The aim of this paper is to characterize the notion of aspect in the workspace and in the joint space for parallel manipulators. In opposite to the serial manipulators, the parallel manipulators can admit not only multiple inverse kinematic solutions, but also multiple direct kinematic solutions. The notion of aspect introduced for serial manipulators in [Borrel 86], and redefined for parallel manipulators with only one inverse kinematic solution in [Wenger 1997], is redefined for general fully parallel manipulators. Two Jacobian matrices appear in the kinematic relations between the joint-rate and the Cartesian-velocity vectors, which are called the "inverse kinematics" and the "direct kinematics" matrices. The study of these matrices allow to respectively define the parallel and the serial singularities. The notion of working modes is introduced to separate inverse kinematic solutions. Thus, we can find out domains of the workspace and the joint space exempt of singularity. Application of this study is the moveability analysis in the workspace of the manipulator as well as path-planing and control. This study is illustrated in this paper with a RR-RRR planar parallel manipulator.
A well known feature of parallel manipulators is the existence of multiple solutions to the direct kinematic problem. That is, the mobile platform can admit several positions and orientations (or configurations) in the workspace for one given set of input joint values [3]. Moreover, parallel manipulators exist with multiple inverse kinematic solutions. This means that the mobile platform can admit several input joint values corresponding to one given configuration of the end-effector. To cope with the existence of multiple inverse kinematic solutions in serial manipulators, the notion of aspects was introduced in [1]. The aspects equal the maximal singularity-free domains in the joint space. For usual industrial serial manipulators, the aspects were found to be the maximal sets in the joint space where there is only one inverse kinematic solution.
A definition of the notion of aspect was given by [2] for parallel manipulators with only one inverse kinematic solution. These aspects were defined as the maximal singularity-free domains in the workspace. For instance, this definition can apply to the Stewart platform [4].
First of all, the working modes are introduced to allow the separation of the inverse kinematic solutions. Then, a general definition of the notion of aspect is given for all fully parallel manipulators. The new aspects are the maximal singularity-free domains of the Cartesian product of the workspace with the joint space.
A possible use of these aspects are the determination of the best working mode. It allows to achieve complex task in the workspace or to make pathplaning without collision. As a matter of fact, currently, the parallel manipulators possessing multiple inverse kinematic solutions evolve only in one working mode. For a given working mode, the aspect associated is different. It is possible to choose one or several working modes to execute the tasks expected in the maximal workspace of the manipulator.
In this paragraph, some definitions permitting to introduce the general notion of aspect are quoted.
Definition 1 A fully parallel manipulator is a mechanism that includes as many elementary kinematic chains as the mobile platform does admit degrees of freedom. Moreover, every elementary kinematic chain possesses only one actuated joint (prismatic, pivot or kneecap). Besides, no segment of an elementary kinematic chain can be linked to more than two bodies [3].
In this study, kinematic chains are always independent. This condition is necessary to find the working modes. Also, the elementary kinematic chains can be called “legs of the manipulator” [5].
For a manipulator, the relation permitting the connection of input values (q) with output values (X) is the following
This definition can be applied to serial or parallel manipulators. Differentiating equation ( 1) with respect to time leads to the velocity model Where w is the scalar angular-velocity and ċ is the two-dimensional velocity vector of the operational point of the moving platform for the planar manipulator. For the spherical and the spatial manipulator, w is the three-dimensional angular velocity-vector of the moving platform. And ċ is the three-dimensional velocity vector of the operational point of the moving platform for the spatial manipulator. Moreover, A and B are respectively the directkinematics and the inverse-kinematics matrices of the manipulator. A singularity occurs whenever A or B, (or both) that can no longer be inverted. Three types of singularities exist [6]:
Parallel singularities occur when the determinant of the direct kinematics matrix A vanishes. The corresponding singular configurations are located inside the workspace. They are particularly undesirable because the manipulator can not resist any force and control is lost.
Serial singularities occur when the determinant of the inverse kinematics matrix B vanishes. When the manipulator is in such a singularity, there is a direction along which no Cartesian velocity can be produced.
The multiple inverse kinematic solutions induce multiple postures for each leg.
Definition 2 A posture changing trajectory is equivalent to a trajectory between two inverse kinematic solutions.
The multiple direct kinematic solutions induce multiple assembling modes for the mobile platform.
Definition 3 An assembling mode changing trajectory is equivalent to a trajectory between two direct kinematic solutions.
As an example, the 3-RRR planar parallel manipulator (the first joints are actuated joints), a posture changing trajectory exists between two inverse kinematic solutions (Fig. 1) and an assembling mode trajectory exits between two direct kinematic solutions (Fig. 2). In these trajectories, the mobile platform can meet a singular configuration.
The working modes are defined for fully parallel manipulators (Def. 1). From this definition, the inversekinematic matrix is always diagonal. For a manipulator with n degrees of freedom, the inverse kinematic mat
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