
Research Article


Postural Balance of Humanoid Step Stance via Hybrid Space Formulation 

Zengshi Chen
and
Weiwei Yong



ABSTRACT

This research studies the postural balance of a step stance through modeling, control, simulation and experimentation. The mathematical model of a five segment humanoid subject is developed. The redundant multivariable system is projected onto a hybrid (Cartesian/joint) space with three degrees of freedom to make control easier. Based on the decoupled nonlinear system, a linear feedback control is designed to regulate the perturbation error. With the Lagrange method, a novel optimal bias control as a function of the system state, velocity and acceleration is designed to counteract the system main dynamics. The related balancing motion is performed by a human with markers on a force plate. The motion trajectories and the ground reaction forces are recorded. Comparison of the experimental and simulation results show that the proposed modeling and control strategy is capable of replicating the step stance balancing process. This research will promote understanding of mechanism of step stance postural balance for elder citizens and assist the relevant prosthesis design.





Received: November 13, 2011;
Accepted: March 16, 2012;
Published: June 20, 2012


INTRODUCTION
Human posture has been a popular research area. Advanced filters have been
used for human posture recognition (Tahir et al.,
2007). A mechanism based on neural network is developed to classify human
body postures (Tahir et al., 2006). The challenge
to postural balance control is created when humans walk, run and stand. A human
is an inherently unstable system unless a control action is continuously implemented
(Winter, 1987). The capability of balance degenerates
due to virtually all neuromusculoskeletal disorders (Byl,
1992). A pathology may not be obvious because the Central Nervous System
(CNS) has ability to compensate for the loss of the function (Winter,
1995). The three major sensory systems are used in balance and posture control:
The vision system, the “gyro” vestibular system and the somatosensory
system (Horak et al., 1990). Due to the three
separated sensory systems, a certain degree of redundancy can be used when one
or two of the systems fails (Horak et al., 1990).
Most of the work on balance have studied the function of each sensory system
and the use of the redundancy if a system is dysfunctional (Diener
et al., 1984).
The slow mobility of the elder at home is often involved in balance control
(Committee Annual Report, 2004). Research on the fall
of the elder covers many areas. Smart sensors are integrated to detect the falls
of the elderly (Sixsmith and Johnson, 2004; Fu
et al., 2008; Wu and Xue, 2008; Zigel
et al., 2009; Doukas and Maglogiannis, 2011).
Balance protheses are designed to prevent falls ( Wall and
Weinberg, 2003; Shi et al., 2009). Appropriate
methods are developed to assess fall risk of the elder (Greene
et al., 2010). However, using dynamics, control and simulation for
fall research is scarce.
A human, modeled in the sagittal plane for a variety of research aims, can
achieve a postural balance either in the Single Support Stance (SSS) (Kubica
et al., 1995) or in the step stance (SS) (Rohlmann
et al., 2001). Some human locomotion such as starting or termination
of sitting (Rohlmann et al., 2001), getting onto
or off the bed and termination of walking or running (Mu,
2004) ends up with balancing a SS posture. Lifting with a SS is a routine
human manual task (Kollmitzer et al., 2002).
In sports, a basketball player after a jump shot (Christgau,
1999) or a soccer kicker after heading a ball (Vogelsinger,
1970) or a skater gliding on a skate after propulsion of his leg (Allinger
et al., 1997) often has to bring his state to a stable SS poise.
Balancing SS postures for patients with locomotion disability is difficult (Patla
et al., 1995). Ritual bow in SS was popular in the Chinese society
(Chai and Chai, 1966). Study of those SS postural balances
in locomotion, activity, sports and pathology can generate knowledge on improving
performance, reducing injuries and facilitating gait rehabilitation.
The balance control of an open threelink human model in the SSS was reported
(Koozekanani et al., 1983). The constrained motion
of a fivelink biped was studied (Ceranowicz et al.,
1980). The sway motion of a threelink biped in the SS was tested (Hemami
and Wyman, 1979). In reality, SS motion satisfying a constraint always exists.
In the SS, the biped can be modeled by the joint space orientation of its torso
and the two Cartesian space variables that are the two hip translational positions.
This strategy has been previously applied to a SS jump (Chen,
2006). It provides the minimal but sufficient number of the state variables
to describe the SS movement of the biped.
The SS postural balance by modeling, simulation and experimentation is studied. The task is to bring the bipedal SS posture to a reference position without prescribing the entire trajectory except the target position and with the motion rigorously constrained to an invariant subspace (the space spanned by the motions of the subject for which the motion constraint is satisfied).
The model: A sagittal five link human subject in a step stance posture
on flat ground is modeled. The subject has five segments: The two identical
legs, the two identical thighs and a torso. The subject has the two ankle joints,
the two knee joints and the two hip joints. Every joint is equipped with a purely
rotational and frictionless actuator. The feet are assumed to be massless. Let
m_{i}, G_{i}, I_{i}, X_{i} and θ_{i}
be the mass, the gravity, the moment of inertia, the center of gravity and the
orientation of the ith link, respectively. Let τ_{i}, F_{ih}
and F_{iv} be the torque, the horizontal force and the vertical force
acting on the ith joint, respectively. Let τ = (τ_{1} τ_{2}
τ_{3} τ_{4} τ_{5} τ_{6})^{T}
where superscript T means “transpose”. Let θ = (θ_{1}
θ_{2} θ_{3} θ_{4} θ_{5})^{T}.
A schematic representation of the model is shown in Fig. 1a.
The markers located on the subject for a SS stance posture balance experiment
is shown in Fig. 1b. The detailed model is shown in Fig.
2. The detailed explanation of the parameters shown in Fig.
1 and 2 (Chen, 2006). In SS, the
subject has three degrees of freedom. Once the subject controls them, the whole
system is under control. Therefore, three state variables are enough to describe
the system. The Cartesian and joint space formulation is effective. As shown
by Chen (2006), this formulation consists of the hip position
variables from the Cartesian space and the torso orientation from the joint
space and the state equation is:

Fig. 1: 
(a) The kinematic model of a fivelink human model, (b) The
locations of the nine markers on the human subject 

Fig. 2: 
The parameters of the biped when the segments are viewed separately 
The controller: Equation 1 is linearized around an equilibrium point. A linear feedback is used to regulate the tracking error. An optimal nonlinear feedforward control is used to cancel out the main dynamics. A step stance posture is balanced. The system trajectories, the ground reaction force and the conservation of the constraints are investigated.
Linear feedback controller: With the hip trajectory and the inverse
kinematics, the joint angle profile for the SS (Chen, 2006).
Let τ_{s} be the static bias torque when the subject is stationary
with q = q_{s},
= 0_{3x1} and
= 0_{3x1 }where 0_{3x1} is a matrix of 3 rows and 1 column.
When the subject is perturbed from its equilibrium, the state variables, their
higher order derivatives and the torques are expressed in the two parts: the
static part with the subscript s and the disturbed part with the subscript p
as q = q_{s}+q_{p},
=
and
=
where τ = τ_{s}+τ_{p} is the bias torque when
the dynamics is considered. By Taylor expansion, the perturbation equation is
written as:
Where:
and:
I_{3x3} is a 3x3 identity matrix. 0_{3x3} and 0_{3x6} are the 3x3 and 3x6 zero matrices, respectively. The desired state equation of the decoupled system is: where:
and:
One can select the appropriate eigenvalues r_{i1}, and r_{i2}
with i = 1 to 3 to obtain the desired system response. The linear control of
τ_{p} = K_{p}(qq_{s})+K_{v}()is
selected where K_{p} and K_{v} are the position and velocity
gain matrices of 6x3, respectively. Comparing Eq. 4 and 5,
one has the desired gain matrices in terms of:
Optimal bias control: When the subject is stationary,
=
0 and _{s} = 0. However, dynamics exists before a subject converges
to a stable SS posture. Therefore, unlike the conventional bias method, the
velocity and acceleration in the bias torque compensation is taken into account.
Defining the dynamic bias torque with the denotation of τ_{d}.
Since there are more joint torques than the segments of the subject, τ_{d}
is not unique. However, a unique solution exists to minimize a criterion function
such as:
The dynamic rotational equations of the subject are:
where, A_{ij} for i or j = 1 to 3 is defined by Chen
(2006). The dynamic translational equations of the subject are:
is the function of
and .
Solving Eq. 9 for F_{2}, F_{3}, F_{4},
F_{5} and F_{6} in terms of F_{1} renders:
where, D_{1}, D_{2} and D are defined by Chen
(2006). Assuming F_{1} is an arbitrary constant vector and D_{1}F_{1}+D_{2}
= D_{3} one can minimize the criterion function of:
with the constraint equation of Dτ_{d}D_{3} = 0_{5x1}. Let the Lagrange equation be:
where, λ is a vector of five Lagrange multipliers corresponding to the five constraint equations. Then, the equilibrium point of the subject from the local minimization could be found as:
Multiplying Eq. 10 by D and solving it for λ enders:
Substituting Eq. 13 into Eq. 11 renders:
Now:
is an explicit function of F_{1} and it can be minimized with respect to F_{1}. The criterion function becomes: With the symmetrical property of D^{T}_{1} (DD^{T})^{1}D_{1}, F_{1 }is found from the partial differentiation of:
with respect to F_{1}. One has:
Equation 14 becomes:
τ_{d} will be used as τ_{s}. RESULTS
Simulation parameters: The horizontal distance of the two feet is taken
as 0.2655 cm. The values of the parameters are shown in Table
1. The equilibrium point of the subject is taken as q_{s} = [0.1328
(m) 1.0431 (m) 0 (rad)]^{T},
= (0 0 0)^{T} and
= (0 0 0)^{T}. The related dynamic bias torque in the unit of N*m is
τ_{s} = (9.67 21.73 1.2 1.2 1.63 6.64). The poles of each decoupled
system are selected as r_{i1} = r_{i2}. This results in:
A_{1} = 0_{3x3}
Table 1: 
Parameters of the subject 


