Mechanical Error Modelling

5. Mechanical Design and Modelling

5.11 Mechanical Error Modelling

5.10.3 Force and Torque Propagation

The knowledge of the force and torque exerted on a cutting tool may be used to determine the force and torque propagation throughout a machining structure. The link-wise torque and force propagation is given by:

^𝑖+1𝑓_𝑖+1= ^𝑖+1_𝑖𝑅 ^𝑖𝑓_𝑖 (5.46) ^𝑖+1𝑛_𝑖+1= ^𝑖+1_𝑖𝑅 ^𝑖𝑛_𝑖+^𝑖+1_𝑖𝑃 ×^𝑖+1𝑓_𝑖+1 (5.47)

Where fi+1 and ni+1 are the force and torque exerted on link i+1 by link i. The position (P) and rotation (R) matrices are derived directly from the HTM‟s of individual modules. For an expected cutting force the joint actuation torques and forces are obtainable. The actuation torque required by a rotary axis is determined by equation 5.48 while the actuation force for a linear axis is calculated by equation 5.49, where ^𝑖𝑍 _𝑖 is the joint axis unit vector.

𝜏_𝑖 = 𝑛^𝑖 _𝑖^{𝑇 𝑖}𝑍 _𝑖 (5.48) 𝜏_𝑖 = 𝑓^𝑖 _𝑖^{𝑇 𝑖}𝑍 _𝑖 (5.49)

5.11 Mechanical Error Modelling 5.11.1. First Order Errors

The ability of a MRM to accurately position a cutting tool relative to a work piece is an essential criterion in evaluating the feasibility of implementing this technology. Tool positioning errors are comprised of first and second order components [63]. First order errors result in dimensional inaccuracies in machined components. These errors are attributed to three factors:

(i) Geometric positioning errors between adjacent interfaces.

(ii) Static errors: deflections due to forces on modules, excluding impulse forces.

(iii) Thermal expansion/contraction errors (not considered in this research).

(iv) Mechanical backlash between mating components (discussed in Section 8.8).

Geometric Positioning/ Assembly Errors

a. b. c.

Figure 5.38: MRM Assembly Errors for Interfacing Module Pairs

a. Concatenation error b. Edge offset error c. Skewness error

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5.11 Mechanical Error Modelling

Geometric positioning errors are introduced into an MRM structure during module assembly/

reconfiguration. Although the geometric errors may be very small for properly designed interfaces and connectors with reasonable design tolerances, the accumulation of these minute errors across a significant number of connected modules may be noteworthy.

The first step in the modelling of the accumulated assembly error in an MRM is the identification of the three types of assembly errors illustrated in Figure 5.38. The first error, called the concatenation error EC is defined by equation 5.50, where LI is the length of an integrated structure, and LM is the length of a similar structure having been created out of two modules instead of one integral piece.

𝐸𝐶 = 𝐿𝑀 – 𝐿𝐼 (5.50) The second type of error is an edge offset errors EO 1 or EO 2 if more than one edge is offset. The third error called “Skewness” defined by the angle ε, is a measure of the rotation of module two about the geometric centre of module interfaces one and two.

The second step in the modelling of the accumulated assembly error is the relation of module- wise assembly errors to the reference frames placed on the interfaces of adjacent modules.

According to the kinematic modelling method established in Section 5.5.2 the reference frames placed on the mating interfaces should perfectly coincide. The errors EC, EO1, EO2 and ε represent the error in aligning these frames. The error may be kinematically represented by the HTM of equation 5.51 which implements the X-Y-Z Euler angle convention. The mapping of errors EC, EO1, EO2 and ε to Euler parameters x, y, z and α, β, γ is dependent on the orientation of reference frame i+1. Errors must be methodically mapped to corresponding axes on frame i+1 for the successful pairing of errors with Euler parameters.

𝐸_{𝑖+1←𝑖} =

𝑐𝛼𝑐𝛽 𝑐𝛼𝑠𝛽𝑠𝛾 − 𝑠𝛼𝑐𝛾 𝑐𝛼𝑠𝛽𝑐𝛾 + 𝑠𝛼𝑠𝛾 𝑠𝛼𝑐𝛽 𝑠𝛼𝑠𝛽𝑠𝛾 + 𝑐𝛼𝑐𝛾 𝑠𝛼𝑠𝛽𝑐𝛾 − 𝑐𝛼𝑠𝛾

−𝑠𝛽

0 𝑐𝛽𝑠𝛾

0 𝑐𝛽𝑐𝛾

0

𝑥 𝑦𝑧 1

(5.51)

The total system transformation matrix, including assembly errors is calculated by equation 5.9.

