# Planning Speed Mode of All-Wheel Drive Autonomous Vehicles Considering Complete Constraint Set

^{*}

## Abstract

**:**

## 1. Introduction

**Models.**The problem of smoothed speed arose as a need for high-quality control preventing the appearance of sudden loads and unstable transients. Some studies as [3] attempted to compose speed plans based on cubic polynomials in such a way as to ensure the continuity of accelerations at the transition nodes under conditions of minimizing the jerk. However, the function inflections did not occur directly in the nodes, which led to some excesses of the preset limits followed by the need for complicating the model to compensate for this fluctuation effect. This approach allows obtaining continuous speed values at any point, considering the initial and final values.

**Objective functions**determine the planning and control strategies. Most studies proceed from the control strategy regarding minimizing the motion time while keeping the vehicle within roadway boundaries [9]. Thus, in [10], the time-optimal trajectory around a racetrack is obtained by solving a minimum lap time problem (MLTP) for a racing vehicle. In [11], the path planning strategy includes preventing rear-end collisions during overtaking while minimizing traveling time.

**Controllers**. Different types of controllers are used for both planning and tracking. The class of controllers based on MPC can be considered the most used that allows obtaining high-quality forecasts due to a combination of hard and soft constraints [5]. Another popular class is the LQR-type controllers. For increasing performance and overcoming physical limitations, a technique can be represented by the augmented Lagrangian framework [6], which refers to iterative LQR (ILQR) and Constrained Iterative LQR (CILQR), respectively. In [12], the sliding mode control (SMC) calculates the total driving force for longitudinal control. In [7], the reference path is followed by low-level tracking using PID controllers. In [13], a learning-based Interaction Point Model (IPM) describes the interaction between agents. In [9], the Nonlinear Model Predictive Control (NMPC) strategy was aimed at controlling a small-scale car model for autonomous racing competitions. Study [6] presented a hierarchical framework with neural physics-driven models to enable the online planning and tracking of minimum-time maneuvers. A lateral speed prediction model for high-level motion planning was considered with economic nonlinear model predictive control (E-NMPC). In [1], a linear parameter-varying (LPV) MPC was deployed for AV trajectory tracking and compared with the linear MPC. The outcomes satisfy the processing rate and high-precision criteria, including safely avoiding obstacles. In [14], the path-following control strategy was based on the linear quadratic regulator (LQR) to compare the performance of models. In [12], the MPC controller was used to calculate the steering wheel angle and the total yaw moment for lateral control, and the sliding mode control (SMC) was used to calculate the total driving force for longitudinal control.

**ADAS**. Note that the problem of speed planning is typical not only for AVs but also for ADAS systems. In [5], it was proposed to form a speed profile for functioning the adaptive cruise control (ACC) based on MPC. The study [15] considered a preview servo-loop speed control algorithm to achieve smooth, accurate, and computationally inexpensive speed tracking for connected automated vehicles (CAVs). A classic PID with an optimal controller was implemented in an automated vehicle platform for lowering speed-tracking errors, mitigating operations, and smoothing brake/throttle activations. The paper [11] proposed a system for speed planning using MPC to estimate the vehicle overtake safety while controlling the maneuver for a dynamic vehicle model.

**Actuators**. Study [9] includes a simple drivetrain model in the optimization problem to limit the lateral and longitudinal forces acting on a car. An Autonomous Emergency Braking Pedestrian (AEB-P) was introduced in [19] to prevent collisions between vehicles and pedestrians. The emergency brake planner generates vehicle deceleration followed by tracking the trajectory, where the PI-controller was adapted to provide the optimum braking force. In [15], the brake/throttle control laws were introduced in five parts: three feedback controls of system states and two feedforward items previewing road slope and target speed. Study [20] proposes a clothoid-curve-based trajectory tracking control method to solve the problem of tracking errors caused by the discontinuous curvature of the control curve calculated by simple algorithms. The parameters of clothoid reference curves are tied to the vehicle’s safe motion constraints.

