Energetic analysis and experiments of earthworm-like locomotion with compliant surfaces

Our earthworm robot, shown in figure 1, is composed of four cells and weighs 0.333 kg. Its length is 160 mm and it has an effective stroke L_s of approximately 11.5 mm. The robot has passive clamps which increase the normal contact forces with its environment, thus allowing the robot to climb over large slopes and even vertically when fitted with rigid contacts. A compliant sponge-like material (cut from a household sponge) is attached to the tips of the passive clamps in order to incorporate compliance into the legs. The prototype used in the experiments reported here crawls at a low speed and completes a full cycle (the motion of all four cells) in approximately 2.5 s (Zarrouk and Shoham 2012b).

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The tangential compliance and COF of the sponge material used on the legs were measured in a set of independent experiments conducted with a parallel robot (Simaan and Shoham 2001) and nano-17 force sensor set up. In these tests, the robot end-effector with the sample material to be tested descends vertically to contact a rigid surface until the normal force reaches 2 N. Subsequently, the sample is moved horizontally through a distance of 5 mm (see Zarrouk and Shoham 2012b). The tangential force increases almost linearly as a function of the displacement until the latter reaches approximately 2.5 mm. At this stage, sliding occurs and, as a result, the tangential force decreases by approximately 30%. From this data, we estimated the static and kinetic COF to be 0.7 and 0.5, respectively. However, the static COF value is not used in the analysis nor the post-processing of the experimental results since it is not needed for the calculation of the frictional losses. The tangential compliance of a single sponge computed as the slope of tangential force versus tangential deflection curve, is approximately 0.49 N mm⁻¹, giving the overall tangential compliance of a cell of 0.98 N mm⁻¹ (each cell has two contacts). Note that in our analysis and post-processing of the experimental data, we assume this value to be the same for all cells of the worm robot.

In the experiments presented here, the robot was operated in a horizontal canal as well as in a sloped canal at 30° and 45°. Each experiment was executed for more than 10 cycles to produce enough measured data for post-processing. As annotated in figure 1, each cell was instrumented with an infrared marker and tracked using Vicon positioning system. The latter includes eight cameras and produces sub-millimetre accuracy (Zarrouk et al 2012a). With the Vicon system update rate of 120 Hz, each experiment yielded around 300 measurement points.

The robot contacts the surface at discrete points at the leg tips, which maybe compliant, as is the case for the prototype used in the present investigation. In addition, or alternatively, the compliance may arise from the environment, but in either case, we assume that the compliance effects remain local, that is, the friction force at cell/contact 'i' results in tangential deflection at the same cell only. It is emphasized that the energy analysis discussed in our work focuses on the tangential compliance effects: the Earthworm robot has no active clamping devices and therefore cannot change the normal forces it applies to the environment. The energy losses are therefore due to sliding over the surface only.

To explain the basic mechanics and energy losses of earthworm locomotion on a frictional compliant surface, we consider the position of a single cell during a cycle. In a locomotion cycle, in general, each cell makes one step forward and n−1 steps backward and hence, the individual cell position during a full cycle takes the general profile illustrated in figure 2. The motion of the active cell is characterized by a 'sticking' stage in which the cell moves while sticking to the surface (from beginning of step to onset of sliding), and a forward sliding stage (from onset of sliding to end of step), but no backward sliding. An inactive cell sticking to the surface while the active cell is actuated to advance may experience sliding backward once the maximum tangential deflection (of the leg or environment) is reached. The net effect is the loss of locomotion efficiency as detailed in our previous work (Zarrouk et al 2012a) and energy losses associated with sliding of the cells.

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In this section, we will illustrate the use of the analytical results obtained in prior work to carry out a basic design study for an earthworm robotic device with the number of cells as the main design parameter. Based on the analysis in (Zarrouk et al 2012c), it is possible to determine the energy required for locomotion per cycle while the distance travelled can be calculated from our previous work in (Zarrouk et al 2012a). The numerical parameter values that are used in this section are based on our experimental earthworm-like robot (described in section 2.1). Specifically, the stroke of the robot is 11.5 mm, the weight of the robot is 0.333 kg which we assume to be proportional to the number of the four cells, giving the weight of 0.0833 kg/cell. As noted earlier, the tangential compliance is 0.98 N mm⁻¹ and the coefficient of friction for forward and backward sliding is taken as 0.5, while the normal passive clamping force is estimated at approximately 3.4 N. In table 1, we summarize the computed energy requirements based on (Zarrouk et al 2012c) and the travel distance during a cycle, for earthworms with varying number of cells (n = 3, ..., 6). For three cells, the robot's advance is just over half of the stroke when moving horizontally and it fails to progress on a 45° incline. While increasing the number of cells increases the progress per cycle, which is beneficial, it also significantly increases the energy requirements.

