The Dynamics of Fixed Base and Free-Floating Robotic Manipulator

The Dynamics of Fixed Base and Free-Floating Robotic Manipulator

The Dynamics of Fixed Base and Free-Floating Robotic Manipulator

Ravindra Biradar1, M.B.Kiran2

1M.Tech (CIM) Student, Department of Mechanical Engineering, Dayananda Sagar College of Engineering, Bangalore-560078

2Professor, Department of Mechanical Engineering, Dayananda Sagar College of Engineering, Bangalore-560078

Abstract- Dynamic modeling means deriving equations that explicitly describes the relationship between force and motion in a system. To be able to control a robot manipulator as required by its operation, it is important to consider the dynamic model in design of the control algorithm and simulation of motion.

In general there are two approaches available; the Euler-Lagrange formulation and the Newton-Euler formulation. This thesis investigates the Lagrange-Euler method in detail. A complete derivation of the method is done using two degree of freedom serial manipulator with revolute joints in the presence and absence of gravitational force. The mathematical model for the Dynamic behaviour of the two degree of freedom manipulator is developed. The dynamic parameters of the system are estimated, and the validity of the resulting dynamic model is verified by several simulations, that describe the dynamic response of the manipulator to input actuator torques. A suggestion for future work is performing thorough dynamic parameter identification. An improved model can ultimately be implemented in the controller of the manipulator, and optimized for a specific job task.

Keywords-Torque, Dynamics, manipulator, force, motion

I-INTRODUCTION

During working cycle a manipulator must accelerate, move at constant speed and decelerate. The time varying position and orientation of the manipulator is termed as its dynamic behaviour. Time varying torques is applied at the joints to balance out internal and external force.

Dynamics is a huge field of study devoted to studying the forces required to cause motion [1], [2]. The dynamic motion of the manipulator arm in a robotic system is produced by the torques generated by the actuators. This relationship between the input torques and the time rates of change of the robot arm components configurations, represent the dynamic modeling of the robotic system which is concerned with the derivation of the equations of motion of the manipulator as a function of the forces and moments acting on. So, the dynamic modeling of a robot manipulator consists of finding the mapping between the forces exerted on the structures and the joint positions, velocities and accelerations [3]. A good model has to satisfy two conflicting objectives. It must include enough detail to represent the real behaviour of the robot with sufficient accuracy, and it should permit an efficient, stable evaluation not only of the model equations but also of their derivatives that are needed in optimization. The availability of the dynamic model is very useful for mechanical design of the structure, choice of actuators, determination of control strategies, and computer simulation manipulator motion.

A robot manipulator is basically a positioning device. To control the position we must know the dynamic properties of the manipulator in order to know how much force to exert on it to cause it to move: too little force and the manipulator is slow to react; too much force and the arm may crash into objects or oscillate about its desired position.

A significant amount of research has been reported concerning optimal trajectory planning using evolutionary methods for an industrial manipulator system. Research made the use of cubic spline curves to generate the trajectory between the intermediate points of the path. The problem of kinematics is solved for two-degree-of-freedom linear industrial manipulators. The Newton-Euler technique is used for the formulation of the dynamic equations of the manipulator. The effectiveness of the proposed method is verified through MATLAB simulations in [7], it has shown a trajectory generation for a two link robot manipulator without the use of the inverse Jacobian matrix, in which the cubic spline approach was employed.

Robotic manipulators will play important roles in future space missions. The control of such space manipulators poses planning and control problems not found in terrestrial fixed-base manipulators due to the dynamic coupling between space manipulators and their spacecraft. A number of control techniques for such systems have been proposed; these schemes can be classified in three categories. In the first category, spacecraft position and attitude are controlled by reaction jets to compensate for any manipulator dynamic forces exerted on the spacecraft. In this case, control laws for earth-bound manipulators can be used, but the utility of such systems may be limited because manipulator motions can both saturate the reaction jet system and consume relatively large amounts of attitude control fuel, limiting the useful life of the system [10], [11], and [14]. In the second category, the spacecraft attitude is controlled, although not its translation, by using reaction wheels or attitude control jets [12]. The control of these systems is somewhat more complicated than that of the first category, although a technique called the Virtual Manipulator (VM) can be used to simplify the problem [10], [11], [12], [13], and [14]. The third proposed category assumes a free-floating system in order to conserve fuel or electrical power [10], [15], and [16]. Such a system permits the spacecraft to move freely in response to manipulator motions. These too can be modelled using the VM approach [15]. Work was done on control algorithm for free-floating space manipulators in [14]. It was shown that nearly and algorithm that can be applied to conventional

fixed-based manipulator can be directly applied to free- floating manipulators, with few additional conditions.

