This contribution investigates local differential techniques for estimating optical flow and its derivatives based on the brightness change constraint. By using the tensor calculus representation we build the Taylor expansion of the gray-value derivatives as well as of the optical flow in a spatiotemporal neighborhood. Such a formulation simplifies a unifying framework for all existing local differential approaches and allows to derive new systems of equations to estimate the optical flow alld its derivatives. We also tested various optical flow estimation approaches on real image sequences recorded by a calibrated camera fixed on the arm of a robot. By moving the arm of the robot along a precisely defined trajectory we can determine the true displacement rate of scene surface elements projected into the image plane and compare it quantitatively with the results of different optical flow estimators.
Dynamic environments have obstacles that unpredictably appear, disappear, or move. We present the first sampling-based replanning algorithm that is asymptotically optimal and single-query (designed for situation in which a priori offline computation is unavailable). Our algorithm, RRTX, refines and repairs the same search-graph over the entire duration of navigation (in contrast to previous single-query replanning algorithms that prune and then regrow some or all of the search-tree). Whenever obstacles change and/or the robot moves, a graph rewiring cascade quickly remodels the existing search-graph and repairs its shortest-path-to-goal sub-tree to reflect the new information. Both graph and tree are built directly in the robot’s state-space; thus, the resulting plan(s) respect the kinematics of the robot and continue to improve during navigation. RRTX is probabilistically complete and makes no distinction between local and global planning, yet it reacts quickly enough for real-time high-speed navigation through unpredictably changing environments. Low information transfer time is essential for enabling RRTX to react quickly in dynamic environments; we prove that the information transfer time required to inform a graph of size n about an ε-cost decrease is O(n log n) for RRTX—faster than other current asymptotically optimal single-query algorithms (we prove RRT* is Ω ( normaln ( normaln log 0.25em normaln ) 1 false/ normalD ) and RRT# is ω (n log2 n)). In static environments RRTX has the same amortized runtime as RRT and RRT*, Θ(log n), and is faster than RRT#, ω (log2 n). In order to achieve O(log n) iteration time, each node maintains a set of O(log n) expected neighbors, and the search-graph maintains ε-consistency for a predefined ε. Experiments and simulations confirm our theoretical analysis and demonstrate that RRTX is useful in both static and dynamic environments.
In (Otte and Correll 2013) we present C-FOREST, a parallelization framework for single-query samplingbased shortest-path planning algorithms. C-FOREST has been observed to have super linear speedup on many problems, e.g., paths of quality Ltarget are found 350X faster by 64 CPUs working in parallel than by 1 CPU. In (Otte and Correll 2013) C-FOREST is tested in conjunction with the RRT* algorithm. In the current work we perform additional experiments that show C-FOREST provides similar advantages when used conjunction with the SPRT algorithm. This reinforces our original claim that C-FOREST is generally applicable to a wide range of sampling based motion planning algorithms.
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