Minimum Energy Trajectories Research Articles

Tiltwing eVTOL Transition Trajectory Optimization

This paper studies transition trajectory optimization for a tiltwing electrical vertical takeoff and landing (eVTOL) aircraft configuration and evaluates the benefits of equipping it with fluid injection active flow control (AFC). Using a three-degree-of-freedom model of the aircraft’s longitudinal dynamics, both hover-to-cruise (forward) and cruise-to-hover (backward) transition are investigated for a range of center-of-gravity (CG) positions to determine minimum-energy trajectories that achieve zero altitude variation while avoiding wing aerodynamic stall. It is demonstrated that constant-altitude transition can be achieved in the forward direction (with or without AFC), whereas the use of AFC in backward transition reduces the total altitude change by 24%. A comparative constant-altitude backward transition solution using a different airfoil with a higher stall limit suggests that a redesign of the tail wing of this unique tiltwing aircraft could better leverage the benefits of using AFC in this application. Additionally, it is also shown that, in the case of forward transition, AFC can be used to decrease the total transition time and increase the range of CG positions where constant-altitude transition can occur.

Minimum-time and minimum-energy trajectory planning under dynamic constraints for a quadrotor

This article proposes an approach for optimizing the trajectory planning of quadrotors in structured environments, taking into consideration the trade-off between task execution time and energy consumption. The constraints taken into account are geometrical (boundary conditions in position and orientation, obstacle avoidance), kinematic (boundary conditions in velocity, linear velocity, and acceleration limits of the system), and dynamic (limit capacities in torque of the actuators, constraints related to the under-actuation of the system). The approach extends the Random Trajectory Profile Approach (RPA) to the case of quadrotors. The results of this approach are validated through experimental tests and compared with those in the literature, demonstrating its effectiveness in offline and online scenarios. The proposed approach offers a promising solution for achieving optimal trajectory planning of quadrotors in structured environments.

A real-time capable method for planning minimum energy trajectories for one degree-of-freedom mechatronic systems

The synthesis of optimal motion profiles has shown to be a successful and virtually inexpensive solution for enhancing energy efficiency of mechatronic systems. A typical application is the design of point-to-point motion profiles for one-degree-of-freedom mechatronic systems. This paper proposes a new method for designing minimum energy trajectories for servo-actuated systems. The problem is solved by exploiting the knowledge of the structure of the optimal solution. That allows to solve the motion design problem by solving a set of nonlinear equations and, if needed, some basic optimization procedures, formulated after some suitable continuity conditions.The herein proposed method applies to systems with and without energy regeneration capability, for maximum adaptability to most industrial applications. The method also handles jerk, acceleration, and velocity constraints, which are typical requirements in many practical applications. The number of equations and thereby the computational time depends on the number of active constraints and on whether negative power is dissipated or regenerated. Overall, the method results to be suitable for Real-Time applications, also in the most challenging case in which all the constraints are active and the system cannot regenerate negative electric power. The accuracy and the effectiveness of the planning method is tested numerically, by comparing the solution to the one obtained by a general purpose optimal control solver and then also experimentally using a lab prototype.

Particle Swarm Optimization-based co-state initialization for low-thrust minimum-fuel trajectory optimization

In this paper, Particle Swarm Optimization with energy-to-fuel continuation is proposed for initializing the co-state variables for low-thrust minimum-fuel trajectory optimization problems in the circular restricted three-body problem. Particle Swarm Optimization performs a search of the solution space by minimizing the weighted sum of squares of the two-point boundary-value problem final boundary condition residuals for the minimum-energy problem. Next, an energy-to-fuel homotopy is employed to transition the minimum-energy trajectory to a minimum-fuel trajectory, starting from the generated guess. The proposed methodology is applied to two low-thrust transfer problems in the Earth–Moon system: a transfer from a geostationary transfer orbit to an L1 halo orbit, and a transfer from an L2 halo orbit to an L1 halo orbit. The resulting minimum-fuel trajectories are validated with the literature. It is demonstrated that the methodology can successfully generate guesses for the initial co-state variables which converge to a solution for both scenarios. A strategically chosen particle swarm size is shown to improve the efficiency of the methodology. The proposed approach is of simple implementation, can be easily extended to other trajectory optimization problems, facilitates the discovery of multiple candidate trajectories, and does not require a user-provided guess, all of which are advantageous features for the preliminary phase of mission design.

