# Optimal Strokes of Low Reynolds Number Linked-Sphere Swimmers
###### Abstract  
Optimal gait design is important for micro-organisms and micro-robots that propel themselves in a fluid environment in the absence of external force or torque. The simplest models of shape changes are those that comprise a series of linked-spheres that can change their separation and/or their sizes. We examine the dynamics of three existing linked-sphere types of modeling swimmers in low Reynolds number Newtonian fluids using asymptotic analysis, and obtain their optimal swimming strokes by solving the Euler-Lagrange equation using the shooting method. The numerical results reveal that (1) with the minimal 2 degrees of freedom in shape deformations, the model swimmer adopting the mixed shape deformation modes strategy is more efficient than those with a single-mode of shape deformation modes, and (2) the swimming efficiency mostly decreases as the number of spheres increases, indicating that more degrees of freedom in shape deformations might not be a good strategy in optimal gait design in low Reynolds number locomotion. low reynolds number swimming; self-propulsion; robotic swimmers; optimal strokes; asymptotic analysis; shooting method +
Footnote †: journal: Applied Physics Letters  
which make existing therapeutic and diagnostic procedures less invasive [9]. Different LRN swimming models and micro-robots have been designed since [PERSON]'s two-hinge model was advanced [1]. In particular, various linked-sphere types of models have appeared, since their simple geometry permits both analytical and computational results [10; 11; 12; 13; 14; 15; 16; 17]. These analytical and numerical results have greatly inspired the designs of micro robotic devices, for example, swimmers with 2 and 3 rotatory cylinders have been built to study the hydrodynamic interaction between a wall and an active swimmer [18; 19]. Other micro-robots inspired by analytical/numerical works include the Quadroar swimmer, which consists of rotating disks and a linear actuator [20], as well as the [PERSON]'s two-hinge model [21]. An important problem in LRN swimming is to find the optimized swimming stroke of the micro-swimmer, either (1) with respect to time, i.e., the stroke in one swimming cycle that moves the cell farthest, or (2) with respect to energy, i.e., among all strokes with designated starting and end points, find the one that consumes the least energy. These are usually called the _time optimal control_ and the _energy optimal control_ problems, respectively. Both optimal problems have attracted substantial interest in optimization as well as geometry [22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33]. Moreover, recently techniques from machine learning have been applied to linked-sphere types of model gait design, which allows incorporating environmental influences on the micro-swimmer's swimming behavior, including noise and a frictional medium [34]. Here by investigating the optimal strokes of a group of linked-sphere types of LRN modeling swimmers, we study the efficiencies of propelling mechanisms at LRN of different types of shape deformations. We start from three linked-sphere swimmers-Najafi-Golestanian (NG) 3-sphere accordion model (Figure 1a) [10; 11; 12], _pushmepulyou_ (PMPY) 2-sphere model (Figure 1b) [15], and the volume-exchange (VE) 3-sphere model (Figure 1c) [17]. All three models have only 2 degrees of freedom in their shape deformations, which is a minimal requirement that enables the swimmer to propel itself at LRN, according to the _scallop theorem_[1]. In particular, the shape deformations in the three models can be generalized as body elongation and/or mass transportation, and we investigate the efficiencies of these two shape deformation modes on the swimming performances. Then we generalize our results into a linear chain of spheres.