Path-following Method Research Articles

Adaptive coordination control strategy for path-following of DDAEV with neural network based moving weight coefficients

This paper presents an adaptive coordination control strategy for path-following of distributed drive autonomous electric vehicles (DDAEV). A model predictive control (MPC) algorithm is used to realize path-following through autonomous steering, where the prediction time is adaptive in relation to different driving conditions. Due to the dynamic characteristic of distributed drive vehicle, the differential torque control is also utilized based on the deviation of path to realize path-following. In order to make full use of the advantages of these two path-following methods, coordination control of autonomous steering and differential steering is adopted to improve the transient response speed, flexibility of steering system and path-following performance by setting moving weight coefficients based on neural network. Results of CarSim-Simulink co-simulation and real vehicle experiment both verify that the proposed coordination control can obtain steering flexibility, as well as tracking accuracy and reliability under various driving conditions.

Direction Control and Adaptive Path Following of 3-D Snake-Like Robot Motion.

This work investigates direction control and path following of a 3-D snake-like robot. In order to control such robots accurately, this work researches the relationships between its phase offsets of pitch joints and directions. A new direction control method is proposed for the robot based on these relationships. An adaptive path-following algorithm based on the line-of-sight guidance law is proposed and combined with the direction control method to steer the robot to move forward and along desired paths. Simulation and experimental results are presented to demonstrate the performances of the proposed 3-D model and control methods. They well outperform the classical and commonly used path-following method.

On the structure of regularization paths for piecewise differentiable regularization terms

Regularization is used in many different areas of optimization when solutions are sought which not only minimize a given function, but also possess a certain degree of regularity. Popular applications are image denoising, sparse regression and machine learning. Since the choice of the regularization parameter is crucial but often difficult, path-following methods are used to approximate the entire regularization path, i.e., the set of all possible solutions for all regularization parameters. Due to their nature, the development of these methods requires structural results about the regularization path. The goal of this article is to derive these results for the case of a smooth objective function which is penalized by a piecewise differentiable regularization term. We do this by treating regularization as a multiobjective optimization problem. Our results suggest that even in this general case, the regularization path is piecewise smooth. Moreover, our theory allows for a classification of the nonsmooth features that occur in between smooth parts. This is demonstrated in two applications, namely support-vector machines and exact penalty methods.

Optimal Path Homotopy For Univariate Polynomials

The goal of this paper is to study the path-following method for univariate polynomials. We propose to study the complexity and condition properties when the Newton method is applied as a correction operator. Then we study the geodesics and properties of the condition metric along those curves. Last, we compute approximations of geodesics and study how the condition number varies with the quality of the approximation.

Trajectory Optimization with Optimization-Based Dynamics

We present a framework for bi-level trajectory optimization in which a system’s dynamics are encoded as the solution to a constrained optimization problem and smooth gradients of this lower-level problem are passed to an upper-level trajectory optimizer. This optimization-based dynamics representation enables constraint handling, additional variables, and non-smooth behavior to be abstracted away from the upper-level optimizer, and allows classical unconstrained optimizers to synthesize trajectories for more complex systems. We provide a path-following method for efficient evaluation of constrained dynamics and utilize the implicit-function theorem to compute smooth gradients of this representation. We demonstrate the framework by modeling systems from locomotion, aerospace, and manipulation domains including: acrobot with joint limits, cart-pole subject to Coulomb friction, Raibert hopper, rocket landing with thrust limits, and planar-push task with optimization-based dynamics and then optimize trajectories using iterative LQR.

Path-following methods for unstable structural responses induced by strain softening: a critical review

Path-following methods for unstable structural responses induced by strain softening: a critical review

Primal-dual path following method for nonlinear semi-infinite programs with semi-definite constraints

In this paper, we propose a primal-dual path following method for nonlinear semi-infinite semi-definite programs with infinitely many convex inequality constraints, called SISDP for short. A straightforward approach to the SISDP is to use classical methods for semi-infinite programs such as discretization and exchange methods and solve a sequence of (nonlinear) semi-definite programs (SDPs). However, it is often too demanding to find exact solutions of SDPs. In contrast, our approach does not rely on solving SDPs accurately but approximately following a path leading to a solution, which is formed on the intersection of the semi-infinite feasible region and the interior of the semi-definite feasible region. Specifically, we first present a prototype path-following method and show its global weak* convergence to a Karush-Kuhn-Tucker point of the SISDP under some mild assumptions. Next, to achieve fast local convergence, we integrate a two-step sequential quadratic programming method equipped with the Monteiro-Zhang scaling technique into the prototype method. We prove two-step superlinear convergence of the resulting algorithm using Alizadeh-Hareberly-Overton-like, Nesterov-Todd, and Helmberg-Rendle-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro scaling directions. Finally, we conduct some numerical experiments to demonstrate the efficiency of the proposed method through comparison with a discretization method that solves SDPs obtained by finite relaxation of the SISDP.

