Algebraic Constraints Research Articles

Computer-Aided Design, Multibody Dynamic Modeling, and Motion Control Analysis of a Quadcopter System for Delivery Applications

This paper elaborates on the modeling and control of an Unmanned Aerial Vehicle (UAV) for delivery purposes, thereby integrating computer-aided design, multibody dynamic modeling, and motion control analysis in a unified framework. The UAV system designed in this study and utilized for item delivery has a quadcopter structure composed of four arms connected to a central trunk. In this investigation, the proposed design of the delivery drone is systematically modeled employing the multibody approach, while SIMSCAPE MULTIBODY is the software used for performing the dynamic analysis and for devising the final design of the control system. To this end, starting from the CAD model designed using SOLIDWORKS, the control system of the quadcopter is developed by performing dynamic simulations in the MATLAB/SIMULINK environment. Additionally, another fundamental contribution of this paper is the analytical derivation of the nonlinear set of algebraic constraint equations peculiar to the present multibody system, which characterizes the kinematics of the delivery drone and describes the relative angular velocity imposed between two rigid bodies as nonholonomic constraints. Furthermore, as discussed in detail in this paper, the choice of the propulsion system and the design of the individual components heavily depends on the structural and functional needs of the UAV under study. On the other hand, the control system devised in this work is based on cascaded Proportional-Integral-Derivative (PID) controllers, which are suitable for achieving different maneuvers that are fundamental for the motion control of the delivery drone. Therefore, the final performance of the UAV system is a consequence of the regulation of the feedback parameters that characterize the PID controllers. In this respect, the paper presents the refining of the parameters characterizing the PID controllers by using both an internal MATLAB tool, which automatically tunes the controller gains of single-input single-output systems, and by observing the resulting transient behavior of the UAV system, which is obtained through extensive dynamical simulations. The set of numerical results found in this investigation demonstrates the high performance of the dynamical behavior of the UAV system designed in this work.

Stable Big Bang formation for Einstein’s equations: The complete sub-critical regime

