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Abstract— Thanks to its simplicity and robustness, time delay

control (TDC) has been recognized as a simple and yet effective alternative to robot model-based controls and/or intelligent controls. An inclusive and enhanced formulation of TDC for robust control of robot manipulators is presented in this paper. The proposed formulation consists of three intuitive terms: i) time delay estimation (TDE), inherited from the original TDC, for cancellation of mostly continuous nonlinearities; ii) nonlinear desired error dynamics (i.e., a ‘mass’–‘nonlinear damper’– ‘nonlinear spring’ system) injection term; and iii) a TDE error correction term based on a nonlinear sliding surface. The proposed TDC formulation has an inclusive structure: depending on the gain/parameter set chosen, the proposed formulation can become Hsia’s formulation; Jin’s formulations, including a type of terminal sliding mode control; a sliding mode control with a switching signum function; or a novel enhanced formulation. Experimental comparisons were made using a PUMA-type robot manipulator with various parameter sets for the proposed control. Among them, the highest position tracking accuracy was obtained by using a terminal sliding desired error dynamics with a terminal sliding correction term.

Index Terms—time delay control (TDC), time delay estimation (TDE), inclusive formulation, nonlinear desired error dynamics, nonlinear sliding surface, robot manipulators, model-free control.

Manuscript received October 12, 2015. This work was supported in part by

the 2015 Research Fund (1.150048.01) of UNIST (Ulsan National Institute of Science and Technology), in part by the Translational Research Center for Rehabilitation Robots (#NRCTR-EX17012), National Rehabilitation Center, Ministry of Health and Welfare, Korea, and in part by the Ministry of Trade, Industry & Energy (MOTIE, Korea) under Industrial Technology Innovation Program (No. 10067184, Development of armored robot systems for personal protections of rescuemen and emergency management operations in the composite disaster site). Corresponding author: Sang Hoon Kang (e-mail: sanghkang@unist.ac.kr).

M. Jin is with the Korea Institute of Robot and Convergence, Pohang 37666, Republic of Korea (e-mail: mulimkim@kiro.re.kr).

S. H. Kang is with the Robotics and Rehabilitation Engineering Laboratory, Department of System Design and Control Engineering, UNIST, Ulsan 44919, Republic of Korea, and with the Department of Physical Medicine and Rehabilitation, Northwestern University, Chicago, IL 60611. (corresponding author to provide phone: +82-52-217-2729; fax: +82-52-217-2709; e-mail: sanghkang@unist.ac. kr).

P. H. Chang is with the Department of Robotics Engineering, Daegu Gyeongbuk Institute of Science and Technology, Daegu 42988, Republic of Korea (e-mail: phchang@dgist.ac.kr).

J. Lee is with the Department of Advanced Robotics, Istituto Italiano di Tecnologia, Genoa 16163, Italy (e-mail: jinoh.lee@iit.it).

I. INTRODUCTION

OTION control of robot manipulators is a highly challenging task. Robot dynamics is highly nonlinear and

strongly coupled because of the nonlinear terms, including gravity, Coriolis and centrifugal torque, and nonlinear friction and other disturbances. Identifying the parameters of the dynamics could be a cumbersome task, and the number of parameters of the model (e.g., the inertia and mass of each segment) that need to be estimated generally increases with the increase of the degree-of-freedom (DOF) of the robot. Most controllers based on robot dynamics [4] are highly complicated, because of the necessity of computation of the nonlinear terms of the robot dynamics, and may be sensitive to changes in the parameters of the robot dynamics. To name a few, computed torque control, computed-torque-like control, sliding mode control (SMC), and adaptive control with robot nominal model are the ones based on robot dynamics model. To alleviate the need of a robot dynamics model, intelligent control techniques (e.g., fuzzy control and neural networks) have been developed, as a model-free control, by obtaining “black box” models of robot manipulator dynamics [5-9]. To use intelligent controls, however, one needs to tune a number of gains/parameters of its own that may affect the control performance. Thus, for practicality, there is a need for a simple yet effective control scheme.

An alternative to the aforementioned controllers could be the time delay control (TDC) [9-16] using time delay estimation (TDE) to compensate for complex robot dynamics without a robot dynamics model and its parameters [13]. TDE requires only the most recent past control input to the robot and the acceleration for estimation and compensation of the robot dynamics (including uncertainties and unknown dynamics, and disturbances [9-15], [17]), based on the assumption that the dynamics and disturbances are at least piecewise continuous. Of course, TDE needs the order and structure of the robot dynamics but not the parameters or a dynamics equation. With compensation/cancellation of robot dynamics using TDE, it is possible to specify the desired error dynamics (DED) explicitly and inject them as closed-loop dynamics behavior [10], [11], [13], [14], which were mostly linear dynamics. Because of the simplicity of its structure, model-freeness (independence), numerical efficiency, and robustness against unknown and uncertain dynamics and disturbances, TDC has been utilized to control various systems, including robot manipulators [10-16],

Robust Control of Robot Manipulators Using Inclusive and Enhanced Time Delay Control

Maolin Jin, Member, IEEE, Sang Hoon Kang, Member, IEEE, Pyung Hun Chang, Member, IEEE, and Jinoh Lee, Member, IEEE

M

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[18-21]. Overall, TDE estimates the robot dynamics and disturbance

closely. However, discontinuous and/or abruptly changing dynamics (e.g., Coulomb friction and stiction) can degrade TDE performance. From the view point of TDE, continuous nonlinear terms are classified as soft nonlinearities, and discontinuous terms such as Coulomb friction and stiction are classified as hard nonlinearities [9], [17]. Friction disturbance accounts for as much as 30% of the maximum motor torque and behaves nonlinearly even at low velocities [22-25]. It has been reported that pulse-type TDE errors due to hard nonlinearity (e.g., sudden changes in the magnitude and polarity of Coulomb friction at velocity reversal) can increase position tracking error [26]. To improve TDC, in conjunction with TDE, various schemes (e.g., SMC [27], ideal velocity feedback [17], [28], internal model control [29], terminal sliding mode (TSM) [9], and nonlinear damping [30], [31]) have been exploited.

