adaptive control of robotic manipulators with unified...

Adaptive control of robotic manipulators with unified motion constraints

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Li, Mingming, Li, Yanan, Ge, Shuzhi Sam and Lee, Tong Heng (2016) Adaptive control of robotic manipulators with unified motion constraints. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 (1). pp. 184-194. ISSN 2168-2216

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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS 1

Adaptive Control of Robotic Manipulators withUnified Motion Constraints

Mingming Li, Yanan Li,Member, IEEE Shuzhi Sam Ge,Fellow, IEEE Tong Heng Lee,Senior Member, IEEE

Abstract—In this paper, we present an adaptive control ofrobotic manipulators with parametric uncertainties and motionconstraints. Position and velocity constraints are consideredand they are unified and converted into the constraint of thenominal input. An adaptive neural network control is developedto achieve trajectory tracking, while the problems of motionconstraints are addressed by considering the saturation effect ofthe nominal input. The uniform boundedness of all closed-loopsignals is verified through Lyapunov analysis. Simulation andexperiment results on a 2 DOF robotic manipulator demonstratethe effectiveness of the proposed method.

Index Terms—robotic manipulator; adaptive control; unifiedmotion constraints; input saturation; neural network appr oxi-mation

I. I NTRODUCTION

I N robotic applications with unstructured environments,especially those related to human-robot interaction [1],

[2], [3], [4], [5], robots must be subject to certain motionconstraints, which typically include position and velocityconstraints. In the example of teleoperated surgical robots,they must be maneuvered in constrained task spaces, oftenthrough a narrow entry portal into the patient’s body [6]. Inanother example of human-robot collaboration, a robot sharesa common workspace with a human, so its velocity must beconstrained within a predefined bound to guarantee the safetyof both the robot and the human. An impact test is conductedin [7] to evaluate endangerment to human beings caused bycollision with robots moving at different velocities. Resultshave shown that the head injury criterion (HIC), which isused to quantify the injury level of human beings, significantlyincreases as the impact velocity of robots becomes larger.

In the literature, research effort has been made on controlof robotic manipulators with motion constraints. A boundedcontrol for set-point regulation problem of robotic manipula-tors with joint velocity constraints is proposed in [8]. In [9], atrajectory tracking control of robotic manipulators is achievedon a surface with joint velocity constraints. A unified quadraticprogramming formulation based dynamical system approachis proposed in [10] to solve the joint torque optimizationproblem of physically constrained redundant manipulators,but it requires complete knowledge of the system dynamics.In [11], [12], adaptive control of robotic manipulators withposition (output) constraints is developed by employing the

M. Li, S. S. Ge and T. H. Lee are with the Department of Electrical andComputer Engineering, and the Social Robotics Lab, Interactive and DigitalMedia Institute (IDMI), National University of Singapore,Singapore 117576(e-mail: li [email protected] ; [email protected]; [email protected]).

Y. Li is with Department of Bioengineering, Imperial College London,London, UK SW7 2AZ (e-mail: [email protected].)

barrier Lyapunov function (BLF), in the presence of paramet-ric uncertainties and external disturbances. In [13], optimalcontrol of robotic manipulators with joint position constraintsis achieved using adaptive dynamic programming. Researchesof task space region control are proposed in [14], [15], [16]and adaptive control of robot subject to position constraints isachieved in [17] by defining the objective functions in the formof a set of inequalities. However, only position constraintsare considered in the designed region control of [17] asthese objective functions (the inequalities) are constructed withrespect to only the robot’s task space positions.

Alternatively, since a robotic manipulator can be consideredas a nonlinear system subject to some physical properties [18],nonlinear control with consideration of state constraintsmaybe applied [19], [20], [21]. Neural network based predictivecontrol is proposed in [22], [23] with system identificationimplemented using historical data and off-line training whilepredictive control with online estimation proposed in [24],[25], [26] is only applicable to single-input-single-output(SISO) system. Recently, a robust adaptive neural trackingcontrol with integral BLF [27] is proposed in [28] with integralBLFs for a class of SISO strict-feedback nonlinear systemsunder state constraints while in [29], adaptive control subjectto full state constraints is achieved for a more general SISOpure-feedback nonlinear system, with simulation results on asingle-link robot. To tackle the uncertainties in multi-input-multi-output (MIMO) systems like robots, neural networks[30], [31] or fuzzy logic systems (FLS) [32], [33], [34] areemployed as online model-free approximators. In [35], anadaptive fault tolerant control is derived for a class of input andstate constrained MIMO nonlinear systems, where the stateconstraints are formulated as a constraint of the state vector’snorm. Therefore, it implies that the state constraints havetobe symmetric. On the other hand, in [36], [37], prescribedperformance adaptive neural network control is developed toconstrain the error signals within a prescribed region, whichis shaped by a performance function. This method may beextended to handle the problem of motion constraints, but itis not straightforward to define the corresponding performancefunction which is related to the tracking error and is requiredto be exponentially decaying.

In this paper, an adaptive neural network control is proposedfor robotic manipulators with unknown dynamics and motionconstraints of position and velocity. This method is differentfrom previous methods since it is able to directly handlevarious types of motion constraints that are independentlyandexplicitly defined. Specifically, a unified framework is estab-lished to convert motion constraints into an input saturation


problem with uncertainties incorporated in both the nominalcontrol input and saturation law, which is different from theinput saturation problem in [38]. By using the GL matrixand operator [39], an adaptive neural network approximatorisdesigned to address the issue of unknown system dynamics. ByLyapunov analysis, the boundedness of all closed-loop signalsis shown to be guaranteed while the motion constraints arenot violated. Simulations and experiments are conducted toillustrate the efficacy of the proposed method.

