A novel kinematic model and its compensation approach for structural parameter of rotating joints series robot

Abstract

The pose accuracy is important performance index of industrial robot. In order to improve the pose accuracy, it is necessary to compensate the kinematic parameters of the robot. A novel approach for kinematic parameter compensation of robot based on the parameter solving of the ruled surface model is developed by establishing a model containing link error and kinematic pair error. The fitting axis of each joint is obtained by fitting the discrete points. Then, the fitting axes of the adjacent joints is used to identify the link errors and the link errors are compensated. This approach has been tested and verified on the LR7-710 robot. The test results show that the position error of the arc trajectory is reduced from 0.4580 to 0.0801 mm.

Introduction

Path planning is the key issue for robots. ¹ The primary problem is to improve the pose accuracy of robot. Generally, link error and kinematic pair error play a major role in the pose accuracy. As a result, the robot has a high repetitive positioning accuracy, but a low absolute positioning accuracy.^2,3 In order to improve the absolute positioning accuracy, it is necessary to compensate the kinematic parameters.^4–8

A kinematic model must first be established for compensation. The most commonly used kinematic model are DH model. ⁹ However, the DH model is discontinuous when the axes of two adjacent kinematic pairs are parallel. In this case, the modified DH model proposed to eliminate discontinuity by adding a parameter. ¹⁰ Due to the additional parameter, the parameters identification complexity of model also increases. Stone and Sanderson apply Monte Carlo simulation techniques to gain relationship between manufacturing errors and robot positioning errors, and propose S-Model to eliminate errors due to manufacturing. ¹¹ To solve the singularities of DH between consecutive parallel joints, Zhuang et al. propose a six-parameter CPC model to make it closer to the DH model. ¹² Due to the constraints of joint twists, the traditional methods based on the POE are not minimal. Yang et al. propose a minimal kinematic model for serial robot calibration using POE formula, whose parameters are completely independent and minimal. ¹³ In these models, they do not consider the effect of kinematic pair error on link error. Since the link is installed on the kinematic pair, the kinematic pair error will be transmitted through the link. And the link error ultimately affects the accuracy of robot.

The actual trajectory of the end link can be obtained by measurement. And the measurement obtains the six kinematic parameters of the end link. It contains three translations and three rotations.^14–17 As the number of joints increases, the equations for parameter identification will also increase, which will increase the difficulty of identification. In order to reduce the difficulty of parameter identification, many scholars put forward a variety of identification methods.^18,19 Renders et al. propose a new technique for the calibration of robots base on a maximum likelihood approach for the identification of geometrical errors. ²⁰ But this approach does not take into account the non-geometrical errors. Since robot motion involves error and noise, the classic methods are not stable. So, Ohtsuki et al.  propose a method to absorb an error of real and virtual angles by three kinds of learning trees on the identification of robot parameters. ²¹ Yu et al.  propose a Bayesian method to acquire the motion intention in a human-robot and neural networks are used to compensate the uncertainties. And the experiment shows that the approach is feasible. ²² To improve the absolute positioning accuracy of robot. Luo et al. propose the pre-identification of the kinematic parameter deviations of robot was achieved by using the Levenberg-Marquardt algorithm. ²³ However, due to large number of parameters, these approaches may result in non-homogeneity and low convergence.

To further improve the identification accuracy and efficiency, a new error model of robot including link errors and kinematic pair errors is presented in this paper. In this model, the direction of the kinematic trajectory is represented by the spherical image of the ruled surface. The position of the kinematic trajectory is represented by the waistline parameter of the ruled surface. The kinematic parameter compensation based on this model can overcome the low convergence and non-homogeneity of the common methods.

The end trajectory error of 6R serial robot

Due to the geometric shape error and the size error of the mating surface of the robot parts, the kinematic pair error is universal. For revolute pairs, the center line of the rotating shaft has a deviation in rotation. And the link is installed on a revolute pair, and the error of the revolute pair will affect the trajectory of the end link. For 6R serial robots, the end trajectory error is the superposition of the errors of six joints.

The vector equation of the kinematic trajectory on a revolute pair

As shown in Figure 1, in the real mechanism, the error of the real motion axis of kinematic pair $R_i$ relative to the ideal axis can be expressed by five parameters.^24–27 Here the angle of rotation is ideal. So the motion trajectory of the kinematic pair can be represented by a ruled surface. Then, the coordinate frame $\{O_i;X_i,Y_i,Z_i\}$ is established on the ideal axis. The ideal axis is expressed as a vector $r_{\mathrm{oi}},l_{\mathrm{oi}}$ in the coordinate frame $O_i$ . The description of the real axis of motion in the ideal coordinate system is as follows:

$$r_{P^{`}i}=r_{oi}+M_i{l}_{oi}{l}_{P^{`}i}=M_i{l}_{oi}$$

(1)

Figure 1.

Actual motion of the revolute pair.

