01 stiffness of a hexapod

7/31/2019 01 Stiffness of a Hexapod

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A METHOD FOR EVALUATING THE STIFFNESS OF A HEXAPOD

MACHINE TOOL SUPPORT STRUCTURE

Erik Rebeck and Guangming ZhangDepartment of Mechanical Engineering and Institute for Systems Research

University of Maryland

College Park, MD 20742

ABSTRACT: This paper describes a new method for evaluating the contribution of a hexapodframe structure to the machines overall stiffness. A stiffness model that includes the contributions

of the actuators is first presented. Use of such a model to obtain a good estimate of the stiffnesscontribution of the support structure to the overall machine is then discussed. Finally, the procedure

is demonstrated using a finite element analysis on a portion of a prototype milling machine toollocated at the National Institute for Standards and Technology (NIST)

1.

INTRODUCTION

A hexapod machine tool is a Stewart platform [12] based machine. It consists of six actuated serialchains, or struts, connected directly to both the base and end-effector in such a manner that the end-

effector can be controlled in all six degrees of freedom. The parallel nature of a hexapod provides it

with several potential advantages over serial mechanisms, such as conventional machining centers orindustrial robotic arms. These advantages include high stiffness to weight ratio, small moving mass,good positioning accuracy and symmetric design. The primary disadvantages of these parallel

mechanisms are their relatively small and complex workspace and the complexity in their controland operation. Because of its potential advantages, hexapods are currently being developed as high

performance / high speed machining centers.

The work presented in this paper is based on the analysis of a prototype Ingersoll OctahedralHexapod milling machine located at the National Institute for Standards and Technology (NIST).

The structure of this machine is pictured in Figure 1. This is a fairly large machine toolapproximately 4 meters in height. The milling tool is mounted in a spindle at the bottom of the

movable spindle platform, which is the hexapods end effector. Movement and orientation of thespindle platform is accomplished with the extension and retraction of the actuated struts connecting

the spindle platform to the support structure (base). The struts are essentially telescoping arms, andare connected to the spindle platform and support structure through spherical joints. The support

structure consists of an octahedral space frame and a worktable.

1Certain 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.


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Figure 1. NIST Ingersoll Octahedral Hexapod Structure

Machine tool stiffness refers to the stiffness between the tool point and the workpiece [1]. In thispaper, applied loads, measured deflections, and stiffness values of the spindle platform are relative

to the tool point position. The stiffness of a machine tool is critical because it determines the abilityof the machine to maintain its desired position under the loads applied during material removal, and

thus the machining accuracy. This paper focuses on analyzing the contributions of a hexapodssupport structure to its overall stiffness. Typically a machine tools support structure is very rigid.

However, designers are always interested in reducing the amount of material used, and thus totalcost, without sacrificing the performance of the machine. A method for evaluating the stiffness of a

particular support structure design is therefore required to obtain a good stiffness to weight ratio.

Support structures are often complex. Because of this, they are often evaluated using finite elementanalysis. Displacement of the joints attached to support structure relative to the base coordinate

system due to platform loading can be used to determine the resulting deflection of the platform.

This would typically require a forward kinematic solution [14]. In this paper a method of analyzingthe support structure joint deflections through the struts is presented that only requires an inversekinematic solution is presented.

KINEMATICS OF A HEXAPOD

When using an inverse kinematic solution the platform position and orientation are given and used

to determine the lengths of the struts. With a forward kinematic problem the strut lengths are givenand the platform position and orientation are determined. The use of forward kinematics for the

evaluation of parallel mechanisms is much more difficult to formulate and poses undetermined

solutions [3, 6, 14]. Inverse kinematics provides a more convenient means for analysis.

Inverse kinematic procedures are well established and have been used by researchers in the study of

parallel mechanisms [3-7, 10]. Coordinate systems are attached to the fixed base and movableplatform. In the study of hexapods, support structure joint locations are defined in the base

coordinate system while platform joint coordinates are defined in the platform coordinate system.Platform joint coordinates are then converted into base coordinates using a transformation matrix.

Strut vectors are then determined by subtracting the platform joint coordinates (in base coordinates)from the support structure joint coordinates. Base and platform coordinate systems can be chosen

arbitrarily. In this study the base coordinate system was attached to the center of the worktablesurface and the platform coordinate systems was attached to the position of the tool post (Figure 2).

