# Adaptive Integral Sliding Mode Stabilization of Nonholonomic Drift-Free Systems

Authors: Waseem Abbasi; Fazal ur Rehman
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

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.


---

## Body

## 1. 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 [5–8], time-varying techniques [9–12], adaptive techniques [13, 14], and sliding mode control technique [15–20]. 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 [21–24].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. Section2 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.


## 2. Problem Formulation

### 2.1. Mathematical Model of Nonholonomic System

The kinematic model for a drift-free nonholonomic system is given as(1)x˙=∑i=1mGixui,x∈Rn,where Gix are linearly independent vector fields on Rn, ui are locally bounded in t, 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.


### 2.2. Problem Statement

Given a desired set pointxdes∈Rn, construct a feedback strategy in presence of the control ui:Rn→R,i=1,2,…,m, so that the desired set point xdes is an attractive set for (1), such that xt;0,x0→xdes, as t→∞ for any initial condition.Generally, by appropriate translation of coordinate system,xdes=0 can be achieved.


### 2.3. Some Assumptions

For steering control problem, the systems described by (1) must satisfy the following conditions:(P1)
The vector fieldsG1x,…,Gm(x) are linearly independent.(P2)
System (1) satisfies the Lie algebra rank condition (LARC) for accessibility, where Lie algebra, LG1,…,Gmx, spans Rn at each point x∈Rn.


## 2.1. Mathematical Model of Nonholonomic System

The kinematic model for a drift-free nonholonomic system is given as(1)x˙=∑i=1mGixui,x∈Rn,where Gix are linearly independent vector fields on Rn, ui are locally bounded in t, 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.


## 2.2. Problem Statement

Given a desired set pointxdes∈Rn, construct a feedback strategy in presence of the control ui:Rn→R,i=1,2,…,m, so that the desired set point xdes is an attractive set for (1), such that xt;0,x0→xdes, as t→∞ for any initial condition.Generally, by appropriate translation of coordinate system,xdes=0 can be achieved.


## 2.3. Some Assumptions

For steering control problem, the systems described by (1) must satisfy the following conditions:(P1)
The vector fieldsG1x,…,Gm(x) are linearly independent.(P2)
System (1) satisfies the Lie algebra rank condition (LARC) for accessibility, where Lie algebra, LG1,…,Gmx, spans Rn at each point x∈Rn.


## 3. The Proposed Control Algorithm

Step 1.
Write system (1) in the following form:(2)x˙1=g1x,u,x˙2=g2x,u,⋮x˙n-1=gn-1x,u,x˙n=gnx,u,where gi:Rn×Rm→R are nonlinear functions.Step 2.
Using the input transformation, transform system (2) into the following form:(3)x˙1=h1x,x˙2=h2x,⋮x˙n-1=hn-1x,x˙n=v,where hi:Rn×Rm→R are nonlinear function and v is the new input.
After some manipulation, system (3) can be rewritten as (4)x˙1=x2+F1,x˙2=x3+F2,⋮x˙n-1=xn+Fn-1,x˙n=v,where Fi=-xi+1+hi(x).Step 3.
Assume thatFi are uncertainties in the system. Let F^i,i=1,…,n be an estimate of Fi,i=1,…,n-1, respectively. Apply the function approximation technique [27] to represent Fi and their estimates F^i as Fi=wiTφi(t) and F^i=w^iTφi(t).
φ i ( t ) = φ i 1 ( t ) φ i 2 ( t ) ⋯ φ i n ( t ) T is the function of basis vector and wi=wi1wi2⋯winT is a vector of weightings. Let w^i=w^i1w^i2⋯w^inT be estimate of wi=wi1wi2⋯winT. Therefore, we can estimate Fi by estimating the weight vector wi; that is, F^i=w^iTφi(t). Define w~i=wi-w^i; then system (4) can be written as(5)x˙1=x2+w^1Tφ1t+w~1Tφ1t,x˙2=x3+w^2Tφ1t+w~2Tφ1t,⋮x˙n-1=xn+w^n-1Tφn-1t+w~n-1Tφn-1t,x˙n=v.Step 4.
Choose the nominal system for (5) as(6)x˙1=x2,x˙2=x3,⋮x˙n-1=xn,x˙n=v0.Step 5.
Define the sliding surface for nominal system (6) as (7)s0=x1+∑i=2n-1cixi+xn,where ci>0 are chosen in such a way that s0 becomes Hurwitz polynomial. Then (8)s˙0=x˙1+∑i=2n-1cix˙i+x˙n=x2+∑i=2n-1cixi+1+v0.Choose(9)v0=-x2-∑i=2n-1cixi+1-ksign⁡s0,k>0.We have s˙0=-ksign⁡(s0). Therefore, nominal system (6) is asymptotically stable.Step 6.
Define the sliding surface for system (5) as(10)s=s0+z=x1+∑i=2n-1cixi+xn+z,where z is an integral term. To avoid the reaching phase, choose z(0) such that s(0)=0. Choose v=v0+vs, where v0 is the nominal input and vs is compensator term. Then (11)s˙=x˙1+∑i=2n-1cix˙i+x˙n+z˙=x2+w^iTφ1t+w~iTφ2t+∑i=2n-1cixi+1+w^iTφit+w~iTφit+v0+vs+z˙=x2+∑i=2n-1cixi+1+v0+vs+z˙+∑i=1n-1w^iTφit+w~iTφit,where c1=1.Step 7.
Choose a Lyapunov function as(12)V=12s2+12∑i=1n-1w~iTw~i.Design the adaptive laws for w~i&w^i,i=1,…,n and compute vs such that V˙<0.Theorem 1.
Choose a Lyapunov function as(13)V=12s2+12∑i=1n-1w~iTw~i.The following adaptive laws for w~i &  w^i and the value of vs will guarantee the time derivative of V in (13) to be strictly negative (i.e., V˙<0)(14)z˙=-x2-∑i=2n-1cixi+1-v0,vs=-∑i=1n-1ciw^iTφit-ks,w~˙i=-cisφit-kiw~i,w^˙i=-w~˙i,where k and ki>0,i=1,…,n-1.Proof.
Since(15)V˙=ss˙+∑i=1n-1w~iTw~˙i=sx2+w^1Tφ1t+w~1Tφ1t+∑i=2n-1cixi+1+w^iTφit+w~iTφit+v0+vs+z˙+∑i=1n-1w~iTw~˙i=sx2+∑i=1n-1cixi+1+ciw^iTφit+v0+vs+z˙+∑i=1n-1w~iTw~˙i+cisφit,by using(16)z˙=-x2-∑i=2n-1cixi+1-v0,vs=-∑i=1n-1ciw^iTφit-ks,w~˙i=-cisφit-kiw~i,w^˙i=-w~˙i,where k and ki>0,i=1,…,n-1, we have(17)V˙=-ks2-∑i=1n-1kiw~iTw~i.Choosing(18)kn=min⁡k,k1,…,kn-1,we have(19)V˙≤-kns2+∑i=1n-1w~iTw~i.Using the chosen Lyapunov function (20)V=12s2+12∑i=1n-1w~iTw~i,we can write(21)V˙≤-2knV,V˙≤-αVβ,where α=2kn and β=1.
From this we conclude thats  &  w~i→0, i=1,…,n. Since s→0, therefore x→0.In the following section we illustrate the above algorithm by applying it to three different nonholonomic drift-free systems.


