Delay-Dependent Stability in Uncalibrated Image-Based Dynamic Visual Servoing Robotic System

This paper addresses the stability problem of uncalibrated image-based visual servoing robotic systems. Both the visual feedback delay and the uncalibrated visual parameters can be the sources of instability for visual servoing robotic systems. To eliminate the negative effects caused by kinematic uncertainties and delays, we propose an adaptive controller including the delay-affected Jacobian matrix and design an adaptive law accordingly. Besides, the delay-dependent stability conditions are provided to show the relationship between the system stability and the delayed time in order to obtain less conservative results. A LyapunovKrasovskii functional is constructed, and a rigorously mathematic proof is given. Finally, the simulation results are presented to show the effectiveness of the proposed control scheme.


Introduction
For human beings, vision is an important sensory channel.Through visual sensors, robots also can monitor the circumstance and perform the tasks.Nowadays, the advanced visual processing techniques and high-speed image processors make vision-based robot systems capable of handling dynamical tasks, and the vision-based control has been applied to many industrial robot systems.It has become the mainstream of robot control.
Vision-based control can be traced back to 1980s [1].Look-and-move is one of the early vision-based technologies [1][2][3].In this approach, two nested loops run simultaneously: the visual loop is the external loop and the joint-space loop is the internal loop.Due to the sensitivities to disturbances and errors, the look-and-move architecture is not suitable for high-performance control tasks [4].As an alternative, the visual servo (VS) technique is proposed [5].This control architecture directly generates the control inputs using the visual information.Such a simple and direct structure is favorable for high-speed servoing tasks.Considerable visual servoing approaches have been investigated for various robot systems and from many different aspects.Figure 1 shows two typical structures of the visual servoing control.
In the existing literature, there are two challenges in the field of visual servoing control: (a) the difficulties of calibration and (b) the image feedback signals of inferior quality.
The calibration of visual servoing systems includes the camera calibration, kinematic calibration, and dynamic calibration.For the sake of identifying unknown or uncertain system parameters, periodical and high-accurate calibration work usually is required, which is tedious and demanding.Without such calibration work, the system models cannot be accurately characterized and the closed-loop visual servoing systems could be unstable.To avoid such calibration work, the uncalibrated control approaches are proposed [6][7][8][9][10].Some work [6][7][8] investigates the approaches with robust controllers for eliminating the negative effects of calibration errors of the system model, and the uncertain system parameters in the above work are replaced with the approximated ones.As for the case of unknown parameters or time-varying parameters, adaptive control techniques are proposed [11,12].To handle such parameters, in these methods, adaptive laws are designed to update them online.
As another cause of system instability, the visual signals of inferior quality are also nonnegligible.Generally speaking, noise and delays in the visual signals are the main inducements.In this paper, we consider delays as the main reason for inferior image signals.As we know, the visual signal flows are expected to be synchronized with other system signals.However, asynchronization could happen due to many reasons including the limitation of image processing [13][14][15][16][17] and restriction of visual signal transmission [18].Some early research studies the instability problem caused by image-processing (or image-sampling) delays [1,2,13].These early efforts focused on reducing the image sampling time through a parallel or pipelined approach [2,[14][15][16][17].With the development of the advanced image processor chips, such a problem has been resolved to a large extent.With the wide application of visual servo, its patterns are becoming various.The connections between visual sensors and controllers can be wireless or Internet, which also means the visual feedback path can be a source of delays due to the transmission block in the intercomponent information exchanges [1,2,13,18].Improving the speed and the reliability of communication links is a straightforward way to address the issue.But it inevitably leads to the increase of the cost.Consequently, designing proper control schemes to handle with delays is an alternative.In the IBVS control scheme design, the delay problem is studied by [19][20][21][22][23]. Using average joint angle values of the past and present moment to replace the present joint angle, [19] obtains predicted image feature position values by the Jacobian matrix to cope with the delay problem.One common flaw of the aforementioned methods is that they require the accurate knowledge of system parameters, and the acquirement of such information is based on calibration.
The two challenges make a visual servoing robotic system become typical complex industrial systems.This is because the mainstream noncalibration techniques usually require accurate image signals to compensate for the parametric errors or to update the unknown parameters.Under the delayed image feedback loop, there is no accurate synchronized visual feedback available.In this context, the control of such systems is of high nonlinearity and complexity.Consequently, it is worthwhile and challenging to be investigated.This paper therefore will concentrate on the influence of visual transmission delays upon the uncalibrated visual servoing robotic systems.
In the literature of this area, [21] presents an online calibration method to overcome the time delay problem.Inoue and Hirai [22] design a two-layer controller called STP to compensate for the delays and the concept of virtual trajectory is introduced.Gao and Su [23] employ local fitting Jacobian matrix based on polynomial fitting to obtain more accurate Jacobian estimation and image precompensation for uncalibrated IBVS robotic system.Unfortunately, the controller design in the above literature is based on kinematics and fails to consider the dynamics of robots.It is well known that the dynamics of robot systems plays an important role in the stability, especially in the case of high speed.Much progress has been made in the aspect of the uncalibrated dynamic-based visual servoing systems control without delay effects [24][25][26][27][28][29][30][31].As for the uncalibrated dynamic-based visual servoing systems with the delay effects, the relevant work focuses on the area of the distributed cooperative control [32][33][34][35].Liu and Chopra [33] study an adaptive control algorithm to guarantee task-space synchronization of networked robotic manipulators in the presence of dynamic uncertainties and time-varying communication delays.Wang [34] investigates the problem of synchronization of networked robotic systems with kinematic and dynamic uncertainties in the case of nonuniform constant communication delays.Liang et al. [35] address cooperative tracking control problem of networked robotic manipulators in the presence of communication delays under strongly connected directed graphs.However, the above work considers the delays in the interagent information exchanges  2 Complexity rather than delays existing in the visual feedback of single dynamic-based visual servoing robotic system.To the best of the authors' knowledge, there is little literature considering the time delay problem in an uncalibrated dynamic-based visual servoing robotic system without using image-space velocity measurements.To address the aforementioned issues, the following problems are expected to be addressed.First, the modeling of the system.Delays, noncalibration, and velocity measurements, these contributing factors, need to simultaneously be included in the modeling of the system.Second, the handling of the time-varying parameters and how to avoid using image velocity measurements.Third, the delaydependent stability conditions are expected to be given for obtaining less conservation.The paper is organized as follows.Section 2 gives some preliminary knowledge used throughout the paper.In Section 3, the kinematic and dynamic models of dynamicbased visual servoing robotics systems are formulated.In Section 4, the main results of this paper, the controller design, and the adaptive laws are proposed to address the stability problem of the uncalibrated dynamic-based visual servoing robotic system with visual feedback delays.In Section 5, rigorous stability analyses are provided.Section 6 presents simulation results to show the effectiveness of the proposed control scheme.Section 7 concludes the paper.

