Falling Cat Robot Lands on its Feet

9 Pages·0.545 MB·English

by  Shields Ben, Robertson William S.P. et al.

Falling Cat Robot Lands on its Feet
Ben Shields, William S. P. Robertson, Natalie Redmond
Ross Jobson, Rian Visser, Zebb Prime, Ben Cazzolato
University of Adelaide, Australia
[email protected]
Abstract Yeadon [1984], citing Batterman [1974], remarks that
the cat uses a combination of the legs-in-legs-out twist
Cats are renowned for their ability to always
and the ‘hula’ twist, in which the rear body and legs
landontheirfeet. Whentheirbodyisdropped
circumductsusingspinalbendinginanoppositesenseto
withnoinitialnetangularmomentum,theyare
thefrontbody. Asapointofdifference,theself-righting
abletorotateintheairusingavarietyofmech-
ofafallingrabbitusesthetwo-axestwist,whichrequires
anisms, including the use of a variable differ-
largeamountsofspinaltwist[Yeadon,1984]whichisnot
encebetweenthemomentofinertiaofthefront
observed in a falling cat [Kane and Scher, 1969].
andbackportionsoftheirbodyduetothemo-
Thefirstdynamicmodeltotakeintoaccountthephys-
tion of their legs. The aim of this project is to
iologicalconstraintsofthebiologicalsystemwasbyKane
buildaroboticfallingcatthancanrightitselfin
and Scher [1969], who remarked that his model achieves
the air not dissimilar in appearance and mech-
verismilitude with a model of a cat ‘at the expense of
anism to a biological cat. This paper describes
simplicity’, although changes in moment of inertia due
the first steps of creating a simplified model
to the varying positions of the legs are not considered.
and designing a simulation to demonstrate the
Kane’s model is notable in that the torso is constrained
mechanismofaprototype. Theoryispresented
to bending motion only, and cannot twist, and this has
to calculate input torques to achieve rotation
beenthebasisformostsimulatedmodelssince. Modern
fromanydropangleor(sufficientlylarge)drop
work has largely focused on trajectory planning and op-
height,andSimMechanicsisusedtoverifyand
timisation for falling cat models which use variations of
simulate the model.
Kane’smodel[WengandNishimura,2002;GeandChen,
2007; Iwai and Matsunaka, 2012; Takahashi, 2012], but
1 Introduction the most complex simulation model proposed to date is
a biomechanical model with 16 degrees of freedom and
The self-righting behaviour of a falling cat (and other
10 joint torque inputs [Arabyan and Tsai, 1998].
animals)iswellknown,andthereareanumberofexpla-
Despite the development of a number of simulated
nations on the mechanisms by which a falling cat rights
models of varying complexity, no robotic systems in the
itself. Yeadon [1984] lists three mechanisms by which
literaturehavebeenbuilttospecificallyimplementsome
a body composed of multiple links achieves changes of
or all of the mechanisms of a self-righting falling cat.
orientation in flight, using any combination of: (a) the
Maintainingacat-likedesignandmotionandproducing
varyingmomentsofinertiatwist,(b)thetwo-axestwist,
a working physical model will distinguish this project
and/or (c) the ‘hula’ twist. Early consideration of a
from prior work. The aim of this paper is to present a
falling cat proposed the moments of inertia twist as the
mechanically-simple model of a falling cat which is be-
main mechanism [Benton, 1912], in which the extension
ing developed into a physical prototype. The prototype
andretractionofthelegschangethemomentsofinertia
is under construction and its details will be reported at
while the spine twists. This ‘legs-in-legs-out’ method is
a future date.
stillcommonlyusedasanexplanationoftheproblemto-
day[Stewart,2011]. However,ithasrecentlybeenshown This paper is structured into the following sections:
for a reasonable model of such a system with typical the simplified cat model is presented, followed by the
cat-like physical parameters that the maximum amount derivation of a rigid-body dynamic model of the system
of rotation that can be achieved using this method is thattakesintoaccountvariablemomentsofinertia. This
◦
around 55 [Kaufman, 2013]. is followed by an analysis of the timing stages required
(a) Catinitiallyupside-down. (b) Frontlegsretractedtoincrease (c) Holds position and both halves
positiverotationoffront;backlegs continuetorotate.
extended to reduce negative rota-
tionofrear.
