Dynamic Simulation of Underwater Vehicle Manipulator Systems

B.M. Oscar De Silva201070588 - oscar.desilva@mun.ca

Memorial UniversityFaculty of Engineering

Engineering 9095: Introduction to Marine Cybernetics

Abstract

Control of Underwater vehicle manipulator systems (UVMS) are limited to human intervened tele operation in almost all practicalapplications. Extending these missions to autonomous or semi autonomous operation requires accurate dynamic modeling of thesystem. Such modeling assists system design and controller testing of autonomous UVMS. The study performs a dynamic analysisto capture relative contributions of manipulator inertial forces, static forces and hydrodynamic forces which constitute as the overallcoupling forces on the vehicle. A system with a three degree of freedom manipulator was modeled using a recursive Newton eulerapproach. Hydrodynamic effects and coupling effects between vehicle and manipulator were introduced using models proposedby previous research. The system was simulated for a task with increasing speeds. It was observed that all dynamic Forcesintroduce significant coupling in high speeds. At low speeds only static and hydrodynamic moments has significant effects. Furtherimprovements to the model are discussed with future work on the study.

Keywords: Autonomous underwater vehicles, Dynamic modeling, Hydrodynamics, Manipulators, Recursive Newton EulerAlgorithm

1. Introduction

Underwater vehicle manipulator systems (UVMS) are vastlyemployed for underwater research and survey. Manipula-tor tasks carried underwater range from inspection of welds,pipeline, and oceanography to active tasks such as cable burial,sampling, obstacle removal etc.. Missions carried underwatermostly employ tele operated ROVs where a human controls themanipulator tasks while overcoming the unwanted movementsof the vehicle due to manipulator handling operation.

An improvement to this would be semi autonomous controlwhere an assistive controller overcomes the unintended mo-tions while a human operator performs strategic task planningremotely. A fully autonomous case in practical applicationswould be a far cry, due to the various complexities involveddrawing from mission planning, fault tolerance and adaptationto dynamic environment. Whatever the case an accurate modelof the UVMS would serve as a base for implementing variouscontrollers in achieving fully autonomous operation.

Accurate modeling of UVMS systems poses many chal-lenges. Underwater vehicles are typically modeled using semiempirical methods to include hydrodynamic effects ,thrusterdynamics and other approximations of environmental effects.Addition of a manipulator makes it very difficult to use stan-dard system identification methods to capture the full model.So many previous studies employ theoretical extension of ma-nipulator dynamics to a fluid environment. The complexity ofthe model used with respect to capturing various effects fromrigid body dynamics/ hydrodynamics to environmental forces;

depends on the performance level which is desired from the sys-tem. Based on the levels of performance or control required, themodeling of the system can be relaxed to exclude certain effectsbased on order of significance.

The work presented performs a dynamic modeling of an 6DOF Odin equipped with a 3 DOF manipulator fully captur-ing possible contributions from various factors. The case studyreports the contribution of various forces involved to the perfor-mance of the system.

2. Background

The study can be broadly defined as solving an n body prob-lem in a fluid environment. Solving of the forward dynamicproblem is necessary for simulation, while feed forward con-trol, feed back linearization methods require solving of the in-verse dynamic problem. This require efficient solving meth-ods which are both extendable to n DOFs and can incorporateexternal forces applied on the system in a fluid environment.The methods widely used are the Recursive Newton Euler Al-gorithm(RNE), Composite Rigid Body Algorithm(CRBA) andArticulated Body Algorithm(ABA) all based on Newton-Eulerformulation[1]. The classic Lagrangian formulation was con-sidered computationally expensive with O(n4). While mostNewton -Euler methods presented fall in to O(n) with differ-ent performances based on number of DOFs and nature of theproblem[1]. The main obstacle in forming the model using themethods discussed above, is the approximation of the externalforces introduced by the fluid.

Preprint submitted to Dr.Ralf Bachmayer December 23, 2010

Several papers has addressed modeling of hydrodynamics inUVMS[2] [3] [4]. Levesque et al. and McMillan et al. usesconstant hydrodynamic coefficients to estimate viscous dragand added mass on the manipulator but lacks experimental vali-dation of the models. McClain et al. proposes a semi empiricalformulation with variable coefficients with experimental valida-tion of the results to a 1 DOF case[4]. Mahesh et al.[5] proposesadaptive techniques for discrete time approximations of the fullnonlinear model but lacks experimental validation of the simu-lations.

