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Research on Kinematics and Stability of a Bionic Wall-Climbing Hexapod Robot

Research on Kinematics and Stability of a Bionic Wall-Climbing Hexapod Robot Hindawi Applied Bionics and Biomechanics Volume 2019, Article ID 6146214, 17 pages https://doi.org/10.1155/2019/6146214 Research Article Research on Kinematics and Stability of a Bionic Wall-Climbing Hexapod Robot Shoulin Xu, Bin He , and Heming Hu Department of Control Science and Engineering, Tongji University, Shanghai 201804, China Correspondence should be addressed to Bin He; hebin@tongji.edu.cn Received 22 August 2018; Accepted 17 January 2019; Published 1 April 2019 Academic Editor: Craig P. McGowan Copyright © 2019 Shoulin Xu 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. Wall-climbing hexapod robot as a bionic robot has become a focus for extensive research, due to a wide range of practical applications. The most contribution of this paper is to analyze the kinematics and stability of a wall-climbing hexapod robot, so as to provide a theoretical basis for the stable walking and control of the robot on the wall. Firstly, the kinematics model of the wall-climbing hexapod robot is established based on the D-H method. Then, in order to keep the robot from tipping over, the stability of the wall-climbing hexapod robot is analyzed in depth, obtaining the critical condition which makes the robot to tip over. Afterward, the kinematics simulation of the wall-climbing hexapod robot is operated to analyze motion performances. Finally, the experiments are used to validate the proposed kinematics model and stability. The experimental results show that the kinematics model and stability condition of the wall-climbing hexapod robot are correct. 1. Introduction and the control of hexapod robots is simple than that of the eight-legged robots. Multilegged wall-climbing robot is a hybrid serial parallel With the development of robot technology, the application of mechanism. Many studies have studied the kinematics of a robot has not been limited to the industrial field and gradu- walking robot as a parallel mechanism. Howard et al. [14] ally moved to more fields, such as service [1], medical treat- proposed a kinematics model of a walking machine which ment [2], and cleaning [3]. Wall-climbing robot as a bionic was equivalent to a parallel mechanism and solved the robot which movement flexibility, a variety of irregular ter- rain adaptability, and can cross obstacles, which can be inverse kinematics of the robot. Shkolnik and Tedrake [15] studied the Jacobian matrices of both the body and the swing widely used in the fields of building, traffic and disaster relief legs of a quadruped robot. García-López et al. [16] presented to complete testing, flaw detection, cleaning, rescue and other operations [4, 5]. a new kinematics model of a single leg of a hexapod robot, and the trajectory generation is implemented. To evaluate At present, the main adhesive methods of wall-climbing the leg movement performance, a simulator was developed robot include magnetic adhesion [6], adhesion of adhesive in order to analyze the trajectory. Campa et al. [17] presented materials [7, 8], and vacuum adhesion [9]. The motion mech- a procedure for computing the forward and inverse kinemat- anism of robot mainly includes legged type [10, 11], crawler type [12], and frame type [13]. Legged-type robots offer ics models of the hexapod robot. Xin et al. [18] proposed an extended hierarchical kinematic modeling method to derive strong obstacle crossing abilities and wall adaptability. The the kinematic equations of the proposed hexapod robot. bearing capacity of crawler type is strong, but the turning is According to the kinematics model, the geometrical parame- difficult. Frame structure is simple, but the motion is not con- ters of the leg are optimized utilizing a comprehensive objec- tinuous. In legged robots, the stability of hexapod robots is stronger than that of the biped robots and quadruped robots, tive function that considers both dexterity and payload. 2 Applied Bionics and Biomechanics According to Soyguder and Alli [19], given the kinematics Joint 3 method of a hexapod robot was realized for walking, run- Robot body ning, and bounding gaits, the developed kinematic makes Joint 2 both the system control easy and the system performance is Joint 1 improved by decreasing the run time. Multilegged wall-climbing robot has a strong adaptability to the complex environment, but because of its foot end inde- pendent of each other, only to choose the appropriate land- ing point to ensure that the robot does not tipping over. Hip Therefore, it is very important to study the stability of the robot, which is also an important reference for the design of the robot. With the deep research on the stability technol- Calf ogy of the foot robot, the stability theory is maturing and the igh stability of the robot can be judged by various stability methods. Liu and Jiang [20] focused on the discussion of Suction cup the stability of the bionic hexapod robot in the horizontal plane and on the slope by using the center of gravity of the Figure 1: The CAD model of the wall-climbing hexapod robot. projection method. Long et al. [21] proposed an improved force-angle stability margin measure method for a radial symmetrical hexapod robot under dynamic conditions. Roy and Pratihar [22] resented stability analysis based on normal- are parallel to the body. The hip and body are also connected via a revolute joint but is perpendicular to the body. The con- ized energy stability margin that is performed for turning motion of the robot with four duty factors for different angu- nection between the calf and the thigh is defined as joint 3, the connection between the thigh and the hip is defined as lar strokes. Gui et al. [23] proposed a criterion called force- angle stability which is used to measure the performance of joint 2, and the connection between the hip and the body is the robot which runs in complex environment with different defined as joint 1. When the wall-climbing hexapod robot adheres on the gaits. Zhang et al. [24] presented the static stability of two kinds of tripod gait; when the step length of the robot meets wall, it can be seen as a parallel mechanism and each leg can be equivalent to a three-link serial mechanism. Next, certain conditions, the state of robot motion is statically stable. Sandoval-Castro et al. [25] proposed the normalized we first establish a forward kinematics model of one leg, energy stability margin (SNE) criterion to analyze the and the geometry of one leg is shown in Figure 2. Here, the D-H method is used to establish the forward robot stability. In this paper, the most contribution is to establish the kinematics of the wall-climbing hexapod robot. Supposing that l is the length of the joint i, d is the offset between the kinematics model and stability condition of a wall-climbing i i hexapod robot to provide a theoretical basis for the stable joint i − 1 and the joint i (moving joint), α is the twist angle walking and control of the wall-climbing hexapod robot. of the joint i, and θ is the angle between the joints (revolving The remainder of the paper is organized as follows. In Section joints). The specific D-H parameters for one leg are shown in 2, establishing the kinematics model of the wall-climbing Table 1. hexapod robot based on the geometric model and the D-H From the D-H method and Table 1 yields method of the robot. In Section 3, the static stability condi- tion of the wall-climbing hexapod robot is established, so as c c −c s s l c c + c l c + l to obtain the critical condition of the robot tipping over. In 1 23 1 23 1 3 1 23 1 2 2 1 Section 4, the kinematics simulation of the wall-climbing s c −s s −c l s c + s l c + l 1 23 1 23 1 3 1 23 1 2 2 1 hexapod robot is carried out. In Section 5, the proposed kine- T = , matics model and stability condition are correct which are s c 0 l s + l s 23 23 3 23 2 2 validated by the experiments. 