This paper establishes a new framework for repetitive control of uncertain robot manipulators via operator-theoretic robust stabilization. After applying the inverse dynamics approach to robot manipulators, by which the relevant nonlinear input/output behavior is converted to a linear time-invariant (LTI) equation, we take the repetitive control approach. Even though such a repetitive controller is known to achieve high performances for periodic reference inputs, it is quite difficult to derive the stability analysis for the resulting closed-loop systems in a rigorous fashion. To solve this difficulty, we construct an operator-theoretic approach to the repetitive control treatment, and show that the closed-loop systems are exponentially stable if and only if the spectrum radius of the relevant monodromy operator is less than 1. Based on the necessary and sufficient condition, we develop a guideline to take the relevant control parameters. Finally, some experiment results are given to demonstrate the overall arguments developed in this paper.

This paper develops a controller synthesis method for ensuring an admissible bound of external forces on biped robots in a desired level. We first introduce the authors’ preceding results on the norm-based stability criterion for a biped walking constructed on its linear inverted pendulum model (LIPM). More precisely, an induced norm can be taken to formulate the fact that the balance for a biped robot is achieved if its zero moment point (ZMP) always stays in the supporting region at each step. Based on this norm-based criterion, we aim at making the maximum energy of external forces admissible for balancing the biped robot be a pregiven desired bound γ(> 0). To achieve this objective, a robust controller is designed through the linear matrix inequality (LMI)-based approach. More importantly, a necessary and sufficient condition for the existence of a robust controller leading to the desired bound is characterized by some LMI conditions. The effectiveness of the overall arguments is validated through some comparative simulation results of a biped walking robot with external forces.

This paper proposes a new framework for the computed torque method (CTM) of robot manipulators without velocity measurements. We first introduce the Luenberger-observer-based CTM with only position measurements. We then clarify that the external disturbance affects not only the tracking performances with respect to the plant but also the estimation accuracies relevant to the state observer. To address this problem, we establish a new architecture for the so-called dual-loop control scheme, by which both the tracking performances and estimation accuracies can be simultaneously improved, in contrast to its existing structure. A guideline for taking control parameters corresponding to the proposed control structure is also provided with respect to the stabilization of the overall closed-loop systems. Finally, simulation and experimental results are provided to demonstrate the validity and practical feasibility of the developed structure.

This article proposes a new stability criterion for biped walking systems on the linear inverted pendulum model, in which the dynamic relationship between the center of mass (CoM) and the zero moment point (ZMP) is dealt with. More precisely, based on the fact that a biped walking robot is stable if its ZMP is always located in the supporting region, we consider whether the ZMP error between its reference and real values stays inside a certain area to guarantee the stability condition. To this end, a norm-based new stability criterion is introduced, in which the temporal supremum of the ZMP error is concerned. Regarding the applicability of the stability criterion, we propose two control approaches to biped walking systems with the consideration of the norm-based stability criterion. In other words, full-state and observer-based feedback control approaches are analyzed in this article, and the initial CoM conditions with respect to the stability criterion are obtained for the two control approaches. We call the sets derived by such conditions the stability regions. Toward a more practical significance, we also deal with the effect of unknown disturbances and sensor noises on the stability of biped walking systems. More importantly, even though computing stability regions intrinsically involves an infinite number of linear inequalities, all the stability regions are shown to be explicitly obtained through only finite numbers of computations in this article. Finally, some simulation results are provided to demonstrate the validity as well as the practical applicability of the developed computation methods.

Motivated by the fact that a number of present state estimations require some presumed conditions and could not lead to a desired accuracy when they are applied to real systems, this paper is concerned with providing a new framework for the state estimation. We first introduce some existing methods of state estimations and describe their weaknesses for unknown bounded persistent elements. Aiming at taking into account more practical situations of real systems, which cannot be treated by the existing methods, this paper provides a new state estimation method by using the l1 optimal control theory. More precisely, the new state estimation method called the l1optimal state estimation considers unknown bounded persistent elements such as external disturbances and measurement noises, which often occur in the systems and make the estimation difficult. The problem of minimizing the effect of the bounded persistent elements on the corresponding state estimation error could be mathematically formulated by using the arguments on l1optimal state estimation introduced in this paper. Finally, the effectiveness of the l1optimal state estimation is demonstrated through some simulation results associated with the center of mass (CoM) estimation for a bipedal robot on its linear inverted pendulum model (LIPM).

