CN107627299A - A kind of kinematic parameter errors scaling method of rope driving parallel robot - Google Patents

Abstract

The invention discloses a method for calibrating the kinematics parameter error of a rope-driven parallel robot, comprising: 1. establishing a kinematics model of the rope-driven parallel robot and a kinematic parameter error model of the rope-driven parallel robot; Identification, 3. Optimizing the positional relationship between the reference coordinate system of the calibration and attitude measuring device relative to the reference coordinate system of the cable-driven parallel robot, 4. Combining with iterative optimization algorithm to jointly calibrate the kinematic parameter error. The invention can accurately calibrate the kinematics parameter error of the cable-driven parallel robot, thereby improving the kinematics precision of the cable-driven parallel robot.

Description

A Kinematic Parameter Error Calibration Method for a Cable-Driven Parallel Robot

technical field

The invention relates to the field of robot kinematics parameter calibration, in particular to a method for calibrating errors of kinematics parameters of a cable-driven parallel robot based on pose measurement.

Background technique

The mechanical structure of the robot will inevitably produce some structural parameter errors in the process of processing and assembly, and these structural parameter errors will lead to errors in the calculation results of the kinematic model of the robot. Therefore, in the robot production process, it is necessary to calibrate the kinematic parameter error of the robot.

Commonly used calibration tools include position and attitude measuring devices such as cameras and laser interferometers, which can measure the position and attitude of the moving platform at the end of the robot actuator. The measuring device uses its own coordinate system as the reference coordinate system to measure the pose of the robot end effector, so the relative positional relationship between the robot reference coordinate system and the measuring device reference coordinate system must be measured before calibrating the robot kinematics parameter error. However, due to the error between the reference measurement position of the measuring device and the robot structure, there is an error between the measured relative position relationship and the actual position relationship, which reduces the error accuracy of the final calibrated robot kinematics parameters. Many robots need to be calibrated in the industrial production process. The traditional method requires manual calibration of the positional relationship of the measuring device, resulting in slow calibration speed, cumbersome calibration process, and low production efficiency. Therefore, there is an urgent need for a calibration method that can be automatically calibrated and has higher precision, in order to improve production efficiency and expand economic benefits.

Contents of the invention

In order to overcome the deficiencies of the prior art, the present invention provides a method for calibrating the kinematic parameter error of a rope-driven parallel robot, in order to accurately calibrate the kinematic parameter error of the rope-driven parallel robot, thereby improving the kinematics of the rope-driven parallel robot precision.

The present invention adopts following technical scheme in order to achieve the above-mentioned purpose of the invention:

The present invention relates to a kinematic parameter error calibration method of a cable-driven parallel robot, which is applied in the calibration process of the cable-driven parallel robot, and a pose measuring device is arranged on one side of the cable-driven parallel robot; the characteristic is that the The kinematic parameter error calibration method is carried out according to the following steps:

Step 1. Establish the reference coordinate system O _s of the cable-driven parallel robot, the reference coordinate system O _p on the moving platform at the end of the robot, and the reference coordinate system of the pose measurement device within the measurement working range of the pose measurement device O _c , and using the reference coordinate system O _c of the pose measurement device as the world coordinate system;

Step 2, establishing a kinematics model of the rope-driven parallel robot;

Step 2.1. For a cable-driven parallel robot with n degrees of freedom output by an m cable-driven device, the theoretical pose of the terminal moving platform in the reference coordinate system O _s of the cable-driven parallel robot is expressed as X _s =[P _s Φ _s ] ^T , where P _s represents the position of the end moving platform, and Φ _s represents the attitude of the end moving platform;

Let b _i represent the position of the i-th rope driving device on the reference coordinate system O _p on the moving platform at the end of the robot, and a _i represent the position of the rope output point on the reference coordinate system O _s on the robot platform; i = 1...m;

Use equation (1) to express the closed-chain equation of a single rope drive:

l _i ＝a _i -P _s -R _sp (Φ _s )b _i (1)

In formula (1), l _i represents the rope vector from the rope output point of the i-th rope driving device to the rope connection point of the end moving platform, and R _sp represents the reference coordinate system _Op on the end moving platform of the robot to all The rotation matrix of the reference frame O _s of the rope-driven parallel robot;

