Practical and asymptotic stabilization of chained systems by the transverse function control approach

A control approach for practical and asymptotic stabilization of driftless controllable control systems with perturbations is proposed. This type of systems naturally appears when addressing the trajectory stabilization problem of driftless control systems on Lie groups. The objective of the approach is to provide practical stability whatever the perturbation ---e.g. practical stability of any trajectory in the state space---, and asymptotic stability or convergence to zero of the error variables when the perturbation term is zero or tends to zero. A general framework is presented in this paper and a control solution is proposed for the class of chained systems

Feedback control of the general two-trailers system with the Transverse Function approach

The so-called "general two-trailers system" is a nonholonomic system composed of a controlled unicycle-like vehicle and two passive trailers with off-axle hitching. It is not differentially flat and cannot be transformed into the chained form system. Methods developed for this latter class of systems thus do not apply. The Transverse Function (TF) approach is here used to solve the trajectory tracking problem for this system. The proposed control solution yields practical stabilization of any reference motion, whether it is or is not feasible. Practical stabilization of non-feasible trajectories in the case of non-differently flat systems is of particular interest due partly to the difficulty of planning and calculating desired feasible state reference motions. The method is illustrated by simulation results which show that, in addition to the unconditional practical stabilization property evoked above, asymptotic stabilization of feasible and persistently exciting motions can also be achieved with the same performance as local stabilizers derived from a linear approximation of the tracking-error equations

Trajectory tracking for nonholonomic systems. Theoretical background and applications

The problem of stabilizing reference trajectories for nonholonomic systems, often referred to as the trajectory tracking problem in the literature on mobile robots, is addressed. The first sections of this report set the theoretical background of the problem, with a focus on controllable driftless systems which are invariant on a Lie group. The interest of the differential geometry framework here adopted comes from the possibility of taking advantage of ubiquitous symmetry properties involved in the motion of mechanical bodies. Theoretical difficulties and impossibilities which set inevitable limits to what is achievable with feedback control are surveyed, and basic control design tools and techniques are recast within the approach here considered. A general method based on the so-called Transverse Function approach --developed by the authors--, yielding feedback controls which unconditionnally achieve the {\em practical} stabilization of arbitrary reference trajectories, including fixed points and non-admissible trajectories, is recalled. This property singles the proposed solution out of the abundant literature devoted to the subject. It is here complemented with novel results showing how the more common property of asymptotic stabilization of persistently exciting admissible trajectories can also be granted with this type of control. The last section of the report concerns the application of the approach to unicycle-type and car-like vehicles. The versatility and potentialities of the Transverse Function (TF) control approach are illustrated via simulations involving various reference trajectory properties, and a few complementary control issues are addressed. One of them concerns the possiblity of using control degrees of freedom to limit the vehicle's velocity inputs and the number of transient maneuvers associated with the reduction of initially large tracking errors. Another issue, illustrated by the car example, is related to possible extensions of the approach to systems which are not invariant on a Lie group

Exponential Stabilization of Driftless Nonlinear Control Systems

This dissertation lays the foundation for practical exponential stabilization of driftless control systems. Driftless systems have the form, xdot = X1(x)u1 + .... + Xm(x)um, x ∈ ℝ^n Such systems arise when modeling mechanical systems with nonholonomic constraints. In engineering applications it is often required to maintain the mechanical system around a desired configuration. This task is treated as a stabilization problem where the desired configuration is made an asymptotically stable equilibrium point. The control design is carried out on an approximate system. The approximation process yields a nilpotent set of input vector fields which, in a special coordinate system, are homogeneous with respect to a non-standard dilation. Even though the approximation can be given a coordinate-free interpretation, the homogeneous structure is useful to exploit: the feedbacks are required to be homogeneous functions and thus preserve the homogeneous structure in the closed-loop system. The stability achieved is called p-exponential stability. The closed-loop system is stable and the equilibrium point is exponentially attractive. This extended notion of exponential stability is required since the feedback, and hence the closed-loop system, is not Lipschitz. However, it is shown that the convergence rate of a Lipschitz closed-loop driftless system cannot be bounded by an exponential envelope. The synthesis methods generate feedbacks which are smooth on ℝ^n \ {0}. The solutions of the closed-loop system are proven to be unique in this case. In addition, the control inputs for many driftless systems are velocities. For this class of systems it is more appropriate for the control law to specify actuator forces instead of velocities. We have extended the kinematic velocity controllers to controllers which command forces and still p-exponentially stabilize the system. Perhaps the ultimate justification of the methods proposed in this thesis are the experimental results. The experiments demonstrate the superior convergence performance of the p-exponential stabilizers versus traditional smooth feedbacks. The experiments also highlight the importance of transformation conditioning in the feedbacks. Other design issues, such as scaling the measured states to eliminate hunting, are discussed. The methods in this thesis bring the practical control of strongly nonlinear systems one step closer

Control of mechanical systems on Lie groups based on vertically transverse functions

"The transverse function approach to control provides a unified setting to deal with practical stabilization and tracking of arbitrary trajectories for controllable driftless systems. Controllers derived from that approach offer advantages over those based on more classical techniques for control of nonholonomic systems. Nevertheless, its extension to more general classes, such as critical underactuated mechanical systems, is not immediate. The present paper explores a possible extension by developing a framework that allows one to cast point stabilization problems for (left-invariant) second-order systems on Lie groups, including simple mechanical systems. The approach is based on "vertical transversality," a property exhibited by derivatives of transverse functions. In this paper, we lay out the theoretical foundations of our approach and present an example to illustrate some of its features.

