Sensor Location Effect on Flexible Robot Control

8/6/2019 Sensor Location Effect on Flexible Robot Control

1/6

Sensor Location Effect on Flexible Robo t C ontrolA . Green

De pa r tme n t of Mechanical a nd Ae ros pa c eEng ine e r ingCarle ton Univers i tyOttawa, ON, K1 S 5B6, C a na [email protected]

AbstractThe effect of sensors collocated and noncollocatedwith robot actuators is simulated using a controlstrategy for a j la ib le robot . A 12.6m x 12.6m squaretrajectory is tracked with a Jacobian transpose controllaw. Results are initially obtained for nonadaptivecontrol then fo r fuzry logic system (FLS) adaptivecontrol. Sensors collocated with join t actuators do notcaph rrejlaur e and nonminimum phase response whilea noncollocated sensor captures inherent nonminimumphase response causing control action delays. Timedelay in the feedback loop simulates nonminimumphase response and time delay in the fee d forwardcontrol loop provides correction. Results demon strateminimal correction of nonminimum phase response ornonadaptive control but significantly improved resultswith FLS adaptive control for collocated andnoncollocated sensors.

1 IntroductionOperational problems with space robots relate toseveral factors, one most importantly being structuralflexibility and subsequently significant difficultieswith the control systems, especially, for positioncontrol. Elastic vibrations of the links coupled withlarge rotations and nonlinear dynam ics is the primarycause. Also, sensor location adds complication inaccurately detecting and controlling endpoint position.This paper dem onstrates the effect of sensor locationon end point tracking of a square trajectory 12.6m x12.6111 by a two-link flexible robot. The dominantassumed mode of vibration for an Euler-Bernoullicantilever beam is coupled with rigid-link dynamics toform an Euler-Lagrange inverse flexible dynamicsrobot model. Initially, tracking results are obtainedwith nonadaptive control then, further results areobtained with fuzzy logic system (FLS) adaptivecontrol. Square trajectory tracking studies for rigid andflexible dynamics models were presented in previouswork [Z], [4], [6], [7], [SI. Banerjee and Singhoseobtained excellent results using an input shapingmethod to reduce residual vibrations coupled with an

J. Z. Sa s ia de kD e p a r t m e n t ofMechanical a n d AerospaceEng ine e r ing

Carle ton Univers i tyO t t a w a , O N , K I S 5B6, C a n a d a

jsas@ccs .car le ton.ca

inverse kinematics control scheme for bo th linear andnonlinear control laws and a recursive order-nalgorithm for a two-link flexible robot [2]. Sasiadekand Srinivasan applied model reference adaptivecontrol (MRAC) for position and vibration control of asingle-link flexible robot using modal expansion todetermine the first three significant vibration modes[lo]. They demonstrate use of the modal expansionmethod within an MRAC strategy to further reduceerrors and decrease settling time o f transient responsesto step inputs. Various results have been obtainedusing fuzzy logic to con trol robots [4], [6], [SI.

2 Flexible Robot

Fig. I . Flexible RobotThe flexible robot shown in Fig. 1 has planar motionand vibration modes with gravity and friction effectsneglected. The robot parameters are taken fromBanerjee and Singhose [ 2 ] .

2.1 Rigid DynamicsRigid-link robot dynam ics are derived using the Euler-Lagrange formulation [6], [7]. For an independent setof generalized coordinates, q , = q, ...., n, the totalkinetic and potential energies, T an d U, are defined bythe Lagrangian.

L ( q i , q i ) = T - U , i = 1, ...._n (1)The equations of motion are given by:

Fourth International Workshop on RobotMotion and Control, June 17-20,2004 253


2/6

d / d f (aL/d4i) - aL/aqi =4 , i = 1, . , (2)when subjected to a generalized force,Fi, acting on ageneralized coordinate qi. The kinetic and potentialgiven by (3) and (4).

[101:(9)az/ax2 ( E l a2u ( x , l ) / ax2 )dx+m(x ) a2u ( * . r ) / a r2

energies for two rigid links of length L, = L2 = L are = P(X> t )The normal modes Oi must satisfy (10) and its

i = I ,...,n;j = I ...,, ; n = 2; r = 2i=lThe rigid dynamics matrix equations are given by:

where'c = M ( e > e +c r e , e ) e ( 5 )

qL2/3)+m22 (~3+cosB2)m2L2 (!/3+.oEs42)4 3

are the rigid dynam ics inertia and coupling matricesrespectively.