Step 2.2, obtain the inverse kinematics equation of the rope-driven parallel robot according to formula (2):

||l _i || ₂ ＝(a _i -P _s -R _sp (Φ _s )b _i ) ^T (a _i -P _s -R _sp (Φ _s )b _i ),i＝1,2,…, m (2)

From formula (2), the rope length vector of the rope-driven parallel robot is l=[||l ₁ || ₂ ||l ₂ || ₂ …||l _i || ₂ …||l _m || ₂ ] ^T ;

Step 3, establishing a kinematic parameter error model of the cable-driven parallel robot;

Step 3.1, let the robot kinematics error parameters caused by machining and assembly factors be expressed as Wherein, b _i represents the position error of the i-th rope drive device on the reference coordinate system O _p on the end-moving platform of the robot, and a _i represents the reference coordinate system of the output point of the rope on the robot platform The position error over O _s , Indicates that the unit rotation angle of the motor encoder corresponds to the output length error of the rope, and there are 7m kinematic parameter errors to be identified Cal=[Cal ₁ Cal ₂ ... Cal _i ... Cal _m ] ^T ;

Step 3.2, using the pose measuring device to measure the poses of N groups of terminal moving platforms, and using the poses measured by the j group to obtain the error of the motor encoder output angle θ _i shown in formula (3)

In formula (3), {θ _m,j } represents the actual rotation angles fed back by the m motor encoders obtained from the j-th group of measured poses, and l ^j represents the rope length vector;

Step 4, obtain the optimization objective equation shown in formula (4) by formula (3):

In formula (4), e _θ represents the error equation of the output angles of the m motor encoders obtained from the poses of the N groups of terminal moving platforms, and has:

Formula (4) is optimized and solved by the least square method, and the kinematic parameter error identification model shown in formula (5) is obtained:

In formula (5), is the Jacobian matrix of the error equation e _θ to the kinematic parameter error Cal, and W _t represents the normalized matrix of the kinematic parameter error Cal;

Step 5, calibrate the positional relationship of the reference coordinate system _Oc of the attitude measuring device relative to the reference coordinate system _Os of the cable-driven parallel robot;

Step 5.1. Let the actual measured pose of the terminal moving platform be expressed as X _cs =[P _cs Φ _cs ] ^T , where P _cs represents the reference coordinate system O _s of the cable-driven parallel robot in the pose measurement The position in the reference coordinate system O _c of the device, Φ _cs represents the attitude of the reference coordinate system O _s of the cable-driven parallel robot in the reference coordinate system O _c of the pose measurement device; then use formula (6) to express the The position P _c of the terminal moving platform in the reference coordinate system O _c of the pose measurement device:

P _c ＝R _cs (Φ _cs )P _s +P _cs (6)

In formula (6), R _cs represents the rotation matrix of the reference coordinate system O _s of the cable-driven parallel robot relative to the reference coordinate system O _c of the pose measuring device;

Step 5.2, Simplify Formula (6), so as to use Formula (7) to establish the position relationship equation between the reference coordinate system _Oc of the pose measuring device and the reference coordinate system _Os of the cable-driven parallel robot:

Utilize formula (8) to establish the position relationship optimization equation of the reference coordinate system O _c of the pose measuring device and the reference coordinate system O _s of the cable-driven parallel robot, and then use the LM algorithm to optimize and solve formula (8) to obtain the position Parameter P _cs :

Step 5.3, let P _c -P _cs =A, P _s =B, get R _cs (Φ _cs )B=A according to formula (6), then use formula (9) to get the reference coordinate system O _c of the pose measurement device With the attitude relationship equation of the reference coordinate system O _s of the cable-driven parallel robot, the LM algorithm is used to optimize and solve the formula (9), and the attitude parameter Φ _cs is obtained:

R _cs (Φ _cs )＝AB ^T (B·B ^T ) ^-1 (9)

Step 6. Jointly calibrate the kinematic parameter error;

Step 6.1, using formula (10) to calculate the mean square error SD of the pose measuring device to measure the poses of N groups of terminal moving platforms:

In formula (10), _Xc represents the actual pose of the robot terminal moving platform measured by the pose measuring device in the reference coordinate system _Oc of the pose measuring device, Indicates the theoretical pose of the moving platform at the end of the robot in the reference coordinate system _Oc of the pose measurement device;

