Center of pressure (terrestrial locomotion)

In terrestrial robot locomotion, the center of pressure is the ground point summarizing the normal load shared by the active contacts. With contact positions $(x_i,y_i)$ and nonnegative normal reactions $N_i$, its coordinates are the weighted averages $x_P={\sum }_i{N}_i{x}_i/{\sum }_i{N}_i$ and $y_P={\sum }_i{N}_i{y}_i/{\sum }_i{N}_i$, provided the total normal force is positive. It lies in the current support polygon, but is a contact-force quantity rather than the robot's center of mass position.^[ref]

Force-sensor coordinates matter when reconstructing this point. Suppose a sensor origin lies height $h$ above a horizontal contact plane, with parallel coordinate axes. If it reports force $(F_x,F_y,F_z)$ and moment $(M_x,M_y,M_z)$, the plane point is located at $(x_P,y_P,-h)$ relative to that origin. Moment translation givesThe ground-plane transformation of a measured contact wrench must therefore precede interpreting moment/force ratios as a contact location.

For an original sensor example, take $h=0.05\mathrm{m}$, force $(30,-20,600)\mathrm{N}$, and measured roll/pitch moments $(17,-49.5)\mathrm{N}\mathrm{m}$. The reconstructed point is $(0.080,0.030)\mathrm{m}$. Ignoring the origin height would instead give $(0.0825,0.02833)\mathrm{m}$. Those millimetre differences are coordinate effects, not a change in the physical pressure distribution.

During motion, the contact point need not equal the horizontal projection of the center of mass. The source's constant-height, negligible-angular-momentum-rate model relates their difference to horizontal acceleration; those assumptions must be established before using that simplified relation.^[ref] A robot-foot experiment can compare known applied loads with the sensor's translated force/moment measurements. When $F_z$ approaches zero, reconstruction becomes sensitive to moment error; at zero normal load the normal-pressure point is undefined. A residual yaw couple may remain, as explained in residual contact torque at the center of pressure.

A ground-plane transformation of a measured contact wrench combines a change of coordinate axes with a change of the point about which moments are reported. A six-component wrench is incomplete without both pieces of metadata. Drake's primary SpatialForce reference explicitly tracks the application point and expressed-in frame separately from the stored force and torque values.^[ref]

Let $R$ rotate sensor-frame components into ground-frame components, and let $p$ be the ground-frame vector from ground origin $O$ to sensor origin $S$. For measured force $f_S$ and moment $M_S$ about $S$, the equivalent result about $O$ isThe plus sign uses the stated vector from the new origin to the old one. A shift expressed with the opposite vector uses the opposite sign. The offset and force must be expressed in the same axes before taking the cross product.

For a planar normal-pressure reconstruction with $f_{G,z}>0$, computeThe special parallel-axis height correction in center of pressure is recovered with $p=(0,0,h)$ and $R=I$.

For an original rotated-sensor example, use a 90-degree rotation about $z$, sensor readings $f_S=(10,0,100)\mathrm{N}$ and $M_S=(1,2,3)\mathrm{N}\mathrm{m}$, and $p=(0.05,-0.02,0.10)\mathrm{m}$. Rotation gives $f_G=(0,10,100)\mathrm{N}$ and $R{M}_S=(-2,1,3)\mathrm{N}\mathrm{m}$. The translated moment is $M_O=(-5,-4,3.5)\mathrm{N}\mathrm{m}$, yielding $(x_P,y_P)=(0.040,-0.050)\mathrm{m}$. The yaw moment remaining at that point is $3.1\mathrm{N}\mathrm{m}$, so translating to the centroid does not erase the entire wrench.

A bench check applies known loads at known ground locations and repeats after changing sensor orientation. Geometry, tare loads, units and axis signs must all be included. A correct rotation with the wrong moment origin can still give an incorrect pressure location; a small normal load also amplifies reconstruction uncertainty.