Active Physical Interaction Control for Aerial Manipulator Based on External Wrench Estimation

This article investigates the active physical interaction problem of an aerial manipulator in terms of capability, reliability, and costs. Then, an active physical interaction control architecture is presented for the aerial manipulator to achieve both stable motion and interaction behaviors with external wrench estimation. Specifically, an external wrench estimator in absence of the acceleration and wrench measurements is designed to regulate the interaction point of the aerial manipulator with a minimal sensor condition. Next, a force tracking impedance control strategy with variable stiffness is presented to guarantee the contact stability and force tracking of the aerial manipulator with uncertain contact targets. Further, utilizing the knowledge of prescribed performance and terminal sliding mode surface, a pose controller is proposed to implement the dynamic response speed and accuracy control of the aerial manipulator, which provides a prerequisite for the realization of reliable physical interaction tasks. The stability of the proposed control architecture is analyzed through Lyapunov tools. Moreover, the feasibility and performance of the proposed control architecture are validated via simulations and real-world contact experiments.

behaviors with external wrench estimation.Specifically, an external wrench estimator in absence of the acceleration and wrench measurements is designed to regulate the interaction point of the aerial manipulator with a minimal sensor condition.Next, a force tracking impedance control strategy with variable stiffness is presented to guarantee the contact stability and force tracking of the aerial manipulator with uncertain contact targets.Further, utilizing the knowledge of prescribed performance and terminal sliding mode surface, a pose controller is proposed to implement the dynamic response speed and accuracy control of the aerial manipulator, which provides a prerequisite for the realization of reliable physical interaction tasks.The stability of the proposed control architecture is analyzed through Lyapunov tools.Moreover, the feasibility and performance of the proposed control architecture are validated via simulations and real-world contact experiments.
Index Terms-Aerial manipulator, aerial physical interaction, external wrench estimation, prescribed performance.

