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Design and control of Power Supplies (PSs) feeding the magnets of a Synchrotron Light Source have to match severe specifications; high accuracy in the range of ppm in output current tracking is required for the correct operation of the magnets, while a Power Factor (PF) close to the unit is demanded at the input section due to the high power involved. In this paper an advanced control strategy is presented for a particular kind of Quadrupole Magnet Power Supply, where variable output current has to be imposed. The case of the "switch-mode" multilevel power converter for booster quadrupole magnets of the DIAMOND synchrotron radiation facility under construction at the Harwell Chilton Science Campus, Didcot, has been considered. High accuracy in the tracking of the desired output current reference is reached by means of a digital internal model-based controller. A multivariable controller is adopted in order to ensure current balancing between the stages of the multilevel converter. Front-end topology selection, proper dimensioning and control design are exploited to guarantee high power factor and low harmonic distortion of the input currents, and to avoid low-frequency components related to the quadrupole magnets' oscillating currents. For this purpose, confined oscillatory behavior imposed to the voltage of the DC-link capacitors plays a key role. Simulations and experimental validations are reported that confirm the expected results. [ABSTRACT FROM AUTHOR]

This paper gives an overview of a framework for analyzing hybrid dynamical systems. The emphasis is on modeling assumptions that guarantee robustness. These conditions lead to a general invariance principle and to results on the existence of smooth Lyapunov functions (converse theorems) for hybrid systems. In turn, the stability analysis tools motivate novel hybrid control algorithms for nonlinear systems. [ABSTRACT FROM AUTHOR]

This chapter presents a structural analysis approach for the design of fault tolerant estimation algorithms. The general fault tolerance problem setting is first given, and structural analysis is presented in the component based modeling frame. An original condition for structural observability is developed, which is constructive, since it allows to identify those Data Flow Diagrams by which unknown variables can be estimated, both in healthy and in faulty conditions. The link with two basic dependability concepts, namely critical faults and reliability is shown. [ABSTRACT FROM AUTHOR]

The Hamiltonian approach has turned out to be an effective tool for modeling, system analysis and controller design in the lumped parameter case. There exist also several extensions to the distributed parameter case. This contribution presents a class of extended distributed parameter Hamiltonian systems, which preserves some useful properties of the well known class of Port Controlled Hamiltonian systems with Dissipation. In addition, special ports are introduced to take the boundary conditions into account. Finally, an introductory example and the example of a piezoelectric structure, a problem with two physical domains, show, how one can use the presented approach for modeling and design. [ABSTRACT FROM AUTHOR]

In this chapter, we present the scenario approach, an innovative technology for solving convex optimization problems with an infinite number of constraints. This technology relies on random sampling of constraints, and provides a powerful means for solving a variety of design problems in systems and control. Specifically, the virtues of this approach are here illustrated by focusing on optimal control design in presence of input saturation constraints. [ABSTRACT FROM AUTHOR]

The goal of this paper is to propose a unique vision able to frame a number of results recently proposed in literature to tackle problems of output regulation for nonlinear systems. This is achieved by introducing the so-called asymptotic internal model property as the crucial property which, if fulfilled, leads to the design of the regulator for a fairly general class of nonlinear systems satisfying a proper minimum-phase condition. It is shown that recent frameworks based upon the use of nonlinear high-gain and adaptive observer techniques for the regulator design can be cast in this setting. A recently proposed technique for output regulation without immersion is also framed in these terms. [ABSTRACT FROM AUTHOR]

This paper surveys recent work of the author with several collaborators, principally Feng Lin, Weilin Wang, and Tae-Sic Yoo; they are kindly acknowledged. Decentralized control of discrete event systems, where local controllers cannot explicitly communicate in real-time, is considered in the first part of the paper. Then the problem of real-time communication among a set of local discrete-event controllers (or diagnosers) is discussed. The writing is descriptive and is meant to inform the reader about important conceptual issues and some recently-completed or on-going research efforts. [ABSTRACT FROM AUTHOR]

