A bio-inspired jumping robot: Modeling, simulation, design, and experimental results

Mechatronics 23 (2013) 1123–1140

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Mechatronics journal homepage: www.elsevier.com/locate/mechatronics

A bio-inspired jumping robot: Modeling, simulation, design, and experimental results Jun Zhang ⇑, Guangming Song, Yuya Li, Guifang Qiao, Aiguo Song, Aimin Wang School of Instrument Science and Engineering, Southeast University, Nanjing 210096, China

a r t i c l e

i n f o

Article history: Received 11 April 2013 Accepted 4 September 2013 Available online 12 October 2013 Keywords: Jumping robot Modeling Simulation Self-righting Steering Takeoff angle adjusting

a b s t r a c t This paper presents the design of a jumping robot inspired by jumping locomotion of locusts. The mechanisms of jumping, self-righting, steering, and takeoff angle adjusting are modeled and simulated firstly. Then the 3D model of the robot is designed and a prototype of the robot is fabricated. An eccentric cam with quick return characteristics is used by the jumping mechanism to compress torsion springs for energy storing and to trigger the springs for a quick release of energy. The self-righting, steering, and takeoff angle adjusting capabilities of the robot are achieved by adding a rotatable pole leg. The pole leg can prop up the body of the robot when it falls down. The pole leg can also steer the robot to turn step by step. By adjusting its center of mass (COM) using the pole leg with an additional weight, the robot can jump at different takeoff angles. A 9 cm  7 cm  12 cm, 154 g jumping robot prototype is implemented. The fundamental characteristics of the robot are tested. Experimental results show that the constructed robot can jump more than 88 cm high at a takeoff angle of 80.33°. The robot rotates about 277° in the air during jumping. The robot can self-right when it falls down to its left, right, and front sides in 9 s, 9 s, and 26 s respectively. The robot can steer 360° in 42 s with 14 steps, about 25.7° per step. Its takeoff angle ranges between 80.33° and 86.92°. The robot can continuously jump to overcome stairs and jump forward in outdoor environments with self-righting and steering. The experimental results are compared with the simulation results. The differences between them are explained. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Locomotion is the basic ability for mobile robots working in unstructured environments. Different locomotion patterns have different terrain adapting capabilities. Wheeled robots are effective in rolling on the flat surfaces. Legged robots have more robust locomotion capabilities in the cluster environments than wheeled robots. Robots with jumping gaits are more efficient in traversing rough terrain than those with rolling and walking gaits. Jumping robots can leap over obstacles several times taller than themselves. With the capabilities of quickly overcoming obstacles and avoiding risks, jumping robots can be applied in many fields such as planetary exploration [1–6], search and rescue [7–9], reconnaissance [10], and entertainment [11]. Some jumping robots with continuously hopping gaits only can jump a little height and need to keep balance dynamically [12–15]. Others with discrete jumping gaits can jump higher than their bodies [16–19]. Landing with stable posture and adjusting postures of their bodies for continuously jumping are difficult for discrete jumping robots. Self-righting, steering, and takeoff angle

⇑ Corresponding author. Tel.: +86 2583790895. E-mail address: [email protected] (J. Zhang). 0957-4158/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechatronics.2013.09.005

adjusting capabilities are still challenging problems in jumping robot design, especially for robots with small size and few degrees of freedom. These capabilities are the basic requirements for jumping robots to implement continuous locomotion. There are two types of self-righting methods used by the existing jumping robots, the passive self-righting method and the active self-righting method. An egg-shaped shell with low center of mass (COM) is used to make the robot self-right after landing in [1]. In [3], the imbalance between the spherical top and the relatively heavy, cone-shaped base of the robot causes it to right itself automatically. In [7], two hemispherical cages are mounted on the outer surface of two wheels to make the robot passively self-right. In [20], the authors propose a conceptual robot that has a spherical body with 12 legs equally distributed over its surface. The spherical body and the multi-legs enable the robot to self-right passively. Robots in [21–23] can self-right passively by using a cage around their body. In [24–26], the robots self-right passively by symmetrical body structures. In [27,28], the robot is equipped with a large wing-like mechanism for passive stabilization. A cylinder-shaped robot [29] is capable of self-righting using a cylindrical frame. There are also some jumping robots landing passively stable for their low hopping height with big feet [30–32]. Jumping robots with active self-righting capability usually have several stable landing states. They use arms or legs to self-right

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actively. In [2], the second generation of JPL robot uses a triangular prism to make the robot only have three possible landing states. Then it uses flaps to make the robot roll onto its back face. Finally, the robot rotates a large flap to force itself toward an upright configuration. In [33], the robot uses two legs to prop up its body when it falling down forward or backward after landing. The authors propose a symmetric and adjustable arm suspension mechanism for an actively stabilized single-legged hopping robot [34]. With the arm mechanism, the robot can self-right and hop continuously. In [35], the authors propose a 3D inverted pendulum jumping robot model. The robot can balance and self-right dynamically by the exchange of angular momentum. The advantages of the passive self-righting method are simple and easy. The egg-like shell and the cage can make the robot roll passively after landing and protect the robot from crashing on the ground. But this method is mainly suitable on the planar ground. The robot also cannot be controlled accurately for passive self-righting and passive rolling. The active self-righting robots are more robust than passive self-righting. But they need more driving mechanisms than passive ones. Jumping robots adopt several methods for steering such as hybrid-structured jumping robots with legs and wheels or robots with rotatable body. The first and second generation of JPL robots [1], the Sandia robot [3], the Jollbot [21], and EPFL robot [23] can rotate their bodies to steer. The third generation of JPL robot [2], the rescue robot [7], the Scout [7], the Scoutrobot [25], and the jumping robot in [36] use two wheels to steer. The wheeled-Leg robot [37], the AirHopper [38], and the SandFlea [10] steer with four wheels. In [26], the Mini-Whegs steers by using simplified universal joints between the wheel-legs and its body. The MSU robot [33] uses a reduction gear to steer. Few jumping robots are found to have the capability of takeoff angle adjusting. The third generation of the JPL robot [2] can adjust its takeoff angle by using a takeoff angle adjustment system. But it cannot self-right. The takeoff angle of the wheeled-Leg robot [37] can be adjusted by regulating the angle between its rear legs and the ground during takeoff. The EPFL robot [23] can adjust its takeoff angle by choosing a different pre-load angle for the torsion springs, but it is manually adjusted. The SandFlea [10] can control its takeoff angles by a foldable leg under its body. In [39], the robot can jump vertically or take off at a constant angle by adjusting the friction forces on its two feet. Few jumping robots have all the capabilities of self-righting, steering, and takeoff angle adjusting for continuously jumping locomotion. The aim of this paper is to design a jumping robot with all these capabilities by only a few driving mechanisms. In the nature, legged animals can easily realize standing state recovery and direction adjusting. Such capabilities enable the animals to adapt to complicated environments. Inspired by the legged animals, a new jumping robot with a motor driven pole leg is designed. The pole leg can be used to implement self-righting, steering, and takeoff angle adjusting. There are two motors in this robot. One is used for jumping and the other is used for self-righting, steering, and takeoff angle adjusting. The operator observes the postures of the robot and controls the robot through a wireless sensor network. The relevant methods used for self-righting, steering, and takeoff angle adjusting are presented in our previous work [40]. This paper presents the detailed modeling, simulation, design, and experimental results of the robot. The main contribution of this work is the self-righting, steering, and takeoff angle adjusting methods using a pole leg driven by only one motor. The rest of this paper is organized as follows. Section 2 gives the models and numerical simulation results of the jumping, selfrighting, steering, and takeoff angle adjusting mechanisms. The mechanisms design and fabrication are presented in Section 3. The experimental results on performances tests of jumping,

self-righting, steering, takeoff angle adjusting, continuous jumping, and outdoor locomotion of the robot are given in Section 4. Concluding remarks and future work are given in Section 5. 2. Modeling and simulation 2.1. Jumping modeling and simulation Four-bar mechanism only has one degree of freedom. It has been applied in many robot mechanisms such as leg system of the quadruped robot [38] and parallel manipulators [41,42]. The four-bar mechanism model adopted in [16,19,26,28] and the sixbar mechanism model used in [2,36] are two commonly used models in jumping mechanism design. In our design, the four-bar mechanism model is selected. Assuming that the right and left sides of the robot are symmetrical, the jumping model of the robot is simplified in the x–y plane as shown in Fig. 1. The four-bar mechanism consists of a body, a main leg, an assistant leg, and a lower leg. A torsion spring is mounted between the body and the main leg. The lengths ri of the bars, the distances li between the COM of the bars and the joints, and the masses of the bars and body are shown in Fig. 1. hi are variables for describing the positions and orientations of the bars and the body. It is assumed that there is no slippage between the lower leg and the ground. The positions Pi (xci, yci) of the COM of the bars and the body are as follows

