Mathematical Biosciences 237 (2012) 28–37

Contents lists available at SciVerse ScienceDirect

Mathematical Biosciences

journal homepage: www.elsevier .com/locate /mbs

A simple strategy for jumping straight up

Hooshang Hemami ⇑, Bostwick F. WymanDepartment of Electrical and Computer Engineering, The Ohio State University, Columbus, OH 43210, United StatesDepartment of Mathematics, The Ohio State University, Columbus, OH 43210, United States

a r t i c l e i n f o

Article history:Received 18 October 2011Received in revised form 29 February 2012Accepted 2 March 2012Available online 14 March 2012

Keywords:Ballistic movementStraight jumpingConstrained dynamicsForces of constraintKinematicsStability

0025-5564/$ - see front matter � 2012 Elsevier Inc. Ahttp://dx.doi.org/10.1016/j.mbs.2012.03.003

⇑ Corresponding author. Tel.: +1 614 292 2848; faxE-mail address: hemami.1@osu.edu (H. Hemami).

a b s t r a c t

Jumping from a stationary standing position into the air is a transition from a constrained motion in con-tact with the ground to an unconstrained system not in contact with the ground. A simple case of thejump, as it applies to humans, robots and humanoids, is studied in this paper. The dynamics of the con-strained rigid body are expanded to define a larger system that accommodates the jump. The formulationis applied to a four-link, three-dimensional system in order to articulate the ballistic motion involved. Theactivity of the muscular system and the role of the major sagittal muscle groups are demonstrated.

The control strategy, involving state feedback and central feed forward signals, is formulated and com-puter simulations are presented to assess the feasibility of the formulations, the strategy and the jump.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Ballistic movements are an interesting class of human move-ments which have been studied for some time [1–3]. Ballisticmovements range from the simple jump considered here to sacc-adic eye movements in complex human activities that involvethrowing, catching and intercepting objects. Jumps are a specialclass of movements [2] where the system, in an original con-strained state space, is propelled into a larger state space with asmaller number of constraints and performs a desired maneuver.According to Ganong [4], the lateral portions of the cerebellar cor-tex and the cerebellum [5] seem to be involved in the jump and inall fast saccadic and ballistic movements. However, the specificfunctions and involvement of the motor cortex and the cerebellumare not known or precisely articulated. One aspect of the jump, themechanism of transition into a larger state space, is formulated,analyzed and explored here. The constraints, prior to the initiationof the jump, could be maintained by the inhibitory activity of thepurkinje cells in the cerebellum. The release of the purkinje cellsto disinhibition could be the mechanism that launches the systemto the larger state space. The jumping phenomenon, as discussedhere, entails a maneuver for the constrained system to acquire suf-ficient kinetic energy to overcome the force of gravity and get pro-pelled upward into space. This entails a quick and large activationof the agonist muscles involved in the motion and a simultaneousdecrease in the activation of the antagonist muscles in order to

ll rights reserved.

: +1 614 292 7596.

exert large forces to the system. This induced large initial velocityaccounts for the needed kinetic energy of the system.

Philosophically speaking, most of what humans do is to makeand break contact with the environment and external objects.Jumping is a complex form of breaking contact with the environ-ment. The physical coupling or interaction takes place primarilythrough the forces of contact with no gripping or holding involved.For ease of presentation and communication, we describe every-thing in terms of human movement, muscles and activation mech-anisms, [1,6]. However, from a scientific point of view, thedevelopments and results presented here are for humanoids, ro-bots and bipedal systems. The dynamics of ballistic movements,and the involvement of the central nervous system (CNS), the mus-cles and the feet are not well understood. The approach here is toconsider a system with large gains and a one-segment foot.

In the past, three methods have been utilized to solve the verti-cal jump problem: experimental studies such as the one by Hassonand colleagues [7], parameter optimization of a musculoskeletalsystem [8–10], and optimal control [11–13]. The musculoskeletalsystem could also be studied via learning methods [14–16]. In con-trast, the problem is formulated here in terms of four positionstates involved in the initial squatting position and four initialvelocities as a result of the implementation of the ballistic move-ment. The ballistic movement, in brief, converts a high level ofelastic energy [17] to kinetic energy. The transition from the stand-ing position to the squatting position is not considered here [18].Three phases of the jumping maneuver are considered: the squat-ting position held by co-activation of the agonist and antagonistmuscles, the lift-off phase initiated by injecting the system with

Input

state

state

state

Γ1

Fig. 2. The block diagram for the three phases of the jump.

H. Hemami, B.F. Wyman / Mathematical Biosciences 237 (2012) 28–37 29

large initial angular velocities to propel the system upward and theforces of contact and support going to zero, and, finally, the stablejump upward in the air. Falling back to the ground and the recov-ery maneuver to maintain balance after contact with the groundare not considered [19,20].

The physical system is a simple, four-link biped with the feet asone rigid body. The legs together, the thighs together and the torsoare the other three links. The three-dimensional dynamics of thissystem, in comparison with other methods [21,22,9], are modularand computer-adapted. The muscular system is presented as anagonist–antagonist co-activated system [23–25], and it is modularand computer-adaptable. It roughly approximates a high gainsystem [26–28]. The analysis is intended to provide a mechanismto explore the involvement of large muscular fibers, spinal reflexes[29], and higher levels of the CNS. Neural encoding of thesignals [30–32], and preliminary implementations [33,34] are notconsidered.

In the equations of the system, the role of the feet are repre-sented by the induced motion of the heels. First, the dynamics ofthe four-rigid body system with fifteen pairs of muscle-like actua-tors are briefly presented in Section 2. The implementation of thejump is presented in Section 3. Simulations and comparisons arepresented in Section 4. Discussions and conclusions are in Section5. References and appendices follow.

