Introducing GyroSym: a single-wheel robot

Abstract

In this paper, GyroSym, a compact highly-stable single-wheel robot is introduced. Symmetrical flywheels have been used as its stabilization mechanism to keep the wheel upright. Thanks to the type of the flywheel used, the dynamic equations of the robot is simplified, the torque needed to tilt the spin axis is reduced, and the space is drastically minimized. Besides, a centrifugal clutch is used to smooth the motion of the robot in forward motion. Two controllers, fuzzy and fuzzy-Padé, were implemented based on their ability to stabilize the robot. Although both prevented the GyroSym from falling over, neither can entirely cancel out the effects of nonlinear behaviors that is present in the response, represented as high-frequency low domain noise. However, fuzzy-Padé demonstrated more efficient functionality, response, and mathematical simplicity which made it a perfect choice for stabilizing the GyroSym. Experimental results of a prototype have validated that of the mathematical analysis and simulations.

Appendices

Appendix A

1.1 Lagrangian of GyroSym

The GyroSym consists of three subsystems which are the flywheels/motors, the pendulum, and the wheel encompassing all other components. Each of these three subsystems has kinetic and potential energy. The Lagrangian presented in Eq. (7) is the subtraction of the summation of kinetic energies from potential energies of the subsystems. The summation of the kinetic energies is calculated using the general equation for kinetic energy shown in Eq. (25).

$$ L = T - V $$

(23)

$$ T = \mathop \sum \limits_{j = 1}^{ num of\;subsys} \frac{1}{2}mV_{j}^{T} V_{j} + \frac{1}{2}\omega_{j}^{T} I_{j} \omega_{j} $$

(24)

$$ \begin{aligned} T &= 0.5m_{ring} \dot{X}^{2} + 0.5m_{ring} \dot{Y}^{2} + 0.5m_{ring} \dot{Z}^{2}\\ & \quad +\,0.5I_{xx} \left( {\dot{\varphi } - \dot{\psi }{\text{s}}\left( \theta \right)} \right)^{2} \\ & \quad + \;0.5I_{yy} \dot{\theta }^{2} + 0.5I_{zz} \dot{\psi }^{2} {\text{c}}\left( \theta \right)^{2} \\ & \quad+ \;0.5m_{p} \left( {\dot{X} - r_{PCG} {\text{c}}\left( \theta \right){\text{c}}\left( \psi \right)\left( {\dot{\theta }{\text{c}}\left( \beta \right) + \dot{\psi }{\text{c}}\left( \theta \right){\text{s}}\left( \beta \right)} \right)} \right. \\ & \quad - \;r_{PCG} {\text{s}}\left( \psi \right)\left( {\dot{\beta }{\text{c}}\left( \beta \right) - \dot{\psi }{\text{s}}\left( \theta \right){\text{c}}\left( \beta \right)} \right) \\ & \quad \left. { + \;r_{PCG} s\left( \theta \right)c\left( \psi \right)\left( {\dot{\beta }s\left( \beta \right) - \dot{\psi }s\left( \theta \right)s\left( \beta \right)} \right)} \right)^{2} \\ & \quad+ \;0.5m_{P} \left( {\dot{Y} - r_{PCG} c\left( \theta \right)s\left( \psi \right)\left( {\dot{\theta }c\left( \beta \right) + \dot{\psi }c\left( \beta \right)s\left( \beta \right)} \right)} \right. \\ & \quad -\,r_{PCG} c\left( \psi \right)\left( {\dot{\beta }c\left( \beta \right) + \dot{\psi }s\left( \theta \right)c\left( \beta \right)} \right) \\ & \quad \left. { +\,r_{PCG} s\left( \theta \right)s\left( \psi \right)\left( {\dot{\beta }s\left( \beta \right) - \dot{\psi }s\left( \theta \right)s\left( \beta \right)} \right)} \right)^{2} \\ & \quad + \;0.5m_{P} \left( {\dot{Z} + r_{PCG} s\left( \theta \right)\left( {\dot{\theta }c\left( \beta \right) + \dot{\psi }c\left( \theta \right)s\left( \beta \right)} \right)} \right. \\ & \quad \left. { +\,r_{PCG} c\left( \theta \right)\left( {\dot{\beta }s\left( \beta \right) - \dot{\psi }s\left( \theta \right)s\left( \beta \right)} \right)} \right)^{2} \\ & \quad + \;0.5I_{{f_{{x^{\prime } x^{\prime } }} }} \left( {\dot{\psi }\left( { - s\left( \theta \right)c\left( \alpha \right) - c\left( \theta \right)s\left( \alpha \right)c\left( \beta \right)} \right)} \right. \\ & \quad \left. { +\,\dot{\theta }s\left( \alpha \right)s\left( \beta \right) + \dot{\beta }c\left( \alpha \right) + \dot{\Omega }} \right)^{2} \\ & \quad + \;0.5I_{{f_{{y^{\prime } y^{\prime } }} }} \left( {\dot{\psi }c\left( \theta \right)s\left( \beta \right) + \dot{\theta }c\left( \beta \right) + \dot{\alpha }} \right) \\ & \quad \left( {\dot{\psi }c\left( \theta \right)s\left( \beta \right) + \dot{\theta }c\left( \beta \right) + \dot{\alpha }} \right) \\ & \quad + \;0.5I_{{f_{{z^{\prime } z^{\prime } }} }} \left( {\dot{\psi }\left( { - s\left( \theta \right)s\left( \alpha \right) + c\left( \theta \right)c\left( \alpha \right)c\left( \beta \right)} \right)} \right. \\ & \quad \left. { -\,\dot{\theta }c\left( \alpha \right)s\left( \beta \right) + \dot{\beta }s\left( \alpha \right)} \right)^{2} \\ \end{aligned} $$

