On Modeling and Simulation of 6 Degrees of Freedom Stewart Platform Mechanism Using Multibody Dynamics Approach

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1 ECCOMS Multibody Dynamics July, 2013, University of Zagreb, Croatia On Modeling and Simulation of 6 Degrees of Freedom Stewart Platform Mechanism Using Multibody Dynamics roach yman El-Badawy 1, haled Youssef Mechanical Engineering deartment, l-zhar University, Cairo, Egyt and German University in Cairo-GUC, New Cairo City - Main Entrance of l Tagamoa l hames, Egyt: ayman.elbadawy@guc.edu.eg German University in Cairo-GUC, New Cairo City - Main Entrance of l Tagamoa l hames, Egyt: khaled.nabih@guc.edu.eg bstract The forward kinematics and dynamic analysis of arallel maniulators is highly comlicated due its closed loo structure and kinematic constraints. This intricacy makes the dynamics study and control for such maniulators need more effort in order to redict its dynamic behavior. One of the most famous arallel maniulators is 6 degrees of freedom Stewart latform mechanism (SPM) (6-6 tye) which consists of fixed base late and uer mobile latform where the two lates are connected through 6 extensible legs. This study concentrates on three main oints: First, modeling of Stewart latform mechanism (6-6 tye) using non-linear multibody dynamics and second, solving its forward kinematics by determining the solely exact ose of the uer mobile latform given the dislacement of the 6 extensible legs. Third, obtaining the reaction forces induced in each joint of the mechanism due to the given motion utilizing that each link is treated as a rigid body and not as a oint mass. eywords: Stewart latform mechanism, dynamic euations, arallel maniulators, forward kinematics 1 Introduction Parallel maniulators have many advantages over serial ones. The closed chain kinematics of arallel maniulators can result in greater structural rigidity, and therefore greater accuracy than oen chain robots [1]. One of the most famous arallel maniulators is Stewart latform mechanism. It is usually either 6 or 3 degrees of freedom mechanisms, where the difference is that 3 degrees of freedom mechanisms, the translation and orientation of the uer latform are deendent on each other (i.e. we cannot change the dislacement of the end effector without changing the orientation accordingly). On the other hand the 6 degrees of freedom mechanism is the most general one where the orientation and dislacement of the uer latform are indeendent. The six degrees of freedom Stewart latform mechanism is used in many alications worldwide such as flight simulators [2], manufacturing CNC machines [3], satellite dish ositioning [4] and even in ositioning device for high recision surgical tools [5]. It consists of a fixed base late and mobile uer late which are connected together with 6 extensible legs. The forward kinematics of Stewart mechanism is determining the osition and orientation of the uer latform for the given 6 legs length. This roblem; unlike serial maniulators; has no known closed form solution for the most general 6 degrees of freedom Stewart latform maniulator (with six joints on the base and six on the mobile latform) [6]. Forward kinematics is a very imortant ste in designing a maniulator, since we can redict the kinematic and dynamic behavior of the mechanism without having to send time and money in creating a rototye of the maniulator. On the other hand, inverse kinematics of Stewart mechanism is determining the 6 legs coordinate for a given osition and orientation of the mobile latform. The inverse kinematics is the key ste for control alications. lthough it aears to be simle and easy to study the dynamics erformance of Stewart latform mechanism, however its forward kinematics analysis is tremendously comlicated due its structure that rovides its mechanical stiffness [7]. The forward kinematics of Stewart mechanism has been studied in several research investigations; nevertheless, it has never been fully solved [8]. Since 1965, when D. Stewart had roosed the 6 degrees of freedom mechanism [2], many researches had ut their effort in order to solve for forward and inverse kinematics. For the sake of simlification, many research tried to solve 751

