Seminar at Loughborough University

also we cab consider as that say prof dr mircea orasanu and prof horia orasanu the followings
LAGRANGIAN AND MECHANICS
Author Horia Orasanu
ABSTRACT
There are many instances in which the basic physics is known (or postulated), and the behavior of a complex system is to be determined

1 INTRODUCTION
A typical example is that in which there are too many particles for the problem to be tractable in terms of single-particle equations, and too few for a statistical analysis to apply. In such situations, use of a computer may furnish information on enough specific cases for the general behavior of the system to be discernable. If the basis physics is postulated but not known, a computer simulation can relate the theory to observations on complex systems and thus test the theory.
A wide range of numerical techniques are available, from simple searches to sophisticated methods, such as annealing, an algorithm for finding global minima which was inspired by an actual physical process. Some of the techniques are repetitive applications of deterministic equations. Others invoke stochastic processes (using “random” numbers), to focus on the important features.
While most applications of such simulations yield expected results, surprises do occur. This is analogous to an unexpected result from an experiment. Either the simulation/experiment went wrong (usual) or a new aspect of nature has been uncovered (rare). Examples of the latter are the identification of constants of motion in chaotic systems and the discovery of runaway motion in the drift and diffusion of ions in gas. Such discoveries are followed by “proper” theories and “proper” experiments, but the computer plays a vital role in the research.
The research of Professor Gatland involves data analysis and the mathematical modeling and simulation of microscopic physical processes. These activities encompass both research and instruction.
In this lecture we recall the definitions of autonomous and non autonomous Dynamical Systems as well as their different concepts of attractors. After that we introduce the different notions of robustness of attractors under perturbation (Upper semicontinuity, Lower semicontinuity, Topological structural stability and Structural stability) and give conditions on the dynamical systems so that robustness is attained. We show that enforcing the appropriately defined virtual holonomic constraints for the configuration variables implies that the robot converges to and follows a desired geometric path. Numerical simulations and experimental rMethods
In this section, we derive the kinematic model along with the dynamic equations of motion of the snake robot in a Lagrangian framework. Moreover, we use partial feedback linearization to write the model in a simpler form for model-based control design.
In order to perform control design, we need to write the governing equations of the system in an implementable way. This is often done by choosing a local coordinate chart and writing the system equations with respect to (w.r.t.) these coordinates. According to the illustration of the snake robot in Figure Figure1,1, we choose the vector of the generalized coordinates of the N-link snake robot as x = [q1,q2,…,qN−1,θN,px,py]T ∈ ℝN+2, where qi with i∈{1,…,N−1} denotes the ith joint angle, θN denotes the head angle, and the pair (px,py) describes the position of the CM of the robot w.r.t. the global x−y axes. Since the robot is not subject to nonholonomic velocity constraints, the vector of the generalized velocities is defined as x˙=[q˙1,q˙2,…,q˙N−1,θ˙N,p˙x,p˙y]T∈RN+2. Using these coordinates, it is possible to specify the kinematic map of the robot. In this paper, we denote the first N elements of the vector x, i.e. (q1,…,qN−1,θN), as the angular coordinates, and the corresponding dynamics as the angular dynamics of the system.

Figure 1
An illustration of the N -link snake robot. Kinematic parameters of the snake robot.

