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Bionic Mechanism and Kinematics Analysis of Hopping Robot Inspired by Locust Jumping
Journal of Bionic Engineering 8 (2011) 429–439
Bionic Mechanism and Kinematics Analysis of Hopping Robot Inspired by Locust Jumping Diansheng Chen, Junmao Yin, Kai Zhao, Wanjun Zheng, Tianmiao Wang Robotic Institute, Beihang University, Beijing 100191, P. R. China
Abstract A flexible-rigid hopping mechanism which is inspired by the locust jumping was proposed, and its kinematic characteristics were analyzed. A series of experiments were conducted to observe locust morphology and jumping process. According to classic mechanics, the jumping process analysis was conducted to build the relationship of the locust jumping parameters. The take-off phase was divided into four stages in detail. Based on the biological observation and kinematics analysis, a mechanical model was proposed to simulate locust jumping. The forces of the flexible-rigid hopping mechanism at each stage were analyzed. The kinematic analysis using pseudo-rigid-body model was described by D-H method. It is confirmed that the proposed bionic mechanism has the similar performance as the locust hind leg in hopping. Moreover, the jumping angle which decides the jumping process was discussed, and its relation with other parameters was established. A calculation case analysis corroborated the method. The results of this paper show that the proposed bionic mechanism which is inspired by the locust hind limb has an excellent kinematics performance, which can provide a foundation for design and motion planning of the hopping robot. Keywords: hopping robot, flexible-rigid mechanism, bionic mechanism, kinematics Copyright © 2011, Jilin University. Published by Elsevier Limited and Science Press. All rights reserved. doi: 10.1016/S1672-6529(11)60048-6
1 Introduction Hopping robot belong to a kind of ground mobile robot. Such robot must have the ability of adaption to unstructured environment, especially to avoid dangers and overleap obstacles agilely and precisely. Many researchers conducted a variety of studies on ground mobile robots with different forms of movement. At present, the terrain adaptability of ground mobile robot has already been largely improved, but the obstacle negotiating performance is still unsatisfactory due to the limitation of the locomotion mode. Biologic hopping locomotion can achieve a certain height and distance in a moment, which enables animals to overcome obstacles and uneven terrains. Therefore, the study on hopping robot, which developed from nature observation, has received much attention in recent years. The exploration of hopping movement will expand the application fields of mobile robot. Raibert and Pratt developed one-legged, biped, and quadruped running and hopping robots based on Raibert’s pneumatic model, all of which have telescopCorresponding author: Diansheng Chen E-mail: [email protected]
ing legs and can achieve different running motions[1–4]. Fiorini et al. constructed three series of hopping robots based on single or six bars spring mechanism. Their robots are characterized by discontinuous motion, i.e. the robots need to stop after each hop to adjust posture, get recharged and orientate themselves[5,6]. Brown and Zeglin proposed a bow leg hopper which has a bow-like resilient leg and adopt a continuous hopping mechanism[7,8]. All of these robots depend on traditional spring energy storage mechanism, which is considered to lead to a low efficiency. Consequently, a novel model was established by taking inspirations from animals, whose ingenious physical structure and locomotion mode have become gradually adaptive to unstructured and unknown environment after billions of years of evolution. Therefore, the study of bionic hopping movement mechanism has been a hot spot in recent years. By incorporating biologically-inspired dynamics into design, Hiraguchi et al. developed a cricket robot with legs actuated by artificial muscles[9]. Kovac et al. developed an EPFL jumper looking like a locust. The versatile jumping mechanism
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can negotiate obstacles with more than 27 times of its own size (5 cm), and the takeoff angle, jumping height and force profile are adjustable[10–12]. Niiyama et al. developed a pneumatically actuated bipedal robot named Mowgli which can hop to a height of over 50% of its size and land steadily[13]. In view of the overall situation, these studies use traditional driver and elastic spring to mimic the jumping performance of insect or animal. This may lead to a low efficiency of energy storage which is caused by the limitation of traditional mechanism and the linearity of drivers and springs. Therefore, it is significant to innovate new mechanisms with high efficiency of energy storage. In this paper, we simplify and establish a hopping mechanism with flexible-rigid model inspired by locust hind leg, and analyze the statics and takeoff characteristics. The proposed mechanism is proved to have a high efficiency. The results of this paper can provide a foundation for the design and kinematics plan of hopping robot.
2 Locust observation and analysis The jumping capabilities of four species are listed in Table 1[14]. Obviously, locust has excellent hopping performance. By taking experiment condition into account, we chose locust as the research object. It is not only because the locust has appropriate size, which makes image capture and observation more easily, but also the hind leg of locust keeps in vertical plane in hopping locomotion, which makes the study more feasible.
2.1 Locust morphology A set of experiments were designed to investigate the morphology of locust. Three locusts, Locusta migratoria manilensis, were captured from the lawn of our campus. As Fig. 1 shows, a locust has 3 pairs of limbs, which are different in size. The fore limb and the mid limb used for walking are small and weak, while the hind limb is large and strong enough to achieve jumping locomotion. A hind leg of locust consists of 4 distinct sections: coxa, femur, tibiae and tarsus, which are connected by joints. The dimension parameters and mass of three locusts were measured (Table 2). The centroid is measured between the base of mid limb and hind limb, and bilaterally symmetrical from the vertical view. A hopping process can be decomposed into several phases including walking (or running), taking-off, flying certain height and distance, and landing on the ground. Jumping locomotion depends on the cooperation of skeleton, muscle, tendon and femur. Skeleton is equal to lever, and joint is fulcrum. Muscle and tendon offer power. The folded hind limbs stretch quickly that thrust against the ground to actuate the locust body in jumping. In addition, the light and rigid tibia and the flexible tarsus also have important effects. As Heitler[15] proposed, a remarkable jump should satisfy the requirement that the legs must thrust against the ground with a lot of force and develop this force quickly. If that is too slow, legs only stretch before the force thrusting against the ground reaches the maximum[15]. The analysis of the thrust will be presented in Section 4.
Table 1 The jumping capabilities of several species Frog
Locust
15
23
30
300
Distant (cm)
30
50
70
900
200
12
22
5
1.2
3.2
Compared to its size (times) í1
Initial velocity (m·s )
Kangaroo
°
®
° ® ° ¯
Flea
® ¯
Parameters Height (cm)
°¯
Fig. 1 Locust morphology. Table 2 Dimension parameters of locusts Locust 1
Locust 2
Mass (g)
0.22
0.21
0.23
Body length (mm)
33.25
31.29
34.05
Femur
Tibiae
Tarsus
Femur
Hind limb (mm)
13.27
11.97
3.10
13.03
Mid limb (mm)
5.10
4.61
2.34
Fore limb (mm)
4.55
3.98
1.80
Locust 3
Tarsus
Femur
Tibiae
Tarsus
3.64
13.25
11.99
3.55
5.17
Tibiae 12.1 6 4.80
2.91
5.19
4.82
2.85
4.45
3.56
2.49
4.65
4.21
2.41
Chen et al.: Bionic Mechanism and Kinematics Analysis of Hopping Robot Inspired by Locust Jumping
Fig. 2 shows the knee of a locust hind limb. The coxa is a 2 DOF joint, which can achieve the flexion/extension and adduction/abduction of the femur with 2 pairs of muscle. The tibia extensor and flexor contribute greatly to jumping, which makes knee stretch and contract respectively. Flexor tendon grows round the femur lump (equal to beam point). As shown in Fig. 2, the dense black half-moon shaped region is the external view of a pair of springs, and Heitler[15] named it as semi-lunar process. This process is only found in hind legs[15]. The lever ratio of extensor to tibia terminal is approximately 1:35. In addition, the flexible multi-tarsus is attached with flexible suckers which can prevent slip and grasp the ground. All the structures are specialized. Therefore, leg structure of locust has jump predominance.
431
Fig. 3 Experimental platform.
200 150 100 50 0
0
50
100
150
200 x(t) (mm)
250
300
350
400
y (mm)
Fig. 4 The images of the jump phase.
Fig. 2 The knee of locust hind limb[12].
