A Machine-Learning Approach for Time-Optimal Trajectory Generation for UAV's

Time-optimal trajectory planning for mechanical systems has been widely studied thus far because of its effectiveness to reduce the time required for many industrial tasks. However, in general, because the time-optimal trajectory is generated based on ideal dynamics of controlled systems, it may be difficult to implement the obtained trajectory to actual systems with vibration modes that are neglected in the trajectory generation. In other words, high-speed motion based on the time-optimal trajectory may cause the vibration of mechanical systems. We propose to employ the contouring control, which enables to reduce controller gain magnitudes while contouring performance are maintained, for the presents a method of generating the time-optimal trajectory for machine tool feed drive systems. Next, the contouring controller is applied to implement the optimal trajectory to the actual machine tool system. The effectiveness of the contouring controller for implementation of the optimal trajectory is demonstrated by comparative experiments with the conventional method.

In this work, a time-optimal trajectory generation approach is developed for the multiple way-point navigation of the quadrotor based on the nonuniform rational B-spline (NURBS) curve and linear programming. To facilitate this development, the dynamic model of the quadrotor is formulated first. Then, the geometric trajectory regarding multiple way-point navigation is constructed based on the NURBS curve. With the constructed geometric trajectory, a time-optimal interpolation problem is imposed considering the velocity, acceleration, and jerk constraints. This optimization problem is solved in two steps. In the first step, a preliminary result is obtained by solving a linear programming problem without jerk constraints. Then by introducing properly relaxed jerk constraints, a second linear programming problem is formulated based on the preliminarily obtained result, and the time-optimal problem can be fully solved in this way. Subsequently, a nonlinear trajectory tracking controller is developed to track the generated trajectory. The feasibilities of the proposed trajectory generation approach as well as the tracking controller are verified through both simulations and real-time experiments. With enhanced computational efficiency, the proposed approach can generate trajectory for an indoor environment with the smooth acceleration profile and moderate velocity V≈1 m/s in real-time, while guaranteeing velocity, acceleration, and jerk constraints: Vmax=1 m/s, Amax=2 m/s2, and Jmax=5 m/s3. In such a case, the trajectory tracking controller can closely track the reference trajectory with cross-tracking error less than 0.05 m.

The specific control problem addressed in the paper is the design of controllers that ensure asymptotic tracking of output reference trajectories for overactuated systems. The redundancy in the input space is used to optimize the steady-state performance of the closed-loop system. Since the considered plant models are not square, the control input and state trajectories in steady state that enforce perfect tracking are not unique. The proposed methodology employs a continuous parameterization of all possible steady state input trajectories, that is used for optimizing the selection of the steady state input, while providing perfect tracking and fulfillment of constraints on the magnitude of the control input. An application to altitude and velocity tracking control for an hypersonic vehicle employing air-breathing (scramjet) propulsion is presented and discussed

Challenges in motion planning for multiple quadrotors in complex environments lie in overall flight efficiency and the avoidance of obstacles, deadlock, and collisions among themselves. In this paper, we present a gradient-free trajectory generation method for multiple quadrotors in dynamic obstacle-dense environments with the consideration of time consumption. A model predictive control (MPC)-based approach for each quadrotor is proposed to achieve distributed and asynchronous cooperative motion planning. First, the motion primitives of each quadrotor are formulated as the boundary state constrained primitives (BSCPs) which are constructed with jerk limited trajectory (JLT) generation method, a boundary value problem (BVP) solver, to obtain time-optimal trajectories. They are then approximated with a neural network (NN), pre-trained using this solver to reduce the computational burden. The NN is used for fast evaluation with the guidance of a navigation function during optimization to guarantee flight safety without deadlock. Finally, the reference trajectories are generated using the same BVP solver. Our simulation and experimental results demonstrate the superior performance of the proposed method.

