Proposed Trajectory Optimization Research Articles

Trajectory optimization for maximization of energy efficiency with dynamic cluster and wireless power for UAV‐assisted maritime communication

AbstractNowadays, the digital development of marine ranching requires a communication system with wide coverage, high transmission rate and stable communication links. It is known that fixed‐wing unmanned aerial vehicles (UAVs) have great advantages in long‐range applications. They have the potential to serve as low‐altitude communication platforms for maritime communication. In this study, with a developmental perspective, considering the intense growth of marine terminals in the future, a new clustering algorithm applied to cluster nonorthogonal multiple access (C‐NOMA) is proposed and its advantages are investigated. In addition, considering the limited energy of marine terminals, combining the wireless power communication (WPC) technology for the UAV to charge terminals, the charging and communication time are optimized with the Lagrange multiplier method and the bisection search method. After completing the above optimization content of charging and communication, combined with the optimization results, it is found the trajectory that maximizes the energy efficiency of the UAV with the convex optimization technique. Experimental results show that the proposed clustering algorithm has good throughput performance, better fairness and lower algorithm complexity, and the proposed trajectory optimization scheme has better energy efficiency.

Trajectory optimization for collaborative recovery of parafoil systems using unmanned vessel

Traditional trajectory optimization methods for the parafoil system set fixed-point landing as the objective. However, in recent payload fairing recovery missions, the recovery system comprising two parafoils is collaboratively recovered by a mobile vehicle, posing new challenges to the current trajectory optimization technique. In order to recover two parafoil systems autonomously with an unmanned surface vessel, this paper presents a trajectory optimization framework composed of three following component processes consecutively. Firstly, a feasibility judgment algorithm based on reachable boundary estimation is designed to determine the possibility of recovering two parafoil systems. Secondly, the decoupled-then-simultaneous strategy is proposed to enhance the convergence of solving the collaborative recovery problem. Thirdly, the finite-element collocation approach is utilized to convert the formulated trajectory optimization problems into nonlinear programming (NLP) problems, which are solved by a highly efficient NLP solver. Simulation results show that the proposed trajectory optimization framework can efficiently generate the optimal trajectory for recovering two parafoil systems with a vessel.

Trajectory optimization method based on ellipse model for dynamic motion control of piezoelectric transducer in an optical resonance cavity

External-cavity diode laser (ECDL) can continuously change the output wavelength by scanning its optical resonance cavity (ORC), which is a piezoelectric transducer (PZT) actuated mechanical system. However, because of the inherent hysteresis characteristic of PZT and limited system bandwidth, the dynamic response of the ORC system exhibits both the nonlinearity and vibration while high-frequency triangular wave scanning, thereby resulting in the degradation of the output wavelength performance of the ECDL. As a result, a trajectory optimization method is presented to improve the scanning characteristics of the ORC system. Firstly, the hysteresis behavior of PZT is described by the elliptic model to generate a compensation voltage for the PZT. Secondly, the cusp region of the compensation voltage is optimized by a frequency domain objective function to greatly reduce the high-order harmonic components of the compensation voltage. Finally, an optimal voltage trajectory for driving the ORC system is obtained. The experimental results demonstrate the validity of our proposed trajectory optimization method for simultaneously suppressing the nonlinearity and vibration of the ORC system within the scanning frequency of 100 Hz.

Whole-Body Dynamics-Based Aerial Fall Trajectory Optimization and Landing Control for Humanoid Robot.

When humanoid robots work in human environments, falls are inevitable due to the complexity of such environments. Current research on humanoid robot falls has mainly focused on falls on the ground, with little research on humanoid robots falling from the air. In this paper, we employ an extended state variable formulation that directly maps from the high-level motion strategy space to the full-body joint space to optimize the falling trajectory in order to protect the robot when falling from the air. In order to mitigate the impact force generated by the robot's fall, during the aerial phase, we employ simple proportion differentiation (PD) control. In the landing phase, we optimize the optimal contact force at the contact point using the centroidal dynamics model. Based on the contact force, the changes to the end-effector positions are solved using a dual spring-damper model. In the simulation experiments, we conduct three comparative experiments, and the simulation results demonstrate that the robot can safely fall 1.5 m from the ground at a pitch angle of 45°. Finally, we experimentally validate the methods on an actual robot by performing a side-fall experiment. The experimental results show that the proposed trajectory optimization and motion control methods can provide excellent shock absorption for the impact generated when a robot falls.