2.2 Observation of jumping An experimental apparatus is set up to observe the jumping process, especially for the takeoff phase. As shown in Fig. 3, the apparatus consists of a high-speed camera (FASTCAM 1280 PCI, Photron, USA), a light source, PC, observation table and background (coordinate paper). The high-speed camera can capture rapid movement with 500 fps (frame per second), so that the whole jumping process can be recorded and stored in PC. During experiment, the temperature was kept at about 20 ÛC to ensure the locust in an active status. The images of takeoff trajectory and locust posture are shown in Fig. 4. The time interval of each image was 4 ms. The trajectory curve is shown in Fig. 5. 2.2.1 Jump process analysis The body center of locust moves as projectile. As we know, in the analytic form of classical mechanics, the
200 180 160 140 120 100 80 60 40 20 0
0
50
100
150
200 250 x (mm)
300
350
400
450
Fig. 5 The trajectory curve of jumping.
projectile is well described by law of particle movement and law of conservation of energy. Thus the locust jumping performance can be described as Eq. (1). x v0 t cos D , ° ® 1 2 °¯ y v0 t sin D 2 gt ,
(1)
where v0 denotes initial velocity, Į is takeoff angle, t is jumping time, x and y denote the height and horizontal distance respectively, g is gravity. Thus, the maximum jump height ymax and distance xmax are given by ° xmax ® °¯ ymax
(v02 sin 2D ) g , (v0 sin D )2 2 g .
(2)
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Eq. (2) shows that the takeoff speed and angle affect the jump height and distance. With a certain speed, it can be proved that the jumping achieves xmax when the takeoff angle Į is about 45Û. As the function of projectile trajectory is given by y
x tan D
350
(3)
250 y = y(t)
200 150 100 50
This derivation ignores the air resistance, so it usually applies to the case with a relatively low speed. However, the takeoff speed of locust may be very high that it is uncertain whether the locust case adapts the equations. Therefore, to get a precise result, the derived curve F(x) = yƍ(x) and second derived curve G(x) = yƎ(x) of fitting trajectory curve are given in the Fig. 6. It shows the case that the takeoff angle is 45Û and the jumping process corresponds to the law of particle movement. Therefore, based on Eq. (3), the conservation of energy is given by Eq. (4) and energy loss Es reveals the error in Fig. 6.
(a)
0
0
V
3.0
F(x), G(x)
150 t (ms)
200
250
sqrt(Vx2 , Vy2 )
2.0 Vx = xƍ(t)
1.5 1.0
Vy = yƍ(t)
0.5 (b) 0.0
0
0.005
Fig. 6 The derived curve F(x) and second derived curve G(x) of trajectory curve.
100
2.5
(4)
where v is defined to be the velocity when the height is h, m denotes the body mass. Fig. 7 shows the curves of jumping height and horizontal distance vs. the time. According to the takeoff angle, initial velocity and Eq. (2), the ymax is calculated to be 216.2 mm, and then the loss of energy Es is also calculated. Locust jumping is approximately a projectile motion, and the trajectory is a quadratic curve. The energy loss in jumping process is tested by experiments. The experimental analysis on the locust jumping parameters and energy loss can provide reference basis for future study on kinematics planning of bio-hopping robot.
50
3.5
50
100
150 t (ms)
200
250
300
The acceleration in the x direction
0.000 Acceleration ( mm·msí2)
1 2 1 mv0 3 mgh ! mv 2 ! Es , 2 2
x = x(t)
300 s ( mm )
1 gx 2 . 2 v02 cos 2 D
400
í0.005 í0.010
The acceleration in the y direction
í0.015 í0.020 í0.025 í0.030 The sum acceleration in the x and y direction
í0.035 í0.040
0
50
100
150 t (ms)
(c) 200
250
300
Fig. 7 The relationship of the jumping height and horizontal distance vs. time. The takeoff initial speed is 3.2 m·sí1. The horizontal speed is reduced slowly, but the vertical velocity decreases rapidly. The air resistance makes the acceleration decreases rapidly besides gravity.
2.2.2 Division of takeoff phase From rest to takeoff, the joint angles are measured using the captured images in Fig. 8. The angles data of each limb are given in Table 3. The takeoff phase can be disintegrated into 4 stages with movement coordination and angles of hind legs joints, as shown in Fig. 9. Stage 1: Initial contraction and flexion of joints. The knees of hind legs bend to a location, mainly for adjusting the leg posture, and the body is up and down like simplified five linkages slider-crank mechanism. Full flexion is an essential pre-requisite for jumping, in which the flexor and extensor muscles contract together.
Chen et al.: Bionic Mechanism and Kinematics Analysis of Hopping Robot Inspired by Locust Jumping
433
Fig. 8 The joint angles calibration on the image. Table 3 The joints angles in takeoff process Video time
Body-horizontal
Tarsus-horizontal
Tibiae-tarsus
Femur-Tibiae
Body-femur
0 ms
10.47
0.00
29.39
5.91
146.05
4 ms
10.32
0.00
27.09
3.95
146.55
8 ms
11.10
0.00
26.54
4.16
146.82
12 ms
11.24
0.00
25.89
4.01
146.87
16 ms
11.38
0.00
24.46
3.72
145.88
20 ms
11.37
0.00
26.52
3.87
145.96
24 ms
11.60
0.00
26.45
4.03
145.96
28 ms
12.70
0.00
28.94
5.18
143.53
32 ms
12.70
0.00
40.27
7.09
134.11
36 ms
8.71
0.00
52.22
24.60
143.67
40 ms
í3.18
0.00
73.55
71.68
í178.06
44 ms
í12.44
40.49
147.94
142.63
í132.38
48 ms
í22.75
46.12
142.75
135.63
í118.24
52 ms
í11.03
56.03
170.77
143.07
í140.64
Fig. 9 The 4 stages of locust takeoff.
Stage 2: Contraction and coordination of the flexor and extensor muscles. In this stage, the leg extension is triggered by a sudden relaxation of the flexor muscle, in which the flexor muscles extend rapidly and forcefully. The joints of hind legs move coordinately and tarsal limb thrusts against ground to produce an initial speed of body with certain direction using the energy stored in semi-lunar. Stage 3: Prolonging the thrust and enhancing the jumping. When the joint is extended to a certain extent, the body attains a certain speed. Then the flexible tarsal begin to deformation and stretch, so that keep the limb in
contact with the ground, which makes the hind legs thrust continually, and enhances the effect of jumping. Stage 4: Takeoff from the ground. During takeoff, the initial full fold of hind legs is prerequisite; the triggering of the flexor muscle and coordinated movement at the first acceleration phase decide whether the jumping is successful or not; while the elastic deformation of tarsal leg makes the second acceleration and prolongs the thrust. Finally the body takes off from the ground in accordance with regular exercise determined by Eq. (1). Based on the physiological mechanism of locust movement, the flexible-rigid mechanical model is built and the force/torque analysis of joints is completed.
3 Modeling and analysis of bionic mechanism inspired by locust 3.1 A mechanical model of locust Based on the locust morphology and jumping observation, a mechanical model is put forward to simulate locust jumping. From the viewpoint of jump function, much importance is attached to the design of hind leg
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bionic mechanism. Fig. 10 shows the model of the hopping robot considered in this paper. Considering that the semi-lunar process of knee joint is a flexible structure, it is replaced with a linear spring to store energy. Moreover a flexible link takes the place of the tarsus. The analysis of kinematic properties will be described in section 4.
ijE į
ijF
F
Fe
E
Fs
OL
Ff
lEF lDE
ș
D įD
Extensor muscle
Tendon Femur
Joint
y
Extensor muscle for pretarsus
Trochanter
lCD x o
Tibiae Coxa
Fg
Pretarsus
A Tarsus
Fm
C
Ffr
B
Fig. 11 The free-body diagram of locust knee and tibiae. Fe
Fs
Ff M1
M0 Fg
Ffr
Fig. 10 The model of the hopping robot.
