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Control Design for a Biped by Using Actuated Angles as Functions of Non-Actuated One
Copyright @ IFAC Control Systems Design, Bratislava, Slovak Republic, 2000
CONTROL DESIGN FOR A BIPED BY USING ACTUATED ANGLES AS FUNCTIONS OF NON-ACTUATED ONE Yannick Aoustin, Alexander Formal'sky· Inslitut of Mechanics, University, 1, Milcburinsky
Irccyn, U.M.R. 6597 1 rue de la Noe, BP 92101 F-44321 Nanles cedex 3, France.
Prospect, Moscow 119899 Russia.
Fax: 33 2 40 37 69 30. rei: 33 2 40376948
Fax: 709593822 15. rei: 70959392628 E-mail: fonnal(Q)inmeclLmsu.su
E-mail: Yannick.Aouslin(Q)in:yn.ec-nantes.fr
Abstract: We consider a biped, which consists of five rigid links, with no feet. Thus, the ankle joint is not actuated. The hip and knee joints are powered only. Gait is composed of alternating single and double support phases. Biped is under actuated during the single support motion. Instantaneous double support is modeled by passive impact equations. We choose the inter-link angles for the four powered articulations as polynomials depending on non-powered ankle angle of the supporting leg. The coefficients of these polynomials are calculated using given initial, desired intermediate and final conditions. Thus the nominal motion of the biped is designed for the known initial conditions. We construct numerically the Poincare's return map and show that the designed nominal motion converges to the cyclic motion asymptotically. Afterwards we hasten the convergence to the cyclic regime, changing the inclination of the torso or the pace length. The computed control torques can be used to track designed nominal trajectory. Copyright © 2000 IFAC
Keywords: robotics, PO controllers, periodic motion, stability analysis, process control, Mechanical engineering, mobile robots, boundary value problem, inverse dynamic problem Our purpose is to obtain a stable generation of gait cycles for a five-link biped robot without feet. We define a nominal motion of the biped. This nominal motion tends to the cycle. We show the asymptotic stability of the cycle numerically, using Poincare's return map. The convergence to the cycle is hastened by changing the inclination of the trunk or the pace length under a linear velocity feedback. The designed nominal trajectory can be tracked with computed control torques.
I. INTRODUCTION Many works are devoted to the control problem of biped robot like (Beletskii, and Chudinov 1977; Chevallereau, et a/.,1998; Goswami, et a!., 1996; Grishin, et al., 1995; Grizzle, et al., 1999; Hirai, et al., 1998; Katoh, and Mori, 1984; Miura, and Shimoyama, 1984; Raibert, 1986, Vukobratovic, et al., 1990). The control low in open loop is designed in some of them. It is important to design control in closed loop, if we want to construct a prototype. Several techniques have been adapted to derme the feedback control, for example, using reference trajectories. In (Chevallereau, et al., 1998) reference trajectory derives from ballistic motion, in (Grishin, et al., 1995) the trajectory is generated by polynomial functions of the time. In (Katoh, and Mori, 1984), it is designed using a Van der Pol's oscillator. In (Miura, and Shimoyama, 1984) the authors adapt at each step the trajectory by a pace length control. In (Hirai, et al., 1998); the authors control the position of "zero moment point" (Vukobratovic, et al., 1990), which is measured by force sensors; they control the pace length as well. Some theoretical works devoted to the walking stable gait use the method of Poincare return map (Goswami, et al., 1996; Grizzle, et al., 1999). In the last paper, authors have developed the Poincare's method to a biped without knees.
2. MATHEMATICAL MODEL Here motion equations of single support and impact equations are described.
2.1 Swing motion equations We consider a biped walking in a vertical xy sagittal plane. It is composed of a trunk and two identical two-
175
For each inter-link angle 0t, the dependence on the angle a is defined here to give a reference trajectory.
link legs (Fig. I). Let us introduce vector X = (x, y, OJ, 02, 0], 0" a l of seven generalized coordinates. It contains two coordinates of the trunk mass center x, y, and five of the legs and the trunk orientation q=(oj, 02, 0], 0" a l, (Fig. la). The angles are positive in the counter clockwise. These variables allow us to describe single support, double support and supportless phases. All links are assumed massive and rigid. All joints are revolute and ideal. Let vector r =(rj, r 2, r j , r 4l describes the torques applied in the hip and knee joints. Let RJ(R J R1y) and R2 (R 2x, R2y) be the forces applied to the leg tips (Fig. Ib). The motion equations of the biped have in swing phase the following form A(q)X+H(q,q)=DGr+DJq)Rj (I)
3.1 Description ofthe gait
Equation (2) specifies that the supporting leg tip does not move. The matrix dimensions are, A(7 x 7), Dl7 x 2), H(7 x 1), Dd7 x 4) and ~(2 x 1), j is the number of the leg (j= 1 or 2). The biped has five degrees of freedom in single support phase.
Two legs 1 and 2 of the biped swap their roles from one half step to the next one. Let the leg 1, which comes in the contact with the ground, has an inelastic impact and does not slide. The leg 2, which was previously in the contact, takes off the ground. In that case, the legs have symmetrical role from one step to the next one. We want to transfer the biped during swing phase from prescribed initial configuration to prescribed fmal one. In these boundary configurations, both legs are on the ground and these both boundary configurations are simultaneously double support configurations. Let be t = 0 and t = T respectively the initial and fmal time of the single support motion. If the boundary configurations of the swing phase are given, the angles aj and ar are prescribed in particular. Here and further indexes i and f correspond respectively to the initial (at t = 0), and fmal (at t = 1) configurations of the biped.
2.2 Equations ofthe passive impact
3.2 Polynomial functions for inter-link angles
){>
(2)
DJq/ X+Hlq,q)=O.
