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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 $F_i\ge 0$, force and moment balance requireHere $(x_c,y_c)$ 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 $(\pm 0.50,\pm 0.30)\mathrm{m}$ 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.^[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.

Simon Stevin was a mathematician and engineer who promoted the practical use of terminating decimals and made important contributions to statics and hydrostatics. His 1585 booklet on decimal arithmetic presented an accessible system for calculation and measurement; he did not invent terminating decimals, which had earlier precedents. His mechanical writings appeared the following year.^[ref] In hydrostatics, the pressure difference in a liquid of constant density depends on vertical depth through $\Delta p=\rho gh$. This explains why equal depths give equal pressures even in vessels of very different shapes, separating pressure from the total amount of liquid.

Statics studies mechanical equilibrium: configurations in which the relevant forces and torques balance. For a rigid body at rest in an inertial frame, the conditions are $\sum \mathbf{F}=0$ and $\sum {\mathbf{\tau }}_O=0$, with all torques calculated about the same point $O$.

A horizontal beam carrying a downward load illustrates why both equations are necessary. Vertical force balance fixes the sum of the support reactions, while torque balance determines how the load is divided between supports. Equilibrium alone does not imply dynamical stability: a perfectly upright inverted pendulum satisfies force and torque balance but falls after a sufficiently small disturbance.