Nonlinear Trajectory Optimization Research Articles

Convex Approach to Real-Time Multiphase Trajectory Optimization for Urban Air Mobility

Electric vertical takeoff and landing technology has emerged as a promising solution to alleviate ground transportation congestion. However, the limited capacity of onboard batteries enforces hard time constraints for vehicle operations in a complex urban environment. Therefore, a constrained, real-time trajectory optimization method is necessary to enable safe and energy-efficient vehicle flight. Unfortunately, most existing trajectory optimization methods suffer from low computational efficiency, unpredictable convergence processes, and strong dependence on a good initial trajectory. In this paper, we employ a successive convex programming approach to solve the multiphase minimum-energy-cost electric vertical takeoff and landing vehicle trajectory optimization problem for urban air mobility applications. The main contribution is the development and validation of two novel convexification methods that find approximate optimal solutions to the multiphase nonlinear trajectory optimization problem in real time by solving a sequence of convex subproblems. Specifically, the first method transforms the original nonlinear problem into a sequence of second-order cone programming problems through a convenient change of variables and lossless convexification, while the second approach achieves a similar goal without the need for control convexification. The resulting convex subproblems can be solved reliably in real time by advanced interior point methods. The proposed methods are demonstrated through numerical simulations of two different urban air mobility scenarios and compared with the solution obtained from GPOPS-II, a state-of-the-art general-purpose optimal control solver. Our proposed sequential convex programming methods can obtain near-optimal solutions with faster computational speed than GPOPS-II.

New Adaptive Mesh Refinement Strategy for Entry Guidance via Sequential Convex Programming

This study focuses on the entry guidance problem, which is a highly nonlinear and constrained optimal control problem. The entry guidance problem is formulated as a nonlinear trajectory optimization problem with six-degrees-of-freedom dynamics, path constraints, and boundary constraints. The formulated problem is discretized and solved using sequential convex programming (SCP) with successive linearization. However, during the discretization process, conventional collocation methods encounter a challenge with high-frequency jittering in the control and bank angle profiles near active path constraints. Because the considered problem especially involves tight path constraints, the solution jittering issue becomes severe. Therefore, this study proposes a novel mesh refinement strategy to address this problem. Unlike conventional mesh refinement strategies based on interpolation error, the proposed approach resolves the convergence and solution jittering issue of the SCP framework by linking the direct and indirect methods with the costate estimation under successive linearization. Along with the development, we introduce a novel costate estimation method and provide proof of its validity. We present numerical simulation results of the proposed method and conduct a comparison with conventional approaches. The results obtained demonstrate successful mitigation of the solution jittering issue and a significant improvement in solution accuracy regarding compliance with the necessary conditions of optimality.

Descent Property in Sequential Second-Order Cone Programming for Nonlinear Trajectory Optimization

Sequential second-order cone programming (SSOCP) is commonly used in aerospace applications for solving nonlinear trajectory optimization problems. The SSOCP possesses good real-time performance. However, one long-standing challenge is its unguaranteed convergence. In this paper, we theoretically analyze the descent property of the penalty function in the SSOCP. Using Karush–Kuhn–Tucker conditions, we obtain two important theoretical results: 1) the penalty function of the original nonlinear problem always descends along the iteration direction; 2) a sufficiently small trust region can decrease the penalty function. Based on these two results, we design an improved trust region shrinking algorithm with theoretically guaranteed convergence. In numerical simulations, we verify the proposed algorithm using a reentry trajectory optimization problem.

Distributed safe trajectory optimization for large-scale spacecraft formation reconfiguration

This paper presents a distributed trajectory optimization algorithm for the reconfiguration of large-scale spacecraft formation, which uses a low thrust propulsion system to safely guide the spacecraft formation to a desired formation. Formation reconfiguration is formulated as a trajectory optimization problem with complex constraints, and the relative motion is accurately described by a nonlinear relative dynamics considering J2 perturbation. The collision avoidance constraint is convexified into a tangent plane to ensure the accuracy of the solution. The resulting nonlinear trajectory optimization problem is solved by applying the hp-adaptive pseudospectral method to convert it into a nonlinear programming problem. In order to overcome the disadvantage of huge computation of centralized algorithm considering collision avoidance, “predicted trajectory” is introduced to transmit information between spacecraft, and the parallel computing is implemented. Finally, a numerical simulation is given to verify the computational efficiency and collision avoidance effectiveness of the proposed distributed algorithm.

