Tatsuya Kai

Theoretical Analysis for a Class of Rheonomous Affine Constraints on Configuration Manifolds—Part I: Fundamental Properties and Integrability/Nonintegrability Conditions

We analyze a class of rheonomous affine constraints defined on configuration manifolds from the viewpoint of integrability/nonintegrability. First, we give the definition of A-rheonomous affine constraints and introduce, geometric representation their. Some fundamental properties of the A-rheonomous affine constrains are also derived. We next define the rheonomous bracket and derive some necessary and sufficient conditions on the respective three cases: complete integrability, partial integrability, and complete nonintegrability for the A-rheonomous affine constrains. Then, we apply the integrability/nonintegrability conditions to some physical examples in order to confirm the effectiveness of our new results.

Theoretical Analysis for a Class of Rheonomous Affine Constraints on Configuration Manifolds—Part II: Foliation Structures and Integrating Algorithms

This paper investigates foliation structures of configuration manifolds and develops integrating algorithms for a class of constraints that contain the time variable, called A-rheonomous affine constrains. We first present some preliminaries on the A-rheonomous affine constrains. Next, theoretical analysis on foliation structures of configuration manifolds is done for the respective three cases where the A-rheonomous affine constrains are completely integrable, partially integrable, and completely nonintegrable. We then propose two types of integrating algorithms in order to calculate independent first integrals for completely integrable and partially integrable A-rheonomous affine constrains. Finally, a physical example is illustrated in order to verify the availability of our new results.

A Gait Generation Method for the Compass-type Biped Robot based on Discrete Mechanics

Abstract In this paper, we consider a new approach based on discrete mechanics to a gait generation problem for the compass-type biped robot. First, both continuous-time and discrete-time models of the compass-type biped robot are derived. We next formulate a discrete gait generation problem for the discrete compass-type biped robot as a finite dimensional nonlinear optimal control problem, and solve it by using the sequential quadratic programming. Then, we propose a transformation method of a discrete-time input into continuous one and apply it to gait generation for the continuous compass-type biped robot. Some simulations are also shown in order to verify the effectiveness of our approach.

A Discrete Mechanics Approach to Gait Generation on Periodically Unlevel Grounds for the Compass-type Biped Robot

This paper addresses a gait generation problem fornthe compass-type biped robot on periodically unlevel grounds.nWe first derive the continuous/discrete compass-type biped robotsn(CCBR/DCBR) via continuous/discrete mechanics, respectively.nNext, we formulate a optimal gait generation problem on periodicallynunlevel grounds for the DCBR as a finite dimensionalnnonlinear optimization problem, and show that a discrete controlninput can be obtained by solving the optimization problemnwith the sequential quadratic programming. Then, we developna transformation method from a discrete control input into ancontinuous zero-order hold input based on the discrete Lagranged’Alembertnprinciple. Finally, we show numerical simulations,nand it turns out that our new method can generate a stable gaitsnon a periodically unlevel ground for the CCBR.

Gait generation on periodically unlevel grounds for the compass-type biped robot via discrete mechanics

This paper addresses a gait generation problem for the compass-type biped robot on periodically unlevel grounds. We first derive the continuous/discrete compass-type biped robots (CCBR/DCBR) via continuous/discrete mechanics, respectively. Next, we formulate a optimal gait generation problem on periodically unlevel grounds for the DCBR as a finite dimensional nonlinear optimization problem, and show that a discrete control input can be obtained by solving the optimization problem with the sequential quadratic programming. Then, we develop a transformation method from a discrete control input into a continuous zero-order hold input based on discrete Lagrange-dAlembert principle. Finally, we show numerical simulations, and it turns out that our new method can generate a stable gaits on a periodically unlevel ground for the CCBR.

Limit Cycle Generation for Multi-Modal and 2-Dimensional Piecewise Affine Control Systems

This paper considers a limit cycle control problem of a multi-modal and 2-dimensional piecewise affine control system. Limit cycle control means a controller design method to generate a limit cycle for given piecewise affine control systems. First, we deal with a limit cycle synthesis problem and derive a new solution of the problem. In addition, theoretical analysis on the rotational direction and the period of a limit cycle is shown. Next, the limit cycle control problem for piecewise affine control system is formulated. Then, we obtain matching conditions such that the piecewise affine control system with the state feedback law corresponds to the reference system which generates a desired limit cycle. Finally, in order to indicate the effectiveness of the new method, a numerical simulation is illustrated. paper. The outline of this paper is as follows. We first consider a limit cycle synthesis problem and derive its new solution in Section II. Some theoretical properties are also shown. Then, in Section III, a formulation of limit cycle control problem is presented, and necessary and sufficient conditions for the problem, which are called matching conditions, are derived. Finally, a numerical simulation is shown in order to confirm the effectiveness of the new method in Section IV.

New hamiltonian formulation of rheonomous affine constraints based on the rheonomous bracket

This paper presents a Hamiltonian formulation technique for rheonomous affine constraints. We first explain some concepts on rheonomous affine constraints and introduce the rheonomous bracket. Next, a complete nonholonomicity condition for the rheonomous affine constraints is shown. Then, we derive the nonholonomic Hamiltonian system with rheonomous affine constraints (NHSRAC) via a transformation and model reduction for the expanded Hamiltonian system defined on the expanded phase space. After that, a passivity condition for the NHSRAC with the control input term and the output equation is investigated. Finally, we show a physical example in order to verify our new results.