Wrench Accuracy for Parallel Manipulators and Interval Dependency

2016 ◽  
Vol 9 (1) ◽  
Author(s):  
Leila Notash
Keyword(s):  
Interval Arithmetic ◽  
Jacobian Matrix ◽  
Parallel Manipulators ◽  
Upper Bounds ◽  
Solution Set ◽  
Lower And Upper Bounds ◽  
Solution Sets ◽  
The Given

In this paper, the wrench accuracy for parallel manipulators is examined under variations in parameters and data. The solution sets of actuator forces/torques are investigated utilizing interval arithmetic (IA). Implementation issues of interval arithmetic to analyze the performance of manipulators are addressed, including the consideration of dependencies in parameters and the design of input vectors to generate the required wrench. Specifically, the effect of the dependency within and among the entries of the Jacobian matrix is studied, and methodologies for reducing and/or eliminating the overestimation of solution set are presented. In addition, the subset of solution set that produces platform wrenches within the required lower and upper bounds is modeled. Furthermore, the formulation of solutions that provide any platform wrench within the defined interval is examined. Intersection of these two sets, if any, results in the given interval platform wrench. Implementation of the methods to identify the solution for actuator forces/torques is presented on example parallel manipulators.

Analytical Methods for Solution Sets of Interval Wrench

2015 ◽  
Author(s):  
Leila Notash
Keyword(s):  
Analytical Methods ◽  
Jacobian Matrix ◽  
Parametric Method ◽  
Parallel Manipulators ◽  
Solution Set ◽  
The Other ◽  
Discrete Method ◽  
Solution Sets ◽  
Dominant Parameter

Methodologies for calculating the solution set of actuator inputs, in the presence of uncertainty/error in parameters/data, are investigated. The enclosure for the vector of actuator torques is formulated utilizing the interval forms of the Jacobian matrix and external wrench. Two analytical methods are utilized to identify the solution set; one method generates the rays that bound the solution set in each orthant, and the other one is based on parameterizing the interval entries of the Jacobian matrix and wrench. For the parametric method, the existence of dominant parameter groups to produce the whole solution set (or a subset of solution set) is examined. Implementation of these methods on example parallel manipulators are presented to identify the solution set for the actuator torques, and the results are verified with the discrete method.

On the Solutions of Interval Systems for Under-Constrained and Redundant Parallel Manipulators

2017 ◽  
Vol 9 (6) ◽  
Author(s):  
Leila Notash
Keyword(s):  
Least Squares ◽  
Jacobian Matrix ◽  
Parallel Manipulators ◽  
Constrained Systems ◽  
Solution Set ◽  
Least Squares Solution ◽  
Interval Linear ◽  
Interval Linear Systems ◽  
Parameter Dependency ◽  
Interval Systems

For under-constrained and redundant parallel manipulators, the actuator inputs are studied with bounded variations in parameters and data. Problem is formulated within the context of force analysis. Discrete and analytical methods for interval linear systems are presented, categorized, and implemented to identify the solution set, as well as the minimum 2-norm least-squares solution set. The notions of parameter dependency and solution subsets are considered. The hyperplanes that bound the solution in each orthant characterize the solution set of manipulators. While the parameterized form of the interval entries of the Jacobian matrix and wrench produce the minimum 2-norm least-squares solution for the under-constrained and over-constrained systems of real matrices and vectors within the interval Jacobian matrix and wrench vector, respectively. Example manipulators are used to present the application of methods for identifying the solution and minimum norm solution sets for actuator forces/torques.

scholarly journals Stability of Switched Server Systems with Constraints on Service-Time and Capacity of Buffers

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Li Wang ◽  
Zhonghe He ◽  
Chi Zhang
Keyword(s):  
Periodic Orbit ◽  
Service Time ◽  
Buffer Capacity ◽  
Time Factors ◽  
Upper Bounds ◽  
Finite Buffer ◽  
Lower And Upper Bounds ◽  
Upper Limit ◽  
Server Systems ◽  
The Given

The execution of emptying policy ensures the convergence of any solution to the system to a unique periodic orbit, which does not impose constraints on service-time and capacity of buffers. Motivated by these problems, in this paper, the service-time-limited policy is first proposed based on the information resulted from the periodic orbit under emptying policy, which imposes lower and upper bounds on emptying time for the queue in each buffer, by introducing lower-limit and upper-limit service-time factors. Furthermore, the execution of service-time-limited policy in the case of finite buffer capacity is considered. Moreover, the notion of feasibility of states under service-time-limited policy is introduced and then the checking condition for feasibility of states is given; that is, the solution does not exceed the buffer capacity within the first cycle of the server. At last, a sufficient condition for determining upper-limit service-time factors ensuring that the given state is feasible is given.

