STRIDE II: A Water Strider-inspired Miniature Robot with Circular Footpads

3.1. Two-dimensional Modelling

To simulate the lift force for a circular footpad, first, the cross-section of a footpad (which will be considered as a pattern with several circles submerged in water as shown in Figure 2), needs to be investigated using the models proposed in [18]. The number of circles that are submerged is equal to the number of concentric circles in the footpad.

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Figure 2.

The side cross-section view of a three-concentric circle footpad. The areas A_b and A_st represent the areas that the footpad deforms because of its weight and the hydrophobicity of the leg material, respectively.

The weight of the water displaced by the footpad is identical to the amount of lift force the footpad generates. In other words, the weight of the water required to fill the volume bounded above by an undisturbed water surface (z = 0 line) and bounded below by the water-air interface is equal to the integral of the areas A_b and A_st along the 3D footpad geometry and is the total lift force. The buoyancy component and the surface tension component of the lift force are proportional to A_b and A_st, respectively [18, 21]. The shape of the air-water interface in the disturbed case is governed by the Young-Laplace equation. It is shown in [18] that this air-water interface profile (h(x)) can be calculated by numerically solving:

ρ g h ( x ) = γ d 2 d x 2 h ( x ) ( 1 + ( d d x h ( x ) ) 2 ) 3 / 2 ,

(1)

with the boundary conditions:

d h d x ( x 0 ) = tan ( θ c + ϕ − π ) ,

(2)

h ( ∞ ) = 0 ,

(3)

d h d x ( ∞ ) = 0.

(4)

In these equations, ρ is the density of water, g is the gravitational constant, γ is the coefficient of the surface tension (0.072 N/m), θ _c is the contact angle of the footpad material, ϕ is the submerge angle of the cross-section, and x₀ = Rsin(ϕ) is the intersection point of the air-water interface and the footpad cross-section where R is the radius of the footpad cross-section. After calculating the surface profile using (1)–(4), the total lift force is calculated by integrating the profile around the centre of the footpad.

Similar to the model in [18], the 2D water surface profile analysis of a cross-section forms the basis of the concentric footpad lift-force calculation. The basic differences are the profile interference due to nearby concentric circles and the change of the coordinate system from Cartesian to polar coordinates. To begin the calculations, an h(x) profile is created using the 2D analysis in [18]. The radius of the material is specified here and a specific h(x) is created by solving the Young-Laplace equation. Next, this profile is modified for several numbers of concentric circles. The concentric circles' radii are compared with pre-defined “infinity” (the distance where the water surface depth reaches its original value - i.e., the dimple disappears and is estimated as 1 cm using [18] and observations) and the various interferences are calculated. A new modified h(x) is created for a given number of concentric circles. Figure 3 shows the water surface profile calculated in MATLAB for a footpad that consists of five concentric circles with four, nine, 13, 17 and 21 mm radii, respectively.

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Figure 3.

The simulation result of the water surface profile h(x) for a footpad with four, nine, 13, 17 and 21 mm radii concentric circles

The modified water surface profile consists of data from the centre of the concentric footpad to the outer undisturbed surface. This modified surface should be integrated along the footpad in order to find the lift forces. A coordinate system transformation is conducted in order to take the integral around the centre of the footpad. Figure 2 shows how this transformation is carried out.

The polar coordinate system, (r^*, θ^*, z^*), where the integral for the lift force of the footpad will be calculated, is placed at the centre of the footpad as shown in Figure 2. z = 0 is the undisturbed water surface. Since the water surface profile only changes with the distance from the centre of the footpad, h(x) will be directly transformed to h(r). Following the coordinate system transformation, a polar integral around the centre of the footpad is performed to calculate the total amount of lift force produced by the footpad:

F l i f t = ∫ 0 2 π ∫ 0 r + ∞ r h ( r ) d r d θ .

(5)

To find the buoyancy component of the lift force separately, the area A_b should be integrated; therefore, the limits of the r integral should be changed. Similarly, for the surface tension component of the lift force, only the area A_t should be considered and the limits of the r integral should be adjusted appropriately.

The simulation results are calculated using the model described above and the experimental results are obtained by loading a single footpad with a platform that can carry a load. Several grams of sugar are placed on the platform above the footpad and the amount of weight that it can carry just before breaking the water surface and sinking is measured.

