lyapunov based control

1050 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICSPART B: CYBERNETICS, VOL. 38, NO. 4, AUGUST 2008

Adaptive Lyapunov-Based Control of a Robotand MassSpring System Undergoing

an Impact CollisionKeith Dupree, Student Member, IEEE, Chien-Hao Liang, Student Member, IEEE,

Guoqiang Hu, Member, IEEE, and Warren E. Dixon, Senior Member, IEEE

AbstractThe control of dynamic systems that undergo animpact collision is both theoretically challenging and of practi-cal importance. An appeal of studying systems that undergo animpact is that short-duration effects such as high stresses, rapiddissipation of energy, and fast acceleration and deceleration maybe achieved from low-energy sources. However, colliding systemspresent a difficult control challenge because the equations ofmotion are different when the system suddenly transitions froma noncontact state to a contact state. In this paper, an adaptivenonlinear controller is designed to regulate the states of twodynamic systems that collide. The academic example of a planarrobot colliding with an unactuated massspring system is used torepresent a broader class of such systems. The control objective isdefined as the desire to command a robot to collide with an unac-tuated system and regulate the mass to a desired compressed statewhile compensating for the unknown constant system parameters.Lyapunov-based methods are used to develop a continuous adap-tive controller that yields asymptotic regulation of the mass androbot links. It is interesting to note that one controller is respon-sible for achieving the control objective when the robot is in freemotion (i.e., decoupled from the massspring system), when thesystems collide, and when the system dynamics are coupled.

Index TermsAdaptive control, backstepping, impactdynamics.

I. INTRODUCTION

ROBOT motion and force control has been well studiedfor several decades (e.g., [1][17]). Most of this paperinvolves designing a controller to position the end-effectorof a robot manipulator constrained to move along a surfaceto a desired point while controlling the force exerted on thesurface. Several techniques to compensate for uncertainty in

Manuscript received August 27, 2007; revised October 30, 2007 andJanuary 8, 2008. This work was supported in part by the National ScienceFoundation under CAREER Award 0547448, in part by the Air Force Officeof Scientific Research under Contract F49620-03-1-0170, in part by ResearchGrant US-3715-05 from the United StatesIsrael Binational Agricultural Re-search and Development Fund, and in part by the Department of Energy(DOE) through Grant DE-FG04-86NE37967 as part of the DOE UniversityResearch Program in Robotics. This paper was recommended by AssociateEditor H. Gao.

K. Dupree, C.-H. Liang, and W. E. Dixon are with the Department ofMechanical and Aerospace Engineering, University of Florida, Gainesville, FL32611 USA (e-mail: [email protected]; [email protected]; [email protected]).

G. Hu was with the Department of Mechanical and Aerospace Engineer-ing, University of Florida, Gainesville, FL 32611 USA. He is now with theDepartment of Mechanical and Nuclear Engineering, Kansas State University,Manhattan, KS 66506 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSMCB.2008.923154

the robot dynamics during contact with the surface have alsobeen reported (e.g., [5], [6], [11], [13], and [17]). The robottypically is required to stay in contact with a static surface;however, techniques have also been developed for the robot toremain in contact with a moving object (e.g., [7][9]). However,the results from this literature have not addressed the problemwhere the robot transitions from a noncontact state to an impactcollision with another dynamic system, with an objective tothen control the coupled dynamic systems.

The control of dynamic systems that undergo an impactcollision is both theoretically challenging and of practical im-portance. If the impact dynamics are not properly modeledand controlled, the impact forces could result in poor systemperformance and instabilities. One difficulty in controllingsystems subject to impacts is that the dynamics are differentwhen the system status transitions from a noncontact stateto a contact state. Another difficulty is measuring the impactforce, which can depend on the geometry of the robot, thegeometry of the environment, and the type of impact. As statedin [18], the appeal of systems with impact conditions is thatshort-duration effects such as high stresses, rapid dissipation ofenergy, and fast acceleration and deceleration may be achievedfrom low-energy sources. Some example current and emergingapplications that motivate this field of research include man-ufacturing, biped walking robots, manipulation of rigid andnonrigid bodies, grasping or catching with a robotic hand, andhumanmachine interaction applications such as robot-assistedrehabilitation.

