Calibration and programming of robots

Abstract

Methods includes calibrating robots without the use of external measurement equipment and copying working programs between un-calibrated robots. Both methods utilize the properties of a closed chain and the relative position of the links in the chain in order to update the kinematic models of the robots.

Claims

1. A method for making a working program for a first robot usable with a second robot, each of the first robot and the second robot having joints connecting two flanges, the two flanges comprising a base flange and a tool flange, the method comprising: obtaining the working program for the first robot; selecting positions or angles based on the working program for the first robot; moving the second robot to the positions or angles to produce a position pair data set; estimating kinematic models of the first robot and the second robot based on a base flange offset and a tool flange center point offset between the first robot and the second robot using the position pair data set; and performing a conversion operation on the working program based on the kinematic models.

2. The method of claim 1, further comprising: making a determination about whether the working program is runnable on the second robot within one or more defined tolerances.

3. The method of claim 2, wherein making the determination comprises making a determination that the working program is not runnable on the second robot within the one or more defined tolerances; and wherein, when the working program is not runnable on the second robot within the one or more predefined tolerances, the method further comprises: selecting additional positions based on the working program; moving the second robot to the additional positions to produce an extended position pair data set; estimating revised kinematic models of the first robot and the second robot using the extended position pair data set; and performing a conversion operation on the working program based on the revised kinematic models.

4. The method of claim 1, further comprising: performing a conversion operation on a second working program based on the kinematic models.

5. The method of claim 1, wherein performing the conversion operation comprises: applying forward kinematics to first position pair data in the working program for a first kinematic model associated with the first robot to produce a first program; applying inverse kinematics to the first program using a second kinematic model for the second robot to produce second position pair data; and completing the conversion operation by replacing the first position pair data in the first working program with the second position pair data to produce a converted working program that is usable with the second robot.

6. The method of claim 1, wherein the kinematic models are based on parameters defining transformations.

7. The method of claim 6, wherein the kinematic models are based on three types of Denavit-Hartenberg parameters.

8. The method of claim 7, wherein the three types of Denavit-Hartenberg parameters comprise Schilling parameters, Parallel variant parameters, and RPY parameters.

9. The method of claim 6, wherein RPY parameters are usable to modulate a last joint of each of the first robot and the second robot.

10. The method of claim 1, wherein the kinematic models are estimated using predetermined models as a starting point.

11. Non-transitory machine-readable data storage storing instructions that are executable to make a working program for a first robot usable with a second robot, each of the first robot and the second robot having joints connecting two flanges, the two flanges comprising a base flange and a tool flange, the instructions being executable to perform operations comprising: obtaining the working program for the first robot; selecting positions or angles based on the working program for the first robot; causing the second robot to move to the positions or angles to produce a position pair data set; estimating kinematic models of the first robot and the second robot based on a base flange offset and a tool flange center point offset between the first robot and the second robot using the position pair data set; and performing a conversion operation on the working program based on the kinematic models.

12. The non-transitory machine-readable data storage of claim 11, wherein the operations comprise: making a determination about whether the working program is runnable on the second robot within one or more defined tolerances.

13. The non-transitory machine-readable data storage of claim 12, wherein making the determination comprises making a determination that the working program is not runnable on the second robot within the one or more defined tolerances; and wherein the operations comprise, when the working program is not runnable on the second robot within the one or more predefined tolerances: selecting additional positions based on the working program; causing the second robot to move to the additional positions to produce an extended position pair data set; estimating revised kinematic models of the first robot and the second robot using the extended position pair data set; and performing a conversion operation on the working program based on the revised kinematic models.

14. The non-transitory machine-readable data storage of claim 11, wherein the operations comprise: performing a conversion operation on a second working program based on the kinematic models.

15. The non-transitory machine-readable data storage of claim 11, wherein performing the conversion operation comprises: applying forward kinematics to first position pair data in the working program for a first kinematic model associated with the first robot to produce a first program; applying inverse kinematics to the first program using a second kinematic model for the second robot to produce second position pair data; and completing the conversion operation by replacing the first position pair data in the first working program with the second position pair data to produce a converted working program that is usable with the second robot.

16. The non-transitory machine-readable data storage of claim 11, wherein the kinematic models are based on parameters defining transformations.

17. The non-transitory machine-readable data storage of claim 16, wherein the kinematic models are based on three types of Denavit-Hartenberg parameters.

18. The non-transitory machine-readable data storage of claim 17, wherein the three types of Denavit-Hartenberg parameters comprise Schilling parameters, Parallel variant parameters, and RPY parameters.

19. The non-transitory machine-readable data storage of claim 16, wherein RPY parameters are usable to modulate a last joint of each of the first robot and the second robot.