**Vehicle model**. Note that most studies in the field of AV motion planning are focused on generating reference curves using relatively simple (often kinematic bicycle) models. However, the motion planner’s high performance does not yet mean high forecast quality. The most important indicators are the prognosis feasibility and ensuring good accuracy in reflecting features of a specific vehicle design. Considering the intensive use of control systems in modern vehicles (for example, traction distribution), it is often not enough to obtain the trajectory and kinematic parameters’ references. Moreover, the desirable parameters themselves may significantly depend on the technologies used in a real vehicle. For example, the distribution of drive torques between the same axle’s wheels can be carried out through a symmetrical differential, through a sport differential (torque vectoring), and by activating the inner wheels’ brakes. All three options will cause different axle drive power, different traction and lateral forces, the tire–road adhesion degree, and, as a result, affect differently on forming the yaw moment. In a classic uncontrolled AWD drivetrain, a wheel with the worst adhesion conditions limits the traction potential. And vice versa, in transmissions design with individual wheel control, the maximum realization of total adhesion potential is possible. In the case of using a standard vehicle model, there will be no difference between these two variants when speed is the planning object. If an extended model is used, including the drive type and redistribution of vertical reactions, then, as shown in our previous articles, the difference between predictions of kinematic parameters may be significant especially for cases with low tire adhesion. Thus, this study’s target consists of composing an inverse vehicle dynamic model that allows estimating physical factors based on the kinematic parameters.

**constraints**is a critical factor in reflecting the realism of forecasting and the optimization procedure’s performance. On the one hand, increasing the number of restrictions complicates the mathematical optimization model; on the other hand, narrowing the boundaries of AV motion parameters contributes to diminishing solution iterations and time costs.

**Safety restrictions**. Most often, this includes speed limits when approaching moving and static obstacles. Within this planning technique, we suppose that a trajectory considering obstacles and speed mode has already been built. Therefore, in the framework of this study, we assume that the AV uses a safe space for realizing a maneuver. As a safety measure, the body roll angle caused by lateral inertia forces is applied. Restricting the maximum roll angle, limitations on lateral acceleration are automatically imposed.

**Planning technique**. Here, the most important requirements concern ensuring the smoothness and unambiguousness of predicted functions. The most common approach implies using polynomials that describe the change in speed concerning time along a trajectory section. However, this does not always lead to a qualitative consistency between the speed and curvature derivatives involved in the formation of kinematic parameters such as jerk and angular acceleration. That is, there may be disruptions in the piecewise representation of these functions. This, in turn, does not contribute to the possibility of using inverse dynamics to determine the force factors acting on the vehicle. In this regard, our method consists of using an inverse approach, when instead of speed its second derivative is directly modeled by smooth functions. In this case, the continuity and smoothness of the vehicle’s linear and angular accelerations will be guaranteed.

## 2. Existing 2.5D Vehicle Dynamics Model

#### 2.1. Model Scheme

#### 2.2. Trajectory-Based Vehicle Kinematics

**Mass center speed**V is directed by the trajectory tangent and may be expressed as follows

_{ζ}, V

_{μ}= unit vectors and speed components of the vehicle local coordinate system ζμ; ${\overrightarrow{u}}_{x}$, ${\overrightarrow{u}}_{y}$, V

_{x}, V

_{y}= unit vectors and speed components of the fixed (global) coordinate system xy.

**absolute speed**V may be decomposed with projections V

_{x}and V

_{ζ}tied through the tangent angle α and central slip angle β. Then,

_{ζ}may be found as follows

_{x}concerning the x-coordinate is

**Lateral Speed**V

_{μ}is geometrically tied with the longitudinal speed V

_{ζ}by the central slip angle β

**Yaw Rate**is the yaw angle ϕ = α – β derivative in the current global coordinates. Thus,

**Angular acceleration**ε is derived from the yaw rate ω concerning time

**Accelerations**in the local vehicle coordinate system ζμ are

**Jerks**in the vehicle coordinate system are

**Velocities and accelerations at the wheels’ centers**are needed for assessing the forces acting in the tire–road contacts. Let us introduce designations

_{2}= identity matrix of dimension 2 × 2, r

_{j}= vector of j-th wheel center’s coordinates in the AV local coordinate system, ${v}_{j}$ = vector of j-th wheel center’s velocities in the wheel local coordinate system, H = rotational matrix, θ

_{j}= angle of j-th wheel turn, and ${\overrightarrow{u}}_{\zeta j}$, ${\overrightarrow{u}}_{\mu j}$ = unit vectors of the j-th wheel local coordinate system.