Preliminaries
Lemma 1.Let X, Y, and F be real matrices with proper dimensions, where F T F ≤ I.For any constant ε > 0, the following holds.
Let ϕ R → R be a uniformly continuous function on 0, ∞ .Suppose that lim t→∞ t 0 ϕ τ dτ exists and is finite.Then, Lemma 3. Consider functional differential equation the zero solution of system ( 8) is uniformly stable.If zero solution of the system is uniformly stable and for any s > 0, ω s > 0 holds, then zero solution of system ( 8) is uniformly asymptotically stable.If zero solution of the system is uniformly asymptotically stable and lim s→∞ u s = ∞, then zero solution of system ( 8) is globally uniformly asymptotically stable.

Kinematics and Dynamics
In this section, we present the mathematical modeling of delayed visual servoing robotic systems with the eye-inhand configuration.In the modeling process, both kinematics and dynamics are considered.To illustrate the kinematics of the system, Figure 2 shows the transformation among different frames.Let y t ∈ ℝ 2 be the coordinates of a feature point's projection on the camera image plane and r ∈ ℝ 3 be the Cartesian coordinates of the feature point w.r.t the robot base frame.Based on the model developed in [7], the mapping between image position y t and the Cartesian position r can be formulated as where z t ∈ ℝ denotes the depth of feature to the camera frame; T c b ∈ ℝ 4×4 denotes the homogeneous transformation matrix from the camera frame to the base frame; Ω i ∈ ℝ 1×4 denotes the ith row of the camera intrinsic parameter matrix Ω (Ω is an intrinsic parameter matrix which is derived from the typical model introduced in [36]).In reality, the feature point is stationary with respect to the robot base and it results in a constant column vector r.
The relationship between z t and r can be formulated by where Ω 3 is the third row of the perspective projection matrix Ω.

