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- 1. ISA Transactions 51 (2012) 74–80 Contents lists available at SciVerse ScienceDirect ISA Transactions journal homepage: www.elsevier.com/locate/isatrans Bilateral control of master–slave manipulators with constant time delay A. Forouzantabara,∗ , H.A. Talebib,1 , A.K. Sedighc,2 a Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran b Department of Electrical Engineering, AmirKabir University of Technology, Tehran, Iran c Department of Electrical Engineering, Khaje Nasir Toosi University of Technology, Tehran, Iran a r t i c l e i n f o Article history: Received 6 November 2010 Received in revised form 8 July 2011 Accepted 27 July 2011 Available online 20 August 2011 Keywords: Robotic Time delay Passivity Teleoperation Position coordination Transparency a b s t r a c t This paper presents a novel teleoperation controller for a nonlinear master–slave robotic system with constant time delay in communication channel. The proposed controller enables the teleoperation system to compensate human and environmental disturbances, while achieving master and slave position coordination in both free motion and contact situation. The current work basically extends the passivity based architecture upon the earlier work of Lee and Spong (2006) [14] to improve position tracking and consequently transparency in the face of disturbances and environmental contacts. The proposed controller employs a PID controller in each side to overcome some limitations of a PD controller and guarantee an improved performance. Moreover, by using Fourier transform and Parseval’s identity in the frequency domain, we demonstrate that this new PID controller preserves the passivity of the system. Simulation and semi-experimental results show that the PID controller tracking performance is superior to that of the PD controller tracking performance in slave/environmental contacts. © 2011 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction Over the past 3 decades, teleoperation technologies have been gradually growing through the world. Teleoperation is used in many applications such as space operation [1], handling of toxic and harmful materials [2], robotic surgery [3] and underwater exploration [4]. Teleoperation can be divided into two main categories, namely, unilateral and bilateral. In unilateral teleoperation, the contact force feedback is not transmitted to the master. In bilateral teleoperation, the remote environment provides some necessary information by many different forms, including audio, visual displays, or tactile through the feedback loop to the master side. However, the contact force feedback (haptic feedback) can provide a better sense of telepresence and as a consequence improve task performances [5]. There are many structures for the bilateral teleoperation sys- tem. Two main structures are two-channel (2CH) architecture [6] and four-channel (4CH) architecture [7,8]. In two-channel struc- ture usually the master position is sent to the slave controller, and the contact force of the slave robot with the environment is directly transmitted to the master. ∗ Corresponding author. Tel.: +98 2144865100; fax: +98 9177013735. E-mail addresses: a.forouzantabar@srbiau.ac.ir, Ahmad.foruzan@gmail.com (A. Forouzantabar), alit@aut.ac.ir (H.A. Talebi), sedigh@kntu.ac.ir (A.K. Sedigh). 1 Tel.: +98 2164543340. 