**Speed and Acceleration vectors**in the center of j-th wheel’s local coordinates

**Rotation angles**of steered wheels can be determined as Ackerman’s angles subject to that the trajectory curvature K is much less than the maximum possible. Thus,

#### 2.3. Vehicle Dynamics

#### 2.3.1. Body Pseudo-Roll

_{b}= mass of the vehicle body, r

_{ψ}= roll lever, g = gravity acceleration constant.

_{ψ}

_{ψ}, then the roll angle will be a linear function of the lateral acceleration

_{ψ}(Figure 2c) can be estimated based on dependencies [21]

_{13}, h

_{24}= ordinates of the roll centers of the front and rear suspensions, respectively (double wishbone suspensions are usually slightly below the wheel-center axis).

_{sf}, k

_{sr}= reduced stiffness of independent front and rear suspensions, respectively.

_{sf,r}= sprung masses distributed on every single front and rear suspensions, respectively, f

_{sf,r}= natural oscillation frequencies of sprung masses (for modern cars f

_{s}= 0.6…1.3 Hz).

#### 2.3.2. Redistribution of Vertical Reactions

_{r}can be partially estimated by omitting kinematic losses (no sliding) and seeing only power losses. However, this requires a normal load value on each wheel. To simplify, we may consider the moments from conditionally paired front and rear wheels. Then,

_{wf(r)}= dynamic radius of front (rear) wheel, f

_{rf(r)}= coefficient of front (rear) rolling resistance.

_{f(r)}, we obtain

_{ζr}, M

_{ζl}= overturning moment from the right and left vehicle sides.

_{b}·g·r

_{ψ}·sin(ψ) ≈ m

_{b}·g·r

_{ψ}·ψ, where the pseudo-roll angle can be preliminarily estimated from the lateral acceleration. The matrix equation of the moment balance relative to the centers of the wheels’ contacts is represented as follows

**Y**is singular because of the vehicle symmetry. Therefore, the pseudo inverse matrix

**Y**

^{+}is used for redistributing vertical reactions caused by the lateral acceleration. Then,

#### 2.3.3. Pseudo-Tires’ Slip Angles

_{ζ}and acceleration a

_{μ}. Since the lateral accelerations and speeds at the wheels’ centers are known by Equation (12), it is possible to evaluate the lateral forces required for stable motion. To do this, briefly consider the main factors affecting the lateral sideslip.

_{μ}= f(δ) is almost linear,

_{μ}, at which the slippage is small.

_{μ}values, at which the slip occurs on a significant part of the tire’s contact area, and the more intensively the larger δ is. At point c, the force F

_{μ}reaches its maximum value according to the adhesion condition, and in segment cd, it is determined by the equality F

_{μmax}= R

_{z}φ

_{μmax}, where φ

_{μmax}is the coefficient of transverse adhesion. Conditionally, in segment 0c, the wheel lateral movement caused by the force F

_{μ}is called the sideslip, and in segment cd it is called the pure slip. The angle value δ

_{c}, at which sliding begins, depends on the tire structure, normal load, coefficient φ

_{μmax}, and other factors. Usually, for a dry road surface δ

_{c}= 12–20°. In terms of kinematics, it does not matter for what reason the velocity vector deviation from the rotation plane occurs; therefore, the angle δ is formed by the components of the wheel’s longitudinal and transverse speeds over the entire range 0d.

