Proceedings of 2014 IEEE International Conference on Mechatronics and Automation August 3 - 6, Tianjin, China
Modeling and Simulation of a Bio-Inspired Symmetrical Jumping Robot Jun Zhang, Kai Ding, Ying Zhang, Xi Yang and Guangming Song School of Instrument Science and Engineering Southeast University Nanjing 210096, China
[email protected] can self-right after landing with an egg-shaped shell around its body and the low center of mass. The cages around the jumping robots [9-11] can help them recover passively. In [12], two hemispherical cages are mounted on the outer surface of two wheels to make the robot passively self-right. The jumping robots [13-14] with symmetrical body structures can self-right passively. The cylinder-shaped robot [15] is capable of self-righting using a cylindrical frame. The jumping robot [16] is equipped with a large wing-like mechanism for passive stabilization after landing. There are also some hopping robots which can land passively stable using their big feet and proper designed controllers [17-19]. The active self-righting robots are more robust than passive self-righting robots. However, they need more driving mechanisms than passive ones. The advantages of the passive self-righting method are simple and easy. The egg-like shell and the cage can protect the robot from crashing during landing on the ground and make the robot roll passively after landing. But this method is mainly suitable for self-righting on the planar ground. The robot also can not be controlled accurately because of the passive rolling after landing. Jumping robots adopt several methods for steering such as robots with rotatable body and hybrid-structured jumping robots with wheels or legs. The first generation of the JPL robot [4] and the EPFL robot [11] can rotate their body relatively to the frames for steering. The rescue robot [12], the Scout [13], the Scoutrobot [14], and the surveillance robot [20] use two wheels to steer. The SandFlea [3], the wheeledLeg robot [21], and the AirHopper [22] steer with four wheels. In [23], the Mini-Whegs steers by using simplified universal joints. The MSU robot [5] adjusts its direction with a reduction gear after landing. Our previous jumping robot [67] also can steer with the proposed pole leg mechanism. In our research project, we plan to design miniature jumping robots as mobile sensor nodes which can be deployed by the unmanned aerial vehicles in unstructured environments. The self-righting and damage avoiding during landing in these environments are very important for this kind of robots. In nature, some creatures use shells to protect themselves from enemies [24]. Clams and tortoises are such kind of animals, and their shells are not regular spherical shapes. This not only can protect their soft bodies but also prevent passive rolling of the body. Inspired by the shape of clams, a novel jumping robot external shape configuration is proposed with two conical shells mounted on the upper and lower sides of the
Abstract – This paper presents the modeling and simulation of a jumping robot with passive self-righting and steering capabilities. Inspired by clams, two conical shells are mounted on the upper and lower sides of the robot’s body symmetrically to prevent the robot from damage during landing. The jumping mechanism of the robot is also designed symmetrically up and down. Therefore, the robot can self-right passively and take off when either its upper shell or lower shell contacts with the ground after landing. A wheel in the body of the robot can be used to steer before takeoff when the springs of the jumping mechanism are compressed. The simplified vertical jumping is modeled and simulated. The 3D model of the robot is designed. The kinematic and dynamic simulations of the jumping and steering are conducted by using ADAMS. The simulation results verify the feasibility of the design concepts of the jumping robot. Index Terms - Jumping robot, Passive self-righting, Steering, Bio-inspired robot, Damage avoiding, Two mass model
I. INTRODUCTION Jumping robots have been investigated by researchers for several decades. Jumping gait is more efficient in traversing rough terrain than 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 in unstructured environments, jumping robots can be applied in many fields such as search and rescue [1], urban warfare [2], and planetary exploration [3]. However, this kind of robots has not been used widely in these fields for some challenging problems in jumping robot design, especially for robots with small size and few degrees of freedom, such as self-righting, steering, and damage avoiding during landing. There are two types of self-righting methods for jumping robots. The first one is the active self-righting method. Jumping robots with the active self-righting capability usually have several stable landing states. They use arms or legs to make the robot recover actively. In [4], the second generation JPL robot uses flaps to make it roll onto its back face and rotates a large flap to force itself toward an upright configuration. In [5], the robot utilizes two legs like the flaps to prop up its body. Our previous jumping robot [6-7] also can actively self-right with the help of a pole leg for propping up its body when it falls down to left or right side after landing. The second self-righting method is the passive selfrighting method. The first generation JPL jumping robot [8]
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Fig. 1. The diagram of the proposed symmetrical jumping robot inspired by clams.
