Improved Uncalibrated Visual Servo Strategy for Hyper-Redundant Manipulators in On-Orbit Automatic Assembly

Featured Application: An uncalibrated visual servo strategy for the on-orbit automatic assembly of an ultra-high-caliber optical system. It is mainly employed to realize the static positioning and dynamic tracking of the desired image features in a harsh space environment when the desired image features required for assembly are known. Abstract: Aiming at on-orbit automatic assembly, an improved uncalibrated visual servo strategy for hyper-redundant manipulators based on projective homography is proposed. This strategy uses an improved homography-based task function with lower dimensions while maintaining its robustness to image defects and noise. This not only improves the real-time performance but also makes the joint space of hyper-redundant manipulator redundant to the homography-based task function. Considering that the kinematic parameters of the manipulator are easily changed in a space environment, the total Jacobian between the task function and manipulator joints is estimated online and used to construct a controller to directly control the manipulator joints without a kinematics model. When designing the controller, the above-mentioned redundancy is exploited to solve the problem of the over-limiting of the joint angles of the manipulator. The KF-SVSF method, which combines the optimality of the Kalman ﬁlter (KF) and the robustness of the smooth variable structure ﬁlter (SVSF), is introduced to the ﬁeld of uncalibrated visual servos for the ﬁrst time to perform the online estimation of the total Jacobian. In addition, the singular value ﬁltering (SVF) method is adopted for the ﬁrst time to avoid the interference caused by the unstable condition number of the estimated total Jacobian. Finally, simulations and experiments verify the superiority of this strategy in terms of its real-time performance and precision. hyper-redundant manipulator, 6 DOF universal robot, smart camera and NI real-time system were used to build the experimental platforms. The parameters of the manipulator and the camera are consistent with the parameters set in the simulations. Experiments for aligning with both static and dynamic assembly objects were conducted to verify the improvement of our strategy. A square target was chosen with a side length of 200 mm and 49 feature points. All experiments performed under an illumination similar to that of actual working conditions to examine the reliability of our strategy.


Introduction
Operating an ultra-high-caliber optical system in high space orbit is one of the key goals of the Chinese space industry and will help scientists to make more detailed observations of the universe [1]. However, it is impossible to transport it to space as a whole or install such a large space system manually in space. Large space structure construction through on-orbit services (OOS) may be the best

Projective Homography
The current frame F and the desired frame F * of an eye-in-hand camera are defined for problems with an eye-in-hand system in which the camera moves with the manipulator. Figure 1 shows the geometric relationship between these two camera frames. Suppose M i = [X i , Y i , Z i ] is the coordinate of the feature point P i in F and M * i = X * i , Y * i , Z * i is the coordinate described by the desired camera, while p i = u i v i 1 and p * i = u * i v * i 1 are the homogeneous pixel coordinates of feature point P i captured by the current and desired cameras, respectively. The geometric relationship between M i and M * i can be expressed as where R ∈ R 3×3 and t ∈ R 3×1 are the orientation matrix and displacement vector between F and F * expressed in F * , respectively.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 28 Suppose each target point i P is on the plane π , * d is the distance between the desired projection center and plane π and * n is the normal vector of π expressed in the desired camera frame * F with length * * 1/ = n d . Based on Equation (1), we can obtain ( ) = H KGK (7) which refers to the mapping between different images projected from the same object to two different image planes. It has been applied in many domains such as motion estimation, 3-D reconstruction [49] and of course the visual servo. Even in an uncalibrated situation, this matrix can be estimated up to a scale β with more than four corresponding image features in different images via where β is independent of the depth ratio α i in (5). Suppose each target point P i is on the plane π, d * is the distance between the desired projection center and plane π and n * is the normal vector of π expressed in the desired camera frame F * with length n * = 1/d * . Based on Equation (1), we can obtain where G is the Euclidean homography matrix. In normalized coordinates, (2) can be expressed as where m i = [X i /Z i , Y i /Z i , 1] and m * i = X * i /Z * i , Y * i /Z * i , 1 . The depth scale is denoted as The following relationship can be obtained from Equations (2), (3) and (5): where K is the internal parameter matrix of the eye-in-hand camera [36]. The homography matrix can be obtained as which refers to the mapping between different images projected from the same object to two different image planes. It has been applied in many domains such as motion estimation, 3-D reconstruction [49] and of course the visual servo. Even in an uncalibrated situation, this matrix can be estimated up to a scale β with more than four corresponding image features in different images viâ where β is independent of the depth ratio α i in (5).
Appl. Sci. 2020, 10, 6968 5 of 26 Because the images are always disturbed by noise during transmission and processing, the exact solution of H cannot be obtained via the noisy measurement image point p i . However, it is possible to obtain the optimal estimated homography matrixĤ in the sense of least squares. The estimated image point of P i in the current image can be obtained viap i =Ĥp * i . The root-mean-squared (RMS) error betweenp i and the true points can be obtained viaε ≈ ε(4/n) 1/2 [38], where n is the number of image features and ε is the RMS error between the measurement point p i and the true point. It is worth noting that the estimated homography matrixĤ can be equivalently considered as being estimated by the matching pointsp i ↔ p * i . Obviously, if n > 4, building the task function with the elements of the homography matrix will have a certain compensation effect for image noise.

