a prototype flexible robot arm-an interdisciplinary undergraduate project

IEEE TRANSACTIONS ON EDUCATION, VOL. 35, NO. 1, FEBRUARY 1992 83

A Prototype Flexible Robot Arm-An Interdisciplinary Undergraduate Project

Hemanshu R . Pota, Member, IEEE

Abstruct- This paper discusses an interdisciplinary undergraduate project to set up a prototype flexible robot link The project consists of modeling, fabrication, and control of the flexible link The combination of already well-known Lagrange and finite element methods seems to be a great advantage in this particular modeling procedure. Lightweight flexible links although known to save a large amount of operational energy are not popular in the industry because of the difficulty in their modeling and control. Following the methodology suggested in this paper, modeling is reduced to a mechanical procedure capable of efficient computer implementation. A simple controller is suggested, the controller is designed using the MATLAB simulation package, and the results are compared graphically. The lessons learned from the project are reported.

I. INTRODU(=TION N this paper, we discuss a final-year undergraduate project I which is highly educative and has a potential for making a

research contribution. A project of this type rightly educates the students in the process of research and gives them a foretaste of things to come. Various aspects of engineering covered by this project are as follows:

1) Lagrangian modeling technique, 2) finite element method, 3) flexible mechanical structures, 4) control theory, and 5) robotics. Let us first introduce the problem. Presently used robots

have rigid and heavy links and a replacement of these heavy rigid links by light flexible links will result in considerable savings in operational energy costs [8]. Moreover, presently envisaged space stations have to be assembled using robots with flexible links as not only the available energy is at a premium but also the payload capacity of the space vehicles is limited. One severe limitation of the flexible link so far has been the uncertainty in its tip position. As most applications involving robots need an accurate tip placement, the industry has preferred rigid links in spite of its very poor payload to link mass ratio. The structural strength of the flexible links is adequate for the sort of payload requirement of commercial robots,.Hence, a good tip position controller would go a long way in making the commercial use of lightweight flexible arms feasible [8].

Manuscript received April 15, 1990; revised March 15, 1991. The author is with the Department of Electrical Engineering, University

College, University of New South Wales Australian Defense Force Academy, Campbell, ACT Australia.

IEEE Log Number 9105121.

A good and reliable tip positioning controller would need the tip position as a feedback signal. There are broadly two difficulties about tip position feedback: one is in actually measuring the tip position but, with the availability of high- precision optical measurement devices, this is no more a problem [ 5 ] , [ l l ] ; secondly the use of this information with great profit is difficult unless the model is known with reasonable accuracy. This difficulty is due to the RHP zeros (as discussed later) and an infinite number of modes of vibration. In this paper, an attempt is made to develop a methodology to model the flexible link robot which is suitable for a control engineer who is not too familiar with the theory of vibration. The model is simple, capable of answering “what if” types of questions, and useful enough for a controller design.

There are numerous ways to model an elastic beam [l] , [4], [5], [19]. The classical model is the following fourth-order Bernoulli-Euler partial differential equation with associated boundary conditions:

84w(x, t ) @ W ( X t ) EI- = - p A b A 8x4 a t 2

where the symbols have the usual meaning [l] . This model, although theoretically sound, is not convenient

for control engineers. We end up with an infinite-dimensional state-space representation which has to be reduced to a finite- dimensional representation. There are other modeling techniques which directly result in finite-dimensional models like the assumed-mode method [l], but they all require a good understanding of the theory of vibration and an endless atten- tion to the boundary conditions. In comparison, the modeling technique presented here is simple and mechanical in its application.

This paper is organized as follows. Section I1 discusses the Lagrange/finite element modeling technique in sufficient detail. Sections I11 and IV are devoted to one-axis-of- freedom 1.5 m long aluminum link modeling and control, respectively. The last section lists the conclusions that can be drawn from the work done so far. One point about the notation: boldface face italic lower-case letters denote a vector and boldface italic upper-case letters denote a matrix.

11. LAGRANGEFINITE ELEMENT MODELING TECHNIQUE The Lagrangian modeling procedure is based on first form-

ing an expression, called the Lagrangian, as a difference between the system kinetic energy (KE) and the system potential energy (PE). The KE and PE are expressed in terms of generalized coordinates. Finite element analysis enters our

0018-9359/92/0200$03.00 0 1992 IEEE

~ ~ _ _ ~ _ _ _ _ _ _ _ ~ ~ __

84 IEEE TRANSACTIONS ON EDUCATION, VOL. 35, NO. 1, FEBRUARY 1992

w’ t

Y y; t

Fig. 1. A flexible link of length 1 and cross section 6 x h.