robot body symmetrically as shown in Fig. 1. The jumping and steering mechanisms are protected by the shells. This conical shape body also can prevent the robot from rolling wildly and becoming out of control after landing. In the next section, we will give the simplified vertical jumping model and simulation results firstly. The robot design and its 3D model will be given in Section III. The kinematic and dynamic simulations of jumping and steering are shown in Section IV. Finally, we will conclude this paper and introduce the future work.
(a) (b) Fig. 2. Vertical jumping model of the proposed jumping robot. (a) The lower springs are compressed and the upper ones are stretched before takeoff. (b) The springs are released and the body begins to rise up for jumping.
the air resistance, and the damping of the springs. One jumping cycle consists of a pre-takeoff (PRT) stage and a post-takeoff (POT) stage. During the PRT stage, the robot has one degree of freedom (DOF). We use z as the independent variable to represent the DOF. During the POT stage, the robot has two degrees of freedom. xf and z are the two independent variables to represent the two DOFs. The robot will take off from the ground when Fz decreases to zero. The dynamic model of jumping is expressed by a Lagrange equation as follows wL d wL ° ( ) ( ) Qi wqi (4) ® dt wqi °L T V ¯
II. VERTICAL JUMPING MODELING AND SIMULATION A. Modeling The simplified vertical jumping model of the robot is shown in Fig. 2. The robot is composed of a body, an upper frame, a lower frame, four extension springs and two sliding ways. The two frames are connected by the sliding ways. The body can slide along the ways. The mass and structure parameters of the body and the shells are shown in Fig. 2. The upper and lower shells of the robot have the same shape. The stiffness coefficient and initial length of the spring are K and D. The deformation of the springs is l when the springs are completely compressed. The positions zci of the center of mass (COM) of the three parts are as follows zc1 z f r / 2 ° (1) ® zc 2 z f r z R / 2 ° ¯ zc 3 z f 3r / 2 2 D R The position zc, the velocity vc, and the acceleration Ac of the COM of the robot is as follows 3 3 ° zc ¦ mi zci / ¦ mi 1 1 ° 3 3 ° (2) ®vc ¦ mi zci / ¦ mi 1 1 ° 3 3 ° zci / ¦ mi ° Ac ¦ mi 1 1 ¯ From (2), we can obtain the ground reaction force on the lower shell of the robot as follows Fz
3
3
1
1
¦ mi zci ¦ mi g
where T is the kinetic energy which includes the translational kinetic energy Ek of the three parts of the robot as follows 1 3 T Ek (5) ¦ mi vci 2 2 1 V is the potential energy which includes the gravitational potential energy Eg of the parts and the elastic potential energy Ee of the extension springs as follows 3 ° Eg ¦ mi gzci (6) 1 ® ° E 2 K ( z D) 2 ¯ e When the springs are triggered, the energy stored in the springs is released and the robot obtains an initial velocity. The robot takes off when the vertical force on the shell of the robot is zero. Using (3), we can calculate the end time of the PRT stage. qi are the generalized coordinates and Qi are the generalized forces. In the PRT stage, the generalized coordinates of qi is z. From (4), we can get a two second-order nonlinear differential equation as follows m2 z m2 g 4 K ( z D) 0 (7) In the POT stage, the generalized coordinates of qi include zf and z. Two second-order nonlinear differential equations also can be derived from (4) as follows
(3)
In order to simplify the modeling and simulation, we do not consider the masses of the sliding ways and the springs, the sliding friction forces between the body and the sliding ways,
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0.008 0.012 0.016 (c) Time (s) Fig. 3. Jumping simulation results of the PRT stage. The changes of the displacement and velocity of the robot’s COM are shown in (a) and (b). (c) is the ground reaction force on the robot.
