Huang Dengpeng, Leyendecker Sigrid
Institute of Applied Dynamics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany.
Soft Robot. 2023 Oct;10(5):897-911. doi: 10.1089/soro.2022.0162. Epub 2023 Mar 28.
In this work, a simulation model for the optimal control of dielectric elastomer actuated flexible multibody dynamics systems is presented. The dielectric elastomer actuator (DEA) behaves like a flexible artificial muscle in soft robotics. It is modeled as an electromechanically coupled geometrically exact beam, where the electric charges serve as control variables. The DEA-beam is integrated as an actuator into multibody systems consisting of rigid and flexible components. The model also represents contact interaction via unilateral constraints between the beam actuator and, for example, a rigid body during the grasping process of a soft robot. With a mathematically concise and physically representative formulation, a reduced free energy function is developed for the electromechanically coupled beam. In the optimal control problem, an objective function is minimized while the electromechanically coupled dynamic balance equations for the multibody system have to be fulfilled together with the complementarity conditions for the contact and boundary conditions. The optimal control problem is solved via a direct transcription method, transforming it into a constrained nonlinear optimization problem. The electromechanically coupled geometrically exact beam is firstly semidiscretized with one-dimensional finite elements and then the multibody dynamics is temporally discretized with a variational integrator leading to the discrete Euler-Lagrange equations, which are further reduced with the null space projection. The discrete Euler-Lagrange equations and the boundary conditions serve as equality constraints, whereas the contact constraints are treated as inequality constraints in the optimization of the discretized objective. The constrained optimization problem is solved using the Interior Point Optimizer solver. The effectiveness of the developed model is demonstrated by three numerical examples, including a cantilever beam, a soft robotic worm, and a soft robotic grasper.
在这项工作中，提出了一种用于介电弹性体驱动的柔性多体动力学系统最优控制的仿真模型。介电弹性体致动器（DEA）在软机器人技术中表现得像一种柔性人造肌肉。它被建模为一个机电耦合的几何精确梁，其中电荷作为控制变量。DEA梁作为一个致动器集成到由刚性和柔性部件组成的多体系统中。该模型还通过梁致动器与例如软机器人抓取过程中的刚体之间的单边约束来表示接触相互作用。通过数学上简洁且物理上具有代表性的公式，为机电耦合梁开发了一个简化的自由能函数。在最优控制问题中，目标函数被最小化，同时多体系统的机电耦合动态平衡方程必须与接触和边界条件的互补条件一起得到满足。最优控制问题通过直接转录方法求解，将其转化为一个约束非线性优化问题。首先用一维有限元对机电耦合的几何精确梁进行半离散化，然后用变分积分器对多体动力学进行时间离散化，得到离散的欧拉 - 拉格朗日方程，再通过零空间投影进一步简化。离散的欧拉 - 拉格朗日方程和边界条件作为等式约束，而接触约束在离散目标的优化中被视为不等式约束。使用内点优化器求解器解决约束优化问题。通过三个数值例子证明了所开发模型的有效性，包括一个悬臂梁、一个软机器人蠕虫和一个软机器人抓手。