We offer an alternative way of evaluating the relative importance of the state coordinates of a nonlinear control system. Our approach is based on making changes of state coordinates to bring the controllability and observability functions into input normal form. These changes of coordinates are done degree by degree and the resulting normal form is unique through terms of degree seven. [ABSTRACT FROM AUTHOR]

We introduce two conceptual models for wireless sensing and control with power-limited sensors and controllers. The limited battery power of the wireless device is captured in the models by imposing hard constraints on either the number of available transmissions the device can make, or on the number of cycles it can stay awake. Such hard constraints can be viewed as a measurement budget, under which estimation or control policies will have to be developed over a given decision horizon. Among the two representative models studied here, the first one is one of optimal scheduling of a finite measurement budget for a Gauss-Markov process over an observation horizon. The second one is an optimal estimation problem where the number of transmissions the wireless sensor can make is limited to a number, M, which is less than the observation horizon, N. It is shown that both problems can be solved by employing dynamic-programming type arguments, and their solutions have a threshold characterization. [ABSTRACT FROM AUTHOR]

Petri nets are a family of powerful discrete event models whose interest has grown, within the automatic control community, in parallel with the development of the theory of discrete event systems. In this tutorial paper our goal is that of giving a flavor, by means of simple examples, of the features that make Petri nets a good model for systems theory and of pointing out at a few open areas for research. We focus on Place/Transitions nets, the simplest Petri net model. In particular we compare Petri nets with automata, and show that the former model has several advantages over the latter, not only because it is more general but also because it offers a better structure that has been used for developing computationally efficient algorithms for analysis and synthesis. [ABSTRACT FROM AUTHOR]

In this paper an internal model based approach to periodic input disturbance suppression for port-Hamiltonian systems is presented; more specifically, an adaptive solution able to deal with unknown periodic signal belonging to a given class is introduced. After an introductive section, the adaptive internal model design procedure is presented in order to solve the input disturbance problem. This theoretical machinery is specialized for the energy-based port-Hamiltonian framework in order to prove the global asymptotical stability of the solution. Finally, in order to clearly point out the effectiveness of the presented design procedure a tracking problem is solved for a robotic manipulator affected by torque ripples. [ABSTRACT FROM AUTHOR]

The minimum phase property is an important notion in systems and control theory. In this paper, a characterization of the minimum phase property of nonlinear control systems in terms of a dissipation inequality is derived. It is shown that this dissipation inequality is equivalent to the classical definition of the minimum phase property in the sense of Byrnes and Isidori, if the control system is affine in the input and the so-called input-output normal form exists. [ABSTRACT FROM AUTHOR]

The extension of application domains of robotics from factories to human environments leads to implementing proper strategies for close interaction between people and robots. In order to avoid dangerous collision, force and vision based control can be used, while tracking human motion during such interaction. [ABSTRACT FROM AUTHOR]

In the average consensus a set of linear systems has to be driven to the same final state which corresponds to the average of their initial states. This mathematical problem can be seen as the simplest example of coordination task and in fact it can be used to model both the control of multiple autonomous vehicles which all have to be driven to the centroid of the initial positions, and to model the decentralized estimation of a quantity from multiple measure coming from distributed sensors. In general we can expect that the performance of a consensus strategy will be strongly related to the amount of information the agents exchange each other. This contribution presents a consensus strategy in which the exchanged data are symbols and not real numbers. This is based on a logarithmic quantizer based state estimator. The stability of this technique is then analyzed. [ABSTRACT FROM AUTHOR]

This chapter presents an overview on the modeling and model-based control of autonomous helicopters. Firstly it introduces some of the platforms and control architectures that has been developed in the last 15 years. Later, the Chapter considers the modeling of the helicopter and the identification techniques. Then, it overviews different linear and non-linear model-based control approaches. This section also includes experiments on the control of the helicopter vertical motion that illustrate the presented techniques and point out the interest of nonlinear analysis methods to study the dynamic behavior of the helicopter. Finally, the Chapter presents open research lines coming from two challenging applications: the autonomous landing in oscillating platforms and the lifting and transporting of a single load with several helicopters. [ABSTRACT FROM AUTHOR]

