Induced and tunable multistability due to nonholonomic constraints

Abstract

Multistability is an area of interest in robotics and locomotion because the ability to achieve multiple configurations or generate multiple gaits allows a single robotic or mechanical system to perform versatile tasks. This multistability is often achieved by adding multistable elements to the system. However, this work finds that two bodies pinned together with a linear rotational spring can exhibit multistable behavior with the introduction of a nonholonomic constraint. Multistable fixed points of the unforced and undamped system are found to correspond to multistable limit cycles with the introduction of damping and periodic forcing, some of which result in fast net turning. This finding has potential implications in understanding the sharp turns executed by biological swimmers and could be exploited to perform efficient turns in low degree of actuation robots.

Appendix

The mass matrix used in the Lagrangian and the equation of motion (3) is

$$\begin{aligned} \varvec{{\mathscr {M}}} = \left[ \begin{array}{cccc} m&{}0&{}-m_2 \, \epsilon \, l \, \sin {\theta _1} &{}-m_2 \, (1-\epsilon ) \, l \, \sin {\theta _2} \\ 0&{}m&{}m_2 \, \epsilon \, l \, \cos {\theta _1}&{}m_2 \, (1-\epsilon ) \, l \, \cos {\theta _2} \\ -m_2 \, \epsilon \, l \, \sin {\theta _1}&{}m_2 \, \epsilon \, l \, \cos {\theta _1}&{}m_2 \, \epsilon ^2 \, l^2+I_1&{}m_2 \, \epsilon \, (1-\epsilon ) \, l \, \cos {(\theta _2-\theta _1)} \\ -m_2 \, (1-\epsilon ) \, l \, \sin {\theta _2}&{}m_2 \, (1-\epsilon ) \, l \, \cos {\theta _2}&{}m_2 \, \epsilon \, (1-\epsilon ) \, l^2 \, \cos {(\theta _2-\theta _1)}&{}m_2 \, (1-\epsilon )^2 \, l+I_2 \end{array} \right] . \end{aligned}$$

(15)

Note that the mass matrix is used in the derivation before the rescaling, so the parameters here have not been rescaled. In the dynamical system (5) $\varvec{{\mathscr {N}}} \dot{\varvec{\xi }} = {\varvec{g}}(\varvec{\xi }) + {\varvec{f}}(t)$, the inertia-like tensor

$$\begin{aligned} \varvec{{\mathscr {N}}} = \begin{bmatrix} 1&{}-\sin {\delta } \left( \epsilon -1 \right) ^{2}&{}-\sin \delta \left( \epsilon -1 \right) ^{2} &{} 0\\ -\sin \delta \left( \epsilon -1 \right) ^{2}&{}\varvec{{\mathscr {N}}}_{2,2}&{}\varvec{{\mathscr {N}}}_{2,3} &{} 0\\ -\sin \delta \left( \epsilon -1 \right) ^{2}&{}\varvec{{\mathscr {N}}}_{2,3} &{}\varvec{{\mathscr {N}}}_{3,3} &{} 0\\ 0 &{} 0 &{} 0 &{} 1 \end{bmatrix} \end{aligned}$$

(16)

where

$$\begin{aligned} \varvec{{\mathscr {N}}}_{2,2}&=4\,\epsilon \, ( \epsilon -1 ) ^{2}\cos ( \delta ) -4\,{\epsilon }^{3}\nonumber \\&\quad + ( 48\,\gamma +7 ) {\epsilon }^{ 2}+ ( -48\,\gamma -3 ) \epsilon +16\,\gamma +1 \end{aligned}$$

(17)

$$\begin{aligned} \varvec{{\mathscr {N}}}_{2,3}&=- ( \epsilon -1 ) ( 16\,{\epsilon }^{2}\gamma -2\,{ \epsilon }^{2}\cos ( \delta )\nonumber \\&\qquad -32\,\epsilon \,\gamma +2\, \epsilon \,\cos ( \delta ) +{\epsilon }^{2}\nonumber \\&\qquad \qquad +\gamma \, r^2 + 16\,\gamma -2\, \epsilon +1 \end{aligned}$$