The remainders of this paper are organized as follows. InSection II, the model of a robotic manipulator with motionconstraints is presented, a general method is introduced toconvert motion constraints into the saturation of the controlinput, and some useful preliminaries are given. In Section III,the adaptive control design is detailed with rigorous Lyapunovstability analysis. Simulation and experiment results arepre-sented in Sections IV and V to demonstrate the effectivenessof the proposed method, followed by some concluding remarksin Section VI.

II. PROBLEM FORMULATION

A. Kinematics and Dynamics

The dynamics of ann degree-of-freedom (DOF) roboticmanipulator can be described as a MIMO nonlinear system asfollows [18]:

M(q)q + C(q, q)q +G(q) = τ (1)

whereq ∈ Rn is the joint position,M(q) ∈ R

n×n is the inertiamatrix, C(q, q)q ∈ R

n is the Coriolis and centrifugal force,G(q) ∈ R

n is the gravitational force, andτ ∈ Rn is the input

joint torque. The joint space dynamics (1) are transformed intothe task space dynamics

Mη(η)η + Cη(η, η)η +Gη(η) = u (2)

via the forward kinematics

η = Ω(q), η =∂Ω

∂qq =: J(q)q (3)

whereη = [η1, η2, · · · , ηm]T is a vector of task variables,Mη

is the inertia matrix,Cη is the Coriolis and centrifugal force,Gη is the gravitational force, andu is the control input in thetask space given as

Mη = J−TMJ−1, Gη = J−TG,

Cη = J−T(C −MJ−1J)J−1, u = J−Tτ (4)

For simplicity, only the non-redundant(m = n) non-singularmanipulators are considered in this paper. Letx1(t) =η, x2(t) = η, we have the dynamics of the robotic manipulatorin the state-space form, as below

x1 = x2,x2 = M−1

η (x1)(u − Cη(x1, x2)x2 −Gη(x1)),y = x1

(5)

Note thatM−1η (x1), Cη(x1, x2) andGη(x1)) are all unknown

matrices. The control objective is to make the position of theend-effectory(t) track the desired trajectoryyd(t). In addition,it is expected to keep all closed-loop signals bounded and

prevent the violation of motion constraints, which will beintroduced in the following.

Assumption 1: [40] For arbitraryt > 0, there existb1 > 0andb2 > 0 such that‖yd(t)‖ ≤ b1 and‖yd(t)‖ ≤ b2 .

B. Motion Constraints

A motion constraint of the end-effector is a kinemat-ic/dynamic constraint related to the variable that describesthe end-effector’s motion status. In this paper, we considertwo typical motion constraints, i.e., position and velocityconstraints.

For the robotic manipulator (5), letyi, ydi, x1i and x2i

denote thei-th element of the vectorsy, yd, x1 and x2 fori = 1, · · · , n, respectively. Motion constraints are defined as

p−i ≤ yi ≤ p+i , andv−i ≤ yi ≤ v+i (6)

for i = 1, 2, · · · , n, where p−i , v−i , p

+i and v+i are known

constant or time-varying limits.Remark 1: Note that the constraints investigated in this pa-

per are given in the form of two-sided inequalities. Comparedwith those considered in [28] and [35], which are describedas bounds of the absolute value/norm of a state variable,respectively, the constraints formulated in (6) are asymmetricand they can be assigned independently as the upper and lowerbounds of the state variable. Therefore, it can account for moregeneral task space constraints in practice and offer greaterflexibility during control design.

Assumption 2: The initial position and velocity of the end-effector satisfyp−i ≤ yi(0) ≤ p+i and v−i ≤ yi(0) ≤ v+i fori = 1, 2, · · · , n.

Based on the above assumption, the two-sided inequalitiesof the motion constraints are converted following the proce-dures below. Choose positive scalarsk1i ,k2i, and k3i, andconsider the scalaryi regulated as

−k1i(yi − p−i ) ≤ yi ≤ −k1i(yi − p+i ) (7)

Considering the following inequalities

yi ≥ −k1i(yi − p−i ) ≥ 0 if yi ≤ p−i , and

yi ≤ −k1i(yi − p+i ) ≤ 0 if yi ≥ p+i (8)

we find thatyi cannot transgress the constraints whenyimin =−k1i(yi−p−i ) andyimax = −k1i(yi−p+i ). Similarly, we canguaranteeyi ∈ [yimin, yimax] if yi satisfies

−k2i(yi − yimin) ≤ yi ≤ −k2i(yi − yimax) (9)

Thus, according to (5), we have

h−pi ≤ x2i ≤ h+

pi (10)

whereh−pi = −k2i(x2i+k1i(x1i−p−i )) andh+

pi = −k2i(x2i+

k1i(x1i − p+i )).Consider the velocity constraint of the end-effector defined

asv−i ≤ x2i ≤ v+i . Similarly to the process of converting theposition constraint, we have

−k3i(x2i − v−i ) ≤ x2i ≤ −k3i(x2i − v+i ) (11)


Thus, x2i is constrained as

h−vi ≤ x2i ≤ h+

vi (12)

whereh−vi = −k3i(x2i−v−i ) andh+

vi = −k3i(x2i−v+i ). Thus,x2i will not violate the velocity constraintv−i ≤ x2i ≤ v+i ifyi(0) = x2i(0) ∈ [v−i , v

+i ].

As both (10) and (12) are with respect tox2, they can becombined as a unified constraint as below

h−i ≤ x2i ≤ h+

i (13)

whereh−i = maxh−

pi, h−vi andh+

i = minh+pi, h

+vi.

Remark 2: In this way, the position and velocity constraintshave been converted into a unified one as (13). In fact, it isclear that one can even incorporate the robot’s accelerationconstraintsa−i ≤ x2i ≤ a+i into (13) if necessary, whichrequires no change in control design as all motion constraintsare compactly represented as (13) and our control is derivedsolely based on it. In addition, such a conversion processcan also be applied to constraints in the joint space whenthe robotic manipulator is subject to different types of jointlimits and a joint space controller is to be designed. Com-paratively, one may extend region control with the unifiedobjective bound in [17] to handle velocity or even accelerationconstraints by adding the corresponding inequalities of the taskspace velocities and accelerations into the objective functions.However, considering these additional constraints with themethod in [17] will require the signalsq and

...q in the resulted

controller as it is derived by taking the time-derivatives of theobjective functions. Considering the fact that the usage ofqor even

...q in control design is generally not desired due to

the difficulties in obtaining their accurate measurements [18],the practicability of this region control method for motionconstraints investigated in this paper may be limited.

