May 25, 27 13:6 WSPC - Proceeings Trim Size: 9in x 6in clawar7 1 GAIT PARAMETER ADAPTATION TO ENVIRONMENTAL PERTURBATIONS IN QUADRUPEDAL ROBOTS E. GARCIA, J. ESTREMERA, P. GONZALEZ DE SANTOS an M. ARMADA Inustrial Automation Institute - CSIC, 285 La Povea, Mari, Spain E-mail: egarcia@iai.csic.es www.iai.csic.es/users/egarcia Quarupeal robots working outoors are very slow robots prone to tumble own in the presence of perturbations. This paper presents a novel gaitaaptation metho that enables walking-machine gaits to autonomously aapt to environmental perturbations, incluing the slope of the terrain, by fining the gait parameters that maximize robot s ynamic stability. Experiments with the SILO4 quarupe robot are presente an show how robot stability is more robust when the propose approach is use for ifferent external forces an sloping terrains. Keywors: Quarupe robot; Gait aaptation; Dynamic stability margin. 1. Introuction Walking robots esigne for fiel an service applications usually perform statically stable gaits, which have been esigne to optimize a static stability margin. Even when a ynamic stability margin is use to control the robot s motion, the gait pattern is a statically-stable one. As a result of this contraiction, the robot will still tumble own when confronte with any perturbing effect. Although an accurate ynamic stability margin measures robot stability, unstable situations are merely observe, not avoie, unless the gait pattern is moifie. Existing ynamic gaits like the trot an the gallop require light, simplifie robot esigns with elastic actuators, which iffer greatly from the heavy-limbe machines 1,2 use for fiel an service applications. Therefore, statically stable robot gaits shoul be moifie base on a more suitable ynamic stability margin to improve gait control. Only a few researchers have attene to partially solve this problem, achieving aaptation to uneven terrain, yet they i not consier robot
May 25, 27 13:6 WSPC - Proceeings Trim Size: 9in x 6in clawar7 2 Leg workspace xc y c1,max y c1,min y c2,max y c2,min (x c2, y ) c2 y c y c x c1,max x c1,min x c3,max (x c1, y ) c1 (x cg, y ) cg (x c4, y ) c4 s x c2,max x c2,min Position of S max NDESM x c4,max (x c3, y ) c3 x c3,min x c4,min y c3,max y c3,min y c4,max y c4,min Fig. 1. Outlining of gait-parameter aaptation for a quarupe in the boy-motion phase of a two-phase iscontinuous gait ynamics nor environmental perturbations. 3,4 This paper presents a new metho for gait aaptation to the environment base on the Normalize Dynamic Energy Stability Margin, S NDESM. 5 The metho maximizes the ynamic stability margin uring the transfer an support phases an thus we get a more robust gait against external perturbations. Section 2 escribes the gait-parameter aaptation approach. Experiments using the SILO4 walking robot (see Figure 5) are escribe in Section 3 to show the improvement of the gait-parameter aaptation metho when external forces are applie. 2. Gait Aaptation Let us assume a quarupe robot walking along the irection of its longituinal boy axis. Let us efine an external, fixe reference frame {x, y, z} an a boy-fixe reference frame, {x c, y c, z c } centere at the robot s center of gravity (CG) so that robot motion is along the x c axis an the z c axis is orthogonal to the boy plane. The terrain can be incline, forming an angle Ψ with the x-axis (pitch angle) an an angle θ with the y-axis (roll angle). The gait optimization proceure is ivie into the following two phases:
May 25, 27 13:6 WSPC - Proceeings Trim Size: 9in x 6in clawar7 3 2.1. First phase: Leg-stroke an foothol calculation In this phase, foot an boy CG positions in the boy plane an the leg stroke that optimize the gait are etermine. Figure 1 outlines a top view of leg workspaces, foothols an leg stroke for a quarupe uring the boymotion phase in a two-phase iscontinuos gait. 5 The four legs are in support, thus propelling the boy forwar. The initial position of the robot has been plotte in thick, soli line, while the final position of the robot has been plotte in thick, ashe line. Let us name x ci an y ci the x c -coorinate an y c -coorinate respectively of foothol i referring to the boy frame {x c, y c, z c } just before the boy-motion phase starts. The CG trajectory has been also plotte (x cg, y cg ), fining the maximum-stability position at the mile of the trajectory, as explaine before. The workspace of leg i is elimite by x cimax, x cimin, y cimax, y cimin. Notice that the length of the boy motion is half the