Optimal Trajectory Generation Research Articles

Trajectory planning and control for drone replacement for multidrone cinematography

This work addresses the problem of trajectory planning and control for a team of drones, so that a vehicle that is following a target as part of a multi-vehicle formation flight can be safely replaced with minimum impact on the formation objectives. An optimal trajectory generation method is presented, which relies on formulating a nonlinear optimization problem with constraints imposed by the physical limitations of the vehicle, the formation, and the surrounding environment, including nonlinear constraints to enforce inter-vehicle collision avoidance and clear visibility of the target. An online planning and control strategy is also implemented through a Nonlinear Model Predictive Control problem, based on the dynamic model of the system, a quadratic objective function, and a set of constraints, that again include collision avoidance and clear visibility. The proposed algorithms are validated in software-in-the-loop simulations that include the autopilot software.

Optimal trajectory generation for time-to-contact based aerial robotic perching

Many biological organisms (e.g. insects, birds, and mammals) rely on the perception of an informational variable called time-to-contact (TTC) to control their motion for various tasks such as avoiding obstacles, landing, or interception. TTC, defined as the required time to contact an object if the current velocity is maintained, has been recently leveraged for robot motion control in various tasks. However, most existing robotic applications of TTC simply control the TTC to be constant or constantly decreasing, without fully exploring the applicability for TTC. In this paper, we propose two-stage TTC based strategies and apply them to aerial robotic perching. With the proposed strategies, we can generate reference trajectories for TTC to realize the non-zero contact velocity required by perching, which is impossible for constant or constantly decreasing TTC strategy, but of critical importance for robust perching performance. We conduct simulations to verify the superiority of the proposed strategies in terms of shorter time for perching and satisfying more constraints. Further, with properly designed controllers, we perform experiments on a palm-size quadcopter to track the planned reference trajectories and realize aerial robotic perching. The research presented in this paper can be readily applied to the control of flying robot for perching with visual feedback, and can inspire more alternative forms of TTC based planning and control for robotic applications.

Path Smoothing Techniques in Robot Navigation: State-of-the-Art, Current and Future Challenges

Robot navigation is an indispensable component of any mobile service robot. Many path planning algorithms generate a path which has many sharp or angular turns. Such paths are not fit for mobile robot as it has to slow down at these sharp turns. These robots could be carrying delicate, dangerous, or precious items and executing these sharp turns may not be feasible kinematically. On the contrary, smooth trajectories are often desired for robot motion and must be generated while considering the static and dynamic obstacles and other constraints like feasible curvature, robot and lane dimensions, and speed. The aim of this paper is to succinctly summarize and review the path smoothing techniques in robot navigation and discuss the challenges and future trends. Both autonomous mobile robots and autonomous vehicles (outdoor robots or self-driving cars) are discussed. The state-of-the-art algorithms are broadly classified into different categories and each approach is introduced briefly with necessary background, merits, and drawbacks. Finally, the paper discusses the current and future challenges in optimal trajectory generation and smoothing research.

Flat trajectory design and tracking with saturation guarantees: a nano-drone application

This paper deals with the problem of trajectory planning and tracking of a quadcopter system based on the property of differential flatness. First, B-spline characterisations of the flat output allow for optimal trajectory generation subject to waypoint constraints, thrust and angle constraints while minimising the trajectory length. Second, the proposed tracking control strategy combines feedback linearisation and nested saturation control via flatness. The control strategy provides bounded inputs (thrust, roll and pitch angles) while ensuring the overall stability of the tracking error dynamics. The control parameters are chosen based on the information of the a priori given reference trajectory. Moreover, conditions for the existence of these parameters are presented. The effectiveness of the trajectory planning and the tracking control design is analysed and validated through simulation and experimental results over a real nano-quadcopter platform, the Crazyflie 2.0.

