Trajectory Optimization Problem Research Articles

Research on Intelligent Optimization of Wellbore Trajectory in Complex Formation

Borehole trajectory optimization is a key issue in oil and gas drilling engineering. The traditional wellbore trajectory design method faces great challenges in optimizing the trajectory length and complexity, and it is difficult to meet the actual engineering requirements. In this paper, the three-stage wellbore trajectory optimization problem is studied, and a multi-objective optimization model including two objective functions of trajectory length and trajectory complexity is constructed. In this paper, an improved multi-objective particle swarm optimization algorithm is proposed, which combines the clustering strategy to improve the diversity of solutions, and enhances the local search ability and global convergence performance of the algorithm through the elite learning strategy. In order to verify the performance of the algorithm, comparative experiments were carried out using classical multi-objective benchmark functions. The results showed that the improved algorithm is superior to the traditional method in terms of diversity and convergence of solutions. Finally, the proposed algorithm was applied to the actual three-stage wellbore trajectory optimization problem. In summary, the research results of this paper provide theoretical support and engineering practice methods for wellbore trajectory optimization, and serve as an important reference for further improving the efficiency and quality of wellbore trajectory design.

Boston Consulting Group Matrix-Based Equilibrium Optimizer for Numerical Optimization and Dynamic Economic Dispatch

Numerous optimization problems exist in the design and operation of power systems, critical for efficient energy use, cost minimization, and system stability. With increasing energy demand and diversifying energy structures, these problems grow increasingly complex. Metaheuristic algorithms have been highlighted for their flexibility and effectiveness in addressing such complex problems. To further explore the theoretical support of metaheuristic algorithms for optimization problems in power systems, this paper proposes a novel algorithm, the Boston Consulting Group Matrix-based Equilibrium Optimizer (BCGEO), which integrates the Equilibrium Optimizer (EO) with the classic economic decision-making model, the Boston Consulting Group Matrix. This matrix is utilized to construct a model for evaluating the potential of individuals, aiding in the rational allocation of computational resources, thereby achieving a better balance between exploration and exploitation. In comparative experiments across various dimensions on CEC2017, the BCGEO demonstrated superior search performance over its peers. Furthermore, in dynamic economic dispatch, the BCGEO has shown strong optimization capabilities and potential in power system optimization problems. Additionally, the experimental results in the spacecraft trajectory optimization problem suggest its potential for broader application across various fields.

Optimal low-thrust orbit transfers connecting Gateway with Earth and Moon

Gateway will represent a primary logistic infrastructure in cislunar space. The identification of efficient orbit transfers capable of connecting Earth, Moon, and Gateway paves the way for enabling refurbishment, servicing, and utilization of this orbiting platform. This study is devoted to determining two-way minimum-time low-thrust orbit transfers that connect both Earth and Moon to Gateway. Backward time propagation is proposed as a very convenient option for trajectories that approach and rendezvous with Gateway, and a unified formulation is introduced for minimum-time orbit transfers, using either forward or backward propagation. Two-way transfers between Gateway and a specified low-altitude lunar orbit are first determined, using an indirect heuristic method, which employs the necessary conditions for optimality and a heuristic algorithm. Second, two-way orbit transfers that connect Earth and Gateway are addressed. Because these trajectories exist under the influence of two major attracting bodies, the underlying optimal control problem is formulated as a multi-arc trajectory optimization problem, involving three different representations for the spacecraft state. Multi-arc optimal control problems are associated with several corner conditions to be enforced at the junction time that separates distinct arcs. However, these conditions are shown to be solvable sequentially for the problem at hand, leveraging implicit costate transformation. This implies that the multi-arc problem has a set of unknown quantities with the same size as that of a single-arc optimal control problem, with apparent advantages of a computational nature. The indirect heuristic method is also applied in this second mission scenario. Low-thrust orbit dynamics is propagated in a high-fidelity dynamical framework, with the use of planetary ephemeris and the inclusion of the simultaneous gravitational action of Sun, Earth, and Moon, along the entire transfer paths, in all cases. The numerical results unequivocally prove that the methodology developed in this research is effective for determining two-way minimum-time low-thrust orbit transfers connecting Earth, Moon, and Gateway.

