Stiffness matrix of manipulators with passive joints:computational aspects

Ecole des Mines de Nantes, 4 rue Alfred-Kastler, Nantes 44307, France
Institut de Recherches en Communications et en Cybernetique de Nantes, 1 rue de la Noe, 44321 Nantes, France

(Submmited to "IEEE Transactions on Robotics", 2011)

I. Introduction

In many applications, manipulator stiffness becomes one of the most important performance measures of a robotic system. In particular, for milling, drilling and other types of machining, the stiffness defines the positioning errors due to interaction between the workpiece and the technological tool. Similarly, in industrial pick-and-place automation, the manipulator stiffness defines admissible velocity/acceleration while approaching the target point, in order to avoid undesirable displacements due to inertia forces. Other examples include medical robots, where elastic deformations of mechanical components under the task load are the primary source of positioning errors.

Numerically, this property is usually described by the stiffness matrix

, which defines a linear relation between the translational/rotational displacement in Cartesian space and the static forces/torques causing this transition (assuming that all of them are small enough). The inverse of

is usually called the compliance matrix and is denoted as

. As it follows from related works, for conservative systems,

semi-definite non-negative symmetrical matrix but in general case its structure may be non-diagonal to represent the coupling between the translation and rotation.

The problem of stiffness matrix computing for different types of manipulators has been in the focus of robotic experts for several decades [1-18]. The existing approaches may be roughly divided into three main groups: (i) the Finite Element Analysis (FEA) [19-23], (ii) the Matrix Structural Analysis (SMA) [24-28], and (iii) the Virtual Joint Method (VJM) [1-2,8,14,29-30]. The most accurate of them is obviously the FEA-based technique but it requires rather high computational expenses. The SMA is less computationally hard due to fairly large structural elements employed (3D flexible beams instead of numerous tiny tetrahedrals and hexahedrals of FEA) but nevertheless it is not convenient for the parametric analysis. And finally, the VJM method is the most attractive in robotic domain since it operates with an extension of the traditional rigid model that is completed by a set of compliant virtual joints (localized springs), which describe elastic properties of the links, joints and actuators. This paper contributes to the VJM technique and focuses on some particularities of the manipulators with passive joints.

For conventional serial manipulators (without passive joints ¹), the VJM approach yields rather simple analytical presentation of the desired stiffness matrix

. Relevant expression

can be found in the work of Salisbury [1] who assumed that the mechanical elasticity is concentrated in actuators and the deflections are small enough to apply linear approximation of the force-deflection relation. Here the matrix

aggregates the stiffness coefficients of all elastic joints, and

is the corresponding kinematic Jacobian. Further, this result was extended by Gosselin for case of parallel manipulators taking into account elasticity of other mechanical elements [2]. More recent publications present VJM-based stiffness analysis for particular case studies, such as various variants of the Stewart–Gough platform, manipulators with US/UPS legs, CaPAMan, Orthoglide, H4 etc. [27-34].

It should be noted that in the majority of related works, the presence of passive joints does not cause any specific computational problems, since these joints are eliminated via geometrical constraints describing the assembling of the relevant parallel architecture [2]. Besides, in most of publications, it is implicitly assumed that the Jacobian

describing influence of the elastic joints on the end-location is non-singular ², i.e.

, to ensure inversion of the related matrix in the modified expression

that always produce non-singular

. It is obvious that the assumption concerning

is completely realistic if the VJM model includes at least a single 6-dimensional virtual spring of a general type (see [38] for details), while it is not realistic that the manipulator stiffness matrix is always non-singular. Hence, common stiffness modelling techniques must be revised with respect to influence of passive joints, which in certain cases can not be straightforwardly eliminated from the kinetostatic equations and, consequently, may cause singularity of

³ .

In this paper, another approach is applied that originates from our publication [30] where the desired stiffness matrix

of size

is extracted from the inverse of a larger matrix, of size

, which additionally includes the passive joint Jacobian

(

is the passive joint number). Advantages of this approach and its ability to produce singular stiffness matrices were confirmed by a number of examples, but explicit analytical solution was not presented. Hence, this work concentrates on analytical computations of the stiffness matrix and also on influence of the passive joints on particular elements of

It is also worth mentioning that some previous works [39] propose (or at least discuss) a trivial solution of the considered problem, which deals with a straightforward modification of the matrix

, in accordance with the passive joint type and geometry (corresponding rows and columns are set to zero). However, as it will be shown below, this straightforward approach gives true results if (and only if) the matrix

is diagonal, but it is not valid in general case where there is a coupling between different types of the elementary virtual springs presented by non-diagonal coefficients.

The remainder of this paper is organized as follows. Section II presents a simple motivation example that confirms the problem non-triviality. Then, Sections III and IV propose relevant analytical solutions for a serial kinematic chain and a parallel manipulator respectively. Section V focuses on computational issues and proposes recursive procedure and a set of corresponding analytical rules. Section VI contains application examples that demonstrate the developed technique advantages. And finally, Section VII summarizes the main results and gives prospective for future work.

¹ It should be mentioned that in this paper passive joints have stiffness equal to zero and they should be distinguished from passive compliant joints studied by other authors, whose stiffness is nonzero.

² It is important to distinguish the conventional kinematic Jacobian , which is computed with respect to actuated coordinates and may be both singular and non-singular, and the Jacobian that is computed with respect to the virtual spring coordinates and is always non-singular. Besides, these Jacobians differ in sizes, which for a standard serial 6-d.o.f. manipulator are respectively and .

³ The main problem with straightforward application of the classical expression

to the manipulators with passive joints is related to the invertibility of the matrix

, which becomes singular if stiffness of some virtual springs is assign to zero (to describe the passive joint properties). In fact, there are two sequential inversions here applied to singular matrices that may be treated similarly to an indeterminate form "1/(1/0)" in calculus that finally produces 0, but cannot be obtained numerically. So, finally, double inversion of singular matrix should produce another singular matrix, but numerical computations obviously fails. To overcome this difficulty and to solve the problem in a rigorous way, our previous publication [30] proposes a dedicated technique that deals with inversion of non-singular matrices of higher dimension and extraction from them relevant 6x6 sub-matrix.

Stiffness matrix of manipulators with passive joints: computational aspects

I. Introduction