8 IEEE International Conerence on Rootics and Automation Pasadena, CA, USA, May 19-3, 8 Speed and Height Control or a Special Class o Running Quadruped Roots Nicholas Cherouvim and Evangelos Papadopoulos, Senior Memer, IEEE Astract In this work a novel control method is presented or controlling the orward speed and apex height o a special class o running quadruped root, with a dimensionless inertia o 1, and one actuator per leg Seeking to minimize the parasitic pitching motion in running, pronking is used as the target gait The control design is ased on the root dynamics, allowing its application to a wide range o roots o the class studied Moreover, the controller adjusts the root speed and height, requiring knowledge only o the root physical parameters The control ensures that negative actuator work during the stance phase is zero, therey reducing the power expenditure Small, o-the-shel DC motors are adequate or the control implementation, while results o application to a detailed root model show good perormance even when including leg mass, oot collision, motor limitations, oot slipping and other actors R I INTRODUCTION ECENTLY, the potential o using legs or eicient locomotion over varied terrain has inspired much research Quadrupeds excite particular interest, as they are stale when stationary, unlike most two legged roots Also, they retain simpler dynamics than roots with six legs There is much work on quadruped control, oth in design and applications, especially using two or more actuators per root leg Raiert controlled a quadruped s orward speed using two actuators per leg, [1] Using delayed eedack control a ounding gait was stailized, though the speed could not e directly set, [] Fuzzy control has een applied to set the speed o a gallop [3], while CPG s have een used on the Tekken quadruped, with three actuators per leg, [4] Although orward speed control has also een achieved using one actuator per leg, trial and error determination o controller parameters is required The Scout II quadruped achieved stale gaits with one actuator per leg, [5], [6], with control parameters ound y trial and error Also, speed setting is reported or a walking gait, [7] It is worth noting that most controllers are applicale to speciic roots Using two actuators per leg rather than one simpliies control design, y adding system inputs However, this also complicates root design, has a severe eect on weight and This work is co-unded y pulic unds (European Social Fund 75% and General Secretariat or Research and Technology 5%) and private unds (Zenon SA), within measure 83 o OpPrComp, 3rd CSP - PENED 3 N Cherouvim is with the Department o Mechanical Engineering, National Technical University o Athens, Athens, 1578, Greece (e-mail: ndcher@mailntuagr, phone: +(3) 1-77-643) E Papadopoulos is with the Department o Mechanical Engineering, National Technical University o Athens, Athens, 1578, Greece (e-mail: egpapado@centralntuagr, phone: +(3) 1-77-144, ax: +(3) 1-77-1455) cost, and increases power supply demands So an important open issue is how to control oth orward speed and attained apex height, using one actuator per leg and with no need or trial and error selection o control parameters In this work, this control issue is studied with the added requirement that the parasitic pitching motion, oten present in quadruped running, shall e minimized To this end, a particular design class o quadruped, descried in Section II, is studied Also, in Section III, pronking is selected as the desired gait, as it does not involve pitching The control is designed in Section IV utilizing analytical root dynamics Further, no actuator negative work is present during stance, which reduces energy expenditure Finally, in Section V, controller perormance is shown or application to a root model including all major real-world characteristics II QUADRUPED ROBOT A Quadruped root design and gait evolution The quadruped root studied has springy legs actuated at the hips, which is the only root actuation, see Fig 1a This is the type o quadruped that the Scout II root elongs to, [5], [6], which has run with a variety o gaits A special class o this root type is studied, or which the ody inertia I, ody mass m and hal the hip spacing, d give a dimensionless inertia o j= I/( md ) = 1 This selection is inspired y the decoupling o the vertical and pitching modes o motion, or j= 1 Murphy ound that when j< 1, the pitching motion in running is stale, while it is unstale