2. System equations

The four-link biped equations can be derived from the free bodydiagrams of four rigid bodies [35,21,36], each having six degrees offreedom, three attitude angles H and three translational coordi-nates of the center of gravity X. Initially, the bodies’ principal axesand the inertial coordinate axes coincide [22,37,38]. The inertialaxes are front, left and vertical. The rotations about these axesare, for easy visualization, roll, pitch and yaw as they apply to anairplane. The derivation of the system equations is briefly summa-rized here (see Appendix A). Pairs of bodies are supposed to be con-nected together at one point. Each connection is described by threeholonomic constraints. A sketch of the system is shown in Fig. 1.The movement of the heel, is expressed as a vector b due to rota-tion of the foot about the toes. The toes contact the ground atthe origin of the inertial coordinate system (ICS) [39].

Using the constraints, one eliminates nine translation coordi-nates. Thus, the system maintains 15 degrees of freedom: 12 atti-tudinal and three translational degrees of freedom. Thetranslational degrees of freedom, measured in the ICS, can be ofany arbitrary or desired point of the body, (e.g., center of gravity

m2g

yz

0

r2

s1

s2

s3

Γ1

Γ2

Γ3

Γ4

m4g

m3g

m1g

r1

r4

r3

x

Fig. 1. The four segment biped with a foot and the support forces C1 at the toes. Thetoes lie at the origin of the inertial coordinate system. The x axis is to the front. The yaxis is to the left. The z axis is vertically upwards.

of any of the four links, center of gravity of the body, the coordi-nates of the center of the head or the coordinates of the heel).We consider, for simplicity, the coordinates of the center of gravityof the foot for the translational motion. In the squatting position,the system has 12 degrees of freedom, and the supporting forceof constraint C1 is a function of the states and input to the system[20]. In jumping, the force of support go to zero and the bipedleaves the ground.

With the help of a projection operator (see Appendix A, Eq. (8)),we map the 24-degree of freedom free body system to the 15-de-gree of freedom system involved in the jump. In turn, the con-straint equation of contact with the ground is described as

X1 ¼ �A1r1 ð1Þ

The squatting position and the lift-off phase involve the same15-dimensional system with C1 going to zero. The block diagramof the system that represents all three phases is given in Fig. 2.In the squatting and lift-off phases, the C1 loop is closed. Duringthe jump, there is no contact with the ground. This means:

C1 ¼ 0

3. Actuation

3.1. Muscle structure

We assume there are 21 pairs of muscle-like actuators, sevenpairs each for controlling pitch, roll and yaw. The seven pairs forpitch control are motivated by dominant human muscles in sagit-tal movement. Of the seven pairs, four activate simple joints. Onepair represents toe flexors and extensors that can rotate the footrelative to the toes. The second pair corresponds to action at theankle (soleus and tibialis anterior). The vasti and bicep femorisare the pair for the knee joint, and gluteals and psoas rotate thetorso relative to the thigh.

The remaining three pairs are bi-articular. A pair representsplantar- and dorsiflexors with origin at the toes and insertion onthe leg. This pair would represent some of the functions of flexorhallucis longus and flexor digitorum longus [40]. The geometryand anatomy of these muscles are illustrated on pages 96 and 97of the cited reference. This pair, like the gastrocnemius, may nothave an obvious and unique antagonist. A second pair is betweenthe foot and the femur, which performs a generalization of the gas-trocnemius function. Finally, a third pair, rectus femoris and ham-strings, connects the leg to the torso.

The muscular structure for roll, i.e., motion in the frontal plane,and yaw, i.e., self rotations of the body, are different than the mus-cular structure for the pitch. We arbitrarily extend the same actu-ation topography to the roll and yaw motions. This means we caninclude up to seven pairs of actuators with analogous connectionsfor roll and yaw motion. We have used six pairs for each of roll,pitch or yaw motion here. We have not included the fifth pair-the biarticular pair between the toes and the leg. The six pairsare further discussed in Appendix B and the simulation section.In natural systems, there exist muscles that cross couple roll,yaw and pitch. We are ignoring this group. The jump here is pri-marily a multiple segment pitch that takes place in the XZ plane

30 H. Hemami, B.F. Wyman / Mathematical Biosciences 237 (2012) 28–37

of the ICS, the sagittal plane. The details of a case with five pairs ofmuscle-like actuators is considered in reference [25].

We assume all of these actuators function similarly and all havethe same moment arm which, for simplicity, is assumed to beunity. Every actuator is driven by tension that stretches the tendonlinearly [41,25]. As a result, linear feedback of the position andvelocity is possible by co-activation, and feeding forward of centralsignals takes place with the same gains. The central input signalsare all trajectories of the angles of the limbs that are either con-stant in equilibrium positions or ramp-like in point-to-pointmaneuvers [42–44]. The velocity feedback is supposed to be linearand a fraction of the gains for position are used for velocityfeedback.

At the beginning of the lift-off phase, the limbs acquire largecoordinated angular velocities due to agonist intensive activity.At the same time, the antagonist level of activity decreases drasti-cally to make the fast movements possible. To sharpen focus on theballistic movement, we assume the increase in angular velocities isinduced by appropriate impulses that increase the angular veloci-ties of the segments instantaneously from zero to their desired val-ues. The effect of input impulses on the system is considered here.