2 752. Youssef,. El-Badawy for 3 degrees of freedom mechanism, where the euation are simler in derivation. In 1988, Lee and Shah [9] solved the forward and inverse kinematics of 3 degrees of freedom mechanism (two for orientation and one for translation), where the base fixed late is connected to the uer mobile late through 3 legs (actuators) only. The euations of motion have been formulated in joint sace using Lagrangian aroach. However it was suggested to be a art of maniulation systems for limited alications. Obviously, the 6 degrees of freedom mechanisms are more caable of higher maniulations. In 2002, Song and won [10] develoed a new direct kinematic formulation of 6 degrees of freedom Stewart-Gough latforms using tetrahedron aroach. The roosed model was 3-6 latform, where the author used a new formulation aroach to easily derive a single constraint euation of the direct kinematics. ccording to the author, the forward kinematic roblem of the 6-6 tye is still a challenge due to the comlicated formulation rocedure and heavy comutational burden. In 2004, Merlet [8] solved the forward kinematics of 6 degrees of freedom Stewart mechanism of the tye 6-6, where the mobile latform is connected the fixed late through 6 extensible links. However, the author did not fully solve for forward kinematics, but solved for all the n ossible oses of the latform, given the six leg lengths of the extensible links. The author declared that this ste is acceted as soon as some method will allow determining which solution among n solutions is the current ose of the robot. ccording to him, no such method is known to date, even for lanar arallel mechanisms. In 2005, Rolland [11] introduced a method for solving the forward kinematics roblem with an exact algebraic method for the general arallel maniulator. In this method the arallel maniulator kinematics is formulated as olynomial euations system where the number of euations is eual to the number of unknowns. s the author mentioned, solving the most general case (6-6 Stewart latform), the rational reresentation comrised a univariate euation of degree 40 and 8 to 12 real solutions were comuted. In 2011, Gonzalez and Lengerke [12] solved for direct and inverse kinematics of Stewart latform alied to offshore cargo transfer simulation. Their model was based on another model as resented in [7] by simlifying the form of the to from 6 vertices to a to of 3 vertices, reducing the number of simultaneous nonlinear euations. The authors concluded that their model also reduces the number of nonlinear euations that should be develoed to find a solution to the direct kinematics. The scoe of this aer is concentrated on three main oints: First, modeling of Stewart latform mechanism (6-6 tye) using non-linear multibody dynamics and second, solving its forward kinematics by determining the solely exact ose of the uer mobile latform given the dislacement of the 6 extensible legs. Third, obtaining the reaction forces induced in each joint of the mechanism due to the given motion utilizing that each link is treated as a rigid body and not as a oint mass. 2 inematic analysis This section discuses the kinematic analysis of SPM by symbolically establishing the coordinate system, orientation matrices and the kinematic constraints due to the sherical and translational joints as well as the driving constraints. 2.1 Coordinate system In our study, a fixed global frame (X-Y-Z) is set at the center of the base late with the Y axis is ointing vertically uward and the X-Z axes are set according to the right hand rule as shown in Fig.1. The SPM consists of a fixed base late, 6 lower legs, 6 extendable uer legs and the uer latform. The total number of the moving links is 13 where each link is described by the location of its center of mass x-y-z and its orientation about fixed global frame X-Y-Z. The generalized coordinates for all the links in Stewart mechanism are: for uer latform [ X, Y, Z,,, ], UL LL for uer legs for lower legs [ X [ X, Y, Y, Z, Z,,,,,, ](i = 1.6), ](i = 1.6),

3 . Youssef,. El-Badawy 753 where X Y, Z,, reresents the location of center of mass for the uer latform in the global frame and, reresents the orientation about global X, Y and Z axes resectively. kewise X, Y, Z,,, and X Y, Z,,, Hence, the generalized coordinate vector,,, for uer leg i and lower leg i resectively. and of size 78. UL LL 2.2 Orientation Matrix The orientation matrices between moving frame and fixed frame is defined by first a rotation about fixed X-axis by theta (θ) degrees, then about fixed Y-axis by beta (β) degrees and finally about fixed Z-axis by gamma (γ) degrees. The basic rotation matrix about fixed X-axis, Y-axis and Z-axis is constructed as: Cβ Cγ Cβ Sγ Sβ where C and S reresent cosine and sine oerators resectively. In this study, there will be one orientation matrix () for each moving link; therefore, there will be a total of 13 matrices. 