Modeling and Control of a Multi degree of Freedom Flexible Joint Manipulator Md. Akhtaruzzaman, Dr. Rini Akmeliawati

Department of Mechatronics Engineering International Islamic University Malaysia

Kuala Lumpur, Malaysia. akhter900@yahoo.com, rakmelia@iiu.edu.my

Teh Wai Yee Faculty of Engineering

Monash University, Malaysia. Selangor Darul Ehsan , Malaysia.

Abstract—The paper describes the model of a two flexible joint manipulator and the development of a nonlinear passivity based control method for that model. For dynamic modeling, the Euler-Language approach is selected to model flexible joint manipulator. Since lightweight materials are employed in the construction of manipulator, the assumption that negligible link deformation no longer applies. A nonlinear passivity-based control method for rigid flexible joint as well as flexible link is proposed here. The main idea of Passivity Based Controller (PBC) is to use energy state of a system to provide stabilization. The PBC is developed to achieve two goals: set point regulation and trajectory tracking. The controller brings the end point from unstable position to a certain stable position. Beside that, the controller are developed to control the position of there end-point in such a way that it tracks the desired trajectory which are tested by simulations.

Keywords-Passivity Based Control; PBC; Euler Lagrange Equation; Flexible Link Manipulator; Flexible Joint manipulator.

I. INTRODUCTION Manipulator mass reduction contributes directly to

achieve a large force to mass ratio. Consequently energy consumption is reduced and better response time can be expected. Since light weight materials are employed in the construction of manipulator, the assumption that negligible link deformation no longer applies. Hence, this will enable manipulators in attaining a higher degree of flexibility. Flexible manipulator is a distributed system; multiple modes are required to derive an accurate mathematical model to describe flexible motion. Due to the fact that a flexible manipulator system is highly nonlinear and complex, therefore it is difficult to control accurately.

A dynamic model of two degree of freedom flexible joint manipulator is done using the Euler-Lagrange approach where both links are assumed to be rigid. The flexible joint manipulator is under actuated because the only actuator available on its first joint. This idea of the rigid link manipulator dynamic modeling will form the foundation towards application of flexible link manipulator dynamics. PBC are designed for rigid flexible joint as well as flexible link. The PBC control objective is achieved by imposing the closed-loop dynamic a certain passivity property [1]. The main idea of PBC is to provide both stabilization and tracking. Hence this is achieved using the energy state of a system so that the PBC would not rely too much on accuracy of the model.

II. PRELIMINARIES

A. Modeling The equation of motion of the manipulator can be derived by number of methods namely Newton Euler equations, Euler-Lagrange equation, Hamilton’s canonical equations and Kane’s Method. Lagrange equation is a common way of deriving equations of motion. Compared with Newton Euler approach, the Lagrange approach has the following advantages [3]:

• It is systematic and simple. • It provides the equation of motion in a compact

analytical form containing the inertia matrix, the matrix in the centrifugal and Coriolis forces and vector of gravitational force.

• It is the most effective for complex mechanical effects such as flexible link deformation.

De Luca et al. [1] presented a close form equation of motion for planar multilink flexible manipulator by using the Lagrange equation approach and assumed mode shapes. Whalen and Sommer [7] proposed mode shape parameter determined experimentally. Wenwei Xu [8] improved the accuracy of the theoretical model [1] by incorporating experimental data. He shows the linearization of model [1] at zero joint angle results at in an unstable system, even though the actual system is stable.

B. Control Vibrations present in the manipulator due to disturbed

link flexibility and concentrated joint elasticity. This is caused because of motors with long shafts or elastic band. In both cases, the system contains additional generalized coordinates that exceed in number the available command input, that make the control more difficult.

The main problem of flexible link is the endpoint’s error as a result of vibration. Several different approaches have been developed for reducing vibration of one-link or two-link flexible arms. Many control schemes can be considered such as Fuzzy control, adaptive control, energy based control, Feedback control, Position/Force control etc.

