Dynamic event-triggered formation control of second-order nonholonomic systems

In this paper,the formation control problem of secondorder nonholonomic mobile robot systems is investigated in a dynamic event-triggered scheme.Event-triggered control protocols combined with persistent excitation(PE)conditions are presented.In event-detecting processes,an inactive time is introduced after each sampling instant,which can ensure a positive minimum sampling interval.To increase the flexibility of the event-triggered scheme,internal dynamic variables are included in event-triggering conditions.Moreover,the dynamic event-triggered scheme plays an important role in increasing the lengths of time intervals between any two consecutive events.In addition,event-triggered control protocols without forward and angular velocities are also presented based on approximate-differentiation(low-pass)filters.The asymptotic convergence results are given based on a nested Matrosov theorem and artificial sampling methods.

Conserved quantities from Lie symmetries for nonholonomic systems

This paper presents a new method to seek the conserved quantity from a Lie symmetry without using either Lagrangians or Hamiltonians for nonholonomic systems. The differential equations of motion of the systems are established. The definition of the Lie symmetrical transformations of the systems is given, which only depends upon the infinitesimal transformations of groups for the generalized coordinates. The conserved quantity is directly constructed in terms of the Lie symmetry of the systems. The condition under which the Lie symmetry can lead to the conserved quantity and the form of the conserved quantity are obtained. Finally, an example is given to illustrate the application of the result.

Mei Symmetry and Conserved Ouantity of Tzénoff Equations for Nonholonomic Systems of Non-Chetaev's Type

Mei symmetry of Tzenoff equations for nonholonomic systems of non-Chetaev＇s type under the infinitesimal transformations of groups is studied. Its definitions and discriminant equations of Mei symmetry are given. Sufficient and necessary condition of Lie symmetry deduced by the Mei symmetry is also given. Hojman conserved quantity of Tzenoff equations for the systems through Lie symmetry in the condition of special Mei symmetry is obtained.

Symmetry of Lagrangians of nonholonomic systems of non-Chetaev's type

This paper studies the symmetry of Lagrangians of nonholonomic systems of non-Chetaev＇s type. First, the definition and the criterion of the symmetry of the system are given. Secondly, it obtains the condition under which there exists a conserved quantity and the form of the conserved quantity. Finally, an example is shown to illustrate the application of the result.

Lie Symmetries and Conserved Quantities of Arbitrary Order Nonholonomic Systems

The invariance of the differential equations under the infinitesimal transformations was used to study the Lie symmetries and conserved quantities of arbitrary order nonholonomic systems. The determining equations, the restriction equations, the structure equation and the form of the conserved quantities were obtained.

A FIELD METHOD FOR SOLVING THE EQUATIONS OF MOTION OF NONHOLONOMIC SYSTEMS

In this paper,the field method for solving the equations of motion of holonomic nonconservative systems is extended to nonholonomic systems with constant mass and with variable mass.Two examples are given to illustrate its application.

Lagrange equations of nonholonomic systems with fractional derivatives

This paper obtains Lagrange equations of nonholonomic systems with fractional derivatives. First, the exchanging relationships between the isochronous variation and the fractional derivatives are derived. Secondly, based on these exchanging relationships, the Hamilton＇s principle is presented for non-conservative systems with fractional derivatives. Thirdly, Lagrange equations of the systems are obtained. Furthermore, the d＇Alembert-Lagrange principle with fractional derivatives is presented, and the Lagrange equations of nonholonomic systems with fractional derivatives are studied. An example is designed to illustrate these results.

Perturbation to Symmetries and Adiabatic Invariants of Nonholonomic System in Terms of Quasi-coordinates

Based on the theory of Lie symmetries and conserved quantities, the exact invariants and adiabatic invariants of nonholonomic system in terms of quasi-coordinates are studied. The perturbation to symmetries for the nonholonomic system in terms of quasi-coordinates under small excitation is discussed. The concept of high-order adiabatic invariant is presented, and the forms of exact invariants and adiabatic invariants as well as the conditions for their existence are given. Then the corresponding inverse problem is studied.

Fractional Action-Like Variational Problem and Its Noether Symmetries for a Nonholonomic System

For an in-depth study on the symmetric properties for nonholonomic non-conservative mechanical systems,the fractional action-like Noether symmetries and conserved quantities for nonholonomic mechanical systems are studied,based on the fractional action-like approach for dynamics modeling proposed by El-Nabulsi.Firstly,the fractional action-like variational problem is established,and the fractional action-like Lagrange equations of holonomic system and the fractional action-like differential equations of motion with multiplier for nonholonomic system are given;secondly,according to the invariance of fractional action-like Hamilton action under infinitesimal transformations of group,the definitions and criteria of fractional action-like Noether symmetric transformations and quasi-symmetric transformations are put forward;finally,the fractional action-like Noether theorems for both holonomic system and nonholonomic system are established,and the relationship between the fractional action-like Noether symmetry and the conserved quantity is given.

Lie symmetry theorem of fractional nonholonomic systems

The Lie symmetry theorem of fractional nonholonomic systems in terms of combined fractional derivatives is estab- lished, and the fractional Lagrange equations are obtained by virtue of the d＇Alembert-Lagrange principle with fractional derivatives. As the Lie symmetry theorem is based on the invariance of differential equations under infinitesimal trans- formations, by introducing the differential operator of infinitesimal generators, the determining equations are obtained. Furthermore, the limit equations, the additional restriction equations, the structural equations, and the conserved quantity of Lie symmetry are acquired. An example is presented to illustrate the application of results.

