Global Stabilisation of Underactuated Mechanical Systems via PID Passivity-Based Control

This paper presents an approach to show global controllability of a class of underactuated mechanical systems. This class of systems include gymnastic robots and the classic cart-pole system. For this class of systems, we can define a connection point, and design a damping controller u(t) with the Lyapunov method to drive all states to the connection point. Further by exploiting the time reversal symmetry of the system and spline interpolation, a controller u(-t) can be obtained to drive the system from the connection point to any other state. The system is thus shown completely controllable, i.e., there exists an admissible trajectory from any given state to any given final state. Swing-up control designs for a pendubot, cart-pole system and triple pendulum are given as illustrative examples.

Control of underactuated mechanical systems via energy shaping is a well-established, robust design technique. Unfortunately, its application is often stymied by the need to solve partial differential equations (PDEs). In this technical note a new, fully constructive, procedure to shape the energy for a class of mechanical systems that obviates the solution of PDEs is proposed. The control law consists of a first stage of partial feedback linearization followed by a simple proportional plus integral controller acting on two new passive outputs. The class of systems for which the procedure is applicable is identified imposing some (directly verifiable) conditions on the systems inertia matrix and its potential energy function. It is shown that these conditions are satisfied by three benchmark examples.

This paper considers the extension of the interconnection and damping assignment passivity‐based control methodology for a class of underactuated mechanical systems with dynamic friction. We present a new damping assignment approach to compensate friction by means of a nonlinear observer. Friction at the actuated joints is assumed to be captured by a bristle deflection model: the Dahl model. Based on the Lyapunov direct method we show that, under some conditions, the overall closed‐loop system is stable and, by invoking the theorem of Barbashin–Krasovskii, we arrive to asymptotic stability conditions. Experiments with an underactuated mechanical system, the Furuta pendulum, show the effectiveness of the proposed scheme when friction is compensated. Copyright © 2010 John Wiley & Sons, Ltd.

Underactuated systems are mechanical systems with fewer actuators than the system degrees of freedom, i.e., some degrees of freedom are not directly driven by actuators. Many benchmark underactuated systems have been developed to study the nature of underactuated dynamics and their possible controller designs. Unlike fully actuated systems where many control strategies are available for an entire class of systems, there are few results applicable to even a small class of underactuated mechanical systems. In this paper, an Adaptive Fuzzy Sliding Mode Controller (AFSMC) is proposed in order to stabilize an inverted pendulum mounted on a ball system, also known as ballbot; this latter an interesting underactuated mechanical system for which the controller designs are a pending task, i.e. the performance for different control strategies could be explored. The performance of our proposal is verified via numerical simulation, showing the robustness against time-varying uncertainties and disturbances.

A synchronization control strategy for a class of underactuated mechanical systems is proposed by using the energy shaping technique, aiming to achieve the required performance of the synchronization motion. A synchronization controller is designed based on the interconnection and damping assignment passivity-based control methodology. It will guarantee that the position tracking errors and the synchronization errors of the underactuated mechanical systems are to converge to zero asymptotically. Experiments on a synchronization control system with two single-inverted pendulums as well as simulations of a synchronization control system consisting of four ball-beam devices are presented to demonstrate the effectiveness of the proposed method.

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.

This paper presents a control design technique which enables approximate reference trajectory tracking for a class of underactuated mechanical systems. The control law comprises two terms. The first involves feedback of the trajectory tracking error in the actuated coordinates. Building on the concept of vibrational control, the second term imposes high-frequency periodic inputs that are modulated by the tracking error in the unactuated coordinates. Under appropriate conditions on the system structure and the commanded trajectory, and with sufficient separation between the time scales of the vibrational forcing and the commanded trajectory, the approach provides convergence in both the actuated and unactuated coordinates. The procedure is first described for a two degree-of-freedom (DOF) system with one input. Generalizing to higher-dimensional, underactuated systems, the approach is then applied to a 4DOF system with two inputs. A final example involves control of a rigid plate that is flapping in a uniform flow, a 3DOF system with one input. More general applications include biomimetic locomotion systems, such as underwater vehicles with articulating fins and flapping wing micro-air vehicles.

ABSTRACTRobust control of under-actuated mechanical systems (UMSs) with model uncertainty is still a challenging problem. For UMSs, the model parametric uncertainties make it difficult to precisely calculate the isolated equilibrium point corresponding to a fixed input. Without an accurate destination state, many set-point control methods cannot eliminate the positioning errors. An improved sliding mode control (ISMC) method is proposed to solve the robust control problem for a class of UMSs with model uncertainty and input disturbance. A balance variable is introduced in the sliding surface design to compensate for the disturbance caused by the inaccurate destination state, and the ISMC method is proposed to make the system state reach the sliding surface in finite time. Linear matrix inequality approach and particle swarm optimisation algorithm are applied to design the sliding mode surface parameters. The simulation results on an UMS are presented to show the effectiveness of the proposed scheme.

