A LOBSTER-INSPIRED HYBRID ACTUATOR
WITH RIGID AND SOFT COMPONENTS

In order to fully capture the extension and compression extension phases of the material, Neo-Hookean model was adopted due to its good fit between test data and computed results. Assuming incompressibility, the strain energy is given by

$$W=\frac{\mu}{2}(I_{1}-1),$$

(1)

where ${\mu}$ is the initial shear modulus and ${I_{1}}$ is the first invariant of three principle stretches (length, width and height) as

$$I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}.$$

(2)

The principle nominal stresses $s_{i}(i=1,2,3)$ can be then obtained as a function of strain energy W, stretches $\lambda_{i}$ and Lagrange multiplier $p$ as

$$s_{i}=\frac{\partial W}{\partial\lambda_{i}}-\frac{p}{\lambda_{i}}.$$

(3)

When pressured air is pumped in, the soft chamber will first experience an radial expansion process to conform to the internal space of rigid shells with no bending occurring, as shown in Fig. 3(a). In width direction, the top and side walls expand into the top wall of a semi-circle with inner radius $r=R-t$, and the average stretch in the top wall can be calculated as $\lambda_{t,2}=(r+t/2)\pi/(b+2t+2a)$. The stretch in length direction of the top wall is negligible due to the constraints from rigid shells before actuating, therefore $\lambda_{t,1}=1$. Stress in length direction will result in an opposing bending torque $M_{\theta}$ around each joint, and the moment equilibrium must be satisfied as:

$$\left\{\begin{aligned} &M_{a}=M_{\theta}-M_{geom}(M_{a}<M_{\theta})\\ &M_{a}=M_{\theta}+M_{f}(M_{a}>M_{\theta}),\end{aligned}\right.$$

(4)

where $M_{geom}$ is an additional torque generated by geometrical constraint as the hybrid actuator can only bend unilaterally, $M_{f}$ is the torque created by friction, and $M_{a}$ is torque induced by air pressure and can be calculated as

	$\displaystyle M_{a}$	$\displaystyle=\int_{0}^{\pi/2}P_{in}(2r\cos\theta_{i})(r_{0}\sin\theta_{i}+t+d)r_{0}\cos\theta_{i}~d\theta$		(5)
		$\displaystyle=\frac{P_{in}r_{0}^{2}(3d\pi+4r_{0}+3\pi t)}{6}$

where $r_{0}$ is the inner radius of soft chamber when inflated and $P_{in}$ is the relative input pressure. The top wall thickness of the semi-circular soft chamber becomes thinner due to stretch in width direction and $t_{0}=t/\lambda_{t,2}$ is denoted, and therefore $r_{0}=R-t_{0}$.

For simplicity, the geometry of the fractured sphere shell is not considered in this part. After the soft chamber fills up the internal space of rigid segments, the hybrid actuator will start to bend with further increasing pressure. At each bending configuration, there are four bending moments involved in the actuation including the pressure-induced torque $M_{a}$, stretch caused torque of top and bottom wall of the soft chamber $M_{top}$ and $M_{bot}$, and friction-induced torque $M_{f}$ with a maximum $M_{f,max}$. While $M_{a}$ acts counter-clockwise around the joint O, $M_{top}$, $M_{bot}$ and $M_{f,max}$ all act clockwise around O (see Fig. 3). The moment equilibrium can be described as

$$M_{a}=M_{\theta}-M_{f}=M_{b}+M_{t}-M_{f}.$$

(6)

For the semi-circular soft chamber, stress in length direction of top wall and bottom wall can be calculated in a similar method as in [17] as

$$s_{t,1}=\mu(\lambda_{t,1}-\frac{1}{\lambda_{t,2}^{2}\lambda_{t,1}^{3}}),$$

(7)

$$s_{b,1}=\mu(\lambda_{b,1}-\frac{1}{\lambda_{b,1}^{3}}).$$

(8)

For the bottom wall, as the stretch in length direction $\lambda_{b,1}$ is a function of the vertical position, a local coordinate $\beta$ is introduced and then it can be written as

$$\lambda_{b,1}=\frac{\beta\theta_{i}+l}{l}.$$

(9)

where $l$ is the distance between two axes and $\theta_{i}$ is the joint angle between two neighboring segments. It follows that

$$M_{b}=\int_{d+t}^{d}s_{b,1}(b+2t)\beta~d\beta.$$

(10)

For the top wall, two more local coordinates $\phi$ and $\tau$ are introduced to define $\lambda_{t,1}$ as

$$\lambda_{t,1}=\frac{\left[{l+d+t+(r_{0}+\tau)\sin\phi}\right]\theta_{i}}{l}$$

(11)

so that

$$M_{t}=2\int_{0}^{t_{0}}\int_{0}^{\frac{\pi}{2}}s_{t,1}\left[(r_{0}+\tau)\sin\phi+t+d\right](r_{0}+\tau)~d\phi d\tau.$$

(12)

A very complicated polynomial solution can be solved for integral 10, and integral 12 can only be solved numerically, which would limit the application of this analytic model in a real-time calculation and further analysis. In this study, two approximations are made to obtain an explicit expression of $M_{t}$ and $M_{b}$.

