Modelling and control strategies for a motorized wheelchair with hybrid locomotion systems

Abstract

Several kinematic and dynamic models of conventional motorized wheelchairs in the literature presume the wheelchair movement only on flat surfaces, disregarding the effects of gravitational forces and rolling friction. In addition, there are no many studies clearly describing how the dynamic properties of a conventional wheelchair and stair-climbing wheelchair with a track mechanism are formulated. However, the design of a good controller generally involves the formulation of a comprehensive wheelchair model. This work contributes to the modelling of the motorized wheelchair with hybrid locomotion systems (MWHLSs), which have wheel and track mechanisms to move the wheelchair, regarding the formulation of kinematic and dynamic models for each locomotion system, considering the effects of gravitational forces and rolling friction, on flat and inclined surfaces and climbing stairs. From the mathematical model for each locomotion system, the gearmotor torque control is used, since the torque provides smooth and precise driving. Thus, the open-loop and closed-loop control techniques, such as control Lyapunov function, Backstepping control, and Proportional-Integral-Derivative controller, are designed due to their well-known characteristics. The robustness analysis of these controllers, taking into account the occurrence of external disturbances, indicates which one best assists wheelchair users, providing a safer and more efficient navigation with the MWHLS.

Appendix

A kinematic model must take into account the constraints that each type of wheel imposes on the locomotion of the MWHLS. We establish kinematic constraints only for the fixed standard wheel, since we consider the castor wheel does not impose any restrictions on movement of the MWHLS. Thus, the kinematic constraints of a chassis configuration are the combination of the constraints imposed by each wheel [24], as shown below:

$$\left[ {\begin{array}{*{20}c} {{\text{Rolling}}\;{\text{Constraints}}} \\ {{\text{Sliding }}\;{\text{Constraints}}} \\ \end{array} } \right]R\left( \varphi \right)\dot{\xi }_{I} = \left[ {\begin{array}{*{20}c} {r\dot{\theta }} \\ 0 \\ \end{array} } \right] ,$$

(35)

Figure 14 describes a fixed standard wheel on local reference frame { $X_{R} , Y_{R}$ }. The wheel position is expressed in polar coordinates by distance $b$ and angle $\alpha$. The angle of the wheel plane relative to the chassis is denoted by $\beta$, which is fixed since this wheel is not steerable [24].

Fig. 14

A fixed standard wheel and its parameters [24]

The rolling constraint for this wheel enforces that all motion along the direction of the wheel plane must be accompanied by the appropriate amount of wheel spin so that there is pure rolling at the contact point [24]:

$$\left[ {\begin{array}{*{20}c} {{ \sin }\left( {\alpha + \beta } \right)} & { - { \cos }\left( {\alpha + \beta } \right)} & { - b{ \cos }\left( \beta \right)} \\ \end{array} } \right]R\left( \varphi \right)\dot{\xi }_{I} = r\dot{\theta } ,$$

(36)

The sliding constraint for this wheel enforces that the component of the wheel motion orthogonal to the wheel plane must be zero [24]:

$$ \left[ {\begin{array}{*{20}c} {{ \cos }\left( {\alpha + \beta } \right)} & { - { \sin }\left( {\alpha + \beta } \right)} & {b{ \sin }\left( \beta \right)} \\ \end{array} } \right]R\left( \varphi \right)\dot{\xi }_{I} = 0 . $$

(37)

Assuming that the centre of rotation is at the midpoint between the wheels or tracks, we may consider [24]:

• Left Wheel/Track: $\alpha = \frac{\pi }{2}$, $\beta = 0$

• Right Wheel/Track: $\alpha = - \frac{\pi }{2}$, $\beta = \pi$

Replacing the Eqs. (36), (37), and the values of $\alpha$ and $\beta$ in the Eq. (35), we obtain:

$$ \left[ {\begin{array}{*{20}c} {{ \sin }\left( { - \frac{\pi }{2} + \pi } \right)} & { - { \cos }\left( { - \frac{\pi }{2} + \pi } \right)} & { - b{ \cos }\left( \pi \right)} \\ {{ \sin }\left( {\frac{\pi }{2} + \pi } \right)} & { - { \cos }\left( {\frac{\pi }{2} + \pi } \right)} & { - b{ \cos }\left( 0 \right)} \\ {{ \cos }\left( {\frac{\pi }{2} + 0} \right)} & { - { \sin }\left( {\frac{\pi }{2} + 0} \right)} & {b{ \sin }\left( 0 \right)} \\ \end{array} } \right]R\left( \varphi \right)\dot{\xi }_{I} = \left[ {\begin{array}{*{20}c} {r\dot{\theta }_{R} } \\ {r\dot{\theta }_{L} } \\ 0 \\ \end{array} } \right] , $$

$$ \dot{\xi }_{I} = R\left( \varphi \right)^{ - 1} \left[ {\begin{array}{*{20}c} 1 & 0 & b \\ 1 & 0 & { - b} \\ 0 & 1 & 0 \\ \end{array} } \right]^{ - 1} \left[ {\begin{array}{*{20}c} {r\dot{\theta }_{R} } \\ {r\dot{\theta }_{L} } \\ 0 \\ \end{array} } \right] , $$

$$ \dot{\xi }_{I} = \left[ {\begin{array}{*{20}c} {\dot{x}_{c} } \\ {\dot{y}_{c} } \\ {\dot{\varphi }_{c} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{ \cos }\left( \varphi \right){ \cos }\left( \phi \right)} & { - { \sin }\left( \varphi \right){ \cos }\left( \phi \right)} & 0 \\ {{ \sin }\left( \varphi \right){ \cos }\left( \phi \right)} & {{ \cos }\left( \varphi \right){ \cos }\left( \phi \right)} & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {1/2} & {1/2} & 0 \\ 0 & 0 & 1 \\ {1/2b} & { - 1/2b} & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {r\dot{\theta }_{R} } \\ {r\dot{\theta }_{L} } \\ 0 \\ \end{array} } \right] . $$

(38)

However, the MWHLS has its centre of rotation displaced from the midpoint between the wheels, then:

$$ \dot{\xi }_{I} = \left[ {\begin{array}{*{20}c} {\dot{x}_{c} } \\ {\dot{y}_{c} } \\ {\dot{\varphi }_{c} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{ \cos }\left( \varphi \right){ \cos }\left( \phi \right)} & { - d{ \sin }\left( \varphi \right){ \cos }\left( \phi \right)} & 0 \\ {{ \sin }\left( \varphi \right){ \cos }\left( \phi \right)} & {d{ \cos }\left( \varphi \right){ \cos }\left( \phi \right)} & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\frac{r}{2}\left( {\dot{\theta }_{R} + \dot{\theta }_{L} } \right)} \\ {\frac{r}{2}\left( {\dot{\theta }_{R} - \dot{\theta }_{L} } \right)} \\ {\frac{r}{2}\left( {\dot{\theta }_{R} - \dot{\theta }_{L} } \right)} \\ \end{array} } \right] . $$

(39)

The kinematic modelling of the MWHLS applying the track locomotion system is similar to the description presented here.