20. The non-transitory machine-readable data storage of claim 11, wherein the kinematic models are estimated using predetermined models as a starting point.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) A further understanding of the nature and advantages of the present invention may be realized by reference to the remaining portions of the specification and the drawings. In the following, preferred embodiments of the invention are explained in more detail with reference to the drawings, wherein

(2) FIG. 1 illustrates Schillings Denavit-Hartenberg notation,

(3) FIG. 2 illustrates Parallel Denavit-Hartenberg notation variant,

(4) FIG. 3 illustrates a closed chain configuration of two robots,

(5) FIG. 4 illustrates an alternative closed chain configuration of two robots,

(6) FIG. 5 schematically shows the transformations of two 6-link robots connected in a closed chain configuration,

(7) FIG. 6 schematically illustrates the representation of an un-calibrated chain of two robots,

(8) FIG. 7 schematically illustrates how a distance can be used to define the scale of a model,

(9) FIG. 8 illustrates an adapter defining a common normal between two rotation axes,

(10) FIG. 9 illustrates where the transformations in table 1 are placed in the system,

(11) FIG. 10 shows a flow diagram illustrating an embodiment of a method of calibrating R robots,

(12) FIG. 11 shows a top level embodiment of a method of calibrating robots,

(13) FIG. 12 illustrates a flowchart for an automatic version of an embodiment of the calibration loop,

(14) FIG. 13 illustrates a flowchart for a manual version of an embodiment of the calibration loop,

(15) FIG. 14 illustrates a flow chart of an embodiment of a method of saving sensor information of non-synchronized robots,

(16) FIG. 15 shows a flow chart of an embodiment of a method of saving sensor information of synchronized robots,

(17) FIG. 16 shows a closed chain, where a new robot is illustrated together with a phantom image of an old robot,

(18) FIG. 17 illustrates where the transformations in table 2 are placed in the system,

(19) FIG. 18 illustrates an embodiment of a flowchart for collecting data pairs to be used in program conversion,

(20) FIG. 19 illustrates an embodiment of a flowchart for program conversion,

(21) FIG. 20 shows a flow diagram of an embodiment of a method for program conversion,

(22) FIG. 21 shows a flow diagram of an embodiment of a method for defining the joint angles, and

(23) FIG. 22 shows a flowchart of an embodiment of a method of performing a program conversion between un-calibrated robots.

DETAILED DESCRIPTION

(24) The present invention will now be described more fully hereinafter with reference to the accompanying drawings, in which exemplary embodiments of the invention are shown. The invention may, however, be embodied in different forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art. Like reference numerals refer to like elements throughout. Like elements will, thus, not be described in detail with respect to the description of each figure.

(25) Classic kinematics is used to model the robots. This makes it possible to transfer information from the robot joints to the Cartesian coordinate system in order to make their joints comparable.

(26) Notation

(27) The used notation is as follows: T.sub.c(θ).sub.r,

(28) which is a homogeneous transformation of a θ rotation clockwise around axis c, where c is x, y or z. T.sub.c(s).sub.t,

(29) which is a homogeneous transformation of s translation along axis c, where c is x, y or z. T.sub.j.sup.i,

(30) which is a homogeneous transformation from i to j. Z.sub.j.sup.i,

(31) which is z-axis unit vector for the homogeneous transformation from i to j. P.sub.j.sup.i,

(32) which is position from i to j.

(33) Robot Representation

(34) The robots are modeled with two types of Denavit-Hartenberg, DH, parameters. The first type of DH uses the Schilling notation (c.f. R. J. Schilling. “Fundamentals of Robotics: Analysis and Control”. Simon & Schuster Trade, 1.sup.st ed. 1996, incorporated in its entirety by reference) to describe a transformation between two non parallel z-axes, since the notation has a singularity when the z-axis is parallel (see equation 1 below).

(35) The second type uses a DH parallel variant to avoid this singularity (see equation 2 below). The parallel DH variant uses the same principles as the original DH, but has a singularity when the first y and the next z-axis, respectively, is parallel. The singularity of Schilling DH is defined by the normal between the first and next z-axis, since the distance to the normal intersection with the first z-axis is used as a parameter, see illustration FIG. 1. The singularity of the parallel DH variant is defined similarly to the Schilling notation, but the singularity is moved by using the normal between the first y and the next z-axis, see FIG. 2.

