# State of the Art: Bipedal Robots for Lower Limb Rehabilitation
## 2 Bipedal Locomotion
### Stability Control  
Stability control is an important part of bipedal locomotion. When people walk on uneven surfaces or are disturbed, the upper limbs and lower limbs ware adjusted accordingly to maintain body balance. Due to the role of inertia, the robot's state is very unstable when it is influenced by outside interference. To allow bipedal robots to be as stable and balanced as humans, researchers created stability control based on sensory reflexes, including the zero moment point (ZMP) reflex, the landing-phase reflex, and the body-posture reflex [52]. The three sensory re-flexes are independent and work when their conditions are satisfied. * ZMP Reflex  
ZMP is an abbreviation for zero moment point. ZMP reflex does not work when ZMP is in a stable area. The ZMP reflex will not be active until ZMP is at a stable area boundary and exceeds the stability range. By changing the corner of the ankle joint, the ZMP can be returned to the stable range. Therefore, the use of ZMP reflex can ensure that ZMP is always in a stable area during walking. The angle of the supporting ankle can be calculated by Equations (1) and (2) [53]:  
\[\Delta\theta_{a}(nT_{s})=\sum_{j=1}^{j=n}\delta\theta_{a}(jT_{s}) \tag{1}\]  
\[\delta\theta_{a}(jT_{s})=\left\{ \begin{array}{c}K_{ac}*d_{ZMP}(jT_{s}),\ F_{ z}>0\\ -K_{as}\Delta\theta_{a}((j-1)T_{s}),\ \ F_{z}=0\end{array} \right. \tag{2}\]  
where \(T_{s}\) is the sampling period, and \(nT_{s}\) is current time. \(d_{ZMP}(jT_{s})\) is the distance between the actual ZMP position and the expected stable area boundary. \(K_{ac}\) and \(K_{as}\) are coefficients. \(F_{z}\) is the force between the supporting foot and the ground. * Landing-Phase Reflex  
When the ground level is uneven, the contact time between the foot and the ground will be corresponding in advance or delay. If the gait is not corrected in time, the robot is likely to fall. Landing-phase reflex can effectively solve this problem. Landing-phase reflex will increase the height  
Figure 3: The mechanical structure of the bipedal locomotion. (**a**) The classic mechanical structure with 12 degrees of freedom; (**b**) The rotating hinge structure. of the leg lift when the contact time between the foot and the ground is advanced. The increase in the height of the leg lift can be calculated according to the Formulas(3) and (4) [53]:  
\[\Delta Z_{f}(nT_{s})=\sum_{j=1}^{j=n}\delta Z_{f}(jT_{s}) \tag{3}\]  
\[\delta Z_{f}(jT_{s})=\left\{ \begin{array}{c}K_{fc}*F_{z},\ 0<F_{z}<Mr\\ -K_{fs}\Delta Z_{f}((j-1)T_{s}),\ F_{z}\geq Mr\end{array} \right. \tag{4}\]  
Similarly, when the contact time between the foot and the ground is delayed, the height of leg lift will be reduced accordingly, as per Equations (5) and (6) [53]:  
\[\Delta Z_{f}(nT_{s})=\sum_{j=1}^{j=n}\delta Z_{f}(jT_{s}) \tag{5}\]  
\[\delta Z_{f}(jT_{s})=\left\{ \begin{array}{c}-h,\ F_{z}=0\\ -K_{fs}\Delta Z_{f}((j-1)T_{s}),\ F_{z}\geq Mr\end{array} \right. \tag{6}\]  
where \(K_{fc}\), \(K_{fs}\) are coefficients, and \(M_{r}\) is estimated body weight of robot. \(h\) is a constant variable. \(F_{z}\) is the force between the supporting foot and the ground. The change of foot height \(\Delta Z_{f}(nT_{s})\) is achieved by controlling the hip and knee joints. \(\bullet\) Body Posture Reflex  
Changes in body posture also affect bipedal locomotion. Reasonable body posture contributes to the stability of the system. When the robot is tilted due to outside interference, the body posture deviates from the expected target. The role of the body posture reflex is to change the hip to adjust the body posture. This ensures the body posture remains in a reasonable range. From the current actual posture to the target posture, the angle of the hip can be calculated from Equations (7) and (8) [53]:  
\[\Delta\theta_{h}(nT_{s})=\sum_{j=1}^{j=n}\delta\theta_{h}(jT_{s}) \tag{7}\]  
\[\delta\theta_{h}(jT_{s})=\left\{ \begin{array}{c}\Delta\theta_{actb}(jT_{s} ),\ F_{z}>0\\ -K_{hs}\Delta\theta_{h}((j-1)T_{s}),\ F_{z}=0\end{array} \right.