# Error Modeling in Distance and Rotation for Self-Calibration of Space Robots on Orbit

Authors: Qingxuan Jia; Shiwei Wang; Gang Chen; Hanxu Sun
Journal: International Journal of Aerospace Engineering
 (2019)

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## Abstract

Vibration and impact of launching, inner and outer pressure difference, and thermal deformation of the space capsule will change the transformation between the pose measurement system and the space robot base. It will be complicated, even hard, to measure and calculate this transformation accurately. Therefore, an error modeling method considering both the distance error and the rotation error of the end-effector is proposed for self-calibration of the space robot on orbit, in order to avoid the drawback of frame transformation. Moreover, according to linear correlation of the columns of the identification matrix, unrecognizable parameters in the distance and rotation error model are removed to eliminate singularity in robot kinematic calibration. Finally simulation tests on a 7-DOF space robot are conducted to verify the effectiveness of the proposed method.


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## Body

## 1. Introduction

Space robots can assist astronauts to reach and expand their maintenance work areas, improving operational efficiency and safety [1], and even can complete on-orbit missions such as spacecraft rendezvous and docking, satellite fault maintenance, and satellite capture, independently [2]. Some of these precise space missions require the space robot to have high end positioning accuracy. Nevertheless, defects in manufacturing and assembly lead to the difference between the actual kinematic parameters and the nominal ones, generally regarded as systematic errors. Additionally, the end positioning accuracy is also affected by random errors, such as environmental changes, gear transmission, and mechanical deformation. Calibration on the ground can remedy the positioning deficiencies caused by these inherent kinematic errors [3, 4]. However, contrast to traditional industrial robots, space robots are subjected to strong vibration and impact with the launch of spacecraft and then confronted with extreme temperature on orbit. These factors will inevitably cause the kinematic parameters of the space robot to change, resulting in a decrease in the end positioning accuracy. Therefore, it is necessary to perform on-orbit kinematic calibration [5].The actual pose of the space robot end-effector can hardly be measured by an external measuring device due to the extreme orbital environment, so the internal sensing system mounted on its end-effector is adopted for measurement during self-calibration. Many researchers have devoted efforts to kinematic self-calibration of robot manipulators. Angulo and Torras [6] developed a neural-network method to recalibrate automatically a commercial robot manipulator after undergoing wear or damage, which has been applied in the REIS robot included in the space station mock-up at Daimler-Benz Aerospace. Liang et al. [7] developed an adaptive self-calibration of hand-eye systems in which a visual-feedback-based self-learning process is used for dynamically and continuously learning the hand-eye transformation through repetitive operation trials. Liu et al. [8] proposed a self-calibration method based on hand-eye vision, which establishes the relative pose error model of the space robot and uses the particle swarm optimization algorithm to identify the kinematic parameters. Yin et al. [9] proposed a vision-based robot self-calibration method, eliminating the need for a robot-based frame and hand-to-eye calibrations. Du et al. [10–13] introduced an inertial measurement unit to estimate the end posture and attaching a position marker to the end-effector to measure the actual position and identified kinematic parameters with different filters to overcome the impact of sensing noise. Especially, for the sake of more accurate and reliable estimation from the sensors, various filter tools such as Kalman filter and particle filter are used in the estimation process and the position estimation always combines with the orientation estimation [14–16]. Zhang et al. [17] realized kinematic calibration based on the local exponential product formula by measuring the end position of the robot with a fixed camera and the plane mark mounted on the end-effector.Works on self-calibration above adopted the absolute pose/position error model for kinematic calibration. They have to describe the end-effector pose errors under the robot base frame, making it inevitable to identify the transformation matrix between the measurement system frame and the robot base frame before calibration. However, this transformation matrix is very complicated to measure and calculate accurately, even hardly possible to obtain in unmanned environments such as on orbit [18]. To avoid the drawback of frame transformation, the distance error of any two positions in robot workspace is applied to calibrate the robot position accuracy indirectly [19, 20]. Roning and Korzun [21] used the criteria of equal distances between the points in the robot space and the task space to perform calibration on the GM Fanuc S-10 robot. Gong et al. [22] used a hybrid noncontact optical sensor mounted on the end-effector, calibrating the 6 degree-of-freedom (DOF) robot based on distance error. Tan et al. [23] made use of the screw theory and the distance error model, considering the initial orientation errors. Gao et al. [24] obtained the linearized equation describing the relationship between the positioning errors and the kinematic errors by differentiating the kinematic equation. Zhang et al. [25] derived a linear model from link parameter errors to squared range difference of the robot end-effector. Zhang et al. [26] proposed a method to directly establish parameter error equations based on relative distance error and identified the parameter errors by employing a hybrid genetic algorithm. Mu et al. [27] synthesized the hand-eye transformation parameters and the robot kinematic parameters during calibration of the system parameters of a flexible measurement system based on spheres’ centre-to-centre distance errors. Shi et al. [28] established the distance error model to connect the robot distance error and the absolute positioning error and validated the error model on a handling robot. Li et al. [29] used two error models, including the position error model and the distance error model, for calibrating Selective Compliance Assembly Robot Arm (SCARA) robots. Yao et al. [30] measured the distance between two points in space to conduct kinematic calibration on a service robot. As for measurement of the distance errors, a laser tracker was always used to measure these errors [31–33], while a CMM (coordinate measuring machine) could be also employed [34]. Joubair and Bonev [35] developed a kinematic calibration method to improve the accuracy of a six-axis serial industrial robot in a specific target workspace, using distance error and sphere constraints. Works above obtained the distance error model by ignoring the linearizing error, however with a lack of the rotation error of the robot end-effector, so as to degrade calibration performance to a certain extent.In this paper, we propose an error modeling method considering both the distance and the rotation error of the end-effector of the space robot. The remainder of this paper is organized as follows. In Section2, the kinematic self-calibration system for a 7-DOF space robot is elaborated. In Section 3, based on the absolute pose error model, we build the mappings from the kinematic errors to the rotation error of the robot end-effector, obtaining the distance and rotation error model. Then, the redundant parameters in this error model are analysed theoretically in Section 4. At last, Sections 5 and 6 give the results of our experiments and conclude this paper, respectively.


## 2. Kinematic Self-Calibration System

Different from kinematic calibration of robot manipulators on the ground, the space robot has large structural size and extreme working environment, making it impossible to measure the end pose of the space robot using external measuring equipment, so its own hand-eye vision system [36–38] with a checkerboard calibration plate is adopted for pose measurement. As shown in Figure 1, a 7-DOF space robot is mounted on the outside of the space capsule, with a hand-eye camera attached to its end. A checkerboard calibration plate is placed on the outside of the space capsule, away from the space robot base, as a target for the hand-eye camera.Figure 1
Kinematic self-calibration system. The space robot and the checkerboard calibration plate are mounted on the outside of the space capsule, and a hand-eye camera is adopted to measure the pose of the checkerboard.As illustrated in Figure2, b denotes the base frame of the space robot, and i,i=1,2,…,7 denotes the reference frame attached to each joint, respectively. t is the end-effector reference frame, which coincides exactly with the reference frame 7. c is the hand-eye camera frame, namely, the measurement system frame, and its transformation matrix with respect to the base frame b is assumed to be Tcb. Therefore, once the poses of the checkerboard in the hand-eye camera frame c are measured, the transformation matrix of the end-effector frame t with respect to the hand-eye camera frame c is obtained, denoted as Ttc. Further, the transformation matrix from the base frame b to the end-effector frame t is calculated as
(1)Ttb=Tcb⋅TtcFigure 2
Skeleton of the coordinate frames for self-calibration. The coordinate frames of the robot base, the checkboard, and all the joints are defined, while the related lengths are marked.However, vibration and impact of launching, inner and outer pressure difference, and thermal deformation of the space capsule will change the pose of the checkerboard with respect to the robot base frame, making it complicated to measure and calculate the transformation matrixTcb between the camera frame and the robot base frame accurately. Therefore, the relative errors including distance and rotation errors of the robot end-effector are analysed.