In table 2, we present the energy of locomotion divided by the distance travelled and the COT. The energy to distance ratio is useful in determining the range of the robot for a given limited amount of energy, whereas the COT allows one to determine the range when the available amount of energy is proportional to the robot weight. Table 2 shows that the COT is nearly the same for any number of cells over horizontal surface. However, the COT decreases as a function of the number of cells for climbing over an incline implying that it would be beneficial to increase the number of cells. On the other hand, the energy per distance ratio suggests a different optimal number of cells depending on the incline: three cells for horizontal locomotion, four cells for 30° incline and six cells for 45° incline.

To verify the analytical results of the energy consumption during a cycle, experiments were performed on a slowly moving four-cell earthworm robot, shown in figure 1, fitted with very compliant contacting pads. The exaggerated compliance of the contacts increases the contact deformations and makes it easier to measure them. The earthworm crawled inside a rigid canal, horizontally as well as up a slope. The only measurements collected during these experiments comprise the absolute positions of the four cells and these are employed to determine the sliding distances of every cell and the energy expenditure by the robot. It is emphasized that no force measurements were collected from individual cells, which would require four to eight force sensors for the robot, but would provide the required information more directly. Thus, the methodology developed to determine the individual cell tip deformations and sliding distances is novel, relying solely on absolute position information of the robot cells together with the compliance of the contact.

We present here the cell positions measured along the canal denoted by coordinate X, as a function of time, obtained for the three aforementioned canal configurations: figure 3 (top, middle, bottom) display the measured positions of the four cells for the robot crawling in the horizontal, 30° and 45° canals, respectively. Indeed, this robot prototype when fitted with compliant contacts is not able to climb slopes beyond 45° as was previously predicted (Zarrouk et al 2012a). A close-up of the measurements over one cycle in horizontal crawling is presented in figure 4 and these results are used in the following section to illustrate the post-processing of the experimental data. We can observe from figure 4 that due to the compliant contacts, the three non-advancing cells retract at the beginning of the step as the load increases on the advancing cell. This effect continues until the advancing cell starts to slide forward (constant load), at which time the non-actuated cells stick and remain sticking till the end of the step. It can also be observed from figure 4 that the amount of retraction of the different cells is not the same for all cells. Specifically, we note that during actuation of cell 2, the other cells (1, 3 and 4) retract by smaller amounts than when any of the other cells are actuated. This can be explained by the fact that the passive clamps of the robot exert somewhat different normal forces on the surface, thereby resulting in different maximum tangential deformations for the different cells. As will be shown in the following sub-sections, determining the maximum tangential deformation will allow estimating when sliding occurs, as well as the sliding distance. With that information, the location of the contact points of the compliant tips can be calculated and then used to evaluate the tangential deformation of the contacts at any time during the locomotion.

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Before the energy expenditure of the robot can be estimated, the measured position data of the cells needs to be mined to identify the missing parameters of the system, in particular the maximum tangential deformations of the compliant contacts in forward and backward directions. For simplicity, we assume these to be equal in magnitude and denote the maximum tangential deflection in either direction by δ_max. In the following, we first describe the procedure employed to identify the value of δ_max of the different cells from experimental data and subsequently use this value to estimate the experimental energy expenditure of the robot.

