Duke University Press has signed a major agreement with ISTEX, a French national licensing program, to make the *Duke Mathematical Journal* (*DMJ*) available to French research institutions.

With this agreement, 112 volumes of content from *DMJ *are made available to millions of users at over 330 French universities, grande écoles, research institutes, and libraries. Published by Duke University Press since its inception in 1935, *DMJ *is one of the world’s leading mathematical journals. *DMJ *emphasizes the most active and influential areas of current mathematics and has several distinguished French mathematicians on its editorial board. The journal has published work by the Fields Medalists Cédric Villani, Ngô Bảo Châu, Jean-Pierre Serre, and Artur Avila.

Since 2012, ISTEX has facilitated the massive acquisition of archives of scientific and mathematical production in all disciplines made available to public institutions of higher education and research in France on one central platform.

David Aymonin, Head of the Bibliographic Agency for Higher Education, says, “All the four partners leading ISTEX are happy to add quality mathematics content from the *Duke Mathematical Journal* to the ISTEX initiative through our arrangement with Duke University Press. We see this partnership as bringing important mathematics scholarship to more researchers throughout France.”

“The Press is delighted to have the opportunity to participate in the ISTEX project by offering content from the *Duke Mathematical Journal*, including eminent French scholars, for use by mathematicians throughout France. We are grateful to TSP Diffusion for their support,” says Cason Lynley, Director of Marketing and Sales at Duke University Press.

*DMJ *content is available to ISTEX institutions on the Project Euclid platform. Read the full announcement.

### Like this:

Like Loading...

here we consider some as say prof dr mircea orasanu and prof horia orasanu as followings

LAGRANGIAN MECHANICS PRINCIPLES

3 RESULTS OF CONSIDERATION

This paper has considered path following control of planar snake robots by using virtual holonomic constraints. The equations of motion of the snake robot were derived using a Lagrangian framework. We then introduced virtual holonomic constraints that defined the geometry of a constraint manifold for the robot. We showed that the constraint manifold can be made positively invariant by a suitable choice of feedback, and we designed an input-output feedback linearizing control law to exponentially stabilize the constraint manifold for the system. We presented simulation and experimental results which validated the theoretical design. In particular, the robot successfully converged to and followed a desired straight path.

As a topic of future work, we aim to prove the practical stability of the desired path with the proposed control approach. Furthermore, a formal proof for boundedness of the solutions of the dynamic compensator remains as a topic of future work. Moreover, application of the proposed control strategy for more complex paths such as curved paths, using different path following guidance laws, remains as a topic of future work.

Complex systems are large interdisciplinary research topics that have been studied by means of a mixed basic theory that mainly derives from physics and computer simulation. Such systems are made of many interacting elementary units that are called “agents”.

The way in which such a system manifests itself cannot be exclusively predicted only by the behavior of individual elements. Its manifestation is also induced by the manner in which the elements relate in order to influence global behavior. The most significant properties of complex systems are emergence, self-organization, adaptability, etc. [1–4].

Examples of complex systems can be found in human societies, brains, the Internet, ecosystems, biological evolution, stock markets, economies and many others [1, 2]. Particularly, polymers are examples of such complex systems. Their forms include a multitude of organizations starting from simple, linear chains of identical structural units and ending with very complex chains consisting of sequences of amino acids that form the building blocks of living fields. One of the most intriguing polymers in nature is DNA, which creates cells by means of a simple, but very elegant language. It is responsible for the remarkable way in which individual cells organize into complex systems, such as organs, which, in turn, form even more complex systems, such as organisms. The study of complex systems can offer a glimpse into the realistic dynamics of polymers and solve certain difficult problems (protein folding) [1–4].

Correspondingly, theoretical models that describe the dynamics of complex systems are sophisticated [1–4]. However, the situation can be standardized taking into account that the complexity of interaction processes imposes various temporal resolution scales, while pattern evolution implies different freedom degrees [5].

In order to develop new theoretical models, we must admit that complex systems displaying chaotic behavior acquire self-similarity (space-time structures seem to appear) in association with strong fluctuations at all possible space-time scales [1–4]. Then, in the case of temporal scales that are large with respect to the inverse of the highest Lyapunov exponent, the deterministic trajectories are replaced by a collection of potential trajectories, while the concept of definite positions by that of probability density. One of the most interesting examples is the collision process in complex systems, a case in which the dynamics of the particles can be described by non-differentiable curves.

Since non-differentiability appears as the universal property of complex systems, it is necessary to construct a non-differentiable physics. Thus, the complexity of the interaction processes is replaced by non-differentiability; accordingly, it is no longer necessary to use the whole classical “arsenal” of quantities from standard physics (differentiable physics).