Largely motivated by manufacturing applications, the studyof systems that undergo an impact collision has historicallytargeted hard-on-hard collisions with an infinitely large andrigid mass. As a result, the impact collision is modeled as adiscontinuous event, where discontinuous controllers have beendeveloped to compensate for the instantaneous velocity jump(e.g., [4] and [19][25]). For example, a discontinuous con-troller was designed in [24] to regulate the impact of a hydraulicactuator with a static environment. A switching controller wasdeveloped in [25] to eliminate the bouncing phenomena asso-ciated with a robot impacting a static surface. In [19], a classof switching controllers was examined for mechanical systemssubject to an algebraic inequality condition and an impact rulerelating the interaction impulse and the velocity. The analysisin [19] utilized a discrete Lyapunov function that required theuse of the Dini derivative to examine the stability of the sys-tem. A hybrid bang-bang impedance/time-delay controller was

1083-4419/$25.00 2008 IEEE

DUPREE et al.: ADAPTIVE LYAPUNOV-BASED CONTROL OF A ROBOT 1051

developed in [20] that establishes a stable contact and achievesthe desired dynamics for contact or noncontact conditions.

Although discontinuous impact models with impulsive ve-locity effects may effectively characterize collisions betweenrigid objects, these models do not capture the effects when arigid body collides with a deformable surface. Existing modelsof contact with a deformable surface typically characterize thedeformation as a linear or nonlinear spring with some boundedstiffness (e.g., [18] and [26][35]). A challenge for the develop-ment of controllers based on these models is that the stiffnessmay not be a priori known or may change through repeateduse (e.g., strain hardening). For example, a two-degree-of-freedom planar manipulator was asymptotically regulated tocontact an infinitely rigid surface with a deformable second linkrepresented by a massless linear spring in [27], where the robotcontroller either required measurement of the interaction forceor the spring stiffness. A reduced-order observer was used in[26] to control the impact between the end-effectors of twocooperating manipulators, where the exact knowledge of therobot dynamics and stiffness of the contact was required. In[33], a backstepping approach is used to asymptotically regulatethe impact of two systems where the exact system dynamics areknown. In [34], a class of continuous energy-based controllerswas developed that achieves global asymptotic stabilization/regulation of an underactuated EulerLagrange system subjectto an elastic contact with finite stiffness provided that the exactknowledge of the impact stiffness is known.

This paper considers a rigid robotic system that undergoes animpact collision with an underactuated deformable massspringsystem. The robot/massspring collision is modeled as a differ-entiable impact, as in the recent work in [18] and [34][36].The dynamic model for both systems and the impact force areassumed to have uncertain parameters. This paper is specificallyfocused on a planar robot colliding with a massspring systemas an academic example of a broader class of such systems.The control objective is defined as the desire to enable arobot to collide with an unactuated system and regulate theresulting coupled massspring robot (MSR) system to a desiredcompressed state despite parametric uncertainty (i.e., unknownmasses and stiffness) throughout the MSR system.

Based on the aforementioned objective, one control strategycould be to implement a controller that regulated the robotto a desired constant set point associated with the desiredmassspring system. The main drawback of such an approachis that the dynamics associated with the impact collision wouldappear as a disturbance to the controller, requiring the controllerto exploit high-gain or high-frequency feedback to ensure sta-bility. In contrast to a regulation control strategy, the controldevelopment in this paper uses integrator backstepping [37] todesign a desired robot trajectory as a virtual control input to themassspring system. The desired robot trajectory is designedbased on a Lyapunov analysis that includes the dynamics of theimpact collision and the coupled motion of the MSR system.A force control input (that can be related to the actual robotjoint torque inputs) is then designed to asymptotically eliminatethe mismatch between the time-varying desired robot trajectoryand the actual trajectory. Unlike some other results in literature,the continuous force controller does not depend on measuring

Fig. 1. MSR system is an academic example of an impact between twodynamic systems.

the impact force or the measurement of other accelerationterms and, due to the adaptive nature, does not depend on theknowledge of the masses or stiffness, although upper boundsare assumed known.

II. DYNAMIC MODEL

The subsequent development is motivated by the academicproblem illustrated in Fig. 1. The dynamic model for the two-link revolute robot depicted in Fig. 1 can be expressed in thejoint space as follows:

M(q)q + C(q, q)q + h(q) = (1)where q(t),q(t),q(t) R2 represent the angular position, thevelocity, and the acceleration of the robot links, respectively,M(q) R22 represents the uncertain inertia matrix, C(q, q) R

22 represents the uncertain centripetal Coriolis effects,h(q) = [h1(q), h2(q)]T R2 represents uncertain conservativeforces (e.g., gravity), and (t) R2 represents the torque con-trol inputs. The Euclidean position of the endpoint of the secondrobot link is denoted by xr(t)

= [xr1(t), xr2(t)]T R2, whichcan be related to the joint space through the following kinematicrelationship:

xr = J(q)q (2)where J(q) R22 denotes the manipulator Jacobian. The un-forced dynamics of the massspring system in Fig. 1 is given by

mxm + ks(xm x0) = 0 (3)where xm(t), xm(t), xm(t) R represent the displacement,the velocity, and the acceleration of the unknown mass m R,x0 R represents the initial undisturbed position of the mass,and ks R represents the unknown stiffness of the spring.