0.3 0.4 0.5 (c) Time (s) Fig. 4. Jumping simulation results of the POT stage. The displacements of the COM and foot of the robot, and distance z are shown in (a). (b) and (c) are the velocity and acceleration of the COM of the robot.
° m1 m2 m3 z f m2 z m1 m2 m3 g 0 (8) ® z f m2 z m2 g 4 K ( z D ) 0 °¯m2 B. Simulation The two stages of jumping are simulated. The parameters of the robot used for simulations are m1 = 0.015 kg, m2 = 0.12 kg, m3 = 0.015 kg, r = 0.01 m, R = 0.035 m, D = 0.045 m, and K = 300 N/m. The initial conditions of the PRT stage are z = 0.015 m and z = 0 m/s. The simulation results of this stage are shown in Fig. 3. zc and vc increase from 0.0485 m to 0.0727 m and 0 m/s to 2.3195 m/s respectively as shown in Fig. 3 (a) and (b). The ground reaction force decreases from the maximum value of 36.29 N to 0 N gradually. At the end time of t = 0.0161 s, the robot begins to lift off. The initial conditions of the POT stage are zf = 0 m, z f =
Although there are three masses in the robot model, it is still a two mass model [25], with m2 of the upper mass and (m1 + m3) of the lower mass. The energy conversion efficiency Șc is as follows Ek 0 Kc (9) Ee 0 where Ek0 is the initial kinetic energy of the robot and Ee0 is the initial elastic potential energy of the spring. The relationship between ratio (m1 + m3) / m2 and Șc is simulated. The simulation results are shown in Fig. 5. Șc drops quickly with the increase of (m1 + m3) / m2. The energy conversion efficiency is lower than 75% when the ratio is 25%, while Șc is near 90% when the ratio is 5%. So, trying to reduce the mass of (m1 + m3) as light as possible will improve the energy
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0 m/s, z = 0.0452 m, and z = 2.8994 m/s. The simulation results of this stage are illustrated in Fig. 4. The COM of the robot rises from 0.0727 m to the maximum of 0.3472 m and then declines to 0.0677 m at the end when zf = 0 m. zf has the same variation trend with zc but has slight fluctuations. z only has slight fluctuation which means the body shakes between the two shells after the robot lifting off. The velocity decreases from 2.3195 m/s to í2.3404 m/s linearly. The acceleration keeps at í9.8 m/s2 and remains unchanged. The simulation results indicate that the jumping process is an ideal projectile motion.
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Fig. 5. Relationship between the energy conversion efficiency and the ratio (m1 + m3) / m2.
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Fig. 7. CAD model of the steering mechanism.
B. Steering Mechanism The steering mechanism is illustrated in Fig. 7. The upper shell and the lower shell are mounted on the upper frame and the lower frame respectively. A steering wheel is driven by the steering motor to adjust the direction of the robot when the lower springs are compressed and the wheel contacts with the ground through the gap on the lower shell. The steering velocity Ȧ is R Z 2S N 2 (11) R1 where N is the rotational speed of the steering motor. The simulation of steering will be given in the next section.
