Mechanism and Machine Theory 71 (2014) 115–125

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Mechanism and Machine Theory

j ourna l homepage: www.e lsev ie r .com/ locate /mechmt

Kinematic nonlinearity analysis in hexapod machine tools:Symmetry and regional accuracy of workspace

Davoud Karimi a, Mohammad Javad Nategh b,⁎a Tarbiat Modares University, Department of Mechanical Engineering, Tehran, Iranb Tarbiat Modares University, Department of Mechanical Engineering, Jalal-e Al-e Ahmad Boulevard, P.O. Box 14115-143, Tehran, Iran

a r t i c l e i n f o

⁎ Corresponding author. Tel./fax: +98 218288439E-mail address: nategh@modares.ac.ir (M.J. Nate

0094-114X/$ – see front matter © 2013 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.mechmachtheory.2013.09.

a b s t r a c t

Article history:Received 19 March 2013Received in revised form 16 September 2013Accepted 20 September 2013Available online 17 October 2013

The kinematic nonlinearity of hexapod machine tools is the source of an error which requiresspecial consideration during toolpath planning and command generation. Analytical formulationfor such an error is presented in this paper. The presented formulation can also be employedin optimal toolpath planning and workpiece setup. The effects of toolpath parameters such as thelength, orientation, and the location of the toolpath together with the upper platform travelspeed are considered in this investigation. Toolpaths are categorized based on their orientationinto vertical toolpaths, horizontal–horizontal toolpaths, horizontal–vertical toolpaths, horizontal-inclined toolpaths, and inclined toolpaths. Variation of the kinematic nonlinearity error and theregional accuracy of the machine's workspace are presented in this paper. Workspace symmetryis studied and it is shown that one-sixth of the machine's workspace can represent the wholeworkspace. Eventually, the presented formulation and the effect of parameters are experimentallyverified.

© 2013 Elsevier Ltd. All rights reserved.

Keywords:Kinematic nonlinearityKinematic errorHexapod machine toolToolpath

1. Introduction

The Gough–Stewart platform mechanism, invented and introduced by Gough [1] and Stewart [2] in the 1960s found anapplication as of the 1990s in machine tool industry known as the Stewart-platform-based or hexapod machine tools. Theprincipal novelty of the mechanism, which has attracted many attentions in the machine tool industrial and academic world, lieswithin its kinematic chain, which is parallel in contrast to the ordinary and common types of machine tools whose kinematicchains are generally serial. The Stewart platform mechanism has some special features like high degrees of freedom, varyingstiffness and dexterity throughout its workspace, existence of singular points inside the workspace, and nonlinear mappingbetween the kinematic input and output spaces. The last feature originates from the nonlinear kinematics of the mechanism.

It is well known that the solution of the forward kinematics of the SPBMT leads to solving a set of nonlinear equations.Although many attempts have been made to solve the set of nonlinear equations, the kinematic nonlinearity of the mechanismhas not received enough attention. Wang et al. [3] considered the nonlinearity and its effect on the interpolation accuracy ofparallel kinematic machine tools based on the Bates and Watts measure of nonlinearity. They demonstrated that the kinematicnonlinearity of the mechanism is the root of interpolation geometric error. Karimi and Nategh [4], considered the effect ofinterpolation parameters on the kinematic nonlinearity of the Stewart platformmechanism. They concluded that the most impactis contributed by the region length. Wang et al. [5] also acknowledged that the nonlinear mapping between the actuators andworkpiece spaces of a three degree of freedom parallel kinematic CNC machine tool leads to a difference between the real andideal trajectories. They used singular values of the inverse of the Jacobian matrix to predict the theoretical error.

Linear mapping between the actuators and workpiece vector space is the characteristic of a CNC interpolation algorithm. Someadvanced interpolators like adaptive interpolators with confined chord error [6,7], or interpolators which observe feed fluctuations

6.gh).