_{δ}= f(R

_{z}) (Figure 3b). For passenger vehicles, k

_{δ}has a maximum value k

_{δ}

_{max}when the force R

_{zopt}is close to that corresponding to the vehicle’s gross weight R

_{z0}. The coefficient k

_{δ}value depends on the coefficient φ

_{μ}

_{max}. In segment 0b, the coefficient k

_{δ}is practically independent of the coefficient φ

_{μ}

_{max}. However, the smaller φ

_{μ}

_{max}, the smaller the angle δ value corresponding to the point d. On a dry road surface with a force R

_{z}corresponding to standards for a given tire, it can be taken equal to δ

_{b}≈ 3–4°. Formula (35) is also valid for the section bc, at which k

_{δ}= f(δ) or k

_{δ}= f(F

_{μ}) (Figure 3c). In this segment, the smaller φ

_{μ}

_{max}, the smaller the values of k

_{δ}.

_{ζ}affects the dependence F

_{μ}= f(δ). With an increase in R

_{ζ}/R

_{z}, to obtain the same angles δ in the traction mode, a smaller force F

_{μ}is needed. In the braking mode, at small R

_{ζ}/R

_{z}, its increase leads to a slight growth in the force F

_{μ}, and at larger R

_{ζ}/R

_{z}—the F

_{μ}decreases. If R

_{ζ}/R

_{z}is small, then the effect of longitudinal forces F

_{μ}= f(δ) is insignificant. The longitudinal forces exert the greatest influence at values of the R

_{ζ}forces that are close in value to the maximum adhesion.

_{max}= R

_{z}φ

_{max}. In this case, the adhesion coefficient φ

_{max}depends on the slip direction of the contact patch relative to the road surface, which coincides with the direction of the reaction R. When sliding in the rotation plane, the adhesion coefficient is φ

_{ζ}, and in the transverse direction φ

_{μ}.

_{δ}by multiplying k

_{δ}

_{max}with several correction factors. With the wheel’s straight-line steady rolling on an even, nondeformable road surface

_{δmax}= the driven wheel’s slip resistance coefficient on the dependence F

_{μ}= f(δ) linear section at the maximum values of the dependences k

_{δ}= f(R

_{z}) or k

_{δ}= f(p

_{a}). q

_{φ}= correction factor considering the slip resistance dependence on the angle δ while moving on roads with different φ (non-linear dependence F

_{μ}= f(δ)), q

_{z}= correction factor considering the effect of normal load deviation from the optimum one, q

_{ζ}= correction factor considering the influence of longitudinal forces on k

_{δ}, and q

_{p}= correction factor considering the effect of the tire’s air pressure p

_{a}deviation from the optimal pressure (supposedly neglected and equals 1).

_{δ0}= the slip resistance factor at a given value of R

_{z}in a linear section F

_{μ}= f(δ), and δ

_{0}= the slip angle corresponding to the transition from a linear section to a nonlinear section (regarding b). Usually, δ

_{0}= 0.025–0.035 rad.

_{μ}= f(δ) corresponds to the steady-state wheel rolling when F

_{μ}= const and the trajectory of the wheel’s center is a straight line. In most cases, this dependence can also be used to study controlled motion at F

_{μ}= f(t).

_{b}≈ 3–4° under a static load and dry surface at the wheel’s driven mode, providing the maximum potential of φ

_{μ}. Since a decrease in φ

_{μ}leads to a reduction in k

_{δ0}, the angle δ

_{b}also drops. Assuming a linear dependence, the initial angle δ

_{0}of the linear zone can be calculated as

#### 2.3.4. Pseudo-Tires’ Lateral Deformations

_{μ}

_{01,2}= the lateral stiffness coefficient at static deformation [22] (generally depends on deformation and tire pressure) and m

_{1}, m

_{2}= gross masses distributed on every single front and rear suspension, correspondingly.

_{S}= the so-called rigidity of the cross-sectional area (supposed to be constant), r

_{w}

_{0}= tire’s free radius, Δr = tire’s radial deformation, h

_{z}

_{0}= tire’s profile height under static load, h

_{z}= tire’s profile dynamic height, and D

_{r}= the diameter of wheel’s rim.

