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A physical rattleback is a toy that can exhibit counter-intuitive behavior when spun on a horizontal plate. Most notably, it can spontaneously reverse its direction of rotation. Using a standard mathematical model of the rattleback, we prove the existence of reversing motion, reversing motion combined with rolling, and orbits that exhibit such behavior repeatedly. [ABSTRACT FROM AUTHOR]

Summary: In this presented work, a systematic adaptive stabilization control strategy is proposed for nonholonomic system with parametric uncertainties and full‐state constraints. To facilitate the handling of state constraints, the original constrained nonholonomic system is converted into a new unconstrained system by state‐dependent function transformations including discontinuous state scalings. The adaptive control algorithm is elaborated by cleverly combing the tuning function design approach with switching control strategy. In stability analysis, it is shown that the designed stabilization control method can realize the desired stabilization control aims and full‐state constraints. Simulation is carried out to demonstrate the effectiveness of the control scheme. [ABSTRACT FROM AUTHOR]

Redundantly constrained mechanisms have – in general – non-uniquely calculated reactions when modeled as rigid multibody systems (MBSs). However, some of the reactions may be unique. An analogous problem of indeterminacy is also present in overactuated MBSs. This paper discusses the constraint-matrix-based method for the uniqueness analysis of the reactions and driving forces (torques) for MBSs with nonholonomic constraints. Four approaches are studied: The rank comparison, SVD, QR, and nullspace methods. The uniqueness criteria are written in a new way. The equivalence of the SVD, QR, and nullspace methods is shown. It is also presented how to check the uniqueness of the selected elements (reactions, driving forces, or more complex combinations) and their individual components. Subsequently, the impact of the driving constraints on the uniqueness of the joint reactions is discussed. Next, the uniqueness analysis using these three methods is extended to perform a newly proposed body-wise analysis instead of the usual constraint-wise analysis. Two examples of spatial systems (one with nonholonomic constraints) are considered to illustrate the approach. Moreover, the computational efficiency of selected methods is analyzed. • A unified formulation of the existing constraint-matrix-based methods is proposed. • Methods for checking the uniqueness of individual force components are presented. • The influence of driving constraints on the uniqueness of reactions is discussed. • The equivalence of constraint analysis methods (SVD, QR, and nullspace) is proven. • A new, body-wise instead of joint-wise, paradigm of uniqueness analysis is devised. [ABSTRACT FROM AUTHOR]

This work proposes and investigates a new model of the rotating rigid body based on the non-twisting frame. Such a frame consists of three mutually orthogonal unit vectors whose rotation rate around one of the three axis remains zero at all times and, thus, is represented by a nonholonomic restriction. Then, the corresponding Lagrange–D'Alembert equations are formulated by employing two descriptions, the first one relying on rotations and a splitting approach, and the second one relying on constrained directors. For vanishing external moments, we prove that the new model possesses conservation laws, i.e., the kinetic energy and two nonholonomic momenta that substantially differ from the holonomic momenta preserved by the standard rigid body model. Additionally, we propose a new specialization of a class of energy–momentum integration schemes that exactly preserves the kinetic energy and the nonholonomic momenta replicating the continuous counterpart. Finally, we present numerical results that show the excellent conservation properties as well as the accuracy for the time-discretized governing equations. [ABSTRACT FROM AUTHOR]

This paper deals with the predefined-time stabilisation problem for a class of uncertain chained-form nonholonomic systems. Based on a novel generalised Lyapunov-like characterisation of predefined-time stability, new predefined-time controllers are introduced for a class of first and second order systems with matched perturbations. Contrary to existing finite-time and fixed-time schemes, an upper bound of the settling time is an easily tuneable control parameter. Then, these results for first and second order systems are used to design a switching strategy, which guarantees the predefined-time stability of the closed-loop system. The switching time and the settling-time can be easily tuned according to the control parameters. A benchmark example is presented to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Investigation of various problems of mechanics and mathematical physics is reduced to the solution of second-order linear differential equations with variable coefficients. In 1986, the American mathematician J. Kovacic proposed an algorithm for solution of a second-order linear differential equation in the case where the solution can be expressed in terms of so-called Liouville functions. If a linear second-order differential equation has no Liouville solutions, the Kovacic algorithm also allows one to ascertain this fact. In this paper, we discuss the application of the Kovacic algorithm to the problem of the motion of a heavy body of revolution on a perfectly rough horizontal plane. The existence of Liouville solutions of the problem is examined for the cases where the rolling body is an infinitely thin disk, a disk of finite thickness, a dynamically symmetric torus, a paraboloid of revolution, and a spindle-shaped body. [ABSTRACT FROM AUTHOR]

