Yoshiro Fukui

Asymptotic stabilization with locally semiconcave control Lyapunov functions on general manifolds

Abstract Asymptotic stabilization on noncontractible manifolds is a difficult control problem. If a configuration space is not a contractible manifold, we need to design a time-varying or discontinuous state feedback control for asymptotic stabilization at the desired equilibrium. For a system defined on Euclidean space, a discontinuous state feedback controller was proposed by Rifford with a semiconcave strict control Lyapunov function (CLF). However, it is difficult to apply Rifford’s controller to stabilization on general manifolds. In this paper, we restrict the assumption of semiconcavity of the CLF to the “local” one, and introduce the disassembled differential of locally semiconcave functions as a generalized derivative of nonsmooth functions. Further, we propose a Rifford–Sontag-type discontinuous static state feedback controller for asymptotic stabilization with the disassembled differential of the locally semiconcave practical CLF (LS-PCLF) by means of sample stability. The controller does not need to calculate limiting subderivative of the LS-PCLF. Moreover, we show that the LS-PCLF, obtained by the minimum projection method, has a special advantage with which one can easily design a controller in the case of the minimum projection method. Finally, we confirm the effectiveness of the proposed method through an example.

Multilayer minimum projection method for nonsmooth strict control Lyapunov function design

Asymptotic stabilization on noncontractible manifolds is known as a difficult control problem. To address this problem, we had proposed the minimum projection method to design nonsmooth control Lyapunov functions. This method, however, has some problems: difficult etale-surjection design, undesirable resulting control Lyapunov functions, etc. In this paper, we propose a new nonsmooth control Lyapunov function design method called the ‘Multilayer minimum projection method’ for nonsmooth control Lyapunov function design on general manifolds. The method considers many simple-structured smooth manifolds associated with the original manifold by etale mappings, and then a function on the original manifold is obtained by projecting control Lyapunov functions defined on the simple-structured manifolds onto the original manifold. In this paper, we prove that the resulting function by the proposed method is a nonsmooth control Lyapunov function on the original manifold. Moreover, we prove that if all control Lyapunov functions defined on simple-structured manifolds are strict, the control Lyapunov function on the original manifold is a strict control Lyapunov function. Finally, the effectiveness of the proposed method and the advantage over the conventional minimum projection method are confirmed by an example.

Automated Linear Function Submission-Based Double Auction as Bottom-up Real-Time Pricing in a Regional Prosumers’ Electricity Network

A linear function submission-based double-auction (LFS-DA) mechanism for a regional electricity network is proposed in this paper. Each agent in the network is equipped with a battery and a generator. Each agent simultaneously becomes a producer and consumer of electricity, i.e., a prosumer and trades electricity in the regional market at a variable price. In the LFS-DA, each agent uses linear demand and supply functions when they submit bids and asks to an auctioneer in the regional market.The LFS-DA can achieve an exact balance between electricity demand and supply for each time slot throughout the learning phase and was shown capable of solving the primal problem of maximizing the social welfare of the network without any central price setter, e.g., a utility or a large electricity company, in contrast with conventional real-time pricing (RTP). This paper presents a clarification of the relationship between the RTP algorithm derived on the basis of a dual decomposition framework and LFS-DA. Specifically, we proved that the changes in the price profile of the LFS-DA mechanism are equal to those achieved by the RTP mechanism derived from the dual decomposition framework except for a constant factor.

Genetics-Based Machine Learning Approach for Rule Acquisition in an AGV Transportation System

We propose an autonomous decentralized method for multiple AGV robots under uncertain delivery requests. Transportation route plans of AGV robots are expected to minimize the transportation time without collisions among the robots in the systems. In our proposed methods, each robot as an agent computes its transportation route by referring to the static path information, and it exchanges its route plan each other. Once collisions are detected, one of the two agents chosen by a negotiation rule modifies its route plan. The rule consists of a condition-part and an action-part, and one rule which matches to the conditions of two agents under negotiation is selected from a set of rules. The rules are generated and improved by a genetic based machine learning approach, where a set of rules is represented symbolically as an individual of genetic algorithms, and fitness of each individual is determined according to the total travel time of the AGVs and the adequacy of the condition-parts of the rules.

