# High-Precision and Modular Decomposition Control for Large Hydraulic Manipulators

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Joint Decomposition and Link Partitioning

^{O}U

_{G}of the coordinate frame G equivalent to O and the matrix

^{E}U

_{F}of the coordinate frame F equivalent to E was derived.

#### 2.2. Kinematic Analysis of the Closed Chains

_{1}, q

_{11}, and q

_{12}, and the position of piston x

_{1}is described as follows:

_{11}and l

_{12}are the lengths of two adjacent links, x

_{11}is the hydraulic cylinder barrel length, and x

_{12}is the piston rod length.

_{1}, $\dot{q}$

_{11}, $\dot{q}$

_{12}and velocity $\dot{x}$

_{1}can be obtained by Equations (7)–(9) as follows:

^{O}V = [0 0 0 0 0 0]

^{T}, z

_{r}= [0 0 0 0 0 1]

^{T}, and z

_{f}= [1 0 0 0 0 0]

^{T}.

^{A}U

^{T}

_{B}represents the transformation matrix of the force and moment vectors in the coordinate system B relative to the coordinate system A:

_{x}, r

_{y}, r

_{z}represent the distance of the coordinate system B concerning A in the directions of x, y, and z.

_{1}to B

_{10}as an example,

^{B1}U

_{B10},

^{B1}R

_{B10},

^{B1}r

_{B1B10×}can be obtained:

#### 2.3. Dynamics Analysis of the Closed Chains

^{A}f

^{net}∈ R

^{3}and

^{A}m

^{net}∈ R

^{3}represent the force vector and the moment vector in frame A.

_{o}as:

_{3}∈ R

^{3×3}is the identity matrix, m

_{A}is the mass of the rigid body, I

_{o}∈ R

^{3×3}is the inertia matrix at the center of mass of the rigid body. v

_{o}∈ R

^{3}and w

_{o}∈ R

^{3}represent the linear velocity vector and the angular velocity vector at the center of mass, respectively. The net force/moment vectors applied to the mass center are denoted by the symbols f

^{net}∈ R

^{3}and m

^{net}∈ R

^{3}.

^{T}F ∈ R

^{6}at a VCP can be obtained through the recursive dynamic calculation of the subsystem. It can be written as:

^{T}

^{1}F and

^{T}

^{2}F at the driving VCP of the open chains 1 and 2 can be expressed as:

_{1}and α

_{2}are two load distribution coefficients, with α

_{1}+ α

_{2}= 1, and ${}^{\mathrm{T}}\eta ={\left[\begin{array}{cccc}{}^{\mathrm{T}}\eta _{fx}& {}^{\mathrm{T}}\eta _{fy}& 0& \begin{array}{ccc}0& 0& {}^{\mathrm{T}}\eta _{mz}\end{array}\end{array}\right]}^{\mathrm{T}}$ is the internal force vector.

#### 2.4. Calculation of Load Distribution Coefficients and Internal Force Vector

_{1}and α

_{2}describe the force distribution between two open-chain links in the x–y plane when

^{T}f is applied to the closed-chain driver VCP.

^{T}f in the coordinate system {T}.

^{T}f when the force vector

^{T}f was applied from the driver VCP of the closed chain to its adjacent subsystem. The force vector −

^{T}f produced the reaction forces −f

_{L}

_{12}and −f

_{L}

_{13}. The reaction forces −f

_{L}

_{12}and −f

_{L}

_{13}, as shown in Figure 5b, were decomposed into the force vectors −

^{T}f

_{L}

_{12}and −

^{T}f

_{L}

_{13}parallel to −

^{T}f. The relationship can be expressed as follows:

_{1}and α

_{2}can be derived as follows:

^{T}η first. The calculation of the internal force vector

^{T}η required to obtain the forces

^{T}η

_{fx}along the x-axis and

^{T}η

_{fy}along the y-axis and the moment

^{T}η

_{mz}along the z-axis.

^{T}η

_{fy}can be derived as follows:

## 3. Subsystem of the Hydraulic Actuator

_{c}be the pressure and f

_{u}be the output force and define the f

_{f}as the friction force. So, we have:

_{f}is proportional to the product of the square root of the spool control signal and the pressure drop:

_{a}and g

_{b}as follows:

_{p}

_{1}, c

_{n}

_{1}, c

_{p}

_{2}, c

_{n}

_{2}are the valve port flow coefficients, P

_{s}, P

_{r}, P

_{a}, and P

_{b}are the system pressure, tank pressure, rod chamber pressure, and rod-less chamber pressure, respectively, v is a symbolic function.