For ( t , x ) ∈ ( 0 , ∞ ) × T D (t,x) \in (0,\infty )\times \mathbb {T}^{\mathfrak {D}} , the generalized Kasner solutions (which we refer to as Kasner solutions for short) are a family of explicit solutions to various Einstein-matter systems that, exceptional cases aside, start out smooth but then develop a Big Bang singularity as t ↓ 0 t \downarrow 0 , i.e., a singularity along an entire spacelike hypersurface, where various curvature scalars blow up monotonically. The family is parameterized by the Kasner exponents q ~ 1 , ⋯ , q ~ D ∈ R \widetilde {q}_1,\cdots ,\widetilde {q}_{\mathfrak {D}} \in \mathbb {R} , which satisfy two algebraic constraints. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be dynamically stable, that is, stable under perturbations of the Kasner initial data, given say at { t = 1 } \lbrace t = 1 \rbrace , as long as the exponents are “sub-critical” in the following sense: max I , J , B = 1 , ⋯ , D I > J { q ~ I + q ~ J − q ~ B } > 1 \underset {\substack {I,J,B=1,\cdots , \mathfrak {D}\\ I > J}}{\max } \{\widetilde {q}_I+\widetilde {q}_J-\widetilde {q}_B\}>1 . Previous works have rigorously shown the dynamic stability of the Kasner Big Bang singularity under stronger assumptions: (1) the Einstein-scalar field system with D = 3 \mathfrak {D}= 3 and q ~ 1 ≈ q ~ 2 ≈ q ~ 3 ≈ 1 / 3 \widetilde {q}_1 \approx \widetilde {q}_2 \approx \widetilde {q}_3 \approx 1/3 , which corresponds to the stability of the Friedmann–Lemaître–Robertson–Walker solution’s Big Bang or (2) the Einstein-vacuum equations for D ≥ 38 \mathfrak {D}\geq 38 with max I = 1 , ⋯ , D | q ~ I | > 1 / 6 \underset {I=1,\cdots ,\mathfrak {D}}{\max } |\widetilde {q}_I| > 1/6 . In this paper, we prove that the Kasner singularity is dynamically stable for all sub-critical Kasner exponents, thereby justifying the heuristics in the literature in the full regime where stable monotonic-type curvature-blowup is expected. We treat in detail the 1 + D 1+\mathfrak {D} -dimensional Einstein-scalar field system for all D ≥ 3 \mathfrak {D}\geq 3 and the 1 + D 1+\mathfrak {D} -dimensional Einstein-vacuum equations for D ≥ 10 \mathfrak {D}\geq 10 ; both of these systems feature non-empty sets of sub-critical Kasner solutions. Moreover, for the Einstein-vacuum equations in 1 + 3 1+3 dimensions, where instabilities are in general expected, we prove that all singular Kasner solutions have dynamically stable Big Bangs under polarized U ( 1 ) U(1) -symmetric perturbations of their initial data. Our results hold for open sets of initial data in Sobolev spaces without symmetry, apart from our work on polarized U ( 1 ) U(1) -symmetric solutions. Our proof relies on a new formulation of Einstein’s equations: we use a constant-mean-curvature foliation, and the unknowns are the scalar field, the lapse, the components of the spatial connection and second fundamental form relative to a Fermi–Walker transported spatial orthonormal frame, and the components of the orthonormal frame vectors with respect to a transported spatial coordinate system. In this formulation, the PDE evolution system for the structure coefficients of the orthonormal frame approximately diagonalizes in a way that sharply reveals the significance of the Kasner exponent sub-criticality condition for the dynamic stability of the flow: the condition leads to the time-integrability of many terms in the equations, at least at the low derivative levels. At the high derivative levels, the solutions that we study can be much more singular with respect to t t , and to handle this difficulty, we use t t -weighted high order energies, and we control non-linear error terms by exploiting monotonicity induced by the t t -weights and interpolating between the singularity-strength of the solution’s low order and high order derivatives. Finally, we note that our formulation of Einstein’s equations highlights the quantities that might generate instabilities outside of the sub-critical regime.

Classical double copy of spinning worldline quantum field theory

We study the classical double copy of massive spinning objects in the worldline quantum field theories (WQFT) formalism. We couple the mathcal{N} = 1 supersymmetric model to a Yang-Mills background to describe the propagation of a spin-half particle interacting with gluons. At the classical level, this model captures physical effects up to linear order in spin. We propose a double copy relation to map the spin tensors to the gravitation side. Enforcing R-symmetry and supersymmetry (SUSY) on the double copy integrands, we find that the gravitational theory is the mathcal{N} = 2 particle coupled to dilaton-gravity (DG). We check the double copy prescription for the eikonal phase up to next-to-leading order and for radiation at leading order in coupling constants, finding that the Grassmann nature of the spin tensor in WQFT plays a crucial role in finding full agreement with direct calculation in the mathcal{N} = 2 model. We show how to deform the SUSY charges of the free theory to include DG. Since the constraints algebra is first class, the worldline model can be quantized, describing the propagation of a massive vector field coupled to DG, in agreement with the literature. In addition, we investigate the double copy without preserving SUSY and R-symmetry, finding that the B-field also couples to the worldline.

Loop quantum gravity of a spherically symmetric scalar field coupled to gravity with a clock

The inclusion of matter fields in spherically symmetric loop quantum gravity has proved problematic at the level of implementing the constraint algebra including the Hamiltonian constraint. Here we consider the system with the introduction of a clock. Using the abelianizaton technique we introduced in previous papers in the case of gravity coupled to matter, the system can be gauge fixed and rewritten in terms of a restricted set of dynamical variables that satisfy simple Poisson bracket relations. This creates a true Hamiltonian and therefore one bypasses the issue of the constraint algebra. We show how loop quantum gravity techniques may be applied for the vacuum case and show that the Hamiltonian system reproduces previous results for the physical space of states and the observables of a Schwarzchild black hole.