C. Preliminaries: Neural Network

It has been shown that the radial basis function neuralnetwork (RBFNN) can approximate an arbitrary continuousfunctiona(Z) over a compact setΩZ ⊂ R

nZ to any accuracy[41] as below

a(Z) = w∗Tφ(Z) + eZ , ∀Z ∈ ΩZ (14)

where Z ∈ RnZ is the input vector,w∗ ∈ R

nw are theoptimal constant weights withnw being the number of n-odes, eZ ∈ R is the functional approximation error, andφ(Z) = [φ(1)(Z), φ(2)(Z), · · · , φ(nw)(Z)] ∈ R

nw are vectorsof Gaussian functions as below

φ(i)(Z) = exp(−(Z − µ(i))T(Z − µ(i))

σ(i)2i

) (15)

with µ(i) being the center of the Gaussian function andσ(i)

being the variance.According to [42], there exist optimal weightsw∗ such

that |eZ | ≤ e∗Z with e∗Z ≥ 0, which can be made arbitrarysmall provided thatnw is sufficiently large. Thus, the optimalweights w∗ are defined such thateZ is minimized for allZ ⊂ ΩZ , i.e.,

w∗ := arg minw∈Rnw

supZ⊂ΩZ

|a(Z)− wTφ(Z)|. (16)

Then, an approximation ofa(Z) can be constructed as

a(Z) = wTφ(Z) (17)

where a(Z) is the approximation ofa(Z) and w ∈ Rnw are

the estimates of the corresponding optimal weightsw∗ definedin (16).

By employing RBFNNs to approximate each of its element,a matrix functionA(Z) ∈ R

n1×n2 is approximated asA(Z),.Following the convention of GL matrices and GL operator forvectors and matrices [39], the expression ofA(Z) is given as

A(Z) =

a11(Z) a12(Z) · · · a1n2(Z)

a21(Z) a22(Z) · · · a2n2(Z)

...... · · ·

...an11(Z) an12(Z) · · · an1n2

(Z)

= WT • Φ

=

wT11φ11 wT

12φ12 · · · wT1nφ1n2

wT21φ21 wT

22φ22 · · · wT2nφ2n2

...... · · ·

...wT

n11φn11 wTn12φn12 · · · wT

n1n2φn1n2

whereW, Φ are the GL matrices and• is the GL operator.Remark 3: In this paper, RBF neural networks are employed

due to its capabilities of approximating any unstructuredsmooth nonlinear functions to arbitrary accuracy over a com-pact set. In fact, one can use other online approximationmethod instead, such as fuzzy logic systems [43], [44], [45],[32]. The online approximator can be further reduced toa regressor function by assuming that the uncertainties arestructured and are linear in parameters [46], which requires aset of model-specific basis functions.

III. C ONTROL DESIGN

To incorporate the constraint (13) into the actual controlinput u, the state equation in (5) is rewritten as

x2 = U + P (x1, x2) (18)

whereU = M−1η (x1)u ∈ R

n is the nominal control input andP (x1, x2) = −M−1

η (x1)(Cη(x1, x2)x2 +Gη(x1)) ∈ Rn. Let

Ui andPi(x1, x2) denote thei-th element of the vectorU andP (x1, x2), respectively. In order to guarantee (13),Ui has tobe saturated as

h−i − Pi(x1, x2) ≤ Ui ≤ h+

i − Pi(x1, x2) (19)

Note that bothU andP (x1, x2) incorporate unknown com-ponents, i.e.,M−1

η (x1), Cη(x1, x2), and Gη(x1). Thus, weconstruct RBFNNs to expressM−1

η andP as below

M−1η = W ∗

MT • ΦM (x1)+ EM (20)

P = W ∗P

T • ΦP (x1, x2)+ EP (21)

whereW ∗M, W ∗

P , ΦM (x1), and ΦP (x1, x2) are GLmatrices formed by optimal neural network weight vectorsW ∗

Mij ∈ RnWM andW ∗

Pi ∈ RnWP , and basis function vectors

φMij ∈ RnWM andφPi ∈ R

nWP , respectively.EM ∈ Rn×n

andEP ∈ Rn are formed by approximation errorseMij and

ePi, respectively. LetEM andEP denote the matrices formedby eMij and ePi, which are the corresponding upper bounds


of eMij and ePi used to deal with the approximation errorslater. Then, unknown matricesM−1

η andP are estimated as

M−1η = WMT • ΦM (x1) (22)

P = WP T • ΦP (x1, x2) (23)

The nominal control input in (18) is estimated as

U = (M−1η − δ)u = (WMT • ΦM (x1) − δ)u (24)

whereδ will be used later to deal with the presence of func-tional approximation errorsEM . Similarly to (19), with theestimated terms (23) and (24), we can obtain the constraintsof the estimated nominal control input as follows:

h−i − Pi(x1, x2) ≤ Ui ≤ h+

i − Pi(x1, x2) (25)

Hereinafter, we will show that the estimated weights in (23)and (24) will exponentially converge at a rate that can becontrolled by properly choosing the design parameters. Inaddition, the functional approximation errors can be madearbitrarily small by using sufficiently large numbers of neurons[41]. Therefore, (25) is valid to prevent the violation of motionconstraints.

