# State Observer Based on an Accelerometer for an Elastic Joint with Nonlinear Friction
## 2 Problem Formulation and Solution  
We consider an elastic joint model with nonlinear friction and stiffness, as described by [7]:  
\[ \begin{split} J_{l}(\theta_{l})\bar{\theta}_{l}+D(\dot{\theta}_{ l}-\theta_{m})-K(\theta_{l},\theta_{m})+C(\theta_{l},\dot{\theta}_{l})+G( \theta_{l})=0,\\ J_{m}\bar{\theta}_{m}+D(\dot{\theta}_{m}-\dot{\theta}_{l})+K( \theta_{l},\theta_{m})+F_{m}(\theta_{m})=\tau.\end{split}  \tag{1}\]  
where \(\theta_{l}\) and \(\theta_{m}\) are the angular positions of the link-side and motor-side, respectively. \(J_{l}(\theta_{l})\) and \(G(\theta_{l})\), which depend on the position of the link, are the link inertia and the gravity term, respectively. While the coriolis and centrifugal term \(C(\theta_{l},\omega_{l})\) depend on the position and velocity of the link, the motor inertia \(J_{m}\) and the damping \(D\) have a constant value. \(\tau\) is the torque input to the motor. As shown in Figure 1, the nonlinear friction on the motor side \(F_{m}(\omega_{m})\) is given by:  
\[F_{m}(\omega_{m})=f_{\
u}\omega_{m}+f_{c}(\mu_{k}+(1-\mu_{k})\text{sech}(\beta \omega_{m}))\tanh(\alpha\omega_{m}) \tag{2}\]  
where \(f_{\
u}\) and \(f_{c}\) are the kinetic coefficients of viscous and coulomb friction, respectively. \(\alpha\), \(\beta\) and \(0<\mu_{k}<1\) are the suitable parameters of the nonlinear friction. Moreover, as shown in [7,9], the stiffness \(K(\theta_{l},\theta_{m})\) with nonlinear characteristics is expressed by  
\[K(\theta_{l},\theta_{m})=\left\{ \begin{array}{ll}-k_{1}\theta_{B}-k_{2} \theta_{B}^{3}-(k_{1}+3k_{2}\theta_{B}^{2})(-\theta_{m}+\theta_{l}-\theta_{B} ),&\text{if }\theta_{m}-\theta_{l}<-\theta_{B}.\\ k_{1}(\theta_{m}-\theta_{l})+k_{2}(\theta_{m}-\theta_{l})^{3},&\text{if } \|\theta_{m}-\theta_{l}\|\leq\theta_{B}.\\ k_{1}\theta_{B}+k_{2}\theta_{B}^{3}+(k_{1}+3k_{2}\theta_{B}^{2})(\theta_{m}- \theta_{l}-\theta_{B}),&\text{if }\theta_{m}-\theta_{l}>\theta_{B}.\\ \end{array} \right. \tag{3}\]  
Figure 1: Example of nonlinear friction: (**a**) static + coulomb + viscous friction, (**b**) static friction, (**c**) coulomb friction, (**d**) viscous friction. where \(k_{1},k_{2}\), and \(\theta_{B}\) are all position numbers. Here, \(k_{1}\) and \(k_{2}\) mean linear and nonlinear stiffness coefficients, respectively. \(\theta_{B}\) is a break-point, which means the physical limit by the torsion degree between the motor and the link. The representation of (1) can be obtained as follows:  
\[\dot{\theta}_{l} =\omega_{l}, \tag{4}\] \[\dot{\omega}_{l} =J_{l}^{-1}(\theta_{l})(D(\omega_{m}-\omega_{l})+K(\theta_{l}, \theta_{m})-C(\theta_{l},\omega_{l})-G(\theta_{l})),\] \[\dot{\theta}_{m} =\omega_{m},\] \[\dot{\omega}_{m} =J_{m}^{-1}(D(\omega_{l}-\omega_{m})-K(\theta_{l},\theta_{m})-F_{ m}(\omega_{m}))+J_{m}^{-1}\tau.\]  
Here, \(\omega_{l}\) and \(\omega_{m}\) are the derivatives of \(\theta_{l}\) and \(\theta_{m}\), which represent the angular velocities of the link and motor, respectively. The final goal of this study is to design a robust state observer that guarantees the asymptotic estimation performance for all the states of the Equation (4). Now, we pose some conditions in order to solve the problem. The position of motor-side \(\theta_{m}\) and the acceleration of link-side \(\dot{\omega}_{l}\) are measurable while the link position \(\theta_{l}\), the link velocity \(\omega_{l}\), and the motor velocity \(\omega_{m}\) are not. The motor inertia \(J_{m}\) and the damping parameter \(D\) and all parameters of the nonlinear friction (2) and stiffness (3) are known. Now, let  
\[x:= \begin{bmatrix}x_{1}&x_{2}&x_{3}&x_{4}\end{bmatrix} ^{\top}:= \begin{bmatrix} \theta_{l}&\omega_{l}&\theta_{m}&\omega_{m}\end{bmatrix} ^{\top}.\]  
Then, the observer design problem considered in this paper can be stated as follows. For the given system (1), find a state observer given in the form  
\[\dot{x}(t)=f(t,\dot{x},\theta_{m},\dot{\omega}_{l},\tau) \tag{5}\]  
such that \(\dot{x}(t)\) converges to \(x(t)\) as time goes to infinity.