Design and Demonstration of a Flying-Squirrel-Inspired Jumping Robot with Two Modes

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Article Design and Demonstration of a Flying-Squirrel-Inspired Jumping Robot with Two Modes

Fei Zhao 1, Wei Wang 1,*, Justyna Wyrwa 1, Jingtao Zhang 1, Wenxin Du 2 and Pengyu Zhong 2

1 Laboratory of Biologically Inspired Mobile Robots, Beihang University, Beijing 100191, China; [email protected] (F.Z.); [email protected] (J.W.); [email protected] (J.Z.) 2 School of Mechanical Engineering & Automation, Beihang University, Beijing 100191, China; [email protected] (W.D.); [email protected] (P.Z.) * Correspondence: [email protected]; Tel.: +86-010-8231-4554

Abstract: The jumping–gliding robot is a kind of locomotion platform with the capabilities to jump on the ground and glide through the air. The jumping of this robot has to juggle the requirements of initial velocity and posture for entry to gliding and progressing on the ground. Inspired by flying squirrels, we proposed the concept of flexible wing-limb blending platform and designed a robot with two jumping modes. The robot can takeoff with different speeds and stances, and adjust aerial posture using the swing of forelimbs. To the best of our knowledge, this is the first miniature and bio-inspired jumping robot that can autonomically change the speeds and stances when takeoff. Experimental results show that the robot can takeoff at about 3 m/s and pitch angle of 0◦ in the mode of jumping for gliding and adjust the pitch angle at the top to 0◦~10◦ by actuating the forelimbs

swing according to the requirement of gliding. In the mode of jumping for progress, the robot can takeoff at about 2 m/s with a pitch angle of 20◦ and then intermittently jump with a distance of

Citation: Zhao, F.; Wang, W.; Wyrwa, 0.37 m of once jump and an average progress speed of 0.2 m/s. The robot presented in this paper J.; Zhang, J.; Du, W.; Zhong, P. Design lays the foundation for the development of ﬂexible wing-limb blending platform, which is capable of and Demonstration of a jumping and gliding. Flying-Squirrel-Inspired Jumping Robot with Two Modes. Appl. Sci. Keywords: biomimetic; ﬂying squirrel; jumping robot; takeoff controllability; pitching adjustment 2021, 11, 3362. https://doi.org/ 10.3390/app11083362

Academic Editor: 1. Introduction Alessandro Gasparetto The combination of jumping and gliding have the advantages of high efﬁciency and low energy consumption. Gliding in the air with the speed generated by the jump can Received: 26 February 2021 overcome relatively large terrain obstacles at a lower energy cost. After gliding to land, it Accepted: 6 April 2021 Published: 8 April 2021 is possible to reach a speciﬁc location quickly and accurately by jumping over the ground in small strides. Robots with this jumping-gliding capability are ideal for long-term search

Publisher’s Note: MDPI stays neutral and rescue missions in unstructured terrain, such as battleﬁelds and disaster zones [1,2]. with regard to jurisdictional claims in Armour et al. [3,4] designed a heavy 700 gr jumping robot of 50 cm size called published maps and institutional afﬁl- “Glumper”. It jumps and deploys membranous wings to glide. Desbiens et al. [5,6] iations. implemented a robot that can gliding by jumping. The robot is mainly composed of a glider and elastic rods and use the cam mechanism to realize energy storage and release of elastic rod. Inspired by the mechanism of insect movement, Kovac et al. [7,8] integrated foldable and rigid wings with the jumping mechanism [9] allowing for jumping–gliding locomotion. This robot, with a mass of 16.5 gr and a size of 50 cm, is able to jump from a height of Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. 2 m and glide 4.5 m at a speed of 3 m/s. Once on level ground, it can jump 0.12 m high This article is an open access article and 0.3 m far. Woodward et al. [10] presented a jumping–gliding system weighing 115 gr, distributed under the terms and which adopted the design of coupled locomotion resulting in a jump height of 3 m and a conditions of the Creative Commons glide distance of only 2 m. Foldable wings and tails were added to the jumping robot by Attribution (CC BY) license (https:// Beck et al. [11–13] inspired by locusts, which enabled the robot to spread the wings at the creativecommons.org/licenses/by/ top and glide. A series of experiments showed the folding wing made of membranes has 4.0/). less effect on the jumping height and is more conducive to lengthening the glide distance.

Appl. Sci. 2021, 11, 3362. https://doi.org/10.3390/app11083362 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 3362 2 of 26

The unchangeability of the takeoff speeds and stances of existing jumping–gliding robots has seriously restricted the improvement of the overall performance. As a matter of fact, most of the miniature jumping robots cannot realize autonomous control of takeoff velocity and posture as well [14,15]. Energy storage and release mechanisms of most robots that use elastic elements to achieve intermittent jumps have definite limits on the initial position of the elastic elements, so they cannot control the takeoff speed. The special case is the jumping robot with two critical states of energy storage proposed by Jung et al. [16]. Although the jumping robots driven by chemical energy or other energy [17–19] can adjust the takeoff velocity, their application scenarios are very limited on account of a complicated theoretical model and control method. The jumping robots developed by Kovac et al. [7] and Chai et al. [20] can fulfilled the adjustment of takeoff posture by changing the linking bar length and gear transmission ratio respectively, but they must be reassembled. Zhang et al. [21] designed a jumping robot with the capability of changing the takeoff stance slightly by adjusting counterweight. At present, jumping robots that can achieve wide-range adjustment of takeoff posture include those developed by Fiorini et al. [22] and Miao et al. [23]. They all actively change the body posture before takeoff using the support structure. Some studies indicate that the aerial pitching posture of the robot can be adjusted effectively by the swing of the rod with large moment of inertia. Libby et al. [24] added a rod that could swing up and down for the wheeled robot and realized the attitude adjustment of the robot when crossing obstacles, falling, and swooping. Zhao et al. [25,26] used a single-DOF tail to adjust aerial pitching posture of a jumping robot. Liu et al. [27] improved the aerial pitch stability of their kangaroo-like jumping robot by controlling a rod capable of swinging. Different from the locust, which is frequently considered as the bionic object of the jumping–gliding robots, flying squirrels do not rely on extra wings to glide through the air, but rather use their expansive patagium composed of the skin that spans from the squirrel’s neck to its forelimbs and back to its hind limbs [28–30]. With the adjustment of limbs and tail, they can generally glide 50 m, the longest record of 150 m [31–36]. Because of the flexibility and scalability of the patagium, flying squirrels also can jump on the ground like ordinary squirrels without restraints on limbs [37–39]. These features of flying squirrels provide a lot of bionics inspiration for the design of jump–gliding robots. This paper presents the biomimetic design and development of a prototype of flying squirrel inspired jumping robot. The robot has two modes of jumping for gliding and jumping for progress, which lays a foundation for the design and development of subsequent jumping-gliding robots. The major contributions of this research can be summarized into two aspects. First, we observed different jumping modes and analyzed the movement characteristics of flying squirrels, especially the change of hind limbs joint angles, which have not been mentioned in the literature of flying squirrels before. Second, we designed a new spring charging and releasing mechanism using a single drive to achieve controllable energy storage and release. Furthermore, we developed a jumping robot with two jumping modes by adopting the mechanism. To the best of our knowledge, it is the first miniature bionic jumping robot that has the capabilities of changing the speeds and stances autonomically at takeoff. The rest of this paper is organized as follows. First, the design goal of the robot is defined in Section2. Then, we collect and analysis jumping information of the flying squirrel in Section3. Based on the information, a jumping robot is designed in Section4 . The requirement for aerial posture of the robot after taking off is analyzed, and the dynamic model of adjusting the pitching by the swing of the forelimbs is established and simulated in Section5. Finally, we present the prototype and experimental results for pitching adjustment and demonstrate the two different jumping modes abilities of the robot in Section6. Appl. Sci. 20212021,, 11,, 3362x FOR PEER REVIEW 3 of 2628

2. Design Goal Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 28 Inspired by by the the special special body body shape shape and and movement movement pattern pattern of flying of ﬂying squirrel, squirrel, we pro- we posedproposed a conceptual a conceptual design design of flexible of ﬂexible wing wing-limb-limb blending blending platform, platform, whose whose structural structural form isform shown is shown in Figure in Figure 1, including1, including body, body, forelimbs, forelimbs, hind hind limbs limbs,, and and wing wing membranes. TheThe wing2. Design membranes Goal can be divided into main membrane and tail membrane. The forelimbs and hind Inspired limbs by theare speciallocated body on the shape sides and of movement the body, pattern on the of one flying hand, squirrel, as a source we pro- of power inposed the jumpinga jumping conceptual movement, design of flexibleand on wing the other- limb blending hand, they platform, are able whose to unfold structural and form support the wingis shown membranes. in Figure 1, TheThe including main main body,membrane forelimbs, is is attached hind limbs to , theand limbs.wing membranes. It does not The hinder the wing membranes can be divided into main membrane and tail membrane. The forelimbs stretch of limbs much when jumping because of its scalability, and it can be unfolded to and hind limbs are located on the sides of the body, on the one hand, as a source of power generate aerodynamic liftlift whenwhen gliding.gliding. TheThe tail tail membrane membrane is is behind behind the the body body and and plays plays a in the jumping movement, and on the other hand, they are able to unfold and support the role in adjusting posture during gliding. awing role membranes.in adjusting The posture main duringmembrane gliding. is attached to the limbs. It does not hinder the stretch of limbs much when jumping because of its scalability, and it can be unfolded to generate aerodynamic lift when gliding.Body The tail membrane is behind the body and plays a role in adjusting posture during gliding.

Body Forelimb

Hind limb Forelimb

Main membrane Hind limb

Main membrane

Tail membrane

Tail membrane Figure 1. Structural form of flexible flexible wing wing–limb–limb blending platform. Figure 1. Structural form of flexible wing–limb blending platform. Figure 22 showsshows thethe locomotionlocomotion conceivingconceiving ofof flexibleflexible wing-limbwing-limb blendingblending platform.platform. JumpingFigure into 2 shows the air the as locomotio fast as possiblen conceiving at the of flexible edge of wing thethe- hathpace,limbhathpace, blending unfolding platform. the main membraneJumping into at the the air top, as fastentering as possible steady steady-state at- statethe edge gliding gliding of the after after hathpace, a a while, unfolding folding the the the main main main mem- mem- branemembrane after at landing, the top, enteringand subsequentlysubsequently steady-state progressing gliding after ona while, ground folding by jumping.the main mem- According to thisbrane assumption, after landing, the and flexible flexible subsequently wing wing-limb -progressinglimb blending on ground platform by jumping.need needss to According have two to different jumpingthis assumption, modes, modes, the namely namely flexible the the wing mode mode-limb of of blending jumping platform for for gliding need ands and to havethe mode two different of jumping for jumping modes, namely the mode of jumping for gliding and the mode of jumping for progress. The mode of jumping for gliding should provide sufficientsufficient initial velocity and progress. The mode of jumping for gliding should provide sufficient initial velocity and suitable aerial posture for subsequent gliding. The mode of jumping for progress should satisfysuitable the aerial needs posture of small for subsequent step forward gliding. on theThe ground.mode of Therefore,jumping for the progress research should objective of satisfysatisfy thethe needs needs of of small small step step forward forward on the on ground. the ground. Therefore, Therefore, the research the research objective objective of of this paper is to design a flying squirrel like jumping robot with two jumping modes, which thisthis paper is is to to design design a flying a flying squirrel squirrel like likejumping jumping robot robot with two with jumping two jumping modes, whichmodes, which cancan realize jumping jumping for for gliding gliding and and jumping jumping for progress for progress with different with different takeoff speeds takeoff speeds andand stances and and lays lays a foundationa foundation for for the design the design and development and development of flexible of flexibleflexiblewing–limb wing–limbwing –limb blendingblending platform. platform.