or j> 1 When, j= 1 the vertical and pitching modes are decoupled [8] This inspires the selection o this root class, which acilitates a decoupled approach to pitching control and avoids triggering o pitching dynamics y other DOF A dimensionless inertia o 1 is simply achieved y proper hip placement or mass redistriution BACK STANCE Fig 1 Quadruped root and () gait phases FLIGHT DOUBLE STANCE () FRONT STANCE The gaits reerred to are the pronk, in which all the root legs are always in phase, and the ound, in which the ack pair o legs is in phase, as is the ront pair The ound may involve up to our motion phases, see Fig 1, while the pronk has only the doule stance and light phases Next, a simpliied planar model used or control design is shown 978-1-444-1647-9/8/$5 8 IEEE 85
For the control evaluation in Section V, a second detailed model, including all major real-world properties, is used B Simpliied model or control design The simpliied planar model is shown in Fig The ody has its center o mass (CoM) at its geometrical center, and is supported on two springy legs The legs each have total mass m l, inertia I l, and are actuated y torques τ, τ at the hips Each model leg has twice the mass, inertia and spring stiness o a root leg and includes viscous riction, o viscous coeicient Tale 1 shows quantities used The inluence o the leg mass on stance phase dynamics is negligile, and or the doule stance phase a Lagrangian approach is used or the model dynamics, using ody Cartesian coordinates, x, y, and pitch, θ, as generalized variales The dynamics in terms o x, y, θ has a lengthy orm, ut oservation shows that certain complex expressions o x, y, θ, actually represent the leg angles γ, γ, and leg lengths l, l, see Fig Sustituting these expressions, the dynamics assumes a more compact orm The dynamics is shown in (1) to (3), while l, l, γ, γ are given as expressions o x, y, θ in (4) to (7) mx + kl ( l)sin γ + kl ( l )sinγ l sinγ (1) l sinγ = τ cos γ / l τ cos γ / l my kl ( l) cos γ kl ( l )cosγ + l cosγ + l cosγ = mg τ sin γ / l τ sin γ / l I θ + d kcos( γ θ)( L l) d kcos( γ θ)( L l ) d cos( γ θ) l + d cos( γ θ) l = (3) ( dsin( γ θ) l ) τ / l ( dsin( γ θ) + l ) τ / l 1 γ = tan ( y dsin θ, xt + dcos θ x) (4) 1 γ = tan ( y + dsin θ, xt dcos θ x) (5) l = ( x+ xt + dcos θ ) + ( y dsin θ ) (6) l = ( x+ xt dcos θ ) + ( y+ dsin θ ) (7) where x t is the position o the ack oot, x t is the position o the ront oot during the doule stance phase The doule stance dynamics aove also yields the dynamics or the remaining stance phases y removing non pertinent terms In light, the system CoM perorms a allistic motion Also, the angular momentum o the system o the ody and two legs, with respect to the system CoM, is conserved: HO= D 1γ+ D γ + D 3 θ= const (8) where D 1, D, D 3 are given y: D1= ( Ilm + l1mm 1 ( m m1) l1ml mcos( γ γ ) dl1m mlsin( γ θ)) m (9) D= D1+ ( l1ml mcos( γ γ ) + dl1m mlsin( γ θ)) m D = I+ d m dl m sin( γ θ) + dl m sin( γ θ)) 3 l 1 l 1 l where l 1 is the leg CoM to hip distance () x γ τ d m, I m l, I l Fig Planar quadruped root model l y γ τ m l, I l TABLE 1 VARIABLES AND INDICES USED IN THE WORK x CoM horizontal position L leg rest length y CoM vertical position viscous riction coeicient θ ody pitch angle g acceleration o gravity l leg length m l leg mass γ leg asolute angle I l leg inertia γ sum sum o leg asolute angles γ ack leg angle γ di dierence o leg asolute angles γ ront leg angle k leg spring stiness τ hip torque m ody mass T st stance duration m total root mass as index: ront leg I ody inertia as index: ack leg d hip joint to CoM distance td as index: value at touchdown The root motion during the doule stance phase is governed y (1) to (3), and y similarly derived equations or the ack and ront stance phases During light, the CoM motion is allistic and the pitch motion is constrained y (8) The orm o the model dynamics still remains complex or analysis due to nonlinearities and its discrete phase nature To enale the analytical manipulation o the dynamics or the derivation o the control, certain hypotheses are made which also appear in the literature [1,1,13] These are laid out elow and reerred to in the text Modeling hypotheses: H1: The contriution o the dissipation orces to the stance phase orward dynamics, see (1), is negligile H: To predict the change in orward speed during a stance phase, and also to compute energy dissipation due to viscous riction in the legs, the ehavior o the springy legs in stance is