Robust Compliant Motion Control of Robot With ... - Semantic Scholar

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 1, JANUARY 2008

Robust Compliant Motion Control of Robot With Nonlinear Friction Using Time-Delay Estimation Maolin Jin, Student Member, IEEE, Sang Hoon Kang, and Pyung Hun Chang, Member, IEEE

Abstract—A simple robust compliant-motion-control technique is presented for a robot manipulator with nonlinear friction. The control technique incorporates both time-delay-estimation technique and ideal velocity feedback; the former is used to cancel out soft nonlinearities, and the latter serves to reduce the effect of hard nonlinearities, including Coulomb friction and stiction. The proposed controller has a simple structure and yet provides good online friction compensation without modeling friction. The robustness of the proposed method has been confirmed through comparisons with other controllers in 2-DOF SCARA-type industrial robot experiments. Index Terms—Compliant motion control, friction, hard nonlinearity, impedance control, time-delay control (TDC).

N OMENCLATURE M(θ) ∈  ˙ ∈ n V(θ, θ) G(θ) ∈ n F, D ∈ n Fv ∈ n Fc , Fst , ∈ n τ , τs , ∈ n Md ∈ n×n Kd ∈ n×n Bd ∈ n×n θd , θ˙d , θ¨d ∈ n n×n

˙ θ¨ ∈ n θ, θ, M, Γ ∈ n×n Ke ∈ n×n θe ∈ n τsn ∈ n ˆ • •t−L {•}ii {•}i

Inertia. Coriolis and centrifugal torque. Gravitational force. Friction term and disturbance. Viscous friction. Coulomb friction and stiction. Actuator torque and interaction torque. Desired mass. Desired spring. Desired damper. Desired position, desired velocity, and desired acceleration, respectively. Joint angle, joint velocity, and joint acceleration, respectively. Positive definite diagonal gain matrices. Environment stiffness. Environment position. Maximum bound of sensor noise. Estimated value of •. Time delayed value of •. ith diagonal element of the matrix •. ith element of the vector •.

Manuscript received September 30, 2006; revised July 5, 2007. This work was supported in part by the SRC/ERC program of MOST/KOSEF and by the second stage of the Brain Korea21 project, in 2007. The authors are with the Robotics and Control Laboratory, Department of Mechanical Engineering, Korea Advanced Institute of Science & Technology (KAIST), Daejeon 305 701, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2007.906132

I. I NTRODUCTION

I

N THE USE of industrial robots and service robots, both free-space motions and contact tasks are involved in many applications including assembling [1], polishing [2], deburring [3], pushing [4], power-assisting [5], and human–robot interactions [6]–[8]. A robot is required not only to track trajectories exactly in free space but also to provide desired compliance in contact tasks to carry out these applications. As a reliable means for these applications, the robust compliant motion control [1], [9], which is based on a mechanical-impedance concept [10]–[12], is a good candidate. In compliant motions, target impedance should be properly selected. For instance, lowstiffness target impedance is required for safety when the robot unexpectedly encounters an unstructured environment while traveling in free space. To implement target impedance, various methods have been proposed over the last two decades [1], [13]–[15]. The computed torque method [1] compensates the nonlinear terms of robot dynamics (including friction) by using a robot model. Robust methods, such as adaptive control [13], variable-structure control [14], and sliding-mode control [15], take into account the modeling error of robot dynamics based on computed torque method. However, implementation of these schemes is highly complicated and computationally demanding due to the calculation of the nonlinear robot model. It is noteworthy that time-delay control [16]–[19] provides a simple but robust scheme that compensates the nonlinear terms in robot dynamics using time delay estimation (TDE) without modeling. Internal-force-based impedance control (IFBIC) [20]–[22] is devised by injecting target impedance after direct cancellation of uncertainties using TDE; however, little attention has been paid to hard nonlinearities, such as Coulomb friction and stiction, which may degrade the control performance as it accounts for nearly 30% of the maximum motor torque in robots [23]. Accordingly, this paper presents a simple robust compliant motion control for robots in the presence of nonlinear friction, including Coulomb friction and stiction. The nonlinear terms in robot dynamics are classified into two categories from the TDE viewpoint: soft nonlinearities (gravity, Coriolis and centrifugal torque, viscous friction, disturbance, and interaction torque) and hard nonlinearities (Coulomb friction, stiction, and inertial force uncertainties). TDE is used to cancel soft nonlinearity, and ideal velocity feedback, which is inspired by natural admittance control (NAC) [24], [25], is used for hard nonlinearities. As a result, the proposed control does not require the modeling of robot dynamics, yet possesses improved robustness over IFBIC

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JIN et al.: ROBUST COMPLIANT MOTION CONTROL OF ROBOT WITH NONLINEAR FRICTION USING TDE

or NAC. The target-damping design and the meaning of the proposed control gains are discussed. Experimental realization of a compliant motion is presented using a 2-DOF SCARA-type robot. Friction compensation is a very significant issue and has attracted research activities in the motion-control area. Let us briefly introduce relatively recent research works [26]–[33] only to the extent that is relevant in comparing with the proposed control. Model-based friction-compensation techniques, such as a modified Southward’s method [26], the Lugre friction model [27], and the decomposition-based friction estimation [28] require prior experimental identification, and the control performance of these offline estimation methods may decrease when the friction effects vary during the robot operations. The adaptive compensation methods, which are introduced in [29] and [30], factor in the modeling error of Lugre friction model, but the complex prior experimental identification remains. Joint-torque sensory feedback [31] compensates friction without the dynamic models, but its implementation may be limited by the cost of torque sensors. Compared with the aforementioned methods [26]–[31], the proposed control does not require the identification of the friction model, nor does it require an expensive joint-torque sensor. There is another method called adaptive friction compensation (AFC) [32], [33]. AFC first tunes proportional-integral-derivative (PID) control as well as possible; it then updates the friction polynomial coefficient regarding the control input as friction-error signal; finally, it adds the estimated friction to the control input to compensate nonlinear friction. AFC is similar to the proposed control in that it does not need friction identification or expensive torque sensor. Hence, a comparison between the proposed method and AFC is given: The proposed control provides a more systematic, easier, and more robust friction-compensation method than AFC. This paper is organized as follows: In Section II, the new impedance control with hard nonlinearities is presented. Section III discusses on the target-impedance design and gain tuning. Section IV presents comparisons between the proposed and other control methods, such as NAC and IFBIC. In Section V, a comparison between the proposed method and AFC is given. Section VI concludes the paper. II. I MPEDANCE C ONTROL W ITH H ARD N ONLINEARITIES

