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

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 [5–7], continuous time-varying feedback [8–10], and hybrid feedback [11, 12]. The stabilization problems of NCF with parameter uncertainties and perturbation terms have also been attacked in recent years [13–17]; 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 𝑥̇=𝑔₀𝑥̇₀=𝑢₀,(1)(𝑥)𝑢₀+𝑔₁(𝑥)𝑢₁,(2) where 𝑥₀∈ℜ,𝑥∈ℜ^𝑛 are state variables and 𝑢₀∈ℜ,𝑢₁∈ℜ are control inputs. The control vector fields 𝑔₀(𝑥)∈ℜ^𝑛,𝑔₁(𝑥)∈ℜ^𝑛 are supposed to have the following forms: 𝑔₀(𝑥)=𝐴𝑥+𝑅₂(𝑥),𝑔₁(𝑥)=𝑏+𝑅₁(𝑥),(3) where 𝐴=⎡⎢⎢⎢⎢⎢⎢⎢⎣⋮0100⋮0010⋱…………⋮0001⋮0000⎤⎥⎥⎥⎥⎥⎥⎥⎦,𝑏=⎡⎢⎢⎢⎢⎢⎢⎢⎣⋮1000⎤⎥⎥⎥⎥⎥⎥⎥⎦,(4)𝑅₁(𝑥)∈ℜ^𝑛 denotes the first-or higher-order residual term of 𝑥 and 𝑅₂(𝑥)∈ℜ^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)‖‖₂≤𝑟₁‖𝑥‖₂,‖‖𝑅₂(𝑥)‖‖₂≤𝑟₂‖𝑥‖²²(5) in the compact set Ω={𝑥∶‖𝑥‖₂≤𝑟}∈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)𝑒^−𝑘0𝑡≠0(∀𝑥₀(0)≠0,0≤𝑡<∞).

Substituting (6) into (2) results 𝑥̇=−𝑘₀𝑥₀𝐴𝑥+𝑅₂(𝑥)+𝑏+𝑅₁(𝑥)𝑢₁.(7)

Inspired by the well-known 𝜎-process [5], we introduce the following discontinuous state transformation:𝑦=𝑇⁻¹𝑥₀𝑥,𝑥=𝑇𝑥₀𝑦(8) with𝑇𝑇₋₁𝑥₀=𝑥^𝑚0diag1,𝑥₀,𝑥²⁰,…,𝑥^0𝑛−1,𝑥₀=𝑥^0−𝑚diag1,𝑥⁰⁻¹,𝑥⁰⁻²,…,𝑥^{0−(𝑛−1)},(9) and 𝑚 a positive integer to be determined.

Remark 3. The discontinuous coordinate transformation (8)-(9) is a generalization of the ordinary 𝜎−process proposed in [5] with 𝑚=0 for NCF. It is seen in what follows that the term 𝑥^𝑚0 with 𝑚>0 is crucial for the controller design of the approximate NCF.

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

The dynamics of the transformed state 𝑦 can be derived as 𝑦̇=𝑇⁻¹𝑥₀𝑥̇+𝑑=−𝑘₀𝑑𝑡𝑇⁻¹𝑥₀𝑥𝑥₀𝑇⁻¹𝑥₀𝐴𝑇𝑥₀𝑦+𝑇⁻¹𝑥₀𝑏𝑢₁+𝑇⁻¹𝑥₀−𝑘₀𝑥₀𝑅₂+𝑅₁𝑢₁+𝑑𝑑𝑡𝑇⁻¹𝑥₀𝑇𝑥₀𝑦.(10)

Direct calculation reveals that 𝑇𝑥⁻¹𝑥₀𝑏=𝑥^0−𝑚𝑏,₀𝑑𝑇⁻¹𝑥₀𝐴𝑇𝑥₀=𝐴,𝑑𝑡𝑇=𝑘⁻¹₀𝑥₀𝑇𝑥₀diag{𝑚,𝑚+1,𝑚+2,…,𝑚+𝑛−1}.(11)

Substituting the above identities into (10) results in𝑦̇=𝐴₁𝑦+𝑥^0−𝑚𝑏𝑢₁+𝑇⁻¹𝑥₀−𝑘₀𝑥₀𝑅₂+𝑅₁𝑢₁,(12) where 𝐴₁=𝑘₀(−𝐴+diag{𝑚,𝑚+1,𝑚+2,…,𝑚+𝑛−1}).(13)

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

The second control input is designed as 𝑢₁=−𝑥^𝑚0𝐾𝑦=−𝑥^𝑚0𝐾𝑇⁻¹𝑥₀𝑥,(14) where 𝐾=[𝑘₁,𝑘₂,…,𝑘_𝑛] is a control gain row vector such that 𝐴₁−𝑏𝐾 is Hurwitz.