Vision-Based Reactive Planning for Aggressive Target Tracking While Avoiding Collisions and Occlusions

In this letter, we investigate the online generation of optimal trajectories for target tracking with a quadrotor while satisfying a set of image-based and actuation constraints. We consider a quadrotor equipped with a camera (either down or front-looking) with limited field of view. The aim is to follow in a smooth but reactive way a moving target while avoiding obstacles in the environment and occlusions in the image space. We propose vision-based approaches based on multiobjective optimization, especially with the occlusion constraint formulation. We design an online replanning strategy inspired from model predictive control that successively solves a nonlinear optimization problem. The problem is formulated as a nonlinear program (NLP) using differential flatness and finite parametrization with B-Splines. This allows a resolution by sequential quadratic programming (SQP) at a rate of 30 Hz. The robustness and reactivity of the replanning algorithm are demonstrated through realistic simulation results. Experiments validating the performance with a real quadrotor are also presented.

Warping model predictive control for application in control of a real airborne wind energy system

Fast online generation of feasible and optimal reference trajectories is crucial in tracking model predictive control, especially for stability and optimality in presence of a time varying parameter. In this paper, in order to circumvent the operational efforts of handling a discrete set of precomputed trajectories and switching between them, time warping of a single trajectory is proposed as an alternative concept. In particular, the conceptual ideas of warping theory are presented and illustrated based on the example of a tethered kite system for airborne wind energy. In detail, for warpable systems, feasibility and optimality of trajectories are discussed. Subsequently, the full algorithm of a nonlinear model predictive control implementation based on warping a single precomputed reference is presented. Finally, the warping algorithm is applied to the airborne wind energy system. Simulation results in presence of real world perturbations are evaluated and compared.

Optimal Trajectory Generation and Design of an 8-DoF Compliant Biped Robot for Walk on Inclined Ground

Robots with joint or link compliance are gaining importance due to their inherent safety. In this paper four different designs of compliant shanks of an 8 degrees of freedom (DoF) biped robot are compared to get the least energy consumed during walk. Different leg and pelvis trajectories of the biped during walk are generated by varying the gait parameters. The deflection of the compliant links during walk is modeled using a finite element method (FEM). Simulations are carried out in which the optimal walking trajectory that consumes least energy is found by using a genetic algorithm (GA). Balance is ensured during walk by ensuring that the trajectory always satisfies the zero moment point (ZMP) criteria. The optimal trajectories obtained from the simulations were then experimentally evaluated for a compliant link biped robot. The motion of the different joints of the biped during the experiment was tracked using a vision system and the performance of the rigid link and compliant link bipeds compared. The best compliant link design was finally chosen from the four different designs.

An Efficient Algorithm for Optimal Trajectory Generation for Heterogeneous Multi-Agent Systems in Non-Convex Environments

Generating optimal trajectories for a large numbers of vehicles in real-time becomes extremely challenging when including realistic dynamic models and time-varying obstacle constraints. We present a novel method to efficiently produce time-optimal collision-free trajectories in complex non-convex maze-like environments while enforcing nonlinear constraints on velocity, acceleration, jerk, and snap. The approach combines two complementary numerical techniques for optimal control: level set reachability analysis and pseudospectral orthogonal collocation. Applied in a multi-agent prioritized planning framework, the methodology allows for heterogeneous sizes, dynamics, endpoint conditions, and individualized optimization objectives, while achieving linear scaling in computation time relative to the number of vehicles. Simulation examples are shown with up to 26 static obstacles and 32 quadcopters in a tightly constrained space with average computation times of 1–3.5 s per vehicle, depending on scenario complexity.

Development of a new tool for constrained conflict resolution

In this paper, the problem of finding the optimal trajectory of an aircraft through a specified airspace for a given start and goal position is put forward. The application of sequential programming to optimal trajectory generation is reported with the selection of the grid size and the space boundaries. A totally discrete approach is adopted in contrast to the other models, which use continuous time but resort to diseretization in the computation. The component of the mathematical formulation of the optimization criteria is considered and the procedures used within the control algorithm on the solution are also proposed. Realistic constraints such as turning rate and spatial layout are incorporated into the system model building before applying the sequential programming technique to be used in solving the problem. Simulation examples for multiple aircrafts are presented to demonstrate the use of the proposed trajectory planner and certain optimal control inputs have been acquired.

