Effect of Spine Motion on Mobility in Quadruped Running

CHEN Dongliang*,LIU Qi,DONG Litao,WANG Hong,and ZHANG Qun

College of Mechanical and Electrical Engineering,Harbin Engineering University,Harbin 150001,China

1 Introduction*

Legged mammals exhibit impressive performance in real nature world.For example,they have good mobility in rough terrain.In the field of legged robots,the quadruped robot is a good trade-off between mobility and the complexity of structure and control.So,many robotics researchers take a strong interest in quadruped robots.

Early attempts are mostly in designing statically stable quadruped robots(walking quadruped robots).Because the mobility of walking quadruped robots is not satisfactory,attention is gradually turned to dynamically stable quadruped robots(running quadruped robots).In the 1980s,RAIBERT[1]set the stage with his groundbreaking work on dynamic legged robots by introducing a“three-part”controller for stabilizing running on his one-,two-,and four-legged machines.Afterwards,many researches on leg structure design,running gait and effective control method are carried out to improve running quadruped robots’performance.YU,et al[2],found that the stability of the system with nonlinear spring leg was better than that of the system with linear spring leg.XU,et al[3],developed a gait learning approach based on evolution algorithm.By using the approach,quadruped robots can learn locomotion in an unknown environment.So far,there are many different kinds of running quadruped robots[4–8].But these quadruped robots are very similar in morphology and feature a stiff body with four compliant legs.The research on body structure design of quadruped robots is ignored.

Spine motion is usually apparent in galloping quadruped mammals(especially the cheetah).Biologists have conducted a lot of researches on spine motion of animals.GRAY[9]found a primary function of spine motion was the kinematic extension of legs,which could effectively increase leg length.He also discussed how increased leg length due to spine motion increased running speed.Refs.[10–11]present spine motion can supply extra power for high speed.SCHILLING,et al[12],studied the effect of spine motion on enhancing systems’stability.ALEXANDER,et al[13],and BERTRAM,et al[14],found spine motion had the ability to improve energy efficiency.These researches show spine motion has many important effects on animals’running performance.

Inspired by the findings on spine motion of animals,robotics researchers have also done some work on spine motion of quadruped robots.LEESER[15]built a planar quadruped robot with two spinal driving joints.With the robot,he explored the role of spine motion on legs’performance.He found that the two spinal driving joints could improve legs’performance and suggested that spine motion could modify the impedance characteristic between the front and rear legs.?ULHA,et al[16],proposed a simplified sagittal plane model of quadruped mammals.In the body of the simplified model,there is a spinal driving joint.They investigated how the bounding gait could be achieved for the simplified model.KHORAMSHAHI,et al[17],designed a quadruped robot with a spinal driving joint.Through experiments with the robot,they demonstrated that spine motion had the ability to improve systems’stability and energy efficiency.These researches show spine motion also has many important effects on quadruped robots’running performance and is worthy to be further investigated.

In the study about spine motion of quadruped robots,the running gait is very important for the results.HOYT,et al[18],showed animals naturally selected the rotary gallop gait for high speed running.And observing rotary galloping animals,spine motion is very obvious.A detailed description of the rotary gallop gait is provided by HILDEBRAND[19].By pairing both the front and rear legs,the rotary gallop gait can be simplified to the bounding gait.By using the bounding gait,planar quadruped robots and planar simplified models of quadruped mammals can achieve high running speed.And only in running with the bounding gait,can spine motion be presented.Therefore,the bounding gait is usually chosen in many researches on spine motion of quadruped robots[15–17].

An effective controller is necessary for planar quadruped robots and planar simplified models of quadruped mammals to achieve the bounding gait.To our best knowledge,there are three kinds of controllers:open-loop controller,feedback controller and biological controller.The open-loop controller is used in Refs.[5,16].A detailed description of the feedback controller is introduced in Refs.[7,15],and Refs.[17,20]detailedly present the biological controller.Among the three kinds of controllers,the open-loop controller is the most simple.The open-loop controller features a simple control law with a suitably designed mechanical system.The stable motion is purely the result of the controller interaction through the passive dynamics of the mechanical system.So,the open-loop controller is very suitable for the research on spine motion of quadruped robots.