The walk consists of an alternating single and double support phases. We assume that the double support phase is instantaneous and modeled by passive impact equations. An impact appears, when the swing leg touches the ground. Let be an impact passive, absolutely inelastic, and without legs sliding. In that case, the ground reactions are impulsive forces. They are described by c;.. functions of Dirac. The impact equations can be obtained by integration of (I) during the infinitesimal time of instantaneous impact, A(q)( k+ = DJq)/R (3)
x- )
Inter-link angles 0t (t = I, ..., 4) for four powered . articulations are chosen here as polynomial functions. In experiments of several authors with bipeds (see, for example, (Grishin, et al., 1995» and in our biped simulations, the angle a of the supporting leg shin relative to the vertical changes strictly monotonically. (Fig. 4 confirms this for our problem.) Therefore, we use angle a instead of time t as independent variable for these polynomials. Thus we choose 0t = otCa) (t= J, ..., 4). We want to obtain a desired fmal configuration of the biped before an impact. Therefore, the segment [aj, atf is divided into two parts [aj, a"] and [a", ajJ. The angles 0t are defmed as polynomials of fifth order in the first segment: 3 2 oda}=atO + at/a + ao a + at] a + at~ a~+ at5 ~ (6)
J
Here q describes the configuration of the biped at the instant of double support; 1Rj is the impulsive reaction in the swing (before an impact) leg j. X - and x+ are the velocity vectors just before and just after impact respectively. The velocity of the supporting leg tip before an impact is equal to zero, D;(q)X- =0, b~j (4)
The first and second time derivatives of these four polynomial functions 0t(a) are defmed such as, 8t (a) = o/(a) a , 8t (a) = o/(a)a + o/'(a)a 2 (7) Here 'means derivation with respect to variable a.
The swing leg j after an impact becomes supporting one. Therefore, its tip velocity becomes zero after impact, DJ (q )X+ = O. (5)
Let in second segment [a', aJ the joint angles do not change and the whole mechanism turns around the point S as a rigid body until it hits the supporting surface i.e., 0t (a) 0t (a") = 0t (aJ) = const, a" =s; a:S aJ (8)
The linear system (3), (5) has an unique solution X + , JRj, because its corresponding matrix is non-singular. The vertical component of the transferring leg tip velocity just after impact has to be directed upwards.
=
Such a fixation of the fmal configuration before an impact helps to reduce in experiments (Grishin, et al., 1995) the error in biped configuration at the beginning of
3. REFERENCE TRAJECTORY FOR INTER-LINK ANGLES
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therefore the total kinetic momentum changes according to the equation ,j =- Mg( Xc - X S ) . Here M is total biped
the next step. The absolute angular velocities of all five links in the interval {a', aft are equal to a . Let be SJa'} = SJa'} = O. Then it follows from (7)
mass, g gravity acceleration, Xc abscissa of biped mass center, Xs abscissa of the point S. Coordinate Xc depends on the angles a,5t (t= J, ..., 4). But 8t = 8da) (t=J, ..., 4) are known functions. Thus we obtain second equation:
that, 5/ (a")
if
= 0,
8/'(a')
=0
(t= J, ..., 4),
(9)
a :t= 0 at the instant, when a=a'
,j
Due to the equalities (9) functions
() t (a) and
= -Mg[xc(a) -
(12)
Here xc(a) is known function of the angle a. The second order system (11), (12) is used here instead of equations (I), (2). Initial value a for this system can be calculated with formula (11), if initial values aj and aj are known. We design further the solutions of system (11), (12) numerically, assuming that a solution exists and is unique over a sufficiently large time interval. After the solving of the system (11), (12) the joint torques Ft (t = J, ..., 4) can be found from the equations (1), (2). System (11), (12) follows also from (1), (2) of course.
consequently the torques in the joints are continuous in the time. The configuration is called as intermediate one, when
a = a;n/=(a; + a·)12. Let us specify the initial, intermediate, and fmal configurations, initial angular velocities. It means that the values ai, af, aj, 8daJ, Sera;), 5daint), 5daf) (10) are given (t = J ,... , 4). All 6 coefficients forpolynomiaI5,(a), (t= J, ..., 4) (6) can be calculated using 4 values, 5daJ,
xs]
4. PERIODIC REGIME
SJa j }/a; ,
Here we design cyclic motion of the biped, when all double support configurations are the same.
8t (aint), 8t(~) from (10) and 2 conditions (9).
4. J Way to design periodic gait 3.3 Way to design swing motion
Let the initial and final configurations of single support motion coincide, but the legs are swapped. Then, the following equalities can be deduced for the given interlink angles (10) and angle a:
All functions 8t (a) can be given, if the values (10) are prescribed. In this case, a semi-inverse problem for the single support motion design appears, because the part of coordinates (inter-link angles) is given as the functions of the angle a, but it is required to obtain the angle a from motion equations. The single support motion can be found from equations (1), (2). But we use here other (it seems simpler) way. Denote by a the total kinetic momentum around the ankle joint S. It follows from the definition of the total kinetic momentum that it is linear combination of the angular velocities a, St (l = J, ... , 4) with the coefficients, which are functions of the angles 8t (t = J, ... , 4). But 8t = 8t (a), St = 8/(a)a (l = J ,..., 4) are known functions. Therefore, we obtain a = j( a)a . Here fray is function, which can be found analytically. It depends on the biped parameters. Note, that j(a}=const in the interval (a', aft, when the configuration of the biped does not change and this constant is total inertia moment of the biped around the ankle joint S. Simple differential equation follows from above equality: a =a/j(a)
5la) =8.(a j },8Ja) = 81 (a j }+8J(a j }
(13)
8J(a)=-8J(aj),8.(aj}=8laj}' af= aj +8laJ+8J(aJ-84(aJ
Before an impact St = 0 (t = J, ... , 4). Therefore, we can consider angular velocity
a- (that is a
j )
of the biped
before an impact as an unique input parameter for impact equations (3), (5). If value aj is given, then, using given terminal configuration, we deduce algebraically from impact equations (3), (5) all five angular velocities aj , SJa) (t
=
J, ..., 4) at the beginning of the next single
support phase. After, using known initial, desired intermediate and terminal configurations, we fmd polynomials (6), and then single support motion, using equations (11), (12). At the end of this motion we calculate new angular velocity af ' etc.
(11)
Nine scalar equations (3), (5) contain nine unknown variables, which form vectors k+, I Rj , and velocity athat is aj . The equalities (3), (5) are linear with respect
Ankle joint S is passive one and there is no torque acting at this joint. The external force is the gravity only,
to all these ten variables. The set of the solutions of
177
equations (3), (5) depends on one parameter
af
.