3.2 The statics and performance analysis of hind leg mechanism Statics studies the relationship between forces when objects keep balance. Considering there is a flexible deformation of the tarsal, in this section we establish a rigid-flexible model, conducts statics and dynamics analyses, and discusses the mechanics performance. As shown in Fig. 10, a hind leg mechanism is built in accordance with the structure and function of locust hind legs. In Fig. 10, the femur is fixed. And the free-body diagram of locust knee and tibiae is shown in Fig. 11. In Fig. 11, points A, B, C, D, E, F, OL denote the front toe, the contact point with ground, the pastern, the action point of knee extension force, the connection point of the energy storage spring and tibia, the action point of knee flexion force and the supporting point of knee, respectively; Ffr and Fg are respectively defined as the friction and reaction forces with ground; lCD, lDE, lEF are the lengths of CD, DE, EF; įD, ijE are the constant angles between CD and DE, and between EF and DE correspondingly; and į is the angle between EF and
vertical direction. In addition, mt is the mass of free-body; vf is the speed; ijF and ș are the angles between Fe and EF, Ff and DE, respectively. In the stage of contraction, the knee flexion force Ff is generated by the contraction of tibia flexion muscle, meanwhile the tibia extension muscle stretches which causes the extension force Fe. Therefore, Fe is much smaller than Ff. According to equilibrium relation of force and torque, the force balance equations of tibia in X and Y directions are written as Fe sin(G M F ) Ff sin(M E G T ) mt vfc Fs Ffr cos(M E G G D ),
(5)
Fe cos(G M F ) Ff cos(M E G T ) Fg cos(M E G G D ), (6) where Fs denote the spring force. The torque balance equation is given by Fe lEF sin M F Ffr (lDE cos(M E G ) lCD cos(M E G G D )) Ff lDE sin T
M E Z c, (7)
where ME denotes the torque of point E, and Ȧ is angular speed. After rearrangement we can derive Ff as Ff
1 Fe lEF sin M F Ffr (lDE cos(M E G ) lDE sin T
lCD cos(M E G G D )) M E Z c .
(8)
For the convenience of analysis, the above equation is rewritten as
Chen et al.: Bionic Mechanism and Kinematics Analysis of Hopping Robot Inspired by Locust Jumping
Ff
Fe lEF sin M F Ffr lDE cos(M E G ) lDE sin T lDE sin T Ffr lCD cos(M E G G D ) M EZ c . lDE sin T lDE sin T
(9)
At the early stage of contraction, Fe and Ȧƍ are small. In the process of contraction, ș and įE increase with the decrease in ijF. What more, lEF is shorter than half of lDE. Consequently, the first term of Eq. (9) can be neglected so as to ensure that the flexion muscle is contracted. The last term also can be ignored for the reason that the contraction phase is very slow and tibia is thin and light. From the second and the third terms in Eq. (9), it can be seen that Ff mainly overcomes Ffr, the friction force between tarsus and the ground. At this stage, locust’s body is mainly supported by fore and mid limbs, so Ffr, which is proportional to Fg, is bit small. In a word, a small knee flexion force is able to fulfill the motion. In addition, no external force acts on tarsus. Thus, flexible deformation is relatively slight enough to be overlooked. The coxa and femur are connected by a hinge with 2 DOF. In case of į > 0, femur move forward, otherwise it moves backward. Just as the analysis in section 2.2.2, the initial bending contraction is the precondition for hopping. After that, the stage of compressing spring is to store energy. At the beginning, ijF decreases to the minimum and ș increases to the maximum. Then, Fe increases while ijF and ș keep constant. According to Eq. (5), Fs increases and the spring contracts to afford force enough to keep balance. At the same time, the energy for hopping is stored. Meanwhile, the angle between the tibia and femur changes from 25Û to approximately 0Û. The above analysis is in accordance with the biology and morphology study of locusts in Ref. [16]. 3.3 Analysis of parameters at takeoff phase As is analyzed in section 2.2.1, the jumping performance directly depends on the takeoff angle and initial velocity. Therefore, joints coordination and takeoff posture are two key parameters. As shown in Fig. 12, we define Ci, Ei, Gi, Oi (i = 0, 1, …) as the points of tarsus,
'LGx 't
c LGx
435
knee, coxa and centroid at position i, and use iji, și, įi (i = 0, 1, …) to denote the angles of joints C, E and G. C0, E0, G0, and O0 represent the original posture.
M C0
M C1
Fig. 12 The takeoff posture of the rigid bionic mechanism.
Then the velocity of the centroid VO can be expressed by JJJJK VO vOx i vOy j or VO VG ZOG u Gi Oi , (10) where VG denote the velocity of the coxa. The angular JJJJK velocity ȦOG is generated by torque of coxa. The Gi Oi is a vector from coxa point Gi to centroid Oi, which influences jumping angle of the centroid and the jumping performance. VG can be calculated by VG
'H Gy 'LGx i j, 't 't
(11)
where ǻt is the corresponding change of time; ǻLGx and ǻHGy are the changes of the distance and height of point G, which can be respectively derived as 'LGx
'H Gy
lCE M1 sin(
M1
M0 ) 2 T T lEG (T1 T 0 ) sin(G 0 1 0 ), 2 lCE M1 cos(
M1 2
M0 )
lEG (T1 T 0 ) cos(G 0
T1 T 0 2
),
(13)
where lCE, lEG denote the length of CE, EG correspondingly. When ǻt is small enough, the difference equations of Eqs. (12) and (13) can be derived as
M 1 M0 ) lCE M1 M1c cos( 1 M0 ) 2 2 2 T1 T 0 1 T T ) lEG (T1 T 0 ) (T1 T 0 )c cos(G 0 1 0 ), lEG (T1 T 0 )c sin(G 0 2 2 2
lCE M1c sin(
(12)
M1
(14)
Journal of Bionic Engineering (2011) Vol.8 No.4
436 'H Gy
c H Gy
't
M 1 M0 ) lCE M1 M1c sin( 1 M0 ) 2 2 2 T1 T 0 1 T T ) lEG (T1 T 0 ) (T1 T 0 )c sin(G 0 1 0 ). lEG (T1 T 0 )c cos(G 0 2 2 2
lCE M1c cos(
M1
The control of direction relies on the adjustment of the reflection between ijƍ and șƍ. So the bearing of velocity at point G results in
DG
actan
c H Gy
actan
c LGx
'H Gy 'LGx
k1 k2 sin(G 0
T1 T 0
) 2 actan , T T k1 k2 cos(G 0 1 0 ) 2
lCE M1 sin(
M1
M0 ), k2 lEG (T1 T 0 ). 2 By Eq. (10), the center of mass off angle is
where k1
W i 3 actan
vOy vOx
Fa 2 (3l a ), (0 d a d l ), 6 EI
3.4 Advantages of bionic mechanism In summary, the rigid-flexible mechanism of hind leg has advantages in jumping movement. At takeoff preparation stage, a minor force Ff can finish the posture adjustment. At the energy storage stage, Ff and Fe compress the spring and keep the mechanism in balance. At the first acceleration stage, the knee extension torque generated by Fe keeps increase, meanwhile the torque generated by elastic force of spring has the same direction, which ensures the fast reaction. At the moment before jumping, the flexible deformation of tarsus provides a second acceleration, which prolongs contact time with ground and improve the energy efficiency.
4 Kinematic analysis
(i 3 0,1, 2,...).
All the analysis above is made on the assumption that tarsus is fixed to the ground. In fact, the tarsal limb gradually adjusts its location, and the contraction of tibia causes the tarsus getting close to the body. Then the tarsal foot gradually begins to play a major role in supporting the body, so as to get prepared for jump. At second stage of acceleration in takeoff phase, the torque produced by tarsus and its motion are considered making the greatest contribution to jump movement, as shown in Fig. 13. It should be noted that the two stage acceleration is not strictly separated by time, but only a matter of the influence on the jump. Based on the athletic physiology study of locusts, the tarsus can be approximately simplified to a cantilever beam. The deflection Y can be written as Y
4.1 Pseudo-rigid body modeling The model of unilateral hind limb and the body of locust is simplified as Fig. 14. Focused on motion in the sagittal plane, this section builds the D-H systems and analyzes the kinematics of flexible-rigid model with the method of pseudo-rigid body modeling. First, the model can be described by D-H method. The homogeneous transformations of linkages are given as
0 1
ªcos(T1 (t )) sin(T1 (t )) « sin(T (t )) cos(T (t )) 1 1 « 0 0 « «¬ 0 0
0 0 1 0
1 2
ªcos(T 2 (t )) sin(T 2 (t )) « sin(T (t )) cos(T (t )) 2 2 « 0 0 « 0 0 ¬«
0 lo12 º 0 0 » », 1 0 » 0 1 »¼
T
T
x (t ) º y (t ) » », 0 » 1 »¼
(16)
where E and I are the elastic modulus and inertia moment; F, a, l denote force, the force position and length of beam, respectively.
