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Compliance motion control of the hydraulic dual-arm manipulator with adaptive mass estimation of unknown object

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Abstract

Given the limited operating ability of a single robotic arm, dual-arm collaborative operations have become increasingly prominent. Compared with the electrically driven dual-arm manipulator, due to the unknown heavy load, difficulty in measuring contact forces, and control complexity during the closed-chain object transportation task, the hydraulic dual-arm manipulator (HDM) faces more difficulty in accurately tracking the desired motion trajectory, which may cause object deformation or even breakage. To overcome this problem, a compliance motion control method is proposed in this paper for the HDM. The mass parameter of the unknown object is obtained by using an adaptive method based on velocity error. Due to the difficulty in obtaining the actual internal force of the object, the pressure signal from the pressure sensor of the hydraulic system is used to estimate the contact force at the end-effector (EE) of two hydraulic manipulators (HMs). Further, the estimated contact force is used to calculate the actual internal force on the object. Then, a compliance motion controller is designed for HDM closed-chain collaboration. The position and internal force errors of the object are reduced by the feedback of the position, velocity, and internal force errors of the object to achieve the effect of the compliance motion of the HDM, i.e., to reduce the motion error and internal force of the object. The required velocity and force at the EE of the two HMs, including the position and internal force errors of the object, are inputted into separate position controllers. In addition, the position controllers of the two individual HMs are designed to enable precise motion control by using the virtual decomposition control method. Finally, comparative experiments are carried out on a hydraulic dual-arm test bench. The proposed method is validated by the experimental results, which demonstrate improved object position accuracy and reduced internal force.

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Abbreviations

AIFE:

Average internal force error

APE:

Average position error

CMC:

Compliance motion control

DH:

Denavit–Hartenberg

DOF:

Degree-of-freedom

EDCM:

Electrically driven cooperative manipulator

EE:

End-effector

FK:

Forward kinematics

HDM:

Hydraulic dual-arm manipulator

HM:

Hydraulic manipulator

IK:

Inverse kinematics

LC:

Linear cylinder

MIFE:

Maximum internal force error

MPE:

Maximum position error

PC:

Position control

SC:

Swing cylinder

VDC:

Virtual decomposition controller

c p1, c p2, c n1, c n2 :

Flow coefficients of the valve

C i :

Coriolis and centrifugal matrix of the ith HM

C in :

Filter cut-off frequency matrix

C O :

Coriolis and centrifugal matrix of the object

D :

Displacement of the SC

e O :

Position/orientation error feedback term of the object

in e ave, in e max :

Average and maximum internal force errors, respectively

in e k :

Internal force error at the kth sampling point

p e ave, p e max :

Average and maximum position errors, respectively

p e k :

Position error at the kth sampling point

f cr :

Required output force of the hydraulic cylinder

f f :

Friction force of the hydraulic cylinder obtained from the identification

f pij :

jth element of Fpi

f pr :

Required driven force

ḟ pr :

Derivative of the required driven force

\(^{{T_1}}{\boldsymbol{F}}{,^{{T_2}}}{\boldsymbol{F}}\) :

Force vectors in frames {T1} and {T2}, respectively

O F*:

Net force vector of the object

F ci :

Estimated contact force vector of the ith HM

F fi :

Friction force vector of the hydraulic cylinder

F i :

External force vector applied to the ith EE

\(^{{T_1}}{{\boldsymbol{F}}_{{\rm{pc}}}}{,^{{T_2}}}{{\boldsymbol{F}}_{{\rm{pc}}}}\) :

Desired external force vectors in frames {T1} and {T2} of PC, respectively

\(^O{\boldsymbol{F}}_{{\rm{pc}}}^ * \) :

Desired net force of the object in PC

F pi :

Output force vector calculated from the pressure of the cylinder

\(^{{T_1}}{{\boldsymbol{F}}_{\rm{r}}}{,^{{T_2}}}{{\boldsymbol{F}}_{\rm{r}}}\) :

Required force vectors in frames {T1} and {T2}, respectively

\(^O{\boldsymbol{F}}_{\rm{r}}^ * \) :

Required net force vector of the object

G i :

Gravity vector of the ith HM

G O :

Gravity vector of the object

i :

Serial number of the HM

I 6 :

Sixth-order identity matrix

j :

Serial number of the joint (or actuator)

J hij :

jth element of Jhi

J hi :

Mapping matrix that converts the output force vector of the cylinder into the joint torque vector

J i :

Jacobian matrix of the ith HM

k :

Sampling moment

k f :

Force error gain of the cylinder

k o :

Orientation error gain of the object

k p :

Position error gain of the object

k v :

Velocity error gain of the cylinder

K in :

Internal force gain matrix

K O :

Velocity error gain matrix

l 1, l 2 :

Lengths of two links for the closed chain

M i :

Inertial matrix of the ith HM

M O :

Inertial matrix of the object

p a, p b :

Pressure of the two chambers of the cylinder

p r :

Tank pressure

p s :

System pressure

q :

Joint angle

\({\dot q}\) :

Actual joint velocity

q i :

Initial joint angle

q p :

Pendulum angle of the SC

\({{\dot q}_{\rm{r}}}\) :

Required joint velocity or angular velocity of the SC

q i, \({{{\boldsymbol{\dot q}}}_i},{{{\boldsymbol{\ddot q}}}_i}\) :

Joint angle, velocity, and acceleration vectors of the ith HM, respectively

O R W :

Rotation matrix from frame {O} to {W}

\({s_{{O_\gamma }}}\) :

γth element of sO

s O :

Auxiliary vector for the object adaptive parameter update

S a, S b :

Areas of the cap-side and rod-side of the LC, respectively

t :

Time

t 0 :

Initialization time of the HDM system

T :