Step 6.2, initialize the number of iterations iter=1, kinematic parameter error Cal, and the maximum number of iterations is itermax;

Step 6.3, use step 5 to calculate the relative relationship between the reference coordinate system O _c of the pose measuring device and the reference coordinate system O _s of the cable-driven parallel robot for the iter time

Step 6.4, using the least squares method to calibrate the kinematic parameter error Cal ^iter for the iter time;

Step 6.5, assign Cal ^iter + Cal to Cali ^iter , assign iter+1 to iter;

Step 6.6. If SD<<δ and iter<iter max, where δ is a minimum value, the calibration ends and ^{Cali iter} is output; if iter>iter max, the loop is terminated; if SD>δ and iter<iter max, then Return to step 6.3.

Compared with the prior art, the beneficial effects of the present invention are reflected in:

1. The present invention establishes a kinematic parameter error model, then uses a calibration algorithm to calibrate the relative positional relationship between the robot reference coordinate system and the measuring device reference coordinate system, and then uses an optimization algorithm to identify the kinematic parameter error, thereby improving the rope-driven parallel connection. Calibration accuracy of kinematic parameter error of the robot; due to measurement error in a calibration process, an iterative optimization algorithm is used to iteratively optimize the calibration process of kinematic parameter error, so that the final kinematic error is minimized and greatly improved Parameter error calibration accuracy. In the whole calibration process, the algorithm can be carried out automatically without manual intervention, which greatly improves the calibration efficiency.

2. The present invention establishes the kinematics parameter error model of the cable-driven parallel robot, and uses an optimization algorithm to identify the error, so that the calibration result of the parameter error is more accurate;

3. The present invention utilizes an optimization algorithm to calibrate the relative positional relationship between the self-reference coordinate system of the orientation measurement device and the reference coordinate system of the parallel robot, which greatly improves the measurement accuracy of the orientation measurement device.

4. The present invention adopts an iterative parameter error identification method to iteratively identify the relative positional relationship between the self-reference coordinate system of the calibrated pose measurement device and the parallel robot’s own reference coordinate system and the identified kinematic parameter errors. The entire calibration process can be automatically Execution without manual intervention greatly improves calibration efficiency and kinematic parameter error identification accuracy.

Description of drawings

Fig. 1 is a schematic diagram of the reference coordinate system in the calibration process of the present invention.

detailed description

In this embodiment, a cable-driven parallel robot kinematic parameter error calibration method based on position and attitude measurement is applied to the calibration process of the cable-driven parallel robot, and a position and attitude measurement device is installed on the side of the cable-driven parallel robot; specifically In other words, the error calibration method includes: a parallel robot kinematics parameter error identification model, a position and attitude measurement device coordinate system calibration module, and a robot kinematics parameter error identification module.

There are kinematic parameter errors in the cable-driven parallel robot, and the kinematic parameter error model of the parallel robot is established according to the kinematic model of the parallel robot; the parameter error identification model is obtained according to the kinematic error parameter model and the error measured by the position and attitude measurement device, and this parameter is used to Error identification model for parameter identification.

The coordinate system calibration model of the position and attitude measurement device establishes the world coordinate system with the position and attitude measurement device, establishes the reference coordinate system with the rope-driven parallel robot, and solves the calibration according to the terminal position and attitude of the parallel robot measured by the position measurement device and the position and attitude in the reference coordinate system The relative position relationship between the reference coordinate system and the world coordinate system.

The robot kinematic parameter error identification module is based on the parallel robot parameter error identification model and the coordinate system calibration module of the position and attitude measurement device, and uses the parameter identification optimization algorithm to identify the kinematic error parameters.

In this embodiment, a method for calibrating kinematic parameter errors of a rope-driven parallel robot is applied in the calibration process of a rope-driven parallel robot, and a pose measuring device is provided on one side of the rope-driven parallel robot. It is a general-purpose position and attitude measurement device, which can be used to measure the motion error of the end moving platform; during the calibration process, it can identify the kinematics parameter error and improve the kinematics accuracy of the cable-driven parallel robot; specifically, the motion The error calibration method of scientific parameters is carried out according to the following steps:

Step 1. Within the measurement working range of the pose measuring device, the measuring device generally refers to the device that can measure the pose of the moving platform at the end of the rope-driven robot. Laser interferometers, 6D cameras and motion capture systems can be used, and the accuracy of the measuring device is generally required should be much higher than the precision of the robot itself; establish the reference coordinate system O _s of the rope-driven parallel robot, the reference coordinate system O _p on the moving platform at the end of the robot, and the reference coordinate system O _s can be established on the rope-driven device of the rope-driven parallel robot At the output point of the rope, the origin of the reference system and the output point of the rope coincide to facilitate the establishment of kinematic equations; the reference coordinate system O _p on the moving platform at the end of the robot is established at the geometric center point of the moving platform, which is convenient for establishing a kinematics model; The reference coordinate system O _c of the measuring device, and the reference coordinate system O _c of the pose measuring device is used as the world coordinate system; the reference coordinate system O _c is set by the measuring device, and is generally marked on the measuring device;

Step 2, establishing the kinematics model of the rope-driven parallel robot;

Step 2.1. For a cable-driven parallel robot with n degrees of freedom output by an m cable-driven device, the theoretical pose of the terminal moving platform in the reference coordinate system O _s of the cable-driven parallel robot is expressed as X _s = [P _s Φ _s ] ^T , where P _s represents the position of the end moving platform, Φ _s represents the attitude of the end moving platform;

As shown in Fig. 1, let b _i denote the position of the i-th rope drive device on the reference coordinate system O _p on the end-moving platform of the robot, and a _i denote the position of the rope output point on the reference coordinate system O _s on the robot platform ;i=1...m;

Use equation (1) to express the closed-chain equation of a single rope drive:

l _i ＝a _i -P _s -R _sp (Φ _s )b _i (1)

In formula (1), l _i represents the rope vector from the rope output point of the i-th rope drive device to the rope connection point of the end moving platform, and R _sp represents the reference coordinate system O _p on the end moving platform of the robot to the rope-driven parallel robot The rotation matrix of the reference frame O _s ;

Step 2.2, obtain the inverse kinematics equation of the rope-driven parallel robot according to formula (2):

||l _i || ₂ ＝(a _i -P _s -R _sp (Φ _s )b _i ) ^T (a _i -P _s -R _sp (Φ _s )b _i ),i＝1,2,…, m (2)

From formula (2), the rope length vector of the rope-driven parallel robot is l＝[||l ₁ || ₂ ||l ₂ || ₂ …||l _i || ₂ …||l _m || ₂ ] ^T ;

Step 3, establishing the kinematic parameter error model of the rope-driven parallel robot;

Step 3.1, let the robot kinematics error parameters caused by machining and assembly factors be expressed as Among them, b _i represents the position error of the i-th rope drive device on the reference coordinate system O _p on the robot end moving platform, a _i represents the position error of the rope output point on the reference coordinate system O _s on the robot platform, Indicates the output length error of the rope corresponding to the unit rotation angle of the motor encoder. Generally, there are 3 parameters in b _i and a _i respectively, If there is one parameter, there are 7m kinematic parameter errors to be identified Cal=[Cal ₁ Cal ₂ ... Cal _i ... Cal _m ] ^T ;

Step 3.2. Use the pose measuring device to measure the poses of N groups of terminal moving platforms. In order to ensure the robustness of parameter identification, the number of measurement data N is required to be much larger than the number of identification parameters; each measuring device can provide 1 equation , so N×m>>7m, in order to ensure the calibration speed, N=20 can be made; and the error of the motor encoder output angle θ _i shown in equation (3) can be obtained by using the pose measured by the jth group

In formula (3), {θ _m,j } represents the actual rotation angles fed back by the m motor encoders obtained from the j-th group of measured poses, and l ^j represents the rope length vector;

Step 4, obtain the optimization objective equation shown in formula (4) by formula (3):

In formula (4), e _θ represents the error equation of the output angles of m motor encoders obtained from the poses of N groups of terminal moving platforms, and has:

Formula (4) is optimized and solved by the least square method, and the kinematic parameter error identification model shown in formula (5) is obtained:

In formula (5), is the Jacobian matrix of the error equation e _θ to the kinematic parameter error Cal, and W _t represents the normalized matrix of the kinematic parameter error Cal;

Step 5, calibrate the positional relationship of the reference coordinate system _Oc of the attitude measuring device relative to the reference coordinate system _Os of the cable-driven parallel robot;