(b) (b) (c) (c) (a) (a)

(e) (d) (e) (d)

(a) Jumping for gliding (b) Unfolding membrane (c) Steady-state gliding (d) Folding membrane (e) Progressing (a) Jumping for gliding (b) Unfolding membrane (c) Steady-state gliding (d) Folding membrane (e) Progressing FigureFigure 2. 2. LocomotionLocomotion conceiving conceiving of flexible of ﬂexible wing wing–limb–limb blending blending platform. platform. Figure 2. Locomotion conceiving of flexible wing–limb blending platform. Appl.Appl. Sci.Sci. 202120202121,,, 11 1111,,, 3362 xx FORFOR PEERPEER REVIEWREVIEW 444 of ofof 26 2828

3.3. AnalysisAnalysis ofof JumpingJumping MechanismMechanism ofof FlyingFlying SquirrelSquirrel TheThe free free activity activity inside inside confined confined spaces spaces of of an an adult adult male male southern southern flying flying squirrel squirrel (Glaucomys(Glaucom(Glaucomys volans) volans) was was recorded recorded using using a a Photron Photron Fastcom Fastcom Mini Mini UX100 UX100 high-speed high--speedspeed camera. cam-cam- Sequentialera.era. SequentialSequential images imagesimages of the ofof thethe flying flyingflying squirrel squirrelsquirrel during duringduring jumping jumpingjumping were werewere captured capturedcaptured at atat 2000 20002000 frames framesframes perper second. second. Figures 33 and and4 4exhibit exhibit the the motional motional sequence sequence of of jumping jumping for for gliding gliding and and jumpingjumpingjumping forforfor progressprogressprogress ofofof thethethe flyingflyingflying squirrel,squirrel,squirrel, respectively.respectively.respectively. AsAsAs shownshownshown ininin FigureFigure3 3,, the the flying flying squirrelsquirrelsquirrel firstfirstfirst curlscurlcurlss upup itsits hindhind limbslimbs andand supportssupportss thethethe bodybody withwith itsitsits forelimbs,forelimbs,forelimbs, thenthenthen leans leanleanss ◦ forwardforwardforward atatat thethethe angleangleangle ofofof approximatelyapproximatelyapproximately 000° betweenbetween thethe bodybody andand thethe horizontalhorizontal line.linee.. Next,Nextext,, ititit graduallygradually stretchesstretchstretcheses itsitsits hindhind limbslimbs untiluntil leavingleaving thethe hathpace. hathpace. As shown in FigureFigure4 4,, whenwhen jumpingjumping for for progress, progress, the the flying flying squirrel squirrel likewise likewise uses use thess thethe stretching stretchingstretching of of hindof hindhind limbs limbslimbs to takeoff,toto takeoff,takeoff, but butbut the thethe stance stancestance is markedly isis markedlymarkedly different: different:different: The angleThe angle between between the body the body and the and horizontal the hori- ◦ linezontal is approximatelyline is approximately 20 . 20°.

FigureFigure 3.3. Motional sequence sequence of of jumping jumping for for gliding gliding of of the the flying ﬂying squirrel. squirrel. The The white white dots dots are are the the markersmarkers onon thethe jointsjoints ofof thethe hindhind limb,limb, andand thethe redred lineslines denotedenote thethe linkageslinkages betweenbetween thethe joints.joints.

FigureFigure 4.4. MotionalMotional sequence of jumping jumping for for progress progress of of the the flying ﬂying squirrel. squirrel. The The white white dots dots are are the the markersmarkers onon thethe jointsjoints ofof thethe hindhind limb,limb, andand thethe redred lineslines denotedenote thethe linkageslinkages betweenbetween thethe joints.joints.

SimilarSimilar to to most most jumping jumping animals, animals, the the hind hind limbs limbs of ﬂying of flying squirrels squ areirrelsirrels the are are main the the source main main ofsourcesource power ofof inpowerpower jumping inin jumpingjumping movement movementmovement [40]. As [40].[40]. shown AsAs shownshown in Figure inin 5 Figure, they are5, they made are up made of femur, up of tibioﬁbular,femur,femur, tibiofibular,tibiofibular, tarsal, and tarsaltarsal phalanx,,, and phalanx, including including hip joint, hip knee joint, joint, knee anklejoint, ankle joint and joint toe and joint. toe Thejoint.joint. bones TheThe bonesbones are attached areare attachedattached to muscles toto musclesmuscles that thatthat can cancan drive drivedrive the tthe legs legs for for movement. movement. The The femoral femoral musclemuscle cancan drivedrive thethe kneeknee jointjoint rotation,rotation, whichwhich providesprovides thethe mainmain powerpower forfor thethe legs.legs. TheThe gastrocnemiusgastrocnemius musclemuscle can drivedrive the ankleankle joint rotation, which gives part of the power. TheThe hiphip jointjoint isis connectedconnected toto thethe bodybody andand cancan transfertransfer thethe forceforce generatedgenerated byby stretchingstretching Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 28

Appl. Sci. 20212021,, 11,, 3362x FOR PEER REVIEW 5 of 2628

the hind limbs to the body. The toe joint acts as a takeoff steering and landing buffer. In addition, the femur and the tibiofibular are connected with the patagium and are able to the hind limbs to the body. The toe joint acts as a takeoff steering and landing buffer. In deploythe hind the limbs patagium to the along body. wit Theh abduction toe joint acts of the as ahip takeoff joint steeringin the gliding and landing phase. buffer. In addition, the femur and the tibiofibular tibioﬁbular are connected with the patagium and are able to deploy the patagium along wit withh abduction of the hip joint in the gliding phase.

Hip joint Femoral muscle Hip jointFemur Femoral muscle Gastrocnemius muscleFemur Gastrocnemius muscle Knee joint Ankle joint TibiofibularKnee joint Ankle joint Tibiofibular

Tarsal Toe joint Phalanx

Figure 5. PhysiologicalTarsal Toe structure joint Pofhalanx hind limbs of flying squirrels. FigureIn 5. order Physiological to facilitate structure the analysis of hind oflimbs the of movement flying flying squirrels. characteristics of the animal’s hind limbs in the two jumping modes, the joints of hind limbs are simplified into revolute pairs In order order to facilitate the analysis of the movement movement characteristics of the animal’s hind with only one degree of freedom, as shown in Figure 6, where  represents the angle limbs in in the the two two jumping jumping modes, modes, the the joints joints of hind of hind limbs limbs are simplified are simplified0 into revolute into revolute pairs betweenpairs with the onlyphalanx one and degree the horizontal of freedom, line, as shown represents in Figure the6 angle, where of toeβ0 joint,represents  rep- the with only one degree of freedom, as shown in1 Figure 6, where 0 represents the2 angle angle between the phalanx and the horizontal line, β1 represents the angle of toe joint, β2 resentsbetween the the angle phalanx of ankle and joint,the horizontal 3 represents line, the r epresentsangle of knee the anglejoint, ofand toe  joint,4 represents  rep- represents the angle of ankle joint, β3 represents1 the angle of knee joint, and β4 represents2 thethe angle angle of of hip hip joint. joint. Reflective Reflective paint paint was was used used to to mark mark each each joint joint of of the the hind hind limbs limbs of of the the resents the angle of ankle joint, 3 represents the angle of knee joint, and 4 represents flyingflying squirrel. squirrel. Then, Then, under under the the same same conditions, conditions, we we made made the the flying flying squirrel squirrel repeat repeat the the the angle of hip joint. Reflective paint was used to mark each joint of the hind limbs of the jumpingjumping for for gliding gliding from from a a hathpace hathpace with with a a height height of of 2 2 m m and and the the jumping jumping for for progress progress flying squirrel. Then, under the same conditions, we made the flying squirrel repeat the onon the the ground ground for for times times,, respectively. respectively. I Inn addition, addition, the the motion motion process process was was recorded recorded with with jumping for gliding from a hathpace with a height of 2 m and the jumping for progress thethe high high-speed-speed camera. camera. MATLAB MATLAB software software was was used used to to extract extract the the joint joint coordinates coordinates of of the the on the ground for times, respectively. In addition, the motion process was recorded with hindhind limbs limbs on thethe motionmotion imagesimages according according to to the the markers, markers, calculate calculate the the angle angle of each of each joint, the high-speed camera. MATLAB software was used to extract the joint coordinates of the jointand, takeand thetake average the average of the of five the repetitions. five repetitions. Finally, Finally, the change the change curves curves of the angleof the of angle each hind limbs on the motion images according to the markers, calculate the angle of each ofjoint each in joint the wholein the motionalwhole motional sequence sequence were plotted, were plotted, as shown as shown in Figures in Figures7 and8. 7 and 8. joint, and take the average of the five repetitions. Finally, the change curves of the angle of each joint in the whole motionalBody sequence were plotted, as shown in Figures 7 and 8.  Hip joint 4 Body

Hip joint 4 3 Knee joint 3 Knee joint

2 Ankle joint

2 1 Ankle joint  Toe joint 0 Horizontal line 1  Toe joint 0 Horizontal line FigureFigure 6. 6. SchematicSchematic of of joints joints of of hind hind limbs limbs of of flying ﬂying squirrel. squirrel.

