Time-integral variational principles offer a theoretical perspective and practical advantages. They provide a concept for understanding the behavior of dynamical systems and a means to develop governing equations of motion. In analytical mechanics, Hamilton’s principle is the most prominent variational principle and it is commonly connected to the classic Lagrange equations. This study extends those ideas by constructing time-integral variational principles from Nielsen’s form of Lagrange’s equations. The paper begins with a new development of Nielsen’s form, after which a set of new variational principles are developed and explained, followed by two example applications.

A comprehensive study of the dynamic behavior of a translating, fluid-draining, S-shaped pipe is presented. The equations of motion are rigorously developed using a version of Lagrange’s equations that befits variable-mass systems. The equations are then analytically integrated to reveal integrals of motion that offer insights into the system’s behavior. This problem and solution is accessible to students and teachers, providing practice with mathematical modeling, differential and integral calculus, momentum balance, and phase space portraits.

The relationship between the Hamiltonian and Lagrangean functions in analytical mechanics is a type of duality. The two functions, while distinct, are both descriptive functions encoding the behavior of the same dynamical system. One difference is that the Lagrangean naturally appears as one investigates the fundamental equation of classical dynamics. It is not that way for the Hamiltonian. The Hamiltonian comes after Lagrange's equations have been fully formed, most commonly through a Legendre transform of the Lagrangean function. We revisit the Legendre transform approach and offer a more refined geometrical interpretation than what is commonly shown.

This work develops a factorized version of the partial-update Schmidt-Kalman filter: a partial-update filter that operates on the covariance matrix modified Cholesky factors U (the upper triangular factor) and D (a diagonal factor) rather than on the error covariance matrix P. Effectively, this new formulation combines the well-known numerical stability properties of the UD factorized Kalman filter with the enhanced tolerance to high nonlinearities and uncertainties of the partial-update filter. Two versions of the UD partial-update filter are presented: one for sequential measurement processing, delivering the most computationally efficient update, and another for the more convenient but slightly more expensive batch measurement update. Additionally, an implementation of the partial update for the multiplicative extended Kalman filter and the associated quaternion attitude representation is provided. The efficacy of the UD partial-update filter is demonstrated via numerical simulations and hardware experiments; the results show that the combined UD partial-update filter provides a more robust filtering capability than when either component is used individually.

This document contains a concise and unified reference for one of the existing mechanizations of the UD Kalman filter. The associated matrix algorithms are also included along with the corresponding references.

It is most common to construct the Hamiltonian function and Hamilton's canonical equations through a Legendre transformation of the Lagrangean function or through the central equation. These common perspectives, however, seem abstract and detached from classical analytical dynamics. A new and different approach is presented in which the Hamiltonian function is created as one investigates d'Alembert's equation of motion. This formulation directly ties the Hamiltonian function and Hamilton's canonical equations to the root of classical analytical dynamics more than any other approach.

Classic techniques have been established to characterize N × N proper orthogonal matrices using the N-dimensional Euler’s theorem and the Cayley transform. These techniques provide separate descriptions of N-dimensional orientation in terms of the constituent principal rotations or a minimum-parameter representation. The two descriptions can be linked by the canonical form of the extended Rodrigues parameters. This form is developed into a new minimum-parameter representation that directly links to the principal rotations. The new representation is solved using analytic and geometric approaches for N = 3 and N = 4, and numerical solutions are found for N= 5. In fact multiple solutions, which are related geometrically by different coordinatizations of the principal planes, have been found. The new parameters represent a projection of the principal rotations onto the planes formed by the body coordinates.

In this paper we concern ourselves with modified versions of the traditional brachistochrone and tautochrone problems. In the modified version of each problem the constant gravity model is replaced with an attractive inverse square law, consequently we name these the 1/r2 brachistochrone and 1/r2 tautochrone problems. With regard to the 1/r2 brachistochrone problem, we show that the shape of the minimizing curve is formally constructed from an infinite series of elliptic integrals, and we use a numerical optimal control technique to generate the trajectories. The 1/r2 tautochrone problem is solved using fractional calculus together with Lagrange’s rule for tautochronous curves.