Neural Network Based Modelling of Time Optimal Interpolation of Non-linear Trajectory

Non linear curve fitting for execution of milling operation on Computer Numerical Control machining is always challenging. This paper focuses on fitting nonlinear curve and generation of time optimal trajectory to perform slot milling operation using NX CAM. Virtual simulation of trajectory is tested with various experimentation conditions and executed using Siemens 828D controller. Simulation and actual experimentation result is then compared with Multilayered Perceptron based Neural Network and optimum cutting parameters for time optimal trajectory is calculated.

Reentry Trajectory Optimization for a Hypersonic Vehicle Based on an Improved Adaptive Fireworks Algorithm

Generation of optimal reentry trajectory for a hypersonic vehicle (HV) satisfying both boundary conditions and path constraints is a challenging task. As a relatively new swarm intelligent algorithm, an adaptive fireworks algorithm (AFWA) has exhibited promising performance on some optimization problems. However, with respect to the optimal reentry trajectory generation under constraints, the AFWA may fall into local optimum, since the individuals including fireworks and sparks are not well informed by the whole swarm. In this paper, we propose an improved AFWA to generate the optimal reentry trajectory under constraints. First, via the Chebyshev polynomial interpolation, the trajectory optimization problem with infinite dimensions is transformed to a nonlinear programming problem (NLP) with finite dimension, and the scope of angle of attack (AOA) is obtained by path constraints to reduce the difficulty of the optimization. To solve the problem, an improved AFWA with a new mutation strategy is developed, where the fireworks can learn from more individuals by the new mutation operator. This strategy significantly enhances the interactions between the fireworks and sparks and thus increases the diversity of population and improves the global search capability. Besides, a constraint-handling technique based on an adaptive penalty function and distance measure is developed to deal with multiple constraints. The numerical simulations of two reentry scenarios for HV demonstrate the validity and effectiveness of the proposed improved AFWA optimization method, when compared with other optimization methods.

Task-Constrained Trajectory Planning of Free-Floating Space-Robotic Systems Using Convex Optimization

This paper addresses the trajectory-planning problem for a free-floating space-robotic system for which there are no actuations on the base spacecraft. The mission considered is that the end effector is subject to task compliance constraints and the whole system is kinematically redundant such that it can incorporate various objects such as collision avoidance. Based on the conservation of the system moment, trajectory planning is formulated as a convex quadratic programming problem in the joint space. The attitude disturbance on the base spacecraft due to the dynamic coupling between the manipulator and the base is shown to be a convex function of the design variables and is incorporated in the objective function to be minimized. Physical constraints including obstacle avoidance and the bounds on the joint angles, joint velocities, and joint accelerations are directly included in the developed framework. The proposed quadratic programming formulation allows for rapid generation of optimal trajectories, which brings robustness to the planning. Numerical simulations for a 10-degree-of-freedom space manipulator validate the efficacy of the proposed approach and demonstrate the method’s potential for real-time applications.

Warping NMPC for Online Generation and Tracking of Optimal Trajectories

Generation of feasible and optimal reference trajectories is crucial in tracking Nonlinear Model Predictive Control. Especially, for stability and optimality in presence of a time varying parameter, adaptation of the tracking trajectory has to be implemented. General approaches are real-time generation of trajectories or switching between a discrete set of precomputed trajectories. In order to circumvent the operational efforts of these methods for a special type of dynamical systems, we propose time warping as an alternative approach. This algorithm implements online generation of tracking trajectories by warping a single precomputed reference. In detail, warpable systems, feasibility and optimality of trajectories and the controller implementation are discussed. Finally, as an application example, simulation results of a tethered kite system for airborne wind energy generation are presented.

