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1

Trajectory Planning for Mobile Manipulators with Control Constraints

Pająk Grzegorz, Pająk Iwona

Pomiary Automatyka Robotyka

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2023

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R. 27, nr 2

21--30

EN

This paper presents the method of trajectory planning for mobile manipulators considering limitations resulting from capabilities of robotic system actuators. The fulfillment of control constraints is achieved by introducing virtual control scaling of the robot trajectory in the limited periods of time. Such an approach allows researchers to obtain the trajectories fulfilling control constraints without significantly increasing the time of task execution. The proposed method generates sub-optimal trajectories maximizing the manipulability measure of the robot arm, preserves mechanical and collision avoidance limitations and can be used in real-time trajectory planning. The effectiveness of the presented solution is confirmed by computer simulations involving a mobile manipulator with parameters corresponding to KUKA youBot.

PL

W pracy przedstawiono metodę planowania trajektorii dla manipulatorów mobilnych uwzględniającą ograniczenia wynikające z możliwości układów napędowych robota. Spełnienie ograniczeń na sterowana zostało osiągnięte poprzez wprowadzenie wirtualnego sterowania skalującego trajektorię robota w ograniczonych przedziałach czasu. Takie podejście pozwoliło na uzyskanie trajektorii spełniających ograniczenia na sterowania bez znaczącego wydłużenia czasu realizacji zadania. Zaproponowana metoda generuje sub-optymalne trajektorie maksymalizując miarę manipulowalności ramienia robota, zachowuje ograniczenia mechaniczne oraz warunki unikania kolizji i może być zastosowana do planowania trajektorii w czasie rzeczywistym. Skuteczność zaproponowanego rozwiązania została potwierdzona symulacjami komputerowymi wykonanymi z użyciem mobilnego manipulatora o parametrach odpowiadających robotowi KUKA youBot.

2

Swing-up problem of an inverted pendulum - energy space approach

Balcerzak M.

Mechanics and Mechanical Engineering

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2018

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Vol. 22, nr 1

33--40

EN

This paper describes a novel, energy space based approach to the swing-up of an inverted pendulum. The details of the swing-up problem have been described. Equations of the velocity-controlled have been presented. Design of the controller based on energy space notion has been elaborated. The control algorithm takes into account state constraints and control signal constraints. Parameters of the controller have been optimized by means of the Differential Evolution method. A numerical simulation of the inverted pendulum driven by the proposed controller has been conducted, its results have been presented and elaborated. The paper confirms that the proposed method results in a simple and effective swing-up algorithm for a velocity-controlled inverted pendulum with state constraints and control signal constraints.

3

Collision-Free Trajectory Planning for Mobile Manipulators Subject to Control Constraints

Pajak G., Pajak I.

Archive of Mechanical Engineering

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2014

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Vol. LXI, nr 1

35--55

EN

A method of planning collision-free trajectory for a mobile manipulator tracking a line section path is presented. The reference trajectory of a mobile platform is not needed, mechanical and control constraints are taken into account. The method is based on a penalty function approach and a redundancy resolution at the acceleration level. Nonholonomic constraints in a Pfaffian form are explicitly incorporated to the control algorithm. The problem is shown to be equivalent to some point-to-point control problem whose solution may be easier determined. The motion of the mobile manipulator is planned in order to maximise the manipulability measure, thus to avoid manipulator singularities. A computer example involving a mobile manipulator consisting of a nonholonomic platform (2,0) class and a 3 DOF RPR type holonomic manipulator operating in a three-dimensional task space is also presented.

PL

W pracy opisano tok postępowania podczas budowy modeli symulacyjnych z wykorzystaniem programu SolidWorks i Matlab/Simulink. Tworzenie modelu symulacyjnego przebiega etapami, to znaczy najpierw opracowywany jest model geometryczny w programie SolidWorks, następnie dzięki możliwości wymiany danych, model CAD jest implementowany w środowisku obliczeniowym Matlab/Simulink. Modele SimMechanics pozwalają na śledzenie wielu parametrów, np. trajektorii, prędkości, czy przyspieszeń dowolnych elementów układu złożonego. W pracy, jako przykłady modeli symulacyjnych opracowanych zgodnie z zaprezentowana metoda, pokazano modele laboratoryjnego żurawia samochodowego oraz żurawia leśnego. Modele te umożliwiają wizualizacje zadanego - za pomocą wymuszeń kinematycznych - cyklu pracy.

