Active Disturbance Rejection Contouring Control of Robotic Excavators with Output Constraints and Sliding Mode Observer

Abstract

This paper proposes an active disturbance rejection contouring control scheme for robotic excavators suffering from model uncertainties, external disturbances, and unmeasurable states. A sliding mode observer (SMO) is firstly designed to precisely estimate both joint velocities and lumped uncertainties and disturbances. These estimations are then fed back into the main controller which is constructed based on the task coordinate frame (TCF) approach. Furthermore, to meet the requirements of high-accuracy control performance, the barrier Lyapunov function (BLF) is utilized in the control design together with the previous techniques, which guarantees the stability of the whole system. Finally, numerical simulation is conducted with a high-reliability excavator model to verify the effectiveness of the proposed control algorithm under various operating conditions. In future work, further practical problems will be conducted to realize the application of robotic excavators in construction.

1. Introduction

For a long time, excavators have played an important role in construction sites due to their mobility and versatility to complete earth-moving tasks. However, most excavators are manually operated, which not only challenges unskilled laborers but also causes occupational safety hazards. Furthermore, during excavators’ operation, repetitive and linear motions are continuously conducted, which is appropriate for robotic applications [1]. Therefore, in recent years, robotic excavators have been researched and developed by utilizing real-time algorithms for perception, task planning, motion planning, and control with input data supplied by multiple proprioceptive sensors, LiDAR, and cameras [2,3]. Among them, the position control algorithm is important to complete tasks, i.e., guaranteeing the tracking performance under difficulties such as system nonlinearities, model uncertainties, and external disturbances [4].

In recent years, together with other energy-saving and recovery purposes, many position control algorithms have been developed for robotic excavators, which vary from model-free approaches as proportional-integral (PI) control [5,6,7] and proportional-integral-derivative (PID) control [8], and model-based control approaches as backstepping control (BC) [9], sliding mode control (SMC) [7], and model predictive control (MPC) [8,10] with optimization performance [11]. Anyways, to increase the control performance which usually relies on excavation system model accuracy, data-driven approaches have also been proposed to re-construct the system model based on learning models with training progress [12,13,14]. However, only focusing on the tracking error as the ultimate performance index is not appropriate in the case of excavator operation since the work quality almost relies on the contouring performance. To deal with this problem, many contouring control approaches have been developed. For example, in [15,16], based on the kinematic transformation for the tracking error, the cross-coupled control (CCC) methods were developed together with the main position tracking control, which provided simple structures, but the final control performance was not reliable and accurate since the excavator model information was not utilized. In contrast, this information was integrated into the task coordinate frame (TCF) approach [4], which provided a general framework for further model-based control algorithms in order to separately regulate the contouring, tangential, and orientation errors. In [17], an MPC was also developed for the contouring control problem of excavators, which relies on the lifting-linearization method with the Koopman operator. Based on the linearized data-driven model of the excavator, the optimal control algorithm was derived to minimize the pre-defined cost function including contouring performance with other criteria. However, since this method requires many computational efforts, the realization of real-time performance in practical conditions will be a challenge.

Like the developments in model-based control approaches, uncertainties and disturbances must be considered in the control design for robotic excavators since these difficulties always exist and deteriorate the system performance during interaction with the environment. Therefore, uncertainties and disturbances attenuation techniques are important to guarantee high-accuracy control performance. Overall, two main approaches have been applied for general robotic applications, i.e., adaptive control and disturbance observer (DOB). For the adaptive control, it was originally proposed to deal with structured uncertainties, i.e., known uncertainty functions with unknown slow-varying parameters in linear regression form [18,19]. Moreover, this method was further extended to the adaptive mechanism with intelligent approximators such as neural networks (NNs) [20,21] or fuzzy logic systems (FLSs) [22,23], which can handle unstructured uncertainties. However, these approaches were not designed to deal with external disturbances due to the states-dependent characteristics. In contrast, disturbance observers (DOBs) have been proposed to deal with not only model uncertainties, but also external disturbances based on the assumption that their derivatives are bounded. Recent developments can be considered as the nonlinear disturbance observer (NDOB) [24,25] and time delay estimation (TDE) [26,27,28] which require states and their derivatives information to achieve high accuracy disturbance estimation performance. Overcoming this requirement, the extended-state observer (ESO) was developed to estimate both lumped disturbances and system states based on the augmented state mechanism [4,29]. However, the order of this observer was quite high, which may cause computational burdens and slow down the transient responses. A potential solution to deal with these difficulties is the sliding mode observer (SMO) which considers a similar problem, i.e., state estimation, based on the switching mechanism of the estimation errors [30]. Therefore, fast responses in the estimates can be guaranteed with robustness to changes in system dynamics. However, to the best of the authors’ knowledge, the SMO has not been applied for states and disturbances estimation in general robotic applications, especially excavator control design.

