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Dottorato di Ricerca in Ingegneria Meccanica e Gestionale
XXVII Ciclo
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Il programma di dual PhD nato nel 2012 tra il Dipartimento di Meccanica, Matematica e
Management del Politecnico di Bari e la NYU-Poly consiste in un percorso di dottorato di ricerca in cotutela in Ingegneria Meccanica e Gestionale. Il dual PhD nasce con lo scopo di sviluppare un tema di ricerca congiuntamente strutturato che sfrutti il potenziale di ricerca e l’offerta formativa di entrambi gli istituti, rafforzando in questo modo le collaborazioni gia’ esistenti tra due Universita’ partner dell’H2CU. Il programma prevede che lo studente superi il concorso di ammissione in entrambi gli istituti, secondo le regole prestabilite nelle rispettive Universita’, e che consegua i crediti necessari al completamento del percorso accademico, ottenendo infine il titolo di Dottore di Ricerca presso il Politecnico di Bari e di Ph.D presso il Politechnic Institue della NYU. Questo programma assume un’ importanza particolare specialmente per i temi di ricerca che coinvolgono piu’ discipline, come quello presente. Nel corso dei tre anni, infatti, la dottoranda alternera’ i suoi periodi di formazione e ricerca tra l’istituo italiano e quello americano, con lo scopo di ottenere le competenze necessarie sulle diverse discipline oggetto di studio, traendo massimo vantaggio dall’esperienza nei diversi campi delle due universita’ coinvolte. In particolare, la formazione della dottoranda si e’ basata su materie quali dinamica analitica, modelli e dinamica del contatto, metodi di ottimizzazione (offerti dal Politecnico di
Bari), in aggiunta a metodi numerici, ottimizzazione numerica, robotica e controllo (offerti dalla
NYU-Poly e NYU Courant Institute of Mathematics). In aggiunta ad un’ampia ed intensa formazione, la cotutela permette soprattutto di sviluppare i diversi aspetti della ricerca in maniera complementare presso le due Universita’, sfuttando le strutture, competenze, e collaborazioni esterne offerte sia in Italia che negli USA.
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Journals
 Mummolo, C., and Kim, J.H., “Passive and dynamic gait measures for biped mechanism: formulation and simulation analysis,”
Robotica , Vol. 31, n. 4, pp. 555-572, 2013.
 Mummolo, C., Mangialardi, L., and Kim, J.H., “Quantifying Dynamic Characteristics of
Human Wakling for Comprehensive Gait Cycle,” Journal of Biomechanical Engineering , Vol.
135, n. 9, pp. 091006, 2013.
 Mummolo, C., Mangialardi, L., and Kim, J.H., “Balance Stability Characterization of Loaded
Walking Using Dynamic Gait Measure,”
Journal of Biomechanical Engineering , submitted, under review.
Conferences
 Mummolo, C., Mangialardi, M., and Kim, J.H., “Experimental analysis for passive and dynamic gait measures of biped walking,”
ASME 2012 Dynamic Systems and Control
Conference , October 17-19, 2012, Ft. Lauderdale, FL, USA.
 Mummolo, C., Joo, C.B., and Kim, J.H., “Dynamic Gait Measure for Biped Walking of
Robots and Humans,” (poster presentation) Dynamic Walking Conference , May 21-24, 2012,
Pensacola Beach, Florida, USA.
Presentations

Mummolo, C., Mangialardi, M., and Kim, J.H., “Quantifying Passive and Dynamic Nature of
Human Walking to Assist the Design of Lower Limbs Exoskeletons,” (poster presentation)
BiHRI Summer School 2012 - Biomechanics in Human-Robot Interaction , Castle of Gargonza
(Arezzo, Italy), July 9-14, 2012.
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Introduction and thesis outline 5
PART I
Chapter 1: Passive and dynamic gait measures for biped mechanism
1.1
Biped system model and dynamics
1.2
Gait modeling
1.3
Stability and Dynamic Gait Measure formulation
1.4
Optimality and Passive Gait Measure formulation
1.5
Human-like and robotic walking simulation results
1.6
PGM and DGM results: human vs. Robotic gait
1.7
Concluding remarks
Chapter 2: Quantifying dynamic characteristics of human walking
2.1
Models and method
2.2
Dynamic Gait Measure for comprehensive gait cycle
2.3
Results
2.4
Discussion and concluding remarks
Chapter 3: Balance stability of loaded walking using Dynamic Gait Measure
3.1
Methods and experiments
3.2
Results and statistics
3.3
Discussion and concluding remarks
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54
56
60
62
PART II
Chapter 4: Optimal trajectory planning through contact detection– On going work
4.1
Objective and method
4.2
Preliminary examples
65
66
67
68
Chapter 5: Future work
5.1 Passive and dynamic gait measures applications on robotic walking and
69 clinical gait analysis.
5.2
General formulation for contact detection to general form of optimal motion planning under general constraints.
5.3
Implementation of new method into optimal walking motion planning and validation with the proposed indexes.
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12
16
17
21
25
30
34
36
38
44
47
50
References 70
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Seeking the fundamental principles of movement of living systems is an old scientific question that for many years has interested researchers in the fields of biomechanics, biomimetics and robotics. The understanding of locomotion and mechanics of moving systems includes the analysis of both mechanical (e.g., robots, mechanisms, machine components, etc.) and biological systems (e.g., humans, animals, insects, etc.) (Figure 1). In biomimetics, studying the structure and function of biological systems found in nature serves as model for the design of engineering systems and modern technologies; on the other hand, the current research trend in biomechanics shows that the mechanical modeling of robots can be useful when studying more complicated biological systems, such as the human body. Researchers investigate locomotion of humans and other animals when designing robots whose motion and physical construction mirror those of living things.
Mechanical principles of bipedal walking have been studied by many researchers for a long time, and are still one of the active subjects in the field of robotics and biomechanics. Researchers investigate locomotion of humans and other animals when designing robots whose motion and physical construction mirror those of living things. The balance and dynamics of walking, hopping, and running have been studied extensively in biomechanics in order to gain a better understanding of human and animal locomotion. The human body forms with its limbs, head, and torso a well balanced walking machine that performs periodic and energy-efficient gait. For this reason, the development of humanoid robots tends to mimic humans' performance in terms of design and motion control. As a consequence, such biped robots are one of the major topics of robotics research, which also have high potentials for future applications. Also, from a purely scientific point of view, bipedal locomotion is a complex topic that opens unsolved basic research problem in several fields such as nonlinear dynamics, controls, biomechanics, optimization theory and algorithm, optimal motion planning, energetics, and so on. This multidisciplinary feature of locomotion research, along with the number of interesting applications, has made walking one of the most popular topic of this decade and widely spread all over the world.
In line with the aforementioned active research trend on bipedal locomotion, this thesis will also contribute to the research in more than one field (Figure 2). This work can be structured in two parts. In the first part, this work will investigate fundamental aspects of biped locomotion that can be applied to the walking of robot, human, and their intersections, such as the humanexoskeleton system. In the second part a novel motion planning algorithm will be studied for the detection and optimization of contact events and contact forces in a fully predictive manner.
The scope of this research will spread from the pure scientific to the engineering and clinical prospective. We aim to add contributions in each of these three aspects of bipedal walking research, by developing new tools for the gait analysis of human and robots, proposing new engineering design framework for novel walking-aid technologies, and contributing to the scientific research in the fields dynamic modeling, optimization and simulation of multi-body systems. By applying mechanical, engineering, and biological principles to living systems we can
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gain a better knowledge of real world phenomena and develop new solutions that can help improve nature, our health, and our living.
Figure 1. Research field overview.
Figure 2. Fields of investigation of the thesis work.
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Understanding and mimicking human gait is essential for design and control of biped walking robots. The unique characteristics of normal human gait are described as passive dynamic walking, whereas general human gait is neither completely passive nor always dynamic. To study various walking motions, it is important to quantify the different levels of passivity and dynamicity, which have not been addressed in the current literature. In this article, we introduce the initial formulations of Passive Gait Measure (PGM) and Dynamic Gait Measure (DGM) that quantify passivity and dynamicity, respectively, of a given biped walking motion, and the proposed formulations will be demonstrated for proof-of-concepts using gait simulation and analysis. The PGM is associated with the optimality of natural human walking, where the passivity weight functions are proposed and incorporated in the minimization of a physiologically inspired weighted actuator torques. The PGM then measures the relative contribution of the stance ankle actuation. The DGM is associated with the gait stability, and quantifies the effects of inertia in terms of the Zero-Moment Point (ZMP) and the ground projection of center of mass
(GCOM). In addition, the DGM takes into account the stance foot dimension and the relative threshold between static and dynamic walking. As examples, both human-like and robotic walking motions during single support phase are generated for a planar biped system using the passivity weights and proper gait parameters. The calculated PGM values show more passive nature of human-like walking as compared with the robotic walking. The DGM results verify the dynamic nature of normal human walking with anthropomorphic foot dimension. In general, the
DGMs for human-like walking are greater than those for robotic walking. The resulting DGMs also demonstrate their dependence on the stance foot dimension as well as the walking motion; for a given walking motion, smaller foot dimension results in increased dynamicity. Future work on experimental validation and demonstration will involve actual walking robots and human subjects. The proposed results will benefit the human gait studies and the development of walking robots.
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a b c i
NOMENCLATURE i1 A i
0 p i
0 s i j R i
T , T
Homogeneous transformation matrix for two adjacent coordinate frames
Position vector of the i th
local frame in terms of the global frame
Position vector of the i th
Single support initial and final time instants, respectively
Foot rear dimension
point mass in terms of the global frame
Rotation matrix between two local frames i and j
Foot front dimension
Ratio of distance of i th
point mass from previous joint over the entire link length l i d Root mean square distance between ZMP and GCOM in x direction over one step period
Root mean square distance between ZMP and GCOM lim d lim
DGM
DS
DW
Dynamic Gait Measure
Double support
Dynamic walking l l fh
FSR
J
K i t
L
Foot height
Foot support region
GCOM Ground projection of center of mass
GCOM lim
Maximum GCOM displacement within FSR
Cost function for optimization
Least upper bound of DGM value i th
link’s length
Distance of torso mass m
2
from the pelvis
Time rate of change of angular momentum of i th
link about its COM mhi m i
PGM sh sl
SS
SW v w i x i
, y i x i
, y i
Malleolus position of rear foot at initial time instant -T i th
point mass
Passive Gait Measure
Step height at mid-stance
Step length
Single support
Static walking
Walking speed
Weight function x , y coordinates of i th
joint, respectively, in terms of the global frame x , y coordinates of i th
link’s COM, respectively, in terms of the global frame
ZMP
, i
, , i
Zero moment point
   
Coefficients of polynomial functions i
 i
 i ,max
Generalized coordinates for i th
Maximum actuator torque for i
joint angle th
joint

1
 tot
 i
Root mean square value of the stance ankle torque
Root mean square value of the total actuation
Generalized torque for i th
joint
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INTRODUCTION
Human locomotion is a controlled and complicated process, but to analyze its dynamics, investigating simpler systems is essential. Various simplified biped models have been introduced in the literature [1]. The most widely accepted is the Inverted Pendulum Model (IPM, [3]; Figure
1) that exploits the simple inverted pendulum analogy. The IPM proposes that it is mechanically beneficial for the stance leg to behave like an inverted pendulum, prescribing a circular arc for the center of mass (COM) located at the hip [4]. Although many other different theories have been developed [1, 3, 4, 5], the IPM serves as a basis for many recent gait models. More advanced models have been developed based on the IPM to generate human-like trajectories, such as
TMIPM ([6]; Figure 1), GCIPM [7], MMIPM ([8]; Figure 1), 3DLIPM [9] and PIPM [10].
Figure 4: Simple walking models based on inverted pendulum—IPM (top, [3]), TMIPM (bottom left, [6]), and MMIPM (bottom right [8]).
The inverted pendulum analogy is usually adopted for systems that perform Dynamic Walking
(DW), in which the motion is largely dictated by the passive dynamics of the limbs. It requires minimal actuation to sustain the periodic behavior of the gait [4]. The motion of a dynamic walker is influenced significantly by the gravitational and inertial characteristics of the system, rather than being imposed by the controller [11]. Research work on DW focuses on different types of walkers; they can be divided into unactuated (fully passive) walkers [12] and actuated walkers, such as efficient actuator-assisted walkers [13, 14] and actuated limit cycle walkers [15,
16]. The passive dynamic walking principle was originally introduced by McGeer [12], where the simple passive dynamic walker can walk down a slope with no controlled actuation, but under the power of gravity alone; it has been demonstrated that some anthropomorphic legged mechanisms exhibit stable, human-like walking on a range of shallow slopes, with no controlled actuation. McGeer's work inspired many researchers who used the passive dynamic walker as a starting point to build walking robots that are mainly passive, but minimally actuated in order to
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walk on level ground [17, 18]. For instance, Collins et al. [13] presented passive dynamic walkers which can walk on level ground with small active power sources in substitution of gravity
(Figure 5). Using a natural dynamics-based control rather than a model-based trajectory control, their robot results in a very natural human-like gait with less energy consumption [14]. Also, it has been found that, humans try to correct any tendency to fall during normal walking while minimizing the energy consumption, rather than following a specific trajectory for global progression [5]. Similarly, the DW of robots is not dictated substantially by the controller, but is rather determined by the natural limb dynamics [11].
Figure 5: Passive Dynamic Walkers.
Unlike passive dynamic walking robots, some other models are developed for active control based on Zero Moment Point (ZMP). The ZMP is the point on the foot/ground contact surface where the net ground reaction forces are applied, while the tipping moment—the tangential component of the ground reaction moment—is zero [19]. The Foot Support Region (FSR; also called support base, stability region, foot contact area, etc.) is delimited by the points of the system in contact with the ground, which corresponds to the surface of the convex hull linking the contact points together. The ZMP must always reside within the FSR, and whenever the calculated ZMP falls outside this region, it should be an indication of non-physical behavior [20].
The concept of the ZMP is useful in understanding dynamic stability and also for monitoring and controlling walking robots [21]. To ensure dynamic stability based on the ZMP, usually an ideal trajectory for ZMP is planned and followed by a feedback controller [22], resulting in unnatural gaits with inefficient locomotive actuation. However, the ZMP control approach is useful for robotic gait with varying walking speeds and step sizes [23].
In contrast to DW, the gait is called Static Walking (SW) if the ground projection of the center of mass (GCOM) stays within the FSR to maintain static stability [24]. However, SW, usually performed by humanoid robots, results in unnatural motion with low speed (0.15 m/s - 0.4 m/s) and small step length (0.2 m - 0.5 m) [6, 8, 22, 25]. It is known that natural human gaits are generally not statically stable, but typically consist of phases in which the GCOM leaves the FSR, while the dynamics and the momentum of the segments are used to keep the gait dynamically stable, i.e., ZMP is kept inside the FSR [19, 20, 26].
Actual robotic biped systems, as well as humans, are much more articulated than the IPM. Thus the motion generation and control problems of biped walking usually include kinematic and
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actuation redundancies. To resolve the redundancy, optimization methods have been widely used in the literature where various cost functions are introduced, such as energy consumption, actuator torques, reference trajectory error, and their combinations [1], [2], [27-36].
As seen in the broad literature as above, the natural human walking is characterized as “passive” and “dynamic.” Passivity is associated with the actuation control, while dynamicity is associated with static equilibrium. However, normal human walking is neither fully passive nor constantly dynamic, and the quantification of the level of passivity and dynamicity has not been rigorously investigated in the literature. In this article, we propose initial formulations of quantitative mechanical measures of passivity and dynamicity of biped walking, based on the aforementioned scientific motivations. The passivity is consistent with the optimality of natural walking, where the gait is generated by minimizing physiologically inspired weighted actuator torques. The passivity is described in terms of the relative contribution of the stance ankle actuator torque. The dynamicity is associated with the gait stability, where the inertia effect is represented in term of the ZMP and GCOM. Using this approach, both human and robotic walking motions with various stance foot dimensions are generated for comparison, physical interpretations, and proofof-concept demonstrations. The introduction and demonstration of such initial concepts through computer simulation in this research will provide theoretical foundations for future experimental validation of the proposed measures. This study will eventually provide valuable insights in exploring and understanding fundamental principles of human biped walking. The proposed measures can also be used as criteria for design and control of walking robots.
In the next section, the kinematic and dynamic models of a planar biped system are explained, followed by the description of the gait model used in the current problem. Next, the gait stability in terms of the GCOM and the ZMP is described, and the Dynamic Gait Measure is defined.
Then, optimization problem is formulated along with passivity weights based actuation cost function inspired by physiologic human energy consumption. The Passive Gait Measure is then introduced. Numerical results of biped walking for human and robot will be demonstrated and analyzed with the measures of optimality, passivity, and dynamicity, followed by a discussion about the future work on experimental validation.
1.1 BIPED SYSTEM MODEL AND DYNAMICS
For the purpose of this research as described earlier, we focus on one step, single support (SS) phase during the periodic motion of a biped walking. Therefore, the impulse at the heel contact is not considered in formulating the dynamics. The SS phase analysis is identified in this research because the passivity and dynamicity of the double support (DS) phase are not as important as in the SS phase due to its relatively short duration [37] and less dynamic nature—usually both
GCOM and ZMP exist within the FSR [20]; this is also consistent with many SS phase models described earlier (e.g., [37]). For simplicity, a two-dimensional (2-D) planar biped system is modeled where the motion is confined in the sagittal plane (Figure 2). Generally, both the passive and dynamic nature of biped walking is most significant in sagittal plane compared with those in frontal plane. Since the main progression of normal walking occurs in the sagittal plane with major momentum in the forward walking direction, the range of lateral motion and the variance of the GCOM in the frontal plane are much smaller than that in the sagittal plane. In fact, it has been
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shown that the displacements of the GCOM and ZMP in the frontal plane during normal human walking are very small compared with those in the sagittal plane [20, 38]. Furthermore, the lateral movement during normal walking has negligible effect on the sagittal-plane motion and it is usually assumed that the two plane motions are decoupled, due to, for instance, the minor pelvic sway motion [17] (approximately 4 degrees [8, 52]), and very small coupled (vertical) displacements [10]. For this reason, many other gait models in the literature as seen above have been analyzed in sagittal plane (e.g., [10, 17]), which is consistent with our proposed approach and validates its feasibility.
The proposed model can be regarded as an extension of the concept of the IPM to a multiplemasses model in which the stance foot is assumed to be in a fixed full contact with the ground, while the hips and the upper-body act like an inverted pendulum. While modeling the role of the foot, particularly the changes of the ankle position during the SS phase with full foot and toe contacts, can provide detailed gait realization, one benefit of using a simplified model, as in this case, is that the underlying principles and the physical interpretations can be demonstrated and analyzed rather directly for proof-of-concepts, compared with more detailed complex models.
For this reason, as described previously, many gait research work in the literature used various simplified models based on full foot contact assumption without detailed foot movement or minor change in ankle positions during SS; examples include IPMs and their extensions [6, 7, 8, 26], which is consistent with our proposed model. In particular, more explicit assumptions on the full foot contact during SS phase can be found in the literature [17, 39]. In addition, this assumption is shown to be valid from kinematic gait data [40] indicating that the stance foot ankle position change occurs in a short time duration (10% of the total gait cycle) and its height increases by a very small amount (e.g., approximately 3 cm from the full foot contact initial position to the toe contact final position at the end of the SS phase). The biped system is a 4-link kinematic openloop chain with four revolute joints and four point masses:

