Trajectory optimization of flexible link manipulators in point-to-point motion

Robotica ◽  
2008 ◽  
Vol 27 (6) ◽  
pp. 825-840 ◽  
Author(s):  
M. H. Korayem ◽  
A. Nikoobin ◽  
V. Azimirad
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimal Trajectory ◽  
Planning Problem ◽  
Flexible Manipulators ◽  
Flexible Link ◽  
Maximum Payload ◽  
Point Motion ◽  
Point To Point

SUMMARYThe aim of this paper is to determine the optimal trajectory and maximum payload of flexible link manipulators in point-to-point motion. The method starts with deriving the dynamic equations of flexible manipulators using combined Euler–Lagrange formulation and assumed modes method. Then the trajectory planning problem is defined as a general form of optimal control problem. The computational methods to solve this problem are classified as indirect and direct techniques. This work is based on the indirect solution of open-loop optimal control problem. Because of the offline nature of the method, many difficulties like system nonlinearities and all types of constraints can be catered for and implemented easily. By using the Pontryagin's minimum principle, the obtained optimality conditions lead to a standard form of a two-point boundary value problem solved by the available command in MATLAB®. In order to determine the optimal trajectory a computational algorithm is presented for a known payload and the other one is then developed to find the maximum payload trajectory. The optimal trajectory and corresponding input control obtained from this method can be used as a reference signal and feedforward command in control structure of flexible manipulators. In order to clarify the method, derivation of the equations for a planar two-link manipulator is presented in detail. A number of simulation tests are performed and optimal paths with minimum effort, minimum effort-speed, maximum payload, and minimum vibration are obtained. The obtained results illustrate the power and efficiency of the method to solve the different path planning problems and overcome the high nonlinearity nature of the problems.

Optimal Motion Planning of Manipulators With Elastic Links and Joints in Generalized Point-to-Point Task

2009 ◽  
Author(s):  
H. N. Rahimi ◽  
M. H. Korayem ◽  
A. Nikoobin
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Minimum Principle ◽  
Linear Equations ◽  
Dynamic Modelling ◽  
Lagrange Formulation ◽  
Point Motion ◽  
Optimal Motion Planning ◽  
Point To Point

Finding optimal trajectory is critical in several applications for robots from payload transport between two given states in a prescribed time such that a cost functional is minimized. This paper is concerned with the path planning of flexible robotic arms in point-to-point motion, based on indirect solution of optimal control problem. Dynamic modelling technique based on the combined Euler–Lagrange formulation and assumed modes method is applied, then by implementing the Pontryagin’s minimum principle; necessary conditions for optimality are derived. Nonlinear states and control constraints are treated without any simplifications or transforming them into sequences of systems with linear equations. By these means, the modelling of the complete optimal control problem and the accompanying boundary value problem is automated to a great extent. Finally, the performance of method is illustrated through the computer simulations.

Maximum load carrying capacity of mobile manipulators: optimal control approach

Robotica ◽  
2009 ◽  
Vol 27 (1) ◽  
pp. 147-159 ◽  
Author(s):  
M. H. Korayem ◽  
A. Nikoobin ◽  
V. Azimirad
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Carrying Capacity ◽  
Nonholonomic Constraints ◽  
Maximum Load ◽  
Mobile Manipulators ◽  
Load Carrying Capacity ◽  
Maximum Payload ◽  
Load Carrying

