Impulsive mechanical systems exhibit discontinuous jumps in their state, and when such jumps are triggered by spatial events, the geometry of the impact surface carries information about the controllability of the hybrid dynamics. For mechanical systems defined on principal G-bundles, two qualitatively distinct types of impacts arise: interior impacts, associated with events on the shape space, and exterior impacts, associated with events on the fibers. A key distinction is that interior impacts preserve the mechanical connection, whereas exterior impacts generally do not. In this paper, we exploit this distinction by allowing actuation through exterior impacts. We study the pendulum-on-a-cart system, derive controlled reset laws induced by moving-wall impacts, and analyze the resulting periodic motions. Our results show that reset action alone does not provide a convincing stabilizing regime, whereas the addition of dissipation in the continuous flow yields exponentially stable periodic behavior for suitable feedback gains.

The nonholonomic integrator is a famous example in feedback design - although it is small-time locally controllable to the origin, no continuous feedback law exists. Therefore, any stabilizing feedback laws must be either time-varying or discontinuous. A previously studied discontinuous feedback law stabilizes initial conditions lying between two paraboloids and has a sliding mode on the xy-plane. We investigate the effect of introducing delays into this discontinuous feedback law. To a first-order analysis, the lag causes the sliding mode of the xy-plane to bifurcate into two switching regions where the resulting dynamics can be interpreted as a hybrid dynamical system with hysteresis. Counterintuitively, the presence of a delay can actually have a positive effect on both the size of the basin of attraction and the convergence rate of the controller. We also consider the natural generalization of the nonholonomic integrator to higher dimensions.

This work generalizes the classical metriplectic formalism to model Hamiltonian systems with nonconservative dissipation. Classical metriplectic representations allow for the description of energy conservation and production of entropy via a suitable selection of an entropy function and a bilinear symmetric metric. By relaxing the Casimir invariance requirement of the entropy function, this paper shows that the generalized formalism induces the free energy analogous to thermodynamics. The monotonic change of free energy can serve as a more precise criterion than mechanical energy or entropy alone. This paper provides examples of the generalized metriplectic system in a 2-dimensional Hamiltonian system and SO(3). This paper also provides a bilevel convex optimization approach for the identification of the metriplectic system given measurements of the system.

Hybrid dynamical systems are systems which undergo both continuous and discrete transitions. The Bolza problem from optimal control theory was applied to these systems and a hybrid version of Pontryagin's maximum principle was presented. This hybrid maximum principle was presented to emphasize its geometric nature which made its study amenable to the tools of geometric mechanics and symplectic geometry. One explicit benefit of this geometric approach was that the symplectic structure (and hence the induced volume) was preserved. This allowed for a hybrid analog of caustics and conjugate points. Additionally, an introductory analysis of singular solutions (beating and Zeno) was discussed geometrically. This work concluded on a biological example where beating can occur.

Hybrid systems are dynamical systems with continuous-time and discrete-time components in their dynamics. When hybrid systems are defined on a principal bundle we are able to define two classes of impacts for the discrete-time transition of the dynamics: interior impacts and exterior impacts. In this paper we define hybrid systems on principal bundles, study the underlying geometry on the switching surface where impacts occur and we find conditions for which both exterior and interior impacts are preserved by the mechanical connection induced in the principal bundle.

Hybrid dynamical systems are systems which undergo both continuous and discrete transitions. As typical in dynamical analysis, an essential goal is to study the long-term behavior of these systems. In this work, we present two different novel approaches for studying these systems. The first approach is based on constructing an analog of the Frobenius-Perron (transport) operator for hybrid systems. Rather than tracking the evolution of a single trajectory, this operator encodes the asymptotic nature of an ensemble of trajectories. The second approach presented applies to an important subclass of hybrid systems, mechanical impact systems. We develop an analog of Lie-Poisson(-Suslov) reduction for left-invariant impact systems on Lie groups. In addition to the Hamiltonian (and constraints) being left-invariant, the impact surface must also be a right coset of a normal subgroup. This procedure allows a reduction from a 2n-dimensional system to an (n+1)-dimensional one. We conclude the paper by presenting numerical results on a diverse array of applications.

Optimal control is ubiquitous in many fields of engineering. A common technique to find candidate solutions is via Pontryagin's maximum principle. An unfortunate aspect of this method is that the dimension of system doubles. When the system evolves on a Lie group and the system is invariant under left (or right) translations, Lie-Poisson reduction can be applied to eliminate half of the dimensions (and returning the dimension of the problem to the back to the original number). Hybrid control systems are an extension of (continuous) control systems by allowing for sudden changes to the state. Examples of such systems include the bouncing ball - the velocity instantaneously jumps during a bounce, the thermostat - controls switch to on or off, and a sailboat undergoing tacking. The goal of this work is to extend the idea of Lie-Poisson reduction to the optimal control of these systems. If n is the dimension of the original system, 2n is the dimension of the system produced by the maximum principle. In the case of classical Lie-Poisson reduction, the dimension drops back down to n. This, unfortunately, is impossible in hybrid systems as there must be an auxiliary variable encoding whether or not an event occurs. As such, the analogous hybrid Lie-Poisson reduction results in a n+1 dimensional system. The purpose of this work is to develop and present this technique.

