Modeling and Experimental Validation of a

Modeling and Experimental Validation of a Generalized Stewart Platform by BondGraph Method İbrahim Yildiz, Vasfi Emre Ömürlü & Ahmet Sağirli

Arabian Journal for Science and Engineering ISSN 1319-8025 Arab J Sci Eng DOI 10.1007/s13369-012-0484-y

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Author's personal copy Arab J Sci Eng DOI 10.1007/s13369-012-0484-y

RESEARCH ARTICLE - MECHANICAL ENGINEERING

Modeling and Experimental Validation of a Generalized Stewart Platform by Bond-Graph Method ˙ Ibrahim Yildiz · Vasfi Emre Ömürlü · Ahmet Sa˘girli

Received: 8 June 2011 / Accepted: 7 January 2012 © King Fahd University of Petroleum and Minerals 2012

Abstract This paper represents modeling and experimental validation of a Stewart platform manipulator by BondGraph method. Dynamic model includes all dynamic and gravity effects, linear motor dynamics as well as viscous friction at the joints. Following the modeling of actuation system and of main structure, merging of these two is accomplished. Linear DC motors are utilized and are modeled as the main part of the actuation system. Since the overall system consists of high nonlinearity originated from geometric nonlinearity and gyroscopic forces, resultant derivative causality problem caused by rigidly coupled inertia elements is addressed and consequential nonlinear system state-space equations are presented. Stability of the model is investigated by observing the variations of the system matrix eigen values which are utilized from the state-space equations. Four different trajectories are applied to the Bond-Graph model and to the experimental setup for validation purposes. Satisfactory close coordination between simulation and experimental system is achieved. Keywords Stewart platform manipulator · Modeling · Bond-Graph method · Simulation and experiment · Model validation ˙I. Yildiz (B) · A. Sa˘girli Department of Mechanical Engineering, Faculty of Mechanical Engineering, Yildiz Technical University, Be¸sikta¸s, Istanbul 34349, Turkey e-mail: [email protected] A. Sa˘girli e-mail: [email protected] V. E. Ömürlü Department of Mechatronics Engineering, Faculty of Mechanical Engineering, Yildiz Technical University, Be¸sikta¸s, Istanbul 34349, Turkey e-mail: [email protected]

Abbreviations Pi Connection point of the leg to the mobile plate Bi Connection point of the leg to the static plate Li Length of the leg mi Mass of the leg mp Mass of the mobile plate Inertia of the mobile plate IP Ibi Inertia of the leg Radius of the mobile plate rP r B Radius of the static plate Angle of the leg θi λ Angle between the neighbor connection points of the legs of the mobile plate  Angle between the neighbor connection points of the legs of the static plate εP Angle between the connection points of the legs and the x axis of the mobile plate ε B Angle between the connection points of the legs and the x axis of the static plate

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1 Introduction Dynamic models of parallel mechanisms are more complicated than serial mechanisms because of their close kinematic chain. Nevertheless, dynamic models are important for control of parallel mechanisms. Many studies have been presented on kinematics of parallel manipulators in recent years, but dynamics of parallel manipulators are comparatively few. A Stewart platform is a six-degree-of-freedom mechanism with two plates mounted by six linearly actuated legs and has been first proposed as a flight simulator in 1965 by D. Stewart [1]. Many applications, related to physical simulation of spatial motion, has been put into realization. Present study mentions obtaining dynamic formulation of a Stewart platform using Bond-Graph method and verifying the model with an experimental system. Bond-Graph method has been established as a new approach to model, analyze and control various dynamical systems by Prof. Henry Paynter in 1961. Many researchers have worked on Bond-Graph modeling of mechanical systems. Karnopp [2], Rosenborg [2] and Thoma [3] developed Bond-Graph techniques for hydraulic, mechatronic and thermodynamic systems. Margolis [4] performed a comprehensive research on Bond-Graph techniques whose results are applicable to the robotics and spatially moving objects. Allen et al. [5] developed a Bond-Graph technique obtaining Lagrange and Hamilton functions by modeling systems with high degree of nonlinearity. The method was based on defining dependent velocities of the system with general coordinates and local coordinates. Bos et al. [6] studied a systematic modeling method of spatially moving mechanisms with Bond-Graph technique. Fahrenthold et al. [7] worked on modeling mechanical systems with vector Bond-Graph method. Karnopp [8] used Lagrange and Hamilton equations for solving the derivative causality problem on geometrically nonlinear systems. The inverse dynamics of a Stewart platform mechanism is solved using Newton–Euler method by Do and Yang [9] assuming symmetrical and thin legs and frictionless joints. Dasgupta and Mruthyunjaya [10] solved the inverse dynamics of a system including all the dynamical and gravitational effects by using Newton–Euler approach. Geng et al. [11] developed motion equations for a system using Lagrangian equations. A similar equation system was developed by Liu et al. [12] for control purposes. Sagirli et al. [13,14] applied kinematic vector Bond-Graph method to a telescopic rotary crane, obtained state-space formulation of this mechanism and solved derivative causality problem by reducing the inertial elements to the independent velocity ports using virtual inertia and a gyristor element. Another approach for dynamic modeling of parallel mechanisms was introduced by Wang and Gosselin [15] which was based on the principle of virtual work. Kane’s equations were also used for modeling of a

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Stewart platform mechanism and for any other parallel mechanisms [16]. Combination of Newton–Euler method with the Lagrange Formulation was applied to describe the dynamic model of a Stewart platform mechanism by Guo and Li [17]. Another Lagrange Formulation was applied to Stewart platform mechanism for designing an adaptive control law for multi-body systems and Lyapunov stability theory are used to formulate a nonlinear feedback control law [19]. In this study, Bond-Graph modeling of a Stewart platform mechanism (SPM) is addressed including all dynamical and gravitational effects. The nonlinear model includes actuator dynamics and viscous friction at the joints. The inverse kinematics equations are utilized in modeling of the mechanism. Stability of dynamic model is investigated and dynamic model is validated using the experimental setup so that obtained state-space representation can be used for control system design for further research.