(a)
(b)
(c)
(d)
Fig. 13 The distortion and torque produced by tarsus.
(15)
Fig. 14 The D-H systems of the proposed mechanism.
Chen et al.: Bionic Mechanism and Kinematics Analysis of Hopping Robot Inspired by Locust Jumping
2 3
ªcos(ʌ T3 (t )) sin(ʌ T3 (t )) « sin(ʌ T (t )) cos(ʌ T (t )) 3 3 « 0 0 « «¬ 0 0
0 lo 23 º 0 0 » », 1 0 » 0 1 »¼
3 4
ªcos(ʌ T 4 (t )) sin(ʌ T 4 (t )) « sin(ʌ T (t )) cos(ʌ T (t )) 4 4 « 0 0 « 0 0 ¬«
0 lo 34 º 0 0 » », 1 0 » 0 1 »¼
T
T
4 5
T
0 i
T
T 12T " i 1iT
0 1
ª1 «0 « «0 « ¬0
0 0 lo 45 º 1 0 0 »» , 0 1 0 » » 0 0 1 ¼
ª nx ox «n o « y y «0 0 « ¬0 0
In Eq. (17) (x(t), y(t)) denote the centroid trajectory and și denotes the rotation angle of joint i. The lo(i)(i+1) is defined as the distance between two adjacent joints oi and oi+1. Then, it is significant to reconsider the elastic stretch of knee joint and the flexible deformation of tarsus (Fig. 14). The stiffness of the spring is k. Approximately, the tarsus is assumed to be equal cross section bar. Therefore, according to Eq. (16), the transformation equations should be modified as ªcos(ʌ T3 (t )) sin(ʌ T3 (t )) « sin(ʌ T (t )) cos(ʌ T (t )) 3 3 2 « T = 3 « 0 0 « 0 0 ¬ ª1 « «0 4 5T = « «0 « «¬0
0 0
0 lo 23 k 'x º » 0 0 » , (18) » 1 0 » 0 1 ¼
º » Fa (3l a ) » ». 6 EI » 0 » »¼ 1 lo 45
2
1 0 0 1 0 0
(Table 3). Therefore, the transformation from the joint space q(ș1, ș2, ș3, ș4) to motion space x of model is the motion equation x = x(q). Then the relationship between the generalized speed x i (including translational speed vi and angular velocity Ȧi) of each component and the generalized velocity q of joints is obtained as x 3 J (q)q , namely, ª vi º «Ȧ » ¬ i¼
0 pxi º 0 p yi »» (i 1, 2,...,5). (17) 1 pzi » » 0 1 ¼
(19)
Finally, the posture of each joint can be derived according to Eq. (17). 4.2 Description of limb posture and speed
It is easier to ascertain 01T according to the experimental result in section 2. 01T indicates the jumping performance such as height, horizontal distance and body posture. The relationship between the rotation angle of joint and time series has been measured
437
ª J l1 J l 2 «J ¬ a1 J a 2
ªT1 º « » " J li º «T2 » (i 1, 2,...,5), " J ai »¼ « # » «» «¬Ti »¼
where J(q) denotes the generalized velocity Jacobin. Based on the vector product method, because the model only consists of the rotary joints, Ji is calculated by Ji
ª zi u 0i R i pn º « ». zi «¬ »¼
(20)
4.3 Calculation case analysis From the experimental data in Table 3, the joint space can be got from takeoff preparation to takeoff instance. The trajectory curve in Fig. 5 indicates the centroid position (x(t), y(t)). Lo12 and law represent the lengths from centroid to femur and abdomen respectively, Lo12, Lo34, Lo45 represent the lengths of linkages between two adjacent joints, all of which can be obtained from Table 2. Assume that k is equal to 1 and joint shrinkage ǻx is 1.5×10í3 m. Moreover, the F, E and I are estimated to be 0.03 N, 1 MPa and 10í6, respectively. As Fig. 13 indicates, the force position is in the middle of the rod, that is a = 0.5lo45. The homogeneous coordinate transformation can be calculated through Eqs. (17) to (19). Then x(q) is acquired according to the following equation xi (n) 3 0iT (n, 4) (n 3 1, 2,3; i 3 1, 2,...,5) .
In MATLAB software, the position matrix can be worked out at any time. At 24 ms, for example, the position matrix x(q(24)) irrespective of flexibility and xƍ(q(24)) taking flexibility into account are as
x (q(24))
ª35.5000 «31.6765 « «19.4318 « «30.1404 «¬ 27.0404
2.8000 0 º 2.0311 0 »» 7.1457 0 » , » 1.7972 0 » 1.7983 0 »¼
Journal of Bionic Engineering (2011) Vol.8 No.4
438 ª35.5000 «31.6765 « «19.4332 « «30.1418 «¬ 27.0418
x c(q (24))
2.8000 0 º 2.0311 0 »» 7.1451 0 » . » 1.7967 0 » 1.7884 0 »¼
In consideration of flexibility, the posture with respect to time is depicted in Fig. 15. The flexible-rigid mechanism has n = 5 joints, and the i pn0 3 0i R i pn of joint i (i 3 1, 2,..., n) is calculated in 24 ms as below p = ª¬ 1 p50
2
= ª¬ 10 R 1 p5
p50 0 2
3
p50
4
p50
5
R 2 p5
0 3
R 3 p5
p50 º¼ 0 4
R 4 p5
0 5
R 5 p5 º¼
ª34.2523 28.4451 20.5774 30.1423 27.0423º = «« 9.7422 10.3340 2.2907 1.7877 1.7803 »» . «¬ 0 »¼ 0 0 0 0
Then, according to Eq. (20), at this time, the velocity Jacobin matrix can be expressed by
J
1.7803 º ª 9.7422 10.3340 2.2907 1.7877 « 34.2523 28.4451 20.5774 30.1423 27.0423» « » « 0 » 0 0 0 0 « ». 0 0 0 0 « 0 » « 0 » 0 0 0 0 « » 1 1 1 1 «¬ 1 »¼
Similarly, the Jacobin matrix at any time can be calculated and then the linear velocity of each joint can be computed with the angular velocity. 40
Acknowledgement This work is financially supported by the National Natural Science Foundation of China (Grant No. 51075014).
References [1]
http://www.ai.mit.edu/projects/leglab/robots/robots.html [2]
Kedzior K (eds). Theory and Practice of Robots and Ma-
t = 52 ms
nipulators, Proceedings of RoManSy'81, 1983, 352–367. [4]
y (mm)
[5]
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Conference, Snowmass, CO, USA, 1999, 153–158.
t = 40 ms
[6]
Fiorini P, Burdick J. The development of hopping capabili-
[7]
Brown B, Zeglin G. The bow leg hopping robot. Proceed-
t = 36 ms
5
ties for small robots. Autonomous Robot, 2003, 14, 239–254.
t = 0, 4, Ă, 32 ms 20
30
40 x (mm)
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Fiorini P, Hayati S, Heverly M, Gensler J. A hopping robot for planetary exploration. Proceedings of IEEE Aerospace
t = 44 ms
10
Pratt G. Legged robots at MIT – What’s new since Raibert. Clawer News, 2000, 4, 4–7.
t = 48 ms 20
10
Raibert M H. Dynamic stability and resonance in a one-legged hopping machine. In: Morecki A, Bianchi G,
25
0
Raibert M H. Legged Robots that Balance, The MIT Press, Cambridge, MA, USA, 1985.
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0
MIT Leg Laboratory. Running & Hopping Robots, [2011-3-10],
[3]
Centre of mass Joint point Body and limbs Abdomen
35
ture and speed on jumping performance. A flexible-rigid mixed model was established to simulate locust jumping. Then the statics and the motional characteristics of flexible-rigid mechanism at takeoff phase were analyzed. At the same time, the kinematics of flexible-rigid mixed model was described with the method of pseudo-rigid body modeling, which achieves a unified description from the joint space to posture space of the takeoff process. The above analysis provides a foundation for the further design and kinematic planning of flexible hopping robot. The highlight of this paper is the rigid-flexible model and its kinematics analysis, which is the groundwork for further study of rigid-flexible mixture dynamics. Further study will concentrate on the relationship between force and the jumping performance.
