The Structure,Design,and Closed-Loop Motion Control of a Differential Drive Soft Robot

Abstract

This article presents the structure, design, and motion control of an inchworm inspired pneumatic soft robot, which can perform differential movement. This robot mainly consists of two columns of pneumatic multi-airbags (actuators), one sensor, one baseboard, front feet, and rear feet. According to the different inflation time of left and right actuators, the robot can perform both linear and turning movements. The actuators of this robot are composed of multiple airbags, and the design of the airbags is analyzed. To deal with the nonlinear performance of the soft robot, we use radial basis function neural networks to train the turning ability of this robot on three different surfaces and create a mathematical model among coefficient of friction, deflection angle, and inflation time. Then, we establish the closed-loop automatic control model using three-axis electronic compass sensor. Finally, the automatic control model is verified by linear and turning movement experiments. According to the experiment, the robot can finish the linear and turning movements under the closed-loop control system.

Introduction

A soft robot is largely constructed from soft materials. The research on it is only in the initial stage. Most traditional robots are constructed from stiff materials. Compared with traditional rigid robots, soft robots are adaptable and more animal like in their capabilities. They can change their structures as the environment changes. It is possible to design medical robots with soft materials, for the operation or rehabilitation of human body.^1,2 Most of the soft robots are bionic, which are designed to adapt to the complex and varied natural environment.^3,4

Most of the soft crawling robots are capable of performing both linear and turning movements,⁵ and some even have the ability of jumping, rolling, and swimming.^6,7 Many of the soft crawling robots come from the natural creature.^8,9 In the nature, the organisms are basically composed of soft materials such as muscles and skin. These flexible materials can store a large amount of elastic potential energy, which can be useful for adapting to the variety of living environments.¹⁰ Among the soft crawling robots,¹¹ pneumatic soft crawling robots have their particular advantages, such as fast moving speed, strong carrying capacity, and large deformation. Fei designed one kind of deformable spherical modular soft robot which can perform linear movement.^12,13 Shepherd et al. designed a pneumatic crawling robot with four feet.¹⁴ It contains the main body and four feet, and each part can be inflated independently. Multi-gait movement of this robot can be realized by changing the inflation sequences. On the basis of this study, Majidi integrated the air pumps, solenoid valves, and the flexible part of the robot into the body of the robot and created an untethered soft robot.¹⁵

In the article, we design a multi-airbag differential drive soft crawling robot, inspired by the inchworm's movement pattern. This robot consists of two columns of paralleled airbags, one baseboard, front feet, rear feet, and one sensor. The design of the actuators is analyzed. This robot can perform both linear and turning movements, according to the paralleled structure and the different control patterns. The linear movement pattern and the turning movement pattern are presented. Because of its turning motion ability, it can avoid the obstacles. Because of its soft structure, it has the potential to be applied to the complex and varied natural environment. This robot is made of silicone rubber, which is extremely nonlinear. To realize the automatic control of this soft crawling robot, we use radial basis function (RBF) neural networks¹⁶ to study the turning ability of this robot and create the mathematical model among inflation time, coefficient of friction, and deflection angle. We establish the closed-loop control model with the three-axis electronic compass sensor (HMC5883L) and the RBF model. Finally, the RBF neural network model is verified by experiments of linear and angle-specified turning movements.

Structure Design and Locomotion Principle

The movement pattern of this soft crawling robot is inspired by inchworm. In the nature, inchworm uses the front legs as anchors when the body bends, so that the center of the body moves forward. And when the body stretches, inchworm uses its back legs anchoring the ground, which also makes the center of the body move forward. In that case, inchworm uses its front and back legs sequentially as anchors to make a crawling movement. The design idea of the front and rear feet comes from the front and back legs of the inchworm. Front and rear feet are used as anchors sequentially to realize the locomotion of the soft robot.