2 Bond-Graph Modeling of a Stewart Platform Mechanism Modeling with Bond-Graph method allows a pictorial representation of internal dynamics for both linear and nonlinear systems. This visuality facilitates understanding of motions and tracking of energy conversions in the system in addition to mathematical description. Another goal of modeling with Bond-Graph method is formulating the dynamics of both linear and nonlinear system with state-space representation which allows one to investigate the stability of a dynamic system and observing many variables which cannot be measured in physical systems. Therefore, representing a system in state-space form is important for control approaches and Bond-Graph method allows one to formulate dynamics of nonlinear systems in state-space form in a more formal way. A Stewart platform mechanism is a well-known nonlinear structure and many studies have been performed for mathematical modeling of this parallel mechanism excluding the Bond-Graph approach. Nonlinear state equations can be employed to analyze the system over the workspace of the mechanism which covers all possible motion scenarios of the system so that whole dynamic shift can be scanned for stability. 2.1 Assumptions and Presentation of the Stewart Platform Mechanism The experimental setup of the 3×3 Stewart platform and the physical model of a generalized SPM are shown in Fig. 1. The modeled mechanism has six linear actuators (L 1,...,6 ). Each actuator is attached to the upper platform(P1,...,6 ) and to the lower platform (B1,...,6 ) by spherical joints which combines two semi-universal joints and a roller bearing shown in

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2 is Ax

Axis 3

Fig. 1 3×3 Experimental setup, upper joint connection and the physical model of a generalized Stewart platform mechanism

Mo to r

1

2 to r Mo

Axis 1

zP

P3

P4

m 3 L3 Ib3

mp Ip rp

P2 yP

λ

P1

εp

{P}

L2 xP

m4 Ib 4 L 4

P5 θ3

B3 θ4

L 6 θ2

L5 m 5 Ib5

θ5

rB

{B}

εB

Λ

xB

m1 L1 I b1

B2

yB

B4

θ1

m6 Ib6

B1

θ6

B6

B5

Fig. 1. Dimensions of the mechanism is illustrated by Fig. 2. SI unit system were used to calculate and simulate mathematical model of the system from derived equations. Smooth pattern demonstrates the legs of the Platform and it also shows the length of legs while actuator length is minimum in Fig. 2. The term “Leg Length” includes actuator lengths (max 100 mm) and lengths of connection elements that are placed between actuators and joints. Dotted pattern represents three degree of freedom (DOF) joints. Dashed pattern which is placed on both upper and lower part of the figure shows the mobile and stationary plates of the SPM, respectively. Upper part of the SPM performs following motions:

m2 Ib 2

P6 zB

40 mm 150 mm

350 mm 317 mm

175 mm 32 mm

Fig. 2 Dimensions of the Experimental Setup

• Linear motion along x P , y P and z P axis; tx , t y , tz , respectively. Linear displacement ranges are ±50 mm on axes x P and y P , 400 ± 50 mm on axis z P . • Rotation around x P , y P and z P axis; γ , ϑ, υ, respectively. Angular displacements are all limited to ±0.78 rad. or 45◦ .

The physical model of SPM is illustrated in Fig. 1. Upper platform is connected to the actuators by spherical joints and lower platform is connected to the actuators by spherical joints as well. It is assumed that,

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• Center of gravity of upper platform is located at the geometrical center of the mobile frame. • Center of gravity of actuators is located about a distance of the 2/3 of leg length from the lower connection point. • External force and moment effects acting from the user are neglected. • Bearing frictions and gravitational forces are present on the system. In Fig. 1, upper platform with the attached {P} coordinate system is performing linear and angular motions with linear and angular velocities, ˙t and ω, respectively. The attached linear actuators are performing both linear and angular velocities, L˙ and θ˙ , respectively. The parallel interaction between actuators causes angular velocities on each leg.

assume significant role on modeling of an SPM according to energy flow. Conversion between translational velocity of actuators and translational and rotational velocity of mobile (upper) plate of an SPM must be formulated initially. After obtaining these relations, conversion between velocities of mobile plate and angular velocities of legs caused by motion of the mobile plate must be formulated. Each leg length can be calculated by using Eq. (4):  − →  Li =  S i  .

(4)

Then, unit vectors of legs are: − → − → s i = S i /L i .

(5)

Linear velocity along a leg can be defined as: 2.2 Kinematics of the SPM

− → ˙ → L˙ i = − s i· Si

According to the physical model of a generalized SPM shown in Fig. 1, inverse kinematics solution can be formulated as: → − → − → − → qi+ t − bi (1) Si =− − → − → where S i denotes the leg vector, t is the position of mobile − → plate of the SPM with respect to base frame {B}, b i is the position vector of lower connection points of legs with → respect to base frame, − q i is the position vector of upper connection points of legs with respect to base frame and this vector can be expressed as: − → → q i = R− pi

(2)

In Eq. (2), R is 3×3 rotation matrix that ensures to identify the vectors with respect to {B} which is defined with respect → to {P}. − p i is the position vector of upper connection points of legs with respect to {P}. Rotation matrix can be defined as: ⎡

cos(υ) cos(ϑ) ⎢ ⎢ R=⎢ ⎢ − sin(υ) cos(ϑ) ⎣ sin(ϑ)

cos(υ) sin(ϑ) sin(γ ) + sin(υ) cos(γ ) cos(υ) cos(γ ) − sin(υ) sin(ϑ) sin(γ ) − cos(ϑ) sin(γ )

⎤ sin(υ) sin(γ ) − cos(γ ) ⎥ sin(ϑ) cos(υ) ⎥ sin(υ) sin(ϑ) cos(γ ) ⎥ ⎥ ⎦ + cos(υ) sin(γ )

cos(ϑ) cos(γ )

(6)

− → ˙ where S is the velocity of upper connection point of each leg and it can be expressed as: − → ˙ → q i + ˙t. Si =ω×−

(7)

And also, angular velocity of the actuators can be expressed as:

− → ˙ → ˙θ i = − s i × S i /L i (8) Translational/rotational velocities along/about the actuators can be expressed in terms of the upper plate velocities by extracting these velocities from Eqs. (6) and (8).  L˙ i = si z qi y − si y qi z si x qi z − si z qi x si y qi x − si x qi y si x si y si z ⎡ ⎤ ωx ⎢ ωy ⎥ ⎢ ⎥ ⎢ ωz ⎥ ⎥ ·⎢ ⎢ t˙x ⎥ ⎢ ⎥ ⎣ t˙y ⎦ t˙z

(9)

(3) ⎡

2.3 Kinematic and Dynamic Bond-Graph Model of the SPM It is important to understand operating sequence of an SPM before modeling of the system. First, a voltage input is applied to the actuators to initiate motion. This input voltage is converted to translational velocity by actuators. Then, this translational velocity is converted to rotational and translational velocities on mobile plate. This motion of mobile plate causes angular motion on each leg. Therefore, velocity conversions

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⎤ θ˙i x 1 θ˙ i = ⎣ θi˙y ⎦ = Li ˙ θi z ⎤ ⎡ −si y qi x −si z qi x 0 −si z si y si y qi y + si z qi z ⎣ · −si x qi y si x qi x + si z qi z −si z qi y si z 0 −si x ⎦ −si x qi z −si y qi z si x qi x + si y qi y −si y si x 0 ⎡ ⎤ ωx ⎢ ωy ⎥ ⎢ ⎥ ⎢ ωz ⎥ ⎥ ·⎢ (10) ⎢ t˙x ⎥ ⎢ ⎥ ⎣ t˙y ⎦ t˙z

