Nonlinear Bilateral Adaptive Impedance Control with ...

Journal of Dynamic Systems, Measurement, and Control

Nonlinear Bilateral Adaptive Impedance Control with Applications in Telesurgery and Telerehabilitation Mojtaba Sharifi Department of Mechanical Engineering, Sharif University of Technology Azadi St., P.O. Box: 11155-9567, Tehran, Iran [email protected] Saeed Behzadipour 1 Department of Mechanical Engineering, Djawad Movaffaghian Research Center in Neuro-Rehab Technologies, Sharif University of Technology Azadi St., P.O. Box: 11155-9567, Tehran, Iran [email protected] Hassan Salarieh Department of Mechanical Engineering, Sharif University of Technology Azadi St., P.O. Box: 11155-9567, Tehran, Iran [email protected]

ABSTRACT A bilateral nonlinear adaptive impedance controller is proposed for the control of multi-DOF teleoperation systems. In this controller, instead of conventional position and/or force tracking, the impedance of the nonlinear teleoperation system is controlled. The controller provides asymptotic tracking of two impedance models in Cartesian coordinates for the master and slave robots. The proposed bilateral controller can be used in different medical applications such as tele-surgery and tele-rehabilitation where the impedance of the robot in interaction with human subject is of great importance. The parameters of the two impedance models can be adjusted according to the application and corresponding objectives and requirements. The dynamic uncertainties are considered in the model of the master and slave robots. The stability and the tracking performance of the system are proved via a Lyapunov analysis. Moreover, the 1

Corresponding author.

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adaptation laws are proposed in the joint space for reducing the computational complexity however the controller and the stability proof are all presented in Cartesian coordinates. Using simulations on a two DOFs robots, the effectiveness of the proposed controller is investigated in telesurgery and telerehabilitation operations.

1. INTRODUCTION

Teleoperation systems enable human to interact with environments that are too inaccessible, too confined, or too hazardous. They have been widely used in different areas such as minimally invasive tele-surgery [1], tele-rehabilitation [2, 3] and telesonography [4] systems, nuclear waste site and radioactive material management [5], outer space and undersea exploration [6, 7], and construction/forestry machines of the excavator type [8]. As shown in Fig. 1, a teleoperation system consists of a user-interface (master) robot, a tele-operated (slave) robot, a human operator, an environment, and communication channels. Teleoperation performance is greatly improved when the haptic force feedback of the interaction point between the slave robot and the remote environment is provided to the human operator using the master robot. The control strategy of these systems is called bilateral as the information flows in two directions between the operator and the remote environment [9]. In the bilateral teleoperation robotic systems, the human operator applies force ( f h ) on the master robot to control the position of the slave robot ( x s ) in order to perform a task in the remote environment. If the slave robot perfectly tracks the master robot’s position trajectory

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( xm ) and the master robot ideally realizes the slave-environment contact force ( f e ) to the human operator, the system is called transparent. Various control strategies have been proposed for linear teleoperation systems (with one-DOF) [10-17]. Among them, the 4- channel architecture is the most successful in terms of fulfilling transparency [12, 13]. However, this control scheme assumes perfect knowledge of the linear dynamics of the master and the slave robots. In other words, these controllers need the exact model of the robots dynamics that may be unavailable in a realistic system due to uncertainties. Therefore, the stability and transparency are hard to guarantee in practice. A few adaptive controllers [18, 19] have been presented in the literature for linear master and slave models where the slave and the environment dynamics are considered to be uncertain. Moreover, to do complex tasks in multi-directional space, multi-DOF nonlinear systems are required instead of one-DOF linear ones. Accordingly, designing a nonlinear bilateral controller for uncertain nonlinear teleoperation systems is a desired objective for researchers in recent years. However, the control for master and slave robots with nonlinear models is more challenging in comparison with those with linear models. In this context, some adaptive controllers [20, 21] were proposed for nonlinear master and slave dynamics by considering linear operator and environment dynamics instead of sensing their interaction forces applied to the system. On the other hand, there are some works on the synchronization of positions and velocities of the master and slave robots using PD [22] and adaptive [23] controllers considering a time delay in the communication channel. In these controllers [22, 23], the objective is state 3


synchronization that is achieved in the absence of external force (the case of free motion), and the simultaneous position and force tracking performances cannot be provided. To achieve the position and force tracking performances for multi-DOF systems, Ryu and Kwon [24] and Hung et al. [25] designed two nonlinear adaptive controllers. As a modification on one of these controllers [24], Liu and Tavakoli [26, 27] developed two nonlinear adaptive 4-channel controllers. In the last mentioned adaptive controllers [26, 27] with position and force tracking objectives, the acceleration of each robot in addition to the velocity and position signals should be measured and used in the control law of the other robot. In this paper, we developed a new bilateral nonlinear adaptive impedance controller instead of previously mentioned bilateral position, force or combined position-force controllers. Therefore, the corresponding background and literature of the impedance/admittance control is briefly mentioned. The impedance/admittance control has recently found an important role in different physical human-robot interactions such as robot assisted rehabilitation and haptic systems. In this context, a virtual impedance/admittance model, that is the desired virtual dynamical relationship between the motion and the external forces or moments acting on the system, is realized. Many interactive tasks that could not be handled well by pure position or force control can be realized by impedance/admittance control theory [28-30]. According to the early innovative works on impedance and compliance control [31, 32], many contributions were presented in this area, including robust impedance control [33] and hybrid impedance control [34]. In addition, the adaptive impedance 4


control [28-30, 35] approximation-based impedance control [36] and adaptive admittance control in [37-39] have the advantage of decreasing the need for precise mathematical modeling of the robot and environment using adaptive control scheme. It should be noted that the above mentioned works [28-39] on the impedance and/or admittance control were performed on a single robot manipulator and not for a teleoperation (telerobotic) system including more than one robot. There are some works on employing the impedance control context for the bilateral teleoperation systems. Rubio et al. [40] and Dubey et al. [41] proposed two control algorithms with varying impedance parameters in order to increase stability and enhance tracking performance. Love and Book [42] realized a virtual master workspace with an impedance model, and reduced the operator’s fatigue during a task by regulating the damping ratio of the impedance model. Moreover, Cho and Park [43] proposed a bilateral impedance controller for linear (one-DOF) master and slave robots using a robust control scheme and Garcia-Valdovinos et al. [44] designed an observer for this controller [43]. In another work, Abbott and Okamura [45] developed impedance and admittance type bilateral controllers considering an impedance/admittance model for both master and slave robots that has only a damping element (without mass and stiffness elements). As a result, the lack of a nonlinear bilateral controller for implementing a generalized impedance/admittance scheme to uncertain nonlinear (multi-DOF) teleoperation systems is evident in the literature. Such controller will definitely require a comprehensive nonlinear stability analysis before it can be implemented in actual systems. 5


In this paper, a new Nonlinear Bilateral Adaptive Impedance Controller (NBAIC) is proposed that has the following characteristics and advantages: 1. A general nonlinear form of bilateral impedance control is designed for multiDOF teleoperation systems by defining two distinct desired impedance models for the master and slave robots, 2. Accordingly, instead of two conventional control strategies (position tracking and force tracking), the impedance/admittance (virtual dynamics) of the teleoperation system is considered as the control objective, 3. The controller can be used in different applications such as telesurgery and telerehabilitation by adjusting the target impedance parameters, 4. The proposed controller is robust against dynamic uncertainties of the system, 5. The signals of robots’ acceleration are not used in the controller due to defining the reference impedance models and controller design. However, in the recent works [26, 27, 46] on bilateral adaptive control, these signals are required because direct position and force tracking of nonlinear system are pursued (conventional bilateral control strategy), and 6. Unlike the previous adaptive bilateral controllers designed in Cartesian coordinates [26, 27, 46], the adaptation laws are proposed in the joint space for reducing the computational complexity however the controller and impedance models are developed in Cartesian coordinates. It should be noted that the characteristics and applications of this new nonlinear bilateral adaptive impedance controller are novel in the context of teleoperation 6


systems. In other words, this work is an extension of the adaptive impedance control methods (such as [28-30, 35, 37, 38]) to bilateral control by realizing two virtual environments (two impedance models). The paper is structured as follows: In section 2, the nonlinear model of a multiDOF uncertain teleoperation system is presented. In section 3, the bilateral nonlinear adaptive impedance controller is proposed. The desired impedance models of the master and slave robots are introduced and discussed in section 3.1. Control laws of the master and slave robots are designed and presented in section 3.2. In section 3.3, the closed-loop dynamics of the teleoperation system using the proposed bilateral adaptive impedance controller is obtained. Moreover, the adaptation laws of the master and slave robots are described in section 3.4. The stability and convergence of the teleoperation system are proved in section 3.5 using the Lyapunov theorem. The simulations of the teleoperation system and controller in telesurgery (Sec. 4.1) and telerehabilitation (Sec. 4.2) applications are presented and discussed in section 4 to show the effectiveness of the proposed bilateral control method. Finally, the concluding remarks are mentioned in section 5.