Accounting for assembly errors is essential during machine calibration, as it is possible for these errors to be compensated for by the machine controller as opposed to the reassembly of the machine tool. The total system transformation matrix, including assembly errors is given by equation 5.52.

^{𝑊𝑜𝑟𝑘} _{𝑇𝑜𝑜𝑙}𝑇𝐸= 𝑀₁𝐸_1←2𝑀₂𝐸_2←3… 𝑀_𝑛−1𝐸_{𝑛−1←𝑛} (5.52) Static Deflections

Static deflections are caused by operational forces being transmitted throughout the machine structure. These errors contribute to the total first order error and result in geometric inaccuracies in machined parts. The geometric effect of static deflections can be mathematically accounted for in individual modules by altering the HTM that describes the spatial relationship, and orientation of the interface reference frames with regard to each other. Deflections may either cause or contribute to: (i) a linear offset in reference frames or (ii) a rotation of one reference frame relative to the other.

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5.11 Mechanical Error Modelling

a. b.

Figure 5.39: Effect of Static Deflections on Kinematic Models

a. A static deflection causing an offset and rotation of frame 2 b. A static deflection causing a deflection of frame 2

Figure 5.39 illustrates the effect of static deflections on the relative position and orientation of reference frames with regard to each other. Static deflections may be determined by established analytical methods or Finite Element Analysis. Once determined, the linear offsets and angles of rotations may be related to Euler parameters (x, y, z) and (γ, β, α). The new x, y and z Euler parameters will describe the origin of frame 2‟ with reference to frame 1; while the angles γ, α and β will describe the orientation of frame 2‟ relative to frame 1. These parameters are then substituted into equation 5.8 to obtain a new HTM for the module 𝑀_𝑛^′ . The total system transformation matrix, including assembly errors and static deflections is therefore given by:

^{𝑊𝑜𝑟𝑘} _{𝑇𝑜𝑜𝑙}𝑇𝐸_{𝑡𝑜𝑡𝑎𝑙} = 𝑀₁^′𝐸_1←2𝑀₂^′𝐸_2←3… 𝑀_𝑛−1^′ 𝐸_{𝑛−1←𝑛} (5.53) The position error vector that characterises the entire first order error in a machine tool is calculated by equation 5.54. The position vector of the tool tip is with relation to the closest reference frame on an associated tool holding module:

𝑃𝐸_{𝑡𝑜𝑡𝑎𝑙} = ^{𝑊𝑜𝑟𝑘𝑡𝑎𝑏𝑙𝑒}_{𝑇𝑜𝑜𝑙}𝑇𝐸_{𝑡𝑜𝑡𝑎𝑙}

𝑥_{𝑡𝑜𝑜𝑙 𝑡𝑖𝑝} 𝑦_{𝑡𝑜𝑜𝑙 𝑡𝑖𝑝}

𝑥_{𝑡𝑜𝑜𝑙 𝑡𝑖𝑝} −^{𝑊𝑜𝑟𝑘𝑡𝑎𝑏𝑙𝑒}_{𝑇𝑜𝑜𝑙} 𝑇

𝑥_{𝑡𝑜𝑜𝑙 𝑡𝑖𝑝} 𝑦_{𝑡𝑜𝑜𝑙 𝑡𝑖𝑝}

𝑧_{𝑡𝑜𝑜𝑙 𝑡𝑖𝑝} (5.54)

𝑊𝑜𝑟𝑘 𝑡𝑎𝑏𝑙𝑒𝑇𝑜𝑜𝑙

5.11.2 Second Order Errors

Second order errors are a result of vibrations induced by the impact of a cutting tool‟s teeth with the work surface, regenerative chatter and other vibrations within a machine. These errors are classified as dynamic - E(t) - and generally affect the surface integrity of a machined component, machine tool wear, and tool breakage. The magnitude of a second order error depends on module stiffness and damping properties, as well the mass distribution in the machine. These errors are specific to the physical attributes of individual mechanical assemblies, and are difficult to model in a generic manner [63].

2 2’

1 d

θ

1 2 2’

d

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