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We propose an approach to enhance the overall detection performance in unmanned aerial reconnaissance with imaging sensors. This is achieved by generating near global optimal UAV trajectories utilizing previously developed sensor performance models. Here, we are taking into account UAV, sensor and environment characteristics as well as mission objectives. The approach can be briefly summarized as follows: First, we apply coverage path planning to determine the sensor footprint path within a predefined recon area. Second, the sensor performance model is utilized to calculate discretized two-dimensional perception maps for each footprint, which is a state-dependent representation of the detection performance. For each perception map, we identify suitable sensor positions of high detection performance. From here, we systematically connect these positions of consecutive perception maps to create a transition graph. With use of optimal control, we assign costs (e.g. recon duration) to these connections while taking several requirements and constraints into account. Dynamic programming is utilized to determine the optimal path in the graph with the lowest cost. Finally, we smooth the path by cubic spline motion primitives, which leads to the reference flight trajectory for the UAV. We evaluate our approach through a series of simulated vehicle detection reconnaissance tasks with a fixed wing UAV. Our approach improves the detection performance by 3% compared to a trajectory generated with a local optimal greedy optimization and the reconnaissance time is reduced by 20%. In comparison with a primitive trajectory, the improvement in detection performance is even up to 10% with an associated increase in reconnaissance time of at least 15%.

Trajectory optimization problem for hypersonic vehicles has received wide attention as its high speed and large flight range. The strong nonlinear characteristic of the ascent phase aerodynamics makes the trajectory optimization problem difficult to be solved by the optimal control theory. In this paper, an improved chicken swarm optimization (ICSO) algorithm is proposed to optimize the hypersonic vehicle ascent trajectory. To overcome the obstacle of premature convergence, three improvement strategies are put forward. To be specific, the updating laws of roosters are modified by the average position of roosters, and the difference of the optimal solution between two adjacent iterations is used to calculate the mutated particle instead of the gradient. Meanwhile, the uniform mutation operator is used to get rid of the local minimum. The convergence analysis of the proposed ICSO is provided subsequently. To handle constraints, an improved adaptive penalty method is put forward. The comparison results show that the proposed ICSO outperforms CSO and PSO on benchmark functions of CEC2014. Finally, the trajectory optimization results for a generic hypersonic vehicle, in compare with the open-source optimization software PSOPT, are put forward to demonstrate the feasibility and effectiveness of the proposed method. The results of 50 independent runs show that the improved adaptive penalty function method is effective in constraints handling.

A safe and rapid formation generation is of great importance for the cooperation performance of the multiple unmanned helicopters. The trajectory optimization and control is the key problem in the process of the formation generation. To achieve this goal, a novel safe formation generation strategy is proposed through trajectory optimization and tracking control. First, the rapidly-exploring random tree (RRT) algorithm is applied to obtain the initial guess to satisfy the requirement of rapid solution. Then, the Gauss pseudospectral method is used to transform the optimal control problem to the nonlinear programming (NLP) problem and the sequence quadratic programming (SQP) method is utilized to gain the optimal trajectory combined with the initial guess. Second, a finite-time sliding mode controller is proposed to ensure the finite time trajectory tracking in the presence of model parameter uncertainties and unknown external disturbances. Finally, the numerical simulation is provided to show the effectiveness of the proposed formation generation strategy for unmanned helicopters formation.

This paper addresses trajectory optimization for hypersonic vehicles under atmospheric and aerodynamic uncertainties using techniques from desensitized optimal control (DOC), wherein open-loop optimal controls are obtained by minimizing the sum of the standard objective function and a first-order penalty on trajectory variations due to parametric uncertainty. The proposed approach is demonstrated via numerical simulations of a minimum-final-time Earth reentry trajectory for an X-33 vehicle with an uncertain atmospheric scale height and drag coefficient. Monte Carlo simulations indicate that dispersions in the final position footprint and the final energy can be significantly reduced without closed-loop control and with little tradeoff in the performance metric set for the trajectory.