Let U0 ∈ Rn and U0i denote the unsaturated nominal

control and its i-th element, respectively. From (25), therelationship betweenUi andU0i is given as

Ui =

h−i − Pi(x1, x2), if U0i < h−

i − Pi(x1, x2)

h+i − Pi(x1, x2), if U0i > h+

i − Pi(x1, x2)U0i, otherwise

(26)

To achieve the control objective, one needs to properly designU0, which will be introduced later. Then, according to (24)and (26) and taking the inverse ofWMT • ΦM (x1) − δ,we have

u′ = (WMT • ΦM (x1) − δ)−1U (27)

Note that, in (27), the estimated termWMT •ΦM (x1)−δmay be ill-conditioned or even singular. Thus, we performsingular value decomposition [47] with this term and have

WMT • ΦM (x1) − δ = UsΣsVTs (28)

whereUs and Vs ∈ Rn×n are two orthogonal matrices, and

Σs ∈ Rn×n is a diagonal matrix formed by the singular values

σsi of the matrixWMT • ΦM (x1) − δ. Then, instead offorming the pseudoinverse ofWMT • ΦM (x1) − δ bydirectly replacing every nonzero diagonal entries with theirreciprocals, we defineΣ+

s = diag(σ+s1, · · · , σ

+sn) where

σ+si =

0, if 0 ≤ σsi < σσ−1si , otherwise

(29)

for i = 1, · · · , n, and σ is a small positive design parame-ter. According to the above definition, the actual control isobtained as below

u = VsΣ+s U

Ts U (30)

Now, we proceed to developU0 in (26) following the back-stepping techniques. Firstly, we define the error variablesz1 = y− yd = x1− yd andz2 = x2−α1, whereα1 ∈ R

n is a

virtual control variable. Considering system dynamics (5), thetime derivative ofz1 is

z1 = z2 + α1 − yd (31)

The virtual control variable is designed as

α1 = yd − L1z1 (32)

whereL1 = LT1 > 0. Substituting (32) into (31), we have

z1 = z2 − L1z1 (33)

According to (5), the time derivative ofz2 is

z2 = x2 − α1 = U + P (x1, x2)− α1 (34)

whereα1 = yd−L1z1. To analyze the saturation effect of theestimated nominal control (26), the following auxiliary designsystem is given

ξ =

−L21ξ−|zT

2L2∆U|+0.5∆UT∆U

‖ξ‖2 ξ+∆U , ‖ξ‖ ≥ χ

0, ‖ξ‖ < χ(35)

where∆U = U − U0, L2 = diag(l21, · · · , l2n) = LT2 > 0,

L21 = LT21 > 0, χ is a small positive design parameter, and

ξ ∈ Rn is the state of the auxiliary design system. LetL20 =

LT20 > 0, the unsaturated nominal controlU0 is given as

U0 = −L−12 z1 − L−1

2 L20(z2 − ξ)

−WP T • ΦP (x1, x2)+ α1

(36)

The termδ and the update laws for vectorsWPi, WMij inGL matricesWP , WM are designed as:

δij = −sgn(z2il2iuj)sij (37)˙WPi = Λi(φPi(x1, x2)z2il2i − βiWPi) (38)˙WMij = Γij(φMij(x1)ujz2il2i − γijWMij) (39)

where sgn(·) is the sign function,sij is a constant gain thatsatisfiessij ≥ eMij , Λi ∈ R

nWP×nWP ,Λi = ΛT

i > 0, Γij ∈R

nWM×nWM ,Γij = ΓT

ij > 0, andβi, γij > 0.Remark 4: [48] If the use of the discontinuous sign function

sgn(z2il2iuj) in (37) is undesired, an alternativeδ can bechosen asδij = −s0ijf(z2il2iuj), wheres0ij is a positiveconstant andf(z2il2iuj) is a continuous odd function ofz2il2iuj. In such a case, it is clear thatzT

2L2(EMu + δu) ≤∑n

i=1

∑n

j=1 |z2il2iuj |(eMij − s0ijf(z2il2iuj)) ≤ 0 for allz2il2iuj outside the region

Ω0 = z2il2uj|sgn(z2il2iuj)f(z2il2iuj) <eMij

s0ij

which can be made arbitrarily small by increasing the valueof the design constants0ij .

Theorem 1: Consider the robotic manipulator (5) satisfyingAssumptions 1-2, with the actual control input (30) and neuralnetwork weight update laws (38) and (39). For bounded initialconditions, the closed-loop signalsz1, z2, ξ, WM andWP are uniformly bounded. In addition, the tracking errorz1remains within the compact setΩz1 and it will eventually andexponentially converge to the steady state compact setΩs tobe defined in the following proof. Finally, motion constraints(6) are not violated.


Proof 1: Consider a Lyapunov function candidate

V ∗ =1

2zT1z1 +

1

2ξTξ +

1

2zT2L2z2 (40)

In the following derivations, let us first consider the case when‖ξ‖ ≥ χ in (35) while the other condition‖ξ‖ < χ will bediscussed later. Invoking (33) to (35), we have

V = −zT1L1z1 + zT

1z2 − ξTL21ξ −12∆UT∆U + ξT∆U

−|zT2L2∆U |+ zT

2L2(U + P (x1, x2)− α1)(41)

As − 12∆UT∆U + ξT∆U ≤ 1

2ξTξ, we have

V ≤ −zT1L1z1 + zT

1z2 − ξT(L21 −12I)ξ

−|zT2L2∆U |+ zT

2L2(U + P (x1, x2)− α1)(42)

whereI denotes the identity matrix with a proper dimension.As U = U + U − U = U0 +∆U + U − U , we obtain

V ∗ ≤ −zT1L1z1 + zT

1z2 − ξT(L21 −12I)ξ

+zT2L2(U0 + U − U + P (x1, x2)− α1)

(43)

Substituting (36) into (43), we have

V ∗ ≤ −zT1L1z1 − ξT(L21 −

12I)ξ − zT

2L20(z2 − ξ)

+zT2L2EP − zT

2L2WP T • ΦP (x1, x2)

+zT2L2(U − U)

(44)

whereWP = WP − W ∗P . Consider

U − U = −WMT • ΦM (x1)u+ (EMu+ δu) (45)

where WM = WM − W ∗M. Substituting (37) into

the term zT2L2(EMu + δu), we havezT

2L2(EMu + δu) ≤∑n

i=1

∑nj=1 |z2il2iuj|(eMij − sgn(z2il2iuj)sij) ≤ 0.