## 4. Application Examples

### 4.1. The Unicycle Model

A unicycle model or a two-wheel car model, shown in Figure1, 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](22)θ˙x˙y˙=100u1+0cos⁡θsin⁡θu2.Introducing a new set of state variables x=def[x1,x2,x3]T=[θ,x,y]T the kinematics model (22) can be written as(23)x˙1x˙2x˙3=100u1+0cos⁡x1sin⁡x1u2or(24)x˙=G1xu1+G2xu2,x∈R3,where(25)G1x=100,G2x=0cos⁡x1sin⁡x1.The kinematics model (22) satisfies the following assumptions: (P1)
The vector fieldsG1(x) and G2(x) are linearly independent.(P2)
System (24) satisfies the Lie algebra rank condition (LARC) for accessibility, where the Lie algebra, L(G1,G2)(x), spans R3 at each point x∈R3.To verify property (P2), it is sufficient to calculate the following Lie bracket of G1(x) and G2(x):(26)G3x=defG1,G2x=0-sin⁡x1cos⁡x1.Then the LARC condition, namely, span⁡(G1,G2,G3)(x)=R3,∀x∈R3, is satisfied.Figure 1
The unicycle model.


#### 4.1.1. Application of the Proposed Algorithm to the Unicycle Model

Step 1.
The unicycle model given (24) can be rewritten as(27)x˙1=u1,x˙2=cos⁡x1u2,x˙3=sin⁡x1u2.Step 2.
Chooseu1=v and u2=x3/cos⁡x1, where, x1≠π/2; then system (27) becomes(28)x˙1=v,x˙2=x3,x˙3=x3tan⁡x1.After some manipulation the above-mentioned system can be written as(29)x˙2=x3,x˙3=x1+F,x˙1=v,where F=-x1+x3tan⁡x1.Step 3.
AssumeF as an uncertainty and let F^ be an estimate of F. The estimate of F by function approximating technique [27] is F=wTφ. Then F^=w^Tφ and system (29) can be written as(30)x˙2=x3,x˙3=x1+w^Tφ+w~Tφ,x˙1=v.Step 4.
Choose the nominal system for (30) as(31)x˙2=x3,x˙3=x1,x˙1=v0.Step 5.
Define the sliding surface for nominal system (31) as(32)s0=x2+2x3+x1.Then (33)s˙0=x˙2+2x˙3+x˙1=x3+2x1+v0.By choosing(34)v0=-x3-2x1-ksign⁡s0,k>0,we have(35)s˙0=-ksign⁡s0.Therefore, nominal system (31) is asymptotically stable.Step 6.
Define the sliding surface for system (30) as(36)s=s0+z=x2+2x3+x1+z.Choose v=v0+vs.
Then(37)s˙=x˙1+2x˙3+x˙2+z˙=x3+2x2+2w^Tφ+2w~Tφ+v0+vs+z˙.Step 7.
The adaptive laws forw~,w^ and the value of vs are as follows:(38)z˙=-x3-2x2-v0,vs=-2w^Tφ-ks,w~˙=-2sφ-k1w~,w^˙=-w~˙,where k and k1>0.
Give(39)V˙=-ks2-k1w~Tw~,where V=1/2s2+1/2w~Tw~.
Choosing(40)k2=min⁡k,k1,we have(41)V˙≤-k2s2+w~Tw~.Using the chosen Lyapunov function we can write(42)V˙≤-2k2V,V˙≤-αVβ,where α=2k2 and β=1.
From this we conclude thats&w~→0. Since s→0, therefore x→0.
Simulation results are shown in Figure4.


