1-s2.0-S0921889014002309-main

Robotics and Autonomous Systems 64 (2015) 84–99

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Robotics and Autonomous Systems

journal homepage: www.elsevier.com/locate/robot

Adaptive control of underactuated robots with unmodeled dynamicsKim-Doang Nguyen ∗, Harry DankowiczDepartment of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 W. Green Street, Urbana, IL 61801, USA

h i g h l i g h t s

• An adaptive control formulation for underactuated robotic systems with unmodeled dynamics is proposed.• The design allows for fast estimation while guaranteeing bounded deviation from a nonadaptive reference system.• The proposed formulation is independent of detailed information about the system model.• The proof of stability is established by the analysis based on input–output maps.• The system’s robustness to measurement noise and time delay is demonstrated.

a r t i c l e i n f o

Article history:Received 12 October 2013Received in revised form3 June 2014Accepted 13 October 2014Available online 13 November 2014

Keywords:Adaptive controlUnderactuated robotsUnmodeled dynamicsMoving platforms

a b s t r a c t

This paper develops an adaptive controller for underactuated robotic systemswith unmodeled dynamics.The control scheme is motivated by the applications of manipulators operating on dynamic platforms.The design decouples the system’s adaptation and control loops to allow for fast estimation rates, whileguaranteeing bounded deviation from a nonadaptive reference system. The proposed formulation isindependent of detailed information about the system model. The control scheme is tested in differenttrajectory-tracking scenarios: (i) a manipulator installed on a ship operating in a high-sea state withuncertain environmental disturbances and (ii) a mobile manipulator moving across a rough terrain ofunknown geometry. The simulation results illustrate the tracking performance of the proposed controlalgorithm, its ability to deal with unmodeled dynamics, and its robustness to measurement noise andtime delay, while maintaining smooth control signals.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

Robot manipulators are widely used in industry and have longbeen considered as testbeds for research in nonlinear control the-ory. Early work on adaptive control of fixed-base manipulatorswas mostly based on model-reference adaptive-control architec-tures [1–3], the linear-in-parameter property of dynamic structure[4–6] and the passivity of rigid robot dynamics [7–9]. Under rel-evant assumptions on the robots’ dynamic properties, these con-trollers were demonstrated to estimate successfully certain typesof unknown model parameters and achieve desired performance.

The extension of these classical control schemes to the con-text of a manipulator installed on a dynamic platform, however,is challenging and still an ongoing research area. Such systems arevery popular in space robots, mobile manipulators, as well as un-derwater and offshore robotic systems [10–12]. Manipulators are

∗ Corresponding author.E-mail address: [email protected] (K.-D. Nguyen).

http://dx.doi.org/10.1016/j.robot.2014.10.0090921-8890/© 2014 Elsevier B.V. All rights reserved.

used in space applications to performmaintenance operation, con-struction of structures, scientific experiments, or to collect debris.Mobile manipulators have a growing range of applications fromexploration, such as the Mars rovers, to rescue missions, deacti-vation of explosive devices, and removal of hazardous materials.Recently, offshore andunderwater platformshave become the newapplication territories for robotic technology, especially in rapidly-changing and challenging environments [13,14]. Examples includemoving loads, maintenance, construction, and other unmannedtasks on ships, seaborne platforms, and underwater vehicles.

When the actuators driving certain degrees of freedom of therobotic systems in the aforementioned scenarios are turned off, therobotic systems become underactuated. In addition, the dynamicsof the manipulator and the platform are mutually coupled due toconservation of momentum. This adds tremendous challenges ascompared with fixed-based manipulators [15]. Firstly, the equa-tions of motion for the unactuated degrees of freedom now act asconstraints on the control design. As these are intrinsically non-holonomic, it is not possible to solve for the underactuated statesin terms of the controlled states. Consequently, model-reductionmethods fail to reduce the system dimension [16]. Moreover,

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K.-D. Nguyen, H. Dankowicz / Robotics and Autonomous Systems 64 (2015) 84–99 85

according to Brockett’s theorem [17], it is impossible to asymptoti-cally stabilize a nonholonomic system to an equilibrium point by acontinuous and time-invariant state-feedback control law, despiteits controllability. Secondly, with complicated platform structures,or unknown terrain geometries in the case ofmobilemanipulators,or in challenging environments, including manipulators operatingon ships and offshore platforms in high sea states, the platform dy-namics are usually unmodeled and add large inertial terms anddisturbances to the description of the manipulator dynamics.Example control designs include switching schemes based onsupport-vector-machine regression [18], a combination of fuzzyand backstepping control [19], adaptive control based on estima-tion of a bounded parameterization of the unknown dynamics[20,21], and adaptive variable structure control [22].

The most popular methods for controlling underactuatedmoving-base manipulators involve adopting the adaptive-controlframeworks for fixed-base manipulators. Recent examples of con-trol schemes in this category for free-floating space manipulatorsinclude [23] which constructs a dynamically equivalent model forparameterization and control of the original systemdynamics, [24]which assumes passivity and other structure properties to elimi-nate the need for measuring the platform acceleration, [25] whichproposes an adaptive controller based on the reaction dynam-ics between the active and passive parts in the system, and [26]which combines a recursive formulation of control torque and theplatform’s reference velocity and acceleration to achieve desir-able tracking performance. For mobile manipulators, work in [27]designs an interaction control scheme, which is composed of anadaptation algorithm and an input–output linearizing controller. Atrajectory and force tracking control problem is addressed in [28],which presents an adaptive controller based on a suitably reduceddynamic model to control a system with some unknown iner-tia parameters. Work in [29] employs a Frenét-like descriptionto transform the kinematics of nonholonomic mobile platformsto generic driftless dynamics, which are then stabilized by twocontrol schemes for the kinematics and dynamics, respectively.In the presence of disturbances, [30] combines a regressor-basedadaptive scheme with an estimator for the disturbance to improvetracking performance of mobile manipulators. Work in [31] devel-ops an image-based visual controller for mobile manipulators totrack objects in three-dimensional space. The adaptive laws are de-signed based on the assumption that the robot dynamics can beparameterized linearly in terms of the unknown parameters in themodel. In the context of marine robotics, an adaptive controller,which assumes passivity in the robot structure, and an adaptive-sliding control scheme are presented in [32] to compensate formodel uncertainties. The analysis in [33,34] decomposes the sys-tem dynamics into control elements of individual bodies to derivea modular controller for manipulators mounted to an underwa-ter vehicle. The control scheme in [35] allows underwater vehicle-manipulator systems to track both a prescribed sub-region as wellas uncertain tasks.

The adaptive-control algorithms in [23–35] rely on the linear-in-parameter property of Lagrangian systems. Specifically, theyexploit the dynamic structure and the passivity of rigid robot dy-namics to factor the model description in terms of a regressionmatrix and a vector of unknown parameters, and proceed to im-plement adaptive laws to estimate these parameters. While thelinear-in-parameter property is an acceptable model parameter-ization for many fixed-based manipulators, it is not applicableto others, such as robots with complicated link and/or joint ge-ometries, unknown lengths, or with nonlinear mass distribution,stiffness, or damping. Furthermore, these controllers require con-struction of a well-defined and complicated model regressionmatrix, which involves correct selection of the joint velocity coeffi-cientmatrix from among several options [36]. The use of themodel

regressionmatrix, whichmust contain no uncertainty, also implieshigh dependence of the control algorithms on system modeling.

In moving-base manipulators, the reliance on the linear-in-parameter property may further undermine system performance.Firstly, accurate modeling of the platform dynamics is muchmore challenging than modeling the manipulator. Platforms areoften complex structures with many uncalibrated parameters anduncertainties. In addition, moving-platform robots often operatein challenging environments such as space, underwater, offshore,across uneven terrains, or on ships operating in high seas [13,14,37–40]. Even when the ship stands still, modeling of the ship’sstructure is a very difficult task. Constructing the ship’s equationof motion under influence of waves, ocean currents, and wind iseven more challenging, if not impossible. The treatment in [14]avoids these problems by assuming that the oscillations of theship are known a priori for all time. This assumption is eliminatedin [38,40] by two intriguing methods for predicting the ship’smotion, including an auto-regressive predictor and a predictor thatsuperposes a series of sinusoidal waves. However, these methodsrequire re-calibration of the parameters in the algorithms fordifferent sea locations. In addition, the prediction accuracy canonly be achieved with advanced sensors, such as wave camerasand sensors thatmeasure interaction forces on the ship fromwavesand wind [40]. Similarly, in the case of mobile manipulators, mostadaptive controllers are formulated with the assumption that therobots are moving across perfectly even terrain.

In this paper, we design an adaptive controller which is inde-pendent of system modeling and can achieve desired tracking formanipulators that operate on an underactuated dynamic platform.The control scheme proposed here is inspired by the work in [41],which proposed the use of a filter in the control input of a refer-ence model adaptive controller to improve the robustness of a lin-ear single input system. The architecture decouples the estimationloop from the control loop to facilitate a significant increase in therate of estimation and adaptation, without a corresponding loss ofrobustness. Following the design method presented in [41], thispaper develops a control scheme for an underactuated systemconsisting of a manipulator mounted on a moving platform withunmodeled dynamics. The proposed controller employs a fastadaptation scheme while maintaining bounded deviation from anonadaptive reference system. In particular, the control design istolerant of timedelays in the control loop, andmaintains clean con-trol channels even in the presence ofmeasurement noise due to theuse of a low-pass filter structure in the control input. Tuning of thefilter also allows for shaping the nominal response and enhancingthe time-delay margin.

