= Quadrupedal robot
A quadrupedal robot is a <robot> with four legs used for support and locomotion. Its <gait> determines which feet contact the ground and when; having four legs does not imply that all four feet remain planted. Leg geometry, contact sensing and control of <forces> connect a desired body motion to feasible ground reactions.
An original <statics> calculation explains one advantage and one limitation of multiple feet. Assume point contacts on a horizontal floor, vertical gravity only, no contact moments and no acceleration. For robot mass $m$ and upward foot forces $F_i\geq0$, force and moment balance require
$$
\sum_iF_i=mg,\qquad
\sum_iF_i x_i=mgx_c,\qquad
\sum_iF_i y_i=mgy_c.
$$
Here $(x_c,y_c)$ is the vertical projection of the <center of mass>. Dividing by $mg$ expresses this point as a <convex combination> of foot locations. It must lie in their <convex hull>, the <support polygon>. A point on its boundary has no positive geometric margin against tipping in the corresponding direction.
For example, feet at $(\pm0.50,\pm0.30)\,\mathrm m$ surround a centered mass projection. Remove the front-left contact $(0.50,0.30)\,\mathrm m$: the origin now lies on the diagonal edge of the remaining triangle. The balance equations load the opposite two feet at $mg/2$ each and leave the third unloaded. Merely retaining three contacts has not created a positive margin. Shifting the mass projection to $(-0.10,-0.10)\,\mathrm m$ places it strictly inside that triangle, permitting three positive vertical reactions.
Dynamic running and jumping require a different balance including acceleration and <angular momentum>. An airborne robot has no ground <support polygon> at all. In the 2015 MIT Cheetah demonstration, onboard <lidar> estimated obstacle size and distance; control adjusted the approach strides and planned motor forces for a running jump.<MIT cheetah robot lands the running jump by Jennifer Chu>[See, run, jump: onboard sensing, stride adjustment and jump-force planning]{ref} The apparatus demonstrates coordinated perception and dynamic contact control, rather than a static support rule.
For a controlled walking test, record foot-contact times and body motion while changing one planned contact at a time. The geometric criterion above checks only the stipulated static model. <Friction>, actuator limits, terrain inclination and motion can make a geometrically supported pose physically infeasible. Conversely, a dynamic maneuver can pass through a pose that could not be maintained at rest.
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