Class Of Underactuated Mechanical Systems Research Articles

Non-linear sliding mode surfaces for a class of underactuated mechanical systems

This study presents a new non-linear sliding surface to control a class of non-minimum phase underactuated mechanical systems, taking into account uncertainties in their physical parameters. The non-linear surface is designed through a fictitious output, which provides the minimum-phase property and allows to prove stability using Lyapunov theory. The non-linear surface is based on the fictitious output and augmented with a non-linear external controller designed using the Lyapunov theory. The present approach assures exponential stability of the equilibrium point and robust stability to parametric uncertainties, avoiding the appearance of non-desired phenomena, as limit cycles. Two pendulum-like examples inside the class are thoroughly analysed and solved, that is, the pendulum on a cart and the inertia wheel pendulum. Performance, time response and parametric robustness are shown through simulations.

Constructive Immersion and Invariance Stabilization for a Class of Underactuated Mechanical Systems

In this paper a constructive approach to the stabilization of a desired equilibrium for a class of underactuated mechanical systems via the Immersion & Invariance methodology (I&I) is proposed. The design procedure shows that cases of mechanical systems with underactuation degree greater than one are included. This work generalizes the results recently reported by the authors, where an approach to obviate the solution of the corresponding PDEs for a class of nonlinear systems was proposed. Finally, our approach is successfully applied to the inertia wheel pendulum system and an interesting connection with the Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) is revealed.

Control of underactuated mechanical systems via passive velocity field control: Application to snakeboard and 3D rigid body

In this paper we consider the application of a new formulation of passive velocity field control (PVFC) to the control problem of a class of underactuated mechanical systems (UAMS). In our previous report synthesizing the desired velocity vector fields for PVFC by the decoupling vector fields, we were able to prove that control torques may be successfully obtained and the solvability of the synthesizing problem of PVFC for the UAMS may be guaranteed. We newly develop this control design methodology and propose an explicit feedback solution to the control problems of the UAMS, such as snakeboard, underactuated 3D rigid bodies. Several simulations show the validity of the proposed control methodology.

Control of a Class of Underactuated Mechanical Systems Using Sliding Modes

In this paper, we present a sliding mode control algorithm to robustly stabilize a class of underactuated mechanical systems that are not linearly controllable and violate Brockett's necessary condition for smooth asymptotic stabilization of the equilibrium, with parametric uncertainties. In defining the class of systems, a few simplifying assumptions are made on the structure of the dynamics; in particular, the damping forces are assumed to be linear in velocities. We first propose a switching surface design for this class of systems, and subsequently, a switched algorithm to reach this surface in finite time using conventional and higher order sliding mode controllers. The stability of the closed-loop system is investigated with an undefined relative degree of the sliding functions. The controller gains are designed such that the controller stabilizes the actual system with parametric uncertainty. The proposed control algorithm is applied to two benchmark problems: a mobile robot and an underactuated underwater vehicle. Simulation results are presented to validate the proposed scheme.

Can we make a robot ballerina perform a pirouette? Orbital stabilization of periodic motions of underactuated mechanical systems

This paper provides an introduction to several problems and techniques related to controlling periodic motions of dynamical systems. In particular, we consider planning periodic motions and designing feedback controllers for orbital stabilization. We review classical and recent design methods based on the Poincaré first-return map and the transverse linearization. We begin with general nonlinear systems and then specialize to a class of underactuated mechanical systems for which a particularly rich structure allows many of the problems to be solved analytically.

Master-slave telecontrol of a class of underactuated mechanical systems with communication time-delay

This paper deals with the problem of time-delay telecontrol of a class of underactuated mechanical systems (UMS) in a master-slave configuration. This problem has been solved by designing some discontinuous causal compensators which have proved by formal stability analysis to guarantee position coordination of the mechanisms and to allow good force reflection to the master side, satisfying in this way classical objectives in the telecontrol of systems. Moreover, the whole closed-loop system behaves stable and results robust to parametric uncertainties and external disturbances. The performance of the proposed control scheme is visualized through experiments developed in a local network.

Control of a class of underactuated mechanical systems

In this paper a new control methodology for underactuated mechanical systems is proposed. The basic idea of this method is to combine passive velocity field control with decoupling vector field. In order that the underactuated mechanical systems can be stabilized at the desired position after settling on the desired velocity vector field, novel control strategies are proposed. Proposed control strategies are applied to the underactuated planar three-link manipulator and underactuated planar body. Simulations demonstrate the usefulness and effectiveness of the proposed control methodology.

