Effect of Geometrical Parameters on Workspace ...

International Conference on Multi Body Dynamics 2011 Vijayawada, India. pp. 281–300

Effect of Geometrical Parameters on Workspace Envelope & Stiffness of a Parallel Manipulator – Stewart Platform Nitin Johri Dept. of Mechanical Engg., FET, Gurukul Kangri University, Haridwar, (U.K.), India [email protected]; [email protected] © K L University 2011 Abstract. Parallel Manipulators have been increasingly studied and developed in last two decades from theoretical view point and practical applications. Advances in computer technology and control techniques have made it possible to make use of parallel manipulators for industrial applications. Parallel manipulators have better rigidity, better accuracy and better mass to payload ratios when compared to serial manipulators but then these manipulators have much smaller workspace with abundance of singularities. In this work 6-DOF (UPS) Stewart platform is considered for study of effect of dimensional parameters such as size of fixed plate and mobile plate and the link lengths on dexterity. Overall dexterity has been evaluated by averaging the condition number of link Jacobian matrix computed over the entire work space. It is observed that change in dimension of mobile plate and link lengths have considerable effect on overall dexterity when compared to change in fixed plate dimensions. Although global conditioning that is average condition number has been used as criteria for optimization before, this study points towards providing a basis for establishing a sensitivity index for different variants of parallel manipulators on the basis of effect of dimensional parameters on performance indices. Keywords.

1. Introduction A robot is a machine capable of physical motion for interacting with the environment. Physical interactions include manipulation, locomotion, and any other tasks changing the state of the environment or the state of the robot relative to the environment. A robot has some form of mechanisms for performing a class of tasks. A rich variety of robot mechanisms has been developed in the last few decades. K L University

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There are basically 2 types of primitive connections between a pair links, as shown in the figure (side): The first is a prismatic joint where the pair of links make a translational displacement along a fixed axis. In other words, one link slides on the other along a straight line. Therefore it is called a sliding joint. The second type of primitive joint is a revolute joint where a pair of links rotate about a fixed axis. This type of joint is often referred to as a hinge, articulated, or rotational joint.

On combining these two types of primitive joints, we can create many useful mechanisms for robot manipulation & locomotion. These two types of primitive joints are simple to build & are well grounded in engineering design. Most of the robots that have been built are combination of only these two types. Parallel Linkages Primitive joints can be arranged in parallel as well as in series. Parallel mechanisms represent a family of devices based on a closed kinematic architecture. This is in contrast to serial mechanisms, which are comprised of a chain-like series joints and links in an open kinematic architecture. The closed architecture of parallel mechanisms offers certain benefits and disadvantages. A particular subset of parallel mechanisms is known as Stewart platforms (or hexapods). The classic Stewart platform is based on a set of six independently actuated struts. The struts translate through sphere joints which reside on a stationery platform. Each strut terminates at a ball joint residing on a mobile tool platform. All joints allow passive rotation. The effect of this kinematic architecture is that the tool platform can be controlled to move in axes. Figure 1 (next page) shows the Stewart mechanism [[2] D. Stewart 1965], which consists of a moving platform, a fixed base, and six powered cylinders connecting the moving platform to the base frame. The position and orientation of the moving platform are determined by the six independent actuators. The load acting on the moving platform is born by the six “arms.” Therefore, the load capacity is generally large, and dynamic response is fast for this type of robot mechanisms. However, this mechanism has spherical joints, a different type of joints than the primitive joints we considered initially. Parallel robots offer benefits in the areas of high speed, high stiffness and improved accuracy & offer a great deal of potential, especially within the areas of manufacturing & factory automation. Improvement of the performance of such robots involves an in-depth analysis of all aspects of the robot system, such as control strategies, mechanical design and mathematical modeling. The integration of design (synthesis) techniques with analysis techniques offers the opportunity for improved parallel manipulators in future. (Effect of Geometrical Parameters on Dexterity of a Parallel Manipulator-Stewart Platform) being a step in this direction.

Effect of Geometrical Parameters on Workspace Envelope

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Figure 1. Stewart mechanism parallel-link robot.

