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Journal of Robotics
Volume 2011 (2011), Article ID 697309, 8 pages
http://dx.doi.org/10.1155/2011/697309
Research Article

Local Exponential Regulation of Nonholonomic Systems in Approximate Chained Form with Applications to Off-Axle Tractor-Trailers

The Seventh Research Division, BeiHang University, Beijing 100191, China

Received 21 January 2011; Revised 29 April 2011; Accepted 2 June 2011

Academic Editor: G. Muscato

Copyright © 2011 Bao-Li Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Most of drift-less nonholonomic systems cannot be exactly converted to an nonholonomic chained form, a wealth of design tools developed for the control of nonholonomic chained form are thus not directly applicable to such systems. Nevertheless, there exists a class of systems that may be locally approximated by the nonholonomic chained form around certain equilibrium points. In this work, we propose a discontinuous and a smooth time-varying control laws respectively for the approximated nonholonomic chained form, guaranteeing local exponential convergence of state to the desired equilibrium point. An tractor towing off-axle trailers is taken as an example to illustrate the approaches.

1. Introduction

The so-called nonholonomic chained form (NCF) has motivated many research activities for about twenty years [1]. Several features such as flatness [2, 3], homogeneity, and nilpotency make the NCF especially attractive to work with. These properties have been used for designing control laws to achieve several control objects such as point stabilization and trajectory tracking. Concerning the point stabilization problem of NCF, which is difficult due to Brokett's well-known obstruction [4], a number of approaches have been developed, which may be roughly classified into discontinuous time-invariant feedback [57], continuous time-varying feedback [810], and hybrid feedback [11, 12]. The stabilization problems of NCF with parameter uncertainties and perturbation terms have also been attacked in recent years [1317]; however, most of these researches require that the perturbation terms satisfy certain cascaded conditions, which may be very restrictive and thus rule out many interesting examples such as the tractor-trailers with off-axle hitching [18] and the ball-plate systems [19]. It is also mentioned that the dynamics of many nonholonomic driftless systems can be approximated by NCF locally around certain equilibrium points. In [18], a time-varying continuous stabilizing scheme was proposed for such approximate NCF, achieving local exponential stability of the closed-loop system around the selected equilibrium point.

In this paper, we consider the local exponential regulation problem of a class of nonholonomic systems convertible to the approximate NCF. By employing a discontinuous and/or a smooth time-varying coordinate transformations, the approximate NCF is converted to linear perturbed ones with the perturbation terms being second or higher orders of the converted states; then a discontinuous time-invariant and/or a smooth time-varying control laws are derived respectively, guaranteeing that the state of the approximate NCF converges to zero exponentially, provided the norm of an initial state is sufficiently small. Compared with the control law presented in [18] which is continuous but not differentiable, the time-varying control law proposed in this paper is smooth and can be easily extended to deal with input dynamics.

The paper is organized as follows. Section 2 defines a class of systems that can be approximated by NCF. In Section 3, a discontinuous time-invariant and a smooth time-varying controllers are developed to stabilize the approximate NCF. In Section 4, a tractor-trailer with off-axle hitching is taken as an example to illustrate the effectiveness of the proposed controllers. Section 5 concludes the paper.

2. A Class of Approximated Chained Forms

Consider the following nonlinear system represented by ̇𝑥0=𝑢0,(1)̇𝑥=𝑔0(𝑥)𝑢0+𝑔1(𝑥)𝑢1,(2) where 𝑥0,𝑥𝑛 are state variables and 𝑢0,𝑢1 are control inputs. The control vector fields 𝑔0(𝑥)𝑛,𝑔1(𝑥)𝑛 are supposed to have the following forms: 𝑔0(𝑥)=𝐴𝑥+𝑅2(𝑥),𝑔1(𝑥)=𝑏+𝑅1(𝑥),(3) where 1000𝐴=0000100001000010,𝑏=,(4)𝑅1(𝑥)𝑛 denotes the first-or higher-order residual term of 𝑥 and 𝑅2(𝑥)𝑛 the second or higher-order residual term of 𝑥 in the state domain 𝐷; or, say more precisely, there exist three positive constants 𝑟,𝑟1, and 𝑟2 such that 𝑅1(𝑥),𝑅2(𝑥) are bounded by 𝑅1(𝑥)2𝑟1𝑥2,𝑅2(𝑥)2𝑟2𝑥22(5) in the compact set Ω={𝑥𝑥2𝑟}𝐷.