2. NONLINEAR MODEL OF AN UNCERTAIN TELEOPERATION SYSTEM

The n-DOF master and slave robot manipulators are modeled in the joint space as follows [47-49]:

M q,m (qm ) qm  Cq,m (qm , qm ) qm  Gq,m (qm )  Fq,m (qm )  τ m  τ h

(1)

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M q,s (qs ) qs  Cq,s (qs , q s ) qs  Gq,s (q s )  Fq,s (q s )  τ s  τe where q m and qs 

n1

are the joint positions, M q,m (qm ) and M q,s (qs ) 

inertia or mass matrices, Cq,m (qm , qm ) and Cq,s (qs , qs )  centrifugal terms, Gq,m (qm ) and Gq,s (qs )  Fq,s (qs ) 

n1

n1

nn

(2) nn

are the

includes the Coriolis and

are the gravity terms, Fq,m (qm ) and

are the viscous and coulomb friction terms, and τ m and τ s 

n1

are the

vectors of the control torques (originated from the actuators) of the master and the slave robots, respectively. Also, τh  applies to the master robot and τe 

n1

n1

is the external torque that the operator

is the external torque that the environment

applies to the slave robot. The dynamics of the robots in the Cartesian space is expressed as:

M x,m (qm ) xm  Cx,m (qm , qm ) xm  Gx,m (qm )  Fx,m (qm )  fm  fh

(3)

M x,s (qs ) xs  Cx,s (qs , q s ) xs  Gx,s (q s )  Fx,s (q s )  fs  fe

(4)

where xm and xs 

61

are the Cartesian coordinates of the master and slave robots’

end-effectors, respectively. f h and fe 

61

are the external interaction forces that the

operator applies to the master, and the environment applies to the slave, respectively. Considering the subscript i  m for the master robot and i  s for the slave robot, the kinematic transformations between the joint and Cartesian spaces are expressed as:

xi  i (qi ),

xi  J i (qi ) qi ,

xi  J i (qi ) qi  J i (qi ) qi

(5)

8


where Ji (qi )   d i (qi ) d qi  

6n

is the Jacobian matrix. The relations between the

matrices of dynamic models in the joint space (Eqs. (1) and (2)) and in the Cartesian space ones (Eqs. (3) and (4)) for non-redundant and non-singular manipulators are:

M x,i (qi )  J i T M q,i (qi ) J i 1





Cx,i (qi , qi )  J i T Cq,i (qi , qi )  M q,i (qi ) J i 1 J i J i 1 Gx,i (qi )  J i T Gq,i (qi ) , f i  J i T τ i ,

Fx,i (qi )  J i T Fq,i (qi )

f h  J m T τ h ,

(6)

f e  J s T τ e

Using i  m and/or i  s the above relations correspond to master and slave, respectively. The matrices expressed in dynamic model equations (1) to (4) have the following properties [48-50]: Property 1. Inertia matrices M q,i (qi ) and M x,i (qi ) are symmetric and positive definite. Property 2. Matrices M q,i (qi )  2Cq,i (qi , qi ) and M x,i (qi )  2Cx,i (qi , qi ) are skew symmetric. Property 3. The left side of Eqs. (1) to (4) can be linearly parameterized in terms of the model parameters such that:

M q,i (qi ) φ1,i  Cq,i (qi , qi ) φ2,i  Gq,i (qi )  Fq,i (qi )  Yq,i (φ1,i , φ2,i , qi , qi ) αq,i

(7)

M x,i (qi ) ψ1,i  Cx,i (qi , qi ) ψ 2,i  Gx,i (qi )  Fx,i (qi )  Yx,i (ψ1,i , ψ 2,i , qi , qi ) αx,i (8) where α q ,i and α x ,i are the vectors of unknown parameters of the robot when parameterization is done in the joint space and Cartesian space, respectively. The regressor matrices Yq ,i and Yx,i contain known functions and φ1,i , φ 2,i , ψ 1,i and ψ 2,i are the known vectors.

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3. NONLINEAR BILATERAL ADAPTIVE IMPEDANCE CONTROL

The block diagram representation of the teleoperation system and transmitted signals in communication channels are shown in Fig. 2. As shown in this figure, the state variables (position and velocity) and the interaction force signals of the master robot (in the local site) are transmitted to the slave (remote site) by two communication channels and the interaction force of the slave robot is transmitted to the master by another channel of the teleoperation system.

3.1. Impedance Models of the Master and Slave The desired impedance model of the master robot’s end-effector expresses the desired dynamical relationship between the human operator and master robot that is defined in Cartesian space as:

mm xmodm  cm xmodm  km (xmodm  x0 )  fh  kf fe

(9)

where xmodm is the response position vector of the master impedance model, and x 0 is the position that corresponds to the natural length of the virtual stiffness element ( k m ). In other words, x 0 is the desired position that the master robot should hold in the absence of interaction forces (when fh  fe  0 ). As mentioned before, f h is the interaction force that the operator applies to the master robot and fe is the interaction force that the environment applies to the slave robot. The force scaling factor kf in Eq. (9) used for the transmitted interaction force f e from the slave to the master (in the

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local site). mm , cm and k m are the desired virtual mass, damping, and stiffness coefficients of the master impedance model, respectively. In order to have different characteristics in different directions, these three parameters can be defined in matrix form. According to Eq. (9), when the master robot reaches its objective of tracking the master impedance model response ( xmodm ), the operator senses the environment interaction force ( f e ) and also the desired impedance elements ( mm , cm and k m ). In other words, the human operator’s perception is that he/she is interacting with the environment using a virtual tool with desired dynamics mm , cm and k m as shown in Fig. 3. It should be mentioned that the master impedance model (9) is constructed and solved in the local site to obtain the desired trajectory ( xmodm ) of the master robot. Accordingly, as shown in Fig. 2, the measurement of the interaction force ( fe ) between the environment and the slave should be transferred from the remote site to the local site using a communication channel to be used in the master impedance model (9). The desired impedance model of the slave robot’s end-effector also represents the desired dynamics between the environment and slave robot during the tracking of the master robot trajectory:

ms xmods  cs xmods  ks xmods  fe

(10)

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where ms , cs and k s are the desired virtual mass, damping, and stiffness coefficients of the slave impedance model in physical interaction with environment. Also, x mod s is the error vector of slave impedance response with respect to the master position that is defined as:

xmods  xmods  kpxm

(11)

where kp is the scaling factor corresponds to the transmitted position and velocity vectors of the master to the slave. It should be mentioned that the position ( x m ), velocity ( x m ) and acceleration ( x m ) of master robot are required to obtain the desired slave impedance model response ( x mod s , x mod s , x mod s ), which will be used in the bilateral adaptive controller. Since a direct measurement of the master robot acceleration ( x m ) is challenging, it is estimated with good accuracy when the master robot mimics its impedance model (9), as:

xm est

c k 1  fh  kf fe   m xm  m (xm  x0 ) mm mm mm

(12)

In Eq. (12), the master robot acceleration is estimated in terms of the interaction forces, master position and velocity that are available signals according to the master impedance model (9). Also, when the master robot trajectory ( x m ) asymptotically converges to the master impedance model response ( x modm ), Eq. (12) has a higher accuracy in obtaining the master acceleration. Also, the acceleration estimation 12


accuracy can be improved by specifying the same initial position and velocity for the master robot and its corresponding impedance model (9). It is known that the slave impedance model (10) is constructed and solved in the remote site to obtain the desired trajectory ( xmod s ) of slave robot. According to Eq. (10), the master robot’s position, velocity and acceleration ( x m , x m and x m ) are required in the slave impedance model. However, using the estimation of master robot acceleration (12), the measurement of the interaction force ( f h ) between the operator and master together with the master robot’s position and velocity ( x m and x m ) should be transferred from the local site to the remote site using two communication channels (instead of x m , x m and x m ), as shown in Fig. 2. Due to Eq. (10), by increasing the slave impedance parameters ( ms , cs and k s ), the tracking error of slave ( xmod ) with respect to master is decreased (the slave s

becomes more accurate in tracking of the master trajectory). On the other hand, by decreasing these impedance parameters in (10), the slave becomes more flexible in its interaction with environment and consequently has higher tracking error ( xmods ). Accordingly, two impedance models are employed in this methodology. The first one is the master reference model (9) that the operator perceives, and the second one is the slave reference model (10) that is the flexibility of the slave robot in its physical interactions with the environment. These concepts are illustrated in Fig. 4. The proposed bilateral impedance control strategy approaches an exact position tracking and/or exact force tracking by adjustment of the parameters in two impedance 13


models (9) and (10). In master impedance model (9), by choosing small values for the impedance parameters ( mm , cm and k m ), the left side of Eq.(9) is small due to the boundedness of position signals xmodm and x0 . Accordingly, the right side of Eq. (9) should also be small ( (fh  kf fe )  0 ); therefore, the force tracking (reflecting) performance is approximately achieved ( fh  kf fe ). On the other hand, by increasing the master impedance parameters ( mm , cm and k m ), the force tracking performance is not obtained and the operator (with f h ) senses the impedance elements ( mm , cm and k m ) in addition to the scaled environment force ( kf fe ). Also, due to the boundedness of the right side of Eq. (10) that is fe , the left side of this equation is also bounded. Consequently, by utilizing large values for the slave impedance parameters ( ms , cs and k s ) in the left side of Eq. (10), the desired slave position error with respect to the scaled master trajectory ( xmods  xmods  kp xm ) becomes small. In other words, the position tracking performance ( xmods  kp xm ) is approximately obtained using large impedance parameters for the slave. On the other hand, if the parameters of the slave impedance are decreased, the flexibility of the slave with respect to the master trajectory is increased and the tracking error ( x mod s ) becomes large. As a result, to have the transparency condition in which position and force tracking performances are provided ( xmods  kp xm and fh  kf fe ), the master

14


impedance parameters should be decreased (for force reflecting) and the slave impedance parameters should be increased (for position tracking). Accordingly, in telesurgery applications in which the slave tracking accuracy is more important in comparison with the slave flexibility (compliance) in the environment interactions, the slave impedance parameters ( ms , cs and k s ) should be increased as much as needed. Also, in telerehabilitation and telesonography applications for which the slave robot should have higher compliance in its interaction with the patient (environment), the slave impedance parameters should be decreased. In other words, in telerehabilitation systems that the slave robot moves the patient limbs with a limited flexibility, the patient is permitted to deviate from the initially designed (master) trajectory using the slave impedance model (10). This performance is approximately required in telesonography systems where the slave robot should show a flexible interaction with the patient tissue or limb.

3.2. Nonlinear Control Laws of the Master and Slave Robots The general structure of the proposed bilateral impedance controller is shown in Fig. 5 and their details are given in this section. As mentioned earlier, there are two impedance models (9) and (10) that their responses are tracked by adaptive controllers of master and slave robots. The dynamic models of master and slave robots are considered to have fully parametric uncertainties. Also, modeling the human operator and the remote environment is not required because their interaction forces ( f h , f e ) applied to robots are directly measured using force/torque sensor.

