A Novel Velocity-Based Control in a Sensor Space for Parallel Manipulators
2. Delta-Type Parallel Robots
2.1. Forward and Inverse Kinematic Model of the Delta Robot
2.2. Computation of the Delta Robot’s Jacobian
3. Camera-Space Manipulation with a Linear Camera Model
4. CSM-Based Velocity Control
5. Materials and Methods
5.3. Position Measuring System
5.4. System Operation
- The first set of experiments consisted of a series of static positioning tasks, i.e., using the proposed control, the robot tracked a static target placed randomly inside the robot’s workspace. Ten tasks were executed, each repeated 3 times, for a total of 30 trials. Once the robot’s end-effector was within 2 mm of the target position, the robot held its position for 50 control cycles, and the task’s mean squared error was computed. During these experiments, the control gain matrix () was chosen of the form where I is the identity matrix and ; this value was tuned heuristically.
- For the second set of experiments, the target was placed on a conveyor belt (approximately in the middle of the belt, width-wise) running at a constant speed. That is, the target moved following a linear trajectory referred to as “constant speed linear trajectory”. Three different speeds were used: 7, 9.5, and 12 cm/s.Additionally, three different control gain matrices () were tested, of the form , where . The value of k was chosen as large as possible while maintaining no osculations on the task’s positioning response.For each speed, the task was performed 10 times with each of the possible matrices. Finally, each task was also carried out under two conditions regarding the target’s velocity () compensation in the control law; (1) an estimate obtained by means of a Kalman filter was used, and (2) no estimation was used (the compensation was set to zero) yielding a simpler implementation but producing a larger error. However, this error can be reduced by increasing the control gain. In each case, the robot’s tracking error was measured.
- For the last set of experiments, the target was moved freehand along different trajectories inside the robot’s workspace. This experiment was referred to as “freehanded trajectory”. These trajectories were: a circle, a square, an eight shape, a lemniscate, a zig-zag, and a decreasing spiral. The control gain matrix () was chosen to be a diagonal matrix with a value of 2.7 on its non-zero terms.
- We obtained the coordinates of the objective point in pixels () and loaded the vision parameters previously estimated during the “pre-plan”.
- The program entered the control cycle, setting a convergence criterion of error in each camera coordinate () less than 1 pixel.
- The centroid of the visual marker attached to the robot’s end-effector () was obtained and the error was computed (22).
- We performed (19) to obtain the speeds to be injected into the robot’s controller.
- We repeated until the convergence was obtained.
6. Results and Discussion
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A.1. Target Velocity Estimation in Camera Space via Kalman Filter
Appendix A.2. Target Velocity Estimation
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|Conveyor Speed||RMS Tracking Error (mm)|
|Trajectory||RMS Tracking Error (mm)||Final Error (mm)|
|Control Cycle Time [ms]||Static Error (mm)||Std Dev (mm)||Tracking Error (mm) @ vel (mm/s)||Method|
|242||1.11||0.35||NA||Gonzalez et al. |
|8.33||4.21||2||[email protected]||Visual Servo—Trashlosheros |
|1.4||4||1||NA||Visual Servo—Özgür |
|57.3||1.48||0.43||NA||LCSM—Lara et al. |
|48||1.09||0.19||[email protected], [email protected], [email protected]||CSM—Vel|
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Loredo, A.; Maya, M.; González, A.; Cardenas, A.; Gonzalez-Galvan, E.; Piovesan, D. A Novel Velocity-Based Control in a Sensor Space for Parallel Manipulators. Sensors 2022, 22, 7323. https://doi.org/10.3390/s22197323
Loredo A, Maya M, González A, Cardenas A, Gonzalez-Galvan E, Piovesan D. A Novel Velocity-Based Control in a Sensor Space for Parallel Manipulators. Sensors. 2022; 22(19):7323. https://doi.org/10.3390/s22197323Chicago/Turabian Style
Loredo, Antonio, Mauro Maya, Alejandro González, Antonio Cardenas, Emilio Gonzalez-Galvan, and Davide Piovesan. 2022. "A Novel Velocity-Based Control in a Sensor Space for Parallel Manipulators" Sensors 22, no. 19: 7323. https://doi.org/10.3390/s22197323