Beyond that, for further improvement of excavator control accuracy, i.e., guarantee predefined control criteria, many techniques which have been widely applied in robotic fields can be investigated as prescribed performance function (PPF) [31,32], barrier Lyapunov function (BLF) [29,33], and their variations [34,35]. However, until now, the application of these advanced techniques in the control design of robotic excavators was quite limited. In the authors′ previous work [4,36], the BLF is adopted to guarantee high accuracy contouring control performance and only PID is applied for the remaining performance indices, which does not fully cover the operation of the whole system. Therefore, to entirely specify the task quality, it is necessary to apply constraint techniques for all the criteria, i.e., contouring error, tangential error, and orientation error when synthesizing the control signal.

Based on the above-mentioned analysis, in this paper, an active disturbance rejection contouring control is proposed for robotic excavators. The main contribution relies on the development and implementation of an SMO, which is firstly designed to effectively estimate both system states and lumped disturbances and uncertainties during the operation of robotic excavators. These estimates are fed back into the main controller, which is designed based on the TCF approach to separate the tracking error into contouring error and the other error components. Finally, the prescribed performance is guaranteed for all system criteria based on the BLF. Simulation with a high-reliability excavator model is conducted to verify the proposed control performance.

The paper is organized as follows: In Section 2, the problem formulation is described. Section 3 explains the proposed SMO design. In Section 4, the design of the proposed control algorithm is presented. Numerical simulation is discussed in Section 5. Finally, Section 6 concludes this work.

2. Problem Formulation

The schematic diagram of the investigated excavator model is described in Figure 1. The operation is completed by the motions of three links including the boom, arm, and bucket. The end-effector position is presented by coordinates $(x,y,\varphi )$. For simplicity, the swing motion is omitted here. The excavator kinematics are clearly described in [4].

Based on Lagrangian mechanics, the excavator dynamics are expressed as

$$\mathit{M}\left(\mathit{q}\right)\ddot{\mathit{q}}+\mathit{V}\left(\mathit{q},\dot{\mathit{q}}\right)+\mathit{G}\left(\mathit{q}\right)+{\mathit{\tau }}_f={\mathit{J}}_a^T\mathit{u}$$

(1)

where $\mathit{q}\in R^3$ is the vector of joint angles; $\mathit{M}(\mathit{q})\in R^{3\times 3}$, $\mathit{V}(\mathit{q},\dot{\mathit{q}})\in R^3$, and $\mathit{G}(\mathit{q})\in R^3$ denotes the inertial matrix, centrifugal and Coriolis force, and gravity, respectively; $\mathit{J}$ and ${\mathit{J}}_a$ are Jacobian matrices; ${\mathit{\tau }}_f={\mathit{J}}^T(\mathit{q}){\mathit{F}}_d+{\mathit{J}}_a{}^T{\mathit{F}}_f$ indicates the lumped disturbances and uncertainties where ${\mathit{F}}_d\in R^3$ is the external torque/force vector acting on the bucket tips and ${\mathit{F}}_f$ represents uncertainties in the actuation system; $u$ is the control signal, i.e., hydraulic actuation force.