CAN WE MAKE A ROBOT BALLERINA PERFORM A PIROUETTE? ORBITAL STABILIZATION OF PERIODIC MOTIONS OF UNDERACTUATED MECHANICAL SYSTEMS

This paper provides an introduction to several problems and techniques related to controlling periodic motions of dynamical systems. In particular, we define and discuss problems of motion planning and orbit planning, analysis methods such as the classical Poincaré first-return map and the transverse linearization, and exponentially orbitally stabilizing control designs. We begin with general nonlinear systems, and then specialize to a class of underactuated mechanical systems for which a particularly rich structure allows many of the problems to be solved analytically. The paper concludes with a discussion of numerical issues related to control design via periodic Riccati equations.

Structure design of two types of sliding-mode controllers for a class of under-actuated mechanical systems

On the basis of sliding-mode control, two sliding-mode controller models based on incremental hierarchical structure and aggregated hierarchical structure for a class of under-actuated systems are presented. The design steps of the two types of sliding-mode controllers and the principle of choosing parameters are given. At the same time, to guarantee the system's stability, two determinant theorems are presented. Then, by theoretical analysis, the two types of sliding-mode controllers are proved to be globally stable in the sense that all signals involved are bounded. The simulation results show the validity of the methods. Therefore an academic foundation for the development of high-dimension under-actuated mechanical systems is provided.

CAN WE MAKE A ROBOT BALLERINA PERFORM A PIROUETTE? ORBITAL STABILIZATION OF PERIODIC MOTIONS OF UNDERACTUATED MECHANICAL SYSTEMS1

This paper provides an introduction to several problems and techniques related to controlling periodic motions of dynamical systems. In particular, we define and discuss problems of motion planning and orbit planning, analysis methods such as the classical Poincaré first-return map and the transverse linearization, and exponentially orbitally stabilizing control designs. We begin with general nonlinear systems, and then specialize to a class of underactuated mechanical systems for which a particularly rich structure allows many of the problems to be solved analytically. The paper concludes with a discussion of numerical issues related to control design via periodic Riccati equations.

Controllability of a class of underactuated mechanical systems with symmetry

In this paper we develop results based on geometric mechanics to study the controllability of a class of controlled under-actuated left invariant mechanical systems on Lie groups. We exploit the invariance of the controlled nonlinear dynamics to the group action (symmetry) to derive a set of reduced dynamics for the system. We first present sufficient conditions for the controllability of the reduced dynamics. We prove conditions (boundedness of coadjoint orbits and existence of a radially unbounded Lyapunov function) under which the drift vector field (of the reduced system) is weakly positively Poisson stable (WPPS). The WPPS nature of the drift vector field along with the Lie algebra rank condition is used to show controllability of the reduced system. We then extend these results to the unreduced dynamics, considering separately the cases when the symmetry group is compact and noncompact. In the noncompact case, under further assumptions of equilibrium controllability, sufficient conditions for controllability of the unreduced dynamics are derived. Jetpuck (hovercraft) and underwater vehicles are used as mechanical systems to motivate our work and illustrate our theoretical results.

Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment

We consider the application of a formulation of passivity-based control (PBC), known as interconnection and damping assignment (IDA) to the problem of stabilization of underactuated mechanical systems, which requires the modification of both the potential and the kinetic energies. Our main contribution is the characterization of a class of systems for which IDA-PBC yields a smooth asymptotically stabilizing controller with a guaranteed domain of attraction. The class is given in terms of solvability of certain partial differential equations. One important feature of IDA-PBC, stemming from its Hamiltonian formulation, is that it provides new degrees of freedom for the solution of these equations. Using this additional freedom, we are able to show that the method of "controlled Lagrangians"-in its original formulation-may be viewed as a special case of our approach. As illustrations we design asymptotically stabilizing IDA-PBCs for the classical ball and beam system and a novel inertia wheel pendulum.

Hybrid control of the Pendubot

Swing up and balance control are two interesting control problems for the Pendubot. Many studies have been conducted for swing up control of the Pendubot. A few results have been reported for feedback stabilization of the Pendubot. In this paper, we apply a new hybrid controller for feedback stabilization of the Pendubot. It is well-known that it is impossible to use smooth feedback to stabilize a class of underactuated mechanical systems around their equilibra, even locally. Various nonsmooth controllers have been presented for feedback stabilization of this type of system. However, most of the studies are either based on theoretical proofs or simulations. There is a strong need for experimental study. The Pendubot arises as a special test bed for this purpose. This experimental study has a particular interest for feedback stabilization of underactuated mechanical systems that are not feedback stabilizable using smooth control.

