Multiphase Trajectory Optimization Research Articles

Multi-Phase Trajectory Optimization for Alpine Skiers Using an Improved Retractable Body Model

Multi-Phase Trajectory Optimization for Alpine Skiers Using an Improved Retractable Body Model

Reentry blackout reachable set footprint prediction using multi-phase trajectory optimization

Blackout emerges in the reentry phase of reusable launch vehicles (RLV). Therein, large uncertainties exist in the telemetry signals of RLV, leading to potential safety problems. To facilitate predicting possible ranges of RLV final position when leaving blackout, this paper proposes a modified approach for computing reachable set footprint (RSF). A multi-phase trajectory optimization method is applied to simplified dynamics of RLV. Specifically, partial final boundary conditions are additionally supplemented to the first phase to exploit the intermediate state information during blackout. On this basis, RSF is predicted via solving a series of trajectory optimization problem by sequential convex programming. RSF with additional state information from different altitude are compared in numerical cases. Simulation results show that there exists a suitable range to update RSF using intermediate information. The decision altitude of updating RSF is determined for the exemplary RLV.

Real-Time Sequential Conic Optimization for Multi-Phase Rocket Landing Guidance

We introduce a multi-phase rocket landing guidance framework that can handle nonlinear dynamics and does not mandate any additional mixed-integer or nonconvex constraints to handle discrete temporal events/switching. To achieve this, we first introduce sequential conic optimization (seco), a new paradigm for solving nonconvex optimal control problems that is entirely devoid of matrix factorizations and inversions. This framework combines sequential convex programming (SCP) and first-order conic optimization and can solve unified multi-phase trajectory optimization problems in real-time. The novel features of this framework are: (1) time-interval dilation, which enables multi-phase trajectory optimization with free-transition-time; (2) single-crossing compound state-triggered constraints, which are entirely convex if the trigger and constraint conditions are convex; (3) virtual state, which is a new approach to handling artificial infeasibility in SCP methods that preserves the shapes of the constraint sets; and, (4) the use of the proportional-integral projected gradient method (pipg), a high-performance first-order conic optimization solver, in tandem with the penalized trust region (ptr) SCP algorithm. We demonstrate the efficacy and real-time capability of seco by solving a relevant multi-phase rocket landing guidance problem with nonlinear dynamics and convex constraints only, and observe that our solver is 2.7 times faster than a state-of-the-art convex optimization solver.

Multi-phase trajectory optimization of aerospace vehicle using sequential penalized convex relaxation

This paper presents a novel methodology for solving the multi-phase trajectories of aerospace vehicles in the framework of convex optimization. As a result of the inherent non-smooth, non-linear, and staged features of the problem, the application of convex optimization is confronted by three categories of tough non-convex terms, non-convex functions, phase linkage constraints, and free endpoints. To overcome these difficulties, a combination of a time-projection approach, and a sequential relaxation and penalization method is developed in this paper. Firstly, free singularity points from linkage constraints and endpoints are eliminated by projecting the time history onto a normalized time interval. Subsequently, the resulting fixed-time problem is equivalently converted in to a pre-semi-definite form and sequentially relaxed as an approximated convex optimization problem. Furthermore, to increase the convergence of the proposed method, a penalization term is included to control the search direction toward the true solution. Analyses are performed to ensure that the convergence and robustness of the proposed algorithm is well guaranteed. A numerical comparison of the results with those from the state-of-art pseudo-spectral nonlinear programming solver GPOPS suggests that the algorithm outputs similar trajectories, but in only one tenth of the computation time.

Surrogate-based entire trajectory optimization for full space mission from launch to reentry

A surrogate-based optimization method is proposed in this paper to realize the optimization of the entire trajectory from launch to reentry. Different from traditional strategy of multiphase trajectory optimization that directly utilizes the original dynamical model, a surrogate model is substituted for the original time-consuming model to improve the iterative efficiency and decrease the computational cost. The procedures of modeling, solving, and optimizing in each substage are addressed, including air-launch ascent trajectory optimization, orbital maneuver planning, deorbit braking, and atmospheric reentry. Instead of optimizing every substage separately and setting interfaces by experience, in this study, substages are patched as an entire system in advance and then the surrogate model is utilized to replace it to perform a systematic optimization. Additionally, the Fourier series is applied to fit the air launch stage during the iteration. Numerical simulations show that the proposed method is acceptable for the pre-design of a full space mission and the results are optimal.