(25)

And the summation of potential energies is shown in Eq. (26).

$$ \begin{aligned} V & = V_{CG of ring} + V_{CG of pendulum} \\ V_{CG of ring} & = m_{ring} gZ \\ V_{CG of pendulum} &= m_{p} g\left( {Z - r_{PCG} \cos \left( \beta \right)\cos \left( \theta \right)} \right) \\ & \quad \to V = m_{ring} gZ\\ & + m_{p} g\left( {Z - r_{PCG} \cos \left( \beta \right)\cos \left( \theta \right)} \right) \\ \end{aligned} $$

(26)

Appendix B

2.1 General constraints of GyroSym

According to the Euler–Lagrange’s equation in the system constraints, should be rewritten in the standard form as shown in Eq. (27).

$$ \mathop \sum \limits_{k = 1}^{9} a_{ik} \dot{q}_{k} + b_{j} = 0 \quad j = 1,2,3 :{\text{Standard}}\;form\;for\;the\;constraints $$

(27)

$$ \begin{aligned} & - R_{ring} \sin \left( \psi \right)\sin \left( \theta \right)\dot{\psi } + R_{ring} \cos \left( \theta \right)\cos \left( \psi \right)\dot{\theta } \\ & + R_{ring} \sin \left( \psi \right)\dot{\varphi } + 0\dot{\beta } + 0\dot{\alpha } + 0\dot{\varOmega } \\ & - \dot{X} + 0\dot{Y} + 0\dot{Z} + 0 = 0 \\ \end{aligned} $$

(28)

$$ \begin{aligned} & R_{ring} \cos \left( \psi \right)\sin \left( \theta \right)\dot{\psi } + R_{ring} \cos \left( \theta \right)\sin \left( \psi \right)\dot{\theta } \\ & - R_{ring} \cos \left( \psi \right)\dot{\varphi } + 0\dot{\beta } + 0\dot{\alpha } + 0\dot{\varOmega } + 0\dot{X} - \dot{Y} + 0\dot{Z} + 0 = 0 \\ \end{aligned} $$

(29)

$$ \begin{aligned} & 0\dot{\psi } + R_{ring} \sin \left( \theta \right)\dot{\theta } + 0\dot{\varphi } + 0\dot{\beta } + 0\dot{\alpha } + 0\dot{\varOmega } \\ & + 0\dot{X} + 0\dot{Y} + \dot{Z} + 0 = 0 \\ \end{aligned} $$

(30)

According to the standard form of constraints in Euler–Lagrange’s equations, the coefficients of Lagrange’s multipliers are shown in (31).

$$ \begin{aligned} a_{11} & = - R_{ring} \sin \left( \psi \right)\sin \left( \theta \right),a_{12} = R_{ring} \cos \left( \theta \right)\cos \left( \psi \right), \\ a_{13} & = R_{ring} \sin \left( \psi \right) \\ a_{14} & = a_{15} = a_{16} = 0,a_{17} = - 1,a_{18} = a_{19} = 0,b_{1} = 0 \\ a_{21} & = R_{ring} \cos \left( \psi \right)\sin \left( \theta \right),a_{22} = R_{ring} \cos \left( \theta \right)\sin \left( \psi \right), \\ a_{23} & = - R_{ring} \cos \left( \psi \right) \\ a_{24} & = a_{25} = a_{26} = a_{27} = 0,a_{28} = - 1,a_{29} = 0,b_{2} = 0 \\ a_{31} & = 0,a_{32} = R_{ring} \sin \left( \theta \right),a_{33} = a_{34} = a_{35} = a_{36} \\ & = a_{37} = a_{38} = 0,a_{39} = 1,b_{3} = 0 \\ \end{aligned} $$

