Samsung robot in different workspace locations and with different .... The experiments are performed on a 6-axe Samsung. Faraman .... P is at the edge of the.
Modeling of Orientation Repeatability Phenomena for Industrial Robots Diala DANDASH, Jean-François Brethé, Eric Vasselin and Dimitri Lefebvre Groupe de Recherche en Electrotechnique et Automatique du Havre (GREAH) UFR Sciences et Techniques, Université du Havre, BP540, 76058 Le Havre, France Abstract— In this paper, a new method for the estimation of orientation repeatability is proposed for industrial manipulator robots. First, we compute repeatability in different locations of the workspace using the experimental angular covariance matrix and the stochastic ellipsoid modeling. Then we display experimental results about the measurement of orientation repeatability for an industrial Samsung robot in different workspace locations and with different loads. Computed and experimental repeatability are compared. We analyse the incidence of load and workspace location on orientation repeatability and bring additional results to the existing literature.
INTRODUCTION Many experts have worked on the topic of precision robots and wrote standards ISO 9283 [1] and ANSI R15.05 [2]. Those standards define a number of performance criteria. Accuracy and repeatability of position and orientation are two of these criteria. The ISO9283 standard describes a process to estimate the position and orientation repeatabilities. On the contrary, in the ANSI R15.05 standard, an estimation for position repeatability is given but nothing concerning the orientation repeatability. This paper focuses on industrial manipulator robot orientation repeatability. We did not find any paper in the scientific literature concerning experiments performed on industrial robots to estimate orientation repeatability. Nevertheless this seems to be an important criterion when minute assembly tasks have to be achieved, because the induced displacements on the part extremities can be more important than robot pose repeatability. In the first section, the usual definitions of position and orientation repeatability are recalled. In the second section, an innovative method for evaluating orientation repeatability is detailed. For this purpose, the angular covariance matrix is estimated and the stochastic ellipsoid theory is used to compute the orientation repeatability. In the third section, an experimental measure of orientation repeatability is performed using the usual stationary cube method but with six micrometers. In the fourth section, computed and experimental repeatability are compared.
I. DEFINITIONS There are several criteria concerning precision detailed in ISO9283: accuracy and repeatability of position and orientation but also accuracy and repeatability of distance. We are here interested more specifically in the orientation repeatability. The estimation is based on the following process. The robot endpoint is commanded to go to a specific position called the target and come back and this cycle is redone 30 times in the same conditions. When the robot control achieves the target, the position and orientation of the robot tool are measured. In the stationary cube method, the tool is a steel cube. Of course, the different poses are not exactly the desired pose. So there are differences in the position and orientation of the cube. These different poses constitute a cloud of points and the criteria proposed in the standard proceed from this cloud and the desired pose. Accuracy is the distance between the mean of points and the target and this definition is valid for position or orientation. This depends then on the coordinate system used. The pose accuracy is the difference between commanded position and the barycenter of the points. It includes: position accuracy : difference between the commanded position of the robot endpoint and the barycenter of achieved positions as display in fig.1. It is given by: With APx, APy et APz are the accuracies of position along the axes x, y and z. Orientation accuracy: difference between the commanded orientation of the robot tool (or cube) and the barycenter of orientation achieved positions as display in fig.2. It is given by:
, and are averages of angular orientations for the same poses repeated n times. , and are angular coordinates of the commanded position.
The pose repeatability of a robot measures the variability or dispersion of the poses around the mean of the poses. For a definite pose it is expressed by: Repeatability of position: it measures the dispersion between final points when the target is the same and the move is repeated several times as shown in fig1. It is defined by:
Where the random variable L is the distance from each point to the barycentre of the set. This random variable L has a mean and a standard deviation SL. Repeatability of orientation: It is defined on the ISO 9283 standard as the range of angular variations ±Sa, ±Sb, ±Sc around the mean values , and as display in fig2. It is defined by: RPa= ±3Sa= RPb= ±3Sb= RPc= ±3Sc= a, b, c indicate a characteristic orientation around axes x, y, z. Sa, Sb and Sc standard deviations related to the three angular coordinates of the achieved position.