approximated y the simple mass-spring model The mass is equal to the root mass, and the spring constant is equal to the sum o the root s leg spring constants: ml = k( L l) mg (1) H3: In the stance dynamics, the leg lengths l, l are considered equal to the leg rest length L, or terms which involve the input torques τ, τ Gait speciic hypotheses: H4: In the pronk the ack and ront legs are almost parallel, so the dierence etween the two angles is small H5: For reasonale speeds, the leg angles and ody pitch angle, are small enough or trigonometric small angle simpliications Using the aove hypotheses, the dynamics is urther manipulated First, or a gait o speed x, the leg angle evolution through stance is as predicted y Raiert [1]: γi = γi, td xt L (11) where i=,, γ itd, is the leg touchdown angle, and time t l θ 86
counts rom each leg touchdown Writing the leg lengths as unctions o y, θ, γ, γ, see Fig, and using H3, H5, (11), the doule stance dynamics in (1) to (3) ecomes: mx + k γ( L l) + k γ ( L l ) + τ / L+ τ / L= (1) my + y + k y= mg + k Lcos (( γsum, td / ) ( xt (13) / L) ) cos ( γdi, td / ) I θ + d θ + d k θ = τ τ (14) + k L d sin (( γsum, td /) ( xt / L) ) sin ( γdi, td /) where γ sum, td = γ, td + γ, td, and γ di, td = γ, td γ, td Finally, using H4, the angular momentum in (8) can e written as: H = ( I+ d m ) l θ+ ( Ilm + l1mm 1 ( m m1))( γ+ γ )/ m (15) The simpliied orm o (15) may e simply integrated, and can thereore predict ody pitch orientation To summarize, the root model is now descried y (1) to (14) during the doule stance phase During the light phase, the CoM motion is allistic and the pitch is constrained y (15) Below, the duration o the stance phase T st is considered known, and can e ound y approximately solving the vertical dynamics in (13), while neglecting riction and torque terms Section III ollows, to explain the advantages o pronking as the desired gait III PRONKING AS THE DESIRED GAIT The pronking gait is essentially ounding with no ody pitching, and has only the light and doule stance phases Though pronking may reduce in practice to ounding with very limited pitching, it does oer advantages in control design, as will e explained in this Section In a twodimensional analysis, the trotting and pronking gaits have essentially the same dynamics, while trotting is more common in nature However, pronking was preerred in this analysis it is tranerrale in principle to the 3D prolem On the contrary, studing the trotting gait in a two dimensional analysis does not provide a solution to the 3D prolem To egin with, the root leg touchdown angles γ td,, γ, td, are known to e important inputs in any gait and are easily set y leg positioning during light, so they are oten used in control [1,3,5,6,1] One issue that arises is identiying the discreet inluence o the ack and ront leg touchdown angles on the dynamics, as they appear in coupled orms in the general case For the special case o pronking it will e shown that the sum and dierence o the touchdown angles, γ sum, td, γ di, td, may e distinctly associated with the vertical and pitching motion modes First, i γ di, td is zero as in ideal pronking, then (13) shows the vertical motion to e dependant on the sum o the touchdown angles, γ sum, td Also, or motions close to pronking, it can e seen rom (14) that γ di, td has the stronger inluence on the pitching motion, as small deviations o its value rom zero determine the existence o the whole term it elongs to The distinct eect o the inputs γ sum, td, di, td γ in pronking is apparent, and this decoupling ehavior acilitates the control approach in Section IV, where the motion modes are studied individually IV CONTROL DESIGN The control inputs to the root system are the sum and dierence o the leg angles at touchdown, γ sum, td, γ di, td, and the actuator torque applied during the stance phase The inputs γ sum, td, γ di, td are not directly controlled in the physical root, ut are ed to the actuator controllers that servo the legs to the desired touchdown positions in light For control design purposes the actuator torques during stance are chosen to e constant or the duration o the phase, and to e equal at the ack and ront hips: τ = τ = τ = const( in stance) (16) Thereore, the three control inputs to the root system are the leg angle inputs, γ sum, td, γ di, td, and the constant torque, τ, applied in stance The root dynamics used or the control derivation is that presented in Section II, in (1) to (15), with generalized