Keywords

1 Introduction

Hydraulic excavator is one of the commonly used engineering machinery, which has the characteristics of typical function and strong adaptability. Since China’s reform and opening up, in order to meet the national infrastructure, mining, engineering water conservancy and other fields, the country’s research and exploration of hydraulic excavators and other engineering sports machinery and industrial development support is very strong. At present, China has become the international largest producer and consumer of hydraulic excavators, and the market is increasingly demanding the working performance of excavators. After the development of traditional hydraulic excavators has matured, the academic research direction of excavators has begun to move towards unmanned and autonomous. In the research process of excavator autonomous operation, the two main problems are motion planning and control accuracy. In the above two problems, the research on the control accuracy is the premise of the motion planning problem. The problem that the control accuracy cannot reach the standard increases the huge safety hazard when the excavator carries out the autonomous operation. At present, domestic and foreign scholars and insiders have initially formed a scale for the research on the motion control accuracy of hydraulic excavators during unmanned operation. He et al. [1] used the adaptive control method to control the flow of the boom hydraulic cylinder. The simulation results are compared with the conventional PID control, and the simulation speed and accuracy are improved. Wang et al. [2] used the cross-coupled pre-compensation (CCP) method to combine its compensation with the nonlinear proportional integral controller of each actuator to achieve coordinated control of unmanned excavators. The model predictive control method was utilized by Zheng et al. [3] to track the predefined trajectory under diverse working conditions. The results show that the model predictive control can provide more accurate feedforward control force under complicated working situations, achieve accurate trajectory tracking, effectively minimize control delay caused by high inertia conditions, and has better robustness. Hanh et al. [4] proposed a fuzzy self-tuning controller based on neural network to control a small electro-hydraulic excavator. The neural network method is used to tune the output signal online, which has certain anti-interference ability and adaptability to improve the tracking performance of the system. Kadu et al. [5] proposed a time-delay sliding mode control strategy for the motion control of unmanned excavators so that the working device performs well under the influence of external disturbances and uncertainties. Precision control approach for industrial hydraulic excavator robot movements developed by Lee et al. [6] based on data-driven model inversion. To boost the system’s learning speed, a modular-driven model inversion control technique is proposed. When compared to PI control, simulation and experiment demonstrate the control system’s accuracy.

In this paper, the motion control accuracy of the self-operated excavator is studied from the perspective of the motion control of the valve-controlled asymmetric hydraulic cylinder. The online compensation control of the excavator’s valve-controlled asymmetric hydraulic cylinder is carried out by using the method of online identification and compensation, and the modified unmanned excavator is used for excavation experiment verification.

2 Establishment of Mathematical Model for Working Device

2.1 The Establishment of Kinematic Model for Working Equipment

The kinematic model is shown in Fig. 1.

Fig. 1
2 schematics. A. Forward kinematic model of the D H method. B. Inverse kinematic model of the D H method. For both, the x, y, and z directions are marked. theta 0, theta 1, theta 2, and theta 3 are mentioned.

Kinematic model of working device

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Where \(\theta_{{\text{i}}}\) and \(\dot{\theta }_{{\text{i}}}\) is joint angle and joint angular velocity. The joint torque of the excavator manipulator is positive in the counterclockwise direction.

The positive kinematics relationship is shown in Eq. (1), and the inverse kinematics relationship is shown in Eq. (2). The unspecified parameters in the formula can be derived from the geometric relationship. In [7] Koivo gave a detailed description of kinematics.

$$ \left\{ \begin{gathered} x = c_{0} (a_{0} + a_{1} c_{1} + a_{2} c_{12} + a_{3} c_{123} ) \hfill \\ y = s_{0} (a_{0} + a_{1} c_{1} + a_{2} c_{12} + a_{3} c_{123} ) \hfill \\ z = a_{1} s_{1} + a_{2} s_{12} + a_{3} s_{123} + d_{0} \hfill \\ \xi = \theta_{1} + \theta_{2} + \theta_{3} \hfill \\ \end{gathered} \right. $$
(1)
$$ \left\{ \begin{gathered} \theta_{0} = \arctan \frac{y}{x} \hfill \\ \theta_{1} = \alpha + \beta + \gamma \hfill \\ \theta_{2} = \pi - \cos^{ - 1} \frac{{a_{1}^{2} + a_{2}^{2} - BJ^{2} }}{{2a_{1} a_{2} }} \hfill \\ \theta_{3} = \xi - \theta_{1} - \theta_{2} \hfill \\ \end{gathered} \right. $$
(2)

