Task Space Outer-Loop Integrated DOB-Based Admittance Control of an Industrial Robot

Admittance control can improve robot performance and robustness in interactive tasks but is still limited by stability when implemented on low-admittance hardware, such as position-controlled industrial robots. This limits applications that require payload, reach, or positioning accuracy. While the idealized reference admittance behavior would be stable with any passive environment (provided positive damping), real robots can be unstable, especially in high-stiffness environments. Thus, instability comes from deviation from the ideal reference model, due to either inner-loop bandwidth, time delay, or other model error. To improve the accuracy of rendered dynamics and reduce contact forces, a novel integrated disturbance observer (DOB)-based admittance control method is proposed. This method does not require access to the robot’s inner-loop position control; instead, it is designed and built around it in task space. The task space multisensor information, i.e., the velocity command, measured output velocity, and the force/torque (F/T) sensor measurement are integrated to estimate and robustly suppress the disturbances. Theoretical analyses and experiments on the actual robot show that the proposed method is able to improve admittance tracking accuracy and reduce contact forces even at higher admittance.

Abstract-Admittance control can improve robot performance and robustness in interactive tasks but is still limited by stability when implemented on low-admittance hardware, such as position-controlled industrial robots.This limits applications that require payload, reach, or positioning accuracy.While the idealized reference admittance behavior would be stable with any passive environment (provided positive damping), real robots can be unstable, especially in high-stiffness environments.Thus, instability comes from deviation from the ideal reference model, due to either inner-loop bandwidth, time delay, or other model error.To improve the accuracy of rendered dynamics and reduce contact forces, a novel integrated disturbance observer (DOB)based admittance control method is proposed.This method does not require access to the robot's inner-loop position control; instead, it is designed and built around it in task space.The task space multisensor information, i.e., the velocity command, measured output velocity, and the force/torque (F/T) sensor measurement are integrated to estimate and robustly suppress the disturbances.Theoretical analyses and experiments on the actual robot show that the proposed method is able to improve admittance tracking accuracy and reduce contact forces even at higher admittance.

Signals
θ, θ, θ Joint space motor-side acceleration, velocity, and position.q, q, q Joint space link-side acceleration, velocity, and position.θi , θ i I. INTRODUCTION P HYSICAL interaction with the environment or humans requires robots to safely render a range of dynamics.Therefore, safety is critical in interactive control, as the unknown environment dynamics couple with the robot, changing effective dynamics and potentially causing oscillation or instability [1], [2].Moreover, contact transitions or collisions can also cause higher transient forces [3].These safety concerns are often stronger in contact with high-stiffness environments.For these reasons, lower admittance hardware is often used, introducing compliance in the actuators [4], [5], flange [6], end-effector, human interface [7], or environment [8].This has improved intrinsic safety properties but typically comes at the cost of payload, reach, or positioning accuracy.
For many industrial applications, the robot payload involves bulky [9], [10] or heavy [11], [12] objects.In such cases, an industrial robot can provide a higher payload capacity or reach.Realizing interactive control on such systems is typically done via admittance control [13].Admittance control on industrial robots means that the intrinsic robot admittance is lower-the robot has a higher physical stiffness and inertia.Moreover, the inner-loop position control further decreases the admittance.
Admittance control involves a linear reference admittance model, typically a mass damper, possibly including a stiffness [13].If the robot perfectly rendered these dynamics, stability interacting with any passive environment would be trivial, even with high, pure stiffness environments, requiring only positive damping and inertia.From the observed stability challenges, many authors have identified deviation from the ideal reference as the source of stability issues.Early work identified higher order dynamics [14], from, e.g., gearbox resonance.Time delay [15] and model uncertainty [16] are also deviations from the idealized reference admittance model.
The above approaches have been developed for manipulators with a joint torque control interface, and moreover, they require knowledge of nonlinear robot dynamics.However, it is not possible to directly apply these approaches for admittance control of typical industrial robots with "blackbox" inner-loop proportional derivative (PD) position control system.
We were the first to propose an alternative idea of designing and utilizing a DOB in task space admittance control for a typical industrial robot [45], [46].With an aim of increasing admittance and improving contact stability, the DOB in [45] was designed to cancel the disturbances, whereas the same DOB in [46] was designed to amplify disturbances using positive velocity feedback inner-loop shaping approach.However, the F/T measurements were not utilized in DOB designs in these approaches.The robot was modeled as a single mass system, and theoretical analyses and experiments for robot model identification were not conducted.This article's motivation is that, for typical industrial robots, the degradation in inner-loop position control accuracy can lead to undesired motions at the end-effector, potentially leading to higher peak contact forces and compromised contact stability.Therefore, the design goal of this article is toward improving the accuracy of the robot's admittance to user/environment forces to make it easier to manipulate the robot, reduce contact force with the environment, keep contact stability, and have minimal oscillation in free space or contact.To achieve this, a task space outer-loop integrated DOB-based admittance control system (OIDOBt) for typical industrial robots is proposed.The foremost advantage of the proposed method lies in its ability to bypass the requirement of accessing the robot's inner-loop position control, which includes both the motion controller and the underlying robot dynamics.Instead, the proposed method is designed and built around the inner-loop position controller in task space.
The OIDOBt exploits the availability of task space multisensor information, i.e., velocity command, measured robot velocity and the inverse model of the closed velocity control loop, and the force/torque (F/T) sensor measurements, which are integrated to estimate and robustly suppress the effects of disturbances.Utilization of F/T sensor measurements in the design comes with an important characteristic of suppressing the high-frequency contact dynamics, thus significantly improving contact stability.In addition, the first-of-its-kind linear modeling approaches of the robot dynamics and admittance control architecture in task space are presented and experimentally validated.Moreover, a technique for designing the reduced-order OIDOBt nominal model that includes both robot and payload dynamics is introduced.
The proposed admittance control method is simple to implement on the existing industrial robots.In addition to thorough theoretical evaluations, various experiments involving both autonomous robot operation and co-manipulation tasks are conducted on the actual 6-DOF industrial robot to verify the effectiveness of the proposed method.
The rest of this article is organized as follows.Section II introduces the target robot system, derives the linear dynamic models, and discusses the control problem.The proposed admittance control architecture is presented in Section III, while Section IV focuses on closed-loop analysis of the control system.Verification through experiments is then conducted in Section V. Finally, the conclusion is given in Section VI.

