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This paper is part of a research project started wity [xc2] and [xc3] aiming to use Hamiltonian methods to study second order conditions in optimal control. It is the opinion of the authors that a complete understanding of the problem in Hamiltonian terms will lead to a deeper insight into the synthesis problem. [ABSTRACT FROM AUTHOR]

We present a new coordinated control law for a group of vehicles in the plane that stabilizes an arbitrary desired group shape. The control law is derived for an arbitrary shape using models of tensegrity structures which are spatial networks of interconnected struts and cables. The symmetries in the coupled system and the energy-momentum method are used to investigate stability of relative equilibria corresponding to steady translations of the prescribed rigid shape. [ABSTRACT FROM AUTHOR]

In this paper we present some additional results regarding the pumping-damping strategy for swinging up a pendulum introduced in [3]. Here, the family of energy functions is enlarged and the corresponding pumping-damping functions are proposed giving rise to new smooth controllers that swing up and stabilize the pendulum. Furthermore, a generalization of the stability criterion is introduced for this larger class of controllers. [ABSTRACT FROM AUTHOR]

Hamiltonian control systems [11, 6] are the systems described by Hamilton's canonical equations which represent general physical systems. Recently portcontrolled Hamiltonian systems are introduced as a generalization of Hamiltonian systems [3] They can represent not only ordinary mechanical, electrical and electro-mechanical systems, but also nonholonomic systems [4]. The special structure of physical systems allows us to utilize the passivity which they innately possess and a lot of fruitful results were obtained so far. These methods are so called passivity based control [8]. [ABSTRACT FROM AUTHOR]

In this paper we analyze the equations of motion of a submerged rigid body. Our motivation is based on recent developments done in trajectory design for this problem. Our goal is to relate some properties of singular extremals to the existence of decoupling vector fields. The ideas displayed in this paper can be viewed as a starting point to a geometric formulation of the trajectory design problem for mechanical systems with potential and external forces. [ABSTRACT FROM AUTHOR]

A particular family of Hamiltonian functions is considered. Such functions are quadratic in the moment variables and arise in spaceflight mechanics when the averaged system of energy minimizing trajectories of the Kepler equation is computed. An important issue of perturbation theory and averaging is to provide integrable approximations of nonlinear systems. It turns out that such integrability properties hold here. [ABSTRACT FROM AUTHOR]

It is said that "human hand" is an agent of the brain. This might attract attention from many prominent robot engineers and researchers who eventually attempted to design multi-fingered robot hands that mimic human hands. In the history of development of multi-fingered robot hands (see the literature [1] ∼ [5]), a variety of sophisticated robot hands designed and make are indeed reported. However, most of them have not yet been used widely in practice such as assembly tasks and other automation lines in place of human hands. The most important reason of this must be owing to the high cost of manufacturing such multi-fingered hands with many joints together with expensive sensing devices such as tactile and/or force sensors, which can not redeem human potentials of flexibility and versatility in execution of a variety of tasks. In fact, multi-fingered robot hands were used only in open-loop control (see [2]) and the importance of sensory feedback was not discussed in the literature until around the year of 2000 (see [12]). This paper firstly introduces a mathematical model of full dynamics of planar but vertical motion a rigid object grasped by a pair of two and three d.o.f fingers with soft and deformable tips whose shape is hemispherical. The behavior of the soft finger tips is lumped-parameterized by assuming that the soft material is distributively composed of massless springs with spring constant k (stiffness constant per unit area) and dampers in parallel. [ABSTRACT FROM AUTHOR]

This paper presents a general modeling method to construct a virtual Lagrangian for infinite-dimensional systems. If the system is self-adjoint in the sense of the Fretchet derivative, there exists some Lagrangian for a stationary condition of variational problems. A system having such a Lagrangian can be formulated as a field port-Lagrangian system by using a Stokes-Dirac structure on a variational complex. However, it is unknown whether any infinite-dimensional system can be expressed as an Euler-Lagrange equation. Then, we introduce a virtual Lagrangian with a homotopy operator. The virtual Lagrangian defines a self-adjoint subsystem, which is realized by a cancellation of non-self-adjoint error subsystems. [ABSTRACT FROM AUTHOR]