Effects of Bio-Inspired Wing Dihedral Variations on Dynamic Soaring Performance of Unmanned Aerial Vehicles

On the basis of a self-developed albatross imitation unmanned aerial vehicle (UAV), three different dihedral angle configurations for the wing’s mid and outer sections are explored: fixed at −50°, fixed at −5°, and varying arbitrarily between −50° and −5°. By solving the optimal loitering dynamic soaring trajectory optimization problem for each configuration, the effect of dihedral angle variation on the dynamic soaring performance of the bio-inspired wings is investigated. The results indicate that under all three configurations, the UAV achieves energy-neutral flight in specific wind field environments. Compared to the fixed dihedral angle of −5°, the UAV demonstrated superior dynamic soaring performance when the dihedral angle was fixed at −50°. When the dihedral angle varied dynamically, the UAV outperformed both fixed configurations across all relevant parameters. Specifically, compared to the fixed dihedral angle of −5°, the total energy increased by 25.43%, and the minimum required wind gradient decreased by 15.56%. Similarly, compared to the fixed dihedral angle of −50°, the total energy increased by 2.52%, and the minimum required wind gradient decreased by 2.07%. These findings suggest that the use of variable dihedral angle technology in bio-inspired UAV wings can significantly enhance dynamic soaring performance and provide theoretical support for the design of morphing wings with superior dynamic soaring capabilities.

Fast Nonlinear Model Predictive Control Combining Online Trajectory Optimization and Value Function Regression

ABSTRACTModel predictive control (MPC) is a well‐developed method capable of handling complex control tasks. Implementation of an MPC requires solution of a deterministic finite horizon nonlinear optimal control (OC) problem. OC problems can be solved globally, explicitly expressing the optimal policy (and value function) as a function of the present state, or, locally, using online trajectory optimization, generating a solution that is only valid for the present state. For nonlinear problems however, neither is possible analytically, and the latter, finding a local solution online, is usually preferred. The difficulty with online trajectory optimization is that the solution must be available within a single sampling period. The main parameter affecting the computational demand of MPC is the length of the prediction horizon. The goal in this work is to reduce the length of the prediction horizon of a model predictive controller (MPC) to reduce computation time whilst preserving optimality guarantees. To this end, we propose approximation of the trajectory optimization problem for MPC by learning a finite horizon value function. The approximated value function is inserted into a truncated trajectory optimization problem so that the MPC can be attained with a reduced prediction horizon, ergo reduced computational load. By sampling the value function in a goal oriented way, we show that an effective approximate value function can be found by including both the function value and the gradients of the value function. The result is an accurate approximate MPC which leverages learning methodologies to reduce computational cost while still accounting for constraints.

Energy-efficient trajectory optimization algorithm based on K-medoids clustering and gradient-based optimizer for multi-UAV-assisted mobile edge computing systems

The mobile edge computing system supported by multiple unmanned aerial vehicles (UAVs) has gained significant interest over the last few decades due to its flexibility and ability to enhance communication coverage. In this system, the UAVs function as edge servers to offer computing services to Internet of Things devices (IoTDs), and if they are located distant from those devices, a significant amount of energy is consumed while data is transmitted. Therefore, optimizing UAVs’ trajectories is an indispensable process to minimize overall energy consumption in this system. This problem is difficult to solve because it requires multiple considerations, including the number and placement of stop points (SPs), their order, and the association between SPs and UAVs. A few studies in the literature have been presented to address all of these aspects; nevertheless, the majority of them suffer from slow convergence speed, stagnation in local optima, and expensive computational costs. Therefore, this study presents a new trajectory optimization algorithm, namely ITPA-GBOKM, based on a newly proposed transfer-based encoding mechanism, gradient-based optimizer, and K-Medoids Clustering algorithm to tackle this problem more accurately. The K-medoid clustering algorithm is used to achieve better association between UAVs and SPs since it is less sensitive to outliers than the K-means clustering algorithm; the transfer function-based encoding mechanism is used to efficiently define this problem’s solutions and manage the number of SPs; and GBO is utilized to search for the best SPs that could minimize overall energy consumption, including that consumed by UAVs and IoTDs. The proposed ITPA-GBOKM is evaluated using 13 instances with several IoTDs ranging from 60 to 700 to show its effectiveness in dealing with the trajectory optimization problem at small, medium, and large scales. Furthermore, it is compared to several rival optimizers using a variety of performance metrics, including average fitness, multiple comparison test, Wilcoxon rank sum test, standard deviation, Friedman mean rank, and convergence speed, to show its superiority. The experimental results indicates that this algorithm is capable of producing significantly different and superior results compared to the rival algorithms.

