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

1. IntroductionDesigning 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. 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.

2. Problem Formulation

2.1. Mathematical Model of Nonholonomic System. The kinematic model for a drift-free nonholonomic system is given as

[??] = [m.summation over (i=1)] [G.sub.i] (x) [u.sub.i], x [member of] [R.sup.n], (1)

where [G.sub.i](x) are linearly independent vector fields on [R.sup.n], [u.sub.i] are locally bounded in t, and piece-wise continuous control functions are defined on the interval [0, [infinity]). 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 point [x.sub.des] [member of] [R.sup.n], construct a feedback strategy in presence of the control [u.sub.i] : [R.sup.n] [right arrow] R, i = 1,2, ..., m, so that the desired set point [x.sub.des] is an attractive set for (1), such that x(t; 0, [x.sub.0]) [right arrow] [x.sub.des], as t [right arrow] [infinity] for any initial condition.

Generally, by appropriate translation of coordinate system, [x.sub.des] = 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 fields [G.sub.1] (x), ..., [G.sub.m](x) are linearly independent.

(P2) System (1) satisfies the Lie algebra rank condition (LARC) for accessibility, where Lie algebra, L([G.sub.1], ..., [G.sub.m])(x), spans [R.sup.n] at each point x [member of] [R.sup.n].

3. The Proposed Control Algorithm

Step 1. Write system (1) in the following form:

[mathematical expression not reproducible], (2)

where [g.sub.i] : [R.sup.n] x [R.sup.m] [right arrow] R are nonlinear functions.

Step 2. Using the input transformation, transform system (2) into the following form:

[mathematical expression not reproducible], (3)

where [h.sub.i] : [R.sup.n] x [R.sup.m] [right arrow] R are nonlinear function and v is the new input.

After some manipulation, system (3) can be rewritten as

[mathematical expression not reproducible], (4)

where [F.sub.i] = -[x.sub.i+1] + [h.sub.i](x).

Step 3. Assume that [F.sub.i] are uncertainties in the system. Let [F.sub.i], i = 1, ..., n be an estimate of [F.sub.i], i = 1, ..., n - 1, respectively. Apply the function approximation technique [27] to represent [F.sub.i] and their estimates [mathematical expression not reproducible].

[[phi].sub.i](t) = [[[phi].sub.i1](t) [[phi].sub.i2](t) ... [[phi].sub.in](t)].sup.T] is the function of basis vector and [w.sub.i] = [[[w.sub.i1] [w.sub.i2] ... [w.sub.in]].sup.T] is a vector of weightings. Let [mathematical expression not reproducible] be estimate of [w.sub.i] = [[[w.sub.i1] [w.sub.i2] ... [w.sub.in]].sup.T]. Therefore, we can estimate [F.sub.i] by estimating the weight vector [w.sub.i]; that is, [mathematical expression not reproducible]. Define [mathematical expression not reproducible]; then system (4) can be written as

[mathematical expression not reproducible]. (5)

Step 4. Choose the nominal system for (5) as

[mathematical expression not reproducible]. (6)

Step 5. Define the sliding surface for nominal system (6) as

[s.sub.0] = [x.sub.1] + [n-1.summation over (i=2)] [c.sub.i][x.sub.i] + [x.sub.n], (7)

where [c.sub.i] > 0 are chosen in such a way that [s.sub.0] becomes Hurwitz polynomial. Then

[mathematical expression not reproducible]. (8)

Choose

[mathematical expression not reproducible]. (9)

We have [[??].sub.0] = -k sign([s.sub.0]). Therefore, nominal system (6) is asymptotically stable.

Step 6. Define the sliding surface for system (5) as

S = [s.sub.0] + z = [x.sub.1] + [n-1.summation over (i=2)] [c.sub.i][x.sub.i] + [x.sub.n] + z, (10)

where z is an integral term. To avoid the reaching phase, choose z(0) such that s(0) = 0. Choose v = [v.sub.0] + [v.sub.s], where [v.sub.0] is the nominal input and vs is compensator term. Then

[mathematical expression not reproducible], (11)

where [c.sub.1] = 1.

