Transverse Function Approach Research Articles

Smooth State Feedback Control of a New Nonholonomic Manipulator Coping With Singularities

This paper presents a smooth state feedback control algorithm for set point stabilization of a three-link planar nonholonomic manipulator (NHM). Compared to known control algorithms for such a manipulator, the one presented here is able to control the NHM in its singular positions. In the control algorithm synthesis, we consider the kinematic model of the NHM. Our control algorithm solution is based on an extension of models state variables. Smooth point stabilization is achieved by implementing the transverse function approach. The control algorithm synthesis is performed with utilization of a Lie group representation. Because the Lie algebra associated with the NHM is not nilpotent, we apply a homogeneous nilpotent approximation. Analytical results have been reinforced by simulations and experiments, which were executed using a newly built three-link planar NHM. To the best of our knowledge, this manipulator is the third of its kind in the world, but due to its distinct kinematic structure, it is a unique mechanism.

Feedback Control of a Motorized Skateboard

This paper develops a feedback control solution for the automatization of a motorized skateboard , also called symmetric snakeboard in the robotics and automatic control literature. Controlling this system is particularly challenging due to the association of kinematic and dynamic nonintegrable constraints to which it is subjected. The transverse function approach, combined with backstepping, is used to derive dynamic feedback laws that ensure practical stabilization of arbitrary reference trajectories in the Cartesian space. Simulation results illustrate the proposed approach.

Rough Linearization By One-Dimensional Rough Paths

In this paper, we propose a linearization method of nonlinear control systems by using rough path analysis. The linearization is achieved by the addition of rough signals, which are designed by functions with unbounded variations such as a sinusoidal wave having infinitely-large frequency Therefore, the proposed method is considered to be an extension of the analysis of Lie bracket motions such as the approximation algorithms by Liu and Sussmann in 1990’s, or, the transverse function approach by Morin et al. in 2000’s. Note, however, that, the characteristic feature of the method is to derive controllable linear approximate models for nonholonomic systems even if no Lie bracket motion is generated. Instead, another motions appear based on the similar effects of Wong-Zakai correction terms in stochastic analysis, provided that any stochastic signals are never dealt with in this paper. The validity of the method is confirmed by a few simple examples of linearization problems for nonlinear control systems.

Transverse function approach to practical stabilisation of underactuated surface vessels with modelling uncertainties and unknown disturbances

This study addresses the problem of stabilising a desired trajectory (also referred to as the trajectory tracking problem) for underactuated surface vessels. The desired trajectory does not need to be a particular type (e.g. feasible trajectory) and can be any sufficiently smooth curve parameterised by time. To overcome difficulties caused by underactuation, the authors apply the transverse function approach to introduce an additional control input in backstepping design procedure. They give the calculation of transverse functions accordingly. To compensate for the effects of the external disturbances, they employ disturbance observer to estimate the unknown time-varying disturbances. Subsequently, disturbance-observer-based tracking control is designed that achieves practical stabilisation of any smooth reference trajectory (i.e. the vessel position and orientation tracking errors converge to a small neighbourhood of zero). A noticeable feature of the design technique is that the desired reference trajectory allows to be feasible or non-feasible. Besides the nominal case where exact knowledge of the vessel model is available, they also deal with scenarios wherein modelling uncertainties are present. The proposed control is robust not only to modelling uncertainties but also to external disturbances. Simulation studies are performed to demonstrate the effectiveness of the proposed design technique.

