A geometry- and muscle-based control architecture for synthesising biological movement

A key problem for biological motor control is to establish a link between an idea of a movement and the generation of a set of muscle-stimulating signals that lead to the movement execution. The number of signals to generate is thereby larger than the body’s mechanical degrees of freedom in which the idea of the movement may be easily expressed, as the movement is actually executed in this space. A mathematical formulation that provides a solving link is presented in this paper in the form of a layered, hierarchical control architecture. It is meant to synthesise a wide range of complex three-dimensional muscle-driven movements. The control architecture consists of a ‘conceptional layer’, where the movement is planned, a ‘structural layer’, where the muscles are stimulated, and between both an additional ‘transformational layer’, where the muscle-joint redundancy is resolved. We demonstrate the operativeness by simulating human stance and squatting in a three-dimensional digital human model (DHM). The DHM considers 20 angular DoFs and 36 Hill-type muscle–tendon units (MTUs) and is exposed to gravity, while its feet contact the ground via reversible stick–slip interactions. The control architecture continuously stimulates all MTUs (‘structural layer’) based on a high-level, torque-based task formulation within its ‘conceptional layer’. Desired states of joint angles (postural plan) are fed to two mid-level joint controllers in the ‘transformational layer’. The ‘transformational layer’ communicates with the biophysical structures in the ‘structural layer’ by providing direct MTU stimulation contributions and further input signals for low-level MTU controllers. Thereby, the redundancy of the MTU stimulations with respect to the joint angles is resolved, i.e. a link between plan and execution is established, by exploiting some properties of the biophysical structures modelled. The resulting joint torques generated by the MTUs via their moment arms are fed back to the conceptional layer, closing the high-level control loop. Within our mathematical formulations of the Jacobian matrix-based layer transformations, we identify the crucial information for the redundancy solution to be the muscle moment arms, the stiffness relations of muscle and tendon tissue within the muscle model, and the length–stimulation relation of the muscle activation dynamics. The present control architecture allows the straightforward feeding of conceptional movement task formulations to MTUs. With this approach, the problem of movement planning is eased, as solely the mechanical system has to be considered in the conceptional plan. Supplementary Information The online version supplementary material available at 10.1007/s00422-020-00856-4.

of muscles which relates the neuronal stimulation to muscular activity A(t) ∈ R n MTU that drives the muscle model (Haeufle et al., 2014). The muscles produce forces F MTU (t) ∈ R n MTU that act on the rigid bodies of the skeletal system. The resultant joint torques F MTU depend on the respective moment arms ∂L MTU ∂Q . In combination with external forces, this results in a movement of the DoFs Q(t) ∈ R n DoF of the body.
The musculoskeletal model allmin consists of n RGB = 15 rigid bodies (see Table 1). The rigid bodies are connected via 14 joints (see Table 2) including n DoF = 20 degrees of freedom. Each Degree of Freedom (DoF) (except for the wrist) is controlled by an Agonistic-Antagonistic Setup (AAS) beeing congruent with the Elementary Biological Drive (EBD) as described by Schmitt et al. (2019). The musculoskeletal model is actuated by n MTU = 36 Muscle Tendon Units (MTU) (see Table 4 and Figure 1a for first impression).
The model is implemented in C/C++ code within our in-house multi-body simulation code demoa.

The Multibody System
The skeletal system is modeled as a chain of rigid bodies, connected by rotational joints and described by differential equations. The resulting DoFs Q(t) = [q 1 (t), . . . , q n DoF (t)] T ∈ R n DoF of these rotational joints describe the movement of the rigid bodies over time and are referred to as generalized coordinates. For the equations of motion, a Lagrangian formulation with the generalized coordinates Q(t) as state variables is realized, which can be set up algorithmically, e.g. as described by Legnani et al. (1996). The evaluation of this algorithm leads to the differential equation of motion of the rigid body system in the form where M ∈ R n DoF ×n DoF is the mass matrix, C ∈ R n DoF is a vector of gravitational, centrifugal and Coriolis forces and F ∈ R nDoF is a vector of forces (internal and external) acting on the mechanical part of the system.
Hereby F includes forces, e.g. due to contact of the body to the environment (external), as well as forces of the biological structures, such as muscles, joint limitations (internal).

Joint limitations
The joint limitations are modeled as linear one-sided spring-damper elements, acting directly on the respective DoF: with the lower and upper threshold angles q l/u , corresponding to the respective Range of Motion (RoM) (

Muscles
The muscles are modeled as lumped muscles, i.e. they represent a multitude of anatomical muscles and motor units. A list of all included muscle elements can be found in Table 4.
The MTU structure is modeled using an extended Hill-type muscle model as described in Haeufle et al. (2014) with muscle activation dynamics as introduced by Hatze (1977) and simplified by Rockenfeller and Günther (2018). Herein, the muscles are activated using a 1 st order differential equation of normalized calcium ion concentration (Rockenfeller et al., 2014)γ and a nonlinear mapping onto the muscles activation with (γ(t), l CE (t)) = (γ(t) · ρ(l CE )) ν and ρ(l CE ) = opt · l CE lopt = γ c · ρ 0 · l CE lopt . The parameter values are chosen muscle non specifically and are given in Table 5.
The muscle model is a macroscopic model consisting of four elements: the Contractile Element (CE), the Parallel Elastic Element (PEE), the Serial Elastic Element (SEE) and Serial Damping Element (SDE), as illustrated in Figure 1b. Herein, the muscle fibers and their contraction dynamics are described by a contractile element (CE) representing the cross-bridge-cycle of the myosin heads and a parallel elastic element (PEE) representing the passive connective tissue in the muscle belly. The viscoelastic properties of tendons are approximated using a series elastic element (SEE) and a serial damping element (SDE).
The inputs to the muscle model are the length of the MTU l MTU , the contraction velocity of the MTUl mtu and the muscular activity a. The output of the muscle model is a one-dimensional muscle force f MTU . This force drives the movement of the skeletal system.
For the routing of the muscle path around the joints, deflection ellipses are implemented as described by Hammer et al. (2019). The muscle path can move within these ellipses and is deflected as soon as it touches the boundary. For the investigations presented here, we set the length of both half-axes of all ellipses to zero, resulting in fixed via points. The position of these points can be found in Table 3. The resulting moment arms translate the muscle force F MTU to generalized forces F MTU acting on the DoFs of the system All in all, the governing model dependencies for all muscles i = 1, ..., n are: where Q = {q i } n DoF i=1 denotes a generalized state vector that contains all joint angles and contain the muscle forces, the joint limitation forces and the external forces, respectively.  Table 1: List of all bodies included in the model with their mechanical properties with m: mass, r x ,r y : radius in x and y direction, h z : height in z direction, d 1 : distance proximal joint to the body's center of mass and d 2 : distance center of mass to distal joint. The spine body has an underlying curvature based on Kitazaki and Griffin (1997