## 2 The Development of Typical Rotating Motion Mechanisms
### Linkage Mechanism  
#### Four-Bar Linkage  
1. Crank-rocker mechanism  
The planar crank rocker linkage mechanism is widely used to convert continuous rotational motion into oscillatory motion [95]. The crank rocker mechanism mainly comprises four components: crank, connecting rod, rocker, and frame [96]. The crank can also rotate under certain conditions when the joystick is used as the active component, as shown in Figure 41. The crank rocker mechanism has the advantages of a simple structure, strong load-bearing capacity, and high reliability in converting rotational motion. It is widely used in fields such as internal combustion engines [97; 98], stamping machines [99], and biomimetic machinery [100]. However, its disadvantages of discontinuous motion, vibration noise, and ample space occupation limit its application in high speed and high precision scenarios. [PERSON] et al. [102] designed and constructed a novel crank rocker mechanism based on a coil spring, which utilises a long free-rotating arm with one end fixed and the other end mounted with an airfoil and provides the necessary sine drive through a crank rocker mechanism based on a coil spring. Figure 42[102] is a schematic diagram of the mechanism, and Figure 43[102] is the experimental platform of the mechanism. The mechanism is based on a four-bar crank rocker mechanism, where the crank is connected to the rocker (oscillating arm) through a non-extendable steel rope, and external torque is provided through a coil spring. The rotation of the crank is transmitted to the rocker through a steel rope, causing the rocker to produce an approximately sinusoidal oscillation motion. The spiral spring design ensures the mechanism's light weight and reliability during long-distance transmission. By using spiral springs and carbon fibre materials, the inertia of the mechanism is reduced, and the response speed is improved. Capable of precise control of oscillation frequency and amplitude, suitable for various experimental conditions. 2. Hyperbolic handle mechanism  
The double crank mechanism consists of two cranks, two connecting rods, and a frame. Both cranks can perform a full rotation motion. When the driving crank rotates  
Figure 41: Schematic diagram of crank rocker mechanism [101]. Figure 42: Geometric representation of the mechanism. M and M\({}^{\prime}\) represent the crank position, while P and P\({}^{\prime}\) represent the corresponding rocker positions at the initial time and time \(t\). Figure 43: **(a)** The top view of the mechanism is fixed on the test bench for characterisation; **(b)** the motor with crank is installed; **(c)** the rocker arm/swing arm is connected to the airfoil through a shaft. The illustration shows the spiral spring inside the spring housing. at a constant speed, the driven crank rotates at a variable speed. The motion condition is that the shortest rod is the frame, and both connecting rods (cranks) can make a full rotation [103]. The structural diagram is shown in Figure 44.  
3. Elastic inside link [PERSON]. [105] proposed a new concept of a flexible rotary joint with low axial drift, high support stiffness, and an extensive range of motion, as shown in Figure 45. This concept is based on a spiral shell, with a portion rotating along its rotational direction. The opposite region gradually increases, resulting in a constant reaction torque. The prototype of this concept has been used to demonstrate the ability of various neutral and stable flexible linkages that can exhibit a wide range of motion with extremely low driving forces. 2.6.2. Double Spherical Linkage Mechanism [PERSON] et al. [106] were inspired by [PERSON] and proposed a super-constrained double ball linkage mechanism, as shown in Figure 46. The 6R over-limit linkage mechanism can be composed of a crank or a hyperbolic crank linkage. The connecting rod and rotating joint replace the panel and the crease increase mode. Figure 44: Schematic diagram of hyperbolic handle mechanism [104].