The proposed controller is implemented for two example un-deractuated robotic systems in two trajectory-tracking contexts:(1) a manipulator mounted on a ship operating in a high-sea stateunder uncertain environmental disturbances on the ship dynam-ics fromwind, waves, and ocean currents; and (2) a mobile manip-ulator moving across a rough terrain of unknown geometry. Thefirst task is used to assess the tracking performance when the plat-form motions contribute large inertia and nonlinearity to the ma-nipulator dynamics. The second task demonstrates the proposedcontroller’s effectiveness when themanipulator dynamics aremu-tually coupled with the platform dynamics, whose high-frequencymotions are induced by both the manipulator motions and traver-sal across a rough terrain via a suspension system. The controlobjectives in these two tasks are achieved under both velocity-measurement noise and time delay in the control signal.

The remainder of this paper is organized as follows. A templatedynamic model of an underactuated robotic system is describedin Section 2. The text presents a broad-strokes description of apopular approach for adaptive control of such systems, includingits potential shortcomings. The proposed adaptive control design

86 K.-D. Nguyen, H. Dankowicz / Robotics and Autonomous Systems 64 (2015) 84–99

is introduced and analyzed in Section 3. Section 4 illustratesthe trajectory-tracking performance of the control design in thecontext of a robot armmounted on a shipwith uncertain dynamics,including in the presence of actuator delay and measurementnoise. Section 5 considers the implementation of the proposedcontroller in a mobile manipulator whose platform dynamics aredisturbed by motion across an uneven, and unknown, terrain,with the addition of a nonholonomic constraint on the platformkinematics. A concluding discussion in Section 6 reviews theadvantages of the proposed design and points to open problemsin its characterization.

2. Dynamic model of underactuated robotic systems

Let a superscribed dot denote differentiation with respect totime t . In the absence of nonholonomic constraints on the systemkinematics, the dynamics of a robotic manipulator mounted ona platform, and with several unactuated degrees of freedom, aregoverned by equations of motion of the form [15]Maa(q) Mau(q)MT

au(q) Muu(q)

M(q)

q +

Ca(q, q)Cu(q, q)

q +

Ga(q)Gu(q)

=

u0

+

Daa(t)Duu(t)

, (1)

where qT =qTa , q

Tu

, the n generalized coordinates contained in

the column vector qa describe the actuated degrees of freedom,and them generalized coordinates contained in the column vectorqu describe the unactuated degrees of freedom of the roboticsystem. The column vectors u, Daa and Duu contain the time-dependent control-input torques and bounded time-dependentunknown disturbances to the actuated and unactuated degreesof freedom, respectively. The inertia matrices Maa, Muu and Mare all positive-definite, symmetric, and bounded. The remainingterms include Coriolis and centripetal effects, gravity and otherconservative forces, as well as dissipative and velocity-dependentmechanisms.

A popular adaptive approach (e.g., [24,25,30]) for controllingthe system described above is based on the so-called linear-in-parameter property of Lagrangian systems, which here takes theformMaa(q) Mau(q)MT

au(q) Muu(q)

qr +

Ca(q, q)Cu(q, q)

qr +

Ga(q)Gu(q)

=

Ya(q, q, qr , qr)Yu(q, q, qr , qr)

a (2)

for some vector a of model parameters, and for arbitrary referencetrajectories qTr = [qTar , q

Tur ] of the actuated and unactuated general-

ized coordinates of the system. In practice, the model parameterscontained in a are assumed to be unknown and, therefore, to beestimated in real-time by an estimator a. In this case, the regres-sion matrices Ya(q, q, qr , qr) and Yu(q, q, qr , qr) are assumed to beknown functions of the actual and reference dynamics for all time.The fact that the left-hand side of Eq. (1) and the left-hand side ofEq. (2) are analogous leads to the design of the control input in thegeneral form

u = upd + Ya(q, q, qr , qr)a, (3)

where upd represents proportional–derivative, negative-feedbackcontrol terms that are responsible for repressing disturbances, anddesigned for specific control tasks and performance tuning. Thetime history of the parameter estimate a is governed by adap-tive laws, designed with the tracking error as feedback in order to

compensate for the nonlinearity in the model. For example, [24]considers the task of controlling a free-floating space manipulatorin its task space. Here, the reference trajectory is the output of a ref-erence systemwhose input includes a sliding-variable formulationof the end-effector tracking error. The nonadaptive control input,upd, in Eq. (3) is taken to be proportional to this sliding variable.Furthermore, the adaptive law is designed as follows:

˙a = −Γ

Ya(q, q, qr , qr)Yu(q, q, qr , qr)

T(q − qr) (4)

withΓ being an adaptation gain. More detailed analysis, for exam-ple in [24,25,30], shows that if an adaptive law drives a in Eq. (3) toa in Eq. (2), thenwhen u in (3) is substituted in (1), qa will track qar .

Two fundamental assumptions are used in the formulationof the adaptive control strategy described above, namely (i) theexistence of a factorization of the left-hand side of Eq. (2) in termsof a product of a matrix and a vector of model parameters, and(ii) that the matrix is known and the vector can be estimated. Forfixed-base manipulators, these assumptions often apply, i.e., theunknown model parameters may be collected in a vector and thedynamics appropriately factored.

When the unknown parameters appear as nonlinear terms, fac-torization is no longer possible. Even in cases where factorizationis possible in principle, the complexity of the geometry and massdistribution of the overall manipulator-and-platform system maymake it prohibitively difficult to construct the regression matricesYa and Yu, especially when the number of degrees of freedom islarge. This is particularly common for moving-base manipulators,mounted on platformswhose structuremay change in time and in-volve many uncalibrated parameters. In this case, the design of thecontroller for the manipulator requires detailed information aboutthe geometry and mass distribution of the platform, which maynot be available or even predictable, as changes to the latter areimposed by other control loops.

In the absence of certain knowledge of the regression matri-ces Ya and Yu, uncertainty associated with large variations in theinertial and geometric properties of the platform enters the con-trol problem as disturbances that are not accommodated withinthe adaptive control design. Examples include significant platformmotions driven by unknown environmental factors, such as un-even terrain for ground vehicles, or high sea states and uncer-tain wind and current conditions for manipulators based on shipsor off-shore platforms, all of which may vary with location andtime. In these cases, disturbance rejection must be handled by thedesign of upd, typically by using very large numerical values forthe proportional and derivative gains, with likely loss of systemrobustness.

To address these observations, the control design in this paperavoids the linear-in-parameter factorization entirely. Here, allmodeling terms in Eq. (1), including Maa, Mau, Muu, Ca, Cu, Ga, Gu,Daa, and Duu, are considered unknown functions of time. Beforederiving the control scheme, the equations of motion in Eq. (1) areconverted to a reduced form to demonstrate the reasoning behindthe controller design that follows. Specifically, by solving for qu inthe bottom component of Eq. (1), we get

qu = M−1uu (q)

−MT

au(q)qa − Cu(q, q)q − Gu(q) + Duu(t). (5)

Substitution in the top component of Eq. (1) then yields

Ma(q)qa + Na(q, q) = u + Da(q, t), (6)

where

Ma(q) = Maa(q) − Mau(q)M−1uu (q)MT

au(q). (7)

Na(q, q) = Ca(q, q)q + Ga(q) − Mau(q)M−1uu (q)

× (Cu(q, q)q + Gu(q)) , (8)


Da(q, t) = Daa(t) − Mau(q)M−1uu (q)Duu(t). (9)

SinceMaa(q) Mau(q)MT

au(q) Muu(q)

= |Ma(q)| · |Muu(q)| (10)

(see, e.g., [42]), it follows that Ma must be invertible. Moreover,it is trivial to show that M−1

a is a symmetric positive definitematrix. In addition, there exist bounding constant ml and mh suchthat

0 < mlI ≤ M−1a (q) ≤ mhI, for all q ∈ Rm+n. (11)

As seen in Eq. (6), the coefficients governing the dynamics inthe actuated degrees of freedom depend on the time-histories ofthe unactuated degrees of freedom (and vice versa). Nevertheless,in the next sections, we design an adaptive controller that isindependent of any detailed knowledge of the system model, andproceed to demonstrate successful estimation of the unknownmodel coefficients, aswell as close tracking of the actuated degreesof freedom along the corresponding desired trajectories.

3. Adaptive controller for underactuated robots

Let the desired time histories for the robot’s actuated degreesof freedom be described by the vector-valued function qad(t) andsuppose that this is bounded with bounded derivatives. In thedefinition of the auxiliary kinematic variable (cf. the sliding controlformulation in [5])

r , qa − qad + λ (qa − qad) , (12)

the choice λ > 0 ensures that Eq. (12) is an exponentially stablesystem for qa. Indeed, as long as the controller drives r to a neigh-borhood of 0, the joint trajectory qa converges to a neighborhoodof qad exponentially fast.

Let Am be a Hurwitz matrix and set µ , M−1a (q)u. It follows by

substitution of Eqs. (6) and (12) into the time derivative of Eq. (12)that

r(t) = Amr(t) + µ(t) + η(t, ζ (t)), (13)

where ζ , (r, qa, qu, qu) and the nonlinearity η(t, ζ (t)) is obtainedfrom

η(t, ζ ) , M−1a (q)

Da(q, t) − Na(q, qa, qu)

− qad(t)

+ λ(qa − qad(t)) − Amr (14)

with qa = r + qad(t) − λ (qa − qad(t)). Here,

η(t, 0) = M−1a (0)

Da(0, t) − Na(0, qad(t) + λqad(t), 0)

− qad(t) + λ2qad(t) (15)

whose norm is bounded for all t by some constant Z provided thatsimilar expectations are placed on the disturbances Da(0, t). Wesimilarly restrict attention to disturbances that guarantee that

∥ζ∥∞ ≤ ξ ⇒

∂η(t, ζ )

∂t

∞

≤ dηt (ξ) < ∞,∂η(t, ζ )

∂ζ

∞

≤ dηζ(ξ) < ∞

(16)

for arbitrary ξ and all t . The objective of the following sectionsis to establish an adaptive control formulation for u that ensurespredictable performance bounds for the system response and thecontrol input.