I. INTRODUCTION
I n the past, aerial robots were considered to lack the capability of active operation, and then mainly adopted for passive monitoring tasks.With the advancement of aerial robot technology, the research field of aerial robots has gradually expanded to implement active physical interaction with the environment.This novel aerial robot system is generally stated as an aerial manipulator, which has numerous potential applications, such as load transportation [1], object grasping [2], infrastructure inspection [3], physical interaction task for pressing an emergency switch in industrial accidents [4], and other maintenance work in hard-to-reach places [5].
Maintaining stable physical interaction with the environment is a very important application and research area on aerial manipulators.The research challenges include system stability throughout the interaction and the regulation of the desired forces in contact.To solve these issues, hybrid force/position and impedance controllers have recently been investigated for aerial physical interaction to be able to control the position and interaction force, and simultaneously preserve the stability of the entire system [6].The core idea of hybrid force/position control is to realize position and force control in unconstrained and constrained spaces, respectively [7], [8].One limitation of this control method is the need to select a priori matrix that determines the degrees of freedom for the position and force control.Therefore, the hybrid force/position controllers generally work for smooth and known environments.
Alternatively, the impedance control strategy allows aerial manipulators to move in both constrained and unconstrained directions within a unified framework [9], [10], [11], [12], [13].The core idea of impedance control is to regard physical interaction with the environment acting as an impedance model, and then a dynamic relationship between the position of the end-effector of the aerial manipulator and the contact force can be established based on the impedance model.The principal restraint of traditional impedance controllers is the fixed impedance model parameter, yet varied tasks need diverse impedance behaviors to achieve the required task performance level.The approaches proposed in the literature to address this restraint mainly include changing the position command [14] or the stiffness of the impedance control [9], [15], [16], [17], [18], [19], [20].In addition, in order to perform active physical interaction tasks, a favorable pose controller is crucial for the aerial manipulator.Namely, the pose controller is expected to design for realizing the dynamic response speed and accuracy control of the aerial manipulator.Therefore, the design idea of prescribed performance [21], [22], [23], [24], [25] and terminal sliding mode surface [25], [26] can be utilized for the aerial manipulator to guarantee the predefined transient/steady-state performance and converge in finite time.
Another critical problem for aerial interaction is the measurements of the external wrenches acting on the body of the aerial manipulator.To solve this problem, force/torque sensors can be utilized to provide reliable measurements.In [11], a force/torque sensor was installed for an aerial manipulator to provide feedback to the admittance controller.However, using force/torque sensors could increase the cost and weight of the aerial manipulator and reduce its total flight time.In [27] and [28], observer-based force-sensorless impedance control approaches were investigated for robot manipulators to implement stable interaction and force tracking.Therefore, another alternative and feasible solution is to use a wrench estimator, which can provide enough accurate wrench estimation if the estimator is correctly designed and the required measurements are taken.Furthermore, utilizing the wrench estimation can increase the flexibility of the controller and adjust the interaction point of the aerial manipulator.In [29] and [30], a Lyapunov-based disturbance observer was presented to estimate the external wrenches, where the linear acceleration was utilized to calculate the interaction forces.In [31], both momentum-based and acceleration-based methods were investigated for a flying robot to estimate the external wrenches acting on the system.The designed wrench estimators in [29], [30], and [31] relied on the necessary acceleration measurement information, however in practical applications, the measurement of the acceleration may be unreliable.Therefore, it is expected that the wrench estimator is designed without relying on unreliable acceleration measurements.In [3] and [32], a generalized momentum approach-based external wrench estimator was employed for aerial manipulators without the acceleration measurements, where only the linear and angular velocity measurements are needed.
According to the above discussions, this article focuses on the physical contact inspection task and aims at designing an active physical interaction control architecture for the aerial manipulator to achieve both stable motion and interaction behaviors with external wrench estimation.The dynamics of the aerial manipulator is modeled and analyzed.Then, the active physical interaction control architecture, including external wrench estimator, force tracking impedance control (FTIC) with variable stiffness, and pose controller, are proposed for physical contact inspection in terms of capability, reliability, and costs.To confirm the feasibility and performance of the proposed control architecture, simulations and real-world interaction experiments of the aerial manipulator are performed.The main contributions of this article are listed as follows: 1) The active physical interaction control architecture, including external wrench estimator, FTIC with variable stiffness, and pose controller, is presented for the aerial manipulator to implement both stable aerial motion and interaction behaviors with external wrench estimation.2) Compared with the solution in [29], [30], and [31], the external wrench estimator is designed without acceleration and wrench measurements, and this is relatively suitable for practical applications.3) Considering the uncertain contact targets, the FTIC strategy with variable stiffness based on the wrench estimator is presented to guarantee the contact stability and force tracking of the aerial manipulator.4) Utilizing the knowledge of prescribed performance and terminal sliding mode surface, the pose controller is proposed for the aerial manipulator to implement the predefined transient/steady-state performance and converge in finite time.The rest of this article is organized as follows.Section II introduces the dynamics model of the aerial manipulator.Section III presents an active physical interaction control architecture, including external wrench estimator, FTIC with variable stiffness, and pose controller.Simulations and real-world experiments reveal the feasibility of the proposed control architecture in Sections IV and V, respectively.Finally, Section VI concludes this article.

