Biped Walking: Compass Gait, Five-Link Walker and Poincaré Stability

Both walkers use $L$, $M$ and $\sqrt{L/g}$ as length, mass and time scales. Angles are absolute and clockwise from upward vertical.

Compass gait. The two leg masses and hip mass give the nonlinear double-pendulum mass matrix, centrifugal terms and gravity vector. Classical RK4 integrates swing. A forward, descending foot-contact crossing is located with 30 bisections of a reintegrated partial step; the coincident-foot crossing is excluded. Conservation of angular momentum gives the plastic impact velocities, followed by leg exchange. Shooting uses the three post-impact section coordinates $(q_{\rm stance},\dot q_{\rm stance},\dot q_{\rm swing})$. Multiple seeds include a shallow-slope start. Damped Newton backtracks until a forward-contact trajectory exists and its fixed-point residual decreases. Central differences of complete event-to-event integrations give the 3 × 3 Jacobian. A scaled cubic characteristic polynomial is solved with Aberth–Ehrlich iteration, and a polynomial residual check validates its three eigenvalues. Goswami et al., 1998, §§2–3.

Five links. With $q=(q_s,q_f,q_T,q_{f2},q_{s2})$, COM locations are $\sum_j C_{bj}(\sin q_j,\cos q_j)$. Thus $K=C^T\operatorname{diag}(m)C$, $D_{ij}=K_{ij}\cos(q_i-q_j)+I_{c,i}\delta_{ij}$, $h_i=\sum_jK_{ij}\sin(q_i-q_j)\dot q_j^2$, and $G_i=-g_j\sin q_i$, where $g_j=\sum_bm_bC_{bj}$. These give all five equations $D\ddot q+h+G=Bu$ with four internal joint torques. Reference-axis inertias are converted by $I_c=I_{\rm joint}-mp^2$.

Impact uses the complete 7-DOF floating model: $\dot q_e^+=\dot q_e^-+D_e^{-1}J^T\Lambda$, $JD_e^{-1}J^T\Lambda=-J\dot q_e^-$. Both impulse components and released-foot velocity are checked, and all five angular coordinates are relabeled.

A quartic Hermite posture curve uses phase $s=((q_s+q_f)/2+a)/(2a)$. Its endpoint positions swap legs; its initial tangent is the relabeled full impact of the final tangent, normalized to the same phase rate. A quartic knee-flexion bump vanishes with its first derivative at both ends. This enforces hybrid invariance, rather than resetting velocity onto an incompatible trajectory.

Let $A=\mathbf1^TDq_s$, $\sigma=A\dot s$, and $\zeta=\sigma^2/2$. The exact five-link zero dynamics are $\dot\sigma=\sum_jg_j\sin q_j$ and $d\zeta/ds=A\dot\sigma$. Composite Simpson integration yields $W(s)$. Full impact gives $\delta=\sigma^+/\sigma^-$ and $P(\zeta)=\delta^2\zeta+W(1)$. Newton solves $P(\zeta)-\zeta=0$; its Jacobian is the scalar $\delta^2$.

The implemented input-output linearizing controller solves the full five dynamic equations together with four prescribed output accelerations. The ideal finite-time law uses $v=-k^2\operatorname{sign}(y)|y|^{1/3}-k\operatorname{sign}(\dot y)|\dot y|^{1/2}$. Locally, when settling occurs before impact, eight transverse multipliers vanish and the remaining multiplier is the HZD value. Westervelt et al., 2003, §§III–V.

Joint torques and total COM acceleration give actuator loading and ground reaction. On the zero manifold these are affine in pre-impact $\zeta$; intersecting their inequalities with $\zeta(s)>0$ produces the displayed momentum bounds. Friction includes the impact impulse ratio.

The five-link admissible interval also enforces compressive slipless impact, upward release of the old foot, descending forward touchdown, swing clearance, knee limits, and above-ground joint geometry. Violating any geometric or impact constraint empties the interval. Margins and successful recovery require the nominal fixed point to lie inside the resulting interval.

Planar rigid links, ideal revolute joints, identical legs, no ankle torque, and instantaneous slipless plastic impact. The old foot releases without impulse. The five-link surface is level; slope applies only to the passive compass gait.

The five-link dimensions and mass distribution scale homothetically from Westervelt et al., Table I: torso, two femurs and two tibias. Their inertia convention is checked against the equation-details supplement. The constructed posture family retains all five link masses and inertias; it does not reproduce the paper's optimized Bézier coefficients. Its fixed stance-knee shaping bump is 0.08 rad peak additional flexion.

Five-link orbital stability assumes the stated ideal finite-time feedback, exact state knowledge and an invertible decoupling matrix. The robustness interval is restricted to momentum disturbances on the zero manifold. The compass follows Goswami's imaginary massless foot-retraction assumption; it is not a physically rigid-foot scuff model.

Standard resolution uses 1441 five-link phase nodes and Simpson quadrature; Fine uses 2881. All extrema and admissible bounds are evaluated on that full grid. Seventy-three PCHIP samples are retained for plots and playback, so pictures interpolate the solved data and can miss very narrow features. The compass uses RK4 steps of $0.015\sqrt{L/g}$ (Fine: $0.0075\sqrt{L/g}$), event refinement, and Jacobian perturbations of 10⁻⁵ in normalized coordinates. Compare Standard and Fine near a boundary.

A stable multiplier answers a local orbital question. It does not establish a global basin, lateral balance, or robustness to arbitrary pushes, terrain changes or posture errors. The eight ideal transverse zeros are a finite-time-feedback result, not measured hardware multipliers. Torque bounds and disturbance limits apply to the nominal zero-manifold dynamics; feedback transients, saturation and finite motor bandwidth can invalidate them. Sampled ground-force and torque inequalities are numerical screening bounds, not certified continuous-time margins.

The solver seeks a symmetric one-step fixed point. It can report that orbit as unstable even where a multi-step or chaotic gait exists. A compass return that fails within six gravitational time scales is treated as unavailable. HZD momentum bounds are inapplicable in compass mode and are shown as gaps.

The published compass transition near 4.37° is approximate: this nonlinear implementation gives a leading eigenvalue near −0.979 there. Its reference check allows a 0.03 difference from −1; no equation or coefficient is adjusted to force the plotted threshold.

The momentum margins are unavailable when the nominal gait violates any contact, impact or geometric requirement. An inadmissible disturbance is followed by gaps in the recovery chart. At zero compass disturbance, the converged fixed point is reused for all twelve returns. Angle inputs use exact compact radian defaults; the unit menu also displays degrees.