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B. Robot Dynamics and TDE The dynamics equation of an n-DOF robot manipulator in joint-space coordinates is given by ˙ + G(θ) + F + D − τs . τ = M(θ)θ¨ + V(θ, θ)

Introducing a constant matrix M, another expression of (4) is obtained as follows: ˙ θ) ¨ τ = Mθ¨ + N(θ, θ,

These terms can be classified into two categories: soft nonlinearities and hard nonlinearities, which are as follows: ˙ θ) ¨ = S(θ, θ) ˙ + H(θ, θ, ˙ θ) ¨ N(θ, θ,

(7)

˙ = V(θ, θ) ˙ + G(θ) + Fv (θ) ˙ + D − τs S(θ, θ)
 ˙ θ) ¨ = Fc (θ) ˙ + Fst (θ, θ) ˙ + M(θ) − M θ. ¨ H(θ, θ,

(8)

(1)

When the robot is performing free-space motion, sensed torque τs is zero, and (1) is reduced to the well-known error dynamics of motion control; it is expressed by ˙ + KP (θd − θ) = 0 θ¨d − θ¨ + KV (θ˙d − θ)

(9)

Soft nonlinearities are described in terms of mathematically well-defined functions resulting from the manipulator geometry [34]. Theoretically, when L is sufficiently small, these effects can be estimated by TDE and expressed by ˙ ˙ = S(θ, θ) ˙ t−L ∼ ˆ θ) S(θ, = S(θ, θ).

(10)

In practice, the smallest achievable L is the sampling period in digital implementation. A digital control system behaves reasonably close to the continuous system if the sampling rate is faster than 30 times the bandwidth [35]. Hence, with L smaller than this level, these effects can be estimated by TDE. Hard nonlinearities, on the other hand, are mathematically characterized as discontinuous functions [34]. The term [M(θ) − M]θ¨ in (9) should be considered as hard nonlinearities, because the acceleration is directly affected by discontinuous frictions and may be discontinuous. Consequently, this class of nonlinearities are not compatible with the TDE technique ˙ θ). ¨ ˙ θ) ¨ = H(θ, θ, ˙ θ) ¨ t−L = H(θ, θ, ˆ H(θ, θ,

¨ + Bd (θ˙d − θ) ˙ + Kd (θd − θ) + τs = 0. Md (θ¨d − θ)

(5)

˙ θ) ¨ includes all nonlinear terms where N(θ, θ,
 ˙ θ) ¨ = M(θ)−M θ¨ + V(θ, θ) ˙ + G(θ) + F + D − τs . N(θ, θ, (6)

A. Control Objective The control objective is to make a robot achieve the following target-impedance dynamics:

(4)

(11)

C. Control Law Derivation With Hard Nonlinearities Equation (5) can be rewritten using (7) as ˙ + H(θ, θ, ˙ θ) ¨ τ = Mθ¨ + S(θ, θ)

(12)

and the control input is designed as follows: ˙ + H(θ, ˙ θ) ¨ ˆ θ) ˆ τ = Mu + S(θ, θ,

(2)

(13)

where

with KV = M−1 d Bd ; ∆

KP = M−1 d Kd . ∆

(3)

  ˙ ˙ u = θ¨d + M−1 (θ − θ) + B ( θ − θ) + τ K d d d d s . d

(14)

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To gain further insight into the ideal velocity feedback, the integral sliding surface [36] was considered as follows:    ˙ ˙ + K (θ − θ) + B ( θ − θ) dt. τ s = θ¨d − θ¨ + M−1 s d d d d d (22) ˙ and (21) is expressed by Then, (22) reduces to s = θ˙ideal − θ,  τ = τt−L − Mθ¨t−L + M θ¨d + M−1 d    ˙ + Γs . (23) × τs + Kd (θd − θ) + Bd (θ˙d − θ) Fig. 1. Ideal velocity-feedback-compensation concept.

˙ + H(θ, ˙ θ) ¨ is obtained from (10)–(12) and is ˆ θ) ˆ In (13), S(θ, θ, given by ˙ θ) ¨ t−L ˙ + H(θ, ˙ θ) ¨ = S(θ, θ) ˙ t−L + H(θ, θ, ˆ θ) ˆ S(θ, θ, ˙ θ) ¨ t−L = τt−L − Mθ¨t−L . ˙ t−L + H(θ, θ, S(θ, θ) Here, TDE error ε is defined as   −1 ∆ ˙ θ) ¨ − H(θ, θ, ˙ θ) ¨ t−L . H(θ, θ, ε=M

(15) (16)

(17)

Then, with the combination of (10), (12), (13), (15), and (17), ε is expressed as ¨ ε = u − θ.

(18)

Impedance error dynamics is obtained from (14) and (18) as ¨ + Bd (θ˙d − θ) ˙ + Kd (θd − θ) + τs = Md ε. Md (θ¨d − θ) (19) Equation (19) clearly shows the impact of the TDE error on the target impedance dynamics: ε causes the resulting dynamics to deviate from the target impedance dynamics. ˙ To suppress ε, ideal velocity feedback term, Γ · (θ˙ideal − θ), ˙ is introduced, with θideal defined here as    ∆ ˙ ˙ K dt. (θ − θ) + B ( θ − θ) + τ θ˙ideal = θ¨d + M−1 d d d d s d (20) As illustrated in Fig. 1, θ˙ideal is the velocity that would be ˙ on the other hand, achieved if (1) was realized without ε. θ, is the velocity under ε due to hard nonlinearities. Hence, θ˙ − θ˙ideal represents the net effect due to ε; the term Γ · (θ˙ideal − ˙ functions to counteract it. Combining previous formulations, θ) the control law is proposed by τ − Mθ¨ t−L t−L

τ=

canceling soft nonlinearities

   ˙ τs + Kd (θd − θ) + Bd (θ˙d − θ) + M θ¨d + M−1 d 

 injecting target impedance dynamics

+

˙ MΓ(θ˙ideal − θ)

  suppressing the effect of hard nonlinearities

.