Considering input constraints and parameter uncertainty of the nonholonomic wheeled mobile robots, a new control law has been designed in this paper to solve the tracking and stabilization control problems simultaneously. The asymptotic convergence of the stabilization or tracking errors is achieved by a smooth controller with time-varying feedback parameters subject to set conditions. A geometric-based parameter design strategy is employed to overcome the difficulties resulting from input constraints. With the parameter design strategy, the designed parameters satisfy the set conditions and the control inputs stay in the constrained domain. Experimental studies have been conducted to verify the effectiveness of the proposed control law. [ABSTRACT FROM AUTHOR]

In this work, the design, analysis, and implementation of an algorithm for automatic parallel parking for a nonholonomic mobile robot is presented. The mobile robot is a four-wheeled scaled vehicle and it is assumed that there is space limitation for the parking maneuver. The main objective was to design a parallel parking path trajectory avoiding collisions. We designed a fuzzy PD+I controller for driving the error generated between the real position and the previously generated objective position to the origin. We presented simulations results to validate the analysis and demonstrating how the fuzzy controller solved the tracking problem for the derived path trajectory to follow. [ABSTRACT FROM AUTHOR]

A large proportion of constrained mechanical systems result in nonlinear ordinary differential equations, for which it is quite difficult to find analytical solutions. The initial motions method proposed by Whittaker is effective to deal with such problems for various constrained mechanical systems, including the nonholonomic systems discussed in the first part of this paper, where in addition to differential equations of motion, nonholonomic constraints apply. The final equations of motion for these systems are obtained in the form of corresponding power series. Also, an alternative, direct method to determine the initial values of higher-order derivatives q¨0,q⃛0,… is proposed, being different from that of Whittaker. The second part of this work analyzes the stability of equilibrium of less complex, nonholonomic mechanical systems represented by gradient systems. We discuss the stability of equilibrium of such systems based on the properties of the gradient system. The advantage of this novel method is its avoidance of the difficulty of directly establishing Lyapunov functions aimed at such unsteady nonlinear systems. Finally, these theoretical considerations are illustrated through four examples. [ABSTRACT FROM AUTHOR]

A stable control strategy is proposed via energy attenuation to realise the position control for a second-order nonholonomic planar four-link active-passive-active-active (APAA) underactuated mechanical system. Since a first-order nonholonomic planar passive-active-active (PAA) system can be stabilised at the target position, we control the first link based on the principle of energy attenuation to target value and the planar APAA systemis reduced to be a first-order nonholonomic PAA system, where the target value is selected to ensure the target position comes within the reachable area of the PAA system. Meanwhile, we divide the control of the reduced-order PAA systeminto two stages based on the holonomic characteristic of planar acrobot, and the target angles of which are obtained according to the target position by using the simulated annealing algorithm. Each stage controllers of the reduced-order system are designed according to its control target. Finally, the validity of the control strategy is demonstrated via the simulations. [ABSTRACT FROM AUTHOR]

This paper presents a position and posture control strategy for a n-link planar underactuated manipulator with passive second joint in horizontal plane. The n-link planar underactuated mechanical manipulator is a second-order nonholonomic system, the control objective is to move the end-effector to a given position with a desired posture. The whole control process is divided into n-2 stages. In each stage, the first link is maintained at its initial states unchanged, there exists an angle constraint between the passive link and one of the active links. Based on the angle constraints, the target angles of the control objective are calculated by using genetic algorithm. The controllers of each stages are designed, respectively, to achieve the control objective of one of the active links. Finally, taking a 5-link planar underactuated mechanical manipulator, for example, the simulation results demonstrate the validity of the proposed control method. [ABSTRACT FROM AUTHOR]