2. Analysis of Effect of Geometrical Parameters on Dexterity of Stewart Platform This approach utilizes some fundamental analyzing techniques for a robotic manipulator viz. Inverse kinematics, modeling an articulated tool-head, workspace analysis (lower & upper) envelope, stiffness analysis; writing of MATLAB codes based on these concepts [[1] V. De Sapio, Sandia National Laboratories [1998]] & analyzing effects of change in geometric dimensions of mobile & stationery platforms on workspace envelope of manipulator & stiffness of struts (arms). 2.1 Inverse kinematics To study the effect of Geometrical Parameters on Dexterity of Stewart Platform, first it will be necessary to establish a set of coordinate frames. As with conventional robots, a Base frame and a Tool (or operational top) will be defined. While our choice of these frames is arbitrary we will use a convenient frame layout shown in figure 3. The Base frame lies on the plane defined by the six sphere joint centers (stationary platform). The operational frame lies on the plane defined by the three ball joint centers (mobile tool platform). In the case this Hexel machine, (taking): (a = dimension of sides on mobile platform, c = dimension of sides on fixed platform, & d = dimension of comer sides on fixed platform for connection of struts (arms)) Taking a = 300 mm c = 572 mm d = 244 mm

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Figure 2. 6–3 node Stewart platform.

Figure 3. Geometric layouts of upper and lower platforms.

We can now define position vectors representing the six base plate vertices in the Base frame [taking co-ordinate frame of reference at the centre of both the upper & lower platform] BP ti (i = 1 to 6), and the three tool plate vertices in the tool frame T P (i = 1 to 3) ti So we have, (vertices for tool platform (for the given values of platform dimension)): T

xt1 = 86.60 mm


285

yt1 = 150 mm

T

xt2 = 173.20 mm

T

yt2 = 0

T

xt3 = 86.6 mm

T

yt3 = −150 mm

T

zti = 0 (i = 1 to 3)

T

& similarly, (vertices for Base/stationery platform): xb1 = 400.68 mm

B

yb1 = 122 mm

B

xb2 = 94.69 mm

B

yb2 = 408 mm

B

xb3 = −306 mm

B

yb3 = −286 mm

B

xb4 = −94.69 mm

B

yb4 = 408 mm

B

xb5 = 400.68 mm

B

yb5 = −122 mm

B

zbi = 0 (i = 1 to 6)

B

If we take the operational frame to be at the end of a tool head (instead of being on the plane defined by the three ball joint centers) the representation of the three vertices is merely shifted in Z direction (below). This simply introduces a Z component for the three points, equal to the shift distance in the negative Z direction. In the case of the Hexel machine the shift distance is 305.1 mm nominally. T

zt1 = Tzt2 = Tzt3 = −305.1 mm

We now need to express the three tool plate vertices with respect to the Base frame to obtain BP ti (i = 1 to 3). To do this we need to define a homogeneous transformation matrix which represents the position and orientation of the Tool tie embedded with the Base frame, BTT . This transform will be a function of the x, y, z position values of the tool and three orientation parameters represented in the Base frame. A typical way of describing orientation as an ordered triplet utilizes angle set parameters α, β, γ , We may either use a fixed set or a Euler set. We will use a

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Figure 4. 6–3 node Stewart problem with tool head frame defined as the operational frame.

Z − Y − Z Euler set. The rotation matrix associated with this parameterization of orientation is, B T R(α, β, γ )

⎡

⎤

⎢ =⎣

⎥ − sin α ⎦ sin α cos β sin γ − sin β cos γ sin β sin γ + sin α cos β cos γ cos α cos β (1)

cos β cos γ + sin α sin β sin γ sin α sin β cos γ − cos β sin γ cos α sin β cos α sin γ

cos α cos γ

and, B T T (x, y, z, α, β, γ )

⎡

⎤ cos β cos γ + sin α sin β sin γ sin α sin β cos γ − cos β sin γ cos α sin β x ⎢ cos α sin γ cos α cos γ − sin α y⎥ ⎢ ⎥ =⎢ ⎥ ⎣sin α cos β sin γ − sin β cos γ sin β sin γ + sin α cos β cos γ cos α cos β z⎦ 0

0

0

1 (2)

