Tethered robots play a pivotal role in specialized environments such as disaster response and underground exploration, where their stable power supply and reliable communication offer unparalleled advantages. However, their motion planning is severely constrained by tether length limitations and entanglement risks, posing significant challenges to achieving optimal path planning. To address these challenges, this study introduces CDT-TCS (Convex Dissection Topology-based Tethered Configuration Search), a novel algorithm that leverages CDT Encoding as a homotopy invariant to represent topological states of paths. By integrating algebraic topology with geometric optimization, CDT-TCS efficiently computes the complete set of optimal feasible configurations for tethered robots at all positions in 2D environments through a single computation. Building on this foundation, we further propose three application-specific algorithms: i) CDT-TPP for optimal tethered path planning, ii) CDT-TMV for multi-goal visiting with tether constraints, iii) CDT-UTPP for distance-optimal path planning of untethered robots. All theoretical results and propositions underlying these algorithms are rigorously proven and thoroughly discussed in this paper. Extensive simulations demonstrate that the proposed algorithms significantly outperform state-of-the-art methods in their respective problem domains. Furthermore, real-world experiments on robotic platforms validate the practicality and engineering value of the proposed framework.

This research focuses on trajectory planning problems for autonomous vehicles utilizing numerical optimal control techniques. The study reformulates the constrained optimization problem into a nonlinear programming problem, incorporating explicit collision avoidance constraints. We present three novel, exact formulations to describe collision constraints. The first formulation is derived from a proposition concerning the separation of a point and a convex set. We prove the separating proposition through De Morgan's laws. Then, leveraging the hyperplane separation theorem we propose two efficient reformulations. Compared with the existing dual formulations and the first formulation, they significantly reduce the number of auxiliary variables to be optimized and inequality constraints within the nonlinear programming problem. Finally, the efficacy of the proposed formulations is demonstrated in the context of typical autonomous parking scenarios compared with state of the art. For generality, we design three initial guesses to assess the computational effort required for convergence to solutions when using the different collision formulations. The results illustrate that the scheme employing De Morgan's laws performs equally well with those utilizing dual formulations, while the other two schemes based on hyperplane separation theorem exhibit the added benefit of requiring lower computational resources.

This paper introduces a differentiable representation for optimization of boustrophedon path plans in convex polygons, explores an additional parameter of these path plans that can be optimized, discusses the properties of this representation that can be leveraged during the optimization process, and shows that the previously published attempt at optimization of these path plans was too coarse to be practically useful. Experiments were conducted to show that this differentiable representation can reproduce the same scores from transitional discrete representations of boustrophedon path plans with high fidelity. Finally, optimization via gradient descent was attempted, but found to fail because the search space is far more non-convex than was previously considered in the literature. The wide range of applications for boustrophedon path plans means that this work has the potential to improve path planning efficiency in numerous areas of robotics including mapping and search tasks using uncrewed aerial systems, environmental sampling tasks using uncrewed marine vehicles, and agricultural tasks using ground vehicles, among numerous others applications.

The Dijkstra algorithm is a classic path planning method, which in a discrete graph space, can start from a specified source node and find the shortest path between the source node and all other nodes in the graph. However, to the best of our knowledge, there is no effective method that achieves a function similar to that of the Dijkstra's algorithm in a continuous space. In this study, an optimal path planning algorithm called convex dissection topology (CDT)-Dijkstra is developed, which can quickly compute the global optimal path from one point to all other points in a 2D continuous space. CDT-Dijkstra is mainly divided into two stages: SetInit and GetGoal. In SetInit, the algorithm can quickly obtain the optimal CDT encoding set of all the cut lines based on the initial point x_{init}. In GetGoal, the algorithm can return the global optimal path of any goal point at an extremely high speed. In this study, we propose and prove the planning principle of considering only the points on the cutlines, thus reducing the state space of the distance optimal path planning task from 2D to 1D. In addition, we propose a fast method to find the optimal path in a homogeneous class and theoretically prove the correctness of the method. Finally, by testing in a series of environments, the experimental results demonstrate that CDT-Dijkstra not only plans the optimal path from all points at once, but also has a significant advantage over advanced algorithms considering certain complex tasks.