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[K ]

Fig. 3 Kinematic vector Bond-Graph model of one leg of the Stewart platform mechanism

θi 1..3

MTF

•ω

ω

θi

1

[K ] Li 1..3

ω

MTF

•ω

Li

•

θi

•

0

0

[K ]

[K ]

θi

MTF

•t

Li 4..6

θi 4..6

•t

1 •

•

t

t

Li

Li

MTF

IP

[ ]

Rθ •

θi

•

θi

1

•ω

θi

K1θ..i 3 MTF

ω

ω

1

[K ] Li 1..3

MTF

•ω

Li

•

θi

RL

0

0

•

θi

Iu

•

•

•t

θi

[K ] θi 4..6

MTF

•

[K ] Li 4..6

1

•

t

t

MTF

Li

Li 1

•

Li

•

•t

Li

Li

mu

mP Fig. 4 Vectorial dynamic Bond-Graph model of the mechanism for the legs

2.3.1 Kinematic Bond-graph Model of the SPM It is possible to write Eqs. (9) and (10) as: ⎡ − ⎤ → ω  Li Li ⎦ ⎣ − L˙ i = K 1,...,3 K 4,...,6 → ˙

  t

(11)

K Li

⎡ − ⎤ → ω  θi − → ˙ θi ⎦. θ i = K 1,...,3 K 4,...,6 · ⎣ − → ˙

  t

(12)

K θi

In Eqs. (11) and (12), the angular velocity of the upper platform multiplies by first, second and third columns of K Li and K θi , same as the linear velocity of the upper platform multiplies by fourth, fifth and sixth columns of K Li and K θi . Consequently, K 1,...,3 signifies the effects of angular velocity of the upper platform on the leg velocities and also K 4,...,6 represents the effects of translational velocity of the upper platform on the leg velocities. Kinematic vectorial BondGraph model of the mechanism for one leg which is obtained from Eqs. (11) and (12) is given in Fig. 3. In Fig. 3, kinematic model is identified for just one leg (i = 1, . . . , 6). In this figure, “0” port and “1” port introduces serial and parallel energy ports, respectively. Double arrows

indicate energy flow direction of the system which carries vectorial quantities. According to this definition, energy input, from the right side of Fig. 3, generates a linear motion on leg. This linear motion energy is separated into two parts in “0” port. Upper arrow indicates the energy that causes the mobile platform to make angular motion. Lower arrow also shows the energy which causes the mobile platform to make translational motion. Modulated transformer (MTF) element helps to convert one type of motion to another. In Fig. 3, motion type of bonds are velocity which leads to a conclusion that MTF elements convert translational velocity to rotational velocity, or translational velocity to translational velocity (depends on the calculation of axes). Each leg has a contribution to the translational and angular motion of the mobile plate in “1” port. In the left side of Fig. 3, motion of the mobile plate creates an angular motion on each leg with the same process. 2.3.2 Dynamic Bond-Graph Model of the SPM Attaching the physical elements of the system to the kinematic model, dynamic Bond-Graph model of the Stewart platform mechanism can be obtained. Vectorial dynamic Bond-Graph model of the mechanism for one leg is shown in

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EJS : MJR

θi 1..3

Rθ •

M θi

θi

Mω ω

[K ]

MTF

•ω

θi •

M θi

θi

1

M θi •t

θi

M

ω θi

[] [K ] θi 4..6

MTF

1

ω

MI MG

FI

Fθti •

M

ω Li

RL

MTF

• ω

t

t F G

FLit •

FRi

Li •

[G ]

•

t

Li 1..3

FLi

•

1

[K ]

ω

ω

~ I

0

M θi

ω

0

[K ] Li 4..6

Li FLi

FLi

•

Li 1

•

Li

Fsi

•t

Li

MTF

t

Fig. 5 Bond-Graph model of entire system after reducing the inertial elements

Fig. 4 in which Rθ and R L indicates the viscous friction on revolute and prismatic joints, respectively. Iu and I p represents inertia of the legs and the upper platform, respectively. m u and m p also symbolizes the mass of legs and the upper platform, respectively. When two or more inertial elements are attached to “1” port, system state variables become dependent and this causes derivative causality problem. This means, independent velocities of these elements becomes dependent velocities. This problem results in an implicit differential equation and may cause numerical problems in solution. This problem can be solved by transforming the inertial elements of dependent velocities to the ports of independent velocities. Therefore, the inertial elements should be reduced to the ports of independent velocity ports as a virtual inertia matrix and gyristor element. Dependent velocities, independent velocities and inertial velocities of the system should be defined before reducing the system. Translational and rotational velocity of legs can be chosen as dependent velocities. Therefore, the upper platform translational and rotational velocities can be chosen as independent velocities. All dependent and independent velocities can be chosen as inertial velocities. Dependent velocities can be expressed as:   → − → − → − → − → − → ˙ ˙ ˙ ˙ ˙ ˙ T q˙ b = L˙ 1 − θ 1 L˙ 2 θ 2 L˙ 3 θ 3 L˙ 4 θ 4 L˙ 5 θ 5 L˙ 6 θ 6 (13) − → ˙ In Eq. (13), θ i is a 3-element column matrix. Therefore, q˙ b is a 24-element column matrix. Independent velocities can be defined as:  q˙ i = ωx

ωy

ωz

t˙x

t˙y

t˙z

T

(14)

and inertial velocities can be expressed as,  q˙ I =

q˙ b q˙ i



123

(15)

The relationship between independent and inertial velocities is T I matrix and can be defined as: q˙ I = T I · q˙ i

(16)

All inertial elements of the system should be defined with a single inertia matrix before obtaining virtual inertial matrix and gyristor element. Inertial elements should be arranged using inertial velocity matrix. Inertia matrix (I) of the system is presented in [18]. The virtual inertial matrix and the gyristor element can be expressed as: ˜ = [T I ]T · [I] · [T I ] [ I]

(17)

and [G] = [T I ]T · [I] · [ T˙ I ].