## 3. Error Modeling in Distance and Rotation

### 3.1. Distance Error Modeling

As the basis of error modeling, kinematic modeling is aimed at describing the relation between any two adjoining link coordinate frames with as few parameters as possible. But inappropriate kinematic parameters might not meet three fundamental principles, namely, completeness, model continuity, and minimality [39]. Various methods of kinematic modeling for robot manipulators have been proposed; e.g., the classic DH method [40] uses only 4 parameters to describing the relation between any two adjoining link coordinate frames, while the MDH (modified DH) method [41] uses 5 parameters. There are more such as the CPC (complete and parametrically continuous) method [42] and zero reference model method [43].Without loss of generality, we assume that the number of kinematic parameters used to describe the relation between any two adjoining link coordinate frames isp, and the DOF of the space robot is n, then all the kinematic parameters of the space robot can be recorded as Ω∈Rnp×1, while one robot configuration is denoted as Θ∈Rn.Definition 1.
Kinematic model: the kinematic model of the space robot relates the configurationΘ and the kinematic parameters Ω to the end-effector pose Ψ∈R6×1 through a function f as
(2)Ψ=fΩ,ΘThe robot configurations are not exactly known due to the existence of sensor noise. Similarly, defects in manufacturing and assembly result in kinematic errors. The difference between the nominal valueΩ and the exact one Ω∗ is assumed to be δΩ, such as Ω∗=Ω+δΩ. It is noteworthy that errors in Θ can be also included in kinematic errors δΩ.Proposition 1.
Absolute pose error model [39]: the differential error of the end-effector pose is the product of the generalized Jacobian matrix and the kinematic errors, such as
(3)δΨ=JΩ,ΘδΩ,where JΩ,Θ∈R6×np is the generalized Jacobian matrix, also termed as the identification matrix, determined by the kinematic parameters Ω and the configuration Θ. δΨ∈R6×1 denotes the differential error of the end-effector pose, composed of 3 differential translations along the x-y-z axis and 3 differential rotations around the x-y-z axis, respectively.In Figure3, as the robot configuration changes from ΘA to ΘB, AB⇀ and A′B′⇀ are the corresponding nominal and actual displacements, respectively, whereas AA′⇀ and BB′⇀ are the end-effector position offset displacements under these two configurations. According to the absolute error model in Equation (3),
(4)AA′⇀=JdΩ,ΘAδΩ,BB′⇀=JdΩ,ΘBδΩ,where Jd∈R3×np is the position-related part of the identification matrix.Proposition 2.
Distance error model [20]: as the robot configuration changes from ΘA to ΘB, the measurable distance error scalar is the product of the identification matrix and the kinematic errors, such as
(5)ΔlBA=AB⇀AB⇀⋅JdΩ,ΘB−JdΩ,ΘA⋅δΩ,where ΔlBA=A′B′⇀−AB⇀ denotes the measurable distance error scalar. AB⇀ and A′B′⇀ are the norms of the vectors AB⇀ and A′B′⇀, respectively, the former of which is calculated by nominal forward kinematics, while the latter can be measured by the hand-eye camera system. Both the values have nothing to do with their reference frames. The identification matrix of distance error model is obtained as AB⇀/AB⇀⋅JdΩ,ΘB−JdΩ,ΘA.Figure 3
Schematic of the relative pose error model. The robot moves from the configurationΘA to ΘB, and its nominal end-effector pose changes from the frame A to B.


### 3.2. Propositions of Differential Rotation

Definition 2.
General rotation transformation: suppose thatK=kx,ky,kz is a unit vector over origin, the rotation matrix around K with the angle θ can be obtained as
(6)RK,θ=kxkxversθ+cθkykxversθ−kzsθkzkxversθ+kysθkxkyversθ+kzsθkykyversθ+cθkxkyversθ−kzsθkxkzversθ−kysθkykzversθ+kxsθkzkzversθ+cθ,where sθ=sinθ, cθ=cosθ, and versθ=1−cosθ.Proposition 3.
Equivalent angle and axis of rotation [40]: any rotation matrix R can be expressed as a rotation around a certain axis K with a certain angle θ. The axis K is termed as equivalent axis of rotation while θ is the equivalent angle of rotation.It should be noted that the equivalent angle and axis of rotation have the following three important properties.(a)
For a certain rotation matrixR, it may have more than one set of equivalent angle and axis of rotation. Actually, K,θ is equivalent to −K,−θ and even equivalent to K,θ+2nπ, n∈Z. Therefore, the value of θ is always forced to locate in 0,π.
(b)
Whenθ is small, the rotation axis K will be hard to be computed because of singularity.
(c)
K⋅RT=K, where RT denotes the transpose of R. The proof is simple and omitted.Proposition 4.
Differential rotation by equivalent angle and axis of rotation: any differential rotation transformationΔR can always be regarded as a differential rotation around a certain axis K=kx,ky,kz with the rotation angle δθ, such as
(7)ΔRK,δθ=1−kzδθkyδθkzδθ1−kxδθ−kyδθkxδθ1.Proof.
δθ is so small that sinδθ=δθ, cosδθ=1, and versδθ=0. Then, Equation (7) can be obtained by substituting K and δθ into Equation (6).Proposition 5.
Differential rotation by 3-dimensional differential rotation angles [44]: any differential rotation transformation ΔR can always be regarded as differential rotation around the axes x, y, and z in turn. Suppose that the 3-dimensional differential rotation angles are ξ=ξx,ξy,ξzT, then
(8)ΔRξ=skewξ+I3,where I3 denotes a 3×3 identity matrix, and the function skew is used to create a skew symmetric matrix, which is defined as
(9)skewv=0−v3v2v30−v1−v2v10.Proposition 6.
If the equivalent angle and axis of rotation of a certain differential rotation matrixΔR is δθ and K=kx,ky,kz, and its 3-dimensional differential rotation angles are ξ=ξx,ξy,ξzT, then
(10)δθ=K⋅ξ.Proof.
It can be obtained by Equations (7) and (8) as
(11)ξx=kxδθ,ξy=kyδθ,ξz=kzδθ.Considering thatK is a unit vector, then
(12)kx2+ky2+kz2=1.Thus, Equation (10) can be obtained by
(13)δθ=kx2+ky2+kz2δθ=kxkxδθ+kykyδθ+kzkzδθ=kxξx+kyξy+kzξz=K⋅ξ.Proposition 7.
Differential rotation by the rotation matrix [44]: if the rotation matrix R′ is the rotation matrix R transformed by a differential rotation ΔR, then
(14)ΔR=R′⋅RT−I3.It should be noted thatΔR is calculated with respect to the reference frame of R. And the differential rotation with respect to the frame of R, termed as ΔRT, can be obtained as
(15)ΔRT=RT⋅R′−I3.The two kinds of differential rotation matrix satisfy(16)ΔR⋅R=R⋅ΔRT.


### 3.3. Rotation Error Modeling

We can avoid identifying the transformation matrix between the measurement system frame and the robot base frame by using the distance error. This is true for equivalent angle of rotation to describe the variation of the end-effector orientation.As shown in Figure3, the matrixes RBA, RB′A′, RA′A, and RB′B denote the rotation between any two frames of A, B, A′, and B′. Since kinematic errors are small, RBA and RB′A′ are very close to each other. In other words, RBAT⋅RB′A′ meets the definition of differential rotation.Proposition 8.
Suppose thatξA, ξB, and ξR represent 3-dimensional differential rotation angles of RA′A, RB′B, and RBAT⋅RB′A′, respectively, then
(17)ξR=ξB−RBATξA.Proof.
According to Proposition7, we can obtain the differential rotation matrix as
(18)ΔAT=RA0T.RA′0−I3,ΔBT=RB0T.RB′0−I3.Then, the differential rotation matrix betweenRBA and RB′A′ can be calculated as
(19)ΔRT=RBAT⋅RB′A′−I3=RBAT⋅RAA′⋅RBA⋅RB′B−I3=RBAT⋅ΔAT+I3T⋅RBA⋅ΔBT+I3−I3=RBAT⋅ΔATT+I3⋅RBA⋅ΔBT+I3−I3=RBAT⋅ΔATT⋅RBA⋅ΔBT+RBAT⋅ΔATT⋅RBA+ΔBT.Equation (17) can be obtained by substituting ξA, ξB, and ξR into Equation (19) and then simplifying it.Suppose that the equivalent angle and axis of rotation corresponding toRBA are θBA and KBA, while those of RB′A′ are θB′A′ and KB′A′. Both KBA and KB′A′ are 1×3 vectors.According to Equation (3), we can obtain
(20)ξA=JrΩ,ΘAδΩ,ξB=JrΩ,ΘBδΩ,where Jr∈R3×np is the orientation-related part of the identification matrix. It should be noted that the identification matrix in Equation (20) is calculated with respect to the end-effector reference frame.Proposition 9.
Rotation error model: as the robot configuration changes fromΘA to ΘB, the measurable rotation error scalar is the product of the identification matrix and the kinematic errors, such as
(21)Δθr=KBA⋅JrΩ,ΘB−JrΩ,ΘA⋅δΩ,where Δθr=θB′A′−θBA is the measurable rotation error scalar, and the identification matrix of the rotation error model is KBA⋅JrΩ,ΘB−JrΩ,ΘA.Proof.
Substituting Equation (20) into Equation (17), we can obtain
(22)ξR=JrΩ,ΘB−RBAT⋅JrΩ,ΘA⋅δΩ.Similar to the derivation of distance error, we make the rotational axisKBA coincide with KB′A′ and the starting point of rotation matrix RBA coincide with that of RB′A′ as shown in Figure 4, where the point H is the projection of B′ onto the flat AOB, while the point G satisfies ∠AOG=∠A′OB′. The measurable equivalent rotation error can be obtained as
(23)Δθr=θB′A′−θBA=∠A′OB′−∠AOB=∠AOG−∠AOB=∠GOB.Figure 4
Derivation of the rotation error model. The rotational axisKBA is forced to coincide with KB′A′, and the starting points of rotation matrix RBA and RB′A′ are also forced to coincide.∠HOB is the projection of ∠BOB′ onto the flat AOB, which is actually the equivalent angle of rotation δθr of differential rotation ΔRT around the axis KBA, such as ∠HOB=δθr. Ignoring the linearizing error, we can obtain ∠GOB≈∠HOB, and then
(24)Δθr=δθrAccording to Proposition6, substitute Equation (22) into (24), then
(25)Δθr=δθr=KBA⋅ξR=KBA⋅JrΩ,ΘB−RBAT⋅JrΩ,ΘA⋅δΩ.According to the third property of the equivalent angle and axis of rotation,(26)KBA⋅RBAT=KBA.Finally, the rotation error model in Equation (21) can be obtained by substituting Equation (26) into (25).


