Principle of Virtual Work

A system is in static equilibrium exactly when the total work done by active forces over any allowed tiny virtual displacement is zero.

The principle of virtual work is a fundamental principle of analytical mechanics: imagine the system undergoes a hypothetical, infinitesimal displacement (a virtual displacement) that's consistent with its constraints — if the total work done by all the active forces over this displacement (the virtual work) sums to zero, the system is in static equilibrium. Johann Bernoulli gave it a systematic statement in correspondence with Varignon in 1715, and Lagrange later built analytical mechanics on top of it. Its advantage is that the reaction forces from ideal constraints — such as the internal forces at a joint — do zero virtual work, so they never need to be solved for individually. The most-used result in robotics is τ = Jᵀ(θ)F: when the end effector applies a force or wrench F to the outside world, the required joint torque τ equals the transpose of the Jacobian matrix J times F, based on the equality of virtual work (or power) done at the joints and at the end effector. Force control, impedance control, gravity compensation, and converting a legged robot's foot contact force f into joint torques via τ = J_cᵀf all rest on this relationship.

ExampleA planar two-link arm needs to push horizontally against a wall with 10 N at its end effector, in some given pose: substituting F = (10, 0) into τ = JᵀF directly gives the torque each of the two joints must output, with no need to analyze the internal forces within the links.

Also called

Principle of Virtual Displacements, Virtual Velocity Principle

Related

Jacobian Matrix · Wrench · Statics · Gravity Compensation · Contact Jacobian · Euler-Lagrange Equations

Sources

Wikipedia: Virtual work
Modern Robotics (Lynch & Park), Sec. 5.2 Statics of open chains