00 0 1 2. Kinematics Model 2.1. Kinematics Model of One Leg. First, a wall-climbing hexa- where pod robot is designed, and the CAD model [26] is shown as Figure 1. The shape of the body frame of the wall-climbing hexapod robot is regular octagon, which is installed with c = cos θ , the communication, control, energy, and other circuit sys- i i tems. The wall-climbing hexapod robot is designed with six s = sin θ , i i legs, and each leg consists of four components: suction cup, s = sin θ + θ , calf, thigh, and hip. The suction cup and the calf are con- ij i j nected by a spherical joint, while the calf and the thigh and c = cos θ + θ ij i j the thigh and the hip are connected by revolute joints, which Applied Bionics and Biomechanics 3 that is c n + s n c o + s o c a + s a c p + s p 1 x 1 y 1 x 1 y 1 x 1 y 1 x 1 y −s n + c n −s o + c o −s a + c a −s p + c p 1 x 1 y 1 x 1 y 1 x 1 y 1 x 1 y x y 2 3 n o a p z z z z z 0 00 0 1 c −s 0 l c + l c + l 23 23 3 23 2 2 1 00 −10 s c 0 l s + l s 23 23 2 23 2 2 00 0 1 Figure 2: The D-H coordinate system of one leg. Taking the second row and fourth column of the two Table 1: The D-H parameters of one leg. matrices in equation (7), obtains l d α θ i i i i 00 0 −s p + c p =0 8 1 x 1 y ° 0 0 l θ 1 2 l θ 2 3 By equation (8), yields 00 0 θ = arccos 9 So the coordinates of the end of the one leg are given as 2 2 p + p x y l c c + c l c + l x 3 1 23 1 2 2 1 0 1 −1 If the two sides of equation (5) are left multiply T 0 −1 p = 3 l s c + s l c + l T , obtains y 3 1 23 1 2 2 1 0 1 l s + l s z 3 23 2 2 1 −10 −1 0 1 −10 −1 0 1 2 3 2 3 T T ∗ T = T T ∗ T ∗ T ∗ T ∗ T = T ∗ T, 2 1 4 2 1 1 2 3 4 3 4 In the following, on the basis of the kinematics model of one leg, the inverse kinematics of the one leg is analyzed. Supposing that is n o a p f n f o f a c c p + s p + s p − l c x x x x 1 1 1 2 1 x 1 y 2 z 1 2 0 0 0 0 n o a p y y y y 0 0 T = , 4 f n f o f a −s c p + s p + c p + l s 2 2 2 2 1 x 1 y 2 z 1 2 0 0 0 n o a p z z z z f n f o f a s p − c p 00 0 1 3 3 3 1 x 1 y 0 0 00 0 1 c −s 0 l c + l 3 3 3 3 2 0 0 1 2 3 T = T ∗ T ∗ T ∗ T 4 1 2 3 4 s c 0 l s 3 3 3 3 = , 0 −1 00 1 0 Left multiply T on both sides of equation (4) yields 00 0 1 0 −1 0 0 −1 0 1 2 3 1 2 3 T ∗ T = T ∗ T ∗ T ∗ T ∗ T = T ∗ T ∗ T, 6 11 1 4 1 1 2 3 4 2 3 4 4 Applied Bionics and Biomechanics where f n = c c n + s n + s n , 2 1 x 1 y 2 z z f n = −s c n + s n + c n , 12 2 2 1 x 1 y 2 z f n = s n − c n 3 1 x 1 y Taking the first row and fourth column, the second row and fourth column of the two matrices in equation (11) obtains c c p + s p + s p − l c = l c + l , 2 1 x 1 y 2 z 1 2 3 3 2 0 0 0 −s c p + s p + c p + l s = l s , Figure 3: The body coordinate system and the leg coordinate 2 1 x 1 y 2 z 1 2 3 3 0 0 0 system. Solving equation (13) yields where X is the body coordinate system, r is the transforma- tion matrix of the position coordinate system, R is the trans- L ∗ p + l ∗ sin θ ∗ N z 3 3 0 formation matrix of the direction coordinate system, and X θ = − arctan , l ∗ p ∗ sin θ − L ∗ N is the single leg base system. 3 z 3 From Figure 3, yields 2 2 2 l + l − p − M 2 3 z θ = arccos∙ − π 2 ∗ l ∗ l 2 3 −350 r = 0 , 18 Thus, obtains x0 θ = arccos , 2 2 p + p x0 y0 cos θ −sin θ 0 L ∗ p + l ∗ sin θ ∗ N z0 3 3 θ = − arctan , 2 R = sin θ cos θ 0 l ∗ p ∗ sin θ − L ∗ N 3 z0 3 2 2 00 1 l + l − p − M 2 3 z0 θ = arccos − π, 2 ∗ l ∗ l 0 −10 2 3 = 10 0 where 00 1 L = l + l ∗ cos −θ , 2 3 3 M = p − l ∗ cos θ + p − l ∗ sin θ , x0 1 1 y0 1 1 Substituting equations (18) and (19) into equation (17) obtains N = M X = r + R ∗ X −350 0 −10 l c c + c l c + l 3 1 23 1 2 2 1 2.2. Kinematics Model of the Body. On the basis of estab- lishing the forward kinematics of one leg, next, by coordi- = + ∗ 0 10 0 l s c + s l c + l 3 1 23 1 2 2 1 nate transformation, the position relation between the center of the body and the end of the leg is obtained. 0 00 1 l s + l s 3 23 2 2 The position relation between the body coordinate system −l s c − s l c + l − 350 3 1 23 1 2 2 1 and the leg coordinate system is shown in Figure 3. The coordinates of the end of the leg in the body coordi- l c c + c l c + l 3 1 23 1 2 2 1 nate system can be obtained by coordinate transformation; the coordinate transformation is given as l s c + l s 3 1 23 2 2 X = r + R ∗ X , 17 20 0 Applied Bionics and Biomechanics 5 E1 E1 Q2 Q1 F1 Figure 5: The tension and compression model of the suction cup. Figure 4: Force deformation diagram of the suction cup. Thus, yields p −l s c − s l c + l − 350 x 3 1 23 1 2 2 1 G6 = 21 p l c c + c l c + l 3 1 23 1 2 2 1 p l s + l s z 3 23 2 2 G2 3. Static Stability Analysis When wall-climbing hexapod robot walks on a vertical wall, it will be removed from the wall due to gravity. Through six G4 vacuum suction cups which are fixed at the foot, the suction cup adhered on the wall, depending on the pressure differ- ence between the inside and outside of the sucker. At this time, the traditional ZMP stability criterion [26–30] has no Figure 6: Force of state overturning loading. projection of the center of gravity on the contact surface; it is impossible to determine whether the robot is in stable state by means of projection and support. The wall-climbing hexa- force acting on the central axis of the suction cup). Only pod robot has three kinds of instability on the vertical wall, the suction cup is not turned over; the robot will be safe the vertical tipping instability, lateral tipping instability, and to walk on the wall. When the end of the suction cup is fall instability, respectively. The analysis of the three unstable subjected to the force acting on the central axis of the suc- states will help us to choose the appropriate adhesion force, tion cup, the load is reversed vertically. The upper half of while maintaining the robot safe and stable operation under the suction cup is pulled and the lower half is squeezed, the premise of as much as possible to reduce the robot’s as shown in Figure 4. When the critical state of the suction energy consumption. This paper assumes that the robot is cup is turned over, the suction cup appears on the E E 1 2 always walking at low speed and at constant speed, regardless line where the force is greatest. As the bulge continues to of acceleration. The triangular gait is the most common gait grow, the suction cup will leak first from the point M, of the robot. Under the triangular gait, the robot has the causing the entire suction cup to tip over, as shown in highest walking speed and efficiency. Thus, the stability of Figure 5. the robot is analyzed under the triangular gait. Under the tilting load F , the rod will deflect γ degrees 3.1. Analysis of the Adhesive Force at the Foot of the Robot. A downward. At this time, it will cause the suction cup corre- robot walks on the wall; each end of the suction cup is sub- sponding uplift and then change the effective adhesion area jected to a tilting load which is parallel to the wall (the S of the suction cup. Supposing that D is inside diameter 1 6 Applied Bionics and Biomechanics of the suction cup and D = 150 mm and D is outside diam- this time, γ =28 2 degrees. In a very small range, the deflec- 1 2 eter of the suction cup and D = 250 mm, the relation tion angle γ and the end force F are approximated as a linear between the effective adhesion area S and the deflection angle function; its relation is γ is obtained as follows: F =68 30γ 23 2 2 2 D D D 2 2 1 S = π − − The analysis shows that the force F is due to the gravity 2 2 2 acting on the body, as shown in Figure 6. The forces at the three suction cups are G , G , and G , 2 4 6 2 2 D D γ γ 2 1 which satisfy the following equation: + − + D sin D sin 1 1 2 2 2 2 = 49062 50 − 150 G cos 45 − θ + G cos 45 + θ = G cos θ, 6 4 2 γ γ G sin 45 − θ + G = G sin 45 + θ + G sin θ, ∗ 100 + 15625 − 75 + 150 sin sin 6 4 2 2 2 G l + Gl sin 45 − θ = G l sin 45 + 2θ 6 2 The suction cup test shows that when the maximum F is 1926 N, the rod reaches the maximum deflection angle, at Solving equation (24) obtains cos θ − sin θ sin θ cos θ − sin θ sin θ cos θ 1 2 2 1 1 2 2 G = G , cos θ − sin θ sin θ cos θ − sin θ cos θ + sin θ sin θ cos θ 2 2 3 2 2 2 G = G , sin θ cos θ − sin θ sin θ cos θ − sin θ sin θ cos θ 3 1 1 2 2 1 G = G , where θ =45+ θ, θ =45 − θ, θ =45+ 2θ, A = sin θ cos θ − sin θ sin θ cos θ − sin θ sin θ cos θ + sin θ cos θ 1 3 1 2 3 2 1 1 To sum up, with the change of robot pose, the effective The following is a detailed analysis of the status b and e, adhesion area of the suction cup will change. When the pres- as shown in Figure 9(a). The support polygons in the triangu- sure difference between the inside and outside of the suction lar gait are shown in Figures 9(b) and 9(c). cup is a fixed value, the adhesive force on each suction cup A longitudinal overturning instability analysis is per- can be changed. The relations between the deflection angle formed on the b status support with the axis 24 as the tilt- of the leg and the effective adhesion area S of the suction ing axis: cup are shown in Figure 7. 