INTRODUCTION
The reason why a reconfigurable Stewart platform with a variable geometry
has not as yet been successfully implemented, is the lack of an efficient methodology
for determining the optimum geometry for the prescribed task at hand (DuPlessis
and Snyman, 2006; Wavering, 1998; Zhang
and Bi, 2006). The original problem being the development of a reconfigurable
Stewart platform suitable for various applications; contour generation and vibration
isolation are the two applications chosen in this study.
From the time the potential of Stewart platform as a six DOF mechanism was
identified by Stewart (1965) for flight simulation,
researchers have constantly tried the Parallel Kinematic Machines (PKM) in various
other fields. Combining the advantages of PKM and industrial robots to solve
the problems of Machine tool applications, a new concept for machine tool application
in the name of Stewart platform was identified (Dasgupta
and Mruthyunjaya, 2000).
Presently commercial hexapods, such as Octahedral Hexapod from Ingersoll Milling
Machine, Geodetic from Geodetic Technology Ltd., are mission specific and there
is not much modular capabilities allowed for the enduser to choose between
structural rigidity and dexterity (Ji and Song, 1998;
Ji and Li, 1999; Xi, 2001).
Modular systems will enable technology obsolescence to be countered, new technologies
to be exploited and changing operational demand to be supported, in a costeffective
way. The added advantages of having modular configurations are (IMing,
2001):
• 
Shortening of system development cycle 
• 
Rapid design, fabrication and low cost 
• 
Meeting the changing operational demand and less maintenance 
The need now is a reconfigurable configuration of Stewart platform developed with at least two applications in mind. While facing the demand for reconfigurability much of research needs to be done. This study presents the initial investigations done for the contour generation application only, from the ongoing research on the problems mentioned beforehand.
Stewart platform has been studied for possible use in multiaxis machine tools
for quite some time (DuPlessis and Snyman, 2006; Dasgupta
and Mruthyunjaya, 2000; Gosselin and Angeles, 1990).
Figure 2 shows Stewart platform as a machine tool. Later it
was proposed to place the Hexapod in the design office along with the plotter
(Matar, 1999) which would reinforce the ethos of a responsive
and robust machine. It would enable us to adapt to a variety of modeling demands
without incorporating large setup overloads. A multiaxis structure enables
the tool cutting path to take the most efficient route for maximum clearance
which would equate in real time saving up to 20% in machining.
A lot of research work (ElKhasawneh and Ferreira, 1999;
Tsai, 1999; Svinin et al.,
2001; Huang and Schimmels, 1998) has taken place
in order to model the stiffness of the Stewart platform. The stiffness variation
for the straight line and circular contours (Pugazhenthi
et al., 2002) was studied and was found to yield good results based
on the dynamics of the Stewart platform. The stiffness increases as the height
of the tool path increases and is independent of its location in the xy plane.
For complex tool paths involving nonplanar cuts this is not valid. The model
needs to be amended to suit the demands of the complex tool paths. Apart from
that the illposed nature of inverse problem the noise of the observed data
leads to deviations in solutions (JingLei and Zhijian,
2011). So a thorough study on the stiffness of the Stewart platform is performed
for different contours including space contours like Logarithmic spiral. The
developed mathematical model for the maximum stiffness of Stewart platform is
implemented for different trajectories within the workspace and the trajectory
with maximum stiffness for different contours are identified. The leg length
variations for each trajectory were found to validate the exactness of the algorithm
in determining the maximum stiffness trajectory. With a continuously data moving
object management systems have been attempted, though precisely mot applicable
to this research problem (Ghajary and Alesheikh, 2008).
A mathematical model was also constructed to realize the automatic positioning
of the workpiece (Wang et al., 2001). This solves
the problem of identifying the pose with which a workpiece should be presented
however, if stiffness is considered, the problem still remains in the form of
the machine tool posture. So, an attempt is made to develop a stiffness model
based on the kinematics of the Stewart platform. This brings down the complexity
of the stiffness analysis and saves considerable amount of time from being built
into the algorithm. Kumar et al. (2005, 2009)
have presented a similar analysis for various contours. The developed model
is now used to study the parameters affecting reconfigurability of the Stewart
platform. Of the various parameters studied the effect of ‘configuration’
is presented in this study.
Spiral trajectory: Archimedes studied the first and simplest spiral
and it bears his name: the Archimedean spiral. An Archimedean spiral trajectory
is shown in the Fig. 1. It is generated when a traveling object
P moves at constant speed v on a pole that in turn, rotates uniformly around
one of its points, at angular speed w. The importance of the spiral trajectory
in machine tool applications is predominant in the case of pocket machining
(Bieterman and Sandstrom, 2003).
The problems faced with the use of conventional machining tool path or trajectory, to mention a few:
• 
Machine axis drive capabilities get concentrated near corners
and other highcurvature path segments 
• 
This results in more required machining time, unnecessary cutting tool
wear and wear and tear on the whole machine 
If the initial position of the point on the pole (measured from rotation center
is indicated with r_{o} and the initial angle with q, the expressions
become (Matar, 1999):
From Eq.1 and 2, deriving t from the second
equation and replacing it in the first one, the equation of the curve described
by the object becomes
where, r represents the distance from the rotation centre and q the angle counted starting from the initial position.
Figure 2a shows the conventional paralleloffset tool path. Using smooth lowcurvature contours for the tool path solves this problem. Spiral trajectory stands as the best candidate in this case. Lower curvature distributes available acceleration along the path and decreases machining time. Figure 2b shows the Spiral tool path in pocket machining. Saving wear and tear on machine and tool is the additional benefit obtained which gives extended tool life when cutting hard metals.
This novel method of introducing the lowcurvature contours is made practical
only with the help of six Degrees of Freedom (DOF) mechanisms like Stewart platform.
Following a similar analysis for the simulation as done for other trajectories,
the moving platform is allowed to follow a spiral path represented by Eq.
4. Since the radius n (θ) and the angle θ are proportional for
the simplest spiral (Qualls and Pimentel, 1999), the
Spiral of Archimedes, the expression developed for displacement along the spiral
trajectory is given by:
where, n  a * β (a  constant) β  the cutter angle position with
respect to the tool cutter (i.e., β = (feed rate/radius) * time).