The design of the soft robot

The differential drive soft robot consists of two parallel columns of multi-airbags, one sensor, top layer, middle layer, bottom layer, front feet, and rear feet. As shown in Figure 1, the top layer, bottom layer, and the multi-airbags are made of silicone rubber. The top layer, middle layer, and bottom layer constitute the baseboard. In Figure 2, the function of the middle layer is to prevent the baseboard expanding, so it is made of nonductile flexible materials, such as paper. The front and rear feet are used as anchors when making a movement, and they are made of high frictional materials such as foam tape (VHB-3M), which is made of polyacrylate, produced by 3M company.

FIG. 1.

The explosive view of differential drive pneumatic soft robot.

FIG. 2.

Cross section of two airbags, (a) before inflating and (b) after inflating.

In Figure 1, the three-axis electronic compass sensor is embedded in the middle of the actuators with silicone rubber. During the movement, the middle section will keep the sensor at a relatively stable state.

The differential device contains two columns of multi-airbags, and each of them is an actuator. There are 11 airbags in one actuator. Each airbag of one actuator is connected by the gas slot which is in the bottom of the airbags (Fig. 1). There are two tubes, and each tube is inserted into the middle airbag of the actuator. While inflating, the gas through the tubes reaches the middle airbag of the actuator and then reaches each airbag through the gas slot. At the same time, each airbag expands with the increasing pressure of the air chamber to drive the robot bends. The purpose of the middle layer is to prevent the baseboard expanding with the air chamber, but only bend with the actuator. The diagram of the inflated airbags is shown in Figure 2b; the length of the baseboard remains constant after inflating because of the paper layer. When the bending angles of the two actuators are the same, the differential drive robot makes a linear movement, otherwise the differential drive robot makes a turning motion.

The design of the airbags

The actuators of this robot are composed of multiple airbags (PneuNets^17,18). The bending of the actuator depends on the inflating of the airbags. Thus, the design of the airbags is important. Figure 2 is the cross section of the airbags. The design of the airbags is based on the Yeoh model,¹⁹ which is widely used on incompressible materials. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} W = {C_{10}} \left( {{I_1} - 3} \right) + {C_{01}}{ \left( {{I_2} - 3} \right) ^2} , \tag{1} \end{align*} \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} E = \int Wdv , \tag{2} \end{align*} \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} B = \left( { I + { \frac { \partial \left( u \right) } { \partial \left( { x , y , z } \right) } } } \right) { \left( { I + { \frac { \partial \left( u \right) } { \partial \left( { x , y , z } \right) } } } \right) ^T } , \tag { 3 } \end{align*} \end{document}

where W is the strain energy density function. E is the strain energy. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${C_{10}}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${C_{01}}$$ \end{document} are the material constants.¹⁹ I₁ and I₂ are the strain invariants. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${I_1} = {I_2} = tr \left( B \right)$$ \end{document} , B is the Cauchy green strain tensor. I is the identity matrix. u is the displacement of the airbag's inflated wall which is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$u = \left( {{u_x} , {u_y} , {u_z}} \right)$$ \end{document} .

To get the displacement of any point in the inflated wall, we study one of the airbags. In Figure 2b, h is the maximum displacement of the airbag's inflated wall, and the coordinate origin \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \rm{o}}$$ \end{document} is in the center of the airbag. We assume that the thickness of the wall is constant before or after inflating. The coordinate of any point on the wall is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\left( {x , y , z} \right)$$ \end{document} , and after inflating the coordinate of the point can be expressed as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\left( {{x^ , } , {y^ , } , {z^ , }} \right)$$ \end{document} . \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \left( { { x\prime } , { y\prime } , { z\prime } } \right) = \left( { x , y + hf \left( { \frac { z } { a } } \right) f \left( { \frac { x } { b } } \right) , z } \right) , \tag { 4 } \end{align*} \end{document}

where a is the height of the wall. b is the width of the wall. And \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$f ( )$$ \end{document} is the monotonically decreasing function, and the range of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$f ( )$$ \end{document} is between 0 and 1. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$f ( )$$ \end{document} meets the following conditions. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} f \left( 0 \right) = 1 , f \left( 1 \right) = 0 , f{ \left( 0 \right) \prime } = 0. \tag{5} \end{align*} \end{document}

