Med. & Biol. Eng. & Comput,, 1978, 16, 250-255
Control of biped locomotion J. L. H u n t
V. L a t h a m
Department of Electronics, University of Southampton, Southampton SO9 5NH, England
A b s t r a c t - - T h e control o f biped locomotion is investigated by computer simulation of a simple model of a biped. A method of control is suggested which attempts to ensure that the motion of the biped resembles human locomotion, even in the presence of large disturbances. The results of computer simulations performed using the suggested control scheme are described. These simulations confirm that the suggested control scheme achieves the aim of approximating human locomotion. K e y w o r d s - - B i p e d , Control Locomotion
1 Introduction
Trns paper describes a simple model of a biped, and proposes a means of controlling the motion of this model. The model, controlled as described below, has been simulated on a digital computer, and the motion of the biped in a typical simulation is described. The controls applied to the model are chosen to produce a motion resembling human locomotion, even in the presence of large disturbances, e.g. stairs, wind etc. The model is largely based on the work described by GUmNA et al. (1974). Gubina describes a method of controlling a biped such that it is dynamically stable, both in terms of body posture and gait periodicity, as defined by VUKOBRATOVlC et al. (1970). The work described below considers an additional factor. As well as ensuring dynamic stability, the controls produce a motion of the biped which is a much better approximation to human locomotion than the motions obtained by previous workers. Comparisons are made between human locomotion and the motion of the models developed by Gubina and the present authors.
body, the hinges being considered frictionless. The motion of the biped is controlled continuously by
I~
Fig. 1 Model of a biped
J~
2 The model
The model of a biped is shown in Fig. 1. It is the same as the model used by Gubina, with the addition of an extra control input provided by a moment applied at the ankle. The motion of the biped is confined to the sagittal plane. The biped consists of a rigid body, mass m and moment of inertia about the centre of mass J', supported by a massless leg of variable length. The leg is hinged to the ground and the First Received
mg
'
step
step 1
5
-
.1
.2:3
2
~"-2-1 -~, .5 .6 time (s)
:7
.'~ .9 1-0
Fig. 2 Stabilisation of trunk angle
0140-0118/78/0742-0250 $1.50/0 9 IFMBE: 1978
250
Medical & Biological Engineering & Computing
May 1978
moments applied at the ankle (Ma') and the hip ( M 2 ' ) and by a force applied along the leg (F'). F o r the biped to walk it is necessary to provide some means of taking steps. This is achieved by moving the leg instantaneously through an angle u~ whenever it is desired to simulate the change from support by one leg to support by the other, which occurs during h u m a n locomotion. The leg-angle increment us is variable, thus providing a discrete control on step length. The equations of m o t i o n for the system shown in Fig. l can be derived by application of Lagrange's method:
equations of m o t i o n derived below. To render the equations of m o t i o n amenable to step 1
I
dE
6 -5
I [
")-
3
- -
x sin (~i - ~2) - g r sin ~i = M2 - m i
4~)-Ir$~
- / g sin ~2 = m l
2
/: = Uo
.
.
.
.
.
.
.
.
.
(4)
F = F'/m
.
.
.
.
.
.
.
.
.
(5)
M1 = M(/m
.
.
.
.
.
.
.
.
(6)
M2 = M2'/m
.
.
.
.
.
.
.
.
(7)
lro ~
. . . . . . . .
(8)
M1 = ul + W1
. . . . . . . .
(9)
M2 = ua+W2
.
.
.
.
.
.
(10)
W2 is chosen to eliminate the effect of F and M~ on the m o t i o n of the body when it is vertical (i.e. when ~b2 = 0). The required value for W.z can be determined by substituting (M2, ~b2, ~b2, ~2) = (W2, 0, 0, 0) into eqns. 1-3: I472 =
r I F sin ~bl + IM1 cos ~
(11)
r + l cos ~bi
Wo is chosen such that it is sufficient to maintain the body at constant height by counteracting gravity: Wo = g
.
.
.
.
.
.
.
.
.
.
9
10
(12)
These values of Wo and W2 simplify the linearised Medical & Biological Engineering & Computing
.
.
.
.
.
.
.
.
= M2-MI
.
(13)
9
(14) (15)
+ ( J + l z) ~ z - I g ( k 2 = M 2
Eqns. 14 and 15 can be solved for qSl, ~b2: ~ i = b2 (Ji - c 2 ~ 2 - E t l 2 + H M 1
~2 = d 2 ~ + 6 . ~
9
. . . . . . .
(16) 07)
where the constants are defined by E2 = ( J + r o l + 1 2 ) / J r o 2, G = (ro + l ) / J r o ,
3 Control o f the m o d e l The controlling force a n d moments F, Mx, M2 are composed of two parts, a bias c o m p o n e n t Wo, 1411, W2 and a controlling component Uo, u~, u2: F = Uo + 141o
.
r2~l+rol~z-gr(~i
(3)
J = J'/m
.
-8
analysis eqns. 1-3 can be linearised about the point of equilibrium (r, f, ~bl, q~x, ~b2, 4 2 ) = (r0, 0, 0, 0, 0, 0):
In these equations some of the parameters have been normalised with respect to the body mass m such that:
.
.
(2)
sin (~x - ~b2)
. . . . . .
.
