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Modeling, analysis, and experiments of a pause-and-leapjumping robot during takeoff, flight, and landing buffering

Citation for published version:Yang, Q, Zhao, J, Zhang, Z, Chang, B & Liu, X 2019, 'Modeling, analysis, and experiments of a pause-and-leap jumping robot during takeoff, flight, and landing buffering', Journal of Mechanical Science andTechnology, vol. 33, no. 10, pp. 4963-4979. https://doi.org/10.1007/s12206-019-0936-3

Digital Object Identifier (DOI):10.1007/s12206-019-0936-3

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Document Version:Peer reviewed version

Published In:Journal of Mechanical Science and Technology

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Download date: 20. Apr. 2022

Modeling, analysis, and experiments of a pause-and-leap jumping robot during

takeoff, flight, and landing buffering

Qi Yang1, Jing Zhao1, Ziqiang Zhang1*, Bin Chang1, Xingkun Liu2 1 College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing100124, China

2 Computer Science Department, Heriot-Watt University, Edinburgh EH14 4AS, UK

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract Jumping robots has a strong ability to overcome obstacles, and they are suitable for complex terrain environments. However, for the

most jumping robots, especially pause-and-leap jumping robots, take-off parameters can’t be changed accurately, and stability of flight and landing phases need to be improved. In this paper, based on observations of locust’s jumping process, leg mechanisms, including one degree of freedom (DOF) jumping leg and series buffering leg, are designed. Then dynamic models for the take-off, flight and landing buffering phases are established combining with Lagrange/Newton-Euler dynamic modeling methods and conservation of momentum moment. For the former, effects of structural parameters, including position of the driving spring, absolute position of the center of mass and stiffness coefficient of the driving spring, on the take-off velocity and acceleration can be obtained, and the take-off parameters can be changed accurately. For the flight phase, relationship between relative position of the center of mass and the flight stability is revealed. For the latter, effects of two buffering modes on supporting forces and energy storage capacity are analyzed. Based on the parameters determined by the above modeling method, a prototype is developed, and experiment is conducted to verify rationality of the modeling method. The experiment results show the prototype can acquire accurate take-off parameters, and achieve stable flight and landing buff-ering process. This study provides a useful reference for design and control of jumping robots.

Keywords: Locust-inspired robot; Dynamic model; Different jumping phases; Performance analysis; Experiment ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

Bio-inspired jumping robot has the ability to move in an environment with large obstacles [1]-[3]. For example, the robot “Handle” [4] and “Atlas” [5] designed by Boston Dynamics have a prominent jumping ability. With the de-velopment of manufacturing technology, the jumping robot is developing towards the direction of microminiaturization, which further widens the application field of jumping robots [6].

The existing jumping robots can be divided into two types: continuous and pause-and-leap jumping robots. Compared with continuous jumping robots, pause-and-leap jumping robots have abundant motion modes and are suita-ble for complex terrain. Therefore, many experts have stud-

ied them. The weight of the locust-inspired pause-and-leap jumping robot designed by Zaitsev et al [7], [8] is 23 g, and the maximum jumping height of the robot is 3.35 m. Ngu-

yen et al [9] designed a pause-and-leap jumping robot weighing only 7 g, and the robot can surmount obstacles 14 times its own height. In addition, Zhang et al [10], [11] de-signed a bio-inspired jumping robot with a jumping height of more than 88 cm. In order to achieve fast jumping, leg mechanism is the focus in the design of jumping robot. Since the jumping process of the robot generally includes three phases: take-off, flight and landing buffering phase [12], the leg mechanism consists of jumping leg and buffer-ing leg. The jumping leg of creature is typical serial mecha-nism [13], and some researchers have designed jumping robots with bionic serial jumping leg, which have good jumping performance. The jumping leg structures of bio-inspired robot “Kenken” [14], [15] and “Mowgli” [16] are series mechanism, and they can simulate the jumping pro-cess of creature approximately. In order to make the jump-ing leg have advantages of simple structure and high stiff-ness, one DOF mechanism is introduced into the design of jumping leg of robot [17]. The jumping leg of the flea robot designed by Noh et al [18] is a four-bar mechanism, and is driven by SMA actuator. The weight of the robot is 1.1 g, and the maximum jump height can exceed 30 times of its own height. In addition, Gui et al of Beihang Unversity also

† This paper was recommended for publication in revised form by Associate Editor 000 000.

*Corresponding author. Tel.: +86-010-67391684 E-mail address: zzq06520011@163.com

© KSME & Springer 2019

studied leg structure of jumping robot. For the design of buffering leg, unlike continuous jumping robot, the buffer-ing leg needs to be designed separately for the pause-and-leap jumping robot to absorb energy [9]. In order to achieve good buffering performance, many researchers have also studied it from the point of view of structure design and performance analysis based on different landing and buffer-ing modes [19]-[21].

For the abovementioned jumping robots with different leg mechanisms, how to reveal the relationship between jumping parameters and jumping performance is one of the research focuses. There are many researchers have studied modeling method of jumping process for jumping robots. For the take-off process, Choi et al [22] proposed two dy-namic modeling methods for jumping robots in the take-off phase using the principle of energy conversion, and correct-ness of the methods was verified by simulation and experi-ment. In order to study the jumping performance of water-jumping robot, Yang et al [23] established theoretical force models for the take-off process of the water-jumping robot, and a series of experiments were conducted to measure actual forces. For the flight phase, Wang et al [24] founded a dynamic model of a frog-inspired jumping robot for the flight process using the principle of virtual work, and simu-lation was conducted to verify correctness of the dynamic equation. Zhang et al [25] used the Lagrange dynamic mod-eling method to establish a dynamic model for the flight phase of a jumping robot, and the joint torque of the robot can be determined under the known motion law by solving the inverse dynamic equation. For modeling of the landing buffering phase, based on Hertz contact theory and non-linear damping theory, Hu [26] et al established a continu-ous contact and collision model between leg and ground of a frog-inspired jumping robot. Then, a rigid-flexible cou-pling dynamic equation of the system was established using Lagrange method and effect of some initial condition pa-rameters during landing process on motion stability was analyzed.

As can be seen from the above research that most of the existing research focuses on how to achieve fast take-off. Since pause-and-leap jumping robot needs to adjust its pos-ture for the next jumping after completing a jumping period, flight and landing buffering phase cannot be ignored, and there is a lack of modeling method for the take-off, flight and landing buffering phases considering coupling relation-ship of these different jumping phases. Therefore, the most existing micro pause-and-leap jumping robot can’t change the take-off parameters accurately, and has a problem of poor stability of flight and landing phases. This limits the application of jumping robots [27].

In order to solve the above problems, the dynamic model for the take-off, flight and landing buffering phases of the robot is established, and the effect of structural parameters on jumping performance of the robot is analyzed in detail. The accurate take-off parameters and stable jumping pro-

cess of the prototype indirectly prove correctness of the dynamic modeling method. This provides a theoretical basis for the design of robots with good jumping performance.