Improved Strategy for the Uncalibrated Visual Servo of a Hyper-Redundant Space Manipulator
In this section, our improved strategy as shown in Figure 2c will be described in detail. The first part details the new task function and provides related proofs. Then, the more robust Kalman filter-smooth variable structure filter (KF-SVSF) method will be introduced to estimate the total Jacobian. Finally, the estimated total Jacobian, processed via SVF to reduce the disturbance caused by an unstable conditional number, is used to build a controller to control the manipulator joints directly for both static positioning and dynamic tracking while limiting the joint angle. , building the task function with the elements of the homography matrix will have a certain compensation effect for image noise.

Improved Strategy for the Uncalibrated Visual Servo of a Hyper-Redundant Space Manipulator
In this section, our improved strategy as shown in Figure 2c will be described in detail. The first part details the new task function and provides related proofs. Then, the more robust Kalman filtersmooth variable structure filter (KF-SVSF) method will be introduced to estimate the total Jacobian. Finally, the estimated total Jacobian, processed via SVF to reduce the disturbance caused by an unstable conditional number, is used to build a controller to control the manipulator joints directly for both static positioning and dynamic tracking while limiting the joint angle.

Improved Homography-Based Task Function
In the original PHUVS strategy proposed by Zeyu Gong [23], the homography-based task function is defined asĥ whereĤ i is the ith row of the estimated homography matrix. The error vector of the system is denoted by where h 0 is constructed by stacking the rows of the identity matrix I 3×3 . The purpose of our strategy can be considered to be making the current camera frame F coincide with F * . To guarantee this, the constraint det(Ĥ) = 1 is defined to obtain the unique projective homography matrixĤ between the current and desired feature points in each iteration. It can be proved that e = 0 if and only if rotation matrix R = I 3×3 and position vector t = 0 [23]. In our strategy, the constraintĥ 9 = 1 is defined, where h 9 is the last element of the homography matrix and a direct linear transformation (DLT) is introduced to estimate the homography matrix. First, it is necessary to normalize the two images, respectively, as follows:

