sensors Article

Trajectory Correction and Locomotion Analysis of a Hexapod Walking Robot with Semi-Round Rigid Feet Yaguang Zhu 1, *, Bo Jin 2 , Yongsheng Wu 1 , Tong Guo 1 and Xiangmo Zhao 3 1 2 3

*

Key Laboratory of Road Construction Technology and Equipment of MOE, Chang’an University, Xi’an 710064, China; [email protected] (Y.W.); [email protected] (T.G.) State Key Laboratory of Fluid Power & Mechatronic Systems, Zhejiang University, Hangzhou 310028, China; [email protected] School of Information Engineering, Chang’an University, Xi’an 710064, China; [email protected] Correspondence: [email protected]; Tel.: +86-187-9285-2585

Academic Editor: Vittorio M. N. Passaro Received: 29 May 2016; Accepted: 24 August 2016; Published: 31 August 2016

Abstract: Aimed at solving the misplaced body trajectory problem caused by the rolling of semi-round rigid feet when a robot is walking, a legged kinematic trajectory correction methodology based on the Least Squares Support Vector Machine (LS-SVM) is proposed. The concept of ideal foothold is put forward for the three-dimensional kinematic model modification of a robot leg, and the deviation value between the ideal foothold and real foothold is analyzed. The forward/inverse kinematic solutions between the ideal foothold and joint angular vectors are formulated and the problem of direct/inverse kinematic nonlinear mapping is solved by using the LS-SVM. Compared with the previous approximation method, this correction methodology has better accuracy and faster calculation speed with regards to inverse kinematics solutions. Experiments on a leg platform and a hexapod walking robot are conducted with multi-sensors for the analysis of foot tip trajectory, base joint vibration, contact force impact, direction deviation, and power consumption, respectively. The comparative analysis shows that the trajectory correction methodology can effectively correct the joint trajectory, thus eliminating the contact force influence of semi-round rigid feet, significantly improving the locomotion of the walking robot and reducing the total power consumption of the system. Keywords: hexapod; semi-round rigid foot; kinematics; trajectory correction; sensors system

1. Introduction Multi-legged walking machines, compared to those with wheeled or tracked locomotion, are widely recognized as a much more effective and efficient form of transportation vehicle, especially on complex and unstructured terrains. Hexapod robots, one such type of legged walking machines, generally have superior performance than those with fewer legs in terms of less complexity of the control method, more statically stable walking, and faster walking speed [1,2]. Therefore, in recent years, multi-legged walking robots have been extensively researched and are noted for their good environmental adaptability and movement flexibility. The potential application terrain for use of these robots is mainly unstructured surface environments [3]. In order to have sufficient adaptive capacity for different conditions, the design of the robotic leg and foot is becoming particularly important. According to current research on multi-legged walking robots, foot designs, on the basis of the different linkages between the leg and foot tip, can be divided into two main groups; non-articulated feet and passive ankle feet. The non-articulated foot group can be farther divided into flat foot, rounded flat foot, (semi)-round foot, and so on. The robots Dante II [4], AMBLER [5], and ROWER [6] have been built with flat foot designs, while the G. E walking Truck [7] Sensors 2016, 16, 1392; doi:10.3390/s16091392

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and Airbug [8] robots adopted rounded flat foot designs. Other robots such as MASCHA [9], Silex [10], LauronIII [11], TUM [12], TARRY II [13], and so on use (semi)-round rigid feet. However, the robots with a passive ankle foot design have a large difference in the number of degrees of freedom (DOF). PV II [14], TITAN [15], and OSU [16] adopted a passive ankle-joint with a single DOF, while the COMET II [17] and SILO4 [18] adopted a foot structure with an ankle-joint, which has two revolute joints. The flat feet of HAMLET [19] are connected with the body by a spherical hinge with three DOF. Overall, the foot with a passive ankle-joint is complex in its design and manufacturing, and the cost is also correspondingly higher. In addition, the trajectory of robots with movable ankle-joints and flat feet is also limited by foothold. The small angle between the foot trajectory and the surface will likely cause impact with the ground, or even stumbling. The non-articulated foot with a flat or round structure also has great limitations in its adaptability to complex terrain. By contrast, a (semi)-round rigid foot has significant advantages in applications, so it is widely used in many robot prototypes [20–22]. When designing the size of a semi-round rigid foot, using a smaller radius can cause the foot to sink into a surface, and should be considered because the walking robot usually has to adapt to a variety of complex surface environments, such as rugged or soft ground. This problem can be effectively alleviated by increasing the radius of the foot, but on hard ground a large-radius foot will change the contact point between the foot and the ground when it is in the supporting phase, thus affecting the trajectory. Because multi-legged robots have a parallel closed chain structure in the standing state, the nonconformity of body trajectory error caused by the semi-round foot of each leg will eventually lead to sliding between the foot and the ground, and affect the stability of the robot. To address the problem above, Chen et al. proposed a trajectory correction method [23], where the predefined trajectory of the robotic foot tip is modified according to the information from force feedback. In this way, the robot can adapt to the current terrain at any time and it can relieve the running deviation. In another paper [24], when the running deviation rate of the robot reaches a gate value, the robot can amend it by adjusting gait or changing direction. Wei et al. [25] proposed a method to measure the deviation degree of the body during movement by using machine vision. More specifically, they used characteristic points of machine vision to track external parameters of the camera during movement between two frames obtained by pose estimation technology. On this basis, calculation of the deviation degree was completed. Kwon et al. [26] proposed an adaptive trajectory generation method for quadruped robots with semicircular feet to control body speed and heading. The adaptive gait patterns are changed by the sequential modulation of the locomotion period and the stride per step, which are determined by the desired body speed and heading commands. The researchers above have achieved the correction from the closed-loop control of the trajectory, but the problem of deviation between the actual position and the desired trajectory of the robot's feet have not been solved. Guardabrazo et al. [27] had proposed a method to solve it. However the kinematic model they established was mainly used to analyze the motion of the robot with mechanical legs imitating insects in a two-dimensional working plane. Although the problem of a semi-round rigid foot in a three-dimensional space has been investigated, a complete kinematic relationship is missing. Due to the special structure of the semi-round foot, the deviation of each leg is different and it will further lead to interference between the legs, as well as more energy consumption. Therefore, it is necessary to propose a correction algorithm for the kinematics. In order to find a better way to solve the problem of deviation caused by a semi-round foot and to improve the accuracy of the inverse kinematic solution, the trajectory deviation of the body and leg caused by a semi-round foot is studied. According to the kinematic analysis, a correction methodology for forward/inverse kinematic solutions in three-dimensional space between the relative position of the semi-round foot and joint trajectory is proposed. A nonlinear mapping relationship between foot position and joint angle is approached by constructing the optimal linear regression function, and then the inverse kinematic relationship of a robot with semi-round foot is realized based on LS-SVM. Using the existing hexapod walking robot platform with multi-sensors, a series of related experiments are carried out, and the validity of the

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Sensors 2016, 16, 1392 3 of 21 correction methodology for improving foot stress, body trajectory, foot slip, and energy consumption is verified. carried out, and the validity of the correction methodology for improving foot stress, body The remainder of and thisenergy paper consumption is organizedisasverified. follows: Section 2 provides a brief introduction to trajectory, foot slip, the robotThe system. Section 3 presents the modeling and theSection methods in detail. Section 4 describes remainder of this paper is organized as follows: 2 provides a brief introduction to the experiments and discusses the results. Section 5 summarizes and concludes the paper. the robot system. Section 3 presents the modeling and the methods in detail. Section 4 describes the

experiments and discusses the results. Section 5 summarizes and concludes the paper.

2. Description of the Structure and Sensor System 2. Description of the Structure and Sensor System

The structure of a robotic leg should not only imitate the structure of legged animals (such as Thespiders), structurebut of ashould robotic also leg should not only imitatesystem the structure of legged animals insects and consider its power and constraints [28]. In(such this as paper, insects and spiders), but should also consider its power system and constraints [28]. In this paper, a type of leg structure with a mammalian configuration is adopted, since mammal legs requirea less of leg structure with a mammalian configuration is adopted, since mammal legs require less joint type torques to support the body [29]. Mammalian legs usually have three joints (base-joint, hip-joint, joint torques to support the body [29]. Mammalian legs usually have three joints (base-joint, and knee-joint). The base-joint and the hip-joint are located higher than knee-joint. Its characteristics of hip-joint, and knee-joint). The base-joint and the hip-joint are located higher than knee-joint. Its low energy consumption and large loading are more suitable for outdoor tasks. The mechanical model characteristics of low energy consumption and large loading are more suitable for outdoor tasks. of theThe robot is shown in Figure Each leg consists of1a. three linkages which are connected by a hip mechanical model of the 1a. robot is shown in Figure Each leg consists of three linkages which joint are andconnected a knee joint. leg mechanism is attached the body via a base joint. a semi-round by a The hip joint and a knee joint. The legtomechanism is attached to theHere, body via a base foot with a certain radius is used. The leg variables are shown in Figure 1b. The robot can joint. Here, a semi-round foot with a certain radius is used. The leg variables are shown in Figureachieve 1b. translation andcan rotation three-dimensional space in its workspacespace [30].in its workspace [30]. The robot achieveintranslation and rotation in three-dimensional

(a)

(b)

(c)

(d)

Figure 1. Structure and sensor system of the robot: (a) Mechanical model of the hexapod walking

Figure 1. Structure and sensor system of the robot: (a) Mechanical model of the hexapod walking robot; robot; (b) Reference coordinate system and corresponding joint variable of the three joint mechanical (b) Reference coordinate system and corresponding joint variable of the three joint mechanical leg; leg; (c) Distribution of the sensors; (d) Diagram of the control architecture. (c) Distribution of the sensors; (d) Diagram of the control architecture.

