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Muscle coordination of maximum-speed pedaling
PIk SOO21-9290(96)00188-l ELSEVIER
MUSCLE
COORDINATION
Christine
C. Raasch.*?
OF MAXIMUM-SPEED
Felix E. Zajac,*?$
Baoming
PEDALING
Ma5 and William
S. Levine9
* Rehabilitation R & D Center (153). Veterans Affairs Palo Alto Health Care System, Palo Alto. CA 94304-1200. U.S.A.; t Departments of Mechanical Engineering and $ Functional Restoration, Stanford University, Stanford? CA 94305-4021, U.S.A.: and fi Electrical Engineering Department, University of Maryland, College Park. MD 20742. U.S.A. Abstract-A simulation based on a forward dynamical musculoskeletal model was computed from an optimal control algorithm to understand uni- and bi-articular muscle coordination of maximum-speed startup pedaling. The muscle excitations. pedal reaction forces, and crank and pedal kinematics of the simulation agreed with measurements from subjects. Over the crank cycle, uniarticular hip and knee extensor muscles provide 55% of the propulsive energy, even though 27% of the amount they produce in the downstroke is absorbed in the upstroke. Only 44% of the energy produced by these muscles during downstroke is delivered to the crank directly. The other 56% is delivered to the limb segments. and then transferred to the crank by the ankle plantarflexors. The plantarflexors, especially soleus, also prevent knee hyperextension, by slowing the knee extension being produced during downstroke by the other muscles. including hamstrings. Hamstrings and rectus femoris make smooth pedaling possible by propelling the crank through the stroke transitions. Other simulations showed that pedaling
can be performed well by partitioning all the muscles in a leg into two pairs of phase-controlled alternating functional groups, with each group also alternating with its contralateral counterpart. In this scheme, the uniarticular hip/knee extensor muscles (one group) are excited during downstroke, and the uniarticular hip/knee flexor muscles (the alternating group) during upstorke. The ankle dorsiflexor and rectus femoris muscles (one group of the other pair) are excited near the transition from upstroke to downstroke. and the ankle plantarflexors and hamstrings muscles (the alternating group) during the downstroke to upstroke these alternating functional muscle groups might represent a centrally generated but also other locomotor tasks as well. Published by Elsevier Science Ltd Ke~~v~~r~f.~
Lower
limb:
Locomotion;
Coordination:
Biarticular
INTRODUCTION
Pedaling, though a constrained lower limb task, seems to require complex muscle coordination, as evidenced by recorded electromyographic (EMG) patterns (Jorge and Hull, 1986; Ryan and Gregor, 1992). Because the relation between the EMG and force is uncertain during nonisometric cyclic contractions, how muscles coordinate the delivery of energy to the crank still remains elusive. Also, some potentially important muscles are difficult to study experimentally (e.g. iliopsoas). The function of muscles in pedaling is controversial
because no analysis of individual muscle con&ibutions, based on forward dynamic simulation, has been performed. Analysis of net joint torques suggests that the hip and knee extensor muscles produce much of the propulsive energy (27 and 39%, respectively; Ericson et al., 1986), in agreement with EMG activity recorded from the uniarticular hip and knee extensor muscles (Ericson. 1988). ‘Simulations produced by net muscle joint torques have shown that the net hip joint torque primarily delivers energy to the limb segments, which is then transferred to the crank by the net ankle joint torque (Fregly and Zajac. 1996). The knee torque supplies energy directly to the crank at the beginning and end of the downstroke. However. because of the action of biarticular
in ,fitzu/ /iwn 12 Norember 1996. Address correspondence to: Felix E, Zajac. R & D Center (153), VA Palo Alto Health Care Ave., Palo Alto. CA 94304-1200. 1J.S.A. Rcwived
Ph.D., System,
Rehabilitation 3801 Miranda
muscle;
Computer
transition,
We conclude
that
primitive for not only pedaling model;
Biomechanics.
muscles, individual muscles may deliver energy to the crank differently. Previous studies of biarticular muscles (Andrews, 1987; Gregor el al., 1985; Hull and Hawkins, 1990; Ingen Schenau, 1990; Ingen Schenau et d., 1992) have wide-ranging conclusions as to their functional role in pedaling, with various explanations for the so-called ‘paradoxial’ co-activation of biarticular and anatomically opposed uniarticular muscles. We therefore generated a forward simulation of muscle coordination of pedaling, compatible with kinematic, kinetic, and EMG measurements, to understand the fundamental muscle coordination principles associated with the delivery of energy to the crank. Maximum-speed pedaling was chosen because it has a clearly defined goal. which allows us to produce a simulation using optimal control. and because this goal requires transferring as much energy as possible from the muscles to the crank subject to the specific frictional and inertial load encountered at the crank.
METHODS
The linkage dynamics were modeled by two legs, each moving in a parasagittal plane with its hip center stationary and its foot rigidly attached to the pedal, with the two pedals connected by the crank. Each leg was thus modeled as three segments: a thigh, a shank, and a foot (which includes the ergometer pedal). All joints were assumed to be pin joints, except the knee, which was assumed to be a one degree-of-freedom planar joint with the two translational degrees-of-freedom constrained to
follow a path that is a function of knee rotation (Yamaguchi and Zajac, 1989). As an attachment site for the quadriceps muscles, the patella was constrained to follow a path that is a function of knee angle (Delp et ul., 1990). The load produced by the ergometer flywheel and friction band was modeled using an ‘effective’ inertial and frictional load (Fregly, 1993). The flywheel inertia and friction matched those of a standard Monarch ergometer with a 20 N frictional resistance. The linkage dynamical equations were produced with SD/Fast software (Symbolic Dynamics, Mountain View, CA). Passive ligamentous structures at the joints were modeled as damped, non-linear torsional springs (Davy and Audu, 1987) except the ankle was made somewhat stiffer at the boundaries of its normal range of motion to prevent excessive ankle dorsiflexion in the first downstroke, and the knee a little less viscous to match better passive knee experimental data (Hatze, 1976). Muscle contraction dynamics were based on a generic Hill-type model (Zajac, 1989), with passive damping added to make the force-velocity curve invertible (Schutte er a/., 1993). Each individual muscle/tendon was modeled by scaling the generic model with four parameters (Table 1; ‘Zajac, 1989). Using SIMM (MusculoGraphics Inc., Evanston, IL) (Delp er ul., 1990; see also Hoy pr u/.. 1990) muscles were partitioned into muscle ‘sets’ (Table 1). Four muscle sets (GMAX, IL, HAM, and SOL) consisted of two muscles. One muscle set (VAS) included three compartments with different slack lengths
Table
1. Muscle/tendon
(Table 1; Schutte et ul., 1993j, representing the large area of origin for this muscle, to match better maximum knee extensor joint torque profiles (Eijden et [I/.. 1987: Lindahl ef ~1.. 1969; Murray et ~1.. 1977). Where necessary- ‘via points’ were included in muscle-tendon paths to preclude muscle-tendons from traversing through bone (Delp et ul., 1990). Peak isometric force in each *equivalent’ muscle was chosen so that its torque-angle curve matched the summed torque profiles generated by SIMM of all anatomically similar lower limb muscles (e.g. peak force of the muscle set TA was adjusted to match the summed torque-angle capability of all dorsiflexor muscles combined). Then, all peak forces were increased by 25% (see Table 1) so the model could accelerate the crank as quickly as the subjects. The computed muscle coordination pattern that maximizes crank speed (see below) was, however, found to be insensitive to this 25% increase. Similar models of musculoskeletal geometry have been used to study posture (Kuo and Zajac, 1993) and electrical stimulation-induced pedaling (Schutte et al., 1993). Activation dynamics were modeled by a first-order differential equation with an activation time constant of 11 ms and deactivation time constant of 68 ms (e.g. Winters and Stark, 1988). An optimal control problem was formulated and solved to find the muscle excitation trajectories. The state equations (2)-(4) were based on the models of the linkage dynamics, the muscle contraction dynamics, and the muscle activation dynamics. The performance measure
parameters
and muscle
partitioning Pennation KW
Muscle set equivalent muscle(s) GMAX Adductor magnus Gluteus maximus HAM Medial hamstrings Biceps femoris long head RF (rectus femoris) VAS (vastus) Part 1 Part 2 Part 3 GAS (gastrocnemius) SOL Soleus Other olantarflexor TA (tibia&s anterior) BFsh (biceps femoris short head) IL Ihacus Psoas
1250 1250
0.131 0.144
0.260 0.145
1288 1312 914
0.080 0.109 0.084
0.359 0.341 0.346
2125 2125 2125 2225
0.087 0.087 0.087 0.045
0.239 0.146 0.061 0.408
17
3549 3250 1375 502
0.030 0.031 0.098 0.173
0.268 0.310 0.273 0.100
25 12 5 23
788 625
0.100 0.104
0.130 0.130
8
5 15
0
3
Nofe: The 32 muscles of each leg that can contribute significantly to sagittal-plane motion, based on SIMM, were represented by 15 ‘equivalent’ muscles. These 15 muscles were further combined into nine muscle sets, with muscles in each set receiving the same excitation signal. Adduetor magnus, not commonly included in experimental studies of cycling because it is impossible to study using surface EMG, was included because it has a substantial hip extensor moment which can contribute to pedaling. F,,,,, is the peak isometric force the muscle can develop; &,,O, optimal muscle fiber length, is the length of the fibers when FmO occurs; &, tendon slack length, is the length of tendon beyond which force can be sustained; and pennation is the effective angle of the muscle fibers with respect to tendon when F,,,O occurs. Individual F,,,O values were adjusted as described in the text. The other muscle parameters are identical to those in SIMM.