But this
By solving algebraic equation (15) we calculated cyclic regime for given above configurations (16), cZ j = Ue = - 2.3696 s", T = 0.5238 s (17) The curves in Figs 3, 4 present the evolution of the angle 8" for example, and a during found cyclic gait. The graphic of the angle 0, has horizontal part at the end of swing phase according to conditions (8). Angle a decreases strictly monotonically.
set can be parameterized by any variable from these ten, for example, by angular velocity a+ that is cZ j • Let us denote initial velocity cZ j of n-th swing phase u., We can consider a quantity of values u. and obtain numerically (after the swing phase and impact) the next quantity of values u.+,. By this way it is possible to construct on the plane (un' u.+) the Poincare's return map (poincare, 19161954) Un = p(u,J (14)
+,
if
The function (14) represents the evolution of the biped before the start of swing phase to before the start of the next swing phase. An equilibrium point U e of the transfonnation (14) U e = p(u,J (15)
1\ \ 1---
''t,
•
u
"
/
.. .. .. .
Fig. 3: Ol(t)
gives us the cyclic motion. It is possible to find this looked for value U e with numerically constructed graphic of the function (14) or solving algebraic equation (15) by Newton's method, for example. We consider the cyclic motion as desired gait. One-dimensional parameterization in our problem is possible due to conditions (8). But it is possible due to conditions (9) also, if a'=af We can choose as a parameter an angular velocity of any biped link at the start of the swing phase.
~!-.-+.--b--+.--b-+-+.--t,
s
Fig. 4: a(t)
In Fig. 5, the cyclic gait in the plane ('I' ,tit) is shown ('I' is the angle between the trunk and the horizontal).
Fig. 5: Cyclic gait, the plane ('I',tit).
4.2 Numerical design ofperiodic gait 5. STABILITY OF THE CYCLE AND HASTENING OF THE CONVERGENCE TO IT
Let the lengths of the thighs and the shins equal OAm, however their masses be different: 8 Kg, for the thigh and 6 Kg, for the shin. The length of the trunk is 0.625 m and its mass equals 27 Kg. Consider the following parameters of the initial, intermediate and fmal configurations: a,=0.157,af~0.3840,
Here it is shown, using Poincare's map, that cycle is asymptotically orbitally stable relative to nominal motion. After we quicken the convergence to the cycle, changing the inclination of the trunk or the pace length.
a'=- 0.330, 8/a,.)=2.93,
8ia,.)=- 2.932, 8ia,.)=- 0.2782, 8/a)=8la)=8/a)=8la)
=-
8la,.)~
2.915,
8ia)~8ia)~0.541, 8la)~2.670,
5.1 Stability ofthe periodic gait
2.402,
The numerically constructed graphic of the function (14) is shown in Fig. 6. We can fmd on the intersection of this graphic with the bissectrice of the first quadrant the equilibrium point u, (-2.3696 s", see (17» for periodic motion and can see if this motion is stable. The point u, on the Poincare's map in the Fig. 6 is asymptotically stable and the attraction region is -3.27 s" < U <1.60 s". If aj <-3.27s", the biped falls down back; if cZj >J.67s"',
(16) 8ia)~3.2J.
The initial, intennediate and fmal configurations are represented by Fig. 2.
than the vertical component of the ground reaction becomes zero. Let us assume that a solution of system (11), (12) depends continuously on the velocity aj • (Note, that the coefficients of polynomials (6) depend on this velocity continuously.) When the swing leg tip hits the ground, its velocity vector is not tangential to the support surface, but
Fig. 2: Initial, intennediate and final configurations.
178
is directed strictly under it. Using this fact, it is possible to prove that the cyclic motion is asymptotically orbitally stable in the sense of Lyapunov.
(18). We will obtain the transformation, un+J=q[u,. Ve + C (u n- u.))
5.2 As it is shown in Fig. 6, the transitional process converges to the point (ue' u) "slowly", because the graphic ofthe function (14) is "close" to the bissectrice.
(22)
·2 .2 ·2. 25
J'-.....
·2
af--+-+--l--1r--+--I--.J-+-J
~
,
-2.3
S·I
nonlinear
U" .. , S·'. 2 3
U.o/ .,
following
'"
•
, ,
-2 .•
blssec rice .,/
'2J--+---1f-+--+~~~~~t'f---J /~V' '2.2j---,;-;-f-+--+-'f-+~~+--U--1
·2
u. ~ ·2 4==f==l==4==F=~:.-.j.-+-+--U---1
-2.5
f".-.
"'"
v,
V
-35
'2+--+-+-+V7/*--+l-+i~-+l---l
-3..
·3.3
-3.2
·3, I
·3
~
-2.8
I'-
-2.8
-2.7
v.
·2 ef--+-~::;j<:"':........jf-*-+++--U--1 ·3
-3.1..
/
.3.2
.../
·3
·2.8
·2.5
·2'
.2;.2
·1.8
U,,+2
Un
S·}
Fig. 7: Un+J= q (U e, v,J The Poincare's map for nonlinear function (22) with c~ a/b=1.3 s is presented in Fig. 8. The graphic of this function is "more" horizontal than with c = O. Due to this the transitional process converges to the point (u e , ue) faster than without corrector.
·1.
Fig. 6: Poincare's map, Unt/ = p(u,J.
5.3 Convergence hastening to the cycle by the correction ofthe torso orientation S·I
Here we try to quicken the convergence to the cyclic regime, using as a parameter the angle 0](a) of the trunk inclination at the end of the swing phase. Let us denote o](a) as Vn, and consider new function: Un+J = q(u", v,J (18)
-l..t--+--+-.,-:--t-r---l-+/~' +---l "f--_+-_-+-~bl""'SS<>f"'CI"-,rI",,'C,,,,,e 1"-/_"-+--1 /'
.,.,t--+--+---t-~-l---+---j
." ·H -2.7
If variable V n equals its nominal value V e (-2.6704, see (16», the functions (14) and (18) coincide. The graphic of the function (18) with Un = Ue = -2.3696 S·I is constructed in Fig. 7. The both curves in the Figs 6 and 7 allow us to design a linear approximation of the function (18) in a domain around the point (u e, Ye), ~ Un+1 = a ~ u n+ b ~vn (~un = Un- Ue , ~vn = V n- Ye) (19)
.... , ./ ·2.6
-2.•
-1.2
-I
-2
S
Fig. 8: Poincare's map, un+J=q{u",
-1 .•
Un
V.