Schrödinger from Wheeler–DeWitt: The issues of time and inner product in canonical quantum gravity

The wave-function in quantum gravity is supposed to obey the Wheeler–DeWitt (WDW) equation, however there is neither a satisfactory probability interpretation nor a successful solution to the problem of time in the WDW framework. To gain some insight on these issues we compare quantization of ordinary systems, first in the usual way having the Schrödinger equation and second by promoting them as parametrized theories by introducing embedding coordinate fields, which yields first class constraints and the WDW equation. We observe that the time evolution in the WDW framework can be described with respect to the embedding coordinates, where the WDW equation becomes Schrödinger like, i.e. it involves first order timelike functional derivatives. Moreover, the equivalence with the ordinary quantization procedure determines a suitable Hilbert space with a viable probability interpretation. We then apply the same construction to general relativity by adding embedding fields without any prior coordinate choice. The reparametrized general relativity has two different types of diffeomorphism invariance, which arises from world-volume and target-space reparametrizations. As in the case of ordinary systems, the time evolution can be described with respect to the embedding fields and the WDW equation becomes Schrödinger like; the construction is almost identical to an ordinary parametrized field theory in terms of time evolution and Hilbert space structure. However, this time, the constraint algebra enforces the wave-function to be in a subspace of states annihilated by an operator that can be identified as the Hamiltonian. The implications of these results for the canonical quantization program, and in particular for the minisuperspace quantum cosmology, are discussed.

Model reduction for constrained mechanical systems via spectral submanifolds

Dynamical systems are often subject to algebraic constraints in conjunction with their governing ordinary differential equations. In particular, multibody systems are commonly subject to configuration constraints that define kinematic compatibility between the motion of different bodies. A full-scale numerical simulation of such constrained problems is challenging, making reduced-order models (ROMs) of paramount importance. In this work, we show how to use spectral submanifolds (SSMs) to construct rigorous ROMs for mechanical systems with configuration constraints. These SSM-based ROMs enable the direct extraction of backbone curves and forced response curves and facilitate efficient bifurcation analysis. We demonstrate the effectiveness of this SSM-based reduction procedure on several examples of varying complexity, including nonlinear finite-element models of multibody systems. We also provide an open-source implementation of the proposed method that also contains all details of our numerical examples.

A Single-Loop Framework for the Reliability-Based Control Co-Design Problem in the Dynamic System

When solving the control co-design (CCD) problem using the simultaneous strategy in a deterministic manner, the uncertainty stemming from the stochastic design variables is ignored, and might have a negative influence on the performance of the dynamic system. In attempting to overcome the undesirable effect of the uncertainty, this research investigates the reliability-based control co-design (RB-CCD) problem and presents a single-loop framework for RB-CCD based on the modified RB-CCD model and single-loop approach (SLA). Specifically, the modified model is deduced by introducing additional design variables and equality constraints (state equations and algebraic equality constraints) so as to transform the probabilistic constraints into inequality constraints. Meanwhile, to enhance the solution efficiency, SLA transforms the modified RB-CCD model into an equivalent single-loop deterministic CCD model by incorporating the approximate reliability information of the stochastic design variables into the deterministic optimization. Finally, a numerical example and an engineering example are implemented to verify the feasibility and effectiveness of the single-loop RB-CCD optimization framework. The results demonstrate that the suggested single-loop framework dramatically improves the reliability of the dynamic system, and significantly increases the solving efficiency without compromising accuracy.

Complex Eigenvalue Analysis of Multibody Problems via Sparsity-Preserving Krylov–Schur Iterations

In this work, we discuss the numerical challenges involved in the computation of the complex eigenvalues of damped multi-flexible-body problems. Aiming at the highest generality, the candidate method must be able to deal with arbitrary rigid body modes (free–free mechanisms), arbitrary algebraic constraints, and must be able to exploit the sparsity pattern of Jacobians of large systems. We propose a custom implementation of the Krylov–Schur method, proving its robustness and its accuracy in a variety of different complex test cases.