This is due to the fact that it is usually not straightforward to integrate the anisotropic external dissipative forces, i.e. ground friction forces, acting on these complex robotic structures into their Euler-Lagrange equations of motion. However, ground friction forces have been proved to play a fundamental role in snake robot locomotion (see, e.g. [16]). In this paper, we derive the equations of motion of the snake robot in a Lagrangian framework, i.e. treating the robot as a whole and performing the analysis using a Lagrangian function, which is simple to follow and better suited for studying advanced mechanical phenomena such as elastic link deformations [25], which might be insightful for future research challenges on snake robots. Moreover, we integrate the anisotropic friction forces into these equations using the Jacobian matrices of the links, which gives a straightforward mapping of these forces for the equations of motion.
Snake robots are a class of simple mechanical systems, where the Lagrangian L(qa,x˙) is defined as the difference between the kinetic energy K(qa,x˙) and potential energy 𝒫(x) of the system [26]. Since the planar snake robot is not subject to any potential field, i.e. −∇𝒫(x) = 0, we may write the Lagrangian equal to the kinetic energy, which is the sum of the translational and the rotational kinetic energy of the robot:
L(qa,x˙)=K(qa,x˙)=12m∑i=1N(p˙2x,i+p˙2y,i)+12J∑i=1Nθ˙2i
(9)
where m and J denote the uniformly distributed mass and moment of inertia of the links, respectively. Using the Lagrangian function (9), we write the Euler-Lagrange equations of motion of the controlled system as
ddt[∂L(qa,x˙)∂x˙i]−∂L(qa,x˙)∂xi=(B(x)τ−τf)i
(10)
where i∈{1,…,N+2}, B(x) = [ej] ∈ ℝ(N+2)×(N−1) is the full column rank actuator configuration matrix, where ej denotes the jth standard basis vector in ℝN+2. Moreover, B(x)τ ∈ ℝN+2 with τ = [τ1,…,τN−1]T ∈ ℝN−1 stands for the generalized forces resulting from the control inputs. Furthermore, τf=[τ1f,…,τN+2f]T∈RN+2 denotes viscous and Coulomb friction forces acting on (N+2) DOF of the system. The controlled Euler-Lagrange equations (10) can also be written in the form of a second-order differential equation as
M(qa)x¨+C(x,x˙)x˙=B(x)τ−τf
(11)
where M(qa) ∈ ℝ(N+2)×(N+2) is the positive definite symmetric inertia matrix, C(x,x˙)x˙∈RN+2 denotes the generalized Coriolis and centripetal forces, and the right-hand side terms denote the external forces acting on the system. The fact that the inertia matrix is only a function of the directly actuated shape variables qa is a direct consequence of the invariance of the Lagrangian
The geometry of the problem
The (N+2)-dimensional configuration space of the snake robot is denoted as 𝒬 = 𝒮 × 𝒢, which is composed of the shape space and a Lie group which is freely and properly acting on the configuration space. In particular, the shape variables, i.e. qa=(q1,…,qN−1), which define the internal configuration of the robot and which we have direct control on, take values in . Moreover, the position variables, i.e. qu=(θN,px,py), which are passive DOF of the system, lie in . The velocity space of the robot is defined as the differentiable (2N+4)-dimensional tangent bundle of as T𝒬 = 𝕋N × ℝN+4, where 𝕋N denotes the N-torus in which the angular coordinates live. The free Lagrangian function of the robot ℒ:T𝒬 → ℝ is invariant under the given action of on . The coupling between the shape and the position variables causes the net displacement of the position variables, according to the cyclic motion of the shape variables, i.e. the gait pattern. Note that for simplicity of presentation, in this paper, we consider local representation of T𝒬 embedded in an (2N+4)-dimensional open subset of the Euclidean space ℝ2N+4. To this end, we separate the dynamic equations of the robot given by (11) into two subsets by taking x = [qa,qu]T ∈ ℝN+2, with qa ∈ ℝN−1 and qu ∈ ℝ3 which were defined in the subsection describing the geometry of the problem:
m11(qa)q¨a+m12(qa)q¨u+h1(x,x˙)=ψ∈RN−1
(20)
m21(qa)q¨a+m22(qa)q¨u+h2(x,x˙)=03×1∈R3
(21)
where m11 ∈ ℝ(N−1)×(N−1), m12 ∈ ℝ(N−1)×3, m21 ∈ ℝ3×(N−1), and m22 ∈ ℝ3×3 denote the corresponding submatrices of the inertia matrix, and 03×1 = [0,0,0]T ∈ ℝ3. Furthermore, h1(x,x˙)∈RN−1 and h2(x,x˙)∈R3 include all the contributions of the Coriolis, centripetal, and friction forces. Moreover, ψ ∈ ℝN−1 denotes the non-zero part of the vector of control forces, i.e. B(x)τ = [ψ,03×1]T ∈ ℝN+2. From (21), we have
q¨u=−m−122(h2+m21q¨a)∈R3
(22)
Substituting (22) into (20) yields
(m11−m12m−122m21)q¨a−(m12m−122)h2+h1=ψ
(23)
For linearizing the dynamics of the directly actuated DOF, we apply the global transformation of the vector of control inputs as
ψ=(m11−m12m−122m21)ϑ−(m12m−122)h2+h1
(24)
where 𝜗 = [𝜗1,𝜗2,…,𝜗N−1]T ∈ ℝN−1 is the vector of new control inputs. Consequently, the dynamic model (20)-(21) can be written in the following partially feedback linearized form
q¨a=ϑ∈RN−1
(25)
q¨u=D(x,x˙)+C(qa)ϑ∈R3
(26)
with
D(x,x˙)=−m−122(qa)h2(x,x˙)=[fθN,fx,fy]T∈R3
(27)
C(qa)==−m−122(qa)m21(qa)[βi(qa),0,0]T∈R3×(N−1)
(28)
Figure 3
The states of the dynamic compensator in simulations. The states of the dynamic compensator remain bounded.

Figure 4
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
Head angle tracking error in simulations. The head angle tracking error converges exponentially to zeroIn 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).
Results and discussion
The results of the experiments are illustrated in Figures Figures8,8, ,9,9, ,10,10, ,11,11, ,12,12, ,13.13. In particular, Figure Figure88 shows that the solution of the dynamic compensator remained bounded. Figure Figure99 shows that the joints of the robot tracked the sinusoidal reference angles provided by the constraint functions (45) and that the tracking error converged to a neighbourhood of the origin. As discussed above, this is probably due to the modification of the joint controller (68) due to the lack of velocity measurements in the lab. Figure Figure1010 shows that the head angle of the robot tracked the reference head angle defined by the constraint function (46) and that the tracking error converged to a neighbourhood of the origin. Figure Figure1111 shows the motion of the CM of the robot in the x-y plane, which converged to and followed the desired path. Figure Figure1212 compares the motion of the CM during the simulation and the experiment, which were performed using the same controller parameters in order to obtain comparable results. In particular, from Figure Figure12,12, it can be seen that the physical snake and the simulated snake follow almost the same path. However, due to precise measurement and a more accurate joint control law for the simulated snake, the path following error converges to a smaller neighbourhood of the origin. Figure Figure1313 shows screenshots from a video recording of the experiment.

Figure 8
The solution of the dynamic compensator in experiments. The solution of the dynamic compensator remains bounded during the experiments.

Figure 9
Joint angles tracking the reference joint angles during the experiments. The joints of the robot track the sinusoidal motions (above). The tracking errors converge to a neighbourhood of zero during the experiment (below).

Figure 10
Head angle tracking error during the experiments. The head angle tracks the reference head angle (above). The head angle tracking error converges to a neighbourhood of zero (below).

Figure 11
The motion of the center of mass in the x – y plane during the experiments. The position of the CM of the robot (blue) converges to and follows the desired straight path (the x-axis).

Figure 12
Comparison between experiments and simulations. Comparison of the convergence of the cross-track error during simulations (red) and experiments (blue).
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14. Ma S, Ohmam

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