-7
Fig. 3 Stabilisation of hip height
l/~ sin (fix - if2) + l r ~ i cos (~i - if2)+ ( d + l 2) ~2 + 21#6~ cos ( ~ -
.
(1)
+ 2 r f ~ , + rl~b2 2
-- ~2)
.
-Z '5 "6 time (s)
i~+ l(~2 sin (~bx- ~b2)- rq~x2 - / ~ 2 2 COS (1~1 ~2)
r 2 ~, + r l ~ 2 c o s ( ~ 1
2
8
9;I
+ g COS ~b~ = F
step
I
.9
H = 1/(ro2+rol), b 2 = g / ( r o + l), c 2 = 12 g / J r o
and d 2 = gl/J.
F r o m eqn. 16 it can be seen that, for q~l to be independent of ~bt, ~b2 and u2, (18)
Wx = E u a / H + c 2 ~ 2 / H - b 2 q ~ / H
Substituting eqns. 9 a n d 18 into eqn. 16 gives ~l = nul
.
.
.
.
.
.
.
.
.
(19)
The overall control strategy is based on measurements of h u m a n locomotion (MORRAY, 1967; LAMOREUX, 1971). A feature of h u m a n locomotion is that the trunk remains at an almost constant angle to the vertical ( + 2~ a n d that the hip moves almost horizontally ( + 5 ~ ) and with a n almost constant forward v e l o c i t y ( + 5 ~ ) . M a n y authors have suggested that this is because a fundamental criterion of h u m a n locomotion is the minimisation of energy M a y 1978
251
expenditure (PE1ZER, 1969; BARD and RALSTON, so the actuators must supply some net energy to 1959; INMAN, 1968). Keeping the potential and kinetic energies of the trunk constant is one obvious way of reducing energy expenditure. To alter the energy level of the trunk would require work to be done by the muscles of the leg, which would be lost as heat when the trunk returns to its original energy level, as it must do if the gait is periodic. It should be noted that even if the energy level of the trunk is h d d constant, energy is still consumed during locomotion due to the structure of the legs. A vertical force at the hip is always necessary to oppose the force of gravity, and therefore the controlling force and moments must have some nonzero value. As the trunk moves forward the leg length and angles change and so work is done. The total energy level of the system remains constant and so any work performed by one actuator must be absorbed by another. Owing to the nature of muscle any work absorbed in this manner is dissipated as heat, and step 1
maintain locomotion. In view of the measurements outlined above it was decided to try and control the biped so as to maintain ~b2=~t, h = r o , V = v o , where h is the height of the hip above ground level, ro some constant desired height, v the forward velocity of the hip, Vo some constant desired velocity and ~t some constant angle. It was also decided to try and control the motion of the biped so that, during a single step, ~1 varied by equal amounts on either side of the vertical. Because of the configuration of the leg in the model this has no direct analogue in human locomotion. As described above the bias moments were chosen such that r is controlled by Uo, ~1 by ul and if2 by u2. It was therefore decided to use the control input Uo to control h, u~ to control v and u2 to control if2. The form of the continuous control inputs are then defined by:
step 2
(
uo = hl(h-ro)+h2h ul = hs(vo-v)
1.4
. .
/22 = h s ( ~ 2 - 0 0 q - h 6
1.2
'
.
.
. .
~2
.
.
.
.
.
.
.
(20)
.
.
(21)
.
.
.
.
(22)
The change in the controls /20 and /21 and the addition of the control u2 are the major differences between the control scheme used by Gubina and the control scheme described in this paper. Now h = r cos ~bl and v = f sin ~'1 +rt~1 cos ~bl. Substituting these values into eqns. 20-22 and linearising about (ro, 0, 0, 0, 0, 0) gives
1.0
Uo = h l ( r - r a ) + h 2 r
.
.
.
.
.
(23)
.
2~
0
.1
.2
.3
.4
.5
-5
:7
.~
.9 1.0
ul = ha(vo-ro
~l)
.
.
.
.
.
.
(24)
.
time (s) u2 = hs(~z-~)+h6~)z
Fig. 4 Stabilisation of forward velocity of hip
.
.
.
.
.
(25)
~,(rod/s)
3
_1-0'
1 --5
-5' -'4
-:3
-:2
-'1
:1
"2
"3
'4
.'5
e, (rad)
Fig. 5 Stabil/sation of leg angle
252
M e d i c a l & Biological Engineering & C o m p u t i n g
M a y 1978
where rd is some desired value for r in the vicinity of r = r o. F r o m eqns. 13 and 23 r-h2r-hl
r = - h i ra
If each step takes a fixed time T, and the angle fix is incremented by an angle us(k) at the end of each step, then the full equation for the leg is:
[0 '1 [']
g~ =
x~-
-a 2
or
r=-hlrd/(s2-h2s-hx)
(26)
. . . . .
The poles of eqn. 26, 21 and 22, can be defined at will by choosing the feedback gains: hi = -~,1~2
. . . . . . . .
(27)
h2 = ~1+~,2
. . . . . . . .