To realize the design of the proposed SMO and main controller, the system dynamics are expressed in the following form:

$$\begin{array}{l}{\dot{\mathit{x}}}_1={\mathit{x}}_2\\ {\dot{\mathit{x}}}_2=\mathit{f}({\mathit{x}}_1,{\mathit{x}}_2)+\mathit{g}({\mathit{x}}_1)\mathit{u}+\mathit{d}\end{array}$$

(2)

where ${\mathit{x}}_1=\mathit{q},\hspace{0.17em}$ ${\mathit{x}}_2=\dot{\mathit{q}},$ $\mathit{f}({\mathit{x}}_1,{\mathit{x}}_2)=-\mathit{M}{({\mathit{x}}_1)}^{-1}[\mathit{V}({\mathit{x}}_1,{\mathit{x}}_2)+\mathit{G}({\mathit{x}}_1)],$$\hspace{0.17em}\mathit{g}({\mathit{x}}_1)=\mathit{M}{({\mathit{x}}_1)}^{-1}{\mathit{J}}_a{({\mathit{x}}_1)}^T,$ and $\mathit{d}=-\mathit{M}{({\mathit{x}}_1)}^{-1}{\mathit{\tau }}_f$.

Assumption 1.

The vector of lumped uncertainties and disturbances is bounded, i.e., $\Vert d\Vert \le {\Delta }_d$ where ${\Delta }_d$ is a positive constant.

Assumption 2.

The Lipschitz condition holds for the nonlinear term $\mathit{f}({\mathit{x}}_1,{\mathit{x}}_2)$ as follows:

$$\Vert \mathit{f}({\mathit{x}}_1,{\mathit{x}}_2)-\mathit{f}({\mathit{x}}_1,{\mathit{x}}_2+\Delta {\mathit{x}}_2)\Vert \le {\alpha }_f\Vert \Delta {\mathit{x}}_2\Vert$$

(3)

where ${\alpha }_f$ is a positive constant.

Remark 1.

Assumption 1 is usually applied for sliding mode techniques, which is reasonable since the physical control system cannot handle unbounded uncertainties and disturbances. Besides, Assumption 2 is necessary to overcome the effect of nonlinearities on the observation and control performance.

Remark 2.

With joint angle measurement, the problem is how to construct an observer that effectively estimates both joint velocities and lumped uncertainties and disturbances in the excavation system. Based on that, the control algorithm needs to be derived to guarantee high-accuracy control performance, especially contouring control performance with prescribed performance.

3. SMO Design

The proposed SMO is expressed as follows:

$$\{\begin{array}{l}{\dot{\mathit{s}}}_1={\mathit{s}}_2+\mathit{v}\\ {\dot{\mathit{s}}}_2=\mathit{f}({\mathit{x}}_1,{\mathit{s}}_2)+\mathit{g}({\mathit{x}}_1)\mathit{u}+\omega \mathit{v}\end{array}$$

(4)

where ${\mathit{s}}_1$ and ${\mathit{s}}_2$ are the states of the observer, and $\omega$ is the observer gain. The symbol $v$ denotes a switching function of the observation error, which is defined by

$$\mathit{v}=\{\begin{array}{l}\frac{\eta ({\mathit{x}}_1-{\mathit{s}}_1)}{\Vert {\mathit{x}}_1-{\mathit{s}}_1\Vert } \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{if}\Vert {\mathit{x}}_1-{\mathit{s}}_1\Vert \ne 0\hspace{0.17em}\hspace{0.17em}\\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{otherwise}\end{array}\hspace{0.17em}$$

(5)

where $\eta$ is a positive parameter.

Theorem 1.

Based on Assumptions 1 and 2, the proposed SMO (4) ensures arbitrarily bounded estimation performance for the system (2) if the observer parameters are selected with large enough values.

Proof of Theorem 1.

