Commun Nonlinear Sci Numer Simulat

Transcription

1 Commun Nonlinear Sci Numer Simulat 14 (2009) Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: Dynamic modeling of a Stewart platform using the generalized momentum approach António M. Lopes Unidade de Integração de Sistemas e Processos Automatizados, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, Porto, Portugal article info abstract Article history: Received 1 August 2008 Received in revised form 11 December 2008 Accepted 3 January 2009 Available online 11 January 2009 PACS: f Ln Keywords: Parallel manipulators Dynamics Robotics Dynamic modeling of parallel manipulators presents an inherent complexity, mainly due to system closed-loop structure and kinematic constraints. In this paper, an approach based on the manipulator generalized momentum is explored and applied to the dynamic modeling of a Stewart platform. he generalized momentum is used to compute the kinetic component of the generalized force acting on each manipulator rigid body. Analytic expressions for the rigid bodies inertia and Coriolis and centripetal terms matrices are obtained, which can be added, as they are expressed in the same frame. Gravitational part of the generalized force is obtained using the manipulator potential energy. he computational load of the dynamic modes evaluated, measured by the number of arithmetic operations involved in the computation of the inertia and Coriolis and centripetal terms matrices. It is shown the model obtained using the proposed approach presents a low computational load. his could be an important advantage if fast simulation or model-based real-time control are envisaged. Ó 2009 Elsevier.V. All rights reserved. 1. Introduction he dynamic model of a parallel manipulator operated in free space can be mathematically represented, in the Cartesian space, by a system of nonlinear differential equations that may be written in matrix form as: IðxÞ x þ Vðx; _xþ _x þ GðxÞ ¼f I(x) being the inertia matrix, Vðx; _xþ the Coriolis and centripetal terms matrix, G(x) a vector of gravitational generalized forces, x the generalized position of the moving platform (end-effector) and f the controlled generalized force applied on the end-effector. hus, f ¼ J ðxþs where s is the generalized force developed by the actuators and J(x) is a jacobian matrix. Generally speaking, the dynamic model can play an important role in system simulation and control. In the first case, the manipulator trajectory (position, velocity and acceleration) is to be computed for the given actuating forces or torques (direct dynamics). In the second case, the actuators forces or torques required to generate a given manipulator trajectory should be calculated (inverse dynamics). Mainly in this case, the efficiency of the computation for the manipulator dynamics is of paramount importance, as manipulator real-time contros usually necessary [1]. ð1þ ð2þ address: aml@fe.up.pt /$ - see front matter Ó 2009 Elsevier.V. All rights reserved. doi: /j.cnsns

2 3390 A.M. Lopes / Commun Nonlinear Sci Numer Simulat 14 (2009) Nomenclature i List of symbols ~a a skew-symmetric matrix representing operator [a] {}, {P}, {C i }, {S i } right-handed Cartesian frames attached to base, moving platform, cylinder i and piston i (origin located at centre of mass) b C distance from the cylinder centre of mass to point i b i position of point i relative to frame {} i, P i kinematic chains connecting points, to the base and moving platform b S distance from the piston centre of mass to point P i f generalized force acting on the end-effector f c(kin) kinetic component of the generalized force acting on a body C i f Ci ðkinþðtraþj ; S i fsi ðkinþðtraþj kinetic component of the generalized force applied to the cylinder, or piston, due to its translation, expressed in frame {} P f Ci ðkinþðtraþj ; P f Si ðkinþðtraþj kinetic component of the generalized force applied to {P}, due to cylinder, or piston, translation, expressed in frame {} C ifci ðkinþðrotþj ; S if Si ðkinþðrotþj kinetic component of the generalized force applied to cylinder, or piston, due to its rotation, expressed in frame {} P f Ci ðkinþðrotþj ; P f Si ðkinþðrotþj kinetic component of the generalized force applied to {P}, due to cylinder, or piston, rotation, expressed in frame {} P f PðgraÞjjE ; P f Ci ðgraþj je ; P f Si ðgraþj je gravitational component of the generalized force acting on {P}, due to moving platform, cylinder h and piston P f PðkinÞj ¼ P F PðkinÞj P M PðkinÞj kinetic component of the generalized force acting on {P}, due to moving platform motion, expressed in frame {} P F PðkinÞj force vector acting on moving platform G(x) vector of gravitational generalized forces H c angular momentum about the centre of mass of body c H P j ; H Ci j ; H Si j angular momentum of the moving platform, cylinder and piston, about the centre of mass, written in frame {} I(x) inertia matrix I c inertia matrix of a rigid body I c(rot) rotationanertia matrix I c(tra) translationanertia matrix I P j moving platform inertia matrix, written in frame {} I PðrotÞj ; I Ci ðrotþj ; I Si ðrotþj inertia matrix of the rotating moving platform, cylinder, or piston, about frames {P}, {C i }, {S i }, expressed in frame {} I