The antiwindup and override design techniques have been developed in industrial design practice for controlling plants with input and output constraints. At present, various analysis and synthesis techniques are well established. These techniques have shown to be both effective and efficient for plants of dominant low order, which are either asymptotically stable or at least marginally stable. [ABSTRACT FROM AUTHOR]

Often one desires to control a nonlinear dynamical system in an optimal way taking constraints on the states and inputs directly into account. Classically this problem falls into the field of optimal control. Often, however, it is difficult, if not impossible, to find a closed solution of the corresponding Hamilton-Jacobi-Bellman equation. One possible control strategy that overcomes this problem is model predictive control. In model predictive control the solution of the Hamilton-Jacobi-Bellman equation is avoided by repeatedly solving an open-loop optimal control problem for the current state, which is a considerably simpler task, and applying the resulting control open-loop for a short time. [ABSTRACT FROM AUTHOR]

The aim of this chapter is the presentation of a technique for the design of stabilizing dynamic output feedback controllers for discrete-time linear systems with rate and amplitude saturating actuators. Two synthesis objectives are considered: the maximization of the domain of attraction of the closed-loop system under some time-domain performance constraints; and the guarantee that the trajectories of the system are bounded in the presence of L2-bounded disturbances, while ensuring an overbound for the L2 gain from the disturbance to the regulated output. [ABSTRACT FROM AUTHOR]

This chapter deals with two problems on stabilization of linear discrete systems by static feedbacks which are bounded and time-delayed, namely global asymptotic stabilization and finite gain lp-stabilizability, p ∈ [1,∞]. Furthermore, an explicit construction of the corresponding feedback laws is given. The feedback laws constructed also result in a closedloop system that is globally asymptotically stable. [ABSTRACT FROM AUTHOR]

We are concerned here with the stabilization of a linear time invariant system subject to actuator saturation via decentralized control while using linear time invariant dynamic controllers. When there exists no actuator saturation, i.e. when we consider just linear time invariant systems, it is known that global stabilization can be done via decentralized control while using linear time invariant dynamic controllers only if the so-called decentralized fixed modes of it are all in the open left half complex plane. [ABSTRACT FROM AUTHOR]

In the present contribution, the control synthesis for state and input constrained systems for linear systems via set-valued techniques will be presented. The results will be developed for both uncertain and gain scheduled constrained control problems, where the difference between the two is the knowledge of the uncertain parameter entering the system at each time instant in the gain scheduled case. It will be shown how slew-rate constrained control problems can be recast into standard constrained control problems by means of an extended system. To present such technique, an overview of some of the existing results with some numerical algorithms to derive the stability conditions and the control will be given. [ABSTRACT FROM AUTHOR]

In this chapter we address several case studies using LMI optimization methods for designing output feedback control laws to achieve regional performance and stability of linear control systems with input saturation. Algorithms are developed for minimizing the upper bound on the regional L2 gain for exogenous inputs with L2 norm bounded by a given value, and for minimizing this upper bound with a guaranteed reachable set or domain of attraction. Based on the structure of the optimization problems, using the projection lemma, the output feedback controller synthesis is cast as a convex optimization over linear matrix inequalities. The problems are studied in a general setting where the only requirement on the linear plant is detectability and stabilizability. [ABSTRACT FROM AUTHOR]

In the past few years, control of Linear Parameter Varying Systems (LPV) has been the object of considerable attention, as a way of formalizing the intuitively appealing idea of gain scheduling control for nonlinear systems. However, currently available LPV techniques are both computationally demanding and (potentially) very conservative. In this chapter we propose to address these difficulties by combining Receding Horizon and Control Lyapunov Functions techniques in a risk-adjusted framework. The resulting controllers are guaranteed to stabilize the plant and have computational complexity that increases polynomially, rather than exponentially, with the prediction horizon. [ABSTRACT FROM AUTHOR]

