stiffness analysis and kinematic modeling

7/22/2019 Stiffness Analysis and Kinematic Modeling

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STIFFNESS ANALYSIS AND KINEMATIC MODELING OF STEWART

PLATFORM FOR MACHINING APPLICATIONS

Satheesh G. Kumar, Bikshapathi M, Nagarajan T, Srinivasa Y. G.PEIL Laboratory, Mechanical Engineering, IIT Madras, Chennai, India

Abstract: Today, the parallel manipulator robots play a very predominant role in Manufacturing Technology, because of their high stiffness, high payload capacity and simple structure. To machine complex shape of objects,

the conventional machining equipments are not sufficient. For the improving product quality and reducing the product cost use of parallel manipulator is mandatory and the Stewart platform comes as a forerunner because of its

6-dof capability. In this paper a concise kinematic modeling of a Stewart platform and stiffness analysis is done. A

MATLAB code was written implementing the Maximum stiffness model for the different trajectories within the

workspace. The trajectory with maximum stiffness for different contours is obtained using the model.

Keywords: stiffness, Stewart platform, trajectory

Introduction: Stiffness is one of the important considerations in the design of Stewart platform based Parallel

Kinematic Machines (PKM). The stiffness of the Stewart platform manipulator is not constant throughout itsworkspace. It depends on the position and orientation of the mobile platform. The stiffness varies with the direction

in which it is computed, the posture (or configuration) of the manipulator. The stiffness matrix can be found out by

using inverse kinematics through the position analysis and Jacobian matrix. A lot of research work has taken placein order to model the stiffness of the Stewart platform. The developed model is used for the benefit of less power

consumption in machining operations without going for high-powered configurations and complex designs in

Stewart platform.

This work focuses on stiffness variations of Stewart platform, varying positions and orientation of the moving

platform offline for different trajectories within the workspace. An optimal solution is obtained for maximum

stiffness with respect to each trajectory.

Stewart Platform as a Machine Tool: The most important properties of the industrial robots are the high speed,

fast acceleration and a large working area compared with machine tools. Parallel kinematics combines successfully

these properties. The kinematics is very simple, and it consists of three sections• Struts (legs), which can change their length generally ball screws are used as these struts.

• One fixed and a movable platform. The fixed platform serves as a rigid frame, which does not deflect under

load, and this platform contains the drive mechanism. The moving platform gives the space to the mainspindle, gripper or any other unit.

• Joints connect the kegs and transfer the load energy.

The advantages of the parallel kinematics are reduction of moving masses and a higher stiffness. Bending moments

do not take effect or they are minimized, therefore a small diameter strut can be used to make a stiff structure. Struts

under compression have a tendency to buckle, usually it sets limit to the load. The disadvantages are the relativelysmall rotating range of the moving platform and due to the kinematics of the hexapod even a simple linear

movement requires the simultaneous control of 6 axes.

INVERSE KINEMATICSPosition Analysis Of A Stewart Platform: The given fig 1 shows a spatial 6-dof, 6 SPS parallel manipulator

having a six identical limbs connecting the moving platform to the fixed base through spherical joints at points B i and Ai, for i = 1,2…6 respectively. First, we derive the location of each limb in terms of the location of the moving

platform. The vector loop equation can be written as

(1)

where ai =[aix, aiy, 0]T denotes the position vector of Ai with respect to the fixed frame, B bi=[biu,biv,0]T denotes the position vector of Bi with respect to the moving frame, bi=[bix,biy,biz]

T represents the vector B bi expressed in the

i i i ia d s P b+ = +



fixed frame (i.e., bi = AR B B bi), where si is a unit vector pointing from Ai to Bi and di is the length of limb i. Solving

for si from the above equation,

(2)

where, (3)

For the inverse kinematics problem, the position vector P and rotation matrix AR B of frame B with respect to A aregiven and the limb length di, i = 1, 2 ...6 are found to be,

(4)

For i = 1, 2...6. Hence, corresponding to each given location of the moving platform, there are generally two

possible solutions for each limb. However, the negative limb length is physically not feasible.

Jacobian Analysis of Parallel Manipulators: An important limitation of a parallel manipulator is that singular

configuration may exist within its workspace where the manipulator gains one or more degrees of freedom andtherefore loses its stiffness completely, which is undesirable. This property is included in the developed model using

the Jacobian matrix. From the above fig 1, the output vector is given by ∆. . . . . . .

2 61 3 4 5[ , , , , , ]T q d d d d d d = , and the

input vector is described by the vector of the centroid P and the angular velocity of the moving platform:

(5)

The Jacobian matrix can be derived by formulating a velocity loop-closure equation for the ith limb can be written as

(6)

Differentiating the above equation with respect to time yields

(7)

where, bi and Si denote the vector and a unit vector along , respectively, and denotes the angular velocityof the ith limb with respect to the fixed frame A. The above equation can be written in matrix form, which yields a

scalar equation i.e.,

(8)

Hence the overall Jacobian matrix is given by

= (9)

i ii

i

P b a s

d

+ −=

Fig.1: General Stewart platform Fig.2: Top view of Stewart platform

A B

i B i id P R b a= + −

[ ] [ ] A B T A B

i B i i B i id P R b a P R b a= ± + − + −

[ , ]T p B X V w

•

=

i i i iOP PB OA A B+ = +

.