Data-driven dynamic relatively optimal control

We show how the recent works on data-driven open-loop minimum-energy control for linear systems can be exploited to obtain closed-loop control laws in the form of linear dynamic controllers that are relatively optimal. Besides being stabilizing, they achieve the optimal minimum-energy trajectory when the initial condition is the same as the open-loop optimal control problem. The order of the controller is N−n, where N is the length of the optimal open-loop trajectory, and n is the order of the system. The same idea can be used for obtaining a relatively optimal controller, entirely based on data, from open-loop trajectories starting from up to n linearly independent initial conditions.

How the Wind Can Be Leveraged for Saving Energy in a Truck-Drone Delivery System

In this work, we investigate the impact of the wind in a drone-based delivery system. For the first time, to the best of our knowledge, we adapt the trajectory of the drone to the wind. We consider a truck-drone tandem delivery system. The drone actively reacts to the wind adopting the “most tailwind” trajectory available between the truck’s path and the delivery. The truck moves on a predefined route and carries the drone close to the delivery point. We propose the Minimum-energy Drone-trajectory Problem (MDP) which aims, when the wind affects the delivery area, at planning minimum-energy trajectories for the drone to serve the customers starting from and returning to the truck. We then propose two algorithms that optimally solve MDP under two different routes of the truck. We also analytically study the feasibility of sending drones with limited battery to deliver packages. Finally, we first numerically compare our algorithms on randomly generated synthetic and real data, and then we evaluate our model simulating the drone’s flight in the BlueSky simulator.

Autonomous drone charging station planning through solar energy harnessing for zero-emission operations

• Solar energy from building envelopes can extend UAVs coverage and reliability in last-mile delivery applications. • This integration can omit GHG emissions in parcel delivery while improving building energy efficiency. • A multi-objective optimization model is proposed that maximizes UAV coverage and minimizes the total cost of energy and decarbonization. • A flexible energy use model is applied to a digital twin to generate minimum-energy 3D trajectories. • RI-SHOT multi-objective optimization algorithm is used to solve all the delivery demands. • GHG emissions for the entire UAV charging network are omitted compared to a grid-connected operation. The introduction of Unmanned Arial Vehicles (UAVs) in smart city operations is considered a sustainable technological solution due to the promised significant greenhouse gas emission reductions. This study developed an integrated multi-objective charging infrastructure coverage optimization model that integrates UAV-based operations with solar energy harnessing from building envelopes. This model maximizes UAVs’ coverage range and minimizes the total cost of energy and decarbonization. The model is based on a flexible energy use model for UAVs calibrated to experimental measurements to generate a minimum-energy trajectory. We also developed an origin-destination (O-D) demand geo-referenced in a digital-twin model to replicate real-world operation. Overall, 12,532 simulated daily missions in a large-sized city are modelled. The results show that the proposed system can eliminate GHG emissions. Furthermore, the system can significantly reduce the decarbonization price through associated savings and excess generated electricity. The proposed approach demonstrates avenues to advance smart cities and capitalize on renewable energy.

Energy-saving trajectory planning by using Fourier sine series and tracking control for a mechatronic elevator system

In this paper, Lagrangian method is proposed to formulate dynamic model for a mechatronic elevator system, which includes mechanical and electrical parts. Lagrange equations, i.e. the torque balance equation and voltage balance equation, are derived from the kinetic energy, potential energy and virtual work of the mechatronic system. From the dynamic equations, the energy balance equation is obtained including electromagnetic energy, electrical dissipation energy, mechanical kinetic energy, mechanical dissipation energy, and the mechanical potential energy. The minimum-energy trajectory by using Fourier sine series (FSS) and power series are found by real-coded genetic algorithm. From simulation results, it is found that FSS with less order can find the minimum input energy more quickly than the power series. The four-degree (4-D) FSS trajectory is given as a reference one for the adaptive tracking control. It is found that the proposed adaptive controller has good tracking performance for the 4-D FSS angular displacement and velocity.