2.3 inematic Constraints Sθ Sβ Cγ Cθ Sγ Sθ Sβ Sγ Cθ Cγ Sθ Cβ Cθ Sβ Cγ Sθ Sγ Cθ Sβ Sγ Sθ Cγ Cθ Cβ In this section, the kinematic constraints due to the sherical and translational joints are develoed. The osition vectors are reresented with resect to the global X-Y-Z axis. The local frames are attached at the center of gravity of each leg with y-axis lies along the length of the leg and the x-axis is erendicular on it and ointing outward as shown in Fig.1. The z-axis is set according to the right hand rule. The uer sherical joint connects the to moving latform with the uer leg of the actuator at sot T i. The local osition of oint T i is located on the to latform with an angle α i measured counter clock wise from the ositive fixed X-axis. The osition vector of T i is reresented with resect to two coordinate systems as shown grahically in Fig.1. First, it was reresented with resect to the global fixed frame as (R P +S T/P). Second, it was reresented with resect to the global fixed frame as (R +S ). Therefore, the kinematic constraint due to sherical joints on to late is: ST X Y Z x y z T / Pi T / Pi T / Pi X Y Z ST x y z 0 (1) (2) Figure 1. Uer and lower leg of SPM

4 754. Youssef,. El-Badawy where and, x x y T Pi T / P i z /, and are the coordinates of oint T / T P i reresented in the local frame attached to uer latform i, y and z are the coordinates of oint T i reresented in the local frame attached to uer leg i. The lower sherical joint connects the base late with the lower leg of the actuator at sot B i. The osition of oint B i is located on the base late with an angle Φ i measured counter clock wise from the ositive fixed X-axis. The osition vector of B i is reresented with resect to two coordinate systems as shown grahically in Fig.1. First, it was reresented with resect to the fixed frame as (R ). Second, it was reresented with resect to the global fixed frame as (R +S ). Therefore the kinematic constraint due to sherical joints on lower late is: SL SL X Y Z X Y Z x y z 0 where (i = 1 6), X, Y and Z are the coordinates of base oint B i reresented in the global frame and, z are the coordinates of base oint B i reresented in the local frame of leg i. x, y and The translational joint allows relative translation between lower leg and uer leg i along local y-axis only (i.e.it does not allow relative translation along local x and z axes); lso, it revents relative rotation of the two bodies about x, y and z axis. Thus, the translational joint eliminates five degrees of freedom from uer and lower leg i [13]. s resented in [14], the three constraints due to elimination of relative rotation between air of legs can be stated mathematically as:, TR T [ ] 0, (i = 1 6) (4) In order to establish the algebraic euations that describe the constraint due to no relative translation along local x and z axes, a unit vector (v) is defined on local frame that is attached to lower leg i as shown in Fig.1, where. v nother vector T is defined between two oints, which are the center of masses of uer leg i where, T and T Ux Uy Uz Lx Ly. To revent the translation along local x axis, first, the unit vector ) axis. Then, the dot roduct between vectors v and is set to be eual to zero. ( v) z ( ) 0 Therefore the no translational kinematic constraint in local x-axis is: TX Lz (3) and lower leg, (5) i li (v is oriented with an angle of 90 o around local z where (i = 1 6), R is 90 o orientation matrix around local z-axis and, Z is the coordinates of the center of mass of uer leg reresented in the global frame axis, and ui is the coordinates of the center of mass of lower leg reresented in the global frame axis. To revent the translation along local z axis, the same rocedure was alied as: Tz TX (R v )) ( ) 0 ( z ui li (R v )) ( ) 0 ( x ui li (6) (7) (8)

5 . Youssef,. El-Badawy 755 Combining all the kinematic constraint due to the translational joint will lead to: [ ] 0 T TR TX TZ (9) 2.4 Driving Constraints The driving constraint describes the motion of the moving bodies in any mechanism. In Stewart mechanism, the linear actuators drive the uer legs with a well defined function of time. Therefore, at any time, the length between the center of masses of the uer and lower legs are known. Hence, in algebraic form, the driving constraint φ D between uer and lower leg can be described as: (10) ui li where, ( t ) is an exlicit known function in time for each leg i and the other arameters are defined in the above section. c i D (( v ( )) c ( t 0 i ) 2.5 Comlete constraint vector The number of constraint euations is seventy two. These constraints are resented as: 1) Three kinematic constraints in each higher sherical joint, which connects the moving uer late with the higher legs. It revents the relative translation in X, Y and Z axes between the higher leg i and the uer latform of a total eighteen. 