Spindle Platform

Struts

Space FrameWorktable


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Figure 2. Inverse Kinematic Coordinate Systems

STRUT BASED STIFFNESS MODEL

A schematic of a hexapod stiffness model that accounts for the contributions of the struts is shown in

Figure 3. Such a model is used to determine the stiffness of the platform in the direction of anapplied load P. P can be either a directional force with an equivalent directional stiffness or a

moment with an equivalent angular stiffness. Several methods can be used to determine the stiffnessof the hexapod machine tool in terms of the stiffness of the struts. The stiffness model formulated in

this research is based on the principle of virtual work. Use of virtual work for analysis of hexapodsand similar mechanisms has been used in [3, 8, 10, 13, 14]. Hexapod stiffness problems are

typically formulated using the Jacobian matrix. In this research however a more classical structuralanalysis approach was taken [2]. This was done to provide a clear insight into the parameters

involved.

Figure 3. Schematic of a Strut-Based Hexapod Stiffness Model

Virtual work equates the external work done on a structure to the internal work done in the structure.

External work(WE) done on a structure is created when an external load (Pi) moves due through a

displacement (di) in the direction of the applied load.

ks,1ks,6ks,2

kh,s

...

P

P

Spindle Platform

Support Joints

Platform Joints

x y

u

z

v

w


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where P is the force applied to the spindle platform and dis the displacement of the spindle platform

in the direction of the applied force P at the point of force application. Internal work(WI) done on astructure is created when stresses in the structure move through their corresponding strains. To

develop this model, assume that the struts can be modeled as simple truss elements with a constantcross sectional area. Truss elements can only transmit axial loads. Bending is assumed negligible.

If strains in the struts are due to externally applied loads the internal work done by a single strut is

given by:

where is the axial stress, is the axial strain, Vis the volume of the strut,A is cross sectional area,l is length, Fis the axial load, andEis the modulus of elasticity. Noting that the quantity (k =AE/l)

is equal to the stiffness of a truss element, the internal work in the structure is given by:

where the subscript i refers to the strut number. Note that because the quantity of stiffness (k) was

identified, the assumption of truss elements can now be dropped. Any value or equation can be usedto define the axial stiffness, but the condition that negligible moments can be transferred through the

elements still applies.

By equating internal and external work and specifying P as a unit force:

Because the stiffness of the hexapod is given by:

and combining equations (4) and (5), the stiffness of the hexapod is given by:

Where Fi is the axial load transmitted through the ith

strut due to a unit force in the direction of theapplied load P and ki is the stiffness of the i

thstrut. It is noted that joint friction can produce

moments that can result in some bending of the actuators. This is certainly true of the spherical

joints used in the prototype machine at NIST. However, moments transferred by joint friction aretypically small. It can also be shown that bending of a telescoping actuators due to applied momentsdoes not significantly affect the length of the actuator, and therefore has a negligible effect on the

position, and thus stiffness, of the machine [11].

This model requires that the forces transmitted through the struts due to an applied unit load at thetool point be determined. Strut forces can be determined using a free body diagram of the spindle

platform and are solved by equating forces and moments in the x, y, and z directions to obtain staticequilibrium (Figure 4). Forces are always in the direction of the strut unit vector, positive when the

strut is in tension.

=

=6

1

2

1

i i

i

p

k

Fk

k

F

l

AE

FdlA

AE

F

A

FdlAdVW

l lV

StrutI

22

, )()( =

=

===

=

=6

1

2

i i

i

Ik

FW

PddPWn

nnE ==

=

==6

1

2

i i

i

k

FdPd

dd

Pkp

1==

(1)

(2)

(3)

(5)

(6)

(4)


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Figure 4. Static Equilibrium of the Spindle Platform

In the above equation si is the unit vector of the ith

strut in the base coordinate system, which is

determined using inverse kinematics. Summing forces in the x, y, and z directions produces thefollowing equations.

Moments are calculated about the position of the tool post where the force or moment P is applied.The vectors from the position of the tool post to the ball joint centers (ri) in the base coordinate

system are first calculated using a rotation matrix. Moments about the tool point position are thengiven by:

Equations (8) and (9) provide the six equations that are needed to solve for the six strut forces.