To simplify the function, we construct the function \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$f \left( x \right) = 1 - {x^2}$$ \end{document} , which meets the above conditions. Then the displacement of any point in the wall can be expressed as the following equation. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \left( { { u_x } , { u_y } , { u_z } } \right) = \left( { 0 , hf \left( { \frac { z } { a } } \right) f \left( { \frac { x } { b } } \right) , 0 } \right) . \tag { 6 } \end{align*} \end{document}

With all the functions above, the strain energy E can be obtained. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} E = & {C_{01}} \left[ {{{2048} \over {1575}} \left( {{{{a^2}} \over {{b^2}}} + {{{b^2}} \over {{a^2}}}} \right) + {{2048} \over {11025}}} \right] {{{h^4}d} \over {ab}} \\ & + {C_{10}}{{32} \over {45}} \left( {{a \over b} + {b \over a}} \right) {h^2}d. \tag{7} \end{align*} \end{document}

The functional relationship between the deformation h and the air chamber's pressure P can be calculated using virtual work principle. The strain energy of the inflated wall equals to the work of the pressure along the displacement of the Y-axis. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \frac { dE \left( h \right) } { dh } } \delta h = P { \frac { dV \left( h \right) } { dh } } \delta h , \tag { 8 } \end{align*} \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} V \left( h \right) = hab \ _0^1f \left( x \right) dx , \tag{9} \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$V \left( h \right)$$ \end{document} is the increased volume of the air chamber after inflating. According to (5), (7), and (8), we can obtain the equation between the pressure P and the deformation h. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} P = { C_ { 01 } } \left[ { { \frac { 6144 } { 525 } } \left( { { \frac { { a^2 } } { { b^2 } } } + { \frac { { b^2 } } { { a^2 } } } } \right) + { \frac { 6144 } { 3675 } } } \right] { \frac { { h^3 } d } { { a^2 } { b^2 } } } . \tag { 10 } \end{align*} \end{document}

In this Equation (10), the deformation h increases with the pressure P. To achieve a larger deformation h and a smaller pressure P, a smaller thickness d is desirable. However, decreasing the thickness d will reduce the strength of the walls and reduce the maximum acceptable pressure. The thickness of the wall is selected as 2 mm. In addition, the deformation h increases with the height and the width of the inflated wall. However, to reduce the dimension of the airbag, a height of 12 mm and a width of 20 mm are selected. The deformation of the airbags causes the bending angle of the actuator. The distance k between the airbags reduces the expansion effect of the airbags, so a smaller distance k is desirable, and a distance of 1 mm is selected. The specific parameters of the airbag are summarized and shown in Table 1.

Table 1.

Parameters of the Airbags

Parameters	Value
The height of the airbag a (mm)	12
The width of the airbag b (mm)	20
The thickness of the wall d (mm)	2
The distance between the airbags k (mm)	1
The no. of one actuator airbags	11

The movement patterns

The differential drive soft robot can not only move forward but also turn left and right. The different movement patterns depend on the different inflation sequences.

The linear movement pattern

The differential drive soft robot can move forward when the two actuators bend simultaneously. The structure of front and rear feet is shown in Figure 1. The front feet and rear feet are both two rectangular rubber plates, and the size of the feet comes from experiments. The front feet are attached under the baseboard. The rear feet are attached at the back of the robot, and the angle between the baseboard and the rear feet is 60°. In Figure 3, there are five modes in a linear movement cycle. State A shows the original position of the robot, in which the front and rear feet are at the deflated state. State B is the inflating process. During the inflating process, the contact area between the front feet and the ground is larger compared with the rear feet. In that case, the friction force of front feet is larger than the friction force of rear feet. Because of the larger friction force, the front feet are used as anchors. And the rear feet move forward with the bending of the actuators until the inflating process is finished. C shows the inflated state, at which the rear feet fully touch the ground and the robot meets the maximum bending curvature. However, the front feet and the ground are in line contact. At the deflating process D, the friction force of the rear feet is larger compared with the front feet. During the deflating process, the rear feet are used as anchors due to the larger static friction force. And the front feet move forward with the stretching of the actuators. State E shows the final position of the robot. The forward moving distance is H in one linear movement cycle. In general, this robot uses its front feet and rear feet sequentially as anchors when it bends and stretches.