FigureCombinedCombined 6. Schematic with with of jointsthe the motional motionalof hind limbs sequence sequence of flying of of jumpingsquirrel. jumping for for gliding gliding of of the the flying ﬂying squirrel, squirrel, itit can can be be seen seen from from Figure Figure 77 that:that: β0,, β2, and, andβ3 are are nearly nearly unchanged unchanged in thein the preparation prepara- Combined with the motional sequence of jumping for gliding of the flying squirrel, stage of 0~0.03 s; β0 remains the same while β2 and β3 increase quickly with the identical tionit can stage be seen of 0~0.0 from3 sFigure; remains7 that: the, same, andwhile are and nearly unchanged increase quickly in the withprepara- the rate of change 54 rad/s in the takeoff stage of 0.03~0.065 s; β0 increase quickly while β2 identicaltionand stageβ3 grow rate of 0~0.0of with change3 slowers; 54 rad/sremains rate inin thethe the launch takeoffsame while stagestage of 0.065~0.0750.03~0.06 and 5increase s; s; β2 andincrease quicklyβ3 reach withquickly thethe maximum at 0.075 s. Throughout the stage, in addition, β keeps increasing and β almost whileidentical rate and of change grow 54 with rad/s slower in the rate takeoff in the stage launch of 0.03~0.06 stage4 of5 0.065~0.075s; increase s; 1 quickly and keeps decreasing with the angle of change of about 85◦, which signify the ﬂying squirrel’s whilebody leans and forward grow considerably with slower when rate it jumps in the for launch gliding. stage Combined of 0.065~0.075 with the s; motional and Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 28

3 reach the maximum at 0.075 s. Throughout the stage, in addition, 4 keeps increas-

ing and 1 almost keeps decreasing with the angle of change of about 85°, which signify the flying squirrel’s body leans forward considerably when it jumps for gliding. Com- bined with the motional sequence of jumping for progress of the flying squirrel, it can be

seen from Figure 8 that: 2 and increase quickly with the identical rate of change 28 rad/s in the takeoff stage of 0.02~0.045 s; and increase and decrease by about 40°, respectively. The comparison between Figures 7 and 8 indicate that the change trends of Appl. Sci. 2021, 11, 3362 6 of 26 angle of hind limbs joints when the animal jump for progress are similar to that when jumping for gliding, but the initial value and the range are obviously different. The above analysis shows that: Whether jumping for gliding or progress, the flying sequencesquirrel both of jumping rapidly forextend progress the knee of the and ﬂying ankle squirrel, joint to it generate can be seen takeoff from force, Figure and8 that: the βchange2 and βtrends3 increase of angle quickly of withknee the and identical ankle joints rate of are change almost 28 rad/ssynchronized; in the takeoff the rate stage and of ◦ 0.02~0.045range of change s; β4 and of angleβ1 increase of knee and and decrease ankle joints by about are greater 40 , respectively. when it jumps The comparisonfor gliding, betweenwhich can Figures produce7 and greater8 indicate force that and the speed change for trends takeoff; of the angle angle of hind between limbs body joints and when the thehorizontal animal jumpline is for smaller progress when are itsimilar jumps to for that gliding, when jumping which gives for gliding, it greater but horizontal the initial valuespeed andto glide. the range are obviously different.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 28

Figure 7. Angles of each joint of the ﬂyingflying squirrel’s hindhind limbslimbs whenwhen jumpingjumping forfor gliding.gliding.

Figure 8. Angles of each joint of the ﬂying squirrel’s hind limbs when jumping for progress. Figure 8. Angles of each joint of the flying squirrel’s hind limbs when jumping for progress. The above analysis shows that: Whether jumping for gliding or progress, the ﬂying 4. Design of Jumping Robot squirrel both rapidly extend the knee and ankle joint to generate takeoff force, and the change4.1. Hind trends Limbs of angle of knee and ankle joints are almost synchronized; the rate and range of changeAccording of angle to ofthe knee above and analysis ankle joints of the are movement greater when characteristics it jumps for of gliding, flying which squirrels can’ producehind limbs, greater a parallel force andfour- speedbar linkage for takeoff; mechan theism angle with between springs bodyis used and tothe mimic horizontal the leg linestructure is smaller of flying when squirrels. it jumps forAs gliding,shown in which Figure gives 9, the it greater linkage horizontal rod ac , speedbd , tode glide., ae

are connected by hinges successively, where the length lab and lde are equal; the

length lae and lbd are equal; the ratio of the length , , and lbc is designed according to the length ratio of the flying squirrel’s femur, tibiofibular and tarsal, approxi-

mately lde: l bd : l bc  1:1:1. The angles 2 (corresponding to the angle of ankle joint)

and 3 (corresponding to the angle of knee joint) are set to have a constant difference,

namely 2320 . Using the linkage rod ae as a frame, the parallel four-bar linkage mechanism abde can ensure the angles and have the same rate of change.

lae e a  lde V0

xad O lab 3 d

b lbd 2

FT lbc

T c

Figure 9. Schematic of structure of hind limbs.

A spring with a stiffness coefficient of k is connected between points a and d ,

whose initial length and stretched length are x0 and xad , respectively. Ignoring the mass Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 28

Figure 8. Angles of each joint of the flying squirrel’s hind limbs when jumping for progress. Appl. Sci. 2021, 11, 3362 7 of 26 4. Design of Jumping Robot 4.1. Hind Limbs 4. DesignAccording of Jumping to the above Robot analysis of the movement characteristics of flying squirrels’ hind4.1. Hindlimbs, Limbs a parallel four-bar linkage mechanism with springs is used to mimic the leg structure of flying squirrels. As shown in Figure 9, the linkage rod ac , bd , de , ae According to the above analysis of the movement characteristics of ﬂying squirrels’ arehind connected limbs, a parallel by hinges four-bar successive linkagely, where mechanism the length with springslab and is l usedde are to equal;mimic the

lengthleg structure lae and of flyinglbd are squirrels. equal; the As ratio shown of the in Figurelength9 , the, linkage, and rod lbc acis, designedbd, de, ae areac- cordingconnected to the by hingeslength successively,ratio of the flying where squirrel’s the length femur,lab and tibiofibularlde are equal; and the tarsal, length approxi-lae and lbd are equal; the ratio of the length lde, lbd, and lbc is designed according to the length ratio mately lde: l bd : l bc  1:1:1. The angles 2 (corresponding to the angle of ankle joint) of the flying squirrel’s femur, tibiofibular and tarsal, approximately lde : lbd : lbc = 1 : 1 : 1. andThe angles3 (correspondingβ2 (corresponding to the toangle the of angle knee of joint) ankle are joint) set to and haveβ3 (correspondinga constant difference, to the angle of knee joint) are set to have a constant difference, namely β = β + 20◦. Using namely 2320 . Using the linkage rod ae as a frame, the parallel2 3 four-bar link- the linkage rod ae as a frame, the parallel four-bar linkage mechanism abde can ensure the age mechanism abde can ensure the angles and have the same rate of change. angles β2 and β3 have the same rate of change.

lae e a  lde V0

xad O lab 3 d

b lbd 2

FT lbc

T c

FigureFigure 9. 9. SchematicSchematic of of structure structure of of hind hind limbs. limbs.

AA spring with aa stiffnessstiffness coefﬁcientcoefficient ofofk isk connectedis connected between between points pointsa and a d and, whose d , initial length and stretched length are x0 and xad, respectively. Ignoring the mass of each whose initial length and stretched length are x0 and xad , respectively. Ignoring the mass rod and spring, we can assume that the takeoff force FT passes through the center of mass (point O) of the entire robot. In order to determine the structural parameters of the hind limbs, we assume that the elastic potential energy is all converted into kinetic energy without dissipative losses during the takeoff process [13]. Although this assumption does not reﬂect real-life conditions, it allows for a simple and clear view of the relationship between structural parameters and energy, which is convenient for the initial design. The energy equation is thus 1 1 kn(x − x )2 = mV2 (1) 2 ad 0 2 0 where n represents the number of springs connected in parallel, m represents the mass of entire robot, and V0 represents the initial velocity after takeoff. According to the geometric relationship, xad can be acquired as

β x = 2l cos 3 (2) ad de 2 ◦ ◦ where β3 is set to minimum value 30 and maximum value 120 , referring to the variation ◦ range of angle of the ﬂying squirrel’s knee joint. When β3 is 30 , the spring is stretched to its maximum length. In this case, the corresponding V0 is the initial velocity in the mode of jumping for gliding. According to the average initial velocity of ﬂying squirrels as they glide [35], the corresponding V0 is set to 3 m/s, and then the length of linkages can be obtained from (1) and (2). The resulting structure of hind limbs is shown in Figure 10. Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 28

of each rod and spring, we can assume that the takeoff force FT passes through the center of mass (point O ) of the entire robot. In order to determine the structural parameters of the hind limbs, we assume that the elastic potential energy is all converted into kinetic energy without dissipative losses during the takeoff process [13]. Although this assumption does not reflect real-life conditions, it allows for a simple and clear view of the relationship between structural parameters and energy, which is convenient for the initial design. The energy equation is thus

1122 kn() xad  x00 mV (1) 22 where n represents the number of springs connected in parallel, m represents the mass

of entire robot, and V0 represents the initial velocity after takeoff. According to the geo-

metric relationship, xad can be acquired as

3 xlad 2 de cos (2) 2

where 3 is set to minimum value 30° and maximum value 120°, referring to the variation range of angle of the flying squirrel’s knee joint. When is 30°, the spring is stretched to its maximum length. In this case, the corresponding is the initial velocity in the mode of jumping for gliding. According to the average initial velocity of flying squirrels as they glide [35], the corresponding is set to 3 m/s, and then the length of Appl. Sci. 2021, 11, 3362 8 of 26 linkages can be obtained from (1) and (2). The resulting structure of hind limbs is shown in Figure 10.

FFigureigure 10. 10. 3D3D model model of of hind hind limbs limbs of of the the robot. robot.

Takeoff angle θT (see Figure9) is deﬁned as the angle between the takeoff force FT and Takeoff angle T (see Figure 9) is defined as the angle between the takeoff force the horizontal line. In order to ensure that the hind limbs do not skid during takeoff, the and the horizontal line. In order to ensure that the hind limbs do not skid during horizontal component of the takeoff force FT should be less than or equal to the maximum takeoff,static sliding the horizontal friction between component hind of limbs the takeoff and the force ground, namelyshould be less than or equal to the maximum static sliding friction between hind limbs and the ground, namely FT cos θT ≤ µ0FT sin θT (3)

FFTTTTcos 0 sin  (3) where µ0 is coefﬁcient of static friction. Equation (3) is further organized to obtain where  is coefficient of static friction. Equation (3) is further organized to obtain 0 1 µ0 ≥ (4) tan 1θT 0  (4) Equation (4) shows that the coefﬁcient oftan static friction between hind limbs and the T ground and the takeoff angle will affect whether the robot can successfully jump. Note that the direction of the takeoff force is perpendicular to the linkage rod ac, and according to the geometric relationship, the takeoff angle can be acquired as

◦ θT = 260 − β3 − γ (5)

where γ (see Figure9) is the angle between the linkage rod ae and the horizontal line. It can be seen from (5) that the takeoff angle θT is related to β3 and γ. For different jumping modes, β3 and γ have different values. Appropriate materials should be selected at the contact surface between the hind limbs and the ground to avoid skidding according to (4), after the takeoff angle is determined.

4.2. The Mechanism of Loading and Releasing of Spring During takeoff, the movement of the hind limbs are driven by the spring, and the energy storage and releasing of the spring are controlled by the loading and releasing mechanism. In order to meet the different requirements of the takeoff speed of the different jumping modes and considering the influence of the overall mass on the jumping performance, the loading and releasing mechanism of the spring should have the following three characteristics besides being able to completely release the stored energy in an instant: 1. The loading degree is controllable; 2. releasing time is controllable; and 3. loading and releasing are driven by the same actuator. Therefore, we designed a mechanism that can realize controllable loading and releasing of the spring, as shown in Figure 11. It is composed of the transmission gear, transmission shaft, cylindrical cam, spring plugs, one-way bearing, ratchet, pawl, winding pulley, restriction spring, and wires. The transmission gear and cylindrical cam are fixed on the transmission shaft. One end of the plugs is in contact with the curved profile of the cylindrical cam, and the other end is inserted into the through hole of the ratchet. The ratchet is mounted on the transmission shaft by the one-way bearing. The winding pulley can rotate freely around the transmission shaft, and its axial movement is constrained by the restriction spring. The wires are wound around Appl. Sci. 2021, 11, 3362 9 of 26

Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 28

the winding pulley and connected to the end of the hind limbs. The overall structure is shown in Figure 11a.