Distributed Nonlinear Trajectory Optimization for Multi-Robot Motion Planning

This work presents a method for multi-robot coordination based on a novel distributed nonlinear model predictive control (NMPC) formulation for trajectory optimization and its modified version to mitigate the effects of packet losses and delays in the communication among the robots. Our algorithms consider that each robot is equipped with an onboard computation unit to solve a local control problem and communicate with neighboring autonomous robots via a wireless network. The difference between the two proposed methods is in the way the robots exchange information to coordinate. The information exchange can occur in a following: 1) synchronous or 2) asynchronous fashion. By relying on the theory of the nonconvex alternating direction method of multipliers (ADMM), we show that the proposed solutions converge to a (local) solution of the centralized problem. For both algorithms, the communication exchange preserves the safety of the robots; that is, collisions with neighboring autonomous robots are prevented. The proposed approaches can be applied to various multi-robot scenarios and robot models. In this work, we assess our methods, both in simulation and with experiments, for the coordination of a team of autonomous vehicles in the following: 1) an unsupervised intersection crossing and 2) the platooning scenarios.

Hierarchical Velocity Control Considering Traffic Signal Timings for Connected Vehicles

Velocity profile optimization using upcoming signal information of urban vehicles is one of the main eco-driving techniques, which can significantly enhance the capability of reducing idling fuel loss and trip time. This paper proposes a hierarchical velocity control system considering different drive preferences for connected and automated vehicles to improve overall efficiency under the vehicle-to-X (V2X) environment. Two major contributions are (i) control-oriented formulation of task planning in multiple road segments, which is amiable to numerical computation and can consider different drive preferences; and (ii) solving the nonlinear velocity trajectory optimization to realize real-time control in the lower level. Simulation results indicate that energy-saving and computational efficiency are improved using the proposed control system. The solution algorithm is further demonstrated under a hardware-in-the-loop simulation.

Efficient online motion planning method for the robotic arm to pick-up moving objects smoothly with temporal constraints

This paper proposes a new efficient online motion planning method for the manipulator to grasp moving objects smoothly. The algorithm framework consists of front-end pathfinding and back-end nonlinear trajectory optimization. The sample-based pathfinding algorithm can predict the intersection location and find a safe initial path in the dynamic environment. The gradient-based continuous-time trajectory optimization method converts the Euclidean Signed Distance Field information into the joint space, uses the B-spline curve to represent the joint trajectories, optimizes the initial path using convex hull characteristics of B-spline, and generates a smooth and kinematics feasible trajectory in joint space. Finally, we use an adjustment method of iterative trajectory fragment length to guarantee kinodynamic feasibility. Simulation comparisons and real-world experiments verify our planning framework’s high performance.

Learning to solve sequential physical reasoning problems from a scene image

In this article, we propose deep visual reasoning, which is a convolutional recurrent neural network that predicts discrete action sequences from an initial scene image for sequential manipulation problems that arise, for example, in task and motion planning (TAMP). Typical TAMP problems are formalized by combining reasoning on a symbolic, discrete level (e.g., first-order logic) with continuous motion planning such as nonlinear trajectory optimization. The action sequences represent the discrete decisions on a symbolic level, which, in turn, parameterize a nonlinear trajectory optimization problem. Owing to the great combinatorial complexity of possible discrete action sequences, a large number of optimization/motion planning problems have to be solved to find a solution, which limits the scalability of these approaches. To circumvent this combinatorial complexity, we introduce deep visual reasoning: based on a segmented initial image of the scene, a neural network directly predicts promising discrete action sequences such that ideally only one motion planning problem has to be solved to find a solution to the overall TAMP problem. Our method generalizes to scenes with many and varying numbers of objects, although being trained on only two objects at a time. This is possible by encoding the objects of the scene and the goal in (segmented) images as input to the neural network, instead of a fixed feature vector. We show that the framework can not only handle kinematic problems such as pick-and-place (as typical in TAMP), but also tool-use scenarios for planar pushing under quasi-static dynamic models. Here, the image-based representation enables generalization to other shapes than during training. Results show runtime improvements of several orders of magnitudes by, in many cases, removing the need to search over the discrete action sequences.

