A method is presented for formulating and numerically integrating ordinary differential equations (ODEs) of motion for holonomically constrained multibody systems. Tangent space coordinates are defined as independent generalized coordinates that serve as state variables in the formulation, yielding ODEs of motion. Orthogonal dependent coordinates are used to enforce kinematic constraints at position, velocity, and acceleration levels. Criteria that assure accuracy of constraint satisfaction and well conditioning of the reduced mass matrix in the equations of motion are used as the basis for redefining local coordinates on the constraint manifold, as needed, transparent to the user and at minimal computational cost. The formulation is developed for holonomically constrained multibody models that are based on essentially any form of generalized coordinates. A spinning top with Euler parameter orientation coordinates is used as a model problem to analytically reduce Euler's equations of motion to ODEs. Numerical results using a fourth-order Nystrom integrator verify that accurate results are obtained, satisfying position, velocity, and acceleration constraints to computer precision. A computational algorithm for implementing the approach with state-of-the-art explicit numerical integrators is presented and used in solution of three examples, one planar and two spatial. Performance of the method in satisfying all three forms of kinematic constraint, based on error tolerances embedded in the formulation, is verified.