(21)

The sliding surface (22) has an initial value of zero (s(t = 0) = 0). The s trajectory represents a time-varying measure of impedance error [15], [36]. According to Barbalat’s Lemma, as the sliding surface s(t) → 0, it can be expected that s˙ (t) → 0, which implies achieving desired impedance (1) as time → ∞; in other words, tracking the ideal velocity is tracking the ideal impedance. It also implies that the robot follows the motion trajectory well in free space according to (2) and (3). Proof of the stability of the control is given in the Appendix. D. Simplicity of the Proposed Control The controller (21) is composed of three elements that have clear meaning: a soft-nonlinearity cancellation element, a hardnonlinearity suppression element, and a target dynamics injection element. The controller is easy to implement because calculations of complex robot dynamics, as well as that of nonlinear friction, are unnecessary. Only two gain matrices, M and Γ, must be tuned for the control law. III. T ARGET -D AMPING D ESIGN AND G AIN T UNING A. Target-Damping Design A critically damped condition is applied to obtain the fastest sliding surface with no overshoot. With free-space motion, τs = 0 in (1), and the critically damped condition of the target dynamics is {Bd }ii = 2 {Md }−1 ii {Kd }ii .

(24)

When the robot comes into contact with the environment, a highly overdamped target dynamics is required to obtain the critically damped response of overall dynamics with high environmental stiffness. If the environmental force/torque is modeled as τs = Ke (θe − θ)

(25)

then (1) can be written as   −1 ¨ ˙ = 0. Ke (θe −θ)+Kd (θd −θ)+Bd (θ˙d − θ) θ¨d − θ+M d (26)

JIN et al.: ROBUST COMPLIANT MOTION CONTROL OF ROBOT WITH NONLINEAR FRICTION USING TDE

The elements of damping Bd should be chosen as {Bd }ii = 2 {Md }−1 ii ({Kd }ii + {Ke }ii ) = 2ζi {Md }−1 ii {Kd }ii .

(27)

Thus, in the viewpoint of the target dynamics, ζ = 1 in free space, and ζ > 1 in constrained space for contact performance. Incidentally, the following judgment condition is used to determine whether the robot is performing free-space motion or constrained space motion: if |{τs }i | ≤ (τsn )i ,

then free space

if |{τs }i | > (τsn )i ,

then constrained space.

(28)

The best performance was obtained by using the diagonal elements of M(θ) directly in simulations without noise. For real systems, the tuning of M is dependent on the diagonal elements of M(θ) and the noise of the system. The effect of the noise may be amplified due to the calculation of θ¨t−L by numerical differentiation introduced in [18]. Fortunately, it is possible to attenuate noise without explicitly using an additional low-pass filter by lowering M as follows. A simpler formulation of (21) is

where

(29)

  ˙ τs + Kd (θd − θ) + Bd (θ˙d − θ) v = θ¨d + M−1 d ˙ + Γ(θ˙ideal − θ).

The ideal velocity feedback gain Γ may be regarded as the slope of the boundary layer of the saturation characteristic approximating the signum function transfer characteristic of the sliding-mode control. In this regard, (23) can be compared with the sliding-mode control (33) and (35), as follows:

  ˙ + Ksgn(s) (33) + Bd (θ˙d − θ) where [sgn(s)]i = sgn(si ). The sliding condition of (33) is expressed by

  −1

˙ θ) ¨

. (34) Kii > M ∆H(θ, θ, i

To reduce the chattering problem, sgn(s) is replaced by a saturation function in (33) as τ = τt−L − Mθ¨t−L 
+ M θ¨d + M−1 d τs + Kd (θd − θ)

(30)

(31)

where τ denotes the input to the filter, and τ f is the output from the filter. Substituting (30) into (31), the following filtered control law is obtained by f + λ (1 + λ )−1 M(v − θ¨t−L ). τ f = τt−L

C. Meaning of Γ

  ˙ + Ksat(s, Φ) + Bd (θ˙d − θ)

If a digital low-pass filter with the cutoff frequency λ is adopted, the control law can be modified as follows: f τ f = λ (1 + λ )−1 τ + (1 + λ )−1 τt−L (λ = λL)

increase Γ as much as possible to make the last term in (21) function effectively. In practice, M can also be tuned without regard to system parameters. It is possible to begin with a small positive initial value of M and, then, increase the diagonal elements to tune the system performance.

τ = τt−L − Mθ¨t−L 
+ M θ¨d + M−1 d τs + Kd (θd − θ)

B. Gain Tuning of M

τ = τt−L − Mθ¨t−L + Mv

261

(32)

Comparing (30) with (32) clearly shows that lowering M has the same effect as using a first-order low-pass filter. In a 2-DOF SCARA robot experiment with a sampling frequency of 1000 Hz, M11 = 0.15 M11 , and M22 = 0.1 M22 were tuned; in other words, the cutoff frequency of the noise was set at approximately 176 Hz for the first joint and 111 Hz for the second joint, according to (32). When M is increased over the above value, there are no further improvements of errors; in contrast, loud sounds from the robot joints were heard. Incidentally, when M is lowered for practical use to filter the noise, the hard-nonlinearity suppression effect of the last term in (21) would be reduced. In this case, one can

(35) where

 sat(s, Φ)i =

sgn(si , Φi ), si /Φi ,

if |si | ≥ Φi . if |si | < Φi

(36)