Robust optimal transition maneuvers control for tail‐sitter unmanned aerial vehicles

AbstractOptimal transition has been widely studied for tail‐sitter unmanned aerial vehicles (UAVs). However, since the derived optimal control can only be applied in an open‐loop form, unmodeled dynamics and external disturbances cannot be restrained. Aiming at this problem, this article presents a method to obtain optimal transition maneuvers in the presence of these uncertainties. The approach includes two steps: nonlinear trajectory optimization and robust trajectory tracking. For the trajectory optimization, the minimum altitude variation front and back transition trajectories are derived. Different from existing optimal transition studies, considerable actuator margins are reserved in this step. Thus the tail‐sitter possesses control efforts to perform closed‐loop corrections for uncertainties. For the trajectory tracking, a robust controller in the form of feedforward plus feedback is proposed. The feedforward term directly employs the derived optimal control to improve the dynamic response of the system. The feedback term adopts linear control and acceleration‐based compensator to place desired poles and restrain all uncertainties. To the authors' knowledge, this article is the first time to study the tail‐sitter robust transition control method considering the transition optimality and actuator margins. It is proved that the tracking errors converge into a specified neighborhood of the origin in a finite time. Simulation studies corroborate the performance of the optimal transition trajectories and the robust trajectory tracking controller.

Online Reentry Trajectory Optimization Using Modified Sequential Convex Programming for Hypersonic Vehicle

In this article, a highly nonlinear trajectory optimization problem for reentry vehicles is rapidly solved by the proposed modified sequential convex programming (MSCP) method. A logarithm linearization approach is proposed to decouple the dynamics of high order and nonconvex path constraints of heat rate, dynamic pressure, and normal load. Next, the model including the no-fly zone constraint, and the nonconvex objective function are convexified with first-order Taylor series expansion. Subsequently, the continuous-time optimal problem is converted to an equivalent finite-dimensional sequential convex programming (SCP) problem. Moreover, a compensation term is added to the state equations to maintain the feasibility of the reformulated problem. Consequently, to guarantee the optimality of the solution, a penalty term with respect to the compensation term is added to the objective function. In the end, an online optimization scheme with MSCP is proposed. Compared to a general proposed optimal control solver, numerical simulations are conducted to verify the optimality and fast convergence of MSCP. Furthermore, online optimization simulations demonstrate the validity of the proposed online scheme for pop-up no-fly zone and mission temporary changes.

Jumping over obstacles with MIT Cheetah 2

This paper presents a planning framework for jumping over obstacles with quadruped robots. The framework accomplishes planning via a structured predictive control strategy that combines the use of heterogeneous simplified models over different prediction time scales. A receding multi-horizon predictive controller coordinates the approach before the jump using a kinematic point-mass model. Consideration of the optimal value function over different planning horizons enables the system to select an appropriate number of steps to take before jumping. The jumping motion is then tailored to the sensed obstacle by solving a nonlinear trajectory optimization problem. The solution of this problem online is enabled by exploiting the analyticity of the flow map for a planar bounding template model under polynomial inputs. By planning with this combination of models, MIT Cheetah 2 is shown to autonomously jump over obstacles up to 40 cm in height during high-speed bounding. Untethered results showcase the ability of the method to automatically adapt to obstacles of different heights and placements in a single trial.

