Dynamic and Stiffness Modeling of New 3DOF PKM for High Speed Machining Application

Dynamic and Stiffness Modeling of New 3DOF PKM for High Speed Machining Application

Hailemariam Nigus

Automobile Engineering Department Federal TVET Institute

Addis Ababa,Ethiopia

Hassen Nigatu

Mechanical engineering Department Adama Science and Technology University

Adama, Ethiopia

Muralidhar Reddy Gurala Mechanical Enigineering Department

Adama Science and Technology University

Adama,Ethiopia

Abstractmany applications in the field of production automation, such as machining, assembly and material loading require machines that are capable of high speed, accelerationand rigidity.This paper mainly addresses the issue of dynamic and stiffness formulation of a three prismatic-revolute-spherical PKM (3-PRS PKM) using screw theoryand virtual work approach. First the dynamic formulation is performed and later the stiffness equation is derived using virtual work principle. In order to build up the stiffness and dynamics model, kinematics, Jacobian, Hessian and Finite Element Analysis are also performed as the basis.The study of robot dynamics is necessary for its mechanical design and synthesis, providing the information on the force that must be resisted by joints, links and actuators while the study of stiffness is highly demanded to predict the rigidity performance of the PKM. So that, the inverse dynamics is developed with capable of calculating the force along the driving link direction. Also, Analytical stiffness model, a function of Jacobian matrix and components stiffness matrix, is obtained first using the principle of virtual work. Stiffness model is also a six by six dimensional matrix and can provide the information of actuation and constraint stiffness simultaneously.

Keywords: PKMs, Hessian, 3PRS, Dynamics, stiffness analysis, FEA

1. INTRODUCTION

   A parallel platform manipulator (PPM) is a device whose end- effector is attached to the ground via multiple serial chains that provides closed kinematic loops in the system for better load handling capacity and stiffness. It is these qualities that make it applicable in a wide range of applications ranging from flight simulators to high speed milling machines. However, it is these closed kinematic loops that increase the complexity of their analysis to a great extent(Y.G Li).

   An important advantage of parallel platform manipulator is that its superior structural rigidity renders it as the better choice over serial chain manipulator (SCM) for moving heavy loads and high precision machining tasks. The advantages include very high accuracy, better stiffness ratio, more payload capacityand better inertial distribution among others(24-41).

   Y.G Li, T.Huang and H.T .Liu[1] a general approach for formulating dynamics of lower mobility parallel manipulators. Moreover, Joshi, S., Liu, H., Chetwynd, D.G.,Li,Z., [2]formulated the generalized jacobian analysis of lower mobility manipulators, Dressler, I., Robertson. And Johansson, R.[3] has designed the accuracy of kinematic and dynamic models of a Gantry-Tau parallel kinematic robot, Li Y G, Song Y M, Feng Z Y, et al.[8] has derived the Inverse dynamics of 3-RPS parallel mechanism by Newton-Euler formulation in Chinese and Liu, H., Huang, T.,Chetwynd, D.G.[11-20], An Approach for Acceleration Analysis of Lower Mobility Parallel Manipulators.

   Since the industrial revolution, there has been an ever increasing to improve product quality and reduce manufacturing cost parallel kinematic machines are applied on various area such as, For high speed and high precision machining center, for machining purpose, air plane simulators and For pick and place, assembly and carrying load purpose in higher industries like aircraft, electronics and automotive factories and robotic assistance surgery. For the successful implementation of parallel manipulators in different engineering and technological areas studying the motion and rigidity has no any equivalent substitution. As it is already discussed before the dynamics and stiffness analysis will be studied in this paper as follows.

2. CAD MODEL KINEMATIC DESCRIPTION

   The architecture of 3PRS PKMs is showing on fig.1 which is composed of moving platform, a fixed base the three supporting limbs with identical configurations. Each limb connects the fixed base to the moving platform by prismatic, revolute and spherical joint respectively and the prismatic joint is the active joint which is actuated by the linear actuator servo motor.