Multiobjective trajectory optimization of the wall‐building robot based on RBF–NSGA‐II in an uncertain viscoelastic contact environment

AbstractTo solve the problem of poor masonry quality of traditional wall‐building robots in an uncertain viscoelastic contact environment while reducing energy consumption, reducing contact forces with the environment, and improving work efficiency and smoothness, a segmented multiobjective trajectory optimization method is proposed based on radial basis function (RBF) and nondominated sorting genetic algorithm II (NSGA‐II). The method divides the motion trajectory into the free motion segment and the masonry segment. In the masonry segment, the compensation variable is introduced at the brick‐stopping position, and the values of design variables are obtained by Latin hypercube sampling. The relationship between the objective functions and the design variables is established by using an RBF substitution model. The optimal design is carried out by the NSGA‐II, and the compromise solution is obtained by using the technique for order preference by similarity to an ideal solution algorithm. On this basis, a multiobjective trajectory optimization method based on seven times nonuniform B‐spline curves is proposed for the free motion segment. According to the performance indicators, such as operation efficiency, trajectory smoothness, and energy consumption, the compromise solution is again sought and obtained. Finally, the proposed trajectory optimization method is compared with the standard gate‐shaped trajectory planning method. The results show that after trajectory optimization, the masonry efficiency of the wall‐building robot is improved by 28.36%, and the energy consumption and trajectory smoothness are reduced by 28.68% and 93.81%, respectively. At the same time, the contact force with the environment is reduced by 12.26%, and the masonry error is reduced from 2.67 to 0.13 mm. These results can contribute to the construction of walls and improve the masonry quality of bricks while considering other performance indicators.

Polytopal approximation for thrust magnitude constraints in rocket trajectory optimization

This paper proposes a linear programming approach for onboard optimization of rocket trajectories, motivated by the need for reliable and computationally efficient trajectory optimization methods in real-time applications. To reformulate rocket trajectory optimization problems within a linear programming framework, a polytopal approximation method is developed. The polytopal approximation method approximates the thrust magnitude constraint using a set of artificial variables and linear constraints. Compared with a state-of-the-art approximation method, the polytopal approximation method demonstrates a higher accuracy-to-size ratio, enabling rocket trajectory optimization problems to be cast into linear subproblems without significantly increasing problem sizes. As linear subproblems can be readily solved by linear programming solvers, the proposed trajectory optimization approach inherits the solver’s reliability and computational efficiency, making it potentially useful for critical onboard applications. An ascent trajectory optimization problem is provided as an illustrative example to demonstrate the effectiveness of the proposed linear programming approach.

Entry trajectory optimization for hypersonic vehicles based on convex programming and neural network

Trajectory optimization is important in achieving long-range atmospheric entry hypersonic vehicles. However, the trajectory optimization problem for atmospheric entry of hypersonic vehicles is characterized by strong nonlinearity, parameter uncertainties and multiple constraints. This study proposes a novel online trajectory optimization method for hypersonic vehicles based on convex programming and a feedforward neural network. A sequential second-order cone programming (SOCP) method is obtained to describe the trajectory optimization problem after the Gauss pseudo-spectral discretization. Subsequently, multiple optimal trajectories under aerodynamic uncertainties are generated offline and classified as the training and validation datasets. Then, a multilayer feedforward neural network is trained using these datasets and to output the optimal control command online. This method yields approximately 95% shorter computation time compared with the offline SOCP method. Considering the existence of the aerodynamic uncertainties, three terminal states calculated by this method are all smaller than 4.1%. In conclusion, the proposed trajectory optimization method can provide a high-precision, robust entry trajectory for hypersonic vehicles efficiently.

