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Dynamic modelling for implementation of a right turn in bipedal walking
1. Rtomrchenrcs Vol. 19. No. 3. pp Rintcd
195-206. 1986.
0021-9290~86 S3.M) + .oO
in Great Bnlam
g
DYNAMIC
MODELLING FOR IMPLEMENTATION RIGHT TURN IN BIPEDAL WALKING
1986 Pcrgamon
Press Ltd.
OF A
BEN-RENCHEN Bell Communications Research. Red Bank, NJ 07701, U.S.A. and
MARGARET J.HINES* and HOOSHANG
HEMAbUt
*Department of Anatomy and TDepartment of Electrical Engineering, The Ohio State University, Columbus, OH 43210, U.S.A. Abstract-This paper deals with the development ofa conceptual model for the control of a multilink biped during a turning maneuver. The skeletal model is a seven link biped for which the equations of motion are derived. A set of lower limb muscles are idealized by simple force actuators with no co-contraction of agonist-antagonist muscle pairs. A nonlinear control scheme is proposed to guide the model along the desired trajectory and to control ground reaction forces. The input to the system isa desired set of trajectories as functions of time and the patterns of desired ground reaction forces in a turn. One set of such inputs are inferred from the existing literature. With this input, the nonlinear control strategy allows computation of muscular forces needed for the turning maneuver.
1. INTRODUCI'ION
This paper deals with the development of a conceptual model for the control of a seven link biped in a turning maneuver. The ground reaction forces and the trajectories of motion must be known in order to implement the turn. The ability to predict the actuator forces and to simulate human locomotion could provide a basis for the design of prostheses and orthoses used by the physically impaired and help the development of effective muscular and neural functional electrical stimulators. Turning is one of the common activities of humans. Work on turning has been reported by Kane and Scher (1970), Morawski (1973), Kulwicki et 01. (1962) while Murray et al. (1964), Seireg and Arvikar (1975), Mena et al. (1981) and Bresler and Frankel (1980) have studied normal walking. The development of a mathematical model for the musculo-skeletal system capable of evaluating muscle and joint reaction forces for different static postures was studied by Seireg and Arvikar (1973, 1975). However, the force patterns they reported were obtained under the conditions that the system is in equilibrium at all times and inertia forces and moment are neglected. Most papers on dynamic analysis of human locomotion use ideal torque generators and ignore muscle actions. There are single and double stance phases involved in turning. Thus the motion of turning may be considered as a sequence of constrained and unconstrained movements as suggested by Hemami and Wyman (1979). In order to use the model in the bipedal model, it is necessary to have a complete set of kinematic trajectories and ground reaction force patterns. This inforReceived23 October 1984;in revised/arm
12 July
1985.
mation is inferred from the existing literature and is discussed in section 2 while section 3 discusses the force actuator model used in this study. Actuator forces are assumed to be directed along straight lines joining the corresponding muscle points of origin and insertion. In section 4 the Newton-Euler state space formulation developed by Hemami (1980) is used to set up the equations of motion of this system. Many control strategies such as linear feedback control, nonlinear feedback control (Hemami and Zheng, 1984) and model-referenced adaptive control (Dubowsky and DesForges, 1979; Koivo and Guo. 1983) can be employed to compensate for possible deviations of the system from the desired trajectories and to stabilize the system. A nonlinear control strategy is considered in section 5. Based on the control strategy, the actuator forces are determined in section 6. 2. DEVELOPIMENTOFK~NEMATICTRAJECTOR~E!SAND COSNECilONCONSTRAlNTS
The model used consists of seven links (Fig. 1) in order lo approximate locomotion characteristics similar to those of the lower extremities of the human body. The foot, the leg and the thigh each are considered as a single rigid body. Consequently three connected rigid bodies represent the lower limb on one side. Another set of three rigid bodies define the other limb. A seventh rigid body represents the pelvis, trunk, hands, neck and head. The mass, length, moment of inertia and center of gravity parameters are taken from Drillis ef al. (1984), and Drillis and Contini (1966) and are listed in Table I. The procedure for making a right turn involves several phases, that according to Hartrum (1973) and Saunders el al. (1953), are performed sequentially, 195
B.-R. CHEN. M. J. HINES and H. HEMAMI
196
Link 4
Frontal Plow View
Soqlttal Plow View
Fig. 2. The skeletal biped model. Fig. 1. The seven link biped model and the body coordinate of the sixth link.
although in a real human turn some phases may overlap. Initially both feet are on the ground. In turning to the right, the support gradually shifts to the left leg so that the right foot can be lifted. The next step is the rotation of the pelvis in the transverse plane (xy plane in Fig. 2) about a longitudinal axis (z axis). A rotation of the pelvis in the frontal plane (xz plane) about a frontal axis (y axis) follows. The knee of the right limb or extremity bends to allow the foot to lift. The right limb turns and is lowered until the foot touches the ground. At this moment, both feet again are on the ground. The support gradually shifts from the left limb to the right limb. The next phases include knee flexion to lift the left leg. Moving, turning and lowering of the left foot phases follow until the left foot touches the ground and the right turn is completed. From the experimental data, it has been determined that it takes about 3-5 s to make a complete slow right turn. Figure 3 shows the swing and stance phases of both feet during turning. This figure is composed from the average of twenty trials by two individuals making slow turns. In this study the duration of turning is taken to be 4.3 s. For the model shown in Fig. 1, the number of rotational degrees of freedom of the hip, knee and ankle joints are respectively three, one and two. In single stance phase with the left foot on the ground, the system has a total of twelve degrees of freedom. In double stance phase, five holonomic constraints are
Right Leq Timebec)
Time bet)
Fig. 3. The different phases of both feet of biped model during turning.
