= Swing-foot clearance
<Swing-foot clearance> is the separation between a robot's swinging foot and the terrain or obstacles it must pass. It concerns the whole <swing phase>, not just the maximum foot height. For a finite foot, use the relevant lowest material surface or full collision geometry; the height of a chosen ankle coordinate can differ from actual clearance.
An original point-foot model isolates vertical geometry. Let a planned foot position be $(x(s),z(s))$ and terrain height be $H(x)$, with swing parameter $s\in[0,1]$. Define
$$
c(s)=z(s)-H(x(s)).
$$
Ground contact at the beginning and end normally gives zero endpoint clearance, so a positive lower bound must be assigned to the obstacle-crossing part of the path rather than incorrectly demanded throughout the closed contact-to-contact interval.
Use the constructed path
$$
x(s)=0.20(3s^2-2s^3)\,\mathrm m,\qquad
z(s)=h\sin^2(\pi s).
$$
Suppose a flat-topped obstacle of height $15\,\mathrm{mm}$ occupies the horizontal range corresponding to $s\in[1/4,3/4]$, namely $31.25$ to $168.75\,\mathrm{mm}$. With $h=40\,\mathrm{mm}$, the lowest foot height over this interval is $h/2=20\,\mathrm{mm}$, attained at its edges. The minimum nominal obstacle clearance is therefore only $5\,\mathrm{mm}$ even though peak foot height is $40\,\mathrm{mm}$.
A <robust swing-foot clearance margin> includes specified error bounds. If actual foot height can be $3\,\mathrm{mm}$ below its plan and actual obstacle height $4\,\mathrm{mm}$ above the model, the guaranteed clearance is at least
$$
5-3-4=-2\,\mathrm{mm}.
$$
Thus the nominally clear path permits collision within those bounds. Guaranteeing at least $5\,\mathrm{mm}$ above this obstacle under the same errors requires
$$
\frac h2-15-3-4\ge5\quad\text{in millimetres},\qquad h\ge54\,\mathrm{mm}.
$$
The bounds add because both deviations can reduce clearance at once; they are deterministic limits, not independent <standard uncertainties> to combine by root-sum-square.
Greater height changes the leg's motion requirements. For a fixed swing duration $T_w$, this path has maximum vertical speed $\pi h/T_w$ and accelerations proportional to $h/T_w^2$. Clearance, timing, <inverse kinematics> and actuator constraints must therefore be checked together. A terrain discontinuity, horizontal position error or non-point foot additionally requires the corresponding geometry rather than just this vertical formula.
The primary planning text treats clearance as a possible planning objective and distinguishes local geometric constraints from dynamic ones.<Robotic Systems by Kris Hauser>[Chapter 8, Stating the motion planning problem]{ref} A proposed robot test measures the foot surface and obstacle geometry in a shared reference frame, records the trajectory throughout the crossing, and compares the minimum separation with the stated bounds. Observing a high midpoint alone does not test the minimum-clearance claim.
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