Feedforward and feedback dynamic trot gait control for a quadrued walking vehicle Ryo Kurazume University ofokyo 7--1, Roongi, Minato-ku, okyo, Jaan Shigeo Hirose Kan Yoneda okyo Institute of echnology -1-1, Ookayama, Meguro-ku, okyo, Jaan Abstract o realize dynamically stable walking for a quadrued walking robot, the combination of the trajectory lanning of the body and leg osition (feedforward control) and the adative attitude control using sensory information (feedback control) is indisensable. In this aer, we initially roose a new trajectory lanning for the stable trot gait named 3Dswaycomensation trajectory, and show that this trajectory has lower energy consumtion than the conventional sway trajectory that the authors have roosed. Net, as the adative attitude control method used during the - leg suorting hase of the trot gait, we consider four methods: a) rotation of body along the diagonal line between suorting feet, b) translation of body along the erendicular line between suorting feet, c) vertical swing motion of recovering legs, and d) horizontal swing motion of recovering legs. he stabilization efficiency of each method is verified through comuter simulation and the daming eeriment using a quadrued walking robot, IAN-VIII. Furthermore, the dynamic trot gait control that combines the feedforward control based ontheroosed 3D sway comensation trajectory and the adative feedback control using body translation and vertical motion of swing legs is develoed, and the walking eeriment on rough terrain using IAN-VIII is carried out. 1 Introduction o increase the walking seed and the energy efficiency of a quadrued walking vehicle, it is indisensable to realize dynamically stable walking that has a hase when the vehicle body is suorted by less than legs and the body attitude becomes unstable. Within a trot [],[3],[],[7], ace[1], and bound [13],[1],[] gaits that are fundamental dynamically stable gaits, the authors have studied the trot gait in aticular. his gait is attractive because it has a close affinity to the crawl gait, which is one of the standard statically stable gaits, and can be classified as a safety gait" [1] that avoids comlete tumbling by touching a swing leg to ground. hen, the authors have roosed a generalized trot gait [],[3] that can smoothly shift from the crawl gait to the trot gait in roortion to walking seed, and an intermittent trot gait [] that makes the diagonal legs as swing and suort legs simultaneously to reduce dynamic effect of the recovering swing legs on the body. Furthermore, the sway comensation trajectory of the vehicle body [], which uses lateral body motion to kee a zero momentum oint (ZMP) on a diagonal line between the suort legs, was roosed and the effectiveness of this trajectory control to realize a dynamically stable walk was verified through walking eeriments using IAN-IV and IAN-VI. In the environment where the unknown roughness or inclination eists, however, dynamically stable walking is difficult using feedforward gait control such as the sway comensation trajectory only, and the adative feedback control system that stabilizes the body attitude using mounted attitude sensors, gyroscoe sensors, etc. is requisite. In this regard, for the realization of the ractical dynamically stable trot gait for a quadrued walking vehicle, authors believe that the combination of feedforward control based on the sway comensation trajectory and feedback control based on the adative attitude control is one ossible effective control method. he basic ideas underlying this feedforward and feedback dynamic trot gait control are as follows: First, the state of the system is generously transferred adjacent to the unstable equilibrium oint by off-line, feedforward gait lanning such as the sway comensation trajectory. hen, the remaining small deviation from the equilibrium oint is adatively comensated by a simle, linear feedback control system. In this aer, first, we roose the new sway trajectory of the vehicle body named 3D sway comensation trajectory that uses lateral, longitudinal, and vertical motion of the vehicle body to kee ZMP on the diag-
onal line of the suort legs. With the roosed trajectory, it is ossible to imrove the energy efficiency against the conventional sway comensation trajectory that controls the osition of ZMP by the lateral motion only. Net, as the adative feedback control method, several attitude control methods using the body translation and swing leg waving motion are roosed. he control erformance of each method is eamined through comuter simulation and daming control eeriments in the two-leg suorting hase using the quadrued walking vehicle named IAN-VIII [11]. Furthermore, the feedforward and feedback dynamic trot gait control system that combines the 3D sway comensation trajectory and the adative body osition and swing leg motion control is develoed and the walking eeriment on rough terrain using IAN- VIII is carried out. 3D sway comensation trajectory.1 Conventional sway comensation trajectory First, we show the formulation of the conventional sway comensation trajectory. Here, we consider a vehicle as a mass oint at( g ;y g ;z g ). If the ground is flat and the height of the body from the ground, z, is constant, the osition of ZMP ( z ;y z ; ) is given as z y z = g y g A ẍg ÿ g (1) where, A = zg. Net, the diagonal line of the suort g legs is defined as cos + sin y= d () hen, in order for ZMP to kee on this line, the center of gravity has to be satisfied cos ( g Aẍ g ) + sin (y g Aÿ g )=d (3) Assuming that the osition of the body along the ais is eressed as g = + vt + 1 at () By substituting this equation into Eq.