Hexapod Gait Stability: Slopes, Support Polygon and ZMP

For each leg, local phase is $q_i=\operatorname{mod}(q-\phi_i,1)$ and stance is $q_i<\beta$. Presets use duty factors ½, ⅔ and ⅚. The ripple preset cycles three opposite-side pairs; custom phases allow other conventions. Feet move backwards by the stance stroke at constant speed relative to uniform body translation. Swing returns them with a sinusoidal-squared lift.

An orthonormal terrain frame rotates the footprint by inclination and uphill azimuth. Each foothold retains its own normal offset. The horizontal convex hull of the stance contacts defines McGhee–Frank's support polygon. Static margin is the shortest horizontal distance from the vertical COM projection to its boundary, positive inside and negative outside. Distances outside corners are measured to the boundary segments.

For each actual 3D hull edge, let $\hat a$ be its unit direction and $l=(I-\hat a\hat a^T)(p-C)$. The following height expression is derived from a circle about an inclined axis. During rotation about that edge the largest attainable COM height increase is $\Delta h=\|l\|\sqrt{1-\hat a_z^2}+l_z$. The Messuri–Klein gravitational barrier is the minimum $mg\Delta h$ over edges. An edge already carrying an outward gravity moment has zero remaining barrier.

Use $F=m(g-a_C)$ and the entered net COM moment $N$. Following Papadopoulos and Rey (1996), equations (5), (9)–(15), the edge force is $f_i=F-\hat a(\hat a\cdot F)+(l\times\hat a)(N\cdot\hat a)/\|l\|^2$. The signed angle between $l$ and $f_i$ is obtained with atan2; counterclockwise hull ordering makes inward reserve positive. The original measure is $\|F\|\min_i\theta_i$, with angles in radians (dimensionless in SI), so its displayed unit is force. Periodic COM sway uses one harmonic per axis, with acceleration equal to minus squared frequency times displacement.

For the reference plane through the origin with unit normal $n$, $p_{\rm ZMP}=n\times(C\times F+N)/(n\cdot F)$. Its signed margin uses the orthogonal projections of the stance feet onto that plane. The tangential moment residual independently checks the zero-moment condition.

The body and payload are represented by total mass and a prescribed COM. The body frame follows the reference slope, with no solved attitude motion. All scheduled stance contacts are maintained without slipping; legs can reach their specified footholds. Per-leg heights repeat each cycle.

Foothold offsets, prescribed accelerations and net COM moments use terrain axes: +x forward, +y left and +z normal away from the reference plane. Positive foothold height is away from that plane. Moments follow the right-hand rule and equal applied moments minus the rate of angular momentum at the COM, including limb effects when known. Prescribed acceleration is held fixed as contact phases vary; it is a frozen loading case rather than an integrated accelerating trajectory. Standard gravity is 9.80665 m/s².

Cycle extrema use 1080 uniform intervals plus touchdown and liftoff and their one-sided limits at 10⁻⁸ cycle. The same 3D equations are evaluated at every sample; the plotted lines do not smooth over contact changes. A narrow interior extremum may lie slightly below the sampled minimum, so inspect and refine the phase when reserve approaches zero. Hulls are reused only while the same contacts remain active; their common stance translation leaves hull ordering unchanged. The energy margin includes gravity only and excludes kinetic energy and impact work. On noncoplanar footholds there is no unique physical support-plane ZMP: the reference-plane value and its margin are diagnostics, not a contact-wrench feasibility test. The original 1996 force-angle candidate-edge construction uses the horizontal contact hull with actual 3D edge endpoints. Positive reserve does not prove feasible friction forces, sufficient actuator torque, attainable leg reach, disturbance recovery or successful touchdown. Local surface normals, soil settlement, foot area, compliant load sharing and controller dynamics are not solved.