Control of Dynamic Gaits for a Quadrupedal Robot

Control of Dynamic Gaits for a Quarupeal Robot Christian Gehring, Stelian Coros, Marco Hutter, Michael Bloesch, Markus A. Hoepflinger an Rolan Siegwart Autonomous Systems Laboratory, ETH Zurich, Switzerlan, gehrinch@ethz.ch Disney Research Zurich, Switzerlan Abstract Quarupeal animals move through their environments with unmatche agility an grace. An important part of this is the ability to choose between ifferent gaits in orer to travel optimally at a certain spee or to robustly eal with unanticipate perturbations. In this paper, we present a control framework for a quarupeal robot that is capable of locomoting using several gaits. We emonstrate the flexibility of the algorithm by performing experiments on StarlETH, a recently-evelope quarupeal robot. We implement controllers for a static walk, a walking trot, an a running trot, an show that smooth transitions between them can be performe. Using this control strategy, StarlETH is able to trot unassiste in 3D space with spees of up to.7m/s, it can ynamically navigate over unperceive 5-cm high obstacles an it can recover from significant external pushes. I. INTRODUCTION Legge robots are better suite for rough terrain locomotion than their wheele or tracke counterparts. As a result, they have the potential of being use for a wier variety of tasks. The rawback of multi-legge systems, however, is that they are more complex, inherently unstable an therefore more ifficult to control. In aition, appropriate control methos nee to be robust to unplanne isturbances because the environments, in general, are only partially observable. Statically stable solutions for this problem rely on position control algorithms an have been stuie extensively [1], [2]. However, they have not yet been shown to prouce motions that are as agile as the motions observe in nature [3]. To ate, consierable progress has been mae towars briging the gap between the skill sets of legge robotic systems an that of real animals. In constrast to Boston Dynamic s LittleDog [4], which is only capable of static walking, its larger counterpart, BigDog [5], looks more agile an life-like an is capable of a variety of locomotion behaviors: staning up, squatting own, walking, trotting an bouning. Stable an robust locomotion has been emonstrate on this platform, but the exact etails of the employe control algorithms are not known. More information is available regaring the control scheme of the IIT s HyQ [6], another hyraulically actuate quarupe, which was recently shown trotting robustly by employing a simple virtual moel control approach for each leg [7]. Recent progress has also been mae in simulation, where it is possible to ecouple the control laws from the limitations of specific harware platforms. For instance, Coros et al. [8] This research was supporte by the Swiss National Science Founation through the National Centre of Competence in Research Robotics. v, v, ψ x y motion generator r F q, q b b position control motion controller torque control q, q j τ j low-level controller I q, q, τ, Fig. 1. The control framework (blue) computes the esire joint positions, velocities, an torques base on the esire walking spee an irection, whereas the low-level controller (green) generates the esire motor current from these outputs. escribe a control framework that was successfully use to control a broa range of ynamic gaits for a og-like simulate quarupe, an Krasny an Orin [9] evelope a control algorithm for galloping quarupes. At a high level, successful control strategies are base on the observations note by Raibert [1], who showe the importance of two essential ingreients: control of the boy through the stance hips an foot placement control for balance recovery. For this work we use a set of conceptually similar ieas in orer to control StarlETH (Springy Tetrapo with Articulate Robotic Legs), a recently evelope, electrically riven quarupe robot [11]. StarlETH s weight of 23kg an leg length of.4m correspon to the imensions of a meium size og, an it uses an actuation scheme base on highly compliant series-elastic actuators that enable torque control. Our aim is to increase StarlETH s repertoire of motions to inclue faster, more life-like ynamic gaits. To this en we buil on the framework escribe by Coros et al. [8]. The control scheme combines several simple