(28)
0
(35)
u,(k) &(t - k T )
0
where k is an integer (k = 1, 2 . . . . . n), x~ = (~j, q~l) r and &(t) is a Dirac delta function. The discrete version of eqn. 35 is: x~(k + 1) =
xs(k) C223
kC21
(36)
us(k) C21 j
where
Similarly, from eqns. 17 and 25,
Cll = C2z = c o s a T
~ 2 - G.h6 6 2 - (d 2 + Ghs) ~2 = - Gh5 ot
c12 = s i n a T / a
or
~2 = - G h s a / [ s 2 - G . h 6 s - ( d 2
+Ghs)]
The form for u A k ) is:
The poles of eqn. 29, 2s and 26, are defined, by: hs
=
(-~,5~,6-d2)/G
.
h6 = (2s+26)/m
.
.
.
.
(30)
.
. . . . . . .
= ha Hvo
or
~
= haHvo/(s+hsroH)
u~(k) = ha {r ( k T ) - r
+ h,{dp~ (k T ) - vl } + ~J~
(37)
(31)
F r o m eqns. 19~and 24: t~l +ha ro n ~ l
Ce~ = - a s i n a T
(29)
(32)
. . . . .
The pole of eqn. 32, 28, is defined by:
~b~is the desired leg angle excursion during one step. If it is desired that ff~ varies equal a m o u n t s either side of the vertical then the desired value of ~ at the end of a step (i.e. at t = k T ) is ~s/2. Assuming that the continuous controls are successful in keeping h a n d v constant, ~b~and the desired angular velocity of the leg at t = k T , v~, are given by: ff~ = 2 tan-l(Vo T / 2 r o )
hs = - 2s/(ro H)
vl = Vo cos2(qJ~/2)/ro
The poles 2t, 22, 2s, 26, 28 must all be in the negative halfplane to ensure stability. If the controls defined above are assumed to control the height and velocity of the hip as desired, then ~1 = --2VO2 sin ~bl COSa ~bl/ro 2
(33)
(38)
. . . . . .
(39)
The eigenvalues of the discrete system, ,t3 a n d 24, are related to the feedback gains by: h3 = 1-2324.
. . . . . . . . .
(40)
h , = - [ ( 1 +23 24) c o s a T - 2 3 - 2 4 ] / ( a s i n a T )
(41)
F o r stability 23 and 24 must lie inside the unit circle.
If eqn. 33 is linearised about ~ = 0 then ~l = -a2ffl
. . . . .
. . . . . . . .
(34)
4 Computer integration of equations of motion The stability criteria outlined above are,
where a z = 2vo2/ro 2.
of
1 -9
"9"
E d;2 "8"
.t:
'8
31.7.
_1.7
E
~1'5. 0
i
.~,
.3
.~.
.s
o
Fig 6 Height and velocity of hip during human locomotion
Medical & Biological Engineering & Computing
.i
.2
.~
.2.
.~ ~-
time(s)
time (s)
Fig 7 Height and velocity of hip with control of height and velocity
May 1978
253
course, only valid for the linearised model. The stability of the complete nonlinear model can only be tested by using a digital computer to integrate the equations of motion (eqns. 1-3), while applying the controls defined by eqns. 20-22. The computer program must also implement the instantaneous change of leg angle at t = k T. It should be noted that the energy of the system must not change as the leg angle is instantaneously changed. This can be assured if the following conditions are satisfied, where k T+ denotes the time immediately following the change, and k T_ the time immediately before the change:
r247
= q)l(kT_)-u,(k)
~b2(kT+) = ~2(kT_)
. . . . .
(42)
. . . . . . .
(43)
r(kT+) = r ( k T _ ) cos gpl(kT+)/cos (ka(kT_) .
(44)
dpt(k T+ ) = i'(k T_ ) sin us(k)/r(k T+ ) + r ( k T _ ) c~l(kT_) cosu,(k)/r(kT+) q~2(kT+) = q~2(kT_)
. . . . . . .
(45) (46)
i'(k T+) = i'(k T_) cos u,(k) -r(kT_) ~(kT_)
sin u,(k)
(47)
The computer thus restarts the numerical integration of eqns. 1-3 whenever t = k T, using the new values defined by eqns. 42-47 as initial conditions. To maintain the desired similarity to human locomotion the computer also restarts integration if ~b1 becomes greater than 0.7 tad or less than - 0 - 7 rad. This is done to avoid possible large excursions of ~b~, which would result in the leg length r becoming unrealistically large. 5 Simulation results
The ability of the control scheme suggested above in controlling the locomotion of the model in the presence of disturbances has been tested by simulation. To simulate disturbances the initial conditions for the first step can be given any desired values. The computer then performs the integration of eqns. 1-3 after initially making an instantaneous
change in the position of the leg in accordance with eqns. 42-47. A large number of simulations have been performed using many different initial conditions. In every case the control system has been able to achieve the same stable gait, which is a very good approximation to the desired gait, i.e. ~2 = ~, h = ro, v = Vo, ~bt(T) = - ~ a ( 0 ) . The results of a typical simulation are shown in Figs. 2-5. The system parameters and initial conditions for this simulation were: 1 = 0.35 m; J = 0.1 m2; ~ = 0 rad; ro=0.85m; vo=l.6m/s; T=0.5s; 21=-5; 22 = - 1 0 ; 2a = 0.1; 24 = 0.1; 2s = - 1 0 ; 9~6
28 = - 1 0 ;
=-10;
4~,(o) = o.o; ~(o)
r(0)=
0.5;
/'(0)=
(bl(0)=
= 0.5; q~(o) = 0.5.