1.
Translate the feature points as p i = T 1 p i and p * i = T * 1 p * i to ensure that the centroid of these feature points is at the origin; 2.
Scale feature points as p i = T 2 p i and p * i = T * 2 p * i to make their average distance from the origin equal to On the basis of the properties of homogeneous coordinates, homogeneous simultaneous equations can be constructed with transformed pixel coordinates as where A is the coefficient matrix and H i is the ith row of H. The purpose of the above normalization operation is to prevent the coefficient matrix A from becoming ill-conditioned owing to image noise. This solves the problem of numerical instability. Then, the least squares solution of equation (11) can be calculated via singular value decomposition. Then, we conduct an inverse transform on H as where H is the estimated homography between the current and desired feature points. The homography H can be scaled at any scale. Thus, H is scaled asĤ = H/ h 9 and the new task function is constructed aŝ Appl. Sci. 2020, 10, 6968 7 of 26 whereĥ reflects the eight degrees of freedom of the homography matrix and further reduces the dimensions of the system. The error vector of system is defined as where e = 0 if and only if the rotation matrix R = 0 and the position vector t = 0. Its sufficiency and necessity can be proved as follows.

2.
Sufficiency: It is obvious thatĤ = I and H = βI if e = 0. From Equation (7), we have Taking two feature points, the coordinates of these points in F and F * can be described as Obviously, regardless of in which frame this is expressed, the distance between P i and P j is constant. That is, This means that β = 1, which implies H = I and G = I. Thus, R = I and t = 0. This new task function takes full advantage of the property that the homography matrix has eight degrees of freedom. This not only reduces the dimensions of the task function but also simplifies the state space of the visual servo system that needs to be estimated online, so the real-time performance of system is further improved. Meanwhile, this causes the total Jacobian J t ∈ R 8×9 , which maps the task function and joint space of the hyper-redundant manipulator, to become redundant. Therefore, this redundancy can be used to finish some secondary tasks such as limiting the joint angle without a manipulator kinematic model during the servo process. Obviously, this improves the robustness of system to changes in the kinematic parameters of manipulators.

Remark 1.
A flaw in the inhomogeneous solution of the homography matrix is that if the true value of h 9 is 0 or very close to 0, the estimated homography matrix becomes very unstable. In fact, this situation rarely occurs in the visual servo. In static positioning tasks, the task function generally changes near to the desired gradient direction. As long asĥ 9 is not equal to 0 at the initial position, its value will never become 0 during the servo process. In tracking tasks, the task function is always close to the desired value.

Online Estimation of Total Jacobian
The kinematic parameters of the space manipulator change easily in the harsh space environment. To avoid the repeated calibration of the complex hyper-redundant manipulator, the total Jacobian between the task function and the joints of the manipulator is used to construct the controller. The expression of the total Jacobian is ∂ĥ 1 /∂q 1 · · · ∂ĥ 1 /∂q 9 . . . . . . . . . Therefore, the accuracy of the estimation of the total Jacobian is directly related to the system performance.
The uncalibrated visual servo system of the nine degree-of-freedom (9 DOF) manipulator is modeled as where and J ti is the ith row of the estimated total Jacobian J t .
q K 8×72 is constructed by the joint velocity of the manipulator, and Z K can be considered as the change rate of task functionĥ. It is worth noting that this model is a linear approximation of the visual servo system, and it causes modeling uncertainty. Meanwhile, because the homography-based task function is used to compensate for the image noise, the statistical characteristics of the measurement noise of the task function become difficult to obtain as well. Therefore, the more robust KF-SVSF method [46][47][48] is introduced into the visual servo to improve the system accuracy.
This method combines the optimality of the KF method with the robustness of the SVSF method. As shown in Figure 3, it introduces the concept of an optimal smoothing variable boundary layer, which is an estimation of the impact of modeling uncertainty and noise. The KF-SVSF method seeks a balance between accuracy and robustness by introducing a saturation limit into the calculated optimal boundary layer. By switching between KF gain and SVSF gain, the optimality of KF is maintained within the limit, while the robustness and stability of SVSF remain outside the limited range.
If 1 ψ ψ + > K S , the SVSF gain is switched to provide a robust estimate: where 8 1 ψ × ∈  is the standard boundary layer in SVSF, and this can be adjusted according to the diagonal elements of the calculated smoothing boundary layer matrix 1 ψ + K . Then, we complete the subsequent total Jacobian estimation process: 6. Calculate the a posteriori state error covariance matrix: 7. Obtain the estimated state vector 8. Split the state vector into the estimated Jacobian t J as: Remark 2: When carrying out dynamic tracking tasks, the expanded total Jacobian matrix represents the Jacobian between the gradient of the task function and the velocity of the target and is important for the construction of a dynamic tracking controller.