(3), we get cos ( Aa + vt + 1 at ) + sin (y g Aÿ g )=d () he solution of this differential equation, y g, is given as y g = C y 1 e t A + y C e t A + y a t + a y 1 t + ay (6) From the boundary condition for the continuity of the trajectory ( _y g;t= = _y g;t= =;y g;t= = y g;t= ), each coefficient is determined as C y 1 = A cot +(1 a e A )v (e A e (7) A ) C y = A cot +(1 a e A )v (e A e (8) A ) a y = a cot (9) a y 1 = v cot (1) a y = cot + d csc (11) = d cos + 1 ( A (e A + e A +) (e A e )a A ) v (1) Where, is a walking cycle. his trajectory of the center of gravity, whichisgiven as Eqs.() and (6), is defined as the (conventional) sway comensation trajectory [].. Eansion for longitudinal motion he conventional sway comensation trajectory mentioned above can be eanded to include a sway toward the longitudinal direction. First, Eq.(3) is decomosed into two equations for the and y directions, and each solution trajectory is assumedtobegiven as Eq.(6) and g = C 1 e t A + C e t A + a t + a 1 t + a (13) By substituting the boundary condition about the continuity of the trajectory, the following equations with two arameters a and a 1 are derived. C 1 = ( + A )a +a 1 L 8(e A 1) C = ( A )a +a 1 L 8(e A 1) (1) () C y = 1 A cot a +(1 e A )a 1 (e A e (16) A ) C y = A cot a +(1 e A )a 1 (e A e (17) A )
a y = a cot (18) a y 1 = a 1 cot (19) a y = a cot + d csc () a = d cos +1 ( A (e A + e A +) (e A e A ) (1) Where, L is the body stroke inonewalking cycle..3 Eansion for vertical motion )a a 1 Moreover, the above equations can be eanded to the form including a sway in the vertical direction. Considering A = zg,we assume the trajectory to g+ zg the vertical direction, z g,isgiven as z g = C z 1 e t A + C z e t A + Ag () Where, A is an arbitrary constant. By substituting the boundary condition about the continuity of trajectory, coefficients with the arameter A are derived as C z C z 1 = Ag H 1+e A = (Ag H)e A 1+e A (3) () Where, H is the body height att =;. We define these eansions of conventional sway comensation trajectory toward longitudinal and vertical direction as the 3D sway comensation trajectory".. Energy efficiency of the 3D sway comensation trajectory he energy consumtion of a walking vehicle is affected by many factors such as the mass of body and legs, the configuration of the degrees of freedom, the trajectory of body and legs, the negative ower at each actuator, etc [9],[1]. In this aer, however, the trajectory that minimizes the sum of squared acceleration through the entire trajectory is considered. ρ = Z (ẍ g +ÿ g + z g )dt () Here, a regular walk (a = ) is considered for simlicity. Simulation results that minimize the sum of squared acceleration are shown in able 1. Here, = able 1: Minimum of squared acceleration lateral lateral and lateral, only longitudinal longitudinal, and vertical ρ.63.33.331 a 1..17.17 A...7 foot osition diagonal line.1. -...1. g -. -.1 y g z g. moving direction foot ositions moving direction foot osition rajectories with longitudinal motion and with vertical motion lateral motion only (conventional trajectory) with vertical motion.1. g -...1. lateral motion only (conventional trajectory) and with longitudinal motion Figure 1: rajectories of vehicle body 1[s];H =:[m];l =:[m]; =3[deg], and d =. Figure 1 shows the obtained body trajectories for the conventional sway comensation trajectory, the eansion in the longitudinal direction, and the longitudinal and vertical direction, resectively. In Fig.1, the uer figure shows the trajectories rojected on the ground (-y lane), and the lower figure shows the trajectories rojected on the -z lane from the lateral direction in which the vertical ais is magnifiedtimes. From these figures, it is verified that the sum of the squared acceleration can be reduced by swaying not only in the lateral direction but also in the longitudinal and vertical directions.. Comuter simulation using the dynamic motion simulator, ADAMS o verify the energy efficiency of the 3D sway comensation trajectory derived in section. in the actual robot system, the comuter simulation using the dynamic motion simulator, ADAMS, is carried out. he configuration of degrees of freedom, weight, etc.
