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.

The so-called

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 [

The paper is organized as follows. Section

Consider the following nonlinear system represented by

System (

Without loss of generality, it is specially assumed in (

It is noted that the approximate NCF (

The approximate NCF represents a large class of nonholonomic systems that cannot be converted to NCF in which

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 (

The control law for the first control input is designed as

Substituting (

Inspired by the well-known

The discontinuous coordinate transformation (

The transformation matrix

The dynamics of the transformed state

Direct calculation reveals that

Substituting the above identities into (

As

The second control input is designed as

The closed-loop system of (

System (

In view of (

As

The above analysis is summarized as the following proposition.

Suppose that

It is obvious that

Proposition

In the next subsection, the controller (

The control law for the first control input is designed as

Let

Now we introduce the following smooth time-varying state transformation:

As

The dynamics of the transformed state

Simple calculation reveals that

Substituting the above identities into (

The second control input is designed as

The closed-loop system of (

In view of (

As

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

Suppose that

It is obvious that

Compared with the approach presented in [

Consider a tractor-trailer with a wheeled mobile tractor towing

A tractor towing

The kinematic equation of the tractor is

The kinematic relations of trailer

Select

The control object can be stated as design control law

To apply Propositions

Suppose that

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

As

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

Under the following coordinate and input transformations:

In the state region

The geometric parameters are set to

The simulation is implemented for two initial states

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

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

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.

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.