The sensors, especially the camera, attitude sensor, and force sensor, play an important role in Theintelligent sensors, especially the camera, attitude sensor, of and force sensor, play an important rolefor in the the robotics field [31,32]. The distribution the multi-sensor system of the robot observing motion state is shown Figure 1c. BLi, HLi,multi-sensor and KLi (BRi,system HRi, and KRirobot for the leg) intelligent robotics field [31,32]. Theindistribution of the of the forith observing respectively represent the base-joint, onand eachKRi left for (right) and respectively LF, LM, motion state is shown in Figure 1c. BLi,hip-joint, HLi, andand KLiknee-joint (BRi, HRi, the leg, ith leg) and LH (RF, RM, and RH) respectively represent the front leg, middle leg, and hind leg on the(RF, left RM, represent the base-joint, hip-joint, and knee-joint on each left (right) leg, and LF, LM, and LH (right) side. Each joint is composed of a DC motor, a high reduction rate gear system, and has anjoint and RH) respectively represent the front leg, middle leg, and hind leg on the left (right) side. Each

is composed of a DC motor, a high reduction rate gear system, and has an integrated encoder, which is

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used for detecting the position of the joint angle. Contact sensors (CoSLi or CoSRi) installed on the semi-round foot tips (SRLi or SRRi) are used for monitoring the gait of the robot. In order to reflect the energy consumption of each leg in different gaits, current detection modules (CuSLi or CuSRi) are used. The arrows represent current flow direction. The attitude sensor (AS) is installed in the body to monitor the posture of the robot body. And then, the data processed by the wireless module transmits to a host computer to generate commands. Finally, the commands are send to the slave board Cortex-M4 to control the motion of each leg and to monitor the state of each sensor. The diagram of the control architecture is shown in Figure 1d. 3. Trajectory Correction Methodology The deviation caused by semi-round rigid foot not only occurs in the vertical direction, but also the horizontal direction. For multi-legged parallel mechanisms, these inaccurate displacements will cause deviation between the real posture and ideal posture of the body, causing the walking robot to not move smoothly. In addition, each supporting leg of the robot will lead to different deviations when they are actuated at the same time. This will not only lead to interference of supporting legs for its redundant DOFs, but also waste a lot of system energy. When this situation persists, slipping will occur and will influence the stability of the robot [33]. Therefore, it is extremely necessary to put forward kinematic correction methodology according to the special structure of the semi-round rigid foot. 3.1. Kinematic Analysis Each leg can be regarded as a manipulator with three rotating joints attached to a stationary base. It includes two parallel joints, the hip-joint θ 2 and the knee-joint θ 3 , which is connected to the body through a base-joint θ 1 . Therefore, the establishment of the kinematic model and the derivation of the kinematic equation can follow traditional robot technology and methods. The kinematic model of this paper is obtained by defining the reference coordinate system using the Denavit-Hartenberg method [34]. The model of the leg structure with three joints is shown in Figure 1b, in which the reference coordinate and the corresponding joint variables are marked. The base coordinate system řpiq piq 0 pO0 ´ xyzq is located on the static robot body. The connection parameters of the D-H model are listed in Table 1. Table 1. Denavit-Hartenberg parameters. Link j 1 2 3

αi(j-1) /˝ 0 90˝ 0

θij /˝

ai(j-1) /m 0 0 L2

´30˝ ~60˝

3)/´60˝ ~30˝

(i = 1, 2, (i = 4, 5, 6) ´240˝ ~0˝ (i = 2, 3, 4, 5)/0˝ ~240˝ (i = 1, 6) ´120˝ ~0˝ (i = 2, 3, 4, 5)/0˝ ~120˝ (i = 1, 6)

dij /m 0 0 0

ř Therefore, under the base reference coordinate system 0 pO0 ´ xyzq of the leg, three joint angles are already known and foot position vector 0tip Pˆ can be obtained by the forward kinematic equation, as follows: » fi » fi 0 P ˆ C1 C23 L3 ` C1 C2 L2 tip x — ffi — ffi — 0 Pˆ ffi 0 ˆ — S1 C23 L3 ` S1 C2 L2 ffi, (1) tip P “ — tip y ffi “ – fl – fl 0 P ˆ S23 L3 ` S2 L2 tip z in which, Si “ sinθi , Ci “ cosθi , Sij “ sinpθi ` θj q, Cij “ cospθi ` θj q, θi and θj are the mean joint angles of ith and jth, respectively.

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 Here, the algebraic method  tip0 Pˆy is used to solve the inverse kinematic problem. The plus symbol is     arctan used for the front-leg of themulti-legged robot, and the minus symbol is used for the other leg: 0 ˆ   1 P tip x   $ ˜0 ¸ ˆy 0 ˆ 2 0 ˆ2 0 ˆ2 P 0 ˆ2 2 2 ’  ’ P  tip0 Pˆx2 tip  P ’ tip y θ1 “ arctantip x tip Py  tip Pz  L2  L3 ’ ’  arctan . 0 P (2)   arcsin ’ ˆ 0 ˆ ’ tip x 0 P ’ 2 ˆ 2  0 Pˆ 2  0 Pˆ 2 Pz ’ 2L tipb ’ x 2 tip tip y tip z ’ ’ 0 P 0 ˆ2 ˆ2 0 0ˆ 2 2` 0 0 P 2 00 P 2 2 L2 & ˆ2 ˆˆy2 ` tip tip y ` tip Px xˆ  tip P 3 θ “ arcsin tiptipP P Pˆ 2z`L22 L2 L´ x btip y tip z 3 . (2) ` arctan 2 0 P ˆz ’ ’  3   arccos 2L2 0 Pˆx22 L`L0 Pˆy2 ` 0 Pˆz2 . tip ’ tip tip  ’ 2 tip 3 ’ ’ ’ 0 P 2 ` 0 P 2 ` 0 P 2 ´ L2 ´ L2 ’ ˆ ˆ ˆ ’ 2 3 tip x tip y tip z ’ ’ . ’ % θ3 “ ˘ arccos 2L2 L3 3.2. Kinematics Correction of a Single Leg with a Semi-Round Rigid Foot real trajectory body with theRigid theoretical trajectory when the foot is 3.2. The Kinematics Correctionof of athe Single Legcoincides with a Semi-Round Foot regarded as a point and there is no slipping. In theory, it can follow the ideal trajectory very well, but The real trajectory of the body coincides with the theoretical trajectory when the foot is regarded it is not practical for the mechanical structure. However, when the foot structure is semi-round, even as a point and there is no slipping. In theory, it can follow the ideal trajectory very well, but it is not though there is no slipping, the contact point between the foot and ground will be changing during practical for the mechanical structure. However, when the foot structure is semi-round, even though the movement, which will cause deviation from the preset trajectory. Hence, in order to eliminate there is no slipping, the contact point between the foot and ground will be changing during the this deviation, the joint angle needs to be corrected. Because the robot is a closed kinematic chain, movement, which will cause deviation from the preset trajectory. Hence, in order to eliminate these deviations will lead to the error between the actual posture and the ideal posture of the robot this deviation, the joint angle needs to be corrected. Because the robot is a closed kinematic chain, body. Figure 2 is a side view of a single leg with a semi-round rigid foot. For ease of analysis, the these deviations will lead to the error between the actual posture and the ideal posture of the robot body. concept of ideal foothold is proposed. On the horizontal surface, when the axial direction of the 3rd Figure 2 is a side view of a single leg with a semi-round rigid foot. For ease of analysis, the concept of linkage of leg is perpendicular to the surface of the ground (shown in Figure 2 as a dashed line), the ideal foothold is proposed. On the horizontal surface, when the axial direction of the 3rd linkage of leg contact point of the horizontal surface and semi-round rigid foot on the ground is the ideal foothold is perpendicular to the surface of the ground (shown in Figure 2 as a dashed line), the contact point of TI, and this point on the semi-round rigid foot is called foot reference point TP. For the real foothold the horizontal surface and semi-round rigid foot on the ground is the ideal foothold TI , and this point of the support leg at any posture, the only ideal foothold can be calculated by neglecting any small on the semi-round rigid foot is called foot reference point TP . For the real foothold of the support leg slippage between the foot and the ground, supposed as pure rolling. It is assumed that the origin of at any posture, the only ideal foothold can be calculated by neglecting any small slippage between the base-joint coordinate is Or, the spherical center of the semi-round foot is Ot, and the real contact the foot and the ground, supposed as pure rolling. It is assumed that the origin of the base-joint point of the semi-round rigid foot and the ground is TG, which is shown in Figure 2. The angle coordinate is Or , the spherical center of the semi-round foot is Ot , and the real contact point of the between link 3 and the perpendicular of the horizontal plane is ϕ. The location of the ideal foothold semi-round rigid foot and the ground is T , which is shown in Figure 2. The angle between link 3 and in the base-joint coordinate system can beG calculated by forward kinematics, as long as each joint the perpendicular of the horizontal plane is ϕ. The location of the ideal foothold in the base-joint angle vector is known. coordinate system can be calculated by forward kinematics, as long as each joint angle vector is known.