2 Tel.: +98 218846 2175x317. In bilateral teleoperation, there are two main objectives that ensure a close coupling between the human operator/master robot and slave robot. The first goal is that the slave robot tracks the position of the master robot and the other is that the force, that occurs when the slave contacts with the remote environment, accurately transferred to the master. When these conditions are met, the bilateral teleoperation system is called a transparent system. Lawrence [8] has shown that there is a tradeoff between the stability and transparency, the improvement of one will deteriorate the other. The delay existing in the network teleoperation system can destabilize the closed-loop system and degrade transparency. Most previous studies on stability were based on the passivity formalism, such as scattering theory [9] and wave variables [10]. The key point for these approaches is to passify the non-passive communication medium with time delay. Although transparency of these two approaches is poor, the stability is robust against the communication delay and called the delay-dependent stability. A comprehensive survey on the delay compensation methods can be found in [11]. Chopra and Spong [12] proposed a new architecture which builds upon the scattering theory by using additional position control on both the master and slave sides. This new architecture has an improved position tracking and comparable force tracking abilities than the traditional teleoperator model of [9,10]. In [13], Lee and Spong, introduced a PD-based controller scheme for the teleoperation system that keeps position coordination 0019-0578/$ – see front matter © 2011 ISA. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.isatra.2011.07.005
- 2. A. Forouzantabar et al. / ISA Transactions 51 (2012) 74–80 75 and ensures the passivity of the closed-loop system. The main drawback of this structure is that the backward and forward communication delays must be exactly known and symmetric. Therefore, they removed these aforementioned restrictions in their recent works. They used the controller passivity concept, the Lyapunov–Krasovskii technique, and Parseval’s identity, to passify the combination of the delayed communication and control blocks altogether since, the delays are finite constants and an upper bound for the round-trip delay is known [14,15]. Nuno et al. [16] showed that it is possible to control a bilateral teleoperation with a simple PD-like controller and achieve stable behaviour under specific condition on control parameters. According to the complexity of the communication network, the backward and forward delays are not only time-varying but also asymmetric. In [17,18], two different methods based on the PD controller have been presented to address these problems. The method in [18] uses a Lyapunov–Krasovskii functional to derive the delay-dependent stability criteria, which is given in the linear- matrix-inequality (LMI) form. Ryu et al. [19,20] proposed a passive bilateral control scheme for the teleoperation system with time- varying delay, which is composed of a Passivity Observer and a Passivity Controller. This controller guarantees the passivity of bilateral teleoperation under some condition, independent of the amount and variation of time-delay in communication channel. In [21], the authors also extended the previously proposed controller in [14]. The main difference is the use of stable neural network in each side to approximate the unknown nonlinear functions in the robot dynamics and enhance the master–slave tracking performance in the face of different initial conditions and environmental contacts. The new neural network controller preserved the passivity of the overall system. In this paper, the passivity based architecture of [14] is ex- tended to improve position tracking and consequently trans- parency in the face of environmental contacts. In this regard, a PID controller is employed in each side to overcome some limitations of PD controllers such as disturbance rejection. The key feature of the proposed PID controller is that it preserves the control passivity of the teleoperation system. For this purpose, we will use Fourier transform, Parseval’s identity, and the Schur complement to show that the proposed PID controller with additional dissipation term will preserve the passivity of the system under some mild condi- tion, since the time delays are constant. The majority of the passiv- ity demonstration is done in the frequency domain. The rest of this paper is organized as follows. Section 2 describes passive bilateral teleoperation structure with constant time delay introduced by Lee and Spong [14]. In Section 3, we describe the new control architecture based on the PID controller and demonstrate its passivity. Furthermore, we used a Nicosia observer [23] to estimate the human hand forces when the master does not have force sensor. Section 4 shows the simulation and experimental results. And finally Section 5 draws conclusions and gives some suggestions for future works. 