Cost Sensitive Trajectory Generation for Time Constrained Aircraft Route Planning

This paper presents a data-driven approach towards time-optimal trajectory generation for Unmanned Aerial Vehicles (UAV's) using a machine-learned trajectory generation mechanism for point-to-point time-optimal trajectories on-the-fly. To train this machine-learned black box trajectory generator off-line, a model-based optimization problem is first constructed for point-to-point time-optimal trajectory generation, with physical constraints on inputs, states, and rates. The formulated optimization problem is then solved off-line for a range of initial and terminal flight states to generate point-to-point data-sets that consist of the optimal state and input trajectories. This information is compressed by parameterizing the input and state trajectories using a set of basis functions. This data is then used to train the neural network-based trajectory planner. The output of the neural network is the basis function coefficient sets for the state and input trajectories (and the total flight time) which can then be used to reconstruct the flight trajectory. Once the neural network is trained, the data-driven on-board trajectory generator is ready to be deployed on the UAV for on-board planning. This approach is demonstrated for two scenarios: (1) the input to the neural network being the initial and terminal flight states and (2) the input to the neural network being initial and terminal flight states as well as physical constraints. To validate the performance of the machine-learned black-box trajectory generator, the root mean squared error between the neural network generated trajectories and the trajectories obtained from solving the optimization problem directly is statistically evaluated. These trajectories are also tested for violation of path constraints (which are not included explicitly in the training or input to the black box planner) by evaluating the mean constraint violation for each path-constrained variable.

The analysis of an aircraft terrain-following (TF) problem and a numerical method for the optimal trajectory generation are the subjects of this paper. The TF problem is formulated as an optimal control problem that combines short flight time and path-following objectives. The aircraft dynamics are represented by point-mass equations of motion in a vertical plane. The problem is first studied analytically to reveal the bang-bang nature of the optimal thrust control in most cases. Then, an inverse dynamics approach is employed to solve the problem numerically. Numerical solutions are provided to support the analysis and demonstrate the features of the problem.

An effective approach for non-singular trajectory generation of a 5-DOF hybrid machining robot

<div>In this dissertation, methods for real-time trajectory generation and autonomous obstacle avoidance for fixed-wing and quad-rotor unmanned aerial vehicles (UAV) are studied. A key challenge for such trajectory generation is the high computation time required to plan a new path to safely maneuver around obstacles instantaneously. Therefore, methods for rapid generation of obstacle avoidance trajectory are explored. The high computation time is a result of the computationally intensive algorithms used to generate trajectories for real-time object avoidance. Recent studies have shown that custom solvers have been developed that are able to solve the problem with a lower computation time however these designs are limited to small sized problems or are proprietary. Additionally, for a swarm problem, which is an area of high interest, as the number of agents increases the problem size increases and in turn creates further computational challenges. A solution to these challenges will allow for UAVs to be used in autonomous missions robust to environmental uncertainties.</div><div><br></div><div>In this study, a trajectory generation problem posed as an optimal control problem is solved using a sequential convex programming approach; a nonlinear programming algorithm, for which custom solver is used. First, a method for feasible trajectory generation for fast-paced obstacle-rich environments is presented for the case of fixed-wing UAVs. Next, a problem of trajectory generation for fixed-wing and quad-rotor UAVs is defined such that starting from an initial state a UAV moves forward along the direction of flight while avoiding obstacles and remaining close to a reference path. The problem is solved within the framework of finite-horizon model predictive control. Finally, the problem of trajectory generation is extended to a swarm of quad-rotors where each UAV in a swarm has a reference path to fly along. Utilizing a centralized approach, a swarm scenario with moving targets is studied in two different cases in an attempt to lower the solution time; the first, solve the entire swarm problem at once, and the second, solve iteratively for a UAV in the swarm while considering trajectories of other UAVs as fixed.</div><div><br></div><div>Results show that a feasible trajectory for a fixed-wing UAV can be obtained within tens of milliseconds. Moreover, the obtained feasible trajectories can be used as initial guesses to the optimal solvers to speed up the solution of optimal trajectories. The methods explored demonstrated the ability for rapid feasible trajectory generation allowing for safe obstacle avoidance, which may be used in the case an optimal trajectory solution is not available. A comparative study between a dynamic and a kinematic model shows that the dynamic model provides better trajectories including aggressive trajectories around obstacles compared to the kinematic counterpart for fixed-wing UAVs, despite having approximately the same computational demands. Whereas, for the case of quad-rotor UAVs, the kinematic model takes almost half the solution time than with a reduced dynamic model, despite having approximately the similar range of values for the cost function. When extended to a swarm, solving the problem for each UAV is four to seven times computationally cheaper than solving the swarm as a whole. With the improved computation time for trajectory generation for a swarm of quad-rotors using centralized approach, the problem is now reasonably scalable, which opens up the possibility to increase the number of agents in a swarm using high-end computing machines for real-time applications. Overall, a custom solver jointly with a sequential convex programming approach solves an optimization problem in a low computation time.</div>