Where $r_{P^{}`i}$ is the position vector describing the real motion trajectory in the ideal frame $O_i$ and $l_{P^{}`i}$ is the direction vector describing the real motion trajectory in the ideal frame $O_i$ . Where $M_i$ can be expressed as follows:

$$M_i=\left[\begin{array}{ccc}c{\alpha }_i\hfill & -s{\alpha }_i\hfill & 0\hfill \\ \multicolumn{1}{c}{s{\alpha }_i}& c{\alpha }_i\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \end{array}\right]\times \left[\begin{array}{ccc}c{\beta }_i\hfill & 0\hfill & s{\beta }_i\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill \\ \multicolumn{1}{c}{-s{\beta }_i}& 0\hfill & c{\beta }_i\hfill \end{array}\right]\times \left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& c{\gamma }_i\hfill & -s{\gamma }_i\hfill \\ \multicolumn{1}{c}{0}& s{\gamma }_i\hfill & c{\gamma }_i\hfill \end{array}\right]$$

(2)

Where letters c and s stand for cosine and sine.

When the link $L_{i+1}$ is installed on the revolute pair $R_i$ , the link moves around the instantaneous axis $L_i^i$ . The pose of the point $P_{i+1}$ on the link $L_{i+1}$ in the coordinate frame $O_i$ is represented by $r_{\mathrm{Pi}+1}$ and $l_{\mathrm{Pi}+1}$ . With the motion of the link $L_{i+1}$ , the trajectory of the point $P_{i+1}$ in frame $O_i$ can be described as a space curve ${\Gamma }_{\mathrm{Pi}+1}$ , the vector equation can be written as:

$$\left[\begin{array}{c}{r}_{i+1}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]=\left[\begin{array}{c}{T}_i^{i+1}\hfill \end{array}\right]\left[\begin{array}{c}{r}_i\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]$$

(3)

$R_i$ is the rotation matrix from frame i+1 to frame i. The rotation matrix is as follows:

$$\left[T_i^{i+1}\right]=\left[\begin{array}{cc}\left[R_i^{i+1}\right]\hfill & r_{O(i+1)}\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right]$$

(4)

The DH parameter expression of equation (4) is as follows:

$$\left[T_i^{i+1}\right]=\left[\begin{array}{cccc}c{\theta }_i\hfill & -s{\theta }_{i}c{\alpha }_i\hfill & s{\theta }_{i}s{\alpha }_i\hfill & a_{i}c{\theta }_i\hfill \\ \multicolumn{1}{c}{s{\theta }_i}& c{\theta }_{i}c{\alpha }_i\hfill & -s{\alpha }_{i}c{\theta }_i\hfill & a_{i}s{\theta }_i\hfill \\ \multicolumn{1}{c}{0}& s{\alpha }_i\hfill & c{\alpha }_i\hfill & d_i\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$$

(5)

Where ${\theta }_i$ is the driving parameters of the revolute pairs.

In equation (3), where $a_i,{\alpha }_i,d_i$ are real link parameters with errors. When the link rotates around the revolute pair, the error of the revolute pair can be transmitted through the link, which affects the trajectory of the points and lines on the link. Therefore, the point $P_{i+1}$ on the link is described in the coordinate frame $\{O_i;X_i,Y_i,Z_i\}$ as follows:

$$r_{P`i+1}=r_{P`i}+r_{\mathrm{Pi}+1}{l}_{P`i+1}=l_{P`i}+l_{\mathrm{Pi}+1}$$

(6)

The discrete trajectory of the link $L_{i+1}$ forms the ruled surface ${\Sigma }_{L_{i+1}}$ in the coordinate frame $i$ . The discrete form of the vector equation of the ruled surface ${\Sigma }_{L_i}$ can be expressed as follows:

$${\Sigma }_{L_i}^{(t)}:r_{L_i}^{(t)}={\rho }_i^{\left(t\right)}+{\lambda }_{\rho }{l}_i^{\left(\mathrm{t}\right)};t=1,\dots ,n$$

(7)

Where ${\rho }_i^{\left(t\right)}$ and $l_i^{\left(\mathrm{t}\right)}$ are the discrete position vector and discrete spherical image on the ruled surface ${\Sigma }_{L_i}^{(t)}$ .

Kinematics model of 6R serial robot

As shown in Figure 2, the motion coordinate $\{O_i;i_i,j_i,k_i\}$ ( $i$ =1, 2, …, 6) is established on the six revolute pairs respectively.

Figure 2.

Kinematics model of 6R serial robot.