4

Planning of collision-free trajectory for mobile manipulators

Pająk G.

International Journal of Applied Mechanics and Engineering

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2013

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Vol. 18, no. 2

475--489

EN

A method of planning sub-optimal trajectory for a mobile manipulator working in the environment including obstacles is presented. The path of the end-effector is defined as a curve that can be parameterized by any scaling parameter, the reference trajectory of a mobile platform is not needed. Constraints connected with the existence of mechanical limits for a given manipulator configuration, collision avoidance conditions and control constraints are considered. The motion of the mobile manipulator is planned in order to maximize the manipulability measure, thus to avoid manipulator singularities. The method is based on a penalty function approach and a redundancy resolution at the acceleration level. A computer example involving a mobile manipulator consisting of a nonholonomic platform and a SCARA type holonomic manipulator operating in a two-dimensional task space is also presented.

5

Optimal Control of a Drying Process with Avoiding Cracks

Galant A., Grossmann C., Scheffler M., Wensch J.

Schedae Informaticae

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2012

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Vol. 21

81--105

EN

The paper deals with the numerical treatment of the optimal control of drying of materials which may lead to cracks. The drying process is controlled by temperature, velocity and humidity of the surrounding air. The state equations dene the humidity and temperature distribution within a simpli ed wood specimen for given controls. The elasticity equation describes the internal stresses under humidity and temperature changes. To avoid cracks these internal stresses have to be limited. The related constraints are treated by smoothed exact barrier-penalty techniques. The objective functional of the optimal control problem is of tracking type. Further it contains a quadratic regularization by an energy term for the control variables (surrounding air) and barrier-penalty terms. The necessary optimality conditions of the auxiliary problem form a coupled system of nonlinear equations in appropriate function spaces. This optimality system is given by the state equations and the related adjoint equations, but also by an approximate projection onto the admissible set of controls by means of barrier-penalty terms. This system is discretized by nite elements and treated iteratively for given controls. The optimal control itself is performed by quasi-Newton techniques.

6

Identification of matrix parameters in elliptic PDEs

Deckelnick K., Hinze M.

Control and Cybernetics

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2011

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Vol. 40, no 4

957-969

EN

In the present work we treat the inverse problem of identifying the matrix-valued diffusion coefficient of an elliptic PDE from multiple interior measurements with the help of techniques from PDE constrained optimization. We prove existence of solutions using the concept of H-convergence and employ variational discretization for the discrete approximation of solutions. Using a discrete version of H-convergence we are able to establish the strong convergence of the discrete solutions. Finally we present some numerical results.

7

A new stopping criterion for iterative solvers for control optimal problems

Krumbiegel K., Rosch A.

Archives of Control Sciences

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2008

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Vol. 18, no. 1

17-42

EN

Linear quadratic optimal control problems governed by PDEs with pointwise control constraints are considered. We derive error estimates for feasible and infeasible controls of the problem. Based on this theory an error estimator is constructed for different discretization schemes. Moreo ver, we establish the estimator as a stopping criterion for several optimization methods. Furthermore, additional errors caused by solving the linear systems are discussed. The theory is illustrated by numerical examples.

8

Regularity and stability of optimal controls of nonstationary Navier-Stokes equations

Wachsmuth D.

Control and Cybernetics

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2005

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Vol. 34, no 2

387-409

EN

The regularity and stability of optimal controls of nonstationary Navier-Stokes equations are investigated. Under suitable assumptions every control satisfying first-order necessary conditions is shown to be a continuous function in both space and time. Moreover, the behaviour of a locally optimal control under certain perturbations of the cost functional and the state equation is investigated. Lipschitz stability is proven provided a second-order sufficient optimality condition holds.

9

Analysis of the Lagrange-SQP-Newton method for the control of a phase field equation

Heinkenschloss M., Troeltzsch F.

Control and Cybernetics

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1999

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Vol. 28, no 2

177-211

EN

This paper investigates the local convergence of the Lagrange-SQP-Newton method applied to an optimal control problem governed by a phase field equation with distributed control. The phase field equation is a system of two semilinear parabolic differential equations. Stability analysis of optimization problems and regularity results for parabolic differential equations are used to proof convergence of the controls with respect to the L[sup 2](Q) norm and with respect to the L[sup infinity](Q) norm.