Chapter 4 Simulation Results
5.4 Experimental Results with Tip-Mass Loading
The payload is considered as the system disturbance in these demonstrations.
The experimental results of rigid controller whose poles designed in -0.5 and -2 for regulation problem are shown in Fig. 5.4.1 to Fig. 5.4.3., the tip position with 0.102
Kg payload reaches the reference goal with much vibration effect. Then, Fig.5.4.4 to Fig. 5.4.7 (Fig.5.4.8 to Fig. 5.4.11) show the robustness of the sliding mode control against 0.102 Kg payload variation with sliding layer parameters ε =0.01 andσ =6 (ε =0.005 andσ =7). The vibration phenomenon is suppressed in the process of regulation. The performance of sliding mode control is alike the non-payload ones but exist greater steady state errors.
0 2 4 6 8 10 12 14 16
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
t(s)
hub angle(rad)
Rigid Control
Fig. 5.4.1 Hub angle regulation in rigid control (mt =0.102kg)
0 2 4 6 8 10 12 14 16
Fig. 5.4.2 Tip position regulation in rigid control (mt =0.102kg)
0 5 10 15 20
Rigid Control Rigid Control
Rigid Control Rigid Control
Fig. 5.4.3 Four states response in rigid control (mt =0.102kg)
0 2 4 6 8 10 12 14 16 18 20 0
0.05 0.1 0.15 0.2 0.25
t(s)
hub angle(rad)
SMC
Fig. 5.4.4 Hub angle regulation in SMC (mt =0.102kg, ε =0.01andσ =6)
0 2 4 6 8 10 12 14 16 18 20
0 0.05 0.1 0.15 0.2 0.25
t(s)
Tip position(m)
SMC
Fig. 5.4.5 Tip position regulation in SMC (mt =0.102kg, ε =0.01andσ =6)
0 5 10 15 20
0 2 4 6 8 10 12 14 16 18 20 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35
t(s)
hub angle(rad)
SMC
Fig. 5.4.8 Hub angle regulation in SMC (mt =0.102kg, ε =0.005andσ = 7)
0 2 4 6 8 10 12 14 16 18 20
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
t(s)
Tip position(m)
SMC
Fig. 5.4.5 Tip position regulation in SMC (mt =0.102kg, ε =0.005andσ = 7)
0 5 10 15 20
The results of sliding mode controller compared with rigid controller obviously indicated the robustness of sliding-mode controller against the variation of payload in these demonstrations.
Chapter 6 Conclusions
A physical sliding mode control is investigated in this thesis to deal with the vibration problem of the tip position control of a single-link flexible arm. Based on the linear model derived by the FEM method, the disturbances related to the higher order modes of the flexible arm and the uncertainties resulted from the structure and payload variations can be successfully suppressed by the sliding mode control with sliding surface designed by Lyapunov method. The simulation and experimental results demonstrate that the sliding mode controller not only requires small control torque but also highly reduces the vibration. Besides, the robustness of the flexible arm controlled by sliding mode method is also verified by applying an impulse disturbance and tip-mass variation. Because the real system may be not enough to be described by the 2-element FEM Euler-Bernoulli beam, in the future the system model might be derived by a more complicated analytic approach. For the better control result in our future work, the experimental accuracy should be improved by reducing the errors from the voltage shifting effect in strain gauges feedback.
Reference
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76, pp. 212-232, Mar. 1988.
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6, Nov., 2000.
[5] G. G. Hastings and Wayne J. Book, “A Linear Dynamic Model for Flexible Robotic Manipulators,” IEEE trans. on Control Systems, Feb. 1987.
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[7] M. L. James, “Vibration of Mechanical and Structural Systems: with Microcomputer Applications,” Harper & Row, 1993.
[8] J. F. Jansen, “Control and Analysis of a Single-Link Flexible Beam with Experimental Verification,” Oak Ridge National Laboratory, Dec. 1992.
[9] J. L. Junkins and Y. Kim, “Introduction to Dynamics and Control of Flexible Structures,” AIAA, pp. 199-219, 1993.
[10] Z. H. Luo, “Direct Strain Feedback Control of Flexible Robot Arms: New Theoretical and Experimental Results,” IEEE trans. on Automatic Control, Vol.
38, No. 11, Nov., 1993.
[11] W. T. Thomson and M. D. Dahleh, “Theory of Vibration with Applications 5th edition,” Prentice Hall, pp. 292-295, 1998.
[12] M. O. Tokhi, Z. Mohamed, S. G. M. Amin, and R. Mamat, “Dynamic Characterisation of a Flexible Manipulator System: Theory and Experiments,”
IEEE. 2000.
[13] R. L. Wells, J. K. Schueller, and J. Tlusty, “Feedforward and Feedback Control of a Flexible Robotic Arm,” IEEE trans. on Control Systems, Jan. 1990.
