Determining feasibility of a set of nonlinear inequality constraints

1982 ◽  
pp. 137-148 ◽  
Author(s):  
Robert B. Schnabel
Keyword(s):  
Inequality Constraints ◽  
Nonlinear Inequality ◽  
Nonlinear Inequality Constraints

Optimum Path Planning for Mechanical Manipulators

1981 ◽  
Vol 103 (2) ◽  
pp. 142-151 ◽  
Author(s):  
J. Y. S. Luh ◽  
C. S. Lin
Keyword(s):  
Path Planning ◽  
Feasible Solution ◽  
Computing Time ◽  
Inequality Constraints ◽  
Programming Algorithm ◽  
Planning Problem ◽  
Linear Inequality Constraints ◽  
Nonlinear Inequality ◽  
Angular Velocities ◽  
Nonlinear Inequality Constraints

To assure a successful completion of an assigned task without interruption, such as the collision with fixtures, the hand of a mechanical manipulator often travels along a preplanned path. An advantage of requiring the path to be composed of straight-line segments in Cartesian coordinates is to provide a capability for controlled interaction with objects on a moving conveyor. This paper presents a method of obtaining a time schedule of velocities and accelerations along the path that the manipulator may adopt to obtain a minimum traveling time, under the constraints of composite Cartesian limit on linear and angular velocities and accelerations. Because of the involvement of a linear performance index and a large number of nonlinear inequality constraints, which are generated from physical limitations, the “method of approximate programming (MAP)” is applied. Depending on the initial choice of a feasible solution, the iterated feasible solution, however, does not converge to the optimum feasible point, but is often entrapped at some other point of the boundary of the constraint set. To overcome the obstacle, MAP is modified so that the feasible solution of each of the iterated linear programming problems is shifted to the boundaries corresponding to the original, linear inequality constraints. To reduce the computing time, a “direct approximate programming algorithm (DAPA)” is developed, implemented and shown to converge to optimum feasible solution for the path planning problem. Programs in FORTRAN language have been written for both the modified MAP and DAPA, and are illustrated by a numerical example for the purpose of comparison.

Optimal Path Planning of Redundant Cooperative Robots Under Equality and Inequality Constraints

2003 ◽  
Author(s):  
Ali Hosseini ◽  
Mehdi Keshmiri
Keyword(s):  
Path Planning ◽  
Optimization Problem ◽  
Search Algorithm ◽  
Minimum Principle ◽  
Optimal Path ◽  
Inequality Constraints ◽  
Optimal Path Planning ◽  
Nonlinear Inequality ◽  
Nonlinear Inequality Constraints ◽  
Cooperative Manipulators

Using kinematic resolution, the optimal path planning for two redundant cooperative manipulators carrying a solid object on a desired trajectory is studied. The optimization problem is first solved with no constraint. Consequently, the nonlinear inequality constraints, which model obstacles, are added to the problem. The formulation has been derived using Pontryagin Minimum Principle and results in a Two Point Boundary Value Problem (TPBVP). The problem is solved for a cooperative manipulator system consisting of two 3-DOF serial robots jointly carrying an object and the results are compared with those obtained from a search algorithm. Defining the obstacles in workspace as functions of joint space coordinates, the inequality constrained optimization problem is solved for the cooperative manipulators.

scholarly journals On finding a point in the relative interior of a polyhedral set and degeneracy in geometric programming

1978 ◽  
Vol 20 (4) ◽  
pp. 487-494 ◽  
Author(s):  
T. R. Jefferson ◽  
C. H. Scott
Keyword(s):  
Objective Function ◽  
Geometric Programming ◽  
Dual Problem ◽  
Optimization Theory ◽  
Inequality Constraints ◽  
Linear Constraints ◽  
Relative Interior ◽  
Nonlinear Inequality ◽  
Nonlinear Inequality Constraints ◽  
Nonlinear Objective Function

AbstractGeometric programming is now a well-established branch of optimization theory which has its origin in the analysis of posynomial programs. Geometric programming transforms a mathematical program with nonlinear objective function and nonlinear inequality constraints into a dual problem with nonlinear objective function and linear constraints. Although the dual problem is potentially simpler to solve, there are certain computational difficulties to be overcome. The gradient of the dual objective function is not defined for components whose values are zero. Moreover, certain dual variables may be constrained to be zero (geometric programming degeneracy).To resolve these problems, a means to find a solution in the relative interior of a set of linear equalities and inequalities is developed. It is then applied to the analysis of dual geometric programs.

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