variales x, y, θ The aim in this Section is to compute the control inputs γ sum, td, γ di, td, τ, given the state o the root at lito rom the ground Once the inputs are computed, the hip actuators position the legs in light Then, at touchdown, the actuators apply a torque deined y the control input τ In the irst three parts o this Section, each o the orward, vertical and pitching modes o motion is studied From the study o each motion mode, a single control design equation involving the control inputs is derived The ourth part o the Section explains how the three control design equations lead to the computation o the three control inputs Details o the root eedack are given in susection E Here, it is relevant to say that leg lengths, leg angles, and the ody pitch are measured, and the rates o change are also known From this data, the root state at lito is known, using root geometry An example is to compute x at lito or the case where the ack leg lits o last, see Fig : x = l sinγ l γ cosγ d θ sinθ (17) lo, lo, lo, lo, lo, lo lo lo A Forward motion mode The orward motion is studied or the pronking gait with the desired orward speed x des A constraint on the control inputs must e derived that ensures that the root orward speed is the desired In Section II, the orward dynamics are given in (1) Using (16) and H4 in (1) provides: mx + k γ ( L l) + τ / L= (18) Due to H4, the sum o the touchdown angles γ sum, td in the pronking gait is equal to twice one o γ td,, γ, td Also, utilizing H6, and integrating (18) once, the orward speed x i + 1 ater stance can e computed, given the speed eore stance, x i, as a unction o the control inputs γ sum, td, τ : T st x = i+ 1 k ( γ sum, td xdest L)( L l ) dt m τtst L m + x i (19) In (19), the irst term on the right represents the eect o the leg spring orces on the speed, while the second term descries the eect o the hip torques The value o the control input γ sum, td is now considered as the sum o a 87
nominal value plus some ias, γ sum, td, ias : γsum, td = ( x destst / L) + γsum, td, ias () The ias γ sum, td, ias can e ound y using () in (19), and oserving the symmetry o the leg evolutions during stance, as predicted y (1) We then otain: γ = 4 τ T m L( l + gt ) (1) ( ) sum, td, ias st, td st where l td, is the leg compression rate just ater touchdown By integrating light dynamics, it is possile to express l td, as a unction o the control input γ sum, td Thereore: l td, = ( γ sumtd, ) () We have oserved, when applying the control, that sustituting the desired orward speed in () with the actual speed o the root, x, attracts the root to the desired speed A similar oservation is made in [1], which inspired us here Equation () is the irst control design equation Oserving (1), (), shows the control design equation to e a constraint etween the two control inputs γ sum, td, τ B Vertical motion mode The apex height during light is controlled indirectly The quantity directly controlled is the amount o energy added during stance, via the hip torques This results in a control on the apex height, as will e explained To this end, consider the ody energy at the light apex: Eap = ( m x ) + m g y+ ( I θ ) (3) I the orward speed is controlled to the desired value and the pitching to zero, as in pronking, then rom (3) the apex height is controlled y regulating the energy So, since the controllale quantity is the energy inserted in each stance, it is sought to show that inserting the same pre-calculated energy, in every stance, controls the ody s energy This is indeed the case i the losses o energy or a desired pronking gait can e calculated, and an amount o energy equal to these losses is inserted in each stance Then, i the root has a higher energy than that o the desired gait, it will have a greater apex height or the same controlled speed Hence the losses due to the leg viscous riction will surpass the compensated pre-calculated energy losses, and the system energy decreases In the opposite case, the apex height will increase, till it stailizes at the desired level Now the energy losses or the desired pronking gait remain to e calculated so as to achieve the desired apex height, as explained aove The ody energy losses, E l, in one cycle, are due to three causes, irstly the leg viscous riction during stance, secondly the leg positioning in light and thirdly the acceleration o unsprung leg mass at lito Comining these, the ody energy losses are expressed as: T st + El = l ( )/ (( ) ( ) )/ dt m