_{a}and S

_{b}represent the rod chamber area and the rod-less chamber area, respectively, c represents the piston rod displacement, and c

_{m}represents the stroke of the piston rod. Consider

_{cr}is the output force, and u

_{f}can be written as:

_{fd}is determined according to the Equation (44)

_{fd}can be rewritten according to Equation (52) when the condition of Equation (51) is satisfied:

## 4. Adaptive Design of the Inertial Parameters

_{B}

_{1}∈ R

^{6×6}, K

_{B}

_{10}∈ R

^{6×6}, K

_{B}

_{11}∈ R

^{6×6}, and K

_{B}

_{12}∈ R

^{6×6}are the positive definite gain matrices $\widehat{\theta}$

_{B}

_{1}∈ R

^{13}, $\widehat{\theta}$

_{B}

_{10}∈ R

^{13}, $\widehat{\theta}$

_{B}

_{11}∈ R

^{13}, and $\widehat{\theta}$

_{B}

_{12}∈ R

^{13}. $\widehat{\theta}$ is an estimated parameter matrix containing mass, moment of inertia, product of inertia, and other parameters.

_{B}

_{1γ}is the parameter updating gain. $\overline{\theta}$

_{B}

_{1γ}and $\underset{\_}{\theta}$

_{B}

_{1γ}are the upper and lower limits of $\widehat{\theta}$

_{B}

_{1γ}, respectively.

_{Bi}can be expressed as:

## 5. Stability Analysis of the System

_{∞}, and y(t) as a virtual function of L

_{2}. The L

_{2}-L

_{∞}stability and convergence of the whole system can be guaranteed when each subsystem has the required stability and convergence. Define a scalar term corresponding to each VCP, called virtual power flow (VPF). VPF is the inner product of the velocity error and the force error and defines the dynamic interaction between subsystems. VPF plays a crucial role in the virtual stability of subsystems and ensures the L

_{2}-L

_{∞}stability of the whole system.

_{∞}, and y(t) as a virtual function of L

_{2}, where 0 ≤ γ

_{s}≤ ∞, P and Q are two diagonally positive definite matrices, Φ and Ψ are coordinate systems placed at the driven cut point, P

_{A}and P

_{C}are VPFs. The inner product of the error of the force/moment vector and the error of the linear/angular velocity vector is known as P

_{A}:

^{B}

^{1}V

_{r}−

^{B}

^{1}V and

^{B}

^{10}V

_{r}−

^{B}

^{10}V of the first open chain are chosen as virtual functions of L

_{2}and L

_{∞}. The non-negative adjoint function of the first open chain is:

_{B}

_{1}and v

_{B}

_{10}are non-negative adjoint functions of the two links of the open chain 1:

_{B}

_{1}and v

_{B}

_{10}in relation to time is obtained with:

_{1}in relation to time is:

^{T}Vr −

^{T}V)

^{T}:

_{ur}– f

_{u}), which prevents the open chain’s stability.

_{c}according to the hydraulic dynamics and control equations as follows:

_{c}satisfies the condition as follows:

^{B}

^{11}V

_{r}−

^{B}

^{11}V,

^{B}

^{12}V

_{r}−

^{B}

^{12}V, and f

_{cr}− f

_{c}as virtual functions for both L

_{2}and L

_{∞}. The second open chain’s non-negative adjoint function is redefined as follows:

## 6. Simulation and Analysis

#### 6.1. Simulation Platform

#### 6.2. Method Validation in a Simulation without Disturbances

#### 6.3. Method Validation in a Simulation with External Disturbances

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Name | Description and Unit |

