## 1 Introduction
### Kinematic Analysis of the Spatial RRURR Electric Releasing Mechanism
The input performance index of each chain can be solved separately [16; 17; 18]:  
\[\gamma_{I}=\frac{|\mathfrak{S}_{T}\circ\mathfrak{S}_{I}|}{|\mathfrak{S}_{T} \circ\mathfrak{S}_{I}|_{max}} \tag{6}\]  
where \(\mathfrak{S}_{I}\) is the input motion screw of each chain, \(\mathfrak{S}_{T}\) is the transmission-force screw of the corresponding chain, and \(|\mathfrak{S}_{T}\circ\mathfrak{S}_{I}|\) represents instantaneous power. The input motion/force transmission performance index of a parallel mechanism is expressed as the minimum value of those among all its chains. Figure 8: Difference between a parallel mechanism with two sub-chains and a single-degree-of-freedom single-loop serial mechanism. (**a**) Schematic diagram of the motion/force transmission path of a parallel mechanism, including its corresponding input drives and output. (**b**) Schematic diagram of the motion/force transmission path of a single-degree-of-freedom single-loop serial mechanism, including its corresponding input drive and output joint. Unlike parallel mechanisms, a single-degree-of-freedom single-loop serial mechanism has only one input drive and transmits motion/force to the end joint for output, as shown in Figure 8b. To apply the motion/force transmission performance index of a parallel mechanism to the motion/force transmission performance index of a single-degree-of-freedom single-loop mechanism, three equivalent matching methods are proposed, as follows:  
1. A single-degree-of-freedom single-loop serial mechanism can be considered a special parallel mechanism with two sub-chains and one input drive through choosing a connecting bar to be the moving platform, as shown in Figure 9a. The input drive, \(R_{A}\), transmits the motion/force to the moving platform through sub-chain 1. 2. To transmit the motion/force to the output joint, \(R_{N}\), an equivalent method for the moving/fixed platform inversion is proposed, as shown in Figure 9b. After inversion, motion/force is transmitted from the new fixed platform to the output joint, \(R_{N}\), through the inverted sub-chain 2. 3. To make both sub-chains have drive, an equivalent method for chain-driving transference is proposed, generating a virtual drive, \(R_{I+1}\), through the kinematical solution of the loop. 1.1 Conceptual Framework for Constructing the Overall Motion/Force Transmission Performance Index of the Single-Degree-of-Freedom Single-Loop Mechanisms  
The conceptual framework for constructing the overall motion/force transmission performance index of a single-degree-of-freedom single-loop mechanism is shown as follows:  
1. Choose a connecting bar to be the moving platform and split the single degree-of-freedom single loop serial mechanism into two sub-chains. This mechanism can be regarded as a special parallel mechanism, which has two sub-chains and one input drive. The motion/force is transmitted to the moving platform by the input drive through sub-chain 1, as shown in Figure 9a;
2. Inverting the moving/fixed platforms, joint \(R_{\text{I+1}}\) becomes the drive of inverted sub-chain 2. Joint \(R_{\text{I+1}}\) transmits the driving force to the new moving platform through the inverted sub-chain 2, which is the output of the end of the serial single-loop mechanism, as shown in Figure 9b;  
Figure 9: Consideration of a single-degree-of-freedom single-loop serial mechanism as a special parallel mechanism, constructing the overall motion/force transmission performance index through moving/fixed platform inversion and driving transference. (**a**) Schematic diagram of motion/force transmission path after splitting of a single-degree-of-freedom single-loop serial mechanism into two sub-chains. (**b**) Schematic diagram of motion/force transmission path after inverting the moving/fixed platforms and transferring drive. Driving transference is shown in Figure 9b. The motion/force is transmitted along sub-chain 1 and the moving platform to joint \(R_{\text{I+1}}\).