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.

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.

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.

Reserve of characteristic inclusion for interval linear systems of relations

2021 ◽  
pp. 61-85
Author(s):  
Ирина Александровна Шарая ◽  
Сергей Петрович Шарый
Keyword(s):  
Linear Systems ◽  
Interval Arithmetic ◽  
Quantitative Measure ◽  
Solution Set ◽  
Special Cases ◽  
Interval Linear ◽  
Strict Inequalities ◽  
Interval Linear Systems ◽  
Interval Systems ◽  
Linear Systems Of Equations

В работе рассматриваются интервальные линейные включения Cx ⊆ d в полной интервальной арифметике Каухера. Вводится количественная мера выполнения этого включения, названная “резервом включения”, исследуются ее свойства и приложения. Показано, что резерв включения оказывается полезным инструментом при изучении АЕ-решений и кванторных решений интервальных линейных систем уравнений и неравенств. В частности, использование резерва включения помогает при определении положения точки относительно множества решений, при исследовании пустоты множества решений или его внутренности и т.п In this paper, we consider interval linear inclusions Cx ⊆ d in the Kaucher complete interval arithmetic. These inclusions are important both on their own and because they provide equivalent and useful descriptions for the so-called quantifier solutions and AE-solutions to interval systems of linear algebraic relations of the form Ax σ b , where A is an interval m × n -matrix, x ∈ R , b is an interval m -vector, and σ ∈ {= , ≤ , ≥} . In other words, these are interval systems in which equations and non-strict inequalities can be mixed. Considering the inclusion Cx ⊆ d in the Kaucher complete interval arithmetic allows studing simultaneously and in a uniform way all the different special cases of quantifier solutions and AE-solutions of interval systems of linear relations, as well as using interval analysis methods. A quantitative measure, called the “inclusion reserve”, is introduced to characterize how strong the inclusion Cx ⊆ d is fulfilled. In our work, we investigate its properties and applications. It is shown that the inclusion reserve turns out to be a useful tool in the study of AE-solutions and quantifier solutions of interval linear systems of equations and inequalities. In particular, the use of the inclusion reserve helps to determine the position of a point relative to a solution set, in investigating whether the solution set is empty or not, whether a point is in the interior of the solution set, etc

scholarly journals Enclosures for the solution set of parametric interval linear systems

2012 ◽  
Vol 22 (3) ◽  
pp. 561-574 ◽  
Author(s):  
Milan Hladík
Keyword(s):  
Linear Systems ◽  
Efficient Algorithm ◽  
Numerical Experiments ◽  
Solution Set ◽  
Systems Of Equations ◽  
Interval Linear ◽  
Interval Linear Systems ◽  
Linear Systems Of Equations

Abstract We investigate parametric interval linear systems of equations. The main result is a generalization of the Bauer–Skeel and the Hansen–Bliek–Rohn bounds for this case, comparing and refinement of both. We show that the latter bounds are not provable better, and that they are also sometimes too pessimistic. The presented form of both methods is suitable for combining them into one to get a more efficient algorithm. Some numerical experiments are carried out to illustrate performances of the methods.

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 Solution Set for Positive Wire Tension With Uncertainty in Wire-Actuated Parallel Manipulators

2016 ◽  
Vol 8 (4) ◽  
Author(s):  
Leila Notash
Keyword(s):  
Jacobian Matrix ◽  
Parallel Manipulators ◽  
Solution Set ◽  
Design Parameters ◽  
Negative Solution ◽  
Wire Tension

The solution for positive wire tension vector in the presence of uncertainties in design parameters and error in data is investigated for parallel manipulators. The minimum 2-norm non-negative solution and enclosures for the vector of wire tensions are formulated utilizing the perturbed and the interval forms of Jacobian matrix and platform wrench. Methodologies for calculating the minimum 2-norm non-negative solution set of wire tension vector, for interval Jacobian matrix and interval external wrench, are presented. Example parallel manipulators are simulated to investigate the implementation and effectiveness of these methodologies while relating their results.

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