A unified framework for measuring interplane and intraplane coupling in spatial biped robots

Abstract

While fully actuated biped robots can control each degree of freedom independently, coupling must be used during phases of underactuation to maintain dynamic stability. The amount of coupling can be quantified via velocity decomposition, which partitions the system’s velocities into controlled and uncontrolled directions. Used previously on planar mechanical systems with one degree of underactuation, the decomposition becomes non-unique with multiple degrees of underactuation. This paper extends the velocity decomposition formulation to mechanical systems with several degrees of underactuation, applying it to a spatial biped robot for the first time. Two measures of dynamic coupling are introduced: intraplane coupling between the controlled and uncontrolled directions in the same plane and interplane coupling between the controlled directions in the sagittal/frontal plane and uncontrolled direction in the frontal/sagittal plane. Comparative studies show that the dynamic coupling of planar and spatial gaits of five-link biped models is very different at slow speeds, suggesting that the mechanisms of slow walking are fundamentally different and that planar models are not adequate to analyze slow-speed walking. Also, weak interplane coupling in slower gaits suggests that the 3D dynamics of the spatial biped are largely decoupled at slow speeds, such that distinct control strategies may be adequate to stabilize the sagittal and frontal planes separately. Strong interplane coupling at faster speeds suggests that the full 3D dynamics must be considered together for stabilization.

Appendices

Appendix A: Coupling coefficients for mechanical systems underactuated by one control

The general forms of the A scalar coefficients are

$$\begin{aligned} {\mathbf {A}}_1^{a} ({\mathbf {Y}}_{\! q},{\mathbf {Y}}_{\! p})&= \left( \frac{\partial Y_{\! p}^k}{\partial q_i}Y_{\! q}^i +\varGamma _{\! ij}^k Y_{\! q}^i Y_{\! p}^j \right) M_{kl}Y_{\! a}^{l}, \nonumber \\ {\mathbf {A}}_2^{a} ({\mathbf {Y}}_{\! q},{\mathbf {Y}}^{\! \perp })&= \left( \frac{\partial Y^{\! \perp k}}{\partial q_i}Y_{\! q}^i +\varGamma _{\! ij}^k Y_{\! q}^i Y^{\! \perp j} \right) M_{kl}Y_{\! a}^{l}, \nonumber \\ {\mathbf {A}}_3^{a} ({\mathbf {Y}}^{\! \perp },{\mathbf {Y}}_{\! p})&= \left( \frac{\partial Y_{\! p}^k}{\partial q_i}Y^{\! \perp i} +\varGamma _{\! ij}^k Y^{\! \perp i} Y_{\! p}^j \right) M_{kl}Y_{\! a}^{l}, \nonumber \\ A_4^{a} ({\mathbf {Y}}^{\! \perp },{\mathbf {Y}}^{\! \perp })&= \left( \frac{\partial Y^{\! \perp k}}{\partial q_i}Y^{\! \perp i} +\varGamma _{\! ij}^k Y^{\! \perp i} Y^{\! \perp j} \right) M_{kl}Y_{\! a}^{l}, \nonumber \\ A_5^a( {\mathcal {V}}, {\mathbf {Y}}_{\! a})&= \frac{\partial {\mathcal {V}}}{\partial q_l} Y_{\! a}^{l}. \end{aligned}$$

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where ${\mathbf {A}}_1 \in {\mathbb {R}}^{n_a \times n_a}$, ${\mathbf {A}}_2 \in {\mathbb {R}}^{n_a}$, ${\mathbf {A}}_3 \in {\mathbb {R}}^{n_a}$, $A_4 \in {\mathbb {R}}$, $A_5 \in {\mathbb {R}}$.

The general forms of the B scalar coefficients are

$$\begin{aligned} {\mathbf {B}}_1 ({\mathbf {Y}}_{\! q},{\mathbf {Y}}_{\! p})&= \left( \frac{\partial Y_{\! p}^k}{\partial q_i}Y_{\! q}^i +\varGamma _{\! ij}^k Y_{\! q}^i Y_{\! p}^j \right) M_{kl}Y^{\! \perp l}, \nonumber \\ {\mathbf {B}}_2 ({\mathbf {Y}}_{\! q},{\mathbf {Y}}^{\! \perp })&= \left( \frac{\partial Y^{\! \perp k}}{\partial q_i}Y_{\! q}^i +\varGamma _{\! ij}^k Y_{\! q}^i Y^{\! \perp j} \right) M_{kl}Y^{\! \perp l}, \nonumber \\ {\mathbf {B}}_3 ({\mathbf {Y}}^{\! \perp },{\mathbf {Y}}_{\! p})&= \left( \frac{\partial Y_{\! p}^k}{\partial q_i}Y_{\! r}^{\! \perp i} +\varGamma _{\! ij}^k Y^{\! \perp i} Y_{\! p}^j \right) M_{kl}Y^{\! \perp l}, \nonumber \\ B_4 ({\mathbf {Y}}^{\! \perp },{\mathbf {Y}}^{\! \perp })&= \left( \frac{\partial Y^{\! \perp k}}{\partial q_i}Y^{\! \perp i} +\varGamma _{\! ij}^k Y^{\! \perp i} Y^{\! \perp j} \right) M_{kl}Y^{\! \perp l}, \nonumber \\ B_5( {\mathcal {V}}, {\mathbf {Y}}^{\! \perp } )&= \frac{\partial {\mathcal {V}}}{\partial q_l} Y^{{\! \perp }l}. \end{aligned}$$

