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International Journal of
eISSN: 2573-2838

Biosensors & Bioelectronics

Correspondence:

Received: January 01, 1970 | Published: ,

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Abstract

The article considers an electromechanical model of an exoskeleton with three links, for which a system of differential equations of motion is written and inverse and direct problems of dynamics are solved. The angles between the links that specify anthropoid motion are figured out analytically. The electric drive controlling torques are found as a result of solving the inverse dynamics problem. The found torques are approximated with stepwise piecewise-constant functions simulating the impulse control of the exoskeleton motion. The solution of the direct problem of dynamics is carried out and the link rotation angles as functions of time are found. The results of the numerical solution of the system of differential equations are compared with the initial motion of the links. It has been found that the results of simulation with impulse control are in good agreement with the original motion. The total energy expenditures have been calculated. The exoskeleton simulation, taking into account the electric drive impacts also has been carried out. For this purpose, a system of differential equations of motion has been compiled; a numerical solution of the Cauchy problem for the compiled system has been carried out. The significance of the electric drive impact on the dynamics of the mechanism has been confirmed.

Keywords: exoskeleton, electric drives, local coordinate systems, control torques

Introduction

Designing mathematical model of three-link exoskeleton is an actual line of research, since this model can prove to be more precise and lifelike than the models with the lower number of links. A more precise model can facilitate the understanding of interaction between exoskeleton and human body. It can also help developers in designing more effective and comfortable exoskeletons. The three-link model can take into account a more complex geometry of exoskeleton links and can describe their interaction during exoskeleton motion more precisely. It can enhance the predictive capabilities of the model, and help designers in creating more accurate and comfortable exoskeletons. It can also encourage scientific research in mechanics and biomechanics, which can lead to further enhancing of exoskeleton technologies and creating more effective solutions for people with disabilities. The exoskeleton models, previously developed by the authors, are presented in the papers.1–4 The issues of simulating exoskeletons and anthropoid mechanisms with various actuators, including electric drives, are covered in the papers.5–10 The studies are focused on some exoskeleton applications. These papers provide relevance and importance of exoskeleton development.11,12

Material and methods

Electromechanical model description for a three-link exoskeleton with the angles calculated between the links

For identifying the mechanism behavior patterns while building the system of differential equations of motion, consider a three-link model with the angles calculated between the links and four systems of coordinates – one of which is absolute and three of which are local (Figure 1). Let's introduce the absolute right-hand Cartesian coordinate system A 0 xyz MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGimaaqabaGccaWG4bGaamyEaiaadQhaaaa@3BBE@ with the plane x A 0 Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaWGbb WaaSbaaSqaaiaaicdaaeqaaOGaamywaaaa@3A9F@ , in which the exoskeleton motion takes place (Figure 1). Let's introduce the mobile local system of coordinates A 0 x 1 y 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGimaaqabaGccaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaamyE amaaBaaaleaacaaIXaaabeaaaaa@3C97@ , which is fixed to the first bottom link A 0 A 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGimaaqabaGccaWGbbWaaSbaaSqaaiaaigdaaeqaaaaa@3A71@ , for describing its motion. The mobile axis A x 0 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgba WcbaGaaGimaaqabaGccaWG4bWaaSbaaSqaaiaaigdaaeqaaaaa@3AA9@  is directed along the link; the axis A 0 y 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGimaaqabaGccaWG5bWaaSbaaSqaaiaaigdaaeqaaOWaaSba aSqaaaqabaaaaa@3ADF@ is introduced based on the right-hand basis. The local system of coordinates for the second and the third link is introduced in a similar way. The exoskeleton model consists of three absolutely rigid weighty links simulated by rods (Figure 1). The links are coupled with cylindrical hinges at the points A 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgba WcbaGaaGimaaqabaaaaa@38BB@ , A 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgba WcbaGaaGimaaqabaaaaa@38BB@ , and A 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgba WcbaGaaGimaaqabaaaaa@38BB@ . The cylindrical hinge at the point A 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgba WcbaGaaGimaaqabaaaaa@38BB@ is firmly fixed to the supporting surface. The lengths of the links are as follows: A 0 A 1 = l 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGimaaqabaGccaWGbbWaaSbaaSqaaiaaigdaaeqaaOGaeyyp a0JaamiBamaaBaaaleaacaaIXaaabeaaaaa@3D59@ , A 1 A 2 = l 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGymaaqabaGccaWGbbWaaSbaaSqaaiaaikdaaeqaaOGaeyyp a0JaamiBamaaBaaaleaacaaIYaaabeaaaaa@3D5C@ , A 2 A = 3 l 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGOmaaqabaGccaWGbbWaaSraaSqaaiaaiodaaeqaaOGaeyyp a0JaamiBamaaBaaaleaacaaIZaaabeaaaaa@3D60@ . The links are linear rods; they are considered unchanged during mechanism motion, and under any applied forces. The mass of the first link A 0 A 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGimaaqabaGccaWGbbWaaSbaaSqaaiaaigdaaeqaaaaa@3A71@ equals to m 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2gadaWgaa WcbaGaaGymaaqabaaaaa@38E7@ , the mass of the second link A 1 A 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGymaaqabaGccaWGbbWaaSbaaSqaaiaaikdaaeqaaaaa@3A73@ equals to m 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2gadaWgaa WcbaGaaGOmaaqabaaaaa@38E8@ , the mass of the third link A 2 A 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGOmaaqabaGccaWGbbWaaSbaaSqaaiaaiodaaeqaaaaa@3A75@ equals to m 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2gadaWgaa WcbaGaaG4maaqabaaaaa@38E9@ . Since in the selected mechanism model the links are assumed to be absolutely rigid weighty rods, their moments of inertia are calculated as those for the homogeneous solid rods. The moment of inertia of the link A 0 A 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGimaaqabaGccaWGbbWaaSraaSqaaiaaigdaaeqaaaaa@3A72@ about the axis passing through its beginning, the point A 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgba WcbaGaaGimaaqabaaaaa@38BB@ perpendicular to the motion plane x 1 A 0 Y 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaaGymaaqabaGccaWGbbWaaSbaaSqaaiaaicdaaeqaaOGaamyw amaaBaaaleaacaaIXaaabeaaaaa@3C77@ , is denoted I 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMeadaWgaa WcbaGaaGymaaqabaaaaa@38C3@ . The moments of inertia of the second and the third links are denoted I 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMeadaWgaa WcbaGaaGOmaaqabaaaaa@38C4@ and I 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMeadaWgaa WcbaGaaGOmaaqabaaaaa@38C4@ respectively. The first link A 0 A 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGimaaqabaGccaWGbbWaaSbaaSqaaiaaigdaaeqaaaaa@3A71@ performs rotational motion about the axis in the cylindrical hinge A 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgba WcbaGaaGimaaqabaaaaa@38BB@ . The link position depends on just one parameter and is determined by the angle φ 1 ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeA8aQnaaBa aaleaacaaIXaaabeaakmaabmaabaaeaaaaaaaaa8qacaWG0baapaGa ayjkaiaawMcaaaaa@3C6D@ . Let's use M 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eadaWgaa WcbaGaaGymaaqabaaaaa@38C7@ to denote the controlling torque developed in the hinge A 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgba WcbaGaaGimaaqabaaaaa@38BB@ . The position of the second link A 1 A 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGymaaqabaGccaWGbbWaaSbaaSqaaiaaikdaaeqaaaaa@3A73@ that performs the plane-parallel motion depends on the motion of the point M 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eadaWgaa WcbaGaaGOmaaqabaaaaa@38C8@ , which is regarded as a pole, and on one additional parameter, the angle φ 2 ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeA8aQnaaBa aaleaacaaIYaaabeaakmaabmaabaaeaaaaaaaaa8qacaWG0baapaGa ayjkaiaawMcaaaaa@3C6E@ . The same is true for the third link. The hinge A 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGOmaaqabaaaaa@38BC@ is considered as a pole. The link position also depends on the angle φ 3 ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeA8aQnaaBa aaleaacaaIZaaabeaakmaabmaabaaeaaaaaaaaa8qacaWG0baapaGa ayjkaiaawMcaaaaa@3C6F@ . Let's use M 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eadaWgaa WcbaGaaGOmaaqabaaaaa@38C8@ and M 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eadaWgaa WcbaGaaGOmaaqabaaaaa@38C8@ to denote controlling torque at the hinge A 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGymaaqabaaaaa@38BB@ , and at the hinge A 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGOmaaqabaaaaa@38BC@ respectively (Figure 1).