8 xc1 > > > > > yc1 > > > > > > xc2 > > > > > yc2 > > > yc3 > > > > > xc4 > > > > > > > > > > yc4 > > > :

¼ xf þ l1 cos h1 ¼ yf þ l1 sin h1 ¼ xf þ r 1 cos h1 þ l2 cos ðh1 þ h2 Þ ¼ yf þ r 1 sin h1 þ l2 sin ðh1 þ h2 Þ ¼ xf þ r 0 cos h1 þ l3 cos ðh1 þ h3 Þ ¼ yf þ r 0 sin h1 þ l3 sin ðh1 þ h3 Þ

ð1Þ

¼ xf þ r 0 cos h1 þ r 3 cos ðh1 þ h3 Þ l4 cos ðh1 þ h3 þ h4 þ aÞ ¼ yf þ r 0 sin h1 þ r 3 sin ðh1 þ h3 Þ l4 sin ðh1 þ h3 þ h4 þ aÞ

where xf and yf are the coordinates of the end point of the lower leg which can be seen as the foot of the robot. The position (xc, yc) of the COM of the robot is as follows

8 , 4 4 X X > > > > x ¼ m x mi c i ci > < 1 1 , 4 4 > X > > y ¼ Xm y > mi > i ci : c 1

1

Fig. 1. Jumping model of the robot.

ð2Þ

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The orientations Ri of the bars and the body are as follows

8 R1 > > > < R2 > > R3 > : R4

¼ h1 ¼ h1 þ h2

ð3Þ

¼ h1 þ h3 ¼ h1 þ h3 þ h4  p

In order to simplify the problem we assume the four-bar mechanism is a parallelogram. So we have r1 = r4, r2 = r3, h2 = h3, and h3 + h4 = p. The jumping process consists of the pre-takeoff (PRT) stage and the post-takeoff (POT) stage. During the PRT stage, the robot has two degrees of freedom (DOF). We use h1 and h2 as independent variables to represent the two DOFs. During the POT stage, the robot has four degrees of freedom. h1, h2, xf, and yf are the four independent variables to represent the four DOFs. From (1) and (2), we can get the velocities and accelerations of the COM of the robot as follows

8 , 4 4 X X > > > > V ¼ mi x_ ci mi > < x 1 1 , 4 4 > X X > > > Vy ¼ _ m mi y > i ci : 1

1

1

4 4 > X X > > > €ci  mi y mi g : Fy ¼ 1

ð5Þ

Ee ¼



h

2

¼

1 K 2



h2

2 p

ð9Þ

3

where K is the stiffness coefficient of the torsion spring. When the spring is triggered, the energy stored in the spring is released and the robot obtains an initial velocity. The robot takes off when the vertical force on the foot of the robot is zero. Using the equation of Fy in (6), we can calculate the end time of the PRT stage. In the PRT stage, the generalized coordinates of qi are [h1, h2]. From (7), we can get the two second-order nonlinear differential equations as follows

ð10Þ

where M22 is the mass matrix, C21 is the centrifugal force, coriolis force, and the gravity force vector, Q21 is the joint torque vector. The elements of the matrix and vector in (10) are shown in Appendix A.1. In the POT stage, the generalized coordinates of qi are [xf, yf, h1, h2]. From (7), we also can get the four second-order nonlinear differential equations as follows

€ 41 þ CðH; H _Þ MðHÞ44 H 41 ¼ Q 41

ð6Þ

1

In order to simplify the modeling and simulation, we do not consider the joint friction forces of the four-bar mechanism, the air friction force, and the damping of the spring. Assuming that there is no slippage between the lower leg and the ground during the PRT stage, the contact point between the lower leg and the ground is an ideal hinge. The robot will take off from the ground when Fy in (6) decreases to zero. The dynamic model of jumping is expressed by a Lagrange equation as follows @L @ q_ i

1

1 K p2 2

ð11Þ

1

8 4 X > > > F mi €xci x ¼ > <

d dt

> > :

€ 21 þ CðH; H _Þ MðHÞ22 H 21 ¼ Q 21

From (5), we can get the forces on the lower leg of the robot from the ground as follows

(  

8 4 X > > < Eg ¼ mi gyci

ð4Þ

1

8 , 4 4 X X > > > > Ax ¼ mi €xci mi > < 1 1 , 4 4 > X X > > > Ay ¼ €ci mi y mi > :

relationship with the compression angle h of the torsion spring, h4 = h + p/6. Because h2 = h3, and h3 + h4 = p, so h = 5p/6  h2. h2 decreases from 5p/6 to p/3 when h increases from 0 rad to p/2 during the stretching of the spring. Eg and Ee are as follows



  @L @qi

¼ Qi

ð7Þ

L¼T V where T is the kinetic energy which includes the translational kinetic energy Ek and rotational kinetic energy Et of the four bars as follows

8 4 X > > 1 > mi v 2ci > < Ek ¼ 2 1

4 > X > > > J i x2i : Et ¼ 12

ð8Þ

1

V is the potential energy which includes the gravitational potential energy Eg of the bars and the elastic potential energy Ee of the torsion spring. qi are the generalized coordinates and Qi are the generalized forces. The free position of the torsion spring is p/2 when there is no energy stored in the spring. The equilibrium position of the spring is 0 rad when it is fully loaded. As shown in Fig. 1, h4 has the

where M44 is the mass matrix, C41 is the centrifugal force, coriolis force, and the gravity force vector, Q41 is the joint force and torque vector. The elements of the matrix and vector in (11) are shown in Appendix A.2. The two stages of jumping are simulated. The parameters of the robot are m1 = 0.013 kg, m2 = 0.0014 kg, m3 = 0.012 kg, m4 = 0.1 kg, r0 = 0.06 m, r1 = 0.03 m, r2 = 0.095 m, r3 = 0.095 m, l1 = 0.045 m, l2 = 0.0475 m, l4 = 0.02 m, a = 0.868 rad, J1 = 3.51e5 kg m2, J2 = 4.30e6 kg m2, J3 = 3.61e5 kg m2, J4 = 5.67e5 kg m2, and K = 2.1585 N m/rad. The initial conditions of the PRT stage are h1 = 0 rad, h_ 1 = 0 rad/s, h2 = 5p/6, and h_ 2 = 0 rad/s. The simulation results of the PRT stage are shown in Fig. 2. Fig. 2(a) shows the changes of h1 and h2. h1 increases slowly while h2 decreases gradually. This indicates that the front part of the robot prepares to take off and the torsion spring stretches to release energy. The velocities of h1 and h2 are shown in Fig. 2(b). The displacements of the COM of the robot are shown in Fig. 2(c). The velocities of the COM are shown in Fig. 2(d). The accelerations of the COM are shown in Fig. 2(e). The reaction forces on the lower leg from the ground are shown in Fig. 2(f). From Fig. 2(f), we can find the vertical force Fy is zero at t = 0.01904 s. So we can get the angles of h1 = 0.625 rad and h2 = 1.687 rad, the velocities of h1 and h2 are 63.328 rad/s and 105.92 rad/s, the displacements of the COM Xc = 0.00533 m, and Yc = 0.10968 m, the velocities of the COM Vx = 0.668 m/s, Vy = 5.41027 m/s, and Vc = 5.451 m/s at t = 0.01904 s. The takeoff angle is arctan(Vy/Vx) = 82.961°. The translational kinetic energy and rotational kinetic energy of the robot are 1.9010 J and 0.1585 J respectively as calculated. The increment of the gravitational potential energy is 0.1375 J. So the amount of the increased energy is 2.20 J. h2 decreases from 5p/6 to 1.687 rad. The amount of the decreased elastic potential energy is 2.22 J which is about the same amount of the increased energy. The spring is not totally stretched and the leg of the robot is not completely extended before taking off from the ground because h2 decreases from 5p/6 to 1.687 rad. The total elastic energy stored in the spring is 2.6629 J. Only 83.4% of the energy is released from the spring. This

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100 θ1 θ2

Angle (rad)

2.5 2 1.5 1 0.5 0

0

0.005

0.01

0.015

Angular velocity (rad/s)

3

50

-50 -100 -150

0.02

0

0.005

0.01

0.015

Time (s)

Time (s)

(a) Changes of θ 1 and θ 2.

(b) Velocity of θ 1 and θ 2.