(d) Front legs extended to reduce (e)Holdspositionuntilrequiredro- (f) Backlegsarenowextendedand
negative rotation; back legs retract tationisachieved. bracedforimpact.
toincreasepositiverotation.
Figure 1: Drawings of a falling cat denoting key points in the trajectory. Inspired by high speed camera footage [Destin,
2012].
to active the desired action of righting the cat during mately equal density, ‘feet’ are added which act as large
its fall. Finally, a SimMechanics model is presented to point masses to increase the possible change of moment
test the theory, and the paper concludes with a brief of inertia.
discussion of the robotic system under development. The robotic cat model therefore consists of a body in
two halves that can rotate around their common axis (a
2 Simplified cat model ‘rigidspine’),andeachbodyhasanattachedlegthatcan
◦
extendto90 andretractasneeded. Aschematicofthe
For the purposes of a robotic system a number of sim-
catrobotisshowninFigure2. Theaxisofthespinewas
plifications were required to reduce the complexity of
located roughly through the centre of the robot to sim-
the mechanical design from that of a biological feline.
plify the calculation of the dynamics. Three actuators
A physical cat uses a combination of spine flexion, per-
are required in total to control relative angle between
haps some spinal twist, and leg extension to manipulate
the front and back body halves and between each set of
its moments of inertia, as discussed in the introduction.
legs and the body. Finally, note that for simplification
The aim of the project is to build a robotic cat, yet the
of the analysis the head is omitted.
complexity of the biological system precludes a simple
mechanical design (cf. the comments of Kane and Scher
3 Modelling of system dynamics
[1969] on simplicity quoted prior). For the first itera-
tion of the project, therefore, a radical simplification is Intheprevioussectionthemechanismbywhichafalling
made to only model the legs-in-legs-out behaviour of a cat rights itself was presented. This section develops a
cat since it has the easiest mechanical design. While model by which this action can be emulated. With ref-
the work of Kaufman [2013] suggests this approach is erence to Figure 2, Table 1 defines the variables used
impossible, another alteration is made to the cat design in the following sections. The values given for the vari-
◦
to allow a full 180 rotation to be achieved. Instead of ablesareindicativeoftheroboticsystemcurrentlyunder
legs which extend out from the body all with approxi- construction; this section introduces a method by which
yˆ yˆ
zˆ xˆ
L L
2 1
2r
BACK FRONT
r
O
2r m2 m1
r
W r3 r4
2 W
2
θ θ
B F
m3 L L
m
4
W W
(a) Sideview. (b) Frontview.
Figure 2: Schematicandgeometryoftheroboticcatmodel.
these values can be chosen to fulfil a certain design cri- (decreasingθ)decreasesmomentofinertiaandincreases
teria. angular velocity. When a cat is falling, it retracts its
◦
front legs (θ = 0 ) and rotates quickly clockwise, while
3.1 Conservation of angular momentum
◦
the back legs remain extended (θ = 90 ) and rotate
and theory of rotation
slowly anticlockwise, due to the conservation of angu-
It can be considered that a cat separates its body into larmomentum. Thecatthenreversesthepositionofits
two separate rotational axes for the front and back por- legs by extending its front legs and rotates slowly anti-
tions of its body. Assuming that the cat falls from rest, clockwise and retracts its back legs and rotates quickly
theinitialnetangularmomentumiszeroandthesumof clockwise. This motion is shown in Figure 1.