Recently, nonlinear sliding surfaces have been adopted as the DED of TDC, in place of the commonly used linear dynamics. A TSM was used for the TDC’s DED injection term to design a fast and terminal sliding DED [26], [30]. Because the TSM speeds up the convergence rate near the equilibrium point, a properly designed terminal sliding DED has been shown to improve the position tracking performance of TDC in comparison to TDC with linear DED [26], [32]. The adoption of a hyperbolic tangential sliding mode (SM) as a DED has been shown to alleviate the negative effects of velocity saturation and noise due to numerical differentiation [33]. In short, advanced control schemes that combine TDE with various SMs [26], [30], [32], [33] have shown promising results. Thus, properly designed nonlinear DED may further improve the tracking performance of TDC.

However, during the reaching phase before the SM occurs, controllers with a SM [26], [30], [32], [33] (and many other conventional SMCs) may not guarantee the robustness to parameter variations and disturbances [34-37]. Furthermore, those utilizing a SM [26], [27] may have discontinuous chattering.

The goals of this paper are to propose an enhanced TDC formulation that can include many other previously proposed TDC formulations; and to examine the performance and properties of the novel TDC formulation in comparison to other controllers, including Hsia’s TDC formulation [12], Jin’s TDC formulation [17], a SMC with a switching signum function, and another TDC formulation by Jin [9], which is a type of TSM control with experiments using a PUMA-type robot manipulator. The enhancement was made with the adoption of i) a novel nonlinear DED based on a ‘mass’–‘nonlinear damper’–‘nonlinear spring’ system, which makes it possible to utilize not only full second-order linear dynamics but also certain types of nonlinear second-order dynamics as a DED, unlike the controllers used in [26], [30], [32], [33]; and ii) a nonlinear residual energy absorber for correction of the TDE error based on a nonlinear integral SM (full-order SM) expecting no reaching phase [34], [35], [37], [38] and no chattering.

II. BRIEF REVIEW OF TIME DELAY CONTROL

TDC will be briefly reviewed with the introduction of the TDC with linear desired error dynamics [11], [12].

A. Robot Manipulator Dynamics

n-DOF robot manipulator dynamics can be described as follows:

( ) ( , ) ( ) ( , ) d M q q C q q q G q F q q τ τ . (1)

The dynamics in (1) can be rewritten as follows with the introduction of M (Table I):

( , , ) Mq N q q q τ , (2)

where ( , , ) [ ( ) ] ( , ) ( ) ( , ) d N q q q M q M q C q q q G q F q q τ . (3)

Note that ( , , )N q q q includes all the uncertain and unknown

dynamics, including the friction and other disturbance torques.

B. Approximation of Uncertainties by the Delayed Dynamics

The idea of time delay estimation can be summarized as follows. Assuming that ( , , )N q q q is piecewise continuous, for

a sufficiently small time L, the value of ( , , )N q q q at time t may

be close to that of ( , , )N q q q at time t-L [10-12], [14], [15]:

( )( , , ) ( , , )t t LN q q q N q q q . (4)

L is usually set to be the sampling period in digital implementation. From the robot dynamics in (2), one can have the following equality:

TABLE I NOMENCLATURE

Symbol Variable Name

, , nq q q joint position, velocity, and acceleration

M(q)Rn×n generalized inertia matrix

( , )C q q Rn×n Coriolis/centripetal matrix

G(q), ( , )F q q Rn gravitational torque and friction disturbance torque

τd, τRn disturbance torque and joint torque n nM positive constant diagonal matrix

●(t), ●(t-L) values of ● at times t and t-L LR estimation time delay

qdRn desired trajectory

eRn position error

KD, KPRn×n diagonal coefficients matrices

uh, unRn auxiliary control signal [1] (also known as the new control input [2], [3])

nε time delay estimation (TDE) error

idealq Rn ideal velocity

α, β Rn exponent vectors

ei, ie R ith joint position error and its time derivative

sgn(•) signum function sRn integral sliding variable

λRn×n constant diagonal gain matrix whose diagonal elements are non-negative (λii ≥ 0; i=1, 2, …, n)

γRn exponent vector whose elements are non-negative (γi ≥ 0; i=1, 2, …, n)

ξR finite positive constant

λminR λ’s smallest diagonal element

ωR cutoff frequency

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( , , ) t L t L t L N q q q τ Mq . (5)

From (4) and (5), one can obtain the TDE of ( , , )N q q q ,

ˆ ( , , )N q q q , as follows:

ˆ ( , , ) ( , , )

t t L

t L t L

N q q q N q q q

τ Mq

, (6)

It is clear from (6) that, to obtain ˆ ( , , )N q q q , the most recent

torque τ(t-L) and acceleration t Lq are all that needed instead of

the complex robot dynamics, (3). As the estimation time delay (L) becomes smaller, TDE in (6) becomes more accurate.

C. Time Delay Control with Linear Desired Error Dynamics (Hsia’s Formulation)

1) Desired error dynamics The following linear second-order DED was suggested by

Hsia and his colleagues and utilized for TDC [11], [12]:

D Pt t t e K e K e 0 , (7)

where e(t) qd(t) − q(t). (8)

KD (=diag(KD11, KD22, …, KDnn)) and KP (=diag(KP11, KP22, …, KPnn)) can be designed to meet the desired performance (e.g., settling time, damping ratio). Note that the DED, (7), is a decoupled dynamics. 2) Time Delay Control formulation

The TDC with the linear DED in (7) can be designed as follows:

Time Delay Estimation

t h t t L t L τ Mu τ Mq

with (9)

D Ph t d t t t u q K e K e . (10)