Link 1 represents the stance leg, which has length l
1
and is pinned at the ankle joint during
SS. The point mass m
1
represents the leg's total mass. The stance leg is modeled as a single rigid body, since the knee angle is almost constant during most of the SS phase and its flexion is typically very small [30, 38].

Link 2 represents the upper body that connects the two legs, where the distance between the hip joints l
2
should be zero in the sagittal plane. (For modeling purpose in this paper, a very small length is assigned for l
2
.). The total upper body mass m
2
includes head, torso, and arms, and its COM is located at a distance l t
above the pelvis.

Link 3 and 4 represents the upper and lower swinging leg, with point mass m
3
and m
4
, respectively.
The link lengths and the mass distribution have been chosen according to human anthropometry data (Table 1; [38]). The point masses are located at each link’s COM, and thus the inertial parameters (mass and moments/products of inertia) in terms of the Denavit-Hartenberg local coordinate frame provide reasonable approximations for the joint-space dynamics (described below), particularly within the given ranges of momentum and inertia for normal walking. This approach is also in line with many robotic and human walking models in the literature, where the point mass assumption is shown to be valid [6, 41, 42] and widely adopted (e.g., [6, 7, 8, 9, 43]).
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Figure 2: Biped system in the sagittal plane.
The global and local reference frames for each link are attached and the mechanism’s four degrees of freedom (DoF),
   
are represented according to the standard Denavit-
Hartenberg kinematic convention (Figure 2). The global reference frame is located at the ankle joint of the stance foot. The 4
 1 position vector of each link’s end point in terms of the global frame can be calculated using homogeneous transformation as follows:
0 p i
 0
A
1
...
i

1
A p i i i
 i

1,2,3,4

(1) where i

1 A i
is the 4 4 homogeneous transformation matrix for two adjacent coordinate frames in terms of the Denavit-Hartenberg parameters [44], which can be simplified for our model as follows: i

1
A i
 





 cos sin
0
0
 i
 i
 sin
 i
0 l i cos cos
 i
0 l i sin
 i
 i
0
0
1
0
0
1






(2)
Then the global position x and y coordinates of each end point are as follows (in this article, is used to represent either 4

1 or 2

1 vector, depending on the context):
0 p i
Stance leg hip: 0 p
1
Swing leg hip: 0 p
2

 x
1
   l
1 l
1 cos

1 sin

1 x
2
   l
1 l
1 cos sin

1

1



 l
2 l
2 cos sin


 
1
 
1
2
2





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Swing leg knee: 0 p
3
 x
  l
1 l
1 cos

1 sin

1
 l
2
 l
2 cos sin

 
2



 
1

2
 l
3 l
3 cos sin

  
  
1 3
3

 


Swing leg ankle:
0 p
4
  
 l
1 l
1 cos sin

1

1

 l
2 l
2 cos sin

  
1
 
1
2
2



 l
3 l
3 cos sin

   
1
  
2 3
  
1 2 3


 l
4 l
4 cos sin

    
1 2 3
   
1 2 4
4





(3)
Here, the trajectory of the stance leg hip p
1
is a circular arc of radius l
1
, similar to the IPM.
The positions of the point masses
0 s i
 0
A s i i
 i

1,2,3,4

can be obtained in a similar way as follows:
(4) where s i
is the local position of the COM of link i expressed in terms of i th
local frame. The time derivatives, such as velocities and accelerations, are calculated using forward iteration starting from the ground (link 0), which has the velocity and acceleration under the gravitational field as:
ω
0
 ω
0
υ 0 and υ
0
 g
T .
The dynamics of the biped system is modeled using the recursive Newton-Euler equations of motion [44]. The actuator torque
τ i
acting on the i th
link is calculated as the axial component of the moment n i
exerted on link i by link i1, as follows:
τ i

 i i
  i R z

(5) where j R i
is the rotation matrix between two local frames i and j , and z
0
 
0,0,1

T
. The moment i term is calculated using backward iteration as follows: i Rn
0 i
 i R i

1 i
 i R i

1
 i i

1 Rn
0 i

1

1 i

1


 i
 i

1
0
R F
0 i i
  i

1 Rf
0 i

1
  i R p
0 i
 i R s
0 i
  i RF
0 i

 i RN
0 i
(6) where
 f i
is the force exerted on link i by link i1 at the origin of local frame i1 to support link i and the links above;

F i
is the total external force exerted on link i at the COM. Thus, referring to its own coordinate system, Newton's equation is i i
 m i i i
, where a i
is the linear acceleration of the COM of link i . The linear velocity v i
of link i COM, the angular velocity ω i
of link i , and the accelerations are calculated using the forward iterative kinematic equations: i i

 i i
  i i
  i i
  i i
  i i

 i R v i i i
R ν
0 i

 i Rω
R ω i
 i R i
0

1 i i
  i R p
0 i
 

1 R ω i

1
 z
0
 i i

Rω
0 i
 i

1
  i i

1

Rω
0 i
 i R p
0 i
 z
0
 i
  
 i R i

1
 i

1 R ν
0 i

1

(7) i i
 i R i

1 i

1 i

1
 z
0
 i
15

 i
N i
is the total external moment exerted on link i . The Euler’s equation is
R N i

 i
R I R i
 i
R ω i
  i i
  i R I R i
 i R ω i

, where I i is the inertia matrix of link i about its COM with reference to the global frame.
Since no applied load other than the gravity is considered in the current problem, the backward iteration with i = 4 , ..., 1 starts by setting f n

1

0 and n n

1

0 , where n = 4 is the number of the links.
1.2 GAIT MODELING
The gait cycle is the period of time between two identical events in the walking process [2]. A complete gait cycle is divided into two periods: stance phase and swing phase of the same leg.
The left swing phase corresponds to the right SS phase and vice-versa, where its duration constitutes approximately 40 percent of the gait cycle (Figure 3). During an entire gait cycle the body performs two strides. The stride length is the distance covered by the same foot during a step. The SS is initiated with the trailing foot toe off and it lasts until its next ground contact (heel strike). The step length sl is the distance between the two feet at the beginning and at the end of a step. Let the SS phase duration be within the time interval t

,

. When the average speed of the hips is given by v , the step period 2 T is defined by
T
 sl
2 v
(8)
For simplicity, we assume that the forward progression of the hips during the DS phase is infinitesimal [37]. In the sagittal plane, foot dimensions are foot height fh and foot length a + b
(Figure 4). It has been shown that the swinging foot starts the SS with its heel raised, where the distance between the malleolus height and the ground ( mhi + fh ) is around 0.17 m [40].
Moreover, during this phase, the stance foot is assumed to be completely fixed to the floor as described above, while the swinging foot motion should take into account the ground clearance.
During the swinging, the trailing foot ankle tends to minimize dorsiflexion. Also, a certain foot elevation is necessary to deal with uneven nature of the floor. For these reasons, imposing the ground clearance has provided valid models in the literature [6, 8]. In the current gait problem, this clearance is modeled by introducing the step height sh at the mid-stance.
16

Figure 3: Gait cycle (Source: Inman et al. [45]).
Figure 4: Biped system model at instants t
 
T and t

T .
1.3 STABILITY AND DYNAMIC GAIT MEASURE
In this article, according to the assumptions described and justified previously, we consider the stability of the biped system during SS phase only, where the stance foot is in full contact with the ground while the swinging leg performs the stride. In 2-D sagittal plane, the stability region is constituted by the FSR, x

[a , b ] in x -axis, and lateral stability is not considered here. Two commonly used ground reference points—GCOM and ZMP—are adopted as criteria in defining two types of gait: SW and DW [26]. When the ZMP resides within the FSR (Figure 5), the gait is dynamically balanced [19], in which case the ZMP coincides with the center of pressure [20, 46].
ZMP x
 
(9)
17

Theoretically, the ZMP criterion cannot be satisfied in limiting cases of edge or point contact between the foot and the ground [19, 20, 24]. In this theoretical event when the foot-ground contact occurs through edge or point, the center of pressure [46] that exists within the FSR or other inertia-based ground reference points that exist outside of the FSR, such as the Foot-
Rotation Indicator (FRI; [24]) or the Fictitious Zero Moment Point (FZMP; [19]), can be used for the stability criteria [1]. On the other hand, in reality, the foot is not ideally rigid but deformable, and any rotational inclination will transform the edge into a new surface [19]. Thus physically the foot-ground contact always occurs through a finite (small or large) surface within which the ZMP exists, and therefore the ZMP criterion is always satisfied during robotic and human walking.
This is also evident from many experimental observations of ZMP (e.g., [20]) during both SS and
DS phases of walking.
The position of the ZMP is calculated as follows:
ZMP x

  i

 g


  m y x i

L m y i
 g
 (10) where x y x y i
, , , i i i of joint angles
are the position and acceleration of the i th point mass m i
, represented in terms

. Here, the time rate of change of angular momentum of each link about the z i axis ( L ) is not included in the ZMP calculation due to the aforementioned point-mass assumption of the proposed biped model, in which the influence of L about each link’s COM during walking has been shown to be negligible in ZMP calculation [6, 41, 42], particularly in the sagittal plane.
The stability criterion for SW refers to the GCOM position within the FSR at all time instants.
Whenever the GCOM leaves the FSR, there is a presence of a statically unbalanced moment on the foot, which causes its rotation about a point on the FSR boundary [24]. The GCOM is calculated by considering only each link’s mass, while inertia terms are not taken into account.
Thus, GCOM position is obtained by removing the acceleration terms from the ZMP formula, as follows (Figure 5):
GCOM x


 m x g i i m g i
(11)
If a robot is stationary, it is shown that the center of pressure coincides with the GCOM [24].
Whether or not the GCOM should stay within the FSR depends on the gait. As described earlier,
SW, which is usually performed by biped robots, requires that the GCOM exists within the FSR at all times, while the DW does not.
18

COM
Gravity and inertia forces
Ground reaction forces
FSR x ground level
GCOM ZMP a b
Figure 5: ZMP and GCOM in sagittal plane.
From definitions, the main difference between the ZMP and the GCOM is the effect of inertia
(assuming gravity is the only externally applied force). Therefore, the distance between the ZMP and the GCOM at each time provides a measure of how dynamic the current gait is. To address the distance in x direction over the time duration, the root mean square (RMS) of the distance between ZMP and GCOM is used, which can provide a measure of dynamicity: d

 T

T
    

2 dt
2 T
(12)
Although the RMS distance quantifies certain degree of dynamicity by providing the effects of inertia, it lacks the information of the relative stability due to the dynamicity. To incorporate the relative stability into the measure of dynamicity, the RMS distance of a gait should be compared with that of a SW. In particular, the RMS distance for the limiting case of the SW will provide a border between the SW and DW. In the case where the system is at rest, the ZMP and GCOM are equal, and the RMS distance becomes zero. It has been shown from our simulation experiments
(and also in the results section) that, for SW of the 2-D system, the ZMP position is almost at x =
0, and the GCOM follows a near-linear monotonic curve along the time from rear end to front end of the FSR. (The validity of these assumptions in 3-D will be studied in future research.) Thus, given a foot size a and b , the RMS distance for the limiting static gait can be approximated as follows: d lim

 T

T

   2
GCOM lim, x t 
 dt

2 T where
GCOM lim, x
   a
 b  t

T
  a
2 T a
2   2
3
(13)
(14)
19

Finally, to represent the relative measure of dynamicity for a given gait motion and foot dimension, we define the dimensionless Dynamic Gait Measure (DGM) as the ratio of the two
RMS distances, as follows:
 x x
, ,
  d d lim



