5 research outputs found
On the Approximated Solution of a Special Type of Nonlinear Third-Order Matrix Ordinary Differential Problem
[EN] Matrix differential equations are at the heart of many science and engineering problems. In this paper, a procedure based on higher-order matrix splines is proposed to provide the approximated numerical solution of special nonlinear third-order matrix differential equations, having the form Y-(3)(x)=f(x,Y(x)). Some numerical test problems are also included, whose solutions are computed by our method.This research was partially funded by the European Regional Development Fund (ERDF) and the Spanish Ministerio de Economia y Competitividad Grant TIN2017-89314-PDefez Candel, E.; Ibáñez González, JJ.; Alonso Abalos, JM.; Tung, MM.; Real-Herraiz, TP. (2021). On the Approximated Solution of a Special Type of Nonlinear Third-Order Matrix Ordinary Differential Problem. Mathematics. 9(18):1-17. https://doi.org/10.3390/math9182262S11791
Systematic construction of efficient six-stage fifth-order explicit Runge-Kutta embedded pairs without standard simplifying assumptions
This thesis examines methodologies and software to construct explicit
Runge-Kutta (ERK) pairs for solving initial value problems (IVPs) by
constructing efficient six-stage fifth-order ERK pairs without
standard simplifying assumptions. The problem of whether efficient
higher-order ERK pairs can be constructed algebraically without the
standard simplifying assumptions dates back to at least the 1960s,
with Cassity's complete solution of the six-stage fifth-order order
conditions. Although RK methods based on the six-stage fifth-order
order conditions have been widely studied and have continuing
practical importance, prior to this thesis, the aforementioned
complete solution to these order conditions has no published usage
beyond the original series of publications by Cassity in the 1960s.
The complete solution of six-stage fifth-order ERK order conditions
published by Cassity in 1969 is not in a formulation that can easily
be used for practical purposes, such as a software implementation.
However, it is shown in this thesis that when the order conditions are
solved and formulated appropriately using a computer algebra system
(CAS), the generated code can be used for practical purposes and the
complete solution is readily extended to ERK pairs. The condensed
matrix form of the order conditions introduced by Cassity in 1969 is
shown to be an ideal methodology, which probably has wider
applicability, for solving order conditions using a CAS. The software
package OCSage developed for this thesis, in order to solve the order
conditions and study the properties of the resulting methods, is built
on top of the Sage CAS.
However, in order to effectively determine that the constructed ERK
pairs without standard simplifying assumptions are in fact efficient
by some well-defined criteria, the process of selecting the
coefficients of ERK pairs is re-examined in conjunction with a
sufficient amount of performance data. The pythODE software package
developed for this thesis is used to generate a large amount of
performance data from a large selection of candidate ERK pairs found
using OCSage. In particular, it is shown that there is unlikely to be
a well-defined methodology for selecting optimal pairs for
general-purpose use, other than avoiding poor choices of certain
properties and ensuring the error coefficients are as small as
possible. However, for IVPs from celestial mechanics, there are
obvious optimal pairs that have specific values of a small subset of
the principal error coefficients (PECs). Statements seen in the
literature that the best that can be done is treating all PECs equally
do not necessarily apply to at least some broad classes of IVPs. By
choosing ERK pairs based on specific values of individual PECs, not
only are ERK pairs that are 20-30% more efficient than comparable
published pairs found for test sets of IVPs from celestial mechanics,
but the variation in performance between the best and worst ERK pairs
that otherwise would seem to have similar properties is reduced from a
factor of 2 down to as low as 15%. Based on observations of the small
number of IVPs of other classes in common IVP test sets, there are
other classes of IVPs that have different optimal values of the PECs.
A more general contribution of this thesis is that it specifically
demonstrates how specialized software tools and a larger amount of
performance data than is typical can support novel empirical insights
into numerical methods
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Efficient and accurate partial derivatives of rigid-body dynamics for trajectory optimization
In the past few years, trajectory optimization has increasingly been used for motion planning in dynamic-legged robots, like humanoids, quadrupeds, and manipulators. Optimization-based closed-loop controllers like Model-Predictive Control solve trajectory optimization problems sequentially in a receding time horizon fashion. However, the partial derivatives of the rigid-body dynamics used for optimization take almost 90% of the run-time for the optimization problem. Hence, efficiency and accuracy for computing the derivatives determine how fast the optimization algorithms can converge. Conventionally, numerical methods like Finite-Difference and automated Chain-Rule methods called Automatic Differentiation (AD) have been used to compute the derivatives. However, numerical difference results in approximate derivatives, and hence compromises the accuracy, while AD methods often lead to increased run times due to the large amount of matrix-vector products in the algorithms. This work provides analytical recursive expressions and algorithms to compute the first and second-order partial derivatives of rigid-body dynamics, for robotic models with open-loop connectivity trees, multi-DoF joints, and external forces in the form of contacts. Featherstone's Spatial Vector Algebra (SVA) has been extensively used to derive the underlying analytical expressions. Firstly, the first-order partial derivatives of Inverse Dynamics are presented. These derivatives are then extended to derive the second-order derivatives of Inverse Dynamics by extending SVA for tensor objects. Then, efficient tricks to compute the first/second-order derivatives of Forward Dynamics using all the previously derived derivatives are presented. Then, related model-based recursive algorithms are developed and compared in run-time with Finite-Difference, Complex-Step, Automatic Differentiation, and manual chain-rule methods. For first-order Inverse-Dynamics derivatives, run-time analysis with the state-of-the-art library Pinocchio (in C++) shows improvements up to 2x for the ATLAS humanoid. Run-time comparison of the Inverse-Dynamics and Forward-Dynamics second-order derivatives with the AD approach shows speed-ups up to 11x, and 4x respectively. Open-source implementation in Pinocchio and Featherstone's library are also provided for all the developed algorithms. The resulting derivatives show machine precision accuracy for the first-order, and up to 10⁻¹⁰ for the second-order derivatives when compared with the Complex-Step method. The derivatives of the impact dynamics used to model the impulsive interaction of the robot with a rigid surface are also presented. Finally, a Multi-Shooting Differential Dynamic Programming (DDP) optimizer is used for solving a bounding gait optimization problem for a 2D 7-DoF planar quadruped model of the MIT Mini-Cheetah. Here, for the first time, both the first and the second-order derivatives are computed using analytical methods for the different modes -stance, impact, and flight for the quadruped. The Quasi-Newton (QN) method used to approximate the State Transition Matrix shows speed-ups up to 10x over the full DDP approach.Aerospace Engineerin
Astronautics
Many people have had and still have misconceptions about the basic principle of rocket propulsion. Here is a comment of an unknown editorial writer of the renowned New York Times from January 13, 1920, about the pioneer of US astronautics, Robert Goddard, who at that time was carrying out the ?rst experiments with liquid propulsion engines:
Professor Goddard … does not know the relation of action to reaction, and of the need to have something better than a vacuum against which to react – to say that would be absurd. Of course he only seems to lack the knowledge ladled out daily in high schools