The most common way of dealing with constraints in control systems is to ignore them, pretend that the system is linear, and fix things up in a more-or-less ad-hoc fashion after performing a linear design. There are some rather systematic ways of ‘fixing things up', including certain anti-windup techniques, but logically they are an afterthought, a way of dealing with the nuisance of constraints after the central work has been done. Until recently, things could not be done differently, because almost all the control design techniques available to us were linear techniques.3 [ABSTRACT FROM AUTHOR]

In this contribution, explicit model predictive control is applied to the benchmark problems. Emphasis is put on the reduction of complexity both for the off-line and on-line computation. For the pendulum, a control-invariant set is computed first. By using this set in the subsequent optimal control problem, a simple solution can be attained which guarantees invariance of the resulting closed-loop system by construction. The computation of such invariant sets for systems with a partially constrained state-space is discussed. [ABSTRACT FROM AUTHOR]

This chapter addresses the problems of design of anti-windup schemes for linear systems subject to amplitude limitations in actuator and sensor. The design of anti-windup loops, using both measured and estimated variables, is considered in order to enlarge the region of stability of the closed-loop system, whereas some performance in terms of bounded controlled output is guaranteed. Based on the modelling of the closed-loop saturated system including the observer loop as a linear system with dead-zone nested nonlinearities, constructive stability conditions are formulated in terms of linear matrix inequalities by using Finsler's Lemma. [ABSTRACT FROM AUTHOR]

This chapter describes the theory and application of a ‘decoupled' approach to anti-windup compensation. One of the salient features of this approach is that the anti-windup problem can be posed in a manner which decouples the nominal linear closed-loop from the nonlinear stability problem associated with actuator saturation and the anti-windup compensator. This process also reveals a logical and intuitive performance criterion which one can optimise by means of linear matrix inequalities. Details of how the anti-windup problem can be solved using various forms of compensator are given, together with some robustness considerations. Initially, the problem is posed for globally asymptotically stable plants, although this condition is later relaxed to allow the consideration of plants with exponentially unstable modes. Finally, the results are demonstrated on a multivariable aircraft model and the benchmark inverted pendulum. [ABSTRACT FROM AUTHOR]

This chapter proposes an anti-windup design for linear systems in the absence of precise knowledge of the actuator nonlinearity. The actuator input output characteristic is assumed to reside in a nonlinear sector bounded by two convex/concave piecewise linear curves. For the closed-loop system with a given anti-windup compensation gain matrix, stability properties, both in the absence and in the presence of external disturbances are characterized in terms of matrix inequalities. The estimation of the domain of attraction with a contractive invariant ellipsoid as well as the assessment of the disturbance tolerance and disturbance rejection properties of the closed-loop system are then formulated and solved as LMI optimization problems. The design of the anti-windup compensation gain matrix for a large domain of attraction and/or a large disturbance tolerance/rejection capability is then carried out by viewing the anti-windup compensation gain as an additional free parameter in these optimization problems. Numerical examples as well as an experimental setup are used to illustrate the proposed analysis and design approach and its effectiveness. [ABSTRACT FROM AUTHOR]

In the Advanced Tokamak (AT) operating mode of the DIII-D tokamak, an integrated multivariable controller takes into account highly coupled influences of plasma equilibrium shape, profile, and stability control. Time-scale separation in the system allows a multiloop design: the inner loop closed by the nominal vertical controller designed to control a linear exponentially unstable plant and the outer loop closed by the nominal shape controller designed to control a nonlinear stabilized plant. Due to actuator constraints, the nominal vertical controller fails to stabilize the vertical position of the plasma inside the tokamak when large or fast disturbances are present or when the references coming from the shape controller change suddenly. Anti-windup synthesis is proposed to find a nonlinear modification of the nominal vertical controller that prevents vertical instability and undesirable oscillations but leaves the inner loop unmodified when there is no input saturation. [ABSTRACT FROM AUTHOR]