Muscle coordination
of maximum-speed pedaling
591
forces. Thus, a muscle can generute mechanical energy to the crank and/or the limb (e.g. when the net curve in Fig. 5 is positive), or a muscle can absorb energy from the crank and/or the limb (e.g. when the ner curve in Fig. 5 is negative), or a muscle can transfer energy between the crank and the limb (e.g. when the limb and crank curves are opposite in Fig. 5), perhaps while also either generating or absorbing energy (Fregly and Zajac, 1996). Nine normal adult males (age 27 k 3.4 yr, height 1.8 k 0.04 m, and weight 714 & 59 N) gave informed consent prior to participation. The protocol for the experiments was approved according to the relevant laws and regulations of Stanford University. With trunk in an upright orientation (15’ backward lean from vertical), upper body secured to a backboard by straps, feet attached to the pedals by cleats and toe clips, and with arms on the chest, the subjects pedaled an instrumented ergometer against a 20 N frictional load from rest to their maximum achievable speed as quickly as possible. Angular velocity was displayed to the subjects via a digital meter. Subjects were asked to relax as much as possible before starting, with the crank near horizontal with the left leg forward, which was within 10’ of the initial position used in the model. Subjects first warmed up by pedaling at 60 rpm for 2 mins. Subjects u 2 UM, = Lu aMl ” [CIU + [Cl ... cJT-j, rested for 5 mins, practiced the maximal startup task a-M (4) once, then rested for another 5 mins. Before data collec[u - a”].c2, u -c a”, tion, subjects were asked if they felt fatigued, and their where q is a vector of generalized coordinates (e.g. crank pulse rate was checked to assure it was within 10 right and left pedal angles); lM and uM are beats min- ’ of the rate prior to the practice trial. A secaw1e chk, vectors of muscle length and activation, respectively; ond maximum-speed trial was then performed and data M(q) and RM are the system mass and moment arm collected for 10 s starting just prior to the ‘go’ signal matrices, respectively; F”(q, 4, l”, a”) is a vector of given to the subject. All subjects reached their maximum muscle forces; G(q), V(q, d), and D(q, 4) are vectors of speed within 5 s. gravity, motion-dependent, and frictional terms, respecCrank and pedal angles were measured with optical tively. cl and c2 are related to muscle activation/deactiencoders, and normal and fore-aft pedal reaction forces vation time constants, with cl = &r - r& and from both feet with pedal dynamometers (Newmiller CL = T&p Since muscle activation uM is monotonic in et al., 1988) Crank angle data was filtered (Butterworth the control a, while the cost function and other equations third-order zero lag filter with a cutoff frequency of are not a function of the control, the optimal controls are 15 Hz), allowing velocity to be estimated through differ‘bang-bang’. For comparison with experimental data (see entiation. Radial and tangential crank forces were calbelow), the optimal controls were converted to functions culated by transforming the pedal reaction forces to of crank angle rather than time. These controls provided crank coordinates, then low-passed filtered (cutoff frea robust initial guess for new simulations (e.g. with one or quency 25 Hz). Bipolar EMG electrodes with preamplimore biarticular muscles not used). fiers were attached to the right leg of each subject over From the trajectories generated by solving the optimal the bellies of five muscles: GAS (medial), TA, VAS (mecontrol problem, the power contributed by each muscu- dial), RF, and HAM (biceps femoris long head). All data lar and non-muscular force to each segment, as well as were sampled at 1000 Hz. In post-processing, the EMG their contribution to segmental accelerations and intersignals were demeaned and rectified. EMG on/off times nal and pedal reaction forces, was computed (Fregly and were selected with an automated waveform processing Zajac, 1996). The segments were grouped as the cmnk program (Datapac, RUN Technologies Inc., Laguna (flywheel, freewheel, chainring, crank arms) and the limb Niguel, CA) using the following procedure: first a base(foot segment, shank, and thigh of the ipsilateral leg). The line EMG level was determined from the relaxed portion net mechanical energy produced by a muscle over a cycle of the trial, and a threshold for excitation set to at least (the amount generated minus the amount absorbed) was 3 SD above this value; on-off times were determined calculated by integrating the net power from the muscle using a minimum on or off duration of 55 ms. The events (tendon force times musculotendon velocity, net curve in selected were checked manually, and on/off durations Fig. 5) over the crank cycle. Note that this is mechanical adjusted as necessary (& 20 ms) to best capture the EMG energy (work), not metabolic energy (which would in- bursts in the first five cycles. These were converted to clude muscle heat as well). A muscle can instantaneously on/off crank angles. For comparison with the model, all contribute to the power of a segment (or accelerate a seg- data were averaged among subjects as a function of crank ment), even when it does not touch (e.g. the crank), by angle on a cycle-by-cycle basis (i.e. first cycle, second means of its contribution to the intersegmental reaction cycle, etc.). chosen for this maximum speed pedaling task was crank progress in a fixed time [Equation (l), i.e. proportional to average pedaling speed]. Thus, the optimal control problem solved was the dual of the fixed-progress minimumtime problem considered in earlier work (Levine et aZ., 1989). The controls (excitations to the muscle sets) which maximized crank progress were found using a constrained optimal control algorithm (Ma and Levine, 1993), which includes the FSQP optimization routines (Zhou and Tits, Univ. Maryland). The initial state for the simulation was found by allowing the musculoskeletal model to come to rest with all muscles off. The optimal control problem can then be stated as follows: find the controls (muscle excitations) u (subject to the bounds 0 < a < 1) that maximize the crank progress, i.e. the cost function
RESULTS
800
The pedaling simulation was found to reproduce the major features of maximal startup pedaling exhibited by the subjects. For the first two cycles, the crank angular velocity of the model is generally within L- 1 S.D. of the subjects (Fig. 1). Simulation and experimental crank forces (Fig. 2) are similar, except the simulation produces a radial crank force near BDC that is more outwarddirected (i.e. negative). The crank phases corresponding to the onset/offset times of the simulated muscle excitation signals are consistent with the estimated on/off phasing of the muscles in the subjects [Fig. 3(b)], derived from EMG signals [Fig. 3(a)]. However, the simulated onset/offset times are later, except for RF offtime. Simulations were run to assessthe sensitivity to the on/off phasing of the muscles. Using EMG phasing [Fig. 3(b)] to excite the dynamical model [Equation (2)44)] instead of the optimal controls. simulated performance decreased but crank velocity was still within 1 S.D. of the subjects. Analysis of the net mechanical energy produced by the muscles over the cycle (computed from the simulation; Fig. 4) showed that VAS is the primary contributor, providing 34% of the total (225 J). GMAX contributes more than twice as much energy (21%) as HAM (9%) though the two have similar force-generating capabilities (Table l), because GMAX’s moment arm at the hip is greater than HAM’s VAS generates energy to both the crank (70 J) and the limb (43 J) throughout most of downstroke (Fig. 5 : O-160., all VAS curves are > O), but then absorbs considerable energy (43 J) from the crank during the upstroke while it relaxes (i.e. after excitation ceases, 1W32OO; net and crank curves are both < 0 and about equal). GMAX generates energy (54 J) mostly to the limb in downstroke (Fig. 5, net and limb curves of
400
Tangential Force Simulation VA Subjects
-800
Radial Force - Simulation m Subjects l
-1200 O0 TDC
9o”
180’= E3DC Crank angle
270
360° TDC
Fig. 2. Right crank forces computed from experimental data (gray and cross-hatched regions, ave k 1 SD.) and generated by the optimal control simulation of pedaling (solid and dashed lines) in the first cycle. Downstroke is from 0 to 180’: upstroke from 188 to 360”. Positive tangential crank force accelerates the crank and positive radial crank force is directed toward the crank axis of rotation. Left crank forces (not shown) are similar but 180’ out-of-phase with the right crank forces.
a HAM RF VAS GAS TA ,
b
,
1
I
I
O0
9o”
180’
270”
360°
O0 TDC
9o”
270°
360° TDC
HAM RF VAS GAS TA SOL BFsh IL GMAX
0’ -1 60°
’
,
,
I
360°
540°
720°
’ O0
180 Crank
angle
Fig. 1. Crank angular velocity measured in the subjects (gray, ave & 1 SD.) and generated by the optimal control simmation of pedaling [solid tine). The right crank arm is vertical at 0 (top dead center, TDC) and 180’ (bottom dead center. BDC). The simulation and the subjects started cranking from rest with the right crank approximately 90’ before TDC 0.e. - 9W). Detailed analysis (e.g. Figs 2-7) focused on the first complete cycle (&3W) of the right leg, where simulation and experiment matched most closely. Regardless. the biomechanics and muscle coordination of subsequent cycles were found to be similar to the first cycle.