Controller Design
Several advanced controllers based on the estimated Jacobian can be used to realize a wellbehaved uncalibrated visual servo; for example, a nonlinear controller with a disturbance observer and a disturbance-rejection controller with an extended-state observer [25]  First, the smooth variable boundary layer can be obtained as follows: 1.
Calculate the a priori state space:X 2.
Calculate the a priori state error covariance matrix: 3.
Calculate the residuals vector:Ẑ

4.
Calculate the smoothing boundary layer ψ K+1 : where Q K and R K+1 are the process noise covariance matrix and measurement noise covariance matrix, respectively; E K+1 is the diagonal matrix constructed with E K+1 ; and the calculated optimal boundary layer ψ K+1 of the uncalibrated visual servo system is the diagonal matrix

5.
Compare the calculated boundary layer ψ K+1 with the threshold ψ S , which is based on the specific conditions of the system, and switch the appropriate filter gain: If ψ K+1 ≤ ψ S , the KF gain is switched to provide an optimal estimate: If ψ K+1 > ψ S , the SVSF gain is switched to provide a robust estimate: where ψ ∈ R 8×1 is the standard boundary layer in SVSF, and this can be adjusted according to the diagonal elements of the calculated smoothing boundary layer matrix ψ K+1 .
Then, we complete the subsequent total Jacobian estimation process: 6. Calculate the a posteriori state error covariance matrix: 7. Obtain the estimated state vectorX K+1|K+1 : 8. Split the state vectorX K+1|K+1 into the estimated Jacobian J t as: Remark 2. When carrying out dynamic tracking tasks, the expanded total Jacobian matrix J e = [J t , ∂h/∂t] 8×10 is estimated online, where ∂h/∂t represents the Jacobian between the gradient of the task function and the velocity of the target and is important for the construction of a dynamic tracking controller.

Controller Design
Several advanced controllers based on the estimated Jacobian can be used to realize a well-behaved uncalibrated visual servo; for example, a nonlinear controller with a disturbance observer and a disturbance-rejection controller with an extended-state observer [25]. However, the proportional controller is still the most widely used and developed method in visual servos [22,23,26,29,[31][32][33][34][35][36][37][38]. In addition, its simplicity is beneficial for us to verify this robust strategy.
Before constructing the proportional controller, two problems need to be solved. The first is the disturbance caused by the unstable conditional number of the estimated total Jacobian. Through a large number of simulations, it is found that the estimated Jacobian of the homography-based task function tends to present an unstable conditional number and causes drastic changes of the manipulator velocity in certain directions, resulting in severe vibrations and serious errors. Therefore, the SVF method is introduced to deal with this problem [45]. Singular value decomposition is performed on the estimated total Jacobian J t of the hyper-redundant manipulator according to where σ i is the singular value of the total Jacobian J t and u i and v i are the ith columns of U and V, respectively. The filtering function h σ 0 ,υ (σ) of the singular value σ i proposed in the SVF method can maintain the stability of the smallest singular value of the estimated total Jacobian. It is defined as where σ 0 is the minimum acceptable singular value and υ is a scaling factor that regulates the filtering shape of function h σ 0 ,υ (σ). If σ 0 < υ and υσ 0 < 2, h σ 0 ,υ (σ) can stay monotonic. The pseudo-inverse of the total Jacobian J t can be developed via the SVF method as The restriction on the minimum singular value guarantees the stability of the conditional number of the estimated total Jacobian. Thus, the disturbance caused by the unstable conditional number can be reduced without generating much additional error in the algorithms.
Another problem is that the manipulator joint angles may overrun. If the joint angle is not limited, it may cause a failure of the visual servo or even cause damage to the space manipulator. At this time, the redundancy of the joint space of the hyper-redundant manipulator resulting from the improved homography-based task function can be used to prevent this from happening. The gradient projection based on the redundant Jacobian can be adopted to avoid the joint limit [41,42]. When the joints move away from the limit of joint, allowing the joints to move freely can reduce unnecessary energy consumption. The gradient function is defined as On the basis of these optimizations, the controllers for static positioning and dynamic tracking are designed, respectively, as follows: 1.
Static positioning controller: where (J S ) K is the processed total Jacobian in the kth period and λ is the proportional gain.