Fig. 6. CAD model of the symmetrical jumping mechanism.
conversion efficiency of the jumping robot. This is in accordance with the analyses results of the jumping robot model in [26]. III. JUMPING ROBOT DESIGN
A. Jumping Mechanism The 3D model of the jumping mechanism is shown in Fig. 6. It is composed of an upper frame, a lower frame, a body frame, three sliding ways, three upper extension springs, three lower extension springs, a DC motor, a gear box with an incomplete gear and a shaft-gear, a winch, two pulleys, and two pulling ropes. The body frame can be driven to slide along the sliding ways downward when the motor rotates clockwise and the lower polling rope twines around the winch. During this process, the lower springs will be compressed while the upper springs will be stretched for elastic energy storing. The deformation ¨l of the springs, the winding length ¨r of the rope, and the radius rr of the winch have the relationships as follows n 'l 'r 2S rr 2 (10) n1 where n1 is the total number of the teeth of the shaft-gear, and n2 is the number of the rest teeth of the incomplete gear. The motor rotates anticlockwise to compress the upper springs and stretch the lower springs when the robot body is reversed and the upper frame is contacting with the ground. Therefore, the symmetrical jumping mechanism can let the robot jump without considering which side contacts with the ground after landing. The kinematic and dynamic simulations of jumping with two shells mounted up and down of the robot body will be given in Section IV.
IV. KINEMATIC AND DYNAMIC SIMULATION The kinematics and dynamics of the designed 3D model of the proposed robot are simulated in Automatic Dynamic Analysis of Mechanical Systems (ADAMS). The jumping and steering simulations are conducted respectively.
Fig. 8. Sequences of jumping simulation. The robot jumps from left side to right side with the takeoff velocity of v0 and takeoff angle of ș.
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Fig. 10. Scenario of steering simulation.
(b)
(c) Fig. 9. Simulation results of jumping. (a) and (b) are the changes of the displacements in x and y directions. (c) is the reaction forces on x and y directions.
Fig. 11. Steering simulation results. X, Y, and Z are the displacements of the robot in the three directions during steering.
A. Jumping Simulation A simplified model of the robot is used to simulate its jumping performances. The stiffness coefficients of the two extension springs up and down are set as 600 N/m which is equal to the values used in vertical jumping simulations. The damping coefficients of the springs are set as 2.8Eí002 N·s/m. The preload forces of the upper and lower springs are set as í18 N and 18 N respectively. The static and dynamic coefficients of the friction forces between the robot and ground are set as 0.3 and 0.1 respectively. The sequences of jumping in the simulation are illustrated in Fig. 8. The robot jumps from left side to right side with the takeoff velocity v0 and takeoff angle of ș. The simulation results of jumping are depicted in Fig. 9. The displacement in x direction shown in Fig. 9 (a) is about 0.16 m. The maximum jumping height shown in Fig. 9 (b) is about 0.18m. So, v0 and ș are about 1.92 m/s and 77.5° respectively as calculated. The ground reaction forces in x and y directions are shown in Fig. 9 (c). Fx and Fy increase sharply from zero to their maximum values 15.7 N and 86.6 N respectively and decrease rapidly to zero at 0.011 s. After t = 0.011 s, the forces remain unchanged which means that the robot lifts off the ground. B. Steering Simulation The proposed steering method is also simulated by using a simplified model as illustrated in Fig. 10. The simulation