Attitude tracking of a rigid spacecraft using two internal torques

The attitude-tracking control of a rigid spacecraft using only two internal torques is addressed. First, a given reference trajectory is classified as feasible or unfeasible according to the preservation or violation of the momentum conservation law. The dynamics of the attitude-tracking error is then formulated on the attitude manifold SO(3) with the angular momentum of the actuators as inputs. Given the Lie group structure of SO(3), the transverse function approach is utilized to design an attitude-tracking law ensuring asymptotically ultimately bounded tracking error for any reference trajectory. For feasible reference trajectories satisfying certain persistence conditions, asymptotic tracking is achieved by constructing an asymptotically stable zero dynamics for the closed-loop system. To deliver control torques to the actuator command signals, steering laws are designed for two reaction wheels, two single-gimbal control moment gyros mounted in parallel, and one variable-speed control moment gyro, respectively. The resulting control law can be applied to a spacecraft with different kinds of momentum actuators but underactuated for tasks ranging from bounded tracking of a generic trajectory, three-axis earth pointing to line-of-sight pointing, etc. Numerical examples are presented to verify the effectiveness of the proposed method.

Control of a motorboat with the Transverse Function approach

The Transverse Function (TF) control approach is applied to the control of a generic motorboat endowed with a surge force along the stern-bow direction and a torque actuation to modify the boat’s orientation. With respect to more conventional methods this approach allows for uniform practical stabilization of any reference position/orientation trajectory. This includes fixed-points in the absence of sea-currents, whose asymptotic stabilization cannot be achieved by classical feedback controllers, and non-feasible trajectories, i.e. trajectories that are not solutions to the system’s motion equations and thus cannot be stabilized asymptotically.

An Elementary Result On The Uniform Stability Of A Class Of Continuous Autonomous Systems

This paper presents an elementary result on the uniform stability of an equilibrium point for a class of autonomous systems that have an exponentially decreasing factor in their dynamics. The mapping that defines the dynamics may be not Lipschitz so uniqueness of solutions is not assured. Such a class of systems may arise in the stability analysis of some closed-loop control systems, as is the case that leads to this problem, namely, the application of the vertically transverse function approach to the practical point-stabilization of underactuated mechanical systems evolving on Lie groups.

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.

Motion Control of Vehicles with Trailers Using Transverse Function Approach

This paper is focused on analysis of the control solution using the transverse function approach. The controller considered here is designed for a nonholonomic vehicle with on-axle passive trailers. The main problem investigated is the optimal parametrization of the transverse functions in order to ensure low sensitivity to the measurement noise and high tracking accuracy. Theoretical analysis referring to transverse function scaling using dilation is illustrated by results of extensive numerical simulations. Taking into account these results the controller properties are considered. Finally, possibility of the controller implementation is discussed.

Experiment of Dynamic Feedback of the Trident Snake Robot with Transverse Function Approach

Experiment of Dynamic Feedback of the Trident Snake Robot with Transverse Function Approach

Control of a unicycle-like robot with trailers using transverse function approach

Abstract The paper presents the application of a smooth kinematic algorithm to control multi-body vehicle which consists of the unicyclelike tractor with three trailers. The controller takes advantage of the transverse functions and properties of the IV-order two input chained system. The derivation of the algorithm is presented in details. In order to improve the performance of the controller in the real application a selected tuning techniques are discussed. The properties of the closed-loop control system are examined based on results of numerical simulations concerning the point stabilization and trajectory tracking tasks.

Feedback control of a redundant wheeled snake mechanism using transverse functions SO(4)

The Transverse Function (TF) approach is applied to the control of a nonholonomic three-segments/snake-like wheeled mechanism, similar to the planar low-dimensional version of Hirose's Active Cord Mechanism (ACM) previously studied by the authors with the same control approach, but with two additional internal degrees of freedom (d.o.f.) whose actuation yields more flexible and efficient control solutions. From a theoretical point of view, these complementary d.o.f. modify the Control Lie Algebra of the system so that only first-order Lie brackets of the control vector fields are needed to satisfy the Lie Algebra Rank Condition (LARC). The fact that four independent (angular velocity) control inputs are used also implies for this system the existence of Transverse Functions (TF) defined on the six-dimensional special orthogonal group(4). Several examples of mechanisms whose control involve TF defined on(3) have been pointed out in the past. Beyond the specific control problem addressed here, a motivation for the present study is to illustrate for the first time how functions defined on the larger set(4) can be determined and used for the control of a physical system. This study is complemented with recalls concerning the parametrization of(4) by pairs of isoclinic quaternions and with the derivation of complementary differential calculus relations associated with this parametrization.

