Time-optimal Path Research Articles

Sigmoid angle-arc curves: Enhancing robot time-optimal path parameterization for high-order smooth motion

Trajectory planning is crucial in the motion planning of robots, where finding the time-optimal path parameterization (TOPP) of a given path subject to kinodynamic constraints is an important component of trajectory planning. The tangential discontinuity at the intersection of continuous line segments limits the speed of trajectory planning and can easily cause jitter and over-constraint phenomena. Smooth transitions at corners can be achieved by inserting parameter spline curves. However, due to the insensitivity of parameter spline curves to arc length, their performance in the application of the TOPP algorithm, which relies on the higher-order robot kinematics smoothness (i.e., the function q(s) of the configuration space to the Cartesian space), fails to meet expectations.A smoothing method suitable for the TOPP algorithm is proposed: Sigmoid Angle-Arc Curve (SAAC). This curve exhibits excellent performance in smooth corner transitions of the TOPP algorithm and is parameterized using arc length. The curvature and geometry of its curves can be expressed analytically in terms of arc lengths. Compared with the traditional B-spline method and the symmetric Euler spiral blending (SE-spiral), SAAC can provide smoother C2 robot kinematics characteristics. Using the TOPP algorithm based on SAAC can significantly enhance the robustness of the TOPP algorithm, significantly reduce jerks, and reduce the time required for movement.

Minimum Distance and Minimum Time Optimal Path Planning With Bioinspired Machine Learning Algorithms for Faulty Unmanned Air Vehicles

Minimum Distance and Minimum Time Optimal Path Planning With Bioinspired Machine Learning Algorithms for Faulty Unmanned Air Vehicles

Time-Optimal Path Planning in a Constant Wind for Uncrewed Aerial Vehicles Using Dubins Set Classification

Time-Optimal Path Planning in a Constant Wind for Uncrewed Aerial Vehicles Using Dubins Set Classification

Classification of Time-Optimal Paths Under an External Force Based on Jacobi Stability in Finsler Space

Classification of Time-Optimal Paths Under an External Force Based on Jacobi Stability in Finsler Space

Time-Optimal Rendezvous for Cooperative UUV Swarms Using Hamilton-Jacobi Partial Differential Equations

A new approach to time-optimal path planning for cooperative swarms of underwater vehicles in the presence of ocean currents is presented. Specifically, the determination of a time-optimal rendezvous among a swarm of vehicles is carried out by solving a family of Hamilton-Jacobi partial differential equations. The resulting algorithm is fast, globally time-optimal, and scales linearly with the number of vehicles. The method is validated through several examples for which optimal trajectories are known from the calculus of variations. Additional examples are also included.

Time-Optimal Path Tracking for Dual-Arm Free-Floating Space Manipulator System Using Convex Programming

In this paper, the time-optimal path tracking problem of the dual-arm free-floating space manipulator (FFSM) system is solved by using convex programming. Firstly, we construct a continuous nonlinear optimization problem that considers system dynamics and constraints along the path, including torque, velocity and acceleration. Subsequently, the original optimization problem is transformed into a convex optimal control problem by using scalar path coordinate and nonlinear variables. To characterize the problem in a more manageable form, we apply the direct transcription method and discretize the problem into a sparse convex optimization program. Numerical results demonstrate that the proposed approach efficiently determines the trajectory of the FFSM along the preset path. Meanwhile, the base's velocity can be limited to a small region, which is an essential requirement for ensuring the stability of the satellite. Furthermore, an experiment is conducted using GLUON robots. The tracking error between the actual and preset pose of the target verifies the feasibility of the proposed method.

Sampling-based time-optimal path parameterization with jerk constraints for robotic manipulation

In this paper, a sampling-based time-optimal path parameterization (S-TOPP) method is proposed to address time-optimal trajectory planning problems with bounded jerks. The key insight of S-TOPP is that a tree of feasible nodes connected by edges is established to find a time-optimal trajectory on the temporal dimension. The tree establishment process includes two major phases at each stage, namely sampling and one-step backtracking. In the sampling phase, a new sampling strategy integrating historical information is proposed to obtain superior samples whereby fewer samples can be controlled automatically, reducing the calculation loss. In one-step backtracking phase, a “lazy” strategy is used to lazily skip constraint-checking when evaluating local connections, enabling S-TOPP to avoid checking the vast majority of nodes that have no chance of being in an optimal trajectory. Simulations and real-world experiments validate the feasibility and practicability of S-TOPP. Results show that S-TOPP is an effective solution to jerk-bounded time-optimal trajectory planning, with features that are more in line with the needs of the practical tasks compared with other methods.