Complexity
Combining with (7), the derivative of (6) w.r.time t satisfies where y is short for y t .It should be noticed that T c b can be divided into two parts: T e b (the forward robot kinematics) and T c e (homogeneous matrix from the camera to the end effector).Due to the eye-in-hand configuration, T c e is a constant matrix.Then, one has where R denotes the rotation matrix, P denotes the translation vector, and q denotes the joint position.For more details, see [7].By letting the matrix L be the left 3 × 3 submatrix of ΩT c e , L 0 ∈ ℝ 2×3 be the 1st and the 2nd rows of L and L 3 ∈ ℝ 1×3 be the 3rd row of L, one can derive the mapping from joint velocities to image velocities as follows: The nonlinear mapping J q introduced in ( 10) is an important matrix in IBVS, which is known as Jacobian matrix [37,38].The differential of (7) w.r.t.time t satisfies The dynamics of robots can be given with Euler-Lagrange equation as follows [39]: where u is the n × 1 vector formed by joint input of the manipulator; H q t is the n × n positive-define and symmetric inertia matrix; C q t , q t ∈ ℝ n×n is a skew-symmetric matrix such that for any proper dimensional vector ψ, ψ T C q t , q t ψ = 0 13 On the left side of (12), the first term is inertia force, the second term is the Coriolis and centrifugal forces, and the last term g q is the gravitational force.
Remark 1. From Figure 2, it intuitively can be seen that the estimation of Jacobian matrices determined by homogeneous transformation matrices (T c b , T e b , and T c e ) is directly affected by the delayed visual feedbacks.The complexity of the system mainly lies in the highly nonlinear relationship between delayed image states and joint states.
To facilitate analysis, we present Figure 3 to show the closed-loop structure of a typically delayed VS robotic system.