+
C
(un - ud)
It follows from (20) that if Au.~ 0, then ~v.~ 0 i.e., the terminal inclination of the trunk tends to the terminal inclination of the cyclic motion. It means that the gait tends to cyclic one as well.
The values a=0.75 and b= - 0.5775'1 are chosen. (Note, that value a is strictly less than one.) Now we quicken the convergence, using the following corrector: ~vn = c ~un (20)
5.4 Convergence hastening to the cycle by the correction ofthe pace length Now let us try to quicken the convergence to the cyclic regime, using as a parameter the angle oj(ap between the legs, but not the angle 8lap. The angle oj(a) is connected directly with the pace length, if the angles ola), oia) do not change. Denote oj(a) as W m and consider the function like (18), Und = r(u", w,J. Using the same procedure as in the previous section, we obtain the corrector like (20): W n = C !1u n• The corresponding Poincare's map with c=O,23 s is shown in the Fig. 9 (we =-0.5411, see data (16»:
where c is looked for constant. The relation (20) means that we change the desired trunk inclination at the end of swing phase as a function of the deviation of the velocity Q: i at the start of swing phase. Let us substitute the relation (20) into formula (19) !1 Un+J = (a+bc)!1 Un (21) If c= - a/b , then ~un+ J (21) becomes zero. Now let us substitute the expression (20) into formula
179
There are several levels of an "adaptation" in our control strategy. We design for each actuated inter-link angle the reference trajectory which is function of the non-actuated ankle angle. Each of these reference trajectories depends in particular on the desired final configuration i.e., for example, on the fmal torso inclination and the pace length. We choose one of these two parameters to hasten the convergence of the reference trajectory to the cyclic regime. In our control, this parameter depends linearly on the initial deviation of the angular velocity of the supporting leg shin. Designed by this way nominal reference trajectory tends asymptotically to the cyclic regime. Computed control torques track designed nominal trajectory.
,'.8;,--...,---,--,-...,---,--,----::l Un-+/
s -1
..... "
.21---+--+--1---+--+----,1"--'-"--;
.2.21---+_-+-_l'-'i~SSc..:.e+t,--nc..:.'c.;jL-t/_/_"I----; Ue
.//
.2.F:::::f:;:;;;:*==F=~===l==f===J
1/
//'
.261---+--+---,I"'---H--+--t----; ./ ..,.., ,28'1---+--¥---!--+I---+---!----1
..,,/./"
,3'1---->f"----+--t---tI--+--t-.-, '3.2'/'/ -32
-3
ue ·28
·26
·2'
-1.8 .22 _1 ·2 S Un
Fig. 9: Poincare's map, un+/=r[U,. We + C (Un - u.J) The Poincare's curve is more horizontal in this case than in the case before, and consequently the convergence is better.
REFERENCES Beletskii V.V., Chudinov P.S., "Parametric Optimization in the Problem of Bipedal Locomotion," Izv. AN SSSR. Mekhanika Tverdogo Tela [Mechanics ofSolids}, N°. 1, pp. 25-35, 1977. Chevallereau C., Fonnal'sky A.M., Perrin B., "Low energy cost reference trajectories for a biped robot", IEEE Int. Conf on Robotics and Automation. Leuven, Belgium, pp. 1088-1094, May 1998. Goswami A., Espiau B., Keramane A., "Limit Cycles and their Stability in a Passive Bipedal Gait. IEEE Int. Conf on Robotics and Automaion, Minneapolis, MN., pp. 246-251, April 1996. Grishin AA, Formal'sky A.M., Lensky A.V, Zhitomirsky S.V., "Dynamic Walking of Two Biped Vehicles", 9-th World Congress on the Theory of Machines and Mechanisms, Vol 3, pp. 2308-2312,1995. Grizzle J.W., Abba G., Plestan F., "Asymptotically Stable Walking for Biped Robots: Analysis via System with Impulse Effects", I 999,personal communication. Hirai K., Hirose M., Haikava Y., Takenaka T., 'The Development of Honda Humanoid Robot", In Proc. of IEEE Int. Conf on Robotics and Automaion, Leuven, pp. 1321-1326, May 1998. Katoh R., Mori M., "Control Method of Biped Locomotion Giving Asymptotic Stability of Trajectory", Automatica, Vol. 20, No. 4, pp. 405-414,1984. Miura H., Shimoyama 1., "Dynamic Walk of a Biped", Int. J. of Robotics Research, Vol. 3, No. 2, pp. 60-74,1984. Poincare H., "Oeuvres completes", I1 vol., Cauthier-Vil/ars Paris 1916-1954, n&impression Jacques Gabay, Paris 1987-1997. Raibert M.H., "Legged Robots that Balance", MIT Press. Cambridge, MA 1986. Vukobratovic M., Borovac B., Sur1a D., Stokic D., "Biped Locomotion," Scientific Fundamentals of Robotics 7, SpringerVerlag , 345p, 1990.
6. TRACKING OF THE REFERENCE TRAJECTORY Excluding from the system (1) the ground reaction, we obtain five equations with five independent variables 0/, oz, 03, 040 a. Four of them contain the inter-link control torques. One equation, corresponding to the theorem about total kinetic momentum changing, does not contain them. Let us substitute the derivative cl from the last equation into four other equations and put: 8= gd+Kl8d_8)+Kp(Od_0) (23) Here 0 is column (4xl), Kpo Kv are diagonal matrices (4x4) of positive feedback gains. Let each element of the desired trajectory vectors Od, 8 d , gd be defmed by formulas (6)-(9). The following second order equation (equivalent to the system (11), (12)) is satisfied along the desired trajectory: ii =-[Mg[xda)-xs ]+ !(a)/(a)a 2 ]
/
F(a) (24)
We substitute derivative (24) into second expression (7). After we can compute four inter-link torques. They depend on the state variables and biped parameters only. The coefficients of the polynomials (6) are calculated under the conditions 8 d (0) =8(0), 8d (0) = 8(0) (see (10)). Then the identity (j(t)=8 d (t) follows from (23). It means that the desired trajectory is tracked ideally, if the biped parameters and state variables are known. It is possible also to compute the torques, substituting into four equations directly the desired polynomials (6).