The proposed control (23) is equivalent to the sliding-mode control (35) in the inside of the boundary layer if we let Γii = Kii /Φii . In the steady state, s of the sliding controller (23) will eventually go inside of the boundary layer to smoothen the control discontinuity. Thus, the best accuracy by (35) is identical to that of the proposed control (23), but the proposed control has less gain than the sliding-mode control (35). As Γs can provide a filterlike structure, tuning Γ involves a tradeoff problem between precision and robustness. The elements of Γ are set with highest possible values without control chatter. IV. E XPERIMENTAL S TUDIES A. Experimental Setup The robot used in the experiment is a 2-DOF SCARAtype robot, as shown in Fig. 2. The lengths of the two links are l1 = 0.35 m and l2 = 0.29 m; and their masses are

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Fig. 2. SCARA-type robot system and environment.

m1 = 11.17 kg and m2 = 6.82 kg. The distance from the joint axis to the center of mass for each link is L1 = 0.30 m and L2 = 0.18 m; the moment of inertia about the joint axis is I1 = 1.0 kg · m2 and I2 = 0.224 kg · m2 . At joint 1, an ac servo motor with a stall torque of 2.39 N · m is used to transmit power through a harmonic drive with a gear-reduction ratio of 100 : 1. At joint 2, a motor with a stall torque of 0.92 N · m is used with a gear-reduction ratio 80 : 1. Each joint has a resolver attached at its shaft to sense the angular displacement with the resolution of 4096 pulses/rev. A force sensor (ATI Gamma SI-130-10) is attached at the end-effector. An aluminum plate with a stiffness of about 150 000 N/m is used for the environment. The implementation of the controller was made in QNX, a real-time operating system, with a sampling frequency of 1 kHz. B. Experiment on 1-DOF Arm The proposed controller was compared with NAC (37), IFBIC (38), and simplified velocity feedback (SVF) (39) for a simple single-DOF robot arm with nonlinear friction using the first joint of the SCARA robot ˙ + Gv (θ˙cmd − θ) ˙ τ = Kd (θd − θ) + Bd (θ˙d − θ) (37)  ˙ where θ˙cmd = Md−1 [τs + Kd (θd − θ) + Bd (θ˙d − θ)]dt. τ = τt−L − Mθ¨t−L 
+ M θ¨d + Md−1 τs + Kd (θd − θ)  ˙ + Bd (θ˙d − θ) .

(38)

˙ τ = Ks (θ˙ideal − θ).

(39)

1) Tasks: The experiment is designed as depicted in Fig. 3: The robot is commanded—as indicated with the dotted line—initially to move very fast in free space repeatedly (t = 0–6 s); then, to make an impact against the environment at time 9 s; finally, to go through the wall (t = 12–20 s). The final task is to emulate a situation in which a position command input is given without details of environment position. The controlled robot is then checked whether it is compliant enough not to harm the environment or the robot itself. A fifth polynomial

Fig. 3. Desired trajectory (the aluminum plate located at 0.15 rad is indicated with a dash-dotted line). TABLE I END POINTS OF THE DESIRED TRAJECTORY

trajectory is used for the 15 path segments listed in Table I, where both the initial position and the final position of each segment are listed along with the initial time and the final time. The velocity and the acceleration at the beginning and end of each segment are set to zero. The aluminum plate located at 0.15 rad is indicated with a dash-dotted line. 2) Target Dynamics and Gains: The identical target impedance was selected for the controllers. For free-space motion, the parameters were selected to obtain critically damped error dynamics as Md = 1 kg · m2 , Bd = 20 N · ms/rad, and Kd = 100 N · m/rad. For constrained motion, the same stiffness is used, but the target damping is set for a highly overdamped condition by tuning to Bd = 120 N · ms/rad. The controller gains are best tuned as Gv = 98.0 N · ms/rad for NAC, M = 0.15 kg · m2 for IFBIC, Ks = 85.0 N · ms/rad for SVF, and M = 0.15 kg · m2 , Γ = 20.0/(rads) for the proposed control. 3) Experimental Result: Experimental results are arranged in Figs. 4 and 5 and Table II. The proposed control shows the smallest error in free space in Fig. 5(a), while NAC, IFBIC, and SVF reveal large tracking errors due to Coulomb friction whenever the arm velocity becomes zero, crossing from positive velocity to negative, or vice versa. The proposed control displays the fastest response to sensed torque and shows the smallest impact torque in Table II (the sensed torque at t = 9 s in Fig. 4). The fastest response comes as a result of the combi˙ TDE immediately cancels nation of TDE and Γ · (θ˙ideal − θ): soft nonlinearity, such as viscous friction, and ideal velocity feedback is immediately activated at the moment of collision, when hard nonlinearities suddenly becomes dominant, owing to the abrupt change of the sign of the velocity. The proposed control shows the smallest impedance error in Fig. 5(b). Compared with NAC, the proposed control cancels soft nonlinearities directly, such as viscous friction. Compared with IFBIC, the proposed control pushes hard to make the robot to

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Fig. 4. Position and force response of the controllers. (a), (c), (e), and (g): Position response of NAC, IFBIC, SVF, and the proposed control, respectively. (b), (d), (f), and (h): Force response of NAC, IFBIC, SVF, and the proposed control, respectively. In free space motion (t = 0–8 s), the proposed control shows the smallest tracking error. Impact occurs at time 9 s, and the proposed control shows the smallest impact torque. In constrained trajectory motion (t = 12–20 s), all of the controllers show similar responses.

follow target dynamics when the robot dynamics contains hard nonlinearities. Compared with SVF, it provides more accurate control. Using (15), (16), and (22), the proposed control (21) can be rewritten as ˙ + H(θ, ˙ θ) ¨ + M˙s + MΓs. ˆ θ) ˆ θ, τ = Mθ¨ + S(θ,

(40)

SVF (39) can be rewritten as τ = Ks s.