Assumption 1: We assume that xr1(t) and xm(t) can bebounded as follows:

xr xr1(t) xm(t) xm (4)where xr R is a known constant that is determined by theminimum coordinate of the robot along the X1-axis, and xm R is a known positive constant. The lower bound assumptionfor xr1(t) is based on the geometry of the robot, and the upperbound assumption for xm(t) is based on the physical fact thatthe mass is attached by the spring to some object, and the masswill not be able to move past that object.


In the following, the contact model is considered as an elasticcontact with finite stiffness. An impact between the second linkof the robot and the springmass system occurs when xr1(t) xm(t). The impact will yield equal and opposite force reactionsbetween the robot and the massspring system. Specifically, theimpact force acting on the mass, represented by Fm(xr, xm) R, is assumed to have the following form [18], [36]:

Fm = KI(xr1 xm) (5)where KI R represents an unknown positive stiffness con-stant, and (xr, xm) R is defined as

={ 1 xr1 xm

0 xr1 < xm.(6)

The impact force acting on the robot links produces a torque,denoted by d(xr, xm, q) R2, as follows:

d = KI(xr1 xm)[l1 sin(q1) + l2 sin(q2 + q1)

l2 sin(q2 + q1)

](7)

where l1, l2 R denote the robot link lengths. Based on (1),(3), and (5)(7), the dynamic model for the MSR system can beexpressed as follows:

M(q)q + C(q, q)q + h(q) d = mxm + ks(xm x0) =Fm. (8)

After premultiplying the robot dynamics by the inverse of theJacobian transpose and utilizing (2), the dynamics in (8) can berewritten as

M(xr)xr + C(xr, xr)xr + h(xr) +[Fm0

]=F (9)

mxm + ks(xm x0) =Fm (10)where F (t) = JT (q)(t) R2 denotes the manipulatorforce. The dynamic model in (9) exhibits the following prop-erties that will be utilized in the subsequent analysis.

Assumption 2: During the subsequent control development,we assume that the minimum singular value of J(q) isgreater than a known small positive constant > 0, such thatmax{J1(q)} is known a priori, and hence, all kinematicsingularities are always avoided.

Assumption 3: We assume that the mass of the massspringsystem can be upper and lower bounded as

ml < m < mu

where ml,mu R denote known positive bounding constants.The unknown stiffness constants KI and ks are also assumedto be bounded as follows:

K

< KI < K ks< ks < ks (11)

where K

, K , ks, ks R denote known positive bounding

constants.Property 1: The inertia matrix M(xr) is symmetric, posi-

tive definite, and can be lower and upper bounded as follows[38], [39]:

a12 T M a22, R2

where a1, a2 R are positive constants.

Property 2: The following skew-symmetric relationship issatisfied [38], [39]:

T(

12, M(xr) C(xr, xr)

) = 0, R2. (12)

Property 3: The robot dynamics given in (9) can be linearlyparameterized as follows [38], [39]:

Y (xr, xr, xr) = M(xr)xr + C(xr, xr)xr + h(xr) +[Fm0

]

where Rp contains the constant unknown system parame-ters, and Y (xr, xr, xr) R2p denotes the known regressionmatrix.

III. CONTROL DEVELOPMENT

As previously stated, the control objective is defined as thedesire to enable a robot to collide with an unactuated systemand regulate the resulting coupled MSR system to a desiredcompressed state despite parametric uncertainty (i.e., unknownmasses and stiffness) throughout the MSR system. To achievethis objective, the control development is based on integratorbackstepping [37] to design a desired robot trajectory as avirtual control input to the massspring system. The desiredrobot trajectory is designed based on a Lyapunov analysisthat includes the dynamics of the impact collision and thecoupled motion of the MSR system. A force control input (thatcan be related to the actual robot joint torque inputs) is thendesigned to asymptotically eliminate the mismatch between thetime-varying desired robot trajectory and the actual trajectory.As is typical with integrator backstepping methods, the forcecontroller depends on the time derivative of the desired robottrajectory (i.e., the virtual control input to the massspringsystem). Taking the derivative of the desired trajectory couldlead to unmeasurable higher order terms (i.e., acceleration). Thesubsequent development exploits the hyperbolic filter structuredeveloped in [40] and [41] to overcome the problem of injectinghigher order terms in the controller and to facilitate the de-velopment of sufficient gain conditions used in the subsequentstability analysis.