Primal-Dual Path-Following Methods and the Trust-Region Updating Strategy for Linear Programming with Noisy Data

In this article, we consider the primal-dual path-following method and the trust-region updating strategy for the standard linear programming problem. For the rank-deficient problem with the small noisy data, we also give the preprocessing method based on the QR decomposition with column pivoting. Then, we prove the global convergence of the new method when the initial point is strictly primal-dual feasible. Finally, for some rank-deficient problems with or without the small noisy data from the NETLIB collection, we compare it with other two popular interior-point methods, i.e. the subroutine pathfollow.m and the built-in subroutine linprog.m of the MATLAB environment. Numerical results show that the new method is more robust than the other two methods for the rank-deficient problem with the small noise data.

A path-following inexact Newton method for PDE-constrained optimal control in BV

We study a PDE-constrained optimal control problem that involves functions of bounded variation as controls and includes the TV seminorm of the control in the objective. We apply a path-following inexact Newton method to the problems that arise from smoothing the TV seminorm and adding an H^1 regularization. We prove in an infinite-dimensional setting that, first, the solutions of these auxiliary problems converge to the solution of the original problem and, second, that an inexact Newton method enjoys fast local convergence when applied to a reformulation of the auxiliary optimality systems in which the control appears as implicit function of the adjoint state. We show convergence of a Finite Element approximation, provide a globalized preconditioned inexact Newton method as solver for the discretized auxiliary problems, and embed it into an inexact path-following scheme. We construct a two-dimensional test problem with fully explicit solution and present numerical results to illustrate the accuracy and robustness of the approach.

Evaluation of stiffeners effects on buckling and post-buckling of laminated panels

In aerospace applications, panel structures are bounded by stiffeners that enforce the loaded edge a constant load or displacement. Depending on the type of these attachments and stiffeners, the real boundary condition could be a combination of loads or displacements. Therefore, the accurate assessment of the effects of these boundary conditions on the mechanical behavior of the plate structures is vital for designers and engineers. In this research, those effects are investigated. Stiffeners with various material properties are modeled, and the post-buckling behaviors of pamel structures are evaluated. The Carrera unified Formulation is recalled and, according to it, the structural theory approximation order and the strain-displacement assumptions, opportunely. Thus, in this paper different nonlinear strain assumptions are considered in the CUF with the total Lagrangian framework. In this regard, the Newton-Raphson linearization scheme is employed with the path-following method based on the arc-length constraint. Several numerical assessments of post-buckling in composite plates are conducted. The von Kármán theory and some modified strain-displacement relationships with various nonlinear terms are compared with the full Green-Lagrange nonlinear model based on the CUF plate model. A number of interesting conclusions regarding the efficiency of the presented model are highlighted. The findings of this research also demonstrate that the full Green-Lagrange nonlinear model based on the CUF plate model can predict the nonlinear behavior of laminated composite plates efficiently and accurately that can be a basis to evaluate the effectiveness of different structural theory approximation orders and strain-displacement assumptions.

Non-linear vibration analysis of visco-elastically damped composite structures by multilevel finite elements and asymptotic numerical method

Composite materials and structures are inherently in-homogeneous across multiple scales. Multi-scale modelling offers opportunities to apprehend the coupling of material behaviour and characteristics from the micro- to meso- and macro-scales. In this paper, a multi-scale finite element method (FE2) is proposed to compute the modal properties of visco-elastic heterogeneous composite materials in terms of damping frequencies and modal loss factors. In the proposed FE2-based vibration analysis, two finite elements (FE) calculations are carried out in a nested manner, one at the macro-scale and the other at the micro-scale. Unlike conventional analysis, the developed analysis does not require homogenized constitutive properties because these are derived from the micro-scale FE simulations at the representative volume element (RVE) level. The non-linearity at the micro-scale is accounted by using a frequency dependent Young’s modulus. The Asymptotic Numerical Method (ANM) and its automatic differentiation is used to solve the non-linear numerical problem. ANM consists of solving an analytical non-linear problem with a path-following (or continuation) method associated with a high-order perturbation technique. Compared with existing methods, the originality of the proposed approach lies in its ability to account for the frequency dependence of Young’s modulus in visco-elastic microstructure. Using the automatic differentiation makes the proposed approach enough flexible and generic to deal with damped and undamped vibration analyses of composite materials structures.