15


Now, the adaptive control laws designed for the master and slave robots are presented. For this purpose, the sliding surfaces used in the bilateral controller are defined for the master and slave robots as:

Sm  xm  1,m xm , S s  xs  1,s xs

(13)

where xm  x m  x modm and x s  x s  x mods are the vectors of the master and slave position error with respect to the responses of their impedance models (9) and (10). 1,m and

1,s are positive constant parameters that guarantee the stability of the sliding surfaces (13). Now, we introduce the master and slave reference velocities as additional variables:

xr,m  xmodm  1,mxm , xr,s  xmods  1,s xs

(14)

The sliding surfaces can be reformulated as:

Sm  xm  xr,m , Ss  xs  xr,s

(15)

Also, the master and slave reference accelerations are obtained from Eq. (14) as:

xr,m  xmodm  1,mxm , xr,s  xmods  1,s xs

(16)

where xmodm is obtained from master impedance model (9). Also, due to Eq. (11),

xmods  xmods  kp xm where xmods is obtained from the slave impedance model (10). However, the exact master acceleration xm is not available and it is estimated by xm est in Eq. (12). This acceleration estimation may has an error as xm  xm est  xm which is rationally assumed to be bounded. Thus, the estimated variable of

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xmods

est

 xmod  kp xm est should be used instead of its real value xmod in Eq. (16) to s s

obtain the estimated slave reference acceleration as:





xr,s est  xmods  1,s xs  xmods  kp xm est  1,s xs est

(17)

which can be reformulated in terms of estimation error of master acceleration ( xm ) and the exact slave reference acceleration ( xr ,s in Eq. (16)) as:

xr ,s est  xmods   kp xm  kp xm   1,s x s  xmods  kp xm  1,s x s  xr ,s  kp xm

(18)

Now, the nonlinear adaptive control laws of the master and slave robots for tracking their impedance models are defined in Cartesian coordinates, respectively, as:

fm  Mˆ x,m (q m )  xr ,m  2,m S m   Cˆ x,m (q m , q m ) xr ,m  Gˆ x,m (q m )  Fˆx,m (q m )  fh





(19)

f s  Mˆ x,s (q s ) xr ,s est  2,s S s  Cˆ x ,s (q s , q s ) xr ,s  Gˆ x, s (q s )  Fˆx,s (q s )  fe   s sgn(S s ) Accent



(20)

denotes the estimated values of matrices, vectors and scalars. 2,m and 2,s

are two positive constants that guarantee the stability and convergence of the teleoperation system, which will be determined using the Lyapunov theorem.  s in the control law of the slave robot (20) is a positive constant parameter, and it will be shown that

 s sgn(Ss )

provides the robustness of the bilateral controller against the bounded

estimation error of the master robot’s acceleration ( xm ).

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The actual control input of robots applied in the joint space by motor torques is obtained using Eq. (6) in (19) and (20), as:

τm  J

T m

τs  Js

T









ˆ (q ) xr ,m   Sm  Cˆ x,m (qm , qm ) xr ,m  M 2, m  x,m m   Gˆ (q )  Fˆ (q )  f  x,m m x ,m m h    Mˆ (q ) x  ˆ r , s est  2, s S s  Cx, s (q s , q s ) xr , s   x, s s  ˆ  ˆ  Gx,s (q s )  Fx,s (q s )  fe   s sgn(S s ) 

(21)

(22)

With the purpose of employing parameterization in the joint space, the master and slave input torques ( τ m and τ s ) are expressed in terms of joint space matrices and vectors using Eq. (6) as:

τ m  Mˆ q,m J m 1  xr ,m  2,m S m 





 Cˆq,m  Mˆ q,m J m 1 J m J m 1 xr ,m  Gˆ q,m  Fˆq,m  J mT f h



τ s  Mˆ q , s J s 1 xr ,s est  2,s S s





(23)



 Cˆq, s  Mˆ q, s J s 1 J s J s 1 xr ,s  Gˆ q,s  Fˆq,s  J sT fe  J sT  s sgn(S s )

(24)

The above relations for the motor input torques can be written as:

τ m  Mˆ q,m v1,m  Cˆq,m v2,m  Gˆq,m  Fˆq,m  J mT fh

(25)

τ s  Mˆ q,s v1,s  Cˆq,s v2,s  Gˆq,s  Fˆq,s  J sT fe  J sT  s sgn(Ss )

(26)

18


where v1,m , v2,m , v1,s and v2,s are known vectors and do not contain any estimated parameters of the system’ dynamics. These vectors are obtained for the master and slave robots as follow:

v1,m  J m1  xr,m  2,m Sm   J m1J m J m1xr,m , v2,m  J m1 xr,m





v1,s  J s 1 xr,s est  2,s Ss  J s 1J s J s 1xr,s , v2,s  J s 1 xr,s

(27)

(28)

According to Property 3, the adaptive control laws of teleoperation system in the joint space (Eqs. (25) and (26)) are linearly parameterized as:

τ m  Yq,m αˆ q,m  J mT fh τ s  Yq,s αˆ q,s  J sT fe  J sT  s sgn(Ss )

(29)

(30)

where Yq,m αˆ q,m and Yq, s αˆ q, s are linear parameterizations of the first four terms of Eq. (25) and (26), respectively. Also, α q ,m and α q ,s are the vectors of actual parameters of the master and slave robots, respectively, and αˆ q ,m and αˆ q , s are their estimations. Yq,m and Yq,s are the joint space regressor matrices for the master and slave robots, respectively. As a result, the linear parameterization for each robot is expressed as:

Yq,i αq,i  M q,i v1,i  Cq,i v2,i  Gq,i  Fq,i

(31)

where v1,i and v2,i were obtained in Eqs. (27) and (28) (using i  m and i  s the above relation is defined for the master and the slave, respectively).

19


3.3. The Closed-loop Dynamics of the Teleoperation System In this section, the closed-loop dynamics of the master and slave robots using the proposed bilateral adaptive impedance controller is obtained. For this purpose, the control laws (19) and (20) are replaced in the teleoperation system dynamics (3) and (4) that yields:

M x ,m  x m  xr ,m    2,m M x ,m S m

   Cˆ   Gˆ

 Mˆ x ,m  M x ,m





x

r ,m

    Fˆ

 2,m S m 

x ,m

 Cx , m x r , m  Cx , m S m

x ,m

 Gx ,m

x ,m

 Fx ,m

(32)



M x , s x s  xr , s est   2, s M x , s S s

   Cˆ   Gˆ

 Mˆ x, s  M x, s

x

r ,s

    Fˆ

 2, s S s 

x,s

 Cx , s x r , s  C x , s S s

x,s

 Gx, s

x,s

(33)



 Fx, s   s sgn(S s )

According to Property 3 and using Eqs. (27), (28) and (31), the above equations can be rewritten as:

M x,m  xm  xr ,m    2,m M x,m Sm  Cx,mSm  J m T Yq,m αq,m





M x,s x s  xr ,s est   2,s M x,s S s  Cx,sS s  J s T Yq,s αq,s   s sgn(Ss )

(34)

(35)

20


where αq,m  αˆ q,m  αq,m and αq,s  αˆ q,s  αq,s are the error vectors of master and slave parameter estimations, respectively. Using Eq. (18) for the estimated value of the slave reference acceleration ( xr ,s est ) in Eq. (35), yields to:

M x,s  xs  xr ,s   kp M x,s xm   2,s M x,s Ss  Cx,sSs  J s T Yq,s αq,s   s sgn(Ss ) (36) where xm  xm est  xm is the bounded estimation error of master acceleration. Substituting the time derivative of Eq. (15) in Eqs. (34) and (36) yields the final error dynamics or closed-loop dynamics of the teleoperation system:

M x,m Sm   2,m M x,m Sm  Cx,mSm  J m T Yq,m αq,m

(37)

M x,s Ss   2,s M x,s Ss  Cx,s Ss  J s T Yq,s αq,s  kp M x,s xm   s sgn(Ss )

(38)

3.4. Composite Adaptation Laws A kind of composite adaptation with a bounded forgetting gain is used in the proposed bilateral controller, in which the estimation error ( ei ) is used in addition to the tracking error ( S i ). Now, the estimation error for each of the master and slave robots is defined. The master or slave robot dynamics equation in the joint space is expressed as follows:

M q,i (qi ) qi  Cq,i (qi , qi ) qi  Gq,i (qi )  Fq,i (qi )  τq,i

(39)

21


where in this paper, for the master: i  m and τq,m  τ m  τ h , and also for the slave:

i  s and τq,s  τ s  τ e . Then, the above dynamics passes through a first-order filter that has the following Laplace form:

W ( s) 

 s

(40)

where w(t )   e  t is its impulse response in time domain. Both sides of Eq. (39) are convolved by w(t ) to have:



t

0

w(t  r )  M q,i qi  Cq,i qi  Gq,i  Fq,i  dr   w(t  r ) τq,i (r ) dr t

0

(41)

The first term in the left side of (41) is obtained using the partial integration, as:

t

 w(t  r )  M 0

q ,i

qi  dr  d  w(t  r ) M q ,i  qi dr  0 dr 

w(t  r ) M q ,i qi 0   t

t

(42)

t

w(0) M q ,i qi  w(t ) M q ,i (qi (0))qi (0)    w(t  r ) M q ,i qi  w(t  r ) M q ,i qi dr 0 As a result, the left side of Eq. (41) is obtained as:

Wq,i (qi , qi ) αq,i  y i (t )

(43)

where Wq,i is the filtered version of Yq ,i in Eq. (7) with φ1,i  qi and φ2,i  qi . Moreover, due to the filtering and partial integration (42), the elements of matrix Wq,i are only functions of the available measurements of q i and q i , and the acceleration

22


( q i ) is not required. Also, Eq. (43) is the linear parameterization of the filtered dynamics (41), and yi (t ) is the filtered torque defined as:

t

 w(t  r ) τ 0

q ,i

(r ) dr  y i (t )

(44)

Equation (43) can be written for αˆ q,i (the estimated value of α q,i ) as:

Wq,i (qi , qi ) αˆ q,i  yˆ i (t )

(45)

Then, the estimation error ( ei ) for each robot is defined as:

ei  yˆ i (t )  yi (t )

(46)