Winding Spring Ratchet Transmission Transmission pulley plugs shaft gear Cylindrical cam

Wires Pawl (a)

One-way Restriction spring bearing Rotating counterclockwise Rotating clockwise

(b) (c)

FigureFigure 11. 11.Composition Composition and and principle principle of of the the mechanism mechanism of of loading loading and and releasing releasing of of spring. spring. ((aa)) Overall Overall structure;structure; ((bb)) loading;load- (c) releasing.ing; (c) releasing.

It is assumed that the plugs are inserted into the circumferential distribution holes of theDifferent winding from pulley the undertriggering the mechanism extrusion of of themost cylindrical miniature jumping cam at the robots, initial this moment. com- Whenpletely the independent transmission design gear of drives loading the and cylindrical releasing camallows to the rotate springs clockwise to be released through in the transmissionan arbitrary state shaft, of the energy ratchet storage, rotates which in the realizes same the direction active control and at of the takeoff same speed. speed In with theaddition, cylindrical the combination cam under theof one action-way of bearing the one-way and cylindrical bearing. cam Furthermore, makes it possible the winding to pulleymeet the also driving rotates requirements clockwise driven of the mechanism by the plugs, only thus using winding a single themotor. wires to compress the structureThe transmission of the hind system limbs, is shown which in realizesFigure 12. the The loading output ofshaft springs of the andmotor storage trans- of energy,fers the as torque shown to in the Figure transmission 11b. When gear the through transmission the two gear-stage drives reduction the cylindrical gears, which cam todrives rotate the counterclockwise transmission shaft through to rotate the clockwise transmission or counterclockwise. shaft, the ratchet The stays main stillparame- due to theters pawl, of each which piece causes of gear the are plugs shown to in quickly Table A withdraw1 (see Appendix from the A). hole The intransmission the pulley. ratio In this case,of the the gears pulley system loses is its rotational constraint and instantly loosens the wires to release springs of hind limbs, as shown in Figure 11c. Under the constraint of the restriction mdz1 z spring, continuing counterclockwisei rotation  of the  cylindrical  4 cam causes the plugs(6) to be reinserted into the pulley, ready for the next loadingzz and releasing. dm2 Different from the triggering mechanism of most miniature jumping robots, this completelywhere m independent and d represent design the of loading speed of and motor releasing and transmission allows the springs gear, respectively; to be released

inz anm , arbitraryz1 , z2 , and state zd of represent energy storage, the number which of realizesteeth of the the motor active gear, control gear of 1, takeoffgear 2, and speed. transmission gear, respectively. Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 28

It is assumed that, during loading, the tension on the wires is Fr , and its direction

is the same as that of the takeoff force FT (see Figure 9); the spring force of hind limbs is

Fs . Ignoring the mass of each rod and spring, and according to the torque balance equation, the relation between and is expressed as

3 Fr[ l ac l ae sin( T   180 )]  F s l de sin (7) 2

where Fs k() x ad x0 . Combined with (2) and (5), (7) can be further arranged as

3 k( lde sin30 x sin ) F  2 (8) r 2sin80 sin(80 ) 3 Appl. Sci. 2021, 11, 3362 10 of 26 Thus, the torque that the motor needs to provide is

nFrr d Tm  (9) In addition, the combination of one-way bearing2 andi cylindrical cam makes it possible to meet the driving requirements of the mechanism only using a single motor. whereThe n transmission represents the system number is shown of springs in Figure connected 12. The in output parallel, shaft and of thedr motor represent transferss the thediameter torque of to the the winding transmission pulley. gear According through to the (8) two-stage and (9), when reduction the structural gears, which parameters drives theand transmissionsprings are determine shaft to rotated, the clockwise torque that or the counterclockwise. motor needs to Theprovide main is parameters related to th ofe each piece of gear are shown in Table A1 (see AppendixA). The transmission ratio of the angle 3 . The known conditions are lde  38mm , k  0.52N/mm , x0  35mm , gears system is n  2 and d  26mm . Then, the ωrelationz betweenz the motor’s torque and the angle r i = m = 1 · d = 4 (6) is presented in Figure 13. The maximalωd availablezm z2 torque of the selected motor is 0.16 whereNm whenωm andoperatedωd represent by the 4.8 the V speed Li-Po of battery. motor andIt can transmission be seen that gear, the respectively;maximal torquezm, zin1, zthe2, and curvezd represent is roughly the 0.033 number Nm, of indicating teeth of the that motor there gear, is muchgear 1, more gear 2, available and transmission potential gear,power respectively. using this transmission system.

Motor

Transmission gear Motor gear

Transmission shaft

Gear 1

Gear 2 Figure 12. The transmission system.

It is assumed that, during loading, the tension on the wires is Fr, and its direction is the same as that of the takeoff force FT (see Figure9); the spring force of hind limbs is Fs. Ignoring the mass of each rod and spring, and according to the torque balance equation, the relation between Fr and Fs is expressed as

β F [l − l sin(θ + γ − 180◦)] = F l sin 3 (7) r ac ae T s de 2

where Fs = k(xad − x0). Combined with (2) and (5), (7) can be further arranged as

β3 k(lde sin β3 − x0 sin 2 ) Fr = ◦ ◦ (8) 2 sin 80 − sin(80 − β3)

Thus, the torque that the motor needs to provide is

nF d T = r r (9) m 2i

where n represents the number of springs connected in parallel, and dr represents the diameter of the winding pulley. According to (8) and (9), when the structural parameters and springs are determined, the torque that the motor needs to provide is related to the angle β3. The known conditions are lde = 38mm, k = 0.52N/mm, x0 = 35mm, n = 2 Appl. Sci. 2021, 11, 3362 11 of 26

and dr = 26mm. Then, the relation between the motor’s torque and the angle β3 is presented in Figure 13. The maximal available torque of the selected motor is 0.16 Nm when operated by the 4.8 V Li-Po battery. It can be seen that the maximal torque in the Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 28 curve is roughly 0.033 Nm, indicating that there is much more available potential power using this transmission system.

Figure 13. The relation between the motor’s torque and the angle β3. Figure 13. The relation between the motor’s torque and the angle 3 . 4.3. Forelimbs and Overall Structure 4.3. ForelimbsIn order and to Overall meet the Structure different requirements of the takeoff stances of the different jumpingIn order modes, to meet the forelimbs the different should requirements not only support of the the takeoff body stances during of takeoff, the different but also jumpingadjust the modes, body the posture. forelimbs Thus, should the forelimbs not only aresupport connected the body to a during motor fixedtakeoff, to thebut bodyalso adjustframe the and body are driven posture. by Thus, the motor the forelimb to swings relativeare connected to the body to a motor frame. fixed to the body frameThe and 3Dare model driven of by the the overall motor structureto swing relative is presented to the in body Figure frame. 14. The hind limbs are fixedlyThe connected 3D model with of the the overall body frame. structure The is loading presented and releasingin Figure mechanism14. The hind of limbs the spring are fixedlycan realize connected controllable with the compression body frame. of hind The limbs.loading The and forelimbs releasing can mechanism adjust the of angle the springbetween can the realize body controllable frame and the compression horizontal lineof hind to change limbs.the The body forelimbs stance. can The adjust controller the angleis used between to control the thebody rotation frame ofand the the motors, horizontal and theline sensor to change is used the tobody detect stance. the number The con- of trollerturns ofis theused pulley to control and the the body rotation posture. of the The motors, battery and powers the sensor the whole is used system. to detect the number of turns of the pulley and the body posture. The battery powers the whole system. 4.4. The Working Process BeforeController the start of jumping, the hindLoading limbs and are releasing at fully extended position. Once ready to jump,and the sensors controllerMotor first of the controls forelimbs themechanism motor ofof the spring hind limbs to rotate counterclockwise for 0.5 s, so that the spring plugs are inserted into the hole of the winding pulley, and then Battery waits for the command. When the controller receives the commandHind of thelimbs mode of jumping for gliding, it first controls the motor of the hind limbs to rotate clockwise until the sensors detect that the number of turns of the pulley is ng, and then stops the rotation. Then the motor of the forelimbs is rotated until the sensors detect that the angle between the body frame and the horizontal line is ϕg. When the controller receives the command of the mode of jumping for progress, it first controls the motor of the hind limbs to rotate clockwise until the sensors detect that the numberBody frame of turns of the pulley is np, and then stops the rotation. Then the motor of the forelimbs is rotated until the sensors detect that the angle between the body frame and the horizontal line is ϕp. After the stance adjustment is completed, the controller controls the motor of the hind limbs to rotate 0.1 s counterclockwise, so that the spring plugs are withdrawn from the hole of the winding pulley, and the springs of the Forelimbs

Figure 14. 3D model of the overall structure of the robot.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 28

Figure 13. The relation between the motor’s torque and the angle 3 .

4.3. Forelimbs and Overall Structure In order to meet the different requirements of the takeoff stances of the different jumping modes, the forelimbs should not only support the body during takeoff, but also adjust the body posture. Thus, the forelimbs are connected to a motor fixed to the body frame and are driven by the motor to swing relative to the body frame. The 3D model of the overall structure is presented in Figure 14. The hind limbs are Appl. Sci. 2021, 11, 3362 12 of 26 Appl. Sci. 2021, 11, x FOR PEER REVIEWfixedly connected with the body frame. The loading and releasing mechanism of13 of the 28 spring can realize controllable compression of hind limbs. The forelimbs can adjust the angle between the body frame and the horizontal line to change the body stance. The con- hindtroller limbs is used are to released control quickly the rotation to complete of the themotors, corresponding and the sensor jumping. is used Its workﬂowto detect the is shownnumber4.4. The in Workingof Figure turns 15ofProcess .the pulley and the body posture. The battery powers the whole system. Before the start of jumping, the hind limbs are at fully extended position. Once ready to jump,Controller the controller first controls the Lmotoroading andof the releasing hind limbs to rotate counterclockwise for 0.5 s,and so sensors that theM springotor of theplugs forelimbs are insertedmechanism into of thethe spring hole of the winding pulley, and then waitsBattery for the command. When the controller receives the command of the mode of jumping for gliding, it first controls the motor of the hind limbs to rotateHind limbsclockwise until the

sensors detect that the number of turns of the pulley is ng , and then stops the rotation. Then the motor of the forelimbs is rotated until the sensors detect that the angle between

the body frame and the horizontal line is  g . When the controller receives the command of the mode of jumping for progress, it first controls the motor of the hind limbs to rotate

clockwise until the sensors detectBody frame that the number of turns of the pulley is n p , and then stops the rotation. Then the motor of the forelimbs is rotated until the sensors detect that

the angle between the body frame and the horizontal line is  p . After the stance adjustment is completed, the controller controls the motor of the hind limbs to rotate 0.1 s counterclockwise, so thatForelimbs the spring plugs are withdrawn from the hole of the winding pulley, and the springs of the hind limbs are released quickly to complete the corresponding jumping. Its workflow is shown in Figure 15. FigureFigure 14.14. 3D3D modelmodel ofof thethe overalloverall structurestructure ofof thethe robot.robot.