Step 5.1, let the actual measured pose of the terminal moving platform be expressed as X _cs =[P _cs Φ _cs ] ^T , where P _cs represents the reference coordinate system O of the cable-driven parallel robot in the reference coordinate system O of the _pose measurement device The position in _c , Φ _cs represents the attitude of the reference coordinate system O _s of the cable-driven parallel robot in the reference coordinate system O _c of the pose measurement device; then use formula (6) to express the reference of the terminal moving platform in the pose measurement device Position P _{c in coordinate system O c :}

P _c ＝R _cs (Φ _cs )P _s +P _cs (6)

In formula (6), R _cs represents the rotation matrix of the reference coordinate system O _s of the cable-driven parallel robot relative to the reference coordinate system O _c of the pose measurement device;

Step 5.2, since R _cs is a homogeneous matrix, there is R _cs (Φ _cs ) ^T R _cs (Φ _cs )=1, transform formula (6), and use formula (7) to establish the reference coordinate system of the pose measurement device The position relationship equation between O _c and the reference coordinate system O _s of the cable-driven parallel robot:

Use formula (8) to establish the position relationship optimization equation of the reference coordinate system O _c of the pose measurement device and the reference coordinate system O _s of the cable-driven parallel robot, and then use the LM algorithm to optimize and solve formula (8) to obtain the position parameter P _cs :

Step 5.3, let P _c -P _cs =A, P _s =B, get R _cs (Φ _cs )B=A according to formula (6), then use formula (9) to get the reference coordinate system O _c of the pose measurement device With the attitude relationship equation of the reference coordinate system O _s of the cable-driven parallel robot, the LM algorithm is used to optimize and solve the formula (9), and the attitude parameter Φ _cs is obtained:

R _cs (Φ _cs )＝AB ^T (B·B ^T ) ^-1 (9)

Step 6. Jointly calibrate the kinematic parameter error;

Step 6.1, using formula (10) to calculate the mean square error SD of the pose measuring device to measure the poses of N groups of terminal moving platforms:

In formula (10), _Xc represents the actual pose of the robot terminal moving platform measured by the pose measuring device in the reference coordinate system _Oc of the pose measuring device, Indicates the theoretical pose of the moving platform at the end of the robot in the reference coordinate system _Oc of the pose measurement device;

Step 6.2, initialize iteration number iter=1, kinematic parameter error Cal, maximum iteration number is itermax, can take value iter max=10;

Step 6.3, use step 5 to calculate the relative relationship between the reference coordinate system O _c of the pose measurement device and the reference coordinate system O _s of the cable-driven parallel robot for the iter time

Step 6.4, using the least squares method to calibrate the kinematic parameter error Cal ^iter for the iter time;

Step 6.5, assign Cal ^iter + Cal to Cali ^iter , assign iter+1 to iter;

Step 6.6. If SD<<δ and iter<iter max, where δ is a minimum value, take δ=10 ^-6 ; then the calibration ends and output Cali ^iter ; if iter>iter max, terminate the loop; if SD>δ And iter<iter max, return to step 6.3.

Claims (1)

1. a kind of kinematic parameter errors scaling method of rope driving parallel robot, it is to be applied to rope driving parallel manipulator In the calibration process of people, and rope driving parallel robot side is provided with pose measuring apparatus；It is it is characterized in that described Kinematic parameter errors scaling method is carried out according to lower step：

Step 1, in the measurement working range of the pose measuring apparatus, establish the reference of rope driving parallel robot Coordinate system O_s, the reference frame O on robot end's moving platform_p, the reference frame O of pose measuring apparatus_c, and by described in The reference frame O of pose measuring apparatus_cAs world coordinate system；

Step 2, the kinematics model for establishing the rope driving parallel robot；

Step 2.1, drive parallel robot for the rope of the n frees degree exported for m wire drives, the end is dynamic flat Reference frame O of the platform in rope driving parallel robot_sIn theoretical pose be expressed as X_s=[P_s Φ_s]^T, wherein P_s Represent the position of the end moving platform, Φ_sRepresent the posture of the end moving platform；

Make b_iRepresent reference frame O of i-th wire drive on robot end's moving platform_pOn position, a_iTable Show reference frame O of the rope output point on the robot platform_sOn position；I=1 ... m；

The closed chain equation of single wire drive is represented using formula (1)：

l_i=a_i-P_s-R_sp(Φ_s)b_i (1)