Hybrid (and impact) systems are dynamical systems experiencing both continuous and discrete transitions. In this work, we derive necessary and sufficient conditions for when a given differential form is invariant, with special attention paid to the case of the existence of invariant volumes. Particular attention is given to impact systems where the continuous dynamics are Lagrangian and subject to nonholonomic constraints. A celebrated result for volume-preserving dynamical systems is Poincaré recurrence. In order to be recurrent, trajectories need to exist for long periods of time, which can be controlled in continuous-time systems through e.g. compactness. For hybrid systems, an additional mechanism can occur which breaks long-time existence: Zeno (infinitely many discrete transitions in a finite amount of time). We demonstrate that the existence of a smooth invariant volume severely inhibits Zeno behavior; hybrid systems with the “boundary identity property” along with an invariant volume-form have almost no Zeno trajectories (although Zeno trajectories can still exist). This leads to the result that many billiards (e.g. the classical point, the rolling disk, and the rolling ball) are recurrent independent on the shape of the compact table-top.

We revisit a textbook example of a singularly perturbed nonlinear boundary-value problem. Unexpectedly, it shows a wealth of phenomena that seem to have been overlooked previously, including a pitchfork bifurcation in the number of solutions as one varies the small parameter, and transcendentally small terms in the initial conditions that can be calculated by elementary means. Based on our own classroom experience, we believe this problem could provide an enjoyable workout for students in courses on perturbation methods, applied dynamical systems, or numerical analysis.

This paper proposes a convolution structure for learning SE(3)-equivariant features from 3D point clouds. It can be viewed as an equivariant version of kernel point convolutions (KPConv), a widely used convolution form to process point cloud data. Compared with existing equivariant networks, our design is simple, lightweight, fast, and easy to be integrated with existing task-specific point cloud learning pipelines. We achieve these desirable properties by combining group convolutions and quotient representations. Specifically, we discretize SO(3) to finite groups for their simplicity while using SO(2) as the stabilizer subgroup to form spherical quotient feature fields to save computations. We also propose a permutation layer to recover SO(3) features from spherical features to preserve the capacity to distinguish rotations. Experiments show that our method achieves comparable or superior performance in various tasks, including object classification, pose estimation, and keypoint-matching, while consuming much less memory and running faster than existing work. The proposed method can foster the development of equivariant models for real-world applications based on point clouds.

A refined microstructure consisting of bcc precipitates embedded in an ordered B2 matrix has been observed in the refractory high entropy alloy (RHEA) AlMo0.5NbTa0.5TiZr, resembling an "inverted superalloy-like" micro-structure. A planar {100} bcc/B2 interface exists in this state. Coarsening of the microstructure occurs after aging at 1000 degrees C for 6 h, resulting in a faceted interface. In this case, the primary plane of the interface is parallel to a {100} plane, while the step associated with the facet is parallel to a {110} plane. Misfit dislocations at the interface were observed on the {110} interface plane, with Burgers vector of a/2 111bcc, which extend into the bcc phase, and have an average spacing of 13.7 +/- 0.6 nm. No lattice invariant deformation (LID) normal to the primary {110} planes was observed, such that the transformation between the B2 and bcc phases constitutes a pure expansion.

The intersection of machine learning and dynamical systems has generated considerable interest recently. Neural Ordinary Differential Equations (NODEs) represent a rich overlap between these fields. In this paper, we develop a continuous time neural network approach based on Delay Differential Equations (DDEs). Our model uses the adjoint sensitivity method to learn the model parameters and delay directly from data. Our approach is inspired by that of NODEs and extends earlier neural DDE models, which have assumed that the value of the delay is known a priori. We perform a sensitivity analysis on our proposed approach and demonstrate its ability to learn DDE parameters from benchmark systems. We conclude our discussion with potential future directions and applications.

Geometric optimal control utilizes tools from differential geometry to analyze the structure of a problem to determine the control and state trajectories to reach a desired outcome while minimizing some cost function. For a controlled mechanical system, the control usually manifests as an external force which, if conservative, can be added to the Hamiltonian. In this letter, we focus on mechanical systems with controls added to the symplectic form rather than the Hamiltonian. In practice, this translates to controlling the magnetic field for an electrically charged system. We develop a basic theory deriving necessary conditions for optimality of such a system subjected to nonholonomic constraints. We consider the representative example of a magnetically charged Chaplygin Sleigh, whose resulting optimal control problem is completely integrable.

