We prove that any invariant algebraic set of a given polynomial vector field can be algebraically represented by one polynomial and a finite set of its successive Lie derivatives. This so-called differential radical characterization relies on a sound abstraction of the reachable set of solutions by the smallest variety that contains it. The characterization leads to a differential radical invariant proof rule that is sound and complete, which implies that invariance of algebraic equations over real-closed fields is decidable. Furthermore, the problem of generating invariant varieties is shown to be as hard as minimizing the rank of a symbolic matrix, and is therefore NP-hard. We investigate symbolic linear algebra tools based on Gaussian elimination to efficiently automate the generation. The approach can, e.g., generate nontrivial algebraic invariant equations capturing the airplane behavior during take-off or landing in longitudinal motion.

Abstract-Nowadays, robots interact more frequently with a dynamic environment outside limited manufacturing sites and in close proximity with humans. Thus, safety of motion and obstacle avoidance are vital safety features of such robots. We formally study two safety properties of avoiding both stationary and moving obstacles: (i) passive safety, which ensures that no collisions can happen while the robot moves, and (ii) the stronger passive friendly safety in which the robot further maintains sufficient maneuvering distance for obstacles to avoid collision as well. We use hybrid system models and theorem proving techniques that describe and formally verify the robot's discrete control decisions along with its continuous, physical motion. Moreover, we formally prove that safety can still be guaranteed despite location and actuator uncertainty.

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The safety of mobile robots in dynamic environments is predicated on making sure that they do not collide with obstacles. In support of such safety arguments, we analyze and formally verify a series of increasingly powerful safety properties of controllers for avoiding both stationary and moving obstacles: (i) static safety, which ensures that no collisions can happen with stationary obstacles, (ii) passive safety, which ensures that no collisions can happen with stationary or moving obstacles while the robot moves, (iii) the stronger passive friendly safety in which the robot further maintains sufficient maneuvering distance for obstacles to avoid collision as well, and (iv) passive orientation safety, which allows for imperfect sensor coverage of the robot, i. e., the robot is aware that not everything in its environment will be visible. We complement these provably correct safety properties with liveness properties: we prove that provably safe motion is flexible enough to let the robot still navigate waypoints and pass intersections. We use hybrid system models and theorem proving techniques that describe and formally verify the robot's discrete control decisions along with its continuous, physical motion. Moreover, we formally prove that safety can still be guaranteed despite sensor uncertainty and actuator perturbation, and when control choices for more aggressive maneuvers are introduced. Our verification results are generic in the sense that they are not limited to the particular choices of one specific control algorithm but identify conditions that make them simultaneously apply to a broad class of control algorithms.