builing blocks. An inverte penulum moel computes esire foot fall locations, PD controllers regulate the motions of the legs an virtual forces are use to continuously moulate the position an orientation of the main boy. In aition to iscussing the changes neee to apply the control scheme to StarlETH, we escribe an improve metho for istributing virtual forces to the stance legs. We emonstrate the flexibility of the escribe framework by generating walking an trotting controllers that are robust to pushes an significant, unanticipate variations in the terrain. In aition, we show that our metho can prouce smooth gait transitions that epen on the walking spee, resulting in increase agility. II. CONTROL FRAMEWORK The goal of our control framework is to provie quarupe robots with the ability to move through their environments c

while robustly ealing with perturbations ue to external pushes or unperceive variations in the terrain. In orer to allow the robots to be steere, the esire forwar spee v x, lateral spee v y or turning rate ψ are treate as high-level parameters that can be moifie at any time. Figure 1 shows the ifferent builing blocks of our control system. Our control framework (blue) computes esire joint positions q an torques τ that are passe to the low-level controller (green). The latter consiers the ynamics of the actuators an regulates the motor currents I. The motion generator an motion controller moules shown in Fig. 1 illustrate the two core questions aresse by our metho: how o we generate appropriate motion objectives for the whole-boy system (Section II-B), an how o we best achieve them (Section II-C)? Before we iscuss in etail these two components, we first escribe the characteristics of the plant. A. The Plant Our quarupe robot, StarlETH, has four articulate legs with three actuate egrees-of-freeom (DoF) each: hip abuction/auction (HAA), hip flexion/extension (HFE), an knee flexion/extension (KFE). The mechanical system therefore has 12 actuate DoFs an 18 DoFs in total. The controller has access to all joint angles q j R 12, joint velocities q j, as well as to the pose q b R 6 an velocities q b of the main boy, which are estimate by an Extene Kalman filter that fuses IMU ata an leg kinematics [12]. The minimal coorinates of the free-floating robot are thus given by q = [q b, q j ] T. Aitionally, pressure sensors in the feet inicate whether the legs are in contact with the groun. By thresholing the pressure reaings, a boolean contact flag for each leg (c flag {, 1}) is available for the control algorithm. The control framework escribe in this paper outputs esire joint angles q j for the swing legs an esire joint torques τ for the stance legs, as the series-elastic actuators employe by StarlETH enable torque control. The high compliance of the system, however, requires sophisticate lowlevel torque an position controllers in orer to cope with the resulting low control banwith. We therefore use the lowlevel control system escribe by Hutter et al. [13] to generate esirable motor currents. B. Motion Generation The motions of the legs an the main boy are escribe in our framework either in the inertial (worl) frame I or in the boy frame B that is locate at the center of the main boy, namely in the center of the HAA joints. The main frame s x-axis is aligne with the robot s heaing irection, which we also refer to as the sagittal irection. The vertical irection is collinear with the z-axis of the inertial frame. The y-axis of the main frame enotes the robot s coronal irection. 1) Terrain: To plan the locations of the foothols, it is essential to know where the groun is locate in the inertial frame. Since the estimate vertical position of the robot can rift, an we restrict ourselves from using any external sensors, we estimate the groun height h g by filtering the vertical LH LF RF RH.25.5 φ.75 1 Fig. 2. Gait graph for a walking trot: the black bar efines the stance phase of the left hin (LH), left front (LF), right front (RF), an right hin (RH) leg, respectively. position of the stance feet, Ir F,z, expresse in the inertial frame: h g (t) = N=4 i= c flagi ( I r Fi,z α + h g (t t) (1 α)), (1) where α =.2 is the parameter of a first orer filter. 