The last two terms (τ(t-L) – t LMq ), the TDE term, in the right-

hand side (RHS) of (9) compensate for the nonlinear robot dynamics; and the first term,

hMu , in the RHS of (9) injects

DED to the closed-loop system (i.e., the robot manipulator

under TDC). Thus, hMu is called the injection term. One can see from (9) and (10) that TDC is simple in that it does not require robot dynamics model and its parameters. Instead, it utilizes recent past control input, τ(t-L), and acceleration, t Lq ,

to compensate for the robot dynamics (3). In this context, the rather recent Model-Free Control (MFC) [39], [40] may be associated with the Hsia’s TDC formulation (Appendix C). Hereafter, the TDC with linear DED, given in (9) and (10), will be called Hsia’s formulation. 3) Closed-loop dynamics and Time Delay Estimation error

Subtracting the RHS of (9) from the left-hand side (LHS) of (2), and solving for

h u q yields

1 ( , , ) ( )h t t Lt t Lt

u q M N q q q τ Mq . (11)

Substitution of (6) into (11) yields

1 ( , , ) ( , , )h t t t Lt

u q M N q q q N q q q . (12)

The RHS of (12) is defined as TDE error ε ( 1[ ( , , )

tM N q q q

( , , ) ] t LN q q q ). With the definition of the TDE error, from

(12), one can easily obtain the following equation:

th t t u q ε , (13)

implying, with the application of TDE in (9), the robot dynamics (2) became a linear decoupled second-order dynamic system subject to ε, and the system can be controlled by designing the auxiliary control signal uh (e.g.,(10)) [1-3], [41]. From (8) and (10), it can be easily derived that

t t t D tPth u q e K e K e . (14)

Substituting (14) into (12) yields

1 ( , , ) ( , , )t t t tD P t L

e K e K e M N q q q N q q q . (15)

Because RHS of (15) is the TDE error, (15) becomes

t t P tD t e K e K e ε , (16)

which is the closed-loop resulting error dynamics of the robot manipulator under the Hsia’s formulation. The TDE error ε causes the resulting dynamics (16) to deviate from the DED in (7). With a sufficient small sampling time L, the ε is close to 0, if

( , , )N q q q is continuous; however, if ( , , )N q q q is discontinuous, ε cannot be ignored. For instance, at the velocity reversal, a pulse-type sudden increase in position tracking error was found, because ε increased due to the sudden polarity change of discontinuous joint friction (i.e., in this case,

( , , )N q q q is discontinuous at the velocity reversal) [26].

III. INCLUSIVE AND ENHANCED FORMULATION OF TDC

A novel formulation of TDC will be proposed to enhance the position tracking performance of TDC with a nonlinear DED and a correcting term reducing the TDE error due to discontinuous dynamics. It will be proved that the proposed TDC formulation is inclusive by showing that each of the previous TDC formulations is a special case of the proposed formulation.

A. Proposed Nonlinear Desired Error Dynamics

We propose a nonlinear DED as below:

( ) ( )t D Pt t α βe K sig e K sig e 0 , (17)

where 1

1 1( ) [ sgn( ), , sgn( )]n Tn ne e e eαsig e , (18)

1

1 1( ) [ sgn( ), , sgn( )]n Tn ne e e eβsig e . (19)

All elements of α (=[α1, α2, …, αn ]T) and β (=[β1, β2, …, βn]T) are positive values (i.e., αi > 0, βi > 0 (i = 1, 2, …, n)). Similar to (7), the novel DED in (17) is also a decoupled dynamics.

If α = 1n and β = 1n, the proposed DED is identical to (7), the conventional linear DED. Here, 1n = [1, 1, …, 1]T Rn. If not,

the DED represents an equivalent ‘mass’–‘nonlinear damper’– ‘nonlinear spring’ model. Phase portraits of the DED for several α and β show that the equilibrium points are either stable node or stable focus, depending on α and β (Fig. 1).

One can check the stability of the DED given in (17) with the construction of a Lyapunov function (V), the total energy of the DED, as follows:

1 12

1

0.5 1 i

n

Pii ii t i tti

V e K e

. (20)

The time derivative of V can be obtained as follows:

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1

1

1

1

1

sgn( )

sig( )

sig( )

i

i

i

i

n

Piii t i t i t i t i tin

Piii t i t i tin

Diii t i tin

Dii i ti

tV e e e K e e

e e K e

e K e

K e

. (21)

Because 0V with (17) implies e≡0 and e ≡0, with positive KDiis and KPiis, the equilibrium (e,e )=(0,0) of (17) is globally asymptotically stable by LaSalle’s invariant set theorem.

B. Control Law Design

1) TDC with nonlinear desired error dynamics Similar to the Hsia’s formulation, control torque can be

designed as follows:

ˆ

t n t t τ Mu N , (22)

where

( ) ( )D Pn t d t t t α βu q K sig e K sig e . (23)

In comparison with uh of the Hsia’s formulation in (10), the proposed control uses un to realize the proposed nonlinear DED. Thus, with the combination of (6), (22), and (23), the control law is expressed as

ˆ( ) ( )D Pt d t t t t

α βτ M q K sig e K sig e N . (24)

Substitution of the control input (24) into robot dynamics (2), and simple manipulations yield the closed-loop dynamics:

( ) ( )D Pt t t t α βe K sig e K sig e ε , (25)

indicating that the closed-loop dynamics is perturbed by ε. As mentioned in Section II.C.3), the TDE error ε can be increased with discontinuity in ( , , )N q q q . Thus, this control may need a

term suppressing the effect of ε on position error. 2) Addition of TDE error correction

If we define

( ) ( )D P dt α βs e K sig e K sig e , (26)

having a zero initial value (i.e., s(t=0)=0), then one can easily find that the closed-loop dynamics (25) can be rewritten as

t ts ε . (27)

Again, one can see that the closed-loop error dynamics is disturbed by the TDE error (ε).

If the DED in (17) is perfectly realized, one can obtain the following relation from (17).

