Nonholonomic Dynamical Systems Research Articles

A Geometric Derivation of the Governing Equations of Motion of Nonholonomic Dynamic Systems

A Geometric Derivation of the Governing Equations of Motion of Nonholonomic Dynamic Systems

Hamilton–Jacobi theory for nonholonomic and forced hybrid mechanical systems

A hybrid system is a system whose dynamics is given by a mixture of both continuous and discrete transitions. In particular, these systems can be utilized to describe the dynamics of a mechanical system with impacts. Based on the approach by Clark [Invariant measures, geometry and control of hybrid and nonholonomic dynamical systems, Ph.D thesis, University of Mirchigan (2020)], we develop a geometric Hamilton–Jacobi theory for forced and nonholonomic hybrid dynamical systems. We state the corresponding Hamilton–Jacobi equations for these classes of systems and apply our results to analyze some examples.

Lie-group modeling and simulation of a spherical robot, actuated by a yoke–pendulum system, rolling over a flat surface without slipping

The present paper aims at introducing a mathematical model of a spherical robot expressed in the language of Lie-group theory. Since the main component of motion is rotational, the space SO(3)3 of three-dimensional rotations plays a prominent role in its formulation. Because of friction to the ground, rotation of the external shell results in translational motion. Rolling without slipping implies a constraint on the tangential velocity of the robot at the contact point to the ground which makes it a non-holonomic dynamical system. The mathematical model is obtained upon writing a Lagrangian function that describes the mechanical system and by the Hamilton minimal-action principle modified through d’Alembert virtual work principle to account for non-conservative control actions as well as frictional reactions. The result of the modeling appears as a series of non-holonomic Euler–Poincaré equations of dynamics plus a series of auxiliary equations of reconstruction and advection type. A short discussion on the numerical simulation of such mathematical model complements the main analytic-mechanic development.

Adaptive passivity-based control of an autonomous underwater vehicle

An Autonomous Underwater Vehicle (AUV) is a nonlinear nonholonomic dynamical system. Tracking control of AUVs in the presence of disturbances and uncertainties is a challenging problem. In this paper, an adaptive passivity-based algorithm is proposed for tracking control of an AUV. Also, a disturbance estimator has been used to attenuate the effects of unknown disturbances. A proper adaptive rule is designed to estimate unknown parameters and disturbances. The stability of the proposed method is analyzed using passivity properties and the Lyapunov theory. Obtained comparative results show the advantages of the proposed Adaptive Computed Torque Method ACTM plus disturbance estimator.

On the Local Planners in the RRT* for Dynamical Systems and Their Reusability for Compound Cost Functionals

In this article, we study properties of local planners for nonholonomic dynamical systems to achieve asymptotic global optimality in a sampling-based planner, namely, the RRT*. Formal results are presented concerning length optimal trajectories under a nonholonomic metric, more specifically, an analysis about necessary or sufficient conditions for the local planners is provided. The analysis makes use of the case of finding time-optimal trajectories for differential drive robots. Later, relevant concepts for complex robotic systems such as detachability and time dominance are introduced. Detachability is a property over the cost functional such that, if respected, it allows the reusability of previously designed local planners for simpler systems into more complex ones. The introduced concepts are further illustrated through a robotic system comprised of a mobile manipulator robot equipped with a camera in the arm’s end-effector, whose task is to move between states maintaining a desired object within its field of view. Experiments in a physical mobile manipulator robot are shown, which validate the proposed theoretical modeling.

Nonholonomic dynamics of the Twistcar vehicle: asymptotic analysis and hybrid dynamics of frictional skidding

The Twistcar vehicle is a classic example of a nonholonomic dynamical system. The vehicle model consists of two rigid links connected by an actuated rotary joint and supported by wheeled axles, where nonholonomic constraints are assumed to impose no skidding of the wheels. Recent experimental measurements conducted with a robotic Twistcar prototype have shown disagreements with previous theoretical analyses. In particular, significant skidding has been observed, in addition to discrepancies with respect to theoretical predictions of divergence in oscillations of the vehicle’s speed and orientation, as well as direction reversal depending on the vehicle’s structure. The goal of our research is to resolve this disagreement by generalizing the theoretical analysis. First, we extend previous asymptotic analysis by incorporating the effects of links’ inertia and oscillation amplitude of the input angle on the direction of net motion. Next, we formulate the vehicle’s hybrid dynamics under frictional bounds and skid-state transitions. Using numerical analysis, we obtain optimal values for the vehicle’s mean speed and energetic cost-of-transport as a function of the input frequency. Our results improve the agreement between theory and experiments and suggest directions for further experimental investigation.