This topic was developed within scale relativity theory (SRT) [6,7] and non-standard scale relativity theory (NSSRT) [8–22]. In this case, we assume that the movements of complex system entities take place on continuous, but non-differentiable, curves (fractal curves), so that all physical phenomena involved in the dynamics depend not only on space-time coordinates, but also on space-time scale resolution. From such a perspective, physical quantities describing the dynamics of complex systems may be considered fractal functions [6,7]. Moreover, the entities of the complex system may be reduced to and identified with their own trajectories, so that the complex system will behave as a special fluid lacking interaction (via their geodesics in a non-differentiable (fractal) space). We have called such fluid a “fractal fluid” [8–22].

In the present paper, we shall introduce new concepts, like non-differentiable entropy, informational non-differentiable entropy, informational non-differentiable energy, etc., in the NSSRT approach (the scale relativity theory with an arbitrary constant fractal dimension). Based on a fractal potential, which is the “source” of the non-differentiability of trajectories of the complex system entities, we establish the relationships among non-differentiable entropy. The correlation fractal potential-non-differentiable entropy implies uncertainty relations in the hydrodynamic representation, while the correlation of informational non-differentiable entropy/informational non-differentiable energy implies specific uncertainty relations through a maximization principle of the informational non-differentiable entropy and for a constant value of the informational non-differentiable energy. The constant value of the informational non-differentiable energy made explicit for the harmonic oscillator induces a quantification condition. We note that there exists a large class of complex systems that take smooth trajectories. However, the analysis of the dynamics of these classes is reducible to the above-mentioned statements by neglecting their fractality.

Head angle tracking error in simulations. The head angle tracking error converges exponentially to zero.

Joint angles tracking the reference joint angles in simulations. The joints of the robot track the sinusoidal motions (above). The joint tracking errors converge exponentially to zero (below).

Figure 5

Figure 6

The motion of the center of mass in the x – y plane in simulations. The position of the CM of the robot (blue) converges to and follows the desired straight line path (the x-axis).

Figure 7

Joint angles tracking the reference joint angles in simulations in the presence of measurement noise. The joints of the robot track the sinusoidal motions in the presence of measurement noise (above). The joint tracking errors converge exponentially .

The experiment was carried out using the snake robot Wheeko [16]. The robot, which is shown in Figure Figure2,2, has 10 identical joint modules, i.e. N=11 links. Each joint module is equipped with a set of passive wheels which give the robot anisotropic ground friction properties during motion on flat surfaces. The wheels are able to slip sideways and thus do not introduce nonholonomic velocity constraints in the system. Each joint is driven by a Hitec servo motor (HS-5955TG; Hitec RCD USA, Inc., Poway, CA, USA), and the joint angles are measured using magnetic rotary encoders. The motion of the snake robot was measured using a camera-based motion capture system from OptiTrack of type Flex 13 (NaturalPoint, Inc., Corvallis, OR, USA). The system consists of 16 cameras which are sampled at 120 frames per second and which allow reflective markers to be tracked on a submillimetre level. During the experiment, reflective markers were mounted on the head link of the snake robot in order to measure the position (px,N,py,N) and orientation (θN) of the head. These measurements were combined with the measured joint angles (q1,…,qN−1) of the snake robot in order to measure the absolute link angles (1) and the position of the CM (px,py) of the robot. In order to obtain the derivatives of the reference head angle (46), we used the same technique as in the simulations, i.e. passing ΦN through a low-pass filter of the form (67). The parameters of the low-pass filter were set to ωn=π/2 and ψf=1.

In the following, we elaborate on a few adjustments that were made in the implemented path following controller in order to comply with the particular properties and capabilities of the physical snake robot employed in the experiment. We conjecture that these adjustments only marginally affected the overall motion of the robot. The successful path following behaviour of the robot demonstrated below supports this claim. Since the experimental setup only provided measurements of the joint angles and the position and orientation of the head link, we chose to implement the joint controller in (57) as

𝜗i = −kpyi

(68)

where i∈{1,…,10}. We conjecture that eliminating the joint angular velocity terms from (57) did not significantly change the dynamic behaviour of the system since the joint motion was relatively slow during the experiment. The main consequence of excluding the velocity terms from (57) is that we potentially introduce a steady-state error in the tracking of the joint angles. Consequently, since with the joint control law (68) the derivative terms in (63) are identically zero, they need not to be linearized in the head angle dynamics by the dynamic compensator. As the result, we implemented the dynamic compensator of the form

ϕ¨o=(∑N−1i=1βi)−1(−fθN+d2ΦN−kp,θNyN−kd,θNy˙N)−kpϕo−kdϕ˙o

(69)

where the controller gains were kp,θN = 20, kd,θN = 1, kp=10, and kd=5. We saturated the joint angle offset ϕo according to ϕo∈[−π/6,π/6], in order to keep the joint reference angles within reasonable bounds w.r.t the maximum allowable joint angles of the physical snake robot. Moreover, from Figure Figure2,2, it can be seen that the head link of the physical snake robot does not touch the ground since the ground contact points occur at the location of the joints. As a results, we implemented (69) with fθN ≡ 0. The solutions of the dynamic compensator (69) were obtained by numerical integration in LabVIEW which was used as the development environment.

