Albatross Optimized Flight Physics: A Natural Extremum Seeking System?

By Sameer Pokhrel and Sameh A. Eisa

Albatrosses belong to a fascinating group of birds known as soaring birds. The physics of albatross flight has captured the interest of biologists, physicists, engineers, and applied mathematicians because of a magnificent flight maneuver known as dynamic soaring (DS), which enables the birds to fly while exerting little to no energy; they barely need to flap their wings and instead acquire lift from the wind itself [5]. Albatross optimized flight has thus fascinated onlookers for centuries, with observations that date back to Leonardo da Vinci [9] and Lord Rayleigh [8].

Figure 1 and Animation 1 both depict the albatross DS maneuver, which typically consists of a four-phase cycle: (i) Windward climb, (ii) high-altitude turn, (iii) leeward descent, and (iv) low-altitude turn. One can ideally consider the DS cycle as energy neutral or near neutral. During the cycle, a soaring bird flies into the headwind and gains lift due to wind shear: a physical property that occurs when wind speeds change significantly with altitude (often above the ocean). The resulting lift helps the bird gain height or altitude. When it can no longer sustain this height increase, the bird performs a high-altitude turn and starts to descend — trading gained potential energy for kinetic energy. Upon reaching the lowest possible altitude, it performs a low-altitude turn and begins another DS cycle. Each DS cycle allows soaring birds to travel a distance without utilizing much energy. 

Figure 1. An albatross or other soaring bird conducts a dynamic soaring (DS) maneuver (black trajectory) in the presence of a wind spatial distribution (green arrows). Wind speed changes considerably with height, creating the wind shear/gradient that is a prerequisite for DS. The bird moves into the headwind to acquire lift, which allows the bird to gain height up to a certain point. The bird then turns and glides/descends, trading the gained potential energy with kinetic energy to increase its velocity and momentum. At a low altitude, energized by the gained velocity/momentum, the bird then turns into the headwind and repeats the DS cycle. Figure courtesy of the authors.

Multiple studies have experimentally observed and validated DS [10]. From a physical and engineering point of view, the energy neutrality—or near neutrality—implies that DS is a conservative-like flying technique, which is extremely rare (if not completely unique); the energy from the wind balances out the energy that is traditionally lost in flight dynamic systems due to the non-conservative drag force.

For decades, researchers configured DS as an optimal control and/or dynamic optimization problem [5]. This configuration requires accurate dynamical system modeling of the albatross or mimicking object, accurate wind profile models, and a defined mathematical formulation of an objective function that aims to conserve energy and minimize its dissipation. The solution of such an optimal control problem is the DS trajectory that is taken—or should be taken—by the bird or mimicking object. If we solve the DS problem, we can then design an elegant and extremely energy-efficient class of unmanned aerial vehicles (UAVs) that mimic soaring birds. Such an achievement would be a miraculous technological feat. 

The equations of motion that represent DS [2, 4, 7] for a bird with mass \(m\) and wing area \(S\) are as follows:

\[\dot{x} =V \cos\gamma \cos\psi,\]

\[\dot{y} =V \cos\gamma \sin\psi-W,\]

\[\dot{z} = V \sin\gamma,\]

\[m \dot{V} = -D -mg\sin\gamma+m\dot{W} \cos\gamma \sin\psi,\tag1\]

\[mV\dot{\gamma} = L \cos \phi - mg \cos \gamma - m \dot{W} \sin \gamma \sin \psi,\]

\[mV \dot{\psi }\cos \gamma  = L \sin \phi + m \dot{W} \cos \psi.\]

Here, \(x\), \(y\), and \(z\) represent the bird’s position in the east, north, and upwards frame of reference. \(V\), \(\gamma\), \(\psi\), and \(\phi\) are respectively the velocity, flight path angle, heading angle, and roll angle. The bird is subject to a generated lift force \(L\) and drag force \(D\), and the wind environment has a density \(\rho\), velocity \(W\), and shear gradient \(\dot{W}\). It also changes with altitude and causes wind shear that follows the logistic wind profile model, which is parameterized by freestream wind speed \(W_0\), shear layer thickness \(\delta\), and altitude \(z_m\) that corresponds to the middle of the shear layer. The green arrows in Figure 1 depict this wind profile model, which can be represented by a sigmoid model of the form

\[W(z)=\frac{W_0}{1+e^{-(z-z_m)/\delta}}.\tag2\]

Parameters of the system \((1)\) are based on existing albatross parameters [2, 4, 7].

To generate the optimal trajectory of DS in Figure 1—based on decades of previous efforts in the literature—we must do the following: (i) Assume the presence of wind shear, subject to a wind profile model like \((2)\) or other documented models [5]; and (ii) impose a set of path, state, and control constraints that allow an optimal control (or dynamic optimization) algorithm or solver to numerically determine the DS trajectory for either the bird or a mimicking UAV. For example, use of the powerful multiple-phase optimal control solver GPOPS-II [6]—which utilizes hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming—necessitates the satisfaction and incorporation of the following constraints and bounds into the software algorithm:

\[V_{\min}<V<V_{\max}, \psi_{\min}<\psi<\psi_{\max},\]

\[\gamma_{\min}<\gamma<\gamma_{\max}, x_{\min}<x<x_{\max},\]

\[y_{\min}<y<y_{\max}, z_{\min}<z<z_{\max},\tag3\]

\[\phi_{\min}<\phi<\phi_{\max}, C_{L_{\min}}<C_L<C_L{_{\max}},\]

where \(()_{\min}\) and \(()_{\max}\) respectively denote minimum and maximum values. One must also consider an expression for the objective function in the optimal control solver.