P(tra) translationanertia matrix of the moving platform J(x) jacobian matrix J A jacobian matrix relating Euler angles derivatives and angular velocity J i jacobian matrix relating cylinder linear velocity and moving platform generalized velocity J C kinematic jacobian matrix relating active joints velocities and moving platform generalized velocity jacobian matrix relating leg angular velocity and moving platform generalized velocity J E Euler angles jacobian matrix, relating active joints velocities and moving platform generalized velocity (linear velocity and Euler angles derivatives) J Gi jacobian matrix relating piston linear velocity and moving platform generalized velocity l ¼ ½l 1 l 2... l 6 Š ; l _ position and velocity of the active joints coordinate (length) of the prismatic joint i ^li versor of vector m P, m C, m S mass of the moving platform, cylinder and piston P M PðkinÞj moment about the origin of {P} acting on the moving platform, expressed in frame {} P c potential energy of a rigid body p Ci j ; p Si j position of the cylinder, or piston, centre of mass, relative to frame {} P p i j position of point P i relative to frame {} q c generalized momentum of a rigid body Q c linear momentum q P j generalized momentum of the moving platform, expressed in frame {} Q P j ; Q Ci j ; Q Si j linear momentum of the moving platform, cylinder and piston, written in frame {} r, r P base and moving platform radius R P ; R Ci rotation matrix of frame {P}, or frame {C i }, relative to frame {} matrix transformation defined by ¼ diagð½i J A ŠÞ generalized velocity (linear and angular) of a rigid body u c

3 A.M. Lopes / Commun Nonlinear Sci Numer Simulat 14 (2009) Vðx; _xþ v c, x c x Coriolis and centripetal terms matrix rigid body linear and angular velocities h generalized position i (position and orientation) x P jje ¼ x PðposÞj x PðoÞj E generalized position of frame {P} relative to frame {} (orientation is given by the Euler angles system) x PðposÞj ¼ ½x P y P z P Š position of the origin of frame {P} relative to frame {} (expressed in frame {}) x PðoÞjE ¼ ½w P h P u P Š Euler angles system representing orientation of frame {P} relative to {} h i _x P jje ¼ _x PðposÞj _x PðoÞj E generalized velocity of the moving platform, expressed in the Euler angles system (linear velocity and Euler angles derivatives) h i generalized velocity of the moving platform, expressed in frame {} (linear and angular velocities) _x P j ¼ _x PðposÞj x P j _x PðposÞj v P j linear velocity of moving platform relative to frame {} _x PðoÞjE Euler angles time derivatives /, / P separation half-angles between kinematic chains connecting points, at base and moving platform s P(kin) actuating forces corresponding to s forces developed by the actuators x P j ; x li j angular velocity of the moving platform, or leg (cylinder or piston), relative to frame {} Dynamic modeling of parallel manipulators presents an inherent complexity, mainly due to system closed-loop structure and kinematic constraints. Several approaches have been applied to the dynamic analysis of parallel manipulators, the Newton Euler and the Lagrange methods being the most popular ones. he Newton Euler approach uses the free body diagrams of the rigid bodies. Do and Yang [2] and Reboulet and erthomieu [3] use this method on the dynamic modeling of a Stewart platform. Ji [4] presents a study on the influence of leg inertia on the dynamic model of a Stewart platform. Dasgupta and Mruthyunjaya [5] used the Newton Euler approach to develop a closed-form dynamic model of the Stewart platform. his method was also used by Khalil and Ibrahim [6], Riebe and Ulbrich [7], and Guo and Li [8], among others. he Lagrange method describes the dynamics of a mechanical system from the concepts of work and energy. Nguyen and Pooran [9] use this method to model a Stewart platform, modeling the legs as point masses. Lebret et al. [10] follow an approach similar to the one used by Nguyen and Pooran [9]. Lagrange s method was also used by Gregório and Parenti-Castelli [11] and Caccavale et al. [12], for example. hese methods use classical mechanics principles, as is the case for all the approaches found in the literature, namely the ones based on the principle of virtual work [13,14], screw theory [15], recursive matrix method [16], and Hamilton s principle [17]. hus, all approaches are equivalent as they are describing the same physical system [18]. All methods lead to equivalent dynamic equations, although these equations present different levels of complexity and associated computational loads [1]. Minimize the number of operations involved in the computation of the manipulator dynamic model has been the main goal of recent proposed techniques [1,16,19 21]. In this paper, the author presents a new approach to the dynamic modeling of a six degrees-of-freedom (dof) Stewart platform: the use of the generalized momentum concept [22]. his method is used to compute the kinetic component of the generalized force acting on each manipulator rigid body. Analytic expressions for the rigid bodies inertia and Coriolis Piston P 4 l 3 P 3 x P z P P 2 P Moving platform yp P 1 l 4 P 5 P 6 l 2 l 1 Cylinder 3 l 5 l 6 2 z 4 ase x y Fig. 1. Stewart platform kinematic structure.