The dynamics of the 15-dimensional system (Appendix A, Eq.(12)) is symbolically written as

J15ð _WÞ ¼ fr þ Nr þ Gr þ ðHcÞ0C1 ð2Þ

The inputs to the system are the moment of force vector Nr:

Nr ¼ Na þ Nb þ Nc

where Na is for stability (feedback), Nb is for point to point move-ment and equilibrium stance (feedforward), and Nc is for gravitycompensation:

Na ¼ �KH� LX

Nb ¼ KH ðdesiredÞNc ¼ �Gr ð3Þ

For jumping, large impulsive Na is applied to induce the properangular and linear velocities to the segments and the footrespectively.

3.2. Implementation of the jump

The strategy for jumping is to synergistically rotate the limbsfast, i.e., induce a ballistic movement There are six steps in thejump. All six have the above block diagram of Fig. 2 in common.

� The system gets to a squatting (flexed) position from an initialvertical stance and can remain in this equilibrium state forsome time.� The muscular system, in a very short time, i.e., in the order of

50 ms, energizes the system to a set of initial velocities.� The system elongates to an almost fully extended form, the

forces of contact decrease to zero and the system leaves theground.� The airborne phase continues until the system stops its vertical

climb.

The strategy for jumping is to derive impulsive forces that inducethe desired initial velocities. Two cases are considered here.

Case 1. The starting time is assumed to be

t ¼ 0

The initial impulsive input torques Nd are given. The correspondingimpulsive constraint force Cd, and the jump in the initial velocities,Wd have to be computed. For this purpose, in Eq. (2) the terms fr andGr are ignored, and the equation of the contact constraint, Eq. (1), is

differentiated twice. The latter two set of equations are simulta-neously solved for the �Wd and Cd The latter values are inserted inthe original equations of motion and integrated from 0� to 0þ.The result is the initial velocities after the impulses have dissipated.The position states do not change.

Case 2. The desired initial velocities, i.e. Wð0Þ0þ are specified.The difference between these specified velocities and Wð0Þ0� canbe used to derive the impulsive constraint force Cd and the substi-tution of the latter in the impulsive equations of motion rendersthe needed impulsive torques Nd.

4. Computer simulation of the simple jump

Two sets of simulations are presented. The first set assumesimpulsive forces can be applied to the system. The actual impulsiveforces for this case are only computed in order to arrive at the stateof the system at t ¼ 0þ sec, namely, after the impulse is appliedand dissipated. The computed and plotted contact forces that ap-pear in the figures reflect the effect of the induced velocities afterthe application of the impulsive forces.

In the second set of simulations, the impulses are approximatedby finite pulses of high amplitude, and, consequently, the forces aremore physical and influence the behavior of the system over the fi-nite time of the application of the pulses.

4.1. Impulsive inputs

We consider the system with six pairs of actuators for each atti-tude angle instead of seven. For pitch, we ignore the bi-articularpair around the ankle. The physical parameters of the four-link sys-tem to be simulated here are given in Appendix C. At the initialsquatting position the angles are, respectively ½�3=8p;p=4;�p=3;p=3� rad for the foot, leg, thigh and torso. The center of gravity ofthe foot lies at ½�0:1386;0;0:0574�m. In this initial position, thecenter of gravity of the body lies at ½�0:1589;0;1:2826�m.

The system’s muscular input for the initial burst of velocity isapproximated by impulses that produce different initial angularvelocities in the sagittal plane. The linear position feedback gains,for the 18 pair of muscle-like actuators, by co-activation of the ago-nist and antagonist muscles are given in the appendix. The velocityfeedback gains are assumed to be a fraction, 25%, of the positiongains.

The four initial velocities for the foot, leg, thigh and torso anglesare: ½2:70;�2:70;3:6;�2:70�m/s. With these initial velocities, thecenter of gravity of the foot also acquires an instantaneous velocityof ½0:1560;0; 0:3737�m/s. The impulsive constraint force is com-puted to be ½�14:8850;0:000;222:5888� N. The jump is successfuland the height is about 0.30 m, which is perhaps average forhumans. The results of the simulation, for a duration of 0.75 s,are given in the following three figures.

Fig. 3 shows the toe support forces as functions of time. Thehorizontal and vertical forces of support are plotted. It is assumedthat the ground has high friction, i.e., the horizontal force is about50% of the vertical force and the feet do not slip. Once the feet leavethe ground, the forces of support are zero. Fig. 4 shows the anglesand angular velocities of the leg, thigh and the torso all going tozero at about 0.5 s

Fig. 5 shows the coordinates of the center of gravity of thewhole body and the translation of the center of gravity of thefoot as functions of time. At about t = 0.6 s, the vertical climbstops and the center of gravity is above the center of the unflexedfoot.

A comparison of the contact forces (vertical and fore-aft) withother investigations [13,9,7] shows that the measured verticalforce trajectory is more bell-shaped and the fore-aft one is more

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

20

40

60

80

100

120

140

time (seconds)

horiz

onta

l γ (N

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

200

400

600

800

1000

1200

time (seconds)

verti

cal γ

(N)

Fig. 3. The horizontal and vertical components of the force of support as functions of time in squatting for about 140 ms, for lift-off for about 120 ms and then during theairborne phase.

0 0.2 0.4 0.6 0.8−1.5

−1

−0.5

0

time (seconds)

angl

e 1

(rad)

0 0.2 0.4 0.6 0.8−5

0

5

time (seconds)

ang

. vel

. 1 (r

ad/s

ec)

0 0.2 0.4 0.6 0.80

0.5

1

time (seconds)

angl

e 2

(rad)

0 0.2 0.4 0.6 0.8−4

−2

0

ang.

vel

. 2 (

rad/

sec)

time (seconds)

0 0.2 0.4 0.6 0.8−1.5

−1

−0.5

0

angl

e 3

(rad)

time (seconds)0 0.2 0.4 0.6 0.8

−5

0

5

time (seconds)

ang.

vel

. 3 (r

ad/s

ec)

0 0.2 0.4 0.6 0.80

0.5

1

1.5

time (seconds)

angl

e 4(

rad)

0 0.2 0.4 0.6 0.8−6

−4

−2

0

time (seconds)

ang.

vel

. 4 (r

ad/s

ec)

Fig. 4. The four pitch angles, relative to the vertical, of the foot, leg, the thigh and the torso, and the corresponding angular velocities during the lift-off and airborne phases ofthe jump.