A multiphase DMOC‐based trajectory optimization method

SummaryDiscrete mechanics and optimal control (DMOC) is a methodology that takes advantage of variational structure to solve certain optimal control problems for mechanical systems. This paper proposes to combine a multiphase strategy with the original DMOC method, resulting in a new multiphase DMOC (MDMOC) method and making optimal trajectory generation more efficient. The advantages of the proposed method are demonstrated mathematically, and in addition, a quadrotor, unmanned aerial vehicle, simulation example is presented to show its superiority over the DMOC method. Furthermore, to show its potential application, an arbitrarily chosen controller was used to track the desired trajectory generated by MDMOC. This new MDMOC methodology can also be applied to other mechanical systems such as mobile robots and underwater gliders.

Maximizing airborne delay at no extra fuel cost by means of linear holding

This paper introduces a linear holding strategy based on prior works on cruise speed reduction, aimed at performing airborne delay at no extra fuel cost, as a complementary strategy to current ground and airborne holding strategies. Firstly, the equivalent speed concept is extended to climb and descent phases through an analysis of fuel consumption and speed from aircraft performance data. This gives an insight of the feasibility to implement the concept, differentiating the case where the cruise flight level initially requested is kept and the case where it can be changed before departure in order to maximize the linear holding time. Illustrative examples are given, where typical flights are simulated using an optimal trajectory generation tool where linear holding is maximized while keeping constant the initially planned fuel. Finally, the effects of linear holding are thoroughly assessed in terms of the vertical trajectory profiles, range of feasible speed intervals and trade-offs between fuel and time. Results show that the airborne delay increases significantly with nearly 3-fold time for short-haul flights and 2-fold for mid-hauls to the cases in prior works.

Optimization of dynamic soaring maneuvers to enhance endurance of a versatile UAV

Dynamic soaring is a process of acquiring energy available in atmospheric wind shears and is commonly exhibited by soaring birds to perform long distance flights. This paper aims to demonstrate a viable algorithm which can be implemented in near real time environment to formulate optimal trajectories for dynamic soaring maneuvers for a small scale Unmanned Aerial Vehicle (UAV). The objective is to harness maximum energy from atmosphere wind shear to improve loiter time for Intelligence, Surveillance and Reconnaissance (ISR) missions. Three-dimensional point-mass UAV equations of motion and linear wind gradient profile are used to model flight dynamics. Utilizing UAV states, controls, operational constraints, initial and terminal conditions that enforce a periodic flight, dynamic soaring problem is formulated as an optimal control problem. Optimized trajectories of the maneuver are subsequently generated employing pseudo spectral techniques against distant UAV performance parameters. The discussion also encompasses the requirement for generation of optimal trajectories for dynamic soaring in real time environment and the ability of the proposed algorithm for speedy solution generation. Coupled with the fact that dynamic soaring is all about immediately utilizing the available energy from the wind shear encountered, the proposed algorithm promises its viability for practical on board implementations requiring computation of trajectories in near real time.

Optimal trajectory generation algorithm for serial and parallel manipulators

In this paper, an Optimal Trajectory Generation Algorithm (OTGA) is developed for generating minimum-time smooth motion trajectories for serial and parallel manipulators. OTGA is divided into two phases. The first phase encompasses derivation of minimum-time optimal trajectory using cubic spline due to its less vibration and overshoot characteristics. Although cubic splines are widely used in robotics, velocity and acceleration ripples in the first & last knots can worsen manipulator trajectory. The second phase includes changing cubic spline interpolation in the first and last knots of optimized trajectory with 7th order polynomial for having zero jerk at the beginning and end points of trajectory. Performing this modification eliminate undesired worsening in the trajectory and provide smoother start and stop of joint motions. Particle Swarm Optimization (PSO) is chosen as optimization algorithm because of its easy implementation and successful optimization performance. OTGA has been tested in simulation for PUMA robot and results are compared with algorithms proposed by earlier authors. In addition, a discrete-time PID control scheme for PUMA robot is designed for comparing energy consumption of OTGA with algorithms developed by previous authors. Comparison results illustrated that OTGA is the better trajectory generation algorithm than the others.