4 3392 A.M. Lopes / Commun Nonlinear Sci Numer Simulat 14 (2009) and centripetal terms matrices are obtained, which can be added, as they can be expressed in the same frame. Gravitational part of the generalized force is obtained using the manipulator potential energy. he computational load of the dynamic modes evaluated, measured by the number of arithmetic operations involved in the computation of the inertia and Coriolis and centripetal terms matrices. It is shown the model obtained using the proposed approach presents a low computational load. his could be an important advantage if fast simulation or model-based real-time control are envisaged. he paper is organized as follows. Section 2 introduces the Stewart platform parallel manipulator. Section 3 presents the manipulator dynamic model using the generalized momentum approach. In Section 4, the computational effort of the dynamic modes evaluated. A numerical simulation of the manipulator inverse dynamics is presented in Section 5. Conclusions are drawn in Section Stewart platform kinematic structure A Stewart platform comprises a fixed platform (base) and a moving (payload) platform, linked together by six independent, identical, open kinematic chains (Fig. 1). Each chain (leg) comprises a cylinder and a piston (or spindle) that are connected together by a prismatic joint,. he upper end of each leg is connected to the moving platform by a spherical joint whereas the lower end is connected to the fixed base by a universal joint. Points i and P i are the connecting points to the base and moving platforms, respectively. hey are located at the vertices of two semi-regular hexagons inscribed in circumferences of radius r and r P. he separation angles between points 1 and 6, 2 and 3, and 4 and 5 are denoted by 2/.Ina similar way, the separation angles between points P 1 and P 2, P 3 and P 4, and P 5 and P 6 are denoted by 2/ P. For kinematic modeling purposes, two frames, {P} and {}, are attached to the moving and base platforms, respectively. Its origins are the platforms centres of mass. he generalized position of frame {P} relative to frame {} may be represented by the vector: h i x P jje ¼ ½x P y P z P w P h P u P Š ¼ x PðposÞj x ð3þ PðoÞj E where x PðposÞj ¼ ½x P y P z P Š is the position of the origin of frame {P} relative to frame {}, and x PðoÞjE ¼ ½w P h P u P Š defines an Euler angles system representing orientation of frame {P} relative to {}. he used Euler angles system corresponds to the basic rotations [23]: w P about z P ; h P about the rotated axis y P 0 ;andu P about the rotated axis x P 00. he rotation matrix is given by: 2 3 Cw P Ch P Cw P Sh P Su P Sw P Cu P Cw P Sh P Cu P þ Sw P Su P 6 7 R P ¼ 4 5 ð4þ Sw P Ch P Sw P Sh P Su P þ Cw P Cu P Sw P Sh P Cu P Cw P Su P Sh P Ch P Su P Ch P Cu P S() and C() correspond to the sine and cosine functions, respectively. he manipulator position and velocity kinematic models are known, being obtainable from the geometrical analysis of the kinematics chains. he velocity kinematics is represented by the Euler angles jacobian matrix, J E, or the kinematic jacobian, J C. hese jacobians relate the velocities of the active joints (actuators) to the generalized velocity of the moving platform: " # _x _l ¼ J E PðposÞj _x P jje ¼ J E ð5þ with _l ¼ J C _x P j ¼ J C _x PðoÞjE " # _x PðposÞj x P j h i _l ¼ _ l1 _ l2... _ l6 ð7þ x P j ¼ J A _x PðoÞjE ð8þ and [23] Sw P Ch P Cw P 6 7 J A ¼ 4 0 Cw P Ch P Sw P 5 ð9þ 1 0 Sh P ð6þ Vectors _x PðposÞj v P j and x P j represent the linear and angular velocity of the moving platform relative to {}, and _x PðoÞjE represents the Euler angles time derivative. 3. Dynamic modeling using the generalized momentum approach he generalized momentum of a rigid body, q c, may be obtained using the following general expression: q c ¼ I c u c ð10þ