180° BDC Crank angle
Fig. 3. EMGs and optimal muscle excitation signals for the right leg in the first cycle. (a] Typical unprocessed EMGs. Scale indicates maximum EMG magnitude recorded, normalized to + 1 and - 1, respectively, for each muscle. (b) Average on/off phasmg of muscles after processing the EMG (gray bar indicates muscle is ‘on’; tines indicate & 1 SD.) compared with the on/off muscle phasing of the optimal control simulation of pedaling (black bar indicates the muscle is ‘on’). EMGs were not recorded for SOL. BFsh. IL, and GMAX, In later cycles. as ct-ank speed rises. both EMGs and simulated excitation signals advance in crank phasing.
Muscle
coordination
of maximum-speed
599
pedaling
w
+200
- -
ow
VAS GMAX Fig. 4. Mechanical
energy
SOL
IL
HAM
BFsh
produced by the muscles the first cycle (0.6 s).
TA
net crank
RF ‘GAS’
of the right
leg over
GMAX are both > 0 and about equal), but absorbs some energy (12 J) from the crank while it relaxes in upstroke (both ner and crank curves are < 0 and about equal). Energy generated to the limb in downstroke by VAS and GMAX is transferred to the crank by SOL (91 J) and GAS (36 J) (Fig. 5: SOL and GAS, &l W, crank curve is > 0 and about equal to jirrrb curve but opposite in sign). SOL and GAS also generate some energy to the crank (20 and 6 J; area under each net curve is > 0). Biarticular GAS, in transferring energy from the limb to the crank in the downstroke, acts primarily as an ankle extensor since its ankle-to-knee moment arm ratio is high in pedaling (Ericson, 1988; Spoor et al., 1990). TA acts as an antagonist to SOL, mostly transferring energy to the crank in the last part of upstroke. IL and BFsh both generate energy to the crank throughout upstroke (25 and 12 J; Fig. 5, 18&36P, aer curve and crank curves are both > 0 and about equal). However, as evidenced by the negative tangential crank force in the simulated upstroke (Fig. 2, 18&36P, solid line). the energy supplied by IL, BFsh, and other sources in the upstroke is insufficient to overcome the energy removed from the crank by the leg. This energy removal is mostly by the relaxing VAS and GMAX (Fig. 5, 180-36v, each crank curve is < O), and somewhat by gravity and the joint/ligament passive forces (not shown). Gravity and motion-dependent forces were found to transfer a small amount of energy between the crank and the limb (see also Fregly and Zajac, 1996) with the amount increasing as crank speed rose. HAM and RF produced peak power near BDC and TDC, respectively (Fig, 5). Before BDC and during the early excitation of HAM (100-1607, HAM transfers energy (8 J) from the crank to the limb while generating energy to the limb (3 J). Afterwards (160-19v), HAM generates energy to both the crank and the limb (4 and 3 J). While still excited during upstroke (19&25P), HAM mainly generates energy directly to the crank (11 J). When first excited during upstroke, RF generates energy to the crank (5 J, 25&300’; Fig. 5). Later RF generates energy to both the crank and the limb (1 and 2 J,
BFsh HAM
I O0 TDC
1 90°
, 160° BDC Crank angle
, 270’
J
360’ TDC
Fig. 5. Distribution of muscle mechanical power of the right leg in the first cycle. Downstroke is from 0 to 180°; upstroke from I80 to 360°. GMAX mostly generates energy to the limb in the downstroke (i.e during &180°, solid and dotted lines of GMAX are both > 0 and about equal; thus the integral of each power curve, which is energy, is > 0). VAS generates energy directly to the crank near TDC (i.e. near W, solid and dashed lines of VAS are both > 0 and about equal), then more to the limb later in downstroke (i.e. during 135160”, dotted line increases, dashed line decreases), and absorbs energy from the crank in the upstroke (i.e. during 18@320’, solid and dashed lines are both < 0 and about equal). SOL, GAS, and TA mainly transfer energy between the limb and the crank (i.e. dashed and dotted lines are about equal but opposite in sign). BFsh and IL generate energy to the crank in the upstroke. HAM and RF have complex patterns, but primarily contribute energy near BDC and TDC, respectively.
30&3307, and finally to just the limb (5 J, 33@-107. The energy generated to the limb by RF is transferred to the crank by TA and motion-dependent forces. DISCUSSION
The nominal muscle coordination pattern [Fig. 3(b)] was found to be insensitive to model parameters (e.g. peak isometric active force, activation/deactivation time constants, and initial state). Also, the simulation produced by using the on/off phasing of the measured EMGs is virtually the same as that produced by the nominal optimal muscle coordination pattern. (The difference is
600
perhperhaps due to inaccuracies in how muscle activation dynamics are modeled.) We conclude, therefore, that the nominal optimal control solution simulates well how muscles coordinate energy flow to the crank during startup pedaling. The higher peak radial crank force near BDC in the nominal simulation of pedaling (Fig. 2), which is the most notable difference between the simulation and the experimental data, may be due to subjects de-exciting their muscles slightly differently from the way assumed in the model and in the simulation. For example, Sandercock and Heckman (1996) have shown that the force measured during relaxation of the cat soleus muscle undergoing locomotor-like shortening is less than the force predicted by a Hi&type model (such as the one used in our study). When we included their suggestion that muscle activation during relaxation is velocity-dependent, peak radial force decreased by 31% (369 N) with little change in pedaling speed ( + 0.1%) or the optimal excitation pattern. Another possibility is that subjects de-excite VAS and GMAX sooner, since these muscles contribute significantly to the peak radial force (found by analyzing the contribution of each muscle to the crank force), but contribute little to tangential force near BDC. Analysis of the nominal simulation, which does not penalize coordination patterns producing high joint reaction forces, showed that high joint reaction forces occur when radial force peaks (e.g. the knee compressive reaction force of the simulation is 5.8 times body weight then). Other simulations show large reductions in both knee compressive force ( - 20%) and radial crank force ( - 32%) if VAS and GMAX are de-excited slightly earlier (VAS = 10 ms and GMAX = 25 ms earlier), though other changes in de-excitation of these muscles seem to be equally effective (e.g. by gradually de-exciting VAS and GMAX). In any case, the changes in the model necessary to reduce the discrepancy in radial force do not significantly affect the coordination pattern required to generate energy to the crank. Excitation of the plantarflexor muscles (including GAS) must be coordinated with the uniarticular knee and hip muscles (VAS and GMAX), which are the muscles producing most of the energy in the downstroke (i.e. 86% by VAS and GMAX; consistent with conclusions derived from net joint torques; Ericson et ul., 1986). About 55% of the energy produced by VAS and GMAX is not delivered directly to the crank but is, instead, generated to the limb segments. If the plantarflexors were not co-excited during downstroke, this energy would go towards accelerating the limbs (dorsiflexing the ankle, and hyperextending the knee, as discussed below) rather than being transferred to accelerate the crank. The plantarflexors (again including GAS) also transfer to the crank energy delivered to the limb by HAM in late downstroke (Fig. 5, 90-18W, area under limb curve), which is significant in comparison with the energy generated to the limb by VAS and GMAX (14 J by HAM; 27 J by VAS; 26 J by GMAX between 90 and 18W). Also, since HAM and VAS are co-excited in late downstroke and produce nearly opposing joint torques at the knee, the resultant torque by these muscles is the hip extensor joint torque produced by HAM. This explains why
Fregly and Zajac (1996) found only the net hip joint torque, and not the net knee joint torque, to deliver energy to the limb in downstroke. Thus, caution must be exercised in conjecturing on muscle coordination from net joint torque data. Though HAM is often labeled a ‘knee flexor’ in pedaling (e.g. in Gregor er u/., 1985), we believe it does not prevent knee hyperextension in late downstroke; instead. the plantarflexors seem to be the responsible muscles. In fact, even though HAM develops a knee flexor torque in late downstroke, it assists in the attempt to produce knee hyperextension since it accelerates the knee towards extension then (Fig. 6, 10&180’, HAM is > O), consistent with previous analyses (Andrews, 1987; Carlsoo and Molbech, 1966). The reason is because the moment produced by HAM at the hip in late downstroke is larger than at the knee, and also because limb geometry in downstroke favors transformation of hip torque into knee joint acceleration more than knee torque. This casts doubt on the idea that HAM activation allows the knee extensors to do more work by remaining active longer (Ingen Schenau, 1990), since the net action of HAM is also to extend the knee. In contrast, the plantarflexors (including SOL) act to accelerate the knee towards flexion (Fig. 6, 10&18@‘, SOL and GAS are ~0). The action of the plantarfiexors (specifically SOL) to accelerate the knee towards flexion, and the HAM to accelerate the knee towards extension, is due to the dynamic coupling by reaction forces (Zajac and Gordon, 1989) The dominant function of HAM and RF seems to be to effect smooth stroke transitions, which are times in the cycle when the total bilateral energy transfer to the crank
O0
9o”
180° Crank
270°
360°
angle
Fig. 6. Angular acceleration of the knee caused by SOL, GAS, VAS, HAM and BFsh. Knee angular acceleration is defined as the angular acceleration of the shank relative to the thigh. Positive acceleration is in the direction of knee extension. HAM accelerates the knee towards extension almost as strongly as VAS in the 135lgOQ region. In constrast, SOL and GAS accelerate the knee towards flexion then, which prevents knee hyperextension from occurring near BDC. Not shown, but also sign&ant, are accelerations due to GMAX (approximately the same as-VAS throughout downstroke) and motion-denendent fora% (similar to SOL from 90 to 1807. The knee is moving inio extension f&m 330 to 16S” (top box, knee &tending) and being accelerated toward flexion from 100 to 24s’ (bottom box, negatiue knee uccel.).