For the experiments, four different footpads with different numbers of concentric circles and different diameters were tried (8–42 mm, 14–42 mm, 8-26-42 mm and 20-32-42 mm). The diameter of the outer concentric circle is kept at 42 mm to keep the area bounded. The dimensions for these footpads are found by running simulations to maximize different parameters (Surface Tension/Buoyancy, Surface Tension, Total Lift × Surface Tension/Buoyancy, and Total Lift, respectively). The simulation and experimental results are shown in Table 1. The results show that the simulated lift force values are slightly lower than the experimental ones, except for the last footpad with three concentric circles with diameters of 20-32-42 mm. This overestimation is caused by the interference calculation part of the 3D model, which calculates where interference occurs and trims the interfered with parts. However, in reality the interference of the water surface profiles causes the slope of the water surface to be 0 at the intersections of the profiles seen in Figure 3, which would mean lower water surface levels at the intersection points, and hence which would cause an increase in the total lift force. On the other hand, the simulation assumes that the cross-sections of the footpads are circular, but in reality the cross-sections are rectangular. The sharp edges at the ends of the rectangles may break the water surface before the maximum lift force is achieved, creating a competing factor. These sharp edges are points where the pressure on the water surface increases. As these pressure localization points get closer to each other, they are grouped on a smaller area on the water surface. Hence, as the concentric circles are brought closer to each other, the actual payload capacity is unable match to the simulation results (which do not consider the sharp edges as a factor). This behaviour causes the water surface to break more easily than expected, which is the reason for the overestimation in the last row of Table 1.

The footpad with diameters 20-32-42 mm is chosen as the best design since it maximizes the lift force, which is the most significant quantity to be optimized. The total lift capacity of a 12 footpad robot would, therefore, be approximately 55 grams - more than necessary to carry on-board electronics, a 3.5 Volt battery, two miniature DC motors and two leg rotation mechanisms with a safety factor of approximately two.

The simulation results in [18] show that the surface tension-based lift force is directly correlated with the contact angle of the supporting leg material. However, the advantage of being hydrophobic diminishes for contact angles above 120 degrees, above which the lift force is approximately the same for any angle. The main reason behind this is that the surface tension force is limited by material properties if the material is hydrophilic or else is limited with a surface tension coefficient (0.072 n/m) if the material is hydrophobic. Hence, for materials with a contact angle above 120 degrees, the surface tension per unit length is almost equal to 0.072 N/m [24]. Therefore, the footpads are coated with a hydrophobic coating (Cytonix, WX2100™) with a contact angle of 145 degrees to obtain the best available surface tension force. The hydrophobicity of the leg enables us to maximize the lift force for a given geometry. This contact angle of 145 degrees is used in the simulations.

Similar to a fisher spider [25], three physical mechanisms are potentially responsible for the drag forces acting on the robot:

4.1.3. Hydrodynamic Drag

A moving, partially submerged body creates viscous drag consisting of a normal pressure component (form drag), a shear stress component (skin friction drag) and a trailing vortex wake component (interference drag). These three drag forces act over the whole of the submerged section of the body [27].

In the case of STRIDE II, the estimated velocity of the robot is well below c_min; thus, the capillary-gravity wave drag is not expected to contribute to the overall resistance to the robot's motion. In [24], it is shown that the maximum surface tension drag achieved before meeting water surface-breaking conditions is significantly less than the hydrodynamic drag. Therefore, it is asserted that hydrodynamic drag dominates the lateral resistance force on the robot.

The hydrodynamic drag is given in [27] as:

D H = 1 / 2 ρ A C D U 2 ,

(7)

where ρ is the density of water, A is the reference area, C_D is the drag coefficient including all contributions from the form drag, the skin friction drag and the interference drag, and U is the velocity of the robot with respect to the water. It is known that C_D varies as a function of the Reynolds number, R_e, of the object [27], which generally combines the fluid properties of density and viscosity, as well as the object's velocity and characteristic length. Although R_e, and thus C_D can vary greatly, the variation within a practical range of interest is usually small, so that C_D is often treated as a constant [28]. Usually, Eq. (7) is applicable with C_D as a constant when R_e > 1000.

It is worth noting that C_D is always associated with a particular surface area, A [29]. Here, this area is defined as the orthographic projection of the dimple's frontal area on a plane perpendicular to the direction of motion, which is:

A = h l

(8)

for the cylindrical wire-legs, where h is the dimple depth and l is the length of the wire-legs; or:

A = ∑ i h ( D i + d i )

(9)

for the footpads, where D_i and d_i are the outer and inner diameters of each concentric circle of the footpads, respectively. Figure 4 shows the defined reference areas of a cylindrical wire-leg and a footpad with three concentric circles.