System (1)-(2) is called the approximate NCF if (3)–(5) are satisfied.

Remark 1. Without loss of generality, it is specially assumed in (4) that {𝐴,𝑏} is in the canonical controllable form. For the controllable pair {𝐴,𝑏} not in this form, one can always find a linear state transformation to convert it to this form.

Remark 2. It is noted that the approximate NCF (1)-(2) is not flat with certain defects [2] and thus difficult to control.

The approximate NCF represents a large class of nonholonomic systems that cannot be converted to NCF in which 𝑅(𝑥)=0. The examples of approximate NCF include tractor-trailers with off-axle hitching [18] and the ball-plate systems [19].

3. Local Exponential Regulation of the Approximate NCF

In this section, a discontinuous and a smooth time-varying control laws are derived to solve the local exponential regulation problem of the approximate NCF defined in (1)–(5).

3.1. Local Exponential Regulation of the Approximate NCF for 𝑥0(0)0

The control law for the first control input is designed as 𝑢0=𝑘0𝑥0,(6) with 𝑘0>0, so that 𝑥0(𝑡)=𝑥0(0)𝑒𝑘0𝑡0(𝑥0(0)0,0𝑡<).

Substituting (6) into (2) results ̇𝑥=𝑘0𝑥0𝐴𝑥+𝑅2+(𝑥)𝑏+𝑅1𝑢(𝑥)1.(7)

Inspired by the well-known 𝜎-process [5], we introduce the following discontinuous state transformation:𝑦=𝑇1𝑥0𝑥𝑥,𝑥=𝑇0𝑦(8) with𝑇𝑥0=𝑥𝑚0diag1,𝑥0,𝑥20,,𝑥0𝑛1,𝑇1𝑥0=𝑥0𝑚diag1,𝑥01,𝑥02,,𝑥0(𝑛1),(9) and 𝑚 a positive integer to be determined.

Remark 3. The discontinuous coordinate transformation (8)-(9) is a generalization of the ordinary 𝜎process proposed in [5] with 𝑚=0 for NCF. It is seen in what follows that the term 𝑥𝑚0 with 𝑚>0 is crucial for the controller design of the approximate NCF.

The transformation matrix 𝑇(𝑥0) is clearly nonsingular for 𝑥0(0)0,0𝑡<.

The dynamics of the transformed state 𝑦 can be derived as ̇𝑦=𝑇1𝑥0𝑑̇𝑥+𝑇𝑑𝑡1𝑥0𝑥=𝑘0𝑥0𝑇1𝑥0𝑥𝐴𝑇0𝑦+𝑇1𝑥0𝑏𝑢1+𝑇1𝑥0𝑘0𝑥0𝑅2+𝑅1𝑢1+𝑑𝑇𝑑𝑡1𝑥0𝑇𝑥0𝑦.(10)

Direct calculation reveals that 𝑇1𝑥0𝑏=𝑥0𝑚𝑥𝑏,0𝑇1𝑥0𝑥𝐴𝑇0𝑑=𝐴,𝑇𝑑𝑡1𝑥0𝑇𝑥0=𝑘0diag{𝑚,𝑚+1,𝑚+2,,𝑚+𝑛1}.(11)

Substituting the above identities into (10) results iṅ𝑦=𝐴1𝑦+𝑥0𝑚𝑏𝑢1+𝑇1𝑥0𝑘0𝑥0𝑅2+𝑅1𝑢1,(12) where 𝐴1=𝑘0(𝐴+diag{𝑚,𝑚+1,𝑚+2,,𝑚+𝑛1}).(13)

Remark 4. As {𝐴,𝑏} is controllable, so is {𝐴1,𝑏}; hence, the eigenvalues of 𝐴1𝑏𝐾 can be arbitrarily assigned by selecting the control gain 𝐾.

The second control input is designed as 𝑢1=𝑥𝑚0𝐾𝑦=𝑥𝑚0𝐾𝑇1𝑥0𝑥,(14) where 𝐾=[𝑘1,𝑘2,,𝑘𝑛] is a control gain row vector such that 𝐴1𝑏𝐾 is Hurwitz.

