Mirror therapy is a standard technique of rehabilitation for recovering motor and vision abilities of stroke patients, especially in the case of asymmetric limb function. To enhance traditional mirror therapy, robotic mirror therapy (RMT) has been proposed over the past decade, allowing for assisted bimanual coordination of paretic (affected) and contralateral (healthy) limbs. However, state-of-the-art RMT platforms predominantly target mirrored motions of trajectories, largely limited to 2-D motions. In this paper, an RMT platform is proposed, which can facilitate the patient to practice virtual activities of daily living (ADL) and thus enhance their independence. Two similar (but mirrored) 3D virtual environments are created in which the patients operate robots with both their limbs to complete ADL (such as writing and eating) with the assistance of the therapist. The recovery level of the patient is continuously assessed by monitoring their ability to track assigned trajectories. The patient's robots are programmed to assist the patient in following these trajectories based on this recovery level. In this paper, the framework to dynamically monitor recovery level and accordingly provide assistance is developed along with the nonlinear controller design to ensure position tracking, force control, and stability. Proof-of-concept studies are conducted with both 3D trajectory tracking and ADL. The results demonstrate the potential use of the proposed system to enhance the recovery of the patients.

This paper presents a method that convexifies state-constrained optimal control problems in the control-input space. The proposed method enables convex programming methods to find the globally optimal solution even if costs and control constraints are non-convex in control and convex in state, dynamics is non-affine in control and convex in state, and state constraints are convex in state. Under the above conditions, generic methods do not guarantee to find optimal solutions, but the proposed method does. The proposed approach is demonstrated in a sixteen-dimensional navigation example. 

This paper considers the problem of optimizing robot navigation with respect to a time-varying objective encoded into a navigation density function. We are interested in designing state feedback control laws that lead to an almost everywhere stabilization of the closed-loop system to an equilibrium point while navigating a region optimally and safely (that is, the transient leading to the final equilibrium point is optimal and satisfies safety constraints). Though this problem has been studied in literature within many different communities, it still remains a challenging non-convex control problem. In our approach, under certain assumptions on the time-varying navigation density, we use Koopman and Perron-Frobenius Operator theoretic tools to transform the problem into a convex one in infinite dimensional decision variables. In particular, the cost function and the safety constraints in the transformed formulation become linear in these functional variables. Finally, we present some numerical examples to illustrate our approach, as well as discuss the current limitations and future extensions of our framework to accommodate a wider range of robotics applications.

The paper is about the computation of the principal spectrum of the Koopman operator (i.e., eigenvalues and eigenfunctions). The principal eigenfunctions of the Koopman operator are the ones with the corresponding eigenvalues equal to the eigenvalues of the linearization of the nonlinear system at an equilibrium point. The main contribution of this paper is to provide a novel approach for computing the principal eigenfunctions using a path-integral formula. Furthermore, we provide conditions based on the stability property of the dynamical system and the eigenvalues of the linearization towards computing the principal eigenfunction using the path-integral formula. Further, we provide a Deep Neural Network framework that utilizes our proposed path-integral approach for eigenfunction computation in high-dimension systems. Finally, we present simulation results for the computation of principal eigenfunction and demonstrate their application for determining the stable and unstable manifolds and constructing the Lyapunov function.

Modern data-driven techniques have rapidly progressed beyond modelling and systems identification, with a growing interest in learning high-level dynamical properties of a system, such as safe-set invariance, reachability, input-to-state stability etc. In this paper, we propose a novel supervised Deep Learning technique for constructing Lyapunov certificates, by leveraging Koopman Operator theory-based numerical tools (Extended Dynamic Mode Decomposition and Generalized Laplace Analysis) to robustly and efficiently generate explicit ground truth data for training. This is in stark contrast to existing Deep Learning methods where the loss functions plainly penalize Lyapunov condition violation in the absence of labelled data for direct regression. Furthermore, our approach leads to a linear parameterization of Lyapunov candidate functions, making them more interpretable compared to standard DNN-based architecture. We demonstrate and validate our approach numerically using 2-dimensional and 10-dimensional examples.

Although Agapanthus africanus (L.) Hoffm. is one of the most popular ornamental species in both hemispheres, it has an extremely restricted wild occurrence (Cape province, South Africa). This contradiction between gener-alized ornamental application and natural distribution was the basis for the analytical approach adopted in the present work. We hypothesized that characteristic features of the cuticular waxes were adopted by this species to help it cope with severe dehydration associated with marine salinity on account of the short distance of the wild populations to the sea. A comprehensive morpho-anatomical, histological and physical-chemical analysis was performed on the epicuticular and intracuticular layers of the adaxial and abaxial surfaces of leaves of specimens of A. africanus. The adaxial epicuticular surface is hydrophilic and the abaxial epicuticular surface exhibits globally hydrophobic behavior. The main chemical compounds detected in the wax layers of both surfaces of the leaf are the short-chain monocaprylin monoglyceride (C8), and very long-chain 1-hexacosanol (C26) and 1-octa-cosanol (C28) alcohols. While monocaprylin is particularly abundant in the intracuticular layers, the epicuticular adaxial surface revealed the highest concentration of both alcohols. We demonstrate that the smart combination of these two classes of molecules with opposite water affinity endows the A. africanus leaf cuticle with a unique water management system combining the efficient entrapment of water in the disordered alpha-gel phase formed by monocaprylin and the high resistance to water transport provided by ordered domains composed of tightly packed, all-trans alkyl chains of the above pair of alcohols. The remarkable structural similarity existing between the monocaprylin alpha-gel and the mucilage of algae is an evidence of the terrestrialization process.

Modern data-driven techniques have rapidly progressed beyond modelling and systems identification, with a growing interest in learning high-level dynamical properties of a system, such as safe-set invariance, reachability, input-to-state stability etc. In this paper, we propose a novel supervised Deep Learning technique for constructing Lyapunov certificates, by leveraging Koopman Operator theory-based numerical tools (Extended Dynamic Mode Decomposition and Generalized Laplace Analysis) to robustly and efficiently generate explicit ground truth data for training. This is in stark contrast to existing Deep Learning methods where the loss functions plainly penalize Lyapunov condition violation in the absence of labelled data for direct regression. Furthermore, our approach leads to a linear parameterization of Lyapunov candidate functions, making them more interpretable compared to standard DNN-based architecture. We demonstrate and validate our approach numerically using 2-dimensional and 10-dimensional examples.