Outer-Loop Admittance and Motion Control Dual Improvement via a Multi-Function Observer

Safe environment contact, and high-performance motion control are typically conflicting design goals. Admittance control can improve safety and stability in contact with a stiff environment but remains challenging on industrial robots. Typically, high-performance motion control is achieved by low-admittance systems, which can give high transient forces or instability in contact with high-stiffness environments. This article proposes a linear admittance control framework from which a multifunction observer (MOB)-based control scheme that succeeds in directly improving the motion control accuracy by suppressing disturbances, while achieving better loop shaping in the outer-loop admittance control is designed. By using the task space force and position measurement of the robot, combined with linearized position-controlled robot and payload models to design the MOB, the outer-loop controller can render improved interactive dynamics. In addition, a methodology to design the proposed MOB based on the reduced-order model is developed. Furthermore, the bounded-magnitude frequency-domain uncertainty in the linear model is identified at a variety of robot poses. Theoretical evaluations and experiments verify the effectiveness of the proposed MOB-based control method, in contact with a very stiff environment and with a 7-kg payload.


Outer-Loop Admittance and Motion Control Dual
Improvement via a Multi-Function Observer Kangwagye Samuel , Member, IEEE, Kevin Haninger , Member, IEEE, Roberto Oboe , Fellow, IEEE, and Sehoon Oh , Senior Member, IEEE Abstract-Safe environment contact, and highperformance motion control are typically conflicting design goals.Admittance control can improve safety and stability in contact with a stiff environment but remains challenging on industrial robots.Typically, high-performance motion control is achieved by low-admittance systems, which can give high transient forces or instability in contact with high-stiffness environments.This article proposes a linear admittance control framework from which a multifunction observer (MOB)-based control scheme that succeeds in directly improving the motion control accuracy by suppressing disturbances, while achieving better loop shaping in the outer-loop admittance control is designed.By using the task space force and position measurement of the robot, combined with linearized position-controlled robot and payload models to design the MOB, the outer-loop controller can render improved interactive dynamics.In addition, a methodology to design the proposed MOB based on the reduced-order model is developed.Furthermore, the bounded-magnitude frequency-domain uncertainty in the linear model is identified at a variety of robot poses.Theoretical evaluations and experiments verify the effectiveness of the proposed MOB-based control method, in contact with a very stiff environment and with a 7-kg payload.