Consequently, the estimation error can be rewritten in terms of the parameter estimation error as:

ei  Wq,i (qi , qi ) αq,i

(47)

Now, the bounded gain forgetting (BGF) composite adaptation laws of master and slave are defined as:

where Ri 

nn

αˆ q,m   Pm (t )T Yq,mT J m 1Sm  Wq,mT Rm em 

(48)

αˆ q,s   Ps (t )T Yq,sT J s 1Ss  Wq,sT Rs es 

(49)

(for master i  m or slave i  s ) is a positive definite weighting matrix

that indicates the cost of estimation error ei (having the parameters information) in the adaptation law. For simplicity, Ri is chosen as: 23


Ri  ai I nn

(50)

where I nn is the unity matrix with the same dimensions as Ri , and ai is a positive constant. Also, Pi (t ) in (48) and (49) is the adaptation gain that is a uniformly positive definite matrix determined by an exponentially forgetting least-squares update rule as:

d 1 Pi (t )   i (t ) Pi 1  Wq ,iTWq,i dt

(51)

that has the following variable forgetting factor:

i (t )  0,i 1  Pi k0,i 

(52)

where 0,i and k0,i are two positive constants specifying the maximum forgetting rate and the upper bound of the gain matrix norm ( Pi ), respectively. The gain update rule (51) guarantees that  t  0 , i (t )  0 , and Pi (t )  k0,i I for any signal of Wi [49]. Moreover, if Wi is positive definite (persistently excited),  1,i  0 such that  t  0 ,

i (t )  1,i [49]. It should be mentioned that Eqs. (48) and (49) express the adaptation laws in joint space to reduce the complexity of calculations; however the closed-loop dynamics and the control laws are presented in Cartesian coordinates. In the following Lyapunov stability proof, it will be shown that the tracking error converges to zero independent from persistent excitation condition in parameter estimation.

24


3.5. Proof of Lyapunov Stability and Tracking Convergence A Lyapunov function candidate in terms of tracking errors ( S m and S s ) and parameter estimation errors ( α q ,m and α q ,s ) is used to prove the stability and convergence of the teleoperation system:

V



1 T Sm M x,m S m  STs M x,s S s  αqT,m Pm (t )1 αq,m  αqT, s Ps (t )1 αq, s 2



(53)

The Lyapunov function terms are proposed in two different coordinates, the first two terms (tracking errors) are in Cartesian coordinates while the last two terms (parameters estimation errors) correspond to the joint space parameterization. This is because the adaptation laws are given in joint space to reduce the complexity of their calculations in comparison with previous works [26, 27, 46] that present the adaptation law in Cartesian space. Since the inertia matrix M x ,i and the adaptation gain matrix Pi are uniformly positive definite, the Lyapunov function is positive definite ( V  0 ). The time derivative of V is found as:

1 1     V  STm  M x ,m S m  M x ,m S m   STs  M x ,s S s  M x ,s S s  2 2     1 d  αˆ q ,mT Pm (t ) 1 α q ,m  α q ,mT Pm (t ) 1 α q ,m 2 dt 1 d  αˆ q , sT Ps (t ) 1 α q , s  α q , sT Ps (t ) 1 α q , s 2 dt







(54)



25


where αq,i  αˆ q,i , because αq,i  αˆ q,i  αq,i and α q ,i is the constant vector of actual parameters; thus αq,i  0 . Substituting the final error dynamics of master and slave robots (Eqs. (37) and (38)) in the time derivative of V , yields:

V   2,m STm M x ,m S m  2, s STs M x , s S s 1 1  STm  M x ,m  2Cx ,m  S m  STs  M x , s  2Cx , s  S s 2 2 1 d  S mT J m T Yq ,m α q ,m  αˆ q ,mT Pm (t ) 1 α q ,m  α q ,mT Pm (t ) 1 α q ,m 2 dt 1 d  S sT J s T Yq , s α q , s  αˆ q , sT Ps (t ) 1 α q , s  α q , sT Ps (t ) 1 α q , s 2 dt T  S s  kp M x , s x m   s sgn(S s ) 







(55)



According to Property 2 of robot dynamics that M x,i  2Cx,i is skew symmetric, Eq. (55) simplifies to:

V   2,m STm M x ,m S m  2,s STs M x , s S s 1 d  S mT J m T Yq ,m α q ,m  αˆ q ,mT Pm (t ) 1 α q ,m  α q ,mT  Pm (t ) 1  α q ,m 2 dt 1 d  S sT J s T Yq , s α q , s  αˆ q , sT Ps (t ) 1 α q , s  α q , sT  Ps (t ) 1  α q ,s 2 dt  S sT  kp M x , s x m   s sgn(S s ) 

(56)

Then, using the bounded gain forgetting (BGF) composite adaptation laws (48) and (49), we have:

26


V   2,m STm M x ,m S m  2, s STs M x, s S s 1 d  emT RmTWm α q ,m  α q ,mT  Pm (t ) 1  α q ,m 2 dt 1 d  e sT RsTW α q , s  α q , sT  Ps (t ) 1  α q , s 2 dt  S sT  kp M x , s x m   s sgn(S s ) 

(57)

Also, using the exponentially forgetting least-squares update rule (51) for the adaptation gain Pi (t ) , the time derivative of V is obtained as:

V   2,m STm M x ,m S m  2, s STs M x , s S s

 m (t ) 1 α q ,mT Pm (t ) 1 α q ,m  α q ,mTWmTWm α q ,m 2 2  (t ) 1  e sT RsTW α q , s  s α q , sT Ps (t )1 α q ,s  α q ,sTWsTWs α q ,s 2 2  S sT  kp M x , s x m   s sgn(S s ) 

 emT RmTWm α q ,m 

(58)

After that, substituting the relation of the estimation error (47) and Eq. (50), we have:

V   2,m STm M x ,m S m  2,s STs M x ,s S s

 (t ) 1    am   α q ,mTWmTWm α q ,m  m α q ,mT Pm (t ) 1 α q ,m 2 2   (t ) 1    as   α q , sTWsTWs α q , s  s α q , sT Ps (t ) 1 α q , s 2 2 

(59)

 S sT  kp M x, s x m   s sgn(S s ) 

In order to have robustness against bounded estimation error of master robot acceleration  xm  , the positive constant parameter

s

should satisfy the following

component-wise inequality:

 s  kp M x,s xm   s

(60) 27


where  s is a positive constant, and . T



gives the infinity norm (maximum norm) of a

vector, i.e.: if r =[r1 , r2 , ... , ri ] is a vector with

i elements: r  = max (r1 , r2 , ... , ri ) . It

should be noted that the actual parameters in M x,s are uncertain in this adaptive control strategy and the error of acceleration estimation  x m  is also unknown; however to have the stability property, the maximum estimated value of M x,s xm should be bounded and its bound, as mentioned in Eq. (54), should be less than other words,

s

s



. In

should be chosen as large as needed (using a trial and error method)

such that the stability of the system is provided; which means inequality (60) is satisfied. Then, the time derivative of Lyapunov function (59) can be written as:

V   2,m STm M x ,m S m  2, s STs M x ,s S s

 (t ) 1    am   α q ,mTWmTWm α q ,m  m α q ,mT Pm (t ) 1 α q ,m 2 2   (t ) 1    as   α q , sTWsTWs α q , s  s α q , sT Ps (t ) 1 α q , s 2 2   s Ss 1

(61)

By choosing ai  1 2 and noting that M x ,i and Pi are positive definite and according to the property of update rule (  t  0 , i (t )  0 ), it is concluded that V is negative definite. Using the Lyapunov theorem and Barbalat’s lemma [49], and according to the positive definiteness of V and negative definiteness of V , the stability of teleoperation system and the convergence to Sm  0 and Ss  0 are proved. Since the dynamics of

28


master and slave sliding surfaces S m and S s are stable (see Eq. (13)), the convergence of tracking errors to zero ( x m  0 and x s  0 ) on the surface of Sm  0 and Ss  0 are proved as t   . Also, as mentioned before for the update rules (51) and (52), Pi (t )  k0,i I and the positive definiteness of Wi (which means the persistent excitation of Wi ) guarantee that

i (t )  1,i  0 . Thus, one can write:

i (t ) αq,iT Pi (t )1 αq,i  1,i (t ) αq,iT αq,i k0,i

(62)

Under this condition, the convergence of i (t ) αq,iT Pi (t )1 αq,i  0 leads to αq,i  0 , which implies the convergence of parameter estimation. Therefore, it is obtained that

i (t )  1,i  0 is the persistent excitation condition for the convergence of parameter estimation ( αq,i  0 ). A positive constant  0,i is defined as  0,i  min  22,i , 1,i  for each of master ( i  m ) and slave ( i  s ). Also, by defining  0  min  0,m , 1,s  and using Eqs. (53) and (61), it can be written for the Lyapunov function that:

V (t )  0 V (t )  0

(63)

Thus, it is concluded that the value of Lyapunov function converges to zero as: V (t ) V (0) e0t . As a result, the exponential convergence of tracking error S i and

estimation error αq,i to zero is proved due to the definition of Lyapunov function (48). Before illustrating the simulation results, it should be mentioned that the “sgn” function in the slave control law (20) or (24) may lead to undesired discontinuities and 29


chattering in the input torques. This function is usually replaced in practice by an approximate continuous alternatives such as tangent hyperbolic or saturation functions. In this study, tanh(400 S s ) is used instead of discontinuous sgn(S s ) .

