Joseph Duffy

A forward displacement analysis of a class of Stewart Platforms

A closed-form forward displacement analysis is performed for a Stewart Platform-type of parallel mechanism, whose six legs meet in a pair-wise fashion at three points in the top and base platforms. The six legs and two platforms of this mechanism together form an octahedron. An eighth-degree polynomial in the square of the tan-half-angle that measures the elevation of a triangular face of the octahedron relative to the base triangle is derived. Each of three vertices of the base triangle of the octahedron is modeled by a spherical four-bar mechanism, and the polynomial is obtained by eliminating a pair of tan-half-angular displacements from the displacement equations of the three spherical four-bar mechanisms. The result are verified numerically by performing a reverse displacement analysis and displaying real solutions on a graphic system. It is clear that there are a maximum of eight reflected pairs of real assembly configurations of the octahedral form of the Stewart Platform.

The fallacy of modern hybrid control theory that is based on orthogonal complements of twist and wrench spaces

Cet editorial se veut etre un argument pour les chercheurs dans le domaine des systemes robotises, pour reconsiderer la theorie de la «commande-hybride» de la commande simultanee de la force et du mouvement

Special configurations of spatial mechanisms and robot arms

Abstract A novel general method is developed for determining special configurations of all single-loop mechanisms. The results are illustrated by means of numerical examples. In addition special configurations of robot arms are investigated, and a simple result which has important practical application is deduced; “ when all the screws representing the instantaneous motion of the joints of a robot arm are reciprocal to a common screw or screw then the end effector cannot have free instantaneous movement parallel to the axis or axes of the reciprocal screws ”.

An application of screw algebra to the acceleration analysis of serial chains

Abstract It is well known that the velocity of the end effector of a serial chain can be written as a linear combination of the screws that represent, with respect to an arbitrary coordinate system, the kinematic pairs that join the end effector with the body used as the reference frame. In this paper, it is shown that the acceleration of a point in the end effector of the serial chain can be expressed in terms of the direction and moment parts of the same screw coordinates.; Further, is if the angular velocity of the end effector vanishes, then its acceleration can be written as a relatively simple expression involving only the complete screw coordinates themselves and their Lie products. Thus, the present work can be regarded in one sense as an extension of screw theory into the acceleration analysis of linkages. Furthermore, since the equations obtained here are tensorial, the present work can be also regarded as a step toward a coordinate free dynamic analysis, as suggested by Chevallier [Mech. Mach. Theory 26, 613–627 (1991)].

A displacement analysis of the general spatial 7-link, 7R mechanism

Abstract An input-output equation of degree 32 in the tan-half-angle of the output angular displacement is expressed in the form of a 16 × 16 determinant equated to zero for the single-loop, single-degree-of-freedom linkage with seven links connected by seven turning pairs arbitrarily oriented in space. An algorithm is developed for computing values of the remaining angular displacements of the mechanism. The results are verified by numerical examples. The derivation of the input-output equation and the development of the algorithm are outlined before proceeding with the detailed algebraic analysis. The detailed analysis is performed using a unified theory for the analysis of spatial mechanisms [1,2] which was developed using a treatise on spherical trigonometry by Todhunter and Leatham [3].

Screw theory and higher order kinematic analysis of open serial and closed chains

In a pair of recent contributions by the authors of this paper, it has been shown that screw theory can be successfully employed in the acceleration analysis of open and closed spatial chains. In this contribution, a novel method for the velocity, acceleration, and jerk analysis of spatial chains is introduced. The method is based in obtaining recursive expressions for the velocity, acceleration, and jerk of the end effector of a serial manipulator. This new method facilitates greatly the derivation of some well known results concerning the velocity and acceleration analyses of spatial chains. Furthermore, the method provides proofs of several new results for the jerk analysis which were not possible using methods developed previously. In addition, the contribution shows how the results of open serial chains can be applied to closed chains and parallel manipulators. Finally, the method can be easily generalized to higher order analyses.

An Optimization Approach to the Determination of the Boundaries of Manipulator Workspaces

An optimization approach to computing the boundaries of the workspaces of planar manipulators is presented. This numerical method consists of finding a suitable radiating point in the output coordinate space and then determining the points of intersection of a representative pencil of rays, emanating from the radiating point, with the boundary of the accessible set. This is done by application of a novel constrained optimization approach that has the considerable advantage that it may easily be automated. The method is illustrated by its application to two planar mechanisms, namely a planar Stewart platform and a planar redundantly controlled serial manipulator. In addition to the exterior boundaries of the workspace, interior curves that represent configurations at which controllability and mobility may be limited, are also mapped. The optimization methodology, implemented here for the planar case, may readily be extended to spatial Stewart platforms.

A forward and reverse displacement analysis of a 6-DOF in-parallel manipulator

Abstract A kinematic analysis of a 6 degree of freedom (DOF) parallel manipulator which has three legs, mounted on moving sliders is described. Therefore, the 6 DOF are provided by the slider displacements and by changing the leg lengths. This manipulator structure provides a large functional workspace. A forward displacement analysis is performed by using a decomposition method and reduces to the one non-linear equation solution by using an interaction method. Two distinct reverse displacement analyses are given. A manipulator workspace determination and a method for the specification of the platform position in space.

A forward and reverse displacement analysis of an in-parallel spherical manipulator

Abstract A displacement analysis is performed for an in-parallel spherical manipulator. Forward and reverse displacement analyses are performed. Two distinct reverse displacement analyses are given: a manipulator workspace determination and a platform determination is given. The displacement analysis is performed by using a decomposition method.

Stiffness mappings employing different frames of reference

Abstract The stiffness mapping matrix for a planar compliant mechanism is analyzed using two different reference frames. The first is rigidly attached to the fixed body of the coupling, whilst the second is attached to the moving body of the coupling. It was found that, in general, these matrices are asymmetric when the coupling is loaded, and that one is the transpose of the other. This is an important result and can be considered an extension of the work done by Dimentberg [1] , who derived a symmetrical stiffness mapping for an unloaded coupling. Additionally, a third frame of reference which produces a symmetric mapping is examined and found to be identical to the Hessian matrix obtained from the second differentials of the elastic potential energy of the system.