Tube Stochastic Optimal Control for Nonlinear Constrained Trajectory Optimization Problems

Recent low-thrust space missions have highlighted the importance of designing trajectories that are robust against uncertainties. In its complete form, this process is formulated as a nonlinear constrained stochastic optimal control problem. This problem is among the most complex in control theory, and no practically applicable method to low-thrust trajectory optimization problems has been proposed to date. This paper presents a new algorithm to solve stochastic optimal control problems with nonlinear systems and constraints. The proposed algorithm uses the unscented transform to convert a stochastic optimal control problem into a deterministic problem, which is then solved by trajectory optimization methods such as differential dynamic programming. Two numerical examples, one of which applies the proposed method to low-thrust trajectory design, illustrate that it automatically introduces margins that improve robustness. Finally, Monte Carlo simulations are used to evaluate the robustness and optimality of the solution.

Multi-sensor fusion localization algorithm for outdoor mobile robot

In this paper, the multi-sensor fusion positioning technology of outdoor mobile robot is studied, and a real-time outdoor positioning algorithm combining GPS and lidar odometry is proposed. Lidar odometry based on 3D lidar for point cloud feature matching is realized. At the same time, a covariance matrix reflecting the local positioning information of the lidar odometry is constructed in real time according to the matching distance and the absolute median difference, and the accuracy factor, the circular probability error, etc. are used as the standard GPS. The positioning effect is evaluated to measure the respective positioning effects of the lidar odometry and GPS. On this basis, the GPS is used as the global factor in the pose structure, and the lidar odometry is used as the local factor to construct the pose observation constraint, and the global nonlinear trajectory optimization is carried out to realize the outdoor real-time positioning of the mobile robot. Experimental results and data analysis verify the effectiveness and practicability of the proposed method.

An Iterative Method Based on the Marginalized Particle Filter for Nonlinear B-Spline Data Approximation and Trajectory Optimization

The B-spline function representation is commonly used for data approximation and trajectory definition, but filter-based methods for NWLS approximation are restricted to a bounded definition range. We present an algorithm termed NRBA for an iterative NWLS approximation of an unbounded set of data points by a B-spline function. NRBA is based on a MPF, in which a KF solves the linear subproblem optimally while a PF deals with nonlinear approximation goals. NRBA can adjust the bounded definition range of the approximating B-spline function during run-time such that, regardless of the initially chosen definition range, all data points can be processed. In numerical experiments, NRBA achieves approximation results close to those of the Levenberg–Marquardt algorithm. An NWLS approximation problem is a nonlinear optimization problem. The direct trajectory optimization approach also leads to a nonlinear problem. The computational effort of most solution methods grows exponentially with the trajectory length. We demonstrate how NRBA can be applied for a multiobjective trajectory optimization for a BEV in order to determine an energy-efficient velocity trajectory. With NRBA, the effort increases only linearly with the processed data points and the trajectory length.

Rapid trajectory optimization for hypersonic entry using a pseudospectral-convex algorithm

Aiming at improving the autonomy of hypersonic entry vehicles, a rapid trajectory optimization algorithm, which has the potential to be implemented online and onboard, is proposed in this paper. The nonlinear and nonconvex hypersonic entry trajectory optimization problem is transformed into a series of convex subproblems through a proper combination of the pseudospectral method and an improved successive convexification method; thus, the high discretization accuracy of the pseudospectral method and the fast and deterministic convergence properties of the convex-optimization-based algorithm can be simultaneously exploited. The resulting subproblems can be solved efficiently by matured interior-point methods, and the solution converges rapidly by adopting a novel dynamic trust-region updating approach. The optimality of the solution is verified by the optimal control theory. The effectiveness of the algorithm is demonstrated by numerical experiments.

Solving Trajectory Optimization Problems in the Presence of Probabilistic Constraints.

The objective of this paper is to present an approximation-based strategy for solving the problem of nonlinear trajectory optimization with the consideration of probabilistic constraints. The proposed method defines a smooth and differentiable function to replace probabilistic constraints by the deterministic ones, thereby converting the chance-constrained trajectory optimization model into a parametric nonlinear programming model. In addition, it is proved that the approximation function and the corresponding approximation set will converge to that of the original problem. Furthermore, the optimal solution of the approximated model is ensured to converge to the optimal solution of the original problem. Numerical results, obtained from a new chance-constrained space vehicle trajectory optimization model and a 3-D unmanned vehicle trajectory smoothing problem, verify the feasibility and effectiveness of the proposed approach. Comparative studies were also carried out to show the proposed design can yield good performance and outperform other typical chance-constrained optimization techniques investigated in this paper.