The closed-loop system of (12) and (14) becomes 𝐴̇𝑦=1𝑏𝐾𝑦𝑇1𝑥0𝑘0𝑥0𝑅2+𝑅1𝑥𝑚0=𝐴𝐾𝑦1𝑏𝐾𝑦+𝑅,(15) where 𝑅=𝑇1𝑥0𝑘0𝑥0𝑅2+𝐾𝑦𝑥𝑚0𝑅1.(16)

System (15) is a linear stable one perturbed by a residual term 𝑅. If 𝑅 can be shown to be second or higher order of 𝑦, then (15) is locally exponential stable.

In view of (5), the converted residual term 𝑅 is bounded by 𝑅2𝑇12𝑘0𝑥0𝑅2+𝐾𝑥𝑚0𝑦𝑅12𝑇12𝑟2𝑘0||𝑥0||𝑥22+𝑟1𝐾2||𝑥0||𝑚𝑥2𝑦2𝑟2𝑘0||𝑥0||𝑇12𝑇22+𝑟1𝐾2||𝑥0||𝑚𝑇12𝑇2𝑦22||𝑥=0||𝑦22(17) with (|𝑥0|) defined as ||𝑥0||𝑟2𝑘0||𝑥max0||𝑚+1,||𝑥0||𝑚𝑛+2||𝑥max1,0||2(𝑛1)+𝑟1𝐾2||𝑥max0||𝑚,||𝑥0||𝑚(𝑛1)||𝑥×max1,0||𝑛1.(18)

As |𝑥0||𝑥0(0)|, (|𝑥0|) is thus bounded uniformly with 𝑡 provided 𝑚(𝑛1)0. In view of the facts that 𝐴1𝑏𝐾 is Hurwitz and lim𝑦20(𝑅2/𝑦2)=0, system (15) is thus locally exponential stable by Lyapunov indirect approach [20].

The above analysis is summarized as the following proposition.

Proposition 1. Suppose that 0<|𝑥0(0)|<, 𝑘0>0, 𝑚n1, 𝐾 is selected such that 𝐴1𝑏𝐾 is Hurwitz then the following control law 𝑢0=𝑘0𝑥0,𝑢1=𝑥𝑚0𝐾𝑦=𝑥𝑚0𝐾𝑇1𝑥0𝑥(19) guarantees that the states 𝑥0(𝑡), 𝑢0(𝑡) globally converge to zero and 𝑥(𝑡), 𝑢1(𝑡) converge to zero exponentially for a sufficiently small 𝑦(0)2.

Proof. It is obvious that 𝑥0(𝑡)=𝑥0(0)𝑒𝑘0𝑡,𝑢0(𝑡)=𝑘0𝑥0(𝑡) globally converge to zero exponentially. As 𝐴1𝑏𝐾 is Hurwitz and 𝑅2(|𝑥0|)𝑦22 with (|𝑥0|) uniformly bounded with 𝑡, the closed-loop system (15) is locally exponential stable, implying that 𝑦(𝑡),𝑥(𝑡)=𝑇(𝑥0(𝑡))𝑦(𝑡) and 𝑢1(𝑡)=𝑥𝑚0(𝑡)𝐾𝑦(𝑡) are all convergent to zero exponentially for a sufficiently small 𝑦(0)2.

Proposition 1 is only applicable for 𝑥0(0)0. In the case of 𝑥0(0)=0, the proposed control law fails to work as the transformation matrix 𝑇(𝑥0) becomes singular. This problem may be solved by introducing a switching mechanism that first drives 𝑥0 away from zero in finite time and then switches to the control law (19) to achieve local exponential regulation for an arbitrarily 𝑥0(0) and a sufficiently small 𝑥(0)2. However, such switching control law is discontinuous and may cause problems when the velocity input dynamics is included in the model since the discontinuities of velocity inputs lead to infinite accelerations.

In the next subsection, the controller (19) is modified to be smooth and time varying for an arbitrary 𝑥0(0) so that the acceleration signals are bounded.

3.2. Local Exponential Regulation of the Approximate NCF for an Arbitrary 𝑥0(0)

The control law for the first control input is designed as 𝑢0=𝑘0𝛼(𝑡)𝑘0𝑥0𝛼(𝑡)(20) with 𝛼(𝑡)=𝛼0𝑒𝑘0𝑡, 𝛼00, 𝑘0>𝑘0>0.

