2016 First International Conference on Micro and Nano Technologies, Modelling and Simulation
Application of Kane's Method for Dynamic Modeling of Rotary Inverted Pendulum System Mukhtar Fatihu Hamza
Hwa Jen Yap
Department of Mechanical Engineering University of Malaya, 50603 Kuala Lumpur, Malaysia. Email:
[email protected]
Department of Mechanical Engineering University of Malaya, 50603 Kuala Lumpur, Malaysia. Email:
[email protected]
Imtiaz Ahmed Choudhury
Abdulbasid Ismail Isa
Department of Mechanical Engineering University of Malaya, 50603 Kuala Lumpur, Malaysia. Email:
[email protected]
Department of Electrical and Electronics Engineering Sokoto State polytechnic Sokoto, Nigeria Email:
[email protected]
picture of classical experimental set-up for RIP is shown in Figure 1 (a). It consists of two optical encoders for measuring the pendulum’s and arm’s angles respectively. It also comprised of the data acquisition device for collecting the information from the encoders and give it to the computer. The data acquisition device also received the control signal from the computer and give it to the power amplifier for the amplification of the signal before feeding to the motor as illustrated in figure 1 (b). The control objectives of the RIP can be categorized into four: 1. Controlling the pendulum from downward stable position to upward unstable position known as Swing-up control. 2. Regulating the pendulum to remain at the unstable position known as stabilization control. 3. The switching between swing-up control and stabilization control known as switching control. 4. Controlling the RIP in such a way that the arm tracks a desired time varying trajectory while the pendulum remains at unstable position known as trajectory tracking control[6]. The modelling of RIP focuses on the kinematic and/or dynamic [7]. Many types of RIP have been developed together with their mathematical model [8-10]. NewtonEular, Lagrage-Eular and Lagrange was used to develop the nonlinear mathematical model of RIP [4, 7, 8]. Lagrange multiplier was used in the derivation using Lagrange method, while the calculation of redundant forces was involved in Newton method [11-13]. As a result, Newton-Eular, Lagrage-Eular and Lagrange methods requisite complicated and tedious formulation for a large multi-body system [7]. Consequently, they likely led to an inefficient computation. Kane's method can be regarded as an alternative method of modelling. This method does not require the calculation of multipliers or redundant forces. Kane method is based on the partial velocities of the constituents of the system[14]. Hence, Kane's method is more efficient than Lagrange and Newton-Euler methods in terms of computation. In this paper, a nonlinear dynamical equations of the RIP are derived using Kane's
Abstract - Rotary inverted pendulum (RIP) is under-actuated mechanical system which is inherently nonlinear and unstable. RIP is known widely as experimental setup for testing different kind of control algorithms. Most of literatures used Newton-Euler or Lagrange methods to find the dynamic equation of RIP. Thus, this paper, described a development of nonlinear dynamical equations of the RIP system using Kane's method. The simulink model of RIP was developed based on the derived equations. Simulation study was carried out and the results indicated that, the RIP system is inherently nonlinear and unstable. Comparisons between the Euler Lagrange and Kane's methods were carried out using Matlabsimulink, which demonstrate the advantage of Kane's method. It is realized that the difficulties and limitations in the previous dynamic equation of RIP proposed in literature are eliminated. Kane's method can be regarded as an alternative method for finding the dynamic model of the systems. This method does not require the calculation of multipliers or redundant forces which some time add complexity to the model. Keywords - Rotary inverted pendulum (RIP); Kane's method; Dynamic modelling; Newton-Euler; Lagrange methods
I. INTRODUCTION The Furuta pendulum also known as Rotary inverted pendulum (RIP) was proposed in 1992 [1]. RIP is in the class of under-actuated mechanical systems with two degree-of-freedom (DOF). The RIP exhibits some challenging and interesting properties, such as nonlinearities and instability[1-3]. These features make RIP attract high attention from researchers and it is known widely as experimental setup for testing both linear and nonlinear control algorithms[4]. The RIP system consists of pendulum, rotational arm and a servomotor system which drives the output gear. RIP has many important real applications like robotics, aerospace vehicles, pointing control, and marine vehicles[5]. In addition, when the pendulum of RIP is at hanging position, it represents real model of the simplified industry crane application. The
978-1-5090-2406-3/16 $31.00 © 2016 IEEE DOI 10.1109/MNTMSim.2016.18
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method. The simulink model is developed based on the derived equations. Comparisons between the Lagrange and kane's methods was carried out using Matlab simulink. Simulations was carried out to study the nonlinear behavior of the RIP.