I. INTRODUCTION
T O ALLOW industrial robots to have high performance in both free space motion and contact with semistructured environments, it is desired that a single system can improve motion tracking accuracy while keeping low-force overshoot and stability in contact.Using admittance control in the outer loop, in combination with an inner-loop position control, can provide a safe and intuitive physical interaction between an industrial robot and a stiff environment.This is because admittance control modulates the robot's impedance based on the force feedback from the environment, making the robot respond in a compliant way [1].Conversely, inner-loop motion control is used to control the position and orientation of the robot, ensuring high accuracy and bandwidth such that the robot moves in a smooth and controlled manner [2,Sec.8].This is particularly important in cases when the robot is in close proximity to the contact environment.Therefore, accuracy of the motion control in free space, admittance rendering accuracy, contact stability, and reduced peak contact forces are all key in improving the performance and safe interaction of the robot with intermittent high-stiffness environment.
For typical industrial robots, such as the COMAU RACER robot in Fig. 1, the position controller is fixed and cannot be accessed externally for design enhancement [3].Consequently, the potential for control improvement primarily lies in the outerloop feedback, such as the admittance control loop.However, this approach also has some limitations as follows.
1) Ensuring the stability of the outer-loop admittance control is a challenging task, as the sensitivity characteristics of the inner-loop system are already established by the design of the inner-loop motion control, which includes factors, such as bandwidth and low intrinsic admittance.Moreover, the limitation highlighted previously can also be attributed to model uncertainties that arise due to unaccounted contact dynamics, which may be induced by the external environment during contact.
2) The performance of the admittance control is subject to the manipulator's dynamics, which may be challenging to obtain in practice [4].Moreover, it is difficult to obtain accurate model of the dynamics of the manipulator and reflect it in the admittance control algorithm design.3) Admittance control can lead to limited accuracy of the final position and orientation of the manipulator, especially in challenging environments with high levels of 0278-0046 © 2023 IEEE.Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
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disturbance [5].Note that, it is difficult to reject external disturbances, particularly when the disturbances are unpredictable or highly variable.A disturbance observer (DOB) [6], [7], [8] is commonly employed to robustly suppress the effects of disturbances and improve motion control effectiveness.DOBs have been utilized in motion control systems for both single and multi-DOF configurations [9].They have also demonstrated performance improvement in force control [8] and environmental contact scenarios [10], [11], [12].Moreover, DOBs are typically designed in joint space and implemented to each joint independently to enhance joint position control in both rigid and flexible joint robots (FJRs) [13], [14], [15].However, DOB design for joint position control is not possible, and there is no literature on DOB utilization in task space admittance control for typical industrial manipulators.In addition, advanced admittance control approaches, such as passivity-based techniques [16], neural network approaches [17], sliding-mode-like position controllers [18], and adaptive fuzzy control approaches [19], are available.However, these are designed for manipulators with joint torque control interfaces.
Samuel et al. proposed a first of its kind idea of designing and utilizing the DOB in task space admittance control for a 6-DOF position-controlled industrial robot [20], [21], where the DOB was built around the inner-loop position controller in task space.Aiming at increasing admittance and improving contact stability, the DOB in [20] was designed to cancel the disturbances.Alternatively, to increase the inner-loop admittance, the estimated disturbances were scaled and added to the robot velocity input [21].However, these DOBs did not incorporate force/torque (F/T) sensor measurements in their design.Moreover, the DOB nominal model adopted was of second-order nature, resulting in many parameters to be identified.Furthermore, the robot was modeled as a single mass system, and theoretical analyses and experiments for robot model identification were insufficient.
The motivation of this research stems from the observation that for typical industrial robots, improving the outer admittance control loop has limitations, and the degradation in the innerloop position control accuracy can lead to undesired motions at the end-effector, thereby increasing the peak contact forces and affecting contact stability.In addition, high-frequency contact forces are generated when the robot's end-effector makes contact with stiff surfaces, which result in oscillations.This is due to the heavy structure and dynamic characteristics of the robot and the time delays between the internal and external controllers.If not suppressed, these oscillations can deteriorate contact stability.
To address these issues, a linear control framework is proposed, which is designed and built around the inner-loop position controller in task space.With this framework, there is no need to access the inner-loop position controller directly, but the position control accuracy is improved and the performance of the outer-loop admittance control is indirectly enhanced.As a result, higher admittance can be rendered, making the robot lighter during manipulation in free space, while reducing peak contact forces and maintaining stability during end-effector contact with high-stiffness surfaces.The contributions of this article are summarized as follows.1) A novel linear model and analysis framework for multi-DOF admittance control is developed.Dynamic characteristics of a manipulator with the task space admittance control is modeled and analyzed using linear system framework, which also considers the model uncertainty induced by a fixed inner joint position control at different robot poses.Assuming that the inner position control is properly designed, a multi-DOF task space admittancemotion control problem can be decomposed into multiple simple single-DOF force servo control problem by deriving equivalent linear task space models.2) A multifunction observer (MOB) is proposed to enhance the robustness and contact stability of manipulator admittance control.Utilizing the developed linear framework, a MOB is designed and implemented in task space as an outer loop, constructed around the fixed position control loop.To this end, the motion control accuracy is directly improved by estimating and suppressing disturbances and cancelling payload dynamics.Simultaneously, admittance control is indirectly enhanced by suppressing high-frequency forces arising from contact dynamics and payload resonance, resulting in reduced peak contact forces and improved contact stability.3) As part of MOB design, a reduced-order MOB nominal model and its tuning procedure are developed.This exploits advantages of the developed linear framework where characteristics of the inner position control loop are investigated to derive the reduced-order model.The derived model offers advantages of reduced parameters to be identified, simplicity, and ease of implementation on the robot software platform.In this article, a COMAU RACER industrial robot (see Fig. 1), featuring fixed inner-loop position control and an outer-loop admittance control using end-effector F/T sensor measurements for feedback, is utilized as the basis for developing the proposed method.Theoretical analyses and experiments are conducted to validate its effectiveness.When compared with standard/basic admittance control, this approach demonstrates improved motion control accuracy, increased robot admittance that can be safely rendered, enhanced contact stability, and reduced peak contact forces.
The rest of this article is organized as follows.Section II presents the description and modeling of the target system.Section III includes the design and analysis of the proposed control method.Closed loop performance analysis is given in Section IV while the experiments are provided in Section V. Finally, Section VI concludes this article.

II. TASK SPACE FORCE/MOTION-CONTROLLED ROBOT SYSTEM DESCRIPTION AND MODELING
Fig. 1 illustrates a position-controlled industrial robot making contact with a stiff environment under admittance control in the outer loop.Its typical block diagram is presented in Fig. 2, dividing the system into joint space, composed of the robot and fixed position controller, and the task space, which contains endeffector attachments and outer-loop admittance controller.Here, J (q) is the Jacobian matrix, F /τ is the vector of force/torque induced from the environment and payload, τ s is the vector of joint torques generated by the joint flexibility, τ c is the vector of controlled torque input, θ and q are joint and link velocity vectors.
Fig. 2 shows a conventional framework of task space admittance control with inner loop joint space position control.A fixed inner-loop cascaded position/velocity/current controller, C θ, θ,I , tracks the motion command, X i , in task space, supplied by the admittance controller, A, to cause motion of the robot, with velocity, V , which can be measured in task space.The F/T sensor and the payload/gripper are fixed to the robot's end-effector, which in turn makes contact with environment at the gripper tip.The generated force, F , physically acts on the robot from the contact surface, and is measured through the F/T sensor and utilized for outer loop feedback.
Note that only joints 1/2/3 are considered in Fig. 2 because, 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 peak contact forces, whereas the rotary motion, primarily joints 4/5/6, contribute less to peak contact forces.
In the following sections, the modeling of the conventional framework in Fig. 2 is presented.A novel framework, in which the multi-DOF joint space position control problem is simplified into a single-DOF linear control problem, is developed and then combined with outer-loop admittance control.From there, the factors that limit control performance are analyzed based on the derived linear control framework, and a remedy is proposed.