A. Control ObjectiveThe control objective is to regulate the states of an uncertain

dynamic system (i.e., a two-link planar robot) that has animpact collision with another uncertain dynamic system (i.e.,a massspring). A regulation error, denoted by e(t) R3, isdefined to quantify this objective as

e = [ em eTr ]T

where er(t)= [er1, er2]T R2 and em(t) R denote the reg-

ulation error for the endpoint of the second link of the robot andmassspring system (see Fig. 1), respectively, and are definedas follows:

er = xrd xr em = xmd xm. (13)


In (13), xmd R denotes the constant known desired positionof the mass, and xrd(t)

= [xrd1(t), xrd2]T R2 denotes thedesired position of the endpoint of the second link of the robot.The subsequent development is based on the assumption thatq(t), q(t), xm(t), and xm(t) are measurable, and that xr(t)and xr(t) can be obtained from q(t) and q(t). To facilitatethe subsequent control design and stability analysis, filteredtracking errors,1 denoted by m(t) R and rr(t) R2, aredefined as follows [40], [41]:

m = em + 1 tanh(em) + 2 tanh(ef )rr = er + er (14)

where , 1, 2 R are positive constant gains, and ef (t) Ris designed as follows [40], [41]:

ef = 3 tanh(ef ) + 2 tanh(em) k1 cosh2(ef )m (15)where k1 R is a positive constant control gain, and 3 R isa positive constant filter gain. The filtered tracking error rr(t)is introduced to reduce the terms in the Lyapunov analysis[i.e., rr(t) can be used in lieu of including er(t) and er(t)].The filtered tracking error m(t) and the auxiliary signal ef (t)are introduced to eliminate dependence on acceleration in thesubsequently designed force controller [42].

B. Closed-Loop Error System

By taking the time derivative of mm(t) and utilizing (5),(10), (13), and (14), the following open-loop error system canbe obtained:

mm = Ydd KI(xr1 xm)+ 2m cosh2(ef )ef + 1m cosh2(em)em. (16)

In (16), Yd(xm) = (xm xo) and d = ks. To facilitate thesubsequent analysis, the following notation is introduced [40]:

Ydd =YdKIK1I d = Ydkdk

=KI(xm xo)[ksKI

]. (17)

After using (14) and (15), the expression in (16) can be rewrit-ten as follows:

mm = Ydd + KI(xrd1 xr1)+KIxm KIxrd1 + 2mk1m (18)

where (em, ef , m, t) R is an auxiliary term defined as =1m cosh2(em) (m 1 tanh(em))

12m cosh2(em) tanh(ef )+ 2m cosh2(ef ) (3 tanh(ef ))+ 2m cosh2(ef ) (2 tanh(em)) . (19)

The motivation for the introduction of the filter signals m(t)and ef (t) and the selective grouping of the terms in (19) allow

1The term ltered tracking error is used to indicate that the filter input [e.g.,er(t)] is equal to a low-pass filtered version of the output [e.g., rr(t)].

the development of the following linear inequality (versus aquadratic inequality):

|| 1z (20)where 1 R is a positive bounding constant, and z(t) R3 isdefined as follows:

z = [ m tanh(em) tanh(ef ) ] . (21)That is, the use of hyperbolic functions in the developmentof m(t) and ef (t) allows the linear inequality in (20) to bedeveloped; without the hyperbolic functions, the bound wouldbe quadratic. Feedback terms in the controller can be usedto damp out terms that are bounded by a linear function ofthe states without restricting the domain of the stability resultas demonstrated in the subsequent stability analysis. If thehyperbolic terms had not been used in the filter structure, thebound in (20) would have been quadratic, potentially limitingthe domain of the stability result (i.e., a semiglobal result).

Based on (18) and the subsequent stability analysis, thedesired robot link position is designed as follows:

xrd1 =Yddk + xm + k2 tanh(em) k1k2 cosh2(ef ) tanh(ef )

xrd2 = . (22)In (22), R is an appropriate positive constant (i.e., isselected, so the robot will impact the massspring system in thevertical direction), k2 R is a positive constant control gain,and the control gain k1 R is defined as

k1 =1ml

(3 + kn121

) (23)where kn1 R is a positive constant nonlinear damping gain.The parameter estimate dk(t) R in (22) is generated by theadaptive update law, i.e.,

dk = proj(Ydm). (24)

In (24), R is a positive constant, and proj() denotes a suf-ficiently smooth projection algorithm [43] utilized to guaranteethat dk(t) can be bounded as follows:

dk dk dk (25)where dk, dk R denote known constant lower and upperbounds for dk(t), respectively.

