Determination of the initial conditions by solving boundary value problem method for period-one walking of a passive biped walking robots

Robotica ◽  
2015 ◽  
Vol 35 (1) ◽  
pp. 166-188 ◽  
Author(s):  
Masoumeh Safartoobi ◽  
Morteza Dardel ◽  
Mohammad Hassan Ghasemi ◽  
Hamidreza Mohammadi Daniali
Keyword(s):  
Boundary Value Problem ◽  
Limit Cycles ◽  
Initial Conditions ◽  
Boundary Value ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Basin Of Attraction ◽  
Initial Guess ◽  
Walking Robots ◽  
Nonlinear Equations Of Motion

SUMMARYWith regard to the small basin of attraction of the passive limit cycles, it is important to start from a proper initial condition for stable walking. The present study investigates the passive dynamic behaviors of two-dimensional bipedal walkers of a compass gait model with different foot shapes. In order to find proper initial conditions for stable and unstable period-one gait limit cycles, a method based on solving the nonlinear equations of motion is presented as a boundary value problem (BVP). An initial guess is required to solve the related BVP that is obtained by solving an initial value problem (IVP). For parametric analysis purposes, a continuation method is applied. Simulation results reveal two, period-one gait cycles and the effects of parameters variation for all models.

Basin of Attraction of the Simplest Walking Model

2001 ◽  
Author(s):  
A. L. Schwab ◽  
M. Wisse
Keyword(s):  
Equations Of Motion ◽  
Basin Of Attraction ◽  
Mapping Method ◽  
Walking Robots ◽  
Natural Energy ◽  
Walking Motion ◽  
The Stability ◽  
The Basin Of Attraction ◽  
General Method ◽  
Small Disturbances

Abstract Passive dynamic walking is an important development for walking robots, supplying natural, energy-efficient motions. In practice, the cyclic gait of passive dynamic prototypes appears to be stable, only for small disturbances. Therefore, in this paper we research the basin of attraction of the cyclic walking motion for the simplest walking model. Furthermore, we present a general method for deriving the equations of motion and impact equations for the analysis of multibody systems, as in walking models. Application of the cell mapping method shows the basin of attraction to be a small, thin area. It is shown that the basin of attraction is not directly related to the stability of the cyclic motion.

Landing Posture Control of a Generalized Twin-Body System Via Methods of Input-Output Linearization and Computed Torque

2007 ◽  
Author(s):  
Yi-Ling Yang ◽  
Paul C.-P. Chao ◽  
Cheng-Kuo Sung
Keyword(s):  
Structural Damage ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Posture Control ◽  
Body System ◽  
Input Output ◽  
Bipedal Robot ◽  
Nonlinear Equations Of Motion ◽  
Computed Torque ◽  
Input Output Linearization

This study is dedicated to achieve landing posture control of a generalized twin-body system using the methods of input-output linearization and computed torque. The twin-body system is a simplified model of bipedal robot, and the success in landing posture control would prevent structural damage. To the end, the dynamic equations are built based on Newton-Euler formulation. The technique of input-output linearization is next applied to the original nonlinear equations of motion, which is followed by adopting the method of computed torque to achieve desired landing postures. While designing the controller, system singularities are circumvented by choosing controllable set of initial conditions and stable landing postures. There are two uncontrollable postures that are immovable under input torques or/and the coupling centripetal and Coriolis forces. Finally, simulation results show that the designed controller is capable of performing desired landing posture control.

Method for calculating the perturbed trajectoryof a two-impulse flight between the halo orbit in the vicinity of the L2 point of the Sun — Earth systemand the near-lunar orbit

2020 ◽  
Author(s):  
Zhou Rui
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Continuation Method ◽  
Initial Approximation ◽  
Halo Orbit ◽  
Lunar Orbit ◽  
The Sun ◽  
Zonal Harmonic ◽  
The Earth ◽  
Parameter Continuation

The paper introduces a new method for solving the problem of calculating the perturbed trajectory of a two-impulse flight between a near-lunar orbit and a halo orbit in the vicinity of the L2 point of the Sun — Earth system. Unlike traditional numerical methods, this method has better convergence. Accelerations from the gravitational forces of the Earth, the Moon and the Sun as point masses and acceleration from the second zonal harmonic of the geopotential are taken into account at all sections of the trajectory. The calculation of the flight path is reduced to solving a two-point boundary value problem for a system of ordinary differential equations. The developed method is based on the parameter continuation method and does not require the choice of an initial approximation for solving the boundary value problem. The last section of the paper provides examples and results of the analysis based on this method.

