Piecewise Linear Path Research Articles

Kernel Error Path Algorithm.

Tuning the values of kernel parameters plays a vital role in the performance of kernel methods. Kernel path algorithms have been proposed for several important learning algorithms, including support vector machine and kernelized Lasso, which can fit the piecewise nonlinear solutions of kernel methods with respect to the kernel parameter in a continuous space. Although the error path algorithms have been proposed to ensure that the model with the minimum cross validation (CV) error can be found, which is usually the ultimate goal of model selection, they are limited to piecewise linear solution paths. To address this problem, in this article, we extend the classic error path algorithm to the nonlinear kernel solution paths and propose a new kernel error path algorithm (KEP) that can find the global optimal kernel parameter with the minimum CV error. Specifically, we first prove that error functions of binary classification and regression problems are piecewise constant or smooth w.r.t. the kernel parameter. Then, we propose KEP for support vector machine and kernelized Lasso and prove that it guarantees to find the model with the minimum CV error within the whole range of kernel parameter values. Experimental results on various datasets show that our KEP can find the model with minimum CV error with less time consumption. Finally, it would have better generalization error on the test set, compared with grid search and random search.

Fast GPU-based Two-way Continuous Collision Handling

Step-and-project is a popular method to simulate non-penetrating deformable bodies in physically based animation. The strategy is to first integrate the system in time without considering contacts and then resolve potential intersections, striking a good balance between plausibility and efficiency. However, existing methods can be defective and unsafe when using large time steps, taking risks of failure or demanding repetitive collision testing and resolving that severely degrade performance. In this article, we propose a novel two-way method for fast and reliable continuous collision handling. Our method launches an optimization from both ends of the intermediate time-integrated state and the previous intersection-free state. It progressively generates a piecewise linear path and eventually obtains a feasible solution for the next time step. The algorithm efficiently alternates between a forward step and a backward step until the result is conditionally converged. Thanks to a set of unified volume-based contact constraints, our method offers flexible and reliable handling of various codimensional deformable bodies, including volumetric bodies, cloth, hair, and sand. Experimental results demonstrate the safety, robustness, physical fidelity, and numerical efficiency of our method, making it particularly suitable for scenarios involving large deformations or large time steps.

The Accessibility Problem for Geometric Rough Differential Equations

We show how to use geometric arguments to prove that the terminal solution to a rough differential equation driven by a geometric rough path can be obtained by driving the same equation by a piecewise linear path. For this purpose, we combine some results of the seminal work of Sussmann on orbits of vector fields [1] with the rough calculus on manifolds developed by Cass, Litterer and Lyons in [2].

UAV Path Planning in 3-D Constrained Environments Based on Layered Essential Visibility Graphs

In this paper, an optimal path search methodology for UAVs flying in 3D environments is presented, taking into account the presence of obstacles and other constraints deriving from flight dynamics. The so-called Essential Visibility Graph (EVG) is extended to the 3D case by describing the obstacles using a finite number of parallel planar layers at different altitudes. The resulting graph, called the Layered Essential Visibility Graph (LEVG), is based on an efficient branching algorithm and it is made up of a reduced number of nodes and edges thus assuring a limited computational burden. Once the optimal piece-wise linear path has been identified over the LEVG, aircraft performance related constraints, formulated in terms of turn and pull-up radii limits, can be taken into account via a smoothing procedure based on a 3D extension of Dubins' paradigm. This way an optimal flyable 3D path can be obtained. The implementation of a specific integer programming formulation within the graph search process ensures the full compliance of the optimal smoothed trajectory with the environmental constraints. The effectiveness of the proposed methodology is proved by means of numerical tests in complex operational scenarios over a real terrain morphology and an urban environment.