A Lagrangian treatment of various forms of the rigid-body equations of motion is presented in this paper, including the most general expressions, which are the Boltzmann-Hamel equations. One key result that enables the derivations is the expression for the Hamel coefficients for the special case of rotational motion of a rigid body. The Hamel coefficients naturally arise in the Lagrange equations for quasi-coordinates. Another key result that enables the derivations is the expression for additional Hamel coefficients that arise when the translational-velocity vector of the mass center is coordinatized (expressed) along body-fixed axes. One interesting discovery is that the Boltzmann-Hamel equations are often misrepresented in standard textbooks. The misrepresentation stems from the fact that care is not exercised to distinguish the functional forms of the kinetic-energy expression.

The Partial-Update Kalman Filter is a recent development that, in a very simple fashion, extends the uncertainty and nonlinearities which the extended or unscented Kalman filters are able to tolerate. However, to date its application has been limited to filters using a full covariance matrix representations or through a wasteful mapping between square root and full covariance representations. In this paper, the Square Root Partial-Update Kalman Filter is developed and demonstrated. This new approach benefits from both the numerical stability inherent in the square root filter and the robustness of the partial update while operating directly on the square root representation of the uncertainty. The paper details the algorithm and implementation and provides a nonlinear filtering example to demonstrate the increased robustness of the partial update form of the square root filter.

No AccessEngineering NotesNielsen, Tzénoff, and Other Equation Forms for Variable-Mass SystemsJohn E. HurtadoJohn E. HurtadoTexas A&M University, College Station, Texas 77843-3141*Professor, Department of Aerospace Engineering and Associate Dean for Academic Affairs, College of Engineering; .Search for more papers by this authorPublished Online:26 Nov 2018https://doi.org/10.2514/1.G003392SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail About References [1] Papastavridis J. G., Analytical Mechanics, Oxford Univ. Press, New York, 2002, p. 280, 879–911. Google Scholar[2] Teodorescu P. P., Mechanical Systems, Classical Models, Vol. III, Analytical Mechanics, Springer, New York, 2002, pp. 88–93. Google Scholar[3] Tzénoff I. V., “Über eine neue Form der Gleichungen der analytischen Dynamik,” Doklady Akademii Nauk UdSSSR, Vol. 89, No. 1, 1953, pp. 21–24. Google Scholar[4] Dolaptschiew B., “Über die verallgemeinerte Form der Lagrangeschen Gleichungen, welche auch die Behandlung von nicht-holonomen mechanischen Systemen gestattet,” ZAMP—Journal of Applied Mathematics and Physics, Vol. 17, No. 3, 1966, pp. 443–449. doi: https://doi.org/10.1007BF01594537 Google Scholar[5] Ghori Q. K., “Generalized Equations of Nonholonomic Systems of Variable Mass,” ZAMM—Journal of Applied Mathematics and Mechanics, Vol. 65, No. 7, 1985, pp. 321–324. doi: https://doi.org/10.1002/(ISSN)1521-4001 CrossrefGoogle Scholar[6] Zekovi D. N., “Dynamics of Mechanical Systems with Nonlinear Nonholonomic Constraints—II Differential Equations of Motion,” ZAMM—Journal of Applied Mathematics and Mechanics, Vol. 91, No. 11, 2011, pp. 899–922. doi: https://doi.org/10.1002/zamm.v91.11 CrossrefGoogle Scholar[7] Hurtado J. E., “Analytical Dynamics of Variable-Mass Systems,” Journal of Guidance, Control, and Dynamics, Vol. 41, No. 3, No. 2018, pp. 701–709. doi: https://doi.org/10.2514/1.G002917 JGCODS 0731-5090 LinkGoogle Scholar[8] Fradlin B. N. and Roshchupkin L. D., “The Equations of Dynamics, Review,” Soviet Applied Mechanics, Vol. 9, No. 1, 1973, pp.1–7. doi: https://doi.org/10.1007/BF00888691 SOAMBT 0038-5298 CrossrefGoogle Scholar Previous article Next article FiguresReferencesRelatedDetails What's Popular Volume 42, Number 1January 2019 CrossmarkInformationCopyright © 2018 by John E. Hurtado. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the ISSN 0731-5090 (print) or 1533-3884 (online) to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp. TopicsEnergyEnergy FormsEnergy Forms, Production and ConversionMechanical and Structural VibrationsStructural Design and DevelopmentStructural EngineeringStructural Kinematics and DynamicsStructures, Design and Test KeywordsKinetic EnergyRotational KinematicsFree Body DiagramPDF Received16 October 2017Accepted6 September 2018Published online26 November 2018

