Construction of periodic adapted orthonormal frames on closed space curves

The construction of continuous adapted orthonormal frames along C1 closed–loop spatial curves is addressed. Such frames are important in the design of periodic spatial rigid–body motions along smooth closed paths. The construction is illustrated through the simplest non–trivial context — namely, C1 closed loops defined by a single Pythagorean–hodograph (PH) quintic space curve of a prescribed total arc length. It is shown that such curves comprise a two–parameter family, dependent on two angular variables, and they degenerate to planar curves when these parameters differ by an integer multiple of π. The desired frame is constructed through a rotation applied to the normal–plane vectors of the Euler–Rodrigues frame, so as to interpolate a given initial/final frame orientation. A general solution for periodic adapted frames of minimal twist on C1 closed–loop PH curves is possible, although this incurs transcendental terms. However, the C1 closed–loop PH quintics admit particularly simple rational periodic adapted frames

C2 interpolation of spatial data subject to arc-length constraints using Pythagorean-hodograph quintic splines

International audienceIn order to reconstruct spatial curves from discrete electronic sensor data, two alternative C2 Pythagorean-hodograph (PH) quintic spline formulations are proposed, interpolating given spatial data subject to prescribed constraints on the arc length of each spline segment. The first approach is concerned with the interpolation of a sequence of points, while the second addresses the interpolation of derivatives only (without spatial localization). The special structure of PH curves allows the arc-length conditions to be expressed as algebraic constraints on the curve coefficients. The C2 PH quintic splines are thus defined through minimization of a quadratic function subject to quadratic constraints, and a close starting approximation to the desired solution is identified in order to facilitate efficient construction by iterative methods. The C2 PH spline constructions are illustrated by several computed examples

Smooth path planning with Pythagorean-hodoghraph spline curves geometric design and motion control

This thesis addresses two significative problems regarding autonomous systems, namely path and trajectory planning. Path planning deals with finding a suitable path from a start to a goal position by exploiting a given representation of the environment. Trajectory planning schemes govern the motion along the path by generating appropriate reference (path) points. We propose a two-step approach for the construction of planar smooth collision-free navigation paths. Obstacle avoidance techniques that rely on classical data structures are initially considered for the identification of piecewise linear paths that do not intersect with the obstacles of a given scenario. In the second step of the scheme we rely on spline interpolation algorithms with tension parameters to provide a smooth planar control strategy. In particular, we consider Pythagorean–hodograph (PH) curves, since they provide an exact computation of fundamental geometric quantities. The vertices of the previously produced piecewise linear paths are interpolated by using a G1 or G2 interpolation scheme with tension based on PH splines. In both cases, a strategy based on the asymptotic analysis of the interpolation scheme is developed in order to get an automatic selection of the tension parameters. To completely describe the motion along the path we present a configurable trajectory planning strategy for the offline definition of time-dependent C2 piece-wise quintic feedrates. When PH spline curves are considered, the corresponding accurate and efficient CNC interpolator algorithms can be exploited

Smooth path planning with Pythagorean-hodoghraph spline curves geometric design and motion control

This thesis addresses two significative problems regarding autonomous systems, namely path and trajectory planning. Path planning deals with finding a suitable path from a start to a goal position by exploiting a given representation of the environment. Trajectory planning schemes govern the motion along the path by generating appropriate reference (path) points. We propose a two-step approach for the construction of planar smooth collision-free navigation paths. Obstacle avoidance techniques that rely on classical data structures are initially considered for the identification of piecewise linear paths that do not intersect with the obstacles of a given scenario. In the second step of the scheme we rely on spline interpolation algorithms with tension parameters to provide a smooth planar control strategy. In particular, we consider Pythagorean\u2013hodograph (PH) curves, since they provide an exact computation of fundamental geometric quantities. The vertices of the previously produced piecewise linear paths are interpolated by using a G1 or G2 interpolation scheme with tension based on PH splines. In both cases, a strategy based on the asymptotic analysis of the interpolation scheme is developed in order to get an automatic selection of the tension parameters. To completely describe the motion along the path we present a configurable trajectory planning strategy for the offline definition of time-dependent C2 piece-wise quintic feedrates. When PH spline curves are considered, the corresponding accurate and efficient CNC interpolator algorithms can be exploited