Consequently, T Pti = B T T (x, y, z, α, β, γ ) Pti

B

(3)

Now a description of the strut lengths in obtained merely by taking the magnitudes of the vector differences between corresponding base plate vertices. 1 = (Bxt1 − Bxb1 )2 + (Byt1 − Byb1 )2 + Bzt1 2 2 = (Bxt1 − Bxb2 )2 + (Byt1 − Byb2 )2 + Bzt1 2

Effect of Geometrical Parameters on Workspace Envelope (Bxt2 − Bxb3 )2 + (Byt2 − Byb3 )2 + Bzt2 2 4 = (Bxt2 − Bxb4 )2 + (Byt2 − Byb4 )2 + Bzt2 2 5 = (Bxt3 − Bxb5 )2 + (Byt3 − Byb5 )2 + Bzt2 2 6 = (Bxt3 − Bxb6 )2 + (Byt3 − Byb6 )2 + Bzt2 2

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3 =

(4)

An inverse kinematic model of the Hexel hexapod has been generated in the Deneb Envision software (see figure 5). A MATLAB program based on this model is written. 2.2 Modeling an articulated tool head An articulated tool head offers a means of increasing the dexterity of a standard Stewart mechanism. In this case we will be characterizing a two-axis tool head. We will take the operational frame to be at the end of the two-axis tool head (see figure 6). Figure 7 describes the configuration of joints on the two-axis tool head. We can find the transform representing the tool head frame {T } in the ball plate frame, {∗ }. ∗ TT

= Rz (θ1 ) · D(120, 0, 125.1) · Rz (−90) · Rz (θ2 ) · D(−120, −180, 0) · Rz (90) ⎡ cos θ 1 cos θ 2

⎢ sin θ1 cos θ 2 ∗ ⎢ T T =⎣ − sin θ2 0

− sin θ 1 cos θ 1 0 0

cos θ 1 sin θ2 sin θ sin θ 2 cos θ2 0

(5) ⎤

(180 sin θ2 − 120 cos θ2 + 120) cos θ1 (180 sin θ2 − 120 cos θ2 + 120) sin θ1 ⎥ ⎥ (180 cos θ2 + 120 sin θ2 + 125.1) ⎦ 1

(6) Alternately, we can use the Denavit-Hartenberg convention. The DenavitHartenberg parameters are, Modified Denavit-Hartenberg approach. We can input these parameters into the Denavit-Hartenberg transform matrices, defined as, i−1

iT

= Rx (αi−1 ) · Dx (ai−1 ) · Rz (θi ) · Dz (di )

(7)

Multiplying out, ∗T T = ( 01 T )( 12 T )( 23 T ) yields the same result arrived at previously. The locations of the bifurcated ball joint centers in the tool head frame, Tpti (i = 1 to 3), are then, T

pti = T∗ T ∗pti

(8)

where, T ∗T

= ∗TT −1

(9)

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Figure 5. 6–3 node Stewart platform modeled in Deneb Envision TR (inverse kinematics).

Figure 6. 6–3 node Stewart platform with two-axis tool head frame. Table 1. Denavit-Hartenberg parameters. i 1 2 3

ai−1 Link Length 0 120 −120

αi−1 Link Twist 0◦ −90◦ 90◦

Di Joint Offset 125.1 0 180

θi Joint Angle θ1 θ2 0

and, ⎡∗ ⎤ xti ⎢∗ ⎥ ∗ ⎢ pti = ⎣ yti ⎥ ⎦ ∗z ti

(10)


289

Figure 7. Tool head kinematics.

Figure 8. Tool head kinematics (Denavit-Hartenberg).

are the constant position vectors representing the bifurcated ball joint centers in the ball plate frame {∗ }, calculated earlier as, ∗

xt2 = 86.60 mm

∗

yt1 = 150 mm

∗

xt2 = 173.20 mm

∗

yt2 = 0 mm

∗

xt3 = 86.6 mm

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Figure 9. Stewart platform with articulated tool head modeled in Deneb Envision TR. ∗

yt3 = −150 mm

∗

zti = 0 (i = 1 to 3)