= ==

= ==

= ==

===

===

===

6

1

6

1

,,,,,

6

1

6

1

6

1

,,,,,

6

1

6

1

6

1

,,,,,

6

1

)()(

)()(

)()(

i

i

i

ziiyixixiyi

i

z

i

i

i

yiixizizixi

i

y

i

i

i

xiiyiziziyi

i

x

FJFrsrsM

FJFrsrsM

FJFrsrsM

krF

jrF

irF

==

zi

yi

xi

iiii

s

ss

FF

,

,

,

sF

=

=

=

=

=

=

6

1

,

6

1

,

6

1

,

i

iziz

i

iyiy

i

ixix

FsP

FsP

FsP

(7)

(8)

(9)

P

Fi

ri

Spindle Platform

Tool Post

Ball Joints


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With the strut forces known the strut stiffness model is complete. Recall that under the conditionsused to develop the stiffness model using virtual work, P must be either a unit force or moment.

Stiffness of a hexapod varies both with position and orientation of the platform. This is due to the

reconfiguration of the struts and the fact that stiffness of the six struts typically depends on theirlength, and thus the position and orientation of the spindle platform. The forces acting on the

individual struts depend both on the position and orientation of the spindle platform and on thedirection of loading (P). As a result, for a given position and orientation of the platform, stiffness

will vary with direction. A thorough stiffness study of the support structure therefore requiresanalysis be performed at several locations within the workspace and in multiple directions at these

locations. Stiffness mapping across sections of the workspace is typically used to visualize stiffnessvariations [4, 5, 6, 8, 10].

SUPPORT STRUCTURE ANALYSIS

The deflected position of the tool post due to the support structure requires that the deflections of the

six ball joints attached to the support structure, which are 3rd

order vectors, be known. Thesedeflections are measured in the base coordinate system relative to the measurement location.

Determining the resulting deflection at the tool point due to these joint deflections would require aforward-kinematic solution. The actuator lengths and deflected base joint locations are known, but

the position of the platform is not. However, only the deflections in the direction of the strut axisdirections are important for this mechanism [11], reducing the information required for a good

approximate solution to six scalar quantities. With this simplification an inverse kinematics solutioncan be used to obtain a good approximation of the hexapods stiffness.

The claim that only support joint deflections in the direction of the struts are of importance can be

demonstrated in a fairly simple manner. For a static problem, only changes in length of the six struts

affect the position of the platform. This is illustrated in Figure 5. If the strut has a constant lengththe base joint can be moved along a spherical surface centered at the platform joint center withoutaffecting the position of the small ball joint, and thus tool point. When small deflections are

assumed, deflections of the large ball joint in a direction perpendicular to the strut axis (r) nearlyfollow this path, producing a negligible effect on the length of the strut.

=

6

5

4

3

2

1

,6,5,4,3,2,1

,6,5,4,3,2,1

,6,5,4,3,2,1

,6,5,4,3,2,1

,6,5,4,3,2,1

,6,5,4,3,2,1

F

F

F

F

F

F

JJJJJJ

JJJJJJ

JJJJJJ

ssssss

ssssss

ssssss

M

M

M

P

P

P

zzzzzz

yyyyyy

xxxxxx

zzzzzz

yyyyyy

xxxxxx

z

y

x

z

y

x

{ } [ ]{ }FAP =

{ } [ ] { }PAF 1=

(10)


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Figure 5. Effect of Support Joint Deflections Perpendicular to the Strut Axis

Changes in the orientation of the struts due to deflections of the support joints are negligibly small

considering the relative magnitudes of the strut length (meters) and joint deflections (-meters). Thedeflection of the support joints in the direction of the strut axis (a) can therefore be estimated bytaking the scalar product of the support joint deflection (sj=[sj-x sj-y sj-z]T) and the undeflectedstrut unit vector:

In the above equation the subscript i refers to the ith

strut/support joint. When only the deflections

of the large ball joint centers (due to loads applied to the space frame) in the direction of the strutaxis are considered, these deflections can be described by equivalent strut stiffness values. This is

accomplished by dividing the force applied through the strut by the strut axial deflection of its balljoint center:

These equivalent strut stiffness values can then be inserted into the strut based hexapod stiffness

model to determine the hexapod stiffness contributed by the space frame.