This paper addresses reactive generation of step time and location of biped robots for balance recovery against a severe push. Key idea is to reformulate the balance recovery problem into a tracking problem for "hybrid" inverted pendulum model of the biped, where taking a new step implicitly yields a discrete jump of the tracking error. This interpretation offers a Lyapunov-based approach to reactive step generation, which is possibly more intuitive and easier to analyze than large-scaled or nonlinear optimization-based approaches. With the continuous error dynamics for the divergent component of motion (DCM), our strategy for step generation is to decrease the "post-step" Lyapunov level for DCM error at each walking cycle, until it eventually becomes smaller than a threshold so that no more footstep needs to be adjusted. We show that implementation of this idea while obeying physical constraints can be done by employing a hybrid tracking controller (together with a reference model) as our reactive step generator, consisting of a simple DCM-based continuous controller and a small-sized quadratic programming-based discrete controller. The validity of the proposed scheme is verified by simulation results.

The aim of this paper is to construct a theoretical framework for stability analysis relevant to standing balance of humanoids on top of the linear inverted pendulum model, in which their dynamics between the center of mass (CoM) and the zero moment point (ZMP) is dealt with. Based on the well-known sufficient condition that the contact between the ground and the support leg is stable if the corresponding ZMP is always inside the supporting region, this paper aims at characterizing three types of the associated stability regions. More precisely, assuming no external force disturbances affecting the motion of the humanoids, the stability region of the initial CoM position and velocity values can be explicitly computed by solving a finite number of linear inequalities. The stability regions of time-invariant force disturbances such as impulsive force and constant force disturbances are also dealt with in this paper, where the former is exactly obtained through a finite number of linear inequalities while the latter is approximately derived by using an idea of truncation. Furthermore, time-varying force disturbances of finite energy and finite amplitude are concerned with, and their maximum admissible${l} _{2}$and${l} _{\infty }$norms are computed in this paper, where the former can be exactly obtained by solving the discrete-time Lyapunov equation while the latter is approximately derived through an idea of truncation. It is further shown for both the truncation ideas that the approximately obtained stability regions converge to the exact stability regions with an exponential order of${N}$, where${N}$is the truncation parameter. Finally, the effectiveness of the computation methods proposed in this paper is demonstrated through some simulation results.

In this paper, we address the problem of generating a trajectory of the zero-moment point (ZMP) and the rate of angular momentum for a bipedal robot in the sagittal plane, with which the balance of the robot is recovered from external push. Unlike most previous works that adjusted a pre-designed ZMP or solved (possibly too heavy) nonlinear optimization problems, our main purpose is to develop a fully reactive step generator in the sense that (a) no pre-calculation of nominal trajectory is required, and (b) the algorithm is simple enough to operate in real time, only by utilizing the current state of the robot. For the design, it is seen by reinterpreting the centroidal dynamics in the hybrid model framework that the balance recovery problem can be recast as the problem of stabilizing a hybrid-type (linear) inverted pendulum model. On the basis of the concept of the divergent component of motion, a simple hybrid control law is then constructed to stabilize the hybrid system, which serves as a step generator that automatically determines where and when to step. This paper briefly sketches a mathematical proof on the performance of the proposed generator from a control-theoretic perspective, which is also supported by simulation results.