2) Timing: Quarupe gaits are to a large extent efine by the foot fall pattern an the uration of the gait cycle T s. In our implementation, this is controlle by the Gait Pattern, which explicitly efines the role of each leg at any moment in time. Legs that are in swing moe nee to safely reach the next foothol location in orer to ensure that the robot can move at the esire spee or that it can recover balance. In contrast, the legs that are in stance moe must help satisfy the motion objectives of the main boy in a coorinate way. The Gait Pattern efines the sequence of swing an stance moes for each leg with respect to the time-normalize strie phase φ [, 1], as illustrate in Figure 2. The white areas inicate the fraction of the strie when a leg is in swing phase, which is characterize by the relative timing of the lift-off an touch-own events. The ark areas inicate that a leg is in stance moe. In aition to informing the controller of whether a leg is in swing or stance, the gait pattern is use to estimate the amount of time left before a leg shoul transition to the next moe. This information is useful as it helps the controller anticipate how the support polygon will change in the near future an plan accoringly. The stance phase φ st [, 1] of a leg inicates the time normalize progress mae uring the stance moe. The swing phase of a leg, φ sw, etermines the amount of time left before the next foot touch-own event, an it is set to 1 if the stance phase φ st >. We efine the rule use to etermine if a leg is in stance moe ι st {, 1} as: { 1 if cflag (φ ι st = sw >.9) (2) φ sw < otherwise The first case employe in the equation above ensures that legs are free to transition to stance moe earlier than preicte in orer to support the main boy, if early contacts are etecte. The swing moe ι sw {, 1} is efine as ι sw = ι st. We introuce another variable, the groune flag g flag = ι st c flag {, 1}, to select the appropriate low-level controller. The flag is only true if the leg is, an shoul be, in contact with the groun. In this case it is safe to apply torque control at all the joints of the leg, incluing the knee. 3) Swing Leg Configuration: Appropriate foot placement control for the swing legs can provie the robot with the ability to recover balance when it is pushe, or when it encounters

unanticipate variations in the terrain. Our foot placement algorithm currently consiers each leg inepenently of the others. At every control cycle we calculate, for each swing leg, an appropriate foothol position. Subsequently, we plan a trajectory for the foot in orer to ensure that the target stepping location is reache safely. This results in esire swing foot positions at every moment moment in time. The target foothol location I r F is compute relative to the HAA joint I r H : Ir HF = I r fb HF + I r ff HF, (3) where I r fb HF is a feeback term preicte by an inverte penulum moel [14], an I r ff HF is a feeforwar step length that epens on the robot s esire spee. This formulation is similar to the one escribe in [15]. We use a slightly moifie version of the inverte penulum preiction in orer to compute the feeback component of the stepping location: Ir fb HF = η( I v ref I v h ) g, (4) where h = I r H,z h g is the current height of the hip with respect to the groun, I v = I v x + I v y is the esire velocity, g is the gravitational acceleration, an I v ref is an estimate reference linear velocity, an η is a scaling parameter that was set to 1.2 for all our experiments. This particular form of the feeback term ensures that, when moving at the esire spee, only the feeforwar component of the step location is use. Consequently, only ifferences between the current spee an the esire spee are taken into account by the feeback component. In practice we notice that the feeback component of the step can be too large when the robot is mostly rotating about the yaw axis. For this reason, we compute the estimate reference velocity use in the equation above as the average between the leg s hip velocity an that of the boy s COM: Iv ref = 1 2 ( Iv H + I v CoM ). (5) The feeforwar component of the stepping location is compute as half the istance the CoM is expecte to travel uring the stance uration t st that is efine by the Gait Pattern: Ir ff HF = 1 2 I v t st. (6) We can optionally a an aitional offset, B r HF, to the feeforwar stepping location in orer to control the with of the steps that are taken. This is particularly useful for slower gaits such as the static walk. The stepping offset I r HF