Let 𝑒0(𝑡)=𝑥0(𝑡)𝛼(𝑡); then ̇𝑒0(𝑡)=𝑢0(𝑡)̇𝛼(𝑡)=𝑘0𝑒0(𝑡), so that 𝑒0(𝑡)=𝑒0(0)𝑒𝑘0𝑡, 𝑥0(𝑡)=𝛼(𝑡)+𝑒0(𝑡)=𝛼(𝑡)+𝑒0(0)𝑒𝑘0𝑡, 𝑢0(𝑡)=𝑘0𝛼(𝑡)𝑘0𝑒0(0)𝑒𝑘0𝑡, and 𝑒0(𝑡)/𝛼(𝑡)=(𝑒0(0)/𝛼0)𝑒(𝑘0𝑘0)𝑡 are all globally convergent to zero exponentially.

Now we introduce the following smooth time-varying state transformation:𝑦=𝑇1(𝛼)𝑥,𝑥=𝑇(𝛼)𝑦(21) with 𝑇(𝛼)=𝛼𝑚diag1,𝛼,𝛼2,,𝛼𝑛1,𝑇1(𝛼)=𝛼𝑚diag1,𝛼1,𝛼2,,𝛼(𝑛1),(22) and 𝑚 a positive integer to be determined.

As 𝛼00, the transformation matrix 𝑇(𝛼) is clearly nonsingular for all 0𝑡<.

The dynamics of the transformed state 𝑦 can be derived as ̇𝑦=𝑇1𝑑(𝛼)̇𝑥+𝑇𝑑𝑡1𝑥(𝛼)=𝑢0𝑇1(𝛼)𝐴𝑇(𝛼)𝑦+𝑇1(𝛼)𝑏𝑢1+𝑇1𝑢(𝛼)0𝑅2+𝑅1𝑢1+𝑑𝑇𝑑𝑡1(𝛼)𝑇(𝛼)𝑦=𝑘0𝛼1+𝑘0𝑒0𝑘0𝛼𝑇1(𝛼)𝐴𝑇(𝛼)𝑦+𝑇1(𝛼)𝑏𝑢1+𝑇1(𝛼)𝑘0𝛼1+𝑘0𝑒0𝑘0𝛼𝑅2+𝑅1𝑢1+𝑑𝑇𝑑𝑡1(𝛼)𝑇(𝛼)𝑦.(23)

Simple calculation reveals that 𝑇1(𝛼)𝑏=𝛼𝑚𝑏,𝛼𝑇1𝑑(𝛼)𝐴𝑇(𝛼)=𝐴,𝑇𝑑𝑡1(𝛼)𝑇(𝛼)=𝑘0diag{𝑚,𝑚+1,𝑚+2,,𝑚+𝑛1}.(24)

Substituting the above identities into (23) results in ̇𝑦=𝐴11+𝑘0𝑒0𝑘0𝛼𝑦+𝛼𝑚𝑏𝑢1+𝑇1𝑥0𝑘0𝛼1+𝑘0𝑒0𝑘0𝛼𝑅2+𝑅1𝑢1,(25) where 𝐴1 is defined in (13).

The second control input is designed as 𝑢1=𝛼𝑚𝐾𝑦=𝛼𝑚𝐾𝑇1(𝛼)𝑥,(26) where 𝐾=[𝑘1,𝑘2,,𝑘𝑛] is a control gain row vector selected such that 𝐴1𝑏𝐾 is Hurwitz.

The closed-loop system of (25) and (26) becomes 𝐴̇𝑦=1𝑏𝐾+𝑘0𝑒0𝑘0𝛼𝐴1𝑦𝑇1𝑘(𝛼)0𝛼1+𝑘0𝑒0𝑘0𝛼𝑅2+𝑅1𝛼𝑚=𝐴𝐾𝑦1𝑏𝐾+𝑘0𝑒0𝑘0𝛼𝐴1𝑦+𝑅,(27) where 𝑅=𝑇1𝑘(𝛼)0𝛼1+𝑘0𝑒0𝑘0𝛼𝑅2+𝑅1𝛼𝑚𝐾𝑦.(28)

