Methodology for Integrated Design Optimization of Actuation Systems for Exoskeletons

Abstract

The engineering of actuation systems for active exoskeletons presents a significant challenge due to the stringent demands for mass reduction and compactness, coupled with complex specifications for actuator dynamics and stroke length. This challenge is met with a model-based methodology. Models for human body, exoskeleton and parametric actuation systems are derived and coupled. Beginning with an inverse dynamics human body simulation, loads in human joints are estimated, and the corresponding support torques are derived. Under the assumption of a control law ensuring these support torques, an optimization problem is stated to determine actuation system parameters such as the number of stator coils and number of battery cells. Lastly, results from the optimization are validated using sophisticated models. The methodology is applied to an exemplary exoskeleton and compared to an approach derived from previous studies.

1. Introduction

Physical stresses in human bodies occur in various fields, e.g., in health care and industrial production. To reduce the risk of musculoskeletal disorders, exoskeletons are applied. With active exoskeletons, support forces are actively controlled, ensuring adaptive human support. An adequate design of the actuation system plays an important role in the exoskeleton’s functionality. Predominantly, electric actuation systems are applied in exoskeletons [1]. Improvements in power densities of electric actuators and batteries enabled the overcoming of earlier limitations [2]. On the other hand, stringent requirements on the actuation system still demand for a structured design process. Therefore, various authors have contributed their work to this field. In the following, the general idea of an integrated design methodology is briefly introduced. Further, the state of the art in actuation system design for exoskeletons is outlined.

Methodologies for the design of mechatronic systems and their components are the subject of ongoing research. Aiming at covering system interdependencies, there is a tendency to integrate models from various components of the considered system, as well as the control approach into the design process. Fundamentals of such integrated design approaches are discussed in [3]. They are applied for the design and control of various mechatronic systems, e.g., flexible manipulators [4], artificial bipedal creatures [5] and drones [6]. An integrated design methodology for various components of an actuation system (power electronics, electric machine, gearbox) and parametrization of a linear control system is presented by Roos [7]. The work addresses an actuation system dimensioning in a general industrial context. Hence, load and motion requirements are considered as inputs for the design methodology.

An integrated design approach for exoskeletons is outlined by Sposito et al. [8] and Villotti et al. [9]. Both of them find that the development of exoskeletons always has to be complemented by prototype studies. Hence, any computational design approach can only be part of a larger methodology. Nonetheless, they underline the importance and usefulness of these approaches for the design and understanding of exoskeletons.

Waldhof et al. [10] aim at personalizing actuation systems of exoskeletons. An individual scalable human arm model and artificially generated motion data are applied to find human joint torques and positions. Based on this, torque, acceleration and mass requirements for an actuation system are derived. The actuation system, i.e., electric machine and battery, is sized based on scaling equations, whereby effects of variations in the geometrical data of the electric machine on the requirements are mapped and a sizing is conducted. Effects of the masses and inertia of the actuation system on the human body are not quantified.

A selection methodology for the choice of electric machines from a catalog, while the gearbox remains fixed, is provided by Kavalieros et al. [11]. A lower-limb exoskeleton is regarded. A corresponding model of the human body is augmented by masses from the exoskeleton and actuation system. Together with motion data, torque requirements for the electric machine are defined and a selection is conducted. Static models are applied and no ground reaction forces are considered.

Mechanical actuator dynamics are explicitly considered in the methodology presented by Aftab et al. [12]. Load requirements are derived from the literature. Based on load and acceleration requirements, viable machine and gearbox combinations are selected. Optimality criteria (peak power, energy efficiency) are applied afterwards to select the final combination.

Di Natali et al. [13] developed a methodology for the evaluation of actuators in existing exoskeletons. They state that the interaction of human, exoskeleton and actuation systems in methodologies for actuation system optimization is usually regarded with large approximations. Therefore, their approach starts with the acquisition of actuator motion data followed by a simulation-based and experimental analysis of the actuation system.

A linear dynamic model of the interaction between actuator, exoskeleton and human is applied by Barjuei et al. [14]. In this case, the bandwidth of a control loop from a reference human support torque to a measured human support torque serves as a function to be maximized. Depending on the chosen electric machine and gearbox combination, different bandwidths are possible. This approach considers the weight and inertia of the actuator in an approximated way.

Most approaches consider some kind of human body model to determine load and motion requirements. The interaction between human, exoskeleton and the actuation system is regarded in a simple way. Electromagnetic actuator models are mostly derived from simple relations and interpolations of catalog data. A connection to deeper insights in actuator dimensioning is only provided in [10]. Further, most contributions do not give approaches for the inertia-compensating control of the actuators in the exoskeleton, limiting comparability between the actuation system’s configurations. An ideal integrated design methodology would be able to predict the effects of the application of varying actuation systems in the application of an exoskeleton while integrating a detailed electromagnetic actuator model covering relevant design parameters. This may serve as a basis for computational optimization. It is concluded that integrated design methodologies for actuation systems for exoskeletons are still lacking.

The methodology presented in this article aims at contributing to the integrated design of actuation systems for exoskeletons. Therefore, a process is established, which guides the actuation system design from the determination of support torques towards controlling the system formulation and system optimization. It is based on the repetitive application of models for human bodies, exoskeletons and actuation systems and a general formulation for their coupling.

The contribution of this article is the integration of detailed models into a methodology for the integrated design of actuation systems for exoskeletons. For the estimation of human body loads, multibody models including ground reaction forces and torques are regarded. The mechanical interaction between the human and exoskeleton is modeled based on algebraic constraints, allowing for movement between human and exoskeleton. The parametric electromagnetic actuator model covers voltage and generalized force equations based on finite element analyses. A control approach is developed, allowing comparable performance for every configuration of the actuation system. Finally, a minimization problem is formulated for the determination of actuation system parameters. The integration of the abovementioned models into a methodology for an exoskeleton actuation system design is a novelty of our work.

A specific actuation system is considered, i.e., the general structure (type and number of actuators) has already been determined. Componentwise, it is mainly the actuators that are regarded; the battery is additionally considered. It can be applied to optimize actuation systems for an individual human, thereby enabling individual support or rehabilitation or (via the consideration of standardized human body models) for a group of humans.

2. Materials and Methods

As presented in Figure 1, the methodology consists of four phases. In the first phase, support torques are defined using a kinetic human body model and, optionally, a model for ground reaction forces and torques (GRF&T), motion tracking data and external load information to describe an exemplary motion task. The application of a model for GRF&T is only required when lower limbs are supported. With this, an inverse dynamic simulation is conducted, and information regarding the loads inside the human body, represented by human joint torques, is gathered. This enables the definition of support torques ${\mathit{\tau }}_{\sup ,\mathrm{hum}}$ for the human body to be supported by the exoskeleton. In the second phase, an optimization of the actuation system is conducted. Here, a parametric model of the actuation system is required. An emphasis of this contribution lies in a traction cable-based actuation system. The extension to directly connected rotational actuators is straightforward and is demonstrated in Section 2.2.10. Based on an optimization procedure, an optimal set of actuation system parameters is calculated. Since the full model for GRF&T is computationally expensive, an approximation is introduced for this step. Further, in this step, a control approach is defined, which enables the precise definition of human support torques. In the third step, a validation for the support torques is conducted. This ensures that support torques from the first phase are approximately achieved despite the approximation of GRF&T applied in the second phase. The methodology is concluded with the fourth phase. Here, the control law developed for the actuation system’s optimization in the second phase shall be applied in the real exoskeleton. Since this work is dedicated to the actuation system optimization, the implementation of the control law on a prototype is beyond the scope. Nonetheless, this completes the claim of a methodology for integrated design optimization to be utilized in design and use.

2.1. Example

A symmetrical lifting task conducted by a healthy male subject (age: 29, height: 1.78 m, weight: 74 kg, lifting weights: 5 kg per hand, duration: approx. 10 s, load applied between $t_{\mathrm{load},\mathrm{start}}=4.7\mathrm{s}$ and $t_{\mathrm{load},\mathrm{end}}=7.2\mathrm{s}$) serves as an example for the methodology. Motion tracking is based on an inertial motion tracking system XSens MTW Awinda with sampling frequency $f_s=60\mathrm{Hz}$. A second-order Butterworth filter with a cutoff frequency of 6 Hz and zero-phase forward and reverse filtering is applied. Frames of the lifting task and ground reaction forces are presented in Figure 2. With the methodology, the actuation system of a combined lower back and hip exoskeleton is designed based on this motion to reduce flexion–extension torques in both hip joints.

The exoskeleton with a blank actuation system is depicted in Figure 3. Flexion–extension degrees of freedom (DOFs) of upper legs are actively actuated while (lower) back DOFs are passively supported using a back brace made from spring steel. An actuation system with six linear actuators is considered. All actuators are mechanically connected with cables to leg links, which are connected with rotational DOFs to the central hip belt. Red lines indicate the routing of the cables. Both outmost actuators act on their opposing upper leg, canceling unwanted lateral flexion–extension torques on back DOFs. Relevant mechanical properties of the exoskeleton may be found in the Supplementary Materials.

2.2. Modeling

Mechanical multibody models are created for human, exoskeleton and the actuation system. Electrical modeling is conducted for the actuation system. For the mechanical models, coordinate frames are defined with homogeneous transformation matrices ${\mathit{T}}_{\mathrm{origin}}^{\mathrm{destination}}$, where origin refers to the coordinate frame {origin} and destination to the frame {destination}. Details on this may be found in [16]. An introduction to constraint multibody modeling may be found in [4].

