= Residual contact torque at the center of pressure
{title2=$M_z^{P}=M_z-x_PF_y+y_PF_x$}
A <residual contact torque at the center of pressure> is the moment left after translating a planar contact's resultant <force> to its normal-pressure centroid. Let the resultant <wrench (mechanics)> at an origin in $z=0$ be <force> $F=(F_x,F_y,N)$ and moment $M=(M_x,M_y,M_z)$, with $N>0$. Put $r_P=(x_P,y_P,0)$, where $x_P=-M_y/N$ and $y_P=M_x/N$. Translation gives
$$
M^P=M-r_P\times F
=\left(0,0,M_z-x_PF_y+y_PF_x\right).
$$
The two horizontal moment components vanish, but the yaw component need not. The zero-moment-point construction for flat robot contact removes moments perpendicular to the surface normal; it does not assert that every component is zero.<Underactuated Robotics by Russ Tedrake>[Highly-articulated Legged Robots, An aside: Zero-moment point derivation]{ref}
A useful invariant makes the limitation exact. Since $F\cdot(r_P\times F)=0$,
$$
F\cdot M=F\cdot M^P=N M_z^P.
$$
If this scalar is nonzero, no change of the moment origin can eliminate the entire moment vector. Replacing the whole contact by only one <force> at the <center of pressure> would discard that information.
For an original two-contact example, put equal normal loads of $100\,\mathrm N$ at $(+0.10,0)$ and $(-0.10,0)$ metres. Add tangential <forces> $(0,+15,0)\,\mathrm N$ at the positive position and $(0,-15,0)\,\mathrm N$ at the negative position. The total <force> is $(0,0,200)\,\mathrm N$, the <pressure> centroid is the origin, and the opposing tangential <forces> give a pure yaw couple of $3\,\mathrm{N\,m}$. A friction coefficient of at least $0.15$ permits the stipulated tangential magnitudes at these loads.
A normal-load-only reconstruction would identify the correct centroid while missing the couple. A sensor measuring all <force> and moment components can distinguish the two situations. The origin, surface orientation and axis signs must be retained when comparing measurements. Near zero normal load, the centroid reconstruction itself becomes singular, independently of whether a tangential moment remains.
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