leg stroke, that is, s = 2. Therefore, maximizing s is achieve by maximizing, which can be expresse as a function of front legs foothols: (x c1, x c2 ) = x c2 x c1. (1) The two rear legs foothols can be expresse in terms of the two front legs foothols: x c3 = x c3min + x c2 x c1 (2) x c4 = x c3min + 2x c2 2x c1. (3) The stroke pitch in the y c -axis irection, P y, provies the following relation to foothol y c components: y c2 = y c1 P y (4) y c4 = y c3 P y. (5) To obtain the optimize gait, x c1, y c1, x c2, y c3, x cg an y cg have to be obtaine so that S NDESM is maximum at the mile of the boy-motion phase. This conition will equal the probabilities of losing robot stability when a rear leg is lifte or when a front leg is lifte, thus ecreasing the overall probability of tumbling. Therefore, S NDESM shoul be expresse as a function of foothols an CG position in the boy reference frame, that is (see Ref. 6 for a more etaile explanation): S NDESM = F(x c1, y c1, x c2, y c3, x cg, y cg, z cg ). (6) Taking into consieration that in this first phase of the optimization process the CG height is consiere constant, z cg = h. S NDESM has
May 25, 27 13:6 WSPC - Proceeings Trim Size: 9in x 6in clawar7 4.5.4.3.2 x c1 x c2 x c3 x c4 max S NDESM.6.4.2 y c1, y c3 y c2, y c4 max S NDESM (m).1 (m).1.2.2.3.4.4.6.5 2 1 1 2 Ψ (eg) (a).8 2 1 1 2 θ (eg) Fig. 2. Results of first-phase gait optimization for sloping terrain (a) Effect of pitch angle, (b) Effect of roll angle. (b) to be maximum when the boy CG is locate at the mile of the CG trajectory; therefore the objective function is: ( J 1 = F x c1, y c1, x c2, y c3, x cg + ) 2, y cg. (7) However, another objective function exists, because the leg stroke must be maximum too: with the following constraints: J 2 = (x c1, x c2 ) (8) x c1min x c1 x c1max (9) x c2min x c2 x c2max (1) y c1min y c1 y c1max (11) y c3min y c3 y c3max (12) 4x c1 3x c2 ξ 1 x c1max (13) Constraints (9) to (12) are given by the leg-workspace limits, while constraint (13) avois leg-workspace overlapping, where ξ 1 is a constant. To solve the multi-objective problem, the ε-constraint Metho is use. 7 The problem has been solve numerically for the SILO4 robot in an iterative manner starting from an initial estimate of foot positions x 1 an x 2. Figures 2 an 3 show results of the first phase optimization for the meium-size
May 25, 27 13:6 WSPC - Proceeings Trim Size: 9in x 6in clawar7 5.5.4.3.2 x c1 x c2 x c3 x c4 max S NDESM.6.4.2 y c1, y c3 y c2, y c4 max S NDESM.1 (m) (m).1.2.2.4.3.4.6.5 2 1 1 2 F x (N).8 2 1 1 2 F y (N) (a) Fig. 3. Results of first-phase gait optimization for external forces (a) x component of force, (b) y component of force. quarupe robot SILO4. Figure 2 shows foothols, leg stroke an maximum S NDESM obtaine for a robot walking on sloping terrain. As the terrain angle increases the robot has to moify foothol x c an y c components linearly to ecrease the leg stroke (see ) an increase SNDESM max. Similar results have been foun at the first phase optimization when external forces are applie (see Figures 3(a) an (b)). Note that the irect consequence of the linear reuction of leg stroke for increasing perturbations is a restriction in leg workspace. Thus, the bigger the isturbance, the bigger the reuction of leg workspace. (b) 2.2. Secon phase: CG-height calculation When, as a result of the first phase of the optimization approach the leg stroke is reuce to aapt the walk to a steppe groun or an external force an enhance stability uring the boy-motion phase, stability is reuce uring the transfer phase. This problem can be solve by reucing the CG height. Thus, the following conition is state: F (h) S NDESM = (14) where SNDESM stans for a constant value of the gait stability margin in nominal conitions, that is, over flat terrain when there are no external isturbances. The function F(h) is the result from the first phase of the
May 25, 27 13:6 WSPC - Proceeings Trim Size: 9in x 6in clawar7 6 h (m).25.2.15.1 Fig. 4. force. 5 5 1 1 Ψ (eg) 15 15 θ (eg) h (m).25.2.15.1 5 1 1 F y (N) 15 2 F x (N) (a) (b) Results of secon-phase gait optimization for (a) sloping terrain, (b) external Leg 4 Leg 3 M I Leg 2 F I mg Leg 1 Fig. 5. Quarupe robot SILO4 pulling a 5-N loa optimization proceure, where every gait parameter has been obtaine. Therefore, the only variable at this phase is CG height (h). This problem is a nonlinear equation of a single variable that has been solve by leastsquares optimization. Figure 4 shows results of CG height for the SILO4 walking (a) in sloping terrain an (b) when external forces are applie. 3. Experiments with SILO4 The improvement of gait stability obtaine by the use of the gaitoptimization metho herein propose has been prove experimentally using the SILO4 robot 5,8 shown in Figure 5. A real service application has been simulate. The SILO4 robot walks while it pulls a loa, thus being affecte by an external force F x = 5 N. As a consequence of the perturbation the robot finally tumbles own. Then, the experiment is repeate using the gait-optimization approach, which moifies the gait pattern for the -5-N