3.2. Walking Stability Analysis of Robot. Here, supposing that the wall-climbing hexapod robot which selects a general tri- F ∗ l + G ∗ cos α ∗ l > G ∗ h, 6 6 G z24 angular gait walks on the wall, as shown in Figure 8, it mainly includes the initial state represented by a and f and four >0 113 ∗ sin α − 0 352 ∗ cos α intermediate states of b, c, d, and e. Applied Bionics and Biomechanics 7 4 4 4 × 10 × 10 × 10 4.95 4.9 4.95 4.9 4.9 4.8 4.85 4.85 4.8 4.8 4.7 4.75 4.75 4.7 4.6 4.7 4.65 4.65 4.5 4.6 4.6 4.55 4.55 4.4 4.5 4.5 4.45 4.45 4.3 0 123456789 10 0 123456789 10 0 123456789 10 θ (degree) θ (degree) θ (degree) (a) (b) (c) Figure 7: (a) Relation between the deflection angle θ and the effective adhesion area S of the leg 2. (b) Relation between the deflection angle θ and the effective adhesion area S of the leg 4. (c) Relation between the deflection angle θ and the effective adhesion area S of the leg 6. 1 1 6 1 4 3 Supporting leg Supporting leg Swinging leg Supporting leg Swinging leg Swinging leg (a) (b) (c) 1 1 1 6 6 Supporting leg Supporting leg Supporting leg Swinging leg Swinging leg Swinging leg (d) (e) (f) Figure 8: A general triangular gait of robot walks on the wall. Next, a longitudinal overturning instability analysis is where G is the total gravity of a robot and a load. G is the z24 performed on the e status support with the axis 35 as gravity which causes the robot to rotate vertically around the the tilting axis: axis 24. G is the gravity which causes the robot to rotate z35 vertically around the axis 35. F is the adhesive force and F = F = F = F = F = F = F. l is the distance 1 2 3 4 5 6 G from the center of gravity to the tilting axis of a robot. l F ∗ l + G ∗ cos α ∗ l > G ∗ h, 1 1 G z35 is the distance from the center of the suction cup to the tilting axis. h is the distance between the centroid of robot >0 116 ∗ sin α − 0 294 ∗ cos α, and the wall. α is the inclination angle of the wall. S (mm ) S (mm ) S (mm ) 8 Applied Bionics and Biomechanics (a) (b) (c) Figure 9: (a) Force analysis of the robot. (b) b status support of the robot. (c) e status support of the robot. A lateral overturning instability analysis is performed on 0.5 the b status support with the axis 46 as the tilting axis: 0.4 0.3 F ∗ l + G ∗ cos α ∗ l > G ∗ h, 2 2 G h46 0.2 0.1 >0 010 ∗ sin α − 0 420 ∗ cos α, –0.1 where G is the gravity which causes the robot to rotate ver- h46 –0.2 tically around the axis 46. –0.3 When the e status support takes the axis 13 as the tilting axis, the lateral overturning instability analysis shows that the –0.4 component of gravity will not cause the robot to lateral over- –0.5 0 10 20 30 40 50 60 70 80 90 turning instability. α (degree) For b status support, to ensure that the robot does not slide on the wall, the balance conditions that need to be sat- Figure 10: Instability critical curve of triangular gait. isfied are as follows: μ ∗ F + F + F + G ∗ cos α > G ∗ sin α, 2 4 6 4. Model Simulations >0 417 ∗ sin α − 0 333 ∗ cos α 4.1. Foot Trajectory of the Swinging Leg. In the triangular gait, the coordinates of the starting point, the highest point, and the falling point of the foot of the robot are (97.24, 551.49, For e status support, to ensure that the robot does not -150) mm, (0, 599.95, -91.93) mm, and (-97.24, 551.49, slide on the wall, the balance conditions that need to be sat- -150) mm, respectively. The speed of the starting point is isfied are as follows: (0, 0, 0) mm/s, and the acceleration is (0, 0, 0) mm/s . The rate of the falling point is (0, 0, 0) mm/s, and the acceleration μ ∗ F + F + F + G ∗ cos α > G ∗ sin α, 2 1 3 5 is (0, 0, 0) mm/s . The swing time is 4 s. Then, the foot trajec- F tory of swinging leg is >0 417 ∗ sin α − 0 333 ∗ cos α, 5 4 3 xt = −1 1395t +11 3953t − 30 3875t +97 2400, where μ is the friction coefficient between the wall and the 5 4 3 yt = −0 7572t +9 0863t − 36 3450t +48 4600t + 551 4900, suction cup, and μ =0 8. 6 5 4 3 zt = −0 9073t +10 8881t − 43 5525t +58 0700t − 150 0000 Based on the above analysis, Figure 10 shows the instabil- ity critical curve of triangular gait. In Figure 10, the blue 32 curve is the b status support longitudinal tipping instability curve, crossing the axis x at 72.29 degrees. The red curve is By equation (32), obtaining that the foot trajectory of the e status support longitudinal tipping instability curve, swinging leg is shown in Figure 11. crossing the axis x at 68.4 degrees. The green curve is the b status support lateral overturning instability curve, crossing 4.2. Foot Trajectory of the Supporting Leg. On the triangular the axis x at 88.67 degrees. The blue-green curve is the critical gait of the supporting leg, the intersection between the sup- curve of the sliding instability of the robot, crossing the axis x porting leg and body as the origin, and establish the coordi- at 38.52 degrees. nate system. The coordinates of the starting point and the F/G Applied Bionics and Biomechanics 9 100 0 50 −25 0 −50 −50 −75 −100 −100 012 34 t (s) t (s) (a) (b) 100 600 −50 −100 t (s) t (s) (c) (d) 50 100 25 50 −25 −50 −100 −50 t (s) t (s) (e) (f) Figure 11: Continued. Speed (mm/s) Acceleration (mm/s ) Displacement (mm) 2 Displacement (mm) Speed (mm/s) Acceleration (mm/s ) y (mm) 10 Applied Bionics and Biomechanics −660 100 680 50 −700 −720 −50 −740 −100 t (s) t (s) (g) (h) −650 −680 −710 −740 −50 −100 −500 −100 t (s) (i) (j) Figure 11: (a) The position of foot in the x direction. (b) The speed of foot in the x direction. (c) The acceleration of foot in the x direction. (d) The position of foot in the y direction. (e) The speed of foot in the y direction. (f) The acceleration of foot in the y direction. (g) The position of foot in the z direction. (h) The speed of foot in the z direction. (i) The acceleration of foot in the z direction. (j) Foot trajectory of swinging leg. falling point of the foot of the robot are (-97.24, 551.49, -150) 4.3. Motion Simulation of the Wall-Climbing Hexapod Robot. mm and (97.24, 551.49, -150) mm, respectively. The speed of According to the foot trajectory planning and inverse kine- the starting point is (0, 0, 0) mm/s, and the acceleration is (0, matics, the relation between joint angle and time is obtained. 0, 0) mm/s . The rate of the falling point is (0, 0, 0) mm/s, and Then, using the spline curve to drive the joint motion, the the acceleration is (0, 0, 0) mm/s . The swing time is 4 s. driving function is shown in Figure 13. Then, the foot trajectory of supporting leg is Moreover, on the basis of the driving function, obtaining that the joint angle and torque change with time are shown in 5 4 3 xt =1 1395t − 11 3953t +30 3875t − 97 2400, Figures 15 and 16. Meanwhile, motion simulation of the cen- ter of gravity displacement in the triangular gait is shown in yt = 551 4900, Figure 17. zt = −150 0000 By Figure 15, it is obtained that the angular changes of each joint are continuous and gentle, without any angle mutation. In Figure 16, six legs are divided into 2 groups in the triangle gait, of which legs 1, 3, and 5 are one group By equation (33), obtaining that the foot trajectory of and legs 2, 4, and 6 are another group. The torque variation supporting leg is shown in Figure 12. x (mm) Displacement (mm) Acceleration (mm/s ) z (mm) Speed (mm/s) Applied Bionics and Biomechanics 11 100 100 50 75 −50 25 −100 01 2 3 4 t (s) t (s) (a) (b) 100 580 50 575 0 570 −50 565 −100 560 t (s) t (s) (c) (d) −1 −1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 t (s) t (s) (e) (f) Figure 12: Continued. Acceleration (mm/s ) Speed (mm/s) Displacement (mm) 2 Displacement (mm) Speed (mm/s) Acceleration (mm/s ) y (mm) 12 Applied Bionics and Biomechanics −551 1 −552 0.5 −553 −554 −0.5 −555 −1 01234 01234 t (s) t (s) (g) (h) −553 0.5 −554 −555 −0.5 −1 −100 t (s) (i) (j) Figure 12: (a) The position of foot in the x direction. (b) The speed of foot in the x direction. (c) The acceleration of foot in the x direction. (d) The position of foot in the y direction. (e) The speed of foot in the y direction. (f) The acceleration of foot in the y direction. (g) The position of foot in the z direction. (h) The speed of foot in the z direction. (i) The acceleration of foot in the z direction. (j) Foot trajectory of the supporting leg. of legs 1, 3, and 