Time-Optimal Trajectory Planning of Six-Axis Manipulators Based on the Improved Direct Collocation Method with FMU

Abstract

The trajectory planning method with dynamics is the key to improving the motion performance of manipulators. The optimal control method (OCM) is a key technology to solve optimal problems with dynamics. There are direct and indirect methods in OCM; indirect methods are difficult to apply to engineering applications, and so direct methods are widely applied instead. The direct collocation method (DCM) is a technology in OCM to transform an optimal control problem (OCP) to a nonlinear problem (NLP), so that plenty of solvers can be used directly. However, the general DCM, for which it has been found that the explicit form of the right-hand-side (RHS) functions of state equations of the complex system in the OCP is hard to derive, is limited to solving the OCP of three-axis manipulators. This paper proposes an improved DCM to solve the OCP of six-axis manipulators, which can find the solution of the time-optimal trajectory for the motion of six-axis manipulators based on the improved DCM. The proposed method derives the RHS equations implicitly by introducing a Functional Mock-up Unit (FMU), which simplifies the representation of the RHS equations as a black-box model, so that the DCM can be applied to the OCP of six-axis manipulators. A simulation case of a three-axis manipulator accomplished in a related study works as a reference compared with our improved method to verify the solution consistence between the DCM using the explicit RHS equations or using the implicit RHS equations, and the loss of computational efficiency is acceptable. In the meantime, a simulation solution and an experiment of six-axis manipulators, which is a novel advancement, are presented to validate the proposed method.

Sequential Optimal Trajectory Planning Scheme for Robotic Manipulators along Specified Path Based on Direct Collocation Method

“…In this part, we want to present the PP problem as an OC problem. The general OC problem presentation is given as follows referring to [10,21,23].…”

“…As a result, DC methods can be applied to discretize the OC problem into an NLP problem. The Hermite-Simpson (H-S) collocation method is used here, and the NLP problem presentation after discretization referring to [10,21,23] is given as below.…”

“…x denotes the system states which refers to the joint position q and the joint velocity q in the system of robotic manipulators. ẋ denotes the differential of the system states which refers to the joint velocity q and the joint acceleration q. f FMU is a black box function constructed by MWorks and denotes the forward dynamics of the robotic system [21]. Functional Mock-up Unit (FMU) is used to name the black box function.…”

“…For the latter, since O. V. Stryk et al proposed a numerical solution framework for OC problems, numerous methods have been derived. Z. Xiong et al integrated the framework with a Functional Mock-up Unit (FMU), accomplishing optimal PTP TP for six-axis robotic manipulators [21]. However, the above technology is limited to solve the PTP TP problem.…”

Energy Efficient Path and Trajectory Optimization of Manipulators With Task Deadline Constraints

“…proposed an improved method based on the conventional direct collocation method using Functional Mock-up Units compared to particle swarm optimization [22] Nonoyama et al, developed a model for the joint robot placement and motion planning problem using metaheuristic algorithms such as Genetic Algorithms and Particle Swarm Optimization methods to create a more precise robot motion trajectory, resulting in an energy-efficient robot configuration [23].…”