Leg Leg 1 (a) Rotation of body (b) ranslation of body Joint Leg 3 Leg Joint 1 Joint 3 Figure : Simulation model for ADAMS able : Comarison of energy consumtion lateral lateral and lateral, straight only longitudinal longitudinal, and vertical E [J] 1. 11.6 18.6 96.8 ffl.3 1.9 1.8 1.6 of a quadrued walking vehicle model for comuter simulation is the same as the IAN-VIII [11] develoed in our laboratory. Figure show the simulation model. he sum of ower consumtion at each joint, E, and secific resistance, ffl, [6] are shown in able. In the simulation, the walking cycle is 1 [s], walking seed is. [m/s] and duty factor is.. he columns show the results of the conventional sway comensation trajectory, the trajectory including lateral and longitudinal sway, the trajectory including lateral, longitudinal, and vertical sway, and straight line with no sway control. he trajectories that minimize the sum of squared acceleration derived in section. are used for the 3D sway comensation trajectories. Note that in case that the body moves on a straight line with no sway control, the body attitude cannot be maintained to be arallel to the ground, and the swing leg touches the ground uneectedly. From these results, ower consumtion and secific resistance can be imroved by the3dsway comensation trajectory that minimizes the sum of squared acceleration. 3 Adative attitude control As mentioned above, in the environment where the unknown roughness or inclination eists, dynamically stable walking is difficult using only the feedforward gait control such as the 3D sway comensation trajec- (c) Vertical motion of swing legs (d) Horizontal motion of swing legs Figure 3: Four attitude control methods tory. In addition, for eamle, when an etended trot gait is alied, the body attitude sometimes becomes unstable and oscillates due to the dynamic effect of the recovering motion of the swing legs []. herefore, a feedback control system using mounted attitude sensors, gyroscoe sensors, etc. which adatively corrects body attitude are required. In this section, we roose the attitude control method during -leg suorting hase using the body translation and rotation, and swing leg waving motion. 3.1 Attitude control methods and comuter simulation o roduce the required moment for attitude recovery in the two-leg suorting hase, the method using rotation of the body along the diagonal line between the suort legs as shown in Fig.3(a) has been roosed [16]. he required moment for attitude recovery, however, can be roduced by the translation of body osition as shown in Fig.3(b). Furthermore, though the recovery motion of swing legs has been considered as disturbance that induces the oscillation of body [], by controlling the recovery ath aroriately, swing legs can be used for the daming control of body attitude. From the above discussion, four attitude control methods as shown in Fig.3 are considered in this aer. (a) Rotation of body along the diagonal line between the suort legs. (b) ranslation of body along the direction erendicular to the diagonal line between the suort legs. (c) Vertical motion of swing legs during recovery. (d) Horizontal motion of swing legs during recovery. First, each method is simlified as a three-link model corresonding to a suort leg, a body, and a swing leg as shown in Fig., and motion equations of each model and otimum linear regulators are designed to reress the oscillation of the body. Fig. shows e-
