Force-Symmetric Type Two-Channel Bilateral Control System Based on Higher-Order Disturbance Observer

In a bilateral control system, reducing the information-transmission channel between the master and slave systems has advantages, such as suppressing data traffic, reducing stored data, and simplifying the control-system design. The position/force hybrid-control-type bilateral control system can realize stable and highly transparent bilateral control; however, it requires four transmission channels between the master and slave systems to transmit the position and force information. Several types of conventional two-channel controllers are difficult to operate and contact with hard environments. A two-channel controller with a performance equivalent to that of a four-channel controller exists, but it still requires local position control loops. Therefore, this study proposes a force-symmetric type two-channel bilateral control system based on force control systems with robust acceleration controllers using second-order disturbance observers (DOBs) among higher-order DOBs. Bilateral control systems have two control goals: synchronization of the position of the master-slave system, which is a position-related control goal, and artificial realization of the law of action-reaction between the master-slave system, which is a force-related control goal. Methods using low-order DOBs, such as zero- and first-order DOBs, cannot achieve the position-related control goal. In contrast, the proposed method can achieve both the position and force control goals simultaneously, even though it only has force transmission channels and control loops. This study also showed that the performance of the proposed two-channel bilateral control system is equivalent to that of a four-channel bilateral control system that transmits both position and force information. The effectiveness of the proposed method was confirmed through experiments.


I. INTRODUCTION A. BACKGROUND OF THIS STUDY
In recent years, many countries have experienced declining birth rates and an aging population.Therefore, the decrease in the working population and loss of trained skills are becoming serious challenges.To solve these problems, it is important to replace human functions and behaviors with robots.For example, automation technology using robots is expected not only in the manufacturing industry but also in the agricultural [1], nursing care, and medical fields [2], and various studies The associate editor coordinating the review of this manuscript and approving it for publication was Bidyadhar Subudhi .
are being conducted on these applications.In addition to automation, robots are expected to perform delicate tasks equivalent to those performed by humans, even in the absence of humans This can be achieved by operating the robots from remote locations.
Various bilateral control systems have been actively studied in recent years as one of such technologies [3], [4], [5], [6].The bilateral control system comprises two pairs of robots: master and slave robots.The objective is to realize position synchronization and force feedback (artificial realization of the laws of action and reaction) between these robots.Therefore, an operator can operate the target environment while obtaining force feedback.Bilateral control methods, such as position-symmetric, force-reflecting [7], parallel [8], and impedance-control [9] types, have been investigated.Bilateral control systems have been modeled, designed, and analyzed as two-terminal pair circuits [7].Transparency is a widely used performance index that originates from a two-terminal-pair circuit model for bilateral control systems [10].
Among these, a four-channel bilateral controller [11], [12], [13] can achieve high transparency.In particular, a four-channel bilateral controller based on acceleration control [14], [15], [16] using a disturbance observer (DOB) [17], [18] can achieve high transparency and stable contact motion, even in a stiff environment.Because DOB-based acceleration control enables not only robust control but also the decoupling control of multi-degree-of-freedom (MDoF) systems [16], [18], [19], a four-channel bilateral control system based on an acceleration controller can be regarded as a high-performance and useful bilateral control system.The four-channel bilateral controller based on the acceleration controller is also known to be excellent in terms of operationality and reproducibility [15], which are performance indices extended from transparency.To obtain clear haptic feedback, bilateral control is expected to be applied in telemedicine, remote communication, and human substitution in extreme environments.
Furthermore, bilateral control is being extended and studied, not only between one-to-one systems, but also for multilateral control systems that transmit force between many-to-many systems [20], [21], [22].Research is also being conducted to extend bilateral control to provide force feedback between systems with different structures, such as mobile robots [23], [24], [25], and between systems with different ranges of motion [26], [27].In addition, bilateral control can be applied to motion-copying systems [28], [29] to save and reproduce human motions.A motion-copying system is a type of robot-motion programming system that uses direct human teaching via bilateral control.By recording motion through bilateral control, human motions, and reaction forces can be separately estimated and controlled from the target environment.Consequently, human motion, including force level, can be reproduced with high accuracy.
In a bilateral control system, reducing the informationtransmission channel between the master and slave systems provides advantages such as suppressing data traffic, reducing stored data, and simplifying the control-system design.However, in conventional two-channel bilateral control systems, contact with the environment via position control may make stable contact motion difficult.In addition, the position/force hybrid-control-type bilateral control system can realize stable and highly transparent bilateral control; however, it requires four transmission channels between the master and slave systems to transmit the position and force information.To reduce data traffic, two-channel bilateral control based on the impedance-field expression of the four-channel bilateral control has been proposed [30], [31]; however, its performance and stability are considered to be equivalent to those of the original four-channel system.A bilateral controller using a transfer-function-based approach was proposed as another two-channel bilateral controller [32].This method realizes a two-channel bilateral controller using position-information transmission.In conventional research, a position-based two-channel controller has shown better operationality than a four-channel controller based on hybrid control.However, the former has low reproducibility [32].A two-channel bilateral control system based on admittance control and its applications have been proposed [33].In this system, only the force information is transmitted between the master and slave systems, and its performance is equivalent to that of four-channel bilateral control.However, a local position-control loop still exists in a master-slave system, which complicates the outlook of the control system.

B. AIMS OF THIS STUDY
In contrast, this study proposes a force-symmetric twochannel bilateral control system based on higher-order DOB (HDOB) [34], [35], [36].The proposed two-channel bilateral control system is designed based on a robust acceleration control-based force control system using second-order DOBs among higher-order DOBs.This study shows that the proposed method can achieve not only the force-control goal but also the position-control goal, whereas methods using lowerorder DOBs, such as zero-and first-order DOBs, cannot achieve synchronization of the master-slave system position.This study also shows that the performance of the proposed method is equivalent to that of a four-channel bilateral control system that transmits position information.The effectiveness of the proposed method is confirmed through bilateral control experiments in the joint space and workspace.