Minimum-Time Trajectory Generation for Wheeled Mobile Systems Using Bézier Curves with Constraints on Velocity, Acceleration and Jerk.

This paper considers the problem of minimum-time smooth trajectory planning for wheeled mobile robots. The smooth path is defined by several Bézier curves and the calculated velocity profiles on individual segments are minimum-time with continuous velocity and acceleration in the joints. We describe a novel solution for the construction of a 5th order Bézier curve that enables a simple and intuitive parameterization. The proposed trajectory optimization considers environment space constraints and constraints on the velocity, acceleration, and jerk. The operation of the trajectory planning algorithm has been demonstrated in two simulations: on a racetrack and in a warehouse environment. Therefore, we have shown that the proposed path construction and trajectory generation algorithm can be applied to a constrained environment and can also be used in real-world driving scenarios.

Multiphase Overtaking Maneuver Planning for Autonomous Ground Vehicles Via a Desensitized Trajectory Optimization Approach

This article studies the problem of trajectory optimization for autonomous ground vehicles with the consideration of irregularly placed on-road obstacles and multiple maneuver phases. By introducing a series of event sequences, a new multiphase constrained optimal control formulation is constructed to describe the automatic overtaking process. Although existing trajectory optimization techniques can be applied to address the constructed problem, they may suffer from poor or premature convergence issues due to the complexity of the mission formulation. Thus, to offer an effective alternative, a novel desensitized trajectory optimization method is designed and implemented to explore the optimal overtaking maneuver for the AGVs. The proposed method applies a double layer structure, where an enhanced intelligent optimization method is used in the outer layer such that the main inner optimization routine can be boosted by starting at a better reference solution. The algorithm convergence as well as the solution optimality conditions are theoretically analyzed. Numerical results are provided to illustrate the validity of the established formulation. Comparative case studies were executed to demonstrate the quality of the obtained solution and the enhanced performance of the proposed trajectory optimization method.

Trajectory optimization of an electric vehicle with minimum energy consumption using inverse dynamics model and servo constraints

Trajectory planning is a crucial aspect of autonomous driving for energy conservation ans savings, especially in electric vehicles (EVs). A particular vehicle inverse dynamics model is necessary to calculate the required torque and steering control for following the target trajectory. Consequently, the energy consumption of an EV can be minimized through optimal trajectory planning. This study proposes a method of optimizing an EV’s trajectory to reduce energy consumption. First, the vehicle inverse dynamics model is developed using a semi-recursive method for modeling multibody dynamics. The suspension, steering, and braking system’s dynamic properties are considered and modeled. Second, the servo constraints are formulated within given vehicle states and kinematic constraints to address the inverse dynamics of an EV. Third, a procedure for optimizing trajectories has been developed using an inverse dynamics model, servo constraints, and the Ritz method. The results indicate that the optimal trajectory uses the least amount of energy. The effects of the trajectory on energy consumption is also investigated. The necessary torques are applied to the left and right wheels, as well as the steering system’s control input. A precise inverse dynamics model and servo constraints are used in self-driving EVs’ proposed trajectory optimization method.

Energy Aware Trajectory Optimization of Solar Powered AUVs for Optical Underwater Sensor Networks

Visible light communication (VLC) provides an alternative underwater wireless connectivity solution with its low latency and high data rates albeit at relatively shorter distances in the order of tens of meters. In the context of underwater sensor networks (USNs), VLC is particularly suitable to establish connectivity between “data mule” autonomous underwater vehicles (AUVs) and sensor nodes since communications is enabled only when the sensor node and mule AUV are in close proximity. In this paper, we consider a USN scenario where a solar-powered AUV gathers data from the sensor nodes using VLC signaling. We formulate a three-dimensional trajectory optimization for solar-powered AUVs with the goal of maximizing the harvested energy under constraints imposed by the data transmission. The optimization constraints include the minimum required data transfer rate, therefore a corresponding transmission distance, between the sensors and the AUV. We formulate the problem as a bilevel optimization problem. The lower-level objective function is in the form of traveling salesman problem which determines the optimum sequence order of the sensor nodes to be visited while the upper-level objective function is the optimization of the trajectory between each pair of adjacent nodes for the given order of node visits. Our numerical results demonstrate that the proposed trajectory significantly prolongs the mission time and autonomous operation of the AUV without the need to return to home base. Furthermore, since the proposed trajectory optimization is reactive to ocean currents, it brings reductions in the energy consumption of the AUVs.