(18)

$$\begin{aligned}&\varvec{{\mathscr {N}}}_{3,3}=- \left( \epsilon -1 \right) \left( 16\,{\epsilon }^{2}\gamma +{\epsilon } ^{2}-32\,\epsilon \,\gamma \right. \nonumber \\&\qquad \quad \left. -2\,\epsilon + \gamma \, r^2+16\,\gamma +1 \right) . \end{aligned}$$

(19)

For the special case where $c=0$, $A=0$, fixed points can be defined for this system by solving $\varvec{{\mathscr {N}}}^{-1} \, {\varvec{g}}={\varvec{0}}$. For nonsingular $\varvec{{\mathscr {F}}}$, this simplifies to ${\varvec{g}}={\varvec{0}}$.

At a fixed point $\dot{\varvec{\xi }}={\varvec{0}}$ and ${\dot{\delta }}=\omega _2=0$, requiring that any solution to 5 must result in ${\varvec{g}}=0$. Both ${\varvec{g}}_1=0$ and ${\varvec{g}}_2=0$ can be solved by setting

$$\begin{aligned} \delta ^*=\cos ^{-1} \left( {\frac{\epsilon \, \left( \epsilon -2 \right) }{{ \epsilon }^{2}-2\,\epsilon +1}} \right) , \end{aligned}$$

(20)

which gives a turning gait, or

$$\begin{aligned} \omega _1^*=0, \end{aligned}$$

(21)

which results in straight-line motion. The equation ${\varvec{g}}_4=0$ is satisfied if $\omega _2=0$. The remaining equation ${\varvec{g}}_3=0$ can be satisfied for infinitely many adjacent $(u,\omega )$ pairs, resulting in non-isolated fixed points extending away from the constant energy manifold. Evaluating the dynamics reduced to the energy manifold results in isolated fixed points which can then be analyzed for stability. This reduction is performed by defining the system energy

$$\begin{aligned} E={\mathscr {T}}({\varvec{q}},\dot{{\varvec{q}}})+{\mathscr {V}}({\varvec{q}}) \end{aligned}$$

(22)

which, rewritten in $\varvec{\xi }$ coordinates, is

$$\begin{aligned} {E}^{\prime }=\frac{E}{m \, l^2}=\frac{1}{2} \left( u_x^2+b \, u_x+c \right) \end{aligned}$$

(23)

for

$$\begin{aligned} b&=m_2^{\prime } (\epsilon -1)(\omega _1+\omega _2)\sin {\delta } \end{aligned}$$

(24)

and

$$\begin{aligned} c&=\left( (I_1^{\prime }+\frac{m_1^{\prime } \epsilon ^2}{4}+m_2^{\prime } \epsilon ^2) \omega _1^2 \right. \nonumber \\&\quad + \left( I_2^{\prime }+\frac{m_2^{\prime } (\epsilon -1)^2}{4} \right) (\omega _1+{\dot{\delta }})^2\nonumber \\&\quad \left. -m_2^{\prime } (\epsilon -1)(\omega _1+\omega _2)(\omega _1 \epsilon \cos {\delta })\right) . \end{aligned}$$

(25)

With energy rescaled, we now once again drop the $^{\prime }$ superscripts and work only with the rescaled variables. The longitudinal velocity can be written as a function of $(\omega _1, \omega _2, \delta )$ and the energy. The reduced dynamical system is then obtained from the last three equations of (5) where $u_x$ is substituted by a function of the other three state variables, $u_x(\omega _1, \omega _2, \delta ; E)$. Suppose ${\varvec{H}}(\varvec{\xi }) = \varvec{{\mathscr {N}}}^{-1} {\varvec{h}}(\varvec{\xi }) \in {\mathbb {R}}^4$. Setting ${\varvec{H}}_r(\varvec{\xi }; E) = [{\varvec{H}}_2(\varvec{\xi }), {\varvec{H}}_3(\varvec{\xi }), {\varvec{H}}_4(\varvec{\xi })]^T$, which excludes the first component of ${\varvec{H}}(\varvec{\xi })$, the reduced dynamical system is

$$\begin{aligned} \dot{\varvec{\xi }}_r = {\varvec{H}}_r(\varvec{\xi }_r; E) \end{aligned}$$