2.
Dynamic tracking controller: where (e t ) K = (∂h/∂t)∆t is the estimation of the task function variation in the kth period caused by target movement.

Simulations
This section describes several simulations of our improved strategy. The performances of this strategy in aligning with both static and dynamic assembly objects are compared with several original strategies. Their abilities of avoiding the joint limit and handling an unstable conditional number of estimated total Jacobian are verified as well.

System Description
The following simulation adopted the model of a 9 DOF hyper-redundant manipulator with an eye-in-hand camera. This 9 DOF manipulator was independently designed and manufactured by our laboratory. Figure 4 shows the mechanical structure and schematic of this manipulator, and Table 1 gives the Denavit-Hartenberg parameters of the manipulator.

Simulations
This section describes several simulations of our improved strategy. The performances of this strategy in aligning with both static and dynamic assembly objects are compared with several original strategies. Their abilities of avoiding the joint limit and handling an unstable conditional number of estimated total Jacobian are verified as well.

System Description
The following simulation adopted the model of a 9 DOF hyper-redundant manipulator with an eye-in-hand camera. This 9 DOF manipulator was independently designed and manufactured by our laboratory. Figure 4 shows the mechanical structure and schematic of this manipulator, and Table 1 gives the Denavit-Hartenberg parameters of the manipulator. Table 1. Denavit-Hartenberg parameters of the nine degree-of-freedom (9 DOF) manipulator.
q9 −90° 165.5 0 ±90°   A five-megapixel industrial camera was mounted at the end-effector, and its intrinsic parameter was To simplify the system model, the camera frame F was considered to coincide with the end-effector frame F e .

Simulations and Discussions of Aligning with Static Assembly Objects
To align with static objects, the various errors after the system reaches the steady state are the most reasonable index for comparing the performances of different uncalibrated visual servo strategies. Several tests were performed with different noise levels to check the robustness of the IBUVS, original PHUVS, improved PHUVS with KF method and improved PHUVS with KF-SVSF method.  Table 2 shows the errors of all the above tests. The improved PHUVS with the KF-SVSF method performs better than the IBUVS, original PHUVS and improved PHUVS with the KF. Sometimes, the IBUVS does not converge. Figure 5 shows the detailed test results of the improved PHUVS with the KF-SVSF method for general motion with a noise mean of |V| = 0.5.  SVSF 0.117 0.003 0.906 0.242 0.014 2.235 0.304 0.029 3 SVSF 0.169 0.006 0.798 0.228 0.015 2.482 0.363 0.031 3.396