A. Joint Space to Task Space Modeling of Motion-Controlled Robot
The joint space part of Fig. 2, i.e., the robot with the fixed position controller, is considered here to derive the equivalent task space model of the position-controlled robot.

1) Robot Description and Modeling:
The 6-DOF CO-MAU RACER-7-1.4industrial robot in Fig. 1 is equipped with harmonic drives and thus, the effects of joint flexibility can deteriorate position control accuracy [22], [23].Due to this, the robot can be modeled in joint space as a flexible joint robot (FJR) system [14], [22], as shown in Fig. 2 (red), consisting of linear motor-side dynamics, nonlinear link-side dynamics, and the joint flexibility between them.
The time domain joint space equations that govern the FJR in Fig. 2 (red color) are derived as τ s = M l (q)q + B l (q) q + H(q, q) (1) defined as nonlinear link-side dynamics in (1) and linear motorside dynamics in (2), where M l (q) and B l (q) are the link-side mass and damping matrices, M m and B m are the motor-side mass and damping matrices, q and θ are link and joint position vectors, respectively.The vector H(q, q) combines Coriolis and gravity forces that depend on q and q.Note that, considering the robot in Fig. 1, Coriolis forces are small and ignored while gravity is compensated in the position controller.The residues of these forces, including the nonlinear friction not accounted for in (2) are considered as part of force disturbances, d F , (see Fig. 3), to be estimated and robustly suppressed by the proposed MOB.

2) Fixed Position Controller Modeling:
The joint space motion controller, C θ, θ,I , shown by orange color in Fig. 2 is collectively modeled as a PD position control, as shown in the following: where θ i = J −1 X i is the vector of joint space position command and μ p and μ i are proportional and integral gain matrices.
It is desired to model the entire robot system in Fig. 2 in task space since the joint space position controller and robot dynamics are closed.Moreover the motion command, X i , is Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
supplied in task space and the robot velocity, V , is measured in task space.Since the task space load-side and motor-side velocities are given by V l = J (q) q and V = J (q) θ, then Substituting ( 4), ( 5) in ( 1), (2), and (3), premultiplying both sides of the result by J −T , and finally ignoring the J terms since the rate of change of Jacobian is too small, gives where ( 6), ( 7), ( 8) are, respectively, analogous to (1), ( 2), (3).
Note in this article that, the FJR concept has been extended to task space formulation in ( 6) and ( 7) as compared to the current conventional joint space representations only [14].
To improve the control performance, advanced motion/admittance controllers can be designed utilizing the complex models in ( 6) and ( 7), such as [24].However, the resulting controllers are complex and difficult to deploy on the software platforms of the closed position control structure of the industrial robots, and their performance is affected by significant model uncertainty [25].
On the other hand, the position-controlled robot system in Fig. 1 offers one advantage of availability of command input, X i /V i , and measured output, X/V , all in task space.Thus, closed-loop nonparametric system identification can be performed to obtain an empirical task space linear transfer function model, which can be utilized for controller design.With this motivation, equivalent linear transfer functions for ( 6), (7), and (8) are derived next.

B. Development of Simplified Linear Model of Position-Controlled Manipulator
In this section, an equivalent linear system modeling employing the diagonalization strategy is developed.
Since ( 6) is purely inertial-damper relation, it is linearized about a fixed robot pose and remodeled as a mass-damper system.In this regard, the linearization of (6) and diagonalization of the whole system is conducted, based on the following assumptions.
1) The effects of joint flexibility are small and linear [26].
2) The joint control is decoupled because a high gear ratio drivetrain and lower accelerations are used [27,Sec.8.3]. 3) Since configuration in the task space coordinates decouples the joint dynamics, each task space DOF is considered to be independent, so the task space models in ( 6), (7), and ( 8) can be considered to be diagonal [28].Thus, taking the Laplace transform of ( 7), (8), and linearized (6), the following diagonal matrices of the equivalent linear transfer functions are obtained: for the linear Cartesian DOFs 1/2/3, where s, and j = 1, 2, 3.The subsystems in ( 9), (10), and ( 11) are task space linear transfer function equivalents of link-side dynamics in (6), motor-side dynamics in (7), and position controller in (8), respectively.Since ( 9), (10), and ( 11) are diagonal, a single-DOF is analyzed in this article and the index j is ignored for simplicity, from now onward.Fig. 3 illustrates ( 9), (10), and ( 11) and the outer admittance control loop for a single-DOF.Where the subsystems C, R, Γ d , P , and E are the velocity controller in (11), robot dynamics, time delays between the external and internal robot controllers, payload, and environment, respectively.While signals F r , F c , V m , F p , and F env are task space virtual desired force, robot control input, measured robot velocity, payload forces, and environment force, respectively.d V and d F are velocity input and force disturbances, respectively.
The robot dynamics, R, is constructed as two masses R m and R l with stiffness, K s = k s /s, between them, forming a twomass system, as seen in Fig. 3 (right-hand side figure).Thus, this can be reduced as a single transfer function from F c to V as With the simplified model in (12), the measured task space robot velocity, V , can be utilized to couple the F/T sensor, payload, and the environment to the robot.