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Figure 4.

(a) Schematics of the defined reference area of a cylindrical wire-leg: A = hl; (b) Schematic of the defined reference area of a footpad with three concentric circles: A = Σ _i h(D_i + d_i). U denotes the direction of velocity of the wire-leg or the footpad.

Note that the hydrodynamic drag has a simple quadratic relationship with U². In the next section, experiments are performed to show the existence of this relationship and thus demonstrate the dominance of the hydrodynamic drag.

As mentioned earlier, when the robot is in motion, the drag coefficient C_D stays constant if R_e > 1000. Experiments are performed to measure this constant value of C_D.

A passive robot platform with four sets of supporting footpads is built to conduct the experiments. In each experiment, an initial velocity is applied to the platform and its displacement with respect to time is measured using a video-tracking technique, shown in Figure 5(a). The results were then used to obtain the mean velocity and acceleration information by calculating the first and second derivatives in which the Savitzky-Golay filters were applied to smooth the data [30]. Figure 5(b) illustrates the experimental data acceleration with respect to velocity. The matching performance of the quadratic curve fitting implies a quadratic relationship between the acceleration and the velocity. Therefore, according to Eq. (7), it is reasonable to assert that the hydrodynamic drag serves as the primary resistance force for the footpads.

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Figure 5.

(a) Plot of experimental displacement data with respect to time. Savitzky-Golay filters were applied to smooth the data. (b) Plot of experimental data of acceleration with respect to velocity. The fitting model is y = ax² and the R-square value of the fit is 0.9974. This quadratic relationship justifies the dominance of the hydrodynamic drag in the resistance forces of the footpads.

Since, in the experiments, no propulsion force is applied to the platform and the hydrodynamic drag is considered to be the only source of resistance, substitute Eq. (7) into Eq. (6):

m a = F D = 1 / 2 ρ A C D U 2 .

(10)

Thus:

C D = 2 m a ρ A U 2 .

(11)

Given a from the experimental data and A calculated using Eq. (9), C_D can be directly computed. Figure 6 shows the variance of C_D with respect to R_e. It is shown that C_D varies slightly in the experimental velocity range (corresponding R_e range: 1671–2833), which overlaps with the estimated velocity range of the robot. By averaging the results shown in Figure 6, 0.155 is chosen as an estimated value of the drag coefficient for the supporting footpads.

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Figure 6.

Plot of the experimental drag coefficient with respect to the Reynolds number. 0.155 is chosen to be an estimate of the drag coefficient for the supporting footpads of STRIDE II.

As discussed before, the calculation of the resistance force of the robot, F_R, in Eq. (6), becomes fairly simple based on Eq. (11) and the experimental evaluation of C_D. But the analysis of the propulsion force, F_P, remains less clear, as the operating range of the rowing wire-legs' velocity is measured to range from 250 mm/s to 755 mm/s, which exceeds c_min ≈ 0.23 m/s. This fact implies that the capillary-gravity wave drag may become one of the sources of F_P. However, Bush et al. [15] conducted experiments with high-speed video and particle-tracking and concluded that the water striders transfer momentum to the underlying fluid not primarily through capillary waves but rather through hemispherical vortices shed by their rowing legs.

To obtain knowledge of the terminal velocity of the robot, a hydrodynamic drag force of a general form D = 1/2ρAC _D U² is assumed to act as F_P, where A is calculated as in Eq. (8) and C _D is chosen as 0.09, an experimental drag coefficient for a streamlined half-body which closely resembles the shape of the dimple associated with the wire-legs [27]. Let a in Eq. (6) be set to zero and substitute the expressions of F_P and F_R into 6:

1 / 2 ρ A w C D w U w 2 − 1 / 2 ρ A f C D f U t 2 = 0 ,

(12)

where U_w is the velocity of the rowing wire-legs with respect to the robot body, U_t is the terminal velocity of the robot, A_w and A_f are the reference areas of the wire-legs and the footpads, respectively, and C_Dw and C_Df are the drag coefficients of the wire-legs and the footpads, respectively. Equation (12) can be rearranged as:

U t = A w C D w A f C D f U w .