FIG. 3.

The schematic diagram of the linear movement pattern. (A) The original state; (B) the inflating process; (C) the inflated state; (D) the deflating process; and (E) the deflated state.

The turning movement pattern

The differential drive soft robot makes a turning movement because of the different bending angles of the two actuators. To make a most efficient turning movement, only one of the two actuators is inflated or deflated during the turning movement process. Figure 4 is the schematic diagram of the right-turning movement pattern. State B is the inflating process of the left actuator. The front feet anchor the ground due to the higher friction force. And the left actuator meets the maximum bending curvature at the state C. At the deflating process D, the friction force of the left rear foot is higher compared with the right rear foot. Then, the robot deflects toward the lower frictional side and makes a right turning movement. With the method of independently inflating, the robot can get the maximum deflection angle. From the original state A to the finial state E, the robot makes a right-turning angle of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \rm{ \alpha }}$$ \end{document} .

FIG. 4.

The schematic diagram of the turning movement pattern. (A) The original state; (B) the inflating process of the left actuator; (C) the inflated state; (D) the deflating process of left actuator; and (E) the deflated state.

The RBF Control of Differential Drive Soft Robot

We establish the closed-loop control model to control the linear movement and the turning movement using three-axis electronic compass sensor and the RBF model. The purpose of the linear movement control is to keep the robot moving in the same direction. When the robot deflects from the original direction, it has the ability to self-correct. And the turning movement control is to make the robot turn to the specified deflection angle with the maximum speed. The RBF model is needed in the linear movement control and the turning movement control.

RBF is an artificial neural network that uses RBFs as activations, which is widely used to deal with the nonlinear partial differential equations. The differential drive soft robot is mainly made of silicone rubber, the expansion model of which is extremely nonlinear and complex. Instead of studying the nonlinear expansion mathematical model, we use RBF neural networks to study the turning ability of this robot on three different surfaces and create the mathematical mapping among frictional coefficient, deflection angle, and inflation time. Inflation time is essential for the closed-loop control of the robot, and to get the appropriate inflation time automatically, the working condition (frictional coefficients) and the azimuth angle are needed.

Figure 5 is the closed-loop control circuit of the differential drive soft robot, which mainly contains power source (DC12V), air pump (30 L/min), reduction valve, electromagnetic valves (MAC-35A), microcontrollers (Arduino UNO), and the soft robot. We use two two-position three-way solenoid valves to control two actuators' inflation and deflation. The inflation and deflation of the actuators are independent.

FIG. 5.

The closed-loop control system of differential drive soft robot.

RBF neural networks

RBF networks have three layers as follows: an input layer, a hidden layer, and a linear output layer. The input layer consists of the input vector, the hidden layer contains the nonlinear RBF activation function, and the output layer is a scalar function of the input vector.^20,21

In RBF model, the neuron number of the input layer is n, represents the dimension of the input vector n, and the dimension of the output layer is m. From the input layer to the hidden layer, the RBF is commonly taken to be Gaussian: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { u_i } = exp \left( { - { \frac { { { \left( { x - { c_i } } \right) } ^2 } } { 2 \sigma _i^2 } } } \right) , \left( { i = 1 , 2 , \cdots , g } \right) , \tag { 11 } \end{align*} \end{document}

where x is the input vector, c_i is the center vector for neuron i, u_i is the output of the neuron i, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \sigma _i}$$ \end{document} is the variance for neuron i, and g is the neuron number of the hidden layer.

The output of the networks is: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {y_j} = \mathop \sum \limits_{i = 1}^g {w_{ij}}{u_i} - { \theta _j} , \left( {j = 1 , 2 , \cdots , m} \right) , \tag{12} \end{align*} \end{document}

where y_j is the output of neuron j in output layer, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${w_{ij}}$$ \end{document} is the weight between hidden layer and output layer, and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \theta _j}$$ \end{document} is the threshold value of the output layer.