2.2 Flexible DynamicsAchieving accurate tracking control of a two-link robotis compounded by deformation of its flexible links,associated flexural vibrations and nonminimum phaseresponse. For a suitable control strategy the robotmodel must capture the flexible dynamics and residualvibrations must he damped while compensating fornonminimum phase response. Com bining -assumedmodes of vibration for an Euler-Bernoulli beam withrigid-link dynamics captures flexural and nonlinearmultibody interactions of a two-link flexible robotderived in Euler-Lagrange equation form suitable forsimulation. Extensive literature exists on the dynamicsand control of flexible robots using assumed modes[5], [7], [XI, [9], [IO]. This approach accommodatesconfiguration changes during operation, whereas,natural modes must be continually recomputed.Approximate deformation of an elastic beam subjectedto transverse vibrations is given by:

where, Qi( x ) is the assumed mode shape. The shapefunction u ( x , I ) ubstitutes into the Euler-Lagrangedynamics ( I ) and (2), [ l 11. For an Euler-Bernoullibeam with flexural rigidity E l and loadp(x,,tJuniformly distributed, the equation of motion is given

boundary conditions.(10)" 2(.I+:) - m i m ( x N i = o

For which, the solution is given by (8). Elastic kineticand potential energies are given by (1 1) and (12).

Combining rigid and flexible terms the matrixequations are given by:

= M ( q ) q + c ( i1,q)q + K q (13)M comprises rigid and flexible link elements, Ccomprises rigid and elastic Coriolis and centrifugaleffects and K is a stiffness matrix. The generalizedcoordinate vector q comprises joint angles and flexiblelink deformations. In calculating the assumed modes,small elastic deformations is an underlying assumptionwhere second-order terms of interacting elastic mo descan he neglected and orthogonal assumed modessimplifies (13). Omitting elastic Coriolis andcentrifugal components gives C, in (5) . Full dynamicsequations for a multi-degree-of-freedom (do0manipulator have been derived previously [3].

2.3 Cantilever Assumed ModesFrom transverse beam vibration theory, cantilevermode shapes are given by, [IO]:Oc i (x) = cosh hcjx- os hCix k , (sinh hcix- in h,x)where (14)

kCi= cosh, i~tcoshh,L/s inh, i~+sinhh~iLfor which, h,,L = ( i - 0 . 5 ) ~ = I ,......, ar enumerically approximated roots of the characteristicequation cos(h&cosh(h& + 1 = 0 and modalfrequencies given by:

wei = ( h C i L ) 2 & G (15)Proportional and derivative (PD) gains for thedominant assumed mode are given by:K , = diag[m:, a:,] =diag[150.79 150.791 (16)an d

254


3/6

K,,=diag[250c1 2


4/6

Delay time from joint 1 to endpoint for two linklengths, i.e. 9m , is given by:(23)9t --=0.0312sdl - 2RX~33

Delay time from joint 2 to endpoint for one link lengthof 4.5m is given by:4. 5 (24)=-- - 0.0156sd 2 288.33

Average trajectory simulation time is402s for 16000simulation steps (ss) at 0.001 step size, i.e. 0.0252s perstep.: Simulation delay time for joint 1:0.03120.0252dl = = 1 . 2 3 8 ~ ~ (25).'. Simulation delay time for Joint 2:0.0156

- 0 .619s ~ (26 )d 2 = 0 . 0 2 5 2 -For simulation purposes the time delay is implementedin the position and velocity feedback loops at themaximum delay time d,, i.e. worst case.

6 Simulation ResultsTrajectories shown in Figs. 5(a), @), (c) and 6(a), (b),(c) track clockwise starting at bottom l e k Fo rnonadaptive control and collocated sensors, Fig. 5(a)shows pronounced overshoots occurring at eachdirection switch caused by flexibility. Fig. 5(b) showsa magnified view of transients at the first directionswitch for collocated sensors that sense only jointrotations. The two adjacent transients depictnonminimum phase responses for time delays in thefeedback loops to simulate a noncollocated sensor andnonminimum phase response and corrective time-delaycontrol action. Fig. 5(c) is zoomed on the transientpeak to clarify the distinction between transients.Similarly, Figs. 6(a), @), (c) show trajectories and firstdirection switch transients for FLS adaptive control.Comparison of the two sets of results shows there isconsiderable difference between transient responsesfor collocated and noncollocated sensors innonadaptive control than obtained with FLS adaptivecontrol. The effectiveness of corrective time-delaycontrol action is minimal for nonadaptive control butclose to the collocated sensors trajectory for FLSadaptive control. The results obtained using FLSadaptive control demonstrates a significant reductionin vibration and significantly improved results forcorrective time-delay control action as shown in Figs.Matlab/SimulinkTM,Control Systems and Fuzzy LogicToolboxes were used fo r simulations.6(a), @), ( 4 .