Simulations and Discussions of Aligning with Dynamic Assembly Objects
The mean square error (MSE) is chosen as an index to compare the performances of different strategies for aligning with dynamic assembly objects. It represents the average image error between the current and desired image features in the servo process, and is defined as where ∆I i is the image error at the ith iteration and N is the number of iterations. The same target is adopted to move in planar and 3D trajectories. Both 2D motion in a double leaf rose path and 3D motion in a triple leaf rose path with different noise levels are simulated. These different paths of the target center are denoted as follows: 450 + 300 cos(3ωN∆T) sin(ωN∆T) 800 + 300 cos(ωN∆T) sin(3ωN∆T) 820 450 + 250 sin(3ωN∆T) sin(ωN∆T) 800 + 300 cos(ωN∆T) sin(3ωN∆T) 820 + vN∆T where ω = π/30 rad·s −1 is the angular velocity in the X − Y plane, v = 2.5 mm·s −1 is the translation velocity along the Z axis, ∆T = 0.02 s is the sampling period and N is the number of iterations. The threshold of the KF-SVSF boundary layer was defined as ψ S = diag 128 42 1360 42 128 1360 16 16 Table 3 gives the MSEs of all the above tests. All of these MSEs are the average of 100 experiments. Figures 6 and 7 show the image errors of the different strategies for the double leaf rose and triple leaf rose paths with a noise mean of |V| = 0.5, and Figures 8 and 9 show the camera trajectories of the different strategies. The IBUVS is the least robust method to image noise, and its tracking precision is unacceptable. The improved PHUVS, which directly controls manipulator joints, has better performance than the original PHUVS, as shown in Figure 2a. Meanwhile, the improved PHUVS with the KF-SVSF method shows better tracking precision than the improved PHUVS with the KF. The abilities of the strategies to avoid the joint limit and handle an unstable conditional number are examined as well. The joint limit was set as 90 ±  , and the minimum acceptable singular value of the total Jacobian was set as . Figure 10 shows the unlimited and limited joint angles for these two trajectories. Figure 11 shows the logarithm of the minimum singular value min        The abilities of the strategies to avoid the joint limit and handle an unstable conditional number are examined as well. The joint limit was set as ±90 • , and the minimum acceptable singular value of the total Jacobian was set as σ 0 = 10 −4 . Figure 10 shows the unlimited and limited joint angles for these two trajectories. Figure 11 shows the logarithm of the minimum singular value σ min of the estimated total Jacobian (J t ) K and processed Jacobian (J S ) K in each period. In each tracking trajectory, all joint angles of the super-redundant manipulator are limited to the acceptable range. When unstable condition numbers are caused by the appearance of excessively small minimum singular values, the minimum singular value σ min of the processed Jacobian remains above the threshold σ 0 = 10 −4 .

Evaluation of Real-Time Performance
Several experiments of systems with different homography-based task functions are carried out. The time of consumed (TOC) is the most appropriate indicator to measure real-time performance. The TOC-H which is the time consumed on the homography matrix estimation, and the TOC-C, which includes the time consumed for Jacobian estimation and control output calculation, are selected as indicators to compare the real-time performance of the system. A PC with an Intel(R) Xeon(R) E5-1620 CPU at 3.5 GHz, 64-b operating system was chosen to run all experiments. The average values of TOC-H and TOC-C in each experiment with more than 10,000 control periods are shown in Table 2. The experiment in the article is only presented to compare the real-time performance of the two strategies. In practical space applications, this part of the function may be implemented using FPGA. Corresponding designs and improvements will be made to meet computational efficiency requirements.
As shown in Table 4 and Figure 12, the TOC-H and TOC-C of the improved homography-based task function are always lower than the original homography-based task function with different numbers of feature points. According to the comparison of the results, the improved homography-based task functions with smaller dimensions show more convincing real-time performance, as expected. Inaccurate trajectory caused by an unstable conditional number in a triple leaf rose path. (d) Logarithm of the minimum singular value of the total Jacobian in a triple leaf rose path.

Evaluation of Real-Time Performance
Several experiments of systems with different homography-based task functions are carried out. The time of consumed (TOC) is the most appropriate indicator to measure real-time performance. The TOC-H which is the time consumed on the homography matrix estimation, and the TOC-C, which includes the time consumed for Jacobian estimation and control output calculation, are selected as indicators to compare the real-time performance of the system. A PC with an Intel(R) Xeon(R) E5-1620 CPU at 3.5 GHz, 64-b operating system was chosen to run all experiments. The average values of TOC-H and TOC-C in each experiment with more than 10,000 control periods are shown in Table 2. The experiment in the article is only presented to compare the real-time performance of the two strategies. In practical space applications, this part of the function may be implemented using FPGA. Corresponding designs and improvements will be made to meet computational efficiency requirements.
As shown in Table 4 and Figure 12, the TOC-H and TOC-C of the improved homography-based task function are always lower than the original homography-based task function with different numbers of feature points. According to the comparison of the results, the improved homographybased task functions with smaller dimensions show more convincing real-time performance, as expected.