H. Hemami, B.F. Wyman / Mathematical Biosciences 237 (2012) 28–37 31

gradual. The reason for the sharp rise in both force trajectories isdue to the impulsive, rather than gradual and sustained, muscularinputs. Similarly, at the trailing end, i.e., the moment of leaving theground, we have rigid body contact, while the other investigationsuse spring-like and visco-elastic structures. Also, we have assumed

discontinuities in the angular velocities. Since the contact forcesare functions of the state (positions and velocities) and the inputs,there are discontinuities in the forces of contact. The same phe-nomenon happens in the next simulation because the input forcesare square-pulse shaped and discontinuous.

32 H. Hemami, B.F. Wyman / Mathematical Biosciences 237 (2012) 28–37

4.2. Pulse inputs for jumping

With the above desired initial velocities and impulsive con-straint forces, one can compute the input sagittal torques

Nc ¼ ½59:6840;�84:1700;74:9272;�45:9776� Nm

0 0.2 0.4 0.6 0.8−0.16

−0.14

−0.12

−0.1

−0.08

time (seconds)

x of

cog

of b

ody

(m)

0 0.2 0.4 0.6 0.8−0.18

−0.16

−0.14

−0.12

−0.1

−0.08

time (seconds)

x of

cog

of f

oot (

m)

Fig. 5. The motion of the center of gravity and

0 0.1 0.2 0.3−150

−100

−50

0

50

100

time (s

horiz

onta

l γ (m

)

0 0.1 0.2 0.30

500

1000

1500

2000

2500

3000

time (s

verti

cal γ

(m)

Fig. 6. The fore-aft and the vertical components

The above torques are rounded to ½60;�84;75;�46� Nm. If themuscular system consists of only four pairs at all the joints, thecorresponding torques at the toes, ankle, knee and hip are½5;�55;29;�46� Nm.

The joint activities specified by the above four numbers seem tobe reasonable for humans. The contribution from the foot flexor is

0 0.2 0.4 0.6 0.8

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

time (seconds)

z of

cog

of b

ody

(m)

0 200 400 600 8000.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

time (seconds)

z of

cog

of f

oot (

m)

the center of the foot in the sagittal plane.

0.4 0.5 0.6 0.7econds)

0.4 0.5 0.6 0.7econds)

of the force of support as functions of time.

H. Hemami, B.F. Wyman / Mathematical Biosciences 237 (2012) 28–37 33

the smallest. The knee joint forces (from the vasti) are next. The an-kle joint forces (the soleus group) are the largest.

We consider isotonic contraction of muscles ([4], p. 40). The im-pulses are distributed as constant torques over an interval of 50ms.For strong action, such responses could potentially last for 100 ms([4], p. 38). We assume a moment arm of unity. Therefore, the four

0 0.2 0.4 0.6 0.8−1.5

−1

−0.5

0

time (seconds)

angl

e 1

(rad)

0 0.2 0.4 0.6 0.80

0.5

1

time (seconds)

angl

e 2

(rad)

0 0.2 0.4 0.6 0.8−1.5

−1

−0.5

0

angl

e 3

(rad)

time (seconds)

0 0.2 0.4 0.6 0.80

0.5

1

1.5

time (seconds)

angl

e 4

(rad)

Fig. 7. The four pitch angles, relative to the vertical, of the foot, leg, thigh and to

0 0.2 0.4 0.6 0.8−0.12

−0.11

−0.1

−0.09

−0.08

−0.07

time (seconds)

x of

cog

of b

ody

(m)

0 0.2 0.4 0.6 0.8−0.14

−0.12

−0.1

−0.08

−0.06

time (seconds)

x of

cog

of f

oot (

m)

Fig. 8. The motion of the center of gravity and

required constant forces are ½1200;�1680;1500;�920� N. Thetranslation to torques at the joints amounts to ½100;�1100;580;�920� Nm.

The results of the simulated jump with these continuous pulse-like forces are presented below. Fig. 6 shows the contact forces asfunctions of time. The fore-aft and the vertical forces of support are

0 0.2 0.4 0.6 0.8−2

0

2

4

time (seconds)

ang

. vel

. 1 (r

ad/s

ec)

0 0.2 0.4 0.6 0.8−4

−2

0

2

ang.

vel

. 2 (r

ad/s

ec)

time (seconds)

0 0.2 0.4 0.6 0.8−2

0

2

4

time (seconds)

ang.

vel

. 3 (r

ad/s

ec)

0 0.2 0.4 0.6 0.8−4

−2

0

2

time (seconds)

ang.

vel

. 4 (r

ad/s

ec)

rso, and the corresponding angular velocities during for pulse force inputs.

0 0.2 0.4 0.6 0.8

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

time (seconds)

z of

cog

of b

ody

(m)

0 200 400 600 8000.05

0.1

0.15

0.2

0.25

time (seconds)

z of

cog

of f

oot (

m)

the center of the foot in the sagittal plane.