SUMMARYIn this paper, finding the maximum load carrying capacity of mobile manipulators for a given two-end-point task is formulated as an optimal control problem. The solution methods of this problem are broadly classified as indirect and direct. This work is based on the indirect solution which solves the optimization problem explicitly. In fixed-base manipulators, the maximum allowable load is limited mainly by their joint actuator capacity constraints. But when the manipulators are mounted on the mobile bases, the redundancy resolution and nonholonomic constraints are added to the problem. The concept of holonomic and nonholonomic constraints is described, and the extended Jacobian matrix and additional kinematic constraints are used to solve the extra DOFs of the system. Using the Pontryagin's minimum principle, optimality conditions for carrying the maximum payload in point-to-point motion are obtained which leads to the bang-bang control. There are some difficulties in satisfying the obtained optimality conditions, so an approach is presented to improve the formulation which leads to the two-point boundary value problem (TPBVP) solvable with available commands in different softwares. Then, an algorithm is developed to find the maximum payload and corresponding optimal path on the basis of the solution of TPBVP. One advantage of the proposed method is obtaining the maximum payload trajectory for every considered objective function. It means that other objectives can be achieved in addition to maximize the payload. For the sake of comparison with previous results in the literature, simulation tests are performed for a two-link wheeled mobile manipulator. The reasonable agreement is observed between the results, and the superiority of the method is illustrated. Then, simulations are performed for a PUMA arm mounted on a linear tracked base and the results are discussed. Finally, the effect of final time on the maximum payload is investigated, and it is shown that the approach presented is also able to solve the time-optimal control problem successfully.

Energy efficiency path planning for a quadrotor aerial vehicle

2021 ◽  
pp. 014233122110585
Author(s):  
Fouad Yacef ◽  
Nassim Rizoug ◽  
Laid Degaa ◽  
Omar Bouhali ◽  
Mustapha Hamerlain
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Weather Conditions ◽  
Optimization Method ◽  
Control Approach ◽  
Mission Success ◽  
Aerial Vehicle ◽  
Vehicle Trajectory ◽  
Point To Point

Unmanned aerial vehicles are used today in many real-world applications. In all these applications, the vehicle endurance (flight time) is an important constraint that affects mission success. This study investigates the limitations of embedded energy for a quadrotor aerial vehicle. We consider a quadrotor simple tasked to travel from an initial hover configuration to a final hover configuration. In order to have a precise approximation of the consumed energy, we propose a power consumption model with battery dynamic, motor dynamic, and rotor efficiency function. We then introduce an optimization algorithm to minimize the energy consumption during quadrotor aerial vehicle mission. The proposed algorithm is based on an optimal control problem formulated for the quadrotor model and solved using nonlinear programming. In the optimal control problem, we seek to find control inputs (rotor velocity) and vehicle trajectory between initial and final configurations that minimize the consumed energy during a point-to-point mission. We extensively test in simulation experiments the proposed algorithm under normal and windy weather conditions. We compare the proposed optimization method with a nonlinear adaptive control approach to highlight the saved amount of energy.

An efficient stochastic approach for robust time-optimal trajectory planning of robotic manipulators under limited actuation

Robotica ◽  
2017 ◽  
Vol 35 (12) ◽  
pp. 2400-2417 ◽  
Author(s):  
Ming-Yong Zhao ◽  
Xiao-Shan Gao ◽  
Qiang Zhang
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Trajectory Planning ◽  
Optimal Trajectory ◽  
Original Problem ◽  
Robotic Manipulators ◽  
Linear Optimal Control ◽  
Time Optimal ◽  
Optimal Trajectory Planning

SUMMARYThis paper focuses on the problem of robust time-optimal trajectory planning of robotic manipulators to track a given path under a probabilistic limited actuation, that is, the probability for the actuation to be limited is no less than a given bound κ. We give a general and practical method to reduce the probabilistic constraints to a set of deterministic constraints and show that the deterministic constraints are equivalent to a set of linear constraints under certain conditions. As a result, the original problem is reduced to a linear optimal control problem which can be solved with linear programming in polynomial time. In the case of κ = 1, the original problem is proved to be equivalent to the linear optimal control problem. Overall, a very general, practical, and efficient algorithm is given to solve the above problem and numerical simulation results are used to show the effectiveness of the method.

