Validated global multiobjective optimization of trajectories in nonlinear dynamical systems

Abstract

We introduce a new approach for global multiobjective optimization of trajectories in continuous nonlinear dynamical systems that can provide rigorous, arbitrarily tight bounds on the objective values and state paths realized by (Pareto-)optimal trajectories. By controlling all sources of error, our resulting method is the first global trajectory optimization method that can reliably handle nonconvex nonlinear dynamical systems with substantial instabilities, such as the notoriously ill-behaved multi-body gravitational systems governing interplanetary space trajectories. Rigorous finite-dimensional global optimization methods based on space partitioning (branch and bound) do not directly extend to infinite-dimensional problems of trajectory optimization, lacking a way to exhaustively partition an infinite-dimensional space. Thus existing generic methods for deterministic global trajectory optimization rely on direct discretization of the control variables, if not also the state variables. While the resulting errors may prove inconsequential for relatively stable (conservative/dissipative) systems, they severely influence results in unstable systems that arise in many aerospace applications, and whose chaotic sensitivities offer great potential for inexpensive trajectory control. In order to achieve higher accuracy, current programs for interplanetary trajectory optimization typically use problem-specific control parameterizations with local optimization methods (commonly, multiple shooting with sequential quadratic programming), combined with stochastic or expert-guided sampling to seek global optimality. This approach substantially relies on pre-existing intuition about the character of optimal solutions, and provides no guarantees on the global optimality of solutions obtained. The requirements for expert guidance and judgment of uncertainties tend to drive up costs and restrict innovation for the trajectory solutions that play a crucial role in early conceptual design for deep space missions. The thesis takes a new approach to avoid unaccountable discretization errors. Using a specially designed exhaustive partition of the (finite-dimensional) state space into subregions, we construct a finite transition graph between these subregions, such that each trajectory of interest maps to a finite path (transition sequence) in the graph, where each transition trajectory lies in a local state space neighborhood of its corresponding subregions. For any such path, the cost of any corresponding trajectory can be bounded below by the sum of lower bounds on the cost of each stepwise transition. Provided that the transition bounds converge to exact bounds with increasing refinement of the state space partition, an adaptive refinement can produce asymptotically convergent bounds on optimal trajectories. We compute a lower bound on each stepwise transition between state subregions by a novel "interval linearization" technique that simultaneously considers all possible trajectories between two subregions that lie within a local neighborhood. This technique first linearizes the dynamics on the local neighborhood, and replaces the remaining nonlinear terms by interval enclosures of their values over the neighborhood. We then derive a nonlinear system-of-equations solution to a corresponding pointwise generalized linear optimal control problem with time-varying coefficients. Finally, using interval methods, we compute enclosures to the solutions of these equations as the coefficients for the nonlinear terms range over the previously computed enclosures on the neighborhood. This technique effectively confines the difficulties of the infinite-dimensional trajectory space to a local neighborhood, where they can be contained by rigorous approximation. While our approach can in principle be applied to compute a complete optimal control policy over the entire state space for a given target, practical efficiency in most cases demands adaptive restriction of the state space to trajectories between particular start and goal subregions. We introduce a bidirectional "bounded path" algorithm, generalizing efficient graph shortest path algorithms, which permits simultaneously identifying the shortest path(s) in the transition graph-to direct adaptive refinement-and identifying state space subregions whose intersecting path bounds exceed a threshold-to prune subregions that cannot intersect optimal trajectories. By expanding a generally nonconvex dynamical flow to a finite graph admitting this Dijkstra-like search procedure, the transition graph may be seen as "unfolding" the state space to leverage some of the same efficiencies as level-set methods for convex dynamical systems. The structure of our method yields additional practical advantages. It is the first global trajectory optimization method indifferent to the forms of the optimal controls, requiring no prior knowledge and dealing naturally with unbounded controls, singular arcs, and certain types of control constraints. By augmenting the state space to represent additional objective functions, it can provide adaptive sampling enclosures of a bounded Pareto front, directly according to the refinement of the state space and independent of further user input. Finally, its persistent data structures built on state space decomposition can provide reusable "maps" indicating regions of interest, that can jump-start refinement for related trajectory optimization problems with small variations in their defining parameters, as may readily arise in engineering design. We demonstrate the behavior of our method first on two simple trajectory optimization problems (single- and multiobjective) for illustrative purposes, and then on two more complex problems (single- and multiobjective) related to current problems of interest in astrodynamics and robotics (respectively). In each case, our results prove consistent with known or strongly conjectured solutions for these problems obtained from highly problem-specific analysis, and overcome the apparent limitations of a benchmark direct multiple shooting method. We also discuss the potential for our method to address important open problems in spaceflight trajectory optimization, given future work to improve the scalability of our implementation.