Robotics and Mechanisms Laboratory

Kinematic Redundancy in Planar Parallel Manipulators

Parallel manipulators have been broadly studied. Certain characteristics, when compared to serial manipulators, such as high pay-load-to-weight ratio, high rigidity, high accuracy and high-speed motion have made them useful mechanisms for certain applications.

On the other hand, they suffer from relatively small workspaces, complex kinematics, low maneuverability and a high number of singular configurations. Many research have been conducted on improving the capabilities of parallel manipulators [merlet-book-2006, gosselin-angeles-1988, carretero-podhorodeski-nahon-gosselin-2000-jmd, liu-jin-gao-2000, hassan-notash-2004-mmt, tsai-zhou-2005, hao-merlet-2005, alici-shirinzadeh-2006]. The majority of studies on parallel manipulators have concentrated on non-redundant manipulators. Redundancy in parallel manipulators, unlike serial manipulators [maciejewski-klein-1985,nakamura-hanafusa-yoshikawa-1987], has only been introduced in [lee-kim-1993] and [merlet-1996]. Redundancy can be divided to: actuation redundancy and kinematic redundancy. Actuation redundancy can be explained as replacing existing passive joints of a manipulator by active ones. Consequently, actuation redundancy does not change the mobility or workspace of a manipulator but may cause singularity reduction [jing-gosselin-2004]. Kinematic redundancy, on the other hand, adds to the mobility and degrees of freedom (DOF) of parallel manipulators. Kinematic redundancy occurs when extra active joints and links (if needed) are added to manipulators. As a result, kinematically redundant manipulators need more controlling variables than needed for a set of specified tasks [jing-gosselin-2004].

In the present work, the 3-RPRR¹ , a new 6-DOF kinematically planar parallel manipulator is introduced first and its inverse displacement problem (IDP) is explained and illustrated. Then, Jacobian matrices of the manipulator are derived and all types of its kinematic singularities are obtained and their geometrical interpretations illustrated. Thereafter, the reachable and dexterous workspaces of the 3-RPRR manipulator are compared to the ones of its original non-redundant 3-PRR manipulator. Next, a geometrical method is proposed to measure the closeness of both 3-RPRR and 3-PRR manipulators to singular configurations. Using this measure as a cost function, a local optimisation method is used to illustrate how to determine the optimal actuation scheme.

 

¹ The terminology used is the following. A 3-RPRR mechanism indicates that the end-effector is connected to the base by three serial kinematic chains (limbs), each consisting of two active (and therefore underlined) joints, one revolute joint (R) and one prismatic joint (P), respectively, connected to the base, followed by two passive revolute joints, the second of which connects the limb to the end-effector.

In this work, four kinematically redundant manipulators have been analysed. 3-RPRR, a new kinematically redundant planar parallel manipulator with six degrees of freedom, is presented. First the manipulator is introduced and its inverse displacement problem discussed. Then, all types of the singularities of the 3-RPRR manipulator are analysed and demonstrated. Thereafter, the reachable and dexterous workspaces are obtained and compared to the non-redundant 3-PRR manipulator. Finally, based on a geometrical measure of proximity to singular configurations, actuation schemes for the manipulators are obtained. It is shown that the proposed manipulator is capable of following a path while avoiding the singularities.