Simulation
Col de la Madeleine Descent Optimization
Minimum time descent simulation of the Col de la Madeleine on a road bike, extended with rolling dynamics, aerodynamic tuck as a control variable, and an energy expenditure constraint enabling Pareto-efficient speed-recovery trade-off analysis.
Links & Resources
The Course
The 25 km descent of the Col de la Madeleine, featured in the Tour de France queen stage, transitions directly into the final ascent of the Col de la Loze. A rider arriving at the top of the Madeleine with a gap must choose how much energy to spend on the descent to defend or extend that advantage before the final climb begins. This is a multi-objective problem: minimize descent time, but preserve capacity for the climb.
Model Modifications
Adapting the minimum lap time simulator to descending required three additions. Rolling dynamics were introduced to account for the effect of leaning into corners on effective radius and stability. Aerodynamic tuck was added as a third control variable alongside propulsive/braking force and steering angle, allowing the rider to trade metabolic pedalling efficiency for a more favorable drag position. Metabolic efficiency is further penalized at high cadence, representing the point at which coasting in supertuck becomes more efficient than generating small surplus propulsive power.
The modified model has 5 degrees of freedom: longitudinal and lateral position, yaw, energy level, and roll. On a descent discretized into 2500 segments, optimization involves 20,000 variables.
Multi-Objective Framing
Minimizing time while allowing recovery is a Pareto front problem. For simplicity, the recovery objective is converted into a hard constraint: a maximum total mechanical energy expenditure of 350 kJ, corresponding to approximately 420 kcal at 20% metabolic efficiency, or roughly 230 W average. The optimizer then finds the fastest trajectory subject to this energy budget.
The animation shows the optimized trajectory and pacing plan under this constraint, including the timing of tuck adoption, braking zones, and cornering radii.
Scope
With accurate rider and bicycle characterization, descending simulation of this type connects directly to race strategy preparation. The same framework extends to motor sport applications where driver endurance and tyre degradation enter the energy constraint.