Passivity-based control of underactuated systems with non-integrable state-dependent matched disturbances

This work investigates the control problem of discrete-time underactuated mechanical systems with fixed input-delay and matched disturbances. A new control strategy is proposed, which builds upon a discrete-time implementation of the interconnection-and-damping-assignment passivity-based control (IDA-PBC) and extends it in two ways: the disturbances are estimated adaptively; the input-delay is compensated with a recursive algorithm. The resulting control law is constructed from IDA-PBC without solving any additional partial-differential-equation (PDE). Stability conditions are discussed and compared to alternative designs. Numerical simulations for the ball-on-beam system and for the Acrobot system demonstrate the effectiveness of the proposed approach.

Passivity-based tracking control of the underactuated linear mechanical systems is investigated in this paper. As our main contribution, the matching condition is decreased into two equations and an adjustable gain (damping gain) is introduced into the controller by setting the desired closed-loop system properly. Stability of the closed-loop system is proved based on passivity of the system. Furthermore, as examples, tracking control of 2-DOF Acrobot and 2-DOF Pendubot are studied. The systems are linearized at their equilibriums and the passivity-based controller design method is applied to the linearized systems. Matching conditions are solved and the design procedures of associate controllers for the two robots are provided. The simulation results show that the designed controllers can realize asymptotical tracking for the given desired trajectories.

Passivity-Based Control is a widely recognised control design methodology that has been successfully applied to solve regulation control problems for Port-Controlled Hamiltonian systems. However, the problem of stabilising time-varying trajectories is a topic that still requires further research, in particular for underactuated systems, i.e. systems with fewer control inputs than generalised coordinates. The purpose of this paper is to contribute to the consolidation of this control technique to solve tracking control problems for underactuated Port-Controlled Hamiltonian systems by identifying a novel class of systems for which it is possible to solve the tracking problem. This class is characterised by a quadratic Hamiltonian function and state-dependent interconnection and damping matrices that generate quadratic nonlinearities. It is shown that the desired time-varying trajectories exhibit asymptotic stability properties by applying well-known and simple Lyapunov-based results from the nonlinear systems theory. The usefulness of the contribution is illustrated in a numerical setting by approaching two case studies of practical interest, namely: the three-dimensional Lorenz oscillator system and the model of a surface-mounted Permanent Magnet Synchronous Motor.

This work investigates the energy shaping control of a class of underactuated mechanical systems subject to state‐dependent linearly parameterized matched disturbances. To this end, a new integral interconnection and damping‐assignment passivity‐based design is proposed. The controller design is developed for systems subject to either momenta‐dependent disturbances or position‐dependent disturbances or to both simultaneously. The effectiveness of the proposed approach is demonstrated with numerical simulations on three examples: an Acrobot system with nonlinear friction; a ball‐on‐beam system with constant and position‐dependent disturbances; a disk‐on‐disk system with a complete set of state‐dependent disturbances. In all examples, the proposed controller shows better performance compared to previous implementations.

An underactuated mechanical system with $$n~(n\ge 3)$$ degree of freedoms (DOFs) is a complicated nonlinear system. This paper develops a new strategy to solve the nonlinear stabilizing control problem for this kind of mechanical systems. First, we introduce a coupled relationship between control torques. It changes the n-DOF underactuated system into a cascade-connected system, which has a 2-DOF driven subsystem and a $$(n-2)$$ -DOF stable driving subsystem. And then, we analyze the passivity of the driven subsystem and discuss how to design an passivity-based controller that stabilizes the driven subsystem at the origin. Finally, the stabilization of the n-DOF underactuated system is achieved by the triangle lemma. Our proposed strategy transforms the stabilization of the n-DOF underactuated system into that of the 2-DOF driven subsystem. This makes the structure of the control system simple and also makes the problem of stabilizing a multi-DOF underactuated system easy to handle. As an application of the strategy, we give detailed statements of using it to achieve the global stabilization of a 3-DOF underactuated mechanical system called spring-coupled horizontal three-link underactuated manipulator. Simulation results demonstrate its validity.

This work investigates the tracking control problem for underactuated mechanical systems. To this end, we develop an extension of the dynamic tube Model Predictive Control (MPC) approach by combining an MPC design, an ancillary energy shaping controller constructed with the Interconnection and Damping Assignment Passivity-Based Control methodology, and an analytical expression of the dynamic tube. In addition, we extend the proposed approach by including the adaptive compensation of a class of unknown disturbances. The stability analysis is presented by employing a Lyapunov approach. The effectiveness of the proposed controller is demonstrated with simulations on two underactuated systems: a two-mass-spring-damper system with uncertain damping and either linear or nonlinear spring; an inertia-wheel-pendulum with unmodeled disturbances.

This paper presents the design of a robust controller for stabilization of the 2-DoF torsion system, which is an underactuated system, characterized by a higher number of degrees of freedom than the number of actuators. The output of the given servo dynamical system with two flexible torsional couplings should be positioned properly with minimum vibrations and response time. The control scheme presented here, to achieve the control requirement, is higher order sliding mode control with genetic algorithm tuning based on the hierarchical sliding mode. This control scheme provides robustness to matched disturbances, a finite time convergence like its conventional counterpart, along with the ability to overcome the problem of chattering. In addition, the choice of the hierarchical sliding mode has the added benefit of having a simpler subsystem-wise sliding mode design, which makes it more convenient to apply to the underactuated system. Genetic algorithm is applied to optimize the controller parameters. The study of the performance of the proposed scheme is based on bounded matched disturbance, randomly varying viscous damping coefficient, parameter uncertainties, and varying frictional force. MATLAB simulations have been carried out to verify the above proposition.KeywordsHigher order sliding mode controlHierarchical sliding modeGenetic algorithmTorsion systemMatched uncertainties