Fig.2 orientation accuracy and repeatability
Note: this figure can be applied to and . II. COMPUTATION OF ORIENTATION REPEATABILITY FROM THE COVARIANCE MATRIX
When the robot moves from the initial posture to the final posture, the six robot axe values change. This movement can be modeled from a transformation matrix describing also the movement from an initial reference base to the final reference base. The consecutive transformation matrices can be computed using the Khalil-Kleinfinger method. All the information about the final pose (position and orientation) is summed up in the homogenous transformation matrix. We choose to use the Roll, Pitch and Yaw angles for the orientation of the tool. The rotations are done consecutively around x-axis (roll), y-axis (pitch) and finally z-axis (yaw) as displayed in fig.3.
Fig 3 RPY angles Fig1. position accuracy and repeatability
The experiments are performed on a 6-axe Samsung Faraman robot displayed in fig.4. The kinematic architecture is hybrid because there is a parallelogram for the second and third axes. Once the homogeneous transformation matrix is obtained: M = T01 * T12 * T23 * T34 * T45 * T56 The joint small variations affect the final position and orientation of the robot tool.
So two Jacobian matrices are necessary to link these joint variations to the position and orientation variations. If the position is computed with the function ( x, y, z ) = f (θ1 , θ 2 ,..., θ 6 ) then the position Jacobian
J ori
function is derived and we obtain:
1 ∂g ∂θ × 1 + g 2 1 ∂h = × ∂θ 1 + h 2 ∂u × 1 ∂θ 1 + u 2
This orientation Jacobian matrix maps the angular small with the tool joint variations ( dθ1 , dθ 2 ,..., dθ 6 ) orientation variations ( dR, dP, dY ) with the formula:
(dR, dT , dY )T = J ori × (dθ1 , dθ 2 ,..., dθ 6 )T This derivation is complex and requires the help of symbolic calculus as available in the Matlab software using the functionality diff ( h, θi ) . We have proved in a previous paper that the angular variation dΘ can be modeled with a Gaussian distribution [3]. As the 6 axes have independent control, the angular position random functions are independent. So dΘ is a Gaussian vector whose covariance matrix is D. The density w of the orientation variation vector
d Ω = (dR, dP, dY )T is the following:
−1 w(d Ω) = K exp × d ΩT × C −1 × d Ω 2
Fig4 Samsung robot structure
The orientation information lies in the 3x3 upper left extracted matrix and RPY angles can be computed by the formulas:
Where the constant k is computed by normalizing the density function:
−1
∫∫∫ K exp 2 × d Ω
T
× C −1 × d Ω = 1
Let D be the angular covariance matrix, it is given as follows: D= Where
are the variances of the random
variables ( dθ1 , dθ 2 ,..., dθ 6 ) . So another orientation Jacobian matrix J ori introduced:
must be
The angular covariance matrix D can be estimated using one micrometer and one axis at a time, the other axes being blocked as displayed in fig.5.
And it is clear that the orientation repeatability can be directly obtained computing the square root of the diagonal values. This gives the following results for the orientation repeatability of the Samsung robot for a given location. For instance, for the location P1 in the workspace center, the orientation covariance matrix is displayed in table I for a 3 kg load and in table II for a 6 kg load.