( ) ( )D Pideal t d t t t α βq q K sig e K sig e . (28)

Note that because (28) holds only when ε is 0, the LHS angular acceleration term is not the same as the real acceleration ( q ),

Fig. 1. The phase portrait of the proposed nonlinear error dynamics, 20sig( ) 100sig( ) 0,e e e with various α and β. If α =1.0 and β=1.0 the error dynamics

becomes linear error dynamics. Dashed lines shown in the center plot showing the phase portrait with α=95/100 and β=95/105.

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but it is an idealized case acceleration. Thus, to avoid confusion, for the idealized case acceleration, we use idealq denoting ideal acceleration. By integrating both sides of (28), one can obtain the ideal velocity ( idealq ), representing the velocity of the closed-loop system that perfectly follows the DED, as follows:

( ) ( )ideal d D P dt α βq q K sig e K sig e . (29)

By comparing (29) with (26), one can see that the following holds.

t ideal t t s q q . (30)

Thus, from (30), one implication of the sliding variable s is the difference between the ideal velocity and the real robot manipulator velocity.

Based on this idea, to suppress the TDE error (ε), a correcting term M λ·sig(s)γ is added to the control input in (22), as

( )

( ) ( ) ( )

t n t t t L t L

D Pd t t t t

t L t L

γ

α β γ

τ M u λ sig s τ Mq

M q K sig e K sig e λ sig s

τ Mq

.(31)

The λ is the coefficient of nonlinear damping for the ideal velocity feedback (see (30)). The damping term λ·sig(s)γ absorbs the residual energy due to the TDE error as a counteracting force while reducing the difference between the real and the ideal velocity (i.e., making the behavior of the closed-loop dynamics close to that of the DED).

Substituting the control input (31) into the robot dynamics (2), we obtain the closed-loop dynamics as

( )t t t γs λ sig s ε . (32)

Because, in (32), s – the LHS of the DED (see (17) and (26)) – can be regarded as the output of a stable first-order filter with input ε [42], it is expected that the effect of ε on s (and consequently on tracking error, e) will be attenuated compared with (27).

The ε is bounded (i.e., ||ε|| < ξ), if M satisfies the well- known stability condition of TDC [11], [17], [26] given below:

1 1 I M M , (33)

because ( , , )N q q q is the sum of continuous terms and

bounded discontinuous terms [9], [26]. The stability criterion can be obtained easily by following the stability analysis in [9], [17], [26].

C. Discussion of the Proposed TDC Formulation

1) Properties of desired error dynamics We can further examine the nonlinear DED (17), which

includes the linear DED (7). Finite-time stable DED can be obtained, if the following

conditions (proof in Appendix A) are met for αi and βi (i = 1, 2, …, n) [43]

0 < αi < 1, and (34) βi = αi / (2−αi). (35)

For instance, if we set αi = 95/100, βi that satisfies (35) is 95/105. With these values, one can see that the finite-time convergence of the nonlinear DED is achieved (Fig. 2) [43]. The convergence speed of the finite-time stable nonlinear DED is faster than that of the asymptotically stable linear DED

requiring infinite time for the convergence of error (e) to 0 (Fig. 2). The phase portrait of the DED with αi = 95/100 and βi = 95/105 is slightly twisted compared with that of the linear DED (center plot of Fig. 1). Moreover, for αi < 1 and βi < 1 (i = 1, 2,

…, n), the nonlinear damping and spring terms ( ( )DαK sig e and

( )PβK sig e ) are more effective than the linear damping and

spring terms – in the linear DED – around the neighborhood of the equilibrium point (i.e., | ie | < 1 and |ei| < 1). The fractional

power terms ( i

ie and i

ie

) amplify ie and ei significantly in

that neighborhood without increasing KD and KP compared with the linear feedback in (10). This property can avoid sticking of the system states near the ei = 0 and/or 0ie ,

helping the system reach the sliding surface. If αi > 1 and βi > 1 (i = 1, 2, …, n), the convergence speed of

the DED becomes slower than that of the linear DED (dash-dot line in Fig. 2).

Note that, even with α=0 and β=0, the DED is stable (see Section III.A). However, the ‘nonlinear damping’ and ‘nonlinear spring’ can induce chattering, because those two terms become KDsgn( e ) and KPsgn(e), respectively. 2) Terms of the proposed TDC

Similarly to the Hsia’s formulation (in Section II.C.2)), in the

proposed control in (31), τ(t-L)− t LMq is the TDE term;

n tMu is the injection term (i.e., injecting DED in (17)). For

the proposed control, there is one more term: the correcting term (i.e., M λ·sig(s)γ) that corrects/suppresses the TDE error by utilizing the ideal velocity feedback.

The elements of the proposed controller, given in (23) and (31), have their own clear physical meaning. The DED is described by a ‘mass’–‘nonlinear damper’–‘nonlinear spring’ system with the nonlinear damping KDsig( e )α and the nonlinear spring KPsig(e)β. The TDE error is corrected by another nonlinear damping term λ·sig(s)γ (=λ·sig (

ideal q q )γ)

to achieve DED. 3) Closed-loop dynamics change with γ

One can examine the resulting closed-loop dynamics (32) with various γ values. First, because sig(s)γ is continuous for γ ≠ 0, there is no chattering unlike the sgn(s).

If we let γ = 0, the closed-loop dynamics becomes

Fig. 2. Comparisons of behaviors of desired error dynamics.

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( )t t t s λ sgn s ε . (36)

For (36), with the Lyapunov function Vs (= 0.5sTs), if λmin > ξ, (37)

theoretically, si (i = 1, 2, …, n) converges to the sliding surface (Appendix B), provided that there is no chattering [9].

If all γi are positive constants, with Vs, each element of s is bounded as follows (Appendix B)

1

min iis , (i =1, 2, …, n) (38)

indicating that increasing λmin can reduce the effect of the TDE error, ξ. In particular, if (37) – the sliding condition when γ = 0 – holds, the bound for si in (38) becomes smaller than unity.