The geometric formulation of motion of the first-order linear homogenous scleronomous nonholonomic system subjected to active forces is studied with the nonholonomic mapping theory. The quasi-Newton law, the quasi-momentum theorem, and the second kind Lagrange equation of dynamical systems are obtained in the Riemann-Cartan configuration spaces. By the nonholonomic mapping, a Euclidean configuration space or a Riemann configuration space of a dynamical system can be mapped into a Riemann-Cartan configuration space with torsion. The differential equations of motion of the dynamical system can be obtained in its Riemann-Cartan configuration space by the quasi-Newton law or the quasi-momentum theorem. For a constrained system, the differential equations of motion in its Riemann-Cartan configuration space may be simpler than the equations in its Euclidean configuration space or its Riemann configuration space. Therefore, the nonholonomic mapping theory can solve some constrained problems, which are difficult to be solved by the traditional analytical mechanics method. Three examples are given to illustrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

In order to realize three-dimensional stable control of underactuated autonomous underwater vehicles (AUVs), this study analyses nonlinear hydrodynamic characteristics of AUVs based on the mathematical Taylor series. The nonholonomic control system properties of AUVs are investigated using nonholonomic system theory under three-dimensional control input. The constraint of underactuated AUVs is proved not to be integrated by the local integrability theorem, and the controllability of underactuated AUVs is verified through small-time local controllability (STLC). In order to simplify the trigonometric terms in underactuated AUV motion functions, quaternion theory is applied to transform the function to develop a continuous time-varying controller. Results of a simulation experiment show that this control law is effective in achieving three-dimensional stabilization from arbitrary initial positions. [ABSTRACT FROM AUTHOR]

This paper discusses the nonholonomic robotic systems whose motion constraints assume the Pfaffian form, and the equations of motion are represented by driftless control systems with outputs. By reference to the end point map of such a control system, we define the system's Jacobian and study Jacobian motion-planning algorithms. A new Lagrange-type Jacobian inverse, referred to as the General Lagrangian Jacobian Inverse (GLJI), is designed as the solution of an optimal control problem with a Lagrange-type objective function. Singularities of GLJI are examined. A special choice of the objective function illustrates features of GLJI. A new motion-planning algorithm based on GLJI is proposed. Theoretical arguments are illustrated with a motion-planning problem of a space robot. [ABSTRACT FROM PUBLISHER]

Equations describing the rolling of a spherical ball on a horizontal surface are obtained, the motion being activated by an internal rotor driven by a battery mechanism. The rotor is modeled as a point mass mounted inside a spherical shell and caused to move in a prescribed circular orbit relative to the shell. The system is described in terms of four independent dimensionless parameters. The equations governing the angular momentum of the ball relative to the point of contact with the plane constitute a six-dimensional, nonholonomic, nonautonomous dynamical system with cubic nonlinearity. This system is decoupled from a subsidiary system that describes the trajectories of the center of the ball. Numerical integration of these equations for prescribed values of the parameters and initial conditions reveals a tendency toward chaotic behavior as the radius of the circular orbit of the point mass increases (other parameters being held constant). It is further shown that there is a range of values of the initial angular velocity of the shell for which chaotic trajectories are realized while contact between the shell and the plane is maintained. The predicted behavior has been observed in our experiments. [ABSTRACT FROM AUTHOR]

Control of nonholonomic spherical mobile robot is a generalization of the classical ball-plate problem which is still challenging in robotic researches. In this paper, point stabilization of a nonholonomic spherical mobile robot actuated by two internal rotors is investigated. Since every kinematic trajectory is not always dynamically realizable for the spherical robot driven by two actuators, the mathematical model of the robot is derived based on the angular momentum conservation principle. The controllability of the robot is evaluated based on the obtained model and the uncontrollable configurations as well as their geometrical meaning are specified. To simultaneous control of position and orientation of the robot, a nonlinear model predictive control (NMPC) is developed for the first time and the stability analysis is performed through using Lyapunov stability theorem. The performance of the designed control system is assessed through computer simulations in different test conditions. The simulation results show the significant performance of the proposed NMPC in stabilization of the spherical shell from every initial configuration to every desired position and orientation even in the uncontrollable region. Considering additive bounded noises, the robust stabilization of the nonholonomic spherical robot by the NMPC is also assessed in simulations. [ABSTRACT FROM AUTHOR]