This article presents a simple stabilizing control algorithm for a class of underactuated mechanical systems for two degrees of freedom (2DoF). In this respect, the adaptive sliding mode based strategy is proposed for the considered class. The controller, along with the adapted laws, is decided in such a way that the time derivative of a Lyapunov function grows negative. The proposed control technique has been applied to three classic benchmark 2DOF systems, the acrobot, pendoubt, and the cart-pole system. The effectiveness of the proposed strategy is proved in the light of simulation results for all said systems considered as an illustrative example.

In this work, we present a quasi-linear parameter varying (quasi-LPV) control design approach for a class of underactuated mechanical systems in cascaded normal form. By incorporating linear system theory, robust control techniques, and linear matrix inequality (LMI) tools, a trajectory-based approach is employed to prove semi-global asymptotic stability of the closed-loop systems. The proposed controller can be applied to a wide class of mechanical systems with various degrees of underactuation. The simple structure of the controller facilitates its straightforward implementation for a variety of applications when compared to existing approaches. The controller is implemented for the translational oscillator with a rotational actuator (TORA) system and an underactuated quadrotor model to illustrate its broad applicability and effectiveness.

The nonlinear dynamk model of the underactuated mechanical systems is buillt with Lagrangian method. Several properties of the system model such as the positive definite symmetric inertia matrix and the passivity are analyzed. A passivity-based control method is proposed for the underactuated mechanical system. The unlderactuated doublependulum- type overhead crane system is used to validate the proposed control algorithm. Simulation rlesults illustrate the complex system dynamics of the double-pendlulum-type overhead crane and the validity of proposed control algorithm.

The development of a nonlinear controller of stabilization of underactuated mechanical systems (UMSs) is a challenging endeavor due to a larger number of output variables to be controlled than the control input space. This paper proposes an adaptive sliding mode control based on a higher-order nonlinear disturbance observer (HONDO) for stabilizing the rotational pendulum (RP) system falling under the class of UMSs. Firstly, the HONDO is designed in such a way that it can improve accuracy in estimations with its incremental order. As a result, the proposed controller obtained from the sliding surface which is developed with system’s states and estimations, forces the states attaining the sliding mode and hence keeps them to their origin forever against disturbances. To achieve this, the sliding coefficients are obtained using inertia matrix of the system. The zero dynamics is stabilized by the proposed controller. This alleviates the chattering problem in the control input. Finally, numerical performance on the underactuated RP model is analyzed to show the efficiency of the proposed controller and it is compared with the established control technique found in the literature.

A strategy to design and implement a robust controller for a class of underactuated mechanical systems, with two degrees of freedom, which solves the problems of regulation and trajectory tracking, is proposed. This control strategy considers the partial measurement of the state vector and the presence of parametric uncertainties in the plant; these conditions are common in the implementation of a control system. The strategy is based on the use of robust finite time convergence observers to estimate the unmeasured state variables, unknown disturbances, and other signals needed for the control system implementation. The performance of the control strategy is illustrated numerically and experimentally.

This work investigates the energy shaping control of a class of underactuated mechanical systems with first‐order actuator dynamics and subject to both matched and unmatched constant additive disturbances. To this end, a new nonlinear control law which includes two independent integral actions is presented. The controller design is outlined for systems with first‐order actuator dynamics, and also for systems with direct actuation. The effectiveness of the proposed approach is demonstrated with numerical simulations on an inertia wheel pendulum and on a ball‐on‐beam system, both actuated by electric DC motors and subject to constant disturbances.

Since its introduction in the late 1980s, passivity-based control (PBC) has proven to be successful in controlling many robotic systems. The connection between stability and passivity theory is the most attractive feature of controllers designed using this methodology. However, the need to solve nonlinear partial differential equations (PDE) in closed-form has been a major challenge in applying PBC to general robotic systems. Here, we introduce a systematic approach to design controllers for a class of underactuated mechanical systems based on interconnection and damping assignment. Exploiting the universal approximation capability of neural networks, we formulate a data-driven optimisation problem that discovers solutions to the required PDEs automatically. Our approach does not destroy the passivity structure, preserving the inherent stability properties. We demonstrate the efficacy of our framework on two benchmark problems: the inertia wheel pendulum and the ball and beam system.