Running period of the trajectory

u fr :

Relevant term of the valve control signal

u v :

Control signal of the valve

\(^{{E_i}}{{\boldsymbol{U}}_O}\) :

Velocity transformation matrix from frame {O} to {Ei}

\(^W{{\boldsymbol{U}}_{{O_i}}}\) :

Velocity transformation matrix from frame {Oi} to {W}

\(^O{{\boldsymbol{U}}_{{T_1}}}{,^O}{{\boldsymbol{U}}_{{T_2}}}\) :

Velocity transformation matrix from frames {T1} and {T2} to {W}, respectively

O U W :

Velocity transformation matrix from frame {W} to {O}

O v :

Actual linear velocity vector of frame {O} expressed in frame {O}

O V :

Actual velocity vector of the object

\(^{{T_1}}{\boldsymbol{V}}{,^{{T_2}}}{\boldsymbol{V}}\) :

Velocity vectors of frames {T1} and {T2}, respectively

O V d :

Desired velocity vector of the object

\(^{{O_i}}{{\boldsymbol{V}}_{{E_i}}}\) :

Velocity vector from frame {Ei} to {Oi}

O V pc :

Desired velocity of the object of PC

W V O :

Velocity vector from frame {O} to {W}

O V r :

Required velocity vector of the object

\(^{{T_1}}{{\boldsymbol{V}}_{{\rm{pc}}}}{,^{{T_2}}}{{\boldsymbol{V}}_{{\rm{pc}}}}\) :

Desired velocity vectors of frames {T1} and {T2} in PC, respectively

\(^{{T_1}}{{\boldsymbol{V}}_{\rm{r}}}{,^{{T_2}}}{{\boldsymbol{V}}_{\rm{r}}}\) :

Required velocity vectors of frames {T1} and {T2}, respectively

x ip :

Initial length of the cylinder

x p :

Displacement of the cylinder

ẋ p :

Actual cylinder velocity

x pl, ẋ pl :

Actual displacement and velocity of the LC, respectively

ẋ plr :

Required velocity of the LC

x ps, ẋ ps :

Actual displacement and velocity of the SC, respectively

ẋ psr :

Required velocity of the SC

x s :

Stroke of the LC

x :

Actual position vector from frame {O} to {W}

ẋ :

Actual linear velocity vector from frame {O} to {W}

x d :

Desired position vector from frame {O} to {W}

ẋ d :

Desired linear velocity vector from frame {O} to {W}

X :

Coordinate value of the trajectory on the x-axis

X O :

Initial x-axis coordinate value of the object

Y i :

Regression matrix of the ith HM

Y O :

Regression matrix of the object

\({{\boldsymbol{Y}}_{{O_{\rm{r}}}}}\) :

Required regression matrix of the object

Z :

Coordinate value of the trajectory on the z-axis

Z O :

Initial z-axis coordinate value of the object

α 1, α 2 :

Two load distribution factors

β :

Oil effective bulk modulus

γ :

Sequence number of the element of the parameter vector

η, η d :

Actual and desired internal force vectors, respectively

ῆ, ῆ d :

Filtered actual and desired internal force vectors, respectively

\({\boldsymbol{\dot \tilde \eta }},{{{\boldsymbol{\dot \tilde \eta }}}_{\rm{d}}}\) :

Derivative of the filtered actual and desired internal force vectors, respectively

\(\theta _{O\gamma }^ - ,\,\,\theta _{O\gamma }^ + \) :

Adaptive lower bound and upper bounds of element \({{\hat \theta }_{O\gamma }}\), respectively

\({{\hat \theta }_{O\gamma }}\) :

γth element of the parameter vector \({{{\boldsymbol{\hat \theta }}}_O}\)

θ i :

Inertial parameter vector of the ith HM

\({{{\boldsymbol{\hat \theta }}}_{{\rm{i}}O}}\) :

Adaptive initial value parameter vector of the object

θ O :

Inertial parameter vector of the object

\({{{\boldsymbol{\hat \theta }}}_O}\) :

Adaptive estimate of θO

\({\boldsymbol{\theta }}_O^ - ,\,{\boldsymbol{\theta }}_O^ + \) :

Adaptive lower bound and upper bound vectors, respectively

μ O :

Part of a quaternion representing the orientation error of the object

ζ :

Adaptive function

\({\rho _{O\gamma }}\) :

Adaptive gain

\({{\boldsymbol{\rho }}_O}\) :

Adaptive gain vector

ω :

Actual angular velocity vector of the object from frame {O} to {W}

O ω :

Actual angular velocity vector of the object

ω d :

Desired angular velocity vector of the object from frame {O} to {W}

Ω ld :

Load distribution matrix

Ω m :

Internal force mapping matrix

κ :

Adaptive switching function

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 52075055 and U21A20124), and the Strategic Basic Product Project from the Ministry of Industry and Information Technology, China (Grant No. TC220H064).

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Authors and Affiliations

  1. State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing, 400044, China

    Bolin Sun & Min Cheng

  2. Key Laboratory of Conveyance and Equipment (Ministry of Education), East China Jiaotong University, Nanchang, 330013, China

    Ruqi Ding

  3. State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou, 310027, China

    Bing Xu

Authors
  1. Bolin Sun
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  2. Min Cheng
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  3. Ruqi Ding
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  4. Bing Xu
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Correspondence to Ruqi Ding.

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Sun, B., Cheng, M., Ding, R. et al. Compliance motion control of the hydraulic dual-arm manipulator with adaptive mass estimation of unknown object. Front. Mech. Eng. 19, 7 (2024). https://doi.org/10.1007/s11465-023-0773-z

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  • DOI: https://doi.org/10.1007/s11465-023-0773-z

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