Step 7. Choose a Lyapunov function as

[mathematical expression not reproducible]. (12)

Design the adaptive laws for [[??].sub.i] & [[??].sub.i], i = 1, ..., n and compute [v.sub.s] such that [??] < 0.

Theorem 1. Choose a Lyapunov function as

[mathematical expression not reproducible]. (13)

The following adaptive laws for [mathematical expression not reproducible] and the value of [v.sub.s] will guarantee the time derivative of V in (13) to be strictly negative (i.e., [??] < 0)

[mathematical expression not reproducible], (14)

where k and [k.sub.i] > 0, i = 1, ..., n - 1.

Proof. Since

[mathematical expression not reproducible], (15)

by using

[mathematical expression not reproducible], (16)

where k and [k.sub.i] > 0, i = 1, ..., n - 1, we have

[mathematical expression not reproducible]. (17)

Choosing

[k.sub.n] = min (k, [k.sub.1], ..., [k.sub.n-1]), (18)

we have

[mathematical expression not reproducible]. (19)

Using the chosen Lyapunov function

[mathematical expression not reproducible], (20)

we can write

[mathematical expression not reproducible], (21)

where [alpha] = 2[k.sub.n] and [beta] = 1.

From this we conclude that s & [[??].sub.i] [right arrow] 0 i = 1, ..., n. Since s [right arrow] 0, therefore x [right arrow] 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 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]

[mathematical expression not reproducible]. (22)

Introducing a new set of state variables [mathematical expression not reproducible] the kinematics model (22) can be written as

[mathematical expression not reproducible] (23)

or

[??] = [G.sub.1] (x)[u.sub.1] + [G.sub.2] (x)[u.sub.2], x [member of] [R.sup.3], (24)

where

[mathematical expression not reproducible]. (25)

The kinematics model (22) satisfies the following assumptions:

(P1) The vector fields [G.sub.1](x) and [G.sub.2](x) are linearly independent.

(P2) System (24) satisfies the Lie algebra rank condition (LARC) for accessibility, where the Lie algebra, L([G.sub.1], [G.sub.2])(x), spans [R.sup.3] at each point x [member of] [R.sup.3].

To verify property (P2), it is sufficient to calculate the following Lie bracket of [G.sub.1](x) and [G.sub.2](x):

[mathematical expression not reproducible]. (26)

Then the LARC condition, namely, span([G.sub.1], [G.sub.2], [G.sub.3])(x) = [R.sup.3], [for all] x [member of] [R.sup.3], is satisfied.

4.1.1. Application of the Proposed Algorithm to the Unicycle Model

Step 1. The unicycle model given (24) can be rewritten as

[mathematical expression not reproducible]. (27)

Step 2. Choose [u.sub.1] = v and [u.sub.2] = [x.sub.3]/cos [x.sub.1], where, [x.sub.1] [not equal to] [pi]/2; then system (27) becomes

[mathematical expression not reproducible]. (28)

After some manipulation the above-mentioned system can be written as

[mathematical expression not reproducible], (29)

where F = -[x.sub.1] + [x.sub.3] tan [x.sub.1].

Step 3. Assume F as an uncertainty and let [??] be an estimate of F. The estimate of F by function approximating technique [27] is F = [w.sup.T][phi]. Then [mathematical expression not reproducible] and system (29) can be written as

[mathematical expression not reproducible]. (30)

Step 4. Choose the nominal system for (30) as

[mathematical expression not reproducible]. (31)

Step 5. Define the sliding surface for nominal system (31) as

[s.sub.0] = [x.sub.2] + 2[x.sub.3] + [x.sub.1]. (32)

Then

[mathematical expression not reproducible]. (33)

By choosing

[v.sub.0] = -[x.sub.3] - 2[x.sub.1] - k sign ([s.sub.0]), k > 0, (34)

we have

[[??].sub.0] = -k sign ([s.sub.0]). (35)

Therefore, nominal system (31) is asymptotically stable.

Step 6. Define the sliding surface for system (30) as

s = [s.sub.0] + z = [x.sub.2] + 2[x.sub.3] + [x.sub.1] + z. (36)

Choose v = [v.sub.0] + [v.sub.s].