3.1. Nonadaptive reference system

Let k > 0 denote the bandwidth of the first-order low-pass filterk/(s+k). For an arbitrary smooth function v : R → Rm+n, considerthe system Rv obtained by appending

µ(t) = M−1a

v(t)

u(t) (17)

and

u(t) = −kµ(t) − kηt, ζ (t)

, u(0) = 0 (18)

to Eqs. (12) and (13). Let Φv be the unique solution to the initial-value problem

∂

∂tΦv(t, t ′) = −kM−1

a

v(t)

Φv(t, t ′), Φv(t ′, t ′) = I. (19)

It follows that

u(t) = Fv[η(t, ζ (t))] , −k t

0Φv(t, t ′)η(t ′, ζ (t ′)) dt ′ (20)

and

µ(t) = Dv[η(t, ζ (t))] , M−1a

v(t)

Fv[η(t, ζ (t))] (21)

in terms of the linear input–output maps Fv and Dv . As shown inAppendix A,

bF , supv

∥Fv∥L1 < ∞. (22)

Now, letI be the identitymap andH be the linear input–outputmap corresponding to the transfer function (sI − Am)−1. It followsthat

r(t) =H ◦ (I + Dv)

[η(t, ζ (t))] + H[r(0)δ(t)], (23)

where b1 , supv ∥H◦(I + Dv) ∥L1 < ∞ and∥H[r(0)δ(t)]∥L∞<

∞. In the special case that r(0) = 0, rearranging the terms in Eqs.(13) and (18) yields

r(s) = (sI − Am)−1 (u(s) + ηu(s)) , (24)

where

u(s) = −k

s + kηu(s) (25)

and ηu(t) ,M−1

a

v(t)

−Iu(t)+η

t, ζ (t)

. In this case, it follows

that

∥r∥L∞≤s(sI − Am)−1

L1

1s + k

L1

∥ηu∥L∞

≤2k

(mh + 1)bF + 1

∥η∥L∞

, (26)

i.e., that

b1 ≤2k

(mh + 1)bF + 1

. (27)

Let Z and Lρref be defined as in Eqs. (B.1) and (B.5) in Appendix B,respectively. Since b1 → 0 uniformly in v as k → ∞, it followsthat there exists a K , such that the stability condition

b1 <ρref − ∥H[r(0)δ(t)]∥L∞

Lρref ρref + Z(28)

is satisfied for some ρref > ∥H[r(0)δ(t)]∥L∞provided that k > K .

Then, from the remarks following Lemma B.1 and the result ofLemmaB.2 in Appendix B,we conclude that if ∥r(0)∥∞ < ρref , thenthe bounds∥r∥L∞

< ρref and∥u∥L∞< bF (Lρref ρref +Z)must hold

forRv , independently of v. In particular, this conclusionmust holdfor the nonadaptive reference system obtained by substituting theactual time histories q(t) for v(t), and we arrive at


Theorem 3.1. Consider the nonadaptive reference system

rref (t) = Amrref (t) + M−1a (q(t))uref (t) + η(t, ζref (t)), (29)

uref (t) = −kM−1

a (q(t))uref (t) + ηt, ζref (t)

, (30)

where rref (0) = r0, uref (0) = 0, q = (qa, qu), ζref = (rref , qa,ref ,qu, qu), qa,ref = rref + qad(t) − λ

qa,ref − qad(t)

, and qa,ref (0) =

qa(0). For sufficiently large k, there exists a positive scalar ρref ,such that ∥rref (0)∥∞ < ρref implies that ∥rref ∥L∞

< ρref and∥uref ∥L∞

< bF (Lρref ρref + Z).

From Eqs. (23) and (27), it follows that

∥r(t) − H[r(0)δ(t)]∥L∞= O

k−1 (31)

for k → ∞. We obtain

Lemma 3.1. The response of Rv converges to eAmt r0 when k → ∞.

Lemma 3.1 implies that the response of the nonadaptive refer-ence system will converge to a neighborhood of 0 exponentiallyfast. The size of the neighborhood is inversely proportional to thefilter bandwidth k.

3.2. Design of the adaptation laws and the state predictor

By Theorem 3.1 and the remarks following Lemma B.1 inAppendix B, it follows that, in the nonadaptive reference sys-tem, and provided that ∥rref (0)∥∞ < ρref , the parameterizationη(t, ζref (t)) = θref (t)∥rref ,t∥L∞

+ σref (t) holds for all t , in termsof a pair of continuous, piecewise-differentiable and uniformlybounded functions θref and σref . Equivalently,

rref = Amrref + uref + θref ∥rref ,t∥L∞+ σref (32)

and

uref = −kuref + θref ∥rref ,t∥L∞

+ σref, (33)

where

σref , σref + (M−1a (q) − I)uref (34)

is similarly bounded.We proceed to consider an adaptive control design for the

original dynamics in Eqs. (12) and (13) in lieu of Eq. (18). Analogousto Eqs. (32) and (33), consider the state predictor

˙r = Amr + u + θ∥rt∥L∞+ σ + Asp r, r(0) = r0, (35)

and control design

u = −ku + θ∥rt∥L∞

+ σ, u(0) = 0, (36)

where r , r − r is the prediction error and Asp is a Hurwitz matrixof loop-shaping parameters thatmay be tuned to reject oscillationscaused by high-frequency disturbances or noise, as well as tomaker converge to 0 faster. Here, θ and σ model adaptive estimates forθ and σ , governed by the projection-based laws

˙θ = Γ Proj

θ , −Pr∥rt∥L∞

; θb, ε, θ (0) = θ0, (37)

˙σ = Γ Projσ , −Pr; σb, ε

, σ (0) = σ0, (38)

in terms of the adaptation gain Γ ∈ R+, and the positive-definite,symmetric matrix P , obtained as the solution to the Lyapunovequation A⊤

spP+PAsp = −I. As defined in Appendix C, the projectionoperator Proj(·, ·; ·, ·), ensures that ∥θ (t)∥∞ ≤ θb and ∥σ (t)∥∞ ≤

σb provided that θ0 and σ0 satisfy these same bounds.

4. Performance bounds

In this section, we prove that the state and control input ofthe proposed control system governed by Eqs. (12)–(13) and (35)–(38) follow those of the nonadaptive reference system Rq closely,provided that the bandwidth k, the adaptation gain Γ , the scalar λ,and the bounds θb and σb are chosen appropriately. In particular,we prove the following theorem:

Theorem 4.1. Suppose that ρref > ∥r(0)∥∞ and that ρref and ksatisfy the condition (28), and choose λ ≥ 1. Then, for ν ≪ 1, thereexist θb, σb, and C > 0, such that

∥r − r∥L∞≤ ν, ∥rref − r∥L∞

= O(ν),

∥uref − u∥L∞= O(ν),

(39)

provided that Γ ν2≥ C.

Proof. Suppose that ν > 0 is given. Since ∥rref (0) − r(0)∥∞ =

0 < 1 and ∥uref (0) − u(0)∥∞ = 0, it follows by continuitythat there exists a τ > 0, such that ∥(rref − r)τ∥L∞

< 1 and∥(uref − u)τ∥L∞

< ∞. Theorem 3.1 implies that

∥rτ∥L∞< ρref + 1, ∥uτ∥L∞

< ∞. (40)

By the remarks following Lemma C.1 in Appendix C, there exist θb,σb, and C > 0 (independent of τ ), such that

∥rτ∥L∞≤C/Γ , (41)

which does not exceed ν provided that Γ ν2≥ C . It further follows

from Eq. (12) and the definition of qa,ref in Theorem 3.1 that

∥(qa,ref − q)τ∥L∞≤ Q1∥(rref − r)τ∥L∞

, (42)

where Q1 = ∥(s + λ)−1∥L1 = 1/λ ≤ 1. It follows that

∥(ζref − ζ )τ∥L∞≤ ∥(rref − r)τ∥L∞

(43)

and, consequently,

∥(ηref − η)τ∥L∞≤ dηζ

ρref (ρref )

∥(ζref − ζ )τ∥L∞

≤ Lρref ∥(rref − r)τ∥L∞. (44)

Eqs. (13), (29), (30), (35) and (36) imply that

rref − r = HMq

uref − u

+ ηref − η

, (45)

where

uref − u = Fq

ηref − η + Asp r − ˙r

= Fq[ηref − η] + Fq[Asp r] + k

Fq ◦ Mq + I

[r]. (46)

These result in the bounds

∥(rref − r)τ∥L∞≤ b1Lρref ∥(rref − r)τ∥L∞

+ b2∥rτ∥L∞(47)

and

∥(uref − u)τ∥L∞≤ bF Lρref ∥(rref − r)τ∥L∞

+ b3∥rτ∥L∞, (48)

where

b2 , supv∈Rm+n

∥H ◦ Mv ◦Fv ◦ Asp + k(Fv ◦ Mv + I)

∥L1 (49)

and

b3 , supv∈Rm+n

∥Fv ◦ Asp + k(Fv ◦ Mv + I)∥L1 (50)


Fig. 1. Block diagram of the proposed control design.

are both finite. The stability condition in (28) implies that 1 −

b1Lρref > 0. We conclude that

∥(rref − r)τ∥L∞≤

b21 − b1Lρref

∥rτ∥L∞(51)

and

∥(uref − u)τ∥L∞≤

b2bF Lρref

1 − b1Lρref

+ b3

∥rτ∥L∞

. (52)

The claim follows by choosing ν and Γ such that the product in Eq.(51) is strictly less than 1. �

4.1. Properties of the proposed control formulation

We note that the components of the proposed control schemein Eqs. (12) and (35)–(38) do not require any modeling knowledgeof the system in Eq. (1). The uncertain nonlinearity η in Eq. (14) ofthe robotic system is estimated via the fast adaptation laws (37)and (38). The controller then drives the sliding variable r definedin Eq. (12) close to zero so that the actuated degrees of freedomconverge toward their desired values. The presence of the low-pass filters in the adaptive control signal in Eq. (36) implies thatthe proposed controller aims for only partial compensation of theunknown nonlinearity, in order to maintain clean control chan-nels with only low-frequency content. This contradicts traditionaladaptive-control algorithms that are always designed toward per-fect compensation of the robot nonlinearity.

