Mathematical Problems in Engineering

Volume 2017, Article ID 6841972, 16 pages

https://doi.org/10.1155/2017/6841972

## Kinematics, Dynamics, and Optimal Control of Pneumatic Hexapod Robot

^{1}School of Mechanical Electronic & Information Engineering, China University of Mining and Technology (Beijing), Beijing 100083, China^{2}Mechanical & Electrical Engineering School, Beijing Information Science & Technology University, Beijing 100192, China

Correspondence should be addressed to Long Bai; moc.621@nj6130gnoliab

Received 8 August 2016; Revised 16 January 2017; Accepted 12 February 2017; Published 9 March 2017

Academic Editor: Francisco Valero

Copyright © 2017 Long Bai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Pneumatic hexapod robot is driven by inert gas carried by itself, which has board application prospect in rescue operation of disaster conditions containing flammable gas. Cruising ability is main constraint for practical engineering application which is influenced by kinematics and dynamics character. The matrix operators and pseudospectral method are used to solve dynamics modeling and numerical calculation problem of robot under straight line walking. Kinematics model is numerically solved and relationship of body, joints, and drive cylinders is obtained. With dynamics model and kinematics boundary conditions, the optimal input gas pressure of leg swing and body moving in one step is obtained by pseudospectral method. According to action character of magnetic valve, calculation results of control inputs satisfy engineering design requirements, and cruising ability under finite gas is obtained.

#### 1. Introduction

In recent years, more and more robots are used in industrial accident’s detect and rescue operation. The commonly used robots include motor and hydraulic drive types, but they are not suited for some close space accident environment which fills with flammable gas, such as gas explosion accident of coal mine, because electric devices of them may lead to secondary explosion. The pneumatic robot is driven by inert gas and is convenient to be controlled, which is widely used in industrial and medical domains. Verrelst et al. [1] designed a pneumatic biped robot, which verifies feasibility of using pneumatic system as power source, Lavoie and Desbiens [2] designed a cockroach type pneumatic hexapod robot, Morimoto et al. [3] designed a rehabilitation used soft touch manipulator by soft cylinder and obtained a high working accuracy, Qiu et al. [4] designed a pipe inspection robot by soft cylinders, Diez et al. [5] designed a neural rehabilitation pneumatic robot, Low et al. [6] explored a soft pneumatic massager used in joints auxiliary motion, and Ramsauer et al. [7] explored an error detection using pneumatic Stewart platform.

With these backgrounds, a natural antiexplosion pneumatic hexapod robot (PHR) which is driven by inert gas is designed in this exploration. However, cruising ability is a big influence in robot’s engineer application, for the carried gas’s volume is limited by self-weight of robot. The cruising ability of PHR is measured by straight line walking distance limited by product of volume and pressure of carried gas. During straight line walking, the same characters of each gait decide they have same gas consumption, so cruising ability problem changes to be calculation of distance and gas consumption of one step. Gas consumption of one step is defined as product of cylinder’s volume and drive pressure. The cylinder volume is known, and pressure is influenced by dynamics character of robot. The optimal control method is used to calculate minimum drive pressure.

In the last few years, there are many explorations on optimal control problems of hexapod robots. Sliva and Machado [8] reviewed optimization method used in legged robots; energy/power optimal control objective functions are listed out; Sanz-Merodio et al. [9] explored energy consumption of mammal and insect type robots and concluded that leg dynamics accounts for most energy consumption; Chen et al. [10] designed an insect type hexapod robot and leg has a series mechanism type; the optimal control of leg swing is solved by pseudospectral method; Roy et al. [11–14] explored kinematic dynamics and optimal control problems of hexapod robot; the hexapod robot is driven by electric motor, so it has a series mechanism type; Luneckas et al. [15] analyzed hexapod robot’s energy consumption by motion of body and step height; Deng et al. [16] explored energy reducing problem of hexapod robot by kinematics analysis; Gonzalez de Santos et al. [17] explored minimization of hexapod robot in irregular terrain; the optimal analysis is based on statically stable gait; Jin et al. [18] explored hexapod walking robot’s power consumption optimization problem by torque distribution algorithm and parameters include duty factor, stride length, bogy height, and foot trajectory lateral offset; Zhu et al. [19] explored optimal design of hexapod robot with kinematic model.

Fundamentally, PHR is a parallel mechanism, most of optimal control explorations of it are static, or simplify it as serial mechanism, so the real dynamics character cannot faithfully represent it. The first reason is that complex dynamics character of parallel mechanism makes it difficult to use triangle functions to calculate it, and complex triangle and antitriangle transformations will lead to unsolvable model. Secondly, complex nonlinear characters need optimal control algorithm that has high calculation accuracy and stability, but classic algorithms such as Runge-Kutta method do not satisfy these two characters.

According to references of dynamics modeling by Lie group [20, 21] and optimal control with pseudospectral method [22, 23], the matrix and vector operators can avoid triangle and antitriangle transformations which makes dynamics modeling easier. Pseudospectral method is a global numerical method which has high stability and is widely used in many domains, many engineering problems are solved successfully [24, 25]. So, in this exploration, matrix and vector operators are used as units for dynamic modeling, and optimal control problem is solved by pseudospectral method. The control inputs curves which satisfy pneumatic control characters are obtained, and then cruising ability calculation method is built at last, which offers a reference for the improvement of robot.

#### 2. The Mechanism and Gait of Pneumatic Hexapod Robot

PHR is a biorobot, so it has two design schemes, insect type in Figure 1 and mammal type in Figure 2. Many people like insect type, but it has some problems. Firstly, it walks along direction by triangle gait of hip; joints bear great yawing forces which will lead to joints’ rapid abrasion and to not being suited for engineering application. Secondly, it has a big width, so crosswise passing ability is restricted. Thirdly, the realization of straight line walking needs combination motion of three joints, which is more difficult to realize by pneumatic system and has high gas consumption. The mammal type PHR does not have these problems. The nitrogen gas bottle is in trunk; the maximum pressure can reach 15 MPa. The high pressure gas is decompressed to 1 MPa by PRV (pressure reducing valve), and then gas can be decomposed to different low pressures from 0.15 MPa to 0.8 MPa by DP (duplex pieces). All the magnetic valves and control devices can be packaged in box which is convenient for antiexplosion design. Each leg is composed of shank and thigh which are driven by cylinder.