Minimum-time Trajectory Research Articles

Smooth Trajectory Generation for Predefined Path With Pseudo Spectral Method

In view of smooth trajectory generation for a 3-axis machine tool, many methods have been presented. Among them, the optimal control based method is increasingly concerned, because it is considered to be able to make full use of kinematic abilities of machine tools. Under the unified framework of optimal control, the feedrate can be adjusted flexibly by adding or removing axial constraints and tangential constraints. But the problem of smooth trajectory generation (PSTG) based on optimal control for a machine tool is not easily to be solved. In this article, to efficiently solve the PSTG, it is divided into two sub-problems: the problem of minimum time trajectory planning (PMTTP) and the pseudo problem of smooth trajectory generation (PPSTG). Since both sub-problems are convex, the existence of unique solutions can be guaranteed. Then, the PMTTP and the PPSTG are transformed into nonlinear programming (NLP) problems with radau-pseudo-spectral (RPM) method successively. Due to convexity, the two NLP problems can be efficiently solved with mature optimization methods. In addition, the RPM method allows two sub-problems to have different Legendre-Gauss-Radau (LGR) points, thereby further saving computational costs. Finally, three different predefined paths are employed to test the proposed method, and simulation results show the effectiveness of proposed method.

Nonprehensile Manipulation: a Trajectory-Planning Perspective

This article discusses nonprehensile manipulation of an asymmetric object using a robotic manipulator from a motion planning point of view. Four different aspects of the problem will be analyzed:object stability, motion planning, manipulator control, and experimental validation. Specifically, via an analysis of marginal stability of an object resting on a moving tray, the work establishes the critical accelerations of the manipulator's end-effector, below which the object's stability is guaranteed. These critical accelerations guide the design of the end-effector's motion for successful nonprehensile manipulation of the object. In particular, we propose two methods to formulate polynomial asymmetric s-curve trajectories such that the end-effector completes its motion in minimum time. In one method, the trajectory is divided into segments whose time intervals are then computed via a recursive algorithm. In the other method, we formulate an optimization problem and design the minimum-time trajectory by balancing the tradeoff between the travel time and actuator effort. A series of experiments with a robotic arm is designed to validate and compare these motion planning methods in the context of nonprehensile manipulation. In addition, the experimental results demonstrate the advantages of the asymmetric s-curve motion profiles over the traditional symmetric s-curves.

Real-Time Optimal Control for Spacecraft Orbit Transfer via Multiscale Deep Neural Networks

This study is motivated by the requirement of on-board trajectory optimization with guaranteed convergence and real-time performance for optimal spacecraft orbit transfers. To this end, a real-time optimal control approach is proposed using deep learning technologies to obtain minimum-time trajectories of solar sail spacecraft for orbit transfer missions. First, the minimum-time two-dimensional orbit transfer problem is solved by an indirect method, and the costate normalization technique is introduced to increase the probability of finding the optimal solutions. Second, by making novel use of deep learning technologies, three deep neural networks are designed and trained offline by the obtained optimal solutions to generate the guidance commands in real time during flight. Consequently, the long-standing difficulty of on-board trajectory generation is resolved. Then, an interactive network training strategy is presented to increase the success rate of finding optima by supplying good initial guesses for the indirect method. Moreover, a multiscale network cooperation strategy is designed to deal with the recognition deficiency of deep neural networks (DNNs) with small input values, which helps achieve highly precise control of terminal orbit insertion. Numerical simulations are given to substantiate the efficiency of these techniques, and illustrate the effectiveness and robustness of the proposed DNN-based trajectory control for future on-board applications.

Minimizing the Degradation of Triple Junction Solar Array of a Spacecraft during Geostationary Orbit Insertion

A method of reducing the radiation power degradation of spacecraft solar array during combined geostationary orbit insertion using a booster and an electric propulsion system is considered in this paper. The essence of the method is to optimize the shape of the transfer trajectory and the perigee argument of the intermediate orbit. The maximum principle is applied to the problem of optimizing the SA electrical power at the end of 15 year spacecraft’s operational lifetime (EOL). For this, the equation for the current SA power and the constraint on this power at the EOL is added to the equations of spacecraft motion. The closed-form solution to the adjoint equation to SA power is obtained. Calculation of SA radiation degradation was carried out using the models of charged particle fluxes of Earth’s radiation belts, AE8 MAX and AP8 MAX. To increase the EOL SA power by 0.16–0.66% of the SA power at the beginning of the transfer depending on the parameters of intermediate orbits. The additional characteristic velocity increased relative to the minimum time trajectories by 13–1087 m/s depending on the parameters of intermediate orbits.

