Integral passivity‐based control of underactuated mechanical systems with actuator dynamics and 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.

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.

In this book, we collected recent results on the control of underactuated mechanical systems subject to internal uncertainties and external disturbances. The strategy developed is so universal that it is not restricted to a specific system but a large class of underactuated systems. Several benchmark systems are studied in this book, including detailed literature review, system dynamics derivation, control problem formulation, and simulation verification. The control strategy developed in chapter 4 is able to stabilize all these benchmark systems with satisfactory performance regardless of the underactuated dynamics and various uncertainties. The book is written as a text suitable for graduate students in the advanced course for the control of underactuated systems. It also provides valuable tools for researchers and practicing engineers working on the control of underactuated mechanical systems.

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.

An open loop vibrational control for underactuated mechanical systems 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 incorporate fully 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 the averaging analysis. A systematic way of obtaining an averaged system for the underactuated system via a coordinate transformation is developed. A manipulator in the space where no gravitation exists can be vibrationally controlled in the event of actuator failure. A control design example of a 2R planar manipulator with a free joint with no brake is provided.

Passivity-based control of underactuated mechanical systems with nonlinear friction effects is analyzed. The port-controlled-Hamiltonian equations of the mechanical system are extended by incorporating a novel dynamic friction model, whose internal states are included as generalized coordinates. The extended Hamiltonian model preserves passivity properties of the LuGre model for the velocity-force pair, with no extra conditions on the friction model parameters. Simulation results are presented for an underactuated double pendulum.

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.

A Lyapunov direct method is proposed for a class of underactuated, mechanical systems. The direct method is derived in general for systems having n degrees of freedom of which only m < n are actuated. The applications consist of a class of systems where the elements of the mass/inertia matrix and the gravitational forces/torques are either constants or functions of a single generalized position variable and where n is two and m is one. The time derivative of the candidate Lyapunov function produces a relation that is solved via a matching method. Some of the matching equations consist of linear differential and partial differential equations. It is shown for this class of systems, that the solutions of these linear differential and partial differential equations necessary for assuring asymptotic stability can be evaluated numerically as part of the feedback process. Examples are presented involving an inverted pendulum cart and an inertia wheel pendulum.

In this chapter an adaptive sliding mode controller for the Furuta pendulum is proposed. The Furuta pendulum is a class of underactuated mechanical systems commonly used in many control systems laboratories due to its complex stabilization which allows the analysis and design of different nonlinear and multivariable controllers that are useful in some fields such as aerospace and robotics. Sliding mode control has been extensively used in the control of mechanical systems as an alternative to other nonlinear control strategies such as backstepping, passivity based control etc. The design and implementation of an adaptive sliding mode controller for this kind of system is explained in this chapter, along with other sliding mode controller variations such as second order sliding mode (SOSMC) and PD plus sliding mode control (PD \(+\) SMC) in order to compare their performance under different system conditions. These control techniques are developed using the Lyapunov stability theorem and the variable structure design procedure to obtain asymptotically stable system trajectories. In this chapter the adaptive sliding mode consist of a sliding mode control law with an adaptive gain that makes the controller more flexible and reliable than other sliding mode control (SMC) algorithms and nonlinear control strategies. The adaptive sliding mode control (ASMC) of the Furuta pendulum, and the other SMC strategies shown in this chapter, are derived according to the Furuta’s pendulum dynamic equations making the sliding variables, position errors and velocity errors reach the zero value in a specified reaching time. The main reason of deriving two well known sliding mode control strategy apart from the proposed control strategy of this chapter (adaptive sliding mode control) is for comparison purposes and to evince the advantages and disadvantages of adaptive sliding mode control over other sliding mode control strategies for the stabilization of the Furuta pendulum. A chattering analysis of the three SMC variations is done, to examine the response of the system, and to test the performance of the ASMC in comparison with the other control strategies explained in this chapter.

In this paper we discuss the partial feedback linearization control of underactuated mechanical systems. We consider an n degree of freedom system having m actuated, or active, degrees of freedom and l=n-m unactuated, or passive, degrees of freedom. It is known that the portion of the dynamics corresponding to the active degrees of freedom may be linearized by nonlinear feedback. In this paper we show, alternatively, that the portion of the dynamics corresponding to the passive degrees of freedom may be linearized by nonlinear feedback under a condition that we call strong inertial coupling. We derive and analyze the resulting zero dynamics which are crucial to an understanding of the response of the overall system. Simulation results are presented showing the performance of two link underactuated robots under partial feedback linearization control. >

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.

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 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 &amp; Sons, Ltd.

This paper introduces an adaptive robust control approach tailored for underactuated mechanical systems encountering matched and mismatched uncertainty, employing a constraint-following methodology. The control strategy unfolds in two phases: initially, a nominal control scheme is devised neglecting uncertainty and deviations in initial conditions from constraints. Subsequently, uncertainty is categorized into matched and mismatched components, ensuring that mismatched uncertainties remain unobservable. Leveraging the structural characteristics of the uncertainty bound, a novel segmented adaptive law is proposed and seamlessly integrated into the adaptive robust control framework. By employing the Lyapunov minimax approach, the method ensures uniform boundedness and uniform ultimate boundedness simultaneously, thereby ensuring approximate adherence to constraints for underactuated mechanical systems facing both matched and mismatched uncertainties alongside initial condition deviations.

A non-linear Euler-Lagrange (EL) controller is proposed for underactuated mechanical Euler-Lagrange systems. The rationale is that mimicking the process structure in the controller will lead to improved performance. The tuning of the controller can be rather intuitive as it can be seen as an auxiliary mechanical system that is connected to the process, also called the main system. The tuning methodology used here finds its roots in the vibration absorber technology: transfer the energy from the main to the auxiliary system and dissipate the energy in the additional structure. Energy, in the form of the Hamiltonian, is used to examine stability of the compound system using Lyapunov’s direct method. To evaluate the performance of the controller, the Hamiltonian of the main system is used. A planar manipulator with two and three degrees-of-freedom (DOF) shows the effectivity of the proposed tuning method