Mathematical models of exosceleton. Dynamics, strength, control (Q2122118)

The present book deals with several aspects from biomechanics. First seven chapters address the contact problem for joints and the strain and stress distributions for bones. Remaining chapters address the human body segmental dynamics. The ordinary differential equations (ODE) systems of the motion equations are obtained by the Lagrange method, taking the kinetic and potential energies into account. In detail, the book is divided into 15 chapters. Each of the 15 chapters ends with references. Chapter 1 (\textit{Statistical model of destruction of a human musculoskeletal system}) concerns with the kinematics of a crack propagation. While in the one-dimensional case, the analytical and numerical solutions of the Fokker-Planck-Kolmogorov (FPK) equation are presented, in the two-dimensional and three-dimensional cases, the FPK equation is numerically solved. Hence, the probability of the crack to reach the joint boundary is evaluated and discussed. It is also estimated the mathematical expectation of the cracks size along each of the coordinates and the time of its occurrence. Chapter 2 (\textit{A model of changing of physical-mechanical properties and accumulations of damages in joints of a person}) deals with the contact problem of interaction of spherical surfaces modeling members of a hip joint of a human being. The authors consider a modified Hertz problem to study the influence of damage accumulation in the contacting bodies on the characteristics of the site of the contact. Chapter 3 (\textit{Propagation of nonstationary waves in elastic inhomogeneous media}) deals with the ray method. The eikonal equation in terms of the phase function is derived from the PDE system \[ \sigma_{,x}= \rho V_{,t}\quad \sigma_{,t}= E V_{,x}, \] where the stress \(\sigma\) obeys the Hooke law, \(\rho\) stands for the density, \(E\) stands for the Young modulus, and \(V\) denotes the particle velocity. Then, the phase function is explicitly determined in the semispace \(x\geq 1\). Next, a system is derived to describe the change in the wave intensity (vortex free and equivoluminal) in an inhomogeneous elastic medium with initial stresses, the parameters of the internal geometry of the wave front, and the beam along which it occurs. In Chapter 4 (\textit{Model of propagation of the impact impulse in the human lower extremity}), some considerations are made about the limb deformability that should be included in the coupled system of equations of motion describing the leg-segmental dynamics. Chapter 5 (\textit{Problems of strength at loading multilayer bones of the human}) deals with the bending moments for (1) a two-sheeted tubular bone, under a concentrated load at the cross section, with an internal layer consisting of a spongy soft substance, and (2) a trizonal rod representing a three layered limb, constituted by bone, periosteum, and muscle, under stretching. Chapter 6 (\textit{Determination of deformations and stresses in a system consisting of an arbitrary number of thick-walled spheres under an external load}) is concerned in analyzing the radial dependence of spherical hinge-joints. First, the strain and stress tensors for the single-layer sphere model are introduced, under given external and internal pressures. Then, the \(n-1\) internal pressures are determined for the \(n\)-layer hollow sphere model, under external pressure. Two practical examples, for the uniform ball (\(n=1\)) and for 2-layer sphere model, are given with selected numerical parameters from the literature. Chapter 7 (\textit{Computation of the dimensions of the irreversible destruction of two ellipsoids in the contact zone}) deals with the Hertz problem. The area of contact and, in particular, the penetration depth are analytically and numerically studied. Chapter 8 (\textit{2-D and 3-D models of anthropomorphic robot and exoskeleton with links of variable length}) addresses the motion equations to describe the leg-segmental dynamics with links of variable length. After the authors review their work, the ODE system is introduced for a two-segmental leg model, with the knee being assumed a fixed hinge and the applied force being axial, due to the Lagrange method, taking the kinetic and potential energies, namely \(T=(m_1n_1^2+m_2)(\dot l_1^2+l_1^2\dot\varphi_1^2)/2\) and \(\Pi =(m_1n_1+m_2)gl_1\sin(\varphi_1)\), into account. The variables are the foot angle \(\varphi_1\) (the foot point being assumed fix on