that the states and the control effort converge to zero and have settling time of 4 sec and 0.8 sec.Figures 5(a) and 5(b) show simulation results of the front wheel car model and represent that the states and the control effort converge to zero and have settling time of 6 sec and 1 sec.Figures 6(a) and 6(b) show simulation results for the car with trailer model and represent that the states and control effort converge to zero and have settling time of 10 sec and 0.4 sec.Simulation results show the effectiveness of the proposed scheme.