1) End-effector Attachments and Outer-Loop Admittance Control:
The payload and the environment in Fig. 3 are modeled as pure inertia and a stiff spring, respectively, as shown in the following: where M p is payload mass and K e is environment stiffness.The desired admittance, A(s), is designed as a mass-damper system of mass, M a , and damping, B a , as shown in the following: To conclude, the linear model in Fig. 3 allow the use of frequency-domain analysis of contact resonance, contact stability, and rendered admittance.Moreover, with the aid of diagonalization strategy, a complex mixed joint-task space model in Fig. 2 has been transformed into a simple single-DOF task space linear force servo control system, as illustrated in Fig. 3 [20], [21].The diagonalization assumption is experimentally verified in Section V-B1.

C. Analysis of Factors That Deteriorate Control Performance Based on Developed Linear Model
The position control performance may be affected by the following.
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1) Force disturbances, d F :
For the case of multi-DOF industrial robots, disturbances include joint nonlinear friction torques, gear cogging, and mechanical vibrations due higher order dynamics [29].In addition, ( 9) is a linearized system, moreover, ( 9), (10), and (11) were derived from ( 6), (7), and (8) with the assumption that the joints dynamics are decoupled.However, there are residual nonlinearities and coupling between DOFs in task space due to imperfections in joint-level control [30], these are also considered here as force disturbances.Another factor is model error, which is caused by the task space model change in different robot poses.This is significant, particularly due to the position of the elbow joint, resulting in higher order dynamics or changes in motion control bandwidth.
To account for this model variation, various poses [see Fig. 9(a)] are considered while measuring the model parameters.However, there may be other residuals of the model error unaccounted for, these too are considered here as disturbances.Also, forces acting externally on the other parts of the robot, such as the joints and links for example due to collision with external objects can cause deviations from the expected motion control performance of the robot.All the abovementioned are collectively called force disturbances, to be estimated and robustly suppressed.
2) Equivalent velocity input disturbances, d V : These are perturbations and discrepancies in the inverse kinematics and joint synchronisation, which result in the joint-level velocities not matching the desired task-space velocity.
3) Contact dynamics and payload induced forces: Interaction with high-stiffness environments is challenging in practice, due to high-frequency contact dynamics and resonances due to heavy payload.This limits the range of admittance, which can be rendered and causes the difference between rendered and idealized dynamics.In addition, payload generates gravity and acceleration forces, F p , in low and mid frequencies, respectively.If not suppressed, the effects of all these forces can lead to undesired robot motions in free space or during contact.
The abovementioned problems can be solved by increasing the motion controller gain [7] or using robust control methods [12].However, these methods require access to the innerloop system, which is fixed and inaccessible for typical manipulators.To mitigate this, a method that does not require access to or change the inner-loop system for its design is proposed next.

A. Proposed Control Architecture
The proposed MOB-based control method that is implemented in task space as an outer loop, built around the fixed velocity control loop is presented in Fig. 4 (magenta color).Here, T n , C n , and P n are nominal models of the closed-loop velocitycontrolled robot, velocity controller, and payload, respectively.The MOB output, d, is obtained by utilizing known task space input and measured output velocities, F/T sensor measurement, and the nominal models of the closed inner-loop system and the velocity controllers to form the component shaping filters, as shown in the following: where D, V • , and F m are the Laplace transforms of d, V • , and F m , respectively.The Q-filter is designed as to make the nominal models proper, where ω Q is the Q-filter cutoff frequency.Thus, the multifunctions of the proposed MOB are explained from Fig. 4 and (15) as follows.n V m is designed to reject the low and mid-frequency payload forces that can cause deviation from the desired position/velocity.To visualize these functions, V i is derived from Fig. 4 as where T = CR/(1 + CR).Substitute ( 16) in (15), and let T = T n and Q = 1 to get showing that the MOB output d consists of the estimated velocity disturbance, force disturbances, and payload forces.These are all suppressed to improve motion control accuracy.

