dynamics-based control of a one-legged hopping robot …sangho/hyon_imech03_kenken.pdf ·...

83

Dynamics-based control of a one-legged hopping robot

S H Hyon1 T Emura1 and T Mita21Department of Mechatronics and Precision Engineering Graduate School of Engineering Tohoku UniversitySendai Japan

2Department of Mechanical and Control Engineering Graduate School of Science and Engineering Tokyo UniversityTokyo Japan

Abstract This paper proposes a new model of a one-legged hopping robot The one-legged hoppingrobot is useful in realizing rapid movement such as that of a running animal Although it has asimple leg mechanism the dynamics are not simple and require non-linear complex analysis Thismeans that it is not easy to derive a controller for stable hopping in a systematic way Therefore adynamics-based approach was introduced where the controller is empirically derived based on charac-teristic dynamics A prototype of the one-legged hopping robot was fabricated and a precise simulatorof the robot including actuator dynamics was constructed to examine the usefulness of the proposeddynamics model Applying the constructed simulator to the prototype the robot succeeded in planarone-legged hopping

Keywords hopping robot biomechanics simulator dynamics-based control

NOTATION

A1 A2 areas of cylinder chamber (m2)Cip Cep internal and external leakage

coe cients (m4 skg)Ctp total leakage coe cients

(m4 skg)fr f

strajectory parameters for r and h

(Hz)Ff friction force of cylinder (N)FL load force of cylinder (N)g gravity constant (ms2)i i

1 i2 constant current about vertical

speed control (mA)ic constant current about vertical

speed control (mA)K1 K2 feedback gain about position

control for joints 1 and 2K

f feedback gain about forwardspeed control

Ki

current gain of servovalveKp feedback gain about body pitch

controlK

static steady ow gain of servovalve

The MS was received on 28 August 2002 and was accepted after revisionfor publication on 20 November 2002 Corresponding author Department of Mechatronics and PrecisionEngineering Graduate School of Engineering Tohoku UniversityAramaki-Aza-Aobo 01 Aobaku Sendai 980-8579 Japan

I05502 copy IMechE 2003 Proc Instn Mech Engrs Vol 217 Part I J Systems and Control Engineering

mc equivalent mass of cylinder

piston rod and oil (kg)PL load pressure of servovalve (Pa)Ps supply pressure (Pa)P1 P2 pressures of cylinder chamber

(Pa)q1 q2 q3 joint angles (deg)q1 q2 desired time trajectories of joints

1 and 2 (deg)Q output ow of servovalve

(Lmin)Qin

Qout ow into and out of cylinder(Lmin)

QL (dynamic) load ow of

servovalve (Lmin)Q

static static load ow of servovalve(Lmin)

r virtual leg length (m)rmax maximum virtual leg length (m)

rret virtual leg length for retraction

(m)r desired time trajectory of virtual

leg length at ight phase (m)Ts stance time (s)V V1 V2 V10 V20 volumes of cylinder chamber

(m3)Win Wout weight owrates into and out of

the servovalvex piston displacement in section 3

or hip position of robot insections 4 and 5 (m)

84 S H HYON T EMURA AND T MITA

x2a desired forward speed (ms)z vertical hip position (m)z2a desired vertical lift-o speed

(ms)

be e ective bulk modulus (Pa)h damping coe cient of servovalveh

b nominal touchdown angle ofvirtual leg (deg)

hf desired touchdown angle ofvirtual leg (deg)

h desired time trajectory of virtualleg angle at ight phase (deg)

r r0 mass density of oil

f damping coe cient of servovalvew pitch angle of body (deg)wd desired body pitch angle (deg)vn natural frequency of servovalve

(rads)

1 INTRODUCTION

In a gravity eld animals must bear ground force thatis greater than twice their weight Moreover they receivea large impulse at touchdown As many biomechanistshave indicated tendons play an important role in run-ning or jumping motion They store kinetic energy aspotential energy during stance and also absorb theimpulse at touchdown

To realize a running robot introducing such springycharacteristics into leg design is quite natural Pioneer-ing work was done by Raibert and co-workers Theydeveloped one-legged biped and quadruped runningrobots all of them had springy legs of the telescopictype and realized various running motions [1 ](httpwwwaimiteduprojectsleglab) Ahmadi andBuehler developed an electrically driven one-leggedrobot which had spring at the hip joint [2 ] and realizedenergy-e cient running using a modi ed Raibertcontroller [3 ] (httpwwwcimmcgillcaarlweb)

Since Raibertrsquos controller is based on uncontrolleddynamics of the springy leg it does not require muchcontrol e ort and yields a quite natural running gait Toembed uncontrolled dynamics into the control algor-ithm Raibert made an assumption of decoupling ofrotary motion of the body from telescopic motion of theleg Therefore for applying his controller to one-leggedhopping the mechanical model itself needs to be care-fully designed to satisfy these assumptions the leg shouldbe su ciently light compared to the body and the CM(centre of mass) of the body should be positioned closelyto the hip joint

Animal legs in nature however are of an articulatedtype the CM of the body is set o the hip joint As their

I05502 copy IMechE 2003Proc Instn Mech Engrs Vol 217 Part I J Systems and Control Engineering

dynamics have a strong non-linear coupling analysis ofuncontrolled dynamics is very di cult and there are fewtheoretical studies [4 ] moreover to the best of theauthorsrsquo knowledge there have been no successfulexperiments in hopping for this type of robot except fortwo examples Monopod [5 ] and Uniroo [6 ] studied byRaibert and co-workers In order to realize robotic run-ning copying that of animals it is important to establisha general control method but it is also important toinvent several new mechanical models inspired bybiology studies such as the examples above and to dem-onstrate successful experimental results Success inexperiments should be a great basis for developing amore sophisticated controller