Recently, the investigation of design approaches to nonlinear control, which can thoroughly exploit the structure and the properties of the physical systems, has been given much attention due to the designed controllers with relatively simple form and effective operation. And it has been shown that for the mechanical and electromechnical systems, by incorporating the Hamiltonian structure into passivity-based control design techniques, the controllers based on physical considerations can be obtained [1],[2],[3]. This design idea has been successfully realized alized in the control of power systems and a lot of achievements have been gotten (see e.g. [1]-[9). In [4], the fact that the power system with excitation control is a Hamiltonian structure is exploited by selecting a storage function constructed by system potential and kinetic energy and an appropriate passivating output on the basis of physical arguments. Along this research line, the excitation control problem for multi-machine power systems has been investigated in [5],[6]. Moreover, an adaptive L2 disturbance attenuation controller based on Hamiltonian structure has been provided for power systems in [7]. [ABSTRACT FROM AUTHOR]

A geodesic-based formulation of geometric PD tracking control for fully actuated mechanical systems is derived from the Riemannian metric. The region of stability is determined directly from the magnitude of the system's injectivity radius and, for a restricted set of control problems, the locus of cut points about a desired reference point in the manifold. Exponential stability is proven for controlled motion along a geodesic, yielding a particularly simple, yet elegant, methodology for control design. [ABSTRACT FROM AUTHOR]

We describe different strategies for using a semi-classical controller to engineer Hamiltonians for quantum systems to solve control problems such as quantum state or process engineering and optimization of observables. [ABSTRACT FROM AUTHOR]

An open problem, set by Yu. Orlov in his contribution to the volume "Open Problems in Mathematical Systems and Control Theory", V. Blondel, A. Megretski Eds., 2004, regards regularization of optimal control-affine problems with control-independent state-quadratic cost. It is asked whether the infima of the regularized (by adding squared L2-norm of controls) functionals converge to the infimum of the original functional? We show that this question can be resolved by an elementary argument. We claim that one should study minimizing sequences of the original functional, rather than its infimum. We advocate the relevance of this question, formulating it via notion of order of singularity of an optimal problem. We study this question and provide computations of the order of singularity for arbitrary singular linear-quadratic problem and also for some classes of nonlinear control-affine problems. Some open problems are set. [ABSTRACT FROM AUTHOR]

We propose a class of second order mechanical systems on Grassmann manifolds that converge to the dominant eigenspace of a given symmetric matrix. Such second order flows for principal subspace analysis are derived from a modification of the familiar Euler-Lagrange equation by inserting a suitable damping term. The kinetic energy of the system is defined by the Riemannian metric on the Grassmannian while the potential energy is given by a trace function, that is defined by the symmetric matrix. Convergence of the algorithm to the dominant subspace is shown for generic initial conditions and a comparison with the Oja flow is made. [ABSTRACT FROM AUTHOR]

In this paper, a new control strategy for a class of 1-generator nonholonomic systems with drift is proposed: 1$$ \begin{gathered} y_1^{(k_1 )} = u_1 \hfill \\ y_2^{(k_2 )} = u_2 \hfill \\ y_3^{(k_3 )} = \ell _{31} (y_3 ,...,y_3^{(k_3 )} ) \hfill \\ + \ell _{32} (y_2 ,...,y_2^{(k_2 - 1)} ) \times \ell _g (u_1 ,y_1 ,...,y_1^{(k_1 - 1)} ) \hfill \\ \vdots \hfill \\ y_n^{(k_n )} = \ell _{n1} (y_n ,...,y_n^{(k_n )} ) \hfill \\ + \ell _{n2} (y_{n - 1} ,...,y_{n - 1}^{(k_{n - 1} - 1)} ) \times \ell _g (u_1 ,y_1 ,...,y_1^{(k_1 - 1)} ), \hfill \\ \end{gathered} $$ where each ℓ*(·) represents a homogeneous linear term. This system is an asymmetric affine nonholonomic system of the form x=f(x)+g(x)u with 1-generator controllability structure. including such as high-order chained form [5], underactuated surface vessel [4] and so on. [ABSTRACT FROM AUTHOR]