2) Indirect Admittance Control Performance Improvement:
The terms 15) that correspond to gold color in Fig. 4 are designed to suppress highfrequency contact and payload forces during force control.The admittance control improvement is indirect since the admittance controller A(s) is not utilized in design of the F m component in the proposed MOB.
Note that C −1 n has a high-pass filter structure.It converts force into velocity as well as both gain and high-pass frequency shaping of the F/T sensor measurements and forces induced by the payload resonance during force control.Thus, only the unwanted high-frequency force characteristics, such as dynamics induced by contact with a stiff object, are estimated and suppressed to reduce peak contact forces and improve contact stability.

1) Design and Limitation of the Full-Order MOB Nominal
Model: From the MOB structure in Fig. 4, the full-order nominal Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
model can be designed as the closed-loop dynamics of the velocity control loop, as shown in the following: where C n (s) and R n (s) are the nominal models of C(s) and R(s) in ( 11) and ( 12), respectively.In conventional DOB design [6], [7], the nominal model is designed with same order as the actual model.This is easy when the actual model is of 1st or 2nd order.However, designing (18) using C(s) and R(s) in ( 11) and ( 12) results into a complex system with many tuning parameters, which are laborious to identify, moreover, these increase model uncertainty.In addition, implementation of the MOB with (18) to the existing robot platform is also challenging.To mitigate the above mentioned challenges, a reduced-order nominal model is proposed next.
2) Proposed Reduced-Order Nominal Model: To begin with, R n (s) is first designed as follows.R(s) in ( 12) 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 to give where Consider a typical controller tuning as C n (s) = k pn + k in /s, and let k c be the overall gain, substituting all these and ( 19) in (18) gives In typical controller design, it is desired from (20) that k pn s + k in = M n s + B n .Employing this design goal, the proposed reduced-order model, T n (s), is derived from (20) as where τ n = 1/k c is a task space parameter and is designed by the following criteria.Substitute C n (s) = k pn + k in /s and ( 19) in (18), rearrange the result and equate it to (21) to get Lastly, consider the low frequency gain in (22) to get From ( 23), the task space parameter τ n can be calculated as the ratio of total damping to integral coefficients.
To summarize, T n (s) in ( 21) is a first-order LPF, which is a reduced-order version of T * n (s) in (18).Compared to (18) and conventional 1-DOF DOB models of mass-damper structure [7], the proposed model in (21) has a LPF structure with one parameter-the time constant, τ n .Thus, the parameter identification burden has been significantly reduced and the controller with MOB is simple and can be easily implemented on the existing software platforms.The proposed MOB model is less susceptible to model uncertainty, as proven experimentally in Section V-C.Further, τ n in ( 23) can be interpreted in terms of bandwidth as ω n = k in /(2πB n ) Hz for easy tuning during experiments.Note that M pn ≤ τ n k p to have stable zeros in 3) Theoretical Validation of the Proposed Model: Initial τ n is computed using the values of B n and k in as τ n = 0.0187.To this end, the bode magnitude of T n (s) is plotted when τ n = [0.0187,0.0289, 0.0350] s −1 , together with the actual measured FRFs, as shown in Fig. 5.It can be observed that the proposed T n (s) is robust against model uncertainty by attenuating the effects of joint flexibility and nonlinearity since the antiresonance and resonance modes are inside its bandwidth.Moreover, τ n can be further increased to 0.0350 and achieve good admittance rendering performance (see Fig. 7-top).

IV. CLOSED LOOP PERFORMANCE ANALYSIS
The numerical transfer function utilized for analyses in this section are determined experimentally and given in (29).
In addition, C n = C, P = P n = 1/4 s, which is appropriate for the COMAU robot since its rated maximum payload is 7 kg, and A = 1/(8s + 800) as commonly used [28].

A. Motion and Admittance Control Accuracy Improvement Analysis With the Proposed Method
From Fig. 4, perfect motion tracking is achieved when innerloop dynamics, i.e., transfer function from V r to V equals nominal model, and when all the disturbances are cancelled.To check this, consider a virtual force F r which makes the robot move in free space with velocity V .The controlled robot velocity is derived when Q(s) = 1, as shown in the following: From ( 24), the nominal performance is preserved, i.e., V /V r = T n , and disturbances, d V and d F are fully rejected.Thus, the motion control accuracy is improved and the perfect desired admittance tracking is achieved in the low frequency range where |T n (s)| = 0 dB.

