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1
Content available 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
Content available An itae optimal sliding mode controller for systems with control signal and velocity limitations
Pietrala Mateusz, Leśniewski Piotr, Bartoszewicz Andrzej
Acta Mechanica et Automatica
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2023
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Vol. 17, no. 2
230--238
EN
In this paper, a sliding mode controller, which can be applied for second-order systems, is designed. Robustness to external disturbances, finite regulation time and a good system’s behaviour are required for a sliding mode controller. In order to achieve the first two of these three goals, a non-linear, time-varying switching curve is introduced. The representative point (state vector) belongs to this line from the very beginning of the control process, which results in elimination of the reaching phase. The stable sliding motion along the switching curve is provided. Natural limitations such as control signal and system’s velocity constraints will be taken into account. In order to satisfy them, the sliding line parameters will be properly selected. However, a good dynamical behaviour of the system has to be provided. In order to achieve that, the integral time absolute error (ITAE) quality index will be introduced and minimised. The simulation example will verify theoretical considerations.
3
Content available Matlab implementation of direct and indirect shooting methods to solve an optimal control problem with state constraints
Karbowski Andrzej
Journal of Automation Mobile Robotics and Intelligent Systems
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2021
|
Vol. 15, No. 1
43--50
EN
The paper presents a general procedure to solve nume‐ rically optimal control problems with state constraints. It is used in the case, when the simple time discretization of the state equations and expressing the optimal cont‐ rol problem as a nonlinear mathematical programming problem is too coarse. It is based on using in turn two multiple shooting BVP approaches: direct and indirect. The paper is supplementary to the earlier author’s paper on direct and indirect shooting methods, presenting the theory underlying both approaches. The same example is considered here and brought to an end, that is two full listings of two MATLAB codes are shown.
4
Content available Adaptive, nonlinear state transformation-based control of motion in presence of hard constraints
Kabziński J., Mosiołek P.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2020
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Vol. 68, nr 5
963–-971
EN
The paper describes a nonlinear controller design technique applied to a servo drive in the presence of hard state constraints. The approach presented is based on nonlinear state-space transformation and adaptive backstepping. It allows us to impose hard constraints on the state variables directly and to achieve asymptotic tracking of any reference trajectory inside the constraints, despite unknown plant parameters. Two control schemes (with and without integral action) are derived, investigated and then compared. Several examples demonstrate the main features of the design procedure and prove that it may be applied in case of motion control problems in electric drive automation.
5
Content available remote 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.
6
Content available The problem of state constraints in designing the discrete time sliding mode controller
Jaskuła M., Pietrala M., Leśniewski P.
Pomiary Automatyka Robotyka
|
2017
|
R. 21, nr 4
15--22
EN
In this paper we study the problem of state constraints in discrete time sliding mode control. We present a sufficient condition for the strategy that drives the representative point monotonically to the sliding hyperplane in finite time. The advantage of this strategy is that disturbances do not have to fulfill the matching conditions. Our approach is based on the so-called reaching law technique.
PL
W artykule przeanalizowany został problem ograniczenia zmiennych stanu w dyskretnym sterowaniu ślizgowym. Do zaprojektowania regulatora zastosowano regułę osiągania ruchu ślizgowego. Zaprezentowano warunek dostateczny na monotoniczną zbieżność stanu obiektu do płaszczyzny ślizgowej w skończonym czasie. Zaletą przedstawionej metody jest to, że zakłócenia nie muszą spełniać warunków dopasowania.
7
Content available remote On some new phenomena in state-constrained optimal control if ODEs as well as PDEs are involved
Pesch H. J., Rund A., Wahl V., Wendl S.
Control and Cybernetics
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2010
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Vol. 39, no 3
647-660
EN
In this paper we investigate a new class of optimal control problems with ODE as well as PDE constraints. We would like to call them "hypersonic rocket car problems", since they were inspired, on the one hand, by the well known rocket car problem from the early days of ODE optimal control, on the other hand by a recently investigated flight path trajectory optimization problem for a hypersonic aircraft. The hypersonic rocket car problems mimic the latter's coupling structure, yet in a strongly simplified form. They can therefore be seen as prototypes of ODE-PDE control problems. Due to their relative simplicity they allow to a certain degree to obtain analytical solutions and insights into the structure of the adjoints, which would currently be unthinkable with complex real life problems. Our main aim is to derive and verify the necessary optimality conditions. Most of the obtained results bear a lot of similarities with state constrained ODE optimal control problems, yet we also observed some new phenomena.
8
Content available remote Revisiting the analysis of optimal control problems with several state constraints
Bonnans J. F., Hermant A.
Control and Cybernetics
|
2009
|
Vol. 38, no 4A
1021-1052
EN
This paper improves the results of and gives shorter proofs for the analysis of state constrained optimal control problems than presented by the authors in Bonnans and Hermant (2009b), concerning second order optimality conditions and the well-posedness of the shooting algorithm. The hypothesis for the second order necessary conditions is weaker, and the main results are obtained without reduction to the normal form used in that reference, and without analysis of high order regularity results for the control. In addition, we provide some numerical illustration. The essential tool is the use of the "alternative optimality system".
