Mechanism and Machine Theory 64 (2013) 251–261

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Mechanism and Machine Theory

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Dynamic modeling and trajectory planning for a mobilespherical robot with a 3Dof inner mechanism

Mahmood Reza Azizi⁎, Davood NaderiMechanical Engineering Department, Engineering Faculty, Bu Ali Sina University, Hamedan, Iran

a r t i c l e i n f o

⁎ Corresponding author. Tel.: +98 9171920384.E-mail addresses: mhr_azizi@yahoo.com (M.R. Azi

0094-114X/$ – see front matter © 2013 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.mechmachtheory.2013.02.

a b s t r a c t

Article history:Received 16 July 2011Received in revised form 18 October 2012Accepted 13 February 2013Available online 15 March 2013

A spherical robot consists of a spherical shell and an inner mechanism. In this paper, a newinner mechanism with three independent actuators is presented. The robot motion is based onangular momentum conservation principle and the system is nonholonomic. The kinematicand dynamic equations of the whole system are derived by using Euler parameters and Kane'smethod. The singularity does not occur in the equations of inverse dynamic because of usingEuler parameters instead of Euler angles. By adding one degree of freedom to the innermechanism, the trajectory planning of the mobile spherical robot is possible for every desiredtrajectory. The trajectory planning is done for linear and circular arc trajectories. The robotstarts its motion from first point with zero velocity to the end point on the desired trajectoryand stops at the end point. The computer simulation of robot motion is done. The comparisonof the numerical solution with computer simulation shows good agreement.

© 2013 Elsevier Ltd. All rights reserved.

Keywords:Spherical robotKane's methodTrajectory planningEuler parameters

1. Introduction

Nowadays the mobile robots are used in industry, scientific researches, space exploration and military. The locomotion isgenerated in various ways like sliding, creeping, rolling, jumping and using legs or wheels. The stability of mobile robots is animportant issue in locomotion. A spherical robot always has stable locomotion on a flat surface. There is no risk of tip over orrollover for a spherical robot because of its external shape. The spherical shell protects the inner mechanism, control system andbatteries. The spherical robot can continue to move after collision with obstacles or falling from height [1]. The inner mechanismsof spherical robots generally generate the motion in two different ways: 1 — Using conservation principle of angular momentum.2— Changing the position of mass center of the whole system. In [2] a spherical robot and its kinematic and dynamic equations ofmotion are presented. The motion of sphere is generated by moving a vehicle on the inner surface of the spherical shell. Themobility of the spherical robot may generate by an unbalancing inner mechanism. The inner mechanism has a wheel, which rollson the inner surface of the sphere [3]. In [4] a pendulum is used as an inner mechanism. The pendulum rotates by two actuatorsabout two perpendicular axes. The position of mass center of whole system changes by changing the position of the pendulum. Amechanism for generating motion that presented in [5] leads the system to be holonomic. The presented mechanism decreasesthe maneuverability of the robot. In [6,7] unbalanced masses are used as an inner mechanism to generate the motion of the robot.The trajectory planning has been done for straight line and circular trajectory. Four unbalanced masses are used to generate themotion of the robot and an algorithmic motion planning is presented in [8]. Equations of motion of the spherical robot that ispresented in [4] are derived using simplified Boltzmann–Hamel equation and trajectory planning of the robot is done on linearand circular trajectory [9]. A PI controller is designed to control the robot motion on straight line trajectory and PI controllerparameters are optimized using genetic algorithm [10]. In [11] a mechanism with two pendulums is used for changing the center

zi), d_naderi@basu.ac.ir (D. Naderi).

ll rights reserved.004

Nomenclature

ijq Rotation quaternion of frame {i} with respect to frame {j}jiq

i Quaternion conjugatejω→

if g Angular velocity vector of frame {i} described in frame {j}jα→

if g Angular acceleration vector of frame {i} described in frame {j}iV→

I Velocity vector of contact point of the spherei r→

G=I Position vector of geometric center of the sphere relative to contact point described in frame {i}jV→

i Linear velocity vector of mass center of the part i described in frame {j}ja→

i Linear acceleration vector of mass center of the part i described in frame {j}jω→

i Angular velocity vector of part i described in frame {j}jα→

i Angular acceleration vector of part i described in frame {j}θx,θy,θz Rotation angle of rotors with respect to spherical shellΩx,Ωy,Ωz Angular velocity of rotors of actuators relative to spherical shell_Ωx; _Ωy; _Ωz Angular acceleration of rotors of actuators relative to spherical shellx,y Components of position vector of center of the sphere with respect to frame {1}_x; _y Components of velocity vector of center of the sphere with respect to frame {1}€x; €y Components of acceleration vector of center of the sphere with respect to frame {1}u→