(13)

Since A_w and A_f can be calculated using Eq. (8) and Eq. (9), and C_Dw and C_Df are, respectively, determined as 0.09 and 0.155, a quantitative relationship between U_t and U_w can be obtained:

U t = 0.103 U w .

(14)

Using this equation, the terminal velocity of the robot can be estimated given the velocity of the rowing wire-legs, which can be further calculated from the gear motors' speed (a controllable parameter).

Experiments were performed to acquire velocity data of the robot and the wire-legs. By video observation, the motor speed was first measured and was then used to calculate U_w. U_t was directly obtained by video-tracking, as shown by Figure 7. Table 2 shows all the data from five different measurements, from which an approximate agreement with Eq. (14) is revealed. These experiments are able to justify the conclusion of Bush et al. [15] that capillary waves do not play an essential role in the propulsion, although the waves are still observable due to the rowing action.

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Figure 7.

Plot of robot velocity with respect to time in one of the robot velocity measurements. A terminal velocity of 36.6 mm/s can be obtained from this plot.

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Table 2.

Comparisons between experimental U_t and U_w values

Exp. #	Uw (mm/s)	Ut (mm/s)	Ut/Uw
1	286.1	28.6	0.1000
2	304.5	32.9	0.1080
3	507.9	48.9	0.0963
4	592.6	53.3	0.0899
5	617.7	59.1	0.0957
6	742.5	71.5	0.0963

5.1. Footpad Fabrication

It is noticed that, in order to obtain a high payload capacity, it is more desirable to use multiple relatively smaller footpads than a few large ones, as support from multiple loci is beneficial for a robot's static balance. This requires that the multiple footpads should be properly arranged in a well-balanced and space-saving configuration. Here, a tripod design as shown in Figure 8 is adopted. Each tripod consists of three concentric-circular footpads connected to a tri-beam with three ball and socket joints. When the tripod is mounted on the robot with the centre of the tri-beam fixed, the ball and socket joints are able to self-align each footpad parallel to the water surface. The design of the distances between each footpad complies with the careful-spacing to preserve 95% of the generated lift force [18].

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Figure 8.

Sketch of the tripod design for the footpads. Three ball and socket joints are employed to implement self-alignment.

The footpads are cut using a laser engraver (GCC, Venus) from a 1 mm thick Delrin® sheet and then coated with a super-hydrophobic material, fluorothane (Cytonix, WX2100™). All the other parts, including the tri-beam, the ball and socket joint and the truss are also formed by the laser engraver. Assembly is done by gluing.

In order to generate an elliptical-like trajectory for the rowing wire-legs, a crank-rocker four-bar mechanism is designed and used on STRIDE II, as shown in Figure 9(a). When the crank is continuously driven by the miniature gear motor, an elliptical-like motion is produced at the bottom vertex of the coupler triangle where the rowing wire-leg is mounted. The dimension design (Figure 9(b)) of the four-bar mechanism obeys three principles: (1) the contact between the rowing wire-legs and the water surface is maximized as much as possible so that more propulsion can be produced; (2) the rowing wire-legs will not break the water surface so that the supporting footpads will not be affected by the disturbance on the water surface; (3) Grashof's law is not violated: the sum of the lengths of the shortest and longest link is not greater than the sum of the lengths of the remaining two links. Figure 9(c) shows the simulation of the designed trajectory of the rowing wire-legs. When the rowing wire-leg is pushed down to 3 mm deep under the water surface, the length of contact can reach up to around 35 mm, which is about 2.5 times longer than that of the previous STRIDE [19].

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Figure 9.

(a) Sketch of the four-bar actuating mechanism with a miniature gear motor. (b) Significant dimensions (unit: mm) of the four-bar actuator. (c) Simulation of the trajectory of the rowing wire-leg. The origin is fixed at the locus of the gear motor.

The classical velocity analysis of a four-bar mechanism can provide the knowledge of the velocity of the rowing wire-legs. When the speed of the gear motor is set to be 150–450 rpm (which is experimentally decided as a proper range), the average linear velocity of the rowing wire-legs along the direction of the robot's motion while they stay in contact with the water surface is calculated to range from 250 mm/s to 755 mm/s.

The frame and all the joint parts of the actuating mechanism are fabricated using a rapid prototyping machine (3D Systems, Inc., Invision HR). The 20 mm long rowing wire-legs are formed by Teflon® coated stainless steel wires (diameter: 0.33 mm). The fabricated actuating mechanism, including the gear motor, weighs around 2.9 grams.

