# Design and Control of a Pneumatic Muscle Servo Drive Applied to a 6-DoF Parallel Manipulator
## 2 Materials and Methods  
### Research Methodology  
A pneumatic muscle is a type of single-acting cylinder that, under the influence of increased internal pressure, increases its volume by expanding its diameter while simultaneously reducing its length. The change in cylinder length is treated as the working motion [1]. The primary physical quantities describing the behavior of the muscle in a steady state are the internal pressure \(p\), the muscle length \(x\), and the force generated by the muscle \(F\). Currently, pneumatic muscles are primarily offered by two companies: since 1987, the Shadow Robot Company has been involved in the production and sale of muscles based on the McKibben design, while in 1999, the well-known pneumatic equipment manufacturer Festo introduced to the market pneumatic muscles than can be customized for individual applications. They fundamentally differ in construction from the McKibben muscle. In the Festo design, the braid and rubber bladder are bonded along the entire length of the muscle. Figure 1 illustrates views of the pneumatic muscle developed by the authors at an internal pressure of \(p=0.6\) MPa and at zero internal pressure.  
The first stage of pneumatic muscle analysis often involves determining their static characteristics in the form of isobaric, isotonic, and isometric characteristics. The next stage is to determine the dynamic characteristics in the form of the muscle length-time relationship in response to a step change in internal pressure and with a constant muscle load force. The methodology for their determination and the characterization of muscles based on the McKibben design, Festo muscles, and many original designs have been presented, among others, as described earlier [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13]. In the article [6], [PERSON] and colleagues presented a research methodology and determined the static characteristics--isobaric, isotonic, and isometric--of Festo pneumatic (Esslingen am Neckar, Germany) muscles and Shadow Robot Company (London, UK) muscles.In publications describing static muscle models, research mainly focuses on determining the relationship between internal pressure, muscle length, and the force generated by the muscle in relation to the muscle's geometric parameters. Studies have also been conducted to determine dynamic models of [PERSON] and Festo muscles, as well as of other structures developed in various research centers, and on the development of position and force control methods using the presented mathematical models. These issues have been extensively discussed in works such as [14; 15; 16; 17; 18; 19]. Due to significant friction between the fibers of the braid and between the braid and the rubber bladder, hysteresis occurs in muscles based on the McKibben design. This phenomenon applies to both muscle shortening and  
Figure 1: The developed pneumatic muscle.  
the force generated by it. In the case of shortening, hysteresis manifests as the difference between the contraction value at a given internal pressure and constant load force when increasing the internal pressure and the contraction value at the same internal pressure and constant load force when decreasing the pressure. When analyzing the force generated by the muscle, hysteresis refers to the difference between the force produced at a given internal pressure and constant contraction when pressure increases and the force produced at the same internal pressure and constant contraction when pressure decreases. This issue is the subject of numerous studies, including [20; 21; 22; 23; 24; 25; 26].