In nearly all control applications, the output signal of the compensator cannot be transferred to the system with unlimited amplitudes and arbitrarily fast. Such limitations in magnitude and rate can lead to performance degradations and they may even cause an unstable behaviour of the closed loop. These undesired effects of actuator saturation are called windup. The problem most intensively investigated is that of amplitude saturation. The prevention of windup effects due to input magnitude restrictions is the subject of a great variety of papers. The references in [3], [8] and [26] give a good overview of the existing literature. [ABSTRACT FROM AUTHOR]

In human walking, the swing leg moves backward just prior to ground contact, i.e. the relative angle between the thighs is decreasing. We hypothesize that this swing leg retraction may have a positive effect on gait stability, because similar effects have been reported in passive dynamic walking models, in running models, and in robot juggling. For this study, we use a simple inverted pendulum model for the stance leg. The swing leg is assumed to accurately follow a time-based trajectory. The model walks down a shallow slope for energy input which is balanced by the impact losses at heel strike. With this model we show that a mild retraction speed indeed improves stability, while gaits without a retraction phase (the swing leg keeps moving forward) are consistently unstable. By walking with shorter steps or on a steeper slope, the range of stable retraction speeds increases, suggesting a better robustness. An optimization of the swing leg trajectory of a more realistic model also consistently comes up with a retraction phase, and indeed our prototype demonstrates a retraction phase as well. The conclusions of this paper are twofold; (1) use a mild swing leg retraction speed for better stability, and (2) walking faster is easier. [ABSTRACT FROM AUTHOR]

Walking, running or jumping are special cases of articulated motions which rely heavily on contact forces for their accomplishment. This central role of the contact forces is widely recognized now, but it is rarely connected to the structure of the dynamics of articulated motion. Indeed, this dynamics is generally considered as a complex nonlinear black-box without any specific structure, or its structure is only partly uncovered. We propose here to precise this structure and show in details how it shapes the movements that an articulated system might realize. Some propositions are made then to improve the design of control laws for walking, running, jumping or free-floating motions. [ABSTRACT FROM AUTHOR]

Mechanical properties of complex biological systems are non-linear, e.g. the force-velocity-length relation of muscles, activation dynamics, and the geometric arrangement of antagonistic pair of muscles. The control of such systems is a highly demanding task. Therefore, the question arises whether these mechanical properties of a muscle-skeletal system itself are able to support or guarantee for the stability of a desired movement, indicating self-stability. Self-stability of single joint biological systems were studied based on eigenvalues of the equation of motions and the basins of attraction were analysed using Lyapunov functions. In general, we found selfstability in single muscle contractions (e.g. frog, rat, cui), in human arm and leg movements, the human spine and even in the co-ordination of complex movements such as tennis or basketball. It seems that self-stability may be a general design criterion not only for the mechanical properties of biological systems but also for motor control. [ABSTRACT FROM AUTHOR]

Stability of biped walking is an important characteristic of legged locomotion. Whereas clinical investigations often relate increased variability to decreased stability, there are only few studies examining stability aspects directly. On the other hand, various papers from the field of robotics are dedicated to the question: how can the stability of legged locomotor systems be quantified? Particularly, when it comes to realizing fast motions in robots, the question of maintaining dynamic stability is of utmost importance. The current paper presents a theoretical comparison of several measures for dynamic stability — namely Floquet multipliers and Local Divergence Exponents (LDE). The sensitivity of these parameters to changes in speed of human treadmill locomotion is investigated. Experimental results show that two different types of stability with respect to speed dependence seem to exist. Short term LDE and Floquet multipliers consider the stability over a period of one stride, which seems to be optimal at intermediate walking speeds. Long term LDE quantify stability of movement trajectories over multiple strides. This type of stability decreases with speed and may be one reason for changing gaits from walking to running at a certain speed value. [ABSTRACT FROM AUTHOR]