the ground) and the total leg length \(l_1\) to simulate extension-contraction. Next, two angles and two lengths are addressed, and subsequently the general 2D case of \(n\) angles and \(n\) segments is discussed, keeping the pin joint on the toe/ankle. Then, the above models are revisited for \(\alpha\) masses per segment to simulate musculoskeletal structure. Similarly, the 3D case is modeled. Finally, the joint moments, for the 3D model with five segments, are illustrated by the inverse dynamics analysis, considering the available data on the angular displacements. Chapter 9 (\textit{Methods for compiling differential equations of motion of an exoskeleton with variable link length}) addresses the motion equations for two models of exoskeleton units of one segment moving through one angle, where the segment is constituted by (1) one weighted rod with fixed length and a weightless section of variable length; and (2) only one rod with three masses. The \(n\)-segmental model is also analyzed. The 3D model, studied in Chapter 8, is revisited. Finally, a numerical example is given for the above two one-segmental models to illustrate the angular displacements. In Chapters 10 (\textit{Modeling the dynamics of an anthropomorphic mechanism with eleven movable links of variable length}) and 11 (\textit{Model of twelve-link exoskeleton}), approximated sagittal-plane models with constant moments of inertia are studied for 11 segments (two feet, two shanks, two thighs, two forearm-hands, two upper arms and one head-trunk) and 12 segments (using two head-trunks instead), respectively. For the 11-segmental model, the angular displacement \(\varphi_3\) (of the knee joint of the supporting leg) is graphically illustrated for both direct observation by interpolation methods and direct dynamics analysis by numerical simulation. For the 12-segmental model, the angular displacements \(\varphi_3\) and \(\varphi_4\) (of the hip joint of the portable leg) are numerically and graphically illustrated. Chapter 12 (\textit{Nonlinear dynamics of a framed structure in case of biped gait: Chaotization and self-organization}) is concerned with the 2D biped circular gait for a two-segmental model with 3 masses (two masses on each leg and the third one on the (hip) joint). The ODE system of the motion equations is obtained and linearized to \[ \begin{cases} \ddot \theta_{ns} = a_1\theta_{ns}+a_2\theta_s\\ \ddot \theta_{s} = a_3\theta_{ns}+a_4\theta_s \end{cases} \] for reaching an analytical solution. Finally, the gait cycle is analyzed. In Chapter 13 (\textit{Simulation of the synchronization of human feet and exoskeleton in the process of motion}), the author analyses the oscillation pattern of the angular displacements over time of the lower limbs that act as pendulums suspended from a common point based on a mass of the HAT unit (i.e. head-arms-trunk and pelvis). The hip joint has one degree of freedom (DOF) for translational motion in the horizontal direction. First, each leg is modeled as one segment. Second, each leg is modeled as two segments linked by the knee joint. Numerical simulations are graphically shown. Then, the generalization for an arbitrary motion, with 3 DOF, of the suspension point, linked with a rod to each 1-segmental leg, is discussed. Chapter 14 (\textit{Mathematical modeling of human body stability on one leg}) addresses the potential energy for a 2D 4-segmental model (feet, shanks, thighs and HAT), under springs located in the joints. The state stability, when standing on the feet, is studied via the Sylvester criterion. The stability areas, at one and at two fixed arguments, are graphically shown. Finally, the generalization for \(n\)-segmental model is discussed. In Chapter 15 (\textit{Synthesis of exoskeletons based on the results obtained in the simulation of the human musculoskeletal system}), the kinetic energy and the resulting ODE system are given, for one single segment \(AB\), constituted by two weighty sections with fixed lengths, \(\overline{AC}=l_{11}\) and \(\overline{DB}=l_{12}\), and an in-between weightless section with variable length \(\overline{CD}=\xi_{1}\), with firstly 1 DOF and secondly 2 DOF motion over a pin joint on the toe/ankle. Next, the toe/ankle is assumed to move either in the corresponding 2D or 3D case. Finally, the inverse dynamics analysis as well as the numerical simulation to direct dynamics analysis are both extended to a 5-segmental model in 2D. This book is intended to young researchers in biomechanics.