Figure 1: Shape interpolation from a cow to a duck to a torus via convolutional Wasserstein barycenters on a 100×100×100 grid, using the method at the beginning of §7. AbstractThis paper introduces a new class of algorithms for optimization problems involving optimal transportation over geometric domains. Our main contribution is to show that optimal transportation can be made tractable over large domains used in graphics, such as images and triangle meshes, improving performance by orders of magnitude compared to previous work. To this end, we approximate optimal transportation distances using entropic regularization. The resulting objective contains a geodesic distance-based kernel that can be approximated with the heat kernel. This approach leads to simple iterative numerical schemes with linear convergence, in which each iteration only requires Gaussian convolution or the solution of a sparse, pre-factored linear system. We demonstrate the versatility and efficiency of our method on tasks including reflectance interpolation, color transfer, and geometry processing.

Figure 1: We provide an interactive system for users to design, optimize and fabricate multicopters. We explore the design space to allow multicopter design with free-form geometry and nonstandard motor positions and directions. Based on user-specified metrics, our system optimizes the copter geometry and suggests a valid controller. Left: a multicopter example with free-form geometry, various motor heights and different propeller sizes. Middle: a pentacopter with optimized motor positions and orientations, allowing 30% increase in payload. Right: a classic hexacopter with three pairs of coaxial propellers. AbstractWe present an interactive system for computational design, optimization, and fabrication of multicopters. Our computational approach allows non-experts to design, explore, and evaluate a wide range of different multicopters. We provide users with an intuitive interface for assembling a multicopter from a collection of components (e.g., propellers, motors, and carbon fiber rods). Our algorithm interactively optimizes shape and controller parameters of the current design to ensure its proper operation. In addition, we allow incorporating a variety of other metrics (such as payload, battery usage, size, and cost) into the design process and exploring tradeoffs between them. We show the efficacy of our method and system by designing, optimizing, fabricating, and operating multicopters with complex geometries and propeller configurations. We also demonstrate the ability of our optimization algorithm to improve the multicopter performance under different metrics.

Underwater soft robots are challenging to model and control because of their high degrees of freedom and their intricate coupling with water. In this paper, we present a method that leverages the recent development in differentiable simulation coupled with a differentiable, analytical hydrodynamic model to assist with the modeling and control of an underwater soft robot. We apply this method to Starfish, a customized soft robot design that is easy to fabricate and intuitive to manipulate. Our method starts with data obtained from the real robot and alternates between simulation and experiments. Specifically, the simulation step uses gradients from a differentiable simulator to run system identification and trajectory optimization, and the experiment step executes the optimized trajectory on the robot to collect new data to be fed into simulation. Our demonstration on Starfish shows that proper usage of gradients from a differentiable simulator not only narrows down its simulation-to-reality gap but also improves the performance of an open-loop controller in real experiments.

We present a novel, fast differentiable simulator for soft-body learning and control applications. Existing differentiable soft-body simulators can be classified into two categories based on their time integration methods: Simulators using explicit timestepping schemes require tiny timesteps to avoid numerical instabilities in gradient computation, and simulators using implicit time integration typically compute gradients by employing the adjoint method and solving the expensive linearized dynamics. Inspired by Projective Dynamics ( PD ), we present Differentiable Projective Dynamics ( DiffPD ), an efficient differentiable soft-body simulator based on PD with implicit time integration. The key idea in DiffPD is to speed up backpropagation by exploiting the prefactorized Cholesky decomposition in forward PD simulation. In terms of contact handling, DiffPD supports two types of contacts: a penalty-based model describing contact and friction forces and a complementarity-based model enforcing non-penetration conditions and static friction. We evaluate the performance of DiffPD and observe it is 4–19 times faster compared with the standard Newton’s method in various applications including system identification, inverse design problems, trajectory optimization, and closed-loop control. We also apply DiffPD in a reality-to-simulation ( real-to-sim ) example with contact and collisions and show its capability of reconstructing a digital twin of real-world scenes.

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