# Torque Allocation of Hybrid Electric Trucks for Drivability and Transient Emissions Reduction
## 1 Introduction
Coming back to hybrid vehicles, mathematical models of both conventional and mild hybrid powertrain were developed in [23] through the integration of a conventional manual transmission-equipped powertrain and an electric motor as a secondary power source, in order to study the performance of partial power-on gear shifts through the implementation of torque hole-filling by the electric motor during gearshifts. Besides the modelling of mechanical components, an appropriate type model can have relevant effects on simulated dynamics, as they have also a considerable influence on damping effects inside the powertrain. [24] demonstrated, using appropriate identification techniques aimed at describing tyre torsional dynamics in the frequency range 10-50 Hz, that damping is more accurately represented if modelled as hysteretic and not as viscous when the tyre load is low. Conversely, [25] proposed three different tyre models, e.g., modelled as a simple torsional spring, adopting a linear slip model, or assuming a fixed relaxation length; these last two models introduced nonlinearity, as an instantaneous slip ratio was considered. The relaxation length was estimated from the steady state tyre properties in [26] and included wheel load and slip dependencies. The adoption of this parameter is necessary for the description of interactions between tyres and roads, as all the other chassis components influence vehicle dynamic properties through tyre contact forces and torques. A lumped LuGre tyre ground contact model was modelled in [27] and in [28], where the developed dynamic friction tyre model was able to accurately capture the transient behaviour of the friction force observed during transitions between braking and acceleration phases. Other researchers have instead used software for tyre dynamics to compute the forces acting on the tyre and hence on the steel wheel for other purposes, such as fatigue assessment [29]. The parallel mild hybrid architecture of a heavy commercial vehicle developed for this research study is similar to the one adopted in [30], but with the improvement of implementing three different control strategies for the direct control of vehicle acceleration, vehicle speed control, and torque control, with superposed active damping of drive train oscillations. The torque distribution between the two power sources is managed as a different task from the regulation task (oscillation damping), adopting a specific control algorithm that does not require the implementation of an optimisation problem. Moreover, as regards the linear model used for the design of the controller, an accurate tyre model with a fixed relaxation length is adopted--thus decoupling the kinematic behaviour of the driving rear types from the free rolling condition, as carried out in [25]. The developed model should also be able to reduce drivetrain vibrations; different works addressing this aim can be found in literature. In [31], a control for vibration reduction was developed for hybrid vehicles and experimentally validated, with two different controllers that reduced both the vibrations due to engine torque ripple and ones related to the driveline. Conversely, the root locus technique [32] has been adopted for anti-jerk control designs capable of preserving the driveline from oscillations. To improve NVH performance, a magnetic differential for vehicle drivetrains was developed in [33], while a controller for driving the engine--in order to actively damp driveline oscillations--was proposed in [34] to improve drivability and passenger comfort. The present paper aims to propose an accurate nonlinear vehicle model and a model-based torque allocation control to achieve both improved vehicle drivability and pollutant emissions performance.