Fig. 2. The ith element.

methodology here because we use it to find expressions for KE and PE. Once this is done, we form the Lagrangian and use the Language-Euler equations to describe the system dynamics [l], as explained below.

Let T 2 total system kinetic energy v 2 total system potential energy

then the Langranian L is defined as

L ~ T - v

and the differential equations describing the dynamics of an n-degree-of-freedom system are given by

where

qr , = 1’2, . . . , n are the generalized coordinates and

Fig. 3. One-axis-of-freedom flexible arm.

V , where

ith element PE = iqTKiq;

12 61; -12 61; K ; = - [ El; 61i 41: -61; 21;

1; -12 -61; 12 -61; 61i 21: -61; 41; 156 221; 54 -131; 221; 41’ 131; -31: 54 131; 156 -221;

-131i -31: -221i 41;

and A; f cross- section area of the ith-element p A mass per unit volume of the ith-element E k Young’s modulus 1; The matrices Mi and K ; are known as mass matrix and

stiffness matrix, respectively. After repeating the same process of finding Mi and K ; for all n elements, we have

area moment of inertia of the ith-element.

T f total system KE = f qrMiqi V total system PE = i qTK;qi.

Note that each vector q; consists of four variables and, Q r , T = 1,2 , . . . , n are the generalized external forces.

Next we discuss how to find the KE and PE of a link given its geometry, specific weight, and the Young’s modulus, such as shown in Fig. 1. First we divide the link into n elements. The elements are sequentially numbered i = 1, .. . , n and

Fig. 2. Now consider one such element and deflections in the a-direction only.

to maintain elemental continuity, we must ensure that the adjoining elements have the same value for the common deflection variables. For example, elements 1 and 2 deflection and the rate of deflection will be constrained as

w22 = w21

every element has two nodes j and ( j + l), as shown in e22 = -921

With this in view, we define a vector q of dimension (271 + 2).

Let 1; 2 length of the ith element wj; deflection at node j of the ith element B j ; slope of the beam at node j of the ith element. Note that normally j = i and j + 1 = i + 1, that is, the first

element has nodes 1 and 2, the second element has nodes 2 and 3, and so on. Now using a standard result of the finite element analysis [3], we get the following expressions for the ith-element KE and PE:

T; 2 ith element KE = fqTM;qi

A q =

POTA A PROTOTYPE FLEXIBLE ROBOT ARM 85

TABLE I CHARACTERISTIC FREQUENCIES IN W/s

Tip mass in Two Three Four kg elements elements elements

0.0 21.42 19.73 44.06 46.28

100.90

0.229 6.88 6.82 28.39 28.25

85.45

1.0 3.49 3.48 25.78 25.46

82.95

19.58 45.59

102.74 195.36

6.82 28.19 83.65

176.95 3.48

25.41 80.67

174.76

With this definition of the vector q, we can rewrite the expressions for the system KE and PE as

1 - T = -qTMq 2 1 2

v = -qTKq

where the matrices k and K are computed accordingly, this being a straightforward manipulation of Mi and K ; matrices and is known as the element assembly procedure in the finite element literature [6]. Boundary conditions are incorporated by appropriately constraining the variables in q. Normally, q is rearranged and partitioned such that

where the variables in qa are active and the ones in qc are constrained. In our example which corresponds to a cantilever,

9 c = [;;I = [ : I . This enables us to write expressions for KE and PE after dropping the coefficients of qc, giving

(2a)

(2b)

1 2 1 2

T = -4:Maaqa

v = -q:Kaaqa.

At this stage, we perform another operation known as mass condensation. First we define two vectors q1 and q2 such that

Note that (2a) and (2b) can be rearranged and written as

ffequcncy in rad

mquency in rad

Fig. 4. Bode ploi of the transfer function B(s) /T(s ) .