After substituting (22) into (18), the closed-loop error systemfor m(t) can be obtained as follows:

mm =KI(xrd1 xr1) + KI(xm xm)+ KIk1k2 cosh2(ef ) tanh(ef ) + YdkdkKIk2 tanh(em) + 2mk1m. (26)

In (26), the parameter estimation error dk(t) R is defined asdk = dk dk.

The open-loop robot error system can be obtained by takingthe time derivative of rr(t) and premultiplying by the robot


inertia matrix as follows:

M rr = Yrr Crr F (27)

where (9), (13), and (14) were utilized, and

Yrr = Mxrd + Mer + h + Cxrd

+ Cxrd +[KI(xr1 xm)

0

] Cxr (28)

where Yr(xr, xr, xm, xm, ef , m, t) R2P denotes a knownregression matrix, and r RP denotes an unknown constantparameter vector. See Appendix A for a linearly parameter-izable expression for M(xr)xrd(t) that does not depend onacceleration terms. Based on (27) and the subsequent stabilityanalysis, the robot force control input is designed as follows:

F = Yr r + er + k3rr (29)

where k3 R is a positive constant control gain, and r(t) R

P is an estimate for r generated by the following adaptiveupdate law:

r = proj

(rY Tr rr

). (30)

In (30), r RPP is a positive-definite constant diagonaladaptation gain matrix, and proj() denotes a projection algo-rithm utilized to guarantee that the ith element of r(t) can bebounded as follows:

ri ri riwhere ri, ri R denote known constant lower and upperbounds for each element of r(t), respectively.

The closed-loop error system for rr(t) can be obtained aftersubstituting (29) into (27) as follows:

M rr = Yr r k3rr Crr er. (31)

In (31), the parameter estimation error r(t) RP is defined as

r = r r. (32)

Remark 1: Based on (29), the control torque input can beexpressed as follows:

= JT (Yr r + er + k3rr) (33)

where J(q) denotes the manipulator Jacobian introduced in (2).

IV. STABILITY ANALYSIS

Theorem: The controller given by (22), (24), (29), and (30)ensures asymptotic regulation of the MSR system in the sensethat the massspring system converges to the desired set point,and the robot links converge to the desired trajectory as follows:

|em(t)| 0 er(t) 0, as t

provided that k1, k2, and kn1 are selected sufficiently large(see Appendix B), and the following sufficient gain condition

is satisfied:

2 > max{

1

, (xm + |xr |)2}

2K

4(34)

where xm , xr , K , and are defined in (4), (11), and (14),respectively.

Proof: See Appendix B.Remark 2: The sufficient gain condition in (34) indicates

that as KI becomes infinitely large, 2 must also grow infinitelylarge. See the classic discussion on this issue given in [44]. Inthis result, we only consider contact with surfaces with finiteKI . In the experimental results for this paper, the actual valuesfor 2 were selected much lower than the sufcient condition in(34) indicates, as is typical in nonlinear control designs.

Remark 3: As is typical in the literature, the controller de-veloped in (22), (24), (29), and (30) is based on the underlyingassumption that arbitrarily large (but finite) control authority isavailable. A potential disadvantage of the controller is that thegain conditions developed in Appendix B and in (34) indicatethat kn1, k2, and 2, respectively, should be selected suffi-ciently large. As demonstrated by the subsequent experimentalresults, the gains may be selected much lower in practice (i.e.,the gain conditions are the result of a conservative Lyapunovanalysis). However, the subsequent experimental section alsoillustrates that even when the gain conditions are violated, largeinitial conditions and a high stiffness coefficient result in a high-gain controller that initially saturates the actuators. The controltorque in the experiment was artificially saturated to reducethe magnitude of the impact to protect a capacitance probefrom contact by excessive bending of the aluminum rod/springassembly. Research that can limit the required control torquefor systems that undergo an impact collision seems to be aninteresting open problem.

V. EXPERIMENTAL RESULTS

The results in this section are not intended to represent thebest possible results that can be obtained by the controller.Different control gains and initial conditions will yield differentresults. The motivation for the experimental results in thissection is to demonstrate the capability of the controller to yielda certain level of performance (see Figs. 412).