   The considered machine is 3-DOF PKM, which can be showed mathematically by using mobility criterion.

   j

   M 6n j 1 fi , where M is the DOF, n is

   i1

   number of links in the system, j is number of joint and fi

   is the number of DOF of the ith joints. For this

   manipulator n=8,j=9,fi=3 for spherical joint and fi=1 for

   Where s

   i, j

   is a unit vector along the joint Jth joint of the

   prismatic and revolute joints. Therefore the above equation will give as the system has 3 DOF.

   M 68 9 1 311 3 3DOF (1)

   The vectors and reference frames are described also in the figure for the sake of analysis, as shown in the figure 2 a fixed Cartesian reference coordinate frame P (x, y, z) is attached at the center point O of the fixed triangle base platform C1C2C3. And the moving coordinate system O(u, v, w) is attached on the moving platform a point O Which is a center of Q1Q2Q3. For simplicity without losing generality, let x-axis be aligned toward OC3 and the u-axis pointing along the direction

   3

   of O'Q .

   ith limb. These five screws form five system for which a one system of reciprocal screw exists. This reciprocal screw lies on the intersection of the two planes. The first plane is perpendicular for the prismatic joint axis and the second plane is containing both the revolute and spherical joint. This reciprocal screw denoted as sr1,i is

   zero pitch screw passing through the center of spherical joint and parallel to s2,i .

   sr 2,i

   From the above expression R1,R2,R3 are the vector of fixed base platform from the center point O to Ci and r1,r2, r3are the vectors of the moving platform from the center point O to Qi.

   l , s

   s3,i

   o '

   s1,i

   i 3,i

   p

   s2,i

   y

   O

   figure 1: 3D CAD model of the machine

3. VELOCITY ANALYSIS

   In this section the velocity analysis of 3PRS have done by applying the principle of screw theory. There are five joint screws associated with each limb. The first joint is the only actuated joint and the remaining joints are

   x

   figure 2: 2D schematic diagram of the mechanism

   r s

   i

   passive. The instantaneous twist of the moving platform

   can be expressed as a linear combination of five screws.

   s

   2,i

   r1,i s

   (10)

   sp h s s s s s (8)

   2,i

   Where

   1,i

   1,i

   2,i

   2,i

   3,i

   3,i

   4,i

   4,i

   5,i

   5,i

   By taking the inner product of both sides of the

   instantaneous with the constraint wrench we found the

   s

   s1

   , s

   ri li s2

   , s

   ri s3

   constraint Jacobian (JC ) .

   1,i

   0 2,i s

   3,i

   s

   s T

   r s T

   s

   2

   3

   2,1

   1 2,1

   T

   s

   ri s4

   , s

   ri s5

   (9)

   J C

   s2,2

   r s

   T

   2

   3

   p/>

   2,2

   (11)

   s

   4,i

   4

   5,i

   5

   T

   s

   2,3

   r s

   T

   2,3

   This matrix represents the constraint imposed by the revolute joint. An additional basis screw which is reciprocal to the passive joint of the ith limb can be identified as zero pitch

   The Hessian matrix can be found from the derivative of screws in the Lie bracket form. This yields

   screw passing through the center of spherical joint. This

   0 $ta,1,i $ta,2,i $ta,1,i $ta,3,i $ta,1,i $ta,n ,i

   reciprocal screw represents wrench of actuation and it is

   i

   normal to the previous system. This can be expressed by

   0 0 $ta,2,i $ta,3,i $ta,2,i $ta,n ,i

   i

   (20)

   ri s3.i

   Ha,i

   s (12)

   r 2,i s

   0 0

   0 $ta,n 1,i $ta,n ,i

   3.i

   i i

   Take the orthogonal product of this reciprocal wrench for both

   0 0

   0 0

   sides of the twist screw. Then we

   0 $tc,1,i $tc,2,i $tc,1,i $tc,3,i $tc,1,i $tc,6n ,i

   found J .

   i

   x 0 0

   $

   $

   $ $

   sp s

   s

   s

   h

   (13)

   tc,2,i tc,3,i tc,2,i tc,6ni ,i

   (21)

   r 2,i

   1,i

   r 2,i i

   Hc,i

   0 0

   0 $tc,5n ,i $tc,6n ,i

   s T

   r s

   1

   T

   i i

   3,1

   T

   3,1 T

   0 0

   0 0

   J x s3,1

   r2 s3,1

   (14)

   s

   3

   T

   3,1

   r s

   T

   3,1

   $

   $ $ $

   Equation 13 can be rewrite three times, once for each

   ta,1,i tc,1,i

   ta,1,i tc,6ni ,i

   limb and it yields.