### 4.2. The Front Wheel Car Model

A front wheel car model, shown in Figure2, is basically a four-dimensional nonholonomic system having two inputs and four states with depth-two Lie bracket. A front wheel car kinematic model [6] can be defined as(43)ψ˙x˙θ˙y˙=1000u1+0cos⁡θ1ltan⁡ψsin⁡θu2.Assuming that l=1 and introducing a new set of state variables x=def(x1,x2,x3,x4)=(ψ,x,y,θ) the kinematics model (43) can be written as(44)x˙1x˙2x˙3x˙4=1000u1+0cos⁡x4sin⁡x4tan⁡x1u2or(45)x˙=G1xu1+G2xu2,x∈R4,where G1x=1000andG2(x)=0cos⁡x4sin⁡x4tan⁡x1.Figure 2
The front wheel car model.The kinematics model (45) satisfies the following assumptions:(P1)
The vector fieldsG1xandG2(x) are linearly independent.(P2)
System (45) satisfies the Lie algebra rank condition (LARC) for accessibility, where the Lie algebra, L(G1,G2)(x), spans R4 at each point x∈R4.To verify property (P2), it is sufficient to calculate the following Lie brackets of G1(x)&G2(x):(46)G3x=defG1,G2x=000sec⁡x12,G4x=defG2,G3x=0-sin⁡x4sec⁡x12cos⁡x4sec⁡x120,which satisfy the LARC condition: span⁡G1,G2,G3,G4x=R4,  ∀x∈R4.


#### 4.2.1. Application of the Proposed Algorithm to the Front Wheel Car Model

Step 1.
The front wheel car model as given in (45) can be rewritten as (47)x˙1=u1,x˙2=cos⁡x4u2,x˙3=sin⁡x4u2,x˙4=tan⁡x1u2.Step 2.
Chooseu1=v and u2=x3/cos⁡x4, where x4≠π/2, and then system (47) becomes(48)x˙1=v,x˙2=x3,x˙3=x3tan⁡x4,x˙4=x3tan⁡x1sec⁡x4,which can be rewritten as(49)x˙2=x3,x˙3=x4+F3,x˙4=x1+F4,x˙1=v,where(50)F3=-x4+x3tan⁡x4,F4=-x1+x3tan⁡x1sec⁡x4.Step 3.
TreatFi,i=3,4 as uncertainties and let F^i,i=3,4 be an estimate of Fi,i=3,4, respectively. Using function approximation technique [27], we can approximate Fi,i=3,4 as F3=w3Tφ3, F4=w4Tφ4. Then F^3=w^3Tφ3 and F^4=w^4Tφ4.
Then system (49) can be written as(51)x˙1=x3,x˙3=x4+w^3Tφ3+w~3Tφ3,x˙4=x2+w^4Tφ4+w~4Tφ4,x˙2=v.Step 4.
Choose the nominal system for (51) as(52)x˙1=x3,x˙3=x4,x˙4=x2,x˙2=v0.Step 5.
Define the sliding surface for nominal system (52) as(53)s0=x1+3x3+3x4+x2.Then(54)s˙0=x˙1+3x˙3+3x˙4+x˙2=x3+3x4+3x2+v0.By choosing(55)v0=-x3-3x4-3x2-ksign⁡s0,k>0,we have(56)s˙0=-ksign⁡s0.Therefore, nominal system (52) is asymptotically stable.Step 6.
Define the sliding surface for system (51) as(57)s=s0+z=x1+3x3+3x4+x2+z.Choose v=v0+vs.
Then(58)s˙=x˙1+3x˙3+3x˙4+x˙2+z˙=x3+3x4+3w^3Tφ3+3w~3Tφ3+3x2+3w^4Tφ4+3w~4Tφ4+v0+vs+z˙.Step 7.
The following adaptive laws forw~i,w^i,i=3,4 and the value of vs are chosen as(59)z˙=-x3-3x4-3x2-v0,vs=-3w^3Tφ3-3w^4Tφ4-ks,w~˙3=-3sφ3-k1w~3T,w^˙3=-w~˙3,w~˙4=-3sφ4-k2w~4T,w^˙4=-w~˙4,where k,k1andk2>0.
Give(60)V˙=-ks2-k1w~3Tw~3-k2w~4Tw~4,where (61)V=12s2+12w~3Tw~3+12w~4Tw~4.Choosing(62)k3=min⁡k,k1,k2,we have(63)V˙≤-k3s2+k1w~3Tw~3+k2w~4Tw~4.Using the chosen Lyapunov function we can write(64)V˙≤-2k3V,V˙≤-αVβ,where α=2k3 and β=1.
From this we conclude thats,w~3,w~4→0. Since s→0, therefore x→0.
Simulation results are shown in Figure5.


### 4.3. The Mobile Robot with Trailer Model

A car with trailer model, shown in Figure3, 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(65)x˙1=cos⁡x3cos⁡x4u1,x˙2=cos⁡x3sin⁡x4u1,x˙3=u2,x˙4=1lsin⁡x3u1,x˙5=1dsin⁡x4-x5cos⁡x3u1.By assuming l=d=1, system (65) can be written in the following standard form:(66)x˙=G1xu1+G2xu2,x∈R5,where(67)G1x=cos⁡x3cos⁡x4cos⁡x3sin⁡x40sin⁡x3cos⁡x3sin⁡x4-x5,G2x=00100.Figure 3
The mobile robot with trailer model.Figure 4
(a) Time response of the system states corresponding to initial condition(x1(0),x2(0),x3(0))=(2,-1,1). (b) Control input v.
(a) (b)Figure 5
(a) Time response of the system states corresponding to initial condition(x1(0),…,x4(0))=(-2,1,-1,3). (b) Control input v.
(a) (b)It can be verified that system (66) satisfies the following assumptions which are necessary for steering problem.(P1)
The vectorsGi(x),i=1,2 are linearly independent and have no singular point for all x∈M⊆R5, where M is some manifold in R5.(P2)
System (66) satisfies the Lie algebraic rank condition (LARC) for controllability, where the Lie algebra, L(G1,G2)(x), spans R5 at each point x∈M⊆R5: that is, span⁡G1x,G2x,…,G5x=R5,∀x∈M.To verify (P1) and (P2), calculate the linearly independent Lie brackets.(68)G3x=defG1,G2x=sin⁡x3cos⁡x4sin⁡x3sin⁡x40-cos⁡x3sin⁡x3sin⁡x4-x5,G4x=defG1,G1,G2x=-sin⁡x4cos⁡x400cos⁡x4-x5,G5x=defG1,G1,G1,G2x=-sin⁡x3cos⁡x4-sin⁡x3sin⁡x400-sin⁡x3sin⁡x4-x5+cos⁡x3.If the motion of system is restricted to manifold,(69)M=defx∈R5:xi<π2,i=3,4.Then the Lie algebra rank condition, namely, span⁡G1x,G2x,…,G5x=R5,∀x∈M, is satisfied, hence guaranteeing that system (66) satisfies conditions (P1) and (P2) on the surface M.