Smooth minimum time trajectory planning with minimal feed fluctuation

Smooth and effective feedrate profile is very important for high-speed and high-precision machining. In this paper, a two-stage feedrate scheduling scheme (TFSS) is proposed to efficiently generate the smooth feedrate profile. The minimum time trajectory planning (MTTP) is firstly employed to generate the initial feedrate profile considering constraints of both the kinematics and the chord error. To improve the computational efficiency, the MTTP is transformed into a problem of linear programming by applying direct transcription method. Then, all the sharp corners along the initial feedrate profile are identified, especially those that are interacted by each other. The jerk-limited method of eliminating interaction (JMEI) at sharp corners is applied to smooth the final feedrate profile. Finally, the improved feed correction polynomials are applied to reduce the fluctuations owing to non-arc-length parameterization. To illustrate the validity and rationality of the proposed scheme, two curves were employed to verify the proposed method. The simulation and experimental results demonstrate the efficiency of the proposed scheme for the nonuniform rational basis spline (NURBS) tool paths.

Toward Time-Optimal Trajectory Planning for Autonomous Ship Maneuvering in Close-Range Encounters

Ship intelligence has been a hot topic in recent years. How to achieve autonomous maneuvers in a complex marine environment in a safe, efficient, and low-cost manner is a fundamental task that ocean engineers face. This paper presents a two-stage trajectory planning scheme to address the minimum-time maneuvering problem in close-range encounters. The scheme is robust and versatile, as it can deal with the complex spatial variability, such as sea current, state constraints, marine traffic, and physical constraints, of close-range maneuvering. In the first stage, a directed graph with variable length is generated according to the sea current distribution. A wavefront search is applied on the graph to explore the reachability, the cost of state constraints, and the risk of collision. After a discrete solution has been found, the second stage involves searching for a smooth solution. A Bezier curve based parameter optimization approach is proposed to get rid of limited moving directions in the directed graph and explore around the discrete path. The result will be a near-optimal, smooth path. The proposed scheme has been tested to solve the Zermelo's ship steering problem and several other close-range maneuvering problems. The results demonstrate that the scheme is efficient in generating smoothed minimum-time trajectories for surface vessels when maneuvering in close-range encounters.

Global Trajectory Planning Based on DWMR Dynamics in Circular C-space

This paper proposes an efficient near time-optimal trajectory planning algorithm for differential wheeled mobile robots(DWMRs) in a circular C-space under constraints on robot’s kinematics, dynamics, and motor’s voltage. This problem is considered to be complex if there exist lots of obstacles to pass and robot’s dynamics including motor’s dynamics should be considered. A* algorithm is incorporated into the modified TOTP (Time Optimal Trajectory Planning) algorithm [1] to find near minimum time trajectory without any homotopy class by defining a node as an obstacle with direction information. In addition, TOTP algorithm is improved so that the proposed algorithm can be applied to the environment without any assumption about the distance between obstacles. Simulations show that the proposed algorithm is applied in an environment where multiple obstacles exist. An experiment shows that the generated trajectory by the proposed algorithm is well followed by the robot.

Trajectory Optimization of a Low-Thrust Geostationary Orbit Insertion for Total Ionizing Dose Decrease

The paper considers a method of the dose exposure decrease from the charged particles of Earth’s radiation belts (ERBs) affecting a reusable orbital transfer vehicle, which is launched from a low circular orbit into geostationary orbit using a nuclear electric propulsion system. The main idea of the method consists in numerical continuation of a solution of the minimum time problem with respect to an ionizing radiation dose accumulated at the end of a transfer. To do this, equations of motion of the orbital transfer vehicle are supplemented by an additional equation for the radiation dose, and the boundary condition for the dose at the right end is introduced. In calculating the dose, the AE8/AP8 MIN, AE8/AP8 MAX, and AE9/AP9 models of fluxes of charged particles of ERBs were used. By changing the insertion trajectory shape, it became possible to lower the radiation dose by 25–38% relative to the minimum time trajectory. At the same time, the transfer time increased no more than by 7% of the minimum time of launching into geostationary orbit, and the characteristic velocity expenses increased by 320–560 m/s.