34 H. Hemami, B.F. Wyman / Mathematical Biosciences 237 (2012) 28–37

plotted. The vertical ground reaction forces (GRF) rise to about fourtimes the body weight. This is not physiologically possible, and isdue to the inaccuracies of our foot model, the muscular modelingand the approximations of the ballistic movement implementationwe have used. Fig. 7 shows the angles and angular velocities of theleg, thigh and the torso all going to zero at about 0.5 s

Fig. 8 shows the sagittal coordinates of the center of gravity ofthe whole body as well as the translational coordinates of the cen-ter of gravity of the foot as functions of time. At about t ¼ 0:6 s, thevertical climb stops and the center of gravity is above the center ofthe unflexed foot.

5. Discussion and conclusions

We have formulated a very simple model of a jump in a three-dimensional skeletal model that is amenable to computer imple-mentation. The muscular system could be more complete andphysiologically more acceptable with utilization of muscle modelsfrom [9,6], inter-muscular coordination [13], better representa-tions of the structure of the foot, modeling of multi-joint muscleconnections, a visco-elastic structure for the ground-foot contactrather than rigid body contact, and higher level human strategies[7].

Considerable research effort is needed to extend our under-standing to all jumps as discussed in ([2], Chapter 21). Of interestare the jumps that involve vision and throwing as described byThach [5]. Understanding the core concepts of kinesiology [45] in-volved in all jumps and a comparison of the kinesiological param-eters may also shed light on the CNS and the cerebellum’sinvolvement in jumping. The research effort may allow testing ofcentral motor programs, coordinated action, [39] and other exper-imentally derived hypotheses and conjectures about the CNS[46,47] by computational models. We have already introducedthe ‘‘set’’ concept from references ([48,4], p. 151), in this paper.In preparation for jumping or throwing a ball, a stable equilibriumpoint with no roll or yaw motion, is desirable. The cerebellum’sinvolvement in bringing about and establishing the equilibriumposition can be understood in terms of the ‘‘set’’ concepts. Thetransition from agonist–antagonist co-activation in the equilib-rium position to the coordinated agonist activation, in order to in-duce the initial velocities, merits further research. This researchcould potentially describe the role of the cerebellum in this activitymore precisely.

The formulation here can be expanded to more complex skele-tal models based on the interconnection of such three-link mod-ules. Specifically the torso, and the arms can be modeled withsuch models and their contribution to jumping better understood.

Acknowledgement

The authors are very grateful to professor H. C. Ko, professoremeritus and former chair of electrical engineering at The OhioState University for his vision, generosity, dedication and supportof this work. The authors acknowledge the assistance of the com-puter facilities personnel of the Department of Electrical and Com-puter Engineering at The Ohio State University, specifically VincentJuodvalkis, Donald S. Gibb, Aaron Aufderheide and Edwin Lim. Theauthors are very grateful to the reviewer for the time and effortspent on the paper and for the very constructive, thorough and illu-minating comments, suggestions and improvements.

Appendix A Four rigid body system

Let X;V ;H and X be, respectively, the translational and rota-tional states of a free rigid body [49,50]. Let K be a 3-vector of force

acting a point C on the body whose coordinates are vector R in BCS.In connection with vector R, we define the skew symmetric 3 by 3matrix �R [51]. The vectors of force Gi and Ni are, respectively, thegravity vector and the vector of control inputs in the ith rigid body.The equations of motion of the four rigid body system [52] can bederived as follows.

Let the first rigid body with index 1 be connected to a platformat point a1, and to the rigid body with index 2 at point b1. Similarly,the points of contact of body 2 to body 1 and body 3 are, respec-tively, designated as a2 and b2. The point of contact of body 3 tobody 2 is designated as a3. Let the origin of the inertial coordinatesystem be at 0, and the point a1, where the platform is supportedby the ground.

Let the coordinates of points a1 and b1 in the rigid body coordi-nate system be r1 and s1, respectively. Let the coordinates of pointsa2 and b2 be, respectively, r2 and s2 in the second body’s coordinatesystem bcs2. Let the force C1 act on body 1 at point a1, and force(�C2) act on body 1 at point b1. It follows that force C2 acts on body2 at point a2.

The equations of motion for the four free rigid bodies are asfollows

The first body corresponds to the feet together.

_X1 ¼ V1

M1_V1 ¼ G1 þC1 � C2

_H1 ¼ B1ðH1ÞX1

J1_X1 ¼ f ðX1Þ þ N1 þ �r1A01C1 � �s1A01C2 ð4Þ

The second body represents the two legs.

_X2 ¼ V2

M2_V2 ¼ G2 þC2 � C3

_H2 ¼ B2ðH2ÞX2

J2_X2 ¼ f ðX2Þ þ N2 þ �r2A02C2 � �s2A02C3 ð5Þ

The third body represents the two thighs.

_X3 ¼ V3

M3_V3 ¼ G3 þC3

_H3 ¼ B3ðH3ÞX3

J3_X3 ¼ f ðX3Þ þ N3 þ �r3A03C3 ð6Þ

Finally, body 4 represents the torso and the rest of the body

_X4 ¼ V4

M4_V4 ¼ G4 þC4

_H4 ¼ B4ðH4ÞX4

J4_X4 ¼ f ðX4Þ þ N4 þ �r4A04C4 ð7Þ

Let Z be the 24-dimensional vector of translational and rotationalpositions states:

Z ¼ ½X01;X02;X

03;X

04;H

01;H

02;H

03;H

04�0

Let

H ¼ ½H01;H02;H

03;H

04�0

Let

U ¼ ½X01;H0�0

Let the BCS be along the principal coordinate axes of the body. Let,in the initial vertical stance, all the four body coordinate systems bealigned with the ICS. Let Ai be the transformation from the BCS tothe ICS for body i,

Table 1Definition and numerical values.

