Darboux frame

Darboux frame

A Darboux frame is a set of three vectors that are associated with a point on a part surface to be machined. These vectors include the unit normal vector, nP, to the part surface, as well as two mutually orthogonal principal unit tangent vectors, t1.P and t2.P, to the part surface. The defining characteristic of a Darboux frame is that the unit normal vector is equal to the cross product of the two principal unit tangent vectors.From: Generation of Surfaces [2019]

Generic Gear Shapes

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Published in Stephen P. Radzevich, Theory of Gearing, 2018

Stephen P. Radzevich

The straight line segment of an actual axial profile of the generic gear surface can be tilted at an angle, ϑgs, as schematically shown in Figure 21.9. The angle, ϑgs, measured in the counterclockwise direction, is considered positive. The orientation of the Darboux frame, ngst1.gst2.gs, of the actual generic gear surface in relation to the Darboux frame, ngat1.gat2.ga, of the desired generic gear surface is specified by the angle, ϑbs. The trihedron, ngst1.gst2.gs, is turned about the unit vector, t1.ga, in a counterclockwise direction through the angle, ϑgs (see Figure 21.9a).

Models for ground vehicle control on nonplanar surfaces

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Journal Information

Published in Vehicle System Dynamics, 2023

Thomas Fork, H. Eric Tseng, Francesco Borrelli

For instance, we could consider a surface parameterised by a 3D centreline curve and lateral offset, yielding a ribbon-like shape for the resulting road, such as in [9,10,12,24]. Mathematically, this takes the form: where is the tangent vector of the curve and is a the lateral offset direction. The Frenet frame [18] is the case where both are restricted to a plane; we can allow 3D shape by relaxing this assumption, for instance with the Darboux frame [21, pg. 261]. Alternatively, one can specify these vectors directly, i.e. with Tait-Bryan angles [14]

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Generic Gear Shapes

Models for ground vehicle control on nonplanar surfaces