For this purpose a new mechanical model of a doghindlimb was proposed and a prototype of a one-leggedhopping robot was designed as the rst step towards thegoal of dynamic quadruped running [7 ] This robot hasan articulated leg composed of three links it uses twohydraulic actuators as muscles and linear springs as atendon However the controller for hopping cannot bederived easily in a systematic (analytic) way because therobot has such complex non-linear dynamics Thereforeas the rst attempt the controller was designed empiri-cally based on the analysis of characteristic dynamics ofthe robot To do this a precise simulator includingactuator dynamics was constructed

In this paper after describing the new leg model andhardware design of the robot details of the simulatorand the controller are presented Using the dynamics-based controller the robot has succeeded in producingplanar one-legged hopping

This paper is organized as follows Section 2 describesthe design and hardware of the one-legged hoppingrobot Kenken Section 3 describes a precise simulator inwhich actuator dynamics are modelled in detailSection 4 presents a dynamics-based controller for stableone-legged hopping Section 5 gives results of simulationand experiment Section 6 concludes the paper

2 DEVELOPING THE ONE-LEGGED HOPPINGROBOT KENKEN

21 Proposal of a new mechanical model of the hindlimb

Alexander modelled a muscle and tendon system of aleg as a serial connection of inelastic actuator and aspring and studied the role of a tendon during runningIn reference [8 ] he showed that a unique running speedexists where the length of muscle is maintained andmechanical energy is preserved In this case the mechan-ical work for running is not done by the muscle but bythe elastic tendon [9 ] The Achilles tendon in particularis known to have large elasticity and the ability to storeup to 35 per cent of mechanical energy in human orkangaroo running [10]

85DYNAMICS-BASED CONTROL OF A ONE-LEGGED HOPPING ROBOT

Assigning mechanical work only to the actuators isnot an e ective strategy for design of a robot capable ofrunning which needs more energy than walking Insteadit is quite e ective to use an energy storing mechanismlike a spring If doing so the simplest and most directimplementation will be the linear massndashspring modeladopted in Raibertrsquos robots [1 ] For studying the mech-anism of animal running this model is very tractablealso many biomechanics researchers have dealt with thismodel For example McMahon and co-workers use thismodel to discuss the relationship between running speedand the spring constant in real animals [11 12]

However there are some reasons why real animalsadopt an articulated leg and not such a telescopic oneThe practical advantages of an articulated-type leg canbe summarized below

1 Large reachability range more clearance betweenthe foot and ground

2 Passive leg retraction during ight (provided if thelink parameters are appropriately chosen)

3 Simple structure it connects two ends of links withthe rotary joint easy construction

In particular advantage 2 is helpful for energy-e cientrunning and has already been seen in the passive walkingrobot [13]

There are several design solutions needed to make anarticulated leg springy Arranging the mechanism ofserial connection of an inelastic actuator and a springto each joint is one possibility Using this design Prattand Pratt realized natural walking of a biped robot [14]However linkage of multiple passive joints with mean-ingful periodic running motion is such a di cult problemthat there are no successful examples so far

Instead of considering multiple passive joints only theankle joint was considered here because the largest ten-dons are at this joint Figure 1 shows extensors of anankle joint which play the main role in running Amongthem this study speci cally examined arrangement ofthe largest muscle groups named gastrocnemius andplantaris Then the new leg model shown in Fig 2 wasproposed It is a planar one-legged robot that has anarticulated leg composed of three links It has two activejoints the hip and knee and one passive joint the ankle

Fig 1 Extensor muscle groups of the ankle joint


Fig 2 One-legged robot with a new leg mechanism

there is no actuator at its foot (this means that the toecan rotate on the ground during the stance phase actingas a free pivot)

The most distinctive feature of this model is thearrangement of the leg spring The leg spring is attachedbetween the thigh and heel parallel to the shankArrangement of the leg spring in this way creates thefollowing two important e ects during hopping

1 During stance holding the knee joint enables the legspring to absorb a large impulse at touchdown andto transfer its kinetic energy to potential energy forthe next stride Extending the knee yields an extradisplacement of the spring hence it adds potentialenergy to the spring (Fig 3)

2 During ight the spring constitutes a member of thelsquoparallel four-bar linkagersquo because compressive forceto the spring is not so large at that durationConsequently it enables passive retraction and exten-sion of the leg provided the inertia of the links ischosen appropriately (Fig 4)

Prilutsky et al show by experiments with cats thatthere is a transfer of energy between the ankle and kneejoints via the gastrocnemius and plantaris they concludethat this facilitates an energy-e cient running motion[15] This model elucidates how the ankle tendon oper-ates on the run in combination with proximal jointmotions On the other hand the spring arrangement inUniroo [6 ] is the same as the soleus in Fig 1

Fig 3 Active energy pumping


Fig 4 Passive leg retraction

Figure 5 illustrates one stride of the hopping motiontogether with each instant of phase transition (touch-down bottom lift-o apex)

22 Hardware overview

Figure 6 shows Kenken the prototype one-legged hop-ping robot that realized the leg model proposed in theprevious section The main speci cations are given inTable 1

A medium-sized dog of about 05 m in hip height wasselected as a target model A length of all links (including

Fig 5 Illustration of one hopping stride

Fig 6 One-legged hopping robotmdashKenken


a foot) was chosen to be the same 018 m for tractabilityof the kinematics This corresponds to the range of theleg length from 031 to 054 m for ideal parallel four-barlinkage The mass of the leg is relatively large 36 kgIndividual masses of thigh shank and foot are 242 075and 043 kg respectively The CM o set from the hipjoint is about 010 m For the leg spring two tensilecoil springs are installed in parallel between the thighand heel The initial value of the spring constant wasdetermined by simple energy analysis

Since running requires relatively high energy andalways accompanies shock against the ground jointactuation is the most critical problem for hardware


Table 1 Main speci cations of Kenken

Parameter Unit Value

Total mass kg 1326Body mass (including boom) kg 966 (05)Leg mass kg 360Body length m 085Thigh shank foot length m 018Toe length m 005Leg length (maximum) m 052Leg length (minimum) m 031Maximum stride m 052Body inertia around hip kg m2 046Leg inertia around hip (maximum) kg m2 013Leg inertia around hip (minimum) kg m2 007Leg spring coe cient (each) Nm 10 000Length of moment arm m 006

