Constrained Trajectory Optimization Research Articles

Constrained trajectory optimization and force control for UAVs with universal jamming grippers

This study presents a novel framework that integrates the universal jamming gripper (UG) with unmanned aerial vehicles (UAVs) to enable automated grasping with no human operator in the loop. Grounded in the principles of granular jamming, the UG exhibits remarkable adaptability and proficiency, navigating the complexities of soft aerial grasping with enhanced robustness and versatility. Central to this integration is a uniquely formulated constrained trajectory optimization using model predictive control, coupled with a robust force control strategy, increasing the level of automation and operational reliability in aerial grasping. This control structure, while simple, is a powerful tool for various applications, ranging from material handling to disaster response, and marks an advancement toward genuine autonomy in aerial manipulation tasks. The key contribution of this research is the combination of a UG with a suitable control strategy, that can be kept relatively straightforward thanks to the mechanical intelligence built into the UG. The algorithm is validated through numerical simulations and virtual experiments.

Safety Constrained Trajectory Optimization for Completion Time Minimization for UAV Communications

Safety Constrained Trajectory Optimization for Completion Time Minimization for UAV Communications

Gaussian Distribution–Based Control Vector Parameterization Method for Constrained Hypersonic Vehicle Reentry Trajectory Optimization

Gaussian Distribution–Based Control Vector Parameterization Method for Constrained Hypersonic Vehicle Reentry Trajectory Optimization

Hybrid Method for Constrained and Unconstrained Trajectory Optimization of Space Transportation

Hybrid Method for Constrained and Unconstrained Trajectory Optimization of Space Transportation

Subspace Structured Neural Network for Rapid Trajectory Optimization

In this study, finite horizon constrained trajectory optimization is tackled by using Artificial Neural Network with an embedded subspace manifold. The resulting network takes advantage of the reduced dimension search space guided by a bio-inspired motion rule. The input nodes of the network are interpreted as collocation points over the time domain transcribed by a pseudospectral discretization method. The activation function for each node is the inverse of the dynamical system. The weights and biases to be optimized in the network are analogous to the parameters of the motion rule. The network is optimized during training by minimizing an augmented loss function where the constraints are considered penalties. The proposed method is simulated in a collision avoidance trajectory planning problem of a mobile robot with two driving wheels and an attitude slewing maneuver problem of an asymmetric rigid body spacecraft.

Geometrically Constrained Trajectory Optimization for Multicopters

In this article, we present an optimization-based framework for multicopter trajectory planning subject to geometrical configuration constraints and user-defined dynamic constraints. The basis of the framework is a novel trajectory representation built upon our novel optimality conditions for unconstrained control effort minimization. We design linear-complexity operations on this representation to conduct spatial–temporal deformation under various planning requirements. Smooth maps are utilized to exactly eliminate geometrical constraints in a lightweight fashion. A variety of state-input constraints are supported by the decoupling of dense constraint evaluation from sparse parameterization and the backward differentiation of flatness map. As a result, this framework transforms a generally constrained multicopter planning problem into an unconstrained optimization that can be solved reliably and efficiently. Our framework bridges the gaps among solution quality, planning efficiency, and constraint fidelity for a multicopter with limited resources and maneuvering capability. Its generality and robustness are both demonstrated by applications to different flight tasks. Extensive simulations and benchmarks are also conducted to show its capability of generating high-quality solutions while retaining the computation speed against other specialized methods by orders of magnitude.

Vector Trajectory Method for Obstacle Avoidance Constrained Planetary Landing Trajectory Optimization

Considering the requirement of avoiding obstacles in different planetary landing missions, this article presents a generalized method dealing with the obstacle avoidance constrained landing trajectory optimization problem with vectorized strategy, i.e., vector trajectory method (VTM). By introducing a trajectory direction constraint with vectorized description, the character of the obstacle avoidance trajectory can be more specifically depicted. Moreover, by satisfying proper trajectory direction constraint, the lander’s particular performances, such as the observation to the target or probability of a safe flight, can be improved. To this end, the VTM and relevant theorem are first developed, revealing the relationship between the two aspects of obstacle avoidance constraint (i.e., distance constraint and trajectory direction constraint). Then, both aspects of constraints are uniformly transformed as the auxiliary angle magnitude constraint with the vectorized strategy, which significantly simplifies the solving procedure of obstacle avoidance constrained trajectory optimization problem. Finally, the proposed VTM is illustrated in detail through the examples with the planetary landing background of atmospheric entry and powered descent landing.

