Archive for March 4th, 2008

Mathematical modelling of the slithering motion

March 4, 2008

Guo and Mahadevan (Yes, the same Mahadevan) have a paper in the latest issue of PNAS titled Limbless undulatory propulsion on land, which describes, mathematically, the slithering motion of snakes and worms on land:

We analyze the lateral undulatory motion of a natural or artificial snake or other slender organism that “swims” on land by propagating retrograde flexural waves. The governing equations for the planar lateral undulation of a thin filament that interacts frictionally with its environment lead to an incomplete system. Closures accounting for the forces generated by the internal muscles and the interaction of the filament with its environment lead to a nonlinear boundary value problem, which we solve using a combination of analytical and numerical methods. We find that the primary determinant of the shape of the organism is its interaction with the external environment, whereas the speed of the organism is determined primarily by the internal muscular forces, consistent with prior qualitative observations. Our model also allows us to pose and solve a variety of optimization problems such as those associated with maximum speed and mechanical efficiency, thus defining the performance envelope of this mode of locomotion.

The paper even makes a reference to Solomon’s proverb of the “way of a serpent on the rock”!

The crucial issue seems to be the following:

At first glance, motion by lateral undulation seems paradoxical; the animal glides forward continuously at a constant velocity, tangential to itself everywhere, despite the fact that the only forces in that direction are due to friction, which constantly retards this motion. The resolution of this apparent paradox is clear: propulsion arises due to the in-plane lateral forces generated when the organism braces the sides of its body against the medium or substrate in the presence of an anisotropy of resistance to lateral and longitudinal motion of the slender, curved body.

Guo and Mahadevan go on to describe the specific problem that is solved in their paper:

Locomotion, which is achieved through the appropriate coupling of endogenous dynamics to exogenous dynamics, involves four components: (i) the endogenous dynamics of force production by muscle, (ii) the exogenous dynamics due to the interaction of the organism with its environment, (iii) the consideration of linear and angular momentum balance in the body–environment system, and finally (iv) the proprioceptive feedback that involves sensorimotor coupling as the organism responds to the forces on and in it. Here we will focus on the first three components by considering the steady lateral undulatory movements of a snake or similar organism on a solid substrate to determine its shape and speed.

The paper ends with a nice, short summary of their results (and, on a promising note that there are many more questions to be answered):

A combination of analysis and numerical simulation of the model yields a number of results which we summarize: the normalized shape of the organism is determined primarily by the interaction of the organism with its external environment, whereas the speed and energetics of locomotion are determined by the internally generated periodic active moment generated by muscle contraction. These results are consistent with prior qualitative experimental observations. In addition, we can define parts of the performance envelope of undulatory propulsion on land by posing and solving some simple optimization problems for the maximization of the efficiency and velocity of the organism. However, our analysis is limited to a consideration of steady movement without proprioceptive feedback, which effect is crucial in understanding how the organism responds to the forces that it senses, so that many questions remain for further study.

Take a look!

Well, what time is it? 3 am at PhD comics!

March 4, 2008

If it is 3 am, and if your friends are safe and asleep, and if the phone in the lab rings, who should be there to pick it up? Here is the message approved by Prof. Smith 🙂 For those of you who are not following the US presidential electoral politics, the inspiration for the comics is here!