B. Force Tracking During Contact, Admittance Rendering, and Contact Stability Analysis
Hereafter, the proposed method in Fig. 4 is abbreviated as MOB w/ F m .Its performance is compared with No MOB (basic method when Q = 0) and MOB w/o F m , i.e., when the F m term is not considered in the MOB design in (15), which has the same structure as classical DOB [6], [7] but different due to the closed-loop nominal model T n (s), and implemented around the position controller in the outer loop.
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The following transfer functions are derived from Fig. 4, ignoring the variable s for compactness: where 25) is the reference force tracking during contact with a stiff surface and ( 26) is the admittance transfer function.
To mathematically analyze the force tracking during contact and admittance rendering, let P = P n = 0 and consider the No MOB (when Q = 0) and the proposed method (MOB w/ F m when Q = 1) cases, ( 25) and ( 26) become Therefore, comparing ( 27) and ( 28), it can be observed with the proposed method in (28) that, the time delays are eliminated while the robot dynamics are nominalized.This result means that the effects of time delays and variations in actual robot dynamics are eliminated.Therefore, nominal transfer functions T n (s) and C n (s) in ( 28) can be tuned to facilitate further increase of the admittance controller gain.Thus, higher admittance can be rendered while reducing the peak contact forces and improving contact stability.On the other hand, with the No MOB method, the actual dynamics and time delays limit admittance rendering, and affect contact stability.Furthermore, to evaluate the influence of K e , τ n , and ω Q , the bode magnitudes of ( 25) are plotted in Fig. 6 while the magnitude and phase characteristics of (26) are plotted in Fig. 7, both for various values of K e , τ n , and ω Q .The same τ n values in Fig. 5 are utilized while K e values are selected to represent very stiff surfaces.From Fig. 6, the MOB w/ F m method is able to suppress the joint flexibility amplified by the high-stiff surfaces, i.e., when K e = 5 × 10 5 N/m.The flexibility can be observed by two resonant peaks in the No MOB case.Moreover, the proposed method exhibits the least resonant forces on all the contact surfaces, improves force tracking accuracy, and increases control bandwidth.Increasing the Q-filter bandwidth from ω Q = 7.5 Hz to ω Q = 15 Hz when K e = 2 × 10 5 N/m and τ n = 0.0289 s −1 (ω n = 5.5 Hz) does not show significant increase in the peak resonant forces.On the other hand, increasing τ n increases the peak contact forces for both MOB w/ and w/o F m , however, the peak forces for MOB w/ F m are less than that of MOB w/o F m .
For the case of admittance control performance (see Fig.   is improved and admittance control bandwidth is increased at higher peak resonant forces (see Fig. 6).This is attributed to the design characteristic of the proposed method where the high-frequency large resonant forces produced are suppressed by the C −1 n F m component to improve the outer-loop admittance control.
To analyze contact stability, the passivity approach is utilized, where a system is passive if ∠Y (s) lies between −90°and 90°, is often used as a stability condition in admittance control [1], [21].As shown in Fig. 7 (bottom), the proposed method improves contact stability by increasing the frequency at which passivity is violated at a higher frequency, which often leads to high resonance in contact [31].Furthermore, setting the Q-filter cutoff frequency less than that of the nominal model, i.e., ω Q = 4 Hz < ω n = 4.5 Hz (τ n = 0.035 s −1 )-gray color in Fig. 7, deteriorates desired admittance tracking in high frequency range and significantly degrades contact stability.Therefore, the design condition for ω Q is that, it should be set much larger than the nominal model cutoff frequency.

A. Setup of the Robot and Apparatus for Experiments
The experiment is setup, as illustrated in Fig. 8.The B&R APC 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 the task space command signal in (32), which are subsequently transmitted to the Linux PC over ROS.

B. Task Space Motion Analysis, Linear Model Validation and Identification for the Position-Controlled Robot
To conduct intuitive nonparametric task space time domain analysis and frequency domain model identification, an excitation signal with a specified band pass spectrum is utilized.In this case, a Schroeder multisine [32,Sec.5] [33] whose frequency increases from 1∼30 Hz from 0∼10 s, is supplied to the robot under position control as the desired position, X i .Six robot poses are considered, as illustrated in Fig. 9(a) for two reasons: to cover all range of robot operation space and to account for uncertainty due to robot model variations, which is crucial for accurate model identification.For each pose, five experiments are conducted in each of the x/y/z-directions.The desired and measured position information are then differentiated using a LPF with high cutoff frequency to obtain the corresponding velocity information.

1) Diagonal and Off-Diagonal Robot Motion Analysis:
The task space time domain measured velocity information is presented in Fig. 9(b)-(d) where both diagonal and off-axis velocities are plotted for the excitation in each axis.It is observed that the amplitudes of the off-axis motions are very small compared to the diagonal motions for the entire excitation period.This result is further quantified by calculating the root-meansquare (rms) values of the time data and presented in Fig. 9(e) where the off-axis rms values are very small to cause a significant effect on diagonal motion.Note for the y-axis excitation that, the off-axis amplitude from ≈ 6.5 s become larger but still much lower than the diagonal axes.In actual robot applications, such as payload co-manipulation and assembly tasks with human guidance [28], the robot excitation speeds are much lower than in this case, which means that the off-axis speeds would be minimal and their gain would be very small.To this the off-diagonal terms are very small and can be ignored.This validates the diagonalization strategy proposed in Section II-B, thus, qualifies the linear models in ( 9), (10), and (11) for MOB design and analysis.Moreover, the residual coupling effects not accounted for at this level, part of which is the off-diagonal motions exhibited in Fig. 9, are robustly suppressed by the proposed MOB.