Velocity at landing should be within a safe limit, to make soft landing possible. For this the thrust steering angle should be controlled to optimize the fuel consumption. This paper deals with designing soft landing trajectory, from lunar parking orbit to the surface of Moon. To ensure vertical landing using minimum fuel, the landing trajectory design is done in four phases: Hohmann transfer phase, power descend phase, attitude maneuver and vertical descend phase. The fuel optimization is done in the power descend phase. Then the attitude of the probe is adjusted and the probe is made vertical during the attitude maneuver. In final vertical descend phase the steering angle is kept as 90 degree to land the probe vertically. Optimal soft landing trajectory is designed using indirect approach of optimization. The indirect approach uses, Pontryagin's minimum principle to transfer optimal control problem to Two Point Boundary Value (TPBV) problem, and to express the control variable as a function of costate variables. Then the state and costate equations are integrated at optimum control angle, with the initial costate found by the different optimization techniques, to get the optimal trajectory. Two optimization techniques are used in this paper: Control Random Search (CRS) and Differential Evolution (DE). The former optimization techniques are compared with optimal trajectory designed by indirect approach. The simulation results show that CRS has taken 10000 iterations in a simulation time of 120 minutes to converge while DE converged with 2000 iteration in 30 minutes of CPU time. Even though the velocity obtained after power descent phase and optimum landing mass are almost same in both technique, the time of convergence of CRS is four times that of DE. i.e., DE is found superior to CRS, since it converges faster and load taken by system is less. The problem of soft landing is extended to design a fuel optimal three dimensional trajectory using DE technique.

When operating autonomous surface vessels in uncertain environments with dynamic obstacles, planning safe trajectories and evaluating collision risk is key to navigating safely. In order to perform these tasks, it is important to have a computationally efficient and adaptable obstacle representation to allow for quick and robust predictions of the obstacle trajectory. This paper presents a novel space-time obstacle representation, which is able to predict the reachable set for a dynamic obstacle under uncertainty. This is done by projecting the area occupied by the obstacle forward in time, using a set of velocities representing the possible maneuvers that the obstacle may take. Under some mild assumptions, we show how the space-time obstacle can be implemented in a computationally efficient way, using both convex polytopes and ellipsoids. Additionally, we show how the space-time obstacle representation can be used for risk assessment, collision avoidance and trajectory planning for autonomous surface vessels.

Force modulation of robotic manipulators has been extensively studied for several decades but is not yet commonly used in safety-critical applications due to a lack of accurate interaction contact modeling and weak performance guarantees - a large proportion of them concerning the modulation of interaction forces. This study presents a high-level framework for simultaneous trajectory optimization and force control of the interaction between manipulator and soft environments. Sliding friction and normal contact force are taken into account. The dynamics of the soft contact model and the manipulator dynamics are simultaneously incorporated in the trajectory optimizer to generate desired motion and force profiles. A constraint optimization framework based on Differential Dynamic Programming and Alternative Direction Method of Multipliers has been employed to generate optimal control input and high-dimensional state trajectories. Experimental validation of the model performance is conducted on a soft substrate with known material properties using Cartesian space force control mode. Results show a comparison of ground truth and predicted model based contact force states for a few cartesian motions and the validity range of the friction model. Potential applications include high-level task planning of medical tasks involving manipulation of compliant, delicate, and deformable tissues., Comment: 7 pages, Submitted to IROS 2020 (Accepted for publication)

Vehicles nowadays are equipped with several assistance functions for e.g. Cruise Control and Lane Keeping Assist. Classical lateral control approaches such as pure pursuit, Stanley used for lane keeping assist provide good path tracking precision. However the operational domain for these approaches is limited i.e. highway driving. An adaptation of control parameters can increase the operational domain, but its difficult to tune classical approaches for the whole range of automated driving maneuvers. Apart from these classical approaches, optimization approaches are also used for lateral trajectory stabilization. The optimization based approaches have a wider operational domain, but the feasibility and real time execution remain some open issues. In this paper, we combine a classical approach and an optimization method for lateral trajectory stabilization. We present a method to optimize the control input calculated using a classical control approach i.e. pure pursuit based on a performance criteria. The performance criteria weighs the precision and comfort requirements. A non- linear optimization based on Differential Dynamic Programming (DDP) is used to solve the optimization problem. The calculated optimal trajectory is finally evaluated using a line search method to ensure the convergence and to verify the optimization policy. The approach is demonstrated on a full scale automated truck prototype and the experimental results are discussed.

When planning and predicting a flight trajectory, uncertainties are inherent both in the current values of various influencing factors and in their evolution. These uncertainties can turn initially “optimized” trajectories into impossible or at least less attractive solutions later when it is executed. In order to support a more robust trajectory planning for Continuous Descent Operations (CDO), this paper investigates how variations of those influencing factors impact the trajectory optimization fidelity. For this purpose, a set of optimal trajectories are generated for each of the variations and their sensitivities are analyzed. The trajectory optimization is formalized as a multi-phase optimal control problem and is numerically solved with the Legendre-Gauss Pseudospectral Method (LGPM). An iterative process is proposed to determine the required number of collocation points to grant a pre-set level of convergence. Case studies are carried out for International Standard Atmosphere (ISA) baseline conditions as well as for wind and temperature variations as relevant representatives of the weather prediction uncertainties. The numerical simulation results show shifts from the reference trajectory depending on wind and temperature variation. Uncertain wind speed caused a larger solution space and more variation in fuel burn than temperature errors. The designed solution spaces, especially the earliest and latest ToD locations, give pilots and air traffic controllers a good reference where their aircraft is expected to match best CDO goals under the individually prevailing uncertainties. We believe that such additional flight envelope information should complement current vertical path displays in glass cockpits to foster robust flight planning and execution.