2. Jumping process of locust

The experiment was conducted first to understand the whole jumping process of locust. The experimental object is Oriental Migratory Locust, which is shown in Fig. 1. Locust has three pairs of legs, namely front leg, middle leg and hind leg. Front leg and middle leg are mainly used for walk-ing, supporting the body and landing buffer. Hind leg is mutated leg, which is mainly used for jumping [17].

Fig. 2 shows the take-off process of locust. It can be seen from Fig. 2 that in the whole take-off process, angle be-tween tibia and femur (φ2) of hind leg increases rapidly to

Fig. 1. Physiological structure of locust.

(a)

(b)

Fig. 2. Take-off process of locust. (a) Parameters definition; (b) Take-off sequence of locust. realize jumping. Suppose coordinate origin of coordinate system o2-x2y2z2 (Fig. 2 (a)) coincides with contact point between jumping leg and ground. The trajectory of center of mass of locust is shown in Fig. 3, and it is approximate to a straight line in the coordinate system o2-x2y2z2.

Fig. 4 (a) shows the landing buffering process of the lo-cust. The experiment results show that front legs and middle

legs land on the ground first and play a major role. Depend-ing on landing posture and landing velocity, the abdomen and wings may play a supporting role to reduce landing impact. Fig. 4 (b) shows the change of velocity of the center of mass along y2 axis from the moment the legs touch the ground to the moment with the velocity of center of mass being zero for the first time.

Fig. 3. Terminal trajectory of hind leg.

(a)

(b)

Fig. 4. Landing buffering process of locust. (a) Landing buffering sequence of locust; (b) Change of velocity along y2 axis.

Fig. 5. Four-bar jumping leg. 3. Modeling and performance analysis of the robot for take-off, flight and landing buffering phases

3.1 Mechanism design

3.1.1 Design for jumping leg

In order to make the jumping leg of the bio-inspired robot have characteristics of high structural stiffness and simple control, a four-bar mechanism is designed as jumping leg instead of a multi-DOF series biomimetic mechanism. Fig. 5 shows the four-bar jumping leg of robot. Link A1A2, A2A3, A3A4 and A4A1 form a four-bar mechanism. The length of link B1A3 is used to simulate the tibia of locust, which is the main structure to achieve fast take-off. The spring is used to store energy. Then the optimization on length of each link for the jumping leg is conducted to make the posture of trunk and the tip trajectory of the jumping leg close to the motion law of locusts.

In Fig. 5, A0B0C0 is the hind leg of locust. The coordinate origin of coordinate system o0-x0y0z0 coincides with A0. The direction of the x0 axis and y0 axis are along horizontal and vertical directions, respectively. The coordinate origin of coordinate system o1-x1y1z1 coincides with A1. The direction of x1 coincides with A1A2, and the direction of y1 coincides with A1A2. The position vector of these two coordinate sys-tems is (a, b, 0). The optimization target should include the following two parts: (a) the tip trajectory of four-bar jump-ing leg should be consistent with the tip trajectory of hind leg of locust in the take-off process with coordinate system o0-x0y0z0 and o1-x1y1z1 are fixed coordinate system; (b) the change of trunk posture should be as small as possible to achieve good jumping stability.

According to kinematic analysis, the end of equivalent tibia link (point B1) in coordinate system o1-x1y1z1 can be shown as

( )

( )01 2 3 5 3 2 1

01 2 3 5 3 2

cos cos

sin sin

x l l l

y l l

β β β

β β β

= − − − +

= − − (1)

where, li is the length of each link (li is shown in Fig. 5. In particular, l5 is the length of B1A3), β2 is the angle be-tween A3A4 and A2A3, and β3 is the angle between A2A3 and A1A2. Because the four-bar mechanism has only one DOF, β2 and β3 have coupling relationship. According to the posi-tion relation of four-bar mechanism, the coupling relation-ship can be written as

( )

( )

21 2 3 4 3 2

2 22 3 4 3 2 3

cos cos

sin sin

l l l

l l l

β β β

β β β

− − −

+ − − = (2)

Then the position of B1 in fixed coordinate system o0-x0y0z0 can be shown as

( ) ( )( ) ( )

0

0

01

01

0

cos / 2 sin / 2 0sin / 2 cos / 2 0

0 0 1 0 0

xy

x ay b

π θ π θπ θ π θ

=

+ − + + + +

(3)

where, θ is the angle between x1 axis and vertical direction, which reflects the attitude of the coordinate system o1-x1y1z1 relative to the o0-x0y0z0. Because β2 or β3 does not have intu-itive physical meaning, the new constraint equations need to be given in order to obtain the tip trajectory. The angle be-tween B1A3 and vertical direction can be written as

1 2 3β π θ β β= − + + (4)

where, β1 can be used as a constraint according to the take-off process of locust, which reflects the swing range of the tibia link. When the optimization parameters are determined, the tip trajectory can also be determined. In order to make the tip trajectory of four-bar jumping leg be consistent with that of hind leg of locust, the optimization objective func-tion can be expressed as

( ) ( )2 21 0 1 0 1

0

n

i i i ii

F x x y y=

= − + − ∑ (5)

where, (x1i, y1i) means the discrete points of the tip trajecto-ry of hind leg of locust.

When o2-x2y2z2 is fixed coordinate system, the center of mass can be written as

( )( )( )( )

2 1 2 5 1

6 1 2 3 4

2 1 1 5 1

6 1 2 3 4

cos cos

sin

sin sin

cos

c

c

x l l

l

y l l

l

β β β

β β β β

β β β

β β β β

= − + −

− + +

= − + + − + +

(6)

where, β4 is the angle between A2A3 and OA2. According to the experiment result, the trajectory of center of mass of locust is approximately a straight line in the take-off phase. So the position of center of mass of robot (xci, yci) in the initial state and the ending state is known. Although there are two equations in Eq. (6) and two unknowns (β2 and β3), β2 and β3 have a coupling relationship, which is shown in Eq. (2). Therefore, β2 and β3 can be determined by optimiz-ing method, which can be written:

( ) ( )2 2 2

1 0 01

cj c j cj c jj

G x x y y=

= − + − ∑ (7)

β2 can take value in a given range, and β3 can be solved

by Eq. (2). The set of β2 and β3 corresponding to the mini-mum value of the G1 is the solution of the equation. At this time, the optimization objective function for the posture of trunk can be written as

2 Initial EndF W W= − (8)

where, W=β1+β2+β3-2π. According to the above analysis, the optimization objective function is

1 21 2

1 max 2 max

F FFF F

ρ ρ− −

= + (9)

where, ρi (i=1, 2) is the proportional coefficient, and ρ1+ρ2=1. The unknown parameters includes li (i=1,2,…5), θ, a and b. Because there are many optimization parameters, the genetic algorithm is used.

After determining the optimized parameters and objective function, the constraint conditions should be determined, which are shown in Table 1.

Take ρ1=ρ2=0.5, the optimization results are shown in Ta-ble 2. The tip trajectory of the four-bar jumping leg is ap-proximately a line in coordinate system o0-x0y0, which is consistent with that of the hind leg of the locust. It is helpful for the robot to take off rapidly.

