Robot control
Complementarity constraint
Definition
A complementarity constraint requires two non-negative quantities to have a zero product, so at least one of them must be zero. Robotics uses this either-or structure to represent contact states such as positive separation with no contact force or zero separation with a possible force.
Also known as: Complementarity condition
Updated
Encode an either-or physical condition
For non-negative quantities a and b, complementarity is commonly written as a times b equals zero. This allows a to be positive only when b is zero, and vice versa. In rigid contact, a can represent separation distance and b a normal contact force. A separated body has no normal force; a body carrying normal force has zero separation.
Similar conditions can represent sticking versus sliding limits, actuator saturation, or other mutually exclusive regimes. Unlike a hand-written if statement, the relation can appear inside a continuous optimisation problem together with dynamics and task costs.
Contact modes need not be ordered in advance
Locomotion and contact-rich manipulation involve making, maintaining, and breaking contacts. A planner that enumerates every discrete contact sequence can grow rapidly with the number of possible contacts.
Posa, Cantu, and Tedrake formulate rigid-body trajectory optimisation as a mathematical program with complementarity constraints. Their method jointly searches state, control, and contact forces instead of requiring an a priori mode ordering. The paper demonstrates the approach on manipulation, planar walking, and bipedal running examples.
Complementarity is the mathematical relation; contact-implicit optimization is a broader planning approach that may use that relation, a smoothing, a relaxation, or another contact representation.
Exact physical logic creates numerical difficulty
A complementarity constraint is non-convex and violates regularity assumptions used by many standard nonlinear-programming analyses. A solver may depend strongly on its initial trajectory, scaling, relaxation schedule, and local minimum. A small complementarity residual is not enough if dynamics or friction constraints are still violated.
The contact model also remains an approximation. Rigid unilateral contact and a friction cone do not capture every compliant surface, impact transient, deformation, or uncertain coefficient. Complementarity can avoid explicit mode enumeration, but it does not make contact timing globally optimal or the resulting motion automatically executable on hardware.
Sources
Related terms
Contact-implicit optimization
Contact-implicit optimization plans motion while allowing contact events and forces to emerge from contact constraints in the optimization. It avoids requiring every contact transition to be fixed in a predefined mode sequence.
Trajectory optimization
Trajectory optimization finds a time-varying motion, and often control inputs, that minimizes an objective while satisfying specified constraints. Robot applications can include geometric, kinematic, and dynamic constraints.
Friction cone
A friction cone is the set of contact forces permitted by a Coulomb friction model at a contact that pushes but does not pull. The allowable tangential force magnitude is bounded by the normal force multiplied by a friction coefficient.
Contact-rich manipulation
Contact-rich manipulation is robotic manipulation in which making, maintaining, breaking, or sliding contacts is central to completing the task. The robot must reason about forces and changing contact modes, not only collision-free motion.