We present a method for learning multi-stage tasks from demonstrations by learning the logical structure and atomic propositions of a consistent linear temporal logic (LTL) formula. The learner is given successful but potentially suboptimal demonstrations, where the demonstrator is optimizing a cost function while satisfying the LTL formula, and the cost function is uncertain to the learner. Our algorithm uses the Karush-Kuhn-Tucker (KKT) optimality conditions of the demonstrations together with a counterexample-guided falsification strategy to learn the atomic proposition parameters and logical structure of the LTL formula, respectively. We provide theoretical guarantees on the conservativeness of the recovered atomic proposition sets, as well as completeness in the search for finding an LTL formula consistent with the demonstrations. We evaluate our method on high-dimensional nonlinear systems by learning LTL formulas explaining multi-stage tasks on a simulated 7-DOF arm and a quadrotor, and show that it outperforms competing methods for learning LTL formulas from positive examples. Finally, we demonstrate that our approach can learn a real-world multi-stage tabletop manipulation task on a physical 7-DOF Kuka iiwa arm.

我们提出了一种通过学习一致线性时间逻辑 (LTL) 公式的逻辑结构和原子命题来从演示中学习多阶段任务的方法。学习者得到成功但可能次优的演示，其中演示者在满足 LTL 公式的同时优化成本函数，而成本函数对学习者来说是不确定的。我们的算法使用演示的 Karush-Kuhn-Tucker (KKT) 最优条件以及反例引导的证伪策略来分别学习 LTL 公式的原子命题参数和逻辑结构。我们为恢复的原子命题集的保守性以及寻找与演示一致的 LTL 公式的完整性提供了理论保证。我们通过学习解释模拟 7 自由度臂和四旋翼飞行器上的多级任务的 LTL 公式来评估我们在高维非线性系统上的方法，并表明它优于从正例中学习 LTL 公式的竞争方法。最后，我们证明了我们的方法可以在物理 7 自由度 Kuka iiwa 手臂上学习真实世界的多阶段桌面操作任务。