Structured Linearization of Discrete Mechanical Systems for Analysis and Optimal Control

Variational integrators are well-suited for simulation of mechanical systems because they preserve mechanical quantities about a system such as momentum, or its change if external forcing is involved, and holonomic constraints. While they are not energy-preserving they do exhibit long-time stable energy behavior. However, variational integrators often simulate mechanical system dynamics by solving an implicit difference equation at each time step, one that is moreover expressed purely in terms of configurations at different time steps. This paper formulates the first- and second-order linearizations of a variational integrator in a manner that is amenable to control analysis and synthesis, creating a bridge between existing analysis and optimal control tools for discrete dynamic systems and variational integrators for mechanical systems in generalized coordinates with forcing and holonomic constraints. The forced pendulum is used to illustrate the technique. A second example solves the discrete LQR problem to find a locally stabilizing controller for a 40 DOF system with 6 constraints.Comment: 13 page

Structured Linearization of Discrete Mechanical Systems for Analysis and Optimal Control

Variational integrators are well-suited for simulation of mechanical systems because they preserve mechanical quantities about a system such as momentum, or its change if external forcing is involved, and holonomic constraints. While they are not energy-preserving they do exhibit long-time stable energy behavior. However, variational integrators often simulate mechanical system dynamics by solving an implicit difference equation at each time step, one that is moreover expressed purely in terms of configurations at different time steps. This paper formulates the first- and second-order linearizations of a variational integrator in a manner that is amenable to control analysis and synthesis, creating a bridge between existing analysis and optimal control tools for discrete dynamic systems and variational integrators for mechanical systems in generalized coordinates with forcing and holonomic constraints. The forced pendulum is used to illustrate the technique. A second example solves the discrete LQR problem to find a locally stabilizing controller for a 40 DOF system with 6 constraints.Comment: 13 page

GroomGen: A High-Quality Generative Hair Model Using Hierarchical Latent Representations

Despite recent successes in hair acquisition that fits a high-dimensional hair model to a specific input subject, generative hair models, which establish general embedding spaces for encoding, editing, and sampling diverse hairstyles, are way less explored. In this paper, we present GroomGen, the first generative model designed for hair geometry composed of highly-detailed dense strands. Our approach is motivated by two key ideas. First, we construct hair latent spaces covering both individual strands and hairstyles. The latent spaces are compact, expressive, and well-constrained for high-quality and diverse sampling. Second, we adopt a hierarchical hair representation that parameterizes a complete hair model to three levels: single strands, sparse guide hairs, and complete dense hairs. This representation is critical to the compactness of latent spaces, the robustness of training, and the efficiency of inference. Based on this hierarchical latent representation, our proposed pipeline consists of a strand-VAE and a hairstyle-VAE that encode an individual strand and a set of guide hairs to their respective latent spaces, and a hybrid densification step that populates sparse guide hairs to a dense hair model. GroomGen not only enables novel hairstyle sampling and plausible hairstyle interpolation, but also supports interactive editing of complex hairstyles, or can serve as strong data-driven prior for hairstyle reconstruction from images. We demonstrate the superiority of our approach with qualitative examples of diverse sampled hairstyles and quantitative evaluation of generation quality regarding every single component and the entire pipeline.Comment: SIGGRAPH Asia 202

Differential Equation Models and Numerical Methods for Reverse Engineering Genetic Regulatory Networks

This dissertation develops and analyzes differential equation-based mathematical models and efficient numerical methods and algorithms for genetic regulatory network identification. The primary objectives of the dissertation are to design, analyze, and test a general variational framework and numerical methods for seeking its approximate solutions for reverse engineering genetic regulatory networks from microarray datasets using the approach based on differential equation modeling. In the proposed variational framework, no structure assumption on the genetic network is presumed, instead, the network is solely determined by the microarray profile of the network components and is identified through a well chosen variational principle which minimizes a biological energy functional. The variational principle serves not only as a selection criterion to pick up the right biological solution of the underlying differential equation model but also provide an effective mathematical characterization of the small-world property of genetic regulatory networks which has been observed in lab experiments. Five specific models within the variational framework and efficient numerical methods and algorithms for computing their solutions are proposed and analyzed in the dissertation. Model validations using both synthetic network datasets and real world subnetwork datasets of Saccharomyces cerevisiae (yeast) and E. Coli are done on all five proposed variational models and a performance comparison vs some existing genetic regulatory network identification methods is also provided. As microarray data is typically noisy, in order to take into account the noise effect in the mathematical models, we propose a new approach based on stochastic differential equation modeling and generalize the deterministic variational framework to a stochastic variational framework which relies on stochastic optimization. Numerical algorithms are also proposed for computing solutions of the stochastic variational models. To address the important issue of post-processing computed networks to reflect the small-world property of underlying genetic regulatory networks, a novel threshholding technique based on the Random Matrix Theory is proposed and tested on various synthetic network datasets

DeepSketchHair: Deep Sketch-based 3D Hair Modeling

We present sketchhair, a deep learning based tool for interactive modeling of 3D hair from 2D sketches. Given a 3D bust model as reference, our sketching system takes as input a user-drawn sketch (consisting of hair contour and a few strokes indicating the hair growing direction within a hair region), and automatically generates a 3D hair model, which matches the input sketch both globally and locally. The key enablers of our system are two carefully designed neural networks, namely, S2ONet, which converts an input sketch to a dense 2D hair orientation field; and O2VNet, which maps the 2D orientation field to a 3D vector field. Our system also supports hair editing with additional sketches in new views. This is enabled by another deep neural network, V2VNet, which updates the 3D vector field with respect to the new sketches. All the three networks are trained with synthetic data generated from a 3D hairstyle database. We demonstrate the effectiveness and expressiveness of our tool using a variety of hairstyles and also compare our method with prior art

Solution of the Kirchhoff-Plateau problem

The Kirchhoff-Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a liquid film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. We establish the existence of an equilibrium shape that minimizes the total energy of the system under the physical constraint of non-interpenetration of matter, but allowing for points on the surface of the bounding loop to come into contact. In our treatment, the bounding loop retains a finite cross-sectional thickness and a nonvanishing volume, while the liquid film is represented by a set with finite two-dimensional Hausdorff measure. Moreover, the region where the liquid film touches the surface of the bounding loop is not prescribed a priori. Our mathematical results substantiate the physical relevance of the chosen model. Indeed, no matter how strong is the competition between surface tension and the elastic response of the filament, the system is always able to adjust to achieve a configuration that complies with the physical constraints encountered in experiments