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

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.

Graphical abstract

Keywords

hydraulic dual-arm manipulator / compliance motion control / unknown object / adaptive mass estimation / nonlinear control

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Nomenclature

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
Variables
cp1, cp2, cn1, cn2	Flow coefficients of the valve
Ci	Coriolis and centrifugal matrix of the ith HM
Cin	Filter cut-off frequency matrix
{\mathit{C}}_O	Coriolis and centrifugal matrix of the object
D	Displacement of the SC
{\mathit{e}}_O	Position/orientation error feedback term of the object
{}^{}\text{in}{e}_{\text{ave}}, {}^{}\text{in}{e}_{max}	Average and maximum internal force errors, respectively
{}^{}\text{in}{e}_k	Internal force error at the kth sampling point
{}^{}\text{p}{e}_{\text{ave}}, {}^{}\text{p}{e}_{max}	Average and maximum position errors, respectively
{}^{}\text{p}{e}_k	Position error at the kth sampling point
fcr	Required output force of the hydraulic cylinder
ff	Friction force of the hydraulic cylinder obtained from the identification
fpij	jth element of Fpi
fpr	Required driven force
{\dot{f}}_{\text{pr}}	Derivative of the required driven force
{}^{}{T}_{\text{1}}\mathit{F}, {}^{}{T}_{\text{2}}\mathit{F}	Force vectors in frames {T1} and {T2}, respectively
{}^{}O{\mathit{F}}^{\ast }	Net force vector of the object
Fci	Estimated contact force vector of the ith HM
Ffi	Friction force vector of the hydraulic cylinder
Fi	External force vector applied to the ith EE
{}_{}^{}{T}_1{\mathit{F}}_{\text{pc}}, {}_{}^{}{T}_2{\mathit{F}}_{\text{pc}}	Desired external force vectors in frames {T1} and {T2} of PC, respectively
{}_{}^{}O{\mathit{F}}_{\text{pc}}^{\ast }	Desired net force of the object in PC
Fpi	Output force vector calculated from the pressure of the cylinder
{}^{}{T}_{\text{1}}{\mathit{F}}_{\text{r}}, {}^{}{T}_{\text{2}}{\mathit{F}}_{\text{r}}	Required force vectors in frames {T1} and {T2}, respectively
{}^{}O{\mathit{F}}_{\text{r}}^{\ast }	Required net force vector of the object
Gi	Gravity vector of the ith HM
{\mathit{G}}_O	Gravity vector of the object
i	Serial number of the HM
I6	Sixth-order identity matrix
j	Serial number of the joint (or actuator)
Jhij	jth element of Jhi
Jhi	Mapping matrix that converts the output force vector of the cylinder into the joint torque vector
Ji	Jacobian matrix of the ith HM
k	Sampling moment
kf	Force error gain of the cylinder
ko	Orientation error gain of the object
kp	Position error gain of the object
kv	Velocity error gain of the cylinder
Kin	Internal force gain matrix
{\mathit{K}}_O	Velocity error gain matrix
l1, l2	Lengths of two links for the closed chain
Mi	Inertial matrix of the ith HM
{\mathit{M}}_O	Inertial matrix of the object
pa, pb	Pressure of the two chambers of the cylinder
pr	Tank pressure
ps	System pressure
q	Joint angle
\dot{q}	Actual joint velocity
qi	Initial joint angle
qp	Pendulum angle of the SC
{\dot{q}}_{\text{r}}	Required joint velocity or angular velocity of the SC
{\mathit{q}}_i, {\dot{\mathit{q}}}_i, {\ddot{\mathit{q}}}_i	Joint angle, velocity, and acceleration vectors of the ith HM, respectively
{}^{}O{\mathit{R}}_W	Rotation matrix from frame {O} to {W}
s_{O\gamma }	γth element of {\mathit{s}}_O
{\mathit{s}}_O	Auxiliary vector for the object adaptive parameter update
Sa, Sb	Areas of the cap-side and rod-side of the LC, respectively
t	Time
t0	Initialization time of the HDM system
T	Running period of the trajectory
ufr	Relevant term of the valve control signal
uv	Control signal of the valve,
{}^{}{E}_i{\mathit{U}}_O	Velocity transformation matrix from frame {O} to {Ei}