Application and improvement of a direct method optimization approach for battery electric railway vehicle operation

While the largely electrified rail network allows for direct utilization of renewable energy sources, there is still a considerable share of diesel-powered trains operating on non- and partly electrified tracks. To replace these, the more sustainable alternatives such as battery electric railway vehicles need to present a viable option with sufficient range. This paper aims to adapt and improve an existing optimization algorithm, previously used with diesel-powered trains, for the operation of battery electric railway vehicles. In this new approach, battery control is optimized alongside train control, utilizing a direct method solver to find the minimum energy trajectory. Furthermore, a detailed train model is implemented that is designed for operation on partly electrified tracks. To yield a highly accurate, yet also sufficiently fast algorithm, a numerical analysis is conducted and the parameters of the algorithm are determined accordingly. Finally, the application of the adapted algorithm on a use case in Germany shows that both velocity profile and control adapt in a way that minimizes utilization of the battery. The results indicate that the proposed algorithm presents a reliable and robust method to obtain minimum energy controls for battery electric railway vehicles with any electrification pattern.

On the Trajectory Planning for Energy Efficiency in Industrial Robotic Systems †

In this paper, we present an approach for the minimum-energy trajectory planning in industrial robotic systems. The method is based on the dynamic and electro-mechanical modeling of one-degree-of-freedom systems and the derivation of the energy formulation for standard point-to-point trajectories, as, for instance, trapezoidal and cycloidal speed profiles. The proposed approach is experimentally validated on two robotic systems, namely a linear axis of a Cartesian manipulator built in the 1990’s, and a test bench composed of two servomotors directly connected or coupled by means of a planetary gear. During the tests, the electrical power expended by the systems is measured and integrated over time to compute the energy consumption for each trajectory. Despite the limitations of the energy measurement systems, the results reveal a trend in agreement with the theoretical calculations, showing the possibility of applying the method for enhancing the performance of industrial robotic systems in terms of energy consumption in point-to-point motions.

Missions to Earth and Mars using a lunar launch facility

Many space agencies are working on solutions for manned operations on the Moon, both on its surface and in cislunar space. In such a scenario, with stable human presence, a launch facility would be needed and launch vehicles could take advantage of the benefits of the lunar environment. In the present paper missions to Earth (to a geostationary orbit) and to Mars (to an areostationary orbit), from a lunar launch facility, has been studied. The trajectories are analysed by using an ephemeris model, taking into account four attractive bodies (Earth, Moon, Sun and Mars). In the first phase of the two missions, the launcher places the spacecraft on a circular lunar orbit by using liquid, solid or hybrid propulsion systems. The latter solution is analysed assuming in situ propellant production using aluminum and oxygen, elements in which the moon is very rich. In the second phase of both the missions, a minimum energy trajectory is studied, which, exploiting the Luni-Solar perturbations, allows the spacecraft to escape the gravitational field of the Moon. In the mission to Earth, this manoeuver puts the spacecraft on a very high and inclined elliptical orbit. The last phase involves a circularization manoeuvre towards the GEO and transfers are studied for two different initial inclinations, using an electric engine. In the mission to Mars, the escape impulse is used to leave both the Earth-Moon and Sun-Earth systems. The interplanetary phase is analysed for several transfer final times and considering an electric engine with variable specific impulse. Finally, the performance of a lunar launch facility is compared to that of a traditional Earth facility, showing great advantages are obtained in terms of payload ratio.