In this paper we present a local motion planning law called trigonometric switch inputs which can steer the chained form system to the final positions, at least locally, around the initial positions. This method steers the system step by step instead of steering all states in one step. The advantages of trigonometric switch inputs law are that the motion trajectories are quite smooth and have less oscillation and lower computational costs, all of which is beneficial for the application of the time scale transformation technique and improvement of motion efficiency of the system. A two-wheeled mobile robot system is steered by this new motion planning law to illustrate the practical application. Finally, simulations with the time scale transformation technique and experiments with the mobile robot verify the feasibility and effectiveness of trigonometric switch inputs law. [ABSTRACT FROM AUTHOR]

This paper proposes a novel recursive terminal sliding mode structure for tracking control of third-order chained-form nonholonomic systems in the presence of the unknown external disturbances. Finite-time convergence of the disturbance approximation error is guaranteed using the designed disturbance observer. Under the proposed terminal sliding model tracking control technique, the finite-time convergence of the states of the closed-loop system is guaranteed via Lyapunov analysis. A new reaching control law is proposed to guarantee the existence of the sliding mode around the recursive TSM surface in a finite-time. Simulation results are illustrated on a benchmark example of third-order chained-form nonholonomic systems: a wheeled mobile robot. The results demonstrate that the proposed control technique achieves promising tracking performance for nonholonomic systems. [ABSTRACT FROM AUTHOR]

Some aspects of the geometry, kinematics, and dynamics of a three-section robotic vehicle with a front steerable wheel are studied. The constraints between the wheels and the flat ground are assumed nonholonomic. The vehicle moves in a narrow L-shaped corridor. A path for the characteristic points of the sections of the robot is designed. A dynamic model of the system is developed. The maximum possible dimensions of the robot that allow its unimpeded and non-stop motion are determined. The kinetostatic analysis of the load on a three-section vehicle moving along a planned path is modeled. The holonomic and nonholonomic constraint reactions between the wheels and the ground and in the joints between the sections are determined [ABSTRACT FROM AUTHOR]

A stabilization controller is addressed for an asymmetric underactuated surface ship subject to input saturation in this work. Prior to control design, the considered ship system is converted into a cascade nonlinear system with a minimum phase by applying state and input transformations. And by introducing a circular trajectory, the investigated stabilization problem is successfully expressed into a tracking issue. Under these preparatory works, finite-time and saturated tracking control laws are proposed by converting the obtained tracking error system into an augmented new system and applying Nussbaum function. It has been demonstrated in stability analysis that the designed controllers can enable system states to converge to zero by choosing a sufficiently small radius of the circle, and meanwhile can also achieve the desired input saturation. The simulation results illustrate the validity of the presented control algorithm. • The underactuated ship is converted into a nonholonomic system for the first time. • The relay switching technique is used to reduce the complexity of control design. • The Nussbaum technique is used to ensure the boundedness of input. [ABSTRACT FROM AUTHOR]

This paper introduces a new algorithm of trajectory reproduction and trajectory tracking for nonholonomic systems. The endogenous configuration space approach is employed as a guideline in the algorithm derivation. The derivation uses a trajectory reproduction error, which is an integral of the difference between the resultant trajectory and the desired trajectory over the motion horizon. Such a definition of the error allows to solve both the trajectory reproduction as well as the trajectory tracking problem. Considerable attention in the paper has been paid to the implementation aspects of the algorithm. The nonparametric approach is used together with a higher order of the integration method. The algorithm efficiency is illustrated with computer simulations accomplished for two nonholonomic systems: the dynamics of the double pendulum with a passive joint, and the kinematics of the unicycle. [ABSTRACT FROM AUTHOR]

