Rational Parametrization of Linear Pentapod’s Singularity Variety and the Distance to It

Abstract

A linear pentapod is a parallel manipulator with five collinear anchor points on the motion platform (end-effector), which are connected via SPS legs to the base. This manipulator has five controllable degrees-of-freedom and the remaining one is a free rotation around the motion platform axis (which in fact is an axial spindle). In this paper we present a rational parametrization of the singularity variety of the linear pentapod. Moreover we compute the shortest distance to this rational variety with respect to a suitable metric. Kinematically this distance can be interpreted as the radius of the maximal singularity free-sphere. Moreover we compare the result with the radius of the maximal singularity free-sphere in the position workspace and the orientation workspace, respectively.

Variational Path Optimization of Linear Pentapods with a Simple Singularity Variety

“…Such an inner product should be in a way that one can derive kinematic information from. It turns out that it is possible to define a metric tensor (and consequently an inner product) in such a way that it implies the metric described in [14] (see Eq. 1.3).…”

“…From this latter characterization the following algebraic one can be obtained (cf. [14]): There exists a bijection between the configuration space of a linear pentapod and all points (u 1 , u 2 , u 3 , u 4 , u 5 , u 6 ) ∈ R 6 located on the singular quadric Γ : u 1 2 + u 2 2 + u 3 2 = 1, where − → i = (u 1 , u 2 , u 3 ) determines the orientation of the linear platform and − → p = (u 4 , u 5 , u 6 ) its position. Then the set of all singular robot configurations is obtained as the intersection of Γ Figure 1.…”

“…From [14] it was already known that the determination of the local extrema of the singularitydistance function is a polynomial problem of degree 80, which can be relaxed to a problem of degree 28 by expanding the transformation group from Euclidean to equiform motions by omitting the normalizing condition Γ. As the obtained distance of the relaxed problem is less or equal to the distance of the original problem, it can be used as the radius of a guaranteed singularity-free ball.…”

Linear Pentapods with a Simple Singularity Variety

“…Note that this assumption can always be made without loss of generality as the fixed/moving frame can always be chosen in a way that the first base/platform anchor point is its origin. Moreover, a rational parametrization of the singularity loci Γ ∩ Σ was given by the authors in [6].…”

“…In [6] it is proven that for a generic linear pentapod, the computation of the maximal singularity-free zone in the position/orientation workspace (with respect to the Euclidean/spherical metric) leads over to the solution of a polynomial of degree 6 In contrast the determination of the closest singular pose (cf. Fig.…”