Weakly Noether Symmetry for Nonholonomic Systems of Chetaev's Type

In the paper [J. of Beijing Institute of Technology 26 （2006） 285] the authors provided the definition of weakly Noether symmetry. We now discuss the weakly Noether symmetry for non-holonomic system of Chetaev＇s type, and present expressions of three kinds of conserved quantities by weakly Noether symmetry. Finally, the application of this new result is shown by a practical example.

Form invariance and conserved quantity for weakly nonholonomic system

The form invariance and the conserved quantity for a weakly nonholonomic system （WNS） are studied. The WNS is a nonholonomic system （NS） whose constraint equations contain a small parameter. The differential equations of motion of the system are established. The definition and the criterion of form invariance of the system are given. The conserved quantity deduced from the form invariance is obtained. Finally, an illustrative example is shown.

Non-Noether Conserved Quantity for Relativistic Nonholonomic System with Variable Mass

Using form invariance under special infinitesimal transformations in which time is not variable, the non-Noether conserved quantity of the relativistic nonholonomic system with variable mass is studied. The differential equations of motion of the system are established. The definition and criterion of the form invariance of the system under infinitesimal transformations are studied. The necessary and sufficient. condition under which the form invariance is a Lie symmetry is given. The condition under which the form invariance can be led to a non-Noether. conserved quantity and the form of the conserved quantity are obtained. Finally, an example is given to illustrate the application of the result.

NOETHER'S THEOREM FOR NONHOLONOMIC SYSTEMS OF NON-CHETAEV'S TYPE WITH UNILATERAL CONSTRAINTS IN EVENT SPACE

To study the Noether's theorem of nonholonomic systems of non_Chetaev's type with unilateral constraints in event space, firstly, the principle of D'Alembert_Lagrange for the systems with unilateral constraints in event space is presented, secondly, the Noether's theorem and the Noether's inverse theorem for the nonholonomic systems of non_Chetaev's type with unilateral constraints in event space are studied and obtained, which is based upon the invariance of the differential variational principle under the infinitesimal transformations of group, finally, an example is given to illustrate the application of the result.

A New type of conserved quantity deduced from Mei symmetry of nonholonomic systems in terms of quasi-coordinates

This paper studies the new type of conserved quantity which is directly induced by Mei symmetry of nonholonomic systems in terms of quasi-coordinates. A coordination function is introduced, and the conditions for the existence of the new conserved quantities as well as their forms are proposed. Some special cases are given to illustrate the generalized significance of the new type conserved quantity. Finally, an illustrated example is given to show the application of the nonholonomic system＇s results.

Form invariance and approximate conserved quantity of Appell equations for a weakly nonholonomic system

A weakly nonholonomic system is a nonholonomic system whose constraint equations contain a small parameter. The form invariance and the approximate conserved quantity of the Appell equations for a weakly nonholonomic system are studied. The Appell equations for the weakly nonholonomic system are established, and the definition and the criterion of form invariance of the system are given. The structural equation of form invariance for the weakly nonholonomic system and the approximate conserved quantity deduced from the form invariance of the system are obtained. Finally, an example is given to illustrate the application of the results.

Quasi-canonicalization for linear homogeneous nonholonomic systems

For conservative linear homogeneous nonholonomic systems, there exists a cotangent bundle with the symplectic structure dπμ∧ dξμ, in which the motion equations of the system can be written into the form of the canonical equations by the set of quasi-coordinates πμand quasi-momenta ξμ. The key to construct this cotangent bundle is to define a set of suitable quasi-coordinates πμby a first-order linear mapping, so that the reduced configuration space of the system is a Riemann space with no torsion. The Hamilton–Jacobi method for linear homogeneous nonholonomic systems is studied as an application of the quasi-canonicalization. The Hamilton–Jacobi method can be applied not only to Chaplygin nonholonomic systems, but also to non-Chaplygin nonholonomic systems. Two examples are given to illustrate the effectiveness of the quasi-canonicalization and the Hamilton–Jacobi method.

Hojman conserved quantity for nonholonomic systems of unilateral non-Chetaev type in the event space

Hojman conserved quantities deduced from the special Lie symmetry, the Noether symmetry and the form invariance for a nonholonomic system of the unilateral non-Chetacv type in the event space are investigated. The differential equations of motion of the system above are established. The criteria of the Lie symmetry, the Noether symmetry and the form invariance are given and the relations between them are obtained. The Hojman conserved quantities are gained by which the Hojman theorem is extended and applied to the nonholonomic system of the unilateral non-Chetacv type in the event space. An example is given to illustrate the application of the results.

Characteristic Functional Structure of Infinitesimal Symmetry Transformations for Nonholonomic System

It was proved that velocity-dependent infinitesima l symmetry transformations of nonholonomic systems have a characteristic functio nal structure, which could be formulated by means of an auxiliary symmetry tra nsformation function and is manifestly dependent upon constants of motion of th e system. An example was given to illustrate the applicability of the results.

On the stability of equilibria of nonholonomic systems with nonlinear constraints

Lyapunov＇s first method, extended by V. V. Kozlov to nonlinear mechani- cal systems, is applied to the study of the instability of the position of equilibrium of a mechanical system moving in the field of conservative and dissipative forces. The mo- tion of the system is limited by ideal nonlinear nonholonomic constraints. Five cases determined by the relationship between the degree of the first nontrivial polynomials in Maclaurin＇s series for the potential energy and the functions that can be generated from the equations of nonlinear nonholonomic constraints are analyzed. In the three eases, the theorem on the instability of the position of equilibrium of nonholonomic systems with linear homogeneous constraints （V. V. Kozlov （1986）） is generalized to the case of nonlin- ear nonhomogeneous constraints. In the other two cases, new theorems are set extending the result from V. V. Kozlov （1994） to nonholonomic systems with nonlinear constraints.