In view of (5), the converted residual term 𝑅 can be shown to be bounded by 𝑅2𝑇12𝑘0|||||𝛼|1+𝑘0𝑒0𝑘0𝛼||||𝑅22+𝑇12𝐾2|𝛼|𝑚𝑦2𝑅12𝑇12𝑟2𝑘0|||||𝛼|1+𝑘0𝑒0𝑘0𝛼||||𝑥22+𝑇12𝑟1𝐾2|𝛼|𝑚𝑥2𝑦2𝑟2𝑘0||||1+𝑘0𝑒0𝑘0𝛼||||𝑇|𝛼|12𝑇22𝑦22+𝑟1𝐾2|𝛼|𝑚𝑇12𝑇2𝑦22=𝛼,𝑒0𝑦22(29) with (𝛼,𝑒0) defined as𝛼,𝑒0𝑟2𝑘0||||1+𝑘0𝑒0𝑘0𝛼||||max|𝛼|𝑚+1,|𝛼|𝑚𝑛+2×max1,|𝛼|2(𝑛1)+𝑟1𝐾2|max𝛼|𝑚,|𝛼|𝑚(𝑛1)max1,|𝛼|𝑛1.(30)

As 𝛼, 𝑒0/𝛼=(𝑒0/𝛼0)𝑒(𝑘0𝑘0)𝑡 are both bounded uniformly with 𝑡, and (𝑥0,𝑒0) is thus uniformly bounded provided 𝑚(𝑛1)0. Since 𝐴1𝑏𝐾 is Hurwitz and 𝑒0/𝛼 converges to zero exponentially, system ̇𝑦=(𝐴1𝑏𝐾+(𝑘0𝑒0/𝑘0𝛼)𝐴1)𝑦 is globally exponential stable, and hence the perturbed system ̇𝑦=(𝐴2+(𝑘0𝑒0/𝑘0𝛼)𝐴1)𝑦+𝑅 is locally exponential stable by Lyapunov indirect approach [20].

Based on the above analysis, we arrive at the following results.

Proposition 2. Suppose that 𝛼=𝛼(𝑡)=𝛼0𝑒𝑘0𝑡, 𝛼00, 𝑘0>𝑘0>0, 𝑚𝑛1, 𝐾 is selected such that 𝐴1𝑏𝐾 is Hurwitz, then the following control law 𝑢0=𝑘0𝛼𝑘0𝑥0𝛼,𝑢1=𝛼𝑚𝐾𝑦(31) guarantees that the states 𝑥0(𝑡), 𝑢0(𝑡) globally converge to zero exponentially and 𝑥(𝑡), 𝑢1(𝑡) converge to zero exponentially for a sufficiently small 𝑦(0)2.

Proof. It is obvious that 𝑥0(𝑡),𝑢0(𝑡) globally converge to zero exponentially. As 𝐴1𝑏𝐾 is Hurwitz and 𝑅2(𝑥0,𝑒0)𝑦22 with (𝑥0,𝑒0) uniformly bounded with 𝑡, the closed-loop system (27) is locally exponential stable, implying that 𝑦(𝑡),𝑥(𝑡)=𝑇(𝛼(𝑡))𝑦(𝑡) and 𝑢1(𝑡)=𝛼𝑚(𝑡)𝐾𝑦(𝑡) are all convergent to zero exponentially for a sufficiently small 𝑦(0)2.

Remark 5. Compared with the approach presented in [18] where the control law is continuous but not differentiable, the proposed control law (31) in this paper is smooth time varying and hence can be easily extended to include input dynamics of the approximate NCF (1)-(2) by one-step back-steeping.

4. An Example: Local Exponential Regulation of an Off-Axle Tractor-Trailer

Consider a tractor-trailer with a wheeled mobile tractor towing 𝑛 off-axle wheeled trailers shown in Figure 1, where (𝑥𝑖,𝑦𝑖,𝜃𝑖) denote the position and orientation of body 𝑖(𝑖=0,1,2,,𝑛), (𝑣𝑖,𝜔𝑖=̇𝜃𝑖) denote the linear and angular velocities of body 𝑖(𝑖=0,1,2,,𝑛), 𝛽𝑖=𝜃𝑖1𝜃𝑖(𝑖=1,2,3,,𝑛) represent the difference of orientation angles between body 𝑖 and body 𝑖1. 𝑃𝑖(𝑖=0,1,2,) is the center point on the wheel axle of body 𝑖 and 𝑄𝑖1(𝑖=1,2,,𝑛) the connection point of body 𝑖 and body 𝑖1. The distance between 𝑃𝑖 and 𝑄𝑖 is 𝑑𝑖, and the distance between 𝑃𝑖 and 𝑄𝑖1 is 𝑓𝑖.