(36) Choosing the right notation for each joint element results in a representation of the robot without any DH parameter singularities:
Φ.sub.DH(θ,d,α,a)≡T.sub.z(θ).sub.rT.sub.z(d).sub.tT.sub.x(α).sub.rT.sub.x(a).sub.t  (1)
Ψ.sub.DH(β,b,α,a)≡T.sub.x(β).sub.rT.sub.x(b).sub.tT.sub.x(α).sub.rT.sub.x(a).sub.t  (2)
Closed Chain Model

(37) The representation combines each element in the chain to one model. The model reflects the physical characteristic of the closed chain, and makes it possible to relate each element to each other. Depending on the equipment used to connect the robots, different approaches may be used. In the basic installation, the tool flanges are connected via a stiff solid adapter 2, and the base flanges are connected with a stiff solid adapter 4 and all the joints in the chain are represented by DH parameters as described above. This has the advantages as being a minimal complete representation describing the joints' rotational axes in the system relative to each other, see illustration FIG. 3 showing two robots 6 and 8, connected to each other in a closed chain via the adapters 2 and 4. In FIG. 4 is illustrated an alternative closed chain configuration of two robots 6, 8, which are connected at their tool flanges via the adaptor 2. The base flanges 10 are mounted on a surface, from which distance between the base flanges 10 can be determined, which is enough information in order to “close the chain”. In FIGS. 3 and 4 are illustrated two industrial robots 6 and 8, of the type UR5 manufactured by Universal Robots.

(38) Advanced Chain Combinations

(39) A more advanced adapter can be used to connect the tool and base flanges 10. Preferably, the model is configured to reflect the used equipment. An example of an advanced way to connect the base flanges 10 of the robots 6, 8, could include a prismatic joint that can displace the robots 6, 8 with a known distance. Another example of adding a more advanced joint to the chain could be done with a ball-bar or an additional joint. The ball-bar is a common known tool for calibration of single robot (c.f. M. R. Driels: Using passive end-point motion constraints to calibrate robot manipulators. Journal of Dynamic Systems, Measurement, and Control, 115(3): 560-566, 1993—hereby incorporated by reference in its entirety). This method works by mounting one of the ball-bar ends to a fixed location and the other end to the robot. This means that all observations are done relative to the fixed location and therefore limited by it. The movement is also limited due to the fixed location. When the equipment is used in the context of the present invention, each end is mounted on each robot 6, 8 and the measurements will therefore not be limited to a fixed location. This gives the possibility to take measurements that relate to the whole workspace of the robots 6, 8.

(40) Some robots have multiple limbs. By treating them as individual robot arms, the method according to the invention, which is described in the present patent specification, can be used to calibrate these limbs without additional active measurement equipment.

(41) The method according to the invention is based on the closed chain principle, as illustrated in FIG. 6. By transferring this principle to a model, one can achieve a calibration method where feedback is embedded, without the need for additional measuring equipment. Then modeling it mathematically, the model can be adjusted to fulfil the principle of the closed chain and thus calibrate the mathematical model of the physical joints of the robots 6, 8. Mathematically, an error is calculated in the model, by opening the chain and then calculating the difference between two joint ends, see illustration in FIG. 7. The robots are not disconnected physically in order to determine the error—this is a purely mathematical trick. The process of adjustment of the kinematic models is done with a Newton-Raphson approach by collecting enough observations from the robots 6, 8 in order to describe the whole workspace, in a system of equations that guide the solver to the right solution. Preferably, so many observations—sensor information—are collected that an over-determined system of equations is achieved, thereby leading to a stable convergence to the right solution.

(42) Least Squares Function

(43) The—preferably over-determined—system of equations is solved by a least squares function that represents the squared error. In equation 3, the transformation represents the error for the joints in the open chain as illustrated in FIG. 7:
Error≡T.sub.0.sup.{circumflex over (0)}=T.sub.0.sup.1T.sub.1.sup.2 . . . T.sub.n-1.sup.nT.sub.n.sup.{circumflex over (0)}  (3)

(44) The position error is taken into account by calculating the squared length of the Error's translation part, see equation 4:
ePos.sup.2≡∥p.sub.0.sup.{circumflex over (0)}∥.sup.2=p.sub.0.sup.{circumflex over (0)}.Math.p.sub.0.sup.{circumflex over (0)}  (4)

(45) The rotation error is calculated by the axis angle notation AA (c.f. Robert J. Schilling: Fundamentals of Robotics: Analysis and Control. Simon & Schuster Trade, 1.sup.st edition, 1996). The error of angle is found by calculating the AA from the Error rotation part, see equation 5:
{angle,{right arrow over (axis)}}=AA(R.sub.0.sup.{circumflex over (0)})eRot≡angle  (5)

(46) The resulting least squares equation is a squared sum of equations 4 and 5 for N observations, see equation 6:

(47) $\begin{array}{cc}{\mathrm{eSum}}^2=\underset{n=0}{\overset{N-1}{.Math.}}{\mathrm{ePos}}_n^2+{\mathrm{eRot}}_n^2& \left(6\right)\end{array}$
representing the error gap between the chain ends as illustrated in FIG. 6.
Constraints

(48) In order to scale the model properly it is necessary to have a known distance or distance change in the model. This can be illustrated by trigonometry as in FIG. 7 where the angles of the triangles are the same but the scale is different. A way to estimate the scale factor of the model is by implementing a model where a known distance can be defined and fixed. Or design a distance that can be fixed in the used model. Another possibility is to use statistical information about the expected dimensions of the robots and use it to regulate the scale guess. This can for example be done by a Tikhonov regulation.