The generalized coordinates clearly describing the mechanism position in the plane are the angles between the corresponding axes of coordinates Figure 1: φ 1 ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeA8aQnaaBa aaleaacaaIXaaabeaakmaabmaabaaeaaaaaaaaa8qacaWG0baapaGa ayjkaiaawMcaaaaa@3C6D@ , φ 2 ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeA8aQnaaBa aaleaacaaIXaaabeaakmaabmaabaaeaaaaaaaaa8qacaWG0baapaGa ayjkaiaawMcaaaaa@3C6D@ , φ 3 ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeA8aQnaaBa aaleaacaaIXaaabeaakmaabmaabaaeaaaaaaaaa8qacaWG0baapaGa ayjkaiaawMcaaaaa@3C6D@ . The considered system has three degrees of freedom. Three independent drives should be used for implementing the controlled motion, i.e. to control the rotation angle of each link. Electric motors are used in the study. The electric motors, controlling the angular coordinates, are coupled with reduction gears decreasing the turnovers and increasing the torques.

Figure 1 The model of the mechanism with three mobile links and local systems of coordinates, moving in the vertical plane.

The kinetic energy of the mechanism has been recorded, which includes the motion energy of the links A 0 A 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGimaaqabaGccaWGbbWaaSbaaSqaaiaaigdaaeqaaaaa@3A71@ , A 1 A 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGymaaqabaGccaWGbbWaaSbaaSqaaiaaikdaaeqaaaaa@3A73@ , and A 2 A 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGOmaaqabaGccaWGbbWaaSbaaSqaaiaaiodaaeqaaaaa@3A75@ :