the angular momenta must also equal zero due to con-
3.2 Moments of inertia
servation of momentum. This is represented as
The cross-section of the cat’s body is comprised of a
I ω +I ω =0, (1)
F F B B rectangle and a semi-circle, as shown in Figure 2. Some
whereI andI arethemomentsofinertiaofthefront assumptionswheremadeindevelopingtheinertiamodel
F B
and back halves, respectively, and ω and ω are the of the system. The front and back bodies are assumed
F B
corresponding angular velocities. Therefore in order to to rotate about point O shown in the figure. (For the
control the individual rotations of the front and back robotic system under development, this can be adjusted
halves of the cat, their respective angular velocities can somewhat using additional weights.) The pairs of legs
be controlled by adjusting the ratio of moments of iner- on each half are assumed to move as one, which also
tia, I /I and I /I : simplifies mechanical design as a single actuator can be
F B B F
usedforeachlegpair. Thelegsaredesignedtohavelow
I I
ω =− Fω , ω =− Bω . (2) mass and therefore low moment of inertia, and the feet
B F F B
I I
B F are designed to have high mass and a correspondingly
Considering this in terms of a cat’s body, extending the high moment of inertia when the legs are fully extended
◦ ◦
legs (increasing θ to 90 ) increases the moment of iner- to 90 . The feet are assumed to be point masses at
tiaandreducesangularvelocity,whileretractingthelegs the end of the legs, which are modelled as slender rods.
Table 1: Variablesusedinmodellingtheroboticfallingcat. Valuesshownareindicativeandmaybevariedtosuit.
Symbol Value Description
m 2.75kg Mass of front body
1
m 1.95kg Mass of back body
2
m 0.4kg Combined point mass of feet
3
m 0.074kg Combined mass of legs
4
L 13mm Length of the legs
W 3mm Width of the legs
2r 13mm Width and height of the body
◦ ◦ ◦
θ , θ 0 to 90 Angle of the legs from the vertical (legs retracted for θ=0 )
F B
I 0.007kgm2 Moment of inertia of front body
OF
I 0.005kgm2 Moment of inertia of back body
OB
I 0.016kgm2 to 0.0477kgm2 Moment of inertia of front body with legs and feet from θ=0◦ to 90◦
T
I 0.014kgm2 to 0.0456kgm2 Moment of inertia of back body with legs and feet from θ=0◦ to 90◦
B
Finally,thecentralshaftaroundwhichthebodiesrotate
0.05
is assumed to have negligible moment of inertia.
The total moments of inertia around the zˆaxis of the
front and back body are given by, respectively,
)
4m 0.04
I =I +I +I , (3) (
F OF LF FF
a Front body
I =I +I +I . (4) ti
B OB LB FB r
ne 0.03
where IOF, IOB are the moments of inertia due to the ofi Back body
bodies, I , I due to legs, and I , I due to the
LF LB FF FB t
n
feet. e
m 0.02
The front and back bodies each respectively have a o
M
constant moment of inertia around the zˆaxis given by
IOF = 12Cm1r2+ 23Sm1r2, (5) 0.01
I = 1Cm r2+ 2Sm r2, (6) 0 15 30 45 60 75 90
OB 2 2 3 2
Leg rotation (degrees)
where S and C are the cross-sectional area ratios of the
bodyfromtherectangleandthesemi-circlerespectively: Figure 3: Change in Moment of Inertia with respect to leg
rotation.
A 2r2 4
rect
S = = = , (7)
A +A 2r2+πr2/2 4+π
rect circ and distance magnitudes of the feet and legs, r and r ,
3 4
and are
π
C =1−S = 4+π. (8) rx =r− 21W, ry =r+ 12W, (12)
The front and back feet are assumed to be point masses r2 =r2+(r +Lsinθ)2, (13)
3 x y
with moments of inertia, respectively,
r2 =r2+ r + 1Lsinθ 2. (14)
4 x y 2
IFF =m3r32, IFB =m3r32. (9) Figure 3 demonstrat!es the change(cid:1) in the moment of
Finally,themomentsofinertiaduetothefrontandback inertia of each body due to rotation of the legs.
legs (without feet) are given by, respectively,
3.3 Angular velocity requirements
m L2 m L2 The overall fall time t can be calculated using
I = 4 θ +m r2, I = 4 θ +m r2, (10) tot
LF 12 4 4 TB 12 4 4 parabolic equations of motion, where
where Lθ is the projection of the leg length in the xˆ–yˆ ttot = 2d/g, (15)
plane, including the square top face of width W, p
anddisthedistancefallenandg istheaccelerationdue
L =(L−W)sinθ+W. (11) to gravity.