7. CONCLUSION
180
CONTROL DESIGN FOR A BIPED BY USING ACTUATED ANGLES AS FUNCTIONS OF NON-ACTUATED ONE Yannick Aoustin, Alexander Formal'sky· Inslitut of Mechanics, University, 1, Milcburinsky
Irccyn, U.M.R. 6597 1 rue de la Noe, BP 92101 F-44321 Nanles cedex 3, France.
Prospect, Moscow 119899 Russia.
Fax: 33 2 40 37 69 30. rei: 33 2 40376948
Fax: 709593822 15. rei: 70959392628 E-mail: fonnal(Q)inmeclLmsu.su
E-mail: Yannick.Aouslin(Q)in:yn.ec-nantes.fr
Abstract: We consider a biped, which consists of five rigid links, with no feet. Thus, the ankle joint is not actuated. The hip and knee joints are powered only. Gait is composed of alternating single and double support phases. Biped is under actuated during the single support motion. Instantaneous double support is modeled by passive impact equations. We choose the inter-link angles for the four powered articulations as polynomials depending on non-powered ankle angle of the supporting leg. The coefficients of these polynomials are calculated using given initial, desired intermediate and final conditions. Thus the nominal motion of the biped is designed for the known initial conditions. We construct numerically the Poincare's return map and show that the designed nominal motion converges to the cyclic motion asymptotically. Afterwards we hasten the convergence to the cyclic regime, changing the inclination of the torso or the pace length. The computed control torques can be used to track designed nominal trajectory. Copyright © 2000 IFAC
Keywords: robotics, PO controllers, periodic motion, stability analysis, process control, Mechanical engineering, mobile robots, boundary value problem, inverse dynamic problem Our purpose is to obtain a stable generation of gait cycles for a five-link biped robot without feet. We define a nominal motion of the biped. This nominal motion tends to the cycle. We show the asymptotic stability of the cycle numerically, using Poincare's return map. The convergence to the cycle is hastened by changing the inclination of the trunk or the pace length under a linear velocity feedback. The designed nominal trajectory can be tracked with computed control torques.
I. INTRODUCTION Many works are devoted to the control problem of biped robot like (Beletskii, and Chudinov 1977; Chevallereau, et a/.,1998; Goswami, et a!., 1996; Grishin, et al., 1995; Grizzle, et al., 1999; Hirai, et al., 1998; Katoh, and Mori, 1984; Miura, and Shimoyama, 1984; Raibert, 1986, Vukobratovic, et al., 1990). The control low in open loop is designed in some of them. It is important to design control in closed loop, if we want to construct a prototype. Several techniques have been adapted to derme the feedback control, for example, using reference trajectories. In (Chevallereau, et al., 1998) reference trajectory derives from ballistic motion, in (Grishin, et al., 1995) the trajectory is generated by polynomial functions of the time. In (Katoh, and Mori, 1984), it is designed using a Van der Pol's oscillator. In (Miura, and Shimoyama, 1984) the authors adapt at each step the trajectory by a pace length control. In (Hirai, et al., 1998); the authors control the position of "zero moment point" (Vukobratovic, et al., 1990), which is measured by force sensors; they control the pace length as well. Some theoretical works devoted to the walking stable gait use the method of Poincare return map (Goswami, et al., 1996; Grizzle, et al., 1999). In the last paper, authors have developed the Poincare's method to a biped without knees.
2. MATHEMATICAL MODEL Here motion equations of single support and impact equations are described.
2.1 Swing motion equations We consider a biped walking in a vertical xy sagittal plane. It is composed of a trunk and two identical two-
175
For each inter-link angle 0t, the dependence on the angle a is defined here to give a reference trajectory.
link legs (Fig. I). Let us introduce vector X = (x, y, OJ, 02, 0], 0" a l of seven generalized coordinates. It contains two coordinates of the trunk mass center x, y, and five of the legs and the trunk orientation q=(oj, 02, 0], 0" a l, (Fig. la). The angles are positive in the counter clockwise. These variables allow us to describe single support, double support and supportless phases. All links are assumed massive and rigid. All joints are revolute and ideal. Let vector r =(rj, r 2, r j , r 4l describes the torques applied in the hip and knee joints. Let RJ(R J R1y) and R2 (R 2x, R2y) be the forces applied to the leg tips (Fig. Ib). The motion equations of the biped have in swing phase the following form A(q)X+H(q,q)=DGr+DJq)Rj (I)
3.1 Description ofthe gait
Equation (2) specifies that the supporting leg tip does not move. The matrix dimensions are, A(7 x 7), Dl7 x 2), H(7 x 1), Dd7 x 4) and ~(2 x 1), j is the number of the leg (j= 1 or 2). The biped has five degrees of freedom in single support phase.
Two legs 1 and 2 of the biped swap their roles from one half step to the next one. Let the leg 1, which comes in the contact with the ground, has an inelastic impact and does not slide. The leg 2, which was previously in the contact, takes off the ground. In that case, the legs have symmetrical role from one step to the next one. We want to transfer the biped during swing phase from prescribed initial configuration to prescribed fmal one. In these boundary configurations, both legs are on the ground and these both boundary configurations are simultaneously double support configurations. Let be t = 0 and t = T respectively the initial and fmal time of the single support motion. If the boundary configurations of the swing phase are given, the angles aj and ar are prescribed in particular. Here and further indexes i and f correspond respectively to the initial (at t = 0), and fmal (at t = 1) configurations of the biped.
2.2 Equations ofthe passive impact
3.2 Polynomial functions for inter-link angles
){>
(2)
DJq/ X+Hlq,q)=O.