No AccessEngineering NoteHamilton's Principle for Variable-Mass SystemsJohn E. HurtadoJohn E. HurtadoTexas A&M University, College Station, Texas 77843-3141Published Online:15 Oct 2018https://doi.org/10.2514/1.G003340SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail About References [1] Papastavridis J. G., Analytical Mechanics, Oxford Univ. Press, New York, 2002, pp. 301–323, 386, 418–427, 461–469, 877, 935–937, 948–957, 961, 972, 1062–1069. Google Scholar[2] Hurtado J. E., "Analytical Dynamics of Variable-Mass Systems," Journal of Guidance, Control, and Dynamics, Vol. 41, No. 3, 2018, pp. 701–709. doi:https://doi.org/10.2514/1.G002917 JGCODS 0731-5090 LinkGoogle Scholar[3] Schaub H. and Junkins J. L., Analytical Mechanics of Space Systems, AIAA, Reston, VA, 2003, pp. 251–254. LinkGoogle Scholar[4] Rosenberg R. M., Analytical Dynamics of Discrete Systems, Plenum, New York, 1977, pp. 139–146, 167–174. CrossrefGoogle Scholar[5] McIver D. B., "Hamilton's Principle for Systems of Changing Mass," Journal of Engineering Mathematics, Vol. 7, No. 3, 1973, pp. 249–261. doi:https://doi.org/10.1007/BF01535286 JLEMAU 0022-0833 CrossrefGoogle Scholar[6] Casetta L. and Pesce C. P., "The Generalized Hamilton's Principle for a Non-Material Volume," Acta Mechanica, Vol. 224, No. 4, 2013, pp. 919–924. doi:https://doi.org/10.1007/s00707-012-0807-9 AMHCAP 0001-5970 CrossrefGoogle Scholar[7] Irschik H. and Holl H. J., "The Equations of Lagrange Written for a Non-Material Volume," Acta Mechanica, Vol. 153, Nos. 3–4, 2002, pp. 231–248. doi:https://doi.org/10.1007/BF01177454 AMHCAP 0001-5970 CrossrefGoogle Scholar[8] Zhao R. and Yu K., "Hamilton's Law of Variable Mass System and Time Finite Element Formulations for Time-Varying Structures Based on the Law," International Journal for Numerical Methods in Engineering, Vol. 99, No. 10, 2014, pp. 711–736. doi:https://doi.org/10.1002/nme.v99.10 IJNMBH 0029-5981 CrossrefGoogle Scholar[9] Guttner W. C. and Pesce C. P., "On Hamilton's Principle for Discrete Systems of Variable Mass and the Corresponding Lagrange's Equations," Journal of the Brazilian Society of Mechanical Sciences and Engineering, Vol. 39, No. 6, 2017, pp. 1969–1976. doi:https://doi.org/10.1007/s40430-016-0625-4 CrossrefGoogle Scholar[10] Cayley A., "On a Class of Dynamical Problems," Proceedings of the Royal Society of London, Vol. 8, Jan. 1856, pp. 506–511. doi:https://doi.org/10.1098/rspl.1856.0133 PRSLAZ 0370-1662 CrossrefGoogle Scholar[11] Cveticanin L., "Conservation Laws in Systems with Variable Mass," Journal of Applied Mechanics, Vol. 60, No. 4, 1993, pp. 954–958. doi:https://doi.org/10.1115/1.2901007 JAMCAV 0021-8936 CrossrefGoogle Scholar[12] Pesce C. P., "The Application of Lagrange Equations to Mechanical Systems with Mass Explicitly Dependent on Position," Journal of Applied Mechanics, Vol. 70, No. 5, 2003, pp. 751–756. doi:https://doi.org/10.1115/1.1601249 JAMCAV 0021-8936 CrossrefGoogle Scholar[13] Meirovitch L., Methods of Analytical Dynamics, McGraw–Hill, New York, 1970, pp. 59–64, 68–69. Google Scholar[14] Frederick D. and Chang T. S., Continuum Mechanics, Scientific Publishers, Cambridge, MA, 1965, pp. 3–28, Chap. 1. Google Scholar[15] Goldstein H., Classical Mechanics, Addison-Wesley, Reading, MA, 1959, pp. 350–355. Google Scholar[16] Junkins J. L. and Kim Y., Introduction to Dynamics and Control of Flexible Structures, AIAA, Washington, D.C., 1993, pp. 144–148. LinkGoogle Scholar[17] Meirovitch L., "General Motion of a Variable-Mass Flexible Rocket with Internal Flow," Journal of Spacecraft and Rockets, Vol. 7, No. 2, 1970, pp. 186–195. doi:https://doi.org/10.2514/3.29897 JSCRAG 0022-4650 LinkGoogle Scholar Previous article Next article FiguresReferencesRelatedDetailsCited byLagrangian Derivation of Variable-Mass Equations of Motion using an Arbitrary Attitude Parameterization9 October 2020 | The Journal of the Astronautical Sciences, Vol. 67, No. 4 What's Popular Volume 41, Number 12December 2018 CrossmarkInformationCopyright © 2018 by John E. Hurtado. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the ISSN 0731-5090 (print) or 1533-3884 (online) to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp. TopicsEquations of Fluid DynamicsFluid DynamicsMechanical and Structural VibrationsStructural Design and DevelopmentStructural EngineeringStructural Kinematics and DynamicsStructures, Design and Test KeywordsLongitudinal VibrationsReynolds Transport TheoremClassical MechanicsBody KinematicsSlender BodyElasticityPDF Received19 September 2017Accepted24 June 2018Published online15 October 2018

