Control Of Nonholonomic Systems Research Articles

Relay Switching–Based Event‐Triggered Finite‐Time Tracking Control for Nonholonomic Systems: Theory and Experiment

ABSTRACTThis paper investigates the event‐triggered finite‐time trajectory tracking controller based on a type of chained‐form nonholonomic systems under unknown disturbances by utilizing relay switching technology. A key design idea is that the entire control design is grouped into two separated control stages to produce different finite‐time tracking controllers with correspondingly event‐triggered control rules. By using some nonlinear design methods, a relay switching–based event‐triggered finite‐time tracking control method is adopted to assure that the resulting closed‐loop tracking error system converges to an arbitrarily small neighborhood around zero within a finite time. All closed‐loop error system states keep bounded throughout the entire process. Simulation and experiment results demonstrate the rationality of the designed tracking control strategy.

Adaptive neural network control for nonholonomic systems with time-varying asymmetric constraints and iISS inverse dynamics

Adaptive neural network control for nonholonomic systems with time-varying asymmetric constraints and iISS inverse dynamics

Smooth Time-Varying Feedback Control of Nonholonomic Systems with Applications to the Attitude Control of Underactuated Spacecraft

This paper studies the control of chained nonholonomic systems by using smooth time-varying feedback. First, by using a time-varying state transformation, the chained nonholonomic system is transformed into a linear time-varying system. Second, with the aid of some properties of a class of parametric Lyapunov equations and time-varying systems theory, smooth time-varying state feedback controller is developed to ensure the exponential convergence of both the system states and controls. Different from existing results, the design parameters in the proposed control strategy are independent of the initial conditions of the system. Third, as an application of the proposed smooth time-varying feedback, the attitude control problem for underactuated axisymmetric spacecraft is addressed. Finally, a numerical simulation is presented to validate the effectiveness of the proposed method.

FAS-Based Anti-Disturbance Stabilization Control of Nonholonomic Systems: Theory and Experiment

FAS-Based Anti-Disturbance Stabilization Control of Nonholonomic Systems: Theory and Experiment

Adaptive Multiple-Surface Sliding Mode Control of Nonholonomic Systems With Matched and Unmatched Uncertainties

The problem of stabilizing a class of nonholonomic systems in chained form affected by both matched and unmatched uncertainties is addressed in this paper. The proposed design methodology is based on a discontinuous transformation of the perturbed nonholonomic system to which an adaptive multiple-surface sliding mode technique is applied. The generation of a sliding mode allows to eliminate the effect of matched uncertainties, while a suitable function approximation technique enables to deal with the residual uncertainties, which are unmatched. The control problem is solved by choosing a particular sliding manifold upon which a second order sliding mode is enforced via a continuous control with discontinuous derivative. A positive feature of the present proposal, apart from the fact of being capable of dealing with the presence of both matched and unmatched uncertainties, is that no knowledge of the bounds of the unmatched uncertainty terms is required. Moreover, the fact of producing a continuous control makes the proposed approach particularly appropriate in nonholonomic applications, such as those of mechanical nature.

Adaptive neural network control for nonholonomic systems with partial/full or without state constraints

In this paper, we present a unifying neural networks-based adaptive control framework for nonholonomic systems with partial/full state constraints or without constraint requirements. By considering the tan-type barrier Lyapunov function, constraint requirements are handled for the x0-subsystem. A novel nonlinear state-dependent function is constructed to meet asymmetric time-varying state constraints for the x-subsystem. Combined with backstepping method, a coordinate transformation is designed to transform the original constrained system into unconstrained system. Then, we put forward the dynamic surface control to eliminate feasibility conditions of virtual controllers. In addition, neural networks can be introduced to compensate for the uncertainties of nonholonomic systems. It is demonstrated that without modifying control structure the adaptive neural networks controller guarantees the stability of nonholonomic systems, while asymmetric time-varying state constraints can be satisfied. Simulation examples of mobile robot systems further validate the efficacy of the devised algorithm.