Denote the error term $\tilde{\mathit{s}}=\mathit{x}-\mathit{s}$. Subtracting both sides of (4) from those of (2), the error dynamics are expressed as follows:

$$\begin{array}{l}{\dot{\tilde{\mathit{s}}}}_1={\tilde{\mathit{s}}}_2-\mathit{v}\\ {\dot{\tilde{\mathit{s}}}}_2=\tilde{\mathit{f}}+\mathit{d}-\omega \mathit{v}\end{array}$$

(6)

Define a Lyapunov function $V_{\mathit{s}1}=1/2{{\tilde{\mathit{s}}}_1}^T{\tilde{\mathit{s}}}_1$. Differentiating it and substituting (5) into, it becomes

$${\dot{V}}_{\mathit{s}1}={\tilde{\mathit{s}}}_1({\tilde{\mathit{s}}}_2-v)\le -\Vert {\tilde{\mathit{s}}}_1\Vert (\eta -\Vert {\tilde{\mathit{s}}}_2\Vert )$$

(7)

where the observer parameter $\eta$ is selected with a large enough value so that there exists a positive constant ${\eta }_0$ satisfying $\Vert {\tilde{\mathit{s}}}_2\Vert \le \eta -{\eta }_0$. Therefore, one obtains

$${\dot{V}}_{s1}\le -\sqrt{2}{\eta }_0\sqrt{V_{s1}}$$

(8)

Based on this, ${\mathit{s}}_1$ converges to zero in finite time. After that, from the first equation of (6), it is observed that the switching function $\mathit{v}$ is equivalent to the error ${\tilde{\mathit{s}}}_2$. Substituting it into the second equation of (6), the equivalent error dynamics are given by

$${\dot{\tilde{\mathit{s}}}}_2=\tilde{\mathit{f}}+\mathit{d}-\omega {\tilde{\mathit{s}}}_2$$

(9)

Another Lyapunov function is designed as $V_{s2}=1/2{\tilde{\mathit{s}}}_2{}^T{\tilde{\mathit{s}}}_2$. The derivative of it is derived by

$$\begin{array}{l}{\dot{V}}_{\mathit{s}2}={{\tilde{\mathit{s}}}_2}^T(\tilde{\mathit{f}}+\mathit{d}-\omega {\tilde{\mathit{s}}}_2)\\ \le -\omega {\tilde{\mathit{s}}}_2{}^T{\tilde{\mathit{s}}}_2+{\tilde{\mathit{s}}}_2{}^T{\tilde{\mathit{s}}}_2+\frac{1}{2}{\tilde{\mathit{f}}}^T\tilde{\mathit{f}}+\frac{1}{2}{d}^{T}d\end{array}$$

(10)

Combining with Assumptions 1 and 2, the following inequality holds

$$\begin{array}{l}{\dot{V}}_{\mathit{s}2}\le -\left(\omega -\frac{1}{2}{\alpha }_f{}^2-1\right){\tilde{\mathit{s}}}_2{}^T{\tilde{\mathit{s}}}_2+\frac{1}{2}{\Delta }_d{}^2\\ \le -{\lambda }_{s2}{V}_{s2}+{\Delta }_{s2}\end{array}$$

(11)

where ${\lambda }_{s2}=(2\omega -{\alpha }_f{}^2-2)$ and ${\Delta }_{s2}=0.5{\Delta }_d{}^2$.

From this, the proof of Theorem 1 is completed. □

The joint velocity and disturbance estimations are expressed as follows:

$$\begin{array}{l}{\hat{\mathit{x}}}_2={\mathit{s}}_2+\mathit{v}\\ \hat{\mathit{d}}={\omega }_1\mathit{v}\end{array}$$

(12)

The first estimation is derived since ${\hat{\mathit{x}}}_2={\dot{\mathit{s}}}_1$ converges to ${\mathit{x}}_2$ after the convergence time of ${\tilde{\mathit{s}}}_1$. Similarly, $\mathit{d}$ can be estimated by $\hat{\mathit{d}}$ when comparing (2) and (4).

Remark 3.

A modified switching function is designed to attenuate the chattering effects on the estimated values in (12) as

$$v_{\mathrm{mod}}=\frac{\eta ({\mathit{x}}_1-{\mathit{s}}_1)}{\Vert {\mathit{x}}_1-{\mathit{s}}_1\Vert +{\epsilon }_v}$$

(13)

where${\epsilon }_v$is a positive constant.