If all elements of γ are in between zero and unity (0 < γi < 1; i =1, 2, …, n), the s dynamics becomes a TSM [44]. If (37) holds and 0 < γi < 1, as shown in (38), decreasing γi can make the bound of |si| smaller without increasing the coefficient of the nonlinear correcting term (i.e., λmin). In other words, ξ can be further attenuated with the decrease of γis, making the correcting term practically valuable. In addition, the sig(•)γ can effectively increase the effect of the correcting term near the sliding surface (|si| < 1) because the fractional power of |si| substantially increases near the region. For instance, if γi = 0.4

and si = 0.0001, ( ) iisig s

= 0.0251. Thus, λ needs to satisfy (37) to take full advantage of the TSM.

If γ =1n, the s dynamics becomes linear. If all elements of γ are larger than unity (γi > 1; i=1, 2, …, n), the s dynamics converges fast when |si| > 1, and converges slowly when |si| < 1 (i =1, 2, …, n). Therefore, various reaching conditions can be made by changing γ.

Note that, because si (i=1, 2, …, n) is bounded with the bound given in (38) and has zero initial condition, si will stay in the boundary from the initial time instance. In other words, practically, the initial deviation of the s from the sliding surface (s = 0) does not depend on the initial condition unlike other controllers [26], [30], [32], [33] that have a reaching phase and may not be insensitive to parameter variations and disturbances during the phase [34-37]. 4) Inclusiveness of the proposed TDC

The proposed formulation of TDC can include many other existing TDC formulations.

• If λ = 0, and α = β = 1n, the proposed control becomes the Hsia’s formulation [12] in (9) and (10).

• If all diagonal elements of λ have positive values (λii > 0; i =1, 2, …, n) and α = β = γ = 1n, the proposed control becomes the Jin’s position control formulation of TDC in [17]:

ˆ

D Pt d t t t t t τ M q K e K e λs N . (39)

• If all diagonal elements of λ have positive values (λii > 0; i =1, 2, …, n), α = β = 1n, and γ = 0, the proposed control becomes a SMC with integral sliding surface and TDE, most probably having chattering due to the signum function, as

ˆ( )D Pt d t t t t t

τ M q K e K e λ sgn s N . (40)

• If all diagonal elements of λ have positive values (λii > 0; i =1, 2, …, n), α = β = 1n, and all elements of γ are in between 0 and 1 (0 < γi <1; i = 1, 2, …, n), the proposed control becomes another TDC formulation of Jin (a type of TSM control) [9]:

ˆ( )D Pt d t t t t t

γτ M q K e K e λ sig s N . (41)

Thus, the proposed TDC formulation is indeed inclusive.

D. Low Pass Filtering with Lowering of M

To reduce the noise due to acceleration and other signals, a low pass filter may be needed. A possible candidate is a first- order digital low pass filter with the cutoff frequency ω. With the application of the filter, (24) changes as follows:

*

( )( )tt n t t L t L

γτ M u λ sig s q τ , and (42)

τ(t)= [ωʹ/(1 + ωʹ)] *tτ + [1/(1 + ωʹ)]τ(t−L) (ωʹ = ωL), (43)

where τ* denotes input to the filter. Substituting (42) into (43)yields

( )1 ( )tt n t t L t L

γτ M u λ sig s q τ . (44)

Thus, lowering of M is equivalent to the use of a low pass filter. Further, there is no additional dynamics introduced with the low pass filter.

E. Tuning Procedure

To implement the proposed controller, we need to tune M , KD, KP, λ, α, β, and γ. Note that L is usually set to be the sampling time for discrete time domain implementation. The gains can be tuned through the following procedure.

1. Set α = β = γ = 1n and λ=0, design a stable linear DED by specifying KD and KP, and tune the M only (i.e., tune Hsia’s formulation [12]) by increasing diagonal elements of the M from a small value until there is no decrease in error with vibrations (and noisy sound).

2. Maintain γ = 1n and tune λ by increasing its diagonal elements from 0, while checking the control performance (e.g., error magnitude and unwanted vibrations).

3. Change the linear reaching condition to the TSM by decreasing positive γi from 1 (i = 1, 2, …, n) with the observation of error magnitude and unwanted vibrations.

4. Twist the DED by decreasing αi and βi from 1 (i = 1, 2, …, n). For the sake of simplicity, βi can be calculated from (35). If vibrations are observed, then one stops decreasing elements of the exponent vectors.

Remark: For the experimental studies in Section IV, KD = 20•I and KP =100•I – locating double pole at -10 for each DOF for linear DED – were used. This selection allows us to compare the experimental results of this study (see Section IV) with many previous studies using the same KD and KP at least qualitatively [9], [11], [13], [26], [29].

IV. EXPERIMENTAL STUDIES

A. Experimental Setup

A Samsung Faraman-AT2 robot manipulator is used in the experiment (Fig. 3). The maximum continuous torques were 0.637 Nm, 0.637 Nm, and 0.319 Nm for joint 1, 2, and 3, respectively. The gear-reduction ratio and the encoder resolution of all those three joints were 120:1 and 2048 pulses/rev, respectively. Resolution of each robot joint was 3.66 × 10-4 °. Angular velocity and acceleration of the three joints were computed by a simple numerical differentiation

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[12]. The controller was operated in Linux-RTAI, a real-time operating system environment. Sampling time was 0.001 s, and, accordingly, the time delay L was set to be 0.001 s.

B. Comparison of Different Parameter Sets

Experimental comparisons were made using the robot with various parameter sets of the proposed control. The desired trajectory of joint 1, 2, and 3 consists of four fifth-order polynomial functions of time (Fig. 4). The parameters of the three joints, and the root-mean-square (RMS) value of those three joints’ position tracking error and the %RMS difference (defined in Table II) are listed in Table II.

By following the tuning procedure described in Section III.E, the proposed controller was tuned experimentally as follows. First, M was tuned and set to be diag(0.3818, 0.30545, 0.15296) kg·m2 (dotted line in Fig. 5).