The closed-loop system of (12) and (14) becomes 𝑦̇==𝐴𝐴₁−𝑏𝐾𝑦−𝑇⁻¹𝑥₀𝑘₀𝑥₀𝑅₂+𝑅₁𝑥^𝑚0𝐾𝑦₁−𝑏𝐾𝑦+𝑅,(15) where 𝑅=−𝑇⁻¹𝑥₀𝑘₀𝑥₀𝑅₂+𝐾𝑦𝑥^𝑚0𝑅₁.(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 ‖𝑅‖₂≤‖‖𝑇⁻¹‖‖₂‖‖𝑘₀𝑥₀𝑅₂+𝐾𝑥^𝑚0𝑦𝑅₁‖‖₂≤≤‖‖𝑇𝑟⁻¹₂‖‖₂𝑟₂𝑘₀||𝑥₀||‖𝑥‖²²+𝑟₁‖𝐾‖₂||𝑥₀||^𝑚‖𝑥‖₂‖𝑦‖₂𝑘₀||𝑥₀||‖‖𝑇⁻¹‖‖₂‖𝑇‖²²+𝑟₁‖𝐾‖₂||𝑥₀||^𝑚‖‖𝑇⁻¹‖‖₂‖𝑇‖₂‖𝑦‖²²=ℎ||𝑥₀||‖𝑦‖²²(17) with ℎ(|𝑥₀|) defined as ℎ||𝑥₀||≅+𝑟₂𝑟𝑘₀₁max||𝑥₀||^𝑚+1,||𝑥₀||^{𝑚−𝑛+2}max1,||𝑥⁰||^2(𝑛−1)×m‖a𝐾x‖²1,m||𝑥a₀x||𝑥₀||^𝑚,||𝑥₀||^{𝑚−(𝑛−1)}||^𝑛−1.(18)

As |𝑥₀|≤|𝑥₀(0)|, ℎ(|𝑥₀|) is thus bounded uniformly with 𝑡 provided 𝑚−(𝑛−1)≥0. In view of the facts that 𝐴₁−𝑏𝐾 is Hurwitz and lim_{‖𝑦‖2→0}(‖𝑅‖₂/‖𝑦‖₂)=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, 𝑚≥n−1, 𝐾 is selected such that 𝐴₁−𝑏𝐾 is Hurwitz then the following control law 𝑢₀=−𝑘₀𝑥₀,𝑢₁=−𝑥^𝑚0𝐾𝑦=−𝑥^𝑚0𝐾𝑇⁻¹𝑥₀𝑥(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)𝑒^−𝑘0𝑡,𝑢₀(𝑡)=−𝑘₀𝑥₀(𝑡) globally converge to zero exponentially. As 𝐴₁−𝑏𝐾 is Hurwitz and ‖𝑅‖₂≤ℎ(|𝑥₀|)‖𝑦‖²² with ℎ(|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)=−𝑥^𝑚0(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)−𝑘₀𝑥₀−𝛼(𝑡)(20) with 𝛼(𝑡)=𝛼₀𝑒^−𝑘0𝑡, 𝛼₀≠0, 𝑘₀>𝑘₀>0.

Let 𝑒₀(𝑡)=𝑥₀(𝑡)−𝛼(𝑡); then 𝑒₀̇(𝑡)=𝑢₀(𝑡)−𝛼̇(𝑡)=−𝑘₀𝑒₀(𝑡), so that 𝑒₀(𝑡)=𝑒₀(0)𝑒^−𝑘0𝑡, 𝑥₀(𝑡)=𝛼(𝑡)+𝑒₀(𝑡)=𝛼(𝑡)+𝑒₀(0)𝑒^−𝑘0𝑡, 𝑢₀(𝑡)=−𝑘₀𝛼(𝑡)−𝑘₀𝑒₀(0)𝑒^−𝑘0𝑡, and 𝑒₀(𝑡)/𝛼(𝑡)=(𝑒₀(0)/𝛼₀)𝑒^{−(𝑘0−𝑘0)𝑡} are all globally convergent to zero exponentially.