4. Control Design

Based on the estimates from the SMO, the main controller is constructed with the Lyapunov function in order to guarantee prescribed performance in all error components, which is different from the authors’ previous work. The control diagram is shown in Figure 2.

The control signal is proposed as follows:

$$\begin{array}{l}\mathit{u}={\mathit{J}}_a^{-T}(\mathit{q})(\mathit{V}(\mathit{q},\hat{\dot{q}})+\mathit{G}(\mathit{q})+\mathit{M}(\mathit{q})\hat{\mathit{d}}+{\mathit{u}}_1)\\ {\mathit{u}}_1=-\mathit{M}(\mathit{q}){\mathit{J}}^{-1}(\mathit{q})\dot{\mathit{J}}(\mathit{q},\hat{\dot{q}})\hat{\dot{q}}-\mathit{M}(\mathit{q}){\mathit{J}}^{-1}(\mathit{q}){\mathit{T}}^{-1}{\mathit{u}}_2+\mathit{M}(\mathit{q}){\mathit{T}}^{-1}{\ddot{\mathit{r}}}_d\end{array}$$

(14)

where ${\mathit{r}}_d$ denotes the reference trajectory of the end-effector $\mathit{r}=(\mathit{x},y,\varphi )$, ${\mathit{u}}_2$ is an auxiliary control signal to be designed later, and $\mathit{T}$ is the task coordinate transformation matrix, which denotes the following relation:

$$\mathit{\epsilon }=\mathit{Te}$$

(15)

where $\mathit{\epsilon }={[{\epsilon }_n,{\epsilon }_t,{\epsilon }_{\varphi }]}^T$ is a new presentation of the original tracking error $\mathit{e}={\mathit{r}}_d-\mathit{r}$, which includes the normal, tangential, and orientation tracking error components, respectively. The transformation matrix $\mathit{T}$ is calculated based on the slope angle $\theta$ of the desired contour as shown in Figure 3.

In the case that linear contour is adopted, i.e., $\mathit{T}$ is constant, from (14) and (15), the error dynamics is derived as

$$\ddot{\mathit{\epsilon }}={\mathit{u}}_2+{\mathit{f}}_d$$

(16)

where $f_d$ is the lumped error components including velocity and disturbance estimation errors of the SMO, which is bounded, i.e., $\Vert f_d\Vert \le {\Delta }_f,{\Delta }_f>0$.

The output constraint is pre-defined as follows:

$$\begin{array}{l}-b_n(t)\le {\epsilon }_n\le b_n(t)\\ -b_t(t)\le {\epsilon }_t\le b_t(t)\\ -b_{\varphi }(t)\le {\epsilon }_{\varphi }\le b_{\varphi }(t)\end{array}$$

(17)

where $b_i(t)$ are predefined limits.

To maintain these prescribed performances during operation, the auxiliary control signal ${\mathit{u}}_2$ is designed by

$$u_{2i}={\dot{\alpha }}_i-\frac{{\epsilon }_i}{b_i{(t)}^2-{\epsilon }_i{}^2}+k_{2i}{z}_i\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(i=n,t,\varphi )$$

(18)

where $z_i={\alpha }_i-{\dot{\epsilon }}_i$, ${\alpha }_i=-(k_{1i}+{\lambda }_i(t)){\epsilon }_i$, ${\lambda }_i(t)=\{{\dot{b}}_i{(t)}^2/b_i{(t)}^2$$+\beta \}$, $k_{1i}$ and $k_{2i}$ are control gains, and $\beta$ is a positive constant.

Theorem 2.

For the system (1), the proposed control algorithm (14), (18) together with the SMO (4), (12) guarantees prescribed performance on both contouring, tangential, and orientation error components under the effects of lumped uncertainties and disturbances.

Proof of Theorem 2.