Clothoid-Based Path Planning for a Formation of Fixed-Wing UAVs

Unmanned aerial vehicles (UAVs) are playing an increasingly crucial role in many applications such as search and rescue, delivery services, and military operations. However, one of the significant challenges in this area is to plan efficient and safe trajectories for UAV formations. This paper presents an optimization procedure for trajectory planning for fixed-wing UAV formations using graph theory and clothoid curves. The proposed planning strategy consists of two main steps. Firstly, the geometric optimization of paths is carried out using graphs for each UAV, providing piece-wise linear paths whose smooth connections are made with clothoids. Secondly, the geometric paths are transformed into time-dependent trajectories, optimizing the assigned aircraft speeds to avoid collisions by solving a mixed-integer optimal control problem for each UAV of the flight formation. The proposed method is effective in achieving suboptimal paths while ensuring collision avoidance between aircraft. A sensitivity analysis of the main parameters of the algorithm was conducted in ideal conditions, highlighting the possibility of decreasing the length of the optimal path by about 4.19%, increasing the number of points used in the discretization and showing a maximum path length reduction of about 10% compared with the average solution obtained with a similar algorithm using a graph based on random directions. Furthermore, the use of clothoids, whose parameters depend on the UAV performance constraints, provides smoother connections, giving a significant improvement over traditional straight-line or circular trajectories in terms of flight dynamics compliance and trajectory tracking capabilities. The method can be applied to various UAV formation scenarios, making it a versatile and practical tool for mission planning.

A probabilistic model for fast-to-evaluate 2D crack path prediction in heterogeneous materials

This paper is devoted to the construction of a new fast-to-evaluate model for the prediction of 2D crack paths in concrete-like microstructures. The model generates piecewise linear cracks paths with segmentation points selected using a Markov chain model. The Markov chain kernel involves local indicators of mechanical interest and its parameters are learnt from numerical full-field 2D simulations of cracking using a cohesive-volumetric finite element solver called XPER. This model does not include any mechanical elements. It is the database, derived from the XPER crack, that contains the mechanical information and optimizes the probabilistic model. The resulting model exhibits a drastic improvement of CPU time in comparison to simulations from XPER.

Local smoothing of the piecewise linear tool path in a CNC system for high-speed machining

The basic algorithms of trajectory smoothing in control systems of machine tools and industrial robots in the case of linear-piece trajectory assignment are considered. The method of local contour smoothing by cubic B-spline in three-axis interpolation, which together with the algorithm of quasi-optimal feed planning with jerk limiting allowed to signifi cantly increase the machining productivity, is discussed in detail. The insertion of splines takes into account userdefi ned geometric constraints and converts the original trajectory into a composite curve that has geometric continuity G2, providing continuity of accelerations for each link of the machine. The defi nition of geometric continuity G1 and G2 is discussed, and its effect on the quality of the machined surface.

LMBOPT: a limited memory method for bound-constrained optimization

Recently, Neumaier and Azmi gave a comprehensive convergence theory for a generic algorithm for bound constrained optimization problems with a continuously differentiable objective function. The algorithm combines an active set strategy with a gradient-free line search CLS along a piecewise linear search path defined by directions chosen to reduce zigzagging. This paper describes LMBOPT, an efficient implementation of this scheme. It employs new limited memory techniques for computing the search directions, improves CLS by adding various safeguards relevant when finite precision arithmetic is used, and adds many practical enhancements in other details. The paper compares LMBOPT and several other solvers on the unconstrained and bound constrained problems from the CUTEst collection and makes recommendations on which solver to use and when. Depending on the problem class, the problem dimension, and the precise goal, the best solvers are LMBOPT, ASACG, and LMBFG-EIG-MS.

UAV Trajectory and Communication Co-Design: Flexible Path Discretization and Path Compression

The performance optimization of UAV communication systems requires the joint design of UAV trajectory and communication efficiently. To tackle the challenge of infinite design variables arising from the continuous-time UAV trajectory optimization, a commonly adopted approach in the existing literature is by approximating the UAV trajectory with piecewise-linear path segments connected via a finite number of waypoints in three-dimensional (3D) space. However, this approach may still incur prohibitive computational complexity in practice when the UAV flight period/distance becomes long, as the distance between consecutive waypoints needs to be kept sufficiently small to retain high approximation accuracy. To resolve this fundamental issue, we propose in this paper a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">new</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">general</i> framework for UAV trajectory and communication co-design with flexible number of waypoint optimization variables (called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">designable</i> waypoints) or their <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sub-path</i> representations. First, we propose a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">flexible path discretization</i> scheme that optimizes only a number of selected waypoints (designable waypoints) along the UAV path for complexity reduction, while all the designable and non-designable waypoints are used in calculating the approximated communication utility along the UAV trajectory for ensuring high trajectory discretization accuracy. Next, we propose a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">path compression</i> scheme, which treats the UAV trajectory as a signal and compresses its path representation based on the basis decomposition. Specifically, the UAV 3D path is first decomposed into three one-dimensional (1D) sub-paths and each sub-path is then approximated by superimposing a number of selected basis paths (which are generally less than the number of designable waypoints) weighted by their corresponding path coefficients, thus further reducing the path design complexity. Finally, we provide a case study on UAV trajectory design for aerial data harvesting from distributed sensors, and numerically show that the proposed flexible path discretization and path compression schemes can significantly reduce the UAV trajectory design complexity yet achieve favorable rate performance as compared to conventional path/time discretization schemes.

