Control Of Underactuated Mechanical Systems Research Articles

Second‐order sliding mode control of underactuated mechanical systems I: Local stabilization with application to an inverted pendulum

AbstractSecond‐order sliding mode control synthesis is developed for underactuated mechanical systems, operating under uncertainty conditions. In order to locally stabilize an underactuated system around an unstable equilibrium, an output is specified in such a way that the corresponding zero dynamics is locally asymptotically stable. Then, the desired stability property of the closed‐loop system is provided by applying a quasihomogeneous second‐order sliding mode controller, driving the system to the zero dynamics manifold in finite time. Although the present synthesis exhibits an infinite number of switches on a finite time interval, it does not rely on the generation of first‐order sliding modes, while providing robustness features similar to those possessed by their standard sliding mode counterparts. A second‐order sliding mode appears on the zero dynamics manifold which is of co‐dimension greater than the control space dimension. Performance issues of the proposed synthesis are illustrated in numerical and experimental studies of a cart–pendulum system. Copyright © 2007 John Wiley & Sons, Ltd.

Second‐order sliding mode control of underactuated mechanical systems II: Orbital stabilization of an inverted pendulum with application to swing up/balancing control

AbstractOrbital stabilization of an underactuated cart–pendulum system is under study. The quasihomogeneous control synthesis is utilized to design a second‐order sliding mode controller that drives the actuated cart to a periodic reference orbit in finite time, while the non‐actuated pendulum produces bounded oscillations. A modified Van der Pol oscillator is introduced into the synthesis as an asymptotic generator of the periodic motion. The resulting closed‐loop system is capable of moving from one orbit to another by simply changing the parameters of the Van der Pol modification. Performance issues of the proposed synthesis are illustrated in numerical and experimental studies of the swing up/balancing control problem of moving a pendulum, located on an actuated cart, from its stable downward position to the unstable inverted position and stabilizing it about the vertical. Copyright © 2007 John Wiley & Sons, Ltd.

Control of underactuated mechanical systems with servo-constraints

This paper deals with a class of controlled mechanical systems in which the number of control inputs, equal to the number of desired system outputs, is smaller than the number of degrees of freedom. The related inverse dynamics control problem, i.e., the determination of control input strategy that force the underactuated system to complete the partly specified motion, is a challenging task. In the present formulation, the desired system outputs, expressed in terms of the system states, are treated as servo-constraints on the system, and the problem is viewed from the constrained motion perspective. Mixed orthogonal-tangent realization of the constraints by the available control reactions is stated, and a specialized methodology for solving the “singular” control problem is developed. The governing equations are manipulated to index three differential-algebraic equations, and a simple numerical code for solving the equations is proposed. The feedforward control law obtained as a solution to these equations can then be enhanced by a closed-loop control strategy with feedback of the actual servo-constraint violations to provide stable tracking of the reference motion in the presence of perturbations and modeling uncertainties. An overhead trolley crane executing a load-prescribed motion serves as an illustration. Some results of numerical simulations are reported.

受動速度場制御を用いた劣駆動メカニカルシステムの制御

受動速度場制御を用いた劣駆動メカニカルシステムの制御

Non-linear Control for Underactuated Mechanical Systems

7R20. Non-linear Control for Underactuated Mechanical Systems. - I Fantoni and R Lozano (UMR CNRS 6599, Univ de Technologie de Compiegne, BP 20529, Compiegne, 60205, France). Springer-Verlag London Ltd, Surrey, UK. 2002. 295 pp. ISBN 1-85233-423-1. $109.00. Reviewed by SC Sinha (Dept of Mech Eng, Auburn Univ, 202 Ross Hall, Auburn AL 36849-5341).This is an application-oriented book that is intended for engineers, graduate students, and researchers who are interested in the design of nonlinear controllers for underactuated mechanical systems. Underactuated systems are defined as systems with lesser number of independent control actuators than the degrees of freedom to be controlled. Readers with a background in intermediate level dynamics and control should have no trouble following the material. After an introduction in Chapter 1, some definitions and background material are presented in Chapter 2. These include Lyapunov stability concepts, Krasovskii-LaSalle invariance principle, passivity criteria, and controllability condition. The next nine chapters deal with the development of control algorithms for a number of underactuated mechanical systems, most of which have served as academic benchmarks. In Chapter 1, an energy-based nonlinear controller is developed for the classical problem of cart-inverted pendulum. Simulations and experimental results are also included. Similar analysis is presented in Chapter 4 that deals with a convey-crane system. Using the passivity property, an energy-based control law is proposed for a pendubot system in Chapter 5.In Chapter 6, modeling, analysis, and control of the Furuta pendulum (named after K Furuta of Tokyo Institute of Technology) is discussed. The reaction wheel pendulum is presented in Chapter 7. Chapters 8 and 9 are devoted to the design of control algorithms for planar robots. In Chapter 8, the authors present analysis and control of two as well as three link planar robots with flexible joints while in Chapter 9, the problem of a planar robot with two prismatic and one revolute (PPR) joints is considered. It is to be observed that the PPR system has four degrees of freedom and only three control inputs. The control strategy, once again, is based on an energy approach and passivity properties of the system. The ball-beam control is described in Chapter 10, and in this case, the control force is assumed to be acting on the ball rather than the beam. The controller for a simple hovercraft model is designed in Chapter 11. Chapter 12 deals with the control problem associated with a planar vertical take-off and landing (PVTOL) aircraft. A Lyapunov function using the forwarding technique is used to obtain the control law. The last three chapters are devoted to the modeling and control of helicopters. In Chapter 13, after some general considerations, a simplified model called the helicopter-platform model is analyzed. This model has three degrees of freedom with two control inputs. The nonlinear control strategy guarantees an asymptotic tracking of the desired trajectories. A Lagrangian formulation of the helicopter dynamics is presented in Chapter 14 and a Lyapunov approach is used to design a tracking controller. The same problem is discussed in Chapter 15 via the Newtonian approach. In this case, the control algorithm was developed using the backstepping and Lyapunov techniques. In summary, Non-linear Control for Underactuated Mechanical Systems is an excellent source for the development of control strategies for an important class of problems, viz, the underactuated mechanical systems. The book is clearly written, and the ideas are conveyed well by the authors. This book is a must for engineering graduate students, researchers, teachers, as well as practicing engineers. It is strongly recommended for individuals and libraries.

