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Fast Stiffness Estimation using Acceleration-based Impedance Control and its Application to Bilateral Control*
Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th9-14, World Congress Control The Federation of Toulouse, France, July 2017 The International International Federation of Automatic Automatic Control Available online at www.sciencedirect.com The International Federation of Automatic Control Toulouse, Toulouse, France, France, July July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 12059–12064
Fast Stiffness Estimation using Fast Stiffness Estimation using Fast Stiffness Impedance Estimation Control using and Acceleration-based Acceleration-based Impedance Control ⋆and Acceleration-based Impedance Control and its Application to Bilateral Control its Application to Bilateral Control ⋆⋆ its Application to Bilateral Control ∗
Daisuke Yashiro ∗ Daisuke Yashiro ∗∗ Daisuke Daisuke Yashiro Yashiro ∗ Department of Electrical and Electronic Engineering, Mie ∗ ∗ Department of Electrical and Electronic Engineering, Mie of and Engineering, Japan (e-mail: [email protected]) ∗ Department DepartmentMie, of Electrical Electrical and Electronic Electronic Engineering, Mie Mie Mie, Japan (e-mail: [email protected]) Mie, Japan (e-mail: [email protected]) Mie, Japan (e-mail: [email protected])
University, University, University, University,
Abstract: Communication delay between a master and a slave robot destabilizes a bilateral Abstract: Communication delay between aa master and a slave robot destabilizes aa bilateral Abstract: Communication delay and robot destabilizes control system. Recent researches showed that an adaptive controller that dynamically deterAbstract: Communication delay between between a master master and aa slave slave robotthat destabilizes a bilateral bilateral control system. Recent researches showed that an adaptive controller dynamically detercontrol system. Recent researches showed that an adaptive controller that dynamically determines the master’s controller gain according to the stiffness of the contact object is effective in control system. Recent researches showed that an adaptive controller thatobject dynamically determines the master’s controller gain according to the stiffness of the contact is effective in mines the master’s controller gain according to the stiffness of the contact object is effective in improving stability. As the stability depends on the convergence speed of the estimated stiffness, mines the master’s controller gain according to the the convergence stiffness of the contact object is effective in improving stability. As the stability depends on speed of the estimated stiffness, improving stability. As the stability depends on the convergence speed of the estimated stiffness, this paper utilizes an acceleration-based impedance control for fast stiffness estimation. The improving stability. As acceleration-based the stability depends on the convergence speed of the estimated stiffness, this paper utilizes an impedance control for fast stiffness estimation. The this paper utilizes an for stiffness estimation. The validity of the proposed estimation methodimpedance is verified control by simulations. The convergence speed of this paper utilizes an acceleration-based acceleration-based impedance control for fast fast The stiffness estimation. The validity of the proposed estimation method is verified by simulations. convergence speed of validity of the proposed estimation method is verified by simulations. The convergence speed of the estimated stiffness is improved, and the control performance of bilateral control is improved validity of the stiffness proposedisestimation is verified by simulations. The convergence speed of the estimated improved, method and the control performance of bilateral control is improved the estimated accordingly. the estimated stiffness stiffness is is improved, improved, and and the the control control performance performance of of bilateral bilateral control control is is improved improved accordingly. accordingly. accordingly. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Force control, motor control, position control, time delay, state observers, adaptive Keywords: Force control, control, position Keywords: Force control, motor motor control, position control, control, time time delay, delay, state state observers, observers, adaptive adaptive control, state estimation, impedance control Keywords: Force control, motor control, position control, time delay, state observers, adaptive control, state estimation, impedance control control, state estimation, impedance control control, state estimation, impedance control 1. INTRODUCTION in this approach is low. On the other hand, Nyquist 1. in this approach is low. On the other Nyquist 1. INTRODUCTION INTRODUCTION in is On other hand, Nyquist theorem-based design methods have beenhand, proposed as 1. INTRODUCTION in this this approach approach is low. low. On the the other hand, Nyquist design methods have been proposed as theorem-based design methods have been proposed as necessary and sufficient condition-based approaches. These The da Vinci Surgical System (Intuitive Surgical, Inc.) or theorem-based theorem-based design condition-based methods have been proposed as necessary and sufficient approaches. These The da Vinci Surgical System (Intuitive Surgical, Inc.) or necessary and sufficient condition-based approaches. These methods fall into two categories in terms of a delay comThe da Vinci Surgical System (Intuitive Surgical, Inc.) or the PackBot (iRobot Corporation) has drawn attention necessary and sufficient condition-based approaches. These The da Vinci Surgical System (Intuitive Surgical, Inc.) or fall into two categories in terms of aa delay the Corporation) has methods fall into two categories in terms compensator. One type a delay andcomthe the PackBot (iRobot Corporation) has drawn drawn attention as a PackBot practical (iRobot application of teleoperated robots.attention The da methods methods fall into twouses categories in compensator terms of of a delay delay comthe PackBot (iRobot Corporation) has drawn attention pensator. One type uses a delay compensator and the as a practical application of teleoperated robots. The da pensator. One type uses a delay compensator and the other does not use a delay compensator. Adaptive Smith as a practical application of teleoperated robots. The da Vinci is a medical robot used for performing minimally pensator. One type uses a delay compensator and the as a practical application of teleoperated robots. The da other does not use aa delay compensator. Adaptive Smith Vinci is aa medical robot used formore performing minimally other does not use delay compensator. Adaptive Smith predictor-based methods provided in Yashiro et al. (2010) Vinci is medical robot used for performing minimally invasive endoscopic surgeries, and than 3000 of these other does not use a delay compensator. Adaptive Smith Vinci is a medical robot used for performing minimally methods provided in Yashiro et al. (2010) invasive endoscopic surgeries, and more 3000 predictor-based methods provided in (2010) or communication disturbance methods invasive endoscopic surgeries, and more than than 3000 of of these these robots are used in the world at present. The PackBot was predictor-based predictor-based methods providedobserver-based in Yashiro Yashiro et et al. al. (2010) invasive endoscopic surgeries, and more than 3000 of these or communication disturbance observer-based methods robots are used in the world at present. The PackBot was or communication disturbance observer-based methods provided in Sabanovic et al. (2010) have been proposed robots are used in the world at present. The PackBot was used for the probe of Fukushima I Nuclear Power Plant or communication disturbance observer-based methods robots are used in the world at present. The PackBot was provided in Sabanovic et al. (2010) have been proposed used for probe Fukushima II Nuclear Power Plant provided in (2010) delay compensator-based A been delay proposed compenused for the the probe ofJapan Fukushima Nuclear Power Plant as after the Great Eastof Earthquake in 2011. However, provided in Sabanovic Sabanovic et et al. al.approaches. (2010) have have been proposed used for the probe of Fukushima I Nuclear Power Plant as delay compensator-based approaches. A delay compenafter the Great East Japan Earthquake in 2011. However, as delay compensator-based approaches. A delay compensator is applied to a position control system, but not to after the Great East Japan Earthquake in 2011. However, as a human operator who manipulates the da Vinci or as delay compensator-based approaches. A delay compenafter the Great East Japan Earthquake inthe 2011. However, sator is applied to a position control system, but notisto as a human operator who manipulates da Vinci or is applied to a position control system, but asator force control system. As a bilateral control system a as a human operator who manipulates the da Vinci or the PackBot does not feel a reaction force from a contact is control appliedsystem. to a position control control system, system but not notisto to as aPackBot human does operator whoa manipulates the da aVinci or sator a force As a bilateral a the not feel reaction force from contact a force control system. As a bilateral control system is hybrid of a position control system and a force control the PackBot does not feel a reaction force from a contact object, achieving sensitive motions such as grasping of a force of control system.control As a bilateral control system is aa the PackBot does not feel a motions reaction such force as from a contact hybrid aa whole position system and aa force control object, achieving sensitive grasping of hybrid of position control system and force control system, the of the control system is not compenobject, achieving sensitive motions such as grasping of brittle materials is difficult. Hence, some researchers have of a whole position control system and a force control object, achievingissensitive motionssome suchresearchers as grasping of hybrid of the control not compenbrittle have system, the whole of the control system is not compensated by the a delay