Fig. 3: 
The four ground reaction forces in the step stance postural
balance 
Simulation results: Figure 3 shows that the ground
reactions forces (GRFs) converge to their equilibrium points. In the entire
response, F_{1h}≤0, F_{6h}≥0, F_{1v}≥0 and
F_{6v}≥0. Figure 4 shows that the position variables
converge to their static values. The Cartesian space variables h_{x}
and h_{y}, are monotonically increasing. The joint space variable, θ_{3},
after experiencing a slight overshoot, converges. The hip is moving forward
and upward. The torso rotates clockwise first and then counterclockwise to the
vertical stance. Figure 5 shows the trajectories of the six
joint torques. Their magnitudes can be generated by human beings and are hence
reasonable.

Fig. 4: 
The simulated trajectories of the state variables: h_{x},
h_{y} and θ_{3} in the Cartesian and joint space 

Fig. 5: 
The simulated trajectories of the state variables: h_{x},
h_{y} and θ_{3} in the Cartesian and joint space 

Fig. 6: 
The trajectories of the joint torques: τ_{1},
τ_{2}, τ_{3}, τ_{4}, τ_{5}
and τ_{6} 
Figure 6 shows the hip trajectory o f the subject in the
sagittal plane. Apparently, the hip is moving in a nonlinear curve.

Fig. 7: 
The simulated trajectories of the joint angles in the joint
space: θ_{1}, θ_{2}, θ_{3}, θ_{4},
θ_{5} and θ_{6} 

Fig. 8: 
The constrained horizontal and vertical distances of the feet
of the subject 
Figure 7 shows the convergence of the five angular state
variables in the joint space to their equilibrium values. Figure
8 shows that the physical constraints between the feet are maintained in
the entire response. Figure 9 shows the sequential stick diagrams
of the subject that is being brought to the equilibrium point from the initial
condition.
Experimental results: In this experiment, the step stance postural balance
is performed by a subject who lands in SS (the two feet touch the ground simultaneously)
on the force plates and stabilizes her postimpact posture. An Optotrak 3020
system and two Bertec force platforms were used to collect the kinematic and
GRF data from a young, healthy female subject.

Fig. 9: 
The stick diagram of the subject in the step stance postural
balance 
The two tracking markers (infrared lightemitting diodes) were attached to
the skin over specific boney landmarks on the hip and the shoulder. The marker
locations are shown in Fig. 1b. As the subject moved, the
Optotrak system tracked the marker positions and simultaneously calculated precise
threedimensional data for each. The sampling frequency was 100 Hz for the motion
data and 1000 Hz for the GRFs. The data was filtered by a low pass Butterworth
filter with the cutoff frequency of 6 Hz for motion and 15 Hz for GRFs. Each
link of the subject is approximated as a rigid body. Assuming the coordinates
of the markers on the hip as (h_{x}, h_{y}) and on the shoulder
as (s_{x}, s_{y}), one computes the rotational angle of the
torso as:
The subject is instructed to land on the force platforms in SS and bring her
posture to a reference position from the postimpact stance. Six experiments
were done. After each impact, a natural postural balance motion is executed
by the subject. It is noticed that the balance motions performed by the subject
are similar to the motion in the simulation. That is, with the initial velocity,
the subject moves her hip upward and forward while rotating her torso counterclockwise.
The subject does not have any slippage during postural stabilization. Figure
10 demonstrates the recorded ground reaction forces. Ignoring impact impulses,
one notices that the measured ground reaction forces take the similar overall
pattern as the simulated ones. After landing, almost instantaneously, F_{h1}
and F_{h6} fall to their minima and F_{v1} and F_{v6}
increase to their maxima. Then, both F_{h1} and F_{h6} increase.
F_{h1} stays below zero and converges to a negative value. F_{h6}
crosses zero to converge to a positive constant.

Fig. 10: 
Measured ground reaction forces in a step stance after a landing. 