Rated actuator force at 14 MPa N 2200Rated actuator speed at 14 MPa ms 221

realization Therefore the authors introduced powerfulhydraulics and developed a small lightweight servo-actuator directly mounted with an industry servovalve(Fig 7) In this sense the authors gave priority to sti -ness over autonomy The actuator force must be zerofor the passive leg retraction during ight mentionedin section 21 However this is di cult for a commer-cial ow-control servovalve it was abandoned at theprototype stage

The experimental set-up is similar to that of RaibertA tether boom constrains the robot to the sagittal planeand measures the horizontal position vertical positionand pitch angle of the body via three optical encodersIt also carries hydraulic hoses a signal line and a dcline An aluminium box attached to the rear includes aninterface circuit servo-ampli er and signal conditioningcircuit which were hand-made taking the impulses andvibrations into account The control program is writtenin MATLABSIMULINK code and runs in a singletimer task with a 05 ms sampling period

Fig 7 Hydraulic servo-actuator of Kenken


3 BUILDING A PRECISE SIMULATOR

It is di cult to derive the controller in a systematic waybecause the robot has the quite complex non-lineardynamics described below

1 Under-actuated sytem The robot has an articulatedleg with a passive joint It has no actuator at its sole

2 Hybrid system The dynamics of the robot changedrastically while the robot transits through to a stancephase ight phase touchdown etc

3 Periodic system Linearization around equilibriumpoints and stabilization has no meaning instead theirorbital stability is needed

Therefore as the rst attempt the controller wasdesigned empirically based on analysis of the character-istic dynamics of the robot A precise simulator wasconstructed for this purpose

MATLAB-based graphical programming was intro-duced for exibility and rapid prototyping of the con-troller Speci cally the mechanical model of the robotand contact model with the ground was constructedusing DADS also the hydraulic actuator was modelledas a general electrohydraulic system using SIMULINKFigure 8 shows a block diagram of the simulator Thedashed line indicates the simulation model describedbelow If the simulation model is replaced by an actualmechanical model through the sensor inputoutputinterface the controllers developed and estimated on thesimulator can be applied immediately to the actualrobot

31 Modelling of a link system

The link model of the robot was constructed withDADS which enables graphical modelling of themechanical system while simplifying dynamic motionsimulation Its interior mechanism is not a lsquoblack boxrsquobut is based on Haugrsquos book [16 ] and is easy to follow

Mechanical parts are modelled as simple rigid linkshaving actual robot parameters Although Kenken is aplanar hopping robot it moves in constrained three-dimensional space Therefore a three-dimensionalmodel was used a tether boom was also modelled as inthe experimental set-up

A contact model is realized as a point-segment contactelement of DADS which was set between the tip of thesole and ground where Youngrsquos modulus the frictioncoe cient and the restitution coe cient are set appropri-ately For contact modelling details see the paper ofHaug et al [17 ]

32 Modelling of a hydraulic servosystem

The actuating system of Kenken is a typical electro-hydraulic servosystem shown in Fig 9 As mentioned


Fig 8 Block diagram of a simulator

Fig 9 Hydraulic system

before the controller is derived based on the character-istic dynamics The most important dynamics arelsquouncontrolledrsquo (or lsquopassiversquo) dynamics which means thatthe dynamics appear when applied torques are set tozero This is however a problem for the present robotbecause a ow-controlled hydraulic servo-actuator hasless force controllability (the active joints are very sti )(The authors refer readers to reference [18] for the forcecontrol using a ow-controlled hydraulic servovalveReference [19] should also be referred to where an inter-esting force controllable actuator composed of a springand hydraulic actuator in series is described) Therefore


the control input is de ned as the input currents to theservo-ampli er instead of the actuator force If doingso actuator dynamics exert a large in uence on the totaldynamics of the robot In this way a su ciently precisemodel of the actuator was built

321 Servovalve dynamics

Dynamic characteristics of the servovalve are given asthe transfer function for input current i to output owQ

Q (s)

i(s)=

Kstatic

v2ns2+2fvns+v2n

(1)

where vn and f are the natural frequency and damping

coe cients of the servovalve respectively and Kstatic isthe static gain On the other hand for a given pressuresupply Ps and load pressure PL the static load ow Qstaticis written as

Qstatic=Kiatilde Ps shy sign(i )PL (2)

where Ki

is the current gainThese equations can be combined in the block diagram

of Fig 10 This system calculates the (dynamic) load ow QL for a given input current i and load pressure PL Note that the dynamic part of the servovalve [equation(1)] is implemented as a current lter All the servovalveparameters can be obtained from the catalogue [20]

322 Continuity equation of hydraulic ow

The continuity equation for the cylinder is written as

Win shy Wout=grdV

dt+gV

r

be

dP

dt(3)

where W represents the weight owrates into and out ofa valve g is a gravity constant r is the mass densityV is the volume of the chamber and b

e is the e ective


Fig 10 Servovalve dynamics

bulk modulus [21] In general r is a function of tem-perature and pressure however if temperature is sup-posed to be constant during experimentation r isexpressed as a linear function of pressure

r=r0+

r0b

eP (4)

where r0 is the initial value Considering that W=grQyields

Qin shy Qout=dV

dt+

V

beV

dP

dt(5)

where the quadratic term of r is neglectedFor the cylinder of Fig 11 the load ow Q

L becomes

QL=

1

2(Q1

+Q2)

=1

2 CCip (P1 shy P2)+CepP1+dV1dt

+V1be

V1dP1dt D

+1

2 CCip(P1

shy P2) shy C

epP

2shy

dV2dt

shyV2b

eV

2dP2dt D

(6)

where Cip and C

ep are the internal leakage coe cientand external leakage coe cient Using piston displace-ment x the volumes of each chamber are written as

V1=V10+A1x (7)

