# New Indices for Evaluating Vibration Characteristics of Flexible-Joint Robots
## 3 Theoretical Development  
### Equations of Motion  
The kinematic scheme of a robot with \(n\) links and \(n\) revolute joints is plotted in Figure 2. The position vector of the \(i\)th joint in the inertial frame \(\{OXYZ\}\) is denoted by \(\textbf{P}_{i}\). A local body frame \(\{o_{i}x_{i}y_{i}z_{i}\}\) is established on each of the links with the origin at the \(i\)th joint. A tool is mounted on the robot's end-effector to complete the process and the tool tip is denoted as \(E\), with its position vector written in the inertial frame as **E**. Then the external force and torque acting on the tool from the workpiece are denoted as **F** and \(\boldsymbol{\tau}\) respectively. Additionally, to describe the process force and torque, a work frame \(\{o^{\prime}x^{\prime}y^{\prime}z^{\prime}\}\) is established and fixed on the part.  
Figure 1: Mobile robotic assembly system.  
Figure 2: Kinematic scheme.  
In this work, it is assumed that the robot links are rigid and that the joints can be modeled as linearly elastic torsional springs with small deformations in series with the joint actuators [34]. To evaluate the elastic displacement of the tool, we develop a structural model of the robot to analyze the forced vibrations of the joints in a given configuration denoted by the input joint angles as \(\mathbf{q}=[\theta_{1}\), \(\theta_{2}\),..., \(\theta_{n}]^{T}\) as follows [28]:  
\[\mathbf{M}(\mathbf{q})\Delta\mathbf{\ddot{q}}+\mathbf{C}(\mathbf{q})\Delta \mathbf{\dot{q}}+\mathbf{K}\Delta\mathbf{q}=\mathbf{J}_{v}^{T}\mathbf{F}+ \mathbf{J}_{w}^{T}\mathbf{\tau} \tag{1}\]  
where \(\Delta\mathbf{q}\) is the elastic joint deflection from their equilibrium values \(\mathbf{q}_{c}\) and represents the torsional deflections of the joints, \(\mathbf{M}(\mathbf{q})\) is the \(n\times n\) symmetric generalized inertial matrix and is configuration dependent, \(\mathbf{K}\) is the diagonal stiffness matrix with the diagonal entries equal to the torsional stiffness of the corresponding joints, and \(\mathbf{C}\) is a damping matrix. For a complete formulation, matrix \(\mathbf{C}\) should include the effect of the Coriolis and centrifugal terms related to the rigid-body motion on the elastic deformations. But this work is mainly focused on the robotic percussive riveting operations where the robot moves to a required spot and then stops there during the riveting. In other words, the robot has no rigid-body motion when its end-effector is subject to dynamic loading. Thus, it can be assumed that the coupling between the vibration and the rigid-body motion can be neglected, and then matrix \(\mathbf{C}\) here only includes the structural damping. The \(3\times n\) matrices \(\mathbf{J}_{v}\) and \(\mathbf{J}_{w}\) can be obtained from the Jacobian matrix \(\mathbf{J}\) that describes the mapping from joint rates to the twist of the end-effector as below:  
\[\mathbf{J}=\left[\begin{array}{c}\mathbf{J}_{v}\\ \mathbf{J}_{w}\end{array}\right] \tag{2}\]  
where \(\mathbf{J}_{v}=\frac{\partial\mathbf{v}}{\partial\mathbf{q}}\), \(\mathbf{J}_{w}=\frac{\partial\mathbf{w}}{\partial\mathbf{q}}\) in which \(\mathbf{\dot{q}}\) is the velocity of the tool tip, and \(\mathbf{\omega}\) is the angular velocity of the tool. We assume that the joint deflections are small, then the linear displacement of the tool tip can be obtained as:  
\[\Delta\mathbf{E}=\mathbf{J}_{v}\Delta\mathbf{q} \tag{3}\]  
From the perspective of process analysis, Equation (3) gives the tool misalignment due to the structural deformation caused by the process force and torque acting on the robot.