C. CONTRIBUTIONS OF THIS STUDY
The contributions of this study are summarized as follows.

Contribution 1: Implementation of bilateral control with only force transmission, force control, and HDOB
In addition to methods such as the position-symmetric type, force-reflecting type [7], force reverse type [9], force-projection type [37], [38], parallel type [8], and impedance-control type [9] even the latest bilateral control systems based on four-channel control systems [19], [31], [33], [39], include position control and/or position information transmission channels in the conventional bilateral control systems.Thus, conventional bilateral control design strategies hold the fixed notion that position control must be used for position synchronization to achieve high-performance bilateral control.
One of the contributions of this study is to break this stereotype and show that position and force synchronization can be achieved by force control systems only, without using position control and position information transmission channels, by constructing a robust acceleration control system using a second-order DOB among the HDOBs.
Since DOB was first proposed in the 1980s [40], various studies and applications have been attempted.However, most of these studies, including the most recent studies, are on or using zero-order DOB [41], [42].Although some recent studies consider higher-order disturbances [43], [44], [45], most of them aim to improve the tracking performance by improving the disturbance suppression characteristics.In contrast, no studies have applied HDOBs to realize synchronization of the positions of master and slave robots in bilateral control systems, as in this study.

Contribution 2: Elimination of position transmission channels and position control loops in bilateral control
Related to the first contribution, removing the position transmission channel is expected to improve stability in time-delay systems where communication delays occur between master and slave systems.In addition, as with the method proposed in this study, the position feedback loop is present even in the latest bilateral control systems [33], [46], which do not have a position transmission channel and have the same performance as a four-channel type control system.However, if the position control loop can be eliminated, it is expected to improve the affinity with the sensorless control of servo motors.[47], [48], [49].The demand for sensorless drives for servo motors in industrial applications is still increasing for the purpose of miniaturization, cost reduction, and reduction of the failure rate, and research on sensorless control is also active.It is also desirable to be able to perform position-sensorless control in bilateral control.
Another contribution of the proposed two-channel bilateral control is that it is expected to improve the stability of the time-delay system by eliminating the need for a position transmission loop.The advantages of the proposed method in this regard are suggested in Appendix A. Furthermore, because a position control loop is not required, it can be expected that the system will be less susceptible to the deterioration in position control performance that accompanies the application of the position-sensorless control of the servo motor.

Contribution 3: Achievement of bilateral-control performance equivalent to that of a four-channel structure by a two-channel structure
To confirm the effect obtained in Contribution 2, it is first necessary to show that the proposed two-channel bilateral control system exhibits the same performance and stability as the conventional four-channel bilateral control system under ideal conditions with no time delay and using a position sensor.
Therefore, as the third contribution of this study, it is demonstrated that the proposed two-channel control system has the same performance as the conventional four-channel control system in terms of transparency and operability/reproducibility, which are the performance indicators of bilateral control.In terms of stability, this study also confirmed that the proposed method has the same degree of stability as the conventional method.
Fig. 1 shows a conceptual representation of the proposed method.

D. ORGANIZATION OF THIS PAPER
The remainder of this paper is organized as follows Section II describes robust acceleration control using zero-order and higher-order (n-th order) DOBs.Section III describes how to configure the conventional and proposed two-channel bilateral controllers based on acceleration control using HDOBs in the joint space and workspace.In Section IV, the performance and stability of the conventional four-channel bilateral controller, two-channel bilateral controllers with lower-order DOBs, and proposed two-channel bilateral controller are analyzed and compared.To confirm the effectiveness of the proposed method, Section V describes the experiments of bilateral control systems in the joint space and workspace.Finally, Section VI concludes the paper.As a supplement, Appendix A provides a primitive method and discussion for applying the proposed two-channel bilateral controller to a time-delay system.

II. HIGH-ORDER DISTURBANCE OBSERVERS
This section describes high-order (nth-order) DOBs.First, the most basic DOB, the zeroth-order DOB, is described, followed by the higher-order DOBs.

A. ZEROTH-ORDER DOB
The equation of motion of the rotary actuator is expressed by the following equation: where In ( 1) to (3), q, τ , J n , k t , i, and δ are the joint angle of the manipulators, joint torque, nominal inertia, torque coefficient, current, and parameter variations, respectively.The superscripts ref, fric, grav, cori, dis, and load represent the reference value, friction, gravitational force, Coriolis term, disturbance, and load torque excluding the external torque, respectively.The subscript n is the nominal value.
To estimate and compensate for the disturbance torque, a zeroth-order DOB [17], [18] is constructed based on the following disturbance model: Equation ( 4) indicates that the disturbance is modeled as a step signal.The zeroth-order disturbance is estimated using the zeroth-order DOB as follows: q = g pd s + g pd sq = L pd (s)sq (7) where τ dis 0 and q represent the estimated zeroth-order disturbance and the estimated velocity through pseudodifferentiation, respectively.L dis 0th (s) and L pd (s) are low-pass filters (LPFs) for disturbance estimation and pseudodifferentiation, which suppress noise in the differentiation of sensor values, respectively.d 0th 0 and g pd are bandwidths of the LPF in DOB and pseudo-differentiation, respectively.Fig. 2 shows a block diagram of an actuator with a zerothorder DOB.If the reference torque is given as τ ref = J n qref , the dynamics of the actuator with a zeroth-order DOB in the acceleration dimension can be rewritten as where H dis 0th (s) represents a high-pass filter resulting from the zeroth-order disturbance compensation.As shown in (8), if the pole (bandwidth) of the observer filter d 0th 0 is set to a sufficiently high value, the disturbance is eliminated from the acceleration response.Consequently, the acceleration response coincides with the acceleration reference value, and acceleration control is achieved [18].A block diagram of the actuator with disturbance compensation by a zeroth-order DOB is shown in Fig. 3.

B. HIGHER-ORDER DOBS
In contrast to the zeroth-order DOB, a higher-order (nthorder) disturbance can be modeled using the following equation: The above equation represents the (n+1)-th-order time derivative of the disturbance equal to zero.The nth-order disturbance τ dis hn is estimated using the higher-order (nthorder) DOB as follows: where In (11), L dis nth (s) represents an LPF for a higher-order disturbance estimation.Fig. 4 shows a block diagram of the higher-order (n-th-order) DOB.
In the case where n is 2, a model of the second-order disturbance is described as follows: This equation indicates that the third-order time derivative of the disturbance torque is zero.This implies that the disturbance torque is modeled as a signal of a quadratic function of time.The second-order disturbance τ dis h2 is estimated using a second-order DOB, as follows: Fig. 5 shows a block diagram of an actuator with a secondorder DOB.
The dynamics of the actuator with a higher-order DOB in the acceleration dimension can be rewritten as: where H dis nth (s) represents a high-pass filter resulting from the higher-order-disturbance compensation.Fig. 6 shows an equivalent block diagram of an actuator with disturbance suppression by the nth-order HDOB.
In the case where n=2, the dynamics of the actuator with a second-order DOB in the acceleration dimension can be rewritten as From ( 16), and (18), as in the case of the zeroth-order DOB, the disturbance effect is eliminated if the bandwidth of the LPFs in the HDOB is sufficiently high.Consequently, acceleration control can be achieved by using HDOBs.