Second, the diagonal elements of λ increased (dashed line in Fig. 5). Comparison of the RMS error of Exp01 (Hsia’s formulation in (9) and (10)) and that of Exp02 (Jin’s formulation in (39)) shows the effectiveness of the correcting term, and the tendency agrees with the free space motion control performance reported in [17].

Third, γis decreased from 1. Examination of RMS errors obtained from Exp02 to Exp05 (Table II) shows that, with the gradual reduction of γis from 1, the RMS error decreased. If γi < 0.4 (i=1, 2, …, n), however, vibration with noisy sound was observed without further decrease of the RMS error (dash- dotted line showing the results of Exp05 in Fig. 5).

Fig. 3. A PUMA-type robot manipulator (Samsung Faraman-AT2).

Fig. 4. Desired trajectory (four 5th-order polynomial functions of time inseries) of joint 1, 2, and 3.

Fig. 5. Comparison of tracking errors with various parameters. Exp01, theHsia’s formulation (dotted line): λ=0, α=13 and β=13; Exp02 (dashed line): λ=diag(20, 20, 20), α=13, β=13, and γ=13; Exp05 (dash-dotted line): λ=diag(20,20, 20), α=13, β=13, and γ=0.4•13; and Exp08 (solid line): λ=diag(20, 20, 20),α=95/100•13, β=95/105•13, and γ=0.4•13. The proposed formulation is asuperset of the existing controls [9], [12], [17]. If we properly twist the errordynamics and the correcting term with tuning of α, β and γ, the trackingperformance can be substantially improved.

Fig. 6. Tracking error with too small α and β (dashed line) and with thoseachieved the smallest RMS error (solid line). Decreasing α and β below acertain value could induce vibration.

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Fourth, we slightly twisted the DED with the tuning of α and β, and obtained better tracking performance (by comparing Exp05, Exp06, and Exp08). With the parameters (λ = diag(20, 20, 20), α = 95/100•13, β = 95/105•13, and γ = 0.4•13) guaranteeing the finite-time stable terminal sliding DED and terminal sliding correcting term (Exp08; solid line in Fig. 5), the smallest RMS error was observed. A severe vibration was observed with too small αi and βi (Fig. 6). Thus, the exponents α, β, and γ need careful tuning.

Additional experiments Exp11–13 show that, if αi >1 and βi >1 (i = 1, 2, …, n), the RMS error increased substantially with the small increase of exponents.

One of the Jin’s formulation [9] in (41), which used linear DED with the nonlinear correcting term, achieved the 2nd smallest RMS error. Even compared with the RMS error of one of the Jin’s formulation [9], a special case of the proposed formulation, the proposed control achieved significant decrease of RMS error (more than 53%; Table II), indicating substantial improvement. At the velocity reversal (3 and 9 s), position error increased due to the rapidly changing friction (Fig. 5).

One can easily see that the control performance with the nonlinear DED (Exp09) was improved in comparison with that with the linear DED (Exp01), showing the usefulness of the nonlinear DED.

C. Robustness Against the Increase of Speed

Robustness against the increase of speed was tested without any further fine tuning of gains from those used in Exp08. With the 50% increase in speed (reaching the robot’s limit), RMS error of each joint was maintained at a reasonable magnitude (1st joint: 0.0017°; 2nd joint: 0.0025°; 3rd joint: 0.0030°), indicating the robustness of the proposed control. It also shows that the fine tuning of gains for each different trajectory is not necessary.

D. Discussion

As shown in (31), the proposed inclusive and enhanced TDC does not use the knowledge of M(q) and ( , , )N q q q – which

includes uncertainties in inertial parameters (M(q)− M ), Coriolis and centrifugal torque terms ( ( , )C q q q ), and nonlinear

joint friction disturbance ( ( , )F q q ) – of the robot being

controlled. The robot we used is a PUMA-type robot having substantial amount of highly nonlinear (even at low velocity), uncertain, and time-, velocity-, and temperature-varying joint friction disturbance [22-25], and coupled and highly nonlinear dynamics. Thus, the aforementioned experimental results proved the robustness against parameter variation and unknown disturbances, including one of the most often encountered friction disturbance, to some extent. The results of analysis, simulations, and experiments about the robustness of previous TDC formulations [9-12], [14], [15], [17], [27] can be similarly applied to the proposed formulation, because the proposed formulation includes all those controllers.

Seemingly, there are two more parameters (α and β) to tune for the proposed control compared with the formulation achieved the 2nd smallest RMS error, but, with the finite-time stable DED condition ((34) and (35)) relating α and β, the number of additional parameters becomes one (e.g., α) with the limited parameter space, (0, 1).

Although the tuning procedure given in Section III.E may seem not very solid, it took less than an hour for the gain tuning of the PUMA-type robot, showing the practicality of the procedure. It would, however, be worth to develop a more solid procedure as a further study.

The RMS errors of many different TDC formulations are given in Table II. Thus, depending on the required precision/ accuracy to complete a task given to a robot that one wants to use, one may be able to choose a suitable controller for the task with Table II. In other words, these promising results with the proposed control do not necessarily mean we always have to use the proposed control with twisted nonlinear DED. In stiffness control or in impedance control (though these are not the position control), the stiffness terms are desired variables of reference model (typically chosen as linear and decoupled ones) and are not gains of the controller [17]. The TDE error

TABLE II ROOT MEAN SQUARE (RMS) ERROR WITH VARIOUS PARAMETER SETS

No. KDii KPii λii αi βi γi

RMS Error (%RMS Differencea) Remark

1st joint 2nd joint 3rd joint

Exp01 20 100 0 1 1 – 0.0139° (1290%) 0.0377° (2118%) 0.0478° (2712%) [12]

Exp02 20 100 20 1 1 1 0.0045° (350%) 0.0100° (488%) 0.0135° (694%) [17]

Exp03 20 100 20 1 1 0.8 0.0029° (190%) 0.0065° (282%) 0.0082° (382%)

Exp04 20 100 20 1 1 0.6 0.0019° (90%) 0.0038° (124%) 0.0048° (182%)