   Hac,i

   (22)

   J x sp Jq q Where the inverse Jacobian is

   J q is:-

   $ta,n ,i $tc,1,i $ta,n ,i $tc,6n ,i

   s T s 0 0

   i

   i i

   J q

   3,1

   0

   1,1

   s3,2

   T

   s

   0

   1,2

   T

   (15)

   A Ji qi qi

   Hi qi This equation puts the screw derivative

   0 0 s T s

   ( * * ) and its coefficient side by side. i.e. qi is

   3,3

   1,3

   coefficient and

   H is screw derivative in Lie bracket form.

   The actuation Jacobian, is responsible to relating the i

   Cartesian velocity with joint rate J

   , is a function of

   And the overall Hessian matrix can be written in the form of

   q

   J and J .

   H Ha,i

   Hac,i

   H

   (23)

   x a

   J J a

   0

   Where,

   J J x

   (16)

   i

   c,i

   J

   J

   a

   c q

   The matrix in eq. (23) is generalized Hessian with

   Equation 16 is a generalized Jacobian that relates the velocity of joint rate to the velocity of the moving platform.:

4. ACCELERATION ANALYSIS

   For the acceleration analysis of this 3PRS parallel manipulator we used a new approach [11] named Hessian matrix to achieve an explicit and compact form equation.

   Ja,i [sta,1,i sta,2,i , sta,3,i sta,4,i sta,5,i ] (17)

   i

   b ls n

   6 6 6 matrix for each limb contain both the actuation and constraint joints and eq. (20, 21, 22) are the element of general Hessian matrix.

   This generalized matrix will help us to solve the dynamics of the generalized lower DOF parallel manipulators in efficient, compact and robust way and it is believed solving the Hessian in this form

   will simplified the dynamics work in unbelievable status.

5. DYNAMICS FORMULATION

   This work comes with applying screw mathematics in order to utilize the advantage of Robust behavior of virtual work for

   Jc,i

   stc1,i

   3,i

   n1,i

   1,i for i=1,2,3 (18)

   solving inverse dynamics of symmetrically configured PKM. Gravity of the moving platform and the pth (1 p ni 1 )

   Where

   J c,i is constraint Jacobian for a limb.

   link of the ith ( i 1, 2,,l ) limb can be expressed as follow

   s s

   $ mC g , $

   mp,i g

   (24)

   Here s

   3,i 5.i

   and s

   Rs

   for i=1,2,3

   wg 0 wg , p,i 0

   4,i s s

   5,i

   2,i

   3,i

   5.i

   where g is the vector of acceleration of the gravity, mC

   and

   Ji

   Ja,i

   Jc,i

   , Jacobian for each limb. (19)

   mp,i is the mass of the moving platform and the pth link of the ith limb respectively.

   According to Eulers Theorem, the inertial forces of the moving platform can be expressed as

   platform, twist of the pth link of the ith limb can be rewritten in the variational form as follows

   $Ct T0$t , q J $t , $Cta, p,i JC, p,i $t

   (27)

   $wI mC $Ct $Ct mC $Ct (25) Where

   Substituting Eqs(24), (25), (26) and (27) into Eq(27), the dynamics model can be constructed

   mC

   mC E

   0 , $

   = A J

   $ vC

   D(r)$t H(r, $t )$t G(r) E(r)

   (28)

   0

   I

   C

   Ct C C t

6. STIFFNESS MODELING

   0 vC 0

   JC JT0 , J

   , $Ct 0

   Under the assumption that the platform and the machine frame

   0 0

   where IC RIC0 RT and IC 0 denotes the inertial matrix of the moving platform about the mass centre described in the reference coordinate system and local coordinate system respectively, R is the rotation matrix of the local coordinate system with respect to the reference coordinate system.