The vector equation from coordinate frame $i+1$ to coordinate frame $i$ can be expressed as equation (6). According to the kinematics model, in the coordinate frame $\{O_i;i_i,j_i,k_i\}$ , the trajectory of the end of the link 6 can be expressed as follows:

$$r_{p6}=\sum _{i=1}^6{r}_{\mathrm{pi}};l_{p6}=\sum _{i=1}^6{l}_{\mathrm{pi}}$$

(8)

Since the error of the kinematic pair is much smaller than the error of the link, the error described by the equation (1) can be approximately described as the link rotating around the ideal axis $l_0$ of the kinematic pair. Then the equation can be expressed as follows:

$$\left[\begin{array}{c}{r}_{\mathrm{PF}}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]=\left[\underset{i=1}{\overset{6}{\Pi }}{T}_i^{i+1}\right]\left[\begin{array}{c}{r}_{\mathrm{PF}6}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]\left[\begin{array}{c}{l}_F\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]=\left[\underset{i=1}{\overset{6}{\Pi }}{T}_i^{i+1}\right]\left[\begin{array}{c}{l}_{F6}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]$$

(9)

Where $r_{\mathrm{PF}6}$ represents the coordinates of point $P_{F6}$ and $l_{F6}$ represents the unit direction vector of $L_{F6}$ in the coordinate frame $\{O_6;i_6,j_6,k_6\}$ .

The method of the kinematic parameter identification for robot

Equations (3)–(5) indicate that the revolute pair error will cause a position error at the end point of the link. The parameters of the axis can be obtained by fitting the motion trajectory. The distance between two adjacent axes is the real rod length. After the real rod length is obtained, the kinematic parameters of the robot are compensated by inverse solution of the equation (9) to obtain higher trajectory accuracy.

The identification process of kinematic parameters

A measuring tool with three tapered points is fixed on the end of the robot, as shown in Figure 3. And the coordinates of the three tapered points are used to fit the coordinates of the end points.

Figure 3.

The measuring tool.

The measurement of the tapered point at the end link

In order to obtain more accurate tapered point coordinates, four measurements are made for each tapered point. Then, the sphere fitting is performed on the four coordinates, and the fitting sphere center is used to represent the real coordinates of the tapered point. The calculation process is as follows:

$$r_{\mathrm{Hu}}^{\left(t\right)}={[x_{\mathrm{Hu}}^{\left(t\right)},y_{\mathrm{Hu}}^{\left(t\right)},z_{\mathrm{Hu}}^{\left(t\right)}]}^T$$

(10)

$$r_{\mathrm{Hu}}^{\left(t\right)}={\left[A_u^{\left(t\right)}\right]}^{-1}\left[H_u^{\left(t\right)}\right]$$

(11)

Where the $u$ represents the number of the tapered point.

Matrixes $\left[A_u^{\left(t\right)}\right]$ and $\left[H_u^{\left(t\right)}\right]$ are written as:

$$\left[A_u^{\left(t\right)}\right]=\left[\begin{array}{ccc}{x}_{u2}^{\left(t\right)}-x_{u1}^{\left(t\right)}\hfill & y_{u2}^{\left(t\right)}-y_{u1}^{\left(t\right)}\hfill & z_{u2}^{\left(t\right)}-z_{u1}^{\left(t\right)}\hfill \\ \multicolumn{1}{c}{x_{u3}^{\left(t\right)}-x_{u1}^{\left(t\right)}}& y_{u3}^{\left(t\right)}-y_{u1}^{\left(t\right)}\hfill & z_{u3}^{\left(t\right)}-z_{u1}^{\left(t\right)}\hfill \\ \multicolumn{1}{c}{x_{u4}^{\left(t\right)}-x_{u1}^{\left(t\right)}}& y_{u4}^{\left(t\right)}-y_{u1}^{\left(t\right)}\hfill & z_{u4}^{\left(t\right)}-z_{u1}^{\left(t\right)}\hfill \end{array}\right]$$

(12)

$$\left[H_u^{\left(t\right)}\right]=\left[\begin{array}{c}(x_{u2}^{\left(t\right)}-x_{u1}^{\left(t\right)})(x_{u2}^{\left(t\right)}+x_{u1}^{\left(t\right)})+(y_{u2}^{\left(t\right)}-y_{u1}^{\left(t\right)})(y_{u2}^{\left(t\right)}+y_{u1}^{\left(t\right)})+(z_{u2}^{\left(t\right)}-z_{u1}^{\left(t\right)})(z_{u2}^{\left(t\right)}+z_{u1}^{\left(t\right)})\hfill \\ \multicolumn{1}{c}{(x_{u3}^{\left(t\right)}-x_{u1}^{\left(t\right)})(x_{u3}^{\left(t\right)}+x_{u1}^{\left(t\right)})+(y_{u3}^{\left(t\right)}-y_{u1}^{\left(t\right)})(y_{u3}^{\left(t\right)}+y_{u1}^{\left(t\right)})+(z_{u3}^{\left(t\right)}-z_{u1}^{\left(t\right)})(z_{u3}^{\left(t\right)}+z_{u1}^{\left(t\right)})}\\ \multicolumn{1}{c}{(x_{u4}^{\left(t\right)}-x_{u1}^{\left(t\right)})(x_{u4}^{\left(t\right)}+x_{u1}^{\left(t\right)})+(y_{u4}^{\left(t\right)}-y_{u1}^{\left(t\right)})(y_{u4}^{\left(t\right)}+y_{u1}^{\left(t\right)})+(z_{u4}^{\left(t\right)}-z_{u1}^{\left(t\right)})(z_{u4}^{\left(t\right)}+z_{u1}^{\left(t\right)})}\end{array}\right]$$