Then

[mathematical expression not reproducible]. (37)

Step 7. The adaptive laws for [??], [??] and the value of [v.sub.s] are as follows:

[mathematical expression not reproducible], (38)

where k and [k.sub.1] > 0.

Give

[mathematical expression not reproducible], (39)

where V = (1/2)[s.sup.2] + (1/2)[[??].sup.T][??].

Choosing

[k.sub.2] = min (k, [k.sub.1]), (40)

we have

[mathematical expression not reproducible]. (41)

Using the chosen Lyapunov function we can write

[mathematical expression not reproducible], (42)

where [alpha] = 2[k.sub.2] and [beta] = 1.

From this we conclude that s & [??] [right arrow] 0. Since s [right arrow] 0, therefore x [right arrow] 0.

Simulation results are shown in Figure 4.

4.2. 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 depth-two Lie bracket. A front wheel car kinematic model [6] can be defined as

[mathematical expression not reproducible]. (43)

Assuming that l = 1 and introducing a new set of state variables [mathematical expression not reproducible] the kinematics model (43) can be written as

[mathematical expression not reproducible] (44)

or

[??] = [G.sub.1] (x) [u.sub.1] + [G.sub.2] (x) [u.sub.2], x [member of] [R.sup.4], (45)

where [mathematical expression not reproducible].

The kinematics model (45) satisfies the following assumptions:

(P1) The vector fields [G.sub.1](x) and [G.sub.2](x) are linearly independent.

(P2) System (45) satisfies the Lie algebra rank condition (LARC) for accessibility, where the Lie algebra, L([G.sub.1],[G.sub.2])(x), spans [R.sup.4] at each point x [member of] [R.sup.4].

To verify property (P2), it is sufficient to calculate the following Lie brackets of [G.sub.1](x) & [G.sub.2](x):

[mathematical expression not reproducible]. (46)

which satisfy the LARC condition: span([G.sub.1], [G.sub.2], [G.sub.3], [G.sub.4])(x) = [R.sup.4], [for all]x [member of] [R.sup.4].

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

[mathematical expression not reproducible]. (47)

Step 2. Choose [u.sub.1] = v and [u.sub.2] = [x.sub.3]/cos [x.sub.4], where [x.sub.4] [not equal to] [pi]/2, and then system (47) becomes

[mathematical expression not reproducible], (48)

which can be rewritten as

[mathematical expression not reproducible], (49)

where

[F.sub.3] = -[x.sub.4] + [x.sub.3] tan [x.sub.4], [F.sub.4] = -[x.sub.1] + [x.sub.3] tan [x.sub.1] sec [x.sub.4]. (50)

Step 3. Treat [F.sub.i], i = 3,4 as uncertainties and let [[??].sub.i], i = 3,4 be an estimate of [F.sub.i], i = 3,4, respectively. Using function approximation technique [27], we can approximate [mathematical expression not reproducible]. Then [mathematical expression not reproducible].

Then system (49) can be written as

[mathematical expression not reproducible]. (51)

Step 4. Choose the nominal system for (51) as

[mathematical expression not reproducible]. (52)

Step 5. Define the sliding surface for nominal system (52) as

[s.sub.0] = [x.sub.1] + 3[x.sub.3] + 3[x.sub.4] + [x.sub.2]. (53)

Then

[mathematical expression not reproducible]. (54)

By choosing

[v.sub.0] = -[x.sub.3] - 3[x.sub.4] - 3[x.sub.2] - k sign ([s.sub.0]), k > 0, (55)

we have

[[??].sub.0] = - sign ([s.sub.0]). (56)

Therefore, nominal system (52) is asymptotically stable.

Step 6. Define the sliding surface for system (51) as

s = [s.sub.0] + z = [x.sub.1] + 3[x.sub.3] + 3[x.sub.4] + [x.sub.2] + z. (57)

Choose V = [v.sub.0] + [v.sub.s].

Then

[mathematical expression not reproducible]. (58)

Step 7. The following adaptive laws for [mathematical expression not reproducible] and the value of [v.sub.s] are chosen as

[mathematical expression not reproducible], (59)

where k, [k.sub.1] and [k.sub.2] > 0.

