the mathematical model and direct kinematics solution analysis of delta parallel robot

The mathematical model and direct kinematics solution analysis of Delta parallelrobot

Jingjun ZhangScientific Research Office

Hebei University of EngineeringHandan,China

[email protected]

Lihong Shi Ruizhen Gao Chaoyang LianMechanical design and theory

Hebei University of EngineeringHebei, P.R. China

[email protected]

Abstract—By analyzing the mechanism of Delta parallel robotand setting up coordinate system on both static platform andmoving platform, then the constraint equation and Jacobmatrix can be gained. Combined with the Multi-body of Deltaparallel robot built with Pro/E, and then the model istransferred to ADAMS by MECHANISM/Pro. The inversesolution and spline curves and spline function can be obtainedthrough simulation at first, by using the result of spline curvesand spline function can obtain the direct solution. This way ismuch easier, faster and accurately than the ways of NumericalSolution and Analytical Method. It is proved that the resultscan be put into practice.

Keywords- Delta parallel robot; mathematical model; ADAMSsoftware; simulation; direct solution

I. INTRODUCTION

Delta parallel robot is a highly commercial used parallelrobot which has a 3-DOF translational and high speed. It wasput forward by Doctor Calve early in 1985[1].The parallelrobot has many advantages, such as big loading ability, highprecision, inverse solution easier, minor error, smalldeadweight load ratio, good dynamic performance, controleasy. [2][3] Proposed several ways of inverse problemsolution. The ways of getting direct solution are NumericalSolution and Analytical Method, which are complex, butusing the ADAMS software to get the forward solution andinverse solution is much easier. It will make any mistakes ifthe model of the robot is right [4][5][6][7].

In this paper, the thought of using ADAMS is to get theresults, proposing an improved solution for direct solution.

II. BUILT MULTI-BODY OF DELTA PARALLEL ROBOT

A. The mechanism constraint equationDelta parallel robot is made up of one static platform,

one moving platform, three driving rods and three drivenmove parallelograms branch chains (Fig.1 is its structurediagram).The three edges of the static platform have thesame kinematic chain join to the moving platform. Eachkinematic chain is a parallelograms branch chain close-loopwhich is joined with two driven rods by four spherical hinge,this close-loop and one driving rod with rotation jointconstitute a series mechanism, the driving rod is fixed to thestatic platform, the driving rod swing repeat by the drive of

determined by these three kinematics chains. The movingplatform has 3-DOF translational, which means that it canmove only along to the direction of x, y, z -axis in the spacecoordinate and can't rotate around the x, y, z- axis.

As shown in Figure 1, the O is the center of the staticplatform, and the O′ is the center of the moving platform,

ii BA are the driving rods with the length of 1l , iiCB are

the driven rods with the length of 2l .For calculatingexpediently, setting up coordinate system on both staticplatform and moving platform named XYZO − and

ZYXO ′′′−′ , 1θ � 2θ � 3θ are the field angle between

the Z -axis and ii BA .

Figure 1. Kinematics diagram of Delta parallel robot.

Let iOA= R , iOC

= r , the coordinate of center

O′ in coordinate system XYZO − isTz]y [x , and then

the position vector of iA in XYZO − are:_____________________________978-1-4244-4520-2/09/$25.00 ©2009 IEEE

Authorized licensed use limited to: UNIVERSIDADE FEDERAL DO CEARA. Downloaded on March 19,2010 at 19:35:49 EDT from IEEE Xplore. Restrictions apply.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

0sincos

i

i

io RR

a αα

, among them πα

634 −

=i

i

� i =1,2,3�, as the same time the position vector of iC in coordinate system XYZO − can also be gained :