For the hybrid actuator presented, the range of the principle stretch in length direction is approximately 1 to 1.6, in which the nonlinear characteristic of silicone rubber is not obvious. Effective Young’s modulus component $E_{1}$ and $E_{2}$ are introduced to linearize the stress-strain relationship and it is proved in the calibration part to be a reasonable fit. With this approximation, stress component $s_{t,1}$ and $s_{b,1}$ in Eqn. (7) and Eqn. (8) become

$$s_{t,1}=E_{t,1}(\lambda_{t,1}-1)+E_{t,2},$$

(13)

$$s_{b,1}=E_{b,1}(\lambda_{b,1}-1)+E_{b,2}.$$

(14)

Another approximation was made by replacing the semi-circular top wall with a rectangular wall of the same width and length, and the top of this rectangular is located at a distance $H$ from the joint $O$, which makes $\lambda_{t,1}$ irrelevant with the local coordinates $\phi$ and $\tau$ as

$$\lambda_{t,1}=\frac{\beta\theta_{i}+l}{l}.$$

(15)

By substitution of Eqn. (14) into Eqn. (10), and Eqn. (15), (13) into Eqn. (12), torques generated by material stretch can be deduced in a simple form which is provided in the appendix part. $H$ can be determined by empirically comparing the value at different joint angles computed with the exact (12) and approximation (22). For the hybrid actuator in this paper with $(a,b,t,l,R)=(0.5,10,1.5,8,8)mm$, it is found that approximation expression (22) with $H$=7 can accurately capture the relationship between joint angle $\theta_{i}$, ranging from 0 to 30 degrees, and stretch-induced torque of top wall $M_{t}$ with an approximation error less than 4%. The accuracy of this approximation is further confirmed in calibration part and details are provided in Appendix A. Therefore, using a flax wall with $H=7mm$ instead of the semi-circular wall of the inflated soft chamber is adopted as a valid approximation of (12) and is used to calculate the stretch-induced torque of top wall for all the rest of this study.

The hybrid actuator has a hinge-style bending motion, and the total bending angle is the summation of bending angles between two neighboring segments. Due to the relatively small mass of this actuator (less than 18g for an 11-segment configuration), we do not consider the gravity and assume a smooth bending, which is consistent with experimental observations. Combining Eqns. (5), (6), (21) and (22), the total bending angle can be then expressed as

	$\displaystyle\theta$	$\displaystyle=(n-1)\theta_{i}$		(16)
		$\displaystyle=\frac{K_{1}}{K_{3}}P_{in}-\frac{K_{2}}{K_{3}}$

where ${K_{1}}$, ${K_{2}}$ and ${K_{3}}$ are parameters only related to actuator geometry and friction limit as

	$\displaystyle{K_{1}}$	$\displaystyle=\frac{1}{6}r_{0}^{2}(3d\pi+4r_{0}+3\pi t)$		(17)
	$\displaystyle{K_{2}}$	$\displaystyle=\frac{E_{t,2}(r_{0}+t_{0}/2)}{2}\left[H^{2}-{(H-t_{0})}^{2}\right]$
		$\displaystyle+\frac{E_{t,2}(r_{0}+t_{0}/2)}{2}\left[H^{2}-{(H-t_{0})}^{2}\right]+M_{f,max}$
	$\displaystyle{K_{3}}$	$\displaystyle=\frac{E_{b,1}(b+2t)}{3l}\left[{(d+t)}^{3}-d^{3}\right]$
		$\displaystyle+\frac{E_{t,1}(r_{0}+t_{0}/2)}{3l}\left[H^{3}-{(H-t_{0})}^{3}\right].$

Denote $P_{t}=[X_{t},Y_{t},Z_{t}]^{T}$ the vector of the actuator tip in the base frame, then the forward kinematics can be given by

$$\begin{split}P_{t}=\begin{bmatrix}\sum_{i=1}^{n}l_{i}\cos(i\theta_{i})\\ \sum_{i=1}^{n}l_{i}\sin(i\theta_{i})\\ 0\end{bmatrix},\end{split}$$

(18)

where $l_{i}$ is the length of link i.

An expression to estimate the blocked force can be developed with the previously derived model for bending angle. In this analysis, the hybrid actuator is assumed to be constrained at 0 bending angles. As the pressure is further increased after the soft chamber fills up all the internal space of the rigid segments, the moment equilibrium around the nearest joint to the tip segment must be satisfied as

$$M_{a}-M_{f,max}-M_{\theta=0}=Fl^{\star}$$

(19)

where $M_{\theta=0}$ is the torque induced by the soft material at 0 bending angles, and $l^{\star}$ is the distance between the actuator tip and the nearest joint.