C. Euler Lagrange equation The Euler Lagrange (EL) equations were normally used

to describe the dynamics of mechanical systems such as manipulator. These equations can also be used to describe

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2009 Second International Conference on Computer and Electrical Engineering

978-0-7695-3925-6/09 $26.00 © 2009 IEEE

DOI 10.1109/ICCEE.2009.100

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the dynamic of the other physical system such as electrical systems.

EL equation can be derived using Newton’s second law. A Lagrange function is defined as the difference between the kinetic energies, , and the potential energies,

of the system. , (1) For typical N degree of freedom dynamic system the Lagrange function is, (2)

is the generalized external force corresponding to the vector of generalized coordinates . In a general context, three types of external force can be considered: the action of controls, dissipative and the interaction of the system with its environment. Thus, (3) Control input linearly as where is constant and is the control vector, is dissipative force where is the Rayleigh dissipation, is external signal that models the effect of disturbances. Hence the EL equation can be written in the equivalent form, (4) It is defined as EL system which is characterized by its EL parameters, , , , , , .

D. Passivity of EL System From an energetic viewpoint, a passive system is a

system that can not store more energy than is supplied to it. The difference between stored energy and supplied energy is being the dissipated energy. The following is defined the passivity and output strict passivity (OSP).

A system is passivity if it is dissipative with supply rate, , . The system is OSP if it dissipative with supply rate, , where 0.

E. Passivity Based Control Since EL system is passive, therefore PBC is most

suitable for them. The main idea of PBC is to use the energy of a system to provide stabilization. The close loop system can be seen as an interconnection between the plant and the EL controller. The EL controller will act as structure preserving passive controllers. The feedback interconnection of two passive EL systems yields also a passive EL system. Furthermore, if the both EL system are OSP, the resulting system is also OSP.

Figure 1. Fedback interconnection of passive system

If the overall energy balance is positive, in the sense that the energy generated by one subsystem is dissipated by the other one, the closed loop will be stable in an input-output sense [1]. This property constitutes the basis of PBC.

III. FLEXIBLE JOINT MANIPULATOR The flexible joint manipulator can be divided into two

main classes: Under-actuated and fully-actuated. A system is the under-actuated if it has control input which is less than the number of degree of freedom. On the other hand, if the number of DOF is equal to available control input, the system is labeled as fully-actuated.

A two link flexible joint manipulator considered as under-actuated as shown in figure 2 whereby there is no control input acting on the second link. Both links moves on a horizontal plane with assumption that the links are rigid, i.e. there is neither vibration nor deformation on the two links. Elasticity of the joint is provided by using the spring between the two links. The angle equal to zero when the spring is fully relaxed and links are collinear. Due to under-actuation of link 2, the controller has to bring the link 1 to desired angle meanwhile the angle of link 2 also is required to be zero as fast as possible.

The following assumptions are made in order to construct a model for the flexible manipulator shown in figure 2.

• All the links are rigid that there is no link deformation.

• Negligible friction • Mass of two links are uniform • Gravitation potential is absent because manipulator

motion is constructed to planar movement.

Figure 2. The planner two link flexible joint manipulator

The controller has to achieve two goals: a) set point regulation, bring the end point from unstable position to a certain stable position and b) trajectory tracking, control the position of their end-point in such a way that it tracks the desired trajectory.