Figs. 2 - 4 s h o w t h e t i m e v a r i a t i o n o f (b=, h a n d v,
respectively. Fig. 5 shows a phase trajectory from which it can be seen that ~b1 settles down to approximately equal positive and negative excursions. The dotted lines show instantaneous changes in ~b~ and q~l which occur at the beginning of each step. The solid lines show changes during a step. The numbers indicate points reached in the trajectory at successive moments in time. Fig. 6 shows the variation in height and velocity during human locomotion, obtained from data given in MURRAY (1967). Fig. 7 shows the variation once a stable gait has been achieved using the control scheme suggested in this paper. Fig. 8 shows the variations when using the control scheme suggested by Gubina, the system parameters being the same as those used for the simulation shown in Figs. 2-5 and 7. The peak-to-peak variations of height and velocity shown in Fig. 8 match the values for human locomotion shown in Fig. 6 fairly well. However, the mean value of forward velocity is too low. If a simulation is performed in which the mean forward velocity is more realistic, again using the controls suggested by Gubina, the resulting height and velocity are as shown in Fig. 9. The system parameters and initial conditions for this simulation are the same as for the simulation shown in Fig. 8
i A
E
"g'
1.4
....
9.......
......
1.2
~ 1-0" >
J
8
o
:1
:2
.3
J .~,
0
-5
time(s)
Fig. 8 Height and velocity of hip using controls suggested by Gubina
254
.1
-2
.3 .4 time(s)
.5
.6
.7
Fig. 9 Height and velocity of hip usmg controls suggested by Gubina, with system parameters changed so as to increase mean velocity
Medical & Biological Engineering & Computing
May 1978
except for: r o = 1.0m, T = 0 . 6 7 5 s , 21 = - 2 0 . Fig. 9 shows that the peak-to-peak variations in height and velocity need to be unrealistically large if the mean height and velocity are to be realistic. Figs. 6-9 show that the control scheme suggested in this paper achieves a closer approximation to human locomotion than can be obtained using the controls suggested by Gubina. 6 Conclusions
The control scheme described in this paper achieves the desired objective of providing a good approximation to certain, rather limited, factors in human locomotion. The simplified model of a biped can maintain the desired gait in the presence of large disturbances. To do this, however, the controlling moments and forces can become unrealistically large. Also the ground reaction forces can be unrealistic. It is possible for the vertical reaction force to become negative and for the magnitude of the horizontal reaction force to exceed the magnitude of the vertical reaction force. It is expected that limiting the values of controlling forces and moments, and also ground reaction forces, will limit the range of disturbances the control system will be able to accommodate. The model of a biped is also much simplified. In the authors' opinion the model will have to be extended to include massive legs, consisting of thr~e
segments jointed at the knee and ankle, before the results obtained from simulations, can usefully be applied to prosthetics, orthotics, etc. Work is in progress on the development of such a model and an accompanying control scheme, which includes the improvements outlined above.
Acknowledgments--One of us (JLH) is indebted to the
Science Research Council for a studentship during the period of this work. References BARD G. and RALSTON H. J. (1959) Measurement of
energy expenditure during ambulation with special reference to evaluation of assistive devices, Arch. o f Phys. & Med. Rehabil., 40, 415-420. GUBINA F. et al. (1974). On the dynamic stability of biped locomotion, IEEE Trans., BME 21, 102-108. INMANV. T. (1968) Conservation of energy in ambulation. Bull. Prosthetics Res., 10--9,26-35. LAMOREUXL. W. (1971) Kinematic measurements in the study of human walking, t~ull. Prosthetics Res. 10-5, 3-84. MURRAYM. P. (1967) Gait as a total pattern of movement. Am. J. Phys. Med., 46, 290-333. PEIZER E. et al. (1969) Human locomotion. Bull. Prosthetics Res., 10-12, 48-104. VUKOBRATOVICM. et al. (1970) On the stability of biped locomotion. IEEE Trans. BME 17, 25-36.
Commande de locomotion d'un bip~de Sommaire--Cet essai enqu6te sur la commande de locomotion d'un bip~de par simulation d'ordinateur d'un mod61e simple de bip6de. On propose une m6thode de commande, qui s'efforce de garantir que le mouvement d'un bip6de ressemble b. celui d'un humain, mfime en pr6sence de troubles importants. On d6crit les r6sultats des simulations d'ordinateur obtenues, en utilisant le programme de comrnande propos6. Ces simulations confirment que le programme de commande propos6 parvient h se rapprocher de celui de la locomotion humaine.
Die Steuerung der ZweifOssorfortbewegung Zusammenfassnng--Die Steuerung der Zweifi~Berfortbewegung wird durch Computersimulation eines einfachen Modells eines ZweifiiBers erforscht. Es wird eine Steuerungsmethode vorgeschlagen, dutch die der Versuch gemacht wird zu beweisen, dab die Bewegung eines ZweifiiBers der menschlichen Fortbewegung ~,hnlich ist, und zwar sogar dann, wenn beachtliche St6rungen vorherrschen. Es werden die Resultate der Computersimulationunter Verwendung der vorgeschlagenen Steuermethode besehrieben. Die Simulationen bestatigen, dab das vorgeschlagene Steuerschema das Ziel erreicht, menschtiche Fortbewegung nachzuahmen.