We chose the look-ahead distance of the path following controller as Δ=1.4 m. The initial values for the configuration variables of the snake robot were qi=0 rad, θN=−π/2 rad, px=0.3 m, and py=1.7 m, i.e. the snake robot was initially headed towards the desired path (the x-axis), and the initial distance from the CM to the desired path was 1.7 m. Furthermore, the parameters of the constraint functions for the joint angles, i.e. (45), were α=π/6, η=70πt/180, and δ=36π/180, while the ground friction coefficients were ct=1 and cn=10 (i.e identical to the simulation parameters).Figure 12

Comparison between experiments and simulations. Comparison of the convergence of the cross-track error during simulations (red) and experiments (blue).

Figure 13

An image of the motion of the robot during the experiments. The robot converges to and follows the desired path.

2. Hallmarks of Non-Differentiability

Let us assume that the motion of complex system entities takes place on fractal curves (continuous, but non-differentiable). A manifold that is compatible with such movement defines a fractal space. The fractal nature of space generates the breaking of differential time reflection invariance. In such a context, the usual definitions of the derivative of a given function with respect to time [6,7],

dFdt=limΔt→0+F(t+Δt)−F(t)Δt=limΔt→0−F(t)−F(t−Δt)Δt

(1)

are equivalent in the differentiable case. The passage from one to the other is performed via Δt → − Δt transformation (time reflection invariance at the infinitesimal level). In the non-differentiable case, (dQ+dt) and (dQ−dt) are defined as explicit functions of t and dt,

dQdt+limΔt→0+Q(t,t+Δt)−Q(t,Δt)Δt

and:

dQdt=limΔt→0−Q(t,Δt)−Q(t,t−Δt)Δt

(2)

The sign (+) corresponds to the forward process, while (−) corresponds to the backward process. Then, in space coordinates dX, we can write [6,7]:

dX±=dx±+dξ±=v±dt+dξ±

(3)

with v± the forward and backward mean speeds,

v+=dx+dt=limΔt→0+⟨X(t+Δt)−X(t)Δt⟩v−=dx−dt=limΔt→0−⟨X(t)+X(t−Δt)Δt⟩

(4)

and dξ± a measure of non-differentiability (a fluctuation induced by the fractal properties of trajectory) having the average:

⟨dξ±⟩=0,

(5)

where the symbol 〈〉 defines the mean value.

While the speed-concept is classically a single concept, if space is a fractal, then we must introduce two speeds (v+ and v−), instead of one. These “two-values” of the speed vector represent a specific consequence of non-differentiability that has no standard counterpart (according to differential physics).

However, we cannot favor v+ as compared to v−. The only solution is to consider both the forward (dt > 0) and backward (dt < 0) processes. Then, it is necessary to introduce the complex speed [6,7]:

Vˆ=v++v−2−iv+−v−2=dx++dx−2dt−idx+−dx−2dt=VD−iVF,VD=v++v−2,VF=v+−v−2

(6)

If VD is differentiable and resolution scale (dt) speed independent, then VF is non-differentiable and resolution scale (dt) speed dependent.

Using the notations dx± = d±x, Equation (6) becomes:

Vˆ=(d++d−2dt−id+−d−2dt)x

(7)

This enables us to define the operator:

dˆdt=d+−d−2dt−id+−d−2dt

(8)

Let us now assume that the fractal curve is immersed in a three-dimensional space and that X of components Xi (i = 1, 2, 3) is the position vector of a point on the curve. Let us also consider a function f(X, t) and the following series expansion up to the second order:

df=f(Xi+dXi,t+dt)−f(Xi,dt)=(∂∂XidXi+∂∂tdt)f(Xi,t)+12(∂∂XidXi+∂∂tdt)2f(Xi,t)

(9)

Using notations, dXi±=d±Xi

, the forward and backward average values of this relation take the form:

⟨d±f⟩=⟨∂f∂tdt⟩+⟨∇f⋅d±X⟩+12⟨∂2f∂t2(dt)2⟩++⟨∂2f∂Xi∂td±Xidt⟩+12⟨∂2f∂Xi∂Xld±Xid±Xl⟩

(10)

We shall stipulate the following: the mean values of function f and its derivatives coincide with themselves, and the differentials d±Xi and dt are independent. Therefore, the averages of their products coincide with the product of averages. Thus, Equation (10) becomes:

d±f=∂f∂tdt+∇f⟨d±X⟩+12∂2f∂t2⟨(dt)2⟩++∂2f∂Xi∂t⟨d±Xidt⟩+12∂2f∂Xi∂Xl⟨d±Xid±Xl⟩

(11)

or more, using Equation (3),

d±f=∂f∂tdt+∇fd±x+12∂2f∂t2(dt)2+∂2f∂Xi∂td±xidt++12∂2f∂Xi∂Xl(d±xid±xl+⟨dξi±dξl±⟩),i,l=1,2,3,

(12)

where the quantities ⟨d±xid±ξl⟩, ⟨d±ξid±xl⟩ are null based on the Relation (5) and also on the above property referring to a product mean.