Muscle
coordination
of maximum-speed
is lowest. However, to accomplish this goal, HAM and RF utilize complex energetic mechanisms (e.g. Fig. 5), perhaps contributing to their heretofore controversial role in pedaling. For example, during downstroke, HAM produces energy, except at first when its tendon stretches more than its muscle fibers shorten (i.e. the muscle/tendon complex absorbs energy), Just before BDC when the hip is still extending but the knee is flexing (160-lSO’), HAM generates energy to both the crank and the limb (Fig. 5, 16&lfW, all curves of HAM are > 0; see also Carlsoo and Molbech, 1966). After BDC in the beginning of upstroke, although the hip is flexing, HAM continues to generate power to the crank (Fig. 5, 18@27W, net and trunk curves are both > 0 and about equal). The net effect of this energy production by HAM near BDC is to help prevent freewheeling (i.e. the decoupling of the crank from the load). Analogously, RF contributes substantially to crank propulsion near TDC. This is consistent with the notion that HAM and RF help achieve an optimally directed reaction force at the pedal (i.e. to maximize the tangential crank force) by distributing power among the joints (Ingen Schenau, 1990), which, according to the power analysis performed in our work, is useful because it generates power to the crank over an important (albeit limited) region of the cycle (i.e. the stroke transition regions). Other simulations by us (Raasch, 1995) demonstrate that replacement of a biartitular muscle with one of its equivalent joint torques can produce higher average pedaling speed with new excitation patterns, but this causes the crank to slow and/or freewheel during stroke transitions. In steady-state pedaling, where smooth cranking is very highly desired, biarticular muscles may play an even more critical role at the stroke transitions. Pervasive in theories of neural control of locomotion is the notion of a simple control (e.g. pattern generator) that, once triggered, can produce the fundamental coordination (i.e. phasing of muscles) needed to locomote (Grillner, 1975, 1981; Pearson, 1993; Rossignol et ul., 1988). Our data, experimental and simulation together, suggest that startup pedaling can be achieved with just two phase-controlled signals, where all the muscles in a leg are partitioned into two pairs of functional groups [Fig. 7(a)] which alternate with their contralateral counterparts. ln this scheme, the two alternating muscle pair (E group: groups of the Extensor/Flexor VAS/GMAX; F group: IL/BFsh) are excited during the downstroke and upstroke [Fig. 7(a)]. The primary function of this pair is to generate energy to both the limb and the crank [Fig. 7(b)]. The two alternating muscle groups of the Top/Bottom pair (T group: RF/TA; B group: HAM/SOL/GAS) are excited through TDC and BDC [Fig. 7(a)]. The muscles of this pair transfer energy from the limb to the crank and provide propulsion through the stroke transitions [Fig. 7(b)]. The simulation produced by this simple phasing strategy, with maximum excitation of the muscles, performs near-optimally with similar kinematics (e.g. average speed slows by less than 4%). Other simulations suggest that this phase-controlled strategy for muscle coordination can effect steady pedaling (e.g. at 60 rpm if the muscle groups are excited less than maximally).
601
pedaling
a Extensor Flexor
-
m E/F Pair
TOP Bottom
I 0
, 9o”
! 180’
, 270’
T/B Pair I 360Q
b Extensor 0 Flexor
Top
:..e.,,
l ..’
1
O0
TDC
,
9o”
1
1 80° BDC Crank angle
270°
f
360’ TDC
Fig. 7. Near replication of the biomechanics of maximum-speed startup pedaling by a simple muscle coordination strategy. (a) Phasing of the excitation of the four functional groups of muscles. Each muscle in the leg is classified into one of these four groups. The Extensor group (E group: VAS/GMAX) and the Flexor group (F group: lL/BFsh) form the E/F pair. The Top (T group: RF/TA) and the Bottom (B group: HAM/SOL/GAS) groups form the T/B pair. The two groups of each pair alternate in crank phasing, and each group alternates with the homologous group of the other leg. (b) Distribution of mechanical power produced by the four muscle groups. These power curves are almost identical to the summed power curves from the optimal simulation (e.g. nef curve for the Extensor group is similar to the sum of the net curves of VAS and GMAX in Fig. 6).
Control of motor tasks with simple strategies by partitioning muscles into alternating functional groups is not unique to pedaling (McCollum, 1993). Similar groups have been identified in animal locomotion (Grillner, 1981), and in studies of neurologically impaired humans (Perry ec ul., 1978) and maintenance of posture (ankle strategy; Nashner and McCollum, 1985). A single parameter (e.g. hip angle) is sufficient to phase-control the muscle groups, as in animal locomotion (Grillner and Rossignol, 1978). Thus, this simple control of human muscle groups may represent a centrally generated primitive for not only pedaling but walking as well, though such a primitive would surely be modulated by sensorimotor control mechanisms to overcome unexpected crank loads and other uncertainties (Prochazka, 1989).
.4[,~~~~~~~edge~ne~lrs~Thiswork was supported by NIH grant NSl7662 and the Rehabilitation R & D Service of the Department of Veterans Affairs (VAj, and is based on the Ph.D. dissertation of C.C.R. (Raasch, 1995). We thank Steve Kautz and Lena Ting for their comments on the manuscript. REFERENCES Andrews, J. G. (1987) The functional roles of the hamstrings and quadriceps during cycling: Lombard’s Paradox revisited. J. Biomechanics
20, 565-515.
Carlsoo, S. and Molbech, S. (1966) The functions of certain two-joint muscles in a closed muscular chain. Actu Morphoi. Neerl. Stand. 6, 317386. Davy, D. T. and Audu, M. L. (1987) A dynamic optimization technique for predicting muscle forces in the swing phase of gait. J. Biomechunks 20, 187-201. Delp, S. L., Loan, J. P., Hay, M. G., Zajac, F. E., Topp, E. L. and Rosen. J. M. ( 1990) An interactive graphics-based model of the lower extremity to study orthopaedic surgical procedures. IEEE. Trans. Biomed. Engng 31,157-167. Eijden, T. M. v., Weijs, W. A., Kouwenhoven, E. and Verburg, J. (1987) Forces acting on the patella during maximal voluntary contraction of the quadriceps femoris muscle at different knee flexiomextension angles. Acta Anat. (Basel) 129, 31@314. Ericson, M. 0. (1988) Muscular function during ergometer cycling. Scund.
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Ericson. M. 0.. Bratt. A.. Nisell. R.. Arborelms. U. P. and Ekholm, J. (1986) Power output and work in different muscle groups during ergometer cycling. Eur. J. Appl. Physiol. 55, 229-235. Fregly, B. .I. (1993) The significance of crank load dynamics to steadystate pedaling biomechanics: an experimental and computer modeling study. Ph.D. dissertation, Stanford University, Stanford. CA. Fregly, B, J. and Zajac, F. E. (1996) A state-space analysis of mechanical energy generation, absorption, and transfer during pedaling. J. Biomechanics 29, 81-90. Gregor. R. J., Cavanagh, P. R. and LaFortune, M. (1985) Knee flexor moments during propulsion in cycling-a creative solution to Lombard’s Paradox. J. Biomechanics 18, 307-316. Grillner, S. (1975) Locomotion in vertebrates: central mechanisms and reflex interaction. Physiol. Reu. 55, 247-304. Grillner, S. (1981) Control of locomotion in bipeds, tetrapods, and fish. In Handbook ofPhysiology, Vol. II, Part 2 (Edited by Brookhart, J., Mountcastle, V. and Brooks, V.). pp. 1179-1236. American Physioiogical Society, Bethesda, MD. Grillner, S. and Rossignal, S. (1978) On the initiation of the swing phase of locomotion in chronic spinal cats. Brain Res. 146, 269-277. Hatze, H. (1976) The complete optimization of human motion. Math. Biosci. 28, 999135. Hoy. M, G.. Zajac, F. E. and Gordon, M. E. (1990) A musculoskeletal model of the human lower extremity: the effect of muscle, tendon, and moment arm on the moment-angle relationship of musculotendon actuators at the hip, knee, and ankle. J. Biomechanics 23, lS7.16Y, Hull. M. L. and Hawkins, D. A. (1990) Analysis of muscular work in multisegmental movements: application to cycling. In Multiple .$~n.sc~eSrstenzs (Edited by Winters, J. and Woo. S-Y.), pp. 621-638. Springer, New York. Ingen Schenau, G. J. v. (1990) On the action of biarticular muscles, a review, Netherlands J. Zoology 40, 521-540. Ingen Schenau, G. J. v., Boots, P. J., de Groot, G., Snackers, R. J. and Woensel, W. W. v. (1992) The constrained control of force and position in multi-joint movements. Neuroscience 46, 197-207. Jorge. M. and Hull, M. L. (1986) Analysis of EMG measurements during bicycle pedalling. J. Biomechanics 19, 683-694.