Modeling and simulation of the n body problem in UVMSis studied incorporating the hydrodynamic modeling discussedabove. [3] uses a Articulated Body Algorithm(ABA) for effi-cient simulation. [5] uses spacecraft attitude dynamics programNBOD2 in his simulations.[6] proposes a model formulationusing Kanes method. This allows relaxed approaches for ex-plicit formulation of dynamic equations with straight forwardmeans of adding external forces to the system.

Control of UVMS is studied by [7] where feed forward meth-ods are proposed. With accurate model developments, [4] usesfeedforward decoupling control to experimentally verify perfor-mance improvements. Methods not relying on accurate modelsare also proposed; Mahesh et al. uses adaptive controllers forcoordinated control of UVMS [5].

Reference Hydrodynamic model System modelLevesque et al.[2] Cd = 1.1 -McMillan et al.[3] Cd = 1.2,Cm = 1 ABAMcClain et al.[4] Cd,Cm = f (s/D) RENAMahesh et al.[5] parameter adaptation NBOD2Tarn et al.[6] Cd = 1.1,Cm = constant KANE

Table 1: Summary of main modeling methods proposed by previous re-searchers. ABA, RENA,.. are n body problem solving methods discussed insection 2

3. Model Development

In this study, the model is developed using an RecursiveNewton euler algorithm. This enables computation of the forceat the vehicle manipulator joint, which is used in decouplingcontrol strategies proposed by [4]. The hydrodynamic approx-imation is based on [3] constant coefficient models. AlthoughMcClain et al. proposes a more accurate semi empirical modelthis was not extendable to a n DOF case with the available ex-perimental data.

3.1. Underwater vehicle modelingThe modeling method assumes the underwater vehicle is

modeled using standard model identification methods. Un-like Kanes method and lagrangian formulation where manip-ulator and the vehicle is modeled together; this study usesthe available underwater vehicle model and adds the couplingforcesF joint caused by the addition of the manipulator as distur-bances to the available equation of motion.

Equation 1 also termed the submarine equation captures thefull dynamic model of the Underwater vehicle. The Equation of

motion is expressed in the body fixed frame attached at the Cen-ter of Buoyancy of the vehicle. The notations used and termsare described in appendix A [8].

Mvv + Cvv + DRB(v)v + Fg + F joint = control (1)

3.2. Manipulator modeling

The mathematical model of the manipulator is reported inequation2. A rigid body model of the manipulator was intro-duced with additional hydrodynamic forces hydro which addsthe torques the joint motors should provide for the additionalhydrodynamic effects. The term Fmobilebase captures the addi-tional inertial force due to accelerations of the vehicle.

Mvq + C(q) + Fg(q) + hydro + mobilebase = q(control) (2)

3.3. Vehicle-Manipulator coupling force modeling

The Coupling forces introduced in Equations 1 & 2 are mod-eled using a Recursive Newton Euler Approach. Forces actingon each link of the manipulator is summed and propagated tothe base. Forces and momenta at a general link i can be repre-sented by figure 1.

Figure 1: Forces acting on a link used for the recursive Newton euler formula-tion

The calculation of individual forces and moments illustratedabove are summarized as follows;Rigid Body forces

1. GravityThe Gravity effect is considered using equation 3.

~f gi = m~g (3)

2. Hydrostatic byoyancyThe buoyancy effect is considered using equation 4.

~Fhydb i = V~g (4)2

3. Link inertiaIndividual link inertia effects are captured by equation 5.

~f ic = mi ~aic~nic = Ii ~ic ~ic Ii ~ic

(5)

4. Mobile base effectThe additional inertia force due to base acceleration is con-sidered using equation 6.

~f ic = mi ~abase~nic = Ii ~base ~base Ii ~base

(6)

Hydrodynamic forcesThe hydrodynamic forces are calculated for each link usingBlade Element theory[4]. The link is divided to 5 elements dlialong its length and the total hydrodynamic force at the linksbase joint is taken as the summation of individual hydrody-namic forces acting on each element (equation7). McClain etal.[4] reports considering 4 elements is sufficient to converge ata good approximation in his studies. The hydrodynamic forcesconsidered at each blade element were the Added massFhyda ,Viscous dragFhydd , and Fluid acceleration Fhyd f .