(13)

Where ( $x_{\mathrm{uv}`}^{\left(t\right)}{y}_{\mathrm{uv}`}^{\left(t\right)}{z}_{\mathrm{uv}}^{\left(t\right)}$ ) is the coordinate of $H_{\mathrm{uv}}$ (v =1, 2, 3, 4) measured by the coordinate measuring arm.

The calculation of the center point at the end link

The coordinates $r_c^{\left(t\right)}$ of the center point at the end link can be determined by the coordinates of three tapered points. The calculation process is as follows:

$$r_c^{\left(t\right)}={\left[x_{c`}^{\left(t\right)}{y}_{c`}^{\left(t\right)}{z}_c^{\left(t\right)}\right]}^T=-{\left[C_u^{\left(t\right)}\right]}^{-1}\left[D_u^{\left(t\right)}\right]$$

(14)

Where $\left[C_u^{\left(t\right)}\right]$ and $\left[D_u^{\left(t\right)}\right]$ are represented as follows

$$\begin{array}{c}\left[C_u^{\left(t\right)}\right]=\left[\begin{array}{ccc}a1\hfill & b1\hfill & c1\hfill \\ \multicolumn{1}{c}{a2}& b2\hfill & c2\hfill \\ \multicolumn{1}{c}{a3}& b3\hfill & c3\hfill \end{array}\right]\hfill \\ a1=y_{H1}^{\left(t\right)}(z_{H2}^{\left(t\right)}-z_{H1}^{\left(t\right)})+y_{H2}^{\left(t\right)}(z_{H3}^{\left(t\right)}-z_{H1}^{\left(t\right)})+y_{H1}^{\left(t\right)}(z_{H2}^{\left(t\right)}-z_{H1}^{\left(t\right)})\hfill \\ b1=z_{H1}^{\left(t\right)}(x_{H2}^{\left(t\right)}-x_{H1}^{\left(t\right)})+z_{H2}^{\left(t\right)}(x_{H3}^{\left(t\right)}-x_{H1}^{\left(t\right)})+z_{H1}^{\left(t\right)}(x_{H2}^{\left(t\right)}-x_{H1}^{\left(t\right)})\hfill \\ c1=x_{H1}^{\left(t\right)}(y_{H2}^{\left(t\right)}-y_{H1}^{\left(t\right)})+x_{H2}^{\left(t\right)}(y_{H3}^{\left(t\right)}-y_{H1}^{\left(t\right)})+x_{H1}^{\left(t\right)}(y_{H2}^{\left(t\right)}-y_{H1}^{\left(t\right)})\hfill \\ a2=2(x_{H2}^{\left(t\right)}-x_{H1}^{\left(t\right)})\hfill \\ b2=2(y_{H2}^{\left(t\right)}-y_{H1}^{\left(t\right)})\hfill \\ c2=2(z_{H2}^{\left(t\right)}-z_{H1}^{\left(t\right)})\hfill \\ a3=2(x_{H3}^{\left(t\right)}-x_{H1}^{\left(t\right)})\hfill \\ b3=2(y_{H3}^{\left(t\right)}-y_{H1}^{\left(t\right)})\hfill \\ c3=2(z_{H3}^{\left(t\right)}-z_{H1}^{\left(t\right)})\hfill \end{array}$$

(15)

$$\left[D_u^{\left(t\right)}\right]=\left[\begin{array}{c}{x}_{H1}^{\left(t\right)}(y_{H3}^{\left(t\right)}{z}_{H2}^{\left(t\right)}-y_{H2}^{\left(t\right)}{z}_{H3}^{\left(t\right)})+x_{H2}^{\left(t\right)}(y_{H1}^{\left(t\right)}{z}_{H3}^{\left(t\right)}-y_{H3}^{\left(t\right)}{z}_{H1}^{\left(t\right)})+x_{H3}^{\left(t\right)}(y_{H2}^{\left(t\right)}{z}_{H1}^{\left(t\right)}-y_{H1}^{\left(t\right)}{z}_{H2}^{\left(t\right)})\hfill \\ \multicolumn{1}{c}{(x_{H1}^{\left(t\right)}+x_{H2}^{\left(t\right)})(x_{H1}^{\left(t\right)}-x_{H2}^{\left(t\right)})+(y_{H1}^{\left(t\right)}+y_{H2}^{\left(t\right)})(y_{H1}^{\left(t\right)}-y_{H2}^{\left(t\right)})+(z_{H1}^{\left(t\right)}+z_{H2}^{\left(t\right)})(z_{H1}^{\left(t\right)}-z_{H2}^{\left(t\right)})}\\ \multicolumn{1}{c}{(x_{H1}^{\left(t\right)}+x_{H3}^{\left(t\right)})(x_{H1}^{\left(t\right)}-x_{H3}^{\left(t\right)})+(y_{H1}^{\left(t\right)}+y_{H3}^{\left(t\right)})(y_{H1}^{\left(t\right)}-y_{H3}^{\left(t\right)})+(z_{H1}^{\left(t\right)}+z_{H3}^{\left(t\right)})(z_{H1}^{\left(t\right)}-z_{H3}^{\left(t\right)})}\end{array}\right]$$