Multiphase Trajectory Optimization of a Lunar Return Mission to an LEO Space Station

Lunar exploration architecture can be made more flexible and reliable with the support of a low-Earth orbit (LEO) space station. This study therefore evaluated a proposed hybrid optimization scheme to design the entire trajectory of a reusable spacecraft starting from trans-Earth injection (EI) at the perilune and ending at an LEO space station. As such a trajectory has multiple constraints and multiple dynamical models, it is divided into the trans-Earth phase, aerocapture phase, and postatmospheric phase. The optimization scheme is performed at two levels: sublevel and top level. At the sublevel, two novel pseudo rules are proposed to optimize the trans-Earth trajectory so that it satisfies the coplanar constraints of the space station. Then, in the aerocapture phase, the bank angle is optimized to satisfy the mission constraints, and in the atmospheric phase, the one-impulsive maneuver is performed and optimized to insert the spacecraft into the target space station orbit. The multiple phases are connected to each other by boundary conditions where the terminal state of the previous phase is transformed into the initial state of the following phase. At the top level, the vacuum perigee height is selected as the mission design variable based on problem characteristics analysis and a hybrid optimization scheme is conducted to minimize the total velocity increment. The simulation results demonstrate that the proposed hybrid optimization method is effective for the design of an entire trajectory with acceptable velocity cost which is less than that in the previous study. The coplanar constraints of the space station and other mission constraints in each phase are also satisfied. Furthermore, the proposed trajectory design method is shown to be applicable to a reusable spacecraft returning to an LEO space station parked in any arbitrary orbital plane.

Multi-phase trajectory optimization for an aerial-aquatic vehicle considering the influence of navigation error

The environment-induced multi-phase trajectory optimization problem is studied in this paper, and the underwater target tracking task is focused on. The task is finished by an aerial-aquatic coaxial eight-rotor vehicle and is divided into two phases, i.e., the diving phase and the underwater navigation phase. The dynamic model and constraints on angular velocity of rotor in each phase are established to understand the motion characteristic. Then the model of navigation information and terrain matching are contained in the trajectory optimization model to reflect the influence of underwater navigation error on the quality of trajectory. Correspondingly, the forms of collision detection and cost function are changed to adapt to the inaccurate navigation information. To obtain the trajectory with the minimum terminal position error, an improved teach & learn-based optimization (ITLBO) algorithm is developed to strengthen the influence of individual historical optimal solution. Besides, Chebyshev collocation points are applied to determine the locations of control variables. Simulation results demonstrate that the established navigation error-based trajectory optimization model can reflect the real situation of multi-phase task. Especially, it is able to calculate the collision probability between the vehicle and the obstacle when GPS is unavailable underwater, thus ensuring the safety of underwater navigation. Compare to other common effective algorithms, the proposed ITLBO algorithm is in general more suitable for solving this problem because it is swarm-based and can obtain good solution without worrying about the inappropriate values of user-defined parameters.

Predicted Step Viability: a stability criterion for biped gait

The formulation of a stability criterion is crucial to perform safe, versatile and efficient biped gaits for robots. We here present a stability criterion for non-cyclic gait synthesis, the Predicted Step Viability, inspired by human gait and N-Step Capturability. The Predicted Step Viability defines the constraints of the current step such that future steps will be able to guarantee convergence to a stable point in finite time. In this way, it is based on the prediction of future viable steps to ensure stability. The criterion was implemented using multiphase trajectory optimization on two biped models, the Compass Gait and the five-link model RABBIT. The Compass Gait was simulated with different model parameters and gait patterns including a random non-periodic one. The five-link model was tested with linear and random reference gait patterns. The Predicted Step Viability criterion successfully generated stable non-periodic gaits under a variety of conditions. Moreover, it is possible to prescribe any gait pattern completely uncoupled from the stability criterion. If it were impossible to follow the prescribed pattern without falling, the controller would give it up, maintaining stability.

Multiphase trajectory optimization for gun-launched glide guided projectiles

In this paper, an approach that considered both the continuous and discrete variables using the Gauss pseudospectral method is presented for multiphase trajectory optimization of gun-launched glide guided projectiles. The entire projectile trajectory is divided into four flight phases that connect to each other sequentially to ensure continuity. Continuous variables are utilized to describe the projectile’s state and control. Meanwhile, discrete variables are proposed to express the initial flight path angle and switching times between different flight phases. The trajectory optimization problem is posed as a constrained multiphase nonlinear optimal control problem, which is then transcribed into a nonlinear programming problem using the Gauss pseudospectral method and is finally solved using the interior point algorithm. Simulation results demonstrate the feasibility and effectiveness of this approach, as well as a detailed analysis of the key features of the trajectory parameters.