4. SIMULATIONS

In this section, the proposed controller is evaluated by simulations on a two-DOF robot with one Revolute and one Prismatic joint, as shown in Fig. 6. Each of the master and slave robots is equipped with a force sensor to measure the externally applied force and moment. The robots along with the proposed bilateral controller are modeled in Simulink-Matlab software. According to Fig. 6, the end-effector can move in the plane of x  z . The position of the end-effector in the joint space of the robots is expressed as

qm  1 l2 m and q s  1 l2  s for master and slave, respectively, where 1 is the T

T

revolute joint position and l z is the prismatic joint position. The Cartesian position of the end-effector for each robot is defined as xi   x

z  i . The Jacobian matrix of the T

master ( i  m ) and slave ( i  s ) robots is then found as:

   l  l  sin(1 ) Ji   1 2   l1  l2  cos(1 )

cos(1 )   sin(1 )  i

(64)

Employing the Lagrange method, the dynamic equations of each robot are obtained as:

30


I  I  m l 1

2

2 1 1g





 m2  l1  l2  l2 g  1  2m2  l1  l2  l2 g  l21 2



 m1l1g  m2  l1  l2  l2 g  g cos(1 )  c11  1 sgn(1 )  1 m2 l2  m2  l1  l2  l2 g 12  m2 g sin(1 )  c2l2  2 sgn(l2 )   2

(65)

(66)

where the subscript 1 denotes link 1 and joint 1 (revolute), and subscript 2 is used for link 2 and joint 2 (prismatic). Also, constant parameters I1 , I 2 , m1 , m2 , l1 , l1g , l2 g are the inertia and dimensional properties of link 1 and 2 of the robots. Moreover, c1 and c2 are the viscous friction coefficients of joints 1 and 2, respectively. Similarly, 1 and  2 are the coulomb friction coefficients of joints 1 and 2, respectively. 1 and  2 are the applied torques on the revolute ( 1 ) and prismatic ( l2 ) joints, respectively. By arranging Eqs. (65) and (66) in the standard form of dynamics presented in the joint space, the matrices and vectors in Eqs. (1) and (2) are obtained as:

 I  I  m l 2  m  l  l  l 2 1 1g 2 1 2 2g 1 2 M q ,i (qi )    0   m2  l1  l2  l2 g  l2 Cq ,i (qi , qi )    m2  l1  l2  l2 g 1 



0     m2  i

m2  l1  l2  l2 g 1    0 

(67) i



 m1l1g  m2  l1  l2  l2 g  g cos(1 )    Gq ,i (qi )      m2 g sin(1 )  i

c11  1 sgn(1 )  1    Fq ,i (qi )   ,     i int   i   2 i c2l2  2 sgn(l2 )  i 31


where for the master  int m   h and for the slave  int s   e . Moreover, the inertia matrix can be rewritten in a compact form of:

 M11  m2l22  2m2  l1  l2 g  l2  M q ,i (qi )    0 

0    m2  i

(68)

where M11  I1  I 2  m1l1g2  m2 l1  l2 g  is a constant parameter. 2

Now, the linear parameterization of the robot dynamics in the joint space that was used in (29), (30) and (31) is found as:

αq   M11 m2 m2  l1  l2 g  m1 gl1g

m2 g

Considering the known vectors v1  v11

m2 g  l1  l2 g 

c1

c2

1

T

2  (69)

v12  and v2  v21 v22  in Eq. (31), the T

T

elements of the 2 10 regressor matrix Yq are defined as:

Y1,1  v11 ,





Y2,1  0,

Y1,2  l2 2 v11  l2 l2 v21  1 v22 ,

Y2,2  v12  l21 v21 ,

Y1,3  2 l2 v11  l2 v21  1 v22 ,

Y2,3  1 v21 ,

Y1,4  cos(1 ) ,

Y2,4  0,

Y1,5  l2 cos(1 ) ,

Y2,5  sin(1 ) ,

Y1,6  cos(1 ) ,

Y2,6  0,

Y1,7  1 ,

Y2,7  0,

Y1,8  0,

Y2,8  l2 ,

Y1,9  sgn(1 ) ,

Y2,9  0,

Y1,10  0,

Y2,10  sgn(l2 ) ,

(70)

32


The parameters of robots’ kinematics and dynamics (67), control laws ((19) and (20)) and adaptation laws ((48) and (49)) that are used in simulations are given in Table 1. The parameters of the control laws are adjusted such that each controller has its minimum tracking errors ( x m , x s ). Similarly, the parameters of the adaptation laws are chosen using trial and error such that the minimum RMS of estimation errors ( em  Wq ,mαq ,m , e s  Wq , s αq , s ) is obtained.

4.1. Telesurgery Application: Position and Force Tracking Objectives In the first step of simulations, a telesurgery operation on a soft tissue is performed with transparent condition (position and force tracking performances). To simulate the teleoperation system, the interaction forces should be produced according to the dynamics models of the human operator and the environment. These models are presented by the following general forms [20, 21, 26, 27, 46, 51, 52]:

fh  fh  (M h xm  Ch xm  Kh (xm  x0 ))

(71)

fe  M e xs  Ce xs  Ke (xs  x0 )

(72)

where f h is the active force of human operator’s hand generated by the muscles.

M h  mh I , M e  me I , Ch  ch I , Ce  ce I , Kh  kh I and Ke  ke I are diagonal matrices corresponding to the mass (inertia), damping and stiffness of the operator and environment, respectively. These dynamic parameters were presented in [26, 27, 46] for a human operator (surgeon) and soft tissue environment that are listed in Table 2.

33


Also, the active force of human operator is considered to be a combination of some harmonic functions (like realistic interaction forces presented in experimental studies [28, 29]):

f hx  f hz  30  25cos(t )  5cos(6 t ) N

(73)

In the transparent condition, as discussed in Sec. 3.1, parameters of the master impedance model (9) are chosen to be small (to obtain force tracking performance) and the parameters of the slave impedance model (10) are chosen to be large (to obtain position tracking performance). Also, the position and force scaling factors are considered to be kp  1 and kf  1 in order to provide the exact tissue force for the surgeon and to have equal trajectories during the operation. The values of these impedance parameters and scaling factors are given in Table 3 for telesurgery application. Moreover, using these parameters, the master impedance model (9) has two negative real poles ( 2 , 2 ), and the slave impedance model (10) has two negative real poles ( 20 , 20 ). In other words, according to the standard expression of linear second order systems, the master impedance model has a natural frequency of 2 rad/sec and a damping ratio of 1 . Also, the slave impedance model has a natural

frequency of 20 rad/sec and a damping ratio of 1 . Employing the proposed bilateral impedance controller, the following results are obtained. The interaction force between the surgeon (operator) and the master robot ( f h ), together with the interaction force between the slave robot and the soft tissue environment ( f e ) both in x direction of the Cartesian space are shown in Fig. 7.

34


As it is seen in Fig. 7, the force tracking (reflecting) performance is approximately achieved ( fh  kf fe ) in the proposed bilateral teleoperation system. This performance is a consequence of choosing small values for the master impedance parameters ( mm ,

cm and k m ), as discussed in Sec. 3.1. The force tracking error ( fh  kf fe ) in

x direction is

shown in Fig. 8. As it is seen in Fig. 8, the maximum value of force error is 4.1 N (less than 10% of the maximum interaction force, i.e. 42.5 N in Fig. 7) and it occurred at the initial instant. Also, the mean value of force error is 0.22 N (0.5% of the maximum interaction force, i.e. 42.5 N in Fig. 7). It should be mentioned that a small force error is seen throughout the motion (Fig. 8). This remaining small force error ( fh  kf fe ) is originated from the small master impedance dynamics ( mm xmodm  cm xmodm  km (xmodm  x0 ) ) that determines the difference between the scaled interaction forces ( f h and kf fe ) according to Eq. (9). Figure 9 shows the position responses of master and slave impedance models together with the master and slave robots’ trajectories in x direction. The initial position of the master robot is considered to be xm0  xm0 , zm0   0.62, 0.62 m that has T

T

an initial error with respect to the initial value of the master impedance model response T xmodm0   xmodm0 , zmodm0   0.6, 0.6 m . Also, the initial position of the slave robot is T

considered to be xs 0  xs 0 , zs 0   0.64, 0.64 m that has an initial error with respect to T

T

the initial position of the master robot.

35


Figure 9 demonstrates that the position tracking performance is achieved in the teleoperation system, which is the result of choosing large values for the slave impedance parameters ( ms , cs and k s ), as discussed in Sec. 3.1. Also, the velocities and accelerations of the master and slave end-effectors and their impedance models are shown in Fig. 9. The absolute position tracking error between master and slave robots ( x s  kp xm ) in x  z plane is shown in Fig. 10. Also, the absolute position tracking 2 errors of the master and slave robots with respect to their corresponding impedance models ( xm  xmodm , x s  xmods ) are shown in Fig. 10. 2

2

In Fig. 10, the initial (maximum) value of the absolute tracking error between the master and slave robots is 2.83 102 m , i.e. 58% of the motion range of master robot in x direction ( 4.85 102 m due to Fig. 9). However, the mean value of the tracking error

between the master and slave is 7.60 104 m , i.e. 1.5% of the range of master motion in x direction that is 4.85 102 m (Fig. 9). This tracking error has two sources: 1. the

desired slave error with respect to master xmods  (xmods  kp xm ) that is tried to be small by choosing large parameters for the slave impedance model (10), and 2. the slave tracking error ( x s  x s  xmods ) that converges to zero and originated from the performance of slave controller, which is shown in Fig. 10. Also, the master tracking error ( xm  xm  xmodm ) is illustrated in Fig. 10 and indicates the performance of the master controller in asymptotic convergence of the tracking error. Moreover, the mean values of the master and the slave absolute tracking errors with respect to their 36


corresponding impedance models (9) and (10) are 7.24 104 m and 7.37 104 m , respectively, as illustrated in Fig. 10. To elaborate more on the performance of the bilateral adaptive controller of master and slave in asymptotic tracking, Fig. 11 shows the (2-norm) values of error dynamics S m