Through the above mathematical manipulations, we have

V ∗ ≤ −zT1L1z1 − ξT(L21 −

12I)ξ − zT

2L20(z2 − ξ)

+zT2L2EP − zT

2L2WP T • ΦP (x1, x2)

−zT2L2WMT • Φ(x1)u

(46)

To investigate the convergence of the error signalsWP andWM, the augmented Lyapunov function candidate is givenas

V = V ∗ +1

2

n∑

i=1

W TPiΛ

−1i WPi

+1

2

n∑

i=1

n∑

j=1

W TMijΓ

−1ij WMij (47)

It is easy to find that

zT2L20ξ ≤

1

2σzT

2z2 +1

2σ−1ξTLT

20L20ξ (48)

zT2L2EP ≤

1

2zT2z2 +

1

2‖L2EP ‖

2 (49)

whereσ > 0. Considering (48), (49) and the update laws (38),(39), we have

V ≤ −zT1L1z1 − ξT(L21 − 0.5I − 0.5σ−1LT

20L20)ξ

−zT2(L20 − (0.5σ + 0.5)I)z2 −

∑n

i=1 βiWTPiWPi

−∑i=n

i=1

∑j=nj=1 γijW

TMijWMij +

12‖L2EP ‖

2

(50)

Based on the facts

W TPiWPi ≥

1

2‖WPi‖

2 −1

2‖W ∗

Pi‖2 (51)

W TMijWMij ≥

1

2‖WMij‖

2 −1

2‖W ∗

Mij‖2 (52)

we obtain

V ≤ −zT1L1z1 − ξT(L21 − 0.5I − 0.5σ−1LT

20L20)ξ

−zT2 (L20 − (0.5σ + 0.5)I)z2 +

1

2‖L2EP ‖

2

−

n∑

i=1

βi

2‖WPi‖

2 −

i=n∑

i=1

j=n∑

j=1

γij2‖WMij‖

2

+

n∑

i=1

βi

2‖W ∗

Pi‖2 +

i=n∑

i=1

j=n∑

j=1

γij2‖W ∗

Mij‖2

≤ −ρV + ζ (53)

ρ = min(

2λmin(L1), 2λmin(L21), 2λmin(L20L−12 ),

min(βi

λmax(Λ−1i )

),min(γij

λmax(Γ−1ij )

))

(54)

ζ =n∑

i=1

βi

2‖W ∗

Pi‖2+

n∑

i=1

n∑

j=1

γij2‖W ∗

Mij‖2+

1

2‖L2EP ‖

2

(55)

with L20 = L20 − (0.5σ + 0.5)I and L21 = L21 − 0.5I −0.5σ−1LT

20L20. To ensureρ > 0, the design parametersβi >0, γij > 0, σ > 0, L2 = LT

2 > 0, L20 = LT20 > 0 andL21 =

LT21 > 0 are to satisfy the following conditions

λmin(L1) > 0, λmin(L21) > 0, λmin(L20) > 0 (56)

According to Lemma 1.2 i) in [49], (53) indicates that all theclosed-loop signalsz1, z2, ξ, WM andWP are uniform-ly bounded. Particularly, the tracking errorz1 is uniformlybounded by a compact setΩz1 derived as follows:

Multiplying (53) by eρt yields

ddt(V (t)eρt) ≤ ζeρt. (57)

By integrating (57) over[0, t], we obtain

0 ≤ V (t) ≤(

V (0)−ζ

ρ

)

e−ρt +ζ

ρ(58)

From (58), it is easy to find that

0 ≤ V (t) ≤(

V (0)−ζ

ρ

)

e−ρt +ζ

ρ≤ V (0) +

ζ

ρ. (59)

According to (47), we have12zT1z1 ≤ V (t). Thus,

‖z1(t)‖2 ≤ 2(V (0) +

ζ

ρ). (60)

Hence, the tracking errorz1 is uniformly bounded by thecompact set

Ωz1 = z1(t)|‖z1(t)‖ ≤

√

2(V (0) +ζ

ρ) (61)

In addition, according to (59) and Lemma 1.2 ii) in [49],we can see thatz1 is uniformly ultimately bounded and it


exponentially converges to the following steady state compactset

Ωs = x(t)| limt→∞

‖z1(t)‖ =

√

2ζ

ρ (62)

From (54) and (55), the size ofΩs can be reduced to besufficiently small by designingρ such that it is large enough.In addition, from (58), the convergence speed ofz1 can alsobe boosted with a largeρ. In other words, the robot’s end-effector can be kept inside the constrained region at steadystate by simply designingρ that is large enough.

Remark 5: The stateξ in the auxiliary design system (35)indicates whether there exists saturation of the nominal input.In particular,‖ξ‖ ≥ χ means that there exists saturation ofU0, which has been considered in the above stability analysis.If there is no saturation, we have‖ξ‖ < χ and∆U = 0, i.e.,U = U0, and thus the nominal input is bounded. Therefore,following (40) to (55), it is easy to show thatTheorem 1 isstill valid [38].

Remark 6: To guarantee that the approximation property ofRBFNNs (20) and (21) holds, their inputs have to be in thecorresponding compact sets, i.e.,x1 ∈ Ωx1

and [xT1 , x

T2 ]

T ∈Ωx1,x2

, respectively. FromTheorem 1 andProof 1, it is clearthat the ultimate bound ofz1 can be made arbitrarily smallby properly choosing the parameters such asL1 and L20.Therefore,x1 and [xT

1 , xT2 ]

T can be guaranteed to be in thecorresponding compact sets.