II. DYNAMICS MODELING
To achieve active physical interaction with the environment, an aerial manipulator consisting of a quadrotor aerial robot with a rigidly attached tool is considered in this section, and the schematic view of the coordinate frames is shown in Fig. 1.To simplify the aerial manipulator system model, its body is assumed to be rigid.Let I : {O I − x I y I z I } denote the inertial world frame and z I represents the vertical direction.B : {O B − x B y B z B } is the body frame of the quadrotor platform with origin O B in the center-of-mass of the quadrotor platform with the tool.E : {O E − x E y E z E } represents the tool frame and the interaction point coincides in origin O E .For a general vector/matrix ν, the notion ν with = {B, E} represents the vector/matrix ν expressed in frame ; ν and ν denote the estimated value and error of ν, respectively; ν d is the desired value of ν, and ν x denotes the variable ν in the x-direction.The position p and linear velocity v expressed in I are denoted as p = [x, y, z] T and v = [v x , v y , v z ] T , respectively.Then, the quadrotor attitude is defined as Φ B = [φ, θ, ψ] T .Further, the rotation matrix R B ∈ SO(3) from B to I is expressed as where C (•) and S (•) denote the trigonometric function cos(•) and sin(•), respectively.
Let ω and ω B denote the angular velocity expressed in I and B, respectively.Then, the Euler angle rate ΦB can be converted into ω and ω B as ) where T (Φ B ) ∈ R 3×3 is the transformation matrix between the time derivative of Φ B and correspondent ω.Φ B ∈ (−π/2, π/2) is presumed to guarantee the existence of the inverse of T (Φ B ).
Considering the aerial manipulator achieves physical interaction with the environment, the dynamics of the system can be modeled by using the Newton-Euler method as [29] where m u and J = diag(I φ , I θ , I ψ ) denote the total mass and the inertia matrix, respectively.g denotes the gravitational constant, and e 3 = [0, 0, 1] T .The symbol × denotes the crossproduct.T = 4 i=1 T i ∈ R is the total thrust of the quadrotor, where T i , i = 1, . . ., 4 is the thrust of each rotor.
B ∈ R 3 are the external force and torque on the centerof-mass of the aerial manipulator due to the force and torque exerted by the external environment on the tool.
Folding (3) and its time-derivative into (4) yields the following dynamics model as where is the Coriolis and centrifugal matrix, where S(•) denotes the skew-symmetric matrix operator and T , the dynamics model can be expressed in matrix form as where is the positive definite inertia matrix, C(q, q) ∈ R 6×6 is the Coriolis and centrifugal term, and G(q) ∈ R 6 is the gravity force vector. and T is the external wrench acting on the system center-of-mass given by [11] T represents the interaction wrench between the environment and the end of the tool, and p B E is the position of the end of the tool in B.
Due to the underactuation characteristic of the quadrotor, the generalized inputs u(1) and u(2) can be used to compute the desired roll and pitch angles as Moreover, the rotation of the rotors produces the thrust and torque of the system.Therefore, the connection between the rotor speed and the thrust/torque can be shown as where Ω 2 i , i = 1, . . ., 4 is the square of the ith rotor speed.c T and c M denote the thrust and torque coefficients, and d represents the wheelbase length of the aerial manipulator.

III. CONTROL ARCHITECTURE DESIGN
In this section, the control architecture is designed for the aerial manipulator to achieve active physical interaction with the surrounding environment, which is composed of the external wrench estimator, FTIC with variable stiffness, and pose controller, as shown in Fig. 2.