compensator. The system methodsis without delay brittle materials isadifficult. difficult. Hence, some researchers have- system, tried tomaterials transmitis reaction Hence, force tosome a human operator system, the whole of the control system is not compenbrittle materials difficult. Hence, researchers have by aa delay compensator. The methods without delay tried to transmit aa reaction force to aa human operator -- sated sated by delay compensator. The methods without compensator fall into two categories: a fixed controller and tried to transmit reaction force to human operator see Hokayem and Spong (2006). This kind of teleoperation by a delay compensator. The methods without delay delay tried to transmit a reaction force to kind a human operator - sated compensator fall into two categories: a fixed controller and see Hokayem and Spong (2006). This of teleoperation compensator fall into two categories: a fixed controller and an adaptive controller. A four-channel controller provided see Hokayem and Spong (2006). This kind of teleoperation is known as bilateral teleoperation or bilateral control. compensator fall into two categories: a fixed controller and see Hokayem and Spongteleoperation (2006). This kind of teleoperation an adaptive controller. A four-channel controller provided is known as bilateral or bilateral control. an adaptive controller. A four-channel controller provided in Iida and Ohnishi (2004) and a three-channel controller is known as bilateral teleoperation or bilateral control. Bilateral control is a hybrid of position control and force an adaptive controller. A four-channel controllercontroller provided is known control as bilateral teleoperation or control bilateraland control. in Iida and Ohnishi (2004) and a three-channel Bilateral is a hybrid of position force in Iida Ohnishi (2004) and aa three-channel controller in Kubo et al. (2007) been proposed as fixed Bilateral control is aa hybrid hybrid ofmaster position control and force provided control. Acontrol slave robot tracks aof robot whenand a human in Iida and and Ohnishi (2004) andhave three-channel controller Bilateral is position control force provided in Kubo et al. (2007) have been proposed as fixed control. A slave robot tracks aa master robot when aaposition human provided in Kubo et al. (2007) have been proposed as controller-based approaches. However, the performance of control. A slave robot tracks master robot when human operator manipulates the master robot. This is in Kuboapproaches. et al. (2007)However, have been proposed as fixed fixed control. Amanipulates slave robot tracks a master robotThis when aposition human provided controller-based the performance of operator the master robot. is controller-based approaches. However, the performance of force control is low in the case of a four-channel controller, operator manipulates the master robot. This is position control. When the slave robot contacts with an object, controller-based approaches. However, the performance of operator manipulates the robot mastercontacts robot. This is position force control is low in the case of aa four-channel controller, control. When the slave with an object, force control is low in the case of four-channel controller, and the performance of position control is low in the case of control. When the slave robot contacts with an object, the human operator feels a reaction force from the object. force control is low inofthe case ofcontrol a four-channel controller, control. When the slave robot contacts with the an object, and the performance position is low in the case of the human operator feels a reaction force from object. the performance of position control is low in the case aand three-channel controller -see Yashiro and Ohnishi (2011). the human operator feels a reaction force from the object. This is force control. However, a bilateral control system the performance of position controland is low in the(2011). case of of the human operator feels a reaction force from the system object. and a three-channel controller -see Yashiro Ohnishi This is force control. However, a bilateral control aa three-channel controller -see Yashiro and Ohnishi (2011). This is force force control. However, a bilateral bilateral control becomes unstable when a feedback loop control in the system three-channel controller -see Yashiro and Ohnishi (2011). This is control. However, a system Recently, a method that modifies position controller gain becomes unstable aa feedback becomes unstable when when delay. feedback loop loop in in the the system system Recently, includes communication aa has method modifies position gain Recently, method that modifies position controller gain becomes unstable when a feedback loop in the system adaptively beenthat researched one ofcontroller the adaptive includes communication delay. Recently, a has method that modifies as position controller gain includes communication delay. adaptively been researched as one of the adaptive adaptively has been researched as one of the adaptive includes communication delay. approaches -see Kitamura et al. (2009). Two kinds of approaches are presently adopted to de- controller-based adaptively has been researched as one of the adaptive approaches Kitamura et al. (2009). Two of are adopted de- controller-based controller-based approaches -see Kitamura et Yashiro et al. (2015) proves-see that high performance is Two kinds of approaches approaches are presently presently adopted to to design akinds bilateral control system with communication approaches -see Kitamura et al. al. (2009). (2009). Two of approaches are presently adopted to de- controller-based Yashiro et al. (2015) proves that high performance is sign aakinds bilateral control system with communication deYashiro et al. (2015) proves that high performance is achieved if the position controller gain in a master robot sign bilateral control system with communication delay. One approach uses a sufficient condition for stability, et al. position (2015) proves thatgain high performance is sign a bilateral control system with communication de- Yashiro achieved if the controller in a master robot lay. One approach uses a sufficient condition for stability, achieved if the position controller gain in a master robot is the same as the stiffness of an object that contacts with lay. One approach uses a sufficient condition for stability, whereas the other approach uses a necessary and sufficient achieved if the position controller gain in a master robot lay. One approach uses a sufficient condition for stability, is the same same as and the stiffness stiffness of an an object thatmethod contacts with whereas approach uses aa necessary and the as the of object that contacts with ais slave robot, a recursive least square is used whereas the other approach uses necessary and sufficient sufficient conditionthe forother stability. Small gain theorem-based design is the same as and the stiffness of an object thatmethod contacts with whereas the other approach uses a necessary and sufficient a slave robot, a recursive least square is used condition for stability. Small gain theorem-based design a slave robot, and a recursive least square method is used to estimate the stiffness of the contact object. However condition for stability. Small gain theorem-based design methods provided in Polushin et al.theorem-based (2007) and passivity a slave robot, and a recursive least square method is used condition for stability. Small gain design to estimate the stiffness of the contact object. However methods provided in et and to estimate the the object. However a contact object isof it is difficult select methods provided in Polushin Polushin et al. al. (2007) (2007) and passivity passivity theorem-based design methods provided in Hannaford and when to estimate the stiffness stiffness of hard, the contact contact object. to However methods provided in Polushin et al. (2007) and passivity when aa contact object is hard, it is difficult to select theorem-based design methods provided in Hannaford and when contact object is hard, it is difficult to small forgetting factor from the viewpoint of stability of theorem-based design methods provided in Hannaford and Ryu (2002) have been proposed as sufficient conditiona contact factor objectfrom is hard, it is difficult to select select theorem-based design methods provided in Hannaford and when small forgetting the viewpoint of stability of Ryu (2002) have been proposed as sufficient conditionsmall forgetting factor from the viewpoint of stability of estimation system. As a result, the convergence speed Ryu (2002) have been proposed as sufficient conditionbased approaches. However, control performance achieved small forgetting factor from the viewpoint of stability of Ryu (2002) have been proposed asperformance sufficient conditionestimation system. As a result, the convergence speed based approaches. However, control achieved estimation system. As a result, the convergence speed of an estimated stiffness becomes slow, and it causes an based approaches. However, control performance achieved ⋆ This work was supported by JSPS KAKENHI Grant Number estimation system. As a result, the convergence speed based approaches. However, control performance achieved of an estimated stiffness becomes slow, and it causes an ⋆ of an stiffness becomes slow, and it an unignorable chattering when the slave robot This work ⋆ 15H05529. work was was supported supported by by JSPS JSPS KAKENHI KAKENHI Grant Grant Number Number of an estimated estimated stiffness becomes slow, andcontacts it causes causeswith an ⋆ This unignorable chattering when the slave robot contacts with This work was supported by JSPS KAKENHI Grant Number unignorable chattering when the slave robot contacts with 15H05529. 15H05529. unignorable chattering when the slave robot contacts with
15H05529. Copyright © 2017, 2017 IFAC 12565 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 12565 Copyright ©under 2017 responsibility IFAC 12565 Peer review of International Federation of Automatic Control. Copyright © 2017 IFAC 12565 10.1016/j.ifacol.2017.08.2126
Proceedings of the 20th IFAC World Congress 12060 Daisuke Yashiro et al. / IFAC PapersOnLine 50-1 (2017) 12059–12064 Toulouse, France, July 9-14, 2017
Table 1. Notation.