added to the system. Three of these constraints are translational and the remaining two are rotational constraints. Therefore, the total system degrees of freedom are seven. The twelve degrees of freedom of the skeletal system are defined here by twelve angles denoted as 4 . . . , dl 2. The correspondence of these angles to ai &atomicaI model is given in Table 2. In this study the trajectories of +,-4,2 as a function of time are obtained from a sequence of photographs of the turning motion from Muybridge (1979), Kish (1978) and Tabor (1981). The discrete points on the trajectory are interpolated to obtain continuous time functions
Table 1. Parameters of the seven link biped model Link number
3 2 1 8
(Pelvis) (Left thigh) (Left leg) (Left foot) (Trunk)
5 6 7
(Right thigh) (Right leg) (Right foot)
4
Moments of inertia Ii3 Ii2
Mass Mi
Ii,
6.1 7.0 3.1 1.0
0.049 0.089 0.105 0.004
0.049 0.089 0.105 0.004
0.0067 0.0276 0.0088 0.0015
42 7.0 3.1 1.0
t.75 0.089 0.105 0.004
1.75 0.089 0.105 0.004
0.4 0.0276 0.0088 0.0015
Ii
Ki
0.165 0.247 0.267 0.123 0.43 0.184 0.235 0.118
0.165 0.184 0.235 0.118 0.37 0.247 0.267 0.123
c3
0.27 0.25 0.264 0.26 0.252 0.25 0.264 0.26
d
0.095 0.172 0.125 0.11 0.276 0.172 0.125 0.11
Implementation of a right turn in bipedal walking
197
Table 2. The anatomical correspondence of the angles I$,-&,~
Leg
The function of the angle
The anatomical correspondence Lateral and medial rotation of the hip joint Abduction and adduction of the hip joint Flexion and extension of the hip joint Flexion and extension of the knee joint Lateral and medial rotation of the ankle joint Flexion and extension of the ankle joint
Right
Lateral and medial rotation of the hip joint Abduction and adduction of the hip joint Flexion and extension of the hip joint Flexion and extension of the knee joint Lateral and medial rotation of the ankle joint Flexion and extension of the ankle joint
by power series approximations discussed by Hemami and Farnsworth (1977). These power series are differentiated in order to derive the corresponding angular velocities and accelerations. The results are shown in Figs 4a and b where the angles are expressed in degrees. The desired ground reaction forces cannot be inferred from the above photographic trajectories of motion. These forces are approximated here by those provided by Vukobratovic (1975) for a normal walk (Fig. 5). In order to verify the kinematic trajectories 41,. . . 94129 the kinematic equations of the biped model are used to generate a perspective of the turning motion on a graphic display (Fig. 6). The Euler angles corresponding to pitch, roll and self rotation as defined by Chow and Jacobson (1972) can be related to 4 = [r$i, . . , 4,,]’ as follows. Let Bi = [Oil, @il. t&]’ be the vector of these rotational Euler angles of body i, and let the relative rotations of the limbs be about axes that are parallel to the principal axes of the bodies. With these assumptions, the set of independent variables 4 is related to the Euler angles as follows
&2 = 49 063= 41, 071= 4% 872 = 0
073= 412
(1)
whereB,i = e2i = &, = 4, due to the ankle and knee joints of the left leg not being allowed to rotate along the z-axis of their body coordinates. If we assume that the upper body (link 4) always remains in an erect posture during turning, then eqj = 0. We also assume that 0i2 = 0,* = 0 during the entire motion.
3.
ACTUATOR DYNAMICS
Table 3, taken from anatomical literature; i.e. Rasch and Burke (1978), Carlsoo (1972). Braus (1954). lists the muscles which are considered to be important in turning. The frontal and sagittal views of the attachment of these muscle-like actuators for the lower limbs are shown in Fig. 7. It should be noted that since this is a sagittal view only the lateral one-half of the hamstrings and quadriceps groups are represented in Fig. 7. The approximate points of origin and insertion of muscles are obtained from Rasch and Burke (1978)and Braus (1954) and are listed in Table 3 where the subscript indicates the particular body coordinate system. The origin of each body coordinate system is located at its center of mass. The lower limb actuators have been idealized by lines joining the muscle point of origin to the point of insertion and are assumed to produce tensile forces only. For some two-joint muscles such as the rectus femoris and gastrocnemius, a straight line representation is not possible due to an interposing structure. In this case, two line segments are used to circumvent the interposing structure, and approximate the shape. The intricate and highly nonlinear dynamics of the muscles are not included in the study. The length variation of a muscle is defined as the difference between the length of a muscle and its reference length, the length measured when the model stands on the ground in an
B.-R. CHEN. M. J.
198
HINESand
4.
erect posture. The reference length defined here is different from physiological rest length because the turning here starts from an erect posture. Based on the turning sequences shown in Fig. 6, the length variations of all the actuators can be derived as functions of time and are shown in Figs 8 and 9.
H. HEMAMI
THE
DYNAMICS OF THE SYSTEM
The Newton-Euler state space formulation developed by Hemami (1980) is used to derive the equations of motion. In Fig. 1, each of the knee joints is assumed to have 4 3 %y
+a 0
!!Gq--Jqfy--J
L&I 0
I
2 3 Time(&)
4
8,
I
4
j4
1”
-4 0
l
2 3 Time(w)
4
I
0
2 3 Time (set)
4
2 3 Time (see)
4
Ir%d 0
0
I
Time (set)
2
3
4
Timetsec)
I
-401
I
-0-’ Time (set)
8 4
10 -4
lsi!id
0 Timebc)
4
I
2 3 Time (set)
4
Time (see)
0 LZII 0 Time (set)
Time (set)
2 3 T ims (ret I
Timebecc)
Time (ret)
Time (reel
Fig. 4. (a) Desired kinematic
trajectories
I
in the turning maneuver.
4
Implementation of a right turn in bipedal walking
i6r
1
I
0
1
2 3 Time (ret)
J
I
Time (set)
Time(sec)
Time (set)
Time (set)
Timetsec)
Time Ised
3 I
'xl
L 0
-’
I
2 3 Timetsecl
4
J
a N
4
\" y
0
$ D
-4 -8 -12
0
I
2
3
4
Time (secl
8
0
E LEiiEl
-4 B
$0
4
s a
0
*o
0
-12 -8m 0
I2
3
4
-4 0
I2
Time(sec1
IO
0
-10
8, m 40
0
12
3
4
01234
Time (set)
40
$ ,” 3 D
20 0 -20
N
3 Time (set)
0
I
2
3
o0
I2
3 Time (set)
4
12 4,
6 0 -6 R:-l 0 I2
3
Time Ised
TimeCsec)
Time(sec)
Time (set)
4
4
Timefsec)
Fig. 4. (b) Desired kinematic
trajectories in the turning maneuver.
degree of freedom and each of the ankle joints is assumed to have two rotational degrees of freedom. There are 22 actuators, as listed in Table 2. Using all the forces discussed above, the equation of motion of the biped system can be obtained. The actual only one rotational
4
degrees of freedom of this system are twelve for the single stance phase and seven for the double stance phase according to Hemami (1980) and Hemami and Zheng (1984). The projection transformations for reducing the
B.-R. CHEW
200
M. J. HINES
and H. HEMAMI
0
Qnl I-’
-8 -16
x
0
40
3
4
2
3
4
2
3
4
40
20
30
5
0
r;
20
-20 -40
2
IO pEl 0
2
3
Lsyll! OO
4
450 300
c 250
200
I50 50 III
r;
loo 0
I
2
3
mi
4
OO
Time bed Fig. 5. The desired ground
t=o
t =0.3
147
VI.1
PI.5
t 4.9-
JlhBul
ta4.3
1’3.9
t=3.5
ts3.1
t =2.7
t=2.3
Fig. 6. The sequencesof turning 90” to the right.
dimensionality of the system for the three different cases are discussed separately. A detailed derivation is provided in Chen (1985). Case 1 Single stance phase with left foot on the ground. The first step is to reduce the Euler-Newton equations to smaller space of Euler angles Bi (i = 1, . . , 7) and angular velocities w (i = 1, . . . , 7) expressed in the body coordinate system. (X. %) in terms of (0, W) where W is the components of the angular velocity
reaction force patterns.
expressed in a body coordinate system. It is shown in Hemami (1980) that the corresponding constraint forces can be eliminated due to this reduction. In the second step of projection, the state space (0, I+) is projected to (6.4). From equation (1) symbolically written as
the reduced equations of the seven link biped system for this case are W)6
+ H(4,d) + GM) + E(& &I- = T
(3)
where D(b) is the inertial matrix, H(& 4) represents Coriolis and centrifugal terms, G(4) denotes the gravity loading term and I?($, 4) describes the effect of the contact forces y of the right foot. A detailed derivation of equation (3) is given in Chen (1985). Case 2 Single stance phase with the right foot on the ground. Following the same two step projection for the single stance phase with the left foot on the ground results in D’(f#+$ + H’(#, &) + G’(#) + E’(#. &,r where
= T’ (4)
Left
Right
Leg
L9 LIO Lll
L5 L6 L7 L8
LI L2 L3 L4
R9 RIO Rll
lliacus Rectus fcmoris Gluteus maximus Biceps femoris (long head) Vastus lateralis Gluteus medius Adductor longus Biceps femoris (short head) Gastrocnemius Tibialis anterior Soleus
Iliacus (IL) Rectus femoris (RF) Gluteus maximus (GM) Biceps femoris (BFLH) (long head) Vastus lateralis (VAL) Gluteus medius (GME) Adductor longus (AD) Bicepts femoris (BFSH) (short head) Gastrocnemius (GA) Tibialis anterior (TA) Soleus (SO)
RI R2 R3 R4
RS R6 R7 RS
Muscle
No.
