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The authors have declared that no competing interests exist.

Sperm cells undergo a wide variety of swimming patterns by a beating flagellum to maintain high speed regardless of the rheological and physical properties of the background fluid. In this work, we develop and control a soft robotic sperm that undergoes controllable switching between swimming modes like biological sperm cells. The soft robotic sperm consists of a magnetic head and an ultra-thin flexible flagellum, and is actuated using external magnetic fields. We observe that out-of-plane wobbling of the head results in helical wave propagation along the flagellum, whereas in-plane wobbling achieves planar wave propagation. Our theoretical predictions and experimental results show the ability of the soft robotic sperm to change its swimming speed by tuning the beating frequency of its flagellum and the propulsion pattern. The average speed of the soft robotic sperm increases by factors of 2 and 1.2 in fluids with viscosity of 1 Pa.s and 5 Pa.s at relatively low actuation frequencies, respectively, when they switch between planar to helical flagellar propulsion.

Sperm is a self-motile cell with approximately 40-50 μm in length. Different beat patterns of sperm cells can be observed by varying the viscosity of the background fluid [

Switching between propulsion modes necessitates fabrication of soft microrobots with a controllable deformation pattern. This fabrication requirement has been demonstrated by Huang

(a) Scanning electron microscopy images of the robotic sperm indicate similar morphology to that of biological sperm cells. The robotic sperm consists of a magnetic head and a flexible flagellum, and is fabricated using polystyrene. Magnetic particles are impeded into the head to provide magnetization (M). (b) Planar flagellar propulsion is achieved by applying in-plane uniform field (B) along direction of motion with a sinusoidally varying orthogonal component. (c) Wobbling of the head of the robotic sperm is achieved by applying out-of-plane field.

The robotic sperm used in this study is an electrospun beaded fiber (see _{t}, length _{t}, and modulus of elasticity _{r}, _{r}, _{r}), as shown in _{y}(_{z}(_{r}- and _{r}-axis, respectively. With this vector, the moving Frenet-Serret frame can be expressed as follows:
_{r} are the length and linear velocity of the robot in its frame of reference, respectively. The planar or helical wave propagation along the tail is achieved via magneto-elastic coupling between the tail and the magnetic torque generated by the magnetic field. The propagation of these waves along the tail is responsible for the propulsion with translational velocity _{r} and rotational velocity _{r}. We assume small local deformations and use local rotations based on Frenet-Serret coordinates (_{i}(_{r}-direction (_{r}-direction (_{i}(_{r} and _{g}, _{t}, _{n}, and _{b} are the local forces along the local tangent, normal, and bi-normal directions throughout the elastic tail, as demonstrated in _{t}, _{n}, _{b} denote the tangential, normal, and bi-normal local force coefficients of the tail in the local Frenet-Serret coordinates, respectively [_{m} and _{m} are the magnetic force and magnetic torque exerted on the magnetic dipole of the robotic sperm, and _{g} and _{g} are the force and torque due to gravity and given by
_{r} and _{disp} are the mass of the robot and the displaced mass of the medium by the robot, respectively, _{r} is the rotation matrix from robot’s frame of reference to laboratory frame of reference. In (_{f} and _{f} denote the following drag force and torque:
_{h} denotes local cross product as follows:
_{cov}, _{cov}, _{cov}} denote the position of center of volume of the head with respect to the center of mass of the robotic sperm. Further, _{h} and _{h} are diagonal resistance matrices for blunt prolate spheroids and given by
_{x} and _{yz} are given by
_{x} and _{yz} are given by

Motion and deformation of the tail are observed using side- and top-view cameras and microscopic lenses. The soft robotic sperm is contained inside a deep chamber of silicone oil with viscosity of 1 Pa.s and 5 Pa.s. The chamber is located at the common center of the eight coils. The robot has tail diameter 2_{t}, length _{t}, and modulus of elasticity

We examine a group of robotic sperm samples suspended in an observation chamber containing silicone oil (Calsil IP 5.000, Caldic, Rotterdam, The Netherlands) with viscosities of 1 Pa.s and 5 Pa.s. The samples are observed using two microscopic systems to determine the swimming patterns from top- and side-view of the chamber, as shown in