Master y(t) y(t) ˙ y¨(t) y[k] p(t) I s
continuous-time scalar signal derivative value of y(t) derivative value of y(t) ˙ discrete-time representation of y(t) continuous-time vector signal unit matrix Laplace operator
+
a hard object. This paper therefore utilizes an accelerationbased impedance control. The impedance control makes it possible to select small forgetting factor even if a contact object is hard. Simulation results show that the convergence speed of the estimated stiffness is improved, and an unignorable chattering is suppressed accordingly when the slave robot contacts with a hard object. This paper is organized as follows: in Section 2, a model of a bilateral control system is described, and a controller design problem is formulated. In Section 3, the accelerationbased impedance control and the algorithm to estimate object’s stiffness are introduced. In Section 4, the validity of the stiffness estimation using the impedance control is verified by simulations. Finally, conclusions are presented in Section 5. Table 1 lists the notations used in this paper. 2. MODELING AND PROBLEM FORMULATION In Section 2.1, a model of a bilateral control system with communication delay is described. And in Section 2.2, an issue to design a controller is formulated.
A bilateral control system with communication delay consists of a master robot and a slave robot. The dynamics of the master robot and the slave robot that includes a plant and a four-channel controller is described in Eq. (1) and Eq. (2)
ˆs (t)} x ¨ref s (t) = ks {xm (t − d) − x √ +2 ks {x˙ m (t − d) − x ˆ˙ s (t)} +fm (t − d) + fˆs (t).
+
+ ++ +
Fig. 1. Bilateral control system with communication delay. where ke [N/m]，de [kg/s]，and xe [m] denote the stiffness of a contact object, viscosity of a contact object, and an initial contact position, respectively. Fig. 1 shows the block diagram of the bilateral control system with communication delay d, where e−ds represents the Laplace domain’s description of communication delay √ d, Ci := ki + 2 ki s, and fidis := fi + fielse (i = m, s). fielse [N] includes friction, gravity, and so on. The controller shown in Fig. 1 is called a four-channel controller as four kinds of control signals are transmitted between the master robot and the slave robot. A controller that does not transmit xs from the slave side to the master side is called a three-channel controller, and a controller that does not transmit xs and fm between the master side and the slave side is called a two-channel controller.
As Yashiro et al. (2016) proves that the system is unstable if |ke − km | is extremely large, this paper design a master’s position controller that achieves km ≈ ke . The problem of designing a controller for the bilateral control system shown in Fig. 1 is written as follows: Proposition 1. Derive the position controller gain km that stabilizes the system described in Eq. (1)–Eq. (3) and achieves Eq. (4) and Eq. (5) xm (t) − xs (t) → 0,
(1)
fm (t) + fs (t) → 0.
(4) (5)
3. CONTROLLER DESIGN (2)
where xi (t) [m], x ˆs (t) [m], fi (t) [N], fˆs (t) [N], d [s], ki [s−2 ] (i = m, s) denote a position of the master/slave robot, a reference position of the slave robot, an external force applied to the master/slave robot, a reference external force applied to the slave robot, an one-way delay between the master and the slave, and a proportional gain of the master/slave position controller, respectively. In addition, √ 2 ki [s−1 ] represents a differential gain of the position controllers. The dynamics of a contact motion between the slave robot and an object is defined in Eq. (3) fs (t) = −ke (xs (t) − xe ) − de x˙ s (t).
+
+ + +
2.2 Problem Formulation
2.1 Modeling of Bilateral Control System
x ¨ref xs (t − d) − xm (t)} m (t) = km {ˆ √ +2 km {x ˆ˙ s (t − d) − x˙ m (t)} +fm (t) + fˆs (t − d),
Slave
(3)
In Section 3.1 and Section 3.2, the disturbance observer (DOB) based acceleration control and the accelerationbased impedance control are introduced, respectively. In Section 3.3, a stiffness estimation method using a recursive least square method (RLSM) with a schmitt trigger (ST) is introduced. 3.1 Acceleration Control The DOB based acceleration control system Pi is shown in Fig. 2, where i = m, s. m and mn are a mass of the robot and a nominal mass of the robot, respectively. firef is a force which is generated by a robot. A DOB is utilized to suppress the influence of fidis . A reaction force observer
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+
12061
+ +
+ DOB
RFOB
Fig. 2. Plant model Pi .
_
+
+ +
+
+ +
+
DOB
Fig. 5. Estimation method of ke .
RFOB
Fˆs (ms2 + Ze )Zr =− ˆs (ms2 + Ze ) + Zr X
Fig. 3. Plant model P˜s .
Virtual Robot
(7)
ˆ s → −Zr . where Zr = kr +dr s. In Eq. (7), if Ze → ∞, Fˆs /X 3.3 Stiffness Estimation
Real Robot
Fig. 4. Concept of the virtual slave robot. (RFOB) provided in Murakami et al. (1993) is utilized to estimate fi . Fact 2. If a RFOB works ideally, the transfer function Fs /Xs is obtained in Eq. (6) Fs = −Ze (6) Xs where Ze := ke + de s. In Eq. (6), if Ze → ∞, Fs /Xs → −∞.
Schmitt Trigger Fig. 5 shows the concept of an object’s stiffness estimation. We proceed to derive the estimated value of ke defined by kˆe [N/m] under the assumptions xe > xs > 0 and de = 0. The position xe on the slave side that achieves −fˆs [k] = fe1 > 0 is defined as the contact position. If −fˆs [k] ≥ fe1 > 0 is satisfied, the state c transitions from 0 to 1. On the other hand, if −fˆs [k] ≤ fe2 is satisfied, the state c transitions from 1 to 0. Two kinds of thresholds fe1 and fe2 are used to change the state c, and this method is called the ST. An effect of ST is to suppress the chattering that occurs when the slave robot contacts with a hard object. Although a large fe1 reduces the influence of noise, estimation delay is increased. Besides, although a small value of fe2 reduces chattering, c may not transition from 1 to 0 even if the slave does not contact with an object. Recursive Least Square Method kˆe is obtained as follows:
The problem of deriving
Proposition 4.