Three six-degree-of-freedom dynamic descent convex optimization guidance methods

A key technology for soft landing of powered descent of aircraft is to solve the six-degree-of-freedom powered descent guidance problem in real time, which can be described as the fuel-saving trajectory optimization problem under multiple constraints. Three optimization models are established by selecting three independent variables: flight time, time substitution variable, and trajectory height. The original trajectory optimization problem is transformed into a solvable form of sequential convex optimization for iterative solution, forming three online guidance methods. The three guidance methods are compared in terms of convergence, real-time performance, optimality, and solution accuracy. The results show that all three guidance methods can solve the six-degree-of-freedom powered descent problem. The guidance method with flight time as the independent variable has the shortest calculation time and the least fuel consumption, but the powered descent flight time needs to be determined in advance. The guidance methods based on the other two types of independent variables can optimize the powered descent flight time, but they are all suboptimal solutions, and the calculation time is significantly increased. The solution accuracy of the three methods is similar under the same number of discrete points. If sequential convex optimization is used as the online guidance scheme for powered descent, how to determine the optimal flight time, approach the optimal solution for fuel, and further shorten the calculation time still needs to be further studied.

Kinodynamic Model-Based UAV Trajectory Optimization for Wireless Communication Support of Internet of Vehicles in Smart Cities

Unmanned aerial vehicles (UAVs) are utilized for wireless communication support of Internet of Intelligent Vehicles (IoVs). Intelligent vehicles (IVs) need vehicle-to-vehicle (V2V) and vehicle-to-everything (V2X) wireless communication for real-time perception knowledge exchange and dynamic environment modeling for safe autonomous driving and mission accomplishment. UAVs autonomously navigate through dense urban areas to provide aerial line-of-sight (LoS) communication links for IoVs. Real-time UAV trajectory design is required for minimum energy consumption and maximum channel performance. However, this is multidisciplinary research including (1) dynamic-aware kinematic (kinodynamic) planning by considering UAVs’ motion and nonholonomic constraints; (2) channel modeling and channel performance improvement in future wireless networks (i.e., beyond 5G and 6G) that are limited to beamforming to LoS links with the aid of reconfigurable intelligent surfaces (RISs); and (3) real-time obstacle-free crash avoidance 3D trajectory optimization in dense urban areas by modeling obstacles and LoS paths in convex programming. Modeling and solving this multilateral problem in real-time are computationally prohibitive unless extensive computational and overhead processing costs are imposed. To pave the path for computationally efficient yet feasible real-time trajectory optimization, this paper presents UAV kinodynamic modeling. Then, it proposes a convex trajectory optimization problem with the developed linear kinodynamic models. The optimality and smoothness of the trajectory optimization problem are improved by utilizing model predictive control and quadratic state feedback control. Simulation results are provided to validate the methodology.

Joint Computation Offloading and Trajectory Optimization for Edge Computing UAV: A KNN-DDPG Algorithm

Unmanned aerial vehicles (UAVs) are widely used to improve the coverage and communication quality of wireless networks and assist mobile edge computing (MEC) due to their flexible deployments. However, the UAV-assisted MEC systems also face challenges in terms of computation offloading and trajectory planning in the dynamic environment. This paper employs deep reinforcement learning to jointly optimize the computation offloading and trajectory planning for UAV-assisted MEC system. Specifically, this paper investigates a general scenario where multiple pieces of user equipment (UE) offload tasks to a UAV equipped with a MEC server to collaborate on a complex job. By fully considering UAV and UE movement, computation offloading ratio, and blocked relations, a joint computation offloading and trajectory optimization problem is formulated to minimize the maximum computational delay. Due to the non-convex nature of the problem, it is converted into a Markov decision process, and solved by the deep deterministic policy gradient (DDPG) algorithm. To enhance the exploration capability and stability of DDPG, the K-nearest neighbor (KNN) algorithm is employed, namely KNN-DDPG. Moreover, the prioritized experience replay algorithm, where the constant learning rate is replaced by the decaying learning rate, is utilized to enhance the converge. To validate the effectiveness and superiority of the proposed algorithm, KNN-DDPG is compared with the benchmark DDPG algorithm. Simulation results demonstrate that KNN-DDPG can converge and achieve 3.23% delay reduction compared to DDPG.