An electric vehicle charging load prediction model for different functional areas based on multithreaded acceleration

This paper proposes an electric vehicle (EV) charging load prediction model for different functional areas based on multithreaded technology. This model comprehensively incorporates the preference characteristics of EV users in charging behavior and mode, as well as accounting for diverse travel purposes. It is worth emphasizing that a comprehensive analysis of the spatiotemporal travel characteristics exhibited by users during different seasons and on working days versus rest days is provided. In terms of temporal scale, refined probability models are employed to accurately fit the user's travel time variables. In terms of the spatial scale, the initial position and spatial transfer probability of users are analyzed. Subsequently, leveraging graph theory, a regional transportation network model is established. On this basis, a novel improved time optimal path decision-making algorithm is proposed by holistically considering factors such as road grade, saturation, traffic conditions, one-way/two-way streets, and delay at signalized intersections. Additionally, the impact of temperature on the upper limit capacity of batteries is analyzed, as well as driving/air conditioning energy consumption models are established. It is noteworthy that a two-stage charging power variation model is introduced, enhancing the precision of charging power calculation. Finally, the effectiveness of the method is validated through case analysis. The results demonstrate a significant disparity in load demand among various functional areas, and the spatiotemporal distribution of load demand is significantly influenced by seasons and working days versus rest days. Furthermore, when employing 4 threads, the proposed method achieves a speedup ratio exceeding 2.5 compared to conventional serial methods.

Time-optimal path tracking for cooperative manipulators: A convex optimization approach

This paper studies the time-optimal path tracking problem for a team of cooperating robotic manipulators carrying an object. Considering the problem for rigidly grasped objects, we show that it can be cast as a convex optimization problem and solved efficiently with a guarantee of optimality. When formulating the problem, we avoid using a particular wrench distribution and exploit the full actuation available to the system. Then, we consider the problem for grasps using frictional forces and show that this problem also, under a force-closure grasp assumption, can be formulated as a convex optimization problem and solved efficiently and to optimality. To ensure a firm grasp, internal forces have been taken into account in this approach.

Time-optimal path planning and tracking based on nonlinear model predictive control and its application on automatic berthing

Autonomous shipping has been identified as the way forward in maritime transport. There are numerous technologies being developed over the past decades for maritime autonomous surface ships, including advanced automatic manoeuvring control. As one of the essential indices to evaluate ships’ manoeuvrability, the time-optimal problem remains a challenging issue in the process of path planning and path tracking, which has not been properly addressed for the maritime autonomous surface ships. To fill this knowledge gap, we propose a novel time-optimal path planning and tracking control method based on nonlinear model predictive control (MPC) and spatial reformulation. The proposed time-optimal controller is designed as a two-level hierarchical controller to plan the time-optimal path in spatial coordinates and track it in temporal coordinates when considering the low-speed manoeuvring model. At the high level, the nonlinear MPC based on a spatial formulation is used to generate a planned path of minimum manoeuvring time for a manoeuvring ship. At the low level, the optimal planning trajectory is interpolated by a constant time iteration and the vessel tracks the time-optimal planning trajectory by implementing the nonlinear MPC based on a temporal formulation. As an advanced predictive controller, the nonlinear MPC optimises the berthing path to a minimum time mathematically at the planning stage and compensates for the uncertain disturbance during the tracking process. To validate the time-optimal control method, a typical scenario of low-speed berthing operation is analysed. The results of the case study show that the proposed control method can be successfully applied to time-optimal manoeuvring problems.

Time-Optimal Path Planning in an Evolving Ocean Wave Field Based on Reachability Theory

We consider the navigation of surface vessels in an evolving ocean wave field, where the detrimental extreme waves are treated as moving and deforming obstacles. We propose a new time-optimal path planning algorithm with avoidance of such obstacles based on the reachability theory. In our framework, the mathematical optimization problem is boiled down to: 1) forward propagation of reachable set described by a variational inequation (VI), which is numerically solved through the level set method combined with a prediction-correction scheme and 2) a backtracking procedure to find the optimal path. The new path planner is tested in a number of cases including one with realistic ocean waves. We demonstrate that our algorithm can provide correct paths avoiding high waves and achieve multiple optimal (equivalent in time) paths when available. Finally, the framework can be straightforwardly extended to other problems involving collision avoidance with moving and deforming obstacles.

