Hybrid robust disturbance rejection position-force control to adjust the motion of mobile manipulator in automatic aluminum foundry recycling industries

Figures & data

Figure 1. Motion example of the mobile manipulator showing the tasks that must be completed before grasping the object, during the contact, and after carrying it.

Render of the operative scenario for the mobile manipulator in different times of task development.

Figure 2. Scheme of the robot with all its frames.

Render of the mobile manipulator with all its joints reference frameworks.

Figure 3. Operating diagram for the MR considering the definition of the hybrid system used in this study.

Figure 4. Employed algorithm flow.

Table 1. Initial gain’s condition of the robotic device.

	Joint
Gain name	ATV	q1	q2	q3	q4	q5	q6	q7
KP	2.8	0.0078	0.0078	0.0078	0.0078	24.4800	23.0400	4.3200
KD	5.6	19.4400	19.6344	19.4400	19.4400	8.1600	7.6800	1.4400
KI	0.1	0.0000	0.0000	0.0000	0.0000	0.4352	0.4096	0.0768
KP	2.8	0.0039	0.0039	0.0039	0.0039	24.4800	23.0400	4.3200
KD	5.8	12.9600	13.0896	12.9600	12.9600	8.1600	7.6800	1.4400
KI	0.00005	0.0000	0.0000	0.0000	0.0000	0.4352	0.4096	0.0768
KII	0.00001	0.0000	0.0000	0.0000	0.0000	0.0218	0.0164	0.0038

Figure 5. Trajectory tracking for each DoF of the Kuka robot.

Figure 6. Temporal evolution of the error for each DoF of the Kuka robot.

Table 2. Comparison of the mean square error values for the different control laws at each stage for the Kuka robot.

	Joint MSE
Stage	q1	q2	q3	q4	q5	q6	q7	Total
ST1 - ADRC	6.855 ${\times }^{-05}$	9.203${\times }^{-06}$	8.695${\times }^{-05}$	4.703${\times }^{-05}$	1.227${\times }^{-05}$	7.802${\times }^{01}$	3.058${\times }^{-07}$	1.627${\times }^{02}$
ST1 - PID	5.391${\times }^{-05}$	1.764${\times }^{-05}$	6.820${\times }^{-05}$	5.001${\times }^{-05}$	3.444${\times }^{-05}$	7.769${\times }^{01}$	1.052${\times }^{-06}$	7.769${\times }^{01}$
ST2 - ADRC	4.048${\times }^{-02}$	9.413${\times }^{01}$	9.004${\times }^{-02}$	2.120${\times }^{01}$	1.085${\times }^{-02}$	4.726${\times }^{01}$	1.941${\times }^{-04}$	1.627${\times }^{02}$
ST2 - PID	7.661${\times }^{-02}$	9.799${\times }^{01}$	7.100${\times }^{-02}$	2.152${\times }^{01}$	4.342${\times }^{-03}$	4.615${\times }^{01}$	1.862${\times }^{-04}$	1.658${\times }^{02}$
ST3 - ADRC	1.706${\times }^{-01}$	9.677${\times }^{02}$	2.935${\times }^{-02}$	4.689${\times }^{01}$	1.165${\times }^{-01}$	1.594${\times }^{-02}$	3.086${\times }^{-02}$	1.015${\times }^{03}$
ST3 - PID	1.905${\times }^{-01}$	9.936${\times }^{02}$	1.848${\times }^{-02}$	4.950${\times }^{01}$	1.868${\times }^{-01}$	1.706${\times }^{-02}$	1.039${\times }^{-01}$	1.044${\times }^{03}$

Table 3. Comparison of the error integral values for the different control laws at each stage for the Kuka robot.

	Joint EI
Stage	q1	q2	q3	q4	q5	q6	q7	Total
ST1 - ADRC	2.056${\times }^{-03}$	2.761${\times }^{-04}$	2.609${\times }^{-03}$	1.411${\times }^{-03}$	3.680${\times }^{-04}$	2.341${\times }^{03}$	9.173${\times }^{-06}$	2.341${\times }^{03}$
ST1 - PID	1.617${\times }^{-03}$	5.292${\times }^{-04}$	2.046${\times }^{-03}$	1.500${\times }^{-03}$	1.033${\times }^{-03}$	2.331${\times }^{03}$	3.157${\times }^{-05}$	2.331${\times }^{03}$
ST2 - ADRC	1.214	2.824${\times }^{03}$	2.701	6.359${\times }^{02}$	3.256${\times }^{-01}$	1.418${\times }^{03}$	5.824${\times }^{-03}$	4.882${\times }^{03}$
ST2 - PID	2.298	2.940${\times }^{03}$	2.130	6.456${\times }^{02}$	1.303${\times }^{-01}$	1.385${\times }^{03}$	5.586${\times }^{-03}$	4.974${\times }^{03}$
ST3 - ADRC	5.117	2.903${\times }^{04}$	8.804${\times }^{-01}$	1.407${\times }^{03}$	3.496	4.782${\times }^{-01}$	9.259${\times }^{-01}$	3.045${\times }^{04}$
ST3 - PID	5.715	2.981${\times }^{04}$	5.544${\times }^{-01}$	1.485${\times }^{03}$	5.603	5.119${\times }^{-01}$	3.117	3.131${\times }^{04}$