The concept of path homotopy has received widely attention in the field of path planning in recent years. In this article, a homotopy invariant based on convex dissection for a two-dimensional bounded Euclidean space is developed, which can efficiently encode all homotopy path classes between any two points. Thereafter, the optimal path planning task consists of two steps: (i) search for the homotopy path class that may contain the optimal path, and (ii) obtain the shortest homotopy path in this class. Furthermore, an optimal path planning algorithm called CDT-RRT* (Rapidly-exploring Random Tree Star based on Convex Division Topology) is proposed. We designed an efficient sampling formula for CDT-RRT*, which gives it a tendency to actively explore unknown homotopy classes, and incorporated the principles of the Elastic Band algorithm to obtain the shortest path in each class. Through a series of experiments, it was determined that the performance of the proposed algorithm is comparable with state-of-the-art path planning algorithms. Hence, the application significance of the developed homotopy invariant in the field of path planning was verified.

As a core part of autonomous driving systems, motion planning has received extensive attention from academia and industry. However, real-time trajectory planning capable of spatial-temporal joint optimization is challenged by nonholonomic dynamics, particularly in the presence of unstructured environments and dynamic obstacles. To bridge the gap, we propose a real-time trajectory optimization method that can generate a high-quality whole-body trajectory under arbitrary environmental constraints. By leveraging the differential flatness property of car-like robots, we simplify the trajectory representation and analytically formulate the planning problem while maintaining the feasibility of the nonholonomic dynamics. Moreover, we achieve efficient obstacle avoidance with a safe driving corridor for unmodelled obstacles and signed distance approximations for dynamic moving objects. We present comprehensive benchmarks with State-of-the-Art methods, demonstrating the significance of the proposed method in terms of efficiency and trajectory quality. Real-world experiments verify the practicality of our algorithm. We will release our codes for the research community

This paper is proposed to efficiently provide a convex approximation for the probabilistic reachable set of a dynamic system in the face of uncertainties. When the uncertainties are not limited to bounded ones, it may be impossible to find a bounded reachable set of the system. Instead, we turn to find a probabilistic reachable set that bounds system states with confidence. A data-driven approach of Kernel Density Estimator (KDE) accelerated by Fast Fourier Transform (FFT) is customized to model the uncertainties and obtain the probabilistic reachable set efficiently. However, the irregular or non-convex shape of the probabilistic reachable set refrains it from practice. For the sake of real applications, we formulate an optimization problem as Mixed Integer Nonlinear Programming (MINLP) whose solution accounts for an optimal $n$-sided convex polygon to approximate the probabilistic reachable set. A heuristic algorithm is then developed to solve the MINLP efficiently while ensuring accuracy. The results of comprehensive case studies demonstrate the near-optimality, accuracy, efficiency, and robustness enjoyed by the proposed algorithm. The benefits of this work pave the way for its promising applications to safety-critical real-time motion planning of uncertain systems.

Modelling stockpile is a key factor of a project economic and operation in mining, because not all the mined ores are not able to mill for many reasons. Further, the financial value of the ore in the stockpile needs to be reflected on the balance sheet. Therefore, automatically tracking the frontiers of the stockpile facilitates the mine scheduling engineers to calculate the tonnage of the ore remaining in the stockpile. This paper suggests how the dynamic of stockpile shape changes caused by dumping and reclaiming operations can be inferred using polygon models. The presented work also demonstrates how the geometry of stockpiles can be inferred in the absence of reclaimed bucket information, in which case the reclaim polygons are established using the diggers GPS positional data at the time of truck loading. This work further compares two polygon models for creating 2D shapes.

One of the most important objectives in motion planning is finding a feasible path from the starting point to a goal without collision with obstacles. The obstacles are either closed doors, which can be opened and removed by the robot, or are obstacles that cannot be passed which can be ignored by the robot with a penalty. Usually, there is no feasible path for some navigation. Recently, some researchers have focused on finding a path for the robot by minimizing the number of removed obstacles. For instance, in Stilman and Kuffner's paper [Stilman and Kuffner, 2005], the robot is able to move the obstacles around and clear its movement space.

The Mondrian process represents an elegant and powerful approach for space partition modelling. However, as it restricts the partitions to be axis-aligned, its modelling flexibility is limited. In this work, we propose a self-consistent Binary Space Partitioning (BSP)-Tree process to generalize the Mondrian process. The BSP-Tree process is an almost surely right continuous Markov jump process that allows uniformly distributed oblique cuts in a two-dimensional convex polygon. The BSP-Tree process can also be extended using a non-uniform probability measure to generate direction differentiated cuts. The process is also self-consistent, maintaining distributional invariance under a restricted subdomain. We use Conditional-Sequential Monte Carlo for inference using the tree structure as the high-dimensional variable. The BSP-Tree process's performance on synthetic data partitioning and relational modelling demonstrates clear inferential improvements over the standard Mondrian process and other related methods.