In formula (1), l_iRepresent the rope output point of the i-th wire drive to the rope of the rope tie point of the end moving platform Suo Xiangliang, R_spRepresent the reference frame O on robot end's moving platform_pTo the ginseng of rope driving parallel robot Examine coordinate system O_sSpin matrix；

Step 2.2, the inverse kinematics equation of rope driving parallel robot is obtained according to formula (2)：

||l_i||₂=(a_i-P_s-R_sp(Φ_s)b_i)^T(a_i-P_s-R_sp(Φ_s)b_i), i=1,2 ..., m (2)

By formula (2) obtain the rope lengths vector of rope driving parallel robot for l=[| | l₁||₂||l₂||₂…||l_i| |₂…||l_m||₂]^T；

Step 3, the kinematic parameter errors model for establishing the rope driving parallel robot；

Step 3.1, order robot kinematics' error parameter as caused by being machined and assemble factor are expressed asWherein, b_iRepresent reference of i-th wire drive on robot end's moving platform Coordinate system O_pOn site error, a_iReference frame O of the rope output point on the robot platform described in representing_sOn Site error,Represent that motor encoder unit turn angle corresponds to the output error in length of rope, then need the fortune recognized Dynamic parameter error of learning shares 7m Cal=[Cal₁ Cal₂ … Cal_i … Cal_m]^T；

Step 3.2, the pose using pose measuring apparatus measurement N group end moving platforms, and utilize the pose of jth group measurement Try to achieve the motor encoder output angle θ as shown in formula (3)_iError

<mrow> <msub> <mi>e</mi> <mrow> <msub> <mi>&amp;theta;</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo>{</mo> <msub> <mi>&amp;theta;</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>}</mo> <mo>-</mo> <mo>|</mo> <mo>|</mo> <msup> <mi>l</mi> <mi>j</mi> </msup> <mo>|</mo> <msub> <mo>|</mo> <mn>2</mn> </msub> <mo>/</mo> <msubsup> <mi>r</mi> <mrow> <mi>p</mi> <mi>m</mi> </mrow> <mi>i</mi> </msubsup> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mi>m</mi> <mo>;</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mi>N</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>

In formula (3), { θ_m,jThe actual rotation angle of m motor encoder feedback tried to achieve of the pose that is measured with jth group of expression, l^j The rope lengths vector that the pose that expression is measured with jth group is tried to achieve；

Step 4, the optimization aim equation as shown in formula (4) obtained as formula (3)：

<mrow> <mi>min</mi> <mi> </mi> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msubsup> <mi>e</mi> <mi>&amp;theta;</mi> <mi>T</mi> </msubsup> <msub> <mi>e</mi> <mi>&amp;theta;</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>

In formula (4), e_θRepresent the error side for the m motor encoder output angle that the pose of the N group ends moving platform is tried to achieve Journey, and have：

Solution is optimized to formula (4) using least square method, obtains the kinematic parameter errors identification mould as shown in formula (5) Type：

<mrow> <msubsup> <mi>J</mi> <msub> <mi>e</mi> <mi>&amp;theta;</mi> </msub> <mi>T</mi> </msubsup> <msub> <mi>W</mi> <mi>t</mi> </msub> <msub> <mi>J</mi> <msub> <mi>e</mi> <mi>&amp;theta;</mi> </msub> </msub> <mi>C</mi> <mi>a</mi> <mi>l</mi> <mo>=</mo> <msubsup> <mi>J</mi> <msub> <mi>e</mi> <mi>&amp;theta;</mi> </msub> <mi>T</mi> </msubsup> <msub> <mi>W</mi> <mi>t</mi> </msub> <msub> <mi>e</mi> <mi>&amp;theta;</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>

In formula (5),For error equation e_θTo kinematic parameter errors Cal Jacobian matrix, W_tRepresent kinematic parameter errors Cal normalization matrix；

Step 5, the reference frame O for demarcating pose measuring apparatus_cThe reference frame of relatively described rope driving parallel robot O_sPosition relationship；