T= T A 0 A 1 + T A 1 A 2 + T A 2 A 3 = = 1 2 [ ( I 1 + I 2 + I 3 +( m 2 + m 3 ) l 1 2 + m 3 l 2 2 +( m 2 +2 m 3 ) l 1 l 2 cos ϕ 2 + m 3 l 2 l 3 cos ϕ 3 + + m 3 l 1 l 3 cos( ϕ 2 + ϕ 3 ) ) ϕ ˙ 1 2 + ( I 2 + I 3 + m 3 l 2 2 + m 3 l 2 l 3 cos ϕ 3 ) ϕ ˙ 2 2 + I 3 ϕ ˙ 3 2 + +( 2 I 2 +2 I 3 +2 m 3 l 2 2 +( m 2 +2 m 3 ) l 1 l 2 cos ϕ 2 + 2 m 3 l 2 l 3 cos ϕ 3 + m 3 l 1 l 3 cos( ϕ 2 + ϕ 3 ) ) ϕ ˙ 1 ϕ ˙ 2 + +( 2 I 3 + m 3 l 2 l 3 cos ϕ 3 + m 3 l 1 l 3 cos( ϕ 2 + ϕ 3 ) ) ϕ ˙ 1 ϕ ˙ 3 + ( 2 I 3 + m 3 l 2 l 3 cos ϕ 3 ) ϕ ˙ 2 ϕ ˙ 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGub Gaeyypa0JaamivamaaBaaaleaacaWGbbWaaSbaaWqaaiaaicdaaeqa aSGaamyqamaaBaaameaacaaIXaaabeaaaSqabaGccqGHRaWkcaWGub WaaSbaaSqaaiaadgeadaWgaaadbaGaaGymaaqabaWccaWGbbWaaSba aWqaaiaaikdaaeqaaaWcbeaakiabgUcaRiaadsfadaWgaaWcbaGaam yqamaaBaaameaacaaIYaaabeaaliaadgeadaWgaaadbaGaaG4maaqa baaaleqaaOGaeyypa0dabaGaeyypa0ZaaSaaaeaacaaIXaaabaGaaG OmaaaadaWabaqaamaabeaabaGaamysamaaBaaaleaacaaIXaaabeaa kiabgUcaRiaadMeadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcaWGjb WaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIaaiikaiaad2gadaWgaaWc baGaaGOmaaqabaGccqGHRaWkcaWGTbWaaSbaaSqaaiaaiodaaeqaaO GaaiykaiaadYgadaqhaaWcbaGaaGymaaqaaiaaikdaaaGccqGHRaWk caWGTbWaaSbaaSqaaiaaiodaaeqaaOGaamiBamaaDaaaleaacaaIYa aabaGaaGOmaaaakiabgUcaRiaacIcacaWGTbWaaSbaaSqaaiaaikda aeqaaOGaey4kaSIaaGOmaiaad2gadaWgaaWcbaGaaG4maaqabaGcca GGPaGaamiBamaaBaaaleaacaaIXaaabeaakiaadYgadaWgaaWcbaGa aGOmaaqabaGcciGGJbGaai4BaiaacohacqaHvpGzdaWgaaWcbaGaaG OmaaqabaGccqGHRaWkcaWGTbWaaSbaaSqaaiaaiodaaeqaaOGaamiB amaaBaaaleaacaaIYaaabeaakiaadYgadaWgaaWcbaGaaG4maaqaba GcciGGJbGaai4BaiaacohacqaHvpGzdaWgaaWcbaGaaG4maaqabaGc cqGHRaWkaiaawIcaaaGaay5waaaabaWaaeGaaeaacqGHRaWkcaWGTb WaaSbaaSqaaiaaiodaaeqaaOGaamiBamaaBaaaleaacaaIXaaabeaa kiaadYgadaWgaaWcbaGaaG4maaqabaGcciGGJbGaai4Baiaacohaca GGOaGaeqy1dy2aaSbaaSqaaiaaikdaaeqaaOGaey4kaSIaeqy1dy2a aSbaaSqaaiaaiodaaeqaaOGaaiykaaGaayzkaaGafqy1dyMbaiaada qhaaWcbaGaaGymaaqaaiaaikdaaaGccqGHRaWkaeaadaqadaqaaiaa dMeadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcaWGjbWaaSbaaSqaai aaiodaaeqaaOGaey4kaSIaamyBamaaBaaaleaacaaIZaaabeaakiaa dYgadaqhaaWcbaGaaGOmaaqaaiaaikdaaaGccqGHRaWkcaWGTbWaaS baaSqaaiaaiodaaeqaaOGaamiBamaaBaaaleaacaaIYaaabeaakiaa dYgadaWgaaWcbaGaaG4maaqabaGcciGGJbGaai4BaiaacohacqaHvp GzdaWgaaWcbaGaaG4maaqabaaakiaawIcacaGLPaaacuaHvpGzgaGa amaaDaaaleaacaaIYaaabaGaaGOmaaaakiabgUcaRiaadMeadaWgaa WcbaGaaG4maaqabaGccuaHvpGzgaGaamaaDaaaleaacaaIZaaabaGa aGOmaaaakiabgUcaRaqaaiabgUcaRmaabeaabaGaaGOmaiaadMeada WgaaWcbaGaaGOmaaqabaGccqGHRaWkcaaIYaGaamysamaaBaaaleaa caaIZaaabeaakiabgUcaRiaaikdacaWGTbWaaSbaaSqaaiaaiodaae qaaOGaamiBamaaDaaaleaacaaIYaaabaGaaGOmaaaakiabgUcaRiaa cIcacaWGTbWaaSbaaSqaaiaaikdaaeqaaOGaey4kaSIaaGOmaiaad2 gadaWgaaWcbaGaaG4maaqabaGccaGGPaGaamiBamaaBaaaleaacaaI XaaabeaakiaadYgadaWgaaWcbaGaaGOmaaqabaGcciGGJbGaai4Bai aacohacqaHvpGzdaWgaaWcbaGaaGOmaaqabaGccqGHRaWkaiaawIca aaqaamaabiaabaGaaGOmaiaad2gadaWgaaWcbaGaaG4maaqabaGcca WGSbWaaSbaaSqaaiaaikdaaeqaaOGaamiBamaaBaaaleaacaaIZaaa beaakiGacogacaGGVbGaai4Caiabew9aMnaaBaaaleaacaaIZaaabe aakiabgUcaRiaad2gadaWgaaWcbaGaaG4maaqabaGccaWGSbWaaSba aSqaaiaaigdaaeqaaOGaamiBamaaBaaaleaacaaIZaaabeaakiGaco