Optimal Periodic Harvesting Strategy for the General Autonomous System of Single Specie

2012 ◽  
Vol 518-523 ◽  
pp. 237-240 ◽  
Author(s):  
Zai Le He ◽  
Xiang Qing Zhao ◽  
Hai Ling Lu
Keyword(s):  
Optimal Control ◽  
Growth Rate ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimal Trajectory ◽  
Autonomous System ◽  
Rate Function ◽  
Growth Rate Function

We study a optimal control problem-optimal periodic harvesting strategy for the general autonomous system of single specie. Suppose the growth rate function f(x) is C^2, x* satisfies f'(x*)=0. We proved that if f"(x*)<0，then x* is the optimal trajectory, otherwise, x* is not the optimal trajectory.

Trajectory planning with mid-air collision avoidance for quadrotor unmanned aerial vehicles

2021 ◽  
pp. 095441002110440
Author(s):  
Yuhang Jiang ◽  
Shiqiang Hu ◽  
Christopher J Damaren
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Unmanned Aerial Vehicles ◽  
Collision Avoidance ◽  
Trajectory Planning ◽  
Pseudospectral Method ◽  
Planning Problem ◽  
Aerial Vehicles ◽  
Optimal Control Algorithms

Flight collision between unmanned aerial vehicles (UAVs) in mid-air poses a potential risk to flight safety in low-altitude airspace. This article transforms the problem of collision avoidance between quadrotor UAVs into a trajectory-planning problem using optimal control algorithms, therefore achieving both robustness and efficiency. Specifically, the pseudospectral method is introduced to solve the raised optimal control problem, while the generated optimal trajectory is precisely followed by a feedback controller. It is worth noting that the contributions of this article also include the introduction of the normalized relative coordinate, so that UAVs can obtain collision-free trajectories more conveniently in real time. The collision-free trajectories for a classical scenario of collision avoidance between two UAVs are given in the simulation part by both solving the optimal control problem and querying the prior results. The scalability of the proposed method is also verified in the simulation part by solving a collision avoidance problem among multiple UAVs.

scholarly journals Parametric Approach to Trajectory Tracking Control of Robot Manipulators

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Shijie Zhang ◽  
Yi Ning
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Trajectory Tracking ◽  
Tracking Control ◽  
Optimal Trajectory ◽  
Open Loop ◽  
Parametric Approach ◽  
Trajectory Tracking Control ◽  
Loop Control

The mathematic description of the trajectory of robot manipulators with the optimal trajectory tracking problem is formulated as an optimal control problem, and a parametric approach is proposed for the optimal trajectory tracking control problem. The optimal control problem is first solved as an open loop optimal control problem by using a time scaling transform and the control parameterization method. Then, by virtue of the relationship between the optimal open loop control and the optimal closed loop control along the optimal trajectory, a practical method is presented to calculate an approximate optimal feedback gain matrix, without having to solve an optimal control problem involving the complex Riccati-like matrix differential equation coupled with the original system dynamics. Simulation results of 2-link robot manipulator are presented to show the effectiveness of the proposed method.

scholarly journals A Trajectory Planning Method for Autonomous Valet Parking via Solving an Optimal Control Problem

Sensors ◽  
2020 ◽  
Vol 20 (22) ◽  
pp. 6435
Author(s):  
Chen Chen ◽  
Bing Wu ◽  
Liang Xuan ◽  
Jian Chen ◽  
Tianxiang Wang ◽  
...  
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Trajectory Planning ◽  
Pseudospectral Method ◽  
Initial Guess ◽  
Planning Method ◽  
Planning Problem ◽  
Starting Position ◽  
Dynamic Obstacle Avoidance

