Correlated with the trend of increasing degrees of freedom in robotic systems is a similar trend of rising interest in spatio-temporal systems described by partial differential equations (PDEs) among the robotics and control communities. These systems often exhibit dramatic under-actuation, high dimensionality, bifurcations, and multimodal instabilities. Their control represents many of the current-day challenges facing the robotics and automation communities. Not only are these systems challenging to control, but the design of their actuation is an NP-hard problem on its own. Recent methods either discretize the space before optimization, or apply tools from linear systems theory under restrictive linearity assumptions in order to arrive at a control solution. This manuscript provides a novel sampling-based stochastic optimization framework based entirely in Hilbert spaces suitable for the general class of semi-linear SPDEs which describes many systems in robotics and applied physics. This framework is utilized for simultaneous policy optimization and actuator co-design optimization. The resulting algorithm is based on variational optimization, and performs joint episodic optimization of the feedback control law and the actuation design over episodes. We study first and second order systems, and in doing so, extend several results to the case of second order SPDEs. Finally, we demonstrate the efficacy of the proposed approach with several simulated experiments on a variety of SPDEs in robotics and applied physics including an infinite degree-of-freedom soft robotic manipulator.

与机器人系统自由度增加的趋势相关的是机器人和控制社区对由偏微分方程 (PDE) 描述的时空系统越来越感兴趣的类似趋势。这些系统通常表现出严重的驱动不足、高维、分叉和多模态不稳定性。他们的控制代表了机器人和自动化社区目前面临的许多挑战。这些系统不仅难以控制，而且它们的驱动设计本身就是一个 NP-hard 问题。最近的方法要么在优化之前离散空间，要么在限制性线性假设下应用线性系统理论中的工具，以获得控制解决方案。半线性SPDE 描述了机器人和应用物理学中的许多系统。该框架用于同时进行策略优化和执行器协同设计优化。由此产生的算法基于变分优化，并在情节上执行反馈控制律和致动设计的联合情节优化。我们研究一阶和二阶系统，并在此过程中将几个结果扩展到二阶 SPDE 的情况。最后，我们通过在机器人和应用物理学中的各种 SPDE 上的几个模拟实验证明了所提出的方法的有效性，包括无限自由度的软机器人机械手。