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.

The problem of finding an optimal aircraft trajectory in close to real time is considered. Using multiple models for different parts of the trajectory, where each model is tailored for the particular conditions, an efficient method has been developed which solves the trajectory optimization problem in a few seconds making interactive use feasible. The different models are used in order to accurately model flight at different conditions such as constant Mach number, constant calibrated airspeed, or constant altitude, using a minimum number of state variables and state equations for each condition. The modeling used and the numerical implementation is described in some detail. A user interface for computing optimal aircraft trajectories using a standard web-browser over the Internet has also been developed. To further illustrate the interactiveness of the developed computational procedures, a mobile Internet implementation has been developed where a cellular phone supporting the Wireless Application Protocol can be used to compute an optimal aircraft trajectory. Copyright

This paper presents a new approach of planar trajectory generation for automated in-hand dexterous manipulation of miniaturized objects. The proposed method aims at improving the efficiency of the previous method [17] to be able to perform real-time in-hand manipulation trajectories generation. The main idea behind this new method is the representation of the configuration space as a set of stable rotations instead of stable grasps as it is usually done. The consequence of this representation is a more compact space that encapsulates more information. The developed algorithm that uses this approach is able to generate optimal trajectories to manipulate complex objects in less than 0.1 s, which represents a reduction in the processing time between 102 and 105 compared to the previous method.

A quantitative comparison of different approaches to control the trajectory of an autonomous vehicle can be achieved by measuring the overall performance of the whole system in closed loop. This work compares two trajectory generation and control approaches for autonomous driving in simulation. One approach is based on a low frequency check of geometric safety distances and a high frequency check of control deviations. The other approach minimizes the probability of colliding with obstacles by explicitly considering actuator uncertainties and solving a chance constraint optimization problem. The comparison is based on systematically applying actuator inaccuracies and searching for event combinations leading to collisions. We demonstrate that the control approach incorporating uncertainties performs better in one class of critical scenarios although its error model is different to the applied inaccuracies.

Any obstacle-free path planning algorithm, in general, gives a sequence of waypoints that connect start and goal positions by a sequence of straight lines, which does not ensure the smoothness and the dynamic feasibility to maneuver the MAV. Kinodynamic-based motion planning is one of the ways to impose dynamic feasibility in planning. However, kinodynamic motion planning is not an optimal solution due to high computational demands for real-time applications. Thus, we explore path planning followed by kinodynamic smoothing while ensuring the dynamic feasibility of MAV. The main difference in the proposed technique is not to use kinodynamic planning when finding a feasible path, but rather to apply kinodynamic smoothing along the obtained feasible path. We have chosen a geometric-based path planning algorithm “RRT*” as the path finding algorithm. In the proposed technique, we modified the original RRT* introducing an adaptive search space and a steering function that helps to increase the consistency of the planner. Moreover, we propose a multiple RRT* that generates a set of desired paths. The optimal path from the generated paths is selected based on a cost function. Afterwards, we apply kinodynamic smoothing that will result in a dynamically feasible as well as obstacle-free path. Thereafter, a b-spline-based trajectory is generated to maneuver the vehicle autonomously in unknown environments. Finally, we have tested the proposed technique in various simulated environments. According to the experiment results, we were able to speed up the path planning task by 1.3 times when using the proposed multiple RRT* over the original RRT*.

