# A Review of Foot-Terrain Interaction Mechanics for Heavy-Duty Legged Robots
## 2 Supporting Feet of Heavy-Duty Legged Robots
### Spherical Supporting Foot Configurations  
In 2010, the DFKI Robot Innovation Center at the University of Bremen designed a highly adaptable free-climbing robot (Space Climber) for the steep slopes of lunar craters. The robot feet are mechanisms similar to eagle claws, as shown in Figure 12a. The feet can effectively improve the terrain adhesion ability of the detector [49]. In 2012, to overcome the limited mobility of detectors in unstructured environments, such as obstacles, normal steep slopes, and steep slopes of fine-grained soil, the Space Climber feet were changed to spherical shapes [50]. The updated robot feet are shown in Figure 12b. The spherical feet can withstand collisions with hard surfaces or obstacles and provide an increased contact area when sinking into the soil. The spherical feet have the natural advantage of having the same tangential performance in different directions of motion. Figure 11: Robots of LS3 and SCalf series: (**a**) LS3 [45]; (**b**) SCalf-I Robot [46]; (**c**) SCalf-II Robot [47]; and (**d**) SCalf-III Robot [48]. Figure 10: HexbotIV robot: (**a**) prototype [43]; (**b**) parallel mechanism leg [44]. TITAN XIII is a quadruped robot, as shown in Figure 13a. Each leg has three degrees of freedom (DOFs). The body mass is 5.65 kg, and the load is 5.0 kg. Its feet shapes are spherical. It can move on rough and irregular terrains by selecting suitable footholds and changing the robot's posture [51]. Ohtsuka S invented terrain-adaptive feet for the TITAN XIII robot. Its feet can passively adapt to rough terrain, including bumps and tilts, while ensuring a stable foothold [52]. The SCOUT II robot [53] is a quadruped robot that can achieve a jumping gait, as shown in Figure 13b. Each leg has two degrees of freedom: a driving hip joint and a linear spring. The stable mobile control strategy can be achieved when it walks at a maximum speed of 0.9 m/s to 1.2 m/s. The dynamically stable legged robots lay greater emphasis on their terrain adaptability, making their movement more stable. The feet of the robot adopt hemispherical shapes to achieve the point contact between the foot and the terrain, which improves the stability of the robot in different terrains. Based on the above statement, it can be very easy to come to a conclusion. The advantages of the spherical foot are that the mechanical structure design is simple, the tangential forces in all directions are equal, and it is very suitable for walking in soft soil or deserts. Meanwhile, the disadvantage of the spherical foot is that the contact between the foot and the terrain is point contact, with a relatively small contact area and low terrain friction, which is not conducive to the robot's smooth walking. When the velocity angle of the foot tip coincides with the attitude angle, the stress distribution models of the spherical foot can be obtained, as shown in Equations (2) and (3). Then  
\[F_{\text{N}}=\left(K\delta^{m}(t)+C\delta^{n}(t)\delta^{p}(t)\right)\sin\eta \tag{2}\]  
Figure 12: Space Climber: (**a**) original robot [49]; (**b**) updated robot [50]. Figure 13: Robots of TITAN-XIII and SCOUT II: (**a**) TITAN-XIII [51]; (**b**) SCOUT II [53]. \[F_{\rm T}=\left\{ \begin{array}{l}\mu F_{\rm N}\\ -\left(K\delta^{m}(t)+C\delta^{n}(t)\dot{\delta}^{p}(t)\right)\cos\eta\geq\mu F_ {\rm N}\\ \\ F_{\rm T}=\left\{\begin{array}{l}-\left(K\delta^{m}(t)+C\delta^{n}(t)\dot{ \delta}^{p}(t)\right)\cos\eta\\ -\left(K\delta^{m}(t)+C\delta^{n}(t)\dot{\delta}^{p}(t)\right)\cos\eta<\mu F_ {\rm N}\\ \end{array}\right.\end{array} \right. \tag{3a}\] \[F_{\rm T}=\left\{ \begin{array}{l}-\left(K\delta^{m}(t)+C\delta^{n}(t)\dot{ \delta}^{p}(t)\right)\cos\eta\\ -\left(K\delta^{m}(t)+C\delta^{n}(t)\dot{\delta}^{p}(t)\right)\cos\eta<\mu F_ {\rm N}\\ \end{array} \right. \tag{3b}\]  
where \(K\) is the spring coefficient, \(C\) is the damping coefficient, \(\mu\) is the friction coefficient, \(m\) and \(q\) are the parameters to be identified, and \(n\) is the model parameter.