Now we introduce the following smooth time-varying state transformation:𝑦=𝑇⁻¹(𝛼)𝑥,𝑥=𝑇(𝛼)𝑦(21) with 𝑇𝑇₋₁(𝛼)=𝛼^𝑚diag1,𝛼,𝛼²,…,𝛼^𝑛−1,(𝛼)=𝛼^−𝑚diag1,𝛼⁻¹,𝛼⁻²,…,𝛼^{−(𝑛−1)},(22) and 𝑚 a positive integer to be determined.

As 𝛼₀≠0, the transformation matrix 𝑇(𝛼) is clearly nonsingular for all 0≤𝑡<∞.

The dynamics of the transformed state 𝑦 can be derived as 𝑦̇=𝑇⁻¹(𝛼)𝑥̇+𝑑𝑑𝑡𝑇⁻¹=𝑢₀(𝛼)𝑥𝑇⁻¹(𝛼)𝐴𝑇(𝛼)𝑦+𝑇⁻¹(𝛼)𝑏𝑢₁+𝑇⁻¹(𝛼)𝑢₀𝑅₂+𝑅₁𝑢₁+𝑑𝑑𝑡𝑇⁻¹=−𝑘₀(𝛼)𝑇(𝛼)𝑦𝛼1+𝑘₀𝑒₀𝑘₀𝛼𝑇⁻¹(𝛼)𝐴𝑇(𝛼)𝑦+𝑇⁻¹(𝛼)𝑏𝑢₁+𝑇⁻¹(𝛼)−𝑘₀𝛼1+𝑘₀𝑒₀𝑘₀𝛼𝑅²+𝑅₁𝑢₁+𝑑𝑑𝑡𝑇⁻¹(𝛼)𝑇(𝛼)𝑦.(23)

Simple calculation reveals that 𝛼𝑇𝑇⁻¹(₋₁𝛼)𝑏=𝛼^−𝑚𝑏,𝑑(𝛼)𝐴𝑇(𝛼)=𝐴,𝑑𝑡𝑇⁻¹=𝑘₀(𝛼)𝑇(𝛼)diag{𝑚,𝑚+1,𝑚+2,…,𝑚+𝑛−1}.(24)

Substituting the above identities into (23) results in 𝑦̇=𝐴₁1+𝑘₀𝑒₀𝑘₀𝛼𝑦+𝛼^−𝑚𝑏𝑢₁+𝑇⁻¹𝑥₀−𝑘₀𝛼1+𝑘₀𝑒₀𝑘₀𝛼𝑅²+𝑅₁𝑢₁,(25) where 𝐴₁ is defined in (13).

The second control input is designed as 𝑢₁=−𝛼^𝑚𝐾𝑦=−𝛼^𝑚𝐾𝑇⁻¹(𝛼)𝑥,(26) where 𝐾=[𝑘₁,𝑘₂,…,𝑘_𝑛] is a control gain row vector selected such that 𝐴₁−𝑏𝐾 is Hurwitz.

The closed-loop system of (25) and (26) becomes 𝑦̇=𝐴₁−𝑏𝐾+𝑘₀𝑒₀−𝑇₋₁𝑘₀𝛼𝐴¹𝑦(𝛼)𝑘₀𝛼1+𝑘₀𝑒₀𝑘₀𝛼𝑅²=𝐴+𝑅₁𝛼^𝑚𝐾𝑦₁−𝑏𝐾+𝑘₀𝑒₀𝑘₀𝛼𝐴¹𝑦+𝑅^∗,(27) where 𝑅^∗=−𝑇⁻¹(𝛼)𝑘₀𝛼1+𝑘₀𝑒₀𝑘₀𝛼𝑅²+𝑅₁𝛼^𝑚𝐾𝑦.(28)

In view of (5), the converted residual term 𝑅^∗ can be shown to be bounded by ‖𝑅^∗‖₂≤‖‖𝑇⁻¹‖‖₂𝑘₀|𝛼|||||1+𝑘₀𝑒₀𝑘₀𝛼||||‖‖𝑅²‖‖₂+‖‖𝑇⁻¹‖‖₂‖𝐾‖₂|𝛼|^𝑚‖𝑦‖₂‖‖𝑅₁‖‖₂≤‖‖𝑇⁻¹‖‖₂𝑟₂𝑘₀|𝛼|||||1+𝑘₀𝑒₀𝑘₀𝛼||||‖𝑥‖²²+‖‖𝑇⁻¹‖‖₂𝑟₁‖𝐾‖₂|𝛼|^𝑚‖𝑥‖₂‖𝑦‖₂≤𝑟₂𝑘₀||||1+𝑘₀𝑒₀𝑘₀𝛼|||||𝛼|‖‖𝑇⁻¹‖‖₂‖𝑇‖²²‖𝑦‖²²+𝑟₁‖𝐾‖₂|𝛼|^𝑚‖‖𝑇⁻¹‖‖₂‖𝑇‖₂‖𝑦‖²²=ℎ𝛼,𝑒₀‖𝑦‖²²(29) with ℎ(𝛼,𝑒₀) defined asℎ≅𝛼𝑟₂,𝑒₀𝑘₀||||1+𝑘₀𝑒₀𝑘₀𝛼||||max|𝛼|^𝑚+1×max1,|𝛼|_2(𝑛−1),|𝛼|^{𝑚−𝑛+2}+𝑟₁‖𝐾‖₂max|𝛼|^𝑚,|𝛼|^{𝑚−(𝑛−1)}max1,|𝛼|^𝑛−1.(30)