Table
(m)
point
of the muscle
- 0.136)’ 0.12)’ - 0.069)’ 0, - 0.206))
(- 0.052, 0. - 0.206)’ (0.0. -0.115y ( - 0.02, 0. - 0.2 15)’
(0.034. 0, (0, 0.165. (0, 0.065, ( - 0.052,
(0.08, 0.165. 0.085)* (0.08, 0.165. 0.0R5)4 ( - 0.08. 0.165, 0)4 (-0.08, 0.165. 0)4
( - 0.052, 0. - 0.206)s (0.0. -0.115)” (-0.02.0, -0.215)b
(0.034, 0, - 0.136)’ (0, -0.165, 12)’ (0, -0.065. -0.069)’ ( - 0.052, 0, - 0.206)s
(0.08, - 0.165, 0.085)* (0.08, - 0.165, 0.085). (- 0.08, - 0.165, 0). (-0.08. -0.165, 0)’
Origin
3. Parameters
(- 0.018. ( - 0.049, ( - 0.038,
(0.02, 0, (0. 0. (0, 0. ( - 0.06, 0. 0)’ 0.0)’ 0, 0)’
0.183)r 0.076)’ 0.076)’ 0, 0.155)’
(0. 0, 0.33)’ (0.02. 0. 0.183)x (0. 0, - 0.032)’ ( - 0.056, 0. 0.215)’
( - 0.018, 0. 0)’ ( - 0.049. 0, 0)’ ( - 0.038.0,O)’
(0.02, 0, 0. 183)6 (0, - 0.052, O.O46)5 (0, 0. - 0.076)5 ( - 0.06. 0. 0.155y
(0, 0, 0.33)5 (0.02, 0. 0.183)6 (0, 0, - 0.032)’ (- 0.056, 0, 0.21 5)6
point
of the lower
(m)
Insertion
actuators
0, - 0.226)’
Midpoint
0. - 0.226)’
0, 0.155)6
( - 0.06,
0, 0.155)’
(0.069, 0. - 0.226)s (0, 0.069, 0.15)’
-_-
(0.069.
---
( - 0.06,
---
(0.069, 0, - 0.226)’ (0, - 0.069. 0. I5)5
(0.069,
---
extremities length
0.545 0.167 0.0794
O.lR45 0.274 0.216 0.121
0.24Y2 0 5x3 0.23 0.452
0.545 0.167 0.07’) 4
0.1845 0.214 0.216 0.121
0.2492 0.583 0.23 0.452
(m)
Reference
B.-R. CHEN, M. J. HINESand H. HEMAMI
202
Right Leg in the Sogittal Plane
posed. The iower extremities collectively have three major functions: 1. To perform and guide the turning motion. 2. To control the ground reaction forces. 3. To stabilize the posture during the motion. Each function requires a system of forces. Let T, be the vector of inputs which is generated to control the ground reaction forces when contact constraints are in effect. Let Tz be the vector of forces generated to guide the model motion and let T3 be the vector of forces to stabilize the system during the motion. For the single stance phase with the left foot on the ground, then
Ripht Leg in the Frontol Plana
Fig. 7. A musculo-skeletal planar model for the lower
T = T1 -I-T2 -I-T,
extremities. Case 3
(5)
and
Double stance phase. The dynamic equations for the double stance phase have already been included in equations (3) and (4). The constraint force r of equation (3) acts on the system while both feet are on the ground. Thus, the equations of motion for the seven link biped model with either one or two feet on the ground are formulated. These equations are essential to the design of the control strategy and to the digital computer simulations of the turning motion. 5. CONTROL STRATEGY
Due to the complexity of both the system and the desired motion, a nonlinear control strategy is pro-
T, = W,
(6)
i,r
where E(& 4) is defined in equation (3). TZ can be computed by the inverse plant method. Assume that the components of f)(4), H(4.4), G(4) and E(4,4) can be accurately computed, then
where the subscript d denotes the desired trajectories. In order to stabilize the system, T, is generated from position and velocity feedback as Ts = W) [Kv(dd -4) + Kp(& - $0
0.03 &ccps
-003 0
(8)
Therefore, the total inputs which are generated by
D 0’2
*DO2-
(7)
j-2 = WJ)& + W#+ d) + GM)
I
2
3
0
4
I
2
3
4
0
I
femaris
2
003
DO2 Gluteus hkuimus
Ttbmlts
Biceps Femoris LLong lieou~
Gostrocnemius
Glutsus Me&s
-0030
I
2
3
Time (set) Fig. 8. Muscle length variations of the left leg.
4
Anlerkw
3
4
Implementation
002
002
203
of a right turn in bipedal walking
‘-_11E_:i
0
0 vasfusLaerahs -002 H 0
I2
3
4
-002
I2
3
4
0
I2
3
4
Time (set) Fig. 9.
Muscle length variations of the right leg.
actuators are T = w4
[id + WA
- 44 + K,Wd - 411
+ HC#h4, + w#4 + w.
Substituting renders
the above equation
(9)
dw.
in equation
of each link in the biped system from that of the other links. If the desired eigenvalues associated with (&, di) are li, and Liz, the diagonal elements of matrices K, and K, are K,(i, i) = li, + li,
(13)
K,(i, i) = Li, . Ait.
(4d-;i)+Ko(&-&+Kp(#J~-#)
=O.
Equation (IO) represents a linear system. The block diagram is shown in Fig. 10. By selecting diagonal matrices K, and K,. one is able to decouple the motion
The decoupled motion of the ith link is then described by A$i + (Ai, + Li,)A&i + i-i, ’ li, A4i = 0
igi”~-piiql
_I q-
\J Inverse
4 +
Plonr
PIOCM
t t
73 \,
\/
'LKp I +d
=
Desired +d Stores 4 d
_
(11)
(10)
Fig. IO. The control block diagram of the system.
(12)
204
B.-R. CHEN. M. J. HINESand H. HEMAMI
where
By assigning the eigenvalues to the system with negative real parts, the gain matrices K, and K, can be obtained to stabilize the system. One set of values used for the computations reported in this paper is given in Table 4. The same methods can be used to generate inputs by muscles for the single stance phase with the right foot on the ground. 6.
