neural computation for adaptive gait control of the quadruped over rough terrain

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  • FYocding of thc 1992 EEE Intomitiad Conference QI R ~ b t i ~ and Autanrtim

    h a ? - May 1992 Neural Computation for Adaptive Gait Control of

    the Quadruped over Rough Terrain

    Byungeui Min and Zeungnam Bien Dep. of Electrical Engineering, KAIST

    P.O.BOX 150, Chungyangni, Seoul, KOREA

    Abstract An adaptive gait control problem for a quadruped is formalized and an adaptive gait algorithm for eliminating dead-lock positions is proposed. The proposed method is based on the concept of a neural optimization network. The desired footholds are taken as the output variables of the neural circuit, and differential equations which tend to minimize the energy of the network are derived. An efficient method to make the quadruped statically stable is ad- dressed. Several conditions for satisfying the geometric limits of reachable volumes are introduced. The mobility is improved by maximizing the stride which depends on the direction of motion and the configuration of the quadruped.

    I. Introduction Even though the distinctive characteristic of legged machines is good off-load mobility, the complexity involved in the design of gaits over rough terrain has limitted the attempts to increase adaptibil- ity to terrain irregularities; Kugushev and Jaroshevskij[3] intro- duced a nonperiodic gait termed as a free gait. and subsequently McGhee and Iswandhi[2] relined the algorithm. More recently, Pal and Jayarajan[4] suggested a free gait algorithm for the quadruped based on a heuristic graph search. Hirose[l] proposed the three diagonal principles which provide a criterion for foot positioning while avoiding the deadlock. However, the solutions proposed to date to solve the adaptive-gait-control problem suffer from one or more of the following difficulties: 0 Due to the complexity of the vehicle-terrain system, the mod-

    eling and the development of mathematicalequation descrip- tion are hard problems.

    0 It is difficult to describe rough terrain in a mathematical form. Hence many researchers parcelled the terrain into rectangular cells. But this discrete scheme may result in combinatorial complexity.

    0 Since the algorithms mainly depend on heuristic approaches, more efforts for improving computational efficiency are re- quired.

    0 The lack of preparation for adapting to an advanced motion

    The adaptive-gait-control algorithm proposed in this paper allevi- ates these difficulties by a systematic approach based on neural optimization network[6.7]. As can be seen in the works[6]-[9], the neural optimization network provides computational efficiency es- pecially for the problems in which either the system is

    trace decreases the mobility.

    not easily modeled or there are many variables in modeling. To complete the design of a gait, the following three steps

    should be incorporated. The first step is the design of a sequence of leg movements for which a gait diagram is widely used. The sec- ond is the design of a motion trace which means the time-dependent position and orientation of the moving body coordinate frame. The final step is the selection of footholds.

    In the paper, we focus our major attention on the problem of foothold selection. The direction of motion of the second step is assumed to be given by the higher level in control architecture [1.2.5]. Then the maximum stride which can be traversed in the current configuration is calculated. For the design step of a gait diagram, we use the canonical crab gait which renders the optimal longitudinal gait stability margin for a given crab angle in case of a quadruped.

    II. Preliminaries The purpose of this section is to present some preliminary material which supports the development of the algorithm in the next sec- tion. Definitions of terminologies and notational conventions used in the paper are given in the following: DEFINITION 1: The crab angle a is the angle from the longitu- dinal axis to the direction of motion, which has the positive mea- sure in the direction of counterclockwise. DEFINITION 2: A normalized time t is the time normalized with respect to the cycle time. In this paper, we will denote tp (i) and r,(i) as normalized times at which leg-i is placed and lifted, respectively. Further, we will denote ti(i) and tl'(i) as nor- malized times immediately before leg-i is placed and immediately after leg-i is lifted, respectively. According to this definition, the normalized cycle time is unity. DEFINITION 3: Thefront longitudinal stability margin s: and the rear longitudinul stability margin Sf. are the distances from the vertical projection of the center of gravity to the front and rear boundaries of the support pattern. respectively, as measured in the direction of x-axis. The shorter of these two is the longitudinal stability margin Sr. The longitudinal gait stability margin S for a periodic gait is the minimum Sr over an entire cycle of locomo- tion. DEFINITION 4: The symbols f, r. 7, r are used to indicate legs relative to a reference leg. The letters f and r denote a front and a

    2612 0-8186-2720-4/92 $3.00 81992 IEEE

    Authorized licensed use limited to: Khajeh Nasir Toosi University of Technology. Downloaded on December 21, 2009 at 05:55 from IEEE Xplore. Restrictions apply.

  • Authorized licensed use limited to: Khajeh Nasir Toosi University of Technology. Downloaded on December 21, 2009 at 05:55 from IEEE Xplore. Restrictions apply.