Table 1 Constraint conditions of optimization parameters

l1-l5 (mm) θ (º) β1 (º) a (mm) b (mm)

[0, 20] [-5, 5] [75, 125] [-15, 15] [-5, 5]

Table 2 Mechanism parameters of four-bar jumping leg

l1/ m l2/ m l3/ m l4/ m l5/ m

9.7×10-3 9×10-3 3.4×10-3 12.9×10-3 18.7×10-3

Fig. 6. Structure of buffering leg.

3.1.2 Design for buffering leg

The bionic design principle is used for the buffering leg. The buffering leg includes the femur link and tibia link, and the length of each link is proportional to the leg of locust. Each part is connected by hip, knee and ankle joint respec-tively, and each joint can be reduced to an equivalent rota-tion pair. In order to be able to store energy to achieve buff-ering effect, two springs are used to limit movements of the hip joint and knee joint, and the connection points are shown in Fig. 6. 3.2 Dynamic modeling and mechanical property analysis for the robot during take-off process

The spring is used to achieve energy storage and rapid re-lease. One end of the spring is connected to the link A2A3 at point H1, and the other end is connected to the link A1A4 at point H2 (Fig. 5). The contact point B1 between the jumping leg and the ground can be simplified as a rotating pair.

3.2.1 Kinematic modeling

In Fig.5, the expression of the center of mass in the coor-dinate system o2-x2y2z2 can be shown in Eq. (6), and (xc, yc) is a series of discrete value. When initial position of the center of mass and take-off direction angle are determined, value of the coordinate of the center of mass during the take-off process can be obtained.

Then, according to Eq. (6), the coupling relationship be-tween β6 (the angle between link A4A3 and A4A1), β5 (the angle between link A1A4 and A1A2), β3 and β2 in the quadri-lateral A1A2A3A4 can be expressed as

( )2 2 2

6 1 22 2arccos f f f f f f

f f

A A B C B Cf

A Bβ β

+ − + = − = +

(10)

( )2 2 6 4 3 65 2 2

1

cos( ) cosarccos l l l fl

β β ββ β + + −

= =

(11)

( )3 2 5 6 3 22 fβ π β β β β= − − − = (12)

where, 4 2 22 sinfA l l β= ⋅ ⋅ ⋅ ; 3 4 4 2 22 2 cosfB l l l l β= ⋅ ⋅ − ⋅ ⋅ ⋅ ;

2 2 2 23 4 2 1 3 2 42 cosfC l l l l l lβ= ⋅ ⋅ ⋅ + − − − .

When coordinates of the center of mass during the take-

off process are known, insert Eqs. (10)-(12) into Eq. (6), and the unknowns contained in Eq. (6) are β1 and β2. At this time, all of the joint angles of the robot during the take-off process can be obtained.

Since the initial coordinate of the center of mass can be obtained by experiments, each joint angle of the jumping robot in initial state can be solved by Eqs. (6)-(12). In order to obtain the coordinates of the center of mass during the whole take-off process, the take-off direction angle (the angle between the direction of external forces the trunk suffered and x2 axis) of the robot should be determined on condition that the initial coordinate of the center of mass is known. The expression of the take-off direction angle can be written as

22 0 23 0

22 0 23 0

=arctan y y

x x

F F mgF F− −

− −

+ −Ψ

+ (13)

where, m is mass of the robot. (F22x-0, F22y-0) and (F23x-0, F23y-

0) are forces of the link A2A3 and A1A4 acting on the trunk in initial state respectively. The static force analysis of the jumping leg is conducted to obtain (F22x-0, F22y-0) and (F23x-0, F23y-0). Suppose the i-th link (i=1-4) subjects forces Fi of the other links which connect with it on it. The link may also suffer external force Fti and spring driving force Fsi. The force and moment balance equations can be shown as

1 2 3

1 1 1

n n n

ji pti qsi ij p q

+ m= = =

+ +∑ ∑ ∑ 0F F F g = (14)

( ) ( ) ( )1 2 3

1 1 1

n n n

ji ji pti pti qsi qsi i ij p q

m= = =

× + × + × ×∑ ∑ ∑ 0r F r F r F + r g =

(15)

where, r is position vectors in the coordinate system O2-x2y2. In particular, for the 1-th link, F11 is support force of the ground to the jumping leg. For the trunk link, F1t4 is equiva-lent take-off force. Then F22-0 and F23-0 can be obtained ac-cording to Eqs. (14)-(15), and the take-off direction angle ψ can be solved by Eq. (13). Since take-off time of the robot is very short, it is generally considered that the robot moves along the take-off direction line during the take-off process without deviation. Therefore, the relationship between xc and yc during take-off process can be written as

( )0 0tanc c c cy x x y− −= − ⋅ Ψ + (16)

where, (xc-0, yc-0) is the initial coordinate of the center of mass O in the coordinate system o2-x2y2.

Insert Eq. (6) into Eq. (16), and the relationship between β1 and β2 can be obtained as follow

( )2 2 2

1 4 22 2arccos2

f f f f f f

f f

W V U U V Wf

U Vπβ β

− + − = − − = +

(17)

where,

tanfU u v= − ⋅ ψ ; tanfV w u= − ⋅ ψ ; tanfW s= ⋅ ψ ;

( )6 2 3 7 2 2 5cos cosu l l lβ β β β= + + − + ;

( )6 2 3 7 2 2sin sinv l lβ β β β= − + + + ;

( )6 2 3 7 2 2sin sinw l lβ β β β= + + − ;

00 tan

OO

ys x −−= −

Ψ.

where, β7 is the angle between A1A2 and OA2 in Fig.5.

Through the analysis above, the joint angles of the robot during take-off process can be obtained.

3.2.2 Dynamic modeling

In order to obtain dynamic performance of the jumping leg during the take-off process, it is necessary to establish dynamic model for the jumping leg. According to Eqs. (10)-(12) and Eq.(17), the joint angles of the robot during take-off process can be expressed as functions of β2. Therefore, the angular velocity of each joint can be expressed as func-tions of β2 and 2β , and their formulas are as follow

2β= ⋅ Qβ (18)

where,

3 4 5 1

Tβ β β β =

β ;

( ) ( ) ( ) ( )1 2 2 2 3 2 4 2

2 2 2 2

Tdf df df df

d d d dβ β β ββ β β β

=

Q .

Assuming that position of the center of mass of each link

is at the link’s geometric center, and the expression of ve-locity of the center of mass of each link for the jumping leg can be shown as

= ⋅ V J Θ (19)

where,

1 1 2 2 3 3 4 4

T

x y x y x y x yv v v v v v v v = V ;

1 2 3 4

Tθ θ θ θ = Θ ;

where, J is coefficient matrix of the velocity. vix and viy

(i=1~4) are the velocity of the center of mass of the i-th link

along x2 and y2 axis respectively. θ1 (θ2, θ3, θ4) is the angle between B1A3 (A2A3, A1A4, OA1) and the positive direction of x2 axis. θi (i=1~4) is function of β2, and their expressions can be shown as

1 1 2πθ β= + (20)

2 1 2 2πθ β β= + − (21)

3 1 6 2πθ β β= − + (22)

4 1 5 6 832πθ β β β β= − − − + (23)

where, β8 is the angle between A1A2 and OA1 in Fig.5.

