Stabilization control of non-holonomic systems with application to rider-motorcycle systems

A design method is presented for fuzzy control to stabilize a system subject to non-holonomic constraints. Although a time-varying feedback law has been proposed for stabilizing non-holonomic systems, it results in slow convergence. By contrast, the current study employs fuzzy control designed with sliding modes to stabilize non-holonomic systems. The non-holonomic system becomes input-output linearizable via a proper choice of output function. Fuzzy control is formulated to become a class of variable structure system (VSS) control. Moreover, it is designed with the aid of sliding modes to ensure stability. A rider-motorcycle system, assuming rolling contact between tyres and ground, can be treated as a non-holonomic system. Owing to the motorcycle's static natural instability, stabilization control becomes an important task while riding. Using the proposed method, both hands-off and hands-on rider control are investigated. The effects of stabilization control exerted by the rider are examined according to simulation results.     

Formation Control and Tracking for Co-operative Robots with Non-holonomic Constraints

This paper mainly addresses formation control problem of non-holonomic systems in an optimized manner. Instead of using linearization to solve this problem approximately, we designed control laws with guaranteed global convergence by leveraging nonlinear transformations. Under this nonlinear transformation, consensus of non-holonomic robots can be converted into a stabilization problem, to which optimal treatment applies. This concept is then extended to the formation control and tracking problem for a team of robots following leader-follower strategy. A trajectory generator prescribes the feasible motion of virtual reference robot, a decentralized control law is used for each robot to track the reference while maintaining the formation. The asymptotic convergence of follower robots to the position and orientation of the reference robot is ensured using the Lyapunov function which is also generated using chained form differential equations. In order to witness the efficacy of the scheme, simulations results are presented for Unicycle and Car-like robots.     

Cooperative control design for non-holonomic chained-form systems

Consensus and formation control problems for multiple non-holonomic chained-form systems are solved in this paper. For consensus problem, based on cascaded structure of the chained-form systems, it amounts to solving two consensus subproblems of two linear subsystems transformed from the original system. With the obtained consensus protocols and the method of virtual structure, decentralised formation controllers can then be designed. According to different desired motion patterns of the entire group, both the formation tracking and formation stabilisation problems can be considered. The significance of this paper lies in adapting theories from non-autonomous cascaded systems for cooperative control design for non-holonomic chained-form systems. A unique feature of our proposed solution is that all states can be cooperatively controlled to achieve the desired references for non-holonomic chained-form system. Simulation results are included to illustrate the effectiveness of the proposed methods in solving cooperative control problems of non-holonomic chained-form systems.     

Learning optimal trajectories for non-holonomic systems

Many advanced robotic systems are subject to non-holonomic constraints, e.g. wheeled mobile robots, space manipulators and multifingered robot hands. Steering these mechanisms between configurations in the presence of perturbations is a difficult problem. In fact, the divide et impera strategy (first plan a trajectory, then track it by feedback) has a fundamental drawback in this case: due to the peculiar control properties of non-holonomic systems, smooth feedback cannot provide tracking of the whole trajectory. As a result, it would be necessary to give up either accuracy in the final positioning or predictability of the actual motion. We pursue here a different approach which does not rely on a separation between planning and control. Based on the learning control paradigm, a robust steering scheme is devised for systems which can be put in chained form, a canonical structure for non-holonomic systems. By overparametrizing the control law, other performance goals can be met, typically expressed as cost functions to be minimized along the trajectory. As a case study, we consider the generation of robust optimal trajectories for a car-like mobile robot, with criteria such as total length, maximum steering angle, distance from workspace obstacles, or error with respect to an offline planned trajectory.     

Output consensus for multiple non-holonomic systems under directed communication topology

In this paper, the problem of output consensus for multiple non-holonomic systems in chained form has been investigated. First, an output consensus controller under the strongly connected communication topology is devised by two steps, where a time-varying control strategy and the backstepping design technique are employed. Then, the results are extended to the general directed topology case via graph decomposition, in which the input-to-state stability theory plays a critical role. We prove that the proposed controller can achieve the semi-global output consensus among multiple non-holonomic systems, provided that the interaction graph contains a spanning tree. Finally, numerical examples are provided to illustrate the effectiveness of the designed controller.     

Structure of optimized generalized coordinates partitioned vectors for holonomic and non-holonomic systems

The generalized coordinates partitioning is a well-known procedure that can be applied in the framework of a numerical integration of the DAE systems. However, although the procedure proves to be a very useful tool, it is known that an optimization algorithm for the coordinates partitioning is needed to obtain the best performance. In the paper, the optimized partitioning of the generalized coordinates is revisited in the context of a numerical forward dynamics of the holonomic and non-holonomic multibody systems. After a short presentation of the geometric background of the optimized coordinates partitioning, a structure of the optimally partitioned vectors is discussed on the basis of a gradient analysis of the separate constraint sub-manifolds at the configuration and the velocity levels when holonomic and non-holonomic constraints are present in the system. It is shown that, for holonomic systems, the vectors of optimally partitioned coordinates have the same structure for the generalized positions and velocities. On the contrary, in the case of non-holonomic systems, the optimally partitioned coordinates generally differ at the configuration and the velocity levels. The conclusions of the paper are illustrated within the framework of the presented numerical example.     

