| We consider the problem of generating feedback motion plans for nonlinear systems that are guaranteed to succeed in the complex constraints.Artificial potential function method is widely applied to solve the feedback motion planning problem,this method unites an attractive function and repulsive functions as an artificial potential function that all the state will move along the gradient direction of the artificial potential function.The absence of the large region of attraction for general nonlinear systems and the artificial potential function will collapse into the local minimum easily under complex constraints.Therefore,the artificial potential function method is mainly used to figure out the feedback motion planning problem for the linear or weekly nonlinear system in the simple constraints.In order to solve the feedback motion planning problem generally,we split the problem into two parts:computation of the sequence of reachable sets for the local system and sequential composition of such sequences in the original system.The main contributions of the paper are as follows:(1)Reachable sets are modeled by Liouville’s equation,and there is a measureinvariant property in such equation during the computation of reachable sets.We build a general model of the reachable set.(2)Based on LaSalle,s invariant set theory,a subset of the state space is said invariant if the inclusion of the state at some times implies the inclusion in the future.In order to compute infinite-horizon reachable sets when the invariant set theory is satisfied,we address the problem by Liouville’s equation and the measure invariant property to perform a general explanation for invariant sets and reachable sets:backward reachable set in infinite-horizon can be used to approximate the region of attraction and forward robust reachable set in infinite-horizon is the minimal robust positively invariant set.Convex optimization is applied to compute the forward and backward invariant sets.Inder to reduce the conservation of existing algorithms for invariant set computations,the parameterized methods and robust closed-loop are proposed.(3)We address the problem by combining approaches from Interval Taylor Methods,Generalized Moment Problem,Christoffel-Darboux kernel function in order to perform an efficient method for computing finite-horizon reachable sets of nonlinear systems.Firstly,the restricted measure of a set is computed by GMP.Secondly,the restricted measure is transported by the interval vector depend on Liouville’s equation and measure-invariant property.Finally,the Christoffel-Darboux kernel is applied to recover the reachable set at any sample time,the support set of each reachable set is recovered by Lebesgue measure and Dirac measure respectively.(4)Solving feedback motion planning problem via backward and forward funnels(sequence of reachable sets).In order to solve the feedback motion planning problem in static constraints environment,a randomly sampled tree of backward funnels is used to cover the entire controllable subset of the state space which also satisfies constraints;In order to solve the feedback motion planning problem in time-varying constraints,an efficient partial pruning technique of the graph is proposed to avoid dynamic constraints.In order to reduce the cost of global computation,when the initial set is known,a robust direct trajectory optimization method based on the robust forward funnel is proposed to compute local feedback motion planning by explicitly taking into account the effect of uncertainties;When parts of the information are unknown,a receding horizon planning method based on the robust forward funnel.We demonstrate and validate our methods in three applications of spacecraft mission. |