3.2 Acceleration-based Impedance Control The acceleration-based impedance control system P˜s is shown in Fig. 3. x ˜s ,f˜s , and zr are a position of the virtual slave robot, an external force applied to the virtual slave robot, and impedance of the slave robot, respectively. Fig. 4 shows the concept of the virtual slave robot. The ¨ acceleration of the virtual robot x ˜s is controlled by x ¨ref s . The virtual robot is connected to the real slave robot through the spring and damper. The spring constant and damper constant are defined by kr and dr , respectively. Fact 3. If a DOB and RFOB works ideally, the transfer ˆ s is obtained in Eq. (7) function Fˆs /X
(if c = 0) kˆe = 0,
(8)
(if c = 1) kˆe = arg min
k ∑
ˆe ≥0 k i=1
( )2 λk−i kˆe [i]x′s [i] + fs′ [i] ,
(9)
where λ (0 < λ < 1) is a forgetting factor, x′s [i] := x ˆs [i] − xe + xo [m], and fs′ [i] := fˆs [i] + fe2 [N]. xo (> 0) [m] is used to avoid the divergence of kˆe . Although a large xo reduces the influence of noise, it causes an estimation error. The solution of Eq. (9) is derived from Eq. (10)
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Proceedings of the 20th IFAC World Congress 12062 Daisuke Yashiro et al. / IFAC PapersOnLine 50-1 (2017) 12059–12064 Toulouse, France, July 9-14, 2017
paper designed an adaptive controller that dynamically determines the master’s controller gain depending on the stiffness of a contact object. As the stability depends on the convergence speed of the estimated stiffness, this paper utilized an acceleration-based impedance control for fast stiffness estimation. Simulation results showed that fast stiffness estimation is achieved and the stability of bilateral control system is improved accordingly.
Table 2. Parameters for simulation.
case case case case case case
1 2 3 4 5 6
Pˆs
x ˆs
fˆs
λ
Zr
Ps Ps P˜s P˜s P˜s P˜s
xs xs x ˜s x ˜s x ˜s x ˜s
fs fs f˜s f˜s f˜s f˜s
0.99 0.9 0.99 0.9 0.99 0.9
10000+500s 10000+500s 2500+100s 2500+100s
ACKNOWLEDGEMENTS
kˆe [k] = kˆe [k − 1] x′ [k]kˆe [k − 1] + fs′ [k] γ[k − 1]x′s [k], − s ′ (10) λ + xs [k]γ[k − 1]x′s [k] } { 1 γ[k − 1]x′2 s [k]γ[k − 1] . (11) γ[k] = γ[k − 1] − λ λ + x′s [k]γ[k − 1]x′s [k] This algorithm is well known as a RLSM that uses a forgetting factor. Although a small value of λ improves convergence speed, it may cause the divergence of kˆe .
This work was supported by JSPS KAKENHI Grant Number 15H05529. REFERENCES
Hannaford, B. and Ryu, J.H. (2002). Time-domain passivity control of haptic interfaces. IEEE Transactions on Robotics and Automation, 18(1), 1 –10. Hokayem, P.F. and Spong, M.W. (2006). Bilateral teleoperation:an historical survey. AUTOMATICA, 42(12), 2035–2057. 4. SIMULATION Iida, W. and Ohnishi, K. (2004). Reproducibility and The validity of the stiffness estimation using the acceleration- operationality in bilateral teleoperation. In the IEEE International Workshop on Advanced Motion Control, based impedance control is verified by simulations. 217–222. Kitamura, K., Yashiro, D., and Ohnishi, K. (2009). Bilat4.1 Setup eral control using estimated environmental stiffness as the master position gain. In 35th Annual Conference of Bilateral control using the system shown in Fig. 1 is IEEE Industrial Electronics Society, 2981–2986. simulated. The control period, the packet-sending interval Kubo, R., Natori, K., Iiyama, N., and Ohnishi, K. (2007). between the master and the slave, and one-way delay d are Performace analysis of a three-channel control architecset to 1ms, 1ms, and 100ms, respectively. One free motion ture for bilateral teleoperation with time delay. IEEJ (1s∼5s, Ze = 0 + 0s), one contact motion with a piece of Transactions on Industry Applications, 127(12), 1224– sponge (10s∼15s, Ze = 400+10s), and one contact motion 1230. with a block of aluminum (20s∼25s, Ze = 30000+10s)) are Murakami, T., Yu, F., and Ohnishi, K. (1993). Torque tested. In the simulation, six cases are compared as shown sensorless control in multidegree-of-freedom manipulain Table 2. Parameters m, km (t), ks , x0 , fe1 , and fe2 are set tor. IEEE Transaction on Industrial Electronics, 40(2), to 0.5, kˆe (t − d), 900, 0.001, 1.0, and 0.2, respectively. 259–265. Polushin, I.G., Liu, P.X., and Lung, C. (2007). A force4.2 Result reflection algorithm for improved transparency in bilateral teleoperation with communication delay. IEEE Figs. 6–11 show the simulation results of case 1–6. (a) Transactions on Mechatronics, 12(3), 361–374. denotes the time responses of ke (dash line) and kˆe (solid Sabanovic, A., Ohnishi, K., Yashiro, D., Sabanovic, N., and Baran, E.A. (2010). Motion control systems with line). (b) denotes the time responses of xm (dash line) and network delay. AUTOMATIKA, 51(2), 119–126. xs (solid line). (c) denotes the time responses of fm (dash line) and −fs (solid line). When λ = 0.99 (Fig. 6, Fig. 8, Yashiro, D., Hieno, T., Yubai, K., and Komada, S. (2015). Design of master’s position controller for bilateral conFig. 10), the convergence speed of kˆe is slow. And the slow trol system with time delay. IEEJ Transaction on estimation causes a large oscillation of position responses Industry Applications, 135(3). ˆ and force responses. Fig. 7(a) shows that ke is in a state Yashiro, D. and Ohnishi, K. (2011). Performance analof frenzied vibration in the case of λ = 0.9. And the ysis of bilateral control system with communication position responses and force responses are influenced by bandwidth constraint. IEEE Transaction on Industrial the oscillation of kˆe . On the other hand, Fig. 9(a) and Electronics, 58(2), 436–443. Fig. 11(a) show that oscillation of kˆe is suppressed by Yashiro, D., Tian, D., and Ohnishi, K. (2010). Central the effect of impedance control. And the oscillations of controller based hybrid control with communication position responses and force responses are also suppressed. delay compensator. In 36th IEEE Annual Conference Fig. 9(b) and Fig. 11(b) show that the smaller the kr , the of Industrial Electronics Society, 3129–3134. larger the position tracking error. Yashiro, D., Yubai, K., and Komada, S. (2016). Fast estimation of environment’s stiffness for bilateral control 5. CONCLUSION systems with communication delay. IEEJ Journal of Industry Applications, 5(6), 422–428. Communication delay between a master robot and a slave robot destabilizes bilateral control systems. Hence, this 12568
Proceedings of the 20th IFAC World Congress Daisuke Yashiro et al. / IFAC PapersOnLine 50-1 (2017) 12059–12064 Toulouse, France, July 9-14, 2017
(a) ke (t), kˆe (t − d).
(b) xm (t), xs (t).
(c) fm (t), −fs (t).
(b) xm (t), xs (t).
(c) fm (t), −fs (t).
(b) xm (t), xs (t).
(c) fm (t), −fs (t).
(b) xm (t), xs (t).
(c) fm (t), −fs (t).
Fig. 6. Simulation results of case 1.
(a) ke (t), kˆe (t − d). Fig. 7. Simulation results of case 2.
(a) ke (t), kˆe (t − d). Fig. 8. Simulation results of case 3.
(a) ke (t), kˆe (t − d). Fig. 9. Simulation results of case 4.
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Proceedings of the 20th IFAC World Congress 12064 Daisuke Yashiro et al. / IFAC PapersOnLine 50-1 (2017) 12059–12064 Toulouse, France, July 9-14, 2017
(a) ke (t), kˆe (t − d).
(b) xm (t), xs (t).
(c) fm (t), −fs (t).
(b) xm (t), xs (t).
(c) fm (t), −fs (t).
Fig. 10. Simulation results of case 5.
(a) ke (t), kˆe (t − d). Fig. 11. Simulation results of case 6.