gacaGGVbGaai4CaiaacIcacqaHvpGzdaWgaaWcbaGaaGOmaaqabaGc cqGHRaWkcqaHvpGzdaWgaaWcbaGaaG4maaqabaGccaGGPaaacaGLPa aacuaHvpGzgaGaamaaBaaaleaacaaIXaaabeaakiqbew9aMzaacaWa aSbaaSqaaiaaikdaaeqaaOGaey4kaScabaGaey4kaSYaaeWaaeaaca aIYaGaamysamaaBaaaleaacaaIZaaabeaakiabgUcaRiaad2gadaWg aaWcbaGaaG4maaqabaGccaWGSbWaaSbaaSqaaiaaikdaaeqaaOGaam iBamaaBaaaleaacaaIZaaabeaakiGacogacaGGVbGaai4Caiabew9a MnaaBaaaleaacaaIZaaabeaakiabgUcaRiaad2gadaWgaaWcbaGaaG 4maaqabaGccaWGSbWaaSbaaSqaaiaaigdaaeqaaOGaamiBamaaBaaa leaacaaIZaaabeaakiGacogacaGGVbGaai4CaiaacIcacqaHvpGzda WgaaWcbaGaaGOmaaqabaGccqGHRaWkcqaHvpGzdaWgaaWcbaGaaG4m aaqabaGccaGGPaaacaGLOaGaayzkaaGafqy1dyMbaiaadaWgaaWcba GaaGymaaqabaGccuaHvpGzgaGaamaaBaaaleaacaaIZaaabeaakiab gUcaRmaadiaabaWaaeWaaeaacaaIYaGaamysamaaBaaaleaacaaIZa aabeaakiabgUcaRiaad2gadaWgaaWcbaGaaG4maaqabaGccaWGSbWa aSbaaSqaaiaaikdaaeqaaOGaamiBamaaBaaaleaacaaIZaaabeaaki GacogacaGGVbGaai4Caiabew9aMnaaBaaaleaacaaIZaaabeaaaOGa ayjkaiaawMcaaiqbew9aMzaacaWaaSbaaSqaaiaaikdaaeqaaOGafq y1dyMbaiaadaWgaaWcbaGaaG4maaqabaaakiaaw2faaaaaaa@3B64@   (1)

The system of differential equations of motion for the three-link exoskeleton model has been composed using the software for building the system of differential equations of motion

( I 1 + I 2 + I 3 +( m 2 + m 3 ) l 1 2 + m 3 l 2 2 +( m 2 +2 m 3 ) l 1 l 2 cos ϕ 2 + m 3 l 3 ( l 2 cos ϕ 3 + l 1 cos( ϕ 2 + ϕ 3 ) ) ) ϕ ¨ 1 + +( I 2 + I 3 + m 3 l 2 2 +( 1 2 m 2 + m 3 ) l 1 l 2 cos ϕ 2 + m 3 l 3 ( l 2 cos ϕ 3 + 1 2 l 1 cos( ϕ 2 + ϕ 3 ) ) ) ϕ ¨ 2 + +( I 3 + 1 2 m 3 l 2 l 3 cos ϕ 3 + 1 2 m 3 l 1 l 3 cos( ϕ 2 + ϕ 3 ) ) ϕ ¨ 3 l 1 ( ( 1 2 m 2 + m 3 ) l 2 ( sin ϕ 2 )+ 1 2 m 3 l 3 sin( ϕ 2 + ϕ 3 ) ) ϕ ˙ 2 2 1 2 m 3 l 3 ( l 2 ( sin ϕ 3 )+ l 1 sin( ϕ 2 + ϕ 3 ) ) ϕ ˙ 3 2 [ ( m 2 +2 m 3 ) l 1 l 2 ( sin ϕ 2 ) m 3 l 1 l 3 sin( ϕ 2 + ϕ 3 ) ] ϕ ˙ 1 ϕ ˙ 2 m 3 l 3 ( l 2 ( sin ϕ 3 )+ l 1 sin( ϕ 2 + ϕ 3 ) ) ϕ ˙ 1 ϕ ˙ 3 m 3 l 3 ( l 2 ( sin ϕ 3 )+ l 1 sin( ϕ 2 + ϕ 3 ) ) ϕ ˙ 2 ϕ ˙ 3 + +g [ ( 1 2 m 1 + m 2 + m 3 ) l 1 cos ϕ 1 +( 1 2 m 2 + m 3 ) l 2 cos( ϕ 1 + ϕ 2 ) + 1 2 m 3 l 3 cos( ϕ 1 + ϕ 2 + ϕ 3 ) ]= M 1 M 2 , ( I 2 + I 3 + m 3 l 2 2 +( 1 2 m 2 + m 3 ) l 1 l 2 cos ϕ 2 + m 3 l 3 ( l 2 cos ϕ 3 + 1 2 l 1 cos( ϕ 2 + ϕ 3 ) ) ) ϕ ¨ 1 + +( I 2 + I 3 + m 3 l 2 2 + m 3 l 2 l 3 cos ϕ 3 ) ϕ ¨ 2 +( I 3 + 1 2 m 3 l 2 l 3 cos ϕ 3 ) ϕ ¨ 3 + + l 1 ( ( 1 2 m 2 + m 3 ) l 2 ( sin ϕ 2 )+ 1 2 m 3 l 3 sin( ϕ 2 + ϕ 3 ) ) ϕ ˙ 1 2 1 2 m 3 l 2 l 3 ( sin ϕ 3 ) ϕ ˙ 3 2 m 3 l 2 l 3 ( sin ϕ 3 ) ϕ ˙ 1 ϕ ˙ 3 m 3 l 2 l 3 ( sin ϕ 3 ) ϕ ˙ 2 ϕ ˙ 3 +g [ ( 1 2 m 2 + m 3 ) l 2 cos( ϕ 1 + ϕ 2 ) + 1 2 m 3 l 3 cos( ϕ 1 + ϕ 2 + ϕ 3 ) ]= M 2 M 3 , ( I 3 + 1 2 m 3 l 2 l 3 cos ϕ 3 + 1 2 m 3 l 1 l 3 cos( ϕ 2 + ϕ 3 ) ) ϕ ¨ 1 +( I 3 + 1 2 m 3 l 2 l 3 cos ϕ 3 ) ϕ ¨ 2 + I 3 ϕ ¨ 3 + + 1 2 m 3 l 3 ( l 2 sin ϕ 3 + l 1 sin( ϕ 2 + ϕ 3 ) ) ϕ ˙ 1 2 + 1 2 m 3 l 2 l 3 ( sin ϕ 3 ) ϕ ˙ 2 2 + + m 3 l 2 l 3 ( sin ϕ 3 ) ϕ ˙ 1 ϕ ˙ 2 + 1 2 g m 3 l 3 cos( ϕ 1 + ϕ 2 + ϕ 3 )= M 3 . 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Thus, the system of differential equations of motion in the form of Lagrange equations of the second kind that form the mathematical model of the three-link exoskeleton has been created. Thereafter, this system is used for the simulation of mechanism motion.