I I

A simple controller with compensator in the feedfonvard path. Fig. 5.

l+KDs

Fig. 6. A simple controller with compensator in the feedback path

where Mij and K;j are derived after appropriate rearrange- ment of Ma, and K,,, respectively. From this, after using a mass condensation procedure given in [2] we can write

( 3 4 1 2

T = -qTMql

V = Iq ' fKq , 2

where

K = K11 - K12KT2K21. (4b)

Now as we have the KE and PE expressions (3a) and (3b) we can write the system dynamic equations using (1) for an unloaded cantilever beam as

(5) Mq, + Kq, = 0.

Let us summarize the Lagrange/finite element technique as follows.

86 IEEE TRANSACTIONS ON EDUCATION, VOL. 35, NO. 1, FEBRUARY 1992

TABLE I1 PERFORMANCE OF FEEDFORWARD CONFIGURATION OF FIG. 5

.......... ................. ............... Peak Settling ............... Torque Time in i

................... ................. ...............

S. No. K 71 72 in N . m ............... ............... .............. S

1.0 0.15 10 0.01 150 1 2.0 0.13 10 0.05 30 1 3.0 0.25 5 0.05 25 1 4.0 1.1 1 0.05 30 oscillating 5.0 0.11 10 0.001 1000 1

0 0.5 1 1.5 2 4.5 -10

0 0.5 1 15 2 Timc (sec.) Time (sec.)

Fig. 7. Step response of the feedfonvard configuration of Fig. 5.

2, I 4 , 1

.....................

8 0.5 ................... P r 0 .

4.5 1 2 3 Time (sec.) Time (sec.)

Fig. 8. Step response of the feedback configuration of Fig. 6.

1) Divide the link into n elements. 2) Form the mass and stiffness matrix Mi and Ki, i =

3) Form the matrices &f and K as discussed in the element

4) Incorporate the boundary conditions to get Ma, and

5 ) Perform mass-condensation using (4a) and (4b) to get

6) Use (1) to get the system dynamic equation. To demonstrate the simplicity of this procedure in ac-

counting for payload, input torque, and changing boundary conditions, we consider the one-axis-of-freedom link next.

1 , 2 , . . * , n.

assembly procedure.

Kaa .

M and K .

111. AN EXAMPLE: ONE-AXIS-OF-FREEDOM LINK MODELING

The methodology developed in the previous section will be illustrated by the means of developing the model of the prototy e flexible beam such as shown in Fig. 3.

i from the origin.

KE and PE as

..F Let I

For small displacements qi we can easily write the system

= [Zl, 12, . . . . In] where Ei equals the distance of node

T q+Ze] M[k+Zs] ]

1 2

PE V = - q T K q

-T with IH = motor mass-moment of inertia, I , 4 IH + I Mi, and MI A Mi.

The Lagrangian can be written as 1 2

L = - pee2 + iTMq + qTM1b - qTKq].

Using (1) to get the system dynamic equations, we get

I,@ + ~ T i j = T

M q + M 1 @ + Kq = 0 (6)

TABLE I11 PERFORMANCE OF FEEDFORWARD CONFIGURATION OF FIG. 6

Peak Settling Torque Time in

S. No. li h-0 in N . m S

1 .o 4 0.5 4 2 2.0 10 0.4 10 2 3.0 5 0.5 5 2 4.0 4 0.6 4 1.5 5.0 4 0.65 5 1.5 6.0 5 0.8 5 3 7.0 7 0.7 7 2

where T is the applied motor torque. To include the effect of

the tip mass mt we simply add the term ;mt ( Z O + 4") to the kinetic energy expression, where 1 is the link length and qn is the node n deflection. Now consider an aluminum alloy flexible link with the following parameters:

1 = 1.5 m,

IH = 0.023 kg . m2

link mass, m f pAZ = 0.686 kg,

I = 8.31934E - 11 m4. E = 68.947573 + 9 Pa

In Table I ,we give a comparison of characteristics frequencies of the motor plus the flexible arm system obtained for different element divisions and different payloads.

The additional effort required to get characteristic frequencies for different payloads, different links, etc., is very little. It is only a matter of changing a parameter and then reexecuting a small subroutine.

Let us write the state-space model for three-element division of the link with no tip-mass.

Define

21 & e , 2 2 A 41, 2 3 A q 2 , 24 5 q3 A '

25 = 8, 26 41, 2 7 42, 28 42

X T = [XI $2 * * ' 281.