Body-related frames are specified as {mod,i,frame number}, where mod represents either hum (human body model), exo (exoskeleton model) or load, and i designates the body number. The mechanical modeling relies on standardized frames per body {..,0}, {..,1} and, optionally, frame {..,2}, which is depicted in Figure 4a. Frames {..,1} and {..,2} are utilized to define the locations of joints, with the former utilized for connections to parent bodies and the latter for child bodies in the kinematic chain. Frame {..,0} represents the root frame for each body. When no frame number is stated, frame {..,0} is chosen. The ground is represented by the x-z plane of the global frame {glo}. Rotations are described with relative ${\mathrm{x}\text{-}\mathrm{y}}^{\prime }{\text{-}\mathrm{z}}^{″}$ Tait–Bryan angles [17], depicted in Figure 4b. Generalized coordinates are summarized in ${\mathit{q}}_{\mathrm{mod}}$. Mechanical loads in the direction of either human or exoskeleton degrees of freedom acting on body i are designated with ${(·)}_{..,\mathrm{mod},i}$ or ${(·)}_{..,\mathrm{mod},i,\mathrm{tbax}}$, where tbax is a Tait–Bryan rotation axis. Forces in the coordinate frame {frame} acting on the origin of the frame {mod,i} are designated with ${}^{\mathrm{frame}}\mathit{f}_{..,\mathrm{mod},i}$ or ${}^{\mathrm{frame}}\mathit{f}_{..,\mathrm{mod},i,\mathrm{ax}}$ and torques with ${}^{\mathrm{frame}}\ell _{..,\mathrm{mod},i}$ or ${}^{\mathrm{frame}}\ell _{..,\mathrm{mod},i,\mathrm{ax}}$, where ax represents an axis in frame {mod,i} along which the force or torque acts. When {frame} and {mod,i} are the same, ${}^{\mathrm{frame}}f_{\mathrm{frame},\mathrm{ax}}$ and its rotational counterpart reduce to a scalar. Position vectors in frame {frame} are designated with ${}^{\mathrm{frame}}(·)$. In Figure 2a, the coordinate frame {hum,1} and the direction of a flexion–extension hip torque ${\tau }_{\mathrm{hum},14,x}$ are presented.

2.2.1. Human Body Model

Biomechanical modeling is essential in determining loads in the human body. As commonly assumed, the human body is represented by rigid body segments that are connected by rotational joints, forming a multibody model with a tree structure [18]. Muscle modeling is omitted in this approach. Hence, it is assumed that loads and efforts within the human body are adequately represented by rotational joint torques. The human body model possesses eighteen rotational joints with three DOFs per joint. The root joint is located at the hips segment and possesses six DOFs. Hence, the human body model has 60 DOFs represented in the corresponding vector ${\mathit{q}}_{\mathrm{hum}}$. In the applied approach, kinematic parameters, i.e., location of joints, are adopted from the motion-tracking human body model [19]. Designations of relevant body segments are summarized in

Table 1. Relevant human body segments.

No.	Human Body Segment	Tracking Segments [19]
1	Hips	Pelvis
2	Spine	L5, L3, T12
9	Hand right	RightHand
13	Hand left	LeftHand
14	Upper leg right	RightUpperLeg
15	Lower leg right	RightLowerLeg
17	Upper leg left	LeftUpperLeg
16	Foot right	RightFoot, RightToe
19	Foot left	LeftFoot, LeftToe

. When a human body model segment consists of multiple tracking model segments, the first tracking model segment defines the DOFs considered for motion data. The calculation of the inverse dynamics of this model was conducted in OpenSim 4.4 [20]. To avoid ambiguities with coupled models, loads from this model alone are designated with ${(·)}_{\mathrm{uc},\mathrm{hum},..}$.

Kinetic parameters, i.e., mass, center of gravity and inertia tensor, for each body segment are derived from the scaling and subsequent body part segmentation of a default body model or a body scan while retaining the kinematics derived from the motion-tracking human body model. This approach enables a vivid visualization of the modeled system. The resulting parameters are listed in the Supplementary Materials.

2.2.2. Ground Reaction Forces and Torques

The root joint of the human body model is located on the hips segment; hence, the introduction of external loads is necessary to represent GRF&T and subsequent lower-limb joint torques. The two-legged stand is mechanically overdetermined; therefore, the determination of GRF&T requires some kind of heuristic. What most approaches have in common is that they aim at eliminating artificial root joint loads. In [15], 14 contact points per foot are introduced, which receive ground reaction forces under friction constraints and are solved using a minimization procedure. This is further validated in [21,22] and shows sufficiently good results over a large variety of motions. Therefore, it is applied in this methodology and designated as a full GRF&T model. Referring to [15], the distance threshold is chosen as 0.03 m, the velocity threshold as 1 m/s, the maximum force per contact point as 40% of the gravitational force acting on the human body and the coulomb friction coefficient as 0.5. The vector of the resulting generalized forces from this GRF&T model is referred to as ${\mathit{\tau }}_{\mathrm{grf}\&\mathrm{t},\mathrm{full}}$.

The application of a numerically expensive minimization procedure to determine GRF&T in the context of the optimization and control of exoskeletons appears to be a burden. A fast approach is required. This approach depends in general on the problem considered. For a symmetrical lifting task, it can approximately be assumed that flexion–extension hip torques ${\tau }_{\mathrm{grf}\&\mathrm{t},\mathrm{hum}14,x}$ and ${\tau }_{\mathrm{grf}\&\mathrm{t},\mathrm{hum}17,x}$ are dominated by the torque around the local hips segment’s x-axis ${}^{\mathrm{hum},1}\ell _{\mathrm{hum},1,x}$ and are equally distributed between both hip joints. Hence, only the elimination of this root torque by ground reaction is considered in this case. With this, ground reaction torques in hip flexion–extension DOFs are

$$\begin{array}{c}\hfill {\mathit{\tau }}_{\mathrm{grf}\&\mathrm{t}}={\left[\begin{array}{ccccc}\mathbf{0}& {\tau }_{\mathrm{grf}\&\mathrm{t},\mathrm{hum},14,x}& \mathbf{0}& {\tau }_{\mathrm{grf}\&\mathrm{t},\mathrm{hum},17,x}& \mathbf{0}\end{array}\right]}^{\top }\end{array}$$

(1)

$$\begin{array}{c}\hfill {\tau }_{\mathrm{grf}\&\mathrm{t},\mathrm{hum},14,x}={\tau }_{\mathrm{grf}\&\mathrm{t},\mathrm{hum},17,x}={\displaystyle \frac{1}{2}}{}^{\mathrm{hum},1}\ell _{\mathrm{hum},1,x}.\end{array}$$

(2)

Note that these torques only model the effect of ground reactions on described hip DOFs.

2.2.3. External Loads

External loads are required to model the task of lifting objects. They are assumed to be attached to the left and right hand and to be cubes with constant homogeneous density ${\gamma }_{\mathrm{load},i}=8000{\mathrm{kg}/\mathrm{m}}^3$, adjustable mass $m_{\mathrm{load},i}$ and following edge length $a_{\mathrm{load},i}$ where index i is the body number of either the left or the right hand. Mass and inertia properties are according their shape. The connection to a hand is expressed by

$$\begin{array}{cc}\hfill {\mathit{q}}_{\mathrm{load},i}& =\left[\begin{array}{c}{}^{\mathrm{glo}}\mathit{p}_{\mathrm{load},i}\left({\mathit{q}}_{\mathrm{hum}}\right)\\ {\beta }_2\left({\mathit{T}}_{\mathrm{hum},1}^{\mathrm{glo}}\right)\end{array}\right]\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{}^{\mathrm{glo}}\mathit{p}_{\mathrm{load},i}\\ 1\end{array}\right]& ={\mathit{T}}_{\mathrm{hum},i}^{\mathrm{glo}}\left[\begin{array}{c}{}^{\mathrm{hum},i}\mathit{p}_{\mathrm{load},i}\\ 1\end{array}\right]\hfill \end{array}$$

(4)

where ${}^{\mathrm{hum},\mathrm{i}}\mathit{p}_{\mathrm{load},i}\left({\mathit{q}}_{\mathrm{hum}}\right)$ represents a position vector of a suitable connection point in proximity to the left resp. right hand, ${\mathit{q}}_{\mathrm{load},i}$ the generalized coordinates describing position and orientation of the cube in global coordinates. Further, ${\beta }_2$ represents the rotation around the y-axis in the destination frame of its transformation matrix argument.

Forces and torques are required for the load bodies to conduct this motion. They are defined by the equation of motion

$$\begin{array}{c}\hfill {\mathit{M}}_{\mathrm{load},i}{\ddot{\mathit{q}}}_{\mathrm{load},i}+{\mathit{k}}_{\mathrm{load},i}={\mathit{g}}_{\mathrm{load},i}+{\tilde{\mathit{\tau }}}_{\mathrm{load},i},\end{array}$$

(5)

where ${\mathit{M}}_{\mathrm{mod}}$ designates a mass matrix; ${\mathit{k}}_{\mathrm{mod}}$, a vector of coriolis and centrifugal generalized forces; ${\mathit{g}}_{\mathrm{mod}}$, a vector of generalized gravitational forces; and ${\tilde{\mathit{\tau }}}_{\mathrm{load},i}$, the vector of external forces that represent interaction forces with both hands. Equation (5) is only required to be fulfilled within a time interval $t\in \left[\begin{array}{cc}{t}_{\mathrm{load},\mathrm{start}},& t_{\mathrm{load},\mathrm{end}}\end{array}\right]$. The consequential load vector is

$${\mathit{\tau }}_{\mathrm{load}}=\sum _{i\in \{9;13\}}\left[\begin{array}{cc}{\mathit{J}}_{\mathrm{T},\mathrm{hum},i}^{\top }& {\tilde{\mathit{J}}}_{\mathrm{R},\mathrm{hum},i}^{\top }\end{array}\right]{\tilde{\mathit{\tau }}}_{\mathrm{load},i}\left(H(t-t_{\mathrm{load},\mathrm{start}})-H(t-t_{\mathrm{load},\mathrm{end}})\right),$$

(6)

where H designates the Heaviside function, ${\mathit{J}}_{\mathrm{T},\mathrm{hum},i}$ is the translational Jacobian matrix of human body segment i at the attachment point, and ${\tilde{\mathit{J}}}_{\mathrm{R},\mathrm{hum},i}$ is the rotational Jacobian matrix reduced to its second row, such that it projects torques around the y-axis in frame {glo} on the generalized coordinates of the human body model.

