Hypothetical neural control of human bipedal walking with voluntary modulation

Abstract

A hypothetical neuromusculoskeletal model is developed to simulate human normal walking and its modulated behaviors. A small set of neural periodic patterns drive spinal muscle synergies which in turn lead to specific pattern of muscle activation and supraspinal feedback systems maintain postural balance during walking. Then, the model demonstrates modulated behaviors by superimposing voluntary perturbations on the underlying walking pattern. Motions of kicking a ball and obstacle avoidance during walking are simulated as examples. The superposition of the new pulse command to a set of invariant pulses representing spino-locomotor is sufficient to achieve the coordinated behaviors. Also, forward bent walking motion is demonstrated by applying similar superposition. The composition of activations avoids a complicated computation of motor program for a specific task and presents a simple control scheme for different walking patterns.

Appendices

Appendix A: Anthropometry used to model a human body

The length of each segment is represented with respect to the total body’s height h _t (=180 cm) and mass m _t (=80 kg) based on [41]. trunk, thigh, shank, and foot masses are respectively 0.678 m _t, 0.1 m _t, 0.047 m _t, and 0.015 m _t; their moments of inertia are respectively 0.031 m _t h ²_t , 6.262 × 10⁻⁴ m _t h ²_t , 2.566 × 10⁻⁴ m _t h ²_t , and 4.976 × 10⁻⁶ m _t h ²_t ; their lengths are respectively 0.47 h _t, 0.245 h _t, and 0.246 h _t; their COM distances from lower end are respectively 0.235h _t, 0.1389 h _t, and 0.1395 h _t; foot is modeled as a triangle with height 0.039 h _t and length 0.152 h _t; foot COM is located at 0.0195 h _t high from bottom and 0.0304 h _t ahead from heel.

Appendix B: Foot interaction with the ground

The vertical ground reaction force is modeled by:

$$ F_{gy}^{i}=(K_{gy}(f_{gy}(x^{i})-y^{i})- B_{gy}\dot{y}^{i})\cdot1\left[f_{gy}(x^{i})-y^{i},0\right]_{+} $$

(29)

where (x ⁱ, y ⁱ) indicates the positions of either heel or toe with i = heel, toe. f _gy (x ⁱ) represents the ground profile as a function of x ⁱ.

If the toe or heel reaches zero horizontal velocity, the horizontal reaction force is modeled by a spring and damper system as long as the horizontal reaction force is smaller than the maximal friction force.

$$ F_{gx}^{i}=(K_{gx}(x_{o}^{i}-x^{i})- B_{gx}\dot{x}^{i})\cdot1\left[f_{gy}(x^{i})-y^{i},0\right]_{+}\,\, \hbox{if}\left|F_{gx}^{i}\right|\leq\left|\mu_{s}F_{gy}^{i}\right|,\quad i=\hbox{heel, toe} $$

(30)

where x ⁱ _o is a location where either heel or toe touches the ground initially and μ _s is the static frictional coefficient.

Otherwise, the horizontal reaction force is modeled as a dynamic friction force.

$$ F_{gx}^{i}=-\mu_{k}F_{gy}^{i}\hbox{sgn}(\dot{x}^{i}) $$

(31)

where μ _k is the dynamic frictional coefficient.

Appendix C: Simulation parameter values

1.1 C.1. Normal walking

• Transmission neural delays

Closed-loop transmission delays are conservatively taken to be 60, 70, and 80 ms for long-loop response to and from the hip, knee, ankle, respectively based on 50 m/s neural conduction velocity, and five synaptic delays of less than 1 ms. Therefore, ${\mathbf{T}}_{\rm spr} + {\mathbf{T}}_{\rm sp} + {\mathbf{T}}_{\rm pr} = {\left[{\begin{array}{*{20}c} {80}& {70}& {60}\\ \end{array}} \right]}^{\rm T}.$ For simulation, T _spr = T _sp + T _pr, T _sp = T _pr are assumed. EC(s) also lags neural signals.

• Foot interaction to the ground

$$K_{gy} = 30,000, \enspace B_{{gy}} = 500, \enspace K_{{gx}} = 10,000, \enspace B_{{gx}} = 1,000; \enspace \mu_{k} = 0.6, \enspace \mu_{s} = 1.2.$$

• Spinal pattern generator

$$ f_{{\rm PG}} = 1.3\quad \hbox{and} \quad m_{{\rm PG}} = 1.2. $$

$$ {\mathbf{W}}_{\rm PG} = {\left[{\begin{array}{*{20}c} {0.3}& {0}& {0}& {0}& {0.8}& {0}& {0}& {0}& {0}\\ {0}& {0.29}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0.4}& {0}& {0.64}& {0}& {0.9}& {0}& {0.35}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0.4}& {0}& {0}& {0.3}\\ {0}& {0.1}& {0}& {0.8}& {0}& {0}& {0.4}& {0.1}& {0}\\ \end{array}} \right]}^{\rm T} $$

(Table 2).