(18)

The derivative of relation matrix TI is needed to calculate the gyristor element in Eq. (18). It is preferred to derivate this matrix during simulation of complete model of the system. In the left side of Fig. 4, inertial effects due to angular motion of legs and in the right side, newtonian forces due to speed of legs are reduced to virtual inertial matrix and the gyristor element using TI matrix that is obtained from Eq. (16). After reducing the inertial elements to two ports, Bond-Graph model of the system is shown in Fig. 5. 2.3.2.1 Modeling Gyroscopic Forces Figure 5 shows the complete Bond-Graph model of mechanical part of the system for just one leg. The gyroscopic forces acting on the platform is expressed by one modulated gyrator element. A detailed view of this element is shown in Fig. 6. The inertial elements which are shown in Fig. 6 are defined in the virtual inertial matrix. In Fig. 6, flow equations can be expressed as: Mx = Hz ω y − Hy ωz

(19)

M y = −Hz ωx + Hx ωz

(20)

Mz = −Hx ω y + Hy ωx

(21)

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ωx

MG

Ix Hy

ωy ωy I y ωy

Y

Hx Iz ωz ωz

ωy

La

My

1

VL i a

Vexi ia

Y

Hz MGY

ωx ωx

MG

ωx Mx 1

Kt TF

Vsi ia

0

•

Li Fsi

VR i a

1

Ra

ωz M z

Fig. 8 Bond-Graph model of a linear DC actuator

Fig. 6 Detailed view of one modulated gyrator element

where, Hx = I x ωx , Hy = I y ω y , Hz = Iz ωz . Each input energy rows in Fig. 6 represents the rotational velocity of the mobile platform. Mx , M y and Mz in Eqs. (19, 20, 21) express the gyroscopic moment acting on the mobile platform. Hx , Hy and Hz indicates both the gain of modulated gyrator (MGY) element and the interaction between inertial velocities. 2.3.2.2 Modeling the Actuation Mechanism Actuation of the system is based on linear DC actuators which are chosen as legs of the experimental system. A schematic diagram of a linear DC actuator is shown on Fig. 7. Figure 8 shows the Bond-Graph model of the linear DC actuator in which Ra and L a indicates the actuator resistance and inductance, respectively. Values of these variables are obtained from the catalog of linear DC motors which were also used for experimental setup. Additionally, Vexi and K t refers to the actuation voltage and force constant of the actuator, respectively. Complete Bond-Graph model of the mechanism is shown in Fig. 9 after the addition of the actuation part. 2.3.2.3 Model of the Entire System and State-Space Equations The gravitational effect on legs and the upper platform

are modeled by effort sources, and the actuation inputs of the legs are modeled by effort sources in Fig. 9. Right side of the picture shows the translational motion and the left side shows angular motion of the leg. All elements with subscript “i” refers to the six legs in this paper. State equations can be expressed as: ⎡

⎤ ⎡ ⎤ M Ix ωx ⎢ ωy ⎥ ⎢ ⎥ ⎢ ⎥   ⎢ MIy ⎥ −1 ⎢ ⎥ ⎢ ⎥ d ⎢ ωz ⎥ ⎢ M Iz ⎥ = I˜ ⎢ ⎥ ⎢ ⎥ dt ⎢ t˙x ⎥ ⎢ FI x ⎥ ⎣ t˙y ⎦ ⎣ FI ⎦ y t˙z FI z d (i a(i) ) = L a VL(i) , (i = 1, . . . , 6) dt

(22)

The constitutive law of the gyrator element can be expressed as: ⎡ ⎤ ⎡ ⎤ MG x ωx ⎢ ωy ⎥ ⎢ MG y ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ MG z ⎥ ⎢ ⎥ = [G] ⎢ ωz ⎥ (23) ⎢ t˙x ⎥ ⎢ FG ⎥ x ⎥ ⎢ ⎥ ⎢ ⎣ t˙x ⎦ ⎣ FG ⎦ y t˙x FG z The right side of the Eq. (22) is created with the terms of dependent variables which must be expressed with the terms of state variables to find state-space equations. From the “1” junctions of the mechanical part of the Bond-Graph model

Fig. 7 Linear DC actuator schematic

La

Ra

Kt

mu

Vexi

RL

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Author's personal copy Arab J Sci Eng Motion of Mobile Platform

Angular Motion of Leg

Linear motion of Leg DC Motor

EJS : MJR

[ ]

Rθ

K MTF

•ω

θi

•

M θi

θi

1

M θi •t

S f = m u g.

θi

2L i cos(θ iz ) 3

ω θi

[K ] θi 4..6

MTF

ω MI MG

FI

Fθti •

M

t

t F G

•

t Ft

• ω

FRi

Li •

[G ]

•

FLit •

0

[K ]

La

RL

MTF

FLi

•

1

t

[K ] ω Li

Electrical Part

Mechanical Part

Li 1..3

ω

1

ω

[~I ]

0

M θi

ω M

θi

•

M θi

Mω ω

θi 1..3

Li FLi

FLi

1

•

Li

Fsi

Kt TF

Vsi

Li

MTF

Sf = m u g. sin(θ iz )

t

Vexi

0

ia VR

•t

Li 4..6

ia

VL

•

Li

ia

Se = Vexi

ia Ra

Sf = m p g

Fig. 9 Bond-Graph model of the system after adding the actuation part

in Fig. 9, inner equations are obtained as: − − − → →ω →ω − → − → MI = M L(i) − M θ(i) − M G − M ω 6

6

i=1

(24)

i=1

⎧ ⎪

(25)

In Eqs. (24) and (25), M G and F G terms are effort variables of the gyrator element which are illustrated in Fig. 9 as [G]. − → − →ω M L(i) and F tL(i) terms are moments and forces that are generated by the actuators, respectively. The reaction moment and the reaction force of upper platform acting on legs are − → − → − → indicated by M ωθ(i) and F tθ(i) , respectively. In Eq. (24), M ω shows the gyroscopical moments of upper platform. Open form of Eq. (24) can be expressed as: L(i) K 11 FL(i)



i=1

⎧ ⎪

−

6 ⎪ ⎨ 

⎪ i=1 ⎪ ⎩

θ(i)

θ(i)

K 11 K 12

− (Hz ω y − Hy ωz ) MIy =

6 

−

i=1

6  

⎡ (i) ⎤⎫ ⎪ M  − ⎬  ⎢ θ x ⎥⎪ [→ ω] θ(i) ⎢ (i) ⎥ K 13 ⎣ Mθ y ⎦ − G [1,:] − → ˙ ⎪ [ t ] ⎪ (i) ⎭ Mθ z (26)

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⎪ i=1 ⎪ ⎩

⎤⎫ Mθ(i)x ⎪  − ⎪ ⎢ ⎥⎬ [→ ω] θ(i) ⎢ (i) ⎥ K 33 ⎣ Mθ y ⎦ − G [3,:] − → ˙ ⎪ [ t ] ⎪ (i) ⎭ Mθ z ⎡

θ(i)

θ(i)

K 31 K 32

− (−Hx ω y + Hy ωx )

(28)

and open form of Eq. (25) is also expressed as,

FI x =

6 

L(i)

(K 14 FL(i) )

⎧ ⎡ (i) ⎤⎫ ⎪ ⎪ M  − ⎪ 6 ⎨ ⎬  ⎢ θ x ⎥⎪  [→ ω] θ(i) θ(i) θ(i) ⎢ (i) ⎥ − K 14 K 15 K 16 ⎣ Mθ y ⎦ − G [4,:] − → ˙ ⎪ ⎪ [ t ] ⎪ i=1 ⎪ ⎩ (i) ⎭ Mθ z (29)