2.2 The Establishment of Dynamic Model of Working Device

The working device of the autonomous excavator has 4 degrees of freedom. The dynamic model is established as shown in Fig. 2. The model mainly includes the boom, the arm, the bucket and its corresponding action cylinder. The influence of system friction and external load on the dynamic characteristics of the system is ignored and the mass of each hydraulic cylinder is equivalent to the working device.

Fig. 2
A photograph of an autonomous excavator mainly includes the boom, the arm, the bucket, and its corresponding action cylinder. f 1, f 2, f 3, tau 1, tau 2, tau 3, phi 1, phi 2, and phi are mentioned.

Dynamic model of the working device of autonomous excavator

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The joint torque of the manipulator can be expressed as follows:

$$ \tau = M(\theta )\ddot{\theta } + c(\theta ,\dot{\theta })\dot{\theta } + G(\theta ) $$
(3)

where \({\varvec{\tau}}\) is the joint moment (N m), \( {\varvec{M}}\,\epsilon \,{\text{R}}^{3 \times 3}\) is the inertia matrix of rigid body (kg m2), \({\varvec{c}}\,\epsilon \,{\text{R}}^{3 \times 3}\) is centripetal force matrix (N m), \({\varvec{G}}\,\epsilon \,{\text{R}}^{3 \times 3}\) is the gravity matrix (N m).

3 Online Parameter Identification and Compensation Terms

3.1 Gravity Identification Compensation Method

The manipulator of the excavator is heavy. The dynamic features of the boom raising, arm, and bucket are strongly influenced during the operation. If the excavator boom, arm, and bucket online gravity compensation computation is performed, the parameters, \(\tau_{1}\), \(\tau_{2}\), \(\tau_{3}\), m1, m2, m3, l1, l2, l3, \(\phi_{1}\), \(\phi_{2}\), \(\phi_{3}\) must be obtained. This work completes the calculation of kinetic parameters using the techniques outlined below.

The pressure sensor is used in the experiment to measure the hydraulic cylinder’s driving force f1, f2, f3 under different attitudes and no-load conditions. The measured value of the pressure sensor at no load is almost equal to the external interference caused by the gravity of the working device, that is fg1, fg2, fg3. In this instantaneous state, the excavator manipulator’s driving force is as follows:

$$ \left\{ \begin{gathered} f_{1} = f_{g1} \hfill \\ f_{2} = f_{g2} \hfill \\ f_{3} = f_{g3} \hfill \\ \end{gathered} \right. $$
(4)

At this time, the dynamic equation of the three joint torques \(\tau^{\prime} = \left[ {\begin{array}{*{20}l} {\tau^{\prime}_{1} } & {\tau^{\prime}_{2} } & {\tau^{\prime}_{3} } \\ \end{array} } \right]^{T}\) of the manipulator generated by gravity is shown as follows:

$$ \begin{aligned} \left[ {\begin{array}{*{20}l} {\tau _{1}^{\prime } } \hfill \\ {\tau _{2}^{\prime } } \hfill \\ {\tau _{3}^{\prime } } \hfill \\ \end{array} } \right] & = \left[ {\begin{array}{*{20}l} {m_{1} l_{1} c_{1} + m_{2} a_{1} + m_{3} a_{1} } \hfill & {m_{2} l_{2} c_{2} + m_{3} a_{2} } \hfill & {m_{3} l_{3} c_{3} } \hfill \\ 0 \hfill & {m_{2} l_{2} c_{2} + m_{3} a_{2} } \hfill & {m_{3} l_{3} c_{3} } \hfill \\ 0 \hfill & 0 \hfill & {m_{3} l_{3} c_{3} } \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {c\theta _{1} } \hfill \\ {c\theta _{{12}} } \hfill \\ {c\theta _{{123}} } \hfill \\ \end{array} } \right]g \\ & \quad - \left[ {\begin{array}{*{20}l} {m_{1} l_{1} s_{1} } \hfill & {m_{2} l_{2} s_{2} } \hfill & {m_{3} l_{3} s_{3} } \hfill \\ 0 \hfill & {m_{2} l_{2} s_{2} } \hfill & {m_{3} l_{3} s_{3} } \hfill \\ 0 \hfill & 0 \hfill & {m_{3} l_{3} s_{3} } \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {s\theta _{1} } \hfill\\ {s\theta _{{12}} } \hfill\\ {s\theta _{{123}} } \hfill\\ \end{array} } \right]g \\ \end{aligned} $$
(5)

