3D Bio-Inspired Hexapod: Blender Walking Robot with CPG Gaits, Inverse Kinematics and Stability Margin

Setting up the lab bench...

Presets

Live readouts

Graphs

Gait diagram: coloured = foot on the ground, last 6 s

Joint angles of the selected leg (degrees)

Static stability margin (mm); below 0 the robot tips

Electrical power (W)

CPG phase-locking error (rad)

Numerical checks

running...

How to use

The robot starts walking a tripod gait. Click a foot (2D) or use the leg selector to follow one leg in the inverse-kinematics view. Try: (1) Self-organising CPG: the phase dials start scrambled and lock within a second; lower the coupling K to 0.5 and press Kick phases. (2) Switch gait while walking and watch the stability-margin graph: tripod is the fastest and has the smallest margin, wave has the largest. (3) Lost a leg: tripod falls, wave keeps walking (press Stand up after a fall). (4) Move the payload forward until the centre of mass leaves the green polygon. (5) Raise the terrain roughness and the swing clearance together. Drag to orbit, wheel to zoom. Space pauses, R restarts, arrow keys steer (up/down speed, left/right turn).

The 3D page uses a Blender-built hexapod (carbon-fibre chassis, anodised aluminium trim, servo housings, rubber feet, sensor head) whose three joints per leg are driven by the same inverse kinematics; the support polygon, centre of mass and footprints are drawn on the terrain.

The science

Why bio-robotics? Insects, crabs and lizards walk over ground where wheels fail. Bio-robotics copies the control principles of animals rather than their shape: a rhythm generator in the nervous system (a central pattern generator, CPG), a few repeatable gaits, and fast leg kinematics. This lab is a hexapod (six-legged) robot with three joints per leg.

Central pattern generator. Each leg has a phase oscillator θ_i that runs at the stepping frequency f. Neighbouring oscillators are coupled so that they settle at the phase offsets φ_i of the chosen gait (Kuramoto-type model):

dθ_i/dt = 2πf + K Σ_j sin(θ_j − θ_i + φ_i − φ_j)

The coupling graph is a ladder: a chain on each side plus a rung per leg pair. For two oscillators with the same target phase the difference ψ obeys dψ/dt = −2K sin ψ, solved exactly by tan(ψ/2) = tan(ψ₀/2) e^−2Kt; the page tests the integrator against this formula. A larger K locks faster and recovers from a disturbance sooner; K = 0 leaves every leg running free. Real CPGs are neural circuits in the spinal cord or the ventral nerve cord; coupled phase oscillators are the standard reduced model.

Gaits. The fraction of a cycle a foot spends on the ground is the duty factor β. A leg is in stance while its phase fraction u = (θ/2π mod 1) is below β and in swing otherwise. Tripod (β = 0.5): two alternating triples of legs; fastest, smallest stability margin. Ripple (β = 2/3) and wave (β = 5/6): lift-off travels along each side from rear to front; slower and steadier. Pronk (all offsets 0): all legs swing together, so nothing carries the body.

Foot placement. While a foot is planted it stays fixed in the world, so in the body frame it moves with velocity −(v + ω × r). At lift-off the swing target is chosen so that the foot lands half a stance displacement ahead of its home point (the Raibert rule):

p_touchdown = p_home + (v + ω × p_home) · βT/2, T = 1/f

Swing follows a cosine blend from the lift-off point to the target with a sine bump of height h (the swing foot clearance).

Inverse kinematics. Each leg has a coxa (yaw), a femur and a tibia of lengths L₁ = 0.10, L₂ = 0.25 and L₃ = 0.40 m. For a foot at (x, y, z) in the leg frame (x outward, z up):

α = atan2(y, x), r = √(x²+y²) − L₁, d = √(r²+z²)
cos γ = (L₂²+L₃²−d²)/(2L₂L₃), δ = −(π − γ)
β = atan2(z, r) + arccos((L₂²+d²−L₃²)/(2L₂d))

The knee is always above the line hip to foot ("elbow up"). The target is reachable when |L₂−L₃| < d < L₂+L₃; the blue annulus in the leg view is that workspace, and a red dot means the target was clamped.

Stability. The support polygon is the convex hull of the feet on the ground. The static stability margin is the shortest distance from the centre of mass (COM) to its edges (positive inside). The robot is declared fallen when fewer than three feet touch the ground or the margin stays negative for 0.45 s. Vertical foot loads are the minimum-norm solution of ΣF_i = mg and ΣF_i(r_i − r_COM) = 0 with F_i ≥ 0 (active set).

Torque, power and cost of transport. A stance leg carrying F_z needs joint torques τ = J^TF with J the foot Jacobian. Electrical power is the copper loss plus mechanical work, with an assumed motor constant k_m = 2.4 N m/√W:

P = Σ_joints (τ/k_m)² + Σ_joints |τ ω|, CoT = P / (m g v)

Honest simplifications. The body is kinematic (the commanded velocity is followed with a 0.35 s lag; no inertia, slipping or foot friction limits), loads are quasi-static and purely vertical, the body stays level on bumpy ground (no pitch or roll), and servo behaviour is reduced to one motor constant. Mass (14 kg) and link lengths are round example numbers, not a specific product. Real insects also use sensory feedback and stored energy in the muscles, which this model leaves out.

Checks run in the page. (1) IK followed by forward kinematics on thousands of random targets; (2) the two-oscillator closed form; (3) six oscillators from random phases locking to the tripod pattern; (4) distance walked equals the commanded speed times time, and planted feet stay fixed after the joint-angle round trip; (5) load sharing sums to the weight with zero moment, and joint power (J^TF)·dq/dt equals foot force times finite-difference foot velocity.