Iterative Robust Ellipsoid Fitting Based on M-Estimator With Geometry Radius Constraint

The ellipsoid fitting technique has been popularly used in many application scenarios, especially in triaxial sensor calibration in recent years. The conventional method to achieve ellipsoid fitting is the pseudo-inverse technique with algebraic constraints to obtain proper fitting results. This technique ensures the least mean square (LMS) error relative to all measured data and prevents trivial solutions. However, this approach lacks robustness against outliers resulting in nonideal fitting performance in real calibration situations. The M-estimator technique with several robust cost functions is adopted and applied in this research to attenuate the outlier’s effect to offer a more accurate fitting performance. In contrast, the constraint with geometric meaning is also applied and examined to prevent trivial solutions. These two methods are achieved by using the iterative constrained gradient descent (G.D.) method to approach the proper ellipsoid fitting. The performances are validated through synthesis and real measured data. Through calibrated experiments, the proposed methods are found to be able to increase robustness against outliers and prevent parameters of ellipsoid degeneration toward trivial solutions.

Semi-explicit multilinear modeling of a [formula omitted] open-loop controlled PV inverter in [formula omitted]-frame

This paper introduces semi-explicit multilinear modeling to ac power systems, including voltage source converters. Firstly, the novel semi-explicit multilinear modeling class is explained. Then this class is applied to a small example of PV inverter connected to an infinite bus. Some part of the nonlinear current reference calculation is expressed as a multilinear algebraic constraint. The formulation of this multilinear algebraic constraint allowed modeling the whole nonlinear inverter as a semi-explicit multilinear model. The novel modeling approach is compared with common nonlinear time-based simulation in Simulink. The results show a good approximation of the nonlinear model by its semi-explicit multilinear model where the relative error is below 10−3%. This low error motivates the further usage of the semi-explicit multilinear modeling class for power system analysis.

SEQUOIA: A sequential algorithm providing feasibility guarantees for constrained optimization*

Real-time optimal control applications rely on quickly solving nonlinear programs (NLPs) with many constraints. It is usually challenging to design a safe controller that guarantees feasibility of the returned points even in the case of early termination. We present a novel algorithm called SEQUOIA that allows users to choose the maximum optimality gap tolerated and implement sub-optimal, feasible decisions faster in time-critical scenarios. Typical constrained optimization methods approach the feasible set from its exterior, hence the point returned in the case of early termination is not guaranteed to be fit for purpose because critical constraints might be violated. Our algorithm guarantees a feasible point is returned after any iteration and an incrementally better point can be easily constructed. Additionally, we developed the theoretical framework behind our algorithm and show some key mathematical properties. The core idea is to store exact upper and lower bounds for the optimal cost and iteratively solve feasibility problems in order to tighten these bounds. To solve the feasibility problems we use a residual-based approach and we can provide an infeasibility detection feature during the first iteration. We illustrate our implementation on a constrained optimization benchmark as well as a maximum radius orbit transfer optimal control problem. Solving this type of dynamic optimization problems is difficult because of the high number of equality constraints generated from the discretization of dynamic and algebraic path constraints. The numerical experiments show that one can reduce the computational time by more than 40% if the allowed optimality gap is increased from 10−15 to 10−10 and the overall solve time can be decreased by up to two orders of magnitude when compared to a quadratic penalty method (QPM).

Structural electroneutrality in Onsager–Stefan–Maxwell transport with charged species

We present a method to embed local electroneutrality within Onsager–Stefan–Maxwell electrolytic-transport models, circumventing their formulation as differential systems with an algebraic constraint. Flux-explicit transport laws are developed for general multicomponent electrolytes, in which the ionic conductivity, component diffusivities, and transference numbers relate to Stefan–Maxwell coefficients through invertible matrix calculations. A construction we call a ‘salt–charge basis’ implements Guggenheim’s transformation of species electrochemical potentials into combinations describing a minimal set of neutral components, leaving a unique combination associated with electricity. Defining conjugate component concentrations and fluxes that preserve the structures of the Gibbs function and energy dissipation retains symmetric Onsager reciprocal relations. The framework reproduces Newman’s constitutive laws for binary electrolytes and the Pollard–Newman laws for molten salts; we also propose laws for salt solutions in two-solvent blends, such as lithium-ion-battery electrolytes. Finally, we simulate a potentiostatic Hull cell containing a non-ideal binary electrolyte with concentration-dependent properties.