θ
30 3.5
Front body 3
)
rad/s 20 ratio 2.5
( a
y ti
cit 10 ner 2 If/Ib Ib/If
Velo ofi 1.5
ular 0 ment 1
ng Mo
A
−10 Back body 0.5

0 t1 t2 t3 t4 t5 t6 0 t1 t2 t3 t4 t5 t6
Time Time
Figure 4: Angularvelocityprofileovertheperiodofmotion. Figure 5: Momentofinertiaratiowithrespecttotime.
Table 2: Timeintervalsanddescriptionofmotion.
During rotation, the legs of both bodies are extended
and retracted at different time intervals. This changes Interval Calculation Description
theratioofthemomentofinertiabetweenthefrontand
t 2d/g Total time in the air
back body IF/IB and is used according to (2) to adjust tot
the relative angular velocity between the halves. p Front legs are retracted
t Variable
The model has been divided into six key time inter- 1 Back legs remain extended
vals marked by times t to t . The first time step t is
1 6 1 Front legs remain retracted
variable as it can be changed to suit the requirement of t −t 1(t −3t )
2 1 2 tot 1 Back legs remain extended
themotorcontrollingthelegrotation. Thesetimeinter-
valsaredisplayedinTable2,alongwithadescriptionof t −t 1t Front legs are half extended
the motion during that interval. The other time steps 3 2 2 1 Back legs are half retracted
are determined automatically based on the overall drop
Front legs are extended
time,sincetheactionsovertheentirefallaresymmetric. t4−t3 t3−t2 Back legs are retracted
Thesetimingintervalscorrespondtoanangularvelocity
Front legs remain extended
profile for the two halves of the cat as shown in Fig- t −t t −t
5 4 2 1 Back legs remain retracted
ure4. Thevelocityprofilesarerelatedusing (2)andthe
velocity magnitudes must be calculated to ensure that Front legs remain extended
t −t t
the correct rotation ensues. 6 5 1 Back legs are extended
Table 3 demonstrates the required time of each inter-
val for a drop height of d = 2m and a leg retraction Table 3: Approximatetimestepsfora2mdropheight.
time t =0.1s. These values will be used in the follow-
1
ing analysis as typical. Using these time intervals, and t t t t t t
1 2 3 4 5 6
the description from Table 1, an overall description of 0.1s 0.27s 0.32s 0.37s 0.54s 0.64s
theratioofmomentsofinertiaI /I andI /I canbe
B F F B
established. Figure5showshowtheseratioschangeover Table 4: Momentofinertiaratioforeachtimeinterval.
time during a fall.
Given a series of time periods and ratios of moment Time Interval Ratio
of inertia as a function of time, it is now possible to
t1 γ1 =IF,45◦/IB,90◦
calculate the necessary angular velocities to achieve the t2−t1 γ2 =IF,0◦/IB,90◦
desired rotation. These calculations are simplified by t3−t2 γ3 =IF,22.5◦/IB,67.5◦
assuming a piece-wise model for the moment of inertia t4−t3 γ4 =IB,25.5◦/IF,67.5◦
ratios by taking an average of the moment of inertia t5−t4 γ5 =IB,0◦/IF,90◦
over each time interval. These averages are displayed in t6−t5 γ6 =IB,45◦/IF,90◦
Table 4.
ItcanbeseeninTable4thatforhalfthetimetheratio
40
isdefinedasI /I andfortheotherhalfitistheinverse
F B
IB/IF. This is because the model has been constructed Back legs
suchthateachbodywill‘control’therotationforhalfof s) Front body
the fall time, using the calculations of angular velocity d/ 20
a
in (2), where ω is ‘controlled’ in the first half and ω r
B F (
is controlled in the second. y
t
Fromthis,simultaneousequationscanbeconstructed ci 0
o
representing the motion of each body during the overall el
V
falltime. Wecanequatetheoverallrotationtoanangle r Back body
a
φ, which is the required angle of rotation to reach an ul −20
g
upright position, resulting in n
A
Front legs
φ=ω c −ω c , (16)
1 1 2 2 −40
φ=−ω1c3+ω2c4, (17) 0 t1 t2 t3 t4 t5 t6
Time
whereω andω arethepeakangularvelocitiesinrad/s
1 2
for the front and back body respectively, and time con-
stants c(·) are given by Figure 6: Angularvelocityprofileovertheperiodofmotion.