12570
ScienceDirect
IFAC PapersOnLine 50-1 (2017) 12059–12064
Fast Stiffness Estimation using Fast Stiffness Estimation using Fast Stiffness Impedance Estimation Control using and Acceleration-based Acceleration-based Impedance Control ⋆and Acceleration-based Impedance Control and its Application to Bilateral Control its Application to Bilateral Control ⋆⋆ its Application to Bilateral Control ∗
Daisuke Yashiro ∗ Daisuke Yashiro ∗∗ Daisuke Daisuke Yashiro Yashiro ∗ Department of Electrical and Electronic Engineering, Mie ∗ ∗ Department of Electrical and Electronic Engineering, Mie of and Engineering, Japan (e-mail: [email protected]) ∗ Department DepartmentMie, of Electrical Electrical and Electronic Electronic Engineering, Mie Mie Mie, Japan (e-mail: [email protected]) Mie, Japan (e-mail: [email protected]) Mie, Japan (e-mail: [email protected])
University, University, University, University,
Abstract: Communication delay between a master and a slave robot destabilizes a bilateral Abstract: Communication delay between aa master and a slave robot destabilizes aa bilateral Abstract: Communication delay and robot destabilizes control system. Recent researches showed that an adaptive controller that dynamically deterAbstract: Communication delay between between a master master and aa slave slave robotthat destabilizes a bilateral bilateral control system. Recent researches showed that an adaptive controller dynamically detercontrol system. Recent researches showed that an adaptive controller that dynamically determines the master’s controller gain according to the stiffness of the contact object is effective in control system. Recent researches showed that an adaptive controller thatobject dynamically determines the master’s controller gain according to the stiffness of the contact is effective in mines the master’s controller gain according to the stiffness of the contact object is effective in improving stability. As the stability depends on the convergence speed of the estimated stiffness, mines the master’s controller gain according to the the convergence stiffness of the contact object is effective in improving stability. As the stability depends on speed of the estimated stiffness, improving stability. As the stability depends on the convergence speed of the estimated stiffness, this paper utilizes an acceleration-based impedance control for fast stiffness estimation. The improving stability. As acceleration-based the stability depends on the convergence speed of the estimated stiffness, this paper utilizes an impedance control for fast stiffness estimation. The this paper utilizes an for stiffness estimation. The validity of the proposed estimation methodimpedance is verified control by simulations. The convergence speed of this paper utilizes an acceleration-based acceleration-based impedance control for fast fast The stiffness estimation. The validity of the proposed estimation method is verified by simulations. convergence speed of validity of the proposed estimation method is verified by simulations. The convergence speed of the estimated stiffness is improved, and the control performance of bilateral control is improved validity of the stiffness proposedisestimation is verified by simulations. The convergence speed of the estimated improved, method and the control performance of bilateral control is improved the estimated accordingly. the estimated stiffness stiffness is is improved, improved, and and the the control control performance performance of of bilateral bilateral control control is is improved improved accordingly. accordingly. accordingly. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Force control, motor control, position control, time delay, state observers, adaptive Keywords: Force control, control, position Keywords: Force control, motor motor control, position control, control, time time delay, delay, state state observers, observers, adaptive adaptive control, state estimation, impedance control Keywords: Force control, motor control, position control, time delay, state observers, adaptive control, state estimation, impedance control control, state estimation, impedance control control, state estimation, impedance control 1. INTRODUCTION in this approach is low. On the other hand, Nyquist 1. in this approach is low. On the other Nyquist 1. INTRODUCTION INTRODUCTION in is On other hand, Nyquist theorem-based design methods have beenhand, proposed as 1. INTRODUCTION in this this approach approach is low. low. On the the other hand, Nyquist design methods have been proposed as theorem-based design methods have been proposed as necessary and sufficient condition-based approaches. These The da Vinci Surgical System (Intuitive Surgical, Inc.) or theorem-based theorem-based design condition-based methods have been proposed as necessary and sufficient approaches. These The da Vinci Surgical System (Intuitive Surgical, Inc.) or necessary and sufficient condition-based approaches. These methods fall into two categories in terms of a delay comThe da Vinci Surgical System (Intuitive Surgical, Inc.) or the PackBot (iRobot Corporation) has drawn attention necessary and sufficient condition-based approaches. These The da Vinci Surgical System (Intuitive Surgical, Inc.) or fall into two categories in terms of aa delay the Corporation) has methods fall into two categories in terms compensator. One type a delay andcomthe the PackBot (iRobot Corporation) has drawn drawn attention as a PackBot practical (iRobot application of teleoperated robots.attention The da methods methods fall into twouses categories in compensator terms of of a delay delay comthe PackBot (iRobot Corporation) has drawn attention pensator. One type uses a delay compensator and the as a practical application of teleoperated robots. The da pensator. One type uses a delay compensator and the other does not use a delay compensator. Adaptive Smith as a practical application of teleoperated robots. The da Vinci is a medical robot used for performing minimally pensator. One type uses a delay compensator and the as a practical application of teleoperated robots. The da other does not use aa delay compensator. Adaptive Smith Vinci is aa medical robot used formore performing minimally other does not use delay compensator. Adaptive Smith predictor-based methods provided in Yashiro et al. (2010) Vinci is medical robot used for performing minimally invasive endoscopic surgeries, and than 3000 of these other does not use a delay compensator. Adaptive Smith Vinci is a medical robot used for performing minimally methods provided in Yashiro et al. (2010) invasive endoscopic surgeries, and more 3000 predictor-based methods provided in (2010) or communication disturbance methods invasive endoscopic surgeries, and more than than 3000 of of these these robots are used in the world at present. The PackBot was predictor-based predictor-based methods providedobserver-based in Yashiro Yashiro et et al. al. (2010) invasive endoscopic surgeries, and more than 3000 of these or communication disturbance observer-based methods robots are used in the world at present. The PackBot was or communication disturbance observer-based methods provided in Sabanovic et al. (2010) have been proposed robots are used in the world at present. The PackBot was used for the probe of Fukushima I Nuclear Power Plant or communication disturbance observer-based methods robots are used in the world at present. The PackBot was provided in Sabanovic et al. (2010) have been proposed used for probe Fukushima II Nuclear Power Plant provided in (2010) delay compensator-based A been delay proposed compenused for the the probe ofJapan Fukushima Nuclear Power Plant as after the Great Eastof Earthquake in 2011. However, provided in Sabanovic Sabanovic et et al. al.approaches. (2010) have have been proposed used for the probe of Fukushima I Nuclear Power Plant as delay compensator-based approaches. A delay compenafter the Great East Japan Earthquake in 2011. However, as delay compensator-based approaches. A delay compensator is applied to a position control system, but not to after the Great East Japan Earthquake in 2011. However, as a human operator who manipulates the da Vinci or as delay compensator-based approaches. A delay compenafter the Great East Japan Earthquake inthe 2011. However, sator is applied to a position control system, but notisto as a human operator who manipulates da Vinci or is applied to a position control system, but asator force control system. As a bilateral control system a as a human operator who manipulates the da Vinci or the PackBot does not feel a reaction force from a contact is control appliedsystem. to a position control control system, system but not notisto to as aPackBot human does operator whoa manipulates the da aVinci or sator a force As a bilateral a the not feel reaction force from contact a force control system. As a bilateral control system is hybrid of a position control system and a force control the PackBot does not feel a reaction force from a contact object, achieving sensitive motions such as grasping of a force of control system.control As a bilateral control system is aa the PackBot does not feel a motions reaction such force as from a contact hybrid aa whole position system and aa force control object, achieving sensitive grasping of hybrid of position control system and force control system, the of the control system is not compenobject, achieving sensitive motions such as grasping of brittle materials is difficult. Hence, some researchers have of a whole position control system and a force control object, achievingissensitive motionssome suchresearchers as grasping of hybrid of the control not compenbrittle have system, the whole of the control system is not compensated by the a delay compensator. The system methodsis without delay brittle materials isadifficult. difficult. Hence, some researchers have- system, tried tomaterials transmitis reaction Hence, force tosome a human operator system, the whole of the control system is not