Time-optimal Trajectory Research Articles

Goal state driven trajectory optimization

Many applications demand a dynamical system to reach a goal state under kinematic and dynamic (i.e., kinodynamic) constraints. Moreover, industrial robots perform such motions over and over again and therefore demand efficiency, i.e., optimal motion. In many applications, the initial state may not be constrained and can be taken as an additional variable for optimization. The semi-stochastic kinodynamic planning (SKIP) algorithm presented in this paper is a novel method for trajectory optimization of a fully actuated dynamic system to reach a goal state under kinodynamic constraints. The basic principle of the algorithm is the parameterization of the motion trajectory to a vector in a high-dimensional space. The kinematic and dynamic constraints are formulated in terms of time and the trajectory parameters vector. That is, the constraints define a time-varying domain in the high dimensional parameters space. We propose a semi stochastic technique that finds a feasible set of parameters satisfying the constraints within the time interval dedicated to task completion. The algorithm chooses the optimal solution based on a given cost function. Statistical analysis shows the probability to find a solution if one exists. For simulations, we found a time-optimal trajectory for a 6R manipulator to hit a disk in a desired state.

On Higher Order Inverse Kinematics Methods in Time-Optimal Trajectory Planning for Kinematically Redundant Manipulators

Time-optimal motion control will only find industrial applications if the optimal motions can actually be performed by standard industrial robots. This is not ensured by any optimal motion planning scheme proposed up to now. The limiting aspect rendering all these schemes impractical is the insufficient continuity of the motion trajectories. In this paper, a time-optimal path following along a predefined end-effector path is addressed for kinematically redundant robots, where nonredundant robots are included as special cases. As prerequisite explicit expressions for the higher order inverse kinematics are presented. Kinematic redundancy is resolved and exploited within the trajectory planning using the joint space decomposition and a novel pseudoinverse-based solution of the higher order inverse kinematics. The approaches are demonstrated for two examples of kinematically redundant manipulators performing time-optimal motions along prescribed end-effector paths in compliance with technological constraints. The optimization results are experimentally validated.

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.

Neural Network Based Modelling of Time Optimal Interpolation of Non-linear Trajectory

Non linear curve fitting for execution of milling operation on Computer Numerical Control machining is always challenging. This paper focuses on fitting nonlinear curve and generation of time optimal trajectory to perform slot milling operation using NX CAM. Virtual simulation of trajectory is tested with various experimentation conditions and executed using Siemens 828D controller. Simulation and actual experimentation result is then compared with Multilayered Perceptron based Neural Network and optimum cutting parameters for time optimal trajectory is calculated.

Time Optimal Turbocharger Testing in Gas Stands with a Known Map

Turbocharger maps are used in design, evaluation and optimization of engine system operation to represent the turbo operation in different scenarios. To construct such a map, the turbo is tested in a gas flow test bench, called gas stand. Turbo testing is a time and resource consuming experimental process. The turbo is tested in a selected number of test points for different turbo rotational speeds, where the temperatures in the turbo have to be stationary when the measurements that constitute the map are acquired. In this paper, optimal control is used to find the most time efficient pattern of test conditions, and the optimal control strategy to traverse between them. A heat transfer model, describing the heat transfer between the compressor, bearing house, and turbine, is presented and validated against measured data. A direct collocation method is used to find time optimal control trajectories between the specified test points in the map. The method objective is to find the least time consuming control strategy which brings the turbo from one test point to the next, while ensuring thermal equilibrium at the final time. The results suggest that this method reduces turbocharger testing time with a factor higher than 60. The improvements can be further increased, with over 70 times, if a traveling salesman problem is solved to find the optimal route through the turbo map. The described method would be able to map a 43 points turbo map in 22 minutes, including a 5 minute warm-up phase.