0.02

6

0.15

Horizontal displacement Xc Vertical displacement Yc

Horizontal velocity Vx Vertical velocity Vy Velocity of COM Vc

5

0.1

Velocity (m/s)

Displacement (m)

ω1 ω2

0

0.05

0

4 3 2 1 0 -1

-0.05

0

0.005

0.01

0.015

-2

0.02

0.01

0.015

(c) Displacements of the COM of the robot.

(d) Velocities of the COM of the robot.

0.02

40

200 150

Horizontal force Fx Vertical force Fy

30

Horizontal acceleration Ax Vertical acceleration Ay

Force (N)

Acceleration (m/s 2)

0.005

Time (s)

250

100 50

20 10 0

0 -50

0

Time (s)

0

0.005

0.01

0.015

0.02

-10

0

0.005

0.01

0.015

0.02

Time (s)

Time (s)

(e) Accelerations of the COM of the robot.

(f) Forces on the foot of the robot from the ground.

Fig. 2. Simulation results of the pre-takeoff stage of the jumping robot. The robot takes off from the ground when the vertical reaction force Fy decreases to 0. We can get h1, h2, velocity of h1 and h2 as the initial conditions of the POT stage at the end time of the PRT stage when Fy = 0.

is the so called ‘‘premature lift-off’’ as introduced in [2], and talked in [19,43]. The POT stage is divided into two processes by the critical time when h2 = p/3. During the first process, h2 decreases from 1.687 rad to p/3 while K is 2.1585 N m/rad. During the second process, the leg of the robot stops extending and h2 is p/3 and remains unchanged. The initial conditions of the first process during POT stage are xf = 0 m, yf = 0 m, h1 = 0.625 rad, x_ f = 0 m/s, y_ f = 0 m/s, h_ 1 = 63.328 rad/s, h2 = 1.687 rad, and h_ 2 = 105.92 rad/s. The simulation results are shown in Fig. 3. Fig. 3(a) shows the changes of h1 and h2 after takeoff. h1 and h2 have the similar changing trends as the PRT stage. At the end of this process, h2 decreases to p/3 while h1 increases to 1.044 rad. The angular velocities of h1 and h2 are 141.765 rad/s and 212.85 rad/s as shown in Fig. 3(b). The displacements of xf and yf are 0.0078 m and 0.0058 m as shown in Fig. 3(c). The velocities of xf and yf are 3.723 m/s and 0.892 m/s as shown in Fig. 3(d). The initial conditions of the second process during POT stage are xf = 0.0078 m, x_ f = 3.7225 m/s, yf = 0.0058 m, y_ f = 0.8915 m/s, h1 = 1.044 rad, h_ 1 = 141.765 rad/s, h2 = p/3 rad, and _h2 = 212.85 rad/s. K = 1000 N m/rad is used as the stiffness coefficient of the stretched spring to simulate the unchanged angle of h. The simulation results are shown in Fig. 4. Fig. 4(a) shows that h1

increases gradually while h2 remains p/3 unchanged. h1 decreases to 17.4 rad which shows that the body of the robot rotates 2.77 rounds anticlockwise in the air during this process. The displacements of the COM of the robot are shown in Fig. 4(b). It is obvious that the trajectory of the COM is a parabolic curve. The jumping height and distance are about 1.61 m and 0.79 m respectively. 0.79 m means that the robot jumps to the direction of x. The velocities in Fig. 4(c) and accelerations in Fig. 4(d) of the COM of the robot also show that the trajectory of the COM is a parabolic curve. 2.2. Self-righting principle and modeling Some animals prop their body up using legs or head when falling down. For example, tortoises use their head to prop up and turn their body to the normal position when the body is turned upside down [44]. Beetles somersault with the aid of contra lateral or diagonal legs for self-righting [45]. A locust placed upside down on a flat surface uses a predictable sequence of leg movements to right itself as proposed in [46]. Inspired by this, a self-recovery mechanism is modeled for our jumping robot as shown in Fig. 5. The four-bar mechanism of the robot is simplified as a rectangle. A pole leg is added on the front side of the robot. The leg can rotate in the

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J. Zhang et al. / Mechatronics 23 (2013) 1123–1140 200

2

Angular velocity (rad/s)

Angle (rad)

θ1 θ2 1.5

1

0.5

0

0.5

1

1.5

2

2.5

3

3.5

ω1 ω2

0 -100 -200 -300

4

x 10

Time (s)

100

-3

0

0.5

2

2.5

3

3.5

4

x 10

-3

(b) Velocity of θ 1 and θ 2. 4

0.005

Velocity (m/s)

0.01

Displacement (m)

1.5

Time (s)

(a) Changes of θ 1 and θ 2.

Xf Yf

0

-0.005

-0.01

1

0

0.5

1

1.5

2

2.5

3

3.5

Time (s)

0

-2

-4

4

x 10

Vxf Vyf

2

-3

0

0.5

1

1.5

2

2.5

3

3.5

(c) Displacement of the foot of the robot.

4

x 10

Time (s)

-3

(d) Velocity of the foot of the robot.

Fig. 3. Simulation results of the first process during post-takeoff stage. h2 decreases to p/3 which is the end time of this process.

2

5

θ1 θ2

Displacement (m)

Angle (rad)

0 -5 -10 -15 -20

0

0.2

0.4

0.6

0.8

0.5 0

1

1.2

-1

1.4

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

Time (s)

(a) Changes of θ 1 and θ 2.

(b) Displacements of the COM of the robot.

1.4

5

Acceleration (m/s 2 )

Horizontal velocity Vx Vertical velocity Vy

4

Velocity (m/s)

1

-0.5

6

2 0 -2 -4 -6

Horizontal displacement Xc Vertical displacement Yc

1.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Horizontal acceleration Ax Vertical acceleration Ay 0

-5

-10

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

Time (s)

(c) Velocities of the COM of the robot.

(d) Accelerations of the COM of the robot.

1.4

Fig. 4. Simulation results of the second process during post-takeoff stage of the jumping robot. The trajectory of the COM is a parabolic curve.

vertical plane y–z. The leg with an appropriate length can prop up the body of the robot when it falls down. The self-righting process is composed of two stages, i.e. actively propping (AP) as shown in Fig. 5(a) and passively self-righting (PS) as shown in Fig. 5(b). The AP stage is the process when the robot uses the pole leg to prop its body up step by step. The pole leg rotates and the end point of the pole leg contacts the ground. When the leg rotates continuously, the body of the robot will be propped up step by step. The

critical time is when the projection of the COM of the robot is on the contact line between the robot and the ground. After this moment, the pole leg will stop to rotate. The robot enters the PS stage. It will passively self-right under its gravity and rotational inertia. From the model of the self-righting shown in Fig. 5(a), we have the relationship between h1–4 and h5 as follows

r5 sin ðh5 þ h14 Þ ¼ r 14 sin h14

ð12Þ

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J. Zhang et al. / Mechatronics 23 (2013) 1123–1140

(a) Actively propping stage.

(b) Passively self-righting stage.

Fig. 5. Model of the self-righting mechanism (front view of the robot).

where h1–4 is the angle between the body and the ground, h5 is the rotation angle of the pole leg. From (12) we have



h14 ¼ arctan½sin h5 =ðr14 =r 5  cos h5 Þ h_ 14 ¼ r5 cos ðh14 þ h5 Þh_ 5 =½r 14 cos h14  r5 cos ðh14 þ h5 Þ ð13Þ

The total mass of the body and the three bars is m1–4. The mass of the pole leg is m5 = 0.002 kg. Other parameters in Fig. 5 are h = 0.14 m, h0 = 0.09 m, AB = 0.07 m, b = 0.371 rad, r1–4 = 0.0966 m, r5 = 0.11 m, l5 = 0.055 m. The rotational speed should be controlled properly to overcome the resistance torque of the robot during self-righting and avoid the robot falling down again after selfrighting. The rotational speed of the pole leg is chosen as 12°/s so as to decide the rotational speed of the body and the reaction forces on the contact point of the pole leg from the ground. The torque changes of the driving motor of the pole leg also need to be decided to help us choose a proper motor.

The initial conditions of the AP stage are h1–4 = b = 0.371 rad, h5 = 2.447 rad. We can get the changes of h1–4 and h5 as shown in Fig. 6(a). h1–4 increases nearly linearly while h5 decreases linearly. h1–4 is p/2 when t = 9.284 s which is the end time of the AP stage. The changes of the velocity of h1–4 and h5 are shown in Fig. 6(b). x1–4 increases to 12°/s at the end time. Based on the principle of torque balance, the reaction forces F1x and F1y on the pole leg and F2x and F2y on the body of the robot can be obtained as follows

8 m14 gr14 cos h14  F 1y ½r14 cos h14  r 5 cos ðh14 þ h5 Þ ¼ 0 > > > > > > < m14 gr5 cos ðh14 þ h5 Þþ F 2y ½r 14 cos h14  r5 cos ðh14 þ h5 Þ ¼ 0 > > > > F 1x ¼ dF 1y > > : F 2x ¼ dF 2y ð14Þ where d is the sliding friction coefficient. The changes of these forces are illustrated in Fig. 6(c). The torque T acting on the output shaft of the driving motor can be obtained as 20

3

Angle (rad)

2

Angular velocity (°/s)

θ1-4 θ5

1 0 -1 -2

0

2

4

6

8

(9.284,12) 10

-10

-20

10

ω1-4 ω5

0

0

2

4

6

Time (s)

Time (s)

(a) Changes of θ1-4 and θ5.