Two four-bar actuators with miniature gear motors (Precision Microdrives™, 206–101), a custom control board and a polymer Li-Ion battery (Powerizer, PL-651628-2C) take up the bulk of the weight of the robot body. To maximize the payload of the robot, the robot body itself and the connections between the supporting footpads and the body are designed to be as light and simple as possible. A 70 mm × 44 mm × 1 mm carbon fibre sheet is used as the main body of the robot. The two actuators are symmetrically attached on the two edges of the body by gluing. The control board and the battery are placed in the middle of the body and their positions are adjusted so that the centre of mass and the geometrical centre of the robot are co-located with each other. This guarantees that any unnecessary roll motion or motion of the robot can be avoided while the robot runs. Four sets of footpads are connected to the robot body, with four 50 mm long cantilevers made of carbon fibre beams with a small connector at each end. Figure 1 shows the fabricated STRIDE II, which weighs 21.75 grams.

5.4. On-board Electronics

To achieve more effective control, a custom control board is developed for STRIDE II. It includes a microcontroller (Microchip, PIC18LF2520), a motor driver (Freescale, MPC17C724), a voltage regulator (Analogic, AAT3221), an IR receiver (Vishay, TSOP36236), a timer (ST TS555), a trimming potentiometer (Bourns, 3223) and several resistors and capacitors. It is capable of power management, motor control and IR communication. The PCB is designed with the EAGLE layout editor (CadSoft Computer Inc.). The usage of SMD packaging for all the electronics helps to decrease the weight and volume of the board and hence maximize the payload of the robot.

5.5. Robot Experiments

The robot is controlled by using an IR remote. Figures 10(a) and 10(b), respectively, show a forward linear motion and a right-turning motion of the robot with speeds of 71.5 mm/s and 0.21 rad/s, respectively. No feedback control is applied; the voltage values for the motors that enable the robot to move in a straight line or with a predefined curvature are tuned beforehand.

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Figure 10.

Photo snapshots of STRIDE II in motion: (a) Forward linear motion at a speed of 71.5 mm/s. (b) Right turn at an angular speed of 0.21 rad/s.

The robot's power consumption and efficiency are calculated and presented in Table 3. Here, the motor efficiency is taken into consideration to obtain the input mechanical power to the actuators. Therefore, the calculated robot efficiency is based on the input mechanical power and output mechanical power rather than the input electrical power. The motors used have a fairly low efficiency (2–6% at the 150–450 rpm range), leading to excess power usage. This can be improved by using higher efficiency, lightweight motors. It is noted that the input and output mechanical power are in the order of microwatts, which implies that this robot can be operated at very low power.

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Table 3.

Experimental Robot Power Consumption and Efficiency Results

Robot	Input Mech.	Output Mech.	Robot
Speed (mm/s)	Power (μW)	Power (μW)	Efficiency
28.6	38.08	7.903	20.8%
32.9	53.92	12.03	22.3%
48.9	320.0	39.50	12.3%
53.3	319.6	51.16	16.0%
59.1	428.4	69.74	16.3%
71.5	641.5	123.6	19.3%

As mentioned in Section 3, the utilization of concentric circular footpads rather than the wire-legs used in STRIDE [24] resulted in an increase in lift force per unit area from 1/756 grams/mm² to 1/302 grams/mm². In addition to the increase in the total area of the lift force-generating mechanism, this improvement resulted in a robot capable of carrying 55 grams (whereas STRIDE was only able to carry 9.3 grams). The self-aligning nature of the footpads also had a positive impact on improving the lift force; STRIDE suffered from fabrication imperfections that caused the wire-legs to be slightly different from one another, resulting in a lower payload capacity. The utilization of DC motors with four-bar mechanisms in STRIDE II allowed us to keep the elliptical trajectory of the rowing legs but eliminated the need for the high actuation voltage of the piezoelectric unimorphs used in STRIDE. Therefore, we were able to make an untethered, on-board powered robot working with lithium-polymer batteries. The resulting new version was able to move faster on a straight line, achieving a speed of 7.1 cm/s compared to the 3 cm/s straight-line speed of STRIDE. However, the addition of the footpads and on-board electronics resulted in a heavier, wider robot, with a slower turning velocity of 0.21 rad/s compared to the 0.5 rad/s turning speed of STRIDE.