[14] H. Yang, J. Hong, and Z. Yu, “Dynamic modeling of a flexible hub-beam system with a tip mass,” Journal of Sound and Vibration 266, pp. 759-774, 2003.
[15] 陳永平， 張俊林 ，“可變結構控制設計”，全華科技圖書股份有限公司，2002.
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Appendix A
Local Mass Matrix and Stiffness Matrix
For the Euler-Bernoulli beam introduced in Chapter 2, its local mass matrix in (2.2.9) and stiffness matrix in (2.2.8) are respectively given as
[ ]
i jk =∫
h j(x) k(x)dxwhere φj
( )
x represents the j-th shape function. It is known that the shape functionsare admissible and expressed as the following Hermite-cubic polynomial functions:
( )
which as been shown in (2.2.14).
Finite element modeling of a structure may be considered to be many fold local application of the assumed modes method wherein linear combinations of the locally valid φi
( )
x s are used to represent the deflection shape over a portion (finite element) of the structure, and with the element equations and boundary constraints beingassembled to form a global model of the structural system. For the general expressions of the Euler-Bernoulli beam elements. Substituting the four shape functions (A.3) into (A.1) and (A.2), the mass matrix Mi and the stiffness matrix Ki of the i-th element are obtained as
⎥⎥
Then, N-element model of flexible arm could be derived with the above result.
Appendix B
Measurement of Young’s Modulus
Young’s modulus E is an important parameter for the precise flexible arm model, and it is measured by the frequencies analysis in this thesis. At first, the flexible arm dynamic equation is described as
t p
where p is external force which is assumed to be zero. Then, let the deformation w represented as
( ) ( ) (x t =w x wt−φ)
w , cos (B.2)
Substituting (B.2) into (B.1) yields
2 0
Because the flexible arm is uniform in shape and property, its area moment of inertia I and Young’s modulus E are constant. Rewrite (B.3) as
4 0
The general solution of (B.4) is
( )x Ae x Ae x Aei x Ae i x
w = 1 λ + 2 −λ + 3 λ + 4 −λ (B.5)
or
( )x C x C x C x C x
w = 1sinhλ + 2coshλ + 3sinλ + 4cosλ (B.6)
The boundary conditions at the clamped end, x=0, are
( )0 0 0
On the other hand, at the free end, x=l, the boundary conditions are
0
Evaluating the boundary conditions at x=0 and x=l leads to the following homogeneous system of algebraic equations
⎥⎥
For the set of homogeneous equations to have a nontrivial solution, the determinant of the coefficients must vanish. It follows that the determinant vanishes if and only if λ is such that the following condition holds
0 l 1
l λ + =
λ cosh
cos (B.10)
which is recognized as the characteristic equation. The explicit expression for the for the roots of this characteristic equation is not simple, so these roots must be determined using some numerical method, generally yielding an infinite set of eigenvalues λr(r=1,2,L). The first few eigenvalues are given approximately as
follows
Therefore, the approximate values of the first three natural frequencies are obtained as follows
Based on (B.13), the Young’s modulus E could be found by getting the first several order resonant nature frequencies of the flexible arm.
Appendix C
Design Sliding Surface by Lyapunov Method
There are several methods to design the sliding surface. The Lyapunov method which is developed from the aspect of the energy convergence is the easiest one inside
them. Assume the original system is d
Bu Ax
x& = + + (C.1)
where d is disturbance. In order to use Lyapunov method, by pole placement method
the feedback matrix K should be obtained first so that all the eigenvalues of the matrix BK
A
As = − (C.2)
are lied in the left half plane of s domain. Then, the input is designed as v
Kx
u=− + (C.3)
(C.1) is rearranged as
r r m
sx Bv Bd B d
A
x& = + + + (C.4)
where
d is matched disturbance m
is mismatched disturbance
r
r
d
is the null space of matrix B B
For every positive-definite matrix Q, The positive-definite matrix P is surely uniquely exist and satisfying the Lyapunov equation below.
Q PA P
ATs + s =− (C.5)
According to (3.3.2.13), the P matrix is calculated and analyzed for the system stability, define
( )
x x PxV = T (C.6)
Substituting (C.1) and (C.4) into (C.6) yields
( )
x xTQx xTPB(
v dm)
xTPBrdrV& =− +2 + +2 (C.7)
In equation (C.7), if the mismatched disturbance is neglect and the condition is included, (C.7) is written as
=0 Px BT
( )
x =−x Qx≤0V& T (C.8)
The equal appears when x is equal to zero, V
( )
x is a Lyapunov function. The system is stable because that the infinity of x tends to become zero. If the mismatched disturbance is neglected, the stable system occurs when the sliding surface was chosen asdr
=0
=
=Cx B Px
s T (C.9)
In the same time, the control goal is accomplished.