Link 3 1 Link Joint 1 Link 1 (a) Rotation of body Joint Link 3 1 Joint Joint 1 (b) ranslation of body able 3: Comarison of ower consumtion for attitude control method a b c d sum of ower [w].7 1. 1. 1. 1 Joint Joint 1 1 Joint Joint 1 Attitude sensor (c) Vertical motion of swing legs (d) Horizontal motion of swing legs Figure : Analysis models 1 8 6-6 8 1 ime [sec.] 8 6 1 1 6 6 8 1 ime [sec.] (a) Rotation of body 1 8 6-6 8 1 ime [sec.] 8 6 8 1 ime [sec.] (c) Vertical motion of swing legs [mm] 1 - - 1-1 - 6 8 1 ime [sec.] 6 8 1 ime [sec.] (b) ranslation of body 1 8 6-6 8 1 ime [sec.] 8 6 1 6 8 1 ime [sec.] (d) Horizontal motion of swing legs Figure : Simulation results amles of comuter simulation of the designed otimum linear regulators when the initial conditions are ffi 1 =[deg:],ffi =[deg:], and =[deg:] From Fig.(a) using the rotation of body, the body has to be inclined u to toward the inclination direction to recover the inclination of the suort leg. Fig.(b) using the translation of body shows that maimum body movement to recover the inclination of the suort leg is. [m] hough this body motion can be eecuted by IAN-VIII, the inclination of the suort leg becomes vibrational. Net, in Fig.(c) and (d), the maimum angles of the swing legs become 8 and 7, resectively. hus, it is difficult to stabilize the body attitude using the effect of vertical and horizontal motion of the swing legs only because of the limitation of the actual movable joint angles. Gyrosensors Figure 6: -legged walking robot, IAN-VIII o comare the energy efficiency for these methods, the sum of ower consumtion was calculated. able 3 shows the comarison of ower consumtion for each method. he method (a) was the least ower consumtion. On the other hand, the method (b) was the worst and it needed ten times larger ower consumtion than method (a). 3. Daming control eeriment in two-leg suorting hase he daming control eeriment using four attitude control methods roosed in Section 3.1 was carried out with the quadrued walking robot named IAN- VIII shown in Fig.6. his robot is equied with a comuter board (Pentium MHz, Jaan Data System), AD/DA boards, Ethernet card, silicon disk, 3- aes attitude sensor, (Macube, Jaan Aviation Electronics), and two gyrosensors (Gyrostar, Murata). In the eeriment, the body of IAN-VIII standing with two suort legs was tilted by an eternal force alied to the body by ushing by hand. he body's attitude return to the stable state was measured for each control method after the hand is released. he ankle of IAN-VIII is restricted mechanically to let the sole be arallel to the body. hus, as shown in Fig.7, the body attitude returns to the stable state without an attitude control if the inclination angle is smaller than about 8. he moment for the recovery of body inclination is roduced by the soles. For the urose of increasing the dynamic effect of the
1 1 - -1-6 8 1 ime [sec.] Figure 7: Eerimental result (no sensor) 1 1-1 - 1 3 6 ime [sec.] 6 - - -6 1 3 6 ime [sec.] 1 - -1-1 [mm] 1 - -1-6 8 1 ime [sec.] 6 8 1 ime [sec.] (a) Rotation of body (b) ranslation of body 3 (a) Recovering ath is erendicular to ground (b) Recovering ath is inclined with resect to ground. Figure 8: Recovering ath of swing legs 1 - -1-6 8 1 ime [sec.] 1 - -1 6 8 1 ime [sec.] (c) Vertical motion of swing legs 1 1 1 - -1-1 - -1-6 8 1 ime [sec.] 6 8 1 ime [sec.] (d) Horizontal motion of swing legs swing legs to the body, the swing legs are stretched laterally in the middle of the return ath as shown in Fig.8(b) and the inertia of the legs is increased. Inclination angle of the suort leg along the diagonal line between the suort legs, ffi 1, and control variables ffi and for each control method are shown in Fig.9. Here, ffi in Fig.9(a) is the body rotation angle, in Fig.9(b) is the body dislacement, and ffi in Figs.9(c) and Fig.9(d) is the angle of joint in Fig. calculated from the amount of swing motion. Fig.1 shows the daming control eeriment using the vertical motion of the swing legs. In this eeriment, the swing legs are swung u during the recovery of body attitude. From the eerimental results in Fig.9(a) using the rotation of body, the swing leg of the leaning side contacted the ground