Trajectory optimization of spacecraft autonomous far-distance rapid rendezvous based on deep reinforcement learning

This paper investigates the application of Deep Reinforcement Learning (DRL) in the trajectory optimization of spacecraft far-distance rapid rendezvous, and uses the most advanced DRL method Proximal Policy Optimization (PPO) to solve the continuous high-thrust minimum-fuel trajectory optimization problem. The space J2 perturbation was considered, its impact on the spacecraft’s on-orbit operation and trajectory design was analyzed, and the effectiveness and accuracy of the proposed method were verified in two far-distance rapid rendezvous missions. In order to ensure the safety of the subsequent close-range operation phase, a safe area reward framework is proposed, and sparse and dense safe area reward functions are designed. The dense safe area reward function significantly improves the training efficiency of the algorithm on the basis of ensuring terminal performance. In addition, the modeling and analysis of possible uncertainties in the spacecraft’s orbit operation, including observation uncertainty, state uncertainty and control uncertainty, is carried out to verify the performance of the proposed method through simulation. For uncertainties, the closed-loop performance of the policy is also evaluated by performing Monte Carlo simulations. The results show that the PPO algorithm can effectively deal with the rendezvous problem in uncertainty environments. These preliminary results demonstrate the great potential of the DRL method in achieving autonomous guidance of spacecraft.

Sequential convex programming without penalty function for reentry trajectory optimization problem

Sequential convex programming (SCP) has been extensively utilized in reentry trajectory optimization due to its high computational efficiency. However, the current SCP approaches primarily rely on penalty function, where the selection of the penalty function weight presents a significant challenge. In this paper, an improved trust region shrinking SCP algorithm is proposed that separates the treatment of the objective function and constraint violation without the need for selecting penalty function weight and introduction of slack variables. Firstly, from the perspective of multi-objective optimization, the filter and acceptance condition are introduced to ensure that the proposed algorithm converges to feasible solutions and then to the optimal solution based on switching condition and sufficient condition. Then an effective feasibility restoration phase is proposed to address infeasibility of subproblems without introducing slack variables, while ensuring the robustness of the proposed algorithm. Additionally, a theoretical analysis is provided to guarantee the convergence of the algorithm. Finally, simulations are conducted to verify that the proposed algorithm demonstrates a 69.54% improvement in average solution time and stronger robustness compared to basic trust region shrinking SCP algorithm. Simultaneously, the proposed algorithm also demonstrates an advantage in solving speed compared to a particular advanced penalty function-based SCP algorithm.

Edge-Intelligence-Powered Joint Computation Offloading and Unmanned Aerial Vehicle Trajectory Optimization Strategy

UAV-based air-ground integrated networks offer a significant benefit in terms of providing ubiquitous communications and computing services for Internet of Things (IoT) devices. With the empowerment of edge intelligence (EI) technology, they can efficiently deploy various intelligent IoT applications. However, the trajectory of UAVs can significantly affect the quality of service (QoS) and resource optimization decisions. Joint computation offloading and UAV trajectory optimization bring many challenges, including coupled decision variables, information uncertainty, and long-term queue delay constraints. Therefore, this paper introduces an air-ground integrated architecture with EI and proposes a TD3-based joint computation offloading and UAV trajectory optimization (TCOTO) algorithm. Specifically, we use the principle of the TD3 algorithm to transform the original problem into a cumulative reward maximization problem in deep reinforcement learning (DRL) to obtain the UAV trajectory and offloading strategy. Additionally, the Lyapunov framework is used to convert the original long-term optimization problem into a deterministic short-term time-slot problem to ensure the long-term stability of the UAV queue. Based on the simulation results, it can be concluded that our novel TD3-based algorithm effectively solves the joint computation offloading and UAV trajectory optimization problems. The proposed algorithm improves the performance of the system energy efficiency by 3.77%, 22.90%, and 67.62%, respectively, compared to the other three benchmark schemes.

Path signatures for diversity in probabilistic trajectory optimisation

Motion planning can be cast as a trajectory optimisation problem where a cost is minimised as a function of the trajectory being generated. In complex environments with several obstacles and complicated geometry, this optimisation problem is usually difficult to solve and prone to local minima. However, recent advancements in computing hardware allow for parallel trajectory optimisation where multiple solutions are obtained simultaneously, each initialised from a different starting point. Unfortunately, without a strategy preventing two solutions to collapse on each other, naive parallel optimisation can suffer from mode collapse diminishing the efficiency of the approach and the likelihood of finding a global solution. In this paper, we leverage on recent advances in the theory of rough paths to devise an algorithm for parallel trajectory optimisation that promotes diversity over the range of solutions, therefore avoiding mode collapses and achieving better global properties. Our approach builds on path signatures and Hilbert space representations of trajectories and connects parallel variational inference for trajectory estimation with diversity-promoting kernels. We empirically demonstrate that this strategy achieves lower average costs than competing alternatives on a range of problems, from 2D navigation to robotic manipulators operating in cluttered environments.

