Rychlé heuristické metody numerického řešení úlohy inverzní kinematiky
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Při zkoumání rozsáhlých robotických soustav pro potřeby firmy ŠKODA AUTO a.s. bylvytvořen software, který je zaměřen na kontrolu robotů a robotových standardů používanýchv koncernu Volkswagen AG. Součástí je i matematická knihovna, která řeší několik základníinženýrských úloh. Jedná se o přímou a inverzní kinematickou úlohu v robotice a o řešení soustavlineárních rovnic více proměnných. Při zkoumání numerických metod v rámci programováníuvedených úloh byly vyvinuty dva zcela nové algoritmy. Jedná se o metodu relaxace úhlu, která jeurčena pro řešení soustav lineárních rovnic. Může být také nepřímo použita pro řešení inverzníkinematické úlohy v robotice. Druhý algoritmus je přímo určen pro řešení inverzní kinematické úlohyv robotice. V tomto případě se jedná o metodu relaxace délky.Metoda relaxace úhlu je diskutována z hlediska základního principu a numerických vlastnostípři výpočtu soustav lineárních rovnic. Je zkoumána rychlost konvergence, závislost na číslupodmíněnosti a schopnost řešit obecné soustavy lineárních rovnic pro různé podoby matice soustavy.V rámci jednotlivých experimentů se metoda porovnává s výsledky známých numerických metod.Ukazuje se, že metoda relaxace úhlu konverguje k výsledkům podobným jako Mooreova-Penroseovapseudoinverze i v případě singulární matice soustavy. Tato vlastnost je vhodná pro řešení inverzníkinematické úlohy v robotice, protože se při výpočtu matice soustavy dynamicky mění v závislosti napostavení kinematické struktury robota.Aby bylo možné metodu relaxace úhlu použít pro výpočet inverzní kinematické úlohyv robotice, vychází práce z přístupů, které převádí tuto problematiku do podoby soustav lineárníchrovnic. Jelikož inverzní kinematická úloha v robotice vede na soustavy nelineárních rovnic, jedná seo přístupy, které linearizují řešení ve vybraném pracovním bodě pomocí Jacobiho matice, tzn.Newtonova metoda, inverze Jacobiho matice a metoda Levenberg-Marquardt. Všechny tři přístupy jsouodzkoušeny na kinematické struktuře planárního manipulátoru a robotu KUKA KR210 R2700 EXTRA.Vzniklé soustavy lineárních rovnic jsou řešeny standardní cestou pomocí Mooreovy-Penroseovypseudoinverze a zároveň metodou relaxace úhlu. Výsledky obou přístupů jsou vyhodnocenya porovnány.Na závěr práce je diskutována metoda relaxace délky, kterou lze zařadit do skupinyheuristických metod. Metoda je popsána z hlediska jejího principu a porovnána se známýmiheuristickými metodami CCD a FABRIK.
As part of a research of large robotic systems for the needs of SKODA AUTO, a.s. companya inspectional software was developed. The software focuses on a control of robots and programmingstandards used by Volkswagen AG. The software includes a mathematical library which solves severalbasic engineering tasks. It is a problem of forward and inverse kinematics in robotics and a solution ofsystems of linear equations. This work introduces two completely new algorithms which weredeveloped during the programming of the mathematical library. It is angle relaxation method which isdesigned for solving of systems of linear equations. It can also be used indirectly to solve inversekinematics in robotics. The second algorithm is designed directly for solving of inverse kinematics inrobotics. In this case the work talks about length relaxation method.The angle relaxation method is discussed from the point of view of basic principle andnumerical properties. A speed of convergence, a dependence on the condition number and an ability tosolve general systems of linear equations for different forms of matrices were examined. Results ofangle relaxation method were compared with results of known numerical methods. It appears that theangle relaxation method converges to results similar to results of Moore-Penrose pseudoinverse. Thisproperty is suitable for the solving of inverse kinematics in robotics because the system matrix isdynamically changed during a calculation depending on a position of the kinematic structure of therobot.In order to use the angle relaxation method to calculate the inverse kinematics in robotics, thework is based on approaches which transform the problem into systems of linear equations. The inversekinematics in robotics leads primarily to systems of nonlinear equations. For this reason approacheswere chosen which linearize the equation using a Jacobian matrix, i.e. Newton's method, inverseJacobian method and Levenberg-Marquardt method. All three approaches were tested on the kinematicstructure of the planar manipulator and the robot KUKA KR210 R2700 EXTRA. Systems of linearequations were solved by a standard way using Moore-Penrose pseudoinverse and at the same time bythe angle relaxation method. Results of both approaches were evaluated and compared.At the end of this work the length relaxation method is discussed which can be included in thegroup of heuristic methods. The length relaxation method is described from the point of view of basicprinciple and compared with known heuristic methods CCD and FABRIK.
As part of a research of large robotic systems for the needs of SKODA AUTO, a.s. companya inspectional software was developed. The software focuses on a control of robots and programmingstandards used by Volkswagen AG. The software includes a mathematical library which solves severalbasic engineering tasks. It is a problem of forward and inverse kinematics in robotics and a solution ofsystems of linear equations. This work introduces two completely new algorithms which weredeveloped during the programming of the mathematical library. It is angle relaxation method which isdesigned for solving of systems of linear equations. It can also be used indirectly to solve inversekinematics in robotics. The second algorithm is designed directly for solving of inverse kinematics inrobotics. In this case the work talks about length relaxation method.The angle relaxation method is discussed from the point of view of basic principle andnumerical properties. A speed of convergence, a dependence on the condition number and an ability tosolve general systems of linear equations for different forms of matrices were examined. Results ofangle relaxation method were compared with results of known numerical methods. It appears that theangle relaxation method converges to results similar to results of Moore-Penrose pseudoinverse. Thisproperty is suitable for the solving of inverse kinematics in robotics because the system matrix isdynamically changed during a calculation depending on a position of the kinematic structure of therobot.In order to use the angle relaxation method to calculate the inverse kinematics in robotics, thework is based on approaches which transform the problem into systems of linear equations. The inversekinematics in robotics leads primarily to systems of nonlinear equations. For this reason approacheswere chosen which linearize the equation using a Jacobian matrix, i.e. Newton's method, inverseJacobian method and Levenberg-Marquardt method. All three approaches were tested on the kinematicstructure of the planar manipulator and the robot KUKA KR210 R2700 EXTRA. Systems of linearequations were solved by a standard way using Moore-Penrose pseudoinverse and at the same time bythe angle relaxation method. Results of both approaches were evaluated and compared.At the end of this work the length relaxation method is discussed which can be included in thegroup of heuristic methods. The length relaxation method is described from the point of view of basicprinciple and compared with known heuristic methods CCD and FABRIK.
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Subject(s)
soustavy lineárních rovnic, inverzní kinematická úloha, pseudoinverze, robotika, systems of linear equations, inverse kinematics, pseudoinverse, robotics