Mixtures of Gaussian Processes for Robot Motion Planning Using Stochastic Trajectory Optimization

Robot motion planning methods based on trajectory optimization can efficiently generate feasible and optimal trajectories by minimizing a suitable cost function, even in high-dimensional spaces. However, the main drawback of these methods lies in their proneness to infeasible local minima, especially in complex environments. To mitigate this issue, we propose a novel motion planning method that represents trajectories as samples from a mixture of continuous-time Gaussian processes (MGP) and employs stochastic optimization in order to update the MGP parameters in a cost-minimizing manner. The contributions of the proposed trajectory optimization method arise from the introduced mixture representation and stochastic gradient estimation, dominantly enabling better exploration of the trajectory space and including nondifferentiable optimizing costs. We evaluated the proposed method in multiple simulation benchmarks featuring 7 degree-of-freedom (DOF) robot arms and a 10DOF mobile manipulator. We also conducted a real-world experiment with a 14DOF dual-arm robot. The experimental results demonstrated that the proposed method achieves higher success rate than several state-of-the-art methods, while the advantages stemming from MGPs and stochastic optimization, like trajectory smoothness, support of nondifferentiable cost functions, multiple trajectory solutions, and the ability to tackle high-dimensional planning problems, are inherently kept.

Multi-impulse smooth trajectory design for long-range rendezvous with an orbiting target using multi-objective non-dominated sorting genetic algorithm

This paper develops a methodology to generate Pareto optimal trajectories for long-range rendezvous of a servicing satellite with a moving target, in an on-orbit servicing mission. The methodology employs a multi-impulse shape-based trajectory planning algorithm for in-plane orbit transfer, based on the two-body problem. We first derive the necessary and sufficient conditions that determine the set of smooth impulsive trajectories connecting the servicing satellite to the orbiting target. The Pareto optimal trajectories from this set are then obtained using a constrained multi-objective optimization algorithm developed based on the Non-dominated Sorting Genetic Algorithm-II (NSGA-II). In a mission, an optimal solution from the Pareto frontier set may be selected based on the mission requirements. Transfer time and control effort are the two Pareto cost functions that are considered in the multi-objective optimization. To reduce the risk of collision in populated orbits and to remain in an orbital regime, we include restrictions on orbital elements as part of the constraints. Further, a maximum available impulse is considered as an upper-bound for velocity changes in an impulsive trajectory. The number of impulses along with the location of the first impulse in the parking orbit and the orbital parameters of the intermediate orbits form the set of design variables. The key advantage of the proposed trajectory optimization methodology compared to its counterparts using continuous thrust is the significant reduction of the number of design variables. Finally, we demonstrate the superiority of the developed trajectory planner by comparing its results with those obtained from another multi-objective evolutionary algorithm called the Multi-Objective Genetic Algorithm and an optimal Lambert approach based on single-objective optimization.

Optimization Control of Cooperative Trajectory towards Dual Arms Based on Time-Varying Constrained Output State

Aiming at the impact and disturbance of dual-arm robots in the process of coordinated transportation, a dual-arm cooperative trajectory optimization control based on time-varying constrained output state is proposed. According to the constraint relationship of the end-effector trajectory of the dual-arm coordinated transportation, the joint space trajectory mathematical model of the dual-arm coordinated transportation was established by using the master-slave construction method. Based on the time impact optimization index of joint trajectory, a multiobjective nonlinear equation is established. Using random probability distribution to extract the interpolation features of nonuniform quintic B-spline trajectory, the feature optimization target is selected, and the Newton numerical algorithm is used for iterative optimization. At the same time, it is combined with an elite retention genetic algorithm to further optimize the target. Based on the disturbance and tracking problem, a PD control method based on time-varying constrained output state is proposed, and the control law is designed. Its convergence is verified by establishing the Lyapunov function equation and asymmetric term. The trajectory optimization results show that the proposed trajectory optimization method can increase the individual diversity and enhance the individual local optimization, thus avoiding the premature impact of the elite retention genetic algorithm. Finally, the proposed control method is simulated on the platform of Gazebo; compared with the traditional PD control method, the results show that the proposed control algorithm has high robustness, and the rationality of the coordinated trajectory control method is verified by the double-arm handling experiment.