Evaluation of System Performance
As shown in Figure 13, a 9 DOF hyper-redundant manipulator, 6 DOF universal robot, smart camera and NI real-time system were used to build the experimental platforms. The parameters of the manipulator and the camera are consistent with the parameters set in the simulations. Experiments for aligning with both static and dynamic assembly objects were conducted to verify the improvement of our strategy. A square target was chosen with a side length of 200 mm and 49 feature points. All experiments were performed under an illumination similar to that of actual working conditions to examine the reliability of our strategy.

Evaluation of System Performance
As shown in Figure 13, a 9 DOF hyper-redundant manipulator, 6 DOF universal robot, smart camera and NI real-time system were used to build the experimental platforms. The parameters of the manipulator and the camera are consistent with the parameters set in the simulations. Experiments for aligning with both static and dynamic assembly objects were conducted to verify the improvement of our strategy. A square target was chosen with a side length of 200 mm and 49 feature points.
All experiments were performed under an illumination similar to that of actual working conditions to examine the reliability of our strategy.  Figure 13. Experiment platform and target.

Static Positioning Experiment
Considering that, in the actual work scene, manipulators often perform general movements to align with assembly targets, the initial pose error of the positioning experiment was set as  Table 5 gives the various steady-state errors of different strategies in positioning experiments, and these results are the average of five repeated experiments. Although the noise of image transmission and processing has a certain impact on the accuracy of target recognition, the improved PHUVS with the KF-SVSF method clearly performs better than other strategies in each comparison. This verifies the superiority of our improved strategy. Figure 14 is the change curve of the proposed improved PHUVS in terms of the image error and task function error. Figure 15 shows the angular velocity of each joint of the manipulator.

Static Positioning Experiment
Considering that, in the actual work scene, manipulators often perform general movements to align with assembly targets, the initial pose error of the positioning experiment was set as  Table 5 gives the various steady-state errors of different strategies in positioning experiments, and these results are the average of five repeated experiments. Although the noise of image transmission and processing has a certain impact on the accuracy of target recognition, the improved PHUVS with the KF-SVSF method clearly performs better than other strategies in each comparison. This verifies the superiority of our improved strategy. Figure 14 is the change curve of the proposed improved PHUVS in terms of the image error and task function error. Figure 15 shows the angular velocity of each joint of the manipulator. Table 5. Steady-state error of different strategies in the static positioning experiment. ∆χ is in millimeters, ∆θ is in degrees and ∆I is in pixels.

Dynamic Tracking Experiment
In the dynamic tracking experiment, the target was fixed to the end-effector of a universal robot and taken to move along different paths. To ensure reliability, both complex planar and 3-D paths were tested. The planar and 3D paths of the target were set as ω ω ω ω ω ω ω ω