In the formula, c1 = cos \(\phi_{1}\), c2 = cos \(\phi_{2}\), s1 = sin \(\phi_{1}\), s2 = sin \(\phi_{2}\), cθ1 = cosθ1, cθ12 = cos(θ1 + θ2), sθ1 = sinθ1, sθ12 = sin(θ1 + θ2). \(\left[ {\begin{array}{*{20}l} {\tau^{\prime}_{1} } & {\tau^{\prime}_{2} } & {\tau^{\prime}_{3} } \\ \end{array} } \right]^{T}\) is the driving torque acting on the boom, arm and bucket only under the action of gravity, m1, m2, m3 are the mass of the boom, arm and bucket respectively.

The driving torque is generated by the hydraulic cylinder’s driving force, and the force arm serves as an intermediary medium for the conversion of the driving force and torque. The geometric method is used in Fig. 3 to derive the relationship between the driving force arm or the equivalent length of each cylinder and the length of the cylinder.

Fig. 3
A photograph of a working excavator mainly includes the boom, the arm, the bucket, and the corresponding action cylinder. Points A to L, E 1, E 2, I a, I c, and I c are mentioned.

Structure analysis diagram of working device

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The driving force arms of E1, E2, and E3, each consisting of three joints, can be derived from geometric relationships

According to θ1, θ2, θ3 derived from inverse kinematics, the external forces fg1, fg2, fg3 caused by self-weight are calculated as follows:

$$ \left\{ \begin{gathered} f_{g1} = \tau^{\prime}_{1} /E_{1} \hfill \\ f_{g2} = \tau^{\prime}_{2} /E_{2} \hfill \\ f_{g3} = \tau^{\prime}_{3} /E_{3} \hfill \\ \end{gathered} \right. $$
(6)

And then the displacement y1, y2, y3 and driving force f1, f2, f3 of the boom, arm, and bucket hydraulic cylinders produced in the preceding phases are used to compute the joint angles θ1, θ2, θ3 and driving torque \(\tau^{\prime}_{1}\), \(\tau^{\prime}_{2}\), \(\tau^{\prime}_{3}\). The least square approach is used to calculate and identify the parameters in Eq. (5). The parameter identification process is as Fig. 4.

Fig. 4
A block diagram of the parameter identification process. The work arm toque bance formula block takes inputs from displacement and pressure transducers and is followed by the least squares parameter identification, data processing, and online compensation dynamics system of the working arm.

Derivation process diagram of relevant parameters

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All the parameters in the \({\text{R}}^{3 \times 3}\) matrix of Eq. (5). can be derived. The obtained matrix is processed to obtain the required parameters m1, m2, m3, l1, l2, l3 etc. The online gravity compensation torque can be obtained by substituting the parameter identification results into formula Eq. (5).

3.2 Online Centripetal Force Term and Inertial Force Term Compensation

The inertia force and centripetal force generated by the large gravity of the manipulator affect the motion control accuracy of the hydraulic cylinder, so it is necessary to compensate the centripetal force and inertia force online.

The inertial force term as:

$$ M(\theta ) = \sum\limits_{i = 1}^{N} {{\text{(m}}_{i} } J_{v}^{{i^{T} }} J_{v}^{i} + I_{i} J_{w}^{{i^{T} }} J_{w}^{i} ) = \left[ {\begin{array}{*{20}l} {M_{11} } & {M_{12} } & {M_{13} } \\ {M_{21} } & {M_{22} } & {M_{23} } \\ {M_{31} } & {M_{32} } & {M_{33} } \\ \end{array} } \right] $$
(7)

The centripetal force term as:

$$ c(\theta ,\dot{\theta }) = C_{ijk} = \frac{1}{2}\sum\limits_{jk} {\left( {\frac{{\partial M_{ij} }}{{\partial \theta_{k} }} + \frac{{\partial M_{ik} }}{{\partial \theta_{j} }} - \frac{{\partial M_{jk} }}{{\partial \theta_{i} }}} \right)} \dot{\theta }_{j} \dot{\theta }_{k} $$
(8)

The relevant parameters identified in the previous section are substituted into Eqs. (7) and (8) to obtain the specific inertial force term and centrifugal force term.