Give

[mathematical expression not reproducible], (60)

where

[mathematical expression not reproducible]. (61)

Choosing

[k.sub.3] = min (k, [k.sub.1], [k.sub.2]), (62)

we have

[mathematical expression not reproducible]. (63)

Using the chosen Lyapunov function we can write

[mathematical expression not reproducible], (64)

where [alpha] = 2[k.sub.3] and [beta] = 1.

From this we conclude that [mathematical expression not reproducible]. Since s [right arrow] 0, therefore x [right arrow] 0.

Simulation results are shown in Figure 5.

4.3. 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

[mathematical expression not reproducible]. (65)

By assuming l = d = 1, system (65) can be written in the following standard form:

[??] = [G.sub.1] (x) [u.sub.1] + [G.sub.2] (x) [u.sub.2], x [member of] [R.sup.5], (66)

where

[mathematical expression not reproducible]. (67)

It can be verified that system (66) satisfies the following assumptions which are necessary for steering problem.

(P1) The vectors [G.sub.i](x), i = 1,2 are linearly independent and have no singular point for all x [member of] M [subset or equal to] [R.sup.5], where M is some manifold in [R.sup.5].

(P2) System (66) satisfies the Lie algebraic rank condition (LARC) for controllability, where the Lie algebra, L([G.sub.1],[G.sub.2])(x), spans [R.sup.5] at each point x [member of] M [subset or equal to] [R.sup.5]: that is, span([G.sub.1](x),[G.sub.2](x), ..., [G.sub.5](x)) = [R.sup.5], [for all]x [member of] M.

To verify (P1) and (P2), calculate the linearly independent Lie brackets.

[mathematical expression not reproducible]. (68)

If the motion of system is restricted to manifold,

[mathematical expression not reproducible]. (69)

Then the Lie algebra rank condition, namely, span([G.sub.1](x), [G.sub.2](x), ..., [G.sub.5](x)) = [R.sup.5], [for all]x [member of] 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

[mathematical expression not reproducible]. (70)

Step 2. Choose [u.sub.1] = [x.sub.2]/ cos [x.sub.3] cos [x.sub.4] and [u.sub.2] = v.

And, [x.sub.3], [x.sub.4] [not equal to] [pi]/2. Then system (70) can be written as

[mathematical expression not reproducible]. (71)

Step 3. Assume [F.sub.i], i = 2,4,5 as uncertainties and let [[??].sub.i], i = 2,4,5 be an estimate of [F.sub.i], i = 2,4, 5, respectively. Approximate [mathematical expression not reproducible] 2,4, 5, respectively. Then system (71) can be written as

[mathematical expression not reproducible]. (72)

Step 4. Choose the nominal system for (72) as

[mathematical expression not reproducible]. (73)

Step 5. Define the Hurwitz sliding surface for nominal system (73) as

[s.sub.0] = [x.sub.1] + 4[x.sub.2] + 6[x.sub.4] + 4[x.sub.5] + [x.sub.3]. (74)

Then

[mathematical expression not reproducible]. (75)

By choosing

[v.sub.0] = -[x.sub.2] - 4[x.sub.4] - 6[x.sub.5] - 4[x.sub.3] - k sign ([s.sub.0]), k > 0, (76)

we have

[[??].sub.0] = - sign ([s.sub.0]). (77)

Therefore, nominal system (73) is asymptotically stable.

Step 6. Define the sliding surface for system (72) as

s = [s.sub.0] + z = [x.sub.1] + 4[x.sub.2] + 6[x.sub.4] + 4[x.sub.5] + [x.sub.3] + z. (78)

And choose v = [v.sub.0] + [v.sub.s].

Then

[mathematical expression not reproducible]. (79)

Step 7. The following adaptive laws for [mathematical expression not reproducible] and the value of [v.sub.s]

[mathematical expression not reproducible], (80)

with k and [k.sub.i] > 0, i = 1,2, 3, result in

[mathematical expression not reproducible], (81)

where

[mathematical expression not reproducible]. (82)

Choosing

[k.sub.4] = min (k,[k.sub.1], [k.sub.2], [k.sub.3]), (83)

we have

[mathematical expression not reproducible]. (84)

Using the chosen Lyapunov function we can write

[mathematical expression not reproducible], (85)

where [alpha] = 2[k.sub.4] and [beta] = 1.