5 A.M. Lopes / Commun Nonlinear Sci Numer Simulat 14 (2009) Vector u c represents the generalized velocity (linear and angular) of the body and I c is its inertia matrix. Vectors q c and u c, and inertia matrix I c must be expressed in the same referential. Eq. (10) may also be written as: q c ¼ Q c H c ¼ I cðtraþ 0 0 I cðrotþ v c x c where Q c is the linear momentum vector due to rigid body translation, and H c is the angular momentum vector due to body rotation. I c(tra) is the translationanertia matrix, and I c(rot) the rotationanertia matrix. v c and x c are the body linear and angular velocities. he kinetic component of the generalized force acting on the body can be computed from the time derivative of Eq. (10): ð11þ f cðkinþ ¼ _q c ¼ _ I c u c þ I c _u c ð12þ with force and momentum expressed in the same frame Moving platform modeling he linear momentum of the moving platform, written in frame {}, may be obtained from the following expression: Q P j ¼ m P v P j ¼ I PðtraÞ v P j ð13þ I P(tra) is the translationanertia matrix of the moving platform, I PðtraÞ ¼ diagð½m P m P m P ŠÞ ð14þ m P being its mass. he angular momentum about the mobile platform centre of mass, written in frame {}, is: H P j ¼ I PðrotÞj x P j ð15þ I PðrotÞj represents the rotationanertia matrix of the moving platform, expressed in the base frame {}. he inertia matrix of a rigid body is constant when expressed in a frame that is fixed relative to that body. Furthermore if the frame axes coincide with the principal directions of inertia of the body, then alnertia products are zero and the inertia matrix is diagonal. herefore, the rotationanertia matrix of the moving platform, when expressed in frame {P}, may be written as: I PðrotÞjP ¼ diag I Pxx I Pyy I Pzz ð16þ his inertia matrix can be written in frame {} using the following transformation [24]: I PðrotÞj ¼ R P I PðrotÞjP R P ð17þ he generalized momentum of the moving platform about its centre of mass, expressed in frame {}, can be obtained from the simultaneous use of Eqs. (13) and (15): q P j ¼ I " # PðtraÞ 0 v P j ð18þ 0 I PðrotÞj x P j where I P j ¼ I PðtraÞ 0 0 I PðrotÞj ð19þ is the moving platform inertia matrix written in the base frame {}. P i P p i P xppos ( ) b C p Ci z i y b i x Fig. 2. Position of the centre of mass of the cylinder i.

6 3394 A.M. Lopes / Commun Nonlinear Sci Numer Simulat 14 (2009) he combination of Eqs. (8) and (15) results into: H P j ¼ I PðrotÞj J A _x PðoÞjE ð20þ Accordingly, Eq. (18) may be rewritten as: q P j ¼ I " # PðtraÞ 0 I 0 v P j 0 I PðrotÞj 0 J A _x PðoÞjE q P j ¼ I P j _x P jje ð22þ ð21þ being a matrix transformation defined by: ¼ I 0 0 J A he time derivative of Eq. (22) results into: ð23þ P f PðkinÞj ¼ _q P j ¼ d dt I P j _x P þ I jje P j x P jje ð24þ P f PðkinÞj is the kinetic component of the generalized force acting on {P} due to the moving platform motion, expressed in frame {}. he corresponding actuating forces, s P(kin), may be computed from the following relation: where s PðkinÞ ¼ J C P f PðkinÞ j ð25þ h i ð26þ P f PðkinÞj ¼ P F PðkinÞj P M PðkinÞj Vector P F PðkinÞj represents the force vector acting on the centre of mass of the moving platform, and P M PðkinÞj represents the moment vector acting on the moving platform, expressed in the base frame, {}. From Eq. (24) it can be concluded that two matrices playing the roles of the inertia matrix and the Coriolis and centripetal terms matrix are: I P j ð27þ d dt I P j ð28þ It must be emphasized that these matrices do not have the properties of inertia or Coriolis and centripetal terms matrices and therefore should not, strictly, be named as such. Nevertheless, throughout the paper the names inertia matrix and Coriolis and centripetal terms matrix may be used if there is no risk of misunderstanding Cylinder modeling If the centre of mass of each cylinder is located at a constant distance, b C, from the cylinder to base platform connecting point, i,(fig. 2), then its position relative to frame {} is: p Ci j ¼ b C ^ þ b i ð29þ where, ^li ¼ k k ¼ ¼ x PðposÞj þ P p i j b i ð30þ ð31þ he linear velocity of the cylinder centre of mass, _p Ci j, relative to {} and expressed in the same frame, may be computed as: _p Ci j ¼ x li j b C ^ ð32þ where x li j represents the leg angular velocity, which can be found from: x li j ¼ v P j þ x P j P p i j ð33þ As the leg (both the cylinder and piston) cannot rotate along its own axis, the angular velocity along ^ is always zero, and vectors and x li j are always perpendicular. his enables Eq. (33) to be rewritten as:

7 A.M. Lopes / Commun Nonlinear Sci Numer Simulat 14 (2009) or, x li j ¼ 1 h i l l i v P j þ x P j P p i j i x li j " # v P j ¼ x P j ð34þ ð35þ where jacobian is given by: h ¼ ~ i li ~ li P ~p i j ð36þ li ¼ ð37þ l i and, for a given vector a ¼ ½a x a y a z Š, 2 0 a z a y 3 6 ~a ¼ 4 a z 0 7 a x 5 ð38þ a y a x 0 On the other hand, Eq. (32) can be rewritten as: " # _p Ci j ¼ J i v P j x P j where the jacobian J i is given by: h J i ¼ b C ~^l i ~ li b C ~^l i ~ i li P ~p i j ð39þ ð40þ he linear momentum of each cylinder, Q Ci j, can be represented in frame {} as: Q Ci j ¼ m C _p Ci j ð41þ where m C is the cylinder mass. Introducing jacobian J i and matrix transformation in the previous equation results into: Q Ci j ¼ m C J i _x P jje ð42þ he kinetic component of the force applied to the cylinder due to its translation and expressed in {} can be obtained from the time derivative of Eq. (42): C ifci ðkinþðtraþj ¼ _ Q Ci j ¼ m C d dt ðj i Þ _x P jje þ m C J i x P jje ð43þ When Eq. (43) is multiplied by J i, the kinetic component of the force applied to {P} due to each cylinder translation is obtained in frame {}: P f Ci ðkinþðtraþj ¼ J i C i fci ðkinþðtraþj ¼ m C J i d dt ðj i Þ _x P jje þ m C J i J i x P jje ð44þ P i P p i P b S xppos ( ) p Si z i y b i x Fig. 3. Position of the centre of mass of the piston i.

8 3396 A.M. Lopes / Commun Nonlinear Sci Numer Simulat 14 (2009) he inertia matrix and the Coriolis and centripetal terms matrix of the translating cylinder being: m C J i J i m C J i d J dt i ð45þ ð46þ hese matrices represent the inertia matrix and the Coriolis and centripetal terms matrix of a virtual moving platform that is equivalent to each translating cylinder. On the other hand, the angular momentum of each cylinder about its centre of mass can be represented in frame {} by: H Ci j ¼ I Ci ðrotþj x li j ð47þ It is convenient to express the inertia matrix of the rotating cylinder in a frame fixed to the cylinder itself, {C i } {x Ci ; y Ci ; z Ci }. So, I Ci ðrotþj ¼ R Ci I Ci ðrotþj Ci R C i ð48þ where R Ci is the orientation matrix of each cylinder frame, {C i }, relative to base frame, {}. Cylinder frames were chosen in the following way: axis x Ci coincides with the leg axis and points towards P i, axis y Ci is perpendicular to x Ci and always parallel to the base plane, this condition being possible given the existence of a universal joint at i, that negates any rotation along its own axis; axis z Ci completes the referential following the right hand rule, and its projection along axis z is always positive. Frame origin is located at the cylinder centre of mass. hus, matrix R Ci becomes: R Ci ¼ x Ci y Ci z Ci ð49þ where x Ci ¼ ^ ð50þ y Ci ¼ y pffiffiffiffiffiffiffiffiffi l pffiffiffiffiffiffiffiffiffi ix 0 ð51þ l 2 ix þl2 iy l 2 ix þl2 iy z Ci ¼ x Ci y Ci ð52þ So, the inertia matrices of the cylinders can be written as: I Ci ðrotþj Ci ¼ diag I Cxx I Cyy I Czz ð53þ where I Cxx, I Cyy and I Czz are the cylinders moments of inertia expressed in its own frame. Introducing jacobian and matrix transformation in Eq. (47) results into: H Ci j ¼ I Ci ðrotþj _x P jje ð54þ he kinetic component of the generalized force applied to the cylinder, due to its rotation and expressed in {} can be obtained from the time derivative of Eq. (54): C i f Ci ðkinþðrotþj ¼ _ H Ci j ¼ d dt ði C i ðrotþj Þ _x P jje þ I Ci ðrotþj x P jje ð55þ When Eq. (55) is pre-multiplied by J D i the kinetic component of the generalized force applied to {P} due to each cylinder rotation is obtained in frame {}: P f Ci ðkinþðrotþj ¼ J D i C i fci ðkinþðrotþj ¼ J D i d dt ði C i ðrotþj Þ _x P jje ð56þ þ J D i I Ci ðrotþj x P jje he inertia matrix and the Coriolis and centripetal terms matrix of the rotating cylinder may be written as: J D i I Ci ðrotþj J D i d I Ci ðrotþj dt ð57þ ð58þ hese matrices represent the inertia matrix and the Coriolis and centripetal terms matrix of a virtual moving platform that is equivalent to each rotating cylinder.

