Inverse Approach to Vehicle Generalized Parameters for Individual Drives

Abstract

In this paper, a method is presented to determine the individual tire slippages and wheel circumferential forces/torques for vehicles with individually-driven wheels. The method is based on a system of parameters of individual wheels mathematically linked to the vehicle generalized parameters. The paper demonstrates an inverse approach on how the generalized parameters can be used to determine individual wheel parameters when the vehicle generalized parameters are given for a single, generalized wheel, whose kinematics and dynamics are equivalent to those of the vehicle. The method can be applied to improve terrain mobility and energy efficiency of autonomous ground vehicles, including planet rovers whose guidance systems do not take into consideration that the tire slippages and circumferential forces are different or should be different by determining individual parameters of the e-motors to provide maximum mobility or energy efficiency in a straight line motion.

1 Introduction

The distribution of forces among the wheels of has a critical impact on the performance of multi-wheel drive vehicles, especially off-road where terrain conditions under the wheels may be very different. For vehicles with a mechanical driveline system, the configuration of the driveline determines the nature of the power split. In electric vehicles with individual wheel motors, the power split is determined by control of the wheel motors. Different methods have been studied for such control, based on factors such as stability, reducing battery usage, and utilizing most efficient regions of the motor [1,2,3,4].

However, navigation and guidance systems of autonomous vehicles (including planet rovers) usually do not take in consideration that the tire slippages and wheel circumferential forces are different or should be different and therefore treat a vehicle as a single wheel. For application to such systems, this paper’s approach is to consider that single wheel as the generalized wheel in a system of generalized parameters, and then splits the given generalized parameters into individual parameters of the wheels with e-motors to provide max mobility or energy efficiency. The method is designed to be applied as part of a two-step process in which a vehicle’s navigation system determines an assigned velocity and the generalized parameters are used to determine an optimal split of wheel forces which provide the best mobility or energy efficiency while fulfilling the required vehicle straight line motion. Therefore, an inverse scenario is simulated where the vehicle velocity or acceleration is assigned, and the wheel forces and tire slippages need to be computed.

Generalized parameters represent a reduced set of parameters or time-variant states taking the place of and replacing those of individual wheels for a group of wheels or the entire vehicle [5]. The generalized parameters include various properties of the wheel dynamics such as tire slippage and rolling radii. Each driving wheel (i.e., a wheel loaded with torque) has a theoretical velocity ${V}_{t}$ which is lower than its actual velocity ${V}_{x}$ because of the tire slip ${s}_{\delta }$:

$$ V_{x} = V_{ti}^{{{^{\prime}}\left( {^{\prime\prime}} \right)}} \left( {1 - s_{\delta i}^{{{^{\prime}}\left( {^{\prime\prime}} \right)}} } \right) $$

(1)

Here, ‘ and ‘’ indicate left and right wheels and $i$ indicates a pair of left and right wheels (i = 1, n). The theoretical velocity for a wheel is determined from its angular velocity ${\omega }_{w}$

$$ V_{ti}^{{{^{\prime}}\left( {^{\prime\prime}} \right)}} = \omega_{wi}^{{{^{\prime}}\left( {^{\prime\prime}} \right)}} r_{wi}^{{0{^{\prime}}\left( {^{\prime\prime}} \right)}} $$

(2)

where ${r}_{w}^{0}$ is the tire rolling radius in the driven mode (at zero wheel torque) which depends on the tire inflation pressure and normal reaction, which can influence the theoretical velocities at different wheels of a vehicle. Slippages of different tires at the left and right wheels can be different due to the different terrain conditions and driveline system characteristics. However, as seen from Eq. (1), ${V}_{x}$ in Eq. (1) is the same for all the wheels and known as the actual vehicle velocity. Similarly to the tire slippages, the vehicle generalized slippage ${s}_{\delta a}$ characterizes the velocity drop from the vehicle’s theoretical velocity, ${V}_{a}$, to ${V}_{x}$, which as mentioned earlier is the same for all wheels

$$ V_{x} = V_{a} \left( {1 - s_{\delta a} } \right) = \omega_{0} r_{a}^{0} \left( {1 - s_{\delta a} } \right) $$

(3)

where ${r}_{a}^{0}$ is rolling radius of a hypothetical/generalized wheel in the driven mode. The generalized wheel has actual linear velocity ${V}_{x}$ and rotates with angular velocity ${\omega }_{0}$. A set of mathematical equations based on the vehicle’s driveline system outline can be derived to link the generalized parameters of the vehicle to their individual wheels. For example, for a 4 × 2 vehicle in straight line motion with a mechanical driveline using an open differential and without wheel hub gears, the generalized slippage of the drive axle can be derived from the following equations. Equations (1) and (2) can be re-written as

$$ V_{x} = \omega_{w}^{^{\prime\prime}} r_{w}^{0^{\prime\prime}} \left( {1 - s_{\delta }^{{^{\prime\prime}}} } \right) = \omega_{w}^{\prime} r_{w}^{0^{\prime}} \left( {1 - s_{\delta }^{\prime} } \right) $$