Model predictive control of non-holonomic systems: Beyond differential-drive vehicles

Non-holonomic systems are of immense practical value and increasingly subject to automation. However, controlling them accurately, e.g., when parking vehicles, is challenging for automatic control methods, including model predictive control (MPC). Combining results from MPC theory and sub-Riemannian geometry in the form of homogeneous nilpotent system approximations, this paper proposes the first comprehensive, ready-to-apply design procedure for MPC controllers to steer controllable, driftless non-holonomic systems into given setpoints. It can be ascertained that the resulting controllers nominally asymptotically stabilize the setpoint for a large-enough prediction horizon. The design procedure is exemplarily applied to four systems, including the kinematic car and a differentially driven mobile robot with up to two trailers. The controllers use a non-quadratic cost function tailored to the non-holonomic kinematics. Novelly, it is proven that a quadratic cost employed in an otherwise similar controller is insufficient to reliably asymptotically stabilize the closed loop for all considered vehicles. Since quadratic costs are the conventional choice in control, this highlights the relevance of the findings. To the knowledge of the authors, it is the first time that MPC controllers of the proposed structure are formulated and applied to non-holonomic systems beyond very simple ones, in particular (partly) on hardware.

Optimal Control of Nonholonomic Systems via Magnetic Fields

Geometric optimal control utilizes tools from differential geometry to analyze the structure of a problem to determine the control and state trajectories to reach a desired outcome while minimizing some cost function. For a controlled mechanical system, the control usually manifests as an external force which, if conservative, can be added to the Hamiltonian. In this letter, we focus on mechanical systems with controls added to the symplectic form rather than the Hamiltonian. In practice, this translates to controlling the magnetic field for an electrically charged system. We develop a basic theory deriving necessary conditions for optimality of such a system subjected to nonholonomic constraints. We consider the representative example of a magnetically charged Chaplygin Sleigh, whose resulting optimal control problem is completely integrable.

A δ-Persistently-Exciting Formation Controller for Non-Holonomic Systems Over Directed Graphs

We study a formation control problem of nonholonomic vehicles which consists in making a group of them gather around a given rendezvous set-point and acquire a common orientation. This task may be regarded as part of a more complex maneuver, e.g., requiring the robots to advance in a scouting mission on a path composed of straight lines and occasional turns. The control approach relies on addressing separately the problems of stabilization (on the plane) and of orientation consensus. For the former we use individual controllers involving smooth time-varying terms and for the latter we use distributed consensus control, under the assumption that the robots form a directed graph that contains a spanning tree.

Output feedback control for nonholonomic systems with nonvanishing disturbances and asymmetric time-varying constraints

In this paper, an adaptive output feedback control scheme is introduced for nonholonomic systems with nonliner uncertainties, nonvanishing disturbances, and output constraints. A tan-type barrier Lyapunov function is considered to address asymmetric time-varying output constraints. We define the disturbance as the generalized system state. The extended state observer is proposed for uncertain external disturbance. A new estimator is designed for unknown parameter and states. It is shown that under the backstepping technique the output feedback adaptive controller ensures the stability of the closed loop system, while the asymmetric time-varying output constraints are not violated. The simulation results of mobile robot system are provided to validate the effectiveness of the presented control approach.

Sampled-Data Integral Control of Nonholonomic Systems With Unknown Constant Disturbances

This note studies the asymptotic regulation problem for a class of nonholonomic systems with unknown constant disturbances, where system nonlinearities admit a lower-triangular form. First, following a new state transformation, the considered systems with unknown constant disturbances are converted into a class of auxiliary systems without disturbances. Second, two sampled-data controllers are designed for the auxiliary systems by using the high-gain method. Third, it is shown that with the introduced state transformation, the desired sampled-data <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">integral</i> controllers are obtained to asymptotically regulate the considered systems. The proposed control strategy is applied to asymptotically regulate a unicycle type mobile robot model.