Generalized velocity vectormi Mass of the part i of the robotj ˜R→

i Inertia force vector of part i described in frame {j}j ˜M→

i Inertia moment vector of part i described in frame {j}jR→

i Impressed force vector of part i described in frame {j}jM→

i Impressed moment vector of part i described in frame {j}jM→

rel i Impressed moment of actuator i described in frame {j}Tx,Ty,Tz Motors torquekV→j

i jth Partial linear velocity vector of part i described in frame {k}kω→j

i jth Partial angular velocity vector of part i described in frame {k}kω→j

rel i jth partial relative angular velocity of part i with respect to sphere described in frame {k}Fi ith generalized forceFi∗ ith generalized inertia force

RS Radius of the spherical shelljIi Moment of inertia matrix of part i about axes of frame {j}j l→

i Position vector of mass center of part i relative to center of the sphere in frame {j}RC Radius of the circular trajectoryϕ Traversed angle by position vector of sphere with respect to center of circular trajectoryV Magnitude of velocity of geometric center of the spherical shell_V Magnitude of acceleration of geometric center of the spherical shell

252 M.R. Azizi, D. Naderi / Mechanism and Machine Theory 64 (2013) 251–261

of gravity of robot. These pendulums can also move along their axis of rotation. Trajectory planning of this robot is studied onstraight line and circular trajectory. The presented robot in [12] has two actuators and its motion is generated based onconservation principle of angular momentum. Design and experimental results and motion planning of the spherical robot arepresented. In [13] the motion planning and control and in [14] the kinematic equations are derived using Euler parameters for thespherical robot presented in [12].

As mentioned, many investigations have been done on spherical robot recently but because of complexity of mathematicalmodel, fewer researchers studied trajectory planning of this kind of robot. Therefore in this paper, a spherical robot with a newinner mechanism is presented. The inner mechanism consists of three independent actuators, in which their axes of rotations areorthogonal to each other. Using three actuators simplify the robot motion planning. This mechanism enables the robot to track alldesired trajectory with good precision.

2. Spherical robot and its inner mechanism

A simple configuration of three actuators as shown in Fig. 1 has been proposed as the inner mechanism. The spherical robotconsists of spherical shell (part 1), three rotors (part 5, 6, and 7) and three counter weights to balance the rotor weights (part 2, 3,4). Generally, the spherical robot's motion is generated by two actuators. The inner mechanism could be improved by increasingthe number of actuators. In this case, it is assumed that all the inner parts of the robot are assembled such that the mass center ofrobot coincides with the geometrical center of sphere and the dynamic equations of robot are simplified. This feature makes it

Fig 1. The spherical shell and the configuration of actuators.

253M.R. Azizi, D. Naderi / Mechanism and Machine Theory 64 (2013) 251–261

possible to stop the robot at any desired point and omits the vibration behavior of the system. For rolling the sphere on thesurface, the two components of angular velocity about two perpendicular axes parallel to the ground is needed. If the innermechanism consists of two actuators and one of the rotation axes of actuators coincides with the vertical axis, the motionplanning is failed. The case is eliminated by using three actuators in inner mechanism.