II. SYSTEM MODELING AND PROBLEM DESCRIPTION
In this section, the multi-DOF robot is introduced and modeled.Moreover, standard/basic linear admittance control architecture is derived and its limitations are identified.
A. Robot and Its Basic Admittance Control Method 1) Robot System: Fig. 1 presents the target system, which is a 6-DOF robot manipulator with a fixed inner-loop position/velocity controller.An F/T sensor is rigidly coupled to the robot flange, and depending on the application, the gripper or payload can be fixed on the F/T sensor.External force acts on the robot from the user and/or environment, physically acting on the robot dynamics and being measured through the F/T sensor.An inner-loop task space motion controller tracks the motion command supplied by the admittance controller.Admittance control is realized about a compliant frame, typically the robot's tool center point (TCP), where the total control is diagonal and each DOF is independent.The force measurements are transformed into this frame, and the robot motion is commanded in this frame.The control of Fig. 1 is applied to the linear Cartesian DOFs, primarily from joints 1/2/3, as the linear motion typically generates larger contact forces, whereas the rotary motion, primarily joints 4/5/6, are making small motions.
2) Linear Task Space Modeling of the Motion-Controlled Robot in Frequency Domain: To begin with, the robot in Fig. 1 is equipped with harmonic drives, and the effects of joint flexibility can deteriorate control accuracy [47], [48].Hence, the robot in Fig. 1 can be modeled in joint space as an FJR system [33], as shown by the robot dynamics in Fig. 2. Note that the task space outer admittance control loop, which computes the command position, θ i , is ignored in Fig. 2. C θ , θ ,I is the cascaded motion controller with hierarchical position/velocity/current control loops, τ ′ c is the control input vector, τ s is the vector of joint torque generated by the flexibility of the joint, d τ is the torque disturbance vector, τ /F is the vector of the feedback torque/force from the F/T sensor to the robot, J is the Jacobian matrix, and θ and q are motor-and link-side velocity vectors, respectively.
To this end, the time-domain equations that govern Fig. 2 are derived as shown in the following: These are defined as motion controller in (1) collectively modeled as a PD position control, linear motor-side dynamics in (2), and nonlinear link-side dynamics in (3).The effect of joint flexibility is analyzed experimentally in Section V-B1 to validate the robot model in Fig. 2 for (2) and (3).µ p and µ i are proportional and integral gain matrices, respectively, M m and B m are motor-side mass and damping matrices, respectively, and M l (q) and B l (q) are link-side mass and damping matrices, respectively.The vector H(q, q) in (3) combines centrifugal, Coriolis, and gravity forces that depend on q and q.Note that, considering the robot in Fig. 1, the motion controller in (1) is designed with gravity compensation feature, and thus, this term is ignored in this article for simplicity.In addition, centrifugal and Coriolis forces are small and can be ignored considering the low speeds in this intended application of the robot.However, the residues of gravity forces that may be present and the effects of Coriolis and centrifugal forces that may be caused by high-speed/jerky motions of the robot, including the nonlinear joint friction, are all considered as part of torque/force disturbances, d τ /d F , shown in (2) (see d F in Fig. 3), to be estimated and robustly suppressed by the method proposed in Section III.
Since the task space desired velocity for the robot in Fig. 1 is V i = J(q) θi , the task space load-and motor-side velocities are V l = J(q) q and V = J(q) θ, respectively, such that Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Substitute ( 4) and ( 5) in ( 1)-( 3), premultiply both sides of the result by J −T , and ignore the J terms to get the task space equivalent dynamics as shown in the following: where ( 6)-( 8) are, respectively, analogous to (1)- (3).
Note that the J terms in ( 6)-( 8) are omitted to simplify the formulation and reduce computational complexity.This is because the acceleration and higher order derivatives in the task space are not explicitly required or controlled in admittance control [49].Moreover, in the target application in this article such as the co-manipulation tasks in Exp 4 (Section V-D.5), the robot is manipulated at very low speeds, and thus, the terms with J are very small and can be ignored.
The effects of joint flexibility are assumed to be small and linear [50], and the inner-loop joint control is assumed to be decoupled because a high gear ratio drivetrain and lower accelerations are used [51,Sec. 8.3].Thus, the nonlinear dynamics in ( 8) is linearized about a fixed robot pose and remodeled as a system of mass-damper type.To this end, the equivalent frequency domain of ( 6), (7), and linearized (8) is obtained by taking their Laplace transforms, and the resulting linear transfer functions are given in the following for a single DOF: These are task space PI velocity controller, C(s), motorside dynamics, R m (s), link-side dynamics, R l (s), and joint flexibility dynamics, K s (s) with k s as the stiffness constant.
Since the Cartesian configuration decouples the joint dynamics, each task space DOF is considered to be independent, so the task space models in ( 9) and ( 10) can be considered to be diagonal, and thus, a single DOF is analyzed in this article, as shown in Fig. 3.This diagonalization assumption is experimentally verified in Section V-D.3.
As shown in Fig. 3 (right), the robot dynamics, R(s), can be interpreted as a linear two-mass system composed of masses R m (s) and R l (s), and the finite stiffness K s (s) between them.This two-mass system can be further reduced as the transfer function from F c to V , resulting in the single transfer function of robot dynamics R(s) given as follows: where V and F c are the Laplace transforms of V and F c , respectively.With the simplified model in (11), it is possible to utilize the measured task space motor-side robot velocity, V , to couple the F/T sensor, payload, and the environment to the robot, as shown in Fig. 3.The task space model may change in different robot poses, and therefore, to account for model variation, various poses (see Fig. 11) are considered while measuring the model parameters in Section V-B.Moreover, the linear model in (11) allows the use of frequency-domain analysis of contact resonance and rendered admittance.The effectiveness of these assumptions is tested, and the linear model derived in (11) is validated experimentally in Section V-B.3.
3) Related Literature About Linear Modeling of Fixed Motion-Controlled Industrial Robot: The concept of linear task space robot modeling proposed in Section II-A2 was first introduced by Haninger et al. [12], where the robot dynamics, R(s), was modeled as a single mass system and the diagonalization assumption was utilized.However, the physical impact of forces, F, on the robot and the effects of joint flexibility were ignored, and the detailed system identification was not conducted.The task space linearized single mass robot model was further utilized in [45] and [46], and the proposed control architectures were designed based on this model.The physical impact of forces, F, on the robot was considered.In addition, experimental results for the proposed control architectures utilizing the linearized models were conducted on an actual robot and results agreed with the theoretical analyses.However, in all these studies, the effect of joint flexibility was ignored and the models were not experimentally validated.Thus, the improvement in this article is considering the joint flexibility and detailed derivations of the linear models in Section II-A2 and the validation of the linear models using experiments on the actual robot in Section V-B.
4) Basic Linear Task Space Admittance Control Architecture: The payload/environment can be attached to the F/T sensor, as shown in Fig. 1.This is graphically shown in Fig. 3, where the F/T sensor, S F (s), and the environment, E(s), are modeled as stiff springs, whereas the payload, P(s), is modeled as pure inertia, as shown in the following: where K f is F/T sensor stiffness, K e is environment stiffness, and M p is payload mass.The human model is not considered in this analysis as typical human arm stiffnesses are <3 N/mm [52], so are typically dominated by environment stiffness (>50 N/mm in this article).The effect of F/T sensor stiffness, K f , is significant for heavy payload applications such as 50-kg manipulation with the COMAU AURA robot in [12], and thus, the F/T sensor dynamics can be considered, as shown by option I in Fig. 3.For light payload applications, the effect of K f is insignificant and can be ignored, and modeling is shown by option II in Fig. 3.
The basic admittance control of Fig. 1 is realized by the outer feedback of the F/T sensor measurements, F m , as shown in Fig. 3 for a single DOF in task space [12].An analog low-pass filter (LPF) cutoff frequency in the F/T amplifier is set to the maximum (2500 Hz in this case) to reduce force measurement delay that may cause instability [13].Force can be supplied to the robot in two ways: either through the external force, F ext , or the virtual force reference, F r .Note that the external force, F ext , is composed of human, F hum , and environment, F env , forces.The error from the reference and measured forces passes through the admittance controller, A(s), to generate the velocity command, V i , which is supplied to the closed-loop velocity-controlled robot with controller C(s) and linearized dynamics R(s).The robot then moves with velocity, V , and thus generates the force, F, which is transmitted to the payload as well as feedback to the robot itself.The disturbances into the system are lumped and defined in force medium by d F , and d (s) = e −st d represents the input time delays, i.e., the communication delays between the external and internal robot controllers.The desired admittance, A(s), is designed as a mass-damper system, as shown in the following: where M a and B a are admittance mass and damping coefficients, respectively.

