This paper presents a new approach to semi-autonomous vehicle hazard avoidance and stability control, based on the design and selective enforcement of constraints. This differs from traditional approaches that rely on the planning and tracking of paths. This emphasis on constraints facilitates "minimally-invasive" control for human-machine systems; instead of forcing a human operator to follow an automation-determined path, the constraint-based approach identifies safe homotopies, and allows the operator to navigate freely within them, introducing control action only as necessary to ensure that the vehicle does not violate safety constraints. The method evaluates candidate homotopies based on "restrictiveness", rather than traditional measures of path goodness, and designs and enforces requisite constraints on the human's control commands to ensure that the vehicle never leaves the controllable subset of a desired homotopy. Identification of these homotopic classes in off-road environments is performed using geometric constructs. The goodness of competing homotopies and their associated constraints is then characterized using geometric heuristics. Finally, input limits satisfying homotopy and vehicle dynamic constraints are enforced using threat-based feedback mechanisms to ensure that the vehicle avoids collisions and instability while preserving the human operator's situational awareness and mental models. The methods developed in this work are shown in simulation and experimentally demonstrated in safe, high-speed teleoperation of an unmanned ground vehicle.