t t −t
c = 1 +(t −t )+ 3 2,
1 2 1
2 2 4 SimMechanics model
t −t t −t
c = 4 3γ +γ (t −t )+ 6 5γ ,
2 4 5 5 4 6
3 3
(18)
c = t1γ +γ (t −t )+ t3−t2γ , SimMechanics is a multi-body simulation environment
3 1 2 2 1 3
3 3 for 3D mechanical systems. The systems are modelled
c = t4−t3 +(t −t )+ t6−t5. using blocks representing bodies, joints, constraints,
4 5 4
2 2 and force elements. SimMechanics then formulates and
solvestheequationsofmotionforthecompletedynamic
These time constants are chosen heuristically because
system. Themodelcanbeparameterisedusingvariables
the moments of inertia change with angle. As will be
and expressions defined within Matlab and the control
seen in the simulation results in the next section (Fig-
system for the bodies can be defined within Simulink.
ure 9), the resultant velocity profile is reasonably close
ThepurposeofSimMechanicsistoverifythemodelcal-
to that desired.
culations and provide a simulation of the expected sys-
Equations 16 and 17 can be expressed in matrix form
tem dynamics.
as
c −c ω φ The body of the cat has two separate rotational axis
(cid:20)−c13 c42(cid:21)(cid:20)ω12(cid:21)=(cid:20)φ(cid:21), (19) for the front and back parts of its body. The simulation
environmentdoesnotaccountfortheconservationofan-
and therefore the required angular velocities can be gular momentum for the body as a whole, and therefore
found with the front and back body have to be analysed separately.
ω c −c −1 φ Before a dynamic analysis can be undertaken, the ge-
(cid:20)ω12(cid:21)=(cid:20)−c13 c42(cid:21) (cid:20)φ(cid:21). (20) ometry and material for the system needed to be estab-
lished. Thiswasachievedbyusingblocksforthebodies,
Theangularvelocityofthelegscanbeconsideredsep- joints and constraints, and applying the required speci-
aratelyfromthemainbody. Whenthelegsareextended fications. SimMechanics is linked to the code developed
◦
or retracted it is required that they rotate 90 in t sec- for the conceptual design, so if changes are made to the
1
onds. Usinganexamplevalueof0.1sfort ,therequired design,itcandirectlyupdatedtothemodel. Theblocks
1
average velocity of the legs is 15.7rad/s. The velocity weredefinedtohaveauniformdensity,whichwasestab-
profile of the legs in conjunction with the body veloci- lishedfromtherequiredmassandshapegeometry. Uni-
ties is shown in Figure 6. form density is a simplification on the real model, as it
Having calculated desired (and approximate) velocity willhavelocalisedmassfrominternalcomponents. Sim-
profilesfortherobot,inthenextsectionadynamicsim- Mechanics then applies this density in accordance with
ulationispresentedtoverifythatthemotionoftherobot the geometry of the SimMechanics model to determine
is as expected. the overall mass moment of inertia of each element.
4.1 Torque inputs
In Section 3, a model was presented that could achieve
the desired rotational behaviour following a set of angu- 15
lar velocity profiles. For the SimMechanics and robotic
model,theinputsintothesystemaremotortorques. For 10
simplicityanopenloopcontrolmethodologywaschosen,
such that torque profiles would be selected to achieve
5
)
known velocity profiles. m
N
It was expected that knowing the angular velocity re- ( 0
e
quirements, the appropriate acceleration and therefore u
q
torque could be determine for each time interval of the oe −5
T
motion. However, while the legs are extending or re-
Front legs Back legs
tracting the moment of inertia is changing as a func- −10
tion of time. If the torque is set at a constant level, Body
theimposedaccelerationwillthereforechangeduetoits
−15
inversely proportional relationship with the moment of 0 t1 t2 t3 t4 t5 t6
inertia. This is not ideal as the velocity profiles were
Time
initially set up assuming a constant acceleration; this is
alsothereasonheuristictimeconstantswererequiredin
Figure 7: Torqueinputsintothelegsandbody,discretised
Equation 18. To overcome this issue, each time inter-
every25msand10msrespectively,toachievethe
val was divided into smaller increments, and the torque resultsshowninFigure9.
ineachincrementwasestimatedtoapproximatethede-
sired acceleration profile. Using the results of Figure 3,
anapproximationforthemomentofinertiacouldbees-
tablished, and the torque would be re-set to correlate
with the required angular acceleration.
