*2.2. Four-DoF Mechanical Steering System Sub-Model*

As mentioned above, the DDAS system maintains the mechanical steering system. The four degrees of freedom model of the steering system is shown in Figure 5. The corresponding dynamic equations of the steering system are shown as follows [11,12]:

$$J\_{\mathbb{C}}\ddot{\delta}\_{\mathbb{S}\mathbb{W}} + B\_{\mathbb{C}}\dot{\delta}\_{\mathbb{S}\mathbb{W}} + K\_{\mathbb{C}} \left(\delta\_{\mathbb{S}\mathbb{W}} - \frac{Y\_{R}}{r\_{P}}\right) = T\_{\mathbb{S}\mathbb{W}}\tag{5}$$

$$A\_R \ddot{Y}\_R + B\_R \dot{Y}\_R + \eta\_F \frac{K\_C}{r\_P} (\frac{Y\_R}{r\_P} - \delta\_{\rm sw}) + CF\_R + \eta\_B \left(\frac{T\_{KL1}}{N\_{L1}} + \frac{T\_{KL2}}{N\_{L2}}\right) = 0\tag{6}$$

$$J\_{FW1}\ddot{\delta}\_{FW1} + B\_{FW1}\dot{\delta}\_{FW1} + C F\_{FW1} + A T\_1 = T\_{KL1} \tag{7}$$

$$
\dot{\delta}\_{\text{FW2}} \ddot{\delta}\_{\text{FW2}} + B\_{\text{FW2}} \dot{\delta}\_{\text{FW2}} + C \text{F}\_{\text{FW2}} + A \text{T}\_2 = T\_{\text{KL}2} \tag{8}
$$

$$T\_{KL1} = K\_{SL1} \left(\frac{Y\_R}{N\_{L1}} - \delta\_{FW1}\right) \tag{9}$$

$$T\_{\rm KL2} = K\_{\rm SL2} \left( \frac{\Upsilon\_R}{N\_{L2}} - \delta\_{\rm FWHM} \right) \tag{10}$$

$$T\_{\rm SC} = K\_{\rm C} \left( \delta\_{\rm sw} - \frac{\mathcal{Y}\_{\rm R}}{r\_P} \right) \tag{11}$$

where *Tsw* is the steering wheel torque, *BC* and *JC* are the damping of the steering column and equivalent inertia of steering wheel and column, δ*sw* is the steering wheel angle, *MR* and *BR* are the mass and damping of the rack, *YR* is the displacement of the rack, *rp* is the radius of the pinion, η*<sup>f</sup>* and η*<sup>B</sup>* are the forward transmitting efficiency and backward transmitting efficiency of the steering gear respectively, *KC* is the torsional stiffness of the torsion bar, *CFR* is the Coulomb friction of the gear and rack [13], *NLi* is the ratio of the rack transfer displacement to knuckle angular displacement, *JFWi* and *BFWi*(*i* = 1, 2) are the inertia of the road wheels round their kingpin and damping of kingpin, *CFFWi* is the coulomb friction caused by the left front wheel and the right front wheel rotating the kingpin, and the specific calculation formula is shown in the literature [14], *TKLi* is the total torque from the kingpins of the left front wheel and the right front wheel, δ*FWi* is the steering angle of front wheels, *KSLi* is the torsional stiffness of the kingpin of the front wheels.

**Figure 5.** 4-DoF mechanical steering system model.

*ATi* represents the alignment torque of each front wheel around the kingpin, which is mainly composed of the following four parts [15,16]:

$$M\_{sy} = F\_y \cdot r\_w \sin \pi \cos \sigma \tag{12}$$

$$M\_{\infty} = F\_{\text{x}} \cdot r\_{\sigma} \cos \tau \cos \sigma \tag{13}$$

$$M\_{\overline{z}\overline{z}} = M\_{\overline{t}\overline{z}} \cdot \cos \tau \cos \sigma \tag{14}$$

$$M\_{\mathbb{S}\mathbb{Z}} = F\_{\mathbb{Z}^\*} \cos \pi \sin \sigma \sin \delta\_{\mathbb{FW}} \cos \sigma (r\_{\mathcal{o}} + r\_{\mathcal{W}} \tan \sigma) \tag{15}$$

where *Msy* is the alignment torque generated by the lateral force of the tire, *Msx* is the alignment torque generated by the longitudinal force of the tire, *Mzz* is the component of the self-alignment torque around the kingpin, *Msz* is the alignment torque generated by the normal force of the front axle, *Mtz* is the self-alignment torque of the tire, *Fx* is the longitudinal force of the wheels, *Fy* is the lateral force of the wheels, *Fz* is the normal force of the wheels, τ is the kingpin caster angle, σ is the kingpin inclination angle.

#### *2.3. Wheel Rotation Dynamic Sub-Model*

The wheel rotation dynamic model is shown in Figure 6.

**Figure 6.** Wheel rotation dynamic model.

The rotation dynamic equation of each driving wheel can be established as follows:

$$I\_{\rm av}\dot{\omega}\_{\rm i} = T\_{\rm i} - F\_{\rm xi}r\_{\rm av} \tag{16}$$

where *Iw* is the moment of inertia of the wheels, ω*i*(*i* = 1, 2, 3, 4) is the angular velocity of the wheels, *Ti*(*i* = 1, 2, 3, 4) is the driving torque of the wheels.