At last, the input layer and the output layer can be expressed as (13). According to the input vector, we can get the output vector. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { y_j } = \mathop \sum \limits_ { i = 1 } ^g { w_ { ij } } exp \left( { - { \frac { { { \left( { x - { c_i } } \right) } ^2 } } { 2 \sigma _i^2 } } } \right) - { \theta _j } . \tag { 13 } \end{align*} \end{document}

RBF networks are typically trained by a two-step algorithm. The first step is unsupervised, and we choose K-means clustering method.²² The second training step is supervised, and we use the least squares method²³ to get the weight between the hidden layer and the output layer.

The control of differential drive soft robot

The RBF model of differential drive soft robot contains input layer, hidden layer, and output layer. The input vector contains the deflection angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \rm{ \alpha }}$$ \end{document} and the frictional coefficient \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \rm{ \mu }}$$ \end{document} between robot's feet and the ground, and the output vector is the inflation time t. According to the RBF model, we can obtain the inflation time by putting the frictional coefficient and the detected deflection angle into the model (Fig. 8). The RBF model is used in both linear movement control and turning movement control.

To collect the experimental data, we experiment on three different surfaces: the acrylonitrile butadiene styrene (ABS) plastic, the plywood, and the glass, as it is shown in Figure 6. Different surfaces have different frictional coefficients, which are 0.6, 0.7, and 0.8 by measuring. This RBF model can be applied to a variety of working environments with different roughness by adding the frictional coefficient as one of the inputs. We collect K groups of experimental samples by controlling the different inflation time and different surfaces and obtaining the corresponding deflection angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \rm{ \alpha }}_k}$$ \end{document} . We get the center vector of Gaussian kernel function c_i, the variance \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \sigma _i}$$ \end{document} , and the weight between hidden layer and output layer w_i, by training the experimental samples, as we discussed in the RBF neural networks section. Figure 7 shows the experimental data and RBF training data among deflection angle, frictional coefficient, and inflation time. The input vectors are the deflection angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \rm{ \alpha }}$$ \end{document} and the frictional coefficient \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \rm{ \mu }}$$ \end{document} . The output vector is inflation time t, which is also shown in Figure 8. The red circles in Figure 7 are the experimental data, as the figure shows, the RBF training data can match the experimental data in a good way.

FIG. 6.

Three different surfaces, (a) the glass, (b) the plywood, and (c) the ABS plastic.

FIG. 7.

Experimental data (red circles) and RBF training data (surface) among deflection angle, frictional coefficient, and inflation time, (a) left turn data, (b) right turn data. RBF, radial basis function. Color images available online at www.liebertpub.com/soro

FIG. 8.

The RBF control logic diagram of differential drive soft robot.

According to the experiment, the maximum deflection angle is recorded as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \rm{ \alpha }}_{max}}$$ \end{document} . The corresponding inflation time is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${t_{max}}$$ \end{document} . Under this inflation time, one of the rear feet can fully touch the ground, which produces the maximum deflection angle. To improve the efficiency and reduce the inside pressure of the actuators, all the inflation time should be smaller than \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${t_{max}}$$ \end{document} , which is selected as 1 s in Figure 7.

The control of linear movement

The purpose of linear movement control is to keep the robot moving toward the initial direction. The robot can self-correct its azimuth angle when it deflects from the initial direction.

Step A: using the three-axis electronic compass sensor to record the initial azimuth angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \oslash _0}$$ \end{document} .

Step B: using the air pump to inflate the actuators synchronously with the maximum inflation time \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${t_{max}}$$ \end{document} and then deflating synchronously to finish a linear movement cycle. After several cycles, using the three-axis electronic compass sensor to check the azimuth angle and record the azimuth angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \oslash _{new}}$$ \end{document} . If the robot deflects from the initial azimuth angle, we can get the inflation times of the self-correction. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} N = floor \left( { { \frac { \left\vert { { { \oslash } _0 } - { \oslash _ { new } } } \right\vert } { { { \rm { \alpha } } _ { max } } } } + 1 } \right) , \tag { 14 } \end{align*} \end{document}

where N is the inflation times of the self-correction phase, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$floor ( )$$ \end{document} is a round down function, which is shown in the “Function” part of Figure 8.