FI y =

6 

L(i)

(K 15 FL(i) )

i=1

⎫ (i) ⎤ Mθ x ⎪  − ⎪ ⎬ ⎢ ⎥ [→ ω] (i) ⎥ θ(i) θ(i) θ(i) − G − M K 21 K 22 K 23 ⎢ [2,:] − → ˙ ⎣ θ y ⎦⎪ ⎪ [ t ] ⎪ i=1 ⎪ ⎩ (i) ⎭ Mθ z − (−Hz ωx + Hx ωz ) (27) ⎧ ⎪ 6 ⎪ ⎨ 

6 ⎪ ⎨ 

i=1

L(i) (K 12 FL(i) )

i=1

L(i)

(K 13 FL(i) )

i=1

i=1

6 6   − → − →t − →t − → FI = F L(i) − F θ(i) − F G − m p g

M Ix =

M Iz =

6 

⎡

−

⎧ ⎪ 6 ⎪ ⎨  ⎪ i=1 ⎪ ⎩

⎤⎫ Mθ(i)x ⎪  − ⎪ ⎬ ⎢ ⎥ [→ ω] (i) ⎥ θ(i) − G M K 26 ⎢ [5,:] − → ˙ ⎣ θ y ⎦⎪ [ t ] ⎪ (i) ⎭ Mθ z (30) ⎡

θ(i)

θ(i)

K 24 K 25

Author's personal copy Arab J Sci Eng

FI z =

6  i=1

−

From Eqs. (24), (25), (36), M Ix , M I y , M Iz , FIx , FI y , FIz , VL(i) terms can be expressed by the terms of state variables.

L(i) (K 16 FL(i) )

⎧ ⎪ ⎪ 6  ⎨

⎫ ⎪ ⎪ ⎢ ⎥⎬ θ(i) ⎢ (i) ⎥ K 36 ⎣ Mθ y ⎦ ⎪ ⎪ (i) ⎭ Mθ z ⎡

θ(i)

θ(i)

K 34 K 35

(i) ⎤

⎪ ⎪ ⎩  − [→ ω] − G [6,:] − − m pg → ˙ [ t ] i=1

mx mx mx mx M Ix = ωx E ωx + ω y E ωy + ωz E ωz + t˙x E tmx x + t˙y E t y

Mθ x

+ t˙z E tmx z + my

(31)

The G [1,:] term in Eq. (26) shows the first row of gyra(i) tor matrix. Leg forces (FL(i) ) and leg moments (Mθ ) are unknown variables of these equations. These unknown variables can be found by writing the constitutive law of “0” energy ports. All unknown variables should be expressed in terms of state variables. Leg moments can be given as:  − → ω (i) θ(i) , (i = 1, . . . , 6) (32) Mθ x = Rθ θ˙(i)x = Rθ K [1,:] − → ˙ [ t ] = Rθ θ˙(i)y

 − → ω θ(i) = Rθ K [2,:] − , (i = 1, . . . , 6) → ˙ [ t ]

(i)

Mθ y = Rθ θ˙(i)y − m u g

2L (i) 3

2L (i) cos(θi z ), (i = 1, . . . , 6) 3 Leg forces can also be given as:  − → ω L(i) − m u g. sin(θi z ) FL(i) = Fs(i) − R L K − → ˙ [ t ]

my

my

my

6 

my

my

my

(i a(n) E i(n) + TL(n) + Tθ(n) )

(39)

n=1 mz mz mz mz M Iz = ωx E ωx + ω y E ωy + ωz E ωz + t˙x E tmz x + t˙y E t y

+ t˙z E tmz z +

6 

mz mz mz (i a(n) E i(n) + TL(n) + Tθ(n) )

(40)

n=1 Fx Fx Fx FIx = ωx E ωx + ω y E ωy + ωz E ωz + t˙x E tFxx + t˙y E tFyx 6 

Fx Fx Fx (i a(n) E i(n) + TL(n) + Tθ(n) )

(41)

n=1 Fy

Fy

Fy

Fy

Fy

FI y = ωx E ωx + ω y E ωy + ωz E ωz + t˙x E t x + t˙y E t y Fy

+ t˙z E t z +

6 

Fy

Fy

Fy

(i a(n) E i(n) + TL(n) + Tθ(n) )

(42)

n=1

(34) Fz Fz Fz FIz = ωx E ωx + ω y E ωy + ωz E ωz + t˙x E tFxz + t˙y E tFyz

(35)

Fz + t˙z E tFz z − Tmp +

6 

Fz Fz Fz (i a(n) E i(n) + TL(n) + Tθ(n) )

n=1

(43)

(36)

where, Vs(i)

my

+ t˙z E t z +

(33)

Actuating forces of the legs (Fs(i) ) is also an unknown variable in Eq. (35) and it can be expressed by using the BondGraph model of Figs. 8 and 9. VL(i) = Vex(i) − Vs(i) − V R(i) , (i = 1, . . . , 6)

(38)

M I y = ωx E ωx + ω y E ωy + ωz E ωz + t˙x E t x + t˙y E t y

+ t˙z E tFz x +

 − → ω θ(i) cos(θi z ) = Rθ K [3,:] − → ˙ [ t ]

− mu g

mx mx mx (i a(n) E i(n) + TL(n) + Tθ(n) )

n=1

my

(i) Mθ y

6 

v(i) v(i) v(i) v(i) v(i) VL(i) = ωx E ωx + ω y E ωy + ωz E ωz + t˙x E t x + t˙y E t y v(i)

v(i)

v(i) + t˙z E t z + i a(i) E i(i) + Tex , (i = 1, . . . , 6)

(44) Open forms of terms E and T are provided in [18]. Equations (38), (39), (40), (41), (42), (43), and (44) should be plugged into Eq. (22).

