Optimal Control of Longitudinal Motion of an Elastic Rod Using Boundary Forces

Gavrikov, Alexander; Kostin, Georgy

doi:10.1134/s1064230721050099

Cited by 6 publications

(8 citation statements)

References 22 publications

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“…This representation of the momentum density p and the normal forces s automatically satisfies (3). By excluding p and s from consideration, the state of the system is determined by two variables, kinematic v and dynamic r.…”

Section: Dynamic Potential In An Equivalent Ibvp Statementmentioning

confidence: 99%

“…The forced motions of an elastic rod with one interval of constancy of the distributed force, that is N = 1, are equivalent to motions of a rod controlled by external boundary forces f ± and was described in [3]. Here, we suppose that N > 1 and M > 1.…”

Section: Solvability Of the System Of Edge Constraintsmentioning

confidence: 99%

“…However, it has intrinsic limitations since a finite number of inputs is used to control a continuum system. For a rod, one of the limitations is a minimal control time [2], [3].…”

Section: Introductionmentioning

confidence: 99%

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Optimal Control of Longitudinal Motions for an Elastic Rod with Distributed Forces

Kostin¹,

Gavrikov²

2022

Preprint

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The study is devoted to mathematical modeling and optimal control design of longitudinal motions of a rectilinear elastic rod. The control inputs are a force, which is normal to the cross section and distributed piecewise constantly along the rod's axis, as well as two external lumped loads at the ends. It is assumed that the intervals of constancy in the normal force have equal length. Given initial and terminal states with a fixed time horizon, the optimal control problem is to minimize the mean mechanical energy stored in the rod. To solve the problem, two unknown functions are introduced: the dynamical potential and the longitudinal displacements. As a result, the initial-boundary value problem is reformulated in a weak form, in which constitutive relations are given as an integral quadratic equation. The unknown functions are both continuous in the new statement. For the uniform rod, they are found as linear combinations of traveling waves. In this case, all conditions on continuity as well as boundary, initial, and terminal constraints form a linear algebraic system with respect to the traveling waves and control functions. The minimal controllability time is found from the solvability condition for this algebraic system. After resolving the system, remaining free variables are used to optimize the cost functional. Thus, the original control problem is reduced to a one-dimensional variational problem. The Euler-Lagrange necessary condition yields a linear system of ordinary differential equations with constant coefficients supplemented by essential and natural boundary conditions. Therefore, the exact optimal control law and the corresponding dynamic and kinematic fields are found explicitly. Finally, the energy properties of the optimal solution are analyzed. KeywordsOptimal control • dynamic systems • model-based control • distributed parameters. * The study has been done under financial support of the Russian Science Foundation (grant 21-11-00151).

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Section: Dynamic Potential In An Equivalent Ibvp Statementmentioning

confidence: 99%

Section: Solvability Of the System Of Edge Constraintsmentioning

confidence: 99%

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Optimal Control of Longitudinal Motions for an Elastic Rod with Distributed Forces

Kostin¹,

Gavrikov²

2022

Preprint

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“…This is an inverse problem which we formulate as a control problem. In case of low Reynolds number rods, we neglect the inertia terms and consider the following optimisation problem: reducing the discrepancy between the centreline x and an objective position given by the observed data x g = (x g , y g , z g ), by optimisation of the external force f and torque l in ( 29) and (30).…”

Section: The Optimal Control Problemmentioning

confidence: 99%

“…In the context of optimal control of a rod or beam, the solution existence of optimal control of the longitudinal vibration of a viscoelastic rod by either a contact force or distributed force is discussed in [73], and the mean mechanical energy minimised by a boundary force is studied, using the methods of the calculus of variations [30], maximum principle [70] and Ritz method [51]; minimisation of the mean square deviation of the Timoshenko beam is investigated by controlling a distributed force [71] or by the angular acceleration [83], and singularity of its solution is discussed in [69]; optimal control of transverse vibration of Euler-Bernoulli beam is introduced in [79]. We will consider displacement tracking of the Cosserat rod in this paper, which, to the best of our knowledge, has not been studied before.…”

Section: Introductionmentioning

confidence: 99%

A monolithic optimal control method for displacement tracking of Cosserat rod with application to reconstruction of C. elegans locomotion

Wang

Ranner

et al. 2022

Comput Mech

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This article considers an inverse problem for a Cosserat rod where we are given only the position of the centreline of the rod and must solve for external forces and torques as well as the orientation of the cross sections of the centreline. We formulate the inverse problem as an optimal control problem using the position of the centreline as an objective function with the external force and torque as control variables, with meaningful regularisation of the orientations. A monolithic, implicit numerical scheme is proposed in the sense that primal and adjoint equations are solved in a fully-coupled manner and all the nonlinear coefficients of the governing partial differential equations are updated to the current state variables. The forward formulation, determining rod configuration from external forces and torques, is first validated by a numerical benchmark; the solvability and stability of the inverse problem are then tested using data from forward simulations. The proposed optimal control method is motivated by reconstruction of the orientations of a rod’s cross sections, with its centreline being captured through imaging protocols. As a case study, we take the locomotion of the nematode, Caenorhabditis elegans. In this study we take laboratory data for its centreline and infer its cross-section orientation (muscle locations) with the control force and torque being interpreted as the reaction force, activated by C. elegans’ muscles, from the surrounding fluids. This method thus combines the mathematical modelling and laboratory data to study the locomotion of C. elegans, which gives us insights into the potential anatomical orientation of the worm beyond what can be observed through the laboratory data. The paper is completed with several additional remarks explaining the theoretical and technical details of the model.

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Energy-Optimal Control by Boundary Forces for Longitudinal Vibrations of an Elastic Rod

Kostin

Gavrikov

2023

Lecture Notes in Mechanical Engineering

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Optimal Control of Longitudinal Motion of an Elastic Rod Using Boundary Forces

Cited by 6 publications

References 22 publications

Optimal Control of Longitudinal Motions for an Elastic Rod with Distributed Forces

Optimal Control of Longitudinal Motions for an Elastic Rod with Distributed Forces

A monolithic optimal control method for displacement tracking of Cosserat rod with application to reconstruction of C. elegans locomotion

Energy-Optimal Control by Boundary Forces for Longitudinal Vibrations of an Elastic Rod

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