From this we conclude that [mathematical expression not reproducible]. Since s [right arrow] 0, therefore x [right arrow] 0. Simulation results are shown in Figures 4-6 for different initial conditions.

5. Simulation Results

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 1sec. 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.

http://dx.doi.org/10.1155/2016/9617283

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] R. W. Brockett, "Asymptotic stability and feedback stabilization," in Differential Geometric Control Theory, R. S. Millman and H. J. Sussmann, Eds., vol. 27, pp. 2961-2963, Birkhaauser, Boston, Mass, USA, 1983.

[2] R. M. Murray, G. Walsh, and S. S. Sastry, "Stabilization and tracking for nonholonomic control systems using time-varying state feedback," in Proceedings of the IFAC Symposium on Nonlinear Control Systems Design 1992, pp. 109-114, June 1992.

[3] A. Astolfi, "Discontinuous control of nonholonomic systems," Systems and Control Letters, vol. 27, no. 1, pp. 37-45, 1996.

[4] Y. Liu and H. Yu, "A survey of underactuated mechanical systems," IET Control Theory and Applications, vol. 7, no. 7, pp. 921-935, 2013.

[5] S. K. Shah and H. G. Tanner, "Control of stochastic unicycle-type robots," in Proceedings of the IEEE International Conference on Robotics and Automation (ICRA '15), pp. 389-394, May 2015.

[6] R. M. Murray and S. S. Sastry, "Nonholonomic motion planning: steering using sinusoids," IEEE Transactions on Automatic Control, vol. 38, no. 5, pp. 700-716, 1993.

[7] S. S. Ge, Z. Wang, and T. H. Lee, "Adaptive stabilization of uncertain nonholonomic systems by state and output feedback," Automatica, vol. 39, no. 8, pp. 1451-1460, 2003.

[8] W. Pasillas-Lepine and W. Respondek, "Conversion of the kinematics of the n-trailer system into Kumpera-Ruiz normal form and motion planning through the singular locus," in Proceedings of the 38th IEEE Conference on Decision and Control (CDC '99), vol. 3, pp. 2914-2919, Phoenix, Ariz, USA, December 1999.

[9] V. I. Utkin, Sliding Modes in Control and Optimization, Springer Science & Business Media, Berlin, Germany, 2013.

[10] Z. Xi, G. Feng, Z. P. Jiang, and D. Cheng, "Output feedback exponential stabilization of uncertain chained systems," Journal of the Franklin Institute, vol. 344, no. 1, pp. 36-57, 2007.

[11] A. Donaire, J. G. Romero, T. Perez, and R. Ortega, "Smooth stabilisation of nonholonomic robots subject to disturbances," in Proceedings of the IEEE International Conference on Robotics and Automation (ICRA '15), pp. 4385-4390, IEEE, Seattle, Wash, USA, May 2015.

[12] C. Gruber and M. Hofbaur, "Remarks on the classification of wheeled mobile robots," Mechanical Sciences, vol. 7, no. 1, pp. 93-105, 2016.

[13] J. Zhang and Y. Liu, "Adaptive stabilization of a class of high-order uncertain nonholonomic systems with unknown control coefficients," International Journal of Adaptive Control and Signal Processing, vol. 27, no. 5, pp. 368-385, 2013.

[14] L.-Y. Sun, S. Tong, and Y. Liu, "Adaptive backstepping sliding mode Hcontrol of static var compensator," IEEE Transactions on Control Systems Technology, vol. 19, no. 5, pp. 1178-1185, 2011.

[15] D. Liberzon, Switching in Systems and Control, Springer Science & Business Media, 2012.

[16] S. Mobayen, "Finite-time tracking control of chained-form nonholonomic systems with external disturbances based on recursive terminal sliding mode method," Nonlinear Dynamics, vol. 80, no. 1-2, pp. 669-683, 2015.