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=′

0sincos

i

i

oi rr

c αα

, and the same πα

634 −

=i

i

� i =1,2,3�,by geometric relation, the position vector of

iB in coordinate system XYZO − can be expressed as�

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−++

=

i

ii

ii

i

llRlR

Bθ

αθαθ

cossin)sin(cos)sin(

1

1

1

Let coordinate of vector OO ′ is [ ]To xyzC = in

coordinate system XYZO − , then iCO′ can be expressed

as:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡++

=z

yrxr

c i

i

io αα

sincos

,

So, based on 2lBC ii =, can be derived

[ ][ ]

22

21

21

21

)cos(

sin)sin(

cos)sin(

lzl

yrlR

xrlR

i

ii

ii

=−−

+−−+

+−−+

θ

αθ

αθ

(1),

Letbe

[ ][ ]

0)cos(

sin)sin(

cos)sin(

22

21

21

21

=−−−

+−−+

+−−+=

lzl

yrlR

xrlRF

i

ii

iii

θ

αθ

αθ (2),

After derivation calculus of T to above (1) the Jac

ob

matrix can be gained: BAJ 1−= , among them ,

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=

zF

yF

xF

zF

yF

xF

zF

yF

xF

A

333

222

111

,

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

∂∂

−=

3

3

2

2

1

1

θ

θ

θ

F

F

F

B

]cos)sin[( 1 xrlRxF

iii −−+=

∂∂ αθ

�

]sin)sin[( 1 yrlRyF

iii −−+=

∂∂ αθ

�

ii lz

zF θcos1+=∂∂

, Letbe

ii rlRM αθ cos)sin( 1 −+= ,

ii rlRN αθ sin)sin( 1 −+= ,

ilP θcos1−=�then�

ii

iiii

i

i

i

i

i

i

i

i

i

i

i

i

i

i

lrRzlsylxl

PPFN

NFM

MFF

θθαθαθ

θθθθ

cos)(sinsincoscoscos

11

11

−+−−−=

∂∂

∂∂

+∂∂

∂∂

+∂∂

∂∂

=∂∂

B. The mechanism of Delta parallel robot Pro/E is powerful software, especially in designing

parametric feature modeling of 3D design, feature modeling not only describes the information of the geometric shape but also highly expresses the function information of the products, and Pro/E software has so powerful assembling function that the designed parts have beautiful shape and good visibility. Using the Pro/E software to build the 3D model of Delta parallel robot can avoid repeating modeling work for assembling the same part .According to the above-described frame of Delta parallel robot, built the 3D show as Figure 2.


Figure 2. Delta parallel robot mechanism

The grey platform is static platform, the red platform ismoving, the green rods are three driving rods, and the greyrods are six driven rods. The detailed parameters are shownin Tabs 1.

TABLE I. PARAMETERS OF DELTA PARALLEL ROBOT MECHANISM

Structure parameters length/mmdriving rod 500driven rod 1139

1r 2402r 70

1r�circumradius of static platform�

2r �circumradius of moving platformMECHANISM/Promodule is the interface between

ADAMS software and Pro/ e software, the two use seamlessinterface and the user can according to the model define themechanical system and do simulation of kinematics anddynamics without exiting the application environment, andthe model can also be transferred to ADAMS /View for thefollowing comprehensive kinematics analysis. The Deltamodel shown as in Figure 3.

Figure 3. Delta parallel robot mechanism model

III. DIRECT SOLUTION OF DELTA PARALLEL ROBOT

The position solving problem of parallel robot includingtwo facts: the inverse solution and the forward solution[8],generally speaking, the inverse solution is much easier toget[9], that to say known the position parameters of movingplatform to seek the position parameters of the input joints,on the contrary, the forward solution is known the positionparameters of input joints to seek the position parameters ofmoving platform which is hard, [10] proposed a geometricsolution of direct solution of Delta parallel robot , but usingthe ADAMS software to solve the forward solution is quiteeasy. The following Figure 4 is the flow chart of directsolution.