A. Kinematic analysis Refer to the figure 2, the two links are numbered as 1 2 respectively. A set of generalized

coordinate namely . The displacement of the centre of mass of the links is

obtained as follows:

cossin (5)

cos cos sin sin (6)

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Velocity and angular velocity of individual link in relation to joint velocities are computed. Linear velocity of the link,

sincos sin 0cos 0 (7)

Where

sin sin sincos cos cos (8)

Angular velocity of the link: 1 0 (9) 1 1 (10) Linear velocity and angular velocity can be rewrite as following: (11) (12) Where 1,2,3 …

and are called as Jacobian Matrix. The Jacobian matrix for 2 DOF flexible joint derived as follows,

sin 0cos 0sin sin sin cos cos cos

1 01 1 (13)

B. Dynamic Analysis 1) Kinetic energy The expression for kinetic energy is written in terms of

velocity and angle of each link. Thus the total kinetic energy of links is represented by ∑ ∑

(14) Where is known as manipulator inertia tensor and

matrix, ∑ . Let , component of the inertia tensor . The equation also can be written as, ∑ ∑ (15)

2) Potential energy A spring is added between the links in order to create a

source of potential energy. So total potential energy,

0 00 (16) Here is stiffness of the spring. It is considered that the actuator exert joint torque, τ, will

work on the link 1 only. Thus net generalized force, 0 (17)

3) Euler lagrange formulation

Using the EL formulation, the N degree of freedom manipulator is generally described by the EL equation. ,

, The inertia matrix,

2

(18)

is 2 2 matrix and it is symmetric positive. Carioles and centrifugal force matrix,

, 0 (19)

Stiffness matrix of the spring, The parameters are set as, (20) (21) (22) Assume the system has zero internal damping and external (disturbance) force, that is, 0 0. So the dynamic equation becomes, , (23)

C. Model Analysis 1) Notable of properties of system There are important property of dynamic model is

presented which will be useful for designing control algorithms: Skew-symmetry of matrix 2 , sin 2 0 11 0 (24)

The skew symmetry property is satisfied using, 2 , 0 (25)

2) Linearization of the system In order to investigate the controllability of the system,

motion equation is linearized around the equilibrium point, 0, 0 .

0 1 0 00 0 00 0 0 10 0 000

(26) The controllability matrix has the full rank. Hence, the

linearized system is always controllable which also means that the poles can be placed wherever desired. Furthermore, a full state feedback controller can be applied to stabilize the system. The linearized model become uncontrollable if k=0 (spring attached).

3) Passivity of system The total energy of the system is given by

(27)

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The system with as input and as output is passivity. This proves that the system is passivity as stated in the definition.

D. Control Design The dynamic equation of the flexible joint manipulator

is expressed in the form EL equation which is nonlinear and has passivity property. A controller is also developed by taking benefit of the passivity property. So the resulting system is a passive system.

PBC controller is used to bring the links from the unstable position to other equilibrium position and then stabilize the links about that equilibrium. It achieves the goal by assigning a desired energy function and introducing the damping into the system.

1) Set point regulation To solve the problem of set point regulation of flexible

joint manipulator it needs to consider the two things, Energy shaping stage and Damping injection. In EL parameter the sub-index are used for open loop and close loop respectively. The EL parameters are modified as ∑, , 0, , , 0, in order to achieve the control objectives. The close loop EL parameters are given in terms of as, ∑ , , , 0 , , , 0 (28) Here 0, 0 . To ensure the system is strictly passive, the dissipation terms, R1 and R2 are considered. The corresponding close loop dynamic equation is, 0 (29) The potential spring is only present between two links, so, (30) should be zero to achieve the control objective. Let written as the second subsystem in the space form as and (31)

The subsystem is negative feedback interconnected to the plant depicted as the figure bellow.

Figure 3. Interconnection between plant and controller

a) Stability analysis The total energy of the close-loop system,

(32) Here 2 , is skew symmetric, thus:

(33) For some 0, then This proves that the q will be converge into a state desired equilibrium .

b) Simulation result The system is studied by taking the parameter as follows:

TABLE I. PARAMETER SETTING FOR FLEXIBLE JOINT MANIPULATOR

Symbol Unit Physical Value 0.5/ 0.2 0.1

0.1 1

Initial position and velocity: , 0 and , 0 Thus the parameters are computed using the equation no. (20), (21), (22). 1.0313, 1.0063, 0.0123 The controller gain is set as, Test1: ( 1 1), Test2: ( 5 1), Test3: ( 1 5), Test4:( 1 0.1).