Start

Motor of hind limbs rotating 0.5s counterclockwise

Receiving command N N

Mode of Motor of hind Motor of Y Winding pulley Y The angle Y jumping for limbs rotating forelimbs rotating n laps  gliding clockwise g rotating g

Mode of Motor of hind Motor of Y Winding pulley Y The angle Y jumping for limbs rotating forelimbs rotating n laps progress clockwise p rotating  p

N N N Motor of hind limbs rotating 0.1s counterclockwise

End

FigureFigure 15.15.Workﬂow Workflow of of the the robot. robot.

AccordingAccording toto thethe initialinitial angleangle ofof thethe kneeknee jointjoint andand thethe angleangle betweenbetween thethe bodybody andand the ground of the ﬂying squirrel in different jumping modes, when the pulley rotates ng the ground of the flying squirrel in different◦ jumping◦ modes, when the pulley rotates laps and np laps, the angle β3 are set as 30 and 60 , respectively, and the angle ϕg and ϕp ◦ ◦ arelaps set and as 0 and laps, 20 the, respectively. angle 3 are As set shown as 30° in and Figure 60° 16, respectively,, the dotted linesand representthe angle the g initialand  state are of set the as hind 0° and limbs, 20° and, respectively. the solid lines As shown represent in Figure the compressed 16, the dotted state oflines the repre- hind p 0 limbs. The curve cc represents the trajectory of the end of the hind limbs. The length l 0 of sent the initial state of the hind limbs, and the solid lines represent the compressedcc state the curve cc0 can be expressed as of the hind limbs. The curve cc represents the trajectory of the end of the hind limbs. ◦ ◦ The length l of the curve can2πl debe( 120expresse− β3d) cosas 10 cc l 0 = (10) cc 180◦ Appl. Sci. 2021, 11, x FOR PEER REVIEW 14 of 28

2πlde (120 3 )cos10 lcc  (10) 180

Since the wires are connected to the end of the hind limbs, ng and n p can be acquired as

lde cos10 n 1.5 (11) g d r

2lde cos10 Appl. Sci. 2021, 11, 3362 n 1 13 of(12) 26 p 3d r

 a d

3

120 b

c b

lcc

FigureFigure 16. 16.Trajectory Trajectory of of the the end end of of the the hind hind limbs limbs when when loading. loading.

5. PitchingSince the Requirement wires are connected and Adjustment to the end of the hind limbs, ng and np can be acquired as

According to the workflow of the robot, the◦ initial pitch angle is 0° when the robot lde cos 10 takeoff in the mode of jumping forng =gliding. Ideally,≈ 1.5 if the takeoff direction goes through (11) the center of mass, the robot will remaind inr this posture until it enters the glide stage. However, taking into account the changes in the◦ position of the center of mass caused by 2lde cos 10 manufacturing and assembly errors,np = it is likely that≈ there1 will be additional angular veloc- (12) 3dr ities after the robot takeoff, which will lead to changes in the robot’s posture. The aerial 5.posture Pitching plays Requirement a decisive androle Adjustmentin the performance of the gliding. Therefore, it is necessary to defineAccording the aeri toal the posture workflow requirement of the robot, according the initial to the pitch relationship angle is 0between◦ when thegliding robot at- takeofftitude inand the gliding mode performance, of jumping for and gliding. then adjust Ideally, the if posture the takeoff to achieve direction the goes design through goal. the center of mass, the robot will remain in this posture until it enters the glide stage. However,5.1. Pitching taking Requirement into account the changes in the position of the center of mass caused by manufacturingAccording to andthe locomotion assembly errors, conceiving it is likely of flexible that therewing- willlimb be blending additional platform, angular the velocitiesrobot needs after to the unfold robot takeoff,the main which membrane will lead at tothe changes top after in thetakeoff robot’s and posture. then toThe glide. aerial The postureratio of plays the horizontal a decisive distance role in the to performancethe vertical height of the in gliding. the whole Therefore, gliding it process is necessary is called to definethe glide the aerialratio, posturewhich is requirement one of the key according indicators to the of relationship gliding performance between gliding [39]. The attitude larger andthe glidingvalue, the performance, better the gliding and then performance. adjust the posture Therefore, to achieve the aerial the posture design goal.of the robot at the top should meet the requirements for the robot to obtain the maximum glide ratio. 5.1. Pitching Requirement According to the locomotion conceiving of flexible wing-limb blending platform, the robot needs to unfold the main membrane at the top after takeoff and then to glide. The ratio of the horizontal distance to the vertical height in the whole gliding process is called the glide ratio, which is one of the key indicators of gliding performance [39]. The larger the value, the better the gliding performance. Therefore, the aerial posture of the robot at the top should meet the requirements for the robot to obtain the maximum glide ratio. In order to facilitate the analysis, the influence of different pitching posture on the glide ratio is only considered in this paper. As shown in Figure 17, the rectangular coordinate frames {G} is built. Assuming that the robot is located at a point on the gliding trajectory at time t, its position coordinate is (x(t), y(t)); its speed is v(t); the angle between its speed and horizontal line is θ(t); the angle between its body axis BC and horizontal line is φB(t); the angle of attack is α(t). The forces on the robot are the gravity mg, drag force FD(t) and lift force FL(t). The direction of drag force is opposite to the direction of speed, Appl. Sci. 2021, 11, x FOR PEER REVIEW 16 of 28

 CCxt32()()()() CC Cxtyt xt() 0 1 0 2 4  22  x()() t y t  1x23 ( t ) y ( t ) y ( t )   CC 03 22  2 x()() t y t  (18) CCyt32()()()() CC Cxtyt y() t  g  0 2 0 1 4  x22()() t y t   1x32 ( t ) x ( t ) y ( t )   CC03  2 x22()() t y t  S where C  , C  sin2  , C  cos2  , C  sin(2 ) , and 0 m 1 B 2 B 3 B

C4  cos(2B ) . It is the second-order ordinary differential equation set of xt() and yt(). The ODE45 solver [43] in MATLAB is used for numerical simulation with parame- 2 3 2 1 ters: m  0.08kg , g  9.8m·s ,  1.2kg·m , S  0.04m , v(0) 3ms· , and Appl. Sci. 2021, 11, 3362 the gliding trajectory with different pitch angle can be obtained, as shown in Figure14 18. of 26 It can be seen from Figure 18 that, when the robot glides from 2 m height at the initial horizontal velocity of 3 m/s, the glide distance will be farthest 4.04 m if the pitch angle is 0°, andand thethe directioncorresponding of lift glide force ratio is perpendicular will be maximal to the 2.02. direction When ofthe speed. pitch angle According increases to the to D’Alembert’s10°, the glide distance principle, decreases the equilibrium to 3.69 m, equations which is of91.3% forces of canthe bemaximum expressed distance. as How- ever, when the pitch angle increases to 20°, 30°, and 40°, the glide distance decreases to .. 73.5%, 56.4%, and 43.5%mx of(t) the = −maximumFD(t) cos θdistance,(t) + FL( trespectively.) sin θ(t) Therefore, in order to .. (13) obtain a large glide distancemy(t) =and− mgglide+ ratio,FD(t) thesin θpitch(t) + angleFL(t) ofcos theθ( trobot) should be within 0°~10° when it reaches the top in the mode of jumping for gliding.

Ft() y L C Ft() D  ()t B ()t B  ()t vt()

mg Gliding trajectory

G x FigureFigure 17.17. ForcesForces onon thethe robotrobot whenwhen gliding.gliding.

According to the theory of aerodynamics, FD(t) and FL(t) can be acquired as

1 2 FD(t) = 2 CDSρv (t) 1 2 (14) FL(t) = 2 CLSρv (t)

where CD and CL represent drag coefficient and lift coefficient, respectively; S and ρ denote the area of the wing membranes and the density of air, respectively. The relationship between drag, lift coefficients, and the attack angle can be expressed as [41]

C = 2 sin2 α(t) D (15) CL = 2 sin α(t) cos α(t)

Assuming that the robot can use its tail membrane to achieve pitching moment balance like a ﬂying squirrel during gliding [42], the pitch angle φB(t) remains constant, and

φB = α(t) − θ(t) (16)

Since, .  ( ) cos θ(t) = x t  q . 2 . 2  x (t)+y (t) . ( ) (17) sin θ(t) = y t  q . 2 . 2  x (t)+y (t) Equation (13) can be further arranged as

 . 3 . . 2 .. C0C1x (t)+C0(C2−C4)x(t)y (t)  x(t) = q  . 2 . 2  x (t)+y (t)  . 2 . . 3  x (t)y(t)+y (t)  − 1 C C  2 0 3 q . 2 . 2  x (t)+y (t) . 3 . 2 . (18) .. C0C2y (t)+C0(C1+C4)x (t)y(t)  y(t) = −g + q  . 2 . 2  x (t)+y (t)  . . .  3( )+ ( ) 2( )  − 1 C C x t x t y t  2 0 3 q . 2 . 2  x (t)+y (t) Appl. Sci. 2021, 11, 3362 15 of 26

ρS 2 2 where C0 = − m , C1 = sin φB, C2 = cos φB, C3 = sin(2φB), and C4 = cos(2φB). It is the second-order ordinary differential equation set of x(t) and y(t). The ODE45 solver [43] in MATLAB is used for numerical simulation with parameters: m = 0.08kg, g = 9.8m·s−2, ρ = 1.2kg·m−3, S = 0.04m2, v(0) = 3m·s−1, and the gliding trajectory with different pitch angle can be obtained, as shown in Figure 18. It can be seen from Figure 18 that, when the robot glides from 2 m height at the initial horizontal velocity of 3 m/s, the glide distance will be farthest 4.04 m if the pitch angle is 0◦, and the corresponding glide ratio will be maximal 2.02. When the pitch angle increases to 10◦, the glide distance decreases to 3.69 m, which is 91.3% of the maximum distance. However, when the pitch angle increases to 20◦, 30◦, and 40◦, the glide distance decreases to 73.5%, 56.4%, and 43.5% of the maximum distance, respectively. Therefore, in order to obtain a large glide distance and glide ratio, Appl. Sci. 2021, 11, x FOR PEER REVIEW ◦ ◦ 17 of 28 the pitch angle of the robot should be within 0 ~10 when it reaches the top in the mode of jumping for gliding.

FigureFigure 18. SimulationSimulation results results for gliding gliding trajectory.

