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Dynamic single actuator robot climbing a chute

Period-doubling bifurcations: analysis and experiments

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Abstract

The planar mechanism analyzed in this paper, called DSAC for Dynamic, Single Actuated Climber, comprises only two links connected by a single oscillating actuator. This simple open loop motion propels the robot stably between two vertical walls. We explore the local orbital stability of the DSAC mechanism. Using the Poincaré map, we reduce the analyzed dimension and find the stable regions while varying the control inputs and mechanism’s parameters. Moreover, in response to a continuous change of a parameter of the mechanism, the symmetric and steady stable gait of the mechanism gradually evolves through a regime of period doubling bifurcations. This investigation includes numerical approximation of the local stability, and basin of attraction. Finally, the paper reports experimental results of open-loop, stable climbing in a planar, reduced gravity environment undergoing bifurcations which correlate well to the numerical analysis.

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Notes

  1. In fact it is possible to only use five parameters (\(I_1+m_2 l^2_1\), \(I_2\), \(m_2 b_2 l_1\), \(m_1 b_1 + m_2 l_1\), and \(m_2 b_2\)) instead of the full set of seven parameters (\(m_1\), \(m_2\), \(l_1\), \(I_1\), \(I_2\), \(b_1\), and \(b_2\)) used here (c.f. [9]).

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Acknowledgments

The author would like to thank Matthew T. Mason, Howie Choset, Kevin Lynch, and Andy Ruina for their guidance and suggestions and Siyuan Feng for his help with the design and experiments.

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Correspondence to Amir Degani.

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Supplementary material 1 (MPG 15836 kb)

Appendix: Equations of motion

Appendix: Equations of motion

This section will derive the general equations of motion of the three phases: flight, impact, and stance.

Free flight phase Using the Lagrange method the energy must be first found. For that, the kinematics including the position of the two masses, then velocities are found—see Fig. 1b for symbols.

$$\begin{aligned} r_{m_1}&= \left[\begin{array}{cc} x+b_1 \sin \theta \\ y-b_1 \cos \theta \end{array}\right],\nonumber \\ r_{m_2}&= \left[\begin{array}{cc} x+l_1 \sin \theta -b_2 \sin (\theta +\phi )\\ y-l_1 \sin \theta +b_2 \sin (\theta +\phi ) \end{array}\right], \end{aligned}$$
(5)

where x, y, \(\theta \) and \(\phi \) are time dependent, i.e., x(t), y(t), \(\theta (t)\) and \(\phi (t)\).

Velocities of masses:

$$\begin{aligned} v_{m_1}&= \frac{d r_{m_1}}{d t}= \left[\begin{array}{cc} \dot{x}+ b_1 \cos \theta \, \dot{\theta }\\ \dot{y}+ b_1 \sin \theta \, \dot{\theta }\\ \end{array}\right], \nonumber \\ v_{m_2}&= \frac{d r_{m_2}}{d t}= \left[\begin{array}{cc} \dot{x}+l_1 \cos \theta \, \dot{\theta }- l_2 \cos (\theta -\phi ) (\dot{\theta }- \dot{\phi })\\ \dot{y}+l_1 \sin \theta \,\dot{\theta }- l_2 \sin (\theta -\phi ) (\dot{\theta }- \dot{\phi })\\ \end{array}\right]. \end{aligned}$$
(6)

The Lagrangian is written as \(L=T-V\) where the kinetic and potential energies are:

$$ T= \frac{1}{2} \left( m_1 v_{m_1}^2 + m_2 v_{m_2}^2 + I_1 \dot{\theta }^2+ I_2 (\dot{\theta }+\dot{\phi })^2 \right) , $$
(7)
$$ V= -m_1 g r_{m_1}-m_2 g r_{m_2}. $$
(8)

Next, the Lagrange equation is used to find the equations of motion

$$\begin{aligned} \frac{d}{dt} \frac{\partial L}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i}=0 \end{aligned}.$$
(9)