V2=V

20shy A

2x (8)

where V10 and V20 are initial volumes On the other hand

Fig 11 Single-ended cylinder x is the piston displacementA

1 A2 are piston areas V

1 V2 are volumes P

1 P2

are pressures and Q1 Q

2 are owrates of eachchamber


using the load pressure PL the pressures of eachchamber can be written as

P1=Ps+PL

2(9)

P2=

Ps shy PL2

(10)

Substituting equations (7) to (10) into equation (6) therelationship between the load ow and load pressurebecomes

QL=1

2(A1+A2)xb +CtpPL

+1

4be[V10+V20+(A1 shy A2)x]Pb L (11)

where Ctp

=Cip

+Cep

2 are the total leakage coe cientsThis equation represents hydraulic compressibility andis expressed by the block diagram in Fig 12

323 Equation of motion for cylinder dynamics

Let mc be the equivalent mass of the cylinder piston rodand hydraulic oil The equation of motion for thecylinder becomes

mc x+Ff+FL=A1P1 shy A2P2 (12)

or using equations (9) and (10)

mc x+Ff+FL=Ps2

(A1 shy A2)+PL2

(A1+A2) (13)

where Ff is the friction of the cylinder and F

L is the load

324 Total system

The above subdynamics are combined to obtain Fig 13A given input current and cylinder rod position and vel-ocity gives the cylinder force Parameters of the servo-valve are borrowed from reference [20] Some constantsare obtained from identi cation tests in section 33


Fig 12 Hydraulic compressibility

Fig 13 Block diagram of the servo-actuator (hip actuator only)

33 Actuator identi cation

Due to the piston seal and other elements the hydraulicactuator has a relatively large friction It is not realisticto ignore the friction term The simple method used toidentify friction in reference [18] is used Rearrangingequation (12) gives

Ff=A

1P

1shy A

2P

2shy F

Lshy mx (14)

where ˆ denotes an ideal value If pressure accelerationand load are known exactly friction can be identi edOther parameters such as be and Ctp were identi edusing the least-squares method

Figure 14 shows the results of simulation and anexperiment of a driving test (tracking to a sinusoidalcurve) after identi cation This gure shows that the


actuator dynamics is su ciently and precisely modelledon the simulator

4 DYNAMICS-BASED CONTROLLER FORSTABLE HOPPING

In this section the controller for stable hopping isderived based on the basic characteristic dynamics of therobot The controller is in the form of a nite statemachine (FSM) ie one hopping stride is divided intoseveral discrete states Then the transition (switching)rule and the control law for each state are derived Thecontinuous states of the system transit according to thisFSM If the controllers and parameters are well chosenperiodic hopping gaits are expected to appear

The coordinates of the robot are de ned in Fig 15


Fig 14 Simulation results (solid line) and experimental results (dashed line) of a load driving test of theservo-actuator x

1 is piston displacement x1D is piston velocity f

1 means applied force and i1 is

input current

Fig 15 Coordinates ( denotes actuated joints J denotesthe passive joint)

The controlled variables are (xb zb w ) the forward speedvertical speed and attitude of the body in the sagittalplane The virtual leg length r and angle h which areused in the control at ight are also de ned in the gureControl inputs i1 and i2 the input currents to the hipactuator and the knee actuator respectively

Below characteristic dynamics of the simulated robotare explored in section 41 Then the control law for thestance phase and ight phase are developed in sections42 and 43 respectively They are combined with theFSM in section 44

41 Characteristic dynamics of the model

Using the precise simulator developed in section 3characteristic dynamics are explored

Firstly passive dynamics of free fall are investigatedThe robot is set to the nominal con guration in whichthe foot is below the CM and both control inputs are


set to zero (i1=0 i2=0) Then the following behaviourwas observed (Fig 16)

(a) At the instant of touchdown a large impulsiveground force makes the body pitch forwardsuddenly

(b) From touchdown to the bottom the body pitchesforward (wb lt0) because of the negative reactionmoment

(c) After maximum extension of the leg spring the bodypitches backward (wb gt0)

(d) When the vertical reaction force becomes zero therobot lifts o with positive angular momentum

Fig 16 Passive motion


Whether it lifts o forward or backward depends on thefoot placement at touchdown This fact is utilized in theforward speed control in section 43

Next actuated dynamics were investigated becausethese are the basis for control actions which are selectedaccording to the system state Here the robot is set tothe nominal con guration with its foot contactingthe ground Then the constant inputs are fed to oneactuator (Fig 17)

(e) If extending the hip actuator with the knee xed(i1gt0 and i2=0) the leg spring is extended andbackward body pitching (wb gt0) occurs

(f ) If extending the knee actuator with the hip xed(i2gt0 and i1=0) the leg sring is extended andforward body pitching (wb lt0) occurs

(g) If shortening the hip actuator with the knee xed(i1lt0 and i2=0) the spring buckles (of course thisis not allowed)

(h) If shortening the knee actuator with the hip xed(i2lt0 and i1=0) undesirable oscillation andchattering occurs

Although there are some quantitative di erences fromthe initial con guration and the amount of the inputsthe qualitative behaviour is identical

42 Control during the stance phase

Since the robot has only two inputs three coordinatescannot be controlled independently Based on obser-vations of the characteristic dynamics above the attitudeand the vertical speed of the body are controlled as

Fig 17 Actuated motion


follows during the stance phase

i1=G shy Kp(w shy w d) if w aring w d

0 else(15)

i2=Gic if qb3 0 zb aring zbd rltrmax0 else

(16)

where Kp is a position gain w

d is a desired attitude zbd isa desired vertical speed i

c is a constant current and rmax

is the maximal virtual leg lengthEquations (15) and (16) mean that the hip actuator

controls body pitch by a feedback law which is executedlsquoonly whenrsquo the pitch angle is lower than a speci edvalue while the knee actuator controls vertical speedand also suppresses the backward body pitch by givinga constant input which is exerted lsquoonly whenrsquo maximumspring extension occurs