Online gait generator for lower limb exoskeleton robots: Suitable for level ground, slopes, stairs, and obstacle avoidance

Minimal energy control using artificial intelligence techniques is developed in this paper. A traditional feedforward neural network is used as the controller. Through learning, the controller can generate trajectory along a pre-defined path. The learning strategy is called recurrent averaging learning. It takes the average of initial states and final states after a cycle of training and sets this value as the new initial and final states for next training cycle. By including the energy criterion in the cost function, this technique can generate a minimal-energy walking gait and still follow the reference trajectory.

In this paper, the infinite horizon optimal robust guaranteed cost control of continuous-time uncertain nonlinear systems is investigated using neural-network-based online solution of Hamilton-Jacobi-Bellman (HJB) equation. By establishing an appropriate bounded function and defining a modified cost function, the optimal robust guaranteed cost control problem is transformed into an optimal control problem. It can be observed that the optimal cost function of the nominal system is nothing but the optimal guaranteed cost of the original uncertain system. A critic neural network is constructed to facilitate the solution of the modified HJB equation corresponding to the nominal system. More importantly, an additional stabilizing term is introduced for helping to verify the stability, which reinforces the updating process of the weight vector and reduces the requirement of an initial stabilizing control. The uniform ultimate boundedness of the closed-loop system is analyzed by using the Lyapunov approach as well. Two simulation examples are provided to verify the effectiveness of the present control approach.

This paper presents a trajectory generation approach for a 7-DOF biped robot on level ground. Simultaneously rotation of feet in double support phase is considered which leads to high-speed and more similar to human being walking. The zero moment point (ZMP) stability criterion is used to ensure the stability of the bipedal walking robot. Since ZMP trajectory in human walking does not stay fixed, it needs to be a straight line shaped forward ZMP trajectory to have a natural walk. A genetic algorithm based method is proposed to obtain key parameters in trajectory generation such that the ZMP follows a predefined trajectory while minimizing power consumption. Simulation results demonstrate the effectiveness of the proposed method.

A major objective of next generation reusable launch vehicle (RLV) programs includes significant improvements in vehicle safety, reliability, and operational costs. In this paper, trajectory generation on approach/ landing (A&L) for RLVs using motion primitives (MPs) and neighboring optimal control (NOC) is discussed. The proposed trajectory generation approach at A&L phase is based on an MP scheme which consists of trims and maneuvers. From an initial point to a given touchdown point, all feasible trajectories that satisfy certain constraints are generated and saved into a trajectory database. An optimal trajectory can then be found off-line by using Dijkstra's algorithm. After perturbations are imposed on the initial states of the off-line optimal trajectory, it is reshaped into a neighboring feasible trajectory on-line by using NOC approach. At this point, a neighboring feasible trajectory existence theorem (NFTET) is investigated and its proof is provided as well. The results show that the vehicle with stuck effectors can be recovered from failures in real time. Finally, robustness issues on NOC approach are briefly discussed.

The present paper presents a trajectory generation approach for a nonlinear dynamic inversion controller using second order error dynamics. The trajectory generation module provides an approach for 3D-trajectories with speed contraints including the determination of the trajectory reference point, the relative kinematics and the determination of the first derivatives of the deviation between unmanned aerial system (UAS) and trajectory. The system is presented as part of an integrated modular mission management system, which provides the whole range from basic autopilot commands up to complex flight plan patterns.