Medical & Biological Engineering & Computing
May 1978
255
Control of biped locomotion J. L. H u n t
V. L a t h a m
Department of Electronics, University of Southampton, Southampton SO9 5NH, England
A b s t r a c t - - T h e control o f biped locomotion is investigated by computer simulation of a simple model of a biped. A method of control is suggested which attempts to ensure that the motion of the biped resembles human locomotion, even in the presence of large disturbances. The results of computer simulations performed using the suggested control scheme are described. These simulations confirm that the suggested control scheme achieves the aim of approximating human locomotion. K e y w o r d s - - B i p e d , Control Locomotion
1 Introduction
Trns paper describes a simple model of a biped, and proposes a means of controlling the motion of this model. The model, controlled as described below, has been simulated on a digital computer, and the motion of the biped in a typical simulation is described. The controls applied to the model are chosen to produce a motion resembling human locomotion, even in the presence of large disturbances, e.g. stairs, wind etc. The model is largely based on the work described by GUmNA et al. (1974). Gubina describes a method of controlling a biped such that it is dynamically stable, both in terms of body posture and gait periodicity, as defined by VUKOBRATOVlC et al. (1970). The work described below considers an additional factor. As well as ensuring dynamic stability, the controls produce a motion of the biped which is a much better approximation to human locomotion than the motions obtained by previous workers. Comparisons are made between human locomotion and the motion of the models developed by Gubina and the present authors.
body, the hinges being considered frictionless. The motion of the biped is controlled continuously by
I~
Fig. 1 Model of a biped
J~
2 The model
The model of a biped is shown in Fig. 1. It is the same as the model used by Gubina, with the addition of an extra control input provided by a moment applied at the ankle. The motion of the biped is confined to the sagittal plane. The biped consists of a rigid body, mass m and moment of inertia about the centre of mass J', supported by a massless leg of variable length. The leg is hinged to the ground and the First Received
mg
'
step
step 1
5
-
.1
.2:3
2
~"-2-1 -~, .5 .6 time (s)
:7
.'~ .9 1-0
Fig. 2 Stabilisation of trunk angle
0140-0118/78/0742-0250 $1.50/0 9 IFMBE: 1978
250
Medical & Biological Engineering & Computing
May 1978
moments applied at the ankle (Ma') and the hip ( M 2 ' ) and by a force applied along the leg (F'). F o r the biped to walk it is necessary to provide some means of taking steps. This is achieved by moving the leg instantaneously through an angle u~ whenever it is desired to simulate the change from support by one leg to support by the other, which occurs during h u m a n locomotion. The leg-angle increment us is variable, thus providing a discrete control on step length. The equations of m o t i o n for the system shown in Fig. l can be derived by application of Lagrange's method:
equations of m o t i o n derived below. To render the equations of m o t i o n amenable to step 1
I
dE
6 -5
I [
")-
3
- -
x sin (~i - ~2) - g r sin ~i = M2 - m i
4~)-Ir$~
- / g sin ~2 = m l
2
/: = Uo
.
.
.
.
.
.
.
.
.
(4)
F = F'/m
.
.
.
.
.
.
.
.
.
(5)
M1 = M(/m
.
.
.
.
.
.
.
.
(6)
M2 = M2'/m
.
.
.
.
.
.
.
.
(7)
lro ~
. . . . . . . .
(8)
M1 = ul + W1
. . . . . . . .
(9)
M2 = ua+W2
.
.
.
.
.
.
(10)
W2 is chosen to eliminate the effect of F and M~ on the m o t i o n of the body when it is vertical (i.e. when ~b2 = 0). The required value for W.z can be determined by substituting (M2, ~b2, ~b2, ~2) = (W2, 0, 0, 0) into eqns. 1-3: I472 =
r I F sin ~bl + IM1 cos ~
(11)
r + l cos ~bi
Wo is chosen such that it is sufficient to maintain the body at constant height by counteracting gravity: Wo = g
.
.
.
.
.
.
.
.
.
.
9
10
(12)
These values of Wo and W2 simplify the linearised Medical & Biological Engineering & Computing
.
.
.
.
.
.
.
.
= M2-MI
.
(13)
9
(14) (15)
+ ( J + l z) ~ z - I g ( k 2 = M 2
Eqns. 14 and 15 can be solved for qSl, ~b2: ~ i = b2 (Ji - c 2 ~ 2 - E t l 2 + H M 1
~2 = d 2 ~ + 6 . ~
9
. . . . . . .
(16) 07)
where the constants are defined by E2 = ( J + r o l + 1 2 ) / J r o 2, G = (ro + l ) / J r o ,
3 Control o f the m o d e l The controlling force a n d moments F, Mx, M2 are composed of two parts, a bias c o m p o n e n t Wo, 1411, W2 and a controlling component Uo, u~, u2: F = Uo + 141o
.
r2~l+rol~z-gr(~i
(3)
J = J'/m
.
-8
analysis eqns. 1-3 can be linearised about the point of equilibrium (r, f, ~bl, q~x, ~b2, 4 2 ) = (r0, 0, 0, 0, 0, 0):
In these equations some of the parameters have been normalised with respect to the body mass m such that:
.
.
(2)
sin (~x - ~b2)
. . . . . .
.