{}^{}W{\mathit{U}}_{O_i}	Velocity transformation matrix from frame {Oi} to {W}
{}^{}O{\mathit{U}}_{T_1}, {}^{}O{\mathit{U}}_{T_{\text{2}}}	Velocity transformation matrix from frames {T1} and {T2} to {W}, respectively
{}^{}O{\mathit{U}}_W	Velocity transformation matrix from frame {W} to {O}
{}^{}O\mathit{v}	Actual linear velocity vector of frame {O} expressed in frame {O}
{}_{}^{}O\mathit{V}	Actual velocity vector of the object
{}^{}{T}_{\text{1}}\mathit{V}, {}^{}{T}_{\text{2}}\mathit{V}	Velocity vectors of frames {T1} and {T2}, respectively
{}_{}^{}O{\mathit{V}}_{\text{d}}	Desired velocity vector of the object
{}^{}{O}_i{\mathit{V}}_{E_i}	Velocity vector from frame {Ei} to {Oi}
{}_{}^{}O{\mathit{V}}_{\text{pc}}	Desired velocity of the object of PC
{}^{}W{\mathit{V}}_O	Velocity vector from frame {O} to {W}
{}_{}^{}O{\mathit{V}}_{\text{r}}	Required velocity vector of the object
{}_{}^{}{T}_1{\mathit{V}}_{\text{pc}}, {}_{}^{}{T}_2{\mathit{V}}_{\text{pc}}	Desired velocity vectors of frames {T1} and {T2} in PC, respectively
{}^{}{T}_{\text{1}}{\mathit{V}}_{\text{r}}, {}^{}{T}_2{\mathit{V}}_{\text{r}}	Required velocity vectors of frames {T1} and {T2}, respectively
xip	Initial length of the cylinder
xp	Displacement of the cylinder
{\dot{x}}_{\text{p}}	Actual cylinder velocity
x_{\text{pl}}, {\dot{x}}_{\text{pl}}	Actual displacement and velocity of the LC, respectively
{\dot{x}}_{\text{plr}}	Required velocity of the LC
x_{\text{ps}}, {\dot{x}}_{\text{ps}}	Actual displacement and velocity of the SC, respectively
{\dot{x}}_{\text{psr}}	Required velocity of the SC
xs	Stroke of the LC
x	Actual position vector from frame {O} to {W}
\dot{\mathit{x}}	Actual linear velocity vector from frame {O} to {W}
xd	Desired position vector from frame {O} to {W}
{\dot{\mathit{x}}}_{\text{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
Yi	Regression matrix of the ith HM
{\mathit{Y}}_O	Regression matrix of the object
{\mathit{Y}}_{O_{\mathrm{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
α1, α2	Two load distribution factors
\beta	Oil effective bulk modulus
\gamma	Sequence number of the element of the parameter vector
\text{η}, {\text{η}}_{\text{d}}	Actual and desired internal force vectors, respectively
\widetilde{\text{η}}, {\widetilde{\text{η}}}_{\text{d}}	Filtered actual and desired internal force vectors, respectively
\dot{\widetilde{\text{η}}}, {\dot{\widetilde{\text{η}}}}_{\mathrm{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 {\hat{\mathit{\theta }}}_O
{\mathit{\theta }}_i	Inertial parameter vector of the ith HM
{\hat{\mathit{\theta }}}_{\text{i}O}	Adaptive initial value parameter vector of the object
{\mathit{\theta }}_O	Inertial parameter vector of the object
{\hat{\mathit{\theta }}}_O	Adaptive estimate of {\mathit{\theta }}_O
{\mathit{\theta }}_O^{-}, {\mathit{\theta }}_O^{+}	Adaptive lower bound and upper bound vectors, respectively
{\mathit{\mu }}_O	Part of a quaternion representing the orientation error of the object
\zeta	Adaptive function
{\rho }_{O\gamma }	Adaptive gain
{\mathit{\rho }}_O	Adaptive gain vector
\text{ω}	Actual angular velocity vector of the object from frame {O} to {W}
{}_{}^{}O\text{ω}	Actual angular velocity vector of the object
{\text{ω}}_{\text{d}}	Desired angular velocity vector of the object from frame {O} to {W}
{\mathbf{\Omega }}_{\text{ld}}	Load distribution matrix
{\mathbf{\Omega }}_{\text{m}}	Internal force mapping matrix
κ	Adaptive switching function

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).

Conflict of Interest

The authors declare that they have no conflict of interest.

RIGHTS & PERMISSIONS

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