## 3 Simultaneous Joint Stiffness and Kinematic Parameters  
In a robot static configuration, a robot joint torque causes a twist deformation about a rotation shaft (that represents the entire drive train from the motor to the associated robot link). Therefore, the shaft can be considered as a torsional spring in the compliance modeling. This study investigates the characteristics of a torsion spring because they are related to modeling of rotational joint compliance. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & & & & & & \\ \hline i & \(\alpha_{i-1}\) (deg) & a\({}_{i-1}\) (m) & \(\beta_{i-1}\) (deg) & b\({}_{i-1}\) (m) & d\({}_{i}\) (deg) & \(\theta_{i}\) (deg) \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & \(\theta_{1}\) \\
2 & 90 & 0.515 & - & - & 0 & \(\theta_{2}\) \\
3 & 0 & 1.6 & 0 & - & 0 & \(\theta_{3}\) \\
4 & 90 & 0.35 & - & - & 1.9 & \(\theta_{4}\) \\
5 & \(-90\) & 0 & - & - & 0 & \(\theta_{5}\) \\
6 & 90 & 0 & - & - & 0.445 & \(\theta_{6}\) \\ T & - & \(-0.45\) & - & \(0.11\) & 0.930 & - \\ \hline \hline \multicolumn{8}{c}{**D-H Parameters of the Main Open Chain**} \\ \hline L5 (m) & 0.8 & L4 (m) & 1.6 & L3 (m) & 0.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Nominal D-H parameters of a Hyundai robot HH800. Figure 1: Kinematic model of the Hyundai robot HH800. The characteristics of torsion springs are basically presented by non-linear functions: for example, \(\tau=k_{1}\star\Delta\theta_{c}+k_{2}\star(\Delta\tau_{c})^{3}\), where \(\tau\) is the spring torque, \(\Delta\theta_{c}\) is the spring rotational deformation, and \(k_{1}\) and \(k_{2}\) are the coefficients. When the robot joint deformation is small, the linear part becomes dominant. Now, we can assume that the functional relationship between the joint torque and its deformation is linear in this calibration process. Assuming that the compliance of the robot links is small and can be neglected, therefore, the elastic errors are mostly dependent on the compliance of joints under the link self-gravity and external payload. Accepting that the robot joint deformations can be expressed by a linear function of the joints torque, the deformation of the \(i\)th joint can be expressed by the effective torques \(\tau_{i}\):  
\[\Delta\theta_{ci}=\frac{\tau_{i}}{k_{i}}=\tau_{i}c_{i} \tag{4}\]  
where \(k_{i}\) is the joint stiffness value of the \(i\)th joint. The Cartesian position errors due to small joint deflections can be modeled as:  
\[\Delta P_{c}=J_{\theta}\Delta\theta_{c}=(J_{\theta}\tau)C \tag{5}\]  
The compliance vector is denoted by \(C=\left[ \begin{array}{ccccc}c_{1}&c_{2}&\cdots&c_{n}&\end{array} \right]^{T}\). The deflection vector is \(\Delta\theta_{c}=\left[ \begin{array}{ccccc}&\Delta\theta_{c1}&\Delta\theta _{c2}&\cdots&\Delta\theta_{\alpha t}&\end{array} \right]^{T}\). At the equilibrium position, \(\tau=diag(\tau_{1},\tau_{2},\cdots,\tau_{n})\) is the effective torque of robot joints. The real position vector of the robot end-effector can be expressed as:  
\[P_{real}=P_{kin}+\Delta P_{kin}+\Delta P_{c}+\Delta P_{extra} \tag{6}\]  
where \(P_{kin}\) is the end effector position that is calculated by the kinematic parameters. \(\Delta P_{kin}\), \(\Delta P_{c}\), and \(\Delta P_{extra}\) are the position errors due to the kinematic parameter errors, joint deflections, and the residual errors due to the unmodeled sources, respectively. [PERSON] and [PERSON] [5] presented a method for simultaneous identifying the kinematic and compliance parameter of a robot:  
\[ \begin{array}{rcl}\Delta X&=&\Delta P_{kin}+\Delta P_{c}\\ &=&J\Delta\phi+J_{\theta}\Delta\theta_{c}\\ &=&J\Delta\phi+J_{\theta}C\tau\\ &=&\left[\begin{array}{ccccc}J&J_{\theta}\tau&\end{array}\right]\left[ \begin{array}{ccccc}\Delta\phi&\\ C&\end{array}\right]\\ &=&J_{\Phi}\Delta\Phi\end{array}  \tag{7}\]  
where \(\Delta X\) is a (3\(\times\)1) vector of three position errors of the robot end-effector. \(J\) is a (3\(\times\)p) matrix that relates the column vectors \(\Delta X\) and \(\Delta P_{kin}\) (\(p=27\) is the total number of kinematic parameters).