Or

 OrTG

 OrTP Ot

R  OtTG TG

 OrTI

 OtTP

TP  

TI

Figure2.2.Side Sideview viewofofaasingle singleleg legwith withaasemi-round semi-roundrigid rigidfoot. foot. Figure

According to the assumption that there is no relative slipping between the semi-round rigid foot and the ground, then:

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According to the assumption that there is no relative slipping between the semi-round rigid foot and the ground, then: ˇá ˇ ˇÝÝᡠˇ ˇ ˇÝ ˇ ˇ ˇ ˇ (3) ˇ Λ ˇ “ ˇTG TI ˇ “ ˇTŐ G TP , ˇ ˇ ˇ in which, ˇTŐ G TP is a circle between the foot reference point T P and the real contact point T G . After vector analysis of Figure 2, the foot reference position is obtained: »

fi C1 C23 pL3 ´ Rq ` C1 C2 L2 ÝÝÝá — ffi Or Ot “ – S1 C23 pL3 ´ Rq ` S1 C2 L2 fl, S23 pL3 ´ Rq ` S2 L2 and real foothold is:

»

fi C1 C23 pL3 ´ Rq ` C1 C2 L2 ´ R ÝÝÝá — ffi Or TG “ – S1 C23 pL3 ´ Rq ` S1 C2 L2 fl, S23 pL3 ´ Rq ` S2 L2

As for ideal foothold, we have: in which:

(4)

(5)

ÝÝá ÝÝÝá á Ý Or TI “ Or TG ` Λ ,

(6)

fi C1 C23 pL3 ´ Rq ` C1 C2 L2 ´ R ÝÝá — ffi Or TI “ – S1 C23 pL3 ´ Rq ` S1 C2 L2 ` Λy fl , S23 pL3 ´ Rq ` S2 L2 ` Λz

(7)

»

because of:

” ÝÝÝá Ot TP “ C1 C23 R

S1 C23 R

” ÝÝÝá Ot TG “ ´R

0

0

S23 R ıT

ıT

,

(8)

.

(9)

Therefore, the angle ϕ between the 3rd linkage and the perpendicular of the horizontal plane can be obtained by Equations (8) and (9): ÝÝÝá ÝÝÝá ´C1 C23 R2 Ot TP ¨ Ot TG “ ´C1 C23 , cosϕ “ ÝÝÝÝá ÝÝÝÝá “ R2 |Ot TP ||Ot TG |

(10)

ñ ϕ “ arccos p´C1 C23 q

(11)

ˇá ˇ ˇÝ ˇ ˇ Λ ˇ “ R ¨ ϕ.

(12)

because:

á Ý ÝÝÝá Usually, Λx “ 0, Λ and Ot TP are coplanar, so: ” á Ý L “ 0

Λy

Λz

S1 C23 ϕR

«

ıT “

0

b

2 ` S2 C 2 S23 1 23

S23 ϕR b

2 ` S2 C 2 S23 1 23

ffT .

(13)

Hence, in the base-joint coordinate system, the kinematics solution of the ideal foothold is obtained: » fi » fi 0T ˆ C1 C23 pL3 ´ Rq ` C1 C2 L2 ´ R Ix ÝÝá — ffi — ffi Or TI “ – 0 TˆIy fl “ – S1 C23 pL3 ´ Rq ` S1 C2 L2 ` Λy fl. (14) 0T ˆ S23 pL3 ´ Rq ` S2 L2 ` Λz Iz In the same way, when the position of the ideal foothold is known, each joint angle vector can be obtained by the inverse kinematics solution.

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The front view of a single leg with a semi-round rigid foot is shown in Figure 3, which shows that: 0T ˆ ´ Λy Iy 0T ˆ `R Ix

“ tanθ1 .

(15)

0T ˆ ´ Λy Iy . 0T ˆ `R Ix

(16)

From Equation (15), we can obtain: θ11 “ arctan Because of Equation (6), we have: » — — –

0T ˆ Ix 0T ˆ Iy 0T ˆ Iz

fi

»

— ffi ffi “ — — fl –

0 P ˆ tip x ´ C1 C23 R ´ R 0 P ˆ tip y ´ S1 C23 R ` Λy 0 P ˆ tip z

´ S23 R ` Λz

fi

» 0 TˆIx ` C1 C23 R ` R ffi — 0T ffi ˆ ` S1 C23 R ´ Λy ffi ñ 0tip Pˆ “ — Iy – fl 0T ˆ ` S23 R ´ Λz Iz

fi ffi ffi . fl

(17)

Substituting Equation (17) into Equation (2), the modified joint angles θ21 and θ31 can be obtained. However, it is difficult to solve because Λx , Λy , and ϕ all have a relationship with θ21 and θ31 . Here, it is approximately solved by using uncorrected joint angles. According to Equation (13), we have: á Ý Λ « r 0

S1 C23 ϕR b

2

S23 ϕR

2 2

b

S23 ` S1 C23

2

2 2

S23 ` S1 C23

sT ,

(18)

¯2 ´ ˘2 ` ˘2 TˆIz ´ Λz ` S23 R ` 0 TˆIy ´ Λy ` S1 C23 R ` 0 TˆIx ` R ` C1 C23 R ` L22 ´ L23 1 c θ2 “ arcsin ¯2 ´ ` ˘ ` ˘ 0T ˆ ´ Λz ` S23 R 2 ` 0 Tˆ ´ Λy ` S1 C23 R ` 0 Tˆ ` R ` C1 C23 R 2 2L2 Iz Iy Ix , c´ ¯2 ˘ ` 0T ˆ ´ Λy ` S1 C23 R ` 0 Tˆ ` R ` C1 C23 R 2 Iy Ix ´arctan 0T ˆ ´ Λz ` S23 R Iz `0

`0 θ31

“ ˘arccos

TˆIz ´ Λz ` S23 R

˘2

´ `

0T ˆ Iy

´ Λy ` S1 C23 R 2L2 L3

¯2 `

`0

TˆIx ` R ` C1 C23 R

˘2

´ L22 ´ L23

,

(19)

(20)

` ˘ ` ˘ where, Si “ sinθ i , Ci “ cosθ i , Sij “ sin θ i ` θ j , and Cij “ cos θ i ` θ j , in which θ i and θ j are the i-th and j-th joint-angles before correction. Upon substituting Equation (18) into Equations (19) and (20), the amended inverse kinematics relationship of a semi-round rigid foot is achieved. For a multi-legged robot, when it is used for the foreleg, Equation (19) has a positive sign, and when it is used for a rear leg, the equation has a negative sign.

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x

Or y

 OtTP O t  OtTG

TP

TI  y TG 3. Front view single leg leg with rigid rigid foot. foot. FigureFigure 3. Front view of of a asingle witha semi-round a semi-round

3.3. Semi-Round Rigid Foot Trajectory Correction Algorithm Based on LS-SVM

3.3. Semi-Round Rigid Foot Trajectory Correction Algorithm Based on LS-SVM

The correction algorithm for the foot trajectory can eliminate the effect of semi-round rigid feet

on the robot.algorithm However, the previous inverse kinematic is obtained by the The correction for the foot trajectory canalgorithm eliminate the effect of approximation semi-round rigid feet algorithm according to Equations (16)–(20), which has fast computational speed but the accuracy is on the robot. However, the previous inverse kinematic algorithm is obtained by the approximation not high, so the trajectory correction algorithm for a semi-round rigid foot based on the least squares algorithmsupport according Equations (16)–(20), which has fast the vectortomachine (LS-SVM) is proposed to solve this computational problem. The basicspeed idea ofbut SVM is accuracy is not high, so the trajectory correction algorithm for a semi-round rigid foot based on mapping the input vector to a high dimensional feature space by using nonlinear transformation, the least and constructing an optimal(LS-SVM) decision function in this space [35].this When constructing optimal squares support vector machine is proposed to solve problem. Thethe basic idea of SVM decision function, the structural risk minimization principle is used and the point multiplication in is mapping the input vector to a high dimensional feature space by using nonlinear transformation, the high-dimensional feature space is replaced by using the kernel function of the original space. and constructing an optimal decision function in, this space [35]. When constructing the optimal , ∈ , ∈ : Assuming a given training sample decision function, the structural risk minimization principle is used and the point multiplication in the y( x)  w , ( x)  b , (21) high-dimensional feature space is replaced by using the kernel function of the original space. . 〉 represents the sample point multiplication, vector of the original where 〈.a, given Assuming training txk , yk ukN“1and , xk P∈Rn , yisk the P Rweight : is a nonlinear mapping of samples from the original space to the high weighted space. (∙): → @ D dimensional feature space, and the linear function ypxq regression “ w, ϕpxq ` isb,constructed in this space. (21) LS-SVM [36] is an extension of standard SVM. The optimization index adopts squared terms @ D nh equation the in-equation of standard SVMand are replaced constraints. That of means where ., and . represents theconstraints point multiplication, w P Rby is the weight vector the original the quadratic programming problems transform into problems of the linear equation set solution. n k weighted space. ϕ p¨q : R Ñ R is a nonlinear mapping of samples from the original space to the high This method simplifies the complexity of the calculation and accelerates the solving process. The dimensional feature space, and the linear regression function is constructed in this space. optimization problem of LS-SVM can be described as: LS-SVM [36] is an extension of standard SVM. TheN optimization index adopts squared terms 2 1 1  w are   replaced lim  ( w , e ) SVM and the in-equation constraints of standard That means  e 2 . by equation constraints.(22) w ,e 2 2 k 1 k the quadratic programming problems transform into problems of the linear equation set solution. constraint condition is: This method The simplifies the complexity of the calculation and accelerates the solving process. yk ( x)  w , be  ( x kdescribed )  b  e k , k as:  1,  , N , (23) The optimization problem of LS-SVM can where ∈ is error variable, and ≥ 0 is a regular constant.ÿ A smaller N 1 1 2 caused by noise. limϑpw, eq “ k w k ` γ ek2 . w,e function: 2 2 Introducing the Lagrangian k “1

can avoid over-fitting

(22)

N

The constraint condition ( wis: , b , e ;  )   ( w , e )    k  w ,  ( xk )  b  e k  y k  , k 1 @ D yk pxq “ w, ϕpxk q ` b ` ek , k “ 1, . . . , N,

(24)

(23)

where ek P R is error variable, and γ ě 0 is a regular constant. A smaller γ can avoid over-fitting caused by noise. Introducing the Lagrangian function: Γpw, b, e; αq “ ϑpw, eq ´

N ÿ

αk

@

D ( w, ϕpxk q ` b ` ek ´ yk ,

k “1

in which, ak P R are Lagrange multipliers, so the constraint conditions become:

(24)

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$ N ř BΓ ’ ’ αk ϕpxk q “ 0Ñw “ ’ ’ ’ Bw ’ k “1 ’ ’ ’ N ’ ř BΓ ’ ’ αk “ 0 “ 0Ñ & Bb k“1 . ’ BΓ ’ ’ “ 0 Ñ αk “ γek , k “ 1, . . . , N ’ ’ ’ Bek ’ ’ ’ ’ BΓ ’ ’ “ 0 Ñ xw, ϕpxk qy ` b ` ek ´ yk pxq, k “ 1, . . . , N % Bαk

(25)

Those constraints are consistent with the optimal conditions of standard SVM except ak “ γek . The linear equation can be obtained by eliminating the variables w and ek : « ff « ff « ff 0 eT b 0 “ , (26) e Ω ` γ ´1 I α y where, I is a unit matrix with n ˆ n, y = [y1 , . . . ,yN ], e = [1, . . . ,1], α = [α1 , . . . ,αN ]. According to the Merce condition, we have: @ D Ωkl “ ϕpxk q, ϕpxl q “ Ψpxk , xl q, k “ 1, . . . , N. (27) Although the selection criteria of the kernel function Ψ(xk ,xl ) is consistent with the standard SVM, the radial basis function is widely used now, so the linear regression function is: + # k x ´ xk k2 , (28) Ψpx, xk q “ exp ´ 2σ2 in which the term σ is the kernel bandwidth. yk pxq “

N ÿ

αk Ψpx, xk q ` b,

(29)

k“1

where α and b satisfy Equation (26). The forward/inverse kinematic solution of the ideal foot is derived under the base-joint coordinate system in the last section. An approximate solution of inverse kinematics is obtained by mapping the idea location of the foot to the angle-joint. This process can be represented by: ´ ¯ 0 ˆ 0 ˆ 0 ˆ (30) tip Px , tip Py , tip Pz Ñ pθ1 , θ2 , θ3 q In order to solve this nonlinear mapping problem, LS-SVM is utilized to approximate the mapping in this section. In other words, this nonlinear mapping between the ideal location of the foot p0tip Pˆx , 0tip Pˆy , 0tip Pˆz q and the joint angle (θ1 , θ2 , θ3 q is approached by constructing the optimal linear regression function, and then the inverse kinematic solution is found. The forward kinematics model can be obtained directly by Equation (17). However, the result of inverse kinematics is an approximate solution, which can be applied in low precision occasions, but it may cause locomotion error when used in high precision occasions. Therefore, when choosing training samples, the results will be closer to the ideal solution after iteration computation by the approximate method. In this way, the training sample obtained is more accurate, and the accuracy of the training result is higher. From Equation (16) to Equation (20), Λy , Λz , and ϕ are all related to θ21 and θ31 , which makes it difficult to obtain analytical solutions directly. In general, only an approximate solution is achieved according to the uncorrected joint rotation angle. This result is usually substituted into Equations (19) and (20), which are the kinematics inverse solutions of θ21 and θ31 after correction of the semi-round foot. Here, the values of θ21 and θ31 obtained by using the iterative method are substituted into Equations (19) and (20), and so forth, until they are no longer changing. θ21 and θ31 at this time are regarded as joint-angles under ideal mapping in the current position coordinate. According to the

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position coordinate. coordinate. According According to position to the the joint joint parameters parameters LL22 == 15 15 cm cm and and LL33== 15 15 cm, cm, the the input/output input/output joint parameters L2the = 15 cm and L3 = 15kinematics cm, the input/output curveinsurface of4.the forward/inverse curve surface of forward/inverse model are shown Figure N curve surface of the forward/inverse kinematics model are shown in Figure 4. N sets sets of of data data are are kinematics model are shown in Figure 4.Due N sets of base-joint data are selected aswith the correction training sample from it. selected as the training sample from it. to the angle as a precise selected as the training sample from it. Due to the base-joint angle with correction as a precise Due to the base-joint angleofθ11 with correction as a be precise value, only the values of θ21 andmodels θ31 need are to be and need value, only the values values and need to to be solved, solved, so so two two LS-SVM LS-SVM mapping mapping models are value, only the of solved, so twoThen, LS-SVM models are established. Then, theasdesired position points established. the mapping desired trajectory position points are used inputstrajectory of the trained LS-SVM established. Then, the desired trajectory position points are used as inputs of the trained LS-SVM are used as models. inputs ofThe the trained mapping according models. The afterare calculation mapping outputs LS-SVM after calculation to outputs the models the jointaccording angles mapping models. The outputs after calculation according to the models are the joint angles tocorresponding the models aretothe angles corresponding to the desired positions. The processbyofLS-SVM the inverse thejoint desired positions. The process of the inverse kinematics solution is corresponding to the desired positions. The process of the inverse kinematics solution by LS-SVM is shown in Figure 5. by LS-SVM is shown in Figure 5. kinematics solution shown in Figure 5.

(a) (a)

(b) (b)

(c) (c)

(d) (d)

Figure4. 4.Input/output Input/output curved curved surface surface of the forward/inverse to to thethe Figure of forward/inversekinematic kinematicmodel modelaccording according Figure 4. Input/output of the the kinematic model according to θ the correction algorithm: (a) curved Forwardsurface kinematics Z; forward/inverse (b) Forward kinematics X; (c) Inverse kinematics correction algorithm: (a) Forward kinematics Z; (b) (b) Forward Forward kinematics kinematicsX; X;(c) (c)Inverse Inversekinematics kinematicsθ2;2θ; 2 ; correction algorithm: (a) Forward kinematics Z; (d) Inverse kinematics θ3. (d) 3 .3. (d)Inverse Inversekinematics kinematicsθ θ

Figure 5. Process of the inverse kinematics solution by LS-SVM. Figure Process of of the the inverse inverse kinematics kinematics solution Figure 5.5.Process solution by byLS-SVM. LS-SVM.

3.4. Results and Error Analysis 3.4. Results and Error Analysis 3.4. Results andexperiments, Error Analysis In our the step size is 0.10 m, the leg lift height is 0.05 m, the foot radius is In and our the experiments, the step size is gait 0.10 m, the leg lift height is 0.05samples m, the isfoot 0.02Inm land coefficient is 0.6 The N =radius 400, theism our experiments, the step sizeinisone 0.10 m,cycle. the leg liftnumber height of is training 0.05 m, the foot radius is 0.02 0.02 m of and thetraining land coefficient 0.6Cartesian in one gaitcoordinate cycle. The number N = desired 400, the , of training of eachsamples point inisthe input the samples isare and the land coefficient is 0.6 in one gait cycle. The number of training samples is N = 400, the input of each in the desired input of the training samples coordinate `0 is0 ( ˘, ,) obtained trajectory, and the output of are the Cartesian training samples by point multi-iteration. The oftrajectory, the training samples are Cartesian coordinate TˆIxis , Tˆ(Iz , of )each point by in the desired trajectory, and the output of the training samples obtained multi-iteration. The 2 = 0.2. ` ˘ number of test samples is n = 200, the kernel function is a radial basis function, γ = 100, and σ 1 , θ 1 obtained by multi-iteration. The number of test and the output of the training samples is θ 2 2 3 number test samples is n = 200, the kernel function radial γ =kinematics 100, and σof= the 0.2. The footof correction algorithm included trained model is is aused to basis solve function, the inverse 2 = 0.2. The foot samples iscorrection n = 200, the kernel included function trained is a radial basis used function, γ =the 100, and σkinematics The foot algorithm model to solve of the robot with a semi-round rigid foot. Figure 6 shows thatisthe maximum errorsinverse of and obtained correction algorithm included trained model is used to solve the inverse kinematics of obtained the robot robot a semi-round rigid foot. 6 shows maximum of can and by thewith approximate calculation areFigure only 0.008 rad that and the 0.016 rad, but errors the errors be reduced to 1 1 with a semi-round rigid foot. Figure 6 shows of errors θ2 andcan θ3 obtained by to the by the approximate calculation are only 0.008that radthe andmaximum 0.016 rad,errors but the be reduced approximate calculation are only 0.008 rad and 0.016 rad, but the errors can be reduced to 0.003 rad and

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0.005 radrad byand using LS-SVM. the iteration algorithm can provide a better trackinga accuracy, 0.003 0.005 rad byAlthough using LS-SVM. Although the iteration algorithm can provide better 0.003 rad and 0.005 rad by using LS-SVM. Although the iteration algorithm can provide a better thetracking execution cycle isthe 0.27 ms, while after the ms, use while of LS-SVM, this cycle is 0.15 ms. accuracy, execution cycle is 0.27 after the use of LS-SVM, this cycle is 0.15 ms. tracking accuracy, the execution cycle is 0.27 ms, while after the use of LS-SVM, this cycle is 0.15 ms. 0.01 0.01 0.005 0.005 0 0 -0.005 -0.005 -0.01 -0.01 -0.015 -0.015 -0.02 -0.020 0

LS-SVM LS-SVM Approximation Approximation

0.25 0.25

(a) (a)

0.5 t/s 0.5

(b) t/s (b)

0.75 0.75

1 1

Figure 6. Joint angle errors of LS-LSM and the approximation method: (a) The angle error of the hip Figure 6. 6. Joint errors ofofLS-LSM and method:(a) (a)The Theangle angleerror errorofofthe the hip Figure Jointangle angleerror errors LS-LSM and the the approximation approximation method: hip joint; (b) The angle of the knee joint. joint; (b) The angle error of the knee joint. joint; (b) The angle error of the knee joint.