Multidisciplinary integrated design of long-range ballistic missile using PSO algorithm

In the case of the given design variables and constraint functions, this paper is concerned with the rapid overall parameters design of trajectory, propulsion and aerodynamics for long-range ballistic missiles based on the index of the minimum take-off mass. In contrast to the traditional subsystem independent design, this paper adopts the research idea of the combination of the subsystem independent design and the multisystem integration design. Firstly, the trajectory, propulsion and aerodynamics of the subsystem are separately designed by the engineering design, including the design of the minimum energy trajectory, the computation of propulsion system parameters, and the calculation of aerodynamic coefficient and dynamic derivative of the missile by employing the software of missile DATCOM. Then, the uniform design method is used to simplify the constraint conditions and the design variables through the integration design, and the accurate design of the optimized variables would be accomplished by adopting the uniform particle swarm optimization (PSO) algorithm. Finally, the automation design software is written for the three-stage solid ballistic missile. The take-off mass of 29 850 kg is derived by the subsystem independent design, and 20 constraints are reduced by employing the uniform design on the basis of 29 design variables and 32 constraints, and the take-off mass is dropped by 1 850 kg by applying the combination of the uniform design and PSO. The simulation results demonstrate the effectiveness and feasibility of the proposed hybrid optimization technique.

홀로노믹 물류로봇의 다 구간 최소 에너지 수송경로 생성

The minimum-energy multi-section trajectory was generated for holonomic robots. First, minimum-energy trajectory planning on a section for various constraints was performed using the optimal control theory, dynamic models, and a cost function, which represent the energy drawn from the battery. Then, two novel strategies were established to formulate a minimum-energy, multi-section trajectory generation algorithm. Through these strategies on loop arrangement considering the computational load of each trajectory generation algorithm on a section, unknown variables for a multi-section could be found within a practical execution time. The generated trajectory for logistics robots saved 7.18% energy compared with the conventional trajectory using the trapezoidal velocity profile. Further, the actual performance was shown using the resolved-acceleration controller, which was implemented in the preliminary research.

Hamiltonian-based minimum-energy trajectory planning and tracking control for a motor-table system: part II system identification and adaptive tracking control

In this paper, the motor-table system is realized to plan the minimum input energy trajectory (MIET) based on Hamiltonian by minimum input energy control (MIEC) approach. In the motor-table system, the unknown parameters are identified by particle swarm optimization. We can identify the parameters experimentally and the MIEC methods are implemented to find the MIET. The adaptive tracking controller is proposed to track the MIET to overcome the nonlinear friction force and external disturbance. Moreover, trapezoidal trajectory and regulator control are compared with the MIET by the proposed adaptive tracking controller. Finally, the MIET based on the adaptive tracking controller can obtain the minimum energy control input and robustness performance for the motor-table system. The numerical simulations and experimental results are demonstrated the proposed strategy successfully.

Hamiltonian-based minimum-energy trajectory planning and tracking control for a motor-table system: part I minimum-energy methods in trajectory planning

In this paper, we propose the minimum energy control input based on Hamiltonian strategy for the motor-table system. The energy equation of the motor-table system contains of the input energy, dissipation energy, potential energy and output energy, and includes electrical and mechanical energies. The methods of the minimum control effort (MCE), minimum input energy control (MIEC), minimum dissipation energy control (MDEC) and trapezoidal trajectory energy control are compared for the motor-table system. From the numerical results, we can find that the MIEC and MDEC approaches can obtain the same minimum input energy. Traditionally, the MCE approach was regarded as the minimum energy control in the previous study. However, we have demonstrated that the MCE approach is not the minimum energy control in this paper.