constitutes the final esire location for foot placement. However, we nee to provie the robot with a continuous trajectory that ensures that the final foot location can be reache safely. We therefore linearly interpolate between the initial location of the foot at the beginning of the swing phase, an this final target location. To provie enough groun clearance for the foot, we use a pre-efine height trajectory that varies as a function of the swing phase, as shown in Fig. 3a. This trajectory is efine by a spline, an all values are relative to the estimate groun height h g. 4) Stance Leg Configuration: In case a leg is in stance moe accoring to the Gait Pattern, but loses contact with the groun (g flag = 1), we compute a esire foot target that is 1cm lower than the leg s current position, in orer to regain contact with the groun as soon as possible. Otherwise, because there are no kinematic reunancies in the mechanical esign, we o not nee to actively control the pose of the stance legs. 5) Main Boy Configuration: The pose q b an velocity q b of the main boy nee to be controlle in orer to increase robustness, i.e. prevent the robot from tumbling over, an to meet the esire velocity commans. By efault, the esire orientation of the main boy is efine by zero roll an pitch angles, whereas the yaw angle is unconstraine. The esire height of the boy relative to the estimate groun height, h H, is specifie by a spline as a function of the strie phase, an it can be use, for instance, to propel the boy upwars at the right moment in time in anticipation of a flight phase. The esire position of the boy along the sagittal an coronal irections is compute relative to the positions of the feet: N Ir i=1 B = w i(φ) I r Fi N i=1 w, (7) i(φ) where the leg weights w i epen on the strie phase φ as illustrate in Fig. 3b. We compute the esire position base not only on the groune legs, but also base on the swing legs, in orer to get smooth trajectories for the esire position of the boy. With the strategy we implemente, as a groune leg approaches the en of the stance phase (φ st = φ st, ), the boy can start shifting away from it. Similarly, the boy starts shifting towars a swing leg, as it reaches the en of the swing phase (φ sw = φ sw, ) an prepares for laning. This anticipatory behavior is flexible enough to control traitional static gaits, an it allows us to also implement ynamic gaits that are increasingly more agile. The minimal weight w min epens on the gait an is foun experimentally. The generalize esire position of the main boy is given by q b = ( Ir Bx, Ir By, h g + h H (φ),,, ) T, while its esire velocity is q b = (vx, vy,,,, ψ ) T. height of foot above groun [m].8.6.4.2.2.2.4.6.8 1 φ (a) Swing foot trajectory w i 1.8.6.4.2 stance moe swing moe φ st w min φ sw.2.4.6.8 1 φ (b) Weights for balance control Fig. 3. Most of the motion characteristics are escribe with respect to the strie phase φ.

C. Motion Control We use low level position controllers in orer to get fast an precise tracking of the swing leg joint trajectories [13]. For the legs that are in stance moe, we make use of virtual force control, as it is an intuitive an effective metho. However, ue to the particular mechanical setup of the knee joint, we cannot apply torque control to the knee joints of the stance legs unless the knee spring is uner tension. In practice, we etect groun contacts, or lack thereof, at a fast enough rate to employ a hybri control approach, using torque control for the stance legs that are in contact with the groun, an position control otherwise. 1) Position Control: We use position control whenever a leg is not, or shoul not be (g flag = ), in contact with the groun. The esire joint angles q j are obtaine from the esire foot positions through inverse kinematics, an are then passe irectly to the low level controller. 