T
T

    


2
ZMP t

GCOM x t dt
2

2
 ab
 b
2

/ 3
(15)
Thus the DGM is a functional of ZMP trajectory, GCOM trajectory, and foot dimension. Here, it should be noted that the DGM, as a measure of the dynamicity of a gait, depends not only on the balance criteria such as ZMP and GCOM, but also on the stance foot dimension (i.e., FSR). Since the ZMP always exists and satisfies the criterion during robotic and human walking (as discussed above), the definition of DGM as a functional of ZMP trajectory is physically and mathematically valid. For a given walking motion and non-zero foot dimensions, the existence and uniqueness of the DGM can be easily checked from its definition.
From the above definition, it can be shown that the DGM is always positive (or zero) and is bounded from above as follows, as long as the stance foot maintains full contact with the ground during the gait:
0
  (16) where K
 max
  x
( a
2
 ab
 b
2
) / 3 x
 
 max 

4  i

1
1 m i
4  i

1
 

 m i


 i

1  j

1 l j
 l c
 





 a ,
4  i

1
1 m i i
4 

1
 

 m i


 i j

1 

1 l j
 l c
( a
2
 ab
 b
2
) / 3
 





 b 

,
(17) d
 
is the distance between two points, and c i
is the fraction of the position of COM from the previous joint over the entire link length. It should be noted that the least upper bound, or supremum, K , can be calculated from link parameters (length, mass, COM of each link), stance foot dimension, and joint variable limits (here, for simplicity, full extension/flexion is assumed for each joint variable limit). In other words, when the stance foot is in full contact with the ground, the dynamicity of a biped walking is bounded from above by these system parameters, regardless of the maximum actuator torque capacity or muscle strength. Here, K represents inherent potential of the dynamic nature of the gait. In other words, the K value indicates the maximum possible dynamicity of walking for a given biped system and foot dimension.
Physically, larger foot dimension ( a + b ) should have smaller K . It should be noted that while
DGM is dependent on the given gait parameters (e.g., speed and step length) as well as the biped system and foot dimension, K is independent of any gait parameters.
The DGM values can be interpreted according to several cases as follows:

DGM = 0: The ZMP and GCOM are identical at all times. The motion of the system is stationary, or can be regarded as quasi-static.
20


0 < DGM < 1: The GCOM, as well as the ZMP, stays within the FSR during the gait. This indicates the SW. The smaller DGM values imply more statically stable than the larger ones.

DGM = 1: The gait motion is marginally static, and its dynamicity is at the border between

SW and DW.
1
  : The GCOM falls outside of the FSR at some times, while the ZMP stays within the FSR at all times. The larger DGM values imply more dynamic gait than the smaller ones. In other words, the existence of GCOM outside of the FSR during longer duration or the larger distance between the ZMP and GCOM will result in larger DGM.
According to the IPM analysis, larger distance between the GCOM and the ankle joint for a given actuation capacity indicates higher tendency to fall at a time instant. Therefore, the
DGM also indicates a time-global measure of instability during a gait.
In the Results section, the DGM will be calculated and used to evaluate the dynamicity and stability of walking for different sets of walking speeds, step lengths, and stance foot dimensions.
It should be noted here that the proposed DGM formulation can be directly applied for the DS phase, in which case the FSR is the convex hull region formed by the two contacting feet [19, 26].
1.4. OPTIMALITY AND PASSIVE GAIT MEASURE
From the viewpoint of minimum actuations, the idea of passivity in human walking is consistent with the optimality of the nature. The gait generation is formulated as an optimization problem, where the joint kinematic profiles and actuator torques are calculated under given constraints.
Other physical quantities, such as ZMP and GCOM trajectories, can be derived from the optimization solutions. The joint kinematics will be represented using the joint angle vector as follows:
 
1
  
2
  
3
  
4
 

T
(18)
A polynomial function is used to parameterize each joint angle profile. In order to guarantee the
C 2
smoothness condition, i.e., continuous velocity and acceleration, the degree of polynomial should be at least three. From several numerical experiments using the proposed model for various cases, it has been found that higher-order terms of the polynomials are negligible. Thus, each joint is parameterized using a third order polynomial as follows, to provide sufficient smoothness of the joint angle function:

1

2

3

4
 
 
3 t
3  
2 t
2  
1 t
 
0
 
 

 

3
3 t t
3
3 
 

2 t
2 t
2
2

 
1 t
 
0

1 t
 
0
   
3 t
3  
2 t
2  
1 t
 
0 where the coefficients i
, , , i
(i = 1, 2, 3, 4 ) are the optimization variables.
(19)
21

To generate the walking motion that demonstrates (partial) passivity, the actuator torques should be included in the cost function. As discussed earlier, various ways of implementing the actuator torques into the cost function have been developed in the literature; this is also consistent with previous studies suggesting that, for a given progression velocity, human chooses a gait that minimizes the energy consumption [1], [2], [28-36]. The proposed cost function is a weighted squared norm of actuator torque vector function for the time interval [T , T ], as follows:
J
  T

T
 w

2
1 1
 w

2
2 2
 w

2
3 3
 w

2
4 4
 dt (20) where
 i
is the actuator torque at the i th
joint, and w is the corresponding weight function. The i quadratic expression of the actuator torques ensures that opposite signs are not cancelled with each other in the cost function, and that there exists a continuous first derivative for gradientbased optimization. By minimizing this cost function, the walking motion will be generated with as much passivity as possible under the given constraints and gait parameters.
In general, the values of the weight functions w i
depend on the criteria of the problem. Here we propose physiologically-inspired weight functions such that the cost function has characteristics that mimic the human energetics, in particular the heat expenditure, which is known as approximately proportional to the muscle force and inversely proportional to the maximum muscle strength [47, 48, 49, 50, 51]. As a consequence, the relative passivity weight is defined as the weight function value at each joint that is inversely proportional to the squared maximum actuation capacity (
 i ,max
) as follows: w i
 k

2 i ,max
(21) i
4 

1 w i

1 where k is a proportionality constant with dimension of [1/ s ], such that the functional is dimensionless. For numerically sound performance of the optimization algorithm, a property similar to the partition of unity is imposed to the passivity weights. From the data available in the literature [38, 40] for the ankle, knee, and hip joints, the following relationship can be assumed during the SS phase:

2,max

3,max

4,max
2

1,max
(22)
Therefore, the passivity weight vector is calculated as follows: w


0.571,0.143,0.143,0.143

(23)
It can be seen from the proposed cost function and passivity weights that, human gait of minimum energy consumption (thus minimum weighted actuator torques) implies that human tends to use the stronger joints (knee and hips) rather than the weaker one (ankle). This means that the actuator torques are distributed in such a way that the larger torques are exerted at the knee and
22

hip joints and smaller torques at the ankle joint. From multi-objective optimization viewpoint, the proposed passivity weights naturally enforce that the stance ankle torque (with larger weight) is more minimized than other joints (with smaller weights), thus controlling the relative passivity of the joints. The reinforced minimization of ankle joint torque by the proposed passivity weights is also consistent with the concept of IPM (where the unactuated hinge corresponds to the human ankle joint) and many other models in the literature that used minimum ankle torque to generate gait motions [28, 29, 30].
Based on the proposed biped system and gait model, initial/final conditions and gait constraints are imposed to the variables as follows:

Initial and final position along x and y axes of the swinging ankle is calculated by the step length sl chosen for the gait: x
4
  sl ;
   sl ; y
4
   mhi ; y T
4
  
0 (24)

Ground penetration should be avoided. In particular, the step height sh is imposed (within a tolerance

) during the mid-stance to ensure ground clearance, which is a common constraint used in the literature [6, 8]: y t
4
  
0; y
4
   sh
 
(25)

The knee joint angle should be bounded within its range of motion to avoid hyperextension during the swinging motion. Also, the swinging leg is almost fully extended at the time of heel strike, according to Saunders et al. [52]:

4
  
0;

4

0 (26)

The swinging hip joint angle should be bounded within its range of motion. In addition, the hip angle is further bounded as it initiates the swinging phase from toe-off:

3
  i

0 ;

3
 

2
(27)

The torso maintains almost vertical position during SS (Figure 4) due to small oscillations in normal human walking [52]. This constrains link 2 to be always horizontal during SS, which also ensures right-left symmetry assumption for simplicity:

1
   
2
 
(28)

The ZMP-based stability is also considered. In this case, rather than imposing the constraints directly into the optimization algorithm for motion generation, the ZMP values are monitored throughout the motion to ensure that the ZMPs always exist within the FSR boundary.
The stable walking (without falling) is ensured if all the constraints are satisfied. The existence of the optimal solution indicates that the physically consistent gait motion is generated according to the initial and final foot positions while maintaining the ZMP constraints under given actuation capacities. This analysis applies to walking motions with various dynamicity levels, including those for DGM > 1. If the gait with given parameters results in instability (due to either ZMP violation or insufficient actuation capacities), then the optimization algorithm will result in infeasibility and the required walking motion cannot be generated.
23

All constraints are functions of joint angles at each time step functions of optimization variables i
, , , t i
, and thus they are also the
    i
. The cost function and the constraints are implemented into the optimization algorithm by the “NMinimize” command in Mathematica©, which uses merit-based choice between Nelder-Mead algorithm and the differential evolution methods.
Since no walking machine or bio-mechanism can be fully passive (except for walking down a slope), the relative passivity, rather than the absolute passivity, should be considered. As described earlier, based on the IPM analogy and minimization of physiologically-inspired passivity weights, the degree of passivity in biped walking is associated with the actuator torque

1
exerted at the stance ankle. In normal DW, the link dynamics and the gravity contribute to the gait motion, allowing less actuator torques at the ankle joint; while the effect of inertia is relatively small in SW, which requires larger ankle torque.
In order to quantify the contribution of

1
over the time duration, we use the RMS representation as follows:
T 

T

1
 
2 t dt

1

(29)
2 T
However, although the RMS ankle torque provides absolute magnitude, its relative contribution to the gait depends on the dynamics of the whole biped system. In other words, applying same ankle torque for the gait of a small/light system and a large/heavy system will demonstrate different levels of passivity. In order to quantify the relative passivity, the dynamics of the whole biped system should be considered. Since the dynamic effects (mass, moments/products of inertia, gravity, external loads, and momentum) are incorporated into the actuator torques at all joints through the equations of motion, the relative passivity can be described as the ratio between the
RMS of the ankle torque (

1
) and the RMS of the total actuation (
 tot
). Therefore, the dimensionless Passive Gait Measure (PGM) is defined as follows:
PGM
    
1
,
2
, ,
4



1 1 1 tot


T
T


T
T

1
2 dt


2
  
1

2
2

2
3

4
2
 dt
(30) where
 T

T


1
 
2  
2
 
2  
3
 
2  
4
 
2
 dt
 tot

(31)
2 T
The PGM quantifies the relative contribution of

1
with respect to the total actuation of the biped system. It represents the relative passivity for the given gait parameters and biped system model, which can be shown to be bounded as follows:
24

0

PGM

1 (32) where

PGM = 0: Only the stance ankle joint is actively controlled, while all other joint actuator torques are zero at all times.

0 < PGM < 1: The stance ankle joint as well as some or all other joints are actuated. Larger
PGM values indicate that the gait of the system is rather passive, while smaller values indicate that the ankle joint is more actively controlled.

PGM = 1: The stance ankle actuator torque is zero at all times. This is the case of fully passive walking with respect to ankle joint, in which the stance leg acts like an ideal IPM with unactuated hinge.
Since the gait constraints cannot be satisfied with a completely unactuated system, a unique PGM value exists for a given walking motion and foot dimension. In the Results section, the PGM will be calculated and used to evaluate the optimality and passivity of walking for different sets of walking speeds and step lengths.
1.5. HUMAN-LIKE AND ROBOTIC WALKING SIMULATION RESULTS
To demonstrate the PGM and DGM, the gait motions are generated for human-like walking
(which is relatively passive and dynamic) and robotic walking (which is actively controlled and mainly statically stable) using the proposed optimal control scheme and physiologically-inspired weighted torques. Here, for simplicity, we differentiate the human-like walking and robotic walking by different sets of walking speeds and step lengths, although practically they are characterized by many additional features, such as model structures and control methods. The biped system model parameters, such as mass distribution and link lengths, and dynamic parameters are assigned based on a human anthropometry of total body mass 75 kg and height 1.8 m ([38]; Table 1).
Table 1: Model parameters of proposed biped system [38]. link
1
2
3
4 name stance leg length ( m ) mass (Kg) c i
0.90 13.9 hip + torso ( l t
) 0.01+0.4 47.2
0.674
0.5 thigh shank
0.43
0.47
10.6
3.3
0.445
0.405
Mathematica© program is used for computational implementation of the proposed formulations.
For each example, the kinematics, actuation, ZMP, and GCOM are calculated. The proposed
RMS distance, RMS torques, DGM, and PGM are also calculated and analyzed. In particular, to demonstrate the dynamic nature of biped gait as associated with the stability and its dependence on the FSR, the DGM is evaluated for three cases with distinct foot dimensions—normal human foot: ( a, b ) = (10, 20), total length = 30 cm; stilt foot: ( a, b ) = (5, 5), total length = 10 cm; and
25

skiboard: ( a, b ) = (35, 35), total length = 70 cm (Figure 6). Selected result data and some qualitative interpretations are discussed.
Stilt foot Human foot Skiboard
Figure 6: Different foot dimensions.
Human-like Walking
The walking speed ( v ) and step length ( sl ) are given (Table 2), which correspond to the selfselected parameters of normal walking performed by the human model compatible with the proposed biped system. Then the joint profiles (motion), required actuator torques, ZMP trajectory, and GCOM trajectory are calculated as the outputs of the proposed optimization algorithm (Figures 7, 8, 15, and 16).
Table 2: Parameters for human-like walking parameters velocity ( v [m/s]) malleolus height ( mhi
[m]) step length ( sl [m]) step height ( sh [m]) foot height ( fh [m]) value
1.53
0.07
0.5
0.1
0.1
Also, the kinematic profiles of the torso ( m
2
) and the swinging ankle (foot) are plotted as functions of time in forward and vertical directions (Figures 9 and 10). The snapshots of the result (Figure 7) show a near-natural human-like walking motion. For the purpose of this research, where the proof-of-concept demonstrations focus on the difference in DGM and PGM for human and robotic gait motions characterized by different sets of walking speeds and step lengths, rather than on generating a completely realistic human walking motion, partial validation of the results is discussed here, while more rigorous validation is proposed as future work later in this section. Two aspects can be considered for partial validation of the generated human-like walking motion. First, since some of the constraints imposed on the system model and the optimization algorithm are based on the realistic human gait data, the associated segments naturally ensure the generation of realistic human walking motion. These include the constant stance knee angle in Link 1 [30, 38], the full extension of swinging knee at heel strike [52], the horizontal pelvic global angle of Link 2 in sagittal plane [52], and the walking speed and step length. Second, the qualitative comparison of the results in terms of both the magnitudes and patterns is used as a validation method, which is a widely accepted approach in the literature [38,
53]. The patterns of the joint trajectories (Figure 8-Left), actuator torques (Figure 15), torso trajectory (Figure 9), and swinging foot trajectory (Figure 10) show matching trends with
26

experimentally measured data for real human subjects available in the literature [40, 53]. In addition, the joint angle ranges and magnitudes (Figure 8-Left) for the stance ankle dorsiflexion, stance hip extension, swinging hip flexion, and swinging knee flexion/extension, as part of the standard determinants [52], show very good similarity with the experimental data [53]. These provide sufficient partial validation of the generated human-like walking motion for the given purpose of this research.
Figure 7: Generated human-like walking motion.
Figure 8: Joint angles (left) and velocities (right) at the stance ankle (thick), stance hip (solid), swinging hip (dotted) and swinging knee (dashed) for human-like walking.
27