The matching conditions of controlled Lagrangians and IDA-passivity based control

This paper discusses the matching conditions resulting from the controlled Lagrangians method and the interconnection and damping assignment passivity based control (IDA-PBC) method. Both methods have been presented recently in the literature as means to stabilize a desired equilibrium point of an Euler-Lagrange, respectively Hamiltonian, system. In the context of mechanical systems with symmetry, the original controlled Lagrangians method is reviewed, and an interpretation of the matching assumptions in terms of the matching of kinetic and potential energy is given. Secondly, both methods are applied to the general class of underactuated mechanical systems and it is shown that the controlled Lagrangians method is contained in the IDA-PBC method. The u -method as described in recent papers for the controlled Lagrangians method, transforming the matching conditions (a set of non-linear PDEs) into a set of linear PDEs, is discussed. The method is used to transform the matching conditions obtained in the IDA-PBC method into a set of quadratic and linear PDEs. Finally, the extra freedom obtained in the IDA-PBC method (with respect to the controlled Lagrangians method) is used to discuss the integrability of the closed-loop system. Explicit conditions are derived under which the closed-loop Hamiltonian system is integrable, leading to the introduction of gyroscopic terms.

Stabilization of underactuated mechanical systems: A non-regular back-stepping approach

This paper presents a design framework for the stabilization of a class of underactuated mechanical systems. By utilizing non-regular static state feedbacks, these systems are transformed into a class of non-linear systems with chain structures. Then, controller design is presented by applying the backstepping design technique. The design procedure is applied to an underactuated robotic system and simulation tests are carried out for illustrating the effectiveness of the proposed approach.

Series Expansions for the Evolution of Mechanical Control Systems

This paper presents a series expansion that describes the evolution of a mechanical system starting at rest and subject to a time-varying external force. Mechanical systems are presented as second-order systems on a configuration manifold via the notion of affine connections. The series expansion is derived by exploiting the homogeneity property of mechanical systems and the variations of constant formula. A convergence analysis is obtained using some analytic functions and combinatorial analysis results. This expansion provides a rigorous means of analyzing locomotion gaits in robotics and lays the foundation for the design of motion control algorithms for a large class of underactuated mechanical systems.

Kinematic controllability for decoupled trajectory planning in underactuated mechanical systems

We introduce the notion of kinematic controllability for second-order underactuated mechanical systems. For systems satisfying this property, the problem of planning fast collision-free trajectories between zero velocity states can be decoupled into the computationally simpler problems of path planning for a kinematic system followed by time-optimal time scaling. While this approach is well known for fully actuated systems, until now there has been no way to apply it to underactuated dynamic systems. The results in this paper form the basis for efficient collision-free trajectory planning for a class of underactuated mechanical systems including manipulators and vehicles in space and underwater environments.

Discontinuous feedback control of a special class of underactuated mechanical systems

Discontinuous feedback control of a special class of underactuated mechanical systems

On global properties of passivity-based control of an inverted pendulum

The paper adresses the problem of stabilization of a specific target position of underactuated Lagrangian or Hamiltonian systems. We propose to solve the problem in two steps: first to stabilize a set with the target position being a limit point for all trajectories originating in this set and then to switch to a locally stabilizing controller. We illustrate this approach by the well-known example of inverted pendulum on a cart. Particularly, we design a controller which makes the upright position of the pendulum and zero displacement of the cart a limit point for almost all trajectories. We derive a family of static feedbacks such that any solution of the closed loop system except for those originating on some two-dimensional manifold approaches an arbitrarily small neighbourhood of the target position. The proposed technique is based on the passivity properties of the inverted pendulum. A possible extension to a more general class of underactuated mechanical systems is discussed.

Discontinuous feedback control of a special class of underactuated mechanical systems

We study a control problem for a special class of underactuated mechanical systems, namely for mechanical systems with several directly actuated degrees of freedom and a single unactuated degree of freedom that must be controlled through the system coupling. Specific assumptions are introduced that define this class, which includes many important models of mechanical system examples. The main result of the paper is the construction of a discontinuous nonlinear feedback controller for which the closed loop equilibrium at the origin is made ‘globally attractive’. The control construction approach is introduced in detail, and a proof of attractiveness is presented. The results are applied to control of the planar motion of a rigid body with a single unactuated internal degree of freedom. Copyright © 2000 John Wiley & Sons, Ltd.