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80
Fig. 15 The posture with respect to time.
ings of the 1998 IEEE International Conference on Robotics and Automation, Leuven, Belgium, 1998, 781–786. [8]
Proceedings of the 1998 IEEE International Conference on
5 Conclusions By means of experiment, this paper studied the jumping motion of locust and the effect of takeoff pos-
Zeglin G, Brown B. Control of a bow leg hopping robot. Robotics & Automation, Leuven, Belgium, 1998, 793–798.
[9]
Hiraguchi T, Yamaguchi T, Takahata M. Mechanoreceptors involved in the hindwing-evoked escape behaviour in
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and Systems, St. Louis, USA, 2009, 583–588. [13] Niiyama R, Nagakubo A, Kuniyoshi Y. Mowgli: A bipedal
[10] Kovac M. Bio-Inspired Jumping Locomotion for Miniature
jumping and landing robot with an artificial musculoskeletal
Robotics, PhD Thesis, Swiss Federal Institute of Technology,
system. IEEE International Conference on Robotics and
Switzerland, 2010.
Automation, Roma, Italy, 2007, 10–14.
[11] Kovac M, Fuchs M, Guignard A. A miniature 7g jumping robot. Proceedings of the 2008 IEEE International Conference on Robotics and Automation, Pasadena, CA, USA, 2008, 373–378.
[14] Qin J D. Animal Locomotion, Tsinghua University Press, Beijing, China, 2000. (in Chinese) [15] Heitler W J. How Grasshoppers Jump. [2011-3-10], http://www.st-andrews.ac.uk/~wjh/ jumping
[12] Kovac M, Schlegel M, Zufferey J-C, Floreano D. A minia-
[16] Heitler W J. The locust jump. III. Structural specializations
ture jumping robot with self-recovery capabilities. The 2009
of the metathoracic tibiae. Journal Experimental Biology,
IEEE/RSJ International Conference on Intelligent Robots
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Bionic Mechanism and Kinematics Analysis of Hopping Robot Inspired by Locust Jumping Diansheng Chen, Junmao Yin, Kai Zhao, Wanjun Zheng, Tianmiao Wang Robotic Institute, Beihang University, Beijing 100191, P. R. China
Abstract A flexible-rigid hopping mechanism which is inspired by the locust jumping was proposed, and its kinematic characteristics were analyzed. A series of experiments were conducted to observe locust morphology and jumping process. According to classic mechanics, the jumping process analysis was conducted to build the relationship of the locust jumping parameters. The take-off phase was divided into four stages in detail. Based on the biological observation and kinematics analysis, a mechanical model was proposed to simulate locust jumping. The forces of the flexible-rigid hopping mechanism at each stage were analyzed. The kinematic analysis using pseudo-rigid-body model was described by D-H method. It is confirmed that the proposed bionic mechanism has the similar performance as the locust hind leg in hopping. Moreover, the jumping angle which decides the jumping process was discussed, and its relation with other parameters was established. A calculation case analysis corroborated the method. The results of this paper show that the proposed bionic mechanism which is inspired by the locust hind limb has an excellent kinematics performance, which can provide a foundation for design and motion planning of the hopping robot. Keywords: hopping robot, flexible-rigid mechanism, bionic mechanism, kinematics Copyright © 2011, Jilin University. Published by Elsevier Limited and Science Press. All rights reserved. doi: 10.1016/S1672-6529(11)60048-6
1 Introduction Hopping robot belong to a kind of ground mobile robot. Such robot must have the ability of adaption to unstructured environment, especially to avoid dangers and overleap obstacles agilely and precisely. Many researchers conducted a variety of studies on ground mobile robots with different forms of movement. At present, the terrain adaptability of ground mobile robot has already been largely improved, but the obstacle negotiating performance is still unsatisfactory due to the limitation of the locomotion mode. Biologic hopping locomotion can achieve a certain height and distance in a moment, which enables animals to overcome obstacles and uneven terrains. Therefore, the study on hopping robot, which developed from nature observation, has received much attention in recent years. The exploration of hopping movement will expand the application fields of mobile robot. Raibert and Pratt developed one-legged, biped, and quadruped running and hopping robots based on Raibert’s pneumatic model, all of which have telescopCorresponding author: Diansheng Chen E-mail: [email protected]
ing legs and can achieve different running motions[1–4]. Fiorini et al. constructed three series of hopping robots based on single or six bars spring mechanism. Their robots are characterized by discontinuous motion, i.e. the robots need to stop after each hop to adjust posture, get recharged and orientate themselves[5,6]. Brown and Zeglin proposed a bow leg hopper which has a bow-like resilient leg and adopt a continuous hopping mechanism[7,8]. All of these robots depend on traditional spring energy storage mechanism, which is considered to lead to a low efficiency. Consequently, a novel model was established by taking inspirations from animals, whose ingenious physical structure and locomotion mode have become gradually adaptive to unstructured and unknown environment after billions of years of evolution. Therefore, the study of bionic hopping movement mechanism has been a hot spot in recent years. By incorporating biologically-inspired dynamics into design, Hiraguchi et al. developed a cricket robot with legs actuated by artificial muscles[9]. Kovac et al. developed an EPFL jumper looking like a locust. The versatile jumping mechanism
Journal of Bionic Engineering (2011) Vol.8 No.4
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can negotiate obstacles with more than 27 times of its own size (5 cm), and the takeoff angle, jumping height and force profile are adjustable[10–12]. Niiyama et al. developed a pneumatically actuated bipedal robot named Mowgli which can hop to a height of over 50% of its size and land steadily[13]. In view of the overall situation, these studies use traditional driver and elastic spring to mimic the jumping performance of insect or animal. This may lead to a low efficiency of energy storage which is caused by the limitation of traditional mechanism and the linearity of drivers and springs. Therefore, it is significant to innovate new mechanisms with high efficiency of energy storage. In this paper, we simplify and establish a hopping mechanism with flexible-rigid model inspired by locust hind leg, and analyze the statics and takeoff characteristics. The proposed mechanism is proved to have a high efficiency. The results of this paper can provide a foundation for the design and kinematics plan of hopping robot.
2 Locust observation and analysis The jumping capabilities of four species are listed in Table 1[14]. Obviously, locust has excellent hopping performance. By taking experiment condition into account, we chose locust as the research object. It is not only because the locust has appropriate size, which makes image capture and observation more easily, but also the hind leg of locust keeps in vertical plane in hopping locomotion, which makes the study more feasible.
2.1 Locust morphology A set of experiments were designed to investigate the morphology of locust. Three locusts, Locusta migratoria manilensis, were captured from the lawn of our campus. As Fig. 1 shows, a locust has 3 pairs of limbs, which are different in size. The fore limb and the mid limb used for walking are small and weak, while the hind limb is large and strong enough to achieve jumping locomotion. A hind leg of locust consists of 4 distinct sections: coxa, femur, tibiae and tarsus, which are connected by joints. The dimension parameters and mass of three locusts were measured (Table 2). The centroid is measured between the base of mid limb and hind limb, and bilaterally symmetrical from the vertical view. A hopping process can be decomposed into several phases including walking (or running), taking-off, flying certain height and distance, and landing on the ground. Jumping locomotion depends on the cooperation of skeleton, muscle, tendon and femur. Skeleton is equal to lever, and joint is fulcrum. Muscle and tendon offer power. The folded hind limbs stretch quickly that thrust against the ground to actuate the locust body in jumping. In addition, the light and rigid tibia and the flexible tarsus also have important effects. As Heitler[15] proposed, a remarkable jump should satisfy the requirement that the legs must thrust against the ground with a lot of force and develop this force quickly. If that is too slow, legs only stretch before the force thrusting against the ground reaches the maximum[15]. The analysis of the thrust will be presented in Section 4.