2.2.4. Exoskeleton Model

The exoskeleton and actuation system are mechanically modeled as one multibody system. Electrical modeling of the actuation system is based on the assumption of fundamental harmonic machine behavior and dq0-transformation. An introduction to this may be found in [23].

An exoskeleton consists of $n_{\mathrm{exo}}$ bodies and $n_{\mathrm{act}}$ actuator bodies that are connected by different joints. There may be bodies that do not fulfill rigid body assumptions, increasing modeling complexity. In a simple approach, elasticity forces ${\mathit{\tau }}_{\mathrm{el}}$ can be considered using approximations such as spring stiffness or Euler–Bernoulli beams.

2.2.5. Parametric Formulation of the Actuation System

For the actuation system, actuators and the battery are parametrically modeled. Parameters are reflected in the vector $\mathit{\kappa }$. In this context, actuators consist of an electrical machine and optionally mechanical components such as gearboxes or spools to create linear motion in cable-based actuation. An example is presented in Figure 5. In the linear direct-drive actuator depicted in the figure, linear motion is created without additional components. It will be further considered in the application example. The actuator is flexible, increasing wearing comfort in applications close to the human body [24]. Further, it features low inertia, which is favorable for applications with high accelerations. Its simple construction makes it easily scaleable and manufacturable, and thus, a well-suited exemplary system to consider in the context of this article. The mentioned advantages are at the expense of relatively low efficiency. In this modeling approach, flexibility is disregarded. This is admissible for the application example because elasticity forces from flexible actuators are transferred only into upper-torso body segments, and the hip DOFs are not affected. Further, actuators based on rotational brushless direct-current (BLDC) machines may be applied. In this case, gearboxes or other reduction devices (e.g., twisted strings) are applied. For a cable-based actuation, a spool is necessary in this context to create linear motion.

2.2.6. Electromagnetic Actuator Model

Electromagnetically, the j-th actuator is modeled with

$${\mathit{u}}_{\mathrm{dq}0,j}={\mathit{R}}_j\left(\mathit{\kappa }\right){\mathit{i}}_{\mathrm{dq}0,j}+{\dot{q}}_{\mathrm{act},j}\left[\begin{array}{ccc}0& -p_j& 0\\ p_j& 0& 0\\ 0& 0& 0\end{array}\right]\left({\mathit{L}}_{\mathrm{dq}0,j}\left(\mathit{\kappa }\right){\mathit{i}}_{\mathrm{dq}0,j}+{\mathit{\lambda }}_{\mathrm{pm},j}\left(\mathit{\kappa }\right)\right)+{\mathit{L}}_{\mathrm{dq}0,j}\left(\mathit{\kappa }\right){\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\mathit{i}}_{\mathrm{dq}0,j}.$$

(7)

Here, ${\mathit{u}}_{\mathrm{dq}0,j}$ is the voltage vector in dq0-coordinates, ${\mathit{R}}_j\left(\mathit{\kappa }\right)$ is the resistance matrix, ${\mathit{i}}_{\mathrm{dq}0,j}$ is the current vector in dq0-coordinates, $p_j$ is a conversion factor from mechanical position $q_{\mathrm{act},j}$ to the electrical angle, ${\mathit{L}}_{\mathrm{dq}0,j}\left(\mathit{\kappa }\right)$ is the inductance matrix, and ${\mathit{\lambda }}_{\mathrm{pm}}\left(\mathit{\kappa }\right)$ is the magnetic flux vector due to permanent magnet mover excitation. Fundamental harmonic machine behavior and the absence of saturation and reluctance are assumed. A dependence on the actuation system’s sizing parameter $\mathit{\kappa }$ is restricted on the indicated variables and is determined by how the parameters influence the design of the actuator.

2.2.7. Actuator Operating Limits

For the sizing process, it is crucial to model actuator-operating limits. This is evaluated by the root mean square (RMS) of the current approximated as

$$\begin{array}{c}\hfill i_{\mathrm{RMS},j}=\sqrt{{\displaystyle \frac{1}{2(t_1-t_0)}}{\int }_{t_0}^{t_1}{i}_{q,j}\left(t\right)\mathrm{d}t}\le i_{\mathrm{RMS},\mathrm{nom},j}\end{array}$$

(8)

where $\Delta t=t_1-t_0$ represents a time interval shorter than the actuator’s winding thermal time constant [25] and $i_{\mathrm{RMS},\mathrm{nom},j}$ is the nominal RMS current of the j-th actuator. In case this is shorter than the duration of the exemplary motion task, it is necessary to ensure Ineq. (8) for all admissible $t_0$.

2.2.8. Mechanical Actuator Model

An actuator is mechanically represented by the mover and stator body. It is assumed that the stator is rigidly connected to the exoskeleton’s structure. For this, a homogeneous transformation matrix ${\mathit{T}}_{\mathrm{C},\mathrm{act},j}^{\mathrm{exo},b_{\mathrm{act}}\left(j\right)}$ with an actuator coupling frame {C,act,j} and a body index function $b_{\mathrm{act}}\left(j\right)$ is defined. The stator is mechanically represented by mass and inertia properties.

For the mover body, it is assumed that it possesses either a rotational or a translational DOF $q_{\mathrm{act},j}$. A translational DOF is present when modeling an actuator based on a linear electric machine while a rotational DOF is present when a rotational electric machine is applied. It is mechanically represented by mass and inertia properties. Rotational DOF is chosen around the x-axis of the local frame {exo,$n_{\mathrm{exo}+j}$} and translational in the direction of this x-axis.

The generalized force produced by each actuator is expressed as

$$\begin{array}{c}\hfill {\tau }_{\mathrm{act},j}={\displaystyle \frac{3}{2}}{\lambda }_{\mathrm{pm},\mathrm{d},j}\left(\mathit{\kappa }\right)p_j{i}_{\mathrm{q},j}-k_{\mathrm{v},j}{\dot{q}}_{\mathrm{act},j}-k_{\mathrm{s},j}\mathrm{sgn}\left({\dot{q}}_{\mathrm{act},j}\right),\end{array}$$

(9)

where ${(·)}_{\mathrm{d}}$ refers to an electromagnetic variable in d-axis, and ${(·)}_{\mathrm{q}}$ in q-axis. Further, the viscous friction coefficient

$$\begin{array}{c}\hfill k_{\mathrm{v},j}\left(\mathit{\kappa }\right)=k_{\mathrm{v},\mathrm{gear},j}\left(\mathit{\kappa }\right)+k_{\mathrm{v},\mathrm{various},j}\left(\mathit{\kappa }\right),\end{array}$$

(10)

where $k_{\mathrm{v},\mathrm{gear},j}$ is the viscous friction coefficient of the gear and $k_{\mathrm{v},\mathrm{various},j}$ is the viscous friction coefficient of various sources, e.g., bearing friction and iron losses. The static friction coefficient is represented by $k_{\mathrm{s},j}$.

2.2.9. Cable Routing

Traction cable routing for the j-th actuator is expressed with

$$c_{\mathrm{act},j}=|a_{\mathrm{act},j}\left(q_{\mathrm{act},j}\right)|+\sum _{n=1}^{m_{\sec }}{|}^{\mathrm{glo}}{\mathit{d}}_{j,n}|-a_j+\left(1-\mathrm{sgn}\left(q_{\mathrm{act},j}\right)\right)q_{\mathrm{act},j}=0,$$

(11)

where ${}^{\mathrm{glo}}\mathit{d}_{j,n}$ is the vector of the cable routing section and $a_j$ is the cable length. The length $a_{\mathrm{act},j}\left(q_{\mathrm{act},j}\right)$ is inside the actuator. In the presence of a gearbox, the reduction ratio and the spool radius need to be considered here. For linear direct-drive actuators, it is simply the actuator’s generalized coordinate $q_{\mathrm{act},j}$. Note that the last term is only necessary to avoid ambiguities in the solution for $q_{\mathrm{act},j}$. The vector of the cable routing section ${}^{\mathrm{glo}}\mathit{d}_{j,n}$ is defined as

$$\left[\begin{array}{c}{}^{\mathrm{glo}}\mathit{d}_{j,n}\\ 0\end{array}\right]={\mathit{T}}_{\mathrm{exo},b_{\sec }(j,n+1)}^{\mathrm{glo}}\left[\begin{array}{c}{}^{\mathrm{exo},b_{\sec }(j,n+1)}\mathit{p}_{j,n+1}\\ 1\end{array}\right]-{\mathit{T}}_{\mathrm{exo},b_{\sec }(j,n)}^{\mathrm{glo}}\left[\begin{array}{c}{}^{\mathrm{exo},b_{\sec }(j,n)}\mathit{p}_{j,n}\\ 1\end{array}\right]$$

(12)

with routing point ${}^{\mathrm{exo},b_{\sec }(j,n)}\mathit{p}_{j,n}$. The body index function $b_{\sec }(j,n)$ returns the body number related to the n-th routing point of the j-th actuator. Routing points ${}^{\mathrm{exo},b_{\sec }(j,n)}\mathit{p}_{j,n}$ are hence defined in a local body frame.

2.2.10. Direct Actuator Coupling

An alternative to the application of traction cables, rotational actuators can be coupled directly with exoskeleton bodies. This coupling is for the j-th actuator expressed with

$$\begin{array}{c}\hfill c_{\mathrm{act},j}={\displaystyle \frac{q_{\mathrm{act},j}}{i_{\mathrm{gear},j}}}-{\alpha }_1\left({\mathit{T}}_{\mathrm{exo},b_{\mathrm{rot}}\left(j\right)}^{\mathrm{exo},b_{\mathrm{act}}\left(j\right)}\right)\end{array}$$

(13)

where ${\alpha }_1$ represents the rotation around the x-axis in the Tait–Bryan rotation description of its transformation matrix argument, and $b_{\mathrm{rot}}$ is a body index function representing the body to which the mover of the j-th actuator is coupled.