Dynamic Tracking Experiment
In the dynamic tracking experiment, the target was fixed to the end-effector of a universal robot and taken to move along different paths. To ensure reliability, both complex planar and 3-D paths were tested. The planar and 3D paths of the target were set as where ω = π/6 rad·s −1 is the angular velocity in the X − Y plane, v = 1.25 mm·s −1 is the translation velocity in the Z direction, ∆T = 0.04 s is the sampling period and K is the number of iterations. The threshold of the KF-SVSF boundary layer was defined as ψ S = diag 128 42 1360 42 128 1360 16 16 Table 6 gives the mean square error (MSE) of different strategies tracking two different paths, and these results are the average of five repeated experiments. Figures 16-19 show the image errors of the different strategies, and the joint angle and joint angular velocity of the proposed improved PUHVS with the KF-SVSF method, in the double leaf rose path and the triple leaf rose paths respectively. Obviously, the improved PHUVS with the KF-SVSF method shows better accuracy and stability than other strategies. During the tracking process, the tracking error is always kept in an acceptable range. It is worth mentioning that, due to the adoption of the SVF method, the system smoothly completes the servo process without severe disturbance. It can also be seen that, because the redundancy resulting from the improved homography-based task function is used for the design of the controller to limit the joint angle, there was no case in which the joint angle exceeded the limit, and the entire experiment was successfully completed. leaf rose paths respectively. Obviously, the improved PHUVS with the KF-SVSF method shows better accuracy and stability than other strategies. During the tracking process, the tracking error is always kept in an acceptable range. It is worth mentioning that, due to the adoption of the SVF method, the system smoothly completes the servo process without severe disturbance. It can also be seen that, because the redundancy resulting from the improved homography-based task function is used for the design of the controller to limit the joint angle, there was no case in which the joint angle exceeded the limit, and the entire experiment was successfully completed.  It is worth noting that, in our experiment, we calculated the corresponding joint increment command ∆q = . q · ∆T through the sampling period ∆T instead of continuous joint velocities commands . q to send to the robot. During the execution of each sampling period, joint velocities were automatically planned when moving the increment ∆q. As shown in Figures 17a and 19a, the joint angle was still relatively smooth. However, in Figures 5f, 15, 17b and 19b, high-frequency oscillations were present in joint velocities caused by various noises. If the manipulator were controlled in a speed mode in actual tasks, this might cause vibrations and it might be necessary to add a filter to cut high frequencies. This would add a delay and possibly make this method unsuitable for real-time operations in space. Researchers have noticed this problem and found that PHUVS experiences a much smaller variation amplitude in joint velocities than IBUVS [23]. However, this level of improvement is not sufficient to completely solve the problem, and this is still a shortcoming that needs to be overcome in future applications. It is worth noting that, in our experiment, we calculated the corresponding joint increment command Δ = ⋅Δ  q q T through the sampling period ΔT instead of continuous joint velocities commands  q to send to the robot. During the execution of each sampling period, joint velocities were automatically planned when moving the increment Δ q . As shown in Figures 17a and 19a, the joint angle was still relatively smooth. However, in Figures 5f, 15, 17b and 19b, high-frequency oscillations were present in joint velocities caused by various noises. If the manipulator were controlled in a speed mode in actual tasks, this might cause vibrations and it might be necessary to add a filter to cut high frequencies. This would add a delay and possibly make this method unsuitable for real-time operations in space. Researchers have noticed this problem and found that PHUVS experiences a much smaller variation amplitude in joint velocities than IBUVS [23]. However, this level of improvement is not sufficient to completely solve the problem, and this is still a shortcoming that needs to be overcome in future applications.

Conclusions
This paper proposed an improved uncalibrated visual servo strategy for hyper-redundant manipulators in an on-orbit automatic assembly. An improved homography-based task function with robustness to image noise and better real-time performance was constructed, which made the joint space of the hyper-redundant manipulator redundant to the task function. To improve the system accuracy and avoid the interference caused by the change of manipulator parameters in space, a proportional controller based on the estimated total Jacobian was designed to control the manipulator joints directly without a manipulator Jacobian. The KF-SVSF method, which combines the optimality of the KF method with the robustness of the SVSF method, was introduced to perform the online estimation of the total Jacobian and further reduced the impact of modeling uncertainty and image noise. The SVF method was adopted to reduce the disturbance caused by an unstable conditional number of the estimated total Jacobian without generating much additional error in the algorithms. In addition, the redundancy of the joint space relative to the new task function was exploited to limit the joint angle. Several tasks for aligning with static or dynamic objects were simulated to test the performance of this strategy. The simulation results demonstrated the improved accuracy and robustness of this strategy. The abilities of the system to limit the joint angle and deal with an unstable conditional number were verified as well. The reliability of this strategy was further validated by experiments based on actual working conditions.