Adaptive Fault-Tolerant Control Design for Multi-linked Two-Wheel Drive Mobile Robots

Abstract

This chapter presents the algorithmic design of an adaptive Fault Tolerant Control (FTC) to address several tasks needed for $n$-linked mobile robots subjected to actuator faults and friction phenomena. Firstly, it presents the development of kinematic and dynamic models for $n$-linked wheel drive mobile robot systems. Then, the kinematic model is transformed into a chained form and an approach to generate desired feasible trajectories is proposed. After that, a Lyapunov kinematic control law to control each state of the system and stabilize the tracking error is presented. In practice, the multi-robot system is affected by some disturbances, as for instance friction phenomena. Moreover, actuator faults may occur during the system life. Therefore, an adaptive law was designed to estimate the unknown friction coefficients and faults to adapt the control law online, leading to a multi-design integration-based adaptive controller. The asymptotical stability of the closed-loop is proven applying the Lyapunov theory. Simulations are performed in MATLAB/SIMULINK with different faulty cases, showing the efficiency of this method.

Appendices

8. Appendix

1.1 8.1. Input Transformation Matrix ${\varvec{B}}\left({\varvec{q}}\right)$

The directions of the $n$-robots in the forward motion can be expressed by the following unit vectors

$$ U_{1} = \left[ {cos\theta_{1} ,sin\theta_{1} } \right]^{T} ,U_{2} = \left[ {cos\theta_{2} ,sin\theta_{2} } \right]^{T} ,....,U_{n} = \left[ {cos\theta_{n} ,sin\theta_{n} } \right]^{T} $$

(138)

where $cos{\theta }_{i}$ and $sin{\theta }_{1}$ ($i=\mathrm{1,2},\cdots ,n$) are the components in the X and Y directions, respectively.

The control force and torque vectors generated by actuators are

$$ F_{1} = \frac{{\tau_{1r} + \tau_{1l} }}{{r_{1} }}U_{1} ,F_{2} = \frac{{\tau_{2r} + \tau_{2l} }}{{r_{2} }}U_{2} ,...,F_{N} = \frac{{\tau_{nr} + \tau_{nl} }}{{r_{n} }}U_{n} $$

$$ M_{t1} = \frac{{\tau_{1r} - \tau_{1l} }}{{r_{1} }}b_{1} ,....,M_{t2} = \frac{{\tau_{2r} - \tau_{2l} }}{{r_{2} }}b_{2} ,....M_{tN} = \frac{{\tau_{nr} - \tau_{nl} }}{{r_{n} }}b_{n} $$

(139)

where, the vectors ${M}_{ti}$ ($\mathrm{1,2},\cdots ,n$) are oriented in the direction of Z-axis. The position vector ${r}_{p1}$ can be expressed as

$$ r_{p1} = \left( {x + \mathop \sum \limits_{i = 1}^{n} d_{i} cos\theta_{i} } \right)\hat{I} + \left( {y + \mathop \sum \limits_{i = 1}^{n} d_{i} sin\theta_{i} } \right)\hat{J},....,r_{pn} = \left[ {x,y} \right]^{T} $$

(140)

The variation in position vector ${r}_{p1}$ can be described by

$$ \delta r_{p1} = \left( {\delta x - \mathop \sum \limits_{i = 1}^{n} d_{i} sin\theta_{i} \delta \theta_{i} } \right)\hat{I} + \left( {\delta y + \mathop \sum \limits_{i = 1}^{n} d_{i} cos\theta_{i} \delta \theta_{i} } \right)\hat{J},\,\delta r_{pn} = \left[ {\delta x,\delta y} \right]^{T} $$

(141)

Also, the variation in the total work due to applied generalized forces is given by

$$ \delta W = F_{1}^{T} \delta r_{p1} + F_{2}^{T} \delta r_{p2} + \cdots + F_{N}^{T} \delta r_{pn} + M_{t1} \delta \theta_{1} + M_{t2} \delta \theta_{2} + \cdots + M_{tN} \delta \theta_{n} $$

(142)

where ${\delta r}_{p1}$, ${\delta r}_{p2}$, $\cdots$, ${\delta r}_{pn}$ and $\delta {\theta }_{1}$, $\delta {\theta }_{2}$, $\cdots$, $\delta {\theta }_{n}$ are variations of ${r}_{p1}$, $\cdots$, ${r}_{pn}$, and ${\theta }_{1}$, $\cdots$, ${\theta }_{n}$. Since $q$ is defined by $q={\left[x,y,{\theta }_{n},{\theta }_{n-1},{\dots ,\theta }_{1}\right]}^{T}$, one can rewrite Equation (144) as follows:

$$ \begin{aligned} \delta W & = F_{1}^{T} \cdot \frac{{\partial r_{{p1}} }}{{\partial q}} \cdot \delta q + F_{2}^{T} \cdot \frac{{\partial r_{{p2}} }}{{\partial q}} \cdot \delta q + ... + F_{n}^{T} \cdot \frac{{\partial r_{{pn}} }}{{\partial q}} \cdot \delta q \\ & + M_{{t1}} \cdot \frac{{\partial \theta _{1} }}{{\partial q}} + M_{{t2}} \cdot \frac{{\partial \theta _{2} }}{{\partial q}} + ... + M_{{tn}} \cdot \frac{{\partial \theta _{n} }}{{\partial q}}Q^{T} \cdot \delta q \\ \end{aligned} $$