A model of a two-axis articulated tool head has been generated in the Deneb Envision software (see figure 9). 2.3 Inverse Jacobian For a serial mechanism a closed form solution of the forward kinematics can be obtained, x = f1 (q1 , q2 , . . . , qn ) y = f2 (q1 , q2 , . . . , qn )

(11)

The application of the chain rule yields differentials of x, y . . . as functions of the differentials of qi (i = 1 to n). δx =

∂f 1 ∂f 1 δq1 + · · · + δqn ∂q1 ∂qn

δy =

∂f 2 ∂f 2 δq1 + · · · + δqn ∂q1 ∂qn

· ·

(12)

Dividing both sides by the differential time element & and expressing in matrix form yields, ⎡ ∂f 1 ∂f 1 ⎤ ⎡ ⎤ ⎡ ⎤ ∂q1 · · ∂qn x˙ q1 ˙ ⎢ ⎥ ⎢y˙ ⎥ ⎢ ∂f 2 · · ∂f 2 ⎥ ⎢ · ⎥ ⎢ ⎥ = ⎢ ∂q1 ∂qn ⎥ ∗ ⎢ ⎥ (13) ⎥ ⎣ · ⎦ ⎣·⎦ ⎢ ⎣ · · · · ⎦ qn ˙ · · · · ·


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Where the matrix is defined as the Jacobian, J. The Jacobian facilitates the mapping of the configuration since joint rate vector, ∼ q, into the Cartesian space velocity vector, V ⎡ ∂f 1 ⎤ ∂f 1 · · ∂qn ⎢ ∂q1 ⎥ ⎢ ∂f 2 ∂f 2 ⎥ · · J =⎢ (14) ∂qn ⎥ ⎢ ∂q1 ⎥ ⎣ · ⎦ · · · · · · · ν¯ = J q˙¯

(15)

For a parallel mechanism, like a Stewart platform, we have seen that a closed-form solution of the inverse kinematics can be obtained. 1 = f1 (x, y, z, α, β, γ ) 2 = f2 (x, y, z, α, β, γ ) · · 6 = f6 (x, y, z, α, β, γ )

(16)

The application of the chain rule yields differentials of I i(i = 1 to 6) as functions of the differentials of x, y, . . . , δ1 =

∂f 1 ∂f 1 ∂f 1 ∂f 1 ∂f 1 ∂f 1 δx + δy + δz + δα + δβ + δγ ∂x ∂y ∂z ∂α ∂β ∂γ

δ2 =


· · δ6 =


(17)

Dividing both sides by the differential time element δt and expressing in matrix form yields, ⎡ ∂f 1 ∂f 1 ∂f 1 ∂f 1 ∂f 1 ∂f 1 ⎤ ⎡ ⎤ ⎡ ⎤ ∂x ∂y ∂z ∂α ∂β ∂γ ˙ 1 x˙ ⎥ ⎢ ⎥ ⎢ ∂f 2 ∂f 2 ∂f 2 ∂f 2 ∂f 2 ∂f 2 ⎢˙ ⎥ ⎢ ⎢ y˙ ⎥ ⎥ ∂y ∂z ∂α ∂β ∂γ ∂x ⎢ 2⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ∂f 3 ∂f 3 ∂f 3 ∂f 3 ∂f 3 ∂f 3 ⎥ ⎥ ⎥ ⎢ ⎢˙3 ⎥ ⎢ ∂x ⎢ z˙ ⎥ ⎥ ∂y ∂z ∂α ∂β ∂γ ⎢ ⎥=⎢ ⎢ ⎥ ∗ (18) ⎢˙ ⎥ ⎢ ∂f 4 ∂f 4 ∂f 4 ∂f 4 ∂f 4 ∂f 4 ⎥ ⎥ ⎢ α˙ ⎥ ⎢ 4 ⎥ ⎢ ∂x ⎥ ⎢ ⎥ ∂y ∂z ∂α ∂β ∂γ ⎢˙ ⎥ ⎢ ⎢ ⎥ ⎣5 ⎦ ⎢ ∂f 5 ∂f 5 ∂f 5 ∂f 5 ∂f 5 ∂f 5 ⎥ ⎥ ⎣β˙ ⎦ ∂x ∂y ∂z ∂α ∂β ∂γ ⎦ ⎣ γ˙ ˙6 ∂f 6 ∂f 6 ∂f 6 ∂f 6 ∂f 6 ∂f 6 ∂x