#### 4.3.1. Application of the Proposed Algorithm to the Mobile Robot with Trailer Model

Step 1.
System (65) can be written as(70)x˙1=cos⁡x3cos⁡x4u1,x˙2=cos⁡x3sin⁡x4u1,x˙3=u2,x˙4=sin⁡x3u1,x˙5=sin⁡x4-x5cos⁡x3u1.Step 2.
Chooseu1=x2/cos⁡x3cos⁡x4 and u2=v.
And,x3,x4≠π/2. Then system (70) can be written as(71)x˙1=x2,x˙2=x2tan⁡x4,x˙3=v,x˙4=x2tan⁡x3sec⁡x4,x˙5=x2sin⁡x4-x5sec⁡x4.Step 3.
AssumeFi,i=2,4,5 as uncertainties and let F^i,i=2,4,5 be an estimate of Fi,i=2,4,5, respectively. Approximate Fi=wiTφi,i=2,4,5 and let F^i=w^iTφi,i=2,4,5, respectively. Then system (71) can be written as(72)x˙1=x2,x˙2=x4+w^2Tφ2+w~2Tφ2,x˙4=x5+w^4Tφ4+w~4Tφ4,x˙5=x3+w^5Tφ5+w~5Tφ5,x˙3=v.Step 4.
Choose the nominal system for (72) as(73)x˙1=x2,x˙2=x4,x˙4=x5,x˙5=x3,x˙3=v0.Step 5.
Define the Hurwitz sliding surface for nominal system (73) as(74)s0=x1+4x2+6x4+4x5+x3.Then(75)s˙0=x˙1+4x˙2+6x˙4+4x˙5+x˙3=x2+4x4+6x5+4x3+v0.By choosing(76)v0=-x2-4x4-6x5-4x3-ksign⁡s0,k>0,we have(77)s˙0=-ksign⁡s0.Therefore, nominal system (73) is asymptotically stable.Step 6.
Define the sliding surface for system (72) as(78)s=s0+z=x1+4x2+6x4+4x5+x3+z.And choose v=v0+vs.
Then(79)s˙=x˙1+4x˙2+6x˙4+4x˙5+x˙3+z˙=x2+4x4+4w^2Tφ2+4w~2Tφ2+6x5+6w^4Tφ4+6w~4Tφ4+4x3+4w^5Tφ5+4w~5Tφ5+v0+vs+z˙.Step 7.
The following adaptive laws forw~i&w^i and the value of vs(80)z˙=-x2-4x4-6x5-4x3-v0,vs=-4w^2Tφ2-6w^4Tφ4-4w^5Tφ5-ks,w~˙2=-4sφ2-k1w~2T,w^˙2≈-w~˙2,w~˙4=-6sφ4-k2w~4T,w^˙4≈-w~˙4,w~˙5=-4sφ5-k3w~5T,w^˙5≈-w~˙5, with kandki>0,i=1,2,3, result in(81)V˙=-ks2-k1w~2Tw~2-k2w~4Tw~4-k3w~5Tw~5,where(82)V=12s2+12w~2Tw~2+12w~4Tw~4+12w~5Tw~5.Choosing(83)k4=min⁡k,k1,k2,k3,we have(84)V˙≤-k4s2+w~2Tw~2+w~4Tw~4+w~5Tw~5.Using the chosen Lyapunov function we can write(85)V˙≤-2k4V,V˙≤-αVβ,where α=2k4 and β=1.
From this we conclude thats,w~2T,w~4Tandw~5T→0. Since s→0, therefore x→0. Simulation results are shown in Figures 4–6 for different initial conditions.Figure 6
(a) Time response of the system states corresponding to initial condition(x1(0),…,x5(0))=(2,-1,1,-2,2). (b) Control input v.
(a) (b)


## 4.1. The Unicycle Model

A unicycle model or a two-wheel car model, shown in Figure1, 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](22)θ˙x˙y˙=100u1+0cos⁡θsin⁡θu2.Introducing a new set of state variables x=def[x1,x2,x3]T=[θ,x,y]T the kinematics model (22) can be written as(23)x˙1x˙2x˙3=100u1+0cos⁡x1sin⁡x1u2or(24)x˙=G1xu1+G2xu2,x∈R3,where(25)G1x=100,G2x=0cos⁡x1sin⁡x1.The kinematics model (22) satisfies the following assumptions: (P1)
The vector fieldsG1(x) and G2(x) are linearly independent.(P2)
System (24) satisfies the Lie algebra rank condition (LARC) for accessibility, where the Lie algebra, L(G1,G2)(x), spans R3 at each point x∈R3.To verify property (P2), it is sufficient to calculate the following Lie bracket of G1(x) and G2(x):(26)G3x=defG1,G2x=0-sin⁡x1cos⁡x1.Then the LARC condition, namely, span⁡(G1,G2,G3)(x)=R3,∀x∈R3, is satisfied.Figure 1
The unicycle model.