2) Five kinematic constraints in each rismatic joint since the relative translation along local x and z axes are revented as well as relative rotation around x, y and z axes of a total thirty. 3) Three kinematic constraints in each lower sherical joint, which connects the fixed base late with the lower legs. It revents the relative translation in X, Y and Z axes between lower leg i and fixed base late of a total eighteen. 4) Six driving constraints on each extendable leg. 3 Dynamic analysis This section discusses the general symbolic reresentation of the 6 DOF Stewart latform mechanism using the multibody dynamics aroach as resented in [13]. mathematical model is develoed to construct the differential algebraic euation (DE) which describes the euation of motion of the Stewart mechanism as: where M is the mass matrix, vector, λ is the vector of Lagrange multilier, euations. 3.1 Mass Matrix φ is the constrained Jacobian matrix, is the generalized coordinate acceleration Q is the alied force vector, and δ is the vector of acceleration Since each moving body in SPM (13 links) is described by the location of its center of mass x-y-z and its orientation about fixed global frame (θ, β and γ), therefore, the mass matrix of comlete model SPM consists of a 7878 element matrix where the non diagonal elements are zeroes and the diagonal is the masses and mass inertias of uer latform, uer legs and lower legs ( m, m, m, I, I, I, m, m, m, I, I, I, m, m, m, I, I, I ) x y z ui ui ui xui yui zui li li li xli yli zli where, m are masses of the uer latform, uer leg and lower leg resectively and, m, and li m ui k D M T k ST k T T 0 λ k Q δ SL D T (11) (12)

6 756. Youssef,. El-Badawy I, I x xui and I are the mass moments of inertia of uer latform, uer leg and lower leg around local x-axis and, xli I, and I are the mass moments of inertia of uer latform, uer leg and lower leg around local y-axis and, yli y I yui I, and zli z I zui I are the mass moments of inertia of uer latform, uer leg and lower leg around local z-axis. The Jacobian Matrix 3.2 Jacobian Matrix φ is the artial derivative of the comlete constraint euation matrix φ with resect to the,,. The first row in the Jacobian matrix is the artial derivative of [ Ul LL generalized coordinate vector ] the first constraint euation with resect to each element in the coordinate vector. Therefore, the size of the Jacobian matrix will be eual to (the number of constraint euations number of generalized coordinates).the Jacobian matrix has been develoed using Mathematica. Due to the enormous size of the Jacobian matrix (7278), it will not be resented here. 3.3 Vector of acceleration euations The vector of acceleration euations γ can be obtained as resented in [13]: where, φ is the Jacobian matrix and, and are the generalized velocity and acceleration vectors resectively and, φ is the derivative of each element of the Jacobian matrix with resect to time and, t φ (φ ) 2φ t φ tt γ (13) φ is the second derivative of each element of the comlete constraint vector with resect to time. tt kewise the Jacobian matrix, the vector of acceleration euations is too massive to be resented here. The term 3.4 Lagrange multilier vector λ T in euation (12) is the reaction forces in the mechanism due to the resence of the kinematic constraints, where λ is the Lagrange multilier associated with kinematic constraints [13]. In Stewart mechanism, we have a total of sixty six kinematic constraints as well as six driving constraints. Therefore the term λ is a vector of dimension lied forces vector The alied forces vector Q consists of all the external forces that act on each of the links in Stewart mechanism. In our case, the only external force that acts on all the links is the force due to the gravity g which acts in the negative global Y axis. For the uer latform (14) Q Q 0 m g For the uer leg i (i = 1 6) (15) Q 0 m g For the lower leg i (i = 1 6) (16) 0 m g T T T