III. PROPOSED FORCE SYMMETRIC TYPE TWO-CHANNEL BILATERAL CONTROLLER
In this section, the proposed two-channel bilateral control system is described.First, the conventional four-channel bilateral control system is described, followed by an explanation of the design methodology of the proposed method.

A. CONVENTIONAL BILATERAL CONTROLLERS
The control goals of the bilateral control are expressed by the following equations: where F and X represent the force control and position control parameters, respectively.

1) FOUR-CHANNEL BILATERAL CONTROLLER BASED ON POSITION-FORCE HYBRID CONTROL
From ( 19) and ( 20), bilateral control can be considered as a position/force hybrid control.The four-channel bilateral control achieves force transmission by configuring the position and force hybrid control based on acceleration control in a common and differential modal coordinate space.These control objectives can be expressed using the modal transformation matrix T as follows [28], [29]: where subscript M represents a parameter in the force and position control-modal space.The acceleration reference for hybrid control in the modal space is calculated as follows [28], [29]: where C f ,C p (s), K f , K p , and K v represent the force controller, position controller, force, position, and velocity feedback gains, respectively.Moreover, τ ext represents the estimated value of the reaction-force (torque) observer (RFOB (RTOB)) [17].This study uses zeroth-order (typical) RFOBs.For simplicity, LPF L pd (s) is henceforth treated as L pd (s) = 1, assuming that the bandwidth of L pd (s) is sufficiently wide.From ( 8) and ( 17) to (21), the responses in the modal space are expressed as follows [16], [18]: where qdis F and qdis X represent the equivalent accelerationdimension disturbances in force and position-control systems, respectively.From (24), the responses of the torque and position in the modal space can be obtained as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.= L dis 0th (s) where K p = ω 2 p , and K v = 2ζ ω p .As shown in ( 23) and ( 24), the artificial action-reaction law and position synchronization are realized between the master and slave systems if the bandwidth of the zeroth-order DOB d 0th 0 is set sufficiently high.From the right-hand side of ( 25), the inertial-force term remains in the force response.However, the effect of the inertial force term is reduced if the force feedback gain is sufficiently high.The acceleration reference in the modal space is converted to that in the joint space using the inverse coordinate transformation matrix T −1 , and the acceleration response in the joint space coincides with the acceleration reference, as expressed by the following equation: Fig. 7 shows a block diagram of the four-channel bilateral control system based on the position-force hybrid control.In the figure, the force-control gain c f and position controller c p (s) include the determinants of the transformation matrix T in the inverse coordinate transformation shown in (27):

2) TWO-CHANNEL BILATERAL CONTROLLER USING LOWER ORDER DOBS
First, the position synchronization in a force-symmetric-type two-channel bilateral controller is considered.In this section, the step signal disturbance is assumed as where D dis represents the magnitude of the disturbance.The final values of the acceleration, acceleration reference, and position are defined as follows: Considering the final-value theorem, when a zeroth-order DOB is used, the influence of the step-signal disturbance is removed from the acceleration response, as shown in the following equation: Similarly, when a first-order DOB is used, the influence of the step-signal disturbance can be eliminated from the acceleration response.
However, as shown below, the influence of the disturbance on the position dimension cannot be eliminated using a zeroorder DOB.
Similarly, even when a first-order DOB is used, the effect of the disturbance appears as a steady-state error in the position response.
As described above, even if the same acceleration reference value is input into the master and slave systems, the positions of each system cannot be synchronized using a zero-or first-order DOB in a control system.
The above discussion shows that a force-symmetric-type twochannel bilateral control system cannot be constructed using force control by acceleration control based on lower-order DOBs, such as zeroth-and first-order DOBs.

B. PROPOSED TWO-CHANNEL BILATERAL CONTROLLER
This study used an acceleration controller with a second-order DOB to achieve force-symmetric-type two-channel bilateral control.The second-order DOB can also eliminate the influence of the step signal disturbance from the acceleration response as follows: Using a second-order DOB, the influence of the disturbance is removed from the position response differently from the cases of the zeroth-and first-order DOBs as follows: The acceleration reference value for the force control expressed below is input to the master-slave system.
Owing to the robust acceleration control by the secondorder DOB, the acceleration-response value coincides with the acceleration reference value expressed by the following equation: Consequently, the second-order DOB can eliminate the influence of the disturbance from the position response.Therefore, according to the previous discussion, the synchronization of the positions of the master and slave systems is achieved as follows: Furthermore, the master and slave systems each have a force-control system.Therefore, the acceleration response is expressed by the following equation: Consequently, the force response of the master-slave system can be expressed as follows: As with the four-channel control system, by setting a sufficiently large force-control gain (setting a sufficiently small virtual inertia), the laws of the acting and reacting forces between the master and slave systems can be realized as follows: In this study, the second-order DOBs are designed to have the same disturbance-suppression characteristics as the position-control system of the four-channel bilateral controller shown in (26).In other words, the coefficients of the observer filter are set as follows so that it has three poles: two poles are the same as those of the position-control system of the four-channel bilateral control, and one pole is the same as that of the zeroth-order DOB.
where p i (i = 1, 2, 3) represents the poles of the LPF in the second-order DOB.
Fig. 8 shows a block diagram of the proposed two-channel bilateral controller in the joint space of the actuator.