2.2.11. Battery

The battery contributes significantly to the system’s weight and performance. Electrical and mechanical data of the applied cells LG ICR18650HB2 are presented in [26].

Sizing parameters of the battery are

$$\begin{array}{c}\hfill {\mathit{\kappa }}_{\mathrm{bat}}={\left[\begin{array}{cc}{n}_{\mathrm{ser}}& n_{\mathrm{par}}\end{array}\right]}^{\top },\end{array}$$

(14)

where $n_{\mathrm{ser}}$ represents the number of cells in series and $n_{\mathrm{par}}$ is the number of strands in parallel. The radius of a cell is $r_{\mathrm{cell}}$ and its height is $h_{\mathrm{cell}}$. Mechanically, the battery is represented by a cube with edge lengths $r_{\mathrm{cell}}(n_{\mathrm{par}}{n}_{\mathrm{ser}}+1)$, $h_{\mathrm{cell}}$ and $2r_{\mathrm{cell}}(1+\sin \left({60}^{\circ }\right))$. For the mass, it is estimated that

$$\begin{array}{c}\hfill m_{\mathrm{bat}}=n_{\mathrm{ser}}{n}_{\mathrm{par}}{m}_{\mathrm{cell}}+n_{\mathrm{ser}}{n}_{\mathrm{par}}·10\mathrm{g}+30\mathrm{g},\end{array}$$

(15)

where $m_{\mathrm{cell}}$ is the mass of a cell. Other inertia properties follow accordingly. The battery is assumed to be mounted rigidly at the first exoskeleton body defined with the transformation matrix ${\mathit{T}}_{\mathrm{C},\mathrm{bat}}^{\mathrm{exo},1}$. For actuation system sizing, the battery poses constraints on currents and operating cycles. Constraints on currents may be formulated based on Equation (7) as follows (in the actual implementation, current dynamics ${\mathit{L}}_{\mathrm{dq}0,j}\left(\mathit{\kappa }\right){\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\mathit{i}}_{\mathrm{dq}0,j}$ are assumed to be negligible and disregarded):

$$\begin{array}{cc}\hfill {\mathit{u}}_{\mathrm{dq}0,j}^{\top }{\mathit{u}}_{\mathrm{dq}0,j}& \le u_{\mathrm{bat}}^2\Rightarrow i_{\mathrm{q},\max ,j}\left(u_{\mathrm{bat}}\right)\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill u_{\mathrm{bat}}& =n_{\mathrm{ser}}{u}_{\mathrm{cell}}/s_{\mathrm{bat}}\hfill \end{array}$$

(17)

where the battery safety factor $s_{\mathrm{bat}}=1.1$, and the cell voltage is represented by $u_{\mathrm{cell}}$. Further, operating cycles may be considered with

$$\begin{array}{cc}\hfill E_{\mathrm{cyc}}& ={\int }_0^{t_{\mathrm{end}}}1.5{\mathit{u}}_{\mathrm{dq}0,j}^{\top }{\mathit{i}}_{\mathrm{dq}0,j}\mathrm{d}t-{\displaystyle \frac{E_{\mathrm{bat}}}{n_{\mathrm{cyc}}}}\le 0\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill E_{\mathrm{bat}}& =c_{\mathrm{cell}}{u}_{\mathrm{bat}}{n}_{\mathrm{par}}\hfill \end{array}$$

(19)

where $n_{\mathrm{cyc}}$ is the number of repetition of the cycles of the exemplary movement, $t_{\mathrm{end}}$ is its duration, and $c_{\mathrm{bat}}$ is the battery capacity.

2.2.12. Model Coupling

Besides the already introduced algebraic constraints for actuator mover bodies $c_{\mathrm{act},j}$, additional algebraic constraints ${\mathit{c}}_{\mathrm{exo},m}$ for particular human bodies $b_{\mathrm{C},\mathrm{hum}}\left(m\right)$ and exoskeleton bodies $b_{\mathrm{C},\mathrm{exo}}\left(m\right)$ are necessary to determine the movement of the exoskeleton that is dependent of human movement. The introduction of these constraints leads to a coupling description, enabling relative motion between the exoskeleton and the human. This approach is reviewed by Scherb et al. [27] as one that is well established and well suited to reproduce reality. For the m-th coupling, the definition of the human body coupling frame is

$$\begin{array}{c}\hfill {\mathit{T}}_{\mathrm{C},\mathrm{hum},m}^{\mathrm{glo}}={\mathit{T}}_{\mathrm{hum},b_{\mathrm{C},\mathrm{hum}}\left(m\right)}^{\mathrm{glo}}{\mathit{T}}_{\mathrm{C},\mathrm{hum},m}^{\mathrm{hum},b_{\mathrm{C},\mathrm{hum}}\left(m\right)}\end{array}$$

(20)

and the corresponding exoskeleton coupling frame ${\mathit{T}}_{\mathrm{C},\mathrm{exo},m}^{\mathrm{glo}}$ is necessary. The latter is expressed in the human body coupling frame

$$\begin{array}{c}\hfill {\mathit{T}}_{\mathrm{C},\mathrm{exo},m}^{\mathrm{C},\mathrm{hum},m}={\left({\mathit{T}}_{\mathrm{C},\mathrm{hum},m}^{\mathrm{glo}}\right)}^{-1}{\mathit{T}}_{\mathrm{C},\mathrm{exo},m}^{\mathrm{glo}}.\end{array}$$

(21)

Based on [17], each algebraic coupling constraint is derived as

$$\begin{array}{c}\hfill {\mathit{c}}_{\mathrm{exo},m}={\left(h_w\left({\mathit{T}}_{\mathrm{C},\mathrm{exo},m}^{\mathrm{C},\mathrm{hum},m}\right)\right)}_{w\in W_m}.\end{array}$$

(22)

Here, the vector pose function

$$\begin{array}{c}\hfill \mathit{h}\left({\mathit{T}}_{\mathrm{origin}}^{\mathrm{destination}}\right)={\left[\begin{array}{cc}{}^{\mathrm{destination}}\mathit{p}& \mathit{\alpha }\end{array}\right]}^{\top },\end{array}$$

(23)

where ${}^{\mathrm{destination}}\mathit{p}$ is the x-y-z position vector and $\mathit{\alpha }$ is the Tait–Bryan ${\mathrm{x}\text{-}\mathrm{y}}^{\prime }{\text{-}\mathrm{z}}^{″}$ rotation angle vector of its homogeneous transformation matrix argument $\mathit{T}$. Since it is only required to couple distinct DOFs, an index set $W_m\subseteq \{1,2,3,4,5,6\}$ is defined to select them. Finally, the equations of motion of the coupled system are defined as in [28]

$$\left[\begin{array}{cc}{\mathit{M}}_{\mathrm{hum}}& \mathbf{0}\\ \mathbf{0}& {\mathit{M}}_{\mathrm{exo}}\end{array}\right]\left[\begin{array}{c}{\ddot{\mathit{q}}}_{\mathrm{hum}}\\ {\ddot{\mathit{q}}}_{\mathrm{exo}}\end{array}\right]+\left[\begin{array}{c}{\mathit{k}}_{\mathrm{hum}}\\ {\mathit{k}}_{\mathrm{exo}}\end{array}\right]-{\mathit{C}}^{\top }\mathit{\lambda }=\left[\begin{array}{c}{\mathit{g}}_{\mathrm{hum}}\\ {\mathit{g}}_{\mathrm{exo}}\end{array}\right]+\left[\begin{array}{c}{\mathit{\tau }}_{\mathrm{hum}}\\ {\mathit{\tau }}_{\mathrm{exo}}\end{array}\right]+\left[\begin{array}{c}{\mathit{\tau }}_{\mathrm{grt}\&\mathrm{t},\mathrm{full}}\\ {\mathit{\tau }}_{\mathrm{el}}\end{array}\right]+\left[\begin{array}{c}{\mathit{\tau }}_{\mathrm{load}}\\ \mathbf{0}\end{array}\right]$$

(24)

$$\begin{array}{c}\hfill {\mathit{\tau }}_{\mathrm{exo}}={\left[\begin{array}{cc}\mathbf{0}& {\mathit{\tau }}_{\mathrm{act}}\end{array}\right]}^{\top }\end{array}$$

(25)

$$\begin{array}{c}\hfill {\mathit{C}}^{\top }=\left[\begin{array}{c}{\mathit{C}}_{\mathrm{hum}}^{\top }\\ {\mathit{C}}_{\mathrm{exo}}^{\top }\end{array}\right]=\left[\begin{array}{c}{{\displaystyle \frac{\partial \mathit{c}}{\partial {\mathit{q}}_{\mathrm{hum}}}}}^{\top }\\ {{\displaystyle \frac{\partial \mathit{c}}{\partial {\mathit{q}}_{\mathrm{exo}}}}}^{\top }\end{array}\right]\end{array}$$

(26)

$$\begin{array}{c}\hfill \mathit{c}={\left[\begin{array}{cc}{\mathit{c}}_{\mathrm{exo}}& {\mathit{c}}_{\mathrm{act}}\end{array}\right]}^{\top }=\mathbf{0}\end{array}$$