An Interior-Point Differentiable Path-Following Method to Compute Stationary Equilibria in Stochastic Games

The subgame perfect equilibrium in stationary strategies (SSPE) is the most important solution concept in applications of stochastic games, making it imperative to develop efficient methods to compute an SSPE. For this purpose, this paper develops an interior-point differentiable path-following method (IPM), which establishes a connection between an artificial logarithmic barrier game and the stochastic game of interest by adding a homotopy variable. IPM brings several advantages over the existing methods for stochastic games. On the one hand, IPM provides a bridge between differentiable path-following methods and interior-point methods and remedies several issues of an existing homotopy method called the stochastic linear tracing procedure (SLTP). First, the starting stationary strategy profile can be arbitrarily chosen. Second, IPM does not need switching between different systems of equations. Third, the use of a perturbation term makes IPM applicable to all stochastic games rather than generic games only. Moreover, a well-chosen transformation of variables reduces the number of equations and variables by roughly one half. Numerical results show that the proposed method is more than three times as efficient as SLTP. On the other hand, the stochastic game can be reformulated as a mixed complementarity problem and solved by the PATH solver. We employ the proposed IPM and the PATH solver to compute SSPEs. Numerical results evince that for some stochastic games the PATH solver may fail to find an SSPE, whereas IPM is successful in doing so for all stochastic games, which confirms the reliability and stability of the proposed method. Summary of Contribution: This paper incorporates the interior-point methods into a differentiable path-following method for computing stationary equilibria for stochastic games. This novel method brings excellent computational advantages and remedies several issues with the existing methods for stochastic games. We prove the global convergence of the proposed method and employ this method to solve numerous randomly generated stochastic games with different scales. Numerical results further confirm the high efficiency, stability, and universality of this method for stochastic games.

Federated Learning for Intelligent Resources Allocation in Internet of Things

By using federated learning (FL), multiple Internet-of-Things (IoT) devices can construct a shared learning model without sending raw data to a centralized server. While FL has come a long way, it still has a ways to go. Issues such as heterogeneous user equipment (UEs) and data that is not independently and uniformly distributed are still obstacles. Facilitating a numerous UEs to participate in the learning in each cycle poses a possible problem of the huge communication budget. A weighted adjoining factor is presented to the localized gradient descent, generalizing the present FedAvg to solve these concerns. At the start of each global round, the proposed FL method randomly selects a fraction of the UEs to perform stochastic gradient descent in parallel. Then, we utilize the suggested FL method in cellular IoT to reduce either total power usage or execution duration of FL, in which a straightforward but effective path-following method is constructed for its explanations. At last, obtained simulations on poorly balanced data are presented to show that the presented FL algorithm is superior to FedAvg in terms of performance with respect to fast convergence. Moreover, they show that the suggested algorithm needs significantly less time and energy to train than the FL algorithm does when users contribute heavily to the learning process. These findings provide strong support for the suggested FL algorithm as a potential paradigm change for training mobile IoT networks with limited bandwidth.

大型車の一般路自動走行のための経路計画手法

In order to solve the problem regarding the shortage of truck and bus drivers in recent years, manpower-saving and unmanned operation of commercial vehicles is currently being actively researched and developed. However, when traveling on narrow road widths and curves with large curvatures, such as urban roads, commercial vehicles have larger internal wheel truck differences and overhang swing than passenger cars. Therefore, the path tracking control focusing on CG position tracking applied to passenger cars cannot be applied directly to commercial vehicles. Therefore, the purpose of this study is to develop a path planning and motion control system for automated operation of commercial vehicles in urban areas. In designing the control system, a risk potential field was used to express the driver's sense of danger with respect to obstacles and road boundaries. A simulation is conducted on a course that recostructs a real intersection, and a parameter study is conducted. Finally, the simulation results of this method are compared with those of a general path-following method, and the results show that this method is effective for avoiding obstacles in intersection situations.

A model for type I diabetes in an HIV-infected patient under highly active antiretroviral therapy

Type 1 diabetes (T1D), previously known as juvenile diabetes or insulin-dependent diabetes, is an autoimmune disease characterized by the insufficient (or lack of) production of insulin by the pancreas. Insulin is crucial to maintain blood sugar at healthy levels. High blood sugar damages the body and causes a variety of symptoms, ranging from severe thirst, fatigue, to urinary infections. The cells responsible for the production of insulin are the β-cells. In T1D, these are killed by an abnormal response of the immune system. Specific clones of cytotoxic T-cells invade the pancreatic islets of Langerhans, and eliminate them.T1D diabetes may develop in human immunodeficiency virus (HIV)-infected patients, though in rare situations. In this paper, we propose a cell model for the development of T1D in these patients, after immune restoration, during highly active antiretroviral therapy (HAART). The study includes the derivation of the qualitative properties of the model, and its comprehensive investigation via path-following methods, using the continuation platform COCO. In this way, the main theoretical predictions are verified in detail. Furthermore, this numerical part establishes accurate parameter thresholds to ensure an effective disease treatment in HIV-infected persons to prevent the development of T1D.

