Locomotion and terramechanics for rough terrain

A field robot is only useful if it can get to the work. Mobility on soft soil, snow, mud, rock and slopes is governed by terramechanics, the study of how vehicles and terrain interact, and by the geometry of stability. This lesson introduces the classical Bekker approach to sinkage and resistance, the Mohr-Coulomb model of soil strength that limits traction, the static stability of vehicles on slopes, and the trade-offs between wheels, tracks and legs. The models are simplified and the soil values are illustrative; real designs use measured soil parameters and testing on representative ground. ROB-340 lesson 1 covered mobile robot platforms and odometry on firm floors; here the ground itself becomes part of the problem.

Ground pressure: the first check

The simplest measure of how a vehicle will behave on soft ground is its nominal ground pressure, the weight divided by the contact area. A person standing exerts about 15 to 20 kPa; a loaded agricultural tractor on standard tires often exceeds 100 kPa; tracked vehicles designed for snow or bogs may exert 10 to 30 kPa. Low pressure means less sinkage and less soil compaction, which matters to farmers because compacted soil reduces crop yields for years. Ground pressure is a crude measure, because pressure is not uniform under a tire or track and soils respond nonlinearly, but it is the right first check when choosing a platform.

The Bekker pressure-sinkage relation

In the 1950s and 1960s M. G. Bekker, working on military and later lunar vehicles, proposed an empirical relation between the pressure \( p \) under a plate and its sinkage \( z \) into soil:

Here \( b \) is the smaller dimension of the contact patch (the width of a wheel), \( k_c \) and \( k_\phi \) are cohesive and frictional moduli of deformation, and \( n \) is an exponent, often between about 0.5 and 1.3. The parameters are found by pressing plates of different widths into the soil with a bevameter and fitting the curves. Bekker's work fed directly into the design of the Lunar Roving Vehicle driven on the Moon by Apollo astronauts in 1971 and 1972.

For a rigid wheel of diameter \( D \) and width \( b \) carrying a vertical load \( W \), integrating the pressure around the contact arc gives an approximate sinkage

and a compaction resistance, the horizontal force needed to keep pushing the wheel through the rut it is making,

These formulas show the main design levers. Sinkage falls as wheel diameter and width increase, and the resistance falls rapidly with sinkage because of the power \( n + 1 \). They also explain why soft-soil vehicles often have large, wide wheels and why lowering tire pressure helps: a flexible tire flattens and behaves like a wider, longer contact patch.

# Bekker sinkage and compaction resistance of a rigid wheel (illustrative dry sand values)
import numpy as np
n, kc, kphi = 1.1, 0.99e3, 1528.43e3     # n, kc [N/m^(n+1)], kphi [N/m^(n+2)]
def sinkage(W, b, D):
    return (3 * W / ((3 - n) * (kc + b * kphi) * np.sqrt(D))) ** (2 / (2 * n + 1))
def resistance(W, b, D):
    z = sinkage(W, b, D)
    return b * (kc / b + kphi) * z ** (n + 1) / (n + 1), z
W = 500.0   # load per wheel in N (about 51 kg)
for b, D in [(0.10, 0.40), (0.20, 0.40), (0.20, 0.80), (0.30, 0.80)]:
    R, z = resistance(W, b, D)
    print(f"width {b:.2f} m  diameter {D:.2f} m  sinkage {z*1000:5.1f} mm  resistance {R:5.1f} N  R/W {R/W:.3f}")

width 0.10 m  diameter 0.40 m  sinkage  49.3 mm  resistance 132.0 N  R/W 0.264
width 0.20 m  diameter 0.40 m  sinkage  32.1 mm  resistance 106.4 N  R/W 0.213
width 0.20 m  diameter 0.80 m  sinkage  25.8 mm  resistance  67.5 N  R/W 0.135
width 0.30 m  diameter 0.80 m  sinkage  20.1 mm  resistance  59.5 N  R/W 0.119

Doubling the width cuts the rolling resistance coefficient by about a fifth in this example, from 0.264 to 0.213, and doubling the diameter as well brings it down to 0.135, about half the original value. The model is less accurate for small wheels, for high slip and for soils that are layered or saturated, but it remains a standard first estimate and a way to compare designs.

Soil strength and traction

Sinkage creates resistance; traction comes from the soil's shear strength. Terramechanics uses the Mohr-Coulomb model, in which the maximum shear stress the soil can carry is

where \( c \) is cohesion, \( \sigma \) the normal stress and \( \phi \) the internal friction angle. Integrating over a contact area \( A \) under total load \( W \) gives the maximum gross thrust \( H_{max} = A c + W \tan \phi \). The drawbar pull, the net force available to climb, tow or push, is thrust minus resistance: \( DP = H - R \). Dry sand has almost no cohesion, so thrust depends on weight and friction angle; clay has cohesion, so a larger contact area helps even when the vehicle is light. Thrust only develops with slip, the difference between wheel surface speed and vehicle speed, usually expressed as \( i = 1 - v/(r\omega) \). Janosi and Hanamoto's relation models the build-up of shear stress with shear displacement; in practice many rovers operate at 10 to 30 percent slip on loose ground and become stuck when slip approaches 100 percent.

