Stiffness modeling of a Stewart-platform-based milling machine

Abstract

This paper presents the development of a mathematical model describing the stiffness of a Stewartplatform-based milling machine. Matrix structural analysis is used to derive the stiffness matrix for each of the elements in the model and assemble them into a system-wide stiffness matrix. By incorporating the inverse kinematics of the machine tool, the system model is used to visualize the stiffness variation over the mill’s workspace. Estimation of the system parameters is conducted through experimental stiffness measurements. Computer simulation is used to demonstrate how the developed stiffness model suggests an optimization process for tool-path planning. INTRODUCTION Competitive pressures compel machine tool designers to continually search for equipment that can deliver higher accuracies and material removal rates. The search for new and innovative machine tool designs has led to an exploration of the capabilities of parallel mechanisms such as Stewart platforms (1965). Typically, these mechanisms consist of a moveable platform connected to a rigid base through multiple, identically jointed and extensible struts. The unique characteristics of high stiffness and high speed, combined with versatile contouring capabilities have made parallel mechanisms good potential candidates for the machine tool industry to advance machining performance. In this paper, a stiffness model of a Stewart-platformbased milling machine is presented. This model uses traditional matrix structural analysis in conjunction with preliminary experimental stiffness results from the Ingersoll Octahedral-Hexapod located at the National Institute of Standards and Technology (NIST). The use of matrix structural analysis differs from the traditional approach to robotic stiffness modeling taken by Gosselin (1990), which relies on the calculation of a Stewart platform’s Jacobian. The objective of this stiffness model is to provide an understanding of how the stiffness of the machine tool changes as a function of its workspace. This can be accomplished using a mapping algorithm. It can also be combined with numerical control programs to examine stiffness variation as the machine follows a tool path. It should be pointed out that the work presented in this paper demonstrates the importance of using the system stiffness model to maximize the machine’s stiffness during operation. INGERSOLL OCTAHEDRAL-HEXAPOD The NIST Ingersoll Octahedral-Hexapod is a prototype Stewart-platform-based milling machine. As illustrated in Figure 1, the machine tool consists of a small platform that supports the spindle motor and tool connected to a space frame structure through six identical struts. These struts consist of two spherical joints connected by a servo driven ball screw. The spindle 1 Certain commercial equipment, instruments, or materials are identified in this paper to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose. platform is translated and rotated by changing the length of the ball screw in each strut. The use of six struts imparts six degrees of freedom to the milling machine. This allows the Hexapod to machine complex contoured surfaces common in areas such as tool and die manufacturing and the aerospace industry. The unique geometry of the Hexapod means that machining forces are primarily distributed as axial loads throughout the structure, greatly increasing stiffness. This also provides for the use of lighter components, allowing higher feed rates and reduces the need for heavy duty foundations necessary for traditional machine tools. FIGURE 1. NIST INGERSOLL OCTAHEDRALHEXAPOD DEVELOPMENT OF STIFFNESS MODEL The Hexapod stiffness model is based on matrix structural analysis (1991), which models structures as a combination of elements and nodes. Eqn. (1) describes the basic relationship in this method of analysis, where {F} is a force vector applied to the nodes of the structure and {x} is the displacement of the nodes due to those applied forces. [k] is the structure stiffness matrix that relates the two vectors. The stiffness matrix depends on the nature of the elements in the structure, whether they are truss or frame elements, their geometric orientation and connectivity. { } [ ] { } F k x = ⋅ (1) In this study, the Hexapod stiffness model relies on truss elements. These elements consist of two nodes and are only capable of linear deformation along their length, as assumed. The deflection of a truss element in its local coordinate frame is thus described by Eqn. (2), where i and j represent the two nodes of the element and {F} and {u} are the external forces and nodal displacements respectively. The element is assumed to lie along the local x axis. This matrix is scaled by the cross-sectional area of the element, A, the elastic modulus, E, and the length of the element L. F