It has long been the dream to build robots which could walk and run with ease. To date, the stance phase of walking robots has been characterized by the use of either straight, rigid legs, as is the case of passive walkers, or by the use of articulated, kinematically-driven legs. In contrast, the design of most hopping or running robots is based on compliant legs which exhibit quite natural behavior during locomotion. [ABSTRACT FROM AUTHOR]

The spring loaded inverted pendulum model (SLIP) has been shown to accurately model sagittal plane locomotion for a variety of legged animals. Tuned appropriately, the model exhibits passively stable periodic gaits using either fixed leg touchdown angle or swing-leg retraction protocols. In this work, we investigate the relevance of the model in insect locomotion and develop a simple feedback control law to enlarge the basin of stability and produce stable periodic gaits for both the point mass and rigid body models. Control is applied once per stance phase through appropriate choice of the leg touchdown angle. The control law is unique in that stabilization is achieved solely through direct observation of the leg angle and body orientation, rather than through feedback of system positions, velocities, and orientation. [ABSTRACT FROM AUTHOR]

The paper considers the use of sum of squares techniques in nonlinear model predictive control. To be more precise, sum of squares techniques are used to solve at each sampling instant a finite horizon optimal control problem which arises in nonlinear model predictive control for discrete time polynomial systems. The combination of nonlinear model predictive control and sum of squares techniques is motivated by the successful application of semidefinite programming in linear model predictive control. The advantages and disadvantages of applying sum of squares techniques to nonlinear model predictive control are illustrated on a small example. [ABSTRACT FROM AUTHOR]

We present velocity-based stability margins for fast bipedal walking that are sufficient conditions for stability, allow comparison between different walking algorithms, are measurable and computable, and are meaningful. While not completely necessary conditions, they are tighter necessary conditions than several previously proposed stability margins. The stability margins we present take into consideration a biped's Center of Mass position and velocity, the reachable region of its swing leg, the time required to swing its swing leg, and the amount of internal angular momentum available for capturing balance. They predict the opportunity for the biped to place its swing leg in such a way that it can continue walking without falling down. We present methods for estimating these stability margins by using simple models of walking such as an inverted pendulum model and the Linear Inverted Pendulum model. We show that by considering the Center of Mass location with respect to the Center of Pressure on the foot, these estimates are easily computable. Finally, we show through simulation experiments on a 12 degree-of-freedom distributed-mass lower-body biped that these estimates are useful for analyzing and controlling bipedal walking. [ABSTRACT FROM AUTHOR]

This paper develops a class of bipedal running controllers based on the hybrid zero dynamics (HZD) framework and discusses related experiments conducted in September 2004 in Grenoble, France. In these experiments, RABBIT, a five-link, four-actuator, planar bipedal robot, executed six consecutive running steps. The observed gait was remarkably human-like, having long stride lengths (approx. 50 cm or 36% of body length), flight phases of significant duration (approx. 100 ms or 25% of step duration), an upright posture, and an average forward rate of 0.6 m/s. A video is available at [7, 17]. In the time allotted for experiments, stability of the gait could not be validated. To put the results into context, background information on hybrid robot modeling, control philosophy, and gait optimization techniques accompany final experimental observations. An additional discussion about some unmodeled dynamic and geometric effects that contributed to implementation difficulties is given. [ABSTRACT FROM AUTHOR]