Static stability on slopes

Even with enough traction, a robot can tip over. For a rigid vehicle standing still on a slope, the static stability limit is reached when the line of action of the weight through the center of mass passes outside the support polygon formed by the wheel or foot contacts. For a vehicle with track width \( t \) and center of mass at height \( h \) above the ground, centered between the wheels, the lateral tip-over angle is

The longitudinal limit uses the horizontal distance from the center of mass to the front or rear axle instead of half the track. Dynamic effects such as turning, bumps and payload sway reduce the usable angle, so designers apply a margin, and many vehicles monitor roll and pitch with an inertial measurement unit and stop or reverse when a threshold is approached. Stability measures such as the force-angle stability margin of Papadopoulos and Rey extend the idea to moving robots and manipulators carrying loads.

# Static tip-over angles and a stability margin check
import numpy as np
def tip_angles(track, wheelbase, h, x_cg_from_rear):
    lateral = np.degrees(np.arctan((track / 2) / h))
    downhill_front = np.degrees(np.arctan((wheelbase - x_cg_from_rear) / h))   # nose down
    downhill_rear = np.degrees(np.arctan(x_cg_from_rear / h))                  # nose up
    return lateral, downhill_front, downhill_rear
designs = {"low and wide": (0.90, 1.10, 0.35, 0.55),
           "tall sensor mast": (0.90, 1.10, 0.70, 0.55),
           "rear-heavy payload": (0.90, 1.10, 0.45, 0.35)}
margin = 0.5   # allow only half the static angle in operation (illustrative policy)
for name, (t, L, h, x) in designs.items():
    lat, fwd, back = tip_angles(t, L, h, x)
    print(f"{name:18s} lateral {lat:4.1f}  nose-down {fwd:4.1f}  nose-up {back:4.1f}  "
          f"operating limit {margin*min(lat, fwd, back):4.1f} deg")

low and wide       lateral 52.1  nose-down 57.5  nose-up 57.5  operating limit 26.1 deg
tall sensor mast   lateral 32.7  nose-down 38.2  nose-up 38.2  operating limit 16.4 deg
rear-heavy payload lateral 45.0  nose-down 59.0  nose-up 37.9  operating limit 18.9 deg

Raising the center of mass by mounting a tall sensor mast cuts the lateral limit by about 20 degrees, and moving payload rearward makes nose-up tipping on a climb the critical case. Field robots that need tall sensors often fold them down for slopes, or carry batteries low in the chassis to compensate.

Wheels, tracks, legs and hybrids

Wheels are efficient, mechanically simple and fast on firm ground, and most field robots use them. Skid-steer wheeled platforms are robust but scrub their tires when turning, which wastes energy and damages turf and crops; Ackermann steering and independently steered wheels avoid this at the cost of complexity. Rocker-bogie suspensions, used on NASA's Mars rovers since Sojourner, keep all six wheels in contact over obstacles without springs, so a rover can climb rocks roughly the size of its wheel diameter while keeping the body level.

Tracks spread load over a long contact length, giving low ground pressure and good traction in snow, mud and sand, and they cross gaps and ditches well. They are heavier and less efficient than wheels on hard ground, they wear and can be thrown off, and skid steering a track causes high turning resistance and ground damage. Rubber tracks are now common on agricultural and utility robots.

Legs choose their footholds, so they can step over obstacles, climb stairs and cross terrain that would stop a wheel. Quadruped robots such as Boston Dynamics' Spot and ANYbotics' ANYmal are used commercially for inspection at industrial plants, offshore platforms and mines. Legs are more complex and less energy efficient over distance, they need sophisticated control, and their small feet can sink in soft ground. Hybrid designs add wheels to legs, combining efficient rolling with the ability to step, and some planetary concepts use wheel-walking, in which wheels and suspension joints are moved in sequence to inch through soft soil where ordinary driving would dig in.

Terrain assessment closes the loop between perception and mobility. Robots estimate traversability from geometry (slope, step height, roughness from lidar or stereo), from appearance (classifying grass, sand, gravel and water with learned models) and from proprioception (motor current, wheel slip and vibration while driving). Learned traversability models trained on the robot's own driving experience have become common, but they must be validated for the seasons and soils the robot will meet. When a rover detects high slip, it can stop, reverse and choose another route, as the Mars rovers do after Spirit became permanently embedded in soft soil in 2009.