RESULTS
AND DISCUSSIONS
Based on the desired kinematic data and ground reaction forces, the control torques are calculated by using equation (9), and are plotted in Fig. 11. The calculation of T involves two stages. (1) For a given set of initial angular positions (6)(O) and angular velocities 4(O), and with the desired &(t), 4,,(t) and &(t), equation (10) is solved for the actual state 4(t) and &(I). Since the desired kinematic trajectories are not analytical functions of time, this solution is arrived at by digital computer simulation. (2) The desired trajectories and these calculated actual trajectories are used in equation (9) to compute T. The next step is to compute actuator forces from the computed torques T. There are more unknowns than equations in computing actuator forces. This implies that many feasible solutions are possible. One therefore has to set up a criterion to choose a suitable
solution from the many possibilities. The performance criterion used here is to minimize the sum of squares of forces required to produce these torques. It is assumed here that coactivation of muscles is not admissible. The computed force patterns of the actuators which are listed in Table 3 are shown in Figs 12 and 13. The actuator forces are very small at the initial, middle, and final phases of motion because of the two limb support. All actuator forces go to zero at the final state of the system because a symmetric double static stance with zero bias torque is assumed. All the actuator forces are positive and in reasonable range, The magnitudes of forces are relatively small which is due to the slow turning motion. Additional techniques can assist in deriving unique solutions to the muscle force distribution. Moments of force and mechanical power (Winter, 1983a) or energy absorption and generation criteria in muscles (Winter, 1983b) can be added to the formulation as additional constraints in order to arrive at more physiologically acceptable force patterns. The concept of relegation of function to arbitrary groups of muscles (Wongchaisuwat, 1985) can also simplify the solution of the force problem. In this formulation the functions of stability, trajectory control, and force of contact (or support) control can be relegated to different groups of muscles. The minimization of the sum of squares of forces implies that all muscles contribute to the turning motion uniquely throughout the time interval of
Table 4. Diagonal matrices K, and K, used for computation of forces. Eigenvalues -4 and -4 result from feedbacks k, = 8 and k, = 16. Eigenvalues - 5 and - 5 result from k, = 10 and k, = 25 K, = diagonal matrix (8. 8, 8, 8, 8, 8, IO, 10, 10, 10, 10, 10) K, = diagonal matrix (16, 16, 16, 16, 16, 25, 25, 25, 25, 25, 25)
Time (set) Fig. 11. The input torques of the system.
Implementation
of a right turn in bipedal walking
Fig. 12. Actuator
205
force patterns of the left leg
Time (set) Fig. 13. Actuator
force patterns of the right leg.
interest. An alternative strategy is to minimize a weighted sum of squares of forces’where the weight would allow larger muscles to bear larger loads. The latter strategy can also be applied to different nonoverlapping phases of motion in order to shift major load bearing from one group of muscles to another. Antigravity muscles that control weight bearing appear to have such a relegated function. Since there is no convenient experimental method for direct quantitative measurement of muscle forces, EMG data may be used as qualitative measures. As in Pedotti et al. (1978) the force patterns do not only depend on a particular criterion but also depend on the kinematic data and the ground reaction forces. Therefore, different actuator torques and EMG patterns may result for different kinematic data. To date,
applicable EMG data are not available for checking the correctness of the proposed control strategy and criterion. Thus in this study a feasible set of forces are specified that can be utilized as a guide in electrical or neural muscle stimulation, and could serve as a first order approximation in helping the physically impaired regain mobility. This method could also be applied to a system without deep muscles, i.e. the number of actuators could be reduced.
7. SUMMARY
A mathematical model which is capable of quantitative computation of simple force actuators of a seven link biped system was presented. Newton-Euler state
B.-R. CHEN.
206
M. J. HISES and H. HEMAMI
space formulation was used to derive the equations of motion. This formulation is convenient for including arbitrary actuators and Lyapunov stability considerations as in Hemami and Chen (1984).
parametrtc model for biped locomotion kinematics. Ph.D. dissertation. The Ohio State University, Columbus. OH. Hemami. H. (19801 A feedback on-off model of biped dynamics. IEEE Trans. Sysf. Man Cybernef. SMC-10,
Nonlinear feedback was employed for stabilizing the system and decoupling the degrees of freedom of the system. The advantage of using nonlinear feedback is that of simplicity. Theoretically, the performance of the biped system with nonlinear feedback is as good as a linear system. This means the errors asymptotically go to zero. A set of kinematic trajectories and ground reaction forces were inferred from the literature. Based on these trajectories and nonlinear control strategy, the required muscle force patterns were obtained. In practice, there are some problems. The masses and moments of inertia are contained in the nonlinear term. For biped systems it is usually very difficult to accurately measure these parameters. Second, a predictor has to be included to account for computation delay. However, a model could be specified with only superficial muscle actuators-those that are more amenable to direct electrical stimulation. This simplified model would have immediate uses in functional electrical stimulation. Finally, one more observation must be made. Though the bipedal system is modelled after human beings, no claim is made that the control structures or the actuator forces are those employed in the human turning maneuver.
Hrmami, H. (1982) A state space model for interconnected
Acknowledyzmenr-This work was supported in part by NSF Grant ECS-820-1240 and in part by the Department of Electrical Engineering, The Ohio State University, Columbus, OH 43210, U.S.A.
REFERENCES Braus, H. (1954) Anafomie des Menschen, Vol. 1. Springer, Berlin. Bresler, B. and Frankel, J. P. (1980) The forces and moments in the leg during level walking. Trans. ASME 72, 27-36. Carlsoo, S. (1972) How Man Mores-Kinesiological Studies and Mefhods. Crane, Russak, New York. Chen, B.-R. (1985) Musculo-skeletal dynamics and multiprocessor control of a biped model in a turning maneuver. Ph.D. dissertation, Department of Electrical Engineering, The Ohio State University. Chow, C. K. and Jacobson, D. H. (1972) Further studies of human locomotion: postural stability and control. Math/ Biosci. IS. 93-108. Contini. R. (1970) Body segment parameters (pathological). Technical Report No. 1584.03,School of Engineering and Science, University of New York. Drillis, R. and Contini, R. (1966) Body segment parameters (pathological). Technical Report NO. 1163.03, School of Engineering and Science, University of New York. Drillis, R., Contini. R. and Bluestein. M. (1984) Body segment parameters. Arrif. Limbs 8.44-66. Dubowskv. S. and DesForges, D. T. (1979)The application of model-referenced adap&e control to robotics manipulators. J. dynam. SJsf. Mrasmr Confrol 101, 193-200. Hartrum, T. C. (1973) Computer implementation of a
376-383.
rigid bodies. IEE.& Trans. Aufomafic Control
AC-27,
376-382.