697309.fig.001
Figure 1: A tractor towing 𝑛 trailer with off-axle hitching.

The kinematic equation of the tractor is ̇𝑥0=𝑣0cos𝜃0,̇𝑦0=𝑣0sin𝜃0,̇𝜃0=𝜔0.(32)

The kinematic relations of trailer 𝑖 can be derived as 𝑣𝑖=𝑣𝑖1cos𝛽𝑖+𝑑𝑖1̇𝜃𝑖1sin𝛽𝑖,̇𝑥𝑖=𝑣𝑖cos𝜃𝑖,̇𝑦𝑖=𝑣𝑖sin𝜃𝑖,̇𝜃𝑖=1𝑓𝑖𝑣𝑖1sin𝛽𝑖𝑑𝑖1̇𝜃𝑖1cos𝛽𝑖.(33)

Select 𝑥=[𝑥0,𝑦0,𝜃0,𝛽1,𝛽2,,𝛽𝑛]𝑇 as the state variables, and 𝑢0=𝑣0cos𝜃0,𝜔0 as the control inputs, the state equation can be derived from (32)-(33) as ̇𝑥0=𝑢0,(34)̇𝑥=𝐴+𝑅2𝑢(𝑥)0+𝑏+𝑅1𝜔(𝑥)0,(35) where 𝑅1,𝑅2 are high-order residual terms satisfying (5) and 1𝐴=010000000000𝑓110000𝑓1𝑑1+1𝑓21𝑓2000𝑎1,𝑛𝑎2,𝑛1𝑓𝑛,𝑑𝑏=011+0𝑓1𝑑0𝑓1𝑑1+1𝑓2𝑏𝑛𝑇,𝑎(36)1,𝑛=(1)𝑛2𝑑𝑛2×𝑑𝑛3×𝑑2×𝑑1𝑓𝑛1×𝑓𝑛2𝑓1𝑑1+𝑛1𝑓𝑛,𝑎2,𝑛=(1)𝑛3𝑑𝑛2×𝑑𝑛3×𝑑3×𝑑2𝑓n1×𝑓𝑛2𝑓2𝑑1+𝑛1𝑓𝑛,𝑏𝑛=(1)𝑛1𝑑𝑛2×𝑑𝑛3×𝑑1×𝑑0𝑓𝑛1×𝑓𝑛2𝑓1𝑑1+𝑛1𝑓𝑛,(37)

The control object can be stated as design control law 𝑢0(),𝜔0() such that the states (𝑥0,𝑦0,𝜃0,𝛽1,𝛽2,,𝛽𝑛) of the closed-loop system (34)-(35) converge to zero exponentially.

To apply Propositions 1 and 2 obtained in Section 3, it is required to verify the controllability of {𝐴,𝑏}.

Lemma 1. Suppose that 𝑑𝑖>0(𝑖=0,1,,𝑛1) and 𝑓𝑖>0(𝑖=1,2,,𝑛), then {𝐴,𝑏} is a controllable pair.

Proof. The lemma can be proved by verifying PBH criterion of linear systems and is omitted here for brevity.

Remark 6. As {𝐴,𝑏} is controllable, it can thus be further converted to the canonical controllable form (4) by a linear transformation so that the tractor-trailers system (34)-(35) can be expressed in approximate NCF (1)-(2).

To illustrate the effectiveness of the proposed control approaches, a tractor towing one trailer is taken as a simulation example. The state equation in this special case can be explicitly obtained as ̇𝑥0=𝑣0cos𝜃0,̇𝑦0=𝑣0sin𝜃0,̇𝜃0=𝜔0,̇𝛽1=𝑐1𝑣0sin𝛽1+1+𝑐2cos𝛽1𝜔0,(38) where 𝑐1=1/𝑓1, 𝑐2=𝑑0/𝑓1.