Kuo, A. D. and Zajac, F. E. (1993) A biomechanical analysis of muscle strength as a limiting factor in standing posture. J. Biomechani~~s 26 (SuppI 11, 137 150. Levine, W. S.. Ma, B. and Za.jac, F. E. (1989) ‘The dynamics and optimal controls for some simplified models of pedaling a stationary bicycle. Proc. 23rd Ann. C’onf on Information Science.5 attd S~.s~wt.~,Baltimore MD, pp. 567-573. ” Lindahl. O., Mown, A. and Ringqvist, I. (1969) Knee extension: measurement of the isometric force in different positions of the knee joint. Acta Orrhop. Stand. 40, 79-85. Ma, B. and Levine. W. S (19Y31 An improved algorithm for solving constrained optimal control problems. Pro<.. 32nd Conf ov Deci.siou and Control, San Antonio. McCollum, G. (I 993) Reciprocal inhibition, synergies, and movements. J. Theoret. Biol. 165, 291-311. Murray, M. P., Baldwin. J. M., Gardner, G. M., Sepic S. B. and Downs, W. J. (1977) Maximum isometric knee flexor and extensor muscle contractions: normal patterns of torque versus time. Whys. Ther. 57, 6377643. Nashner, L. M, and McCollum, G. (1985) The organization of human postural movements: a formal basis and experimental synthesis, Behavioral
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Newmiller, J., Hull. M. L. and Zajac, F. E. (1988) A mechanically decoupled two force component bicycle pedal dynamometer. J. Biomerhanics
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Pearson, K. G. (1993) Common principles of motor control in vertebrates and invertebrates. Ann. Reu. Neurosci. 16, 265-297. Perry, J., Giovan, P., Harris, 1~.J., Montgomery, J. and Azaria. M. (1978) The determinants of muscle action in the hemiparetic lower extremity (and their effect on the examination procedure). Clin. Orthop.
Rel. Res. 131, 71 -.89.
Prochazka, A. (1989) Sensorimotor gain control: a basic strategy of motor systems? Proq. Neurobiol. 33, 281-307. Raasch, C, C. (1995) Coordination of pedaling: functional muscle groups and locomotor strategies. Ph.D. dissertation. Stanford University, Stanford, CA. Rossignol, S., Lund, J. P. and Drew. T. (1988) The role of sensory inputs in regulating patterns of rhythmical movements in higher vertebrates: a comparison between locomotion, respiration, and mastication. In Neural Control oj Rhythmical Movements in Vertebrates (Edited by Cohen, A.V., Rossignol, S. and Grillner, S.). pp. 201-283. Wiley, New York. Ryan. M. and Gregor, R. (1992J EMG profiles of lower extremity muscles during cycling at constant workload and cadence. J. Electromyography
Kinesiology
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Sandercock, T. G. and Heckman, C. J. (1997) Force from cat soleus muscle during imposed locomotor-like movements: experimental data versus Hill-type model predictions. J. &‘enroph#ol. (accepted). Schutte, L, M., Rodgers. M. M., Zajac, F. E. and Glaser, R. M. (1993) Improving the efficacy of electrical stimulation-induced leg cycle ergometry: an analysis based on a dynamic musculoskeletal model. IEEE
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Spoor, C. W., van Leeuwen, J. L., Meskers, C. G., Titulaer, A. F. and Huson, A, (1990) Estimation of instantaneous moment arms oflowerleg muscles. J. Biomechanics 23, 1247-1259. Winters, J. M. and Stark, L. (1988) Estimated mechanical properties of synergistic muscles involved in movements of a variety of human joints. J. Biomechuks 21, 1027-1041. Yamaguchi, G. T. and Zajac, F, E. (1989) A planar model of the knee joint to characterize the knee extensor mechanism. J. Biomechanks 22, l-10. Zajac, F. E. (1989) Muscle and tendon: properties, models, scaling. and application to biomechanics and motor control. CRC Crit. Reu. Biomed. Engng 17, 3599411. Zajac, F. E. and Gordon. M. E. (1989) Determining muscle’s force and action in multi-articular movement. Exert. Sport Sci. Rev. 17, 187-230,
MUSCLE
COORDINATION
Christine
C. Raasch.*?
OF MAXIMUM-SPEED
Felix E. Zajac,*?$
Baoming
PEDALING
Ma5 and William
S. Levine9
* Rehabilitation R & D Center (153). Veterans Affairs Palo Alto Health Care System, Palo Alto. CA 94304-1200. U.S.A.; t Departments of Mechanical Engineering and $ Functional Restoration, Stanford University, Stanford? CA 94305-4021, U.S.A.: and fi Electrical Engineering Department, University of Maryland, College Park. MD 20742. U.S.A. Abstract-A simulation based on a forward dynamical musculoskeletal model was computed from an optimal control algorithm to understand uni- and bi-articular muscle coordination of maximum-speed startup pedaling. The muscle excitations. pedal reaction forces, and crank and pedal kinematics of the simulation agreed with measurements from subjects. Over the crank cycle, uniarticular hip and knee extensor muscles provide 55% of the propulsive energy, even though 27% of the amount they produce in the downstroke is absorbed in the upstroke. Only 44% of the energy produced by these muscles during downstroke is delivered to the crank directly. The other 56% is delivered to the limb segments. and then transferred to the crank by the ankle plantarflexors. The plantarflexors, especially soleus, also prevent knee hyperextension, by slowing the knee extension being produced during downstroke by the other muscles. including hamstrings. Hamstrings and rectus femoris make smooth pedaling possible by propelling the crank through the stroke transitions. Other simulations showed that pedaling
can be performed well by partitioning all the muscles in a leg into two pairs of phase-controlled alternating functional groups, with each group also alternating with its contralateral counterpart. In this scheme, the uniarticular hip/knee extensor muscles (one group) are excited during downstroke, and the uniarticular hip/knee flexor muscles (the alternating group) during upstorke. The ankle dorsiflexor and rectus femoris muscles (one group of the other pair) are excited near the transition from upstroke to downstroke. and the ankle plantarflexors and hamstrings muscles (the alternating group) during the downstroke to upstroke these alternating functional muscle groups might represent a centrally generated but also other locomotor tasks as well. Published by Elsevier Science Ltd Ke~~v~~r~f.~
Lower
limb:
Locomotion;
Coordination:
Biarticular
INTRODUCTION
Pedaling, though a constrained lower limb task, seems to require complex muscle coordination, as evidenced by recorded electromyographic (EMG) patterns (Jorge and Hull, 1986; Ryan and Gregor, 1992). Because the relation between the EMG and force is uncertain during nonisometric cyclic contractions, how muscles coordinate the delivery of energy to the crank still remains elusive. Also, some potentially important muscles are difficult to study experimentally (e.g. iliopsoas). The function of muscles in pedaling is controversial
because no analysis of individual muscle con&ibutions, based on forward dynamic simulation, has been performed. Analysis of net joint torques suggests that the hip and knee extensor muscles produce much of the propulsive energy (27 and 39%, respectively; Ericson et al., 1986), in agreement with EMG activity recorded from the uniarticular hip and knee extensor muscles (Ericson. 1988). ‘Simulations produced by net muscle joint torques have shown that the net hip joint torque primarily delivers energy to the limb segments, which is then transferred to the crank by the net ankle joint torque (Fregly and Zajac. 1996). The knee torque supplies energy directly to the crank at the beginning and end of the downstroke. However. because of the action of biarticular
in ,fitzu/ /iwn 12 Norember 1996. Address correspondence to: Felix E, Zajac. R & D Center (153), VA Palo Alto Health Care Ave., Palo Alto. CA 94304-1200. 1J.S.A. Rcwived
Ph.D., System,
Rehabilitation 3801 Miranda
muscle;
Computer
transition,
We conclude
that
primitive for not only pedaling model;
Biomechanics.
muscles, individual muscles may deliver energy to the crank differently. Previous studies of biarticular muscles (Andrews, 1987; Gregor el al., 1985; Hull and Hawkins, 1990; Ingen Schenau, 1990; Ingen Schenau et d., 1992) have wide-ranging conclusions as to their functional role in pedaling, with various explanations for the so-called ‘paradoxial’ co-activation of biarticular and anatomically opposed uniarticular muscles. We therefore generated a forward simulation of muscle coordination of pedaling, compatible with kinematic, kinetic, and EMG measurements, to understand the fundamental muscle coordination principles associated with the delivery of energy to the crank. Maximum-speed pedaling was chosen because it has a clearly defined goal. which allows us to produce a simulation using optimal control. and because this goal requires transferring as much energy as possible from the muscles to the crank subject to the specific frictional and inertial load encountered at the crank.