In the last decade, research studies on parking planning mainly focused on path planning rather than trajectory planning. The results of trajectory planning are more instructive for a practical parking process. Therefore, this paper proposes a trajectory planning method in which the optimal autonomous valet parking (AVP) trajectory is obtained by solving an optimal control problem. Additionally, a vehicle kinematics model is established with the consideration of dynamic obstacle avoidance and terminal constraints. Then the parking trajectory planning problem is modeled as an optimal control problem, while the parking time and driving distance are set as the cost function. The homotopic method is used for the expansion of obstacle boundaries, and the Gauss pseudospectral method (GPM) is utilized to discretize this optimal control problem into a nonlinear programming (NLP) problem. In order to solve this NLP problem, sequential quadratic programming is applied. Considering that the GPM is insensitive to the initial guess, an online calculation method of vertical parking trajectory is proposed. In this approach, the offline vertical parking trajectory, which is calculated and stored in advance, is taken as the initial guess of the online calculation. The selection of an appropriate initial guess is based on the actual starting position of parking. A small parking lot is selected as the verification scenario of the AVP. In the validation of the algorithm, the parking trajectory planning is divided into two phases, which are simulated and analyzed. Simulation results show that the proposed algorithm is efficient in solving a parking trajectory planning problem. The online calculation time of the vertical parking trajectory is less than 2 s, which meets the real-time requirement.

scholarly journals Multiaircraft Optimal 4D Trajectory Planning Using Logical Constraints

2019 ◽  
Vol 2019 ◽  
pp. 1-15
Author(s):  
Dinesh B. Seenivasan ◽  
Alberto Olivares ◽  
Ernesto Staffetti
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Obstacle Avoidance ◽  
Trajectory Planning ◽  
Model Complexity ◽  
Mixed Integer ◽  
Planning Problem ◽  
Integer Variables ◽  
Logical Constraints

This paper studies the trajectory planning problem for multiple aircraft with logical constraints in disjunctive form which arise in modeling passage through waypoints, distance-based and time-based separation constraints, decision-making processes, conflict resolution policies, no-fly zones, or obstacle or storm avoidance. Enforcing separation between aircraft, passage through waypoints, and obstacle avoidance is especially demanding in terms of modeling efforts. Indeed, in general, separation constraints require the introduction of auxiliary integer variables in the model; for passage constraints, a multiphase optimal control approach is used, and for obstacle avoidance constraints, geometric approximations of the obstacles are introduced. Multiple phases increase model complexity, and the presence of integer variables in the model has the drawback of combinatorial complexity of the corresponding mixed-integer optimal control problem. In this paper, an embedding approach is employed to transform logical constraints in disjunctive form into inequality and equality constraints which involve only continuous auxiliary variables. In this way, the optimal control problem with logical constraints is converted into a smooth optimal control problem which is solved using traditional techniques, thereby reducing the computational complexity of finding the solution. The effectiveness of the approach is demonstrated through several numerical experiments by computing the optimal trajectories of multiple aircraft in converging and intersecting arrival routes with time-based separation constraints, distance-based separation constraints, and operational constraints.

scholarly journals Energy-Optimal Trajectory Planning for Planar UnderactuatedRRRobot Manipulators in the Absence of Gravity

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
John Gregory ◽  
Alberto Olivares ◽  
Ernesto Staffetti
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Calculus Of Variations ◽  
Trajectory Planning ◽  
Integral Form ◽  
Necessary Condition ◽  
Planning Problem ◽  
Control Input ◽  
Revolute Joints

In this paper, we study the trajectory planning problem for planar underactuated robot manipulators with two revolute joints in the absence of gravity. This problem is studied as an optimal control problem in which, given the dynamic model of a planar horizontal robot manipulator with two revolute joints one of which is not actuated, the initial state, and some specifications about the final state of the system, we find the available control input and the resulting trajectory that minimize the energy consumption during the motion. Our method consists in a numerical resolution of a reformulation of the optimal control problem as an unconstrained calculus of variations problem in which the dynamic equations of the mechanical system are regarded as constraints and treated using special derivative multipliers. We solve the resulting calculus of variations problem using a numerical approach based on the Euler-Lagrange necessary condition in integral form in which time is discretized and admissible variations for each variable are approximated using a linear combination of piecewise continuous basis functions of time. The use of the Euler-Lagrange necessary condition in integral form avoids the need for numerical corner conditions and the necessity of patching together solutions between corners.

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