C*10-9 Fig.5 estimation of the angular covariance matrix
The different positions are measured and we obtain statistical series for each actuator. At this stage, it is interesting for a better estimation to use the jump process as explained in detail in [4]. For the Samsung robot, the experimental covariance matrix is estimated for a medium (3 kg) and a high load (6 kg) leading to the following results : D*e-12 0.643
0
0
0
0
-0.048
0.050
0.090
-0.036
0.050
-0.036
0.533
Table I: Covariance matrix for location P1 with a 3 kg load
C*10-9 0.104
- 0.423
0
- 0.423
4121
0.213
0.030
0.213
1.717
0
6.40
0
0
0
0
0
0
5.39
0
0
0
0
0
0
159
0
0
0
0
0
0
35.4
0
0
0
0
0
0
514
The angular covariance matrix D with a 3 kg load
D*e-12 0.937
0
0
0
0
0
0
3.69
0
0
0
0
0
0
7.06
0
0
0
0
0
0
59.2
0
0
0
0
0
0
422
0
0
0
0
0
0
1710
The angular covariance matrix D with a 6 kg load
The covariance matrix C associated with the orientation variations of ( dR, dT , dY ) is obtained by computing :
C = J ori × D × J ori
0.121 -0.048
T
This covariance matrix can be detailed in the following terms:
0.030
Table II: Covariance matrix for location P1 with a 6 kg load
The values of orientation repeatability can be computed in different workspace locations and from angular covariance matrices corresponding to different loads. It is then possible to analyse the effects of load and workspace location on orientation repeatability. This work is inspired by the ISO standard definitions and will allow us to make some comparison with the experimental measured orientation repeatability of next section. III. EXPERIMENTAL APPROACH 3.1 EXPERIMENTAL DEVICE First, to calculate the repeatability index we have chosen targets in the workspace and we measure for each attempt the relative differences in six dimensions: 3 dimensions for the position and 3 dimensions for the orientation. The targets named P1 , P2 , P3 are representative points in the workspace. They are displayed on fig.5. P1 is situated in the workspace center, the point P3 is at the edge of the workspace and the last one P2 is as near as possible as the first axis.
3.2 EXPERIMENTAL RESULTS Repeatability was computed in the 3 different locations in the workspace. Table III and IV display the orientation repeatability values for the points P1 , P2 , P3 and with a low (3 kg) or high load (6kg): Measured Rep. (10-5 rad) Roll Pitch Yaw Point1 Fig. 5. Dimensional characteristics of the Samsung robot
Point2
We measured repeatability using the stationary cube method proposed by the Ford Company. The experimental measurement device consists of two trihedrons. One is an aluminium parallelepiped moving with the robot and is supported by the robot gripper. The other one is fixed on the robot base and supports the measurement device consisting of six micrometers (2 on each side) disposed orthogonally as displayed in fig.6. The six 543-390 Mitutoyo micrometers have a precision error less than 3 µm and their resolution is 1 µm.
Point3
The robot is set up to reach a target point two hundred times. We organize a communication between the robot and the PC so that the 6 micrometers are read once the robot has reached its target. This dialog via the RS232 protocol is very useful to reduce the experiment delay. The setup of the six micrometers is done in such a way that it is possible to compute the position and orientation variations from the six micrometers variations. There is a linear transformation that can be calculated using the Barre de Saint-Venant modeling of small solid movements.
Point3
4.29
5.16
3.95
6.17
8.85
6.71
9.32
11.5
8.95
Table III : Orientation Repeatability values with a 3 kg load
Measured Rep. (10-5 rad) Roll Pitch Yaw Point1 Point2
4.47
6.12
4.73
6.01
7.84
7.19
8.48
8.58
7.98
Table IV : Orientation Repeatability values with a 6 kg load
Is is interesting to notice two important points: First the orientation repeatability varies within the workspace. The values are statistically different when the workspace location is significantly different. It is higher in P2 than in P1 . Concerning the case of P3 , one hypothesis that could explain this result is the parallelogram structure that is working in a unfavorable posture. Secondly the load influence for the same location is not very important because the results are quite the same in a statistical analysis.
IV. DISCUSSIONS We computed the orientation covariance matrix from the angular covariance matrix in the 3 different locations P1 , P2 , P3 and with the 2 loads of 3 kg and 6 kg. All the results are summed up in the table V.
