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Kinematic and dynamic synthesis of a parallel kinematic high speed

drilling machine

Abstract

Typically, the term‘‘high speed drilling’’ is related to spindle capability of high cutting speeds. The suggested high speed drilling machine (HSDM) extends this term to include very fast and accurate point-to-point motions. The new HSDM is composed of a planar parallel mechanism with two linear motors as the inputs. The paper is focused on the kinematic and dynamic synthesis of this parallel kinematic machine (PKM). The kinematic synthesis introduces a new methodology of input motion planning for ideal drilling operation and accurate point-to-point positioning. The dynamic synthesis aims at reducing the input power of the PKM using a spring element.

Keywords: Parallel kinematic machine; High speed drilling; Kinematic and dynamic synthesis

1. Introduction

During the recent years, a large variety of PKMs were introduced by research institutes and by industries. Most, but not all, of these machines were based on the well-known Stewart platform [1] configuration. The advantages of these parallel structures are high nominal load to weight ratio, good positional accuracy and a rigid structure [2]. The main disadvantages of Stewart type PKMs are the small workspace relative to the overall size of the machine and relatively slow operation speed [3,4]. Workspace of a machine tool is defined as the volume where the tip of the tool can move and cut material. The design of a planar Stewart platform was mentioned in [5] as an affordable way of retrofitting non-CNC machines required for plastic moulds machining. The design of the PKM [5] allowed adjustable geometry that could have been optimally reconfigured for any prescribed path. Typically, changing the length of one or more links in a controlled sequence does the adjustment of PKM geometry.

The application of the PKMs with ‘‘constant-length links’’ for the design of machine tools is less common than the type with ‘‘varying-length links’’. An excellent example of a ‘‘constant-length links’’ type of machine is shown in [6]. Renault-Automation Comau has built the machine named ‘‘Urane SX’’. The HSDM described herein utilizes a parallel mechanism with constant-length links.

Drilling operations are well introduced in the literature [7]. An extensive experimental study of highspeed drilling operations for the automotive industry is reported in [8]. Data was collected fromhundreds controlled drilling experiments in order to specify the parameters required for quality drilling. Ideal drilling motions and

guidelines for performing high quality drilling were presented in [9] through theoretical and experimental studies. In the synthesis of the suggested PKM, we follow the suggestions in [9].

The detailed mechanical structures of the proposed new PKM were introduced in [10,11]. One possible configuration of the machine is shown in Fig. 1; it has large workspace, highspeed point-to-point motion and very high drilling speed. The parallel mechanism provides Y, and Z axes motions. The X axis motion is provided by the table. For achieving highspeed performance, two linear motors are used for driving

the mechanism and a highspeed spindle is used for drilling. The purpose of this paper is to describe new kinematic and dynamic synthesis methods that are developed for improving the performance of the machine. Through input motion planning for drilling and point-to-point positioning, the machining error will be reduced and the quality of the finished holes can be greatly improved. By adding a well-tuned spring element to the PKM, the input power can be minimized so that the size the machine and the energy consumption can be reduced. Numerical simulations verify the correctness and effectiveness of the methods presented in this paper. 2. Kinematic and dynamic equations of motion of the PKM module

The schematic diagram of the PKM module is shown in Fig. 2. In consistent with the machine tool conventions, the z-axis is along the direction of tool movement. The PKM module has two inputs (two linear motors) indicated as part 1 and part 6, and one output motion of the tool. The positioning and drilling motion of the PKM module in this application is characterized by (y axis motion for point-to-point positioning) and (z axis motion for drilling). Motion equations for both rigid body and elastic body PKM module are developed. The rigid body equations are used for the synthesis of input motion planning of drilling and input power reduction. The elastic body equations are used for residual vibration control after point-to-point positioning of the tool.

2.1. Equations of motion of the PKM module with rigid links

Using complex-number representation of mechanisms [12], the kinematic equations of the tool unit (indicated as part 3 which includes the platform, the spindle and the tool) are developed as follows. The displacement of the tool is

and

where b is the distance between point B and point C, r is the length of link AB (the lengths of link AB, CD and CE are equal). The velocity of the tool is

where

The acceleration of the tool is

where

The dynamic equations of the PKM module are developed using Lagrange’s equation of the second kind [13] as shown in Eq. (7).

where T is the total kinetic energy of the system; coordinates and velocities;

and

are the generalized

. k is the

is the generalized force corresponding to

number of the independent generalized coordinates of the system. Here, k=2, q1=y1 and q2=y6. After derivation, Eq. (7) can be expressed as

where n is the number of the moving links; inertia of link i;

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