Insert Eqs. (20)-(23) and their derivations into Eq. (19), and vix and viy (i=1~4) are functions of β2 and 2β . Then ac-celeration of each link and angular acceleration of each joint can be expressed as functions of β2, 2β and 2β by deri-vation of Eq. (18) and Eq. (19).

The Lagrange dynamic modeling method is used to es-tablish the dynamic model of the jumping leg. The total potential energy of the jumping leg can be expressed as

( ) ( )4

20

1

12i i s s

iU m gh k l l−

=

= + −∑ (24)

where, mi is the mass of each link. hi is the height of the center of mass of each link in the fixed coordinate system o2-x2y2. ls-0 is the original length of the spring. ls is the length of the spring in the take-off process. The expressions of hi (i=1~4) and ls can be written as

( )51 1 1 2sin

2 hlh fθ β= = (25)

( )22 5 1 2 2 2sin sin

2 hlh l fθ θ β= + = (26)

( ) ( )43 5 3 1 3 3 2sin sin

2 hlh l l fθ θ β= − + = (27)

( ) ( )4 5 3 1 4 3 7 4 4 2sin sin sin hh l l l l fθ θ θ β= − + + = (28)

( )4 2 62 21 8 1 8 2

8

sin2 cos arcsin s

s s s

l ll l l l l

lβ

β −

= + − ⋅ ⋅ ⋅ − (29)

where, ls1 is the length of A2H1, ls2 is the length of A1H2, and l8 is the length of A3H2. The expression of the length of A3H2 can be written as

( ) ( )228 3 4 2 3 4 2 62 coss sl l l l l l l β= + − − − (30)

Eq. (24) can be further expressed as

( )( ) ( ) ( )4

2 1 2 21

i hi k Ui

U m g f f fβ β β=

= ⋅ + =∑ (31)

The total kinetic energy of the jumping leg can be written

as

( )4

2 2

1

1 1 2 2oi i i ix iy

iT J m v vθ

=

= + +

∑ (32)

where, Joi (i=1~4) is the moment of inertia of the i-th link of the jumping leg. Eq. (32) can be further shown as

( ) ( ) ( )4

2 2 2 2 2 21

1 1, , ,2 2oi i i vi T

iT J f m f fϕ β β β β β β

=

= ⋅ + ⋅ =

∑

(33)

According to Eq. (31) and Eq. (33), the potential energy

and kinetic energy of the jumping leg are functions of β2 and 2β . Insert Eq. (31) and Eq. (33) into the Lagrange dy-namic equation, and the following formula can be obtained

2 22

0d dT dT dUdt d dd β ββ

− + =

(34)

Eq. (34) can be further expressed as

( ) ( ) ( )21 2 2 2 2 2 3 2 0G G Gβ β β β β+ + = (35)

Eq. (35) is a two order quadratic nonlinearity differential

equation, and it can be simplified to two first-order differen-tial equations. The expressions can be shown as

( )( )

( )( )

2

2 2 3 222

1 2 1 2

yG G

yG G

ββ β

ββ β

=

= −

(36)

The numerical method is adopted to solve Eq. (36), and

the initial value of β2 and 2β can be solved as

( )( )

2 2 0

2

0

0 0

β β

β− =

=

(37)

Then, the solution of Eq. (35) can be expressed as

( )( )

2 1

2 2

h t

h t

β

β

=

=

(38)

Insert Eq. (38) into Eqs. (10)-(19), and the velocity (ac-

celeration) of each link and angular velocity (angular accel-

eration) of each joint can be obtained during the take-off process.

The Newton-Euler method is adopted to obtain the joint forces. Based on Eqs. (14)-(15), inertial forces Fci and iner-tia moments Mci need to be considered, and the balance equations of the force and moment of each link can be ex-pressed as

1 2 3

1 1 1

n n n

ji pti qsi ci ij p q

+ m= = =

+ + +∑ ∑ ∑ 0F F F F g = (39)

( ) ( ) ( )( )

1 2 3

1 1 1

n n n

ji ji pti pti qsi qsij p q

i ci i cim= = =

× + × + ×

× +

∑ ∑ ∑0

r F r F r F

+r F g + M = (40)

According to the analysis above, all the joint forces of the

jumping leg can be obtained, and the dynamic modeling of the jumping leg is completed.

3.2.3 Performance analysis

According to the dynamic analysis, the main factors af-fecting the jumping performance include the position of the driving spring, the absolute position of the center of mass and the stiffness coefficient of the spring.

The position of the driving spring. The change of the position of point H1 and H2 (Fig.5) may affect the energy storage capacity of the driving spring, and further cause the maximum take-off velocity of the robot different. Suppose that the mass of the robot m=0.2 Kg. The proportion of oth-er links is the same as the data showed in Table. 1, and the length magnify tenfold. The initial position of the center of mass in the coordinate system o2-x2y2 is (4.5×10-2 m, 10.0×10-2 m). The stiffness coefficient of the spring is 4000 N/m, and the original length of the spring is 0.1077 m. Fig.7 shows change of the maximum velocity of the robot and total energy of the driving spring.

Fig.7 (a) and Fig.7 (b) show change of the maximum ve-locity along x2 axis and y2 axis of the robot with the position of the driving spring varies respectively. When the values of ls1 and ls2 decrease (that is in Fig.5, point H1 (H2) is closer to point A3 (A1)), the maximum velocity along x2 axis and y2 axis of the robot increase respectively, which means the robot has better jumping ability. When values of ls1 and ls2 are both zero, the maximum velocity along x2 axis and y2 axis are 4.03 m/s and 8.05 m/s respectively. When the val-ues of ls1 and ls2 are both 0.025 m, the maximum velocity along x2 axis and y2 axis are 1.49 m/s and 0.37 m/s respec-tively. Fig.7 (c) shows total energy of the driving spring, whose change regulation is consistent with that of maxi-mum velocity. When the gravitational potential energy of the robot is approximately equal, the greater the energy, the greater the velocity. As is shown in Fig.7 (c), the maximum energy and minimum energy stored in the driving spring are 5.91 J and 0.16 J respectively.

(a)

(b)

(c)

Fig. 7. Change of the velocity and energy. (a) Change of the maximum velocity along x2 axis of the robot; (b) Change of the maximum veloci-ty along y2 axis of the robot; (c) Change of the total energy of the spring during the take-off phase.

However, the values of ls1 and ls2 are not as small as pos-sible. The values of ls1 and ls2 corresponding to the maxi-mum and minimum value of energy are (0 m, 0 m) and (0.025 m, 0.025 m) respectively. Correspondingly, the ve-locity and acceleration along x2 axis and y2 axis during the take-off process is shown in Fig. 8. In order to analyze the fluctuation of acceleration curve, a fluctuation index (called FI) is defined as following.

