Gated communities: Estados Unidos

The possible models for leg-terrain shear interaction depending on the rigidity of both the foot and the terrain are described in [116]. The interaction of the wheel-leg with soil studied in this research would normally fall into the category of ‘rigid foot-deformable terrain’. This motivated the application of Terramechanics and Terradynamics principles to analytically model the normal penetration and shearing of the legs into the soil as presented in Chapter 4. However, during the post-midstance propulsion phase of the SLS, the rotation of the leg over critically compacted soil can lead to abrupt stick-slip phenomena characteristic of a ‘rigid foot-rigid terrain’ interaction. Such events are not appropriately modelled by the previously mentioned Terramechanics and Terradynamics principles. A variety of models to describe static and dynamic friction regimes in this type of contact have been studied, as presented in [170], with renovated interest in recent years due to the role of dynamic phenomena in engineering applications and to the difficulty of satisfactorily explaining all aspects of such static-dynamic frictional interactions.

Since the main objective of this research is to develop on-line algorithms for wheel-leg-soil interaction analysis, computational efficiency is prioritised over model accuracy. Hence, the contact force between leg and soil during this phase is modelled as a point contact, applying the classic generalised non-linear spring-damper model proposed by Hunt and Crossley [171] to rule the normal contact force (F_N) with Eq. (5.10). Given the effective absence of cohesion and lubrication and the relatively low speeds under consideration, the model is simplified to a linear Hertzian contact by assuming b_HC = 0 and n_HC = 1.

5.1. Wheel-Leg Slip Detection Approaches

F_N = k_HCδnHC + b HCδ

p_˙δq nHC=1,bHC=0

−−−−−−−−−→ F_N = k_HCδ (5.10)

The tangential force can then be modelled by introducing stiction into the classical Coulomb friction model. This implies that during the static regime, i.e. when v_δ = ˙δ = 0 , the force is equal to the external tangential force (FE ) in the opposite direction. When this external

force overcomes the break-away force determined by the normal load and the static friction coefficient (µS), a transition takes place to the dynamic regime, during which the contact

point slips according to the force balance acting on it.

The function ruling the tangential reaction of the terrain in this dynamic regime can take a variety of arbitrary forms depending on the application, possibly varying with velocity to account for viscous effects or to eliminate discontinuities by taking into account the Stribeck effect. However, for the sake of simplicity, and taking into account the previously mentioned assumptions, the dynamic friction is considered to be proportional to the normal force by the dynamic friction coefficient (µ_D). This parameter takes a lower value than the static friction coefficient, producing a discontinuous behaviour of the tangential reaction force (F_{T C}) reflected in Eq. (5.11) that leads to the stick-slip phenomena under study.

FT C =          F_E v = 0 and |F_E| ≤ µ_SF_N µDFNsgn(v) v 6= 0 µ_DF_Nsgn(F_E) otherwise (5.11)

The system is modelled as two bodies, representing the wheel-leg rotation hub with mass MH

and the contact foot with mass m_F. Both bodies are linked by a linear spring with rigidity coefficient kS and uncompressed length lS. This spring represents the combination of contact

soil stiffness and leg compliance, hence adopting the uncompressed length of a wheel-leg’s spoke. As previously noted, the compliance of the wheel-leg used for experimentation is negligible under the loads considered. Therefore, the combined stiffness of the modelled spring can be approximated as fully due to terrain contact compliance.

5.1. Wheel-Leg Slip Detection Approaches

During the static friction regime, also known as stiction, the body representing the contact foot remains fixed to the terrain, only being able to rotate as the leg angle grows beyond midstance in the propulsive SLS phase. This configuration is depicted in Fig. 5.3 (a). As the leg rotates, the position of the wheel-leg hub relative to the foot ([x_H, z_H]) varies accordingly to this rotation and the compression of the spring under the weight attached to the hub (WH = MHg). The force equilibrium on the foot produces a normal force and

an external tangential force due to the reaction from the terrain depending on the spring force (FS) and the angle of the leg.

As long as the external tangential force remains below the static friction force threshold, the wheel-leg stays in this regime. However, as soon as this threshold is surpassed the wheel-leg switches to the dynamic regime depicted in Fig. 5.3 (right). The foot is now not only able to rotate, but also slides horizontally according to the tangential force balance expressed in Eq. (5.12).

m_Fx¨_C = m_Fa_X,C = F_{T C}− F_Ssin θ_LW = F_S(µ_Dcos θ_LW − sin θ_LW) (5.12)

(a) (b)

Figure 5.3: Diagram of the simplified contact model for rotary leg stick-slip in the regimes of (a) static friction and (b) dynamic friction

5.1. Wheel-Leg Slip Detection Approaches

If the horizontal velocity of the foot is cancelled by this force balance (v_X,C = 0) the wheel- leg terrain contact returns to the stiction regime. In both regimes, the spring force can be calculated according to the vertical position of the hub and the relative horizontal position of hub and foot (xH − xC) by applying Eq. (5.13).