A. External Wrench Estimator
To correctly handle the physical interaction of the aerial manipulator with the surrounding environment, the information of the interaction wrench W E between the end of the tool and the environment is required.For this purpose, a force/torque sensor can be utilized to provide a reliable measurement.However, using the sensor could increase the cost and weight of the aerial manipulator.Alternatively, the flexibility of the controller Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.can be increased by utilizing wrench estimation, and then the interaction point of the aerial manipulator can be adjusted.
According to (7), the external wrench on the aerial manipulator W B can be regarded as the influence on the system center-of-mass of the wrench W E imposed by the environment on the end of the tool.Therefore, for convenience, the wrench W B is estimated by an external contact wrench estimator and can be converted from ( 7) to be utilized for establishing the dynamic relationship between the position of the end of the tool and the contact force.The wrench estimator enlightened by the nonlinear disturbance-observer for robotic arms is designed as (10) where T is the estimated wrench of W B , and L will be defined later to guarantee the convergence of the estimator.
By defining the estimator error as W B = W B − Ŵ B and calculating the derivative of W B from (10), and then in the presence of a constant or slowly varying external wrench, the error dynamics is expressed as [29], [30] where 0 6×1 ∈ R 6 is the null vector.It can be concluded that the asymptotic stability of the error dynamics is influenced by the choice of L, generally, the positive definite matrix L can be applied.
Notice that the estimator of (10) needs the information of q, q, and q.In practical applications, the measurement of the acceleration q may be unreliable.Thus, an auxiliary variable is defined by where K( q) is defined later on.From ( 10) and ( 12), it can be given as By choosing where a is a positive estimator gain.Then, from ( 12), (13), and ( 14), the external wrench estimator can be expressed as Theorem 1: For the dynamics ( 6) of the aerial manipulator, the estimation error W B under the external wrench estimator in (15) can converge to a bounded area with the equilibrium state, namely, 15), the derivative of V 1 (t) can be calculated as From (6), it is noted that H(q) is the positive definite and symmetric inertia matrix.Also, H(q) −1 is positive definite and symmetric.Then, for ∀ ∈ R 6 , there exist positive constants h Then, notice that the inequality where which implies the boundedness of the estimator error W B .By regulating k V and c V , the upper bound c V /k V can be set small arbitrarily, and then W B can be set arbitrarily small.
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Furthermore, consider in the presence of a constant or slowly varying external wrench, the derivative of V 1 (t) in ( 16) can be expressed as V1 ≤ −a h W B 2 , which indicates that asymptotic stability of the proposed external wrench estimator in (15) by using Barbalat's lemma [33].In this case, W B asymptotically becomes zero.
Remark 1: In practical applications, the modeling errors would cause a bounded constant error in the external wrench estimator Ŵ B .To remove this error, the steady-state error is estimated during a hovering flight, and then the offset is considered both in the external wrench estimator and in the latter pose controller.
Remark 2: The estimated wrench Ŵ B acting on the aerial manipulator is filtered with a low-pass filter before it is further utilized.Once the estimated wrench Ŵ B is known, the estimated wrench Ŵ E acting on the end of the tool can be computed as

B. FTIC With Variable Stiffness
In this section, a FTIC method with variable stiffness based on the wrench estimator is designed for the translational dynamics of the aerial manipulator to provide a stable physical interaction.The main purpose of the impedance control concept is to regulate a desired dynamic relationship between the contact force and the position of the aerial manipulator.In general, the interaction with a vertical target is considered, and the superscript "x" indicates the variable in the x-direction.
Further, for a human in contact with a target, the contact force is controlled by regulating his arm stiffness, and it indicates that the stiffness is adjusted to the difference between the desired and actual contact force.Then, to imitate the human contact behavior, the general one-dimensional impedance control strategy with variable stiffness is presented as the following: where m d , b d , and k d (t) ∈ R + represent mass, damping, and stiffness of the impedance model, respectively.The modification x dc is defined by x dc = x c − x d , where x d is the desired position trajectory computed via offline trajectory generator and x c is the modified commanded trajectory.e x f is the x-direction force error defined by e , where f x B is the contact force in the x-direction, f E,d is the desired force of the end of the tool, and , and k 0 are the positive control gains.
Note that x d is a constant in contact, substituting (20) into (19), and the following equation is obtained: Considering the environment model with a linear stiffness k e , the interaction force is represented as where x e is the x-direction position when the tool just touches the contact target and no force is imposed.Notice that f x B,d is a constant, the force tracking error and its time-derivative are expressed as In order to analyze the stability of the proposed FTIC in ( 19) with (20), the following Lyapunov function is considered as Consider x → x c if the system has accurate position tracking capability.Then, from ( 21) and ( 22), the derivative of V f can be calculated by Further, Barbalat's Lemma in [33] is utilized to show the stability.Due to Vf ≤ 0 always holds, thus V f is upper bounded.From the definition of V f in (24), ẋ and e f are bounded.Then, computing the derivative of Vf yields which is upper bounded.According to Barbalat's Lemma [33], Vf → 0 as t → ∞, thus ẋ → 0 and e x f → 0 as t → ∞.Therefore, the asymptotic stability of the proposed FTIC is proved.
Remark 3: The desired trajectory x d is defined in this article such that the tool just touches the contact target and no force is imposed.Then, to apply the contact force, the modified commanded trajectory x c is calculated by the target impedance model such that x c > x d .It means that x dc in (20) will not go to zero.
Remark 4: Note that the estimator error under the external wrench estimator can converge to a bounded region and ŴB → W B .Therefore, the estimator force f x B can be utilized to replace the actual contact force f x B in the FTIC in (19).