results of 360° steering are shown in Fig. 11. The rotational speed of the wheel is 5 °/s as set. The radius R2 of the wheel is 0.028 m. The distance R1 between the contacting point of the wheel on the ground and the conical point of the shell is 0.084 m. From (10) we can obtain the steering time in a circle is about 215 s. However, the simulated time in a circle of steering is about 172 s which is less than the calculated result. This is because of the slippage between the shell and the ground in the steering simulation but no consideration in the calculation. V. CONCLUSIONS AND FUTURE WORK This paper presents a novel symmetrical jumping robot model. The symmetrical jumping mechanism and shells can help the robot recover passively, steer, and take off either its upper or lower shell contacts with the ground after landing. The shells also can prevent damage of the robot during landing. The vertical jumping simulation results verify the feasibility of the jumping robot model. The two mass model simulation results indicate that we should make the lower mass as light as possible in the robot prototype implementation to improve its energy conversion efficiency. The kinematic and dynamic simulation results of the 3D model validate the effectiveness of the designed jumping and
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[7] J. Zhang, G. Song, Z. Li, G. Qiao, H. Sun, and A. Song, “Self-righting, steering and takeoff angle adjusting for a jumping robot,” in: Proc. of IEEE International Conference on Intelligent Robots and Systems, Vilamoura, Algarve, Portugal, Oct. 7–12, 2012, pp. 2089–2094. [8] P. Fiorini, S. Hayati, M. Heverly, and J. Gensler, “A hopping robot for planetary exploration,” in: Proc. of IEEE Aerospace Conference, Snowmass, CO., USA, 1999, vol. 2, pp. 153–158. [9] R. Armour, K. Paskins, A. Bowyer, J. F. V. Vincent, and W. Megill, “Jumping robots: a biomimetic solution to locomotion across rough terrain,” Bioinspiratoin and Biomimetics, vol. 2, pp. 65–82, Jun. 2007. [10]D. Aksaray and D. Mavris, “A Trajectory Tracking Algorithm For A Hopping Rotochute Using Surrogate Models,” in: Proc. of 2013 American Control Conference (ACC), Washington, DC, USA, June 17–19, 2013. pp. 2000–2005. [11]M. Kovaþ, M. Schlegel, J. Zufferey, and D. Floreano, “Steerable Miniature Jumping Robot,” Autonomous Robots, vol. 28, no. 3, pp. 295– 306, Apr. 2010. [12]E. Watari, H. Tsukagoshi, T. Tanaka, D. Kimura, and A. Kitagawa, “Development of a throw & collect type rescue inspector,” in: Proc. IEEE International Conference on Robotics and Automation, Rome, Italy, Apr. 10–14, 2007, pp. 2762–2763. [13]S. A. Stoeter and N. Papanikolopoulos, “Kinematic Motion Model for Jumping Scout Robots,” IEEE Transactions on Robotics, vol. 22, no. 2, pp. 398–403, Apr. 2006. [14]D. H. Kim, J. H. Lee, I. Kim, S. H. Noh, and S. K. Oho, “Mechanism, control, and visual management of a jumping robot,” Mechatronics, vol. 18, no. 10, pp. 591–600, Dec. 2008. [15]T. Ho, and S. Lee, “A novel design of a robot that can jump and roll with a single actuator,” in: Proc. of IEEE/RSJ International Conference on Intelligent Robots and Systems, Vilamoura, Algarve, Portugal, Oct. 7–12, 2012, pp. 908–913. [16]F. Li, W. Liu, X. Fu, G. Bonsignori, U. Scarfogliero, C. Stefanini, and P. Dario, “Jumping like an insect: design and dynamic optimization of a jumping mini robot based on bio-mimetic inspiration,” Mechatronics, vol. 22, no. 2, pp. 167–176, Mar. 2012. [17]Z. Xu, T. Lü, and F. Ling, “Trajectory planning of jumping over obstacles for hopping robot,” Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol. 30, no. 4, pp. 327–334, Oct. 2008. [18]T. Wu, T. J. Yeh, and B.H. Hsu, “Trajectory planning of a one-legged robot performing stable hop,” in: Proc. of IEEE/RSJ International Conference on Intelligent Robots and Systems, Taipei, Taiwan, Oct. 18– 22, 2010, pp. 4922–4927. [19]X. Yu and F. Iida, “Minimalistic Models of an Energy-Efficient VerticalHopping Robot,” IEEE Transation on Industrial Electronics, vol. 