5 is the same, and the torque variation of legs Shenzhen Techservo Co. Ltd in China are selected. The suc- 2, 4, and 6 is the same. The torque changes continuously as tion cup which is produced by Japanese SMC Company is the robot runs, but torque changes occur at the time of each selected, of which the model is ZP2-250HTN. The motor transition. By Figure 17, it is obtained that the robot moves model of joint 1 is ST8N40P10V2E, the motor model of joint along the x direction, and the displacement of the robot is 2 is ST8N40P20V2E, and the motor model of joint 3 is 13.7 mm. The robot moves along the y direction, and the dis- ST8N40P10V4E [26]. The retarder models of joints 1, 2, placement of the robot is 1.3 mm. The robot runs with a body and 3 are S042L3-100 and RAD-60-2 [26]. The structural height of 150 mm, and offset is 0.87%; the robot can run parameters of the robot are set as follows: L = 550 mm, smoothly. The robot moves along the z direction, and the dis- L = 450 mm, L = 200 mm, and r = 350 mm. The total 2 3 placement of the robot is 503.7 mm; the average speed is mass of the robot is 72.0 kg. 30.9 mm/s. Next, the kinematics model and stability analysis of the wall-climbing hexapod robot are correct as shown in the experiments. The experimental results indicate that the kine- 5. Experiments matics models of one leg and body are correct; meanwhile, The kinematics and stability of the wall-climbing hexapod the experimental results are in good agreement with the kine- robot have been analyzed on the above. A prototype of the matics simulation results. In addition, the experimental wall-climbing hexapod robot is developed, as shown in results show that the stability conditions of the wall- Figure 18. The motors and the retarders produced by climbing hexapod robot are correct. Figure 19 shows that x (mm) Acceleration (mm/s ) Displacement (mm) z (mm) Speed (mm/s) Applied Bionics and Biomechanics 13 90 100 80 90 70 70 80 60 60 70 50 50 60 40 40 50 0 510 15 0 510 15 0 5 10 15 0 5 10 15 t (s) t (s) t (s) t (s) Joint 1 in leg 3 Joint 1 in leg 4 Joint 1 in leg Joint 1 in leg 1 70 70 80 60 60 50 50 60 40 40 50 30 30 40 20 20 0 510 15 0 5 10 15 0 5 10 15 0 5 10 15 t (s) t (s) t (s) t (s) Joint 1 in leg 5 Joint 1 in leg 6 Joint 2 in leg 1 Joint 2 in leg 2 70 70 70 60 60 60 50 50 50 50 40 40 40 40 30 30 30 20 20 20 20 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 t (s) t (s) t (s) t (s) Joint 2 in leg 5 Joint 2 in leg 4 Joint 2 in leg 6 Joint 2 in leg 3 −60 −60 −60 −60 −70 −70 −70 −70 −80 −80 −80 −80 −90 −90 −90 −90 −100 −100 −100 −100 −110 −110 −110 −110 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 t (s) t (s) t (s) t (s) Joint 3 in leg 3 Joint 3 in leg 4 Joint 3 in leg 1 Joint 3 in leg 2 −60 −60 −70 −70 −80 −80 −90 −90 −100 −100 −110 −110 0 5 10 15 0 5 10 15 t (s) t (s) Joint 3 in leg 6 Joint 3 in leg 5 Figure 13: The driving function of the driving joint. Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) 14 Applied Bionics and Biomechanics Figure 14: Motion simulation of the wall-climbing hexapod robot in triangular gait. 15 15 10 10 5 5 0 0 −5 −5 −5 −10 −10 −10 −15 −15 −15 −20 −20 −20 0 510 15 0 5 10 15 0 5 10 15 t (s) t (s) t (s) Joint angle of leg 1 Joint angle of leg 2 Joint angle of leg 3 Joint angle 1 Joint angle 1 Joint angle 1 Joint angle 2 Joint angle 2 Joint angle 2 Joint angle 3 Joint angle 3 Joint angle 3 20 20 15 15 10 10 5 5 0 0 0 −5 −5 −5 −10 −10 −10 −15 −15 −15 −20 −20 −20 0 5 10 15 0 5 10 15 0 5 10 15 t (s) t (s) t (s) Joint angle of leg 4 Joint angle of leg 5 Joint angle of leg 6 Joint angle 1 Joint angle 1 Joint angle 1 Joint angle 2 Joint angle 2 Joint angle 2 Joint angle 3 Joint angle 3 Joint angle 3 Figure 15: The joint angle change with time in triangular gait. the wall-climbing hexapod robot walks on the vertical wall 6. Conclusions and Future Work with horizontal direction by using the triangular gait. The wall-climbing hexapod robot can walk well on the vertical In this paper, the most contribution is to analyze the kine- wall, and the stability is well. matics and stability condition of the wall-climbing hexapod Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Applied Bionics and Biomechanics 15 0 0 −50 −100 −100 −100 −200 −200 −150 −300 −300 −200 0 510 15 0 510 0 510 15 t (s) t (s) t (s) Joint torque of leg 1 Joint torque of leg 2 Joint torque of leg 3 Joint 1 Joint 1 Joint 1 Joint 2 Joint 2 Joint 2 Joint 3 Joint 3 Joint 3 300 200 0 0 −0 −50 −100 −100 −100 −200 −200 −150 −300 −300 −200 0 510 15 0 510 15 0 510 15 t (s) t (s) t (s) Joint torque of leg 4 Joint torque of leg 5 Joint torque of leg 6 Joint 1 Joint 1 Joint 1 Joint 2 Joint 2 Joint 2 Joint 3 Joint 3 Joint 3 Figure 16: The joint torque change with time in triangular gait. 0.4 the kinematics model of the body is obtained by coordinate 0.3 transformation. Second, to make the robot walk steadily on 0.2 the wall, and no tipping occurs, the static stability of wall- climbing hexapod robot is analyzed, obtaining the critical 0.1 stability condition of the robot. Third, the kinematics simula- tion of the wall-climbing hexapod robot is operated to ana- −0.1 lyze motion performances, obtaining the foot trajectory of −0.2 the swinging and supporting legs. Moreover, the relation between joint angle and time and joint torque and time is −0.3 obtained by simulation. Finally, the experiments are used to −0.4 05 10 15 validate the proposed kinematics model and stability condi- t (s) tions. The experimental results show that the proposed kine- Direction of x axis matics model and stability conditions of the wall-climbing Direction of y axis hexapod robot are correct. Direction of z axis In the future, we can establish a dynamic model of the Figure 17: Motion simulation of the center of gravity displacement wall-climbing hexapod robot, obtaining the relation between change with time in triangular gait. the driving force and acceleration, velocity, and position of the wall-climbing hexapod robot, so as to provide an effective basis for motion control and force control of the wall- robot to provide a theoretical basis for the stable walking and control of the robot. First, the kinematics model of one leg climbing hexapod robot. In addition, the supporting legs is established, on the basis of the kinematics model, the which are deformed are influenced by the gravity of the body. inverse kinematics of the one leg is solved. Meanwhile, The accuracy of motion control of the wall-climbing hexapod Torque (N∙m) Torque (N∙m) Displacement (m) Torque (N∙m) Torque (N∙m) Torque (N∙m) Torque (N∙m) 16 Applied Bionics and Biomechanics (a) (b) Figure 18: (a) The prototype of the wall-climbing hexapod robot. (b) The structure of one leg of the wall-climbing hexapod robot [31]. (a) (b) (c) (d) (e) (f) Figure 19: The wall-climbing hexapod robot walks on vertical wall with horizontal direction by using the triangular gait [31]. robot is thus affected, and the walking trajectory deviation Conflicts of Interest occurs in the process. Thus, the stiffness of the wall- The authors declare that there is no conflict of interests climbing hexapod robot is investigated in the future. regarding the publication of this paper. Data Availability Acknowledgments The data used to support the findings of this study are avail- The work was supported by the National Key R&D Program able from the corresponding author upon request. of China (Grant No. 2018YFB1305300) and the National Applied Bionics and Biomechanics 17 in Proceedings 2007 IEEE International Conference on Robotics Natural Science Foundation of China (Grant Nos. 61825303, and Automation, pp. 4331–4336, Roma, Italy, 2007. U1713215, and 51605334). The authors would like to thank Jiangsu Greenhub Technology Co. Ltd, China, for its techni- [16] M. C. García-López, C. Gorrostieta-Hurtado, E. Vargas-Soto, J. M. Ramos-Arreguín, A. Sotomayor-Olmedo, and J. C. M. cal support. Morales, “Kinematic analysis for trajectory generation in one leg of a hexapod robot,” Procedia Technology, vol. 3, pp. 342– 350, 2012. References [17] R. Campa, J. Bernal, and I. Soto, “Kinematic modeling and control of the hexapod parallel robot,” in 2016 American Con- [1] I. H. 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Research on Kinematics and Stability of a Bionic Wall-Climbing Hexapod Robot

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Copyright © 2019 Shoulin Xu 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.
