Angular momentum-based control of an underactuated orthotic system for crouch-to-stand motion

Abstract

This paper presents an angular momentum-based controller for crouch-to-stand motion of a powered pediatric lower-limb orthosis. The control law is developed using an underactuated triple pendulum model representing the legs of an orthosis-dummy system where the hip and knee joints are actuated but the ankle joint is unpowered. The control law is conceived to drive the angular momentum of the system to zero, thereby bringing the system to a statically balanced upright configuration. The parameters of the dynamic model of the orthosis-dummy system are experimentally identified and used to synthesize the momentum-based controller. Control parameters are selected using closed-loop pole placement of the linearized system via numerical optimization to ensure local closed-loop stability with adequate damping and satisfactory response time without too large controller gains. The controller is applied in simulation to determine the region of viable initial conditions resulting in no knee hyperextension or loss of balance, as determined from a zero-moment point analysis. The controller is then implemented in experiment showing feasibility of the control strategy in practice. Results are compared against a similarly-synthesized linear-quadratic regulator.

Appendices

Appendix 1

When the conditions $L=\dot{L}=\ddot{L}=0$ are satisfied, a double pendulum will be in a statically balanced configuration as presented in Azad and Featherstone (2016). However, when these conditions are satisfied for the triple pendulum, there may still be motion in the system. This appendix derives the relation between the joint velocities when these conditions are satisfied.

The angular momentum L in (13) and its second derivative $\ddot{L}$ in (14) can be rewritten as a linear combination of the link velocities.

$$ \begin{aligned} \left[ {\begin{array}{*{20}c} L \\ {\ddot{L}} \\ \end{array} } \right] & = {\varvec{M}}\left[ {\begin{array}{*{20}c} {\omega_{1} } \\ {\omega_{2} } \\ {\omega_{3} } \\ \end{array} } \right] \\ {\varvec{M}} & = \left[ {\begin{array}{*{20}c} {\varPhi_{2} + L_{1} \varPhi_{3} c_{2} + L_{1} \varPhi_{5} c_{23} } & {g\varPhi_{1} s_{1} } \\ {\varPhi_{4} + L_{1} \varPhi_{3} c_{2} + L_{2} \varPhi_{5} c_{3} } & {g\varPhi_{3} s_{12} } \\ {\varPhi_{6} + L_{1} \varPhi_{5} c_{23} + L_{2} \varPhi_{5} c_{3} } & {g\varPhi_{5} s_{123} } \\ \end{array} } \right]^{\text{T}} \\ \end{aligned} $$

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The nullspace of the matrix M represents the set of link velocities which will result in $L=\ddot{L}=0$. Since M is a function of the states of the system, the nullspace will be as well. Due to the unwieldy size of the expression, the nullspace is presented only for the upright configuration.

$$ \begin{aligned} & \text{null} \left( {{\varvec{M}}|_{{{\varvec{q}} = {\varvec{q}}_{0} }} } \right) = \text{span} \left\{ {\left[ {\begin{array}{*{20}c} {n_{1} } \\ {n_{2} } \\ {n_{3} } \\ \end{array} } \right]} \right\} \\ & n_{1} = \varPhi_{3} \varPhi_{6} - \varPhi_{4} \varPhi_{5} - L_{2} \varPhi_{5}^{2} + L_{2} \varPhi_{3} \varPhi_{5} \\ & n_{2} = \varPhi_{2} \varPhi_{5} - \varPhi_{1} \varPhi_{6} + L_{1} \varPhi_{5}^{2} \\ & \quad \quad - L_{1} \varPhi_{1} \varPhi_{5} - L_{2} \varPhi_{1} \varPhi_{5} + L_{1} \varPhi_{3} \varPhi_{5} \\ & n_{3} = \varPhi_{1} \varPhi_{4} - \varPhi_{2} \varPhi_{3} - L_{1} \varPhi_{3}^{2} \\ & \quad \quad + L_{1} \varPhi_{1} \varPhi_{3} + L_{2} \varPhi_{1} \varPhi_{5} - L_{1} \varPhi_{3} \varPhi_{5} \\ \end{aligned} $$

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This means link velocities of the form ω₁  =  α·n₁, ω₂  =  α·n₂, and ω₃  =  α·n₃ for any $\alpha \in {\mathbb{R}}$ will hold true if and only if $L=\ddot{L}=0$, at least for the upright configuration. For the system in this paper, this gives necessary and sufficient conditions for $L=\ddot{L}=0$ being joint velocities related by $\dot{q}_2=-2.3243\cdot\dot{q}_1 \,\,\text{and}\,\, \dot{q}_3=1.7993\cdot\dot{q}_1$.

Appendix 2

The controller is synthesized based on pole placement of the linearized closed-loop system using numerical optimization. The state matrix of this system takes the form

$$ \begin{aligned} & {\varvec{A}} = \left[ {\begin{array}{*{20}c} \mathbf{0} & {{\varvec{I}}_{3} } \\ {{\hat{\varvec{K}}}} & {{\hat{\varvec{D}}}} \\ \end{array} } \right] \\ & {\hat{\varvec{K}}} = \left[ {\begin{array}{*{20}c} {k_{11} } & {k_{12} } & {k_{13} } \\ {k_{21} } & {k_{22} } & {k_{23} } \\ {k_{31} } & {k_{32} } & {k_{33} } \\ \end{array} } \right]\quad {\hat{\varvec{D}}} = \left[ {\begin{array}{*{20}c} {d_{11} } & {d_{12} } & {d_{13} } \\ {d_{21} } & {d_{22} } & {d_{23} } \\ {d_{31} } & {d_{32} } & {d_{33} } \\ \end{array} } \right] \\ \end{aligned} $$

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where I₃ is the 3  ×  3 identity matrix and the elements of the matrices ${{\hat{\varvec{K}}}}$ and ${{\hat{\varvec{D}}}}$ are affine in the control parameters. The full state matrix in terms of system parameters is too large to include in the manuscript. Instead, using the parameters of the system, the elements of the matrices as a function of the MBC gains is provided in Table 7 along with their numerical values for the selected controller.

Table 7 Elements of the linearized system state matrix

The eigenvalues of the state matrix from the numerical optimization are − 49.60, − 44.12, − 35.88, − 2.792, and − 2.786  ±  0.2052i. Since the real components are all strictly negative, the system is locally stable. This means the states will approach the equilibrium point, provided the system configuration and velocities are sufficiently close.