We derive conditions for a nonholonomic system subject to nonlinear constraints (obeying Chetaev's rule) to preserve a smooth volume form. When applied to affine constraints, these conditions dictate that a basic invariant density exists if and only if a certain 1-form is exact and a certain function vanishes (this function automatically vanishes for linear constraints). Moreover, this result can be extended to geodesic flows for arbitrary metric connections and the sufficient condition manifests as integrability of the torsion. As a consequence, volume-preservation of a nonholonomic system is closely related to the torsion of the nonholonomic connection. Examples of nonlinear/affine/linear constraints are considered.

A unique feature of hybrid dynamical systems (systems whose evolution is subject to both continuous- and discrete-time laws) is Zeno trajectories. Usually these trajectories are avoided as they can cause incorrect numerical results as the problem becomes ill-conditioned. However, these are difficult to justifiably avoid as determining when and where they occur is a non-trivial task. It turns out that in optimal control problems, not only can they not be avoided, but are sometimes required in synthesizing the solutions. This work explores the pedagogical example of the bouncing ball to demonstrate the importance of "Zeno control executions."

Why do we move forward when we walk? Our legs undergo periodic motion and thus possess no net change in position; however, our bodies do possess a net change in position and we are propelled forward. From a geometric perspective, this phenomenon of periodic input producing non-periodic output is holonomy. To obtain non-zero holonomy and propel forward, we must alternate which leg is in contact with the ground; a non-zero net motion can be obtained by concatenating arcs that would individually produce no net motion. We develop a framework for computing the holonomy group of hybrid systems and analyze their behavior in the limit as the number of impacts goes to infinity.

This paper presents a control framework on Lie groups by designing the control objective in its Lie algebra. Control on Lie groups is challenging due to its nonlinear nature and difficulties in system parameterization. Existing methods to design the control objective on a Lie group and then derive the gradient for controller design are non-trivial and can result in slow convergence in tracking control. We show that with a proper left-invariant metric, setting the gradient of the cost function as the tracking error in the Lie algebra leads to a quadratic Lyapunov function that enables globally exponential convergence. In the PD control case, we show that our controller can maintain an exponential convergence rate even when the initial error is approaching π in SO(3). We also show the merit of this proposed framework in trajectory optimization. The proposed cost function enables the iterative Linear Quadratic Regulator (iLQR) to converge much faster than the Differential Dynamic Programming (DDP) with a well-adopted cost function when the initial trajectory is poorly initialized on SO(3).

This paper reports on a new error-state Model Predictive Control (MPC) approach to connected matrix Lie groups for robot control. The linearized tracking error dynamics and the linearized equations of motion are derived in the Lie algebra. Moreover, given an initial condition, the linearized tracking error dynamics and equations of motion are globally valid and evolve independently of the system trajectory. By exploiting the symmetry of the problem, the proposed approach shows faster convergence of rotation and position simultaneously than the state-of-the-art geometric variational MPC based on variational-based linearization. Numerical simulation on tracking control of a fully-actuated 3D rigid body dynamics confirms the benefits of the proposed approach compared to the baselines. Furthermore, the proposed MPC is also verified in pose control and locomotion experiments on a quadrupedal robot MIT Mini Cheetah.

Four adult dogs weighing <10 kg presented for the evaluation of severe mitral valve stenosis with clinical signs. Owing to the size of the dogs, a hybrid surgical and interventional approach was utilized for balloon valvuloplasty. A left lateral thoracotomy was performed to allow direct entry through the left atrial wall. Transesophageal echocardiography was utilized for the entirety of the procedure in all dogs, and fluoroscopy was additionally used in two dogs. One dog had mild to moderate intra-operative bleeding from the left atrial wall during the procedure, but no other intra-operative complications were observed. No dogs developed a clinically relevant amount of worsened mitral regurgitation. Based on mitral leaflet mobility and transmitral flow profiles, there was perceived improvement in all four dogs. One dog died 6 h after extubation due to respiratory arrest. The remaining dogs survived to discharge and had resolution of clinical signs at home and discontinuation of heart failure medications. One dog died of an unknown cause at five months and another developed atrial fibrillation, and the owners elected to euthanize at ten months after the procedure. One dog continues to do well six months after the procedure as of the time of this writing. Hybrid balloon valvuloplasty can be a viable management option for small breed dogs with severe mitral stenosis exhibiting clinical signs, and both transesophageal echocardiography and fluoroscopy can be used intra-operatively to assist in successful procedural outcomes.