Figure 7. Temporal evolution of the error for the PLMR.

Figure 9. Temporal evolution of the control for each DoF of the Kuka robot.

Figure 8. Temporal evolution of the control for the PLMR.

Table 4. Comparison of the mean square control values for the different control laws at each stage for the Kuka robot.

	Joint MSC
Stage	q1	q2	q3	q4	q6	q7		Total
ST1 - ADRC	5.493${\times }^{-04}$	9.529${\times }^{-05}$	6.831${\times }^{-04}$	3.275${\times }^{-04}$	3.875${\times }^{-05}$	1.123${\times }^{02}$	1.570e-08	1.123${\times }^{02}$
ST1 - PID	4.895${\times }^{-04}$	2.555${\times }^{-04}$	4.892${\times }^{-04}$	4.037${\times }^{-04}$	1.017${\times }^{-04}$	1.124${\times }^{02}$	1.963e-08	1.124${\times }^{02}$
ST2 - ADRC	2.477${\times }^{-01}$	7.711${\times }^{01}$	7.072${\times }^{-01}$	6.472${\times }^{01}$	3.598${\times }^{-02}$	5.435${\times }^{01}$	8.407${\times }^{-06}$	1.972${\times }^{02}$
ST2 - PID	5.568${\times }^{-01}$	7.902${\times }^{01}$	5.841${\times }^{-01}$	7.202${\times }^{01}$	1.406${\times }^{-02}$	5.347${\times }^{01}$	9.859${\times }^{-06}$	2.057${\times }^{02}$
ST3 - ADRC	1.286	2.796${\times }^{02}$	2.272${\times }^{-01}$	1.594${\times }^{02}$	3.462${\times }^{-01}$	3.210${\times }^{-02}$	7.916${\times }^{-04}$	4.410${\times }^{02}$
ST3 - PID	1.516	2.793${\times }^{02}$	1.380${\times }^{-01}$	1.808${\times }^{02}$	5.782${\times }^{-01}$	4.509${\times }^{-02}$	2.814${\times }^{-03}$	4.624${\times }^{02}$

Table 5. Comparison of the control integral values for the different control laws at each stage for the Kuka robot.

	Joint CI
Stage	q1	q2	q3	q4	q6	q7		Total
ST1 - ADRC	1.648${\times }^{-02}$	2.859${\times }^{-03}$	2.049${\times }^{-02}$	9.824${\times }^{-03}$	1.163${\times }^{-03}$	3.369${\times }^{03}$	4.709${\times }^{-07}$	3.369${\times }^{03}$
ST1 - PID	1.468${\times }^{-02}$	7.664${\times }^{-03}$	1.468${\times }^{-02}$	1.211${\times }^{-02}$	3.051${\times }^{-03}$	3.372${\times }^{03}$	5.888${\times }^{-07}$	3.372${\times }^{03}$
ST2 - ADRC	7.430	2.313${\times }^{03}$	2.122${\times }^{01}$	1.942${\times }^{03}$	1.079	1.631${\times }^{03}$	2.522${\times }^{-04}$	5.915${\times }^{03}$
ST2 - PID	1.670${\times }^{01}$	2.370${\times }^{03}$	1.752${\times }^{01}$	2.161${\times }^{03}$	4.218${\times }^{-01}$	1.604${\times }^{03}$	2.958${\times }^{-04}$	6.170${\times }^{03}$
ST3 - ADRC	3.857${\times }^{01}$	8.388${\times }^{03}$	6.815	4.783${\times }^{03}$	1.039${\times }^{01}$	9.629${\times }^{-01}$	2.375${\times }^{-02}$	1.323${\times }^{04}$
ST3 - PID	4.547${\times }^{01}$	8.380${\times }^{03}$	4.139	5.423${\times }^{03}$	1.735${\times }^{01}$	1.353	8.442${\times }^{-02}$	1.387${\times }^{04}$

Data availability statement

The data that support the findings of this study are available from the corresponding author, [author initials], upon reasonable request.