Polytopal approximation for thrust magnitude constraints in rocket trajectory optimization

Joint resource and trajectory optimization in a UAV-enabled dual-function radar-communication network

Ascent trajectory optimization and explicit guidance problems for a satellite launch vehicle with yaw maneuver in a 3-dimension are considered. The trajectory optimization problem with boundary conditions is formulated as a nonlinear programming problem by parameterizing the inertial pitch and yaw attitude control variables, and is solved by using the SQP algorithm. The flight constraints such as gravity-turn and range safety conditions are imposed. An explicit inertial guidance algorithm in the exoatmospheric phase is also presented. The guidance algorithm provides steering command and time-to-go value directly using the current states of the vehicle and the desired orbit insertion conditions. The liquid propelled Delta 2910 launch vehicle is used as a numerical model.

Numerous computer models for river basin planning and management have been developed and used extensively since the mid‐1970s. Early developments have relied on the use of network flow algorithms (NFA), due mainly to higher execution speed than the standard Simplex solvers. However, subsequent efforts to include proper modeling of hydraulic and hydrologic constraints introduced iterative schemes into the NFA‐based models, which diminished the initial advantages in execution speed and which also caused concerns over the accuracy of the convergence schemes. Hence full‐blown commercial linear programming (LP) solvers were introduced as a replacement to the iterative solution strategy of the NFA approach. This paper demonstrates one possible failure to solve a simple allocation problem using the NFA‐based model and shows how this problem can be solved using the standard LP approach. It then identifies cases when even a full‐blown LP approach cannot properly model two critical aspects of river basin management, one related to reservoirs with multiple outflows and the other one related to modeling of hydrologic channel routing. For NFA‐based models the failures are the result of the inability to include relationships between flows on different model components directly into the search process. For the models based on LP solvers, the failures are caused by the fact that integrated reservoir outflow capacity between the starting and the ending storage levels is assumed over the entire length of the assumed time step, while the actual outflow can only take place during the portion of the time step when the storage level is above the invert of the outlet structure.

An improved hp adaptive pseudo-spectral method ia scaling parameters designed for the hypersonic vehicle trajectory optimization problem. In this paper, we take hypersonic vehicle as the research object, consider its ascent phase with various constraints, and establish a multi-constrained climbing segment trajectory optimization model with the shortest ascent time as the objective function. Because the nonlinear programming problem (NLP) obtained by applying the hp adaptive pseudo-spectral method to discretize the trajectory optimization model has high dimensionality, which makes the trajectory optimization problem difficult to solve, we propose an improved hp adaptive pseudo-spectral method. The algorithm uses the improved wild horse optimizer (IWHO) algorithm to adjust the distribution of interpolation nodes and the order of interpolation polynomials in the trajectory optimization subinterval, which avoids unnecessary mesh refinement and reduces the dimensionality of the discretized obtained NLP with a given precision, and improves the efficiency of the trajectory optimization solution. Simulation results show that the trajectories solved by the designed hybrid optimization algorithm satisfy all process constraints and terminal constraints.

Trajectory optimization for lunar soft landing with a Hamiltonian-based adaptive mesh refinement strategy

A general overview of the topic of trajectory optimization is presented. The general trajectory optimization problem is posed, and the two broad classes of indirect and direct methods for solving trajectory optimization problems are discussed. A brief discussion is then given of well‐known software tools to solve trajectory optimization problems. Finally, a brief survey is given of famous trajectory optimization problems that have been solved in aerospace engineering.

Restricted Simplicial Decomposition with Side Constraints is a price decomposition method designed for large‐scale nonlinear problems with a set of linear constraints with special structure and an additional set of linear side constraints. The resultant algorithm iterates by solving a linear subproblem subject to the set of structured constraints and a small nonlinear master problem whose feasible region is defined by the intersection of a simplex and the set of side constraints. The number of variables of the master problem is controlled by a parameter named r. The conditions required for the parameter r and the rules for dropping extreme points of the simplex are established for global convergence of the algorithm. Computational results are presented for nonlinear network problems with side constraints.

SMT solvers combine SAT reasoning with specialized theory solvers either to find a feasible solution to a set of constraints or to prove that no such solution exists. Linear programming (LP) solvers come from the tradition of optimization, and are designed to find feasible solutions that are optimal with respect to some optimization function. Typical LP solvers are designed to solve large systems quickly using floating point arithmetic. Because floating point arithmetic is inexact, rounding errors can lead to incorrect results, making inexact solvers inappropriate for direct use in theorem proving. Previous efforts to leverage such solvers in the context of SMT have concluded that in addition to being potentially unsound, such solvers are too heavyweight to compete in the context of SMT. In this paper, we describe a technique for integrating LP solvers that improves the performance of SMT solvers without compromising correctness. These techniques have been implemented using the SMT solver CVC4 and the LP solver GLPK. Experiments show that this implementation outperforms other state-of-the-art SMT solvers on the QF_LRA SMT-LIB benchmarks and is competitive on the QF_LIA benchmarks.

In this work, we develop a new framework for dynamic network flow problems based on optimal transport theory. We show that the dynamic multicommodity minimum-cost network flow problem can be formulated as a multimarginal optimal transport problem, where the cost function and the constraints on the marginals are associated with a graph structure. By exploiting these structures and building on recent advances in optimal transport theory, we develop an efficient method for such entropy-regularized optimal transport problems. In particular, the graph structure is utilized to efficiently compute the projections needed in the corresponding Sinkhorn iterations, and we arrive at a scheme that is both highly computationally efficient and easy to implement. To illustrate the performance of our algorithm, we compare it with a state-of-the-art linear programming (LP) solver. We achieve good approximations to the solution at least one order of magnitude faster than the LP solver. Finally, we showcase the methodology on a traffic routing problem with a large number of commodities. Funding: This work was supported by KTH Digital Futures, Knut och Alice Wallenbergs Stiftelse [Grants KAW 2018.0349, KAW 2021.0274, the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation], Vetenskapsrådet [Grant 2020-03454], and the National Science Foundation [Grants 1942523 and 2206576].