ll rights reserved.007

116 D. Karimi, M.J. Nategh / Mechanism and Machine Theory 71 (2014) 115–125

like [8] are developed based on the assumption of a linear mapping between the actuators (inputs) and workpiece (outputs) vectorspace. These algorithms are readily applicable to serial kinematic basedmachine tools as thismapping is already linear. However, thisis not true in the case of a parallel kinematic basedmachine tool, which adds an additional complexity to the problem of interpolationin these machines. Besides, as Wang et al demonstrated [5], such a nonlinear mapping brings about a geometrical error while themechanism is moving between two points. Linear toolpaths are very common during machining. In most part programs morecomplex paths are even broken out into linear segments to make it easy for the CNC machine to interpolate them. There are manymachining operations in which linear motions of the tool with respect to the workpiece are required. Among many are drilling,boring, reaming, slot milling, broaching, etc. Based on this fact the axes of a machine tool are strictly required to move straightly.Straightness is a significant characteristic of amachine tool's axis as ISO 230 indicates. Out of straightness in serial-kinematicmachinetools roots mainly in the malfunction of mechanical components of the feed drive system. In parallel-kinematic machine tools,however, the source of this error could be the nonlinear behavior of their kinematics. This error is termed here as kinematic error.Kinematic error is formulated and the effects of parameters are considered based on the presented formulation to study the regionalaccuracy of a hexapod machine's workspace. The symmetry of the workspace is studied and it is shown that for general hexapodmachines the workspace can be represented by 1/6th of its whole workspace. The formulation and the discussion are experimentallyverified.

2. Configuration symmetry in hexapod machine tools

There are six spherical, six universal, and six prismatic joints in a hexapod mechanism as shown in Fig. 1. For the machinemanufactured in the ATMT (advanced technologies in machine tools) laboratory, spherical joints are arranged around a circle onthe upper platform (UP) with the radius of 250.93 mm and the universal joints around a circle on the lower platform (LP) withthe radius of 634.24 mm. These values are obtained through parameter identification after calibration [10].

Both the spherical and universal joints are arranged in pairs each of which are designed to be 120° away from the adjacentpair. The angular distance between each two universal joints are 40° and each two spherical joints are 85° away from each other.The joints shown in Fig. 1b are numbered counterclockwise. It is evident from the arrangement of the spherical joints that thecircle of UP as well as the circle of LP has a 120-degree symmetry around their center points. The mechanism shown in Fig. 1 is atits home position, in which the prismatic joints (pods) are fully retracted. In the home position, the center of the local coordinatesystems is located at the point measured in the global coordinate system which is attached to the center of the circle of LP. All

ba

dc

First principle axis of symmetry

Fig. 1. Hexapodmachine tool table manufactured in ATMT Lab (a) CADmodel, (b) schematic figure of the mechanism, (c) mechanism at AI and (d) mechanism at AII.

117D. Karimi, M.J. Nategh / Mechanism and Machine Theory 71 (2014) 115–125

dimensions are in millimeters. The arrangement of the joints implies a 60-degree symmetry in any x–y plane of workspace at anyspecific z-coordinate.

The symmetry of the mechanism can be visualized from Figs. 1(c), (d) and 2. By the definition any symmetrical points in theworkspace provide the mechanism with an identical configuration. From the physical point of view identical configurations can berealized by examining the pods' lengths. In two identical configurations, the mechanism has the same pod lengths but in differentcombinations. It is evident that each configuration corresponding to a specific point in the workspace can be obtained from theconfiguration determined from its symmetrical point by just reordering the pods' number. Fig. 1(c) and (d) shows the mechanismfrom the top view at two symmetrical points with respect to the first principal axis of symmetry, i.e. points AI = [175 175 550]and AII = [175 − 175 550]. The corresponding lengths of the six pods calculated through the inverse kinematics are lA

I ¼571:24 607:35 625:73 712:61 708:34 585:58½ � and lA

II ¼ 585:58 708:34 712:61 625:73 607:35 571:24½ �, respectively. These twosets of six-pod lengths are permutation of each other. For example the length of the first pod in the first configuration is equal to571.24 mmwhich is the same as the length of the sixth pod in the other configuration, etc.