(27)

where ${\mathit{\tau }}_{\mathrm{hum}}$ are the human joint generalized forces. Given the trajectories of the human’s generalized coordinates and derivatives ${\mathit{q}}_{\mathrm{hum}},{\dot{\mathit{q}}}_{\mathrm{hum}},{\ddot{\mathit{q}}}_{\mathrm{hum}}$ from motion tracking, the solutions for the exoskeleton’s generalized coordinates ${\mathit{q}}_{\mathrm{exo}}$ are found with Equation (25). In Equation (24), interaction forces are represented with Lagrange multipliers $\mathit{\lambda }$. They are found using the last row of Equation (24), hence requiring the inversion of ${\mathit{C}}_{\mathrm{exo}}^{\top }$. Rearrangement leads to

$${\mathit{M}}_{\mathrm{hum}}{\ddot{\mathit{q}}}_{\mathrm{hum}}+{\mathit{k}}_{\mathrm{hum}}=\mathit{B}({\mathit{\tau }}_{\mathrm{exo}}-{\mathit{\tau }}_{\mathrm{dyn},\mathrm{exo}})+{\mathit{g}}_{\mathrm{hum}}+{\mathit{\tau }}_{\mathrm{hum}}+{\mathit{\tau }}_{\mathrm{grt}\&\mathrm{t},\mathrm{full}}+{\mathit{\tau }}_{\mathrm{load}}$$

(28)

$$\begin{array}{cc}\hfill {\mathit{\tau }}_{\mathrm{dyn},\mathrm{exo}}& ={\mathit{M}}_{\mathrm{exo}}{\ddot{\mathit{q}}}_{\mathrm{exo}}+{\mathit{k}}_{\mathrm{exo}}-{\mathit{g}}_{\mathrm{exo}}-{\mathit{\tau }}_{\mathrm{el}}.\hfill \end{array}$$

(29)

It can be seen that the coupling matrix

$$\begin{array}{c}\hfill \mathit{B}=-{\mathit{C}}_{\mathrm{hum}}^{\top }{\mathit{C}}_{\mathrm{exo}}^{-\top }\end{array}$$

(30)

projects loads from exoskeleton DOFs to human DOFs. In the following, this matrix is reduced columnwise and row-wise, such that the domain (i.e., set of generalized loads in exoskekelton DOFs) and image (i.e., set of resulting generalized loads in human DOFs) are varied. The same principle has already been applied with the transposed rotational Jacobian matrix in Equation (6).

2.2.13. Models in Methodology

Referring to Figure 1, different models are applied in the methodology. All mechanical models can be derived from Equation (28). For the first phase, only the human body model including the full GRF&T model is regarded, i.e.,

$$\begin{array}{c}\hfill {\mathit{M}}_{\mathrm{hum}}{\ddot{\mathit{q}}}_{\mathrm{hum}}+{\mathit{k}}_{\mathrm{hum}}={\mathit{g}}_{\mathrm{hum}}+{\mathit{\tau }}_{\mathrm{hum},\mathrm{uc}}+{\mathit{\tau }}_{\mathrm{grt}\&\mathrm{t},\mathrm{full}}+{\mathit{\tau }}_{\mathrm{load}}.\end{array}$$

(31)

and no electric model is considered.

In the second phase, a kinematic human body model and a kinetic exoskeleton model are applied, i.e.,

$$\begin{array}{c}\hfill \mathbf{0}=\mathit{B}({\mathit{\tau }}_{\mathrm{exo}}-{\mathit{\tau }}_{\mathrm{dyn},\mathrm{exo}})+{\mathit{\tau }}_{\mathrm{hum}}+{\mathit{\tau }}_{\mathrm{grt}\&\mathrm{t}}.\end{array}$$

(32)

As can be seen, generalized forces of the exoskeleton are projected on human DOFs. Further, a parametric electric model is considered based on Ineq. (8) and (18); see Section 2.4.

For the third phase, Equation (28) is applied. Additionally, insight into the electric quantities is gained with Equation (7).

2.3. Control

In order to optimize the actuation system, a coordinating control law for the actuators is required. It has to be ensured that actuators work in a coordinated way and that different actuation system configurations are able to generate the same support torque ${\mathit{\tau }}_{\sup ,\mathrm{hum}}$ despite their different masses and inertia. For cable-based actuation, it has to also be ensured that positive tension is kept in the cables. The presented approach is based on an inversion of the constrained dynamics of the exoskeleton. This allows for the precise adjustment of support torques. Based on Equation (28), the following is derived:

$$\begin{array}{c}\hfill {\mathit{\tau }}_{\mathrm{act}}={\tilde{\mathit{B}}}^{+}({\mathit{\tau }}_{\sup ,\mathrm{hum}}+{\mathit{B}}^{*}{\mathit{\tau }}_{\mathrm{dyn},\mathrm{exo}}-{\mathit{\tau }}_{\mathrm{grt}\&\mathrm{t}}^{*})\end{array}$$

(33)

where ${\mathit{B}}^{*}$ represents the coupling matrix reduced row-wise to the required support DOFs; ${\mathit{\tau }}_{\mathrm{grt}\&\mathrm{t}}^{*}$ is a likewise reduced vector for GRF&T from Equation (1); $\tilde{\mathit{B}}$ is additionally, columnwise, reduced to actuator DOFs; and ${(·)}^{+}$ represents a Moore–Penrose pseudoinverse.

To maintain tension in the cable, the following generalized force on the mover body is required:

$$\begin{array}{c}\hfill {\tau }_{\mathrm{act},j}\ge {\tau }_{\mathrm{dyn},\mathrm{exo},n_{\mathrm{exo}+j}}.\end{array}$$

(34)

On the other hand, the generalized force is limited by the current

$$\begin{array}{c}\hfill {\tau }_{\mathrm{act},j}\le {\tau }_{\mathrm{act},j}\left(i_{\mathrm{q},\max ,j}\right).\end{array}$$

(35)

When the actuator’s generalized coordinates are

$$\begin{array}{c}\hfill q_{\mathrm{act},j}\le 0\mathrm{mm}\vee q_{\mathrm{act},j}\ge a_{\mathrm{stroke},j},\end{array}$$

(36)

where $a_{\mathrm{stroke},j}$ is the admissible stroke of the j-th actuator, the actuator forces are adjusted as follows:

$$\begin{array}{c}\hfill {\tau }_{\mathrm{act},j}=-g_{\mathrm{exo},b_{\mathrm{act},j}}.\end{array}$$

(37)

A set $T_{\mathrm{con}}$ is defined, consisting of times when this constraint is active for at least one actuator. If bounds are violated by Equation (33), the values are replaced accordingly and the time t (when this is necessary) is included in the set $T_{\mathrm{con}}$.

This control law can be seen as ensuring that the actuation system performs adequately and is measurable. It is necessary to define support torques ${\mathit{\tau }}_{\sup ,\mathrm{hum}}$. In this methodology, this is the connection to sophisticated exoskeleton control, e.g., based on myoelectricity or other methods of human load estimation. This contribution focuses on actuation system design; hence, a simple definition based on human DOFs is chosen in Section 3.

Integration of Control in the Model

Referring to the model Equations (24)–(27), control laws can be integrated by a substitution of generalized actuator forces ${\mathit{\tau }}_{\mathrm{act}}$. This leads to varying human loads in directions of generalized coordinates ${\mathit{\tau }}_{\mathrm{hum}}$. Additionally,

$${\mathit{\tau }}_{\sup ,\mathrm{hum},\mathrm{actual},\mathrm{approx}}={\mathit{B}}^{*}({\mathit{\tau }}_{\mathrm{act},\mathrm{actual}}-{\mathit{\tau }}_{\mathrm{dyn},\mathrm{exo}})+{\mathit{\tau }}_{\mathrm{grt}\&\mathrm{t}}^{*}$$

(38)

is defined. Here, ${\mathit{\tau }}_{\mathrm{act},\mathrm{actual}}$ is the actual generalized force produced by actuators considering previously stated constraints and ${\mathit{\tau }}_{\sup ,\mathrm{hum},\mathrm{actual},\mathrm{approx}}$ is the approximated actual support torque. It is approximated due to the application of the approximated GRF&T model.

2.4. Optimization

Based on definitions of the support torques, operating time, and parametric models, optimization is applied to achieve desirable system properties. For this, the problem

$$\begin{array}{cc}\hfill \underset{\mathit{\kappa }}{min}& \sum _{j=1}^{n_{\mathrm{act}}}(m_{\mathrm{act},\mathrm{stat},j}+m_{\mathrm{act},\mathrm{mov},j})+m_{\mathrm{bat}}\hfill \\ \hfill & \mathrm{subject}\mathrm{to}\hfill \\ \hfill & \mathrm{Ineq}.\left(8\right)\hfill \\ \hfill & \mathrm{Ineq}.\left(18\right)\hfill \\ \hfill & {\mathit{b}}_{\mathrm{lower}}\le \mathit{\kappa }\le {\mathit{b}}_{\mathrm{upper}}\hfill \\ \hfill & {\mathit{A}}_{\mathrm{ineq}}\mathit{\kappa }\le {\mathit{b}}_{\mathrm{ineq}}\hfill \\ \hfill & {\mathit{A}}_{\mathrm{eq}}\mathit{\kappa }={\mathit{b}}_{\mathrm{eq}}\hfill \\ \hfill & {\int }_0^{t_{\mathrm{end}}}|{\mathit{\tau }}_{\sup ,\mathrm{hum},\mathrm{actual},\mathrm{approx}}-{\mathit{\tau }}_{\sup ,\mathrm{hum}}|f_s\mathrm{d}t\le 0,t\notin T_{\mathrm{con}}\hfill \end{array}$$

(39)

is formulated. Here, $m_{\mathrm{act},\mathrm{stat},j}$ is the stator mass of the j-th actuator, $m_{\mathrm{act},\mathrm{mov},j}$ is the mover mass, ${\mathit{b}}_{\mathrm{lower}}$ is the lower boundary for parameters $\mathit{\kappa }$, ${\mathit{b}}_{\mathrm{upper}}$ is the upper boundary, ${\mathit{A}}_{\mathrm{ineq}}$ is the linear inequality matrix, ${\mathit{b}}_{\mathrm{ineq}}$ is the linear inequality vector, ${\mathit{A}}_{\mathrm{eq}}$ is the linear equality matrix and ${\mathit{b}}_{\mathrm{eq}}$ is the linear equality vector.