(41)

The proposed control has an additional estimation term of nonlinearities of the plants, which reduces the magnitude of the nonlinearities. It also has the derivative term on s, which

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Fig. 5. (a) Comparisons of free-space-motion error. (b) Comparisons of the impedance error. The proposed control shows the smallest tracking error in free space and the smallest impedance error. TABLE II MAXIMUM TRACKING ERRORS AND IMPACT TORQUES

Fig. 6. (a) Interaction torque of different damping parameters. (b) Free-spacemotion error of different gains Γ. (c) Chattering occurs when Γ is very high.

increases the bandwidth and the speed of the response. Consequently, the proposed control shows more robust performance than NAC, IFBIC, and SVF: It shows the smallest error in free-space motion, the smallest impact torque, and the smallest deviation from the target dynamics. The controlled robot is compliant and safe, even when the robot is commanded to go through the wall (t = 12–20 s). The contact force changes according to θd − θwall , and the robot does not move farther, making contact with the environment. All of the controllers have similar responses when the friction effect is not dominant. An inspection of the segment (t = 20–25 s) in Fig. 4 reveals that the stiffness realized by the proposed control is very close to the desired stiffness (K = 4.935/0.049148 = 100.41 N · m/rad). 4) Discussion on Target Dynamics and Gains: The most crucial parameter to contact stability is the damping ratio. In this paper, the target damping is tuned experimentally because the environmental stiffness is unknown. The damping effect is depicted in Fig. 6(a), in which the same tasks are performed by only the damping changed in constrained space using the proposed control. It can be observed that ζ = 6.0(Bd = 120) shows a smooth torque profile and the smallest impact torque, whereas ζ = 3.0(Bd = 60) shows the largest impact torque and a frequent knocking action between the robot and the wall. Increasing the gain of the proposed control reduces the ultimate bound of the error, as shown in Fig. 6(b). A very high gain of Γ, however, will cause the controller to switch across the surface s(t) discontinuously, leading in practice to control chattering. Limit cycles are observed in the sensed torque and the control input in Fig. 6(c) when Γ is too high (Γ = 40).

Fig. 7. The robot end-effector meets stiff environment while drawing a circle. (a) Desired trajectory. (b) Sensed force. (c) Position response of (dashed) NAC, (dash-dot) IFBIC, (dotted) SVF, and the (solid) proposed control, respectively. (d) Impedance error norm (|s| = [(x˙ ideal − x) ˙ 2 + (y˙ ideal − y) ˙ 2 ]0.5 ). The proposed control shows the smallest tracking error in free space and the smallest impedance error.

is shown in Fig. 7(a), in which parts of the Y -trajectory are commanded to go through an aluminum wall. The derivation of the proposed control in operational space is straightforward using the same method introduced in the study in [37]. The final form is described as follows: ¨ t−L ) xd − x Fu = Fu(t−L) + Mx (θ)(¨ + Mx (θ)M−1 d [Kd (xd − x)

C. Meeting Aluminum Wall While Drawing a Circle An application is given through 2-DOF SCARA-type robot in operational space. The robot end-effector is commanded to draw a circle (diameter = 40 mm) in 4 s. The desired trajectory

x˙ ideal

˙ + Fs + Γ(x˙ ideal − x)] ˙ + Bd (x˙ d − x)  ¨ d + M−1 ˙ d − x) ˙ + Fs ] dt x = d [Kd (xd − x) + Bd (x

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Fig. 8. Robot–robot cooperation. (Right) 2-DOF SCARA-type robot is performing an impedance control, and (left) the faraman AT2 robot is performing a position control.

Fig. 10. X − Y position response. (a) NAC. (b) IFBIC. (c) SVF. (d) Proposed control.

Fig. 11. Robot–robot cooperation. (Dashed) Impedance error of NAC, (dashdot) IFBIC, (dotted) SVF, and (solid) the proposed control, respectively.

Fig. 9. Experiment result of the proposed control. (a) Position response. (b) Sensed force.

and τu = JT Fu

(42)

where Mx (θ) = J−T (θ)MJ−1 (θ), and J is Jacobian matrix. The target impedance parameters are chosen as follows: Md = diag(20, 20) kg, and Kd = diag(4500, 2000) N/m. The damping ratio ζ is 1.0 for free-space motion. When the robot makes contact with the wall, the x-axis is not affected by the contact; thus, the x-axis damping ratio is 1.0; the y-axis is constrained, and the y-axis damping ratio is tuned to 6.0 (overdamped). The gains are tuned best as follows: Gv = diag(1960, 1960); Ks = diag(1700, 1700); M = diag(0.15, 0.022); and Γ = diag(400, 400).

The X − Y position responses of the end-effector are shown in Fig. 7(c), where the dash-dotted line indicates the wall. The robot tracks the circle in free space and moves along the stiff aluminum wall when contact occurs at y = 305 mm. In free space, the proposed control shows the smallest error when drawing the circle, whereas NAC shows the largest. In the contact task, the proposed control displays the smallest deviation from the target impedance dynamics in Fig. 7(d). The experiment confirms that the proposed control is also applicable to multi-DOF robots, and that it shows more robust performance than NAC, IFBIC, and SVF. D. Cooperation With the Other Robot The experimental scenario is as follows: The SCARA-type robot end-effector draws a circle in free space and, afterward, stands still before it is pushed by another robot. Then, as shown in Fig. 8, another position-controlled robot (left) pushes and

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Fig. 12. Comparison of the position-control performances. (a) Desired trajectory. (b) Error of joint 1. (c) Error of joint 2. (d) X − Y plot of the position response of the PID. (e) X − Y plot of the position response of the PID with AFC. (f) X − Y plot of the position response of the proposed control.

holds the impedance-controlled robot (right) repeatedly in the y-direction. This is done to simulate a situation in which two robots work together to move or to install an object. In this cooperation, it is often desired for the robot end-effector to move to the task point quickly and correctly and to behave as a low-stiffness spring for a given direction. This is achieved by properly specifying the target dynamics as follows: Md = diag(20, 20) kg, and Kd = diag(2000, 980) N/m. Experimental results are displayed in Figs. 9–11. The robot acts as a soft spring-mass-damper system, and it neither destabilizes the robot nor exerts an excessive force on the other robot. In the free-space task, the proposed control shows the best result when tracking the circle in Fig. 10. In the robot–robot cooperation task, the proposed control shows the smallest x-direction deviation in Fig. 10; and the smallest impedance error in Fig. 11. An inspection of the position and force profiles of the proposed control with the segments (t = 9–12 s and 18–21 s) in Fig. 9 reveals that the proposed control achieved de-

sired stiffness very well when the robots did not move and maintained balance. The actual stiffness in the y-direction is 18.4/0.0189 = 973.5 N/m, which is very close to the desired stiffness of 980 N/m. This experiment illustrates that robot–robot cooperative tasks with suitable compliance can be accomplished successfully under the proposed control.