C. IMPLEMENTATION OF PROPOSED TWO-CHANNEL BILATERAL CONTROLLER IN WORKSPACE OF MANIPULATOR 1) KINEMATICS AND DYNAMICS IN MDOF MANIPULATOR
A motion equation of the joint space in an MDoF manipulator is described as follows: where q, τ , J n , k t , i, and are the joint angle vector of the manipulators, joint torque vector of the manipulators, nominal joint inertia matrix, torque coefficient matrix, current vector, and parameter variation, respectively.The joint torque reference of an acceleration controller in a workspace (task space) is determined using a nominal-inertia matrix, inverse Jacobian (coordinate transformation) matrix, and virtualmass matrix, as follows [16], [18], [19]: where F ref , Ẍref , J −1 aco and M vn are the force reference, acceleration reference in the workspace, the inverse of the Jacobian matrix, and virtual-mass matrix, which can be set arbitrarily, respectively.The dynamics in the workspace of the MDoF manipulator can be derived using the following kinematic relationship: Ẋ = J aco q (61) Ẍ = J aco q + Jaco q (62) where F, X, and H(q) are the force vector, position vector, and function for kinematic description, respectively.The workspace dynamics of the manipulator can be derived from the transformation of the joint-space motion equation (56) using the joint torque reference (59) and the above equations, as follows [16], [18], [19]: where f dis = J T aco −1 τ dis and M n = J aco J −1 n J T aco −1 represent the disturbance force transformed from the disturbance torque in the joint space and an equivalent-mass matrix of the workspace of a manipulator, respectively.

2) HDOB IN WORKSPACE
A disturbance in the workspace, including the force caused by the coordinate transformation, is defined as follows: where Ẍdis is the equivalent acceleration-dimension disturbance in the workspace.Using (65), the workspace dynamics can be rewritten as A higher-order DOB in the workspace can be constructed based on the following equation, similar to the joint-space case: where In (69), L dis nth (s) represents a diagonal matrix with an LPF of the n-th order DOB.The second-order disturbance Fdis h2 in the workspace is estimated using the second-order DOB in the workspace as follows: where G dis 2nd (s) represents a diagonal matrix with G dis 2nd (s) as the disturbance estimation along each axis.By adding the estimated disturbance Fdis h2 to the force reference vector F ref to compensate for the workspace disturbance F dis , the workspace dynamics can be rewritten from (67) as follows: where H dis 2nd (s) represents a diagonal matrix with a high-pass filter resulting from second-order-disturbance compensation in each axis.
As indicated by the above equation, a disturbance in the workspace can be eliminated by setting a sufficiently high observer bandwidth.In addition, decoupling control of each axis can be achieved by setting the virtual-mass matrix M vn as a diagonal matrix.Moreover, if the virtual-mass matrix is set as an identity matrix, the controller becomes an acceleration controller in the workspace (task space) [16], [18], [19] as follows: The force references of the proposed bilateral controller in the workspace are obtained using the second-order DOB as follows: where is the inverse of a diagonal matrix with virtual masses for force control, which corresponds to force-control gains.
Based on the same discussion as in the previous subsection, the synchronization of the master-slave system position and artificial realization of the action-reaction law, which are the control goals of bilateral control, are achieved in the workspace by providing the master-slave system with force reference values (acceleration reference values), as in the above equation.Figs. 9 and 10 show the block diagrams of the four-channel bilateral controller and the proposed two-channel bilateral controller in the workspace of the manipulators.

IV. ANALYSIS
This section presents an analysis of the performance and stability of the proposed method in comparison with a fourchannel-type bilateral controller and a two-channel bilateral controller with lower-order DOBs.parameters used in this analysis are listed in Table 1.

A. PERFORMANCE ANALYSIS AND COMPARISON 1) ANALYSIS BY HYBRID PARAMETERS
The performances of the conventional bilateral control systems and the proposed method are compared based on hybrid parameters.The hybrid parameters are defined as follows [10], [33]: where H yb (s) and its elements represent the hybrid matrix and hybrid parameters, respectively.To achieve ideal bilateral control, the values of the hybrid parameters should be As shown in [10] and [33], hybrid parameters are typically described in terms of velocity-dimension information, considering the similarity between the current and velocity.However, this study describes hybrid parameters using positional-dimension information similar to that in [15].This confirms the performance regarding the synchronization of the position of the control target of the bilateral control.The hybrid parameters of the proposed method are derived based on (18) and (47).In addition, the hybrid parameters of the four-channel method are derived based on ( 21), ( 22), ( 25) and (26).For simplicity, the cutoff frequencies for pseudo-differentiation are sufficiently high in the present study.Additionally, we assume that the effects of disturbances owing to the Coriolis force, gravity, frictional force, and parameter fluctuations are negligibly small; that is, the DOB sufficiently compensates for them, and only the effects of external forces on each system are considered.
Fig. 11 shows the Bode diagrams of the hybrid parameters with respect to the four-channel bilateral controller, force-symmetric-type two-channel bilateral controller with lower-order DOBs, and the proposed two-channel bilateral controller with second-order DOBs.Fig. 11 (a) shows the Bode plot of H 11 (s).All methods have a second-order differential characteristic of 40 dB/dec.This corresponds to the inertial force of the response (operating force).Although the inertial force increases in the high-frequency domain, it exhibits a characteristic of 0 dB or lower in the low-frequency domain, indicating that it has ideal characteristics for bilateral control.Fig. 11 (b) and (c) show the Bode diagrams of H 12 (s) and H 21 (s), respectively.These figures show that all the methods exhibit ideal characteristics in the frequency range of approximately 100 rad/s or less.Fig. 11 (d) shows the Bode diagram of H 22 (s).In contrast to other characteristics, methods using lower-order (zerothand first-order) DOBs exhibit characteristics above 0 dB.This indicates that the force response on the slave side affects the position response when these methods are used.This implies that the synchronization of the positions of the master and slave systems is not achieved when the slave systems come in contact with the environment.This characteristic corresponds to the analysis results shown in the previous section, in which position synchronization was not achieved with low-order DOBs.Conversely, in the four-channel-type method and proposed two-channel-type method, the influence of the external force response on the slave-system side is sufficiently attenuated in the lowerfrequency region, and the frequency characteristics are such that position synchronization is achieved.