Firstly, according to the coordinates of a series of discrete points on the acceleration curve, a straight line L can be obtained using fitting method, and its equation can be writ-ten as

Y A X B= ⋅ + (41)

where,

.

1

1 1

2

1 1 1

n n

i i ii in n n

i i ii i i

x n x yAB

x x y

−

= =

= = =

= ⋅

∑ ∑

∑ ∑ ∑

X represents the value of time, and Y represents the value of acceleration. (xi, yi) (i=1~n) is coordinate of the i-th discrete point in acceleration curve.

Then the sum of the distance from all discrete points on the acceleration curve to the straight line L is defined as the fluctuation index FI, that is

21 1

ni i

i

Ax y BFI

A=

− +=

+∑ (42)

The smaller the value of FI, the smaller the fluctuation of

acceleration curve. When FI equals zero, the acceleration curve is a straight line, and it does not fluctuate. In Fig. 8(a), the fluctuation indexes of two acceleration curves are: FIX1=0.3059, FIY1=0.3261. Similarly, In Fig. 8(b), the fluc-tuation indexes are: FIX2=0.1267, FIY2=0.1276. Therefore, it can be known from Fig. 8 that the greater the velocity of the robot, the larger value of FI, and the greater the fluctuation of its acceleration curve, which is not conducive to the sta-bility of the trunk of the robot.

(a)

(b)

Fig. 8. Change of velocity and acceleration. (a) Change of velocity and acceleration corresponding to the maximum energy; (b) Change of velocity and acceleration corresponding to the minimum energy.

In summary, in order to make the robot have good jump-ing performance (such as large take-off velocity), the values of ls1 and ls2 should be reduced, while in order to make the trunk of the robot have good jumping stability during the take-off and flight process, the values of ls1 and ls2 should be increased.

The absolute position of the center of mass. The initial position of the center of mass in the fixed coordinate system o2-x2y2 (Fig. 5) has an effect on forces the trunk suffered in the initial state, and further causes the take-off direction angle of the robot different. Moreover, the absolute position of the center of mass of the robot can be changed by rotat-ing the robot around point B1 (Fig. 5). Suppose that the val-ues of ls1 and ls2 are both 0.03 m, the stiffness coefficient of the spring is 4000 N/m, and the initial coordinate of the center of mass in the coordinate system o2-x2y2 is (4.5×10-2 m, 10.0×10-2 m). On this basis, two different coordinates of the center of mass are obtained by turning the robot coun-terclockwise to 5o and clockwise to 5o, and they are (0.036 m, 0.104 m) and (0.054 m, 0.0966 m) respectively. The change of velocity and acceleration under different initial posture of the jumping leg can be seen in Fig. 9.

In Fig.9 (a) and Fig.9 (b), the solid lines represent the ve-locity and acceleration along x2 axis, and the dotted lines represent the velocity and acceleration along y2 axis. In addition, longitudinal coordinate represents the rotation angle of the robot relative to its initial attitude. “+” means the robot rotates counterclockwise, and “-” means the robot rotates clockwise in Fig.9. It can be seen from Fig.9 (a) that when the robot rotates counterclockwise, the velocity of the robot along x2 axis decreases, while the velocity along y2 axis increases. This is because the total energy is a constant value under these cases, and the decrease of the velocity along x2 axis may lead to the increase of the velocity along y2 axis for the robot. When the robot rotates 5° counter-clockwise, the maximum velocity along x2 axis and y2 axis are 1.711 m/s and 1.936 m/s respectively, and when the robot rotates 5° clockwise, the maximum velocity along x2 axis and y2 axis are 2.34 m/s and 1.57 m/s respectively. Fig.9 (b) shows the change of acceleration under different initial absolute position of the center of mass. The initial accelerations showed in Fig.9 (b) are all in range [4 m/s2, 10 m/s2], which means the robot can take off normally. It can be seen from Fig. 9(b) that the change law of acceleration is consistent with that of velocity. Fig. 9 (c) shows that when the robot rotates counterclockwise, the take-off direction angle of the robot increases.

It can be seen from the analysis above that by adjusting the initial absolute position of the center of mass, the take-off direction angle and velocity (acceleration) of the robot are changed accurately.

The stiffness coefficient of the spring. Since the jump-ing robot is driven by spring, the stiffness coefficient may affect energy stored in the spring, and further cause jumping performance different. The change of the velocity and

0

-5

+5

(a)

Reletive angle/ °

0

-5

+5

(b)

28.56°

35.79°

43.63°

52.13°

61.28°

(c)

Fig. 9. Change of velocity and acceleration under different initial abso-lute position of the center of mass. (a) Change of velocity under differ-ent initial absolute position of the center of mass; (b) Change of accel-eration under different initial absolute position of the center of mass; (c) Change of takeoff direction angle under different initial absolute position of the center of mass.

acceleration of the robot with different stiffness coefficient of the driving spring is shown in Fig. 10. It can be seen from Fig.10 (a) that when the stiffness coefficient of the driving spring increases, the maximum velocity along x2 axis decreases slightly, while the maximum velocity along y2 axis increases markedly. It can be seen from Fig. 10 (b) that with the increase of the stiffness coefficient of the driv-ing spring, the acceleration along x2 axis and y2 axis in-creases. Therefore, in order to make the robot have good jumping performance, the stiffness coefficient of the driving spring can be increased appropriately.

Multi structural parameters. The analysis above showed the effect of some individual structural parameters on jumping performance. It is essential to clarify the effect of multi structural parameters on jumping performance. Suppose the value of the driving spring is in range [3000 N/m, 4500 N/m], and the relevant angle of the robot is in range [-5°, 5°]. The change of maximum velocity and fluc-tuation index are shown in Fig. 11.

It can be seen from Fig. 11 (a) that when the robot rotates clockwise and the stiffness coefficient of the driving spring decreases, the maximum velocity along x2 axis increases, while the maximum velocity along y2 axis decreases. In Fig. 11 (b), when the robot rotates counterclockwise, the fluctua-tion indexes along x2 axis and y2 axis all show increasing tendency, which is not conductive to the stability of robot. When the stiffness coefficient of the driving spring increas-es, the fluctuation index along x2 axis decreases, while the fluctuation index along y2 axis increases. The maximum and minimum fluctuation index along x2 axis are 1.869 m and 1.248 m respectively, and the maximum and minimum fluc-tuation index along y2 axis are 1.064 m and 0.168 m respec-tively.

Therefore, the relative rotation angle of the robot and the stiffness coefficient of the driving spring can be selected according to the regulation shown in Fig. 11 in order to make the robot have good jumping performance and stabil-ity. Meanwhile, the jumping parameters for the take-off process can be changed accurately from Fig. 11.

(a)

(b)

Fig. 10. Change of velocity and acceleration. (a) Change of velocity with different stiffness coefficient of the driving spring; (b) Change of acceleration with different stiffness coefficient of the driving spring.