V. C OMPARISON W ITH AFC In this section, the proposed control is compared with AFC, a simple effective online friction-compensation technique [32], [33], which has recently been regarded as a promising technique as compared to those in the study in [26]–[31]. As shown in Fig. 12(a), the experiment is designed to track a circle (with a radius of 60 mm) four times in 12 s. The tracking speeds of the second and the third circle are much faster than those of the first and fourth circle.

JIN et al.: ROBUST COMPLIANT MOTION CONTROL OF ROBOT WITH NONLINEAR FRICTION USING TDE

To implement AFC, the method described in [32] and [33] was followed exactly. The PID control was tuned well using [38] (the maximum tracking error of joint 1 using the PID is 0.132◦ for low-speed tracking circle, and 0.315◦ for very high-speed tracking). Moreover, AFC is implemented with the PID. Figs. 12(b) and (c) show that the tracking error of the PID with AFC is nearly 30% of that achieved using only a PID control. This result approximately coincides with the experimental results in [32] (in [32], it is 24%). The proposed control, owing to the impedance-control formulation property, becomes a motion-control formulation when the interaction torque τs is omitted in the target dynamics (1); the simplified motion-control formulation is then

suppression of the effect of hard nonlinearities using ideal velocity feedback. The proposed control turns out to be simple in form and simple to tune; specifying desired dynamics to the robot is all that is necessary for a user to do. One can use (43) for position-control purposes and use (21) for impedancecontrol purposes. The proposed control has shown improved robustness in the experiment with an industrial robot having nonlinear friction and is practical for realizing compliant manipulation. The proposed control provides a more systematic, easier, and more robust friction compensation method than AFC. In addition, the proposed method can be generalized to industrial servo systems other than a robot manipulator. The TDE error ε can be suppressed by other compensation methods.

τ = τt−L − Mθ¨t−L + M   ˙ + KP (θd − θ) + Γ(θ˙ideal − θ) ˙ × θ¨d + KV (θ˙d − θ) (43)  ˙ where θ˙ideal = [θ¨d + KP (θd − θ) + KV (θ˙d − θ)]dt. As shown in Fig. 12(b) and (c), the tracking error of the proposed method is the smallest (approximately 10% of that using PID), which confirms that the proposed method is superior to AFC. When tracking the first and fourth circle, the error profile of AFC and the proposed method seems to be at the same level; however, when tracking the second and the third circle at high speed (t = 4–8 s), the proposed method provides the fastest compensation of friction among the three and shows the best circle tracking, as shown in Fig. 12(d)–(f). AFC attempts to adapt the polynomial coefficients of friction. The adaptation takes quite some sampling time to update the friction parameters due to its property of neural network (see [32]); moreover, the neural network is normally slower than TDE, as discussed in [39]. In contrast, the proposed control directly cancels most of the nonlinearities using TDE and immediately activates the ideal velocity feedback when TDE is not sufficient. The proposed method provides a simpler method for the design of the controllers as compared with AFC. For each joint, AFC must tune three gains of the PID by trial and error, as well as two additional parameters of AFC (α and η) (see [32]). In contrast, the proposed control tunes only two gains, and the target dynamics is very easy to design because it is closely related to a spring-mass-damper system. Consequently, the proposed control can provide an easier, more systematic, and more robust method than AFC for friction compensation; thus, the proposed control may be regarded as a simpler and more effective alternative to AFC.

A PPENDIX S TABILITY A NALYSIS The global uniform ultimate boundedness of the overall system with hard nonlinearities is presented. The closed-loop equation of the proposed control can be obtained by combining (10), (12), (16), (17), (21), and (30), as   ˙d − θ) ˙ τ + K (θ − θ) + B ( θ θ¨d − θ¨ + M−1 s d d d d ˙ = ε, + Γ(θ˙ideal − θ)

or

v − θ¨ = ε.

(44)

Using (22), the closed-loop dynamics (44) is expressed by s˙ + Γs = ε.

(45)

If ε is asymptotically bounded, then s and s˙ in (45) are asymptotically bounded, and consequently, the overall system is bounded-input–bounded-output stable. Therefore, the asymptotic boundedness of ε will be discussed henceforth. The boundedness of ε can be proved in the same manner of the stability proof in [22]. Using (4), (44) gives ¨ M(θ)ε = M(θ)(v − θ) ˙ + G(θ)F + D − τs − τ. = M(θ)v + V(θ, θ)

(46)

A combination of (16), (30), (8), and (9) gives ˙ + G(θ) + F + D − τs M(θ)ε = M(θ)v + V(θ, θ) ˙ t−L − H(θ, θ, ˙ θ) ¨ t−L − Mv − S(θ, θ)

 = M(θ) − M v − M(θ)t−L − M θ¨t−L + ∆

 = M(θ) − M v − M(θ) − M θ¨t−L − [M(θ)t−L − M(θ)] θ¨t−L + ∆

VI. C ONCLUSION A simple robust compliant motion control has been proposed and compared with NAC, IFBIC, and SVF. The proposed control enables analysis and compensation nonlinear terms in robot dynamics including friction without modeling them. The proposed control turns out to be robust, owing to the direct cancellation of soft nonlinearities using TDE as well as the

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(47)

where ˙ ˙ ∆ = S(θ, θ)+F c +Fst −S(θ, θ)t−L −Fc(t−L) −Fst(t−L) . (48) As Fc and Fst are bounded, it is clear that ∆ is bounded for a sufficiently small L. Substituting θ¨t−L = vt−L − εt−L from

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(44) into (47) yields

 M(θ)ε = M(θ) − M v + M(θ) − M (εt−L − vt−L ) − [M(θ)t−L − M(θ)] θ¨t−L + ∆  ε = I − M(θ)−1 M εt−L
 + I − M(θ)−1 M (v − vt−L ) + η1


(49) where   η1 = M(θ)−1 [M(θ) − M(θ)t−L ] θ¨t−L + ∆ .

