Piecewise Linear Parameterization Research Articles

Variational Optimization of Continuous Matrix Product States.

Just as matrix product states represent ground states of one-dimensional quantum spin systems faithfully, continuous matrix product states (cMPS) provide faithful representations of the vacuum of interacting field theories in one spatial dimension. Unlike the quantum spin case, however, for which the density matrix renormalization group and related matrix product state algorithms provide robust algorithms for optimizing the variational states, the optimization of cMPS for systems with inhomogeneous external potentials has been problematic. We resolve this problem by constructing a piecewise linear parameterization of the underlying matrix-valued functions, which enables the calculation of the exact reduced density matrices everywhere in the system by high-order Taylor expansions. This turns the variational cMPS problem into a variational algorithm from which both the energy and its backwards derivative can be calculated exactly and at a cost that scales as the cube of the bond dimension. We illustrate this by finding ground states of interacting bosons in external potentials and by calculating boundary or Casimir energy corrections of continuous many-body systems with open boundary conditions.

Improved GRACE kinematic orbit determination using GPS receiver clock modeling

Kinematic positions of Low Earth Orbiters based on GPS tracking are frequently used as pseudo-observations for single satellite gravity field determination. Unfortunately, the accuracy of the satellite trajectory is partly limited because the receiver synchronization error has to be estimated along with the kinematic coordinates at every observation epoch. We review the requirements for GPS receiver clock modeling in Precise Point Positioning (PPP) and analyze its impact on kinematic orbit determination for the two satellites of the Gravity Recovery and Climate Experiment (GRACE) mission using both simulated and real data. We demonstrate that a piecewise linear parameterization can be used to model the ultra-stable oscillators that drive the GPS receivers on board of the GRACE satellites. Using such a continuous clock model allows position estimation even if the number of usable GPS satellites drops to three and improves the robustness of the solution with respect to outliers. Furthermore, simulations indicate a potential accuracy improvement of the satellite trajectory of at least 40 % in the radial direction and up to 7 % in the along-track and cross-track directions when a 60-s piecewise linear clock model is estimated instead of epoch-wise independent receiver clock offsets. For PPP with real GRACE data, the accuracy evaluation is hampered by the lack of a reference orbit of significantly higher accuracy. However, comparisons with a smooth reduced-dynamic orbit indicate a significant reduction of the high-frequency noise in the radial component of the kinematic orbit.

On Computing Solutions to the Continuous Time Constrained Linear Quadratic Regulator $ $

We propose in this note a method for computing the solution to the infinite horizon continuous-time constrained linear quadratic regulator. The method is based on two main ingredients: a multigrid method for placing a finite number of time intervals, and a piece-wise linear parameterization of the input within the intervals. The input values at the decision-time points and slopes within the time intervals are computed via quadratic programs (QPs). The grids are gradually refined to efficiently improve the accuracy of the solution, and the required matrices and vectors for all QPs are computed offline and stored to improve the online efficiency. Two examples are presented to show the main characteristics of the proposed method.

A novel AR-based robot programming and path planning methodology

This paper discusses the benefits of applying Augmented Reality (AR) to facilitate intuitive robot programming, and presents a novel methodology for planning collision-free paths for an n degree-of-freedom (DOF) manipulator in an unknown environment. The targeted applications are where the end-effector is constrained to move along a visible 3D path/curve, which position is unknown, at a particular orientation with respect to the path, such as arc welding and laser cutting. The methodology is interactive as the human is involved in obtaining the 3D data points of the desired curve to be followed through performing a number of demonstrations, defining the free space relevant to the task, and planning the orientations of the end-effector along the curve. A Piecewise Linear Parameterization (PLP) algorithm is used to parameterize the data points using an interactively generated piecewise linear approximation of the desired curve. A curve learning method based on Bayesian neural networks and reparameterization is used to learn and generate 3D parametric curves from the parameterized data points. Finally, the orientation of the end-effector along the learnt curve is planned with the aid of AR. Two case studies are presented and discussed.