2.5. Specific Actuator Model

It is necessary to determine the influence of actuation system parameters $\mathit{\kappa }$ on the mechanical and electrical models. For the battery, this is described in Equations (15)–(19). For the j-th flexible direct-drive actuator depicted in Figure 5, parameters are

$$\begin{array}{c}\hfill {\mathit{\kappa }}_{\mathrm{act},j}={\left[\begin{array}{cccc}{n}_{\mathrm{seg},\mathrm{stat},j}& n_{\mathrm{seg},\mathrm{mov},j}& a_{1,j}& a_{2,j}\end{array}\right]}^{\top }.\end{array}$$

(40)

In Figure 6, the sizing parameters of appendix lengths $a_{1,j}$ and $a_{2,j}$ are shown for the j-th actuator. Additionally, the number of segments in the stator $n_{\mathrm{seg},\mathrm{stat},j}$ and the number of segments in the mover $n_{\mathrm{seg},\mathrm{mov},j}$ may vary. A mover and a stator segment are depicted in Figure 6. Both have a length of $a_{\mathrm{act},\mathrm{seg}}=24\mathrm{mm}$. A summary of the actuation system parameters, their type, and lower and upper bounds is presented in

Table 2. Actuation system parameters.

Parameter	Lower Bound	Upper Bound	Type
No. stator segments $n_{\mathrm{seg},\mathrm{stat},j}$	4	8	integer
No. mover segments $n_{\mathrm{seg},\mathrm{mov},j}$	5	16	integer
Appendix length $a_{1,j}$	5 mm	400 mm	continuous
Appendix length $a_{2,j}$	5 mm	400 mm	continuous
No. cells in series $n_{\mathrm{ser}}$	6	14	integer
No. cells in parallel $n_{\mathrm{par}}$	1	2	integer

.

The dependence of the electromagnetic actuator model on these parameters is determined in the following. As one stator segment consists of three strands with two air-core coils each, the ohmic resistance for each strand is based on [29]:

$$\begin{array}{c}\hfill R_{\mathrm{seg}}=2·{\displaystyle \frac{{\rho }_{\mathrm{Cu}}}{A_{\mathrm{Cu}}}}{n}_{\mathrm{t}}{a}_{\mathrm{sp}}=0.942\Omega ,\end{array}$$

(41)

where ${\rho }_{\mathrm{Cu}}=1.78\Omega \mathrm{m}$ is the specific resistance of copper, $A_{\mathrm{Cu}}=0.0962{\mathrm{mm}}^2$ is the wire cross-section, $n_{\mathrm{t}}=49$ is the number of turns per coil, and $a_{\mathrm{sp}}=51.84\mathrm{mm}$ is the average length per turn.

Further electromagnetic quantities are gained from static electromagnetic finite element analyses (FEAs) using FEMM 4.2. The following considerations are based on [30]. Due to a changing conductor diameter (from 0.3 mm to 0.35 mm), new calculations are necessary. In the following, two dimensional models exploiting rotational symmetry are regarded. An asymptotic boundary condition [31] is applied to model the unbounded domain around the actuator. Coils are modeled based on the number of turns $n_t$, phase currents, and copper as the material from the program’s library. Magnets are modeled using N50 permanent magnet material and opposing axial polarity. Appendices are modeled as air. The regarded actuator has stator segments $n_{\mathrm{seg},\mathrm{stat},\mathrm{fea}}=6$ and eight mover segments.

The first set of FEAs aims at determining the permanent magnet flux linkage ${\lambda }_{\mathrm{pm},\mathrm{d},j}$ depending on the relative position from stator to mover. Starting with a distance of 24 mm between the closest mover magnet and stator coil, the mover is successively moved towards and inside the stator coils until both the mover and stator are centered axially. This corresponds to a stroke of 192 mm that is conducted in steps of 0.96 mm. No current is applied in this set. In Figure 7, the resulting flux densities from the FEAs with axially centered mover and stator are shown. The linkage between stator coils and permanent magnet flux can clearly be seen.

In Figure 8, permanent magnet flux linkage from FEA ${\lambda }_{\mathrm{pm},\mathrm{d},\mathrm{fea}}$ for varying mover positions is shown. This is calculated based on strand flux linkages and dq0-transformation. A piecewise approximation ${\lambda }_{\mathrm{pm},\mathrm{d}}$ is derived and further detailed in Equation (44).

In a second set of FEAs, self and mutual inductances of each phase are determined. For this, the absence of a mover is assumed. Currents are successively applied in each phase and magnetic fluxes in each phase are evaluated. The results are averaged, divided by the number of stator segments $n_{\mathrm{seg},\mathrm{stat},\mathrm{fea}}$, and transformed into dq0-inductances.

Quantities of the electromagnetic model of the j-th actuator from Section 2.2.6 are derived as follows:

$$\begin{array}{cc}\hfill {\mathit{R}}_j\left(\mathit{\kappa }\right)& =\mathrm{diag}\left(n_{\mathrm{seg},\mathrm{stat},j}\left[\begin{array}{ccc}{R}_{\mathrm{seg}}& R_{\mathrm{seg}}& R_{\mathrm{seg}}\end{array}\right]\right)\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill {\mathit{L}}_{\mathrm{dq}0,j}\left(\mathit{\kappa }\right)& =\mathrm{diag}\left(n_{\mathrm{seg},\mathrm{stat},j}\left[\begin{array}{ccc}0.105& 0.105& 0.0339\end{array}\right]\right)\mathrm{mH}\hfill \end{array}$$

(43)

$${\lambda }_{\mathrm{pm},\mathrm{d},j}=\left\{\begin{array}{cc}{c}_{\mathrm{pm}}{\displaystyle \frac{(a_{\mathrm{r},\mathrm{o}}-a_{\mathrm{s},\mathrm{u}})}{a_{\mathrm{act},\mathrm{seg}}}}\hfill & a_{\mathrm{r},\mathrm{o}}\le a_{\mathrm{s},\mathrm{o}}\wedge a_{\mathrm{r},\mathrm{u}}\le a_{\mathrm{s},\mathrm{u}}\hfill \\ c_{\mathrm{pm}}{n}_{\mathrm{seg},\mathrm{stat},j}\hfill & a_{\mathrm{r},\mathrm{o}}>a_{\mathrm{r},\mathrm{o}}\wedge a_{\mathrm{r},\mathrm{u}}\le a_{\mathrm{s},\mathrm{u}}\hfill \\ c_{\mathrm{pm}}(n_{\mathrm{seg},\mathrm{stat},j}-{\displaystyle \frac{a_{\mathrm{r},\mathrm{u}}-a_{\mathrm{s},\mathrm{u}}}{a_{\mathrm{act},\mathrm{seg}}}})\hfill & a_{\mathrm{r},\mathrm{o}}>a_{\mathrm{s},\mathrm{o}}\wedge a_{\mathrm{r},\mathrm{u}}>a_{\mathrm{s},\mathrm{u}}\hfill \end{array}\right.$$

(44)

$$\begin{array}{cc}\hfill a_{\mathrm{r},\mathrm{o}}& =q_{\mathrm{act},j}+n_{\mathrm{seg},\mathrm{mov},j}{a}_{\mathrm{act},\mathrm{seg}}\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill a_{\mathrm{s},\mathrm{o}}& =a_{1,j}+a_{2,j}+n_{\mathrm{seg},\mathrm{stat},j}{a}_{\mathrm{act},\mathrm{seg}}\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill a_{\mathrm{r},\mathrm{u}}& =q_{\mathrm{act},j}\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill a_{\mathrm{s},\mathrm{u}}& =a_{1,j}\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill c_{\mathrm{pm}}& =\max \left({\displaystyle \frac{{\lambda }_{\mathrm{pm},\mathrm{d},\mathrm{fea}}}{n_{\mathrm{seg},\mathrm{stat},\mathrm{fea}}}}\right)=4.2\mathrm{mWb}.\hfill \end{array}$$

(49)

with the generalized coordinate $q_{\mathrm{act},j}$ depicted in Figure 6.

For the mechanical model, masses and inertia properties are derived using cylindrical approximations with homogeneous densities of the mover and stator. For the stator, it is assumed that it consists of three hollow cylinders, one for each appendix and one for the active section. The mover is assumed to consist of one cylinder. Sizes and densities are presented in

Table 3. Cylindrical representation of actuator bodies.

Parameter	Value
Density appendix	$1000{\mathrm{kg}/\mathrm{m}}^3$
Density active section	$2870{\mathrm{kg}/\mathrm{m}}^3$
Inner radius stator	$6.5\mathrm{mm}$
Outer radius stator	$11.5\mathrm{mm}$
Density mover	$3333{\mathrm{kg}/\mathrm{m}}^3$
Radius mover	$6\mathrm{mm}$

.