-7
Fig. 3 Stabilisation of hip height
l/~ sin (fix - if2) + l r ~ i cos (~i - if2)+ ( d + l 2) ~2 + 21#6~ cos ( ~ -
.
(1)
+ 2 r f ~ , + rl~b2 2
-- ~2)
.
-Z '5 "6 time (s)
i~+ l(~2 sin (~bx- ~b2)- rq~x2 - / ~ 2 2 COS (1~1 ~2)
r 2 ~, + r l ~ 2 c o s ( ~ 1
2
8
9;I
+ g COS ~b~ = F
step
I
.9
H = 1/(ro2+rol), b 2 = g / ( r o + l), c 2 = 12 g / J r o
and d 2 = gl/J.
F r o m eqn. 16 it can be seen that, for q~l to be independent of ~bt, ~b2 and u2, (18)
Wx = E u a / H + c 2 ~ 2 / H - b 2 q ~ / H
Substituting eqns. 9 a n d 18 into eqn. 16 gives ~l = nul
.
.
.
.
.
.
.
.
.
(19)
The overall control strategy is based on measurements of h u m a n locomotion (MORRAY, 1967; LAMOREUX, 1971). A feature of h u m a n locomotion is that the trunk remains at an almost constant angle to the vertical ( + 2~ a n d that the hip moves almost horizontally ( + 5 ~ ) and with a n almost constant forward v e l o c i t y ( + 5 ~ ) . M a n y authors have suggested that this is because a fundamental criterion of h u m a n locomotion is the minimisation of energy M a y 1978
251
expenditure (PE1ZER, 1969; BARD and RALSTON, so the actuators must supply some net energy to 1959; INMAN, 1968). Keeping the potential and kinetic energies of the trunk constant is one obvious way of reducing energy expenditure. To alter the energy level of the trunk would require work to be done by the muscles of the leg, which would be lost as heat when the trunk returns to its original energy level, as it must do if the gait is periodic. It should be noted that even if the energy level of the trunk is h d d constant, energy is still consumed during locomotion due to the structure of the legs. A vertical force at the hip is always necessary to oppose the force of gravity, and therefore the controlling force and moments must have some nonzero value. As the trunk moves forward the leg length and angles change and so work is done. The total energy level of the system remains constant and so any work performed by one actuator must be absorbed by another. Owing to the nature of muscle any work absorbed in this manner is dissipated as heat, and step 1
maintain locomotion. In view of the measurements outlined above it was decided to try and control the biped so as to maintain ~b2=~t, h = r o , V = v o , where h is the height of the hip above ground level, ro some constant desired height, v the forward velocity of the hip, Vo some constant desired velocity and ~t some constant angle. It was also decided to try and control the motion of the biped so that, during a single step, ~1 varied by equal amounts on either side of the vertical. Because of the configuration of the leg in the model this has no direct analogue in human locomotion. As described above the bias moments were chosen such that r is controlled by Uo, ~1 by ul and if2 by u2. It was therefore decided to use the control input Uo to control h, u~ to control v and u2 to control if2. The form of the continuous control inputs are then defined by:
step 2
(
uo = hl(h-ro)+h2h ul = hs(vo-v)
1.4
. .
/22 = h s ( ~ 2 - 0 0 q - h 6
1.2
'
.
.
. .
~2
.
.
.
.
.
.
.
(20)
.
.
(21)
.
.
.
.
(22)
The change in the controls /20 and /21 and the addition of the control u2 are the major differences between the control scheme used by Gubina and the control scheme described in this paper. Now h = r cos ~bl and v = f sin ~'1 +rt~1 cos ~bl. Substituting these values into eqns. 20-22 and linearising about (ro, 0, 0, 0, 0, 0) gives
1.0
Uo = h l ( r - r a ) + h 2 r
.
.
.
.
.
(23)
.
2~
0
.1
.2
.3
.4
.5
-5
:7
.~
.9 1.0
ul = ha(vo-ro
~l)
.
.
.
.
.
.
(24)
.
time (s) u2 = hs(~z-~)+h6~)z
Fig. 4 Stabilisation of forward velocity of hip
.
.
.
.
.
(25)
~,(rod/s)
3
_1-0'
1 --5
-5' -'4
-:3
-:2
-'1
:1
"2
"3
'4
.'5
e, (rad)
Fig. 5 Stabil/sation of leg angle
252
M e d i c a l & Biological Engineering & C o m p u t i n g
M a y 1978
where rd is some desired value for r in the vicinity of r = r o. F r o m eqns. 13 and 23 r-h2r-hl
r = - h i ra
If each step takes a fixed time T, and the angle fix is incremented by an angle us(k) at the end of each step, then the full equation for the leg is:
[0 '1 [']
g~ =
x~-
-a 2
or
r=-hlrd/(s2-h2s-hx)
(26)
. . . . .
The poles of eqn. 26, 21 and 22, can be defined at will by choosing the feedback gains: hi = -~,1~2
. . . . . . . .
(27)
h2 = ~1+~,2
. . . . . . . .