Figure 9: Torso point mass displacement (thick), velocity (solid), and acceleration (dashed) for human-like walking in forward x-direction (left) and vertical y-direction (right).
Figure 10: Swinging ankle displacement (solid) and velocity (dashed) for human-like walking in forward x-direction (left) and vertical y-direction (right).
Robotic Walking
To demonstrate the proposed DGM and PGM with a counter-example to the human-like walking, a mainly-statically-stable robotic walking motion is generated with lower speed and smaller step length. The gait parameters v and sl are chosen based on the typical ranges of speed and step length of humanoid robots available in the literature (0.15 m/s–0.4 m/s and 0.2 m–0.5 m; [6, 8,
22, 25]). The specific gait parameters used in the current problem are listed in Table 3.
Table 3: Parameters for robotic walking. parameters velocity ( v [m/s]) malleolus height ( mhi
[m]) step length ( sl [m]) step height ( sh [m]) foot height ( fh [m]) value
0.3
0.07
0.35
0.1
0.1
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Again, the joint profiles, required actuator torques, ZMP, GCOM, and kinematic profiles of the torso and the swinging foot are calculated as the outputs of the proposed algorithm (Figures 11–
16). Compared with the results of the human-like walking, the notable difference is in the ankle actuator torque. The ankle actuator torque for the robotic walking is much larger than that of human-like walking, indicating that the statically stable walking requires more active control at the ankle joint. This large ankle torque is also in line with the less dynamic nature of the SW, where the contribution of the gravity force to the ankle moment is small due to the short distance between the ankle and the GCOM. Further analysis will be discussed along with the calculated
PGM and DGM values in the next subsection.
Figure 11: Generated robotic walking motion.
Figure 12: Joint angles (left) and velocities (right) at the stance ankle (thick), stance hip (solid), swinging hip (dotted) and swinging knee (dashed) for robotic walking.
29

Figure 13: Torso point mass displacement (thick), velocity (solid), and acceleration (dashed) for robotic walking in forward x-direction (left) and vertical y-direction (right).
Figure 14: Swinging ankle displacement (solid) and velocity (dashed) for robotic walking in forward x-direction (left) and vertical y-direction (right).
1.6 PGM AND DGM RESULTS: HUMAN VS. ROBOTIC GAIT
Based on the calculated results for the walking motion (Table 4), the problems of “optimality and passivity” and “stability and dynamicity” are analyzed.
Table 4. Cost function value, PGM, RMS distance, upper bound K , and DGM.
Cost function value
PGM
RMS distance d [m]
Human-like Robotic
7.162
0.712
0.139
DGM stilt foot (10 [cm]) 4.388
DGM human foot (30 [cm]) 1.294
DGM skiboard (70 [cm]) 0.627
41.644
0.2
0.068
2.150
0.634
0.307
K = 33.08
K = 11.05
K = 6.21
30

Optimality and passivity
The walking motion of each case is obtained through the optimality of the passivity-weighted actuator torques under the given gait parameters and constraints. The required joint actuator torques for both gaits are consistent with the generated motion (Figure 15). Although the patterns for both gaits are similar, the human-like walking requires less actuator torques overall. In particular, the ankle actuator torque for the human-like walking is much less than that of the robotic walking, indicating that the human gait is more passive than the robotic gait, from the perspective of the IPM. The natural selection of the passivity weights in the cost function based on the physiologic energy consumption leads to the gait motion that is generated according to reinforced minimization of the ankle torque. Therefore, the proposed passivity weights, along with the resulting actuator torques for the human-like walking, support the IPM analogy.
Figure 15: Actuator torque at the stance ankle (thick), stance hip (solid), swinging hip (dotted), and swinging knee (dashed) for human-like (left) and robotic (right) walking.
The difference in the actuation is also evident from the resulting cost function value, which is much smaller for the human-like walking (7.162) than for the robotic walking (41.644). Since the proposed cost function is formulated on the analogy of the physiologic heat dissipation, it represents a measure of actuation effort or a dimensionless approximation of energy consumption.
Therefore, these cost function values also represent a dimensionless measure of the cost of transport for each walking motion, which is defined as the energy consumption per unit weight and unit distance travelled [54, 55, 56]. The comparison of the cost function values also suggests that the natural human walking with self-selected speed and step length demonstrates less total actuation (energy consumption) compared with the gait with lower speed and smaller step length.
This result partially verifies another optimality of the human-like walking in terms of speed [2,
31].
Note that the cost function value indicates the overall optimality and a rough measure of total passivity, but it cannot be used to indicate the relative passivity of the ankle joint in general. The resulting PGM values for both the human-like walking (0.712) and the robotic walking (0.2) provide the quantified measures of the relative passivity, where large PGM indicates higher passivity and vice versa. While the cost of transport has a positive relationship with the
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physiology-inspired cost function, it is not necessarily coupled with PGM for general gait motions. This is because if the stance ankle actuation is small while the other joint actuations are large (due to additional task requirements or environmental constraints), the walking is relatively passive, but the cost function and the cost of transport maintain large values; similar arguments can be made for the opposite case. However, for naturally coordinated walking motions—such as normal human walking—to which all body segments are dedicated, the whole body actuations are minimal and are associated with the stance ankle actuation; in these cases, the PGM is also consistent with the cost of transport. Since the PGM represents relative contribution of the ankle torque compared with the total actuations, the results suggest and verify that normal human walking is more passive (at the ankle) than robotic walking.
Stability and dynamicity
The transition of the system's COM in the forward direction is consistent with the calculated
GCOM trajectories. Although the ZMP trajectories for both gaits are similar in term of the magnitude and the shape, the GCOM trajectories show significant difference (Figure 16). This is also evident from the calculated RMS distances. The similar ZMP trajectories, which are located close to the ankle joint ( x = 0), are due to the compensation of large and small inertia for large and small GCOM magnitude, respectively; this feature is also consistent with the experimentally measured data for human subjects [20] and robotic control methods [6, 8, 22], which serves as partial validation of our simulated results.
Figure 16: ZMP (solid) vs. GCOM (dashed) for human-like (left) and robotic (right) walking.
For all foot dimensions (stilt foot, human foot, and skiboard) and for both gaits, the ZMP stays within the FSR; this indicates that the generated motions are physically feasible. (If the ZMP tends to demonstrate large magnitude, then the ZMP constraint should be additionally implemented into the optimization algorithm to ensure physical consistency.) The GCOM of the human-like walking is maintained within the skiboard FSR, while it falls outside of FSR during certain period for stilt and human foot. On the other hand, the GCOMs of robotic walking with both skiboard and human foot are maintained within the FSR. The GCOM falls outside of only the stilt foot FSR for robotic walking. The results indicate that providing a larger support to the stance foot, for example with a skiboard, ensures the GCOM to stay within the FSR (thus
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statically stable) during the entire SS for both the human-like walking and robotic walking. This is seen from the real-world experience where a human with skiboard on does not fall within the sagittal plane. On the other hand, a smaller FSR (e.g., stilt foot) makes the GCOM falls outside of the foot/ground contact region for most of the time, indicating that the dynamic effect is dominant for both the human-like walking and robotic walking. The average human foot shows DW for human-like walking parameters, and SW for robotic walking parameters.
The physically valid analysis from above can be quantified more rigorously by the calculated
DGM for each case, where the DGM of 1 is the threshold between SW and DW. A DGM greater than 1 indicates DW, while less than 1 indicates SW. The calculated K values—the upper bounds of DGMs
—indicate maximum inherent potentials of the dynamicity of walking for each foot dimension. It can be seen that larger foot dimensions result in smaller K values (stilt K = 33.08, human foot K = 11.05, and skiboard K = 6.21), which is consistent with its physical interpretation.
Overall, the DGMs for human-like walking are larger than those for the robotic walking. The larger the DGM, the more dynamic the walking is. For example, the human gait with stilt foot is more dynamic than the robotic gait with stilt foot and the human gait with human foot. Similarly, as the DGM approaches to zero, the gait becomes more statically stable. The robotic gait with skiboard is more statically stable than the robotic gait with human foot and the human gait with skiboard. The DGM values for human foot support base verifies the DW ( DGM > 1) for humanlike walking with average speed, and SW ( DGM < 1) for robotic walking—a result that is consistent with the experiments in the literature [25]. The results can also be used to predict the characteristics of the DGM in more detailed human gait realization. The increased DGMs for decreased foot dimensions indicate that, if the stance foot contact is further segmented to include the heel and toe contacts that have smaller contact surfaces, the overall DGM will be larger than the above value. This indicates that the above DGM result for full foot contact can be considered as the minimum value for the given gait, which represents a valuable quantitative measure that can be used to compare different gait parameters and foot dimensions. In addition, the increased
DGM for human gait with full foot and toe contacts will demonstrate a valid and even convincing dynamic nature of normal human walking as compared with the robotic walking. Overall, it can be seen that the DGM provides a single reliable measure for the dynamicity level of the given gait parameters and foot dimension.
As discussed above, the resulting small ankle torque for human-like walking is also in line with the more dynamic nature of human gait. In fact, the dynamicity and the passivity of biped walking are related with each other, particularly for normal human walking. This is because usually the passivity is achieved by the contribution of the gravity force and momentum. Before the heel strike of normal human walking, the moment about the ankle joint due to the gravity force increases as the GCOM moves farther away from the ankle location. Then the acquired momentum leads to a reduced required ankle actuation after the toe-off of the swinging leg. Since the ZMP is located close to the ankle joint in human-like walking (under the current simple model), the passivity will increase as the dynamicity increases. Although the dynamicity and the passivity are usually coupled in normal human walking, it is not necessarily true for general gait motions such as unnatural human walking and robotic walking. In general, the DGM and PGM are independent and both should be evaluated independently. As evident from the proposed mathematical formulation, the variables used in each measure are not necessarily dependent on each other. In other words, the GCOM and ZMP are determined by not only the actuator torques,
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but also the initial and boundary conditions of the system and any applied external loads (if exists) other than the gravity. Also, depending on the foot dimension, step length, and walking speed, the coupling between the DGM and PGM can diminish. For example, walking with small ankle actuation for small step length and large foot dimension will result in high passivity but low dynamicity. On the other hand, it is also possible to exert large ankle actuation for unnatural walking with large step length and small foot dimension, which will result in low passivity and high dynamicity. It can be stated that, particularly for unnatural human (e.g., due to injury or heavy backpack) or robotic walking, the correlations between the PGM and DGM are weaker than those for normal human walking.
1.7 CONCLUDING REMARKS
The research in this article was focused on initial concepts, mathematical formulations, physical analyses, and computer simulations of the proposed DGM and PGM, based on the supporting scientific motivations. The proof-of-concepts results demonstrated the validity and reliability of the proposed initial concepts in 2-D sagittal plane, and promise that the PGM and DGM concepts can be well applied to more detailed models. On the other hand, an extended future research including the experiments for actual walking machines and human subjects is required to demonstrate the broad and general applicability of the concepts and to validate their proof-ofpractice in realistic gait. In addition, the experimental analysis can also provide feedback for refinement and improvement of the DGM and PGM formulations. However, experimental measurements of the required parameters involve several major challenges. The ZMP, GCOM
(both for DGM calculation), and the inverse-dynamics-based joint actuator torques (for PGM calculation) require the measurements of motion kinematics, ground reaction forces and moments, and body segment inertial parameters. While the measurements of motion kinematics and ground reaction forces/moments are relatively straightforward by using motion capture camera systems and force plates, respectively, the estimation of inertial parameters of the complex 3-D high-DoF biped system (real robots and humans) requires another in-depth research. Although the conventional linear methods of parameter estimation have been extended to human/humanoid systems (e.g., [57]), a nonlinear approach has to be established for more accurate identification of inertial parameters of multi-segmental systems. In addition, more detailed gait realization will require more detailed and extensive models of the foot with additional DoFs at the toe-ball joint and the contact dynamics for the transition from full foot to toe contact, which will result in the ankle position change and time-varying foot contact surface during SS; these problems are not completely resolved in the current gait literature. Therefore, the development of this framework, along with the effort required for experimental design and set-up, poses a new set of extensive research problems, which will be addressed in our future work.
Normal human walking is usually characterized as passive and dynamic. Nevertheless, human walking is neither fully passive nor constantly dynamic, and the quantification of the level of passivity and dynamicity has not been rigorously addressed in the literature. In this article, we proposed initial formulations of two quantitative measures—PGM and DGM—of biped walking.
The passivity weight functions were introduced and incorporated in the actuation cost to generate walking motions. Using a simple planar biped model, two different walking motions—humanlike and robotic—were generated from given walking speeds and step lengths. The human-like
34