Combining Eqn. (5), (22), (21) and (19), an explicit relationship between output tip force and input pressure can be get as

	$\displaystyle F$	$\displaystyle=\frac{M_{a}-M_{f,max}-M_{\theta=0}}{l^{*}}$		(20)
		$\displaystyle=\frac{K_{1}}{l_{*}}P_{in}-\frac{K_{2}}{l_{*}}$

where ${K_{1}}$ and ${K_{2}}$ are parameters only related to actuator geometry and friction limit, as discussed in section 3.3.

To demonstrate the mechanical performance of this proposed hybrid actuator and validate the proposed analytic model, hybrid actuators with six configurations, from 7 to 12 segments, were fabricated and an evaluation platform was developed for the characterization tests. The platform (see Fig. 4) contained a 6-axis force/torque sensor which was mounted onto a three-axis adjustable aluminum frame to measure the force at the tip of the actuator. The measured data was displayed through a graphical interface (GUI) written in LabView. Pressure readings were obtained through a microcontroller (Arduino Mega) and displayed to the user to correspond the pressure in the soft actuator to the force it could produce. The hybrid actuator’s proximal tip with air inlet was clamped in a rigid fixture, emulating a boundary condition as fixed. In the bending motion test, the distal tip of the actuator was free to bend in the vertical plane, and a high definition camera was mounted on a tripod to observe the actuators from the side. Post-processing of motion photos was conducted with Adobe Illustrator. A 3D printed post was added to this system mounted on top of the 6-axis force/torque sensor to bring contact with the actuator tip and measure the force exerted. The input pressure was increased gradually, and the force exerted by the tip segment was recorded.

Calibration was conducted to determine the material parameter $\mu$ and the maximum torque that friction can generate $M_{f,max}$. Three trials of free bending tests with the 8-segment configuration were conducted, and the prediction obtained from the analytic model was compared with experimental data. It was found that the prediction can fit well with the experimental results with $\mu=0.07MPa$ and $M_{f,max}=5N\cdot mm$. The two approximations were also assessed by comparing the exact and approximated values, and the result validated their accuracy as further shown in Appendix A. Although there were two parameters to be calibrated, they were not relevant. As for different configurations, the material used for fabrication and cross-section geometrical characteristics were the same. Therefore, the same $\mu$ and $M_{f,max}$ value were used in all subsequent studies with good results.

Five different configurations, including 7, 9, 10, 11 and 12 segments, were tested to evaluate the bending capacity. As for each configuration, the proximal tip segment was firmly clamped, hybrid actuators characterized had five different degrees of freedom including 6, 8, 9, 10 and 11. Bending angle is defined as the deflection angle of the distal tip segment as shown in Fig.5(a). Each actuator was pressured three times to bend in free space to check accuracy, and bending motions were captured by the camera mentioned earlier. Pressure increasing was kept at a sufficiently low rate to ensure the actuator in a quasi-static state. Experimental data was collected to compare with the prediction of the analytic model as shown in Fig. 5(c) and (d), and for a clear reading, only results of 7, 9 and 11 were plotted in this part. Although there are some approximations and linearizations in the analytic model, the findings demonstrate that the analytic model can capture the overall trend of the hybrid actuators. Discrepancies between analytic and experimental results are more evident when total bending angle was small, and this is probably due to not take the geometry of the fractured sphere shells into consideration in the analytic models. Both experimental and analytic results show an enhanced bending capacity with increasing segments, and the 11-segment configuration could reach $250^{\circ}$ total bending angle under about 60 kPa. Fig. 6 shows the actuator tip trajectories of 10-segment configuration obtained from experimental tests and prediction of the forward kinematics as presented in Eqn. (18). The position where the proximal tip segment was clamped is denoted the origin (0, 0). It can be seen that the forward kinematics can well describe the trend of the tip trajectory, while some small discrepancies might be caused by the simplifications in analytic models and gravity effects.

When the free bending is constrained, hybrid actuators are capable of exerting forces either at the interaction points along their body or the tip. In the blocked force test, the proximal tip segments of hybrid actuators were clamped while the distal tips were in contact with the silicone pad connected to the force sensor. A constraining platform was positioned on top of hybrid actuators to ensure that actuators were in 0 bending angles during pressurization (see Fig. 5(b)). In total six configurations were tested including 7, 8, 9, 10, 11 and 12 segments, and experimental data was collected with conforming results. In Fig. 5(c) and (d), results of configurations of 7, 9 and 11 segments are presented. These results have demonstrated the ability of the analytic model to predict the exerting tip force for different configurations, and a clear trend shows that tip force increases with increasing pressure after a certain threshold pressure. Also, it is shown that for hybrid actuators with the same rigid segment and soft actuator design, segment number has a small influence on force generation. The input pressure was increased until 130 kPa which was limited by the valve capacity, and for all configurations, they can reach a maximum tip force of about 2.1 N.