Excavator and Bulldozer Lab: Hydraulics, Inverse Kinematics, Digging Force and Blade Push

Loading Blender-built machines...

tip - m, - m bucket breakout - kN stick - kN cycle - production -

stick outstick inswing Lswing R

boom upboom downcurldump

Excavator: kinematics and hydraulics

Drag the bucket in the 3D view (or the dot in the envelope diagram): the analytic inverse kinematics solves boom, stick and bucket; the cylinders move at the speed the pump flow allows.

Swing 0 Bucket tooth direction (absolute) - Boom angle θ1 - Stick angle θ2 - Bucket angle θ3 - Pump flow - Relief pressure -

Digging force and the dig cycle

Bucket penetration depth d - Engine power - Share of engine power to the hydraulics - Dump swing angle -

Bulldozer: drawbar pull and blade load

Cut depth while loading - Push distance D - Gear - Slope (uphill positive) -

Lift capacity and tipping

Swing relative to the tracks - Load hung from the stick pin: radius - Height of the hook above ground -

Rated lift = the smaller of 75 % of the tipping load and 87 % of the hydraulic lifting load (the common industry convention). Over the front the tipping edge is the track end; over the side it is the track side.

Flood levee: earthwork and slope

Levee height h - Crest width - Landside slope (H:1V) - Waterside slope (H:1V) - Levee length - Haul distance (one way) - Trucks in the fleet - Flood water level (fraction of h) -

Experiments: measured against theory

Every row compares two independent routes to the same number (for example a world-space cylinder length against the law of cosines, or a closed-form soil force against a force polygon solved by Gaussian elimination).

The science

Two generic machines, a teaching model. The crawler excavator (about 20 t) and crawler bulldozer (about 24 t) are generic designs modelled in Blender; no manufacturer, model or marking is represented, and all values are textbook or typical orders of magnitude, not data for any product. The topic is construction, flood-barrier building and disaster-relief earthworks (embankments, levees, drainage ditches). Everything here is educational mechanics; no weapons or military content is included.