Hemami, H. and Chen, B.-R. (1984) Stability analysis and input design of a two-link planar biped. Inf. J. Robofics Res. 3, 93100. Hemami. H. and Farnsworth, R. L. (1977) Postural and gait stability of a planar five link biped by simulation. IEEE
Trans. Automatic Control AC-22, 452-457. Hemami, H. and Wyman, B. F. (1979) Modeling and control of constrained dynamic systems with applications to biped locomotion in the frontal plane. IEEE Trans. Aufomafic Confrol AC-24. 526-535. _ Hemami, H. and Zheng. Y.-F. (1984) Dynamics and control of motion on theehe ground and in the air with application to biped robots. J. Robotic Sysf. 1, 101-l 16. Kane, T. R. and Scher. M. F. (1970) Human self-rotation by means of limb movements. J. Biomechanics 3, 39-49. Kish, L. R. (1978) A computer model and display of articulated human movements. MS. Thesis, Electrical Engineering Department, Ohio State University. Koivo, A. J. and Guo, T. H. (1983) Adaptive linear controller for robotic manipulators. IEEE Trans. Aufomafic Control AC-28,
162-171,
Kulwicki, P. V.. Schlei, E. J. and Vergamini, P. L. (1962) Weightless self-rotation techniques. Ohio Technical Report No. TDR 62-129, 6470th Aerospace Medical Research Laboratory, Wright-Patterson Air Force Base. Mena, D., Mansour, D. M. and Simon, S. R. (1981) Analysis and synthesis of human swing leg motion during gait and its clinical application. J. Biomechanics 14, 823-832. Morawski, J. M. (1973) Control systems approach to a skiturn analysis. J. Eiomechanics 6, 267-279. Murrav. M. P.. Droueht. A. B. and Korv. R. C. (1964) Walking patterns ofnormal men. J. BoneJr Surg. h6-A; 335-360. Muybridge, E. (1979) Muybridges Complefe Human and Animal Locomofion. Dover, New York. Pedotti, A., Krishnan, V. V. and Stark, L. (I 978) Optimization of muscle-force sequencing in human locomotion. Mafhl Biosci. 38, 57-76. Rasch, P. J. and Burke, R. K. (1978) Kinesiologg and Applied Anatomy. 6th ed. Lea and Febiger, Philadelphia, PA. Saunders, J. B.. Inman, V. T. and Eberhart, H. D. (1953) The major determinants in normal and pathological gait. J. Bone Jr Surg. 35-A. 543-558. Seireg, A. and Arvikar, R. J. (1973) A mathematical model for evaluation of forces in lower extremities of the musculoskeletal system. J. Biomechonics 6, 313-326. Seireg, A. and Arvikar, R. J. (1975)The prediction of muscular load sharing and joint forces in the lower extremities during walking. J. Biomrchanics 8, 89-102. Tabor, J. (1981) Walking: focus on motion. Backpacker 9, 40-49. Vukobratovic. M. (1975) Lugged Locomotion Robofs and Anrhropomorphic Xfechonisms, Mihailo Pupin Institute, Biograd. Winter, D. A. (1983a) Moments of force and mechanical power in jogging. J. Biomechanics 16, 91-97. Winter, D. A. (1983b) Energy generation and absorption at the ankle and knee during fast, natural, and slow cadences. Clin. Orfhop. Rel. Rex 175. 147-I 54. Wongchaisuwat, C. (1985) Control of constrained motion in natural and robotic systems. Ph.D. dissertation, The Ohio Stale University.
195-206. 1986.
0021-9290~86 S3.M) + .oO
in Great Bnlam
g
DYNAMIC
MODELLING FOR IMPLEMENTATION RIGHT TURN IN BIPEDAL WALKING
1986 Pcrgamon
Press Ltd.
OF A
BEN-RENCHEN Bell Communications Research. Red Bank, NJ 07701, U.S.A. and
MARGARET J.HINES* and HOOSHANG
HEMAbUt
*Department of Anatomy and TDepartment of Electrical Engineering, The Ohio State University, Columbus, OH 43210, U.S.A. Abstract-This paper deals with the development ofa conceptual model for the control of a multilink biped during a turning maneuver. The skeletal model is a seven link biped for which the equations of motion are derived. A set of lower limb muscles are idealized by simple force actuators with no co-contraction of agonist-antagonist muscle pairs. A nonlinear control scheme is proposed to guide the model along the desired trajectory and to control ground reaction forces. The input to the system isa desired set of trajectories as functions of time and the patterns of desired ground reaction forces in a turn. One set of such inputs are inferred from the existing literature. With this input, the nonlinear control strategy allows computation of muscular forces needed for the turning maneuver.
1. INTRODUCI'ION
This paper deals with the development of a conceptual model for the control of a seven link biped in a turning maneuver. The ground reaction forces and the trajectories of motion must be known in order to implement the turn. The ability to predict the actuator forces and to simulate human locomotion could provide a basis for the design of prostheses and orthoses used by the physically impaired and help the development of effective muscular and neural functional electrical stimulators. Turning is one of the common activities of humans. Work on turning has been reported by Kane and Scher (1970), Morawski (1973), Kulwicki et 01. (1962) while Murray et al. (1964), Seireg and Arvikar (1975), Mena et al. (1981) and Bresler and Frankel (1980) have studied normal walking. The development of a mathematical model for the musculo-skeletal system capable of evaluating muscle and joint reaction forces for different static postures was studied by Seireg and Arvikar (1973, 1975). However, the force patterns they reported were obtained under the conditions that the system is in equilibrium at all times and inertia forces and moment are neglected. Most papers on dynamic analysis of human locomotion use ideal torque generators and ignore muscle actions. There are single and double stance phases involved in turning. Thus the motion of turning may be considered as a sequence of constrained and unconstrained movements as suggested by Hemami and Wyman (1979). In order to use the model in the bipedal model, it is necessary to have a complete set of kinematic trajectories and ground reaction force patterns. This inforReceived23 October 1984;in revised/arm
12 July
1985.
mation is inferred from the existing literature and is discussed in section 2 while section 3 discusses the force actuator model used in this study. Actuator forces are assumed to be directed along straight lines joining the corresponding muscle points of origin and insertion. In section 4 the Newton-Euler state space formulation developed by Hemami (1980) is used to set up the equations of motion of this system. Many control strategies such as linear feedback control, nonlinear feedback control (Hemami and Zheng, 1984) and model-referenced adaptive control (Dubowsky and DesForges, 1979; Koivo and Guo. 1983) can be employed to compensate for possible deviations of the system from the desired trajectories and to stabilize the system. A nonlinear control strategy is considered in section 5. Based on the control strategy, the actuator forces are determined in section 6. 2. DEVELOPIMENTOFK~NEMATICTRAJECTOR~E!SAND COSNECilONCONSTRAlNTS
The model used consists of seven links (Fig. 1) in order lo approximate locomotion characteristics similar to those of the lower extremities of the human body. The foot, the leg and the thigh each are considered as a single rigid body. Consequently three connected rigid bodies represent the lower limb on one side. Another set of three rigid bodies define the other limb. A seventh rigid body represents the pelvis, trunk, hands, neck and head. The mass, length, moment of inertia and center of gravity parameters are taken from Drillis ef al. (1984), and Drillis and Contini (1966) and are listed in Table I. The procedure for making a right turn involves several phases, that according to Hartrum (1973) and Saunders el al. (1953), are performed sequentially, 195
B.-R. CHEN. M. J. HINES and H. HEMAMI
196
Link 4
Frontal Plow View
Soqlttal Plow View
Fig. 2. The skeletal biped model. Fig. 1. The seven link biped model and the body coordinate of the sixth link.