A review of Hybrid/combined methods for trajectory optimization of flight vehicles

This work aims to review the proposed trajectory optimization algorithms and their applications in solving flight trajectory planning and optimal control problems. Beside the classification of trajectory optimization algorithms listed in Chai R, Savvaris A, Tsourdos A and Chai S such as the deterministic; the stochastic; and the multi-objective algorithms, this work adds in reviewing the effective and frequently-used approach for solving optimal problems— the hybrid/combined methods. The results of related works show that this hybrid approach is effective and can provide feasible solutions for solving the flight vehicle trajectory optimal design and control problems.

An efficient and direct method for trajectory optimization of robots constrained by contact kinematics and forces

In this work, we propose a trajectory generation method for robotic systems with contact kinematics and force constraints based on optimal control and reachability analysis tools. Normally, the dynamics and constraints of a contact-constrained robot are nonlinear and coupled to each other. Instead of linearizing the model and constraints, we solve the optimal control problem directly to obtain feasible state trajectories and their corresponding control inputs. A tractable optimal control problem is formulated and subsequently addressed by dual approaches, which rely on sampling-based dynamic programming and rigorous reachability analysis tools. In particular, a sampling-based method together with a Partially Observable Markov Decision Process solution approach are used to break down the end-to-end trajectory generation problem by generating a sequence of subregions that the system’s trajectory will have to pass through to reach its final destination. The distinctive characteristic of the proposed trajectory optimization algorithm is its ability to handle the intricate contact constraints, coupled with the system dynamics, in a computationally efficient way. We validate our method using extensive numerical simulations with two legged robots.

Centroidal-momentum-based trajectory generation for legged locomotion

This paper presents a trajectory optimization framework for planning dynamic legged locomotion based on a robot’s centroidal momentum (CM), which is the aggregation of all the links’ momenta at the robot’s Center of Mass (CoM). This new framework is built around CM dynamic model driven by Ground Reaction Forces (GRFs) parameterized with Bézier polynomials. Due to the simple form of CM dynamics, the closed-form solution of the robot’s CM can be obtained by directly integrating the Bézier polynomials of GRFs. The CM can be also calculated from the robot’s generalized coordinates and velocities using Centroidal Momentum Matrices (CMM). For dynamically feasible motions, these CM values should match, thereby providing equality constraints for the proposed trajectory optimization framework. Direct collocation methods are utilized to obtain feasible GRFs and joint trajectories simultaneously under kinematic and dynamic constraint. With the closed-form solutions of CM due to the parameterization of GRFs in the formulation, numerical error induced by collocation methods in the solution of trajectory optimization can be reduced, which is crucial for reliable tracking control when applied to real robotic systems. Using the proposed framework, jumping trajectories of legged robots are obtained in the simulation. Experimental validation of the algorithm is performed on a planar robot testbed, proving the effectiveness of the proposed method in generating dynamic motions of the legged robots.

