arXiv research

Paper provides closed-form time derivatives for rigid body systems.

Geometrically interpolates rigid body motions with initial and terminal twists.

New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.

New equations for rigid body motion on infinite-dimensional spaces of operators.

Study of rigid body displacements in a projective space over dual numbers with geometric interpretations.

A linkage mechanism consists of rigid bodies assembled by joints which can be used to translate and transfer motion from one form in one place to another. In this paper, we are particularly interested in a family of spacial linkage mechanisms which consist of $n$-copies of a rigid body joined together by hinges to form…

Suppose that the initial triangle formed by the three moving masses of the three-body problem is similar to the triangle formed at some later time. We derive a simple integral formula for the overall rotation relating the two triangles. The formula is based on the fact that the space of similarity classes of triangles …

Paper derives and applies a parallel transport equation on Lie groups.

Extended Regge complex for linearized Riemann-Cartan geometry and cohomology.

This monograph describes a Riemannian geometric reduction approach to the three-body problem. The fundamental theorems are presented in the introductory part, whereas their proofs are provided in later chapters where specific topics are analyzed in more detail. The basic idea is to reduce the kinematic and dynamics of …

Study cohomological equation for robotic screw motions on SE(3).

New normalizing flows model molecular crystal structures.

The aim of this note is to present simple proofs of the completeness of Manakov's integrals for a motion of a rigid body fixed at a point in $\mathbb R^n$, as well as for geodesic flows on a class of homogeneous spaces $SO(n)/SO(n_1)\times\dots\times SO(n_r)$.

Fixed points of Minkowski valuations are found in specific ball neighborhoods.

New approach for obstacle avoidance in robotics using learned representations.

Construct intrinsic Langevin dynamics for rigid inclusions on curved surfaces.

Study on curvature measures in non-Euclidean spaces linked to Euclidean geometry.

The decomposition of the space of continuous and translation invariant valuations into a sum of SO(n) irreducible subspaces is obtained. A reformulation of this result in terms of a Hadwiger type theorem for continuous translation invariant and SO(n)-equivariant tensor valuations is also given. As an application, symme…

We propose a sliding surface for systems on the Lie group $SO(3)\times \mathbb{R}^3$ . The sliding surface is shown to be a Lie subgroup. The reduced-order dynamics along the sliding subgroup have an almost globally asymptotically stable equilibrium. The sliding surface is used to design a sliding-mode controller for t…

Louis Poinsot has shown in 1854 that the motion of a rigid body, with one of its points fixed, can be described as the rolling without slipping of one cone, the 'body cone', along another, the 'space cone', with their common vertex at the fixed point. This description has been further refined by the second author in 19…

The paper proves convex bodies are minimal fillings and have Lipschitz-volume rigidity.

In a recent paper (arXiv:math-ph/0609076) the authors investigated the basic global geometry of congruence moduli curves and shape curves of 3-body motions with vanishing angular momentum. Here the study is extended to the case of planary 3-body motions in general. In particular, the results on the separation of the si…

Integrates magnetic geodesic and sub-Riemannian flows on Stefel variety, proving integrability and Lax presentations.

Various neural network based methods are capable of anticipating human body motions from data for a short period of time. What these methods lack are the interpretability and explainability of the network and its results. We propose to use Dynamic Mode Decomposition with delays to represent and anticipate human body mo…

We consider the motion of small bodies in general relativity. The key result captures a sense in which such bodies follow timelike geodesics (or, in the case of charged bodies, Lorentz-force curves). This result clarifies the relationship between approaches that model such bodies as distributions supported on a curve, …

Linking human whole-body motion and natural language is of great interest for the generation of semantic representations of observed human behaviors as well as for the generation of robot behaviors based on natural language input. While there has been a large body of research in this area, most approaches that exist to…

Study of motion control systems on Lie groups with specific geometric constraints.

A new diffusion model generates novel protein backbones without relying on pretrained networks.

Paper compares Lagrangian reduction methods for rigid body systems.

Equations for minimal surfaces from rigid motions in high dimensions.

Integrable geodesics found on special orthogonal group.

We consider a generalization of the notion of a natural mechanical system to the case of additional forces of gyroscopic type. Such forces appear, for example, as a result of global reduction of a natural system with symmetry. We study symmetries in the systems with gyroscopic forces to find out when these systems admi…

Swept Volume (SV), the volume displaced by an object when it is moving along a trajectory, is considered a useful metric for motion planning. First, SV has been used to identify collisions along a trajectory, because it directly measures the amount of space required for an object to move. Second, in sampling-based moti…

Develops computational methods for simulating rigid body dynamics on SO(3).

We consider an SO(4) Euler rigid body with two 'inertia momenta' coinciding. We study it from the point of view of bihamiltonian geometry. We show how to algebraically integrate it by means of the method of separation of variables.

Study examines how body segments respond to random vibrations.

The paper studies rigidity and continuity in nonlinear elasticity on manifolds and hypersurfaces.

The paper studies how points and lines can move while preserving incidences.

We prove complete integrability of the Manakov-type SO(n)-invariant geodesic flows on homogeneous spaces $SO(n)/SO(k_1)\times...\times SO(k_r)$, for any choice of $k_1,...,k_r$, $k_1+...+k_r\le n$. In particular, a new proof of the integrability of a Manakov symmetric rigid body motion around a fixed point is presented…

The paper derives Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.

We present a convolutional network that is equivariant to rigid body motions. The model uses scalar-, vector-, and tensor fields over 3D Euclidean space to represent data, and equivariant convolutions to map between such representations. These SE(3)-equivariant convolutions utilize kernels which are parameterized as a …

In this note we find a generic defining function of projective motion in the 6-dimensional rigid h-space.

In the paper, some concepts of modern differential geometry are used as a basis to develop an invariant theory of mechanical systems, including systems with gyroscopic forces. An interpretation of systems with gyroscopic forces in the form of flows of a given geodesic curvature is proposed. For illustration, the proble…

Diagnostic stroke imaging with C-arm cone-beam computed tomography (CBCT) enables reduction of time-to-therapy for endovascular procedures. However, the prolonged acquisition time compared to helical CT increases the likelihood of rigid patient motion. Rigid motion corrupts the geometry alignment assumed during reconst…

In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean $C^2$-smooth surface in the affine group and the group of rigid motions of the Minkowski plane away from characteristic points and signed geodesic curvature for Euclidean $C^2$-smooth curves on surfaces. We get Gauss-Bonnet theorems i…

We build CSI-Net, a unified Deep Neural Network~(DNN), to learn the representation of WiFi signals. Using CSI-Net, we jointly solved two body characterization problems: biometrics estimation (including body fat, muscle, water, and bone rates) and person recognition. We also demonstrated the application of CSI-Net on tw…

Long-term human motion can be represented as a series of motion modes---motion sequences that capture short-term temporal dynamics---with transitions between them. We leverage this structure and present a novel Motion Transformation Variational Auto-Encoders (MT-VAE) for learning motion sequence generation. Our model j…

A Steiner type formula for continuous translation invariant Minkowski valuations is established. In combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations, this new Steiner type formula is used to obtain a family of Brunn-Minkowski type inequalities for rigid motion intertwi…