### 4.1.1. Application of the Proposed Algorithm to the Unicycle Model

Step 1.
The unicycle model given (24) can be rewritten as(27)x˙1=u1,x˙2=cos⁡x1u2,x˙3=sin⁡x1u2.Step 2.
Chooseu1=v and u2=x3/cos⁡x1, where, x1≠π/2; then system (27) becomes(28)x˙1=v,x˙2=x3,x˙3=x3tan⁡x1.After some manipulation the above-mentioned system can be written as(29)x˙2=x3,x˙3=x1+F,x˙1=v,where F=-x1+x3tan⁡x1.Step 3.
AssumeF as an uncertainty and let F^ be an estimate of F. The estimate of F by function approximating technique [27] is F=wTφ. Then F^=w^Tφ and system (29) can be written as(30)x˙2=x3,x˙3=x1+w^Tφ+w~Tφ,x˙1=v.Step 4.
Choose the nominal system for (30) as(31)x˙2=x3,x˙3=x1,x˙1=v0.Step 5.
Define the sliding surface for nominal system (31) as(32)s0=x2+2x3+x1.Then (33)s˙0=x˙2+2x˙3+x˙1=x3+2x1+v0.By choosing(34)v0=-x3-2x1-ksign⁡s0,k>0,we have(35)s˙0=-ksign⁡s0.Therefore, nominal system (31) is asymptotically stable.Step 6.
Define the sliding surface for system (30) as(36)s=s0+z=x2+2x3+x1+z.Choose v=v0+vs.
Then(37)s˙=x˙1+2x˙3+x˙2+z˙=x3+2x2+2w^Tφ+2w~Tφ+v0+vs+z˙.Step 7.
The adaptive laws forw~,w^ and the value of vs are as follows:(38)z˙=-x3-2x2-v0,vs=-2w^Tφ-ks,w~˙=-2sφ-k1w~,w^˙=-w~˙,where k and k1>0.
Give(39)V˙=-ks2-k1w~Tw~,where V=1/2s2+1/2w~Tw~.
Choosing(40)k2=min⁡k,k1,we have(41)V˙≤-k2s2+w~Tw~.Using the chosen Lyapunov function we can write(42)V˙≤-2k2V,V˙≤-αVβ,where α=2k2 and β=1.
From this we conclude thats&w~→0. Since s→0, therefore x→0.
Simulation results are shown in Figure4.


## 4.1.1. Application of the Proposed Algorithm to the Unicycle Model

Step 1.
The unicycle model given (24) can be rewritten as(27)x˙1=u1,x˙2=cos⁡x1u2,x˙3=sin⁡x1u2.Step 2.
Chooseu1=v and u2=x3/cos⁡x1, where, x1≠π/2; then system (27) becomes(28)x˙1=v,x˙2=x3,x˙3=x3tan⁡x1.After some manipulation the above-mentioned system can be written as(29)x˙2=x3,x˙3=x1+F,x˙1=v,where F=-x1+x3tan⁡x1.Step 3.
AssumeF as an uncertainty and let F^ be an estimate of F. The estimate of F by function approximating technique [27] is F=wTφ. Then F^=w^Tφ and system (29) can be written as(30)x˙2=x3,x˙3=x1+w^Tφ+w~Tφ,x˙1=v.Step 4.
Choose the nominal system for (30) as(31)x˙2=x3,x˙3=x1,x˙1=v0.Step 5.
Define the sliding surface for nominal system (31) as(32)s0=x2+2x3+x1.Then (33)s˙0=x˙2+2x˙3+x˙1=x3+2x1+v0.By choosing(34)v0=-x3-2x1-ksign⁡s0,k>0,we have(35)s˙0=-ksign⁡s0.Therefore, nominal system (31) is asymptotically stable.Step 6.
Define the sliding surface for system (30) as(36)s=s0+z=x2+2x3+x1+z.Choose v=v0+vs.
Then(37)s˙=x˙1+2x˙3+x˙2+z˙=x3+2x2+2w^Tφ+2w~Tφ+v0+vs+z˙.Step 7.
The adaptive laws forw~,w^ and the value of vs are as follows:(38)z˙=-x3-2x2-v0,vs=-2w^Tφ-ks,w~˙=-2sφ-k1w~,w^˙=-w~˙,where k and k1>0.
Give(39)V˙=-ks2-k1w~Tw~,where V=1/2s2+1/2w~Tw~.
Choosing(40)k2=min⁡k,k1,we have(41)V˙≤-k2s2+w~Tw~.Using the chosen Lyapunov function we can write(42)V˙≤-2k2V,V˙≤-αVβ,where α=2k2 and β=1.
From this we conclude thats&w~→0. Since s→0, therefore x→0.
Simulation results are shown in Figure4.


## 4.2. The Front Wheel Car Model

A front wheel car model, shown in Figure2, is basically a four-dimensional nonholonomic system having two inputs and four states with depth-two Lie bracket. A front wheel car kinematic model [6] can be defined as(43)ψ˙x˙θ˙y˙=1000u1+0cos⁡θ1ltan⁡ψsin⁡θu2.Assuming that l=1 and introducing a new set of state variables x=def(x1,x2,x3,x4)=(ψ,x,y,θ) the kinematics model (43) can be written as(44)x˙1x˙2x˙3x˙4=1000u1+0cos⁡x4sin⁡x4tan⁡x1u2or(45)x˙=G1xu1+G2xu2,x∈R4,where G1x=1000andG2(x)=0cos⁡x4sin⁡x4tan⁡x1.Figure 2
The front wheel car model.The kinematics model (45) satisfies the following assumptions:(P1)
The vector fieldsG1xandG2(x) are linearly independent.(P2)
System (45) satisfies the Lie algebra rank condition (LARC) for accessibility, where the Lie algebra, L(G1,G2)(x), spans R4 at each point x∈R4.To verify property (P2), it is sufficient to calculate the following Lie brackets of G1(x)&G2(x):(46)G3x=defG1,G2x=000sec⁡x12,G4x=defG2,G3x=0-sin⁡x4sec⁡x12cos⁡x4sec⁡x120,which satisfy the LARC condition: span⁡G1,G2,G3,G4x=R4,  ∀x∈R4.