(16)

According to the kinematic geometry, three non-collinear points on a link can be used to determine the discrete motion of the link. ²³ So, the tapered points $H_1,H_2$ and $H_3$ are used to create the motion coordinate frame $\{O_H;i_H,j_H,k_H\}$ . This coordinate frame is represented as follows:

$$\begin{array}{c}{r}_{\mathrm{OH}}^{\left(t\right)}=r_c^{\left(t\right)};i_H^{\left(t\right)}=\frac{r_{H2}^{\left(t\right)}-r_{H1}^{\left(t\right)}}{|r_{H2}^{\left(t\right)}-r_{H1}^{\left(t\right)}|};k_H^{\left(t\right)}\hfill \\ =\frac{(r_{H2}^{\left(t\right)}-r_{H1}^{\left(t\right)})\times (r_{H3}^{\left(t\right)}-r_{H1}^{\left(t\right)})}{|(r_{H2}^{\left(t\right)}-r_{H1}^{\left(t\right)})\times (r_{H3}^{\left(t\right)}-r_{H1}^{\left(t\right)})|};\hfill \\ j_H^{\left(t\right)}=k_H^{\left(t\right)}\times i_H^{\left(t\right)};\hfill \end{array}$$

(17)

The discrete motion can be calculated as follows:

$$\{\begin{array}{c}{[x_H^{\left(t\right)},y_H^{\left(t\right)},z_H^{\left(t\right)}]}^T=r_c^{\left(t\right)}\hfill \\ \multicolumn{1}{c}{\{\begin{array}{c}{\alpha }_H^{\left(t\right)}=\mathrm{atan}2(i_{H2}^{\left(t\right)},i_{H1}^{\left(t\right)});\hfill \\ \multicolumn{1}{c}{{\beta }_H^{\left(t\right)}=\mathrm{atan}2(-i_{H3}^{\left(t\right)},\sqrt{{\left(i_{H1}^{\left(t\right)}\right)}^2+{\left(i_{H2}^{\left(t\right)}\right)}^2});}\\ \multicolumn{1}{c}{{\gamma }_H^{\left(t\right)}=\mathrm{atan}2(j_{H3}^{\left(t\right)},k_{H3}^{\left(t\right)})}\end{array}}\end{array}$$

(18)

Where atan2 stand for the double parameters arctangent function. Variables $i_{\mathrm{Hk}}^{\left(t\right)},j_{\mathrm{Hk}}^{\left(t\right)}$ and $k_{\mathrm{Hk}}^{\left(t\right)}$ are the kth elements of vectors $i_H^{\left(t\right)},j_H^{\left(t\right)}$ and $k_H^{\left(t\right)}$ , respectively.

The motion trajectory of a single joint

When the robot rotates with a single joint, the motion trajectory formed by the end of the robot in space can be represented by a discrete spherical image $\{l_i^{\left(t\right)}\}$ , as shown in Figure 4. The discrete points ${\rho }_i^{\left(t\right)}$ of the trajectory can be expressed as follows:

$${\rho }_i^{\left(t\right)}=r_{\mathrm{Pi}}^{\left(t\right)}+b_L^{\left(t\right)}{l}_i^{\left(t\right)}$$

(19)

Figure 4.

Spherical image circle fitting.

The linear trajectory parameters of the real rotary motion are presented as discrete parameter sets $\{l^{(i)}\}$ and $\{R^{(i)}\}$ in the fixed coordinate frame. Then use the rotational motion invariant parameter to describe the motion axis direction and position vector. Where the deflection angle between $\{l^{(i)}\}$ and the fitting spherical image circle is described by the spherical image error. The distance deviation between $\{R^{(i)}\}$ and the fitting axis is described by the distance error from the straight generatrix of the rotary motion to the axis. The axis position and direction parameters of the real rotary motion can be obtained by fitting.

The vector equation of the axis fitting of the kinematic pair

The fitting axis is divided into two steps: axis direction fitting and axis position fitting.