(26)

where $\varvec{\xi }_r = (\omega _1, \omega _2, \delta )$. We then define the fixed points as a function of energy by substituting $u_x(\omega _1,\omega _2,\delta ,E)$ into ${\varvec{g}}_3=0$, which gives an expression which can be solved for $\omega _1^*$ as

$$\begin{aligned} \omega _1^*=\pm \frac{\sqrt{2}}{2 \, m_2} \sqrt{ A \left( B \pm \sqrt{C} \right) } \end{aligned}$$

(27)

for

$$\begin{aligned} A&= \Big ( \Big ( {\epsilon }^{2}-{\frac{\epsilon }{2}}+ {\frac{1}{4}} \Big ) m_{2}-{\frac{3\,{\epsilon }^{2}}{4}}+I_{1}+I_{2} \Big ) ( ( \cos ( \delta ) ) ^{2}\\&-m_{2}\, \epsilon \, ( -1+\epsilon ) \cos ( \delta ) +{\epsilon }^{2} ) ^{-1} ( -1+ \epsilon ) ^{-2}\\ B&=( ( -{\delta }^{2}{\alpha }^{2}+2\,E ) {\epsilon }^{2}+ ( 2\,{\delta }^{2}{\alpha }^{2}- 4\,E )\\&\epsilon -{\delta }^{2}{\alpha }^{2}+2\,E ) {m_{2}}^{2 } ( \cos ( \delta ) ) ^{2} \\&+( -2\,{\alpha } ^{2}\delta \,{\epsilon }^{2}+4\,{\alpha }^{2}\delta \,\epsilon -2\,{\alpha } ^{2}\delta ) {m_{2}}^{2}\sin ( \delta ) \cos ( \delta ) \\&+ ( 4\,{\alpha }^{2}\delta \,{\epsilon }^{2}-4\,{ \alpha }^{2}\delta \,\epsilon ) m_{2}\,\sin ( \delta ) \\ C&= ( {m_{2}}^{2} ( -1+\epsilon ) ^{2} \Big ( \Big ( -{\frac{{\delta }^{2} }{2}}+\delta \Big ) {\alpha }^{2}+E \Big ) \\&\Big ( \Big ( -{\frac{{ \delta }^{2}}{2}}-\delta \Big ) {\alpha }^{2} +E \Big ) ( \cos ( \delta ) ) ^{2} \\&-2\,{\alpha }^{2} ( m_{2}\, ( -1+\epsilon ) ( -1/2\,{\delta }^{2}{\alpha }^{2}+ E ) \sin ( \delta ) \\&-2\,{\alpha }^{2}\delta \,\epsilon ) m_{2}\, ( -1+\epsilon ) \delta \,\cos ( \delta ) \\&+4\,{\alpha }^{2}\delta \, ( m_{2}\, ( -1+\epsilon ) ( -1/2\,{\delta }^{2}{\alpha }^{2}+E ) \epsilon \, \sin \nonumber \\&( \delta ) -{\alpha }^{2}\delta \, ( -1/4\,{m_{2}}^ {2} ( -1+\epsilon ) ^{2} \\&+ ( {\epsilon }^{2}-\epsilon /2+ 1/4 ) m_{2}+1/4\,{\epsilon }^{2}+I_{1}+I_{2} ) ) ) \\&( \cos ( \delta ) ) ^{2}{m_{2}}^{2} ( -1+\epsilon ) ^{2}. \end{aligned}$$

Through straightforward manipulation of equation 23, we can find

$$\begin{aligned} u_x=\frac{1}{2} \left( -b \pm \sqrt{b^2-4 \left( c-2 E \right) } \right) . \end{aligned}$$

(28)

Substituting $b=b(\omega _1=\omega _1^*)$ and $c=c(\omega _1=\omega _1^*)$ into 28 then yields $u_x^*$.