Trajectory generation in relative velocity coordinates using mixed integer linear programming with IHDR guidance

Mixed-integer linear programming (MILP) for trajectory generation of mobile robot suffers from nonlinear constraints due to complex obstacle contours and dynamic environment. In this paper, firstly, we introduce a relative velocity coordinates MILP (RVCs-MILP) for solving the nonlinear constraints problem in the trajectory generation of the target pursuit and multiple-obstacle avoidance (TPMOA). The computational load of the RVCs-MILP does not increase with the complexity of obstacle contour but only relates to the number of the obstacles. It can be applied in real time when the number of the obstacles is small. For the large numbers of obstacles avoidance, further, we propose an IHDR based online learning mechanism. It sets up a "scenario-action mapping" knowledge base by continuously offline training and online updating. For a trajectory generation task, it will search a best match path of the current state in the knowledge base according to the external environments and the state of the robot in real time. Simulations are presented in comparison with the evolution algorithms (EA) and IHDR. The former shows significant improvement in a number of aspects. The latter confirms the validation of the proposed IHDR methods

Fractal analyses of some natural systems

Fractal dimensions are estimated by the box-counting method for real world data sets and for mathematical models of three natural systems. 1 he natural systems are nearshore sea wave profiles, the topography of Shei-pa National Park in Taiwan, and the normalised difference vegetation index (NDV1) image of a fresh fern. I he mathematical models which represent the natural systems utilise multi-frequency sinusoids for the sea waves, a synthetic digital elevation model constructed by the mid-point displacement method for the topography and the Iterated Function System (IFS) codes for the fern leaf. The results show that similar fractal dimensions are obtained for discrete sub-sections of the real and synthetic one-dimensional wave data, whilst different fractal dimensions are obtained for discrete sections of the real and synthetic topographical and fern data. The similarities and differences are interpreted in the context of system evolution which was introduced by Mandelbrot (1977). Finally, the results for the fern images show that use of fractal dimensions can successfully separate void and filled elements of the two-dimensional series