B. Limitations of Basic Admittance Control Method
The robot should be stable on dynamic interaction with the environment.In practice, interaction with high-stiffness environments is challenging, limiting the range of admittance, which can be rendered.Note that if the target admittance dynamics in (13) were perfectly rendered, the system would be trivially stable for an arbitrary stiffness environment provided M a , B a > 0. One explanation why this does not attain on physical systems is the divergence between rendered and idealized dynamics.The limitations include the following.
1) Desired Admittance Rendering Limitation: Since the application goal is co-manipulation tasks, the rendered admittance can be calculated by considering the transfer function from external forces, F ext , to velocity output, V , defined as It is desired that, for a robot carrying payload, the rendered admittance equals the desired admittance, i.e., Y (s) = A(s).To check this condition, Y (s) is derived by considering option II of Fig. 3 for simplicity of analysis as shown in the following: where the Laplace operator s in ( 14) and from herein after is ignored on transfer functions for simplicity.Thus, from ( 14), the desired admittance cannot be rendered to the robot as it is affected by robot and payload dynamics, and time delays.
2) Contact Stability Limitation: This limitation can be analyzed by considering the effect of the stiffness of the contacting surface.To this end, the force tracking transfer function from F r to F in contact, defined as F/F r (s) = G(s), can be utilized.This is derived for Fig. 3 as where L is the open-loop plant given in (37).The Bode magnitudes of (15) are presented in Fig. 4 when the environment stiffness is varying.Parameters utilized are given in ( 42)- (44).It can be observed in Fig. 4 that the stiffness of the contacting surface significantly amplifies contact resonance.
3) Constraint of Inner-Loop Velocity Control: The desired admittance rendering and contact resonance limitations can be mitigated by setting the proportional gain in C(s) very large [24].To this end, (14) becomes but the effect of payload dynamics and time delays is not eliminated by the high gain control.In addition, setting C(s) proportional gain large can reduce the contact resonance in Fig. 4.However, the constraint with inner-loop position/velocity control for industrial robots is that it is closed and cannot be changed.On the other hand, it is hypothesized in this article that a DOB can be implemented in task space outside the fixed position/velocity control loop to cancel robot and payload dynamics, and time delays, thus being a possible solution to all the above limitations.