Quadrotor Thrust Vectoring Control with Time and Jerk Optimal Trajectory Planning in Constant Wind Fields

This paper proposes a trajectory planning and control strategy to optimally visit a given set of waypoints in the presence of wind. First, aerodynamic properties of quadrotors which affect trajectory planning and tracking performance are investigated. Blade flapping, induced and parasitic drag are derived and an extended method to identify all coefficients from flight test data is developed. Then, a three-step approach is suggested to optimize the trajectory. These steps reduce the size of the optimization problem and thereby increase computational efficiency while still guaranteeing near optimal results. The trajectories are optimized for minimal aerodynamic drag and minimal jerk. The derived smooth trajectory generation is compared with traditional trajectory planning consisting of discrete point to point tracking followed by low-pass filtering. The new trajectories yield a clear reduction in maximal needed thrust and in Euler angle aggressiveness. A thrust vectoring controller is designed, which exploits the a priori trajectory information and identified aerodynamic properties. Its performance is compared to a standard PID controller and results show a reduction in tracking delay and an increase in thrust and attitude angle margins, which overall enable faster flight.

A time optimal trajectory planning algorithm for closed-loop multibody systems and its verification by experiments

A time optimal trajectory planning algorithm for closed-loop multibody systems and its verification by experiments

Interactive Robot Programing Using Mixed Reality

Abstract The number of robots in industry is increasing every year, they are used to automate various processes. It is necessary to create a system for intuitive and effective interaction between robot and human. This paper represents a system for interactive programming of industrial robots based on Mixed Reality (MR). Microsoft HoloLens glasses were used for immersion in MR. By the developed system, a user without programming skills can assign a task to the manipulator. The main system parts are intuitive geometric path planning, time-optimal trajectory planning, and simulator based on MR. Via MR users are able to interact with robots and the spatial environment through the highly intuitive interface. The system comprises two KUKA robot with different kinematic models, with and without redundancy. The works show features and advantages of the MR-based system comparing to Augmented and Virtual Reality. Experiments on three different path cases show the system performance.

Time-Optimal Trajectory Planning for Delta Robot Based on Quintic Pythagorean-Hodograph Curves

In this paper, a time-optimal trajectory planning method based on quintic Pythagorean-Hodograph (PH) curves is proposed to realize the smooth and stable high-speed operation of the Delta parallel robot. The trajectory is determined by applying the quintic PH curves to the transition segments in the pick-and-place operation trajectory and the 3-4-5 polynomial motion law to the trajectory. The quintic PH curves are optimized to reduce the cycle time of the pick-and-place operation. In addition, a comparison between different trajectory planning methods has been implemented so as to observe the performance of the obtained results. The MATLAB simulation results reveal that compared with the trajectory planning based on vertical and horizontal motion superposition, the trajectory planning based on quintic PH curves is completed with a shorter motion cycle time and more stable motion performance, with the velocities, accelerations, and jerks in joint space bounded and continuous. Experiments carried out on the prototype also confirm that the trajectory planning based on quintic PH curves has a shorter cycle time, which is of great importance to high-speed operations of Delta parallel robots.

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.

Generating Time Optimal Trajectory from Predefined 4D waypoint Networks

The main purpose of this paper is to develop a trajectory optimization method to generate optimal trajectories that minimize aircraft total trip time between the initial and final waypoint in predefined 4D waypoint networks. In this paper, the 4D waypoint networks only consist of waypoints for climb, cruise and descent approach without the take-off and landing approach phases. The time optimal trajectory is generated for three different lengths of flights (short, medium, and long-haul flight) for two different commercial aircraft and considering zero wind condition. The Results about the presented applications show that by flying a time optimal trajectory, which was found by applying a single source shortest path algorithm (Dijkstra’s algorithm), can lead to the reduction of average travel time by 2.6% with respect to the total trip time.