61, no. 2, pp. 1053–1062, Feb. 2014. [20]G. Song, K. Yin, Y. Zhou, X. Cheng, “A surveillance robot with hopping capabilities for home security,” IEEE Transaction on Consumer Electronics, vol. 55, no. 4, pp. 2034–2039, Nov. 2009. [21]J. A. Smith, James Andrew, I. Poulakakis, M. Trentini, and I. Sharf, “Bounding with active wheels and liftoff angle velocity adjustment,” International Journal of Robotics Research, vol. 29, no. 4, pp. 414–427, Apr. 2010. [22]T. Tanaka, and S. Hirose, “Development of leg-wheel hybrid quadruped “AirHopper”,” in: Proc. IEEE/RSJ International Conference on Intelligent Robots and Systems, Sep. 22–26 2008, pp. 3890–3895. [23]B. G. A. Lambrecht, A. D. Horchler, R. D. Quinn, “A small, insectinspired robot that runs and jumps,” in: Proc. IEEE International Conference on Robotics and Automation, Barcelona, Spain, Apr. 18–22, 2005, pp. 1240–1245. [24]W. Zhang, C. Wu, C. Zhang, and Z. Chen, “Numerical Study of the Mechanical Response of Turtle Shell,” Journal of Bionic Engineering, vol. 9, no. 3, pp. 330–335, Oct. 2012. [25]F. B. Mathis and R. Mukherjee, “Apex Height Control of a Two-Mass Hopping Robot,” in Proc. IEEE International Conference on Robotics and Automation, Karlsruhe, Germany, May 6–10, 2013, pp. 4785–4790. [26]J. Zhao, R. Yang, N. Xi, B. Gao, X. Fan, M. W. Mutka, and L. Xiao, “Development of a miniature self-stabilization jumping robot,” in Proc. IEEE/RSJ International Conference on Intelligent Robots and Systems, St. Louis, USA, Oct. 11–15, 2009, pp. 2217–2222.
Fig. 12. Scenario of miniature symmetrical jumping robots deployment in unstructured environment by using a quadrotor carrier robot.
steering mechanisms and give us guidelines for the prototype design of the robot in the future. The future work will focus on four aspects. (1) The prototype of the robot will be implemented and optimizations of the jumping and steering mechanisms will be conducted. (2) The collision during landing will be modeled and simulated. (3) Multiple jumping robots deployment by using a quadrotor carrier robot as depicted in the scenario of Fig. 12 and the damage avoiding performance will be investigated. (4) We also will study the applications of this kind of robots in unstructured environments for information detection, search and rescue, and bridging communication in wireless sensor networks. ACKNOWLEDGMENT This work was supported in part by the Fundamental Research Funds for the Central Universities under Grant 2242014R20018, Jiangsu Planned Projects for Postdoctoral Research Funds under Grant 1302064B, Natural Science Foundation of China under Grant 61375076, and Natural Science Foundation of Jiangsu Province under Grant BK2011254. REFERENCES [1] S. Dubowsky, S. Kesner, J. Plante, and P. Boston, “Hopping mobility concept for search and rescue robots,” Industrial Robot, vol. 35, no. 3, pp. 238–245, 2008. [2] E. Ackerman, “Boston dynamics sand flea robot demonstrates astonishing jumping skills,” IEEE Spectrum Robot. Blog, Mar. 22, 2012. [3] S. D. Howe, R. C. Obrien, R. M. Ambrosi, B. Gross, J. Katalenich, et al., “The mars hopper: an impulse driven, long range, long-lived mobile platform utilizing in-situ Martian resources,” Acta Astronaut, vol. 69, no. 11-12, pp. 1050–1056, 2011. [4] J. Burdick, and P. Fiorini, “Minimalist jumping robot for celestial exploration,” International Journal of Robotics Research, vol. 22, no. 7, pp. 653–674, 2003. [5] J. Zhao, J. Xu, B. Gao, N. Xi, F.J. Cintrón, M.W. Mutka, L. Xiao, “MSU Jumper: A Single-Motor-Actuated Miniature Steerable Jumping Robot,” IEEE Transactions on Robotics, vol. 29, no. 3, pp. 602–614, Jun. 2013. [6] J. Zhang, G. Song, Y. Li, G. Qiao, A. Song, and A. Wang, “A BioInspired Jumping Robot: Modeling, Simulation, Design, and Experimental Results,” Mechatronics, vol. 23, no. 8, pp. 1123–1140, Dec. 2013.
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