Fig. 1 A 2-DOF planar articulated (RR) arm with fixed base

These include the measurement or estimation of a spacecrafts orientation and the avoidance of dynamic singularities.

Significant amount of work is done on the Dynamic Control of a Space Robot system with no Thrust Jets controlled base [22]. In this paper he discusses dynamic control of a free-flying space robot system where the base attitude is not controlled by thrust jets. Without external forces and moments, the system is governed by linear and angular momentum conservation laws. he first derive the system dynamic formulations in joint space and in inertia space, based on Lagrangian dynamics. Then discuss the fact that dynamics of a space robot system cannot be linearly parameterized, as opposed to the case of a fixed-based robot.

II-BASIC FORMULATION

In this paper we are considering two cases, one for fixed base manipulator and another for the free-floating manipulator, the end-effector is made to move in a circular trajectory figure 1 shows fixed base manipulator, figure 2 shows free-floating manipulator. The time varying torques is compared for both.

1. Assumptions

   In this paper author assume a simple model of a robot satellite which has an articulated manipulator system, in order to clarify the point at issue, they make the following assumptions [19]

   1. It is a 2-DOF manipulator with two links, articulated robotic arm.

   2. Control technique used is FREE-FLOATING.

   3. Mass of the spacecraft is assumed to very high when compared with the mass of the links.

   4. Gravitational force is zero.

   5. Robot arms are not affeced by the friction and disturbance.

   6. There are no mechanical restriction nor external forces and torques, so that momentum conservation,

      and equilibrium of forces and moments, strictly hold true during the operation.

2. Nomenclature

   The symbols are defined as

   torque

   F force

   , q joint angle

   angular velocity

   , angular acceleration

   L Lagrangian

   , L1 length of link 1

   , L2 length of link 2

   mass of link 1 and link 2 linear velocity in x-direction linear velocity in y-direction linear acceleration in x-axis

   linear acceleration in y-direction

   Fig.2 Model of a 2-DOF planar robot without fixed base

   Fig.3 Solving for the joint angles of a two-link planar arm

   b =

   b = –

   =

   Fig.4 Coordinate frames for two-link planar robot

3. fundamental equation

   Inverse kinematics of manipulators, in order to command the end-effector to move to a particular position we need the angles of both the links

   = b

   Trajectory generation, a common way of causing a manipulator to move from here to there in a smooth, controlled fashion is to cause each joint to move as specified by a smooth function of time. Commonly, each joint starts and ends its motion at the same time, so that manipulator motion appears coordinated. Exactly how to compute these motion functions is the problem of trajectory generation. Often, a path is described not only by a desired destination but also by some intermediate locations, or via points, through which the manipulator must pass en route to the destination. In such instances the term spline is sometimes used to refer to a smooth function that passes through a set of via points. In order to force the end-effector to follow a

   cos : = D.

   sin( =

   Velocity Kinematics of manipulator, coordinates (x, y) of the tool are expressed in this coordinate frame as

   x = x2 = +

   y = +

   (1)

   (2)

   (3)

   (4)

   (5)

   (6)

   straight line (or other geometric shape) through space, the desired motion must be converted to an equivalent set of joint motions [2].

   Equations of Motion, The purpose of a robot manipulator is to position and interface its end-effector with the working object. The equations of motion are important to consider in the design of robots, as well as in simulation and animation, and in the design of control algorithms. Equation of motion can be described by a set of differential or difference equations. The equation set consists of two parts, the kinematics equations and the dynamic equation. Robot arm kinematics deals with the geometry of robot arm motion as a function of time (position, velocity, and acceleration) without regards to the forces and moments that cause it.

4. Dynamic Equations

   Dynamics of robot is the study of motion with regard to forces (the study of the relationship between forces/torques

   In order to follow a contour at constant velocity, or at any prescribed velocity, we must know the relationship between the velocity of the tool and the joint velocities. In this case we can differentiate Equations (5) and (6) to obtain

   and motion). A dynamic analysis of a manipulator is useful for the following purposes:

   1. It determines the joint forces and torques required to produce specified end-effector motions (the direct dynamic problem).