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Hindawi Applied Bionics and Biomechanics Volume 2019, Article ID 6146214, 17 pages https://doi.org/10.1155/2019/6146214 Research Article Research on Kinematics and Stability of a Bionic Wall-Climbing Hexapod Robot Shoulin Xu, Bin He , and Heming Hu Department of Control Science and Engineering, Tongji University, Shanghai 201804, China Correspondence should be addressed to Bin He; hebin@tongji.edu.cn Received 22 August 2018; Accepted 17 January 2019; Published 1 April 2019 Academic Editor: Craig P. McGowan Copyright © 2019 Shoulin Xu 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. Wall-climbing hexapod robot as a bionic robot has become a focus for extensive research, due to a wide range of practical applications. The most contribution of this paper is to analyze the kinematics and stability of a wall-climbing hexapod robot, so as to provide a theoretical basis for the stable walking and control of the robot on the wall. Firstly, the kinematics model of the wall-climbing hexapod robot is established based on the D-H method. Then, in order to keep the robot from tipping over, the stability of the wall-climbing hexapod robot is analyzed in depth, obtaining the critical condition which makes the robot to tip over. Afterward, the kinematics simulation of the wall-climbing hexapod robot is operated to analyze motion performances. Finally, the experiments are used to validate the proposed kinematics model and stability. The experimental results show that the kinematics model and stability condition of the wall-climbing hexapod robot are correct. 1. Introduction and the control of hexapod robots is simple than that of the eight-legged robots. Multilegged wall-climbing robot is a hybrid serial parallel With the development of robot technology, the application of mechanism. Many studies have studied the kinematics of a robot has not been limited to the industrial field and gradu- walking robot as a parallel mechanism. Howard et al. [14] ally moved to more fields, such as service [1], medical treat- proposed a kinematics model of a walking machine which ment [2], and cleaning [3]. Wall-climbing robot as a bionic was equivalent to a parallel mechanism and solved the robot which movement flexibility, a variety of irregular ter- rain adaptability, and can cross obstacles, which can be inverse kinematics of the robot. Shkolnik and Tedrake [15] studied the Jacobian matrices of both the body and the swing widely used in the fields of building, traffic and disaster relief legs of a quadruped robot. García-López et al. [16] presented to complete testing, flaw detection, cleaning, rescue and other operations [4, 5]. a new kinematics model of a single leg of a hexapod robot, and the trajectory generation is implemented. To evaluate At present, the main adhesive methods of wall-climbing the leg movement performance, a simulator was developed robot include magnetic adhesion [6], adhesion of adhesive in order to analyze the trajectory. Campa et al. [17] presented materials [7, 8], and vacuum adhesion [9]. The motion mech- a procedure for computing the forward and inverse kinemat- anism of robot mainly includes legged type [10, 11], crawler type [12], and frame type [13]. Legged-type robots offer ics models of the hexapod robot. Xin et al. [18] proposed an extended hierarchical kinematic modeling method to derive strong obstacle crossing abilities and wall adaptability. The the kinematic equations of the proposed hexapod robot. bearing capacity of crawler type is strong, but the turning is According to the kinematics model, the geometrical parame- difficult. Frame structure is simple, but the motion is not con- ters of the leg are optimized utilizing a comprehensive objec- tinuous. In legged robots, the stability of hexapod robots is stronger than that of the biped robots and quadruped robots, tive function that considers both dexterity and payload. 2 Applied Bionics and Biomechanics According to Soyguder and Alli [19], given the kinematics Joint 3 method of a hexapod robot was realized for walking, run- Robot body ning, and bounding gaits, the developed kinematic makes Joint 2 both the system control easy and the system performance is Joint 1 improved by decreasing the run time. Multilegged wall-climbing robot has a strong adaptability to the complex environment, but because of its foot end inde- pendent of each other, only to choose the appropriate land- ing point to ensure that the robot does not tipping over. Hip Therefore, it is very important to study the stability of the robot, which is also an important reference for the design of the robot. With the deep research on the stability technol- Calf ogy of the foot robot, the stability theory is maturing and the igh stability of the robot can be judged by various stability methods. Liu and Jiang [20] focused on the discussion of Suction cup the stability of the bionic hexapod robot in the horizontal plane and on the slope by using the center of gravity of the Figure 1: The CAD model of the wall-climbing hexapod robot. projection method. Long et al. [21] proposed an improved force-angle stability margin measure method for a radial symmetrical hexapod robot under dynamic conditions. Roy and Pratihar [22] resented stability analysis based on normal- are parallel to the body. The hip and body are also connected via a revolute joint but is perpendicular to the body. The con- ized energy stability margin that is performed for turning motion of the robot with four duty factors for different angu- nection between the calf and the thigh is defined as joint 3, the connection between the thigh and the hip is defined as lar strokes. Gui et al. [23] proposed a criterion called force- angle stability which is used to measure the performance of joint 2, and the connection between the hip and the body is the robot which runs in complex environment with different defined as joint 1. When the wall-climbing hexapod robot adheres on the gaits. Zhang et al. [24] presented the static stability of two kinds of tripod gait; when the step length of the robot meets wall, it can be seen as a parallel mechanism and each leg can be equivalent to a three-link serial mechanism. Next, certain conditions, the state of robot motion is statically stable. Sandoval-Castro et al. [25] proposed the normalized we first establish a forward kinematics model of one leg, energy stability margin (SNE) criterion to analyze the and the geometry of one leg is shown in Figure 2. Here, the D-H method is used to establish the forward robot stability. In this paper, the most contribution is to establish the kinematics of the wall-climbing hexapod robot. Supposing that l is the length of the joint i, d is the offset between the kinematics model and stability condition of a wall-climbing i i hexapod robot to provide a theoretical basis for the stable joint i − 1 and the joint i (moving joint), α is the twist angle walking and control of the wall-climbing hexapod robot. of the joint i, and θ is the angle between the joints (revolving The remainder of the paper is organized as follows. In Section joints). The specific D-H parameters for one leg are shown in 2, establishing the kinematics model of the wall-climbing Table 1. hexapod robot based on the geometric model and the D-H From the D-H method and Table 1 yields method of the robot. In Section 3, the static stability condi- tion of the wall-climbing hexapod robot is established, so as c c −c s s l c c + c l c + l to obtain the critical condition of the robot tipping over. In 1 23 1 23 1 3 1 23 1 2 2 1 Section 4, the kinematics simulation of the wall-climbing s c −s s −c l s c + s l c + l 1 23 1 23 1 3 1 23 1 2 2 1 hexapod robot is carried out. In Section 5, the proposed kine- T = , matics model and stability condition are correct which are s c 0 l s + l s 23 23 3 23 2 2 validated by the experiments. 