3. Kinematic modeling

The three coordinates are defined as illustrated in Fig. 2. The frame {1} (x1y1z1) is the inertial reference frame. The origin of theframe {2} (x2y2z2) coincides with the geometric center of the sphere and always remains parallel to frame {1}. The frame {3}(x3y3z3) is the body coordinate and is located to the center of the sphere and rotates with it. The Euler parameters are used todescribe the rotation of the frame {3} relative to the frame {2} [15–18]. The quaternion of Euler parameters are:

23q ¼ 1

3q ¼ q0q→

� �¼ q0 þ q1iþ q2jþ q3k ð1Þ

23q��� ���2 ¼ q20 þ q21 þ q22 þ q23 ¼ 1 ð2Þ

02ω→

f3g

" #¼ 22

3 _q23q ð3Þ

2α→

3f g ¼ddt

2ω 3f gx� �

i2 þddt

2ω 3f gy� �

j2 þddt

2ω 3f gz� �

k2 ð4Þ

The angular velocity and angular acceleration of spherical shell are determined by using Eqs. (3) and (4). The angular velocityand acceleration are as follow:

2 ω→

1 ¼ 2ω→

3f g2α→

1 ¼ 2α→

3f g

ð5Þ

2ω1x2ω1y2ω1z

264

375 ¼

−2q1 2q0 −2q3 2q2−2q2 2q3 2q0 −2q1−2q3 −2q2 2q1 2q0

24

35

_q0_q1_q2_q3

26664

37775 ð6Þ

The velocity and acceleration of the geometric center of the sphere are:

2V→

1 ¼ 2V→

I þ 2ω→

1 � 2 r→

G=I¼ 2ω→

1 � Rs k2 ð7Þ

2 3

Fig. 2. The spherical robot and the related frames.

254 M.R. Azizi, D. Naderi / Mechanism and Machine Theory 64 (2013) 251–261

_x_y

� �¼ 2Rs

−q2 q3 q0 −q1q1 −q0 q3 −q2

� � _q0_q1_q2_q3

664 775 ð8Þ

2a→

1 ¼ ddt

2V1x

� �i2 þ

ddt

2V1y

� �j2 þ

ddt

2V1z

� �k2 ð9Þ

3ω→

1 ¼ 32q

02ω→

1

" #32q ð10Þ

3α→

1 ¼ 32q

02 α→

1

� �32q ð11Þ

Hence, the angular velocity and acceleration of counter weights are equal to angular velocity of spherical shell, we can writethat:

3 ω→

i ¼ 3ω→

1 i ¼ 2;3;4;ð Þ3α→

i ¼ 3α→

1 i ¼ 2;3;4ð Þð12Þ

The angular velocities and accelerations of the rotors of actuators with respect to frame {1}:

3 ω→

5 ¼ 3ω→

1 þΩx i33ω→

6 ¼ 3ω→

1 þΩy j33ω→

7 ¼ 3ω→

1 þΩz k3

ð13Þ

3 α→

5 ¼ 3α→

1 þΩ:

x i3 þ 3ω→

1 �Ωx i33α→

6 ¼ 3α→

1 þΩ:

y j3 þ 3ω→

1 �Ωy j33α→

7 ¼ 3α→

1 þΩ:

z k3 þ 3ω→

1 �Ωz k3

ð14Þ

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The linear velocity and acceleration of mass center of the rotors and counter weights are:

2 V→

i ¼ 2V→

1 þ 23q

3ω→

1 � 3 l→

i

� �23q i ¼ 2;3;4;5;6;7ð Þ ð15Þ

2 a→

i ¼ddt

2Vix Þ i2 þddt

2Viy

� �j2 þ

ddt

2Viz

� �k2 i ¼ 2;3;4;5;6;7ð Þ

�ð16Þ

4. Dynamic modeling

The nine parameters [q0 q1 q2 q3 x y θx θy θz] describe the configuration of the system, but these are not independent of eachother. Eq. (8) expresses two nonholonomic constraints and Eq. (2) expresses one holonomic constraint, therefore the system isnonholonomic and according to Kane's definition of degree of freedom [19], has six DOF. Kane's method is used to derive therobot differential equations of motion. The six generalized velocities are selected in the following form:

u→¼ ⌊ _q1 _q2 _q3 Ωx Ωy Ωz⌋ ð17Þ

In this way, the forces and moments between counter weights and sphere are not impressed forces and do not appear in theequations of motion. Local coordinate frames {4}, {5} and {6} are attached to the rotors at their center of gravity. These frames areparallel to coordinate frame {3} and have one axis aligned with the axis of symmetry of the rotors. The inertia forces andmomentsof these parts are:

2 ˜R→

i ¼ −mi2a→

i i ¼ 1;2;3;4;5;6;7ð Þ ð18Þ

3 ˜M→

i ¼ −3Ii3α→

i−3ω→

i � 3Ii3ω→

i

�i ¼ 1;2;3;4ð Þ

�ð19Þ

The inertia moments of rotors are:

3 ˜M→

5 ¼ −3I53α→

1 þ Ωx i3� �

−3ω→

1 � 3I53ω→

5

� �3 ˜M→

6 ¼ −3I63α→

1 þ Ωy j3� �

−3ω→

1 � 3I63ω→

6

� �3 ˜M→

7 ¼ −3I73α→

1 þ Ωz k3

� �−3ω

→1 � 3I7

3ω→

7

��ð20Þ

The actuators' torque is impressed moment and it is described as Eq. (21):

3 M→

rel1 ¼ Txi3

3M→

rel2¼ Tyj3

3M→

rel3 ¼ Tzk3

ð21Þ

The partial linear and angular velocity according to Kane's method:

2V→j

i ¼∂�2V→

i

�∂uj

i ¼ 1;2;3;4;5;6;7; j ¼ 1;2;3;4;5;6ð Þ ð22Þ

3ω→j

i ¼∂ 3ω

→i

� �∂uj

i ¼ 1;2;3;4;5;6;7; j ¼ 1;2;3;4;5;6ð Þ ð23Þ

The partial relative angular velocities of the rotors with respect to frames {4}, {5} and {6} are:

3 ω→ j

rel 1¼

∂ Ωxi3� �∂uj

j ¼ 1;2;3;4;5;6ð Þ

3 ω→j

rel 2 ¼∂ Ωyj3� �∂uj

j ¼ 1;2;3;4;5;6ð Þ

3 ω→j

rel 3 ¼∂ Ωzk3

� �∂uj

j ¼ 1;2;3;4;5;6ð Þ

ð24Þ

Table 1Numerical value of the spherical robot parameters.

Spherical robot parameters

g I1 SR m1 l→

i

�������� Ii2, Ii3 (i ≥2) Ii1 (i ≥2) mi (i ≥2)

9.81 m/s2 1.6667 kgm2 0.5 m 10 kg 0.35 m 0.05 kgm2 0.1 kgm2 0.5 kg

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The generalized inertia forces and the generalized impressed forces are:

Fj ¼X7i¼1

2R→

i ⋅2V→j

i þX7i¼1

3M→

i ⋅3ω→j

i þX3i¼1

3M→

rel i3ω→j

rel i j ¼ 1;2;3;4;5;6ð Þ ð25Þ

F�j ¼X7i¼1

2 ˜R→

i ⋅2V→j

i þX7i¼1

3 ˜M→

i ⋅3ω→j

i j ¼ 1;2;3;4;5;6ð Þ ð26Þ

The six differential equations of motion for the spherical robot are expressed by Kane's method (Eq. (27)):

Fj þ F�j ¼ 0 j ¼ 1;2;3;4;5;6ð Þ ð27Þ

Fig. 3. Variation of Euler parameters.

Fig. 4. Angular velocity of rotors with respect to sphere.

Fig. 5. Motors torque on linear trajectory.

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5. Spherical robot trajectory planning

The differential equations of motion of the system (Eq. (27)) could be written in the form of Eqs. (28) and (29):

f i q0; q1; q2; q3; _q0; _q1; _q2; _q3; €q0; €q1; €q2; €q3;_Ωx;

_Ωy;_Ωz

� �¼ 0 ; i ¼ 1;2;3ð Þ ð28Þ

f 4 q0; q1; q2; q3; €q0; €q1; €q2; €q3;_Ωx; Tx

� �¼ 0

f 5 q0; q1; q2; q3; €q0; €q1; €q2; €q3;_Ωy; Ty

� �¼ 0

f 6 q0; q1; q2; q3; €q0; €q1; €q2; €q3;_Ωz; Tz

� �¼ 0

ð29Þ

Three second-order differential equations are derived if the trajectory of spherical robot and its rotation about vertical axis areknown:

−€q0q2 þ €q1q3 þ €q2q0−€q3q1 ¼ €x ð30Þ

€q0q1−€q1q0 þ €q2q3−€q3q2 ¼ €y ð31Þ

−€q0q3−€q1q2 þ €q2q1 þ €q3q0 ¼ 2α1z ð32Þ

Eq. (32) describes the rotation of the spherical robot about its vertical axis. Eqs. (30), (31), (32), Eq. (28) (three equations) andEq. (2) can be solved for q0, q1, q2, q3, Ωx, Ωy and Ωz. Finally, the actuators' torque is determined by using Eq. (29).