On the transversality of functions at the core of the transverse function approach to control

"The transverse function approach to control, introduced by Morin and Samson in the early 2000s, is based on functions that are transverse to a set of vector fields in a sense formally similar to, although strictly speaking different from, the classical notion of transversality in differential topology. In this paper, a precise link is established between transversality and the functions used in the transverse function approach. It is first shown that a smooth function f : M -> Q is transverse to a set of vector fields which locally span a distribution D on Q if, and only if, its tangent mapping T f is transverse to D, where D is regarded as a submanifold of the tangent bundle T Q. It is further shown that each of these two conditions is equivalent to transversality of T f to D along the zero section of T M. These results are then used to rigorously state and prove that if M is compact and D is a distribution on Q, then the set of mappings of M into Q that are transverse to D is open in the strong (or "Whitney C (a)-") topology on C (a)(M, Q).

Robust Adaptive Stabilization of Nonholonomic Mobile Robots with Bounded Disturbances

The stabilization problem of nonholonomic mobile robots with unknown system parameters and environmental disturbances is investigated in this paper. Considering the dynamic model and the kinematic model of mobile robots, the transverse function approach is adopted to construct an additional control parameter, so that the closed-loop system is not underactuated. Then the adaptive backstepping method and the parameter projection technique are applied to design the controller to stabilize the system. At last, simulation results demonstrate the effectiveness of our proposed controller schemes

Control with transverse functions and a single generator of underactuated mechanical systems

The control of a class of underactuated mechanical systems on Lie groups is addressed, with the objective of stabilizing, in a practical sense, any (possibly non-admissible) reference trajectory in the configuration space. The present control design method extends a previous result by the authors to systems underactuated by more than one control. For example, it allows to address the case of a 3D-rigid body immersed in a perfect fluid with only three control inputs. The choice of the control parameters is also discussed in relation to the system's zero-dynamics

Nonholonomic Feedback Control Among Moving Obstacles

A feedback controller is developed for navigating a nonholonomic vehicle in an area with multiple stationary and possibly moving obstacles. Among other applications the developed algorithms can be used for automatic parking of a passenger car in a parking lot with complex configuration or a ground robot in cluttered environment. Several approaches are explored which combine nonholonomic systems control based on sliding modes and potential field methods

Vertical transversality and its applications to control of mechanical systems

Tesis (Maestría en Control y Sistemas Dinámicos)"The transverse control approach proposed by Morin and Samson is a technique based on the use of transverse functions to practically stabilize controllable driftless systems. This control technique is able to cope with practical stabilization of admissible trajectories, including fixed points, as well as practical stabilization of non-admissible trajectories. In this thesis we attempt to generalize this technique to the control of second-order systems and, in particular, to the case of mechanical systems described on Lie groups. Within this class one finds mechanical systems subjects to (holonomic and non-holonomic) constraints as well as underactuated mechanical systems. It is important to note that for systems in this class, the drift vector field is required along with the control vector fields to generate the accessibility distribution. We define vertical transversality and we show how transverse functions satisfy vertical transversality, a property that generalizes transversality to second-order systems. By applying the methodology introduced in this thesis to second-order systems one achieves practical stabilization of the configuration variables, namely one ensures that the projection of the state trajectories onto the configuration manifold converge to an arbitrarily small neighborhood, specified in advance, of the desired equilibrium point. Although the approach outlined in this thesis does not constitute a complete extension of Morin and Samson’s approach based on transverse functions, it takes initial steps toward what might constitute an interesting theory for the stabilization of admissible trajectories for second-order systems.""La aplicación de la técnica de control por medio de funciones transversas propuesta por Morin y Samson a sistemas controlables sin derivada como resultado una estabilización práctica de las trayectorias del sistema. Esta técnica trata con estabilización práctica de puntos fijos, trayectorias admisibles e inclusive trayectorias no admisibles. En esta tesis se generaliza la noción de transversalidad para sistemas de segundo orden y se plantea el desarrollo de un método de control para estabilizar sistemas de segundo orden, en particular para sistemas mecánicos que evolucionan en grupos de Lie. Dentro de esta clase de sistemas se encuentran sistemas mecánicos sujetos a restricciones (holonómicas y no holonómicas) como también sistemas mecánicos subactuados. Es importante notar que en esta clase de sistemas el término de deriva se requiere, junto con los campos vectoriales de control, para generar la distribución de accesibilidad. En esta disertación se define transversalidad vertical y se muestra cómo las funciones transversas definen funciones verticalmente transversas. Se presenta además un esquema de control para estabilizar sistemas de segundo orden en grupos de Lie usando funciones verticalmente transversas. Este método asegura estabilización práctica de las variables de configuración del sistema, es decir, la proyección de las trayectorias del sistema a la variedad de configuración converge a una vecindad arbitrariamente pequeña del punto deseado de equilibrio. El esquema expuesto, aún cuando no resuelve por completo el problema de estabilización práctica para sistemas de segundo orden, se presenta como el punto de partida de un esquema que podría llegar a constituir una teoría interesante para la estabilización de trayectorias admisibles para sistemas de segundo orden.