VX-max

VY-max

(a)

FI-AX

FI-AY

(b)

Fig. 11. Change of velocity and fluctuation index. (a) Change of max-imum velocity along x2 axis and y2 axis; (b) Change of fluctuation index of acceleration curve along x2 axis and y2 axis. 3.3 Dynamic modeling and mechanical property analysis for the robot during flight process

3.3.1 Dynamic modeling

In order to distinguish the take-off process from the flight process, it is necessary to judge whether the robot has en-tered the flight process or not. There are two main features for the robot to leave the ground and enter the flight pro-cess: support force of the ground to the robot is zero, and velocity of the center of mass of the trunk for the robot along vertical direction is greater than zero. That is,

4

00

N

y

Fv

≤ >

(43)

where, FN is the support force of the ground to the robot.

When the robot is in the flight phase, the jumping leg re-mains fully extended, and the robot can be equivalent to a rigid body (Fig. 12). The mass of the equivalent rigid body is M, and the moment of inertia is Jo. Ignoring effect of air resistance on the robot, the robot is only affected by gravity after leaving the ground, and action point of gravity is in the equivalent center of mass of the equivalent rigid body. Therefore, motion form of the robot in the flight process

x2

y2

P

Q

X

Y

O P

Q

P

Q

Equivalent center of mass

Trajectory of center of mass

Fig. 12. Schematic diagram for flight phase.

can be regarded as the coupled motion of oblique projection motion of the equivalent center of mass and rotation motion of the equivalent rigid body around the center of mass. Fixed coordinate system O-XY is established by taking ini-tial position of the equivalent center of mass of the robot during the flight process as the coordinate origin. The posi-tive direction of X axis is horizontal to right and the positive direction of Y axis is vertical to upward (Fig. 12).

The position and posture of the equivalent rigid body PQ in the flight process in the coordinate system O-XY can be expressed as

( ) ( )

( ) ( )

( )

( )

0

4 0

4 024 0 0 0

21 2

x

yy

t

t

X t v t tv t

Y t v t t g t t t tg

t dtω

= ⋅ ⋅ = ⋅ − ⋅ ≤ ≤ +

Θ = ∫

(44)

where, t0 is the time when the robot finishes the take-off process and leaves the ground. (v4x(t0), v4y(t0)) is the velocity of the robot along x2 axis and y2 axis when the robot finishes the take-off process and leaves the ground. Θ is the angle between the attitude of the equivalent rigid body PQ during the whole flight process and the initial attitude of PQ. ω(t) is the angular velocity of the equivalent rigid body PQ ro-tating around its center of mass O in the flight process.

t0 and (v4x(t0), v4y(t0)) can be obtained through the analy-sis of the dynamics in the take-off process. In order to ob-tain position and posture of the equivalent rigid body PQ in the coordinate system O-XY during the whole flight process, it is necessary to determine the angular velocity ω(t) of the equivalent rigid body PQ rotating around its center of mass O.

Since the equivalent rigid body PQ is only subjected to gravity in the flight phase, its momentum moment is con-served, which can be expressed as

2

OL rp mrv mr Jω ω= = = = (45)

Therefore, the angular velocity of the equivalent rigid body PQ rotating around its center of mass in the flight phase is a fixed value, which can be obtained by following methods.

The total moment of inertia of the robot in the take-off process can be shown as

( )4

1I ci ci ci

i== × +∑M r F M (46)

where, Fci and Mci are the inertia force and moment of each link, which can be solved by the dynamic analysis in the take-off process.

The angular velocity of the robot rotating around its cen-ter of mass during the take-off process can be written as

( ) ( )'00

0t I

O

M tt dt t t

Jω

−= ≤ ≤

∫ (47)

According to Eq. (46), the angular velocity of the robot

rotating around its center of mass is ω’ (t0) when the robot takes off from the ground. Then, according to the conserva-tion of momentum moment, the angular velocity of the equivalent rigid body PQ in the flight phase can be shown as

( ) ( ) ( )4 0'0 0 0

2 yv t

t t t t tg

ω ω⋅

= ≤ ≤ +

(48)

Insert Eq. (48) into Eq. (44), and the position and posture

of the equivalent rigid body PQ in the coordinate system O-XY can be obtained.

3.3.2 Performance analysis

Suppose coordinate origin of moving coordinate system Ot-XtYt is on the link A1A2 (Fig. 5). The direction of Yt axis coincides with the link A1A2 and points to joint A2, and the direction of Xt axis is vertical to Yt and points to the center of mass O.

Fig. 13 shows relationship between the position of the

Fig. 13. The rotation angle of the trunk.

center of mass in the coordinate system Ot-XtYt and the rota-tion angle of the robot during the flight process. When the center of mass of the robot moves along the positive direc-tion of Xt axis and negative direction of Yt axis, the rotation angle of the trunk during the flight process decreases. The maximum rotation angle of trunk is 547.3o, and the robot rotates one and a half circle in the air, while the minimum rotation angle of trunk is 154.5o. Therefore, in order to make the robot have better motion stability in the flight phase, the position of the center of mass of the robot can be adjusted according to Fig. 13. 3.4 Dynamic modeling and mechanical property analysis for the robot during landing buffering phase

In Fig.6, when the ground is very rough, the end of the buffering leg cannot slide on the ground in the buffering process. According to literature [38], the kinematics of buffering leg can be obtained.

2 2 2 2

2 11 2 2

2

arccos arctan2

b bb

b

l s h l hsl s h

β + + −= +

+ (49)

2 2 2 21 2

21 2

arccos2

b bb

b b

l l s hl l

β π + − −= − (50)

3 2 1b b bβ β β= − (51)

where, lb1 and lb2 are the length of tibia link (AB) and femur link (BC) respectively. h is the distance from the point C to the ground, and s is the projection distance of the buffering leg on the ground.

The mechanical property can be obtained from Eqs. (14)-(15). At this time, the support force the ground to the buff-ering leg can be shown as

0

0

111 1 1 1 11 12

2 3 2 3 21 22 2

sin cossin cos

x

y

sb b b b

b b b b s

FF

Fl l a al l a a F

β ββ β

−

=

−

(52)

where,

( ) ( )11 1 1 1 1 1 1 1 1

1 1

cos sin sin cos

sin ;

b b b b b b b b

b b

a l l

l

β α β β α β

α

= − − ⋅ ⋅ − − ⋅ ⋅

′− ⋅

12 0a = ;

( ) ( )( ) ( )

21 1 1 2 3 3 2

1 1 2 3 3 2

2 2

cos cos sin

sin sin cos

sin

b b b b b b

b b b b b b

b b

a l

l

l

β α α β β

β α α β β

α

= − − − − ⋅ ⋅ − − − − ⋅ ⋅ −

′⋅ ;

( ) ( )22 3 3 3 2 3 3 3

2 3 2

cos sin sin cos

sin .

b b b b b b b

b b b

a l

l l

β α β β α β

α

= − + ⋅ ⋅ + + ⋅

′′⋅ − ⋅

where, βb1 is the angle between tibia link and ground, and βb3 is the angle between femur link and ground. Fs1 and Fs2 are the spring force which act on the femur link and tibia link respectively. αb1 is the angle between spring 1 and tibia link, αb2 is the angle between spring 1 and femur link, and αb3 is the angle between spring 2 and femur link. lb1’is length from A to the connection point of the spring 1 on the tibia link, lb2’is length from C to the connection point of the spring 1 on the femur link, and lb2’’is length from A to the connection point of the spring 1 on the tibia link.