2) ANALYSIS BY REPRODUCIBILITY AND OPERATIONALITY
Next, the performance of the proposed method is analyzed from the perspective of reproducibility and operationality [15], which are the performance indicators of bilateral control using hybrid parameters.The reproducibility P r (s) and operationality P o (s) are defined as follows: q m (79) where Z e (s), D e , and K e represent environmental impedance, viscosity, and stiffness, respectively.Reproducibility represents the degree of environmental impedance reproduced on the master side, and operationality corresponds to the operational force generated on the master side.The ideal characteristic of reproducibility is P r (s) = 1, and that of operationality is P o (s) = 0. Fig. 12 shows the Bode diagrams of the operationalities and reproducibilities of the four-and two-channel-type bilateral controllers using low-order (zeroth-and first-order) DOBs and the proposed method when the environmental stiffness is 500 Nm/rad.Fig. 12 (a) shows that the method using a low-order DOB has better operationality than those of the other methods in the low-frequency domain.However, methods that use low-order DOBs do not achieve ideal reproducibility characteristics in any frequency domain, as shown in Fig. 12 (b).This implies that they cannot provide accurate force/haptic feedback, which is the original purpose of the bilateral control.Conversely, although the proposed method only transmits force information, it exhibits performance characteristics equivalent to those of a four-channel method that uses both position and force information in terms of operationality and reproducibility, as shown in Figs. 12 (a) and (b), respectively.Fig. 13 shows the Bode diagrams of the operationalities and reproducibilities of the four-and two-channel controllers with the proposed method when the environmental stiffnesses are 500, 2500, and 10000 Nm/rad.Fig. 13 (a) and (b) show that for both methods, the lower the stiffness is, the wider the bandwidth over which the ideal characteristics of operationality and reproducibility can be maintained, and vice versa.However, we can observe that the four-channel controller and the proposed two-channel controller have equivalent characteristics in both environmental high-stiffness and lowstiffness cases.
Fig. 14 shows the Bode diagrams of the operationality and reproducibility obtained while varying the parameters in the coefficients of the observer represented by (53) to (55) and the virtual inertia in the force control (inverse of force-control gain) in the proposed two-channel controller.Fig. 14 (a) shows the Bode diagrams of operationality and reproducibility with variations in d 0th 0 in (53) to (55).Fig. 14  (a) shows that as d 0th 0 increases, the operationality decreases slightly.However, the larger d 0th 0 is set, the lower the degree of decrease in operationality is.In contrast, the reproducibility improves when d 0th 0 is set higher.These results show that it is desirable to set d 0th 0 as large as possible.Fig. 14 (b) shows that when ω 2 p (or ω p ) is varied, the trends in operationality and reproducibility are similar to those when d 0th 0 is varied.Therefore, ω 2 p (or ω p ) should also be set as large as possible.
Fig. 14 (c) shows the Bode diagrams of the operationality and reproducibility with the variation of ζ .Compared to d 0th 0 and ω 2 p , ζ has a smaller impact on operationality and reproducibility.However, ζ is a parameter that affects stability of the bilateral control, as discussed in the next subsection.Fig. 14 (d) shows the Bode diagrams of the operationality and reproducibility with the variation of M f .The figure shows that the effect of the variation in M f on reproducibility is marginal.However, the smaller M f is, the better the operationality.A smaller M f indicates that the feedback gain in the force control increases (virtual inertia decreases).Therefore, to improve operability, it is desirable to maintain M f within the allowable range of stability, which has no effect on reproducibility.

B. STABILITY ANALYSIS AND COMPARISON
The stabilities of the conventional and proposed bilateral controllers are analyzed using the following characteristic equation of the transfer function: Fig. 15 shows the transition of the dominant poles of the bilateral-control system when the parameters of the observer are varied in the proposed method.Fig. 15 p is varied, the poles of the bilateral control system deviate from the real axis when ω 2 p is approximately 25100(rad/s) 2 or higher.This may be because 2ζ ω p = p 2 + p 3 is fixed, resulting in an underdamping, which is a characteristic of the second-order system.This result indicates that ζ must be greater than 1.
Fig. 15 (e) and (f) show the transition of the poles of the bilateral-control system when ζ is varied.The same figure shows that the pole moves to the left half-plane, in the range where ζ is larger than approximately 0.03, and the pole of the bilateral-control system exists on the real axis if ζ is set between approximately 1 and 4.04.Similar to the results shown in Fig. 15(c) and (d), ζ may still be appropriate to set to approximately 1 (critical damping).
Fig. 16 shows the root loci of a four-channel controller and a two-channel bilateral-control system based on the proposed method with variations in the force feedback gains (J −1 f and c f ), bandwidth of the RTOB g ext , environmental viscosity D e , and environmental stiffness K e .
Fig. 16(a) shows the root locus with respect to the forcefeedback gain.Fig. 16(a) shows that the bilateral-control system becomes unstable when the virtual inertia of the proposed method exceeds approximately 3.15 × 10 −3 kgm 2 .However, this value of virtual inertia is approximately 25 times the nominal inertia J n of the actuator, and a value lower than this value of virtual inertia may be easily set.The figure also shows that if the virtual inertia is set to a value lower than approximately 1/4640 = 2.16 kgm 2 , or 1.75 times the nominal inertia, the poles of the bilateral-control system move along the real axis.These results indicate that a bilateral-control system based on the proposed method can maintain a sufficiently high degree of stability if the virtual inertia for force control is set to approximately the nominal inertia.
Fig. 16 (b) shows the root locus with respect to the band of the RTOB.This figure shows that the poles of the bilateral-control system with the four-channel controller and the proposed controller shift to the left half-plane when g ext is approximately 40 and 88 rad/s and shift to the real axis at approximately 1700 rad/s.Considering the trends in previous studies, setting the estimated bandwidth of the reaction torque above 88 rad/s is relatively easy.Therefore, the estimated bandwidth of the reaction torque should be set as high as possible, within the range in which the influence of noise can be tolerated.
Fig. 16 (c) shows the root locus for the environmental viscosity.The figure shows that the poles of the bilateral control system with the four-channel and proposed controllers follow almost the same trajectory with respect to the environmental viscosity.However, the proposed method is marginally more stable.Fig. 16 (e) shows the root locus of the environmental stiffness.In the case of this root locus only, the value of the environmental viscosity is set near the limit at which the bilateral control system is stable (D e = 0.25 Nsm/rad), as shown in Fig. 16 (c).As in the case of the root locus for environmental viscosity, the figure shows that the poles of the bilateral-control system with the four-channel and proposed controllers follow almost the same trajectory.However, the proposed method can maintain stability, even in environments with marginally greater stiffness (stiffer environments) than a four-channel controller.This is because the proposed method has no position-information transmission loop between the master and slave systems.
The above results indicate that to improve the performance of the proposed method, d 0th 0 and ω 2 p should be set to high values within their acceptable ranges of stability, and M f should also be set to the smallest value possible.The analysis results also show that the damping ratio should be set to 1.However, these performance-and stability-analysis results are obtained when the LPF parameters in the observer are set based on (53) to (55).

V. EXPERIMENTS
In this section, we describe the experiments conducted using actual equipment to confirm the effectiveness of the proposed method.In addition to bilateral-control experiments using the proposed method, bilateral-control experiments using a four-channel controller and two-channel bilateral-control experiments using lower-order DOBs are compared.