Previous researches on spine motion of quadruped robots establish the foundation for later study.In previous work,some functions of spine motion have been studied and verified with experiments.But the effect of spine motion on mobility in quadruped running has not been investigated systematically and deeply.In this paper,a planar quadruped robot is designed and a simplified sagittal plane model of quadruped mammals is introduced.In the body of both the robot and the simplified model,there is a spinal driving joint.Six group prototype experiments with the robot and six group simulation experiments with the simplified model are carried out.Analyzing the results,we demonstrate that spine motion generated by a spinal driving joint can increase the average running speed(mobility)and find that the intrinsic reason of the speed increase is the improvement of the maximum rear leg horizontal thrust.

2 Prototype and Experiments

2.1 Planar quadruped robot

Fig.1 shows a planar quadruped robot with a spinal driving joint.We designed the robot based on cheetahs’morphology and built it mainly with aluminum tubes.Its mechanical structure in the sagittal plane is depicted in Fig.2.The robot has three main parts:a body,a front leg and a rear leg.The body consists of two same aluminum tubes connected by an articulated joint.The spinal joint is driven by a DC motor.Each leg has three joints:hip,keen and ankle.Hip and keen actuators are bidirectional pneumatic cylinders but ankle is a passive joint with a spring.Table 1 details the parameters of the robot.

Fig.1.Planar quadruped robot with a spinal driving joint

Fig.2.Mechanical structure of the robot in the sagittal plane

Table 1.Parameter values of the robot

The robot has following sensors:a contact switch on each foot to infer ground contacts and an incremental optical encoder on the DC motor to measure its angular displacement.Each actuator is driven by a PWM servo amplifier.The central processing unit(CPU)is a microprocessor STM32F103ZET6.An open-loop control method,developed by ?ULHA,et al[16],is programmed in CPU.

2.2 Experiment environment

Fig.3 shows the experiment environment.The motion of the robot is restricted to the sagittal plane by a mechanism.The restriction mechanism is a spherical pivot fixed to the ground.The spherical pivot and the robot are connected by an aluminum tube with a length of 2m.The restriction mechanism permits the robot to travel on a large circle.In the vertical axis of the spherical pivot,there is an incremental optical encoder which measures the stride length of the robot.The stride time is measured by a timer which is in CPU.Based on the recorded stride lengths and stride times,the average speeds can be calculated.The ground is flat and the static friction coefficient between feet and the ground is large enough that there is no need to consider foot slippage[21].

Fig.3.Experiment environment

2.3 Experiments and results

In the experiment environment,the robot can steadily locomote with the bounding gait.When the spinal driving joint acts,the robot has spine motion;otherwise,the robot has not spine motion.The range of the average speed of the robot with spine motion is from 123(mm?s–1)to 526(mm?s–1).The average speeds of the robot without spine motion vary from 153(mm?s–1)to 482(mm?s–1).Fig.4 presents four snapshots of the robot with spine motion at an average speed of 350(mm?s–1).In a full bounding cycle,the bounding gait has following phases:the rear stance phase(Snapshot 1),the flight phase(Snapshot 2),the front stance phase(Snapshot 3)and the double stance phase(Snapshot 4).The configuration of the body of the robot with spine motion transforms from convex(Snapshot 1)to concave(Snapshot 2 and Snapshot 3)and comes back to convex(Snapshot 4),which is agreement with observations on galloping animals.

To study the effect of spine motion on mobility,six group prototype experiments with the robot were carried out.Every group includes two comparative experiments:“Experiment A”and“Experiment B”.The spinal driving joint acts in Experiment A but does not in Experiment B.Other conditions of Experiment A and Experiment B are the same.The process of each group is as follows.