Efficient computation of the L-infinity norm of descriptor systems

Many applications from industry and technology use models formulated by systems of differential equations. Often, some equations describe additional algebraic constraints. Such systems, referred to as descriptor systems (or singular systems), naturally arise, e.g., in electrical circuit simulation, in multibody dynamics with constraints, or by semidiscretization of certain partial differential equations. A very important characteristic value for a descriptor system is the L∞-norm of its corresponding transfer function. The computation of this norm is essential in robust control, model order reduction, and other applications. The paper summarizes efficient and reliable algorithms for finding L∞-norm, for continuous- and discrete-time descriptor systems, which exploit the underlying Hamiltonian or symplectic structure, respectively. An improved solver has been developed and will be made available in the SLICOT Library. Numerical results and comparisons illustrate the good performance and effectiveness of this solver.

Optimality equivalence and computational advantages of free-floating base dynamics compared to full-body dynamics

Several transcription methods exist for predictive simulations of free-floating base motions like athlete performing twisting somersaults. Following a brief overview of the literature, we proposed two original direct multiple shooting optimal control problems: (1) with a reduced set of equations motion (free-floating base dynamics) and generalized accelerations as controls; and (2) with the dynamics enforced implicitly through algebraic constraints and generalized jerks as controls. To assess their performance (computational time, dynamic consistency, cost), we compared full-body vs free-floating base dynamics and explicit vs implicit formulations of an optimal control problem of a double straight somersault with three twists. We found that free-floating dynamics is relevant when bounds on joint torques are not required since it hastens the convergence up to 10 times and the optimal solutions are similar. Implicit dynamics also speeds up optimization compared to the explicit formulation, but dynamic consistency may be affected. Using joint jerks instead of accelerations as controls mitigates this limitation, especially when the mesh grid densifies.

Constrained Large-Displacement Thermal Analysis

Abstract Two different cases are encountered in the thermal analysis of solids. In the first case, continua are not subject to boundary and motion constraints and all material points experience same displacement-gradient changes as the result of application of thermal loads. In this case, referred to as unconstrained thermal expansion, the thermal load produces uniform stress-free motion within the continuum. In the second case, point displacements due to boundary and motion constraints are restricted, and therefore, continuum points do not move freely when thermal loads are applied. This second case, referred to as constrained thermal expansion, leads to thermal stresses and its study requires proper identification of the independent coordinates which represent expansion degrees-of-freedom. To have objective evaluation and comparison between the two cases of constrained and unconstrained thermal expansion, the reference-configuration geometry is accurately described using the absolute nodal coordinate formulation (ANCF) finite elements. ANCF position-gradient vectors have unique geometric meanings as tangent to coordinate lines, allowing systematic description of the two different cases of unconstrained and constrained thermal expansions using multiplicative decomposition of the matrix of position-gradient vectors. Furthermore, generality of the approach for large-displacement thermal analysis requires using the Lagrange–D'Alembert principle for proper treatment of algebraic constraint equations. Numerical results are presented to compare two different expansion cases, demonstrate use of the new approach, and verify its results by comparing with conventional finite element (FE) approaches.

Calculating voltage feasibility boundaries for power system security assessment

Modern power systems face a grand challenge in grid management due to increased electricity demand, imminent disturbances, and uncertainties associated with renewable generation, which can undermine grid security. The security assessment is directly connected to the robustness of the operating condition and is evaluated by analyzing proximity to the Power Flow (PF) solution space’s boundary. Calculating the location of such a boundary is a computationally challenging task, linked to the PF equations’ non-linear nature, presence of technical constraints, and complex network topology. This paper introduces a general framework to characterize points on the PF solution space boundary in terms of auxiliary variables subject to algebraic constraints. Then we develop an adaptive continuation algorithm to trace 1-dimensional sections of boundary curves which exhibits robust performance and computational tractability. Implementation of the algorithm is described in detail, and its performance is validated on different test networks.