Segment Mass Dim. Dim. Mom. of I. Mom. of I. Mom. of I.Symbol mi ri si i1 i2 i3

Foot 2 �0.15 0.15 0.0193 0.0154 0.0047Leg 6 �0.27 0.24 0.2411 0.21 0.567Thigh 16 �0.25, 0.20 0.2963 0.18 0.1735Trunk 46 �0.28 0.28 2.35 0.900 0.4570

Table 2Unity moment arms for six pairs of muscles.

E ¼

1 �1 0 0 1 00 1 �1 0 0 �10 0 1 �1 �1 00 0 0 1 0 1

26664

37775 ð20Þ

The co-activation gains for the six pairs of muscles isgain =

½50 250 150 150 150 150�

The feedback gains for position in the sagittal plane becomes.

Table 3Position feedback gains in the saggital plane.

Gain ¼

450 �250 �150 0�250 550 �150 �150�150 �150 450 �150

0 �150 �150 300

26664

37775 ð21Þ

This set of gains is extended to roll, pitch and yaw for the 15-dimensionalsystem

H. Hemami, B.F. Wyman / Mathematical Biosciences 237 (2012) 28–37 35

i ¼ 1;2;3;4

Finally, let the heel motion be described, in the ICS system,by vector b. The 12 connection constraints and the 12 attitudeangles

H ¼ ½H01;H02;H

03;H

04�0

define the projection operator that maps the 24-degree of freedomfree body system to the 15-degree of freedom system involved inthe jump, as follows:

X1 ¼ X1

X2 ¼ �A2r2 þ A1s1 þ X1

X3 ¼ �A3r3 þ A2ðs2 � r2Þ þ A1s1 þ X1

X4 ¼ �A4r4 þ A3ðs3 � r3Þ þ A2ðs2 � r2Þ þ A1s1 þ X1

H ¼ H ð8Þ

The projection from Z to U is by the transformation given by Eq. (8).Substitution of Eq. (8) in the equations of the four bodies eliminatesC2;C3;C4X2;X3 and X4. The resulting system has 15 degrees offreedom.

We define, for convenience, two 3� 3 matrices. The identitymatrix is represented as I3, and the null matrix as Z3. For ease offormulation, we represent the 48 equations in a matrix form. Thefollowing vectors and matrices help in this regard.

Let J24�24 be a matrix whose diagonal elements are the eight3� 3 diagonal matrices M1;M2;M3;M4; J1; J2; J3; ;J4. The follow-ing definitions simplify the notation:

Z12 ¼ ½0;0;0;0;0;0;0;0; 0;0; 0;0� ð9Þ

N ¼ ½Z12;N10;N20;N30;N40�0

f ¼ ½Z12; f 10; f 20; f 30; f 40�0

G ¼ ½G10;G20;G30;G40; Z12�0

H4 ¼ ½I3; I3; I3; I3�n4 ¼ ½Z3; Z3Z3; Z3�C ¼ ½C1;C2;C3;C4�0 ð10Þ

Similarly, define HCas:

Hc ¼

I3 �I3 Z3 Z3

Z3 I3 �I3 Z3

Z3 Z3 I3 �I3

Z3 Z3 Z3 I4

�r1A01 ��s1A01 Z3 Z3

Z3 �r2A02 ��s2A02 Z3

Z3 Z3 �r3A03 ��s3A03

Z3 Z3 Z3 �r4A04

26666666666666666664

37777777777777777775

ð11Þ

With these definitions, the 24 equations of motion can be sum-marily written as

J½ _V 0; _X0�0 ¼ N þ f þ Gþ HcC ð12Þ

Differentiating Eq. (8) twice with respect to time, one derives

_V_X

" #¼ Ha

_V1

_X

" #þ Hb ð13Þ

The 24� 15 matrix Ha is

Ha ¼

I3 Z3 Z3 Z3 Z3

I3 �A1�s1 þA2ð�r2Þ Z3 Z3

I3 �A1�s1 �A2ð�s2 � �r2Þ �A3�r3 Z3

I3 �A1�s1 �A2ð�s2 � �r2Þ �A3ð�s3 � �r3Þ �A4�r4

Z3 I3 Z3 Z3 Z3

Z3 Z3 I3 Z3 Z3

Z3 Z3 Z3 I3 Z3

Z3 Z3 Z3 Z3 I3

266666666666664

377777777777775

ð14Þ

The 24-vector Hb is

Hb ¼

03�1

A1ð�X1Þ2ðs1Þ � A2ð�X2Þ2ðr2ÞA1ð�X1Þ2ðs1Þ þ A2ð�X2Þ2ðs2 � r2Þ � A3ð�X3Þ2r3

A1ð�X1Þ2ðs1Þ þ A2ð�X2Þ2ðs2 � r2Þ � A3ð�X3Þ2ðs3 � r3Þ � A4ð�X4Þ2r4

03�1

03�1

03�1

03�1

2666666666666664

3777777777777775

The reduced 15-dimensional equations of the system can nowbe derived in two steps:

� Substitute Eq. (14) in Eq. (12).� Pre-multiply both sides of the resulting equation by the trans-

pose of Ha.

As a consequence of the reduction, C2;C3 and C4 get eliminatedfrom Eq. (12), but C1remains. The equations of motion are

Table 4Position gains in the squatting position.