 − → ω L(i) = L˙ i K i = K Kt , − → ˙ [ t ]

Fs(i) = K t i a(i) ,

d −1 −1 −1 −1 ωx = I˜11 M I x + I˜12 M I y + I˜13 M I z + I˜14 FI x dt + I˜−1 FI y + I˜−1 FI z

(45)

d −1 −1 −1 −1 ω y = I˜21 M I x + I˜22 M I y + I˜23 M I z + I˜24 FI x dt + I˜−1 FI y + I˜−1 FI z

(46)

d −1 −1 −1 −1 ωz = I˜31 M I x + I˜32 M I y + I˜33 M I z + I˜34 FI x dt + I˜−1 FI y + I˜−1 FI z

(47)

15

V R(i) = i a(i) Ra . By writing the Vs(i) , Fs(i) , V R(i) in Eq. (36), actuating forces can be expressed as:  − → ω L(i) K t −i a(i) Ra , (i = 1, . . . , 6) VL(i) = Vex(i) − K − → ˙ [ t ] (37)

25

35

16

26

36

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d −1 −1 −1 −1 M I x + I˜42 M I y + I˜43 M I z + I˜44 FI x t˙x = I˜41 dt + I˜−1 FI y + I˜−1 FI z

(48)

d −1 −1 −1 −1 M I x + I˜52 M I y + I˜53 M I z + I˜54 FI x t˙y = I˜51 dt + I˜−1 FI y + I˜−1 FI z

(49)

d −1 −1 −1 −1 M I x + I˜62 M I y + I˜63 M I z + I˜64 FI x t˙z = I˜61 dt + I˜−1 FI y + I˜−1 FI z

(50)

45

55

65

46

56

66

d (i a(i) ) = L a VL(i) , (i = 1, . . . , 6) dt

(51)

These equation series (45), (46), (47), (48), (49), (50), and (51) can be expressed in matrix form as: ⎤ ωx ⎢ ωy ⎥ ⎢ ⎥ ⎢ω ⎥ ⎢ z⎥ ⎢ t˙ ⎥ ⎢ x ⎥ ⎢ t˙ ⎥ ⎢ y ⎥  ⎥ [A]6×6 [B]6x6 d ⎢ ⎢ t˙z ⎥ ⎢ ⎥= dt ⎢ i a1 ⎥ ⎢ ⎥ [C]6×6  [D]6x6 ⎢ i a2 ⎥ ⎢ ⎥

⎢ i a3 ⎥ ⎢ ⎥ ⎢ i a4 ⎥ ⎢ ⎥ ⎣ i a5 ⎦ i a6 ⎡

 +

[I˜ −1 ]6×6

0

0

[La ]6×6

⎡

⎤ ωx ⎢ ωy ⎥ ⎢ ⎥ ⎢ω ⎥ ⎢ z⎥ ⎢ t˙ ⎥ ⎢ x ⎥ ⎢ t˙ ⎥ ⎢ y ⎥ ⎢ ˙ ⎥ ⎢ tz ⎥ ⎢ ⎥ ⎢ i a1 ⎥ ⎥ ⎢ ⎢ i a2 ⎥ ⎢ ⎥ ⎢ i a3 ⎥ ⎢ ⎥ ⎢ i a4 ⎥ ⎢ ⎥ ⎣ i a5 ⎦ i a6

[U]12×1

(52)

Equation (52) represents the state-space equation of a Stewart platform mechanism. Open form of matrices A, B, C, D, −1 I˜ , L a and U are in [18]. It is very clear in Eq. (52) that state matrix, , does not contain state variables. But it contains variables indirectly related to state variables. Therefore, the state matrix is variable in the workspace of the model.

2.3.3 Simulation Open-loop simulation of the model is performed with a program written in Matlabm-file. State-Space equations, (52), are used for simulation. Derivatives and integrals are calculated in discrete time with a sampling rate of 0.0001 s for obtaining the best accuracy. Simulations are performed on a computer which has Pentium Dual Core 1.73 GHz processor and 2 GB memory with 32-bit operating system. Simulation

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Fig. 10 Locations of the boundary points on spherical workspace of the SPM

results are compared with experimental results for model verification.

2.3.4 Stability Analysis of the Dynamic Model Using State-Space Equations Representing the system in state-space form has several advantages, one of which is analyzing the system using the eigen values of the state matrix. In this model, state matrix is variable along the workspace of the mechanism and it is also time-invariant. Therefore, the location of system roots on complex plane is directly related to the value of each element of the state matrix (). Locations of the system roots are investigated on some several selected locations on the workspace.These points refer to the boundary positions of the mobile platform on each axis and on axis quadrants as shown in Fig. 10. These points are given as input to the model

Author's personal copy Arab J Sci Eng

-25

-15

-5

Fig. 11 System roots in complex plane

and eigen values for each points are calculated from obtained state matrix. Eigen values of the model on these points can be shown in complex plane as in Fig. 11. Table 1 also shows numerical values of the system eigen values for related locations. According to Fig. 11 and Table 1, the system has 12 distinct eigen values on real axis. On each point in workspace, eigenvalues varies with small differences and all eigen values are placed at the left side of the complex plane. Therefore, the model can be called stable in this spherical workspace. Meanwhile, modeled system goes to zero on both x and y axis from any start point in workspace which proves the stability in workspace. One of the advantages of modeling nonlinear systems with Bond-Graph method is to obtain state-space equations. Stability of system can be analyzed by using eigen values of the state matrix. Elements of the state matrix values are usually constant in linear systems. Because of the nonlinearity of the modeled system in this approach, values of state matrix are not constant and are vary depending on the location of the SPM. In Fig. 11, two main clusters of roots are present and one cluster is located near zero and other is located far from zero. Therefore, since large values of roots do not jeopardize the dynamics of the system,the ones which are far from zero have secondary effect on system dynamics. Roots near zero have no complex part and that means model of the system has overdamped behavior. Although, extreme 14 points of the spherical workspace are referred in Fig. 11 and Table 1, other internal locations of the workspace which are not listed on this paper are also checked for stability. However, no unstable roots are observed.

3 The Experimental System Dimensions of the mechanism needed for calculations are shown in Fig. 2. General components of the experimental setup are shown in Fig. 12. The experimental system is initially designed as a joystick to manipulate spatially moving objects. Therefore, lever, force/torque transducer and helicopter part of the illustration is related with this part of the project. As seen on Fig. 12, the system consists of a lever (A), a six-axis force/torque transducer (B), ATI nano25, a motion control card interface (C), NI UMI-7774, a six-axis motion control card (D), NI PCI7356, a PC, linear motor drivers, E210-VF (E), and a 3×3 SP. Lever is rigidly assembled to the force/torque transducer while the transducer is also rigidly assembled to the Stewart platform so that force/torque interaction between the lever and the SP is fully measurable. The force/torque transducer, linear motor drivers and the motion control card’s data signal cables are connected to motion control card interface.The software that is written for this system gets the digital data and converts it to numerical values of forces and torques. The software uses inverse kinematics to calculate the legs’ lengths. Since linear DC actuator drivers are operated under force/torque mode, the ±10 V of input signals are sent to linear motors’ drivers to convert the input to actuating forces. By controlling the motors current in this way, the control capability is transferred to the designer comparing to velocity mode where internal velocity control algorithms of the drivers are heavily used. The encoder data of selectable resolutions (1 mm ≤ 1µm) from the auxiliary encoder output of the driver is used for positional feedback purposes. The

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Author's personal copy −6,663.56