III. DESIGN OF PROPOSED CONTROL METHOD
To resolve the above limitations, an admittance control system that implements a DOB in task space, built around the fixed velocity control loop, is developed in this section.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
A. Proposed Control Architecture Fig. 5 presents the proposed control scheme, which is implemented in task space and outside the velocity control loop, compared to the conventional inner-loop DOB implementations [53], [54].The proposed method is called task space OIDOBt.The OIDOBt exploits the available multisensor information in the basic method for its design.With the proposed method, the velocity command V i in Fig. 3 is modified to include the output of OIDOBt, d, and V r becomes the new auxiliary velocity command, as shown in Fig. 5.

B. Detailed Design of the OIDOBt
The proposed OIDOBt is designed to improve admittance rendering accuracy and contact stability by suppressing the effects of force disturbances, d F .In addition, payload forces and high-frequency forces generated when the robot's end-effector contacts a stiff surface can deteriorate control performance.All these are combined and called lumped disturbances to be estimated as d and robustly suppressed.The force disturbances, d F , typically considered here for the case of the robot in Fig. 1 are nonlinear friction, time delays, mechanical vibrations due to higher order dynamics, coupling between DOFs in task space due to imperfections in jointlevel control [55], [56], residues of gravity forces, and effects of Coriolis and centrifugal forces due to high-speed/jerky motions.Here, it is also considered that the external force acts on the robot system, causing deviation from expected motion control performance.Note that d F is assumed to be generated by an exogenous system given by ḋ F (t) = 0 [24], [57].
1) Lumped Disturbance Estimation: With the proposed method (magenta in Fig. 5), the lumped disturbance estimate, d, is obtained by integrating the velocity input, the measured output velocity and the inverse model, and the F/T sensor information as shown in the following: where D• , V • , and F m are the Laplace transforms of d• , V • , and F m , respectively.The V m and V i terms in (17), which form the conventional DOB structure [24] but different in design of D n [as shown in (19)], estimate the force disturbances and payload forces, whereas the F m part estimates the high-frequency contact forces.
2) Design Criteria and Considerations for the F/T Sensor Measurement Term: The inclusion of the A(1−Q) term in (17) is strategically designed to ensure that the rendered admittance tracks the desired admittance, as elaborated in Section IV-A.The high-pass filter component, 1 − Q, allows high-pass frequency shaping of the F/T sensor measurements, which in turn gives the OIDOBt an extra capability of effectively suppressing the impact of vibrations arising from contact dynamics, thereby enhancing overall contact stability.This feature contributes to the improved performance of the controller in various contact scenarios.Furthermore, the desired admittance, A, plays an important role in not only converting force into velocity but also shaping the force information to accurately track the desired admittance.This dual functionality provides the OIDOBt with a distinct advantage in effectively handling contact interactions.
3) Design of Q-Filter and OIDOBt Nominal Model: The Q-filter is designed as a first-order low-pass filter to make the nominal model proper as shown in the following: where the Q-filter cutoff frequency, ω Q , is determined by the standard empirical tuning method taking into consideration the tradeoff between noise attenuation and latency [54].
The OIDOBt nominal model, D n (s), is designed from the inner-velocity closed-loop dynamics, including the payload suppression function, as shown in the following: where is the inner velocity closedloop model, which is different from the conventional DOB where the nominal model is the open-loop plant [53].
The nominal payload dynamics is designed as P n = 1/M pn s where M pn ≤ M a to have stable poles in (19) and M pn is the nominal value of M p .Note that the suppression of payload dynamics effects in (19) also depends on the desired admittance setting.
The nominal velocity controller dynamics is designed with the same structure as the actual ones in (9) as where k pn and k in are nominal values of k p and k i , respectively.The robot nominal dynamics, R n (s), in (19) can be designed to have the same order as the actual dynamics in (11), as is done with the conventional DOB designs [53].However, (11) has many tuning parameters, making it laborious to identify.To resolve this, a nominal model of reduced order and derived from (11) is proposed as follows.
To begin with, R(s) in ( 11) is simplified by dividing through its numerator and denominator by K s (s).Since k s is relatively large, the terms with K −1 s (s) are small and ignored.The result is further simplified to give R n (s) as shown in the following: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where M n = M m + M l and B n = B m + B l .Thus, R n (s) in ( 21) is a single-mass system, and hence, the number of parameters has been significantly reduced to two.4) ODOBt Design for Comparison: For comparison purposes, the task space outer-loop DOB-based admittance control system (ODOBt) [45], which has a two input single output structure of the standard DOB, i.e., only utilizes the command and measured velocity (in blue of Fig. 5), is designed as However, the ODOBt in ( 22) is slightly different from the DOB in [45] since it considers the payload suppression feature by utilizing D n in (19).This ODOBt method is included in the analyses and experiments that follow to compare with the proposed OIDOBt and the basic methods.

5) Theoretical Validation of the OIDOBt Functionality:
The output of the OIDOBt, d, contains force disturbances, d F , payload forces, and high-frequency contact forces.This is theoretically validated as follows.
First, substitute ( 19) in ( 17) and ( 22) to get Next, to eliminate V i (s) in ( 23) and ( 24), V m (s) is derived from Fig. 5 as shown in the following: where T = C R/(1 + C R). Finally, substitute (25) in ( 23) and (24), and assume that d = 1 and T = T n to get Thus, both outputs of ODOBt and OIDOBt in ( 26) and ( 27) contain the disturbances, contact forces, and payload forces.However, the proposed OIDOBt introduces the high-frequency force feature of A(1 − Q) to enhance contact stability beyond the capabilities of the ODOBt.

