We present a discrete-optimization technique for finding feasible robot arm trajectories that pass through provided 6-DOF Cartesian-space end-effector paths with high accuracy, a problem called pathwise-inverse kinematics. The output from our method consists of a path function of joint-angles that best follows the provided end-effector path function, given some definition of ``best''. Our method, called Stampede, casts the robot motion translation problem as a discrete-space graph-search problem where the nodes in the graph are individually solved for using non-linear optimization; framing the problem in such a way gives rise to a well-structured graph that affords an effective best path calculation using an efficient dynamic-programming algorithm. We present techniques for sampling configuration space, such as diversity sampling and adaptive sampling, to construct the search-space in the graph. Through an evaluation, we show that our approach performs well in finding smooth, feasible, collision-free robot motions that match the input end-effector trace with very high accuracy, while alternative approaches, such as a state-of-the-art per-frame inverse kinematics solver and a global non-linear trajectory-optimization approach, performed unfavorably.
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B-spline Parameterized Joint Optimization of Reconstruction and K-space Sampling Patterns (BJORK) for Accelerated 2D Acquisition
The proposed approach, BJORK, provides a robust and generalizable workflow to jointly optimize non-Cartesian sampling patters and a physics-informed reconstruction. Several approaches, including re-parameterization of trajectories, multi-level optimization, and non-Cartesian unrolled neural networks, are introduced to improve training effect and avoid sub-optimal local minima. The invivo experiments show that the networks and trajectories learned on simulation dataset are transferable to the real acquisition even with different parameter-weighted MRI contrasts and noise-levels, and demonstrate improved image quality compared with previous learning-based and model-based trajectory optimization methods.
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- Award ID(s):
- 1838179
- NSF-PAR ID:
- 10309641
- Date Published:
- Journal Name:
- International Society Magnetic Resonance in Medicine
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Purpose Parallel imaging and compressed sensing reconstructions of large MRI datasets often have a prohibitive computational cost that bottlenecks clinical deployment, especially for three‐dimensional (3D) non‐Cartesian acquisitions. One common approach is to reduce the number of coil channels actively used during reconstruction as in coil compression. While effective for Cartesian imaging, coil compression inherently loses signal energy, producing shading artifacts that compromise image quality for 3D non‐Cartesian imaging. We propose coil sketching, a general and versatile method for computationally‐efficient iterative MR image reconstruction.
Theory and Methods We based our method on randomized sketching algorithms, a type of large‐scale optimization algorithms well established in the fields of machine learning and big data analysis. We adapt the sketching theory to the MRI reconstruction problem via a structured sketching matrix that, similar to coil compression, considers high‐energy virtual coils obtained from principal component analysis. But, unlike coil compression, it also considers random linear combinations of the remaining low‐energy coils, effectively leveraging information from all coils.
Results First, we performed ablation experiments to validate the sketching matrix design on both Cartesian and non‐Cartesian datasets. The resulting design yielded both improved computatioanal efficiency and preserved signal‐to‐noise ratio (SNR) as measured by the inverse g‐factor. Then, we verified the efficacy of our approach on high‐dimensional non‐Cartesian 3D cones datasets, where coil sketching yielded up to three‐fold faster reconstructions with equivalent image quality.
Conclusion Coil sketching is a general and versatile reconstruction framework for computationally fast and memory‐efficient reconstruction.
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We present a virtual element method (VEM)-based topology optimization framework using polyhedral elements, which allows for convenient handling of non-Cartesian design domains in three dimensions. We take full advantage of the VEM properties by creating a unified approach in which the VEM is employed in both the structural and the optimization phases. In the structural problem, the VEM is adopted to solve the three-dimensional elasticity equation. Compared to the finite element method, the VEM does not require numerical integration (when linear elements are used) and is less sensitive to degenerated elements (e.g., ones with skinny faces or small edges). In the optimization problem, we introduce a continuous approximation of material densities using the VEM basis functions. When compared to the standard element-wise constant approximation, the continuous approximation enriches the geometrical representation of structural topologies. Through two numerical examples with exact solutions, we verify the convergence and accuracy of both the VEM approximations of the displacement and material density fields. We also present several design examples involving non-Cartesian domains, demonstrating the main features of the proposed VEM-based topology optimization framework. The source code for a MATLAB implementation of the proposed work, named PolyTop3D, is available in the (electronic) Supplementary Material accompanying this publication.more » « less
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Understanding the geometry of collision-free configuration space (C-free) in the presence of Cartesian-space obstacles is an essential ingredient for collision-free motion planning. While it is possible to check for collisions at a point using standard algorithms, to date no practical method exists for computing C-free regions with rigorous certificates due to the complexity of mapping Cartesian-space obstacles through the kinematics. In this work, we present the first to our knowledge rigorous method for approximately decomposing a rational parametrization of C-free into certified polyhedral regions. Our method, called C-Iris (C-space Iterative Regional Inflation by Semidefinite programming), generates large, convex polytopes in a rational parameterization of the configuration space which are rigorously certified to be collision-free. Such regions have been shown to be useful for both optimization-based and randomized motion planning. Based on convex optimization, our method works in arbitrary dimensions, only makes assumptions about the convexity of the obstacles in the 3D Cartesian space, and is fast enough to scale to realistic problems in manipulation. We demonstrate our algorithm’s ability to fill a non-trivial amount of collision-free C-space in several 2-DOF examples where the C-space can be visualized, as well as the scalability of our algorithm on a 7-DOF KUKA iiwa, a 6-DOF UR3e, and 12-DOF bimanual manipulators. An implementation of our algorithm is open-sourced in Drake . We furthermore provide examples of our algorithm in interactive Python notebooks .
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Near-wall flow simulation remains a central challenge in aerodynamics modelling: Reynolds-averaged Navier–Stokes predictions of separated flows are often inaccurate, and large-eddy simulation (LES) can require prohibitively small near-wall mesh sizes. A deep learning (DL) closure model for LES is developed by introducing untrained neural networks into the governing equations and training in situ for incompressible flows around rectangular prisms at moderate Reynolds numbers. The DL-LES models are trained using adjoint partial differential equation (PDE) optimization methods to match, as closely as possible, direct numerical simulation (DNS) data. They are then evaluated out-of-sample – for aspect ratios, Reynolds numbers and bluff-body geometries not included in the training data – and compared with standard LES models. The DL-LES models outperform these models and are able to achieve accurate LES predictions on a relatively coarse mesh (downsampled from the DNS mesh by factors of four or eight in each Cartesian direction). We study the accuracy of the DL-LES model for predicting the drag coefficient, near-wall and far-field mean flow, and resolved Reynolds stress. A crucial challenge is that the LES quantities of interest are the steady-state flow statistics; for example, a time-averaged velocity component $\langle {u}_i\rangle (x) = \lim _{t \rightarrow \infty } ({1}/{t}) \int _0^t u_i(s,x)\, {\rm d}s$ . Calculating the steady-state flow statistics therefore requires simulating the DL-LES equations over a large number of flow times through the domain. It is a non-trivial question whether an unsteady PDE model with a functional form defined by a deep neural network can remain stable and accurate on $t \in [0, \infty )$ , especially when trained over comparatively short time intervals. Our results demonstrate that the DL-LES models are accurate and stable over long time horizons, which enables the estimation of the steady-state mean velocity, fluctuations and drag coefficient of turbulent flows around bluff bodies relevant to aerodynamics applications.more » « less
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