Residual contact torque at the center of pressure ($M_z^P=M_z-x_P{F}_y+y_P{F}_x$)

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 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 givesThe 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.^[ref]

A useful invariant makes the limitation exact. Since $F\cdot (r_P\times F)=0$,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 $3\mathrm{N}\mathrm{m}$. 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.