   Similarly, the inertial forces of the pth ( 1 p ni 1 ) link of the ith ( i 1, 2,,l ) limb can be expressed as follows

   are rigid, when the platform is subjected to the external wrench [F T , M T ] on the reference point p, where F and M are the external force and torque applied to the platform, the deformation of the limbs will causes the platform

   to experience a twist x [rT , T ] in terms of the translational and rotational deformations along/about the axes of frame. Then, applying the virtual work principle to the

   $ m $ $ m $

   (26)

   platform gives

   wI , p,i p,i Cta, p,i Cta, p,i p,i Cta, p,i

   Where,

   xT T f

   (29)

   Where and f represents the set of deflections and

   m mp,i E

   0 , $

   = A J

   $ = vC , p,i

   reaction force magnitude

   p,i

   0

   I p,i

   Cta, p,i C , p,i C , p ,i t

   p,i

   Jx

   This equation

   xT T f can be re write

   C, p,i

   , p,i p,i p,i a, p,i

   , p,i

   J J T J J L , J

   p,i 0

   xT T f 0 Substitute

   vC , p,i

   0 0

   0

   from the above equation

   ( T Jf T )x 0

   $Cta, p,i T T

   0

   p,i

   Jf 0

   p,i

   where I p,i Rp,i I p,i,0 RT

   and I p,i,0 denotes the inertial

   Taking the transpose yields:

   J T f

   matrix of the link about the mass centre described in the

   Where

   f [ f T , f T ] is the internal wrench vector of limbs,

   reference coordinate system and local coordinate system a c

   respectively,

   Rp,i is the rotation matrix of the local coordinate

   where

   f a and

   fc are the generalized force of the PRS limbs

   system with respect to the reference coordinate system.

   nd the PR limb related to the twist [ T , T ],

   Let $ f T , t T T be the load wrench applied on the a c

   we

   reference point of the moving platform, where f and t is the

   vector of force and moment respectively. By using of the principle of virtual work, we can obtain

   fa is a force which is parallel to the screw axis while the

   fc is parallel to the revolute axis.

   Therefore the virtual work principle can be written

   t we

   w qT $T$

   0 (26)

   X T T f

   rT F T

   T f

   • T f

   a

   c

   a

   c

   where

   l ni 1

   fai

   [ f

   a1 ,

   fa 2 ,

   fa3

   ]T ,

   w $T $ $ $T $ $

   [q , q , q ]T

   Ct wI wg Cta, p,i wI , p,i wg , p,i i1 p1

   a 1 1 3

   T T T

   f [ f , f , f ]T ,

   a c

   ci c1 c2 c3

   where a is the f-dimensional vector composed of the driving

   force coefficients, and c is the vector composed of the

   [c , c , c ]T

   c 1 2 3

   (30)

   constraint force coefficients.

   The equation of velocity of joint, twist of the moving

   fa ka a , fc kc c and

   f kac

   ka 0

   klsa Is the lead screw assembly the combination of serially connected springs such as

   Where

   kac 0 k

   1 1 1 1

   c

   k k k k

   Here ka ka

   and kc

   are known as the component stiffness

   lsa

   ls n sb

   (34)

   matrix of actuation and constraints respectively the formulation of their element

   Where kls , kn and ksb are the stiffness coefficients of lead

   ac

   kx, k J T k J

   (31)

   screw.nut and support bearings respectively. kls

   is the lead

   And the compliance model can be evaluated as

   c k 1

   screw which is the linear function of the limb length and can be defined as

   kls

   AE(L1i L2i )

   L L

   (35 )

   1i 2i

   1. Formulation of kac

      As shown in the Figure 3 above the limb model has To

      Where

      A, E stands for cross sectional area of the lead screw and yangs modular respectively

      formulate kac I group all the parts of an PRS limp in to four

      components: (1) the spherical joint (2) the limp body which is

      L1i

      and L2i are the distance b/n the nut and the supporting bearing located at both ends.

      the fixed lengths link (3) R joint assembly (4) the lead screw

      assembly. as well as analytically convenient, they are sub systems that must realistically be subjected independently to design improvements.