Assessment of the Effect of Energy Consumption on Trajectory Improvement for a Car-like Robot

SummaryReducing the energy consumed by a car-like mobile robot makes it possible to move at a lower cost, yet it takes more working time. This paper proposes an optimization algorithm for trajectories with optimal times and analyzes the consequences of restricting the energy consumed on the trajectory obtained for a car-like robot. When modeling the dynamic behavior of the vehicle, it is necessary to consider its inertial parameters, the behavior of the motor, and the basic properties of the tire in its interaction with the ground. To obtain collision-free, minimum-time trajectories quadratic sequential optimization techniques are used, where the objective function is the time taken by the robot to move between two given configurations. This is subject to constraints relating to the vehicle and tires as well as the energy consumed, which is the basis for this paper. We work with a real random distribution of consumed energy values following a normal Gaussian distribution in order to analyze its influence on the trajectories obtained by the vehicle. The energy consumed, the time taken, the maximum velocity reached, and the distance traveled are analyzed in order to characterize the properties of the trajectories obtained. The proposed algorithm has been applied to 101 examples, showing that the computational times needed to obtain the solutions are always lower than those required to realize the trajectories. The results obtained allow us to reach conclusions about the energy efficiency of the trajectories.

Minimum-time path planning for robot manipulators using path parameter optimization with external force and frictions

This paper presents a new minimum-time trajectory planning method which consists of a desired path in the Cartesian space to a manipulator under external forces subject to the input voltage of the actuators. Firstly, the path is parametrized with an unknown parameter called a path parameter. This parameter is considered a function of time and an unknown parameter vector for optimization. Secondly, the optimization problem is converted into a regular parameter optimization problem, subject to the equations of motion and limitations in angular velocity, angular acceleration, angular jerk, input torques of actuators’, input voltage and final time, respectively. In the presented algorithm, the final time of the task is divided into known partitions, and the final time is an additional unknown variable in the optimization problem. The algorithm attempts to minimize the final time by optimizing the path parameter, thus it is parametrized as a polynomial of time with some unknown parameters. The algorithm can have a smooth input voltage in an allowable range; then all motion parameters and the jerk will remain smooth. Finally, the simulation study shows that the presented approach is efficient in the trajectory planning for a manipulator that wants to follow a Cartesian path. In simulations, the constraints are respected, and all motion variables and path parameters remain smooth.

Minimum-Time Trajectory Generation for Quadrotors in Constrained Environments

In this paper, we present a novel strategy to compute minimum-time trajectories for quadrotors in constrained environments. In particular, we consider the motion in a given flying region with obstacles and take into account the physical limitations of the vehicle. Instead of approaching the optimization problem in its standard time-parameterized formulation, the proposed strategy is based on an appealing reformulation. Transverse coordinates, expressing the distance from a frame path, are used to parameterize the vehicle position and a spatial parameter is used as independent variable. This reformulation allows us to: 1) obtain a fixed horizon problem and 2) easily formulate (fairly complex) position constraints. The effectiveness of the proposed strategy is proven by numerical computations on two different illustrative scenarios. Moreover, the optimal trajectory generated in the second scenario is experimentally executed with a real nanoquadrotor in order to show its feasibility.