Fig. 11: 
The top panel shows the friction coefficient μ_{1}
and its average value. The actual coefficient is μ_{1}. The
bottom panel shows the friction coefficient μ_{6} and its average
value 
After the subject is brought to standstill, the GRFs converge to constants
with F_{h1} 30 N, F_{h6} 35 N, F_{v1} 270 N and F_{v6}
350 N. Both F_{h1} and F_{h6} are less than or equal to 0 in
the beginning of the landing because the inertial momentum of the subject tends
to throw her forward. F_{h1} and F_{h6} provide the countermomentum
to decelerate the subject to rest in the horizontal direction. During the standstill
time, the subject shifts her body weight from her right lower limb to her left
lower limb.
In the simulation, since the initial perturbation is comparatively small with
a slight throwingforward momentum, thereafter, F_{h6} can start from
a value close to zero and F_{h1} starts from a negative value. The phenomenon
of shifting the body weight by the simulated subject is also observed in the
simulation. In the beginning, the majority of the weight is supported by the
left lower limb such as F_{v6} = F_{v1}. In the sequential short
interval, the weight is shifted to the right lower limb such as F_{v1}
= F_{v6}. Finally, the weight is shifted back to the left lower limb
such as F_{v6} = F_{v1}. The coincidence of the results from
the simulation and experiment provides the conviction that the CNS of humans
may be able to choose the minimal number of variables, adopt a simple control
and stabilize human SS postures. In Fig. 11, the top panel
shows F_{1}, the friction coefficient of the right foot with the force
plate over the time and its average value; the bottom panel shows F_{6},
the friction coefficient of the left foot with the force plate over the time
and its average value. Inspecting Fig. 11, one concludes
that the friction coefficients required in the simulation can be provided by
the force plates because the measured forces show that the force plates can
provide the much larger friction coefficients.
DISCUSSION
Most of the existing postural balance research is confined to the following
categories: (1) kinematics and measurement, (2) statics, (3) dynamics with the
simplified model (4) dynamics with the simplified control. As for (1), an example
is the study for the changes in postural sway and strategy elicited by lumbar
extensor muscle fatigue (Madigan et al., 2006).
Wholebody movement and ground reaction force data of twelve healthy male participants
were recorded and used to calculate mean body posture and variability of center
of mass, center of pressure and joint kinematics during quiet standing. As for
(2), an example is the study of the impact of a human with the environment (Zheng
and Hemami, 1984). The contact of the two feet in parallel with the ground
is studied. The transient dynamics in the postural balance is ignored in the
integration. As for (3), an example is the study of sway motion through a damper
or spring model (Dijkstra, 2000). The damper and spring
are connected to a stick representing the human body that itself is oversimplified.
As for (4), an example is the computer simulation of postural balance through
a recursive approach and linearization (Koozekanani et al.,
1983). The human is modeled as a multilink rigid body system with one contact
point on the ground. Only linear control is used for postural balancing.
The proposed method in this paper overcomes the drawbacks of the above approaches. It models, controls and simulates a human SS postural balance. The dynamics model takes into account many physical details of an actual human. The optimal control is based on the rigorous analysis and derivation. The underactuation of the system in the joint space is avoided by modeling the system in the Cartesian and joint space. With the inverse kinematics mapping from the Cartesian space to the joint space, the system during SS postural balance is constrained to an invariant subspace in which the state of the biped starts, roams and converges. Simulations give one the flexibility to study a hypothetical situation, such as bringing a SS posture from one to another. In particular, when the similar postural balance is executed by a real subject, the GRFs from the measurement and the GRFs from the simulation basically comply with each other. Although the subject in the experiment does not exactly bring her posture to the equilibrium set up in the simulation, the values of the converged GRFs in the experiment are still close to the optimal solution. One may infer that humans have a tendency to minimize their energy expenditure when stabilizing and maintaining their postures. The proposed method faithfully reproduces the SS postural balance which is common in sport competition and humanoid daily balance control. The responses of the ground reaction forces tell that the gravity of the body shifts between the lower limbs frequently and easily. Only if an elder is physically still strong or carries a prosthesis may he avoid a fall during those subconscious shifts of the body weight. In SS balance, for the Cartesian and joint space model, the hip cannot be perturbed too far from the equipoise. Otherwise, the inverse kinematics will have no solution. This may hint that an elder should refrain from strong motions in order to avoid a fall. The low friction of the certain floors and the weak vertical ground reactions an elder provides may cause him to slip and stumble. The proposed method has a few disadvantages which should be addressed in the future study. The model under study is confined to the sagittal plane although sometimes a posture is balanced in the threedimensional space. The proposed model is still simple since neither the feet nor the arms are considered. The joint torques instead of the muscle forces are used in the model. Although the optimal bias control is used, the central nerve system control that is closest to the nature is not studied. An actual fall motion has not been simulated.

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