−6,663.40

−6,663.43

−6,663.58 −6,652.20 −6,640.91 −6,639.79 −6,639.45 −3.24

−14.46

−14.47 −27.03

−27.13 −27.41

−27.32 −26.02

−25.83

−3.06 0.365 −0.025 0.025 P14

−3.08 0.365 −0.025 −0.025 P13

−27.41 −25.83

−27.13

−14.46

−3.08 0.365 0.025 −0.025 P12

−6,639.47 −6,639.58 −6,641.09 −3.22

−6,652.20

−6,663.40

−6,663.56 −6,663.43 −6,639.47 −6,639.58 −6,641.09 −3.22

−6,652.20

−6,664.26 −6,664.36

−6,663.58 −6,652.20 −6,640.91 −6,639.79 −6,639.45 −3.24 −14.47

−11.16

−27.32 −26.02

−27.03

−27.07 −29.13

−26.28

−3.06 0.025 P11

−2.29 0.435

0.365

−0.025

0.025

0.025 P10

−6,640.6,6 −6,639.87 −6,637.54 −2.39

−6,655.51

−6,664.27 −6,664.34 −26.13

−27.21

−29.12

−11.16

−2.38 0.435 −0.025 −0.025 P9

−27.21 −26.13

−29.12

−11.16

−2.38 0.435 0.025 −0.025 P8

−6,637.54 −6,639.72 −6,640.81 −2.30

−6,655.51

−6,664.26

−6,664.27 −6,664.34 −6,655.51 −6,639.72 −6,640.81 −2.30

−6,637.54

−6,640.21 −6,640.22

−6,664.36 −6,655.51 −6,639.87 −6,637.54 −2.39

−26.71

−11.16 −26.28

−3.41

−27.07

−3.41

−29.13

−15.47

−2.29 0.435 0.025 0.025 P7

−26.71 0.35 0 0 P6

−6,655.99 −6,640.20 0

−12.56 −28.36

−6,664.42 −6,637.30 −6,640.20 0.45 0 P5

−6,640.6,6

−6,663.23 −6,651.20 −26.46

−6,663.23

−6,640.22

−2.23 −2.23 −26.74 −26.74 −6,664.42

−29.37

−10.68

−2.56

−6,663.89 −6,664.05

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−2.59 −27.37 −25.76 0.4 −0.05 0 P4

−12.56

−6,638.30 0.4

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−6,640.90

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−6,654.11

0.4

−6,663.87

−2.59

0 −0.05 P3

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0.05 0 P2

−27.59 −12.56

z (m)

Roots

−2.61

−25.57

−28.34 0.4 0

4 Results

0.05

3.2 Comparison Between Bond-Graph Model of SPM and Experimental Setup

P1

−6,638.32 −6,639.56 −6,641.17 −2.76

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−28.37 −27.08

−6,654.11

−26.03 −12.56

Four trajectories are applied to both simulation and the experimental system for model verification. In the first trajectory, all motors are powered up at the same time with 3.662 V. For the other scenarios, each of the two adjacent motors are powered up which are connected to the mobile platform at the same joint. These motor couples are 1–6, 2–3 and 4–5, respectively. In the last three scenarios, motor couples are excited with 3.662 V and the remaining motors are powered up with 3.357 V to hold their position. The illustration of these scenarios are shown in Fig. 13.

y (m)

−6,638.32 −6,639.56 −6,641.17 −2.76

−2.78

−6,638.33

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3.1 Scenarios for Experimental Studies and Simulation

Four selected trajectories are applied to both Bond-Graph model and to the experimental system, as in Fig. 13. Responses of the both are recorded for these trajectories. For instance, the upper platform of the SPM is expected to elevate only on the z-axis for the first trajectory in Fig. 13a. Results for this trajectory are shown in Fig. 14. For the second trajectory in Fig. 13b, the mobile platform is expected to make an angular motion about x and y axes. Additionally, it is expected to show some small displacements on x, y and z axes. Results of the second trajectory scenario are shown in comparison graphs in Fig. 15. For the third trajectory in Fig. 13c, upper platform is expected to make an angular motion about x and y axes. Results of the third trajectory scenario (Fig. 13c) are shown in comparison charts in Fig. 16. In the last trajectory scenario (Fig. 13d), mobile plate is expected to perform an angular motion about axis y and small displacements on axes x and z. Results of last motion are shown in Fig. 17. In Figs. 14, 15, 16, 17, L i (i = 1–6) indicates leg lengths, X-Y-Z indicates the linear displacement of mobile plate on x, y and z axes, respectively. A x , A y , A z represent the angular displacement of mobile plate about x, y, and z axes, respectively.

Points

Table 1 Root of the system in boundary points of workspace

software developed in Visual Basic utilizes this data for both closed-loop control and open-loop control of the system. In this study, the system is operated in open-loop control. Some parameters of the experimental system are listed in Table 2.

x (m)

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Arab J Sci Eng

For the comparison graphs of the first trajectory in Fig. 14, expected motion of the mobile plate consists of translation along z direction. Leg lengths are shown on the left side and all are increasing with the same slope with small deviation.

Author's personal copy Arab J Sci Eng Fig. 12 General components of the overall mechatronic system

Table 2 Parameters of the experimental setup of the Stewart platform mechanism Parameter

Value

Mobile platform mass (Mu )

1.387 kg

Motor shaft mass (m u )

0.135 kg

Motor body mass (m d )

0.44 kg

Mobile platform connection points radius (r p )

0.15 m

Stationary platform connection points radius (rb )

0.175 m

Damping coeff. of joints (c f )

0.03 N m s/deg

Motor inductance (L a )

1.5 mH 10  11 N/A

L3

L5

L2

L1

L6

L2

X

X

Y

X

Y

X

Y

L3

L5

L6

L1

L3

L5

L6

L2

L1

L2

L1

L3

L6

L4

L4

Y

L4

L5

L4

Motor resistance (Ra ) Motor force coefficients (K t )

On the right side of the figure, motion of the mobile plate is recorded for the simulation and the experimental system, and both show consistent results with negligible errors. In Fig. 15 which compares the results for the second trajectory, L1 and L6 increases with small deviation and mobile plate performs expected angular and translational displacements. There are small deviations on axes x and y and mobile platform elevates about 27 mm along z axis. Mobile plate also makes angular displacement around x axis about 0.3 rad and around y axis about 0.2 rad as expected. Mobile plate performs the expected motion for the third trajectory in Fig. 16. In this motion, L2 and L3 elevates and other legs are kept stationary. Therefore, mobile plate moves about 30 mm on

(A)

(B)

(C)

(D)

Fig. 13 Simulation and experiment scenarios: a VL1 = VL2 = VL3 = VL4 = VL5 = VL6 = 3, 662V.bVL1 = VL6 = 3, 662V, VL2 = VL3 = VL4 = VL5 = 3, 357V.cVL2 = VL3 = 3, 662V, VL1 = VL4 = VL5 = VL6 = 3, 357V.dVL4 = VL5 = 3, 662V, VL1 = VL2 = VL3 = VL6 = 3, 357V