This paper begins by recalling how a constraint distribution on a configuration manifold induces a Dirac structure together with an implicit Lagrangian system, a construction that is valid even for degenerate Lagrangians. In such degenerate cases, it is shown in this paper that an implicit Hamiltonian system can be constructed by using a generalized Legendre transformation, where the primary constraints are incorporated into a generalized Hamiltonian on the Pontryagin bundle. Some examples of degenerate Lagrangians for L-C circuits, nonholonomic systems, and point vortices illustrate the theory. [ABSTRACT FROM AUTHOR]

The rolling sphere problem on $$ \mathbb{E}^n $$ consists of determining the path of minimal length traced by the point of contact of the oriented unit sphere $$ \mathbb{S}^n $$ as it rolls without slipping between two boundary points of $$ \mathbb{E}^n \times SO_n $$. This problem is extended to the following cases of rolling: ℍn on $$ \mathbb{E}^n ,\mathbb{S}_\rho ^n $$ on $$ \mathbb{S}_\sigma ^n $$, and ℍρn on ℍσn, where σ≠ρ are the radii of the spheres or hyperboloids. The term "rolling" is generalized to an isometric sense: the length of a curve is measured using the Riemannian metric of the stationary manifold while the orientation of the rolling object is described by a matrix from its isometry group. These problems constitute left-invariant optimal control problems on Lie groups, whose Hamiltonian equations reveal certain integrals of motion and show, on the level of Lie algebras, that all of the above problems are governed by a single set of equations. [ABSTRACT FROM AUTHOR]

An energy balance equation with respect to a control contact system provides port outputs which are conjugated to inputs. These conjugate variables are used to define the composition of port contact systems in the framework of contact geometry. We then propose a power-conserving interconnection structure, which generalizes the interconnection by Dirac structures in the Hamiltonian formalism. Furthermore, the composed system is again a port contact system, as illustrated on the example of a gas-piston system undergoing some irreversible transformation. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to apply methods from geometric mechanics to the analysis and control of bipedal robotic walkers. We begin by introducing a generalization of Routhian reduction, functional Routhian Reduction, which allows for the conserved quantities to be functions of the cyclic variables rather than constants. Since bipedal robotic walkers are naturally modeled as hybrid systems, which are inherently nonsmooth, in order to apply this framework to these systems it is necessary to first extend functional Routhian reduction to a hybrid setting. We apply this extension, along with potential shaping and controlled symmetries, to derive a feedback control law that provably results in walking gaits on flat ground for a three-dimensional bipedal walker given walking gaits in two dimensions. [ABSTRACT FROM AUTHOR]

This paper proposes a novel framework to generate optimal gait trajectories for a one-legged hopping robot via iterative learning control. This method generates gait trajectories which are solutions of a class of optimal control problems without using precise knowledge of the plant model. It is expected to produce natural gait movements such as that of a passive walker. Some numerical examples demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper an internal model based approach to implicit Fault Tolerant Control for port-Hamiltonian systems is presented: the main idea is to cast the problem into a regulation problem in presence of input disturbances representing exogenous effects of possible faults; this can be solved following an adaptive internal model based approach. The theoretical machinery exploited is specialized for the energy-based port-Hamiltonian formalism in order to prove the global asymptotical stability of the solution. Finally an application example is presented in order to deeply point out the effectiveness of the design procedure presented: a Fault Tolerant Control problem is solved for a magnetic levitation system affected by periodic voltage disturbances. [ABSTRACT FROM AUTHOR]

