Online footstep planning is essential for bipedal walking robots to be able to walk in the presence of disturbances. Until recently this has been achieved by only optimizing the placement of the footstep, keeping the duration of the step constant. In this paper we introduce a footstep planner capable of optimizing footstep placement and timing in real-time by asynchronously combining two optimizers, which we refer to as asynchronous real-time optimization (ARTO). The first optimizer which runs at approximately 25 Hz, utilizes a fourth-order Runge-Kutta (RK4) method to accurately approximate the dynamics of the linear inverted pendulum (LIP) model for bipedal walking, then uses non-linear optimization to find optimal footsteps and duration at a lower frequency. The second optimizer that runs at approximately 250 Hz, uses analytical gradients derived from the full dynamics of the LIP model and constraint penalty terms to perform gradient descent, which finds approximately optimal footstep placement and timing at a higher frequency. By combining the two optimizers asynchronously, ARTO has the benefits of fast reactions to disturbances from the gradient descent optimizer, accurate solutions that avoid local optima from the RK4 optimizer, and increases the probability that a feasible solution will be found from the two optimizers. Experimentally, we show that ARTO is able to recover from considerably larger pushes and produces feasible solutions to larger reference velocity changes than a standard footstep location optimizer, and outperforms using just the RK4 optimizer alone.

中文翻译：

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双足步行机器人中脚步放置和时序的异步实时优化

在线足迹规划对于双足步行机器人能够在存在干扰的情况下行走至关重要。直到最近，这只是通过优化足迹的位置来实现的，保持步骤的持续时间恒定。在本文中，我们介绍了一种足迹规划器，它能够通过异步组合两个优化器来实时优化足迹放置和时序，我们将其称为异步实时优化 (ARTO)。第一个优化器以大约 25 Hz 的频率运行，利用四阶 Runge-Kutta (RK4) 方法精确近似线性倒立摆 (LIP) 模型的双足行走动力学，然后使用非线性优化来寻找最佳足迹和持续时间在较低的频率。第二个优化器以大约 250 Hz 的频率运行，使用从 LIP 模型的完整动力学和约束惩罚项导出的解析梯度来执行梯度下降，从而在更高的频率下找到近似最佳的脚步放置和时序。通过异步组合两个优化器，ARTO 具有对梯度下降优化器干扰的快速反应、避免 RK4 优化器局部最优的准确解决方案的优点，并增加了从两个优化器中找到可行解决方案的概率。通过实验，我们表明 ARTO 能够从相当大的推动中恢复，并为更大的参考速度变化生成可行的解决方案，而不是标准的足迹位置优化器，并且优于仅使用 RK4 优化器。通过异步组合两个优化器，ARTO 具有对梯度下降优化器干扰的快速反应、避免 RK4 优化器局部最优的准确解决方案的优点，并增加了从两个优化器中找到可行解决方案的概率。通过实验，我们表明 ARTO 能够从相当大的推动中恢复，并为更大的参考速度变化生成可行的解决方案，而不是标准的足迹位置优化器，并且优于仅使用 RK4 优化器。通过异步组合两个优化器，ARTO 具有对梯度下降优化器干扰的快速反应、避免 RK4 优化器局部最优的准确解决方案的优点，并增加了从两个优化器中找到可行解决方案的概率。通过实验，我们表明 ARTO 能够从相当大的推动中恢复，并为更大的参考速度变化生成可行的解决方案，而不是标准的足迹位置优化器，并且优于仅使用 RK4 优化器。