xtd xlo m ylo y lo (4) where y lo, y + lo indicate vertical velocities just eore and just ater lito The three terms in (4) correspond to the three causes o energy loss descried aove Given the state o the root at lito, the last two terms are computale, using (17), and solving light dynamics The irst term o (4) is computed using (1), and will e a unction o the control input γ sum, td ecause o () Thereore, the ody energy losses E l or the next stance phase are a direct unction o the input γ sum, td, given the root state at lito For the system energy to remain at the correct level, the losses E l must equal the energy inserted y the actuators: Tst Tst E = τ ( γ θ) dt+ τ ( γ θ) dt (5) l So, using H5, H6 and (16), in (5), it is: τ = ( El L)/x Tst (6) Using the current speed x in (6), as opposed to the desired value, ensures that the correct energy is inserted in any transient case Equation (6) is the second control design equation Note that, as explained ater (4), E l is a unction o the control input γ sum, td Thereore, given the known root state at lito, the second control design equation is a constraint etween the control inputs γ sum, td, τ This was also the case in the irst control design equation in (), so there is now a system o two design equations with the two unknown inputs γ sum, td, τ The solution process is descried in part our o this section C Pitching motion mode The third control input γ di, td now remains to e determined To remain close to the pronking desired gait, it is necessary to eliminate deviations o the ody pitching velocity rom zero To do this, the evolution o the pitching motion is irst mapped rom the current lito o light phase i through to the touchdown o light phase i+1, as a unction o the control inputs γ sum, td, γ di, td, τ, see Fig 3 θ θ θ θ θ θ θ θ () (c) (d) (e) Fig 3 Sequence o events or deriving the pitching map The control inputs γ sum, td, τ are known rom aove, so the single ree parameter to inluence pitching is the dierence o the leg touchdown angles γ di, td The map is comprised o three parts, the irst through light i, the second through stance and the third through light i+1 For the two light parts, the map is given y (15), while during stance it is given y the doule stance vertical and pitching dynamics in (13), (14), and the ack and ront stance dynamics which may e similarly derived Using this dynamics, the vector u = [ y, θ, y, θ ] T is mapped rom lito o light i to touchdown o light i+1 The vertical motion is controlled independently in part B o this Section Thereore, the control o pitching is implemented y computing input γ di, td, such that zero pitch velocity is otained at the touchdown o light i+1, given u at lito o light i: Su i+1,td = Su (, γdi, td ) = i,lo θ td, i + 1, des (7) where θ td, i + 1, des is the desired pitch velocity at touchdown, set equal to zero, see Fig 3, and the vector S = [,,,1] 88
selects the pitch velocity component o u, and represents the pitching map descried aove The solution process is numerical, ut is ast, as it does not involve dynamics integration, due to the analytically integrale orm o stance dynamics when I 1ms or so, which is acceptale as it is required only once per light Equation (7) is the third control design equation = md A solution on a typical PC takes D Computation o control inputs Now, all three control inputs, γ sum, td, γ di, td, τ, can e computed The sum o the leg touchdown angles, γ sum, td, and the torque to e applied at each hip during stance, τ, are ound irst y comining the irst two control design equations in () and (6) The solution is analytical, ut lengthy, and does not oer urther insight into the motion The solution process requires expanding harmonic unctions o γ sum, td in a second-order Taylor series around the value ( x destst / L) Finally, the third control input, the dierence o the leg touchdown angles, γ di, td, is ound rom (7) E Sensory eedack and control application The sensors required or control are encoders or measuring leg angles and lengths, an inclinometer, and a rate gyro or pitch velocity and orientation Leg rotation speeds and rates o leg compression or extension can e ound y dierentiating sensor output Recently it was shown that using low-cost state o the art sensors, it is possile to otain good measurements o pitch and pitch velocity [13] To give an overview o the control method, it is useul to ollow the root through the execution o one motion cycle, starting with lito, when the root enters