9 A.M. Lopes / Commun Nonlinear Sci Numer Simulat 14 (2009) able 1 Number of arithmetic operations involved in the computation of the Coriolis and centripetal terms matrices. Lagrange Lagrange Koditschek representation Generalized momentum Add. Mul. Div. Add. Mul. Div. Add. Mul. Div. Coriolis and centripetal terms matrix of the mobile platform Coriolis and centripetal terms matrix of each translating cylinder Coriolis and centripetal terms matrix of each rotating cylinder Coriolis and centripetal terms matrix of each translating piston Coriolis and centripetal terms matrix of each rotating piston otal operations 101, , , , , able 2 Manipulator parameters. Para. Value Para. Value (kg m 2 ) Para. Value r m I Pxx 0.2 I Sxx 0.0 kg m 2 r P m I Pyy 0.2 I Syy 0.1 kg m 2 / 15 I Pzz 0.4 I Szz 0.1 kg m 2 / P 0 I Cxx 0.0 b C 0.5 m m P kg I Cyy 0.1 b S 0.5 m m C 0.39 kg I Czz 0.1 m S 0.39 kg 3.3. Piston modeling If the centre of mass of each piston is located at a constant distance, b S, from the piston to moving platform connecting point, P i,(fig. 3), then its position relative to frame {} is: p Si j ¼ b S ^ þ p i j þ x PðposÞj ð59þ he linear velocity of the piston centre of mass, _p Si j, relative to {} and expressed in the same frame, may be computed as: _p Si j ¼ l _ i þ x li j ð b S ^ Þ ð60þ " # _p Si j ¼ J Gi v P j x P j ð61þ where the jacobian J Gi is given by: h J Gi ¼ I b S ~^l i ~ li ði b S ~^l i ~ i li Þ P ~p i j ð62þ he linear momentum of each piston, Q Si j, can be represented in frame {} as: Q Si j ¼ m S _p Si j ð63þ where m S is the piston mass. Introducing jacobian J Gi and matrix transformation in the previous equation results into: Q Si j ¼ m S J Gi _x P jje ð64þ he kinetic component of the force applied to the piston due to its translation and expressed in {} can be obtained from the time derivative of Eq. (64): S ifsi ðkinþðtraþj ¼ _ Q Si j ¼ m S d dt ðj i Þ _x P jje þ m S J i x P jje ð65þ When Eq. (65) is multiplied by J G i, the kinetic component of the force applied to {P} due to each piston translation is obtained in frame {}: P f Si ðkinþðtraþj ¼ J G i S i fsi ðkinþðtraþj ¼ m S J G i d dt ðj G i Þ _x P jje þ m S J G i J Gi x P jje ð66þ

10 3398 A.M. Lopes / Commun Nonlinear Sci Numer Simulat 14 (2009) he inertia matrix and the Coriolis and centripetal terms matrix of the translating piston are: m S J G i J Gi m S J G i d dt ðj G i Þ ð67þ ð68þ Position (mm) Actuator 1 Actuator 2 Actuator 3 Actuator 4 Actuator 5 Actuator ime (s) Velocity (mm/s) Actuator 1 Actuator 2 Actuator 3 Actuator 4 Actuator 5 Actuator ime (s) Acceleration (mm/s 2 ) Actuator 1 Actuator 2 Actuator 3 Actuator 4 Actuator 5 Actuator ime (s) Fig. 4. Actuators trajectories: (a) position; (b) velocity; (c) acceleration.

11 A.M. Lopes / Commun Nonlinear Sci Numer Simulat 14 (2009) which represent the inertia matrix and the Coriolis and centripetal terms matrix of a virtual moving platform that is equivalent to each translating piston. On the other hand, the angular momentum of each piston about its centre of mass can be represented in frame {} by: H Si j ¼ I Si ðrotþj x li j ð69þ I Si ðrotþj ¼ R Si I Si ðrotþj Si R S i ð70þ where R Si is the orientation matrix of each piston frame, {S i }, relative to the base frame, {}. As the relative motion between cylinder and piston is a pure translation, {S i } can be chosen parallel to {C i }, and its origin located at the piston centre of mass. herefore, R Si ¼ R Ci. So, the inertia matrices of the pistons can be written as: I Si ðrotþj Si ¼ diag I Sxx I Syy I Szz ð71þ where I Sxx, I Syy and I Szz are the pistons moments of inertia expressed in its own frame. Introducing jacobian and matrix transformation in Eq. (69) results into: H Si j ¼ I Si ðrotþj _x P jje ð72þ he kinetic component of the generalized force applied to the piston, due to its rotation and expressed in {} can be obtained from the time derivative of Eq. (72): S ifsi ðkinþðrotþj ¼ H _ Si j ¼ d I Si ðrotþj dt _x P jje þ I Si ðrotþj x P jje ð73þ Pre-multiplied by J D i, the kinetic component of the generalized force applied to {P} due to each piston rotation is obtained in frame {}: P f Si ðkinþðrotþj ¼ J D i S i fsi ðkinþðrotþj ¼ J D i d dt ði S i ðrotþj Þ _x P jje ð74þ þ J D i I Si ðrotþj x P jje he inertia matrix and the Coriolis and centripetal terms matrix of the rotating piston will be: J D i I Si ðrotþj J D i d I Si ðrotþj dt ð75þ ð76þ hese matrices represent the inertia matrix and the Coriolis and centripetal terms matrix of a virtual moving platform that is equivalent to each rotating piston Actuator 1 Actuator 2 Actuator 3 Actuator 4 Actuator 5 Actuator 6 Force (N) ime (s) Fig. 5. Developed actuators forces.