It is the goal for future lunar missions to seek and design an optimal soft-landing trajectory of the lunar probe from the lunar satellite orbit to the lunar surface. This study aims at the optimization of three-dimensional trajectory method able to achieve soft precision landing at the desired site with a minimum fuel. The trajectory optimization model is constrained by the use of two types of sets: the positions and velocities of the lander. The Pontryagain maximum principle (PMP) is used to transform the nonlinear optimal control into a two-point boundary value problem(TPBVP) in which the backward integration method is applied to compute the initial value of conjugate variables. The trajectory generation can be formulated as a time-invariant set of equations by a monotonically increasing state variable because the thrust control angles are obtained by conjugate equations. The robustness and optimality for three-dimensional trajectory is shown in a numerical simulation. The results of the new optimization method satisfy the fuel consumption condition with permissible tolerances.

In the paper, the problem associated with optimization of deep oil and gas borehole outlines is posed. Vertically drilled boreholes usually aim at piercing several different producing levels to improve recovery from multiple horizons by combination of their production. An inclined deep borehole has a better luck to meet petroliferous layers than a vertical well, but there are limits to what trajectory of a curvilinear borehole can be drilled and what cost will it absorb. Solutions to these questions can be achieved through the use of methods of differential geometry, nonlinear programming and optimal control. On their basis, the mathematic model of the well trajectory tracking is elaborated. It includes differential equations, objective functional, and constraints, limiting phase variables and controlling function. The objective functionals are selected in the integral (Lagrange) form minimizing the trajectory curvature, length, or cost; the algorithm of their minimization is proposed; typical examples are considered.

Connected and autonomous vehicles (CAV) is a key technology for this century. One of the main challenges is to define a smart interaction behavior of CAV with the other road users. The challenge is mainly raised at conflicting points where path of CAV intersects with the other users. Recent studies show interaction with humans is a big challenge. It not only requires a collision avoidance system but also more communicative behaviors of the CAV. More precisely, pedestrian needs to understand the intention of the incoming CAV whether it will cross first or not according to its speed profile. One way to overcome this issue is to design optimal trajectory control of the CAV that matches with the pedestrian expectation. However, such a design faces the non-linearity of the space sharing constraint. This paper uses sequence modeling based on Petri Net in order to overcome the problem. It also uses Hamiltonian analysis to derive the optimal control. Numerical examples of the communicative behavior of CAV at pedestrian crossing show that the proposed approach provides CAV with a kind of automatic courtesy.

Application of traditional indirect optimization methods to optimal control problems (OCPs) with control and state path constraints is not a straightforward task. However, recent advances in regularization techniques and numerical continuation methods have enabled application of indirect methods to very complex OCPs. This study demonstrates the utility and application of an advanced indirect method, the Uniform Trigonometrization Method (UTM), to a Mars Science Laboratory type entry problem. The objective is to maximize the parachute deployment altitude for a free-time, fixed-final-velocity entry trajectory. For entry vehicles, in addition to the bank angle that is characterized by bang-bang control profiles, there are typically three state path constraints that have to be considered, namely, the dynamic pressure, heat-rate, and g-load. This study shows that the UTM enables simultaneous regularization of the bang-bang control and satisfaction of the state path constraints. Two scenarios with and without state path constraints are considered. The results obtained using the UTM for both of these cases are found to be in excellent agreement with a direct optimization method. Furthermore, an interesting feature emerges in the optimal control profile of the UTM during the initial high-altitude part of the resulting optimal trajectory for the scenario with state path constraints, which has an appealing practical implication.

In order to reduce the operating cost of a flight, a new optimization algorithm based on cost index (CI) that combines the cost of fuel and time is presented to generate the reference cruise flight trajectory. Cruise speed optimization at a constant altitude can be formulated as an optimal control problem solved by Pontryagin's maximum principle and the Hamilton-Jacobi-Bellman (HJB) equation. In the troposphere, the economy cruise speed is approximately linear with the cruising altitude which can be linearly fitted by least squares. The optimal altitude is obtained by taking the aircraft performance and air traffic regulation into account. Simulation results of a Boeing 747–200 model demonstrate that the proposed solution is faster than the search method and more economical than the reference trajectory provided by Eurocontrol's Base of Aircraft Data (BADA).