00 0 1 2. Kinematics Model 2.1. Kinematics Model of One Leg. First, a wall-climbing hexa- where pod robot is designed, and the CAD model [26] is shown as Figure 1. The shape of the body frame of the wall-climbing hexapod robot is regular octagon, which is installed with c = cos θ , the communication, control, energy, and other circuit sys- i i tems. The wall-climbing hexapod robot is designed with six s = sin θ , i i legs, and each leg consists of four components: suction cup, s = sin θ + θ , calf, thigh, and hip. The suction cup and the calf are con- ij i j nected by a spherical joint, while the calf and the thigh and c = cos θ + θ ij i j the thigh and the hip are connected by revolute joints, which Applied Bionics and Biomechanics 3 that is c n + s n c o + s o c a + s a c p + s p 1 x 1 y 1 x 1 y 1 x 1 y 1 x 1 y −s n + c n −s o + c o −s a + c a −s p + c p 1 x 1 y 1 x 1 y 1 x 1 y 1 x 1 y x y 2 3 n o a p z z z z z 0 00 0 1 c −s 0 l c + l c + l 23 23 3 23 2 2 1 00 −10 s c 0 l s + l s 23 23 2 23 2 2 00 0 1 Figure 2: The D-H coordinate system of one leg. Taking the second row and fourth column of the two Table 1: The D-H parameters of one leg. matrices in equation (7), obtains l d α θ i i i i 00 0 −s p + c p =0 8 1 x 1 y ° 0 0 l θ 1 2 l θ 2 3 By equation (8), yields 00 0 θ = arccos 9 So the coordinates of the end of the one leg are given as 2 2 p + p x y l c c + c l c + l x 3 1 23 1 2 2 1 0 1 −1 If the two sides of equation (5) are left multiply T 0 −1 p = 3 l s c + s l c + l T , obtains y 3 1 23 1 2 2 1 0 1 l s + l s z 3 23 2 2 1 −10 −1 0 1 −10 −1 0 1 2 3 2 3 T T ∗ T = T T ∗ T ∗ T ∗ T ∗ T = T ∗ T, 2 1 4 2 1 1 2 3 4 3 4 In the following, on the basis of the kinematics model of one leg, the inverse kinematics of the one leg is analyzed. Supposing that is n o a p f n f o f a c c p + s p + s p − l c x x x x 1 1 1 2 1 x 1 y 2 z 1 2 0 0 0 0 n o a p y y y y 0 0 T = , 4 f n f o f a −s c p + s p + c p + l s 2 2 2 2 1 x 1 y 2 z 1 2 0 0 0 n o a p z z z z f n f o f a s p − c p 00 0 1 3 3 3 1 x 1 y 0 0 00 0 1 c −s 0 l c + l 3 3 3 3 2 0 0 1 2 3 T = T ∗ T ∗ T ∗ T 4 1 2 3 4 s c 0 l s 3 3 3 3 = , 0 −1 00 1 0 Left multiply T on both sides of equation (4) yields 00 0 1 0 −1 0 0 −1 0 1 2 3 1 2 3 T ∗ T = T ∗ T ∗ T ∗ T ∗ T = T ∗ T ∗ T, 6 11 1 4 1 1 2 3 4 2 3 4 4 Applied Bionics and Biomechanics where f n = c c n + s n + s n , 2 1 x 1 y 2 z z f n = −s c n + s n + c n , 12 2 2 1 x 1 y 2 z f n = s n − c n 3 1 x 1 y Taking the first row and fourth column, the second row and fourth column of the two matrices in equation (11) obtains c c p + s p + s p − l c = l c + l , 2 1 x 1 y 2 z 1 2 3 3 2 0 0 0 −s c p + s p + c p + l s = l s , Figure 3: The body coordinate system and the leg coordinate 2 1 x 1 y 2 z 1 2 3 3 0 0 0 system. Solving equation (13) yields where X is the body coordinate system, r is the transforma- tion matrix of the position coordinate system, R is the trans- L ∗ p + l ∗ sin θ ∗ N z 3 3 0 formation matrix of the direction coordinate system, and X θ = − arctan , l ∗ p ∗ sin θ − L ∗ N is the single leg base system. 3 z 3 From Figure 3, yields 2 2 2 l + l − p − M 2 3 z θ = arccos∙ − π 2 ∗ l ∗ l 2 3 −350 r = 0 , 18 Thus, obtains x0 θ = arccos , 2 2 p + p x0 y0 cos θ −sin θ 0 L ∗ p + l ∗ sin θ ∗ N z0 3 3 θ = − arctan , 2 R = sin θ cos θ 0 l ∗ p ∗ sin θ − L ∗ N 3 z0 3 2 2 00 1 l + l − p − M 2 3 z0 θ = arccos − π, 2 ∗ l ∗ l 0 −10 2 3 = 10 0 where 00 1 L = l + l ∗ cos −θ , 2 3 3 M = p − l ∗ cos θ + p − l ∗ sin θ , x0 1 1 y0 1 1 Substituting equations (18) and (19) into equation (17) obtains N = M X = r + R ∗ X −350 0 −10 l c c + c l c + l 3 1 23 1 2 2 1 2.2. Kinematics Model of the Body. On the basis of estab- lishing the forward kinematics of one leg, next, by coordi- = + ∗ 0 10 0 l s c + s l c + l 3 1 23 1 2 2 1 nate transformation, the position relation between the center of the body and the end of the leg is obtained. 0 00 1 l s + l s 3 23 2 2 The position relation between the body coordinate system −l s c − s l c + l − 350 3 1 23 1 2 2 1 and the leg coordinate system is shown in Figure 3. The coordinates of the end of the leg in the body coordi- l c c + c l c + l 3 1 23 1 2 2 1 nate system can be obtained by coordinate transformation; the coordinate transformation is given as l s c + l s 3 1 23 2 2 X = r + R ∗ X , 17 20 0 Applied Bionics and Biomechanics 5 E1 E1 Q2 Q1 F1 Figure 5: The tension and compression model of the suction cup. Figure 4: Force deformation diagram of the suction cup. Thus, yields p −l s c − s l c + l − 350 x 3 1 23 1 2 2 1 G6 = 21 p l c c + c l c + l 3 1 23 1 2 2 1 p l s + l s z 3 23 2 2 G2 3. Static Stability Analysis When wall-climbing hexapod robot walks on a vertical wall, it will be removed from the wall due to gravity. Through six G4 vacuum suction cups which are fixed at the foot, the suction cup adhered on the wall, depending on the pressure differ- ence between the inside and outside of the sucker. At this time, the traditional ZMP stability criterion [26–30] has no Figure 6: Force of state overturning loading. projection of the center of gravity on the contact surface; it is impossible to determine whether the robot is in stable state by means of projection and support. The wall-climbing hexa- force acting on the central axis of the suction cup). Only pod robot has three kinds of instability on the vertical wall, the suction cup is not turned over; the robot will be safe the vertical tipping instability, lateral tipping instability, and to walk on the wall. When the end of the suction cup is fall instability, respectively. The analysis of the three unstable subjected to the force acting on the central axis of the suc- states will help us to choose the appropriate adhesion force, tion cup, the load is reversed vertically. The upper half of while maintaining the robot safe and stable operation under the suction cup is pulled and the lower half is squeezed, the premise of as much as possible to reduce the robot’s as shown in Figure 4. When the critical state of the suction energy consumption. This paper assumes that the robot is cup is turned over, the suction cup appears on the E E 1 2 always walking at low speed and at constant speed, regardless line where the force is greatest. As the bulge continues to of acceleration. The triangular gait is the most common gait grow, the suction cup will leak first from the point M, of the robot. Under the triangular gait, the robot has the causing the entire suction cup to tip over, as shown in highest walking speed and efficiency. Thus, the stability of Figure 5. the robot is analyzed under the triangular gait. Under the tilting load F , the rod will deflect γ degrees 3.1. Analysis of the Adhesive Force at the Foot of the Robot. A downward. At this time, it will cause the suction cup corre- robot walks on the wall; each end of the suction cup is sub- sponding uplift and then change the effective adhesion area jected to a tilting load which is parallel to the wall (the S of the suction cup. Supposing that D is inside diameter 1 6 Applied Bionics and Biomechanics of the suction cup and D = 150 mm and D is outside diam- this time, γ =28 2 degrees. In a very small range, the deflec- 1 2 eter of the suction cup and D = 250 mm, the relation tion angle γ and the end force F are approximated as a linear between the effective adhesion area S and the deflection angle function; its relation is γ is obtained as follows: F =68 30γ 23 2 2 2 D D D 2 2 1 S = π − − The analysis shows that the force F is due to the gravity 2 2 2 acting on the body, as shown in Figure 6. The forces at the three suction cups are G , G , and G , 2 4 6 2 2 D D γ γ 2 1 which satisfy the following equation: + − + D sin D sin 1 1 2 2 2 2 = 49062 50 − 150 G cos 45 − θ + G cos 45 + θ = G cos θ, 6 4 2 γ γ G sin 45 − θ + G = G sin 45 + θ + G sin θ, ∗ 100 + 15625 − 75 + 150 sin sin 6 4 2 2 2 G l + Gl sin 45 − θ = G l sin 45 + 2θ 6 2 The suction cup test shows that when the maximum F is 1926 N, the rod reaches the maximum deflection angle, at Solving equation (24) obtains cos θ − sin θ sin θ cos θ − sin θ sin θ cos θ 1 2 2 1 1 2 2 G = G , cos θ − sin θ sin θ cos θ − sin θ cos θ + sin θ sin θ cos θ 2 2 3 2 2 2 G = G , sin θ cos θ − sin θ sin θ cos θ − sin θ sin θ cos θ 3 1 1 2 2 1 G = G , where θ =45+ θ, θ =45 − θ, θ =45+ 2θ, A = sin θ cos θ − sin θ sin θ cos θ − sin θ sin θ cos θ + sin θ cos θ 1 3 1 2 3 2 1 1 To sum up, with the change of robot pose, the effective The following is a detailed analysis of the status b and e, adhesion area of the suction cup will change. When the pres- as shown in Figure 9(a). The support polygons in the triangu- sure difference between the inside and outside of the suction lar gait are shown in Figures 9(b) and 9(c). cup is a fixed value, the adhesive force on each suction cup A longitudinal overturning instability analysis is per- can be changed. The relations between the deflection angle formed on the b status support with the axis 24 as the tilt- of the leg and the effective adhesion area S of the suction ing axis: cup are shown in Figure 7. 