On the geometric construction of a stabilizing time-invariant state feedback controller for the nonholonomic integrator

The paper presents a rather natural and elementary geometric construction of a stabilizing time-invariant state feedback law for the nonholonomic integrator. The key features of the particular construction are the direct inclusion of certain classical optimality considerations pertaining to the geodesics of the nonholonomic integrator, the confinement of all discontinuities in the feedback law to the z-axis, as well as a uniform exponential convergence result for the closed-loop system. The results of this treatment also have interesting implications for the control of nonholonomic systems in general, e.g., they highlight that for nonholonomic systems even the most natural seeming stabilizing feedback laws may not be amenable to a closed-form expression and may need to be formulated in more elaborate and implicit terms.

Supervised learning of mapping from sensor space to chained form for unknown non-holonomic driftless systems

PurposeThis study aims to propose an offline exploratory method that consists of two stages: first, the authors focus on completing the kinematics model of the system by analyzing the Jacobians in the vicinity of the starting point and deducing a virtual input to effectively navigate the system along the non-holonomic constraint. Second, the authors explore the sensorimotor space in a predetermined pattern and obtain an approximate mapping from sensor space to chained form that facilitates controllability.Design/methodology/approachIn this paper, the authors tackle the controller acquisition problem of unknown sensorimotor model in non-holonomic driftless systems. This feature is interesting to simplify and speed up the process of setting up industrial mobile robots with feedback controllers.FindingsThe authors validate the approach for the test case of the unicycle by controlling the system with time-state control policy. The authors present simulated and experimental results that show the effectiveness of the proposed method, and a comparison with the proximal policy optimization algorithm.Originality/valueThis research indicates clearly that feedback control of non-holonomic systems with uncertain kinematics and unknown sensor configuration is possible.

On the Hamel Coefficients and the Boltzmann\u2013Hamel Equations for the Rigid Body

The Boltzmann–Hamel (BH) equations are central in the dynamics and control of nonholonomic systems described in terms of quasi-velocities. The rigid body is a classical example of such systems, and it is well-known that the BH-equations are the Newton–Euler (NE) equations when described in terms of rigid body twists as quasi-velocities. It is further known that the NE-equations are the Euler–Poincaré, respectively, the reduced Euler–Lagrange equations on SE(3) when using body-fixed or spatial representation of rigid body twists. The connection between these equations are the Hamel coefficients, which are immediately identified as the structure constants of SE(3). However, an explicit coordinate-free derivation has not been presented in the literature. In this paper the Hamel coefficients for the rigid body are derived in a coordinate-free way without resorting to local coordinates describing the rigid body motion. The three most relevant choices of quasi-velocities (body-fixed, spatial, and hybrid representation of rigid body twists) are considered. The corresponding BH-equations are derived explicitly for the rotating and free floating body. Further, the Hamel equations for nonholonomically constrained rigid bodies are discussed, and demonstrated for the inhomogenous ball rolling on a plane.

Stabilizability in optimal control

We extend the classical concepts of sampling and Euler solutions for control systems associated to discontinuous feedbacks by considering also the corresponding costs. In particular, we introduce the notions of Sample and Euler stabilizability to a closed target set mathbf{C}with({p_{0}},W)-regulated cost, for some continuous, state-dependent function W and some constant {p_{0}}>0: it roughly means that we require the existence of a stabilizing feedback K such that all the corresponding sampling and Euler solutions starting from a point z have suitably defined finite costs, bounded above by W(z)/{p_{0}}. Then, we show how the existence of a special, semiconcave Control Lyapunov Function W, called {p_{0}}-Minimum Restraint Function, allows us to construct explicitly such a feedback K. When dynamics and Lagrangian are Lipschitz continuous in the state variable, we prove that K as above can be still obtained if there exists a {p_{0}}-Minimum Restraint Function which is merely Lipschitz continuous. An example on the stabilizability with ({p_{0}},W)-regulated cost of the nonholonomic integrator control system associated to any cost with bounded Lagrangian illustrates the results.