A. Rendering the Desired Admittance
To evaluate the rendered admittance limitation discussed in Section II-B1, the admittance transfer functions, Y (s), for both methods are derived and given in (38) and (39) in the Appendix.Assuming that ω Q is very large such that Q(s) = 1, (38) and (39) become It is observed from ( 28) that the closed-loop robot dynamics and time delays are canceled, and thus, their effect is eliminated with both methods.Furthermore, to eliminate the effects of payload, assume that P n (s) = P(s) and substitute (19) in (28) to cancel out the payload dynamics as shown in the following: Considering the low-frequency range where |T n (s)| = 1 (0 dB), (29) becomes From ( 30), the rendered admittance equals the desired admittance.Therefore, with both methods, the tracking accuracy is greatly improved since the effect of robot and payload dynamics and time delays, can be minimized.The key advantage of this result is that the rendered admittance can be greatly increased via the desired admittance parameters to make the robot lighter in free-space motion.Note that according to (29), the nominal model T n (s) should be designed with a cutoff frequency higher than the overall control bandwidth.This enables an increased frequency at which the desired admittance can be effectively rendered.

B. Reducing Collision Forces and Improving Contact Stability
Contact stability limitation pointed out in Section II-B2 is evaluated here for the proposed method in the frequency domain by force tracking performance and passivity criteria.

1) Force Tracking During Contact With Stiff Environment:
The effect of the stiffness of the contacting surface is evaluated in terms of force tracking accuracy, contact resonance, and control bandwidth.To achieve this, transfer functions from F r to F are derived for Fig. 5 as shown in the following: where ξ 1 is given in (37).The structural difference of OIDOBt from ODOBt is the high-frequency term L(1 − Q) in the denominator.Thus, the effects of high-frequency force dynamics can be reduced by this term.The magnitude plots of ( 15), (31), and (32) are presented in Fig. 6.Compared to basic and ODOBt methods, the OIDOBt method reduces contact resonance, improves tracking accuracy, and increases the control bandwidth (top figure).This is attributed to the L(1 − Q) term added by the force measurement component of the OIDOBt, which can suppress the high-frequency vibrations caused by the contact dynamics.Contact stability is further checked at a range of K e values where the OIDOBt method exhibits the least resonances (left figure) and high control bandwidth (right figure), followed by the ODOBt and last the basic method.
2) Passivity Analysis: Contact stability is checked by utilizing the passivity approach [13], [16], [58].Here, the admittance transfer function of the system with respect to the external deviation is utilized to investigate the passivity Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.condition of a coupled system in a decoupled way in the frequency domain.A system is said to be passive if the phase characteristic of the admittance transfer function lies in the range −90 • to 90 • [13].For the system in Fig. 5, the energy port F to V is utilized, and option II in Fig. 3 is considered.
To this end, the passivity transfer functions from F to V defined as −V /F(s) = ψ(s) are derived and given in ( 46)- (48).Their phase characteristics plotted in Fig. 7 show that the proposed OIDOBt method improves the passivity since the phase has been increased from 9 Hz for basic method to 10.2 Hz, whereas the ODOBt method deteriorates the passivity.

C. Robustness to Payload Mass Variations
The frequency characteristics of the rendered admittance transfer function Y (s) in ( 14), (38), and (39) are plotted in Fig. 8 with payload mass values of 4-and 7-kg (maximum payload rating for the RACER robot), when M pn = 3 kg.Despite the payload variations, the OIDOBt method shows the best performance with the highest bandwidth and phase margin as compared to ODOBt and basic methods.The increased phase margin by the OIDOBt method implies that it is possible to further increase the admittance by adjusting the desired admittance parameters while still ensuring stability.

D. Disturbance Rejection and Nominalization
Ignoring the force feedback loop in Fig. 5, the output velocity, V , of the DOB loop is derived and given in (49).When Q = 1, the robot velocity becomes where V, V r , and D F are the Laplace transforms of V , V r , and d F , respectively.Thus, it is observed from (33) that the nominal performance is preserved and the disturbances are fully suppressed by both OIDOBt and ODOBt.Furthermore, to evaluate the disturbance rejection performance, Bode magnitude plots of the transfer functions from d F to V given in (50) are presented in Fig. 9 (top).It is observed that both the OIDOBt and ODOBt methods effectively suppress disturbances in both low-and high-frequency ranges.In contrast, the basic method shows poor disturbance rejection, particularly in the low-frequency range.

E. Influence of Sensor Noises on Lumped Disturbance Estimation Performance
The effect of F/T sensor and encoder noises on the disturbance estimation performance can be analyzed by the transfer functions from noise inputs, n F and n V , to the estimated lumped disturbance, d, of Fig. 5.These are derived and given in ( 53)-( 56), and their frequency characteristics are presented in Fig. 9 (bottom).Both methods exhibit comparable results.Note from Fig. 9 (bottom left) that the encoder noises present a positive magnitude in the estimated disturbances and their effect is observed to be large from middle to highfrequency regions.However, in practice, the encoder noises in this region are much less and their effect is insignificant in real applications.On the other hand, from Fig. 9 (bottom right), the magnitude of the F/T sensor noises in the estimated disturbances is very small, and thus, the proposed method is robust against F/T sensor noises in all frequency ranges when estimating the disturbances both in free-space motion and during contact.

V. EXPERIMENTAL VALIDATION
To validate the proposed method, various experiments involving both contact with a human and the environment are Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.conducted on the actual robot in various poses.Furthermore, the assumptions made while deriving the linear models ( 9) and ( 10) in Section II-A2 are also validated.