3.2. Walking Stability Analysis of Robot. Here, supposing that the wall-climbing hexapod robot which selects a general tri- F ∗ l + G ∗ cos α ∗ l > G ∗ h, 6 6 G z24 angular gait walks on the wall, as shown in Figure 8, it mainly includes the initial state represented by a and f and four >0 113 ∗ sin α − 0 352 ∗ cos α intermediate states of b, c, d, and e. Applied Bionics and Biomechanics 7 4 4 4 × 10 × 10 × 10 4.95 4.9 4.95 4.9 4.9 4.8 4.85 4.85 4.8 4.8 4.7 4.75 4.75 4.7 4.6 4.7 4.65 4.65 4.5 4.6 4.6 4.55 4.55 4.4 4.5 4.5 4.45 4.45 4.3 0 123456789 10 0 123456789 10 0 123456789 10 θ (degree) θ (degree) θ (degree) (a) (b) (c) Figure 7: (a) Relation between the deflection angle θ and the effective adhesion area S of the leg 2. (b) Relation between the deflection angle θ and the effective adhesion area S of the leg 4. (c) Relation between the deflection angle θ and the effective adhesion area S of the leg 6. 1 1 6 1 4 3 Supporting leg Supporting leg Swinging leg Supporting leg Swinging leg Swinging leg (a) (b) (c) 1 1 1 6 6 Supporting leg Supporting leg Supporting leg Swinging leg Swinging leg Swinging leg (d) (e) (f) Figure 8: A general triangular gait of robot walks on the wall. Next, a longitudinal overturning instability analysis is where G is the total gravity of a robot and a load. G is the z24 performed on the e status support with the axis 35 as gravity which causes the robot to rotate vertically around the the tilting axis: axis 24. G is the gravity which causes the robot to rotate z35 vertically around the axis 35. F is the adhesive force and F = F = F = F = F = F = F. l is the distance 1 2 3 4 5 6 G from the center of gravity to the tilting axis of a robot. l F ∗ l + G ∗ cos α ∗ l > G ∗ h, 1 1 G z35 is the distance from the center of the suction cup to the tilting axis. h is the distance between the centroid of robot >0 116 ∗ sin α − 0 294 ∗ cos α, and the wall. α is the inclination angle of the wall. S (mm ) S (mm ) S (mm ) 8 Applied Bionics and Biomechanics (a) (b) (c) Figure 9: (a) Force analysis of the robot. (b) b status support of the robot. (c) e status support of the robot. A lateral overturning instability analysis is performed on 0.5 the b status support with the axis 46 as the tilting axis: 0.4 0.3 F ∗ l + G ∗ cos α ∗ l > G ∗ h, 2 2 G h46 0.2 0.1 >0 010 ∗ sin α − 0 420 ∗ cos α, –0.1 where G is the gravity which causes the robot to rotate ver- h46 –0.2 tically around the axis 46. –0.3 When the e status support takes the axis 13 as the tilting axis, the lateral overturning instability analysis shows that the –0.4 component of gravity will not cause the robot to lateral over- –0.5 0 10 20 30 40 50 60 70 80 90 turning instability. α (degree) For b status support, to ensure that the robot does not slide on the wall, the balance conditions that need to be sat- Figure 10: Instability critical curve of triangular gait. isfied are as follows: μ ∗ F + F + F + G ∗ cos α > G ∗ sin α, 2 4 6 4. Model Simulations >0 417 ∗ sin α − 0 333 ∗ cos α 4.1. Foot Trajectory of the Swinging Leg. In the triangular gait, the coordinates of the starting point, the highest point, and the falling point of the foot of the robot are (97.24, 551.49, For e status support, to ensure that the robot does not -150) mm, (0, 599.95, -91.93) mm, and (-97.24, 551.49, slide on the wall, the balance conditions that need to be sat- -150) mm, respectively. The speed of the starting point is isfied are as follows: (0, 0, 0) mm/s, and the acceleration is (0, 0, 0) mm/s . The rate of the falling point is (0, 0, 0) mm/s, and the acceleration μ ∗ F + F + F + G ∗ cos α > G ∗ sin α, 2 1 3 5 is (0, 0, 0) mm/s . The swing time is 4 s. Then, the foot trajec- F tory of swinging leg is >0 417 ∗ sin α − 0 333 ∗ cos α, 5 4 3 xt = −1 1395t +11 3953t − 30 3875t +97 2400, where μ is the friction coefficient between the wall and the 5 4 3 yt = −0 7572t +9 0863t − 36 3450t +48 4600t + 551 4900, suction cup, and μ =0 8. 6 5 4 3 zt = −0 9073t +10 8881t − 43 5525t +58 0700t − 150 0000 Based on the above analysis, Figure 10 shows the instabil- ity critical curve of triangular gait. In Figure 10, the blue 32 curve is the b status support longitudinal tipping instability curve, crossing the axis x at 72.29 degrees. The red curve is By equation (32), obtaining that the foot trajectory of the e status support longitudinal tipping instability curve, swinging leg is shown in Figure 11. crossing the axis x at 68.4 degrees. The green curve is the b status support lateral overturning instability curve, crossing 4.2. Foot Trajectory of the Supporting Leg. On the triangular the axis x at 88.67 degrees. The blue-green curve is the critical gait of the supporting leg, the intersection between the sup- curve of the sliding instability of the robot, crossing the axis x porting leg and body as the origin, and establish the coordi- at 38.52 degrees. nate system. The coordinates of the starting point and the F/G Applied Bionics and Biomechanics 9 100 0 50 −25 0 −50 −50 −75 −100 −100 012 34 t (s) t (s) (a) (b) 100 600 −50 −100 t (s) t (s) (c) (d) 50 100 25 50 −25 −50 −100 −50 t (s) t (s) (e) (f) Figure 11: Continued. Speed (mm/s) Acceleration (mm/s ) Displacement (mm) 2 Displacement (mm) Speed (mm/s) Acceleration (mm/s ) y (mm) 10 Applied Bionics and Biomechanics −660 100 680 50 −700 −720 −50 −740 −100 t (s) t (s) (g) (h) −650 −680 −710 −740 −50 −100 −500 −100 t (s) (i) (j) Figure 11: (a) The position of foot in the x direction. (b) The speed of foot in the x direction. (c) The acceleration of foot in the x direction. (d) The position of foot in the y direction. (e) The speed of foot in the y direction. (f) The acceleration of foot in the y direction. (g) The position of foot in the z direction. (h) The speed of foot in the z direction. (i) The acceleration of foot in the z direction. (j) Foot trajectory of swinging leg. falling point of the foot of the robot are (-97.24, 551.49, -150) 4.3. Motion Simulation of the Wall-Climbing Hexapod Robot. mm and (97.24, 551.49, -150) mm, respectively. The speed of According to the foot trajectory planning and inverse kine- the starting point is (0, 0, 0) mm/s, and the acceleration is (0, matics, the relation between joint angle and time is obtained. 0, 0) mm/s . The rate of the falling point is (0, 0, 0) mm/s, and Then, using the spline curve to drive the joint motion, the the acceleration is (0, 0, 0) mm/s . The swing time is 4 s. driving function is shown in Figure 13. Then, the foot trajectory of supporting leg is Moreover, on the basis of the driving function, obtaining that the joint angle and torque change with time are shown in 5 4 3 xt =1 1395t − 11 3953t +30 3875t − 97 2400, Figures 15 and 16. Meanwhile, motion simulation of the cen- ter of gravity displacement in the triangular gait is shown in yt = 551 4900, Figure 17. zt = −150 0000 By Figure 15, it is obtained that the angular changes of each joint are continuous and gentle, without any angle mutation. In Figure 16, six legs are divided into 2 groups in the triangle gait, of which legs 1, 3, and 5 are one group By equation (33), obtaining that the foot trajectory of and legs 2, 4, and 6 are another group. The torque variation supporting leg is shown in Figure 12. x (mm) Displacement (mm) Acceleration (mm/s ) z (mm) Speed (mm/s) Applied Bionics and Biomechanics 11 100 100 50 75 −50 25 −100 01 2 3 4 t (s) t (s) (a) (b) 100 580 50 575 0 570 −50 565 −100 560 t (s) t (s) (c) (d) −1 −1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 t (s) t (s) (e) (f) Figure 12: Continued. Acceleration (mm/s ) Speed (mm/s) Displacement (mm) 2 Displacement (mm) Speed (mm/s) Acceleration (mm/s ) y (mm) 12 Applied Bionics and Biomechanics −551 1 −552 0.5 −553 −554 −0.5 −555 −1 01234 01234 t (s) t (s) (g) (h) −553 0.5 −554 −555 −0.5 −1 −100 t (s) (i) (j) Figure 12: (a) The position of foot in the x direction. (b) The speed of foot in the x direction. (c) The acceleration of foot in the x direction. (d) The position of foot in the y direction. (e) The speed of foot in the y direction. (f) The acceleration of foot in the y direction. (g) The position of foot in the z direction. (h) The speed of foot in the z direction. (i) The acceleration of foot in the z direction. (j) Foot trajectory of the supporting leg. of legs 1, 