Hexapod CPG Gait Generator: Tripod, Ripple and Wave

The engine integrates all 12 Cartesian states of six nonlinear Hopf oscillators in a common rotating frame. With $z_i=x_i+\mathrm{i}y_i$, unit target radius and node phase targets $p_i$:

This is rotated diffusive coupling of the full Hopf states. Coupling acts on both radius and phase, so transient loss of coordination can reduce amplitude. The sum is evaluated in the node-target frame through its mean Cartesian vector, an exact algebraic rearrangement of all six pair contributions. The common rotation is removed by the exact transformation $w_i=z_i e^{-\mathrm{i}2\pi f t}$ before integration and restored afterward. The nonlinear Cartesian coupling and amplitude dynamics remain in the ODE. The Hopf foundation follows Ijspeert (2008); Bai et al. (2019), Eqs. 13–18, motivate the six-node oscillator network. The normalized diffusive coupling and three-link kinematics here are explicit design choices, not a reproduction of their curved-leg controller.

The phase targets $p_i$ are negative touchdown lags in the order LF, LM, LR, RF, RM, RR. Tripod uses lags [0, 1/2, 0, 1/2, 0, 1/2] cycles. Ripple uses [0, 1/3, 2/3, 1/2, 5/6, 1/6]; wave uses [0, 1/6, 1/3, 1/2, 2/3, 5/6]. These explicit sequence choices define this generator. Stance duties are 1/2, 2/3 and 5/6, respectively (Mrva 2014, printed p. 8; Bai et al., pp. 4–5). Ripple here has two overlapping swings, unlike the three-pair tetrapod convention.

Each node's shortest phase change is blended using $s(v)=10v^3-15v^4+6v^5$, with $v$ clamped to 0…1. Pairwise targets remain consistent because they are always differences of node targets. Duty factor is blended by the same ramp. Oscillator states continue through both transitions without resets.

For $u=\mathrm{mod}(\theta_i/2\pi,1)$, stance is $u<\beta$. Stance moves the foot from +stroke/2 to −stroke/2 using $s(u/\beta)$; swing returns it using $s((u-\beta)/(1-\beta))$. Lift is $64H q^3(1-q)^3$ during swing. Stroke and lift are multiplied by $r_i$. Both foot paths have continuous position, velocity and acceleration at lift-off and touchdown. Closed-form coxa yaw and a planar two-link inverse solve give femur and tibia angles. A forward-kinematic closure property verifies that mapping.

Touchdown events satisfy $Q_i=k$ and lift-off events satisfy $Q_i-\beta=k$, where $Q_i=\theta_i/(2\pi)$ and $k$ is an integer. Crossings are located separately for each leg using the same continuous linear phase interpolation as the animation and the exact quintic duty schedule. Intervals are split at ramp boundaries and at stationary points of $Q_i-\beta$ before bracketing roots, including during reversals. Touchdown roots are linear solves; lift-off roots use bisection. Events within 10 ns are coalesced to suppress simultaneous-event roundoff. All six stance flags are evaluated between successive events. The minimum and footfall drawing use this shared event partition; final-cycle averages weight interval durations. Red chart markers identify intervals with zero commanded stance.

Adaptive Dormand–Prince integration uses relative tolerance 10⁻⁶. The saved mesh targets at least 100 samples/s and 160 samples per nominal stride, capped at 6,001 saved times. Relative phase RMS compares each of the five other legs with LF after subtracting its scheduled lag. Joint speeds use central differences on that mesh.

Identical intrinsic frequencies; reciprocal all-to-all coupling normalized by six; initial radii in the unit disk; no sensory feedback. The initial disturbance is a fixed unequal phase pattern scaled by the disturbance input. Three nominal strides are retained after the second ramp.

Each leg has an outward coxa followed by femur and tibia hinges in one vertical plane. Coxa zero points outward; positive yaw is forward on both sides. Femur zero is horizontal outward, positive upward. Tibia zero is straight extension and flexion is negative. Mirror these signs and add servo zero offsets for actual hardware.

The stationary chassis and hip spacing are illustrative. The drawn links and foot trajectories use the entered lengths; chassis half-width is 0.4 times foot reach and front/rear hip spacing is 0.7 times foot reach. The diagram uses an oblique projection.

This is an open-loop rhythm and joint-command generator. Commanded stance is a timing state, not measured contact. Smooth phase transitions do not establish balance: there is no body dynamics, support-polygon margin, contact force, traction, terrain compliance, collision detection, servo saturation, torque, power or load model. It does not predict walking speed; the quintic stance path does not enforce constant ground speed.

All commanded feet must stay more than 1 mm from full folding and full extension. Conservative reach bounds cover the entire stroke and lift. Inverse-trigonometric arguments are clamped only for numerical robustness; invalid geometries are blocked by conditions. Phase reversal is reported when a sampled phase speed becomes nonpositive; lengthen the transition or raise the stride frequency.

Cartesian states give radii and unwrapped phases. The animation linearly interpolates these saved radii and unwrapped phases, then recomputes the foot path and inverse kinematics. Saved output is capped at 6,001 times to keep long programs interactive. Event roots resolve contact crossings within the interpolated phase histories, rather than assigning an entire sample interval one contact state. Event timing accuracy is limited by the ODE solution and phase interpolation; the 10 ns grouping tolerance is not a claim of physical timing accuracy. At the output cap, rapid phase curvature can shift event times and sampled joint-speed peaks can be underestimated, especially during short wave-gait swings. The adaptive ODE tolerance is unchanged. The reported final error is at the end of the three-stride observation window; weak coupling can require longer to lock. No physical balance or hardware experiment is claimed.