(50)

In the discrete-time domain, this can be represented as
 ε(k) = I − M(k)−1 M ε(k − 1) 
¯ η2 (k) + η1 (k) + I − M(k)−1 M

(51)

where   ¨ − 1) + ∆(k) η1 (k) = M(k)−1 [M(k) − M(k − 1)] θ(k (52) and η2 (k) = [v(k) − v(k − 1)] .

(53)

In (51), η1 (k) and η2 (k), from the viewpoint of ε(k), are considered as forcing function, which are bounded for sufficiently small time-delay L. The first-order difference equation, (51), is asymptotically bounded if roots of [I − M(k)−1 M] reside inside a unit circle. Appropriate values for M should be selected to satisfy the stability. Moreover, if the Lyapunov function is considered as V = ˙ = −sT Γs + sT ε. It can 0.5 sT s, the time derivative is then V be determined that if |si | > di ,

˙ < 0, where di = {Γ−1 ε}i . then V

(54)

Because ε is bounded, it can be concluded that s is globally uniformly ultimately bounded with the ultimate bound |si | ≤ di . Increasing Γ in (54) could reduce the tracking error of the ideal velocity. A smaller tracking error of the ideal velocity implies faster performance when achieving the desired impedance. R EFERENCES [1] S. P. Chan and H. C. Liaw, “Generalized impedance control of robot for assembly tasks requiring compliant manipulation,” IEEE Trans. Ind. Electron., vol. 43, no. 4, pp. 453–461, Aug. 1996. [2] L. Basanez and J. Rosell, “Robotic polishing systems,” IEEE Robot. Autom. Mag., vol. 12, no. 3, pp. 35–43, Sep. 2005. [3] H. Kazerooni, “Automated robotic deburring using impedance control,” IEEE Control Syst. Mag., vol. 8, no. 1, pp. 21–25, Feb. 1988. [4] S. Katsura, J. Suzuki, and K. Ohnishi, “Pushing operation by flexible manipulator taking environmental information into account,” IEEE Trans. Ind. Electron., vol. 53, no. 5, pp. 1688–1697, Oct. 2006. [5] S. Hara, “A smooth switching from power-assist control to automatic transfer control and its application to a transfer machine,” IEEE Trans. Ind. Electron., vol. 54, no. 1, pp. 638–650, Feb. 2007.

[6] G. Grunwald, G. Schreiber, A. Albu-Schaffer, and G. Hirzinger, “Programming by touch: The different way of human-robot interaction,” IEEE Trans. Ind. Electron., vol. 50, no. 4, pp. 659–666, Aug. 2003. [7] Y. Matsumoto, S. Katsura, and K. Ohnishi, “Dexterous manipulation in constrained bilateral teleoperation using controlled supporting point,” IEEE Trans. Ind. Electron., vol. 54, no. 2, pp. 1113–1121, Apr. 2007. [8] S. Katsura and K. Ohnishi, “Semiautonomous wheelchair based on quarry of environmental information,” IEEE Trans. Ind. Electron., vol. 53, no. 4, pp. 1373–1382, Jun. 2006. [9] H. Kazerooni, T. B. Sheridan, and P. K. Houpt, “Robust compliant motion for manipulators—Part I: The fundamental concepts of compliant motion,” IEEE J. Robot. Autom., vol. RA-2, no. 2, pp. 83–92, Jun. 1986. [10] N. Hogan, “Impedance control: An approach to manipulator. Part I—Theory,” Trans. ASME J. Dyn. Syst. Meas. Control, vol. 107, no. 1, pp. 1–7, Mar. 1985. [11] N. Hogan, “Impedance control: An approach to manipulator. Part II— Implementation,” Trans. ASME J. Dyn. Syst. Meas. Control, vol. 107, no. 1, pp. 8–16, Mar. 1985. [12] N. Hogan, “Impedance control: An approach to manipulator. Part III— Applications,” Trans. ASME J. Dyn. Syst. Meas. Control, vol. 107, no. 1, pp. 17–24, Mar. 1985. [13] R. Kelly, R. Carelli, M. Amestegui, and R. Ortega, “Adaptive impedance control of robot manipulators,” Int. J. Robot. Autom., vol. 4, no. 3, pp. 134–141, 1989. [14] B. Yao, S. P. Chan, and D. Wang, “Unified formulation of variable structure control schemes for robot manipulators,” IEEE Trans. Autom. Control, vol. 39, no. 2, pp. 371–376, Feb. 1994. [15] Z. Lu, S. Kawamura, and A. A. Goldenburg, “An approach to slidingmode based control,” IEEE Trans. Robot. Autom., vol. 11, no. 5, pp. 754– 759, Oct. 1995. [16] K. Youcef-Toumi and O. Ito, “A time delay controller design for systems with unknown dynamics,” Trans. ASME J. Dyn. Syst. Meas. Control, vol. 112, no. 1, pp. 133–142, Mar. 1990. [17] K. Youcef-Toumi and S. T. Wu, “Input/output linearization using time delay control,” Trans. ASME J. Dyn. Syst. Meas. Control, vol. 114, no. 1, pp. 10–19, 1992. [18] T. C. Hsia, “A new technique for robust control of servo systems,” IEEE Trans. Ind. Electron., vol. 36, no. 1, pp. 1–7, Feb. 1989. [19] T. C. Hsia, T. A. Lasky, and Z. Guo, “Robust independent joint controller design for industrial robot manipulators,” IEEE Trans. Ind. Electron., vol. 38, no. 1, pp. 21–25, Feb. 1991. [20] T. A. Lasky and T. C. Hsia, “On force-tracking impedance control of robot manipulators,” in Proc. IEEE ICRA, Apr. 1991, pp. 274–280. [21] R. C. Bonitz and T. C. Hsia, “Internal force-based impedance control for cooperating manipulators,” IEEE Trans. Robot. Autom., vol. 12, no. 1, pp. 78–89, Feb. 1996. [22] S. Jung, T. C. Hsia, and R. G. Bonitz, “Force tracking impedance control of robot manipulators under unknown environment,” IEEE Trans. Control Syst. Technol., vol. 12, no. 3, pp. 474–483, May 2004. [23] B. Armstrong, “Friction: Experimental determination, modeling, and compensation,” in Proc. IEEE ICRA, Apr. 1988, vol. 3, pp. 1422–1427. [24] W. S. Newman, “Stability and performance limits of interaction controllers,” Trans. ASME J. Dyn. Syst. Meas. Control, vol. 114, no. 4, pp. 563–570, Dec. 1992. [25] M. Dohring and W. S. Newman, “The passivity of natural admittance control implementations,” in Proc. IEEE ICRA, Sep. 2003, vol. 3, pp. 3710–3715. [26] Z.-Q. Mei, Y.-C. Xue, and R.-Q. Yang, “Nonlinear friction compensation in mechatronic servo systems,” Int. J. Adv. Manuf. Technol., vol. 30, no. 7/8, pp. 693–699, Oct. 2006. [27] B. Bona, M. Indri, and N. Smaldone, “Rapid prototyping of a model-based control with friction compensation for a direct-drive robot,” IEEE/ASME Trans. Mechatronics, vol. 11, no. 5, pp. 576–584, Oct. 2006. [28] G. Liu, A. A. Goldenberg, and Y. Zhang, “Precise slow motion control of a direct-drive robot arm with velocity estimation and friction compensation,” Mechatronics, vol. 14, no. 7, pp. 821–834, Sep. 2004. [29] L. Marton and B. Lantos, “Modeling, identification, and compensation of stick-slip friction,” IEEE Trans. Ind. Electron., vol. 54, no. 1, pp. 511– 521, Feb. 2007. [30] W.-F. Xie, “Sliding-mode-observer-based adaptive control for servo actuator with friction,” IEEE Trans. Ind. Electron., vol. 54, no. 3, pp. 1517– 1527, Jun. 2007. [31] F. Aghili and M. Namvar, “Adaptive control of manipulators using uncalibrated joint-torque sensing,” IEEE Trans. Robot., vol. 22, no. 4, pp. 854– 860, Aug. 2006.