(b) Velocity of θ 1-4 and θ 5.

2

0.1

1.5

0.08

8

10

Torque (N*m)

Force (N)

T 1 0.5

F1x F1y F2x F2y

0 -0.5 -1

0

2

4

6

0.06 0.04 0.02

8

10

0

0

2

4

6

8

Time (s)

Time (s)

(c) Reaction forces on the body and the pole leg from ground.

(d) Torque changes of the driving motor of the pole leg.

Fig. 6. Simulation results of self-righting during actively propping stage.

10

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J. Zhang et al. / Mechatronics 23 (2013) 1123–1140

T ¼ F 1x r5 sin ðh14 þ h5 Þ  F 1y r 5 cos ðh14 þ h5 Þ

ð15Þ

From the torque changes curve as shown in Fig. 6(d) we can get the maximum torque as 0.083 N m which will help us to choose an appropriate motor for self-righting. The PS stage begins when the projection of the COM of the robot is on the contact line between the robot and the ground. The two corners of the underside of the body will impact on the ground one by one. The rotational kinetic energy will completely lose after several times of impacts. The model of this process is as follows

(1

J x2 ¼ 12 J 14 x2ð14Þi  EKi 2 14 ð14Þðiþ1Þ EKi ¼ rJ 14 x2ð14Þi

i ¼ 1N

ð16Þ

where J1–4 is the moment of inertia of m1–4, EKi is the decreasing amount of the rotational kinetic energy in the ith time impact, r is a coefficient which represents the decreasing velocity of the rotational kinetic energy. From (16) we have

xð14Þðiþ1Þ ¼ ð1  2rÞ0:5 xð14Þi i ¼ 1    N

ð17Þ

If we assume the robot stops shaking when x(1–4)i < 0.01 rad/s, we can plot the changes of x(1–4)i when r = 0.1, 0.2, 0.3, and 0.4 respectively. The results are shown in Fig. 7. The robot shakes 43 times before stopping when r = 0.1, while 6 times when r = 0.4. The robot may fall down again if it shakes too wildly after AP stage. The most likely falling down happens at the first shaking. So the initial rotational speed of the body in the PS stage should not be too high. The shape of the underside of the body also needs to be designed carefully. Here the underside of the body is designed as an isosceles trapezoid as shown in Fig. 8. The body of the robot rotates around one opposite side AC of the isosceles trapezoid during AP stage. And the other opposite side BD will impact on the ground. The projection point of the COM of the robot on the

ground is O. The distance between O and AC is d1. The distance between D and line OG is d2. When x1–4 = 12°/s, b = 0.371 rad, r1–4 = 0.0966 m, l = 0.09 m, J1–4 = 4.134e4 kg m2 are known, we need to decide the size of the isosceles trapezoid. The most likely falling down happens when the rotational kinetic energy is totally converted to gravity potential energy of the robot. We assume that r is zero which means there is no energy losing during the first impact. If the robot inclines to the critical condition that the projection point of the COM is on the same point of D. From Fig. 8, we have

8 m14 gOD ¼ J 14 x2ð14Þ1 =2 þ m14 gr14 > > > > > 2 02 > d2 ¼ OD2  h > > > < cos s ¼ d2 =ðDE þ EF þ FGÞ > > tan s ¼ DE=l ¼ FG=0:5l > > > > > EF ¼ AB=2 > > : CD ¼ 2ðDE þ EFÞ

From (18) we can get OD = 0.09726 m, d2 = 0.035 m, and CD = 0.08 m. These calculation results can help us design the shape of the underside of the robot. 2.3. Steering method and modeling Animals use their legs to steer during locomotion. Inspired by this, we attempt to use the pole leg to adjust the jumping direction of the robot. The self-righting mechanism also can be used for steering without other additional devices. The principle of steering is shown in Fig. 9. After self-righting, the pole leg rotates continuously to enable steering. The front part of the robot will be propped up and turned on the ground while the feet of the robot touch the ground. The principle of steering can be expressed as

(

q ¼ MN=l _

0

MN ¼ 2r5 arccosðh =r 5 Þ

15

σ =0.1 σ =0.2 σ =0.3 σ =0.4

ω (°/s)

10

ð18Þ

ð19Þ

where l is the radius of steering, MN is the track length of the end point of the pole leg when it touches the ground, q is the angle of

5

0

0

10

20

30

40

50

i Fig. 7. The impact times between the body and the ground during passively selfrighting.

(a) Front view of the robot.

(b) Top view of the robot.

(a) Top view of the robot.

(b) Front view of the robot.

Fig. 8. The isosceles trapezoid shape of the underside surface of the robot.

Fig. 9. The principle of steering. The robot steers angle of q each step in a circular track with a radius of l.

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one steering step, h0 is the height of the point O, r5 is the length of the pole leg. The relationship between q and r5 is shown in Fig. 10 when l = 0.09 m and h0 = 0.09 m are given. The robot cannot steer if r5 < h0 . q increases with the increasing of r5. The maximum of r5 is decided by the balance of the robot. The projection point of COM on the ground must be in the inner part of the contact trapezoid. From the calculation in Section 2.2, we know that the maximum of r5 = OE = 0.09726 m if CD = 0.08 m is fixed. The maximum of r5 is larger if CD is designed longer. 2.4. Takeoff angle adjusting modeling As we can see in the jumping of animals, takeoff angle of jumping can be changed by adjusting the position of body COM relative to the last ground contact point before jumping [47]. The principle of takeoff angle adjusting of our robot is similar to this. The principle of COM adjusting is shown in Figs. 11 and 12. An additional weight (AW) can move with the rotation of the pole leg in the vertical plane of the front side of the robot which can make the COM of the robot in a changing range in y direction. Because the position of the AW only changes in the vertical plane, so xc is constant while yc changes with the moving of the AW. From Figs. 11 and 12, we can obtain the COM of the pole leg and the AW as follows

50

Fig. 12. Model of the takeoff angle adjusting mechanism (front view).

8 xc5 ¼ xf þ r0 > > > < y ¼ y þ r0 c5 f > xc6 ¼ xf þ r0 > > : yc6 ¼ yf þ r0

sin h1 þ r 3 sin ðh1 þ h2 Þ þ b sin ðh1 þ cÞ þ l5 cos h1 cos h1 þ r3 cos ðh1 þ h2 Þ þ b cos ðh1 þ cÞ  l6 sin h1 sin h1 þ r 3 sin ðh1 þ h2 Þ þ b sin ðh1 þ cÞ þ l6 cos h1 ð20Þ

where b is the distance between the rotation shaft of the pole leg and the rotation axis of the torsion spring as shown in Fig. 12. c is the angle between b and r4. l5 and l6 are the projections of l50 and l60 in the x–y plane respectively as follows



40

ρ (°)

cos h1 þ r3 cos ðh1 þ h2 Þ þ b cos ðh1 þ cÞ  l5 sin h1

30

l5 ¼ l50 cos ðb þ h5 Þ l6 ¼ l60 cos ðb þ h5 Þ

ð21Þ

The range of h5 is [h50, p  h50  b], where h50 is as follows

20 0

h50 ¼ arccosðh =r 5 Þ  b

10 0 0.089

0.09

0.091 0.092

0.093 0.094

0.095 0.096 0.097

0.098

r5 (m) Fig. 10. The relationship between one step steering angle q and the length of the pole leg r5.

where h0 is the distance of the rotation shaft of the pole leg and the ground when the torsion spring are fully loaded and the COM is in the lowest position, this is the time when h2 = 5p/6 as shown in Fig. 11.

0

h ¼ r 2 sin h2 þ b sin c

Fig. 11. Model of the takeoff angle adjusting mechanism (right view).

