Brian Ichter

Deterministic Sampling-Based Motion Planning: Optimality, Complexity, and Performance

Probabilistic sampling-based algorithms, such as the probabilistic roadmap (PRM) and the rapidly-exploring random tree (RRT) algorithms, represent one of the most successful approaches to robotic motion planning, due to their strong theoretical properties (in terms of probabilistic completeness or even asymptotic optimality) and remarkable practical performance. Such algorithms are probabilistic in that they compute a path by connecting independent and identically distributed (i.i.d.) random points in the configuration space. Their randomization aspect, however, makes several tasks challenging, including certification for safety-critical applications and use of offline computation to improve real-time execution. Hence, an important open question is whether similar (or better) theoretical guarantees and practical performance could be obtained by considering deterministic, as opposed to random sampling sequences. The objective of this paper is to provide a rigorous answer to this question. The focus is on the PRM algorithm—our results, however, generalize to other batch-processing algorithms such as \(\text {FMT}^*\). Specifically, we first show that PRM, for a certain selection of tuning parameters and deterministic low-dispersion sampling sequences, is deterministically asymptotically optimal, i.e., it returns a path whose cost converges deterministically to the optimal one as the number of points goes to infinity. Second, we characterize the convergence rate, and we find that the factor of sub-optimality can be very explicitly upper-bounded in terms of the \(\ell _2\)-dispersion of the sampling sequence and the connection radius of PRM. Third, we show that an asymptotically optimal version of PRM exists with computational and space complexity arbitrarily close to O(n) (the theoretical lower bound), where n is the number of points in the sequence. This is in stark contrast to the \(O(n\, \log n)\) complexity results for existing asymptotically-optimal probabilistic planners. Finally, through numerical experiments, we show that planning with deterministic low-dispersion sampling generally provides superior performance in terms of path cost and success rate.

Real-time stochastic kinodynamic motion planning via multiobjective search on GPUs

In this paper we present the PUMP (Parallel Uncertainty-aware Multiobjective Planning) algorithm for addressing the stochastic kinodynamic motion planning problem, whereby one seeks a low-cost, dynamically-feasible motion plan subject to a constraint on collision probability (CP). To ensure exhaustive evaluation of candidate motion plans (as needed to tradeoff the competing objectives of performance and safety), PUMP incrementally builds the Pareto front of the problem, accounting for the optimization objective and an approximation of CP. This is performed by a massively parallel multiobjective search, here implemented with a focus on GPUs. Upon termination of the exploration phase, PUMP searches the Pareto set of motion plans to identify the lowest cost solution that is certified to satisfy the CP constraint (according to an asymptotically exact estimator). We introduce a novel particle-based CP approximation scheme, designed for efficient GPU implementation, which accounts for dependencies over the history of a trajectory execution. We present numerical experiments for quadrotor planning wherein PUMP identifies solutions in ∼100 ms, evaluating over one hundred thousand partial plans through the course of its exploration phase. The results show that this multiobjective search achieves a lower motion plan cost, for the same CP constraint, compared to a safety buffer-based search heuristic and repeated RRT trials.

Downwind pre-aligned rotor for a 13.2 mw wind turbine

To alleviate the mass-scaling issues associated with conventional upwind rotors of extreme-scale turbines, a downwind rotor concept is considered that uses coning and curvature to align the non-circumferential loads for a given steady-state condition. This alignment can be pre-set to eliminate downwind blade moments for a given steady-state condition near rated wind speed and to minimize them for other conditions. The alleviation in downwind cantilever loads may enable a reduced structural blade mass as compared with a conventional upwind rotor. Previous quasi-steady scaling analysis indicates that this cantilever load alleviation becomes significant for extreme-scale systems (10-20 MW). To examine the potential impact of this design, Finite Element Analysis (FEA) was conducted for a 13.2 MW rated turbine at steady-state conditions for two rotor configurations with similar power outputs: 1) a conventional upwind rotor with three blades and 2) a downwind pre-aligned rotor (DPAR) with two blades. Based on previous work, the pre-aligned rotor configuration was set based on steady-state loads at a wind speed equal to 1.25 times the rated wind speed. By keeping the blade mass about the same between these two configurations, the rotor mass was reduced by approximately one third for the DPAR configuration. In addition, the average stresses on the blades for a variety of steady-state wind speeds was reduced for the DPAR configuration. However, these results can only be considered to be qualitative in terms of impact on turbine mass and cost. In particular, simulations at non-ideal, extreme and unsteady conditions are needed to determine the viability of this concept.