In addition to the variable torque required to achieve
theangularvelocityprofiles,thereisanadditionaltorque
required to hold the legs in position due to the cen-
tripetal force caused by rotation of the body. During
intervals where the body is accelerating, the magnitude
of this centripetal force varies with time. Again, to ac-
count for this variable torque requirement the velocity (a) t=0s (b) t=0.05s (c) t=0.1s
profilewas brokendown intosmaller timeincrements to
give a more accurate approximation of the centripetal
force in order to be accounted for in the motor selection
for the physical robot.
4.2 Simulation results
(d) t=0.27s (e) t=0.32s (f) t=0.37s
Startingfromasetofsystemvariables(masses, lengths,
etc.), the velocity profiles were generated to achieve the
desired rotational action. These were taken as inputs
into the SimMechanics model, which in turn generated
step-wise torque inputs, shown in Figures 7, to approx-
iate these velocity profiles. The visual representation of
(g) t=0.54s (h) t=0.59s (i) t=0.64s
the SimMechanics model can be seen in Figure 8, with
associated kinematic profiles as shown in Figure 9. It
Figure 8: Screenshots taken from SimMechanics model in-
can be seen that although the stepwise approximation
dicatingchangeintorque.
for the input torques yields a noisy acceleration signal,
the velocities track with the desired profile described in
Section 3.
6
20
)
2ad/s) 500 d/s) 10 n(rad 42
(r ra tio
on 0 y( 0 osi 0
Accelerati −500 Velocit −−2100 AngularP −−−642
0 t1 t2t3t4 t5 t6 0 t1 t2t3t4 t5 t6 0 t1 t2t3t4 t5 t6
Time Time Time
(a) Frontbody,angularacceleration. (b) Frontbody,angularvelocity. (c) Frontbody,angularposition.
6
20
)
2ad/s) 500 d/s) 10 n(rad 42
(r ra tio
on 0 y( 0 osi 0
Accelerati −500 Velocit −−2100 AngularP −−−642
0 t1 t2t3t4 t5 t6 0 t1 t2t3t4 t5 t6 0 t1 t2t3t4 t5 t6
Time Time Time
(d) Backbody,angularacceleration. (e) Backbody,angularvelocity. (f) Backbody,angularposition.
Figure 9: Simulated body pose during a fall, showing angular position, velocity, and acceleration for the front and back
halvesofthecatbody.
5 Physical prototype ronment has been constructed using SimMechanics and
openloopcontrolhasbeenshowntobesufficienttosuc-
The simulation results presented in the previous sec-
cessful right a falling cat.
tion allowed design speculation for the specifications of
the robotic system. An iterative process was conducted Forfuturework,acontrolsystemwillbeconsideredto
considering torque and weight requirements. (Larger drivetheinputstagemoreefficientlywithouttherequire-
weights required larger torques, which required larger ment for interpolating and approximation of the desired
motors, which increased the weight, etc.) It should acceleration. The robotic system is being commissioned
be noted that the technique used to generate the ac- and will be tested shortly.
celeration curves is not applicable to the DC motors
that havebeenchosen, sincetheircontrollersallowadi-
rect displacement input. This fact simplifies the control
Acknowledgments
methodology of the robot significantly.
6 Conclusion and future work
The authors of this paper would like to thank Garry
In this paper, a dynamic model of a flipping cat robot Clarke for advice on the mechanical design of the robot,
hasbeenpresented. Thedesignhasbeensimplifiedtofa- Phil Schmidt and Derek Franklin for their assistance
ciliatetheconstructionofaroboticprototypetodemon- with the electrical design, and Ian Linke and Aubrey
stratethefeasibilityofthesystem. Themodelgenerates SlaterfromtheSchoolofElectricalandElectronicEngi-
velocity profiles to achieve flipping for any (sufficiently neering,UniversityofAdelaide,forgenerouslyproviding
large)fallingheightandinitialangle. Asimulationenvi- access to and assistance with their 3D printer.
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