The condensed mass matrix and stiffness matrix are given as

23.82 380.55 59.48 ; 1 369.8 23.82 -27.94 M=&[

420 -27.94 59.48 122.01

1313 [ 12 -16 -+6]

80 -46 12 K = - 81EI -46 44

POTA: A PROTOTYPE FLEXIBLE ROBOT ARM

1 0 0 0 - 0 1 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -

O O 0 l x

87

TABLE IV PERFORMANCE OF FEEDFORWARD CONFIGURATION AND TIP POSITION FEEDBACK 0; FIG. 11

Peak Settling Gain S. Torque Time in Margin wcg in

NO. K I K 71 72 in N . m s in dB rad

1.0 1.1 0.12 30 0.001 5000 1 1.95 5.52 2.0 1.0 1.5 2 0.01 - 25 1 4.27 5.175 3.0 1.0 1.5 2 0.05 -5 1 1.72 4.73 4.0 1.0 0.5 5 0.05 5 1 4.78 4.74 5.0 1.0 0.25 10 0.05 8 1 5.1 4.78 6.0 1.0 0.25 10 0.01 40 1 7.42 5.32 7.0 1.0 0.15 10 0.01 30 1.5 12.77 5.18

and the state-space representation is

X =

- 0 0 0 0 0 0 0

.o

0 0 0 0

5525 -8113 -1710

-13702

0 0 0 0

-1381 4108

7529 -2133

0 0 0 0

230.2 -1103 1044

-2464

1 -2.17 I ::::;I

y ( t ) = [Z3 0 0 1 0 0 0 O]X; y ( t ) is the tip position in m

O ( t ) = [l 0 0 0 0 0 0 OIX.

The eigenvalues of the A matrix give the characteristic frequencies, and the tip position step response has a back kick testifying for the RHP zero. The RHP zero models the time delay inherent in all flexible links. The applied torque cannot immediately affect the tip position so as the base of the link attached to the motor moves forward, the tip position in a reaction kicks back. As the time delay is represented by exp(-sTI) = (1 - sT1 + s2T,2/2.. -), we can see that the problem in controlling the flexible link system with a RHP zero will be similar to a time-delay system where Tf is negligible. In the next section, we discuss the controller design.

IV. CONTROLLER DESIGN We intend to design a feedback controller for the system for

which a representation was obtained in the previous section. The controller is to be designed such that the closed-loop system has a satisfactory step response. Suppose we need the tip of the link to move by 1 m, then the simplest thing to do is to feedback the hub angle O ( t ) and design a compensator which gives a satisfactory closed-loop response. We can also design a compensator with the tip position y ( t ) as the feedback signal. But, from the analysis in Section 111, we know that the transfer function Y ( s ) / T ( s ) is nonminimum phase and an error in modeling or gain selection will result in an unstable

TABLE V PERFORMANCE OF FEEDBACK CONFIGURATION

AND TIP POSITION FEEDBACK OF FIG. 12

Peak Settling Torque in Time in

S. NO. K I I< Ii-D N . m S

1 .o 0.6 5 0.5 0.5 2 2.0 0.7 10 0.4 0.8 1.5 3.0 0.25 5 0.2 0.3 2 4.0 0.4 5 0.3 0.4 2

system [14], so in order to keep things simple we initially explore the possibility of using only the hub angle O(t) as the feedback signal.

The thing to note is that the steady-state value of the hub angle O(t) and the tip position y ( t ) is the same, meaning that we can control y( t ) by "dead-reckoning." Secondly, by looking at the Bode plot of O(s)/T(s) in Fig. 4 we see that the system poles and zeros alternate on the jw-axis. Note that zero deg in the phase plot is the same as 360 deg. With this in mind, the phase plot in Fig. 4 is confined between +180 deg and -180 deg. This means that the transfer function sO(s) /T(s) is passive [12] and that any PD compensator will stabilize the system [13]. This type of control is also called a colocated control in the literature [14] because the sensor [measuring O(t)] and the actuator are physically located at the same point. The PD compensators can be either in the feedforward path (Fig. 5) or in the feedback path (Fig. 6). Table I1 shows the effect of varying different gains in the feedforward configuration of Fig. 6 and Table 111 shows the same for the feedback configuration of Fig. 7. By comparing the two results, we see that we can either achieve a settling time of 1 s with a maximum torque requirement of 25 N . m (Fig. 7) or a settling line of 1.5 s with a maximum torque requirement of 4 N . m (Fig. 8). The motors we have for the project can supply a peak torque of 15 N - m; hence, we choose the feedback configuration of Fig. 6.