Time-optimal velocity tracking control for differential drive robots

Nonholonomic wheeled mobile robots are often required to implement control algorithms designed for holonomic kinematic systems. This creates a velocity tracking problem for an actual wheeled mobile robot. In this paper, we investigate the issue of tracking a desired velocity in the least amount of time, for a differential drive nonholonomic wheeled mobile robot with torque inputs. The Pontryagin Maximum Principle provides time-optimal controls that must be implemented as open-loop commands to the motors. We propose two discontinuous state-based feedback control laws, such that the associated closed-loop systems track a desired velocity in minimum time. The feedback control laws are rigorously shown to produce only time-optimal trajectories, by constructing a regular synthesis for each control law. The availability of these time-optimal feedback control laws makes re-computation of open-loop time-optimal controls (due to changes in the desired velocity or input disturbances) unnecessary.

Reactive optimal UAV motion planning in a dynamic world

Navigation in a dynamic environment requires reactive actions to avoid collision and navigate safely. This paper deals with optimal and reactive motion planning for unmanned air vehicles in a dynamic world. A virtual space representation is used to formulate and solve the problem and derive real time optimal trajectories. This formulation allows for the construction of different control subspaces from which optimal control laws for the speed, the flight path angle, and the heading angle are derived. The safety margins are translated to the control subspaces as speed and orientation margins. Closed-form reactive optimal solutions are calculated to achieve reactive motion and avoid collision in each subspace. The dynamic and kinematic constraints are taken into account when planning motion in the control subspaces. Simulation results show the effectiveness of the proposed methods.

Time-Optimal Trajectory Planning Based on Dynamics for Differential-Wheeled Mobile Robots With a Geometric Corridor

We propose an efficient time-optimal trajectory planning algorithm for an environment with multiple static circular obstacles, which is based on dynamics for differential-wheeled mobile robots (DWMRs) including actuators. This problem is known to be complex particularly if obstacles are present and if full dynamics including actuators is considered. Given a geometric corridor, the proposed technique defines a portion of trajectory between obstacles as a section which is constituted of an arc, out-curve, and in-curve, each with constant control input satisfying bang–bang principle for time-optimization. Then, it designs time-optimal trajectory composed of multiple sections using common tangent between two tangent circles, in which it handles multiple consecutive obstacles without arcs. This method helps to treat complex environment considering dynamics with DWMR's actuators. Simulation results are shown to validate the efficiency of the proposed algorithm.

Iterative Learning of Feasible Time-optimal Trajectories for Robot Manipulators

Time-optimal trajectories describe the minimum execution time motion along a given geometric path while taking system dynamics and constraints into account. By using a model of the real plant, inputs are provided that ought to yield minimal execution time and good tracking performance. In practice however, due to an imperfect model, the computed inputs might be suboptimal, result in poor tracking or even be infeasible in that they exceed given limits. This paper therefore presents a novel two-step iterative learning approach for industrial robots to find time-optimal, yet feasible trajectories and improve the tracking performance by repeatedly updating the nonlinear robot model and solving a time-optimal path tracking problem. The proposed learning algorithm is experimentally validated on a serial robotic manipulator, which shows that the developed approach results in reduced execution time and increased accuracy.

A swing constrained time-optimal trajectory planning strategy for double pendulum crane systems

In practice, overhead crane systems are widely used and the traditional control methods for a crane system usually treat it as a single pendulum system. However, when the hook mass cannot be ignored or the payload is too large, the crane system may behave more like a double pendulum system, which leads to the fact that traditional control methods are not suitable in this situation. In this paper, we focus on the control problem of a double pendulum crane system and propose a time-optimal trajectory planning method with the consideration of various constraints which can achieve the objectives of both accurate trolley positioning and double pendulum swing suppression. Specifically, the discrete system model is obtained using the discretization technique firstly. Then by deeply analyzing and considering a series of constraints, we formulate a quasiconvex optimization problem. After that, the bisection method is chosen to solve the obtained optimization problem with the corresponding time-optimal trajectory constructed conveniently. A tracking controller is also designed for the double pendulum crane system, which achieves proper trolley tracking performance. At last, both simulation and experimental results are included to illustrate the superior performance of the proposed trajectory planning method.