x-axis). , , are the inertial earth fixed reference is away and perpendicular to the earth frame, in which surface. , , are the arm fixed reference in which is away from fixed point B along the arm length. The pendulum which is attached to free end of the rotating armhas its mass centre at point C. is the pendulum angle (generalized coordinate for pendulum, which is the angle between the pendulum and the vertical z-axis). The pendulum has two plane of symmetry through axis with normal direction and and it can rotate in a plane perpendicular to .The pendulum fixed of reference point away from point A.The are , , in which torque τ is applied at the fixed end of the arm by the motor and the direction of the torque depends on the direction of the voltage applied to the motor. The mathematical model of RIP will be derived using Kane's method based on the following assumption: The system consists of two bodies (pendulum and arm). The pendulum can move from 0 degree to 360 degrees about the arm axis. The arm can move from -75 to 75 degree about z-axis. The position of the whole system is zero (fixed) The motor inductance and friction on the armature are neglected equivalent frictional force of motor/arm is neglected
(a)
Contrary to most literature were they simplify the generated model by assuming that, the pendulum is rotating in a constant plane [15, 16]. In this study, we consider the rotary motion of the arm all together. That is, we assume that the actual plane pendulum is rotating in is different in every instant. This make the developed model to be more complex but with high accuracy.
(b) Figure 1. Quanser RIP experimental setup
II. ROTARY INVERTED PENDULUM MODELLING The RIP is made up of two connected rigid bodies activated by a servomotor system as shown in figure 2. These bodies are rotational arm and pendulum. Both arm and pendulum are one DOF. The arm is attached to the output gear of the motor and it can rotate around the fixed point B. is the arm angle (generalized coordinate for arm which is the angle between the arm and the horizontal
Figure 2. Free body diagram of RIP including the system’s oordinates
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direction
III. KINEMATICS OF RIP There are two generalized speed defined as:
cos
(1) (2) The pendulum angular velocity with respect to the earth fixed frame vectors is as follows: cos sin (3) The position vector of pendulum mass center is: ̅ (4) Therefore, the linear velocity of pendulum's center of mass with respect to the earth fixed reference can be derived as [17]: sin
sin (14)
direction cos direction
and
cos
sin
sin (15) cos
sin cos sin
A. Pendulum dynamic equation direction is the pendulum equation of The equation for motion and it can be express as follows: cos sin cos sin (16) Equation (16) can be written in the following form cos sin cos sin (17) were , and in which the parameter is approximately equal to one for thin rod and is the natural frequency of the pendulum. Based on parallel axis theorem, , ≡ . and The right hand side of equation (17) is the expression of standard pendulum on cart while left hand side is due to the additional rotation of the base on the circular arc.
(5)
IV. DYNAMICS OF RIP The pendulum fixed frame associated with some major point directions on its axis. This is due to the symmetry. So, the angular momentum of the center of mass is: ̅ ̅ ̅ (6) , , are the In which, the angular velocities , in equation (3) respectively. Or coefficients of , simply ̅ ̅ ̅ cos sin (7) were ̅ is moment of inertia of mass centre For thin rod with 2 length, The following are proved [17]: ̅ ̅ , and ̅ 0 (8)
B. Arm dynamic equation TABLE I.
Sym bol m g lp r
Based on Euler's equation for centre of mass and Newton's law, the external force and the moment about the center of mass that are acting on the rigid body can be express as follows [18]:
Jr Rm kt km Kg ηm ηg
(9) (10) This include the sum of reaction force and gravitational force in the directions , and as well as the couple and . Note, we assumed there is no in the directions friction in the pin, therefore there is no couple component direction. The force equilibrium in equation (9) in the has the following components: direction sin 2 cos (11) direction cos sin sin (12) direction sin cos (13) The components of the moment equilibrium for equation (10) are:
PARAMETERS OF RIP
Description Mass of pendulum Acceleration due to gravity Length of pendulum center of mass
Value 0.127 9.82
Unit kg m/s2
0.156
m
Rotary arm length Rotary arm moment of inertia about its center of mass Motor armature resistance Motor current-torque constant Motor back-emf constant High-gear total gear ratio Motor efficiency Gearbox efficiency
0.168
m
0.000998 2.6
Kg.m2 Ω
0.00768 0.00768 70 69 90
N-m/A V/(rad/s) % %
The dynamic equation of arm can be found from its component of moment equilibrium about the fixed point B as follows: cos sin (18) is the equivalent frictional force of motor/arm, where is the transverse moment of inertia for arm at the fixed point B, is the torque applied to the arm by the motor. at their directions defined Substituting for , and in equations (14), (15) and (12) respectively, we have the dynamic equation for arm as follows:
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sin sin 2 Where
cos
sin
.
(19) ,
and
.
.
.
The torque (τ) at the load gear is generated by servo motor and is described by equation (20) [19].
.
58.4 tan
.
.
(21)
. .
.