43 Control during the ight phase

During ight the robot swings the leg to prepare for thenext touchdown It also retracts and extends the leg toreduce inertia for fast leg swinging and prevent stubbingagainst the ground Note that the touchdown angle (footplacement) of the leg is critical to gait stability for robotsthat cannot change their leg lengths arbitrarily and haveno actuator at their foot [22]

Although Kenken has no actuator at its foot it has aknee actuator that can be used to control its leg lengthHowever the knee actuator has already been used forvertical speed control Hence the touchdown leg anglewas used for forward speed control following Raibertrsquosalgorithm [1 ]

In other words the desired touchdown angle hf of the

virtual leg is chosen as

hf =h+Kf (xb shy xb d)+hb (17)

where

h=arcsin A1

r0

xbTs

2 B (18)

Here xbd is a desired forward speed and Ts is the stancetime which decreases as the forward speed increases[12] The constant r0 is the nominal length of the virtualleg and hb is a nominal bias term introduced empiricallyto reduce coupling e ects from the o set of the CMControl parameters are the feedback gain K

f and hb

Having determined the touchdown angle hf the hipactuator swings the leg so that it tracks a smooth refer-ence trajectory h (t) which reaches hf (Fig 18 upper)Also for leg retraction a smooth reference trajectoryr(t) which reaches the nominal length r

0 (Fig 18


Fig 18 Reference trajectories of h and r during ight

below) was given Speci cally the robot tracks the path

h (t)=h

lo+h

f2

+h

loshy h

f2

cos(2ethfst) (19)

r(t)=rlo

+rret

2+

rlo

shy rret

2cos(2ethfrt)

shy (rloshy r

0) (1 shy frt) (20)

where t is the time after lift-o rlo is the leg length atlift-o rret is the leg length of retraction h lo is the lift-o angle and fr and fs determine the speed of conver-gence They are chosen large enough to complete swingand retraction

Then the simple local feedback law can be applied

i1=shy K1(q1 shy q1) (21)


In these equations q1=q

1(r(t) h (t)) q2=q

2(r(t) h (t))are the desired joint trajectories calculated via inversekinematics of the parallel four-bar linkage mechanismand K1 K2 are the position gains

44 Implementation of the controller

Figure 19 shows the FSM which combines the control-lers described in sections 42 and 43 Each discrete stateand the corresponding control law as well as events andcorresponding switching conditions are summarized inTable 2 where sw represents the ono state of the footswitch A situation that does not obey these transitionrules implies falling down

5 SIMULATIONS AND EXPERIMENTS

Hopping simulations and experiments were carried outusing the controller derived in the previous sectionTable 3 shows the initial conditions and control param-


Fig 19 Finite state machine

Table 2 Details of events and control actions

State Control action

HOLD i1=0 i2=0PITCH i1=shy Kp(w shy wd) i2=0EXTEND i1=shy Kp(w shy wd) or 0 i2=icPOSE i1=0 i2=0SWING i

j=shy K

j(q

jshy q

j) ( j=1 2)

Event Condition when event occurs

td_pos sw=1 w shy w dgt0td_neg sw=1 w shy w d aring 0phi_neg w shy wd aring 0bottom qb3 0ext_max r=rmax or zb shy zbdgt0lift o sw=0

Table 3 Initial conditions and control parameters duringexperiments only the desired forward speed xb d isvaried

Unit Value Unit Value

w 0 deg shy 10 Kp mArad 7h0 deg 10 ic mA 05r0 m 047 Kf sm 01z0 m 055 hb deg 10hd deg shy 5 K1 mArad 100zbd ms 2 K2 mArad 100

eters used in both the simulation and experiments Theseparameters were partially tuned on the simulator andthen applied to the experiments Since optimization orlearning control had not yet been applied the controlparameters were tuned by hand

Firstly the simulation results are shown The robotis initially set to be 01 m away from the ground it isthen dropped Figure 20 shows snapshots of animationduring hopping when the desired forward speed xbd wasset to 1 ms Two arrows running from the sole representthe ground reaction forces


Fig 20 Snapshots of DADS animation



Figure 21 depicts time evolutions of each state Notethat xb (denoted by XD in the gures) represents thevelocity of the hip not of the CM Foot SW representsthe state of the foot switch The dashed line in the graphof r and h represents a lsquodummyrsquo trajectory ie a situ-ation where the angles of the ankle and knee are thesame (q3

=q2)

Next experimental results with the same conditionsare shown in Fig 22 Compared with Fig 21 time evol-utions of each state are similar to those of the simulationAfter four steps the forward speed xb converges to thedesired value of 1 ms The hopping period is about 05 sThe pitch angle w is oscillating within 0deg to shy 15deg whichmeans that the robot is stable The two control inputsi1 and i

2 also shift similarly which means that the vari-ation of hydraulic pressure in the simulation and theexperiment are almost identical

However some di erences were found One is thesmall di erence in transition which is not so importantfor control tasks This comes from an unavoidable gapof initial conditions in the experiment Another di er-ence is the hopping height In experiments the hoppingheight converged to a larger value than in the simulationIt is supposed that the di erence arose from the controllaw [equation (16)] which is described in the form offeedforward Therefore it should be more able to controlvertical speed in the form of global feedback as inequation (17)

Fig 21 Simulation results when xb d=10


Fig 22 Experimental results when xb d=10

Experiments were also carried out for various desiredforward speeds For example Figs 23 and 24 showresults for low-speed (xbd

=05 ms) and high-speed hop-ping (xb d=15 ms) It is found that the robot can hopstably at speeds below 1 ms including vertical hoppingin both simulation and experiment In this case pitchangles oscillate around relatively higher valuesHowever at speeds over 15 ms stability deterioratesSpeci cally the forward and vertical speed and bodypitch angle oscillate drastically It is still stable buttracking to a speci c speed becomes di cult Moreoveronce forward speed exceeds 2 ms the leg angle easilyreaches the limit of the movable range and it becomesimpossible to retain gait stability