### 4.2.1. Application of the Proposed Algorithm to the Front Wheel Car Model

Step 1.
The front wheel car model as given in (45) can be rewritten as (47)x˙1=u1,x˙2=cos⁡x4u2,x˙3=sin⁡x4u2,x˙4=tan⁡x1u2.Step 2.
Chooseu1=v and u2=x3/cos⁡x4, where x4≠π/2, and then system (47) becomes(48)x˙1=v,x˙2=x3,x˙3=x3tan⁡x4,x˙4=x3tan⁡x1sec⁡x4,which can be rewritten as(49)x˙2=x3,x˙3=x4+F3,x˙4=x1+F4,x˙1=v,where(50)F3=-x4+x3tan⁡x4,F4=-x1+x3tan⁡x1sec⁡x4.Step 3.
TreatFi,i=3,4 as uncertainties and let F^i,i=3,4 be an estimate of Fi,i=3,4, respectively. Using function approximation technique [27], we can approximate Fi,i=3,4 as F3=w3Tφ3, F4=w4Tφ4. Then F^3=w^3Tφ3 and F^4=w^4Tφ4.
Then system (49) can be written as(51)x˙1=x3,x˙3=x4+w^3Tφ3+w~3Tφ3,x˙4=x2+w^4Tφ4+w~4Tφ4,x˙2=v.Step 4.
Choose the nominal system for (51) as(52)x˙1=x3,x˙3=x4,x˙4=x2,x˙2=v0.Step 5.
Define the sliding surface for nominal system (52) as(53)s0=x1+3x3+3x4+x2.Then(54)s˙0=x˙1+3x˙3+3x˙4+x˙2=x3+3x4+3x2+v0.By choosing(55)v0=-x3-3x4-3x2-ksign⁡s0,k>0,we have(56)s˙0=-ksign⁡s0.Therefore, nominal system (52) is asymptotically stable.Step 6.
Define the sliding surface for system (51) as(57)s=s0+z=x1+3x3+3x4+x2+z.Choose v=v0+vs.
Then(58)s˙=x˙1+3x˙3+3x˙4+x˙2+z˙=x3+3x4+3w^3Tφ3+3w~3Tφ3+3x2+3w^4Tφ4+3w~4Tφ4+v0+vs+z˙.Step 7.
The following adaptive laws forw~i,w^i,i=3,4 and the value of vs are chosen as(59)z˙=-x3-3x4-3x2-v0,vs=-3w^3Tφ3-3w^4Tφ4-ks,w~˙3=-3sφ3-k1w~3T,w^˙3=-w~˙3,w~˙4=-3sφ4-k2w~4T,w^˙4=-w~˙4,where k,k1andk2>0.
Give(60)V˙=-ks2-k1w~3Tw~3-k2w~4Tw~4,where (61)V=12s2+12w~3Tw~3+12w~4Tw~4.Choosing(62)k3=min⁡k,k1,k2,we have(63)V˙≤-k3s2+k1w~3Tw~3+k2w~4Tw~4.Using the chosen Lyapunov function we can write(64)V˙≤-2k3V,V˙≤-αVβ,where α=2k3 and β=1.
From this we conclude thats,w~3,w~4→0. Since s→0, therefore x→0.
Simulation results are shown in Figure5.


## 4.2.1. Application of the Proposed Algorithm to the Front Wheel Car Model

Step 1.
The front wheel car model as given in (45) can be rewritten as (47)x˙1=u1,x˙2=cos⁡x4u2,x˙3=sin⁡x4u2,x˙4=tan⁡x1u2.Step 2.
Chooseu1=v and u2=x3/cos⁡x4, where x4≠π/2, and then system (47) becomes(48)x˙1=v,x˙2=x3,x˙3=x3tan⁡x4,x˙4=x3tan⁡x1sec⁡x4,which can be rewritten as(49)x˙2=x3,x˙3=x4+F3,x˙4=x1+F4,x˙1=v,where(50)F3=-x4+x3tan⁡x4,F4=-x1+x3tan⁡x1sec⁡x4.Step 3.
TreatFi,i=3,4 as uncertainties and let F^i,i=3,4 be an estimate of Fi,i=3,4, respectively. Using function approximation technique [27], we can approximate Fi,i=3,4 as F3=w3Tφ3, F4=w4Tφ4. Then F^3=w^3Tφ3 and F^4=w^4Tφ4.
Then system (49) can be written as(51)x˙1=x3,x˙3=x4+w^3Tφ3+w~3Tφ3,x˙4=x2+w^4Tφ4+w~4Tφ4,x˙2=v.Step 4.
Choose the nominal system for (51) as(52)x˙1=x3,x˙3=x4,x˙4=x2,x˙2=v0.Step 5.
Define the sliding surface for nominal system (52) as(53)s0=x1+3x3+3x4+x2.Then(54)s˙0=x˙1+3x˙3+3x˙4+x˙2=x3+3x4+3x2+v0.By choosing(55)v0=-x3-3x4-3x2-ksign⁡s0,k>0,we have(56)s˙0=-ksign⁡s0.Therefore, nominal system (52) is asymptotically stable.Step 6.
Define the sliding surface for system (51) as(57)s=s0+z=x1+3x3+3x4+x2+z.Choose v=v0+vs.
Then(58)s˙=x˙1+3x˙3+3x˙4+x˙2+z˙=x3+3x4+3w^3Tφ3+3w~3Tφ3+3x2+3w^4Tφ4+3w~4Tφ4+v0+vs+z˙.Step 7.
The following adaptive laws forw~i,w^i,i=3,4 and the value of vs are chosen as(59)z˙=-x3-3x4-3x2-v0,vs=-3w^3Tφ3-3w^4Tφ4-ks,w~˙3=-3sφ3-k1w~3T,w^˙3=-w~˙3,w~˙4=-3sφ4-k2w~4T,w^˙4=-w~˙4,where k,k1andk2>0.
Give(60)V˙=-ks2-k1w~3Tw~3-k2w~4Tw~4,where (61)V=12s2+12w~3Tw~3+12w~4Tw~4.Choosing(62)k3=min⁡k,k1,k2,we have(63)V˙≤-k3s2+k1w~3Tw~3+k2w~4Tw~4.Using the chosen Lyapunov function we can write(64)V˙≤-2k3V,V˙≤-αVβ,where α=2k3 and β=1.
From this we conclude thats,w~3,w~4→0. Since s→0, therefore x→0.
Simulation results are shown in Figure5.


