Stiffness analysis of overconstrained parallel manipulators

https://doi.org/10.1016/j.mechmachtheory.2008.05.017Get rights and content

Abstract

The paper presents a new stiffness modeling method for overconstrained parallel manipulators with flexible links and compliant actuating joints. It is based on a multidimensional lumped-parameter model that replaces the link flexibility by localized 6-dof virtual springs that describe both translational/rotational compliance and the coupling between them. In contrast to other works, the method involves a FEA-based link stiffness evaluation and employs a new solution strategy of the kinetostatic equations for the unloaded manipulator configuration, which allows computing the stiffness matrix for the overconstrained architectures, including singular manipulator postures. The advantages of the developed technique are confirmed by application examples, which deal with comparative stiffness analysis of two translational parallel manipulators of 3-PUU and 3-PRPaR architectures. Accuracy of the proposed approach was evaluated for a case study, which focuses on stiffness analysis of Orthoglide parallel manipulator.

Introduction

Parallel manipulators have become more and more popular in industrial applications, including high-accuracy positioning and high-speed machining [1], [2]. This growing attention is inspired by their essential advantages over serial manipulators, which have already reached the dynamic performance limits (bounded by high masses of the machine components required to support sequential joints, links and actuators). In contrast, parallel manipulators are claimed to offer better accuracy, lower mass/inertia properties, and higher structural rigidity (i.e. stiffness-to-mass ratio) [3]. These features are induced by their specific kinematic structure, which resists the error accumulation in kinematic chains and allows convenient actuators location close to the manipulator base. Besides, the links act in parallel against the external force/torque, eliminating the cantilever-type loading and increasing the manipulator stiffness [4]. The latter makes them attractive for innovative machine-tool architectures [5], [6], [7], but practical utilization of the potential benefits requires development of efficient stiffness analysis techniques, which satisfy the computational speed and accuracy requirements of relevant design procedures [8].

Generally, the stiffness analysis evaluates the effect of the applied external torques and forces on the compliant displacements of the end-effector. Numerically, this property is defined through the “stiffness matrix” K, which gives the relation between the translational/rotational displacement and the static forces/torques causing this transition. The inverse of K is usually called the “compliance matrix” and is denoted as k. As follows from mechanics, K is 6×6 semi-definite non-negative matrix, where structure may be non-diagonal to represent the coupling between the translation and rotation [9]. Besides, this matrix may be not-symmetrical under the static load [10], but standard stiffness analysis focuses on the non-loaded structures.

Similar to other manipulator properties (kinematical, for instance), the stiffness essentially depends on the force/torque direction and on the manipulator configuration. Hence, to provide the designer with integrated performance criteria, various scalar indices are usually computed (such as the best/worst/average stiffness with respect to the rotation or translation). They are typically derived using the singular-value decomposition of K. However, there are still a number of open questions here regarding the significance of these indices for a particular manufacturing task. Besides, since the matrix K varies through the workspace, corresponding global benchmarks must be computed. In some cases, a relevant analysis produces the “stiffness maps”, which describe the end-effector compliance as a function of the manipulator configuration [11], [12], [13].

Several approaches exist for the computation of the stiffness matrix, which differ in the modeling assumptions and computational techniques. They are the finite element analysis (FEA), the matrix structural analysis (MSA), and the virtual joint method (VJM) that is often called the lumped modeling.

The FEA method is proved to be the most accurate and reliable, since the links/joints are modeled with its true dimension and shape [14]. Its accuracy is limited by the discretization step only. However, because of high computational expenses required for the repeated re-meshing, this method is usually applied at the final design stage for the verification and component dimensioning. For example, in [15], a FEA model was used to evaluate the static rigidity and natural frequencies of the T3R1 parallel robot. Also, this method is widely used for validation of other stiffness analysis techniques [16], [17], [18] and for the comparative study [19].

The MSA method is a common technique in mechanical engineering [20], it incorporates the main ideas of the FEA but operates with rather large flexible elements (beams, arcs, cables, etc.). This obviously yields reduction of the computational expenses and, in some cases, allows even obtaining an analytical stiffness matrix. For parallel manipulators, the relevant stiffness model is a combination of flexible beams and nodes, where each beam is defined by two nodes and described by 12×12 stiffness matrix derived from the Euler–Bernoulli presentation. Then, these matrices are “assembled” in accordance with the superposition principle and the manipulator geometry, to produce the desired 6×6 matrix for the whole mechanism. Sometimes this approach is also referred to as the “distributed stiffness” modeling. One of the first examples of MSA application for the problem of interest is the stiffness analysis of a Stewart platform [21], which was performed under the assumption that the links are not subject to bending. This approach was also used in [22], [23] for other manipulators and/or other modeling assumptions. Some resent MSA-based results are obtained for the Delta-type mechanisms [24]. This method gives a reasonable trade-off between the accuracy and computational time, provided that link approximation by the beam elements is realistic. Because it involves rather high-dimensional matrix operations, it is not attractive for the parametric stiffness analysis and analytical modeling.