Instability at higher speeds is related to speed-dependent deviations of the pitch angle at touchdownThe faster the desired speed the larger the leg angle attouchdown because of equation (17) which results inmore nose-down body pitch angle at touchdown Thenthe large error of the pitch angle turns to the large hiptorque and propels the robot strongly during stanceThis process causes the system to diverge

There are two ways to avoid such a deviation of thebody pitch angle at touchdown The simplest way is toreplace the leg by a less massive one This reduces theamplitude of body pitching relative to leg swingingduring the ight phase The other way is to actively con-trol body pitching by installing a tail mechanism into


Fig 23 Experimental results when xbd=05

the body as Uniroo [6 ] does or by introducing somenon-holonomic attitude controls [23]

6 CONCLUSIONS

Inspired by biomechanical studies on animal running anew mechanical model for the hindlimb of a dog wasproposed a one-legged hopping robot Kenken wasdeveloped To derive the controller basic characteristicsof the model were examined using a precise dynamicsimulator Although there is room for improvementespecially for the case of high speed the robot succeededin planar hopping both in simulation and in experimentsIt was found that the leg spring attached as a gastro-cnemius or plantaris enables the robot to producesu cient propulation force by virtue of an energy trans-fer from the knee even if there is no actuator at theangle joint Figure 25 shows snapshots of a hoppingmotion at high speed (2 ms) Although the robot fallsafter a few steps in this high-speed hopping it may bethe rst time that such a realistic and natural gait hasbeen achieved in a machine (not in a computer simu-lation) Consequently simulation and experimentalresults validate the leg model and the controller

Through this paper the authors want to emphasize

1 Investigating characteristic dynamics of the robot is



a direct and e ective way of determining thecontroller

2 Replacing lsquomassiversquo actuators with springs and utiliz-ing passive dynamics minimizes the control e ort

3 A biologically inspired mechanical design necessarilyyields animal-like dynamic behaviour by dictating thepassive dynamics of the model

However the authors do not assert that the controllercan always be found by a dynamics-based aproachbecause the model does not always exhibit meaningfuldynamics for the desired control task In this case howto install passive elements appropriately such as a springor damper into the model arises as an important andinteresting problem Also the larger the size of thesystem the more it becomes di cult to pick up usefulcharacteristic dynamics because it is time consumingTherefore some adaptive or learning scheme should becombined with dynamics-based control This goal is leftfor future work

ACKNOWLEDGEMENT

This work is a part of the Grant-in-Aid for the COEResearch Project supported by the Japanese Ministry ofEducation Science Sports and Culture


Fig 25 Snapshots of Kenken hopping for one stride



REFERENCES

1 Raibert M H Legged Robots That Balance 1985 (MITPress Boston Massachusetts)

2 Ahmadi M and Buehler M The ARL Monopod II run-ning robot control and energetics In Proceedings of theIEEE International Conference on Robotics andAutomation 1999 pp 1689ndash1694

3 Ahmadi M and Buehler M Stable control of a simulatedone-legged running robot with hip and leg complianceIEEE Trans Robotics and Automn 1997 13(1) 96ndash104

4 Saranli U Schwind W J and Koditschek D E Towardthe control of a multi-jointed monopod runner InProceedings of the IEEE International Conference onRobotics and Automation 1998 pp 2676ndash2682

5 Lee W and Raibert M H Control of hoof rolling in anarticulated leg In Proceedings of the IEEE InternationalConference on Robotics and Automation 1991pp 1386ndash1391

6 Zeglin G J Uniroo a one-legged dynamic hopping robotBS thesis Department of Mechanical Engineering MITBoston Massachusetts 1991

7 Hyon S H and Mita T Development of a biologicallyinspired hopping robotmdashKenken In Proceedings of theIEEE International Conference on Robotics andAutomation 2002 pp 3984ndash3991

8 Alexander R M Elastic Mechanisms in Animal Movement1988 (Cambridge University Press)

9 Pennisi E A new view of how leg muscles operate on therun Science 1997 275(21) 1067

10 Alexander R M The maximum forces exerted by animalsJ Expl Biol 1985 115 231ndash238

11 McMahon T A and Cheng G C The mechanics of run-ning how does sti ness couple with speed J Biomechanics1990 23(Suppl 1) 65ndash78

12 Farley C T Glasheen J and McMahon T A Runningsprings speed and animal size J Expl Biol 1993 18571ndash86


13 McGeer T Passive walking with knees In Proceedings ofthe IEEE International Conference on Robotics andAutomation 1990 pp 1640ndash1645

14 Pratt J and Pratt G Intuitive control of a planar bipedalwalking robot In Proceedings of the IEEE InternationalConference on Robotics and Automation 1998pp 2014ndash2021

15 Prilutsky B I Herzog W and Leonard T Transfer ofmechanical energy between ankle and knee joints by gastro-cnemius and plantaris muscles during cat locomotionJ Biomechanics 1996 29(4) 391ndash403

16 Haug E J Computer-Aided Kinematics and Dynamics ofMechanical Systems 1989 (Allyn and Bacon BostonMassachusetts)

17 Haug E J et al Dynamics of mechanical systems withCoulomb friction stiction impact and constraint additionndashdeletion IndashIII Mech Mach Theory 1986 21(5) 401ndash425

18 Alleyne A and Liu R Nonlinear forcepressure trackingof an electro hydraulic actuator Trans ASME J DynamicSystems Measmt Control 2000 122(1) 232ndash237

19 Robinson D W and Pratt G Force controllable hydro-elastic actuator In Proceedings of the IEEE InternationalConference on Robotics and Automation 2000pp 1320ndash1327