The equitable dissemination of COVID-19 vaccinations in lower-income nations presents a crucial worldwide dilemma, necessitating smart allocation techniques to guarantee impartial and effective coverage. This paper introduces a mathematical approach that utilizes Linear Programming (LP) to efficiently distribute vaccination doses among lower-income countries. The model incorporates crucial variables such population magnitude, healthcare infrastructure capability, susceptibility to the virus, financial limitations, and worldwide vaccine availability. The linear programming (LP) model is constructed using decision variables that indicate the quantity of vaccination doses assigned to each country. The goal function seeks to optimize the fair allocation of resources by giving weights to countries according to certain criteria. Constraints are imposed to ensure compliance with global vaccine supply, population coverage targets, healthcare capacity, vulnerability concerns, and budget limits. The non-negativity constraints guarantee that the vaccination allocations will always be greater than or equal to zero. The model is designed to be flexible, taking into account dynamic factors such as emerging viral strains and fluctuating vaccination availability. It highlights the significance of timely data updates to facilitate decision-making. The LP solver, such as the simplex method or interior-point approaches, is used to determine the best solution that achieves a fair distribution while considering logistical limitations. The results obtained from the linear programming (LP) model offer practical and effective guidance for policymakers and health organizations. These insights help in distributing COVID-19 vaccinations in a way that maximizes their impact and reduces inequalities among lower-income countries. The approach recognizes the interdependence of global health and emphasizes the need for coordinated endeavors in addressing the pandemic. This mathematical model is a useful tool in the ongoing efforts to distribute COVID-19 vaccinations fairly and efficiently. It contributes to the worldwide aim of obtaining universal immunity and overcoming the problems posed by the epidemic in lower-icnome countries.

The row pivoting method for linear programming

Trajectory optimization for lunar rover performing vertical takeoff vertical landing maneuvers in the presence of terrain

Differential drive robots are widely used in various scenarios thanks to their straightforward principle, from household service robots to disaster response field robots. The nonholonomic dynamics and possible lateral slip of these robots lead to difficulty in getting feasible and high-quality trajectories. Although there are several types of driving mechanisms for real-world applications, they all share a similar driving principle, which involves controlling the relative motion of independently actuated tracks or wheels to achieve both linear and angular movement. Therefore, a comprehensive trajectory optimization to compute trajectories efficiently for various kinds of differential drive robots is highly desirable. In this paper, we propose a universal trajectory optimization framework, enabling the generation of high-quality trajectories within a restricted computational timeframe for these robots. We introduce a novel trajectory representation based on polynomial parameterization of motion states or their integrals, such as angular and linear velocities, which inherently matches the robots’ motion to the control principle. The trajectory optimization problem is formulated to minimize computation complexity while prioritizing safety and operational efficiency. We then build a full-stack autonomous planning and control system to demonstrate its feasibility and robustness. We conduct extensive simulations and real-world testing in crowded environments with three kinds of differential drive robots to validate the effectiveness of our approach.Note to Practitioners—The Differential drive robot, known for its simple mechanics and high maneuverability, is widely used in many applications. However, current methods have limitations in practice when high-performance motion is needed. Due to the state representation in Cartesian space, path planning makes it difficult to consider nonholonomic constraints directly. The existing trajectory optimization cannot effectively constrain the angular velocity and it is difficult to model forward and backward motion into a continuous trajectory. This paper provides a novel trajectory representation that inherently utilizes the motion performance of differential drive robots, which ensures its universality for different platforms, and reduces the time required to generate trajectories to ensure real-time performance. Based on this, we propose a robust planning and control framework to achieve efficient navigation. We release the source code at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://zju-fast-lab.github.io/DDR-opt/</uri> facilitating expansion and deployment for practitioners. We validate this framework through extensive experiments, demonstrating its capability to navigate challenging environments.

In this work, we propose a trajectory optimization approach for robot navigation in cluttered 3D environments. We represent the robot's geometry as a semialgebraic set defined by polynomial inequalities such that robots with general shapes can be suitably characterized. We exploit the collision-free space directly to construct a graph of free regions, search for the reference path, and allocate each waypoint on the trajectory to a specific region. Then, we incorporate a uniform scaling factor for each free region and formulate a Sums-of-Squares (SOS) optimization problem whose optimal solutions reveal the containment relationship between robots and the free space. The SOS optimization problem is further reformulated to a semidefinite program (SDP), and the collision-free constraints are shown to be equivalent to limiting the scaling factor along the entire trajectory. Next, to solve the trajectory optimization problem with the proposed safety constraints, we derive a guiding direction for updating the robot configuration to decrease the minimum scaling factor by calculating the gradient of the Lagrangian at the primal-dual optimum of the SDP. As a result, this seamlessly facilitates the use of gradient-based methods in efficient solving of the trajectory optimization problem. Through a series of simulations and real-world experiments, the proposed trajectory optimization approach is validated in various challenging scenarios, and the results demonstrate its effectiveness in generating collision-free trajectories in dense and intricate environments.

Trajectory Optimization for Planetary Multi-Point Powered Landing