In matrix form, the Lagrange equation is

$$\begin{aligned} M(q) \ddot{q} + h(q,\dot{q}) = 0, \end{aligned}$$
(10)

where

$$ M(q) = \left( {\begin{array}{*{20}c} {M_{{11}} } & 0 & {M_{{13}} } \\ 0 & {M_{{22}} } & {M_{{23}} } \\ {M_{{31}} } & {M_{{32}} } & {M_{{33}} } \\ \end{array} } \right), $$
(11)
$$\begin{aligned}&M_{11}=M_{22}=m_1+m_2, \nonumber \\&M_{13}=M_{31}+(b_1 m_1 + l_1 m_2) \cos \theta - b_2 m_2 \cos (\theta + \phi ),\nonumber \\&M_{23}=M_{32}=(b_1 m_1 + l_1 m_2) \sin \theta - b_2 m_2 \sin (\theta + \phi ), \nonumber \\&M_{33}=I_1 + I_2 +b_1^2 m_1 + (b_2^2 +l_1^2) m_2 -2 b_2 l_1 m_2 \cos \phi , \end{aligned}$$
(12)

and

$$ h(q,\dot{q})= \left(\begin{array}{l} - b_1 m_1 \sin \theta \, \dot{\theta }^2 + m_2 (-l_1 \sin \theta \, \dot{\theta }^2 + b_2 \sin (\theta -\phi )\ldots\ (\dot{\theta }+ \dot{\phi })^2 - b_2 \cos (\theta + \phi ) \ddot{\phi }\\ g (m_1+m_2) + b_1 m_1 \cos \theta \, \dot{\theta }^2 + m_2 (l_1 \cos \theta \, \dot{\theta }^2 - \ldots\ b_2 \cos (\theta +\phi ) (\dot{\theta }+ \dot{\phi })^2+ b_2 \sin (\theta + \phi ) \ddot{\phi }) \\ g (b_1 m_1 + l_1 m_2) \sin \theta - g b_2 m_2 \sin (\theta + \phi ) + I_2 \ddot{\phi }+\ldots\ b_2 m_2 (l_1 \sin \phi (2 \dot{\theta }+ \dot{\phi }) \dot{\phi }+ (b_2 - l_1 \cos \phi ) \ddot{\phi }\end{array}\right). $$
(13)

Impact phase From conservation of angular momentum around the contact point during impact, and noting that \(\phi \) is constrained, i.e., \(\dot{\phi }^- - \dot{\phi }^+ =0\), the angular velocity after impact is

$$\begin{aligned} \dot{\theta }^+ =\frac{1}{M_{33}^{-}} \Big (M_{13}^{-} \dot{x}^- + M_{23}^{-} \dot{y}^- + M_{33}^{-} \dot{\theta }^- \Big ), \end{aligned}$$
(14)

where, \(M_{ii}^{-}\) are the components of the M Matrix, described in (12) at the state prior to impact, i.e., \(\dot{\theta }^-,\dot{\phi }^-\).

Stance phase The stance phase equations of motion can be decoupled since the leg is in contact with the wall. While keeping the no rebound, no slip assumptions, only the equations of motion for the \(\theta , \dot{\theta }\) must be solved while observing the contact forces to see when they change sign, corresponding to transition to flight phase. The equation of motion for \(\theta , \dot{\theta }\) is the last (third) row of (10).

The contact force is calculated using the Lagrange multipliers method

$$\lambda _{ext}(q,\dot{q}) = \left( A(q) M(q)^{-1} A(q)^T \right) ^{-1} \left( \dot{A}(q) \dot{q} -A(q) M(q)^{-1}h(q,\dot{q}) \right) , $$
(15)

where \(A(q) = \left( \begin{array}{lll} 1&{}0&{}0\\ 0&{}1&{}0 \end{array} \right) \) and M(q) and \(h(q,\dot{q})\) are from (11) and (13).

The above equations of motion can finally be rewritten in terms of dimensionless variables using a characteristic length of \(d_{\text {wall}}\) and characteristic time \(\frac{1}{\omega }\).

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Degani, A. Dynamic single actuator robot climbing a chute. Meccanica 51, 1227–1243 (2016). https://doi.org/10.1007/s11012-015-0286-x

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