This paper uses the ellipsoidal parameters associated with volume moments of inertia of a bounded solid object to construct a motion sweep joining two poses of the solid object, in contrast to earlier works on motion interpolation in SE(3) without taking into account the shape of the moving object. The paper borrows the concept of shape-dependent object norms introduced by Kazerounian and Rastegar [1] and refined by Chirikjian and Zhou [2] to compute as a metric the average of the squared distances (or ASD) among all homologous points of the bounded body between two given poses and seeks to obtain an optimal interpolating motion that minimizes a combination of two ASD distances from each intermediate pose to the two given poses. It is found that the ASD minimizing motion sweep is a novel straight-line motion such that while the centroid of the object follows a straight line, the orientation of the object is constrained so that the ASD metric is minimized. Furthermore, the rotational component can be determined by polar decomposition of the linearly interpolated rotation matrices, scaled by the object's inertial parameters. As an illustration of one of its applications, this motion sweep is repeatedly applied using the de Casteljau algorithm to generate Bézier-like freeform motions, whose paths are in general dependent on the shape of the inertia ellipsoid.

The moving platform of a conventional parallel mechanism is typically connected to the ground link via serial kinematic chains or limbs. When interconnections are created among the limbs, they form coupling sub-chains that can provide additional constraints to the motion of the platform. The resulting complex mechanism offers the potential for improved performance and functionality of the overall mechanism. The basic building block of the parallel mechanism with coupling sub-chain is found to be a two-loop mechanism. This paper presents a screw theory based general method for synthesizing two-loop mechanism by adding an open chain on top of an existing closed chain for a specified combination of rotational and translational degrees of freedom of the end-effector. A comprehensive set of two-loop mechanisms have been presented for various combinations of rotational and translational degrees of freedom.

In recent years, there has been an increasing interest in developing geometric algorithms for kinematic computations. The aim of this paper is to present the notion of kinematic convexity as a key element for a new framework for spherical kinematic geometry that allows for the development of more elegant and efficient algorithms for geometric computations in kinematic applications. The resulting framework, called computational spherical kinematic geometry, is developed by combining the oriented projective geometry with the kinematic geometry of spherical motions. By extending the idea of convexity in affine geometry to an oriented image space of spherical displacements, the notion of kinematic convexity is proposed. A novel application to the collision prediction problem is presented to illustrate the theory developed.

Presents an approach to 5-axis CNC tool path generation for sculptured surface machining with a flat-end cutter. Rational Bezier and B-spline motions are used to plan cutter motions so that an exact representation of the effective cutting shape can be obtained. The exact representation leads to an accurate computation of the scallop curve generated by two adjacent tool paths. Two examples are given to show how this result can be used to accurately plan and verify tool paths for 5-axis CNC milling of sculptured surfaces.

Current practice in CAD/CAM separates the process of geometric shape design from that of CNC (computerised numerical control) tool path generation. Although successful for 3-axis machining, this strategy is not as effective for 5-axis machining of sculptured surfaces due to the difficulty in generating accurate and gouge-free-5-axis tool paths from the CAD models. This paper presents a framework for a new, kinematics-based methodology for geometric modeling that integrates sculptured-surface design with too-path generation for 5-axis CNC machining. In particular, sculptured surfaces are represented directly in terms of rational Bezier and B-spline motions of the cutting tool. The proposed framework brings together the fields of Kinematics and computer aided geometric design as well as the recent results in computer aided motion synthesis and swept volume modeling, and provides a basis for concurrent geometric shape design and manufacture.

This paper deals with joint trajectory synthesis for robot manipulators with consideration of the dynamic response limitations of their actuators. During a robot motion, the higher harmonics present in the actuating torques (forces) may excite the natural modes of oscillation of the robot manipulator. The purpose of this paper is to develop a method for synthesizing joint trajectories that are least susceptible to vibrational excitation. The authors show that a four-harmonic trajectory pattern corresponds to a special class of rational Bezier curves with a set of specially selected weights. The authors study C/sup 3/ conditions for piecing two low-harmonic rational Bezier curve-segments and develop a geometric algorithm for constructing low-harmonic rational spline curves. Finally, the authors discuss how these low-harmonic curves may be used to design joint trajectories that are least susceptible to vibrational excitation.

The calibration of spatial relationships among CAD models of objects used in offline robot programming is discussed. An approach based on a Clifford algebra is used, and simple algorithms for this kinematic localization problem are developed. The position and orientation of each object in a workcell are represented by multivectors of different ranks and the localization problem is converted into that of solving a set of multivector equations. Given locations of points in two frames, an averaging process which yields the best orientation as the root of four linear equations and the best position as a linear combination of the best rotation and the centroids of two point systems is developed. If normal vectors are also sensed, the Clifford algebra representation facilitates definition of a measure of orientation error compatible with the position error.>

The characterization of the relative position of parts in an assembly can be viewed as a generalized robot kinematics problem in which the set of acceptable assemblies is the workspace of the system. The authors show how the Clifford algebra of projective space provides algebraic manifolds characterizing this allowable movement in an assembly. They examine six basic constraints that represent the mating of cylindrical, spherical, and polyhedral features: pure rotation about a line, pure rotation about a point, pure translation along a line, contact of a point with a plane, contact of a plane with a point, and contact of a line with a line. The authors derive parametric and algebraic formulas for the six primitive functional constraints. To illustrate how this theory gives a geometric form to functional constraints, the hyperboloid that defines the planar peg-in-hole constraint is derived. It is shown that the constraint that two pegs fit into two holes is the intersection of two hyperboloids.>

An algebraic formulation of the boundaries of configuration-space obstacles for wrist-partitioned spatial robots is presented. When the end-effector of the robot moves in contact with an obstacle, it is constrained not only by the robot reachability constraints but also by the link-obstacle contact constraints. The reachability constraint is modeled by a chain of two spherical joints, and the contact constraint is a combination of spherical joint and planar joint. Each of these constraints defines a manifold in the 6D space of rigid displacements. Parameterized and algebraic expressions defining these manifolds are obtained using dual quaternions. The obstacle boundary is obtained from the intersection of the manifolds associated with two types of constraints. An example is provided to show how this formulation leads to equations for the boundary of a joint obstacle for a PUMA robot.>

Parameterized equations for the boundaries of joint space obstacles for planar robots are derived. The boundary of a joint obstacle prescribes the link motion of the closed chain formed by the robot in contact with a given obstacle. The theory of kinematic mappings is used to represent the link motion as a set of points in a projective three space. The link positions that are feasible to the arm define a manifold in this space, termed its reachable manifold. The link positions satisfying contact conditions define another manifold in the same space called the contact manifold. Their intersection is mapped to joint space to obtain equations defining the obstacle boundary. While the theory applies to general planar robots, the authors focus on the 3R robot arm. An example is also provided in which the boundaries of joint obstacles are computed.>