C. Pose Controller With Specified Performance
Define p d and p dc are desired position and modification of position, since the modified commanded position p c = p d + p dc has been computed via the FTIC in (19), a pose controller needs to be designed to guarantee the stability and accurate control of the aerial manipulator both in free flight and physical interaction.Thus, the idea and knowledge of prescribed performance and terminal sliding mode can be utilized for the aerial manipulator to implement the dynamic response speed and accuracy control.The diagram of the pose controller is shown in Fig. 3.
Define the position error as e p = p − p c and the attitude error as e Φ B = Φ B − Φ B,c , where Φ B,c is the commanded attitude that can be calculated by (8).Then, the state error can be expressed as e = q − q c , where e = [e T p , e T Φ B ] T and q c = [p T c , Φ T B,c ] T .If each state error converges to the predetermined residual error, then predefined performance can be Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.achieved.The predefined performance is described by where the subscript k = {1, . . ., 6}, δk and δ k are the positive constants, and ρ k (t) is the performance function expressed by where the positive parameters ρ ∞,k and l k are the ultimate bound of the error and convergence rate, respectively.Then, an error transformation is introduced to convert the error e k into an equivalent unconstrained error ε k .The error transformation function T (ε k ) is defined as where T (ε k ) can be selected as which T (ε k ) is strictly increasing and its inverse always exists, and satisfies as −δ k < T (ε k ) < δk .According to the calculations, the inverse of T (ε k ) is given as where ε k is the kth element of the transformed error ε expressed as ε = [ε 1 , . . ., ε 6 ] T .From (31), the dynamics of transformed error ε in the vector form can be written as where the diagonal matrix R = diag(r 1 , . . ., r 6 ) with , and the diagonal matrix Λ = diag(α 1 , . . ., α 6 ) with α k = − ρk /ρ k .
From the external wrench estimator in (15) and Assumption 1, the pose control law of the aerial manipulator is designed as with the adaptive update law as where K 0 is the positive diagonal matrix and β is the estimated value of β. λ and b are the positive constants, and −ς } is a diagonal matrix.Theorem 2: For the dynamics in (6) under the motion control law ( 36)- (38) with adaption law (39) and the external wrench estimator (15), the terminal sliding mode surface s will converge into a bound area in finite time.Then, the state error e will be also stable in finite time and the predefined performance (27) can be guaranteed.
Proof: The proof procedures of Theorem 2 are presented as follows.
Step 1: The terminal sliding mode surface s will be proved to converge into the bounded area with s = 0 in finite time.
Define β = β − β and consider the candidate Lyapunov function Utilizing the control law ( 36)-( 38), the expression of ( 33) can be given as Substituting (41) into (40), it can be obtained as (42) From the adaptive update law (39) and Assumption 1, the derivative of V 2 can further expressed by Notice that the inequality β β = β(β − β) ≤ (β 2 − β2 )/2 always holds.Further, (43) can be derived as Denote , then the derivative of V 2 can be obtained as According to [35], it is proved that the variable s converges into the bounded area satisfying with 0 < θ 0 < 1 in finite time t 1 .
Step 2: The finite-time stabilities of ε, ε, and e in the sliding phase will be proved.
Consider the candidate Lyapunov function V 3 (t) = ε T ε/2.Then, from (34), the derivative of V 3 is given by If ε k ≥ s k + χ 2 with χ 2 > 0, then the derivative of V 3 is obtained as where χ 3 is a positive constant.Then, it can be concluded as According to [36], it can be obtained that the transformed error ε converges into a bounded set ε i ≤ s + χ 2 in finite-time 3 (t 1 ).Then, it can be concluded that the signals s, ε, ε, and β all remain bounded.Further, notice that the function T (ε k ) in (30) remains bounded since the boundedness of ε k .Thus, the predefined performance ( 27) is ensured.
More specifically, the existence of the positive scalar Tk satisfies T (ε k ) ≤ Tk after finite-time t 2 due to ε k ≤ s + χ 2 .Then, the state error e satisfies e k ≤ Tk ρ k (t 2 ).Therefore, the boundedness of e and finite-time convergence can be simultaneously achieved.
Remark 5: It is known that the chattering issue of the sliding mode control generally exists.To alleviate this phenomenon, the external wrench estimator could adjust the modulation gains related to sliding mode control.However, the gain values are modulated to guarantee that the sliding of the sliding mode can be maintained.Moreover, the function sgn(•) may lead to the chattering of the estimation signal, and thus a continuous function m 0 /( m 0 + w 0 ) for a vector m 0 ∈ R n is introduced to take place of sgn(•) in ( 34) and (38), where w 0 is a positive constant [37].