F contact /N

F contact /N

h r /mm

h r /mm

4. Experiments and Discussion 4. 4.Experiments Experimentsand andDiscussion Discussion In order to verify the proposed trajectory correction methodology, a series of experiments were InIn order to verify the trajectory correction methodology,aaseries series ofexperiments experiments were order theproposed proposed methodology, were carried out ontoaverify leg platform and atrajectory hexapod correction walking robot. The experimentalof results and related carried out and aahexapod The experimental experimentalresults resultsand andrelated related carried outon onapresented aleg legplatform platform and hexapod walking walking robot. robot. The discussion are in this section. discussion are discussion arepresented presentedininthis thissection. section. 4.1. Single Leg Platform Tests 4.1. Single 4.1. SingleLeg LegPlatform PlatformTests Tests Experiments on a leg platform with less influence from the factors of the other legs are Experiments onon a leg platform withwith less influence from the factors of thethe other legs are conducted Experiments athat leg platform less influence from thecorrect factors ofactual the other legsbut are conducted and prove the proposed methodology can not only trajectory and prove that the proposed methodology can not only correct the actual trajectory but also improve conducted and prove that the proposed methodology can not only correct the actual trajectory but also improve the leg locomotion. The platform is shown in Figure 7. the leg locomotion. platform is shown thealso legimprove locomotion. The platform isThe shown in Figure 7. in Figure 7.

Figure 7. Leg platform test diagram for locomotion of the walking robot.

Figure 7. Leg platform diagram locomotion the walkingrobot. robot. Figure 7. Leg platform testtest diagram forfor locomotion of of the walking

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θ 2 /rad

θ 3 /rad

The main body of the platform is supported with two legs in a mammal-like configuration, similar thebody mentioned hexapod in Figure with 1. Each leg in consists of three configuration, linkages which are Theto main of the platform is supported two legs a mammal-like similar connected by a hip joint and a knee 1. joint. mechanism attachedwhich to theare body via a base to the mentioned hexapod in Figure EachThe legleg consists of threeislinkages connected by joint. a hip Sinceand the aplatform is designed for locomotion tests, to it is equipped horizontal and rails joint knee joint. The leg mechanism is attached the body viawith a base joint. Since thevertical platform is with corresponding displacement sensors, awith five-axial forceand sensor, andrails LPC2368 based motor designed for locomotion tests, it is equipped horizontal vertical with corresponding controllers. During theaexperiment of a single left leg based is fixedmotor on the supporting platform. displacement sensors, five-axial force sensor,leg, andthe LPC2368 controllers. During the Thus, the forward direction Z left remains X direction is free, and itsthe displacement can be experiment of a single leg, the leg isstationary, fixed on thethe supporting platform. Thus, forward direction obtained the vertical sensor. right leg moves the desired Z remainsby stationary, the Xdisplacement direction is free, and The its displacement can beinobtained by thetrajectory vertical periodically. The jointThe angle obtained the proposed methodology and displacement sensor. rightcurves leg moves in theby desired trajectory correction periodically. The joint angle traditional method for thecorrection foot trajectory test, base-joint vibration test, and contact force curves obtained byare theused proposed methodology and traditional method are used for test, the as shown in the platformvibration test diagram in contact Figure 7. According to the in test above, test the foot trajectory test,leg base-joint test, and force test, as shown theresults leg platform performance of the7.methodology is the observed. The gait parameters used in the experiments are as diagram in Figure According to test results above, the performance of the methodology is follows: the leggait lift parameters height is h used = 0.05inm, the coefficient of is β = 0.6, cycle is s, the observed. The the experiments areland as follows: thethe legstep lift height is T h == 10.05 m, stepcoefficient size is S =of0.1 m, is the body height H = 0.2 thestep radius semi-round rigid foot is the land β= 0.6, the stepiscycle is Tm, = 1and s, the size of is Sthe = 0.1 m, the body height R ==0.02 m.and the radius of the semi-round rigid foot is R = 0.02 m. H 0.2 m, the forward/inverse forward/inverse kinematic According to the kinematic methodology methodology based based on on LS-SVM LS-SVM introduced above, the gait gaitof ofthe therobot robotisisknown, known,the the corresponding joint-angle curves be obtained. Finally, if the corresponding joint-angle curves cancan be obtained. Finally, theythey can can drive the robot to launch a series of experiments. a tripod gait an improved wave gait drive the robot to launch a series of experiments. Here, Here, a tripod gait and anand improved wave gait [37,38] [37,38] are adopted to generate straight joint-angle walking joint-angle curves. The curves corrected curves of each are adopted to generate straight walking curves. The corrected of each joint-angle joint-angle and the uncorrected curves in are a gait cycleinare shown in Figure 8. Since the rotation of the and the uncorrected curves in a gait cycle shown Figure 8. Since the rotation of the base-joint is base-joint isduring unchanged during straight walking, theofangle curve of is the base-joint is omitted. Asgait for unchanged straight walking, the angle curve the base-joint omitted. As for the tripod the tripod gait andwave the improved gait, ifofthe parameters of those are will the same, legs and the improved gait, if the wave parameters those gaits are the same,gaits the legs have athe similar will have a similar motion two kinds gait. Therefore, joint-angle curves in motion under the two kindsunder of gait.the Therefore, theofjoint-angle curvesthe shown in Figure 8 are shown the same Figure 8 aregaits the same for theabove. two gaits mentioned above. for the two mentioned

(a)

(b)

Figure 8. 8 Rotating Rotating angle anglecurves curvesofofdifferent different joints: Rotating angle of hip thejoint; hip joint; (b) Rotating Figure joints: (a) (a) Rotating angle of the (b) Rotating angle angle of the knee joint. of the knee joint.

4.1.1. Foot Trajectory 4.1.1. Foot Tip Tip Trajectory The trajectories trajectories of of the the foot foot tip relative to The tip relative to the the base-joint base-joint are are tested, tested, which which is is shown shown in in Figure Figure 9. 9. The supporting phase and the swing phase appear alternately and periodically. The gait cycle time The supporting phase and the swing phase appear alternately and periodically. The gait cycle time is 11 s, s, and and the the leg leg isis in in swing swing state state from from 00 to to0.4 0.4 s.s. For For the the first first half half of of that that time time period period the the leg leg is is in in is lift-off phase, and in the second half it is in flight phase. The maximum lift height of the leg is 0.05 m lift-off phase, and in the second half it is in flight phase. The maximum lift height of the leg is 0.05 m as designed. designed.From From0.40.4 s 1tos,1the s, robot the robot is in the supporting The displacement in the X as s to is in the supporting phase. phase. The displacement in the X direction direction is shown in Figure 9a. When the leg is in this state, the distance between the lowest of is shown in Figure 9a. When the leg is in this state, the distance between the lowest point of point the foot the foot tip and the base-joint is larger than 0.2 m slightly. The trajectory in the X direction is not tip and the base-joint is larger than 0.2 m slightly. The trajectory in the X direction is not a smootha smooth line, straight but has some fluctuation. That is the actual contact point is not the straight but line, has some fluctuation. That is because thebecause actual contact point is not the design point. design point. Thus, the robot will vibrate up and down during walking. However, with correction Thus, the robot will vibrate up and down during walking. However, with correction the trajectory the trajectory becomes a smooth line. In the Zthe direction, theis problem the in the X becomes a smooth straight line. straight In the Z direction, problem the sameisas insame the Xasdirection, direction, whichinisFigure shown9b. in Figure 9b. With correction, the smoothness of theistrajectory is notably which is shown With correction, the smoothness of the trajectory notably improved. improved. The synthesized trajectory curves in Figure 9c,d are obtained by compounding the

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Z/m

The synthesized trajectory curves in Figure 9c,d are obtained by compounding the displacement of displacement of the X direction and the Z direction. The upper part of the curve represents the swing thephase, X direction and the Z direction. The upper part of the curve represents the swing phase, and the and the lower part indicates the supporting phase. The unsmooth contact will make an lower parton indicates the supporting phase. The unsmooth contact will make an impact on the ground impact the ground and increase the contact force. Moreover, the imbalance body posture will and increase the contact force. Moreover, the imbalance body posture will cause slipping other legs. cause slipping of other legs. With correction, the contact trajectory is a smooth straightofline. The With correction, the contact trajectory is a smooth straight line. The centroid of the robotic body will centroid of the robotic body will be kept constant during movement, so the robot will be more stable be kept constant during movement, so the robot will be more stable during walking. during walking.

(a)

(b)

(c)

(d)

Figure Trajectories of of the thethe base-joint: (a) Displacement in X-direction; (b) Figure 9. 9.Trajectories the foot foot tip tiprelative relativetoto base-joint: (a) Displacement in X-direction; Displacement in Z-direction; (c) Synthesize Trajectory without correction; (d) Synthesize Trajectory (b) Displacement in Z-direction; (c) Synthesize Trajectory without correction; (d) Synthesize Trajectory with correction. with correction.