Setup of the Robot and Apparatus for Experiments
The experiment is set up, as shown in Fig. 10.It shows a 6-DOF COMAU RACER-7-1.4robot with "black-box" PD position control.An ME-Meßsysteme KD6110 F/T sensor with rated limits of 5 kN and 250 Nm is fixed to its flange and the gripper is then rigidly coupled to the F/T sensor.A GSV-8DS amplifier is used, equipped with an internal analog LPF whose cutoff frequency is set to its maximum value of 2500 Hz to reduce force measurement delays.B&R 910 (Linux PC), which is running in Realtime, is responsible for sending task space position commands to the robot controller located in the control unit cabinet, at a frequency of 1250 Hz via ProfiNET.The F/T sensor is connected to the Linux PC over Ethercat.An external PC is utilized for remote access to the Linux PC, to configure and start the external controller.MATLAB running on a laptop is used to compute the transfer function components of command signals in ( 57)-(59), which are subsequently transmitted to the Linux PC over ROS.

B. Task Space Model Measurement and Motion Analysis of Velocity-Controlled Robot
Using Fig. 1, task space transfer functions for the actual controller, actual robot dynamics, and their respective nominal models are experimentally determined by a nonparametric system identification technique.
Under position control, a Schroeder multisine with a frequency range of 0-30 Hz, which is utilized as an excitation signal, is supplied as the desired position, and the position response of the robot is measured by the encoders.To obtain reliable linear models, entire robot operation space coverage and robot model variations are considered by considering six robot poses, as shown in Fig. 11 (top).In every robot pose, five experiments are conducted in each of the x-, y-, and zdirections.The desired and the measured position information  are then differentiated using an LPF with high cutoff frequency to obtain the corresponding velocity information.The frequency response functions (FRFs) from command velocity to the measured velocity are calculated for all the experiments.FRF averages of the x-, y-, and z-axes for all the poses are computed, and their magnitude characteristics are plotted in Fig. 11 (bottom).
1) Joint Flexibility Validation: The effects of joint flexibility considered in Section II-A2 to model the robot as an FJR system in Fig. 2 and (2) and (3) are confirmed by the antiresonance and resonance modes in the measured FRFs in Fig. 11 (bottom).Moreover, these modes are outside the robot velocity control bandwidth.
2) Parameter Determination for Robot and OIDOBt Nominal Models: Using the averaged FRFs, the robot model T (s) and its nominal model T n (s) are then fit, as shown in Fig. 11 (black-dashed-dotted and dotted lines).Here, d T = d CR/(1 + CR), where d = e −st d and C and R are given in ( 9) and (11), respectively, and where C n and R n are given in (20) and (21), respectively.The fit nominal model is capable of attenuating the effects of the antiresonance and resonance modes since they are within its Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.bandwidth.To this end, the estimated numerical models for theoretical analysis are given in ( 42)- (44).Furthermore, the nominal model, T n (s), for (19) utilized in all the experiments is further empirically tuned from the initial one in (44) and given in (45).All of these numerical models are capable of representing the robot in any pose, as they have been fit using the averaged actual FRFs shown in Fig. 11 (bottom).These FRFs were measured based on the robot poses shown in Fig. 11 (top).
3) Off-Axis Motion Analysis for Diagonalization Verification: To validate the diagonalization assumption made in Section II-A2, the Bode magnitudes of the averaged off-axis motions are computed and presented in Fig. 12.It is observed that all these magnitudes are less than those of diagonal axis motions in Fig. 11 (bottom) at all frequencies.Note that, in actual human co-manipulation applications such as gear insertion task in Section V-D5, the robot velocities are much lower than that in Fig. 12, which means that the off-axis speeds would be minimal and their gain would be very small.Thus, the off-diagonal terms are very small and can ignored, the task space linear models in ( 9) and ( 10) can be utilized to represent the motion-controlled robot in theoretical analysis.

C. Stability Against Severe Robot Model Uncertainty
Linear models are used to design the proposed OIDOBt in (17), which is then implemented on the nonlinear robot system in Fig. 1.However, the effective linearized dynamics change depending on the robot's pose, with significant model deviations occurring particularly due to the configuration of the elbow joint.Thus, errors between model and plant behavior may cause instability.In this regard, robustness of the proposed method against severe model uncertainty is validated, considering the six poses shown in Fig. 11 (top).
Consider the robot moving in free space without any payload such that P(s) = 0 and E(s) = 0.This motion is due to the virtual force, F r , whose force tracking transfer function is given in (31) and its characteristic equation is given in (52).Let (s) represent a multiplicative model uncertainty with bounded magnitude on the linear motion control loop T (s) = CR/(1 + CR) such that Robust stability of (31) can be proven by showing that (52) does not approach zero for some bound on the uncertainty | (s)|.Substitute (34) in ( 52) and let d (s) = 1 and Equation ( 31) is stable when (35) is not zero, and thus, a sufficient condition for robust stability is derived from (35) as As one source of model uncertainty is variation in task space motion bandwidth due to changes in the robot's pose, the bandwidth of Q(s) can be set conservatively to meet condition (36).Note that, this stability condition is similar to the result for typical DOBs in motion control [24], [53].
Next, the stability condition in ( 36) is experimentally validated as follows.Considering only the worst uncertainty cases from all the robot poses and axes of motion in Fig. 11 (top) and using the actual measured, T (s), and estimated nominal model, T n (s), FRFs in Fig. 11 (bottom), the inverse model uncertainty is calculated as −1 (s) = T n /(T − T n ) for each pose/axis.Condition (36) is validated from the results in Fig. 13, and thus, the proposed control method is robust against severe robot model variations.