In traditional adaptive controllers for manipulators, the degreeof adaptation, the robustness against time delays and unmodeleddynamics, and the tracking performance are coupled. The adapta-tion rate canbe significantly boosted by raising the adaptation gain,which in turn improves the response performance. However, highadaptation gain will induce high-frequency signals in the controlchannels. As a result, the control system becomes very sensitiveto uncertainties and time delay, i.e., loses robustness. In contrast,in the proposed control architecture depicted in Fig. 1, the controlsignal makes use of the estimate for the unknown nonlinearity topartially compensate for this nonlinearity. A low-pass filter is inte-grated in the control signal to block any high-frequency signal offthe control channels and keep them clean and smooth. This filterstructure decouples the estimation loop from the control loop andallows for arbitrarily large values of the adaptation gain (limitedonly by available hardware). In fact, as suggested by the rigorousanalysis in [41] for a linear, constant-coefficient, single-input sys-tem with an architecture related to that proposed here, the time-delay margin of the control system is likely bounded away from 0,ensuring guaranteed robustness.

The characteristics of the proposed adaptive controller arerepresented by two independent indicators:

1. The adaptation gainΓ , which determines the rate of adaptation,as well as the deviation between the actual control system anda nonadaptive reference system governing the ideal response;

Fig. 2. A pick-and-place manipulator mounted on a ship with uncertain dynamics.

2. The filter bandwidth k, which determines the deviation of theideal response from an exponential decay to 0, as well as thesystem’s ability to tolerate input delay (cf. Section 5.5).

The objectives of a practical implementation based on the de-scribed control design are to choose the parameter values λ,Am, Asp, Γ , and k in order to achieve desirable performancebounds on the prediction error r , as well as to drive r sufficientlyclose to zero so that qa approaches the desired value qad withinsome desired bound. The adaptation gain Γ and the loop-shapinggain Asp decide the rate of adaptation. The matrix Am character-izes proportional–derivative feedback terms for fine-tuning theperformance.

5. Manipulators operating on ships

5.1. Simulation model

We restrict attention in this section to a typical pick-and-placemanipulator operating on a ship in a high sea state, a scenarioalso investigated in [14,38–40]. The system is sketched in Fig. 2,in which w := (w1,w2,w3) is an inertial reference triad (withgravity along the negative w3-axis) and b := (b1, b2, b3) is a triadrigidly attached to the ship. We restrict attention to motions ofthe ship described by (φb, xb, zb), where φb represents the rollingangle of the ship (i.e., the rotation of w about w3 that yields b),and xb and zb represent the displacement of the ship’s center ofmass Cship relative to the inertial reference frame alongw1 andw3,respectively. These motions are caused by the unknown influenceof surfacewinds,waves, and ocean currents. JointA connecting link1 to the ship has twodegrees of freedom represented by two triads:a := (a1, a2, a3) is obtained by rotating b an angle q3 about b3, andl(1) =

l(1)1 , l(1)2 , l(1)3

, attached to link 1, is obtained by rotating a

an angle q1 about a1. The position of the joint A of the manipulatoris represented by the position vector rCshipA := dAb1. The triadl(2) =

l(2)1 , l(2)2 , l(2)3

, attached to link 2, is obtained by rotating

l(1) an angle q2 about l(1)1 . The relative joint angles q1, q2, and q3describe the configuration of the manipulator links relative to theplatform. Points B, C1, C2, and E represent the joint connecting links1 and 2, the centers of mass of links 1 and 2, and the location of thepayload at the end-effector, respectively, such that rAB := L1l

(1)3

and rBE := L2l(2)3 .


We model the platform as a body with mass mship and mo-ment of inertia about the w2 axis equal to Jship. A linear spring–mass–damper model is used to describe partially the dynamicinteraction between the sea and the ship with effective stiffnessand damping coefficient given by Kφ and Cφ for the ship rolling an-gle φb, by Kx and Cx for the displacement xb, and by Kz and Cz forthe displacement zb. In each case, the reference value equals 0. Themasses of links 1 and 2 and the payload at the end-effector arem1,m2 andme, respectively. The links are assumed to be homogeneouscylinderswith radius equal to one fifth of the corresponding length.

With this system set-up, the kinetic energy, the potentialenergy and the generalized nonconservative forces of the sys-tem are shown in Appendix D. The equations of motion are ofthe exact form in Eq. (1) with the actuated degrees of freedomqa = [q1, q2, q3] and the unactuated degrees of freedom qu =

[φb, xb, zb]. In these equations, the ship’s mass and moment of in-ertia matrix, as well as the effective stiffnesses and damping co-efficients along the ship’s degrees of freedom, only appear in theMuu, Cu and Gu components. In this particular case study, on theone hand, since themass and size of the ship are significantly largerthan those of themanipulator, the dominance of these terms in thebottom part of Eq. (1) implies that the motions of the manipulatorhave very little effect on the ship’s dynamics. On the other hand,the ship’smotions enter the top part of Eq. (1) not only via the com-ponents of the mass matrices Maa and Mau, but also through non-linearity contributions to the acceleration-dependent terms withthe same magnitude of influence as the manipulator’s dynamics.

In the numerical results reported below, and in a set of con-sistent units (the SI system is used throughout the paper), thelink lengths, the masses of the two links, the payload at theend-effector, and the acceleration of gravity are given by L1 =

0.25; L2 = 0.2; m1 = 6; m2 = 4; me = 5; g = 9.81. Thedistance between Cship and A is dA = 3. The ship’s effective massand moment of inertia about the w2-axis equal mship = 105 andJship = 1.25 × 105. The effective stiffnesses and damping coef-ficients parameterizing the dynamic interaction between the seaand the ship are given by Kφ = 107, Kx = 3 × 105, Kz =

3 × 105, Cφ = 108, Cx = 2 × 105, and Cz = 2 × 105. The dis-turbances from the environment are assumed to be given by

Duu(t) =

3 × 107 sin(0.5t)3 × 105 sin(0.5t)

−3 × 105 cos(0.5t)

(53)

and Daa(t) = 0. A typical ship motion that results from numericalsimulation is shown in Fig. 3. The peak-to-peak amplitudes of theship’s displacement are approximately 2 m in both the horizontaland vertical directions, and 76° in the rolling angle φb. These mo-tions contribute large unknown time-varying inertias and nonlin-earity terms to the description of the manipulator dynamics.

5.2. Control objectives

We begin by illustrating the performance of the control designwhen the manipulator’s joint angles are tasked to track desiredtrajectories given by

• step inputs with q1,d(t) = q2,d(t) = q3,d(t) ≡ α, for values ofα ∈ [1, 2], with initial conditions q0 = q0 = (0, 0, 0)T ; and

• sinusoidal inputs with q1,d(t) = q2,d(t) = sin 0.4t andq3,d(t) = cos 0.4t with initial conditions q0 = (0.5, 0.5, 0.5)T

and q0 = (0, 0, 0)T .

In the simulations in this paper, the control parameters and thefilters are tuned to the following control objectives:

Fig. 3. Typical oscillations of the ship under prescribed environmental distur-bances.

• In the case of the step inputs, achieve a response settling time(the time required for the response to reach and stay within 2%of the final value) of less than 3 s, withmaximumovershoot lessthan 5% of the desired values.

• In the case of the sinusoidal inputs, achieve a root-mean-square deviation percentage (denoted below by RMSD%) of lessthan 5% under various disturbances, including time delay andmeasurement noise. Here, RMSD% is defined as the percentageof a response’s root-mean-square deviation from its desirabletrajectory over the time history relative to the peak-to-peakamplitude of the desired trajectory.

To achieve these objectives, the parameter λ in Eq. (12) is hereset to 2, the adaptation gain Γ is set to 106, and the trackingresponse characteristics are governed by the design matrix Am =

−diag(30, 20, 15). The filter bandwidth k in (36) is set to 10 (Hz).The matrix of loop-shaping parameters Asp in Eq. (35) is set to0.1

√Γ I. In the projection based adaptive laws in Eqs. (37)–(38),

we set θb = σb = 100, ϵ = 0.1, and θ0 = σ0 = 0.The numerical results below were obtained from a Simulink-

based implementation of the system’s equations of motion and theproposed control scheme. Default Simulink tolerances and settingswere used throughout.

5.3. Performance in ideal working conditions

Figs. 4 and 5 show the manipulator’s response to the proposedcontrol actuation for step inputs with different values of α and forthe sinusoidal input, respectively. As seen in the bottom panels, forboth types of desired trajectories, the control signals are smoothand clean, in spite of the use of high-rate adaptive estimation toaccommodate nonlinearity and model uncertainty while retainingsmall prediction errors. For the step input, themaximumovershootis 4.1%, and the maximum settling time is 2.76 s. As seen in Fig. 4,the system response scales approximately with the size of thestep. This implies predictable responses of the proposed controlsystem when the desired values are varied. For the sinusoidalinput in Fig. 5, the maximum tracking RMSD% is 3.64%. The systemresponses quickly converge to the desired time histories despitethe large inertia and nonlinearity added due to the unmodeleddynamics of the ship. Consistent with the theory discussed inthe previous sections, an increase in the filter bandwidth results


Fig. 4. Performance of the proposed controller under ideal working conditions for various step inputs.