Since dξ± describes the fractal properties of the trajectory with the fractal dimension DF [23], it is natural to impose that (dξ±)DF

is proportional with resolution scale dt [6,7],

(dξ±)DF=2D−−−√dt

(13)

where D is a coefficient of proportionality (for details, see [6,7]). In Nottale’s theory [6,7], D is a coefficient associated with the transition fractal-non-fractal.

Let us focus now on the mean ⟨dξi±dξl±⟩

, which has statistical significance [6,7]. This means that at any point on a fractal path, the local acceleration, ∂tVˆ

, the non-linearly (convective) term, (Vˆ⋅∇)Vˆ, and the dissipative one, D(dt)2DF−1ΔVˆ, are in balance. Therefore, the complex system dynamics can be assimilated with a “rheological” fluid dynamics. Such a dynamics is described by the complex velocity field Vˆ, by the complex acceleration field dˆVˆdt, etc., as well as by the imaginary viscosity type coefficient iD(dt)2DF−1

.

For irrotational motions of the complex system entities:

∇×Vˆ=0,∇×VD=0,∇×VF=0

(21)

Vˆ can be chosen with the form:

Vˆ=−2iD(dt)2DF−1∇lnψ

(22)

where ϕ = ln ψ is the velocity scalar potential. Substituting (22) in (20), we obtain:

dˆVˆdt=−2iD(dt)(2DF)−1[∂∂t−2iD(dt)(2DF)−1(∇lnψ)⋅∇−iD(dt)(2DF)−1Δ](∇lnψ)=0

or more:

dˆVˆdt=−2iD(dt)(2DF)−1{∂∂t(∇lnψ)−i[2D(dt)(2DF)−1(∇lnψ⋅∇)+(∇lnψ)+D(dt)(2DF)−1Δ(∇lnψ)]}=0

(23)

Using the identities [7]:

(∇lnψ)2+Δlnψ=Δψψ∇(Δψψ)=2(∇lnψ⋅∇)(∇lnψ)+Δ(∇lnψ)

the Equation (23) becomes:

dˆVˆdt=−2iD(dt)(2DF)−1∇[∂∂tlnψ−iD(dt)(2DF)−1Δψψ].

This equation can be integrated up to an arbitrary phase factor, which may be set to zero by a suitable choice of phase of ψ and this yields:

D2(dt)(4DF)−2Δψ+iD(dt)(2/DF)−1∂ψ∂t=0.

(24)

Relation (24) is a Schrödinger-type equation. For motions of complex system entities on Peano’s curves, DF = 2, Equation (24) takes the Nottale’s form [6,7]. Moreover, for motions of complex system entities on Peano’s curves at the Compton scale, D=h2m0

(for details, see [6,7]), with ħ the reduced Planck constant and m0 the rest mass of the complex system entities, Relation (24) becomes the standard Schrödinger equation.

If ψ=ρe−−√iS

, with ρ√

the amplitude and S the phase of ψ, the complex velocity field (22) takes the explicit form:

Vˆ=2D(dt)2DF−1∇S−iD(dt)2DF−1∇lnρVD=2D(dt)2DF−1∇SVF=D(dt)2DF−1∇lnρ

(25)

Substituting (25) into (20) and separating the real and the imaginary parts, up to an arbitrary phase factor, which may be set to zero by a suitable choice of the phase of ψ, we obtain:

∂VD∂t+(VD⋅∇)VD=−∇Q∂ρ∂t+∇⋅(ρVD)=0

(26)

with Q the specific fractal potential (specific non-differentiable potential):

Q=−2D2(dt)4DF−2Δρ√ρ√=−V2F2−D(dt)2DF−1∇⋅VF

(27)

The specific fractal potential can simultaneously work with the standard potentials (for instance, an external scalar potential).

The first Equation (26) represents the specific momentum conservation law, while the second Equation (26) exhibits the state density conservation law. Equations (26) and (27) define the fractal hydrodynamics model (FHM).

The following conclusions are obvious:

• Any entity of the complex system is in permanent interaction with the fractal medium through a specific fractal potential.

• The fractal medium is identified with a non-relativistic fractal fluid described by the specific momentum and state density conservation laws (probability density

LikeLike