9
Content available remote On the implicit programming approach in a class of mathematical programs with equilibrium constraints
Outrata J. V., Cervinka M.
Control and Cybernetics
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2009
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Vol. 38, no 4B
1557-1574
EN
In the paper we analyze the influence of implicit programming hypothesis and presence of state constraints on first order optimality conditions to mathematical programs with equilibrium constraints. In the absence of state constraints, we derive sharp stationarity conditions, provided the strong regularity condition holds. In the second part of the paper we suggest an exact penalization of state constraints and test the behavior of standard bundle trust region algorithm on academic examples.
10
Content available remote On the development of Pontryagin's Maximum Principle in the works of A.Ya. Dubovitskii and A.A. Milyutin
Dmitruk A. V.
Control and Cybernetics
|
2009
|
Vol. 38, no 4A
923-957
EN
We give a short review of the development and generalizations of the Pontryagin Maximum Principle, provided in the studies of Dubovitskii and Milyutin in the 1960s and later years.
11
Content available remote On the regularization error of state constrained Neumann control problems
Krumbiegel K., Rosch A.
Control and Cybernetics
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2008
|
Vol. 37, no 2
369-392
EN
A linear elliptic optimal control problem with point-wise state constraints in the interior of the domain is considered. Furthermore, the control is given on the boundary with associated constraints. An artificial distributed control is introduced in the cost functional, in the state equation and in the state constraints. Since there are no control constraints for the artificial control, efficient numerical methods can be easily established. Based on a possible violation of the pure pointwise state constraints, an error estimate for the regularization error is derived. The theoretical results are illustrated by numerical tests.
12
Content available remote Finite element error analysis for state-constrained optimal control of the Stokes equations
Los Reyes J. C. de, Meyer C., Vexler B.
Control and Cybernetics
|
2008
|
Vol. 37, no 2
251-284
EN
An optimal control problem for 2d and 3d Stokes equations is investigated with pointwise inequality constraints on the state and the control. The paper is concerned with the full discretization of the control problem allowing for different types of discretization of both the control and the state. For instance, piecewise linear and continuous approximations of the control are included in the present theory. Under certain assumptions on the L∞-error of the finite element discretization of the state, error estimates for the control are derived which can be seen to be optimal since their order of convergence coincides with the one of the interpolation error. The assumptions of the L∞-finite-eleinent-error can be verified for different numerical settings. Finally the results of two numerical experiments are presented.
13
Content available remote Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints
Meyer C.
Control and Cybernetics
|
2008
|
Vol. 37, no 1
51-83
EN
We consider a linear-quadratic elliptic optimal control problem with pointwise state constraints. The problem is fully discretized using linear ansatz functions for state and control. Based on a Slater-type argument, we investigate the approximation behavior for mesh size tending to zero. The obtained convergence order for the L²-error of the control and for H 1-error of the state is 1 - ε in the two-dimensional case and 1/2 - ε in three dimensions, provided that the domain satisfies certain regularity assumptions. In a second step, a state-constrained problem with additional control constraints is considered. Here, the control is discretized by constant ansatz functions. It is shown that the convergence theory can be adapted to this case yielding the same order of convergence. The theoretical findings are confirmed by numerical examples.
14
Content available remote A characterization of stability and sensitivity properties for state-constrained optimal control
Malanowski K.
Control and Cybernetics
|
2007
|
Vol. 36, no 3
711-726
EN
In a series of the recent papers of the author, it was shown that the solutions and Lagrange multipliers of state-constrained optimal control problems are locally Lipschitz continuous and directionally differentiable functions of the parameter, under usual constraint qualifications and weakened second order conditions. In this paper, it is shown that those conditions are not only sufficient, but also necessary. Thus, they consitute a characterization of Lipschitz stability and sensitivity properties for state-constrained optimal control problems.
15
Content available Application of the classical implicit function theorem in sensitivity analysis of parametric optimal control
Malanowski K.
Control and Cybernetics
|
1998
|
Vol. 27, no 3
335-352
EN
A family {(O[sub h])} of parametric optimal control problems for nonlinear ODEs is considered. The problems are subject to pointwise inequality type state constraints. It is assumed that the reference solution is regular. The original problems (O[sub h]) are substituted by problems [...] subject to equality type constraints with the sets of activity depending on the parameter. Using the classical implicit function theorem, conditions are derived under which stationary points of [...] are Frechet differentiable functions of the parameter. It is shown that, under additional conditions, the stationary points of [...] correspond to the solutions and Lagrange multipliers of (O[sub h]9).
16
Content available remote Relaxing constrained control systems
Frankowska H., Rampazzo F.
Bulletin of the Polish Academy of Sciences. Mathematics
|
1998
|
Vol. 46, no 1
70-81
EN
In this paper we provide a relaxation result for control systems under both equality and inequality constraints involving the state and the control. In particular we show that the Mangasarian-Fromowitz constraint qualification allows to rewrite constrained systems as differential inclusions with locally Lipschitz right-hand side. Then Filippov-Ważewski relaxation theorem may be applied to show that ordinary solutions are dense in the set of relaxed solutions. If, besided agreeing with the above constraints, the state has to remain in a control-independent set K, then we provide a condition on the feasible velocities on the boundary of K to get a relaxation theorem.
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