2. Passive bilateral teleoperation structure with constant time delay In [14], a novel control framework for bilateral teleoperation of a pair of multi-DOF nonlinear robotic systems with constant communication delays was proposed. The proposed bilateral teleoperation framework is shown in Fig. 1. A bilateral teleoperation system which is shown in Fig. 1 consists of five interacting subsystems: the human operator, the master manipulator, the control and communication medium, the slave manipulator and the environment. The human operator commands via a master manipulator by applying a force F1(Fh) to move it with position q1 and velocity ˙q1 which is transmitted to the slave manipulator through the communication medium. A local control (T2) on the slave side drives the slave position q2 and velocity ˙q2 towards the master position and velocity. If the slave contacts a remote environment and/or some external source, the remote force F2(−Fe) is sent back from the slave side and received at the master side as the force or control signal T1. Assuming the absence of friction, gravitational forces and other disturbances, the equation of motion for a master and slave nonlinear robotic system is given as follows [14] M1(q1)¨q1 + C1(q1, ˙q1)˙q1 = T1(t) + F1(t) (1) M2(q2)¨q2 + C2(q2, ˙q2)˙q2 = T2(t) + F2(t) (2) where qi(t) ∈ ℜn are the vector of joint displacements, ˙qi(t) ∈ ℜn are the vector of joint velocities, Ti(t) ∈ ℜn are the control signals, Fi(t) ∈ ℜn represent the human/environmental force, Mi(qi) ∈ ℜn×n are symmetric and positive-definite inertia matrices and Ci(qi, ˙qi) ∈ ℜn×n are the Coriolis/centripetal matrices (i = 1, 2). The Lagrangian robot dynamics enjoys certain fundamental properties [22]. • Property 1: The inertia matrix M(q) is symmetric, positive definite and bounded so that µ1I ≤ M(q) ≤ µ2I where the bounds µi, i = 1, 2 are positive constants. • Property 2: The Coriolis/centripetal matrices can always be selected such that the matrix ˙M(q) − 2C(q, ˙q) is skew- symmetric. The control objectives are designing the controllers Ti(t), i = 1, 2 to achieve these two goals: I. master–slave position coordination: if (F1(t), F2(t)) = 0, qE (t) := q1(t) − q2(t) → 0, t → ∞ (3) II. static force reflection: with (¨q1(t), ¨q2(t), ˙q1(t), ˙q2(t)) → 0 F1(t) → −F2(t) or Fh(t) → Fe(t). (4) The closed-loop teleoperator (1)–(2) is said to satisfy the energetic passivity condition if there exists a finite constant d ∈ ℜ such that for t ≥ 0: ∫ t 0 FT 1 (θ)˙q1(θ) + FT 2 (θ)˙q2(θ) dθ ≥ −d2 . (5) The teleoperator controllers hold the controller passivity if there exists a finite constant c ∈ ℜ such that for t ≥ 0: ∫ t 0 TT 1 (θ)˙q1(θ) + TT 2 (θ)˙q2(θ) dθ ≤ c2 . (6) This means that the energy generated by the master and slave controllers is always bounded [15]. It is shown that for teleoperation system (1)–(2), controller passivity (6) indicate energetic passivity (5). This allows us to investigate the passivity of the closed-loop teleoperator just by checking the controller structure and with no concern about the nonlinear dynamics of the master and slave robots [14]. The following PD-like controllers are proposed to guarantee the master–slave coordination (3), bilateral force reflection (4), and energetic passivity (5). T1(t) = −Kv(˙q1(t) − ˙q2(t − τ2)) − (Kd + Pε)˙q1(t) − Kp(q1(t) − q2(t − τ2)) (7) T2(t) = −Kv(˙q2(t) − ˙q1(t − τ1)) − (Kd + Pε)˙q2(t) − Kp(q2(t) − q1(t − τ1)) (8) where τ1, τ2 ≥ 0 are the finite constant communication delays, Kυ, Kp ∈ ℜn×n are the symmetric and positive-definite PD gains,