The Adaptive Controller Design
In this section, we will investigate the uncalibrated dynamicbased visual servoing robotic system with visual feedback delays and kinematic uncertainties.In our study, the formulation of the uncalibrated VS robotic system is partly based upon the depth-independent Jacobian model developed by 4 Complexity [27].This model allows depth to be time varying so that the visual servoing system can still be stabilized even in the presence of the fast-changing feature image depth.From (10), we can easily split 1/z t from J q and thereby obtain the depth-independent Jacobian matrix D which is given by Additionally, from (11), we define such a vector as follows: Therefore ( 10) and ( 11) can be rewritten as In the uncalibrated dynamic-based visual servoing system, the estimate of Jacobian matrix is usually used as the replacement of unknown exact Jacobian matrix.It can be easily seen from ( 14) and ( 15) that components of the depth-independent Jacobian matrix D and the matrix D 3 can be classified as two categories: the known and the unknown.Known components are R, P, and unknown components are L, L 3 , and r.The estimate of Jacobian matrix can be analytically derived through the linear parameterization [40].From (14), it can be seen that the known and the unknown are coupled.And the coupling of the known and the unknown hinders the linear parameterization of these matrices.The following property is proposed to decouple them.
Property 1.For a vector q ∈ ℝ n×1 , the product Dq can be linearly parameterized as follows: where Y D y, q, q ∈ ℝ 2×p 1 and Y D 3 q, q ∈ ℝ 1×p 1 are regressor matrices which consist of known parameters; θ k ∈ ℝ p 1 ×1 is a vector which consists of unknown parameters; and p 1 denotes the number of unknown parameters, which satisfies p 1 ≤ 36.
Proof 1. Due to the limitation of pages, see proof in Appendix A.
By Property 1, the Jacobian matrix D can be expressed in a linear form: a known matrix (regressor matrix) multiplies an unknown vector.From (12), it can be clearly seen that the regressor matrix Y D y, q, q includes the current image position y t .Unfortunately, the feedback visual signals are delayed as we consider.We may use y t − h to denote the coordinates of delayed feature image position, where h denotes the constant delayed time.In this case, the matrix Y D y, q, q cannot be obtained.Instead, we can only obtain Y D y t − h , q, q .After substituting this regressor matrix including the delayed visual feedback matrix into ( 14) and ( 12), we have where Q is named as delay-affected depth-independent Jacobian matrix.For simplification, we call it delay-affected Jacobian matrix hereafter.The relationship between Q and D is given by Using the delay-affected Jacobian matrix Q and D 3 , we define a new composite Jacobian matrix as where q + denotes the vector which satisfies q + q = 1.Based on all above analyses, we now propose the controller for delay-affected uncalibrated VS robotic systems as follows: where K 1 ∈ ℝ 2×2 and K 2 ∈ ℝ n×n are positive definite symmetric matrices and Ĵd q denotes the estimate of J d q .
Note that the estimate Ĵd q for the new Jacobian matrix is able to obtain from ( 21) by respectively replacing unknown matrices Q and D 3 with their estimates Q and D3 , and it yields Additionally, recalling ( 12) and ( 18) in Property 1, we can easily derive the following linear parameterization form where Y y t − h , y t , q, q is the new regressor matrix including the delayed image state.To obtain θ k , we proposed the following adaptive law: where Γ k is a positive definite symmetric matrix with proper dimensions and Y is short for Y y t − h , y t , q, q .Besides, it is not hard to derive Y m and Y M accordingly.Please refer to Notation in Introduction for the explanation.Additionally, it is also not hard to roughly give the bound of the unknown parameter vector θ k according to the T c e and feature Cartesian coordinates [27].Thereby, we assume that both θ m k and θ M k are known, i.e., θ k ∈ ℒ ∞ .Basing on the above analyses, we can effortlessly know the bound of Ĵd from (24), i.e., Ĵm d and ĴM d can be regarded as known ones.We define Consequently, Ĵd can be expressed in the interval matrix form [41] as follows: where ΔG ij denotes the element at the ith row and the jth column of Δ Ĵd .Likewise, J q is also bounded.J m and J M are given by Hence, J q can be expressed in the interval matrix form where e i ∈ ℝ 2 denotes the column vector whose ith element is 1 and the other element is 0; e j ∈ ℝ n denotes the row vector whose jth element is 1 and others are 0; ΔJ ij denotes the element at the ith row and jth column of ΔJ.
Remark 2. From ( 24), one of the key points in deriving Ĵm From the practical experience, the range scales of θ k actually depend on the (1) the initial value of θ k , which is set artificially and (2) the real value θ k which is unknown.Even if θ k is unknown, we can easily give some estimates of its elements according to some other rough estimates.For more details, please refer to Appendix A.
Proof 2. Combining ( 14), (19), and ( 21), we have where Δθ k = θ k − θ k .Substituting controller ( 22) into ( 12),we have the following closed-loop system, As aforementioned, the fact that Y and Δθ k are all bounded yields the result that for some positive constants B M .Let us consider the following nonnegative Lyapunov-Krasovskii functional candidate, where the employment of the term ▽ follows the typical practice (refer to [42], p118).The time derivative of V along the trajectory of system is given by V x = q T H q q + q T 1 2 H q q + Δy T K 1 zΔy Multiplying q T from left side to both sides of (35) yields Rewriting the (34) and then multiplying Δy T K 1 from the left side of zy + 1/2 zΔy, we have After taking differential of 1/2 Δθ k Γ k Δθ k and invoking (25), it yields Substituting ( 39), (40), and ( 41) into (38), we obtain Likewise, with Lemma 1, the below cross terms yield Besides, from (10), hy T t Γ y y t can be rewritten as hq T J T q Γ y J q q.
Having obtained the results in (43), substituting them to (43) and we have the following inequality:

44
We will analyze the term I and the term II one by one.Firstly, we consider the term I.In this term, both Ĵd and J are time-varying matrices.It should be noted that we assume J m and J M being unknown ones as aforementioned.Using Lemma 1 and ( 29), we can easily derive

