Flatness of musculoskeletal systems under functional electrical stimulation

Abstract

Control of musculoskeletal yy system through functional electrical stimulation (FES) still remains a complex and a challenging process. Indeed, the used musculoskeletal models are often complex and highly nonlinear, which makes their control and inversion (getting appropriate inputs from a desired outputs) very difficult. On the other hand, the system flatness has been proved to be an efficient method for nonlinear system control, since in this technique, the nonlinear system can be controlled more easily through its flat outputs. Therefore, it is very promising to apply this control technique on the musculoskeletal system, to overcome its problems, which has never been explored so far. The aim of this work is to explore the flatness technique and its feasibility on the knee joint musculoskeletal system in dynamic condition, controlled by electrically stimulated quadriceps muscle. A mathematical proof developed in the current work highlights that the two-input musculoskeletal system is flat, where two flat outputs are the muscle stiffness and the knee joint angle. It also shows that the single-input musculoskeletal system is not flat. These results are crucial for flatness-based control of musculoskeletal systems, since this model in literature deals with a single input. Simulation results in open-loop control of two-input system highlight the consistency of the mathematical proof, and the applicability of this technique on the musculoskeletal system, where its simulated outputs fit perfectly with the desired ones if the model is considered perfect. When, one parameter of the system is not well estimated (10% of error), simulations show limits of open-loop control, with a joint angle rms deviation of 4%; hence, the closed-loop control should be considered.

Appendix: Ruled manifold criterion of one-input musculoskeletal model

Considering only one input (α) and u_ch as a constant, we can define the input (10)

$$v = \alpha u_{ch}$$

and parameters

$$u_{c} = u_{ch}; ~c_{1}=\frac{r_{1}}{L_{c0}}; ~c_{2}=-\frac{r_{1}}{J}; ~ c_{3}=\frac{m g d}{J}; ~c_{4}=-\frac{F_{v}}{J}$$

Therefore, the one-input musculoskeletal model can described by Eq. 11, by considering u_c as a constant:

The first stage of ruled manifold criterion consists in expressing the implicit form (21) of the model by eliminating the input control v from Eq. 34b, c. Let us extract the input v from Eq. 34c as follows:

$$ \begin{array}{@{}rcl@{}} && (1+p x_{1} + q x_{2})~\dot x_{2} = F_{0}(x_{3})~v - x_{2}~u_{c} \\ && \qquad\qquad\qquad\qquad\quad \!+ (b x_{1} + a x_{2})~c_{1}~x_{4} \end{array} $$

(35)

$$ \begin{array}{@{}rcl@{}} &&\Leftrightarrow v = \frac{(1+p x_{1} + q x_{2})~\dot x_{2} + x_{2}~u_{c} - (b x_{1} + a x_{2})~c_{1}~x_{4}}{F_{0}(x_{3})}\\ \end{array} $$

(36)

From Eq. 36b, we get

$$ \begin{array}{@{}rcl@{}} (1+p x_{1} + q x_{2})~\dot x_{1} &=& a c_{1} x_{1} x_{4} - \left( x_{1}~(1+p x_{1} + q x_{2}) \right. \\ &&\left. +q x_{2} x_{1}\right)u_{c}+ \left( K_{0}(x_{3})(1+p x_{1} \right.\\ &&\left.+ q x_{2}) - q F_{0}(x_{3}) x_{1}\right)~v \end{array} $$

(37)

Then, by replacing v in this equation, we obtain an equation without control input:

$$ \begin{array}{@{}rcl@{}} &&(1 +p x_{1} + q x_{2})~F_{0}(x_{3})\dot x_{1}\\ &=& a F_{0}(x_{3}) c_{1} x_{1} x_{4} - (x_{1}~(1+p x_{1} + q x_{2}) + q x_{2} x_{1})\\ &&\times F_{0}(x_{3}) u_{c}+ (K_{0}(x_{3})(1+p x_{1} + q x_{2}) - q F_{0}(x_{3}) x_{1})\\ &&\times ((1+p x_{1} + q x_{2})~\dot x_{2}+ x_{2}~u_{c} - (b x_{1} + a x_{2}) c_{1} x_{4}) \end{array} $$

Based on this input elimination and Eq. 34d, e, we define the implicit form of the one-input model as $F(\mathbf {x},\dot {\mathbf { x}})=0$:

$$ \left\{\begin{array}{l} (1 + p x_{1} + q x_{2})~F_{0}(x_{3}) \dot x_{1} + \left( x_{1}~(1+p x_{1} + q x_{2}) + q x_{2} x_{1}\right) F_{0}(x_{3}) u_{c} \\ ~~~- \left( K_{0}(x_{3})(1+p x_{1} + q x_{2}) - q F_{0}(x_{3}) x_{1} \right)~\left( (1 + p x_{1} + q x_{2})~\dot x_{2} + x_{2}~u_{c} - (b x_{1} + a x_{2})~c_{1}~x_{4}\right) \\ ~~~ - a F_{0}(x_{3}) c_{1} x_{1} x_{4} = 0\\ \dot x_{3} - x_{4} = 0\\ \dot x_{4} + c_{2}~x_{2} + c_{3}~\cos(x_{3}) + c_{4}~x_{4} = 0 \end{array}\right. $$

(38)

Let us then explore a necessary condition of flatness by using the negation of ruled manifold criterion (24). Then we have

$$\forall (\lambda,\mu) \in \mathbb{R}^{2n} $$

where λ = (λ₁,λ₂,λ₃,λ₄) and μ = (μ₁,μ₂,μ₃,μ₄).

