This paper aims to provide a new problem formulation of path following for mechanical systems without time parameterization nor guidance laws, namely, we express the control objective as an orbital stabilization problem. It is shown that, it is possible to adapt the immersion and invariance technique to design static state-feedback controllers that solve the problem. In particular, we select the target dynamics adopting the recently introduced Mexican sombrero energy assignment method. To demonstrate the effectiveness of the proposed method we apply it to control underactuated marine surface vessels.
In this note we study the generation of attractive oscillations of a class of mechanical systems with underactuation one. The proposed design consists of two terms, i.e., a partial linearizing state feedback, and an immersion and invariance orbital stabilization controller. The first step is adopted to simplify analysis and design, however, bringing an additional difficulty that the model loses its Euler-Lagrange structure after the collocated pre-feedback. To address this, we propose a constructive solution to the orbital stabilization problem via a smooth controller in an analytic form, and …
The problem of orbital stabilization of underactuated mechanical systems with one passive degree-of-freedom (DOF) is revisited. Virtual holonomic constraints are enforced using a continuous controller; this results in a dense set of closed orbits on a constraint manifold. A desired orbit is selected on the manifold and a Poincare section is constructed at a fixed point on the orbit. The corresponding Poincare map is linearized about the fixed point; this results in a discrete linear time-invariant system. To stabilize the desired orbit, impulsive inputs are applied when the system trajectory c…
A Hybrid passive Linear Inverted Pendulum (HLIP) model is proposed for characterizing, stabilizing and composing periodic orbits for 3D underactuated bipedal walking. Specifically, Period-1 (P1) and Period-2 (P2) orbits are geometrically characterized in the state space of the H-LIP. Stepping controllers are designed for global stabilization of the orbits. Valid ranges of the gains and their optimality are derived. The optimal stepping controller is used to create and stabilize the walking of bipedal robots. An actuated Spring-loaded Inverted Pendulum (aSLIP) model and the underactuated robot …
Transverse linearization-based approaches have become among the most prominent methods for orbitally stabilizing feedback design in regards to (periodic) motions of underactuated mechanical systems. Yet, in an $n$-dimensional state-space, this requires knowledge of a set of $(n-1)$ independent transverse coordinates, which can be nontrivial to find and whose definitions might vary for different motions (trajectories). In this paper, we consider instead a generic set of $excessive$ transverse coordinates which are defined in terms of a particular parameterization of the motion and a projection …
Orbital Stabilization of Nonlinear Systems via the Immersion and Invariance Technique
Laboratoire des Signaux et Systémes, CNRS-CentraleSupélec（信号与系统实验室，法国中央苏黎世高等学院）；Department of Control Systems and Informatics, ITMO University（ITMO大学控制系统与信息系）；Australian Centre for Field Robotics, The University of Sydney（悉尼大学澳大利亚野外机器人中心）；Departamento Académico de Sistemas Digitales, ITAM（墨西哥美洲理工学院数字系统学术部）；Department of Electrical and Electronic Engineering, Imperial College London（伦敦帝国理工学院电气与电子工程系）；Department of Civil Engineering and Computer Science Engineering, University of Rome “Tor Vergata”（罗马“托尔维尔加塔”大学土木与计算机工程科学系）
The task of inducing, via continuous static state-feedback control, an asymptotically stable heteroclinic orbit in a nonlinear control system is considered in this paper. The main motivation comes from the problem of ensuring convergence to a so-called point-to-point maneuver in an underactuated mechanical system. Namely, to a smooth curve in its state--control space, which is consistent with the system dynamics and connects two (linearly) stabilizable equilibrium points. The proposed method uses a particular parameterization, together with a state projection onto the maneuver as to combine tw…
Immersion and Invariance Stabilization of Nonlinear Systems: A Horizontal Contraction Approach
State Key Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang University（浙江大学控制科学与工程学院）；CASY - DEI, University of Bologna（博洛尼亚大学电气、电子与信息技术工程系）；University of Cambridge, Department of Engineering（剑桥大学工程系）；Department of Electrical Engineering and Computer Science, University of Liège（列日大学电气与计算机工程系）
The varied and complex dynamics of real-world systems challenge the formulation of a systematic strategy for designing a stabilizing feedback law. Rather than taking a universal approach, the control strategies developed thus far to handle this problem are specific to the inherent structure of the system under consideration. Therefore, this paper attempts to develop a generalized theory for the design of the stabilizing feedback law wherever possible for a general class of systems, including the systems in standard structured and unstructured forms discussed in the existing literature. The the…
In this work, we present the integrated structure-control design of a 2-DOF underactuated mechanical system, aiming to achieve a periodic motion of the end-effector. The desired behavior is generated via input-output linearization, followed by structural optimization of the zero dynamics. Inspired by recent works on the control-oriented design of multibody systems, we define an optimization problem based on the simulation of the system's response. In particular, relevant model parameters are used to match the reference with a specific orbit of the zero dynamics, while also penalizing the input…
Real-Time Nonlinear MPC via Sequential Quadratic Programming with Structure-Exploiting ADMM and Interior-Point Methods for Underactuated Double-Pendulum Swing-Up
The design of robust orbitally stabilizing feedback is considered. From a known orbitally stabilizing controller for a nominal, disturbance-free system, a robustifying feedback extension is designed utilizing the sliding-mode control (SMC) methodology. The main contribution of the paper is to provide a constructive procedure for designing the time-invariant switching function used in the SMC synthesis. More specifically, its zero-level set (the sliding manifold) is designed using a real Floquet-Lyapunov transformation to locally correspond to an invariant subspace of the Monodromy matrix of a …
This manuscript describes several approaches to tune the parameters of a class of passivity-based controllers for standard nonlinear mechanical systems. In particular, we are interested in controllers that preserve the mechanical system structure in closed-loop. To this end, first, we provide tuning rules for stabilization, i.e., the rate of convergence (exponential stability) and stability margin (input-to-state stability). Then, we provide guidelines to remove the overshoot while prescribing the rise time. Additionally, we propose a methodology to tune the gyroscopic-related parameters. We a…
Stabilization of an underactuated mechanical system may be accomplished by energy shaping. Interconnection and damping assignment passivity-based control is an approach based on total energy shaping by assigning desired kinetic and potential energy to the system. This method requires solving a partial differential equation (PDE) related to he potential energy shaping of the system. In this short paper, we focus on the reformulation of this PDE to be solved easier. For this purpose, under a certain condition that depends on the physical parameters and the controller gains, it is possible to mer…
Stabilizing Controllers for Multi-Input, Singular Control Gain Systems
Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin（德克萨斯大学奥斯汀分校航空航天工程与应用力学）；Systems and Control Engineering, Indian Institute of Technology, Bombay（印度孟买理工学院系统与控制工程）
Nonholonomic mechanical systems encompass a large class of practically interesting robotic structures, such as wheeled mobile robots, space manipulators, and multi-fingered robot hands. However, few results exist on the cooperative control of such systems in a generic, distributed approach. In this work we extend a recently developed distributed Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) method to such systems. More specifically, relying on port-Hamiltonian system modelling for networks of mechanical systems, we propose a full-state stabilization control law for a…