### 3.4. Distance and Rotation Error Modeling

In summary, the distance and rotation error model is obtained by Equations (5) and (21) as
(27)ΔlBA=AB⇀AB⇀⋅JdΩ,ΘB−JdΩ,ΘA⋅δΩ,Δθr=KBA⋅JrΩ,ΘB−JrΩ,ΘA⋅δΩ.So we can obtain the identification matrix of the distance and rotation error model as(28)Jdr=AB⇀AB⇀⋅JdΩ,ΘB−JdΩ,ΘAKBA⋅JrΩ,ΘB−JrΩ,ΘA,Jdr∈R2×np.When there are redundant parameters in the error model, the identification matrix is rank defect, and measurement noises will affect the accuracy and robustness of parameter identification seriously, inferring the necessity of removal of redundant parameters. Redundant parameters in the error model will be discussed in the next section.


## 3.1. Distance Error Modeling

As the basis of error modeling, kinematic modeling is aimed at describing the relation between any two adjoining link coordinate frames with as few parameters as possible. But inappropriate kinematic parameters might not meet three fundamental principles, namely, completeness, model continuity, and minimality [39]. Various methods of kinematic modeling for robot manipulators have been proposed; e.g., the classic DH method [40] uses only 4 parameters to describing the relation between any two adjoining link coordinate frames, while the MDH (modified DH) method [41] uses 5 parameters. There are more such as the CPC (complete and parametrically continuous) method [42] and zero reference model method [43].Without loss of generality, we assume that the number of kinematic parameters used to describe the relation between any two adjoining link coordinate frames isp, and the DOF of the space robot is n, then all the kinematic parameters of the space robot can be recorded as Ω∈Rnp×1, while one robot configuration is denoted as Θ∈Rn.Definition 1.
Kinematic model: the kinematic model of the space robot relates the configurationΘ and the kinematic parameters Ω to the end-effector pose Ψ∈R6×1 through a function f as
(2)Ψ=fΩ,ΘThe robot configurations are not exactly known due to the existence of sensor noise. Similarly, defects in manufacturing and assembly result in kinematic errors. The difference between the nominal valueΩ and the exact one Ω∗ is assumed to be δΩ, such as Ω∗=Ω+δΩ. It is noteworthy that errors in Θ can be also included in kinematic errors δΩ.Proposition 1.
Absolute pose error model [39]: the differential error of the end-effector pose is the product of the generalized Jacobian matrix and the kinematic errors, such as
(3)δΨ=JΩ,ΘδΩ,where JΩ,Θ∈R6×np is the generalized Jacobian matrix, also termed as the identification matrix, determined by the kinematic parameters Ω and the configuration Θ. δΨ∈R6×1 denotes the differential error of the end-effector pose, composed of 3 differential translations along the x-y-z axis and 3 differential rotations around the x-y-z axis, respectively.In Figure3, as the robot configuration changes from ΘA to ΘB, AB⇀ and A′B′⇀ are the corresponding nominal and actual displacements, respectively, whereas AA′⇀ and BB′⇀ are the end-effector position offset displacements under these two configurations. According to the absolute error model in Equation (3),
(4)AA′⇀=JdΩ,ΘAδΩ,BB′⇀=JdΩ,ΘBδΩ,where Jd∈R3×np is the position-related part of the identification matrix.Proposition 2.
Distance error model [20]: as the robot configuration changes from ΘA to ΘB, the measurable distance error scalar is the product of the identification matrix and the kinematic errors, such as
(5)ΔlBA=AB⇀AB⇀⋅JdΩ,ΘB−JdΩ,ΘA⋅δΩ,where ΔlBA=A′B′⇀−AB⇀ denotes the measurable distance error scalar. AB⇀ and A′B′⇀ are the norms of the vectors AB⇀ and A′B′⇀, respectively, the former of which is calculated by nominal forward kinematics, while the latter can be measured by the hand-eye camera system. Both the values have nothing to do with their reference frames. The identification matrix of distance error model is obtained as AB⇀/AB⇀⋅JdΩ,ΘB−JdΩ,ΘA.Figure 3
Schematic of the relative pose error model. The robot moves from the configurationΘA to ΘB, and its nominal end-effector pose changes from the frame A to B.


## 3.2. Propositions of Differential Rotation

Definition 2.
General rotation transformation: suppose thatK=kx,ky,kz is a unit vector over origin, the rotation matrix around K with the angle θ can be obtained as
(6)RK,θ=kxkxversθ+cθkykxversθ−kzsθkzkxversθ+kysθkxkyversθ+kzsθkykyversθ+cθkxkyversθ−kzsθkxkzversθ−kysθkykzversθ+kxsθkzkzversθ+cθ,where sθ=sinθ, cθ=cosθ, and versθ=1−cosθ.Proposition 3.
Equivalent angle and axis of rotation [40]: any rotation matrix R can be expressed as a rotation around a certain axis K with a certain angle θ. The axis K is termed as equivalent axis of rotation while θ is the equivalent angle of rotation.It should be noted that the equivalent angle and axis of rotation have the following three important properties.(a)
For a certain rotation matrixR, it may have more than one set of equivalent angle and axis of rotation. Actually, K,θ is equivalent to −K,−θ and even equivalent to K,θ+2nπ, n∈Z. Therefore, the value of θ is always forced to locate in 0,π.
(b)
Whenθ is small, the rotation axis K will be hard to be computed because of singularity.
(c)
K⋅RT=K, where RT denotes the transpose of R. The proof is simple and omitted.Proposition 4.
Differential rotation by equivalent angle and axis of rotation: any differential rotation transformationΔR can always be regarded as a differential rotation around a certain axis K=kx,ky,kz with the rotation angle δθ, such as
(7)ΔRK,δθ=1−kzδθkyδθkzδθ1−kxδθ−kyδθkxδθ1.Proof.
δθ is so small that sinδθ=δθ, cosδθ=1, and versδθ=0. Then, Equation (7) can be obtained by substituting K and δθ into Equation (6).Proposition 5.
Differential rotation by 3-dimensional differential rotation angles [44]: any differential rotation transformation ΔR can always be regarded as differential rotation around the axes x, y, and z in turn. Suppose that the 3-dimensional differential rotation angles are ξ=ξx,ξy,ξzT, then
(8)ΔRξ=skewξ+I3,where I3 denotes a 3×3 identity matrix, and the function skew is used to create a skew symmetric matrix, which is defined as
(9)skewv=0−v3v2v30−v1−v2v10.Proposition 6.
If the equivalent angle and axis of rotation of a certain differential rotation matrixΔR is δθ and K=kx,ky,kz, and its 3-dimensional differential rotation angles are ξ=ξx,ξy,ξzT, then
(10)δθ=K⋅ξ.Proof.
It can be obtained by Equations (7) and (8) as
(11)ξx=kxδθ,ξy=kyδθ,ξz=kzδθ.Considering thatK is a unit vector, then
(12)kx2+ky2+kz2=1.Thus, Equation (10) can be obtained by
(13)δθ=kx2+ky2+kz2δθ=kxkxδθ+kykyδθ+kzkzδθ=kxξx+kyξy+kzξz=K⋅ξ.Proposition 7.
Differential rotation by the rotation matrix [44]: if the rotation matrix R′ is the rotation matrix R transformed by a differential rotation ΔR, then
(14)ΔR=R′⋅RT−I3.It should be noted thatΔR is calculated with respect to the reference frame of R. And the differential rotation with respect to the frame of R, termed as ΔRT, can be obtained as
(15)ΔRT=RT⋅R′−I3.The two kinds of differential rotation matrix satisfy(16)ΔR⋅R=R⋅ΔRT.


## 3.3. Rotation Error Modeling

We can avoid identifying the transformation matrix between the measurement system frame and the robot base frame by using the distance error. This is true for equivalent angle of rotation to describe the variation of the end-effector orientation.As shown in Figure3, the matrixes RBA, RB′A′, RA′A, and RB′B denote the rotation between any two frames of A, B, A′, and B′. Since kinematic errors are small, RBA and RB′A′ are very close to each other. In other words, RBAT⋅RB′A′ meets the definition of differential rotation.Proposition 8.
Suppose thatξA, ξB, and ξR represent 3-dimensional differential rotation angles of RA′A, RB′B, and RBAT⋅RB′A′, respectively, then
(17)ξR=ξB−RBATξA.Proof.
According to Proposition7, we can obtain the differential rotation matrix as
(18)ΔAT=RA0T.RA′0−I3,ΔBT=RB0T.RB′0−I3.Then, the differential rotation matrix betweenRBA and RB′A′ can be calculated as
(19)ΔRT=RBAT⋅RB′A′−I3=RBAT⋅RAA′⋅RBA⋅RB′B−I3=RBAT⋅ΔAT+I3T⋅RBA⋅ΔBT+I3−I3=RBAT⋅ΔATT+I3⋅RBA⋅ΔBT+I3−I3=RBAT⋅ΔATT⋅RBA⋅ΔBT+RBAT⋅ΔATT⋅RBA+ΔBT.Equation (17) can be obtained by substituting ξA, ξB, and ξR into Equation (19) and then simplifying it.Suppose that the equivalent angle and axis of rotation corresponding toRBA are θBA and KBA, while those of RB′A′ are θB′A′ and KB′A′. Both KBA and KB′A′ are 1×3 vectors.According to Equation (3), we can obtain
(20)ξA=JrΩ,ΘAδΩ,ξB=JrΩ,ΘBδΩ,where Jr∈R3×np is the orientation-related part of the identification matrix. It should be noted that the identification matrix in Equation (20) is calculated with respect to the end-effector reference frame.Proposition 9.
Rotation error model: as the robot configuration changes fromΘA to ΘB, the measurable rotation error scalar is the product of the identification matrix and the kinematic errors, such as
(21)Δθr=KBA⋅JrΩ,ΘB−JrΩ,ΘA⋅δΩ,where Δθr=θB′A′−θBA is the measurable rotation error scalar, and the identification matrix of the rotation error model is KBA⋅JrΩ,ΘB−JrΩ,ΘA.Proof.
Substituting Equation (20) into Equation (17), we can obtain
(22)ξR=JrΩ,ΘB−RBAT⋅JrΩ,ΘA⋅δΩ.Similar to the derivation of distance error, we make the rotational axisKBA coincide with KB′A′ and the starting point of rotation matrix RBA coincide with that of RB′A′ as shown in Figure 4, where the point H is the projection of B′ onto the flat AOB, while the point G satisfies ∠AOG=∠A′OB′. The measurable equivalent rotation error can be obtained as
(23)Δθr=θB′A′−θBA=∠A′OB′−∠AOB=∠AOG−∠AOB=∠GOB.Figure 4
Derivation of the rotation error model. The rotational axisKBA is forced to coincide with KB′A′, and the starting points of rotation matrix RBA and RB′A′ are also forced to coincide.∠HOB is the projection of ∠BOB′ onto the flat AOB, which is actually the equivalent angle of rotation δθr of differential rotation ΔRT around the axis KBA, such as ∠HOB=δθr. Ignoring the linearizing error, we can obtain ∠GOB≈∠HOB, and then
(24)Δθr=δθrAccording to Proposition6, substitute Equation (22) into (24), then
(25)Δθr=δθr=KBA⋅ξR=KBA⋅JrΩ,ΘB−RBAT⋅JrΩ,ΘA⋅δΩ.According to the third property of the equivalent angle and axis of rotation,(26)KBA⋅RBAT=KBA.Finally, the rotation error model in Equation (21) can be obtained by substituting Equation (26) into (25).