The electric motor rotor impact assessment on the dynamics of electromechanical model of the three-link exoskeleton:

Let’s analyze the electric motor rotor impact on the dynamics of the three-link exoskeleton model.2 Consider electromechanical model of the link drives that includes electric motors with reduction gears located at the hinges A i 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaamyAaaqabaGccqGHsislcaaIXaaaaa@3AA0@ (i = 1,2,3). Taking into consideration the rotating rotors of electric motors and their masses, the kinetic energy expression (1) is transformed into the following:

T= 1 2 [ ( I 1 + I R 1 k R 1 2 + I 2 + I 3 +( m 2 + m E 2 + m 3 + m E 3 ) l 1 2 +( m 3 + m E 3 ) l 2 2 + +( m 2 +2 m 3 +2 m E 3 ) l 1 l 2 cos ϕ 2 + m 3 l 2 l 3 cos ϕ 3 + + m 3 l 1 l 3 cos( ϕ 2 + ϕ 3 ) ) ϕ ˙ 1 2 +( I 2 + I R 2 k R 2 2 + I 3 +( m 3 + m E 3 ) l 2 2 + m 3 l 2 l 3 cos ϕ 3 ) ϕ ˙ 2 2 +( I 3 + I R 3 k R 3 2 ) ϕ ˙ 3 2 + +( 2 I 2 +2 I 3 +2( m 3 + m E 3 ) l 2 2 +( m 2 +2 m 3 +2 m E 3 ) l 1 l 2 cos ϕ 2 + 2 m 3 l 2 l 3 cos ϕ 3 + m 3 l 1 l 3 cos( ϕ 2 + ϕ 3 ) ) ϕ ˙ 1 ϕ ˙ 2 + +( 2 I 3 + m 3 l 2 l 3 cos ϕ 3 + m 3 l 1 l 3 cos( ϕ 2 + ϕ 3 ) ) ϕ ˙ 1 ϕ ˙ 3 + ( 2 I 3 + m 3 l 2 l 3 cos ϕ 3 ) ϕ ˙ 2 ϕ ˙ 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaGabeaacaWGub Gaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaWabaqaamaabeaa baGaamysamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadMeadaWgaa WcbaGaamOuamaaBaaameaacaaIXaaabeaaaSqabaGccaWGRbWaa0ba aSqaaiaadkfadaWgaaadbaGaaGymaaqabaaaleaacaaIYaaaaOGaey 4kaSIaamysamaaBaaaleaacaaIYaaabeaakiabgUcaRiaadMeadaWg aaWcbaGaaG4maaqabaGccqGHRaWkcaGGOaGaamyBamaaBaaaleaaca aIYaaabeaakiabgUcaRiaad2gadaWgaaWcbaGaamyramaaBaaameaa caaIYaaabeaaaSqabaGccqGHRaWkcaWGTbWaaSbaaSqaaiaaiodaae qaaOGaey4kaSIaamyBamaaBaaaleaacaWGfbWaaSbaaWqaaiaaioda aeqaaaWcbeaakiaacMcacaWGSbWaa0baaSqaaiaaigdaaeaacaaIYa aaaOGaey4kaSIaaiikaiaad2gadaWgaaWcbaGaaG4maaqabaGccqGH RaWkcaWGTbWaaSbaaSqaaiaadweadaWgaaadbaGaaG4maaqabaaale qaaOGaaiykaiaadYgadaqhaaWcbaGaaGOmaaqaaiaaikdaaaGccqGH RaWkaiaawIcaaaGaay5waaaabaGaey4kaSIaaiikaiaad2gadaWgaa WcbaGaaGOmaaqabaGccqGHRaWkcaaIYaGaamyBamaaBaaaleaacaaI ZaaabeaakiabgUcaRiaaikdacaWGTbWaaSbaaSqaaiaadweadaWgaa adbaGaaG4maaqabaaaleqaaOGaaiykaiaadYgadaWgaaWcbaGaaGym aaqabaGccaWGSbWaaSbaaSqaaiaaikdaaeqaaOGaci4yaiaac+gaca GGZbGaeqy1dy2aaSbaaSqaaiaaikdaaeqaaOGaey4kaSIaamyBamaa BaaaleaacaaIZaaabeaakiaadYgadaWgaaWcbaGaaGOmaaqabaGcca WGSbWaaSbaaSqaaiaaiodaaeqaaOGaci4yaiaac+gacaGGZbGaeqy1 dy2aaSbaaSqaaiaaiodaaeqaaOGaey4kaScabaWaaeGaaeaacqGHRa WkcaWGTbWaaSbaaSqaaiaaiodaaeqaaOGaamiBamaaBaaaleaacaaI XaaabeaakiaadYgadaWgaaWcbaGaaG4maaqabaGcciGGJbGaai4Bai aacohacaGGOaGaeqy1dy2aaSbaaSqaaiaaikdaaeqaaOGaey4kaSIa eqy1dy2aaSbaaSqaaiaaiodaaeqaaOGaaiykaaGaayzkaaGafqy1dy MbaiaadaqhaaWcbaGaaGymaaqaaiaaikdaaaGccqGHRaWkdaqadaqa aiaadMeadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcaWGjbWaaSbaaS qaaiaadkfadaWgaaadbaGaaGOmaaqabaaaleqaaOGaam4AamaaDaaa leaacaWGsbWaaSbaaWqaaiaaikdaaeqaaaWcbaGaaGOmaaaakiabgU caRiaadMeadaWgaaWcbaGaaG4maaqabaGccqGHRaWkcaGGOaGaamyB amaaBaaaleaacaaIZaaabeaakiabgUcaRiaad2gadaWgaaWcbaGaam yramaaBaaameaacaaIZaaabeaaaSqabaGccaGGPaGaamiBamaaDaaa leaacaaIYaaabaGaaGOmaaaakiabgUcaRiaad2gadaWgaaWcbaGaaG 4maaqabaGccaWGSbWaaSbaaSqaaiaaikdaaeqaaOGaamiBamaaBaaa leaacaaIZaaabeaakiGacogacaGGVbGaai4Caiabew9aMnaaBaaale aacaaIZaaabeaaaOGaayjkaiaawMcaaiqbew9aMzaacaWaa0baaSqa aiaaikdaaeaacaaIYaaaaOGaey4kaSYaaeWaaeaacaWGjbWaaSbaaS qaaiaaiodaaeqaaOGaey4kaSIaamysamaaBaaaleaacaWGsbWaaSba aWqaaiaaiodaaeqaaaWcbeaakiaadUgadaqhaaWcbaGaamOuamaaBa aameaacaaIZaaabeaaaSqaaiaaikdaaaaakiaawIcacaGLPaaacuaH vpGzgaGaamaaDaaaleaacaaIZaaabaGaaGOmaaaakiabgUcaRaqaai abgUcaRmaabeaabaGaaGOmaiaadMeadaWgaaWcbaGaaGOmaaqabaGc cqGHRaWkcaaIYaGaamysamaaBaaaleaacaaIZaaabeaakiabgUcaRi aaikdacaGGOaGaamyBamaaBaaaleaacaaIZaaabeaakiabgUcaRiaa d2gadaWgaaWcbaGaamyramaaBaaameaacaaIZaaabeaaaSqabaGcca GGPaGaamiBamaaDaaaleaacaaIYaaabaGaaGOmaaaakiabgUcaRiaa cIcacaWGTbWaaSbaaSqaaiaaikdaaeqaaOGaey4kaSIaaGOmaiaad2 gadaWgaaWcbaGaaG4maaqabaGccqGHRaWkcaaIYaGaamyBamaaBaaa leaacaWGfbWaaSbaaWqaaiaaiodaaeqaaaWcbeaakiaacMcacaWGSb WaaSbaaSqaaiaaigdaaeqaaOGaamiBamaaBaaaleaacaaIYaaabeaa kiGacogacaGGVbGaai4Caiabew9aMnaaBaaaleaacaaIYaaabeaaki abgUcaRaGaayjkaaWaaeGaaeaacaaIYaGaamyBamaaBaaaleaacaaI ZaaabeaakiaadYgadaWgaaWcbaGaaGOmaaqabaGccaWGSbWaaSbaaS qaaiaaiodaaeqaaOGaci4yaiaac+gacaGGZbGaeqy1dy2aaSbaaSqa aiaaiodaaeqaaOGaey4kaSIaamyBamaaBaaaleaacaaIZaaabeaaki aadYgadaWgaaWcbaGaaGymaaqabaGccaWGSbWaaSbaaSqaaiaaioda aeqaaOGaci4yaiaac+gacaGGZbGaaiikaiabew9aMnaaBaaaleaaca aIYaaabeaakiabgUcaRiabew9aMnaaBaaaleaacaaIZaaabeaakiaa cMcaaiaawMcaaiqbew9aMzaacaWaaSbaaSqaaiaaigdaaeqaaOGafq y1dyMbaiaadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkaeaacqGHRaWk daqadaqaaiaaikdacaWGjbWaaSbaaSqaaiaaiodaaeqaaOGaey4kaS IaamyBamaaBaaaleaacaaIZaaabeaakiaadYgadaWgaaWcbaGaaGOm aaqabaGccaWGSbWaaSbaaSqaaiaaiodaaeqaaOGaci4yaiaac+gaca GGZbGaeqy1dy2aaSbaaSqaaiaaiodaaeqaaOGaey4kaSIaamyBamaa BaaaleaacaaIZaaabeaakiaadYgadaWgaaWcbaGaaGymaaqabaGcca WGSbWaaSbaaSqaaiaaiodaaeqaaOGaci4yaiaac+gacaGGZbGaaiik aiabew9aMnaaBaaaleaacaaIYaaabeaakiabgUcaRiabew9aMnaaBa aaleaacaaIZaaabeaakiaacMcaaiaawIcacaGLPaaacuaHvpGzgaGa amaaBaaaleaacaaIXaaabeaakiqbew9aMzaacaWaaSbaaSqaaiaaio daaeqaaOGaey4kaSYaamGaaeaadaqadaqaaiaaikdacaWGjbWaaSba aSqaaiaaiodaaeqaaOGaey4kaSIaamyBamaaBaaaleaacaaIZaaabe aakiaadYgadaWgaaWcbaGaaGOmaaqabaGccaWGSbWaaSbaaSqaaiaa iodaaeqaaOGaci4yaiaac+gacaGGZbGaeqy1dy2aaSbaaSqaaiaaio daaeqaaaGccaGLOaGaayzkaaGafqy1dyMbaiaadaWgaaWcbaGaaGOm aaqabaGccuaHvpGzgaGaamaaBaaaleaacaaIZaaabeaaaOGaayzxaa aaaaa@6296@   (3)

Where I R i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGsbWaaSbaaWqaaiaadMgaaeqaaaWcbeaaaaa@38EE@  is the moment of inertia of the rotor of the electric motor relative to the axis of rotation, k R i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaBa aaleaacaWGsbWaaSbaaWqaaiaadMgaaeqaaaWcbeaaaaa@3910@  is the gear ratio of the gearbox, number 1 refers to the drive in the A 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGimaaqabaaaaa@38BA@ joint, number 2 - in the A 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGymaaqabaaaaa@38BB@ joint, number 3 - in the A 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGOmaaqabaaaaa@38BC@ joint.

Analyzing the structure of kinetic energy expression (3), and comparing it with the expression (1), it can be observed that the differences, in case of taking into account the inertial properties of the electric drives, are limited to extra terms in parentheses. These terms include moments of inertia and masses equal to those of electric drives. The pattern of kinetic energy expression for the mechanism stays the same. The system of differential equations of motion (2) is changed in a similar way. This system is not listed here.