Construction of planar quintic Pythagorean-hodograph curves by control-polygon constraints

In the construction and analysis of a planar Pythagorean–hodograph (PH) quintic curve r(t), t∈[0,1] using the complex representation, it is convenient to invoke a translation/rotation/scaling transformation so r(t) is in canonical form with r(0)=0, r(1)=1 and possesses just two complex degrees of freedom. By choosing two of the five control–polygon legs of a quintic PH curve as these free complex parameters, the remaining three control–polygon legs can be expressed in terms of them and the roots of a quadratic or quartic equation. Consequently, depending on the chosen two control–polygon legs, there exist either two or four distinct quintic PH curves that are consistent with them. A comprehensive analysis of all possible pairs of chosen control polygon legs is developed, and examples are provided to illustrate this control–polygon paradigm for the construction of planar quintic PH curves

Continuous-Time Estimation of Attitude Using B-Splines on Lie Groups

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/140656/1/1.g001149.pd

Path planning of multiple autonomous vehicles

Safe and simultaneous arrival of constant speed, constant altitude UAVs on target is solved by design of paths of equal lengths. The starting point of the solution is the well-known Dubins path which is composed of circular arcs and line segments, thus requiring only one simple manoeuvre - constant rate turn. An explicit bound can be imposed on the rate during the design and the resulting paths are the minimum time solution of the problem. However, transition between arc and line segment entails discontinuous changes in lateral accelerations (latax), making this approach impractical for real fixed wing UAVs. Therefore, the Dubins solution is replaced with clothoid and also a novel one, based on quintic Pythagorean Hodograph (PH) curves, whose latax demand is continuous. The clothoid solution is direct as in the case of the Dubins path. The PH path is chosen for its rational functional form. The clothoid and the PH paths are designed to have lengths close to the lengths of the Dubins paths to stay close to the minimum time solution. To derive the clothoid and the PH paths that way, the Dubins solution is first interpreted in terms of Differential Geometry of curves using the path length and curvature as the key parameters. The curvature of a Dubins path is a piecewise constant and discontinuous function of its path length, which is a differential geometric expression of the discontinuous latax demand involved in transitions between the arc and the line segment. By contrast, the curvature of the PH path is a fifth order polynomial of its path length. This is not only continuous, also has enough design parameters (polynomial coefficients) to meet the latax (curvature) constraints (bounds) and to make the PH solution close to the minimum time one. The offset curves of the PH path are used to design a safety region along each path. The solution is simplified by dividing path planning into two phases. The first phase produces flyable paths while the second phase produces safe paths. Three types of paths are used: Dubins, clothoid and Pythagorean Hodograph (PH). The paths are produced both in 2D and 3D. In two dimensions, the Dubins path is generated using Euclidean and Differential geometric principles. It is shown that the principles of Differential geometry are convenient to generalize the path with the curvature. Due to the lack of curvature continuity of the Dubins path, paths with curvature continuity are considered. In this respect, initially the solution with the Dubins path is extended to produce clothoid path. Latter the PH path is produced using interpolation technique. Flyable paths in three dimensions are produced with the spatial Dubins and PH paths. In the second phase, the flyable paths are tuned for simultaneous arrival on target. The simultaneous arrival is achieved by producing the paths of equal lengths. Two safety conditions: (i) minimum separation distance and (ii) non-intersection of paths at equal distance are defined to maneuver in free space. In a cluttered space, an additional condition, threat detection and avoidance is defined to produce safe paths. The tuning is achieved by increasing the curvature of the paths and by creating an intermediate way-point. Instead of imposing safety constraints, the flyable paths are tested for meeting the constraints. The path is replanned either by creating a new way-point or by increasing the curvature between the way-points under consideration. The path lengths are made equal to that of a reference path.EThOS - Electronic Theses Online ServiceGBUnited Kingdo