As shown in Fig. 2b, there are three principal axes of symmetry and six areas of symmetry. Geometrically, the principalaxes of symmetry lie on the bisectors of adjacent spherical joint pairs. Therefore, the first, second, and third axes ofsymmetry are coincident with the bisectors of 1st/6th, 2nd/3rd, and 4th/5th spherical joints, respectively. Physically, eachprincipal axis together with the z-axis determines a plane with respect to which each pod can be mirrored to obtain itssymmetrical pair at home position. Thus at home position pod 1 is the mirror of pod 6 with respect to the plane characterizedby the first axis of symmetry and the z-axis. Pods 2/3 and 4/5 have the same position with respect to the second and thirdaxes of symmetry, respectively. Any point in the first area of symmetry provides the mechanism with an identicalconfiguration as its symmetrical point in the second and sixth areas. As shown in Fig. 2b, the first and sixth areas aresymmetrical about the first principal axis. Likewise, the first and second areas are symmetrical about the third principal axis,etc. It should be noted here that the mechanism has no symmetry along its z axis. As shown in Fig. 2c, points A and B are

Fig. 2. Areas and axes of symmetry of the mechanism in home position; a) perspective view, b) top view, c) principal and secondary axes of symmetry.

118 D. Karimi, M.J. Nategh / Mechanism and Machine Theory 71 (2014) 115–125

symmetrical with respect to the third principal axis of symmetry. Points B and C are also symmetrical with respect to thesecond principal axis of symmetry.

It can be inferred from Fig. 2c that points A and C should also be symmetrical. The axis of symmetry of these points isillustrated in Fig. 2c. This axis of symmetry is termed here as secondary axis of symmetry. The same rationale can be employed toobtain the symmetrical point of A at the fifth area of symmetry. It should be noted here that there are only three principal axes ofsymmetry. However, for any two points in the second and sixth areas of symmetry there is a unique secondary axis of symmetry.Thus there exists infinite numbers of secondary axes of symmetry for this mechanism.

2.1. Kinematic error model

It is assumed that the UP at the time t is at the point Pr(t), which is located on the actual trajectory. The corresponding point atthe same time (t) on the desired trajectory is denoted by Pd(t). After an infinitesimal variation of time, the UP is traveled to thepoint Pr(t + dt), whose corresponding point on the desired trajectory is shown by Pd(t + dt) in Fig. 3.

In Fig. 3, N(t) is a unit vector in the direction of Pr(t) − Pd(t). Correspondingly, N(t + dt) is a unit vector in the direction ofPr(t + dt) − Pd(t + dt); e(t) and e(t + dt) are the kinematic errors at t and t + dt, respectively. Based on this figure, thefollowing can be written.

e t þ dtð ÞN t þ dtð Þ−e tð ÞN tð Þ ¼ dPr−dPd ð1Þ

Substituting the first term in the Taylor expansion of e(t) and N(t) into Eq. (1), the following relation is obtained.

e tð Þ þ deð Þ N tð Þ þ dNð Þ−e tð ÞN tð Þ ¼ dPr−dPd⇒eNþ edNþ deNþ dedN−eN ¼ dPr−dPd

ð2Þ

Since N(t) is a unit vector, dN(t) can be written as follows:

dN ¼ dθM ð3Þ

dθ is the angle between N(t) and N(t + dt) as shown in Fig. 3; M is a unit vector perpendicular to N. Substituting Eq. (3)

whereinto Eq. (2) yields:

eNþ edθMþ deNþ dedθM−eN ¼ dPr−dPd⇒edθMþ deN ¼ dPr−dPd

ð4Þ

dedθ is a higher order term and is neglected. As N and M are normal to each other, the first term on the left hand side of

whereEq. (4) vanishes when the dot product of this equation by the vector N is taken into consideration, which yields:

de ¼ dPr−dPdð Þ � N⇒de ¼ V−Fð Þ � N dt ð5Þ

V and F are the velocity vectors along the actual and the desired trajectories, respectively. The speed along the desired

wheretrajectory is the programmed feedrate in part programming of CNC machines.

Considering the velocity kinematics of the mechanism [9], Eq. (5) can be rewritten as follows.

e tð Þ ¼ V tð Þ−F tð Þð Þ � N tð Þ ¼ J−1Pr tð Þ � lr tð Þ− J−1

Pd tð Þ � ld tð Þ� �

� N tð Þ ð6Þ

Fig. 3. The actual and desired trajectories.

where

119D. Karimi, M.J. Nategh / Mechanism and Machine Theory 71 (2014) 115–125

JPr−1 is the inverse of the Jacobian matrix at Pr(t) and JPd−1 is the inverse of the Jacobianmatrix at Pd(t). l ˙r is the actual rate of

where

change of the pod's elongation vector at the time (t), and l ˙d is a 6D vector whose elements are the derivatives of the pod length

equation with respect to time. The pod length equation is presented in Eq. (7).