Powered–coasting–powered multi-phase mission reconfiguration method for launch vehicles under power system failures

For the final stage of a launch vehicle with powered–coasting–powered multi-phase flight, this paper investigates the joint reconfiguration method to determine subsequent flight trajectory and degraded orbit after power system failure in the first powered phase. While power system failures due to mass–flow–rate drop have been thoroughly studied in the existing literature, this paper focuses on the more challenging specific impulse drop failures, which result in loss of total launcher energy. This paper aims to design a fault-tolerant guidance method with onboard potential and to analyze the survivability and characteristics of the launcher in the event of such a failure. In the problem modeling, the strong nonlinearity of the orbital elements as terminal constraints is eliminated by analyzing the orbital properties in the orbital coordinate system to establish the primal trajectory optimization problem. For this multi-phase trajectory optimization problem, a joint optimization algorithm for the degradation orbit and the flight trajectory, based on convex optimization and Radau pseudospectral method, is designed to achieve autonomous mission reconfiguration. The algorithm introduces the relaxation–penalization idea, where the terminal constraints are relaxed and penalized in the performance index, allowing the algorithm to adaptively form an optimal degradation circular or elliptical orbit based on the remaining capacity of the launch vehicle. Simulation experiments demonstrate the effectiveness and superiority of the mission reconfiguration algorithm. Finally, the paper presents the first analysis of the optimal allocation of flight time between the first powered phase, the coasting phase, and the second powered phase in a propellant depletion scenario following the launcher power system failure.

A Two Phases Multiobjective Trajectory Optimization Scheme for Multi-UGVs in the Sight of the First Aid Scenario.

Timely delivery of first aid supplies is significant to saving lives when an accident happens. Among the promising solutions provided for such scenarios, the application of unmanned vehicles has attracted ever more attention. However, such scenarios are often very complex, while the existing studies have not fully addressed the trajectory optimization problem of multiple unmanned ground vehicles (multi-UGVs) against the scenario. This study focuses on multi-UGVs trajectory optimization in the sight of first aid supply delivery tasks in mass accidents. A two-stage completely decoupling fuzzy multiobjective optimization strategy is designed. On the first stage, with the proposed timescale involved tridimensional tunneled collision-free trajectory (TITTCT) algorithm, collision-free coarse tunnels are build within a tridimensional coordinate system, respectively, for the UGVs as the corresponding configuration space for a further multiobjective optimization. On the second stage, a fuzzy multiobjective transcription method is designed to solve the decoupled optimal control problem (OCP) within the configuration space with the consideration of priority constrains. Following the two-stage design, the computational time is significantly reduced when achieving an optimal solution of the multi-UGV trajectory planning, which is crucial in a first aid task. In addition, other objectives are optimized with the aspiration level reflected. Simulation studies and experiments have been curried out to testify the effectiveness and the improved computational performance of the proposed design.

Multi-Unmanned Aerial Vehicles Cooperative Trajectory Optimization in the Approaching Stage Based on the Attitude Correction Algorithm

This study investigated the problem of multi-UAVs cooperative trajectory optimization for remote maritime targets in the approach phase. First, based on the precise location information of the cooperative target, a real-time algorithm for correcting UAV attitude angles is proposed to reduce the impact of UAV attitude angle errors and observation system errors on target positioning accuracy. Then, the attitude correction algorithm is integrated into the interacting multiple model-cubature information filter (IMM-CIF) algorithm to achieve the fusion of multi-UAVs observation information. Furthermore, an improved receding horizon optimization (RHO) method is employed to plan the cooperative observation trajectories for UAVs in real time at the target approaching stage. Finally, numerical simulations are conducted to examine the proposed attitude correction and trajectory optimization algorithm, verifying the effectiveness of the proposed method and enhancing the tracking accuracy of the remote target.