Ensemble observability of Bloch equations with unknown population density

We introduce in the paper a novel observability problem for a large population (in the limit, a continuum ensemble) of nonholonomic control systems with unknown population density. We address the problem by focussing on a prototype, namely, the ensemble of Bloch equations which is known for its use of describing the evolution of the bulk magnetization of a collective of non-interacting nuclear spins in a static field modulated by a radio frequency (rf) field. The dynamics of the equations are structurally identical, but show variations in Larmor dispersion and rf inhomogeneity. We assume that the initial state of any individual system (i.e., individual Bloch equation) is unknown and, moreover, the population density of these individual systems is also unknown. Furthermore, we assume that at any time, there is only one scalar measurement output at our disposal. The measurement output integrates a certain observation function, common to all individual systems, over the continuum ensemble. The observability problem we pose in the paper is thus the following: Whether one is able to use the common control input (i.e., the rf field) and the single measurement output to estimate both the initial states of the individual systems and the population density? Amongst other things, we establish a sufficient condition for the ensemble system to be observable: We show that if the common observation function is any harmonic homogeneous polynomial of positive degree, then the ensemble system is observable. The proof of the result utilizes tools from representation theory of Lie algebras. Although the results established in the paper are for the specific ensemble of Bloch equations, the approach developed along the analysis can be generalized to investigate observability of other ensemble systems with single, integrated measurement outputs.

Output feedback control for nonholonomic systems with non-vanishing disturbances

ABSTRACT A new adaptive output feedback controller is given for a class of nonholonomic systems with non-vanishing external disturbances and nonlinear uncertainties. The non-vanishing disturbance is regarded as the generalised system state and an extended state observer (ESO) is designed for the uncertain external disturbance in order to ensure the global asymptotical stability of the closed loop system. A novel observer/estimator is constructed for auxiliary system states and unknown parameter estimates. The integrator backstepping recursive technique based on a construction rule is applied to the design of the output feedback adaptive controller. The simulation example for a mobile robot with velocity and angle velocity measured deviation and disturbance is given to validate the effectiveness of the proposed method.

Uncertain Nonholonomic Systems Control via Sampled-Data Output Feedback of Motor Robot With Two-Dimensional Z-States

Nonholonomic systems with uncertain nonlinearities are sufficiently important to research due to the numerous applications in the real world. It is imperative to find a sampled-data controller for nonholonomic systems due to their implementation within digital computers. It has been proven that under a lower-triangular growth condition, a class of nonholonomic system with a single z-state can be globally stabilized by a sampled-data output feedback controller whose observer and control laws are discrete-time and linear. In this paper, we will further extend the results to the two-dimensional z-states which is more challenging since the boundedness of the change of coordinates is not proved. To solve this problem, we firstly design a controller for the z-system, which can guarantee the boundedness of the change of coordinates after a certain time. From this, we can construct a globally stabilizing output feedback controller for nonholonomic systems. The examples and computer simulations were conducted to show the effectiveness of the proposed controllers for the two-dimensional z-states.

Iterative optimal control syntheses illustrated on the Brockett integrator

In this paper, we investigate computational methods for synthesizing optimal control inputs for nonholonomic control systems. We use the Brockett integrator as a benchmark example to illustrate different aspects of the computational trajectory optimization problem. The main result of this paper is the establishment of a rather attractive iterative scheme for practically constructing optimal control inputs. The functionality and efficiency of the proposed iterative scheme are illustrated on different optimal steering problems centered around the Brockett integrator.

Distributed optimal control for nonholonomic systems with input constraints and uncertain interconnections

This paper studies a distributed optimal tracking control method for nonholonomic mobile mechanical multi-agent systems under complex conditions such as input constraints, the presence of both kinematic and dynamic disturbances, and uncertain interconnections. Initially, novel feed-forward control inputs are proposed to transform the inherently separate systems of kinematics and dynamics into an equivalent integrated system. Successively, an online distributed $$\mathcal {L}_2$$ -bounded optimal control algorithm is designed by utilizing adaptive dynamic programming and the theory of cooperative differential graphical games. In the algorithm, a single neural network instead of three for each agent is chosen, and the online weight-tuning laws for which are designed without identifying uncertain parameters directly or indirectly. Additionally, the optimal control and worst disturbance policies are synchronously updated in only one iterative loop. It is shown that during the convergence of the value functions to the approximate optimal values when the agents perform the algorithm, overall tracking and function approximation errors are uniformly ultimately bounded. Finally, as a successful application of the study, control of the wheeled mobile multi-robot system is discussed through simulations.