4.2.1. WMRs with *δ<sup>M</sup>* = 3

These types of WMRs are omnidirectional in nature, which means that they have independent heading and linear movements. However, the holonomic type with *δ* = (3, 0) provides full mobility and, hence, instantaneous velocity in any direction. The other two categories (*δ* = (2, 1) and *δ* = (1, 2)) are steerable and nonholonomic. They are capable of providing movement in any arbitrary direction, but only after they have steered their wheels to the corresponding configuration. For the problem at hand, the difference between holonomic and nonholonomic types only occurs at the beginning of the path, in which the holonomic type may start path-following instantly, but the nonholonomic types have to rearrange their steerable wheels. Other than this, on a smooth path and heading profile, both types provide the same functionality.

These types of WMRs are capable of changing the direction of their base linear velocity while the base is stationary. This can be instantly performed in the case of holonomic types or in the case of nonholonomic types by changing the direction of the steering wheels over time. This fact allows us to directly control *ψ<sup>v</sup>* and have *S*<sup>∗</sup> as the error states instead of *S*. Consequently, for this type of WMR, the heading and linear movements are independent. Hence, the controller's inputs are both *P<sup>d</sup>* and *θd*, and the body frame *B* is chosen arbitrarily. For the base path-following (Definition 2) of such WMRs, *s* , and *κ<sup>b</sup>* of Proposition 1 are *s* , and *θ <sup>b</sup>* given by Equations (30a) and (30b) in Section 4.1, respectively. There are two viable options for *κv*. If the target states is set to be full states (*s*,*S*∗), then, as mentioned earlier, *κ<sup>v</sup>* should be set as *ψ <sup>d</sup>* given by Equation (30d). The second option is to set (*s*,*S*) as the target states and, therefore, *κ<sup>v</sup>* should be evaluated using *ψ <sup>v</sup>* given by Equation (31a). Table 3 summarize these results.