Fig. 6. experimental measurement tryhedron
3kg
6kg
C*10-9 (rad2m Pt 1 Pt 2 Pt 3
0.121 -0.048 0.050 0.138 0.142 0.037 0.097 0.028 0.066
-0.048 0.090 -0.036 0.142 0.871 0.098 0.028 0.416 -0.059
0.050 -0.036 0.533 0.037 0.098 0.523 0.066 -0.059 0.571
0.104 - 0.423 0.030 0.097 -0.471 0.019 0.058 -0.266 0.001
- 0.423 4121 0.213 -0.471 6.293 0.169 -0266 3.930 -0.322
0.030 0.213 1.717 0.019 0.169 1.713 0.001 -0.322 1.732
Table V : orientation covariance matrices
We have also the experimental measured orientation repeatability of the previous section. So it is then possible to compare the measured and computed orientation repeatability. Results are summed up in the tables VI and VII. Measured Rep. Point 1 Point 2 Point 3
Computed Rep. from C *10-5 rad
4.29
5.16
3.95
3.30
2.85
6.93
6.17
8.85
6.71
3.53
8.85
6.86
9.32
11.5
8.95
2.95
6.12
7.17
Table VI: Orientation Repeatability values with a 3 kg load
Measured Rep. Point 1 Point 2 Point 3
Computed Rep. from C *10-5 rad
4.47
6.12
4.73
3.05
19.2
12.4
6.01
7.84
7.19
2.96
23.7
12.4
8.48
8.58
7.98
2.278
18.8
12.4
Table VII: Orientation Repeatability values with a 6 kg load
The results are in the same range but there is quite a difference between the measured and computed repeatability. At this stage it is not clear if one procedure is better than the other. The differences can be explained by several factors, among them: It is quite difficult to estimate precisely the orientation variations because of the size of the cube. The dimension between the micrometers positions are 30 2 mm and we have to estimate precisely the final resolution on the orientation variations linked with the micrometers resolution. So in the end, the measured orientation variations suffer from a
lack of precision inherent to the procedure itself. If we want more precision, we have to increase the distance between the micrometers to benefit from larger lever arm length in the angles estimation. It is also difficult to estimate precisely the angular covariance matrix with a 6 kg load which is a load superior to the nominal load of the robot. This could explain the bad results on the 5th and 6th axes especially.
V. CONCLUSIONS In this paper, we tried to compute orientation repeatability using stochastic ellipsoid theory. It is possible to estimate an angular covariance matrix and from this matrix, compute the orientation covariance matrix. Then if the ISO definition is chosen with the Roll Pitch and Yaw angles used to define orientation, it is easy to compute orientation repeatability from this orientation covariance matrix. For this, it is necessary to introduce an orientation Jacobian matrix. We setup then an experimental procedure to measure orientation repeatability in 3 different workspace locations and with two different loads. This was done with the stationary cube method of the Ford company and six Mitutoyo micrometers. Hundreds of experiments were performed and the influence of load and workspace location were analysed. In the end, both methods are compared. If the result range are the same, there is still some work to do to improve the precision of the results. This method is interesting because it is possible to compute a priori the orientation repeatability in the whole workspace from one angular covariance matrix that can be estimated experimentally using only one micrometer. We are now working to improve the precision of this promising method and we have given the future investigated directions. REFERENCES [1] ISO, 1998, Manipulating Industrial Robots - Performance criteria and related test methods. [2] Norme ANSI R15.05-1-1990., Point-to-Point and Static Performance Characteristics -Evaluation[3] Brethé, J.-F., Vasselin, E., Lefebvre, D., and Dakyo, B., 2006, “Modelling of repeatability phenomena using the stochastic ellipsoid approach,” Robotica, vol. 24, pp. 477–490. [4] Brethé, J.-F., Vasselin, E., Lefebvre, D., and Dakyo, B., “Determination of the repeatability of a Kuka robot using the stochastic ellipsoid approach,” ICRA05, CIMNE-Barcelone, pp 4350-4355.