Finally, the VJM method, which is also referred to as the “lumped modeling”, is based on the expansion of the traditional rigid model by adding virtual joints (localized springs), which describe the elastic deformations of the manipulator components (links, joints and actuators). This approach originates from the work of Gosselin [25], who evaluated parallel manipulator stiffness taking into account only the actuators compliance and by presenting them as one-dimensional linear springs (the links were assumed to be rigid, and the passive joints to be perfect). Besides, the compliance in all actuated joints was assumed to be equal. The latter allowed reducing the stiffness analysis to the analysis of the condition number of the Jacobian matrix. Further development of VJM allowed taking into account the links flexibility, which were presented as rigid beams supplemented by linear and torsional springs [26]. There are a number of variations and simplifications of the VJM method, which differ in modeling assumptions and numerical techniques. In particular, it was applied to the CaPAMan, Orthoglide and H4 robots, specific variants of Stewart–Gough platform, manipulators with US/UPS legs and other kinematic machines [27], [28], [29], [30], [31], [32]. Generally, the lumped modeling provides acceptable accuracy in short computational time, so it is widely used at the pre-design stage, especially for the analytical parametric analysis. However, it is very hypothetic and operates with simplified stiffness models that are composed of one-dimensional springs that do not take into account the coupling between the rotational and translational deflections. There are also other restrictions, which limit its applications to non-overconstrained mechanisms.

This paper presents a new stiffness modelling method, which combines advantages of the above mentioned approaches. It is based on a multidimensional lumped-parameter model that replaces the link flexibility by localized 6-dof virtual springs that describe both the linear/rotational deflections and the coupling between them. The spring stiffness parameters are evaluated using FEA-modelling to ensure higher accuracy. In addition, it employs a new solution strategy of the kinetostatic equations, which allows computing the stiffness matrix for the overconstrained architectures, including the singular manipulator postures. This gives almost the same accuracy as FEA but with essentially lower computational effort because it eliminates the model re-meshing through the workspace.

Since the developed technique is targeted to the design optimization, it relies on the assumption that the manipulator is located in unloaded equilibrium configuration. This allows evaluating the symmetrical part of the general stiffness matrix while neglecting the skew-symmetrical components, which describe effects caused by the external loading and relevant changes in Jacobian [12], [33], [34]. It is obvious that at the preliminary design stage, which focuses on the conceptual issues (such as comparison of alternative manipulator architectures, defining critical components in the kinematic chains, deciding on the stiffness specifications for the links, etc.), this assumption is practical and reasonable. Besides, for a particular case study presented below, validity of the assumption is justified numerically.

The remainder of this paper is organized as follows. Next section introduces a general methodology of deriving/computing of the kinematic and stiffness model. Section 3 describes the manipulator compliant elements and proposes FEA-based technique for the evaluating their parameters. Section 4 includes application examples, which deal with comparative stiffness analysis of two translational parallel manipulators and demonstrate advantages of the proposed technique. Section 5 summarizes the main contributions of this work and defines future research directions.

Section snippets

Manipulator architecture

Let us consider a general n-dof parallel manipulator, which consists of a mobile platform connected to a fixed base by n identical kinematics chains (Fig. 1). Each chain includes an actuated joint “Ac” (prismatic or rotational) followed by a “Foot” and a “Leg” with a number of passive joints “Ps” inside. Generally, certain geometrical conditions are assumed to be satisfied with respect to the passive joints to eliminate the undesired platform rotations and to achieve stability of desired

Evaluating model parameters

The adopted stiffness model of each kinematic chain includes four compliant components, which are described by one 1-dof spring corresponding to the actuator control loop and three 6-dof springs corresponding to the actuator transmission and the manipulator links (see Fig. 3). Let us present particular techniques for their evaluation.

Application examples

To demonstrate the efficiency of the proposed methodology, let us apply it to the comparative stiffness analysis of 3-dof translational mechanisms that employ the Orthoglide architecture [37], [38]. This problem was previously studied using other techniques [31], [40], but the results were essentially different from those obtained from both the FEA-modeling and from the physical experiments. Thus, this section presents several stiffness models for the Orthoglide (Fig. 5) and compares their

Conclusions

The paper proposes a new systematic method for computing the stiffness matrix of overconstrained parallel manipulators. It is based on a multidimensional lumped model of the flexible links, whose parameters are evaluated via the FEA-modeling and describe both the translational/rotational compliances and the coupling between them. In contrast to previous works, the method employs a new solution strategy of the kinetostatic equations, which considers simultaneously the kinematic and static

Acknowledgement

This work has been partially funded by the European projects NEXT, acronyms for “Next Generation of Productions Systems”, Project No. IP 011815.

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