Performance of GPS-based accelerometry: A simulation experiment

Recent studies have shown that with the availability of high-quality CHAMP and GRACE gravity field models, it is feasible to determine accurate non-gravitational accelerations for low Earth orbiting satellites indirectly from precise GPS satellite-to-satellite observations. Possible applications of this so-called GPS-based accelerometry approach consist of accelerometer calibration and atmospheric density and wind computations. With the growing number of high-quality space-borne GPS receivers, this method could be applied to a large range of satellites. In this paper an extensive simulation study has been carried out, based on real accelerometer data from the GRACE mission, in order to determine the optimal processing strategy and the resulting accuracy of the estimated non-gravitational accelerations. It is shown that the optimal processing strategy consists of a piecewise linear parameterization of the estimated empirical accelerations, together with short 6-h orbit arcs. The GPS-based accelerometry approach makes use of triple-differenced GPS observations and the impact of considering the correlated observation noise was found to be marginal in the presence of other error sources such as GPS ephemeris errors. Using a priori non-gravitational force models improves the recovery of low temporal resolution accelerations, except during huge geomagnetic storms. With this strategy, non-gravitational accelerations can be recovered during high solar activity with an accuracy of better than 10% of the total signal in along-track direction and around 25–40% in cross-track direction, at time resolutions of around 8–20 min. During solar minimum conditions, the relative recovery error will increase to approximately 50% in along-track direction and around 60–70% in cross-track direction, due to the reduced atmospheric drag signal. Unfortunately, GPS-based accelerometry is hardly sensitive in the radial direction.

Discussion on: “H∞ Control of Distributed and Discrete Delay Systems via Discretized Lyapunov Functional”

performance requirement, and design of state feed-backcontrol.Asisnowwellknown,inordertoobtainstability conditions using Lyapunov-Krasovskiifunctional (LKF) approach, it is necessary to use acomplete quadratic LKF, as is done in this article. Asa general quadratic functional involve infinite numberof parameters, some discretization process is neces-sary in order to render conditions in a computableform. A piece-wise linear parameterization such aswidely used in finite element methods seems to be themost natural. However, there is a fundamental dif-ference between these two discretizations: in finiteelement analysis, the objective is purely to approx-imate the solution; in the discretization of LKF, it isalso necessary to guarantee the satisfaction of quad-raticinequalities.Therefore,thediscretizationofLKFis a much more sophisticated process.

Computation of the Infinite Horizon Continuous Time Constrained Linear Quadratic Regulator

We present a method for computing the solution to the infinite horizon continuous-time constrained linear quadratic regulator (CLQR). The method relies on two main features: a multi-grid method for placing a finite number of time intervals, and a piece-wise linear parameterization of the input within the intervals. The input values at the grid points and slopes within the time intervals are computed via quadratic programs (QPs). The grids are gradually refined to efficiently improve the accuracy of the solution, and the required matrices and vectors for all QPs are computed offline and stored to improve the online efficiency.We present two examples, a single-input single-output unstable system and a three-input three-output stable system, to show the main characteristics of the proposed computation method.

Geodesics-based one-to-one parameterization of 3D triangle meshes

Digital geometry is a new data type for multimedia applications. To foster the use of 3D geometry, we introduce a piecewise linear parameterization of 3D surfaces that we can use for texture mapping, morphing, remeshing, and geometry imaging. Our method guarantees one-to-one mapping without foldovers in a geometrically intuitive way.

A digital self-tuning tracking controller for permanent-magnet synchronous motors

A new digital self-tuning tracking controller for the permanent-magnet synchronous motor is presented, providing for either velocity or position trajectory tracking when both the electrical and mechanical parameters of the motor are unknown. A time-scale simplification leads to a discrete-time design model that is reduced order and that evolves in the slower mechanical time scale. A new piecewise-linear parameterization of the motor torque-angle characteristic functions and cogging allows for arbitrarily accurate identification of higher-order harmonics, without requiring more computation than traditional single-term sinusoidal descriptions. Experimental results demonstrate that good performance is achieved even with very poor initial parameter knowledge. >