Online dual robot–human collaboration trajectory generation by convex optimization

For dynamic collision-free trajectory planning in dual-robot and human collaborative tasks, this paper develops an online dual-robot Mutual Collision Avoidance (MCA) scheme based on convex optimization. A novel convex optimization formulation model, named Disciplined Convex programming by Shifting reference paths (DCS), is proposed for solving the single-robot trajectory optimization problem. Furthermore, a new dual-robot trajectory convex optimization algorithm is presented for online adjustment of the dual-robot trajectories according to the collaborative task priority. The overall pipeline, named DCS-MCA, generates collision-free and time-optimal dual-robot trajectories, while prioritizing the task accessibility of the high-priority robot. Simulation experiments demonstrate that DCS exhibits comparable performance to the current state-of-the-art single-robot motion planner, while the DCS-MCA outperforms common algorithms by up to 30% in time optimality for dual-robot collaborative tasks. The feasibility and dynamic performance of the proposed approach are further validated in a real collaborative cell, illustrating its suitability for collaborative dual-robot tasks in moderately dynamic environments.

Trajectory optimization for aerodynamically controlled missiles by chance-constrained sequential convex programming

The flight environment of aerodynamically controlled missiles is full of complexity and uncertainty. To cope with the uncertainty more effectively and enhance the convergence performance in trajectory optimization problems for aerodynamically controlled missiles simultaneously, the chance-constrained sequential convex programming (CC-SCP) algorithm is proposed in this paper. The uncertainty is regarded as the chance constraint, and a smooth and differential approximation function is designed to transform this chance constraint into the constraint that the convex optimization method can handle. Subsequently, the originally non-convex trajectory optimization problem is reformulated into a series of convex optimization subproblems, in which an initial reference trajectory guess generation strategy is proposed, and a theoretical proof of the exact convex relaxation is given to enhance the algorithm's convergence performance and theoretical value, respectively. Numerical simulations are provided to verify the convergence and effectiveness of the CC-SCP algorithm, and the advantages of using the CC-SCP algorithm to cope with the uncertainty are illustrated. Furthermore, comparative simulation examples show that the proposed algorithm possesses a low conservatism, which means the proposed algorithm can obtain a bigger convergence region and a better solution than other current methods when handling the same chance constraints. Finally, the robustness of the algorithm is discussed.

Entry trajectory optimization of lifting-body vehicle by successive difference-of-convex programming

The complexity of the three-dimensional entry trajectory optimization problem has escalated due to the need to liberalize the angle of attack and bank angle as control variables, thereby enhancing the inherent maneuverability and control capabilities of lifting-body vehicles. The difference-of-convex (DC) properties inherent in the constraints of the problem are exploited in this paper. A DC decomposition approach is utilized to address the nonlinear auxiliary control equations, and the DC relaxation technique is applied to resolve iteration infeasibilities arising from Taylor expansion. The dependence on the initial trajectory is diminished by the implementation of an exact penalty method, thus improving the applicability of the methods. Furthermore, a control variable oscillation suppression mechanism has been constructed to tackle the control variable oscillation issues arising from the relaxation of the angle of attack and bank angle. This mechanism effectively suppresses large jumps in the angle of attack and high-frequency oscillations in the bank angle. Two novel successive DC programming methods are proposed: the successive concave-convex procedure and the successive proximal bundle method, functioning independently of trust-region constraints. Numerical experiments have demonstrated that the two proposed successive DC optimization methods exhibit exceptional performance in accuracy, feasibility, adaptability, and low sensitivity to initial values when applied to solving the three-dimensional entry trajectory optimization problem.

The applicability of Zermelo's equation to indirect and direct optimization of commercial aircraft flights

This paper focuses on the applicability of Zermelo's equation within the context of 3D commercial aircraft trajectory optimization problems. The associated optimal control problem includes two singular controls, specifically aerodynamic path angle and throttle setting, and one regular control, specifically heading angle. Using Pontryagin's maximum principle and the direct adjoining method, we show that the optimal heading angle is defined by Zermelo's equation. The significance of the presented analysis is that Zermelo's equation holds for 3D commercial aircraft flights, even in the presence of standard state-inequality constraints and a more general objective function. With the help of Zermelo's equation, the control problem is initially analyzed for a 3D time-optimal climb-phase scenario, which is solved by an indirect approach. Through the analysis of the switching function, we demonstrate the dependency of the optimal aerodynamic path angle on the optimal heading angle. Next, by tackling a 3D time-fuel-optimal free-routing flight, solved by a direct approach, we show that Zermelo's equation can help reduce the overall dimension of the associated nonlinear programming. To successfully handle the initial guess in both indirect and direct problems, it is proposed a diminutive nonlinear programming technique as a fast and robust initializer. This simple-yet-effective initializer provides sufficient information about the optimal controls and state dynamics required for initializing a direct optimization, as well as the optimal switching times and co-states required for initializing an indirect optimization.