Controlling coexisting attractors of a class of non-autonomous dynamical systems

This paper studies a control method for switching stable coexisting attractors of a class of non-autonomous dynamical systems. The central idea is to introduce a continuous path for the system’s trajectory to transition from its original undesired stable attractor to a desired one by varying one of the system parameters according to the information of the desired attractor. The behaviour of the control is studied numerically for both non-smooth and smooth dynamical systems, using a soft-impact and a Duffing oscillators as examples. Special attention is given to identify the parametric regions where the proposed control strategy is applicable by using path-following methods implemented via the continuation platform COCO. It is shown that the proposed control concept can be implemented through either using an external control input or varying a system parameter. Finally, extensive numerical results are presented to validate the proposed control methods.

Sensitivity analysis to geometrical imperfections in shell buckling via a mixed generalized path-following method

An efficient continuation method is proposed for a direct sensitivity analysis of the critical point to geometrical imperfections, e.g. expressed as combination of given shapes, for thin-walled structures prone to buckling. After a finite element discretization, the critical point is defined by a system of nonlinear algebraic equations imposing equilibrium and critical condition according to the null vector method. An arc-length equation is added to follow a path of critical points in the imperfection space. Remarkable novelties are achieved. Firstly, the Jacobian of the extended system is entirely computed analytically by means of a solid-shell model and a strain-based modeling of the geometrical deviation. Moreover, a mixed scheme with independent element-wise stress variables is devised for a more efficient and robust iterative solution compared to the standard null vector method. The mixed algorithm speeds up the sensitivity analysis allowing larger imperfection steps and much fewer factorizations of the condensed stiffness matrix, only one per imperfection in its modified version with constant matrix over the step. Finally, the critical point derivatives with respect to the imperfection parameters are also obtained analytically and can be used to generate a gradient-based critical path for a quick search of the worst-case imperfection.

Optimal step length for the Newton method: case of self-concordant functions

The theoretical foundation of path-following methods is the performance analysis of the (damped) Newton step on the class of self-concordant functions. However, the bounds available in the literature and used in the design of path-following methods are not optimal. In this contribution we use methods of optimal control theory to compute the optimal step length of the Newton method on the class of self-concordant functions, as a function of the initial Newton decrement, and the resulting worst-case decrease of the decrement. The exact bounds are expressed in terms of solutions of ordinary differential equations which cannot be integrated explicitly. We provide approximate numerical and analytic expressions which are accurate enough for use in optimization methods. Consequently, the neighbourhood of the central path in which the iterates of path-following methods are required to stay can be enlarged, enabling faster progress along the central path during each iteration and hence fewer iterations to achieve a given accuracy.

A differentiable path-following method to compute subgame perfect equilibria in stationary strategies in robust stochastic games and its applications

As an effective paradigm to address uncertainty in payoffs and transition probabilities, robust stochastic games have been formulated in the literature. This paper is concerned with the computation of subgame perfect equilibria in stationary strategies (SSPEs) in robust stochastic games. To tackle this problem, we develop in this paper a globally convergent differentiable path-following method by exploiting the structures of the games. Incorporating a logarithmic-barrier term into each player’s payoff function with an extra variable between zero and one, we constitute a logarithmic-barrier robust stochastic game in which each player solves in each state a convex optimization problem. An application of the optimality conditions to the barrier game together with a fixed-point argument yields a polynomial equilibrium system for the barrier game. As a result of this system, we establish the existence of a smooth path that starts from an arbitrary mixed strategy profile and ends at an SSPE as the extra variable descends from one to zero. As an alternative scheme, we make up a convex-quadratic-penalty robust stochastic game and attain a globally convergent convex-quadratic-penalty differentiable path-following method for SSPEs in robust stochastic games. Numerical comparisons show that the logarithmic-barrier path-following method significantly outperforms the convex-quadratic-penalty path-following method. To further evince the value of the proposed methods, we apply the logarithmic-barrier path-following method to solve a supply chain configuration problem and a market entry problem from medical waste recycling.

A Topologically Inspired Path-Following Method With Intermittent State Feedback

Autonomous systems often operate in environments where state feedback may not be available, such as in anti-access and area denial environments. In these environments, it is often required that an agent track a path, despite interruptions in state feedback. As a result, the class of Relay-Explorer problems has emerged, where a switched system approach is used to account for intermittent state feedback. Past work on these problems established a framework for developing dwell-time conditions for stable tracking using these methods. However, existing work only applies to a limited class of reference paths and feedback region geometries. This letter advances a topologically inspired method for guaranteeing re-acquisition of feedback for nearly arbitrary geometries in arbitrary dimensions, all while relaxing the dwell-time conditions and retaining the uniformly ultimately bounded stability result from preceding work. Numerical experiments in the plane demonstrate an increase of hundreds of percentage points—even for fairly generic geometries—in the tracking error the agent could afford, using the proposed method, without sacrificing stability.