Regularization paths of L1‐penalized ROC Curve‐Optimizing Support Vector Machines

The receiver operator characteristic (ROC) curve is one of the most popular tools to evaluate the performance of binary classifiers in a variety of applications. Rakotomamonjy (2004) proposed the ROC‐SVM that directly optimizes the area under the ROC curve instead of the prediction accuracy. In this article, we study the L1‐penalized ROC‐SVM that directly optimizes the ROC curve. We first show that the L1‐penalized ROC‐SVM has piecewise linear regularization paths and then develop an efficient algorithm to compute the entire paths, which greatly facilitates its tuning procedure.

Fractional Line Integral

This paper proposed a definition of the fractional line integral, generalising the concept of the fractional definite integral. The proposal replicated the properties of the classic definite integral, namely the fundamental theorem of integral calculus. It was based on the concept of the fractional anti-derivative used to generalise the Barrow formula. To define the fractional line integral, the Grünwald–Letnikov and Liouville directional derivatives were introduced and their properties described. The integral was defined for a piecewise linear path first and, from it, for any regular curve.

Scalable and Safe Multi-Agent Motion Planning with Nonlinear Dynamics and Bounded Disturbances

We present a scalable and effective multi-agent safe motion planner that enables a group of agents to move to their desired locations while avoiding collisions with obstacles and other agents, with the presence of rich obstacles, high-dimensional, nonlinear, nonholonomic dynamics, actuation limits, and disturbances. We address this problem by finding a piecewise linear path for each agent such that the actual trajectories following these paths are guaranteed to satisfy the reach-and-avoid requirement. We show that the spatial tracking error of the actual trajectories of the controlled agents can be pre-computed for any qualified path that considers the minimum duration of each path segment due to actuation limits. Using these bounds, we find a collision-free path for each agent by solving Mixed Integer-Linear Programs and coordinate agents by using the priority-based search. We demonstrate our method by benchmarking in 2D and 3D scenarios with ground vehicles and quadrotors, respectively, and show improvements over the solving time and the solution quality compared to two state-of-the-art multi-agent motion planners.

Iterative Learning Control for Path-Following Tasks With Performance Optimization

The classical problem setup of iterative learning control (ILC) is to enforce tracking of a reference profile specified at all time points in the fixed task duration. The removal of the time specification releases significant design freedom in how the path is followed but has not been fully exploited in the literature. This article unlocks this extra design freedom by formulating the ILC task description to handle repeated path-following tasks, e.g., welding and laser cutting, which aim at following a given “spatial” path defined in the output space without any temporal information. The general ILC problem is reformulated for ILC design with the inclusion of an additional performance index, and the class of piecewise linear paths is characterized for the reformulated problem setup. A two-stage design framework is proposed to solve the characterized problem and yields a comprehensive algorithm based on an ILC update and a gradient projection update. This algorithm is verified on a gantry robot experimental platform to demonstrate its practical efficacy and robustness against model uncertainty.

Restriction of Transverse Feedback Linearization for Piecewise Linear Paths

This work presents a path-following controller for a unicycle robot. The main contribution of this paper is to demonstrate the restriction of transverse feedback linearization (TFL) to obtuse angles on piecewise linear paths. This restriction is experimentally demonstrated on a Kobuki mobile robot, where it is possible to observe, as a result of the limitation of the TFL, the convergence to another domain of attraction.