(31)

Besides the inherent constraints of the GyroSym which are presented in (31), some extra constraints are added to the system’s dynamics to maneuver the robot in the simulation. These constraints are the movements of the robotic actuators which are applied to the generalized coordinates of the system as an angular velocity. In Simulink, for locomotion of the robot, tilting the flywheel system, and the angular velocity of the flywheels, some functions are given to the coefficients of the constraints equation in its standard format. Within the simulation, these functions will be seen in the behavior of the GyroSym.

$$ \dot{\varphi } - \dot{\beta } = C_{1} \to \dot{\varphi } - \dot{\beta } - C_{1} = 0 $$

(32)

$$ \dot{\alpha } = C_{2} \to \dot{\alpha } - C_{2} = 0 $$

(33)

$$ \dot{\Omega } = C_{3} \to \dot{\varOmega } - C_{3} = 0 $$

(34)

The matrices of the extra coefficients are presented in (35) and (36).

$$ a^{{\prime }} = \left[ {\begin{array}{*{20}c} 0 & 0 & 1 & { - 1} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ \end{array} } \right] $$

(35)

$$ b^{{\prime }} = \left[ {\begin{array}{*{20}c} { - C_{1} } & { - C_{2} } & { - C_{3} } \\ \end{array} } \right]^{T} $$

(36)

The general coefficients of the robot are the combination of the inherent constraints with the extra constraints shown in (37) and (38).

$$ a_{tot} = \left[ {\begin{array}{*{20}c} a & {a^{{\prime }} } \\ \end{array} } \right]^{T} $$

(37)

$$ b_{tot} = \left[ {\begin{array}{*{20}c} b & {b^{{\prime }} } \\ \end{array} } \right]^{T} $$

(38)

Appendix C

3.1 Definition of unknown terms for the integrated multiplier method

The differential equations of GyroSym in the form of Integrated Multiplier Method is presented in (14). The unknown elements are $\left[ M \right]_{9 \times 9}$ and $\left\{ F \right\}_{9 \times 1}$ while other elements are defined in the previous sections. In this appendix, these elements are presented in (39) and (40).

$$ M_{ij} = \frac{{\partial^{2} L}}{{\partial \dot{q}_{i} \partial \dot{q}_{j} }} , \quad i = 1 \ldots 9 \;{\text{and}}\; j = 1 \ldots 9 $$

(39)

$$ \begin{aligned} & I_{xx} = 2I_{yy} = 2I_{zz} = I_{r} = 0.0034 \\ & I_{{f_{{x^{\prime}x^{\prime}}} }} = 2I_{{f_{{y^{\prime}y^{\prime}}} }} = 2I_{{f_{{z^{\prime}z^{\prime}}} }} = I_{f} = 0.0006824 \\ \end{aligned} $$

$$ \begin{aligned} M_{11} & = r_{PCG}^{2} \left( {1 - c^{2} \left( \theta \right)c^{2} \left( \beta \right)} \right)m_{p} + \left( {1 - \frac{1}{2}c^{2} \left( \theta \right)} \right)I_{r} \\ & \quad +\,\left( {s\left( \theta \right)c\left( \theta \right)c\left( \beta \right)c\left( \alpha \right)s\left( \alpha \right)} \right. \\ & \quad \left. +\,\frac{1}{2}\Big( c^{2} \left( \alpha \right)\left( {1 - c^{2} \left( \theta \right) - c^{2} \left( \theta \right)c^{2} \left( \beta \right)} \right)\right.\\ &\qquad \left. + c^{2} \left( \theta \right)c^{2} \left( \beta \right) + 1 \right) \Big)I_{f} \\ \end{aligned} $$

