Kinematic Analysis of Planar Tensegrity 2-X Manipulators

Abstract

Abstract. This paper analyzes the kinematics of planar tensegrity manipulators made of two Snelson's X-shape mechanisms in series. The variable instantaneous center of rotation of each mechanism renders the kinematic analysis of the resulting manipulator more challenging. A general formulation of the direct kinematics is set. A method is proposed to solve the inverse kinematic problem in a symbolic way and up to four inverse kinematic solutions are found. The singularities of the manipulator are shown to divide the joint space into two singularity-free components, showing for the first time a planar positioning manipulator that can be cuspidal. The workspace is determined and plotted for different values of the geometric parameters.

Workspace and Cuspidality Analysis of a 2-X Planar Manipulator

“…A line segment of length l i is defined that links the middle points of the top and base bars of each mechanism i (shown in red dotted line in figure 2). The angle between this line and the direction orthogonal to the base bar, referred to as θ i , is used to define the configuration of mechanism i without ambiguity, assuming that it remains always in its crossed-bar assembly mode [10]. Accordingly, the manipulator configuration can be fully defined with (θ 1 , θ 2 ).…”

“…Manipulator description sides of each mechanism define an isosceles trapezoid, the length l i of the line segment that links the middle points of the top and base bars can be expressed as follows [10]:…”

“…The inverse kinematics is much more challenging to establish and cannot be obtained from (2) easily. A methodology was proposed in [10], which makes it possible to derive a characteristic polynomial of degree four and it was shown that the manipulator may have up to four solutions.…”

“…These boundaries can be obtained from the discriminant of the 4th-order characteristic polynomial derived for the inverse kinematics. This characteristic polynomial in t=tan(φ 1 /2) was derived in [10] and is recalled below, where L has been set to 1 without loss of generality:…”

“…First investigations on the kinematics of such manipulators have proven more challenging than expected, in particular for the solution of the inverse kinematics [10]. This paper is a follow up of the work done in [10]. It focuses on the full workspace analysis of the manipulator and on cuspidality conditions.…”