(143)

where $\delta q$ represents the incremental variation of $q$, and the vector $Q$, which represents the generalized forces that corresponds to generalized system coordinates, can be written as,

$$ Q = \left( {\frac{{\partial r_{p1} }}{\partial q}} \right)^{T} \cdot F_{1} + \nu + \left( {\frac{{\partial r_{pn} }}{\partial q}} \right)^{T} \cdot F_{n} + \left( {\frac{{\partial \theta_{1} }}{\partial q}} \right)^{T} \cdot M_{t1} + \nu + \left( {\frac{{\partial \theta_{n} }}{\partial q}} \right)^{T} \cdot M_{tn} $$

(144)

One can define the injection matrix $B\left(q\right)$ as

$$B\left(q\right)=\frac{\partial Q}{\partial \tau }$$

(145)

where,$\tau$ represents the control torque vector $\tau =\left[{\tau }_{1r},{\tau }_{1l},{\tau }_{2r},{\tau }_{2l},\cdots ,{\tau }_{nr},{\tau }_{nl}\right]$.

9. Appendix

1.1 9.2. Inertia Matrix ${\varvec{M}}\left({\varvec{q}}\right)$ and Matrix of Coriolis Forces ${\varvec{C}}\left({\varvec{q}},\dot{{\varvec{q}}}\right)$ for Robotic System

The position vectors ${C}_{1},\dots ,{C}_{n}$ can be defined by

$$ \begin{aligned} r_{{c1}} = & r_{{p1}} + a_{1} U_{1} \\ & = \left[ {x + dcos\theta _{2} + a_{1} cos\theta _{1} ,y + dsin\theta _{2} + a_{1} sin\theta _{1} } \right]^{T} \\ & r_{{cn}} = r_{{pn}} + a_{n} U_{n} = \left[ {x + a_{n} cos\theta _{n} ,y + a_{n} sin\theta _{n} } \right]^{T} \\ \end{aligned} $$

(146)

The kinematic energy of the system is defined by

$$T\left(q,\dot{q}\right)=\frac{1}{2}{\sum }_{i=0}^{n}\{{I}_{mi}{\dot{\theta }}_{i}^{2}+{m}_{i}{\dot{r}}_{ci}^{T}{\dot{r}}_{ci}\}$$

(147)

Taking the time derivative of ${r}_{c1}....,{r}_{cn}$, to have

$$ \dot{r}_{c1} = \frac{{\partial r_{c1} }}{\partial q} \cdot \dot{q} = T_{1} \cdot \dot{q},...,\dot{r}_{cn} = \frac{{\partial r_{cn} }}{\partial q} \cdot \dot{q} = T_{n} \cdot \dot{q} $$

(148)

where, the matrices, ${T}_{1},\dots ,{T}_{n}$ in (B.3) represents Jacobian matrices. Accordingly, one can rewrite the kinematic energy of robotic system as follows:

$$T\left(q,\dot{q}\right)=\frac{1}{2}{\sum }_{i=0}^{n}\{{{I}_{mi}{\dot{\theta }}_{i}^{2}+m}_{i}{\dot{q}}^{T}{{T}_{i}^{T}T}_{i}\dot{q}\}$$

(149)

where

$$M\left(q\right)=diag\left(\{\mathrm{0,0},{I}_{m1},{I}_{m2},\cdots ,{I}_{mn}\}\right)+{\sum }_{i=0}^{n}{\{m}_{i}{{T}_{i}^{T}T}_{i}\}$$

(150)

where, $M\left(q\right)$ represents inertia matrix of robotic system, which is symmetric and positive-definite matrix.

One can proceed by writing the Lagrange formulation of the robotic system as

$$\frac{d}{dt}\left(\frac{\partial T}{\partial \dot{q}}\right)-\frac{\partial T}{\partial q}=Q+{A}^{T}\left(q\right)\lambda $$

(151)

From (149),

$$\frac{\partial T}{\partial \dot{q}}=M\dot{q}$$

(152)

Then,

$$\frac{d}{dt}\left(\frac{\partial T}{\partial \dot{q}}\right)-\frac{\partial T}{\partial q}=\dot{M}\dot{q}+M\ddot{q}-\frac{\partial T}{\partial q}$$

$$=M\ddot{q}+C\left(q,\dot{q}\right)$$

(153)

where the centripetal Coriolis vector $C\left(q,\dot{q}\right)$ can be expressed as

$$C\left(q,\dot{q}\right)=\dot{M}\dot{q}-\frac{\partial T}{\partial q}$$

(154)