Fig. 5. Performance of the proposed controller under ideal working conditions for sinusoidal inputs.

in improved tracking performance, i.e., reduced tracking RMSD%.However, better tracking is achieved with the trade-off of thesystem robustness, i.e., the control system is less tolerant of timedelays in the control signal.

5.4. Control performance with time delay and measurement noise

Next, we consider the performance of the control system inthe presence of velocity measurement noise, with a time delayof 50 ms in the control signal. To this end, unfiltered, uniformlydistributed noise in the range [−0.15, 0.15] rad/s andwith sampletime of 10 ms was added to the angular velocity measurements.This measurement noise is reflected in the right bottom panel inFig. 6, since the velocity measurements appear explicitly in thedefinition of r . As seen in this figure, without any further tuning,the proposed adaptive controller successfully rejects the noise.The control scheme still maintains desirable tracking performance

with the maximum tracking RMSD% of 4.21%, which is very closeto the previous case with the time delay in the control signals.This demonstrates the robustness of the proposed controller tonoisy measurements. In addition, the control signals are cleanand implementable because all the high-frequency and noisycontributions are completely filtered out of the control channel.

5.5. Robustness

Weproceed to analyze the robustness of the adaptive controllerby investigating its performance in the presence of time delaysin the control input. It is a well-known property of controlsystems that the presence of time delay in the control loop maydestabilize the systems (see, for example, [43]). Furthermore, asdiscussed in [44,45], any unmodeled dynamics can be equivalentlyrepresented by a delay in the plant input. In linear, time-invariantsystems, the phasemargin or, alternatively, the time-delay margin


Fig. 6. Performance of the proposed controller in the presence of 50 ms actuator time delay and velocity measurement noise in the range [−0.15, 0.15] rad/s and withsample time of 0.01 s.

are reliable indicators of the system robustness [46]. In nonlinearcontrol systems, since the phase margin is not computable, weneed an alternative method for checking the system’s robustness,i.e., its ability to tolerate input delay [44,47]. In this paper, wedefine the critical time delay as the maximum input delay forwhich the controller is able to maintain bounded performanceover a given time interval, for each choice of system parametersand desired trajectory qd(t). Consistent with the literature, we usethe critical time delay for some representative trajectory as anindicator of the system robustness.

In order to numerically estimate the critical time delay, we re-place u(t) in Eq. (1), and in the definition of µ preceding equation(13), by u(t − d), and let qd(t) be given by the constant trajectory(1, 1, 1). For each choice of parameter values, define the criticaltime delay dcrit as the largest value of d for which ∥qa(t)∥∞ < 10for t ∈ [0, 20]. Groups of discrete estimates of dcrit are shown inFig. 7 for k = 10, 20, 30, and 40 Hz. For purposes of comparison,the figure includes estimated values of dcrit in the case that Eq. (36)is replaced by the unfiltered form u(t) = −θ (t)∥rt∥L∞

− σ (t),in which case the control structure is that of an indirect model-reference controller, popular in adaptive control of nonlinearsystems. The results show that increasing the filter bandwidth de-teriorates the system robustness to delay. For finite bandwidth, thecritical time delay remains well above zero across the entire rangeof adaptation gains and shows no sign of deteriorating as Γ → ∞.In contrast, for the case of the traditional model-reference con-troller, the critical time delay is close to zero (∼10−3) for the en-tire range of the adaptation gain. This is consistent with the factthat model-reference controllers have zero robustness margin inthe sense of the gap metric [44,48], and are therefore expected toexhibit limited robustness to input delay for any given desired tra-jectory.

6. Mobile manipulators with suspension systems moving onrough terrains

The case study in the previous section considers a practicalsituation of a manipulator mounted on a ship operating in ahigh-sea state under uncertain environmental disturbances. Theunactuated dynamics of the ship and the actuated dynamics of the

Fig. 7. The dependence of the critical time delay on the adaptation rate for differentvalues of filter bandwidth. The values of dcrit for case of finite bandwidth areobtained with a roundoff error of ±5 × 10−4 . In the case of the model-referencecontroller (corresponding to infinite filter bandwidth), the roundoff error is ±5 ×

10−5 .

manipulator are coupled in such a way that the unmodeled shipmotions add very large time-varying inertia and nonlinearity to thedescription of themanipulator dynamics. However, since themassof the manipulator is significantly smaller than that of the ship, itsmotions have little effect on those of the ship.

In this section, we consider the context of amobilemanipulatorfor which the unactuated and actuated dynamics are stronglycoupled. In addition, the platform oscillates with much higherfrequency than in the case of the ship, as the disturbances froma rough terrain are transmitted to the platform via a suspension.In the control design, we again consider the system dynamics asbeing completely unmodeled. A slight twist is the inclusion of anonholonomic constraint on the platform kinematics.

6.1. Dynamics of a mobile manipulator

The model of a mobile manipulator of interest is sketched inFig. 8, in which w := (w1,w2,w3) is an inertial reference triadand the triad b := (b1, b2, b3) is obtained by rotatingw an angleφhabout the vertical axis w3. The position of the platform’s center ofmass B relative to the inertial reference frame is represented by thedisplacements xv and yv along w1 and w2, respectively. The triad


a := (a1, a2, a3), attached to the platform, is obtained by rotatingb an angle φp about b2. The point A represents the joint connectinglink 1 of the manipulator to the platform, such that rBA := LBAa1.The triad l(1) :=

l(1)1 , l(1)2 , l(1)3

, attached to link 1, is obtained by

rotating a an angle q1 about a2. The triad l(2) :=

l(2)1 , l(2)2 , l(2)3

,

attached to link 2, is obtained by rotating l(1) an angle q2 aboutl(1)2 . Points D, C1, C2, and E represent the joint connecting links 1and 2, the centers of mass of links 1 and 2, and the location of thepayload at the end-effector, respectively, such that rAD := L1l

(1)3

and rDE := L2l(2)3 .

From this set-up, relative to the platform, the manipulator hastwo degrees of freedom, represented by the relative joint anglesq1 and q2. The configuration of the platform relative to the inertialreference frame is described by the position coordinates xv and yv ,the heading angle φh, and the pitching angle φp. The platform isattached to the chassis of a rover via a suspension system to smoothoscillations of the platform that might be induced by an uneventerrain. The chassis’ configuration relative to the inertial frame isassumed to be identical to that of the platform other than in itspitching angle, which is here assumed to be an explicit functionof time that is unknown to the control design. Finally, the motionof the chassis is constrained in such a way that the velocity of theplatform’s center of mass is perpendicular to the axis of the drivingwheels, i.e., parallel to b1.

We assume generalized forces associated with the q1 andq2 degrees of freedom, given by the control torques u1 and u2applied at the joints A and D, respectively. Under the assumptionthat the pitching motion of the platform is unactuated, thegeneralized force corresponding to φp equals 0. The generalizedforces corresponding to the remaining degrees of freedom, xv ,yv , and φh, are assumed to be obtained from a matrix productB(qv)uv . Here, the control input uv may be parameterized by twoindependent control signals representing, for example, the torquesapplied to the left and right driving wheels, respectively.

Based on the expressions for the kinetic and potential energies,as well as the nonconservative generalized forces in Appendix D,and in terms of the vector of generalized coordinates q , [qTr , q

Tv ]

T ,where qr , [q1, q2, φp]

T and qv , [xv, yv, φh]T , the equations of

motion of the mobile manipulator then take the formMrr(q) Mrv(q)MT

rv(q) Mvv(q)

qrqv

+

Nrr(q, q)Nvv(q, q)

=

ur

B(qv)uv

+

0

AT (φh)ℓ

+

DrrDpp

, (54)

where ur = [u1, u2, 0]T , A(φh) is the coefficient matrix of qv in thevelocity constraint, and ℓ is the corresponding Lagrangemultiplier.It follows that the systemhas six geometric degrees of freedomandfive dynamic degrees of freedom. In the control design below, theunactuated pitching angle φp is assumed to be a bounded functionof time.

A popular reduction method is employed to convert thelower part of Eq. (54) for the rover’s locomotion to equations inindependent coordinates (see, e.g., [30]). Let s denote the speedof the rover in the direction of motion and consider the vectorv , [s, φh]

T . The allowable motions of the rover may then bedescribed by the relationship

qv = S(φh)v, (55)

where S(φh) is a full-rank matrix, whose columns are a smoothbasis for the null space of A(φh). Since ST (φh)S(φh) is invertible byconstruction, it follows that

v =

ST (φh)S(φh)

−1ST (φh)qv. (56)

Fig. 8. A mobile manipulator mounted on a platform suspended from a chassismoving across an uneven terrain.

Substitution into the equations of motion andmultiplication of thebottom part with ST (φh) then yields

Mrr(q) Mrv(q)S(φh)

ST (φh)MTrv(q) ST (φh)Mvv(q)S(φh)

qrv

+

Nrr(q, q)

ST (φh)Nvv(q, q)

=

uruv

+

Drr

ST (φh)Dpp

, (57)

where the components of uv , ST (φh)B(qa)uv correspond toindependent control inputs for the speed and rate of change ofheading of the rover.