walking results showed better optimality and more passivity than the robotic walking. The relative passivity of each gait was quantified by introducing the PGM, where the PGM value for the human-like walking was greater than that of the robotic walking; this result validated the relatively passive nature of human walking. While the stability of the generated gait motions was characterized by the ZMP and GCOM, the dynamicity was quantified as the proposed DGM. In general, the DGMs for human-like walking were greater than those for robotic walking, verifying the dynamic nature of human gait. The resulting DGMs also demonstrated their dependence on the stance foot dimension as well as the walking motion. For a given walking motion, smaller stance foot dimension resulted in increased dynamicity, and vice versa. Thus, the proposed PGM and DGM provide a single measure of passivity and dynamicity, respectively, of a given biped walking motion in sagittal plane. The extension and application of the proposed concepts to 3-D walking motion will be studied as future research, where a statistically significant number of gait parameter sets will be used to demonstrate these measures. In addition, more accurate data on the mass and inertia parameters will improve the model with more realistic results. Also, future experimental measurements of PGM and DGM with many human subjects will provide normal ranges for these values. The proposed results will benefit the human gait studies and the development of walking robots and prosthetic mechanisms.
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Normal human walking typically consists of phases during which the body is statically unbalanced, while maintaining dynamic stability. Quantifying the dynamic characteristics of human walking can provide better understanding of gait principles. We introduce a novel quantitative index, the Dynamic Gait Measure (DGM), for comprehensive gait cycle. The DGM, quantifies the effects of inertia and the static balance instability in terms of zero-moment point and ground projection of center of mass, and incorporates the time-varying foot support region
(FSR) and the threshold between static and dynamic walking. Also, a framework of determining the DGM from experimental data is introduced, in which the gait cycle segmentation is further refined. A multi-segmental foot model is integrated into a biped system to reconstruct the walking motion from experiments, which demonstrates the time-varying FSR for different subphases. The proof-of-concept results of the DGM from a gait experiment are demonstrated. The
DGM results are analyzed along with other established features and indices of normal human walking. The DGM provides a measure of static balance instability of biped walking during each
(sub-)phase as well as the entire gait cycle. The DGM of normal human walking has the potential to provide some scientific insights in understanding biped walking principles, which can also be useful for their engineering and clinical applications.
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INTRODUCTION
Human body represents a well-balanced walking machine that performs periodic, stable, and energy-efficient gait through highly sophisticated mechanics and control, which are not easily replicable [1-4]. Nevertheless, simplified models usually provide valid analytical and computational tools for understanding the principles of human walking. The most commonly accepted is the inverted pendulum model [2, 5-7], in which the stance leg behaves like an inverted pendulum for a mechanically efficient gait. Although many other different theories have been developed [1, 4, 8, 9], the inverted pendulum model serves as a basis for the development of various simplified biped gait models to analyze or generate human-like gait trajectories [10-16].
Many biped gait models that adopt the inverted pendulum analogy are based on the passive dynamics of the limbs or passive dynamic walking principle [5]. The concept of passive dynamic walking was extensively adopted for developing walking robots [17, 18], and for the analysis and prediction of human walking motions [14, 19-21]. Passive dynamic walking is influenced significantly by the gravitational and inertial effects of the system, rather than active control [22], and requires minimal actuation for periodic gait with preferred speed and step size [23-25].
Although the concept of passive dynamic walking is frequently used as a whole to describe normal walking of humans, the passivity and dynamicity are distinct terms in general. While the passive characteristics of walking are related to actuation control [5], the dynamic characteristics of walking are related to balance stability. The scope of this paper is confined to the dynamic characteristics of human walking, while their relationships with gait passivity are also briefly discussed.
Dynamic walking (DW) is a biped gait that is dynamically stable, but not necessarily statically stable [26, 27], in terms of balance with respect to two ground reference points: zero-moment point (ZMP; [28]) and the ground projection of center of mass (GCOM). For both static and dynamic stability, the ZMP must always reside within the foot support region (FSR), which is the surface of the convex hull formed by the foot/ground contact points; whenever the calculated
ZMP falls outside the FSR, it is an indication of a non-physical behavior [26, 29] or the contacting foot’s tipping-over [28]. Therefore, the ZMP criteria must hold for all phases of gait cycle, as evident from many experimental observations (e.g., [29, 30]). For a finite contact area without relative motion between the contacting foot and the ground, the ZMP also coincides with the center of pressure (COP) [31]. If also the GCOM stays within the FSR during the gait through low speed and limited accelerations in order to minimize inertial effects, the biped system constantly maintains static stability, resulting in a static walking (SW) [26, 27]. Walking with low speed and small step length is usually performed by humanoid robots, resulting in a SW [32] or near-SW [12, 20], or by human subjects with disabilities, pathologies, age, infancy, or obesity, resulting in unnatural walking motions [33-35]. On the other hand, dynamic stability in DW is characterized by the GCOM trajectory that can be extended to outside of the FSR [26] (which does not necessarily imply falling). For this reason, the DW in this context can also be defined in contrast to the existing definition of the SW [5, 26]. It is known that natural human gaits are generally not statically stable, but typically consist of phases in which the GCOM leaves the FSR while the inertia and momentum of the segments are used to keep the gait dynamically stable [26,
29].
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Although normal human walking is characterized as DW, it is not constantly dynamic, according to its definition as seen above. To gain valuable insights in understanding fundamental principles of biped walking based on the aforementioned scientific motivations and for various engineering and clinical applications, it is important to quantify the level of dynamicity of a walking motion.
However, no objective measure currently exists that systematically quantifies the dynamic characteristics of human walking.
In this paper, we propose a novel index for the dynamic characteristics of human walking. The initial concept of Dynamic Gait Measure (DGM) introduced in a previous simulation study (for single-support phase of simplified models with constant FSR assumption) [36] is refined and extended in this research to provide comprehensive formulations for a complete gait cycle of human walking, including both single and double support phases. The DGM, associated with static balance instability, quantifies the effects of inertia in terms of the ZMP and the GCOM, and takes into account the time-varying FSR and the threshold between SW and DW according to their respective definitions. In addition, a framework of evaluating the DGM from gait experimental data is introduced, in which the gait cycle segmentation is further refined into fundamental sub-phases. A multi-segmental foot model is proposed and integrated into a biped system to reconstruct the walking motion from experiments, which demonstrates the time-varying
FSR for different sub-phases. The results for human experimental data are discussed and analyzed along with the previous simulation results and other existing indices.
2.1 MODELS AND METHODS
In this section, the mechanical models for biped walking and gait cycle segmentations are established, and the associated time-varying gait balance criteria are formulated. Based on these models and criteria, comprehensive formulation of the DGM and a framework of its experimental evaluation for a complete gait cycle of human walking are introduced.
Human biped and foot models
To reconstruct the experimental data of human walking within the proposed framework, the corresponding mechanical models need to be established. Since the main progression of walking occurs in the sagittal plane with major momentum in the forward direction, the range of lateral motion and the displacements of the GCOM and the ZMP in the frontal plane during normal walking are relatively small [29, 30]. We also assume that the lateral movement during normal walking has small effect on the sagittal-plane motion (i.e., decoupled) due to, for instance, the minor pelvic sway motion [13, 19, 37] and relatively small coupled vertical displacements [14].
These assumptions are in line with many existing gait models in the literature that have been analyzed in sagittal plane (e.g., [14, 19]). However, it should be noted that even when the GCOM exists within the FSR as viewed in the sagittal plane, it may not exist within the actual FSR as viewed in another plane. The more complicated model in all three planes should be investigated in a future study.
38

Y leg Y y
4
θ
5 ,
τ
5 x
4 l t l
4 m
4
θ
4 ,
τ
4 y
3 x
3 ankle joint heel
2 fh m
1 hl al fl toe-ball joint
1
X tl = l
8 toe link l
3 l
5
θ
3 ,
τ
3 x
2 m
3 m
5
θ
6 ,
τ
6 y
5 y
2 l
2 l
6 x
5 m
2 m
6
θ
7 ,
τ
7 x
6 y
6 y
8 y
7 x
1
θ
2 ,
τ
2 y
1 m
1 l
1
θ
1 ,
τ
1 l
7 m
7 l
8
X x
7
θ
8 , x
8
τ
8 toe link
Figure 1. Human biped and multi-segmental foot models for walking.
The proposed human biped model in sagittal plane, as an extension of the inverted pendulum model, has eight revolute degrees of freedom (Figure 1). Links 1 and 7 represent the foot segments that connect the respective toe-ball (metatarsal phalangeal) and ankle joints. The toe segment of stance leg is modeled as an additional link with a negligible mass. Link 4 represents the upper body that connects the two legs through hip joints, where its center of mass is located at a distance l t
normal to the pelvis, and the distance between the hip joints is zero (however, for modeling purposes, a very small positive number is used) in the sagittal plane. Links 5-8 for the swinging leg have dimensions and mass distributions that are symmetric with respect to those of the stance leg. The length and mass of Link i ( i = 1, …, 8) are indicated as l i
and m i
, respectively.
The center of mass position of each link is defined by the parameter c i
, which is the ratio of distance of Link i 's center of mass measured on the x i
direction from i th joint over the entire link length l i
. The link parameters and inertial parameters are assigned according to the reference human anthropometry data (Table 1; [38]). The global (located at the toe-ball joint of stance foot) and local coordinate frames for each link are attached such that the anatomical joint axes coincide with the z i
revolute joint axes. The actuations are represented by joint actuator torques
 i
( i = 1,
…, 8) as the generalized torques at the corresponding joint angles (  i
, i = 1, …, 8).
Table 1. Parameters for the human biped model.
Body segment Length (m) c i
Mass (Kg) foot shank l l
1
= 0.122 0.5 l
7
= 0.122 0.5
8
= 0.074
-
0.822
0.822
- l
2
= 0.425 0.567 2.636
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thigh l
6
= 0.425 0.433 2.636 l
3
= 0.314 0.567 5.670 l
5
= 0.314 0.433 5.670 upper body l
4
= 0.010 0.5 - l t
= 0.535 - 38.443
To incorporate the significant contribution of inter-segment foot kinematics [39], a multisegmental foot model is introduced in this work, which includes the foot-ankle complex with two main joints under rigid-body assumption (Figure 1). Joints 1 and 2 represent toe-ball and ankle revolute joint, respectively. The proposed multi-segmental foot model is crucial for modeling the time-varying contacts in human walking. While the dimensions of some foot segments (instep length l
1
and foot height fh ) are directly adopted from the literature (Table 2; [38]), the total lengths of foot and its other segments (heel length, arch length, and toe length) are estimated from the calculations on the given geometry and from the reconstruction of the experimental data. The total foot length ( fl ) is calculated as the average of two estimated measures: one is obtained by the geometry of the foot model, where the segment lengths are calculated from the marker positions; the other is estimated from the experimental data of the COP displacement in the x direction, from heel contact to toe-off. Similarly, the heel length ( hl ) is estimated from the x dimension of the foot-ground contact area during the heel contact phase. The toe length ( tl ) and the arch length ( al ) are estimated from the measured positions of the attached markers.
Table 2. Human foot segment dimensions.
Anthropomorphic foot measures Dimensions (m) foot height ( fh ) toe length ( tl ) heel length ( hl ) arch length ( al ) foot length ( fl )
0.0934
0.0740
0.0702
0.0891
0.233
Gait cycle segmentation for dynamic characteristics during one-step period
The gait cycle is the period of time between two identical events in the walking process [3]. A complete gait cycle (Figure 2) is the alternate repetition of stance and swing phases of both legs, during which the lower body performs two strides, one for each leg. The stride length is the distance covered by the same foot (typically between two consecutive heel contacts) during a gait cycle. The step length ( sl ) is the distance between the two feet at the beginning and at the end of a step, which is equal to half of the stride length for normal walking. The complete gait cycle is composed of single support (SS) and double support (DS) phases, where the left swing phase corresponds to the right SS, and vice-versa.
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RHC LTO
DS Right SS
LHC RTO
DS Left SS
RHC
DS
Right stance phase
Left swing phase
Right swing phase
Left stance phase
Figure 2. Gait cycle and phases (figure partially adopted and modified from [40] and [41]).
To characterize the dynamic nature of a gait, which is highly dependent on the FSR at a given instant, human gait cycle is segmented into step motions. A complete one-step period ( T step
) includes one SS and one DS, where left leg is chosen as the swinging leg and right leg as the stance leg. The time interval {0, t step
} of one-step period, from left toe-off (LTO) to right toe-off
(RTO), is segmented here into three fundamental sub-phases, according to the foot-ground contact sequence and gait events [41]. The SS phase is initiated with LTO and lasts until the next left heel contact (LHC), followed by the DS phase, until the right leg is in toe-off configuration.
The legs’ configuration at instant t = t step
will mirror the configuration assumed by the legs at the initial instant t = 0, but the roles of right and left legs are switched.
SS 1 foot-flat
SS 2 – DS toe contact
DS heel contact
R
R
L
L
R
Left toe-off
Left heel contact
Left stance foot-flat
L
R
Right stance foot-flat
0
T
SS 1
Right heel-off t
SS 1 T
SS 2 t
SS 2
= t
SS T
DS
Right toe-off t step
T step
Figure 3. Sub-phase segmentation of one-step period shown for the stance right foot and the landing left foot. The uppercase T represents time duration and the lowercase t represents time instant.
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Let the SS phase duration be within the time interval {0, t
SS
} and the DS phase within { t
SS
, t step
}.
The SS phase is further refined into two sub-phases according to gait events [41]: in SS1 phase, the stance foot (right) is assumed to be flat on the ground and in full contact, until the heel loses contact with the ground; after the right heel-off (RHO), the SS2 phase starts, where the stance foot is in toe contact condition with the ground (Figure 3). The complete gait cycle can be segmented in this approach. The relative time durations of the phases and sub-phases are as follows:
T step

T
SS

T
DS

T
SS 1

T
SS 2
T
SS 1
T
SS



T
SS

T step
and T
SS 2
and T
DS
  

T
DS
) T
SS
  
) T step
(33)
(34)
(35) where

and

are the fractions of the corresponding durations. The values of these parameters can be obtained from experimental data. The criteria adopted for the identification of the three sub-phases are based on ground reaction force (GRF) measurements (magnitude and location on the foot sole) and kinematics data. The zero GRF condition implies that the foot is in swinging phase, while a positive GRF implies that the foot is in contact with the ground. These two conditions are sufficient to identify the left swinging (right SS) phase and the DS phase. The foot-flat condition in the SS1 phase is verified in the time interval {0, t
SS 1
} where the normal ( y ) components of the position and velocity of the stance foot are (almost) constant. At t = t
SS 1
, the stance heel position and velocity start increasing in normal direction; the toe contact starts and lasts until toe-off, { t
SS 1
, t step
}. In this manner, both right and left leg motion sub-phases can be identified from symmetric gait assumption.
Sequence of time-varying FSR and ground reference points
If no external force is applied on the system other than the gravity ( g ), the x component of the
ZMP position in sagittal plane is calculated as follows:
ZMP

 m x i
 y
 i
 i g

 i
 m y x
 g
 i

L
(36) where x y x y i
, , , i i i
are the positions and accelerations of the i th
link’s center of mass, and L is the time rate of change of angular momentum of each link about the z -axis. The ZMP should reside within (excluding the boundaries) the FSR for a dynamically stable gait: ( )

( ) .
Theoretically, the ZMP criterion cannot be satisfied in limiting cases of edge or point contact between the foot and the ground [26, 28, 29], while the COP criterion still holds. However, in reality, the foot is not ideally rigid but deformable, and any rotational inclination will transform the edge into a new surface [28]. Thus, physically, the foot-ground contact always occurs through a finite surface within which the ZMP exists, and therefore the ZMP criterion is always satisfied during normal human walking. Under stationary foot contacts in balanced gait, the ZMP also coincides with the COP, at which the resultant of normal pressure GRFs is exerted and their moment vanishes [29, 31].
The GCOM is another significant ground reference point for balanced gait criteria:
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GCOM


 (37)
Unlike the ZMP, the GCOM trajectories can be extended to outside of the FSR depending on the gait (i.e., SW or DW), as discussed previously.
Based on the proposed multi-segmental foot model, the sequence of time-varying contact areas corresponding to the three gait sub-phases can be identified (Figure 4). The contact area is assumed to be finite and constant within each sub-phase. In the SS1 sub-phase, the contact area’s x dimension is equal to the entire foot length fl . In the SS2 sub-phase, the contact occurs through the toe length tl . In the DS phase, the contact area is composed of the right toe surface tl and the left heel surface hl . These sequential models of contact areas result in a discrete time-varying
FSR depending on the sub-phases. During the SS phase, the FSR coincides with the contact area, while, during the DS phase, it is the convex hull ( sl al ) of the contact areas (Figure 4) between right foot toe-ball joint and left foot ankle at the heel contact (Table 3).
Left swing phase
Y
Z
Right stance phase hl al tl
X fl
0
T
SS 1 t
SS 1
T
SS 2 t
SS 2
= t
SS
T
DS t step
T step
Figure 4. Foot-ground contact areas. The dashed areas indicate the foot contact region; the union of dashed and dotted areas represents the FSR.
Table 3. Contact areas and FSR dimensions.
Phase Contact area FSR
SS 1
SS 2
DS fl tl tl+hl fl tl sl-al
The ZMP trajectories of human gait experiment are calculated using the measured GRF data during each phase (Figure 5). For the SS phase, the x -position of the ZMP coincides with the
COP, and can be calculated directly from the experimental data of the COP during T
SS
:
ZMP
SS 1
  
R
 
(for 0 t t
SS 1
) and ZMP
SS 2 t

R
 
(for t
SS 1
  t
SS
) (38)
The ZMP trajectory during the DS phase can be calculated from its definition using the measured
COP and the GRFs. The foot segments that are in contact with the ground can be assumed to be
43

fixed to the ground with negligible relative motion. Then, the ZMP during the DS phase is identical to the point where the resultant normal GRF ( R p
R

