This paper presents a complete integrability condition for fully rheonomous affine constraints in terms of the rheonomous bracket. We first define fully rheonomous affine constraints and develop geometric representation for them. Next, the rheonomous bracket is explained and some properties of it are derived. We then investigate a necessary and sufficient condition on complete integrability for the fully rheonomous affine constraints based on the rheonomous bracket as an extension of Frobenius’ theorem. The effectiveness and the availability of the new results are also evaluated via an example.
 T. Kai and H. Kimura, “Theoretical Analysis of Affine Constraints on a Configuration Manifold—Part I: Integrability and Nonintegrability Conditions for Affine Constraints and Foliation Structures of a Configuration Manifold,” Transactions of the Society of Instrument and Control Engineers, Vol. 42, No. 3, 2006, pp. 212-221.
 T. Kai, “Mathematical Modelling and Theoretical Analysis of Nonholonomic Kinematic Systems with a Class of Rheonomous Affine Constraints,” Applied Mathematical Modelling, Vol. 36, No. 7, 2012, pp. 3189-3200. doi:10.1016/j.apm.2011.10.015
 T. Kai, “Theoretical Analysis for a Class of Rheonomous Affine Constraints on Configuration Manifolds—Part I: Fundamental Properties and Integrability/Nonintegrability Conditions,” Mathematical Problems in Engineering, Vol. 2012, 2012, Article ID: 543098. doi:10.1155/2012/543098
 T. Kai, “Theoretical Analysis for a Class of Rheonomous Affine Constraints on Configuration Manifolds—Part II: Foliation Structures and Integrating Algorithms,” Mathematical Problems in Engineering, Vol. 2012, 2012, Article ID: 345942. doi:10.1155/2012/345942