IV. SIMULATION STUDY
In this section, the feasibility of the proposed control architecture is validated through the numerical simulations, including stable flight under random disturbances and force tracking with variable environmental stiffness.

B. Stable Flight Under Random Disturbances
In the case of disturbance rejection simulation, the comparative simulations are performed with the nonlinear disturbance observer-based sliding mode control (NDO-SMC) in [39], where the same disturbances are simulated.The parameters of NDO-SMC are set as: The results of the two controllers with disturbance rejection verification are illustrated in Fig. 4. Specifically, the position and attitude state errors under external disturbances are depicted in Fig. 4(a) and (b), respectively, where the state error remains bounded with the predefined performance, and the steady-state error is less than ρ ∞,k .Moreover, Table I shows the mean and variance of absolute error of comparison results.From Fig. 4 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE I MEAN AND VARIANCE OF ABSOLUTE ERROR OF THE SIMULATION COMPARISON RESULTS
and Table I, the method performs better convergence speed and tracking accuracy under external disturbances, and guarantees the state error with predefined performance.

C. Force Tracking With Variable Environmental Stiffness
In another case of simulation of the active physical interaction task, the desired position p d is set as The desired yaw angle also remains zero, and the desired interaction forces are set as  speed and tracking accuracy, where the modified commanded position x c is computed by the FTIC strategy.Then, as displayed in Fig. 5(c) and (d), the state error remains bounded with the predefined performance, and the steady-state error is less than ρ ∞,k , which means that the appropriate adjustment of ρ ∞,k can make the tool reach the desired location for achieving the active physical interaction task.Further, the force regulation in the x-direction is depicted in Fig. 5(e), and it can be obtained that the external wrench estimator has a fine estimation effect for the drastic change of desired contact force, so as to complete the interactive task with the environment.The interaction force has a sharp effect at t = 25 s and 65 s since the end of the tool is still in the position of the previous moment when the environment stiffness k e changes abruptly, but it has settled to the desired contact force in a very short time.This simulation scenario demonstrates that the proposed control architecture can guarantee the contact stability of the aerial manipulator, and the desired contact force tracking of active interaction can be implemented with the variable environmental stiffness.