And the deflection angle of each correction is: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \rm { \alpha } } = \frac { { { \oslash _0 } - { \oslash _ { new } } } } { N } . \tag { 15 } \end{align*} \end{document}

Step C: we put the deflection angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \rm{ \alpha }}$$ \end{document} and the corresponding frictional coefficient \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \rm{ \mu }}$$ \end{document} into the RBF model to get the corresponding inflation time t, as it is shown in Figure 8. Then, robot can self-correct the deflection angle with the inflation time t and inflate N times.

Step D: the robot is continuously moving in the same direction by repeating step B and step C.

The control of turning movement

The control of turning movement is to make the robot turn to the specified deflection angle with the maximum speed.

Step A: we give the robot a specified deflection angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\oslash$$ \end{document} and record the initial azimuth angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \oslash _0}$$ \end{document} and then we can get the final azimuth angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \oslash _{last}}$$ \end{document} : \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \oslash _{last}} = \oslash + { \oslash _0}. \tag{16} \end{align*} \end{document}

Step B: the azimuth angle after k times' deflection is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \oslash _k}$$ \end{document} (k = 0, 1, 2,…, n). When \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \oslash _{last}} - { \oslash _k} > {{ \rm{ \alpha }}_{max}}$$ \end{document} , the robot makes a turning movement with the maximum inflation time \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${t_{max}}$$ \end{document} to guarantee the maximum turning speed (turning motion by inflating one of the actuators). This step is included in the “Function” part of the Figure 8.

Step C: we repeat step B until \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \oslash _{last}} - { \oslash _k} < {x_{max}}$$ \end{document} . When \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \oslash _{last}} - { \oslash _k} < {x_{max}}$$ \end{document} , using RBF model to train the final deflection angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \rm{ \alpha }}$$ \end{document} : \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \rm{ \alpha }} = { \oslash _{last}} - { \oslash_k}. \tag{17} \end{align*} \end{document}

Figure 8 shows that to get the corresponding inflation time t, the deflection angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \rm{ \alpha }}$$ \end{document} and the frictional coefficient \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \rm{ \mu }}$$ \end{document} are needed in the RBF model. The robot finishes the turning movement after the final inflation and deflation cycle.

Experiments and Results

These experiments are carried out under room temperature. The experimental devices include air pump (30 L/min), reduction valve, high speed electromagnetic valves (MAC-35A), control unit (microcontrollers and the printed circuit board), power source (DC24V), the air pipes ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\phi 1{ \rm{mm}}$$ \end{document} ), and the soft robot. We perform these experiments on the ABS plastic surface, the plywood, and the glass surface, respectively, and collect the linear and turning experimental data. Figures 9 and 10 introduce the experimental process of the linear and turning motion. And it is done on the ABS plastic surface as an example.

FIG. 9.

The top view of the linear locomotion on the ABS plastic surface, (a) the initial state, (b) the position after nine cycles' linear movement, (c) the self-correction process, and (d) the position after self-correction process.

FIG. 10.

The top view of the turning motion on the ABS plastic surface, (a) the initial state, (b) the first inflating process, and (c) the final state.

Linear locomotion experiment

Figure 9 shows the linear locomotion process of differential drive soft robot on the ABS plastic surface. The distance of each grid on the experimental platform is 50 mm. In linear movement experiment, left and right actuators are inflated and deflated synchronously. After nine cycles' linear locomotion, the azimuth angle is detected by three-axis electronic compass sensor. If the robot deflects from the initial azimuth angle, the robot produces corrective movement, which is controlled by the control unit. Figure 9a shows the robot at the initial state. Figure 9b shows the position of the robot after nine cycles' linear movement. At this moment, robot deflects from the initial azimuth angle without the use of self-correction. Figure 9c is the self-correcting process with RBF model. Figure 9d shows the position of the robot after self-correcting process, the robot back to the initial azimuth angle.