∂y

∂z

∂α

∂β

∂γ

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Noting the standard Jacobian expression, ν¯ = J q˙¯ or ν¯ = J˙¯

(19)

˙¯ = J −1 ν¯

(20)

⎡ ∂f 1

J −1 =

∂x ⎢ ∂f 2 ⎢ ⎢ ∂x ⎢ ∂f 3 ⎢ ⎢ ∂x ⎢ ∂f 4 ⎢ ⎢ ∂x ⎢ ∂f 5 ⎢ ⎣ ∂x ∂f 6 ∂x

∂f 1 ∂y ∂f 2 ∂y ∂f 3 ∂y ∂f 4 ∂y ∂f 5 ∂y ∂f 6 ∂y

∂f 1 ∂z ∂f 2 ∂z ∂f 3 ∂z ∂f 4 ∂z ∂f 5 ∂z ∂f 6 ∂z

∂f 1 ∂α ∂f 2 ∂α ∂f 3 ∂α ∂f 4 ∂α ∂f 5 ∂α ∂f 6 ∂α

∂f 1 ∂β ∂f 2 ∂β ∂f 3 ∂β ∂f 4 ∂β ∂f 5 ∂β ∂f 6 ∂β

∂f 1 ⎤ ∂γ ∂f 2 ⎥ ⎥ ∂γ ⎥ ∂f 3 ⎥ ⎥ ∂γ ⎥ ∂f 4 ⎥ ⎥ ∂γ ⎥ ⎥ ∂f 5 ⎥ ∂γ ⎦ ∂f 6 ∂γ

(21)

The matrix above is the inverse of the conventional Jacobian. This facilitates the mapping of the Cartesian space velocity vector, v, into the confirmation space strut ¯ displacement rate vector, q¯ = . 2.4 Workspace envelope The workspace evaluation for the hexapod is divided into two parts. The first part seeks to find the workspace boundary associated with limits in strut travel. This is the lower workspace boundary. A program written in MATLAB was used to search points in a rectangular grid to determine if they were inside or outside the workspace boundary, based on one or more strut exceeding their travel limits. orientation of the tool head was maintained in the horizontal position (α = β = γ = 0). The same algorithm can be executed with other orientations of the tool head. A MATLAB Program based on this algorithm is written & executed. Block diagram of the algorithm is shown in figure K, and sample output is shown in figure 10. The second part seeks to find the workspace boundary determined by collisions between the struts and the top platform. This is the upper workspace boundary. A program written in MATLAB was used to search points in a rectangular grid to determine if they were inside or outside the workspace boundary, based on one or more struts exceeding an angle limit between the strut axis and the horizontal plane of the top platform. Here again, a MATLAB Program based on this algorithm is written & executed. A block diagram of the algorithm is shown in figure M, and sample output is shown in figure 11. Based on the above algorithms programs in MATLAB are utilized for seeing the effect of variation in dimensional parameters of fixed & mobile platforms of Stewart platform (values of a, b, & c). The corresponding effect on Lower & upper workspaces are visualized as shown below:


Figure 10. Lower workspace algorithm.

Figure 11. Upper workspace algorithm.

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Figure 12. Lower & upper space envelopes for a = 300 mm, c = 572 mm, d = 244 mm.

2.5 Stiffness Applying the principle of virtual work to an arbitrary mechanism allows us to equate work done in Cartesian space terms to work done in configuration space terms. Specifically, work in Cartesian terms is associated with a Cartesian force/torque vector, F applied at a mechanism’s tool frame and acting through an infinitesimal Cartesian displacement δp, Work in configuration space terms is associated with a configuration space force/torque vector, f , applied at a mechanism’s joints and acting through infinitesimal joint displacements, δq.