FS = kS  lS − q z2 S + x 2 S  = kS  lS− q z2 H + (xH − xC) 2  (5.13)

During simulation, the angular speed of the leg is fixed at ω = 0.63 rad/s, which is the angular speed used during experimentation to achieve 10 cm/s traversal speed in no-slip conditions. The horizontal speed of the wheel-leg hub (v_X,H) is calculated as a function of the given angular speed, the leg angle and the desired simulated slip as described in Sub- Section 5.1.1. The fact that these linear and angular speeds are kept constant imply that the external horizontal force and rotating torque applied on the hub constantly compensate the reaction horizontal force and torque produced by the foot-soil interaction.

The vertical motion of the wheel-leg hub is determined by the balance between the vertical projection of the spring compression force and the weight applied on the hub. Hence, the vertical motion of the hub is ruled by Eq. (5.14). The vertical acceleration (aZ,H) can be

measured by an IMU rigidly attached to the reference frame of the wheel-leg hub.

M_Hz¨_H = M_Ha_Z,H = W_H − F_Scos θ_LW (5.14)

Simulations were carried out using the parameters of the wheel-leg used during experimen- tation for slip levels in the 0-50% range. Soil parameters are varied to analyse the sensitivity of the model to spring stiffness, static friction angle (φ_S = arctan (µ_S)) and dynamic friction angle (φD = arctan (µD)). Each simulation starts in static regime at midstance (θLW = 0)

and finishes at θ_LW = π/n_L.

In each simulation step, the spring force is calculated according to Eq. (5.13) and then used to calculate FE and FN. If currently in the static regime, the external tangential force

5.1. Wheel-Leg Slip Detection Approaches

this threshold is overcome. In such case, the simulation switches to the dynamic regime. Otherwise, the static regime is maintained. Similarly, if the simulation is currently in the dynamic regime and the horizontal speed of the contact point is zero, the simulation returns to the static regime.

The tangential reaction force is calculated depending on the regime of the current and previous simulation steps according to Eq. (5.11). Thereafter, the vertical motion of the hub and, if in the dynamic regime, the horizontal motion of the contact point are updated. The simulation flow diagram is shown in Fig. 5.4.

As the tilt of the leg grows, the ratio between the normal and tangential spring forces on the terrain decreases, entering the dynamic regime. As the foot slips, the spring de- compresses down to the point where the dynamic friction with the soil returns the leg to static regime. The process reiterates as the spring regains enough compression, leading to oscillating vibrations.

Plotting the unbiased vertical acceleration of the wheel-leg hub against the leg angle, as shown in Fig. 5.5, reveals that the amplitude of said vibrations initially increases and then gradually decreases at higher leg angles. The maximum amplitude of these vibrations shows sensitivity both to the level of slip, as seen in Fig. 5.5 (a) and soil parameters of the stick-

Figure 5.4: Flow diagram of the simulation program of the rotating linear spring model with stiction for wheel-leg stick-slip phenomena.

5.1. Wheel-Leg Slip Detection Approaches

slip model, as shown in Fig. 5.5 (b) and (c). Therefore, this magnitude is a potentially suitable indicator of both slip and physical characteristics of the soil.

The relationship between the wheel-leg maximum vibration amplitude over a leg cycle (∆a = max a_Z,H
− min a_Z,H
) and slip can be modelled as the linear fit in the left-hand side of Eq. (5.15), with an average GoF of 0.84 for the simulated parameter ranges.

i(∆a) = a_I,V + b_I,V∆a →    bI,V = aµ+ bµ∆φ a_I,V = c_I + cµ∆φ + cKkS (5.15)

Moreover, the slope (aI,V) and intercept (bI,V) parameters of these linear fits can be related

to the soil parameters used in the simulation. The slope shows a highly linear correlation with the difference between the static and the dynamic friction angles (∆φ = φS − φD).

Regarding the intercept, the planar fit with both the friction angle increment and the spring stiffness as input variables has a high GoF, as shown in Table 5.2.

0 0.2 0.4 0.6 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Slip Influence θ_LW [rad] a z,h [g] i=0.50 i=0.25 i=0.125 0 0.2 0.4 0.6 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Spring Stiffness Influence

θ_LW [rad] az,h [g] k s=1.50 kN/m k s=1.00 kN/m k s=0.50 kN/m 0 0.2 0.4 0.6 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Friction Angle Influence

θ_LW [rad] az,h [g] φ_s=0.70 rad φ_s=0.52 rad φ_s=0.35 rad (a) (b) (c)

Figure 5.5: Acceleration vs. leg angle for (a) different slip levels, (b) different spring stiffness constants and (c) static friction angles