5.2.5.2. Pitching Pitching Adjustment Adjustment 5.2.1.5.2.1. Dynamic Dynamic Model Model InIn this paper,paper, the the swing swing of of the the forelimbs forelimbs is usedis used to adjust to adjust the pitchthe pitch attitude attitude of the of robot. the robot.Because Because the hind the limbs hind nolimbs longer no longer move move relative relative to the to body the afterbody takeoff, after takeoff, the hind the limbshind limbsand the and body the canbody be can regarded be regarded as the sameas the rigid same body. rigid As body. shown As in shown Figure in 19 Figure, the center 19, the of mass of the forelimbs, the body containing the hind limbs, and the entire robot are located center of mass of the forelimbs, the body containing the hind limbs, and the entire robot at point A, point B, and point O, respectively. The mass of the forelimbs and the body are are located at point A , point B , and point O , respectively. The mass of the forelimbs mL and mB, respectively. The forelimbs can swing around point C on the body. The length m m andbetween the body point areA andL pointand C isB ,l respectively.AC. The length The between forelimbs point canB swingand point aroundC is pointlBC. TheC angle between the forelimbs and the horizontal line is φ . The angle between the body and on the body. The length between point and pointL is lAC . The length between the horizontal line is φB. The angle between the body and the forelimbs is θBL. point and point is l . The angle between the forelimbs and the horizontal line The rectangular coordinateBC frame {O} is set up on the point O. In frame {O}, the x isaxis L points. The angle to the between horizontal the direction, body and and the the horizontaly axis points line is to  theB . verticalThe angle direction. between They the → → denote the coordinates for point A and point B in frame {O} as r and r , respectively. body and the forelimbs is BL . OA OB They can be obtained using the relation y → → mL r OA + mB r OB = 0 (19) Body B C   L B BL x O A Forelimbs

Hind limbs

Figure 19. Schematic of the robot after takeoff for dynamics modeling of pitching adjustment. Appl. Sci. 2021, 11, x FOR PEER REVIEW 17 of 28

Appl. Sci. 2021, 11, 3362 16 of 26

The Lagrange equation is used for dynamics modeling, and its expression is

d ∂L ∂L ( . ) − = Qi (i = 1, ··· , n) (20) dt ∂qi ∂qi

where qi is the ith generalized coordinate; Qi is the generalized force corresponding to qi; L is the Lagrange function, which is equal to the difference between the kinetic energy and the potential energy of the system. Igoring the height change in the ascent stage, we only consider the inﬂuence of the swing of the forlimbs on the aerial posture [26]. In this case, the robot’s potential energy is zero. Therefore, the system’s total energy is the sum of the kinetic energy of the forelimbs and body, namely . . 1 . 2 1 → 1 . 2 1 → L = J φ + m r |2 + J φ + m r |2 (21) 2 L L 2 L OA 2 2 B B 2 B OB 2

where JL and JB denote the moment of inertia of the forelimbs and body when their axes of rotation pass through the point C, respectively. Further, (19) and (21) can be combined and arranged as  . 2 . 2 . .  Figure 18. Simulation. results. for gliding2 trajectory.2 1 2 2 mLmB(lACφL + lBCφB + 2lAClBCφLφB cos θBL) L = JLφL + JBφB +  (22) 2 mL + mB 5.2. Pitching Adjustment 5.2.1.Neglecting Dynamic the Model air resistance, Equation (20) can be written as

 d ∂L ∂L In this paper, the swing of( the. ) − forelimbs= τ is used to adjust the pitch attitude of the  dt ∂φL ∂φL robot. Because the hind limbsd no∂L longer∂L move relative to the body after(23) takeoff, the hind ( . ) − = −τ  dt ∂φB limbs and the body can be regarded∂φB as the same rigid body. As shown in Figure 19, the wherecenterτ is of the mass the actuation of the forelimbs, torque from the motor.body Combiningcontaining (22) the and hind (23), welimbs, obtain and the the entire robot dynamicsare located equation at point as A , point B , and point O , respectively. The mass of the forelimbs ( .. .. . 2 and the body are mL and mB , respectively. The forelimbs can swing around point C −MφL − KφB cos θBL − KφB sin θBL = τ .. .. . 2 (24) on the body. The lengthNφB + betweenKφL cos θ BLpoint+ Kφ L sin θandBL = point−τ is lAC . The length between

2 2 point andm LpointmBlAC is lBC .m TheLmBl angle betweenmLmBlAC thelBC forelimbs and the horizontal line where M = JL + , N = J + BC , K = mL+mB B mL+mB mL+mB is DueL . The to the angle nonlinear between coupling the between body φandL and theφB horizontal, it is difﬁcult line to transform is B . (24)The into angle between the an equation only containing φ and τ. Therefore, the state variable θ is used for solving body and the forelimbs isB  . BL by following two steps. First, substituteBL θBL = π − φL − φB into (24) to obtain y Body B C   L B BL x O A Forelimbs

Hind limbs

FigureFigure 19. 19.Schematic Schem ofatic the of robot the afterrobot takeoff after for takeoff dynamics for modeling dynamics of pitchingmodeling adjustment. of pitching adjustment. Appl. Sci. 2021, 11, 3362 17 of 26

. 2 . 2 .. HKφ cos θ − HNφ − Gτ φ = L BL B (25) L E . 2 . 2 .. −HMφ + HKφ cos θ − Dτ φ = L B BL (26) B E where D = M − K cos θ , G = N − K cos θ , H = K sin θ , E = MN − K2 cos2 θ . .. .. BL.. BL BL BL From θBL = −φL − φB, (25) and (26) can be arranged as

. 2 . 2 .. HDφ + HGφ D + G θ = L B + τ (27) BL E E Second, we utilize the conservation of angular momentum to further arrange (27). If the air resistance is negligible, the angular momentum of the total system with respect to the point O is a constant, namely

. . . → → . → → LO = JLφL + r OA × mL r OA + JBφB + r OB × mB r OB . . (28) = G φB − DφL = constant

. . . . . Since θBL = −φB − φL, φL and φB can be solved as . . −Gθ − L φ = BL O (29) L D + G . . −Dθ + L φ = BL O (30) B D + G Substitute (29) and (30) into (27) to obtain

. 2 .. DGHθ + HL2 D + G θ = BL O + τ (31) BL E(D + G) E

For the forelimbs motor actuated by a constant rated voltage supply, its torque τ is . related to its angular speed θBL by . τ = τs(1 − θBL/ω0) (32)

where τs is the motor’s stall torque, and ω0 is the no-load angular speed. Therefore, (31) becomes a second-order ordinary differential equation for θBL, namely

. 2 . .. 2 DGHθBL + HLO D + G θBL θBL = + τs(1 − ) (33) E(D + G) E ω0

Finally, by solving θBL from (33) and combining with (30), the pitch angle of the body can be obtained as . Z −Dθ + L φ = BL O dt + φ (0) (34) B D + G B

where φB(0) is the initial angle between the body and horizontal line.

5.2.2. Numerical Simulation The above dynamics model is simulated using MATLAB to verify the feasibility of adjusting the pitching posture by the forelimbs in this section. Assuming that the additional angular velocity is 95 deg/s when the robot takes off at t = 0 in the mode of jumping for Appl. Sci. 2021, 11, 3362 18 of 26

gliding. The simulation parameters are tabulated in Table1, and the simulation results are shown in Figure 20. It can be seen from Figure 20 that when the forelimbs are not actuated and remain stationary relative to the body, the pitch angle of the body increases linearly under the action of additional angular velocity, reaching 14◦ at 0.15 s (dotted line in Figure 20); when the forelimbs swing away from the body driven by the forelimbs motor with a stall torque of 0.15 N·m and a no-load angular speed of 600 deg/s, the pitch angle of the body ﬁrst increases from 0◦ to 3◦, and then basically stops changing (solid line in Figure 20).

Table 1. MATLAB simulation parameters.

Parameters Value Parameters Value

mB 0.07 kg mL 0.01 kg lBC 0.03 m lAC 0.06 m −5 2 −5 2 JB 6.3 × 10 kg·m JL 3.6 × 10 kg·m τs 0.15 N·m ω0 600 deg/s Appl. Sci. 2021, 11, x FOR PEER REVIEW −4 2 ◦ 21 of 28 LO 8.5 × 10 kg·m /s φB(0) 0 ◦ ◦ θBL(0) 90 φL(0) −90

Figure 20. Simulation results for pitching adjustment under different states of forelimbs.

With the same other simulation parameters, the changes of the pitch angle of the body under different no-load angular speeds of the forelimbs motor are shown in Figure 21. It can be seen from Figure 21 that when the no-load angular speed decreases to 580 deg/s, the pitch angle of the body increases from 0◦ to 4.3◦ within 0~0.15 s; when the no-load angular speed increases to 620 deg/s, the pitch angle of the body ﬁrst increases from 0◦ to 2.2◦, and then gradually decreases to 1.4◦ within 0~0.15 s. The above simulation results show that the pitching posture of the body can be adjusted by driving the forelimbs to swing relative to the body after the robot takeoff. In the case of jumping for gliding with initial angular momentum (counterclockwise rotation), the change of pitching attitude caused by the additional angular velocity can be inhibited by the forelimbs swinging away from the body, and the inhibition effect is related to the no-load angular speed of the forelimbs motor: The greater the angular speed is, the more signiﬁcant the inhibition effect is.

Figure 21. Simulation results for pitching adjustment under different angular speeds of forelimbs.

6. Prototype and Experiment 6.1. Prototype Manufacture In order to reduce the weight of the prototype as much as possible, we used the materials with low density and high strength to make the prototype. The body frame was made of a 2 mm carbon fiber plate, which was laser-cut and had a size of 117 mm × 44 mm. The cam, ratchet, gears, pulley, rods, and brackets were 3D-printed from photosensitive resin materials. The transmission shaft was made of aluminum alloy with a diameter of 4 mm. In addition, the ends of hind limbs were pasted with thin rubber pads to increase the friction between the hind limbs and the ground. The S0090D-R 360° continuous rotating digital motor with a mass of 9 gr and a stall torque of 1.6 kg·cm (4.8 V) was used as the motor of the hind limbs, and the S0025 digital motor with a mass of 3 gr and a rotatable angle of 90° was used as the motor of the forelimbs. They both are produced by Doman® RC Hobby Co. Ltd located in Shenzhen, China. Appl. Sci. 2021, 11, x FOR PEER REVIEW 21 of 28

Appl. Sci. 2021, 11, 3362 19 of 26

Figure 20. Simulation results for pitching adjustment under different states of forelimbs.

Figure 21. Simulation results for pitchingpitching adjustment under different angularangular speedsspeeds ofof forelimbs.forelimbs.