Review of dynamic soaring: technical aspects, nonlinear modeling perspectives and future directions

In this paper, we present a comprehensive and detailed review of dynamic soaring process, and in particular, its application to unmanned aerial vehicles (UAVs). We start by explaining the biological inspiration that comes from soaring birds and how researchers have tried to utilize the dynamic soaring phenomenon/maneuver and apply it to UAVs. We present and discuss the fundamentals of wind shear models in both the linear and nonlinear cases. Moreover, a comprehensive parametric characterization of the key performance parameters for the dynamic soaring maneuver is given. Numerical methods for nonlinear trajectory optimization are summarized and methodologies capable of generating rapid solutions suitable for real-time implementation, are presented. Additionally, the paper introduces mathematical modeling and procedure to generate the optimized dynamic soaring trajectory. Through this paper, a consolidated platform is built, which not only covers technical aspects of advancements made over the passage of time, but also identifies and discusses the existing challenges. These challenges which are encountered by UAVs curtail the potential utility of dynamic soaring. Integrating dynamic soaring with morphology and inclusion of nonlinear control theory in the flight control system are introduced as a possible future research directions that may overcome the existing limitations.

Reentry Trajectory Optimization using Gradient Free Algorithms

In this work, a study on atmospheric reentry trajectory optimization using gradient free optimization techniques is presented. Genetic Algorithm (GA), Pattern Search (PS) and Simulated Annealing (SA) approaches are explored for the considered nonlinear constrained trajectory optimization problem. Aerodynamic performance indicator, (L/D), is used as a control variable to minimize the objective function subject to various terminal and in-flight path constraints. Results and findings on the performance comparison of the considered methods are discussed.

Entry Trajectory Optimization by Second-Order Cone Programming

Convex optimization has found wide applications in recent years due to its unique theoretical advantages and the polynomial-time complexity of state-of-the-art solution algorithms for convex programming. This paper represents an attempt to apply second-order cone programming, a branch of convex optimization, to the class of highly nonlinear trajectory optimization problems in entry flight. The foremost challenge in applying convex optimization in most aerospace engineering problems lies in the nonlinearity and nonconvexity of the problem. Exclusive reliance on linearization does not always work well, as is the case in entry trajectory optimization. This paper focuses on how to formulate realistic, highly constrained entry trajectory optimization problems in a fashion suitable to be solved by second-order cone programming with a combination of successive linearization and relaxation techniques. Rigorous analysis is conducted to support the soundness of the approach. Numerical demonstrations are provided to show the efficacy and effectiveness of the proposed method.

Physics-model-based nonlinear actuator trajectory optimization and safety factor profile feedback control for advanced scenario development in DIII-D

DIII-D experimental results are reported to demonstrate the potential of physics-model-based safety factor profile control for robust and reproducible sustainment of advanced scenarios. In the absence of feedback control, variability in wall conditions and plasma impurities, as well as drifts due to external disturbances, can limit the reproducibility of discharges with simple pre-programmed scenario trajectories. The control architecture utilized is a feedforward + feedback scheme where the feedforward commands are computed off-line and the feedback commands are computed on-line. In this work, a first-principles-driven (FPD), physics-based model of the q profile and normalized beta () dynamics is first embedded into a numerical optimization algorithm to design feedforward actuator trajectories that steer the plasma through the tokamak operating space to reach a desired stationary target state that is characterized by the achieved q profile and . Good agreement between experimental results and simulations demonstrates the accuracy of the models employed for physics-model-based control design. Second, a feedback algorithm for q profile control is designed following an FPD approach, and the ability of the controller to achieve and maintain a target q profile evolution is tested in DIII-D high confinement (H-mode) experiments. The controller is shown to be able to effectively control the q profile when is relatively close to the target, indicating the need for integrated q profile and control to further enhance the ability to achieve robust scenario execution. The ability of an integrated q profile + feedback controller to track a desired target is demonstrated through simulation.