Robust Multi-Robot Trajectory Optimization Using Alternating Direction Method of Multiplier

Wepropose a variant of alternating direction method of multiplier (ADMM) to solve constrained trajectory optimization problems. Our ADMM framework breaks a joint optimization into small sub-problems, leading to a low iteration cost and decentralized parameter updates. Starting from a collision-free initial trajectory, our method inherits the theoretical properties of primal interior point method (P-IPM), i.e., guaranteed collision avoidance and homotopy preservation throughout optimization, while being orders of magnitude faster. We have analyzed the convergence and evaluated our method for time-optimal multi-UAV trajectory optimizations and simultaneous goal-reaching of multiple robot arms, where we take into consider kinematics-, dynamics-limits, and homotopy-preserving collision constraints. Our method highlights an order of magnitude’s speedup, while generating trajectories of comparable qualities as state-of-the-art P-IPM solver.

Constrained Trajectory Optimization With Flexible Final Time for Autonomous Vehicles

Differential dynamic programming (DDP) is a well-recognized method for trajectory optimization due to its fast convergence characteristics. However, the original DDP requires a predefined final time and cannot handle nonlinear constraints in optimization. This prohibits the application of DDP for autonomous vehicles due to the heuristic nature of setting a final time beforehand and the existence of inherent physical limits. This article revisits DDP by dynamically optimizing the final time via the first-order optimality condition of the value function and using augmented Lagrangian method to tackle the issue of nonlinear constraints. The resultant algorithm is termed as flexible final time constrained differential dynamic programming (FFT-CDDP). Extensive numerical simulations for a three-dimensional guidance problem are used to demonstrate the working of FFT-CDDP. The results indicate that the proposed FFT-CDDP provides much higher computational efficiency and stronger robustness against the initial solution guess, compared with the off-the-shelf general purpose optimal control software (GPOPS) algorithm.

Spatially Formulated Connected Automated Vehicle Trajectory Optimization with Infrastructure Assistance

This paper presents a constrained connected automated vehicles (CAVs) trajectory optimization method on curved roads with infrastructure assistance. Specifically, this paper systematically formulates trajectory optimization problems in a spatial domain and a curvilinear coordinate. As an alternative of temporal domain and Cartesian coordinate formulation, our formulation provides the constrained trajectory optimization flexibility to describe complex road geometries, traffic regulations, and road obstacles, which are usually spatially varying rather than temporal varying, with assistances vehicle to infrastructure (V2I) communication. Based on the formulation, we first conducted a mathematical proof on the controllability of our system, to show that our system can be controlled in the spatial domain and curvilinear coordinate. Further, a multiobjective model predictive control (MPC) approach is designed to optimize the trajectories in a rolling horizon fashion and satisfy the collision avoidances, traffic regulations, and vehicle kinematics constraints simultaneously. To verify the control efficiency of our method, multiscenario numerical simulations are conducted. Suggested by the results, our proposed method can provide smooth vehicular trajectories, avoid road obstacles, and simultaneously follow traffic regulations in different scenarios. Moreover, our method is robust to the spatial change of road geometries and other potential disturbances by the road curvature, work zone, and speed limit change.

Time-varying formation dynamics modeling and constrained trajectory optimization of multi-quadrotor UAVs

The formation of multi-quadrotor UAVs (QUAVs) in long flights will encounter many constraints and obstacles in the flight process, so it is necessary to change the formation shape to avoid these constraints. When the flight path of QUAVs is individually optimized, multiple individual dynamic problems will be faced, making the solution complicated and spending a long time. In this paper, as a whole instead of individuals considering the QUAVs, the formation is regarded as a rigid body using the Voronoi graph theory method in the flight process. The constraints of the followed QUAVs are transformed into the constraints of leader QUAV. Therefore, the formation is transformed for different constraints by changing the virtual rigid body structure or shape. A time-varying model is established to facilitate the use of optimization. The energy consumption of the leader UAV within the specified time is minimized as the trajectory optimization objective. The optimization improvement of the end heading and height constraints is proposed. A trajectory optimization method for the leader QUAV based on Gauss pseudospectral method is proposed, transforming the original optimal control problem into the nonlinear programming one. The simulation results demonstrate and verify the proposed method.

State-Space Characterization of Balance Capabilities in Biped Systems with Segmented Feet

The human ability of keeping balance during various locomotion tasks is attributed to our capability of withstanding complex interactions with the environment and coordinating whole-body movements. Despite this, several stability analysis methods are limited by the use of overly simplified biped and foot structures and corresponding contact models. As a result, existing stability criteria tend to be overly restrictive and do not represent the full balance capabilities of complex biped systems. The proposed methodology allows for the characterization of the balance capabilities of general biped models (ranging from reduced-order to whole-body) with segmented feet. Limits of dynamic balance are evaluated by the Boundary of Balance (BoB) and the associated novel balance indicators, both formulated in the Center of Mass (COM) state space. Intermittent heel, flat, and toe contacts are enabled by a contact model that maps discrete contact modes into corresponding center of pressure constraints. For demonstration purposes, the BoB and balance indicators are evaluated for a whole-body biped model with segmented feet representative of the human-like standing posture in the sagittal plane. The BoB is numerically constructed as the set of maximum allowable COM perturbations that the biped can sustain along a prescribed direction. For each point of the BoB, a constrained trajectory optimization algorithm generates the biped’s whole-body trajectory as it recovers from extreme COM velocity perturbations in the anterior–posterior direction. Balance capabilities for the cases of flat and segmented feet are compared, demonstrating the functional role the foot model plays in the limits of postural balance. The state-space evaluation of the BoB and balance indicators allows for a direct comparison between the proposed balance benchmark and existing stability criteria based on reduced-order models [e.g., Linear Inverted Pendulum (LIP)] and their associated stability metrics [e.g., Margin of Stability (MOS)]. The proposed characterization of balance capabilities provides an important benchmarking framework for the stability of general biped/foot systems.

