## 2 Dynamic Modeling and Problem Formulation  
### Dynamic Modeling  
Figure 1 shows the structure of a deep-sea electric oil-filled joint actuator, which is a test joint of the mechanic leg of a deep-sea crawling robot named \"Qilin\". The oil-filled  
joint actuator is mainly composed of a motor, a gear reducer, an angular position sensor, a housing with sealing assembly, and a pressure compensation system.  
The dynamic model of the deep-sea electric oil-filled joint actuator can be described as follows:  
\[J\left(N\ddot{q}\right)=k_{t}u-k_{b}\left(N\dot{q}\right)-T_{v}-\frac{T_{d}}{N}- \frac{T_{f}}{N} \tag{1}\]  
where \(q,\ \dot{q},\ \ddot{q}\) are joint actuator output position, velocity, and acceleration, respectively; \(u\) is the control current of the motor; \(J\), \(k_{t}\), \(k_{b}\), and \(N\) are the moment of inertia, the motor torque constant, the motor viscous friction coefficient, and the gear reduction ratio, respectively; \(T_{v}\), \(T_{d}\), and \(T_{f}\) are oil stirring loss torque, unknown external disturbance torque, and dynamic seal loss torque of the joint actuator output shaft, respectively.  
Define \(\theta_{1}=k_{t}/\left(JN\right)\), \(\theta_{2}=-k_{b}/J\), then the dynamic model (1) of the joint actuator can be expressed as follows:  
\[\ddot{q}=\theta_{1}u+\theta_{2}\dot{q}+\frac{1}{J}\left(-\frac{T_{v}}{N}- \frac{T_{d}}{N^{2}}-\frac{T_{f}}{N^{2}}\right) \tag{2}\]  
As the precise values of \(J\), \(k_{t}\), and \(k_{b}\) are usually unknown in practical applications, the parameters \(\theta_{1}\) and \(\theta_{2}\) can be expressed as the combination of nominal values and uncertain values as follows:  
\[\left\{\begin{array}{ll}\theta_{1}=\theta_{1n}+\Delta\theta_{1}\\ \theta_{2}=\theta_{2n}+\Delta\theta_{2}\end{array}\right. \tag{3}\]  
where \(\theta_{1n}=k_{tn}/\left(J_{n}N\right)\) and \(\theta_{2n}=-k_{bn}/J_{n}\) are determined by the nominal values of the joint actuator (denoted by \(J_{n}\), \(k_{tn}\), and \(k_{bn}\)), while \(\Delta\theta_{1}\) and \(\Delta\theta_{2}\) represent deviations from the nominal parameters of the system model.  
Substituting Equation (3) into (2), the system model can be expressed as follows:  
\[\ddot{q}=\theta_{1n}u+\theta_{2n}\dot{q}+\left[\Delta\theta_{1}u+\Delta\theta _{2}\dot{q}+\frac{1}{J}\left(-\frac{T_{v}}{N}-\frac{T_{d}}{N^{2}}-\frac{T_{f} }{N^{2}}\right)\right] \tag{4}\]  
Defining \(x=\left[x_{1},\ x_{2}\right]^{T}=\left[q,\ \dot{q}\right]^{T}\) as the system state vector, the dynamic Equation (4) can be rewritten in a state-space form:  
\[\left\{\begin{array}{ll}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=\theta_{1n}u+\theta_{2n}x_{2}+d\\ y=x_{1}\end{array}\right. \tag{5}\]  
Figure 1: Structure of a deep-sea electric oil-filled joint actuator. 1—sealing assembly; 2—angular position sensor; 3—output shaft; 4—gear reducer; 5—housing; 6—motor; 7—pressure compensation interface.  
where \(y\) is the output of the joint actuator, and \(d=\Delta\theta_{1}u+\Delta\theta_{2}\dot{q}+\frac{1}{l}\Big{(}-\frac{T_{\rm{e}}}{N} -\frac{T_{\rm{d}}}{N^{2}}-\frac{T_{f}}{N^{2}}\Big{)}\) represents the lumped uncertainties of the joint actuator, including parametric uncertainties and unknown internal and external disturbances.