[17] J.-M. Yang and J.-H. Kim, "Sliding mode control for trajectory tracking of nonholonomic wheeled mobile robots," IEEE Transactions on Robotics and Automation, vol. 15, no. 3, pp. 578-587, 1999.

[18] S. Ding, J. Wang, and W. X. Zheng, "Second-order sliding mode control for nonlinear uncertain systems bounded by positive functions," IEEE Transactions on Industrial Electronics, vol. 62, no. 9, pp. 5899-5909, 2015.

[19] Y. Wu, F. Gao, and Z. Zhang, "Saturated finite-time stabilization of uncertain nonholonomic systems in feedforward-like form and its application," Nonlinear Dynamics, vol. 84, no. 3, pp. 1609-1622, 2016.

[20] S. Ding, A. Levant, and S. Li, "Simple homogeneous sliding-mode controller," Automatica, vol. 67, pp. 22-32, 2016.

[21] H. Chen, B. Li, B. Zhang, and L. Zhang, "Global finite-time partial stabilization for a class of nonholonomic mobile robots subject to input saturation," International Journal of Advanced Robotic Systems, vol. 12, no. 11, article 159, 2015.

[22] Y. Wu, F. Gao, and Z. Liu, "Finite-time state-feedback stabilisation of non-holonomic systems with low-order non-linearities," IET Control Theory & Applications, vol. 9, no. 10, pp. 1553-1560, 2015.

[23] F. Gao and F. Yuan, "Adaptive finite-time stabilization for a class of uncertain high order nonholonomic systems," ISA Transactions, vol. 54, pp. 75-82, 2015.

[24] F. Gao, Y. Wu, and Z. Zhang, "Finite-time stabilization of uncertain nonholonomic systems in feedforward-like form by output feedback," ISA Transactions, vol. 59, pp. 125-132, 2015.

[25] M. Defoort, T. Floquet, A. Kokosy, and W. Perruquetti, "Integral sliding mode control for trajectory tracking of a unicycle type mobile robot," Integrated Computer-Aided Engineering, vol. 13, no. 3, pp. 277-288, 2006.

[26] Z. Tang, J. Zhou, X. Bian, and H. Jia, "Simulation of optimal integral sliding mode controller for the depth control of AUVT in Proceedings of the IEEE International Conference on Information and Automation (ICIA '10), pp. 2379-2383, IEEE, Harbin, China, June 2010.

[27] A. C. Huang, Y. F. Chen, and C. Y. Kai, Adaptive Control of Underactuated Mechanical Systems, World Scientific, 2015.

Waseem Abbasi (1,2) and Fazal ur Rehman (1)

(1) Department of Electrical Engineering, Capital University of Science and Technology (CUST), Kahuta Road, Express Highway, Islamabad 44000, Pakistan

(2) Department of Electrical Engineering, The University of Lahore (UOL), Japan Road, Express Highway, Islamabad 44000, Pakistan

Correspondence should be addressed to Waseem Abbasi; waseemabbasi97@gmail.com

Received 23 June 2016; Revised 28 September 2016; Accepted 19 October 2016

Academic Editor: Muhammad N. Akram

Caption: Figure 1: The unicycle model.

Caption: Figure 2: The front wheel car model.

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

Caption: Figure 4: (a) Time response of the system states corresponding to initial condition ([x.sub.1](0), [x.sub.2](0), [x.sub.3](0)) = (2, -1,1). (b) Control input v.

Caption: Figure 5: (a) Time response of the system states corresponding to initial condition ([x.sub.1](0), ..., [x.sub.4](0)) = (-2,1, -1,3). (b) Control input v.

Caption: Figure 6: (a) Time response of the system states corresponding to initial condition ([x.sub.1](0), ..., [x.sub.5](0)) = (2,-1,1, -2,2). (b) Control input v.

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Title Annotation: | Research Article |
---|---|

Author: | Abbasi, Waseem; Rehman, Fazal ur |

Publication: | Mathematical Problems in Engineering |

Date: | Jan 1, 2016 |

Words: | 4670 |

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