This paper develops a passivity-based robust motion controller for a robot used in prosthetic leg performance studies. The mathematical model of the robot and passive prosthesis corresponds to a three degree-of-freedom, underactuated rigid manipulator. A form of robotic testing of prostheses involves tracking reference trajectories obtained from human gait studies. The robot presented in this paper emulates hip vertical displacement and thigh swing, and we consider a prosthesis with a passive knee for control development. The control objectives are to track commanded hip displacements and thigh angles accurately, even in the presence of parametric uncertainties and large disturbance forces arising from ground contact during the stance phase. We develop a passivity-based controller suitable for an underactuated system and compare it with a simple independent-joint sliding mode controller (IJ-SMC). This paper describes the mathematical model and nominal parameters, derives the passivity-based controller using Lyapunov techniques and reports success in real-time implementation of both controllers, whose advantages and drawbacks are compared.

This paper presents a passivity-based control scheme for the two main axes of a 5 t-overhead crane, which guarantees both tracking of desired trajectories for the crane load and an active damping of crane load oscillations. The passivitybased control is performed by interconnection and damping assignment according to the IDA-PBC approach for underactuated systems. The tracking capabilities concerning desired trajectories for the crane load can be significantly improved by introducing feedforward control based on an inverse system model. Furthermore, a reduced-order disturbance observer is utilised for the compensation of nonlinear friction forces. In this paper, feedforward and feedback control as well as observer based disturbance compensation are adapted to the varying system parameters rope length as well as load mass by gain-scheduling techniques. Thereby, desired trajectories for the crane load position in the 3-dimensional workspace can be tracked independently with high accuracy. Experimental results of an implementation on a 5 t-crane show both excellent tracking performance with maximum tracking errors of 2 cm and a high steady-state accuracy.

Passivity property gives a sense of energy balance. The classical definitions and theorems of passivity in dynamical systems require time invariance and locally Lipschitz functions. However, these conditions are not met in many systems. Characteristic examples are nonautonomous and discontinuous systems due to the presence of Coulomb friction. This paper presents an extended result for the negative feedback connection of two passive nonautonomous systems with set-valued right-hand side based on an invariance-like principle. Such extension is the base of a structural passivity-based control synthesis for underactuated mechanical systems with Coulomb friction. The first step consists of designing the control able to restore the passivity in the considered friction law, achieving stabilization of the system trajectories to a domain with zero velocities. Then, an integral action is included to improve the latter result and perform a tracking over a constant reference (regulation). Finally, the control is designed considering dynamics in the actuation. These control objectives are obtained using fewer control inputs than degrees of freedom, as a result of the underactuated nature of the plant. The presented control strategy is implemented in an earthquake prevention scenario, where a mature seismogenic fault represents the considered frictional underactuated mechanical system. Simulations are performed to show how the seismic energy can be slowly dissipated by tracking a slow reference, thanks to fluid injection far from the fault, accounting also for the slow dynamics of the fluid’s diffusion.

Classical control strategies for robotic systems are based on the idea that feedback control can be used to override the natural dynamics of the machines. Passivity-based control (Pbc) is a branch of nonlinear control theory that follows a similar approach, where the natural dynamics is modified based on the overall energy of the system. This method involves transforming a nonlinear control system, through a suitable control input, into another fictitious system that has desirable stability characteristics. The majority of Pbc techniques require the discovery of a reasonable storage function, which acts as a Lyapunov function candidate that can be used to certify stability. There are several challenges in the design of a suitable storage function, including: 1) what a reasonable choice for the function is for a given control system, and 2) the control synthesis requires a closed-form solution to a set of nonlinear partial differential equations. The latter is in general difficult to overcome, especially for systems with high degrees of freedom, limiting the applicability of Pbc techniques. A machine learning framework that automatically determines the storage function for underactuated robotic systems is introduced in this dissertation. This framework combines the expressive power of neural networks with the systematic methods of the Pbc paradigm, bridging the gap between controllers derived from learning algorithms and nonlinear control theory. A series of experiments demonstrates the efficacy and applicability of this framework for a family of underactuated robots.

In this paper a system of ODEs and a method for its solution are proposed for passivity-based stabilization of a class of underactuated mechanical systems. This method simplifies the PDEs imposed by the passivity-based control via interconnection and damping assignment (IDA-PBC) technique. For a class of mechanical systems satisfying some mathematical properties the PDEs are reduced to a simple set of nonlinear ODEs. A general solution for these equations in the 2-DOF case is presented, through a new parametrization of the closed loop inertia matrix. Two examples illustrate the method, achieving kinetic and potential energy shaping in a straightforward manner. The simulation analysis in comparison with recent studies highlight the advantages of these techniques.

Distributed Control of Heterogeneous Underactuated Mechanical Systems

Passivity-based control of under-actuated mechanical systems with nonlinear friction effects in the generalized coordinates of motion is analyzed in this paper. Nonlinear friction is modeled with a modified LuGre dynamic friction model. The internal states of the dynamic friction model are incorporated as generalized coordinates in a port-controlled Hamiltonian formulation for the complete mechanical system in such a way that all passivity properties of this formulation are preserved for the extended generalized coordinates system. Interconnection and damping assignment passivity-based control laws are developed for the models of two case studies: a building with a magneto-rheological damper and a double pendulum. Simulation results are also presented.

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.