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Work is calculated as the dot product of a force/torque vector with a displacement vector. F¯ · δ p¯ = f¯ · δ q¯

(22)

F¯ T · δ p¯ = f¯T · δ q¯

(23)

F¯ T · J · δ q¯ = f¯T · δ q¯

(24)

F¯ T · J = f¯T

(25)

or

and nothing that δ p¯ = J · δ q, ¯

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Transposing both sides we have, (F¯ T · J )T = f¯ f¯ = J T F¯

(26) (27)

From this we conclude that “actuating” a mechanism with a force/torque vector, F , applied at the tool is equivalent to actuating that mechanism with a force/torque vector, f , applied at joints, when the same amount of virtual work is done in either case. For the hexapod (Stewart Platform) we can relate an applied force, F , at the tool to the resulting axial forces in the struts, f F = J −T f

(28)

Positive values of f indicate struts in tension. Negative values indicate struts in compression Given pure axial loading,


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Figure 15. Total workspace envelope.

ε = δi /i = σ/E = fi /AE fi = (AE/i )δi

(29) (30)

Where E is the elastic modulus of the strut material and A is the cross-sectional area of the strut. In matrix form, ⎡ ⎤ AE/1 0 0 0 0 0 ⎢ 0 0 0 0 0 ⎥ AE/2 ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥ 0 0 0 0 AE/ 3 ⎥ δ ¯ f¯ = ⎢ (31) ⎢ 0 0 0 AE/4 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎦ 0 0 0 AE/5 ⎣ 0 0 0 0 0 0 AE/6 where the matrix is defined as the strut space stiffness matrix, Ks Noting that δ = J −1 δp,

(32)

and, f = Ks J −1 δp F = J −T Ks J −1 δp Kc = J −T Ks J −1 F = Kc δp

(33)

Where Kc is the Cartesian space stiffness matrix. Setting up an eigenvalue problem we can find principal stiffness axes, η´ i and principal stiffness, λi F = Kc δp = λiδp (Kc − λi I6 )δp = 0 [Kc − λi I6 ] = 0; (i = 1 to 6)

(34)

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η´ 1 is in the direction of δpi where the above condition holds. A program has been written in MATLAB which performs the stiffness calculations described in this section. Sample output is shown in figure 16.

Figure 16. Continued.


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Figure 16. Stiffness sample output.

3. Conclusion The sample results obtained show us the effect of variations of dimensions of the 2 platforms of Parallel Mechanism on the workspace and on the correspond-

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ing stresses/forces induced in the struts. This brings us the observation that on decreasing the platform dimensions the workspace is increased but at the cost of higher stresses induced in the struts whereas on increasing the platform dimensions, the reverse is true, i.e. the workspace available is less but we have an advantage of reduced stresses induced in the struts. So we conclude that if our workspace requirement is limited, it is better to use large sized platforms as compared to small stationery & mobile platforms as stresses/forces induced in the struts are comparatively less. This, in turn, can help us in cost cutting as slightly cheaper materials with appropriate/required (less) strength can be used in design & manufacture of struts, which in turn is an important “design criteria”. In addition we have an advantage of having actuators of less capacity. Acknowledgment The concept & MATLAB programs utilized are based on research work on “Some Approaches for Modeling and Analysis of a Parallel Mechanism with Stewart Platform Architecture” [V. De Sapio, Sandia National Laboratories] SANDIA REPORT-SAND98-8242. UC-706, Printed May 1998. Author is greatly indebted to this work & continued guidance & support of my thesis guide “Associate Professor-K. N. Rustagi, ITM University.” References [1] V. De Sapio, [Sandia National Laboratories] SANDIA REPORT-SAND98-8242. UC-706, Printed May 1998. [2] D. Stewart, A Platform with Six Degrees of Freedom, Proceedings of the Institution of Mechanical Enginem, Vol. 180, No. 15, May 1965. [3] J. J. Craig, Introduction to Robotics: Mechanics and Control, Addison-Wesley, second edition, 1989. [4] K. Liu, M. Fitzgerald and F. L. Lewis, Kinematics Analysis of a Stewart Platform Manipulator, IEEE Transactions on Industrial Electronics, Vol. 40, No. 2, April 1993. [5] H. Yang and O. Masory, Vibration Analysis of Stewart-Platform, IEEE Transactions on Industrial Electronics, Vol. 40, No. 2, April 1993. [6] P. I. Corke, A Robotics Toolbox for MATLAB, IEEE Robotics & Automation Magazine, March 1996.