2) Torque Control: The joint torques that nee to be applie through the stance legs are compute in three steps. We first compute virtual forces an torques that shoul ieally act on the main boy in orer to control the robot s posture. These are then optimally istribute to the stance legs given the current kinematic configuration of the robot. Lastly, we map the virtual leg forces F leg to the joint torques by applying Jacobian transpose control: τ = J T F leg. The esire forces an torques that shoul act on the main boy are compute base on the esire pose q b, the current pose q b an their erivatives, as: [ BF ] v y B BT = k p (q b q b ) + k ( q b q b ) + k ff mg B ψ where k p, k, an k ff are the proportional, erivative an fee-forwar gains, respectively an m is the total mass of the robot. The fee-forwar gains improve tracking the esire velocities an compensate for gravity. The esire net virtual force B F B an torque B T B that shoul be applie to the main boy are boune before being istribute to the stance legs, in orer to ensure that the robot oes not apply excessively large forces through the stance legs. In the framework escribe in [8], B F B an B T B are equally istribute to the stance legs. This strategy i not work for our static walking gait. Instea, at each control step, we solve a convex optimization problem with linear constraints in orer to compute esirable contact an friction forces to be applie through the stance feet. More formally, the problem formulation is as follows: v x (8) minimize (Ax b) T S(Ax b) + x T W x (9) subject to Fleg,i n Fmin, n (1) µfleg,i n Fleg,i t µfleg,i n (11) where x = [F T leg,,, F T leg,i, F T leg,m] T, F T leg,i represents the net force to be applie through the i th stance leg an m is the number of legs that are an shoul be groune (g flag = 1). In orer to ensure that the forces applie through the stance legs result in a net force an torque that are as close as possible to the esire values, we compute A an b using: F leg, [ ] I I I F leg,1 r r 1 r m }{{}. A F leg,m }{{} x ( ) F = B, (12) T B }{{} b where r i is the vector between the CoM an the location of the foot of stance leg i. The weighting matrix S traes off the egree to which we want to match the net resulting torque over the net resulting force, an the term x T W x acts as a regularizer that iscourages the use of large virtual forces. The constraints applie ensure that the normal component of the force applie through each leg, Fleg,i n, is strictly positive (no pulling on the groun). In practice we foun that requiring a minimal force Fmin n = 2N to always be applie results in fewer instances where the feet slip. We also restrict the tangential component F t leg,i to remain within an approximate friction cone efine by the assume friction coefficient µ =.8 in orer to avoi slipping. D. Gait Transitions Gaits are mainly characterize by the gait pattern an the strie uration, but several parameters in our ontrol framework have to be ajuste specifically for each gait in orer to increase performance. Fortunately, we have observe that smooth transitions between gaits can be generate by lineary interpolating the iniviual parameter sets. As long as the gait patterns are compatible (there is no smooth transition between trot an gallop, but there is one between walking to trotting, for instance), this approach seems to work well an oes not require aitional parameter tuning. However, it is likely that the resulting transitions may be suboptimal. We efine a time horizon for the interpolation proceure, an the gait transitions are either initiate manually by an operator, or as a function of the esire spee. III. RESULTS Before conucting any experiments on StarlETH, we verifie the control framework in simulation. Here we only iscuss the results obtaine by running the control strategy on the physical robot. Our results are best seen in the accompanying vieo. For more information about the simulation environment an the software package we refer the intereste reaer to Hutter et al. [11]. StarlETH was able to move freely in 3D uring all our experiments, an was not aie by any support structures. A static walk, a walking trot, an a running trot (with flight phase) were successfully implemente on StarlETH. The walking trot reache a top spee of.7m/s on a treamill