The slope-of-a-mountain problem in a cross gravitational wind

In this paper, the Matsumoto slope-of-a-mountain problem is revisited. We present a general model of the slope as a Riemannian manifold, exploiting the influences of both transverse and longitudinal components of the gravitational wind on time-optimal paths. In contrast to the original Matsumoto’s exposition, the cross-gravity additive is taken into account in the equations of motion, while the along-gravity effect is entirely compensated. The purely geometric solutions to the problem are investigated by means of Finsler geometry. Our study explores the properties of the new cross-slope metric obtained in this work, which belongs to the class of general (α,β)-metrics, emphasizing the strong convexity conditions as well as the corresponding time geodesics. Using some relevant examples the developed theory is illustrated, while explaining and comparing the deformations of the Finslerian indicatrices and the behavior of time-minimal paths on the mountain slopes with different approaches to the gravity effect.

Time-Optimal Trajectory Planning for the Manipulator Based on Improved Non-Dominated Sorting Genetic Algorithm II

To address the issues of low efficiency and lengthy running time associated with trajectory planning for 6-degree-of-freedom manipulators, this paper introduces a novel solution that generates a time-optimal path for a manipulator while adhering to its kinematic limitations. The proposed method comprises several stages. Firstly, the kinematics of the manipulator are analyzed. Secondly, the manipulator’s joint-space path points are interpolated via the quintic B-spline curve. Subsequently, the non-dominated sorting genetic algorithm II (NSGA-II) is improved by applying reinforcement learning to optimize its crossover and mutation probabilities, and the variable neighborhood search (VNS) algorithm is integrated to enhance its local search capability. Finally, the joint increments and running time of the manipulator are optimized using the improved NSGA-II, and the time-optimal trajectory is then determined by simulation on MATLAB. Furthermore, compared with other conventional optimization methods, the proposed approach has reduced the total running time by 19.26%, effectively improving the working efficiency of the manipulator.

Differential Flatness of Slider–Pusher Systems for Constrained Time Optimal Collision Free Path Planning

Abstract In this work, we show that the differential kinematics of slider–pusher systems are differentially flat assuming quasi-static behavior and frictionless contact. Second, we demonstrate that the state trajectories are invariant to time-differential transformations of the path parametrizing coordinate. For one this property allows to impose arbitrary velocity profiles on the slider without impacting the geometry of the state trajectory. This property implies that certain path planning problems may be decomposed approximately into a strictly geometric path planning and an auxiliary throughput speed optimization problem. Building on these insights, we elaborate a numerical approach tailored to constrained time optimal collision free path planning and apply it to the slider–pusher system.

Time-optimal path following for non-redundant serial manipulators using an adaptive path-discretization

AbstractThe time-optimal path following (OPF) problem is to find a time evolution along a prescribed path in task space with shortest time duration. Numerical solution algorithms rely on an algorithm-specific (usually equidistant) sampling of the path parameter. This does not account for the dynamics in joint space, that is, the actual motion of the robot, however. Moreover, a well-known problem is that large joint velocities are obtained when approaching singularities, even for slow task space motions. This can be avoided by a sampling in joint space, where the path parameter is replaced by the arc length. Such discretization in task space leads to an adaptive refinement according to the nonlinear forward kinematics and guarantees bounded joint velocities. The adaptive refinement is also beneficial for the numerical solution of the problem. It is shown that this yields trajectories with improved continuity compared to an equidistant sampling. The OPF is reformulated as a second-order cone programming and solved numerically. The approach is demonstrated for a 6-DOF industrial robot following various paths in task space.

Point to point time optimal handling of unmounted rigid objects and liquid-filled containers

Time optimal handling of parts loosely placed at the end-effector (EE) of a robot, known as waiter motion problem, has high practical relevance, which becomes more challenging if the object is a liquid-filled container. The waiter motion problem was often restricted to a time-optimal path following problem, which limits the flexibility and task efficiency. To overcome this restriction, in this paper, the general point to point (PtP) time-optimal motion planning problem is formulated and solved with a multiple shooting method. The formulation includes all relevant dynamic effects (joint friction, contact, motor limits, etc.) assuming rigid behavior of the robot as well as of the object placed at the EE. The trajectory is C2-continuous, avoiding bang–bang behavior of motor torques. The CasADi framework and the Ipopt solver are used for numerical computations. The reported experimental results confirm applicability of the trajectory to real robotic setup in case of rigid objects. Further, handling of containers filled with liquid is addressed. Experiments are compared with numerical results obtained with SPH particle simulation. Experiment and simulation indicate that sloshing effects must be taken into account in the control formulation.