Froma state-space representation point of view, the descriptionof the kinematics in terms of 12 states (six position coordinatesand six velocity coordinates) in Eq. (54) with one nonholonomicvelocity constraint A(φh)qv = 0 has been reduced to a systemof three second-order differential equations and five first-orderequations in eleven states (six position coordinates and fivevelocity coordinates). To accommodate a control design analogousto that in Section 3, we first rearrange Eq. (57) to yield (cf. [28])Maa(q) Mas(q) Mau(q)

MTas(q) Mss(q) Msu(q)

MTau(q) MT

su(q) Muu(q)

qasqu

+

Na(q, q)Ns(q, q)Nu(q, q)

=

uaus0

+

DaaDssDuu

, (58)

where qa , [q1, q2, φh] and qu = φp. Now let

r ,

rars

=

qa − qad + λ(qa − qad)

s − sd

. (59)

Elimination of qu using the bottom part of Eq. (58), and noting theabsence of explicit dependence of the equations of motion on xv

and yv , then again yields

r = Amr + M−1a u + η, r(0) = r0, (60)

for some functions Ma and η(t, ζ ), where ζ T , [rT , χ T] and χ ,

qTa , qTu, q

Tu

. Here, it follows that

∥χt∥L∞≤ max{∥qa,t∥L∞

, ∥qu,t∥L∞, ∥qu,t∥L∞

}. (61)

The definition of r in Eq. (59) implies

∥qa,t∥L∞≤(sI + λ)−1

L1

ra,tL∞

+qad(s) + (sI + λ)−1qa(0) − qad(0)

L∞

. (62)

Now, from Eqs. (61) and (62) together with the observations thatra,tL∞≤ ∥rt∥L∞

and the assumption that the unactuated


Fig. 9. Typical pitch angle of the platform during operation.

degree of freedom is bounded, it follows that there exist positivenumbers Q1 and Q2 such that

∥χt∥L∞≤ Q1∥rt∥L∞

+ Q2, (63)

which is the same inequality as in Eq. (B.3) in Appendix B.The remainder of the control design now proceeds as in

Section 3, bearing in mind the implications of the modified slidingformulation in Eq. (59). In particular, the adaptive controllerestimates and compensates for the nonlinearity η to drive r to aneighborhood of zero, such that qa and s converge to the vicinitiesof the corresponding desired trajectories qad and sd. The latter doesnot, however, guarantee convergence in the position of the rover,since this is not part of the feedback control design.

6.2. Numerical results

In the numerical results reported below, the link lengths and themasses of the twomanipulator links and the payload are L1 = 0.25,L2 = 0.2, m1 = 6, m2 = 5, and me = 5, respectively. Themass mp of the platform equals 20 and the moments of inertiaabout a1, a2 and a3 are 0.75, 1.75 and 2.5, respectively. The distancebetween B and A is LBA = 0.1. The effective stiffness and dampingof the suspension system is Kp = 50 and Cp = 1, respectively.In addition, Daa(t) = 0 and Dss(t) = 0, and the disturbance tothe platform, due to disturbances to the pitch of the chassis fromtraversal across rough terrain and transmitted via the suspensionsystem, is assumed to be given by Duu(t) = 3 sin(5t) + 3 sin(3t).As a result, the pitching angle of the platform will have the typicalmotion given in Fig. 9.

The proposed control scheme is implemented for the systemdynamics in Eqs. (56)–(57), without assuming any detailed knowl-edge of the system model, in order to track a desired trajectorygiven by the time histories q1d(t) = sin(0.4t), q2d(t) = cos(0.4t),φhd(t) = sin(0.5t)+ cos(0.3t), sd = 0.5t , in the presence of veloc-itymeasurement noise in the range [−0.15, 0.15] and a time delayof 50 ms in the control signal. The control architecture is parame-terized by Am = diag(20, 10, 15, 5) and the same values of λ, Γ ,and k as before. The 4 × 4 matrix Asp is set to 0.1

√Γ I. With this

choice the tracking RMSD% is less than 5% for all relevant degreesof freedom.

The results are shown in Fig. 10. The level of measurementnoise is reflected by the right bottom panel for the predictionerror, which is computed directly using the noisy measuredvelocity. Despite the time delay and measurement noise, theadaptive control signals remain smooth because high-frequencysignals are blocked by the low-pass filter. Here, the high-frequencyoscillations in the platform pitching angle, induced by the roughterrain, are evident in the control channels. However, the controlobjective is still met with the RMSD% for the manipulator’s twojoints, as well as for the steering angle and the speed of the roverbeing 4.47%, 4.87%, 3.02% and 2.08%, respectively. It is remarkedthat this performance is achieved not only in the presence ofmeasurement noise, but also with a delay of 50 ms in the controlsignal.

7. Conclusion

Traditional control architectures employ the linear-in-para-meter property of Lagrangian systems to obtain the factorizationin Eq. (2) in terms of a model regression matrix and a vectorof unknown parameters, which is to be estimated by relevantadaptive laws, of which Eq. (4) is an example. The construction ofthe regression matrices in Eq. (2) requires information about thestructure/geometry of the ship such as the position of the ship’scenter of mass, the ship’s moment of inertia, the location of themanipulator relative to the ship center of mass, and so on. If theother dynamic factors, for example the movements of humans orequipments in the ship, is integrated in the equations of motion,then the modeling process to construct such regression matricesmay not be feasible.

In contrast, this paper has described a robust adaptive con-troller, inspired by the framework proposed in [41] with a low-pass filter in the control input, for an underactuated system of arobot installed on a (moving) platform. The controller does not re-quire the construction of regression matrices, while the dynam-ics of the entire system are unmodeled. We restrict our focus tothose robotic systems operating in challenging environments, suchas free-floating space manipulators, robots on ships or off-shoreplatforms, and mobile manipulators moving across rough terrains.As evidenced by fundamental theoretical results on the existenceof computable performance bounds, this framework successfullyseparates the adaptation loop from the control loop, thereby allow-ing for arbitrary increases in the adaptation rate (bounded only byhardware constraints) without sacrificing the system’s robustness,and allows for a predictable transient response with smooth andimplementable control signals. With the introduction of a variabletransformation, inspired by earlier work in [5], and an innovativecontrol design, the formulation is able to compensate for the non-linearity and uncertainties in the dynamic model without assum-ing any knowledge of the system modeling.

Numerical simulations were used to illustrate the controlparadigm in trajectory tracking tasks imposed on two scenarios:(1) a robot arm mounted on a ship operating in a high-sea state,and (2) a mobile manipulator moving across a rough terrain. Inthe first context, the ship has three unactuated degrees of freedom,which are disturbed by unknown environmental factors, e.g., wind,waves, and ocean currents. In the context of the mobile manipula-tor operating on a rough terrain, the disturbances from the unmod-eled geometry of the terrain are transmitted to the platform viathe suspension system of the rover. In contrast to traditional adap-tive controllers, for which the lack of knowledge about the ship dy-namics and rover motion violates basic assumptions of the controldesign, the proposed controller is independent of the systemmod-eling, and therefore especially useful for systems with unmodeleddynamics. The results demonstrate desirable tracking performanceand clean and smooth control signalswith different types of distur-bances, including measurement noise and unknown time delay.

The successful design of the proposed control architecture re-lies upon a key parameterization of the nonlinear contribution tothe robot equations ofmotion in terms of two time-varying param-eters with theL∞ norm of the sliding state as a regressor. The con-troller further employs projection operators in the adaptive laws toimpose bounds on the parameter estimates, and uses a low-passfilter in the control signal to keep the control-signal frequenciesbelow the available control-system bandwidth. In this case, Theo-rem 4.1 implies close agreement between the system response andthe control signal, on the onehand, and the corresponding timehis-tories for a suitably formulated nonadaptive reference system, onthe other hand, provided that the adaptation gain is chosen suffi-ciently large. Any deviation between the system response and thedesired trajectory observed in the numerical results may be traced


Fig. 10. Performance of the proposed controller in the presence of 50 ms actuator time delay and velocity measurement noise in the range [−0.15, 0.15] rad/s and withsample time of 0.01 s.

to the need tomaintain a finite filter bandwidth, in order to guaran-tee robustness. This is the design trade-off between performanceand robustness of the proposed control architecture.

The control scheme proposed in this paper for underactuatedrobotic systems is relatively simple with the state predictor in Eq.(35), the adaptive control signal in Eq. (36), and the adaptation lawsin Eqs. (37) and (38). In addition, the control formulation is inde-pendent of systemmodeling. Hence, the same control structure canbe used for any Lagrangian system, including serial manipulatorsor parallel robots, while current adaptive controllers formanipula-tors require reconstruction of the regressionmatrix for each appli-cation, or accurate estimation of the system model. Furthermore,though the control scheme is demonstrated for a robot mountedon a ship and amobilemanipulator, it can be directly implementedfor free-floating space manipulators whose dynamics have the ex-act form in Eq. (1), as well as any fixed-base manipulators whoseequations of motion are simpler than those considered here.

We finally comment on the observations made regarding thecritical time delay during the two trajectory-tracking tasks con-sidered above. As seen in [41] in the case of a linear, constant-coefficient, single-input system, the basic control architecturesupports the formulation of theoretical lower bounds on the timedelay margin, through the use of a suitably formulated equivalentLTI system. Somedegree of robustness is therefore guaranteed pro-vided that the adaptation gain is sufficiently large. Work is cur-rently underway (cf. [49]) to adapt these theoretical results to themanipulator context. For static andperiodic reference inputs, a sys-tematic analysis of the dependence on the actual critical time de-lay on the control parameters may be obtained, for example, usingtechniques of numerical continuation.Wewill return to these con-siderations in a future publication.