CubeSat constellation management using Ionic Liquid Electrospray Propulsion

The Space Propulsion Laboratory (SPL) of the Massachusetts Institute of Technology (MIT) is developing the Ion Electrospray Propulsion System (iEPS), designed to address a current need in CubeSat technology: miniaturized electric thrusters. These could be used for different applications, ranging from attitude control to interplanetary flights. In this work, performed together with the Space Systems Laboratory of the Costa Rica Institute of Technology (SETEC Lab), we explore a case study in which the iEPS is used for constellation management in Low Earth Orbit (LEO) when integrated in a 3U CubeSat. We analyze how a 180° separation in the Right Ascension of the Ascending Node (RAAN) between two CubeSats (SatA and SatB) starting in the same orbit can be achieved by modifying one of the spacecraft's orbital altitude, resulting in a difference in their rate of nodal precession (defined as the drift rate) due to the J2 effect, and therefore a difference in their relative RAAN. The method consists of SatB increasing its semi-major axis, drifting in a higher orbit with a lower drift rate, and returning to the original semi-major axis once the desired difference in RAAN in achieved relative to the other spacecraft. SatA will stay in its original orbit, using its thruster to compensate for orbital energy loss due to atmospheric drag, therefore demonstrating another application of iEPS for constellation management. Three different simulations were studied, defined as the minimum time trajectory, minimum propellant trajectory and a hybrid trajectory, consisting of reaching a higher altitude orbit, but actively changing the RAAN using the propulsion system instead of drifting. It was observed that the difference in this orbital element could be achieved using 85 g of propellant in as little as 164 days for the minimum time trajectory. The same difference could also be achieved using only 44 g of propellant in 245 days for the minimum propellant trajectory. Furthermore, the results of the hybrid trajectory showed that the goal could be achieved in 161 days, but using 158 g of propellant mass, demonstrating the benefit of using a drift orbit. The results proved the feasibility of implementing iEPS for constellation management using 3U CubeSats in LEO.

The Effect of Gravitational Field on Brachistochrone Problem

The equation of the minimum time trajectory (the brachistochrone) does not contain the gravity acceleration explicitly. The proof of the gravitational field effect on the trajectory is revealed by solving functional for the uniform gravitational field and nonuniform gravitational field for the central gravity in this article. The Findings revealed that the effect of constant gravity acceleration is inversely proportional on the arc length of the cycloid, except at g = 0 m/s2, which means that the trajectory could not be formed without gravity acceleration at a location where a particle are not affected by the gravitational field, whereas in nonuniform gravitational field, the particle’s trajectory is not a cycloid and lies in two quadrant. The curve in first quadrant is a mirror image of the curve in fourth quadrant and vice versa. The difference trajectory between uniform and nonuniform gravitational cases is the proof of the existence of the gravitational field effect.

Minimum-Time Trajectories for Steered Agent With Constraints on Speed, Lateral Acceleration, and Turning Rate

We present time-optimal trajectories for a steered agent with constraints on speed, lateral acceleration, and turning rate for the problem of reaching a point on the plane in minimum time with free terminal heading angle. Both open-loop and state-feedback forms of optimal controls are derived through application of Pontryagin's minimum principle. We apply our results for the single agent to solve a multi-agent coverage problem in which each agent has constraints on speed, lateral acceleration, and turning rate.

Dynamics analysis and time-optimal motion planning for unmanned quadrotor transportation systems

Aerial transportation is enjoying more and more popularity among disaster-oriented tasks, such as fire fighting, delivery of relief supplies, life-saving aid, and so on, which are infeasible for ground robots especially in complex terrains. However, the contradictory between transportation efficiency and payload swing suppression is an annoying problem, which badly limits the application of aerial transportation. In this paper, a time-optimal motion planning (TOMP) scheme is proposed with effective payload antiswing performance. To the best of our knowledge, it is the first minimum-time trajectory planning method designed for unmanned quadrotor transportation systems. Compared with existing methods, the proposed approach presents significant superiority in the sense that both nonlinear dynamics of the system and various constraints are taken into full consideration simultaneously. Specifically, the nonlinear system model is established using Lagrangian mechanics, based on which the augmented system is transformed into a nonlinear affine system regarding the acceleration/jerk as the control input without the need of the linearized system model. After discretization and approximation process, the time-optimal motion planning problem is converted into a standard nonlinear programming problem, wherein various practical constraints are considered, including the bounds of the payload swing, the quadrotor velocity, acceleration, and even jerk. Finally, the nonlinear programming problem is solved by the sequential quadratic programming (SQP) method. Experimental results are exhibited to illustrate the effectiveness and feasibility of the proposed approach.