Minimization of Bounded Cable Tensions in Cable-Based Parallel Manipulators

2007 ◽  
Author(s):  
Mahir Hassan ◽  
Amir Khajepour
Keyword(s):  
Lower Bound ◽  
Numerical Method ◽  
Upper Bound ◽  
Task Force ◽  
Parallel Manipulators ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Alternating Projection ◽  
Alternating Projection Method ◽  
Cable Forces

In this work, the application of the Dykstra’s alternating projection method to find the minimum-2-norm solution for actuator forces is discussed in the case when lower and upper bounds are imposed on the actuator forces. The lower bound is due to specified pretension desired in the cables and the upper bound is due to the maximum allowable forces in the cables. This algorithm presents a systematic numerical method to determine whether or not a solution exists to the cable forces within these bounds and, if it does exist, calculate the minimum-2-norm solution for the cable forces for a given task force. This method is applied to an example 2-DOF translational cable-driven manipulator and a geometrical demonstration is presented.

Properties of the Solution Set for Generalized Equilibrium Problems with Lower and Upper Bounds in FC-Metric Spaces

2013 ◽  
Vol 671-674 ◽  
pp. 1557-1560
Author(s):  
Kai Ting Wen
Keyword(s):  
Metric Spaces ◽  
Equilibrium Problems ◽  
Optimization Problems ◽  
Upper Bounds ◽  
Solution Set ◽  
Lower And Upper Bounds ◽  
Paper Properties ◽  
Special Cases ◽  
Mathematical Problems ◽  
Generalized Equilibrium

The equilibrium problem includes many fundamental mathematical problems, e.g., optimization problems, saddle point problems, fixed point problems, economics problems, comple- mentarity problems, variational inequality problems, mechanics, engineering, and others as special cases. In this paper, properties of the solution set for generalized equilibrium problems with lower and upper bounds in FC-metric spaces are studied. In noncompact setting, we obtain that the solution set for generalized equilibrium problems with lower and upper bounds is nonempty and compact. Our results improve and generalize some recent results in the reference therein.

Investigation of Wrench Accuracy for Parallel Manipulators

2016 ◽  
Author(s):  
Leila Notash
Keyword(s):  
Parallel Manipulators ◽  
Coefficient Matrix ◽  
Solution Set ◽  
Output Vector ◽  
Discrete Method ◽  
Interval Linear ◽  
Closed Line ◽  
Interval Linear Systems ◽  
Real Linear ◽  
The Given

In this paper, the wrench accuracy for parallel manipulators is examined and the solution sets of actuator forces/torques are investigated under variations in parameters and data. The subset of solution set that produces platform wrenches within the required lower and upper bounds are modeled using discrete and analytical methods. In addition, the formulation of the solutions that provide any platform wrench within the defined interval is examined. Intersection of these two sets, if any, results in the given interval platform wrench. Moreover, the dependency among the entries of the interval linear systems and its effect on the solution set is considered. The discrete method is based on the discretization of solution set enclosure and validation at each increment, or the collection of the solutions of real linear relations for the discretized interval coefficient matrix and output vector. The analytical method for each solution set is based on the intersection of the pertinent closed half-spaces or the assembly of closed line segments that encompass the solution. Implementation of the methods to identify the solution for actuator forces/torques is presented on example parallel manipulators.