R p
L
) is applied without any tangential moment:
ZMP ( )
DS

R p  p
( ) ( ) ( )
R L p
R

L p
( ) ( )
L
( )
(for t
SS
  t step
) (39) where the subscripts R and L represents right (rear stance) and left (front stance) foot, respectively.
Y Y
COM COM gravity and inertia forces
X R
R p
R p
R

R
L p
R
L p X
GRF
COP
R
( t )
COP
L
( t )
ZMP ( t ) = COP ( t ) ZMP ( t )
GCOM ( t ) GCOM ( t )
Figure 5. The COP, ZMP, and GCOM in sagittal plane during the SS and DS phases.
2.2 DGM FOR COMPREHENSIVE GAIT CYCLE
As described previously, in contrast to the SW, the DW is characterized by the GCOM trajectory that can be extended to outside of the FSR, in which the increased distance between the GCOM and the ZMP indicates reduced static balance stability. Therefore, the inertia effects, which can be represented by the distance between the ZMP and the GCOM at each time instant (according to their respective formulations), provide a measure of the dynamic characteristics of a given gait in terms of balance. This is also consistent with the significant contribution of inertia effects that defines or characterizes the DW in the literature [5, 22].
To quantify the relative measure of the dynamic characteristics for a given gait motion and foot dimensions, the dimensionless DGM is defined as the ratio of two root mean square (RMS) distances, d and d max
, for every (sub-)phases and the entire cycle:
DGM
 d d max
(40) where the inertia effects in terms of balance are quantified using the following RMS of the distance between the ZMP and the GCOM over the time duration T :
44

d

 T
0

     

2
ZMP t

GCOM t dt
(41)
T
Although the above distance quantifies a certain degree of dynamic characteristics by providing the effects of inertia, it lacks the information of the relative balance stability. In other words, for a given d value, the balance stability or the dynamic characteristics of the gait can be different depending on the FSR. To incorporate the relative stability with respect to the FSR and the
SW/DW threshold into the quantification of the DW, the d for marginal SW, which represents a virtual motion where the GCOM trajectory is at the border of the SW and the DW, should be considered: d lim

 T
0

    


2
ZMP lim t

GCOM lim t dt
(42)
T where ZMP lim
( t
) is a function of time that depends on the biped’s motion, and
GCOM lim
( t ) is the limiting function of the GCOM trajectory in the x direction, both corresponding to the marginal
SW. Since the GCOM trajectory serves as a determinant of whether the given gait is SW or DW, the GCOM lim
( t ) provides their threshold and extends over the maximum allowable bounds within the FSR from the rear edge to the front edge. Here, while the GCOM lim
( t ) can be determined from the FSR, the ZMP lim
( t ) is not unique, and thus d lim
is a functional of ZMP lim
( t ). Since the feasible trajectories of ZMP lim
( t ) must satisfy the FSR criteria at all times, d lim
is bounded from above by: d max

Max
ZMP lim
( )

( ) d lim
(43)
Here, d max
quantifies the maximum inertia effect that a walking motion can achieve during the marginal SW. Once the inertia effect exceeds the value d max
, the GCOM extends outside of the
FSR boundaries, resulting in a DW. The proposed refined segmentation of the SS phase into SS1 and SS2, and the associated time-varying FSR are used to obtain GCOM lim
( t ). For the entire onestep period, GCOM lim
( t ) should represent the marginal SW conditions for each FSR sequence
(Figure 4). Based on the near-linear monotonic trajectories of actual GCOM during normal human walking [29], the limiting GCOM trajectory is formulated as a virtual piecewise-linear function of time from the rear to front boundary of the FSR in each sub-phase:
GCOM lim
( )









GCOM
GCOM
GCOM
SS lim
SS lim
DS lim
1
2
  t
 
 
=
= tl
= (
T
SS 2 t
 t


 fl
T
SS 1 t
 t t

( fl
 tl
T
DS
SS 


(
SS 1
) for 0
) for t sl
 al
SS 1 t t t t
SS
SS
1
) for t
SS
  t step
(44)
45

Although these limiting GCOM trajectories are not usually achieved concurrently in a normal gait, they are feasible at each sub-phase under the constraints of joint ranges of motion. Then for a given FSR (assume rectangular), with bl and bu being its lower and upper bound in the x direction, respectively, it can be shown that the maximum inertia effect achievable in a marginal
SW is given by the following RMS:

0
T / 2   
2 bu GCOM t dt lim
( )
 
T
T
/ 2
  
2 bl GCOM t dt lim
( ) d max

, DGM
SS 2 d
SS 2

SS 2 d max
,
T
DGM
SS
The DGMs of each (sub-)phase provide the corresponding measures at different phases of walking: d
SS 1
DGM
SS 1 
SS 1 d max d
SS
 d
SS max
, and DGM
DS d
DS
 d
DS max
(45)
(46) where, for the proposed FSR sequence, d
SS 1 max

7 fl
12
2
, d
SS 2 max

7 tl
12
2
, d
SS max

T
SS 1
 
2

T
SS
T
SS 2
 
2 max
2
, and d
DS max

7
 al
 sl

2
12
(47)
Finally, the DGM for complete one-step period that represents the dynamic characteristics of a biped walking is:
DGM
 d d max

T
SS 1

0 t step d

     

2
ZMP t

GCOM t dt
2

T
SS 2
 
2

T
DS
 
2
(48)
The DGM is a functional of the ZMP trajectory, the GCOM trajectory, and the foot dimensions, and its formulation provides a systematic approach of determining the dynamic characteristics of a given gait from experimental data. Since the ZMP always exists (so does the GCOM) and satisfies the criterion during human walking, the definition of the DGM is physically and mathematically valid, based on which the existence and uniqueness of the DGM is also trivial.
Here, it should be noted that the DGM depends not only on the balance criteria such as the ZMP and the GCOM, but also on the foot and FSR dimensions, which varies for each (sub-)phase.
From the definition, it can be shown that the DGM is bounded from above:
0

DGM

K (49) where K is a real finite upper bound, whose existence can be proved from the characteristics of the ZMP ( t ), GCOM ( t ), and GCOM lim
( t
) trajectories. The DGM’s denominator depends only on the FSR dimension, resulting in a finite positive number. And since the ZMP always exists within the FSR and the GCOM is bounded by the system parameters (link lengths, foot dimensions, and joint limits), the numerator is bounded from above. This shows that the DGM, and therefore the dynamicity of a biped walking, is bounded from above by a finite value K , regardless of the maximum actuation capacity or muscle strength. Physically, larger foot dimension should have smaller K , and vice versa.
46

The calculated DGM values can be interpreted according to several cases:
•
DGM = 0: The ZMP and the GCOM are identical at all times. The walking motion of the system is stationary [26], or can be regarded as quasi-static.
•
0 < DGM < 1: The GCOM may (for SW; stable static balance) or may not (for DW; unstable static balance) stay within the FSR. Whether a given gait is SW or DW is determined by the actual GCOM trajectory with respect to the FSR. The DGM SS and DGM DS can further provide the corresponding measures at different phases of walking. Generally, larger DGM indicates relatively dynamic walking with greater inertia effects, and vice versa.
•
DGM = 1 (or near 1): The gait is marginally static, and its DGM value and dynamicity are the maximum achievable for a SW.
•
1 < DGM < K : The GCOM falls outside of the FSR at some times, i.e., a statically unstable
DW. The larger DGM value implies more dynamic gait than the smaller one. The existence of the GCOM outside of the FSR during a longer duration or the larger distance between the ZMP and the GCOM will result in a larger DGM.
The DGM can be interpreted as a time-global measure of static instability (which does not necessarily indicate falling), with respect to balance, during a dynamically stable walking. As discussed later, it should also be noted that the proposed characterization approach of the DW is distinct from other instability measures that are dependent on walking speed.
2.3 RESULTS
To demonstrate the proof-of-concepts of the DGM, a set of existing human gait experimental data
[38] is used, in which the kinematics (from markers and high-speed infrared cameras) and GRFs
(from force plates) data for the right leg and the upper body are measured. The total time duration is 1.501 seconds, sampled at a rate of 69.9 frames/second, where the GRFs and the markers’ kinematics are recorded in a total of 106 sampling time frames. The stride interval is 68 frames and 0.972 seconds (Table 4). The right stance phase is identified by the right foot contact ( GRF >
0). The stride and step lengths and times are identified from the x positions of the right heel at two successive right heel contact (RHC) (Table 5; a one-step motion corresponds to half stride).
The left leg motion is derived from the right leg data from cyclic and symmetric gait assumptions, where the left leg swinging motion is assumed to be equal to that of the right leg with an out-ofphase time delay of T step
—i.e., half stride (34 frames) later from the initial RHC—and is shifted in the x direction by the step length. Then both legs data is integrated into the biped model to reconstruct the complete one-step walking motion (Figure 6). The mass distribution, multisegmental foot dimensions, and gait parameters are calculated in this work (Tables 1, 2, and 5) by incorporating the anthropometry of the human subject (total mass 56.7 Kg) into the biped model.
Then the trajectories of the ZMP, the GCOM, and the limiting GCOM are determined using the reconstructed experimental walking data.
Table 4. Derived gait phases and events for the given experiment (stride is between two RHCs).
Gait phase Time duration (sec) Total frames
Stride
Right stance
Right swing
0.972
0.586
0.357
68 {28-95}
42 {28-69}
26 {70-95}
47

Gait events Time instant (sec)
RHC
LTO
RHO
LHC
RTO
0.386
0.5
0.772
0.872
0.972
Frames
28
36
55
62
69
Table 5. Calculated gait parameters (stride is between two RHCs).
Gait parameters Values
Stride length
Stride time
1.405 (m)
0.972 (sec)
Step length ( sl ) 0.703 (m)
Step time 0.487 (sec)
Average speed 1.443 (m/sec)
Cadence 2.053 (step/sec)
Figure 6. One-step human walking motion in sagittal plane reconstructed from the experiment and the calculated ZMP trajectory (vertical bars).
The gait (sub-)phases according to the proposed segmentation are identified for the given experimental data, and are validated against those predicted in the literature [41] (Table 6). The
ZMP and the COP trajectories are calculated from the measured GRF magnitudes and their positions. The positions of the ZMP and the COP at each sampling time are taken into account for the one-step interval, as well as the stance (right) and landing (left) foot motions (Figure 7).
Table 6. Gait segmentation and calculated time durations.
Gait phase
Initial/final instant (sec)
Time duration
(sec)
Total time frames
Gait segments (%)
(Predicted values from [41] are in parenthesis)
48

One-step
DS
SS
SS1
SS2
{0, t step
} T step
{0, 0.472} 0.472
{ t
SS
, t step
}
{0.358,
0.472}
T
DS
0.114
{0, t
SS
} T
SS
{0, 0.358} 0.358
{0, t
SS 1
} T
SS 1
{0, 0.258} 0.258
T
SS 2
{ t
SS 1
, t
SS 2
}
{0.258,
0.358}
0.1
34
8
26
19
7
R foot
-
1 24.2
(24)
 
75.8
(76)
 
72 (75)
1 28 (25)
L foot
COP
ZMP
SS 1 SS 2 DS
Figure 7. ZMP and COP trajectories in the three sub-phases: foot-flat SS1 (circles), toe contact
SS2 (squares), and toe/heel contact DS (rhombs).
The trajectories of ZMP ( t ), GCOM ( t ), and GCOM lim
( t ) on the x axis for the one-step interval {0, t step
} are obtained along with the x dimensions of the FSRs corresponding to the segmented subphases (Figure 8; the initial time of one-step period is set to zero at the foot-flat configuration).
The DGMs are calculated to quantify and demonstrate the dynamic characteristics of the given human walking, and are also compared with the results of simulated human and robotic walking
(during the SS phase only) from the previous study [36] (Table 7; the DGM values for the simulation results in the previous study are recalculated according to the refined formulation proposed in this paper).
49

SS 1 SS 2 DS
FSR DS
FSR SS 2
FSR SS 1
Figure 8. ZMP (thick), limiting GCOM (thin), and GCOM (dashed) trajectories during normal walking.
Table 7. Results of dynamic characteristics. The simulated results [36] are recalculated using the refined formulation in this paper.
Experimental results Simulated results
Human Human Robot
Phase or sub-phase SS SS 1 SS 2 DS step SS SS
RMS distance
DGM
Walking speed (%)
 max
L d
(m) 0.108 0.0812 0.160 0.0987 0.106
0.139 0.068
0.702 0.456 2.825 0.211 0.398 0.607 0.0297
100%
4.97x10
-4
123%
5.32x10
-
4
24%
6.19x10
-
5
2.4 DISCUSSION AND CONCLUDING REMARKS
The quantitative results of the DGM are discussed with respect to the dynamic characteristics of human walking, and are also compared with other relevant indices existing in the literature.
Dynamic characteristics and DGM
From the ground reference point trajectories (Figure 8), it can be observed that the GCOM falls outside the FSRs during the SS phase. More specifically, the GCOM is outside the FSR at the beginning and at the end of foot-flat SS1 phase, and for the entire duration of the toe contact SS2 phase. During the entire DS phase, the GCOM remains inside the FSR. This observation qualitatively indicates that both the SS1 and SS2 (and therefore the complete SS) phases are dynamic phases of walking, while the DS is a static phase. The DGM values (Table 7)
50

quantitatively prove this dynamic nature of normal human walking during the SS phase, with values during all sub-phases of the SS phase greater than that of the DS phase. In particular, the toe contact SS2 phase is the most dynamic phase, resulting in the highest DGM. The experimental SS phase DGM value is similar with that of the human walking simulation. As expected, the DGM for normal human walking is much greater than that of a robotic walking, which typically demonstrates the SW even during the SS phase. In contrast, the DGM for the DS phase of human walking experiment demonstrates relatively static nature during this phase. The calculated DGM values quantitatively confirm the characteristics of normal human walking featuring the periodic sequence of statically stable DS phase and statically unstable (or dynamic)
SS phase [42]. It is also shown that, throughout the sub-phases, smaller FSRs result in greater
DGMs and vice-versa, which is consistent with the previous simulation results [36] of walking with various FSR dimensions (normal human foot, stilt foot, and ski-board). In the case of an abnormal walking for which the single value of DGM may not clearly differentiate itself from that of a normal walking, the DGM
SS
and DGM
DS
can provide distinct measures at different gait phases. Overall, the DGMs calculated from the gait experimental data demonstrate reasonable quantification of the dynamic nature of normal human walking.
Dynamic walking vs. walking speed
The DGM characterizes the DW based on static balance instability (as mentioned previously, this does not necessarily implies falling), in accordance with the relevant definitions in the aforementioned literature. On the other hand, several instability measures, which are dependent
(explicitly or implicitly) on walking speed, are also used for gait analysis, such as gait variability,
Floquet multipliers, and maximum Lyapunov exponents [43-47]. Especially, the maximum
Lyapunov exponent is commonly used as a local dynamic instability measure of a gait cycle, and quantifies the ability of a system to respond continuously to small perturbations. The maximum
Lyapunov exponent measures the exponential attenuation of variability between neighboring kinematic trajectories [45], such that higher values of the exponent result in higher instability
(which does not imply falling); here, it is assumed that every stride is identical, and that the strideto-stride kinematic differences are attributed to small perturbations. While the maximum
Lyapunov exponent is highly dependent on walking speed [44, 45], the DGM is not necessarily.
For instance, a gait with a small step length or a large foot dimension and a very high frequency can result in a small DGM (or even a SW) but a large walking speed; similar arguments can be made for the opposite case. It is also possible to generate a gait of small DGM with low speed if both the step length and the frequency are small, and vice versa. Even for a given speed of normal human walking, larger step lengths correlate with larger DGMs, given that the DS phase is short and the forward speed does not fluctuate too wildly. To illustrate these concepts quantitatively, the long-term maximum finite-time Lyapunov exponent (
 max
L
) for steady-state gait is calculated (Table 7) at the upper-body center of mass for each gait motion, using an experimental regression curve [44] as a function of average walking speed in percentage of the preferred walking speed of a given biped model. The walking speed is 100% for the human experiment at its self-selected speed, while for each simulated gait, the respective preferred walking speed is calculated as 42% of its Froude velocity [45] (the preferred walking speed of the robotic walking is hypothetically estimated from that of a human with same morphology). As expected,

