Design of Multidimensional Nonlinear Predictive Controller for 3D Crane

Maciej Szafrański; Robert Piotrowski

doi:10.14313/jamris-2025-013

.blurhash-client-img { display: none !important; }

Design of Multidimensional Nonlinear Predictive Controller for 3D Crane

Journal of Automation, Mobile Robotics and Intelligent Systems

Volume 19 (2025): Issue 2 (June 2025)

By: Maciej Szafrański and Robert Piotrowski

Open Access

|Jun 2025

Figures & Tables

Table 1.

State and control variables of the mathematical model

Variable	Description
x₁	Position X axis [m]
x₂	Linear velocity X axis [m/s]
x₃	Position Y axis [m]
x₄	Linear velocity Y axis [m/s]
x₅	Angular deviation between lifting rope and Y axis [rad]
x₆	Angular velocity for x₅ [rad/s]
x₇	Angular deviation between the negative direction on the Z axis and the projection of the lifting rope onto the XZ plane [rad]
x₈	Angular velocity for x₇ [rad/s]
x₉	Position Z axis [m]
x₁₀	Linear velocity Z axis [m/s]
u₁	PWM control signal for DC motor in X axis [-]
u₂	PWM control signal for DC motor in Y axis [-]
u₃	PWM control signal for DC motor in Z axis [-]

Table 2.

Constant parameters of the 3D crane model

Parameter	Variable	Value
Load weight	m_c	1 kg
Boogie weight	m_w	1.155 kg
Bus weight	m_s	2.2 kg
Friction force axis X	T_x	100 N
Friction force axis Y	T_y	82 N
Friction force axis Z	T_z	75 N

The structure of 3D crane control system with 5 PID controllers

The structure of 3D crane predictive control system

Table 3.

Proposed gains of PID controllers

No.	PID X	PID Y	PID Z	PID A	PID B
1	kp = 33 ki = 0.4 kd = 0	kp = 41 ki = 0.1 kd = 0	kp = 50 ki = 0.2 kd = 0	kp = 0 ki = 0 kd = 0	kp = 0 ki = 0 kd = 0
2	kp = 33 ki = 0.4 kd = 0	kp = 41 ki = 0.1 kd = 0	kp = 50 ki = 0.2 kd = 0	kp = 10 ki = 0.2 kd = 0	kp = 10 ki = 0.3 kd = 0
3	kp = 33 ki = 0.4 kd = 0.2	kp = 41 ki = 0.1 kd = 0,32	kp = 50 ki = 0.2 kd = 0.1	kp = 15 ki = 0.2 kd = 0.05	kp = 17 ki = 0.3 kd = 0.2

Table 4.

IAE for PID controllers

No.	x₁	x₃	x₉	x₅	x₇	Sum
1	0.3451	0.345	0.253	0.1945	0.3759	1.513
2	0.342	0.3646	0.253	0.1164	0.248	1.324
3	0.3413	0.3788	0.253	0.1077	0.2022	1.283

Table 5.

ISE for PID controllers

No.	x₁	x₃	x₉	x₅	x₇	Sum
1	0.02403	0.04063	0.01387	0.003578	0.01483	0.09693
2	0,02398	0,04076	0,01387	0,00133	0,005404	0,08534
3	0,02402	0,04096	0,01387	0,001256	0,003864	0,08397

Table 6.

Weight values for NMPC controller

No.	w₁	w₂	w₃	w₄	w₅	w₆	w₇	w₈
1	1000	1000	1000	0	0	0.001	0.001	0.001
2	1000	1000	3000	0.03	0.03	0.01	0.01	0.01
3	1000	1000	3000	0.03	0.03	0.0005	0.0005	0.0005
4	1000	1000	3000	1	1	0.001	0.001	0.001

Table 7.

IAE for NMPC controller – tuning the weight values with constant H_p

No.	x₁	x₃	x₉	x₅	x₇	Sum
1	0.3249	0.35	0.3904	0.0874	0.4617	1.614
2	0.3742	0.4531	0.3978	0.1022	0.4505	1.778
3	0.3243	0.3507	0.2591	0.08956	0.4705	1.494
4	0.3227	0.5025	0.2551	0.08911	0.5194	1.689

Table 8.

ISE for NMPC controller – tuning the weight values with constant H_p

No.	x₁	x₃	x₉	x₅	x₇	Sum
1	0.02347	0.04116	0.02141	0.0008966	0.02083	0.1078
2	0.02349	0.04272	0.01805	0.001013	0.01973	0.105
3	0.02347	0.04171	0.01386	0.00092	0.02129	0.1013
4	0.02353	0.0463	0.01375	0.0008992	0.02578	0.1103

Table 9.

IAE for NMPC controller – tuning the control horizon H_p

H_c.	x₁	x₃	x₉	x₅	x₇	Sum
H_c=2	0.3335	0.3589	0.3007	0.09413	0.4805	1.568
H_c=4	0.3243	0.3507	0.2591	0.08956	0.4705	1.494
H_c=6	0.326	0.3552	0.2607	0.08828	0.4496	1.48

Table 10.

ISE for NMPC controller – tuning the control horizon H_p

H_c	x₁	x₃	x₉	x₅	x₇	Sum
H_c=2	0.02313	0.04175	0.01481	0.0009334	0.02222	0.1028
H_c=4	0.02347	0.04171	0.01386	0.00092	0.02129	0.1013
H_c=6	0.0236	0.04202	0.01357	0.0009144	0.01989	0.09999

Obtained waveforms of tracking reference trajectory x₁ for both designed control systems

Obtained waveforms of tracking reference trajectory x₃ for both designed control systems

Obtained waveforms of tracking reference trajectory x₉ for both designed control systems

Obtained waveforms of x₅ control for both designed control systems

Obtained waveforms of x₇ control for both designed control systems

References

Authors

Metrics

Articles in this issue

DOI: https://doi.org/10.14313/jamris-2025-013 | Journal eISSN: 2080-2145 | Journal ISSN: 1897-8649

Journal RSS Feed

Language: English

Page range: 41 - 49

Submitted on: Nov 11, 2023

Accepted on: Feb 8, 2024

Published on: Jun 26, 2025

Published by: Łukasiewicz Research Network – Industrial Research Institute for Automation and Measurements PIAP

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

3D crane,

control systems,

mathematical modelling,

multidimensional controller,

nonlinear MPC controller

Related subjects:

Computer sciences,

Artificial intelligence,

Engineering,

Electrical engineering,

Control engineering, metrology and testing,

Mechanical engineering,

Fundamentals of mechanical engineering

© 2025 Maciej Szafrański, Robert Piotrowski, published by Łukasiewicz Research Network – Industrial Research Institute for Automation and Measurements PIAP
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 19 (2025): Issue 2 (June 2025)

Design of Multidimensional Nonlinear Predictive Controller for 3D Crane

Figures & Tables

Figure 1.

Table 1.

Table 2.

Figure 2.

Figure 3.

Figure 4.

Figure 5.

Figure 6.

Table 3.

Table 4.

Table 5.

Table 6.

Table 7.

Table 8.

Table 9.

Table 10.

Figure 7.

Figure 8.

Figure 9.

Figure 10.

Figure 11.

Paradigm

My account