li tð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXi

2 þ Yi2 þ Zi

2q

ð7Þ

XiYiZi

24

35 ¼

x ˙t þ xA þ Rsið Þx−uix

y ˙t þ yA þ Rsið Þy−uiy

z ˙t þ zA þ Rsið Þz−uiz

264

375 ð8Þ

F ¼ x ˙ y ˙ z ˙½ �T . PA ¼ xA yA zAh iT

is the position vector of the UP at the beginning of the desired trajectory; ui ¼�

whereuix uiy uiz

� T and si are the vectors of ith universal and spherical joints, respectively. The Jacobian matrix of the mechanismtogether with the rotation matrix (R) is described in [9].

Eq. (6) can be rearranged as follows:

de ¼ J−1Pr l ˙

r−l ˙d

� �þ ΔJ−1

P l ˙d

n o� N dt ¼ J−1

Pd l ˙r−l ˙

d

� �þ ΔJ−1

P l ˙r

n o� N dt

⇒de ¼ J−1Pr δ l ˙ þ Δ J−1

P l ˙d

� �� N dt ¼ J−1

Pd δ l ˙ þ Δ J−1P l ˙

r

� �� N dt

ð9Þ

ΔJP−1 = JPr−1 − JPd−1, and δl˙ is a 6D vector whose elements are the derivatives of the pods error equation, known as the pod

whereelongation rate error, which can be written as follows:

δli tð Þ ¼ Δli � fL

� t þ lAi −ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXi

2 þ Yi2 þ Zi

2q

ð10Þ

Δli = (liB − liA), liB and li

A are the lengths of ith pod when the UP is at the end and beginning of the desired trajectory,

whererespectively; f = |F| is the speed of UP moving from A to B; and L is the length of the desired trajectory.

The kinematic error can be derived from Eq. (9), as follows:

e tð Þ ¼Zt

0

J−1Pd δ l ˙ þ ΔJ−1

P l ˙r

� �� N

n odt: ð11Þ

This equation provides the kinematic error at a specific time (t). Of course replacing t with T, which is the total time requiredfor the upper platform to travel all through the trajectory, will provide the kinematic error at the end of the trajectory.

The time at which the maximum kinematic error occurs is of special interest. This time is denoted here by t* and can be foundby maximizing Eq. (6) as the derivative of the kinematic error. Substituting t* into Eq. (11) gives the maximum kinematic error,which is designated here by e*.

3. Regional accuracy and the effect of parameters on the maximum kinematic error

It can be verified through Eqs. (11) and (10) that the kinematic error is a function of the velocity Jacobian which is in turn afunction of the position/orientation of the UP, pod elongation rates, unit vector N, pods elongation rate error, speed of UP travel andthe length of the desired trajectory. For the purpose of simplicity, the orientation of the UP is neglected. Furtherwork is needed to takethe orientation of the UP into account. A desired trajectory is described here by the position of its midpoint (P*) represented by thedistance from the z-axis denoted by r, the length of the trajectory (L), and its orientation represented by two angles φ and γ; the

Fig. 4. The desired trajectory.

Table 1The values of parameters.

From To Step length Number of level

z (mm) 600 900 100 4r (mm) 0 300 20 16θ (deg) 0 60 10 7φ (deg) −90 90 10 19γ (deg) −90 90 30 7L (mm) 50 300 50 6

120 D. Karimi, M.J. Nategh / Mechanism and Machine Theory 71 (2014) 115–125

position of r is determined by the angular distance measured from the positive direction of the x-axis denoted by θ, and itsz-coordinate. These parameters are shown in Fig. 4. The positive signs of angles are considered to be counterclockwise.

To consider the effects of parameters (r, θ,φ, z, γ, and L), different values have been examined for these parameters as presented inTable (1). In this table, the step length is the value by which the parameter range is divided to obtain the number of levels that eachparameter takes during the analysis. By “level” a specified value of the parameter utilized during the analysis is meant.