The walk consists of an alternating single and double support phases. We assume that the double support phase is instantaneous and modeled by passive impact equations. An impact appears, when the swing leg touches the ground. Let be an impact passive, absolutely inelastic, and without legs sliding. In that case, the ground reactions are impulsive forces. They are described by c;.. functions of Dirac. The impact equations can be obtained by integration of (I) during the infinitesimal time of instantaneous impact, A(q)( k+ = DJq)/R (3)
x- )
Inter-link angles 0t (t = I, ..., 4) for four powered . articulations are chosen here as polynomial functions. In experiments of several authors with bipeds (see, for example, (Grishin, et al., 1995» and in our biped simulations, the angle a of the supporting leg shin relative to the vertical changes strictly monotonically. (Fig. 4 confirms this for our problem.) Therefore, we use angle a instead of time t as independent variable for these polynomials. Thus we choose 0t = otCa) (t= J, ..., 4). We want to obtain a desired fmal configuration of the biped before an impact. Therefore, the segment [aj, atf is divided into two parts [aj, a"] and [a", ajJ. The angles 0t are defmed as polynomials of fifth order in the first segment: 3 2 oda}=atO + at/a + ao a + at] a + at~ a~+ at5 ~ (6)
J
Here q describes the configuration of the biped at the instant of double support; 1Rj is the impulsive reaction in the swing (before an impact) leg j. X - and x+ are the velocity vectors just before and just after impact respectively. The velocity of the supporting leg tip before an impact is equal to zero, D;(q)X- =0, b~j (4)
The first and second time derivatives of these four polynomial functions 0t(a) are defmed such as, 8t (a) = o/(a) a , 8t (a) = o/(a)a + o/'(a)a 2 (7) Here 'means derivation with respect to variable a.
The swing leg j after an impact becomes supporting one. Therefore, its tip velocity becomes zero after impact, DJ (q )X+ = O. (5)
Let in second segment [a', aJ the joint angles do not change and the whole mechanism turns around the point S as a rigid body until it hits the supporting surface i.e., 0t (a) 0t (a") = 0t (aJ) = const, a" =s; a:S aJ (8)
The linear system (3), (5) has an unique solution X + , JRj, because its corresponding matrix is non-singular. The vertical component of the transferring leg tip velocity just after impact has to be directed upwards.
=
Such a fixation of the fmal configuration before an impact helps to reduce in experiments (Grishin, et al., 1995) the error in biped configuration at the beginning of
3. REFERENCE TRAJECTORY FOR INTER-LINK ANGLES
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therefore the total kinetic momentum changes according to the equation ,j =- Mg( Xc - X S ) . Here M is total biped
the next step. The absolute angular velocities of all five links in the interval {a', aft are equal to a . Let be SJa'} = SJa'} = O. Then it follows from (7)
mass, g gravity acceleration, Xc abscissa of biped mass center, Xs abscissa of the point S. Coordinate Xc depends on the angles a,5t (t= J, ..., 4). But 8t = 8da) (t=J, ..., 4) are known functions. Thus we obtain second equation:
that, 5/ (a")
if
= 0,
8/'(a')
=0
(t= J, ..., 4),
(9)
a :t= 0 at the instant, when a=a'
,j
Due to the equalities (9) functions
() t (a) and
= -Mg[xc(a) -
(12)
Here xc(a) is known function of the angle a. The second order system (11), (12) is used here instead of equations (I), (2). Initial value a for this system can be calculated with formula (11), if initial values aj and aj are known. We design further the solutions of system (11), (12) numerically, assuming that a solution exists and is unique over a sufficiently large time interval. After the solving of the system (11), (12) the joint torques Ft (t = J, ..., 4) can be found from the equations (1), (2). System (11), (12) follows also from (1), (2) of course.
consequently the torques in the joints are continuous in the time. The configuration is called as intermediate one, when
a = a;n/=(a; + a·)12. Let us specify the initial, intermediate, and fmal configurations, initial angular velocities. It means that the values ai, af, aj, 8daJ, Sera;), 5daint), 5daf) (10) are given (t = J ,... , 4). All 6 coefficients forpolynomiaI5,(a), (t= J, ..., 4) (6) can be calculated using 4 values, 5daJ,
xs]
4. PERIODIC REGIME
SJa j }/a; ,
Here we design cyclic motion of the biped, when all double support configurations are the same.
8t (aint), 8t(~) from (10) and 2 conditions (9).
4. J Way to design periodic gait 3.3 Way to design swing motion
Let the initial and final configurations of single support motion coincide, but the legs are swapped. Then, the following equalities can be deduced for the given interlink angles (10) and angle a:
All functions 8t (a) can be given, if the values (10) are prescribed. In this case, a semi-inverse problem for the single support motion design appears, because the part of coordinates (inter-link angles) is given as the functions of the angle a, but it is required to obtain the angle a from motion equations. The single support motion can be found from equations (1), (2). But we use here other (it seems simpler) way. Denote by a the total kinetic momentum around the ankle joint S. It follows from the definition of the total kinetic momentum that it is linear combination of the angular velocities a, St (l = J, ... , 4) with the coefficients, which are functions of the angles 8t (t = J, ... , 4). But 8t = 8t (a), St = 8/(a)a (l = J ,..., 4) are known functions. Therefore, we obtain a = j( a)a . Here fray is function, which can be found analytically. It depends on the biped parameters. Note, that j(a}=const in the interval (a', aft, when the configuration of the biped does not change and this constant is total inertia moment of the biped around the ankle joint S. Simple differential equation follows from above equality: a =a/j(a)
5la) =8.(a j },8Ja) = 81 (a j }+8J(a j }
(13)
8J(a)=-8J(aj),8.(aj}=8laj}' af= aj +8laJ+8J(aJ-84(aJ
Before an impact St = 0 (t = J, ... , 4). Therefore, we can consider angular velocity
a- (that is a
j )
of the biped
before an impact as an unique input parameter for impact equations (3), (5). If value aj is given, then, using given terminal configuration, we deduce algebraically from impact equations (3), (5) all five angular velocities aj , SJa) (t
=
J, ..., 4) at the beginning of the next single
support phase. After, using known initial, desired intermediate and terminal configurations, we fmd polynomials (6), and then single support motion, using equations (11), (12). At the end of this motion we calculate new angular velocity af ' etc.
(11)
Nine scalar equations (3), (5) contain nine unknown variables, which form vectors k+, I Rj , and velocity athat is aj . The equalities (3), (5) are linear with respect
Ankle joint S is passive one and there is no torque acting at this joint. The external force is the gravity only,
to all these ten variables. The set of the solutions of
177
equations (3), (5) depends on one parameter
af
.