ð22Þ

ð23Þ

Then we can obtain a new dynamic model using the similar method as proposed in Section 2.1 with considering the influence of the pole leg and the AW on the takeoff angle. The parameters of the pole leg and the AW are m5 = 0.002 kg, m6 = 0.0013 kg, r5 = 0.11 m, r6 = 0.015 m, w6 = 0.012 m, l50 = 0.055 m, l60 = 0.068 m, J5 = 8.07e6 kg m2, J6 = 4.33e6 kg m2, b = 0.046 m, and c = 0.1089 rad. From (23) we can obtain h0 = 0.053 m. From (22) we can obtain h50 = 40.23°. From (21) we can get different values of l5 and l6 at different angles of h5. Then we can use the new dynamic model to simulate the takeoff velocities in different values of l5 and l6. The initial conditions are the same ones of the PRT stage in Section 2.1. The simulation results of the influence of the pole leg’s rotation angle on the takeoff velocity are shown in Fig. 13. Vx increases slowly in the negative direction while Vy nearly remains unchanged when h5 increases as shown in Fig. 13(a). The takeoff angle increases from 89.68° to 82.31° when h5 increases from 40.23° to 160.24°. The takeoff angle is smaller than zero which means the robot jumps backward. The robot may jump forward if we choose a heavier AW or a longer r0. The influences of structural parameters on takeoff angle will be investigated in our future work.

J. Zhang et al. / Mechatronics 23 (2013) 1123–1140

where Ev is the kinetic energy, m is the mass of the robot, h is the jumping height, v is the takeoff velocity, c is the takeoff angle. Ee is the elastic energy, K is the stiffness coefficient of the torsion spring, h is the compression angle of the torsion spring. We assume there is no energy loss. According to the law of conservation of energy, Ev = Ee. From (25)–(27) we can get

6

Velocity (m/s)

Vx Vy 4

2

h ¼ ðNKh2 sin 2 cÞ=2 mg 0 40

60

80

100

120

140

160

180

θ5 (°)

(a) Takeoff velocity changes against the rotation angle of the pole leg.

Takeoff angle (°)

-82

-84

-86

-88

-90 40

60

80

100

120

140

160

180

θ5 (°)

(b) Takeoff angle changes against the rotation angle of the pole leg. Fig. 13. Simulation results of the influence of the pole leg’s rotation angle with the additional weight on the takeoff velocity.

3. Robot design and fabrication

Jumping robots are always designed by using various methods to convert other kinds of energies to kinetic energy and gravitational potential energy. All these methods need to release energy quickly to get high energy efficiency. When observing the jumping locomotion of some animals such as kangaroos, frogs, fleas, and locusts, we find that they take off in a very short time. Jumping mechanisms designed by researchers worldwide also have the same characteristic. These mechanisms have a fast return characteristic. This can be expressed as

ð24Þ

where k is the coefficient of return process, t1 is the time used for energy storing, t2 is the time used for energy releasing. The ideal condition is that k is infinity. So we need to get k as large as possible. There are several kinds of methods to implement this function such as the clutch in [2], the eccentric cam firstly introduced by Scarfogliero et al. in [48], the slip-gear system in [26], the one way bearing in [33], the incomplete gear in [36], and non-circular gears [49]. Our jumping mechanism design borrows ideas from the robots in [48,16]. We select the four-bar mechanism and use a DC motor coupled with a reduction gear system to drive an eccentric cam to store energy into torsion springs. In our design, we aim to implement a robot that is 150 g in weight, 12 cm in height and can jump higher than 100 cm at a takeoff angle of 80°. Because of

Ev ¼ mv 2 =2 2

ð26Þ

Ee ¼ Kh2 =2

ð27Þ

ð29Þ

where Dw is rotation angle of the swinging bar, Du is the rotation angle of the cam. The swinging bar swings p/2 when the cam rotates 2p. Then the graphical method termed inversion [50] for cam profile design is used to design the contour line and CAD model of the cam. After choosing the torsion springs and designing the cam, we must select a motor and the reduction system for the jumping mechanism. As shown in Fig. 14, M is the torque on the cam, T is the torque of the torsion springs, F is the force between the cam and the roller, p1 is the distance between the point O0 and the action line of force F, p2 is the distance between the point O and the action line of force F. The relation between M and Dw is expressed as

M ¼ Fp2 ¼ Tp2 =p1 ¼ NK Dwp2 =p1 ¼ 2:16Dwp2 =p1

ð25Þ

h ¼ ðv sin cÞ =2 g

ð28Þ

where N is the number of torsion springs. We choose four torsion springs with K = 0.5396 N m/rad of one spring. The maximum compression angle h is set to 90°. Because m = 150 g, c = 80°, g = 9.8 N/kg, p = 3.14, we can calculate that the ideal jumping height is about 178 cm. After choosing the torsion springs, we need to design the jumping mechanism. The principle of the energy storing and releasing mechanism is shown in Fig. 14. a is the distance between the rotation axis O of the cam and the axis O0 of the swinging bar, rb is the base radius of the cam, rr is the radius of the roller, r0, r1, r2, r3, and r4 are lengths of the four-bar mechanism as proposed before, b and r3 are the lengths of the swinging bar. The axis of the cam is fixed in the body of the robot. The eccentric cam rotates anticlockwise on its axis O to drive the swinging bar to rotate anticlockwise on its axis O0 . The four-bar mechanism compresses the torsion springs. When the critical point reaches the roller, the springs will release for a sudden jump. The lengths of r0, r1, r2, r3, and r4 are the same ones of the simulation model. In order to make the robot to meet the size requirement of our robot design, we choose a = 30 mm, b = 22 mm, rb = 6.5 mm, and rr = 3 mm. We choose the equation of motion of the swinging bar as

Dw ¼ f ðDuÞ ¼ Du=4

3.1. Jumping mechanism

k ¼ t1 =t 2

1131

Fig. 14. Principle of the energy storing and releasing mechanism.

ð30Þ

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where the number of torsion springs is N = 4 and with K = 0.5396 N m/rad of one spring. At different angles of Dw, we can get different p1 and p2 in the CAD model of the cam. The maximum of M is 0.793 N m. A DC motor with reduction mechanism is chosen to drive the cam. Its stall torque is 0.093 N m. We designed a 3-stage reduction gearbox with a reduction ratio of 37. We assume that the efficiency of every stage is 0.8. So the driving torque is 1.762 N m, 2.2 times of M, that is enough to drive the cam. The 3D model of the jumping robot and driving system are shown in Fig. 15. The body, the main leg, the lower leg, and the assistant leg form the four-bar mechanism. The body and the main leg are made of aluminum alloy 2024. The cam is made of Nylon 6 because this kind of material has the advantages of low density and excellent wear resistance. The lower leg and the assistant leg are made of carbon fiber for its very high strength to weight ratio. In order to make the robot lighter and meet the stress requirement, the stresses of the cam, the main leg, the body frame, and the lower leg are simulated and their sizes are optimized. The strength analysis results are shown in Fig. 16. The maximum stresses of the parts are smaller than their yield strengths. The lowest factors of safety in the design are bigger than 1. The strengths of the parts can meet our design requirements. 3.2. Self-righting mechanism A pole leg is used to realize the function of self-righting. A DC motor mounted on the front side of the robot drives the pole leg to rotate. The self-righting process is shown in Fig. 17. There are three possible landing postures, i.e. lying down on the left side, on the right side and on the front side. The left side and the right side are symmetrical. So we only analyze the left and front landing postures here. If the robot falls down on the left side as shown in Fig. 17(d), the self-righting motor rotates clockwise to prop the robot body up to state (g) through (e) and (f). If the robot falls down on the front side