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Fig. 14 Results of Bond-Graph model simulation and experimental setup of the SPM for trajectory 1 (Fig. 13a)

Fig. 15 Results of Bond-Graph model simulation and experimental setup of the SPM for trajectory 2 (Fig. 13b)

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Fig. 16 Results of Bond-Graph model simulation and experimental setup of the SPM for trajectory 3 (Fig. 13c)

Fig. 17 Results of Bond-Graph model simulation and experimental setup of the SPM for trajectory 4 (Fig. 13c)

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z axis and also performs small displacement on x axis and y axis. Angular motion of the mobile plate is, around x axis about −0.4 rad. and y axis about 0.2 rad. In Fig. 17, L4 and L5 are excited causing the mobile plate to perform translational motion on z axis about 38 mm and rotational motion around y axis about −0.4 rad. It is assumed that all leg parameters such as viscous friction, elasticity are equal to each other in the model unlike the experimental setup. It is also assumed that the bearings which are shown in Fig. 1 are all have face-to-face contact but in experimental setup, bearings have spaces on their contact surfaces. These assumptions can be shown to be the cause of the significant error values in the given figures. Both simulation results of Bond-Graph model of SPM and experimental results of SPM shows that all simulation results of leg motions and mobile plate motions overlap the experimental system responses. Therefore, it can be concluded that the Bond-Graph model of the SPM reflects real dynamics of the SPM.

5 Conclusions Dynamic characteristics of mechatronic systems take important place for efficient control. Parallel mechanisms, especially Stewart platform mechanisms, have complex system dynamics and are completely nonlinear including geometric and element-wise nonlinearities. Therefore, in the present study, kinematics and dynamics of a 3×3 Stewart platform mechanism are obtained by Bond-Graph method and nonlinear state-space representation of the system dynamics is developed. An important problematic point of obtaining dynamic equations is the possible derivative causality problem. If there is a disregarded derivative causality problem during calculations, numerical problems and errors occur when simulating the system. Thus, transforming inertial elements of dependent velocities to the ports of independent velocities, derivative causality problem is solved. At the end of the study, a state-space model of the mechanism is provided. Utilizing the state-space representation of the mechanism, eigen values of the system matrix are obtained for 14 boundary locations of the workspace. Stability of the model is discussed referring to the root locations on complex plane and it is proved that the model is stable in the workspace. Simulation of the Stewart platform mechanism is realized by using obtained state-space model of the mechanism and is compared with the experimental setup of the SPM for series of selected leg trajectories. Verification of Bond-Graph model of the SPM is performed using comparison graphs. As a result of satisfactory consistency between simulation and experimental systems, obtained Bond-Graph model and

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nonlinear state-space equations are verified. Thus, the produced mathematical model and the state-space equations can be used to establish further control design studies like LPV control or nonlinear control techniques. Acknowledgments This work is supported by The Scientific and Technological Research Council of Turkey (TUBITAK) grant 3501-105M192.

References 1. Stewart, D.: A platform with six degrees-of-freedom. Proc. Mech. Eng. Part I 180, 371–386 (1965–1966) 2. Karnopp, D.C.; Rosenberg, R.C.: Introduction to Physical System Dynamics. McGraw Hill, New York (1983) 3. Thoma, J.U.: Simulation By Bond Graphs. Springer, Berlin (1990) 4. Margolis, D.; Shim, T.: A Bond Graph model incorporating sensors, actuators, and vehicle dynamics for developing controllers for vehicle safety. J. Frankl. Inst. 338, 21–34 (2001) 5. Allen, R.R.; Dubowsky, S.: Mechanisms as components of dynamic systems: A Bond Graph approach. ASME J. Eng. Ind. 99(1), 104–111 (1977) 6. Bos, A.M.; Tiernego, M.J.L.: Formula manipulating in the Bond Graph modeling and simulation of large mechanical systems. J. Frankl. Inst. 319(1/2), 51–55 (1985) 7. Fahrenthold, E.P.; Worgo, J.D.: Vector bond graph analysis of mechanical systems. Trans. ASME 113, 344–353 (1991) 8. Karnopp, D.: Approach to derivative causality in Bond Graph models of mechanical systems. J. Frankl. Inst. 329(1) (1992) 9. Do, W.Q.D.; Yang, D.C.H.: J. Robot. Syst. 5(3), 209–227 (1988) 10. Dasgupta, B.; Mruthyunjaya, T.S.: Closed-form dynamic equations of the general Stewart Platform through the Newton–Euler approach. Mech. Mach. Theory 33(7), 993–1012 (1998) 11. Geng, Z.; Haynes, L.S.; Lee, J.D.; Carroll, R.L.: On the dynamic model and kinematic analysis of a class of Stewart platforms. Robot. Autono. Syst. 9, 237–254 (1992) 12. Liu, K.; Fitzgerald, M.; Dawson, D.W.; Lewis, F.L.: Modeling and control of a Stewart platform manipulator. In: ASME DSC, Control of Systems with Inexact Dynamic Models, vol. 33, pp. 83–89 (1991) 13. Sagirli, A.; Bogoclu, M.E.; Omurlu, V.E.: Modeling a rotary crane by Bond-Graph Method, analysis of dynamical behavior and load spectrum analysis (Part I). Nonlinear Dyn. 33, 337–351 (2003) 14. Sagirli, A.; Bogoclu, M.E.; Omurlu, V.E.: Modeling a rotary crne by Bond-Graph method, analysis of dynamical behavior and load spectrum analysis (Part II). Nonlinear Dyn. 33, 353–367 (2003) 15. Wang, J.; Gosselin, C.M.: A new approach for the dynamic analysis of parallel manipulators. Mutibody Syst. Dyn. 2, 317–334 (1998) 16. Liu, M.; Li, C.X.; Li, C.N.: Dynamic analysis of the Gough–Stewart platform manipulator. IEEE Trans. Robot. Autom. 16(1), February (2000) 17. Guo, H.B.; Li, H.R.: Dynamic analysis and simulation of a six degree of freedom Stewart platform manipulator. Proc. ImechE Part C 220, 61–72 (2006) 18. Yildiz, I.; Omurlu, V.E.; Sagirli, A.: Dynamic modeling of a generalized Stewart platform by Bond Graph method utilizing a novel spatial visualization technique. Int. Rev. Mech. Eng. 2(5), (2008) 19. Bai, X.; Turner, J.D.; Junkins, J.L.: Dynamic Analysis and Control of a Stewart Platform Using A Novel Automatic Differentiation Method. AIAA/AAS Astrodynamics Specialist Conference and Exhibit 21–24, Keystone, CO, August (2006)