Minimum Energy Trajectory Optimization for Driving Systems of Palletizing Robot Joints

Palletizing robot is widely used in logistics operation. At present, people pay attention to not only the loading capacity and working efficiency of palletizing robots, but also the energy consumption in their working process. This paper takes MD1200-YJ palletizing robot as the research object and studies the problem of low energy consumption optimization of joint driving system from the perspective of trajectory optimization. Firstly, a multifactor dynamic model of palletizing robot is established based on the conventional inverse rigid body dynamic model of the robot, the Stribeck friction model and the spring balance torque model. And then based on joint torque, friction torque, motion parameter, and joule’s law, the useful work model, thermal loss model of joint motor, friction energy consumption model of joint system, and total energy consumption model of palletizing robot are established, and through simulation and experiment, the correctness of the multifactor dynamic model and energy consumption model is verified. Secondly, based on the Fourier series approximation method to construct the joint trajectory expression, the minimum total energy consumption as the optimization objective, with coefficients of Fourier series as optimization variables, the motion parameters of initial and final position, and running time constant as constraint conditions, the genetic algorithm is used to solve the optimization problem. Finally, through the simulation analysis the optimized Fourier series motion law and the 3-4-5 polynomial motion law are comprehensively evaluated to verify the effectiveness of the optimization method. Moreover, it provides the theoretical basis for the follow-up research and points out the direction of improvement.

Path Planning Generation Algorithm for a Class of UAV Multirotor Based on State of Health of Lithium Polymer Battery

Nowadays, it exists path planning strategies dedicated to generate trajectories considering different navigation issues in UAV multirotors, such as 3D navigation in cluttered and uncluttered environments, obstacle avoidance, and path re-planning. Such path generators are mainly based on the dynamics associated to position and orientation of the UAV, and the attenuation of external disturbances as the wind. However, one of the main limitations of these methods is that they do not take into account the relationship between the path planning task and the energy consumption associated with the battery performance or State of Health (SoH). In this work, a path planning generation algorithm that take into account the evolution of the battery performance is presented. First, the computation of the battery SoH is realized by introducing two degradation models. Subsequently, the path planning algorithm is defined as a multi-objective optimization problem where the objective is to find a feasible trajectory between way-points whiles minimizing the energy consumed and the mission final time depending on the variation of the battery SoH. Finally, the proposed path planning algorithm is compared with a classical path generation method based on polynomial functions to evaluate the minimization of the energy consumption. The simulation results demonstrate that the proposed path planning algorithm is able to generate feasible and minimum energy trajectories despite the constraints in the battery SoH.

Optimal trajectory operation of a cogging torque assisted motor driven valve actuator for internal combustion engines

Cogging torque assisted motor drives (CTAMD) are a novel electromechanical valve actuation (EVA) technology for internal combustion engines that have a compact size and do not require conventional valve springs. Despite using an unoptimized position trajectory, the CTAMD was shown to produce the smallest ohmic loss at 6000 RPM when compared with other EVA systems found in literature. This paper presents a minimum-energy trajectory for a CTAMD and validates its improvements. First, an ohmic loss model for the CTAMD is formulated. Next, the optimized trajectory is generated through the use of the Nelder–Mead direct search algorithm. The optimized trajectory is then tested on a previously studied experimental CTAMD setup. Finally, the results of the experiments are analyzed to determine the performance improvements and practical advantages of the optimized trajectory. The optimized trajectory enables a further reduction of ohmic loss and reduces the number of MOSFETs required to drive the CTAMD by 75%.

柔軟構造物のPTP制御のための新たな省エネルギー軌道計画法

柔軟構造物のPTP制御のための新たな省エネルギー軌道計画法

Adaptive Tracking Control of the Minimum-Energy Trajectory for a Mechatronic Motor-Table System

In this paper, a mechatronic motor-table system is realized to plan the minimum input electrical energy trajectory based on the Hamiltonian function. In this system, unknown parameters are identified by particle swarm optimization, and an adaptive tracking controller is designed to track the minimum input electrical energy trajectory to overcome the nonlinear friction and external disturbance. Moreover, trapezoidal trajectory and regulator control are compared with the minimum input electrical energy trajectory by an adaptive tracking controller. Finally, it can be concluded that the minimum input electrical energy trajectory based on the adaptive tracking controller can obtain the minimum input electrical energy and robustness performance for the mechatronic motor-table system. Numerical simulations and experimental results demonstrate the adaptive tracking control strategy successfully in the minimum-energy trajectory.