The performance of the PBC is verified by carrying out simulations and the result shows that test 1 (shown in the figure bellow) achieves steady state faster and less oscillation. Increase or decrease in either R1 or R2 gives more oscillation.

Figure 4. Simulation result for test 1.

2) Trajectory tracking control Besides reshaping the potential energy, it also needs to

reshape the kinetic energy because of imposing a specific pattern to the transformation of potential into kinetic energy. ∑ , , , 0 , , , 0 Here 0, 0, dissipation term is added to ensure the system is strictly passive. The corresponding close loop dynamic equation is, 0 0 (34) because , are zero to achieve the control objective and the equation is similar to the control law in a set point regulator.

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a) Stability analysis The total energy of the close loop system,

(35) For some 0, This proves that the q will be tracks the desired trajectory. The simulation result in figure 5 shows the joint 1 angle is successfully follows desired tracking trajectory.

Figure 5. Simulation result for the motion of Joint 1.

IV. FLEXIBLE LINK MANIPULATOR Since lightweight materials are employed in the

construction of manipulator, the assumption that negligible link deformation no longer applies. Hence, this will enable manipulators in attaining a higher degree of flexibility. An explicit dynamic model of a planar two link flexible manipulator is constructed according to [1]. Both the first and second manipulators are flexible links as shown in figure 6. The following coordinate frames are required for analysis. , : The base reference inertial frame associated to

the ground. , : The rigid body and local reference frame associated to link 1,2 , : The flexible body and local reference frame associated to link 1,2 The rigid motion of the link is described by the joint

angle , while denotes the transversl deflection of link at 0 , .

Figure 6. A planar two-links flexible manipulator [1].

Joint 1 and joint 2 are connected to shafts of actuator 1 and actuator 2 respectively. Actuator 1 is fixed on the ground while actuator 2 is attached on the tip of link 1. Link 2 has a payload at the other end. Control inputs ate available on the both link but there are no control input acting on flexible variable directly and hence the flexible manipulator is an under-actuated system. A set of mechanical parameters

of the arm is required for dynamic equation and the following assumptions are made:

• The deflection of the flexible link are treated small • Negligible friction in the joint • Each link is made up of uniform density • Each link is considered to be an Euler Bernoulli

beam • Fixed free beam boundary conditions for each

beam.

A. Assumed Mode Shape Two links are modeled as Euler Bernoulli beams of

uniform density , and constant flexural rigidity , with deformation , . The partial differential equation can be obtained as, , , 0 (36)

Due to the geometric constraints imposed on these links, the deflection and the slope are zero at the clamped end 0 , while at the free end , the deflection , and slope , are non zero. Hence the two special boundary conditions are: At 0 for all , 0, 0; , 0 Where 1,2. A solution can be found by applying the assumed modes method. , ∑ (37)

Where is the beam deformation shape function of an eigenfunction. is the time varying amplitude of motion with an assumed shape of the link .

B. Kinematics Model The linear velocity and the angular velocity are required

to compute the kinetic energy and the potential energy. The rotation matrix of the rigid joint and rotation matrix of the flexible link at the end-point can be written as follows.

1 1

Where

The displacement of any point along link can be expressed as, , Linear velocity can be expressed as, (38) The angular velocity can be represent as, ∑ ∑ (39)

C. Euler Lagrange Equation Approach Substitution of kinetic and potential energies and their

respective derivatives into the Lagrange equation, ,

Assume the deformation is small. The model is linear with respect to a suitable a set of constant parameter in order

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to eventually produces the resultant dynamic equation in the form of, , (40) Where , , , , , is the vector of generalized coordinates with being the joint angle and the variable associated with , the mode shape j of link i. B is positive definite symmetric matrix,

(41)

H is the vector of Coriolis and centrifugal force, (42)

K is the stiffness diagonal matrix , , , … , . Where 0 and , … , 0.