## 3.4. Distance and Rotation Error Modeling

In summary, the distance and rotation error model is obtained by Equations (5) and (21) as
(27)ΔlBA=AB⇀AB⇀⋅JdΩ,ΘB−JdΩ,ΘA⋅δΩ,Δθr=KBA⋅JrΩ,ΘB−JrΩ,ΘA⋅δΩ.So we can obtain the identification matrix of the distance and rotation error model as(28)Jdr=AB⇀AB⇀⋅JdΩ,ΘB−JdΩ,ΘAKBA⋅JrΩ,ΘB−JrΩ,ΘA,Jdr∈R2×np.When there are redundant parameters in the error model, the identification matrix is rank defect, and measurement noises will affect the accuracy and robustness of parameter identification seriously, inferring the necessity of removal of redundant parameters. Redundant parameters in the error model will be discussed in the next section.


## 4. Parameter Independence Analysis

### 4.1. Kinematic Model by the MDH Method

A modified DH method, termed as MDH method [41], is used for kinematics modeling in this paper, which describes the deviation between two adjacent parallel joints axes with an additional rotation transformation to remedy the incompleteness. The MDH method is a good selection for verifying the proposed error model, because its modeling process is simple and partially overcomes the singularity.The coordinate frame is established by the MDH method as shown in Figure5. The MDH method uses five parameters (let p=5) including Ωi=ai,αi,di,θi,βiT to describe the two adjoining link coordinate frames, and the homogeneous transformation between them is shown as follows:
(29)Aii−1=transx,ai⋅rotx,αi⋅transz,di⋅rotz,θi⋅roty,βi=cθicβi−sθicθisβiaicαisθicβi+sαisβicαicθicαisθisβi−sαicβi−disαisαisθicβi−cαisβisαicθisαisθisβi+cαicβidicαi0001,where rot and trans denote the translation and rotation matrices, respectively, and Aii−1 denotes the homogeneous transformation of joint i with respect to joint i−1. sX means sinX and cX means cosX. The length of the rods shown in Figure 2 is illustrated in Table 1.Figure 5
Coordinate frame established by the MDH method. The angle between they-axis of the adjacent coordinate frames is defined as β, overcoming the singularity in the DH method.Table 1
Nominal length of the rods in the kinematic self-calibration system of the space robot.
Rod
l0
l1 l2 l3 l6 l7
l4l5
lcy
lcz
Length (m)
2.3
0.5
5
6.5
2.55According to the modeling rules of the MDH method, we obtain the nominal kinematic parameters of the 7-DOF space robot as shown in Table2.Table 2
Nominal kinematic parameters of the 7-DOF space robot.
Joint
a (m)
α (°)
d (m)
θ (°)
β (°)
1
0
0
2.3
0
0
2
0
−90
0.5
0
0
3
0
−90
0.5
0
0
4
5
0
0.5
0
0
5
5
0
0.5
0
0
6
0
90
0.5
0
0
7
0
90
0.5
0
0Since the relation between the camera frame and the end-effector has been calibrated on the ground and it is assumed to be unchanged on orbit, the coordinate frame of the end-effector coincides with that of the last joint. Therefore, the end-effector pose of a robot manipulator with n DOFs can be calculated as(30)Ψ=gAn0=g∑i=1nAii−1,where the function g denotes the transformation from a homogeneous matrix to its corresponding 3-dimension positions and Z-Y-X Euler angles.


### 4.2. Absolute Pose Error Model and Its Identifiability

It should be noted that nonsingularity of an error model indicates its identification matrix is column full rank. That is to say, any column of the identification matrix is linearly independent. The kinematic parameters of the robot can be sorted into three groups by its corresponding column in the identification matrix.(a)
Independent parameters, whose corresponding column is linearly independent with each other
(b)
Relative parameters, whose corresponding column is linearly dependent with another one
(c)
Ineffective parameters, whose corresponding column is the zero vector, indicating that they have no effect on the pose error of the robot end-effectorWe differentiate Equation (30) against the kinematic parameters to obtain the absolute pose error model as
(31)δΨ=∑i=1nJiδΩi,where δΩi denotes the kinematic parameter errors of the ith joint, and Ji denotes its corresponding identification matrix.Definition 3.
Local transfer matrix of the MDH method [44]: the local transfer matrix Gi as illustrated in Figure 6 is used to transfer δΩi to the local pose error in its own frame i. It can be calculated as
(32)Gi=cθicβi−disθicβi−sβi00−sθi−dicθi000cθisβi−disθisβicβi000cθicβi0−sβi00−sθi0010cθisβi0cβi0.Definition 4.
Global transfer matrix[44]: then, the local pose error is passed to the next coordinate frame until the last one, and we obtain the pose error of the end-effector with respect to the last coordinate frame. The global transfer matrix from the ith frame to the i+1th one is written as
(33)Fii−1=Rii−1TskewPii−1Rii−1T03×3Rii−1T,where Rii−1 denotes the rotation matrix in the homogeneous matrix Aii−1 and Pii−1 is its translation vector. The error transferring matrix between the two adjoining coordinate frames satisfies
(34)Fi+1i−1=Fi+1iFii−1,Fii=I6.Figure 6
Schematic of error transferring. Kinematic errors are transferred to the end-effector pose by the local transfer matrixGi and the global transfer matrix Fni. That is, Ji=Fni⋅Gi.Parameter independence in the error model can be determined just by analysing the redundant parameters of adjacent joints [42]. In other words, the matrix Ji−1Ji is required to be full column rank to ensure that all parameters are identifiable.Proposition 10.
The full column rank ofJi−1Ji is equivalent to the full column rank of Fii−1Gi−1Gi.Proof.
According to Equations (32), (33), and (34), we can obtain the identification matrix with respect to δΩi−1 and δΩi as
(35)Ji−1=FniFii−1Gi−1,Ji=FniGi.SinceFni has nothing to do with Ωi−1 and Ωi, the full column rank of Ji−1Ji is equivalent to the full column rank of Fii−1Gi−1Gi according to Equation (35). Considering that the expression of Fii−1Gi−1Gi is more simple and just depends on Ωi−1 and Ωi, it becomes easier to analyze the parameter independence in the error model.SupposeHi−1=Fii−1Gi−1 and then
(36)Hi−1=Hi−1aHi−1αHi−1dHi−1θHi−1β,Gi=GiaGiαGidGiθGiβ.Considering thatθi−1 and θi are variables, and the initial value of βi−1 and βi is generally set as zero, we can obtain by Equation (36)
(37)Gi=cθi−disθi000−sθi−dicθi000001000cθi0000−sθi00100010.Next, three typical singularities are discussed to analyze the parameter independence betweenHi−1 and Gi.
(1)
The two adjacent joints are parallel but not collinear, indicating thatαi=0orπ,ai≠0. Hi−1 can be rewritten as
(38)Hi−1=cθicθi−1+sθisθi−1di−di−1cθisθi−1−sθicθi−10±asθi±dcθicθisθi−1−sθicθi−1di−di−1sθisθi−1−cθicθi−10±acθi∓dsθi0−asθi−1±10∓a0cθicθi−1+sθisθi−100±sθi0cθisθi−1−sθicθi−100±cθi000±10.According to Equations (37) and (38), Hi−1d=±Gid, indicating that there is a linear interrelationship between di−1 and di.
(2)
The two adjacent joints are parallel and collinear, indicating thatαi=0orπ,ai=0. Hi−1 can be rewritten as
(39)Hi−1=cθicθi−1+sθisθi−1di−di−1cθisθi−1−sθicθi−100±dcθicθisθi−1−sθicθi−1di−di−1sθisθi−1−cθicθi−100∓dsθi00±1000cθicθi−1+sθisθi−100±sθi0cθisθi−1−sθicθi−100±cθi000±10.According to Equations (37) and (39), Hi−1d=±Gid and Hi−1θ=±Giθ, indicating that there is a linear interrelationship between di−1 and di, and this is true for θi−1 and θi.
(3)
The two adjacent joints are orthogonal withαi=±π/2,ai=0. Hi−1 can be rewritten as
(40)Hi−1=cθicθi−1−disθicθi−1−di−1cθisθi−1sθidicθi0−sθicθi−1−dicθicθi−1+di−1sθisθi−1cθi−disθi00di−1cθi−10000cθicθi−10sθi00−sθicθi−10cθi00000∓1.According to Equations (37) and (40), Hi−1β=∓Giθ, so βi−1 is linearly related to θi.Based on the analysis above, the identifiability of kinematic parameters in the absolute pose error model of the 7-DOF space robot can be obtained and shown in Table3.Table 3
Identifiability of kinematic parameters in the absolute pose error model of the 7-DOF space robot.
Identifiability
Kinematic parameters
Independent parameters
a1,α1,d1,θ1,a2,α2,d2,a3,α3,β3,a4,α4,θ4,β4,a5,α5,θ5,a6,α6,d6, a7,α7,d7,β7
Relative parameters
β1θ2,β2θ3,d3d4,d4d5,β5θ6,β6θ7
Ineffective parameters
NoneOnly the relative parametersθ2,θ3,d5,θ6,θ7 and all the independent parameters can be identified.