METHODS
The linkage dynamics were modeled by two legs, each moving in a parasagittal plane with its hip center stationary and its foot rigidly attached to the pedal, with the two pedals connected by the crank. Each leg was thus modeled as three segments: a thigh, a shank, and a foot (which includes the ergometer pedal). All joints were assumed to be pin joints, except the knee, which was assumed to be a one degree-of-freedom planar joint with the two translational degrees-of-freedom constrained to
follow a path that is a function of knee rotation (Yamaguchi and Zajac, 1989). As an attachment site for the quadriceps muscles, the patella was constrained to follow a path that is a function of knee angle (Delp et ul., 1990). The load produced by the ergometer flywheel and friction band was modeled using an ‘effective’ inertial and frictional load (Fregly, 1993). The flywheel inertia and friction matched those of a standard Monarch ergometer with a 20 N frictional resistance. The linkage dynamical equations were produced with SD/Fast software (Symbolic Dynamics, Mountain View, CA). Passive ligamentous structures at the joints were modeled as damped, non-linear torsional springs (Davy and Audu, 1987) except the ankle was made somewhat stiffer at the boundaries of its normal range of motion to prevent excessive ankle dorsiflexion in the first downstroke, and the knee a little less viscous to match better passive knee experimental data (Hatze, 1976). Muscle contraction dynamics were based on a generic Hill-type model (Zajac, 1989), with passive damping added to make the force-velocity curve invertible (Schutte er a/., 1993). Each individual muscle/tendon was modeled by scaling the generic model with four parameters (Table 1; ‘Zajac, 1989). Using SIMM (MusculoGraphics Inc., Evanston, IL) (Delp er ul., 1990; see also Hoy pr u/.. 1990) muscles were partitioned into muscle ‘sets’ (Table 1). Four muscle sets (GMAX, IL, HAM, and SOL) consisted of two muscles. One muscle set (VAS) included three compartments with different slack lengths
Table
1. Muscle/tendon
(Table 1; Schutte et ul., 1993j, representing the large area of origin for this muscle, to match better maximum knee extensor joint torque profiles (Eijden et [I/.. 1987: Lindahl ef ~1.. 1969; Murray et ~1.. 1977). Where necessary- ‘via points’ were included in muscle-tendon paths to preclude muscle-tendons from traversing through bone (Delp et ul., 1990). Peak isometric force in each *equivalent’ muscle was chosen so that its torque-angle curve matched the summed torque profiles generated by SIMM of all anatomically similar lower limb muscles (e.g. peak force of the muscle set TA was adjusted to match the summed torque-angle capability of all dorsiflexor muscles combined). Then, all peak forces were increased by 25% (see Table 1) so the model could accelerate the crank as quickly as the subjects. The computed muscle coordination pattern that maximizes crank speed (see below) was, however, found to be insensitive to this 25% increase. Similar models of musculoskeletal geometry have been used to study posture (Kuo and Zajac, 1993) and electrical stimulation-induced pedaling (Schutte et al., 1993). Activation dynamics were modeled by a first-order differential equation with an activation time constant of 11 ms and deactivation time constant of 68 ms (e.g. Winters and Stark, 1988). An optimal control problem was formulated and solved to find the muscle excitation trajectories. The state equations (2)-(4) were based on the models of the linkage dynamics, the muscle contraction dynamics, and the muscle activation dynamics. The performance measure
parameters
and muscle
partitioning Pennation KW
Muscle set equivalent muscle(s) GMAX Adductor magnus Gluteus maximus HAM Medial hamstrings Biceps femoris long head RF (rectus femoris) VAS (vastus) Part 1 Part 2 Part 3 GAS (gastrocnemius) SOL Soleus Other olantarflexor TA (tibia&s anterior) BFsh (biceps femoris short head) IL Ihacus Psoas
1250 1250
0.131 0.144
0.260 0.145
1288 1312 914
0.080 0.109 0.084
0.359 0.341 0.346
2125 2125 2125 2225
0.087 0.087 0.087 0.045
0.239 0.146 0.061 0.408
17
3549 3250 1375 502
0.030 0.031 0.098 0.173
0.268 0.310 0.273 0.100
25 12 5 23
788 625
0.100 0.104
0.130 0.130
8
5 15
0
3
Nofe: The 32 muscles of each leg that can contribute significantly to sagittal-plane motion, based on SIMM, were represented by 15 ‘equivalent’ muscles. These 15 muscles were further combined into nine muscle sets, with muscles in each set receiving the same excitation signal. Adduetor magnus, not commonly included in experimental studies of cycling because it is impossible to study using surface EMG, was included because it has a substantial hip extensor moment which can contribute to pedaling. F,,,,, is the peak isometric force the muscle can develop; &,,O, optimal muscle fiber length, is the length of the fibers when FmO occurs; &, tendon slack length, is the length of tendon beyond which force can be sustained; and pennation is the effective angle of the muscle fibers with respect to tendon when F,,,O occurs. Individual F,,,O values were adjusted as described in the text. The other muscle parameters are identical to those in SIMM.
Muscle coordination
of maximum-speed pedaling
591
forces. Thus, a muscle can generute mechanical energy to the crank and/or the limb (e.g. when the net curve in Fig. 5 is positive), or a muscle can absorb energy from the crank and/or the limb (e.g. when the ner curve in Fig. 5 is negative), or a muscle can transfer energy between the crank and the limb (e.g. when the limb and crank curves are opposite in Fig. 5), perhaps while also either generating or absorbing energy (Fregly and Zajac, 1996). Nine normal adult males (age 27 k 3.4 yr, height 1.8 k 0.04 m, and weight 714 & 59 N) gave informed consent prior to participation. The protocol for the experiments was approved according to the relevant laws and regulations of Stanford University. With trunk in an upright orientation (15’ backward lean from vertical), upper body secured to a backboard by straps, feet attached to the pedals by cleats and toe clips, and with arms on the chest, the subjects pedaled an instrumented ergometer against a 20 N frictional load from rest to their maximum achievable speed as quickly as possible. Angular velocity was displayed to the subjects via a digital meter. Subjects were asked to relax as much as possible before starting, with the crank near horizontal with the left leg forward, which was within 10’ of the initial position used in the model. Subjects first warmed up by pedaling at 60 rpm for 2 mins. Subjects u 2 UM, = Lu aMl ” [CIU + [Cl ... cJT-j, rested for 5 mins, practiced the maximal startup task a-M (4) once, then rested for another 5 mins. Before data collec[u - a”].c2, u -c a”, tion, subjects were asked if they felt fatigued, and their where q is a vector of generalized coordinates (e.g. crank pulse rate was checked to assure it was within 10 right and left pedal angles); lM and uM are beats min- ’ of the rate prior to the practice trial. A secaw1e chk, vectors of muscle length and activation, respectively; ond maximum-speed trial was then performed and data M(q) and RM are the system mass and moment arm collected for 10 s starting just prior to the ‘go’ signal matrices, respectively; F”(q, 4, l”, a”) is a vector of given to the subject. All subjects reached their maximum muscle forces; G(q), V(q, d), and D(q, 4) are vectors of speed within 5 s. gravity, motion-dependent, and frictional terms, respecCrank and pedal angles were measured with optical tively. cl and c2 are related to muscle activation/deactiencoders, and normal and fore-aft pedal reaction forces vation time constants, with cl = &r - r& and from both feet with pedal dynamometers (Newmiller CL = T&p Since muscle activation uM is monotonic in et al., 1988) Crank angle data was filtered (Butterworth the control a, while the cost function and other equations third-order zero lag filter with a cutoff frequency of are not a function of the control, the optimal controls are 15 Hz), allowing velocity to be estimated through differ‘bang-bang’. For comparison with experimental data (see entiation. Radial and tangential crank forces were calbelow), the optimal controls were converted to functions culated by transforming the pedal reaction forces to of crank angle rather than time. These controls provided crank coordinates, then low-passed filtered (cutoff frea robust initial guess for new simulations (e.g. with one or quency 25 Hz). Bipolar EMG electrodes with preamplimore biarticular muscles not used). fiers were attached to the right leg of each subject over From the trajectories generated by solving the optimal the bellies of five muscles: GAS (medial), TA, VAS (mecontrol problem, the power contributed by each muscu- dial), RF, and HAM (biceps femoris long head). All data lar and non-muscular force to each segment, as well as were sampled at 1000 Hz. In post-processing, the EMG their contribution to segmental accelerations and intersignals were demeaned and rectified. EMG on/off times nal and pedal reaction forces, was computed (Fregly and were selected with an automated waveform processing Zajac, 1996). The segments were grouped as the cmnk program (Datapac, RUN Technologies Inc., Laguna (flywheel, freewheel, chainring, crank arms) and the limb Niguel, CA) using the following procedure: first a base(foot segment, shank, and thigh of the ipsilateral leg). The line EMG level was determined from the relaxed portion net mechanical energy produced by a muscle over a cycle of the trial, and a threshold for excitation set to at least (the amount generated minus the amount absorbed) was 3 SD above this value; on-off times were determined calculated by integrating the net power from the muscle using a minimum on or off duration of 55 ms. The events (tendon force times musculotendon velocity, net curve in selected were checked manually, and on/off durations Fig. 5) over the crank cycle. Note that this is mechanical adjusted as necessary (& 20 ms) to best capture the EMG energy (work), not metabolic energy (which would in- bursts in the first five cycles. These were converted to clude muscle heat as well). A muscle can instantaneously on/off crank angles. For comparison with the model, all contribute to the power of a segment (or accelerate a seg- data were averaged among subjects as a function of crank ment), even when it does not touch (e.g. the crank), by angle on a cycle-by-cycle basis (i.e. first cycle, second means of its contribution to the intersegmental reaction cycle, etc.). chosen for this maximum speed pedaling task was crank progress in a fixed time [Equation (l), i.e. proportional to average pedaling speed]. Thus, the optimal control problem solved was the dual of the fixed-progress minimumtime problem considered in earlier work (Levine et aZ., 1989). The controls (excitations to the muscle sets) which maximized crank progress were found using a constrained optimal control algorithm (Ma and Levine, 1993), which includes the FSQP optimization routines (Zhou and Tits, Univ. Maryland). The initial state for the simulation was found by allowing the musculoskeletal model to come to rest with all muscles off. The optimal control problem can then be stated as follows: find the controls (muscle excitations) u (subject to the bounds 0 < a < 1) that maximize the crank progress, i.e. the cost function
RESULTS
800
The pedaling simulation was found to reproduce the major features of maximal startup pedaling exhibited by the subjects. For the first two cycles, the crank angular velocity of the model is generally within L- 1 S.D. of the subjects (Fig. 1). Simulation and experimental crank forces (Fig. 2) are similar, except the simulation produces a radial crank force near BDC that is more outwarddirected (i.e. negative). The crank phases corresponding to the onset/offset times of the simulated muscle excitation signals are consistent with the estimated on/off phasing of the muscles in the subjects [Fig. 3(b)], derived from EMG signals [Fig. 3(a)]. However, the simulated onset/offset times are later, except for RF offtime. Simulations were run to assessthe sensitivity to the on/off phasing of the muscles. Using EMG phasing [Fig. 3(b)] to excite the dynamical model [Equation (2)44)] instead of the optimal controls. simulated performance decreased but crank velocity was still within 1 S.D. of the subjects. Analysis of the net mechanical energy produced by the muscles over the cycle (computed from the simulation; Fig. 4) showed that VAS is the primary contributor, providing 34% of the total (225 J). GMAX contributes more than twice as much energy (21%) as HAM (9%) though the two have similar force-generating capabilities (Table l), because GMAX’s moment arm at the hip is greater than HAM’s VAS generates energy to both the crank (70 J) and the limb (43 J) throughout most of downstroke (Fig. 5 : O-160., all VAS curves are > O), but then absorbs considerable energy (43 J) from the crank during the upstroke while it relaxes (i.e. after excitation ceases, 1W32OO; net and crank curves are both < 0 and about equal). GMAX generates energy (54 J) mostly to the limb in downstroke (Fig. 5, net and limb curves of
400
Tangential Force Simulation VA Subjects
-800
Radial Force - Simulation m Subjects l
-1200 O0 TDC
9o”
180’= E3DC Crank angle
270
360° TDC
Fig. 2. Right crank forces computed from experimental data (gray and cross-hatched regions, ave k 1 SD.) and generated by the optimal control simulation of pedaling (solid and dashed lines) in the first cycle. Downstroke is from 0 to 180’: upstroke from 188 to 360”. Positive tangential crank force accelerates the crank and positive radial crank force is directed toward the crank axis of rotation. Left crank forces (not shown) are similar but 180’ out-of-phase with the right crank forces.