Multilayer Minimum Projection Method with Infinite Layers

Abstract Asymptotic stabilization on noncontractible manifolds is a difficult control problem. To address the problem, we had proposed the multilayer minimum projection method to design nonsmooth control Lyapunov functions. The method considers many simple-structured smooth manifolds associated with the original manifold by local diffeomorphisms. However, the case of infinitely many manifolds was not considered. Then, we consider infinitely many simple-structured smooth manifolds associated with the original manifold by local diffeomorphisms in the paper. First of all, we show that the function obtained by “infimum projection” of functions does not become a control Lyapunov function. Then, we suppose that the parameter space of mappings is a compact Hausdorff space. In this case, we prove the fact that the function obtained by the minimum projection becomes a control Lyapunov function. Finally, the advantage of consideration of the infinitely many manifolds is confirmed through an example.

Velocity field control with energy compensation toward therapeutic exercise

Human-machine systems, such as those used for rehabilitation, must be safe for human use when performing a given operational task. Passivity-based controllers such as the passive velocity field control method helps in realizing safe operation of human-machine systems. However, active behavior toward the external environment, including human bodies, is required to realize a given task. Such active behavior is difficult for passivity-based controllers. This study focuses on ensuring that a manipulator behaves passively toward an external force when the kinetic energy is greater than or equal to a given threshold and actively otherwise. We present a newly developed velocity field control method with an energy compensation mechanism. Theoretical analysis shows the properties of the proposed method: when the passivity toward external forces increases the kinetic energy of the augmented system beyond the required limit, the energy converges to a given range for safety rehabilitation without external forces, and the closed-loop system tracks a given desired velocity field without external forces. As a further theoretical discussion, we show the energy flow between the manipulator and human. Numerical simulations demonstrate that (1) the closed-loop system tracks a given desired velocity field, which describes a rehabilitation task, without external forces, (2) the proposed method inhibits the decrease in the kinetic energy of a closed-loop system, and (3) the closed-loop system behaves passively toward external forces such as an unexpected reaction of a human.

Desingularization by minimum projection method and its application to rigid body control

Stabilization on a non-contractible manifold is a difficult control problem. We have proposed the minimum projection method for asymptotic stabilization of nonlinear systems on manifolds. The minimum projection method requires the strict condition for a control Lyapunov function (CLF) on another manifold. In this paper, we relax the condition by desingularization of the function on the other manifold. We consider a rigid body control problem as an application of desingularization based control method. S3 is a double covering space of SO(3). Stabilization on SO(3) is a difficult problem; however, the strong CLF design on S3 is not difficult. Then, we design a strict CLF from the CLF on S3 by desingularization. Finally, we confirm the effectiveness of the desingularization based control method by computer simulation.

Finite-time PD control of robot manipulators with adaptive gravity compensation

We propose a new PD control using adaptive gravity compensation with a finite-time convergence rate for the position control problem in robot manipulators. The control is an extension of the conventional linear PD control with adaptive gravity compensation for robot manipulators. Theoretical analysis showed that the desired position is finite-time stable when the origin is locally asymptotically stable. Simulations with a two-link manipulator showed that the proposed method achieves higher convergence rate and accuracy than the conventional method.

Adaptive trajectory tracking control with finite-time stability for robot manipulators

We propose an adaptive control method with a finitetime convergence rate for the trajectory tracking control problem in robot manipulators. We extend Slotine and Lis conventional adaptive control method for robot manipulators and compare the convergence rates through simulation with one- and two-link manipulators. We also compare the convergence rates through experiments with a one-link manipulator. Both results show that the proposed method converges more quickly than Slotine and Lis method.

Space design for the multilayer minimum projection method in nonsmooth control Lyapunov function design

This paper considers a non-smooth control Lyapunov function (CLF) design problem for global asymptotic stabilization of a system defined on a manifold. This problem can be addressed by the multilayer minimum projection method with singular point assignment, which considers manifolds, called multilayered spaces, different from those where the control system is originally defined. This requires a multilayered space satisfying specific conditions. However, a design method for the required space has not previously been provided. In this paper, we propose a design method for a multilayered space guaranteed to satisfy the required conditions using the concept of a multigraph and apply it in a robot navigation example.