We believe that the historical optimal control and/or dynamic optimization problem configurations of DS do not match the biological behavior of soaring birds and albatrosses. Here, we make several observations to enforce this point: (i) Soaring birds do not require high computational power or non-real time processing to conduct the DS maneuver; (ii) there is no mathematical expression for an a priori objective function that dynamically optimizes their flight physics; and (iii) they can sense their environment and conduct periodic behavior based on that sensing. Driven by these observations, we focused on finding mathematical control systems and models—specifically extremum seeking control (ESC) systems—that naturally describe the biological behavior of soaring birds [1, 3]. Previous studies have rarely (if ever) employed ESC systems to model or describe biological phenomena, especially in bird flight. 

Figure 2. Parallelism between dynamic soaring (DS) and extremum seeking control (ESC) systems. One can consider the (pitching and rolling) variation/perturbation action of the bird as the modulation step. These actions then affect the flight dynamics state space, including the velocity and orientation. The bird senses the change in wind, velocity, and orientation, similar to a measurement of the objective function; ESC systems do not require its mathematical expression. Finally, an assessment of the measurement determines whether or not the energy conservation is met properly. Hence, the bird updating its actions according to the feedback is similar to the demodulation and parameter update steps in ESC. Figure courtesy of the authors.

Real-time ESC systems can steer a given dynamical system that is driven by a periodic perturbation/excitation to the extremum (maximum or minimum) of an objective function for which we do not have its closed form a priori. Instead, ESC systems operate based on access to measurements of the objective function. They can also tolerate less-than-proper models as long as the objective function is accurately measured. We therefore hypothesize that albatross optimized flight physics (DS maneuver) manifests as an extremum seeking system in nature. The bird simply steers itself towards an optimal energy state that minimizes its energy spending (and maximizes its energy gain) based on real-time sensing of wind, height, velocity, and so forth to assess its energy state. Moreover, ESC systems are simple structures with periodic control inputs (actuation) that are easier to generate, learn, and memorize.

To prove the concept of our hypothesis that one can characterize DS as an extremum seeking system, we used two different ESC structures: the classic structure [7] and a control-affine structure [4]. Figure 2 and Animation 2 depict the parallelism between the general philosophy of such ESC structures and DS.

Animation 2 briefly explains our hypothesis and provides a real-time simulation that highlights the strength of our results; given a simple periodic perturbation input, ESC structures perfectly conduct the DS maneuver in real time without the constraints of \((3)\) [4, 7]. Our results are comparable with those of the powerful optimal control optimizer GPOPS-II [6] when it operates offline/in non-real time. 

These findings offer further evidence to support our hypothesis that extremum seeking systems manifest in nature and biological systems, and the physics of albatross flight is one such example. We believe that this result can revolutionize biomimicry in robotics and open the door for further research with applications in systems biology and natural phenomena. Numerous other birds and biological organisms likely manifest and exploit the simple philosophy of ESC systems in similar ways.

References
[1] Ariyur, K.B., & Krstic, M. (2003). Real-time optimization by extremum-seeking control. Hoboken, NJ: John Wiley & Sons.
[2] Bousquet, G.D., Triantafyllou, M.S., & Slotine, J.-J.E. (2017). Optimal dynamic soaring consists of successive shallow arcs. J. R. Soc. Interface, 14(135), 20170496.
[3] Dürr, H.-B., Stanković, M.S., Ebenbauer, C., & Johansson, K.H. (2013). Lie bracket approximation of extremum seeking systems. Automatica, 49(6), 1538-1552.
[4] Eisa, S.A., & Pokhrel, S. (2022). Analyzing and mimicking the optimized flight physics of soaring birds: A differential geometric control and extremum seeking-like approach with real time implementation. Preprint, arXiv:2207.08564.
[5] Mir, I., Eisa, S.A., & Maqsood, A. (2018). Review of dynamic soaring: Technical aspects, nonlinear modeling perspectives and future directions. Nonlin. Dyn., 94, 3117-3144.
[6] Patterson, M.A., & Rao, A.V. (2014). GPOPS-II: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive gaussian quadrature collocation methods and sparse nonlinear programming. ACM Trans. Math. Softw., 41(1), 1-37.
[7] Pokhrel, S. & Eisa, S.A. (2022). A novel hypothesis for how albatrosses optimize their flight physics in real time: An extremum seeking model and control for dynamic soaring. Bioinspir. Biomim., accepted manuscript.
[8] Rayleigh, L. (1883). The soaring of birds. Nature, 27, 534-535.
[9] Richardson, P.L. (2019). Leonardo da Vinci’s discovery of the dynamic soaring by birds in wind shear. Notes Rec., 73(3), 285-301.
[10] Sachs, G., Traugott, J., Nesterova, A.P., & Bonadonna, F. (2013). Experimental verification of dynamic soaring in albatrosses. J. Exp. Biol., 216(22), 4222-4232.

Sameer Pokhrel is a Ph.D. student in the Department of Aerospace Engineering and Engineering Mechanics at the University of Cincinnati, where he works as a research assistant in the Modeling, Dynamics, and Control Lab (MDCL). He received his bachelor's degree in mechanical engineering from Tribhuvan University in Nepal. Pokhrel’s research interests lie in optimization, bio-inspired robotics, nonlinear dynamics, and control theory.

Sameh A. Eisa is an assistant professor of aerospace engineering and engineering mechanics at the University of Cincinnati. He is also the principal investigator of MDCL. Eisa holds a B.Sc. in electrical engineering and a Ph.D. in applied and industrial mathematics, and has postdoctoral and lecturing experience in mechanical and aerospace engineering. He is interested in nonlinear dynamics, control theory, and their applications in bio-inspired systems and renewable energy.