Abstract
Torque vectoring control is one of the most interesting techniques applicable to electric vehicles with multiple motors. Essentially it is the possibility to allocate desired amounts of torque to each motor. With an uneven allocation of torque between left and right sides of the vehicle, a direct yaw moment can be generated and exploited to enhance the vehicle handling behaviour. This allows to enhance vehicle safety, stability and cornering performance. Significant energy efficiency benefits are also achievable, either as a main priority or as a secondary objective when the main target is the vehicle handling behaviour. This Chapter explains the underlying principles of each element of a torque vectoring control framework: reference generator, high level controller, and low level controller. Notable applications are also presented and discussed in detail.
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Notes
- 1.
Also, sometimes R is confused with the distance between instant centre and centre of mass \(\frac{\sqrt{u^2+v^2}}{r}\) (or simply CG in Fig. 3), this is fine for small values of v.
- 2.
\(S_r\) may be roughly between 12 and 20.
- 3.
Again the sideslip angle is assumed small so that \(V \approx u\).
- 4.
The effect of \(a_x\) on the cornering response can be seen in De Novellis et al. (2015a), Figs. 6 and 7.
- 5.
For instance, the yaw rate controller is designed for dry conditions, \(\mu \approx 0.8 - 1\), but the road is wet. Or, even in dry conditions, \(\mu \) happens to be overestimated.
- 6.
Unless \(m V^2 + C_1 a_1 - C_2 a_2=0\).
- 7.
See Table 1 in Tota et al. (2018).
- 8.
For instance, e.g., Manning and Crolla (2007) suggest that the feedforward contribution “tends to reduce the time lag between steer angle input and vehicle reaction”.
- 9.
In other words, a simple “cruise control” logic.
- 10.
In principle the double-track model above could also be used, with potentially any tyre model—however in case of strong nonlinearities and/or if the tyre model accounts for combined interactions, for computational reasons it would be more appropriate to derive a \(M_{z,FF}\) map, as in De Novellis et al. (2013).
- 11.
The interested reader might find interesting to know how to obtain this result from, e.g., Stanford-University (2020).
- 12.
Note that this is not a Differential Riccati Equation (DRE) because of the infinite horizon used in 56 (otherwise the right-hand-side of this equation would be \( - \dot{P}\) instead of 0).
- 13.
Because the single-track vehicle model assumes constant vehicle speed, a gain scheduling approach is often implemented.
- 14.
- 15.
The reference yaw rate is designed based on the steady-state relationship \(r_{ref}=a_y/V\), but in general \(a_x \ne 0\) is also allowed.
- 16.
It should also be noted that compared to a conventional ESC, the torque vectoring controller interventions can be seamlessly and continuously generated, without necessarily decreasing the vehicle speed.
- 17.
- 18.
In iCOMPOSE a dedicated sideslip angle sensor was used, to ensure a reliable measurement of u and v in any condition, so as to be able to fully focus on the design of the controllers.
- 19.
Another example would be a parking manoeuvre, which entails high steering angles, hence high kinematic sideslip angles, but low tyre slips, hence small dynamic sideslip angles.
- 20.
That is, the wheel steer angle divided by the vehicle wheelbase and multiplied by the distance between the rear axle and the point of interest.
- 21.
Obviously this index makes sense only if \(\beta _{ref}\) is defined. As discussed, concurrent yaw rate and sideslip control is possible even without defining \(\beta _{ref}\), by integrating the sideslip angle in the reference yaw rate formulation, as shown in 42.
- 22.
Note that lateral force contributions, \(F_{y_{ij}}\) do not appear in these equations because they cannot be controlled, as extensively discussed in Sect. 4.2.
- 23.
- 24.
These can be easily obtained imposing that the first order derivative of \(P_{loss,{ij}}\) with respect to \(T_{ij}\) must be positive, and that the second order derivative is zero for a positive value of \(T_{ij}\).
- 25.
Note that the left-hand side can be either \(P_{loss,{1j}}(T_{sw},V)+P_{loss,{2j}}(0,V)\) or \(P_{loss,{1j}}(0,V)+P_{loss,{2j}}(T_{sw},V)\) since, as discussed, \(\sigma =0\) and \(\sigma =1\) yield the same power losses for a vehicle side.
- 26.
While \(\sigma =0\) and \(\sigma =1\) are equivalent in terms of power losses, \(\sigma =1\) is preferred for safety reasons since, from a vehicle dynamics point of view, understeer is better than oversteer.
- 27.
It is worth to note that, hypothetically, assigning more than the overall torque demand on the external vehicle side would imply a negative (regenerative) torque on the inner side, which is far from optimal (Lenzo et al., 2017a).
- 28.
Details are in Lenzo et al. (2017a).
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Lenzo, B. (2022). Torque Vectoring Control for Enhancing Vehicle Safety and Energy Efficiency. In: Lenzo, B. (eds) Vehicle Dynamics. CISM International Centre for Mechanical Sciences, vol 603. Springer, Cham. https://doi.org/10.1007/978-3-030-75884-4_4
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