Table 1 The jumping capabilities of several species Frog
Locust
15
23
30
300
Distant (cm)
30
50
70
900
200
12
22
5
1.2
3.2
Compared to its size (times) í1
Initial velocity (m·s )
Kangaroo
°
®
° ® ° ¯
Flea
® ¯
Parameters Height (cm)
°¯
Fig. 1 Locust morphology. Table 2 Dimension parameters of locusts Locust 1
Locust 2
Mass (g)
0.22
0.21
0.23
Body length (mm)
33.25
31.29
34.05
Femur
Tibiae
Tarsus
Femur
Hind limb (mm)
13.27
11.97
3.10
13.03
Mid limb (mm)
5.10
4.61
2.34
Fore limb (mm)
4.55
3.98
1.80
Locust 3
Tarsus
Femur
Tibiae
Tarsus
3.64
13.25
11.99
3.55
5.17
Tibiae 12.1 6 4.80
2.91
5.19
4.82
2.85
4.45
3.56
2.49
4.65
4.21
2.41
Chen et al.: Bionic Mechanism and Kinematics Analysis of Hopping Robot Inspired by Locust Jumping
Fig. 2 shows the knee of a locust hind limb. The coxa is a 2 DOF joint, which can achieve the flexion/extension and adduction/abduction of the femur with 2 pairs of muscle. The tibia extensor and flexor contribute greatly to jumping, which makes knee stretch and contract respectively. Flexor tendon grows round the femur lump (equal to beam point). As shown in Fig. 2, the dense black half-moon shaped region is the external view of a pair of springs, and Heitler[15] named it as semi-lunar process. This process is only found in hind legs[15]. The lever ratio of extensor to tibia terminal is approximately 1:35. In addition, the flexible multi-tarsus is attached with flexible suckers which can prevent slip and grasp the ground. All the structures are specialized. Therefore, leg structure of locust has jump predominance.
431
Fig. 3 Experimental platform.
200 150 100 50 0
0
50
100
150
200 x(t) (mm)
250
300
350
400
y (mm)
Fig. 4 The images of the jump phase.
Fig. 2 The knee of locust hind limb[12].
2.2 Observation of jumping An experimental apparatus is set up to observe the jumping process, especially for the takeoff phase. As shown in Fig. 3, the apparatus consists of a high-speed camera (FASTCAM 1280 PCI, Photron, USA), a light source, PC, observation table and background (coordinate paper). The high-speed camera can capture rapid movement with 500 fps (frame per second), so that the whole jumping process can be recorded and stored in PC. During experiment, the temperature was kept at about 20 ÛC to ensure the locust in an active status. The images of takeoff trajectory and locust posture are shown in Fig. 4. The time interval of each image was 4 ms. The trajectory curve is shown in Fig. 5. 2.2.1 Jump process analysis The body center of locust moves as projectile. As we know, in the analytic form of classical mechanics, the
200 180 160 140 120 100 80 60 40 20 0
0
50
100
150
200 250 x (mm)
300
350
400
450
Fig. 5 The trajectory curve of jumping.
projectile is well described by law of particle movement and law of conservation of energy. Thus the locust jumping performance can be described as Eq. (1). x v0 t cos D , ° ® 1 2 °¯ y v0 t sin D 2 gt ,
(1)
where v0 denotes initial velocity, Į is takeoff angle, t is jumping time, x and y denote the height and horizontal distance respectively, g is gravity. Thus, the maximum jump height ymax and distance xmax are given by ° xmax ® °¯ ymax
(v02 sin 2D ) g , (v0 sin D )2 2 g .
(2)
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Eq. (2) shows that the takeoff speed and angle affect the jump height and distance. With a certain speed, it can be proved that the jumping achieves xmax when the takeoff angle Į is about 45Û. As the function of projectile trajectory is given by y
x tan D
350
(3)
250 y = y(t)
200 150 100 50
This derivation ignores the air resistance, so it usually applies to the case with a relatively low speed. However, the takeoff speed of locust may be very high that it is uncertain whether the locust case adapts the equations. Therefore, to get a precise result, the derived curve F(x) = yƍ(x) and second derived curve G(x) = yƎ(x) of fitting trajectory curve are given in the Fig. 6. It shows the case that the takeoff angle is 45Û and the jumping process corresponds to the law of particle movement. Therefore, based on Eq. (3), the conservation of energy is given by Eq. (4) and energy loss Es reveals the error in Fig. 6.
(a)
0
0
V
3.0
F(x), G(x)
150 t (ms)
200
250
sqrt(Vx2 , Vy2 )
2.0 Vx = xƍ(t)
1.5 1.0
Vy = yƍ(t)
0.5 (b) 0.0
0
0.005
Fig. 6 The derived curve F(x) and second derived curve G(x) of trajectory curve.
100
2.5
(4)
where v is defined to be the velocity when the height is h, m denotes the body mass. Fig. 7 shows the curves of jumping height and horizontal distance vs. the time. According to the takeoff angle, initial velocity and Eq. (2), the ymax is calculated to be 216.2 mm, and then the loss of energy Es is also calculated. Locust jumping is approximately a projectile motion, and the trajectory is a quadratic curve. The energy loss in jumping process is tested by experiments. The experimental analysis on the locust jumping parameters and energy loss can provide reference basis for future study on kinematics planning of bio-hopping robot.
50
3.5
50
100
150 t (ms)
200
250
300
The acceleration in the x direction
0.000 Acceleration ( mm·msí2)
1 2 1 mv0 3 mgh ! mv 2 ! Es , 2 2
x = x(t)
300 s ( mm )
1 gx 2 . 2 v02 cos 2 D
400
í0.005 í0.010
The acceleration in the y direction
í0.015 í0.020 í0.025 í0.030 The sum acceleration in the x and y direction
í0.035 í0.040
0
50
100
150 t (ms)
(c) 200
250
300
Fig. 7 The relationship of the jumping height and horizontal distance vs. time. The takeoff initial speed is 3.2 m·sí1. The horizontal speed is reduced slowly, but the vertical velocity decreases rapidly. The air resistance makes the acceleration decreases rapidly besides gravity.
2.2.2 Division of takeoff phase From rest to takeoff, the joint angles are measured using the captured images in Fig. 8. The angles data of each limb are given in Table 3. The takeoff phase can be disintegrated into 4 stages with movement coordination and angles of hind legs joints, as shown in Fig. 9. Stage 1: Initial contraction and flexion of joints. The knees of hind legs bend to a location, mainly for adjusting the leg posture, and the body is up and down like simplified five linkages slider-crank mechanism. Full flexion is an essential pre-requisite for jumping, in which the flexor and extensor muscles contract together.
Chen et al.: Bionic Mechanism and Kinematics Analysis of Hopping Robot Inspired by Locust Jumping
433
Fig. 8 The joint angles calibration on the image. Table 3 The joints angles in takeoff process Video time
Body-horizontal
Tarsus-horizontal
Tibiae-tarsus
Femur-Tibiae
Body-femur
0 ms
10.47
0.00
29.39
5.91
146.05
4 ms
10.32
0.00
27.09
3.95
146.55
8 ms
11.10
0.00
26.54
4.16
146.82
12 ms
11.24
0.00
25.89
4.01
146.87
16 ms
11.38
0.00
24.46
3.72
145.88
20 ms
11.37
0.00
26.52
3.87
145.96
24 ms
11.60
0.00
26.45
4.03
145.96
28 ms
12.70
0.00
28.94
5.18
143.53
32 ms
12.70
0.00
40.27
7.09
134.11
36 ms
8.71
0.00
52.22
24.60
143.67
40 ms
í3.18
0.00
73.55
71.68
í178.06
44 ms
í12.44
40.49
147.94
142.63
í132.38
48 ms
í22.75
46.12
142.75
135.63
í118.24
52 ms
í11.03
56.03
170.77
143.07
í140.64
Fig. 9 The 4 stages of locust takeoff.
Stage 2: Contraction and coordination of the flexor and extensor muscles. In this stage, the leg extension is triggered by a sudden relaxation of the flexor muscle, in which the flexor muscles extend rapidly and forcefully. The joints of hind legs move coordinately and tarsal limb thrusts against ground to produce an initial speed of body with certain direction using the energy stored in semi-lunar. Stage 3: Prolonging the thrust and enhancing the jumping. When the joint is extended to a certain extent, the body attains a certain speed. Then the flexible tarsal begin to deformation and stretch, so that keep the limb in
contact with the ground, which makes the hind legs thrust continually, and enhances the effect of jumping. Stage 4: Takeoff from the ground. During takeoff, the initial full fold of hind legs is prerequisite; the triggering of the flexor muscle and coordinated movement at the first acceleration phase decide whether the jumping is successful or not; while the elastic deformation of tarsal leg makes the second acceleration and prolongs the thrust. Finally the body takes off from the ground in accordance with regular exercise determined by Eq. (1). Based on the physiological mechanism of locust movement, the flexible-rigid mechanical model is built and the force/torque analysis of joints is completed.