Fig. 1: 
Spiral of archimedes 

Fig. 2: 
Conventional and curvilinear tool paths 
A dataset of the positions and orientations of the centroid of the moving platform
is obtained as a result of simulation. It should be executed throughout the
machining process for a given contour, in order to have maximum stiffness for
the Stewart platform. It also provides the trajectory for the spiral contour
avoiding singular positions. The position and orientation vector of the start
point for maximum stiffness trajectory is:
p_{x }= 0.5000; p_{y} = 0.8660; p_{z
}= 1.0000;
ψ= 15.0000; θ = 10.0000; φ = 5.0000; 
Maximum stiffness value obtained is = 3.0048e+004 N m^{1} for Tsai 66 platform. Since this is an offline process the machining trajectory with maximum stiffness could be acquired from the database of trajectory parameters. An analysis of the starting points of the maximum stiffness trajectory would provide us an idea of the relation between the starting points and the maximum stiffness. This would help us to solve any positioning problems within the workspace. The notations used in this section are tabulated in Table 1.
Influence of configuration: Configuration is chosen as one of the parameter
to study the reconfigurability of the Stewart platform. Four different platforms
are chosen for the analysis: Tsai 66, Tsai 33, PeilPOD 66 and PeilPOD 33.

Fig. 3: 
Stiffness variation for spiral trajectory (Tsai 66) 

Fig. 4: 
Stiffness variation for spiral trajectory (Tsai 33) 
Table 1: 
Notations 

Figure 3 and 4 give the stiffness variation
of the maximum stiffness trajectories for Tsai 66 and Tsai 33 platforms, respectively.

Fig. 5: 
Stiffness variation for spiral trajectory (PeilPOD 33) 

Fig. 6: 
Stiffness variation for spiral trajectory (PeilPOD 66) 
The variation of stiffness within the trajectory of maximum stiffness is more
for Tsai 33 platform.
Similar analysis is done for the PeilPOD configurations also and the results
are presented in Fig. 5 and 6. The parallel
nature of the curves in the plots for the stiffness and leg length displacement
is the characteristic of the 33 platform for trajectories based on space curves.
Table 2 provides a comparison for the stiffness of the maximum
stiffness trajectory, for different configurations.
Table 2: 
Comparison of maximum stiffness for archimedian spiral contour 