### 4.3. Parameter Independence Analysis of the Distance and Rotation Error Model

Equation (28) shows that the identification matrix of the distance and rotation error model is calculated based on that of the absolute pose error model. Specifically, the distance-related identification matrix is based on the position-related one whereas the rotation-related one is based on the orientation-related one. However, it should be noted that the two parts have different coefficients, i.e., AB⇀/AB⇀ and KBA, and that the position-related identification matrix is expressed in the end-effector frame while the orientation-related is expressed in the robot base frame. Accordingly, we discuss the two kinds of identification matrix separately in the following section.(1) The distance-related identification matrix: we can obtain the distance-related identification matrix with respect toΩi by Equations (28) and (33) as
(41)Jdi=Rn0Fnr1−r3iGi=Rn0RniTskewPniRniTGi=Ri0Ri0skew−PniGi,where Fnr1−r3i denotes the first three rows of Fni, and Rn0 is used to convert the related frame of the position-related identification matrix from the end-effector frame to the robot base frame.The position of the end-effector can be written as(42)Pn0=Pi0+Ri0PniFor the first joint,i=1, such as
(43)Jd1=Rn0Rn1TskewPn1Rn1TG1,(44)Pn0=P10+R10Pn1.Substitute Equation (44) into Equation (43), and Equation (44) can be rewritten as
(45)Jd1=R10I3skew−Pn1G1,where R10 can be calculated as Equation (29) and Pn1 is independent of Ω1.We assume that the space robot moves from the configurationΘA to ΘB, and the first joint rotates from θ1A to θ1B, i.e., from Ω1A=a1,α1,d1,θ1A,β1T to Ω1B=a1,α1,d1,θ1B,β1T. The end-effector position changes from PnA1=xA,yA,zAT to PnB1=xB,yB,zBT. Using the Wolfram Mathematica, we can obtain the distance-related identification matrix as
(46)Jd1ΘA⟶ΘB=Pn0B−Pn0AJd1θ1B−Jd1θ1A=00002xBzA+xAzBcos2β1+−xAxB+zAzBsin2β1sinθ1A−θ1B22+−yBzA+yAzBcosβ1+−xByA+xAyBsinβ1sinθ1A−θ1BT.Obviously, the kinematic parametersa1,α1,d1,θ1 are ineffective, for that their corresponding columns are zero shown in Equation (46).For the last joint,i=n and Pnn=01×3, such as
(47)Jdn=Rn0I303×3Gn=Rn0Gnr1−r3,where Rn0 depends on all the kinematic parameters, and so does the coefficient Pn0B−Pn0A, which indicates that the identifiability of JdnΘA⟶ΘB depends on Gnr1−r3. Obviously, from Equation (37), the kinematic parameters θn,βn are ineffective.For theith joint i=2,3,⋯,n−1, the coefficients AB⇀/AB⇀ in Equation (28) and Ri0Ri0skew−Pni in Equation (41) both depend on all the kinematic parameters, so the parameter identifiability of the distance error model is the same with that of the position error model.
(1)
The rotation-related identification matrixWe can obtain the rotation-related identification matrix by Equations (28) and (33) as
(48)Jri=FniGi=03×3RniTGi=RniTGir4−r6For the first joint,i=1,such as
(49)Jr1=Rn1TG1r4−r6=R01Rn0TG1r4−r6=Rn0TR10G1r4−r6where
(50)R10G1r4−r6Ω1=0100−sθ1000−sα1cα1cθ1000cα1sα1cθ1By Equations (28), (49) and (50), we can obtain
(51)Jr1ΘA⟶ΘB=KBAJr1θ1B−Jr1θ1A=KBARn0TRGθ1B−RGθ1A=KBARn0T0000sθ1A−sθ1B0000cθ1B−cθ1Acα10000cθ1B−cθ1Asα1where KBARn0T depends on all the kinematic parameters, and the kinematic parameters a1,α1,d1,θ1 are ineffective.For the last joint,i=n, such as
(52)Jrn=RnnTGnr4−r6=0cθn0000−sθn00100010By Equations (28) and (52), we can obtain
(53)JrnΘA⟶ΘB=KBAJrnθnB−Jr1θnA=KBARn0TRGθnB−RGθnA=KBARn0T0cθnB−cθnA0000sθnA−sθnB00000000By Equation (53), the kinematic parameters an,dn,θn,βn are ineffective. Actually, if both the distance and rotation error are taken into consideration, the ineffective parameters are θn,βn.For theith joint (i=2,3,…,n−1), the coefficients KBA in Equation (28) and RniT in Equation (48) both depend on all the kinematic parameters, so the parameter identifiability of the rotation error model is the same with that of the orientation error model.(2) The distance and rotation error model: in summary, the identifiability of kinematic parameters in the distance and rotation error model of the 7-DOF space robot can be obtained and shown in Table4.Table 4
Identifiability of kinematic parameters in the distance and rotation error model of the 7-DOF space robot.
Identifiability
Kinematic parameters
Independent parameters
a2,α2,d2,a3,α3,β3,a4,α4,θ4,β4,a5,α5,θ5,a6,α6,d6, a7,α7,d7
Relative parameters
β1θ2,β2θ3,d3d4,d4d5,β5θ6
Ineffective parameters
a1,α1,d1,θ1,β6,θ7,β7Sinceβ6 is linearly related to θ7, and θ7 is ineffective, β6 is also ineffective. Therefore, only the relative parameters θ2,θ3,d5,θ6 and all the independent parameters can be identified.


## 4.1. Kinematic Model by the MDH Method

A modified DH method, termed as MDH method [41], is used for kinematics modeling in this paper, which describes the deviation between two adjacent parallel joints axes with an additional rotation transformation to remedy the incompleteness. The MDH method is a good selection for verifying the proposed error model, because its modeling process is simple and partially overcomes the singularity.The coordinate frame is established by the MDH method as shown in Figure5. The MDH method uses five parameters (let p=5) including Ωi=ai,αi,di,θi,βiT to describe the two adjoining link coordinate frames, and the homogeneous transformation between them is shown as follows:
(29)Aii−1=transx,ai⋅rotx,αi⋅transz,di⋅rotz,θi⋅roty,βi=cθicβi−sθicθisβiaicαisθicβi+sαisβicαicθicαisθisβi−sαicβi−disαisαisθicβi−cαisβisαicθisαisθisβi+cαicβidicαi0001,where rot and trans denote the translation and rotation matrices, respectively, and Aii−1 denotes the homogeneous transformation of joint i with respect to joint i−1. sX means sinX and cX means cosX. The length of the rods shown in Figure 2 is illustrated in Table 1.Figure 5
Coordinate frame established by the MDH method. The angle between they-axis of the adjacent coordinate frames is defined as β, overcoming the singularity in the DH method.Table 1
Nominal length of the rods in the kinematic self-calibration system of the space robot.
Rod
l0
l1 l2 l3 l6 l7
l4l5
lcy
lcz
Length (m)
2.3
0.5
5
6.5
2.55According to the modeling rules of the MDH method, we obtain the nominal kinematic parameters of the 7-DOF space robot as shown in Table2.Table 2
Nominal kinematic parameters of the 7-DOF space robot.
Joint
a (m)
α (°)
d (m)
θ (°)
β (°)
1
0
0
2.3
0
0
2
0
−90
0.5
0
0
3
0
−90
0.5
0
0
4
5
0
0.5
0
0
5
5
0
0.5
0
0
6
0
90
0.5
0
0
7
0
90
0.5
0
0Since the relation between the camera frame and the end-effector has been calibrated on the ground and it is assumed to be unchanged on orbit, the coordinate frame of the end-effector coincides with that of the last joint. Therefore, the end-effector pose of a robot manipulator with n DOFs can be calculated as(30)Ψ=gAn0=g∑i=1nAii−1,where the function g denotes the transformation from a homogeneous matrix to its corresponding 3-dimension positions and Z-Y-X Euler angles.