7 Position (meters) Orientation (Radians). Youssef,. El-Badawy Numerical simulation fter obtaining the euations of motion of the mechanical system, the main urose is to solve it at each time ste t in order to determine the state vector which consists of osition, velocity, acceleration, and Lagrange multilier of the multibody dynamics roblem. Since our roblem is subjected to kinematic constraints due to sherical, rismatic and driving constraints, it should satisfy at all the times the following two euations [15]: T M Φ ()λ Q Φ(, t) (,, t) 0 0 (17) (18) Euations (17) and (18) are a set of differential algebraic euations (DE). There are many numerical integration methods to solve for DE such as G-stiff, Newmark and Hilber-Hughes-Taylor (HHT) [16]- [17]. HHT method relies on a set of integration formulas that relates osition and velocity to acceleration based on Taylor series exansion. The HHT method was used in this study since, according to Hussien et al [15], unlike the ODE solver, the second order differential euations that reresent the dynamic euations of the system do not have to be reduced to first order and therefore the imlementation is easier and lead to a smaller dimension roblem. lso, the method has been tested and verified in many engineering alication fields. MTLB code was develoed to solve euations (17) and (18) using HHT method, which reuires the initial ositions of the center of mass, orientations and velocities of all the links. lso, the mass moment of inertia of all the moving arts should be known. ll the initial X, Y and Z coordinates of moving uer latform as well as all the lower and uer legs were referenced with resect to the global frame axis which was fixed in the middle of the lower fixed base late. The uer and lower legs were considered uniform cross section cylinders with the center of mass lies in the middle of each leg. ll the legs have the same mass, length, shae and mass moment of inertia. ll the numerical arameters for this simulation are rovided in endix. 5 Results Considering the inut motion for the 6 legs (Table2 endix ), the following grahs show the osition of the center of mass of the uer latform as well as its 3 Euler orientations according to the used orientation matrix. Fig.2 shows the osition of the uer mobile latform center of mass in the Cartesian coordinates. Since the attached moving frame to the uer latform is exactly in the middle, the X and Z coordinates started from zero and the X oscillates between almost meters and meters. The Y coordinate started from aroximately meters which is the altitude of the uer latform at t=0 before the motion of the actuators. It was observed that the harmonic resonse is reeated every 6.25 seconds or 2π after that the cycle is reeated again. Fig. 3 shows the orientation of the uer mobile latform around the fixed axes according to the rescribed orientation matrix. It was observed that the maximum rotation occurs around the Y axis of value one radian which is aroximately 57 degrees Y X Z me (Sec.) Figure 2. Position of the Platform me (sec.) Figure 3. Orientation of the Platform

8 Z u1 -reaction force comonent (newton) X u1 -reaction force comonent (newton) Y u1 -reaction force comonent (newton) 758. Youssef,. El-Badawy Figures 4, 5 and 6 show the X, Y and Z comonents of the reaction force induced in the uer sherical joint for the first leg that connects the uer leg 1 with the uer mobile latform. It was observed that the Y comonent reaction force started at t=0 (where there was no inertia force yet) with a value of 2.9 Newtons. This value can be exlained as the weight of the uer latform divided by 6 which are the number of uer legs that suort the weight of the latform. The grahs show a very high oscillation until aroximately 1 second; afterward the curve is smooth and eriodic. This numerical error can be exlained due to the fixed ste size used in the imlementation of the HHT numerical techniue, comared to other numerical methods that use variable ste size. ll the rest of the results such as the angular velocities and angular accelerations of all the moving links of SPM are a byroduct of the MTLB code me (sec.) me (sec.) Figure 4. X-Reaction force comonent of the first uer joint Figure 5. Y-Reaction force comonent of the first uer joint me (sec.) Figure 6. Z-Reaction force comonent of the first uer joint 6 Conclusion In this article, the modeling rocess used in this study, which is based on multi body dynamics aroach, is simle and systematic comared to other modeling rocesses such as Lagrange formulation and Newton-Euler. The rocedure described in this aer can be alied to any tye of arallel maniulators while treating all the moving arts as rigid bodies and not oint masses. The forward kinematic roblem for 6-6 tye of SPM has been solved by determining the only exact ose of the uer mobile latform given the linear dislacement of the 6 extensible legs. lso, its dynamic resonse was determined by obtaining the reaction forces induced in each joint of the mechanism due to the given motion utilizing that each link is treated as a rigid body.