L max increases with the walking speed. While the lowest walking speed is associated with the lowest DGM and the lowest

L max , the highest walking speed is associated with the
51

highest
 max
L
and an intermediate DGM. The most dynamic gait, with respect to the definition of the DW, occurs not at the highest speed, but at the preferred walking speed of the real human subject with an intermediate value of
 max
L
. It should be noted that the DGM and the maximum
Lyapunov exponent are not explicitly dependent on the motor controller, and therefore cannot be the indicators of stability in terms of control. These results show that the DGM is not necessarily correlated with the maximum Lyapunov exponent or walking speed, particularly for walking motions with unnatural gait parameters.
Dynamic walking vs. energy consumption
Another index relevant to the dynamic characteristics of a gait is the energy consumption [48], which is also consistent with the cost of transport [49]. For normal human walking, the dynamicity and the passivity (i.e., low energy consumption) are usually coupled with each other, which results in passive dynamic walking, as seen in the literature above [5, 22]. However, this is not necessarily true for general gait motions including unnatural/abnormal human walking and robotic walking, since the energy consumption is associated with total actuations including those that do not directly contribute to the gait progression. In general, the DGM and the energy consumption are independent, since the variables used in each formulation are not necessarily dependent on each other. In other words, the GCOM and the ZMP are determined by not only the joint actuator torques, but also the initial and boundary conditions of the system. Also, depending on the foot dimensions, step length, and walking speed, the coupling between the DGM and the energy consumption can diminish. For example, walking with small ankle actuation for a small step length and a large foot dimension can result in a low energy consumption but low dynamicity. On the other hand, it is also possible to exert large ankle actuation for unnatural walking with a large step length and a small foot dimension, which will result in a high energy consumption and high dynamicity. It can be stated that, particularly for human walking with unnatural compensation (due to injury, disability, or heavy load) or robotic walking, the correlations between the DGM and the energy consumption can be much weaker than those for naturally coordinated normal human walking.
Applications and Future Work
The DGM of normal human walking has the potential to provide some scientific, engineering, and clinical insights in understanding biped walking principles. For example, the DGM values of normal human walking can serve as reference criteria against various walking motions of different human subjects to detect gait abnormality due to pathologies, injuries, aging, or obesity, and to measure the improvement or recovering rate in rehabilitation therapies. They can also be used as performance criteria and goals for design and control of efficient walking machines and assistive exoskeletons. Possible future research to demonstrate the broad and general applicability of the concepts include experiments for a large number of human subjects with various gait parameters (step length and speed) and conditions (e.g., pathologies or load) to identify the reference and abnormal levels of gait dynamicity; and extending the DGM formulation to three-dimensional walking that takes into account the effects of the frontal plane motions and their possible coupling with the sagittal plane.
Concluding Remarks
The DGM, a novel quantitative index for the dynamic characteristics of biped walking, was formulated for comprehensive gait cycle phases. A framework was introduced that determines
52

the DGM from a set of human gait experimental data. The proposed multi-segmental foot model, which was integrated into the biped system to reconstruct the walking motion from experiments, demonstrated the time-varying FSR for different gait sub-phases. The results of the DGM using human gait experimental data were analyzed along with other established features and indices of normal human walking in the literature. Also, the relationships among the DGM, the walking speed, and the energy consumption were discussed. Overall, the DGM provided a reasonable measure of static balance instability of biped walking during each (sub-)phase as well as the entire gait cycle.
53

Evaluating the effects of load carriage on gait balance stability is important in various applications. However, their quantification has not been rigorously addressed in the current literature, partially due to the lack of relevant quantitative indices. The novel Dynamic Gait
Measure (DGM) is used in this paper to characterize the distinct balance stability of loaded walking resulting from the modified gait strategy due to load carriage. The DGM provides a measure of the static balance instability of biped walking by quantifying the relative effects of inertia in terms of the zero moment point and the ground projection of center of mass trajectories with respect to the time-varying foot support region. In this study, the DGM is reformulated in terms of the gait parameters that explicitly reflect the gait strategy in loaded walking, and is evaluated from experiments to quantify the changes in gait balance stability in the sagittal plane caused by load carriage. The combined effects of relevant gait parameters (increased single support duration, reduced inertia effects, and decreased step length) during loaded walking result in decreased DGM values, which indicate that the loaded walking motions are more statically stable due to implementation of a more cautious and guarded gait strategy compared with the unloaded normal walking. The unique characterization capability of the DGM is also evaluated through comparison with other existing reference stability indices (the maximum Floquet multiplier and the margin of stability), which shows that the DGM is the only index that is consistently informative of the presence of the added load.
54

INTRODUCTION
Human walking is frequently accompanied by load carriage, which may result in notably different gait parameters and balance stability from those of natural human walking. The effects of load carriage have been widely studied particularly in terms of gait biomechanical and physiological parameters, such as speed, frequency, step size, support duration, muscle activity, metabolic cost, coordination, and ground reaction forces [1-5]. On the other hand, the combined effects of the individual gait parameters on the balance stability of loaded walking have been addressed only in a few studies within given problem-specific scopes [2,6]. Quantitative indices that characterize the effects of load carriage on gait balance stability would be beneficial in the design and evaluation of wearable and portable systems in various applications, such as military, space exploration, and assistive devices for persons with disabilities [7-10].
For general gait analysis, several stability indices have been introduced in the literature, each within its own specific contexts and perspectives. Most of the current gait stability indices can be divided into two approaches according to their perspectives of quantification [11]: dynamical systems theory and balance criteria.
Gait stability measures derived from dynamical systems theory address the system’s behavior in response to continuous small perturbations on a steady-state gait cycle. Common indices in this category include maximum Floquet multiplier (MaxFM) and maximum Lyapunov exponent [11].
The maximum Lyapunov exponent quantifies the average logarithmic rate of divergence in a steady-state walking after a perturbation [12,13]. Although it has several advantages, such as the flexibility in choosing the reference frame for kinematic data [14,15], the maximum Lyapunov exponent has been shown to be ineffective in quantifying the gait stability variations caused by different load conditions [9]. On the other hand, the Floquet analysis supports the hypothesis that different load conditions cause changes in orbital stability of human walking [9]. The Floquet analysis has largely been used to evaluate the response of the state of a locomotive system to internal or external perturbation [16-18]. This analysis is based on the assumption that the system dynamics is strictly periodic and quantifies the ability of the system to return to the limit cycle of the stable gait pattern after a perturbation is encountered. The rate of convergence/divergence from limit cycle trajectories is calculated by the eigenvalues (Floquet multipliers) of the Jacobian of the Poincare return map. For a limit cycle to be stable, all Floquet multipliers should have magnitude less than one, and the closer the magnitude is to zero, the faster the locomotive system returns to the steady-state orbital gait pattern. The MaxFM is the magnitude of the largest eigenvalue and represents the most unstable point in the stride cycle [9,11,19,20].
Another approach to characterizing gait stability is through balance criteria—also categorized as the biomechanics approach [11]—based on ground reference points, in order to address the thresholds between balanced and falling. Several ground reference points and the associated criteria have been introduced in the literature [21-27]. However, as a single index, the ground reference point data by themselves are not sufficient to quantify the stability characteristics of a given gait, but rather they should be integrated into the associated balance criteria [27]. One common index established within this approach is the spatial margin of stability (MOS), which is defined as the distance between the extrapolated center of mass and the edge of the foot support region (FSR; a convex hull area determined by the foot dimensions and the step size) at any given
55

time instant [28]. The extrapolated center of mass is an extension of the center of mass (COM) position by including the COM velocity and extends the condition for simple static balance (COM projection inside FSR) to dynamic balance required for walking (extrapolated center of mass inside FSR) [11,28]. For given initial conditions (i.e., COM position and velocity) and FSR margins, the MOS quantifies the closeness of a gait system to falling. The minimum MOS indicates the most unstable point and is directly related to the impulse required to cause instability
[28-29].
Although each of the existing gait stability indices has its own perspectives and advantages, none of them has been proven to be effective in characterizing the balance stability of loaded walking; as a matter of fact, we are not aware of such an index in the current literature. In this paper, the
Dynamic Gait Measure (DGM) [30], a novel index for a comprehensive gait cycle, is evaluated from a set of experiments to demonstrate its validity in quantifying the variation of gait balance stability caused by load (backpack) carriage. The evaluation in this study focuses on the stability in the sagittal plane, in which maintaining balance is more challenging and critical than in the frontal plane in loaded walking [1,31,32]. The trajectories of two ground reference points are used in the DGM calculation: the ground projection of center of mass (GCOM) and the zeromoment point (ZMP), which coincides with the center of pressure for a finite contact area without relative motion between the contacting foot and the ground [33]. The DGM quantifies the effects of inertia in terms of the ZMP and GCOM trajectories and incorporates the time-varying FSR and the threshold between static (i.e., statically balanced) and dynamic (i.e., statically unbalanced without falling) walking [19,21,30,34]. Since normal human walking typically consists of phases during which the body is statically unbalanced, the DGM provides a time-global measure of static balance instability during each phase as well as the entire gait cycle. Since the load carriage during walking mainly affects the inertia (due to added mass) and the associated gait strategy
(step length and support duration), which are also the main components of the DGM formulation, this research is based on the hypothesis that the DGM is a valid balance stability index that discriminates the loaded walking conditions. For the quantification and analysis of the effects of load carriage on gait balance stability, the previous DGM formulation is modified accordingly in this research. The distinct balance stability and gait strategies of loaded walking, as compared with unloaded normal walking, are characterized by the DGM using experimental data. Also, such unique discriminative ability of the DGM is demonstrated and validated along with the comparison with the MaxFM and MOS as reference indices.
3.1 METHODS AND EXPERIMENTS
To quantify the effects of load carriage on the balance stability of human walking, the DGM is reformulated in terms of the gait parameters that explicitly reflect the variations in gait strategy due to load carriage. The DGM, along with the existing reference stability indices MaxFM and
MOS, is calculated from experimental data of human walking under loaded and unloaded walking conditions.
Reformulation of Dynamic Gait Measure for Loaded Walking
The DGM is defined as the dimensionless ratio of two root mean square (RMS) distances, d and d max
, for each phase of the gait cycle [30]:
56

DGM
 d d max
(50) where the inertia effects in terms of balance are quantified using the following RMS of the distance between the ZMP and the GCOM in the sagittal plane over the time interval t

[0, ] : d


0
T

      

2
ZMP t GCOM t dt
T
(51)
Although the above time-average distance provides a certain degree of dynamic characteristics by quantifying the effects of inertia, it lacks the information of the relative balance stability. In other words, for a given d value, the gait balance stability can be different depending on the GCOM position relative to the FSR [34]. In fact, the GCOM trajectory serves as a determinant of whether the given gait is static or dynamic walking [19,21,34]. In order to incorporate the threshold between the static and dynamic walking into the quantification of the balance stability characteristics, d max
, which corresponds to the maximum d of a marginal static walking, is implemented into the formulation. Here, a marginal static walking [30] represents a virtual motion where the corresponding GCOM trajectory GCOM lim
( t ) extends in the anterior-posterior direction ( x axis) from the rear edge ( x = bl ) to the front edge ( x = bu ) of the FSR (Figure 1).
Consequently, d max
quantifies the maximum inertia effect that a given gait can achieve during one phase of a marginal static walking, assuming the FSR as a constant finite area within that phase.
Once the inertia effect exceeds the value d max
, the GCOM extends outside of the FSR boundaries, resulting in dynamic walking [21,30]. For a given FSR and gait phase, it can be shown that [30]: d max

 T
0
/ 2   lim
( )

2 bu GCOM t dt
 
T
T
/ 2
  lim
( )

2 bl GCOM t dt

7( bu
 bl )
2
(52)
T 12
For implementation of experimental data into the DGM, the gait cycle is segmented into three sub-phases: single-support (SS) sub-phases SS1 and SS2, where the stance foot is in full and toe contact with the ground, respectively, and double-support (DS) phase (with duration T
DS
), with the trailing leg in toe contact and leading leg in heel contact (Figure 2). The foot-ground contact is modeled as a sequence of finite contact areas, resulting in a discrete time-varying FSR (Figure
2) and the corresponding DGM at each (sub-)phase of walking. Then for a complete step cycle
T
STEP
[30], d STEP max

T
SS 1 d
SS 1 2
( ) max

T d
SS 2
(
SS max
2 2
)

T
DS d
DS
( ) max
2
T
STEP
(53) where d
SS 1 max
, d
SS max
2
, and d
DS max
are the d max
for each (sub-)phase, calculated with the corresponding
FSR dimension—foot length in SS1, toe length in SS2, and the difference between step and arch lengths in DS, respectively (Figure 2). This can be reformulated using the ratio

of SS1
57

duration T
SS 1
with respect to the total SS duration T
SS
and the ratio

of T
SS
with respect to T
STEP
, as follows: d STEP max
 
( d SS 1 2 max
)

(
 
)( d SS max
)
  
)( d DS max
) 2 (54)
The variables d ,

,

, and d
DS max
in the DGM formulation are determined from the given gait strategy, which shows that the DGM, as a comprehensive index, is not only dependent on the inertia effects (quantified by d in the numerator), but also on the relative duration of gait phases and the step and foot sizes (in the denominator), resulting in a time-global indicator of gait static balance instability. Hence, depending on the contributions of the gait phase durations and the step and foot sizes, a gait with small inertia effects can result in a highly dynamic walking (i.e., high static balance instability), and vice versa. In particular, since the step length is generally greater than the foot sizes during human walking, the term (1
 
)( d
DS max
)
2
is the most significant in the denominator. This implies that the most influencing factors for gait static balance instability are the relative SS duration and the step length, as well as d .
Figure 1. Schematic illustration of the DGM concepts using a compliant biped walking model:
(A) a statically unbalanced SS phase and (B) a statically balanced DS phase. The distance | ZMP –
GCOM | represents the inertia effect, while the FSR (shaded areas) dimension | bu – bl | represents the maximum available inertia effect during a marginal static walking, all at a given time instant.
The DGM is the ratio of the time-averaged RMS values derived from these two distances.
58