2.5.1. Cable Length

The traction cable length $a_j=a_{j+3}$ is determined, such that

$$\begin{array}{cc}\hfill \min \left(q_{\mathrm{act},j}\right)& \ge 2.5\mathrm{mm}\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill \min \left(q_{\mathrm{act},j+3}\right)& \ge 2.5\mathrm{mm}.\hfill \end{array}$$

(51)

2.5.2. Actuation System Parameter Vector

Due to the mirrored alignment of the left and right side, there are three pairs of actuators with the same parameters. Hence, it is

$$\begin{array}{c}\hfill {\mathit{\kappa }}_{\mathrm{act},j}={\mathit{\kappa }}_{\mathrm{act},3+j}.\end{array}$$

(52)

Therefore, the vector of the actuation system’s parameter is

$$\begin{array}{c}\hfill \mathit{\kappa }={\left[\begin{array}{cccc}{\mathit{\kappa }}_{\mathrm{act},1}& {\mathit{\kappa }}_{\mathrm{act},2}& {\mathit{\kappa }}_{\mathrm{act},3}& {\mathit{\kappa }}_{\mathrm{bat}}\end{array}\right]}^{\top }.\end{array}$$

(53)

2.5.3. Constraints

As can be seen in Formula (39), there are several constraints for the optimization problem. For Ineq. (8), it is assumed that the nominal RMS current $i_{\mathrm{RMS},\mathrm{nom},j}=0.98\mathrm{A}$, and that the winding thermal time constant is longer than the exemplary motion task. The RMS current is equivalent to a current density of $10$. Further, the time span of the RMS integral is extended beyond the exemplary motion’s duration to $\Delta t=35\mathrm{s}$. For Ineq. (18), the number of cycles $n_{\mathrm{cyc}}=70$. Lower and upper bounds can be derived from Table 2. Linear equality constraints are employed to ensure that a stroke length is fulfilled. This is performed with

$$\begin{array}{cc}\hfill {\mathit{a}}_{\mathrm{act},\mathrm{eq}}{\mathit{\kappa }}_{\mathrm{act},j}& =0.151\mathrm{m}\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill {\mathit{a}}_{\mathrm{act},\mathrm{eq}}& =\left[\begin{array}{ccc}{a}_{\mathrm{act},\mathrm{seg}}-a_{\mathrm{act},\mathrm{seg}}& 1& 1\end{array}\right].\hfill \end{array}$$

(55)

As can be seen, a stroke length of 0.15 m is set. This is due to design space restrictions. It is found that this stroke length is not sufficient for the movers to follow the complete movement. To deal with this, the application of highly elastic elements in traction cables, allowing them to be loose when the mover is in the upper limit, is planned. Therefore, all values for $q_{\mathrm{act},j}$ are limited to 0.151 m.

Linear inequality constraints are employed such that the mover is always longer than the stator:

$$\begin{array}{cc}\hfill {\mathit{a}}_{\mathrm{act},\mathrm{ineq}1}{\mathit{\kappa }}_{\mathrm{act},j}& \le 0\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill {\mathit{a}}_{\mathrm{act},\mathrm{ineq}1}& =\left[\begin{array}{cccc}1& -1& 0& 0\end{array}\right].\hfill \end{array}$$

(57)

This is due to the lower mass of the actuator when a stroke is realized with a mover longer than the stator. Further, for mover sensing, it is required that the mover always covers the middle of the actuator. This is expressed with

$$\begin{array}{cc}\hfill {\mathit{A}}_{\mathrm{act},\mathrm{ineq}2}{\mathit{\kappa }}_{\mathrm{act},j}& \le {\mathit{b}}_{\mathrm{act},\mathrm{ineq}2}\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill {\mathit{A}}_{\mathrm{act},\mathrm{ineq}2}& =\left[\begin{array}{cccc}{a}_{\mathrm{act},\mathrm{seg}}/2& -a_{\mathrm{act},\mathrm{seg}}& 1& 0\\ a_{\mathrm{act},\mathrm{seg}}/2& -a_{\mathrm{act},\mathrm{seg}}& 0& 1\end{array}\right]\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill {\mathit{b}}_{\mathrm{act},\mathrm{ineq}2}& ={\left[\begin{array}{cc}-0.01\mathrm{m}& -0.01\mathrm{m}\end{array}\right]}^{\top }.\hfill \end{array}$$

(60)

Finally, the maximum length is expressed with

$$\begin{array}{cc}\hfill {\mathit{a}}_{\mathrm{act},\mathrm{ineq}3}{\mathit{\kappa }}_{\mathrm{act},j}& \le {\mathit{b}}_{\mathrm{act},\mathrm{ineq}3}\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill {\mathit{a}}_{\mathrm{act},\mathrm{ineq}3}& =\left[\begin{array}{cccc}{a}_{\mathrm{act},\mathrm{seg}}& 0& 1& 1\end{array}\right]\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill {\mathit{b}}_{\mathrm{act},\mathrm{ineq}3}& =0.44\mathrm{m}\hfill \end{array}$$

(63)

These linear formulae are arranged in matrices ${\mathit{A}}_{\mathrm{eq}},{\mathit{A}}_{\mathrm{ineq}}$ and vectors ${\mathit{b}}_{\mathrm{eq}},{\mathit{b}}_{\mathrm{ineq}}$.

2.5.4. Numerical Implementation

Besides the kinetic human body model, the methodology is implemented using Matlab 2021b and its Symbolic Toolbox. Derivatives of generalized coordinates are approximated using the first-order hold assumption. With the models completely defined, Formula (39) is solved using a real-coded mixed-integer genetic algorithm within the function ga. A constraint tolerance of 0.01, a function tolerance of 0.001, and an elite count of 10 are chosen for the algorithm.

3. Results

In the following, the methodology is applied to the example introduced in Section 2.1.

3.1. First Phase: Determination of Support Torques

Referring to Section 2, in the first phase, an analysis of the loads in supported human DOFs during the exemplary motion is conducted. Both flexion–extension hip degrees of freedom with associated torques ${\tau }_{\mathrm{hum},\mathrm{uc},14,x}$ and ${\tau }_{\mathrm{hum},\mathrm{uc},17,x}$ are shown based on Equation (31) in Figure 9. Human support torques are defined as

$$\begin{array}{c}\hfill {\mathit{\tau }}_{\sup ,\mathrm{hum}}=\left\{\begin{array}{cc}{\left[\begin{array}{cc}31\sin \left(\gamma \right)& 31\sin \left(\gamma \right)\end{array}\right]}^{\top }\hfill & t_{\mathrm{load},\mathrm{start}}\le t\le t_{\mathrm{load},\mathrm{end}}\hfill \\ {\left[\begin{array}{cc}0& 0\end{array}\right]}^{\top }\hfill & \mathrm{otherwise}\hfill \end{array}\right.,\end{array}$$

(64)

where $\gamma$ is the angle between the y-axis of frame {hum,1} and the x-z plane of frame {glo}. The application of a load detection is assumed, allowing the support to be activated only during load-handling phases. This function is depicted for the exemplary motion in Figure 9. A support of approx. 20% is achieved.

3.2. Second Phase: Actuation System Optimization

After the definition of support torques, the actuation system is optimized. For this, Formula (39) is solved. The result from this is depicted in Figure 10 and the corresponding parameters may be found in

Table 4. Parameters after optimization.

Parameter	Value
No. stator segments $n_{\mathrm{seg},\mathrm{stat},j}$	6
No. mover segments $n_{\mathrm{seg},\mathrm{mov},j}$	10
Appendix length $a_{1,j}$	89 mm
Appendix length $a_{2,j}$	158 mm
No. cells in series $n_{\mathrm{ser}}$	13
No. cells in parallel $n_{\mathrm{par}}$	1
Actuation system mass	2.57 kg
Exoskeleton mass	5.03 kg

.

In Figure 11a, the support torque ${\tau }_{\sup ,\mathrm{hum},1}$ and approximated actual support torque ${\tau }_{\sup ,\mathrm{hum},\mathrm{actual},\mathrm{approx},1}$ are presented. It can be seen that there are times (belonging to the set $T_{\mathrm{con}}$) when the desired support is not fulfilled. The reason for this lies in the limited actuator stroke and the limitations it is imposing on actuator forces ${\tau }_{\mathrm{act},j}$ according to Equation (37).

In Figure 11b, the current $i_{\mathrm{q},3}$, voltage $u_{\mathrm{q},3}$, and force ${\tau }_{\mathrm{act},3}$ of a resulting actuator during the specified motion are shown. Differences between the course of the current and force are mostly due to varying permanent magnet flux linkages ${\lambda }_{\mathrm{pm},\mathrm{d},3}$.

3.3. Third Phase: Actuation System Validation

Support torques in the second phase are calculated based on the simplified GRF&T model from Equation (2). Hence, it is necessary to validate them based on Equation (28). In the following, the actual support torque

$$\begin{array}{c}\hfill {\mathit{\tau }}_{\sup ,\mathrm{hum},\mathrm{acutal}}={\mathit{\tau }}_{\mathrm{hum},\mathrm{uc}}-{\mathit{\tau }}_{\mathrm{hum}}\end{array}$$

(65)

is regarded in Figure 12. It can be seen that support torques ${\mathit{\tau }}_{\sup ,\mathrm{hum}}$ are approximately achieved in both hip joints. In Figure 12a, at the beginning of support at $t=4.7$ s, it is observed that the support torque is delivered slightly too early. This offset is due to different ways of approximating derivatives of generalized coordinates in inverse dynamics calculations in OpenSim for the human body model and in Matlab for the exoskeleton model.

4. Discussion and Conclusions

An integrated design optimization methodology for actuation systems in exoskeletons is presented. The methodology integrates mostly nonlinear dynamic models for humans, exoskeletons, and actuation systems. Further, a general formulation for the interaction of human and exoskeleton is provided. The generality of the coupling formulation enables the optimization of actuation systems for various exoskeletons. A continuous utilization of these models from the definition of support torques over the derivation of an approach for control, to the formulation of an minimization problem for system design is presented.