Table 3: 
Comparison of maximum stiffness for the logarithmic spiral
contour 

It is observed that the 66 configuration performs better than the 33 configuration
in both cases, though the difference is not more than 4%.
Logarithmic spiral: Logarithmic spiral, also known as equiangular spiral,
is taken for analysis as the next case. The characteristic feature of this curve
is that each line starting in the origin cuts the spiral with the same angle.
In parametric form, provided by Koller (2002), it is
represented by x (θ) = exp (θ) cos (θ), y (θ) = exp (θ)
sin (θ). For logarithmic spiral trajectory the displacement of the centroid
of the moving platform is obtained as:
Where:
• 
r a * β 
• 
β  cutter angle position with respect to the tool cutter (i.e.,
β = (feed rate/ radius) * time). 
• 
α a constant 
Table 3 provides a comparison for the stiffness of the maximum stiffness trajectory, for logarithmic spiral contour. Upon comparison with this result obtained for Archimedian spiral it is observed that the logarithmic spiral performs better than Archimedian spiral.
It is observed that 66 configuration performs better than 33 configuration in both cases, though the difference is not more than 4%. So the designer can prefer 33 configuration unless the loss of 4% stiffness over the 66 configuration is accountable, for a gain on the reconfigurable capability of a developed Stewart platform and increased workspace.
The length of the spiral trajectory taken for simulation is 66 mm at a feed
rate of 1.33 mm sec^{1} for Tsai platforms. The length of the trajectory
for PeilPOD platforms is taken as 25 mm at the same feed rate. The simulation
time for Tsai and PeilPOD platforms are 31 and 12 sec, respectively.

Fig. 7: 
Actuator displacement for spiral trajectory (Tsai 66) 

Fig. 8: 
Actuator displacement for spiral trajectory (Tsai 33) 
It could be seen that the maximum stiffness trajectories have minimum stiffness
variation along the trajectory, except in Fig. 4. These variations
are attributed to the occurrence of singular configurations along the trajectories.
This also shows the importance of the stiffness analysis of different contours
to be treated individually for different platforms, rather than generalizing
the results. A matching trend is observed in the actuator displacement plot
for Tsai 33 configuration shown in Fig. 8 which is unique
in comparison to the plots of other platforms. This happens in order to compensate
for the loss of stiffness when singularity is encountered.
Similarly Fig. 7 and 8 provides actuator
displacements of Tsai 66 and Tsai 33 platforms, respectively, for the maximum
stiffness trajectory. The actuator force requirements for the maximum stiffness
trajectory could be deducted from it using the forcedisplacement correlation.

Fig. 9: 
Developed stewart platform (PeilPOD 33) 

Fig. 10: 
Connections for PeilPOD 33 
All the plots follow the same trend for the spiral trajectory with exception of Fig. 8 although with varying magnitudes for all the legs. The actuator displacement plot for Tsai 33 shown by Fig. 8 following the corresponding stiffness plot shows continuous increase in the displacement. This happens in order to compensate for the reduction in stiffness within its workspace. Similar results are obtained for the PeilPOD configurations too.
The effect of second chosen parameter (anglebetweenlegs) is currently being
simulated to improvise the modular capabilities of the Stewart platform. The
dataset obtained offline from simulation needs to be analyzed for the workspace
demand it places on the platform. More information on the reconfiguration parameters
could be obtained from Kumar et al. (2009).
Experimental setup: A Stewart platform (PeilPOD33) was developed at
IIT Madras to conduct initial studies on the parameters affecting the modularity.
Figure 9 shows the developed Stewart platform, a 33 configuration
with spherical joints at the top and the bottom platform. The actuators used
are microstepper linear actuators with microstepping drive and onboard electronics.
Figure 10 shows the connections for the experimental investigations
to be conducted. The drive’s integrated electronics eliminates the need
to run the motor cabling through the machine, reducing the potential for problems
due to electrical noise.
CONCLUSION
A thorough study on the stiffness of Stewart platform is performed for different contours. The developed mathematical model for maximum stiffness of Stewart platform is implemented for different trajectories within the workspace and the trajectory with maximum stiffness for different contours are identified. Leg length variations for each trajectory were found to validate the exactness of the algorithm in determining the maximum stiffness trajectory.
This method provides us with set of positions and orientations of the moving platform that should be executed throughout the machining process for a given contour in order to have maximum stiffness. The results obtained are to be validated for singularityfree path planning and an attempt has to be made to estimate the computational efficiency of the algorithm.
In general the Tsai platform has relatively lesser stiffness when compared to the PeilPOD owing to its longer leg lengths. For most of the contours the 66 configuration fares better than the 33 configuration. The spiral trajectory which has influenced the manufacturing industries for the benefits, mentioned before is given special treatment in this research work. Archimedian and Logarithmic spirals are compared for stiffness performance. Logarithmic spiral performs better than Archimedian spiral. It is observed that 66 configuration performs better than 33 configuration in both cases, though the difference is not more than 4%. So the designer can prefer 33 configuration unless the loss of 4% stiffness over the 66 configuration is accountable, for a gain on the reconfigurable capability of a developed Stewart platform and increased workspace.