A. EXPERIMENTAL SETUP
This section describes the following experiments conducted to evaluate the proposed method: • Case 1: Bilateral control in joint space • Case 2: Bilateral control in workspace of manipulators.In Case 1, bilateral control experiments are conducted for five control systems, including the proposed method.
• Case 1-5: proposed 2ch.controller In Case 1-4, the position-controller from the perspective of the joint space c p (s) is set to be equal to the second-order DOB parameter 2ω p s + ω 2 p in the proposed method.In other words, the controller in Case 1-4 is a four-channel controller that ignores the reciprocal of the determinant of the coordinate-transformation matrix T, which appears in the inverse coordinate-transformation matrix T −1 shown in (27).This study uses master and slave systems comprising a single-DoF manipulator, as shown in Fig. 17 in Case 1.The manipulators comprise direct-drive motors (µDD motor: MDS-7018, Microtech Laboratory, Inc.).Therefore, this study assumes that the friction in the actuators is negligibly small.The control program is coded using the Advanced Robot Control System: ARCS [50].
Via bilateral control, the operator first performs free motions and then executes a contact motion on the aluminum 45588 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.plate, followed by a contact motion on a rubber ball after moving it.
In Case 2, in the workspace of the parallel two-link manipulator, the four-channel and the proposed two-channel bilateral control are performed as follows: • Case 2-1: 4ch.controller • Case 2-2: proposed 2ch.controller This study uses master and slave systems composed of a parallel two-link manipulator, as shown in Fig. 18 in Case 2. The manipulators also comprise the same direct-drive motors as those in Case 1.The operator first performs free motions in the x-axis direction and then in the y-axis direction via bilateral control.After the free motion, contact motions are performed in the x-axis direction to the plate and block, followed by contact motions in the y-axis direction.
The parameters used in the joint-space experiments (Case 1) are the same as those used in the analysis, and are listed in Table 1.Table 2 lists the experimental parameters in the workspace (Case 2).The other parameters used in Case 2 were the same as those used in the analysis and Case 1, and are listed in Table 1.

B. EXPERIMENTAL RESULTS IN JOINT SPACE (CASE 1)
Fig. 19 shows the experimental results for the two-channel bilateral control with lower-order DOBs.For ease of reading, the waveforms in this section are shown with the reverse sign of the reaction-torque response on the master side.The shaded area (A) represents the area of free motion, the area (B) represents the area of contact motion against the aluminum plate, and the area (C) represents the area of contact motion against the rubber ball.As shown in Fig. 19 (a), the magnitudes of the forces in the master-slave system are consistent when a zeroth-order DOB is used.However, as the analysis shows, position synchronization is not achieved, and the more force is applied, the more the master and slave systems continue to move away from each other.As a result, the control goal of bilateral control is not achieved in the force control system with a zero-order DOB.In addition, as shown in Fig. 19 (b), the response of the control system using the first-order DOB is improved compared to the case using the zero-order DOB.However, as shown in the analysis results, a steady-state error exists in the position response during contact motion.Fig. 19 shows that the control goals of bilateral control cannot be achieved in a two-channeltype control system with lower-order (zero-and first-order) DOBs.
Fig. 20 (a) shows the results of bilateral control with a fourchannel controller.This figure shows that both the position and force responses are good, indicating that the control target of the bilateral control is sufficiently achieved by the fourchannel controller.the same as 2ω p s+ω 2 p of Cases 1-5, the response is oscillatory in area (C) in contact with the rubber ball.The environmental viscosity of the rubber ball is low, and the stability of the four-channel bilateral control is reduced due to the increase in the position-control gains.
In contrast, Fig. 21 shows the response of two-channel bilateral control using the proposed method.The same figure shows that the proposed method exhibits good position and force responses, comparable to those of Cases 1-3, both in the hard environment of the aluminum plate and in the soft elastic environment of the rubber ball.Parameter 2ω p s + ω 2 p in the second-order DOB has the same value as in the four-channel bilateral control for Cases 1-5.However, the response is not oscillatory in region (C) in contact with the rubber ball, indicating that the proposed method maintains a high degree of stability.
In conclusion, the results confirmed that the two-channel bilateral-control system based on the proposed method exhibits a performance equivalent to that of a four-channel control system that uses position information without transmitting it between the master and slave systems.

C. EXPERIMENTAL RESULTS IN WORK SPACE (CASE 2)
Fig. 22 shows the experimental results for Case 2. For ease of reading, the waveforms in this section are shown with the reverse signs of the responses of the reaction force in the y-axis on the master side and in the x-axis on the slave side.the areas of free motion in the x-axis direction, free motion in the y-axis direction, contact motion in the x-axis direction, and contact motion in the y-axis direction, respectively.Fig. 22 (a) and (b) show the x-and y-axis positions and the force responses of the four-channel bilateral control in the workspace, respectively.These figures show that, as in Case 1, the four-channel controller in the workspace sufficiently achieves the bilateral control goal.
Conversely, Fig. 22 (c) and (d) show the position and force responses of the two-channel bilateral control using the proposed method on the x-and y-axes, respectively.The figures show that by configuring second-order DOBs in the workspace, the bilateral control goals are sufficiently achieved in both free and contact motions without transmitting position information, as in the joint-space case (Cases 1-5).In addition, as in Cases 1-5, no position information is transmitted between the master and slave systems; however, the two-channel bilateral control system has the same position and force responsiveness as those of a four-channel bilateral control system.
These results confirm the effectiveness of the proposed two-channel bilateral control system experimentally.

VI. CONCLUSION
This study proposed a force-symmetric two-channel bilateralcontrol system based on a higher-order DOB.In a bilateral control system, reducing the information-transmission channel between the master and slave systems provides advantages, such as suppressing data traffic, reducing stored data, and simplifying the control-system design.
Therefore, this study proposed a two-channel bilateralcontrol system based on robust acceleration control using second-order DOBs among the higher-order DOBs.The results showed that the proposed method can achieve not only the force-control goal, but also the position-control goal, whereas methods using lower-order DOBs, such as  zero-and first-order DOBs, cannot achieve synchronization of the master-slave system position.The performance of the proposed method was equivalent to that of a fourchannel bilateral-control system that transmitted position  information.The effectiveness of the proposed method was confirmed through bilateral control experiments in the joint space and workspace.
Future applications of the proposed two-channel bilateralcontrol method include an extension to the preservation and reproduction control of human motion [29], [33], multilateral control [22], and bilateral control with different motion ranges [25].
The objective of this study was to realize two-channel bilateral control with a higher-order DOB in the ideal state, that is, without a time-delay.Therefore, future work will also focus on a detailed analysis of the performance and stability of a two-channel bilateral-control system when applied to a time-delay system.A primitive and elementary discussion on the application of the proposed two-channel bilateral control to time-delay systems is provided in Appendix A. The results presented in this appendix demonstrate the high control performance and stability potential of the proposed two-channel bilateral-control system for time-delay systems.