Acknowledgments

This work was partially supported by a NASA SBIR Phase Icontract, order no. NNX12CE97P, awarded to CU Aerospace, L.L.C.We gratefully acknowledge Naira Hovakimyan, Evgeny Kharisov,and Enric Xargay for instructing the authors on the overall controlmethodology, and for their feedback and contributions to an earliertechnical report on this topic.

Appendix A. An input–output map

For each smooth function v : R → Rn+m, let Φv : R × R →

Rn×n be the unique solution to the initial-value problem

∂

∂tΦv(t, t ′) = −kM−1

a

v(t)

Φv(t, t ′), Φv(t ′, t ′) = I, (A.1)

where the symmetric matrix Ma is a smooth function of itsargument, and

0 < mlI ≤ M−1a (q) ≤ mhI, for all q ∈ Rn+m. (A.2)

Using the Gronwall lemma, it follows that

trΦT

v (t, t ′)Φv(t, t ′)

≤ ne−2kml(t−t ′) (A.3)

and, consequently, thatΦv,ij(t, t ′) ≤√ne−kml(t−t ′) for i, j = 1, . . . ,

n, independently of v. TheL1 norm of the linear input–outputmap

Fv : η → −k t

0Φv(t, t ′)η(t ′) dt ′ (A.4)

is then given by

∥Fv∥L1 , max1≤i≤n

n

j=1

supt≥t∗,t∗∈R+

t

t∗kΦv,ij(t, t ′)

dt ′

≤n√n

ml, (A.5)

independently of v.Let Mv[η] , M−1

a (v)η. Since the matrix M−1a is bounded, it fol-

lows that the compositionsMv◦Fv andFv◦Mv also have boundednorm, independently of v. Moreover, since ∂Φv(t, t ′)/∂t ′ =

Φv(t, t ′)kM−1a (v(t ′)), it follows by integration by parts that

Fv[η] = −k (Fv ◦ Mv + I) [η], (A.6)

provided that η(0) = 0.


Appendix B. Preliminary parameterizations and bounds

Let τ > 0 be given, and consider a differentiable functionη(t, ζ ) : R × Rp

→ Rn, such that

∥η(t, 0)∥∞ ≤ Z < ∞ (B.1)

and

∥ζ∥∞ ≤ ξ ⇒

∂η(t, ζ )

∂t

∞

≤ dηt (ξ) < ∞,∂η(t, ζ )

∂ζ

∞

≤ dηζ(ξ) < ∞

(B.2)

for arbitrary ξ and t ∈ [0, τ ]. Let ∥ft∥L∞denote the restriction

of the L∞ norm to the interval [0, t], and suppose that thedifferentiable functions r(t) and χ(t) satisfy the inequality

∥χt∥L∞≤ Q1∥rt∥L∞

+ Q2, t ∈ [0, τ ] (B.3)

for some positive constants Q1 and Q2. Let

ρ(ρ) , max{ρ + 1,Q1(ρ + 1) + Q2}, (B.4)

and define

Lρ ,ρ(ρ)

ρdηζ

ρ(ρ)

(B.5)

and ζ (t) ,r(t), χ(t)

. Together with (B.3), the bound ∥rτ∥L∞

≤

ρ < ∞ implies that

∥ζt∥L∞≤ max{∥rt∥L∞

, ∥χt∥L∞} < ρ(ρ) (B.6)

and, using the bounds in (B.1) and (B.2),

∥ηt∥∞ ≤ ∥η(t, ζ (t)) − η(t, 0)

∥∞ + ∥η(t, 0)∥∞ < ρLρ + Z

(B.7)

for t ∈ [0, τ ]. If, in addition, ∥rτ∥L∞≤ dr < ∞, then:

Lemma B.1. There exist continuous, piecewise-differentiable func-tions θ(t) ∈ Rn and σ(t) ∈ Rn such that

ηt, ζ (t)

= θ(t)∥rt∥L∞

+ σ(t), t ∈ [0, τ ]. (B.8)

On this interval, ∥θ(t)∥∞ < Lρ and ∥σ(t)∥∞ < LρQ2 + Z + α forsome arbitrary α > 0, and, within any subinterval of differentiability,∥θ (t)∥∞ < dθ < ∞ and ∥σ (t)∥∞ < dσ < ∞.

Proof. See proof of Lemma A.9.2 in [41]. �

The conclusions of Lemma B.1 hold in the special case thatχ(t) ,

qa(t), qu(t), qu(t)

, where the functions qu(t) and qu(t)

are bounded for all t , and

qa − qad + λ(qa − qad) = r (B.9)

for λ > 0 and some bounded function qad(t) with all boundedderivatives. Indeed, in this case

∥χt∥L∞≤ ∥qa,t∥L∞

+ max{∥qu,t∥L∞, ∥qu,t∥L∞

}

≤ Q1∥rt∥L∞+ Q2, (B.10)

where Q1 = ∥(s + λ)−1∥L1 = λ−1 and

Q2 = ∥(s + λ)−1(qa(0) − qad(0))∥L∞+ ∥qad,t∥L∞

+ max{∥qu,t∥L∞, ∥qu,t∥L∞

}. (B.11)

As a byproduct, it follows that ∥qa,τ∥L∞and ∥qa,τ∥L∞

are bothfinite.

Suppose, instead, that η(t, ζ ) satisfies (B.1) and (B.2), r(t) andχ(t) satisfy (B.3), and

r(t) = A[η(t, ζ (t))] + B[r(0)δ(t)], (B.12)

where ζ (t) ,r(t), χ(t)

, and A and B are linear integral opera-

tors such that ∥A∥L1 = A < ∞, and ∥B[r(0)δ(t)]∥L∞= B < ∞.

Then,

Lemma B.2. The inequalities ∥r(0)∥∞ < ρ and A(ρLρ +Z)+B < ρimply that ∥rτ∥L∞

< ρ .

Proof. Suppose that ∥rτ∥L∞≥ ρ. Then ∥r(0)∥∞ < ρ implies that

there exists a τ ∈ [0, τ ], such that

∥r(t)∥∞ < ρ, ∀t ∈ [0, τ ), and∥rτ∥L∞

= ∥r(τ )∥∞ = ρ.(B.13)

Using (B.12) and (B.7) we obtain

∥rτ∥L∞≤ A∥ητ∥L∞

+ B < ρ, (B.14)

contradicting the original inequality. �

Appendix C. Estimation bounds

Given positive scalars xb and ε, let f : Rn→ R be given by

f : x →∥x∥2

− (xb − ε)2

2εxb − ε2(C.1)

such that f |∥x∥=xb−ε = 0 and f |∥x∥=xb = 1, and define the projectionoperator [50]:

Proj : (x, y; xb, ε) →

y − f (x)

yT

∇f (x)∥∇f (x)∥

∇f (x)

∥∇f (x)∥if f (x) > 0 and yT∇f (x) > 0

y otherwise

(C.2)

for x, y ∈ Rn. Then, following [50], for Γ > 0,

x = Γ Proj(x, y; xb, ε) and∥x(0)∥ ≤ xb ⇒ ∥x(t)∥ ≤ xb, ∀t.

(C.3)

Moreover, provided that ∥x∗∥ ≤ xb − ε,

x − x∗T Proj(x, y; xb, ε) − y

≤ 0. (C.4)

Now let τ > 0 and ε > 0 be given, and suppose that θ(t)and σ(t) are continuous, piecewise-differentiable functions thatsatisfy ∥θ(t)∥∞ < θb − ε < ∞ and ∥σ(t)∥∞ < σb − ε < ∞

for all t ∈ [0, τ ], and, within any subinterval of differentiability,∥θ (t)∥∞ < dθ < ∞ and ∥σ (t)∥∞ < dσ < ∞. Let the symmetric,positive-definite matrix P satisfy the Lyapunov equation A⊤

spP +

PAsp = −Q , for some arbitrary symmetric, positive-definitematrixQ and Hurwitzmatrix Asp. Finally, for some functionµ(t), considerthe initial-value problems

r = Amr + µ + θ∥rt∥L∞+ σ , r(0) = r0 (C.5)

and

˙r = Amr + µ + θ∥rt∥L∞+ σ + Asp r, r(0) = r0, (C.6)

˙θ = Γ Proj

θ , −Pr∥rt∥L∞

; θb, ε, θ (0) = θ0, (C.7)

˙σ = Γ Projσ , −Pr; σb, ε

, σ (0) = σ0, (C.8)

where r , r − r , ∥θ0∥ ≤ θb, and ∥σ0∥ ≤ σb.

Lemma C.1. Let λmin(S) and λmax(S) denote the smallest and largesteigenvalue, respectively, of a positive-definite symmetric matrix S.Then,

∥rτ∥L∞≤

νm

λmin(P)Γ, (C.9)

where

νm , 4(θ2b + σ 2

b ) + 4λmax(P)

λmin(Q )

θbdθ + σbdσ

. (C.10)


Proof. Let θ , θ − θ and σ , σ − σ and define the Lyapunovfunction

V , rTPr +1Γ

(θ T θ + σ T σ ). (C.11)

It follows that

V (0) ≤4Γ

θ2b + σ 2

b

<

νm

Γ. (C.12)

We show by contradiction that

V (t) ≤νm

Γ, ∀t ∈ [0, τ ]. (C.13)

To this end, choose τ ∈ (0, τ ] such that θ and σ are continuous on[0, τ ). Suppose that V (τ ) > νm/Γ and V (τ ) ≥ 0 for some τ < τ .It follows from (C.11) that

∥r(τ )∥2∞

>4

Γ λmin(Q )

θbdθ + σbdσ

. (C.14)

Moreover, by the properties of the projection operators,

V ≤ −rTQ r +2Γ

θ T θ + σ T σ

≤ −∥r∥2∞

λmin(Q ) +4Γ

θbdθ + σbdσ

(C.15)

for all t ∈ [0, τ ], and we arrive at a contradiction by evaluation att = τ . By continuity, V (t) ≤ νm/Γ for all t ∈ [0, τ ]. Eq. (C.11) thenimplies that

∥rτ∥L∞≤

νm

λmin(P)Γ. (C.16)

By repeating this analysis for each subsequent interval of continu-ity of θ and σ , we conclude that (C.16) holds with τ replaced byτ . �

Suppose that the function η(t, ζ ) satisfies (B.1) and (B.2). Letζ (t) :=

r(t), χ(t)

, where

r(t) = Amr(t) + µ(t) + η(t, ζ (t)), r(0) = r0 (C.17)

for some function µ(t). Suppose that r(t) and χ(t) satisfy (B.3),∥rτ∥L∞

< ρ < ∞, and ∥µτ∥L∞< ∞. In this case, Eq. (B.7) im-

plies ∥ητ∥L∞< ∞. Eq. (C.5) and the bounds on θ , σ , θ and σ then

follow from Lemma B.1 and the conclusions of Lemma C.1 againapply.