Total Energy Shaping is a controller design methodology that achieves (asymptotic) stabilization of mechanical systems endowing the closed-loop system with a Lagrangian or Hamiltonian structure with a desired energy function—that qualifies as Lyapunov function for the desired equilibrium. The success of the method relies on the possibility of solving two PDEs which identify the kinetic and potential energy functions that can be assigned to the closed-loop. Particularly troublesome is the PDE associated to the kinetic energy which is nonlinear and non-homogeneous and the solution, that defines the desired inertia matrix, must be positive definite. In this paper we prove that we can eliminate or simplify the forcing term in this PDE modifying the target dynamics and introducing a change of coordinates in the original system. Furthermore, it is shown that, in the particular case of transformation to the Lagrangian coordinates, the possibility of simplifying the PDEs is determined by the interaction between the Coriolis and centrifugal forces and the actuation structure. The example of a pendulum on a cart is used to illustrate the results. [ABSTRACT FROM AUTHOR]

Passivity-based control (PBC) is a generic name given to a family of controller design techniques that achieves system stabilization via the route of passivation, that is, rendering the closed-loop system passive with a desired storage function (that usually qualifies as a Lyapunov function for the stability analysis.) If the passivity property turns out to be output strict, with an output signal with respect to which the system is detectable, then asymptotic stability is ensured. See the monographs [5, 12], and [6] for a recent survey. [ABSTRACT FROM AUTHOR]

It is well known that energy balancing control is stymied by the presence of pervasive dissipation. To overcome this problem in electrical circuits, the authors recently proposed the alternative paradigm of power shaping—where, as suggested by its name, stabilization is achieved shaping a function akin to power instead of the energy function. In this paper we extend this technique to general nonlinear systems and apply it for the stabilization of the benchmark tunnel diode circuit. It is shown that, in contrast with other techniques recently reported in the literature, e.g. piecewise approximation of nonlinearities, power shaping yields a simple linear static state feedback that ensures (robust) global asymptotic stability of the desired equilibrium. [ABSTRACT FROM AUTHOR]

In the present work we are interested on the derivation of power-based passivity properties for a certain class of non-linear mechanical systems. While for general mechanical systems, it is of common use to adopt a storage function related to the system's energy in order to show passivity and stabilize the system on a desired equilibrium point (e.g., IDA-PBC [1]), we want here to obtain similar properties related to the system's power. The motivation arises from the idea that in some engineering applications (satellite orbit motion, aircraft dynamic, etc⋯) seems more sensible to cope with the power flowing into the system instead of the energy that, for stabilization purposes, means to consider the systems's equilibrium the state for which the energy flow-rate (i.e., system's power) achieve its minimum. In this respect, we recall first the power-based description for a certain class of (non)-linear mechanical systems given in [2] and then we give sufficient conditions to obtain power-based passivity properties, provided a suitable choice of port-variables. We conclude with the example of the inverted pendulum on the cart. [ABSTRACT FROM AUTHOR]

Interconnection and damping assignment passivity-based control (IDA-PBC) is a recently developed method for nonlinear controller design. A majority of the examples of IDA-PBC found in literature are from the electro-mechanical domain. To show its applicability in other domains, in this work IDA-PBC is used to design a stabilizing controller for a process control example, the four-tank system. Both simulations and real experiments are provided for this example system. [ABSTRACT FROM AUTHOR]

A bilateral telemanipulator is a robotic system that allows the interaction with remote environments and it is composed by a controlled local robot (the master) and a controlled remote robot (the slave) interconnected through a communication channel. The motion imposed to the master by the human is transmitted to the slave which moves accordingly; when the slave interacts with a remote environment, the interaction force is fed back to the master side in order to improve the perception of the remote environment felt by the user. Passivity theory is a very suitable tool for the implementation of bilateral telemanipulation schemes over delayed communication channels. In passivity based telemanipulation, both master and slave are controlled by means of passive impedance controllers and master and slave sides are interconnected through a scattering based communication channel [1, 8] which allows an exchange of information which is passive independently of any constant communication delay. In this way, the overall telemanipulation system is passive and, consequently, its behavior is stable both in case of free motion and in case of interaction with any passive environment. In [12, 11], a generic framework for geometric telemanipulation of port-Hamiltonian systems [13] has been proposed; master and slave are interconnected through intrinsically passive port-Hamiltonian impedance controllers which allow to shape the energetic behavior of the robots and to achieve desired dynamic properties at master and slave sides. Local and remote sides are interconnected through a scattering based communication channel that allows a lossless exchange of energy. [ABSTRACT FROM AUTHOR]