3.2.1. Initial estimate of δ_max

3.2.2. Instantaneous cell tip deformation

As stated earlier, with the value of the maximum tangential deformation known and with the measured cell positions ${X}^{(i)},i=\mathrm{1...4},$ one can estimate the actual deformation of the cell, i.e., the compliant leg tip, at any instant during the locomotion. First, we note that at the end of the step of the advancing cell, i.e., when it stops moving forward, the tangential deformation of that cell must be δ_max (since the advancing cell slides forward). Accordingly, we can compute the location of the contact point between cell i and the surface, i.e., the location of the leg tip ${X}_{{\rm{C}}}^{(i)},$ at the end of the step as:

$$\begin{eqnarray}&&{X}_{{\rm{C}}}^{(i)}={X}_{{\rm{end}}}^{(i)}-{\delta }_{{\rm{max}}}^{(i)},\end{eqnarray} \tag{ 1 }$$

where ${X}_{{\rm{end}}}^{(i)}$ denotes the position of cell i at the end of the step. Subsequently, the above contact point location will not change unless the distance between the cell and the contact point exceeds δ_max, in which case sliding will occur at the cell. Thus, the criteria to identify whether a cell is sliding or sticking can be formulated as follows:

$$\begin{eqnarray}&&|{X}^{(i)}-{X}_{{\rm{C}}}^{(i)}|\leqslant {\delta }_{{\rm{max}}}^{(i)}\quad \quad {\rm{Sticking}}\\ &&|{X}^{(i)}-{X}_{{\rm{C}}}^{(i)}|\gt {\delta }_{{\rm{max}}}^{(i)}\quad \quad {\rm{Sliding}}\end{eqnarray} \tag{ 2 }$$

and if sliding is detected, the new position of the contact point can be estimated using δ_max as:

$$\begin{eqnarray}&&{X}_{{\rm{C}}}^{(i)}={X}^{(i)}\pm {\delta }_{{\rm{max}}}^{(i)}.\end{eqnarray} \tag{ 3 }$$

The sign in the previous equation is chosen depending on the sliding direction of the particular cell (forward for actuated cells and backward otherwise). Based on equation (3), the contact point locations over two cycles of motion are illustrated in figure 5. Finally, the deformation associated with cell i can be estimated at any time during the locomotion according to the following:

$$\begin{eqnarray}&&{\delta }^{(i)}={X}^{(i)}-{X}_{{\rm{C}}}^{(i)}\quad \quad \;\;{\rm{Sticking}}\\ &&{\delta }^{(i)}={\delta }_{{\rm{max}}}^{(i)}\quad \quad \quad \quad \quad {\rm{Sliding}}\end{eqnarray} \tag{ 4 }$$

with ${X}^{(i)}$ measured, ${X}_{{\rm{C}}}^{(i)}$ computed from (1) for the sticking phase and the value of ${\delta }_{{\rm{max}}}^{(i)},$ identified as per sections 3.2.1 and 3.2.3. Then, the tangential contact force acting on every cell is computed by pre-multiplying the above by the tangential stiffness k_t.

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3.2.3. Final estimate of δ_max

We now refine the initial estimate of the maximum tangential deflection for our robot obtained previously in section 3.3.1. The mathematical basis for the algorithm is provided by applying the principle of linear momentum to the robot over N complete steps of motion, that is:

$$\begin{eqnarray}{\rm{\Delta }}P & = & \displaystyle {\int }_{{\rm{step}}\;1}^{{\rm{step}}\;N}{F}_{{\rm{total}}}{\rm{d}}t\\ & = & \displaystyle {\int }_{{\rm{step}}\;1}^{{\rm{step}}\;N}\left({k}_{{\rm{t}}}\displaystyle \sum _{i=1}^{4}{\delta }^{(i)}-mg\;\mathrm{sin}(\lambda )\right){\rm{d}}t=0,\end{eqnarray} \tag{ 5 }$$

where P is the linear momentum, λ is the slope of the canal and F_total denotes the total external force applied to the device, both quantities along the direction of motion. As well, k_t represents the tangential spring constant of the surface or the leg or their combination and m represents the mass of the robot. Since the robot comes to a stop at the end of each step, the above statement for the change in linear momentum can be further simplified by setting ΔP to zero. The forces acting on the robot are due to the tangential deformation of the compliant contacts between each cell and the surface and the component of robot weight along the direction of motion, i.e., mg sin(λ). With the initial estimate for δ_max and the relationships established in section 3.2.2, the former can be computed explicitly at any instant during the step. Applying equation (5) to the case of horizontal crawling, thereby eliminating the weight, we obtain:

$$\begin{eqnarray}&&\displaystyle {\int }_{{\rm{step}}\;1}^{{\rm{step}}\;N}\left({k}_{{\rm{t}}}\displaystyle \sum _{i=1}^{4}{\delta }^{(i)}\right){\rm{d}}t=0.\end{eqnarray} \tag{ 6 }$$