(49) Collecting Observations

(50) In order to use the method, as described above, is it necessary to collect and save observations from the physical chain in order to imitate it with the mathematical representation. The observations can be collected both manually supervised by an operator or automatically by the robots moving autonomously around in the workspace. It is necessary to collect enough measurements distributed in the whole workspace. This is due to the fact that the calibration can only be assumed to represent the used workspace. As the robots 6, 8 need to be connected and follow each other, it is furthermore preferred that the position regulators of some of the joints can be turned off and be led around by the others. It is necessary to manipulate the chain of robots 6, 8 without any use of inverse kinematics, since the kinematics is not known due to the lack of calibration.

(51) Evaluation

(52) A reliable evaluation of the final calibration uses the estimated mathematical model and compares it with the observations. It is important that these observations are independent from those used in the calibration procedure and distributed throughout the workspaces of the robots. If the observations from the calibration are used to evaluate the calibration, then the result will not be trustworthy as no new information has been added to support the result. Another reliability indication for the quality of the results is to let the calibration adjust on physical known parameters and afterwards compare it with the known values.

(53) Having joints of the robots 6, 8 connected in a closed chain opens the possibility to adjust other parameters than those, which relate to determination of the position. An example of these parameters could be to adjust parameters that relate to the forces in the chain.

(54) Calibration Example

(55) The object of this example is to illustrate and explain how the method according to the invention can be used on specific robots manufactured by the applicant, Universal Robots. The setup includes two UR-6-85-5-A industrial serial robots 6, 8 and two solid passive adapters to connect the tool and base flanges. In FIG. 8 only the solid adapter 2 used to connect the tool flanges is shown. This specific type of robots 6, 8 has six degrees of freedom divided on two revolute joints that are parallel and four which are perpendicular with the previous rotation axis. To make a pure static kinematic calibration of the two robots 6, 8 it is needed to mount them on fixed positions relative to each other. The adapters 2 that close the chain are designed so that the connected end joints do not share the same rotational axis as this would introduce some unwanted model dependencies between the joints. In this example, the tool and base flanges are connected with a known displacement and rotation of 90 degrees in order to optimize the setup to the Schilling DH notation. To define the scale of the closed chain model it is possible to use statistical information about the robots or a fixed dimension in the model. In this implementation it has been chosen to use a fixed known dimension. The known dimension is the length of the common normal 16 between the two rotational axes 18, 20. The length of this normal is directly represented by the DH “a” parameter and can therefore be defined and fixed (see FIG. 8).

(56) Representation

(57) The model of the closed chain includes a transformation for each joint as illustrated in FIG. 9. To use the same representation for both robots, the error calculation changes but still calculating the same error as in equation 7 below:
T.sub.base.sup.wrist3≡T.sub.base.sup.shoulderT.sub.shoulder.sup.elbowT.sub.elbow.sup.wrist1T.sub.wrist1.sup.wrist2T.sub.wrist2.sup.wrist3
Error≡[T.sub.base.sub.6.sup.wrist3.sup.6T.sub.wrist3.sub.6.sup.wrist3.sup.8].sup.−1[T.sub.base.sub.6.sup.base.sup.8T.sub.base.sub.8.sup.wrist3.sup.8]  (7)

(58) The connection between the robots can be represented with an extended version of DH (see equation 8 and 9 below). This representation makes it possible to represent any transformation in a similar way as DH which gives the same advantages.
T.sub.ΦDH(θ.sub.1,d.sub.1,α,a,θ.sub.2,d.sub.2)≡T.sub.z(θ.sub.1).sub.rT.sub.z(d.sub.1).sub.tT.sub.x(α).sub.rT.sub.x(a).sub.tT.sub.z(θ.sub.2).sub.rT.sub.z(d.sub.2).sub.t  (8)
T.sub.ΨDH(β.sub.1,b.sub.1,α,a,β.sub.2,b.sub.2)≡T.sub.x(β.sub.1).sub.rT.sub.x(b.sub.1).sub.tT.sub.x(α).sub.rT.sub.x(a).sub.tT.sub.x(β.sub.2).sub.rT.sub.x(b.sub.2).sub.t  (9)

(59) The parameters for each transformation of the closed chain model are described in table 1 below. The fixed distances are represented in the model with the “a” parameter (see table 1). The same design trick is done for both the connection, the tool and base flanges, both of them can be fixed with a known normal length. As mentioned above it is preferred—but not necessary—to have multiple pre-known parameters in the system. This makes it possible to include one of them as a ground true to evaluate the outcome of the calibration.