The navigation problem for the simplest wheeled robotic vehicle is solved by just measuring kinematical parameters, doing without accelerometers and angular-rate sensors. It is supposed that the steerable-wheel angle sensor has a bias that must be corrected. The navigation parameters are corrected using the GPS. The approach proposed regards the wheeled robot as a system with nonholonomic constraints. The performance of such a navigation system is demonstrated by way of an example [ABSTRACT FROM AUTHOR]

Wheeled mobile robots are a special class of nonholonomic mechanical systems. The mobility of such highly nonlinear systems is restricted due to the presence of nonholonomic constraints of wheels, also the system severe underactuated nature. These conditions generate major difficulties in system stabilization, i.e. to park or reach a given configuration for the overall system. This leads to a challenging control problem for research that is the focus of this paper. In this paper a new method based on time-varying feedbacks has been developed for a Tractor-trailer wheeled robot (TTWR). First kinematic model of the TTWR is obtained. Next, a novel method using timevarying feedbacks is investigated in order to stabilize the TTWR around the origin. The proposed kinematic control algorithm is developed based on switching between two finite-time controllers. Appropriate control algorithms have been designed for each step based on the stability of the closed loop system. Obtained simulation and experimental results show the effectiveness of the proposed control law. [ABSTRACT FROM AUTHOR]

This paper deals with stabilization and tracking control problems defined with respect to a planar mechanical structure similar to Raibert's robot. The proposed control solution is based on formal analysis of the control system on a Lie group. In order to take advantage of Lie group theory a dynamic extension of the robot kinematics is introduced. To cope with non-zero angular momentum the controller based on transverse functions is employed. Properties of the closed-loop control system are investigated based on simulations including practical stabilization at neighborhood of a constant point or a reference trajectory. [ABSTRACT FROM AUTHOR]

The present paper proposes a new feedback control law for a cooperative transportation system comprising two car-like vehicles. The proposed feedback control law controls the two manipulation points, which are revolute joints coupling the two vehicles to a carrier, to follow their parametric curve paths, such as Bezier curves at variable velocities. Thus, the proposed feedback control law makes it possible to specify the motion of the carrier quantitatively using design parameters such as the path shapes and the variable velocity profiles. The convergence of the position of the first manipulation point to its desired position on the first path is guaranteed by linear control theory, and the convergence of the position of the second manipulation point to its desired position on the second path is guaranteed by Lyapunov’s second method. In particular, the design of the Lyapunov functions is facilitated by a new form of differential equations that are equivalent to time differential equations in a chained form into which the kinematic equations of the system are converted. The validity of the proposed feedback control law is verified experimentally. [ABSTRACT FROM AUTHOR]

Consensus problems of multiple nonholonomic systems are considered in this paper. This problem is simplified into consensus problems of two subsystems based on the cascaded structure of nonholonomic chained form systems. Continuous and hybrid distributed controllers have been constructed for these two subsystems respectively based on the theory of cascaded systems. Consensus of multiple nonholonomic chained form systems can be realized using the methodology proposed in this paper no matter whether the group reference signal is persistently exciting or not. Different to previous assumptions on group reference such as persistent excitation or converging to nonzero constant, the condition on the group reference signal have been further relaxed in this paper. Simulation results using Matlab have illustrated the effectiveness of the results presented in this paper. [ABSTRACT FROM AUTHOR]

The kinematics of a two-link mobile robot with three steerable wheels moving in an L-shaped corridor is analyzed. A smooth (with continuous first derivative) path is designed maintaining the optimal maneuverability of the vehicle. The motion of the vehicle along this path is planned. Analytical expressions for the reactions at the contact of the wheels with the ground are given in the general case of motion. The radius of curvature of the programmed path is shown to have a strong influence on the reactions. [ABSTRACT FROM AUTHOR]

Nonholonomic systems describing the rolling of a rigid body on a plane and their relationship with various Poisson structures are considered. The notion of generalized conformally Hamiltonian representation of dynamical systems is introduced. In contrast to linear Poisson structures defined by Lie algebras and used in rigid-body dynamics, the Poisson structures of nonholonomic systems turn out to be nonlinear. They are also degenerate and the Casimir functions for them can be expressed in terms of complicated transcendental functions or not appear at all. [ABSTRACT FROM AUTHOR]