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Appendix B: Velocity decomposition of 4-DOF mechanical system underactuated by 3

This appendix applies velocity decomposition to a 4-DOF mechanical system underactuated by 3 controls. This derivation is kept general and only requires a symmetric inertia matrix of the form

$$\begin{aligned} {\mathbf {M}} = \begin{pmatrix} M_{11} &{} M_{12} &{} M_{13} &{} M_{14} \\ M_{12} &{} M_{22} &{} M_{23} &{} M_{24} \\ M_{13} &{} M_{23} &{} M_{33} &{} M_{34} \\ M_{14} &{} M_{24} &{} M_{34} &{} M_{44} \end{pmatrix}. \end{aligned}$$

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Assuming that the first joint is actuated, the input matrix ${\mathbf {B}}$ is given by

$$\begin{aligned} {\mathbf {B}} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}, \end{aligned}$$

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and the starting list of linearly independant vectors for the uncontrolled directions is

$$\begin{aligned} {\mathbf {F}}_1^\perp = \begin{pmatrix} 0 \\ \alpha \\ 0 \\ 0 \end{pmatrix}, \quad {\mathbf {F}}_2^\perp = \begin{pmatrix} 0 \\ \beta \\ \gamma \\ 0 \end{pmatrix}, \quad {\mathbf {F}}_3^\perp = \begin{pmatrix} 0 \\ \omega \\ \eta \\ \kappa \end{pmatrix}, \end{aligned}$$

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where $\alpha$, $\gamma$, $\kappa \ne 0$ and $\alpha$, $\beta$, $\gamma$, $\omega$, $\eta$, $\kappa \in {\mathbb {R}}$.

The first uncontrolled direction is fairly straightforward to obtain,

$$\begin{aligned} {\mathbf {Y}}_1^\perp = \frac{{\mathbf {F}}_1^\perp }{\sqrt{ ({\mathbf {F}}_1^\perp )^T {\mathbf {M}} {\mathbf {F}}_1^\perp } } = \begin{pmatrix} 0 \\ \frac{\text {sgn}( \alpha )}{\sqrt{M_{22}}} \\ 0 \\ 0 \end{pmatrix}, \end{aligned}$$

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where $\text {sgn}(\cdot )$ is the sign function given by Eq. (30). As mentioned in Sect. 3.3, the first uncontrolled direction is a function of the sign of ${\mathbf {F}}_1^\perp$ but is independent of its magnitude. Thus, without loss of generality, the choice of ${\mathbf {F}}_1^\perp$ can be reduced to $(0,1,0,0)^T$ or $(0,-1,0,0)^T$.

Following the Gram–Schmidt procedure, the second uncontrolled direction is given by

$$\begin{aligned} {\mathbf {Y}}_2^\perp = \frac{\widetilde{{\mathbf {Y}}}_2^\perp }{|| \widetilde{{\mathbf {Y}}}_2^\perp ||_{\mathbf {M}}^2}, \end{aligned}$$

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where

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As mentioned in Sect. 3.3, the Gram–Schmidt procedure flushes out superfluous components from the direction—in this case the $\beta$ entries in red—so the final result is independent of $\beta$. Just like ${\mathbf {Y}}_1^\perp$ was independent of the magnitude of ${\mathbf {F}}_1^\perp$, the final result for ${\mathbf {Y}}_2^\perp$ is independent of the magnitude of ${\mathbf {F}}_2^\perp$,

$$\begin{aligned} {\mathbf {Y}}_2^\perp = \frac{\text {sgn}(\gamma )}{\sqrt{M_{33}-\frac{M_{23}^2}{M_{22}}}} \begin{pmatrix} 0 \\ - \frac{M_{23}}{M_{22}} \\ 1 \\ 0 \end{pmatrix}. \end{aligned}$$

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Similarly, the third uncontrolled direction is given by

$$\begin{aligned} {\mathbf {Y}}_3^\perp = \frac{\widetilde{{\mathbf {Y}}}_3^\perp }{|| \widetilde{{\mathbf {Y}}}_3^\perp ||_{\mathbf {M}}^2}, \end{aligned}$$

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where

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and

$$\begin{aligned} a_1&= \frac{M_{23}(M_{23}M_{24}-M_{22}M_{34})}{M_{22}(M_{22}M_{33}-M_{23}^2)} \nonumber \\ a_2&= \frac{M_{23}M_{24}-M_{22}M_{34}}{M_{23}^2-M_{22}M_{33}}. \end{aligned}$$

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The direction is ultimately a function of the sign of $\kappa$ only.

$$\begin{aligned} {\mathbf {Y}}_3^\perp = a_3 \, \text {sgn}(\kappa ) \begin{pmatrix} 0 \\ \frac{ M_{24} M_{33} - M_{23} M_{34} }{M_{23}^2 - M_{22} M_{33}} \\ \frac{ -M_{23} M_{24} + M_{22} M_{34} }{M_{23}^2 - M_{22} M_{33}} \\ 1 \end{pmatrix}, \end{aligned}$$

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where

$$\begin{aligned} a_3 = \frac{1}{\sqrt{\frac{M_{24}^2 M_{33} - 2 M_{23} M_{24} M_{34} + M_{23}^2 M_{44} + M_{22} (M_{34}^2 - M_{33} M_{44})}{M_{23}^2 - M_{22} M_{33}}}} \end{aligned}$$

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