T
STEP heel length bl bu bl bu bl arch length x bu foot length toe length step length
T
SS 1
T
SS 2
T
DS
Figure 2. Gait cycle segmentation and finite time-varying FSR for human walking.
Experiments
Seven healthy young subjects (all male, age: 29.28 ± 2.14 years, mass: 75.1 ± 8.6 kg, height: 1.75
± 0.04 m) participated in loaded walking trials. They reported no history of gait disorders, and signed informed consent forms that were approved by the Institutional Review Board of the
Korea Advanced Institute of Science and Technology (KAIST) prior to testing. During the test, subjects walked on a 12-m-long, 1-m-wide walkway, with and without a 25-kg backpack, at three different speed levels: self-selected, maximum, and intermediate. The speed for the over-ground walking trials was controlled by a metronome with the frequency corresponding to each gait speed. The kinetic and kinematic data were measured using three force platforms (Accugait,
AMTI, US) and a motion capture system (Hawk, Motion Analysis, US). Detailed experimental protocol and measurements can be found in a previous work [35].
Data Processing
The DGMs are calculated from the kinematic and kinetic experimental data. The COM trajectory and the link lengths are calculated from the kinematic marker data using an eight-degree-offreedom human model in the sagittal plane [30], and the inertial parameters are estimated according to a reference anthropometry data [36]. The gait events of heel strike, heel off, and toe off, as well as the FSR dimensions in the x direction, are identified from the ground reaction forces data and the toe and heel marker trajectories. The ZMP trajectories of the human gait experiments are calculated using the ground reaction forces data in the sagittal plane during each phase of gait. In particular, for the SS phase, the ZMP at each time step is calculated as the ratio of the resultant tangential ground reaction moment and the resultant normal ground reaction force.
For the DS phase, the ZMP is identical to the point where the resultant of the normal ground reaction forces acting on each foot is applied without any tangential moment.
In addition, the MaxFM and the MOS are also calculated from the experiments as reference indices and compared with the DGM results. For the MaxFM, since upper body stability is critical in human walking [11], the trunk state trajectory (vertical position and velocity) is considered for each steady state walking trial. The Poincare section is periodically sampled at the
59

heel strike [11,37], and the Jacobian matrix of the Poincare map is calculated from a least-square algorithm [9,20]. The MOS in the anterior–posterior direction is calculated as the average of the distance between the extrapolated center of mass and the most anterior edge of the FSR at every heel strike [29]. Independent t-tests are performed to analyze the differences in the gait parameters and the gait stability indices between the loaded and unloaded walking trials.
3.2 RESULTS AND STATISTICS
The resulting experimental gait parameters for both loaded and unloaded walking conditions include speed, step length, and the SS duration ratio (Figure 3). The presence of the additional load causes the change in gait parameters at some speed levels. The loaded trials show slower walking speeds in the self-selected and intermediate speed levels ( p = 0.011 and p = 0.014, respectively; Figure 3A), shorter step lengths for the intermediate and maximum speed levels ( p =
0.041 and p = 0.015, respectively; Figure 3B), and smaller SS duration ratios at all speed levels ( p
< 0.001 for all levels of speed; Figure 3C), as compared with those of the unloaded walking trials.
Figure 3. Experimental gait parameters of unloaded and loaded walking trials for each speed level: (A) walking speeds; (B) step lengths; and (C) the ratios of SS phase duration with respect to full step duration. The asterisks (*) indicate statistically significant differences ( p < 0.05).
The effects of load carriage on the balance stability of human walking motions are quantified by the DGM results (Figure 4A), which show significantly lower values (0.440 ± 0.037) than those of the unloaded normal walking (0.535 ± 0.052) throughout all experiments ( p < 0.0001). On the other hand, the MaxFM and the MOS are also calculated from the experiments of loaded
(MaxFM = 0.579 ± 0.238; MOS = 0.107 m ± 0.027 m) and unloaded walking motions (MaxFM =
0.640 ± 0.221; MOS = 0.114 m ± 0.023 m) as two typical reference indices of gait stability
(Figures 4B-C). Throughout all experiments, neither MaxFM nor MOS shows any significant difference between loaded and unloaded walking motions ( p = 0.140 and p = 0.127, respectively).
60

Figure 4. The DGM, MaxFM, and MOS (magnitudes) results from unloaded (squares) and loaded (circles) walking experiments.
In addition, the dispersed speed values among the subjects (Figure 3A) for a given speed level
(self-selected, intermediate, and maximum) should be considered for further insights, since one subject’s self-selected walking speed may be higher than another subject’s intermediate or maximum walking speed, and vice versa. In this regard, the analysis of DGM, MaxFM, and MOS values of loaded and unloaded walking within each comparable speed level (Figure 5 and Table
1) evaluates the consistency of those indices’ discriminative ability at any given speed level.
While the MaxFM (Figure 5B) and the MOS (Figure 5C) do not show any statistically meaningful difference between the unloaded and loaded walking within any speed levels ( p > 0.05 for selfselected, intermediate, and maximum speeds of both indices; Table 1), the DGM (Figure 5A) is the only index that is consistently informative of the presence of the load carriage within each speed level ( p < 0.05; Table 1).
Figure 5. Mean and standard deviation results of the DGM (A), MaxFM (B), and MOS (C) at different speed levels. The asterisks (*) indicate statistically significant differences ( p < 0.05).
Table 1. The p-values from the t-test for the difference between the means of the loaded and unloaded gait stability indices within each speed level. The asterisks (*) indicate the statistically significant differences ( p < 0.05).
61

Self-selected
Intermediate
Maximum
DGM
0.000 (*)
0.000 (*)
0.000 (*)
MaxFM
0.413
0.142
0.515
MOS
0.189
0.887
0.167
3.3 DISCUSSION AND CONCLUDING REMARKS
The DGM quantifies the relative effects of inertia with respect to the time-varying FSR corresponding to gait phases. For a given gait cycle, a larger DGM corresponds to a higher level of static balance instability, and vice versa. In this study, the DGM is used to experimentally quantify the distinct balance stability characteristics resulting from the modified gait strategy due to load carriage.
The variations of the gait strategy due to load carriage are represented by the gait parameters.
While the speeds and step lengths are reduced only at some speed levels (Figure 3A-B), the SS duration ratios are reduced at all speed levels (Figure 3C). The decrease in step length during loaded walking is consistent with the results of previous studies (e.g., [1]). However, as seen above, the balance stability is also dependent on additional parameters, such as the ZMP and
GCOM trajectories and the time-varying FSR. Consequently, not one of those individual gait parameters directly characterizes the gait balance stability and its adaptation to various load conditions.
On the other hand, the DGM as a single index can effectively characterize the state of balance of a given gait pattern by including in its formulation the combined effects of relevant gait parameters
(hence the overall gait strategy). The significantly reduced DGM values throughout all experiments (Figure 4A) of the loaded walking show that the loaded walking motions are more statically stable (i.e., less dynamic) compared with the unloaded normal walking. In general, if all other gait parameters remain the same, a smaller step length would tend to decrease the balance stability due to smaller FSR dimensions [22,34,38]. However, the combined effects of the relevant gait parameters during loaded walking, including the decreased SS duration ratios as well as the reduced step length (Figure 3), result in the decreased DGM values, which may indicate that the subjects tend to increase their balance stability in order to reduce the risk of falling.
In addition, the DGM analysis within each comparable speed level results in consistently and significantly reduced mean values ( p < 0.05) for the loaded walking as compared with those of the unloaded walking (Figure 5A and Table 1). These DGM values represent the combined effects of the step lengths (Figure 3B) and the SS duration ratios (Figure 3C), as well as the inertia effects, on the balance stability by explicitly incorporating them in the formulation. Although the RMS d represents the inertia effects in a given gait, it alone does not take into account the temporal and spatial variables

and d
DS max
, which also affect the overall balance stability of a gait cycle.
Those variables depend on the given gait strategy, but not necessarily only on d . Consequently, for any given speed level, the loaded walking (with ~35% of the body weight in the current experiments) results in a more cautious and guarded gait strategy through increased

, reduced
62

d , and decreased DS d max
(or step length), which manifest themselves as the decreased DGMs.
These results may be related to the effects of coordinated movements on loaded walking [1,2,6], such as foot placement and angular momentum regulation, which are critical to controlling dynamic balance and avoiding falling [11,39,40].
It should be noted here that the DGM is not a direct indicator of the risk of falling. The likelihood of falling would depend on many factors including various ground reference points [22,41], motor control [42], subject’s strength [43], initial and boundary conditions (such as position, velocity, and environmental constraints) [27,37,44-48], etc., which are also related to each other.
Satisfying each of these factors can be a necessary, but not a sufficient, condition to avoid falling
[27]. In this regard, the balance stability characteristics quantified by the DGM are related to the necessary conditions to avoid falling during gait, but are not direct predictors of the fall risk, since static instability does not necessarily imply falling. Rather, the physical interpretation of the
DGM can be well illustrated using its analogy with the rippling of water in a cup. In other words, the DGM value for a given gait is equivalent to, for example, the amount of water that spills out of a cup carried while walking, where the height of the cup and the ripples of the water are analogous to the FSR dimensions in the sagittal plane and the inertia force applied during the gait, respectively. In this analogy, the amount of the spilt water ( the gait static balance instability) depends not only on the rippling of the water inside the cup ( inertia force during the gait, represented by d ), but also on the height of the cup ( the FSR or foot dimensions and step lengths). This dependence of the DGM on the FSR is consistent with the previous simulation study with various gait parameters and foot dimensions [34]. Accordingly, no spilt water due to small ripples compared with the height of the cup indicates static (statically stable) walking, during which the GCOM, as well as the ZMP, trajectories are bounded within the FSR. The above results demonstrate that the combination of the inertia effects and other influencing factors in the gait strategy results in an overall reduced static balance instability in loaded walking, as compared with the unloaded normal walking.
The unique capability of the DGM in quantifying the effects of load carriage on the balance stability of a given gait is also shown from the comparison with other existing indices—MaxFM and MOS—that have been frequently used for gait stability analysis for walking with or without load [2,9,29,37]. Unlike the DGM, the MaxFM and MOS values throughout all experiments
(Figure 4B-C) do not show any statistically meaningful difference ( p > 0.05) between the loaded and unloaded walking motions (Figure 4). The unique ability of the DGM to discriminate loaded walking from unloaded normal walking is also evident within each subjective level of walking speed. The DGM is the only index that is consistently informative of the presence of the added load (Figure 5A), while the MaxFM (Figure 5B) and MOS (Figure 5C) do not show statistically meaningful differences within any speed level (Table 1). This is partially due to the distinct and complementary scopes of the gait stability measured by these indices. As mentioned above, the
MaxFM measures the rate of convergence/divergence from limit cycle trajectories due to small perturbations from one cycle to the next. The MaxFM is mainly used from dynamical systems perspectives to assess the stability of strictly periodic theoretical systems [11], such as passive dynamic walkers under the periodic gait assumption [19]. Although the MaxFM has been analyzed for the stability of loaded walking, its ability to discriminate from unloaded normal walking is generally dependent on the magnitude and location of the load and the gait phase [9], and does not reflect the overall time-varying changes in gait stability. Also, it has been shown
63

that the orbital gait stability in the sagittal plane is preserved during walking with a load up to
30% of the body weight [49]. On the other hand, the MOS provides a measure of dynamic stability from balance perspectives and is based on ground reference points. However, unlike the
DGM, it is not a normalized index, measures only time-local aspects of stability (i.e., dependent on specific gait event or phase considered for the FSR), and therefore is not indicative of the overall balance stability for a complete gait cycle in a time-global sense [28,29]. For these reasons, the MOS concept, which can be applied in principle to both anterior-posterior and medio-lateral stability, proves less useful in practical analyses of the sagittal plane stability [11].
In summary, the experimental results show the validity of the DGM as a single comprehensive index of the balance stability characteristics of a given gait, in which the inertia effect (RMS d ), the time-varying FSR, and the temporal and spatial gait parameters are taken into account concurrently. Such formulation of the DGM proves its capability to properly and uniquely reflect the changes in the walking balance stability induced by load carriage. The comparison with other existing indices also shows that the static balance instability (measured by the DGM) is most affected by the loaded conditions among the aforementioned various aspects of gait stability.
ACKNOWLEDGMENTS
The authors would like to thank Keonyoung Oh at KAIST for his help in data collection.
64

65

Planning optimal walking motion is still a difficult task due to its complex dynamics under unilateral constraints, and its high kinematic and kinetic redundancy. For this reason, today the locomotion research leaves open a number of unsolved research questions in the framework of the optimal motion planning and control. Trajectory planning of walking motion is indeed a typical examples of finding the optimal motion of a complex system periodically in contact with the environment, where the contact forces exchanged between the foot and the ground greatly affect the system dynamics, being the only external forces acting on a biped system, along with gravity and inertia forces. The objective of this problem is to detect and optimize the contact events and their relative magnitudes of redundant systems within the optimization sequential problem, through a fully predictive algorithm. The approach is to obtain the information on the contact dynamics by looking at the dual variables resulting from the NLP and relating them to the lagrange multipliers that represent the contact forces in analytical dynamics of constrained systems. With this approach, for example, we should be able to predict the “on/off” contact condition of two rigid bodies, rather than assigning it by means of bilateral constraints, as it is currently done. The results of this problem will be applied to the human walking, balance, falling simulation. With this extremely novel approach, the resulting formulation will give a substantial contribution in the development of more and more reliable computational framework for the simulation of redundant systems’ dynamics and their interaction with the environment or possible interconnected systems.
66

4.1 OBJECTIVE AND METHOD
Optimal motion planning refers to the problem of solving the dynamics of a redundant system when both motion and the forces that generate it are unknowns. The problem is solved as an optimization problem, through the minimization of a given cost function, resulting in an optimal local solution. In particular, in our approach we are concerned in designing a direct method for the optimal motion planning (as opposed to the indirect optimal control methods), where the trajectory planning problem is translated into a finite dimension constrained Nonlinear
Programming (NLP). In our optimal motion planning formulation the objective is to detect and optimize the contact events and their relative magnitudes within the NLP problem, in a fully predictive algorithm, without requiring a priori information on the events of contact. Therefore we look for an algorithm that concurrently solve for the motion, generalized forces and contact dynamics, resulting in an optimal motion that closely matches the results observed in the real physics.
The original optimal motion planning formulation can be schematized as follows:
Existing approaches to include the contact dynamics in the equations of motion are limited to non fully predictive techniques, which require the knowledge a priori of some characteristics of contact events (time of contact, point of contact, and so on…). On the other hand, our method will allow to design a NLP algorithm corresponding to the original problem from above to concurrently solve for motion, actuation, and contact dynamics. Based on a direct approach for optimal control, we utilize third order B-spline curves for the parameterization of the kinematic and dynamic functions: j
 nc

1  i

0
N i ,3
( ) t initial t t final that allows to define the new discrete optimization variables, i.e. the B-spline control vertex x.
The problem is in this manner translated into a NLP and implemented into a SQP solver, which will provide solution estimates on x and on the dual variables at each major iteration. Within this
67

framework, the novel approach will use the information provided by the dual variables to retrieve information on the contact event. So the objective is to formulate a mapping between the lagrange multipliers in the NLP and the lagrange multipliers in the dynamic problem.
τ 
[ q
( k
( k
))]
T
( k
)
The resulting algorithm will be fully predictive algorithm and will detect the contact events
(where?, when?, how many?) and their magnitudes, without any a priori knowledge.
4.2 PRELIMINARY EXAMPLES
The preliminary results are based on simple static examples, as proof-of-concept of the proposed method and are summarized in the following figure:.
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Future work will see further applications of the biomechanical indexes for gait passivity and dynamicity. As far as the second topic is concerned, the research goal is to extend the formulation of contact detection and optimization, i.e. the relationship between dual variables and lagrange multipliers, to the most general case with general cost function and constraints forms.
Eventually the two topics of bipedal locomotion and optimal motion planning through contacts will result highly interconnected when implementing the new method of contact detection in optimal control to plan the walking motion. The solution resulting from the optimal motion planning could be indeed validated with the proposed biomechanical indexes and with experiments, in order to understand how far the simulated results are from the motion observed in the real physics.
69

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