Control Barrier Functions (CBF) are a powerful tool for designing safety-critical controllers and motion planners. The safety requirements are encoded as a continuously differentiable function that maps from state variables to a real value, in which the sign of its output determines whether safety is violated. In practice, the CBFs can be used to enforce safety by imposing itself as a constraint in a Quadratic Program (QP) solved point-wise in time. However, this approach costs computational resources and could lead to infeasibility in solving the QP. In this paper, we propose a novel motion planning framework that combines sampling-based methods with Linear Quadratic Regulator (LQR) and CBFs. Our approach does not require solving the QPs for control synthesis and avoids explicit collision checking during samplings. Instead, it uses LQR to generate optimal controls and CBF to reject unsafe trajectories. To improve sampling efficiency, we employ the Cross-Entropy Method (CEM) for importance sampling (IS) to sample configurations that will enhance the path with higher probability and store computed optimal gain matrices in a hash table to avoid re-computation during rewiring procedure. We demonstrate the effectiveness of our method on nonlinear control affine systems in simulation.
翻译:控制障碍函数(CBF)是用于设计安全关键控制器和运动规划器的有力工具。安全要求被编码成一个连续可微的函数,该函数从状态变量映射到一个实值,它的输出符号决定了安全是否受到破坏。在实践中,CBF可以通过将其作为约束在时间点上求解二次规划(QP)来实现强制安全。然而,这种方法需要耗费大量的计算资源,可能会导致无法解决QP问题。在本文中,我们提出了一种新的运动规划框架,将采样方法与线性二次调节(LQR)和CBF相结合。我们的方法不需要解决控制合成的QP问题,并避免在采样过程中显式地检测碰撞。相反,它使用LQR来生成最优控制器并使用CBF来拒绝不安全的轨迹。为了提高采样的效率,我们采用交叉熵方法(CEM)进行重要性采样(IS),以采样能够提高路径概率的配置,并将计算的最优增益矩阵存储在哈希表中,以避免在重连过程中重新计算。我们在仿真中演示了该方法在非线性控制仿效系统上的有效性。