Step 5.1, the actual measurement pose of the end moving platform is made to be expressed as X_cs=[P_cs Φ_cs]^T, wherein, P_csRepresent institute State the reference frame O of rope driving parallel robot_sIn the reference frame O of the pose measuring apparatus_cIn position, Φ_csRepresent the reference frame O of rope driving parallel robot_sIn the reference frame O of the pose measuring apparatus_cIn appearance State；Then reference frame O of the end moving platform in the pose measuring apparatus is represented using formula (6)_cIn position P_c：

P_c=R_cs(Φ_cs)P_s+P_cs (6)

In formula (6), R_csRepresent the reference frame O of rope driving parallel robot_sRelative to the ginseng of the pose measuring apparatus Examine coordinate system O_cSpin matrix；

Step 5.2, simplified style (6), so as to establish the reference frame O of pose measuring apparatus using formula (7)_cDriven with the rope The reference frame O of dynamic parallel robot_sPosition relationship equation：

<mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>P</mi> <mi>c</mi> </msub> <mo>-</mo> <msub> <mi>P</mi> <mrow> <mi>c</mi> <mi>s</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mrow> <mo>(</mo> <msub> <mi>P</mi> <mi>c</mi> </msub> <mo>-</mo> <msub> <mi>P</mi> <mrow> <mi>c</mi> <mi>s</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>P</mi> <mi>s</mi> <mi>T</mi> </msubsup> <msub> <mi>P</mi> <mi>s</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>

The reference frame O of pose measuring apparatus is established using formula (8)_cWith the reference coordinate of rope driving parallel robot It is O_sPosition relationship optimization method, recycle L-M algorithms solution is optimized to formula (8), obtain location parameter P_cs：

<mrow> <mi>min</mi> <mi> </mi> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>P</mi> <mrow> <mi>c</mi> <mi>s</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mo>|</mo> <mo>|</mo> <msup> <mrow> <mo>(</mo> <msub> <mi>P</mi> <mi>c</mi> </msub> <mo>-</mo> <msub> <mi>P</mi> <mrow> <mi>c</mi> <mi>s</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mrow> <mo>(</mo> <msub> <mi>P</mi> <mi>c</mi> </msub> <mo>-</mo> <msub> <mi>P</mi> <mrow> <mi>c</mi> <mi>s</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <msubsup> <mi>P</mi> <mi>s</mi> <mi>T</mi> </msubsup> <msub> <mi>P</mi> <mi>s</mi> </msub> <mo>)</mo> </mrow> <mo>|</mo> <msub> <mo>|</mo> <mn>2</mn> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>

Step 5.3, make P_c-P_cs=A, P_s=B, R is obtained according to formula (6)_cs(Φ_cs) B=A, then obtain pose measurement using formula (9) The reference frame O of device_cWith the reference frame O of rope driving parallel robot_sPosture relation equation, recycle L-M algorithms optimize solution to formula (9), obtain attitude parameter Φ_cs：

R_cs(Φ_cs)=AB^T(B·B^T)^-1 (9)

Step 6, combined calibrating kinematic parameter errors；

Step 6.1, the mean square error SD that pose measuring apparatus measures the pose of N group end moving platforms is calculated using formula (10)：

<mrow> <mi>S</mi> <mi>D</mi> <mo>=</mo> <msqrt> <mfrac> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msup> <mi>eX</mi> <mi>T</mi> </msup> <mi>e</mi> <mi>X</mi> </mrow> <mi>N</mi> </mfrac> </msqrt> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>

In formula (10),X_cRepresent the reference frame O in pose measuring apparatus_cIn, pose measuring apparatus measurement Robot end's moving platform attained pose,Represent the reference frame O in pose measuring apparatus_cIn, robot end The theoretical pose of moving platform；

Step 6.2, initialization iterations iter=1, kinematic parameter errors Cal, maximum iteration itermax；

Step 6.3, the reference frame O using i-th ter times calculating pose measuring apparatus of step 5_cIt is in parallel with rope driving The reference frame O of robot_sRelativeness

Step 6.4, utilize i-th ter times demarcation kinematic parameter errors Cal of least square method^iter；

Step 6.5, by Cal^iter+ Cal is assigned to Cal^iter, iter+1 is assigned to iter；

If step 6.6, SD ＜ ＜ δ and iter ＜ itermax, wherein δ are minimum, then demarcation terminates, and exports Cal^iter；Such as Fruit iter ＞ itermax, then terminate circulation；If SD ＞ δ and iter ＜ itermax, return to step 6.3.