Optimal Greedy Control in Reinforcement Learning.

We consider the problem of dimensionality reduction of state space in the variational approach to the optimal control problem, in particular, in the reinforcement learning method. The control problem is described by differential algebraic equations consisting of nonlinear differential equations and algebraic constraint equations interconnected with Lagrange multipliers. The proposed method is based on changing the Lagrange multipliers of one subset based on the Lagrange multipliers of another subset. We present examples of the application of the proposed method in robotics and vibration isolation in transport vehicles. The method is implemented in FRUND-a multibody system dynamics software package.

Perturbations of a class of locally rotationally symmetric cosmologies with applications to dissipative fluids

A gauge invariant perturbation theory, based on the covariant split of spacetime, is used to study first order perturbations on a class of anisotropic cosmological backgrounds. The perturbations as well as the energy-momentum tensor are kept general, giving a system of equations on which different physical situations may be imposed. Through a harmonic decomposition, the system is then transformed to evolution equations in time and algebraic constraints. This result is then applied to dissipative one-component fluids, and on using the simplified acausal Eckart theory the system is reduced to two closed subsystems, governed by four and eight harmonic coefficients for the odd and even sectors respectively. The system is also seen to close in a simplified causal theory. It is then demonstrated, within the Eckart theory, how vorticity can be generated from viscosity.

Metric Gravity in the Hamiltonian Form—Canonical Transformations—Dirac’s Modifications of the Hamilton Method and Integral Invariants of the Metric Gravity

Two different Hamiltonian formulations of the metric gravity are discussed and applied to describe a free gravitational field in the d dimensional Riemann space-time. Theory of canonical transformations, which relates equivalent Hamiltonian formulations of the metric gravity, is investigated in detail. In particular, we have formulated the conditions of canonicity for transformation between the two sets of dynamical variables used in our Hamiltonian formulations of the metric gravity. Such conditions include the ordinary condition of canonicity known in classical Hamilton mechanics, i.e., the exact coincidence of the Poisson (or Laplace) brackets which are determined for both the new and old dynamical Hamiltonian variables. However, in addition to this, any true canonical transformations defined in the metric gravity, which is a constrained dynamical system, must also guarantee the exact conservation of the total Hamiltonians Ht (in both formulations) and preservation of the algebra of first-class constraints. We show that Dirac’s modifications of the classical Hamilton method contain a number of crucial advantages, which provide an obvious superiority of this method in order to develop various non-contradictory Hamiltonian theories of many physical fields, when a number of gauge conditions are also important. Theory of integral invariants and its applications to the Hamiltonian metric gravity are also discussed. For Hamiltonian dynamical systems with first-class constraints this theory leads to a number of peculiarities some of which have been investigated.

Novel Operating Modes for the Charging of Lithium-ion Batteries

Conventional battery simulation tools offer current, voltage, and power operating modes. This article presents General Operating Modes (GOMs), which move beyond these standard modes and allow battery models of any scale to simulate novel operating modes such as constant temperature, constant lithium plating overpotential, and constant concentration. The governing equations of the battery model are solved alongside a single algebraic constraint that determines the current. The operating modes are simulated efficiently and deterministically inside a differential-algebraic equation (DAE) solver, and constraints are satisfied within solver tolerances. We propose a mixed-continuous discrete (aka hybrid) solution to the constrained charging problem, using the GOMs to satisfy charging constraints. This approach enables nonlinear model predictive control (NMPC) to be implementable in real-time while directly using sophisticated physics-based battery models. The approach is demonstrated for three models of various complexity: a thin-film nickel hydroxide electrode model, a Single-Particle (SP) model, and a Porous Electrode Theory (PET) model. The hybrid fast charging algorithm is shown to be slightly suboptimal for the thermal SP model in some cases, which is not of practical importance for NMPC.