# Design and Experimental Development of
## 2 Working Mechanism of the Stiffness-Adjustable Foot System
### Theoretical Analysis of the Variable Stiffness Mechanism  
The PVSU is the most crucial component of the whole foot system because the function of the foot depends on the interactions of each PVSU with the environment. This section introduces the variable stiffness mechanism of a single PVSU. Variable stiffness mechanisms have been more and more widely utilized in vibration isolation devices [27], humanoid robots [28], and rehabilitation devices [29]. Most recent work of variable stiffness mechanisms rely on two techniques. One is to alter the structural or mechanical geometry of an elastic mechanism [30], and the other is to change the elastic modulus of a structure by thermal or electromagnetic stimulation [31]. In this paper, the stiffness-adjusting mechanism of the PVSU is inspired by pneumatic artificial muscles, of which the stiffness can be changed by regulating the air pressure inside the muscles. Figure 3 illustrates the working principle diagram of a single PVSU interacting with the environment. The PVSU includes two major parts: a pneumatic bladder and a stiffness-adjusting mechanism. The stiffness-adjusting mechanism works based on a two-position three-port mechanical reversing valve. The PVSU is mounted underneath the foot sole through the reversing valve. The valve piston is connected with one end of the supporting spring, and the other end of the supporting spring is fixed on the bottom of the pneumatic bladder. The trigger signal of the reversing valve is the external force applied on the pneumatic bladder by the bump. In the normal position, the supply port of the reversing valve is connected with the pneumatic bladder. To trigger the valve, the external force needs to overcome the friction, the restoring spring, as well as the air pressure inside the pneumatic bladder. Once the trigger signal is off, the loaded restoring spring returns the valve to its original position. The theoretical working process of the PVSU is as follows. In the swing phase of walking when there is no contact between the ground and the foot, the reversing valve is in normal position, supplying compressed air into the pneumatic bladder to keep it stiff. After transferring to the stance phase, the pneumatic bladder is compressed once touching the bump on the ground, and the valve piston is driven to move upward. As soon as the piston reaches the working position, the valve is triggered to expel the air out of the pneumatic bladder. Then, the pneumatic bladder becomes soft and generates more compression to comply with the shape of the bump. When the PVSU interacts with the bump, compressions can be generated on any one of the restoring spring, supporting spring and pneumatic bladder. According to which elements are compressed, the entire compression process can be divided into three phases. During these phases, the PVSU has different configurations of series and/or parallel elastic elements, and exerts different stiffness values. Prior to analyzing the stiffness in each phase, several assumptions are made to simplify the modeling problem by referring to the mathematical models of air-supported structures and air springs [32; 33; 34], specifically, (1) the foot sole, the ground and bumps are perfectly rigid and cannot generate deformation under external forces; (2) the damping effects of the elastic elements are negligible considering that the damping factors are much smaller than stiffness factors [35]; (3) the  
Figure 3: Working principle diagram of the pneumatic variable stiffness unit (PVSU). compression of the PVSU is taken as a quasi-static behavior; (4) the static pressure inside the bladder is assumed to remain stable during each phase; and (5) the inflation and deflation, in other words, the changes of air pressure inside the pneumatic bladder are completed instantaneously. The first phase starts when the pneumatic bladder touches the bump and ends when the piston arrives at its working position. During this phase, the restoring spring, supporting spring, and the inflated pneumatic bladder are compressed simultaneously.