The success in a model-based direct discrete-time design for nonlinear sampled-data control systems depends on the availability of a good discrete-time plant model to use for the design. Unfortunately, even if the continuous-time model of a plant is known, we cannot in general compute the exact discrete-time model of the plant, since it requires computing an explicit analytic solution of a nonlinear differential equation. One way to solve the problem of finding a good model is by using an approximate model of the plant. A general framework for stabilization of sampled-data nonlinear systems via their approximate discrete-time models was presented in [11]. It is suggested that approximate discrete-time models can be obtained using various numerical algorithms, such as Runge-Kutta and multistep methods. Yet, to the best of the authors knowledge, almost all available results on this direction view the systems as dissipative systems, whereas for design purpose, there are many systems that are better modeled as Hamiltonian conservative systems. [ABSTRACT FROM AUTHOR]

A key issue when dealing with distributed parameter systems is the solution of the set of partial differential equations that compose the system model. Even if we restrict our analysis to the linear case, it is often impossible to find a closedform solution for this kind of equations, especially in control applications, where the forcing action on the system generates time-varying boundary conditions. Therefore, numerical solvers play a key role as support tools for the analysis of this kind of systems. [ABSTRACT FROM AUTHOR]

In this paper we study the problem of synchronization of networked Lagrangian systems. This problem arises in applications such as the control of multiple robots, formation control of UAV's, mobile sensor networks, Kuramoto oscillators, and teleoperation. We assume that the agents exchange position and velocity information over a network described by an interconnection graph. We consider only linear interconnections on balanced, directed graphs but similar results can also be derived for nonlinear interconnections on undirected graphs. Our results exploit the well-known passivity property of Lagrangian mechanical systems. We demonstrate synchronization on both fixed and switching graphs. In the case of fixed graphs, the passivity property allows us, in addition, to treat the practically important problem of time delay in communication. [ABSTRACT FROM AUTHOR]

This paper presents a survey on Riemannian geometry methods for smooth and nonsmooth constrained optimization. Gradient and subgradient descent algorithms on a Riemannian manifold are discussed. We illustrate the methods by applications from robotics and multi antenna communication. Gradient descent algorithms for dextrous hand grasping and for sphere packing problems on Grassmann manifolds are presented respectively. [ABSTRACT FROM AUTHOR]

This article challenges Bernstein's problem of redundant degrees of freedom (DOF) that remains unsolved from the control-theoretic point of view as well as from the standpoint of both neuro-physiology and robotics. Firstly, a rather simpler but mysterious control problem of movements of human-like multi-joint reaching with excess DOFs is analyzed from Newtonian mechanics and differential geometry. Secondly, another illustrative control problem that seems to be sophisticated and complicated is tackled, which is to find a sensory coordinated control signal for 3-Dimensional stable grasping and object manipulation by a pair of robot fingers with multiple joints under the effect of gravity and nonholonomic constraints. In each illustrative control problem, it is possible to find a simple control signal that renders each corresponding closed-loop dynamics stable on its corresponding equilibrium-point manifold. It is claimed, however, that convergences of solutions of closed-loop dynamics to an equilibrium-point manifold can not be analyzed by using Lyapunov's direct method, because a Lyapunov-like energy form can not be positive definite due to redundancy of DOFs. Instead, a novel definition called "stability on a manifold" based upon the concept of Riemannian distance on the constraint manifold is introduced in both illustrative problems and used in the analysis of convergence of solution trajectories. It is also shown that finiteness of Riemannian metrics plays an important role in evaluation of the performance of control in both problems. [ABSTRACT FROM AUTHOR]