By Jillian Kunze
Figure 1. A group of Adélie penguins, some walking and some resting. Not pictured is the tobogganing behavior of Adélie penguins, in which they lie on their stomachs and propel themselves forward with their feet and flippers. Figure courtesy of
Hubert Neufeld on Unsplash.
Adélie penguins exhibit two main methods of motion as they travel across land: walking and tobogganing (see Figure 1). But how do they decide which variety of locomotion to use in a given scenario? Are they trying to reach their destination as fast as possible, or are they more concerned with saving energy as they navigate the harsh Antarctic landscape?
During the 2023 SIAM Conference on Control and Its Applications, which took place this week in Philadelphia, Pa., Angelia O’Toole of the Naval Postgraduate School delivered a contributed presentation that detailed her investigations into this intriguing avian issue. Along with Mark Karpenko (also of the Naval Postgraduate School), O’Toole developed a dynamic optimization formulation for predicting penguin travel tactics across terrains. “This problem came about during an optimal control project at school,” O’Toole said, describing the serendipitous circumstances that led her to delve into this new area of research over the last two years.
Throughout the talk, O’Toole referenced a 1991 paper that appeared in the Canadian Journal of Zoology titled “To Slide or Stride: When Should Adélie Penguins (Pygoscelis adeliae) Toboggan?” This thirty-year-old study explored whether penguins gauge which method of locomotion to use based on which is the most energetically favorable under the particular circumstances, and the authors spent three days observing the movements of penguins across a section of ground in Antarctica with varying snow depths and slopes. The theoretical work within this study indicated that tobogganing is always more energetically efficient when penguins travel downhill, or even uphill at slopes of less than seven degrees. However, the study authors felt that they must be missing something, since the data showed that real penguins do not actually toboggan so often.
To craft the optimization problem, O’Toole turned to both pen-and-paper and computational methods. Applying Pontryagin’s minimum principle allowed her to optimize a penguin’s trajectory to minimize either energy usage or travel time; this formulation could then be fed into DIDO, an optimal control software in MATLAB. The reference data came from the 1991 Canadian Journal of Zoology paper’s velocity and step data for both waddling and tobogganing. This paper also offered useful equations for energy efficiency, as well as the optimal switching problem between walking and tobogganing. The dynamics for both methods of locomotion had to be captured by different models, since walking penguins only propel themselves with footsteps whereas tobogganing penguins are also able to utilize their flippers as they slide.
Figure 2. The two modeled terrains for Adélie penguins to traverse. Figure courtesy of Angelia O’Toole.
The researchers generated two different terrain functions for the modeled penguins to traverse, with varying heights and slopes across each (see Figure 2). Each terrain had a mixture of downhill and uphill segments, requiring the penguins to switch their method of locomotion at some point. As the penguins move across these features, they want to minimize their time while maximizing their efficiency.
O’Toole first exhibited a candidate solution from DIDO in which time was the driving factor of the optimization (see Figure 3a). Here, a penguin switches from tobogganing to walking at an uphill slope of about 10 degrees. “How do we know the candidate solution is valid?” O’Toole asked. “We’re going to look at verification and validation.” The fact that the propagated values matched the state values, the co-state vector was nonlinear, and the Hamiltonian was constant all lent credence to the validity of the solution.
Figure 3. Optimal locomotion strategies for Adélie penguins across a varying terrain. The blue sections indicate where it is optimal for the penguin to toboggan, and the purple where it is optimal for the penguin to walk. 3a. Candidate solution where time is the driving factor. 3b. Candidate solution where efficiency is the driving factor. Figure courtesy of Angelia O’Toole.
The collaborators also investigated the effect of changing the weight of each locomotive method within the cost function; O’Toole displayed numerical results for simulations in which the efficiency of walking and tobogganing were weighted equally, or where one was weighted twice as heavily as the other. The analysis indicated that the penguins would decide to start walking earlier when walking was more heavily weighted in the cost function, but would start walking later when tobogganing was more heavily weighted.
“We could tell that the driving factor was time,” O’Toole said, based on these results. “But what if we weight efficiency more?” Figure 3b illustrates the optimal control strategy that DIDO produces when the efficiency was weighted significantly more heavily than in Figure 3a. Now, the penguin switches from tobogganing to walking at slopes of around seven degrees, which is a much more similar result to the 1991 Canadian Journal of Zoology paper.
Overall, this work was able to demonstrate that tobogganing is frequently the most efficient method of locomotion for Adélie penguins by validating the findings of an observational research paper through mathematical techniques. “Our results show that penguins care more about energy efficiency, but still care somewhat about time minimization,” O’Toole said. “Maybe this is what the 1991 research paper was missing.”
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Jillian Kunze is the associate editor of SIAM News. |