Predefined-time leader-following consensus for nonholonomic chained-form multiagent dynamic systems

ABSTRACT In addition to stability and accuracy, rapidity is another important index of the performance of control systems. The current fixed time design cannot meet the requirement for rapidity. The design to meet this requirement must be such that the settling time can be given in advance, that is, the so-called predefined-time design. This paper deals with the predefined-time consensus problem for a class of nonholonomic chained-form multiagent dynamic systems with unknown disturbances. First, a distributed observer for each follower is investigated such that the leader state can be estimated by the followers in a predefined time. Based on this observer, a switching consensus tracking controller is proposed to ensure that the tracking errors converge to zero within a predefined time. Compared with the existing finite-time and fixed-time schemes, the upper bound of the settling time is an directly tunable control parameter, and the settling time can be easily tuned by the control parameter. The stability of the closed-loop system is proved by the Lyapunov method. A simulation example of wheeled mobile robots is performed to demonstrate the effectiveness of the proposed controllers.

Distributed fixed-time leader-following consensus tracking control for nonholonomic multi-agent systems with dynamic uncertainties

The problem of distributed fixed-time leader-following consensus tracking control for nonholonomic chained-form dynamic systems with nonlinear disturbances is considered in this paper. Compared with the existing works, this paper considers a class of more general nonholonomic multi-agent systems where both the leader and the followers have nonlinear uncertainties. For undirected graphs, a distributed observer for each follower is investigated such that both the leader state and input can be estimated by the followers in a fixed time. Based on the distributed observers, a consensus tracking controller is proposed to ensure that the tracking errors convergence to zero within a fixed time. Lyapunov analysis proves the stability of the closed-loop system. A simulation example of wheeled mobile robots is given to demonstrate the effectiveness of the proposed controllers.

Motion Planning of a Second-Order Nonholonomic Chained Form System Based on Holonomy Extraction

This paper proposes a motion planning algorithm for dynamic nonholonomic systems represented in a second-order chained form. The proposed approach focuses on the so-called holonomy resulting from a kind of motion that traverses a closed path in a reduced configuration space of the system. According to the author’s literature survey, control approaches that make explicit use of holonomy exist for kinematic nonholonomic systems but does not exist for dynamic nonholonomic systems. However, the second-order chained form system is controllable. Also, the structure of the second-order chained form system analogizes with the one of the first-order chained form for kinematic nonholonomic systems. These survey and perspectives brought a hypothesis that there exists a specific control strategy for extracting holonomy of the second-order chained form system to the author. To verify this hypothesis, this paper shows that the holonomy of the second-order chained form system can be extracted by combining two appropriate pairs of sinusoidal inputs. Then, based on such holonomy extraction, a motion planning algorithm is constructed. Furthermore, the effectiveness is demonstrated through some simulations including an application to an underactuated manipulator.

Visual servoing of dynamic wheeled mobile robots with anti-interference finite-time controllers

PurposeThe finite-time visual servoing control problem is considered for dynamic wheeled mobile robots (WMRs) with unknown control direction and external disturbance.Design/methodology/approachBy using finite-time control method and switching design technique.FindingsFirst, the visual servoing kinematic WMR model is developed, which can be converted to the dynamic chained-form systems by using a state and input feedback transformation. Then, for two decoupled subsystems of the chained-form systems, according to the finite-time stability control theory, a discontinuous three-step switching control strategy is proposed in the presence of uncertain control coefficients and external disturbance.Originality/valueA class of discontinuous anti-interference control method has been presented for the dynamic nonholonomic systems.

Dynamics of non-holonomic systems with stochastic transport.

This paper formulates a variational approach for treating observational uncertainty and/or computational model errors as stochastic transport in dynamical systems governed by action principles under non-holonomic constraints. For this purpose, we derive, analyse and numerically study the example of an unbalanced spherical ball rolling under gravity along a stochastic path. Our approach uses the Hamilton-Pontryagin variational principle, constrained by a stochastic rolling condition, which we show is equivalent to the corresponding stochastic Lagrange-d'Alembert principle. In the example of the rolling ball, the stochasticity represents uncertainty in the observation and/or error in the computational simulation of the angular velocity of rolling. The influence of the stochasticity on the deterministically conserved quantities is investigated both analytically and numerically. Our approach applies to a wide variety of stochastic, non-holonomically constrained systems, because it preserves the mathematical properties inherited from the variational principle.

Dynamics of a rolling robot

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.

Fixed-time regulation control of uncertain nonholonomic systems and its applications

ABSTRACTStabilisation of nonholonomic systems is of great practical importance to the industry. Moreover, fixed-time control is more comfortable than finite-time control since the upper bound of the settling time is independent on the initial system states in a fixed-time control issue and therefore can be estimated in advance. Inspired by the aforementioned two points, we consider the fixed-time stabilisation for a kind of uncertain nonholonomic systems subject to perturbations in this paper. A globally fixed-time stabilisation strategy is proposed by taking advantage of adding one power integrator technique and switching ideal. Under the designed controllers, all states can be regulated to zero before a fixed time and kept zero afterwards. As an application, the fixed-time stabilisation for a class of dynamic nonholonomic systems is also addressed by the combined method of adding one power integrator and terminal sliding-mode control. Three mechanical and academic examples are provided to show the flexibility and effectiveness of the assumptions and control algorithms.