20 Moog Catalogue Type 30 Flow Control ServovalvesAerospace Group Moog Inc ( latest informationhttpwwwservovalvecom)

21 Merrit H E Hydraulic Control Systems 1967 (John WileyNew York)

22 Miura H and Shimoyama I Dynamic walk of a bipedInt J Robotics Res 1984 3 60ndash74

23 Mita T Hyon S H and Nam T K Analytical timeoptimal control solution for a two link planar acrobatwith initial angular momentum IEEE Trans Robotics andAutomn 2001 17(3) 361ndash366


x2a desired forward speed (ms)z vertical hip position (m)z2a desired vertical lift-o speed

(ms)

be e ective bulk modulus (Pa)h damping coe cient of servovalveh

b nominal touchdown angle ofvirtual leg (deg)

hf desired touchdown angle ofvirtual leg (deg)

h desired time trajectory of virtualleg angle at ight phase (deg)

r r0 mass density of oil

f damping coe cient of servovalvew pitch angle of body (deg)wd desired body pitch angle (deg)vn natural frequency of servovalve

(rads)

1 INTRODUCTION

In a gravity eld animals must bear ground force thatis greater than twice their weight Moreover they receivea large impulse at touchdown As many biomechanistshave indicated tendons play an important role in run-ning or jumping motion They store kinetic energy aspotential energy during stance and also absorb theimpulse at touchdown

To realize a running robot introducing such springycharacteristics into leg design is quite natural Pioneer-ing work was done by Raibert and co-workers Theydeveloped one-legged biped and quadruped runningrobots all of them had springy legs of the telescopictype and realized various running motions [1 ](httpwwwaimiteduprojectsleglab) Ahmadi andBuehler developed an electrically driven one-leggedrobot which had spring at the hip joint [2 ] and realizedenergy-e cient running using a modi ed Raibertcontroller [3 ] (httpwwwcimmcgillcaarlweb)

Since Raibertrsquos controller is based on uncontrolleddynamics of the springy leg it does not require muchcontrol e ort and yields a quite natural running gait Toembed uncontrolled dynamics into the control algor-ithm Raibert made an assumption of decoupling ofrotary motion of the body from telescopic motion of theleg Therefore for applying his controller to one-leggedhopping the mechanical model itself needs to be care-fully designed to satisfy these assumptions the leg shouldbe su ciently light compared to the body and the CM(centre of mass) of the body should be positioned closelyto the hip joint

Animal legs in nature however are of an articulatedtype the CM of the body is set o the hip joint As their


dynamics have a strong non-linear coupling analysis ofuncontrolled dynamics is very di cult and there are fewtheoretical studies [4 ] moreover to the best of theauthorsrsquo knowledge there have been no successfulexperiments in hopping for this type of robot except fortwo examples Monopod [5 ] and Uniroo [6 ] studied byRaibert and co-workers In order to realize robotic run-ning copying that of animals it is important to establisha general control method but it is also important toinvent several new mechanical models inspired bybiology studies such as the examples above and to dem-onstrate successful experimental results Success inexperiments should be a great basis for developing amore sophisticated controller

For this purpose a new mechanical model of a doghindlimb was proposed and a prototype of a one-leggedhopping robot was designed as the rst step towards thegoal of dynamic quadruped running [7 ] This robot hasan articulated leg composed of three links it uses twohydraulic actuators as muscles and linear springs as atendon However the controller for hopping cannot bederived easily in a systematic (analytic) way because therobot has such complex non-linear dynamics Thereforeas the rst attempt the controller was designed empiri-cally based on the analysis of characteristic dynamics ofthe robot To do this a precise simulator includingactuator dynamics was constructed

In this paper after describing the new leg model andhardware design of the robot details of the simulatorand the controller are presented Using the dynamics-based controller the robot has succeeded in producingplanar one-legged hopping

This paper is organized as follows Section 2 describesthe design and hardware of the one-legged hoppingrobot Kenken Section 3 describes a precise simulator inwhich actuator dynamics are modelled in detailSection 4 presents a dynamics-based controller for stableone-legged hopping Section 5 gives results of simulationand experiment Section 6 concludes the paper

2 DEVELOPING THE ONE-LEGGED HOPPINGROBOT KENKEN

21 Proposal of a new mechanical model of the hindlimb

Alexander modelled a muscle and tendon system of aleg as a serial connection of inelastic actuator and aspring and studied the role of a tendon during runningIn reference [8 ] he showed that a unique running speedexists where the length of muscle is maintained andmechanical energy is preserved In this case the mechan-ical work for running is not done by the muscle but bythe elastic tendon [9 ] The Achilles tendon in particularis known to have large elasticity and the ability to storeup to 35 per cent of mechanical energy in human orkangaroo running [10]




























































Q (s)

i(s)=

Kstatic

v2ns2+2fvns+v2n

(1)




where Ki





Win shy Wout=grdV

dt+gV

r

be

dP

dt(3)


e is the e ective




r=r0+

r0b

eP (4)


Qin shy Qout=dV

dt+

V

beV

dP

dt(5)


L becomes

QL=

1

2(Q1

+Q2)

=1


+V1be

V1dP1dt D

+1

2 CCip(P1

shy P2) shy C

epP

2shy

dV2dt

shyV2b

eV

2dP2dt D

(6)

where Cip and C


V1=V10+A1x (7)

V2=V

20shy A

2x (8)




1 V2 are volumes P

1 P2





P1=Ps+PL

2(9)

P2=

Ps shy PL2

(10)


QL=1

2(A1+A2)xb +CtpPL

+1

4be[V10+V20+(A1 shy A2)x]Pb L (11)

where Ctp

=Cip

+Cep






mc x+Ff+FL=Ps2

(A1 shy A2)+PL2

(A1+A2) (13)


L is the load

324 Total system







Ff=A

1P

1shy A

2P

2shy F

Lshy mx (14)












input current




























0 else(15)


(16)












where

h=arcsin A1

r0

xbTs

2 B (18)


f and hb


0 (Fig 18




h (t)=h

lo+h

f2

+h

loshy h

f2

cos(2ethfst) (19)

r(t)=rlo

+rret

2+

rlo

shy rret

2cos(2ethfrt)

shy (rloshy r

0) (1 shy frt) (20)






