Automated Kinematics Reasoning for Wheeled Mobile Robots

Abstract

Control schemes for wheeled mobile robots typically assume a specific mobility capability of a drive and implicitly use the drive’s kinematics within its control procedures. This makes it difficult to deal with faults in the drive and to handle drives with diverse geometry and functionality that might even change during operation of a robot. As a consequence, we propose a model-based control scheme that builds upon an automated analysis of a robotic drive and on an on-line deduction of the drive’s kinematics. We achieve this functionality through (1) the introduction of steering-angle independent, generalized variants of the rolling and sliding constraints for wheeled mobile robots and (2) the corresponding reformulation of kinematic analysis. This leads to a computationally efficient algorithm that deduces the (inverse) kinematics of a drive for its mode of operation or failure. Fault tolerant and robust behavior, however, is only one aspect of our control architecture. On-line kinematics analysis enables us to easily handle robots that change in geometry or functionality such as self-configuring modular robot systems and teams of cooperative robots.

Appendix

11.1.1 Qualitative Rolling and Sliding Constraints for an Omni-Directional Mecanum Wheel

It is straight forward to provide qualitative sliding constraints for Mecanum wheels as well. If one considers a mecanum wheel with the geometry as shown in Fig. 11.6, one obtains the standard rolling and sliding constraints as

Fig. 11.6

Mecanum wheel geometry

$$ \begin{array}{clclcllc}r\dot{\varphi }\;\rm cos(\gamma ) = \left[ {\begin{array}{clclcllc}{\rm sin(\alpha + \beta + \gamma )\:} { - \rm cos(\alpha + \beta + \gamma )\quad} { - l\rm cos(\beta + \gamma )} \\\end{array} } \right]\dot{\xi } \hfill \\ = {{{\mathbf{j}}}^T}\dot{\xi }, \end{array} $$

$$ \begin{array}{clclcllc}0 = \left[ {\begin{array}{clclcllc}{\rm cos(\alpha + \beta + \gamma )\:} \quad{\rm sin(\alpha + \beta + \gamma )\:} \quad{l\rm sin(\beta + \gamma )} \\\end{array} } \right]\;\dot{\xi } - r\dot{\varphi }\;\rm sin(\gamma ) - {{\it r}_{\it r}}\mathop{{{{\cdot}\varphi }}}\nolimits_{\it r} \hfill \\ = {{{\mathbf{c}}}^T}\dot{\xi }. \end{array} $$

Note that a fully operational mecanum wheel (actuated or freely spinning) does not impose any constraints on the robot’s movement as the rotation of the individual rolls ensures the sliding constraint 1.32. We express this fact in terms of a qualitative constraint with ${\mathbf{c}}_q^T: = \left[ {\begin{array}{clclcllc}{0\:} & {0\:} & 0 \\\end{array} } \right]$. The rolling constraint is time-invariant as mecanum wheels are typically used without active steering. Therefore, we obtain ${\mathbf{j}}_q^T: = \left[ {\begin{array}{clclcllc}{\rm sin(\alpha + \beta + \gamma )\:} & { - \rm cos(\alpha + \beta + \gamma )\:} & { - l\rm cos(\beta + \gamma )} \\\end{array} } \right]$. However, a blocked mecanum wheel exhibits a behavior, where the individual rolls act like non-actuated standard wheels at $\beta + \gamma + \tfrac{\pi }{2}$. The according qualitative sliding constraint is thus

$$ {\mathbf{c}}_q^T: = \left[ {\begin{array}{clclcllc}{\rm cos(\alpha + \beta + \gamma + \frac{\pi }{2})\:} & {\rm sin(\alpha +\beta + \gamma + \frac{\pi }{2})\:} & {l\rm sin(\beta + \gamma +\frac{\pi }{2})} \\\end{array} } \right].$$

We summarize the matrix entries ${\mathbf{c}}_q^T$ and ${\mathbf{j}}_q^T$ for the qualitative constraint matrices ${{{\mathbf{C}}}_q}$ and J _q in Table 11.2.

Table 11.2 Qualitative rolling and sliding constraints for an omni-directional mecanum wheel