D. Model Analysis The physical parameter for the simulation of the flexible

link is given bellow.

TABLE II. PARAMETER SETTING FOR FLEXIBLE LINK MANIPULATOR Symbol Unit Physical value

0.5 / 0.2

0.1 1

0.0083 0.1

0.0005 10.48 0.481.8 1.82.18 2.1815.91 15.91

Here the nominal payload, the natural frequency /2 . The mode shape coefficients are computed as follows. , 0.186, , 0.215, , 0.657, , 0.560, , 0.883, , 0.069 , 2.641, , 10.853, 0.007, 0.013, 0.033, 0.054 0.002, 0.004, 0.012, 0.016

Here the same parameters, proposed in [1], are used so that, counter checking can be done with available result.

E. Control Design Objective is concentrated to move the rigid mode joint

angles by the desired angles, without neglecting the flexible vibration being suppressed as fast as possible. To achieve this control objective, the passivity based control method is proposed for slow subsystem. The dynamics of the system now become to rigid manipulators. The slow subsystem is still an Euler-Lagrange system and can be expressed as, ∑ , 0, , 0, (43) Here control laws are structural damping and passivity based control and

The parameters, used in simulation, are similar to those parameters set in section 4.4. The controller gain is set as below: 1 00 1 1.5 00 1.5 (44)

Figure 6.The joint 1 motion with damping injection

Figure 6 shows the time history of the joint 1 motion. With the PBC controller, it can be seen that the joint 1 is successfully follows desired tracking trajectory.

CONCLUSION A dynamic equation of 2 DOF flexible joint / link

manipulator is derived using Lagrange approach. Since the dynamic system is a passive EL system, passivity based control (PBC) is most suitable which is designed using the energy shaping pulse damping injection. Two goals, set point regulation and trajectory tracking are achieved by developing the PBC. However, it is assumed that the both light weight links are rigid that is not practical in real world. The idea of the rigid link manipulator dynamic modeling forms the first step towards the application of flexible link manipulator dynamics.

REFERENCES [1] De Luca A., Siciliano B., Closed-Form Dynamic Model of

Planar Multilink Lightweight Robots, IEEE Transaction On system, Man and Cybernetics, July/August 1991, vol. 21, No. 4, pp.826-839

[2] Xu Bo, Hayakawa Y., Control two-link flexible manipulators using controlled Lagrangian method, SICE Annual Conference in Sapporo, August 4-6, 2004

[3] Sciavicco L., Sicilliano B., Modeling and control of robot manipulators, 2nd edition, London: Springer; 2000.

[4] Yoshikawa T., Mutakami H., Hosoda K., Modeling and control of a three degree of freedom manipulator with two flexible links, IEEE Conference on Decision and Control, 1990 December, vol. 4, pp.2532-2537.

[5] Chih-Wu Jen, Valigi P., Theoretical and Experimental Study of Dynamics and Control of a Two link Flexible Robot, System theory, 1996. Proceedings of the twenty Eighth Southeastern Symposium on System Theory, 31 March-2 April 1996 pp. 237-241.

[6] Zhang X. Xu W., Nair S.S., Chellaboina V. PDE Modeling and Control of a Flexible Two Link Manipulator, IEEE Transactions on control system Technology, March 2005, vol 13, No. 2, pp. 301-312.

[7] Whalen M., Sommer III H. J., Modal Analysis and Mode Shape Selection for Modeling an Experimental Two Link Flexible Manipulator, in proc. ASME Winter Annu. Meeting, vol. 75, 1994, pp. 499-509.

[8] Wenwei Xu. Xiaoping Zhang, graduate students. Satish Nair, Professor, Dynamic modeling of a two link Flexible manipulator incorporation experimental data, Proceedings of American Control Conference Arlington, VA june 25-27 2001.

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