For fast motions in biomechanics and robotics, stability and robustness against perturbations are critical issues. The faster a motion the more important it is to exploit the system's natural stability properties for control. The stability of a periodic motion can be measured in terms of the spectral radius of the monodromy matrix. We optimize this stability criterion for a given robot topology, using special purpose optimization methods and leaving the model parameters, actuator inputs, trajectory start values and cycle time free to be determined by the optimization. This approach allows us to create simulations of robots that can move stably without any feedback. In order to analyze the robustness of a resulting periodic motion, we propose two methods, the first of which relies on forward simulations using perturbed start data and parameters while the second is based on the pseudospectra of the matrix. As a new example for a fast open-loop stable motion that has been produced by stability optimization, we present a biped gymnastics robot performing repetitive flip-flops (i.e. back handsprings). A similar model has previously been shown capable of performing open-loop stable running motions and repetitive somersaults. [ABSTRACT FROM AUTHOR]

This paper deals with a methodology to design optimal reference trajectories for walking gaits. This methodology consists of two steps: (i) design a parameterized family of motions, and (ii) determine the optimal parameters giving the motion that minimizes a criterion and satisfies some constraints within this family. This approach is applied to a five link biped, the prototype Rabbit. It has point feet and four actuators which are located in each knee and haunch. Rabbit is underactuated in single support since it has no actuated feet and is overactuated in double support. To take into account this under-actuation, a characteristic of the family of motions considered is that the four actuated joints are prescribed as polynomials in function of the absolute orientation of the stance ankle. There is no impact. The chosen criterion is the integral of the square of torques. Different technological and physical constraints are taken into account to obtain a walking motion. Optimal process is solved considering an order of treatment of constraints, according to their importance on the feasibility of the walking gait. Numerical simulations of walking gaits are presented to illustrate this methodology. [ABSTRACT FROM AUTHOR]

In the next few years considerable effort will be expended to make humanoid robots that can do true dynamic walking, or even running. One may numerically compute a desired gait, e.g. one that has been optimized to be asymptotically stable without feedback. One would normally give the gait as commands to the controllers for the robot joints. However, control system outputs generally differ from the command given, and the faster the command changes with time, the more deviation there is. Iterative learning control (ILC) and repetitive control (RC) aim to fix this problem in situations where a command is repeating or periodic. Since gaits are periodic motions, it is natural to ask whether ILC/RC can be of use in implementing gaits in hardware. These control concepts are no substitutes for feedback control but work in combination with them by adjusting the commands to the feedback controllers from a higher level perspective. It is shown that the gait problem does not precisely fit either the ILC or the RC problem statements. Gait problems are necessarily divided into phases defined by foot strike times, and furthermore the durations of the phases are not the same from cycle to cycle during the learning process. Several methods are suggested to address these issues, and four repetitive control laws are studied numerically. The laws that include both position and velocity error in the updates are seen to be the most effective. It appears that with appropriate refinement, such generalized RC laws could be very helpful in getting hardware to execute desired gaits. [ABSTRACT FROM AUTHOR]

Dynamic walking with two-legged robots is still an unsolved problem of todays robotics research. Beside finding mathematical models for the walking process, suitable mechanical designs and control methods must be found. This paper presents concepts for the latter two points. As biological walking makes use of the elastic properties of e.g. tendons and muscles, a joint design using a pneumatic rotational spring with adjustable stiffness is proposed. Equations to model the spring's dynamics as well as the supporting sensor systems and electronics are presented. For controlling the robot a behaviour-based approach is suggested. [ABSTRACT FROM AUTHOR]

The Lateral Leg Spring model (LLS) was developed by Schmitt and Holmes to model the horizontal-plane dynamics of a running cockroach. The model captures several salient features of real insect locomotion, and demonstrates that horizontal plane locomotion can be passively stabilized by a well-tuned mechanical system, thus requiring minimal neural reflexes. We propose two enhancements to the LLS model. First, we derive the dynamical equations for a more flexible placement of the center of pressure (COP), which enables the model to capture the phase relationship between the body orientation and center-of-mass (COM) heading in a simpler manner than previously possible. Second, we propose a reduced LLS "plant model" and biologically inspired control law that enables the model to follow along a virtual wall, much like antenna-based wall following in cockroaches. [ABSTRACT FROM AUTHOR]