VDC | Virtual decomposition control |

VCP | Virtual cutting point |

VPF | Virtual power flow |

MBC | Model-based control |

CC | Closed chain |

OC | Open chain |

v | Linear velocity in the original frame |

w | Angular velocity in the original frame |

q_{i} | Joint angle (rad) |

x_{i} | Displacement of the cylinders (mm) |

${\dot{q}}_{i}$ | Joint angle velocity (rad/s) |

${\dot{x}}_{i}$ | Velocity of the hydraulic cylinder (mm/s) |

${\dot{q}}_{ir}$ | Required angular velocity (rad/s) |

${\dot{q}}_{id}$ | Desired angular velocity (rad/s) |

s_{a} | Hydraulic cylinder area with the rod (cm^{2}) |

s_{b} | Hydraulic cylinder area without the rod (cm^{2}) |

c | Displacement of the piston rod (mm) |

p_{s} | System pressure (Mpa) |

p_{a} | Hydraulic cylinder pressure with the rod (Mpa) |

P_{b} | Hydraulic cylinder pressure without the rod (Mpa) |

E | Bulk modulus (pa) |

## References

- Henikl, J.; Kemmetmüller, W.; Bader, M.; Kugi, A. Modelling, simulation and identification of a mobile concrete pump. Math. Comput. Model. Dyn. Syst.
**2015**, 21, 180–201. [Google Scholar] [CrossRef] - Luo, S.; Cheng, M.; Ding, R.; Wang, F.; Xu, B.; Chen, B. Human–Robot Shared Control Based on Locally Weighted Intent Prediction for a Teleoperated Hydraulic Manipulator System. IEEE/ASME Trans. Mechatron.
**2022**, 27, 4462–4474. [Google Scholar] [CrossRef] - Cheng, M.; Li, L.; Ding, R.; Xu, B. Real-Time Anti-Saturation Flow Optimization Algorithm of the Redundant Hydraulic Manipulator. Actuators
**2021**, 10, 11. [Google Scholar] [CrossRef] - Konrad, J.; Morten, K. Development of Point-to-Point Path Control in Actuator Space for Hydraulic Knuckle Boom Crane. Actuators
**2020**, 9, 27–46. [Google Scholar] - Chao, Y.; Yujia, W.; Feng, Y. Driving performance of underwater long-arm hydraulic manipulator system for small autonomous underwater vehicle and its positioning precision. Int. J. Adv. Robot. Syst.
**2017**, 14, 1–18. [Google Scholar] - Qiao, L.; Zhao, M.; Wu, C.; Ge, T.; Fan, R.; Zhang, W. Adaptive PID control of robotic manipulators without equality inequality constraints on control gains. Int. J. Robust Nonlinear Control
**2022**, 32, 9742–9760. [Google Scholar] [CrossRef] - Tan, N.; Li, C.; Yu, P.; Ni, F. Two model-free schemes for solving kinematic tracking control of redundant manipulators using CMAC networks. Appl. Soft Comput.
**2022**, 126, 109267. [Google Scholar] [CrossRef] - Hu, S.; Liu, H.; Kang, H.; Ouyang, P.; Liu, Z.; Cui, Z. High Precision Hybrid Torque Control for 4-DOF Redundant Parallel Robots under Variable Load. Actuators
**2023**, 12, 232. [Google Scholar] [CrossRef] - Feng, C.; Chen, W.; Shao, M.; Ni, S. Trajectory Tracking and Adaptive Fuzzy Vibration Control of Multilink Space Manipulators with Experimental Validation. Actuators
**2023**, 12, 138. [Google Scholar] [CrossRef] - Lee, J.; Chang, P.H.; Yu, B.; Jin, M. An Adaptive PID Control for Robot Manipulators under Substantial Payload Variations. IEEE Access.
**2020**, 8, 162261–162270. [Google Scholar] [CrossRef] - Li, S.; He, J.; Li, Y.; Rafique, M.U. Distributed Recurrent Neural Networks for Cooperative Control of Manipulators: A Game-Theoretic Perspective. Appl. Soft Comput.