12 3400 A.M. Lopes / Commun Nonlinear Sci Numer Simulat 14 (2009) It should be noted that rigid bodies inertia and Coriolis and centripetal terms matrices can be added, as they are expressed in the same frame Dynamic model gravitational components Given a general frame {x, y, z}, with z ^g, the potential energy of a rigid body is given by: P c ¼ m c g z c ð77þ where m c is the body mass, g is the modulus of the gravitational acceleration and z c the distance, along z, from the frame origin to the body centre of mass. he gravitational components of the generalized forces acting on {P} can be easily obtained from the potential energy of the different bodies that compose the P x P P jje f PðgraÞjjE ¼ x P P Ci x P jje f Ci ðgraþj je ¼ x P P Si x P jje f Si ðgraþj je ¼ x P jje he three vectors P f PðgraÞjjE ; P f Ci ðgraþj je and P f Si ðgraþj je represent the gravitational components of the generalized forces acting on {P}, expressed using the Euler angles system, due to the moving platform, each cylinder and each piston. herefore, to be added to the kinetic force components, these vectors must be transformed, to be expressed in frame {}. his may be done pre-multiplying the gravitational components force vectors by the matrix transformation Manipulator dynamic equations In previous sections, analytic expressions for the rigid bodies inertia and Coriolis and centripetal terms matrices were obtained. hese matrices can be added, as they are expressed in the same frame, resulting in the system inertia matrix, I je, and system Coriolis and centripetal terms matrix, V je. herefore, the total kinetic component of the generalized force acting on the moving platform, P f ðkinþj, may be easily computed using the following equation: I je x P jje þ V je _x P jje ¼ P f ðkinþj ð81þ In a similar way, P f ðkinþj, could also be obtained using Eqs. (24), (44), (56), (66) and (74): P f PðkinÞj þ P f Ci ðkinþðtraþj þ P f Ci ðkinþðrotþj þ P f Si ðkinþðtraþj þ P f Si ðkinþðrotþj ¼ P f ðkinþj ð82þ his total kinetic component should be added to the total gravitational part of the generalized force acting on the moving platform, P f ðgraþj, which can be obtained using the manipulator potential energy, as in Eqs. (78) (80): 1 P f PðgraÞjjE þ P f Ci ðgraþj je þ P f Si ðgraþj je ¼ P f ðgraþj ð83þ he total generalized force acting on the moving platform, and the corresponding actuating forces, will be: P f j ¼ P f ðkinþj þ P f ðgraþj ð84þ s P ¼ J C P f j ð85þ 4. Computational load of the proposed dynamic model As a means of evaluating efficiency, the number of scalar operations (additions, multiplications and divisions) required for the computation of the inertia and Coriolis and centripetal terms matrices is tabulated. Specification of the model computational load in this manner makes comparison with other models both easy and convenient [25]. hus, the computational efficiency of the proposed dynamic modes compared with the one resulting from applying the Lagrange method, both directly from Lagrangian equation and using the Koditschek representation [10]. As the largest difference between the methods rests on how the Coriolis and centripetal terms matrices are calculated, the models are evaluated by the number of arithmetic operations involved in the computation of these matrices. he results were obtained using the symbolic computational software Maple Ò and are presented in able 1.