   2. It produces a mathematical model which simulates

      = .

      = .

      dX = J(q)d

      Inverse Velocity and Acceleration,

      =

      = +

      (7)

      (8)

      (9)

      (10)

      the motion of the manipulator under various loading conditions (the inverse dynamic problem) and/or control schemes.

   3. It provides a dynamic model for use in the control of the actual manipulator.

III-DYNAMIC FORMULATION

In this section we derive a general set of differential equations that describe the time evolution of mechanical

systems. These are called the Euler-Lagrange equations of motion.

Lagrangian formulation, which describes the behavior of a dynamic system in terms of work and energy stored in the system rather than of forces and moments of the individual members involved. The constraint forces involved in the system are automatically eliminated in the formulation

K1 = m1 L12 12

And the potential energy is given by

P1 = m1 gL1

(15)

(16)

of Lagrangian dynamic equations. The closed-form dynamic equations can be derived systematically in any coordinate system.

For control design purposes, it is necessary to have a

Where g is the magnitude of acceleration due to gravity in the

negative y-axis direction.

For link 2

mathematical model that reveals the dynamical behavior of a system; we derive the dynamical equations of motion for a robot manipulator. Our approach is to derive the kinetic and

x2 = L1 cos1 + L2 cos ( + ) y2 = L2 sin1 + L2 sin ( + )

(17)

(18)

potential energy of the manipulator and then use Lagranges equations of motion.

1. Lagranges Equations of Motion

   Lagranges equation of motion for a conservative system are given by

   Differentiating equation (17) and (18) will give velocity component of link 2 and , from these component, the square of the magnitude of velocity of the end of link 2 is

   =+

   (11) The kinetic energy of link 2 with w2= + and I2= m2 L22

   is

   where q is an n-vector of generalized coordinates qi, is an n- vector of generalized forces i, and the Lagrangian is the difference between the kinetic and potential energies

   K2 = m2v22 + I2 w22

   The potential energy of link 2

   (19)

   L=K-P.

   (12)

   P2 = m2 g L1 S1 + m2 g L2 S12

   (20)

   In our usage, q will be the joint-variable vector, consisting of joint angles i, (in degrees or radians) and joint offsets di (in meters). Then is a vector that has components ni of torque (Newton-meters) corresponding to the joint angles, and fi of force (Newtons) corresponding to the joint offsets. We shall use Lagranges equation to derive the general robot arm dynamics. Let us first get a feel for what is going on by considering some examples.

2. Two degree of freedom manipulator with fixed base

   The Lagrangian requires kinetic and potential energies of the manipulator. The kinetic energy of a rigid body (a link) is given by

   Where C1 , S1 , C12 =cos + ) and S12

   =sin + )

   The Lagrangian L= K – P = K1 + K2 P1 P2 is obtained by Substituting values of equation (15), (16), (19) and (20) in above equation of Lagrangian give the value of L. The derivation is in detail in ([2], [3])

   The Lagrangian-Euler formulation for link1 equation (11) gives the torque at joint 1 as

   (21)

   For link 2 it gives torque at the joint 2 as

   K= mv 2 + Iw2

   (13)

   (22)

   Where v is the linear velocity, w is the angular velocity, m is he mass, and I is the moment of inertia of the rigid body at its centre of mass. linear velocity v1 L1 1, angular velocity w1= 1, moment of inertia I1=

   The Lagrangian value is differentiated w r t and and substituted in equation (21) and differentiated w r t and and substituted in equation (22) which gives

   =H11 +H12 +C1+G1

   (23)

   K1 = m1v12 + I1 w12

   After substituting the values

   (14)

   Where

   H11=

   =H21

   +H22

   +C2+G2

   H12=H21= H22= m2

   C1=- L1L2S2 – C2=

   G1= g

   G2= g

   These coefficients are defined as iterations

   Mii =Hii=effective inertia, Mij =Hij=effective coupling inertia,

   Ci=centrifugal and Coriolis acceleration forces The general form of equation is

   1. The coefficient represents the velocity induced reaction torque at joint i, the first index. The indices j and k are related to velocities of joint j and joint k, whose dynamic interplay induces a reaction torque at joint i. In particular, a term of the orm is the centrifugal force acting at joint i due to velocity at joint j and a term of the form is shown as the Coriolis force acting at joint i due to velocities at joint j and k. In particular

      Coriolis force coefficients generated by the velocities of joint j and joint k and felt at joint i. Coriolis force acting at joint i due to velocities at joint j and joint k is a combination term of +

      .