2

and S s

2

that converge to the sliding

surfaces S m  0 and S s  0 during the first second of motion. This convergence to sliding surfaces ( S m  0 and S s  0 ) was proved using the Lyapunov stability theorem in section 3.5. According to the abovementioned results, the proposed bilateral adaptive impedance controller can achieve their objectives. In other words, the force tracking (reflecting) performance ( fh  kf fe ) is achieved according to Figs. 7 and 8 by choosing small values for the master impedance parameters ( mm , cm and k m ) in Eq. (9). Moreover, the position tracking performance ( x s  kp xm ) is obtained according to Figs. 9 and 10 by choosing large values for the slave impedance parameters ( ms , cs and k s ) in Eq. (10). These two objectives (position and force tracking performances) are obtained by defining a desired impedance objective for each of master and slave robots (one objective for each robot). However, in previous bilateral nonlinear adaptive controllers (such as [26, 27]), the position and force tracking performances are followed simultaneously for each of robots (two objectives for each robot). In addition, the actual and estimation values of the master acceleration ( x m and

x m est ) together with their difference (estimation error xm  xm est  xm ) in x direction 37


of Cartesian coordinates are shown in Fig. 12. The acceleration estimation error is similar in z direction and it is not presented here for the sake of brevity. The initial acceleration of the master robot is specified such that to be approximately the same as its corresponding impedance model (9), by applying the zero forces at the initial moment ( t  0 ) of motion. As a result, the master robot has no significant error in its acceleration at the initial moment in comparison with its impedance model (9). However, as shown in Fig. 12, the acceleration estimation error becomes non-zero after applying the interaction forces and having non-zero response of master impedance model, and due to the initial position error of the master robot with respect to its impedance response. As it is seen, the acceleration estimation error ( x m ) converges to zero during the first 0.1 sec of motion as a result of convergence of master robot trajectory to its impedance model response (Fig. 9). This performance is in accordance with the presented discussion for obtaining the master estimated acceleration x m est by Eq. (12). During this initial short time duration (about 0.1 sec) of motion, the acceleration estimation has a bounded error that its upper bound is about 1.7 m s 2 in each direction of Cartesian space (Fig. 12). This upper bound should be estimated conservatively in experimental studies; however, simulation results help the controller designer to have a better initial estimation. Based on the inequality (60) and the maximum estimated value of M x,s xm



, the value of robust gain

s

in the slave

control law (20) is adjusted on 3.5 as mentioned in Table 1. Due to Figs. 9-11, it is seen that the slave robot can successfully achieve its desired trajectory ( x mod s ) and also the 38


transparency condition is provided well in the presence of bounded transient error of the acceleration estimation ( x m ). Moreover, it is concluded that  s sgn(S s ) in the slave control law (20) (which is replaced by  s tanh(400 Ss ) to avoid chattering) can provided the robustness against the bounded estimation error of the master acceleration ( x m ). It should be mentioned that the upper bound of transient error ( x m ) in the master acceleration estimation decreases by specifying the same initial position and velocity for the master robot and its corresponding impedance model which results in decreasing the initial master position and velocity errors. Under this condition, the accuracy of Eq. (12) in estimating the actual master acceleration using the dynamics of its impedance model increases and the transient value of xm  xm est  xm decreases. The performance of the proposed controller in parameter adaptation is shown in Fig. 13. This figure shows the estimation of ten dynamic parameters introduced in Eq. (69) for the master (Fig. 13 a) and slave (Fig. 13 b) robots. All of the model parameters of the two robots are considered to be completely uncertain. Accordingly, in the adaptation laws (48) and (49) of both robots, an arbitrary initial guess

αˆ q initial  [0.058 0.53 0.43 15.58 5.2 5.3 15 6 2.6 1.2]T is used for the vector of parameters that is completely different from its real value:

αq actual  [0.875 2 0.5 9.81 19.62 4.905 3 15 1.5 2.5]T (in SI units). As it is seen in Fig. 13, the estimation process has almost settled after about seven seconds. As mentioned before in Sec. 3.4 about the BGF composite adaptation 39


laws and in the stability proof (Sec. 3.5), when the persistent excitation conditions

m (t )  1,m  0 and s (t )  1,s  0 are satisfied for both master and slave, the parameter estimation errors ( α q ,m and α q ,s ) converge to zero. Using parameters ai  1 ,

0,i  3.5 , k0,i  100 and Pi initial  33 I in the adaptation laws (48)-(52), the value of i (t ) for the master ( i  m ) and slave ( i  s ) during the motion is obtained and shown in Fig. 14. As it is clear in Fig. 14, i (t ) is always positive during the whole motion and consequently the condition i (t )  1,i  0 is satisfied. Thus, the composite adaptation laws (48) and (49) can estimate the actual parameters of the master and slave robots as their convergence is shown in Fig. 13. This can also be realized from the comparison of the final estimations of the parameters (Fig. 13) and their actual values:

αq actual  [0.875 2 0.5 9.81 19.62 4.905 3 15 1.5 2.5 ]T .

4.2. Telerehabilitation Application: Flexibility of Patient and Force Reflecting Objectives In the second step of simulations, a telerehabilitation operation on the upper limb (arm) of a patient is performed. As mentioned in Sec. 3.1, in telerehabilitation and telesonography applications an adjustable flexibility is desired for the slave robot in its interaction with the patient limb (environment). Accordingly, the parameters ( ms , cs and ks ) of the slave impedance model (10) should be decreased in comparison with the previous telesurgery application (Sec. 4.1). By decreasing these impedance parameters in (10), the slave robot becomes more flexible in its interactions with the environment 40


and consequently has higher tracking errors ( xmods ). In other words, in a telerehabilitation system that the slave robot moves the patients’ limbs, it can have a limited flexibility and the patient is permitted to deviate from the initially designed (master) trajectory according to the slave impedance model (10). In this case, the dynamic models of therapist (human operator) and the patient limb (environment) are presented by the following forms:

fh  fh  (M h xm  Ch xm  Kh (xm  x0 ))

(74)

fe  fe  M e xs  Cexs  Ke (xs  x0 )

(75)

where f h and f e are respectively the active forces of therapist and patient generated by their arm muscles. The dynamic parameters ( M h  mh I , M e  me I , Ch  ch I ,

Ce  ce I , Kh  kh I and Ke  ke I ) of the therapist and patient’s arms can be used from [26, 27, 46] where the human arm parameters are presented in Table 4. Also, the active force of therapist (operator) and patient (environment) are considered to be a combination of some harmonic functions (similar to the experimental studies in [28, 29]):

f hx  f hz  90  75cos(t )  15cos(6 t ) N

(76)

fex  fez  10  8cos(1.5t )  2cos(5 t ) N

(77)

To have force reflecting (tracking) condition, as discussed in Sec. 3.1, the parameters of the master impedance model (9) are chosen to be small (to obtain force tracking performance). Moreover, to have a flexible interaction between the slave robot

41


and the environment, values of the slave impedance parameters in (10) should be chosen to be as small as needed for its flexibility. Also, the position and force scaling factors are considered as kf  1 and kp  1 in order to reflect the exact patient interaction force on the therapist hand and to follow the exact trajectory of the master robot by the slave. The values of these impedance parameters and scaling factors are givrn in Table 5 for telerehabilitation applications. Similar to the previous section, using these parameters, the master impedance model (9) has two negative real poles ( 2 , 2 ), and the slave impedance model (10) has two negative real poles ( 20 , 20 ). Accordingly, the master impedance model has a natural frequency of 2 rad/sec and a damping ratio of 1 , and the slave impedance model has a natural frequency of

20 rad/sec and a damping ratio of 1 . Since in telerehabilitation applications the patient should have flexibility in physical interactions, the slave impedance parameters in Table 5 are decreased 100 times in comparison to the previous section (a telesurgery application). Also, similar to the previous section, the master impedance parameters are chosen to be small to obtain the force reflecting (tracking) performance as discussed in Sec. 3.1. Using the proposed bilateral adaptive impedance controller, the interaction forces between the therapist (operator) and the master robot ( f h ), and the interaction force between the patient (environment) and the slave robot ( kf fe ) in z direction are shown in Fig. 15.

42


As it is seen in Fig. 15, the force tracking (reflecting) performance is approximately achieved ( fh  kf fe ) by choosing small values for the master impedance parameters ( mm , cm and k m ), similar to the previous section. In other words, the therapist has a full haptic force feedback from the patient force. The force tracking error ( fh  kf fe ) in z direction is shown in Fig. 16. As seen in Fig. 16, the mean value of the force error is 0.42 N (less than 0.5% of the maximum interaction force, i.e. 86 N in Fig. 14). The source of the small force error in Fig. 16 is the small master impedance dynamics ( mm xmodm  cm xmodm  km (xmodm  x0 ) ) which determines the difference between the scaled interaction forces ( f h and kf fe ) according to Eq. (9). The position, velocity and acceleration responses of master and slave impedance models, and the master and slave robots’ trajectories in z direction are shown in Fig. 17. According to Fig. 17, the position tracking performance is weak, which is the result of choosing small values for the slave impedance parameters ( ms , cs and k s ) to provide flexibility, as discussed in Sec. 3.1. The absolute position tracking error between master and slave robots ( x s  kp xm ) in x  z plane is shown in Fig. 18. Also, the 2 absolute position tracking error of the master and slave robots with respect to their corresponding impedance models ( kp xm  xmodm , x s  xmods ) are shown in Fig. 18. 2

2

In Fig. 18, the absolute tracking error between the master and slave robots are considerable due to realizing a flexibility for the patient that can deviate from the master trajectory. Therefore, the main part of this tracking error is the desired slave

43


error with respect to the scaled master position xmod  xmod  kp xm . This error becomes s

s

larger in comparison with the previous simulations of telesurgery operation (Sec. 4.1) as a result of choosing smaller parameters for the slave impedance model (10), as discussed in Sec. 3.1. Also, the master and slave tracking errors with respect to their corresponding impedance models ( xm  xm  xmodm , x s  x s  xmods ) in Fig. 18 are small and converge to zero that indicates a suitable performance of master and slave adaptive controllers. In the obtained results for telesurgery and telerahabilitation applications, the master robot trajectory tracks its impedance response ( xm  xmodm , xm  xmodm and

xm  xmodm as shown in Figs. 9 and 17). Therefore, the master robot behaves similar to its desired impedance model (9) in response to the operator and environment forces and can be written as:

mmxm  cmxm  km (xm  x0 ) fh  kf fe

(78)

As a result, the master impedance parameters ( mm , cm and km ) in addition to the position, velocity and acceleration of the master ( x m , x m and x m ) specify the force tracking error (fh  kf fe ) . On the other hand, the human operator can enforce his desired position, velocity and acceleration to the master by applying an appropriate force ( f h ) according to Eq. (78). However, by increasing the absolute values of master position ( x m with respect to x 0 ), velocity ( x m ) and acceleration ( x m ), the force tracking error (fh  kf fe ) also increases due to Eq. (78).

