Why do we like to walk when we move slowly, but run when moving quickly? There are many ways of approaching such a question, but a fairly fundamental start can be made by considering the energetic requirements of gaits. Back in 2006, Manoj Srinivasan and Andy Ruina (at Cornell University) reported a computer optimisation of a ‘minimal biped model’, which searched for the energetically cheapest way of moving the body along. Despite the extreme simplicity of the ‘minimal biped model’ – consisting only of a point mass and massless legs – they (and their computer) discovered walking and running to be energetically optimal at slow and fast speeds, respectively.

But the really interesting bipeds – humans, birds and robots – do deviate quite significantly from the extreme reduction used in the 2006 computer optimisations. Does this matter? How should we walk and run if a little more realism is included? It is this that Srinivasan (now at Ohio State University) starts to approach, by sticking with massless legs and point-mass bodies, but including knees, spring tendons and a range of models for cost due to muscles.

Again, Srinivasan asked a computer to calculate many, many possible ways each biped model could possibly locomote. And then those gaits that allow locomotion at minimum energetic cost for a given speed were found.

Walking at low speeds and running at high speeds were again found. Without sensible constraints, extreme forms of ‘inverted pendulum’ walking and ‘impulsive’ running were found to be best. This means walking with a hugely forceful push-off, followed by a completely passive vault and finishing with another hugely forceful crash as the next foot hits the ground; for running, this means immensely forceful, but brief, vertical forces interspersed with passive ballistic aerial phases between each stance. Of course, neither biology nor engineering ‘likes’ infinite forces, even if they may be, according to extreme reductionist models, theoretically optimal. Adding any one of many sensible constraints or cost functions quickly makes the predicted gaits more realistic, and removes the prediction of huge leg forces.

Interestingly, these findings also stand for a model including a knee, and cost model appropriate for an electric motor (a ‘knee-torque-squared’ cost). So, the energetically sensible way for kneed robots to move is... pretty much like us; expect future C-3POs to walk with a vaulting, inverted pendulum path when slow, and run with ballistic aerial phases when fast.

Unsurprisingly, adding a lossless spring to the model allows gaits to be found that require no work from the muscles; the springs perform all the work required from the legs. However, with the inclusion of an energetic cost associated with the force (as opposed to the work) applied by the muscles, more compliant gaits again become favoured. Srinivasan also describes the family of muscle cost models – within which standard models fall – that predict walking and running, with a gait transition in between. For all these muscle cost models, the gaits minimising the cost of the body motions are mostly indistinguishable from those minimising the cost of the muscle work.

So, walking and running gaits are energetically sensible, and remain so even when further complexity is added to the model bipeds. Also, more realistic gaits (those without infinite forces) can be predicted with many of these additions. But exactly which additions are most realistic – exactly which muscle cost model? And how much tendon elasticity? This remains to be determined.

Fifteen observations on the structure of energy-minimizing gaits in many simple biped models.
J. R. Soc. Interface.
doi: 10.1098/rsif.2009.0544