But this
By solving algebraic equation (15) we calculated cyclic regime for given above configurations (16), cZ j = Ue = - 2.3696 s", T = 0.5238 s (17) The curves in Figs 3, 4 present the evolution of the angle 8" for example, and a during found cyclic gait. The graphic of the angle 0, has horizontal part at the end of swing phase according to conditions (8). Angle a decreases strictly monotonically.
set can be parameterized by any variable from these ten, for example, by angular velocity a+ that is cZ j • Let us denote initial velocity cZ j of n-th swing phase u., We can consider a quantity of values u. and obtain numerically (after the swing phase and impact) the next quantity of values u.+,. By this way it is possible to construct on the plane (un' u.+) the Poincare's return map (poincare, 19161954) Un = p(u,J (14)
+,
if
The function (14) represents the evolution of the biped before the start of swing phase to before the start of the next swing phase. An equilibrium point U e of the transfonnation (14) U e = p(u,J (15)
1\ \ 1---
''t,
•
u
"
/
.. .. .. .
Fig. 3: Ol(t)
gives us the cyclic motion. It is possible to find this looked for value U e with numerically constructed graphic of the function (14) or solving algebraic equation (15) by Newton's method, for example. We consider the cyclic motion as desired gait. One-dimensional parameterization in our problem is possible due to conditions (8). But it is possible due to conditions (9) also, if a'=af We can choose as a parameter an angular velocity of any biped link at the start of the swing phase.
~!-.-+.--b--+.--b-+-+.--t,
s
Fig. 4: a(t)
In Fig. 5, the cyclic gait in the plane ('I' ,tit) is shown ('I' is the angle between the trunk and the horizontal).
Fig. 5: Cyclic gait, the plane ('I',tit).
4.2 Numerical design ofperiodic gait 5. STABILITY OF THE CYCLE AND HASTENING OF THE CONVERGENCE TO IT
Let the lengths of the thighs and the shins equal OAm, however their masses be different: 8 Kg, for the thigh and 6 Kg, for the shin. The length of the trunk is 0.625 m and its mass equals 27 Kg. Consider the following parameters of the initial, intermediate and fmal configurations: a,=0.157,af~0.3840,
Here it is shown, using Poincare's map, that cycle is asymptotically orbitally stable relative to nominal motion. After we quicken the convergence to the cycle, changing the inclination of the trunk or the pace length.
a'=- 0.330, 8/a,.)=2.93,
8ia,.)=- 2.932, 8ia,.)=- 0.2782, 8/a)=8la)=8/a)=8la)
=-
8la,.)~
2.915,
8ia)~8ia)~0.541, 8la)~2.670,
5.1 Stability ofthe periodic gait
2.402,
The numerically constructed graphic of the function (14) is shown in Fig. 6. We can fmd on the intersection of this graphic with the bissectrice of the first quadrant the equilibrium point u, (-2.3696 s", see (17» for periodic motion and can see if this motion is stable. The point u, on the Poincare's map in the Fig. 6 is asymptotically stable and the attraction region is -3.27 s" < U <1.60 s". If aj <-3.27s", the biped falls down back; if cZj >J.67s"',
(16) 8ia)~3.2J.
The initial, intennediate and fmal configurations are represented by Fig. 2.
than the vertical component of the ground reaction becomes zero. Let us assume that a solution of system (11), (12) depends continuously on the velocity aj • (Note, that the coefficients of polynomials (6) depend on this velocity continuously.) When the swing leg tip hits the ground, its velocity vector is not tangential to the support surface, but
Fig. 2: Initial, intennediate and final configurations.
178
is directed strictly under it. Using this fact, it is possible to prove that the cyclic motion is asymptotically orbitally stable in the sense of Lyapunov.
(18). We will obtain the transformation, un+J=q[u,. Ve + C (u n- u.))
5.2 As it is shown in Fig. 6, the transitional process converges to the point (ue' u) "slowly", because the graphic ofthe function (14) is "close" to the bissectrice.
(22)
·2 .2 ·2. 25
J'-.....
·2
af--+-+--l--1r--+--I--.J-+-J
~
,
-2.3
S·I
nonlinear
U" .. , S·'. 2 3
U.o/ .,
following
'"
•
, ,
-2 .•
blssec rice .,/
'2J--+---1f-+--+~~~~~t'f---J /~V' '2.2j---,;-;-f-+--+-'f-+~~+--U--1
·2
u. ~ ·2 4==f==l==4==F=~:.-.j.-+-+--U---1
-2.5
f".-.
"'"
v,
V
-35
'2+--+-+-+V7/*--+l-+i~-+l---l
-3..
·3.3
-3.2
·3, I
·3
~
-2.8
I'-
-2.8
-2.7
v.
·2 ef--+-~::;j<:"':........jf-*-+++--U--1 ·3
-3.1..
/
.3.2
.../
·3
·2.8
·2.5
·2'
.2;.2
·1.8
U,,+2
Un
S·}
Fig. 7: Un+J= q (U e, v,J The Poincare's map for nonlinear function (22) with c~ a/b=1.3 s is presented in Fig. 8. The graphic of this function is "more" horizontal than with c = O. Due to this the transitional process converges to the point (u e , ue) faster than without corrector.
·1.
Fig. 6: Poincare's map, Unt/ = p(u,J.
5.3 Convergence hastening to the cycle by the correction ofthe torso orientation S·I
Here we try to quicken the convergence to the cyclic regime, using as a parameter the angle 0](a) of the trunk inclination at the end of the swing phase. Let us denote o](a) as Vn, and consider new function: Un+J = q(u", v,J (18)
-l..t--+--+-.,-:--t-r---l-+/~' +---l "f--_+-_-+-~bl""'SS<>f"'CI"-,rI",,'C,,,,,e 1"-/_"-+--1 /'
.,.,t--+--+---t-~-l---+---j
." ·H -2.7
If variable V n equals its nominal value V e (-2.6704, see (16», the functions (14) and (18) coincide. The graphic of the function (18) with Un = Ue = -2.3696 S·I is constructed in Fig. 7. The both curves in the Figs 6 and 7 allow us to design a linear approximation of the function (18) in a domain around the point (u e, Ye), ~ Un+1 = a ~ u n+ b ~vn (~un = Un- Ue , ~vn = V n- Ye) (19)
.... , ./ ·2.6
-2.•
-1.2
-I
-2
S
Fig. 8: Poincare's map, un+J=q{u",
-1 .•
Un
V.