(28)
0
(35)
u,(k) &(t - k T )
0
where k is an integer (k = 1, 2 . . . . . n), x~ = (~j, q~l) r and &(t) is a Dirac delta function. The discrete version of eqn. 35 is: x~(k + 1) =
xs(k) C223
kC21
(36)
us(k) C21 j
where
Similarly, from eqns. 17 and 25,
Cll = C2z = c o s a T
~ 2 - G.h6 6 2 - (d 2 + Ghs) ~2 = - Gh5 ot
c12 = s i n a T / a
or
~2 = - G h s a / [ s 2 - G . h 6 s - ( d 2
+Ghs)]
The form for u A k ) is:
The poles of eqn. 29, 2s and 26, are defined, by: hs
=
(-~,5~,6-d2)/G
.
h6 = (2s+26)/m
.
.
.
.
(30)
.
. . . . . . .
= ha Hvo
or
~
= haHvo/(s+hsroH)
u~(k) = ha {r ( k T ) - r
+ h,{dp~ (k T ) - vl } + ~J~
(37)
(31)
F r o m eqns. 19~and 24: t~l +ha ro n ~ l
Ce~ = - a s i n a T
(29)
(32)
. . . . .
The pole of eqn. 32, 28, is defined by:
~b~is the desired leg angle excursion during one step. If it is desired that ff~ varies equal a m o u n t s either side of the vertical then the desired value of ~ at the end of a step (i.e. at t = k T ) is ~s/2. Assuming that the continuous controls are successful in keeping h a n d v constant, ~b~and the desired angular velocity of the leg at t = k T , v~, are given by: ff~ = 2 tan-l(Vo T / 2 r o )
hs = - 2s/(ro H)
vl = Vo cos2(qJ~/2)/ro
The poles 2t, 22, 2s, 26, 28 must all be in the negative halfplane to ensure stability. If the controls defined above are assumed to control the height and velocity of the hip as desired, then ~1 = --2VO2 sin ~bl COSa ~bl/ro 2
(33)
(38)
. . . . . .
(39)
The eigenvalues of the discrete system, ,t3 a n d 24, are related to the feedback gains by: h3 = 1-2324.
. . . . . . . . .
(40)
h , = - [ ( 1 +23 24) c o s a T - 2 3 - 2 4 ] / ( a s i n a T )
(41)
F o r stability 23 and 24 must lie inside the unit circle.
If eqn. 33 is linearised about ~ = 0 then ~l = -a2ffl
. . . . .
. . . . . . . .
(34)
4 Computer integration of equations of motion The stability criteria outlined above are,
where a z = 2vo2/ro 2.
of
1 -9
"9"
E d;2 "8"
.t:
'8
31.7.
_1.7
E
~1'5. 0
i
.~,
.3
.~.
.s
o
Fig 6 Height and velocity of hip during human locomotion
Medical & Biological Engineering & Computing
.i
.2
.~
.2.
.~ ~-
time(s)
time (s)
Fig 7 Height and velocity of hip with control of height and velocity
May 1978
253
course, only valid for the linearised model. The stability of the complete nonlinear model can only be tested by using a digital computer to integrate the equations of motion (eqns. 1-3), while applying the controls defined by eqns. 20-22. The computer program must also implement the instantaneous change of leg angle at t = k T. It should be noted that the energy of the system must not change as the leg angle is instantaneously changed. This can be assured if the following conditions are satisfied, where k T+ denotes the time immediately following the change, and k T_ the time immediately before the change:
r247
= q)l(kT_)-u,(k)
~b2(kT+) = ~2(kT_)
. . . . .
(42)
. . . . . . .
(43)
r(kT+) = r ( k T _ ) cos gpl(kT+)/cos (ka(kT_) .
(44)
dpt(k T+ ) = i'(k T_ ) sin us(k)/r(k T+ ) + r ( k T _ ) c~l(kT_) cosu,(k)/r(kT+) q~2(kT+) = q~2(kT_)
. . . . . . .
(45) (46)
i'(k T+) = i'(k T_) cos u,(k) -r(kT_) ~(kT_)
sin u,(k)
(47)
The computer thus restarts the numerical integration of eqns. 1-3 whenever t = k T, using the new values defined by eqns. 42-47 as initial conditions. To maintain the desired similarity to human locomotion the computer also restarts integration if ~b1 becomes greater than 0.7 tad or less than - 0 - 7 rad. This is done to avoid possible large excursions of ~b~, which would result in the leg length r becoming unrealistically large. 5 Simulation results
The ability of the control scheme suggested above in controlling the locomotion of the model in the presence of disturbances has been tested by simulation. To simulate disturbances the initial conditions for the first step can be given any desired values. The computer then performs the integration of eqns. 1-3 after initially making an instantaneous
change in the position of the leg in accordance with eqns. 42-47. A large number of simulations have been performed using many different initial conditions. In every case the control system has been able to achieve the same stable gait, which is a very good approximation to the desired gait, i.e. ~2 = ~, h = ro, v = Vo, ~bt(T) = - ~ a ( 0 ) . The results of a typical simulation are shown in Figs. 2-5. The system parameters and initial conditions for this simulation were: 1 = 0.35 m; J = 0.1 m2; ~ = 0 rad; ro=0.85m; vo=l.6m/s; T=0.5s; 21=-5; 22 = - 1 0 ; 2a = 0.1; 24 = 0.1; 2s = - 1 0 ; 9~6
28 = - 1 0 ;
=-10;
4~,(o) = o.o; ~(o)
r(0)=
0.5;
/'(0)=
(bl(0)=
= 0.5; q~(o) = 0.5.