Piecewise Linear Path Following for a Unicycle Using Transverse Feedback Linearization

Motion planning and path following together constitute a critical part of the decision-making hierarchy in autonomous ground vehicles. One of the simplest instances of this architecture is when the path planner generates waypoints that define a sequence of collision-free line segments from a start location to goal destination and when the vehicle’s kinematic model is taken to be Dubin’s vehicle. The low-level feedback controller can then be designed by treating the path following problem as a set stabilization problem; one such approach is called transverse feedback linearization (TFL). However, for a Dubin’s vehicle with only one input, the direction of traversal along the path is completely determined by the vehicle’s initial condition. In this article, we provide easily checkable sufficient conditions that only need to be checked once at run time and a simple systematic design procedure, which guarantees that the robot traverses the path while coming within a user-specified distance of the waypoints; the procedure is also empirically verified.

A Trade-off Between Energy Saving and Cycle Time Reduction by Pareto Optimal Corner Smoothing in Industrial Feed Drive Systems

Apart from improving motion accuracy, increasing productivity while reducing computer numerical control machine tools' carbon footprint is vital in the manufacturing industry. Several studies have proposed corner smoothing methods that improve cycle time for piecewise linear paths by exploiting axial limits to achieve time-optimal trajectories. Energy-saving is not considered an objective in these methods. This paper presents a method of generating Pareto optimal corner smoothing trajectories that trade-off the contradicting objectives of minimizing cycle time and energy consumption. The trajectories along linear paths and smoothed corners are respectively described using jerk limited acceleration profiles and kinematic corner smoothing methods. An energy consumption model of an industrial two-axis feed drive system is identified and used in solving the bi-objective optimization problem. From the resulting Pareto frontier, the best trade-off trajectory is selected as the one that minimizes both objectives. The proposed method's effectiveness is vindicated experimentally, where the best trade-off trajectory achieves respectively 65.87 % and 60.44 % of the time and energy-saving potentials.

Sparse Road Network Model for Autonomous Navigation Using Clothoids

To autonomously navigate in traffic roads, an Autonomous Vehicle must take into account perception information, as well as the topological and geometric structure of the environment it is inserted in. Specifically in urban scenarios, the vehicle has to plan its path across intersections, roundabouts and perform lane changes to obey traffic rules. In addition, the planning algorithm must also consider the kinematics constraints of the vehicle and comfort parameters to the passengers. This paper proposes a road network model based on clothoids, which embraces the geometric and topological representation of the environment in a compact data structure. Piecewise linear continuous-curvature paths composed of clothoids, circular arcs, and straight lines are used for this purpose. The proposed approaches are evaluated in an urban scenario composed of curved and straight roads with single and double lanes, roundabouts, and intersections. As a result, a navigation architecture for Autonomous Vehicles was developed using the model, including global planning with continuous-curvature paths.

Path Planning and Real-Time Collision Avoidance Based on the Essential Visibility Graph

This paper deals with a novel procedure to generate optimum flight paths for multiple unmanned aircraft in the presence of obstacles and/or no-fly zones. A real-time collision avoidance algorithm solving the optimization problem as a minimum cost piecewise linear path search within the so-called Essential Visibility Graph (EVG) is first developed. Then, a re-planning procedure updating the EVG over a selected prediction time interval is proposed, accounting for the presence of multiple flying vehicles or movable obstacles. The use of Dubins curves allows obtaining smooth paths, compliant with flight mechanics constraints. In view of possible future applications in hybrid scenarios where both manned and unmanned aircraft share the airspace, visual flight rules compliant with International Civil Aviation Organization (ICAO) Annex II Right of Way were implemented. An extensive campaign of numerical simulations was carried out to test the effectiveness of the proposed technique by setting different operational scenarios of increasing complexity. Results show that the algorithm is always able to identify trajectories compliant with ICAO rules for avoiding collisions and assuring a minimum safety distance as well. Furthermore, the low computational burden suggests that the proposed procedure can be considered a promising approach for real-time applications.

Fast Algorithms for Rank-1 Bimatrix Games

Games with multiplicative payoffs

Speed of propagation for Hamilton–Jacobi equations with multiplicative rough time dependence and convex Hamiltonians

We show that the initial value problem for Hamilton–Jacobi equations with multiplicative rough time dependence, typically stochastic, and convex Hamiltonians satisfies finite speed of propagation. We prove that in general the range of dependence is bounded by a multiple of the length of the “skeleton” of the path, that is a piecewise linear path obtained by connecting the successive extrema of the original one. When the driving path is a Brownian motion, we prove that its skeleton has almost surely finite length. We also discuss the optimality of the estimate.