3 Modeling and analysis of bionic mechanism inspired by locust 3.1 A mechanical model of locust Based on the locust morphology and jumping observation, a mechanical model is put forward to simulate locust jumping. From the viewpoint of jump function, much importance is attached to the design of hind leg
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bionic mechanism. Fig. 10 shows the model of the hopping robot considered in this paper. Considering that the semi-lunar process of knee joint is a flexible structure, it is replaced with a linear spring to store energy. Moreover a flexible link takes the place of the tarsus. The analysis of kinematic properties will be described in section 4.
ijE į
ijF
F
Fe
E
Fs
OL
Ff
lEF lDE
ș
D įD
Extensor muscle
Tendon Femur
Joint
y
Extensor muscle for pretarsus
Trochanter
lCD x o
Tibiae Coxa
Fg
Pretarsus
A Tarsus
Fm
C
Ffr
B
Fig. 11 The free-body diagram of locust knee and tibiae. Fe
Fs
Ff M1
M0 Fg
Ffr
Fig. 10 The model of the hopping robot.
3.2 The statics and performance analysis of hind leg mechanism Statics studies the relationship between forces when objects keep balance. Considering there is a flexible deformation of the tarsal, in this section we establish a rigid-flexible model, conducts statics and dynamics analyses, and discusses the mechanics performance. As shown in Fig. 10, a hind leg mechanism is built in accordance with the structure and function of locust hind legs. In Fig. 10, the femur is fixed. And the free-body diagram of locust knee and tibiae is shown in Fig. 11. In Fig. 11, points A, B, C, D, E, F, OL denote the front toe, the contact point with ground, the pastern, the action point of knee extension force, the connection point of the energy storage spring and tibia, the action point of knee flexion force and the supporting point of knee, respectively; Ffr and Fg are respectively defined as the friction and reaction forces with ground; lCD, lDE, lEF are the lengths of CD, DE, EF; įD, ijE are the constant angles between CD and DE, and between EF and DE correspondingly; and į is the angle between EF and
vertical direction. In addition, mt is the mass of free-body; vf is the speed; ijF and ș are the angles between Fe and EF, Ff and DE, respectively. In the stage of contraction, the knee flexion force Ff is generated by the contraction of tibia flexion muscle, meanwhile the tibia extension muscle stretches which causes the extension force Fe. Therefore, Fe is much smaller than Ff. According to equilibrium relation of force and torque, the force balance equations of tibia in X and Y directions are written as Fe sin(G M F ) Ff sin(M E G T ) mt vfc Fs Ffr cos(M E G G D ),
(5)
Fe cos(G M F ) Ff cos(M E G T ) Fg cos(M E G G D ), (6) where Fs denote the spring force. The torque balance equation is given by Fe lEF sin M F Ffr (lDE cos(M E G ) lCD cos(M E G G D )) Ff lDE sin T
M E Z c, (7)
where ME denotes the torque of point E, and Ȧ is angular speed. After rearrangement we can derive Ff as Ff
1 Fe lEF sin M F Ffr (lDE cos(M E G ) lDE sin T
lCD cos(M E G G D )) M E Z c .
(8)
For the convenience of analysis, the above equation is rewritten as
Chen et al.: Bionic Mechanism and Kinematics Analysis of Hopping Robot Inspired by Locust Jumping
Ff
Fe lEF sin M F Ffr lDE cos(M E G ) lDE sin T lDE sin T Ffr lCD cos(M E G G D ) M EZ c . lDE sin T lDE sin T
(9)
At the early stage of contraction, Fe and Ȧƍ are small. In the process of contraction, ș and įE increase with the decrease in ijF. What more, lEF is shorter than half of lDE. Consequently, the first term of Eq. (9) can be neglected so as to ensure that the flexion muscle is contracted. The last term also can be ignored for the reason that the contraction phase is very slow and tibia is thin and light. From the second and the third terms in Eq. (9), it can be seen that Ff mainly overcomes Ffr, the friction force between tarsus and the ground. At this stage, locust’s body is mainly supported by fore and mid limbs, so Ffr, which is proportional to Fg, is bit small. In a word, a small knee flexion force is able to fulfill the motion. In addition, no external force acts on tarsus. Thus, flexible deformation is relatively slight enough to be overlooked. The coxa and femur are connected by a hinge with 2 DOF. In case of į > 0, femur move forward, otherwise it moves backward. Just as the analysis in section 2.2.2, the initial bending contraction is the precondition for hopping. After that, the stage of compressing spring is to store energy. At the beginning, ijF decreases to the minimum and ș increases to the maximum. Then, Fe increases while ijF and ș keep constant. According to Eq. (5), Fs increases and the spring contracts to afford force enough to keep balance. At the same time, the energy for hopping is stored. Meanwhile, the angle between the tibia and femur changes from 25Û to approximately 0Û. The above analysis is in accordance with the biology and morphology study of locusts in Ref. [16]. 3.3 Analysis of parameters at takeoff phase As is analyzed in section 2.2.1, the jumping performance directly depends on the takeoff angle and initial velocity. Therefore, joints coordination and takeoff posture are two key parameters. As shown in Fig. 12, we define Ci, Ei, Gi, Oi (i = 0, 1, …) as the points of tarsus,
'LGx 't
c LGx
435
knee, coxa and centroid at position i, and use iji, și, įi (i = 0, 1, …) to denote the angles of joints C, E and G. C0, E0, G0, and O0 represent the original posture.
M C0
M C1
Fig. 12 The takeoff posture of the rigid bionic mechanism.
Then the velocity of the centroid VO can be expressed by JJJJK VO vOx i vOy j or VO VG ZOG u Gi Oi , (10) where VG denote the velocity of the coxa. The angular JJJJK velocity ȦOG is generated by torque of coxa. The Gi Oi is a vector from coxa point Gi to centroid Oi, which influences jumping angle of the centroid and the jumping performance. VG can be calculated by VG
'H Gy 'LGx i j, 't 't
(11)
where ǻt is the corresponding change of time; ǻLGx and ǻHGy are the changes of the distance and height of point G, which can be respectively derived as 'LGx
'H Gy
lCE M1 sin(
M1
M0 ) 2 T T lEG (T1 T 0 ) sin(G 0 1 0 ), 2 lCE M1 cos(
M1 2
M0 )
lEG (T1 T 0 ) cos(G 0
T1 T 0 2
),
(13)
where lCE, lEG denote the length of CE, EG correspondingly. When ǻt is small enough, the difference equations of Eqs. (12) and (13) can be derived as
M 1 M0 ) lCE M1 M1c cos( 1 M0 ) 2 2 2 T1 T 0 1 T T ) lEG (T1 T 0 ) (T1 T 0 )c cos(G 0 1 0 ), lEG (T1 T 0 )c sin(G 0 2 2 2
lCE M1c sin(
(12)
M1
(14)
Journal of Bionic Engineering (2011) Vol.8 No.4
436 'H Gy
c H Gy
't
M 1 M0 ) lCE M1 M1c sin( 1 M0 ) 2 2 2 T1 T 0 1 T T ) lEG (T1 T 0 ) (T1 T 0 )c sin(G 0 1 0 ). lEG (T1 T 0 )c cos(G 0 2 2 2
lCE M1c cos(
M1
The control of direction relies on the adjustment of the reflection between ijƍ and șƍ. So the bearing of velocity at point G results in
DG
actan
c H Gy
actan
c LGx
'H Gy 'LGx
k1 k2 sin(G 0
T1 T 0
) 2 actan , T T k1 k2 cos(G 0 1 0 ) 2
lCE M1 sin(
M1
M0 ), k2 lEG (T1 T 0 ). 2 By Eq. (10), the center of mass off angle is
where k1
W i 3 actan
vOy vOx
Fa 2 (3l a ), (0 d a d l ), 6 EI
3.4 Advantages of bionic mechanism In summary, the rigid-flexible mechanism of hind leg has advantages in jumping movement. At takeoff preparation stage, a minor force Ff can finish the posture adjustment. At the energy storage stage, Ff and Fe compress the spring and keep the mechanism in balance. At the first acceleration stage, the knee extension torque generated by Fe keeps increase, meanwhile the torque generated by elastic force of spring has the same direction, which ensures the fast reaction. At the moment before jumping, the flexible deformation of tarsus provides a second acceleration, which prolongs contact time with ground and improve the energy efficiency.