Gain ¼

450 0 0 �250 0 0 �150 0 0 0 0 00 450 0 0 �250 0 0 �150 0 0 0 00 0 450 0 0 �250 0 0 �150 0 0 0�250 0 0 550 0 0 �150 0 0 �150 0 0

0 �250 0 0 550 0 0 �150 0 0 �150 00 0 �250 0 0 550 0 0 �150 0 0 �150�150 0 0 �150 0 0 450 0 0 �150 0 0

0 �150 0 0 �150 0 0 450 0 0 �150 00 0 �150 0 0 �150 0 0 450 0 0 �1500 0 0 �150 0 0 �150 0 0 300 0 00 0 0 0 �150 0 0 �150 0 0 300 00 0 0 0 0 �150 0 0 �150 0 0 300

26666666666666666666666664

37777777777777777777777775

The negative feedback gains for angular velocities: X are 25% of the position gains. In the lift-off and airborne phases, the co-activationgains are reduced by 50%. The position and velocity states, after the impulsive inputs to induce initial velocities is

p ¼ ½�0:1386;0;0:0574; 0;�1:1781; 0;0;0:7854;0;0;�1:0472;0;0;1:0472;0�:

36 H. Hemami, B.F. Wyman / Mathematical Biosciences 237 (2012) 28–37

H0aJHa_W ¼ Ha0ðf þ N þ Gþ HdC1Þ � H0aJHb: ð15Þ

The coefficient of C1 is the 15� 3 matrix Hd

Hd ¼

I3

�r1A01Z3

Z3

Z3

26666664

37777775

ð16Þ

Appendix B. The forces of contact

The forces of contact are computed as follows. The contact con-straint is differentiated twice with respect to time to derive

H0d _W ¼ He ð17Þ

where Hd is given by Eq. (16), and

He ¼ �A1ð�X1Þ2ðr1Þ ð18Þ

From Eqs. (15) and (17), one can solve for _W and C1. This im-plies that when impulsive N is applied to the system, both _Wand C1 are impulsive. As a consequence, angular velocities andtranslational velocity of the foot’s center of gravity are discontinu-ous and jump.

It is worthwhile to also note that the above state space H and xformulation can be transformed to the classical state space of

_H ¼ BðHÞX€H ¼ BðHÞð _XÞ þ @ðBXÞ=@H _H ð19Þ

Appendix C. Physical and control parameters

The physical parameters of the system are given in Table 1.We use six pairs of muscles for control of roll, yaw or pitch: four

single joint pairs and two double joint pairs. The moment arm ma-trix for the six pair is shown in Tables 2–4.

References

[1] A. Hubbard, Homokinetics: Muscular Function in Human Movement, Harperand Brothers Publishers, 1960.

[2] P.J. Rasch, R.K. Burke, Kinesiology and Applied Anatomy, sixth ed. Lea andFebiger, 1978.

[3] J. Basmajian, Muscles Alive. The Williams and Wilkins Company, 1978.

[4] W.F. Ganong, The Nervous System, second ed., LANGE med. Publications, LosAltos, California, 1979.

[5] W.T. Thach, A role for the cerebellum in learning movement coordination,Neurobiology of Learning and Memory 70 (1998) 177.

[6] J. Desmedt, E. Godaux, Ballistic contractions in man – characteristicrecruitment pattern of single motor units of the tibialis anterior muscle, J.Physiol. London 264 (1977) 673.

[7] C. Hasson, E. Dugan, T. Doyle, B. Humphries, R.U. Newton, Neuromechanicalstrategies employed to increase jump height during the initiation of the squatjump, Journal of Electromyography and Kinesiology (1982) 515.

[8] M.G. Pandy, F. Anderson, D. Hull, A parameter optimization approach for theoptimal control of large scale musculoskeletal systems, J. Biomechanics (4)(1992) 450.

[9] F. Anderson, M. Pandy, A dynamic optimization solution for vertical jumping inthree dimensions, Computer Methods in Biomechanics and BiomedicalEngineering (4) (1999) 201.

[10] S. Davarani, H. Hemami, K. Barin, A. Shirazi-Adl, M. Parnianpour, Dynamicstability of spine using stability-based optimization and muscle spindle reflex,IEEE Trans Neural Systems and Rehab. Eng 16 (1) (2007) 106.

[11] E. Sim, The application of optimal control theory to analysis of human jumpingand pedaling, Ph.D. thesis, University of Maryland, 1988.

[12] M.G. Pandy, L. Zajac, Felix, E. Sim, W.S. Levine, An optimal control model formaximum-height human jumping, J. Biomechanics (12) (1990) 1185.

[13] M.G. Pandy, F.E. Zajac, Optimal muscular coordination strategies for jumping, J.Biomechanics (1991) 1.

[14] K. Kawato, M. Furukawa, R. Suzuki, A hierarchical neural network model forcontrol and learning of voluntary movement, Biological Cybernetics 57 (1987)169.

[15] L. Wang, D.L. Alkon, Artificial Neural Networks, IEE, Computer Society, 1993.[16] F. Mussa-Ivaldi, E. Bizzi, Learning Newtonian Mechanics, in: P. Morasso, V.

Sanguineti (Eds.), Advances in Psychology, Self-Organizational ComputationalMaps, vol. 119, North Holland, Elsevier, Amsterdam, 1997, pp. 191–237.

[17] M. Berkemeier, K. Desai, Design of a robot leg with elastic energy storage,comparison to biology and preliminary experimental results, in: Proceedingsof the 1996 International Conference on Robotics and Automation, IEEERobotic Society, 1996, pp. 213–218.

[18] K. Iqbal, B.D. Clymer, H. Hemami, Squatting maneuver of a three-link nine-muscle biped model in the sagittal plane, J. of Automatic Control (1) (1997)113.

[19] H. Hemami, Y.-F. Zheng, Dynamics and control of motion on the ground and inthe air with application to biped robots, Journal of Robotic Systems (1) (1984)101.