compenbrittle materials difficult. Hence, researchers have by aa delay compensator. The methods without delay tried to transmit aa reaction force to aa human operator -- sated sated by delay compensator. The methods without compensator fall into two categories: a fixed controller and tried to transmit reaction force to human operator see Hokayem and Spong (2006). This kind of teleoperation by a delay compensator. The methods without delay delay tried to transmit a reaction force to kind a human operator - sated compensator fall into two categories: a fixed controller and see Hokayem and Spong (2006). This of teleoperation compensator fall into two categories: a fixed controller and an adaptive controller. A four-channel controller provided see Hokayem and Spong (2006). This kind of teleoperation is known as bilateral teleoperation or bilateral control. compensator fall into two categories: a fixed controller and see Hokayem and Spongteleoperation (2006). This kind of teleoperation an adaptive controller. A four-channel controller provided is known as bilateral or bilateral control. an adaptive controller. A four-channel controller provided in Iida and Ohnishi (2004) and a three-channel controller is known as bilateral teleoperation or bilateral control. Bilateral control is a hybrid of position control and force an adaptive controller. A four-channel controllercontroller provided is known control as bilateral teleoperation or control bilateraland control. in Iida and Ohnishi (2004) and a three-channel Bilateral is a hybrid of position force in Iida Ohnishi (2004) and aa three-channel controller in Kubo et al. (2007) been proposed as fixed Bilateral control is aa hybrid hybrid ofmaster position control and force provided control. Acontrol slave robot tracks aof robot whenand a human in Iida and and Ohnishi (2004) andhave three-channel controller Bilateral is position control force provided in Kubo et al. (2007) have been proposed as fixed control. A slave robot tracks aa master robot when aaposition human provided in Kubo et al. (2007) have been proposed as controller-based approaches. However, the performance of control. A slave robot tracks master robot when human operator manipulates the master robot. This is in Kuboapproaches. et al. (2007)However, have been proposed as fixed fixed control. Amanipulates slave robot tracks a master robotThis when aposition human provided controller-based the performance of operator the master robot. is controller-based approaches. However, the performance of force control is low in the case of a four-channel controller, operator manipulates the master robot. This is position control. When the slave robot contacts with an object, controller-based approaches. However, the performance of operator manipulates the robot mastercontacts robot. This is position force control is low in the case of aa four-channel controller, control. When the slave with an object, force control is low in the case of four-channel controller, and the performance of position control is low in the case of control. When the slave robot contacts with an object, the human operator feels a reaction force from the object. force control is low inofthe case ofcontrol a four-channel controller, control. When the slave robot contacts with the an object, and the performance position is low in the case of the human operator feels a reaction force from object. the performance of position control is low in the case aand three-channel controller -see Yashiro and Ohnishi (2011). the human operator feels a reaction force from the object. This is force control. However, a bilateral control system the performance of position controland is low in the(2011). case of of the human operator feels a reaction force from the system object. and a three-channel controller -see Yashiro Ohnishi This is force control. However, a bilateral control aa three-channel controller -see Yashiro and Ohnishi (2011). This is force force control. However, a bilateral bilateral control becomes unstable when a feedback loop control in the system three-channel controller -see Yashiro and Ohnishi (2011). This is control. However, a system Recently, a method that modifies position controller gain becomes unstable aa feedback becomes unstable when when delay. feedback loop loop in in the the system system Recently, includes communication aa has method modifies position gain Recently, method that modifies position controller gain becomes unstable when a feedback loop in the system adaptively beenthat researched one ofcontroller the adaptive includes communication delay. Recently, a has method that modifies as position controller gain includes communication delay. adaptively been researched as one of the adaptive adaptively has been researched as one of the adaptive includes communication delay. approaches -see Kitamura et al. (2009). Two kinds of approaches are presently adopted to de- controller-based adaptively has been researched as one of the adaptive approaches Kitamura et al. (2009). Two of are adopted de- controller-based controller-based approaches -see Kitamura et Yashiro et al. (2015) proves-see that high performance is Two kinds of approaches approaches are presently presently adopted to to design akinds bilateral control system with communication approaches -see Kitamura et al. al. (2009). (2009). Two of approaches are presently adopted to de- controller-based Yashiro et al. (2015) proves that high performance is sign aakinds bilateral control system with communication deYashiro et al. (2015) proves that high performance is achieved if the position controller gain in a master robot sign bilateral control system with communication delay. One approach uses a sufficient condition for stability, et al. position (2015) proves thatgain high performance is sign a bilateral control system with communication de- Yashiro achieved if the controller in a master robot lay. One approach uses a sufficient condition for stability, achieved if the position controller gain in a master robot is the same as the stiffness of an object that contacts with lay. One approach uses a sufficient condition for stability, whereas the other approach uses a necessary and sufficient achieved if the position controller gain in a master robot lay. One approach uses a sufficient condition for stability, is the same same as and the stiffness stiffness of an an object thatmethod contacts with whereas approach uses aa necessary and the as the of object that contacts with ais slave robot, a recursive least square is used whereas the other approach uses necessary and sufficient sufficient conditionthe forother stability. Small gain theorem-based design is the same as and the stiffness of an object thatmethod contacts with whereas the other approach uses a necessary and sufficient a slave robot, a recursive least square is used condition for stability. Small gain theorem-based design a slave robot, and a recursive least square method is used to estimate the stiffness of the contact object. However condition for stability. Small gain theorem-based design methods provided in Polushin et al.theorem-based (2007) and passivity a slave robot, and a recursive least square method is used condition for stability. Small gain design to estimate the stiffness of the contact object. However methods provided in et and to estimate the the object. However a contact object isof it is difficult select methods provided in Polushin Polushin et al. al. (2007) (2007) and passivity passivity theorem-based design methods provided in Hannaford and when to estimate the stiffness stiffness of hard, the contact contact object. to However methods provided in Polushin et al. (2007) and passivity when aa contact object is hard, it is difficult to select theorem-based design methods provided in Hannaford and when contact object is hard, it is difficult to small forgetting factor from the viewpoint of stability of theorem-based design methods provided in Hannaford and Ryu (2002) have been proposed as sufficient conditiona contact factor objectfrom is hard, it is difficult to select select theorem-based design methods provided in Hannaford and when small forgetting the viewpoint of stability of Ryu (2002) have been proposed as sufficient conditionsmall forgetting factor from the viewpoint of stability of estimation system. As a result, the convergence speed Ryu (2002) have been proposed as sufficient conditionbased approaches. However, control performance achieved small forgetting factor from the viewpoint of stability of Ryu (2002) have been proposed asperformance sufficient conditionestimation system. As a result, the convergence speed based approaches. However, control achieved estimation system. As a result, the convergence speed of an estimated stiffness becomes slow, and it causes an based approaches. However, control performance achieved ⋆ This work was supported by JSPS KAKENHI Grant Number estimation system. As a result, the convergence speed based approaches. However, control performance achieved of an estimated stiffness becomes slow, and it causes an ⋆ of an stiffness becomes slow, and it an unignorable chattering when the slave robot This work ⋆ 15H05529. work was was supported supported by by JSPS JSPS KAKENHI KAKENHI Grant Grant Number Number of an estimated estimated stiffness becomes slow, andcontacts it causes causeswith an ⋆ This unignorable chattering when the slave robot contacts with This work was supported by JSPS KAKENHI Grant Number unignorable chattering when the slave robot contacts with 15H05529. 15H05529. unignorable chattering when the slave robot contacts with
15H05529. Copyright © 2017, 2017 IFAC 12565 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 12565 Copyright ©under 2017 responsibility IFAC 12565 Peer review of International Federation of Automatic Control. Copyright © 2017 IFAC 12565 10.1016/j.ifacol.2017.08.2126
Proceedings of the 20th IFAC World Congress 12060 Daisuke Yashiro et al. / IFAC PapersOnLine 50-1 (2017) 12059–12064 Toulouse, France, July 9-14, 2017
Table 1. Notation.