The test bed depicted in Figs. 2 and 3 was developed forexperimental demonstration of the proposed controller. Thetest bed is composed of a massspring system and a two-link robot. The body of the massspring system includes aU-shaped aluminum plate [item (8) in Fig. 2] mounted on anundercarriage with porous carbon air bearings, which enablesthe undercarriage to glide on an air cushion over a glass-coveredaluminum rail. A steel core spring [item (1) in Fig. 2] connectsthe undercarriage to an aluminum frame, and a linear variabledisplacement transducer [LVDT; item (2) in Fig. 2] is usedto measure the position of the undercarriage assembly. Theimpact surface consists of an aluminum plate connected to theundercarriage assembly through a stiff spring mechanism [item(7) in Fig. 2]. A capacitance probe [item (3) in Fig. 2] is usedto measure the deflection of the stiff spring mechanism. Thetwo-link robot [items (46) in Fig. 2] is made of two aluminum


Fig. 2. Top view of the experimental test bed, including (1) spring,(2) LVDT, (3) capacitance probe, (4) link1, (5) motor1, (6) link2, (7) stiff springmechanism, and (8) mass.

Fig. 3. Side view of the experimental test bed.

links, mounted on 240.0 N m (base link) and 20.0 N m (sec-ond link) direct-drive switched reluctance motors. The motorsare controlled through power electronics operating in the torquecontrol mode. The motor resolvers provide rotor position mea-surements with a resolution of 614 400 pulses/revolution, and astandard backward difference algorithm is used to numericallydetermine the velocity from the encoder readings. A Pentium2.8-GHz PC operating under QNX hosts the control algorithm,which was implemented via a custom graphical user interface[45] to facilitate real-time graphing, data logging, and theability to adjust control gains without recompiling the program.Data acquisition and control implementation were performed ata frequency of 2.0 kHz using the ServoToGo I/O board.

The control gains and k3, defined as scalars in (14) and(29), were implemented (with nonconsequential implications tothe stability result) as diagonal gain matrices to provide moreflexibility in the experiment. Specifically, the control gains wereselected as follows:

k1 =0.18k2 =0.9k3 =diag{185, 170}1 =452 =83 =0.01 =diag{60, 90}. (35)

The control gains in (35) were obtained by choosing gainsand then adjusting based on performance. If the responseexhibited a prolonged transient response (compared with theresponse obtained with other gains), the proportional gainswere adjusted. If the response exhibited overshoot, derivativegains were adjusted. Last, to fine-tune the performance, theadaptive gains were adjusted after the feedback gains weretuned as described to yield the best performance. As a resultof a conservative stability analysis, the final gains used may notsatisfy the sufficient gain conditions developed in the theoremproof provided in Appendix B. The subsequent results indicatethat the developed controller can be applied despite the fact thatsome gain conditions are not satisfied. In contrast to the aboveapproach, the control gains could potentially have been adjustedusing more methodical approaches. For example, the nonlinearsystem in [46] was linearized at several operating points, anda linear controller was designed for each point; moreover, thegains were chosen by interpolating or scheduling the linearcontrollers. In [47], a neural network is used to tune the gains ofa PID controller. In [48], a genetic algorithm was used to fine-tune the gains after an initial guess was made by the controllerdesigner. Additionally, in [49], the tuning of a PID controllerfor robot manipulators is discussed.

The adaptation gains were selected as follows:

=90r =diag{4.01 1012, 1.2 107, 0.2, 3.3 1012, 6 106,

0.1, 2.4 1011, 7 105, 0.1, 2.35 1011}.(36)

The adaptation gains r in (36) are used to enable the adaptiveestimate to sufficiently change relative to the large values of theuncertain parameters in r. Smaller adaptation gains could beused to obtain different results. The initial conditions for therobot coordinates and the massspring position were given by(in meters)

[xr1(0) xr2(0) xm(0) ] = [ 0.008 0.481 0.202 ] .

The initial velocities of the robot and the massspring werezero, and the desired massspring position was given by (inmeters)

xmd = 0.232.

That is, the tip of the second link of the robot was initially224 mm from the desired set point and 194 mm from x0 alongthe X1-axis (see Fig. 2). Once the initial impact occurs, therobot is required to depress the spring [item (1) in Fig. 2] tomove the mass 30 mm along the X1-axis.

The massspring and robot errors [i.e., e(t)] are shown inFigs. 4 and 5. The peak steady-state position errors of theendpoint of the second link of the robot along the X1-axis[i.e., |er1(t)|] and along the X2-axis [i.e., |er2(t)|] are0.212 mm and 5.77 m, respectively. The peak steady-state po-sition error of the mass [i.e., |em(t)|] is 2.72 m. The 0.212-mmmaximum steady-state error in |er1(t)| is due to the Yddk(t)term of xrd1(t) in (22), where Yd(xm) is approximately 0.03 m,and dk(t) has a maximum steady-state value of 0.007 (N/m)/


Fig. 4. Massspring and robot errors e(t). (a) Position error of the robot tipalong the X1-axis [i.e., er1(t)]. (b) Position error of the robot tip along theX2-axis [i.e., er2(t)]. (c) Position error of the massspring [i.e., em(t)].

