Optimal Path Smoothing with Log-Aesthetic Curves based on Shortest Distance, Minimum Bending Energy and Curvature Variation Energy

Introduction: Path planning is a process of constructing a desired movement from an initial position into discrete motions which satisfy given constraints before reaching to the nal location. In practice, there are two common formulation used in path planning; either parametric representation (Beziers and BSplines) or nonlinear representation (clothoid generation with Fresnal Integral). Even though cubic Bezier/Bpline parametric representation has been widely used in CAD/CAM practices, it imposes several unwanted characteristics, e.g. complicated curvature/arc-length computation and curvature extrema. Howewer, natural spirals do not have these problems except for the generation of the spiral itself which involves integration. With the advancement of computers, numerical integration can be carried out with minimal e ort while preserving high precision. Runge-Kutta methods can also be to generate spiral [3] which greatly reduces computation time. There are also attempts to represent spirals by means of Beziers [11], however these curves lose their degree of freedom while satisfying curvature monotonicity to mimic spiral. Typical curves used to replace polyline path include Dubins path; the combination of line segments and circular arcs. It is one of the most popular choice in path smoothing [1]. Dubin paths are widely used for path planning but it only satis es G continuity. Highway design and railyway route planning has somewhat similar design procedure as compared to path planning. The underlying property is to satisfy given G data which comprises of position, tangent and curvature at the endpoints. Hickerson [5] stated that highway designers must avoid sudden changes between curves with di erent curvatures or long tangents. He proposed using gradual increase/decrease types of curvatures which is in fact the main feature of spirals. Baass [2] simpli ed G highway design into ve cases using clothoid; (1) Line to circle with a single spiral, (2) Circle to circle forming S-curve with a pair of spirals, (3) Circle to circle forming C-curve with a pair of spirals, (4) Circle to circle with a single spiral, and (5) Line to line with a pair of spiral. These cases are the building block to design highway/railway design. There are many attempts to solve these cases using various types of curves, e.g., Bezier spiral, Pythagorean Hodograph spirals etc. The solution are either unique for given G data or no solution since the proposed method is curve centric. Hence, designers have no option on nding optimum path which minimizes arc-length, bending energy or curvature variation.