      To find the constrained coefficient of stiffness matrix we can

      find as the same fashion of finding the actuation coefficient matrix. Similarly kc diag[kci ] (i 1,2,3) where kci is

      ci

      ka is given in a diagonal matrix i.e. ka diag[kai ] where the bending stiffness coefficient at the platform along the s2i

      (i 1,2,3)

      with

      kai being the axial stiffness coefficient at

      axis of the ith limb Then k can be evaluated by taking

      Bi along the

      s1i

      axis of the ith limb referring to the fig 4.

      reciprocal sum of the bending stiffness coefficient of the fixed length limb, S joint and R joint assembly respectively.

      kai

      Can be modeled by four serially connected springs each

      1 1

      1 1

      (36)

      representing the stiffness of one of the four components such

      kci

      kcl

      kcs

      kcr

      1

      kai

      1

      klsa

      1 1 1

      kl ks kr

      Again the kcs can be evaluated by the configuration and should be evaluated as in the local frame since the spherical

      (32)

      Where klsa , kl , ks and kr are the axial stiff nesses

      parallel to the constraint is to u the coefficient stuffiness is calculated as follows

      coefficients of the lead screw assembly ,the fixed length limb

      ,S joint and R joint assembly respectively.

      1

      kcs

      1

      ku1

      1

      ku 2

      1

      ku3

      (37)

      Note that

      kl , and kr

      are constant and can be evaluated using

      The value of the stiffness will be substitute from table 4.1and

      finite element analysis. (FEA) by the ANSYS workbench which is very convenient to analysis a solid model like PKM. The ks varies with the configuration and should be evaluated as in the local frame since the spherical actuation is parallel to w, the coefficient stuffiness is calculated as follows

      easy to compute the value of kcs .the kcl and kcl can be computed easy by FEA. Using the software ANSYS workbench by applying a 1KN force on the spherical joint which is parallel to the s2i

      1 1

      ks kw1

      1

      kw2

      1

      kw3

   2. Formulation of overall stiffness matrix on tool tip

   Formulation of overall stiffness matrix applied on the center of

   (33) end-effectors is calculated on Eq (36) which is

   The values of Eq (12) can be substituted from Table 1 below.

   ac

   Table 2 The stiffness coefficient of the S joint

   kx,

   k J T k J

   ku1	kv1	kw1	ku 2	kv2	kw2	ku3	kv3	kw3
   31.4	33.3	436	4000	1810	2174	2119	270	588 T

   To find the overall coefficient of stuffiness matrix on tool tip it needs to transform the Eq(31) in to tool tip by the transformation matrix

   E3 C s1i x

   C 0 E

   (38)

   3×3 3

   If let be imposed at the tool tip C and x be the corresponding small deflection twist the overall stiffness matrix about point C can easily be developed by replacing J

   in Eq.(31) with JT that is

   when force applied along x, y and z which explain in detail. In order to evaluate the rigidity of a system we define the rigidity along/about three orthogonal axis of the frame C uvwby the diagonal corresponding element of C.

   C K x

   1/ C(1.1)

   K TC

   T J T kJT

   K y 1/ C(2.,2) K z

   1/ C(3,3)

   C

   Where C is the distance from O to C , s1i denotes the

   Krw

   rC

   2 / C(6,6)

   screw matrix of

   s1i

   ,and E3 denotes a unit matrix of order3.

   c K 1

   x c

   5.2 Comparison with FEA Results

   According to the above analysis, the detailed design was carried out and the stiffness of the virtual prototype was

   Where

   (Fx Fy Fz

   M )T

   	K x (N / m	K y (N / m)	K z (N / m	Krw (x106 Nm / ra
   Ana lytic	25.4	25.4	447.2	5.3
   FE A	25.87	25.87	437.7	5.4

   rC

   T

   x (uvwrC w )

   rC is the maximum radius of the cutting tool specified by the

   k11

   k12

   k13

   spindle head.