like (a), the self-righting motor needs to cooperate with the jumping motor for self-righting. First, the jumping motor rotates to make the robot switch from (a) to (b). Then the self-righting motor rotates a few angles to make the robot switch from (b) to (c). The COM of the robot is out of the contacting triangle ABC as shown in (c). This state is unstable so the robot will fall down to its left side by the force of gravity. Then it switches to the left side posture as shown in (d) and repeats (e), (f), and (g). If the robot falls down on the right side, the self-righting motor only needs to rotate the pole leg anticlockwise for self-righting. When the robot stands up, the self-righting motor will stop and then rotate in the opposite direction until the pole leg rotates to the state as shown in Fig. 17(g). As we simulated in Section 2.2, the maximum torque of the DC motor needed to drive the pole leg is 0.083 N m. A DC motor with a maximum torque of 0.145 N m is selected to drive the pole leg, which is 1.7 times of the needed maximum torque. That is enough to drive the pole leg. 3.3. Steering mechanism The steering angle of one step is analyzed in Section 2.3. The steering angle can be adjusted by the length of r5 as shown in Fig. 10. If we want to change r5, we need to add more mechanisms to achieve this function. The distance between the output shaft of the adjusting motor and the ground h0 has the relationship with angle h2 as shown in (23). Here we choose a fixed length of the pole leg r5 and adjust the height h0 by compressing the four-bar mech0 0 anism. As shown in Fig. 18(a), h2 < h1 when the springs are compressed. From (19), we know that when r5 is fixed, one step steering angle q is larger when h0 is smaller. So the steering angle can be adjusted by the compressing angle of the four-bar mechanism. Effective steering requires cooperation between the selfrighting motor and the jumping motor in order to make the pole leg to contact the ground at the appropriate point. The diagram of one-step of steering is shown in Fig. 18(b). The pole leg rotates clockwise while the front side of the body is propped up. By continuously rotating of the pole leg, the body of the robot will be driven to steer to its left side. 3.4. Takeoff angle adjusting mechanism In our design, we still use the self-righting mechanism to implement the takeoff angle adjusting function. As shown in Fig. 19, an additional weight (AW) is mounted at the end of the pole leg. The height of the COM of the robot can be adjusted when the AW goes up and down by rotating the pole leg. We can get the three angles in three different heights of COM in Fig. 19. They correspond to three takeoff angles. The takeoff angle of the robot can be adjusted between the state in Fig. 19(a) and the state in Fig. 19(c). A prototype of the robot is fabricated finally as shown in Fig. 20. The materials and masses of the parts of the robot are shown in Table 1. The total mass is 153.8 g with a size of 9 cm  7 cm  12 cm. 4. Experimental results and discussion

Fig. 15. (a) 3D model of the robot. (b) 3D model of the driving system.

Several experiments have been done to test the functions of the jumping robot. The jumping, self-righting, steering, takeoff angle adjusting and continuous jumping capabilities were tested respectively. The outdoor continuous forward jumping locomotion performance is also tested. The control module of the robot is a wireless microcontroller. The operator observes the postures of the robot directly and sends control commands to the robot through a wireless sensor network gateway. The experimental results are compared with the simulation results and the differences between them are discussed.

J. Zhang et al. / Mechatronics 23 (2013) 1123–1140

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Fig. 16. Stress distribution of the cam, the main leg, the body frame, and the lower leg, with maximum stress of 6.352e+7 N/m2, 1.455e+7 N/m2, 1.57036e+8 N/m2, and 4.51804e+8 N/m2 respectively and with factor of safety of 2.2, 5.2, 4.8, and 1.37 respectively.

4.1. Jumping The jumping height of this prototype can be increased by installing up to four torsion springs. In this test, four springs are used to get a maximum jumping height. The torque of the motor is 93 N mm. The total mass of the robot is 132 g when we get rid of the self-righting mechanism in this test. Jumping trajectories of the prototype robot have been recorded by a camera as shown in Fig. 21. A board with 80 cm  100 cm grid coordinates behind the robot is used to estimate the position of the robot. The robot jumps about 100 cm in height and traverses 65 cm at a takeoff velocity of 4.485 m/s and a takeoff angle of 80.77°. The red lines with arrow indicate the angle of the lower leg h1 as shown in Fig. 21. This angle is used to represent the rotation angle of the robot. The robot rotates clockwise in the PRT stage and anti-

clockwise in the POT stage which is the same as shown in the simulation results. The robot rotates anticlockwise about 277° in the air which is smaller than the simulation result h1 as shown in Fig. 4(a). This is caused by the deformation of the lower leg under the ground reaction force. The deformation of the lower leg will damp the rotation speed of the robot and reduce the distance between the reaction force and the COM of the robot. This will reduce the rotational kinetic energy. There are also some differences between the model in Section 2.1 and the prototype of the robot in the test such as moment of inertia of the four-bar mechanism. The friction forces on the joints of the four-bar mechanism and damping of the springs also will reduce the rotation velocity of the robot. The translational kinetic energy is 1.3276 J when considering the takeoff velocity of 4.485 m/s. The duration in the air is about 0.9 s. The rotational

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J. Zhang et al. / Mechatronics 23 (2013) 1123–1140

Fig. 17. Self-righting process of the jumping robot. (d)–(g) Selfrighting from left side. (a)–(g) Self-righting from front side.

speed is 5.37 rad/s. The rotational kinetic energy is about 0.0019 J. Only about 49.9% of the elastic energy stored in the spring is converted into kinetic energy of the robot. The current of the motor used for jumping mechanism driving is recorded by a digital multimeter Tektronix DMM4050 when the springs is compressing and stretching and the robot is turned upside down without jumping movement. The result is shown in Fig. 22. The current increases when the cam rotates to compress the torsion springs. When reaching the peak value, the current begins to drops sharply. This shows that the springs stretch quickly. The jumping mechanism has the quick return characteristics as we discussed before. The average current is 66.4 mA. A platform is established to test the reaction forces on the robot before taking off from the ground as shown in Fig. 23(a). A sixdimension force sensor is used to test the reaction forces on x and y directions. A data acquisition card is used to sample the force data. The test results are shown in Fig. 23(b). The force Fy has a downward trend which is similar to the simulation result as shown in Fig. 2(f). The repeatability of jumping is tested on a smooth surface and a rough surface respectively. The results are shown in Table 2. The jumping heights h, jumping distances d, takeoff velocities v, takeoff angles h, and rotation angles R have good repeatability on both surfaces. The landing postures are the left, right, or front side contacting on the ground. The takeoff angles and rotation angles in the air taking off from the smooth surface are smaller than from the rough surface. This is because the slippage on the smooth surface is larger than on the rough surface during takeoff. The COM of the robot will approach its foot taking off from the smooth surface. This means that the taking off angle from smooth surfaces is smaller than from rough surfaces. Also the jumping distance from smooth surfaces will be larger than from rough surfaces. The jumping height, distance, rotational angle, and time in the air are recorded in the jumping test, which are used to estimate the takeoff velocity, takeoff angle, and velocity of h1 after takeoff. The sensing of angle h1 and h2, their velocities, acceleration and tra-

Fig. 18. The diagram of steering. (a) One-step steering angle adjusting by compressing the four-bar mechanism. (b) One-step steering by using the pole leg.

Fig. 19. The takeoff angle adjusting mechanism and the adjusting diagram. (a) Additional weight (AW) on top. (b) AW in middle. (c) AW at bottom.

jectory of the robot in the air by using three-axis gyroscopes and accelerometers will be investigated in our future work which can help us learn more about the dynamic characteristics of the robot. 4.2. Self-righting The performances of self-righting are tested. The sequences of the self-righting after the robot falls down on its left side, right

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J. Zhang et al. / Mechatronics 23 (2013) 1123–1140

Fig. 20. Prototype of the proposed jumping robot.

Table 1 Materials selected for the parts of the robot. Part

Material

Mass (g)

Body frame Main leg 2 Lower legs Assistant leg Cam Torsion springs Pole leg Additional weight 4 Gears Collectors and Shafts Control and Sensor module Onboard battery 2 Motors Total mass of the prototype

Aluminum alloy 2024 Aluminum alloy 2024 Carbon fiber Carbon fiber Nylon 6 Stainless steel Carbon fiber Stainless steel Aluminum alloy 2024 COM or carbon fiber N/A N/A N/A N/A

29.8 8 2.6 0.8 8.2 20.3 1.3 13 8.9 21.6 12.1 4.2 23 153.8

Fig. 21. Jumping trajectory of the prototype robot. The jumping height and distance are 100 cm and 65 cm respectively. The takeoff velocity is about 4.485 m/s. The takeoff angle is about 80.77°. The robot rotates anticlockwise about 277° in the air.

150

Testing result Average current

side, and front side are shown in Fig. 24(a)–(c) respectively. The robot spends 9 s, 9 s, and 26 s on self-righting from its left side and front side respectively. The current changes of the adjusting motor during self-righting from left side are recorded as shown in Fig. 25 which indicates that the current changes of the motor have the same tendency as the torque change simulation result shown in Fig. 6(d). The average current is 40.7 mA. The self-righting time is decided by the rotational speed of the motor. In order to make the self-righting process more stable, we can lower the COM by rotating the jumping motor to compress the torsion springs in a proper angle before self-righting. 4.3. Steering The result of the steering test is shown in Fig. 26. The robot steers anticlockwise 360° in 42s within 14 steps, about 25.7° per step. The current of the motor during the steering process is recorded as shown in Fig. 27. There are 14 peaks of the current which is corresponding to the 14 steps of steering. The average current is 22.4 mA. The steering speed can be adjusted by compressing the torsion springs to adjust the height of the rotation shaft of the motor as we analyzed in Section 3.3. The jumping motor rotates in three states. The steering speeds are 24.33°, 25.7°, and 23.67° per step when the angles between the assistant leg and the ground are 85°, 88.4°, and 95.4° respectively.