Master y(t) y(t) ˙ y¨(t) y[k] p(t) I s
continuous-time scalar signal derivative value of y(t) derivative value of y(t) ˙ discrete-time representation of y(t) continuous-time vector signal unit matrix Laplace operator
+
a hard object. This paper therefore utilizes an accelerationbased impedance control. The impedance control makes it possible to select small forgetting factor even if a contact object is hard. Simulation results show that the convergence speed of the estimated stiffness is improved, and an unignorable chattering is suppressed accordingly when the slave robot contacts with a hard object. This paper is organized as follows: in Section 2, a model of a bilateral control system is described, and a controller design problem is formulated. In Section 3, the accelerationbased impedance control and the algorithm to estimate object’s stiffness are introduced. In Section 4, the validity of the stiffness estimation using the impedance control is verified by simulations. Finally, conclusions are presented in Section 5. Table 1 lists the notations used in this paper. 2. MODELING AND PROBLEM FORMULATION In Section 2.1, a model of a bilateral control system with communication delay is described. And in Section 2.2, an issue to design a controller is formulated.
A bilateral control system with communication delay consists of a master robot and a slave robot. The dynamics of the master robot and the slave robot that includes a plant and a four-channel controller is described in Eq. (1) and Eq. (2)
ˆs (t)} x ¨ref s (t) = ks {xm (t − d) − x √ +2 ks {x˙ m (t − d) − x ˆ˙ s (t)} +fm (t − d) + fˆs (t).
+
+ ++ +
Fig. 1. Bilateral control system with communication delay. where ke [N/m]，de [kg/s]，and xe [m] denote the stiffness of a contact object, viscosity of a contact object, and an initial contact position, respectively. Fig. 1 shows the block diagram of the bilateral control system with communication delay d, where e−ds represents the Laplace domain’s description of communication delay √ d, Ci := ki + 2 ki s, and fidis := fi + fielse (i = m, s). fielse [N] includes friction, gravity, and so on. The controller shown in Fig. 1 is called a four-channel controller as four kinds of control signals are transmitted between the master robot and the slave robot. A controller that does not transmit xs from the slave side to the master side is called a three-channel controller, and a controller that does not transmit xs and fm between the master side and the slave side is called a two-channel controller.
As Yashiro et al. (2016) proves that the system is unstable if |ke − km | is extremely large, this paper design a master’s position controller that achieves km ≈ ke . The problem of designing a controller for the bilateral control system shown in Fig. 1 is written as follows: Proposition 1. Derive the position controller gain km that stabilizes the system described in Eq. (1)–Eq. (3) and achieves Eq. (4) and Eq. (5) xm (t) − xs (t) → 0,
(1)
fm (t) + fs (t) → 0.
(4) (5)
3. CONTROLLER DESIGN (2)
where xi (t) [m], x ˆs (t) [m], fi (t) [N], fˆs (t) [N], d [s], ki [s−2 ] (i = m, s) denote a position of the master/slave robot, a reference position of the slave robot, an external force applied to the master/slave robot, a reference external force applied to the slave robot, an one-way delay between the master and the slave, and a proportional gain of the master/slave position controller, respectively. In addition, √ 2 ki [s−1 ] represents a differential gain of the position controllers. The dynamics of a contact motion between the slave robot and an object is defined in Eq. (3) fs (t) = −ke (xs (t) − xe ) − de x˙ s (t).
+
+ + +
2.2 Problem Formulation
2.1 Modeling of Bilateral Control System
x ¨ref xs (t − d) − xm (t)} m (t) = km {ˆ √ +2 km {x ˆ˙ s (t − d) − x˙ m (t)} +fm (t) + fˆs (t − d),
Slave
(3)
In Section 3.1 and Section 3.2, the disturbance observer (DOB) based acceleration control and the accelerationbased impedance control are introduced, respectively. In Section 3.3, a stiffness estimation method using a recursive least square method (RLSM) with a schmitt trigger (ST) is introduced. 3.1 Acceleration Control The DOB based acceleration control system Pi is shown in Fig. 2, where i = m, s. m and mn are a mass of the robot and a nominal mass of the robot, respectively. firef is a force which is generated by a robot. A DOB is utilized to suppress the influence of fidis . A reaction force observer
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Proceedings of the 20th IFAC World Congress Daisuke Yashiro et al. / IFAC PapersOnLine 50-1 (2017) 12059–12064 Toulouse, France, July 9-14, 2017
+
12061
+ +
+ DOB
RFOB
Fig. 2. Plant model Pi .
_
+
+ +
+
+ +
+
DOB
Fig. 5. Estimation method of ke .
RFOB
Fˆs (ms2 + Ze )Zr =− ˆs (ms2 + Ze ) + Zr X
Fig. 3. Plant model P˜s .
Virtual Robot
(7)
ˆ s → −Zr . where Zr = kr +dr s. In Eq. (7), if Ze → ∞, Fˆs /X 3.3 Stiffness Estimation
Real Robot
Fig. 4. Concept of the virtual slave robot. (RFOB) provided in Murakami et al. (1993) is utilized to estimate fi . Fact 2. If a RFOB works ideally, the transfer function Fs /Xs is obtained in Eq. (6) Fs = −Ze (6) Xs where Ze := ke + de s. In Eq. (6), if Ze → ∞, Fs /Xs → −∞.
Schmitt Trigger Fig. 5 shows the concept of an object’s stiffness estimation. We proceed to derive the estimated value of ke defined by kˆe [N/m] under the assumptions xe > xs > 0 and de = 0. The position xe on the slave side that achieves −fˆs [k] = fe1 > 0 is defined as the contact position. If −fˆs [k] ≥ fe1 > 0 is satisfied, the state c transitions from 0 to 1. On the other hand, if −fˆs [k] ≤ fe2 is satisfied, the state c transitions from 1 to 0. Two kinds of thresholds fe1 and fe2 are used to change the state c, and this method is called the ST. An effect of ST is to suppress the chattering that occurs when the slave robot contacts with a hard object. Although a large fe1 reduces the influence of noise, estimation delay is increased. Besides, although a small value of fe2 reduces chattering, c may not transition from 1 to 0 even if the slave does not contact with an object. Recursive Least Square Method kˆe is obtained as follows:
The problem of deriving
Proposition 4.