- 3. 76 A. Forouzantabar et al. / ISA Transactions 51 (2012) 74–80 Fig. 1. The schematic of the closed-loop teleoperation system [14]. Pε ∈ ℜn×n is an additional damping that guarantee master–slave coordination (3), and Kd ∈ ℜn×n is the dissipation to passify the proportional term in the master and slave controllers. One of the options for Kd is Kd = ¯τrt 2 Kp (9) where ¯τrt ≥ 0 is an upper-bound of the round-trip communication delay which is the sum of the forward (τ1) and backward (τ2) delays. 3. The proposed controller scheme In order to eliminate the steady-state tracking error and improve the disturbance rejection when the slave contact to the environment, we extend a PD controller to a PID controller. The integral term is added so that the system will have no steady- state error in the presence of constant disturbances. The controller design is mainly based on analysing the energy production of the PID controller gains in frequency domain and passifies the effect of these gains. In sequel, the passivity of the presented control methodology is illustrated through rigorous mathematical analysis. Adding an integral term and additional dissipation term Kd2 to (7), (8) control law, the PID control law can be introduced as T′ 1(t) = −Kv(˙q1(t) − ˙q2(t − τ2)) − Kp(q1(t) − q2(t − τ2)) − Ki ∫ t 0 (q1(t) − q2(t − τ2))dt − (Kd + Kd2 + Pε)˙q1(t) (10) T′ 2(t) = −Kv(˙q2(t) − ˙q1(t − τ1)) − Kp(q2(t) − q1(t − τ1)) − Ki ∫ t 0 (q2(t) − q1(t − τ1))dt − (Kd + Kd2 + Pε)˙q2(t). (11) The following theorem shows that the two local controllers T′ 1(t), T′ 2(t) in (10), (11) guarantee energetic passivity (5) of the closed-loop teleoperation system under the condition on Kd2 which is given in (12). Note that Kp, Kv, Ki, Kd and Kd2 are positive-definite diagonal matrices. Theorem 1. Consider the nonlinear bilateral teleoperation system given by (1), (2) with the controllers (10), (11). If condition (9) is hold with the gain Kd2 satisfying Kd2 ≥ 2 cos2 ω(τ1+τ2) 4 ω2 Ki, ∀ω. (12) Then the closed-loop teleoperation system is energetic passive. Proof. The procedure of the passivity proof is similar to that presented in [14] with the difference that our controller has extra integral and dissipation terms. Using Fourier transform, Parseval’s identity and under some mild condition on integral and dissipation gains, it was shown that the presented controller preserve system passivity. Substituting (10), (11) in the definition of controller passivity (6), we obtain ∫ t 0 T /T 1 (θ)˙q1(θ) + T /T 2 (θ)˙q2(θ) dθ ≤ c2 ∀t ≥ 0. (13) Indicating the terms in integral (13) by Pt (t) and substituting from (10), (11) yield Pt (t) := Pd(t) + Pp(t) + Pi(t) − P(t) (14) where Pd(t), Pp(t) and Pi(t) are the powers related to the delayed D-action, delayed P-action (+ dissipation Kd) and delayed I-action (+ dissipation Kd2), respectively and defined by Pd(t) := −˙qT 1 (t)Kv ˙q1(t) + ˙qT 1 (t)Kv ˙q2(t − τ2) − ˙qT 2 (t)Kv ˙q2(t) + ˙qT 2 (t)Kv ˙q1(t − τ1) (15) PP (t) := −˙qT 1 (t)Kd ˙q1(t) − ˙qT 1 (t)Kp(q1(t) − q2(t − τ2)) − ˙qT 2 (t)Kd ˙q2(t) − ˙qT 2 (t)Kp(q2(t) − q1(t − τ1)) (16) Pi(t) := −˙qT 1 (t)Kd2˙q1(t) − ˙qT 1 (t)Ki ∫ t 0 (q1(t) − q2(t − τ2))dt − ˙qT 2 (t)Kd2 ˙q2(t)− ˙qT 2 (t)Ki ∫ t 0 (q2(t) − q1(t − τ1))dt (17) and P(t) is the following positive-definite quadratic form: P(t) := ˙q1(t) ˙q2(t) T [ Pε 0 0 Pε ] ˙q1(t) ˙q2(t) . (18) It is shown in [14] that ∫ t 0 Pd(θ)dθ ≤ −Vv(t) + Vv(0) (19) where Vv(t) ≥ 0 ∀t ≥ 0 is a Lyapunov–Krasovskii functional for delayed systems defined by Vv(t) := 2− i=1 1 2 ∫ 0 −τi ˙qT i (t + θ)Kv ˙qi(t + θ)dθ ≥ 0. (20) Moreover, if condition (9) is satisfied, it is shown in [14] that ∫ t 0 Pp(θ)dθ ≤ −Vp(t) + Vp(0) (21) where VP (t) ≥ 0 ∀t ≥ 0 is defined as Vp(t) := 1 2 qT E (t)KpqE (t) (22) with qE (t) = q1(t) − q2(t).