Turning motion experiment

Figure 10 shows the left-turning motion process of differential drive soft robot on the ABS plastic surface. The aim of this experiment is to make the robot turn to the specified deflection angle with the maximum speed. In this experiment, we give the robot a 30° deflection angle to test the specified deflection angle turning ability. Figure 10a shows the robot at the initial state. Figure 10b shows the first inflation process. At this process, the bending of the left actuator is caused by the inflating of the right actuator. The rear foot of the right side can completely touch the ground to provide maximum friction force. Figure 10c is the final state after the whole turning process.

Results and discussion

The experimental data of the differential soft robot on three different surfaces are shown in Figure 11. Figure 11a shows the linear movement experimental data. The linear movement process includes nine cycle's linear movement and one corrective movement. After nine cycle's linear movement, the differential drive robot slightly deflects to the left side. And then the robot turns back to the initial direction after the self-corrective process. The deflection speed is proportional to the frictional coefficient, which can be seen in Figure 11a. However, the average speed in the initial direction is no big difference among different frictional coefficients. The linear displacement of the robot is about 192 mm after 10 cycles' locomotion (nine linear locomotion cycles and one self-correction cycle). The locomotion time is 19.5 s, and we can get the average moving speed which is 9.85 mm/s (in Fig. 9). And the robot is continuously moving toward the same direction with nine cycles' linear locomotion and one cycle's self-correction.

FIG. 11.

The experimental data of the differential soft robot on different surface, (a) the linear movement experiment and (b) the left-turning movement experiment.

Figure 11b shows the left-turning movement experimental data with different frictional coefficients. This differential drive soft robot has different turning ability in different surfaces. And the turning ability is proportional to the friction force between the robot and the ground surface. It takes the robot 13 cycles to finish the 30° turn on the ABS plastic surface with the frictional coefficient of 0.8. However on the plywood surface and on the glass surface it needs 16 cycles and 17 cycles, respectively. Under these working environments, the average turning speed is from 1.23 to 0.88°/s. This turning movement model has a high turning accuracy, which can be seen in Figure 11b. This RBF model can be applied to a variety of working environments with different roughness, not only the three surfaces.

Conclusion

In this article, a differential drive soft robot is built. It is composed of two columns of paralleled airbags, one baseboard, front feet, rear feet, and one sensor. This robot can perform differential motion, according to the different inflation time of the left and right actuators. Linear movement pattern and turning movement pattern are presented. RBF neural network model is used to train the turning ability of this robot. We collect the experimental data on three different surfaces: the ABS plastic, the plywood, and the glass and create the mathematical model about inflation time, frictional coefficient, and deflection angle. And the closed-loop automatic control model is established by the use of three-axis electronic compass sensor. Finally, the RBF neural networks model is verified by the linear and turning movement experiments. According to the experiment, this linear motion model can keep the robot moving toward the same direction, and the average speed of the linear locomotion is 9.85 mm/s on the ABS surface. In the above experimental condition, the turning movement model has a high accuracy rate, and the average turning speed is from 1.23 to 0.88°/s. Because of the robot's soft structure, it can be applied to the complex and varied natural environment. This robot could also be useful in earthquake rescue and reconnaissance missions where humans or rigid robots are not capable of accessing. Furthermore, to improve the accuracy of RBF neural network model, we will increase the experimental data by performing the experiments on more surfaces with different frictional coefficients. The future work will focus on the integration of this kind of soft robot and apply it to different environments.

Footnotes

Acknowledgment

This work was supported by National Natural Science Foundation of China (grant no. 51475300).

Author Disclosure Statement

No competing financial interests exist.

References

Polygerinos

, Lyne

, Wang

, Nicolini

. Towards a soft pneumatic glove for hand rehabilitation. IEEE/RSJ Int Conf Intell Robots Syst, 2013; 8215:1512–1517.