6. Prototype and Experiment 6.1. Prototype Manufacture 6.1. Prototype Manufacture In order to reduce the weight of the prototype as much as possible, we used the In order to reduce the weight of the prototype as much as possible, we used the ma- materials with low density and high strength to make the prototype. The body frame was terials with low density and high strength to make the prototype. The body frame was made of a 2 mm carbon ﬁber plate, which was laser-cut and had a size of 117 mm × 44 mm. made of a 2 mm carbon fiber plate, which was laser-cut and had a size of 117 mm × 44 The cam, ratchet, gears, pulley, rods, and brackets were 3D-printed from photosensitive mm. The cam, ratchet, gears, pulley, rods, and brackets were 3D-printed from photosen- resin materials. The transmission shaft was made of aluminum alloy with a diameter of sitive resin materials. The transmission shaft was made of aluminum alloy with a diameter 4 mm. In addition, the ends of hind limbs were pasted with thin rubber pads to increase of 4 mm. In addition, the ends of hind limbs were pasted with thin rubber pads to increase the friction between the hind limbs and the ground. the friction between the hind limbs and the ground. The S0090D-R 360◦ continuous rotating digital motor with a mass of 9 gr and a stall The S0090D-R 360° continuous rotating digital motor with a mass of 9 gr and a stall torque of 1.6 kg·cm (4.8 V) was used as the motor of the hind limbs, and the S0025 digital torque of 1.6 kg·cm (4.8 V) was used as the motor of the hind limbs, and the S0025 digital motor with a mass of 3 gr and a rotatable angle of 90◦ was used as the motor of the motor with a mass of 3 gr and a rotatable angle of 90° was used as the motor of the fore- forelimbs. They both are produced by Doman® RC Hobby Co. Ltd located in Shenzhen, ® China.limbs. They The both SS451A are hallproduced sensor by made Doman by Honeywell RC Hobby® Co.Company Ltd located located in Shenz in Morristown,hen, China. New Jersey, USA was used to detect the number of turns of the winding pulley. The MPU6050 inertial measurement unit (IMU) produced by the InvenSense® company located in San Jose, California, USA was used to detect the pitch angle, and it contains a three-axis gyroscope and a three-axis accelerometer, which can measure the attitude information of the robot. We used the HC-05 Bluetooth module for wireless communication, and its effective transmission distance is 10 m. The Arduino Nano board was used as the central processing unit. It integrates an ATmega328 microcontroller with 6 PWM outputs. The size is 18 mm × 45 mm, and the weight is only 7 gr. The total energy for the robot is provided by a 150 mAh LiPo battery with 7.4 V voltage input. The size of the fully assembled prototype is about 180 mm × 50 mm × 110 mm, and the overall weight including the battery and control system is 83 gr. Figure 22a,b shows the jumping robot at fully ﬂexed and fully extended positions, respectively. The materials and weights of components of the prototype are summarized in Table2.

6.2. Jumping Performance In order to analyze the performance of the prototype in different jumping modes, a high-speed camera (Photron Fastcom Mini UX100) was placed perpendicular to the jumping plane to record posture and trajectory changes. The camera was operated at a Appl. Sci. 2021, 11, x FOR PEER REVIEW 22 of 28

The SS451A hall sensor made by Honeywell® Company located in Morristown, New Jer- sey, USA was used to detect the number of turns of the winding pulley. The MPU6050 inertial measurement unit (IMU) produced by the InvenSense® company located in San Jose, California, USA was used to detect the pitch angle, and it contains a three-axis gyroscope and a three-axis accelerometer, which can measure the attitude information of the robot. We used the HC-05 Bluetooth module for wireless communication, and its effective transmission distance is 10 m. The Arduino Nano board was used as the central processing unit. It integrates an ATmega328 microcontroller with 6 PWM outputs. The size is 18 mm × 45 mm, and the weight is only 7 gr. The total energy for the robot is provided by a 150 Appl. Sci. 2021, 11, 3362 mAh LiPo battery with 7.4 V voltage input. The size of the fully assembled prototype20 of 26is about 180 mm × 50 mm × 110 mm, and the overall weight including the battery and control system is 83 gr. Figure 22a,b shows the jumping robot at fully flexed and fully extended positions, respectively. The materials and weights of components of the prototype are × summarizedrecording speed in Table of 1000 2. frames per second. A board with 80 cm 100 cm grid coordinates behind the robot was used to estimate the position of the robot.

(a) (b)

Figure 22. Prototype of thethe jumpingjumping robotrobot inspiredinspired by by ﬂying flying squirrels. squirrels. (a ()a The) The robot robot at at fully fully ﬂexed flexed position; position; (b )(b the) the robot robot at fullyat fully extended extended position. position.

Table 2.2. Materials and weights of the prototype components.components.

ComponentsComponents MaterialMaterial WeightWeight (gr) (gr) BodyBody frame frame CarbonCarbon fiber fiber 1313 Cam,Cam, Gears, Gears, Brackets Brackets PhotosensitivePhotosensitive resin resin 10.510.5 Transmission shaft Aluminum alloy 4.2 TransmissionLimbs shaft PhotosensitiveAluminum resinalloy 144.2 SpringsLimbs Photosensitive Steel resin 2.114 Controller and sensors - 17.2 MotorsSprings -Steel 122.1 ControllerBattery and sensors -- 1017.2 TotalMotor weights - 8312 Battery - 10 6.2.1. The Mode of Jumping for Gliding Total weight 83 According to the workflow of the mode of jumping for gliding, 1.5 turns of the pulley and 0◦ of the angle between the body and the ground were taken as the initial state before 6.2. Jumping Performance jumping. The motion sequence when the forelimbs were not actuated in the process of jumpingIn order is shown to analyze in Figure the performance 23. It can be of seen the fromprototype Figure in 23 different that the jumping prototype modes, leaves a highthe ground-speed camera after 0.06 (Photron s from Fastcom the initial Mini time UX100) with the was velocity placed ofperpendicular 2.91 m/s and to the the takeoff jump- ingangle plane of 51 to◦ ;record the prototype posture reachesand trajectory the highest changes. point The of camera the jump was trajectory operated at at 0.16 a record- s with ingthe speed velocity of of1000 1.82 frames m/s and per thesecond. height A ofboard 21 cm; with the 80 pitch cm × angle 100 cm of grid the prototype coordinates gradually behind theincreases robot fromwas used 0◦ to to 56 estimate◦ within 0~0.16the position s, and of the the rate robot. of change of angle is about 357 deg/s. Due to the continuous increase of the pitch angle, the prototype finally lands in an inverted position at 0.24 s. With the same initial state before jumping, the forelimbs were actuated to swing counterclockwise at the speed of 600 deg/s once the onboard IMU detected the acceleration change of the prototype. The motion sequence when the forelimbs were actuated in the process of jumping is shown in Figure 24. It can be seen from Figure 24 that the prototype leaves the ground after 0.06 s from the initial time with the velocity of 3.07 m/s and the takeoff angle of 31◦; the prototype reaches the highest point of the jump trajectory at 0.16 s with the velocity of 2.63 m/s and the height of 13 cm; the pitch angle of the prototype gradually increases from 0◦ to 9.6◦ within 0~0.16 s, and the rate of change of angle is about 59 deg/s. Due to the impact and the springs of hind limbs, the prototype inevitably oscillates as it lands and ends up on its side. Appl. Sci. 2021, 11, x FOR PEER REVIEW 23 of 28

6.2.1. The Mode of Jumping for Gliding According to the workflow of the mode of jumping for gliding, 1.5 turns of the pulley and 0° of the angle between the body and the ground were taken as the initial state before jumping. The motion sequence when the forelimbs were not actuated in the process of jumping is shown in Figure 23. It can be seen from Figure 23 that the prototype leaves the ground after 0.06 s from the initial time with the velocity of 2.91 m/s and the takeoff angle of 51°; the prototype reaches the highest point of the jump trajectory at 0.16 s with the velocity of 1.82 m/s and the height of 21 cm; the pitch angle of the prototype gradually Appl. Sci. 2021, 11, 3362 increases from 0° to 56° within 0~0.16 s, and the rate of change of angle is about 357 deg/s.21 of 26 Due to the continuous increase of the pitch angle, the prototype finally lands in an inverted position at 0.24 s.

Appl. FigureSci. 2021 23., 11 ,Motion x FOR PEER sequence REVIEW of jumping for gliding when the forelimbs are not actuated. The solid red lines denote the24 of 28 Figure 23. Motion sequence of jumping for gliding when the forelimbs are not actuated. The solid red lines denote the body, body, the dotted red lines denote the forelimbs, and the blue arrows denote the direction of velocity. the dotted red lines denote the forelimbs, and the blue arrows denote the direction of velocity. With the same initial state before jumping, the forelimbs were actuated to swing counterclockwise at the speed of 600 deg/s once the onboard IMU detected the acceleration change of the prototype. The motion sequence when the forelimbs were actuated in the process of jumping is shown in Figure 24. It can be seen from Figure 24 that the prototype leaves the ground after 0.06 s from the initial time with the velocity of 3.07 m/s and the takeoff angle of 31°; the prototype reaches the highest point of the jump trajectory at 0.16 s with the velocity of 2.63 m/s and the height of 13 cm; the pitch angle of the prototype gradually increases from 0° to 9.6° within 0~0.16 s, and the rate of change of angle is about 59 deg/s. Due to the impact and the springs of hind limbs, the prototype inevitably oscillates as it lands and ends up on its side.

Figure 24. Motion sequencesequence ofof jumpingjumping for for gliding gliding when when the the forelimbs forelimbs are are actuated. actuated. The The solid solid red red lines lines denote denote the the body, body, the the dotted red lines denote the forelimbs, and the blue arrows denote the direction of velocity. dotted red lines denote the forelimbs, and the blue arrows denote the direction of velocity.

