Measurement of End-effector Pose Errors and the Cable Profile of Cable-Driven Robot using Monocular Camera

Abstract

Measurement of performance metrics namely repeatability and directional accuracy of an under-actuated cable-driven robot using a monocular camera is proposed. Experiments were conducted on a cable-driven robot used for construction and the results are compared with that from a laser tracker. Least value of about 18 mm was observed at the middle of the workspace and it agreed with the measurement from laser tracker. Using the proposed method it was also possible to measure orientation repeatability considering the end-effector coordinate system as the reference, which is an advantage over using a laser tracker. Since the robot workspace is very large and there are restrictions on feasible measurement volume while using a camera, a method to determine suitable locations of the camera to enable measurement in all of the workspace is also proposed. Results of this analysis show that the camera may be positioned at four feasible regions during measurement over the whole workspace. Finally, a method for measurement of cable profile is also proposed. The position and slope of the cable at two locations were measured using a camera. This was utilized to fit a catenary model such that profile of the cable between these points can be determined. Experiments conducted to determine the cable profile in different regions of workspace showed that there is comparatively greater sagging at the periphery of the workspace which explains the higher positioning errors in the same region.

Appendices

Appendix A: Performance Criteria of Industrial Robots

Repeatability and accuracy are two important metrics used to measure performance criteria of industrial robots. Measurement of repeatability and distance accuracy is explained in this Appendix. Repeatability of robot [6, 21] is measured after defining a cube shown in Fig. 14. Then, the robot is made to move within the shaded region with a given velocity. The movement is to be made in the order of P1, P2, P3, P4 and P5, as shown in Fig. 15.

Fig. 14

Options of plane within cubes for experiments on repeatability measurement [6, 21]

Fig. 15

Poses for repeatability measurement [6, 21]

Let the parameters, x_j, y_j and z_j, are the coordinates of the j^th attained pose. Robot repeatability in position, RP_l, is then computed as

$$ RP_{l}=\bar{l}+3S_{l} $$

(7)

where, $\bar {l}=\frac {1}{n}{\sum }_{j=1}^{n}l_{j}$ and $l_{j}=\sqrt {(x_{j}-\bar {x})^{2}+(y_{j}-\bar {y})^{2}+(z_{j}-\bar {z})^{2}}$. Additionally, $\bar {x}=\frac {1}{n}{\sum }_{j=1}^{n}x_{j}$, $\bar {y}=\frac {1}{n}{\sum }_{j=1}^{n}y_{j}$ and $\bar {z}=\frac {1}{n}{\sum }_{j=1}^{n}z_{j}$ which are the coordinates of the bary-center obtained after repeatedly approaching to a pose n times. Moreover, $S_{l}=\sqrt {\frac {{\sum }_{j=1}^{n}(l_{j}-\bar {l})^{2}}{n-1}}$.

Now, if A_j, B_j and C_j be the roll, pitch and yaw angles of the end-effector of the robot at the j^th pose respectively, robot’s repeatability for orientation corresponding to each rotation is given by

$$ RR_{A}=\pm3S_{A}=\pm3\sqrt{\frac{{\sum}_{j=1}^{n}(A_{j}-\bar{A})}{n-1}} $$

(8)

$$ RR_{B}=\pm3S_{B}=\pm3\sqrt{\frac{{\sum}_{j=1}^{n}(B_{j}-\bar{B})}{n-1}} $$

(9)

$$ RR_{C}=\pm3S_{C}=\pm3\sqrt{\frac{{\sum}_{j=1}^{n}(C_{j}-\bar{C})}{n-1}} $$

(10)

where $\bar {A}=\frac {1}{n}{\sum }_{j=1}^{n}A_{j},\bar {B}=\frac {1}{n}{\sum }_{j=1}^{n}B_{j}$, and $\bar {C}=\frac {1}{n}{\sum }_{j=1}^{n}C_{j}$ in which, $\bar {A}$, $\bar {B}$ and $\bar {C}$ are the mean values of the angles obtained after reaching the orientation n times. Distance accuracy represents the deviation of the measured distance moved by the robot in comparison to the actual distance commanded to the robot. This is represented by the mean of the difference in measured and commanded distances. Directional accuracy AD_P is calculated as follows:

$$ AD_{P}=\bar{D}-D_{C} $$

(11)

where, $\bar {D}=\frac {{\sum }_{j=1}^{j=n}D_{j}}{n}$, D_j = ||p_1j − p_2j||₂ and D_j = ||p_c1 − p_c2||₂. The 3-D coordinates of commanded pose for two points are represented by p_c1 and p_c2. The corresponding measured coordinates at j^th measurement are p_1j and p_2j respectively.

Appendix B: Fiducial Markers

Methods of pose detection using fiducial markers are explained in this Appendix. A Software Development Kit (SDK) called ArUco [37] was used. Details about it are explained next. ArUco is a SDK for generating fiducial markers (Fig. 16a) which is used for artificial reality applications. The markers can be easily detected in an image taken by a camera. If the camera is calibrated before-hand, the pose consisting of information on 3-D position coordinates and 3-D orientation of the marker with respect to the camera can also be estimated. There is option to use a single marker, or an array of markers called board of markers. If an array of markers are used, even if there is occlusion of some of the markers it is still possible to detect the pose of the board when the configuration of the markers is known. Detection of markers in image is shown in Fig. 16b.

Fig. 16

Fiducial markers

In this work, the camera was mounted on a tripod so that the robot’s end-effector is visible. Then the calibration was done to obtain the camera’s internal parameters. ArUco markers was created using the ArUco SDK. Its configuration was saved in a separate file. Given the configuration of the markers and the physical dimension of each marker, the pose of the board of markers from any image of the camera can be obtained. The robot was moved along a particular trajectory. At each pose of the robot, an image of the system of the markers was grabbed. Later, the images were processed using ArUco’s SDK to obtain the pose of the camera. After multiple measurements, the values of repeatability (as defined in Appendix A) were calculated. As discussed earlier in Appendix B, ArUco SDK can create markers which can be easily identified within a given image. They are designed in such a way that each marker has a unique id and can also be uniquely detected. A board of marker has multiple such markers arranged on a planar board which is then printed on a paper. Given the sequence in which the markers are arranged in the board and the marker size, the 2-D pose of each marker with respect to another can be obtained and this is termed as marker configuration. Therefore, given an input image from a calibrated camera in which any part of the board is visible and the marker configuration, the pose of the camera can be estimated.

An alternate method of utilizing the fiducial markers is to use a Marker map [38]. For Marker maps, the markers may not necessarily be pasted on the same planar surface. It can be pasted at any arbitrary orientations. An image containing any two given markers could provide the pose of one with respect to other. Assume there are multiple markers, for e.g., m₁, m₂,...m_n. If multiple images with markers m₁ and m₂, then m₂ and m₃ etc., are present, it is possible to compute the pose of the marker m₂ with respect to the other ones. Similarly, if the whole grid of markers is imaged in this fashion, the mutual pose of each one of the marker with respect to the others is also determined. An application of pose measurement of industrial robots using fiducial markers can be found at [11, 34].