[20] H. Hemami, B. Wyman, Modeling and control of constrained dynamic systemswith application to biped locomotion in the frontal plane, IEEE Trans. onAutomatic Control (4) (1979) 526.

[21] D. Wang, Parallel implementation of multibody dynamics for real-timesimulation, PhD thesis, University of British Columbia, 1991.

[22] F.M.L. Amirouche, Computational Methods in Multi-body Dynamics, seconded., Prentice Hall, Englewood, NJ, 1992.

[23] W. Bizzi, E. Chapple, N. Hogan, Mechanical properties of muscle: implicationsfor motor control, Trends Neurosci. 5 (1982) 395.

[24] C. Abul-Haj, N. Hogan, Functional assessment of control systems forcybernetics elbow prostheses – part ii, application of the technique, IEEETrans. Biomed. Eng. 37 (November) (1990) 1037–1047.

[25] H. Hemami, K. Barin, Y.-C. Pai, Quantitative analysis of a sagittal biped underplatform disturbance, IEEE Trans. Neural Syst. Rehabilitation Eng. (4) (2006)470.

[26] F. Mussa-Ivaldi, N. Hogan, E. Bizzi, Neural mechanisms and geometrical factorssubserving arm posture in humans, J. Neurosci. 5 (1985) 2732.

H. Hemami, B.F. Wyman / Mathematical Biosciences 237 (2012) 28–37 37

[27] P.G. Morasso, M. Schieppati, Can muscle stiffness alone stabilize uprightstanding?, J Neurophysiol. 82 (1999) 1622.

[28] P. Krishnamurthy, F. Khorrami, High gain output feedback control fornonlinear systems based on multiple time series, Syst. Control Lett. 56 (1)(2007) 7.

[29] E. Henneman, Organization of the spinal cord and its reflexes, in: V.Mountcastle (Ed.), Medical Physiology, vol. 1, The C.V. Mosby Company,1980, pp. 762–786.

[30] S.W. Kuffler, J. Nicholls, From Neuron to Brain, a Cellular Approach to theFunction of the Nervous System, first ed., Sinauer Associates, Inc.,1976.

[31] J. Houk, W. Rymer, Neural control of muscle length and tension, in: Handbookof Physiology: The Nervous System II, American Physiol. Society, 1981.

[32] Y. Ivanenko, R. Grasso, V. Macellari, F. Lacquaniti, Spinal control of support andcontact forces, J. Physiol. 87 (2002) 3070.

[33] J. Kim, H. Hemami, Control of a one link arm by burst signal generators, Biol.Cybern. 73 (1995) 37.

[34] J. Kim, H. Hemami, Coordinated three-dimensional motion of the head andtorso bydynamic neural networks, IEEE Trans. Syst. Man Cybern. Part B 28 (5)(1998) 653.

[35] H. Hemami, A general framework for rigid body dynamics, stability andcontrol, Journal of Dynamic Systems Measurement and Control (2002) 241–251.

[36] F. Anderson, M.G. Pandy, Dynamic optimization of human walking, Trans.ASME J. Biomech. Eng. (2001) 381.

[37] W. McCuskey, Introduction to Advanced Dynamics, Addison Wesley Pub.,Reading, Mass, 1959.

[38] E. Whittaker, A Treatise on the Analytical Dynamics of Particles and RigidBodies, Dover Publications, 1944.

[39] R.M. Restak, The Brain, fourth ed., Bantam Books, Toronto, 1984.[40] J.H. Warfel, The Extremities, fourth ed., Lea and Febiger, Philadelphia, 1978.

[41] J. Dinneen, H. Hemami, Stability and movement of a one-link neuro-musculo-skeletal sagittal arm, IEEE Trans. Biomed. Eng. 40 (6) (1993) 541.

[42] H.H. Kornhuber, Motor functions of cerebellum and basal ganglia: thecerebellocortical saccadic (ballistic) clock the cerebellonuclear hold regulatorand the basal ganglia ramp (voluntary speed smooth movement) generator,Kybernetik 8 (4) (1971) 157–162.

[43] H.H. Kornhuber, Cerebral corex, cerebellum, and basal ganglia: andintroduction to their motor functions, Neurosci. III (1973) 267–280.

[44] H. Hemami, B. Dariush, Control of constraint forces and trajectories in a richsensory and actuation environment, Math. Biosci. (2010) 171.

[45] J.L. Hudson, Core concepts in kinesiology. <http://www.csuchico.edu/jhudson/pdf/corecons.pdf>.

[46] E. Evarts, E. Bizzi, R. Burke, M. DeLong, W. Thach Jr., Central control ofmovement, Neurosci. Res. Prog. Bull. (1971) 1–170.

[47] E. Evarts, Changing concepts of central control of movement, Can. J. Physiol. 53(1975) 191.

[48] E. Evarts, Y. Shinoda, S. Wise, Neurophysiological Approaches to Higher BrainFunctions, John Wiley and Sons, 1984.

[49] H. Hemami, B. Dariush, Single rigid body representation, control and stabilityfor robotic applications, in: Proceedings of the IEEE 2000 InternationalConference on Robotics and Automation, Institute of Electronic and ElectricalEngineers, April 2000.

[50] H. Hemami, B. Dariush, A mathematical representation of biorobots andhumanoids for performance assessment, computer simulation and motionanimation, in: Proceedings 6th International Workshop on Advanced MotionControl, IEEE, Japan, 2000, pp. 531–536.

[51] H. Hemami, A. Katbab, Constrained inverted pendulum for evaluating uprightstability, J. Dynam. Syst. Measure. Control (1982) 343.

[52] H. Hemami, A state space model for interconnected rigid bodies, IEEE Trans.Autom. Control (2) (1982) 376.