Two-parameter Motion Research Articles

Instantaneous kinematics of a planar two-link open chain in complex plane

This research aims to investigate the instantaneous kinematics of the terminal link of a planar two-link open chain using the canonical coordinate system of planar two-parameter motion. It is based on the complex-number technique. We review the higher-order instantaneous invariants of this motion of the terminal link with reference to the first and second-order instantaneous invariants. We apply these instantaneous invariants in complex number form to curvature theory. Finally, we give some examples by choice of different parameters and exhibit the comparisons.

A Sweep Surface based on Two-Parameter Motion

We present a new technique for constructing a sweep surface using two-parameter motion. Firstly, a new rational B-spline motion with two parameters is introduced, which is obtained by extending its orientation curve and scaling curve to surface counterparts. A sweep surface is then defined by a single vertexvunder the two-parameter motion and allows to represent different u-directional iso-curves depending on parameter v. Efficient techniques for modeling and editing the surface are achieved by intuitively controlling the two-parameter motion. We demonstrate the effectiveness of our technique with experimental results on modeling and editing a 3D propeller model.

Automatic Fairing of Two-Parameter Rational B-Spline Motion

This paper deals with the problem of automatic fairing (or fine-tuning) of two-parameter rational B-spline spherical and spatial motions. The results presented in this paper extend the previous results on fine-tuning of one-parameter rational B-spline motions. A dual quaternion representation of spatial displacements is employed and the problem of fairing two-parameter motions is studied as a surface fairing problem in the space of dual quaternions. By combining surface fairing techniques from the field of computer aided geometric design with the computer aided synthesis of freeform rational motions, smoother (C3 continuous) two-parameter rational B-spline motions are generated. Several examples are presented to illustrate the effectiveness of the proposed method. Techniques for motion smoothing have important applications in the Cartesian motion planning, camera motion synthesis, and spatial navigation in virtual reality systems. In particular, smoother two-parameter freeform motions have applications in the development of a kinematic based approach to geometric shape design and in five-axis NC tool path planning.

A genetic algorithm for optical flow estimation

This paper illustrates a new optical flow estimation technique that builds upon a genetic algorithm (GA). First, the current frame is segmented into generic shape regions, using only luminance and color information. For each region, a two-parameter motion model is estimated using a GA. The fittest individuals identified at the end of this step are used to initialize the population of the second step of the algorithm, which estimates a six-parameter affine motion model, again using a GA. The proposed method is compared with a multi-resolution version of the well-known Lucas–Kanade differential algorithm. Our simulations demonstrate that, with respect to Lucas–Kanade, it significantly reduces the energy of the motion-compensated residual error.

Computer Aided Geometric Design of Two-Parameter Freeform Motions

This paper brings together the notion of analytically defined two-parameter motion in Theoretical Kinematics and the notion of freeform surfaces in Computer Aided Geometric Design (CAGD) to develop methods for computer aided design of two-parameter freeform motions. In particular, a rational Be´zier representation for two-parameter freeform motions is developed. It has been shown that the trajectory surface of such a motion is a tensor-product rational Be´zier surface and that such a kinematically generated surface has a geometric as well as a kinematic control structure. The results have not only theoretical interest in CAGD and kinematics but also applications in CAD/CAM and Robotics.

Instantaneous Kinematics of a Tangent-Plane in Two-Parameter Space Motion

A tangent-plane undergoing two-parameter motion envelopes a surface called the tangent-plane envelope. Such surfaces can be considered as the envelope of a two-parameter family of planes or ∞2 family of planes. The properties of the tangent-plane motion are characterized through the properties of the spherical image of the normal to the surface it envelopes. This paper presents a methodology to locate a family of planes that envelope surfaces with similar characteristics.

The Canonical Inverse Velocity and Acceleration Solutions of a Planar Two-Link Open Chain

This paper describes the instantaneous kinematics of the terminal link of a planar two-link open chain using the ca nonical coordinate system of planar two-parameter motion. The first- and second-order instantaneous invariants of the motion of the terminal link are presented. Using a systematic procedure, the results are applied to the inverse velocity and acceleration problems of a two-link open chain. The solutions to the inverse kinematic problems are important, for exam ple, in a robot positioning mechanism. Two examples are presented, a revolute-revolute robot positioner and a pris matic-revolute robot positioner.

�ber zweiparametrige ebene Bewegungsvorg�nge

TheBlaschke-Grunwald-map associates to each surface β in the quasielliptic space a planar euclidean two-parameter motion\(\mathfrak{B}\). Using this map we find kinematic meanings of the Gaussian quasielliptic curvature and furthermore some theorems about the asymptotic motions of\(\mathfrak{B}\). The kinematic equivalent of the relation of left-conjugated tangents in points of β is a regular or singular projectivity σ on each poleline of\(\mathfrak{B}\). This projectivities have been studied already byH. J. E. Beth andW. v.d. Woude. We show that in every position the quadratic curvature transformations of almost all one-parameter motions of\(\mathfrak{B}\) are determined by σ.

Closure to “Discussion of ‘Instantaneous Kinematics of a Special Two-Parameter Motion’” (1977, ASME J. Eng. Ind., 99, p. 340)

Closure to “Discussion of ‘Instantaneous Kinematics of a Special Two-Parameter Motion’” (1977, ASME J. Eng. Ind., 99, p. 340)