Exp05 20 100 20 1 1 0.4 0.0017° (70%) 0.0026° (53%) 0.0027° (59%) [9]

Exp06 20 100 20 98/100 98/102 0.4 0.0014° (40%) 0.0021° (24%) 0.0023° (35%)

Exp07 20 100 20 98/100 98/102 1 0.0043° (330%) 0.0105° (518%) 0.0131° (671%)

Exp08 20 100 20 95/100 95/105 0.4 0.0010° (0%) 0.0017° (0%) 0.0017° (0%) *smallest

Exp09 20 100 0 95/100 95/105 – 0.0082° (720%) 0.0238° (1300%) 0.0294° (1629%)

Exp10 20 100 20 95/100 95/105 0.8 0.0023° (130%) 0.0055° (227%) 0.0072° (324%)

Exp11 20 100 20 1.2 1.2 1 0.0094° (840%) 0.0219° (1188%) 0.0271° (1494%)

Exp12 20 100 0 1.2 1.2 – 0.0478° (4680%) 0.1171° (6788%) 0.1287° (7471%)

Exp13 20 100 0 1.5 1.5 – 0.2073° (1973%) 0.4018° (23535%) 0.4129° (24188%) a%RMS Difference = ((RMS Error of ith EXP) - (RMS Error of 8th EXP))/(RMS Error of 8th EXP) ×100

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correcting term may not be required if discontinuities of nonlinear terms are not dominant (e.g., with the use of magnetic bearing). Because of its inclusiveness and simplicity, the proposed formulation may be used for the practical precise position control of robot manipulators. Further, the proposed formulation might be extended to stiffness control, impedance control, and force control for robot manipulators.

V. CONCLUSION

The inclusive and enhanced TDC formulation proposed in this paper inherits the advantages of TDC: it is simple, intuitive, and model-free.

The proposed control consists of three terms: a continuous nonlinearity canceling term using TDE, a DED injecting term, and a TDE error correcting term. The nonlinear DED describes a ‘mass’–‘nonlinear damper’–‘nonlinear spring’ system. The residual energy due to the perturbation of the TDE error can be absorbed by a nonlinear damper utilizing ideal velocity feedback based on an integral SM.

The formulation of proposed control is inclusive: with certain parameters, the proposed formulation becomes Hsia’s formulation, Jin’s formulations including a type of TSM control, and a SMC with switching signum function. The position tracking performance is improved significantly by properly twisting the DED and the correcting dynamics by changing α, β, and γ. Thus, a near future work might be adaptive tuning of α, β, and γ to increase its practicality. It might be possible to extend the proposed TDC formulation to the stiffness control, impedance control, and force control considering the simplicity and inclusiveness of the proposed formulation, which is an on-going research in authors’ laboratory.

APPENDIX

A. Condition for Finite-time Desired Error Dynamics

Lemma 2 of [43] may be summarized as follows: the following control law 1 2

1 1 2 2sig( ) sig( )u l x l x with (45)

0 < ρ1 <1, and (46) ρ2 = 2ρ1/(1 + ρ1), (47) finite-time stabilizes the following double integrator system:

1 2

2

x x

x u

. (48)

Substituting u in (45) into (48) yield

1 2

1 2

2 1 1 2 2sig( ) sig( )

x x

x l x l x

. (49)

By using the lemma, we will show that, if (34) and (35) are met, the desired decoupled error dynamics, (17), is finite-time stable. The DED in (17) can be rewritten as below.

1 2 1

2 1 2

,

( ) ( ) P D

β α

e e e e

e K sig e K sig e

, (50)

where e1 denotes the position error e; and e2 the time derivative of e1 (i.e., velocity error). Because (50) is a decoupled

dynamics, for each DOF, one can rewrite (50) as follows:

1 2 1

2 1 2

, ( )

( ) ( ) , ( 1,2, , )i i

i i i i

i Pii i Dii i

e e e e

e K sig e K sig e i n

, (51)

where e1i and e2i denote ith element of e1 and e2, respectively. By comparing (49) with (51), one can see that, if KPii=l1, KDii=l2, βi = ρ1, and (52) αi = ρ2, (53) (51) is finite-time stable.

With simple mathematical manipulations, (46) and (47) can be rewritten as follows: 0 < 2ρ1/(1 + ρ1) < 1, and (54) ρ1 = ρ2/(2 − ρ2). (55) From (54), it is obvious that 0 < ρ2 < 1. (56) Thus, (46) and (47) are equivalent to (55) and (56). Therefore, from (52), (53), (55), and (56), for the exponents αi and βi, the condition that (51) is finite-time stable is 0 < αi < 1, and βi = αi/ (2 − αi) as given in (34) and (35). □

B. Analysis of the s Dynamics in (32)

For (32), the time derivative of Vs can be obtained as follows:

1

1

min1

min1

( )

( )

i

i

i

i

Ts

n

i i i i iin

i i i i iin

i i iin

i ii

V

s sig s s

s s s

s s s

s s

γs λ sig s ε

. (57)

Thus, if

mini

is , (i = 1, 2, …, n) (58)

then 0sV .

If γ = 0, (57) becomes mins iV s . (59)

Therefore, if (37) is satisfied, 0sV outside of s = 0.

If all γi are positive constants, from (58), the bound of si, (38), can be directly obtained.

C. Hsia’s TDC Formulation and Model-Free Control (MFC)

MFC employs a simple and effective estimation scheme to estimate poorly known parts of the plant as well as of the various possible disturbances [39], [40]. Because the objective and function of the TDE of TDC is similar to those of the MFC’s estimation method, the two schemes are compared. We restrict ourselves, for simplicity, to single-input single-output (SISO) robot manipulators.