      Centrifugal force coefficient at joint i

      + +G=

      1. – n x n mass matrix

         (24)

         generated due to angular velocity at joint j. the centrifugal force acting at joint i due to velocity at joint j is given by

      2. – n x n matrix and is an n x 1 vector of centripetal and Coriolis terms

      3. G- n x 1 vector of gravity terms, and

      4. n x 1 vector of joint torque or forces

         1. It can also be written as

            because the Coriolis force acts at joint i due to velocities of joints j and k and suffix order does not matter.

            , Coriolis force at joint i is not due to joint velocity itself

   2. The term involving gravity g represents the gravity

   + + =

   (25)

   generated moment at joint i. The coefficient Gi is the gravity loading force as joint i due to the links i to n. The

   this is the final form of the robot dynamical equation we have been seeking.

   The units of elements of M(q) corresponding to revolute joint variables qi=i are kg-m2. The units of the elements of M(q) corresponding to prismatic joint variables qi=di are kilograms. The units of elements of V(q) and G(q) corresponding to revolute joint variables are kg-m2/s2. The units of elements of V(q ) and G(q) corresponding to prismatic joint variables are kg-m/s2.

   The final EOM (dynamic model) is

   = + +

   For i= 1, 2,, n

   The physical meaning of above equation is as follows.

   1. The coefficients of the terms in these equations represent inertia. It is known as effective inertia when acceleration of joint i cause a torque at joint i. and coupling inertia when acceleration at joint j causes a torque at joint i. In other words, the coefficient is related to acceleration of the joint and represents inertia loading of the actuator, In summary

      = effective inertia at the joint i where the driving torque acts.

      = coupling inertia between joint i and joint j. It is reaction torque at joint i induced by acceleration at joint j. Reverse applies equally with torque as torque at joint j due to acceleration of joint i.

      Since Tr(A)=Tr(AT), it can be shown that

      gravity term is a function of the current position.

      4. = generalized force applied at joint i due to motion of links.

      5. = joint displacement of joint i.

      6. = velocity of joint i.

      7. =acceleration of joint i.

      In dynamic model equation inertia and gravity terms are significant in manipulator control as they affect positioning accuracy and servo stability, which in turn determine the repeatability of the manipulator. The Coriolis and centrifugal forces are significant for high-speed motion of the manipulator.

3. Two degree of freedom manipulator with no fixed base (Free-floating manipulator)

We have considered smaller systems installed on a free-floating unmanned satellite, considered a manipulator placed on a satellite having its mass and inertia much larger than the manipulator, and considered it similar to fixed based manipulator without considering the gravitational effect. [16] Any movement of the manipulator on satellite will not disturb the position or attitude of the satellite.

The kinetic energy both the links remains same as the one for the fixed base manipulator but the potential energy for both the link will be zero.

K1 = m1 L12 12

K2 = m2v22 + I2 w22 P1=0

P2=0

we find the orientation for every time, consider the cubic polynomial equation [2],

q(t) = a0 + a1t + a2t2 + a3t3

deriving equation (28) will give the joint velocity, and deriving again will give joint acceleration.

The Lagrangian-Euler formulation for link1 gives the torque at joint 1 is given by equation (21) and for link 2 it gives torque at the joint 2 is given by equation (22).for derivation

refer Appendix 1.

(t) = a1 +2a2t + 3a3t2

(t) = 2a2 + 6a3t

(29)

(30)

The Lagrangian value is differentiated w.r.t and and substituted in equation (21) and differentiated wrt and and substituted in equation (22) which gives

To solve these equations we require atleast four constraints, we consider initial and final velocity as zero, and starting and goal-point values.

=H11 +H12 +C1

=H21 +H22 +C2

The general form of equation is

+ =

H11= H12=H21= H22= m2

C1= – L1L2S2 – C2=

IV-CASE STUDY

(26)

(27)

q(0) = qs q(tg) = qg (0) = 0

(tg) = 0

Where qs, qg starting and goal-point position and ts and tg are initial and final time.

Applying constraints in equation (28), (29) and (30) gives following set of equation,

a0 = 0

a1 = 0

a2 =

Let us consider the Inverse kinematic problem, given two link manipulator as shown in the figure 1, having mass, inertia and geometry as shown in table 1, chosen circular trajectory of end-effector as x=a + , y=b + , where r = 0.2, a = 1.2, b = 1.2 and makes this circular rotation from 0 to 2 in 10 seconds.