- 4. A. Forouzantabar et al. / ISA Transactions 51 (2012) 74–80 77 Considering the fact that qi(t), ˙qi(t) = 0 ∀t ∈ (−∞, 0] then, the energy generation by Pi, the power in (17), can be written as ∫ t 0 Pi(λ)dλ = − ∫ ∞ −∞ ˙qT 1 (λ)Kd2˙q1(λ)dλ − ∫ ∞ −∞ ˙qT 2 (λ)Kd2 ˙q2(λ)dλ − ∫ ∞ −∞ ˙qT 1 (λ) [ Ki ∫ t −∞ (q1(ξ) − q2(ξ − τ2))dξ ] dλ − ∫ ∞ −∞ ˙qT 2 (λ) [ Ki ∫ t −∞ (q2(ξ) − q1(ξ − τ1))dξ ] dλ. (23) Now, denoting the Fourier transforms of qi(t), (i = 1, 2) by Vi(jω) = ∞ −∞ qi(t)e−jωt dt = t 0 qi(t)e−jωt dt, and using Parseval’s identity, we have ∫ ∞ −∞ ˙qT 1 (λ)Kd2 ˙q1(λ)dλ = 1 2π ∫ ∞ −∞ (−jω)V∗ 1 (jω)Kd2(jω)V1(jω)dω = 1 2π ∫ ∞ −∞ (ω2 )V∗ 1 (jω)Kd2V1(jω)dω (24) ∫ ∞ −∞ ˙qT l (λ) [ Ki ∫ t −∞ (ql(ξ) − qk(ξ − τk))dξ ] dλ = 1 2π ∫ ∞ −∞ (−jω)V∗ l (jω)Ki 1 jω Vl(jω) + πVl(0)δ(ω) − e−jωτk jω Vk(jω) + πVk(0)δ(ω) dω = − 1 2π ∫ ∞ −∞ V∗ l (jω)Ki[Vl(jω) − e−jωτk Vk(jω)]dω, (l, k) = {(1, 2), (2, 1)} (25) where V∗ i is the complex conjugate transpose of a complex vector Vi. The last equality is obtained by using the fact that ωδ(ω) = 0. Then, substituting (24), (25) in (23) we have ∫ t 0 si(λ)dλ = − 1 2π ∫ ∞ −∞ ω2 V∗ 1 (jω)Kd2V1(jω)dλ − 1 2π ∫ ∞ −∞ ω2 V∗ 2 (jω)Kd2V2(jω)dλ − 1 2π ∫ ∞ −∞ V∗ 1 (jω)Ki e−jωτ2 V2(jω) − V1(jω) dω − 1 2π ∫ ∞ −∞ V∗ 2 (jω)Ki e−jωτ1 V1(jω) − V2(jω) dω = − 1 2π ∫ ∞ −∞ [ ¯V1(jω) ¯V2(jω) ]T H(jω) [ ¯V1(jω) ¯V2(jω) ] dω (26) where, H(jω) ∈ C2n×2n is given by H(jω) = ω2 Kd2 − Ki Ki 2 (ejωτ1 + e−jωτ2 ) Ki 2 (e−jωτ1 + ejωτ2 ) ω2 Kd2 − Ki; (27) for more details see [14]. Since H(jω) is Hermitian, then using the Schur complement, H(jω) is a positive-semidefinite matrix if and only if ω2 Kd2 − Ki ≥ 0 → ω2 Kd2 ≥ Ki (28) and (ω2 Kd2 − Ki) ≥ (e−jωτ1 + ejωτ2 ) 2 (ejωτ1 + e−jωτ2 ) 2 Ki(ω2 Kd2 − Ki)−1 Ki = 1 + cos ω(τ1 + τ2) 2 Ki(ω2 Kd2 − Ki)−1 Ki (29) which is always true if Kd2 ≥ 2 cos2 ω(τ1+τ2) 4 ω2 Ki. (30) Selecting the gain according to (30) guarantees that t 0 Pi(λ)dλ is semi-negative. Since Vv(t) ≥ 0, Vp(t) ≥ 0 and P(t) ≥ 0, ∀t ≥ 0 therefore, by considering inequalities (19), (21) with (18), we have ∫ t 0 T′T 1 (θ)˙q1(θ) + T′T 2 (θ)˙q2(θ) ≤ −Vv(t) + Vv(0) − Vp(t) + Vp(0) − ∫ t 0 P(θ)dθ ≤ Vv(0) + Vp(0) =: c2 (31) with c a finite constant. Thus the controller passivity is proved. Finally, from the fact that, for teleoperation system (1)–(2), con- troller passivity (6) results energetic passivity [14], energetic pas- sivity (5) of the closed-loop teleoperation system is demonstrated. It is also concluded that, if (¨q1(t), ¨q2(t), ˙q1(t), ˙q2(t)) → 0, then from the master–slave robot dynamics (1), (2) and their controls (10), (11), we get F1(t) → −F2(t) → −Kp(q1(t) − q2(t)) − Ki t 0 (q1(t) − q2(t))dt where ˙qi(t − τi) → 0 and qi(t − τi) → qi(t). 3.1. Hand force observer In our master–slave system, master is a two-DOF joystick and has uncoupled motions about the two axes due to its gimbal-based design. In the experiment, the measurements of hand/master forces F1(Fh) are required. Because our master do not have a force sensor, we use a nonlinear state observer to estimate F1(Fh). Now, choosing x1 = q1 and x2 = ˙q1, we write the master dynamics in state-space as ˙x1 = x2 ˙x2 = M−1 1 (q1)(−C1(x1, x2)x2 + T1(t) + F1(t)). (32) The Nicosia observer, which is used to estimate the hand forces F1 or joint velocity ˙q1, described by [23,24]: ˙ˆx1 = ˆx2 + k2e ˙ˆx2 = M−1 1 (x1)(−C1(x1, ˙ˆx1)˙ˆx1 + T1(t) + K1e) e = x1 − ˆx1 (33) where ˆx1 and ˆx2 are the estimated joint position and velocity, e is the output observation error, k2 is a positive scalar constant and K1 is a symmetric positive-definite matrix. This nonlinear observer uses joint position and the portion of the joint torque which comes from the controller to estimate the external applied joint force. It is shown in [23] that the observer is asymptotically stable and the error dynamics is: M1 ¨e + k2M1 ˙e + K1e = F1. (34) In steady state, ¨e = ˙e = 0. In this result, at low frequency operation or in steady-state the hand force is estimated as proportional relationship to position estimation error as ¯F1 = K1e.
- 5. 78 A. Forouzantabar et al. / ISA Transactions 51 (2012) 74–80 Fig. 2. The master and the slave joint positions with the PID controller for 2-DOF robots. 4. Simulation and experiment 4.1. Simulation In order to evaluate the effectiveness of the proposed control scheme in this paper, the controller has been applied to a pair of 2- link planar RR robot arm. The robot dynamics was given in [25]. The lengths of serial links for both master and slave robots are considered as a1 = a2 = 1.3 m, and inertia of the links was taken as m1 = 0.8, m2 = 2.3 kg. To evaluate the system’s contact behaviour, a virtual soft ball is installed in the slave environment at x = 0.5. The ball is modelled like a spring–damper system with the spring and damping gains as 100 N/m and 0.2 Ns/m. The controller gains are chosen such that the close loop system behaves as a critical damping system. In this regard, the gains Kp, Kv and Ki are selected Kv = 40I2×2, Kp = Kv 2 2 = 400I2×2, Ki = I2×2 and the Pε is also set to zero. Note that, Ki has been chosen small so that the third-order error dynamics is close to the second-order error dynamics without this term (i.e., a dominant pole analysis can be performed). The delays were selected as τ1 = τ2 = 1 s. Then, according to condition (9), (12) we choose (Kd, Kd2) = (400I2×2, 5I2×2). Because the observer should have fast response, the observer gains are taken as (K1, k2) = (450I2×2, 45). The scenario is set up such that the operator moves the master robot from the home position in x-direction to (x, y) = (1, 0) in the Cartesian space and returns it to imitate a sinusoidal input. The slave trying to follow these commands is steered towards an obstacle. As a result, force information is generated and sent back to the master. The simulation result for the two joint positions and force tracking are shown in Figs. 2 and 3. The force and position tracking and consequently transparency are satisfactory. The master position in the Cartesian space and the slave contact force is also shown in Fig. 4. As shown in Figs. 3 and 4, when the slave robot contact with the ball at x = 0.5, the contact force is reflected to the master. Moreover, when the slave robot moves in free motion the interaction force is zero. 4.2. Experiment In the experiment, a two-DOF Logitech joystick with force feedback is used as the master. The joystick has uncoupled motions about the two axes due to its gimbal-based design. We use joystick only in x-direction. The slave, which is constructed as a virtual robot, is a 1_DOF robot with prismatic joint. The slave dynamics is Fig. 3. The estimation of human force F1 and the environmental force F2 with the PID