Qdott

qdotT refers to the transpose of the time derivative of a vector of generalized coordinates, typically written as q̇^T. If q(t) is a column vector of generalized coordinates, then q̇(t) = dq/dt is its time derivative, and q̇^T is a 1-by-n row vector. This notation is common in dynamics and control theory where algebraic forms involve quadratic expressions of the velocity.

In many mechanical systems, q̇^T is used to form scalar quantities by multiplying a velocity row vector by a suitable matrix and then by the velocity column vector. A canonical example is the kinetic energy T = 1/2 q̇^T M(q) q̇, where M(q) is the inertia (mass) matrix. If M is symmetric positive definite, this yields a positive scalar energy. The same bilinear form q̇^T M(q) q̇ appears in various energy, dissipation, and control expressions, and it generalizes to other symmetric or structured matrices as needed.

Notes for interpretation and usage: q̇^T is not a product of two independent quantities but the transpose of the velocity vector, used to produce scalars from quadratic forms. It is distinct from the dot product q̇ · q̇, though that dot product equals q̇^T q̇ when the identity matrix is the metric. In programming and notation, q̇^T is often written as q̇^T or qdot.T, depending on the language, with transpose operations implemented accordingly.