Cooperative near-space interceptor mid-course guidance law with terminal handover constraints

2018 ◽  
Vol 233 (6) ◽  
pp. 1960-1976 ◽  
Author(s):  
Haoqiang Zhang ◽  
Shengjing Tang ◽  
Jie Guo
Keyword(s):  
Boundary Value Problem ◽  
Finite Time ◽  
Sliding Mode ◽  
Initial Conditions ◽  
Boundary Value ◽  
Programming Method ◽  
Line Of Sight ◽  
Point Boundary ◽  
Guidance Law ◽  
Bounded Perturbation

A cooperative problem in mid-course guidance phase is addressed in this paper. For providing suitable initial conditions of successful terminal salvo attack, a novel finite-time cooperative mid-course guidance law with terminal handover constraints is proposed. Firstly, a three-dimensional guidance model of mid-course is decoupled in a planar line-of-sight frame as a two-point boundary value problem. The terminal handover constraints which can guarantee an ideal zero effort terminal engagement are proposed and analyzed. Secondly, the design of the cooperative mid-course guidance law is separated into two stages. The acceleration commands along the line-of-sight direction are developed based on the finite-time average-consensus protocol and super-twisting algorithm in the first stage. In the second stage, the model predictive static programming method is adopted to solve the two-point boundary value problem in line-of-sight frame with a known final time from first stage. Furthermore, sliding mode control theory is used in combination with model predictive static programming method to satisfy terminal handover constraints with bounded perturbation. Finally, numerical simulations of two four-interceptor cooperative scenario are carried out to verify the validity of the proposed cooperative guidance law. The simulation results reflect that all the four interceptors can reach their own predictive interception points with specific approach angles simultaneously.

scholarly journals INVESTIGATION OF HYPERELASTIC SOFT SHELLS NONSTATIONARY DYNAMICS PROBLEMS SOLUTION FEATURES

2021 ◽  
Vol 83 (2) ◽  
pp. 151-159
Author(s):  
E.A. Korovaytseva
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value ◽  
Equations Of Motion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Difference Approximation ◽  
Analytical Studies ◽  
Initial Boundary Value

Results of hyperelastic soft shells nonlinear axisymmetric dynamic deforming problems solution algorithm testing are represented in the work. Equations of motion are given in vector-matrix form. For the nonlinear initial-boundary value problem solution an algorithm which lies in reduction of the system of partial differential equations of motion to the system of ordinary differential equations with the help of lines method is developed. At this finite-difference approximation of partial time derivatives is used. The system of ordinary differential equations obtained as a result of this approximation is solved using parameter differentiation method at each time step. The algorithm testing results are represented for the case of pressure uniformly distributed along the meridian of the shell and linearly increasing in time. Three types of elastic potential characterizing shell material are considered: Neo-hookean, Mooney – Rivlin and Yeoh. Features of numerical realization of the algorithm used are pointed out. These features are connected both with the properties of soft shells deforming equations system and with the features of the algorithm itself. The results are compared with analytical solution of the problem considered. Solution behavior at critical pressure value is investigated. Formulations and conclusions given in analytical studies of the problem are clarified. Applicability of the used algorithm to solution of the problems of soft shells dynamic deforming in the range of displacements several times greater than initial dimensions of the shell and deformations much greater than unity is shown. The numerical solution of the initial boundary value problem of nonstationary dynamic deformation of the soft shell is obtained without assumptions about the limitation of displacements and deformations. The results of the calculations are in good agreement with the results of analytical studies of the test problem.

On the existence and uniqueness of a positive solution to a boundary value problem for a nonlinear ordinary differential equation of even order

2021 ◽  
pp. 341-347
Author(s):  
Gusen E. Abduragimov ◽  
Patimat E. Abduragimova ◽  
Madina M. Kuramagomedova
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Ordinary Differential Equation ◽  
Even Order

In the article, we consider a boundary value problem for a nonlinear ordinary differential equation of even order which, obviously, has a trivial solution. Sufficient conditions for the existence and uniqueness of a positive solution to this problem are obtained. With the help of linear transformations of T. Y. Na [T. Y. Na, Computational Methods in Engineering Boundary Value Problems, Acad. Press, NY, 1979, ch. 7], the boundary value problem is reduced to the Cauchy problem, the initial conditions of which make it possible to uniquely determine the transformation parameter. It is shown that the transformations of T. Y. Na uniquely determine the solution of the original problem. In addition, based on the proof of the uniqueness of a positive solution to the boundary value problem, a sufficiently effective non–iterative numerical algorithm for constructing such a solution is obtained. A corresponding example is given.

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