## 4.2. Absolute Pose Error Model and Its Identifiability

It should be noted that nonsingularity of an error model indicates its identification matrix is column full rank. That is to say, any column of the identification matrix is linearly independent. The kinematic parameters of the robot can be sorted into three groups by its corresponding column in the identification matrix.(a)
Independent parameters, whose corresponding column is linearly independent with each other
(b)
Relative parameters, whose corresponding column is linearly dependent with another one
(c)
Ineffective parameters, whose corresponding column is the zero vector, indicating that they have no effect on the pose error of the robot end-effectorWe differentiate Equation (30) against the kinematic parameters to obtain the absolute pose error model as
(31)δΨ=∑i=1nJiδΩi,where δΩi denotes the kinematic parameter errors of the ith joint, and Ji denotes its corresponding identification matrix.Definition 3.
Local transfer matrix of the MDH method [44]: the local transfer matrix Gi as illustrated in Figure 6 is used to transfer δΩi to the local pose error in its own frame i. It can be calculated as
(32)Gi=cθicβi−disθicβi−sβi00−sθi−dicθi000cθisβi−disθisβicβi000cθicβi0−sβi00−sθi0010cθisβi0cβi0.Definition 4.
Global transfer matrix[44]: then, the local pose error is passed to the next coordinate frame until the last one, and we obtain the pose error of the end-effector with respect to the last coordinate frame. The global transfer matrix from the ith frame to the i+1th one is written as
(33)Fii−1=Rii−1TskewPii−1Rii−1T03×3Rii−1T,where Rii−1 denotes the rotation matrix in the homogeneous matrix Aii−1 and Pii−1 is its translation vector. The error transferring matrix between the two adjoining coordinate frames satisfies
(34)Fi+1i−1=Fi+1iFii−1,Fii=I6.Figure 6
Schematic of error transferring. Kinematic errors are transferred to the end-effector pose by the local transfer matrixGi and the global transfer matrix Fni. That is, Ji=Fni⋅Gi.Parameter independence in the error model can be determined just by analysing the redundant parameters of adjacent joints [42]. In other words, the matrix Ji−1Ji is required to be full column rank to ensure that all parameters are identifiable.Proposition 10.
The full column rank ofJi−1Ji is equivalent to the full column rank of Fii−1Gi−1Gi.Proof.
According to Equations (32), (33), and (34), we can obtain the identification matrix with respect to δΩi−1 and δΩi as
(35)Ji−1=FniFii−1Gi−1,Ji=FniGi.SinceFni has nothing to do with Ωi−1 and Ωi, the full column rank of Ji−1Ji is equivalent to the full column rank of Fii−1Gi−1Gi according to Equation (35). Considering that the expression of Fii−1Gi−1Gi is more simple and just depends on Ωi−1 and Ωi, it becomes easier to analyze the parameter independence in the error model.SupposeHi−1=Fii−1Gi−1 and then
(36)Hi−1=Hi−1aHi−1αHi−1dHi−1θHi−1β,Gi=GiaGiαGidGiθGiβ.Considering thatθi−1 and θi are variables, and the initial value of βi−1 and βi is generally set as zero, we can obtain by Equation (36)
(37)Gi=cθi−disθi000−sθi−dicθi000001000cθi0000−sθi00100010.Next, three typical singularities are discussed to analyze the parameter independence betweenHi−1 and Gi.
(1)
The two adjacent joints are parallel but not collinear, indicating thatαi=0orπ,ai≠0. Hi−1 can be rewritten as
(38)Hi−1=cθicθi−1+sθisθi−1di−di−1cθisθi−1−sθicθi−10±asθi±dcθicθisθi−1−sθicθi−1di−di−1sθisθi−1−cθicθi−10±acθi∓dsθi0−asθi−1±10∓a0cθicθi−1+sθisθi−100±sθi0cθisθi−1−sθicθi−100±cθi000±10.According to Equations (37) and (38), Hi−1d=±Gid, indicating that there is a linear interrelationship between di−1 and di.
(2)
The two adjacent joints are parallel and collinear, indicating thatαi=0orπ,ai=0. Hi−1 can be rewritten as
(39)Hi−1=cθicθi−1+sθisθi−1di−di−1cθisθi−1−sθicθi−100±dcθicθisθi−1−sθicθi−1di−di−1sθisθi−1−cθicθi−100∓dsθi00±1000cθicθi−1+sθisθi−100±sθi0cθisθi−1−sθicθi−100±cθi000±10.According to Equations (37) and (39), Hi−1d=±Gid and Hi−1θ=±Giθ, indicating that there is a linear interrelationship between di−1 and di, and this is true for θi−1 and θi.
(3)
The two adjacent joints are orthogonal withαi=±π/2,ai=0. Hi−1 can be rewritten as
(40)Hi−1=cθicθi−1−disθicθi−1−di−1cθisθi−1sθidicθi0−sθicθi−1−dicθicθi−1+di−1sθisθi−1cθi−disθi00di−1cθi−10000cθicθi−10sθi00−sθicθi−10cθi00000∓1.According to Equations (37) and (40), Hi−1β=∓Giθ, so βi−1 is linearly related to θi.Based on the analysis above, the identifiability of kinematic parameters in the absolute pose error model of the 7-DOF space robot can be obtained and shown in Table3.Table 3
Identifiability of kinematic parameters in the absolute pose error model of the 7-DOF space robot.
Identifiability
Kinematic parameters
Independent parameters
a1,α1,d1,θ1,a2,α2,d2,a3,α3,β3,a4,α4,θ4,β4,a5,α5,θ5,a6,α6,d6, a7,α7,d7,β7
Relative parameters
β1θ2,β2θ3,d3d4,d4d5,β5θ6,β6θ7
Ineffective parameters
NoneOnly the relative parametersθ2,θ3,d5,θ6,θ7 and all the independent parameters can be identified.


## 4.3. Parameter Independence Analysis of the Distance and Rotation Error Model

Equation (28) shows that the identification matrix of the distance and rotation error model is calculated based on that of the absolute pose error model. Specifically, the distance-related identification matrix is based on the position-related one whereas the rotation-related one is based on the orientation-related one. However, it should be noted that the two parts have different coefficients, i.e., AB⇀/AB⇀ and KBA, and that the position-related identification matrix is expressed in the end-effector frame while the orientation-related is expressed in the robot base frame. Accordingly, we discuss the two kinds of identification matrix separately in the following section.(1) The distance-related identification matrix: we can obtain the distance-related identification matrix with respect toΩi by Equations (28) and (33) as
(41)Jdi=Rn0Fnr1−r3iGi=Rn0RniTskewPniRniTGi=Ri0Ri0skew−PniGi,where Fnr1−r3i denotes the first three rows of Fni, and Rn0 is used to convert the related frame of the position-related identification matrix from the end-effector frame to the robot base frame.The position of the end-effector can be written as(42)Pn0=Pi0+Ri0PniFor the first joint,i=1, such as
(43)Jd1=Rn0Rn1TskewPn1Rn1TG1,(44)Pn0=P10+R10Pn1.Substitute Equation (44) into Equation (43), and Equation (44) can be rewritten as
(45)Jd1=R10I3skew−Pn1G1,where R10 can be calculated as Equation (29) and Pn1 is independent of Ω1.We assume that the space robot moves from the configurationΘA to ΘB, and the first joint rotates from θ1A to θ1B, i.e., from Ω1A=a1,α1,d1,θ1A,β1T to Ω1B=a1,α1,d1,θ1B,β1T. The end-effector position changes from PnA1=xA,yA,zAT to PnB1=xB,yB,zBT. Using the Wolfram Mathematica, we can obtain the distance-related identification matrix as
(46)Jd1ΘA⟶ΘB=Pn0B−Pn0AJd1θ1B−Jd1θ1A=00002xBzA+xAzBcos2β1+−xAxB+zAzBsin2β1sinθ1A−θ1B22+−yBzA+yAzBcosβ1+−xByA+xAyBsinβ1sinθ1A−θ1BT.Obviously, the kinematic parametersa1,α1,d1,θ1 are ineffective, for that their corresponding columns are zero shown in Equation (46).For the last joint,i=n and Pnn=01×3, such as
(47)Jdn=Rn0I303×3Gn=Rn0Gnr1−r3,where Rn0 depends on all the kinematic parameters, and so does the coefficient Pn0B−Pn0A, which indicates that the identifiability of JdnΘA⟶ΘB depends on Gnr1−r3. Obviously, from Equation (37), the kinematic parameters θn,βn are ineffective.For theith joint i=2,3,⋯,n−1, the coefficients AB⇀/AB⇀ in Equation (28) and Ri0Ri0skew−Pni in Equation (41) both depend on all the kinematic parameters, so the parameter identifiability of the distance error model is the same with that of the position error model.
(1)
The rotation-related identification matrixWe can obtain the rotation-related identification matrix by Equations (28) and (33) as
(48)Jri=FniGi=03×3RniTGi=RniTGir4−r6For the first joint,i=1,such as
(49)Jr1=Rn1TG1r4−r6=R01Rn0TG1r4−r6=Rn0TR10G1r4−r6where
(50)R10G1r4−r6Ω1=0100−sθ1000−sα1cα1cθ1000cα1sα1cθ1By Equations (28), (49) and (50), we can obtain
(51)Jr1ΘA⟶ΘB=KBAJr1θ1B−Jr1θ1A=KBARn0TRGθ1B−RGθ1A=KBARn0T0000sθ1A−sθ1B0000cθ1B−cθ1Acα10000cθ1B−cθ1Asα1where KBARn0T depends on all the kinematic parameters, and the kinematic parameters a1,α1,d1,θ1 are ineffective.For the last joint,i=n, such as
(52)Jrn=RnnTGnr4−r6=0cθn0000−sθn00100010By Equations (28) and (52), we can obtain
(53)JrnΘA⟶ΘB=KBAJrnθnB−Jr1θnA=KBARn0TRGθnB−RGθnA=KBARn0T0cθnB−cθnA0000sθnA−sθnB00000000By Equation (53), the kinematic parameters an,dn,θn,βn are ineffective. Actually, if both the distance and rotation error are taken into consideration, the ineffective parameters are θn,βn.For theith joint (i=2,3,…,n−1), the coefficients KBA in Equation (28) and RniT in Equation (48) both depend on all the kinematic parameters, so the parameter identifiability of the rotation error model is the same with that of the orientation error model.(2) The distance and rotation error model: in summary, the identifiability of kinematic parameters in the distance and rotation error model of the 7-DOF space robot can be obtained and shown in Table4.Table 4
Identifiability of kinematic parameters in the distance and rotation error model of the 7-DOF space robot.
Identifiability
Kinematic parameters
Independent parameters
a2,α2,d2,a3,α3,β3,a4,α4,θ4,β4,a5,α5,θ5,a6,α6,d6, a7,α7,d7
Relative parameters
β1θ2,β2θ3,d3d4,d4d5,β5θ6
Ineffective parameters
a1,α1,d1,θ1,β6,θ7,β7Sinceβ6 is linearly related to θ7, and θ7 is ineffective, β6 is also ineffective. Therefore, only the relative parameters θ2,θ3,d5,θ6 and all the independent parameters can be identified.