Planar forward kinematics. With the boom foot at P₀ and absolute directions φ₁ = θ₁, φ₂ = θ₁+θ₂, φ₃ = φ₂+θ₃: tip = P₀ + L₁e(φ₁) + L₂e(φ₂) + L₃e(φ₃), L = 5.7, 3.0, 1.6 m. The swing turns this vertical plane about the machine axis.
Analytic inverse kinematics. Give the tip and the tooth direction φ₃; the wrist is W = tip − L₃e(φ₃). The law of cosines gives cosθ₂ = (|W−P₀|² − L₁² − L₂²)/(2L₁L₂), with the elbow-up sign (the normal machine pose; elbow-down would put the boom-stick joint below the line to the wrist); then θ₁ = atan2(W_z−P_0z, W_x−P_0x) − atan2(L₂sinθ₂, L₁+L₂cosθ₂) and θ₃ = φ₃ − θ₁ − θ₂. When the answer lies outside a cylinder's stroke, a damped least-squares solver with joint limits finds the nearest reachable pose and the page says which cylinder is at the end of its stroke.
Workspace. The envelope diagram is the set of tip positions swept by all joint angles inside their stroke limits (about 10.7 m reach at ground level, about 6.3 m digging depth, about 6.0 m dump height and 7.8 m maximum tip height for this geometry). Its edges are why a machine this size cannot dig deeper than about 6 m.
Cylinder geometry. Each cylinder joins a pin on the parent link to a pin on the child, at distances a and b from the joint, so l = √(a² + b² − 2ab cosδ), δ = θ/ρ + const. The lever arm is h = dl/dθ = ab sinδ / (ρ l). The bucket uses an equivalent gain ρ = 1.6 for its four-bar linkage (the real H-link and I-link are not solved separately).
Force, torque, speed, power. Cylinder force F = p A (cap side A, rod side A−A_r), joint torque τ = F h, cylinder speed v = Q/A, joint rate ω = v/h, so τω = F v = p Q exactly: hydraulic power equals mechanical power (before losses). Breakout force at the tip = τ divided by the distance from the joint to the tip. The relief valve caps the pressure (here 34.3 MPa by default), so the force saturates there however hard the operator pushes.
Soil cutting: the fundamental earth-moving equation. For a blade or bucket of width w at depth d with rake angle α the soil fails along a plane at angle ρ. Equilibrium of the wedge (weight, cohesion, friction on the plane and on the blade with soil-metal angle δ and adhesion c_a) gives R = w d [ (γd/2 + q)(cotρ + cotα) + c(1 + cotρ cot(ρ+φ)) + c_a(1 − cotα cot(ρ+φ)) ] / [cos(α+δ) + sin(α+δ) cot(ρ+φ)] and the draft (horizontal force) is R sin(α+δ) + c_ad w cotα. The page picks the failure angle ρ that minimises R (the soil takes the weakest wedge). It is a flat-blade, plane-strain model (McKyes 1985, after Reece and Hettiaratchi); for a real bucket an empirical factor 2 is applied here for side walls, teeth and the curved path. Resistance grows with d² for the weight term and with d for cohesion, so doubling the depth more than doubles the load in sand and gravel.
Dig cycle and production. Dig time = bite length / tip speed, where the speed is the smaller of the flow-limited crowd speed and the power-limited speed P/F (P the hydraulic power at the tip). Cycle = dig + 2 swing + dump + fixed; the bank volume per bucket is V_bank = V_bucket f / s (fill factor f, swell factor s) and production = 3600 V_bank E / T, with E = 0.83 (50 min per hour). The measured cycle in the 3D view is compared with this model.
Bulldozer. Drawbar pull is traction-limited, F ≤ μ W, at low speed and power-limited, F = η P / v (a hyperbola), at high speed; the push speed is where the pull curve meets the resistance R = R_roll + R_cut + R_pile + W sinβ (rolling, blade cutting from the equation above, dragging the pile, slope). The blade fills as dV/dx = w d_cut s − λV (gain from cutting minus side spill) until V reaches the blade capacity, giving V(x) = (a/λ)(1 − e^−λx); U blades hold more and spill less than straight blades, and slot dozing (soil banks left as side walls) cuts the spill. Production per hour = V(D) × 3600 / (D/v_fwd + D/v_rev + t_fixed) × E; the curve peaks at about the distance where the blade becomes full. Cross-slope limits are the tip angle atan((b)/(2h_cg)) and the slide angle atan(0.8 μ). Dozers normally push in first or second gear (3.6 or 6.5 km/h); the gear caps the speed even when the engine has power to spare.
Lift capacity and tipping. The load hangs from the stick pin; the machine stays up while the stabilising moment about the tipping edge, Σ m_i g (x_t − x_i), exceeds the overturning moment W (x − x_t). The tipping load is W_tip = M_stab/(g (x − x_t)); the hydraulic load comes from the boom and stick torque capacity minus the self-weight torque. Rated lift = min(0.75 W_tip, 0.87 W_hyd). The tipping edge is the track end over the front and the track side over the side; the tipping edge distance for a swing angle is min(L/2|cosψ|, B/2|sinψ|).
Levee volume and fleet. Cross-section A = ½(b₁+b₂) h with b₁ = b₂ + h(m₁+m₂), volume V = A L. Fill is compacted to a dry density (here 1.85 t/m³), so the borrow (bank) volume is V × ρ_fill/ρ_bank and the loose volume to haul is that times the swell factor. The fleet is limited by the slowest of excavator loading, truck cycles (load + haul + dump + return), dozer spreading and compaction; the number of trucks that keeps the loader busy is the truck cycle divided by the load time, rounded up.
Slope stability. For an infinite slope of dry cohesionless soil FS = tanφ / tanβ; with seepage parallel to the slope and the water table at the surface FS = (γ′/γ_sat) tanφ / tanβ, about half the dry value, which is why saturated levee slopes fail first. A 2H:1V slope in φ = 34 degree soil has FS = 1.35 dry but about 0.69 with seepage. Real design adds cohesion, drainage, filters, a toe berm and checks deep failure surfaces.
Experiments tab. IK to FK round trip; world-space cylinder length against the law of cosines; finite-difference lever arm against the analytic one; power balance p Q against τω; the earth-moving equation against a force-polygon solve with numerically integrated wedge weight; levee volume against a Simpson integral; slope factor of safety against the numerically integrated sliding angle; blade load against RK4 integration.
Simplifications. Planar arm, rigid links; hydraulic losses reduced to one efficiency; no pump control dynamics; soil as a uniform Mohr-Coulomb material; the heightfield terrain is visual and its slumping is a cellular repose-angle smoother, not a soil simulation; bulldozer track slip is not modelled below the traction limit; levee design is a volume and schedule exercise, not a geotechnical design.

Typical references: McKyes, Soil Cutting and Tillage (1985); Reece, The fundamental equation of earth-moving mechanics (1964); Hettiaratchi and Reece (1967); Caterpillar and Komatsu public handbooks for the typical machine orders of magnitude; ISO 6015 (hydraulic excavator digging forces) and ISO 10567 (lift capacity); Craig, Soil Mechanics, and Lambe and Whitman for infinite-slope stability.

Independent educational model of generic earth-moving machinery for construction and disaster-relief earthworks; not affiliated with or endorsed by any company. Numbers are textbook-level approximations, not design values.