~Fhi =

~dF j

~Nhi =

~l j ~dF j (7)

5. Added massThe force due to added mass was considered 0 in the linkaxis direction and maximum in the normal direction hats.In calculating the added mass the relative acceleration ofblade element dli with respect to the fluid is used and anadded mass coefficient of Ca = 1 was used.

~dFhyda j = CaVg(~a fj .s).s (8)6. Viscous drag

Viscous drag on each element was calculated using a con-stant drag coefficient of 1.2. The reference area Ai of eachelement is considered to be the area normal to the incom-ing fluid n of each element. In the calculation the relativevelocity of the link with respect to the fluid was used ~v fi .

~dFhydd j = 0.5CdA j|~v fj |~v fjA j = D ~dr j.n (9)

7. Fluid inertiaFluid inertia is present only if the fluid it self has an ac-celeration w.r.t the inertial frame of reference ~a0f . In thisstudy such acceleration was not introduced so no contribu-tion from this hydrodynamic effect is expected.

~dFhydd j = V~a0f (10)

Adding all these effects to the RNE algorithm yields;

from i=n:1

~f i1i = ~fii+1 mi~g ~f ic + V~g ~f ic ~Fhi

~ni1i = ~nii+1 + (~r

i1ic ~f i1i ) (~riic ~f ii+1) ~nic ~nic

~Nhi + (~ri1ic ~Fhi)end

(11)

The coupling joint force at the vehicle F joint can be calcu-lated by solving the above algorithm for ~f 01 and ~n

01 considering

all the forces and moments.

~F joint =(~f 01~n01

)(12)

The hydrodynamic and mobile base effects on the manipula-tor model ~hyd and ~mobilebase are computed by the same methodincluding only the hydrodynamic, hydrostatic and mobile basecomponents forces in the Equation 11. After computing ~f i1ithe respective torque on the motor ~taui is found.

~hyd + ~mobilebase = ~zi. ~f i1i (13)

This completes the full coupled models of the vehicle (Equation1) and the manipulator (Equation 2).

3.4. Model VerificationThe developed model was verified for hydrodynamic forces

using the experimental data provided by McClain et al. for areference trajectory[4]. The forces generated from the devel-oped model appears good approximation with the experimentaldata of McClain et al.s study.

Figure 2: McClain et al. Experimental results of hydrodynamic torque(left).The torque from the developed model (right)

4. Model Simulation

4.1. Simulated TasksThe developed model was implemented in a Matlab Simulink

environment to simulate behavior in underwater missions. 3manipulator tasks ranging from low 1deg.s1 to high 7.5deg.s1slew rates were set as desired link trajectories (figure 3). The

3

performance of the tracking operation was measured (end ef-fectro tracking error) along with individual contributions of in-ertial joint force F ji, Static joint force F js and hydrodynamicjoint force F jhyd.

0 5 10 1555

60

65

70

75

80

85

90

Joint

varia

ble q2

(deg)

Time(s)

7.5 slew2.5 slew1 slew

Figure 3: The manipulator joint 2 reference trajectories for Task 1, Task 2 andTask 3

4.2. Simulation setupDormian Prince(ode45) variable step solver was used with

0.1 tolerance in an Simulink environment. The results werevisualized through a Virtual Reality system, which was usedfor generating animations of the dynamic simulation and visualverification of results. The detailed model can be found in Ap-pendix A.

A 6DOF Odin vehicle model was used in the study [8]. Thiswas combined with a 3DOF manipulator model for full systemmodeling of the UVMS. All system parameters are presented inAppendix A.

4.3. Feedback controller2 separate feed back controllers were implemented for the

vehicle and the manipulator. Saturation non linearities wereintroduced to the feedback controller output. The simulationassumes ideal sensors and ideal actuators for simplicity. Thevehicle PID was introduced with a station keeping task whilethe manipulator PIDs were given desired joint trajectories ofTask 1,2 and 3.