This paper presents a new method of computing a sub-optimal solution of a continuous-time continuous-space chance-constrained stochastic nonlinear optimal control problem (SNOC) problem. The proposed method involves two steps. The first step is to derive a deterministic nonlinear optimal control problem (DNOC) with convex constraints that are surrogate to the SNOC by using generalized polynomial chaos (gPC) expansion and tools taken from chance-constrained programming. The second step is to solve the DNOC problem using sequential convex programming (SCP) for trajectory generation. We prove that in the unconstrained case, the optimal value of the DNOC converges to that of SNOC asymptotically and that any feasible solution of the constrained DNOC is a feasible solution of the chance-constrained SNOC because the gPC approximation of the random variables converges to the true distribution. The effectiveness of the gPC-SCP method is demonstrated by computing safe trajectories for a second-order planar robot model with multiplicative stochastic uncertainty entering at the input while avoiding collisions with a specified probability.

A rapid trajectory re-plan algorithm for ascent phase of ballistic missiles when power system fault happens is presented in this paper. Firstly, the trajectory re-plan and optimization problem is described as an optimal control problem including the dynamical model, objective function and constraints. Since the problem is nonconvex and cannot be solved accurately and rapidly, various techniques are adopted to transform the original problem to a series of convex problems equivalently. Considering the nonconvex thrust magnitude constraint, lossless convexification is proposed to avoid nonconvexity. And then, successive convexification and some improved strategies are applied in order to convexify the remaining nonconvexities and improve the accuracy and robustness of the algorithm. Finally, simulation tests are conducted and results demonstrate the accuracy and adaptability of the proposed method.

A method of designing optimal parabolic trajectory for long range air-defense interceptor missile with a constraint on capability of error correction in its terminal guidance phase is proposed. In this method a constraint condition on the error correction capability in finite terminal guidance phase for the missile with bounded acceleration is presented to replace the traditional constraint on instantaneous normal acceleration. A minimum time problem under the constraint is solved using the nonlinear programming optimal control theory. Simulation results for some typical scenarios of long range interception show that the optimal trajectories satisfy a series of constraints including not only the error correction capability, but also the constraint on and maximum angle of attack, and the constraints on terminal flight path angle and terminal velocity.

At present, there are many methods for trajectory optimization problems in dealing with finite thrust orbit maneuvers, including indirect, direct, and hybrid methods. In this paper, a real-time optimal guidance method based on self-adaptive hp-Radau pseudo-spectral method which is sampled at free frequency is proposed. The detailed design method is given in the paper. We analyzed the objects and conditions applicable to the guidance method and the stability of the algorithm is proved by mathematical derivation. In this paper, the optimal analysis of the coplanar orbital lifting time is taken as an example. The simulation results show that the introduction of the guidance law ensures the stability and guidance accuracy of the system after the introduction of interference. It is proved that the adaptive real-time on-line closed-loop guidance law based on the hp-Radau pseudo-spectral method has higher computational efficiency and better convergence when dealing with the optimization problems with small number or range of orbital elements.

In this paper, a new stochastic based method for low-thrust trajectory design optimization is proposed. Indirect optimization methods based on optimal control theory (OCT) provide accurate solutions but convergence is difficult. In our new approach, an augmented Gaussian is used to relax the terminal conditions of the Two Point Boundary Value Problem (TPBVP), and model overall impacts of initial guess of costates. Objective function of the low-thrust trajectory design is transformed into a quasi-quadratic problem using quadratic convolution technique. The quasi-quadratic form objective function contains information matrix of the impacts due to varied costates and terminal conditions. A gradient based searching algorithm is then used to search the optimal initial guess. Based on the method, two algorithms are developed for resolving the TPBVP of low thrust trajectory design. Convergence performances of the algorithms are demonstrated through two low-thrust interplanetary trajectory design problems. Possible extension of the algorithm to multi-objective evolutionary optimization is also discussed.

In order to improve the acceleration response of hybrid vehicles, this paper applies Radau pseudo-spectral method to solve the trajectory optimization problem of vehicle acceleration. The mathematical model of engine-generator, battery pack and electric motor is established, and the optimal control problem of vehicle acceleration time is constructed. As for the problem is strongly nonlinear. The problem is transformed into a nonlinear programming problem by Radau pseudo-spectral method, and the numerical solution is carried out. The rotational speed and torque trajectory curves of the engine-generator set and the motor are obtained. The simulation results show that Radau pseudo-spectral method can optimize the working curve of various components of hybrid vehicles and improve response speed.