A. Experimental Setup
The aerial manipulator is developed for the physical contact inspection task, as shown in Fig. 6.The quadrotor aerial platform is chosen and developed in the Ardupilot Omnibus F4.To execute the high-level physical interaction control scheme, an onboard computer Intel Stick STK1AW32SC of 1.44 GHz with 2 GB of RAM is used to handle the control signal.The onboard computer communicates with the Ardupilot via Mavlink.To further obtain the attitude and direction information of the quadrotor, an inertial measurement unit (IMU) MPU6000 is equipped with the Extended Kalman Filter (EKF) processing, and an optical flow unit is utilized to provide a highly dynamic response for the system.A tool made of carbon fiber tube is rigidly mounted to the quadrotor platform body, and the tool length from the quadrotor body origin to the end of the tool is about 0.570 m.The aerial manipulator weighs approximately 0.935 kg.The estimator gain in the experiments is selected as a = 1.25 that is a compromise between convergence of the estimation and its performance.The parameter l k is set as l k = 0.3.
To obtain the position feedback in the GPS-denied environment, an onboard monocular camera with a resolution of 640×480 is installed on the aerial manipulator to compute the relative position between the aerial manipulator and the April-Tag through the open-source AprilTag detection system [38].Moreover, a single-axis force sensor is mounted on the end of the tool to measure the force exerted on a target object.The force output is low-pass filtered, and the measured force is not used as control feedback.

B. Controller Comparisons
To verify the feasibility and performance of the proposed method, experimental comparisons are conducted with the NDO-SMC in [39].The initial position and target position of the aerial manipulator are set as p(0) = [−0.10,−0.15, 2.0] T and p d = [0, 0, 1.5] T .The experimental comparison results of the two controllers are illustrated in Fig. 7. Fig. 7(a) and (b) show the position and attitude errors with prescribed performance, respectively.It can be seen that the error of the proposed method always remains the predefined bound, and the proposed method performs better convergence speed.Then, the mean and variance of absolute error of comparison results are displayed in Table II, and it can be obtained that the proposed method has better tracking accuracy.In this comparative experiment, the proposed method exhibits better tracking accuracy and convergence speed, and ensures the predefined performance.
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C. Point Contact With Force Tracking Evaluation
For this experimental scenario, two different stiffness targets placed vertically are adopted as the physical interaction environments.To validate the feasibility of the proposed physical interaction control architecture with the different stiffness environments, point contact experiments are conducted when the aerial manipulator touches a carton and a whiteboard, respectively.Fig. 8 shows the sequence of images corresponding to the experiments.The aerial manipulator first tracks the commanded trajectory and approaches the target.After approaching, the aerial manipulator touches the target to maintain a desired contact force.
Fig. 9 displays the results of the proposed control architecture when the aerial manipulator touches the whiteboard.Fig. 9(a) illustrates the time response of the x-direction position of the aerial manipulator.From t = 0 to 5.6 s, the aerial manipulator approaches the contact object, and the contact occurs at 5.6 s.It can be obtained that the aerial manipulator can perform a favorable tracking performance under the proposed control architecture.Then, Fig. 9(b) and (c) illustrate the position and attitude tracking errors of the aerial manipulator, and the state error remains bounded with the predefined performance.Fig. 9(d) visualizes the force tracking performance, wherein the force trajectory is set as 0.8 N and it can be seen that the wrench force estimator performs a favorable estimated effect.This experimental scenario shows the effectiveness of the proposed control architecture for achieving the active physical interaction between the aerial manipulator and the environment.Besides, the stable interaction and expected force controlled indicate that  the aerial manipulator with the proposed control architecture can implement the physical contact inspection task.Fig. 10 shows the results of the proposed control architecture when the aerial manipulator touches the carton.The analysis of the experimental results is similar to that in Fig. 9.The purpose of constructing this experimental scenario is to further demonstrate the effectiveness of the proposed control architecture and its applicability to an environment with different stiffness.It means that the proposed control architecture can implement the physical contact inspection task in uncertain environments.