Modeling and adaptive tracking for stochastic nonholonomic constrained mechanical systems

This paper is devoted to the problem of modeling and trajectory tracking for stochastic nonholonomic dynamic systems in the presence of unknown parameters. Prior to tracking controller design, the rigorous derivation of stochastic nonholonomic dynamic model is given. By reasonably introducing so-called internal state vector, a reduced dynamic model, which is suitable for control design, is proposed. Based on the backstepping technique in vector form, an adaptive tracking controller is then derived, guaranteeing that the mean square of the tracking error converges to an arbitrarily small neighborhood of zero by tuning design parameters. The efficiency of the controller is demonstrated by a mechanics system: a vertical mobile wheel in random vibration environment.

Structural aspects of Hamilton–Jacobi theory

In our previous papers [J. F. Cariñena, X. Gràcia, G. Marmo, E. Martínez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometric Hamilton–Jacobi theory, Int. J. Geom. Meth. Mod. Phys. 3 (2006) 1417–1458; Geometric Hamilton–Jacobi theory for nonholonomic dynamical systems, Int. J. Geom. Meth. Mod. Phys. 7 (2010) 431–454] we showed that the Hamilton–Jacobi problem can be regarded as a way to describe a given dynamics on a phase space manifold in terms of a family of dynamics on a lower-dimensional manifold. We also showed how constants of the motion help to solve the Hamilton–Jacobi equation. Here we want to delve into this interpretation by considering the most general case: a dynamical system on a manifold that is described in terms of a family of dynamics (slicing vector fields) on lower-dimensional manifolds. We identify the relevant geometric structures that lead from this decomposition of the dynamics to the classical Hamilton–Jacobi theory, by considering special cases like fibered manifolds and Hamiltonian dynamics, in the symplectic framework and the Poisson one. We also show how a set of functions on a tangent bundle can determine a second-order dynamics for which they are constants of the motion.

Saturated finite-time stabilization of uncertain nonholonomic systems in feedforward-like form and its application

This paper investigates the problem of finite-time stabilization by state feedback for a class of uncertain nonholonomic systems with inputs saturation. Comparing with the existing relevant literature, a distinguishing feature of the systems under investigation is that the x-subsystem is in feedforward-like form. Rigorous design procedure for saturated finite-time state feedback control is presented by using the adding a power integrator and the nested saturation methods. The development of saturated finite-time controller is also presented briefly for a class of dynamic nonholonomic systems in feedforward-like form. An application example for a kinematic hopping robot is provided to illustrate the effectiveness of the proposed approach.

The Phenomenon of Reversal in the Euler–Poincaré–Suslov Nonholonomic Systems

The development of robotics makes it necessary to study the problem of controlling nonholonomic systems (Svinin et al., Regul Chaotic Dyn. 2013; 18(1---2): 126---143, Borisov et al., Regul. Chaotic Dyn. 2013; 18(1---2): 144---158, Ivanova et al., Regul Chaotic Dyn. 2014; 19(1): 140---143). In this paper, the dynamics of nonholonomic systems on Lie groups with a left-invariant kinetic energy and left-invariant constraints are considered. Equations of motion form a closed system of differential equations on the corresponding Lie algebra. In addition, the effect of change in the stability of steady motions of these systems with the direction of motion reversed (the reversal found in rattleback dynamics) is discussed. As an illustration, the rotation of a rigid body with a fixed point and the Suslov nonholonomic constraint as well as the motion of the Chaplygin sleigh is considered.

Finite-time stabilization of a general class of nonholonomic dynamic systems via terminal sliding mode

This paper presents control strategies for finite-time stabilization of a class of nonholonomic dynamic systems with unknown virtual control coefficients and system parameters. The minimal dilation degree technique and the terminal sliding mode control scheme with finite-time convergence are used to design the controllers. The systematic control strategy development involves the introduction of state transformations and the application of recursive terminal sliding mode structure. Depending on whether the system in question can be converted into a time-invariant linear system or not, two control schemes are proposed respectively guaranteeing that system states converge to zero in finite time. The effectiveness and the robust feature of the developed control approaches are testified by two practical examples: the simplified underactuated hovercraft system and the parking problem for a mobile robot of the unicycle type.

Modeling and adaptive motion/force tracking for vertical wheel on rotating table

This paper is devoted to the problem of modeling and adaptive motion/force tracking for a class of nonholonomic dynamic systems with affine constraints (NDSAC): a vertical wheel on a rotating table. Prior to the development of tracking controller, the dynamic model of the wheel in question is derived in a meticulous manner. A continuously differentiable friction model is also considered in the modeling. By exploiting the inherent cascade interconnected structure of the wheel dynamics, an adaptive motion/force tracking controller is presented guaranteeing that the trajectory tracking errors asymptotically converge to zero while the contact force tracking errors can be made small enough by tuning design parameters. Simulation results are provided to validate the effectiveness of the proposed tracking methodology.

Topology and Bifurcations in Nonholonomic Mechanics

This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic, it can be represented in Hamiltonian form with a Lie–Poisson bracket of rank two. This Lie–Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible types of integral manifolds are found and a classification of trajectories on them is presented.