5. Results

Individual results for each tasks were combined to form meanmagnitude plots of all Forces, Moments and Tracking errors. Aresult set for task 1 is reported in Appendix A, while this sectionreports all the results in a summarized form.

5.1. End effector trackingThe results exhibits the increase in error at high task speeds.

The vehicle has moved from its station keeping during opera-tion which has assisted the end effector tracking of the manipu-lator (figure 4).

Task 1 Task 2 Task 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Trac

kin er

ror (m

)

EndEffector ErrorStation keeping

Figure 4: The tracking errors of end effector and station keeping of vehicle forTask 1,2, & 3

5.2. Joint coupling ForcesThe results indicate that at high slew rates (Task 1) Hydro-

dynamic coupling forces dominate while at slow speed opera-tions the Static Coupling forces has significant effect(figure 5).The Hydrodynamic moments appears to be significant in all thecases(figure 6).

Task 1 Task 2 Task 30

50

100

150

200

Coup

ling f

orce

(N)

FstaticFinertialFhydro

Figure 5: The coupling Joint forces on vehicle for Task 1,2, & 3

6. Conclusion

Task Joint Forces Joint Moments7.5deg.s1 20% 27% 53% 4% 12% 84%2.5deg.s1 68% 11% 21% 10% 17% 73%1deg.s1 91% 3% 6% 16% 26% 58%

Table 2: Summary of % contribution to the total joint coupling force and mo-ment for tasks 1,2 & 3

Table 2 summarizes the percentage contribution of individ-ual coupling force and moment effects at the vehicle- manip-ulator joint. The accelerations at the manipulator was signifi-cantly larger than the desired values. This was due to the effects

4

Task 1 Task 2 Task 30

10

20

30

40

50

60

70

80

90

100

Coup

ling M

omen

t (Nm)

MstaticMinertialMhydro

Figure 6: The coupling Joint moments on vehicle for Task 1,2, & 3

brought by the PID controller. These accelerations were trans-lated as inertial forces at the vehical-manipulator junction. Inthe simulation, Hydrodynamic forces and inertial forces domi-nate at high speeds (Task 1). The Hydrodynamic moments re-main the dominating effect in all the cases considered.

This implies that, in developing controllers for high speedoperations, all hydrodynamic, Inertial and static forces at thevehicle manipulator joint should be considered. In systems op-erating in low speeds the consideration of static force effectsand the hydrodynamic moment effects would be sufficient.

The model used assumes perfect station keeping perfor-mance of the vehicle and ideal actuators. The model shouldbe developed to include the noise of the sensors, thruster dy-namics and manipulator joint motor dynamics for improvingthe simulation validity.

Future work on this study would include implementing thediscussed modifications which will capture all dynamic effects.This will be used for testing of new feedforward control strate-gies and parameter adaptive strategies for robust control ofUVMS.*

Appendix A. -Appendix A

Appendix A.1. The under water vehicle model

Mvv + Cvv + DRB(v)v + Fg + F joint = control (A.1)

where;

Mass Matrix Mv captures both the Rigid body mass matrixMRB and the Hydrodynamic added mass matrix MA of thesystem.

Mv(6x6) = MRB + MA (A.2)

Corioli-Centrifugal Matrix Cv captures the coriolis and cen-trifugal forces of both the rigid body and hydrodynamicadded masses.

Cv(6x6) = CRB + CA (A.3)

For low velocity and three plane of symmetry case the ma-trices MA and CA reduces to;

MA(v) = diag{Xu Yv ... Nr

}(6x6)

(A.4)

Damping Matrix DRB(v) includes all the dissipative DragFdrag and Fli f t forces. A simplification is made such thatonly quadratic effects are captured neglecting the couplingdissipative effects yields a diagonal form for DRB(v).

DRB(v) = diag{Xu|u||u| Yv|v||v| ... Nr|r||r|

}(6x6)

(A.5)

Gravity matrix Fg(2) includes the gravity buoyancy effects.Equation A.6 forms the total gravity and buoyancy effectswith rBG being the vector from Center of Buoyancy to Cen-ter of Gravity.