$$ \begin{aligned} M_{12} & = M_{21} = r_{PCG}^{2} c\left( \beta \right)c\left( \theta \right)s\left( \beta \right)m_{p} \\ & \quad +\,\frac{1}{2}s\left( \beta \right)\left( {c\left( \theta \right)c^{2} \left( \alpha \right)c\left( \beta \right) - s\left( \theta \right)c\left( \alpha \right)s\left( \alpha \right) - c\left( \beta \right)c\left( \theta \right)} \right)I_{f} \\ M_{13} & = M_{31} = - s\left( \theta \right)I_{r} \\ M_{14} & = M_{41} = - r_{PCG}^{2} s\left( \theta \right)m_{p} \\ &\quad -\,\frac{1}{2}\left( {s\left( \theta \right)c^{2} \left( \alpha \right) + c\left( \alpha \right)c\left( \theta \right)s\left( \alpha \right)c\left( \beta \right) + s\left( \theta \right)} \right)I_{f} \\ M_{15} & = M_{51} = \frac{1}{2}c\left( \theta \right)s\left( \beta \right)I_{f} \\ M_{16} & = M_{61} = - \left( {s\left( \theta \right)c\left( \alpha \right) + c\left( \theta \right)s\left( \alpha \right)c\left( \beta \right)} \right)I_{f} \\ M_{17} & = M_{71} = r_{PCG} \left( {s\left( \theta \right)c\left( \beta \right)s\left( \psi \right) - s\left( \beta \right)s\left( \psi \right)} \right)m_{p} \\ M_{18} & = M_{81} = - r_{PCG} \left( {s\left( \theta \right)c\left( \beta \right)c\left( \psi \right) + s\left( \beta \right)s\left( \psi \right)} \right)m_{p} \\ M_{22} & = r_{PCG}^{2} c^{2} \left( \beta \right)m_{p} + \frac{1}{2}I_{r} \\ & \quad +\,\frac{1}{2}\left( {2 - c^{2} \left( \beta \right) - c^{2} \left( \alpha \right) + c^{2} \left( \alpha \right)c^{2} \left( \beta \right)} \right)I_{f} \\ M_{24} & = M_{42} = \frac{1}{2}s\left( \alpha \right)s\left( \beta \right)c\left( \alpha \right)I_{f} \\ M_{25} & = M_{52} = \frac{1}{2}c\left( \beta \right)I_{f} \\ M_{26} & = M_{62} = s\left( \alpha \right)s\left( \beta \right)I_{f} \\ M_{27} & = M_{72} = - r_{PCG} c\left( \beta \right)c\left( \theta \right)c\left( \psi \right)m_{p} \\ M_{28} & = M_{82} = - r_{PCG} c\left( \beta \right)c\left( \theta \right)s\left( \psi \right)m_{p} \\ M_{29} & = M_{92} = - r_{PCG} s\left( \theta \right)c\left( \beta \right)m_{p} \\ M_{33} & = I_{r} \\ M_{44} & = r_{PCG}^{2} m_{p} + \frac{1}{2}\left( {1 + c^{2} \left( \alpha \right)} \right)I_{f} \\ M_{46} & = M_{64} = c\left( \alpha \right)I_{f} \\ M_{47} & = M_{74} = r_{PCG} \left( {s\left( \theta \right)c\left( \psi \right)s\left( \beta \right) - s\left( \psi \right)c\left( \beta \right)} \right)m_{p} \\ M_{48} & = M_{84} = r_{PCG} \left( {s\left( \theta \right)s\left( \psi \right)s\left( \beta \right) + c\left( \psi \right)c\left( \beta \right)} \right)m_{p} \\ M_{49} & = M_{94} = r_{PCG} s\left( \beta \right)c\left( \theta \right)m_{p} \\ M_{55} & = \frac{1}{2}I_{f} \\ M_{66} & = I_{f} \\ M_{77} & = m_{ring} + m_{p} \\ M_{88} & = m_{ring} + m_{p} \\ M_{99} & = m_{ring} + m_{p} \\ \end{aligned} $$

$$ \begin{aligned} M_{19} & = M_{91} = M_{23} = M_{32} = M_{34} = M_{43} = M_{35} = M_{53} \\ & = M_{36} = M_{63} = M_{37} = M_{73} = M_{38} \\ & = M_{83} = M_{39} = M_{93} = M_{45} = M_{54} = M_{56} = M_{65} \\ & = M_{57} = M_{75} = M_{58} = M_{85} = M_{59} = M_{95} \\ & = M_{67} = M_{76} = M_{68} = M_{86} = M_{69} = M_{96} = M_{78} \\ & = M_{87} = M_{79} = M_{97} = M_{89} = M_{98} = 0 \\ \end{aligned} $$

$$ \begin{aligned} \left[ M \right]_{9 \times 9} \left\{ {\ddot{q}} \right\}_{9 \times 1} & = \left\{ F \right\}_{9 \times 1} + \left[ a \right]_{9 \times 6}^{T} \left\{ \lambda \right\}_{6 \times 1} \to \left\{ F \right\}_{9 \times 1} \\ & = \left[ M \right]_{9 \times 9} \left\{ {\ddot{q}} \right\}_{9 \times 1} - \left[ a \right]_{9 \times 6}^{T} \left\{ \lambda \right\}_{6 \times 1} \\ & the\; equations\;of\;\left\{ F \right\}_{9 \times 1} \;are \\ & too\;big\;for\; this\;space. \\ \end{aligned} $$

(40)