The axis direction fitting function is expressed as follows:

$$\{\begin{array}{c}{\mathrm{\Delta }}_{\delta }=\underset{a}{min}\underset{1\le i\le n}{max}\{g_l^{\left(i\right)}(a)\}\hfill \\ \multicolumn{1}{c}{g_l^{\left(i\right)}\left(a\right)=\left|\begin{array}{c}\arccos (l_0·l^{\left(i\right)}-{\delta }_0)\hfill \end{array}\right|}\\ \multicolumn{1}{c}{s.t.{\delta }_f\in [0,\pi ),{\theta }_f\in [0,2\pi ),{\delta }_0\in [0,\pi /2]}\\ \multicolumn{1}{c}{a={({\delta }_f,{\theta }_f,{\delta }_0)}^T}\end{array}$$

(20)

Where $l_0$ is the fitting axis direction, ${\delta }_f$ and ${\theta }_f$ are the direction angles of the $l_0$ . As shown in Figure 3, where ${\delta }^{(i)}$ is the angle between the discrete line and the fitting axis, and the ${\delta }_0$ is the half-cone angle of the fitting spherical image circle.

The axis position fitting function is expressed as follows:

$$\{\begin{array}{c}{\mathrm{\Delta }}_r=\underset{b}{\min }\underset{1\le i\le n}{\max }\{g^{(i)}(b)\}\\ g^{(i)}(b)=\left|r^{(i)}-r_0\right|\\ s.t.r_0\in (0,+\infty )\\ a={(x_f,y_f,r_0)}^T\end{array}$$

(21)

Where $r^{(i)}$ is the fitting circle radius of the projection points of the discrete waist points and the $r_0$ is the fitting circle radius of the discrete waist points. The projection point of the discrete waist point on the ∏ plane is shown in Figure 5:

Figure 5.

Striction circle fitting.

The kinematic parameter identification of adjacent axes

The vector equation of the axis $S_a$ of the ruled surface ${\Sigma }_L$ corresponding to the discrete trajectory line cluster $\left\{r_{\rho }^{(t)}\right\}$ can be expressed as follows:

$$r_{S_a}=r_Q+\mu S_a$$

(22)

Where $r_Q$ and $S_a$ have the same meaning as in equation (6).

The equations in discrete form of the fitting axes of the six joints during the measurement process can be expressed as follows:

$$r_{S_i^a}=r_{\mathrm{Qi}}+\mu S_i^a;i=1,\cdots ,6$$

(23)

Where equation (23) is the discrete form of equation (22). The { $r_{\mathrm{Qi}}$ } and { $S_i^a$ } are the projection point of the fitting axis and the direction vector of the fitting axis, respectively.

The distance $r_i$ between any two axes can be calculated from the vector equation of the fitting axis, and this distance is the rod length.

$$r_i=(r_{\mathrm{Qi}}-r_{Q(i+1)})·\frac{S_i^a\times S_{i+1}^a}{|S_i^a\times S_{i+1}^a|}$$

(24)

Where $S_i^a$ and $S_{i+1}^a$ can be calculated from equation (5), and $r_{\mathrm{Qi}}$ and $r_{Q(i+1)}$ can be calculated from equation (6).

The identification approach and example verification

The approach based on the parameter solving of the ruled surface model is used to identify the kinematic parameters of the robot. The real rod length is determined by the identified kinematic parameters, and the robot is compensated for verification.

The LR7-710 robot is used to verify the proposed approach, as shown in Figure 6. The discrete motions of the end link generated by revolute pairs are measured, respectively. During the measurement, each joint is controlled individually, which means that when one joint is measured, the others are kept in the original positions. For each joint, 8 discrete positions are chosen to determine the position and orientation of the axis. And the process is as follows:

• Step 1: Calculate the coordinates $r_{\mathrm{Hu}}^t$ ( $u$  = 1,2,3) of the tapered points according to equations (11)–(13). The detailed data of $r_{\mathrm{Hu}}^t$ of the revolute pair $R_1,R_2$ , …, $R_6$ are shown in Appendix 1.

• Step2: Calculate the center point $r_c^{\left(t\right)}(t=1,2,\dots ,8)$ at the end link according to equations (14)–(16). At the same time, the other three kinematic parameters of the end link are calculated according to equations (17)–(18). The kinematic parameters of the end link of the revolute pair $R_1,R_2$ , …, $R_6$ are shown in Table 1:

• Step 3: Calculate the position and orientation of the axis

• In this step, ideally, since the axis of rotation of the sixth joint is parallel to the central axis. Therefore, the position and orientation of the first five axes and the position and orientation of the sixth axis are calculated independently.

Figure 6.

The environment of experiment.

Table 1.

The kinematic parameters of the end link.