47
where J and G are defined in (33).Then, we consider the term II.In actual visual servoing robotic system, the depth changing velocity is actually bounded.Here, we can reasonably assume that the z is bounded, z < z M .Invoking (31), we have Combining (47), (48), and (32), we can finally have V ≤ 0 in (44), which means that the Lyapunov-Krasovskii functional V never increases its value so that it is upper bounded.From (37), bounded V directly implies that the joint velocity and image error Δy ∈ ℒ 2 ∩ ℒ ∞ .Then the joint acceleration q ∈ ℒ ∞ can be concluded from the closed-loop dynamics (35).Therefore, the joint velocity q is uniformly continuous.Note that it is not hard to derive y ∈ ℒ ∞ from (10) with bounded J T q and q ∈ ℒ 2 ∩ ℒ ∞ .Thereby, we can also conclude that Δy is uniformly continuous.y ∈ ℒ ∞ and y t − h ∈ ℒ ∞ yield y − y t − h ∈ ℒ ∞ and hence we can derive that the image delay error y − y t − h is uniformly continuous.And from (25), it can be derived that θ k ∈ ℒ ∞ .Thereby, θ k is uniformly continuous.Invoking Lemma 2 and Lemma 3, we have lim t→∞ Δy = 0, lim t→∞ Δθ k = 0, and lim t→∞ y − y t − h = 0.This completes the proof.Remark 3. It can be clearly seen that the delay-dependent stability condition is presented in Theorem 1. Stability analyses given by [33][34][35] are delay-independent results, which means the stability conditions impose no constraint on system delays.Hence, their stability results hold with any magnitude of delays.However, in reality, the delays are usually bounded and the delay-dependent results are conservative.To obtain less conservative results, we should consider magnitude of delays.It is significant to the delay stability research due to less conservativeness.Remark 4. In order to fully control 6-DOFs or more degree robots, we need more noncollinear feature points.For instance, three noncollinear feature points should be considered for a 6-DOF manipulator.The scheme proposed in this paper can be effortlessly generalized to the case of multiple feature points by the similar method described in [28].Considering the page limitation, we only present the single feature point case in this paper.

Simulation Results
To show the effectiveness of the control scheme described in (22) and Theorem 1, we conduct the following simulations.
The actual visual parameters are set as follows: f = 0 035 m, u 0 = 280 pixels, v 0 = 250 pixels, k u = 1800 pixels/ m, k v = 1800 pixels/m, and ϑ = π/2 rad, where f is focal length; u 0 and v 0 are coordinates of camera principal point in the image frame; k u and k v denote scale factors along axis x and y, respectively; and ϑ denotes intersection angle between axis u and axis v.The intrinsic matrix Ω therefore can be derived as For the setting of the camera's position and pose, the T c e is set as follows: The gravitational acceleration is set as g = 10 m/s 2 .T e b is time varying and determined by forward kinematics of the manipulator whose parameters are given in Table 1.
From Property 1 and according to the ranges of q i , z, L 0 , L 3 , ∂/∂q Rr + P , in this paper, we may set J M d and J m d as  Besides, we set z M = 0 3 m/s here and K 1 and K 2 are obtained by solving the feasibility problem of (30), (31), and (32) with the solver feasp.In this simulation, we use

Ĵm
Based on all above settings, two simulations are conducted.In the first simulation, the proposed control scheme is used to track the desired position under two different constant delays: h 1 = 98 ms and h 2 = 198 ms.Figures 4(a), 5(a), 6(a), and 7(a) demonstrate the position errors, the position, the velocity, and the trajectory of the feature point on the image plane, respectively.It can be observed that the performance is almost identical even under the different delays, 198 ms and 98 ms.It verifies that the convergence of the system will be achieved once as long as the conditions given in Theorem 1 hold.Besides, Figure 7 also shows better position tracking performance with 98 ms delays than that of 198 ms.To show the convergence of estimates θ k to real values, we partly choose some elements in the vector θ k .Figure 8 shows the profile of estimated parameters from θ k4 to θ k12 .It should be noted that the kinematic parameters θ k converge only when the persistent excitation (P.E.) condition is satisfied.In our simulation, we choose the θ k 0 close to their real values such that these estimated parameters can converge to them.In most cases, the estimated parameters only converge to the true values up to a scale.However, it will not affect the convergence of image errors.
To demonstrate the superiority of the proposed control scheme, we make a comparison between the two control schemes: the scheme 1 and the scheme 2. The scheme 1 is the method proposed in this paper, and the scheme 2 originating from [30] is modified accordingly in this simulation as follows.
From Figures 4(b)-7(b), it can be clearly seen that the performance of scheme 2 in the presence of delay time h = 98 ms is unsatisfying.Abnormal oscillations can be observed, which is caused by the delays.In contrast, the proposed scheme can still guarantee very satisfying control performance, which is not affected too much by delayed signals.In conclusion, the second simulation result shows the superiority of the proposed scheme over existing schemes that can eliminate the negative effect caused by delays to a great extent.