Lyapunov functions are defined as follows:

$$V_i=\frac{1}{2}\ln \left(\frac{b_i^2(t)}{b_i^2(t)-{\epsilon }_i^2}\right)+\frac{1}{2}{z}_i{}^2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(i=n,t,\varphi )$$

(19)

Taking derivatives of them and substituting (16) and (18) into them, one obtains

$$\begin{array}{l}{\dot{V}}_i=\frac{{\epsilon }_i}{b_i^2(t)-{\epsilon }_i^2}\left({\dot{\epsilon }}_i-{\epsilon }_i\frac{{\dot{b}}_i(t)}{b_i(t)}\right)+z_i{\dot{z}}_i\\ =\frac{{\epsilon }_i}{b_i^2(t)-{\epsilon }_i^2}\left({\alpha }_i-{\epsilon }_i\frac{{\dot{b}}_i(t)}{b_i(t)}\right)-\frac{{\epsilon }_i{z}_i}{b_i^2(t)-{\epsilon }_i^2}+z_i({\dot{\alpha }}_i-u_{2i}-f_{di})\\ \le -\frac{k_{1i}{\epsilon }_i{}^2}{b_i^2(t)-{\epsilon }_i^2}-k_{2i}{z}_i^2+z_i{f}_{di}\\ \le -k_{1i}\ln \left(\frac{b_i^2(t)}{b_i^2(t)-{\epsilon }_i^2}\right)-\left(k_{2i}-\frac{1}{2}\right)z_i^2+\frac{1}{2}{f}_{di}^2\\ \le -{\lambda }_i{V}_i+{\Delta }_i\end{array}$$

(20)

where ${\lambda }_i=2\min (k_{1i},k_{2i}-0.5)$ and ${\Delta }_i=0.5{\Delta }_f^2$.

From this, the stability of each error component is proved. Based on that, the stability of the total system is guaranteed. Therefore, the proof is completed. □

5. Numerical Simulation

5.1. Simulation Setup

In this section, a mini-excavator model is utilized to verify the effectiveness of the proposed algorithm, as shown in Figure 4. The simulation is conducted in MATLAB Simulink 2022a environment with sampling times of 1ms for the main controller and observer and 0.1ms for the excavator. Dynamic parameters are given in

Table 1. Model parameter [4], Reprinted/adapted with permission from Ref. [4]. 2021, Elsevier.

Parameter	Description	Value	Unit
$m_1$	Mass of boom	36.863	kg
$m_2$	Mass of arm	13.138	kg
$m_3$	Mass of bucket	9.008	kg
$I_{z1}$	Mass moment of inertial of the boom	7,306,759.708	kgmm2
$I_{z2}$	Mass moment of inertial of the arm	1,178,748.843	kgmm2
$I_{z3}$	Mass moment of inertial of the bucket	348,927.380	kgmm2
$r_{c1}$	Center of mass of the boom	[848.173, 213.106, 2.615] T	mm
$r_{c2}$	Center of mass of the arm	[336.525, 65.798, −0.000] T	mm
$r_{c3}$	Center of mass of the bucket	[210.471, 154.244, 0.082] T	mm
$L_1$	Length of boom	1692	mm
$L_2$	Length of arm	851	mm
$L_3$	Length of bucket	580	mm

To test the control performance, multi-step disturbances, i.e., interaction forces and torques with the environment, are added during the simulation. Moreover, simulation results with different desired contours, i.e., horizontal and slope contours, are also investigated.

.

5.2. Controller for Comparision

Three controllers are investigated to verify the contribution of the paper as follows:

-

The ESO-based contouring controller (ESOC): This controller was constructed in our previous work [4] with the ESO to estimate joint velocities and lumped disturbances. The prescribed control performance is only guaranteed for the contouring error. For the remaining tangential and orientation errors, only the proportional-derivative (PD) control law is applied as follows:

$$u_{2i}=-k_{di}{\dot{\epsilon }}_i-k_{pi}{\epsilon }_i\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(i=t,\varphi )$$

where $k_{di}$ and $k_{pi}$ are the derivative and proportional gains, respectively.

-

The SMO-based contouring controller (SMOC): This controller has the same structure as the ESO-based controller. However, the ESO is replaced by the SMO to verify the proposed observer.