  • position of center of gravity. But only the constant speed locomo- tion is considered, we complete the design of the vehicle's trajec- tory by deciding the stride length which is the traveling distance of vehicle's center in one locomotion cycle. Since the vehicle's speed is directly proportional to the stride, it is desirable to maximize the s ~ d e length. Let Axi(k) be the distance from the initial foothold of leg-i in the k'th period to the rear boundary of the reachable volume.[See Fig.11 Then Axi(k) is given by

    Similarly let Ayi(k) be the distance from the initial foothold of leg-i to the side boundary of the reachable volume in the reverse ydirection of locomotion.

    Ayi (k ) =sign(a) yi(k)

    1 S a 2 0 -1 else

    where sign(a) = (

    Then maximum stride in xdirection is given by

    kJ =min for i =1,2,3,4

    Further ydirectional maximum stride is given by

    where r ( i ) comes of the pre-planned gait diagram for which any gait can be used, e.g. the wave gait[l]. The nomalized time in- stances at 4 lifting events of the canonical crab gait was given in Eq45). But in case that y-directional maximum stride nb, is less than the value A++.. tan a, it is impossible to travel along the given crab axis. Therefore, Eqs.(l2) and (13) should be modified as the followings:

    = min {U, a) for i=1,2,3,4 (14) t l i ) t l i ) . tan 01

    ky= hr- tan a (15)

    B. The mobility In the previous subsection, we maximize the stride of the k'th lo- comotion cycle, which depends on the configuration of the robot at the starting time of the locomotion cycle. By the similar concept, we now determine the desired footholds which give the maximum stride of the (k+l)'th locomotion cycle. For this purpose, we have to know the the crab angle of the (k+l)'th period. As stated earier the crab angle is given by the motion planner and the canonical crab gait is utilized to optimize the stability for a given crab angle stride of the (k+l)'th locomotion cycle. For this purpose, we have to know the the crab angle of the (k+l)'th period. As stated earier the crab angle is given by the motion planner and the canonical crab gait is utilized to optimize the stability for a given crab angle.

    &y + Rm- tlli)/l for i=1,2 m:= {

    ky - R ~ . (1 - rlt(i)p) for i=3,4 (16) w +(-lr . R,,

    2 m , J = ~ ~ + ~ , ~ . t ~ ~ i ~ ~ + ( - 1 ~ - ~ .

    for i=1.23,4 (17) where tlyi) is the normalized lifting time of leg-i in the (k+l)'th period. Ordered pair (mc,m,J) gives the desired foothold of leg-i for maximizing the stride of the (k+l)'th period for the case of even terrain.

    C. Static Stabiliry In this section, an efficient method for maintaining positive

    static stability in one locomotion cycle is introduced. If a line segment lies inside a convex polygon, then the mini" distance from the line segment to the boundaries of the convex polygon is given at the one end of the line segment. Hence, we can state that the minimum stability during the transfer phase of each leg appears immediately after a leg is lifted or immediately before a leg is placed. In other words, to maintain the positive static stability, the vehicle's center should be located above the rear boundary at the time instance of lifting event, and be located below the front bound- ary at the time instance of placing event. As can be seen in the model of Fig.1, the front footholds should always be located above the vehicle's center. Therefore, vehicle's center is always below the front boundary formed out of front legs. By the similar reason, vehicle's center is always above the rear boundary formed out of rear legs. Hence, it is enough to check the stablity condition of the followings: Sgr?(r)l, sf6tdf)], s%ti+(;)] and S$td?)]. The corre- sponding bounaries are given as follows:

    S$t

  • transferred to the rear boundary of the reachable volume should be greater than the traveling distance of vehicle's center which must be traversed in the second support phase[fdi), 11. This can be expressed

    Some arrangements yield

    AX(^) - &y. tdi) + AiT 2 (1 - tdi)). &y

    AT = &y - dxlk) (26)

    (27) From Eq.(25) and Eq.(27), we get the following:

    x ( k ) + Sx(k+l) X ( k ) + Ay for i=1,2,3,4 (28) Similarly we can derive the y-directional constraint of each leg as follows:

    y (k )+Ayly (k+ l ) ly (k )+ Ay for i=1,2,3.4 (29)

    Fig3 Situation of Eq.(19)

    Here t l runs from zero with the intersection point at CLO] to unity with the intersection point at Cdl]. But the stability margin is zero when the intersection occurs at tdf) or t{i). To increase the stabil- ity margin. the interval [tdf), tl+(F)] may be modified as follow:

    (22) rJf) + 01 I tl I qyi) - oI where 01 E [0, 0.5 (I{;) + tdf)}]

    By the similar method. the intersection between the line L(Pik+l), Pi(k+l)) and the c.0.g. trajectory can be calculated by utilizing the following variable:

    xdk+i)- dk+i))(cgq - dk+i))- (&+1)- dk+i))(cf[q- &+I)) (23) ( 4 1 1 - cAOl)(Yip+t+l)- &+I)) -(cfi11- C~o])(xdk+l)- dk+l))

    ti?)+ 0 2 1 t z I tl'

  • and

    To prevent footholds from being selected in the forbidden area, the following energy is defiied.