3.2 Acceleration-based Impedance Control The acceleration-based impedance control system P˜s is shown in Fig. 3. x ˜s ,f˜s , and zr are a position of the virtual slave robot, an external force applied to the virtual slave robot, and impedance of the slave robot, respectively. Fig. 4 shows the concept of the virtual slave robot. The ¨ acceleration of the virtual robot x ˜s is controlled by x ¨ref s . The virtual robot is connected to the real slave robot through the spring and damper. The spring constant and damper constant are defined by kr and dr , respectively. Fact 3. If a DOB and RFOB works ideally, the transfer ˆ s is obtained in Eq. (7) function Fˆs /X
(if c = 0) kˆe = 0,
(8)
(if c = 1) kˆe = arg min
k ∑
ˆe ≥0 k i=1
( )2 λk−i kˆe [i]x′s [i] + fs′ [i] ,
(9)
where λ (0 < λ < 1) is a forgetting factor, x′s [i] := x ˆs [i] − xe + xo [m], and fs′ [i] := fˆs [i] + fe2 [N]. xo (> 0) [m] is used to avoid the divergence of kˆe . Although a large xo reduces the influence of noise, it causes an estimation error. The solution of Eq. (9) is derived from Eq. (10)
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Proceedings of the 20th IFAC World Congress 12062 Daisuke Yashiro et al. / IFAC PapersOnLine 50-1 (2017) 12059–12064 Toulouse, France, July 9-14, 2017
paper designed an adaptive controller that dynamically determines the master’s controller gain depending on the stiffness of a contact object. As the stability depends on the convergence speed of the estimated stiffness, this paper utilized an acceleration-based impedance control for fast stiffness estimation. Simulation results showed that fast stiffness estimation is achieved and the stability of bilateral control system is improved accordingly.
Table 2. Parameters for simulation.
case case case case case case
1 2 3 4 5 6
Pˆs
x ˆs
fˆs
λ
Zr
Ps Ps P˜s P˜s P˜s P˜s
xs xs x ˜s x ˜s x ˜s x ˜s
fs fs f˜s f˜s f˜s f˜s
0.99 0.9 0.99 0.9 0.99 0.9
10000+500s 10000+500s 2500+100s 2500+100s
ACKNOWLEDGEMENTS
kˆe [k] = kˆe [k − 1] x′ [k]kˆe [k − 1] + fs′ [k] γ[k − 1]x′s [k], − s ′ (10) λ + xs [k]γ[k − 1]x′s [k] } { 1 γ[k − 1]x′2 s [k]γ[k − 1] . (11) γ[k] = γ[k − 1] − λ λ + x′s [k]γ[k − 1]x′s [k] This algorithm is well known as a RLSM that uses a forgetting factor. Although a small value of λ improves convergence speed, it may cause the divergence of kˆe .
This work was supported by JSPS KAKENHI Grant Number 15H05529. REFERENCES
Hannaford, B. and Ryu, J.H. (2002). Time-domain passivity control of haptic interfaces. IEEE Transactions on Robotics and Automation, 18(1), 1 –10. Hokayem, P.F. and Spong, M.W. (2006). Bilateral teleoperation:an historical survey. AUTOMATICA, 42(12), 2035–2057. 4. SIMULATION Iida, W. and Ohnishi, K. (2004). Reproducibility and The validity of the stiffness estimation using the acceleration- operationality in bilateral teleoperation. In the IEEE International Workshop on Advanced Motion Control, based impedance control is verified by simulations. 217–222. Kitamura, K., Yashiro, D., and Ohnishi, K. (2009). Bilat4.1 Setup eral control using estimated environmental stiffness as the master position gain. In 35th Annual Conference of Bilateral control using the system shown in Fig. 1 is IEEE Industrial Electronics Society, 2981–2986. simulated. The control period, the packet-sending interval Kubo, R., Natori, K., Iiyama, N., and Ohnishi, K. (2007). between the master and the slave, and one-way delay d are Performace analysis of a three-channel control architecset to 1ms, 1ms, and 100ms, respectively. One free motion ture for bilateral teleoperation with time delay. IEEJ (1s∼5s, Ze = 0 + 0s), one contact motion with a piece of Transactions on Industry Applications, 127(12), 1224– sponge (10s∼15s, Ze = 400+10s), and one contact motion 1230. with a block of aluminum (20s∼25s, Ze = 30000+10s)) are Murakami, T., Yu, F., and Ohnishi, K. (1993). Torque tested. In the simulation, six cases are compared as shown sensorless control in multidegree-of-freedom manipulain Table 2. Parameters m, km (t), ks , x0 , fe1 , and fe2 are set tor. IEEE Transaction on Industrial Electronics, 40(2), to 0.5, kˆe (t − d), 900, 0.001, 1.0, and 0.2, respectively. 259–265. Polushin, I.G., Liu, P.X., and Lung, C. (2007). A force4.2 Result reflection algorithm for improved transparency in bilateral teleoperation with communication delay. IEEE Figs. 6–11 show the simulation results of case 1–6. (a) Transactions on Mechatronics, 12(3), 361–374. denotes the time responses of ke (dash line) and kˆe (solid Sabanovic, A., Ohnishi, K., Yashiro, D., Sabanovic, N., and Baran, E.A. (2010). Motion control systems with line). (b) denotes the time responses of xm (dash line) and network delay. AUTOMATIKA, 51(2), 119–126. xs (solid line). (c) denotes the time responses of fm (dash line) and −fs (solid line). When λ = 0.99 (Fig. 6, Fig. 8, Yashiro, D., Hieno, T., Yubai, K., and Komada, S. (2015). Design of master’s position controller for bilateral conFig. 10), the convergence speed of kˆe is slow. And the slow trol system with time delay. IEEJ Transaction on estimation causes a large oscillation of position responses Industry Applications, 135(3). ˆ and force responses. Fig. 7(a) shows that ke is in a state Yashiro, D. and Ohnishi, K. (2011). Performance analof frenzied vibration in the case of λ = 0.9. And the ysis of bilateral control system with communication position responses and force responses are influenced by bandwidth constraint. IEEE Transaction on Industrial the oscillation of kˆe . On the other hand, Fig. 9(a) and Electronics, 58(2), 436–443. Fig. 11(a) show that oscillation of kˆe is suppressed by Yashiro, D., Tian, D., and Ohnishi, K. (2010). Central the effect of impedance control. And the oscillations of controller based hybrid control with communication position responses and force responses are also suppressed. delay compensator. In 36th IEEE Annual Conference Fig. 9(b) and Fig. 11(b) show that the smaller the kr , the of Industrial Electronics Society, 3129–3134. larger the position tracking error. Yashiro, D., Yubai, K., and Komada, S. (2016). Fast estimation of environment’s stiffness for bilateral control 5. CONCLUSION systems with communication delay. IEEJ Journal of Industry Applications, 5(6), 422–428. Communication delay between a master robot and a slave robot destabilizes bilateral control systems. Hence, this 12568
Proceedings of the 20th IFAC World Congress Daisuke Yashiro et al. / IFAC PapersOnLine 50-1 (2017) 12059–12064 Toulouse, France, July 9-14, 2017
(a) ke (t), kˆe (t − d).
(b) xm (t), xs (t).
(c) fm (t), −fs (t).
(b) xm (t), xs (t).
(c) fm (t), −fs (t).
(b) xm (t), xs (t).
(c) fm (t), −fs (t).
(b) xm (t), xs (t).
(c) fm (t), −fs (t).
Fig. 6. Simulation results of case 1.
(a) ke (t), kˆe (t − d). Fig. 7. Simulation results of case 2.
(a) ke (t), kˆe (t − d). Fig. 8. Simulation results of case 3.
(a) ke (t), kˆe (t − d). Fig. 9. Simulation results of case 4.
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Proceedings of the 20th IFAC World Congress 12064 Daisuke Yashiro et al. / IFAC PapersOnLine 50-1 (2017) 12059–12064 Toulouse, France, July 9-14, 2017
(a) ke (t), kˆe (t − d).
(b) xm (t), xs (t).
(c) fm (t), −fs (t).
(b) xm (t), xs (t).
(c) fm (t), −fs (t).
Fig. 10. Simulation results of case 5.
(a) ke (t), kˆe (t − d). Fig. 11. Simulation results of case 6.
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