44


Therefore, the bound and mean values of force tracking error are related to the selected master impedance parameters and the generated master trajectory by the interaction forces. In the presented simulations, the master impedance parameters are chosen to be small (as listed in Tables 3 and 5) to decrease the force tracking error while the master impedance model stability is also guaranteed. Under this condition, for the common motions of master and consequently slave robots (such as in Figs. 9 and 17), the force tracking error (as shown in Figs. 8 and 16) is small in comparison with the magnitude of each interaction force ( f h and f e as illustrated in Figs. 7 and 15). As mentioned previously, the mean value of the force error in Figs. 8 and 16 is around 0.5% of the maximum interaction forces shown in Figs. 7 and 15.

4.3. Intermittent Contact with Hard Environment

Intermittent contact with hard environment such as bones in robotic telesurgery is of great importance since it may have significant impact on the performance of the bi-lateral control system. To investigate such condition for the proposed controller, the stiffness of the environment (tissue) in model (72) is significantly increased to ke  10000 N/m (as assumed previously in [53]). The other parameters of the human operator and tissue are considered the same as those presented in Table 2  for models (71) and (72). Also, the components of f h as the active force of the human

operator’s hand in Eq. (71) are considered to be fhx  fhz  30  30 cos(t ) N . The initial positions of the robots and their impedance models are the same as those in Sec. 4.1. To 45


simulate a hard tissue contact during a tele-surgery operation, the parameters of impedance models (9) and (10) are adjusted similar to Sec. 4.1 (listed in Table 3) for achieving the transparency condition. As seen in Fig. 19, when the position of the slave end-effector reaches the surface of the hard tissue that is located on ze  0.66 m , the contact starts ( t  0.88 sec ) and the environment interaction force ( f e ) becomes non-zero. As shown in Fig. 19, the force tracking error temporarily becomes larger at the starting moments of this intermittent hard contact and then decreases. However, this force error is small after and before the contact when the salve robot has a free (non-contact) motion based on previous discussions on Eq. (78). The velocity and acceleration values have sudden changes at the starting time of the contact which are not shown for the sake of brevity. Also, the estimation error of the master acceleration that converges to zero before the contact, has bounded jumps (with the absolute maximum value of 1.8 m sec2 ) at the start of contact ( t  0.88 sec ) as illustrated in Fig. 18. However, the controller is robust against this bounded acceleration error as mentioned previously in sections 3.2, 3.5 and 4.1. Accordingly, the proposed controller provides the stability of the teleoperation system during such hard intermittent contact; however, bounded transient force tracking error and acceleration estimation error may occur at the start of this kind of contact.

5. CONCLUSIONS

46


In this paper, a new bilateral nonlinear adaptive impedance controller is proposed for nonlinear multi-DOF teleoperation systems. The controller provides an asymptotic tracking of two desired impedance models for master and slave robots in presence of parametric uncertainties. The reference impedance models define two dynamical relationships between end-effector positions and interaction forces of master and slave robots. The master impedance model determines the haptic sense of the operator in its physical interaction, and the slave impedance model establishes the flexibility of the slave robot in its interactions with environment. In previous nonlinear adaptive controllers, two conventional control objectives that are position and force tracking are followed simultaneously for each robot. As discussed in Sec. 1, reaching these two objectives for each DOF of robot (in Cartesian coordinates) with one control input is difficult. However, in the proposed nonlinear adaptive impedance controller, one control objective (realizing the impedance model) can be provided for each DOF with one control input. Moreover, unlike some previous nonlinear bilateral controllers, the measurement of the acceleration signals of master and slave robots are not required in the control laws. The system dynamics, impedance models, control laws and Lyapunov stability proof are all presented in Cartesian coordinates. However, the BGF composite adaptation laws are defined in the joint space to decrease the computational complexity, especially for higher DOF robots. Accordingly, the terms of the Lyapunov function are found in two different coordinates.

47


The presented simulations on a two DOFs master and slave robots show the performance of the proposed bilateral controller in realizing the desired impedance models. Consequently, the controller can be used in many applications of teleoperation systems such as tele-surgery, tele-sonography and tele-rehabilitation. In each application, the parameters of the two impedance models can be chosen according to the requirements and objectives. For example, in tele-surgery applications, the slave tracking accuracy is more important in comparison with the slave flexibility (compliance). In this case, the slave impedance parameters should be increased as much as needed. Also, in tele-rehabilitation and tele-sonography applications for which the slave robot should have higher compliance in its interaction with the patient (environment), the slave impedance parameters should be decreased. Moreover, the parameters of the master impedance model can be chosen according to the haptic sense, transparency and force reflecting performance of the teleoperation system in each application as it is discussed in Sec. 3.1. It should be mentioned that the existence of bounded time delays in the communication channels of the proposed teleoperation system will affect the stability proof and transparency analysis. These issues may be the subjects of some future works for the systems with communication delays.

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50


[33] Lu, Z., and Goldenberg, A. A., 1995, "Robust Impedance Control and Force Regulation: Theory and Experiments," The International Journal of Robotics Research, 14(3), pp. 225-254. [34] Anderson, R. J., and Spong, M. W., 1988, "Hybrid Impedance Control of Robotic Manipulators," Robotics and Automation, IEEE Journal of, 4(5), pp. 549-556. [35] Lu, W. S., and Meng, Q. H., 1991, "Impedance Control with Adaptation for Robotic Manipulations," Robotics and Automation, IEEE Transactions on, 7(3), pp. 408-415. [36] Chien, M., and Huang, A., 2004, "Adaptive Impedance Control of Robot Manipulators Based on Function Approximation Technique," Robotica, 22(04), pp. 395403. [37] Abdossalami, A., and Sirouspour, S., 2009, "Adaptive Control for Improved Transparency in Haptic Simulations," Haptics, IEEE Transactions on, 2(1), pp. 2-14. [38] Seraji, H., 1994, "Adaptive Admittance Control: An Approach to Explicit Force Control in Compliant Motion," eds., 4, pp. 2705-2712. [39] Tee, K. P., Yan, R., and Li, H., 2010, "Adaptive Admittance Control of a Robot Manipulator under Task Space Constraint," eds., pp. 5181-5186. [40] Rubio, A., Avello, A., and Florez, J., 1999, "Adaptive Impedance Modification of a Master-Slave Manipulator," eds., 3, pp. 1794-1799. [41] Dubey, R. V., Tan Fung, C., and Everett, S. E., 1997, "Variable Damping Impedance Control of a Bilateral Telerobotic System," Control Systems, IEEE, 17(1), pp. 37-45. [42] Love, L. J., and Book, W. J., 2004, "Force Reflecting Teleoperation with Adaptive Impedance Control," Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on, 34(1), pp. 159-165. [43] Cho, H. C., and Park, J. H., 2005, "Stable Bilateral Teleoperation under a Time Delay Using a Robust Impedance Control," Mechatronics, 15(5), pp. 611-625. [44] Garcaa-Valdovinos, L. G., Parra-Vega, V., and Arteaga, M. A., 2007, "ObserverBased Sliding Mode Impedance Control of Bilateral Teleoperation under Constant Unknown Time Delay," Robotics and Autonomous Systems, 55(8), pp. 609-617. [45] Abbott, J. J., and Okamura, A. M., 2007, "Pseudo-Admittance Bilateral Telemanipulation with Guidance Virtual Fixtures," The International Journal of Robotics Research, 26(8), pp. 865-884. [46] Liu, X., Tao, R., and Tavakoli, M., 2014, "Adaptive Control of Uncertain Nonlinear Teleoperation Systems," Mechatronics, 24(1), pp. 66-78. [47] Craig, J. J., 2005, Introduction to Robotics: Mechanics and Control, Pearson/Prentice Hall, [48] Spong, M. W., and Hutchinson, S., 2005, Robot Modeling and Control, Wiley, [49] Slotine, J. J. E., and Li, W., 1991, Applied Nonlinear Control, Prantice-Hall, NJ, Englewood Cliffs. [50] Khatib, O., 1987, "A Unified Approach for Motion and Force Control of Robot Manipulators: The Operational Space Formulation," Robotics and Automation, IEEE Journal of, 3(1), pp. 43-53. [51] Sirouspour, S., and Setoodeh, P., 2005, "Adaptive Nonlinear Teleoperation Control in Multi-Master/Multi-Slave Environments," eds., pp. 1263-1268. 51


[52] Speich, J. E., Shao, L., and Goldfarb, M., 2005, "Modeling the Human Hand as It Interacts with a Telemanipulation System," Mechatronics, 15(9), pp. 1127-1142. [53] Hashtrudi-Zaad, K., and Salcudean, S. E., 1996, "Adaptive Transparent Impedance Reflecting Teleoperation," eds., 2, pp. 1369-1374. Figure Captions List Fig. 1

A schematic diagram of teleoperation systems consist of master robot, a slave robot, a human operator, an environment, and communication channels

Fig. 2

The signal transmissions in the proposed bilateral controller with three communication channels

Fig. 3

The haptic sense of the human operator using the master impedance model (9)

Fig. 4

The concepts of the master and slave impedance models in the proposed bilateral control strategy

Fig. 5

The structure of the bilateral adaptive impedance controller

Fig. 6

The two-DOF robot manipulator with revolute and prismatic joints used for the evaluation of the proposed controller

Fig. 7

Interaction forces between the surgeon and master ( f h ), and between the slave and soft tissue environment ( f e ) in x direction

Fig. 8

The force tracking error ( fh  kf fe ) in x direction

Fig. 9

Position, velocity and acceleration of the master and slave robots together with their impedance model responses in x direction

52


Fig. 10

The absolute position tracking errors in x  z plane: the master error with respect to its impedance model xm  xmod

m

2

(blue dashed line), the

slave error with respect to its impedance model x s  x mod

s

dot line), and the error between master and slave x s  kp xm

(red dash-

2

2

(black solid

line). Fig. 11

The convergence of master and slave error dynamics S m

2

and S s

2

to

the sliding surfaces S m  0 and S s  0 in less than one second Fig. 12

The actual and estimation values of the master acceleration ( x m and

x m est ) together with their difference (estimation error xm  xm est  xm ) in x direction, where the master acceleration is estimated by Eq. (12) Fig. 13

The estimation of ten dynamic parameters for (a) master and (b) slave

Fig. 14

The value of the master and slave forgetting factors m (t ) and  s (t )

Fig. 15

Interaction forces between the therapist and the master robot ( f h ), and between the slave and the patient ( f e ) in z direction

Fig. 16

The force tracking error ( fh  kf fe ) in z direction

Fig. 17

Position, velocity and acceleration of the master and slave robots together with their impedance model responses in z direction

Fig. 18

The absolute position tracking errors in x  z plane: the master error

53


with respect to its impedance model xm  xmod

m

2

(blue dashed line), the

slave error with respect to its impedance model x s  x mod

s

dot line), and the error between master and slave x s  kp xm

2

2

(red dash(black solid

line). Fig. 19

Position and force tracking performance together with master acceleration estimation during the intermittent contact with a hard tissue, the contact starts at t  0.88 sec when the slave end-effector reach to the surface of tissue ( ze  0.66 m ) and ends at t  5.37 sec .