Since our experimental data is discrete, obtained at a sampling frequency f_s , we employ the discretized version of the above, that is:

$$\begin{eqnarray}&&\displaystyle \sum _{j=1}^{M}{k}_{{\rm{t}}}\displaystyle \sum _{i=1}^{4}{\delta }^{(i)}({t}_{j})\displaystyle \frac{1}{{f}_{{\rm{s}}}}=0.\end{eqnarray} \tag{ 7 }$$

Using equation (7) in combination with equations (2)–(4), we search for the optimal values of ${\delta }_{{\rm{max}}}^{(i)},$ starting with the initial estimates obtained in section 3.2.1, and exhaustively searching in the interval [−0.3 mm 0.3 mm] with increments of 0.1, thus producing seven possibilities for each of the four legs. In total, 7⁴ = 2401 possibilities were evaluated and the final solution is the one resulting in the least violation of (7), and minimum variance of F_contact over the cycle time. The 'optimal' set of ${\delta }_{{\rm{max}}}^{(i)}$ was found to be: 2 mm, 1.1 mm, 1.75 mm and 2.15 mm respectively for cells 1, 2, 3 and 4. To provide additional validation of these maximum deflection results, they are used to estimate the mass of the robot, again from the linear momentum principle employed for the two nonzero inclinations used in our experiments. These, together with the mean deformations are reported in table 3 demonstrating that for both slopes, the estimated weight is within 3% of the real weight of the robot. The mean deformation in table 3 is computed according to:

$$\begin{eqnarray}&&{\delta }_{{\rm{mean}}}=\displaystyle \frac{1}{M}\displaystyle \sum _{j=1}^{M}\displaystyle \sum _{i=1}^{4}{\delta }^{(i)}({t}_{j})\end{eqnarray} \tag{ 8 }$$

while the estimated mass is calculated using:

$$\begin{eqnarray}&&m\approx \left(\displaystyle \frac{{k}_{{\rm{t}}}}{g\;\mathrm{sin}(\lambda )}\right)\displaystyle \frac{1}{M}\displaystyle \sum _{j=1}^{M}\displaystyle \sum _{i=1}^{4}{\delta }^{(i)}({t}_{j}).\end{eqnarray} \tag{ 9 }$$

Table 3.  Mean deformation and mass estimates obtained with optimal maximum deflections for different slopes and compared to actual mass of the robot (333 g).

Slope (°)	Mean deformation	Estimated weight (g)	Error (%)
0	−0.008	—	—
30	−1.71	342	2.7
45	−2.28	322	3.3

3.2.4. Cell tip deformations and contact forces

Using our latest estimates of δ_max and equations (1)–(4), we can now determine the tangential deformation profiles of the compliant leg tips for each cell (see figure 6). These results are presented here for the device climbing the 30° uphill. For this test case, cell 2 has the smallest δ_max and the smallest deformation. From the cell tip deformations, it is possible to calculate the forces acting on the cells due to the compliance (contact forces), as well as to determine the sliding distance of the cells both forward and backward. The sliding distance allows one to calculate the energy losses, which together with the work against weight yields the total work done by the worm robot. Interestingly, the duration of sliding forward is shorter than the duration of sliding back (see figure 6, the lower nearly horizontal lines are sliding backwards whereas the higher ones are sliding forward) even though the sliding forward distance is larger (since the robot is advancing). The reason is that sliding forward of a cell occurs only when the cell is advancing; however, sliding back can occur during the whole step when the cell is not active.

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3.2.5. Actuator forces and displacements

The force acting on the actuator (or exerted by the actuator) is the sum of the elastic force acting on the cell and the component of the weight of the cell along the canal. In order to estimate the work done by the actuator, the elastic component of the actuator force must be integrated over the actuator displacement. The actuator displacement can be estimated by subtracting the motion of the actuated cell from one of the non-actuated cells. A slightly more accurate result can be obtained if we subtract the motion of the actuated cell from the average motion of the three non-actuated cells:

$$\begin{eqnarray}&&{X}_{{\rm{A}}}^{(i)}={X}^{(i)}-{X}_{{\rm{beginning}}}^{(i)}+\displaystyle \frac{1}{3}\displaystyle \sum _{j=\mathrm{1:4}}^{j\ne i}({X}_{{\rm{beginning}}}^{(j)}-{X}^{(j)}).\end{eqnarray} \tag{ 10 }$$

With all the ingredients in place, it is possible to estimate the energy consumption of the worm-robot.