(4)

which leads to

$$ \frac{{\omega_{w}^{\prime} }}{{\omega_{w}^{{^{\prime\prime}}} }} = \frac{{r_{w}^{{0{^{\prime\prime}}}} \left( {1 - s_{\delta }^{{^{\prime\prime}}} } \right)}}{{r_{w}^{{0{^{\prime}}}} \left( {1 - s_{\delta }^{\prime} } \right)}} $$

(5)

The relationship between angular velocities of the three links of the open differential is

$$ \omega_{0} = \frac{{\omega_{w}^{\prime} + \omega_{w}^{^{\prime\prime}} }}{2} $$

(6)

Here, ${\omega }_{0}$ is the angular velocity of the differential’s case that is the same as the rotational speed of the generalized wheel. Thus, the generalized wheel is reduced to the case of the axle’s differential. Equations (5) and (6) result in

$$ \omega_{w}^{^{\prime\prime}} = 2\omega_{0} \frac{{r_{w}^{{0{^{\prime}}}} \left( {1 - s_{\delta }^{\prime} } \right)}}{{r_{w}^{{0{^{\prime}}}} \left( {1 - s_{\delta }^{\prime} } \right) + r_{w}^{{0{^{\prime\prime}}}} \left( {1 - s_{\delta }^{{^{\prime\prime}}} } \right)}} $$

(7)

Using Eqs. (3), (4) and (7), the axle generalized slippage is derived as

$$ s_{\delta ai} = 1 - \frac{{\left( {r_{w}^{{0{^{\prime\prime}}}} + r_{w}^{{0{^{\prime}}}} } \right)\left( {1 - s_{\delta }^{\prime} } \right)\left( {1 - s_{\delta }^{{^{\prime\prime}}} } \right)}}{{r_{w}^{{0{^{\prime}}}} \left( {1 - s_{\delta }^{\prime} } \right) + r_{w}^{{0{^{\prime\prime}}}} \left( {1 - s_{\delta }^{{^{\prime\prime}}} } \right)}} $$

(8)

Similar analysis can be used to derive generalized slippage of the vehicle and generalized parameters for other driveline configurations. The generalized parameters mathematically link kinematics of the wheels through the generalized wheel to that of the vehicle. By combining the individual wheels-generalized wheel kinematics equations with vehicle dynamics equations, the wheel and vehicle dynamics can be studied by analyzing and optimizing driveline systems for vehicle mobility and energy efficiency.

This paper introduces generalized parameters for an electric vehicle with individual wheel motors and develops a method to split vehicle generalized parameters modeled for the overall vehicle into parameters of individual wheels. The generalized vehicle parameters are mathematically linked to the individual wheels through introduced distribution factors. The split of individual wheel parameters is based on criteria for maximizing mobility or energy efficiency which can then be applied as reference signals to control the wheel motors.

2 Generalized Parameters of Individual Drives

Slippage proportional factors $\gamma$ are introduced in Eq. (9) to link the vehicle generalized slippage to individual tire slippages.

$$ s_{\delta i}^{{{^{\prime}}\left( {^{\prime\prime}} \right)}} = \gamma_{i}^{{{^{\prime}}\left( {^{\prime\prime}} \right)}} s_{\delta a} $$

(9)

Tire slippages are related to their circumferential forces ${F}_{x}$ through Eq. (10) [6]

$$ F_{xi}^{{{^{\prime}}\left( {^{\prime\prime}} \right)}} = \mu_{pxi}^{{{^{\prime}}\left( {^{\prime\prime}} \right)}} R_{zi}^{{{^{\prime}}\left( {^{\prime\prime}} \right)}} \left\{ {1 - \left( {s_{\delta ci}^{{{^{\prime}}\left( {^{\prime\prime}} \right)}} /2s_{\delta i}^{{{^{\prime}}\left( {^{\prime\prime}} \right)}} } \right)\left[ {1 - \exp \left( { - 2s_{\delta i}^{{{^{\prime}}\left( {^{\prime\prime}} \right)}} /s_{\delta ci}^{{{^{\prime}}\left( {^{\prime\prime}} \right)}} } \right)} \right]} \right\} $$

(10)

where ${\mu }_{px}$ is the peak friction coefficient, ${s}_{\delta c}$ is a characteristic slippage past which the slippage becomes increasingly nonlinear, and ${R}_{z}$ is the wheel normal reaction. Circumferential forces ${F}_{x}$ are produced by the wheel torques ${T}_{w}$ and are related by Eq. (11).

$$ T_{wi}^{{{^{\prime}}\left( {^{\prime\prime}} \right)}} = F_{xi}^{{{^{\prime}}\left( {^{\prime\prime}} \right)}} r_{wi}^{{0{^{\prime}}\left( {^{\prime\prime}} \right)}} $$

(11)

The circumferential force distribution factor $\upsilon$ is introduced to characterize the distribution of the total circumferential force among the driving wheels.