Considering, from the implicit form of the model (38), the following set of equation, which corresponds to F(λ,μ) = 0:

$$ \left\{\begin{array}{l} (1 + p \lambda_{1} + q \lambda_{2})~F_{0}(\lambda_{3}) \mu_{1} + (\lambda_{1}~(1 + p \lambda_{1} + q \lambda_{2}) + q \lambda_{2} \lambda_{1}) F_{0}(\lambda_{3}) u_{c} \\ ~~~- \left( K_{0}(\lambda_{3})(1+p \lambda_{1} + q \lambda_{2}) - q F_{0}(\lambda_{3}) \lambda_{1} \right) \left( (1 + p \lambda_{1} + q \lambda_{2})~\mu_{2} + \lambda_{2}~u_{c} - (b \lambda_{1} + a \lambda_{2})~c_{1}~\lambda_{4}\right) \\ ~~~- a F_{0}(\lambda_{3}) c_{1} \lambda_{1} \lambda_{4} = 0\\ \mu_{3} - \lambda_{4} = 0\\ \mu_{4} + c_{2}~\lambda_{2} + c_{3}~\cos(\lambda_{3}) + c_{4}~\lambda_{4} = 0 \end{array}\right. $$

(39)

Then, $\forall e \in \mathbb {R}$, let us consider the statement F(λ,μ + ew) = 0, where w = (w₁,w₂,w₃,w₄).

$$ \Rightarrow \left\{\begin{array}{l} (1 + p \lambda_{1} + q \lambda_{2})~F_{0}(\lambda_{3}) (\mu_{1} + e w_{1}) + (\lambda_{1}~(1 + p \lambda_{1} + q \lambda_{2}) + q \lambda_{2} \lambda_{1}) F_{0}(\lambda_{3}) u_{c} \\ ~~~ -\left( K_{0}(\lambda_{3})(1+p \lambda_{1} + q \lambda_{2}) - q F_{0}(\lambda_{3}) \lambda_{1} \right) \left( (1 + p \lambda_{1} + q \lambda_{2})~(\mu_{2} + e w_{2}) + \lambda_{2}~u_{c} - (b \lambda_{1} + a \lambda_{2})~c_{1}~\lambda_{4}\right) \\ ~~~ - a F_{0}(\lambda_{3}) c_{1} \lambda_{1} \lambda_{4} = 0\\ \mu_{3} + e w_{3} - \lambda_{4} = 0\\ \mu_{4} + e w_{4} + c_{2}~\lambda_{2} + c_{3}~\cos(\lambda_{3}) + c_{4}~\lambda_{4} = 0 \end{array}\right. $$

(40)

By taking into account that F(λ,μ) = 0 (39), we obtain

$$ \left\{\begin{array}{l} (1 + p \lambda_{1} + q \lambda_{2})~F_{0}(\lambda_{3}) e w_{1} - (K_{0}(\lambda_{3})(1+p \lambda_{1} + q \lambda_{2}) - q F_{0}(\lambda_{3}) \lambda_{1}) (1 + p \lambda_{1} + q \lambda_{2})~e w_{2} = 0\\ e w_{3} = 0\\ e w_{4} = 0 \end{array}\right. $$

(41)

$$ \Rightarrow \left\{\begin{array}{l} e F_{0}(\lambda_{3}) w_{1} = e w_{2} \frac{(K_{0}(\lambda_{3})(1+p \lambda_{1} + q \lambda_{2}) - q F_{0}(\lambda_{3}) \lambda_{1}) (1 + p \lambda_{1} + q \lambda_{2})}{1 + p \lambda_{1} + q \lambda_{2}} = e w_{2} K(\lambda_{1}, \lambda_{2}) \\ e w_{3} = 0\\ e w_{4} = 0 \end{array}\right. $$

(42)

Thus, for all λ₁,λ₂,λ₃, and λ₄, the only possible values of w that satisfy this last statement are (w₁,w₂,w₃,w₄) = (0, 0, 0, 0).

Since w = (w₁,w₂,w₃,w₄) are proved to be equal to zero in order to respect the ruled manifold criterion negation (24), the musculoskeletal system with a single input is thus proved to be not flat.