An SVD-Based Projection Method for Interpolation on \u3cem\u3eSE\u3c/em\u3e(3)

This paper develops a method for generating smooth trajectories for a moving rigid body with specified boundary conditions. Our method involves two key steps: 1) the generation of optimal trajectories in GA+(n), a subgroup of the affine group in IRn and 2) the projection of the trajectories onto SE(3), the Lie group of rigid body displacements. The overall procedure is invariant with respect to both the local coordinates on the manifold and the choice of the inertial frame. The benefits of the method are threefold. First, it is possible to apply any of the variety of well-known efficient techniques to generate optimal curves on GA+(n). Second, the method yields approximations to optimal solutions for general choices of Riemannian metrics on SE(3). Third, from a computational point of view, the method we propose is less expensive than traditional methods

An SVD-Based Projection Method for Interpolation on \u3ci\u3eSE\u3c/i\u3e(3)

This paper develops a method for generating smooth trajectories for a moving rigid body with specified boundary conditions. Our method involves two key steps: 1) the generation of optimal trajectories in GA+(n), a subgroup of the affine group in Rn and 2) the projection of the trajectories onto SE(3), the Lie group of rigid body displacements. The overall procedure is invariant with respect to both the local coordinates on the manifold and the choice of the inertial frame. The benefits of the method are threefold. First, it is possible to apply any of the variety of well-known efficient techniques to generate optimal curves on GA+(n). Second, the method yields approximations to optimal solutions for general choices of Riemannian metrics on SE(3). Third, from a computational point of view, the method we propose is less expensive than traditional methods

Modelling and Control of Space Vehicles with Fuel Slosh Dynamics

Ever since the launch of the early high-efficiency rockets, controlling liquid fuel slosh within a launch vehicle has been a major design concern. Moreover, with today\u27s large and complex spacecraft, a substantial mass of fuel is necessary to place them into orbit and to perform orbital maneuvers. --from the book\u27s introductio

Stabilized Neural Differential Equations for Learning Dynamics with Explicit Constraints

Many successful methods to learn dynamical systems from data have recently been introduced. However, ensuring that the inferred dynamics preserve known constraints, such as conservation laws or restrictions on the allowed system states, remains challenging. We propose stabilized neural differential equations (SNDEs), a method to enforce arbitrary manifold constraints for neural differential equations. Our approach is based on a stabilization term that, when added to the original dynamics, renders the constraint manifold provably asymptotically stable. Due to its simplicity, our method is compatible with all common neural differential equation (NDE) models and broadly applicable. In extensive empirical evaluations, we demonstrate that SNDEs outperform existing methods while broadening the types of constraints that can be incorporated into NDE training.Comment: 22 pages, 8 figures. Accepted at NeurIPS 202

Synchronization of multiple rigid body systems: a survey

The multi-agent system has been a hot topic in the past few decades owing to its lower cost, higher robustness, and higher flexibility. As a particular multi-agent system, the multiple rigid body system received a growing interest since its wide applications in transportation, aerospace, and ocean exploration. Due to the non-Euclidean configuration space of attitudes and the inherent nonlinearity of the dynamics of rigid body systems, synchronization of multiple rigid body systems is quite challenging. This paper aims to present an overview of the recent progress in synchronization of multiple rigid body systems from the view of two fundamental problems. The first problem focuses on attitude synchronization, while the second one focuses on cooperative motion control in that rotation and translation dynamics are coupled. Finally, a summary and future directions are given in the conclusion

An ontology-based approach towards coupling task and path planning for the simulation of manipulation tasks

This work deals with the simulation and the validation of complex manipulation tasks under strong geometric constraints in virtual environments. The targeted applications relate to the industry 4.0 framework; as up-to-date products are more and more integrated and the economic competition increases, industrial companies express the need to validate, from design stage on, not only the static CAD models of their products but also the tasks (e.g., assembly or maintenance) related to their Product Lifecycle Management (PLM). The scientific community looked at this issue from two points of view: - Task planning decomposes a manipulation task to be realized into a sequence of primitive actions (i.e., a task plan) - Path planning computes collision-free trajectories, notably for the manipulated objects. It traditionally uses purely geometric data, which leads to classical limitations (possible high computational processing times, low relevance of the proposed trajectory concerning the task to be performed, or failure); recent works have shown the interest of using higher abstraction level data. Joint task and path planning approaches found in the literature usually perform a classical task planning step, and then check out the feasibility of path planning requests associated with the primitive actions of this task plan. The link between task and path planning has to be improved, notably because of the lack of loopback between the path planning level and the task planning level: - The path planning information used to question the task plan is usually limited to the motion feasibility where richer information such as the relevance or the complexity of the proposed path would be needed; - path planning queries traditionally use purely geometric data and/or “blind” path planning methods (e.g., RRT), and no task-related information is used at the path planning level Our work focuses on using task level information at the path planning level. The path planning algorithm considered is RRT; we chose such a probabilistic algorithm because we consider path planning for the simulation and the validation of complex tasks under strong geometric constraints. We propose an ontology-based approach to use task level information to specify path planning queries for the primitive actions of a task plan. First, we propose an ontology to conceptualize the knowledge about the 3D environment in which the simulated task takes place. The environment where the simulated task takes place is considered as a closed part of 3D Cartesian space cluttered with mobile/fixed obstacles (considered as rigid bodies). It is represented by a digital model relying on a multilayer architecture involving semantic, topologic and geometric data. The originality of the proposed ontology lies in the fact that it conceptualizes heterogeneous knowledge about both the obstacles and the free space models. Second, we exploit this ontology to automatically generate a path planning query associated to each given primitive action of a task plan. Through a reasoning process involving the primitive actions instantiated in the ontology, we are able to infer the start and the goal configurations, as well as task-related geometric constraints. Finally, a multi-level path planner is called to generate the corresponding trajectory. The contributions of this work have been validated by full simulation of several manipulation tasks under strong geometric constraints. The results obtained demonstrate that using task-related information allows better control on the RRT path planning algorithm involved to check the motion feasibility for the primitive actions of a task plan, leading to lower computational time and more relevant trajectories for primitive actions

Differentiable Physics Simulations with Contacts: Do They Have Correct Gradients w.r.t. Position, Velocity and Control?