## 5. Method Verification

The process of calibration simulation is shown in Figure7. Firstly, the measurement configurations are selected from the permissible operating range of joints with the given amount of configurations; then, the actual end-effector poses are calculated with the actual kinematic parameters; finally, taking the encoder noise and the measurement noise into account, the least squares method is adopted to identify the kinematic parameters. It is worthy noted that the actual parameters are calculated by the nominal parameters artificially added with parameter errors, indicating that all these parameters are known for analysis and comparison.Figure 7
Flowchart of calibration simulation. Encoder noise and pose measurement noise are added to achieve more reliable simulation results.Whether it is the absolute pose error model or the proposed distance and rotation error model, the least squares method is a powerful tool to identify the kinematic errors against sensor noises. The specific application of least squares method is as(54)δΨi=Ψ∗−fΩi,Θ,δΩi+1=JΩΩi,Θ+δΨi,Ωi+1=Ωi+δΩi+1,where i=0,1,2,⋯,N, N denotes the total number of iterations, and Ω0 is the initial nominal kinematic parameters. ·+ is the Moore-Penrose inverse of a matrix. The process is iterated until δΩi converges to a small threshold. Finally, the identified kinematic parameters Ωc can be obtained as
(55)Ωc=Ω0+∑i=1NδΩi.For the distance and rotation error model,Ψ∗ is the measured value of the distance and rotation of the space robot, so it is inaccurate due to the existence of sensor noises. The work [39] indicates that a sufficient number of measurements can guarantee the convergence of the above process. For the distance and rotation error model of the space robot, the least number of measurements is the number of all the identifiable kinematic parameters.


### 5.1. Selection of Measurement Configurations

The end-effector poses of the space robot are measured by a hand-eye camera, so the camera has to point to the target checkerboard. Besides, the end-effector of the space robot under the selected measurement configurations has to be close to the checkerboard in order to ensure measurement accuracy.For the sake of convenience and economy, we select one configuration for comparison asΘA and another 51 configurations as ΘB. Actually, the corresponding end-effector position of the configuration ΘA is located 0.5 m above the centre of the checkerboard, while the corresponding end-effector positions of the other 51 configurations are distributed uniformly on the hemispherical surface, whose centre is located at just the corresponding end-effector position of the configuration ΘA, and its radius is 0.5 m. Additionally, the z-axis of the coordinate frames corresponding to all these end-effector poses points to the centre of the checkerboard, as shown in Figure 8.Figure 8
The corresponding end-effector frames of the measurement configurations. A total of 52 configurations are selected as measurement configurations, one of which is defined asΘA, while the other configurations are treated as ΘB.


### 5.2. Sensor Noises and Kinematic Parameter Errors

In practical applications, the actual end-effector pose of a robot can be obtained by the hand-eye camera, while the actual robot configurations can be measured by the encoders. However, in simulation applications, measuring noises and encoder noises are added according to their respective distributions in order to imitate the influence of sensor errors. These measuring noises meet normal distribution as shown in Table5.Table 5
Normal distribution of measuring noises.
Measuring noise
End-effector position (m)
End-effector orientation (°)
Robot configuration (°)
Mean
0
0
0
Standard deviation
0.001
0.01
0.005According to Section4, the transformation matrix between the robot base frame and the frame 1 cannot be identified, so we have to make the assumption that this transformation matrix will not be changed on orbit or can be calibrated in another way. Therefore, errors are added to other kinematic parameters but Ω1. All the kinematic parameter errors δΩ are shown in Table 6.Table 6
Kinematic parameter errors of the 7-DOF space robot.
Joint
δa (m)
δα (°)
δd (m)
δθ (°)
δβ (°)
1
0
0
0
0
0
2
0.01
0.05
0.01
0.05
0.05
3
0.01
0.05
0.01
0.05
0.05
4
0.01
0.05
0.01
0.05
0.05
5
0.01
0.05
0.01
0.05
0.05
6
0.01
0.05
0.01
0.05
0.05
7
0.01
0.05
0.01
0.05
0.05


### 5.3. Result and Analysis

The purpose of robot calibration is to obtain the accurate kinematic parameters representing the robot structure and the exact estimation of the end-effector pose. The least squares method is adopted to obtain the calibrated kinematic parametersΩc, and then the calibration residuals by different kinds of error models are calculated as
(56)ςΩ=Ω∗−Ωc.The calibration residuals by the distance error model, by the distance and rotation error model, and by the absolute pose error model are shown in Tables7-9.Table 7
Calibration residuals by the distance error model.
Joint
ςa (m)
ςα (°)
ςd (m)
ςθ (°)
ςβ (°)
1
0.00000
0.00000
0.00000
0.00000
0.00000
2
0.00054
0.00445
0.00094
0.00038
−0.04111
3
0.00063
0.01097
0.00015
0.03997
0.11269
4
0.00355
0.00632
−0.00861
0.01106
0.15176
5
0.00442
−0.07475
0.02887
0.05620
−0.04298
6
0.00012
−0.20839
−0.00284
−0.01352
−0.02700
7
−0.00194
−0.29798
0.00152
−0.00042
−0.03749Table 8
Calibration residuals by the distance and rotation error model.
Joint
ςa (m)
ςα (°)
ςd (m)
ςθ (°)
ςβ (°)
1
0.00000
0.00000
0.00000
0.00000
0.00000
2
0.00056
0.00050
0.00051
0.00176
−0.04111
3
0.00041
−0.00313
0.00015
0.03701
0.00586
4
0.00081
0.00485
−0.00861
0.01164
0.01368
5
0.00053
−0.00637
0.01005
0.00310
−0.04298
6
0.00048
−0.01963
0.00047
−0.05903
−0.02700
7
−0.00159
−0.01215
0.00045
−0.00042
−0.03749Table 9
Calibration residuals by the absolute pose error model.
Joint
ςa (m)
ςα (°)
ςd (m)
ςθ (°)
ςβ (°)
1
0.00000
0.00000
0.00000
0.00000
0.00000
2
0.00017
0.00140
−0.00024
0.00247
−0.04111
3
0.00016
0.00262
0.00015
0.04160
0.00187
4
−0.00005
0.00014
−0.00861
−0.00283
0.00287
5
−0.00009
0.00569
0.00890
0.00251
−0.04298
6
0.00005
−0.00458
−0.00003
−0.04436
−0.02700
7
−0.00006
0.00457
−0.00003
−0.02722
0.00050Some of the calibration residuals are relatively big in Tables7-9, which are actually relative parameters, so these calibration residuals should be summed up accordingly.For the distance and rotation error model,(57)ςβ1+ςθ2=0.00000+0.00176=0.00176,ςβ2+ςθ3=−0.04111+0.03701=−0.0041,ςd3+ςd4+ςd5=0.00015+−0.00861+0.01005=0.00159,ςβ5−ςθ6=−0.04298−−0.05903=0.01605.For the absolute pose error model,(58)ςβ1+ςθ2=0.00000+0.00247=0.00247,ςβ2+ςθ3=−0.04111+0.04160=0.00049,ςd3+ςd4+ςd5=0.00015+−0.00861+0.00890=0.00044,ςβ5−ςθ6=−0.04298−−0.04436=0.00138,ςβ6−ςθ7=−0.02700−−0.02722=0.00022.Equations (57) and (58) illustrate that once summed up, calibration residuals of these relative parameters will counteract each other, indicating that only some of the relative parameters are necessary to be identified. This is consistent with the analysis of parameter independence, which proves the correctness of the analysis results.Moreover, we come to a conclusion that the distance and rotation error model does better than the distance error model in identification accuracy of the kinematic parameters, but a little weaker than the absolute pose error model.Finally, 500 robot configurations are selected randomly to serve as the validation group. The end-effector position estimate errors of validation configurations are calculated, respectively, with different error models, and maximum and average of these estimate errors are analysed to compare calibration performance.Figure9 gives the end-effector position estimate errors of validation configurations before calibration and those after calibration by three kinds of error models. Figure 10 gives the corresponding histograms of these position errors. The maximum and average of these 500 position errors are shown in Table 10.Figure 9
The end-effector position errors of 500 confirm configurations.Figure 10
The histograms of the end-effector position errors of 500 confirm configurations.Table 10
The statistical characteristics of the end-effector position errors of 500 confirm configurations.
Location
Max (mm)
Mean (mm)
Before calibration
38
19
By distance error
21
11
By distance and rotation error
4.8
2.5
By absolute error
1.1
0.5Whether it is the identification accuracy of the kinematic parameters shown in Tables7-9 or statistical analysis of validation tests in Figures 9 and 10 and Table 10, it verifies the effectiveness of the proposed distance and rotation error model. The rotation error of the robot end-effector is included so as to improve calibration performance. But it should be pointed out that, due to the lack of absoluteness, errors in kinematic parameters of the first (root) joint of the space robot cannot be identified.