3, and 5 is the same, and the torque variation of legs Shenzhen Techservo Co. Ltd in China are selected. The suc- 2, 4, and 6 is the same. The torque changes continuously as tion cup which is produced by Japanese SMC Company is the robot runs, but torque changes occur at the time of each selected, of which the model is ZP2-250HTN. The motor transition. By Figure 17, it is obtained that the robot moves model of joint 1 is ST8N40P10V2E, the motor model of joint along the x direction, and the displacement of the robot is 2 is ST8N40P20V2E, and the motor model of joint 3 is 13.7 mm. The robot moves along the y direction, and the dis- ST8N40P10V4E [26]. The retarder models of joints 1, 2, placement of the robot is 1.3 mm. The robot runs with a body and 3 are S042L3-100 and RAD-60-2 [26]. The structural height of 150 mm, and offset is 0.87%; the robot can run parameters of the robot are set as follows: L = 550 mm, smoothly. The robot moves along the z direction, and the dis- L = 450 mm, L = 200 mm, and r = 350 mm. The total 2 3 placement of the robot is 503.7 mm; the average speed is mass of the robot is 72.0 kg. 30.9 mm/s. Next, the kinematics model and stability analysis of the wall-climbing hexapod robot are correct as shown in the experiments. The experimental results indicate that the kine- 5. Experiments matics models of one leg and body are correct; meanwhile, The kinematics and stability of the wall-climbing hexapod the experimental results are in good agreement with the kine- robot have been analyzed on the above. A prototype of the matics simulation results. In addition, the experimental wall-climbing hexapod robot is developed, as shown in results show that the stability conditions of the wall- Figure 18. The motors and the retarders produced by climbing hexapod robot are correct. Figure 19 shows that x (mm) Acceleration (mm/s ) Displacement (mm) z (mm) Speed (mm/s) Applied Bionics and Biomechanics 13 90 100 80 90 70 70 80 60 60 70 50 50 60 40 40 50 0 510 15 0 510 15 0 5 10 15 0 5 10 15 t (s) t (s) t (s) t (s) Joint 1 in leg 3 Joint 1 in leg 4 Joint 1 in leg Joint 1 in leg 1 70 70 80 60 60 50 50 60 40 40 50 30 30 40 20 20 0 510 15 0 5 10 15 0 5 10 15 0 5 10 15 t (s) t (s) t (s) t (s) Joint 1 in leg 5 Joint 1 in leg 6 Joint 2 in leg 1 Joint 2 in leg 2 70 70 70 60 60 60 50 50 50 50 40 40 40 40 30 30 30 20 20 20 20 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 t (s) t (s) t (s) t (s) Joint 2 in leg 5 Joint 2 in leg 4 Joint 2 in leg 6 Joint 2 in leg 3 −60 −60 −60 −60 −70 −70 −70 −70 −80 −80 −80 −80 −90 −90 −90 −90 −100 −100 −100 −100 −110 −110 −110 −110 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 t (s) t (s) t (s) t (s) Joint 3 in leg 3 Joint 3 in leg 4 Joint 3 in leg 1 Joint 3 in leg 2 −60 −60 −70 −70 −80 −80 −90 −90 −100 −100 −110 −110 0 5 10 15 0 5 10 15 t (s) t (s) Joint 3 in leg 6 Joint 3 in leg 5 Figure 13: The driving function of the driving joint. Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) 14 Applied Bionics and Biomechanics Figure 14: Motion simulation of the wall-climbing hexapod robot in triangular gait. 15 15 10 10 5 5 0 0 −5 −5 −5 −10 −10 −10 −15 −15 −15 −20 −20 −20 0 510 15 0 5 10 15 0 5 10 15 t (s) t (s) t (s) Joint angle of leg 1 Joint angle of leg 2 Joint angle of leg 3 Joint angle 1 Joint angle 1 Joint angle 1 Joint angle 2 Joint angle 2 Joint angle 2 Joint angle 3 Joint angle 3 Joint angle 3 20 20 15 15 10 10 5 5 0 0 0 −5 −5 −5 −10 −10 −10 −15 −15 −15 −20 −20 −20 0 5 10 15 0 5 10 15 0 5 10 15 t (s) t (s) t (s) Joint angle of leg 4 Joint angle of leg 5 Joint angle of leg 6 Joint angle 1 Joint angle 1 Joint angle 1 Joint angle 2 Joint angle 2 Joint angle 2 Joint angle 3 Joint angle 3 Joint angle 3 Figure 15: The joint angle change with time in triangular gait. the wall-climbing hexapod robot walks on the vertical wall 6. Conclusions and Future Work with horizontal direction by using the triangular gait. The wall-climbing hexapod robot can walk well on the vertical In this paper, the most contribution is to analyze the kine- wall, and the stability is well. matics and stability condition of the wall-climbing hexapod Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Angle (degree) Applied Bionics and Biomechanics 15 0 0 −50 −100 −100 −100 −200 −200 −150 −300 −300 −200 0 510 15 0 510 0 510 15 t (s) t (s) t (s) Joint torque of leg 1 Joint torque of leg 2 Joint torque of leg 3 Joint 1 Joint 1 Joint 1 Joint 2 Joint 2 Joint 2 Joint 3 Joint 3 Joint 3 300 200 0 0 −0 −50 −100 −100 −100 −200 −200 −150 −300 −300 −200 0 510 15 0 510 15 0 510 15 t (s) t (s) t (s) Joint torque of leg 4 Joint torque of leg 5 Joint torque of leg 6 Joint 1 Joint 1 Joint 1 Joint 2 Joint 2 Joint 2 Joint 3 Joint 3 Joint 3 Figure 16: The joint torque change with time in triangular gait. 0.4 the kinematics model of the body is obtained by coordinate 0.3 transformation. Second, to make the robot walk steadily on 0.2 the wall, and no tipping occurs, the static stability of wall- climbing hexapod robot is analyzed, obtaining the critical 0.1 stability condition of the robot. Third, the kinematics simula- tion of the wall-climbing hexapod robot is operated to ana- −0.1 lyze motion performances, obtaining the foot trajectory of −0.2 the swinging and supporting legs. Moreover, the relation between joint angle and time and joint torque and time is −0.3 obtained by simulation. Finally, the experiments are used to −0.4 05 10 15 validate the proposed kinematics model and stability condi- t (s) tions. The experimental results show that the proposed kine- Direction of x axis matics model and stability conditions of the wall-climbing Direction of y axis hexapod robot are correct. Direction of z axis In the future, we can establish a dynamic model of the Figure 17: Motion simulation of the center of gravity displacement wall-climbing hexapod robot, obtaining the relation between change with time in triangular gait. the driving force and acceleration, velocity, and position of the wall-climbing hexapod robot, so as to provide an effective basis for motion control and force control of the wall- robot to provide a theoretical basis for the stable walking and control of the robot. First, the kinematics model of one leg climbing hexapod robot. In addition, the supporting legs is established, on the basis of the kinematics model, the which are deformed are influenced by the gravity of the body. inverse kinematics of the one leg is solved. Meanwhile, The accuracy of motion control of the wall-climbing hexapod Torque (N∙m) Torque (N∙m) Displacement (m) Torque (N∙m) Torque (N∙m) Torque (N∙m) Torque (N∙m) 16 Applied Bionics and Biomechanics (a) (b) Figure 18: (a) The prototype of the wall-climbing hexapod robot. (b) The structure of one leg of the wall-climbing hexapod robot [31]. (a) (b) (c) (d) (e) (f) Figure 19: The wall-climbing hexapod robot walks on vertical wall with horizontal direction by using the triangular gait [31]. robot is thus affected, and the walking trajectory deviation Conflicts of Interest occurs in the process. Thus, the stiffness of the wall- The authors declare that there is no conflict of interests climbing hexapod robot is investigated in the future. regarding the publication of this paper. Data Availability Acknowledgments The data used to support the findings of this study are avail- The work was supported by the National Key R&D Program able from the corresponding author upon request. of China (Grant No. 2018YFB1305300) and the National Applied Bionics and Biomechanics 17 in Proceedings 2007 IEEE International Conference on Robotics Natural Science Foundation of China (Grant Nos. 61825303, and Automation, pp. 4331–4336, Roma, Italy, 2007. U1713215, and 51605334). The authors would like to thank Jiangsu Greenhub Technology Co. Ltd, China, for its techni- [16] M. C. García-López, C. Gorrostieta-Hurtado, E. Vargas-Soto, J. M. Ramos-Arreguín, A. Sotomayor-Olmedo, and J. C. M. cal support. Morales, “Kinematic analysis for trajectory generation in one leg of a hexapod robot,” Procedia Technology, vol. 3, pp. 342– 350, 2012. References [17] R. Campa, J. Bernal, and I. Soto, “Kinematic modeling and control of the hexapod parallel robot,” in 2016 American Con- [1] I. H. 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Applied Bionics and BiomechanicsHindawi Publishing Corporation

Published: Apr 1, 2019

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