If the ground is very smooth, the end of the buffering leg may slide on the ground in the buffering process. At this time, βb1, βb2 and βb3 cannot be calculated directly according to Eqs. (49)-(52) because s becomes a variable. The chang-es of joint angles are also related to spring forces. In partic-ular, Fx0 represents the friction of the ground to the buffer-ing leg. Insert Eqs. (49)-(52) into force and moment balance equations, the equations can be simplified

1 1 1 1 11 1 12 2

2 3 2 3 21 1 22 2

sin cossin cos

b b b b s s

b b b b s s

l l a F a Fl l a F a F

µ β βµ β β⋅ ⋅ − ⋅ ⋅ + ⋅

=⋅ ⋅ − ⋅ ⋅ + ⋅

(53)

where, μ is friction coefficient. There is only one unknown number s in Eq. (53), which can be solved by the search algorithm. An example is given to compare the buffering performance with different buffering modes. Suppose lb1=8.92×10-3m, lb2=7.31×10-3m, lb1’=6.24×10-3m, lb2’=5.12×10-3 m, lb2’’=2.19×10-3 m, and μ=0. The spring stiffness coefficients of spring 1 and spring 2 are 70 N/m. The changes of support forces and sliding distance of the end of the buffering leg are shown in Fig. 14(a), and energy storage capacity is shown in Fig. 14(b). In Fig. 14, buffering mode 1 and buffering mode 2 represents the modes with the end of the leg fixed and sliding respectively. When the end of the buffering leg fixed, the force of the ground to the leg is much larger than that with the end of buffering leg sliding (because μ=0, Fx0 is zero). Energy storage capacity is lower with the end of buffering leg sliding. 3.5 Parameter optimization

In order to make the robot have better jumping perfor-mance, optimization on the structural parameters should be carried out. According to the analysis above, the structural parameters consist of the position of the driving spring ls1 and ls2, the relevant angle of the robot Φ, the stiffness coef-ficient of the driving spring K and the position of the center of mass in coordinate system Ot-XtYt (xT, yT). Correspond-ingly, the performance indexes include maximum velocity Vmax, fluctuation index FI and the rotation angle Θ of the trunk in the flight phase. In particular, the relevant angle of the robot Φ and the maximum velocity Vmax can be con-strained in constraint conditions during the process of opti-mization. Thus, the optimization objective function can be

(a)

(b)

Fig. 14. Motion law of buffering leg. (a) Change of suppose forces in buffering process; (b) Energy absorption capacity in the buffering process. written as

1 2 3max max max yx

x y

FIFIfFI FI

η η η Θ= + +

Θ (54)

In Eq. (54), the smaller the fluctuation index along x2 axis

or y2 axis is, and the smaller the rotation angle of the robot in the flight process is, the better the jumping performance is. ηi (i=1~3) is the proportional coefficient, and η1+η2+η3=1. Considering the constraint conditions, the standard form of the constrained optimization problem with inequality con-straints can be expressed as

( ) max

min. .

I EX Y

I E

fs t V V V

ϕ ϕ−

≤ ≤

≤ Φ ≤

(55)

where, [VI, VE] is the range of the maximum velocity of robot, and [φI, φE] is the range of the relevant angle of the robot. The constraint condition of each structural parameter is shown in Table. 3.

Table 3 Constraint conditions of structural parameters

ls1/ls2 (m) K (Nm/rad) Φ (º) xT/yT (m)

[0, 0.03] [3000, 4500] [-10, 10] [-0.04, 0.04]

4. Experiment and results

4.1 Structure design of the robot

The structure model of locust-inspired jumping robot is shown in Fig. 15. The robot consists of three parts: buffer-ing legs, jumping legs and trunk structure, and the hind leg has only one DOF. The motor can drive the cam to rotate. In the initial state, the cam limits movement of the baffle. When the robot is ready to take off, the motor drives the cam rotate, and the cam cannot limit the movement of the baffle. Then the hind legs move under the action of the spring, and the robot can take off.

4.2 Experiment and results

A prototype of the jumping robot is manufactured, which is shown in Fig. 16. The robot consists of trunk, jumping legs and buffering legs. The parameters of the robot are shown in Table 4. According to the optimization results, the

Fig. 15. Structure model of locust-inspired jumping robot.

Fig. 16. Prototype of locust-inspired jumping robot

Table 4 Parameters of locust-inspired jumping robot

Mass/g

Total mass Head Left (right) trunk Middle trunk Buffering leg Jumping leg

183.6 3.6 9.7 10.8 7.4 15.7

Spring stiffness coefficient/(N/m) Lengh/cm Height/cm

Spring of jumping leg Spring 1 of buffering leg Spring 2 of buffering leg

810 70 70 20.5 9.5

(a)

(b)

Fig. 17. Take-off sequence of locust-inspired jumping robot. (a) Take-off sequence (ϕ=78.3°); (b) Take-off sequence (ϕ=47.1°) values of ls1 and ls2 are both 0.025 m, and the position of the center of mass in the coordinate system Ot-XtYt is (0.025 m, -0.01 m).

The high-speed camera (Fastcam, Mini, UX100, Photron, Japan) is used to shoot the jumping process of the jumping robot, and the shooting frequency is 1000 frames / sec. Fig.17 (a) and Fig. 17 (b) show the take-off sequence of locust-inspired jumping robot with take-off direction angle ϕ=73.8°and ϕ=47.1° respectively.

The trajectory of center of mass of the robot is shown in Fig. 18 (a), and the motion law of take-off velocity and trunk posture is shown in Fig. 18 (b). It can be seen from Fig. 18 (a) that the jumping distance of the robot along ver-tical direction are 206.9 mm and 98.4 mm respectively with ϕ=73.8°and ϕ=47.1° when the jumping leg leaves the ground, and the jumping distance along horizontal direction are 149.1 mm and 236.5 mm respectively. And the trajecto-ry is approximately a straight line. When ϕ=73.8°, the ve-locity of the center of mass and the rotation angle of the trunk of the jumping robot are shown in Fig. 18 (b). In par-ticular, the jumping legs leave the ground at 105 ms. It can be seen from Fig. 18 (b) that the rotation angle of the trunk is about 29° during the take-off process, and the horizontal velocity increases gradually in take-off phase and the max-imum value is 1.101 mm/ms. When the jumping legs leave the ground, the horizontal velocity decreases gradually. The horizontal velocity is 0.931 mm/ms at 144ms. Similarly, the vertical velocity continues to increase in take-off phase, and the maximum value is 1.759 mm/ms. After that, when the

(a)

(b)

Fig. 18. Motion law of jumping robot for take-off phase. (a) Trajectory of center of mass of the robot; (b) Motion law of take-off velocity and trunk posture of the robot.