Remark 7: In Theorem 1,the designed unsaturated nominalcontrolU0 and the corresponding weight update laws (38) and(39) lead to the Uniformly Ultimate Boundedness (UUB) ofthe tracking errorz1. It is shown inProof 1 that this ultimatebounds can be made arbitrarily small by increasingρ. It isworth mentioning that the asymptotic convergence result ofz1 can also be obtained within our control design framework.To do this, we can simply add a term−sgn(z2)z2L2Ep to U0

in (36) to cancel the effects of the NN approximation errorEP

in Lyapunov analysis (46) and chooseγij = βi = 0. In thisway, it is clear thatV ≤ 0 andV < 0 for all z1 6= 0. However,this result comes at a cost of possible chattering in the systemdue to usage of the discontinuous sign function.

Remark 8: The proposed method is motivated by controllinga robotic manipulator with motion constraints, but it is alsoapplicable to other systems which are affine in control, e.g.,ocean vessels [50].

IV. SIMULATION

In this section, we simulate a 2-DOFs robotic manipulatormoving in a horizontal plane under different types of motionconstraints. The following general simulation scenarios canreflect a variety of new robotic applications such as robotictool use [51] and human-robot physical collaboration [52],es-pecially in unstructured environments where asymmetric, time-varying and task-dependent motions constraints are usuallyrequired to guarantee the safety of operations. The dynamicmodel of the robotic manipulator is given by Eq. (63) andvalues of the parameters are given in Table I.

The initial positions of the end-effector are given asq(0) =[− 1

2π,12π]

T. The desired trajectory is a circular path in the

TABLE IPARAMETERS OF THE ROBOTIC MANIPULATOR

Parameter Description Valuem1 Mass of link 1 1.1510kgm2 Mass of link 2 0.5755kgl1 = 2lc1 Length of link 1 0.31ml2 = 2lc2 Length of link 2 0.34mI1 Moment of inertia of link 1 0.3kgm2

I2 Moment of inertia of link 2 0.3kgm2

task space, which is described byyd1(t) = 0.5 cos(0.5t) andyd2(t) = 0.5 sin(0.5t). To quantify and evaluate the trackingperformance, the tracking error is defined asei = |yi − ydi|for i = 1, 2. Motion constraints of the robotic manipulator aredefined as−0.6 ≤ y1 ≤ 0.6,−0.6 ≤ y2 ≤ 0.6,−0.4 ≤ y1 ≤0.4 and−0.4 ≤ y2 ≤ 0.4.

The design parameters are chosen ask1 = [1000, 1000]T,k2 = [100, 100]T, k3 = [104, 104]T, L1 = diag[5, 7], L20 =diag[500, 500], L21 = diag[1000, 1000], L2 = diag[1, 2],σ = 200, χ = 0.001, si,j = 0.001 for i, j = 1, 2,ξ(0) = [0.001, 0.001]T and σ = 10−10. For the RBFNNs (22)and (23), the number of nodesnwM

= nwP= 27 = 128.

Γi,j = Λi = 0.01 × I128×128 and βi = γi,j = 0.1 fori, j = 1, 2. The centers of the radial basis functions areevenly distributed in[−1, 1] and their variance is set to be1. The initial weights arewM11 = 0.3× [1, 1, · · · , 1]T ∈ R

128

and wM12 = wM21 = wM22 = wP1 = wP2 = 0.1 ×[1, 1, · · · , 1]T ∈ R

128.A guideline for choosing the above parameters are given as

follows: as the parametersk1, k2, k3 are used when convertinga motion constraint into the constraint of its derivative in(9), (11) and (12), they should be sufficiently large so thatthe converted motion constraints are hard enough and theresulted constrained region will not be over-conservative. Thediagonal matricesL1, L20, L2 are gains for the signalsz1and z2 used in the standard back-stepping control designprocedure. Generally,L1 andL20 should be tuned accordingto the position and velocity errors, respectively andL2 is aglobal scaling factor for these two gains.L21 is a gain for theauxiliary signal designed to handle the saturation effectsofthe nominal input signals, which generally gives satisfactoryresults with the default values. As mentioned inProof 1,all these gains should be properly chosen such that (56) issatisfied. On the other hand, parametersσ, s, χ, σ are used forstability analysis and there is generally no need for tuningthem. As for the learning ratesΛ,Γ, λ, γ, they should bechosen to be large enough to achieve fast and stable onlinelearning. More instructions on designing adaptive RBFNNscan be found in [39] and other related papers.

To investigate the efficacy of the proposed method, com-parative simulation studies are conducted on its transientresponses with and without motion constraints and the resultsare shown in Figs. 1 and 2. Note that the adaptive controlwithout motion constraints in Fig. 2 is also implemented withthe proposed method, where position and velocity constraintsare defined with infinite values.

Figure 1 shows that, under the proposed control, the end-effector of the robot successfully tracks the desired trajectory


M(q) =

[

m1l2c1 +m2(l

21 + l2c2 + 2l1lc2 cos q2) + I1 + I2 m2(l

2c2 + l1lc2 cos q2) + I2

m2(l2c2 + l1lc2 cos q2) + I2 m2l

2c2 + I2

]

,

C(q, q) =

[

−m2l1lc2q2 sin q2 −m2l1lc2(q1 + q2) sin q2m2l1lc2q1 sin q2 0

]

, G(q) = 0 (63)

while not violating the motion constraints. During the transientresponse, the position and velocity of the end-effector areper-fectly limited within the constrained region marked by the pinkdash-dotted lines. More specifically, in Figs. 1(c) and 1(d), wecan observe that the end-effector’s velocity stops increasingwhen it reaches the boundary of the constrained region. Thisdemonstrates that the proposed control can effectively preventthe violation of motion constraints.