Results

Inverse dynamics problem solution

Let's find the controlling torques required for specifying the anthropomorphic motion of the considered three-link exoskeleton. These torques are required for selecting electric drives with reduction gears. For this purpose, let's use the system of differential equations of motion (2). It is assumed that the exoskeleton simulates the shin with the hip of the supporting leg, and the human body. Let's find analytically the angles between the links, i.e. between the axes of the local systems of coordinates, in the form of periodic functions specifying anthropoid motion of the three considered links in the absolute system of coordinates:

φ 1 ( t )=π/2+ a 1 j 1 sin [ f 1 (1cos [ 2πt/T])π/2], φ 2 ( t ) = a 2 ( j 2 cos[ f 2 π(1cos[2πt/T])π/2] /2 j 1 sin [ f 1 (1cos[2πt/T])π/2]), φ 3 ( t ) = a 3 ( j 3 sin[ f 2 π(1+cos[2πt/T])π/2]  ( j 2 cos[ f 2 π(1cos[2πt/T])π/2] /2 j 1 sin [ f 1 (1cos[2πt/T])π/2]). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeea0dXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaOabaiqabaqbaeqabi qaaaqaaabaaaaaaaaapeGaeqOXdO2damaaBaaaleaapeGaaGymaaWd aeqaaOWaaeWaaeaapeGaamiDaaWdaiaawIcacaGLPaaapeGaeyypa0 JaeqiWdaNaai4laiaaikdacqGHRaWkcaWGHbWdamaaBaaaleaapeGa aGymaaWdaeqaaOWdbiaadQgapaWaaSbaaSqaa8qacaaIXaaapaqaba GcpeGaam4CaiaadMgacaWGUbWdamaajicabaWdbiaadAgapaWaaSba aSqaa8qacaaIXaaapaqabaGcpeGaai4eG8aacaGGOaWdbiaaigdaca GGtaIaam4yaiaad+gacaWGZbaapaGaay5waiaawUfaa8qacaaIYaGa eqiWdaNaamiDaiaac+cacaWGubWdaiaac2facaGGPaWdbiabec8aWj aac+cacaaIYaWdaiaac2fapeGaaiilaaWdaeaapeGaeqOXdO2damaa BaaaleaapeGaaGOmaaWdaeqaaOWaaeWaaeaapeGaamiDaaWdaiaawI cacaGLPaaapeGaaeiiaiabg2da9iaadggapaWaaSbaaSqaa8qacaaI YaaapaqabaGccaGGOaWdbiaadQgapaWaaSbaaSqaa8qacaaIYaaapa qabaGcpeGaam4yaiaad+gacaWGZbWdaiaacUfapeGaamOza8aadaWg aaWcbaWdbiaaikdaa8aabeaak8qacaGGtaIaeqiWda3daiaacIcape GaaGymaiaacobicaWGJbGaam4BaiaadohapaGaai4wa8qacaaIYaGa eqiWdaNaamiDaiaac+cacaWGubWdaiaac2facaGGPaWdbiabec8aWj aac+cacaaIYaWdamaajmcabaWdbiaac+cacaaIYaGaai4eGiaadQga paWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaam4CaiaadMgacaWGUb aapaGaayzxaiaawUfaa8qacaWGMbWdamaaBaaaleaapeGaaGymaaWd aeqaaOWdbiaacobipaGaaiika8qacaaIXaGaai4eGiaadogacaWGVb Gaam4Ca8aacaGGBbWdbiaaikdacqaHapaCcaWG0bGaai4laiaadsfa paGaaiyxaiaacMcapeGaeqiWdaNaai4laiaaikdapaGaaiyxaiaacM capeGaaiilaaaaa8aabaWdbiabeA8aQ9aadaWgaaWcbaWdbiaaioda a8aabeaakmaabmaabaWdbiaadshaa8aacaGLOaGaayzkaaWdbiaabc cacqGH9aqpcaWGHbWdamaaBaaaleaapeGaaG4maaWdaeqaaOGaaiik a8qacaWGQbWdamaaBaaaleaapeGaaG4maaWdaeqaaOWdbiaadohaca WGPbGaamOBa8aacaGGBbWdbiaadAgapaWaaSbaaSqaa8qacaaIYaaa paqabaGcpeGaai4eGiabec8aW9aacaGGOaWdbiaaigdacqGHRaWkca WGJbGaam4BaiaadohapaGaai4wa8qacaaIYaGaeqiWdaNaamiDaiaa c+cacaWGubWdaiaac2facaGGPaWdbiabec8aWjaac+cacaaIYaWdai aac2fapeGaaeiiaiaacobiaeaacaGGtaYdaiaacIcapeGaamOAa8aa daWgaaWcbaWdbiaaikdaa8aabeaak8qacaWGJbGaam4Baiaadohapa Gaai4wa8qacaWGMbWdamaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiaa cobicqaHapaCpaGaaiika8qacaaIXaGaai4eGiaadogacaWGVbGaam 4Ca8aacaGGBbWdbiaaikdacqaHapaCcaWG0bGaai4laiaadsfapaGa aiyxaiaacMcapeGaeqiWdaNaai4laiaaikdapaWaaKWiaeaapeGaai 4laiaaikdacaGGtaIaamOAa8aadaWgaaWcbaWdbiaaigdaa8aabeaa k8qacaWGZbGaamyAaiaad6gaa8aacaGLDbGaay5waaWdbiaadAgapa WaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaai4eG8aacaGGOaWdbiaa igdacaGGtaIaam4yaiaad+gacaWGZbWdaiaacUfapeGaaGOmaiabec 8aWjaadshacaGGVaGaamiva8aacaGGDbGaaiyka8qacqaHapaCcaGG VaGaaGOma8aacaGGDbGaaiyka8qacaGGUaaaaaa@02D7@   (4)

Where Т – the walk period, a i , j i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaamyAaaqabaGccaGGSaGaamOAamaaBaaaleaacaWGPbaabeaa kiaacYcaaaa@3C8B@ and  f i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaWgaa WcbaGaamyAaaqabaaaaa@3913@ – the walk parameters (i = 1,…,5).

Let's select the values of mechanism properties corresponding to those of the human shin (subscript 1), human hip (subscript 2), and the human body (subscript 3). The information about these values can be found in the monograph.1 The lengths of the links are as follows: I 1 =0.385m, I 2 =0.477m, I 3 =0.771m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMeadaWgaa WcbaGaaGymaaqabaGccqGH9aqpcaaIWaGaaiOlaiaaiodacaaI4aGa aGynaiaad2gacaGGSaGaamysamaaBaaaleaacaaIYaaabeaakiabg2 da9iaaicdacaGGUaGaaGinaiaaiEdacaaI3aGaaiyBaiaacYcacaWG jbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaaGimaiaac6cacaaI3a GaaG4naiaaigdacaGGTbaaaa@4E93@ . The masses of the links are as follows: m 1 =2.91 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaaIXaaabeaakiabg2da9iaaikdacaGGUaGaaGyoaiaaigda aaa@3BCC@  kg, m 2 = 8.93 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaaIYaaabeaakiabg2da9iaabccacaqG4aGaaeOlaiaabMda caqGZaaaaa@3C62@  kg, m 3 = 28.93 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaaIZaaabeaakiabg2da9iaabccacaqGYaGaaeioaiaab6ca caqG5aGaae4maaaa@3D18@  kg. The moment of inertia of the link is calculated based on the formula for that of the rod about the axis passing perpendicularly through its end. These moments are as follows: I 1 =0.144 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaaIXaaabeaakiabg2da9iaaicdacaGGUaGaaGymaiaaisda caaI0aaaaa@3C5F@  kg·m2, I 2 =0.677 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaaIYaaabeaakiabg2da9iaaicdacaGGUaGaaGOnaiaaiEda caaI3aaaaa@3C6B@  kg·m2, I 3 =5.732 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaaIZaaabeaakiabg2da9iaaiwdacaGGUaGaaG4naiaaioda caaIYaaaaa@3C69@  kg·m2. The acceleration due to gravity is g=9.81 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgacqGH9a qpcaaI5aGaaiOlaiaaiIdacaaIXaaaaa@3BF2@ m/s2. The time span of the single-support step phase, i.e. the half of the walk period is t k =0.36 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadshadaWgaa WcbaGaam4AaaqabaGccqGH9aqpcaaIWaGaaiOlaiaaiodacaaI2aaa aa@3D1C@ s. The parameters of the walk are as follows: a 1 =1, a 2 =0.11, a 3 =0.4, j 1 = j 2 =0.25, j 3 =0.1, f 1 =π/2, f 2 =0.687 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaaGymaaqabaGccqGH9aqpcaaIXaGaaiilaiaadggadaWgaaWc baGaaGOmaaqabaGccqGH9aqpcaaIWaGaaiOlaiaaigdacaaIXaGaai ilaiaadggadaWgaaWcbaGaaG4maaqabaGccqGH9aqpcaaIWaGaaiOl aiaaisdacaGGSaGaamOAamaaBaaaleaacaaIXaaabeaakiabg2da9i aadQgadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcaaIWaGaaiOlaiaa ikdacaaI1aGaaiilaiaadQgadaWgaaWcbaGaaG4maaqabaGccqGH9a qpcaaIWaGaaiOlaiaaigdacaGGSaGaamOzamaaBaaaleaacaaIXaaa beaakiabg2da9iabec8aWjaac+cacaaIYaGaaiilaiaadAgadaWgaa WcbaGaaGOmaaqabaGccqGH9aqpcaaIWaGaaiOlaiaaiAdacaaI4aGa aG4naaaa@63F2@ .

