Sums-of-squares (SOS) optimization is a promising tool to synthesize certifiable controllers, but most examples to date have been limited to relatively simple systems. Here we demonstrate that SOS can synthesize controllers with bounded suboptimal performance for various underactuated robotic systems by finding good approximations of the value function. We summarize a unified SOS framework to synthesize both under- and over- approximations of the value function for continuous-time, control-affine systems, use these approximations to generate suboptimal controllers, and perform regional analysis on the closed-loop system driven by these controllers. We then extend the formulation to handle hybrid systems with contacts. We demonstrate that our method can generate tight under- and over- approximations of the value function with low-degree polynomials, which are used to provide stabilizing controllers for continuous-time systems including the inverted pendulum, the cart-pole, and the 3D quadrotor as well as a hybrid system, the planar pusher. To the best of our knowledge, this is the first time that a SOS-based time-invariant controller can swing up and stabilize a cart-pole, and push the planar slider to the desired pose.

翻译：平方和优化是一种极具前景的可验证控制器综合工具，但现有案例多局限于相对简单的系统。本文证明，通过求解价值函数的良好近似，平方和方法可为各类欠驱动机器人系统综合具有有界次优性能的控制器。我们总结了一个统一的平方和框架，用于连续时间控制仿射系统的价值函数上下近似求解，利用这些近似生成次优控制器，并对这些控制器驱动下的闭环系统进行区域分析。随后我们将该框架扩展至含接触的混合系统。研究表明，该方法可通过低阶多项式生成价值函数的紧致上下近似，为倒立摆、欠驱动摆车、三维四旋翼飞行器等连续时间系统以及平面推杆混合系统提供镇定控制器。据我们所知，这是首次基于平方和方法的时不变控制器能够实现欠驱动摆车的摆起镇定，并将平面滑块推至目标位姿。