because the body was tilted more and more, and large body vibration was generated by the reaction force. Net, for the method using the translation of the body shown in Fig.9(b), though the body attitude was vibrational, it finally converged to horizontal. From the comarison of Fig.9(c) and Fig.7, which are the results for the control method using vertical motion of the swing legs and without an attitude control, erformance of convergence was clearly imroved by the dynamic effect of swinging. However, the swing width of the swing leg is limited in the joint movable area, and thus, recovery from a large tilt angle to horizontal is difficult using only this method. On the other hand, Fig.9(d) which shows using the horizontal motion of the swing legs indicates the convergence erformance is hardly imroved. his is because the dislacement of actual Figure 9: Eerimental results of attitude control for slow trotgait Figure 1: Eeriment of attitude control using swing legs links was smaller than the case using vertical motion of the swing legs, and the dynamic effect is small. hese results suggest that the attitude stabilization erformance might be the highest by combining the vertical motion of the swing legs and translation of the body. Fig.11 shows the eerimental result when these two methods are simly suerimosed. In comarison with Fig.11, Fig.7 and Fig.9(b), both convergence erformance and stability are imroved. Dynamically stable walking eeriment with IAN-VIII he dynamic trot gait control method, which emloys feedforward control using the 3D sway comensation trajectory and feedback control using vertical motion
1 - -1-6 8 1 ime [sec.] 1 1 - -1 6 8 1 ime [sec.] [mm] 1-1 - 6 8 1 ime [sec.] Figure 11: Eerimental result (Combination of translation of body and vertical motion of swing legs) of the swing legs and translation of body, was alied to IAN-VIII, and the walking eeriment on rough terrain was carried out. In the eeriment, IAN-VIII walked with the dynamically stable trot gait in an environment where an unknown and leaning ste eists, and the attitude stabilization erformance was eamined. Fig.1 is a series of hotos of the eeriment, and Fig.13 shows the inclination angle of the suort leg around the diagonal line between the suort legs. In this eeriment, the duty factor is., the walking cycle is 1[s], and the walking velocity is. [m/s]. he hotos on the left in Fig.1 show the results using the 3D sway comensation trajectory and the roosed attitude stabilization control. he hotos on the right show the results using the 3D sway comensation trajectory only. Fig.1 shows the return ath of the swing legs. he horizontal ais is the walking direction, and the vertical ais is the vertical direction. By using swing leg control only in the region where the height of the swing leg from the ground is larger than a secific height ( [cm]), the body vibration caused by the contact of the swing leg to the ground is revented. From Fig.13, dynamically stable walking with small attitude fluctuation can be erformed by the 3D sway comensation trajectory on a flat surface in ο 1[s]. In 1ο[s], one of the right legs runs on the ste. For the case without attitude control, the body gradually inclined and the swing leg contacted the ground uneectedly as shown in the fourth hoto of the right column in Fig.1. However, for the case with the roosed attitude stabilization control, maimum inclination of the body was about 8 degrees. As shown in Fig.11, if the inclination of the body is smaller than about 1 degrees, the body attitude can be recovered by the roosed attitude control method. hus, dynamically stable walking was realized even in an en- (a) attitude control (b) no sensor Figure 1: Dynamically stable walking eeriment on rough terrain vironment where an unknown and leaning ste eists, and the