4.1. Comparison with an Approach Derived from Previous Studies

As presented in Section 1, previous studies show less depth in their integrated models, especially regarding the coupling of human and exoskeleton models as well as actuator models. To compare the suggested methodology with an approach reflecting them, this depth is reduced in the following way:

• A description of coupling between human and exoskeleton models based on rigid coupling assumption [10]. For this, the coupling description in Equation (27) is reduced, such that

  $$\begin{array}{cc}\hfill {\mathit{c}}_{\mathrm{exo},2}& ={\left[\begin{array}{cc}{q}_{\mathrm{exo},7}+q_{\mathrm{hum},7}& q_{\mathrm{exo},8}+q_{\mathrm{hum},9}\end{array}\right]}^{\top }\hfill \end{array}$$

  (66)

  $$\begin{array}{cc}\hfill {\mathit{c}}_{\mathrm{exo},3}& ={\left[\begin{array}{cc}{q}_{\mathrm{exo},9}+q_{\mathrm{hum},43}& q_{\mathrm{exo},10}+q_{\mathrm{hum},45}\end{array}\right]}^{\top }\hfill \end{array}$$

  (67)

  $$\begin{array}{cc}\hfill {\mathit{c}}_{\mathrm{exo},4}& ={\left[\begin{array}{cc}{q}_{\mathrm{exo},11}+q_{\mathrm{hum},52}& q_{\mathrm{exo},12}+q_{\mathrm{hum},54}\end{array}\right]}^{\top },\hfill \end{array}$$

  (68)

  i.e., it is assumed that the exoskeleton’s and human’s joint trajectories are the same. The generalized coordinates $q_{\mathrm{exo},7},q_{\mathrm{exo},8}$ describe the motion of the exoskeleton’s back brace, which is equalized with the corresponding DOFs of the human spine body segment. Further, flexion–extension and the rotation of the human upper leg body segments is equalized with the corresponding exoskeleton leg link DOFs. Further details on the DOF numbering may be found in the OpenSim files provided in the Supplementary Materials.
• An actuator model based on force constant and simplified voltage equation [14]. The electromechanical model in Equation (9) is replaced with a generalized force:

  $$\begin{array}{c}\hfill {\tau }_{\mathrm{act},j}=c_{\mathrm{f},j}{i}_{\mathrm{q},j}\end{array}$$

  (69)

  with the force constant $c_{\mathrm{f},j}=(3/2)c_{\mathrm{pm}}{n}_{\mathrm{seg},\mathrm{stat},j}{p}_j$. The voltage Equation (7) is replaced by a consideration of ohmic resistances and motion-induced voltages:

  $$\begin{array}{cc}\hfill u_{\mathrm{q},j}& =n_{\mathrm{seg},\mathrm{stat},j}{R}_{\mathrm{seg}}{i}_{\mathrm{q},j}+c_{\mathrm{pm}}{n}_{\mathrm{seg},\mathrm{stat},j}{p}_j{\dot{q}}_{\mathrm{act},j}\hfill \end{array}$$

  (70)

  $$\begin{array}{cc}\hfill u_{\mathrm{d},j}& =u_{0,j}=0\mathrm{V}.\hfill \end{array}$$

  (71)
  Based on this and Equation (16), the maximum current $i_{\mathrm{q},\max ,j}$ is calculated.
• Neglect of dynamic influences ${\mathit{\tau }}_{\mathrm{dyn},\mathrm{exo}}$ from exoskeleton on the human body in the control approach [10]. The calculation of the desired generalized actuator forces ${\mathit{\tau }}_{\mathrm{act}}$ is simplified as

  $$\begin{array}{c}\hfill {\mathit{\tau }}_{\mathrm{act}}={\tilde{\mathit{B}}}^{+}{\mathit{\tau }}_{\sup ,\mathrm{hum}}.\end{array}$$

  (72)
  When an actuator is at its stroke limit, i.e., when Formula (36) is true,

  $$\begin{array}{c}\hfill {\tau }_{\mathrm{act},\mathrm{eas},j}=\left\{\begin{array}{c}2\mathrm{N}\hfill \\ c_{\mathrm{f},j}{i}_{\mathrm{q},\max ,j}\hfill & i_{\mathrm{q},j}\ge i_{\mathrm{q},\max ,j}\hfill \\ {\tau }_{\mathrm{act},j}\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

  (73)
• Constraints in the minimization problem in Formula (39) change. Due to the simplified definition of the electromechanical model in Equation (69), it must now be constrained that the mover fully covers the active section of the stator, i.e.,

  $$\begin{array}{cc}\hfill {\mathit{A}}_{\mathrm{act},\mathrm{ineq}2}& =\left[\begin{array}{cccc}{a}_{\mathrm{act},\mathrm{seg}}& -a_{\mathrm{act},\mathrm{seg}}& 1& 0\\ a_{\mathrm{act},\mathrm{seg}}& -a_{\mathrm{act},\mathrm{seg}}& 0& 1\end{array}\right]\hfill \end{array}$$

  (74)

  $$\begin{array}{cc}\hfill {\mathit{b}}_{\mathrm{act},\mathrm{ineq}2}& ={\left[\begin{array}{cc}0\mathrm{m}& 0\mathrm{m}\end{array}\right]}^{\top }.\hfill \end{array}$$

  (75)
  Further, it is necessary to adapt the generalized force constraint, i.e., the last line of Formula (39) is changed to

  $$\begin{array}{c}\hfill {\int }_0^{t_{\mathrm{end}}}{\tau }_{\sup ,\mathrm{hum},1}+{\tau }_{\sup ,\mathrm{hum},2}-\sum _{j=1}^6\left|{\tau }_{\mathrm{act},\mathrm{actual},j}{a}_{\mathrm{act},j}\right|\mathrm{d}t\le 0,t\notin T_{\mathrm{con}}.\end{array}$$

  (76)
  To find a solution for the modified minimization problem, it is required to reduce the support torque requirement:

  $$\begin{array}{c}\hfill {\mathit{\tau }}_{\sup ,\mathrm{hum}}=\left\{\begin{array}{cc}{\left[\begin{array}{cc}29.76\sin \left(\gamma \right)& 29.76\sin \left(\gamma \right)\end{array}\right]}^{\top }\hfill & t_{\mathrm{load},\mathrm{start}}\le t\le t_{\mathrm{load},\mathrm{end}}\hfill \\ {\left[\begin{array}{cc}0& 0\end{array}\right]}^{\top }\hfill & \mathrm{otherwise}\hfill \end{array}\right.\end{array}$$

  (77)

  and to adapt the maximum length constraint in Equation (63):

  $$\begin{array}{cc}\hfill {\mathit{b}}_{\mathrm{act},\mathrm{ineq}3}& =0.46\mathrm{m}.\hfill \end{array}$$

  (78)

The results of this approach are presented in

Table 5. Parameters of approach reflecting previous studies after optimization.

Parameter	Value
No. stator segments $n_{\mathrm{seg},\mathrm{stat},j}$	6
No. mover segments $n_{\mathrm{seg},\mathrm{mov},j}$	13
Appendix length $a_{1,j}$	151 mm
Appendix length $a_{2,j}$	168 mm
No. cells in series $n_{\mathrm{ser}}$	14
No. cells in parallel $n_{\mathrm{par}}$	1
Actuation system mass	2.93 kg
Exoskeleton mass	5.39 kg

. Compared to the results of the approach presented in Section 2 in Table 4, there is an increase in the actuation system’s mass of approx. 14%. The volume of each actuator is increased by approx. 18%. This mostly stems from a lower modeling depth of the actuators. In this section, no detailed electromagnetic modeling is applied. This leads to the constraint that mover and active stator sections of each actuator must always fully overlap. In the design process, this leads to a larger actuator.

The control approach has been adapted in this section according to Equation (72). Actual generalized actuator forces ${\mathit{\tau }}_{\mathrm{act},\mathrm{eas}}$ are applied and validated based on Equation (28). The actual support torque

$$\begin{array}{c}\hfill {\mathit{\tau }}_{\sup ,\mathrm{hum},\mathrm{acutal},\mathrm{eas}}={\mathit{\tau }}_{\mathrm{hum},\mathrm{uc}}-{\mathit{\tau }}_{\mathrm{hum}}\left({\mathit{\tau }}_{\mathrm{act},\mathrm{eas}}\right)\end{array}$$

(79)

is considered. Results of the application of the control law from this section are presented in Figure 13. It can be seen that a significant difference is present. The RMS error from the approximated support torque ${\tau }_{\sup ,\mathrm{hum},\mathrm{actual},\mathrm{approx},1}$ to the actual support torque ${\tau }_{\sup ,\mathrm{hum},\mathrm{acutal},14,\mathrm{x}}$ is 0.91 Nm and to the actual support torque with simple control approach ${\tau }_{\sup ,\mathrm{hum},\mathrm{acutal},\mathrm{eas},14,\mathrm{x}}$ is 1.5 Nm. The observed deviation stems mostly from the neglect of the actuation system and exoskeleton weight, its inertia, and the approximation of kinematics in Equation (72). Since direct-drive actuators are applied, the influence of drive inertia is comparatively low.

Regarding the comparison of the suggested methodology with an approach derived from previous studies, it is concluded that a higher modeling depth increases the suitability for mass and volume reduction as well as the fulfillment of complex motion and force requirements. Further advantages regarding actuation system individualization, which is also possible with the suggested methodology, are described in [10].

4.2. Assumptions

Establishing this methodology required several assumptions. These assumptions are substantiated with work from other authors that is cited within affected sections. The exoskeleton is assumed to be coupled to the human body with distinct idealized DOFs, neglecting additional elastic DOFs. The human body is represented by a multibody model in a tree structure, where GRF&T is calculated based on an optimization approach. The battery is electrically modeled with simple energy balances and the influence of power electronic is disregarded. The consideration of high-frequency pulse width modulation in power electronics would drastically increase the modeling complexity and computational cost. For the application example, losses from power electronics are significantly smaller than ohmic losses in the actuators. For other cases, they could be considered using approximations from [32]. Operating limits of the actuators are considered using RMS current values. A thermal model would allow for deeper insights into this. Since no maximum current constraint is introduced, the possibility of demagnetization is neglected. Further, it is assumed that power electronics are chosen appropriately with regard to current and voltage specifications. The presented methodology can be seen as a solid basis for the implementation of improvements in these fields.

4.3. Conclusion and Future Work

An exemplary application demonstrates the usefulness of the methodology in the design process. It is shown that it enables the identification of problems (stroke length is not sufficient) that are usually realized later in the design process. Further, its advantage in regard to mass and volume minimization and motion requirement fulfillment over an approach derived from previous studies is demonstrated.

In future work, a prototype on the basis of this methodology will be presented. The motor drivers Texas Instruments DRV 83x2 Series will be applied, driving delta-connected strands of each actuator with the required currents. The prototype will enable many real-world validations. For the validation of the control approach, a test bench will be applied such that support torques become measurable. Further, the methodology will be applied for the optimization of BLDC-based actuation for faster movements. This will underline the usefulness of the ability of the methodology to compensate for drive inertia and optimize actuation systems based on this.