May 25, 27 13:6 WSPC - Proceeings Trim Size: 9in x 6in clawar7 7.4 S NDESM (m).3.2.1 5 1 15 2 25 3 35 4 45 Time (s).4 (a) S NDESM (m).3.2.1 5 1 15 2 25 3 35 4 45 Time (s) (b) Fig. 6. Time evolution of S NDESM for the SILO4 robot walking against a 5-N isturbance. (a) Stanar gait, (b) Optimize gait using the propose approach. external force. In this experiment the robot pulls successfully the same loa that mae the robot tumble own in the previous experiment. Figure 6(a) shows the evolution of the stability margin uring the first experiment, when the SILO4 robot walks against the isturbance using a stanar two-phase iscontinuous gait. The stability margin rops uring the transfer phase of the rear legs (because the estabilizing effects of leg transfer an force isturbance combine). Let us compare Figure 6(a) with Figure 6(b) which shows the evolution of the stability margin uring the secon experiment, when the SILO4 robot walks using the results of the gait optimization. As a result of aapting the gait to a -5-N force, the stability margin is symmetrical for the transfer phases of both the rear an the front legs. As a result of the aaptation to a -5-N force, the gait is more stable than in Figure 6(a), increasing the gait stability margin by 7% in this experiment. As shown in the gait optimization results, the use of the propose gait-parameter aaptation approach increases the robot s robustness to external isturbances from a 5-N loa up to a 2-N loa, that is, four times the perturbation in this particular robot. 4. Conclusions A new gait-aaptation metho has been presente in this paper to cope with external perturbations. The metho enables quarupeal gaits to
May 25, 27 13:6 WSPC - Proceeings Trim Size: 9in x 6in clawar7 8 aapt to the slope of the terrain an external forces by fining the gait parameters that maximize robot stability. The resulting foothols of the gait optimization approach are linear functions of external forces an terraininclination angles. This reuces the computation time an enables the application of the gait-optimization metho in real time. Experimental results with the SILO4 robot have shown how robot stability is enhance when the propose approach is use to counteract external forces. The improvement in gait stability margin when applying the propose gait-aaptation metho to the SILO4 robot has been prove to be 7 percent for a -5-N external force. Also, the resulting gait increases the SILO4 robot s robustness to external isturbances from a 5-N loa up to a 2-N loa, that is, four times the perturbation in this particular robot. The propose metho has been shown to be of major relevance for the use of walking robots in fiel an service applications, where robots an environment interact. Acknowlegements This work has been fune by the Spanish Ministry of Eucation an Science through Grant DPI24-5824. References 1. P. Gonzalez e Santos, E. Garcia, J. Estremera an M. Armaa, International Journal of Systems Science 36, 545 (25). 2. T. Doi, R. Hooshima, Y. Fukua, S. Hirose, T. Okamoto an J. Mori, Journal of Robotics an Mechatronics 18, 318 (26). 3. D. Wettergreen an C. Thorpe, Developing planning an reactive control for a hexapo robot, in Proc. IEEE Int. Conf. Robotics an Automation, (Atlanta, Georgia, 1996). 4. H. Tsukagoshi an S. Hirose, Intermittent crawl gait for quarupe walking vehicles on rough terrain, in Int. Conf. Climbing an Walking Robots, (Brussels, Belgium, 1998). 5. P. Gonzalez e Santos, E. Garcia an J. Estremera, Quarupeal Locomotion: An Introuction to the Control of Four-Legge Robots (Springer-Verlag, Lonon, 26). 6. E. Garcia an P. Gonzalez e Santos, IEEE Transactions on Robotics 22, 124 (26). 7. V. Chankong an Y. V. Haimes, Multiobjective Decision MakingTheory an Methoology (Elsevier, New York, 1983). 8. Inustrial Automation Institute, C.S.I.C., Mari, Spain, The SILO4 Walking Robot, (22). Available: http://www.iai.csic.es/users/silo4/.