The next question we ask ourselves is: can we use the tip position y ( t ) in addition to the hub angle O(t) to damp the vibrations of the flexible link and obtain a settling time of less than or equal to 1 s and simultaneously satisfy our torque constraints? An answer to this question was given in the affirmative in [14] and here we try and improve the system response by a simple addition to both feedforward and feedback configurations of Figs. 5 and 6, respectively.

88

100 101 102 103 frequency in rad

100 101 102 103 frequency in rad

Fig. 9. Bode plot of the closed-loop transfer function in Fig. 5.

10' 102 frequency in rad

-A_."

100 103

frequency in rad

Fig. 10. Bode plot of the closed-loop transfer function in Fig. 6.

L I ' I Fig. 11. A controller with tip position feedback with feedforward

configuration.

The bode plots of the closed-loop systems, with tip position as the output, of Figs. 5 and 6 are given in Figs. 9 and 10, respectively. Looking at the plots, we clearly see that we need an integral compensator to offset the steady-state error. With this in view, we have modified the two simple configurations as shown in Figs. 12 and 13. Table IV shows the effect of varying the various gain parameters on the system responses for the system configuration shown in Fig. 11, and Table V shows the same for the system configuration shown in Fig. 12. By comparison,we see that the control configuration of Fig. 11 gives a superior performance with 1 s settling time and a peak torque requirement of 8 N . m. Fig. 14 shows the step

IEEE TRANSACTIONS ON EDUCATION, VOL. 35, NO. 1, FEBRUARY 1992

K MOTOR PLUS ARM

l+KDs I - I I

Fig. 12. A controller with tip position feedback with feedback configuration.

1 2 3

-2 ;L 1 2 3

Time (sec.) Time (sec.)

Fig. 13. Step response of the feedforward configuration of Fig. 11.

-0.5 0 : p

1 1 2 3

Time (sec.) Time (sec.)

Fig. 14. Step response of the feedback configuration of Fig. 12.

response for the configuration in Fig. 12 for case 2, Table V. The step response for the feedforward configuration of Fig. 11 with KI = 1, K = 0.25,q = 10, and 72 = 0.05 is shown in Fig. 13. This response looks satisfactory and is in some ways superior to the other responses; hence, we choose this controller configuration. The design process is very simple and needs only a simulation package like MATLAB to perform the entire analysis and design.

System Hardware During the initial stage of the experimental setup, we have

a brushless dc motor as a torque producing mechanism, an I-section aluminum link 1.5 m long. The tip position is sensed using a UDT photosensitive diode mounted at the base of the link, i.e., at the top of the motor and a laser source is mounted at the tip of the link. The configuration of 68020 microprocessor and 68881 math coprocessor connected via a VME Bus to ADC and DAC cards is used as a digital controller.

V. CONCLUSION

At the conclusion of the project, it was found that the students had assimilated the material remarkably well and had acquired a real feel for the interdisciplinary approach needed to solve engineering problems. During the project, various theoretical issues were also discussed like robust control, model order reduction, infinite-dimensional systems, and their control. From the research point of view, the following con-

POTA: A PROTOTYPE FLEXIBLE ROBOT ARM 89

clusion can be drawn. The combination of Lagrange and finite element techniques seems to have a great potential in modeling flexible robot links. The model is accurate enough to enable a tight controller design. Although a very simple numerical example is selected for illustration, the extension to multiple links and multiple-degree-of-freedom is merely a routine task after understanding the coordinate transformation procedures discussed in [7]. This procedure is much simpler compared to the manipulation of the classical Bernoulli-Euler equation to yield a finite-dimensional state-space representation and is ideal for a designer who does not need to master the intricate details of vibration theory. The simulation shows that all the essential features of the elastic link have been modeled with good accuracy. After having set up a prototype flexible link and analyzing the problems associated with the control, we intend to extend the project in the following ways:

1) use colocated active control methods [9] to get over the RHP zero problem;

2) use the modified output suggested in [lo], [16] so as to eliminate the RHP zeros from the transfer function itself: and

3) use robust H-infinity control design techniques to build in robustness with regard to the tip-mass variation.