Fig. 5. Massspring and robot errors e(t) during the initial 2 s.

(N/m), yielding a 0.21-mm error. All of the other terms in er1(t)are negligible at the steady state.

The input control torques in (33) are shown in Figs. 6 and7. To constrain the impact force to a level that ensured thatthe aluminum plate did not flex to the point of contact withthe capacitance probe, the computed torques are artificiallysaturated. Fig. 6 depicts the computed torques, and Fig. 7depicts the actual torques (solid line) along with the computedtorques (dashed line). The resulting desired trajectory along theX1-axis [i.e., xrd1(t)] is depicted in Fig. 8, and the desiredtrajectory along the X2-axis was chosen as xrd2 = 370 mm.Fig. 9 depicts the value of dk(t) R, and Figs. 1012 depictthe values of r(t) R10. The order of the curves in the plotsis based on the relative scale of the parameter estimates ratherthan the numerical order in r(t). A video of the experiment isprovided in [50].

Fig. 6. Computed control torques JT (q)F (t) for (a) base motor and(b) second-link motor.

Fig. 7. Applied control torques JT (q)F (t) (solid line) versus computedcontrol torques (dashed line) for (a) base motor and (b) second-link motor.

Fig. 8. Computed desired robot trajectory, xrd1(t).


Fig. 9. Unitless parameter estimate dk(t) introduced in (22).

Fig. 10. Estimate for the unknown constant parameter vector r(t).(a) r10(t) = KI , (b) r4(t) = KIms/m, (c) r1(t) = KIm1/m, and(d) r7(t) = KIm2/m, where m1, m2 R denotes the mass of the first andsecond links of the robot, respectively, ms R denotes the mass of the motorconnected to the second link of the robot, and m R denotes the mass of themassspring system.

VI. CONCLUSION

An adaptive nonlinear controller is proven to regulatethe states of a planar robot colliding with an unactuatedmassspring system. The continuous controller yields asymp-totic regulation of the springmass and robot links. New con-trol design, error system development, and stability analysistechniques were required to compensate for the fact that thedynamics changed from an uncoupled state to a coupled state.Experimental results are provided to illustrate the successfulperformance of the controller. Sufficient conditions developedin the stability analysis indicate that the control gains shouldbe selected large enough to minimize the closed-loop steady-state error; however, high gains could result in large torquesfor large initial errors. The high-gain problem is exacerbated inthe developed result because of the presence of the estimated

Fig. 11. Estimate for the unknown constant parameter vector r(t).(a) r5(t) = ksms/m. (b) r2(t) = ksm1/m. (c) r8(t) = ksm2/m.

Fig. 12. Estimate for the unknown constant parameter vector r(t).(a) r6(t) = ms. (b) r3(t) = m1. (c) r9(t) = m2.

impact stiffness coefficient. The experimental results were ob-tained by artificially saturating the torque to prevent damage tothe capacitance probe. These issues point to a need to developcontrollers that account for limited actuation.

APPENDIX AEXPRESSION FOR xrd(t)

Since xrd2 is a constant, the subsequent development is onlyfocused on determining xrd1(t). After using (13), (15), (22),and (24), the first time derivative of xrd1(t) can be determinedas follows:

xrd1=Yd (proj(Ydm))+(dk+1k2 cosh2(em)

)xm

k1k2(sinh2(ef ) + cosh2(ef )

)(3 tanh(ef ) + 2 tanh(em)k1 cosh2(ef )m) .

(37)


Based on the fact that the projection algorithm for dk(t) isdesigned to be sufficiently smooth [43], the expressions in (24)and (37) can be used to determine the second time derivative ofxrd1(t) as follows:

xrd1 =Yd (proj(Ydm))

mm

+(Yd

(proj(Ydm))xm

+ 2proj(Ydm))

xm

2k2 cosh3(em) sinh(em)x2m+(dk + 1 k2 cosh2(em)

)xm

4k1k2 (sinh(ef ) cosh(ef )) e2f k1k2

(sinh2(ef ) + cosh2(ef )

) (3 cosh2(ef ) 2k1 cosh(ef ) sinh(ef )m) ef+ k1k2

(sinh2(ef ) + cosh2(ef )

) (2 cosh2(em)em + k1 cosh2(ef )m) . (38)

After substituting (15) and (16) into (38) for ef (t) and m(t),respectively, and substituting (5) and (8) into (38) for xm(t),the expression for M(xr)xrd(t) in the linear parameteriza-tion in (28) can be determined without requiring accelerationmeasurements.