$R_1$
$x_H^{\left(t\right)}/\mathrm{mm}$	$y_H^{\left(t\right)}/\mathrm{mm}$	$z_H^{\left(t\right)}/\mathrm{mm}$	${\alpha }_H^{\left(t\right)}/°$	${\beta }_H^{\left(t\right)}/°$	${\gamma }_H^{\left(t\right)}/°$
−860.0330	−67.6754	71.9689	−175.4477	−4.8103	−89.9355
−762.2371	29.6052	71.2669	177.8291	−5.3824	−89.9469
−632.3733	76.5108	71.2462	173.1487	−6.4320	−89.9458
−494.6837	63.9432	71.9444	172.6601	−8.2653	−89.9316
−375.1499	−5.4436	73.0003	−179.2064	−11.0795	−89.8775
−296.4199	−119.2096	74.9365	−158.2245	−13.2911	−77.0398
−273.1414	−255.5199	76.3876	−137.0281	−11.6432	89.8709
−309.3696	−389.0032	77.2939	−128.5527	−8.9212	89.9282
$R_2$
$x_H^{\left(t\right)}/\mathrm{mm}$	$y_H^{\left(t\right)}/\mathrm{mm}$	$z_H^{\left(t\right)}/\mathrm{mm}$	${\alpha }_H^{\left(t\right)}/°$	${\beta }_H^{\left(t\right)}/°$	${\gamma }_H^{\left(t\right)}/°$
−552.5120	−163.2729	155.7464	−163.5432	−15.1367	−89.7590
−514.3474	−90.6356	141.5708	−170.0077	−15.1881	−89.8343
−477.4069	−21.3793	113.6670	−177.4323	−13.4189	−89.8815
−443.3643	42.4766	72.4387	174.5338	−9.2996	−89.9183
−413.0802	98.8700	19.2624	166.5472	−2.6834	−89.9485
−387.6968	145.9315	−44.0881	159.3777	5.9667	−89.9753
−367.6623	182.8564	−115.9338	153.5538	15.6462	−90.0006
−353.6031	208.1740	−193.7184	149.5006	25.1506	−90.0261
$R_3$
$x_H^{\left(t\right)}/\mathrm{mm}$	$y_H^{\left(t\right)}/\mathrm{mm}$	$z_H^{\left(t\right)}/\mathrm{mm}$	${\alpha }_H^{\left(t\right)}/°$	${\beta }_H^{\left(t\right)}/°$	${\gamma }_H^{\left(t\right)}/°$
−624.8422	−298.0806	411.2905	−154.4124	−30.7017	89.1146
−589.1969	−230.2087	396.0118	−158.5651	−32.0459	−80.9742
−555.4201	−166.0234	367.3938	−163.2565	−32.3757	−89.2633
−524.2372	−107.1546	326.9101	−168.3415	−31.4498	−89.6227
−496.2913	−55.3262	275.4842	−173.5320	−28.9291	−89.7565
−473.4168	−12.3400	214.3683	−178.4052	−24.4193	−89.8339
−455.4173	20.5999	146.2816	177.5021	−17.8719	−89.8815
−443.2317	42.1105	72.0286	174.6513	−9.2812	−89.9188
$R_4$
$x_H^{\left(t\right)}/\mathrm{mm}$	$y_H^{\left(t\right)}/\mathrm{mm}$	$z_H^{\left(t\right)}/\mathrm{mm}$	${\alpha }_H^{\left(t\right)}/°$	${\beta }_H^{\left(t\right)}/°$	${\gamma }_H^{\left(t\right)}/°$
−442.9558	40.4912	−45.2186	174.8576	5.8910	−90.0852
−479.8546	60.0868	−28.3003	172.9074	3.4623	−90.0480
−495.4706	68.5024	13.1604	172.1436	−1.4204	−90.0012
−480.3434	61.1020	54.8716	172.7514	−6.4424	−89.9549
−443.5068	41.9731	72.3702	174.6006	−9.2889	−89.9188
−406.4927	22.5168	55.3860	176.8707	−7.8766	−89.9207
−391.0340	14.0306	13.9522	178.0322	−2.1528	−90.0010
−406.1730	21.6392	−27.7947	177.0535	3.9049	−90.0827
$R_5$
$x_H^{\left(t\right)}/\mathrm{mm}$	$y_H^{\left(t\right)}/\mathrm{mm}$	$z_H^{\left(t\right)}/\mathrm{mm}$	${\alpha }_H^{\left(t\right)}/°$	${\beta }_H^{\left(t\right)}/°$	${\gamma }_H^{\left(t\right)}/°$
−509.6611	−83.2182	125.0120	−170.7143	−13.5568	89.5890
−486.3424	−38.4425	131.3540	−175.4542	−15.0465	−88.5540
−463.7104	4.3436	115.6612	179.4912	−14.0197	−89.7607
−446.0663	37.2093	80.9994	175.2438	−10.3119	−89.9017
−436.7024	53.9965	33.9286	172.9325	−4.4939	−89.9755
−437.4248	51.4100	−16.7162	173.2392	2.0800	−90.0393
−448.1472	30.1801	−61.6475	176.0543	7.7379	−90.1225
−466.5987	−5.7320	−92.3336	−179.4106	11.1494	−90.3148
$R_6$
$x_H^{\left(t\right)}/\mathrm{mm}$	$y_H^{\left(t\right)}/\mathrm{mm}$	$z_H^{\left(t\right)}/\mathrm{mm}$	${\alpha }_H^{\left(t\right)}/°$	${\beta }_H^{\left(t\right)}/°$	${\gamma }_H^{\left(t\right)}/°$
−435.9483	54.3804	13.7452	172.8555	−1.8832	−90.0007
−436.1644	54.4934	13.8180	172.8316	−1.7951	−90.0006
−436.4252	54.5905	13.6870	172.8381	−1.6836	−90.0011
−436.4886	54.6730	13.4581	172.8620	−1.6177	−90.0011
−436.3900	54.6297	13.2438	172.8988	−1.6332	−90.0013
−436.2012	54.5137	13.1519	172.9237	−1.7190	−90.0009
−436.0020	54.4046	13.2421	172.9198	−1.8254	−90.0011
−435.8969	54.3952	13.3957	172.8857	−1.8815	−90.0007