The above extensions are state-of-the-art research in flexible arm control and also comprehensible by the final- year undergraduate students.

REFERENCES

[ l ] L. Meirovitch, Elements of Vibration Analysis. New York: McGraw- Hill, 1975.

[2] __,Computational Methods in Structural Dynamics. Alphenaanden Rijn, The Netherlands: Sijthoff and Noordhoff, 1980, pp. 370-372.

[3] R. R. Craig, Jr., Structural Dynamics, An Introduction to Computer Methods. New York: Wiley, 1981.

[4] G.G. Hastings and W.J. Book, “A linear dynamic model for flexible robot manipulators,” IEEE Control Syst. Mag., vol. 7, pp. 61-64, Feb. 1987.

(51 E. Schmitz, “Robotic arm control”, Ph.D. dissertation, Aero./Astro., Stanford, Univ., Stanford, CA, 1985.

[6] J. N. Reddy, An Introduction to the Finite-Element Method. New York: McGraw-Hill, 1984, pp. 115-118.

[7] K. S. Fu, R. C. Gonzalez, and C. S. G. Lee, Robotics: Control, Sensing, Vision, and Intelligence. New York: McGraw-Hill, 1987.

[8] G.B. Andeen, Ed., Robot Design Handbook Sydney: SRI Interna- tional, McGraw-Hill, 1988, ch. 15.

[9] J. H. Davis and R. M. Hirschom, “Tracking control of a flexible robot link,” IEEE Trans Automat. Contr., vol. 33, pp. 238-248, Mar. 1988.

[lo] D. Wang and M. Vidyasagar, “Transfer functions for a single flexible link,” in Proc. IEEE Int. Con$ RoboticsAutomat., 1989, pp. 1042- 1047.

[ll] W. J. Wang, S. S. Lu, and C. Hsu, “Experiments on the position control of a one-link flexible robotic arm,” IEEE Trans. Robot. Automat., vol. 5, pp. 373-377, June 1989.

[12] C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties. New York: Academic, 1975.

[I31 W. Gevarter, “Basic relations for control of flexible vehicles,” A M J., vol. 8, no. 4, pp. 666-672, Apr. 1970.

[14] R.H. Cannon and E. Schmitz, “Initial experiments on the end-point control of a flexible one-link robot,” Int. J. Robot. Res., vol. 3, no. 3,

I151 V.A. Spector and H. Flashner, “Sensitivity of structural models for noncollocated control systems,” Trans. ASME, J. Dynam. Syst., Mea- surement, Contr., vol. 111, pp. 646-654, Dec. 1989.

[16] H.R. Pota and M. Vidyasagar, “Passivity of flexible beam transfer functions with modified outputs,’’ in Proc. IEEE Int. Con$ Robot. Automat., Sacramento, CA, Apr. 7-11, 1991.

[I71 R. L. Wells, I. K. Schueller, and J. Tlusty, “Feedforward and feedback control of a flexible robot arm,” IEEE Contr. Syst. Mag., pp. 9-15, Jan. 1990.

[18] F. Pfeiffer, “A feedforward decoupling concept for the control of elastic robots,”J. Robot. Syst., vol. 6, pp. 407-416, Aug. 1989.

[ 191 J. L. Humar, Dynamics of Structures. Englewood Cliffs, NJ: Prentice Hall. 1990.

pp. 62-75, 1984.

Hemanshu R. Pota (S’80-M’Sl-S’82-M’85) received the B.E. degree from Sardar Vallabhbhai Regional College of Engineering and Technology, Surat, India, in 1979, the M.E. degree from the Indian Institute of Science, Bangalore, India, in 1981, and the Ph.D. degree from the University of Newcastle, NSW, Australia, in 1985, all in electrical engineering.

He spent one year at Iowa State University, Ames, as a Post-Doctoral Fellow/Assistant Professor during 198-1986, He then spent a few years at the

Auckland University in New Zealand, and currently he is with the University College, University of New South Wales, Australian Defence Force Academy, Canberra, Australia. He has held summer appointments, during 1990 and 1991, at the Centre for Artificial Intelligence and Robotics, Bangalore, India. His research interests include flexible structure robotics and power system dynamics.

a prototype flexible robot arm-an interdisciplinary undergraduate project

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