Current (mA)

125 100 75 50 25 0

0

5

10

15

20

25

30

35

40

Time (s) Fig. 22. Current of the motor during one cycle of jumping locomotion.

The self-righting motor is controlled by Pulse-Width Modulation (PWM). The speed of the motor can be adjusted by setting the duty ratio of the PWM. So the steering speed also can be changed by adjusting the rotational speed of the motor. To prevent the robot from falling down again, the rotational speed of the pole leg should not be too fast when it touches the ground. For this prototype, the speed of the pole leg needs to be modulated so that it will slow down when approaching and touching the ground.

4.4. Takeoff angle adjusting In order to test the function of takeoff angle adjusting, a 13 g additional weight is mounted at the end of the pole leg. The pole leg rotates to three positions i.e. the additional weight is on top, in middle, and at bottom of the robot. Jumping videos of the robot are recorded by a Casio high speed camera with a frame rate of

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J. Zhang et al. / Mechatronics 23 (2013) 1123–1140 100

Testing result Average current

Current (mA)

80 60 40 20 0 0

2

4

6

(a) Platform for force test.

8

10

12

Time (s) Fig. 25. Current of the motor during self-righting from left side.

60

Fx Fy

Force (N)

40

20

0

-20

0

0.005

0.01

0.015

0.02

0.025

0.03

Time (s)

(b) Forces during jumping. Fig. 23. Reaction force test from the ground during jumping.

420frames/s. The board with 80 cm  100 cm grid coordinates behind the robot is used to estimate its position in the air.

As shown in Fig. 28, the sequences are when the robot at the highest positions during the three jumps. The jumping heights are 88 cm, 91 cm, and 93 cm respectively. The horizontal displacements are 30 cm, 14 cm, and 10 cm respectively when the robot at the highest position in the air. Takeoff velocities and angles are calculated using the jumping heights and distances. The takeoff angles are about 80.33°, 85.6°, and 86.92° when the AW is on the top, in the middle, and at the bottom of the robot. The offset of the COM of the robot caused by the AW when it in the middle position will make the robot rotate along x axis as shown in Fig. 28(b). There is a small drift of the COM out of the x–y plane in the air because the AW is out of the plane and its mass accounts for about 8% of the total mass of the robot. The trajectories of the three jumps on the x–y plane are obtained by estimating the COM of the robot from the videos as

Table 2 Repeatability test of jumping on a rough surface and a smooth surface. Jumping height h, jumping distance d, takeoff velocity v, takeoff angle h, rotation angle R, posture after landing (left, right, or front side contact on the ground) of ten tests results on smooth and rough surfaces are given respectively. Rough surface

Smooth surface

h (cm)

d (cm)

v (m/s)

h (°)

R (°)

Posture

h (cm)

d (cm)

v (m/s)

h (°)

R (°)

Posture

1 2 3 4 5 6 7 8 9 10

100 98 99 98 99 97 96 98 97 98

67 67 67 68 68 65 62 64 64 66

4.49 4.45 4.47 4.45 4.47 4.42 4.39 4.44 4.42 4.44

80.49 80.30 80.40 80.16 80.26 80.49 80.83 80.73 80.63 80.44

450 405 400 440 440 450 500 450 450 450

Left Right Left Left Left Right Front Left Front Front

1 2 3 4 5 6 7 8 9 10

100 99 97 98 99 99 99 100 100 100

73 73 68 76 73 72 71 74 74 73

4.50 4.48 4.43 4.46 4.48 4.48 4.48 4.50 4.50 4.50

79.66 79.56 80.06 79.03 79.56 79.70 79.84 79.52 79.52 79.66

380 390 380 370 380 380 370 380 380 380

Left Front Left Left Right Left Right Left Right Right

Average

98

65.8

4.44

80.47

443.5

N/A

Average

99.1

72.7

4.48

79.61

379

N/A

(a) Self-righting from left side

(b) Self-righting from right side

(c) Self-righting from front side. Fig. 24. Video sequences of self-righting from left, right, and front side of the robot.

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J. Zhang et al. / Mechatronics 23 (2013) 1123–1140 100 AW at bottom AW in middle AW on top

y (cm)

80 60 40 20 0

0

10

20

30

40

50

60

x (cm) Fig. 29. Jumping trajectories of the takeoff angle adjusting tests when the additional weight on top, in middle, and at bottom of the robot respectively. The positions of the robot are recorded by a high speed camera.

leg during takeoff in the practical test. The COM of the robot will move backward a little when the lower leg bends under the reaction force. So the robot will jump further at a smaller takeoff angle than the simulation results. 4.5. Continuous Jumping

Fig. 26. Video sequences of steering in 360°.

50

Testing result Average current

Current (mA)

40 30 20 10 0 -10

0

5

10

15

20

25

30

35

40

In the test of continuous jumping, we let the robot jump to overcome stairs. The results are shown in Fig. 30. Two torsion springs with the total K = 1.0792 N m/rad and a new jumping motor are used. The stall torque of the motor is 77 N mm. The total time used for overcoming the two-step stairs is about 108 s. The average currents of the motors for jumping, self-righting, and steering are 66.4 mA, 40.7 mA, and 22.4 mA respectively as tested before. The spring compressing time for jumping is about 60 s. The maximum self-righting time is about 26 s. The maximum steering time is 21 s. So, the total energy use of one jump is about 1.53 mA h. The 200 mA h lithium battery can provide energy to the robot for jumping more than 131 times continuously.

45

4.6. Outdoor test

Time (s) Fig. 27. Current of the motor during steering.

In order to test the performance of the robot in the outdoor environment, a continuous forward jumping test in an outdoor environment is conducted. The results are shown in Fig. 31. The robot jumps two times with self-righting and steering in 156 s. The forward moving distance is about 2 m. The ground is not as smooth and flat as the testbed used in Section 4.5. The robot does not selfright and steer very well on the outdoor ground. The self-righting and steering mechanisms will be investigated in the future to improve the performance of continuously jumping of the robot. 5. Conclusions

(a) AW on top

(b) AW in middle

(c) AW at bottom

Fig. 28. The video sequences of jumping when the pole leg rotates to three angles and the AW is in three different positions respectively. The takeoff angles are about 80.33°, 86.6°, and 86.92° when the AW is on the top, in the middle, and at the bottom of the robot respectively.

shown in Fig. 29. The takeoff angle ranges between 80.33° and 86.92° which is smaller than the simulated results between 82.31 and 89.68°. This is caused by the deformation of the lower

A new jumping robot with self-righting, steering, and takeoff angle adjusting capabilities is presented in this paper. The jumping model, self-righting model, steering model, and takeoff angle adjusting model are detailed. Based on modeling and simulation, the 3D model of the robot is designed. Jumping is implemented by a motor-driven eccentric cam. A pole leg mechanism is designed to implement self-righting, steering, and takeoff angle adjusting. A prototype of the robot is fabricated to test the validity of the proposed models. Jumping test results show that the energy efficiency is about 49.9% when we compare the kinetic energy of the robot and the elastic energy stored in the springs. 83.4% of the energy is released from the spring as the simulation result indicated. About 33.5% of the elastic energy is lost. The main source of energy loss is frictions on the rotational joints of the robot. Bearings will be used in the

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J. Zhang et al. / Mechatronics 23 (2013) 1123–1140

Fig. 30. The continuous jumping test results. The robot can overcome two steps of stairs in about 108 s.

