Costs for rotational hopping strategy.

<p>The two terms of the energetic collision loss in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194375#pone.0194375.e015" target="_blank">Eq (9)</a> 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 <i>mgh</i>. The parameters are: mass <i>m</i> = 80<i>kg</i>, obstacle height of <i>h</i> = 0.3<i>R</i>, Radius <i>R</i> = 1.05<i>m</i>, moment of inertia <i>I</i> = 0.78<i>kgm</i>², and the take-off angle <i>ϕ</i>* = −0.43<i>rad</i>.</p

Optimality regions of rotational hopping strategy and rolling strategy.

<p>Regions of strategies with least energy loss to overcome an obstacle of height <i>h</i> as a function of the moment of inertia factor <i>α</i> = <i>I</i>/<i>mR</i>², and the Froude number . The results shown are independent of the mass of the wheel.</p

Rotational hopping model.

<p>Generalised coordinates <b><i>q</i></b> = [<i>x</i>, <i>y</i>, <i>ϕ</i>]^<i>T</i> with <i>x</i> the horizontal displacement from origin <i>O</i>, <i>y</i> vertical displacement from origin <i>O</i>, angular position <i>ϕ</i>, and system parameters mass <i>m</i>, moment of inertia <i>I</i>, wheel radius <i>R</i>, eccentricity <i>a</i>, and obstacle height <i>h</i>. (I) State just before angular impulse. The angular velocity of the system is reverted in a collisional event with angular impulse <i>ζ</i>_<i>R</i> from to . The velocity <b><i>v</i></b>_<i>CoM</i> 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 <i>ζ</i>. Note that .</p

Experimental collision loss for rolling and trivial hopping strategies.

<p>(A) Energy loss of rolling strategy and trivial hopping strategy as a function of pre-collision energy for an obstacle height of 0.18<i>R</i>. 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 0.39<i>R</i>. Error bars indicate one standard deviation.</p

Experimental set-up.

<p>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.</p

Optimal locomotion strategy for animal related parameters.

<p>Regions of optimal strategies as a function of body mass <i>m</i>, locomotion speed <i>u</i>_<i>x</i>, and obstacle height <i>h</i>. Parameters are set using the allometric relations (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194375#pone.0194375.e024" target="_blank">17</a>)–(<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194375#pone.0194375.e027" target="_blank">20</a>), which scale the wheel radius <i>R</i> such that it corresponds to the animal leg length, the point mass <i>m</i> corresponds to animal body mass, and the moment of inertia around the centre of mass <i>I</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 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194375#pone.0194375.ref026" target="_blank">26</a>–<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194375#pone.0194375.ref030" target="_blank">30</a>].</p

Prediction error of energetic collision loss for rotational hopping strategy.

<p>The off-centred mass wheel was thrown over the obstacle, and the touchdown position and velocity state was used to predict theoretical loss Δ<i>E</i>_<i>Theor</i>, which was compared to the experimental loss Δ<i>E</i>_<i>Exp</i> to give the prediction error Δ<i>E</i>_<i>Pred</i>. The value is normalized with the total energy at touchdown Δ<i>E</i>_<i>Total</i> = Δ<i>E</i>_<i>Kin</i> + Δ<i>E</i>_<i>Pot</i>.</p

Construction process.

<p>This example illustrates the execution of the third (and last) gene of the genome shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0128444#pone.0128444.t001" target="_blank">Table 1</a>. (a) The initial configuration shows the state after construction of the first two genes, i.e. a passive element is fixed onto an active module (labeled as structure) and a second active element is prepared. (b-c) Rotation of the structure by -90° around the z-axis and 90° around y. (d) Rotation of the new element by -90° around y. (e-f) HMA is applied to the structure and the module is connected on top of the structure according to the position offset parameters Δx and Δy. (g) An end-rotation of the whole agent by 90° around the y-axis is performed. (h) The motor control parameters are assigned to the active elements and the agent can be evaluated.</p

Exploration of the design space.

<p>Each agent can be located in the design space given its number of elements and shape factor (A,B,C). The grey area shows the theoretical limit which can be reached using the cubic modules. The portion of this space that can be covered is restricted by the constraints which apply to the real-world construction process. The colored areas illustrate the parts of the design space that can be reached with different sets of active constraints. All constraints are active in (A), the stability condition is relaxed in (B) and no constraints are active in (C). This subsequently increases the reachable portion of the design space. The solid black markers indicate the distribution of the agents in experiment 1b (A) and experiment 2 (B). Their area is proportional to the number of agents in the bin. The right column (D-I) shows two example morphologies per experiment.</p

Developmental process.

<p>A “mother robot” (A) is used for the automatic assembly of candidate agents from active and passive modules. For the construction process, the robotic manipulator is equipped with a gripper and a glue supplier. Each agent is represented by the information stored in its genome (B). It contains one gene per module, and each gene contains information about the module types, construction parameters and motor control of the agent. A construction sequence encoded by one gene is shown in (C). First, the part of the robot which was encoded by the previous genes is rotated (C1 to C2). Second, the new module (here active) is picked from stock, rotated (C3), and eventually attached on top of the agent (C4).</p