## 4.3. The Mobile Robot with Trailer Model

A car with trailer model, shown in Figure3, 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(65)x˙1=cos⁡x3cos⁡x4u1,x˙2=cos⁡x3sin⁡x4u1,x˙3=u2,x˙4=1lsin⁡x3u1,x˙5=1dsin⁡x4-x5cos⁡x3u1.By assuming l=d=1, system (65) can be written in the following standard form:(66)x˙=G1xu1+G2xu2,x∈R5,where(67)G1x=cos⁡x3cos⁡x4cos⁡x3sin⁡x40sin⁡x3cos⁡x3sin⁡x4-x5,G2x=00100.Figure 3
The mobile robot with trailer model.Figure 4
(a) Time response of the system states corresponding to initial condition(x1(0),x2(0),x3(0))=(2,-1,1). (b) Control input v.
(a) (b)Figure 5
(a) Time response of the system states corresponding to initial condition(x1(0),…,x4(0))=(-2,1,-1,3). (b) Control input v.
(a) (b)It can be verified that system (66) satisfies the following assumptions which are necessary for steering problem.(P1)
The vectorsGi(x),i=1,2 are linearly independent and have no singular point for all x∈M⊆R5, where M is some manifold in R5.(P2)
System (66) satisfies the Lie algebraic rank condition (LARC) for controllability, where the Lie algebra, L(G1,G2)(x), spans R5 at each point x∈M⊆R5: that is, span⁡G1x,G2x,…,G5x=R5,∀x∈M.To verify (P1) and (P2), calculate the linearly independent Lie brackets.(68)G3x=defG1,G2x=sin⁡x3cos⁡x4sin⁡x3sin⁡x40-cos⁡x3sin⁡x3sin⁡x4-x5,G4x=defG1,G1,G2x=-sin⁡x4cos⁡x400cos⁡x4-x5,G5x=defG1,G1,G1,G2x=-sin⁡x3cos⁡x4-sin⁡x3sin⁡x400-sin⁡x3sin⁡x4-x5+cos⁡x3.If the motion of system is restricted to manifold,(69)M=defx∈R5:xi<π2,i=3,4.Then the Lie algebra rank condition, namely, span⁡G1x,G2x,…,G5x=R5,∀x∈M, is satisfied, hence guaranteeing that system (66) satisfies conditions (P1) and (P2) on the surface M.