The position vectors $p_l^i$ of the first five axes can be calculated using equations (14)–(16), and their orientation vectors $d_l^i$ can be calculated using equation (17). The detailed data is shown in Table 2:

Table 2.

The position and orientation of the first five axes.

	Position vectors $p_l^i$			Orientation vectors $d_l^i$
	x/mm	y/mm	z/mm	x/mm	y/mm	z/mm
The 1st joint	−591.6950	−240.1650	74.6314	−0.0000	0.0001	0.0134
The 2nd joint	−567.7010	−199.4880	−320.1740	−0.0022	0.0011	−0.0000
The 3rd joint	−643.6680	−343.3600	−33.1536	−0.0022	0.0011	−0.0000
The 4th joint	−443.2360	41.2968	13.5847	−0.0028	−0.0053	0.0000
The 5th joint	−490.3450	−48.7564	14.3706	−0.0021	0.0011	−0.0000

The first five axes are shown in the following Figure 7 and Figure 8:

Figure 7.

The fitting axes: (a) the axis of the 1st joint, (b) the axis of the 2nd joint, (c) the axis of the 3rd joint, (d) the axis of the 4th joint, and (e) the axis of the 5th joint.

Figure 8.

The first five axes.

The position and orientation of the sixth axis is calculated based on the parameter solving of the ruled surface model. The position $p_l^6$ can be calculated using equation (20) and the orientation $d_l^6$ can be calculated using equation (21). The calculation results are as follows:

$$p_l^6=(-436.1844,54.5066,13.4940{)}^T$$

$$d_l^6=(0.8870,-0.0090,-0.4618{)}^T$$

And the ruled surface formed by the 6th joint is shown in Figure 9:

• Step 4: Calculate the kinematic parameters.

Figure 9.

The ruled surface formed by the 6th joint.

As shown in Figure 2, $a_1,a_2$ and $a_3$ are the distance between adjacent joint axes. The $d_4,a_3$ and the distance between the position vector of the third axis and the fifth axis form a right triangle. Where $a_3$ and $d_4$ are the two right-angled sides of a right triangle. Since, d4 can be calculated by solving right triangle. And d6 can be calculated according to equation (24) . Finally, the real rod length and nominal rod length are shown in the Table 3:

Table 3.

The value of real and nominal rod length.

	The nominal rod length (mm)	The real rod length (mm)
$a_1$	50	49.8177
$a_2$	330	329.9069
$a_3$	50	50.3894
$d_4$	332	331.6903
$d_6$	117	117.4737

• Step 5: Compensate errors by inverse solution of kinematic equation

The identified parameters are used to compensate these errors by inverse solution of the equation (9). After errors compensation is completed, the trajectory error can be calculated before and after compensation by controlling the robot to walk the arc trajectory, as shown in Figure 10:

Figure 10.

The trajectory error before and after compensation.

The results show the trajectory errors are decreased enormously with compensation.

Conclusions

A new approach for kinematic parameter compensation of robot based on the parameter solving of the ruled surface model is developed by establishing a model containing kinematic pair error and link error. The results are summarized as follows.

1. A model containing kinematic pair error and link error is established. The model provides a basis for the identification of robot parameters.

2. A new approach for identification of robot rod length error is proposed. This approach mainly solves from the geometrical aspect. This approach calculates the real axis by fitting the single-joint motion trajectory.

3. Take the LR7-710 robot as an example, using the identification approach proposed in this paper to complete the identification of the rod length error. And the effectiveness of the approach is verified by controlling the robot to walk the arc trajectory. The position error of the arc trajectory is reduced from 0.4580 to 0.0801 mm.