TABLE I PARAMETER SETS FOR DIFFERENT GAITS Parameter Symbol Static Walk Walking Trot Running Trot gait graph strie uration T s[s] 1.5.8.7 min. leg weight for support polygon w min.35.15.15 start of increasing the weight of a swing φ sw,.7.7.7 leg for support polygon start of ecreasing the weight of the stance leg for support polygon φ st,.7.7.7 efault left front swing leg offset Br HF [m] [,.1, ]T [,, ] T [,, ] T efault left hin swing leg offset Br HF [m] [,.14, ]T [,, ] T [,, ] T height of mile of hip AA joints h H [m].39.44.44 virt. force proportional gain k p [5, 64, 6, 4, 2, ] T [, 64, 6, 4, 2, ] T [, 64, 26, 4, 2, ] T virt. force erivative gain k [15, 1, 12, 6, 9, ] T [15, 1, 12, 6, 9, ] T [9, 6, 12, 6, 9, ] T virt. force fee-forwar gain k ff [25,, 1,,, ] T [6,, 1,,, ] T [25,, 1,,, ] T weights for matching the es. virt. forces S iag(1, 1, 1, 1, 1, 5) iag(1, 1,.2, 2, 2, 5) iag(1, 1,.2, 2, 2, 5) weights for reucing joint torques W iag(.1... ) iag(.1... ) iag(.1... ) whose spee was set to match that of the robot, as measure by a motion capture system. A. Parameter Sets We tune the initial gains of the virtual force controller while perturbing the robot as it trie to stan in place. We then ajuste the parameters for the ifferent gaits while the robot was walking or trotting. We foun this process to be intuitive, because a large range of parameters result in successful motions, an the parameters are largely orthogonal as they affect ifferent aspects of the motion objectives or motion control components. The parameters we use for the static walk, walking trot, an running trot are summarize in Table I. B. Robustness The robustness of the control system was examine by asking the robot to walk an trot on flat groun, while introucing unanticipate obstacles up to 5cm high (an eighth of the leg length) as shown in Fig. 5. In aition, we teste the ability of the robot to recover from external pushes. While the uration of the push, the current phase in the locomotion cycle an the push irection can affect the ability of the robot to reject perturbations, we notice that significant pushes are generally hanle well (as shown in the supplementary vieo). The foot placement strategy, in conjunction with the appropriate istribution of virtual forces to the stance legs allowe the robot to successfully recover from various such scenarios. When the robot faile to recover balance, this was typically ue to the HAA joints reaching their joint limits while the legs were in stance moe. Figure 4 presents some relevant ata from one of the push experiments we performe. As seen in the supplementary vieo, StarlETH was pushe in the sagittal irection for a uration of roughly.2s (inicate by the gray area). The first sub-plot in Fig. 4 shows the sagittal position. The robot moves forwar uring the push an soon thereafter steps in orer to recover balance. The secon plot shows the coronal position, where a slight lateral rift is visualize. The following three plots show the net virtual forces for the main boy in the sagittal an coronal irections, as well as the net torque about the y-axis of the robot. The soli lines illustrate the esire virtual forces, whereas the ashe lines show the sum of the contributions of the istribute leg forces. As can be seen, the force istribution favors matching the esire torque over matching the esire forces, as inicate by the input weighting matrix S. When all four legs are in contact with groun, the net istribute forces begin to match both the esire forces an the esire torques, as there are enough egrees of freeom in the system. The influence of the unilateral contact constraints can also be observe in these plots. When only two legs are in contact with the groun, the errors in the istribute coronal force result in the rift observe in the secon plot. The last plot shows the measure (soli) an esire (ashe) joint torque in the knee joint. The swing phase can be clearly ientifie by the zero joint torque. C. Gait Transitions StarlETH can smoothly transition from the static walk to the walking trot if we linearly interpolate between the parameter sets shown in Table I. The uration of the interapolation can be chosen somewhat arbitrarily, but for the results we showe here we use a time perio of 3s. The transition from the trot to the walk takes place over.5s. We notice that the transitions are robust with respect to the exact strie phase when they are initiate, an we therefore o not require them to start at a particular point in the locomotion cycle. To transition between the walking trot an the flying trot we similarly interpolate the parameter sets of the two gaits. IV. CONCLUSION The control framework escribe by Coros et al. [8] was extene to enable our quarupe robot to perform a static walk, a walking trot an a running trot. In aition to etailing the various changes neee to apply this control framework to a real robot, we employe a new force istribution metho, without which the robot was unable to walk.