(60) TABLE-US-00001 TABLE 1 Table describing the adjustable and fixed (the “a” parameter) parameters of the representation of the model for the setup where the robots 6, 8 are displaced with 90 degrees. Description Robot.sub.6 Robot.sub.8 T.sub.wrist3.sub.6.sup.wrist3.sup.8 T.sub.ΦDH (θ.sub.1, d.sub.1, α, â, θ.sub.2, d.sub.2) T.sub.wrist2.sup.wrist3 Φ.sub.DH (θ, d, α, a) Φ.sub.DH (θ, d, α, a) T.sub.wrist1.sup.wrist2 Φ.sub.DH (θ, d, α, a) Φ.sub.DH (θ, d, α, a) T.sub.elbow.sup.wrist1 Ψ.sub.DH (β, b, α, a) Ψ.sub.DH (β, b, α, a) T.sub.shoulder.sup.elbow Ψ.sub.DH (β, b, α, a) Ψ.sub.DH (β, b, α, a) T.sub.base.sup.shoulder Φ.sub.DH (   ,    , α, a) Φ.sub.DH (   ,    , α, a) T.sub.base.sub.6.sup.base.sup.8 T.sub.ΦDH (θ.sub.1, d.sub.1, α, â, θ.sub.2, d.sub.2)

(61) In order to use the calibrated model in the controller of the robots 6, 8, it is necessary to re-compute the transformations into the needed format. Including extraction of useful information from

(62) $T_{{\mathrm{base}}_6}^{{\mathrm{base}}_8}\mathrm{and}{T}_{\mathrm{wrist}{3}_6}^{{\mathrm{wrist}}_8}.$
In this case the Schilling DH notation.

(63) FIG. 10 shows a flow diagram illustrating an embodiment of the method of calibrating R robots, R being a natural number equal to or greater than 2. The illustrated embodiment comprises the steps of: Providing R robots having multiple joints, a base flange and a tool flange. In step 22 a closed chain is formed from the R robots. Then in step 28 the corresponding position pairs (<R.sub.iQ, R.sub.jQ>.sub.m) are collected. Then, in step 30 it is evaluated whether enough data is collected in order to estimate the models. If this is not the case, then step 26, the joint positions of the robots, is changed and the corresponding position pairs (<R.sub.iQ, R.sub.jQ>.sub.m) are collected as indicated by the block 28. Then again in step 30 it is evaluated whether enough data has been collected. If this is not the case, than the steps 26 and 28 are repeated as indicated by the arrow 29. However, if sufficient data has been collected then the models are updated or refined or estimated by using the knowledge of the collected position pair data sets <R.sub.iQ, R.sub.jQ>, as indicated by step 32. These updated models are then saved in step 34, whereby the calibration is completed as indicated by step 36.

(64) FIG. 11 shows a top level overall embodiment of a method of calibrating robots in accordance with the invention. In step 38 the method is started, and then in step 40 the robots are mounted together in a closed kinematic chain, and then in step 42 the calibration loop in accordance with an embodiment of the invention is performed for high uncertainty parameters to prime the full calibration with a starting point. If this loop succeeds, then the full model is determined in step 44. If this succeeds, then the calibration is done, as indicated by the block 46. If, however, any of the steps 42 or 44 fails, as indicated by the arrows 50 and 52, respectively, then the calibration fails, as indicated by the block 48.

(65) FIGS. 12 and 13 show two alternative ways of implementing the blocks 42 and 44 illustrated in the flow diagram of FIG. 12.

(66) FIG. 12 illustrates an embodiment of an automatic version of the calibration loop 42 and 44 in FIG. 11. In step 54 the next target joint angles are generated or loaded, then in step 56 all robots except one are set to force control heading at the target angles. Then in step 58 the last remaining robot is set to heading at target angles with position control. Then in step 60 it is evaluated whether the change in sensor information is larger than a suitably chosen threshold to avoid identical or nearly identical measurements. If this is the case, then the sensor information is saved as indicated by step 62. On the other hand, if the change in sensor information is lower than said suitably chosen threshold, then it is in the next step 64 evaluated whether enough data has been collected in order to proceed with the calibration. If this is the case, then the robots are stopped from moving, as indicated by step 66, whereafter the calibration is performed, i.e. the models are refined, as indicated by step 68. Then in step 70 it is assessed whether the calibration is ok. If this is the case, then the calibration loop is done as indicated by step 72. If the evaluation in step 64 shows that there has not been collected enough data in order to complete the calibration, then it is checked whether the robots are still moving in step 74, and in the affirmative, the steps 60-64 are repeated. If the robots are not moving, then it is in step 76 ascertained whether the robots have stopped unexpectedly; if this is the case, then in step 80 an error check for physical inconveniencies is performed, e.g. if the robots have collided with the surrounding environment or themselves. If the outcome of the evaluation in step 76 is negative, then the steps from 54 are repeated.