Under the following coordinate and input transformations: 𝑥1=𝑐21𝛽1,𝑥2=𝑐1𝛽1+𝑐11+𝑐2𝜃0,𝑥3=𝛽11+𝑐2𝜃0+𝑐11+𝑐2𝑦0,𝑢0=𝑣0cos𝜃0,𝑢1=𝑐21𝑐1𝑣0sin𝛽1+1+𝑐2cos𝛽1𝜔0.(39) the state equation (38) is converted to the following form:̇𝑥3=𝑥2+𝑅23𝑢0+𝑅13𝑢1,̇𝑥2=𝑥1+𝑅22𝑢0+𝑅12𝑢1,̇𝑥1=𝑢1,̇𝑥0=𝑢0,(40) where 𝑅23=𝑐1sin𝛽1cos𝜃0𝛽1+1+𝑐2sin𝜃0cos𝜃0𝜃0+𝑐1𝑐2cos𝛽11sin𝛽11+𝑐2cos𝛽1/cos𝜃0,𝑅13=𝑐2cos𝛽11𝑐21/1+𝑐2cos𝛽1,𝑅22=𝑐21sin𝛽1cos𝜃0𝛽1+𝑐21𝑐21cos𝛽1sin𝛽11+𝑐2cos𝛽1/cos𝜃0,𝑅12=𝑐2𝑐11cos𝛽1/1+𝑐2cos𝛽1.(41)

In the state region 𝐷={(𝑥0,𝑦0,𝜃0,𝛽1)|𝜃0|𝜃0𝑀<𝜋/2,|𝛽1|𝛽1𝑀}, |𝑅2𝑗|(𝑗=2,3) can be shown to be 𝑂((𝜃0,𝛽1)32) and 𝑅1𝑗(𝑗=2,3) to be 𝑂((𝜃0,𝛽1)22).

The geometric parameters are set to 𝑑0=𝑓1=1. The controller parameters are selected as 𝑚=2, 𝑘0=0.2, 𝛼0=10, 𝑘0=2, and 𝐾=[1.92,8.26,14.81] chosen such that the eigenvalues of 𝐴1𝑏𝐾 are assigned to (0.02,0.04,0.06).

The simulation is implemented for two initial states (𝑥0(0),𝑦0(0),𝜃0(0),𝛽1(0))=(10,2,0.2,0.2) and (𝑥0(0),𝑦0(0),𝜃0(0),𝛽1(0))=(0,2,0.2,0.2). For the first initial state, where 𝑥0(0)0, the control law (19) is applied; for the second initial state where 𝑥0(0)=0, the control law (31) is applied. The time plots of state trajectories and geometric paths of the tractor and the trailer are shown in Figures 2 and 3 in respect to the two initial states. It is observed that the proposed control laws successfully regulate the state to zero from initial states and produce nice geometric paths for both the tractor and the trailer.

fig2
Figure 2: Time trajectories of states and geometric paths of the tractor and the trailer starting from the first initial state.
fig3
Figure 3: Time trajectories of states and geometric paths of the tractor and the trailer starting from the second initial state.

5. Conclusion

In this paper, we propose a discontinuous and a smooth time-varying control schemes for a class of nonlinear driftless systems in the approximated nonholonomic chained form, achieving local exponential convergence of state to the desired equilibrium point. The proposed control laws rely on the discontinuous and the smooth time-varying state transformations that convert the system to linear stable one perturbed by two- or higher-order terms of state. An application example of off-axle tractor-trailers is discussed in detail for illustrating the effectiveness of the proposed control approaches.

Acknowledgments

The paper is supported by National Science Foundation of China (no. 60874012). The author would like to thank the Editor and the reviewers for their helpful suggestions and careful review of the paper.