Fig. 6. Variation of Euler parameters.

Fig. 7. Angular velocity of rotors with respect to sphere.

258 M.R. Azizi, D. Naderi / Mechanism and Machine Theory 64 (2013) 251–261

5.1. Linear trajectory planning

The robot is initially at rest at the origin and must track a linear trajectory and stop at point (5 m, 6 m) after 15 s. It is assumedthat the rotation of robot about its vertical axis is zero. The acceleration of robot is determined by using a binomial model and thefollowing assumptions:

x 0ð Þ ¼ 0 x 15ð Þ ¼ 5

_x 0ð Þ ¼ 0 _x 15ð Þ ¼ 5

€x 0ð Þ ¼ 0 €x 15ð Þ ¼ 0

y tð Þ ¼ 65x tð Þ 2ω1z tð Þ ¼ 0

ð33Þ

The numerical parameters of the spherical robot are shown in Table 1.The variation of Euler parameters, rotors angular velocity with respect to sphere and motors torque are determined by using

Adams–Moulton method for solutions of differential equations of motion as shown in Figs. 3 to 5.

5.2. Circular trajectory planning

The desired trajectory is a circular arc with radius Rc=2 m and angle φ=270°. The robot starts from the origin with zerovelocity and negotiates the circular arc and stops at the end point of the arc. The total motion time of the robot is 15 s. It is

Fig. 8. Motors torque on circular arc trajectory.

Fig. 9. Robot linear trajectory.

259M.R. Azizi, D. Naderi / Mechanism and Machine Theory 64 (2013) 251–261

assumed that the only middle ring on spherical shell rolls along the trajectory. The acceleration of robot is determined by using abinomial model and the following assumptions:

V 0ð Þ ¼ 0 _V 0ð Þ ¼ 0

V 15ð Þ ¼ 0 _V 15ð Þ ¼ 0

ϕ tð Þ ¼ − 1RC

∫t

0

V tð Þdt þ ϕ 0ð Þ ϕ 15ð Þ ¼ 3π2

ð34Þ

Considering Eq. (34):

€x ¼ V2

RCcosϕ− _V sinϕ

€y ¼ V2

RCsinϕþ _V cosϕ

ð35Þ

Fig. 10. Robot circular trajectory.

Fig. 11. Magnitude of velocity of geometric center of the sphere on linear trajectory.

260 M.R. Azizi, D. Naderi / Mechanism and Machine Theory 64 (2013) 251–261

Because only the middle ring on the spherical shell rolls along the trajectory:

2α1Z ¼ €φ ¼ −_VRC

ð36Þ

The variation of Euler parameters, rotors angular velocity with respect to sphere and motors torque are shown in Figs. 6 to 8.

6. Simulation

In the previous section trajectory planning on linear and circular trajectory is studied and motors' torque is determined bymeans of differential equations of the robot. Using Msc. Visual Nastran 4D software, verification of the mentioned mathematicalmodel is studied. To achieve this aim, spherical robot model has been created in Solidworks software and exported to Msc. VisualNastran 4D. By setting Time Step to .01 s and using Kutta–Merson integration method in software package, motors' torque onlinear and circular arc trajectory (Figs. 5 and 8) has been exerted on model actuators and robot trajectory has been determined bysimulation. The results show the adequate adaptation between simulation results and the desired trajectory as demonstrated inFigs. 9 to 12.

Fig. 12. Magnitude of velocity of geometric center of the sphere on circular trajectory.

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7. Conclusion

In this paper a new inner mechanism with three independent actuators for spherical robot is presented. Kinematic anddynamic modeling of the spherical robot are derived using Kane's method. For describing the orientation of spherical robot, theEuler parameters are used to avoid singularity. The presented inner mechanism enabled the robot to start its motion from everypoint on the desired trajectory with zero velocity, track the trajectory and stop on another point of it. Simulation has been donefor linear and circular arc trajectories. The inner mechanism makes it possible to plan the robot motion by considering theorientation and position of the robot.