Solutions of Interval Systems for Under-Constrained and Redundant Parallel Manipulators

2017 ◽  
Author(s):  
Leila Notash
Keyword(s):  
Jacobian Matrix ◽  
Parallel Manipulators ◽  
Constrained Systems ◽  
Solution Set ◽  
Least Square ◽  
Data Problem ◽  
Interval Linear ◽  
Interval Linear Systems ◽  
Parameter Dependency ◽  
Interval Systems

For under-constrained and redundant parallel manipulators, the actuator inputs are studied with bounded variations in parameters and data. Problem is formulated within the context of force analysis. Discrete and analytical methods for interval linear systems are presented, categorized and implemented to identify the solution set, as well as the minimum 2-norm least square solution set. The notions of parameter dependency and solution subsets are considered. The hyperplanes that bound the solution in each orthant characterize the solution set of manipulators. While the parameterized form of the interval entries of the Jacobian matrix and wrench produce the minimum 2-norm least square solution for the under-constrained and over-constrained systems of real matrices and vectors within the interval Jacobian matrix and wrench vector, respectively. Example manipulators are used to present the application of methods for identifying the solution and minimum norm solution sets for actuator forces/torques.

Statistical properties of simple types

2000 ◽  
Vol 10 (5) ◽  
pp. 575-594 ◽  
Author(s):  
M. MOCZURAD ◽  
J. TYSZKIEWICZ ◽  
M. ZAIONC
Keyword(s):  
Finite Number ◽  
Lambda Calculus ◽  
Upper Bounds ◽  
Statistical Properties ◽  
Propositional Variable ◽  
Lower And Upper Bounds ◽  
Type Formula ◽  
Ground Types ◽  
Asymptotic Probability ◽  
The Given

We consider types and typed lambda calculus over a finite number of ground types. We are going to investigate the size of the fraction of inhabited types of the given length n against the number of all types of length n. The plan of this paper is to find the limit of that fraction when n → ∞. The answer to this question is equivalent to finding the ‘density’ of inhabited types in the set of all types, or the so-called asymptotic probability of finding an inhabited type in the set of all types. Under the Curry–Howard isomorphism this means finding the density or asymptotic probability of provable intuitionistic propositional formulas in the set of all formulas. For types with one ground type (formulas with one propositional variable), we prove that the limit exists and is equal to 1/2 + √5/10, which is approximately 72.36%. This means that a long random type (formula) has this probability of being inhabited (tautology). We also prove that for every finite number k of ground-type variables, the density of inhabited types is always positive and lies between (4k + 1)/(2k + 1)2 and (3k + 1)/(k + 1)2. Therefore we can easily see that the density is decreasing to 0 with k going to infinity. From the lower and upper bounds presented we can deduce that at least 1/3 of classical tautologies are intuitionistic.

scholarly journals The inverse maximum flow problem under Lk norms

2012 ◽  
Vol 28 (1) ◽  
pp. 59-66
Author(s):  
ADRIAN DEACONU ◽  
◽  
ELEONOR CIUREA ◽  
Keyword(s):  
Polynomial Time ◽  
Flow Problem ◽  
Maximum Flow ◽  
Upper Bounds ◽  
Initial Vector ◽  
Lower And Upper Bounds ◽  
Maximum Flow Problem ◽  
The Given

The problem consists in modifying the lower and upper bounds of a given feasible flow f in a network G so that the given flow becomes a maximum flow in G and the distance between the initial vector of bounds and the modified one measured using Lk norm (k ∈ N∗) is minimum. A fast apriori fesibility test is presented. An algorithm for solving this problem is deduced. Strongly and weakly polynomial time implementations of this algorithm are presented. Some particular cases of the problem are discussed.

scholarly journals Lower and Upper Bounds for the Discrete Bi-Directional Preemptive Conversion Problem with a Constant Price Interval

Algorithms ◽  
2020 ◽  
Vol 13 (2) ◽  
pp. 42
Author(s):  
Michael Schwarz
Keyword(s):  
Competitive Analysis ◽  
Upper Bound ◽  
Competitive Ratio ◽  
Upper Bounds ◽  
Investment Horizon ◽  
Lower And Upper Bounds ◽  
Single Period ◽  
Price Trends ◽  
The Given ◽  
Period Model

In the conversion problem, wealth has to be distributed between two assets with the objective to maximize the wealth at the end of the investment horizon. The bi-directional preemptive conversion problem with a constant price interval is the only problem, of the four main variants of the conversion problem, that has not yet been optimally solved by competitive analysis. Assuming a given number of monotonous price trends called runs, lower and upper bounds for the competitive ratio are given. In this work, the assumption of a given number of runs is rejected and lower and upper bounds for the bi-directional preemptive conversion problem with a constant price interval are given. Furthermore, an algorithm based on error balancing is given which at minimum achieves the given upper bound. It can also be shown that this algorithm is optimal for the single-period model.

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