Computing capture tubes

International audienceA dynamic system can often be described by a state equation ˙x = h(x, u, t)where x ∈ Rn is the state vector, u ∈ Rm is the control vector and h :Rn × Rp × R → Rn is the evolution function. Assume that the control lowu = g (x, t) is known (this can be obtained using control theory), the systembecomes autonomous. If we define f (x, t) = h(x, g (x, t) , t), we get the followingequation.˙x = f (x, t) .The validation of some stability properties of this system is an important anddifficult problem [2] which can be transformed into proving the inconsistency of aconstraint satisfaction problem. For some particular properties and for invariantsystem (i.e., f does not depend on t), it has been shown [1] that the V-stabilityapproach combined interval analysis [3] can solve the problem efficiently. Here,we extend this work to systems where f depends on time

Computing capture tubes

International audienceMany mobile robots such as wheeled robots, boats, or plane are described by nonholonomic differential equations. As a consequence, they have to satisfy some differential constraints such as having a radius of curvature for their trajectory lower than a known value. For this type of robots, it is difficult to prove some properties such as the avoidance of collisions with some moving obstacles. This is even more difficult when the initial condition is not known exactly or when some uncertainties occur. This paper proposes a method to compute an enclosure (a tube) for the trajectory of the robot in situations where a guaranteed interval integration cannot provide any acceptable enclosures. All properties that are satisfied by the tube (such as the non-collision) will also be satisfied by the actual trajectory of the robot