0 0.5 1 1.5 2 2.5 3time (s)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

posi

tion

(m)

y1yd1

(a) y1 andyd1

0 0.5 1 1.5 2 2.5 3time (s)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

posi

tion

(m)

y2

yd2

(b) y2 andyd2

0 0.5 1 1.5 2 2.5 3time (s)

-0.5

0

0.5

velo

city

(m

/s) y1

yd1

(c) y1 and yd1

0 0.5 1 1.5 2 2.5 3time (s)

-0.5

0

0.5

velo

city

(m

/s)

y2

yd2

(d) y2 and yd2

Fig. 1. Robot’s transient response with motion constraints

A more illustrative demonstration of the efficacy of theproposed method can be obtained by comparing above resultswith those in Fig. 2, where no motion constraint is imposedand the simulated situation is reduced to a standard adaptivetracking control problem. As can be seen from Fig. 2, theend-effector also successfully tracks the desired trajectoryquickly. However, since no motion constraint is considered,

the velocities of the end-effector rise to a much higher levelcompared to those in Figs. 1(c) and 1(d) as the system drivenby the designed control tries to track the desired trajectory assoon as possible.

From this comparison, we show that motion constraintscan be incorporated into our control design without causingany degradation to the tracking performance. In addition, forapplications where the overshoot of position and/or velocityin the transient response is undesired, the proposed controlprovides an explicit and direct way to address this problem byadding some reasonable motion constraints into consideration.

0 0.5 1 1.5 2 2.5 3time (s)

0

0.1

0.2

0.3

0.4

0.5

0.6

posi

tion

(m)

y1yd1

(a) y1 andyd1

0 0.5 1 1.5 2 2.5 3time (s)

0

0.1

0.2

0.3

0.4

0.5

0.6

posi

tion

(m)

y2yd2

(b) y2 andyd2

0 0.5 1 1.5 2 2.5 3time (s)

-0.5

0

0.5

1

1.5

2

velo

city

(m

/s) y1

yd1

(c) y1 and yd1

0 0.5 1 1.5 2 2.5 3time (s)

-0.2

-0.1

0

0.1

0.2

0.3

0.4

velo

city

(m

/s)

y2yd2

(d) y2 and yd2

Fig. 2. Robot’s transient response without motion constraints

To further investigate the efficacy of the proposed control,we consider another simulation where the motion constraintsare asymmetric and time-varying. Specifically, we require−0.6 − 0.2 cos(t) ≤ y1(t) ≤ 0.6 − 0.2 sin(t),−0.6 −


0.2 cos(t) ≤ y2(t) ≤ 0.6 − 0.2 sin(t),−0.5 − 0.1 cos(t) ≤y1(t) ≤ 0.5 − 0.1 sin(t) and −0.5 − 0.1 cos(t) ≤ y2(t) ≤0.5− 0.1 sin(t). The simulation results are shown in Fig. 3

0 0.5 1 1.5 2 2.5 3time (s)

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

posi

tion

(m)

y1yd1

(a) y1 andyd1

0 0.5 1 1.5 2 2.5 3time (s)

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

posi

tion

(m) y2

yd2

(b) y2 andyd2

0 0.5 1 1.5 2 2.5 3time (s)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

y1yd1

(c) y1 and yd1

0 0.5 1 1.5 2 2.5 3time (s)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

velo

city

(m

/s)

y2yd2

(d) y2 and yd2

0 0.5 1 1.5 2 2.5 3time (s)

-100

-50

0

50

100

cont

rol i

nput

(N

)

u1u2

(e) Control inputu

Fig. 3. Robot’s transient response with time-varying asymmetric motionconstraints

From Fig. 3, we can observe that excellent tracking perfor-mance can still be achieved with the same configuration ofcontrol parameters when the motion constraints have variedsignificantly. Both of the time-varying asymmetric positionand velocity constraints (marked by the pink dash-dottedcurves) are not violated even when the constrained regionsare shrinking at the beginning of the transient response. Theseresults show the efficacy of the proposed method in handlingtime-varying motion constraints, which corresponds to thecases when motion constraints are a function of time, therobot’s state or its operation environment.

It is worth mentioning that larger design parametersk1,

Motor 2

Link 2

Motor 1

Link 1 Controller 2

Controller 1

Fig. 4. The 2-DOF planar robotic manipulator

k2 and k3 result in harder motion constraints, i.e., the posi-tion/velocity will get nearer to or even stay at the boundariesof the constrained region when it is to escape this region.However, a larger control input is required to keep it atthe boundary, as shown in Fig. 3(e), and mild chattering inthe control input may appear. Therefore, a trade-off betweentracking performance at the boundaries of the constrainedregion and control effort needs to be considered.

V. EXPERIMENT

In this section, the proposed method is further examinedthrough an experiment, which is conducted on a 2-DOF planarrobotic manipulator as shown in Fig. 4. The robot is composedof 2 MAXON motors with two MAXON EPOS270/10 dualloop controllers. The links of the robot are of lengthsl1 =0.14m andl2 = 0.15m. A desktop PC is used to process thecollected data and implement the proposed method.

The initial joint position of the robot isq(0) = [0, π2 ]T

and the desired trajectory of the end-effector isyd(t) =[0.24 cos(0.05πt + π

6 ), 0.24 sin(0.05πt+π6 )]

T for t ∈ [0, 4]s.The motion constraints are defined as−0.245 ≤ y1 ≤0.245,−0.245 ≤ y2 ≤ 0.245,−0.1 ≤ y1 ≤ 0.1 and−0.05 ≤ y2 ≤ 0.05.

