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Approximating Viability Kernels of Non-linear Systems

A typical concern in Robotics is to assess if it is possible to keep a robot inside a set of safe states, e.g., an autonomous car that must stay on the road. That is closely tied with the problem of computing the viability kernel of the system, i.e., the largest set of initial states for which it is guaranteed that the system has controls that keep maintain the trajectories inside the constraint set. The approach in this paper builds on previous work, on linear sampled-data systems. It is based on sampling the boundary of the constraint set, finding the states inside the viability kernel using finite-horizon forward simulation. Our adaptation extends the original algorithm, approximating the viability kernel for some non-linear systems through linearization methods. The non-linear systems here approached are the ones described by first order differential equations with continuous derivatives and convex with respect to the inputs. Existence and uniqueness conditions are also established to ensure adequate results for the whole algorithm. A practical example, with a simple non-linear system, to illustrate the proposed algorithm is also presented.