Figure 4. Flow chart of direct solution

A. Seeking the inverse solution by adding point drivingChoosing the moving platform-CG as the key point

where adding the 3D-point driving, let the manipulator movevertically 25mm/s, the relation equations of the position ofpoint driving and time T are as follows:

Horizontal direction X: 200*sin (5*time)Horizontal direction Y: 200*cos (5*time) Vertical direction Z: 50*sin(5*time)

By adding point driving can realize circular motion ofmanipulator according to the predetermined path. Using theobject measurement function of ADAMS/view can measurethe rotate angle of the driving rods. Simulation of Deltaparallel robot 5s 500steps can get the rotate angle� 1ϕ � 2ϕ � 3ϕ �of 3 moving rods, thecurves shown as in Figure 5.

Figure 5. Rotate angle curves of moving rods

The up three curves are the initial curves and the underthree are the equivalent forms which the number begin with0, the under curves can be used as motion adding to thedriving rods.


B. Kinematics direct solutionAfter getting the inverse solution, then save the measured

curves and they will be send to the post processing module,change the three curves to spline curves, in this way thediscrete date of the driving joint of the Delta parallel robotduring motion state can be gained, the treated spline curvescan be used as driving function, for the known condition toseek the forward solution of Delta parallel robot.

According to the provided spline function by ADAMS,take the discrete date point as known conditions generate todriving function, then according to the driving function thenmoving function which about the angle and time can beadded to the three moving rods. The functions are:

Motion1�AKISPL�time, 0, sp1, 0);Motion1�AKISPL�time, 0, sp2, 0);Motion1�AKISPL�time, 0, sp3, 0);

The point motion can be deleted or set invalid afteradding the moving function to the driving rods, Simulationof Delta parallel robot 5s 300steps again, at this time we canclearly see the trajectory of the manipulator, with thetrajectory visible in Figure 6. And the displacement curves ofmanipulator-CM shown as in Figure 7.

Put the result of trajectory of the manipulator to postprocessing module generated to spline function, the dates ofthe spline function are the right forward solution of themanipulator.

Figure 6. The trajectory of the manipulator

Figure 7. The displacement curves of X, Y, Z

The curves of angular velocity and acceleration ofdriving rods changed with time can also be measured. The

gained curves and dates and the spline curves are consistentwith the measure curves of motion relationship, it is provedthat this method is right and the curves conversion accuracyof ADAMS is very high. Right now we got the right forwardsolution. (The angular velocity shown as in Fig.8)

From the simulation Fig.6 we can see that the trajectoryof the manipulator is the same as the trajectory of formerinverse solution when added point motion. The result is turnout to be right and the way of seeking inverse solution andforward solution by ADAMS is feasible.

Figure 8. The angular velocity curves of driving rods

IV. CONCLUSIONS

In this paper the mathematical model of delta robot isbuilt and then the Jacob matrix can be gained, and a simplemethod of getting the direct solution is introduced. Itneedn’t heavy programming task and can save too muchtime & work force that we use ADAMS to solve theproblems of inverse and direct solution. This method makesimportant significance of development of new type robotand machine tool and can also verify the generality andvalidity made by scientific researchers and expedited thestep of innovation.

ACKNOWLEDGMENT

The authors gratefully acknowledge the supports ofInternational cooperative project of Scientific Office ofHebei Province under No. 07393522D.

REFERENCES

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[3] Pierrot F, Fournnier A, Dauchez P Towards a fully-parallel 6 DOFrobot for high speed applications.Proc.1991 IEEE Int. Conf. onRobotics and Automation,1991:1288�1293.

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[7] Mlynarski T. Position analysis of planar linkages using the method ofmodification of kinematic unit [J]. Mech Mach Theory, 1996, 31(6):831 - 838.

[8] Wang dan, Guo hui, Sun zhili. “Normal and Inverse Pose Solutions to3-RPS Parallel Robot Based on ADAMS” [J].School of MechanicalEngineering & Automation, Northeastern University, 2005� 26(12)�1185-1187.

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the mathematical model and direct kinematics solution analysis of delta parallel robot

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