A nonlinear friction model for the passivity-based control of underactuated mechanical systems

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.

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.

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. >

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.

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 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.

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

On closed-loop vibrational control of underactuated mechanical systems

This paper presents how the temperature dependence of the friction phenomena can be handled in mechanical control systems using state estimation techniques. The method is based on the dynamic friction model, originally introduced by Batista and Carlson (Physical Rev. E; 1996, 1998), where the temperature dependence of friction is described by an un-measurable internal state. According to the author's knowledge this model was not applied before in mechanical control system analysis and design. For the model a state estimator was developed based on the theory of hybrid systems. A robust tracking control algorithm was also developed for mechanical systems with temperature dependent friction. Simulation measurements are provided in the second part of the paper to show the applicability of the theoretical results.

A robust positioning control scheme has been developed using friction parameter observer and recurrent fuzzy neural networks based on the sliding mode control. As a dynamic friction model, the LuGre model is adopted for handling friction compensation because it has been known to capture sufficiently the properties of a nonlinear dynamic friction. A developed friction parameter observer has a simple structure and also well estimates friction parameters of the LuGre friction model. In addition, an approximation method for the system uncertainty is developed using recurrent fuzzy neural networks technology to improve the precision positioning degree. Some simulation and experiment provide the verification on the performance of a proposed robust control scheme.

Due to friction characteristics of clutch, the driveline is prone to cause a judder during vehicle starting, and then to cause the vehicle body to vibrate, which affects driving quality. In order to analyze the judder phenomenon, a nonlinear numerical friction model based on the Gaussian friction model is established in this paper. For the driveline of a front-wheel-drive vehicle, a five-degree-of-freedom (5DOF) lumped parameter model including a nonlinear friction element is established. The complex mode of the driveline during the clutch in slip condition is calculated. The key parameters affecting the driveline stability are analyzed. The self-excited judder and pressure-induced judder of the driveline are numerically simulated, and the corresponding causes are analyzed. The nonlinear friction torque of the clutch is also calculated. Furthermore, the effects of the key parameters such as the torsional stiffness and damping of the clutch and drive shaft suppressing the self-excited judder and pressure-induced judder are numerically studied respectively. Compared with the widely used Karnopp friction model, the nonlinear numerical friction model established in this paper comprehensively includes the stribeck effect in slip and the friction torque characteristics in stick. The phenomena of the judder and stick-slip of the driveline during vehicle starting are more accurately simulated. The simulation results are in good agreement with the experimental results, which verify the accuracy and effectiveness of the dynamic model including the nonlinear friction element established in this paper.

There have not been any suitable friction models to properly solve plasticity problems yet. Most recently, a new dynamic friction model merging the definition expression of the viscosity with the Amontons-Coulomb friction model suggested by the author has been successfully applied in axisymmetric upsetting to predict the contact stresses. In this paper, the dynamic friction model is further used for resolving the problem of plane-strain compression. With element-equilibrium approach, contact stresses in plane-strain compression have been derived. The mathematical expressions of these contact stresses are verified by the previous experimental data with various ratios of instant length to instant height of workpiece and different lubrications. The curves obtained from the dynamic friction model agree with these experimental results very well, not only for contribution of the normal pressure, mean normal pressure, but also for contribution of the friction stress. Comparisons of results of the dynamic friction model, the Amontons-Coulomb friction model, and the constant friction model with the experimental data are also made. It is seen that the predictions by the dynamic friction model show much better solutions to the practical plasticity problems. It is believed that the dynamic friction model could extensively be used to plasticity problems of solids.

This article presents the development of a multiple inner-loop control strategy for improving the performance of hydrostatic actuation systems. In these actuators, the presence of non-linearities associated with pump/motor static friction and backlash, pressure drop in the piping system, and nonlinear friction at the load have a significant effect on the performance and positional accuracy of the system. The effect of nonlinear friction at the pump/motor interface has been overcome by the use of a high gain pump-speed inner-loop control strategy. In this article, the concept of inner-loop control will be extended to target specific sources of performance degradation. Velocity feedback will be incorporated in this manner to decrease the effects of pump backlash and nonlinear friction at the load. Simulation results supported by theoretical analysis indicate that a considerable improvement in performance can be achieved by the implementation of this control strategy.