a HAM RF VAS GAS TA ,
b
,
1
I
I
O0
9o”
180’
270”
360°
O0 TDC
9o”
270°
360° TDC
HAM RF VAS GAS TA SOL BFsh IL GMAX
0’ -1 60°
’
,
,
I
360°
540°
720°
’ O0
180 Crank
angle
Fig. 1. Crank angular velocity measured in the subjects (gray, ave & 1 SD.) and generated by the optimal control simmation of pedaling [solid tine). The right crank arm is vertical at 0 (top dead center, TDC) and 180’ (bottom dead center. BDC). The simulation and the subjects started cranking from rest with the right crank approximately 90’ before TDC 0.e. - 9W). Detailed analysis (e.g. Figs 2-7) focused on the first complete cycle (&3W) of the right leg, where simulation and experiment matched most closely. Regardless. the biomechanics and muscle coordination of subsequent cycles were found to be similar to the first cycle.
180° BDC Crank angle
Fig. 3. EMGs and optimal muscle excitation signals for the right leg in the first cycle. (a] Typical unprocessed EMGs. Scale indicates maximum EMG magnitude recorded, normalized to + 1 and - 1, respectively, for each muscle. (b) Average on/off phasmg of muscles after processing the EMG (gray bar indicates muscle is ‘on’; tines indicate & 1 SD.) compared with the on/off muscle phasing of the optimal control simulation of pedaling (black bar indicates the muscle is ‘on’). EMGs were not recorded for SOL. BFsh. IL, and GMAX, In later cycles. as ct-ank speed rises. both EMGs and simulated excitation signals advance in crank phasing.
Muscle
coordination
of maximum-speed
599
pedaling
w
+200
- -
ow
VAS GMAX Fig. 4. Mechanical
energy
SOL
IL
HAM
BFsh
produced by the muscles the first cycle (0.6 s).
TA
net crank
RF ‘GAS’
of the right
leg over
GMAX are both > 0 and about equal), but absorbs some energy (12 J) from the crank while it relaxes in upstroke (both ner and crank curves are < 0 and about equal). Energy generated to the limb in downstroke by VAS and GMAX is transferred to the crank by SOL (91 J) and GAS (36 J) (Fig. 5: SOL and GAS, &l W, crank curve is > 0 and about equal to jirrrb curve but opposite in sign). SOL and GAS also generate some energy to the crank (20 and 6 J; area under each net curve is > 0). Biarticular GAS, in transferring energy from the limb to the crank in the downstroke, acts primarily as an ankle extensor since its ankle-to-knee moment arm ratio is high in pedaling (Ericson, 1988; Spoor et al., 1990). TA acts as an antagonist to SOL, mostly transferring energy to the crank in the last part of upstroke. IL and BFsh both generate energy to the crank throughout upstroke (25 and 12 J; Fig. 5, 18&36P, aer curve and crank curves are both > 0 and about equal). However, as evidenced by the negative tangential crank force in the simulated upstroke (Fig. 2, 18&36P, solid line). the energy supplied by IL, BFsh, and other sources in the upstroke is insufficient to overcome the energy removed from the crank by the leg. This energy removal is mostly by the relaxing VAS and GMAX (Fig. 5, 180-36v, each crank curve is < O), and somewhat by gravity and the joint/ligament passive forces (not shown). Gravity and motion-dependent forces were found to transfer a small amount of energy between the crank and the limb (see also Fregly and Zajac, 1996) with the amount increasing as crank speed rose. HAM and RF produced peak power near BDC and TDC, respectively (Fig, 5). Before BDC and during the early excitation of HAM (100-1607, HAM transfers energy (8 J) from the crank to the limb while generating energy to the limb (3 J). Afterwards (160-19v), HAM generates energy to both the crank and the limb (4 and 3 J). While still excited during upstroke (19&25P), HAM mainly generates energy directly to the crank (11 J). When first excited during upstroke, RF generates energy to the crank (5 J, 25&300’; Fig. 5). Later RF generates energy to both the crank and the limb (1 and 2 J,
BFsh HAM
I O0 TDC
1 90°
, 160° BDC Crank angle
, 270’
J
360’ TDC
Fig. 5. Distribution of muscle mechanical power of the right leg in the first cycle. Downstroke is from 0 to 180°; upstroke from I80 to 360°. GMAX mostly generates energy to the limb in the downstroke (i.e during &180°, solid and dotted lines of GMAX are both > 0 and about equal; thus the integral of each power curve, which is energy, is > 0). VAS generates energy directly to the crank near TDC (i.e. near W, solid and dashed lines of VAS are both > 0 and about equal), then more to the limb later in downstroke (i.e. during 135160”, dotted line increases, dashed line decreases), and absorbs energy from the crank in the upstroke (i.e. during 18@320’, solid and dashed lines are both < 0 and about equal). SOL, GAS, and TA mainly transfer energy between the limb and the crank (i.e. dashed and dotted lines are about equal but opposite in sign). BFsh and IL generate energy to the crank in the upstroke. HAM and RF have complex patterns, but primarily contribute energy near BDC and TDC, respectively.