4.1.2. Base Joint Vibration 4.1.2. Base Joint Vibration The base-joint is connected with the body rigidly, and the trajectory of the centroid of the robot The base-joint is connected body rigidly, and the of trajectory of the will centroid the robot is is determined by the trajectory with of thethe base-joint. The vibration the base-joint affectof the motion determined by the trajectory of the base-joint. The vibration of the base-joint will affect the motion of the robotic centroid directly. If the vibration is large, the energy consumption of the robot will be of increased the roboticand centroid directly.will If the is large, the energy consumption of the above, robot will the stability bevibration greatly reduced. According to the experiments the be increased and the stability will be greatly reduced. According to the experiments above, the corrected trajectory of the base-joint in the X direction and the un-correction trajectory are corrected tested, trajectory of the base-joint in the X direction and the un-correction trajectory are tested, which which are shown in Figure 10. The ideal base joint trajectory according to the method above shouldare shown Figureline, 10. The base joint trajectory according the method should be aand straight be a in straight sinceideal the robot should go forward along to the Z-axis withabove constant speed no displacement in the X direction. It can be concluded Figure 10speed that the without in line, since the robot should go forward along the Z-axis from with constant andtrajectory no displacement hasItapparent vibration in the Figure X direction andthe thetrajectory value of the deviation is nearly mm at thecorrection X direction. can be concluded from 10 that without correction has5 apparent the time points 0.9 s, 1.8 s, and 2.7 s. That means the robot will suffer periodic shocks while moving. vibration in the X direction and the value of the deviation is nearly 5 mm at the time points 0.9 s, 1.8 s, However, after correction, the base-joint almostwhile kept level, the vibration is small, and the and 2.7 s. That means the robot will suffertrajectory periodicisshocks moving. However, after correction, stability is improved. This result is not only related to the radius of the semi-round foot but also to the base-joint trajectory is almost kept level, the vibration is small, and the stability is improved. the angle between the end-linkage and the ground generated by the foot trajectory planning. In thethe This result is not only related to the radius of the semi-round foot but also to the angle between second half of the support phase, as the angle between the end-linkage and the ground increases, the end-linkage and the ground generated by the foot trajectory planning. In the second half of the support distance of the actual contact point and the ideal contact point becomes farther. This makes the phase, as the angle between the end-linkage and the ground increases, the distance of the actual contact base-joint more shock upwards obviously. The trajectory correction methodology proposed can be point and the ideal contact point becomes farther. This makes the base-joint more shock upwards used for effectively modifying the trajectory of the base-joint. obviously. The trajectory correction methodology proposed can be used for effectively modifying the trajectory of the base-joint.

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205

With correction With correction Without correction

h r/mm

h r/mm

Without correction

200

200

195

195

0

0.5

0

1

1.5 t/s

2

2.5

0.5 1 1.5 2 Figure 10. Base joint vibration of a single leg. t/s of a single leg. Figure 10. Base joint vibration

3

2.5

3

Figure 10. Base joint vibration of a single leg.

4.1.3. Contact Force

4.1.3. Contact Force

During the movement of the hexapod robot, the foot mechanics play a very important role [39]. On unstructured terrain, the robot can adapt to complex conditions by changing its posture, and the On unstructured theof can adapt to the complex conditions by its posture, gait generation strategy the robot according to the tip contact force. study ofrole foot During the terrain, movement ofrobot the hexapod robot, footfoot mechanics play a changing veryThe important [39].and theOn gait generation strategy of the robot according to the foot tip contact force. The study mechanics is beneficial to optimize the gait planning, enhance the stability, and reduce the energy unstructured terrain, the robot can adapt to complex conditions by changing its posture, andof thefoot consumption of the robot during locomotion [40]. In the single leg experiment shown in mechanics is beneficial to optimize the gait planning, enhance the stability, and reduce the energy gait generation strategy of the robot according to the foot tip contact force. The study of foot Figure 7,isof the ideal state is that in which theplanning, contact the foot and theand ground is kept inFigure a consumption the robot locomotion [40]. Inbetween the single leg experiment shown 7, mechanics beneficial toduring optimize the gait enhance the stability, reduce theinenergy critical state. However, if there is a deviation between the real trajectory and the desired trajectory, it robot during locomotion In the leg experiment theconsumption ideal state is of thatthe in which the contact between[40]. the foot and single the ground is kept in a shown critical in state. will lead to interaction with the ground, causing a bigger contact force or even impact, which is the Figure 7,ifthe ideal is that in which the between and the foot and the ground is kept in lead a However, there is astate deviation between thecontact real trajectory the desired trajectory, it will reason for vibrations. If the foot trajectory is designed in a better way, the force will be smaller and critical state. However, if there is a deviation between the real trajectory and the desired trajectory, it to interaction with the ground, causing a biggerthat contact forcestability or evenisimpact, which is the with no sudden changes, which demonstrates the contact enhanced. If there is noreason lead to interaction with the ground, causing in a bigger contact force or even impact, whichand is the forwill vibrations. If the foot trajectory is designed a better way, the force will be smaller force, it means no contact and it is unreasonable as well. The forces with correction and without with reason for vibrations. If the foot trajectory is designed in a better way, the force will be smaller no sudden changes, the there noand force, correction are allwhich tested demonstrates in experiments, that which arecontact shown stability in Figure is 11.enhanced. It illustratesIfthat theisfoot with no sudden changes, which demonstrates that the contact stability is enhanced. If there is no starts at 0.4 s and the force without correction becomes larger, which it means notouching contact the andground it is unreasonable as contact well. The forces with correction and without correction force, it means with no contact and it deviation. is unreasonable as well.0.9 The 1.8 forces with correction without the trajectory At in time points s, and 2.7 the foot forceand reaches a are allcorresponds tested in experiments, which are shown Figure 11. Its,illustrates thats, the starts touching correction are all tested in experiments, which are shown in Figure 11. It illustrates that the foot maximum of 13 N, but with correction, the trajectory deviation becomes smaller. Meanwhile, the the ground at 0.4 s and the contact force without correction becomes larger, which corresponds with starts touching the ground at 0.4 s and the contact force without correction becomes larger, which contact force is also very small, reduced by 70% compared with the uncorrected force, and less than the trajectory deviation. At time points 0.9 s, 1.8 s, and 2.7 s, the force reaches a maximum of 13 N, corresponds theconsidered trajectoryasdeviation. At time points 0.9 s, 1.8 s, and 2.7that s, the reaches a 4 N, whichwith can be a critical contact with the ground. That means the force modification butmaximum with correction, deviation smaller. Meanwhile, the contact force is also of the trajectory by the suggested methodology can deviation reduce thebecomes influencesmaller. on the Meanwhile, contact forcethe of 13 N,the buttrajectory with correction, the becomes trajectory very small, reduced by 70% compared the uncorrected and less 4 N, which can be effectively. great significance for further controlwith of force, the foot force andthan stability [41]. contact force isThis alsohas very small, reducedwith bythe 70% compared the uncorrected force, and less than

4.1.3. Contact During the Force movement of the hexapod robot, the foot mechanics play a very important role [39].

considered as can a critical contact with the ground. that the modification of the trajectory by 4 N, which be considered as a critical contactThat withmeans the ground. That means that the modification 15 theofsuggested methodology can reduce the influence the contact force effectively. This has great the trajectory by the suggested methodology canonreduce thecorrection influence on the contact force With significance thehas further of the and stability [41]. effectively.for This great control significance forfoot the force further control of Without the foot force and stability [41]. correction 1510

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4.2. Walking Tests of the Hexapod Robot After the single leg experiments described above, walking experiments of the hexapod robot were carried out. The overall structure of the hexapod robot system is shown in Figure 1. The axis of the hip-joint is parallel to the forward direction of the robot. The robot consists of its body and six legs. Because the six legs are distributed symmetrically along the two sides of the robotic body, the location of the geometrical center can be regarded as the location of gravity of the robot. As mentioned before, tripod gait and improved wave gait were both adopted to generate joint angle curves in straight walking. The parameters of the following tests are the same as the leg platform tests. The robot walking with a tripod gait is shown in Figure 12a. The blue section represents the swing phase, and the white section represents the supporting phase. In this gait, the legs are divided into two groups (LF, RM, and LH are the first group; RF, LM, and RH are the second group). Two groups alternately appear in supporting state and swing state. Because the landing coefficient is 0.6, the six legs will support at the same time in a moment of 0.2 s in 1 s. The initial state of the robot is shown in snapshot 1, where the six legs are all in the supporting state. In snapshot 2, the first group of legs (marked by red) is in the supporting state and second group is in the swing state. In snapshot 3, the six legs are all in the supporting state, prepared for gait alternation. In snapshot 4, the second group of legs (remarked by blue) is in the supporting state, and first group is in the swing state. At this point, one gait cycle is completed. The diagram of the gait and snapshots are respectively shown in Figure 12b when the robot is walking with improved wave gait. The mechanical legs located in the vertices of the blue polygon are in the supporting state. The rest of the mechanical legs are in the swing state (for instance, in snapshot 1, RH and LM are in the swing state, and RM, RF, LH, and LF are in the supporting state). 4.2.1. Direction Deviation Because of the terrain or slippage of certain legs, the robot will appear to deviate from its direction during movement. In this part, the experiments are carried out on smooth ground, as shown in Figure 12. Thus, the influence of the ground can be excluded. In order to verify the effect of the correction methodology on the deviation, the deviation in the Y and Z directions with and without correction are compared. The deviations of the two gaits with the two methods in the Y-direction are shown in Figure 13a. In the diagram, for the tripod gait without correction, the minimum deviation distance is 0.52 m, the maximum is 0.92 m, and the mean value after 10 tests is 0.72 m. Similarly, for improved wave gait without correction, the values of deviation are slightly smaller, because it has more legs supporting it than with the tripod gait. However, for the gaits with correction, the minimum deviation distance is 0.02 m, the maximum deviation distance is 0.21 m, and the mean value is 0.11 m. The slippingage was very serious and obvious for some feet while walking, when the joint trajectory without correction was used. It caused the robot to deviate from the desired direction and trajectory. After using the correction methodology, the deviation distances of the tripod gait and improved wave gait are reduced on average by 84% and 78%, respectively. The progress in the Z-direction is shown in Figure 13b. The progresses of both gaits with correction are all slightly larger than before. Since the foot slippage is reduced, larger friction is obtained and the locomotion of the robot is dramatically improved. The position deviations under the two methods are shown in Figure 13c.