4 Kinematic analysis
(i 3 0,1, 2,...).
All the analysis above is made on the assumption that tarsus is fixed to the ground. In fact, the tarsal limb gradually adjusts its location, and the contraction of tibia causes the tarsus getting close to the body. Then the tarsal foot gradually begins to play a major role in supporting the body, so as to get prepared for jump. At second stage of acceleration in takeoff phase, the torque produced by tarsus and its motion are considered making the greatest contribution to jump movement, as shown in Fig. 13. It should be noted that the two stage acceleration is not strictly separated by time, but only a matter of the influence on the jump. Based on the athletic physiology study of locusts, the tarsus can be approximately simplified to a cantilever beam. The deflection Y can be written as Y
4.1 Pseudo-rigid body modeling The model of unilateral hind limb and the body of locust is simplified as Fig. 14. Focused on motion in the sagittal plane, this section builds the D-H systems and analyzes the kinematics of flexible-rigid model with the method of pseudo-rigid body modeling. First, the model can be described by D-H method. The homogeneous transformations of linkages are given as
0 1
ªcos(T1 (t )) sin(T1 (t )) « sin(T (t )) cos(T (t )) 1 1 « 0 0 « «¬ 0 0
0 0 1 0
1 2
ªcos(T 2 (t )) sin(T 2 (t )) « sin(T (t )) cos(T (t )) 2 2 « 0 0 « 0 0 ¬«
0 lo12 º 0 0 » », 1 0 » 0 1 »¼
T
T
x (t ) º y (t ) » », 0 » 1 »¼
(16)
where E and I are the elastic modulus and inertia moment; F, a, l denote force, the force position and length of beam, respectively.
(a)
(b)
(c)
(d)
Fig. 13 The distortion and torque produced by tarsus.
(15)
Fig. 14 The D-H systems of the proposed mechanism.
Chen et al.: Bionic Mechanism and Kinematics Analysis of Hopping Robot Inspired by Locust Jumping
2 3
ªcos(ʌ T3 (t )) sin(ʌ T3 (t )) « sin(ʌ T (t )) cos(ʌ T (t )) 3 3 « 0 0 « «¬ 0 0
0 lo 23 º 0 0 » », 1 0 » 0 1 »¼
3 4
ªcos(ʌ T 4 (t )) sin(ʌ T 4 (t )) « sin(ʌ T (t )) cos(ʌ T (t )) 4 4 « 0 0 « 0 0 ¬«
0 lo 34 º 0 0 » », 1 0 » 0 1 »¼
T
T
4 5
T
0 i
T
T 12T " i 1iT
0 1
ª1 «0 « «0 « ¬0
0 0 lo 45 º 1 0 0 »» , 0 1 0 » » 0 0 1 ¼
ª nx ox «n o « y y «0 0 « ¬0 0
In Eq. (17) (x(t), y(t)) denote the centroid trajectory and și denotes the rotation angle of joint i. The lo(i)(i+1) is defined as the distance between two adjacent joints oi and oi+1. Then, it is significant to reconsider the elastic stretch of knee joint and the flexible deformation of tarsus (Fig. 14). The stiffness of the spring is k. Approximately, the tarsus is assumed to be equal cross section bar. Therefore, according to Eq. (16), the transformation equations should be modified as ªcos(ʌ T3 (t )) sin(ʌ T3 (t )) « sin(ʌ T (t )) cos(ʌ T (t )) 3 3 2 « T = 3 « 0 0 « 0 0 ¬ ª1 « «0 4 5T = « «0 « «¬0
0 0
0 lo 23 k 'x º » 0 0 » , (18) » 1 0 » 0 1 ¼
º » Fa (3l a ) » ». 6 EI » 0 » »¼ 1 lo 45
2
1 0 0 1 0 0
(Table 3). Therefore, the transformation from the joint space q(ș1, ș2, ș3, ș4) to motion space x of model is the motion equation x = x(q). Then the relationship between the generalized speed x i (including translational speed vi and angular velocity Ȧi) of each component and the generalized velocity q of joints is obtained as x 3 J (q)q , namely, ª vi º «Ȧ » ¬ i¼
0 pxi º 0 p yi »» (i 1, 2,...,5). (17) 1 pzi » » 0 1 ¼
(19)
Finally, the posture of each joint can be derived according to Eq. (17). 4.2 Description of limb posture and speed
It is easier to ascertain 01T according to the experimental result in section 2. 01T indicates the jumping performance such as height, horizontal distance and body posture. The relationship between the rotation angle of joint and time series has been measured
437
ª J l1 J l 2 «J ¬ a1 J a 2
ªT1 º « » " J li º «T2 » (i 1, 2,...,5), " J ai »¼ « # » «» «¬Ti »¼
where J(q) denotes the generalized velocity Jacobin. Based on the vector product method, because the model only consists of the rotary joints, Ji is calculated by Ji
ª zi u 0i R i pn º « ». zi «¬ »¼
(20)
4.3 Calculation case analysis From the experimental data in Table 3, the joint space can be got from takeoff preparation to takeoff instance. The trajectory curve in Fig. 5 indicates the centroid position (x(t), y(t)). Lo12 and law represent the lengths from centroid to femur and abdomen respectively, Lo12, Lo34, Lo45 represent the lengths of linkages between two adjacent joints, all of which can be obtained from Table 2. Assume that k is equal to 1 and joint shrinkage ǻx is 1.5×10í3 m. Moreover, the F, E and I are estimated to be 0.03 N, 1 MPa and 10í6, respectively. As Fig. 13 indicates, the force position is in the middle of the rod, that is a = 0.5lo45. The homogeneous coordinate transformation can be calculated through Eqs. (17) to (19). Then x(q) is acquired according to the following equation xi (n) 3 0iT (n, 4) (n 3 1, 2,3; i 3 1, 2,...,5) .
In MATLAB software, the position matrix can be worked out at any time. At 24 ms, for example, the position matrix x(q(24)) irrespective of flexibility and xƍ(q(24)) taking flexibility into account are as
x (q(24))
ª35.5000 «31.6765 « «19.4318 « «30.1404 «¬ 27.0404
2.8000 0 º 2.0311 0 »» 7.1457 0 » , » 1.7972 0 » 1.7983 0 »¼
Journal of Bionic Engineering (2011) Vol.8 No.4
438 ª35.5000 «31.6765 « «19.4332 « «30.1418 «¬ 27.0418
x c(q (24))
2.8000 0 º 2.0311 0 »» 7.1451 0 » . » 1.7967 0 » 1.7884 0 »¼
In consideration of flexibility, the posture with respect to time is depicted in Fig. 15. The flexible-rigid mechanism has n = 5 joints, and the i pn0 3 0i R i pn of joint i (i 3 1, 2,..., n) is calculated in 24 ms as below p = ª¬ 1 p50
2
= ª¬ 10 R 1 p5
p50 0 2
3
p50
4
p50
5
R 2 p5
0 3
R 3 p5
p50 º¼ 0 4
R 4 p5
0 5
R 5 p5 º¼
ª34.2523 28.4451 20.5774 30.1423 27.0423º = «« 9.7422 10.3340 2.2907 1.7877 1.7803 »» . «¬ 0 »¼ 0 0 0 0
Then, according to Eq. (20), at this time, the velocity Jacobin matrix can be expressed by
J
1.7803 º ª 9.7422 10.3340 2.2907 1.7877 « 34.2523 28.4451 20.5774 30.1423 27.0423» « » « 0 » 0 0 0 0 « ». 0 0 0 0 « 0 » « 0 » 0 0 0 0 « » 1 1 1 1 «¬ 1 »¼
Similarly, the Jacobin matrix at any time can be calculated and then the linear velocity of each joint can be computed with the angular velocity. 40
Acknowledgement This work is financially supported by the National Natural Science Foundation of China (Grant No. 51075014).
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