(67) FIG. 13 illustrates an embodiment of a manual version of the calibration loop 42 and 44 in FIG. 11. Since many of the steps in the present method are similar to those of the method illustrated in FIG. 12, only the differences will be explained in the following. The essential difference is that all the robots are set to force control in step 81, and then the joint angles are manipulated manually in step 82, before step 60.

(68) FIG. 14 and FIG. 15 show a flow chart for saving sensor information as indicated by step 62 in FIGS. 12 and 13. FIG. 14 illustrates the flow chart for saving sensor information of non-synchronized robots, and FIG. 15 shows one for synchronized robots. In the first step 84 in FIG. 14, the movement of the robots is paused, then in step 86 it is evaluated whether the robots are stopped, and if affirmative, then the sensor information, e.g. joint angles, is saved in step 88, whereafter movement of the robots is continued in step 90. However, if the robots are synchronized, then the sensor information is saved directly, as indicated by the sub step 88 in the flow chart in FIG. 15. This is possible as the synchronization ensures that the information from all the sensors is collected simultaneously. If this is not done simultaneously, the collected dataset does not reflect a closed chain and the measurements will in that case be unusable. If the robots are not synchronized, it is necessary to stop them before the position pairs are collected in order to ensure the consistency.

EXAMPLE OF PROGRAM DUPLICATION

(69) This section will be described as an example of how the closed chain method can be used to duplicate a working program from one robot to another where one or both is un-calibrated. The example takes advantage of the program that is to be duplicated. The program to be duplicated contains way-points of robot configurations. Here a closed chain is obtained by using the way-points to represent the first part of the chain and then—mathematically—close the chain by re-teaching the same way-points with the new robot that is intended to perform the tasks of the first robot. See FIG. 16, where the new robot 6 is illustrated together with a phantom image of the old robot 8.

(70) Data

(71) The generated reference data is based on the program of the robot that is going to be duplicated. The data is collected manually by relearning key way-point positions, which are essential to the task of program. These essential positions may vary in precision depending on the program, but will in most cases be those that are most explicit and easiest to replicate. By relearning these essential positions it is also achieved that the resulting program of the new robot will match in these key positions. If several programs are being duplicated between the same two robots, then those key positions can be reused, thereby leading to a higher level of compliance between them. Depending on the task and the robot tool it may be difficult to determine the correct position and rotation. Especially the rotations may be difficult to replicate correctly if the tool is rotation independent. This can thus reduce the level of knowledge but still be usable as the rotation in these tasks is not that important because of the rotation independence. Depending on the tasks of the robots, the collected data is typically grouped in groups where the essential action of the robot is made. This implies that the relationship between the workspaces of the two robots are only partly known and that correct corresponding configurations can only be provided near the key positions. However, as the other waypoints are not essential for the tasks of the robot it will in the most cases be good enough for the robot executing the tasks of the program. If the program needs to be duplicated back to the original robot, then the same corresponding key positions may be reused. This enables the original program as well as changes in it to be ported back to the original robot. This is particularly interesting if the same program is used in similar robot cells, which thereby enables production to be up-scaled by duplicating the programs without loss of flexibility.

(72) Robot Base Representation

(73) In the present example it is assumed that the same tool is mounted on both robots 6, 8 and that the position of the tool center point is the same (see FIG. 16). Modulation of the transformation between the two base rotation centers of the robots 6, 8 is done by Euler angles, RPY, and 3D position vectors. By using Euler angles, a representation with as few parameters as possible is achieved, which is free of singularities for modest or none adjustment relative to the rotation of displacement of the robots and avoids the need for using a constrained optimization algorithm. The chosen representation is formulated in equation 10 below:
T.sub.RPYxyz(θ.sub.x,θ.sub.y,θ.sub.z,x,y,z)≡T.sub.x(x).sub.tT.sub.y(y).sub.tT.sub.z(z).sub.tT.sub.z(θ.sub.z).sub.rT.sub.y(θ.sub.y).sub.rT.sub.x(θ.sub.x).sub.r  (10)
Robot Representation