although in a real human turn some phases may overlap. Initially both feet are on the ground. In turning to the right, the support gradually shifts to the left leg so that the right foot can be lifted. The next step is the rotation of the pelvis in the transverse plane (xy plane in Fig. 2) about a longitudinal axis (z axis). A rotation of the pelvis in the frontal plane (xz plane) about a frontal axis (y axis) follows. The knee of the right limb or extremity bends to allow the foot to lift. The right limb turns and is lowered until the foot touches the ground. At this moment, both feet again are on the ground. The support gradually shifts from the left limb to the right limb. The next phases include knee flexion to lift the left leg. Moving, turning and lowering of the left foot phases follow until the left foot touches the ground and the right turn is completed. From the experimental data, it has been determined that it takes about 3-5 s to make a complete slow right turn. Figure 3 shows the swing and stance phases of both feet during turning. This figure is composed from the average of twenty trials by two individuals making slow turns. In this study the duration of turning is taken to be 4.3 s. For the model shown in Fig. 1, the number of rotational degrees of freedom of the hip, knee and ankle joints are respectively three, one and two. In single stance phase with the left foot on the ground, the system has a total of twelve degrees of freedom. In double stance phase, five holonomic constraints are
Right Leq Timebec)
Time bet)
Fig. 3. The different phases of both feet of biped model during turning.
added to the system. Three of these constraints are translational and the remaining two are rotational constraints. Therefore, the total system degrees of freedom are seven. The twelve degrees of freedom of the skeletal system are defined here by twelve angles denoted as 4 . . . , dl 2. The correspondence of these angles to ai &atomicaI model is given in Table 2. In this study the trajectories of +,-4,2 as a function of time are obtained from a sequence of photographs of the turning motion from Muybridge (1979), Kish (1978) and Tabor (1981). The discrete points on the trajectory are interpolated to obtain continuous time functions
Table 1. Parameters of the seven link biped model Link number
3 2 1 8
(Pelvis) (Left thigh) (Left leg) (Left foot) (Trunk)
5 6 7
(Right thigh) (Right leg) (Right foot)
4
Moments of inertia Ii3 Ii2
Mass Mi
Ii,
6.1 7.0 3.1 1.0
0.049 0.089 0.105 0.004
0.049 0.089 0.105 0.004
0.0067 0.0276 0.0088 0.0015
42 7.0 3.1 1.0
t.75 0.089 0.105 0.004
1.75 0.089 0.105 0.004
0.4 0.0276 0.0088 0.0015
Ii
Ki
0.165 0.247 0.267 0.123 0.43 0.184 0.235 0.118
0.165 0.184 0.235 0.118 0.37 0.247 0.267 0.123
c3
0.27 0.25 0.264 0.26 0.252 0.25 0.264 0.26
d
0.095 0.172 0.125 0.11 0.276 0.172 0.125 0.11
Implementation of a right turn in bipedal walking
197
Table 2. The anatomical correspondence of the angles I$,-&,~
Leg
The function of the angle
The anatomical correspondence Lateral and medial rotation of the hip joint Abduction and adduction of the hip joint Flexion and extension of the hip joint Flexion and extension of the knee joint Lateral and medial rotation of the ankle joint Flexion and extension of the ankle joint
Right
Lateral and medial rotation of the hip joint Abduction and adduction of the hip joint Flexion and extension of the hip joint Flexion and extension of the knee joint Lateral and medial rotation of the ankle joint Flexion and extension of the ankle joint
by power series approximations discussed by Hemami and Farnsworth (1977). These power series are differentiated in order to derive the corresponding angular velocities and accelerations. The results are shown in Figs 4a and b where the angles are expressed in degrees. The desired ground reaction forces cannot be inferred from the above photographic trajectories of motion. These forces are approximated here by those provided by Vukobratovic (1975) for a normal walk (Fig. 5). In order to verify the kinematic trajectories 41,. . . 94129 the kinematic equations of the biped model are used to generate a perspective of the turning motion on a graphic display (Fig. 6). The Euler angles corresponding to pitch, roll and self rotation as defined by Chow and Jacobson (1972) can be related to 4 = [r$i, . . , 4,,]’ as follows. Let Bi = [Oil, @il. t&]’ be the vector of these rotational Euler angles of body i, and let the relative rotations of the limbs be about axes that are parallel to the principal axes of the bodies. With these assumptions, the set of independent variables 4 is related to the Euler angles as follows
&2 = 49 063= 41, 071= 4% 872 = 0
073= 412
(1)
whereB,i = e2i = &, = 4, due to the ankle and knee joints of the left leg not being allowed to rotate along the z-axis of their body coordinates. If we assume that the upper body (link 4) always remains in an erect posture during turning, then eqj = 0. We also assume that 0i2 = 0,* = 0 during the entire motion.
3.
ACTUATOR DYNAMICS
Table 3, taken from anatomical literature; i.e. Rasch and Burke (1978), Carlsoo (1972). Braus (1954). lists the muscles which are considered to be important in turning. The frontal and sagittal views of the attachment of these muscle-like actuators for the lower limbs are shown in Fig. 7. It should be noted that since this is a sagittal view only the lateral one-half of the hamstrings and quadriceps groups are represented in Fig. 7. The approximate points of origin and insertion of muscles are obtained from Rasch and Burke (1978)and Braus (1954) and are listed in Table 3 where the subscript indicates the particular body coordinate system. The origin of each body coordinate system is located at its center of mass. The lower limb actuators have been idealized by lines joining the muscle point of origin to the point of insertion and are assumed to produce tensile forces only. For some two-joint muscles such as the rectus femoris and gastrocnemius, a straight line representation is not possible due to an interposing structure. In this case, two line segments are used to circumvent the interposing structure, and approximate the shape. The intricate and highly nonlinear dynamics of the muscles are not included in the study. The length variation of a muscle is defined as the difference between the length of a muscle and its reference length, the length measured when the model stands on the ground in an
B.-R. CHEN. M. J.
198
HINESand
4.
erect posture. The reference length defined here is different from physiological rest length because the turning here starts from an erect posture. Based on the turning sequences shown in Fig. 6, the length variations of all the actuators can be derived as functions of time and are shown in Figs 8 and 9.
H. HEMAMI
THE
DYNAMICS OF THE SYSTEM
The Newton-Euler state space formulation developed by Hemami (1980) is used to derive the equations of motion. In Fig. 1, each of the knee joints is assumed to have 4 3 %y
+a 0
!!Gq--Jqfy--J
L&I 0
I
2 3 Time(&)
4
8,
I
4
j4
1”
-4 0
l
2 3 Time(w)
4
I
0
2 3 Time (set)
4
2 3 Time (see)
4
Ir%d 0
0
I
Time (set)
2
3
4
Timetsec)
I
-401
I
-0-’ Time (set)
8 4
10 -4
lsi!id
0 Timebc)
4
I
2 3 Time (set)
4
Time (see)
0 LZII 0 Time (set)
Time (set)
2 3 T ims (ret I
Timebecc)
Time (ret)
Time (reel
Fig. 4. (a) Desired kinematic
trajectories
I
in the turning maneuver.