APPENDIX BTHEOREM PROOF

In the following proof, a Lyapunov function and its deriv-ative are provided. The analysis is then separated into twocasescontact and noncontact. For the noncontact case, thestability analysis indicates that the controller and error signalsare bounded and converge to an arbitrarily small region. Ad-ditional analysis indicates that within this region, contact mustoccur. When contact occurs, a Lyapunov analysis is providedthat illustrates that the MSR system asymptotically convergesto the desired set point.

Proof: Let V (rr, er, em, ef , m, r, dk, t) R denote thefollowing nonnegative radially unbounded function (i.e., aLyapunov function candidate):

V =12rTr Mrr +

12Tr

1r r +

12TdkKI

1dk + k2KI

[ln (cosh(em)) + ln (cosh(ef ))] + 12eTr er +

12m2m.

(39)

The time derivative of (39) can be determined as follows:

V = rTr M rr +12rTr

Mrr + Tr 1r

r

+ k2KI [tanh(em)em + tanh(ef )ef ]

+ TdkKI1 dk + eTr er + mmm. (40)

After using (12), (14), (15), (23), (24), (26), and (30)(32), theexpression in (40) can be rewritten as

V k3rTr rr 1k2KI tanh2(em) 322m k2KI3 tanh2(ef ) kn12122m eTr er + m [KI(xrd1 xr1) + KI(xm xm) + ] . (41)

The expression in (41) will now be examined under two differ-ent scenarios.

Case 1: Noncontact

For this case, the systems are not in contact ( = 0), and (41)can be rewritten as follows:

V k3rTr rr 1k2KI tanh2(em) k2KI3 tanh2(ef ) 322m kn12122m eTr er+ m[KIxrd1 KIxm + ].

Rewriting xrd1(t) and substituting for (em, ef , m, t) yield

V k3rTr rr 1k2KI tanh2(em) k2KI3 tanh2(ef ) 222m

[er2 K |m|er

] [kn12212m 1z|m|] [22m K |m||xm xr1|] . (42)

Completing the squares on the bracketed terms yields

V k3rTr rr 1k2KI tanh2(em) k2KI3 tanh2(ef )

222m [

(er+ K |m|2

)2]+

2K |m|24

[kn12

21

(m +

z2kn121

)2]+

z24kn1

[2

(m +

K |xm xr1|22

)2]+

2K |xm xr1|2

42.

(43)

After upper bounding V (t) by eliminating the three bracketednegative terms in (43), the following inequality is obtained:

V k3rTr rr1k2KI tanh2(em)k2KI3 tanh2(ef )

22m(2

2K

4

)2m+

z242kn1

+2K(xmxr1)2

42.

(44)

Provided that kn1 is selected according to the sufficientcondition

kn1 >1

42 min{1k2K , k2K3, 2}

the expression in (44) can be further reduced as follows:

V 1z2 k3rr2 (

2 2K

4

)

2m +2K(xm xr1)2

42(45)

where 1 R is defined as

1 = min{1k2K , k2K3, 2} 1

42kn1.


Based on (4) in Assumption 1, for the noncontact case

xr xr1 xm xm . (46)

Hence, the expression in (45) can be upper bounded as follows:

V y2 + x (47)

where R is defined as

= min

{1, k3,

(2

2K

4

)}

and y(t) R5 and x R are defined as

y = [ zT rTr ]T x =

2K (xm + |xr |)2

42(48)

where x can be made arbitrarily small by making 2 large.Based on (39) and (47), either y2 x or y2 > x.If y2 > x, then Barbalats lemma [51] can be used toconclude that V (t) 0 since V (t) is lower bounded, V (t) isnegative semidefinite, and V (t) can be shown to be uniformlycontinuous. As V (t) 0, eventually, y2 x. Providedthat the sufficient gain condition in (34) is satisfied (i.e., x m, where m R denotes a known positive constant. Based on(54) and the fact that em(t) will eventually be lower boundedby m in a noncontact condition, the inequality in (53) can besimplified as follows:

k2(tanh(m) k1 cosh2(ef )

) 1

x xmddk + (m + x0)dk. (55)

To further simplify the inequality in (55), an upper bound canbe determined for ef (t). The inequality in (49) along with(21) and (48) can be used to conclude that as the manipulatorapproaches the mass, ef (t) will eventually be upper bounded asfollows:

ef tanh1(

1

x

) f (56)

where f R is a known positive constant. Based on (49) and(56), the control parameter k2 can be selected according to thefollowing sufficient condition to ensure that the robot and themassspring system make contact:

k2 1

x xmddk + (m + x0)dktanh(m) k1 cosh2(f )

(57)

where k1 is chosen as follows:

k1

lyapunov based control

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