Drowsiness at the wheel is one of the leading causes of road fatalities. Driver monitoring systems (DMS) mainly rely on vehicle-based data and drivers' facial information to detect drowsiness. However, the introduction of partially autonomous driving will change the way we drive, letting the vehicle manage the driving task while drivers may be free to engage in non-driving tasks. This calls for new ways of detecting drowsiness, and even sleeping, at the wheel. Here, 22 participants drove for 100 min in a static simulator under level-2 automation on a 2 × 2 motorway. Postural (i. e., pressure and movements) and physiological (i.e., cardiac and respiratory) indicators were continuously recorded, while PERCLOS70 was used to classify drowsiness. The results reveal different physiological and postural signatures for the different states of drowsiness defined. While slight drowsiness is mainly associated with a higher heart rate, slower breathing, and an increased number of movements on the seat, being asleep is characterized by a lower heart rate and a slouched position on the seat. This study points to the relevance of using postural indicators in combination with physiological data to detect driver drowsiness. Focusing on the partially automated vehicle, it explores not only resistance to drowsiness but also sleeping at the wheel.