The Lyapunov first method generalized to the case of nonlinear differential equations is applied to the study of the instability of the equilibrium position of a mechanical system, whose motion is constrained by singular nonholonomic constraints. Starting from the results of S. D. Furta (On the instability of equilibrium position of constrained mechanical systems) three theorems on the instability are formulated. The first theorem considers the case of nonholonomic constraints that do not satisfy the condition of weak nonholonomity. The other two theorems are related to the case of weakly nonholonomic systems. In each of the formulated theorems it is shown that the minimum form of Maclaurin series for the potential energy has not a local minimum. Thus, a contribution has been made to the inversion of Lagrange's theorem. [ABSTRACT FROM AUTHOR]

This paper considers distributed tracking control of multiple nonholonomic chained systems using neighbours’ information. With the aid of the cascade structure of each system and properties of persistently excited signals, distributed state feedback tracking controllers and distributed output feedback tracking controllers are proposed such that the tracking errors exponentially converge to zero. To show applications of the proposed results, formation control of wheeled mobile robots is considered. Distributed controllers are obtained with the aid of the proposed theorems. Simulation results verify the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]

Nonlinear controllability and attitude stabilization are studied for the underactuated nonholonomic dynamics of a rigid spacecraft with one variable-speed control moment gyro (VSCMG), which supplies only two internal torques. Nonlinear controllability theory is used to show that the dynamics are locally controllable from the equilibrium point and thus can be asymptotically stabilized to the equilibrium point via time-invariant piecewise continuous feedback laws or time-periodic continuous feedback laws. Specifically, when the total angular momentum of the spacecraft-VSCMG system is zero, any orientation can be a controllable equilibrium attitude. In this case, the attitude stabilization problem is addressed by designing a kinematic stabilizing law, which is implemented through a nonlinear proportional and derivative controller, using the generalized dynamic inverse (GDI) method. The steady-state instability inherent in the GDI controller is elegantly avoided by appropriately choosing control gains. In order to obtain the command gimbal rate and wheel acceleration from control torques, a simple steering logic is constructed to accommodate the requirements of attitude stabilization and singularity avoidance of the VSCMG. Illustrative numerical examples verify the efficacy of the proposed control strategy. [ABSTRACT FROM AUTHOR]

This study suggests a control Lyapunov-based optimal integral terminal sliding mode control (ITSMC) technique for tracker design of asymmetric nonholonomic robotic systems in the existence of external disturbances. The design procedure is based on the control Lyapunov function (CLF) approach. Hence, the output tracking problem is solved by combining the ITSMC with optimal control. The CLF synthesizes a nonlinear optimal control input for the nominal system. Once the control system's states lie far away from the operating point, it is activated to drive them toward the equilibrium point optimally. However, on the condition that the system perturbations are the main factor, the ITSMC would be designed to take over in the vicinity of the equilibrium point. Accordingly, the control goals, such as robustness and precise control, are warranted in the perturbed system. The usefulness of the suggested method is demonstrated with a wheeled mobile robot via a simulation study. [ABSTRACT FROM AUTHOR]

In this paper, a model-based exponential stabilization of a quadruped robot is studied in bounding motion. The dynamics of the five-link planar underactuated mechanical model of the quadruped robot with four actuated joints system is derived. It is shown that the dynamical equation of the proposed simplified model belongs to a class of second-order nonholonomic mechanical systems which cannot be stabilized by any smooth time-invariant state feedback. Utilizing a coordinate transformation based on the so-called normalized momentum, a robust backstepping control method is presented for the quadruped robot. Both theoretical analysis and numerical simulations show that the robust backstepping controller can stabilize the underactuated quadruped robot so that it could balance on its rear legs and track a desired trajectory. Despite the model parameter uncertainties, the robustness of the controller is maintained. The simulation results show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, Noether symmetry and Mei symmetry of discrete nonholonomic dynamical systems with regular and the irregular lattices are investigated. Firstly, the equations of motion of discrete nonholonomic systems are introduced for regular and irregular lattices. Secondly, for cases of the two lattices, based on the invariance of the Hamiltomian functional under the infinitesimal transformation of time and generalized coordinates, we present the quasi-extremal equation, the discrete analogues of Noether identity, Noether theorems, and the Noether conservation laws of the systems. Thirdly, in cases of the two lattices, we study the Mei symmetry in which we give the discrete analogues of the criterion, the theorem, and the conservative laws of Mei symmetry for the systems. Finally, an example is discussed for the application of the results. [ABSTRACT FROM AUTHOR]