The curves representing the link rotation angles (3), angular velocities, and angular accelerations are presented in the Figure 2. Several frames of the three-link exoskeleton motion animation are given below (Figure 3).

Figure 2 The curves representing rotation angles ϕ 1 , ϕ 2 , ϕ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaS baaSqaaiaaigdaaeqaaOGaaiilaiabew9aMnaaBaaaleaacaaIYaaa beaakiaacYcacqaHvpGzdaWgaaWcbaGaaG4maaqabaaaaa@3F7B@  (rad), angular velocities ϕ ˙ 1 , ϕ ˙ 2 , ϕ ˙ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqy1dyMbai aadaWgaaWcbaGaaGymaaqabaGccaGGSaGafqy1dyMbaiaadaWgaaWc baGaaGOmaaqabaGccaGGSaGafqy1dyMbaiaadaWgaaWcbaGaaG4maa qabaaaaa@3F96@  (rad/s), and angular accelerations ϕ ¨ 1 , ϕ ¨ 2 , ϕ ¨ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqy1dyMbam aadaWgaaWcbaGaaGymaaqabaGccaGGSaGafqy1dyMbamaadaWgaaWc baGaaGOmaaqabaGccaGGSaGafqy1dyMbamaadaWgaaWcbaGaaG4maa qabaaaaa@3F99@  (rad/s2) of the links as functions of time t (s).

Figure 3 Animation frames of the three-link exoskeleton motion specified by the angles ϕ 1 , ϕ 2 , ϕ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaS baaSqaaiaaigdaaeqaaOGaaiilaiabew9aMnaaBaaaleaacaaIYaaa beaakiaacYcacqaHvpGzdaWgaaWcbaGaaG4maaqabaaaaa@3F7B@ based on the formulas (3).

The controlling torques M 1 (t), M 2 (t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eadaWgaa WcbaGaaGymaaqabaGccaGGOaGaamiDaiaacMcacaGGSaGaamytamaa BaaaleaacaaIYaaabeaakiaacIcacaWG0bGaaiykaaaa@3FE9@ and M 3 (t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eadaWgaa WcbaGaaG4maaqabaGccaGGOaGaamiDaiaacMcaaaa@3B25@ for the drives at the hinges A 0 , A 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGimaaqabaGccaGGSaGaamyqamaaBaaaleaacaaIXaaabeaa kiaacYcaaaa@3BDB@ and A 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaaGOmaaqabaaaaa@38BC@ are found by solving the inverse dynamics problem based on the system of equations (2), (Figure 4).

Figure 4 The curves representing controlling torques M 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIXaaabeaaaaa@37B0@ (N·m), M 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIXaaabeaaaaa@37B0@ (N·m), and M 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIXaaabeaaaaa@37B0@ (N·m) as functions of time t (s).

The peak absolute values of controlling torques M 1 =2321.3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eadaWgaa WcbaGaaGymaaqabaGccqGH9aqpcaaIYaGaaG4maiaaikdacaaIXaGa aiOlaiaaiodaaaa@3E36@ N·m, M 2 =1115.44 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eadaWgaa WcbaGaaGOmaaqabaGccqGH9aqpcaaIXaGaaGymaiaaigdacaaI1aGa aiOlaiaaisdacaaI0aaaaa@3EF6@ N·m, and M 3 =326.703 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eadaWgaa WcbaGaaG4maaqabaGccqGH9aqpcaaIZaGaaGOmaiaaiAdacaGGUaGa aG4naiaaicdacaaIZaaaaa@3EFC@ N·m are used for selecting electric motors and reduction gears. The controlling torques M 1 , M 2 , M 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIXaaabeaakiaacYcacaWGnbWaaSbaaSqaaiaaikdaaeqa aOGaaiilaiaad2eadaWgaaWcbaGaaG4maaqabaaaaa@3C99@ are the torques on the output shaft of the reduction gear. It is worth noting, that controlling torques at the hinges corresponding to ankle joint and knee joint became an order of magnitude greater compared to the two-link model. It is related to the addition of body with a mass exceeding the mass of the shin and the mass of the hip taken together.

Using the Figure 4, let’s solve the Cauchy problem for the system of differential equations (2) with controlling torques approximated with step piecewise-specified function. The motion time is broken down into six equal time spans. The controlling torque is assumed to be constant within each time span. The value of controlling torque is calculated as arithmetic mean on the corresponding time span based on the following formula:

M i,γ = t γ1 t γ M i ( t )dt t γ t γ1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGPbGaaiilaiabeo7aNbqabaGccqGH9aqpdaWcaaqaamaa pehabaGaamytamaaBaaaleaacaWGPbaabeaakmaabmaabaGaamiDaa GaayjkaiaawMcaaiaadsgacaWG0baaleaacaWG0bWaaSbaaWqaaiab eo7aNjabgkHiTiaaigdaaeqaaaWcbaGaamiDamaaBaaameaacqaHZo WzaeqaaaqdcqGHRiI8aaGcbaGaamiDamaaBaaaleaacqaHZoWzaeqa aOGaeyOeI0IaamiDamaaBaaaleaacqaHZoWzcqGHsislcaaIXaaabe aaaaGccaGGSaaaaa@5465@   (5)

where i = 1,2,3 is the number of the link controlled by the torque M i ( t ),γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGPbaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaiaa cYcacqaHZoWzaaa@3CC6@  – the ranking variable specifying the time span.

The plots in the form of step functions for the controlling torques found in the process of inverse problem solution Figure 4 are presented in the Figure 5, 6

Thus, the inverse dynamics problem for the three-link exoskeleton model has been solved, and the controlling torques have been found. The found controlling torques are used for the Cauchy problem solution.