References

[1] J. Suomela, T. Ylikorpi, Ball shaped robots: an historical overview and recent development at TKK, Field and Service Robotics 25 (6) (2006) 343–354.[2] A. Bicchi, A. Balluchi, D. Prattichizzo, A. Gorelli, Introducing the “SPHERICLE”: an experimental testbed for research and teaching in nonholonomy,

Proceedings — IEEE International Conference on Robotics and Automation 3 (1997) 2620–2625.[3] A. Halme, T. Schönberg, Y. Wang, Motion control of a spherical mobile robot, International Workshop on Advanced Motion Control, AMC 1 (1996) 259–264.[4] Q. Zhan, T. Zhou, M. Chen, S. Cai, Dynamic trajectory planning of a spherical mobile robot, IEEE International Conference on Robotics, Automation &

Mechatronics (RAM), 2006, pp. 1–6.[5] Y. Ming, D. Zongquan, Y. Xinyi, Y. Weizhen, Introduction hit spherical robot: dynamic modeling and analysis base on decoupled subsystem, IEEE

International Conference on Robotics and Biomimetics, December 2006, pp. 181–186.[6] R. Mukherjee, M.A. Minor, J.T. Pukrushpan, Motion planning for a spherical mobile robot: revisiting the classical ball-plate problem, Journal of Dynamic

Systems, Measurement and Control, Transactions of the ASME 124 (4) (2002) 502–511.[7] R. Mukherjee, M.A. Minor, J.T. Pukrushpan, Simple motion planning for spherobot: a spherical mobile robot, Proceedings of the IEEE Conference on Decision

& Control 3 (December 1999) 2132–2137.[8] A.H. Javadi, P. Mojabi, Introducing August: a novel strategy for an omnidirectional spherical rolling robot, Proceedings — IEEE International Conference on

Robotics and Automation 4 (May 2002) 3527–3533.[9] Z. Qiang, L. Zengbo, C. Yao, A back-stepping based trajectory controller for a non-chained nonholonomic spherical robot, Chinese Journal of Aeronautics 21

(5) (2008) 472–480.[10] Q. Zhang, Q. Jia, H. Sun, Z. Gong, Application of Genetic Algorithm-based PI controller in a spherical robot, IEEE International Conference on Control and

Automation, ICCA, december 2009, pp. 180–184.[11] A. Ghanbari, S. Mahboubi, M.M.S. Fakhrabadi, Design, dynamic modelling and simulation of spherical mobile robot with a novel motion mechanism,

IEEE/ASME International Conference on Mechatronics and Embedded Systems and Applications, MESA, 2010, pp. 434–439.[12] S. Bhattacharya, S.K. Agrawal, Spherical rolling robot: a design and motion planning studies, IEEE Transactions on Robotics and Automation 16 (6)

(December 2000) 835–839.[13] T. Li, Y. Zhang, Y. Zhang, Approach to motion planning for a spherical robot based on differential geometry control theory, Proceedings of theWorld Congress

on Intelligent Control and Automation (WCICA) 2 (June 2006) 8918–8922.[14] V.A. Joshi, R.N. Banavar, R. Hippalgaonkar, Design and analysis of a spherical mobile robot, Mechanism and Machine Theory 45 (2) (2010) 130–136.[15] K.W. Spring, Euler parameters and the use of quaternion algebra in the manipulation of finite rotations: a review, Mechanism and Machine Theory 21 (5)

(May 1986) 365–373.[16] P.E. Nikravesh, Spatial kinematic and dynamic analysis with Euler parameters, in: E. Haug (Ed.), Computer Aided Analysis and Optimization of Mechanical

System Dynamics, Springer-Verlag, Berlin, 1984, pp. 261–281.[17] S. Altmann, Rotations, Quaternions and Double Groups, Dover Publications, England, 2005.[18] R.A. Wehage, Quaternions and Euler parameters—a brief exposition, in: E.J. Haug (Ed.), Computer Aided Analysis and Optimization of Mechanical System

Dynamics, Springer-Verlag, Berlin, 1984, pp. 147–180.[19] T.R. Kane, D.A. Levinson, Dynamic Theory and Applications, McGraw-Hill Book Company, New York, 1985.