Table 2 Parameters for periodic pattern generation

• Spinal segmental inhibition

$$ \theta_{{\rm th,a}} = 0.35; \quad \theta_{{\rm th,k}} = - 0.35; \quad \theta_{{\rm th,h}} = 0.55. $$

$$ {\mathbf{W}}_{{\rm reflex}} = \rho {\left[{\begin{array}{*{20}c} {0}& {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}& {0}& {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0} \\\end{array}} \right]}^{\rm T} $$

where ρ is a sufficient large number (ρ > m _PG).

• Supraspinal system

Cerebro-cerebellar feedback control i _a  = 0.2, i _r  = 100,f = 0.6, g _b  = 0, g _k  = 3,

$$ {\mathbf{W}}_{\rm C} = {\left[{\begin{array}{*{20}c} {0}& {0}& {2}& {- 5}& {6}& {- 1}& {3}& {- 1}& {- 3}\\ \end{array}} \right]}^{\rm T} $$

Estimate of COM $(\hat{x}_{\rm com}): p_{1} = 0.97,\; p_{2} = 0.53,\; p_{3} = {0.14}.$

• Vestibulospinal reflex feedback control

$$ k_{\rm p} = 3, \quad b_{\rm p} = 0.9. $$

$$ {\mathbf{W}}_{\rm ves} = {\left[{\begin{array}{*{20}c} {0.132}& {- 0.092}& {0}& {0}& {0}& {0}& {0.049}& {- 0.054}& {0}\\ \end{array}} \right]}^{\rm T} $$

• Initial positions

$$ \theta_{\rm a} = 0.2, \; \theta_{\rm k} = 0, \; \theta_{\rm h} = - 0.2\quad \hbox{for\; right\; leg};\, \theta_{\rm a} = 0, \; \theta_{\rm k} = - 0.1, \; \theta_{\rm h} = 0.4 \quad \hbox{for\; left\; leg}. $$

• Initial velocities

$$ \dot{\theta}_{\rm a} = (f_{{\rm PG}} + 1)/2, \dot{\theta}_{\rm k} = - (f_{{\rm PG}} + 1)/2, \dot{\theta}_{\rm h} = (f_{{\rm PG}} + 1)/2 \quad \hbox{for \; right \; leg}; $$

$$ \dot{\theta}_{\rm a} = - (f_{{\rm PG}} + 1)/2, \dot{\theta}_{\rm k} = (f_{{\rm PG}} + 1)/2, \dot{\theta}_{\rm h} = - (f_{{\rm PG}} + 1)/2 \quad \hbox{for\; left \; leg}. $$

• Reference signals

$$ u_{\rm ref} = 0.25; \quad \theta_{\rm tr,ref} = 0. $$

1.2 C.2. Forward bent walking

Other parameters are the same as in two except the followings:

• Initial positions

$$ \begin{aligned} \theta_{\rm a} &= 0.2, \; \theta_{\rm k} = 0, \; \theta_{\rm h} = 0.7 \quad\hbox{for\; right\; leg};\\ \theta_{\rm a} &= 0, \; \theta_{\rm k} = - 0.1, \; \theta_{\rm h} = 1.3 \quad \hbox{for\; left\; leg}. \end{aligned} $$

• Reference signals

$$ \theta_{{\rm tr,ref}} = 0.7. $$

• Neuronal network of spinal pattern generation

$$ {\mathbf{W}}_{{\rm PG}} = {\left[{\begin{array}{*{20}c} {0.4}& {0}& {0}& {0}& {0}& {0.1}& {0}& {0.4}& {0.2}\\ {0}& {0.3}& {0.5}& {0}& {0}& {0}& {0}& {0.3}& {0.2}\\ {0.5}& {0}& {0.64}& {0}& {1.0}& {0}& {0.35}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0.1}& {0}& {0.2}& {0}\\ {0.4}& {0.1}& {0}& {0.8}& {0}& {0}& {0.4}& {0.1}& {0}\\ \end{array}} \right]}^{\rm T} $$