VI. CONCLUSION
This article presented an active physical interaction control architecture for the aerial manipulator to achieve both stable motion and interaction behaviors with external wrench estimation.First, the dynamics of the aerial manipulator was modeled and analyzed via the Newton-Euler method.Then, the interaction control architecture was proposed, including the external wrench estimator, the FTIC with variable stiffness, and the pose controller.Specifically, the external wrench estimator was designed without the acceleration measurements, where only the linear and angular velocity measurements were needed.The FTIC method with variable stiffness was presented to ensure the contact stability and force tracking of the aerial manipulator with unknown environmental stiffness.The pose controller was presented to implement the dynamic response speed and accuracy control of the aerial manipulator.Further, the Lyapunov approaches were used to prove the stability of the proposed interaction control architecture.Simulations and real-world experiments of the aerial manipulator were completed to validate the correctness and performance of the proposed interaction control architecture.For further work, the aerial manipulator combined with external sensors (such as cameras, lasers, and force sensors) and advanced position estimation methods could be utilized to achieve aerial maintenance work in high altitudes or extremely dangerous places.

Fig. 1 .
Fig. 1.Schematic view of frames of the aerial manipulator for contact inspection containing the quadrotor aerial robot with the rigidly attached tool.

Fig. 2 .
Fig. 2. Block diagram of the active physical interaction control architecture, combining external wrench estimator, FTIC with variable stiffness, and pose controller.

Fig. 3 .
Fig. 3. Diagram of pose controller of aerial manipulator with prescribed performance.

Fig. 4 .
Fig. 4. Simulation comparison results of the proposed control architecture and the NDO-SMC in [39] for disturbance rejection verification.(a) Time response of the position errors of the aerial manipulator with predefined performance.(b) Time response of the attitude errors of the aerial manipulator with predefined performance.

Fig. 5 .
Fig. 5. Simulation results of the proposed control architecture for the active physical interaction task.(a) Modified commanded position trajectory tracking of the aerial manipulator.(b) Time response of the attitude of the aerial manipulator.(c) Time response of the position errors of the aerial manipulator with predefined performance.(d) Time response of the attitude errors of the aerial manipulator with predefined performance.(e) Time response of the force tracking in the x-direction.

[ 1 . 5 ,
0, 0] T N 35 < t ≤ 55 s [0.5, 0, 0] T N 55 < t ≤ 75 s.The environment stiffness k e is set to abruptly change from 100 to 200 N/m at t = 25 s and from 200 to 100 N/m at t = 65 s.At the initial state, the aerial manipulator hovers at the position p(0) = [0.4,−0.3, 1.2] T m.The results of interaction simulation in the x-direction are depicted in Fig.5.In more detail, Fig.5(a) and (b) illustrate the position and attitude tracking with both favorable convergence

Fig. 7 .
Fig. 7. Experimental comparison results of the proposed control architecture and the NDO-SMC in [39].(a) Time response of the position errors of the aerial manipulator with predefined performance.(b) Time response of the attitude errors of the aerial manipulator with predefined performance.

Fig. 8 .
Fig. 8. Snapshots of experiments when the aerial manipulator touches the whiteboard and carton.(a) Aerial manipulator touches the whiteboard.(b) Aerial manipulator touches the carton.The numbers in the top right corner of each picture imply specific phases such that: 1 the approaching phase of the aerial manipulator, and 2 the stable contact phase of the aerial manipulator.

Fig. 9 .
Fig. 9. Experimental results of the proposed control architecture when the aerial manipulator touches the whiteboard.(a) Time response of the x-direction position of the aerial manipulator.(b) Position errors of the aerial manipulator with predefined performance.(c) Attitude errors of the aerial manipulator with predefined performance.(d) Time response of the force tracking in the x-direction.

Fig. 10 .
Fig. 10.Experimental results of the proposed control architecture when the aerial manipulator touches the carton.(a) Time response of the x-direction position of the aerial manipulator.(b) Position errors of the aerial manipulator with predefined performance.(c) Attitude errors of the aerial manipulator with predefined performance.(d) Time response of the force tracking in the x-direction.

TABLE II MEAN
AND VARIANCE OF ABSOLUTE ERROR OF THE EXPERIMENTAL COMPARISON RESULTS