(22)
Based on the autonomous model in equation (21) and (22), the Matlab Simulink of RIP is developed as shown in figure 3. Initially the pendulum is positioned in an inverted position with very small displacement 0.005 , then it is allowed to fall by applying a pulse signal to the model. The open loop responses for arm and pendulum are shown in Figure 4 and Figure 5 respectively. This response of the Simulink model was revealed to be accurate in comparison with the widely known empirical observations of pendula characteristics [4, 9, 10]. Moreover, the adequate correctness of the simulation model was approved. The response shows that the whole system is nonlinear and unstable.
(20) V. SIMULATIONS
Figure 4. Open loop response for Arm
Figure 3. Nonlinear simulink model of RIP
To study the nonlinear open-loop dynamic behavior of RIP, some simulations in Matlab Simulink were done. The parameters of Table 1 was used [19]. By assuming the equivalent frictional force of motor/arm equal to zero. Substituting the values of the parameters and rearranging equations (17) and (19), the equation of motion of RIP can be found as Figure 5. Open loop response for pendulum
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The open loop response for nonlinear dynamic equation of RIP developed using Kane’s method is compared with the nonlinear dynamic equation of RIP developed using Eular Lagrange method [19] as shown in figure 6. Firstly the pendulum is placed in an inverted position with very small displacement (0.0050) then it is permitted to fall by applying a pulse signal to the model. The simulation result for arm and pendulum are shown in Figure 7 and 8 respectively. The simulation results show that, the kane’s and Lagrange response are similar and their behaviors are closely the same for both pendulum angle and arm angle.
Figure 7.
Open loop response for pendulum
Figure 6. Comparison of the nonlinear models of RIP developed by Kane’s and Eular-Lagrange methods
It can be seen that the for the Kane’s based model, the pendulum falls in about 1.4 second while for Lagrange based model the pendulum falls in more than 1.75 second. Also for the arm, it can be seen that for the Kane’s based model the arm swing from 0 to about 21 degree and comeback to about 9 degrees within 1 to 1.75 second. However, within the same time the Lagrange based model con only swing from 0 to about 23 degrees. This demonstrate that the Kane’s model based is faster in respond compared to the Lagrange based model. This simulation results agree with the theoretical results; therefore, the presented model is valid to some level.
Figure 8.
Open loop response for Arm
VI. CONCLUSION This study, presented a development of nonlinear dynamical equations of the RIP system using Kane's method. The Matlab/simulink model of RIP was developed based on the derived equations. Simulation study was carried out and the result shows that, the RIP system is inherently nonlinear and unstable. ACKNOWLEDGEMENT The authors would like to acknowledge the University of Malaya Postgraduate Research Grant (PPP)-Research (2015A) Ref. No : 4394
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[10] S. Jadlovský and J. Sarnovský, "Modelling of Classical and Rotary Inverted Pendulum Systems–a Generalized Approach," Journal of Electrical Engineering, vol. 64, pp. 12-19, 2013. [11] A. I. Isa and M. F. Hamza, "Effect of sampling time on PID controller design for a heat exchanger system," in Adaptive Science & Technology (ICAST), 2014 IEEE 6th International Conference on, 2014, pp. 1-8. [12] A. I. Isa, N. Magaji, and M. F. Hamza, "Development of Hybrid Fuzzy PD Plus I Controller for Voltage Stability," in 16th International Conference on Mathematical and Computational Methods in Science and Engineering (MACMESE '14), Kuala Lumpur Malaysia, 2014. [13] N. Magaji, M. F. Hamza, and A. Dan-Isa, "Comparison Of Ga And Lqr Tuning Of Static Var Compensator For Damping Oscillations," international Journal Of Advances In Engineering & Technology, 2012. [14] R. D. Komistek, J. B. Stiehl, D. A. Dennis, R. D. Paxson, and R. W. Soutas-Little, "Mathematical model of the lower extremity joint reaction forces using Kane’s method of dynamics," Journal of Biomechanics, vol. 31, pp. 185-189, 1997. [15] P. Ernest and P. Horacek, "Algorithms for control of a rotating pendulum," in Proceeding of the 19th IEEE Mediterranean Conference on Control and Automation (MED’11). Athens: National Technical University, 2011. [16] C. van Kats, "Nonlinear control of a Furuta Rotary inverted pendulum," DCT report, 2004. [17] H. Ashrafiuon and A. M. Whitman, "Closed-loop dynamic analysis of a rotary inverted pendulum for control design," Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME, vol. 134, 2012. [18] S. Rudra, R. Barai, M. Maitra, D. Mandal, S. Ghosh, S. Dam, et al., "Stabilization of Furuta Pendulum: A backstepping based hierarchical sliding Mode approach with disturbance estimation," in Intelligent Systems and Control (ISCO), 2013 7th International Conference on, 2013, pp. 99-105. [19] M. F. Hamza, H. J. Yap, and I. A. Choudhury, "Genetic Algorithm and Particle Swarm Optimization Based Cascade Interval Type 2 Fuzzy PD Controller for Rotary Inverted Pendulum System," Mathematical Problems in Engineering, vol. 2015, 2015.
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