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b)) ((b Figure 12. Diagram and snapshots of different gaits: (a) Tripod gait; (b) Improved wave gait. gait. Figure Figure12. 12.Diagram Diagramand andsnapshots snapshotsof ofdifferent different gaits: gaits: (a) (a)Tripod Tripod gait; gait; (b) (b) Improved Improved wave

5.3 5.3 5.1 5.1 4.9 4.9 4.7 4.7 4.5 4.5 ((aa))

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4.85(c) 4.9 4.95 5 Z/m Figure 13. Walking experiment results of the (hexapod robot: (a) Deviation in Y-direction; (b) c) Progress in Z-direction; (c) Position deviations. I, II, III, IV, respectively represent tripod gait Figure 13.Walking Walking experiment results ofhexapod the hexapod robot: (a) Deviation in Y-direction; (b) Figure 13. results of the robot: (a) Deviation Y-direction; (b) Progress in without correction,experiment improved wave gait without correction, tripod gaitinwith correction, and wave Progress in (c) Z-direction; (c) Position deviations. I, II, III, represent IV, respectively represent tripod gait Z-direction; Position deviations. I, II, III, IV, respectively tripod gait without correction, gait with correction. without correction, wave gaittripod without tripodand gaitwave withgait correction, and wave improved wave gaitimproved without correction, gaitcorrection, with correction, with correction. gait with correction. 4.2.2. Energy Consumption

4.2.2. Energy Consumption system energy consumption of the robot under two different methods during walking is 4.2.2.The Energy Consumption Thebased system consumption of the robot consumption under two different during studied onenergy the analysis above. The energy of eachmethods leg in the robotwalking system is The system energy consumption of the robot under two different methods during walking is studied based on the analysis above. The leg energy consumption evaluated by measuring the current of each during walking. of each leg in the robot system is studied based on the analysis above. The energy consumption of each leg in the robot system is evaluated by measuring the current of each walking. The energetic cost curves of a single legleg in during improved wave gait with two methods are shown in evaluated by measuring the current of each leg during walking. The cost line curves of a single in improved wave gait withThe twosolid methods areFigure shown14a in Figure 14.energetic The dashed represents theleg optimized theoretical value. line in The energetic cost curves of a single leg in improved wave gait with two methods are shown in Figure 14. the Theenergy dashedconsumption line represents optimized theoretical value. The solidalgorithm line in Figure 14a represents of athe single leg under control of the kinematic without Figure 14. The dashed line represents the optimized theoretical value. The solid line in Figure 14a represents energy a single leg under control of the kinematic algorithm without taking foot the shape into consumption consideration.ofThe solid line in Figure 14b represents the energy consumption represents the energy consumption of a single leg under control of the kinematic algorithm without taking foot into consideration. The solidmethodology. line in Figure 14b represents energy by by using a shape semi-round rigid foot correction As can be seenthe from the consumption graph, without taking foot shape into consideration. The solid line in Figure 14b represents the energy consumption using semi-round rigidalgorithm, foot correction methodology. Asofcan seen power from the graph,15 without the use the usea of the modified the maximum value thebe actual exceeds W, while the by using a semi-round rigid foot correction methodology. As can be seen from the graph, without of the modified algorithm, the maximum of the with actualthe power exceeds methodology, 15 W, while the the theoretical theoretical maximum value is 12 W. value However, correction actual the use of the modified algorithm, the maximum value of the actual power exceeds 15 W, while the value isis12 withclose the correction methodology, the actual maximum is maximum power 13W. W,However, which is very to the theoretical value. The trend of the curvepower is more theoretical maximum value is 12 W. However, with the correction methodology, the actual 13 W, which is very close to the theoretical The trend of theofcurve is more consistent with the consistent with the theoretical value. That isvalue. because the slippage the legs at some times will affect maximum power is 13 W, which is very close to the theoretical value. The trend of the curve is more theoretical That is because the the legs some times theredistribution trend of energy the trend ofvalue. energy consumption. Onslippage the otherofhand, theatslippage will will alsoaffect lead to of consistent with the theoretical value. That is because the slippage of the legs at some times will affect consumption. Onand the moment other hand, theleg, slippage will also lead to the redistribution of the contact force the contact force of the and eventually affect energy consumption results ofand the the trend of energy consumption. On the other hand, the slippage will also lead to redistribution of moment of the leg, and eventually energy consumption results the whole system,has so taking whole system, so taking the foot affect shapethe into consideration during the of trajectory planning great the contact force and moment of the leg, and eventually affect the energy consumption results of the the foot shape consideration during the trajectory planning has great significance for stability. significance forinto stability. whole system, so taking the foot shape into consideration during the trajectory planning has great significance for stability. 15 Simulation Experiment Simulation Experiment

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Figure 14. Energeticcost cost curves of a single leg: (a) Without considering the shape of foot; (b) Figure 14. curves of aof single leg: (a) Without considering the shape of foot; Figure 14. Energetic Energetic cost curves a single leg: (a) Without considering the shape(b)ofCorrection foot; (b) Correction algorithm with a semi-round rigid foot. algorithm with a semi-round rigid foot. rigid foot. Correction algorithm with a semi-round

A comparison of system energy consumption between the correction methodology (dashed A comparison comparison ofofsystem systemenergy energy consumption between correction methodology (dashed consumption between thethe correction methodology (dashed line) line) and un-correction (solid line) in one gait cycle time is shown in Figure 15. It is obvious that the line) and un-correction (solid in one cycle time shown Figure15. 15.ItItisisobvious obvious that that the and un-correction (solid line)line) in one gaitgait cycle time is is shown ininFigure maximum power is 31 W without correction and 27.5 W with correction. The average value of the maximum power power is 31 W without correction and 27.5 W with correction. correction. The The average average value value of the energy consumption goes from 25 W to 20 W. That means about 20% of the energy can be saved energy consumption consumptiongoes goesfrom from2525 W to 20 W. That means about 20% of the energy can beduring saved W to 20 W. That means about 20% of the energy can be saved during walking, so the correction methodology not only avoids the force impact caused by sliding of during walking, so the correction methodology notavoids only avoids the force impact caused by sliding of walking, so the correction methodology not only the force impact caused by sliding of the the supporting legs, but also reduces unnecessary energy consumption caused by interference of the supporting but reduces also reduces unnecessary consumption by interference of supporting legs,legs, but also unnecessary energyenergy consumption caused caused by interference of each leg each leg with the semi-round foot. each leg the semi-round foot. with the with semi-round foot. 35 35 30 30

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Figure 15. System energy consumption of the robot. Figure 15. System energy consumption of the robot. Figure 15. System energy consumption of the robot.

5. Conclusions 5. Conclusions 5. Conclusions In this paper, the body trajectory misplacement problem caused by the rolling of a robot foot In this paper, the body trajectory misplacement problem caused by the rolling of a robot foot paper, thestructure body trajectory misplacement problem caused the rollinggeneration, of a robot foot with In a this semi-round is analyzed. The influences of jointbytrajectory footwith tip with a semi-round structure is analyzed. The influences of joint trajectory generation, foot tip a semi-round is analyzed. The influences of joint trajectory generation, foot caused tip trajectory, trajectory, basestructure joint vibration, contact force, body trajectory and energy consumption by the trajectory, base joint vibration, contact force, body trajectory and energy consumption caused by the base joint vibration, contact force,robot body trajectory andare energy consumption caused the semi-round semi-round structure during movement studied. Through the by analysis of the semi-round structure during robot movement are studied. Through the analysis of the structure during robot movement studied. themethodology analysis of the kinematics forward/inverse kinematics of theare robot leg, aThrough correction forforward/inverse a robot with a semi-round forward/inverse kinematics of the robot leg, a correction methodology for a robot with a semi-round of thefoot robot leg, aon correction for a robot with semi-round rigid foot based on has LS-SVM is rigid based LS-SVMmethodology is proposed. According to thea results, the suggested method higher rigid foot based on LS-SVM is proposed. According to the results, the suggested method has higher proposed. According to the results, the suggested than the approximation accuracy than the approximation method used method before, has andhigher has a accuracy faster calculation speed of the accuracy than the approximation method used before, and has a faster calculation speed of the methodkinematic used before, and has faster calculation speed the inverseanalysis kinematic solution than the inverse solution thana the iterative method. Theoflocomotion of the robot obtained inverse kinematic solution than the iterative method. The locomotion analysis of the robot obtained by comparing correction methodology under a semi-round foot based on LS-SVM with a traditional by comparing correction methodology under a semi-round foot based on LS-SVM with a traditional generation indicates that the correction methodology can effectively improve the foot tip trajectory, generation indicates that the correction methodology can effectively improve the foot tip trajectory,

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iterative method. The locomotion analysis of the robot obtained by comparing correction methodology under a semi-round foot based on LS-SVM with a traditional generation indicates that the correction methodology can effectively improve the foot tip trajectory, so as to reduce the influence of the semi-round foot on the vibration of the base-joint and the impact of contact force. The validity of the correction methodology for the semi-round rigid foot is verified by conducting experiments on a hexapod walking robot. Using tripod gait and improved wave gait as examples, the results indicate that the correction methodology can reduce slipping and trajectory deviation of the robot. The current detection also demonstrates that the proposed methodology can reduce the system energy consumption to some extent. Therefore, the correction methodology has great significance for the locomotion performance and the stability of walking robots. The theoretical contributions and novelty of this paper can be summarized as follows. An approach for the misplacement problem caused by rolling of a foot with a semi-round structure is proposed. A solution method based on LS-SVM with higher accuracy and faster calculation speed is also established. Influences of joint trajectory generation, foot tip trajectory, base joint vibration, and contact force caused by the semi-round structure are studied. Finally, a series of experiments are performed on a hexapod robot, and the locomotion of the robot is studied using the proposed method. Slipping, trajectory deviations, and system energy consumption of the robot are all reduced, which validates our theory. In addition, the suggested methodology can be used as an underlying control of the robot architecture and the theory is concise and straightforward in the software, which indicates that force control, gait generation, trajectory planning, and other advanced theories can be applied simultaneously. This means that further improvement can be performed without too much effort. Acknowledgments: This study is supported by The General Program of National Natural Science Foundation of China (No. 51278058), the National Natural Science Foundation of China (No. 51605039), the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2016JQ6066) and the China Postdoctoral Science Foundation (No. 2016M592728). Author Contributions: Yaguang Zhu and Bo jin designed the algorithm. Yaguang Zhu, Yongsheng Wu, and Tong Guo designed and carried out the experiments. Yaguang Zhu analyzed the experimental data and wrote the paper. Xiangmo Zhao gave many meaningful suggestions about the structure of the paper. Conflicts of Interest: The authors declare no conflict of interest.

