Arun D. Mahindrakar

Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one

We consider the problem of (asymptotic) stabilization of mechanical systems with underactuation degree one. A state-feedback design is derived applying the interconnection and damping assignment passivity-based control methodology. Its application relies on the possibility of solving a set of partial differential equations that identify the energy functions that can be assigned to the closed-loop. The following results are established: 1) identification - in terms of some algebraic inequalities - of a subclass of these systems for which the partial differential equations are trivially solved; 2) characterization of all systems which are feedback-equivalent to this subclass; and 3) introduction of a suitable parametrization of the assignable energy functions that provides the designer with a handle to address transient performance and robustness issues. An additional feature of our developments is that the open-loop system need not be described by a port-controlled Hamiltonian (or Lagrangian) model, a situation that arises often in applications due to model reductions or preliminary feedbacks that destroy the structure. The new result is applied to obtain an (almost) globally stabilizing controller for the inertia wheel pendulum, a controller for the chariot with pendulum system that can swing-up the pendulum from any position in the upper half plane and stop the chariot at any desired location, and an (almost) globally stabilizing scheme for the vertical takeoff and landing aircraft with strong input coupling. In all cases we obtain very simple and intuitive solutions that do not rely on, rather unnatural and technique-driven, linearization or decoupling procedures but instead endows the closed-loop system with a Hamiltonian structure with desired potential and kinetic energy functions.

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 Brocketts 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.

Robust Stabilization of a Class of Underactuated Mechanical Systems Using Time Scaling and Lyapunov Redesign

This paper presents a controller-design methodology for a class of underactuated mechanical systems that are affected by parametric uncertainties and external disturbances. The perturbations due to parametric uncertainties are mismatched, whereas those caused by external disturbances are of the matched type. Their effects are canceled by employing a novel strategy that combines time scaling and Lyapunov redesign. The control methodology is applied to a two-wheeled mobile inverted pendulum and a ball-beam system. Along the way, the nonexistence of a smooth control law for point-to-point stabilization of the mobile inverted pendulum is established. Simulation and experimental studies are used to verify the efficacy of the proposed controller-design method.

Constructive immersion and invariance stabilization for a class of underactuated mechanical systems

A constructive approach to stabilize a desired equilibrium for a class of underactuated mechanical systems, which obviates the solution of partial differential equations, is proposed. The Immersion & Invariance methodology is adopted, with the main result formulated in the Port-Hamiltonian framework, for both model and target dynamics. The procedure is applicable to mechanical systems with under-actuation degree larger than one, extending the results recently reported by some of the authors. The approach is successfully applied to two benchmark examples and some basic connections with the interconnection and damping assignment passivity-based control are revealed. An additional contribution of this work is the identification of a class of mechanical systems whose mechanical structure remains invariant under partial feedback linearization.

Mobile Robot Navigation Through a Hardware-Efficient Implementation for Control-Law-Based Construction of Generalized Voronoi Diagram

This paper proposes a navigation control law based on a Lyapunov function for the construction of a generalized Voronoi diagram (GVD), applicable for mobile robots with ultrasonic sensors operating in indoor environments. The proposed control law is a modification of the one in [H. Choset and K. Nagatani, “Topological simultaneous localization and mapping (SLAM): Toward exact localization without explicit localization,” IEEE Trans. Robot. Autom., vol. 17, no. 2, pp. 125-137, Apr. 2001.]. A new definition is proposed for the distance function that renders the proposed modified control law well defined for all points in free space. To follow the trajectory closely, an architecture for the control law based on simultaneous handling of sensors and the motors is proposed. An efficient implementation of the control law using merely a low-end field-programmable gate array and no external memory is presented. Experimental results validate the efficacy of the proposed modified control law. The paper also proves that the proposed modified control law traces the GVD without any approximation. The results are compared with two earlier studies.

Geometric Controllability and Stabilization of Spherical Robot Dynamics

Geometric control of a spherical robot rolling on a horizontal plane with three independent inertia disc actuators is considered in this note. The dynamic model of the spherical robot in the geometric framework is used to establish the strong accessibility and small-time local controllability properties. Smooth stabilizability to an equilibrium fails for the nonholonomic spherical robot. A novel contribution of this note is a smooth, asymptotically stabilizing geometric control law for position and reduced attitude, which corresponds to an equilibrium submanifold of dimension one. From Brocketts condition, this is the best possible dimension of a smoothly stabilized equilibrium submanifold. We also present a novel smooth global tracking controller for tracking position trajectories.

Extending interconnection and damping assignment passivity-based control (IDA-PBC) to underactuated mechanical systems with nonholonomic Pfaffian constraints: The mobile inverted pendulum robot

In this paper, we extend the interconnection and damping assignment passivity-based control strategy to systems subjected to nonholonomic Pfaffian constraints. The results are applied to stabilize the pitch dynamics of an underactuated mobile inverted pendulum (MIP) robot subjected to nonholonomic constraints arising out of no-slip conditions. A novel feature of this paper is that we reduce the kinetic energy partial differential equations and potential energy partial differential equation to a set of ordinary differential equations, that are explicitly solved.

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

Abstract 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.

Position Stabilization and Waypoint Tracking Control of Mobile Inverted Pendulum Robot

The mobile inverted pendulum (MIP) is a mechanical system that presents multiple control challenges. In particular, two objectives, namely desired position control and stabilization of the unstable pendulum-like central body, need to be simultaneously met. In this brief, we propose a novel smooth time-invariant controller to achieve the twin objectives. A feature of the controller design is that it readily extends to achieve waypoint tracking, another interesting task for mobile platforms. To validate the theory developed, an MIP has been indigenously designed and fabricated. Extensive experiments on the MIP have been performed. It has been observed that the system accomplishes position stabilization as well as waypoint tracking with negligible error.

Terminal Sliding Mode Control of a Twin Rotor Multiple-Input Multiple-Output System

Abstract This paper deals with the finite time stabilization of a twin rotor multiple-input multiple-output (TRMS) system. A terminal sliding mode control law is obtained for the linearized model of the system by transforming it to the Brunovsky canonical form. Through experiment, the resulting control law is shown to be stable to disturbances in pitch and yaw.