**2022**, 126, 109267. [Google Scholar] [CrossRef] [PubMed] - Mohammad, M.F.; Siamak, A. Discrete adaptive fuzzy control for asymptotic tracking of robotic manipulators. Nonlinear Dynamic.
**2014**, 78, 2195–2204. [Google Scholar] - Cho, B.; Kim, S.W.; Shin, S.; Oh, J.H.; Park, H.S.; Park, H.W. Energy-Efficient Hydraulic Pump Control for Legged Robots Using Model Predictive Control. IEEE/ASME Trans. Mechatron.
**2023**, 28, 3–14. [Google Scholar] [CrossRef] - Li, J.; Wang, C. Dynamics Modeling and Adaptive Sliding Mode Control of a Hybrid Condenser Cleaning Robot. Actuators
**2022**, 11, 119. [Google Scholar] [CrossRef] - Fu, Z.; Junhui, Z.; Min, C.; Bing, X. A Flow-Limited Rate Control Scheme for the Master–Slave Hydraulic Manipulator. IEEE Trans. Ind. Electron.
**2022**, 69, 4988–4998. [Google Scholar] - Pang, W.; Chai, Y.; Liu, H.; Li, F. Nonlinear Vibration Characteristics of a Hydraulic Manipulator Model: Theory and Experiment. J. Vib. Eng. Technol.
**2023**, 11, 1765–1775. [Google Scholar] [CrossRef] - Park, C.G.; Yoo, S.; Ahn, H.; Kim, J.; Shin, D. A coupled hydraulic and mechanical system simulation for hydraulic excavators. J. Syst. Control Eng.
**2020**, 234, 527–549. [Google Scholar] [CrossRef] - Janošević, D.; Pavlović, J.; Jovanović, V.; Petrović, G. A Numerical and Expremental Analysis of The Dynamics Stability of Hydraulic Excavators. Facta Univ.-Ser. Mech. Eng.
**2018**, 16, 157–170. [Google Scholar] - Kalmari, J.; Backman, J.; Visala, A. Nonlinear model predictive control of hydraulic forestry crane with automatic sway damping. Comput. Electron. Agric.
**2014**, 109, 36–45. [Google Scholar] [CrossRef] - Ruqi, D.; Zhen, W.; Min, C. Model-based Control of the Hydraulic Manipulator for the High-precision Trajectory Tracking. J. Mech. Eng.
**2023**, 59, 1–12. [Google Scholar] - Zhou, S.; Shen, C.; Xia, Y.; Chen, Z.; Zhu, S. Adaptive robust control design for underwater multi-DoF hydraulic manipulator. Ocean Eng.
**2022**, 248, 110822. [Google Scholar] [CrossRef] - Zheng, X.; Zhu, X.; Chen, Z.; Wang, X.; Liang, B.; Liao, Q. An efficient dynamic modeling and simulation method of a cable-constrained synchronous rotating mechanism for continuum space manipulator. Aerosp. Sci. Technol.
**2021**, 119, 107156. [Google Scholar] [CrossRef] - Zhu, W.-H. Virtual Decomposition Control: Toward Hyper Degrees of Freedom Robots; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Zhu, W.H.; Lamarche, T.; Dupuis, E.; Jameux, D.; Barnard, P.; Liu, G. Precision control of modular robot manipulators: The VDC approach with embedded FPGA. IEEE Trans. Robot.
**2013**, 29, 1162–1179. [Google Scholar] [CrossRef] - Koivumäki, J.; Zhu, W.-H.; Mattila, J. Addressing Closed-chain dynamics for high-precision control of hydraulic cylinder actuated manipulators. In Proceedings of the BATH/ASME 2018 Symposium on Fluid Power and Motion Control, Bath, UK, 12–14 September 2018; ASME: New York, NY, USA, 2018; p. V001T01A018. [Google Scholar]
- Koivumäki, J.; Mattilam, J. High performance nonlinear motion/force controller design for redundant hydraulic construction crane automation. Autom. Constr.
**2015**, 51, 59–77. [Google Scholar] [CrossRef] - Petrović, G.R.; Mattila, J. Mathematical modelling and virtual decomposition control of heavy-duty parallel–serial hydraulic manipulators. Mech. Mach. Theory
**2022**, 170, 104680. [Google Scholar] [CrossRef]