Experiment A:

(1)Allow the spinal driving joint to act;

(2)Adjust the average speed to a desired value(200(mm?s–1),250(mm?s–1),300(mm?s–1),350(mm?s–1),400(mm?s–1)and 450(mm?s–1)for the six groups respectively);

(3)Record the average stride length and stride time when the robot steadily bounds at the desired average speed.

Experiment B:

(1)Lock the spinal driving joint;

(2)Keep other conditions the same as Experiment A;

(3)Record the average stride length,stride time and speed when the robot steadily bounds.

Table 2 presents the average stride lengths,stride times and speeds of Experiment A of the six group prototype experiments and Table 3 corresponds to Experiment B.For the fist group,the average speed of Experiment A is 8.7%larger than that of Experiment B.The other five groups correspond to 10.6%,11.1%,12.9%,14.9% and 15.9%respectively.That is to say the average speeds of the robot with spine motion are 8.7%–15.9% larger than those of the robot without spine motion.Therefore,it is obvious that spine motion can allow the robot to bound at higher speeds.

Table 2.Average stride lengths,stride times and speeds of Experiment A of the six group prototype experiments

Table 3.Average stride lengths,stride times and speeds of Experiment B of the six group prototype experiments

GRAY[9]considered that the reason of speed increase due to spine motion was the increase of stride length.In each group,the average stride length of Experiment A(the robot with spine motion)is indeed larger than that of Experiment B(the robot without spine motion).However,the average stride time of Experiment A is also larger than that of Experiment B.So,we thought that the intrinsic reason of speed increase was not the increase of stride length.

The results of the six group prototype experiments demonstrate that spine motion can increase the average running speed and show that the intrinsic reason of speed increase is not the increase of stride length.To analyze the intrinsic reason of speed increase,the emulation technique was used.And in order to highlight the effect of spine motion,a simplified sagittal plane model of quadruped mammals was introduced.

3 Model and Simulations

3.1 Simplified model of quadruped mammals

A simplified sagittal plane model of quadruped mammals is depicted in Fig.5.The simplified model is proposed by ?ULHA,et al[16],and has been used in Refs.[22–23].The body of quadruped mammals is modeled as two same rigid beams connected by an articulated joint,as our robot.The leg systems are represented as massless springs.In the simplified model,there are three active joints:a spinal joint and two hip joints.The leg spring is only passively compressed.

The only difference between the simplified model and our robot is the leg structure.Using simpler leg structure(the spring leg)is better for highlighting the effect of spine motion.The spring leg has previously been used in many Refs.[1,5,15–16,22–25]because of its simple structure and good effect in emulating real animals’leg.

3.2 Simulation environment

The simulation environment is MSC Adams and Matlab-Simulink.Based on the simplified sagittal plane model of quadruped mammals,we built a simulation model in MSC Adams.The simulation model is shown in Fig.6.and Table 4 details the parameters of the simulation model.We chose the values of these parameters based on Ref.[17]and the morphology of the cheetah.The controller of the simulation model is the same as that of our robot.

Fig.6.Simulation model

Table 4.Parameter values of the simulation model

In simulations,the functions of MSC Adams are calculating dynamic equations and recording performance of the simulation model.The control method is programmed in Matlab-Simulink.The ground is modeled as a flat plate with infinite length.We set both the simulation step time and the communication interval between MSC Adams and Matlab-Simulink as 2 ms to ensure numerical accuracy of the results.And we set the static friction coefficient between feet and the ground as 0.9 to avoid foot slippage.

3.3 Simulations and results

In the simulation environment,the simulation model can steadily locomote with the bounding gait,as our robot.When the spinal driving joint acts,the simulation model has spine motion;otherwise,the simulation model has not spine motion.The running conditions of the simulation model are shown in Fig.7.Fig.7(a)and Fig.7(b)correspond to the simulation model with spine motion and the simulation model without spine motion respectively.