This brief investigates a computational framework based on optimal control for addressing the problem of stochastic trajectory optimization with the consideration of chance constraints. This design employs a discretization technique to parameterize uncertain variables and create the trajectory ensemble. Subsequently, the resulting discretized version of the problem is solved by applying standard optimal control solvers. In order to provide reliable gradient information to the optimization algorithm, a smooth and differentiable chance-constraint approximation method is proposed to replace the original probability constraints. The established methodology is implemented to explore the optimal trajectories for a spacecraft entry flight planning scenario with noise-perturbed dynamics and probabilistic constraints. Simulation results and comparative studies demonstrate that the present chance-constraint handling strategy can outperform other existing approaches analyzed in this brief, and this computational framework can produce reliable and less conservative solutions for the chance-constrained stochastic spacecraft trajectory planning problem.

To solve the problem of collision avoidance between UAVs in the process of multi-UAV formation configuration, a new form of safety zone constraints for collision avoidance is proposed, which transforms the safety zone constraints from the circular zone to the half-plane zone. This transformation makes the formation trajectory optimization be a standard quadratic programming problem, and a distributed receding horizon optimization method is used to solve the problem efficiently. Simulation results show that this method can get optimal formation trajectories under cooperative collision avoidance conditions.

Highly constrained trajectory optimization problems are usually difficult to solve. Due to some real-world requirements, a typical trajectory optimization model may need to be formulated containing several objectives. Because of the discontinuity or nonlinearity in the vehicle dynamics and mission objectives, it is challenging to generate a compromised trajectory that can satisfy constraints and optimize objectives. To address the multiobjective trajectory planning problem, this paper applies a specific multiple-shooting discretization technique with the newest NSGA-III optimization algorithm and constructs a new evolutionary optimal control solver. In addition, three constraint handling algorithms are incorporated in this evolutionary optimal control framework. The performance of using different constraint handling strategies is detailed and analyzed. The proposed approach is compared with other well-developed multiobjective techniques. Experimental studies demonstrate that the present method can outperform other evolutionary-based solvers investigated in this paper with respect to convergence ability and distribution of the Pareto-optimal solutions. Therefore, the present evolutionary optimal control solver is more attractive and can offer an alternative for optimizing multiobjective continuous-time trajectory optimization problems.

We consider the problem of generating optimal planar trajectories for marine vehicles taking into account the effect of time-varying ocean currents. This is motivated by the need for economic trajectory generation in long-duration operations, and it becomes even more important in scenarios where the magnitude of the ocean current speed is comparable to the maximum vehicle speed, as illustrated in our experiments. We propose a solution based on a simple motion model, dynamic programming and high-resolution ocean models. Optimal trajectories are generated under the assumption that the vehicle is capable of reaching the destination even when the matching condition is violated in some segments of the path. Numerical results show that the method is appropriate for practical generation of feasible trajectories in spatially constrained areas with strong currents.

This paper proposed a new methodology for optimal decentralized trajectory planning of multiple flying robots, in an obstacle-laden environment by incorporating obstacle avoidance and conflict resolution into one unified optimal control trajectory generation framework. In other words, an optimal trajectory that; firstly, passes through all obstacles, secondly maintains the safe flying zone for each of flying robot could be determined. In development of this path planner a Radau-pseudo spectral optimization method has been employed. The nonlinear point mass equations of motion of the flying vehicle in three dimensional space are utilized and realistic operational constraints influencing on flying vehicle state and control variables are considered via simplified mathematical models. The effectiveness of the proposed approach for solving centralized trajectory optimization problems, has been demonstrated via presenting a case study including two robots forced to fly in a hazardous area with fixed obstacles

Nonlinear optimal control problems are challenging to solve efficiently due to non-convexity. This paper introduces a trajectory optimization approach that achieves realtime performance by combining machine learning to predict optimal trajectories with refinement by quadratic optimization. First, a library of optimal trajectories is calculated offline and used to train a neural network. Online, the neural network predicts a trajectory for a novel initial state and cost function, and this prediction is further optimized by a sparse quadratic programming solver. We apply this approach to a fly-to-target movement problem for an indoor quadrotor. Experiments demonstrate that the technique calculates near-optimal trajectories in a few milliseconds, and generates agile movement that can be tracked more accurately than existing methods.