    4 Ni Eo =E F(IP(k+I) - o ~ J - rij)

    (36) where the notation 1.1 denotes an Euclidean norm, andhri is the number of obstacles located under the reachable volume of leg-i. From the characteristics of the network, we get the energy;

    i=l j4

    E , , = E [ k I o &+1) ~ d v + ; / ~ &+I) vdv]

    (37) i=l

    Combining these energies with corresponding weights, we can cal- culate the total energy as follows:

    (38) Differentiating it with respect to time, and employing the capaci- tance and the resistance, we obtain the network equations as fol- lows:

    E = Wm. E, + Ws. E, + Wv* Ev + WO. Eo + W.. E.

    + W,,. x(k+l) (for i = 1,2,3,4) R (39)

    + W". y(k+l) - (for i = 1.2.3,4) R

    Iv. Simulation Simulation results are shown in Fig.3. where the area inside circle represents the forbidden region which is not suitable for foot place- ment. The geometric limits of reachable volumes in Eqs.(28) and (29) are drawn by the rectangles inside which each desired foot- hold should be placed. The situations of Fig3.(a) and Fig3.(b) are same except initial foot positions. The area and the location of each rectangle depends upon the time instances of each leg's W i g and placing events and the vehicle's traveling distance in the locomo- tion cycle.The crab angle is changed in Fig3.(c). As can be seen in the results, the network always gives positive stability and prevents each foothold from beiig located in forbidden region. In the case of no obstacle, the solution converges to the optimal crab gait. Hence the algorithm is applicable to both even and rough terrains without gait transitions.

    output of the network

    geometric limits of reachable volume ,

    i : I I

    crab angle of current period : 20" crab angle of next period : 20"

    ;b'l="= 87.9[cm], &y=32.O[cm] S=5.l[cm], /3=0.89

    Fig3(a) Simulation result

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  • crab angle of current period : 20 " crab angle of next period : 20' ;by = 102.6 [cml. ;by = 37.3 [cml S = 6.5[~m]. B= 0.89

    Fig3(b) Simulation result

    crab angle of current period : 10" crab angle of next period : 20"

    ;by= 88.7[m]. ;by=15.6[cm] S = 5.5[cm], p= 0.89

    V. Conclusion and Further Study We proved that the neural optimization network can give a good solution for an adaptive-gait control of the quadruped which is considered too complex in modeliig and fiiding solution. In the course of constructing the network, we introduced some new and efficient methods in terms of the stability and the mobility, etc. The mathematical description of the vehicle-terrain system is great- ly simplified by employing the pseudo world coordinate frame as a reference frame. The characteristic of the network may provide a clue to understand the nervous system of living bodies, the flex- ibility and the adaptibility of which are greatly superior to those of the current robots.

    Since the focus of the paper was placed on the problem for- malization using the neural network. the proposed method requires more investigations especially for resolving the local minimum problem, which is generally troublesome in applications. Further, an extension of the algorithm to the problem of global path plan- ning(navigati0n problem) may be expected.

    References [l] Hirose. S., "A study of design and control of a quadruped

    walking vehicle" Int. J. of Robotics Reasearch. Vol. 3. No. 2, 1984.

    [2] McGhee, R.B. and Iswandhi, GI., "Adaptive locomotion of a multilegged robot over rough terrain" IEEE Trans. on Sys- tems, Man and Cybemetics. vo1.9, No.4, 1979.

    [3] E.I.Kugushev and V.S. Jarwhevskij, "Problems of selecting a gait for an integrated locomotion robot" Proc. Fourth Int. Conf. Artificial Intelligence, Tbilish, Georgian SSR, USSR, 789-793, Sept. 1975.

    [4] Prabm K. Pal and K.Jayarayan. "Generation of Free Gait - A Graph Search Approach." IEEE Tramon Robotics and Au- tomation, vo1.7, No.3, June 1991.

    [5] David E.Orin. "Supervisory Control of a Multilegged Robot" Int. J. of Robotics Reasearch. Vol. 1, No. 1,1982.

    [6] J.J. Hopfield and D.W. Tank, "Computing with Neural Cir- cuits: A Model'' Science 233,625-633.1986.

    [7] D.W.Tank and J.J.Hopfield, " Simple Neural Optimization Net- works: An A/D Converter, Signal Decision Circuit and a Lin- ear Programming Circuit" IEEE Trans. Circuits Systems CAS- 33, No.5,533-541,1986.

    [8] K.Tsutgumi and H.Matsumoto,"Neural Computation and Learn- ing Strategy for Manipulator Position Control" Int. Conf. Neu- ral Networks Vol.IV, 525-534.1987.

    [9] J.H. Lee and ZBien, "Collision-free trajectoxy control for mul- tiple robots based on neural optimization network Robotic4 Vol.8. 185-194, 1990.

    [lo] A.Bowyer and J.Adrian, A Programmer's Geomeq, Butter- worth, London, 48-52.1983.

    Fig3(c) Simulation result v v v

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