Arc lengths of rational Pythagorean–hodograph curves

Abstract The present paper outlines a method of shape synthesis using intrinsic geometry to be used for two-dimensional shape optimization problems. It is observed that the shape of a curve can be defined in terms of intrinsic parameters such as the curvature as a function of the arc length. The method of shape synthesis, proposed here, consists of selecting a shape model, defining a set of shape design variables and then evaluating Cartesian coordinates of a curve. A shape model is conceived as a set of continuous piecewise linear segments of the curvature; each segment defined as a function of the arc length. The shape design variables are the values of curvature and/or arc lengths at some of the end-points of the linear segments. The proposed method of shape synthesis and optimization is general in nature. It has been shown how the proposed method can be used to find the optimal shape of a planar Variable Geometry Truss (VGT) manipulator for a pre-specified position and orientation of the end-effector. In conclusion, it can be said that the proposed approach requires fewer design variables as compared to the methods where shape is represented using spline-like functions.

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Path Planning of Multiple UAV’s in an Environment of Restricted Regions

Dynamic Systems and Control, Parts A and B ◽

10.1115/imece2005-79682 ◽

2005 ◽

Author(s):

Madhavan Shanmugavel ◽

Antonios Tsourdos ◽

Rafal Zbikowski ◽

Brian White

Keyword(s):

Path Planning ◽

Path Length ◽

Mathematical Property ◽

Multiple Uavs ◽

Safety Margins ◽

Pythagorean Hodograph ◽

Offset Curves ◽

Safety Constraints ◽

Continuous Curvature ◽

Ph Curve

This paper describes a novel idea of path planning for multiple UAVs (Unmanned Aerial Vehicles). The path planning ensures safe and simultaneous arrival of the UAVs to the target while meeting curvature and safety constraints. Pythagorean Hodograph (PH) curve is used for path planning. The PH curve provides continuous curvature of the paths. The offset curves of the PH paths define safety margins around and along each flight path. The simultaneous arrival is satisfied by generation of paths of equal lengths. This paper highlights the mathematical property — changing path-shape and path-length by manipulating the curvature and utilises this to achieve the following constraints: (i) Generation of paths of equal length, (ii) Achieving maximum bound on curvature, and, (iii) Meeting the safety constraints by offset paths.

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