Solution of the direct problem of dynamics

In the process of the Cauchy problem solution for the system of equations (3), with the controlling torques in the form of step functions shown in the Figure 5, the curves representing the rotation angles, angular velocities, and angular accelerations, presented in the Figure 6, have been obtained.

Figure 5 The plots of controlling torques M 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIXaaabeaaaaa@37B0@ (N·m), M 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIYaaabeaaaaa@37B1@ (N·m), and M 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIZaaabeaaaaa@37B2@ (N·m) in the form of piecewise-specified functions of time t (s).

Figure 6 The Cauchy problem solution for three-link mechanism: rotation angles ϕ 1 , ϕ 2 , ϕ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaS baaSqaaiaaigdaaeqaaOGaaiilaiabew9aMnaaBaaaleaacaaIYaaa beaakiaacYcacqaHvpGzdaWgaaWcbaGaaG4maaqabaaaaa@3F7B@  (rad/s2), angular velocities ϕ ˙ 1 , ϕ ˙ 2 , ϕ ˙ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqy1dyMbai aadaWgaaWcbaGaaGymaaqabaGccaGGSaGafqy1dyMbaiaadaWgaaWc baGaaGOmaaqabaGccaGGSaGafqy1dyMbaiaadaWgaaWcbaGaaG4maa qabaaaaa@3F96@  (rad/s2), and angular accelerations ϕ ¨ 1 , ϕ ¨ 2 , ϕ ¨ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqy1dyMbam aadaWgaaWcbaGaaGymaaqabaGccaGGSaGafqy1dyMbamaadaWgaaWc baGaaGOmaaqabaGccaGGSaGafqy1dyMbamaadaWgaaWcbaGaaG4maa qabaaaaa@3F99@  (rad/s2) of the links as functions of time t (s).

Solving the system of differential equations (2) numerically and comparing the obtained results with the initial motion of the links Figure 2, we can observe good agreement between the link rotation angles and between the angular velocities. The agreement between the angular accelerations is satisfactory. Hence, the impulse control in the form of step functions for the controlling torques Figure 6 is acceptable and can be used to control the link motion.

The energy expenditures during motion of anthropomorphic mechanism with three mobile links are calculated as the work of controlling torques, neglecting resisting forces and energy recuperation during link deceleration:

A= 0 t k ( | M 1 |+| M 2 |+| M 3 | )dt t k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maalaaabaWaa8qCaeaadaqadaqaamaaemaabaGaamytamaaBaaa leaacaaIXaaabeaaaOGaay5bSlaawIa7aiabgUcaRmaaemaabaGaam ytamaaBaaaleaacaaIYaaabeaaaOGaay5bSlaawIa7aiabgUcaRmaa emaabaGaamytamaaBaaaleaacaaIZaaabeaaaOGaay5bSlaawIa7aa GaayjkaiaawMcaaiaadsgacaWG0baaleaacaaIWaaabaGaamiDamaa BaaameaacaWGRbaabeaaa0Gaey4kIipaaOqaaiaadshadaWgaaWcba Gaam4AaaqabaaaaOGaaiilaaaa@53C6@   (6)

where t k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaWGRbaabeaaaaa@380C@  – the mechanism motion time, M 1 , M 2 , M 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIXaaabeaakiaacYcacaWGnbWaaSbaaSqaaiaaikdaaeqa aOGaaiilaiaad2eadaWgaaWcbaGaaG4maaqabaaaaa@3C99@  – the controlling torques delivered by mechanism drives.

The following values have been found in the process of calculating the energy expenditures by the drives performing link rotations, provided that controlling torques are specified in the form of step functions Figure 6: A φ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeacqaHgp GAdaWgaaWcbaGaaGymaaqabaaaaa@3A78@ = 1251.51 J, Aφ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeacqaHgp GAdaWgbaWcbaGaaGOmaaqabaaaaa@3A7A@ = 594.94 J, A φ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeacqaHgp GAdaWgaaWcbaGaaG4maaqabaaaaa@3A7A@ = 170.07 J. The total energy expenditures for the mechanism amounted to A = 2016.51 J.

A solution tailored to the dynamics of electric drives

As a result of the Cauchy problem solution for the system of differential equations of motion, taking into account the electric drives, and applying the same controlling torques presented in the Figure 5, the curves for the link rotations, presented in the Figure 7, have been obtained.

Figure 7 The Cauchy problem solution taking into account the mechanism electric drives: the rotation angles ϕ 1 , ϕ 2 , ϕ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaS baaSqaaiaaigdaaeqaaOGaaiilaiabew9aMnaaBaaaleaacaaIYaaa beaakiaacYcacqaHvpGzdaWgaaWcbaGaaG4maaqabaaaaa@3F7B@  (rad/s2), the angular velocities ϕ ˙ 1 , ϕ ˙ 2 , ϕ ˙ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqy1dyMbai aadaWgaaWcbaGaaGymaaqabaGccaGGSaGafqy1dyMbaiaadaWgaaWc baGaaGOmaaqabaGccaGGSaGafqy1dyMbaiaadaWgaaWcbaGaaG4maa qabaaaaa@3F96@  (rad/s2), and the angular accelerations ϕ ¨ 1 , ϕ ¨ 2 , ϕ ¨ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqy1dyMbam aadaWgaaWcbaGaaGymaaqabaGccaGGSaGafqy1dyMbamaadaWgaaWc baGaaGOmaaqabaGccaGGSaGafqy1dyMbamaadaWgaaWcbaGaaG4maa qabaaaaa@3F99@  (rad/s2) of the links as functions of time t (s).

Comparing the obtained results with the solution neglecting electric drive impact Figure 6, we can observe poor agreement between link rotation angles, and angular velocities.

Discussion

The agreement between angular accelerations is satisfactory, especially at the end of the mechanism motion. Therefore, the impact of electric drives on the mechanism dynamics is significant and it should not be neglected in designing models of exoskeletons and anthropomorphic robots. After obtaining poor simulation results based on controlling torques for the model neglecting actual drives, the authors conducted motion simulation with recalculated torques, taking into account the electric drives, and obtained results that are in good agreement. The curves of these results are not listed in the study, because they are very close to the curves presented in the Figure 7. The total energy expenditures for the mechanism amounted to 2030.12 J. The insignificant increase of the energy expenditures by 13.6s1 J, or by 0.68%, with the significant deviation of numerical solution results for the system of differential equations of motion indicate the instability of the derived system. In this case, the stabilization techniques proposed in the papers.13,14 or alternative control methods, considering and compensating disturbances in the model, should be used to improve the simulation results. For example, a neuro-fuzzy control method could be the option.15,16

Conclusion

The electromechanical model of three-link exoskeleton has been created in the study. The simulation of the model dynamics has been conducted in two cases: neglecting the electric drives, and taking them into account. The importance and the necessity of taking electric drives into account for synthesizing controlling torques has been established. The instability of the system of differential equations of motion has been demonstrated. The stabilization techniques should be applied to it during its numerical solution. The proposed exoskeleton model can find practical application in developing actual exoskeletons, anthropomorphic robots, and robotic arms.

Acknowledgments

The work was supported by the Russian Science Foundation and Smolensk region no. 22-29-20308, https://rscf.ru/en/project/22-29-20308/.

Conflicts of interest

There are no conflicting interests declared by the authors.

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