Fig. 31. The sequences of forward jumping in an outdoor environment.

next generation robot design to improve energy efficiency. The experimental result of the rotation angle of the robot in the air is smaller than the simulation result. The robot model is an ideal rigid model. The deformation of the leg under the reaction force of the ground, the friction force on the joints of the four-bar mechanism, and the damping of the springs are not considered in the simulated model. In the future, these factors will be considered and the model of the robot will be refined to close to the real robot prototype. The force test results on the lower leg of the robot during the pretakeoff stage show a similar changing tendency to the simulation results. The current of the motor tests help us to calculate the energy consumption of jumping. The reaction force test results help us to learn more about the performances of the robot and will help us to optimize the mechanisms of the robot in the future. The self-righting test results verify the effectiveness of the proposed self-righting method. The current of the motor indicates its torque changes have the same tendency with the simulation result. The maximum of the torque is in the beginning of self-righting. This helps us to choose a proper motor for self-righting driving. The steering test video sequences and current of the motor show the steering capability of the robot. The steering speed can be adjusted by the length of the pole leg, the compressing angle of the four-bar mechanism, and the rotational speed of the motor. The

takeoff angle adjusting test result shows the validity of the takeoff angle adjusting principle. The experimental results show that the deformation of the lower leg can cause the offset of the takeoff angles. The continuous jumping capability of the robot is also tested. The robot can overcome two-step stairs in about 108 s. An outdoor forward jumping test result shows the robot can continuously jump about 2 m far in 156 s. The ground is not as smooth and flat as the indoor testbed. The robot fails to self-right or steer sometimes. Although this prototype of the robot has the basic locomotion capabilities to jump continuously, the self-righting and steering functions only can work well on smooth and flat surfaces. The self-righting and steering also need to be controlled by human operators. The future work includes four aspects: (1) The jumping model of the robot will be refined with considering of the friction forces of the joints and damping of the springs. The mechanisms of the robot will be optimized to improve the energy conversion efficiency. (2) Onboard motor current detecting combined with the tilt angle sensing by using a three axis accelerometer will be done to improve the stability of self-righting. (3) The speed of the adjusting motor will be controlled changing against the rotation angle of the pole leg to avoid failure of steering. (4) The rotation of the body in

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J. Zhang et al. / Mechatronics 23 (2013) 1123–1140

the air will be investigated to help us optimize the parameters of the robot to reduce the rotation angle and help the robot to land safely and lower the impact on the ground.

m12 ¼ m21 ¼ 0; m13 ¼ m31 ¼ f½m1 l1 þ m2 ðr 0 þ r 1 Þ þ m3 r0 þ m4 r 0  sin h1 þ ðm2 l2 þ m3 l2

Acknowledgements

þ m4 r3 Þ sin ðh1 þ h2 Þ þ m4 l4 sin ðh1 þ aÞg;

The research reported in this paper was carried out at the Robotic Sensor and Control Lab, School of Instrument Science and Engineering, Southeast University, Nanjing, Jiangsu, China. This work was supported in part by the Scientific Research Foundation of Graduate School of Southeast University under Grant YBJJ-1221, Program for New Century Excellent Talents in University under Grant NCET-10-0330, and Natural Science Foundation of Jiangsu Province under Grant BK2011254.

m14 ¼ m41 ¼ ðm2 l2 þ m3 l2 þ m4 r 3 Þ sin ðh1 þ h2 Þ; m22 ¼ m1 þ m2 þ m3 þ m4 ; m23 ¼ m32 ¼ ½m1 l1 þ m2 ðr0 þ r1 Þ þ m3 r 0 þ m4 r 0  cos h1 þ ðm2 l2 þ m3 l2 þ m4 r3 Þ cos ðh1 þ h2 Þ þ m4 l4 cos ðh1 þ aÞ;

Appendix A

m24 ¼ m42 ¼ ðm2 l2 þ m3 l2 þ m4 r 3 Þ cos ðh1 þ h2 Þ; A.1. The dynamic equations of jumping model in the pre-takeoff stage

M 22 ¼



m11

m12

m21

m22

 ðA1Þ

  2 2 2 m33 ¼ m1 l1 þ m2 ½ðr0 þ r1 Þ2 þ l2  þ m3 r 20 þ l2   2 þ m4 r 20 þ r 23 þ l4 þ 2r0 l4 cos a þ 2½m2 ðr 0 þ r 1 Þl2 þ m3 r 0 l2 þ m4 r0 r 3  cos h2 þ 2m4 r 3 l4 cos ðh2  aÞ þ J 1 þ J 2 þ J 3 þ J 4 ;

where 2

2

m11 ¼ m1 l1 þ m2 ½ðr 0 þ r 1 Þ2 þ l2 þ 2ðr 0 þ r 1 Þl2 cos h2    2 þ m3 r 20 þ l2 þ 2r 0 l2 cos h2 h i 2 þ m4 r 20 þ r 23 þ l4 þ 2r 0 r 3 cos h2 þ 2r 3 l4 cos ðh2  aÞ þ 2r0 l4 cos a þ J 1 þ J 2 þ J3 þ J 4 ;

m22 ¼

þ

2 m3 l2

2

2

¼ m2 l2 þ m3 l2 þ m4 r 23 þ ½m2 ðr 0 þ r1 Þl2 þ m3 r 0 l2 þ m4 r0 r 3  cos h2 þ m4 r 3 l4 cos ðh2  aÞ þ J 2 þ J 3 þ J 4 ; 2

2

m44 ¼ m2 l2 þ m3 l2 þ m4 r 23 þ J 2 þ J 3 þ J 4 ;

m12 ¼ m21 h i   2 2 ¼ m2 l2 þ ðr 0 þ r 1 Þl2 cos h2 þ m3 l2 þ r 0 l2 cos h2

þ m4 r23 þ r0 r 3 cos h2 þ r 3 l4 cos ðh2  aÞ þ J 2 þ J 3 þ J 4 ; 2 m2 l2

m34 ¼ m43

þ

m4 r 23

ðA5Þ

where

c1 ¼ f½m1 l1 þ m2 ðr0 þ r 1 Þ þ m3 r0 þ m4 r 0  cos h1 þ ðm2 l2 þ m3 l2 þ m4 r3 Þ cos ðh1 þ h2 Þ þ m4 l4 cos ðh1 þ aÞgh_ 21  2ðm2 l2 þ m3 l2

þ J2 þ J3 þ J4

C 21 ¼ ½c1 c2 T

C 41 ¼ ½c1 c2 c3 c4 T

ðA2Þ

where

þ m4 r3 Þ cos ðh1 þ h2 Þh_ 1 h_ 2  ðm2 l2 þ m3 l2 þ m4 r 3 Þ cos ðh1 þ h2 Þh_ 22 ; c2 ¼ ðm1 l1 þ m2 ðr 0 þ r 1 Þ þ m3 r 0 þ m4 r0 Þ sin h1 h_ 21  m4 l4 sin ðh1

c1 ¼ ½ðm2 r0 l2 þ m2 r1 l2 þ m3 r 0 l2 þ m4 r 0 r3 Þ sin h2   þ m4 r 3 l4 sin ðh2  aÞ 2h_ 1 h_ 2 þ h_ 2 þ ½m1 l1 þ m2 ðr 0 þ r 1 Þ

þ aÞh_ 21  ðm2 l2 þ þm3 l2 þ m4 r 3 Þ sin ðh1   þ h2 Þ h_ 21 þ 2h_ 1 h_ 2 þ h_ 22 þ ðm1 þ m2 þ m3 þ m4 Þg;

2

þ m3 r 0 þ m4 r0 g cos h1 þ ðm2 l2 þ m3 l2 þ m4 r 3 Þg cos ðh1 þ h2 Þ þ m4 gl4 cos ðh1 þ aÞ; c2 ¼ f½m2 ðr 0 þ r 1 Þl2 þ m3 r 0 l2 þ m4 r 0 r 3  sin h2 þ m4 r 3 l4 sinðh2  aÞgh_ 21 þ ½ðm2 þ m3 Þl2 þ m4 r 3 g cosðh1 þ h2 Þ þ Kðh2  p=3Þ Q 21 ¼ ½q1 q2 T ¼ ½0 0T

ðA3Þ

c3 ¼ f½m2 ðr 0 þ r 1 Þl2 þ m3 r 0 l2 þ m4 r 0 r 3  sin h2 þ m4 r 3 l4 sinðh2    aÞg 2h_ 1 h_ 2 þ h_ 22 þ ½m1 l1 þ m2 ðr0 þ r1 Þ þ m3 r 0 þ m4 r0 g cos h1 þ ðm2 l2 þ m3 l2 þ m4 r 3 Þg cos ðh1 þ h2 Þ þ m4 gl4 cos ðh1 þ aÞ; c4 ¼ f½m2 ðr0 þ r1 Þl2 þ m3 r 0 l2 þ m4 r 0 r 3  sin h2 þ m4 r3 l4 sin ðh2

A.2. The dynamic equations of jumping model in the post-takeoff stage

2 M 44

3

m11

m12

m13

m14

6m 6 21 ¼6 4 m31

m22

m23

m32

m33

m24 7 7 7 m34 5

m41

m42

m43

m44

where

m11 ¼ m1 þ m2 þ m3 þ m4 ;

 aÞgh_ 21 þ ðm2 l2 þ m3 l2 þ m4 r3 Þg cos ðh1 þ h2 Þ þ Kðh2  p=3Þ Q 41 ¼ ½q1 q2 q3 q4 T ¼ ½0 0 0 0T

ðA6Þ

ðA4Þ References [1] Fiorini P, Hayati S, Heverly M, Gensler J. A hopping robot for planetary exploration. In: Proc of IEEE Aeros Conf, vol.2, Pasadena; 1999. p. 153–8. [2] Burdick J, Fiorini P. Minimalist jumping robot for celestial exploration. Int J Robot Res 2003;22(7):653–74.

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