D. Experimental Results
The numerical nominal model, T n (s), utilized in experiments, is given in (45).The allowable range 7.5 ≤ ω Q ≤ 15 Hz is determined empirically using contact control, considering the tradeoff between latency and noise attenuation.The values of M a , B a , K e , and M pn are chosen according to the specific requirements of each experiment.
1) Controller Implementation: The task space position commands for the three methods are derived from Fig. 5 and given in ( 57)-(59).For faster implementation and tuning between test cycles, the component transfer functions in (57)-(59) are discretized with a Tustin transformation in MAT-LAB at a sampling frequency of 1250 Hz on the Laptop PC and sent to the Linux PC (see Fig. 10) using the rosparam command of the ROS package in MATLAB.The Linux PC, which is running in real time, then computes the command signal and sends it to the robot controller.
2) Exp 1: Admittance Rendering in Free Space: Experiments for admittance rendering in free space are conducted to validate the theoretical results presented in Section IV-A.
The gripper of the admittance-controlled robot is fastened to a second robot with a large Velcro loop, as shown in Fig. 14 (top).A motion program is executed on the second robot, leading to repeatable force application on the admittancecontrolled robot.Results are plotted in the middle row of Fig. 14: acceleration-deceleration profile with 6.4-cm/s maximum velocity and a high acceleration and high jerk motion profile with 5.1-and 8.2-cm/s maximum velocities, respectively.Note that the motion in free space with different velocity profiles also considers the dynamic changes in various poses.With these profiles, the tracking accuracy and step response characteristics of the admittance controller can be approximately evaluated as shown by the force response results for all the methods in the middle figure.The reference/desired force (shown as gray-dotted lines) is calculated as the product of the velocity of the robot supplying the external force and the admittance damping parameter, B a .
The OIDOBt method shows the best response and tracking accuracy in all the motion conditions.On the other hand, the basic and ODOBt methods are unable to suppress the effects of nonlinear friction in Fig. 14(a), but ODOBt shows a significant improvement in Fig. 14(b) and (c).Furthermore, the accuracy of force tracking is quantified by calculating the root-meansquare error (RMSE) values of the force responses at steady state.These values are then plotted in the bar chart shown at the bottom of Fig. 14, where OIDOBt exhibits the lowest values.
Thus, the proposed OIDOBt method effectively mitigates the effects of robot dynamics and payload (gripper), confirming the design objective stated in (30).Moreover, this finding indicates that the proposed OIDOBt approach yields minimal deviation from the desired velocity trajectory when compared to the basic and ODOBt methods.
To evaluate the heavy payload compensation performance in low-frequency region, a 5-kg mass is added on the gripper and the experiment is repeated with a 15.95-cm/s velocity, A(s) = 1/(10s + 300), and M pn = 7 kg such that M pn < M a .The results plotted in Fig. 15 show the ODOBt and OIDOBt with similar steady-state payload compensation performance, which is confirmed by their similar RMSE value of 1.74 N.This confirms the low-frequency assumption made on (29) to get (30) and also the low-frequency region of Fig. 8.Moreover, the OIDOBt method demonstrates superior performance compared to the ODOBt and basic methods, even when payload compensation is not factored into the design, due to the additional contribution from the F/T sensor component.High-frequency region performance is discussed in the contact experiments presented next.
3) Exp 2: Contact With Stiff Surface: The contact force control performance is examined in making contact with a pure stiffness environment.This is to validate the theoretical results in Section IV-B.A pure stiff environment, a 3-D-printed plastic beam, is cantilevered into the robot's workspace, as shown in Fig. 16 (top).The robot is starting from a constant position [approximately as shown in Fig. 10-corresponding to poses 1 and 2 in Fig. 11 (top)] for all the methods, commanded with a constant F r to bring the tip of the gripper into contact with the beam.Note that, due to its lightweight (≈2.1 kg), the forces generated by the gripper can be ignored, as they are overshadowed by the dominant contact forces.Two contact points (CPs) with different stiffness, i.e., ≈17.2 N/mm for CP 1 and ≈37.7 N/mm for CP 2, are considered.CP2 generates larger contact forces due to its high stiffness than CP1, and the proposed method is expected to show significant improvement on CP2 due to its A(1− Q) design characteristic.For each CP, contact control experiments are conducted for various ω Q , M a , and B a values, and the results of force responses are plotted in Fig. 16.The force control performance is compared in the sense of peak contact force, settling time, and steady-state force tracking performance, all of which are desired to be reduced.The values of peak contact force and steady-state errors are calculated and given in Table I.
Overall, it is observed that the OIDOBt method demonstrates the highest tracking accuracy with the least steady-state errors (see Table I).The ODOBt also achieves a comparable steady-state error to that of OIDOBt but with an increased settling time.The OIDOBt exhibits the smallest force peaks as admittance is increased in the low-frequency range, which verifies the theoretical results shown in Fig. 6.Reducing the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Q-filter cutoff frequency from 15 to 7.5 Hz increases the settling time for OIDOBt but is able to reduce the tracking error.On the other hand, the ODOBt becomes unstable, and the basic method exhibits uniform vibrations throughout the entire experiment duration.Thus, the proposed OIDOBt method utilizes its characteristic in (31) of damping the high-frequency contact dynamics to reduce first contact forces and maintain contact with a very stiff surface.Note that, the basic and ODOBt methods perform nearly as well as the OIDOBt on CP 1, as they are capable of suppressing the smaller contact forces generated by the compliant surface (CP 1).Therefore, the observed smaller improvements on CP 1 and the substantial improvements on CP 2 align with the expected behavior of the OIDOBt, considering the magnitude of contact forces generated based on surface stiffness.Furthermore, to analyze the effect of high payload and its compensation in the high-frequency range, the contact experiments are repeated on CP 1 with F r = 32 N, while an additional 5-kg mass is attached to the gripper.The results are plotted in Fig. 17 when ω Q = 15 Hz, B a = 300, M pn = 7 kg, and M a = 10 (left), while M a = 8 (right).The OIDOBt exhibits the least peak contact force and faster settling time compared to the ODOBt and basic methods.This confirms the superior high-frequency performance of OIDOBt, as observed theoretically in Figs.7 and 8.
4) Exp 3: High-Speed Force Control: This experiment is conducted to evaluate the capability of the proposed method in suppressing extreme cases of high-frequency contact dynamics.The experimental protocol is shown in Fig. 18 (top).The adjacent robot is set with only a one-way stroke such that it hits the gripper of the experimental robot with an impulse force, F ext , and stops.Two different magnitudes of F ext are applied, and the force responses from the experimental robot are plotted in Fig. 18.
For small collision force (left figure), the basic and ODOBt methods make multiple contacts due to their inability to react to high-speed contact forces, whereas with the OIDOBt, the robot is pushed away by the first contact only, moreover, with the smallest collision force.For large collision force (right figure), all the methods result in single contact due to the large collision force.However, the OIDOBt method has the least peak force (113.23 N) compared to ODOBt (116.42N) and basic (122.27N) methods.Overall, the proposed OIDOBt method is able to reduce the collision forces and shows better damping after collision.complex contact-rich assembly tasks.Here, ω Q = 15 Hz, and the desired admittance is set as low as A(s) = 1/(2.5s+ 250) while maintaining stability during robot co-manipulation due to the contribution of extra dumping from the human arm.The robot is moved by human, starting from a constant position [approximately as shown in Fig. 10-corresponding to poses 1 and 2 in Fig. 11 (top)] for all the methods.a) Insert gear in the shaft: The setup, including the gear, shaft, and robot's guided translational-only motions by the human, is shown in Fig. 19 (top).The human moves the gripper from the resting point (before contact), inserts the gear into the shaft (during contact), and removes the gear (after contact).Results of force responses (left column) and the corresponding robot operation speeds (right column) in the x-, y-, and z-directions are presented in Fig. 19.The proposed OIDOBt method shows one main peak of 61.77N in magnitude and damped behavior after the peak, an indication of more stable contact transitions compared to the basic and ODOBt methods, which exhibit many peaks during contact with the highest at 65.91 and 61.91 N, respectively.Moreover, by observing the forces and the corresponding robot speeds, with the proposed OIDOBt, the robot can be manipulated at higher speeds with less force application compared to basic and ODOBt methods.
b) Insert switch onto the rail: The experimental protocol is given in Fig. 20 (top).Note that, in addition to translational motions of joints 1-3, the rotational motions of the wrist are also activated separately using an admittance controller setting of 1/(s+25).This is due to the complexity of fitting the switch onto the rail, which requires more freedom of motion for the gripper.Force responses (left column) and the corresponding robot speeds (right column) are plotted in Fig. 20.The proposed OIDOBt exhibits smooth and stable contact transitions, and less force is required to manipulate the robot compared to basic and ODOBt methods.The attached video, accessible via this link,1 demonstrates a 3-DOF gear assembly as well as a 6-DOF switch rail assembly applications.

