## 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 \((x,y,\phi)\). 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  
\[\mathbf{M}(\mathbf{q})\ddot{\mathbf{q}}+\mathbf{V}(\mathbf{q},\dot{\mathbf{q}})+\mathbf{G}(\mathbf{q})+\mathbf{ \tau}_{f}=\mathbf{J}_{a}^{T}\mathbf{u} \tag{1}\]  
where \(\mathbf{q}\in R^{3}\) is the vector of joint angles; \(\mathbf{M}(\mathbf{q})\in R^{3\times 3}\), \(\mathbf{V}(\mathbf{q},\dot{\mathbf{q}})\in R^{3}\), and \(\mathbf{G}(\mathbf{q})\in R^{3}\) denotes the inertial matrix, centrifugal and Coriolis force, and gravity, respectively; \(\mathbf{J}\) and \(\mathbf{J}_{a}\) are Jacobian matrices; \(\mathbf{\tau}_{f}=\mathbf{J}^{T}(\mathbf{q})\mathbf{F}_{d}+\mathbf{J}_{a}^{T}\mathbf{F}_{f}\) indicates the lumped disturbances and uncertainties where \(\mathbf{F}_{d}\in R^{3}\) is the external torque/force vector acting on the bucket tips and \(\mathbf{F}_{f}\) 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:  
\[\begin{array}{l}\dot{\mathbf{x}}_{1}=\mathbf{x}_{2}\\ \dot{\mathbf{x}}_{2}=\mathbf{f}(\mathbf{x}_{1},\mathbf{x}_{2})+\mathbf{g}(\mathbf{x}_{1})\mathbf{u}+\mathbf{d} \end{array} \tag{2}\]  
where \(\mathbf{x}_{1}=\mathbf{q}\), \(\mathbf{x}_{2}=\dot{\mathbf{q}}_{r}\mathbf{f}(\mathbf{x}_{1},\mathbf{x}_{2})=-M(\mathbf{x}_{1})^{-1}[ \mathbf{V}(\mathbf{x}_{1},\mathbf{x}_{2})+\mathbf{G}(\mathbf{x}_{1})]\),\(\mathbf{g}(\mathbf{x}_{1})=M(\mathbf{x}_{1})^{-1}\mathbf{J}_{a}(\mathbf{x}_{1})^{T}\), and \(\mathbf{d}=-M(\mathbf{x}_{1})^{-1}\mathbf{\tau}_{f}\).  
**Assumption 1**.: _The vector of lumped uncertainties and disturbances is bounded, i.e., \(\|d\|\leq\Delta_{d}\) where \(\Delta_{d}\) is a positive constant._  
Figure 1: Excavator model [4], Reprinted/adapted with permission from Ref. [4]. 2021, Elsevier.  
**Assumption 2**.: _The Lipschitz condition holds for the nonlinear term \(\mathbf{f}(\mathbf{x}_{1},\mathbf{x}_{2})\) as follows:_  
\[\|\mathbf{f}(\mathbf{x}_{1},\mathbf{x}_{2})-\mathbf{f}(\mathbf{x}_{1},\mathbf{x}_{2}+\Delta\mathbf{x}_{2}) \|\leq\alpha_{f}\|\Delta\mathbf{x}_{2}\| \tag{3}\]  
_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._