## 5.1. Selection of Measurement Configurations

The end-effector poses of the space robot are measured by a hand-eye camera, so the camera has to point to the target checkerboard. Besides, the end-effector of the space robot under the selected measurement configurations has to be close to the checkerboard in order to ensure measurement accuracy.For the sake of convenience and economy, we select one configuration for comparison asΘA and another 51 configurations as ΘB. Actually, the corresponding end-effector position of the configuration ΘA is located 0.5 m above the centre of the checkerboard, while the corresponding end-effector positions of the other 51 configurations are distributed uniformly on the hemispherical surface, whose centre is located at just the corresponding end-effector position of the configuration ΘA, and its radius is 0.5 m. Additionally, the z-axis of the coordinate frames corresponding to all these end-effector poses points to the centre of the checkerboard, as shown in Figure 8.Figure 8
The corresponding end-effector frames of the measurement configurations. A total of 52 configurations are selected as measurement configurations, one of which is defined asΘA, while the other configurations are treated as ΘB.


## 5.2. Sensor Noises and Kinematic Parameter Errors

In practical applications, the actual end-effector pose of a robot can be obtained by the hand-eye camera, while the actual robot configurations can be measured by the encoders. However, in simulation applications, measuring noises and encoder noises are added according to their respective distributions in order to imitate the influence of sensor errors. These measuring noises meet normal distribution as shown in Table5.Table 5
Normal distribution of measuring noises.
Measuring noise
End-effector position (m)
End-effector orientation (°)
Robot configuration (°)
Mean
0
0
0
Standard deviation
0.001
0.01
0.005According to Section4, the transformation matrix between the robot base frame and the frame 1 cannot be identified, so we have to make the assumption that this transformation matrix will not be changed on orbit or can be calibrated in another way. Therefore, errors are added to other kinematic parameters but Ω1. All the kinematic parameter errors δΩ are shown in Table 6.Table 6
Kinematic parameter errors of the 7-DOF space robot.
Joint
δa (m)
δα (°)
δd (m)
δθ (°)
δβ (°)
1
0
0
0
0
0
2
0.01
0.05
0.01
0.05
0.05
3
0.01
0.05
0.01
0.05
0.05
4
0.01
0.05
0.01
0.05
0.05
5
0.01
0.05
0.01
0.05
0.05
6
0.01
0.05
0.01
0.05
0.05
7
0.01
0.05
0.01
0.05
0.05


## 5.3. Result and Analysis

The purpose of robot calibration is to obtain the accurate kinematic parameters representing the robot structure and the exact estimation of the end-effector pose. The least squares method is adopted to obtain the calibrated kinematic parametersΩc, and then the calibration residuals by different kinds of error models are calculated as
(56)ςΩ=Ω∗−Ωc.The calibration residuals by the distance error model, by the distance and rotation error model, and by the absolute pose error model are shown in Tables7-9.Table 7
Calibration residuals by the distance error model.
Joint
ςa (m)
ςα (°)
ςd (m)
ςθ (°)
ςβ (°)
1
0.00000
0.00000
0.00000
0.00000
0.00000
2
0.00054
0.00445
0.00094
0.00038
−0.04111
3
0.00063
0.01097
0.00015
0.03997
0.11269
4
0.00355
0.00632
−0.00861
0.01106
0.15176
5
0.00442
−0.07475
0.02887
0.05620
−0.04298
6
0.00012
−0.20839
−0.00284
−0.01352
−0.02700
7
−0.00194
−0.29798
0.00152
−0.00042
−0.03749Table 8
Calibration residuals by the distance and rotation error model.
Joint
ςa (m)
ςα (°)
ςd (m)
ςθ (°)
ςβ (°)
1
0.00000
0.00000
0.00000
0.00000
0.00000
2
0.00056
0.00050
0.00051
0.00176
−0.04111
3
0.00041
−0.00313
0.00015
0.03701
0.00586
4
0.00081
0.00485
−0.00861
0.01164
0.01368
5
0.00053
−0.00637
0.01005
0.00310
−0.04298
6
0.00048
−0.01963
0.00047
−0.05903
−0.02700
7
−0.00159
−0.01215
0.00045
−0.00042
−0.03749Table 9
Calibration residuals by the absolute pose error model.
Joint
ςa (m)
ςα (°)
ςd (m)
ςθ (°)
ςβ (°)
1
0.00000
0.00000
0.00000
0.00000
0.00000
2
0.00017
0.00140
−0.00024
0.00247
−0.04111
3
0.00016
0.00262
0.00015
0.04160
0.00187
4
−0.00005
0.00014
−0.00861
−0.00283
0.00287
5
−0.00009
0.00569
0.00890
0.00251
−0.04298
6
0.00005
−0.00458
−0.00003
−0.04436
−0.02700
7
−0.00006
0.00457
−0.00003
−0.02722
0.00050Some of the calibration residuals are relatively big in Tables7-9, which are actually relative parameters, so these calibration residuals should be summed up accordingly.For the distance and rotation error model,(57)ςβ1+ςθ2=0.00000+0.00176=0.00176,ςβ2+ςθ3=−0.04111+0.03701=−0.0041,ςd3+ςd4+ςd5=0.00015+−0.00861+0.01005=0.00159,ςβ5−ςθ6=−0.04298−−0.05903=0.01605.For the absolute pose error model,(58)ςβ1+ςθ2=0.00000+0.00247=0.00247,ςβ2+ςθ3=−0.04111+0.04160=0.00049,ςd3+ςd4+ςd5=0.00015+−0.00861+0.00890=0.00044,ςβ5−ςθ6=−0.04298−−0.04436=0.00138,ςβ6−ςθ7=−0.02700−−0.02722=0.00022.Equations (57) and (58) illustrate that once summed up, calibration residuals of these relative parameters will counteract each other, indicating that only some of the relative parameters are necessary to be identified. This is consistent with the analysis of parameter independence, which proves the correctness of the analysis results.Moreover, we come to a conclusion that the distance and rotation error model does better than the distance error model in identification accuracy of the kinematic parameters, but a little weaker than the absolute pose error model.Finally, 500 robot configurations are selected randomly to serve as the validation group. The end-effector position estimate errors of validation configurations are calculated, respectively, with different error models, and maximum and average of these estimate errors are analysed to compare calibration performance.Figure9 gives the end-effector position estimate errors of validation configurations before calibration and those after calibration by three kinds of error models. Figure 10 gives the corresponding histograms of these position errors. The maximum and average of these 500 position errors are shown in Table 10.Figure 9
The end-effector position errors of 500 confirm configurations.Figure 10
The histograms of the end-effector position errors of 500 confirm configurations.Table 10
The statistical characteristics of the end-effector position errors of 500 confirm configurations.
Location
Max (mm)
Mean (mm)
Before calibration
38
19
By distance error
21
11
By distance and rotation error
4.8
2.5
By absolute error
1.1
0.5Whether it is the identification accuracy of the kinematic parameters shown in Tables7-9 or statistical analysis of validation tests in Figures 9 and 10 and Table 10, it verifies the effectiveness of the proposed distance and rotation error model. The rotation error of the robot end-effector is included so as to improve calibration performance. But it should be pointed out that, due to the lack of absoluteness, errors in kinematic parameters of the first (root) joint of the space robot cannot be identified.


## 6. Conclusions

Proposed in this paper is an error model involving both the distance and the rotation error of the space robot end-effector. The error model can avoid identifying the transformation matrix between the measurement system frame and the robot base frame, suitable for self-calibration of the space robot. Besides, identifiable parameters in the distance and rotation error model are confirmed to eliminate singularity in robot kinematic calibration. Finally, we conduct the calibration simulation and compare differences in calibration performance between these models. Statistical results indicate that the proposed error model does better in the accuracy of the robot end-effector position estimate and of the kinematic parameter identification than the only distance error model. However, it still matters that observability of the distance and rotation error model be studied as an indicator of measurement configuration optimization, which will significantly reduce the number of configurations required for calibration. Besides, information fusion provides a powerful tool to deal with uncertainty and external disturbance of pose measurement and application of filtering algorithms in robot calibration is worthy of attention. From the operational point of view, the light conditions to carry on calibration processes should be also taken into consideration. In summary, there are still more future works and challenges to adopt the proposed method in practical application.



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*Source: 8349048-2019-02-17.xml*