ia

i

ip

Fk

,

,

=

iisjia S,, = (11)

(12)

a

r

Support Joint Platform Joint

ks,1ks,6ks,2 kh,s...

a) Struts Only


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Figure 6. Incorporation of Strut and Space Frame Stiffness

The stiffness of the hexapod due to the struts and support structure can be combined using theprinciple of superposition. This is illustrated in Figure 6. Since the deflections of the space frame

and struts are additive they can be treated as springs in series. The hexapod stiffness due to the struts(kh,s) is combined with the hexapod stiffness due to the support structure (kh,p) to produce the overall

hexapod stiffness using the following relationship:

ANALYSIS OF THE PROTOTYPE NIST INGERSOLL SUPPORT STRUCTURE

A finite element model was created to analyze the contributions of part of the prototype NISTIngersoll hexapods space frame to its overall stiffness. The model used in this analysis is pictured

in Figure 7. This model focuses on the upper portion of the space frame. The worktable and tubesconnecting the feet of the hexapod have been removed and replaced with boundary conditions that

prohibit the feet from translation. This simplification is equivalent to stating that the removed partsare infinitely rigid. While this is certainly not true, these components are believed to be sufficiently

rigid to neglect, allowing the complexity of the model to be greatly reduced. Another modificationmade to the actual geometry is that cylindrical surfaces have been extended where the center of the

ball joints would be. This provides a convenient surface for force application.

1

,,

11

+=

phsh

hkk

k (13)

kh,p

kp,1kp,6kp,2 ...

b) Space Frame Only

kh,spks,1

kp,1

ks,6ks,2

kp,6kp,2

...

...

c) Struts and Space Frame


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Figure 7. Space Frame Finite Element Model

Finite element trials were performed for loading in thex,y, andz directions at 14 locations for a total

of 42 trials. At all locations the spindle platform was in a horizontal orientation. The locationschosen were based on previously performed experiments of the hexapods stiffness [4, 5].

Measurement positions were performed within a 60 wedge on the horizontal plane to takeadvantage of z-axis symmetry (Figure 8). Half the measurements were taken at a height of z = 1.268

m and half were taken at a height of z = 0.353 m. The procedure used for each trial was as follows:An inverse kinematic solution was used to calculate the un-deflected strut vectors (Si) and platform

joint vectors (ri) in base coordinates for the particular location.

Figure 8. Hexapod Stiffness Measurement Positions

Feet

Legs

Upper Space Frame Joint Center

Surfaces

Strut Forces

Boundary Conditions

-x

y

z

60

Z= 0.353 m

Z= 1.268 m

X= -0.150 m

X= -0.300 m

8

9

10

1112

1314

1

2

3

45

67


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1) Strut axial loads for the particular position and unit force loading in a particular direction werethen calculated.

2) Strut axial loads were applied to the nodes of the model around the location of the ball jointcenters.

3) The analysis was run and deflections of the ball joint center nodes ( -x -y -z]T)were recorded.

4) Measured deflections were then analyzed using the procedures outlined in this paper.

Results of this finite element analysis are shown in Table 1. In this table the hexapod stiffness due

to deflections of the space frame (kh,p) are shown. The data is presented in two groups. In the firstgroup, directional stiffness values in the x, y, and z directions for positions at a height of z = 1.268 m

are shown. In the second group, directions stiffness values in these directions for positions at aheight of z = 0.353 m. Two observations can be made from examining this data:

1) The directional stiffness along the z-axis is greater in the lower portions of the workspace. It can

be seen that the directional stiffness ranged from 2405 to 2502 N/m at a height of z = 0.353 mand ranged from 2268 to 2353 N/m at a height of z = 1.268 m.

2) When comparing data at a specific platform position (data within a row) the space frame is

nearly three times stiffer in the z-direction and the x and y directions. This is observed at bothtested heights (z = 1.268 and 0.353 m). The reason stiffness is greater in the z-direction is

because loads applied in horizontal direction (x or y for instance) subject the legs of thestructure to cantilever bending. This is illustrated in Figure 9. When forces are applied to the

spindle platform in the z-direction, cantilever bending of the legs is greatly reduced

CONCLUSIONS

A new method for estimating the contributions of a hexapods support structure to its overallstiffness is presented. The advantage of the new approach is that it eliminates the need for a forward

kinematic solution, simplifying support structure stiffness analysis. This was accomplished byrecognizing that only the support joint deflections in the direction of the strut vectors are significant.Under conditions where support joint deflections are relatively small, particularly those

Figure 9. Magnified Deflection of Space Frame Under a Load Applied

in the x Direction


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perpendicular to their respective strut axes, this method provides an effective means of accuratelyevaluating the stiffness of a hexapod support structure.