4
Implementation of a right turn in bipedal walking
i6r
1
I
0
1
2 3 Time (ret)
J
I
Time (set)
Time(sec)
Time (set)
Time (set)
Timetsec)
Time Ised
3 I
'xl
L 0
-’
I
2 3 Timetsecl
4
J
a N
4
\" y
0
$ D
-4 -8 -12
0
I
2
3
4
Time (secl
8
0
E LEiiEl
-4 B
$0
4
s a
0
*o
0
-12 -8m 0
I2
3
4
-4 0
I2
Time(sec1
IO
0
-10
8, m 40
0
12
3
4
01234
Time (set)
40
$ ,” 3 D
20 0 -20
N
3 Time (set)
0
I
2
3
o0
I2
3 Time (set)
4
12 4,
6 0 -6 R:-l 0 I2
3
Time Ised
TimeCsec)
Time(sec)
Time (set)
4
4
Timefsec)
Fig. 4. (b) Desired kinematic
trajectories in the turning maneuver.
degree of freedom and each of the ankle joints is assumed to have two rotational degrees of freedom. There are 22 actuators, as listed in Table 2. Using all the forces discussed above, the equation of motion of the biped system can be obtained. The actual only one rotational
4
degrees of freedom of this system are twelve for the single stance phase and seven for the double stance phase according to Hemami (1980) and Hemami and Zheng (1984). The projection transformations for reducing the
B.-R. CHEW
200
M. J. HINES
and H. HEMAMI
0
Qnl I-’
-8 -16
x
0
40
3
4
2
3
4
2
3
4
40
20
30
5
0
r;
20
-20 -40
2
IO pEl 0
2
3
Lsyll! OO
4
450 300
c 250
200
I50 50 III
r;
loo 0
I
2
3
mi
4
OO
Time bed Fig. 5. The desired ground
t=o
t =0.3
147
VI.1
PI.5
t 4.9-
JlhBul
ta4.3
1’3.9
t=3.5
ts3.1
t =2.7
t=2.3
Fig. 6. The sequencesof turning 90” to the right.
dimensionality of the system for the three different cases are discussed separately. A detailed derivation is provided in Chen (1985). Case 1 Single stance phase with left foot on the ground. The first step is to reduce the Euler-Newton equations to smaller space of Euler angles Bi (i = 1, . . , 7) and angular velocities w (i = 1, . . . , 7) expressed in the body coordinate system. (X. %) in terms of (0, W) where W is the components of the angular velocity
reaction force patterns.
expressed in a body coordinate system. It is shown in Hemami (1980) that the corresponding constraint forces can be eliminated due to this reduction. In the second step of projection, the state space (0, I+) is projected to (6.4). From equation (1) symbolically written as
the reduced equations of the seven link biped system for this case are W)6
+ H(4,d) + GM) + E(& &I- = T
(3)
where D(b) is the inertial matrix, H(& 4) represents Coriolis and centrifugal terms, G(4) denotes the gravity loading term and I?($, 4) describes the effect of the contact forces y of the right foot. A detailed derivation of equation (3) is given in Chen (1985). Case 2 Single stance phase with the right foot on the ground. Following the same two step projection for the single stance phase with the left foot on the ground results in D’(f#+$ + H’(#, &) + G’(#) + E’(#. &,r where
= T’ (4)
Left
Right
Leg
L9 LIO Lll
L5 L6 L7 L8
LI L2 L3 L4
R9 RIO Rll
lliacus Rectus fcmoris Gluteus maximus Biceps femoris (long head) Vastus lateralis Gluteus medius Adductor longus Biceps femoris (short head) Gastrocnemius Tibialis anterior Soleus
Iliacus (IL) Rectus femoris (RF) Gluteus maximus (GM) Biceps femoris (BFLH) (long head) Vastus lateralis (VAL) Gluteus medius (GME) Adductor longus (AD) Bicepts femoris (BFSH) (short head) Gastrocnemius (GA) Tibialis anterior (TA) Soleus (SO)
RI R2 R3 R4
RS R6 R7 RS
Muscle
No.
Table
(m)
point
of the muscle
- 0.136)’ 0.12)’ - 0.069)’ 0, - 0.206))
(- 0.052, 0. - 0.206)’ (0.0. -0.115y ( - 0.02, 0. - 0.2 15)’
(0.034. 0, (0, 0.165. (0, 0.065, ( - 0.052,
(0.08, 0.165. 0.085)* (0.08, 0.165. 0.0R5)4 ( - 0.08. 0.165, 0)4 (-0.08, 0.165. 0)4
( - 0.052, 0. - 0.206)s (0.0. -0.115)” (-0.02.0, -0.215)b
(0.034, 0, - 0.136)’ (0, -0.165, 12)’ (0, -0.065. -0.069)’ ( - 0.052, 0, - 0.206)s
(0.08, - 0.165, 0.085)* (0.08, - 0.165, 0.085). (- 0.08, - 0.165, 0). (-0.08. -0.165, 0)’
Origin
3. Parameters
(- 0.018. ( - 0.049, ( - 0.038,
(0.02, 0, (0. 0. (0, 0. ( - 0.06, 0. 0)’ 0.0)’ 0, 0)’
0.183)r 0.076)’ 0.076)’ 0, 0.155)’
(0. 0, 0.33)’ (0.02. 0. 0.183)x (0. 0, - 0.032)’ ( - 0.056, 0. 0.215)’
( - 0.018, 0. 0)’ ( - 0.049. 0, 0)’ ( - 0.038.0,O)’
(0.02, 0, 0. 183)6 (0, - 0.052, O.O46)5 (0, 0. - 0.076)5 ( - 0.06. 0. 0.155y
(0, 0, 0.33)5 (0.02, 0. 0.183)6 (0, 0, - 0.032)’ (- 0.056, 0, 0.21 5)6
point
of the lower
(m)
Insertion
actuators
0, - 0.226)’
Midpoint
0. - 0.226)’
0, 0.155)6
( - 0.06,
0, 0.155)’
(0.069, 0. - 0.226)s (0, 0.069, 0.15)’
-_-
(0.069.
---
( - 0.06,
---
(0.069, 0, - 0.226)’ (0, - 0.069. 0. I5)5
(0.069,
---
extremities length
0.545 0.167 0.0794
O.lR45 0.274 0.216 0.121
0.24Y2 0 5x3 0.23 0.452
0.545 0.167 0.07’) 4
0.1845 0.214 0.216 0.121
0.2492 0.583 0.23 0.452
(m)
Reference
B.-R. CHEN, M. J. HINESand H. HEMAMI
202
Right Leg in the Sogittal Plane
posed. The iower extremities collectively have three major functions: 1. To perform and guide the turning motion. 2. To control the ground reaction forces. 3. To stabilize the posture during the motion. Each function requires a system of forces. Let T, be the vector of inputs which is generated to control the ground reaction forces when contact constraints are in effect. Let Tz be the vector of forces generated to guide the model motion and let T3 be the vector of forces to stabilize the system during the motion. For the single stance phase with the left foot on the ground, then
Ripht Leg in the Frontol Plana
Fig. 7. A musculo-skeletal planar model for the lower
T = T1 -I-T2 -I-T,
extremities. Case 3
(5)
and
Double stance phase. The dynamic equations for the double stance phase have already been included in equations (3) and (4). The constraint force r of equation (3) acts on the system while both feet are on the ground. Thus, the equations of motion for the seven link biped model with either one or two feet on the ground are formulated. These equations are essential to the design of the control strategy and to the digital computer simulations of the turning motion. 5. CONTROL STRATEGY
Due to the complexity of both the system and the desired motion, a nonlinear control strategy is pro-
T, = W,
(6)
i,r
where E(& 4) is defined in equation (3). TZ can be computed by the inverse plant method. Assume that the components of f)(4), H(4.4), G(4) and E(4,4) can be accurately computed, then
where the subscript d denotes the desired trajectories. In order to stabilize the system, T, is generated from position and velocity feedback as Ts = W) [Kv(dd -4) + Kp(& - $0
0.03 &ccps
-003 0
(8)
Therefore, the total inputs which are generated by
D 0’2
*DO2-
(7)
j-2 = WJ)& + W#+ d) + GM)
I
2
3
0
4
I
2
3
4
0
I
femaris
2
003
DO2 Gluteus hkuimus
Ttbmlts
Biceps Femoris LLong lieou~
Gostrocnemius
Glutsus Me&s
-0030
I
2
3
Time (set) Fig. 8. Muscle length variations of the left leg.