APPENDIX A APPLICATION FOR TIME DELAY SYSTEMS
In this appendix, a primitive and elementary discussion and analysis of the application of two-channel bilateral control based on the proposed method to time-delay systems are presented.First, four-channel and two-channel bilateral controllers based on the impedance-feed expression of the four-channel bilateral controller in time-delay systems are described.Subsequently, the problems of two-channel bilateral control using the proposed method in a time-delay system and their corresponding solutions are described.Through simulations, the effectiveness of the modification of the two-channel bilateral control using the proposed method for the time-delay system is confirmed.

A. FOUR-CHANNEL BILATERAL CONTROLLER AND TWO-CHANNEL BILATERAL CONTROLLER BASED ON IMPEDANCE FIELD EXPRESSION OF FOUR-CHANNEL CONTROLLER
The acceleration references for the four-channel bilateral controller in a time-delay system qref m4chD and qref s4chD are described as where T d1 represents the delay time from the master system to the slave system, and T d2 also represents a delay time from the slave system to the master system.In addition, the acceleration references for a two-channel bilateral controller based on the impedance-field expression of the four-channel bilateral controller [30] qref′ m4chD are qref′ s4chD are described below: Fig. 23 shows a block diagram of the two-channel bilateral controller based on the impedance-field expression of the four-channel controller in a time-delay system.This method can reduce data traffic compared with a four-channel bilateral teleoperation controller [30]; however, the control performance is equivalent to that of the four-channel type.

B. PROPOSED TWO-CHANNEL BILATERAL CONTROLLER BASED ON HDOB FOR TIME-DELAY SYSTEM
If the proposed two-channel bilateral controller is directly applied to a time-delay system, the acceleration-reference value is expressed as follows: However, position synchronization is not achieved using this control system.The same problem occurs when force-based two-channel bilateral control based on admittance control is applied to a time-delay system [33], [51].Therefore, the acceleration reference value of the proposed two-channel bilateral control system is modified, as shown in the following equation: .The above equation indicates that information on the external torque of the master side is transmitted from the master system to the slave system, and the error in the external torque control calculated on the slave-system side is transmitted from the slave-system side to the master-system side.Using (87), position synchronization can be achieved, even in a two-channel bilateral control using the HDOB.Additionally, as reported in the literature [51], transmitting information including the force control gain is unnecessary, and only transmitting information on the dimensions of (the error in) the external torque is necessary.
Fig. 24 shows a block diagram of the modified proposed two-channel bilateral controller for application in a timedelay system.

C. SIMULATIONS
This section describes the simulations used to confirm the effectiveness of the modified proposed method for applying the proposed two-channel bilateral control to a time-delay system.

1) SIMULATION SETUP
This section describes a four-channel controller, two-channel controller based on the impedance-field expression of a four-channel bilateral controller, two-channel controller using the proposed method, and two-channel controller using the modified proposed method in time-delay systems.The simulations are performed based on (81) to (85) and the block diagrams in Fig. 7, 8, 23 and 24.The simulation parameters are listed in Table 3.The other parameters were the same as those used in the analysis and joint-space experiments and are listed in Table 1.To ensure stability, the reciprocal of virtual inertia J −1 f and force feedback gain K f for the force control are set to smaller values than in the experiment without timedelay.Simulations were performed for two sets of time-delay systems, represented by Cases 1 and 2, as listed in Table 3.

2) SIMULATION RESULTS
Fig. 25 shows the simulation results for Case 1 of bilateral teleoperation control using a four-channel controller and the proposed two-channel controller.Fig. 25(a) shows that the stability of the four-channel control system decreases owing to the influence of time delay, resulting in an oscillatory response.Fig. 25(b) shows the simulation results when the two-channel bilateral controller based on the proposed method is directly applied to a time-delay system.The figure shows that although the force response appears to have less vibration and higher stability than the four-channel controller, position synchronization has not been achieved.This is because the acceleration reference values input to the master and slave systems deviate owing to time delay.Fig. 26 shows the simulation results for Case 1 of a two-channel controller based on the impedance-field expression of the four-channel controller and two-channel controller using the modified proposed method.As shown in Fig. 26 (a), the two-channel controller based on the impedance-field expression exhibits the same response as the four-channel controller shown in Fig. 25 (a).This implies that a two-channel controller based on impedance-field expression can reduce data traffic more than a four-channel controller can; however, its performance and stability are essentially the same as that of a four-channel controller.
Fig. 26(b) shows the response of the two-channel controller using the modified proposed method.In contrast to Fig. 25 and Fig. 26 (a), the two-channel controller with the modified proposed method achieves position synchronization and maintains high stability with less vibration.This is considered as an advantage of having no position-information transmission loop between the master and slave systems.
Fig. 27 shows the simulation results for Case 2 (a delay time that is five times larger than that of Case 1) for a two-channel controller based on the impedance-field expression of the four-channel controller and two-channel controller using the modified proposed method.As shown in Fig. 27 (a), the two-channel controller based on the impedance-field representation oscillates completely and is likely to diverge.Conversely, Fig. 27 (b) shows that the two-channel controller based on the modified proposed method maintains stability, even when the delay time is five times longer than that of Case 1.Moreover, both positional synchronization and action-reaction law are achieved artificially in a steady state.Notably, the modified proposed method maintains stability because of the absence of a position-information transmission loop, even though the reaction-torque information on the master side is input to the master system through a round trip between the master and slave systems.
Thus, the effectiveness of the proposed two-channel bilateral controller using HDOBs is demonstrated also in a time-delay system.However, more detailed performance and stability analyses should be conducted in the future.

FIGURE 1 .
FIGURE 1. Conceptual figure of the proposed method in this paper.

FIGURE 2 .
FIGURE 2. Block diagram of a zeroth-order DOB using velocity input.