Conclusions
In this paper, we have proposed a control method for uncalibrated dynamic-based visual servoing robotic systems to Notes: l i denotes link length; α i denotes link twist; d i denotes link offset; q i denotes joint angle; m i denotes link mass; l ci denotes the length between barycenter and its prior joint.Complexity cope with the delay problem existing in the visual feedback loops.To handle the unknown camera intrinsic and extrinsic parameters, we introduced the depth-independent Jacobian matrix and used the linear parameterization to adaptively identify these uncertainties.Then, we took the delays into consideration and constructed a novel matrix called delay-affected Jacobian matrix.Based on the delayaffected Jacobian matrix, we proposed the adaptive controller.To prove the stability of the closed-loop system, the Lyapunov-Krasovskii functional is constructed and delaydependent stability conditions are also provided to obtain less conservative results.Simulation results of the proposed control scheme were presented to show the effectiveness.
To further validate the performance of the proposed scheme, experimental tests on real networked visual servoing robotics systems are expected to be the most appropriate choice and this is also one of our main objectives in the future research.
Besides, because d T is a subset of D, the linearization of d T q can be a direct result of the Property.When p 1 = 36, Y D 3 q, q can be expressed as Y D 3 q, q = 0 1×24 ϱ T A 3 When p 1 < 36, i.e., ρ k is independent of q, Y D 3 q, q and θ k can be obtained by removing corresponding elements.Then, we can derive expression of θ k and Y D 3 q, q for every p 1 < 36.This completes the proof.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.The 3rd row of L J q Jacobian matrix

H q
The inertia matrix of manipulator dynamics C q, q The Coriolis and centrifugal forces Image-based visual servoing

Figure 2 :
Figure 2: Robotic system with the eye-in-hand configuration.

Figure 3 :
Figure 3: The structure of a delayed visual servoing robotic system.

3 53
The feature point's coordinates w.r.t. the base frame are (150, 20) T m.The initial position coordinates on the image plane are (140, 81.44) T , and the desired position coordinates on image plane are (160.7,120.6)T .

Figure 5 : 2 Figure 6 :
Figure 5: Position trajectory of the feature point on the image plane.

Figure 7 :
Figure 7: Position trajectory of the feature point on the image plane.

e 0
Homogeneous transformation matrix from the camera to the end effector r The Cartesian coordinates of the feature point w.r.t. the robot base frame L The matrix consists of the left 3 × 3 part of ΩT c e L The 1st and 2nd rows of L L 3

3
The vector derived from L 3 ∂/∂q Rr + P QThe delay-affected depth-independent Jacobian matrix J dA novel composite Jacobian matrixK 1 , K 2The control gain matricesY D * , Y D 3 *The regressor matrices θ k The unknown parameter vector e i e i ∈ ℝ 2 , the column vector whose ith element is 1 and the other element is 0 e j e i ∈ ℝ n , the row vector whose jth element is 1 and other elements are 0 Symbols ⋅ The standard Euclidean normA T The transposition of matrix A A mThe matrix consists of the minimum elements of AA MThe matrix consists of the maximum elements of AA ijThe element in the ith row and the jth column of matrix A y t − h The visual signal delayed unknown constant duration of h

Table 1 :
Parameters of manipulator in the simulation.

11 Complexity Appendix A. Proof of Property 1
Proof 3. Let L i denote the ith row of L. Recalling (14), we can expand L T i ∂/∂q Rx + P as follows:

Table 2 :
List of notations and symbols.