Fig. 5. StarlETH performs a walking trot while ealing with unperceive obstacles. [m] [m] [N] [N] [Nm] [Nm].2.1.2 sagittal position.2.4.6.8 1 1.2 1.4 1.6 1.8 2 coronal position.4.2.4.6.8 1 1.2 1.4 1.6 1.8 2 1 virtual force in sagittal irection 1.2.4.6.8 1 1.2 1.4 1.6 1.8 2 1 virtual force in coronal irection 1.2.4.6.8 1 1.2 1.4 1.6 1.8 2 5 virtual torque for pitching 5.2.4.6.8 1 1.2 1.4 1.6 1.8 2 4 2 number of groune legs.2.4.6.8 1 1.2 1.4 1.6 1.8 2 2 1 joint torque of left front knee.2.4.6.8 1 1.2 1.4 1.6 1.8 2 time [s] Fig. 4. Experimental results of a push that was applie in sagittal irection uring.2s as inicate by the grey area. The main benefits of the framework that we use are that it is highly moular, that it allows the various parameters to be tunne in an an intuitive way, an that it results in locomotion controllers that are robust to pushes an unexpecte variations in the terrain. As shown in simulation, the parameter space of this control framework is rich enough to also escribe other gaits, such as a pace, boun or gallop [8]. In the future we plan on further extening StarlETH s repertoir of motions by creating controllers an transitions for this new set of gaits. We have not yet performe a quantitative evaluation of the performance an robustness of the control system, an this will be part of future investigations. The clean separation of the motion generator an the motion controller moules will enable us to also compare ifferent control strategies. For instance, ifferent moels can be plugge in for the foot placement component, or the force istribution metho coul be replace by an operational space approach [16] in orer to test the relative merits of the ifferent builing blocks we use. Last but not least, we plan to investigate a systematic way of fining optimal parameter sets on the real system. REFERENCES [1] J. Buchli, M. Kalakrishnan, M. Mistry, P. Pastor, an S. Schaal, Compliant quarupe locomotion over rough terrain, in IEEE/RSJ International Conference on Intelligent Robots an Systems, 29. [2] P. González-e Santos, E. Garcia, an J. Estremera, Quarupeal locomotion: an introuction to the control of four-legge robots. Springer, 26. [3] M. Hilebran, The Quarupeal Gaits of Vertebrates, BioScience, vol. 39, no. 11, pp. 766 775, Dec. 1989. [4] M. P. Murphy, A. Sauners, C. Moreira, A. A. Rizzi, an M. Raibert, The littleog robot, The International Journal of Robotics Research, 21. [5] M. Raibert, Bigog, the rough-terrain quarupe robot, in Proceeings of the 17th IFAC Worl Congress, M. J. Chung, E., vol. 17, no. 1, 28. [6] C. Semini, N. Tsagarakis, E. Guglielmino, M. Focchi, F. Cannella, an D. Calwell, Design of hyq a hyraulically an electrically actuate quarupe robot, Proceeings of the Institution of Mechanical Engineers, Part I: Journal of Systems an Control Engineering, vol. 225, no. 6, pp. 831 849, 211. [7] J. B. I. Havoutis, C. Semini an D. Calwell, Progress in quarupeal trotting with active compliance, Dynamic Walking, 212. [8] S. Coros, A. Karpathy, B. Jones, L. Reveret, an M. van e Panne, Locomotion skills for simulate quarupes, ACM Transactions on Graphics, vol. 3, no. 4, 211. [9] D. Krasny an D. Orin, Evolution of a 3 gallop in a quarupeal moel with biological characteristics, Journal of Intelligent an Robotic Systems, vol. 6, pp. 59 82, 21. [1] M. H. Raibert, Symmetry in running, Science, vol. 231, no. 4743, pp. pp. 1292 1294, 1986. [11] M. Hutter, C. Gehring, M. Bloesch, M. Hoepflinger, C. Remy, an R. Siegwart, StarlETH: a compliant quarupeal robot for fast, efficient, an versatile locomotion, in Proc. of the International Conference on Climbing an Walking Robots (CLAWAR), 212. [12] M. Bloesch, M. Hutter, M. H. Hoepflinger, C. D. Remy, C. Gehring, an R. Siegwart, State estimation for legge robots - consistent fusion of leg kinematics an IMU, Proceeings of Robotics: Science an Systems, 212. [13] M. Hutter, C. D. Remy, M. H. Hoepflinger, an R. Siegwart, High compliant series elastic actuation for the robotic leg ScarlETH, in Int. Conference on Climbing an Walking Robots (CLAWAR), 211. [14] J. E. Pratt an R. Terake, Velocity-base stability margins for fast bipeal walking, Fast Motions in Biomechanics an Robotics, vol. 34, pp. 1 27, 26. [15] M. H. Raibert, Legge Robot that Balance. MIT Press, 1986. [16] M. Hutter, M. Hoepflinger, C. Gehring, M. Bloesch, C. D. Remy, an R. Siegwart, Hybri operational space control for compliant legge systems, in Proceeings of Robotics: Science an Systems, 212.