## 3 Spherical Robotic Positioning Device
### 3D-Modeling  
Although the use of a spherical configuration for the first two frames is a good design choice for improving the physical safety and movement efficiency of the end-effector, deflection can be produced easily as in the case of hinges when two spherical frames are stretched. In addition, when an external force is applied to the device, that is, when the device is decelerated and is stopped, vibration problems can occur in a spherical configuration. These mechanical characteristics reduce the accuracy of the device. Therefore, for reducing error produced by hardware and for achieving sufficient strength and stiffness, the thicknesses and shapes of the spherical frames are decided based on two interpretations:The analysis related to the deflection and stress distribution and modal analysis using the finite element method (FEM). Two interpretations are performed in a configuration that the two spherical frames are stretched completely, as shown in Figure 3, because the closer the configuration of the serial mechanism to a beam is, the larger the deflection that is produced. For performing FEM, all parts are considered as a rigid body.  
The simulations of deflection and stress distribution are performed in conditions that the base frame is fixed on the ground, material of the rigid body is set as aluminum 6061, and an external force with 30 N is applied at the center of the end-effector, which is sufficient force magnitude because the desired payload is 2 kg. The conditions except the external force are applied to the simulation of modal analysis. The result is presented in Figure 3a. The deflection at the red point of the end-effector is 0.664 mm, and the maximum stress distribution is \(1.901\times 10^{7}\) N/m\({}^{2}\). Because the tensile strength of aluminum 6061 is 55.2 MPa, the developed device has adequate strength in the given conditions. The configuration has the natural frequency of 18 Hz, which was thought to be enough because the system will be controlled to move slowly at most a few tens of centimeters.  
The robotic positioning device was manufactured, as shown in Figure 4. All revolute joints in the device are built with the combination of a harmonic drive, an incremental encoder, a proximal sensor, and an actuator. The gear ratios of all harmonic drives are 100:1. The prismatic joint is composed of a ball screw joint and an actuator. The pitch of ball screw joint is 1 mm, and its stroke length is 150 mm. The power, nominal speed and nominal torque of the actuators installed on the first and second revolute joints from the ground are 100 W, 4390 rpm and 222 mNm, respectively. Those of the actuators at the prismatic joint and the last three revolute joints are 70 W, 4860 rpm, and 128 mNm and 30 W, 2940 rpm, and 54.8 mNm, respectively. In addition, the first and second joints are equipped with power-off braking systems to cope with emergencies. The weight of the final positioning device is about 15 kg.  
The workspace of the proposed device is shown in Figure 4b under conditions in which the last three revolute joints are fixed and the prismatic joint is moved between 0 mm and 100 mm from the initial position.  
Figure 3: (**a**) Result of deflection. The deflection at the red point is 0.664 mm and (**b**) Result of stress distribution. The maximum stress distribution is \(1.901\times 10^{7}\) N/m\({}^{2}\).