30&3307, and finally to just the limb (5 J, 33@-107. The energy generated to the limb by RF is transferred to the crank by TA and motion-dependent forces. DISCUSSION
The nominal muscle coordination pattern [Fig. 3(b)] was found to be insensitive to model parameters (e.g. peak isometric active force, activation/deactivation time constants, and initial state). Also, the simulation produced by using the on/off phasing of the measured EMGs is virtually the same as that produced by the nominal optimal muscle coordination pattern. (The difference is
600
perhperhaps due to inaccuracies in how muscle activation dynamics are modeled.) We conclude, therefore, that the nominal optimal control solution simulates well how muscles coordinate energy flow to the crank during startup pedaling. The higher peak radial crank force near BDC in the nominal simulation of pedaling (Fig. 2), which is the most notable difference between the simulation and the experimental data, may be due to subjects de-exciting their muscles slightly differently from the way assumed in the model and in the simulation. For example, Sandercock and Heckman (1996) have shown that the force measured during relaxation of the cat soleus muscle undergoing locomotor-like shortening is less than the force predicted by a Hi&type model (such as the one used in our study). When we included their suggestion that muscle activation during relaxation is velocity-dependent, peak radial force decreased by 31% (369 N) with little change in pedaling speed ( + 0.1%) or the optimal excitation pattern. Another possibility is that subjects de-excite VAS and GMAX sooner, since these muscles contribute significantly to the peak radial force (found by analyzing the contribution of each muscle to the crank force), but contribute little to tangential force near BDC. Analysis of the nominal simulation, which does not penalize coordination patterns producing high joint reaction forces, showed that high joint reaction forces occur when radial force peaks (e.g. the knee compressive reaction force of the simulation is 5.8 times body weight then). Other simulations show large reductions in both knee compressive force ( - 20%) and radial crank force ( - 32%) if VAS and GMAX are de-excited slightly earlier (VAS = 10 ms and GMAX = 25 ms earlier), though other changes in de-excitation of these muscles seem to be equally effective (e.g. by gradually de-exciting VAS and GMAX). In any case, the changes in the model necessary to reduce the discrepancy in radial force do not significantly affect the coordination pattern required to generate energy to the crank. Excitation of the plantarflexor muscles (including GAS) must be coordinated with the uniarticular knee and hip muscles (VAS and GMAX), which are the muscles producing most of the energy in the downstroke (i.e. 86% by VAS and GMAX; consistent with conclusions derived from net joint torques; Ericson et ul., 1986). About 55% of the energy produced by VAS and GMAX is not delivered directly to the crank but is, instead, generated to the limb segments. If the plantarflexors were not co-excited during downstroke, this energy would go towards accelerating the limbs (dorsiflexing the ankle, and hyperextending the knee, as discussed below) rather than being transferred to accelerate the crank. The plantarflexors (again including GAS) also transfer to the crank energy delivered to the limb by HAM in late downstroke (Fig. 5, 90-18W, area under limb curve), which is significant in comparison with the energy generated to the limb by VAS and GMAX (14 J by HAM; 27 J by VAS; 26 J by GMAX between 90 and 18W). Also, since HAM and VAS are co-excited in late downstroke and produce nearly opposing joint torques at the knee, the resultant torque by these muscles is the hip extensor joint torque produced by HAM. This explains why
Fregly and Zajac (1996) found only the net hip joint torque, and not the net knee joint torque, to deliver energy to the limb in downstroke. Thus, caution must be exercised in conjecturing on muscle coordination from net joint torque data. Though HAM is often labeled a ‘knee flexor’ in pedaling (e.g. in Gregor er u/., 1985), we believe it does not prevent knee hyperextension in late downstroke; instead. the plantarflexors seem to be the responsible muscles. In fact, even though HAM develops a knee flexor torque in late downstroke, it assists in the attempt to produce knee hyperextension since it accelerates the knee towards extension then (Fig. 6, 10&180’, HAM is > O), consistent with previous analyses (Andrews, 1987; Carlsoo and Molbech, 1966). The reason is because the moment produced by HAM at the hip in late downstroke is larger than at the knee, and also because limb geometry in downstroke favors transformation of hip torque into knee joint acceleration more than knee torque. This casts doubt on the idea that HAM activation allows the knee extensors to do more work by remaining active longer (Ingen Schenau, 1990), since the net action of HAM is also to extend the knee. In contrast, the plantarflexors (including SOL) act to accelerate the knee towards flexion (Fig. 6, 10&18@‘, SOL and GAS are ~0). The action of the plantarfiexors (specifically SOL) to accelerate the knee towards flexion, and the HAM to accelerate the knee towards extension, is due to the dynamic coupling by reaction forces (Zajac and Gordon, 1989) The dominant function of HAM and RF seems to be to effect smooth stroke transitions, which are times in the cycle when the total bilateral energy transfer to the crank
O0
9o”
180° Crank
270°
360°
angle
Fig. 6. Angular acceleration of the knee caused by SOL, GAS, VAS, HAM and BFsh. Knee angular acceleration is defined as the angular acceleration of the shank relative to the thigh. Positive acceleration is in the direction of knee extension. HAM accelerates the knee towards extension almost as strongly as VAS in the 135lgOQ region. In constrast, SOL and GAS accelerate the knee towards flexion then, which prevents knee hyperextension from occurring near BDC. Not shown, but also sign&ant, are accelerations due to GMAX (approximately the same as-VAS throughout downstroke) and motion-denendent fora% (similar to SOL from 90 to 1807. The knee is moving inio extension f&m 330 to 16S” (top box, knee &tending) and being accelerated toward flexion from 100 to 24s’ (bottom box, negatiue knee uccel.).
Muscle
coordination
of maximum-speed
is lowest. However, to accomplish this goal, HAM and RF utilize complex energetic mechanisms (e.g. Fig. 5), perhaps contributing to their heretofore controversial role in pedaling. For example, during downstroke, HAM produces energy, except at first when its tendon stretches more than its muscle fibers shorten (i.e. the muscle/tendon complex absorbs energy), Just before BDC when the hip is still extending but the knee is flexing (160-lSO’), HAM generates energy to both the crank and the limb (Fig. 5, 16&lfW, all curves of HAM are > 0; see also Carlsoo and Molbech, 1966). After BDC in the beginning of upstroke, although the hip is flexing, HAM continues to generate power to the crank (Fig. 5, 18@27W, net and trunk curves are both > 0 and about equal). The net effect of this energy production by HAM near BDC is to help prevent freewheeling (i.e. the decoupling of the crank from the load). Analogously, RF contributes substantially to crank propulsion near TDC. This is consistent with the notion that HAM and RF help achieve an optimally directed reaction force at the pedal (i.e. to maximize the tangential crank force) by distributing power among the joints (Ingen Schenau, 1990), which, according to the power analysis performed in our work, is useful because it generates power to the crank over an important (albeit limited) region of the cycle (i.e. the stroke transition regions). Other simulations by us (Raasch, 1995) demonstrate that replacement of a biartitular muscle with one of its equivalent joint torques can produce higher average pedaling speed with new excitation patterns, but this causes the crank to slow and/or freewheel during stroke transitions. In steady-state pedaling, where smooth cranking is very highly desired, biarticular muscles may play an even more critical role at the stroke transitions. Pervasive in theories of neural control of locomotion is the notion of a simple control (e.g. pattern generator) that, once triggered, can produce the fundamental coordination (i.e. phasing of muscles) needed to locomote (Grillner, 1975, 1981; Pearson, 1993; Rossignol et ul., 1988). Our data, experimental and simulation together, suggest that startup pedaling can be achieved with just two phase-controlled signals, where all the muscles in a leg are partitioned into two pairs of functional groups [Fig. 7(a)] which alternate with their contralateral counterparts. ln this scheme, the two alternating muscle pair (E group: groups of the Extensor/Flexor VAS/GMAX; F group: IL/BFsh) are excited during the downstroke and upstroke [Fig. 7(a)]. The primary function of this pair is to generate energy to both the limb and the crank [Fig. 7(b)]. The two alternating muscle groups of the Top/Bottom pair (T group: RF/TA; B group: HAM/SOL/GAS) are excited through TDC and BDC [Fig. 7(a)]. The muscles of this pair transfer energy from the limb to the crank and provide propulsion through the stroke transitions [Fig. 7(b)]. The simulation produced by this simple phasing strategy, with maximum excitation of the muscles, performs near-optimally with similar kinematics (e.g. average speed slows by less than 4%). Other simulations suggest that this phase-controlled strategy for muscle coordination can effect steady pedaling (e.g. at 60 rpm if the muscle groups are excited less than maximally).
601
pedaling
a Extensor Flexor
-
m E/F Pair
TOP Bottom
I 0
, 9o”
! 180’
, 270’
T/B Pair I 360Q
b Extensor 0 Flexor
Top
:..e.,,
l ..’
1
O0
TDC
,
9o”
1
1 80° BDC Crank angle
270°
f
360’ TDC
Fig. 7. Near replication of the biomechanics of maximum-speed startup pedaling by a simple muscle coordination strategy. (a) Phasing of the excitation of the four functional groups of muscles. Each muscle in the leg is classified into one of these four groups. The Extensor group (E group: VAS/GMAX) and the Flexor group (F group: lL/BFsh) form the E/F pair. The Top (T group: RF/TA) and the Bottom (B group: HAM/SOL/GAS) groups form the T/B pair. The two groups of each pair alternate in crank phasing, and each group alternates with the homologous group of the other leg. (b) Distribution of mechanical power produced by the four muscle groups. These power curves are almost identical to the summed power curves from the optimal simulation (e.g. nef curve for the Extensor group is similar to the sum of the net curves of VAS and GMAX in Fig. 6).
Control of motor tasks with simple strategies by partitioning muscles into alternating functional groups is not unique to pedaling (McCollum, 1993). Similar groups have been identified in animal locomotion (Grillner, 1981), and in studies of neurologically impaired humans (Perry ec ul., 1978) and maintenance of posture (ankle strategy; Nashner and McCollum, 1985). A single parameter (e.g. hip angle) is sufficient to phase-control the muscle groups, as in animal locomotion (Grillner and Rossignol, 1978). Thus, this simple control of human muscle groups may represent a centrally generated primitive for not only pedaling but walking as well, though such a primitive would surely be modulated by sensorimotor control mechanisms to overcome unexpected crank loads and other uncertainties (Prochazka, 1989).
.4[,~~~~~~~edge~ne~lrs~Thiswork was supported by NIH grant NSl7662 and the Rehabilitation R & D Service of the Department of Veterans Affairs (VAj, and is based on the Ph.D. dissertation of C.C.R. (Raasch, 1995). We thank Steve Kautz and Lena Ting for their comments on the manuscript. REFERENCES Andrews, J. G. (1987) The functional roles of the hamstrings and quadriceps during cycling: Lombard’s Paradox revisited. J. Biomechanics
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