Different runs amounting to 357,504 were conducted for evaluating Eq. (11) at each corresponding t* to obtain the maximumkinematic error. The values of P*, PA, and PB. can be obtained from the considered parameters as follows:

P� ¼ r cos θð Þ r sin θð Þ z 0 0 0½ �T ð12Þ

Fig. 5. The effect of L in the presence of a) r, b) θ, c) φ, d) γ, and e) z on e⁎.

where

121D. Karimi, M.J. Nategh / Mechanism and Machine Theory 71 (2014) 115–125

PA ¼ r cos θð Þ− L2cos γð Þ cos φð Þr sin θð Þ− L

2cos γð Þ sin φð Þzþ L

2sin γð Þ 0 0 0

� �Tð13Þ

PB ¼ r cos θð Þ þ L2cos γð Þ cos φð Þr sin θð Þ þ L

2cos γð Þ sin φð Þz− L

2sin γð Þ 0 0 0

� �Tð14Þ

A and B denote the first point and the last point of the trajectory, respectively.

3.1. The effect of the desired trajectory length

The effect of the trajectory length, L on e* is illustrated in Fig. 5. This figure clearly shows that for all values of θ,φ, and γ an increaseof L always results in the increase of e*. In addition, the effects of r, θ, and φ compared to the effect of L and e* are negligible to a gooddegree of accuracy. Fig. 5d and e shows that at higher values of L the effects of γ and z become more significant.

3.2. The effect of z-coordinate

The surface plot of e* versus z in the presence of r, θ, φ and γ or L = 150 (mm) is illustrated in Fig. 6.It can be seen from Fig. 6a, b and c that e* decreases from the bottom to the top of the workspace for different values of r, θ

and φ. This decrease amounts to about 55%. The effects of r, θ and φ are almost negligible compared to the effect of z Theinteractive effect of z, γ and e* also shows that the increase of z decreases the maximum kinematic error of the actual trajectory asFig. 6d illustrates. On the contrary to the effects of r, θ and φ the effect of γ is more significant than that of z.

More accurate and finishing operations are recommended to be performed in higher portions of the workspace where themaximum kinematic error decreases. The finishing operations are usually carried out at higher cutting speeds, indicating that thehigher portions of the hexapod workspace are more suitable for high speed machining. The same recommendation has alreadybeen presented by Mahboubkhah et al. [11] based on the stiffness and dynamic behavior of a hexapod machine.

3.3. The effect of the midpoint location (r, θ) and the orientation (φ, γ)f the desired trajectory

To discuss the effects of r, φ and γ three different categories of toolpaths consisting of the vertical, the horizontal and theinclined ones are taken into consideration.

The vertical toolpaths are aligned along the z axis of the machine. Such toolpaths are very common in machining operationslike drilling, reaming, broaching, boring, honing etc. In this category of toolpaths γ = ± 90 (Fig. 4). The effect of r and θ is plotted

Fig. 6. The effect of z in the presence of a) r, b) θ, c) φ, and d) γ on e⁎.

122 D. Karimi, M.J. Nategh / Mechanism and Machine Theory 71 (2014) 115–125

in Fig. 7a. This figure shows that e* is almost zero for r = 0which implies that for the vertical toolpaths the kinematic error at thecenter of workspace almost vanishes. The vertical toolpaths coinciding with the center of the hexapod workspace are the onlytoolpaths along which the kinematic of the mechanism exhibits linear behavior. Therefore, it is recommended that the workpiecebe positioned at the center of the UP so that the toolpath coincides with the z axis in operations requiring the UP to movevertically. Fig. 7a also shows that the farther a vertical toolpath from the center of the workspace, the more the maximumkinematic error is. It also shows that the effect of θ is almost negligible around the center of the workspace. However, the farther avertical toolpath from the center of the workspace, the more significant the effect of θ could be. At far distances from the center,the kinematic error decreases by increasing θ so that θ = 60 provides a minimum error.

The surface plots of the maximum kinematic error versus r and θ or horizontal toolpaths are depicted in Fig. 7b to e.Fig. 7b shows that as horizontal toolpaths move away from the center of the workspace on the x-axis (θ = 0), the maximum

error grows most rapidly. The maximum error occurs on the x-axis at the farthest point. On the other hand, on the θ = 60 (deg)line the distance from the midpoint to the center of the workspace has no significant effect on the maximum kinematic error.

Fig. 7c indicates that for horizontal toolpaths in the y-axis direction, the effect of r is dominant. It can also be inferred from thisfigure that the smallest error is expected to occur at the center of the workspace and the largest one is expected to happen at thefarthest point for the highest value of θ. This figure demonstrates that increasing both parameters r and θ increases the maximumkinematic error by about 15% at most.