Motion-Controlled Robot Model:
The frequency response functions (FRFs) from command velocity to the measured velocity are calculated for the averaged x/y/z-axes, and their magnitude characteristics are plotted in Fig. 5 shown by T (s).The effects of joint flexibility considered in Section II-A to model the robot as a FJR system in Fig. 2 and ( 1) and ( 2) are visualized in Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

C. Stability Against Severe Robot Model Uncertainty
As in the DOB [7], errors between model and plant behavior may cause instability.Note that in this article, linear models are used on the nonlinear robot system, however, the effective linearized dynamics change depending on the robot pose.Where large model deviation is especially caused by the pose of elbow joint.To this end, robustness of the proposed method against severe model uncertainty is validated considering the six poses illustrated in Fig. 9(a).
Consider the robot carrying payload and moving in free space (E = 0) due to the virtual force, F r , whose transfer function is given in (25).Since the numerator, i.e., AT P −1 , is stable, and assume Γ d = 1, P = P n , and C = C n , stability of ( 25) can be proven by showing that the denominator does not approach zero for some bound on the uncertainty |Δ(s)|, where Δ(s) is defined as multiplicative model uncertainty with bounded magnitude on the linear motion control loop, T = CR/(1 + CR).Substituting T = (1 + Δ)T n in ξ 2 (s), i.e., the denominator of (25), gives Equation ( 25) is stable when (30) is not zero, thus, a sufficient condition for stability is derived from (30) as where As one source of model uncertainty is variation in task space motion bandwidth due to changes in robot pose, the nominal model time constant τ n and the bandwidth of Q can be set conservatively to meet condition (31).Note that, this stability condition is similar to the result for typical DOBs in motion control when ξ 3 (s) = 0.
Next, the stability condition in ( 31) is experimentally validated as follows.Considering only the worst uncertainty cases from all the robot poses and axes of motion in Fig. 9(a), and using the actual measured, T (s), and estimated, T est (s), FRFs in Fig. 5, the inverse model uncertainty is calculated as Δ −1 = T n /(T − T n ) for each pose and axis.It is validated from the results in Fig. 10 that, the proposed control method is robust against severe robot model variations.

D. Motion and Contact Control Experiments
To implement the proposed control method on the COMAU RACER robot software platform, the task space position command is derived from Fig. 4, as shown in the following: For faster implementation and tuning between test cycles, the transfer functions in (32) are discretized with a Tustin transformation in MATLAB at a sampling frequency of 1250 Hz on the Laptop PC and sent to the Linux PC (see Fig. 8) using the rosparam command of the ROS package in MATLAB.The Linux PC, which is running in realtime, then computes the command signal and sends it to the robot controller.Further, the nominal parameter values of the controller, C n (s) for use in (32) are determined as follows.First, the initial parameters of C n (s) and R n (s) are obtained by fitting the model C n R n /(1 + C n R n ) with the measured FRFs in Fig. 5, where C n (s) = k pn + k in /s and R n (s) = 1/(M n s + B n ), as given in (19).Then, the obtained C n (s) is further fine-tuned experimentally as C n (s) = 10 + 400/s, taking into consideration the objective of suppressing the effects of nonlinearity and uncertainty.The initial value of τ n is determined from (23) and is further fine-tuned empirically to τ n = 0.05 s −1 taking into consideration the tradeoff between fast response and suitable cutoff frequency to suppress the disturbances and nonlinearity.ω Q is empirically tuned as ω Q = 15 Hz taking into consideration the tradeoff between noise reduction and latency.Finally, the parameters of A(s) are selected as per each experiment requirement taking into account the tradeoff between increasing the admittance to make the robot lighter in free space while reducing the peak contact forces and keeping contact with the stiff environment.

1) Direct Motion Control Performance Improvement Validation:
The velocity tracking and disturbance rejection experiments are conducted to validate the direct improvement of motion control accuracy.In motion control experiments, (32) is set as: F m = 0 and A = 1 such that F r = V r .
With the robot in Pose 1, [see Fig. 9(a)], F r = V r is supplied in the x-axis direction at 3 s as step signals of 0.0393 and 0.1185 m/s to test the robot moving at low and high speeds, and the results are plotted in Fig. 11.The proposed method exhibits the best steadystate tracking performance (left-hand side figure) with least rms values (right-hand side figure) as compared to the No MOB (basic) method.This validates the capability of the proposed MOB to suppress the effects of disturbances which deteriorate accuracy of the No MOB method.
Furthermore, an experiment involving disturbances is conducted.The controlled input in (32) is first modified as disturbances d V and d F .Then, with A = 1, F m = 0, and ) is supplied as a fictitious sinusoidal signal of 0.015 m/s magnitude and varying frequencies of [0.5, 3, 6, 10] rad/s, each at 12 s interval.The corresponding robot velocities are recorded and plotted in Fig. 12.Here, (D V + C −1 n D F ) is interpreted as joint-task space velocity synchronization discrepancies for d V and undesired external forces acting on other parts of the robot other than the F/T sensor for d F .Note that 0.015 m/s is utilized for experimental purposes, however, the amplitudes of disturbances is much smaller in real robot operation.It can be observed from the results that the proposed method can reject the disturbances by ≈ 80%, i.e., to an amplitude of 0.003 m/s, which validates the theory in (17).On the other hand, the No MOB (basic method) is unable to reject the disturbances.

