Fig 1.
The motion progression of the three studied strategies to overcome an obstacle. Energetic cost of collision is derived and completely defined for each strategy by wheel radius R, eccentricity a of the wheel, the mass located in the centre of mass m, the moment of inertia around the centre of mass I, the obstacle height h, and the approach velocity ux. (A) Motion progression of the rotational hopping strategy using an off-centred wheel. Hopping is induced by reverting the rotation during stance phase. (B) Motion progression of the trivial hopping strategy using a wheel with centred mass. Hopping is induced by an increase of the kinetic energy in vertical direction. (C) Motion progression of the rolling strategy. The obstacle is overcome by a wheel with centred mass colliding and rolling over it.
Fig 2.
Generalised coordinates q = [x, y, ϕ]T with x the horizontal displacement from origin O, y vertical displacement from origin O, angular position ϕ, and system parameters mass m, moment of inertia I, wheel radius R, eccentricity a, and obstacle height h. (I) State just before angular impulse. The angular velocity of the system is reverted in a collisional event with angular impulse ζR from to
. The velocity vCoM dictated by the rolling motion then leads to a ballistic flight phase. (II) Ballistic flight phase. (III) State just before impulsive energy loss due to impact ζ. Note that
.
Fig 3.
Sketch of the experimental conditions. A wooden and rigid wheel-axle system is placed on top of a ramp, released, and guided towards an obstacle. Depending on the locomotion strategy, the system is either passively negotiating the obstacle or hopping over it by an impulse. The motion is recorded with a motion capturing system using four trackable markers placed on one face of the wheel.
Fig 4.
Costs for rotational hopping strategy.
The two terms of the energetic collision loss in Eq (9) as a function of forward speed. The blue line corresponds to the cost of retraction or rotation at take-off, and the yellow line corresponds to the energetic discrepancy between initial (pre-impulse) and end (post-impact) state. The black dashed line indicates the trivial hopping strategy’s collision loss, equal to mgh. The parameters are: mass m = 80kg, obstacle height of h = 0.3R, Radius R = 1.05m, moment of inertia I = 0.78kgm2, and the take-off angle ϕ* = −0.43rad.
Fig 5.
Optimality regions of rotational hopping strategy and rolling strategy.
Regions of strategies with least energy loss to overcome an obstacle of height h as a function of the moment of inertia factor α = I/mR2, and the Froude number . The results shown are independent of the mass of the wheel.
Fig 6.
Optimal locomotion strategy for animal related parameters.
Regions of optimal strategies as a function of body mass m, locomotion speed ux, and obstacle height h. Parameters are set using the allometric relations (17)–(20), which scale the wheel radius R such that it corresponds to the animal leg length, the point mass m corresponds to animal body mass, and the moment of inertia around the centre of mass I corresponds to leg moment of inertia around the hip. (A) Rolling strategy compared to trivial hopping strategy and their optimal regions. (B) Rolling and rotational hopping strategy and their optimal regions. Walk to hop/run/trot gait transitions for various animals are indicated [26–30].
Fig 7.
Motion progression of experimentally tested strategies.
(A) motion progression of the rolling experiment at an approach speed of 1.8 m/s. (B) motion progression of the trivial hopping experiment at an approach speed of 1.9 m/s. (C) motion progression of the rotational hopping experiment by throwing the wheel. Approach speed corresponds to 2 m/s.
Fig 8.
Experimental collision loss for rolling and trivial hopping strategies.
(A) Energy loss of rolling strategy and trivial hopping strategy as a function of pre-collision energy for an obstacle height of 0.18R. Error bars indicate one standard deviation. (B) Energy loss of rolling strategy and trivial hopping strategy as a function of pre-collision energy for an obstacle height of 0.39R. Error bars indicate one standard deviation.
Fig 9.
Prediction error of energetic collision loss for rotational hopping strategy.
The off-centred mass wheel was thrown over the obstacle, and the touchdown position and velocity state was used to predict theoretical loss ΔETheor, which was compared to the experimental loss ΔEExp to give the prediction error ΔEPred. The value is normalized with the total energy at touchdown ΔETotal = ΔEKin + ΔEPot.