References

  1. I. Kolmanovsky and N. H. McClamroch, “Developments in nonholonomic control problems,” IEEE Control Systems Magazine, vol. 15, no. 6, pp. 20–36, 1995. View at Publisher · View at Google Scholar · View at Scopus
  2. M. Fliess, J. Levine, P. Martin, and P. Rouchon, “Flatness and defect of non-linear systems: introductory theory and examples,” International Journal of Control, vol. 61, no. 6, pp. 1327–1361, 1995. View at Scopus
  3. P. Martin and P. Rouchon, “Any (controllable) driftless system with 3 inputs and 5 states is flat,” Systems and Control Letters, vol. 25, no. 3, pp. 167–173, 1995. View at Scopus
  4. R. W. Brockett, “Asymptotic stability and feedback stabilization,” in Differential Geometric Control Theory, R. W. Brockett, R. S. Millman, and H. J. Sussmann, Eds., pp. 181–191, Birkhauser, Boston, Mass, USA, 1983.
  5. A. Astolfi, “Discontinuous control of nonholonomic systems,” Systems and Control Letters, vol. 27, no. 1, pp. 37–45, 1996. View at Publisher · View at Google Scholar · View at Scopus
  6. Z. Sun and S. S. Ge, “Nonregular feedback linearization: a nonsmooth approach,” IEEE Transactions on Automatic Control, vol. 48, no. 10, pp. 1772–1776, 2003. View at Publisher · View at Google Scholar · View at Scopus
  7. N. Marchand and M. Alamir, “Discontinuous exponential stabilization of chained form systems,” Automatica, vol. 39, no. 2, pp. 343–348, 2003. View at Publisher · View at Google Scholar · View at Scopus
  8. C. Samson, “Control of chained systems application to path following and time-varying point-stabilization of mobile robots,” IEEE Transactions on Automatic Control, vol. 40, no. 1, pp. 64–77, 1995. View at Publisher · View at Google Scholar · View at Scopus
  9. O. J. Sordalen and O. Egeland, “Exponential stabilization of nonholonomic chained systems,” IEEE Transactions on Automatic Control, vol. 40, no. 1, pp. 35–49, 1995. View at Publisher · View at Google Scholar · View at Scopus
  10. P. Morin and C. Samson, “Control of nonlinear chained systems: from the Routh-Hurwitz stability criterion to time-varying exponential stabilizers,” IEEE Transactions on Automatic Control, vol. 45, no. 1, pp. 141–146, 2000. View at Publisher · View at Google Scholar · View at Scopus
  11. C. Prieur and A. Astolfi, “Robust stabilization of chained systems via hybrid control,” IEEE Transactions on Automatic Control, vol. 48, no. 10, pp. 1768–1772, 2003. View at Publisher · View at Google Scholar · View at Scopus
  12. I. Kolmanovsky, M. Reyhanoglu, and N. H. McClamroch, “Switched mode feedback control laws for nonholonomic systems in extended power form,” Systems and Control Letters, vol. 27, no. 1, pp. 29–36, 1996. View at Publisher · View at Google Scholar · View at Scopus
  13. Z. P. Jiang, “Robust exponential regulation of nonholonomic systems with uncertainties,” Automatica, vol. 36, no. 2, pp. 189–209, 2000. View at Publisher · View at Google Scholar · View at Scopus
  14. K. D. Do and J. Pan, “Adaptive global stabilization of nonholonomic systems with strong nonlinear drifts,” Systems and Control Letters, vol. 46, no. 3, pp. 195–205, 2002. View at Publisher · View at Google Scholar · View at Scopus
  15. Z. Xi, G. Feng, Z. P. Jiang, and D. Cheng, “A switching algorithm for global exponential stabilization of uncertain chained systems,” IEEE Transactions on Automatic Control, vol. 48, no. 10, pp. 1793–1798, 2003. View at Publisher · View at Google Scholar · View at Scopus
  16. T. Floquet, J. P. Barbot, and W. Perruquetti, “Higher-order sliding mode stabilization for a class of nonholonomic perturbed systems,” Automatica, vol. 39, no. 6, pp. 1077–1083, 2003. View at Publisher · View at Google Scholar · View at Scopus
  17. E. Valtolina and A. Astolfi, “Local robust regulation of chained systems,” Systems and Control Letters, vol. 49, no. 3, pp. 231–238, 2003. View at Publisher · View at Google Scholar · View at Scopus
  18. D. A. Lizárraga, P. Morin, and C. Samson, “Chained form approximation of a driftless system. Application to the exponential stabilization of the general N-trailer system,” International Journal of Control, vol. 74, no. 16, pp. 1612–1629, 2001. View at Publisher · View at Google Scholar · View at Scopus
  19. H. Date, M. Sampei, M. Ishikawa, and M. Koga, “Simultaneous control of position and orientation for ball-plate manipulation problem based on time-state control form,” IEEE Transactions on Robotics and Automation, vol. 20, no. 3, pp. 465–479, 2004. View at Publisher · View at Google Scholar · View at Scopus
  20. H. K. Khalil, Nonlinear Systems, Prentice-Hall, Upper Saddle River, NJ, USA, 2nd edition, 1996.