Compared to the jumping with the relatively stationarystationary forelimbs, the increase of pitch angle whenwhen jumpingjumping withwith thethe swinging swinging forelimbs forelimbs is is obviously obviously inhibited inhibited as as shown shown in inFigure Figure 24 .2 The4. The pitch pitch angle angle at the at highestthe highest point point of the of trajectory the trajectory is less is than less 10 than◦, which 10°, which meets meetsthe demand the demand of aerial of aerial attitude attitude when when obtaining obtaining excellent excellent gliding gliding performance. performance. In addition, In addition, since the hind limbs are fixed on the body frame, the swing of the forelimbs not only affects the pitching posture of the body, but also changes the direction of the takeoff force, thus leading to the takeoff angle smaller. This change reduces the jumping height and increases the horizontal velocity of the prototype, which is helpful to obtain greater aerodynamic force after the wing membrane is unfolded and improve the gliding performance. Figures 25 and 26 show the variation of pitch angle with respect to time for the five jumps for gliding when the forelimbs are not actuated and the forelimbs swing, respectively. The data in the figures are collected by the onboard IMU, and the data update frequency is 100 Hz. The curves are fitted by polynomial functions in MATLAB software. It can be seen from Figure 25 that the pitch angles of all the five jumps increase from about 0° at the initial point to about 65° at the highest point of the jumping trajectory when the forelimb is not actuated. It can be seen from Figure 26 that when the forelimbs swing counterclockwise, the pitch angles increase only from about 0° to about 10° for all five jumps. It should be noted that the angles shown in Figure 25 do not change linearly as the dotted line in the simulation result Figure 20. The reason is that the forelimbs can rotate slightly during takeoff even though it is not actuated due to inertia. Compared with the solid line in Figure 20, the curves shown in Figure 26 present a trend of decreasing within 0~0.02 s. This is due to the forelimbs swinging at the beginning of the jump when the takeoff force is small, which makes the impact of the forelimbs swinging on the body posture greater than the change in posture caused by the takeoff force that has not passed the center of mass, while in the theoretical analysis and simulation calculation, the entire takeoff process is simplified. The above experimental results show that when the robot jumps in the mode of jumping for gliding, the body’s pitch posture can be effectively adjusted by driving the forelimbs to swing away from the body at an angular velocity of 600 deg/s. This way, the Appl. Sci. 2021, 11, 3362 22 of 26

since the hind limbs are fixed on the body frame, the swing of the forelimbs not only affects the pitching posture of the body, but also changes the direction of the takeoff force, thus leading to the takeoff angle smaller. This change reduces the jumping height and increases the horizontal velocity of the prototype, which is helpful to obtain greater aerodynamic force after the wing membrane is unfolded and improve the gliding performance. Figures 25 and 26 show the variation of pitch angle with respect to time for the five jumps for gliding when the forelimbs are not actuated and the forelimbs swing, respectively. The data in the figures are collected by the onboard IMU, and the data update frequency is 100 Hz. The curves are fitted by polynomial functions in MATLAB software. It can be seen from Figure 25 that the pitch angles of all the five jumps increase from about 0◦ at the initial point to about 65◦ at the highest point of the jumping trajectory when the forelimb is not actuated. It can be seen from Figure 26 that when the forelimbs swing counterclockwise, the pitch angles increase only from about 0◦ to about 10◦ for all five jumps. It should be noted that the angles shown in Figure 25 do not change linearly as the dotted line in the simulation result Figure 20. The reason is that the forelimbs can rotate slightly during takeoff even though it is not actuated due to inertia. Compared with the solid line in Figure 20, the curves shown in Figure 26 present a trend of decreasing within 0~0.02 s. Appl. Sci. 2021, 11, x FOR PEER REVIEW 25 of 28 This is due to the forelimbs swinging at the beginning of the jump when the takeoff force is small, which makes the impact of the forelimbs swinging on the body posture greater than the change in posture caused by the takeoff force that has not passed the center of bodymass,’s while pitch inangle the theoreticalat the highest analysis point and of simulationthe jumping calculation, trajectory theis maintained entire takeoff at process0°~10°, sois simplified.as to meet the requirements of the subsequent gliding movement for the aerial attitude.

FigureFigure 25. ExperimentalExperimental results results for pitching adjustment when the forelimbs are not actuated.

The above experimental results show that when the robot jumps in the mode of jumping for gliding, the body’s pitch posture can be effectively adjusted by driving the forelimbs to swing away from the body at an angular velocity of 600 deg/s. This way, the body’s pitch angle at the highest point of the jumping trajectory is maintained at 0◦~10◦, so as to meet the requirements of the subsequent gliding movement for the aerial attitude.

6.2.2. The Mode of Jumping for Progress According to the workﬂow of the mode of jumping for progress, one turn of the pulley and 20◦ of the angle between the body and the ground were taken as the initial state before jumping. The motion sequence of two intermittent jumps is shown in Figure 27. It can be

Figure 26. Experimental results for pitching adjustment when the forelimbs are actuated.

6.2.2. The Mode of Jumping for Progress According to the workflow of the mode of jumping for progress, one turn of the pulley and 20° of the angle between the body and the ground were taken as the initial state before jumping. The motion sequence of two intermittent jumps is shown in Figure 27. It can be seen from Figure 27 that, the prototype leaves the ground with the velocity of 1.95 m/s and the takeoff angle of 45°; the prototype reaches the highest point of the jump trajectory at 0.13 s with the velocity of 1.35 m/s and the height of 10 cm; the prototype makes a smooth landing and completes its first jump at 0.26 s with a jump distance of 38 cm; the prototype enters the preparation stage for next jumping, and it began to drive the winding Appl. Sci. 2021, 11, x FOR PEER REVIEW 25 of 28

body’s pitch angle at the highest point of the jumping trajectory is maintained at 0°~10°, so as to meet the requirements of the subsequent gliding movement for the aerial attitude.

Appl. Sci. 2021, 11, 3362 23 of 26

seen from Figure 27 that, the prototype leaves the ground with the velocity of 1.95 m/s and the takeoff angle of 45◦; the prototype reaches the highest point of the jump trajectory at 0.13 s with the velocity of 1.35 m/s and the height of 10 cm; the prototype makes a smooth landing and completes its ﬁrst jump at 0.26 s with a jump distance of 38 cm; the prototype enters the preparation stage for next jumping, and it began to drive the winding pulley to rotate and adjust the body posture at 0.82 s; the prototype starts the second jump with the same initial state at 3.37 s; the highest point of the second jump is reached at 3.47 s with the velocity of 1.31 m/s and the height of 9 cm; the prototype completes its second jump at 3.66 s with a jump distance of 36 cm. The total distance of the two jumps is 0.74 m, and the totalFigure time 25. isExperimental 3.66 s, so the results average for pitching progress adjustment speed is when 0.2 m/s. the forelimbs are not actuated.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 26 of 28

pulley to rotate and adjust the body posture at 0.82 s; the prototype starts the second jump with the same initial state at 3.37 s; the highest point of the second jump is reached at 3.47 s with the velocity of 1.31 m/s and the height of 9 cm; the prototype completes its second jump at 3.66 s with a jump distance of 36 cm. The total distance of the two jumps is 0.74 m,FigureFigure and 26.26.the ExperimentalExperi total timemental is results results3.66 s, forfor so pitching pitchingthe average adjustmentadjustment progress whenwhen speed thethe forelimbsforelimbs is 0.2 m/s. areare actuated. actuated.

0s 6.2.2. The Mode of0 .Jumping13s for Progress 0.26s According to the workflow of the mode of jumping for progress, one turn of the pul- 20 20 20 1.95m/s ley and 20° of the angle between1.35m/s the body and the ground were taken as the initial state 10 before jumping.10 The motion sequence of two intermittent10 jumps is shown in Figure 27. It 45° can be seen from Figure 27 that, the prototype leaves the ground with the velocity of 1.95 20 30 40 50m/s60 and70 the80 takeoff angle20 30 of40 45°;50 the60 prototype70 80 reaches20 the30 highest40 50 point60 70 of80 the jump trajectory at 0.13 s with the velocity of 1.35 m/s and the height of 10 cm; the prototype makes 0.82s 1.64s 2.47s a smooth landing and completes its first jump at 0.26 s with a jump distance of 38 cm; the

20 prototype enters20 the preparation stage for next20 jumping, and it began to drive the winding

10 10 10

20 30 40 50 60 70 80 20 30 40 50 60 70 80 20 30 40 50 60 70 80

3.37s 3.47s 3.66s

20 1.92m/s 20 20 1.31m/s 10 10 45° 10

20 30 40 50 60 70 80 20 30 40 50 60 70 80 20 30 40 50 60 70 80

Figure 2 27.7. Motion sequence of jumping jumping for for progress. progress. The The dotted dotted red red lines lines denote denote jumping jumping trajectory, trajectory, and the blue arrows denote the direction of velocity.

7. Conclusions In this paper, we have presented the design, analysis, and experiment of a miniature jumping robot that mimics a flying squirrel and realizes two jumping modes with different takeoff speeds and stances by using a new spring charging and releasing mechanism. According to the design goal of the robot, this paper analyzes the jump characteristics of flying squirrels in different modes and designs each part and workflow of the robot. Additionally, the requirement for aerial pitching posture of the robot after taking off is analyzed, and the dynamic model of adjusting the pitching by the swing of the forelimbs is established and simulated. Finally, a prototype is made and its performance is tested. From the results, the robot can takeoff at about 3 m/s and pitch angle of 0° in the mode of jumping for gliding, and adjust the pitch angle at the top to 0°~10° by actuating the forelimbs swing according to the requirement of gliding; in the mode of jumping for progress, the robot can takeoff at about 2 m/s and pitch angle of 20° and then intermittent jump with the distance of 0.37 m of once jump and the average progress speed of 0.2 m/s. To the best of our knowledge, the robot presented in this paper is the first centimeter- scale bionic jumping robot that has the capabilities of changing the speeds and stances autonomically when takeoff. In the future work, we will further reduce the mass of the robot, increase the takeoff speed and the stability of the aerial posture, and integrate the wing membranes deployable mechanism to complete the design and development of the flexible wing-limb blending platform.

7. Conclusions In this paper, we have presented the design, analysis, and experiment of a miniature jumping robot that mimics a flying squirrel and realizes two jumping modes with different takeoff speeds and stances by using a new spring charging and releasing mechanism. According to the design goal of the robot, this paper analyzes the jump characteristics of flying squirrels in different modes and designs each part and workflow of the robot. Additionally, the requirement for aerial pitching posture of the robot after taking off is analyzed, and the dynamic model of adjusting the pitching by the swing of the forelimbs is established and simulated. Finally, a prototype is made and its performance is tested. From the results, the robot can takeoff at about 3 m/s and pitch angle of 0◦ in the mode of jumping for gliding, and adjust the pitch angle at the top to 0◦~10◦ by actuating the forelimbs swing according to the requirement of gliding; in the mode of jumping for progress, the robot can takeoff at about 2 m/s and pitch angle of 20◦ and then intermittent jump with the distance of 0.37 m of once jump and the average progress speed of 0.2 m/s. To the best of our knowledge, the robot presented in this paper is the first centimeter- scale bionic jumping robot that has the capabilities of changing the speeds and stances autonomically when takeoff. In the future work, we will further reduce the mass of the robot, increase the takeoff speed and the stability of the aerial posture, and integrate the wing membranes deployable mechanism to complete the design and development of the flexible wing-limb blending platform.

Author Contributions: Conceptualization, F.Z. and W.W.; methodology, F.Z.; software, F.Z.; vali- dation, F.Z., J.W., and J.Z.; formal analysis, F.Z., J.W., and J.Z.; investigation, F.Z., W.D., and P.Z.; resources, W.W.; data curation, F.Z., W.D., and P.Z.; writing—original draft preparation, F.Z.; writing— review and editing, F.Z., W.W., and J.W.; visualization, F.Z.; supervision, W.W.; project administration, F.Z.; funding acquisition, W.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by Beijing Natural Science Foundation (3202015) and the Academic Excellence Foundation of BUAA for PhD Students. Institutional Review Board Statement: The flying squirrel used in this study follow the regulations for the Administration of Affairs Concerning Experimental Animals issued by the Institutional Animal Care and Use Committee of Beijing. Conflicts of Interest: The authors declare no conflict of interest.

Appendix A

Table A1. Main parameters of gear system.

Parameters Motor Gear Gear 1 Gear 2 Transmission Gear Number of teeth 20 40 20 40 Module (mm) 0.5 0.5 0.5 0.5 Pressure angle (◦) 20 20 20 20

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