Combined with the previous section, the online compensation model of gravity, centripetal force and inertial force can be obtained. The driving force of the boom, arm and bucket hydraulic cylinder measured by the force sensor during operation is f1, f2, f3. The driving force after the external interference force fe1, fe2, fe3 in the direction of the compensated hydraulic cylinder can be used as the driving force fR1, fR2, fR3 of the hydraulic cylinder under actual conditions, which as:

$$ \left\{ \begin{gathered} f_{R1} = f_{1} - f_{e1} \hfill \\ f_{R2} = f_{2} - f_{e2} \hfill \\ f_{R3} = f_{3} - f_{e3} \hfill \\ \end{gathered} \right. $$
(9)

4 Experiment

4.1 Control Objective

According to the task requirements, trajectory planning was carried out for the autonomous excavator. This section presents the Cartesian trajectory path of the working device bucket tip through the excavation trajectory of the excavator driver. The Cartesian path of the excavator can be transformed through inverse kinematics to obtain the joint angle planning for excavation, as shown in Fig. 5.

Fig. 5
A, B, C, and D. 4 line graphs of angle versus time depict time-domain planning of boom joint angle, arm joint angle, bucket joint angle, and rotary joint angle, respectively. A and D, depict a first-rising then decreasing trend. B and C depict a first decreasing, then rising trend.

Trajectory planning of bucket tooth tip

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The experimental equipment of the self-operated excavator is shown in Fig. 6. The joint angle measurement units at the boom and stick are installed on their respective outer surfaces. The bucket uses a built-in displacement sensor, and the pressure sensor is installed at the inlet and outlet of the hydraulic cylinder pipeline.

Fig. 6
A set of 7 photographs displays the unmanned excavator test equipment, boom pressure sensor, arm pressure sensor, arm angle sensor, boom angle sensor, bucket pressure sensor, and bucket built-in pressure sensor.

Autonomous excavator test platform

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4.2 Experimental Results

In order to verify the effectiveness of the control strategy, the prototype test was carried out using the above-mentioned self-operated excavator. The tracking effect of the experimental results under no-load are shown in Fig. 7. The spatial distance error of the excavator bucket tooth tip is shown in Fig. 8.

Fig. 7
A, B, C, and D. 4 line graphs of angle versus time compare the trends for target joint angle, actual joint angle after compensation, and actual joint angle before compensation. A and D, depict a first rising then decreasing trend. B and C depict a first decreasing, then rising trend.

Tracking under no-load

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Fig. 8
A multiline graph of displacement versus time compares the wavering trends for error after compensation, and error before compensation. The trend of error before compensation has a peak displacement of 2000 millimeters between 10 and 20 seconds.

Cartesian space position error

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In the case of compensation, the average error of the boom joint angle tracking is reduced by 69.72%, the maximum error of the arm joint angle and the bucket joint angle tracking is reduced by 58.44 and 51.05% respectively, and the maximum error of the tooth tip space position tracking is reduced by 57.42%. It can be seen that the on-line compensation control strategy can effectively improve the control accuracy of the working device system of the autonomous operation excavator, and increase the operation accuracy and tracking performance of the manipulator of the autonomous operation excavator.

5 Conclusions

In this paper, a motion control method of excavator autonomous operation based on online identification method is proposed to improve the motion control accuracy of excavator. The motion control accuracy of the autonomous excavator manipulator after compensating the gravity term, centripetal force term and inertial force term is studied. The effectiveness of the motion control strategy is verified by tracking the fitted excavator trajectory. The experiments prove that the accuracy of the excavator’s autonomous operation can be improved by compensating the gravity of the excavator’s working device and the inertial force and centripetal force generated during the movement.