We will find the toque (t) applied at both the joints at different time intervals

a3 = –

the value of a2 and a3 after substituting the values are

a2 = 0.1885

a3 = -0.0125

Equation (28) becomes

Table 1

2R Manipulator parameters[8]

Link	Length (m)	Mass (kg)	C.G (m)	Inertia (kg )
1	1.0	12.456	.773	1.042
2	1.0	12.456	.583	1.042

Let us consider the point to point trajectory tracking where end-effector moves from 0 to 360 degrees in 10 seconds. Here

q(t) = 0.1885(t2) 0.0125(t3)

calculate q(t) for every change of time and substitute in given trajectory

x=a + , y=b + , considering q(t) = , we get x

and y values

The fig.4 generated by MatLab shows the trajectory taken by end-effector.

Now consider the two links and find the initial angle using equation (1) (4). Which gives two cases elbow-up and elbow-down position,

Case 1: = 0.3109 and = 0.7953.

(28)

Case 2: = 1.1056 and = -0.7953

We consider case 2 condition, and we get the following results as shown in the figures. Using the values of x and y find the value of ie ] which actually gives the linear velocity.

x 10-3

6

Angular velocity(rad/sec2)

4

2

0

plot of joint angular velocities for 2R Manipulator

, =

We know the Jacobian and using equation (9) find the value of or , i.e.

=

Fig.6 gives the angular velocity path

Find s (9) to (10)

-2

thetadot1 thetadot2

-4

-6

-8

0 50 100 150 200 250 300 350 400 450 500

No of iterations or time( 10 seconds)

Fig.7 computed using inverse Jacobian

= J(q) +

Where is acceleration which is change in velocity.

x 10-4

Angular Acceleration(rad/sec2)

1.5

1

0.5

plot for joint angular Accelerations for 2R manipulator

Angular velocity 1

Angular velocity 2

= 0

2

1.5

y-axis

1

A 2R manipulator

-0.5

-1

-1.5

0 50 100 150 200 250 300 350 400 450 500

No of iterations or time (10 seconds)

Fig.8 Computed and

0.5

0

0 0.5 1 x-axis

1.5 2

Using equation (23) we find the values of o both the joints.

= (26.2128+14.52*cos( + (5.275+7.26*cos( ) –

2

1.5

Angle(rad)

1

0.5

0

-0.5

-1

Fig.5 Path taken by the end-effector

plot of joints Angles for a 2R Manipulator

theta1 theta2

(7.26*sin(**)-(7.26*sin(*2)+9.81*(22.084 cos( +7.26*C12).

= (5.275+7.26*cos( ) + (7.26*

sin( * 2) +(71.238*C12).

The torque is also calculated without the gravitational force, using equation (26) below shows the equations obtained

= + ) –

(7.26*sin(**)-(7.26*sin(*2)

-1.5

0

50 100 150 200 250 300 350 400 450 500

No of itterations or time( 10 seconds )

Fig.6 Computed using inverse kinematics

= (5.275+7.26*cos( ) + (7.26*

sin( * 2)

240

220

200

Torque(Nm)

180

160

140

120

100

80

60

plot of joint toque for 2R Manipulator

Torque 1

Torque 2

1. The Lagrange-Euler method is used for building the Dynamic equation

2. From the case studied we found that the final angle of both the joints are same as that of initial angle

3. It was found that the initial and final velocities were zero which are as per our requirement. As the velocity was increasing linearly and then again decreasing linearly the acceleration was also varying which also showed the expected results.

4. Torque is much varying when compared in the presence and absence of gravitational force. High torque is required for the same body at particular time to move particular point in the presence of gravitational force

0 50 100 150 200 250 300 350 400 450 500

No of iterations or time( 10 seconds)

Fig.9 Computed and

where less is required in the absence.

These suggested that the key to solving the problem of

1.5

1

x 10-3

plot of joint torques without gravitational force

torque1 torque2

planning and control of space robotic system lies in understanding the fundamental dynamic behavior of the systems.