The design parameters are chosen ask1 = [1, 30]T,k2 = [0.05, 1]T, k3 = [1, 1]T, L1 = diag(30, 30), L20 =diag(130, 12), L21 = diag(500, 500), L2 = diag(2.5, 0.4),σ = 20, χ = 0.001, si,j = 0.001 for i, j = 1, 2,ξ(0) = [0.001, 0.001]T and σ = 10−10. For the RBFNNs(22) and (23), the number of nodesnWM

= nWP= 27 =

128. The centers of the radial basis functions are evenlydistributed in[−1, 1] and their variance is set to be 1.Λi =0.1I128×128, βi = 0.1, Γ1,1 = Γ1,2 = Γ2,1 = 0.05I128×128,Γ2,2 = 0.005I128×128, and γi,j = 0.01 for i, j = 1, 2. Theinitial weights areWM11 = 0.6 × [1, 1, · · · , 1]T ∈ R

128,WM12 = WM21 = WM22 = −0.1 × [1, 1, · · · , 1]T ∈ R

128,andWP1 = WP2 = 0.1× [1, 1, · · · , 1]T ∈ R

128.The transient response of the robot is shown in Fig. 5.

As shown in Figs. 5(a) and 5(b), the end-effector is able totrack the desired trajectory in about 3.5 seconds and positionconstraints (marked by pink dash-dotted lines) are constantlysatisfied. From Figs. 5(c) and 5(d), we can see that the velocity


0 0.5 1 1.5 2 2.5 3 3.5 4time (s)

-0.2

-0.1

0

0.1

0.2po

sitio

n (m

)

y1yd1

(a) y1 andyd1

0 0.5 1 1.5 2 2.5 3 3.5 4time (s)

-0.2

-0.1

0

0.1

0.2

posi

tion

(m)

y2

yd2

(b) y2 andyd2

0 0.5 1 1.5 2 2.5 3 3.5 4time (s)

-0.2

-0.1

0

0.1

0.2

0.3

velo

city

(m

/s)

y1 with velocity constrainty1 without velocity constraint

(c) y1

0 0.5 1 1.5 2 2.5 3 3.5 4time (s)

-0.05

0

0.05

0.1

0.15

velo

city

(m

/s)

y2 with velocity constrainty2 without velocity constraint

(d) y2

Fig. 5. Robot’s transient response with motion constraintsin experiments

of the end-effector is successfully limited within the prescribedconstrained region. For the comparison purpose, we consideranother experiment without velocity constraints. As a result,a larger velocity appears in the beginning of the transientresponse, as also shown in Figs. 5(c) and 5(d), due to the factthat the control tries to drive the end-effector to the desiredposition as fast as possible. These results are in accordance tothe simulation results in the previous section, and illustrate theefficacy of the proposed method. Different from the simulationresults, the velocities in the experiment are not so smooth,which is because they are obtained by differentiating thepositions collected from the encoders. A filter may be designedto obtain a cleaner velocity signal if it is required in someapplications.

VI. CONCLUSION

In this paper, we have investigated the control of a roboticmanipulator with parametric uncertainties and two types ofmotion constraints. A framework has been proposed to convertmotion constraints of different types into a unified constraintof the nominal input, based on which an adaptive neuralnetwork control has been developed. The efficacy of theproposed method has been demonstrated through simulationand experiment studies.

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Mingming Li Mingming Li received the B.Eng.degree in communication engineering from Sun Yat-sen University, Guangzhou, China, in 2013. He iscurrently pursuing his Ph.D. degree with the De-partment of Electrical and Computer Engineering,National University of Singapore, Singapore. Hiscurrent research interests include adaptive control,associative memories and artificial intelligence.

Yanan Li (M’14) received the B.Eng degree incontrol science and engineering and the M.Eng de-gree in control and mechatronics engineering, fromthe Harbin Institute of Technology, China, in 2006and 2009, respectively, and the Ph.D. degree fromthe NUS Graduate School for Integrative Sciencesand Engineering, National University of Singapore,Singapore, in 2013. He has been a Research Scientistat the Institute for Infocomm Research, Agencyfor Science, Technology and Research (A*STAR),Singapore from 2013 to 2015. Currently, he is a

Research Associate at the Department of Bioengineering, Imperial CollegeLondon, UK. His general research interests include physical human-robotinteraction and human-robot collaboration.

Shuzhi Sam Ge(S’90-M’92-SM’99-F’06) receivedthe B.Sc. degree from Beijing University of Aero-nautics and Astronautics, Beijing, China, in 1986,and the Ph.D. degree from the Imperial College ofScience, Technology and Medicine, University ofLondon, London, U.K., in 1993.

He is the Founding Director of the Social RoboticsLaboratory, Interactive Digital Media Institute, Na-tional University of Singapore and on leave withUniversity of Electronic Science and Technology ofChina, Chengdu 610054, China. He has authored or

coauthored six books and more than 400 international journal and conferencepapers. His current research interests include social robotics, multimediafusion, medical robots, and intelligent systems.

Dr. Ge is the Editor-in-Chief of the International Journal of Social Robotics.He has served/been serving as an Associate Editor for a number of flagshipjournals. He also serves as an Editor of the Taylor& Francis Automation andControl Engineering Series. He also served as the Vice President of TechnicalActivities, 2009-2010, and the Vice President for Membership Activities,2011-2012, IEEE Control Systems Society.


Tong Heng Lee (M’90) received the B.A. degree(with first class honors) in engineering tripos fromCambridge University, Cambridge, U.K., in 1980and the Ph.D. degree in electrical engineering fromYale University, New Haven, CT, in 1987.

He is a Professor with the Department of Electri-cal and Computer Engineering, National Universityof Singapore, Singapore, and is currently the Headof the Drives, Power, and Control Systems Groupin this department. He is the Deputy Editor-in-Chiefof the International Federation of Automatic Control

Mechatronics International Journal and serves as the Associate Editor of manyother flagship journals. He has also coauthored three research monographsand holds four patents (two of which are in the technology area of adaptivesystems, and the other two are in the area of intelligent mechatronics). Hisresearch interests are in the areas of adaptive systems, knowledge-basedcontrol, intelligent mechatronics, and computational intelligence.

Dr. Lee received the Cambridge University Charles Baker Prize in Engi-neering.

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