$$ F_{xi}^{{{^{\prime}}\left( {^{\prime\prime}} \right)}} = \nu_{i}^{{{^{\prime}}\left( {^{\prime\prime}} \right)}} F_{{x{\Sigma }}} = \nu_{i}^{{{^{\prime}}\left( {^{\prime\prime}} \right)}} R_{{m{\Sigma }}} $$

(12)

${R}_{m\Sigma }$ is the total resistance to motion which includes forces such as rolling resistance, grade resistance, inertia force, and air drag that impede vehicle motion. The sum of circumferential forces ${F}_{x\Sigma }$ must equal the sum of motion resistance ${R}_{m\Sigma }$ for the vehicle to maintain its assigned speed. Factor $\nu$ is introduced to connect the tires’ rolling radii in the driven mode to the vehicle generalized rolling radius ${r}_{a}^{0}$, derived by relating the sum of wheel torques to the overall resistance to motion multiplied by ${r}_{a}^{0}$.

$$ r_{a}^{0} = \sum \nu_{i}^{{{^{\prime}}\left( {^{\prime\prime}} \right)}} r_{wi}^{{0{^{\prime}}\left( {^{\prime\prime}} \right)}} $$

(13)

Using the above-introduced equations, an inverse approach to vehicle simulation is considered in this paper. ${V}_{x}$ is assigned and generalized slippage can be computed using the actual and theoretical velocities and ground conditions. In this paper, the generalized slippage is treated as a given input generated without individual wheel control. Figure 1 shows generalized slippage for a 5482 kg 4 × 4 vehicle moving on stochastically generated terrain in an offline simulation at a constant speed of 10 mph. The terrain is a sandy loam field with naturally packed soil after tilling with a moisture content of about 10% and is generated to fit the exponential function in Eq. (10). Left and right terrain conditions are the same while the front and rear differ due to soil compaction.

Fig. 1.

Generalized slippage and total circumferential force on stochastic terrain.

Equations (9–13) split the generalized slippage into individual tire slippages which can be optimized for improved performance. Two optimal slippage configurations are considered in this paper, an optimal mobility case and optimal energy efficiency case. Slip energy efficiency is evaluated using Eq. (14); the slip efficiency of a 4×4 vehicle was demonstrated to be maximum when the slips of the front and rear wheels are equal [5].

$$ \eta_{\delta } = \frac{{F_{{x{\Sigma }}} }}{{F_{{x{\Sigma }}} + \mathop \sum \nolimits_{i = 1}^{n} \left( {\frac{{F_{x}^{\prime} s_{\delta i}^{\prime} }}{{1 - s_{\delta i}^{\prime} }} + \frac{{F_{x}^{{{^{\prime\prime}})}} s_{\delta i}^{{^{\prime\prime}}} }}{{1 - s_{\delta i}^{{^{\prime\prime}}} }}} \right)}} $$

(14)

The vehicle’s mobility, considered as the conversion of the potential of the wheel loads into traction, occurs at a different combination of wheel traction forces than the maximum energy efficiency. Equation (15) is a Vehicle Mobility Performance index which was introduced to evaluate mobility relative to a hypothetical case where the complete potential of the wheel load can be converted into traction without any loss of velocity [7].

$$ VMP = \frac{{\mathop \sum \nolimits_{i = 1}^{n} F_{xi}^{\prime} V_{x} + F_{xi}^{{^{\prime\prime}}} V_{x} }}{{{ }\mathop \sum \nolimits_{i = 1}^{n} R_{zi}^{\prime} V_{ti}^{\prime} + R_{zi}^{{^{\prime\prime}}} V_{ti}^{{^{\prime\prime}}} }},{ }i = 1,2 $$

(15)

Lagrangian optimization was used to determine the optimal slippages [7]. The optimal slippages and ${F}_{x}$ forces with their corresponding circumferential forces are shown in Fig. 2. When the slippages are optimized for energy efficiency, it yields a different combination of circumferential forces than is needed for optimal mobility. Circumferential force distribution factors $\nu$ and slippage proportional factors $\gamma$ from Eq. (9) and Eq. (12) needed for the optimal tire slippages are shown in Fig. 3.

Fig. 2.

Optimal slippages and circumferential forces for mobility and energy efficiency.

Fig. 3.

Optimal $\nu$ and $\gamma$ factors for mobility and energy efficiency.

Compared to an equal torque split, the mobility-based power splits results in a mean VMP increase of 0.84% and the energy efficiency-based power split increases slip efficiency by 1.32% averaged over 10 simulation runs, which is a considerable improvement of mobility and energy efficiency.

3 Conclusion

In this paper, an inverse dynamics-based approach was presented which uses a system of generalized parameters to split a given generalized vehicle parameter into individual wheel parameters. The method is designed for optimizing the distribution of wheel forces when the vehicle motion is assigned without consideration of individual slippages (treating the vehicle as a single wheel). By applying conditions for optimal energy efficiency or mobility, the slippage proportional factors and circumferential force distribution factors can be computed which correspond to an optimized distribution of wheel forces and tire slippages that fulfils the assigned motion requirements of the vehicle. The generalized parameters for individual drives will be developed into reference signals for assigning e-motor wheel torques.