In recent years, an increasing amount of work has focused on differentiable physics simulation and has produced a set of open source projects such as Tiny Differentiable Simulator, Nimble Physics, diffTaichi, Brax, Warp, Dojo and DiffCoSim. By making physics simulations end-to-end differentiable, we can perform gradient-based optimization and learning tasks. A majority of differentiable simulators consider collisions and contacts between objects, but they use different contact models for differentiability. In this paper, we overview four kinds of differentiable contact formulations - linear complementarity problems (LCP), convex optimization models, compliant models and position-based dynamics (PBD). We analyze and compare the gradients calculated by these models and show that the gradients are not always correct. We also demonstrate their ability to learn an optimal control strategy by comparing the learned strategies with the optimal strategy in an analytical form. The codebase to reproduce the experiment results is available at https://github.com/DesmondZhong/diff_sim_grads.Comment: 2nd AI4Science Workshop at ICML 202

Static and dynamic global stiffness analysis for automotive pre-design

In order to be worldwide competitive, the automotive industry is constantly challenged to produce higher quality vehicles in the shortest time possible and with the minimum costs of production. Most of the problems with new products derive from poor quality design processes, which often leads to undesired issues in a stage where changes are extremely expensive. During the preliminary design phase, designers have to deal with complex parametric problems where material and geometric characteristics of the car components are unknown. Any change in these parameters might significantly affect the global behaviour of the car. A target which is very sensitive to small variations of the parameters is the noise and vibration response of the vehicle (NVH study), which strictly depends on its global static and dynamic stiffness. In order to find the optimal solution, a lot of configurations exploring all the possible parametric combinations need to be tested. The current state of the art in the automotive design context is still based on standard numerical simulations, which are computationally very expensive when applied to this kind of multidimensional problems. As a consequence, a limited number of configurations is usually analysed, leading to suboptimal products. An alternative is represented by reduced order method (ROM) techniques, which are based on the idea that the essential behaviour of complex systems can be accurately described by simplified low-order models.This thesis proposes a novel extension of the proper generalized decomposi-tion (PGD) method to optimize the design process of a car structure with respect to its global static and dynamic stiffness properties. In particular, the PGD method is coupled with the inertia relief (IR) technique and the inverse power method (IPM) to solve, respectively, the parametric static and dynamic stiffness analysis of an unconstrained car structure and extract its noise and vibrations properties. A main advantage is that, unlike many other ROM methods, the proposed approach does not require any pre-processing phase to collect prior knowledge of the solution. Moreover, the PGD solution is computed with only one offline computation and presents an explicit dependency on the introduced design variables. This allows to compute the solutions at a negligible computational cost and therefore opens the door to fast optimisation studies and real-time visualisations of the results in a pre-defined range of parameters. A novel algebraic approach is also proposed which allows to involve both material and com-plex geometric parameters, such that shape optimisation studies can be performed. In addition, the method is developed in a nonintrusive format, such that an interaction with commercial software is possible, which makes it particularly interesting for industrial applications. Finally, in order to support the designers in the decision-making process, a graphical interface app is developed which allows to visualise in real-time how changes in the design variables affect pre-defined quantities of interest

Structure Preserving and Scalable Simulation of Colliding Systems

Predictive computational tools to study granular materials are important in fields ranging from the geosciences and civil engineering to computer graphics. The simulation of granular materials, however, presents many challenges. The behavior of a granular medium is fundamentally multi-scale, with pair-wise interactions between discrete granules able to influence the continuum-scale evolution of a bulk material. Computational techniques for studying granular materials must therefore contend with this multi-scale nature. This research first addresses both the question of how to accurately model interactions between grains and the question of how to achieve multi-scale simulations of granular materials. We propose a novel rigid body contact model and a time integration technique that, for the first time, are able to simultaneously capture five key features of rigid body impact. We further validate this new model and time integration method by reproducing computationally challenging phenomena from granular physics. We next propose a technique to couple discrete and continuum models of granular materials to one another. This hybrid model reveals a family of possible discretizations suitable for simulation. We derive an explicit integration technique from this framework that is able to capture phenomena previously reserved for discrete treatments, including frictional jamming, while treating bulk regions of the material with a continuum model. To effectively handle the large plastic deformations inherent in the evolution of a granular medium, we further propose a method to dynamically update which regions are treated with a discrete model and which regions are treated with a continuum model. We demonstrate that hybrid simulations of a dynamically evolving granular material are possible and practical, and lay the foundation for further algorithmic development in this space. Finally, as the the tools used in computational science and engineering become progressively more complex, the ability to effectively train students in the field becomes increasingly important. We address the question of how to train students from a computer science background in numerical computation techniques by proposing a new system to automatically vet and identify problems in numerical simulations. This system has been deployed at the undergraduate and graduate level in a course on physical simulation at Columbia University, and has increased both student retention and student satisfaction with the course