Energy–time optimal path planning in dynamic flows: Theory and schemes

We obtain, solve, and verify fundamental differential equations for energy–time path planning in dynamic flows. The equations govern the energy–time reachable sets, optimal paths, and optimal controls for autonomous vehicles navigating to any destination in known dynamic environments, minimizing both energy usage and travel time. Based on Hamilton–Jacobi theory for reachability and the level set method, the resulting methodology computes the Pareto optimal solutions to the multi-objective path planning problem, numerically solving the exact equations governing the evolution of reachability fronts and optimal paths in the augmented energy and physical-space domain. Our approach is applicable to path planning in various dynamic flow environments and energy types. We first validate the methodology through a benchmark case of crossing a steady jet for which we compare our results to semi-analytical optimal energy–time solutions. We then consider unsteady flow environments and solve for energy–time optimal missions in a quasi-geostrophic double-gyre flow field. Results show that our theory and schemes can provide all the energy–time optimal solutions and that these solutions can be strongly influenced by unsteady flow conditions.

Time-optimal obstacle avoidance of autonomous ship based on nonlinear model predictive control

Autonomous shipping has been identified as the way forward in the maritime transport. However, the time-optimal path planning, anti-disturbance trajectory tracking and obstacle avoidance are still ongoing challenging problems, which have not been properly addressed for autonomous ship. To fill the knowledge gap, we propose a novel nonlinear model predictive control (MPC), which integrates the time-optimal path planning, anti-disturbance tracking and obstacle avoidance. The proposed controller is designed as a 2-level hierarchical controller. In the high level, a planned path considering time minimum and obstacle avoidance is generated by nonlinear MPC in spatial formulation. A spatial reformulation is adopted to express manoeuvring time as a function mathematically. In the spatial coordinate, the manoeuvring time is minimised by MPC to generate a reference path for tracking. In the low level, vessel tracks the time-optimal planning trajectory by nonlinear MPC with extended Kalman filter in temporal formulation. The deviation of the tracking path in longitudinal direction is 15% ship length and the deviation in width direction is 1% under disturbances. The obstacle avoidance is implemented by using the proposed control method, and the tolerance of obstacle avoidance is 2L (ship length) to meet the safety requirement.

Optimal Path Planning of Autonomous Marine Vehicles in Stochastic Dynamic Ocean Flows Using a GPU-Accelerated Algorithm

Autonomous marine vehicles play an essential role in many ocean science and engineering applications. Planning time and energy optimal paths for these vehicles to navigate in stochastic dynamic ocean environments are essential to reduce operational costs. In some missions, they must also harvest solar, wind or wave energy (modeled as a stochastic scalar field) and move in optimal paths that minimize net energy consumption. Markov decision processes (MDPs) provide a natural framework for sequential decision making for robotic agents in such environments. However, building a realistic model and solving the modeled MDP becomes computationally expensive in large-scale real-time applications, warranting the need of parallel algorithms and efficient implementation. In this article, we introduce an efficient end-to-end graphical processing unit (GPU)-accelerated algorithm that 1) builds the MDP model (computing transition probabilities and expected one-step rewards) and 2) solves the MDP to compute an optimal policy. We develop methodical and algorithmic solutions to overcome the limited global memory of GPUs by 1) using a dynamic reduced-order representation of the ocean flows; 2) leveraging the sparse nature of the state transition probability matrix; 3) introducing a neighboring subgrid concept; and 4) proving that it is sufficient to use only the stochastic scalar field’s mean to compute the expected one-step rewards for missions involving energy harvesting from the environment, thereby saving memory and reducing the computational effort. We demonstrate the algorithm on a simulated stochastic dynamic environment and highlight that it builds the MDP model and computes the optimal policy 600–1000 times faster than conventional CPU implementations, making it suitable for real-time use.

A Study on the Fuzzy Rule Hunting Strategy of Multiple Unmanned Vehicles Based on the Improved Artificial Potential Field Method

A multiple unmanned vehicle hunting strategy based on the improved artificial potential field method is designed to solve the multiple unmanned vehicle cooperative hunting problem in complex environments. First, the decision layer is designed based on the fuzzy inference system to achieve the stage division and selection of the corresponding strategies. Subsequently, the improved repulsive field function for variable safety distance is designed according to the relative velocity and angle between the unmanned vehicle and the obstacle, to achieve efficient obstacle avoidance of the unmanned vehicle during hunting. Then, a hunting strategy comprising the consistent state, obstacle avoidance, and control costs is designed to obtain the time-optimal path of the unmanned vehicle and achieve fast target hunting by minimizing this cost function. Finally, the effectiveness of the designed hunting strategy is verified via simulations and experiments and then compared with other hunting strategies. Accordingly, it is inferred that the hunting strategy designed in this study improves the efficiency of hunting by 12.97%, in addition to ensuring the obstacle avoidance effect.