Movements of humans are achieved by muscle contractions. Humans are able to perform coordinated movements even in the presence of perturbations from the environment or of the muscles themselves. But which properties of the muscles and the geometry of the joints are responsible for the stability? Does the stability depend on the joint angle? How large are the perturbations, the muscle-skeletal system can cope with before reflexes or controls by the brain are necessary? To answer these questions, we will derive a mathematical model of the muscle-skeletal system without reflexes. We present different mathematical methods to analyze these systems with respect to the stability of movements and thus provide the mathematical tools to answer the above questions. This paper is a companion paper to [13] where the biological applications of the mathematical methods presented in this paper are discussed in more detail. [ABSTRACT FROM AUTHOR]

The objective of this study is to analyze the stability of two control strategies for a planar biped robot. The unexpected rotation of the supporting foot is avoided via the control of the center of pressure or CoP. For the simultaneous control of the joints and of the CoP, the system is under-actuated in the sense that the number of inputs is less than the number of outputs. Thus a control strategy developed for planar robot without actuated ankles can be used in this context. The control law is defined in such a way that only the geometric evolution of the biped configuration is controlled, but not the temporal evolution. The temporal evolution during the geometric tracking is completely defined and can be analyzed through the study of a model with one degree of freedom. Simple conditions, which guarantee the existence of a cyclic motion and the convergence toward this motion, are deduced. These results are illustrated with some simulation results. In the first control strategy, the position of the CoP is tracked precisely, in the second one, only the limits on the CoP position are used to speed-up the convergence to the cyclic motion. [ABSTRACT FROM AUTHOR]

This paper first introduces a multi-locomotion robot with high mobility and then proposes Passive Dynamic Autonomous Control (PDAC) for the comprehensive control method of multiple types of locomotion. PDAC is the method to take advantage of the robot inherent dynamics and to realize natural dynamic motion. We apply PDAC to a biped walk control. On the assumption that the sagittal and lateral motion can be separated and controlled individually, each motion is designed based on the given desired step-length and period. In order to stabilize walking, the landing position control according to the status is designed. In addition, a coupling method between these motions, which makes the period of each motion identical, is proposed. Finally, we show that the multi-locomotion robot realizes the 3-dimensional dynamic walking using the PDAC control. [ABSTRACT FROM AUTHOR]

In this overview paper, we first survey numerical approaches to solve nonlinear optimal control problems, and second, we present our most recent algorithmic developments for real-time optimization in nonlinear model predictive control. [ABSTRACT FROM AUTHOR]

A dynamic model of running-the spring-loaded inverted pendulum (SLIP)-has proven effective in describing the force patterns found in a wide variety of animals and in designing and constructing a number of terrestrial running robots. Climbing or vertical locomotion has, on the other hand, lacked such a simple and powerful model. Climbing robots to date have all been quasi-static in their operation. This paper introduces a one degree of freedom model of a climbing robot used to investigate the power constraints involved with climbing in a dynamic manner. Particular attention is paid to understanding how springs and body dynamics can be exploited to help relieve a limited power/weight ratio and achieve dynamic running and climbing. [ABSTRACT FROM AUTHOR]

An important technique for computing motions for robot systems is to conduct a numerical search for a trajectory that minimizes a physical criteria like energy, control effort, jerk, or time. In this paper, we provide example solutions of these types of optimal control problems, and develop a framework to solve these problems reliably. Our approach uses an efficient solver for both inverse and forward dynamics along with the sensitivity of these quantities used to compute gradients, and a reliable optimal control solver. We give an overview of our algorithms for these elements in this paper. The optimal control solver has been the primary focus of our recent work. This algorithm creates optimal motions in a numerically stable and efficient manner. Similar to sequential quadratic programming for solving finite-dimensional optimization problems, our approach solves the infinite-dimensional problem using a sequence of linear-quadratic optimal control subproblems. Each subproblem is solved efficiently and reliably using the Riccati differential equation. [ABSTRACT FROM AUTHOR]