the light phase At lito the three control inputs are computed, ased on the root state Once the inputs are known, the individual touchdown angles are known, rom γ sum, td, γ di, td, so the legs are positioned in light The leg positioning is accomplished using a simple PD control When the stance phase egins, and given a transmission eiciency o η a, the hip actuators are commanded to apply a constant torque τ a, such that the applied torque is equal to the control input τ : τ a = τ ηa (8) At lito, at the end o stance, the cycle repeats V RESULTS The control approach is evaluated or the real-world prolem, using simulations o a detailed planar root model, reerred to as the test model, using the Working Model D sotware, see Fig 7 Simulations with adequate modeling have een shown to predict real quadruped motion [9] The test model includes oth root and actuator modeling The root has ody mass k = 7 N/m, ody inertia I =1 kg m, leg viscous riction =15 N s/m and leg rest length L =3 m Hal the hip spacing is d =5 m, and the dimensionless inertia is equal m =16 kg, leg spring constant to j= I/ md = 1 The test model simulated also includes nontrivial leg mass, modeling o plastic oot-ground collisions, a DC motor model or the hip actuators, and a Coulom ootground riction model allowing realistic oot slip The DC motors are rom Maxon Motors [1], o 6 W power, with nominal voltage 4 V, and torque constant 59 N m/a Each motor weighs under 4 kg and the gear ratio is 5 Initially, the response to height control is tested The root response is shown in Fig 4, and the desired height trajectory in Fig 4, together with the apex height attained 8 6 4 1 3 θ 15 td (deg/s) 1 5-5 1 3 (c) τ (Nm) 1 4 4 38 36 actual 34 1 3 () θ (deg) 4-1 3 (d) γ,td, γ,td (deg) 1-1 1 (e) 3 1 3 () Fig 4 Root response to height control, at a speed o 8 m/s, orward speed, () height at apex and desired in orange, (c) pitch velocity at touchdown, (d) ody pitch, (e) sum o applied torque on oth ack legs, () ack and ront leg touchdown angles 15 y ap (m) desired The maximum error in steady-state is less than 1cm, and then only or the largest apex height, which gives a ground clearance o 1cm This is where luctuations in the orward speed appear, see Fig 4a It can e seen that the torques applied y the actuators, in Fig 4e, are completely realistic Secondly, results rom ollowing a desired orward speed trajectory are shown in Fig 5 The trajectory is shown in Fig 5a with a solid line, together with the actual speed, or a range o 6 to 15 m/s The pitching motion is not shown or space economy, ut it resemles the response in Fig 4 The desired apex height o 36 cm is maintained, see Fig 5 y (m) 15 36 desired 34 1 3 3 8 6 5 actual 1 3 γ,td, γ,td (deg) 15 1 16 14 1 1 8 6 1 () 3 +15% +1% x = x des -1% -15% 5 4 1 3 6 8 1 1 14 (c) x des (m/s) (d) Fig 5 Root response to speed control, or a constant apex height o h=36 m, orward speed and desired in orange, () CoM height, (c) leg touchdown angles (d) Actual orward speed versus desired value (with crosses) The solid line represents zero error, while dashed lines show deviations o 1% and 15% Gray circles indicate steady-state motion The orward speed o the root at each light apex is plotted versus the desired speed, in Fig 5d Although overall errors o up to aout 15% are oserved, this also includes transient states o motion The size o the errors can 89
e veriied y oserving the orward speed in Fig 5a The gray circles indicate the steady-state motions, where it can e seen that errors are limited to less than 1% The controller maintains a pronking gait, as can e seen y the almost total asence o the pitching motion in Figs 4d, 4c Further, in Figs 4, 5c, the leg touchdown angles used are realistic It is also interesting to portray a small time window o the motion, as in Fig 6 In Fig 6a, the velocity o a root oot relative to the ground is shown and the slip o the oot is visile as it touches the ground, due to the ground-oot riction model Also, Fig 6 depicts a detail o the actuator power Because the applied torque during stance has een chosen to e constant, see (16), there is no negative actuator work during stance This is interpreted as the control working in coordination with the natural dynamics, pumping only the required energy in to the system () Fig 6 Response detail Foot slipping due to riction model, () actuator power or the ack (solid) and ront (dashed) motor In oth plots, the gray line denotes stance with its high value, and light with its