7 Summary and ConclusionsComplexities of link flexibility and classical controldeficiencies are overcome by using an FLS adaptivecontrol strategy for a flexible robot. The FLS adaptsthe control law and significantly reduces. trackingerrors. The effect of a noncollocated sensor andnonminimum phase response is simulated by timedelay. Results show a drastic difference innonminimum phase response over a relatively smallchange in sensor location for nonadaptive control. Thedifference is less for FLS adaptive control. The effectof corrective time-delay control action is minimal fornonadaptive control but significant for FLS adaptivecontrol. The FLS has overriding effectiveness ontracking control regardless of sensor location. Resultsmay represent more complex dynamics and multi-dofsystems

ReferencesH. L. Alexander, "Control of Articulated andDeformable Space S tructures," MachineIntelligence and Autonomy o r AerospaceSystems, edited by E . Heer an d H. Lum, A I MProgress in Astronautics and Aeronautics, AIAA,Washington, DC, 1988.A. K. Banerjee and W. Singhose, "CommandShaping in Tracking Control of a Two-LinkFlexible Robot," AIAA Journal of Guidance,Control and Dynamics, Engineering Note,Vol. 21 , No. 6,pp.1012-1015, Reston, VA, 1998.W. Beres and I. 2.Sasiadek, "Finite ElementDynamic M odel of Mu lti-link FlexibleManipulators," Applied Mathematics andComputer Science, Vol. 5,No. 2, pp. 231-262,ISSI, University of Zielona G6ra, Poland, 1995.C . W. de Silva, Intelligent Control: Furry LogicApplications, CRC Press, Boca Raton, FL, 1995.A . R. Fraser and R. W. Daniel, PerturbationTechniques or Flexible Manipulators, TheKluwer International Series in Engineering andComputer S cience, Vol. 138, Kluwer, Dordrecht,Th e Netherlands, 1991.A. Green and J. Z. Sasiadek, "Methods ofTrajectory Tracking for F lexible Manipulators,"Proceedings of the AIM Guidance, Navigationand Control Conference, Reston, VA, 2002.

25 6


5/6

[7] A. Green and J. 2. Sasiadek, "RobotManipulator Control for Rigid and AssumedMode Flexible Dynamics M odels", Proceedingso the AIAA Guidance. Navigation and ControlConference, Reston, VA, 2003.A. Green and J. Z. Sasiadek, "Adaptive Controlof a Flexible Robot Using Fuzzy Logic," Journalofcuid ance, Control and Dynamics, AIAA, to bepublished.T. G. Mordfin and S. S. K. Tadikonda, "TruthModels for Articulating Flexible MultibodyDynamic Systems", Journal o Guidance,Control and Dynamics, Vol. 23, No.5, pp. 805-811, AIAA, Reston, VA, Sep-Oct 2000.

[8 ]

[ Y ]

NVH Ni M I N z w o p n PL PH p\RI

-50 0 50-Fig. 2. Membership Functions for Input Variables&,andS2 .

[IO] J. Z. Sasiadek and R. Srinivasan, "DynamicModeling and Adaptive Control of a Single-Link Flexible Manipulator", Journal oGuidance, Control and Dynamics, Vol. 12,No. 6, pp . 838-844, AIAA, Reston, VA, 1989.W . T. Thomson, Theory o Vibration withApplications, (2nd ed), Prentice-Hall, UpperSaddle River, NJ, 1981.

[1 ]

0 IFig. 3. Membership Functions for Output Variable h.

sGal"=. Y

Fig. 4. Nonadaptive and FLS Adaptive Control Strategies.

Table 1. Fuzzy Logic System Rule Matrix82NVH NH NL NVL ZERO PVL PL PH PVH

hNV H PMAX PVVH PVH PH PM PH PVH PVVH PMAXNH PVVH PVH PH PM PL PM PH PVH PVVHNL PVH PH PM PL PVL PL PM PH PVHNVL PH PM PL PVL PVVL PVL PL PM PH

ijl ZERO PM PL PVL PVVL ZERO PVVL PVL PL PMPVL PH PM PL PVL PVVL PVL PL PM PHPL PVH PH PM PL PVL PL PM PH PVHPH PVVH PVH PH PM PL PM PH PVH PVVHPVH PMAX PVVH PVH PH PM PH PVH PVVH PMAX

251


6/6

( c )Fig. 6. FLS Adaptive Control

25 8

Sensor Location Effect on Flexible Robot Control

Documents