This work presents a novel motion planning framework, rooted in nonlinear programming theory, that treats uncertain fully and underactuated dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it, and poor robustness and suboptimal performance result if it is not accounted for in a given design. In this work uncertainties are modeled using generalized polynomial chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach enables the inclusion of uncertainty statistics in the nonlinear programming optimization process. As such, the proposed framework allows the user to pose, and answer, new design questions related to uncertain dynamical systems. Specifically, the new framework is explained in the context of forward, inverse, and hybrid dynamics formulations. The forward dynamics formulation, applicable to both fully and underactuated systems, prescribes deterministic actuator inputs that yield uncertain state trajectories. The inverse dynamics formulation is the dual to that of forward dynamics, and is only applicable to fully actuated systems; deterministic state trajectories are prescribed and yield uncertain actuator inputs. The inverse dynamics formulation is more computationally efficient as it requires only algebraic evaluations and completely avoids numerical integration. Finally, the hybrid dynamics formulation is applicable to underactuated systems where it leverages the benefits of inverse dynamics for actuated joints and forward dynamics for unactuated joints; it prescribes actuated state and unactuated input trajectories that yield uncertain unactuated states and uncertain actuated inputs. The benefits of the ability to quantify uncertainty when planning the motion of multibody dynamic systems are illustrated through several case studies. The resulting designs determine optimal motion plans—subject to deterministic and statistical constraints—for all possible systems within the probability space.

SummaryThe ability to learn is crucial for neural network (NN) control as it is able to enhance the overall stability and robustness of control systems. In this study, a composite learning control strategy is proposed for a class of strict‐feedback nonlinear systems with mismatched uncertainties, where raised‐cosine radial basis function NNs with compact supports are applied to approximate system uncertainties. Both online historical data and instantaneous data are utilized to update NN weights. Practical exponential stability of the closed‐loop system is established under a weak excitation condition termed interval excitation. The proposed approach ensures fast parameter convergence, implying an exact estimation of plant uncertainties, without the trajectory of NN inputs being recurrent and the time derivation of plant states. The raised‐cosine radial basis function NNs applied not only reduces computational cost but also facilitates the exact determination of a subregressor activated along any trajectory of NN inputs so that the interval excitation condition is verifiable. Numerical results have verified validity and superiority of the proposed approach.

In this paper a novel approach for aircraft trajectory generation is presented. Based on a pathfinding algorithm in conjunction with a cost function, a concept for a cost-optimized trajectory generator is introduced. The trajectory generator is able to incorporate time constraints along the aircraft’s route, as they will be an integral part of future airspace concepts to overcome the challenges of growing air-traffic. The target value, which is sought to be inimized by the pathfinding algorithm, is the result of the herein described cost function. Contrary to already existing approaches on aircraft trajectory optimization in this conceptual cost function an airline-wide, network centric, cost optimization can be performed, rather than focusing solely on a single aircraft. Evaluation of the cost-sensitive trajectory generator is performed in a fast-time simulation, results are compared to a non-optimizing trajectory generation. Results indicate average cost savings of 0.12% for arrival time variations of �30s at time constrained waypoints and proof general viability of the proposed concept. Further development of the herein presented method is recommended, especially in respect to Overall airline network optimization and disruption management use cases across the various entities of an airline.

Distributed Model Predictive Control with Asymmetric Adaptive Terminal Sets for the Regulation of Large-scale Systems

Higher-order continuous-time implicit systems: Consistency and weak consistency, impulse controllability, geometric concepts, and invertibility properties

This article presents two different approximations to linear infinite-horizon optimal control problems arising in model predictive control. The dynamics are approximated using a Galerkin approach with parametrized state and input trajectories. It is shown that the first approximation represents an upper bound on the optimal cost of the underlying infinite dimensional optimal control problem, whereas the second approximation results in a lower bound. We analyze the convergence of the costs and the corresponding optimizers as the number of basis functions tends to infinity. The results can be used to quantify the approximation quality with respect to the underlying infinite dimensional optimal control problem.

