Humanoid Push Recovery: Capture Point, DCM and Step Placement

The impulse is applied at the CoM: $\Delta\mathbf v=\mathbf J/m$. With $\omega=\sqrt{g/h}$, the horizontal LIP equations are $\ddot{\mathbf c}=\omega^2(\mathbf c-\mathbf p)$, $\boldsymbol\xi=\mathbf c+\dot{\mathbf c}/\omega$, and $\dot{\boldsymbol\xi}=\omega(\boldsymbol\xi-\mathbf p)$. Constant CoP intervals are integrated analytically, including the stable CoM component. These are the constant-height horizontal specialization of Englsberger et al. (2015), and Pratt et al. (2006), pp. 3–4 of the attached paper.

Koolen et al., attached author manuscript pp. 7–15, provide step-time/length constraints, equivalent constant CoP, and backward N-step capturability. This tool extends that construction to identical, fixed-yaw rectangular feet. Let $B=[-a,a]\times[-b,b]$ be the inset CoP rectangle, $L$ the ankle step limit, $q=e^{-\omega T_{min}}$, and $\mathbb D(R)$ a disk of radius $R$. The post-touchdown capture sets obey

Here $\oplus$ denotes a Minkowski sum. This rectangular-set derivation is supplied here; it does not replace the foot by its circumscribed circle.

For the selected first delay $T$, capture within $n\ge1$ steps requires $d(\boldsymbol\xi_0,B)\le e^{-\omega T}(L+R_{n-1})$. No step is needed exactly when $\boldsymbol\xi_0\in B$. Equality at the infinite-step limit is not finite-step capture. The latest first touchdown follows by solving the finite-budget inequality for $T$.

The feasible first ankle locations satisfy $\|\mathbf s\|\le L$ and $d(\mathbf s,e^{\omega T}(\boldsymbol\xi_0-B))\le R_{n-1}$. The map plots these nested sets. The suggested sequence keeps CoP closest to DCM and steps maximally toward the residual until the final shorter step; it attains the minimum count within this model.

Level ground; constant CoM height; a point body with massless legs and feet; no centroidal angular momentum. A horizontal impulse acts at time zero and changes velocity without changing position. Each stance permits any CoP inside the inset rectangular foot. CoP may move instantaneously, and touchdown switches support without changing CoM position or velocity. Feet retain their initial yaw. All horizontal ankle displacements up to the circular reach limit are permitted; no side preference or obstacle map is imposed. The first delay is at least the minimum step interval, and later steps use that interval. Zero foot dimensions with zero inset give the point-foot reference limit.

This is a reduced-order capture calculation, not a whole-body balance controller. Reachability depends only on ankle displacement, so results omit foot collisions, leg crossing, joint configuration, finite swing acceleration, friction, ankle torque/slew limits and terrain. The CoP inset is geometric reserve, not a measured control robustness margin. The displayed sequence uses the boundary of that inset rectangle, and later touchdown times assume the requested minimum interval is achievable.

The no-step verdict concerns the inset support rectangle of the current stance foot, not a two-foot support polygon. Capture means asymptotic stopping: playback ends six LIP time constants after the last touchdown. An infeasible or over-budget case has no touchdown schedule; playback then shows the initial divergence under the best stationary-foot CoP. A dash for a touchdown deadline means none is needed, and dashes in the schedule mean no plan was generated.

Minimum counts, deadlines and footprints are analytic and are not quantized by the picture. Isolated one-step landing points and zero-width support segments are drawn analytically and clipped exactly to the step-reach circle. Landing regions with area use a 121 × 101 signed-distance grid; very thin feasible slivers near the reach boundary may not be visible. Trajectory curves use 301 samples and timing curves 151. No time integration or grid approximation enters reported capture margins or touchdown coordinates. The LIP inset in the drawing is schematic and the foot plan uses a common horizontal scale. The published three-dimensional DCM paper also treats varying height and leg-force shaping, which are outside the constant-height LIP method used here.