VI. CONCLUSION
In this article, an integrated DOB-based admittance control approach (OIDOBt) is introduced.This approach, which is designed based on a linear framework, overcomes the limitation of "black-box" PD position control of typical industrial robots, as it is constructed in the task space as an outer loop and implemented around the fixed position control loop.It has been shown that OIDOBt utilizes its feature of suppressing high-frequency contact forces to reduce peak contact forces and improve contact stability.Moreover, the accuracy of admittance rendering in free space has been significantly improved, allowing higher admittance to be rendered while maintaining contact stability.In addition, the proposed approach can maintain stability in applications involving complex contact-rich tasks.The proposed method has a simple structure and is easy to implement on an industrial robot, turning it into a collaborative robot.Thus, there is a

Fig. 2 .
Fig. 2. Joint space block diagram of Fig. 1 showing the fixed inner-loop motion controller, C θ, θ,I , physical effect of the end-effector forces, F, on the robot, and robot dynamics modeled as the FJR system.

Fig. 3 .
Fig. 3. Block diagram of the linear modeling and basic admittance control architecture of Fig. 1 for a single DOF in task space.The Laplace operator s on the subsystems is ignored for compactness.

Fig. 5 .
Fig. 5. Proposed admittance control architecture for a single DOF in task space, where d is the estimated lumped disturbance, Q(s) is the Q-filter, and D n (s) is the OIDOBt nominal model.

Fig. 6 .
Fig. 6.Bode magnitudes of G(s) (top when K e = 5e5 N/m), resonant peak values (bottom left), and the frequencies at which they occur (bottom right) when the environment stiffness is varying.

Fig. 9 .
Fig. 9. Magnitude plots of d F to V (top) and n F and n V to d (bottom).

Fig. 16 .
Fig. 16.Contact control experimental results with ≈2.1-kg gripper system.Legends in CP 1(b) and CP 1(c) apply to all the subplots above.

Fig. 19 .Fig. 20 .
Fig. 19.Results for inserting the gear in the shaft task showing the x yz forces in the left column and their corresponding robot operation speeds in the right column.

5 )
Exp 4: Co-Manipulation Applications: Application experiments are conducted by utilizing the high-DOF robot in Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Task Space Outer-Loop Integrated DOB-Based Admittance Control of an Industrial Robot Kangwagye Samuel , Member, IEEE, Kevin Haninger , Member, IEEE, Roberto Oboe , Fellow, IEEE, and Sehoon Oh , Senior Member, IEEE

TABLE I PEAK
CONTACT FORCE (F PEAK ) AND STEADY-STATE ERROR (e SS ) VALUES FOR CONTACT CONTROL RESULTS IN FIG.16