References 1.

2. 3. 4. 5. 6. 7. 8.

Manoonpong, P.; Parlitz, U.; Wörgötter, F. Neural control and adaptive neural forward models for insect-like, energy-efficient, and adaptable locomotion of walking machines. Front. Neural Circuits 2013, 7, 709–724. [CrossRef] [PubMed] Tedeschi, F.; Carbone, G. Hexapod Walking Robot Locomotion. In Motion and Operation Planning of Robotic Systems; Springer International Publishing: New York, NY, USA, 2015; pp. 439–468. Pérez, L.; Rodríguez, Í.; Rodríguez, N.; Usamentiaga, N.; García, D.F. Robot Guidance Using Machine Vision Techniques in Industrial Environments: A Comparative Review. Sensors 2016, 16, 335. [CrossRef] [PubMed] Wettergreen, D.; Thorpe, C.; Whittaker, R. Exploring Mount Erebus by walking robot. Robot. Auton. Syst. 1993, 11, 171–185. [CrossRef] Bares, J.E.; Whittaker, W.L. Configuration of autonomous walkers for extreme terrain. Int. J. Robot. Res. 1993, 12, 535–559. [CrossRef] De Santos, P.G.; Armada, M.A.; Jimenez, M.A. Ship building with ROWER. IEEE Robot. Autom. Mag. 2000, 7, 35–43. [CrossRef] Mosher, R.S. Test and evaluation of a versatile walking truck. In Proceedings of the O-Road Mobility Research Symposium, Washington, DC, USA, 1968; pp. 359–379. Berns, K.; Albiez, J.; Kepplin, V.; Hillenbrand, C. Airbug—insect-like machine actuated by fluidic muscle. In Proceedings of the 4th Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines, Karlsruhe, Germany, 2001; pp. 237–244.

Sensors 2016, 16, 1392

9. 10. 11.

12. 13. 14.

15.

16. 17.

18. 19.

20. 21.

22.

23. 24. 25. 26. 27. 28. 29. 30. 31.

20 of 21

Okhotsimski, D.; Platonov, A. Control algorithm of the walking climbing over obstacles. In Proceedings of the 3rd International Joint Conference on Artificial Intelligence, Stanford, CA, USA, 20 August 1973; pp. 317–323. Preumont, A. An investigation of the kinematic control of a six-legged walking robot. Mechatronics 1994, 4, 821–829. [CrossRef] Berns, K.; Kepplin, V.; Dillmann, R. Terrain and obstacles detection for walkingmachines using a stereo-camera-head. In Proceedings of the IEEE 24th Annual Conference on Indeustrial Electronics Society, Aachen, Germany, 4–6 September 1998; pp. 1170–1175. Pfeiffer, F.; Eltze, J.; Weidemann, H.J. The TUM-walking machine. Intell. Autom. Soft Comput. 1995, 1, 307–323. [CrossRef] Frik, M.; Guddat, M.; Karata¸s, M.; Losch, D. Terrain adaptive control of the walking machine Tarry II. In Proceedings of the European Mechanics Colloquiu, Munich, Germany, 1998; pp. 108–115. Hirose, S.; Umetani, Y. Some considerations on a feasible walking mechanism as a terrain vehicle. In Proceedings of the 3rd CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators, Udine, Italy, 12–15 September 1978; pp. 357–375. Hirose, S.; Masui, T.; Kikuchi, H. TITAN III: A quadruped walking vehicle—Its structure and basic characteristic. In Proceedings of the 2nd International Symposium on Robotic Research, Warren, MI, USA, 25–28 March 1985; pp. 325–331. Hirose, S.; Yoneda, K.; Arai, K. Design of prismatic quadruped walking vehicle TITAN VI. In Proceedings of the 5th International Conference on Advanced Robotics, Pisa, Italy, 19–22 June 1991; pp. 723–728. Nonami, K.; Huang, Q.J. Humanitarian mine detection six-legged walking robot comet-II with two manipulators. In Proceedings of the 4th Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines, Pisa, Italy, 19–22 June 1991; pp. 989–996. De Santos, P.G.; Galvez, J.A.; Estremera, J. SILO4—A true walking robot for the comparative study of walking machine techniques. IEEE Robot. Autom. Mag. 2003, 10, 23–32. [CrossRef] Fielding, M.R.; Damaren, C.J.; Dunlop, R. HAMLET: Force/position control hexapod walker—Design and systems. In Proceedings of the IEEE Conference on Control Applications, Mexico City, Mexico, 5–7 September 2001; pp. 984–989. De Santos, P.G.; Cobano, J.A.; Garcia, E.; Estremera, J.; Armada, M.A. A six-legged robot-based system for humanitarian demining missions. Mechatronics 2007, 17, 417–430. [CrossRef] Gómez-Bravo, F.; Aznar, M.J.; Carbone, G. Planning an Optimal Step Motion for a Hybrid Wheeled-Legged Hexapod. In New Advances in Mechanisms, Transmissions and Applications; Springer Netherlands: Dordrecht, The Netherlands, 2014; pp. 207–214. Marzuki, N.A.; Zarain, N. AEDESTROYER a proposed robotic solution to control dengue in Malaysia. In Proceedings of 2014 IEEE International Symposium on Robotics and Manufacturing Automation, Kuala Lumpur, Malaysia, 15–17 December 2014; pp. 242–247. Chen, W.H.; Ren, G.J.; Wang, J.H.; Dong, L. An adaptive locomotion controller for a hexapod robot: CPG, kinematics and force feedback. Sci. China Inf. Sci. 2014, 57, 1–18. [CrossRef] Ren, G.; Chen, W.; Dasgupta, S.; Kolodziejski, C.; Wörgötter, F. Multiple chaotic central pattern generators with learning for legged locomotion and malfunction compensation. Inf. Sci. 2015, 294, 666–682. [CrossRef] Wei, L.; Tian, K.C. An algorithm of machine vision-based yaw-angle estimation. Transducer Microsyst. Technol. 2011, 30, 131–133. Kwon, O.; Park, S. Adaptive dynamic locomotion of quadrupedal robots with semicircular feet on uneven terrain. Int. J. Control Autom. Syst. 2014, 12, 147–155. [CrossRef] Guardabrazo, T.A.; Jimenez, M.A.; de Santos, P.G. Analysing and solving body misplacement problems in walking robots with round rigid feet. Robot. Auton. Syst. 2006, 54, 256–264. [CrossRef] Zielinska, T.; Heng, J. Development of a walking machine: Mechanical design and control problems. Mechatronics 2002, 12, 737–754. [CrossRef] Irawan, A.; Yin, T.Y. Optimizing Hexapod Robot Reconfiguration using Hexa-Quad Transformation. Int. J. Robot. Autom. 2014, 3, 139–150. [CrossRef] Tâche, F.; Pomerleau, F.; Caprari, G.; Siegwart, R.; Bosse, M.; Moser, R. Three-dimensional localization for the MagneBike inspection robot. J. F. Robot. 2011, 28, 180–203. [CrossRef] Chai, X.; Gao, F.; Pan, Y.; Qi, C.; Xu, Y. A Novel Identification Methodology for the Coordinate Relationship between a 3D Vision System and a Legged Robot. Sensors 2015, 15, 9519–9546. [CrossRef] [PubMed]

Sensors 2016, 16, 1392

32. 33.

34. 35.

36. 37. 38. 39.

40. 41.

21 of 21

Arena, P.; Fortuna, L.; Frasca, M. Learning anticipation via spiking networks: application to navigation control. IEEE Trans. Neural Netw. 2009, 20, 202–216. [CrossRef] [PubMed] De Santos, P.G.; Estremera, J.; Garcia, E. Optimizing leg distribution around the body in walking robots. In Proceedings of the 2005 IEEE International Conference on Robotics and Automation, Barcelona, Spain, 18–20 April 2005; pp. 3207–3212. Craig, J.J. Introduction to Robotics Mechanics and Control; Addison-Wesley Publication Company: Boston, MA, USA, 2010; pp. 388–423. Suykens, J.A.K.; Lukas, L.; Vandewalle, J. Sparse approximation using least squares support vector machines. In Proceeding of The 2000 IEEE International Symposium on Circuits and Systems, Geneva, Switzerland, 28–30 May 2000; pp. 757–760. Wang, H.; Hu, D. Comparison of SVM and LS-SVM for regression. In Proceedings of the 2005 IEEE International Conference on Neural Networks and Brain, Beijing, China, 13–15 October 2005; pp. 213–230. Erden, M.S.; Leblebicioglu, ˘ K. Analysis of wave gaits for energy efficiency. Auton. Robot. 2007, 23, 213–230. Lei, Y.F.; Lu, B.Y.; Sun, L.L.; Zhu, M.Q. Research on Bionic Hexapod Robot Based on ARM. Microcomput. Inf. 2008, 24, 217–219. Manoiu-Olaru, S.; Nitulescu, M. Hexapod robot. Virtual models for preliminary studies. In Proceeding of the 2011 15th International Conference on System Theory, Control, and Computing (ICSTCC), Sinaia, Romania, 14–16 October 2011; pp. 1–6. You, B.; Yu, G.X. Hexapod robot foot-end dynamics modeling simulation analysis. J. Eng. Heilongjiang Univ. 2013, 4, 82–96. Zhu, Y.G.; Jin, B.; Li, W. Leg compliance control of a hexapod robot based on improved adaptive control in different environments. J. Cent. South Univ. 2015, 22, 904–913. [CrossRef] © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).