Solving Constrained Trajectory Planning Problems Using Biased Particle Swarm Optimization

Constrained trajectory optimization has been a critical component in the development of advanced guidance and control systems. An improperly planned reference trajectory can be a main cause of poor online control performance. Due to the existence of various mission-related constraints, the feasible solution space of a trajectory optimization model may be restricted to a relatively narrow corridor, thereby easily resulting in local minimum or infeasible solution detection. In this article, we are interested in making an attempt to handle the constrained trajectory design problem using a biased particle swarm optimization approach. The proposed approach reformulates the original problem to an unconstrained multicriterion version by introducing an additional normalized objective reflecting the total amount of constraint violation. Besides, to enhance the progress during the evolutionary process, the algorithm is equipped with a local exploration operation, a novel ε-bias selection method, and an evolution RS. Numerical simulation experiments, obtained from a constrained atmospheric entry trajectory optimization example, are provided to verify the effectiveness of the proposed optimization strategy. Main advantages associated with the proposed method are also highlighted by executing a number of comparative case studies.

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.

Trajectory optimization for Rendezvous and Docking using Nonlinear Model Predictive Control

In this paper, MPC based constrained trajectory optimization algorithms has been applied for rendezvous and docking of two small satellites to achieve precise terminal soft docking with path constraints. A receding horizon MPC has been applied in order to reduce the computational load. The constraints such as thrust magnitude limit, chaser spacecraft positioning with in Line-of-Sight (LOS) while approaching the target spacecraft, and terminal velocity limit for soft-docking are considered. In addition debris avoidance constraints are considered and obstacles are assumed as spheres for simplicity. Nonlinear MPC optimization problem with constraints is solved by Sequential Quadratic Program (SQP) technique. The robustness of this control has been verified using nonlinear dynamics with perturbation model.

Approximate convex hull based scenario truncation for chance constrained trajectory optimization

In this paper, we study chance constrained trajectory optimization of linear systems with general ellipsoidal and polytopic state-input constraints where the constraints must be met with some prescribed confidence level. We use the sampling techniques, specifically scenario approach, due to their generality and tractability compared to the analytical methods. To address the main drawback of scenario approach, which may require large number of samples, we introduce an approximate convex hull-based method to significantly reduce the number of samples. Based on the allowable computational complexity, the prominent samples are selected in a proper mapping and the rest are truncated. The truncation error is later compensated for by adjusting (buffering) the constraint set, so that the satisfaction of constraints with the desired confidence level is still guaranteed. Simulation results confirm the theoretical predictions with solid performance of the proposed method after discarding about 99% of samples from the scenario approach, which remarkably speeds up the online computations.

Trajectory Optimization for Target Localization With Bearing-Only Measurement

This paper considers the problem of two-dimensional constrained trajectory optimization of a point-mass aerial robot for constant-maneuvering target localization using bearing-only measurement. A performance metric that can be utilized in trajectory optimization to maximize target observability is proposed first based on geometric conditions. One-step optimal maneuver that maximizes the observability criterion is then derived analytically for moving targets. The heading angle constraint is also incorporated in the proposed optimal maneuver derivation to support practical application. Numerical simulations with some comparisons are presented to validate the analytical findings.

Trajectory Planning of the Unmanned Aerial Vehicles with Adaptive Convex Optimization Method

Abstract In this paper, a constrained unmanned aerial vehicles trajectory optimization problem is formulated. We present an iterative convex programming algorithm to solve the optimal control problem, which consists of both nonlinear dynamics and path constraints. A hp-adaptive pseudospectral method is introduced to transform the original problem into nonlinear programming. Furthermore, in order to improve the computational ability, nonlinear dynamics and path constraints are convexified by convexification techniques. The trajectory planning problem is formulated as a convex programming problem. We evaluate the proposed strategy by some numerical results. The results indicate that the algorithm can get better performance compared with other typical optimal control solvers.

Solving Multiobjective Constrained Trajectory Optimization Problem by an Extended Evolutionary Algorithm.

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.

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.