As 𝛼, 𝑒₀/𝛼=(𝑒₀/𝛼₀)𝑒^{−(𝑘0−𝑘0)𝑡} are both bounded uniformly with 𝑡, and ℎ(𝑥₀,𝑒₀) is thus uniformly bounded provided 𝑚−(𝑛−1)≥0. Since 𝐴₁−𝑏𝐾 is Hurwitz and 𝑒₀/𝛼 converges to zero exponentially, system 𝑦̇=(𝐴₁−𝑏𝐾+(𝑘₀𝑒₀/𝑘₀𝛼)𝐴₁)𝑦 is globally exponential stable, and hence the perturbed system 𝑦̇=(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)=𝛼₀𝑒^−𝑘0𝑡, 𝛼₀≠0, 𝑘₀>𝑘₀>0, 𝑚≥𝑛−1, 𝐾 is selected such that 𝐴₁−𝑏𝐾 is Hurwitz, then the following control law 𝑢₀=−𝑘₀𝛼−𝑘₀𝑥₀−𝛼,𝑢₁=−𝛼^𝑚𝐾𝑦(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 𝐴₁−𝑏𝐾 is Hurwitz and ‖𝑅^∗‖₂≤ℎ(𝑥₀,𝑒₀)‖𝑦‖²² with ℎ(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)=−𝛼^𝑚(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 (𝑥_𝑖,𝑦_𝑖,𝜃_𝑖) denote the position and orientation of body 𝑖(𝑖=0,1,2,…,𝑛), (𝑣_𝑖,𝜔_𝑖=𝜃̇_𝑖) denote the linear and angular velocities of body 𝑖(𝑖=0,1,2,…,𝑛), 𝛽_𝑖=𝜃_𝑖−1−𝜃_𝑖(𝑖=1,2,3,…,𝑛) represent the difference of orientation angles between body 𝑖 and body 𝑖−1. 𝑃_𝑖(𝑖=0,1,2,…) is the center point on the wheel axle of body 𝑖 and 𝑄_𝑖−1(𝑖=1,2,…,𝑛) the connection point of body 𝑖 and body 𝑖−1. The distance between 𝑃_𝑖 and 𝑄_𝑖 is 𝑑_𝑖, and the distance between 𝑃_𝑖 and 𝑄_𝑖−1 is 𝑓_𝑖.

Figure 1 

A tractor towing 𝑛 trailer with off-axle hitching.

The kinematic equation of the tractor is 𝑥̇₀=𝑣₀cos𝜃₀,𝑦̇₀=𝑣₀sin𝜃₀,𝜃̇₀=𝜔₀.(32)

The kinematic relations of trailer 𝑖 can be derived as 𝑣_𝑖=𝑣_𝑖−1𝑥cȯ_𝑖s𝛽_𝑖+𝑑_𝑖−1𝜃̇_𝑖−1sin𝛽_𝑖,𝑦̇_𝑖=𝑣_𝑖cos𝜃_𝑖,𝜃̇_𝑖=𝑣_𝑖sin𝜃_𝑖,=1𝑓_𝑖𝑣_𝑖−1sin𝛽_𝑖−𝑑_𝑖−1𝜃̇_𝑖−1cos𝛽_𝑖.(33)