0.5

Torque(Nm)

0

-0.5

-1

-1.5

-2

-2.5

0 50 100 150 200 250 300 350 400 450 500

No of iterations or time(10 seconds)

Fig.10 Toque without gravitational force

plot of joint torque when only gravitational force is considered

240

torque1 torque2

220

200

torque(Nm)

180

160

140

120

100

80

60

0 50 100 150 200 250 300 350 400 450 500

No of iterations

Fig.11 Effect of gravitational force on joint torques

IV-CONCLUSION AND SCOPE OF FUTURE WORK

1. Conclusion

   1. In this paper the Dynamic model (EOM) is derived for the two degree of freedom serial manipulator. Two cases were considered, in one the manipulator is under the influence of gravitational force and in another gravitational force was not considered as it is freely floating in the space.

   2. We have also analyzed the behavior of the torques under the influence of various factors like centrifugal force, Coriolis force and inertia.

2. Scope of future work

1. We hope that the results encourage the development of more effective control algorithm for free-floating space manipulator. Further work can be done on the kinematic and dynamics of space manipulators.

2. We can propose generalized Jacobian matrix for space manipulator, taking dynamical interactions between the arm and satellite into account, it is just an initial step for future work on space robots.

3. The dynamic equations obtained can be used to study dynamic behavior of free-flying and free-floating space manipulator study in detail during pre-impact and post impact operations.

4. A possible extension for this work would be implementation of a dynamic model for a 6 DOF space robots.

APPENDIX I

Derivation without the gravitational force, the Lagrangian L= K – P = K1 + K2 P1 P2 is obtained by Substituting values of equation (15), (16), (19) and (20) in above equation

.

L=K1+K2

L =

(A1)

Lagrange-Euler formulation for link1 is given by

(A2)

Diff (A1) w.r.t and

Diff equation A3 w.r.t time

Substituting equation A3 and A4 in A2 will give

Similarly for joint 2

(A3)

(A4)

(A5)

(A6)

(A7)

[12]. Dubowsky, S., and Torres, M., Path Planning for Space Manipulators to Minimize the Use of Attitude Control Fuel, Proc. of the International Symposium on Artificial Intelligence, Robotics and Automation in Space (ISAIRAS), November 1990.

[13]. S. Dubowsky, Advanced Methods for the Dynamic Control of High Performance Robotic Devices and Manipulators with Potential for Applications in Space. NASA Research Grant NAG1-489, June 30, 1984 to June 30, 1987.

[14]. Evangelos Papadopoulos and Steven Dubowsky, On the Nature of Control Algorithm for Free-Floating Space Manipulators, IEEE Transactions on Robotics and Automation, vol. 7, No. 6, December 1991.

[15]. Vafa Z and Dubowsky S., The Kinematics and Dynamics of Space Manipulators: The Virtual Manipulator Approach, The International Journal of Robotics Research, Vol. 9, No. 4, August 1990, pp. 3-21.

[16]. Umetani.Y and Yoshida. K, Experimental Study on Two Dimensional Free-Flying Robot Satellite Model, Proc. of the NASA Conference on Space Telerobotics, Pasadena, CA, January 1989.

[17]. Alex Ellery, Space Robotics: Space-based Manipulators, Surrey Space Centre, Surrey, United Kingdom, pp. 213-216.

[18]. Kyong-Sok Chang, Efficient Algorithms for Articulated Branching Mechanisms: Dynamic Modeling, Control and Simulation, Stanford University, March 2000.

[19]. Umetani.Y and Yoshida. K, Resolved Motion Rate Control of space Manipulator with generalized Jacobian Matrix, IEEE Transaction on robotic and automation, Vol. 5,No. 3, June 1989.

[20]. Bruno Zappa, G.Legnani and Riccardo Adamini, Path Planning Of Free-Flying Space Manipulators: An Exact Solution for Polar Robots, Science Direct, Mechanism and Machine theory, 40 (2005), 806-820.

[21]. Sabri Cetinkune and Wayne J. Book, Flexibility Effects on the Control System Performance of Large Scale Robotic Manipulators, The Journal of the Astronautical Sciences, Vol. 38, No.4, October-December 1990, pp. 531-556. [22]. Yangsheng Xu, Heung-Yeung Shum, Dynamic Control of A Space Robot System with No Thrust Jets Controlled Base, CMU-RI-TR- 9 1-33, August 1991.

[23]. Mathworks, MATLAB – The Language of Technical Computing, Last accessed July 2, 2012, http://www.mathworks.com/products/matlab/.

Substituting equation (A6) and (A7) in equation in (A5)

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