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[32] A. Visioli, R. Adamini, and G. Legnani, “Adaptive friction compensation for industrial robot control,” in Proc. IEEE/ASME Int. Conf. AIM, Jul. 2001, vol. 1, pp. 577–582. [33] F. Jatta, G. Legnani, and A. Visioli, “Friction compensation in hybrid force/velocity control of industrial manipulators,” IEEE Trans. Ind. Electron., vol. 53, no. 2, pp. 604–613, Apr. 2006. [34] J. J. Gonzalez and G. R. Widmann, “Investigation of nonlinearities in the force control of real robots,” IEEE Trans. Syst., Man, Cybern., vol. 22, no. 5, pp. 1183–1193, Sep./Oct. 1992. [35] G. F. Franklin, J. Powell, and M. Workman, Digital Control of Dynamic Systems. Reading, MA: Addison-Wesley, 1998. [36] J.-J. E. Slotine, “Robustness issues in robot control,” in Proc. IEEE ICRA, 1985, vol. 3, pp. 656–661. [37] P. H. Chang, D. S. Kim, and K. C. Park, “Robust force/position control of a robot manipulator using time-delay control,” Control Eng. Pract., vol. 3, no. 9, pp. 1255–1264, Sep. 1995. [38] P. H. Chang and J. H. Jung, “Method for tuning PID controllers applicable to nonlinear systems,” U.S. Patent 6 937 908, Aug. 30, 2005. [39] J.-W. Lee and J.-H. Oh, “Time delay control of nonlinear systems with neural network modeling,” Mechatronics, vol. 7, no. 7, pp. 613–640, Oct. 1997.

Maolin Jin (S’05) was born in Helong, Jilin, China, in 1976. He received the B.S. degree from Yanbian University of Science and Technology, Jilin, in 1999 and the M.S. degree from Korea Advanced Institute of Science and Technology, Daejeon, Korea, in 2004, where he is currently working toward the Ph.D. degree in mechanical engineering. His research interests include robust control of nonlinear plants, robot impedance control, and firefighting robots. Mr. Jin is a Student Member of The Institute of Control, Robotics and System Engineers (ICROS).

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Sang Hoon Kang received the B.S. and M.S. degrees in mechanical engineering from Korea Advanced Institute of Science and Technology, Daejeon, Korea, in 2000 and 2002, respectively, where he is currently working toward the Ph.D. degree in mechanical engineering. His research interests include robust control of nonlinear systems, rehabilitation robots, interaction control, impedance control, and impact control. Mr. Kang is a Student Member of The Korean Society of Mechanical Engineers (KSME).

Pyung Hun Chang (S’86–M’89) received the B.S. and M.S. degrees in mechanical engineering from Seoul National University, Seoul, Korea, in 1974 and 1977, respectively, and the Ph.D. degree in mechanical engineering from the Massachusetts Institute of Technology, Cambridge, MA, in 1987. Since 1987, he has been with the Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daejeon, Korea, where he is currently a Professor. His research interests include the control of redundant robots, robust control of nonlinear plants, observer-based controls, and input-shaping technique. Dr. Chang is a member of American Society of Mechanical Engineers, KSME, and The Korea Society for Precision Engineering (KSPE).