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 \({f : M \longrightarrow Q}\) 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∞-”) topology on C∞(M, Q).

Practical formation control of multiple underactuated ships with limited sensing ranges

This paper presents a constructive method to design cooperative controllers that force a group of N underactuated ships with limited sensing ranges to perform a desired formation, and guarantee no collisions between the ships. These ships do not have an independent actuator in the sway axis. The desired formation is stabilized at any sufficiently smooth reference trajectories, including fixed points and nonadmissible trajectories for the ships. The formation control design is based on several nonlinear coordinate changes, the transverse function approach, the backstepping technique, the Lyapunov direct method, and smooth and p -times differentiable step functions. These functions are introduced and incorporated into novel potential functions to solve the collision avoidance problem without the need of switchings despite the ships’ limited sensing ranges. Simulations illustrate the results.

Practical Formation Control of Multiple Unicycle-Type Mobile Robots with Limited Sensing Ranges

This paper presents a design of cooperative controllers that force a group of N unicycle-type mobile robots with limited sensing ranges to perform a desired tight formation and that guarantee no collisions between any robots in the group. The desired formation can be stabilized at any reference trajectories with bounded time derivatives. The formation control design is based on several nonlinear coordinate changes, the transverse function approach, the backstepping technique, the Lyapunov direct method, and smooth or p??times differentiable step functions. These functions are introduced and incorporated into novel potential functions to solve the collision avoidance problem without the need of switchings despite of the robots' limited sensing ranges. The proposed formation control system is applied to solve a gradient climbing problem.

Practical control of underactuated ships

This paper presents a design of global smooth controllers that achieve the practical stabilization of arbitrary reference trajectories, including fixed points and nonadmissible trajectories for underactuated ships. These ships do not have an independent actuator in the sway axis. The control design is based on several nonlinear coordinate changes, the transverse function approach, the back-stepping technique, the Lyapunov direct method, and utilization of the ship dynamics. Simulation results illustrate the effectiveness of the proposed control design.

Control of Nonholonomic Mobile Robots Based on the Transverse Function Approach

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.

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.

Averaging of zero dynamics for systems controlled by the vertically transverse function approach

In this paper, a method is explored to introduce dissipation in the closed-loop, zerodynamics systems that arise in the vertically transverse function approach (VTFA) recently proposed by the authors. The VTFA is an attempt to extend the transverse function approach (TFA) of Morin and Samson to deal with practical point-stabilization of a class of critical simple mechanical systems on Lie groups. This class comprises systems that are not kinematic reductions and hence fall beyond the scope of application of the TFA as originally formulated. The VTFA gives rise to a nontrivial zero dynamics that ultimately determines the qualitative nature of the trajectories and, in order to constrain the velocity (or “fiber”) coordinates to vanish asymptotically, as required in typical applications, dissipation must be injected into the zero dynamics. Reported below is a possible way to reach that goal based on the adjunction of additional auxiliary inputs, via generalized vertically transverse functions, and the use of nonlinear, high-order averaging theory. Also included is an illustrative example as well as numerical simulations suggesting that the feedback laws herein designed yield promising results.

Stabilization of trajectories for systems on Lie groups. Application to the rolling sphere.

This paper addresses the stabilization of admissible reference trajectories generated with constant inputs for driftless systems on Lie groups. The general expression of the linear approximation of the tracking error system is derived from the system's constants of structure and a necessary condition for the controllability of this approximation is specified in terms of the growth of the filtration of the Lie Algebra generated by the system's vector fields. This condition is illustrated with examples of mechanical systems whose control inputs correspond to velocity variables. By contrast with nonholonomic mobile robots whose kinematic equations can be transformed into the chained form, the linearized system associated with the rolling sphere is never controllable. Consequences of this lack of controllability as for stabilization problems are discussed from a general viewpoint and addressed more specifically for the rolling sphere. Finally, a practical stabilizer for this system based on the transverse function approach is proposed.