# Developing an Assisting Device to Reduce the Vibration on the Hands of Elders
## 3 Proposed Approach
### Dynamic Analysis of Gyroscopic Modeling System  
The modeling concept of gyroscopic hand tremor suppression is shown in Figure 3. It consists of a hand rotating about the wrist joint axis and a gimbal structure that contains a precession axis. To depict the movement, four coordinates are defined as follows. The inertial coordinate \(\{O\}:X_{0}Y_{0}Z_{0}\) with an origin at the central point of the wrist joint; \(Z_{0}\) coincides with the wrist joint axis, and \(Y_{0}\) coincides with the gravity force. Figure 3: Modeling of the gyroscopic system: (**a**) 3D view and (**b**) coordinate-attached analysis. The coordinate attached to a hand \(\{W\}:X_{W}Y_{W}Z_{W}\) located at a central point of the wrist joint; \(Z_{w}\) coincides with \(Z_{0}\) and deviates at \(\theta\) angle from \(X_{0}Y_{0}Z_{0}\). The gimbal frame \(\{G\}:X_{G}Y_{G}Z_{G}\) has a precession axis \(Z\) intersecting with the \(Y\) axis at its central point \(\left(x_{p},y_{p},z_{p}\right)\). The reference coordinate \(\{M\}:X_{M}Y_{M}Z_{M}\) has the same central point as \(X_{G}Y_{G}Z_{G}\) but deviates at a \(\varphi\) angle from its location  
The transformation matrices among the coordinates are denoted as follows. \[X_{G}Y_{G}Z_{G}\overset{Q,R}{\rightarrow}X_{W}Y_{W}Z_{W}\overset{W}{\leftrightarrow }X_{G}Y_{G}Z_{G}\overset{G,R}{\rightarrow}X_{M}Y_{M}Z_{M} \tag{1}\]  
\[\overset{Q}{W}R=\left[ \begin{array}{ccc}\cos\theta&\sin\theta&0\\ -\sin\theta&\cos\theta&0\\ 0&0&1\end{array} \right];\overset{W}{G}R=\left[ \begin{array}{ccc}0&0&-1\\ 0&1&0\\ 1&0&0\end{array} \right];\overset{G}{M}R=\left[ \begin{array}{ccc}\cos\varphi& \sin\varphi&0\\ -\sin\varphi&\cos\varphi&0\\ 0&0&1\end{array} \right] \tag{2}\]  
Later, the human hand is considered as a linear ruler with a mass \(m_{W}\), location of center of gravity (CoG), and angular velocity of hand in \(X_{W}Y_{W}Z_{W}\). A coordinate is computed according to the equations below, where \(K_{W},D_{W},T_{W}^{pert}\) denote the stiffness coefficient, damping ratio, and moment due to the oscillation acting on the hand, respectively. \[p_{w}=\left[x_{w}y_{w}z_{w}\right]^{T} \tag{3}\]  
\[\omega_{w}=\omega_{x_{1}y_{1}z_{1}}=\left[00\dot{\theta}\right]^{T} \tag{4}\]  
The linear velocity of the human hand model can be estimated as in Figure 4.  
\[\dot{p}_{w}\!\!=\left(\dot{p}_{w}\right)_{relative}+\omega_{x_{1}y_{1}z_{1}} \times p_{w}\!\!=\left[-\dot{\theta}y_{w}\dot{\theta}w_{w}0\right]^{T} \tag{5}\]  
where \(\left(\dot{p}_{w}\right)_{relative}\) is relative speed, which has zero value since \(x_{W},y_{W},z_{W}\) are fixed. The kinetic energy of the hand is:  
\[T_{w}=\frac{1}{2}m_{w}\dot{p}_{w}^{2}+\frac{1}{2}\;\omega_{w}^{T}\left[ \begin{array}{ccc}I_{xw}&0&0\\ 0&I_{yw}&0\\ 0&0&I_{zw}\end{array} \right]\omega_{w}\!\!=\frac{1}{2}m_{w}\!\left(x^{2}+y^{2 }\right)\!\dot{\theta}^{2}+\frac{1}{2}I_{zw}\dot{\theta}^{2} \tag{6}\]  
Figure 4: Modeling of system parameters for a human hand. The gravity component can be computed from the transformation of CoG into \(X_{0}\), \(Y_{0}\), \(Z_{0}\). Consequently, the potential energy includes the gravity component and the stiffness of the hand wrist. \[p_{wX_{0}Y_{0}Z_{0}}=R_{1}^{T}p_{w}=[x_{w}\cos\theta-y_{w}\sin\theta x_{w}\sin \theta+y_{w}\cos\theta z_{w}]^{T} \tag{7}\]  
\[V_{w}=m_{w}\dot{g}Y_{0_{w}}+\frac{1}{2}K_{w}\theta^{2}=m_{w}g(x_{w}\sin\theta+ y_{w}\cos\theta)+\frac{1}{2}K_{w}\theta^{2} \tag{8}\]  
Therefore, the Lagrange equation of the hand is:  
\[L_{w}=T_{w}-V_{w}=\frac{1}{2}[m_{w}\Big{(}x^{2}+y^{2}\Big{)}+\frac{1}{2}I_{zw} \dot{\theta}^{2}-m_{w}g(x_{w}\sin\theta+y_{w}\cos\theta)-\frac{1}{2}K_{w} \theta^{2} \tag{9}\]  
Then, for the driving motor, as shown in Figure 5a, which consists of the mass \(m_{m}\), its CoG location on \(Z_{M}\) is \(r_{m}\) in \(X_{M},Y_{M},Z_{M}\). In total, the coordinate of CoG and angular velocity of the motor are:  
\[p_{m}=[x_{p}y_{p}z_{p}]^{T}+R_{2}^{T}R_{3}^{T}\Big{[} \begin{array}{ccc}0&r_{ m}&0\end{array} \Big{]}^{T}=\big{[} \begin{array}{ccc}x_{p}&r_{m}\cos\phi+y_{p}&r_{m} \sin\phi+z_{p}\end{array}