2) Indirect Outer-Loop Admittance Control Improvement
Validation: The indirect improvement in contact force control performance with the proposed method is examined in making contact with a pure stiffness environment.The robot and the contact object (plastic beam) are setup, as illustrated in Fig. 1.The robot is, starting from a constant position for all the methods, commanded with a constant F r to bring the tip of the gripper into contact with surface of the plastic beam.Six experiments in varying conditions of stiffness of the contact surface (CP1 and CP2), reference force, desired admittance, and payload mass, are conducted and results are presented in Fig. 13.Parameters for experiments (a)-(f) are designed, as shown in each of the figure's caption.Note that, in results (a)-(e), due to its light weight (≈ 2.1 kg), the forces generated by the gripper are ignored since they are dominated by contact forces, thus, M pn = 0 kg such that P −1 n = 0 in (32).The first experiment is conducted to verify the reduction in peak contact forces and contact stability improvement.Results are presented in Fig. 13  sensor measurements are not utilized in the MOB design (MOB w/o F m ).This confirms the capability of the proposed method to damp the high-frequency contact dynamics through the F/T sensor measurements.
Contact is then done at a higher stiffness (CP2), with results plotted in (c) and (d), where it can be seen that the oscillation increases relative to the lower stiffness experiments.Both the No MOB and MOB w/o F m have strong oscillation, but the MOB maintains a good settling time and is much better damped.The peak contact forces increase at the lower admittance gains in (d) due to a higher speed contact.For larger driving force, results are plotted in (e), where the same trends remain.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Finally, to investigate the capability of the proposed method to suppress the effects of heavy payload during contact, an additional 5 kg mass is clamped on the gripper to make ≈ 7 kg, i.e., the maximum payload of the COMAU RACER robot.Thus, M pn = 7 kg such that P −1 n = 7 s in (32), and the results are plotted in Fig. 13(f).The proposed MOB exhibits the perfect force tracking in steady state.This result confirms the capability of the proposed method to cancel the effects of heavy payload, such as oscillations during contact.
To summarize, the proposed method exhibits superior performance over other controllers, as indicated by the rms values of force tracking errors at steady-state in Fig. 13(g).This is because, the MOB w/ F m nominalizes the actual robot dynamics, eliminates the time delays, and suppresses the effects of high-frequency contact and payload forces, which is not possible for the No MOB method.This confirms the theory in Section IV-B.

VI. CONCLUSION
This article presented an approach to designing a robust task space admittance control system where a mixed joint-task space complex admittance-motion control problem of a multi-DOF industrial robot was simplified into single-DOF force servo linear control problem.A MOB was then developed and implemented as an outer loop, built around the fixed motion control loop.The proposed approach improved both motion and admittance control performances by reducing contact forces, improving contact stability, and suppressing the effects of heavy payload and disturbances.Moreover, the proposed controller was simple, easy to implement, and required less computation power due to the utilization of the linear models.Future work will explore the utilization of IMU sensors, specifically accelerometers, for estimating the end-effector force instead of an F/T sensor.Compared to F/T sensors, IMUs offer advantages, such as lower cost, lighter weight, ease of implementation, and wider bandwidth.In addition, the current literature lacks sufficient exploration of accelerometer usage in manipulator control, presenting an opportunity for further investigation and research in this field.

Fig. 1 .
Fig. 1.Position-controlled industrial robot: the gripper tip is making contact with a stiff surface under admittance control in the outer loop.

Fig. 2 .
Fig. 2. Representative block diagram of Fig. 1 showing the four subsystems in both task space and joint space for joints 1/2/3.

Fig. 3 .
Fig. 3. Block diagram of the derived linear modeling and basic admittance control architecture for a single DOF in task space.

1 )
Direct Improvement of Motion Control Accuracy: The term T −1 n V m − V i is classical DOB structure for estimating and rejecting disturbances d V and d F during motion control.The component AP −1
7top), when ω Q = 15 Hz and τ n = 0.00350 s −1 (ω n = 4.5 Hz), the proposed MOB w/ F m improves admittance tracking accuracy and increases admittance rendering bandwidth, as compared to MOB w/o F m and No MOB methods.It can be observed with the proposed method that admittance rendering accuracy

Fig. 9 .
Fig. 9. Task space time domain robot system measurement results.(a) Robot poses for system measurement.(b) Averaged velocity measurements from x-axis excitation.(c) Averaged velocity measurements from y-axis excitation.(d) Averaged velocity measurements from z-axis excitation.(e) rms values of x/y/z-axes excitation.
(a)  and (b).The proposed MOB (MOB w/ F m ) is observed to maintain the contact at steady state and has no overshoot as compared to No DOB and when the F/T