Running on a slope: A collision-based analysis to assess the optimal slope.

When running, energy is lost during stance to redirect the center of mass of the body (COM) from downwards to upwards. The present study uses a collision-based approach to analyze how these energy losses change with slope and speed. Therefore, we evaluate separately the average collision angle, i.e. the angle of deviation from perpendicular relationship between the force and velocity vectors, during the absorptive and generative part of stance. Our results show that on the level, the collision angle of the absorptive phase is smaller than the collision angle of the generative phase, suggesting that the collision is generative to overcome energy losses by soft tissues. When running uphill, the collision becomes more and more generative as slope increases because the average upward vertical velocity of the COM becomes greater than on the level. When running downhill at a constant speed, the collision angle decreases during the generative phase and increases during the absorptive phase because the average downward vertical velocity of the COM becomes greater. As a result, the difference between the collision angles of the generative and absorptive phases observed on the level disappears on a shallow negative slope of ∼-6°, where the collision becomes 'pseudo-elastic' and collisional energy losses are minimized. At this 'optimal' slope, the metabolic energy consumption is minimal. On steeper negative slopes, the collision angle during the absorptive phase becomes greater than during the generative phase and the collision is absorptive. At all slopes, the collision becomes more generative when speed increases.