Conformal invariance and conserved quantities for a nonholonomic system of Chetaev's type with variable mass are studied. The conformal factor expressions are derived. The necessary and sufficient conditions are obtained to make the system's conformal invariance Lie symmetrical. The conformal invariance of the weak and strong Lie symmetries for the system is given. The corresponding conserved quantities of the system are derived. Finally, an application of the result is shown with an example. [ABSTRACT FROM AUTHOR]

We propose a method for constructing conformally Hamiltonian systems of dynamical equations whose invariant measure arises from the Hamiltonian equations of motion after a change of variables including a change of time. As an example, we consider the Chaplygin problem of the rolling ball and the Veselova system on the Lie algebra e* (3) and prove their complete equivalence. [ABSTRACT FROM AUTHOR]

A dynamical inverse problem of a nonholonomic non-conservative system in phase space was studied. The differential equations of motion were established for non-conservative and nonholonomic non-conservative systems in phase space , respectively. A first-order ordinary differential equation was obtained by differentiating a known integral of the system with respect to time and introducing the Erugin function. Under two circumstances of which the non-conservative forces only rely on generalized coordinates and only rely on generalized momentum, the algebraic equations for determining the non-conservative forces were obtained by the first-order ordinary differential equation and using the differential equations of motion of the systems. The non-conservative forces of the systems can be determined by solving the above algebraic equations. Some examples were given to illustrate the application of the results. [ABSTRACT FROM AUTHOR]

Abstract: This paper addresses the constrained motion planning problem for nonholonomic systems represented by driftless control systems with output. The problem consists in defining a control function driving the system output to a desirable point at a given time instant, whereas state and control variables remain over the control horizon within prescribed bounds. The state and control constraints are handled by extending the control system with a pair of state equations driven by the violation of constraints, and adding regularizing perturbations. For the regularized system a Jacobian motion planning algorithm is designed, called imbalanced. Solutions of example constrained motion planning problems for the rolling ball illustrate the theoretical concepts. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to study the modeling method for nonholonomic systems with friction by linear complementary problem (LCP). Firstly, the dynamic equation with multipliers for the nonholonomic system with friction is given. Secondly, a standard linear complementary model is established, which describes the normal and tangential characteristics for a nonholonomic system. Thirdly, by using LCP theory, a general criterion is obtained, which can identify the singularity induced by nonideal geometrical constraints in a nonholonomic system. By a typical example of a nonholonomic system with friction, it is revealed how the problem solving for the constraint reaction forces can be transformed into the standard linear complementary problem. The research work may provide reliable theoretical basis for the dynamical simulation of a nonholonomic system with non-smooth factors. [Copyright &y& Elsevier]

Abstract: We propose a new motion planning and simulation scheme for nonholonomic systems in this paper to provide a practical solution for these application problems taking into account of real-time obstacle avoidance and the continuous curvature path generation simultaneously in 3D unknown environment. The proposed motion planning and simulation scheme generates the motion path using a new universal Euler spiral generation algorithm, which is locally optimal based on perceived points of view. The generated Euler spiral solution can be non-symmetrical and easily implemented while maintaining a C 2 continuous. It is therefore more flexible and powerful in dealing with dynamic situations in real-time, compared with current symmetrical Euler spirals solutions. Real-time solutions are particularly important in navigation in unknown environments. The universal Euler spiral algorithm proposed displays a smaller maximum curvature value and smaller mean square curvature value than the conventional symmetrical algorithm in tested cases. Another significant contribution of our work is the new motion planning scheme which extend current 2D based motion planning into three-dimensional (3D) space. In this paper, we have conducted experiments and describe simulation results including 3D motion trajectory modeling for a flight simulation. [ABSTRACT FROM AUTHOR]