For horizontal toolpaths inclined to both x-axis and y-axis with φ b 0 (Fig. 7d), as the UP moves away from the center of theworkspace, the impact of the parameter θ becomesmore significant such that at the edge of the workspace error falls most rapidlywith an increase of θ and consequently the maximum kinematic error occurs at the farthest point on the x-axis and the minimumkinematic error happens at the farthest point on the θ = 60 (deg) line. It can also be shown that r is not significant around theθ = 50 (deg) line, thus along this direction moving away from the center of workspace does not change the error significantly. Incontrast, e* increases by about 10% when moving away from the center along the x-axis.

Fig. 7e illustrates that the significance of parameter r or horizontal toolpaths inclined to both x-axis and y-axis with φ N 0 isdominant. The maximum error, e* increases by about 25% when moving away from the center to the farthest point of theworkspace. As is evident from this figure, e* is maximum at the edge of the workspace and minimum at its center.

Fig. 7. Surface plot of e⁎ versus r and θ for a) vertical tool paths, b) horizontal toolpaths in x-axis direction, c) horizontal toolpaths in y-axis direction, d) horizontaltoolpaths inclined to both x and y axes with φ b 0, and e) horizontal toolpaths inclined to both x and y axes with φ b 0; in all cases L = 150, and z = 700.

123D. Karimi, M.J. Nategh / Mechanism and Machine Theory 71 (2014) 115–125

The surface plots of the maximum kinematic error for inclined toolpaths are illustrated in Fig. 8.Generally, for inclined toolpaths with γ b 0 (Fig. 8a to c), it can be stated that a minimum e* occurs at the farthest point from

the center on the θ = 60 (deg) line.For toolpaths with γ N 0 and φ ≤ 0, Fig. 8d and e illustrates that, an increasing θ increases e*. This effect is stronger at higher

values of r at the edge of the workspace. On the other hand, the curvature is inversely proportional to r and an increase in thedistance from the center of the workspace leads to a decrease in e*. However this effect vanishes on the θ = 60 (deg) line.Therefore, the minimum e* happens at r = 300 and θ = 0 (at the edge of the workspace on the x-axis of the machine) and themaximum one occurs near the center of the workspace. For HITs with positive φ, r lays the same role and is inversely proportionalto e*. It can also be seen from Fig. 8f that for large values of r increasing θ decreases e*. Therefore, the minimum error happens atthe edge of the workspace on the θ = 60 (deg) line while the maximum one occurs at the center of the workspace.

4. Experimental setup and verification

The hexapodmachinemanufactured in ATMT Lab as shown in Fig. 9was used for the experiments. It consists of a Stewart platformmechanism, a PC onwhich an interface (HMI) is installed, a motion controller, and six servo systems including servo drives and servomotors. The interface developed by the authors has been programmed in Visual C# and is capable of interpreting the G-codecommands based on ISO 6983. The motion controller is a real time module and is capable of controlling six servo systemssimultaneously. Commands are input using theG-code standard. TheHMI interprets the command, solves the inverse kinematics, andsends the data corresponding to the six pods' lengths to the motion controller. The motion controller provides the six servo driveswith the appropriate voltage as the servo input command and receives the feedback from the servo encoders to close the positioncontrol loop. The velocity control loop is closed in the servo drive. A P–PI controller was developed for themachine. The P-controller isapplied to the position control loop and the PI controller is applied to the velocity control loop. To measure the position of the UPduring its motion, an optical pair of digital cameras is used. The cameras take a picture from two perpendicular directions from atarget installed on the UP every 1/30 of a second. The side camera as shown in Fig. 9 is carefully aligned normal to the y–z plane of themachine and the front camera is also aligned normal to the x–z plane of themachine. The films recorded by the cameras are processedusing image correlation function embedded in MATLAB to obtain the location of the target every 1/30 of a second.

It should be noted that the image correlation output is obtained in terms of pixels, which is calibrated using the target, whosediameter as shown in Fig. 9b is 24.3 mm.Measuring the number of pixels occupied by the target in the images recorded by the side andfront cameras revealed the fact that the resolution of displacement measurement during the experimental procedure in this researchvaries from 0.4 to 0.7 (mm) for the front camera depending on the zoom of the camera and 0.3 to 0.5 (mm) for the side camera. Theseresolutions are accurate enough regarding the accuracy and repeatability of the structure of the manufactured machine.

