Unsatisfactory Hand-Eye Calibration Results (Loose, Sagging, or Vibrating Calibration Board)

Problem

The hand-eye calibration results are unsatisfactory (bad or marginal), looking like something between these combinations:

Hand-Eye Transform

Hand-Eye Residuals

Projection Verification

Touch Test

Reasonable

Bad

Bad

Bad

Reasonable

Marginal

Poor

Bad
Example of marginal hand-eye calibration results

The Hand-Eye Transformation Matrix appears reasonable and is consistent with what one would expect from rough manual measurements (see table and image below).

Hand-Eye Transformation Matrix

0.01

1.00

0.02

-23.06

1.00

-0.01

0.06

440.37

0.06

0.02

-1.00

1969.85

0.00

0.00

0.00

1.00
../../_images/eye_to_hand_approximate_hand_eye_matrix.png

The calibration residuals are slightly larger than expected.

Residuals

N (Poses)

Rot. avg (°)

Rot. max (°)

Trans. avg (mm)

Trans. max (mm)

16

0.24

0.79

0.68

1.48

For smaller robots, we expect the residuals to in the sub-millimeter range, while for larger robots, the residuals can be in the millimeter range. As for the rotational component of the residuals, we expect them to be in the sub-degree range.

Complete Residuals
Residuals

Pose

Rotation (°)

Translation (mm)

1

0.04

0.78

2

0.74

1.46

3

0.17

0.72

4

0.10

0.72

5

0.15

0.51

6

0.12

0.51

7

0.22

0.60

8

0.11

0.26

9

0.12

0.26

10

0.16

0.54

11

0.27

0.91

12

0.14

0.73

13

0.20

0.58

14

0.79

1.48

15

0.21

0.44

16

0.30

0.34

Verification by Projection technically succeeds, but the resulting error is still too large to be acceptable. One should be very critical of projection verification, as the projection error may seem acceptable even if the hand-eye calibration is incorrect.

../../_images/projection_verification_poor_eth.jpg

The touch test result is unsatisfactory. For example, the tip of the touch test tool does not reach the ArUco marker.

Potential Cause

A common cause of high residuals is flex or movement in the on-arm mounting structure that connects the robot flange to the calibration board, or in the calibration board itself. This can cause the calibration board to still vibrate while the camera captures it or to be slightly shifted across camera captures. Both of these factors directly affect the accuracy of the hand-eye calibration results.

This can occur when:

  • The robot acceleration is too high

  • The delay is too short before capturing to let the calibration board vibration settle

  • The bracket holding the calibration board is 3D-printed, thin, or otherwise insufficiently rigid

  • Screws or fasteners are not properly tightened

  • The mount sags, bends, or shifts as the robot changes orientation

Any mechanical movement during dataset acquisition results in the calibration board appearing in slightly different poses, which directly increases residuals.

Potential Solution

  1. Ensure that the calibration board is completely still during dataset acquisition:

    • Keep the robot acceleration low during the dataset acquisition

    • Add sufficient delay after the robot reaches its target pose before capturing

  2. Ensure that the mounting interface between the robot flange and the calibration board is completely rigid:

    • Use a solid metal bracket or another high-stiffness material

    • Avoid 3D-printed, thin, or long cantilevered designs

    • Verify that all screws, adapters, and extension plates are fully tightened

    • If necessary, shorten the lever arm or reinforce the mount

  3. Repeat the hand-eye calibration using the same dataset.

  4. Verify the hand-eye calibration results (transform, residuals, projection).

  5. Run the touch test (if the hand-eye calibration results are good).

Reproducing the issue (vibrating calibration board)

Good Dataset

A hand-eye calibration was successfully performed using the dataset shown in the images below.

../../_images/hand-eye-calibration-eth-good-dataset.png

The calibration resulted in the following results:

Hand-Eye Transformation Matrix

0.01

1.00

0.02

-24.28

1.00

-0.01

0.06

442.41

0.06

0.02

-1.00

1969.45

0.00

0.00

0.00

1.00
Residuals

N (Poses)

Rot. avg (°)

Rot. max (°)

Trans. avg (mm)

Trans. max (mm)

16

0.06

0.11

0.34

0.65
Complete Residuals
Residuals

Pose

Rotation (°)

Translation (mm)

1

0.03

0.56

2

0.06

0.65

3

0.11

0.34

4

0.06

0.08

5

0.05

0.64

6

0.03

0.06

7

0.07

0.36

8

0.06

0.18

9

0.07

0.14

10

0.04

0.19

11

0.05

0.29

12

0.09

0.34

13

0.06

0.21

14

0.07

0.33

15

0.06

0.46

16

0.06

0.62

Verification by Projection was successful.

../../_images/projection_verification_good_eth.jpg

The Hand-Eye calibration was then verified with a touch test.

The touch test was successful, which confirms that the hand-eye calibration is correct and accurate.

Bad Dataset (vibrating calibration board)

The same dataset was used for hand-eye calibration. However, the acceleration of the robot was intentionally increased to simulate a vibrating calibration board during the dataset acquisition. The delay after the robot reached its target pose before capturing was also intentionally shortened to further simulate the effect of a vibrating calibration board.

The calibration resulted in the following results:

Hand-Eye Transformation Matrix

0.01

1.00

0.02

-23.06

1.00

-0.01

0.06

440.37

0.06

0.02

-1.00

1969.85

0.00

0.00

0.00

1.00
Residuals

N (Poses)

Rot. avg (°)

Rot. max (°)

Trans. avg (mm)

Trans. max (mm)

16

0.24

0.79

0.68

1.48
Complete Residuals
Residuals

Pose

Rotation (°)

Translation (mm)

1

0.04

0.78

2

0.74

1.46

3

0.17

0.72

4

0.10

0.72

5

0.15

0.51

6

0.12

0.51

7

0.22

0.60

8

0.11

0.26

9

0.12

0.26

10

0.16

0.54

11

0.27

0.91

12

0.14

0.73

13

0.20

0.58

14

0.79

1.48

15

0.21

0.44

16

0.30

0.34

Verification by Projection technically succeeded, but the resulting error was still too large to be acceptable. One should be very critical of projection verification, as the projection error may seem acceptable even if the hand-eye calibration is incorrect.

../../_images/projection_verification_poor_eth.jpg

The touch test result was unsatisfactory as the tip of the touch test tool did not reach the ArUco marker.

Comparisons

The results between hand-eye calibration matrices (translation values) and hand-eye calibration residuals are compared in the tables below.

Translation Values in Hand-Eye Transformation Matrix

Dataset

X (mm)

Y (mm)

Z (mm)

Good (Still)

-24.28

442.41

1969.45

Bad (Vibrating)

-23.06

440.37

1969.85

Δ (Good - Bad)

-1.22

2.04

-0.40
Residuals

Dataset

N

Rot. avg (°)

Rot. max (°)

Trans. avg (mm)

Trans. max (mm)

Good (Still)

16

0.06

0.11

0.34

0.65

Bad (Vibrating)

16

0.24

0.79

0.68

1.48