Lyapunov’s first method, extended by V. V. Kozlov to nonlinear mechanical 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 motion 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 cases, 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 nonlinear 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. [ABSTRACT FROM AUTHOR]

Abstract: The first Lyapunov method, extended by V. Kozlov to nonlinear mechanical systems, is applied to the study of the instability of the equilibrium position of a mechanical system moving in the field of potential and dissipative forces. The motion of the system is subject to the action of the ideal linear nonholonomic nonhomogeneous constraints. Five theorems on the instability of the equilibrium position of the above mentioned system are formulated. The theorem formulated in [V.V. Kozlov, On the asymptotic motions of systems with dissipation, J. Appl. Math. Mech. 58 (5) (1994) 787–792], which refers to the instability of the equilibrium position of the holonomic scleronomic mechanical system in the field of potential and dissipative forces, is generalized to the case of nonholonomic systems with linear nonhomogeneous constraints. In other theorems the algebraic criteria of the Kozlov type are transformed into a group of equations required only to have real solutions. The existence of such solutions enables the fulfillment of all conditions related to the initial algebraic criteria. Lastly, a theorem on instability has also been formulated in the case where the matrix of the dissipative function coefficients is singular in the equilibrium position. The results are illustrated by an example. [Copyright &y& Elsevier]

The problem of stabilizing reference trajectories- also referred to as the trajectory tracking problem-for nonholonomic mobile robots is revisited. Theoretical difficulties and impossibilities that set inevitable limits to what is achievable with feedback control are surveyed, and properties of kinematic control models are recalled, with a focus on controllable driftless systems that are invariant on a Lie group. This geometric framework takes advantage of ubiquitous symmetry properties involved in the motion of mechanical bodies. The transverse function approach, a control design method developed by the authors for the past few years, is reviewed. A salient feature of this approach, which singles it out of the abundant literature devoted to the subject, is the obtention of feedback laws that unconditionally achieve the practical stabilization of arbitrary reference trajectories, including fixed points and nonadmissible trajectories. This property is complemented with novel results showing how the more common property of asymptotic stabilization of a large class of admissible trajectories can also be granted with this type of control. Application to unicycle-type and car-like vehicles is presented and illustrated via simulations. Complementary issues (transient maneuvers monitoring, extensions of the approach to systems that are not invariant on a Lie group, etc.) are also addressed with the concern of practicality. [ABSTRACT FROM AUTHOR]

A novel force planning method for an underground articulated robot is presented based on the interaction between soil, structure and motion. The objective is to develop a new swerving manner in relation to soil– structure interaction, which affects the behavior of the underground articulated robot in path tracking. The localization of a shield machine is studied first to make the track superpose with the center of a tunnel. The controllability of the shield machine is realized by consideration of the machine–soil interaction during tunneling. In addition to being an obstacle, the soil can also act as a special kind of actuator to assist steering the shield machine, based on which a general force planning model is established for path tracking. Finally, a simulation is performed to show the real states during excavating. [ABSTRACT FROM AUTHOR]

We introduce a method which allows one to recover the equations of motion of a class of nonholonomic systems by finding instead an unconstrained Hamiltonian system on the full phase space, and to restrict the resulting canonical equations to an appropriate submanifold of phase space. We focus first on the Lagrangian picture of the method and deduce the corresponding Hamiltonian from the Legendre transformation. We illustrate the method with several examples and we discuss its relationship to the Pontryagin maximum principle. [Copyright &y& Elsevier]

Abstract: This paper considers feedback control of a group of nonholonomic dynamic systems with uncertainty. Decentralized cooperative controllers are proposed with the aid of Lyapunov techniques, results of graph theory, and backstepping techniques. Robustness of the control laws with respect to communication delays is analyzed. An application of the proposed results is discussed. Simulation results show the effectiveness of the proposed controllers. [Copyright &y& Elsevier]