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Optimal Design of a Stewart–Gough Platform for Multidirectional 3-D Printing

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Precision Product-Process Design and Optimization

Part of the book series: Lecture Notes on Multidisciplinary Industrial Engineering ((LNMUINEN))

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Abstract

The existing 3-D printing techniques have several disadvantages such as aliasing and difficulty in building around inserts due to limited motions associated with the equipment. The limitation of build direction results in poor surface finish due to aliasing (or layer stair-stepping) and adverse material properties in certain directions which limits use of 3-D printing for many industrial applications. The present study investigates the application of Parallel Kinematic Machines (PKMs) in achieving multidirectional 3-D printing. The proposed architecture addresses some of the limitations of existing Fused Deposition Modelling (FDM)-based 3-D printer by allowing six-axis motions between extruder and platform while building the component. The study explores the application of Stewart–Gough Platform (SGP) further for 3-D printing and illustrates its capability as a viable solution for multi-axis FDM. The design of SGP for multidirectional FDM is realized for optimal dexterity using bulk dexterity index. The study discusses details of the optimization formulation and consequent results associated with the same. A conceptual design of the SGP is subsequently proposed based on the results of the optimization. The proposed SGP-based machine architecture is expected to offer advantages such as improved surface finish and control of directional properties, which signifies push towards freeform fabrication using multidirectional 3-D printing.

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Abbreviations

PKM:

Parallel Kinematic Machine

FDM:

Fused Deposition Modelling

SGP:

Stewart–Gough Platform

AM:

Additive Manufacturing

CAD:

Computer-Aided Design

W/F Ratio:

Workspace-to-Footprint Ratio

DOF:

Degrees of Freedom

GD:

Group Decoupling

SA-PM:

Selectively Actuated Parallel Machine

RPY:

Roll–Pitch–Yaw

KPI:

Kinetostatic Performance Index

GCI:

Global Conditioning Index

GMI:

Global Manipulability Index

ME:

Manipulability Ellipsoid

SVD:

Singular Value Decomposition

SRSGP:

Semi-Regular Stewart–Gough Platform

GA:

Genetic Algorithm

\( v_{a = x,y,z} \) :

Translational velocity of end effector of robot with subscript indicating axis

\( \omega_{a = x,y,z} \) :

Rotational velocity of end effector of robot with subscript indicating axis

\( \theta_{n} \) :

Displacement of actuator with subscript indicating actuator number

{B}:

Base frame of reference of Stewart–Gough platform

{P}:

Platform frame of reference of Stewart–Gough platform

O B :

Origin of base frame

O P :

Origin of platform frame

B i :

Actuator connecting points with subscript i indicating connection point on base

P i :

Actuator connecting points with subscript i indicating connection point on platform

b i :

Vector from centre of base to actuator connecting point i

p i :

Vector from centre of platform to actuator connecting point i

B t :

Vector indicating tool point represented in frame B

B R P :

Standard rotation matrix for rotating vector in frame B to frame P

Α :

Roll angle of platform

Β :

Pitch angle of platform

Γ :

Yaw angle of platform

\( c\alpha \) :

Cosine of angle \( \alpha \)

\( s\alpha \) :

Sine of angle \( \alpha \)

\( l_{i} \) :

Length of strut/leg i

\( ^{B} {\varvec{\Omega}} \) :

Angular velocity matrix in frame B

\( ^{B} {\mathbf{s}}_{i} \) :

Vector along leg i represented in frame B

\( {\mathbf{J}} \) :

Jacobian matrix transforming tool point velocities to actuator velocities

\( {\mathbf{J}}^{ - 1} \) :

Inverse of Jacobian matrix transforming actuator velocities to tool point velocities

Q :

Vector representing current pose of platform

\( \dot{\varvec{q}} \) :

Velocity vector of tool points

\( \dot{\varvec{l}} \) :

Velocity vector of actuators

\( \sigma_{i} ({\mathbf{J}}) \) :

ith singular values of Jacobian matrix

\( \lambda_{i} \left( {{\mathbf{JJ}}^{\text{T}} } \right) \) :

ith eigenvalue of \( {\mathbf{JJ}}^{\text{T}} \)

\( \mu \left( {\mathbf{J}} \right) \) :

Manipulability measure of Stewart–Gough platform

\( \kappa \left( {\mathbf{J}} \right) \) :

Condition number of Jacobian

\( w \) :

Workspace of robot

\( {\text{d}}w \) :

Infinitesimal volume element of workspace

\( r_{B} \) :

Radius of the base circle

\( r_{P} \) :

Radius of the platform circle

\( \phi_{B} \) :

Spacing angle between a set of base passive joints B1–B2, B3–B4, B5–B6

\( \phi_{P} \) :

Spacing angle between a set of base passive joints P1–P2, P3–P4, P5–P6

\( h \) :

Distance between the origins of the base and platform frames when the platform is at neutral position

\( {\mathbf{x}} \) :

Design space vector

\( d_{b1} \) :

Distance between two passive joints in base

\( d_{p1} \) :

Distance between two passive joints in platform

\( n_{\text{CW}} \) :

Number of divisions in Cartesian workspace

\( n_{\text{EW}} \) :

Number of divisions in Euclidean workspace

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology Jodhpur, Jodhpur, 342037, Rajasthan, India

    Shricharana Shastry, Ritwik Avaneesh, K. A. Desai & S. V. Shah

Authors
  1. Shricharana Shastry
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  2. Ritwik Avaneesh
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  3. K. A. Desai
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  4. S. V. Shah
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Corresponding author

Correspondence to K. A. Desai .

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Editors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology Bombay , Mumbai, Maharashtra, India

    Sanjay S. Pande

  2. Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, Assam, India

    Uday S. Dixit

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Shastry, S., Avaneesh, R., Desai, K.A., Shah, S.V. (2018). Optimal Design of a Stewart–Gough Platform for Multidirectional 3-D Printing. In: Pande, S., Dixit, U. (eds) Precision Product-Process Design and Optimization. Lecture Notes on Multidisciplinary Industrial Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-10-8767-7_1

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  • DOI: https://doi.org/10.1007/978-981-10-8767-7_1

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