-

The proposed controller: The controller is similar to the SMO-based controller. Nevertheless, the prescribed performance is applied for all the error components. Depending on the priority, the constraint level can be adjusted.

For a fair comparison, equivalent control parameters are selected for both controllers, as shown in

Table 2. Control parameters.

Controller	Parameters
ESOC	$k_{1n}=100,k_{2n}=100,{\beta }_n=0.1,k_{dt}=50,k_{pt}=200,k_{d\varphi }=100,k_{p\varphi }=500,\omega =200$
SMOC	$\begin{array}{l}{k}_{1n}=100,k_{2n}=100,{\beta }_n=0.1,k_{dt}=50,k_{pt}=200,k_{d\varphi }=100,k_{p\varphi }=500,\\ \omega =200,\eta =150,{\epsilon }_v=0.3\end{array}$
Proposed controller	$\begin{array}{l}{k}_{1n}=k_{1t}=k_{1\varphi }=100,k_{2n}=k_{2t}=k_{2\varphi }=100,{\beta }_n={\beta }_t=0.1,{\beta }_{\varphi }=0.2,\\ \omega =200,\eta =150,{\epsilon }_v=0.3\end{array}$

.

5.3. Simulation Results

5.3.1. Case 1: Horizontal Contour

In this case study, a horizontal desired contour is selected to verify the effectiveness of the proposed controller compared to the ESOC and the SMOC. In Figure 5a, it is observed that the proposed controller and the SMOC provide high accuracy contouring control performance within prescribed bounds while the ESOC cannot achieve that, which shows the superiority of the proposed SMO compared to the ESO. Similar results are described in Figure 5b,c for the tangential errors and contouring errors, respectively. However, only the proposed controller guarantees the prescribed performance in these error components since the model-based control with BLF is applied here compared to the PD control of the ESOC and SMOC. Anyways, the contour shapes are given in Figure 5d, which once again states the dominations of the proposed control algorithm with the other ones.

For further analysis of these results, the disturbance estimation performances of the proposed SMO and ESO in terms of the boom, arm, and bucket are presented in Figure 6. One observes that the SMO in the proposed controller shows the fastest transient responses when abrupt disturbances occur compared to the SMO in the SMOC and the ESO, which explains the ability to keep the errors component inside the bounds of the proposed controller as mentioned above. In the steady-states, the ESO even shows better disturbance estimation compared to the other ones, but the tracking performance cannot be maintained since it already violates constraints at the last transient times.

In addition, Figure 7 describes the high-accuracy velocity estimation performance of the proposed SMO during the operation, which partly contributes to the final tracking performance of the proposed control algorithm in Figure 5.

Finally, the control signals of both controllers are shown in Figure 8. Overall, at the steady state, there is not so much difference in the control signals of the SMO-based controllers and the ESOC. However, in the transient response, since the ESOC reacts slower with changes in disturbance and provides the worst tracking errors, it generates the highest control signal compared to the remaining ones.

5.3.2. Case 2: Sloped Contour

To further investigate the performance of the proposed control algorithm, a sloped contour with 45 degrees is selected in the simulation. Overall, similar results are observed compared to those of Case 1. The contouring error, tangential error, orientation error, and contour shape are described in Figure 9a–d, respectively, which indicates the superiority of the proposed control algorithm.

Moreover, the disturbance and velocity estimation performances are given in Figure 10 and Figure 11, which once again emphasize the importance of the proposed SMO in the final control performance under different operating conditions.

6. Conclusions

In this paper, an SMO-based contouring controller is developed for robotic excavators to achieve high-accuracy contouring control performance during operation. The SMO is designed for the first time to effectively estimate not only the lumped uncertainties and disturbances but also the unmeasurable joint velocities, especially when abrupt disturbances occur, which directly contribute to the final control performances. Furthermore, to guarantee better performance compared to previous works, the BLF technique is integrated into the control design for both error components. Simulation results verify the effectiveness of the proposed observer and controller compared to the ESO and the ESO-based controller. In future work, further practical problems will be considered in this SMO-based contouring control scheme with experiment verification.