Table 1

Hexapod Stiffness Attributed to Space Frame

Position Location (m) Stiffness (N/m)x y z x Y z

1 0.000 0.000 1.268 678 719 23522 -0.150 0.000 1.268 698 661 2324

3 -0.300 0.000 1.268 704 657 22774 -0.130 0.075 1.268 699 663 2324

5 -0.260 0.150 1.268 690 647 22686 -0.075 0.130 1.268 690 666 2337

7 -0.150 0.260 1.268 688 658 2282

8 0.000 0.000 0.353 698 695 2502

9 -0.150 0.000 0.353 705 682 246910 -0.300 0.000 0.353 715 668 2422

11 -0.130 0.075 0.353 703 686 2468

12 -0.260 0.150 0.353 710 669 240513 -0.075 0.130 0.353 718 690 246814 -0.150 0.260 0.353 700 682 2409

ACKNOWLEDGMENTS

The authors would like to thank the National Institute for Standards and Technology and the

National Science Foundation for financial support of this work. Additionally, the Algor FiniteElement Analysis software used in this research was provided by a Society of Manufacturing

Engineers Education Foundation grant. We would also like to thank Johannes Soons, AlbertWavering, and Fred Rudder from NIST for their help and comments. Joe Falco (also from NIST)

created the Pro/ENGINEER model pictured in Figure 1 and used to develop the space frame FEAmodel.

REFERENCES

1. ANSI/ASME B5.54-1992, Methods for Performance Evaluation of Computer Controlled

Machining Centers, ASME, New York, 1993.2. Armenakas, A.E.,Modern Structural Analysis: the Matrix Method Approach, Mcgraw-Hill,

Inc., New York, 1991.3. Bryfogle, M.D., Nyguyen, C.C., Antrazi, S.S., Chiou, P.C., Kinematics and Control of a

Fully Parallel Force Reflecting Hand Controller for Manipulator Teleoperation,Journal of

Robotic Systems, Vol. 10, 1993.4. Clinton, C., Zhang, G., Wavering, A., Stiffness Modeling of a Stewart Platform based

Milling Machine, Transactions of NAMRC XXV, 1997.

5. Clinton, C., Characteristics of a Hexapod Machine Tool, M.S. Thesis, University ofMaryland, College Park, 1998.

6. El-Khasawneh, B.S., Ferreira, P., On Using Parallel Link Manipulators as Machine Tools,

Transactions of NAMRI/SME, Vol. 25, 1997.

7. Fichter, E.F., A Stewart Platform Based Manipulator: General Theory and PracticalConstruction,International journal of Robot Research, Vol. 5, 1986.

8. Gosselin, C., Stiffness Mapping for Parallel Manipulators,IEEE Trans. On Robotics andAutomation, Vol. 6, 1990.


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9. Luh, C., Adkins, F., Haung, E., and Qiu, C., Working Capability Analysis of StewartPlatforms,ASME Journal of Mechanical Design, Vol. 118, 1996.

10. Rudder, F.F., Static Structural Analysis of a Reconfigurable Rigid Platform Supported byElastic Legs, National Institute of Standards and Technology Internal Report, NISTIR 5885,

1996.11. Soons, J.A., On the Geometric and Thermal Errors of a Hexapod Machine Tool, First

European-American Forum on Parallel Kinematic Machines, 1998 (in press)12. Stewart, D., A Platform with Six Degrees of Freedom, Proc. Inst. Mechanical

Engineering, 371-386, 1965.

13. Tahmasebi, F., Tsai, L.W., On the Stiffness of a Novel Six-Degree-of-Freedom ParallelMinimanipulator,Journal of Robotic Systems, Vol. 12, 1995

14. Tsai, L.W.,Robotic Analysis: The Mechanics of Serial and Parallel Manipulators, John-

Wiley & Sons, New York, 1999

01 stiffness of a hexapod

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