(74) The robots 6, 8 are modulated with three types of Denavit-Hartenberg parameters. The two first ones are the same as used in the calibration example above. However, for modulation of the last joint, a RPY notation is chosen in order to ensure that any position and rotation can be modulated (see equation 11 below). This notation is used because none of the other DH notations can represent any transformation which is needed to modulate any tool flange transformation. Like the base representation, the RPY is a good choice as only modest or none rotation is needed to be modeled. By choosing the right notation for each joint, the result is a representation of the robot without any parameter singularities:
RPY.sub.DH(r,p,y,x,y,z)≡T.sub.z(θ.sub.z).sub.rT.sub.y(θ.sub.y).sub.rT.sub.x(θ.sub.z).sub.rT.sub.x(x).sub.tT.sub.y(y).sub.tT.sub.z(z).sub.t  (11)
Tool Representation

(75) The transformation that defines where the tool center point is placed relative to the tool flange of the robot is called T.sub.tf.sup.tcp. This transformation is fixed due to the assumption that the same tool is mounted on both robots 6, 8. Furthermore, any small adjustment, which is needed, can be obtained by the parameters of the last joint. In this example, the scale of the model is not important as long as the relative scales of the robots are the same as the scale of the physical robots 6, 8. This is valid as long as the input and output to the method are pure angles, as angles are independent of the scale of the objects as illustrated in FIG. 7.

(76) Full Model

(77) The representation of the model is illustrated in table 2 below. To avoid linearly dependent parameters in the model, some of them are fixed. The parameter θ and d for the base joint of both robots are fixed because the same adjustment can be done by the T.sub.base.sub.1.sup.base.sup.2 representation. One of the robots' last joint transformation T.sub.5.sup.ft is fixed because both robots share the tool center point T.sub.tf.sup.tcp and any small adjustment can be done through both representations of the joints. Accordingly, said parameters depend on each other. Since the method only receives joint angle information it is necessary to fix a physical length in order to avoid a linear dependency related to the scale of the model or use statistical information. In this case, it has been chosen to use statistical information to regulate the optimization method as this makes the method usable even though the problem is ill-posed/undetermined. The chosen regulation method is based on Tikhonov regulation for ill-posed equations. This enables the least square optimization to select a solution closest to the expected amount of infinity possibilities. The amount N of adjustable parameters for each robot fulfils equation below, for a minimal and complete notation as the T.sub.base.sub.1.sup.base.sup.2 and T.sub.5.sup.ft is reused for both robots R=Number of revolute joints. T=Number of prismatic joints. N=4R+2T+6. In this example R=6 and T=0, which gives N=30.

(78) TABLE-US-00002 TABLE 2 describing the adjustable and fixed parameters of the representation of the model. Description Robot.sub.6 Robot.sub.8 T.sub.tf.sup.tcp fixed T.sub.wrist3.sup.tf RPY.sub.DH (   ,    ,    , RPY.sub.DH (r, p, y, x, y, z)  ,    ,    ) T.sub.wrist2.sup.wrist3 Φ.sub.DH (θ, d, α, a) Φ.sub.DH (θ, d, α, a) T.sub.wrist1.sup.wrist2 Φ.sub.DH (θ, d, α, a) Φ.sub.DH (θ, d, α, a) T.sub.elbow.sup.wrist1 Ψ.sub.DH (β, b, α, a) Ψ.sub.DH (β, b, α, a) T.sub.shoulder.sup.elbow Ψ.sub.DH (β, b, α, a) Ψ.sub.DH (β, b, α, a) T.sub.base.sup.shoulder Φ.sub.DH (   ,    ,α, a) Φ.sub.DH (   ,    ,α, a) T.sub.base.sub.6.sup.base.sup.8 T.sub.RPYxyz (θ.sub.x, θ.sub.y, θ.sub.z, x, y, z)

(79) In FIG. 17 is shown a schematic illustration of where the transformations in table 2 are placed in the system.

(80) In FIG. 18 is shown an embodiment of a flowchart for collecting position pairs to be used in a program conversion as illustrated in FIG. 19. In the following it is assumed that robot j has been replaced by robot i. Each robot has multiple joints, a base flange, and a tool flange. In the first step 92, a program set R.sub.iP is loaded, and then a number of positions or angles R.sub.iQ in accordance with the working program R.sub.iP are chosen in the next step 94. These positions—and thereby the corresponding angles R.sub.iQ are preferably essential way-points in the programmed task. Then in step 96, for all R.sub.iQ.sub.m in the list R.sub.iQ, step 98 is performed. In step 98 the robot R.sub.j is moved to those corresponding positions of R.sub.iQ.sub.m positions, which then are saved in step 100 as a position or angle pair set <R.sub.iQ, R.sub.jQ>.sub.m. As indicated by the block 102, the steps 96-100 are repeated if the loop is not done. However, when the loop is done, the result is a position or angle pair data set <R.sub.iQ, R.sub.jQ>, as indicated by block 104.