+
C
(un - ud)
It follows from (20) that if Au.~ 0, then ~v.~ 0 i.e., the terminal inclination of the trunk tends to the terminal inclination of the cyclic motion. It means that the gait tends to cyclic one as well.
The values a=0.75 and b= - 0.5775'1 are chosen. (Note, that value a is strictly less than one.) Now we quicken the convergence, using the following corrector: ~vn = c ~un (20)
5.4 Convergence hastening to the cycle by the correction ofthe pace length Now let us try to quicken the convergence to the cyclic regime, using as a parameter the angle oj(ap between the legs, but not the angle 8lap. The angle oj(a) is connected directly with the pace length, if the angles ola), oia) do not change. Denote oj(a) as W m and consider the function like (18), Und = r(u", w,J. Using the same procedure as in the previous section, we obtain the corrector like (20): W n = C !1u n• The corresponding Poincare's map with c=O,23 s is shown in the Fig. 9 (we =-0.5411, see data (16»:
where c is looked for constant. The relation (20) means that we change the desired trunk inclination at the end of swing phase as a function of the deviation of the velocity Q: i at the start of swing phase. Let us substitute the relation (20) into formula (19) !1 Un+J = (a+bc)!1 Un (21) If c= - a/b , then ~un+ J (21) becomes zero. Now let us substitute the expression (20) into formula
179
There are several levels of an "adaptation" in our control strategy. We design for each actuated inter-link angle the reference trajectory which is function of the non-actuated ankle angle. Each of these reference trajectories depends in particular on the desired final configuration i.e., for example, on the fmal torso inclination and the pace length. We choose one of these two parameters to hasten the convergence of the reference trajectory to the cyclic regime. In our control, this parameter depends linearly on the initial deviation of the angular velocity of the supporting leg shin. Designed by this way nominal reference trajectory tends asymptotically to the cyclic regime. Computed control torques track designed nominal trajectory.
,'.8;,--...,---,--,-...,---,--,----::l Un-+/
s -1
..... "
.21---+--+--1---+--+----,1"--'-"--;
.2.21---+_-+-_l'-'i~SSc..:.e+t,--nc..:.'c.;jL-t/_/_"I----; Ue
.//
.2.F:::::f:;:;;;:*==F=~===l==f===J
1/
//'
.261---+--+---,I"'---H--+--t----; ./ ..,.., ,28'1---+--¥---!--+I---+---!----1
..,,/./"
,3'1---->f"----+--t---tI--+--t-.-, '3.2'/'/ -32
-3
ue ·28
·26
·2'
-1.8 .22 _1 ·2 S Un
Fig. 9: Poincare's map, un+/=r[U,. We + C (Un - u.J) The Poincare's curve is more horizontal in this case than in the case before, and consequently the convergence is better.
REFERENCES Beletskii V.V., Chudinov P.S., "Parametric Optimization in the Problem of Bipedal Locomotion," Izv. AN SSSR. Mekhanika Tverdogo Tela [Mechanics ofSolids}, N°. 1, pp. 25-35, 1977. Chevallereau C., Fonnal'sky A.M., Perrin B., "Low energy cost reference trajectories for a biped robot", IEEE Int. Conf on Robotics and Automation. Leuven, Belgium, pp. 1088-1094, May 1998. Goswami A., Espiau B., Keramane A., "Limit Cycles and their Stability in a Passive Bipedal Gait. IEEE Int. Conf on Robotics and Automaion, Minneapolis, MN., pp. 246-251, April 1996. Grishin AA, Formal'sky A.M., Lensky A.V, Zhitomirsky S.V., "Dynamic Walking of Two Biped Vehicles", 9-th World Congress on the Theory of Machines and Mechanisms, Vol 3, pp. 2308-2312,1995. Grizzle J.W., Abba G., Plestan F., "Asymptotically Stable Walking for Biped Robots: Analysis via System with Impulse Effects", I 999,personal communication. Hirai K., Hirose M., Haikava Y., Takenaka T., 'The Development of Honda Humanoid Robot", In Proc. of IEEE Int. Conf on Robotics and Automaion, Leuven, pp. 1321-1326, May 1998. Katoh R., Mori M., "Control Method of Biped Locomotion Giving Asymptotic Stability of Trajectory", Automatica, Vol. 20, No. 4, pp. 405-414,1984. Miura H., Shimoyama 1., "Dynamic Walk of a Biped", Int. J. of Robotics Research, Vol. 3, No. 2, pp. 60-74,1984. Poincare H., "Oeuvres completes", I1 vol., Cauthier-Vil/ars Paris 1916-1954, n&impression Jacques Gabay, Paris 1987-1997. Raibert M.H., "Legged Robots that Balance", MIT Press. Cambridge, MA 1986. Vukobratovic M., Borovac B., Sur1a D., Stokic D., "Biped Locomotion," Scientific Fundamentals of Robotics 7, SpringerVerlag , 345p, 1990.
6. TRACKING OF THE REFERENCE TRAJECTORY Excluding from the system (1) the ground reaction, we obtain five equations with five independent variables 0/, oz, 03, 040 a. Four of them contain the inter-link control torques. One equation, corresponding to the theorem about total kinetic momentum changing, does not contain them. Let us substitute the derivative cl from the last equation into four other equations and put: 8= gd+Kl8d_8)+Kp(Od_0) (23) Here 0 is column (4xl), Kpo Kv are diagonal matrices (4x4) of positive feedback gains. Let each element of the desired trajectory vectors Od, 8 d , gd be defmed by formulas (6)-(9). The following second order equation (equivalent to the system (11), (12)) is satisfied along the desired trajectory: ii =-[Mg[xda)-xs ]+ !(a)/(a)a 2 ]
/
F(a) (24)
We substitute derivative (24) into second expression (7). After we can compute four inter-link torques. They depend on the state variables and biped parameters only. The coefficients of the polynomials (6) are calculated under the conditions 8 d (0) =8(0), 8d (0) = 8(0) (see (10)). Then the identity (j(t)=8 d (t) follows from (23). It means that the desired trajectory is tracked ideally, if the biped parameters and state variables are known. It is possible also to compute the torques, substituting into four equations directly the desired polynomials (6).
7. CONCLUSION
180