Trajectory Planning With Shortest Path for Modified Uncalibrated Visual Servoing Based on Projective Homography

In order to improve the robustness and optimize trajectory of projective homography-based uncalibrated visual servoing (PHUVS) proposed in our previous work, an analytical expression of optimal trajectory for camera in projective homography space is proposed in this article, which is totally free of camera parameters and is corresponding to camera’s shortest path in the 3-D space with straight path in translation and minimal geodesic in rotation. The projective homography is computed without scale ambiguity in both planning and tracking stages. The PHUVS controller is modified correspondingly to track the planned trajectory in projective homography space while maintaining superior characteristic of PHUVS under uncalibrated scenario. The simulations and experiments’ results reveal the effectiveness and necessity of the proposed trajectory optimization method in the existence of large initial errors. Note to Practitioners —The state-of-the-art visual-guided robotic technology in industry usually requires system calibration, which is often costly, vulnerable, and challenging for ordinary workers. In our previous work, we offered an uncalibrated visual servo method based on projective homography named projective homography-based uncalibrated visual servoing (PHUVS), which is suitable for plug and play application for eye-in-hand robot visual servo tasks. However, PHUVS suffers from some defects, including undesirable 3-D space motion and local convergence. In this article, we proposed the trajectory planning method along with a modified PHUVS controller to improve the original one from the disadvantages mentioned earlier. This planning method is also calibration-free. With pure image information, a straight-line path in translational motion along with minimal geodesic in rotary motion can be achieved. This approach is capable of extending the range of applications for uncalibrated visual servo technology in robotic tasks, such as assembling, painting, and robotic machining.

Autonomous Robotic Exploration by Incremental Road Map Construction

In this paper, we propose a novel path planning framework for autonomous exploration in unknown environments using a mobile robot. A graph structure is incrementally constructed along with the exploration process. The structure is the road map that represents the topology of the explored environment. To construct the road map, we design a sampling strategy to get random points in the explored environment uniformly. A global path from the current location of the robot to the target area can be found on this road map efficiently. We utilize a lazy collision checking method that only checks the feasibility of the generated global path to improve the planning efficiency. The feasible global path is further optimized with our proposed trajectory optimization method considering the motion constraints of the robot. This mechanism can facilitate the path cost evaluation for the next best view selection. In order to select the next best target region, we propose a utility function that takes into account both the path cost and the information gain of a candidate target region. Moreover, we present a target reselection mechanism to evaluate the target region and reduce the extra path cost. The efficiency and effectiveness of our approach are demonstrated using a mobile robot in both simulation and real experimental studies. Note to Practitioners —This paper is motivated by the efficient exploration problem, which plays a key role in various areas such as the information gathering and environment monitoring. In these applications, the mission length and executing time are often restricted by the battery capacity of the robot. In this paper, an efficient path planning framework is introduced to reduce the path length and exploration time. The robot keeps a road map of the environment to facilitate the path planning. The proposed target selection mechanism helps the robot determine the next best target to explore. Furthermore, the proposed trajectory optimization algorithm helps in reducing the path cost. Overall, this framework enables efficiently and autonomously exploration with a novel path planning framework.

Re-entry trajectory optimization using pigeon inspired optimization based control profiles

In this paper, the entry trajectory optimization problem of lifting type re-entry vehicle with path constraints is solved using Pigeon Inspired Optimization (PIO). Entry trajectory optimization problem involves finding the control profiles, bank angle, and angle of attack to guide the vehicle safely to the destination. The proposed approach parametrizes the bank angle to be a linear function of energy while the angle of attack is considered to be a monotonic function of Mach number. Thus, the problem of finding control profiles is transformed into three parameter search problem. The PIO algorithm is used to find the values of these parameters that minimizes the objective function. The terminal heading angle offset is minimized using traditional bank reversal logic. A new approach is proposed in which the bank angle is modulated to eliminate the oscillations observed in the altitude profile of an entry vehicle with a high lift to drag ratio (L/D). A methodology to satisfy the given load factor constraint is also proposed, as an alternative to traditional penalty factor approach used for incorporating path constraints in PIO algorithm. The proposed approach is further validated by considering sub-cases with different load factor limits and bank angle as the only control variable. The angle of attack profile obtained from the previous case is considered as the nominal profile. The proposed trajectory optimization strategy using PIO algorithm is simulated for Common Aero Vehicle with high L/D ratio (CAV-H) with different load factor constraint limits. The results show that the obtained angle of attack profile minimizes the peak heat rate experienced by the vehicle and bank angle modulation eliminates the oscillations in the altitude profile as well as makes the entry trajectory satisfy the load factor constraint.