FIGURE 3 .
FIGURE 3. Equivalent block diagram of an actuator with disturbance compensation by zeroth-order DOB.

FIGURE 5 .
FIGURE 5. Block diagram of Second-order DOB using velocity input.

FIGURE 6 .
FIGURE 6. Equivalent block diagram of an actuator with disturbance suppression by n-th order DOB.

FIGURE 7 .
FIGURE 7. Block diagram of four-channel bilateral control system based on position/force hybrid control in joint space.

FIGURE 8 .
FIGURE 8. Block diagram of the proposed force-symmetric-type two-channel bilateral control system in joint space.

FIGURE 9 .
FIGURE 9. Block diagram of four-channel bilateral control system based on position/force hybrid control in workspace.

FIGURE 10 .
FIGURE 10.Block diagram of the proposed force-symmetric-type two-channel bilateral control system in workspace.

FIGURE 13 .
FIGURE 13.Reproducibility and operationality of four-channel controller and proposed controller with variations in environmental stiffness K e .(a) Reproducibility.(b) Operationality.

FIGURE 16 .
FIGURE 16.Root locus of conventional and proposed controllers with the variation of force feedback gain, external torque estimation bandwidth, and environmental parameter-variation. with respect to (a) Force feedback gain J −1 f .(a) Estimation bandwidth g ext .(c) Environmental viscosity D e .(d) Environmental stiffness K e .
Fig. 15 shows the transition of the dominant poles of the bilateral-control system when the parameters of the observer are varied in the proposed method.Fig. 15(a) and (b) show the pole transition for the variation of p 1 = d 0th 0 .The figure shows that the poles of the bilateral-control system can exist on the real axis if p 1 = d 0th 0 is set in the range of approximately 770-2374 rad/s.Fig. 15(c) and (d) show the pole transition of the bilateral control system when ω 2 p is varied.However, only the term related to ω 2 p = p 2 p 3 is varied, whereas the term related to 2ζ ω p = p 2 + p 3 is fixed.Fig. 15 (c) and (d) show that when ω 2p is varied, the poles of the bilateral control system deviate from the real axis when ω 2 p is approximately 25100(rad/s) 2 or higher.This may be because 2ζ ω p = p 2 + p 3 is fixed, resulting in an underdamping, which is a characteristic of the second-order system.This result indicates that ζ must be greater than 1.Fig.15(e)and (f) show the transition of the poles of the bilateral-control system when ζ is varied.The same figure shows that the pole moves to the left half-plane, in the range where ζ is larger than approximately 0.03, and the pole of the bilateral-control system exists on the real axis if ζ is set between approximately 1 and 4.04.Similar to the results shown in Fig.15(c) and (d), ζ may still be appropriate to set to approximately 1 (critical damping).Fig.16shows the root loci of a four-channel controller and a two-channel bilateral-control system based on the proposed method with variations in the force feedback gains (J −1

FIGURE 18 .
FIGURE 18. Experimental setup for bilateral control in work space.

FIGURE 21 .
FIGURE 21.Experimental result of the proposed force-symmetric-type two-channel bilateral controller using second-order DOB.
Fig.19shows the experimental results for the two-channel bilateral control with lower-order DOBs.For ease of reading, the waveforms in this section are shown with the reverse sign of the reaction-torque response on the master side.The shaded area (A) represents the area of free motion, the area (B) represents the area of contact motion against the aluminum plate, and the area (C) represents the area of contact motion against the rubber ball.As shown in Fig.19 (a), the magnitudes of the forces in the master-slave system are consistent when a zeroth-order DOB is used.However, as the analysis shows, position synchronization is not achieved, and the more force is applied, the more the master and slave systems continue to move away from each other.As a result, the control goal of bilateral control is not achieved in the force control system with a zero-order DOB.In addition, as shown in Fig.19 (b), the response of the control system using the first-order DOB is improved compared to the case using the zero-order DOB.However, as shown in the analysis results, a steady-state error exists in the position response during contact motion.Fig.19shows that the control goals of bilateral control cannot be achieved in a two-channeltype control system with lower-order (zero-and first-order) DOBs.Fig.20(a)shows the results of bilateral control with a fourchannel controller.This figure shows that both the position and force responses are good, indicating that the control target of the bilateral control is sufficiently achieved by the fourchannel controller.Fig. 20 (b) shows the results for Cases 1-4.As shown in Fig. 20 (b), when the position-control gains in terms of the joint space are set to twice that of Cases 1-3 and

FIGURE 22 .
FIGURE 22. Experimental result of four-channel bilateral controller based on position/force hybrid control and proposed force-symmetric two-channel type bilateral controller in workspace.(a) X-axis responses in four-channel type.(b) Y-axis responses in four-channel type.(c) X-axis responses in proposed two-channel type.(d) Y-axis responses in proposed two-channel type.
Fig.22shows the experimental results for Case 2. For ease of reading, the waveforms in this section are shown with the reverse signs of the responses of the reaction force in the y-axis on the master side and in the x-axis on the slave side.The shaded areas (A), (B), (C), and (D) in the figureindicate

FIGURE 23 .
FIGURE 23.Block diagram of a two-channel bilateral controller based on impedance expression of the four-channel bilateral controller with a time delay[30].

FIGURE 24 .
FIGURE 24.Block diagram of proposed two-channel bilateral control system based on HDOB in time-delay system.

FIGURE 25 .
FIGURE 25.Simulation results of the conventional and proposed two-channel bilateral teleoperation in Case 1.(a) Four-channel bilateral controller (b) The proposed two-channel bilateral controller as-is.

FIGURE 26 .
FIGURE 26.Simulation results of the conventional and proposed two-channel bilateral teleoperation in Case 1.(a) Two-channel bilateral controller based on impedance expression of four-channel bilateral controller.(b) The modified proposed two-channel bilateral controller for a time-delay system.

FIGURE 27 .
FIGURE 27.Simulation results of the conventional and proposed two-channel bilateral teleoperation in Case 2 (time-delay is five times longer).(a) Two-channel bilateral controller based on impedance expression of the four-channel bilateral controller.(b) Modified proposed two-channel bilateral controller.
Fsm = e −T d2 s τ ext Fs = e −T d2 s e −T d1 s τ ext m + τ ext s

TABLE 1 .
Parameters used in analysis and joint space experiments (case 1).
FIGURE 17. Experimental setup for bilateral control in joint space.

TABLE 3 .
Parameters used in joint-space simulations with time-delay.