Optimal capture occasion determination and trajectory generation for space robots grasping tumbling objects

This paper presents an optimal trajectory planning scheme for robotic capturing of a tumbling object. Motion planning of a space robot is much more complex than that of a fixed-based robot, due to the dynamic coupling between the manipulator and its base. In this work, the Path Independent Workspace (PIW), in which no dynamic singularity occurs, and Path Dependent Workspace (PDW) of the space robot are first calculated by the proposed algorithm. The motion equations of the tumbling object are formulated based on the Euler dynamics equations and the quaternion, which are used to predict the long-term motion of a grasping point on the tumbling object. Subsequently, the obtained PIW workspace and predicted motion trajectories are used to plan the trajectory of the end-effector to intercept the grasping point with zero relative velocity (to avoid impact) in an optimal way. In order to avoid dynamic singularity occurring at the capture moment, the optimal capture occasion is first determined by three proposed criterions guaranteeing the capture can be safely, reliably and rapidly performed, then the optimal trajectory of the end-effector is generated minimizing a cost function which acts as a constraint on acceleration magnitude. Simulations are presented to demonstrate the trajectory planning scheme for a space robot with a 3-degree of freedom (DOF) manipulator grasping a tumbling satellite, the results show that the manipulator end-effector can smoothly intercept the grasping point on the tumbling satellite with zero relative velocity.

Optimal 3D trajectory generation in delivering missions under urban constraints for a flying robot

Interest in applying flying robots especially quadcopters for civil applications, in particular for delivering purposes, has dramatically grown in the recent years. In fact, since quadcopters are capable of vertical takeoff and landing, they can be widely employed for nearly any aerial task where a human presence is hazardous or response time is critical. In this regard, quadcopters come to be very beneficial in delivering packages; accordingly, generating an optimal flight trajectory plays a preponderant role for meeting this vision. This paper is concerned with generation of a time-optimal 3D path for a quadcopter under municipal restrictions in delivering tasks. To this end, the flying robot’s dynamics is first modeled through Newton–Euler method. Subsequently, the problem is formulated as a time-optimal control problem such that the urban constraints, which are safe-margins of high-rise buildings located throughout the course, are first modeled and then imposed to the trajectory optimization problem as inequality constraints. After discretizing the trajectory by means of Hermit–Simpson method, the optimal control problem is transformed into a nonlinear programming problem and finally is solved by the direct collocation technique. Extensive simulations demonstrate the efficacy of the proposed method and correspondingly verify the effectiveness of the suggested method in generation of optimum 3D routes while all constraints and mission requirements are satisfied.

Optimal Aircraft Trajectories to Minimize the Radiative Impact of Contrails and CO 2

The rapid growth of air traffic in the Asia Pacific region in the last decade has brought about the need for more sustainable modes of flight. A key initiative is the development of a Next Generation Air Traffic Management (NG-ATM) system which allows aircraft to fly optimal trajectories. Besides fuel- and time-related costs, other considerations for optimal routing include emissions, noise and contrails. Multi-Objective Trajectory Optimisation (MOTO) allows the generation of optimal trajectories with regards to these objectives, with dynamic weights depending on the phase of flight. Contrails are a major contributor to aviation's total Radiative Forcing (RF), being more significant than that of CO2. In particular, when formed in areas of low temperature and high relative humidity, contrails are known to persist for hours, spreading and eventually transitioning into cirrus clouds. Contrails trap heat by reflecting the long-wave infra-red radiation emitted by the earth back to its surface, producing positive RF. However, the albedo of contrails also reflects the incoming shortwave radiation from the sun, resulting in a negative component of RF. The impact of contrails, quantified by its associated RF, is thus not merely a function of environmental parameters but also a function of time. In this paper, a MOTO algorithm is used to generate optimal trajectories that minimize the radiative impact of contrails and CO2, while minimizing flight time and fuel burn. A case study of a transcontinental flight from Paris to Beijing is presented to demonstrate the feasibility of such an algorithm in providing strategic and tactical trajectory optimization capabilities.