The International Journal of Robotics Research

The "optimum" 6R manipulator's singular configurations are used to develop the "best" 4R orientation and 4R regional structures for singularity avoidance. For comparison with the "best" 4R regional structure, the kinematics of a second 4R regional structure is explored. Conceptually, the control algo rithms for the 4R orientation and two 4R regional structures begin in primary mode with three actuator screws apiece. In primary mode the traditional position kinematics approach is used. However, when either the orientation or regional struc ture has difficulty following its Cartesian trajectory, control for that structure goes into secondary mode, where the redundant actuator is called on. The algorithms maintain a sixth-order screw system for an 8R manipulator with bounded actuator rates and well-behaved actuator values. The numerical com putations for an 8R manipulator are comparable to those for the "optimum" 6R manipulator and can be implemented in real time.

The development of useful mathematical techniques for an alyzing swept volumes, together with efficient means of im plementing these methods to produce serviceable models, has important applications to numerically controlled (NC) machin ing, robotics, and motion planning, as well as other areas of automation. In this article a novel approach to swept volumes is delineated—one that fully exploits the intrinsic geometric and group theoretical structure of Euclidean motions in or der to formulate the problem in the context of Lie groups and differential equations.

Precise definitions of sweep and swept volume are given that lead naturally to an associated ordinary differential equation. This sweep differential equation is then shown to be related to the Lie group structure of Euclidean motions and to generate trajectories that completely determine the geometry of swept volumes.

It is demonstrated that the notion of a sweep differential equation leads to criteria that provide useful insights concern ing the geometric and topologic features of swept volumes. Several new results characterizing swept volumes are obtained. For example, a number of simple properties that guarantee that the swept volume is a Cartesian product of elementary mani folds are identified. The criteria obtained may be readily tested with the aid of a computer.

The local nature of inertial, nonlinear, and coupling character istics of manipulators is an impediment to their optimal design. In this article, a method to represent and measure the global and posture-independent measures of inertia, coupling, and nonlinear forces is presented. A new concept, referred to as the global inertia ellipsoid, which represents the global iner tial characteristics of a manipulator, is introduced. The global inertia ellipsoid is obtained by considering a weighted distri bution of the end-effector position over the workspace of the manipulator. Based on this concept, four global parameters, representing the global nonlinear, coupling, and inertial char acteristics of the manipulator, are obtained. The number of global parameters does not change with the degrees of freedom or the complexity of the manipulator structure. A probabilistic or deterministic description of payload range can be included while designing the manipulator. Criteria for the optimal de sign of manipulators are presented in the form of an objective function of the global parameters. The method is illustrated by the design of a 2R manipulator, a 2R manipulator with given payload range, and a 3R manipulator. The extension of the concept of the global inertia ellipsoid to task placement and task planning is discussed.

In this study, global planning of a collision-free trajectory (simultaneous planning of geometric path and its time para meterization) involving potential functions and direct kinematic and dynamic equations of the manipulator is presented. This planning is based on using the necessary conditions of min imum for an integral type criterion. General transversality conditions corresponding to the boundary ones (resulting from the task to be performed) are derived. Thus, closed systems of boundary dependences fully specifying differential equations, which arise from the necessary conditions of extreme for the above criterion, are obtained. As a result, this system renders it possible to reduce the collision-free trajectory planning problem to a two-point boundary value problem. A numerical example involving a manipulator of three-revolute kinematic pairs of the Vth class is considered.

The conditioning index of a serial robotic manipulator is de fined in this article in terms of the reciprocal of its minimum condition number. The condition number of a manipulator is defined, in turn, as that of its Jacobian matrix. Moreover, in defining the Jacobian condition number, a quadratic norm of the Jacobian matrix is needed. However, this norm, or for that matter any other norm, brings about dimensional inho mogeneities. It is shown here that by properly defining the said norm based on a weighting positive definite matrix, the dimensional inhomogeneity is resolved. Manipulators with a conditioning index of 100% are termed isotropic, a six-axis isotropic manipulator being introduced. This manipulator has all its angles between neighboring revolute axes at 90° and all its distances between neighboring axes identical; more over, these distances are identical to the offsets of those axes. The kinematic conditioning of wrist-partitioned manipulators is given due attention, and illustrated with some examples of industrial robots of this type.

It is demonstrated in the article that, for an appropriately chosen output variable, the transfer function for a sufficiently rigid single flexible link is passive. As such, by the passivity theorem, using any strictly passive controller with finite gain will result in an L ₂ -stable system. One approach to designing such a controller is outlined, and the experimental results are presented. The implications of this work are examined, and future research issues are discussed.

A video system is used as a single contactless multipoint dis placement sensor to measure the vibration displacement of a beam. In this article, several schemes are proposed to reduce the image processing time and to increase its accuracy. The vibration displacements of the entire beam are measured with a sampling rate of 60 Hz. From these measured data, the mode shapes and vibration states can be calculated. The video system is shown to have a potential for on-line control and on-line identification of flexible beam systems.

Kinematically redundant manipulators are considered, with a prespecified end-effector task and multiple additional tasks. The number of additional tasks is assumed to be equal to or less than the degree of redundancy. This assumption justifies an equal-priority (or duality) concept for the end-effector task and additional tasks. Based on this concept, expressions for two restricted Jacobian matrices are derived: the restricted manipulator Jacobian and the restricted additional-task Jaco bian. It is shown that generalized inverses of these matrices can be utilized to derive two alternative solutions of instanta neous inverse kinematics that satisfy both tasks simultaneously. The first solution, which is based on the null space of the manipulator Jacobian, has been widely applied in various redundancy-resolution techniques; it is shown that when used in manipulator self-motion tasks, this solution yields cyclic- path motion in joint space. The second solution utilizes the null space of the additional-task Jacobian; it is dual to the first one. Hence, in the particular case of end-effector motion with additional-task maintenance, this solution yields cyclic-path motion as well.

It is further shown that at algorithmic singularities, both restricted Jacobian matrices lose rank simultaneously. Algo rithmic singularities, together with the well-known manipulator singularities, and newly defined additional-task singularities are shown to play an important role in kinematic analysis. The duality concept calls for the definition of a new type of dexterity measure: the additional-task performance measure. The restricted Jacobian matrices are shown to also provide a convenient tool for deriving "restricted" dexterity measures that are directly related to the various types of singularities. Typical end-effector/additional-task combinations have been tested through computer simulation, illustrating the application of the restricted Jacobian matrices to configuration analy sis and path-planning problems of redundant manipulators in constrained-motion tasks.

A small change in a homogeneous transform may be repre sented by either the change in the description vector of the transform or by a differential translation and rotation transfor mation. The differential motion vector describing the differential translation and rotation transformation is in general not equal to the change in the description vector. The calculations in Durrant-Whyte (1987a) are based on the assumption that they are equal and have to be modified accordingly.