Time-optimal trajectory planning for hyper-redundant manipulators in 3D workspaces

Abstract A novel approach is proposed for time-optimal trajectory planning of a hyper-redundant manipulator which is requested to move from an initial configuration to a final configuration in 3D workspaces. The 3D workspace is cluttered with static objects which have known geometry and position. The proposed approach generates a trajectory for the manipulator's end-effector considering simultaneously the kinematical constraints of the manipulator (specifically velocity and acceleration) and the presence of obstacles. The method solves an optimization problem to find the minimum time trajectory to perform the requested tasks. The optimization problem is solved by using a Genetic Algorithm with multiple population. The efficiency of the developed method is investigated and discussed through characteristic simulated experiments concerning a variety of operating environments.

Relaxing Nonholonomic Constraints to Eliminate Chattering From Time-Optimal Control Solutions

This letter presents a new approach to tackle the problem of infinite chattering , observed at bang-singular junctions of the minimum-time trajectories of car-like robots with bounded angular acceleration control. To eliminate chattering from time-optimal control solutions, we propose to relax the ideal nonholonomic constraint of no-skidding. A proof is presented to demonstrate that this relaxation eliminates chattering from the bang-singular junctions. Finally, a solution to the time-optimal synthesis under the condition of small skid angles is provided.

On the Minimum-Time Control Problem for Differential Drive Robots with Bearing Constraints

This paper presents a study of analysis of minimum-time trajectories for a differential drive robot equipped with a fixed and limited field-of-view camera, which must keep a given landmark in view during maneuvers. Previous works have considered the same physical problem and provided a complete analysis/synthesis for the problem of determining the shortest paths. The main difference in the two cost functions (length vs. time) lays on the rotation on the spot. Indeed, this maneuver has zero cost in terms of length and hence leads to a 2D shortest path synthesis. On the other hand, in case of minimum time, the synthesis depends also on the orientations of the vehicle. In other words, the not zero cost of the rotation on the spot maneuvers leads to a 3D minimum-time synthesis. Moreover, the shortest paths have been obtained by exploiting the geometric properties of the extremal arcs, i.e., straight lines, rotations on the spot, logarithmic spirals and involute of circles. Conversely, in terms of time, even if the extremal arcs of the minimum-time control problem are exactly the same, the geometric properties of these arcs change, leading to a completely different analysis and characterization of optimal paths. In this paper, after proving the existence of optimal trajectories and showing the extremal arcs of the problem at hand, we provide the control laws that steer the vehicle along these arcs and the time-cost along each of them. Moreover, this being a crucial step toward numerical implementation, optimal trajectories are proved to be characterized by a finite number of switching points between different extremal arcs, i.e., the concatenations of extremal arcs with infinitely many junction times are shown to violate the optimality conditions.

Trajectory optimization for a ramjet-powered vehicle in ascent phase via the Gauss pseudospectral method

To determine the optimal trajectory for the ascent phase of a general supersonic vehicle propelled by a hydrocarbon-fueled ramjet, a strategy is developed based on the coupling characteristics of a flight-propulsion system. With two different cost functions and a series of necessary constraints, minimum-fuel and minimum-time trajectory problems are formulated. Both problems are transformed into typical continuous Bolza problems, which can be solved via the Gauss pseudospectral method. The results show that along the two optimal trajectories, the vehicle accelerates horizontally at the onset of the ascent process until the dynamic pressure can provide sufficient lift. Efficiency analysis is conducted to reveal the coordination between engine performance and atmosphere characteristics. It is found that the take-off weight influences the optimized results. The difference between the two optimal trajectories narrows as the take-off weight increases.

A star-photon sailcraft mission in the Alpha Centauri system

The recent findings of exoplanets have renewed interest in exploration missions of extra-solar star systems. In this context the binary star system of Alpha Centauri A/B plays a scientifically important role. The current paper aims at investigating the capture and the transfer trajectories from A to B performed by a stellar-photon sailcraft, which gives the best performance in long duration missions. The mission has been divided into four phases and minimum transfer time trajectories have been obtained by using a direct optimisation approach. Finally, a preliminary analysis in terms of Zero Velocity Curves has been carried out.