4
Anlerkw
3
4
Implementation
002
002
203
of a right turn in bipedal walking
‘-_11E_:i
0
0 vasfusLaerahs -002 H 0
I2
3
4
-002
I2
3
4
0
I2
3
4
Time (set) Fig. 9.
Muscle length variations of the right leg.
actuators are T = w4
[id + WA
- 44 + K,Wd - 411
+ HC#h4, + w#4 + w.
Substituting renders
the above equation
(9)
dw.
in equation
of each link in the biped system from that of the other links. If the desired eigenvalues associated with (&, di) are li, and Liz, the diagonal elements of matrices K, and K, are K,(i, i) = li, + li,
(13)
K,(i, i) = Li, . Ait.
(4d-;i)+Ko(&-&+Kp(#J~-#)
=O.
Equation (IO) represents a linear system. The block diagram is shown in Fig. 10. By selecting diagonal matrices K, and K,. one is able to decouple the motion
The decoupled motion of the ith link is then described by A$i + (Ai, + Li,)A&i + i-i, ’ li, A4i = 0
igi”~-piiql
_I q-
\J Inverse
4 +
Plonr
PIOCM
t t
73 \,
\/
'LKp I +d
=
Desired +d Stores 4 d
_
(11)
(10)
Fig. IO. The control block diagram of the system.
(12)
204
B.-R. CHEN. M. J. HINESand H. HEMAMI
where
By assigning the eigenvalues to the system with negative real parts, the gain matrices K, and K, can be obtained to stabilize the system. One set of values used for the computations reported in this paper is given in Table 4. The same methods can be used to generate inputs by muscles for the single stance phase with the right foot on the ground. 6.
RESULTS
AND DISCUSSIONS
Based on the desired kinematic data and ground reaction forces, the control torques are calculated by using equation (9), and are plotted in Fig. 11. The calculation of T involves two stages. (1) For a given set of initial angular positions (6)(O) and angular velocities 4(O), and with the desired &(t), 4,,(t) and &(t), equation (10) is solved for the actual state 4(t) and &(I). Since the desired kinematic trajectories are not analytical functions of time, this solution is arrived at by digital computer simulation. (2) The desired trajectories and these calculated actual trajectories are used in equation (9) to compute T. The next step is to compute actuator forces from the computed torques T. There are more unknowns than equations in computing actuator forces. This implies that many feasible solutions are possible. One therefore has to set up a criterion to choose a suitable
solution from the many possibilities. The performance criterion used here is to minimize the sum of squares of forces required to produce these torques. It is assumed here that coactivation of muscles is not admissible. The computed force patterns of the actuators which are listed in Table 3 are shown in Figs 12 and 13. The actuator forces are very small at the initial, middle, and final phases of motion because of the two limb support. All actuator forces go to zero at the final state of the system because a symmetric double static stance with zero bias torque is assumed. All the actuator forces are positive and in reasonable range, The magnitudes of forces are relatively small which is due to the slow turning motion. Additional techniques can assist in deriving unique solutions to the muscle force distribution. Moments of force and mechanical power (Winter, 1983a) or energy absorption and generation criteria in muscles (Winter, 1983b) can be added to the formulation as additional constraints in order to arrive at more physiologically acceptable force patterns. The concept of relegation of function to arbitrary groups of muscles (Wongchaisuwat, 1985) can also simplify the solution of the force problem. In this formulation the functions of stability, trajectory control, and force of contact (or support) control can be relegated to different groups of muscles. The minimization of the sum of squares of forces implies that all muscles contribute to the turning motion uniquely throughout the time interval of
Table 4. Diagonal matrices K, and K, used for computation of forces. Eigenvalues -4 and -4 result from feedbacks k, = 8 and k, = 16. Eigenvalues - 5 and - 5 result from k, = 10 and k, = 25 K, = diagonal matrix (8. 8, 8, 8, 8, 8, IO, 10, 10, 10, 10, 10) K, = diagonal matrix (16, 16, 16, 16, 16, 25, 25, 25, 25, 25, 25)
Time (set) Fig. 11. The input torques of the system.
Implementation
of a right turn in bipedal walking
Fig. 12. Actuator
205
force patterns of the left leg
Time (set) Fig. 13. Actuator
force patterns of the right leg.
interest. An alternative strategy is to minimize a weighted sum of squares of forces’where the weight would allow larger muscles to bear larger loads. The latter strategy can also be applied to different nonoverlapping phases of motion in order to shift major load bearing from one group of muscles to another. Antigravity muscles that control weight bearing appear to have such a relegated function. Since there is no convenient experimental method for direct quantitative measurement of muscle forces, EMG data may be used as qualitative measures. As in Pedotti et al. (1978) the force patterns do not only depend on a particular criterion but also depend on the kinematic data and the ground reaction forces. Therefore, different actuator torques and EMG patterns may result for different kinematic data. To date,
applicable EMG data are not available for checking the correctness of the proposed control strategy and criterion. Thus in this study a feasible set of forces are specified that can be utilized as a guide in electrical or neural muscle stimulation, and could serve as a first order approximation in helping the physically impaired regain mobility. This method could also be applied to a system without deep muscles, i.e. the number of actuators could be reduced.
7. SUMMARY
A mathematical model which is capable of quantitative computation of simple force actuators of a seven link biped system was presented. Newton-Euler state
B.-R. CHEN.
206
M. J. HISES and H. HEMAMI
space formulation was used to derive the equations of motion. This formulation is convenient for including arbitrary actuators and Lyapunov stability considerations as in Hemami and Chen (1984).
parametrtc model for biped locomotion kinematics. Ph.D. dissertation. The Ohio State University, Columbus. OH. Hemami. H. (19801 A feedback on-off model of biped dynamics. IEEE Trans. Sysf. Man Cybernef. SMC-10,
Nonlinear feedback was employed for stabilizing the system and decoupling the degrees of freedom of the system. The advantage of using nonlinear feedback is that of simplicity. Theoretically, the performance of the biped system with nonlinear feedback is as good as a linear system. This means the errors asymptotically go to zero. A set of kinematic trajectories and ground reaction forces were inferred from the literature. Based on these trajectories and nonlinear control strategy, the required muscle force patterns were obtained. In practice, there are some problems. The masses and moments of inertia are contained in the nonlinear term. For biped systems it is usually very difficult to accurately measure these parameters. Second, a predictor has to be included to account for computation delay. However, a model could be specified with only superficial muscle actuators-those that are more amenable to direct electrical stimulation. This simplified model would have immediate uses in functional electrical stimulation. Finally, one more observation must be made. Though the bipedal system is modelled after human beings, no claim is made that the control structures or the actuator forces are those employed in the human turning maneuver.
Hrmami, H. (1982) A state space model for interconnected
Acknowledyzmenr-This work was supported in part by NSF Grant ECS-820-1240 and in part by the Department of Electrical Engineering, The Ohio State University, Columbus, OH 43210, U.S.A.
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