effectiveness of the roosed attitude stabilization control was confirmed. Conclusion In this aer, the 3D sway comensation trajectory, an eansion of the conventional sway comensation trajectory toward longitudinal and vertical motion, is roosed. he 3D sway comensation trajectory enables keeing ZMP on the diagonal line of the suort legs more efficiently with less energy consumtion. Net, four adative attitude control methods using the body osition and swing leg motion control are roosed, and the daming erformance of each method was comared through comuter simulation and daming control eeriments with IAN-VIII. Furthermore, the feedforward and feedback dynamic trot gait control system that combines the 3D sway comensation trajectory and the adative body osition and swing leg motion control is develoed and the walking eeriment on rough terrain using IAN-
Inclination - -8 3D sway comensation trajectory and attitude control -1 3D sway comensation trajectory only -16 1 3 3 ime [sec.] Figure 13: Body inclination of robot walking dynamically on rough terrain z [m].1 with vertical motion of swing leg.1. [m] Figure 1: Recovering ath of swing leg VIII is carried out. Results of the eeriment showed that this control methodology was very effective for ractical use and could make dynamically stable walking of walking vehicle ossible on rough terrain. References [1] S. Hirose and K. Yoneda,oward the Develoment of Practical Quadrue Walking Vehicles,J. of Robotics and Mechatronics,vol.,No.6,.98-,1993 [] K. Yoneda and S. Hirose,Dynamic and static fusion gait of a quadrued walking vehicle on a winding ath,advanced Robotics,,,.- 136,199 [3] S. Hirose, K. Yoneda, R. Furuya, and. akagi,dynamic and static fusion gait of a quadrued walking vehicle,proc. of IEEE/RSJ Int. Conf. on Intelligent Robots and Systems '89,,,.199-,1989 [] K. Yoneda, H. Iiyama, S Hirose,Intermittent rot Gait of a Quadrued Walking Machine Dynamic Stability Control of an Omnidirectional Walk,Proc. Int. Conf. on Robotics and Automation,,,.3-37,1996 [] S. Hirose,A Study of Design and Control of a Quadrued Walking Vehicle,Int. J. Robotics Research,Vol.3,No.,.113-133,198 [6] G. Gabrielli and I. von Karman,What rice seed?,mechanical Engineering,Vol.7,No.1,.77-781,19 [7] K. Yoneda, H. Iiyama, and S. Hirose,Skyhook susention control of a quadrued walking vehicle,proc. Int. Conf. on Robotics and Automation,,,.999-1,199 [8] H.Kimura, I.Shimoyama and H.Miura, Dynamics in the dynamic walk of a quadrued robot, RSJ. Advanced Robotics, vol., no.3,.83-31, 199 [9] D. W. Marhefka and D. E. Orin,Gait Planning for Energy Efficiency in Walking Machines,Proc. Int. Conf. on Robotics and Automation,.7-8, (1997). [1] K. Arikawa and S. Hirose, Study of Walking Robot for 3 Dimensional errain,proc. of IEEE/RSJ Int. Conf. on Intelligent Robots and Systems '9,.73-78, (199). [11] K. Arikawa and S. Hirose,Develoment of Quadrued Walking Robot IAN-VIII,Proc. of IEEE/RSJ Int. Conf. on Intelligent Robots and Systems '96,,,.8-1,1996 [1] A. Sano and J. Furusho,Dynamically Stable Quadrued Locomotion (A Pace Gait in he COL-3),Proc. of the Int. Sym. on Industrial Robots,,,.3-6,1989 [13] M. H. Raibert,Legged Robots hat Balance,MI Press,,,,1986 [1] J. Furusho, A. Sano, M. Sakaguchi, and K. Honda,Bounce Gait Control of a Quadrued Robot,Proceedings of the Second International Conference on Motion and Vibration Control,Vol. 1,,.198-3,199 [] J. Furusho, A. Sano, M. Sakaguchi and E. Koizumi :,Realization of Bounce Gait in a Quadrued Robot with Articlar-Joint- ye Legs,Proc. Int. Conf. on Robotics and Automation,,,.697-7,199 [16] M. Hiraki,. Emura, Y. Senta, and S. Okada,rotting Gait of a Quadrued Robot Based on Reaction Wheel Model (in Jaanese), Proc. of 1th Conference of the Robotics Society of Jaan,,,.967-968,1996
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