The length (L), the location of the midpoint (r, θ, and z) and the UP travel speed for different trajectories are the factorsinvolved in the experiments. The values of these factors are presented in Table 2.

The γ and φ angles for different toolpaths are presented in Table 3.The analytical and experimental results for e* are compared in Fig. 10. The maximum difference between the experimental and

analytical kinematic errors among the 24 toolpaths occurred for the 15th toolpath. The difference is 0.7 (mm) which is 14%.

Fig. 8. Surface plots of e⁎ versus r and θ for z = 700, L = 150; a) φ b 0, γ b 0, b) φ = 0, γ b 0, c) φ N 0, γ b 0, d) φ b 0, γ b 0, e) φ = 0, γ b 0, f) φ b 0, γ N 0.

Fig. 9. Experimental setup; a) hexapod machine, b) target for the side camera, c) target for the front camera.

124 D. Karimi, M.J. Nategh / Mechanism and Machine Theory 71 (2014) 115–125

5. Conclusion

In the present study, the kinematic error due to the nonlinearity of the mechanism was obtained analytically. The conclusionsare summarized as follows.

• The workspace of a class of the Stewart mechanism in which joints are arranged symmetrically at home position was studiedand it was shown that 1/6 of the workspace can efficiently represent the behavior of the mechanism.

• Increasing the length of the toolpath always increases the maximum kinematic error. Since the toolpath length has nointeraction with the symmetry of the mechanism this conclusion can be generalized to the general Stewart mechanisms.

Table 2Factors and their levels.

Factors Level 1 Level 2 Level 3

L 100 150 200r 0 150 300θ 0 30 60z 600 700 800f 10 15 20

Table 3The values of γ and φ for toolpaths.

Toolpaths γ φ

Horizontal toolpaths in x-axis direction 0 0Horizontal toolpaths in y-axis direction 0 90Horizontal toolpaths inclined to x-axis and y-axis 0 45Vertical toolpaths 90 0Inclined toolpaths 1 −30 0Inclined toolpaths 2 30 60

Fig. 10. Analytical and experimental results of the maximum kinematic errors for different toolpaths.

125D. Karimi, M.J. Nategh / Mechanism and Machine Theory 71 (2014) 115–125

• The kinematic error was shown to decrease by about 55% from the bottom to the top of the workspace. Since the mechanism isnot symmetrical along the z-axis it is expected for the general class of Stewart mechanisms that the kinematic error decreases atelevated levels of the workspace.

• The height of the upper platform position is the most influential parameter among the other parameters specifying the locationof a toolpath.

• More accurate and finishing operations and also high speed machining are recommended to be performed in the higherportions of the workspace of the general class of Stewart mechanisms.

• For the vertical toolpaths, the kinematic error at the center of the workspace is zero. The farther a vertical toolpath is from thecenter of the workspace, the more the maximum kinematic error. The vertical toolpaths produce less kinematic error than othertoolpaths. Equivalently, the mechanism exhibits more linear behavior when moving vertically.

• For horizontal toolpaths in the x-axis direction, the largest error occurs on the x-axis at the farthest point on the workspaceboundary and the smallest one exists on the farthest distance on the θ = 60 (deg) line.

• For horizontal toolpaths in the y-axis direction, the smallest error occurs at the center of the workspace and the largest onehappens at the farthest point for the highest value of θ.

• For horizontal toolpaths inclined to both x and y axes with φ b 0 the largest kinematic error occurs at the farthest point on thex-axis and the smallest one happens at the farthest point on the θ = 60 (deg) line.

• For horizontal toolpaths inclined to both x and y axes with φ N 0 the largest error occurs at the edge of the workspace and thesmallest one happens at its center.

• For inclined toolpaths with γ b 0 the smallest error occurs at the edge of the workspace on the θ = 60 (deg) line and largesterror occurs at the edge of the workspace as well but at lower values of θ when the toolpath approaches the x axis.

• For inclined toolpaths with γ N 0 and φ ≤ 0 the smallest error happens at the edge of the workspace on the x-axis of themachine and the largest one occurs near the center of the workspace.

• For inclined toolpaths with γ N 0 and φ N 0 the smallest error happens at the edge of the workspace on the θ = 60 (deg) linewhile the maximum one occurs at the center of the workspace.

The experimental results verified the presented formulation and the above conclusions.

References

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