(81) In FIG. 19 is shown an embodiment of a flowchart for program conversion, i.e. for copying a working program from robot i to robot j, using the above mentioned collected data set. In step 92 a program R.sub.iP is loaded. Then in step 106, the corresponding position or angle pair data set <R.sub.iQ, R.sub.jQ> is loaded. Then in step 108, the kinematic models M.sub.i, M.sub.j, T.sub.base and T.sub.tcp are estimated using the collected position or angle pair data set <R.sub.iQ, R.sub.jQ> and the closed chain rule. Then in step 110 forward kinematics is applied on all R.sub.iQ in R.sub.iP with M.sub.i, thereby resulting in R.sub.iK. Whereafter in step 112, inverse kinematics is applied on all R.sub.iK with M.sub.j, thereby resulting in R.sub.jQ. Then in step 114 all R.sub.iQ in R.sub.iP are replaced with the corresponding R.sub.jQ resulting in R.sub.jP, whereby the program conversion is completed as indicated by the block 116.

(82) FIG. 20 shows a top level flowchart for program correction. The starting point is a working robot installation performing a task described in its programs as indicated in step 188. Then in step 120 a change in the setup is performed, e.g. in case of a mechanical breakdown. This could be the change of a robot, e.g. replacement of a joint of it, as indicated by block 122, or a replacement of the robot as indicated by block 124, or some other change in the setup as described by block 126. Then in the next step 128—using the new robot part (e.g. whole new robot)—new corresponding joint angles to some of the old important positions in the program are determined. Thereby joint angle pairs data of the old and new robot are determined. These data are then in step 130 used to refine the kinematic models of the old and new robots using the closed chain properties. Whereafter in step 132 the program conversion is performed as described above. If the program correction fails, then the steps 128-132 are repeated. However, if the program correction succeeds, then it is, in step 134, determined whether the program is usable, for example by letting it run on the new robot. This can for example be done by evaluating whether the program can run on the second robot within suitably chosen tolerances.

(83) If the program is usable, then it is in the next step—optional step 136—evaluated whether more programs need to be converted, and if affirmative, then perform the steps 132-136 where the given angle pairs can be adjusted or additional can be added. Otherwise, the program is ready to use, as indicated by block 138 in the illustrated flowchart.

(84) FIG. 21 shows a flowchart illustrating in more details an embodiment of how step 128 in the above described flowchart illustrated in FIG. 20 could be performed. The illustrated method is iterative: First, in step 140, one important waypoint from the program is used, then, in step 142, a new set of joint angles of the new robot is defined by moving it to the right position. Then in step 144, both the old and new set of joint angles is saved as a corresponding pair. Then, in the last step 146, it is determined whether other important waypoints need to be defined. If yes, then the steps 140-146 are repeated. If no, then the angle pairs are used in the next step of updating the models (see step 130 in FIG. 20).

(85) In FIG. 22 is illustrated in more detail an embodiment of how step 132 of correcting/converting a program can be performed. As illustrated, the first step is to load a program 148 to correct and then in the next step, a first waypoint is determined. In step 152 it is evaluated whether the corresponding position is already defined by step 128. If it is not, then, in step 154, forward kinematics is applied using the model representing the old robot, thereby yielding a result on which result inverse kinematics is calculated using the model representing the new robot, as indicated by step 156. Then in step 158 it is evaluated, whether the calculations succeeded. And if yes, then in step 160 the closest solution calculated by inverse kinematics is selected. Then in the next step 162, the old waypoint is replaced with the new calculated one. Finally, in step 164 it is evaluated whether more waypoints are needed, and if yes, then perform the steps 150-164. If the outcome of the evaluation in step 152 is true, then the corresponding joint angles already defined are used, as indicated by step 166.

(86) As illustrated it is by the invention possible to perform program conversions between robots, which are un-calibrated.

LIST OF REFERENCE NUMBERS

(87) In the following is given a list of reference numbers that are used in the detailed description of the invention. 2 tool flange adapter 4 base flange adapter 6, 8 robots 10 base flanges 16 common normal between rotational axes 16, 20 rotational axes 22-36 method steps, calibration 38-52 method steps, calibration 54-82 method steps, automatic and manual calibration loop 84-90 method steps, saving of sensor information 92-104 method steps, collection of angle data pairs 106-116 method steps, program conversion 118-138 method steps, program correction 140-146 method steps, definition of joint angles 148-166 method steps, correction of program