This thesis addresses the design of optimization-based control laws for the case where convergence to a desired set-point, minimization of an arbitrary performance index, or a combination of the two, is the desired objective. The results are developed within the sample-data Model Predictive Control (MPC) framework considering constrained nonlinear continuous-time time-varying dynamical systems. For a given time sampling, a sample-data MPC control strategy consists of i) choosing among all future finite horizon predictions of state and input trajectories of the system the one that minimizes the given performance index, ii) applying the optimal input trajectory to the system until a new time sample is reached, and iii) iterating this process. The performance index is chosen to describe the specific control problem under consideration. In a classic Tracking-MPC framework, where the main goal is to steer the state of the system to a desired steady-state, the performance index is properly chosen to penalize the distance from the current state to a desired one. In order to capture more complex control objectives, in recent years a growing attention has been dedicated to a new class of controllers that goes under the name of Economic-MPC. Here, the term economic is used to stress the fact that the performance index is a general index of interest that we wish to minimize, e.g., economic, which does not denote the distance to a desired set point. This setting makes full use of the potentialities of optimization-based control strategies. Although, it comes with disadvantages. In fact, by choosing an arbitrary performance index, it is difficult to predict, and therefore certify, the evolution of the closed-loop system, which could potentially manifest undersirable behaviors. This thesis provides analysis and certification of a variety of closed-loop behaviors stemming from the use Economic-MPC controllers. A set of tools for design of provably correct MPC controllers is provided for the case where the performance index is of the Tracking-MPC type, purely economic, or a combination of the two. The results focus the certification of both closed-loop economic performance and closed-loop state evolution. The proposed strategies are applied to a range of motion control problems for underactuated vehicles. An MPC controller for Trajectory-Tracking and Path-Following with convergence guarantees is first proposed and then extended, using the results presented on Economic-MPC, to address the control problems of distributed formation keeping, energy efficient trajectory-tracking, and target-following through highly observable trajectories.

A new approach for trajectory optimization of musculoskeletal dynamic models is introduced. The model combines rigid-body and muscle dynamics described with a Hill-type model driven by neural control inputs. The objective is to find input and state trajectories that are optimal with respect to a minimum-effort objective and meet constraints associated with musculoskeletal models. The measure of effort is given by the integral of pairwise average forces of the agonist-antagonist muscles. The concepts of flat parameterization of nonlinear systems and sum-of-squares optimization are combined to yield a method that eliminates the numerous set of dynamic constraints present in collocation methods. With terminal equilibrium, optimization reduces to a feasible linear program, and a recursive feasibility proof is given for more general polynomial optimization cases. The methods of the article can be used as a basis for fast, and efficient solvers for hierarchical, and receding-horizon control schemes. Two simulation examples are included to illustrate the proposed methods.

In earlier work, we proposed to approximate the constrained linear quadratic regulator problem by representing input and state trajectories as linear combinations of basis functions. As a result, a relatively low-dimensional optimization problem is obtained, while at the same time an infinite prediction horizon can be retained, providing benefits regarding computation and closed-loop stability when applied to model predictive control. In the following, we will extend the approach and present an optimization strategy to design basis functions that are tailored to the specific problem instance. We will show on a simulation example that the same approximation quality of the underlying problem can be achieved with fewer basis functions, which reduces the problem dimension and the computational load even further. The optimization of the basis functions can be done offline. We also discuss the constraint-free case and show that, by choosing the basis functions appropriately, the solution of the underlying linear quadratic regulator problem is recovered by the approximation.

This paper presents the formulation of an impedance controller for regulating the contact force with the environment. To achieve an accurate force tracking control, uncertainties in both robot dynamics and the environment require to be addressed. As part of the framework of the proposed force tracking formulation, a neural network is introduced at the desired trajectory to compensate for all uncertainties in an on-line manner. Compensation at the input trajectory leads to a remarkable structural advantage in that no modifications of the internal force controllers are required. Minimizing the objective function of the training signal for a neural network satisfies the desired force tracking performance. A neural network actually compensates for uncertainties at the input trajectory level in an on-line fashion. Simulation results confirm the position and force tracking abilities of a robot manipulator.

An effective approach for non-singular trajectory generation of a 5-DOF hybrid machining robot