An open-loop control for underactuated manipulators using oscillatory inputs: steering capability of an unactuated joint

An open-loop control for underactuated mechanical systems using oscillatory inputs with amplitude and frequency modulations is investigated. Once all actuated joints are moved to their desired positions, oscillatory inputs are applied to an actuated joint to move the remaining unactuated joints. The steering force of the unactuated joints is achieved by utilizing the dynamic coupling between actuated and unactuated joints. Such a dynamic coupling occurs due to the oscillatory motions of an actuated joint. Once the frequency of the oscillatory input is decided, the amplitude is determined by analyzing a time-invariant system, which is derived from the unactuated joint dynamics by the method of averaging. A systematic way, via a generating equation and a coordinate transformation derived from the generating equation, for converting the unactuated joint dynamics into the standard form of averaging is proposed. In the event of an actuator failure in outer space, the failed joint can be steered by adopting the method proposed. Illustrating examples are given. Experimental results are provided.

Simple Mechanical Control Systems with Constraints and Symmetry

We develop tools for studying the control of underactuated mechanical systems that evolve on a configuration space with a principal fiber bundle structure. Taking the viewpoint of affine connection control systems, we derive reduced formulations of the Levi--Civita and the nonholonomic affine connections, along with the symmetric product, in the presence of symmetries and nonholonomic constraints. We note that there are naturally two kinds of connections to be considered here, affine and principal connections, leading to what we term a ``connection within a connection.'' These results are then used to describe controllability tests that are specialized to simple, underactuated mechanical systems on principal fiber bundles, including the notion of fiber configuration controllability. We present examples of the use of these tools in studying the planar rigid body with a variable direction (vectored) thruster and the snakeboard robot.

Control Algorithms Using Affine Connections on Principal Fiber Bundles

In this paper, we develop tools for studying the control of underactuated mechanical systems that evolve on a configuration space with a principal fiber. bundle structure. We utilize the affine connection to descnbe the equations of motion and study the controllability of such systems. The invariance of the system with respect to the action of a Lie group leads to a reduced form of the affine connection. This connection and its corresponding symmetric product, are then used to develop constructive, cyclic control inputs to steer the mechanical system. We explore the development of the equations, based on the original formulation by Bullo, and Its application to the steering of a blimp-like vehicle with directional thrusters.

Vibrational control of underactuated mechanical systems control design through averaging analysis

An open loop vibrational control for an underactuated mechanical system with amplitude and frequency modulation is investigated. Since there is no direct external input to an unactuated joint, the dynamic coupling between the actuated and unactuated joint is utilized for controlling the unactuated joint. Feedback linearization has been performed to fully incorporate the known nonlinearities of the underactuated system considered. The actuated joints are firstly positioned to their desired locations, and then periodic oscillatory inputs are applied to the actuated joints to move the remaining unactuated joints to their target positions. The amplitudes and frequencies of the vibrations introduced are determined through averaging analysis. A systematic way of obtaining an averaged system for the underactuated system via a coordinate transformation is developed. A manipulator in the zero gravity space can be vibrationally controlled in the event of actuator failure. A control design example of the 2R planar manipulator with a free joint with no brake is provided.

Adaptive variable structure set-point control of underactuated robots

Control of underactuated mechanical systems (robots) represents an important class of control problem. In this correspondence, a model-based adaptive variable structure control scheme is introduced, where the uncertainty bounds only depend on the inertia parameters of the system. Global asymptotic stability is established in the Lyapunov sense. Numerical simulations are conducted to validate the theoretical analysis.

Dynamics and control of a class of underactuated mechanical systems

This paper presents a theoretical framework for the dynamics and control of underactuated mechanical systems, defined as systems with fewer inputs than degrees of freedom. Control system formulation of underactuated mechanical systems is addressed and a class of underactuated systems characterized by nonintegrable dynamics relations is identified. Controllability and stabilizability results are derived for this class of underactuated systems. Examples are included to illustrate the results; these examples are of underactuated mechanical systems that are not linearly controllable or smoothly stabilizable.