Marchese

, Katzschmann

, Rus

. A recipe for soft fluidic elastomer robots. Soft Robot, 2015; 27–25.

Jin

, Dong

, Xu

, Liu

, Alici

, Jie

. Soft and smart modular structures actuated by shape memory alloy (SMA) wires as tentacles of soft robots. Smart Mater Struct, 2016; 25:085026.

Lin

, Leisk

, Trimmer

. GoQBot: a caterpillar-inspired soft-bodied rolling robot. Bioinspir Biomim, 2011; 6:26007–26020.

Wang

, Lee

, Rodrigue

, Song

, Chu

, Ahn

. Locomotion of inchworm-inspired robot made of smart soft composite (SSC). Bioinspir Biomim, 2014; 9:046006.

Mao

, Dong

, Jin

, Xu

, Zhang

, Yang

, et al. Gait study and pattern generation of a starfish-like soft robot with flexible rays actuated by SMAs. J Bionic Eng, 2014; 11:400–411.

Kim

, Song

, Ahn

. A turtle-like swimming robot using a smart soft composite (SSC) structure. Smart Mater Struct, 2012; 22:205.

Umedachi

, Trimmer

. Design of a 3D-printed soft robot with posture and steering control. IEEE International Conference on Robotics and Automation (ICRA), Hong Kong, China, 2014; 2874–2879.

Wehner

, Truby

, Fitzgerald

, Mosadegh

, Whitesides

, Lewis

, et al. An integrated design and fabrication strategy for entirely soft autonomous robots. Nature, 2016; 536:7617.

10.

Jenkins

, Chapman

, Bryant

. Bio-inspired online variable recruitment control of fluidic artificial muscles. Smart Mater Struct, 2016; 25:125016.

11.

Correll

, Önal

ÇD

, Liang

, Schoenfeld

, Rus

. Soft autonomous materials—using active elasticity and embedded distributed computation. Springer Berlin Heidelberg, 2014; 79:227–240.

12.

Fei

, Shen

. Nonlinear analysis on moving process of soft robots. Nonlin Dyn, 2013; 73:671–677.

13.

Fei

, Gao

. Nonlinear dynamic modeling on multi-spherical modular soft robots. Nonlin Dyn, 2014; 78:831–838.

14.

Shepherd

, Ilievski

, Choi

, Morin

, Stokes

, Mazzeo

, et al. Multigait soft robot. Proc Natl Acad Sci U S A, 2011; 108:20400–20403.

15.

Majidi

, Shepherd

, Kramer

, Whitesides

, Wood

. Influence of surface traction on soft robot undulation International. J Robot Res, 2013; 32:1577–1584.

16.

Powell

MJD

. Radial Basis Functions for Multivariable Interpolation: A Review (Algorithms for Approximation). New York, NY; Clarendon Press: 1987, pp. 143–167.

17.

Mosadegh

, Polygerinos

, Keplinger

, Wennstedt

, Shepherd

, Gupta

, et al. Soft robotics: pneumatic networks for soft robotics that actuate rapidly. Adv Funct Mater, 2014; 24:2163–2170.

18.

Deimel

, Brock

. A compliant hand based on anovel pneumatic actuator. Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), Karlsruhe, Germany, 2013; 2047–2053.

19.

Yeoh

. Some forms of the strain energy function for rubber. Rubber Chem Technol, 1993; 66:754–771.

20.

Kindelan

, Bernal

, Gonz , et al. Application of the RBF meshless method to the solution of the radiative transport equation. J Comput Phys, 2010; 229:1897–1908.

21.

Seshagiri

, Khalil

. Output feedback control of nonlinear systems using RBF neural networks. IEEE Trans Neural Netw, 2000; 11:69–79.

22.

Kriegel

, Schubert

, Zimek

. The (black) art of runtime evaluation: are we comparing algorithms or implementations?. Know Inf Syst, 2016; 52:1–38.

23.

Bühlmann

, van de Geer

. Statistics for high-dimensional data: methods, theory and applications. J Jpn Stat Soc, 2011; 44:247–249.