Figs. 2 - 4 s h o w t h e t i m e v a r i a t i o n o f (b=, h a n d v,
respectively. Fig. 5 shows a phase trajectory from which it can be seen that ~b1 settles down to approximately equal positive and negative excursions. The dotted lines show instantaneous changes in ~b~ and q~l which occur at the beginning of each step. The solid lines show changes during a step. The numbers indicate points reached in the trajectory at successive moments in time. Fig. 6 shows the variation in height and velocity during human locomotion, obtained from data given in MURRAY (1967). Fig. 7 shows the variation once a stable gait has been achieved using the control scheme suggested in this paper. Fig. 8 shows the variations when using the control scheme suggested by Gubina, the system parameters being the same as those used for the simulation shown in Figs. 2-5 and 7. The peak-to-peak variations of height and velocity shown in Fig. 8 match the values for human locomotion shown in Fig. 6 fairly well. However, the mean value of forward velocity is too low. If a simulation is performed in which the mean forward velocity is more realistic, again using the controls suggested by Gubina, the resulting height and velocity are as shown in Fig. 9. The system parameters and initial conditions for this simulation are the same as for the simulation shown in Fig. 8
i A
E
"g'
1.4
....
9.......
......
1.2
~ 1-0" >
J
8
o
:1
:2
.3
J .~,
0
-5
time(s)
Fig. 8 Height and velocity of hip using controls suggested by Gubina
254
.1
-2
.3 .4 time(s)
.5
.6
.7
Fig. 9 Height and velocity of hip usmg controls suggested by Gubina, with system parameters changed so as to increase mean velocity
Medical & Biological Engineering & Computing
May 1978
except for: r o = 1.0m, T = 0 . 6 7 5 s , 21 = - 2 0 . Fig. 9 shows that the peak-to-peak variations in height and velocity need to be unrealistically large if the mean height and velocity are to be realistic. Figs. 6-9 show that the control scheme suggested in this paper achieves a closer approximation to human locomotion than can be obtained using the controls suggested by Gubina. 6 Conclusions
The control scheme described in this paper achieves the desired objective of providing a good approximation to certain, rather limited, factors in human locomotion. The simplified model of a biped can maintain the desired gait in the presence of large disturbances. To do this, however, the controlling moments and forces can become unrealistically large. Also the ground reaction forces can be unrealistic. It is possible for the vertical reaction force to become negative and for the magnitude of the horizontal reaction force to exceed the magnitude of the vertical reaction force. It is expected that limiting the values of controlling forces and moments, and also ground reaction forces, will limit the range of disturbances the control system will be able to accommodate. The model of a biped is also much simplified. In the authors' opinion the model will have to be extended to include massive legs, consisting of thr~e
segments jointed at the knee and ankle, before the results obtained from simulations, can usefully be applied to prosthetics, orthotics, etc. Work is in progress on the development of such a model and an accompanying control scheme, which includes the improvements outlined above.
Acknowledgments--One of us (JLH) is indebted to the
Science Research Council for a studentship during the period of this work. References BARD G. and RALSTON H. J. (1959) Measurement of
energy expenditure during ambulation with special reference to evaluation of assistive devices, Arch. o f Phys. & Med. Rehabil., 40, 415-420. GUBINA F. et al. (1974). On the dynamic stability of biped locomotion, IEEE Trans., BME 21, 102-108. INMANV. T. (1968) Conservation of energy in ambulation. Bull. Prosthetics Res., 10--9,26-35. LAMOREUXL. W. (1971) Kinematic measurements in the study of human walking, t~ull. Prosthetics Res. 10-5, 3-84. MURRAYM. P. (1967) Gait as a total pattern of movement. Am. J. Phys. Med., 46, 290-333. PEIZER E. et al. (1969) Human locomotion. Bull. Prosthetics Res., 10-12, 48-104. VUKOBRATOVICM. et al. (1970) On the stability of biped locomotion. IEEE Trans. BME 17, 25-36.
Commande de locomotion d'un bip~de Sommaire--Cet essai enqu6te sur la commande de locomotion d'un bip~de par simulation d'ordinateur d'un mod61e simple de bip6de. On propose une m6thode de commande, qui s'efforce de garantir que le mouvement d'un bip6de ressemble b. celui d'un humain, mfime en pr6sence de troubles importants. On d6crit les r6sultats des simulations d'ordinateur obtenues, en utilisant le programme de comrnande propos6. Ces simulations confirment que le programme de commande propos6 parvient h se rapprocher de celui de la locomotion humaine.
Die Steuerung der ZweifOssorfortbewegung Zusammenfassnng--Die Steuerung der Zweifi~Berfortbewegung wird durch Computersimulation eines einfachen Modells eines ZweifiiBers erforscht. Es wird eine Steuerungsmethode vorgeschlagen, dutch die der Versuch gemacht wird zu beweisen, dab die Bewegung eines ZweifiiBers der menschlichen Fortbewegung ~,hnlich ist, und zwar sogar dann, wenn beachtliche St6rungen vorherrschen. Es werden die Resultate der Computersimulationunter Verwendung der vorgeschlagenen Steuermethode besehrieben. Die Simulationen bestatigen, dab das vorgeschlagene Steuerschema das Ziel erreicht, menschtiche Fortbewegung nachzuahmen.
Medical & Biological Engineering & Computing
May 1978
255