This paper revisits computed torque method for robot manipulators and aims at developing its new framework based on the$L_{1}$robustness, in which the$L_{\infty}$norm together with its induced norm is employed to characterize model uncertainties and a performance measure. More precisely, we consider the$L_{1}$robust stability and performance for a given robot manipulator with a computed torque controller. We first show that the modelling errors in the computed torque method can be divided into an exogenous disturbance and a multiplicative model uncertainty, which are bounded in terms of the$L_{\infty}$norm and its induced norm, respectively. It is next shown that the robot manipulator with the computed torque controller can be equivalently represented by an interconnection of a continuous-time linear time-invariant (LTI) nominal plant and a stabilizing controller together with the$L_{\infty}$-induced norm bounded model uncertainty. Based on the interconnected representation, the$L_{1}$robust stability condition and an upper bound of the$L_{1}$performance against the exogenous disturbance with respect to all model uncertainties in a class of a bounded$L_{\infty}$-induced norm are dealt with by using the small-gain theorem. Finally, the effectiveness of the theoretical results is demonstrated through some experiment results.

Based on the liner inverted pendulum model (LIPM) for the dynamics of biped humanoid robots, an analytical method for computing stability regions relevant to standing balance of the biped humanoid robots is introduced in this paper. More precisely, two types of the stability regions are discussed in this paper with the consideration of that the zero moment point (ZMP) should be located in the supporting region to guarantee stable standing of the biped humanoid robots. First, assuming no external disturbances affecting the motion of the biped humanoid robots, the set of the initial values of the center of mass (CoM) position and velocity with which the location of the ZMP is limited to be inside the supporting region can be explicitly obtained by solving a finite number of linear inequalities. Second, two admissible sets of external force disturbances (impulse and finite energy) with which the ZMP does not deviate from the supporting region are characterized by solving finite number of linear inequalities or the discrete-time Lyapunov equation, respectively. The validity and effectiveness of the analytical method proposed in this paper are verified through a simulation result.

Feedback controller for robotic bipedal walking models can be multi-layered, consisting of low-level continuous-time controller and high-level event-based controller. Stimulated by the success in our preceding study that demonstrates the validity of the l∞-induced norm as an adequate performance measure, we suggest a systematic methodology to design optimal event-based feedback controller for bipedal models walking on rough terrains. More precisely, we first assume that the system is already equipped with a low-level continuous-time feedback controller, capable of stable flat-ground-walking, and then formulate the design problem of the high-level event-based feedback control as the l1-optimal control problem for discrete-time linear systems defined on the linearized Poincarè map. In order to validate the proposed methodology, nonlinear dynamic simulations are conducted with a simple biped model walking on rough terrain. The terrain slope randomly varies at each footstep while the magnitude of slope variation is bounded by some maximum value. Simulation results indicate that the optimal system equipped with the proposed controller can successfully overcome a rough terrain on which the original system could not walk and fall.

In this paper, we consider the L1optimal proportional-derivative (PD) controller synthesis by which the L∞gain of trajectory tracking systems can be reduced. The notion of input-to-state stability (ISS), which has been equipped with the L2norm of a vector-valued signal, plays important roles in evaluating the effect of disturbances on the system states. In connection with this, we first redefine the conventional ISS with the L∞norm to deal with bounded persistent disturbances because the disturbances should be mathematically regarded as elements of the L∞space. A tractable model for trajectory tracking control of robot systems is then given by using ideas of extended disturbance and composite error. We next introduce a design method of the L1optimal PD controller for such a model, by which the trajectory tracking system satisfies the redefined ISS and the L∞gain is less than a performance level γ. Finally, we examine the effectiveness of the design method through experimental results for a typical robot manipulator.

Despite successful demonstrations of outdoor walking by a few biped robots, performance measure for such robots has not been formally defined yet. The performance measure should be adequately defined by which one can evaluate how well the robot keeps from falling in the presence of disturbance. If such performance measure is suitably determined, designing a sort of optimal controller for stable outside walking would be possible. This paper firstly suggests to adopt the l∞-induced norm defined on the linearized Poincarè map as a novel performance measure for biped robots, in the presence of bounded persistent disturbance. In order to validate the measure, a nonlinear dynamic simulation is conducted by extending an existing simple model to walk on rough terrain, of which height variance is bounded by some maximum value. The measure exploited for the model is compared with the numerical result obtained from nonlinear dynamic simulation. Finally, an example of application of this measure is provided, which successfully predicts whether the system could overcome upcoming terrain roughness.