An Efficient Approach of Time-Optimal Trajectory Generation for the Fully Autonomous Navigation of the Quadrotor

In this work, a time-optimal trajectory generation approach is developed for the multiple way-point navigation of the quadrotor based on the nonuniform rational B-spline (NURBS) curve and linear programming. To facilitate this development, the dynamic model of the quadrotor is formulated first. Then, the geometric trajectory regarding multiple way-point navigation is constructed based on the NURBS curve. With the constructed geometric trajectory, a time-optimal interpolation problem is imposed considering the velocity, acceleration, and jerk constraints. This optimization problem is solved in two steps. In the first step, a preliminary result is obtained by solving a linear programming problem without jerk constraints. Then by introducing properly relaxed jerk constraints, a second linear programming problem is formulated based on the preliminarily obtained result, and the time-optimal problem can be fully solved in this way. Subsequently, a nonlinear trajectory tracking controller is developed to track the generated trajectory. The feasibilities of the proposed trajectory generation approach as well as the tracking controller are verified through both simulations and real-time experiments. With enhanced computational efficiency, the proposed approach can generate trajectory for an indoor environment with the smooth acceleration profile and moderate velocity V≈1 m/s in real-time, while guaranteeing velocity, acceleration, and jerk constraints: Vmax=1 m/s, Amax=2 m/s2, and Jmax=5 m/s3. In such a case, the trajectory tracking controller can closely track the reference trajectory with cross-tracking error less than 0.05 m.

Multiple-Pursuer/One-Evader Pursuit–Evasion Game in Dynamic Flowfields

In this paper, a reachability-based approach is adopted to deal with the pursuit–evasion differential game between one evader and multiple pursuers in the presence of dynamic environmental disturbances (for example, winds or sea currents). Conditions for the game to be terminated are given in terms of reachable set inclusions. Level set equations are defined and solved to generate the forward reachable sets of the pursuers and the evader. The time-optimal trajectories and the corresponding optimal strategies are subsequently retrieved from these level sets. The pursuers are divided into active pursuers, guards, and redundant pursuers according to their respective roles in the pursuit–evasion game. The proposed scheme is implemented on problems with both simple and realistic time-dependent flowfields, with and without obstacles.

An efficient stochastic approach for robust time-optimal trajectory planning of robotic manipulators under limited actuation

SUMMARYThis paper focuses on the problem of robust time-optimal trajectory planning of robotic manipulators to track a given path under a probabilistic limited actuation, that is, the probability for the actuation to be limited is no less than a given bound κ. We give a general and practical method to reduce the probabilistic constraints to a set of deterministic constraints and show that the deterministic constraints are equivalent to a set of linear constraints under certain conditions. As a result, the original problem is reduced to a linear optimal control problem which can be solved with linear programming in polynomial time. In the case of κ = 1, the original problem is proved to be equivalent to the linear optimal control problem. Overall, a very general, practical, and efficient algorithm is given to solve the above problem and numerical simulation results are used to show the effectiveness of the method.

Essential Properties of Numerical Integration for Time-Optimal Path-Constrained Trajectory Planning

This letter summarizes some known properties and also presents several new properties of the Numerical Integration (NI) method for time-optimal trajectory planning along a specified path. The contribution is that rigorous mathematical proofs of these properties are presented, most of which cannot be found in existing literatures. We first give some properties regarding switch points and accelerating/decelerating curves of the NI method. Then, for the fact that when kinematic constraints are considered, the original version of NI which only considers torque constraints may result in failure of trajectory planning, we give the concrete failure conditions with rigorous mathematical proof. Accordingly, a failure detection algorithm is given in a run-and-test manner. Some simulation results on a unicycle vehicle are provided to verify those presented properties. Note that though those known properties are not discovered first, their mathematical proofs are given first in this letter. The detailed proofs make the theory of NI more complete and help interested readers to gain a thorough understanding of the method.