54


Table Caption List Table 1

The parameter values of the robots, control laws and adaptation laws

Table 2

Dynamic parameters of a human operator (surgeon)’s arm and soft tissue environment

Table 3

Master and slave impedance parameters with scaling factors for telesurgery application

Table 4

Dynamic

parameters

of

the

therapist

(operator)

and

patient

(environment) Table 5

Master and slave impedance parameters together with scaling factors for telerehabilitation application

55


Fig. 1 A schematic diagram of teleoperation systems consist of master robot, a slave robot, a human operator, an environment, and communication channels

56


Fig. 2

The signal transmissions in the proposed bilateral controller with three

communication channels

57


Fig. 3 The haptic sense of the human operator using the master impedance model (9)

58


Fig. 4 The concepts of the master and slave impedance models in the proposed bilateral control strategy

59


Fig. 5 The structure of the bilateral adaptive impedance controller

60


Fig. 6 The two-DOF robot manipulator with revolute and prismatic joints used for the evaluation of the proposed controller

61


Interaction Forces : f h and f e (N)

50

f 40

f

h e

30

20

10

0

-10

0

2

4

6

8

10

12

14

Time (sec)

Fig. 7 Interaction forces between the surgeon and master ( f h ), and between the slave and soft tissue environment ( f e ) in x direction

62


Force Tracking Error: f h-k f f e (N)

2

1.5

1

0.5

0

-0.5

-1

0

2

4

6

8

10

12

14

Time (sec)

Fig. 8 The force tracking error ( fh  kf fe ) in x direction

63


0.67

Position in x direction (m)

0.66 0.65 0.64

Master Imp. Model

0.63

Master Robot

0.62

Slave Imp. Model 0.61 0.6

Slave Robot 0

2

4

6

8

10

12

14

10

12

14

10

12

14

Velocity in x direction (m/s)

Time (sec) 0.08 0.06 0.04 0.02 0 -0.02 -0.04

0

2

4

6

8

2

Acceleration in x direction (m/s )

Time (sec) 0.2

0.1

0

-0.1

-0.2

0

2

4

6

8

Time (sec)

Fig. 9 Position, velocity and acceleration of the master and slave robots together with their impedance model responses in x direction

64


Position Tracking Error (m)

0.03

0.025

Master Error WRT its Imp. Model Slave Error WRT its Imp. Model Error Between Master and Slave

0.02

0.015

0.01

0.005

0

0

2

4

6

8

10

12

14

Time (sec)

Fig. 10 The absolute position tracking errors in x  z plane: the master error with respect to its impedance model xm  xmod respect to its impedance model x s  x mod master and slave x s  kp xm

2

m

s

2

2

(blue dashed line), the slave error with (red dash-dot line), and the error between

(black solid line).

65


Distance to Sliding Surfaces

0.2

||S || : Distance to Master Sliding Surface m 2

0.15

||S || : Distance to Slave Sliding Surface s 2

0.1

0.05

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (sec)

Fig. 11 The convergence of master and slave error dynamics S m

2

and S s

2

to the

sliding surfaces S m  0 and S s  0 in less than one second

66


3

Actual Master Acceleration Estimated Master Acceleration Estimation Error of Master Acceleration

2

Acceleration estimation (m/s )

2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Time (sec)

Fig. 12 The actual and estimation values of the master acceleration ( x m and x m est ) together with their difference (estimation error xm  xm est  xm ) in x direction, where the master acceleration is estimated by Eq. (12)

67


20

 q,m 1

Estimated Parameters of Master Robot

18

 q,m 2

16

 q,m 3

14

 q,m 4

12

 q,m 5

10

 q,m 6

8

 q,m 7

6

 q,m 8

4

 q,m 9

2 0

 q,m 10 0

2

4

6

8

10

12

14

Time (sec)

(a) 20

 q,s 1

Estimated Parameters of Slave Robot

18

 q,s 2

16

 q,s 3

14

 q,s 4

12

 q,s 5

10

 q,s 6

8

 q,s 7

6

 q,s 8

4

 q,s 9

2 0

 q,s 10 0

2

4

6

8

10

12

14

Time (sec)

(b) Fig. 13 The estimation of ten dynamic parameters for (a) master and (b) slave

68


Forgetting Factors of Adaptation Laws

3.5

m

3

s

2.5 2 1.5 1 0.5 0 0

2

4

6

8

10

12

14

Time (sec)

Fig. 14 The value of the master and slave forgetting factors m (t ) and  s (t )

69


Interaction Forces : f h and f e (N)

100

f

80

f

h e

60 40 20 0

-20

0

2

4

6

8

10

12

14

Time (sec)

Fig. 15 Interaction forces between the therapist and the master robot ( f h ), and between the slave and the patient ( f e ) in z direction

70


Force Tracking Error: f h- k f f e (N)

4 3 2 1 0 -1 -2 -3 -4

0

2

4

6

8

10

12

14

Time (sec)

Fig. 16 The force tracking error ( fh  kf fe ) in z direction

71


Position in z direction (m)

0.9 0.85 0.8 0.75 0.7

Master Imp. Model Master Robot Slave Imp. Model Slave Robot

0.65 0.6

0

2

4

6

8

10

12

14

8

10

12

14

8

10

12

14

Time (sec)

Velocity in z direction (m/s)

0.4

0.2

0

-0.2

-0.4

0

2

4

6

2

Acceleration in z direction (m/s )

Time (sec) 1.5 1 0.5 0 -0.5 -1 -1.5

0

2

4

6

Time (sec)

Fig. 17 Position, velocity and acceleration of the master and slave robots together with their impedance model responses in z direction

72


0.035

Position Tracking Error (m)

0.03 0.025 0.02

Master error WRT its Imp. Model Slave error WRT its Imp. Model Error between Master and Slave

0.015 0.01 0.005 0

0

2

4

6

8

10

12

14

Time (sec)

Fig. 18 The absolute position tracking errors in x  z plane: the master error with respect to its impedance model xm  xmod respect to its impedance model x s  x mod master and slave x s  kp xm

2

m

s

2

2

(blue dashed line), the slave error with (red dash-dot line), and the error between

(black solid line).

73


0.67

Position in z direction (m)

0.66 0.65

Contact with hard tissue

0.64 0.63

Master Imp. Model Master Robot Slave Imp. Model Slave Robot

0.62 0.61 0.6

0

1

2

3

4

5

6

7

5

6

7

Force Tracking Error: f h-k f f e (N)

Time (sec) 2

1

0

Contact with hard tissue -1

-2

-3

0

1

2

3

4

Time (sec) 2

Acceleration estimation (m/s )

5

0

-5

Contact

Actual Master Acceleration Estimated Master Acceleration Estimation Error of Master Acceleration

-10 0

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

Fig. 19 Position and force tracking performance together with master acceleration estimation during the intermittent contact with a hard tissue, the contact starts at t  0.88 sec when the slave end-effector reach to the surface of tissue ( ze  0.66 m ) and

ends at t  5.37 sec . 74


Table 1 The parameter values of the robots, control laws and adaptation laws Parameters of the Robots (for master and slave)

Parameters of the Adaptation Laws

Parameters of the Control Laws

m1  4 kg , m2  2 kg I1  0.083 kg.m2 , I 2  0.0415 kg.m2 l1  0.5 m , l2  0.5 m

am  as  1

1,m  1,s  1

0,m  0,s  5

2,m  2,s  60

k0,m  k0,s  100

 s  3.5

l1g  0.25 m , l2 g  0.25 m

Pm initial  Ps initial  33 I66

g  9.81 m/s2

c1  3 N.m.s , c2  15 N.s/m

1  1.5 N.m , 2  2.5 N

75


Table 2 Dynamic parameters of a human operator (surgeon)’s arm and soft tissue environment Parameters of the Surgeon

Parameters of the Soft Tissue

(Operator)

(Environment)

mh  3.25 kg

me  1 kg

ch  20 N.s/m

ce  40 N.s/m

kh  300 N/m

ke  1500 N/m

76


Table 3 Master and slave impedance parameters with scaling factors for the telesurgery application Master impedance

Slave impedance

Position and force

parameters

parameters

scaling factors

mm  1 kg

ms  1000 kg

kp  1

cm  4 N.s/m

cs  40000 N.s/m

kf  1

km  4 N/m

ks  400000 N/m

77


Table 4 Dynamic parameters of the therapist (operator) and patient (environment) Parameters of the Therapist

Parameters of the Patient

(Operator)

(Environment)

mh  3.25 kg

me  3.25 kg

ch  20 N.s/m

ce  20 N.s/m

kh  300 N/m

ke  300 N/m

78


Table 5 Master and slave impedance parameters together with scaling factors for the telerehabilitation application Master impedance

Slave impedance

Position and force

parameters

parameters

scaling factors

mm  0.5 kg

ms  10 kg

kp  1

cm  2 N.s/m

cs  400 N.s/m

kf  1

km  2 N/m

ks  4000 N/m

79