Trajectory tracking control of dynamic non-holonomic systems with unknown dynamics

Trajectory tracking control of the non-holonomic system has attracted lots of interest recently, but little has been done in the tracking problem of the uncertain dynamic non-holonomic systems. The paper deals with the trajectory tracking problem of the dynamic non-holonomic systems with unknown dynamics. New adaptive robust controllers are proposed for globally tracking problem. The controllers of low dimension and no singular points can make the configuration state asymptotically track the desired trajectory. An application to a wheeled mobile robot is presented, and the effectiveness of the approach is verified by simulation. Copyright © 1999 John Wiley & Sons, Ltd.     

On Derivation of Motion Equations for Systems with Non-Holonomic High-Order Program Constraints

The paper develops and discusses the generalization of modeling methods for systems with non-holonomic constraints. The classification of constraints has been revisited and a concept of program constraints introduced. High-order non-holonomic constraints (HONC), as presented in examples, are the generalization of the constraint concept and may, as a constraint class, include many of motion requirements that are put upon mechanical systems. Generalized program motion equations (GPME) that have been derived in the paper can be applied to systems with HONC. Concepts of virtual displacements and a generalized variational principle for high-order constraints are presented. Classical modeling methods for non-holonomic systems based on Lagrange equations with multipliers, Maggi, Appell–Gibbs, Boltzman–Hamel, Chaplygin and others are peculiar cases of GPME. The theory has been illustrated with examples of high-order constraints. Motion equations have been derived for a system subjected to a constraint that programmed a trajectory curvature profile. Efficiency, advantages and disadvantages of GPME have been discussed.     

Geometric properties of projective constraint violation stabilization method for generally constrained multibody systems on manifolds

During numerical forward dynamics of constrained multibody systems, a numerical violation of system kinematical constraints is the important issue that has to be properly treated. In this paper, the stabilized time-integration procedure, whose constraint stabilization step is based on the projection of integration results to underlying constraint manifold via post-integration correction of the selected coordinates is discussed. A selection of the coordinates is based on the optimization algorithm for coordinates partitioning. After discussing geometric background of the optimization algorithm, new formulae for optimized partitioning of the generalized coordinates are derived. Beside in the framework of the proposed stabilization algorithm, the new formulae can be used for other integration applications where coordinates partitioning is needed. Holonomic and non-holonomic systems are analyzed and optimal partitioning at the position and velocity level are considered further. By comparing the proposed stabilization method to other projective algorithms reported in the literature, the geometric and stabilization issues of the method are addressed. A numerical example that illustrates application of the method to constraint violation stabilization of non-holonomic multibody system is reported. An erratum to this article can be found at     

Adaptive stabilization of uncertain dynamic non-holonomic systems

This paper investigates the stabilization problem of uncertain dynamic non-holonomic chained systems with unknown constant inertia parameters. Globally time-varying adaptive stabilizing control laws for such systems are presented. The application of it to a non-holonomic wheeled mobile robot is described. Simulation results show that the proposed approach is effective.     

Lyapunov design of global state and output feedback trackers for non-holonomic control systems

In this paper, new results are obtained for the global tracking of a class of non-holonomic dynamic systems via state and output feedback. The tracking controllers are systematically constructed on the basis of a recursive technique and a full exploitation of the system structure. When disturbances occur in a non-holonomic chained system, it is shown how to modify the controller design procedure to yield robust tracking control laws. The proposed method is demonstrated and discussed by means of a benchmark non-holonomic knife-edge mechanical system.     

A system decomposition method for region tracking control of a non-holonomic mobile robot with dynamic parameter uncertainties

The tracking control problem of non-holonomic mobile robot systems has been extensively investigated in the past decades, however, most of the existing control strategies were developed specifically for the fixed-point tracking. This technical note focuses on the region tracking control for a non-holonomic mobile robot system with parameter uncertainties in the robot dynamics. With the system decomposition and adaptive control method, some restrictions imposed on the angular and linear velocities of the non-holonomic mobile robot in recent literature are removed, enabling to track dynamic trajectories with any values of the angular and line velocities. The proposed adaptive control scheme can simultaneously solve both the regulation and region tracking problems of a non-holonomic mobile robot with one passive wheel and two actuated wheels. By utilizing the designed control laws, the mobile robot system is able to globally reach inside a moving region specified by potential functions whose path can be a circular curve, a straight line, or sinusoidal curve, by using a single adaptive controller. Since the dynamic region can be specified arbitrarily small, the fixed-point tracking can be regarded as a special case of region tracking studied in this paper. Compared with the traditional fixed-point tracking, region tracking has more flexibility and better robustness. Numerical results are presented to show the effectiveness of the designed strategy.     

