7 research outputs found
Unified Spatial Intersection Algorithms Based on Conformal Geometric Algebra
Conformal Geometric Algebra has been introduced into geographic information science as a mathematical theory because of its advantages in terms of uniform multidimensional representation and computation. The traditional intersection computation between two geometric objects of different types is not unified. In this study, we propose algorithms based on Conformal Geometric Algebra to determine the spatial relationships between geographic objects in a unified manner. The unified representation and intersection computation can be realized for geometric objects of different dimensions. Different basic judgment rules are provided for different simple geometries. The algorithms are designed and implemented using MapReduce to improve the efficiency of the algorithms. From the results of several experiments we provide, the correctness and effectiveness of the algorithms can be verified
An Optimized Architecture for CGA Operations and Its Application to a Simulated Robotic Arm
Conformal geometric algebra (CGA) is a new geometric computation tool that is attracting growing attention in many research fields, such as computer graphics, robotics, and computer vision. Regarding the robotic applications, new approaches based on CGA have been proposed to efficiently solve problems as the inverse kinematics and grasping of a robotic arm. The hardware acceleration of CGA operations is required to meet real-time performance requirements in embedded robotic platforms. In this paper, we present a novel embedded coprocessor for accelerating CGA operations in robotic tasks. Two robotic algorithms, namely, inverse kinematics and grasping of a human-arm-like kinematics chain, are used to prove the effectiveness of the proposed approach. The coprocessor natively supports the entire set of CGA operations including both basic operations (products, sums/differences, and unary operations) and complex operations as rigid body motion operations (reflections, rotations, translations, and dilations). The coprocessor prototype is implemented on the Xilinx ML510 development platform as a complete system-on-chip (SoC), integrating both a PowerPC processing core and a CGA coprocessing core on the same Xilinx Virtex-5 FPGA chip. Experimental results show speedups of 78x and 246x for inverse kinematics and grasping algorithms, respectively, with respect to the execution on the PowerPC processor
Contributions to automated realtime underwater navigation
Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution February 2012This dissertation presents three separate–but related–contributions to the art of underwater
navigation. These methods may be used in postprocessing with a human in
the loop, but the overarching goal is to enhance vehicle autonomy, so the emphasis is
on automated approaches that can be used in realtime. The three research threads
are: i) in situ navigation sensor alignment, ii) dead reckoning through the water column,
and iii) model-driven delayed measurement fusion. Contributions to each of
these areas have been demonstrated in simulation, with laboratory data, or in the
field–some have been demonstrated in all three arenas.
The solution to the in situ navigation sensor alignment problem is an asymptotically
stable adaptive identifier formulated using rotors in Geometric Algebra. This
identifier is applied to precisely estimate the unknown alignment between a gyrocompass
and Doppler velocity log, with the goal of improving realtime dead reckoning
navigation. Laboratory and field results show the identifier performs comparably to
previously reported methods using rotation matrices, providing an alignment estimate
that reduces the position residuals between dead reckoning and an external acoustic
positioning system. The Geometric Algebra formulation also encourages a straightforward
interpretation of the identifier as a proportional feedback regulator on the
observable output error. Future applications of the identifier may include alignment
between inertial, visual, and acoustic sensors.
The ability to link the Global Positioning System at the surface to precision dead
reckoning near the seafloor might enable new kinds of missions for autonomous underwater
vehicles. This research introduces a method for dead reckoning through
the water column using water current profile data collected by an onboard acoustic
Doppler current profiler. Overlapping relative current profiles provide information to
simultaneously estimate the vehicle velocity and local ocean current–the vehicle velocity
is then integrated to estimate position. The method is applied to field data using
online bin average, weighted least squares, and recursive least squares implementations.
This demonstrates an autonomous navigation link between the surface and the
seafloor without any dependence on a ship or external acoustic tracking systems. Finally, in many state estimation applications, delayed measurements present an
interesting challenge. Underwater navigation is a particularly compelling case because
of the relatively long delays inherent in all available position measurements. This research
develops a flexible, model-driven approach to delayed measurement fusion in
realtime Kalman filters. Using a priori estimates of delayed measurements as augmented
states minimizes the computational cost of the delay treatment. Managing
the augmented states with time-varying conditional process and measurement models
ensures the approach works within the proven Kalman filter framework–without
altering the filter structure or requiring any ad-hoc adjustments. The end result is
a mathematically principled treatment of the delay that leads to more consistent estimates
with lower error and uncertainty. Field results from dead reckoning aided
by acoustic positioning systems demonstrate the applicability of this approach to
real-world problems in underwater navigation.I have been financially supported by:
the National Defense Science and Engineering Graduate (NDSEG) Fellowship administered
by the American Society for Engineering Education, the Edwin A. Link
Foundation Ocean Engineering and Instrumentation Fellowship, and WHOI Academic
Programs office
Doctor of Philosophy
dissertationWhile boundary representations, such as nonuniform rational B-spline (NURBS) surfaces, have traditionally well served the needs of the modeling community, they have not seen widespread adoption among the wider engineering discipline. There is a common perception that NURBS are slow to evaluate and complex to implement. Whereas computer-aided design commonly deals with surfaces, the engineering community must deal with materials that have thickness. Traditional visualization techniques have avoided NURBS, and there has been little cross-talk between the rich spline approximation community and the larger engineering field. Recently there has been a strong desire to marry the modeling and analysis phases of the iterative design cycle, be it in car design, turbulent flow simulation around an airfoil, or lighting design. Research has demonstrated that employing a single representation throughout the cycle has key advantages. Furthermore, novel manufacturing techniques employing heterogeneous materials require the introduction of volumetric modeling representations. There is little question that fields such as scientific visualization and mechanical engineering could benefit from the powerful approximation properties of splines. In this dissertation, we remove several hurdles to the application of NURBS to problems in engineering and demonstrate how their unique properties can be leveraged to solve problems of interest