1(r(t) h (t)) q2=q











j=shy K

j(q

jshy q

j) ( j=1 2)













=q2)














6 CONCLUSIONS










ACKNOWLEDGEMENT






REFERENCES




















































































Q (s)

i(s)=

Kstatic

v2ns2+2fvns+v2n

(1)




where Ki





Win shy Wout=grdV

dt+gV

r

be

dP

dt(3)


e is the e ective




r=r0+

r0b

eP (4)


Qin shy Qout=dV

dt+

V

beV

dP

dt(5)


L becomes

QL=

1

2(Q1

+Q2)

=1


+V1be

V1dP1dt D

+1

2 CCip(P1

shy P2) shy C

epP

2shy

dV2dt

shyV2b

eV

2dP2dt D

(6)

where Cip and C


V1=V10+A1x (7)

V2=V

20shy A

2x (8)




1 V2 are volumes P

1 P2





P1=Ps+PL

2(9)

P2=

Ps shy PL2

(10)


QL=1

2(A1+A2)xb +CtpPL

+1

4be[V10+V20+(A1 shy A2)x]Pb L (11)

where Ctp

=Cip

+Cep






mc x+Ff+FL=Ps2

(A1 shy A2)+PL2

(A1+A2) (13)


L is the load

324 Total system







Ff=A

1P

1shy A

2P

2shy F

Lshy mx (14)












input current




























0 else(15)


(16)












where

h=arcsin A1

r0

xbTs

2 B (18)


f and hb


0 (Fig 18




h (t)=h

lo+h

f2

+h

loshy h

f2

cos(2ethfst) (19)

r(t)=rlo

+rret

2+

rlo

shy rret

2cos(2ethfrt)

shy (rloshy r

0) (1 shy frt) (20)






1(r(t) h (t)) q2=q











j=shy K

j(q

jshy q

j) ( j=1 2)













=q2)














6 CONCLUSIONS










ACKNOWLEDGEMENT






REFERENCES




























r=r0+

r0b

eP (4)


Qin shy Qout=dV

dt+

V

beV

dP

dt(5)


L becomes

QL=

1

2(Q1

+Q2)

=1


+V1be

V1dP1dt D

+1

2 CCip(P1

shy P2) shy C

epP

2shy

dV2dt

shyV2b

eV

2dP2dt D

(6)

where Cip and C


V1=V10+A1x (7)

V2=V

20shy A

2x (8)




1 V2 are volumes P

1 P2





P1=Ps+PL

2(9)

P2=

Ps shy PL2

(10)


QL=1

2(A1+A2)xb +CtpPL

+1

4be[V10+V20+(A1 shy A2)x]Pb L (11)

where Ctp

=Cip

+Cep






mc x+Ff+FL=Ps2

(A1 shy A2)+PL2

(A1+A2) (13)


L is the load

324 Total system







Ff=A

1P

1shy A

2P

2shy F

Lshy mx (14)












input current




























0 else(15)


(16)












where

h=arcsin A1

r0

xbTs

2 B (18)


f and hb


0 (Fig 18




h (t)=h

lo+h

f2

+h

loshy h

f2

cos(2ethfst) (19)

r(t)=rlo

+rret

2+

rlo

shy rret

2cos(2ethfrt)

shy (rloshy r

0) (1 shy frt) (20)






1(r(t) h (t)) q2=q











j=shy K

j(q

jshy q

j) ( j=1 2)













=q2)














6 CONCLUSIONS










ACKNOWLEDGEMENT






REFERENCES






























Ff=A

1P

1shy A

2P

2shy F

Lshy mx (14)












input current




























0 else(15)


(16)












where

h=arcsin A1

r0

xbTs

2 B (18)


f and hb


0 (Fig 18




h (t)=h

lo+h

f2

+h

loshy h

f2

cos(2ethfst) (19)

r(t)=rlo

+rret

2+

rlo

shy rret

2cos(2ethfrt)

shy (rloshy r

0) (1 shy frt) (20)






1(r(t) h (t)) q2=q











j=shy K

j(q

jshy q

j) ( j=1 2)













=q2)














6 CONCLUSIONS










ACKNOWLEDGEMENT






REFERENCES





























input current




























0 else(15)


(16)












where

h=arcsin A1

r0

xbTs

2 B (18)


f and hb


0 (Fig 18




h (t)=h

lo+h

f2

+h

loshy h

f2

cos(2ethfst) (19)

r(t)=rlo

+rret

2+

rlo

shy rret

2cos(2ethfrt)

shy (rloshy r

0) (1 shy frt) (20)






1(r(t) h (t)) q2=q











j=shy K

j(q

jshy q

j) ( j=1 2)













=q2)














6 CONCLUSIONS










ACKNOWLEDGEMENT






REFERENCES







































0 else(15)


(16)












where

h=arcsin A1

r0

xbTs

2 B (18)


f and hb


0 (Fig 18




h (t)=h

lo+h

f2

+h

loshy h

f2

cos(2ethfst) (19)

r(t)=rlo

+rret

2+

rlo

shy rret

2cos(2ethfrt)

shy (rloshy r

0) (1 shy frt) (20)






1(r(t) h (t)) q2=q











j=shy K

j(q

jshy q

j) ( j=1 2)













=q2)














6 CONCLUSIONS










ACKNOWLEDGEMENT






REFERENCES




























h (t)=h

lo+h

f2

+h

loshy h

f2

cos(2ethfst) (19)

r(t)=rlo

+rret

2+

rlo

shy rret

2cos(2ethfrt)

shy (rloshy r

0) (1 shy frt) (20)






1(r(t) h (t)) q2=q











j=shy K

j(q

jshy q

j) ( j=1 2)













=q2)














6 CONCLUSIONS










ACKNOWLEDGEMENT






REFERENCES






























=q2)














6 CONCLUSIONS










ACKNOWLEDGEMENT






REFERENCES




























6 CONCLUSIONS










ACKNOWLEDGEMENT






REFERENCES





























REFERENCES

























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