The range of the average speed of the simulation model with spine motion is from 443(mm?s–1)to 1863(mm?s–1).The average speeds of the simulation model without spine motion vary from 521(mm?s–1)to 1632(mm?s–1).The average speed of the simulation model with spine motion is generally larger than that of the robot with spine motion.The average speed of the simulation model without spine motion is also larger than that of the robot without spine motion.We though this relationship was mainly caused by the differences between the simulation model and the robot.But note that the differences have no effect on our research,studying the effect of spine motion on mobility.

In order to further investigate the effect of spine motion on mobility,especially to analyze the intrinsic reason of speed increase,six group simulation experiments with the simulation model were conducted.The process of the six group simulation experiments is similar with that of the six group prototype experiments.Every group includes two comparative simulations:“Simulation A”and“Simulation B”.The spinal driving joint acts in Simulation A but does not in Simulation B.Other conditions of Simulation A and Simulation B are the same.The adjusted average speeds of Simulation A are 600(mm?s–1),800(mm?s–1),1000(mm?s–1),1200(mm?s–1),1400(mm?s–1)and 1600(mm?s–1)for the six groups respectively.

Table 5 presents the average stride lengths,stride times and speeds of Simulation A of the six group simulation experiments and Table 6 corresponds to Simulation B.In the six groups,the average speeds of Simulation A are 15%-28.5% larger than those of Simulation B.So,the six group simulation experiments,as the six group prototype experiments,demonstrate that spine motion can increase the average running speed(mobility).Table 5 and Table 6 also show that the average stride lengths and stride times of Simulation A are larger than those of Simulation B respectively,as Table 2 and Table 3.That is to say the intrinsic reason of speed increase due to spine motion can not be got from Table 5 and Table 6.

Table 5.Average stride lengths,stride times and speeds of Simulation A of the six group simulation experiments

Table 6.Average stride lengths,stride times and speeds of Simulation B of the six group simulation experiments

To analyze the intrinsic reason of speed increase,we explored the change relationship between speed and time in the six group simulation experiments.The relation curves between speed and time from 1 s to 2 s of the fifth group,as an example,are plotted in Fig.8.Note that the speed cures of every group are similar.In Fig.8,the area between the two vertical dashed lines is the flight phase of a full bounding cycle.Fig.8(a)and Fig.8(b)correspond to Simulation A and Simulation B respectively.For a full bounding cycle,the maximum speed is in the flight phase(Fig.8).Fig.8 also shows that the flight time and flight speed of Simulation A is much larger than those of Simulation B respectively.

Table 7 presents the average flight times and stride times of Simulation A of the six group simulation experiments and Table 8 corresponds to Simulation B.In the six groups,the percentages of flight time to stride time of Simulation A are 16.4%-18.0% and Simulation B corresponds to 6.2%-7.4%.It is verified that the flight time of the simulation model with spine motion is indeed much longer than that of the simulation model without spine motion.So,the flight time and flight speed have huge contribution to the average running speed.

Table 7.Average flight times and stride times of Simulation A of the six group simulation experiments

Table 8.Average flight times and stride times of Simulation B of the six group simulation experiments

Why are both the flight time and flight speed of the simulation model with spine motion larger than those of the simulation model without spine motion respectively?In a full bounding cycle,the flight phase comes after the rear stance phase(Fig.4).Considered the system’s dynamics in the rear stance phase,the reason may be the horizontal thrust of the rear leg.HUDSON,et al[26],suggested that the horizontal thrust of the rear leg had very important influence on animals’running performance.Therefore,we also explored the change relationship between rear leg horizontal thrust and time in the six group simulation experiments.