Fg(2) = [mgB VgB

rBG mgB]

(A.6)

External Force Matrix Fext all other environmental forces arecaptured in this term which is neglected in the study. Butthe Reaction force and moment at the manipulator-vehiclejoint is added as an Fext in the modeling strategy used.

Control Force Matrixcontrol represents the control forcesfrom thrusters and control surfaces. A thruster driven UVis studied, so the control forces can be mapped to thecontrol thruster RPM signals using a simplified linearizedthruster configuration matrixBv.

v = Bvuc (A.7)

Appendix A.2. Odin vehicle model parameters

r = 0.3m,m = 125Kg, rBG = [000.05]T m

I =

8 0 00 8 00 0 8 Nms2

Xu = Yv = Zw = 62.5Kp = Mq = Nr = 30Xu|u| = Yv|v| = Zw|w| = 48Kp|p| = Mq|q| = Nr|r| = 80

(A.8)

Appendix A.3. 3DOF RRR Manipulator model

Frame i di ai i1 1 0.4 0 90o

2 2 0 0.3 03 3 0 0.3 0

Table A.3: DH notations for 3DOF model

Link parameters;

m1 = 4Kg,m2 = 3Kg,m3 = 3KgD = 0.071m,Cd = 1.2,Ca = 1

(A.9)

5

Appendix A.4. Simulink model

Figure A.7: Simulink model of the system

Appendix A.5. Results generated for task 1

0 5 10 150.3

0.25

0.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

End e

ffecto

r tra

cking

error

(m)

Time(s)

x error

y errorz error

Figure A.8: End effector error for Task 1

References[1] R. Featherstone, D. Orin, Robot dynamics: Equations and algorithms,

in: Robotics and Automation, 2000. Proceedings. ICRA00. IEEE Inter-national Conference on, Vol. 1, 2002, p. 826834.

[2] B. Lvesque, M. J. Richard, Dynamic analysis of a manipulator in a fluidenvironment, The International Journal of Robotics Research 13 (3) (1994)221.

[3] S. McMillan, D. E. Orin, R. B. McGhee, Efficient dynamic simulation ofan underwater vehicle with a robotic manipulator, Systems, Man and Cy-bernetics, IEEE Transactions on 25 (8) (2002) 11941206.

[4] T. W. McLain, S. M. Rock, M. J. Lee, Experiments in the coordinatedcontrol of an underwater arm/vehicle system, Autonomous Robots 3 (2)(1996) 213232.

[5] H. Mahesh, J. Yuh, R. Lakshmi, A coordinated control of an underwatervehicle and robotic manipulator, Journal of Robotic Systems 8 (3) (1991)339370.

[6] T. J. Tarn, G. A. Shoults, S. P. Yang, A dynamic model of an underwa-ter vehicle with a robotic manipulator using kanes method, Autonomousrobots 3 (2) (1996) 269283.

[7] E. Koval, Automatic stabilization system of underwater manipulationrobot, in: OCEANS94.Oceans Engineering for Todays Technology andTomorrows Preservation.Proceedings, Vol. 1, IEEE, 2002.

[8] G. Antonelli, Underwater Robots: Motion and Force Control of Vehicle-Manipulator Systems, 2nd Edition, Springer, 2006.

0 5 10 150

10

20

30

40

50

60

Stati

c joint

Force

/Mome

nt(N,N

m)

Time(s)

XYZKMN

Figure A.9: Static coupling for Task 1

0 5 10 15100

50

0

50

100

150

Inertia

l joint

Force

/Mome

nt(N,N

m)

Time(s)

XYZKMN

Figure A.10: Inertial coupling for Task 1

0 5 10 15400

300

200

100

0

100

200

Hydro

dyna

mic jo

int for

ce(N,N

m)

Time(s)

XYZKMN

Figure A.11: Hydrodynamic coupling for Task 1

6

IntroductionBackgroundModel DevelopmentUnderwater vehicle modelingManipulator modelingVehicle-Manipulator coupling force modelingModel Verification

Model SimulationSimulated TasksSimulation setupFeedback controller

ResultsEnd effector trackingJoint coupling Forces

Conclusion-Appendix AThe under water vehicle modelOdin vehicle model parameters3DOF RRR Manipulator modelSimulink modelResults generated for task 1