p B i i i i i iv b d s d sω ω + × = × +

A Bi ii

PB

. .

x q J X J q=

1 *q x J J J −=

5

1 1 1

2 2 2

3 3 3

4 4 4

5 5

6 6 6

( )

( )

( )

( )

( )

( )

T T

T T

T T

T T

T T

T T

s b s

s b s

s b s

s b s

s b s

s b s

× ×

×

× × ×

iω



Stiffness Modeling: Stewart platforms in particular are increasingly being studied for possible use in multi-axis

machine tools. An important consideration in the design of such machine is their stiffness. For a given design,

stiffness varies with the direction in which it is computed, the posture (or configuration) of the mechanism and the

direction of the actuation or disturbing force. The stiffness matrix for the parallel manipulator given by,

(10)

where, χ

- the axial stiffness of the legs being equal and

JT - the transpose of Jacobian matrix.

The above equation implies that the end-effector output force is related to its deflection by the stiffness matrix K.

Further more, if the limbs are of the same type and the spring constants associated with all the drivelines are of thesame valve (i.e., k 1=k 2=k 3=k 4=k 5=k 6), the stiffness matrix reduces to the form:

(11)

COMPUTER SIMULATION AND DISCUSSION

CASE 1: For the first simulation, the orientation of the moving platform remains constant while the position vector

of moving platform is given a sine wave motion. The displacement of the moving platform for this purpose is given

by: [0,0,1.0 0.2*sin ,0,0,0] X wt ∆ = + .

Conditions: 0ψ = ; 0;θ = 0;φ = P = [0 0 (1.0 + 0.2*sinwt)], where w = 3.0 rad/sec.

The moving platform is moving along Z direction alone without any orientation changes. Fig 3(a) shows the sinewave motion of the moving platform in y-axis and the x-axis shows the time in seconds. Fig 3(b) shows the leg

length variation when the moving platform moves along the Z-direction. Fig 3(c) shows the stiffness variation of the

Stewart platform when moving only in Z-direction without any rotation of moving platform. Fig 3(d) shows the

corresponding actuators deflections.

CASE 2: For the next simulation, the moving platform will follow the circular path, with a radius of 200mm along x

and y directions and at a constant linear velocity of 0.0133m/s. For the circular trajectory displacement is given by,

where r - the radius of the circular machining path trajectory which is 0.2 mβ - the cutter angle position with respect to the tool cutter (i.e. β = (feed rate/ radius) * time.

T K J J χ =

T K kJ J =

L ( m )

K

N / m

D

( x 1 0 - 3 m )

time (s) time (s)

time (s) time (s)

D x

( x 1 0 - 3 m )

(a) (b)

(c) (d)

Fig. 3: Plots for Sine wave trajectory in Z direction

[ *sin , *(1 cos ), , , , ] x y z x p r P r p β β ψ θ φ ∆ = + + −



The orientation of the platform is varied as: ψ = -15 to15 deg; θ = -10 to 15 deg; φ = -15 to 20 deg.

Conditions: Px = 0.8660; Py = 0.5000; Pz= 3.0000; ψ = -10.0000; θ = 15.0000; φ = 0;

Maximum stiffness value is 7.1219e+004 N/m

From the fig 4(a) indicates that the moving platform displacement along the circular trajectory. For the

corresponding circular path the leg length variations are shown in fig 4(b). The leg lengths, which are obtained fromthis simulation, are within range and satisfy the workspace criterion also. Fig 4(c) shows the trajectory, which has

maximum stiffness and the stiffness variation within the trajectory. The maximum stiffness path has minimumstiffness variation along the trajectory. Fig 4(d) shows the variations of actuator’s displacements when it starts at

zero (at time t = 0).

CONCLUSION

Mathematical model has been derived for the Stiffness of Stewart platform. A MATLAB code was written

implementing the Maximum stiffness model for the different trajectories within the workspace. The trajectory with

maximum stiffness for different contours were obtained with the above model, the results of which were shown for two cases. The results obtained with this stiffness model were found to be valid and implementable. By changing the

position and orientation of the moving platform the model was tracked and was found to guarantee valid results

throughout, although no attempt has been made to estimate the computational efficiency of the algorithm. The leg

length variations and the corresponding actuator displacements were also studied and were found to be within the

workspace. The output of this work can be used to get important design information with respect to stiffness of theStewart platform manufactured for machining purposes.

LIST OF REFERENCES

[1] Tsai, L.W., Robotic Analysis: The Mechanics of Serial and Parallel Manipulators, John-Wiley & Sons, New

York, 1999.[2] Bhaskar Dasgupta, T.S Mruthyunjaya. “The Stewart platform manipulator: A review”, Mechanism and Machine

theory, Vol. 35, 15-40, 2000.

[3] S. Pugazhenthi, T.Nagarajan and M.Singaperumal, “Optimal trajectory planning for a hexapod machine tool

during contour machining”, Journal of Mechanical Engineering Science, Vol. 216, 1247-1257, 2002.

D x

x 1 0 - 3 m

L

m

K

N / m

D

( x 1 0 - 3 m )

time s time (s)

time (s) time (s)

(a) (b)

(c) (d)

Fig. 4: Plots for circular trajectory

stiffness analysis and kinematic modeling

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