Optimality Conditions (in Pontryagin Form)

Aronna, M. Soledad; Tonon, Daniela; Boccia, Andrea; Campos, Cédric M.; Mazzola, Marco; Luong, Nguyen Van; Palladino, Michele; Scarinci, Teresa; Silva, Francisco J.

doi:10.1007/978-3-319-60771-9_1

Cited by 4 publications

(3 citation statements)

References 111 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…for all t ∈ [0, T f ] and β ≥ 0, where

is just the vector perpendicular to

. In Appendix C , we show the details of the calculation based on Pontryagin's maximum principle (Aronna et al, 2017 ), showing that this indeed meets the requirements of the optimal control problem described in Equations (47)–(49). Substituting the specific solution described in Equation (52) into the dynamics of Equation (47) while also writing the tangent vector in terms of the angle θ( s, t ) between

and the stimulus direction

, i.e.,

, yields the following dynamical equation:…”

Section: Example Of An Optimal Control Approachmentioning

confidence: 55%

A General 3D Model for Growth Dynamics of Sensory-Growth Systems: From Plants to Robotics

et al. 2020

Self Cite

View full text Add to dashboard Cite

In recent years, there has been a rise in interest in the development of self-growing robotics inspired by the moving-by-growing paradigm of plants. In particular, climbing plants capitalize on their slender structures to successfully negotiate unstructured environments while employing a combination of two classes of growth-driven movements: tropic responses, growing toward or away from an external stimulus, and inherent nastic movements, such as periodic circumnutations, which promote exploration. In order to emulate these complex growth dynamics in a 3D environment, a general and rigorous mathematical framework is required. Here, we develop a general 3D model for rod-like organs adopting the Frenet-Serret frame, providing a useful framework from the standpoint of robotics control. Differential growth drives the dynamics of the organ, governed by both internal and external cues while neglecting elastic responses. We describe the numerical method required to implement this model and perform numerical simulations of a number of key scenarios, showcasing the applicability of our model. In the case of responses to external stimuli, we consider a distant stimulus (such as sunlight and gravity), a point stimulus (a point light source), and a line stimulus that emulates twining of a climbing plant around a support. We also simulate circumnutations, the response to an internal oscillatory cue, associated with search processes. Lastly, we also demonstrate the superposition of the response to an external stimulus and circumnutations. In addition, we consider a simple example illustrating the possible use of an optimal control approach in order to recover tropic dynamics in a way that may be relevant for robotics use. In all, the model presented here is general and robust, paving the way for a deeper understanding of plant response dynamics and also for novel control systems for newly developed self-growing robots.

show abstract

“…for all t ∈ [0, T f ] and β ≥ 0, where

is just the vector perpendicular to

and the stimulus direction

, i.e.,

, yields the following dynamical equation:…”

Section: Example Of An Optimal Control Approachmentioning

confidence: 55%

A General 3D Model for Growth Dynamics of Sensory-Growth Systems: From Plants to Robotics

et al. 2020

Self Cite

View full text Add to dashboard Cite

show abstract

“…49: for all t ∈ [0, T f ] and β ≥ 0, where is just the perpendicular vector to . In Appendix C in the SM we show the details of the calculation based on the Pontryagin’s maximum principle (Aronna et al, 2017), showing that this indeed meets the requirements of the optimal control problem described in Eqs. 46, 47, 48.…”

Section: Example Of An Optimal Control Approachmentioning

confidence: 69%

A general 3D model for growth dynamics of sensory-growth systems: from plants to robotics

Porat

Tedone

Palladino

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

2In recent years there has been a rise in interest in the development of self-growing robotics 3 inspired by the moving-by-growing paradigm of plants. In particular, climbing plants capitalize on 4 their slender structures to successfully negotiate unstructured environments, while employing 5 a combination of two classes of growth-driven movements: tropic responses, which direct 6 growth in the direction of an external stimulus, and inherent nastic movements, such as periodic 7 circumnutations, which promote exploration. In order to emulate these complex growth dynamics 8 in a 3D environment, a general and rigorous mathematical framework is required. Here we 9 develop a general 3D model for rod-like organs adopting the Frenet-Serret frame, providing a 10 useful framework from the standpoint of robotics control. Differential growth drives the dynamics 11 of the organ, governed by both internal and external cues. We describe the numerical method 12 required to implement this model, and perform numerical simulations of a number of key scenarios, 13 showcasing the applicability of our model. In the case of responses to external stimuli, we consider 14 a distant stimulus (such as sunlight and gravity), a point stimulus (a point light source), and a 15 line stimulus which emulates twining of a climbing plant around a support. We also simulate 16 circumnutations, the response to an internal oscillatory cue, associated with search processes. 17 Lastly we also demonstrate the superposition of both the response to an external stimulus 18 together with circumnutations. Lastly we consider a simple example illustrating the possible use of 19 an optimal control approach in order to recover tropic dynamics, in a way which may be relevant 20 for robotics use. In all, the model presented here is general and robust, paving the way for a 21 deeper understanding of plant response dynamics, as well as novel control systems for newly 22 developed self-growing robots. 23

show abstract

“…One prominent approach to find an optimal feedback law is calculating the value function, which can be done by solving either the Bellman equation or the Hamilton-Jacobi-Bellman equation (HJB). Popular numerical solutions to this problem are semi-Lagrangian methods [17,22,66], Domain splitting algorithms [23], variational iterative methods [35], data based methods with Neural Networks [47,49] or Policy Iteration with Galerkin ansatz [36,45].…”

Section: Introductionmentioning

confidence: 99%

Approximating the Stationary Bellman Equation by Hierarchical Tensor Products

Oster

Sallandt

Schneider

2024

JCM

View full text Add to dashboard Cite

We treat infinite horizon optimal control problems by solving the associated stationary Bellman equation numerically to compute the value function and an optimal feedback law. The dynamical systems under consideration are spatial discretizations of non linear parabolic partial differential equations (PDE), which means that the Bellman equation suffers from the curse of dimensionality. Its non linearity is handled by the Policy Iteration algorithm, where the problem is reduced to a sequence of linear equations, which remain the computational bottleneck due to their high dimensions. We reformulate the linearized Bellman equations via the Koopman operator into an operator equation, that is solved using a minimal residual method. Using the Koopman operator we identify a preconditioner for operator equation, which deems essential in our numerical tests. To overcome computational infeasability we use low rank hierarchical tensor product approximation/tree-based tensor formats, in particular tensor trains (TT tensors) and multi-polynomials, together with high-dimensional quadrature, e.g. Monte-Carlo. By controlling a destabilized version of viscous Burgers and a diffusion equation with unstable reaction term numerical evidence is given.

show abstract

Optimality Conditions (in Pontryagin Form)

Cited by 4 publications

References 111 publications

A General 3D Model for Growth Dynamics of Sensory-Growth Systems: From Plants to Robotics

A General 3D Model for Growth Dynamics of Sensory-Growth Systems: From Plants to Robotics

A general 3D model for growth dynamics of sensory-growth systems: from plants to robotics

Approximating the Stationary Bellman Equation by Hierarchical Tensor Products

Contact Info

Product

Resources

About