controller for 2-DOF robots. Fig. 4. The master position in the Cartesian space and the slave contact force. assumed to be M2 = 0.7 kg, C2 = 2 Ns/m. The communication delays and the environment model are the same as before. The virtual slave robot and a ball as environment placed in front of it, is shown in Fig. 5. The integration of the master–slave teleoperation is accomplished through the MATLAB Simulink environment. The scenario is set up such that the operator moves the joystick away from the home position at x = 0 to the position x = 0.8 and returns it to imitate a step input. The slave trying to follow these commands is steered towards an obstacle. As a result, force information is generated and sent back to the master. The controller gains are chosen such that the close loop system behaves as a critical damping system. For this purpose, the control parameters are chosen as Kv = 10, Kp = Kv 2 2 = 25, Ki = 1 and the Pε is also set to zero. Then, According to (9), (12), (Kd, Kd2) = (25, 5) are chosen. The observer gains are taken as (K1, k2) = (250, 100). The experimental results for position and force tracking are shown in Figs. 6 and 7. To study the contribution of the proposed controller, we implement the conventional controller (PD) (7), (8) with the same control parameters. Figs. 8 and 9 show the results. The position tracking performance is not good. However force tracking is slightly better for the PID controller. To sum up, it is clear that the proposed PID controller makes a significant improvement in position tracking and rejects the constant disturbance (environmental contact) as a result, transparency is improved.
- 6. A. Forouzantabar et al. / ISA Transactions 51 (2012) 74–80 79 Fig. 5. Virtual slave robot with a ball as environment in free motion and contact situation. Fig. 6. The master and the slave positions with the PID controller for 1-DOF robots. Fig. 7. The estimated human force F1 and the environmental force F2 with the PID controller for 1-DOF robots. 5. Conclusions A novel PID control architecture for the bilateral teleoperation system with time delay has been proposed in this paper that ensures position coordination and static force reflection in the presence of disturbances and environmental contacts. The new architecture extends a PD controller to a PID controller that allows the teleoperation system to reject the constant disturbance. We have shown that the new PID controller preserve the energetic passivity under the condition on additional dissipation term. Since the master does not have a force sensor, a state observer is used Fig. 8. The master and the slave positions with the PD controller for 1-DOF robots. Fig. 9. The estimated human force F1 and the environmental force F2 with the PD controller for 1-DOF robots. to estimate the human force. The new controller provides better transparency which is measured in terms of position and force tracking ability of the bilateral system. Evaluating the performance of this controller architecture with real slave robot remains for future work. References [1] Imaida T, Yokokohji Y, Doi T, Oda M, Yoshikwa T. Groundspace bilateral teleoperation of ETS-VII robot arm by direct bilateral coupling under 7-s time delay condition. IEEE Trans Robot Autom 2004;20(3):499–511.
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