**Figure 1.**The hydraulic manipulator system studied in this work. (

**a**) the real manipulator studied in this work. (

**b**) the 3D model and hydraulic schematic and the red arrows mean coordinates at joints.

**Figure 2.**Schematic diagram of the virtual decomposition control of the studied hydraulic manipulator. (

**a**) the position of the virtual cutting point in the manipulator. (

**b**) the simple oriented graph of the manipulator structure.

**Figure 3.**Equivalent closed-chain structures in arm1 and arm2. (

**a**) Multi closed chain structures consisted by luffing mechanism, hydraulic cylinder and links. The letters in the figure represent the coordinate systems at the hinge points, (

**b**) Closed-chain structure, (

**c**) Virtual equivalent component in the equivalent closed chain structure.

**Figure 5.**Force analysis of the driving VCP in a closed chain. (

**a**) The forces applied to the closed-chain driver VCP and the arbitrary force/moment vector

^{T}f in the coordinate system. (

**b**) The reaction forces −f

_{L}

_{12}and −f

_{L}

_{13}.

Model | Parameter | Value | Model | Parameter | Value |
---|---|---|---|---|---|

Pressure source | Pressure | 25 [Mpa] | Hydraulic Oil | Volume modulus | 7000 [bar] |

Density | 850 [kg/m^{3}] | ||||

Proportional valve1 | Maximum flow P-A\A-T | 10 [L/min] | Hydraulic | Stroke | 840 [mm] |

Maximum flow P-B\B-T | 10 [L/min] | Cylinder 1 | Viscous friction | 10,000 [N/(m/s)] | |

Proportional valve2 | Maximum flow P-A\A-T | 10 [L/min] | Hydraulic | Stroke | 863 [mm] |

Maximum flow P-B\B-T | 10 [L/min] | Cylinder 2 | Viscous friction | 10,000 [N/(m/s)] | |

Proportional valve3 | Maximum flow P-A\A-T | 10 [L/min] | Hydraulic | Stroke | 700 [mm] |

Maximum flow P-B\B-T | 10 [L/min] | Cylinder 3 | Viscous friction | 10,000 [N/(m/s)] |

Parameter | Arm 1 | Arm 2 | Arm 3 |
---|---|---|---|

Length [m] | 5.550 | 3.75 | 3.7 |

Mass [kg] | 883 | 250 | 190 |

Angle range [deg] | 0°–88° | −180°–0° | −60°–180° |

Cylinder mass [kg] | 55.9 | 48.7 | 32.6 |

Cylinder diameter [mm] | 110 | 100 | 80 |

Piston mass [kg] | 37.1 | 21.3 | 10.6 |

Piston diameter [mm] | 63 | 55 | 45 |

Rotational inertia [kg·m^{2}] | 5562.988 | 723.350 | 427.5 |

Density [kg·m^{3}] | 7700 | 7700 | 7700 |

Damping coefficient [Nm/(rev/min)] | 250 | 250 | 250 |

Parameter | Servo Valve 1 | Servo Valve 2 | Servo Valve 3 |
---|---|---|---|

${E}^{-1}\left[\frac{{\mathrm{m}}^{2}}{\mathrm{N}}\right]$ | 2.6 × 10^{−8} | 2.6 × 10^{−8} | 2.6 × 10^{−8} |

${c}_{p1}\left[\frac{{\mathrm{m}}^{3}}{\mathrm{s}\sqrt{\mathrm{P}\mathrm{a}}}\right]$ | 2.3 × 10^{−7} | 1.2 × 10^{−7} | 6.3 × 10^{−8} |

${c}_{n1}\left[\frac{{\mathrm{m}}^{3}}{\mathrm{s}\sqrt{\mathrm{P}\mathrm{a}}}\right]$ | 2.1 × 10^{−7} | 1.7 × 10^{−7} | 1.8 × 10^{−7} |

${c}_{p2}\left[\frac{{\mathrm{m}}^{3}}{\mathrm{s}\sqrt{\mathrm{P}\mathrm{a}}}\right]$ | 9.3 × 10^{−8} | 1.73 × 10^{−7} | 1.2 × 10^{−7} |

${c}_{n2}\left[\frac{{\mathrm{m}}^{3}}{\mathrm{s}\sqrt{\mathrm{P}\mathrm{a}}}\right]$ | 2.3 × 10^{−8} | 1.7 × 10^{−7} | 4.8 × 10^{−7} |

Method | PID | MBC | VDC | Effect | Compare with PID | Compare with MBC |
---|---|---|---|---|---|---|

Triangular trajectory | 125.3 mm | 15.7 mm | 12 mm | Triangular trajectory | Decrease by 90.4% | Decrease by 23.6% |

Elliptical trajectory | 170.2 mm | 40.2 mm | 22.4 mm | Elliptical trajectory | Decrease by 86.8% | Decrease by 44.3% |

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## Share and Cite

**MDPI and ACS Style**

Ding, R.; Liu, Z.; Li, G.; Deng, Z.
High-Precision and Modular Decomposition Control for Large Hydraulic Manipulators. *Actuators* **2023**, *12*, 405.
https://doi.org/10.3390/act12110405

**AMA Style**

Ding R, Liu Z, Li G, Deng Z.
High-Precision and Modular Decomposition Control for Large Hydraulic Manipulators. *Actuators*. 2023; 12(11):405.
https://doi.org/10.3390/act12110405

**Chicago/Turabian Style**

Ding, Ruqi, Zichen Liu, Gang Li, and Zhikai Deng.
2023. "High-Precision and Modular Decomposition Control for Large Hydraulic Manipulators" *Actuators* 12, no. 11: 405.
https://doi.org/10.3390/act12110405