The equations of motion for variable-mass systems are commonly assembled in part by adding reactive forces (that is, those terms that are a direct consequence of the mass variation) to other externally applied body and surface forces. A different approach is presented in this study, wherein the contributions from mass variation are naturally created from a proper kinematic development. Specifically, analytical dynamics is used to produce a few equivalent mathematical expressions, each of which is able to generate the governing equations of motion for variable-mass systems: one is similar to the Lagrange equations, another is similar to the Appell equations, and a third represents the Kane equations. The expressions allow one to apply the methods of analytical dynamics toward this class of systems.

Multirotors could be used to autonomously perform tasks in search-and-rescue, reconnaissance, or infrastructure-monitoring applications. In these environments, the vehicle may have limited or degraded GPS access. Researchers have investigated methods for simultaneous localization and mapping (SLAM) using on-board vision sensors, allowing vehicles to navigate in GPS-denied environments. In particular, SLAM solutions based on a monocular camera offer low-cost, low-weight, and accurate navigation indoors and outdoors without explicit range limitations. However, a monocular camera is a bearing-only sensor. Additional sensors are required to achieve metric pose estimation and the structure of a scene can only be recovered through camera motion. Because of these challenges, the performance of monocular-based navigation solutions is typically very sensitive to the environment and the vehicle's trajectory. This work proposes an integrated estimation and guidance approach for improving the robustness of monocular SLAM to environmental uncertainty. It is specifically intended for a multirotor carrying a monocular camera, downward-facing rangefinder, and inertial measurement unit (IMU). A guidance maneuver is proposed that takes advantage of the metric rangefinder measurements. When the environmental uncertainty is high, the vehicle simply moves up and down, initializing features with a confident and accurate baseline. In order to demonstrate this technique, a vision-aided navigation solution is implemented which includes a unique consider least squares approach to feature covariance initialization. Features are only initialized if there is enough information to accurately triangulate their position, providing an indirect metric of environmental uncertainty that could be used to signal the guidance maneuver. The navigation filter is validated using hardware and simulated data. Finally, simulations show that the proposed initialization maneuver is a simple, practical, and effective way to improve the robustness of monocular-vision-aided-navigation and could increase the amount of autonomy that GPS-denied multirotors are capable of achieving.