Suppose, in addition, that µ = M[u], where ∥M∥L1 < ∞ and∥uτ∥L∞

= ub < ∞. Then, the conclusions of Lemma C.1 followalso by replacingµ by u in Eqs. (C.5) and (C.6), σ by σ + (M − I) uin Eq. (C.5), and σb by σb +

∥M∥L1 + 1

ub in Eq. (C.8).

Appendix D. Model for simulation

Themechanicalmodel studied in Section 5 is described in termsof the generalized coordinates q1, q2, q3,φb, xb and zb. The equationsof motion are obtained from Lagrange’s equations of the first kindin terms of the total kinetic energy

T = 0.5(J1 + J2 + L21m4 + L22m5 + 2L1L2m2c2)q21+ 0.5(J2 + L22m5)q22 + 0.25

J1 + J1p + J2 + J2p

+ L21m4 + L22m5 − (J1 − J1p + L21m4)c212 + 2L1L2m2c2

− (J2 + J2p − L22m5)c1222 − 2L1L2m2c2122

× q23 +

0.5Jship + L1L2m2c1c12 + 0.25c21

× (J1 + 2L21m4 + J1c32) + 0.25c212(J2 + L22m2 + J2c32)

+ 0.5d2Am3c2b + 0.5c23 (J1ps21 + J2ps212)

+ dAL1c2b s1s3m1 + dAL2c2b s12s3m2 + 0.5s23(J1 + J2+ L21c

2b s

21m4) + 0.5c2b s1s12s

23(2L1L2m2 + L22m5)

+ s2b0.5d2Am3 + dAL1s1s3m1 + dAL2s12s3m2

+ s23(0.5L21s

21m4 + L1L2s1s12m2 + 0.5L22s

212m5)

φ2b

+ 0.5m3(x2b + z2b ) +L1m1c1cb + L2m2c12cb

− (dAm3 + L1m1s1s3 + L2m2s12s3)sbxbφb

−cb(dAm3 + L1m1s1s3 + L2m2s12s3) + (L1c1m1

+ L2m2c12)sbzbφb + c3q3

cb(L1m1s1 + L2m2s12)xb

− (L1m1s1 + L2m2s12)sbzb +0.5(J1 − J1p

+ L21m4)s12 + 0.5(J2 − J2p + L22m5)s1222

+ L1L2m2s122φb+ q1

J2 + L22m5 + L1L2m2c2

q2

+s1(−L2m2cbs2s3 − (L1m1 + L2m2c2)sb)

+ c1((L1m1 + L2m2c2)cbs3 − L2m2s2sb)xb

− (−L1m1cbs1 − L2m2(c2cbs1 + c1cbs2 + c1c2s3sb− s1s2s3sb) − L1m1c1s3sb)zb +

dAL1m1s1

+ dAL2m2s12 + (J1 + J2 + L21m4 + L22m5)s3

+ 2L1L2m2c2s3φb

+ q2

L2−m2s1(cbs2s3 + c2sb)

+ c1(m2c2cbs3 − m2s2sb)xb + L2

−m2c2(cbs1

+ c1s3sb) + s2(−m2c1cb + m2s1s3sb)zb

+ (dAL2m2c1s2 + (J2 + L22m5)s3

+ L2m2c2(dAs1 + L1s3))φb

, (D.1)

the total potential energy

U = g(L1m1c1cb + L2m2c12cb − dA(m1 + m2 + me)sb

+ m3zb) + 0.5(Kφφ2b + Kxx2b + Kzz2b ), (D.2)

and the corresponding nonconservative generalized forces

Fq = [u1 + Daa,1, u2 + Daa,2, u3 + Daa,3, −Cφ φb + Duu,1,

−Cxxb + Duu,2, −Cz zb + Duu,3]. (D.3)

Here, m1 = 0.5m1 + m2 + me, m2 = 0.5m2 + me, m3 = m1 +

m2 + me + mship, m4 = 0.25m1 + m2 + me, m5 = 0.25m2 + me,cb = cosφb, ci = cos qi, cikj = cos(kqi + qj), and so on.

The mechanical model studied in Section 6 is described interms of the generalized coordinates q1, q2, φp, φh, xb and yb. Theequations of motion are obtained from Lagrange’s equations of thefirst kind in terms of the total kinetic energy

T = 0.5(J1 + J2 + L22m5 + L21m4 + 2L1L2m2c2)q21+ 0.5(J2 + L22m5)q22 + 0.25

J1 + J1p + J2 + J2p

+ Jh + L2BAm6 + L22m5 + L21m4 + (Jh + L2BAm6)cp2− (J1 − J1p + L21m4)c12p2 + 2L1L2m2c2− (J2 − J2p + 0.25L22m2)c1222p2 − (L22me + 2L1L2m2)

× c122p2 + 2L1LBAm7(s1 + s1p2) + L2LBAm2

× (s12 + s12p2)φ2h + 0.5

J1 + J2 + Jp + L2BAm6 + L22m5

+ L21m4 + 2L1L2m2c2 + 2LBA(L1m7s1+ L2m2s12)

φ2p + 0.5m8(x2b + y2b) + (J2 + L22m5

+ L1L2m2c2)q1q2 + φp

J1 + J2 + L22m5 + L21m4


+ 2L1L2m2c2 + LBA(L1m7s1 + 0.5L2m2s12)q1

+ (J2 + L22m5 + L1L2m2c2 + 0.5L2LBAm2s12)q2

+ ch

(L1m1c1p + L2m2c12p − LBAm6sp)φp

+ (L1m1c1p + L2m2c12p)q1 + L2m2c12pq2xb

+ (LBAm6cp + L1m1s1p + L2m2s12p)φhyb

+ sh(−LBAm6cp − L1m1s1p + L2m2s12p)φhxb

+ ((L1m1c1p + L2m2c12p − LBAm6sp)φp

+ (L1m1c1p + L2m2c12p)q1 + L2m2c12pq2)yb

(D.4)

the total potential energy

U = 0.5Kpφ2p + g(cp(L1m1c1 + L2m2c12)

− 3LBAmesp), (D.5)

and the corresponding nonconservative generalized forces

Fq =

u1 + Drr,1, u2 + Drr,2, − Cpφp + Drr,3,

b2rw

(ur − ul) + Dpp,1,cosφh

rw(ur + ul) − sinφhℓ

+Dpp,2,sinφh

rw(ur + ul) + cosφhℓ + Dpp,3

. (D.6)

Here, m1 through m5 are defined in the same way as before, andm6 = m1 + m2, m7 = 0.5m1 + m2, m8 = m1 + m2 + me +

mb. The quantity ℓ is the Lagrange multiplier associated with thenonholonomic constraint, ur and ul are the input torques to theright and left wheels, respectively, rw = 0.1 is the driving wheel’sradius, and b = 0.5 is the width of the platform.

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[50] E. Lavretsky, T.E. Gibson, Projection operator in adaptive systems, 2011.arXiv:1112.4232v6 [nlin.AO].

Kim-Doang Nguyen received the B.E. and M.E. degreesfrom Nanyang Technological University, Singapore, in2007 and 2010, respectively. He was mainly involvedin the development of a new generation of inexpen-sive motion-capture systems. He is currently working to-ward the Ph.D. degree in the department of MechanicalScience and Engineering at University of Illinois at Urbana-Champaign. His research interests include robotics, dy-namical systems, control theories, and delay differentialequations.

Harry Dankowicz is a professor of Mechanical Scienceand Engineering at the University of Illinois at Urbana-Champaign. He graduated from KTH Royal Institute ofTechnology in Stockholm, Sweden, with an M.Sc. in Engi-neering Physics in 1991 and, subsequently, from CornellUniversity with a Ph.D. in Theoretical and Applied Me-chanics in 1995. Following a post-doctoral and researchassociate appointment at KTH between 1995 and 1999,he joined the Department of Engineering Science and Me-chanics at Virginia Polytechnic Institute and State Univer-sity, where he remained until 2005. He is a recipient of

several prestigious faculty career awards, including a Junior Investigator Grant fromthe Swedish Foundation for Strategic Research, a CAREER award from the US Na-tional Science Foundation, a PECASE award from the US National Science Founda-tion, and the Fred Merryfield Design Award from the ASEE. He conducts dynamicalsystems research at the intersection of engineering,math and physics. Thiswork in-volves studying a wide range of complex systems that are governed by differentialequations and learning the behavior of those systems through theory and experi-ments. His research efforts further seek to make original and substantial contribu-tions to the development and design of existing or novel devices or methodologiesthat capitalize on system nonlinearities for improved system understanding andperformance.


http://arxiv.org/1112.4232v6

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