Parallelism and Robotics: The Perfect Marriage By R.Theron,F.J.Blanco,B.Curto,V.Moreno and F.J.Garcia University of Salamanca,Spain Rejitha Anand CMPS 5433

INTRODUCTION Robotics is a fast evolving area of scientific study Autonomous robots Programmed to achieve specific objective Goal - formulate a sequence of intelligent movements Need parallel processing techniques to manage heavy computational load Path planning- search for an optimal route from an initial location to a desired one avoiding collisions

TERMINOLOGIES Fourier transform – decomposes or separates a waveform or function into sinusoids of different frequency which sum to the original waveform Discrete Fourier transform(DFT) – numerical algorithm to compute Fourier transform Fast Fourier Transform – is a DFT algorithm which reduces number of computations form N 2 to N log N

TERMINOLOGIES (continued) Convolution – is an integral that expresses the amount of overlap of one function g as it is shifted over another function f, therefore blends one function with another Convolution theorem – states Fourier transform of a convolution is the product of Fourier transforms Master-slave paradigm – master task allocates problem. Slave requests and executes problem from master. Slaves can also send new tasks to manager for allocation to other slaves

KNOWING THE ENVIRONMENT Global planning – requires total knowledge of the environment Configuration space (C-space) represents the workspace of the robot position and orientation of robot defined by a single point Evaluation of C-space is computational intensive

CALCULATION OF C-OBSTACLES Curto and Moreno proposed mathematical formalism through evaluation of a convolution of two functions one describes the robot A and other describes the obstacles B Convolution theorem states Fourier transforms can be used CB the function that describes the C-obstacles

TOWARDS A FASTER EVALUATION Solving the equation  Need to evaluate the integral product of Fourier transforms of N m-r configurations and values of space variables -N is the resolution chosen to describe the discrete workspace - m is the dimension of the c-space -r the Fourier transform dimension  When convolution does not exist integration performed provides new opportunities for parallelization

THREE LEVELS OF PARALLELISM  Fourier transform level(FFT): the Fourier transform algorithm which is inherently parallel is the computational tool to calculate this level  Space variable level: for those variables where convolution can’t be found (x r +1,…,x n ), it is necessary to perform an integration which may be processed in parallel  Configuration variable level: in this case some configuration variables(q 1,…,q r ) the convolution must be performed; trivial parallelization exists for configuration variables where there is no convolution (q r+1,…,q m )

EXAMPLE  Consider a 2D mobile robot has three configuration variables - (x,y) - which describe the position that convolute with the workspace coordinates - Theta – defines the angle of rotation and does not convolute  two levels of parallelism is possible – the Fourier transform level and the configuration variable level ( for the accumulation of convolutions products over the values of theta)

MASTER SLAVE APPROACH The FFT level is exploited. The master asks slaves to perform the computation of a Fourier transform, and they proceed in parallel with the well known FFT algorithm

MASTER SLAVE APPROACH Shows the configuration variable level, where again there is a master and n slave processes. The master asks the slaves to evaluate a slice of the C- space, that is, the convolution product for one value of the configuration variable for each slave. Then the partial results are gathered and the final C-space is built.

MASTER SLAVE APPROACH a more complex model is where three level solution is exploited. Sub master processes organize the communications of partial results, that are then communicated to the master. This method can be seen as a single level nested solution.

A CASE STUDY: IMPLEMENTING THE PARALLEL ALGORITHM  Implemented to evaluate configuration space of a mobile robot moving on a 3D workspace partially occupied with obstacles  Robot has a planar movement – position defined by three configuration variables (x r, y r, theta r )  Expression for C-space evaluation

POSSIBILITIES  First the algorithm consists of iteratively performing a series of basic operations one for each possible robot orientation,theta r, (i.e., the configuration variable level)  Second, each orientation needs to accumulate the convolution products for each plane z (i.e., the spatial variable level)  These calculations are independent, therefore they can be done in parallel. These intermediate results are integrated to form a final result (instance of data parallelism model).

MASTER  Master communicates the tasks that need to be calculated to each of its slaves  After receiving the result (the slice of the corresponding C- space) from a slave the master assigns a new task.  Finally, the master indicates to the slaves that no more calculations are required. During this process the master builds a three-dimensional array representing the C-space.

SLAVES  Need to know which orientation to calculate (determined by the master with the number of task)  The slave builds the needed robot bitmap for that orientation and a z plane and obtains its FFT  Carries out the point-to-point product of this FFT with the transformed workspace that the master has provided  Procedure is repeated a number of times as determined by the highest point of the mobile robot, and the results are added (i.e., the accumulation of the spatial variable level)  Inverse transformation applied to the accumulated result to obtain the portion of the C-space  This intermediate result is in turn, sent to the master, expecting another task to be assigned or a end of execution signal.

RESULTS  Two different platforms used to validate the implementation - A Silicon Graphics Origin 200 computer with four MIPS R10000 processors Mbytes of memory, and four workstations (PII 266Mhz with 64Mbytes of memory) connected by a FastEthernet network  Used MPI to develop both implementations  The results are similar, however, more problems were encountered with the networked workstations because of communication difficulties.

DISCRETIZATIONS In order to work with the FFT algorithm employed for the needed evaluation of 2D Fourier transforms, three discretizations of the 3D workspace have been used: - 64 x 64 x 64, x 128 x and 256 x 256 x 256

Execution times (seconds) for 64 x 64 resolution SequentialParallelRobot height Speedup

Execution times (seconds) for 128 x 128 resolution SequentialParallelRobot height Speedup

Execution times (seconds) for 256 x 256 resolution SequentialParallelRobot height Speedup

CONCLUSION Limitations of test environment prompted development and implementation of a solution optimizing at the configuration variable level Different experiments have been carried out in which the resulting speedups are very acceptable The presented case study provides reasonable results that could be extrapolated to more complex robotic structures More complex structures can hopefully be designed on the lines of the parallel algorithm studied in this paper

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