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Mathematical modelling and control of a nonholonomic spherical robot on a variable-slope inclined plane using terminal sliding mode control

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Abstract

Motion analysis and control of a pendulum-driven spherical robot (PDSR) on an inclined plane with a variable slope is investigated. Firstly, the mathematical model of a PDSR on a variable-slope inclined plane is deduced applying a Lagrangian formulation. Afterwards, in the presence of an unknown external disturbance, the terminal sliding mode control (TSMC) technique is employed to stabilize the robot on the inclined plane, while the plane is still moving. In other words, the terminal sliding mode disturbance observer is used to estimate the unknown disturbance. Based on the disturbance estimation, the TSMC scheme is established to control the single-input and single-output nonlinear system with control singularity and an unknown nonsymmetric control input saturation. In fact, a compound disturbance is defined and estimated, which includes the external disturbance, the control singularity and the unknown input saturation. Simulations are then conducted to validate the proposed approach for motion control of a PDSR on a variable-slope inclined plane with an unknown external disturbance and nonsymmetric input limits.

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Abbreviations

\(\phi \) :

Rotation of the spherical shell w.r.t. the (inclined) plane

\(\beta \) :

Rotation of the main shaft w.r.t. the spherical shell

\(\theta \) :

Instantaneous angle of the pendulum w.r.t. the perpendicular to the (inclined) plane

\(\alpha \) :

Angle of the inclined plane

\(m_\mathrm{p} \) :

Mass of the pendulum

\(m_\mathrm{s} \) :

Mass of the spherical shell

M :

Mass of the whole robot

\(I_\mathrm{A} \) :

Spherical shell moment of inertia about point A (centre)

\(I_\mathrm{C} \) :

Spherical shell moment of inertia about point C (contact point)

\(\rho \) :

Radius of the spherical shell

r :

Radius of the pendulum

\(r_G \) :

Radius of the robot mass centre

\({{\varvec{q}}}\) :

Vector of generalized coordinates, i.e., \({{\varvec{q}}}=\left[ {\phi \quad \theta } \right] ^\mathrm{T}\)

\({{\varvec{Q}}}\) :

Vector of generalized forces, i.e., \({{\varvec{Q}}}=\left[ {{{Q}}_\phi \quad {{Q}}_\theta } \right] ^\mathrm{T}\)

T :

Total kinetic energy of the system

V :

Total potential energy of the system

\({\mathcal {L}}\) :

Lagrangian, i.e., \({\mathcal {L}}=T-V\)

\(T_\mathrm{p} \) :

Kinetic energy of the pendulum

\(T_\mathrm{s} \) :

Kinetic energy of the spherical shell

\({\varvec{\omega }}_\mathrm{s} \) :

Angular velocity of the spherical shell

\({{\varvec{V}}}_\mathrm{A} \) :

Velocity of point A (shell centre)

\({{\varvec{V}}}_\mathrm{C} \) :

Velocity of point C (contact point)

\({{\varvec{V}}}_\mathrm{A/C} \) :

Velocity of point A w.r.t. point C

\({{\varvec{V}}}_\mathrm{p} \) :

Velocity of the pendulum

\({{\varvec{V}}}_\mathrm{p/A} \) :

Velocity of the pendulum w.r.t. point A

x :

Position of the robot on the inclined plane

\(x_0 \) :

Initial position of the robot on the inclined plane

g :

Earth gravitational acceleration

\(\left( {{{\varvec{i}}},{{\varvec{j}}},{{\varvec{k}}}} \right) \) :

Unit vectors in x, y and z directions

W :

Work done by external forces/moments

\(\tau _\mathrm{m} \) :

Torque of the motor

\(\theta _\mathrm{e} \) :

Desired angle for \(\theta \) in the equilibrium state of the robot

\({{\varvec{x}}}\) :

Vector of state variables, i.e., \({{\varvec{x}}}=\left[ {x_1 x_2 \ldots x_n } \right] ^\mathrm{T}\)

u :

Control input

\(u_{\mathrm{min}} ,u_{\mathrm{max}} \) :

Lower and upper limits of the control input

y :

Output of the system

\(y_d \) :

Desired output

d :

Unknown external disturbance

D :

Compound disturbance

\({\widehat{D}}\) :

Estimated compound disturbance

\(\tau ,k,\gamma , \epsilon , p_0, q_0,\) \(p_i, q_i, \alpha _i , \beta _i ,\delta , \mu \) :

Controller and estimator design parameters

\(\hbox {sign}\left( *\right) \) :

Sign function

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Authors and Affiliations

  1. Department of Mechanical Engineering, Centre for Intelligent Machines (CIM), McGill University, Montréal, Canada

    M. Roozegar

  2. School of Mechanical Engineering, University of Tehran, Tehran, Iran

    M. Ayati

  3. Centre for Mechatronics and Intelligent Machines, School of Mechanical Engineering, University of Tehran, Tehran, Iran

    M. J. Mahjoob

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  1. M. Roozegar
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  2. M. Ayati
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  3. M. J. Mahjoob
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Correspondence to M. Roozegar.

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Roozegar, M., Ayati, M. & Mahjoob, M.J. Mathematical modelling and control of a nonholonomic spherical robot on a variable-slope inclined plane using terminal sliding mode control. Nonlinear Dyn 90, 971–981 (2017). https://doi.org/10.1007/s11071-017-3705-9

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