13 A.M. Lopes / Commun Nonlinear Sci Numer Simulat 14 (2009) he dynamic model obtained using the generalized momentum approach is computationally much more efficient and its superiority manifests precisely in the computation of the matrices requiring the largest relative computational effort: the Coriolis and centripetal terms matrices. It should be noted, the proposed approach was used in the dynamic modeling of a 6-dof Stewart platform. Nevertheless, it can be applied to any mechanism. 5. Numerical simulation A 6-dof parallel manipulator presenting the kinematic and dynamic parameters shown in able 2 was considered. A trajectory was specified in task space. he moving platform initial position is P 1 = [0,0,2000,0,0,0] (mm; ). he moving platform is then displaced to point P 2 =[ 100, 200,2500,15, 15,15] (mm; ), and finally it returns to point P 1. hird order trigonometric splines were interpolated between the specified points, in order to obtain continuous and smooth trajectories. Fig. 4 shows the corresponding actuators trajectories. Fig. 5 shows the developed actuators forces, necessary to follow the specified trajectories. Fig. 6 represents the contribution of the moving platform, the six cylinders, and the six pistons to the total forces developed by the actuators. 6. Conclusion Dynamic modeling of parallel manipulators presents an inherent complexity. Despite the intensive study in this topic of robotics, mostly conducted in the last two decades, additional research still has to be done in this area. In this paper, an approach based on the manipulator generalized momentum was explored and applied to the dynamic modeling of a Stewart platform. he generalized momentum is used to compute the kinetic component of the generalized force acting on the moving platform. Analytic expressions for the rigid bodies inertia and Coriolis and centripetal terms matrices are obtained, which can be added, as they are expressed in the same frame. Having these matrices, the kinetic component of the generalized force acting on the moving platform may be easily computed. his component can be added to the gravitational part of the generalized force, which is obtained through the manipulator potential energy. References [1] Zhao Y, Gao F. Inverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual work, Robotica. doi: / S [2] Do W, Yang D. Inverse dynamic analysis and simulation of a platform type of robot. J Robot Syst 1988;5: [3] Reboulet C, erthomieu, Dynamic models of a six degree of freedom parallel manipulators. In: Proceedings of the IEEE international conference on robotics and automation; p [4] Ji Z. Dynamics decomposition for Stewart platforms. ASME J Mech Des 1994;116:67 9. [5] Dasgupta, Mruthyunjaya. A Newton Euler formulation for the inverse dynamics of the Stewart platform manipulator. Mech Mach heory 1998;34: [6] Khalil W, Ibrahim O. General solution for the dynamic modelling of parallel robots. J Intell Robot Syst 2007;49: [7] Riebe S, Ulbrich H. Modelling and online computation of the dynamics of a parallel kinematic with six degrees-of-freedom. Arch Appl Mech 2003;72: [8] Guo H, Li H. Dynamic analysis and simulation of a six degree of freedom Stewart platform manipulator. Proc Inst Mech Eng Part C J Mech Eng Sci 2006;220: [9] Nguyen C, Pooran F. Dynamic analysis of a 6 DOF CKCM robot end-effector for dual-arm telerobot systems. Robot Auton Syst 1989;5: [10] Lebret G, Liu K, Lewis F. Dynamic analysis and control of a Stewart platform manipulator. J Robot Syst 1993;10: [11] Di Gregório R, Parenti-Castelli V. Dynamics of a class of parallel wrists. J Mech Des 2004;126: [12] Caccavale F, Siciliano, Villani L. he tricept robot: dynamics and impedance control. IEEE/ASME rans Mech 2003;8: [13] Staicu S, Liu X-J, Wang J. Inverse dynamics of the HALF parallel manipulator with revolute actuators. Nonlinear Dyn 2007;50:1 12. [14] sai L-W. Solving the inverse dynamics of Stewart Gough manipulator by the principle of virtual work. J Mech Des 2000;122:3 9. [15] Gallardo J, Rico J, Frisoli A, Checcacci D, ergamasco M. Dynamics of parallel manipulators by means of screw theory. Mech Mach heory 2003;38: [16] Staicu S, Zhang D. A novel dynamic modelling approach for parallel mechanisms analysis. Robot Comput Integr Manufact 2008;24: [17] Miller K. Optimal design and modeling of spatial parallel manipulators. Int J Robot Res 2004;23: [18] Yiu YK, Cheng H, Xiong ZH, Liu GF, Li ZX. On the dynamics of parallel manipulators. In: Proceedings of the 2001 IEEE international conference on robotics and automation, Seoul, Korea; p [19] Abdellatif H, Heimann. Computational efficient inverse dynamics of 6-DOF fully parallel manipulators by using the Lagrangian formalism. Mech Mach heory 2009;44: [20] Wang J, Wu J, Wang L, Li. Simplified strategy of the dynamic model of a 6-UPS parallel kinematic machine for real-time control. Mech Mach heory 2007;42: [21] Sokolov A, Xirouchakis P. Dynamics analysis of a 3-DOF parallel manipulator with R-P-S joint structure. Mech Mach heory 2007;42: [22] Lopes A. A computational efficient approach to the dynamic modeling of 6-dof parallel manipulators. In: Proceedings of the ENOC 08; [23] Vukobratovic M, Kircanski M. Kinematics and trajectory synthesis of manipulation robots. Springer-Verlag; [24] orby. Advanced dynamics for engineers. CS College Publishing; [25] Lilly K. Efficient dynamic simulation of robotic mechanisms. Dordrecht: Kluwer Academic; 1993.