The deformation of the tail and the drag forces along the tail are determined using finite-difference discretization (tail is discretized into 250 equally spaced mesh nodes). The partial differential ^{−5} seconds). The velocity of the robotic sperm is obtained by solving (

The measured velocity of the robotic sperm is compared to the theoretical prediction of our model. It is also necessary to compare the deformation of the tail. However, our problem is an ‘initial condition problem’ that requires measuring the precise initial condition (position and orientation of the swimmer) and the initial transient of the electronics (the electromagnetic behavior of the driving system during startup). Therefore, the calculated time-dependent deformation along the tail will not be in a quantitative agreement with the measured deformations of the tail. In addition, geometric aberrations along the elastic tails of the fabricated robotic sperm samples via electrospinning hinders our effort to obtain quantitative agreement between the predicted time-dependent behaviour and measured deformations. Therefore, time-averaged velocity of the robotic sperm is only used to compare our theoretical predictions to experimental results.

We achieve planar flagellar propulsion by applying in-plane periodic magnetic fields. These fields exert a magnetic torque to achieve wobbling of the head and induce a planar wave along the flexible tail. In

The robot swims at an average speed of 18.9 ± 0.5 μm/s (

(a) Helical flagellar propulsion enables the robot to swim at higher speed than planar flagellar propulsion below actuation frequency of 6 Hz (

Now we turn our attention to helical flagellar propulsion by applying out-of-plane magnetic fields. These fields are generated such that the resultant magnetic field at the position of the head follows a cone pointing in the direction of motion.

The robot swims at an average speed of 38.1±0.6 μm/s in silicone oil with viscosity 1 Pa.s, at frequency of 1 Hz (

In order to mimic the unique ability of sperm cells to achieve planar-to-helical transition and conversation from helical-to-planar beating, we apply the two mentioned electromagnetic field patterns on the same robotic sperm to determine its response.

The speed of the robot is increased by a factor of 1.97 after the switch (

The central result from our hydrodynamic model is shown in

Helical flagellar propulsion is more efficient than planar flagellar propulsion at relatively low actuation frequency regardless to the viscosity of the fluid. (a) At low viscosity (

(a) Helical flagellar propulsion enables the robot to swim at higher speed than planar flagellar propulsion till actuation frequency of 2 Hz (

We demonstrate the ability of a soft robotic sperm to change its swimming speed controllably by tuning the actuation frequency and its propulsion modes. This capability enables biological sperm cells to maintain high swimming speed regardless of the rheological properties of the background fluid. Our experimental results and theoretical predictions show that helical flagellar propulsion is more efficient than planar flagellar propulsion for relatively low actuation frequencies. In silicone oil with viscosity of 1 Pa.s, helical flagellar propulsion is twice as fast as planar flagellar propulsion for relatively low actuation frequencies, and in silicone oil with viscosity of 5 Pa.s, helical propulsion achieves higher speed than planar propulsion by a factor of 1.2 for low actuation frequencies. Similar to their biological counterparts, soft robotic sperm has to switch from helical-to-planar flagellar propulsion to avoid relatively large reduction in speed at high actuation frequencies. The ability to switch between propulsion modes to maintain relatively high swimming speed is essential for navigation in bodily fluids (with different viscosities) to perform targeted diagnosis, targeted therapy, and broad biomedical applications [

The robot swims at an average speed of 18.9 ± 0.5 μm/s (

(MP4)

Helical flagellar propulsion enables the robot to swim at higher speed than planar flagellar propulsion below actuation frequency of 6 Hz.

(MP4)

The robot swims at an average speed of 38.1 ± 0.6 μm/s in silicone oil with viscosity 1 Pa.s, at frequency of 1 Hz.

(MP4)

The speed of the robot is increased by a factor of 1.97 after the switch.

(MP4)

The dataset provides the deformation of the tail at different time instants.

(ZIP)

The dataset provides the average velocity of the robotic sperm versus the actuation frequency.

(ZIP)

The dataset provides the deformation of the tail at different time instants.

(ZIP)

The dataset provides the deformation of the tail at different time instants.

(ZIP)

The dataset provides calculated sperm number for various viscosities.

(ZIP)

The dataset provides the average velocity of the robotic sperm versus the actuation frequency.

(ZIP)