   C K 1

   k16 / rc

   evaluated by ANSYS at four typical positions as shown from

   k k k / r

   K

   22 23 26 C

   (39)

   Figure 3, Figure 4, Figure 5 and Figure 6 with 1KN force is

   k33

   k36 / rC

   applied at the tool tip along x, y and z and moment about z

   sym k

   / r

   axis respectively. I can get the deformation easily from the

   66 C

   FEA and the stiffness can get by

   1000

   . It can been seen from

   In order to evaluate the rigidity of a system we define the rigidity along/about three orthogonal axis of the frme C uvw by the diagonal corresponding element of C.

   z

   K x 1/ C(1.1) , K y 1/ C(2.,2) K 1/ C(3,3)

   x

   Table 3 that the estimated results of the mathematical models developed have a good match with those obtained by the FEA in terms of magnitude and distribution as well.

   Table 4 Results obtained by the semi-analytical method and by FEA

   Krw

   rC

   K x

   2 / C(6,6)

   1/ C(1.1)

   The estimated linear stiffness along three orthogonal axis and the torsion stiffness about the w-axis of the C uvw frame, it can be seen that the stiffness distribution are tri-symmetrical

   z

   K y 1/ C(2.,2) K

   1/ C(3,3)

   in nature and the K x and

   K y are similar in magnitude.

   Krw

   rC

   2 / C(6,6)

7. STIFFNESS ANALYSIS

   The stiffness of the 3PRS PKM is evaluated in the decomposing the machine in to limbs and apply a force on the spherical joint to find the actuated and constraint coefficient of stuffiness. With the aid of finite element analysis and numerical evaluated both actuated and constraint stiffness of the limb assembly is evaluated. The overall stiffness of the manipulator on the center of the plate form will be calculated as in Eq (31) indicated. Since in real sense the force/moment is applied on the too tip of the machine, it needs transform the stiffness matrix gained in Eq (31) to the tool tip by the transform matrix Eq (37) it gives a stiffness coefficient matrix on tool tip as shown in the Eq(39).then diagonal values in the stiffness matrix indicates the overall stiffness of the machine

   Figure 3 Deformation with 1KN force imposed at the spindle along x-axis

   model of the lower mobility parallel manipulators can be formulated by two steps

   1. formulation of the generalized Jacobian by

      Figure 4 Deformation with 1KN force imposed at the spindle along z-axis

      Figure 5 Deformation with 1KN force imposed at the spindle moment about z

      1050

      1000

      driving force in N

      950

      900

      850

      linear linear driving force of actuator

      fz1 fz2 fz3

      25

      20

      15

      constrain moment

      10

      5

      0

      -5

      -10

      -15

      								tao1 tao2 tao3

      constraint force of actuating limb

      800

      0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

      time (sec)

      -20

      -25

      0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

      time (sec)

      Figure 6. Linear actuating force

8. CONCLUSION

An approach for the dynamics and stiffness modeling 3PRS parallel mchanism is presented in this paper. With this approach,the analytical expressions of the mass centre velocity/acceleration of the link can be easily derived by using the operation of Lie Algebra and reciprocal screw. With the aid of the generalized Jaccobian and Hessian matrix, the new dynamics model can be applied to solve the generalized forces of actuation and constraint at the same time. The stiffness

Figure 7. Constraint force of active limb

simultaneously taking into account the deflections of the platform in both the free and fixed directions, and

(2) evaluation of the component stiffness matrices using the analytical approach, finite element analysis and/or their combinations, depending upon geometry complicity

of the moving components. In concluding the robust dynamic equation is formulated and this is tested by developing simulation material and verify the result with numerical simulation shown in figure 7 and 8. Regarding the rigidity

analysis with the tool of ANSYS workbench we analysed the machine stiffness and compared the FEA result with the semi- analytical analysis and the estimated stiffness results have a good match with those obtained by the FEA. The result is in range of acceptable error as we can see it from fig.4,5 and 6 of ANSYS output compared with the FEA result in table 1.

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