MFC may be summarized as follows. Instead of the plant model, MFC adopts the continuously updated ultra-local model, which is valid only for very short time interval, as below

q F , (60)

where σ denotes a non-physical constant chosen by the practitioner; and F a time-varying function that includes poorly known parts of the plant as well as of the various possible

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disturbances. Integer ν ≥1 may always be chosen as 1 or 2 based on the selection of the practitioner [39], [40]. With an estimate of F, , MFC may be designed as follows:

1d DM PM IMq K e K e K edt . (61)

Here, KDM, KPM, and KIM denote gains that are tuned based on the desired convergent behavior of error, and the choice of ν [39], [40]. With (61), the closed-loop dynamics becomes

DM PM IMe K e K e K edt F . (62)

From (60), in discrete time domain, with the assumption of sufficiently small time interval, is estimated as follows [39]:

ˆ( ) 1k q k k , (63)

where q̂ k denotes the estimate of q(ν) at the time instant k.

Later, in [40], to avoid the use of noisy q̂ k

, an algebraic identification technique (iterated time integrals during short time interval) is proposed. For instance, if ν=1, for sufficiently small te, was computed as follows:

3( ) 6 2

e

t

e e et tt t t q t d . (64)

From this summary, on the one hand, one can see that MFC differs from TDC in that MFC does not need system order [40]. On the other hand, TDC is similar to MFC when ν=2. One can rewrite (2) as follows:

1 1( , , )q M N q q q M . (65)

If ν=2 in (60), from (60) and (65), one can see that F and α in (60) amount to 1 ( , , )M N q q q and 1M , respectively.

Multiplying 1M on both sides of (6) yields

1 1ˆ ( , , )

t t L t LM N q q q q M , (66)

implying similarity between the MFC’s estimation scheme (63) and TDE in (66) when ν=2 with a difference on the use of acceleration: t Lq in TDE, tq in MFC’s estimation method.

Hsia’s formula, (9) and (10), can be rewritten as

1 1 1( ) [( ) ]D Pt t L t L d t t tM q M q K e K e

. (67)

When ν=2, by comparing (61) and (67), one can again find a similarity:

1t L t Lq M corresponds to , the estimate of

F, and Dd t tq K e −KPe(t) to d DMq K e −KPMe − KIM∫edt. If

KIM=0, the similarity becomes more evident. Furthermore, an alternative TDE method [45] utilizes convolution integrals instead of t Lq to reduce noise, and, in this sense, the

alternative TDE method is similar to the algebraic method (64). Because MFC estimates and compensates for the unknown

and uncertain dynamics and disturbance, such as the TDE error, MFC may have estimation error (i.e., F ) that perturbs the

error dynamics (62), unless F . Thus, it may be possible to

reduce the effect of the estimation error ( F ) by applying

the well-developed TDE error correcting methods introduced in this study. Meanwhile, the idea of the MFC’s algebraic identification method may benefit TDE in reducing noise.

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Maolin Jin (S’06–M’08) received the B.S. degree in material science and mechanical engineering from Yanbian University of Science and Technology, Jilin, China, in 1999, and the M.S. and Ph.D. degrees in mechanical engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea, in 2004 and 2008, respectively. In 2008, he was a Postdoctoral Researcher with the Mechanical Engineering Research Institute, KAIST. He was a Senior Researcher with the Research Institute of Industrial Science and

Technology, Pohang, Republic of Korea. He is currently a Principal Researcher with the Korea Institute of Robot and Convergence (KIRO), Pohang, Republic of Korea. His research interests include robust control of nonlinear plants, time-delay control, robot motion control, electro-hydraulic actuators, winding machines, disaster robotics, and factory automation.

Sang Hoon Kang (M’12) received the B.S. (cum laude), M.S., and Ph.D. degrees in mechanical engineering from KAIST, Daejeon, Republic of Korea, in 2000, 2002, and 2009, respectively.

He was a Research Associate at the Sensory Motor Performance Program (SMPP) of the Rehabilitation Institute of Chicago (RIC), Chicago, IL, and a Postdoctoral Research Fellow at the Department of Physical Medicine and Rehabilitation of Northwestern University, Chicago, IL, from 2010 to 2015. He was also an Instructor at the Department of Biomedical

Engineering, Northwestern University, Evanston, IL, in 2012 and 2013. He is currently an Assistant Professor of the Department of System Design and Control Engineering, Ulsan National Institute of Science and Technology (UNIST), Ulsan, Republic of Korea, and is an Adjunct Assistant Professor at the Department of Physical Medicine and Rehabilitation, Feinberg School of Medicine, Northwestern University, Chicago, IL. His current research interests include rehabilitation robotics and biomechanics of human movement, with an emphasis on rehabilitation medicine.

Pyung Hun Chang (S’86–M’89) received the M.S. and Ph.D. degrees in mechanical engineering from the Massachusetts Institute of Technology, Cambridge, MA, in 1984 and 1987, respectively.

He was a Professor at the KAIST, Daejeon, Republic of Korea, for 24 years. He is currently a Professor of the Robotics Engineering Department of the Daegu Gyeongbuk Institute of Science and Technology, Daegu, Republic of Korea. He has authored more than 80 papers in international journals, 22 papers in domestic journals, and 90 papers in conference proceedings. His current

research interests include rehabilitation robotics, robust control of nonlinear systems including kinematically redundant manipulators, and impedance control of dual-arm manipulators.

Jinoh Lee (S’09–M’12) received the B.S. summa cum laude in mechanical engineering from Hanyang University, Seoul, South Korea, in 2003 and the M.Sc. and the Ph.D. degrees in Mechanical Engineering from KAIST, Daejeon, Republic of Korea, in 2012. He was a Postdoctoral Researcher with the Department of Advanced Robotics (ADVR), Istituto Italiano di Tecnologia (IIT), Genoa, Italy, from 2012 to 2017. He is currently a Research Scientist with the department of ADVR, IIT. His professional is about robotics and control

engineering which include whole-body manipulation of humanoids, robust control of highly nonlinear systems, and compliant robotic system control for safe human–robot interaction. Dr. Lee has participated in Technical Committee member of International Federation of Automatic Control (IFAC), TC4.3 Robotics, since 2014. He is also a member of IEEE Robotics and Automation, Control Systems, and Industrial Electronics Societies.