## 2 The Real 2-DOF CKCM  
This section describes the main components of a real 2-DOF CKCM from the Robotics and Intelligent Control Laboratory (RICL) of the School of Engineering at the Catholic University of America (CUA). This manipulator was built as a prototype to demonstrate the capacity of the high-precision motion of CKCMs for potential applications in space assembly. Figure 1 shows the manipulator, consisting of two links, each composed of a ball screw linear actuator driven by a direct current (DC) motor. The top end of each link is suspended below a fixed platform, and the links are connected to the platform by two pin joints acting as revolute joints. The other ends of the links are coupled together by the same type of revolute joint on which an end-effector is mounted. Two linear voltage differential transformers (LVDT) mounted along the links serve as position sensors to provide real-time information on the link lengths. Table 1 presents the manipulator parameters and the specifications of the ball screw actuators. The LVDT position sensors are precision linear variable differential transformers, encapsulated in a metal housing. An external Alternating Current (AC) source excites the primary, producing magnetic flux within the transducer. Consequently, this induces voltages in the two secondaries, whose magnitude varies with the position of the core. The secondaries are connected in series opposition, producing a total output voltage that is essentially zero when the core is at the electrical center of the unit and increases linearly when the core is displaced in either direction from the electrical center. The transducers can be excited at frequencies varying from 400 Hz to 10 kHz. Table 2 provides the main characteristics of the LVDTs. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Manipulator** & **Description** & **Value** & **Unit** \\
**Parameters** & **Description** & **Value** & **Unit** \\ \hline \(m\) & Link’s total mass & 4.91 & kg \\ \(m_{1}\) & Link’s moving part mass & 0.59 & kg \\ \(d\) & Ground’s horizontal distance & 0.74 & m \\ \(l_{s}\) & Link’s fixed length & 0.26 & m \\ g & Gravitational acceleration constant & 9.81 & m/s\({}^{2}\) \\  & Link’s minimum length & 0.84 & m \\  & Link’s maximum length & 1.22 & m \\ \hline
**Actuator** & **Description** & **Value** & **Unit** \\
**Specifications** & **Description** & **Value** & **Unit** \\ \hline \hline \end{tabular}
\begin{tabular}{c c c} \hline \hline  & Motor & Permanent magnet \\  & Voltage & 24 VDC & VDC \\  & Speed & 3000 RPM & RPM \\  & Diameter & 0.076 & m \\  & Stroke & 0.406 & m \\  & Gear reduction & 10:1 \\  & Max velocity & 1.78 & cm/second \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The manipulator parameters and actuator specifications. Figure 2 illustrates a block diagram of the manipulator control system. Interfacing with the outside world, consisting of the robot manipulator, sensors, amplifiers, and computer processing unit, is performed through a data acquisition system (DAQ). The DAQ in this system is a NI BNC 2120 DAQ board (National Instruments, Austin, TX, USA) that can support two analog inputs and two analog outputs by BNC connections. The voltage of the input and output channels can be selected to be either unipolar (0 to 10 volts) or bipolar (\(\pm\)5 volts or \(\pm\)10 volts). Following the [PERSON] sampling theory and standard control system design guidelines that recommend sampling at least 5-10 times the maximum system frequency [30], the DAQ sampling rate is set to 1 KHz by the compatible software package LabView in the processing unit. This sampling rate is selected because the dominant motion frequencies of our system are below 100 Hz. LabView, version 2019, a graphical programming environment, allows data acquisition and real-time analysis to be performed.