The relation curves between rear leg horizontal thrust and time from 1 s to 2 s of the fifth group,as an example,are plotted in Fig.9.Note that the rear leg horizontal thrust cures of every group are similar.Fig.9(a)and Fig.9(b)correspond to Simulation A and Simulation B respectively.In Fig.9,the non-zero areas are the rear stance phases.Fig.9 shows that the rear leg horizontal thrusts of both Simulation A and Simulation B vary from negative values to positive values in the rear stance phase.In the negative area,the simulation model decelerates and its kinetic energy decreases.In the positive area,the simulation model accelerates and its kinetic energy increases.Therefore,there is an obvious conclusion:the maximum rear leg horizontal thrust is large;the increased kinetic energy is big;both the flight time and flight speed are naturally large.

Fig.9.Relation curves between rear leg horizontal thrust and time of the fifth group simulation experiments

Table 9 presents the maximum rear leg horizontal thrusts of the six group simulation experiments.For the fist group,the maximum rear leg horizontal thrust of Simulation A is 71.1% larger than that of Simulation B.The other five groups correspond to 69.0%,71.3%,68.2%,71.0% and 69.2% respectively.That is to say the maximum rear leg horizontal thrusts of the simulation mode with spine motion are 68.2%-71.3% larger than those of the simulation mode without spine motion.Therefore,it is obvious that spine motion can increase the maximum rear leg horizontal thrust.

Table 9.Maximum rear leg horizontal thrusts of the six group simulation experiments

In conclusion,the intrinsic reason of speed increase due to spine motion is the improvement of the maximum rear leg horizontal thrust.Spine motion is introduced to the robot and the simplified model by adding a spinal driving joint to the stiff body.Spine motion can expand legs’motion space.Thus,the rear leg can make contact with the ground at a more suitable angle which is good for leg performance.So,the maximum rear leg horizontal thrust is naturally large.Under the action of the large rear leg horizontal thrust,the robot and the simplified model can obtain high flight time and flight speed.Because the flight time and flight speed have huge contribution to the average running speed,the robot and the simplified model with spine motion can bound at higher average speeds than the robot and the simplified model without spine motion.Note that the large rear leg horizontal thrust increases the possibility of foot slippage.So,the static friction coefficient between feet and the ground is very important for stable locomotion.Real animals,like the cheetah,usually use their claws to increase the static friction coefficient at high running speeds.

4 Conclusions

(1)Spine motion is introduced to a planar quadruped robot and a simplified sagittal plane model of quadruped mammals by adding a spinal driving joint to the stiff body.Both the robot and the simplified model can run with the bunding gait at high speeds.The spinal driving joint performs well in emulating real animals’spine motion.

(2)The six group prototype experiments with the robot demonstrate spine motion can increase the average running speed(mobility)and show the intrinsic reason of speed increase is not the increase of stride length.

(3)The results of the six group simulation experiments with the simulation model present the flight time and flight speed have huge contribution to the average running speed and the intrinsic reason of speed increase is the improvement of the maximum rear leg horizontal thrust.

[1]RAIBERT M H.Legged robots that balance[M].Cambridge MA:The MIT Press,1986.

[2]YU Haitao,LI Mantian,CAI Hegao.Analysis on the performance of the SLIP runner with nonlinear spring leg[J].Chinese Journal of Mechanical Engineering,2013,26(5):892–899.

[3]XU Tao,CHEN Qijun.Learning gait of quadruped robot without prior knowledge of the environment[J].Chinese Journal of Mechanical Engineering,2012,25(5):1068–1074.

[4]ESTREMERA J,WALDRON K J.Thrust control,stabilization and energetics of a quadruped running robot[J].Int.Journal of Robotics Research,2008,27(10):1135–1151.

[5]POULAKAKIS I,PAPADOPOULOS E,BUELER M.On the stability of the passive dynamics of quadrupedal running with a bounding gait[J].Int.Journal of Robotics Research,2006,25(7):669–687.

[6]SMITH J A,POULAKAKIS I,TRENTINI M,et al.Bounding with active wheels and liftoff angle velocity adjustment[J].Int.Journal of Robotics Research,2009,29(4):414–428.