Select 𝑥=[𝑥₀,𝑦₀,𝜃₀,𝛽₁,𝛽₂,…,𝛽_𝑛]^𝑇 as the state variables, and 𝑢₀=𝑣₀cos𝜃₀,𝜔₀ as the control inputs, the state equation can be derived from (32)-(33) as 𝑥̇=𝐴+𝑅₂𝑥̇₀=𝑢₀,(34)(𝑥)𝑢₀+𝑏+𝑅₁(𝑥)𝜔₀,(35) where 𝑅₁,𝑅₂ are high-order residual terms satisfying (5) and 𝐴=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣000100−00100……00𝑓₁0010…0𝑓₁1+𝑓𝑑¹₂−𝑓1₂⋮0⋮0𝑎⋮_1,𝑛⋮⋱…⋮0𝑎_2,𝑛…−1𝑓_𝑛𝑏=011+𝑑₀⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝑓₁−𝑑⁰𝑓₁1+𝑓𝑑¹₂…𝑏^𝑛^𝑇𝑎,(36)_1,𝑛=(−1)^𝑛−2𝑑_𝑛−2×𝑑_𝑛−3⋯×𝑑₂×𝑑₁𝑎_2,𝑛𝑓_𝑛−1×𝑓_𝑛−2⋯𝑓₁1+𝑑𝑓^𝑛−1_𝑛,=(−1)^𝑛−3𝑑_𝑛−2×𝑑_𝑛−3⋯×𝑑₃×𝑑₂𝑏_𝑛𝑓_n−1×𝑓_𝑛−2⋯𝑓₂1+𝑑𝑓^𝑛−1_𝑛,=(−1)^𝑛−1𝑑_𝑛−2×𝑑_𝑛−3⋯×𝑑₁×𝑑₀𝑓_𝑛−1×𝑓_𝑛−2⋯𝑓₁1+𝑑𝑓^𝑛−1_𝑛,(37)

The control object can be stated as design control law 𝑢₀(⋅),𝜔₀(⋅) such that the states (𝑥₀,𝑦₀,𝜃₀,𝛽₁,𝛽₂,…,𝛽_𝑛) 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 𝑑_𝑖>0(𝑖=0,1,…,𝑛−1) and 𝑓_𝑖>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 𝑥𝑦̇₀̇₀=𝑣₀cos𝜃₀,𝜃=̇₀𝑣₀sin𝜃₀,𝛽̇₁=𝜔₀,=−𝑐₁𝑣₀sin𝛽₁+1+𝑐₂cos𝛽₁𝜔₀,(38) where 𝑐₁=1/𝑓₁, 𝑐₂=𝑑₀/𝑓₁.

Under the following coordinate and input transformations: 𝑥𝑥₁₂=𝑐²¹𝛽₁,𝑥₃=−𝑐₁𝛽₁+𝑐₁1+𝑐₂𝜃₀,𝑢₀=𝛽₁−1+𝑐₂𝜃₀+𝑐₁1+𝑐₂𝑦₀,𝑢₁=𝑣₀cos𝜃₀,=𝑐²¹−𝑐₁𝑣₀sin𝛽₁+1+𝑐₂cos𝛽₁𝜔₀.(39) the state equation (38) is converted to the following form:𝑥𝑥̇̇₃₂=𝑥₂+𝑅₂₃𝑢₀+𝑅₁₃𝑢₁,𝑥̇₁=𝑥₁+𝑅₂₂𝑢₀+𝑅₁₂𝑢₁,𝑥̇₀=𝑢₁,=𝑢₀,(40) where 𝑅₂₃=𝑐₁−csoins𝛽𝜃¹₀−𝛽₁+1+𝑐²csoins𝜃𝜃⁰₀−𝜃₀+𝑐₁𝑐₂cos𝛽₁−1sin𝛽₁1+𝑐₂cos𝛽₁/cos𝜃₀𝑅₁₃,=𝑐²cos𝛽₁−1𝑅₂₂𝑐²¹/1+𝑐₂cos𝛽₁,=𝑐²¹csoins𝛽𝜃¹₀−𝛽₁+𝑐²¹𝑐₂1−cos𝛽₁sin𝛽₁1+𝑐₂cos𝛽₁/cos𝜃₀𝑅₁₂,=𝑐²𝑐₁1−cos𝛽₁/1+𝑐₂cos𝛽₁.(41)

In the state region 𝐷={(𝑥₀,𝑦₀,𝜃₀,𝛽₁)∶|𝜃₀|≤𝜃_0𝑀<𝜋/2,|𝛽₁|≤𝛽_1𝑀}, |𝑅_2𝑗|(𝑗=2,3) can be shown to be 𝑂(‖(𝜃₀,𝛽₁)‖³²) and 𝑅_1𝑗(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 𝐴₁−𝑏𝐾 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.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2 

Time trajectories of states and geometric paths of the tractor and the trailer starting from the first initial state.

(a)

(b)

(c)

(d)

(e)

(f)

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.