low value Now the case o the root carrying a load is studied The load is known, and is two loads o kg placed at 11 cm each side o the ody CoM, preserving mass symmetricity ut changing total root mass and inertia The dimensionless inertia j is now 85, so the root does not strictly elong to the class studied Following the same speed trajectory as in Fig 5, Fig 7a shows good ehavior all the way up to 13 m/s Pitching was a little increased, ut the 15% deviation in the dimensionless inertia did not result in control ailure 16 14 1 1 8 6 4 5 1 15 5 3 () Fig 7 Forward speed o root with j=85 () Working Model D snapshots o running at 11 m/s A urther interesting case is when the root is made to carry asymmetric loads For example, let two loads again e placed at 11 cm each side o the ody CoM, except this time the loads are unequal, with masses o 15 kg and 5 kg With this asymmetrical mass distriution the same speed trajectory as that used or Fig 5 is applied Despite the asymmetry, the control works well up to speeds o 11 m/s However, i the asymmetry is increased y placing loads o 1kg, 3kg instead, the controller eventually ails The root is also tested in the case o modeling error Speciically, the root parameters m, m l,, k, were all increased y 1% compared to the values used y the controller In practice, error in parameter estimation will proaly e smaller Despite this, the root ollowed the desired speed trajectory used or Fig 5, up to speeds o 11 m/s, while pitching varied etween -1 deg and deg Finally, simulations in 3D were also perormed using the ADAMS 3D simulation package The controller response is similar to the Working Model D simulations This result is expected, as the pronking gait studied here is symmetric with respect to the plane o orward motion CONCLUSIONS In this work it has een shown how it is possile to control oth the orward speed and height o a running quadruped, while retaining minimum pitching, using one actuator per leg The control has een designed ased on an insight o the root dynamics and an unveiling o the eect o the leg touchdown angles Extensive simulations showed the control to perorm well, achieving high-speed running with almost no pitching Negative actuator work was asent during stance, which reduced consumed energy y attracting the root to the desired gait without working against the natural dynamics This indicates that this work could contriute to designing lighter and more autonomous roots, as smaller actuators and power supplies could e used REFERENCES [1] M H Raiert, Legged roots that alance, MIT Press, Camridge, MA, 1986 [] Z Zhang, Y Fukuoka and H Kimura, Adaptive running o a quadruped root using delayed eedack control, Proc 5 IEEE Int Con on Rootics and Automation, pp 375-3755, 5 [3] W Marheka, D E Orin, J P Schmiedeler and K J Waldron, Intelligent control o quadruped gallops, IEEE/ASME Transactions On Mechatronics, Vol 8, No 4, pp 446-456, 3 [4] Y Fukuoka, H Kimura and A H Cohen, Adaptive dynamic walking o a quadruped root on irregular terrain ased on iological concepts, The Int Journal o Rootics Research, Vol, No 3-4, pp 187-, 3 [5] S Talei, I Poulakakis, E Papadopoulos and M Buehler, Quadruped root running with a ounding gait, Proc 7 th Int Symp on Experimental Rootic (ISER ), Honolulu, HI, pp81-89, [6] D Papadopoulos and M Buehler, Stale running in a quadruped root with compliant legs, Proc IEEE Int Con Rootics and Automation, San Francisco, CA, pp 444-449, April [7] M Lasa and M Buehler, Dynamic compliant quadruped walking, Proc 1 IEEE Int Con on Rootics and Automation, pp 3153-3158, 1 [8] Murphy K N and Raiert M H, Trotting and Bounding in a Planar Two-legged Model, 5th Symp on Theory and Practice o Roots and Manipulators, A Morecki, G Bianchi, K Kedzior (eds), MIT Press, Camridge MA, pp 411-4, 1984 [9] I Poulakakis, J A Smith and M Buehler, Modeling and experiments o untethered quadrupedal running with a ounding gait: the scout II root, The Int Journal o Rootics Research, Vol 4, No 4, pp 39-56, 5 [1] Maxon Motor AG, wwwmaxonmotorcom [11] N Cherouvim and E Papadopoulos, Single Actuator Control Analysis o a Planar Hopping Root, Proc Rootics: Science and Systems 5, MIT, Camridge, MA, 5 [1] M D Berkemeier, Approximate Return Maps or Quadrupedal Running, Proc 1997 IEEE Int Con on Rootics and Automation, pp 85-81, 1997 [13] J Leavitt, A Sideris and J E Burow, High Bandwidth Tilt Measurement Using Low-Cost Sensors, IEEE/ASME Transactions on Mechatronics, vol 11, no 3, pp 3-37, 6 83