In the Cobotic Hand Controller, we have introduced an admittance display that can render very high impedances (up to its own structural stiffness). This is due to its use of infinitely variable transmissions. While admittance displays typically excel at rendering high impedances, the incorporation of infinitely variable transmissions in the Cobotic Hand Controller allows the stable display of a wide dynamic range, including low impedances. The existence of a display that excels at rendering high-impedance constraints, but has high-fidelity control of low impedances tangent to those constraints, has led us to describe an admittance control architecture not often examined in the haptics community. In this paper, we develop a comprehensive approach that enables rendering of rigid motion constraints while simultaneously preserving the physical integrity of the intended inertial dynamics tangent to those constraints. This is in contrast to conventional impedance-control algorithms that focus primarily on rendering reaction forces along contact normals with constraints. We present this algorithm here, which is general to all admittance displays, and report on its implementation with the Cobotic Hand Controller. We offer examples of rigid bodies and linkages subject to holonomic and/or nonholonomic constraints