Hybrid adaptive control laws solving a path following problem for non-holonomic mobile manipulators

In this paper a general solution to the path following problem for mobile manipulators with non-holonomic mobile platform has been presented. New proposed control algorithms — for mobile manipulators with fully known dynamics or with parametric uncertainty in the dynamics — take into considerations the kinematics as well as the dynamics of the non-holonomic mobile manipulator. The convergence of the control algorithms is proved using the LaSalle's invariance principle.     

Feedback linearization and stabilization of second-order non-holonomic chained systems

This paper presents a theoretical framework for non-regular feedback linearization and stabilization of second-order non-holonomic chained systems. By giving a new criterion for the problem of non-smooth non-regular feedback linearization, it is proved that second-order chained systems are non-regular static state feedback linearizable. A discontinuous control law is obtained based on linear system theory and the inversion technique. The design mechanism is generalized to higher-order non-holonomic chained systems. Simulation studies are carried out to show the effectiveness of the approach.     

Stabilization of non-holonomic chained systems by gain scheduling

In this paper, a simple method the gain scheduling technique is applied for the stabilization of non-holonomic chained systems, such as mobile robots, front-wheel drive cars, and fire trucks. First, the stabilizing non-holonomic chained systems' problem is formulated as a multi-input multi-state (MIMS) linear parameter-varying system. By choosing a suitable and non-zero control input in the new formulated linear parameter-varying system, the gain scheduling techniques can then easily be applied to design the state feedback stabilizing controller. Simulation results show that the proposed methodology has demonstrated superiority over previous methods and results in much simpler trajectories.     

Non-uniform in time stabilization for linear systems and tracking control for non-holonomic systems in chained form

The paper contains certain results concerning non-uniform in time stabilization for linear time-varying systems by means of a linear time-varying feedback controller. These results enable us to present an explicit solution for the state feedback tracking control problem of non-holonomic systems in chained form under weaker hypotheses than those imposed in earlier existing works.     

Cooperative Control of Dynamical Systems With Application to Autonomous Vehicles

In this paper, a new framework based on matrix theory is proposed to analyze and design cooperative controls for a group of individual dynamical systems whose outputs are sensed by or communicated to others in an intermittent, dynamically changing, and local manner. In the framework, sensing/communication is described mathematically by a time-varying matrix whose dimension is equal to the number of dynamical systems in the group and whose elements assume piecewise-constant and binary values. Dynamical systems are generally heterogeneous and can be transformed into a canonical form of different, arbitrary, but finite relative degrees. Utilizing a set of new results on augmentation of irreducible matrices and on lower triangulation of reducible matrices, the framework allows a designer to study how a general local-and-output-feedback cooperative control can determine group behaviors of the dynamical systems and to see how changes of sensing/communication would impact the group behaviors over time. A necessary and sufficient condition on convergence of a multiplicative sequence of reducible row-stochastic (diagonally positive) matrices is explicitly derived, and through simple choices of a gain matrix in the cooperative control law, the overall closed-loop system is shown to exhibit cooperative behaviors (such as single group behavior, multiple group behaviors, adaptive cooperative behavior for the group, and cooperative formation including individual behaviors). Examples, including formation control of nonholonomic systems in the chained form, are used to illustrate the proposed framework.     

Closed-loop form quantization algorithm with bounded accumulative error

This paper proposes a closed-loop form quantization algorithm that guarantees the boundness of accumulative error. The algorithm is particularly useful for mobile robot navigation that is usually implemented on embedded systems. If the wheel command of the mobile robot is given by velocity or positional increments at every control instant that are quantized due to the finite word length of the controller's CPU, the quantization error accumulates to produce large position error. Such an error is critical for wheeled mobile robots or autonomous vehicles with non-holonomic constraints. To solve this problem, a non-error accumulative quantization algorithm with a closedloop form is presented. We can extend it to a generalized form corresponding to the nth-order accumulation. The boundness of the accumulative quantization error is proven via mathematic processes and verified by a series of computer simulations. The proposed method is effective to accurately control the autonomous mobile robot, particularly with embedded systems.     

非完整机械控制系统的模型参考变结构控制

在参数扰动和外部干扰情况下,对非完整机械控制系统设计了变结构模型参考跟踪控制器,基于适当的矩阵分解,非线性控制理论中的输入－输出解耦概念及变结构控制理论,为解决干扰非完整机械控制系统的跟踪问题提出了三级控制设计过程,最后通过一非完整机械系统的例子（在给定平面上运动的垂直轮）的计算机仿真说明了提出方法的优越性。     

Formation-Containment Control Using Dynamic Event-Triggering Mechanism for Multi-Agent Systems

The paper proposes a novel approach for formation-containment control based on a dynamic event-triggering mechanism for multi-agent systems. The leader-leader and follower-follower communications are reduced by utilizing the distributed dynamic event-triggered framework. We consider two separate sets of design parameters: one set comprising control and dynamic event-triggering parameters for the leaders and a second set similar to the first one with different values for the followers. The proposed algorithm includes two novel stages of co-design optimization to simultaneously compute the two sets of parameters. The design optimizations are convex and use the weighted sum approach to enable a structured trade-off between the formation-containment convergence rate and associated communications. Simulations based on non-holonomic mobile robot multi-agent systems quantify the effectiveness of the proposed approach.