With the results of section 3.2, we can now employ the theoretical analysis of energy expenditure to estimate the energy expenditure of our experimental earthworm. In addition, we compare the forward and backward sliding distances per cycle, averaged over the four cells of the robot, as estimated from the experimental data (see figure 7). Two methods of computation of energy consumption are used here: one by taking the perspective of the work done against the resisting forces (energy losses), in particular, work done by sliding friction and against gravity with post-processed results as per section 3.2.4. In the second method, we directly integrate the actuator forces over the actuator displacements for each cell, which are obtained as discussed in section 3.2.5.

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Considering the energy losses of the crawling robot, the energy consumption is comprised of the frictional losses incurred by each cell during sliding (including backward sliding) and the work done against the appropriate component of the weight over the net displacement of the robot in a cycle:

$$\begin{eqnarray}&&{W}_{{\rm{cycle}}}=\displaystyle \sum _{i}{F}_{{\rm{f}}}^{(i)}{\rm{\Delta }}{X}^{(i)}+mg\;\mathrm{sin}(\lambda ){L}_{{\rm{cycle}}},\end{eqnarray} \tag{ 11 }$$

where the friction force F_f can be computed from the deformation, resulting in:

$$\begin{eqnarray}&&{W}_{{\rm{cycle}}}={k}_{{\rm{t}}}\displaystyle \sum _{i}{\delta }_{{\rm{max}}}^{(i)}{\rm{\Delta }}{X}^{(i)}+mg\;\mathrm{sin}(\lambda ){L}_{{\rm{cycle}}}.\end{eqnarray} \tag{ 12 }$$

In the above, ΔX⁽ⁱ⁾ denotes the amount of sliding of cell i while L_cycle represents the measured net displacement of the robot. On the other hand, the work input computed by integrating the actuator force over its extension, for all cells becomes:

$$\begin{eqnarray}&&{W}_{{\rm{cycle}}}={k}_{{\rm{t}}}\displaystyle \sum _{i=1}^{4}\displaystyle {\int }_{0}^{{L}_{{\rm{s}}}}{\delta }^{(i)}{\rm{d}}{X}_{{\rm{A}}}^{(i)}+mg\;\mathrm{sin}(\lambda ){L}_{{\rm{s}}}.\end{eqnarray} \tag{ 13 }$$

For a particular experiment (horizontal canal or sloped canal), equations (11) and (13) were employed to determine the energy consumption for every cycle (approximately ten cycles for each experiment) and the corresponding averages and standard deviations were computed. Figure 7 presents the sliding distances in the forward and backward directions as expected from the theory and compared to the experimental estimates. The robot is sliding both forward and back by more than predicted by the theory. For example, the backward sliding on the horizontal surface is more than 1 mm/cycle whereas no sliding is predicted. The reason behind this is that since some of the clamps are exerting less normal force, they are more likely to slide (both forward and back) resulting in a larger sliding distance. Note that the energy dissipated by sliding is the product of the normal force times COF times the distance. Therefore the clamps that exert less normal force will dissipate less energy when sliding. Figure 8 presents the results for the experimental energy consumption estimates computed with equation (11) (Exp. sliding) and (13) (Exp. actuator). Also included is the theoretical value of energy consumption which differs from the computation of the experimental values in two respects. First, the value of δ_max employed to obtain the theoretical values was taken as the average of δ_max identified for each leg in the experiments. Second, the value of the net displacement of the robot, L_cycle, is evaluated using the analytical predictions (Zarrouk et al 2012a). The net displacement values are summarized in table 4 for the horizontal and two sloped canals.

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Several observations can be made based on figure 8: (i) the two methods of computing the experimental energy consumption are in excellent agreement with each other and exhibit low standard deviations, pointing to good repeatability of the results; (ii) the energy consumption observed in the experiments is in very good agreement with the theoretical predictions, as expected, with discrepancies between the two of approximately 15%; (iii) the energy consumption computed by neglecting the compliance at the contact between the robot and environment progressively underestimates the real consumption as the slope increases. Note that since the total energy consumed is the combination of the energy dissipated in sliding (sliding distance times normal force times COF) together with work against gravity, the energy errors observed in figure 8 are relatively smaller than the errors observed in the estimated sliding distance in figure 7.