0000 Y. Qi et al. / Journal of Mechanical Science and Technology 23 (2019) 1261~1278 1275

jumping leg leaves the ground, the horizontal velocity is reduced to 1.19 mm/ms at 144ms. It can be seen from the analysis above that the jumping parameters for the take-off process of the robot can be changed accurately. Because the parameter selection of the prototype is conducted by guid-ance of the modeling method for a complete jumping pro-cess, this shows that the modeling method has guiding sig-nificance for the robot to acquire accurate take-off parame-ters.

For the whole jumping process, the jumping trajectory of the locust-inspired jumping robot (jumping height and dis-tance) and the change of trunk posture are important per-formances of robot. The complete jumping sequence of the robot is shown in Fig. 19 (a). The trajectory of center of mass and rotation angle of trunk can be seen in Fig. 19 (b). Fig. 19 shows that the maximum jumping height of the ro-bot is 439.91 mm, and the maximum jumping distance is 949.01 mm. Although the trunk of the robot inevitably turns counterclockwise in the jumping process with the four-bar jumping leg, the robot rotates 305° in the air, which is con-sistent with the theoretical value (310°). The angle between the symmetry axis of the robot and the ground is 5.3°, and the robot almost turns a circle. So the robot can achieve good landing posture.

In the landing buffering process, whether the robot col-

(a)

(b)

Fig. 19. Motion law of jumping robot in the whole jumping process. (a) Jumping sequence; (b) Motion law of trajectory of center of mass and trunk posture of robot.

(a)

(b)

Fig. 20. Motion law of bio inspired jumping robot in the landing buff-ering process. (a) Buffering movement sequence; (b) Motion law of velocity along vertical direction and buffering distance of robot. lides with the ground and the change of velocity of the jumping robot are important criterion for judging the buffer-ing performance. The landing buffer of the locust-inspired jumping robot is shown in Fig. 20 (a). The distance between the lowest point of the robot and the ground and the veloci-ty along vertical direction are shown in Fig. 20 (b). In the buffering process, the velocity is zero when the robot moves down to the lowest point. Due to the action of the leg springs, the robot moves upwards for secondary buffer. Then the robot achieves a stable posture after third buffer. The experiment results show that the minimum distance between robot and the ground is 2.34 mm, and the trunk will not collide with the ground. Because the ground is smooth, the movement mode with the end of buffering legs sliding on the ground is used in the buffering process. The stability area is larger with the end of the buffering legs sliding. So the robot will not overturn. 5 Discussions

Table 5 lists the jumping performance of several existing micro bio-inspired jumping robots.

The accurate take-off parameters and the stability of flight and landing phases are one of the key points in the research of pause-and-leap jumping robots. Compared with existing bio-inspired jumping robots shown in Table 5, the biggest advantage of the jumping robot designed by the guidance of the modeling method proposed in this paper is that it can acquire accurate take-off parameters and achieve stable flight and landing buffering process simultaneously. This is because the dynamic models for the take-off phase,

Y. Qi et al. / Journal of Mechanical Science and Technology 00 (2019) 0000~0000

flight phase and landing buffering phase are established combining with Lagrange/Newton-Euler dynamic modeling methods and conservation of momentum moment, and the relationship between structural parameters and jumping performance is revealed. In particular, for the flight process of the robot, according to Eq. (46) and the dynamics analy-sis result of take-off phase, the total inertia moment of the robot at the end of take-off process is not zero, and the ro-bot will inevitably turn over in the flight phase. Therefore, the position of the center of mass in coordinate system Ot-XtYt of the robot can be adjusted according to the regulation showed in Fig. 13 in order to make the robot overturn 310° theoretically in the flight phase and overturn almost a circle. Thus the robot can land stably. In addition, some jumping robots adjust their posture in air through some auxiliary structures, such as wings [8], [32], tails [33]-[35] and some other mechanisms [36], [37]. The robot designed in this paper does not have any auxiliary structure, which makes structure of the robot simple, and allows the robot prepare for the next jumping process conveniently. Furthermore, since the four-bar jumping leg mechanism inevitably over-turns in the flight phase, the research on the six-bar or eight-bar jumping leg can be carried out later to make the robot

Table 5 Performance comparison of existing micro bio-inspired jumping robots

Robot Leg mecha-nism

Controllable take-off?

Good landing buffer?

GRILLO III [28] Four-bar mechanism Yes No research

Jumping robot (Konkuk University)

[9]

Similar-bionic leg No No

Jumping robot (Ort Braude College)

[7]

Bionic serial leg No research No research

Flea- inspired jump-ing mechanism[29]

Four-bar mechanism No No

Jumping robot (Seoul National University)[30]

Two links mechanism

Yes (Jump along the vertical direction)

No research

Jumping Robot (Seoul National University)[31]

Two links mechanism

Yes (Jump along the vertical direction)

No research

Locust-inspired jumper [8]

Bionic serial leg No

Yes (Wing as an auxil-iary mechanism)

Bio-inspired jumping robot studied in this

paper

Four-bar mechanism Yes

Yes (No other auxiliary

institutions)

do not turn over in the flight phase.

6. Conclusions

In order to solve the problem that most of the existing pause-and-leap jumping robots cannot change the jumping parameters for take-off phase accurately and have poor sta-bility for flight and landing buffering phases, the dynamic modeling methods for take-off, flight and landing buffering phases of the robot are put forward in this paper, and the experiment is conducted to verify the rationality of the modeling methods.

(1) The take-off and buffering phases of the locust is ob-served. Based on the take-off and buffering mechanism, a one-DOF four-bar jumping leg and a two-DOF buffering leg of the robot are designed.

(2) The modeling methods for a whole jumping process of the robot are proposed combining with La-grange/Newton-Euler dynamic modeling methods and con-servation of momentum moment. For different jumping phases, the performance analysis is conducted, and the cou-pling relationship between the parameters and the perfor-mance is obtained.

(3) Experiment results show that the prototype developed by the guidance of the modeling methods can achieve con-trollable take-off process and stable flight and landing buff-ering process simultaneously without adding auxiliary mechanism.

The present study provides a theoretical basis and tech-nical guidance for the design of pause-and-leap bio-inspired jumping robots with good jumping performance.

Further extensions to this work include the design of new mechanisms (six-bar or eight-bar one DOF leg mechanism) to prevent robots from overturning in the air. Then the con-trol methods can be studied. Acknowledgments

The authors would like to acknowledge the support from the National Natural Science Foundation of China (Grant No. 51805010), China Postdoctoral Science Foundation funded project (2018M630051), and Beijing Postdoctoral Research Foundation (2018-ZZ-034). References

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Yang Qi is a postgraduate student at the College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, China. He re-ceived his bachelor degree from Beijing University of Technology, China, in 2017. His research interests include bio-inspired jumping robot.

Zhang Ziqiang is currently a post-doctoral at College of Mechanical En-gineering and Applied Electronics Technology, Beijing University of Technology, China. He received his doctorate degree from Beihang Uni-versity, China, in 2017. His research interests include bionic robot.