### 4.3.1. Application of the Proposed Algorithm to the Mobile Robot with Trailer Model

Step 1.
System (65) can be written as(70)x˙1=cos⁡x3cos⁡x4u1,x˙2=cos⁡x3sin⁡x4u1,x˙3=u2,x˙4=sin⁡x3u1,x˙5=sin⁡x4-x5cos⁡x3u1.Step 2.
Chooseu1=x2/cos⁡x3cos⁡x4 and u2=v.
And,x3,x4≠π/2. Then system (70) can be written as(71)x˙1=x2,x˙2=x2tan⁡x4,x˙3=v,x˙4=x2tan⁡x3sec⁡x4,x˙5=x2sin⁡x4-x5sec⁡x4.Step 3.
AssumeFi,i=2,4,5 as uncertainties and let F^i,i=2,4,5 be an estimate of Fi,i=2,4,5, respectively. Approximate Fi=wiTφi,i=2,4,5 and let F^i=w^iTφi,i=2,4,5, respectively. Then system (71) can be written as(72)x˙1=x2,x˙2=x4+w^2Tφ2+w~2Tφ2,x˙4=x5+w^4Tφ4+w~4Tφ4,x˙5=x3+w^5Tφ5+w~5Tφ5,x˙3=v.Step 4.
Choose the nominal system for (72) as(73)x˙1=x2,x˙2=x4,x˙4=x5,x˙5=x3,x˙3=v0.Step 5.
Define the Hurwitz sliding surface for nominal system (73) as(74)s0=x1+4x2+6x4+4x5+x3.Then(75)s˙0=x˙1+4x˙2+6x˙4+4x˙5+x˙3=x2+4x4+6x5+4x3+v0.By choosing(76)v0=-x2-4x4-6x5-4x3-ksign⁡s0,k>0,we have(77)s˙0=-ksign⁡s0.Therefore, nominal system (73) is asymptotically stable.Step 6.
Define the sliding surface for system (72) as(78)s=s0+z=x1+4x2+6x4+4x5+x3+z.And choose v=v0+vs.
Then(79)s˙=x˙1+4x˙2+6x˙4+4x˙5+x˙3+z˙=x2+4x4+4w^2Tφ2+4w~2Tφ2+6x5+6w^4Tφ4+6w~4Tφ4+4x3+4w^5Tφ5+4w~5Tφ5+v0+vs+z˙.Step 7.
The following adaptive laws forw~i&w^i and the value of vs(80)z˙=-x2-4x4-6x5-4x3-v0,vs=-4w^2Tφ2-6w^4Tφ4-4w^5Tφ5-ks,w~˙2=-4sφ2-k1w~2T,w^˙2≈-w~˙2,w~˙4=-6sφ4-k2w~4T,w^˙4≈-w~˙4,w~˙5=-4sφ5-k3w~5T,w^˙5≈-w~˙5, with kandki>0,i=1,2,3, result in(81)V˙=-ks2-k1w~2Tw~2-k2w~4Tw~4-k3w~5Tw~5,where(82)V=12s2+12w~2Tw~2+12w~4Tw~4+12w~5Tw~5.Choosing(83)k4=min⁡k,k1,k2,k3,we have(84)V˙≤-k4s2+w~2Tw~2+w~4Tw~4+w~5Tw~5.Using the chosen Lyapunov function we can write(85)V˙≤-2k4V,V˙≤-αVβ,where α=2k4 and β=1.
From this we conclude thats,w~2T,w~4Tandw~5T→0. Since s→0, therefore x→0. Simulation results are shown in Figures 4–6 for different initial conditions.Figure 6
(a) Time response of the system states corresponding to initial condition(x1(0),…,x5(0))=(2,-1,1,-2,2). (b) Control input v.
(a) (b)


## 4.3.1. Application of the Proposed Algorithm to the Mobile Robot with Trailer Model

Step 1.
System (65) can be written as(70)x˙1=cos⁡x3cos⁡x4u1,x˙2=cos⁡x3sin⁡x4u1,x˙3=u2,x˙4=sin⁡x3u1,x˙5=sin⁡x4-x5cos⁡x3u1.Step 2.
Chooseu1=x2/cos⁡x3cos⁡x4 and u2=v.
And,x3,x4≠π/2. Then system (70) can be written as(71)x˙1=x2,x˙2=x2tan⁡x4,x˙3=v,x˙4=x2tan⁡x3sec⁡x4,x˙5=x2sin⁡x4-x5sec⁡x4.Step 3.
AssumeFi,i=2,4,5 as uncertainties and let F^i,i=2,4,5 be an estimate of Fi,i=2,4,5, respectively. Approximate Fi=wiTφi,i=2,4,5 and let F^i=w^iTφi,i=2,4,5, respectively. Then system (71) can be written as(72)x˙1=x2,x˙2=x4+w^2Tφ2+w~2Tφ2,x˙4=x5+w^4Tφ4+w~4Tφ4,x˙5=x3+w^5Tφ5+w~5Tφ5,x˙3=v.Step 4.
Choose the nominal system for (72) as(73)x˙1=x2,x˙2=x4,x˙4=x5,x˙5=x3,x˙3=v0.Step 5.
Define the Hurwitz sliding surface for nominal system (73) as(74)s0=x1+4x2+6x4+4x5+x3.Then(75)s˙0=x˙1+4x˙2+6x˙4+4x˙5+x˙3=x2+4x4+6x5+4x3+v0.By choosing(76)v0=-x2-4x4-6x5-4x3-ksign⁡s0,k>0,we have(77)s˙0=-ksign⁡s0.Therefore, nominal system (73) is asymptotically stable.Step 6.
Define the sliding surface for system (72) as(78)s=s0+z=x1+4x2+6x4+4x5+x3+z.And choose v=v0+vs.
Then(79)s˙=x˙1+4x˙2+6x˙4+4x˙5+x˙3+z˙=x2+4x4+4w^2Tφ2+4w~2Tφ2+6x5+6w^4Tφ4+6w~4Tφ4+4x3+4w^5Tφ5+4w~5Tφ5+v0+vs+z˙.Step 7.
The following adaptive laws forw~i&w^i and the value of vs(80)z˙=-x2-4x4-6x5-4x3-v0,vs=-4w^2Tφ2-6w^4Tφ4-4w^5Tφ5-ks,w~˙2=-4sφ2-k1w~2T,w^˙2≈-w~˙2,w~˙4=-6sφ4-k2w~4T,w^˙4≈-w~˙4,w~˙5=-4sφ5-k3w~5T,w^˙5≈-w~˙5, with kandki>0,i=1,2,3, result in(81)V˙=-ks2-k1w~2Tw~2-k2w~4Tw~4-k3w~5Tw~5,where(82)V=12s2+12w~2Tw~2+12w~4Tw~4+12w~5Tw~5.Choosing(83)k4=min⁡k,k1,k2,k3,we have(84)V˙≤-k4s2+w~2Tw~2+w~4Tw~4+w~5Tw~5.Using the chosen Lyapunov function we can write(85)V˙≤-2k4V,V˙≤-αVβ,where α=2k4 and β=1.
From this we conclude thats,w~2T,w~4Tandw~5T→0. Since s→0, therefore x→0. Simulation results are shown in Figures 4–6 for different initial conditions.Figure 6
(a) Time response of the system states corresponding to initial condition(x1(0),…,x5(0))=(2,-1,1,-2,2). (b) Control input v.
(a) (b)


## 5. Simulation Results

Figures4(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.


## 6. 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 drift-free 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.



---

*Source: 9617283-2016-11-17.xml*