[7]RAIBERT M H,BLANKESPOOR K,NELSON G,et al.Bigdog,the rough-terrain quadruped robot[C]//Proceedings of the 17th World Congress,Seoul,Korea,July 6–11,2008:10 823–10 825.

[8]SPR?WITZ A,TULEU A,VESPIGNANI M,et al.Towards dynamic trot gait locomotion:design,control,and experiments with Cheetah-cub,a compliant quadruped robot[J].Int.Journal of Robotics Research,2013,32(8):932–950.

[9]GRAY J.How animals move[M].London:Cambridge University Press,1960.

[10]HILDEBRAND M.Motions of the running cheetah and horse[J].Journal of Mammalogy,1959,40(4):481–495.

[11]ENGLISH A W.The functions of the lumbar spine during stepping in the cat[J].Journal of Morphology,1980,165(1):55–66.

[12]SCHILLING N,HACKERT R.Sagittal spine movements of small therian mammals during asymmetrical gaits[J].Journal of Experimental Biology,2006,209(19):3925–3939.

[13]ALEXANDER R,DIMERY N J,KER R F.Elastic structures in the back and their role in galloping in some mammals[J].Journal of Zoology,1985,207(4):467–482.

[14]BERTRAM J E A,GUTMANN A.Motions of the running horse and cheetah revisited:fundamental mechanics of the transverse and rotary gallop[J].Journal of The Royal Society Interface,2009,6:549–559.

[15]LEESER K F.Locomotion experiments on a planar quadruped robot with articulated spine[D].Boston:Massachusetts Institute of Technology,1996.

[16]?ULHA U,SARANLI U.Quadrupedal bounding with an actuated spinal joint[C]//IEEE International Conference on Robotics and Automation,Shanghai,China,May 9–13,2011:1392–1397.

[17]KHORAMSHAHI M,SPR?WITZ A,TULEU A,et al.Benefits of an active spine supported bounding locomotion with a small compliant quadruped robot[C]//IEEE International Conference on Robotics and Automation,Karlsruhe,Germany,May 6–10,2013:3329–3334.

[18]HOYT D F,TAYLOR C R.Gait and the energetics of locomotionin horses[J].Nature,1981,292(5820):239–240.

[19]HILDEBRAND M.Further studies on locomotion of the cheetah[J].Journal of Mammalogy,1961,42(1):84–91.

[20]ZHANG Xiuli,E Mingcheng,ZENG Xiangyu,et al.Adaptive walking of a quadrupedal robot based on layered biological reflexes[J].Chinese Journal of Mechanical Engineering,2012,25(4):654–664.

[21]WANG Zheng,TAN Jianrong,LIU Zhenyu,et al.Modeling compliant non-penetration constraint for VP motion simulation[J].Chinese Journal of Mechanical Engineering,2005,18(2):163–168.

[22]DENG Qi,WANG Shiguang,XU Wei,et al.Quasi passive bounding of a quadruped model with articulated spine[J].Mechanism and Machine Theory,2012,52:232–242.

[23]CAO Qu,POULAKAKIS I.Quadrupedal bounding with a segmented flexible torso:passive stability and feedback control[J].Bioinspiration &Biomimetics,2013,8(4):046007.

[24]SEYFARTH A,GEYER H,GüNTHER M,et al.A movement criterion for running[J].Journal of Biomechanics,2002,35(5):649–655.

[25]GEYER H,SEYFARTH A,BLICKHAN R.Compliant leg behaviour explains basic dynamics of walking and running[J].Proceedings of the Royal Society B:Biological Sciences,2006,273(1603):2861–2867.

[26]HUDSON P E,CORR S A,WILSON A M.High speed galloping in the cheetah(Acinonyx jubatus)and the racing greyhound(Canis familiaris):spatio-temporal and kinetic characteristics[J].Journal of Experimental Biology,2012,215(14):2425–2434.