9 . Youssef,. El-Badawy 759 References [1] Mark W.Song, Seth Hutchinson, M.Vidyasagar, Robot Modeling and Control, John Wiley& Sons, Inc., [2] D. Stewart. " latform with six degrees of freedom", Proc. of institute for Mechanical Engineering, London, Vol. 180, , [3] Y. ng, Y. Chen and H. Jar, Modeling and Control for a Gough-Stewart Platform CNC Machine, Journal of Robotic Systems, Vol. 21, No. 11, (2004). [4] Y. Cheng, G. Ren and S. Dai, The multi-body system modeling of the Gouh-Stewart latform for vibration control, Journal of Sound and Vibration, 271 (2004) [5] J. Troccaz, M. Peshkin and B. Davies, The Use of Localizers, Robots and Synergistic Devices in CS, CVRMed-MRCS 97 Proceedings,1997, [6] H. Sadjadian, H. Taghirad and. Fatehi, Neural Networks roaches for Comuting the Forward inematics of a Redundant Parallel Maniulator, International Journal of Comutational Intelligence Vol. 2 No. 1 (2006) [7] G. Lebret,. u and F. L. Lewis, Dynamics nalysis and Control of a Stewart Platform Maniulator, Journal of Robotics Systems 10(5), (1993). [8] J.-P. Merlet, Solving the Forward inematics of a Gough-Tye Parallel Maniulator with Interval nalysis, The International Journal of Robotics Research, Vol. 23, No. 3, , March [9]. M. Lee, and D..Shah, "inematics analysis of a three-degree-of-freedom in-arallel actuated maniulator," IEEE J. of Robotics and utomation, 4, (1988). [10] S. Song and D. won, New Direct inematic Formulation of 6 D.O.F Stewart-Gough Platforms Using the Tetrahedron roach, Transactions on Control utomation and Systems Engineering Vol. 2, No. 1, January, [11] L. Rolland, Certified solving of the forward kinematics roblem with an exact algebraic method for the general arallel maniulator, dvanced Robotics, Vol. 19, No. 9, (2005). [12] H. Gonzalez, M. Dutra, O. Lengerke, Direct and inverse kinematics of Stewart latform alied to offshore cargo transfer simulation, 13th World Congress in Mechanism and Machine Science, Guanajuato, México, June, [13] E. J. Haug, Comuter ided inematics and Dynamics of Mechanical Systems, llyn and Bacon, [14].. Shabana, Dynamics of Multibody Systems, Cambridge University Press, [15] B. Hussein, D. Negrut and. Shabana, Imlicit and exlicit integration in the solution of the absolute nodal coordinate differential/algebraic euations, Nonlinear Dyn (2008) 54: [16] D. Negrut, G. Ottarsson, R. Ramalli and. Sajdak, On an Imlementation of the Hilber-Hughes-Taylor Method in the Context of Index 3 Differential-lgebraic Euations of Multibody Dynamics, Journal of Comutational and Nonlinear Dynamics, Volume 2, Issue 1, , [17] B. Gavrea, D. Negrut and F. Potra, THE NEWMR INTEGRTION METHOD FOR SIMULTION OF MULTIBODY SYSTEMS: NLYTICL CONSIDERTIONS, SME International Mechanical Engineering Congress and Exosition 2005, IMECE

10 760. Youssef,. El-Badawy endix Lower fixed late (Radius = 0.27 cm) Uer moving late (Radius = cm) Table 1. ngles of sherical joints on uer and lower lates Φ 1 Φ 2 Φ 3 Φ 4 Φ 5 Φ o o o o o o α 1 α 2 α 3 α 4 α 5 α o 70.5 o o o o o Table 2. near actuators functions Leg i Function 0.02Sin(t) -0.06Sin(t) 0.02Sin(t) -0.02Sin(t) 0.08Sin(t) -0.02Sin(t) Table 3. Uer mobile latform initial osition, orientation and numerical arameters Mass = 1.79 g, Ixx = g-m2, Iyy = g-m2, Izz = g-m2 X Y Z θ X β Y γ Z m Table 4. Uer and lower legs initial osition, orientation and numerical arameters