_{0}= tire’s radial static deformation.

_{μ}

_{0}one. As the deformation Δr decreases, it leads to a decrease in lateral stiffness, and vice versa.

#### 2.3.5. Evaluation of Lateral Reactions

_{j}= drive torque, r

_{wj}= dynamic radius, r

_{ej}= effective radius, f

_{rj}= rolling resistance, and I

_{wj}= inertia of rotating masses tied and reduced to a wheel.

_{sy}

_{1}, q

_{sy}

_{3}, q

_{sy}

_{4}= coefficients [23].

_{j}= radial deformation of the j-th wheel, C

_{fzj}= the j-th tire’s radial stiffness, F

_{zj}

_{0}= static load on the j-th wheel, and q

_{fz}

_{1}, q

_{fz}

_{2}= coefficients [23].

_{ref}, D

_{ref}, F

_{ref}= coefficients [23].

_{ζj}on the wheels, and the lateral reactions R

_{μ}are unknown, we need five equations. The first three equations express the balances of forces and moments relative to the vehicle axes. Denote

_{j}, μ

_{j}= coordinates of the j-th wheel center in the vehicle coordinate system.

_{ad}= aerodynamic drag force.

#### 2.4. Variants for Organizing Distributed Traction

#### 2.4.1. Variants of Wheel Torque Control

_{max}is the same for all wheels, that is, an unstable movement mode of any wheel occurs when the geometric sum of the longitudinal and lateral adhesion coefficients approaches the limit. Within the motion planning, we do not consider here the cases of control when the maximum possible adhesions on all wheels are different.

_{fl}, B

_{rl}, it is possible to increase the drive moments of the axles and realize greater traction on the outer wheels.

#### 2.4.2. Distribution Devices

_{f}, R

_{r}of the output shafts’ crown gears (Figure 5b) give the initial asymmetry to the differential mechanism (DM) design. This leads to distributing the gearing torques with a ratio equal to 40/60 [24]. In normal conditions (both output shafts rotate with the same angular speed), the rear axle’s driving shaft transmits about 60% of the total torque, and the front axle −40%, correspondingly. Each crown gear has a friction clutch connecting it with the differential’s carrier. The relative sliding of friction elements generates moments that are redistributed according to the mechanism’s kinematic state. Thus, the output shaft’s torque may be reduced or increased. The clutch packs may be installed with precompression, stipulating the static friction torque by the pressing washers. With a decrease of one shaft’s resistance, its angular speed becomes greater than the carrier’s angular speed. The excessive power flow from an outrunning shaft returns to the carrier by the friction moment and increases the lagging shaft’s torque. A feature of this DM is the dynamic redistribution of friction moments due to the axial component of the gearing reaction. The greater the satellite’s force (Figure 5c), the greater the compression, and in turn, the greater the moment that can be passed by friction clutches.

_{D}, ω

_{f}, and ω

_{r}= angular velocities of the differential carrier, front, and rear output shafts, respectively; and g

_{fr}= R

_{r}/R

_{f}= 60⁄40 = 3⁄2—algebraic ratio from the front drive axle to the rear drive axle.

_{f}, ω

_{r}can be estimated based on the symmetrical DM’s kinematic properties. Assuming that all wheels are rotating in approximately the same mode relative to the tires’ longitudinal slip, we can calculate with sufficient accuracy

_{fl}, ω

_{fr}= angular speeds of the front left and right wheels, respectively, ω

_{rl}, ω

_{rr}= angular speeds of the rear left and right wheels, respectively, i

_{f}= final gear ratio, ${v}_{\zeta j}$ = longitudinal speed of the j-th wheel center in the wheel’s local coordinate system, and r

_{ej}= wheel’s effective radius, j = [1, …, 4].

_{D}, ω

_{f}, ω

_{r}. Note that the friction discs operate in an oily environment, which contributes to a smooth redistribution of output torques without the effects of sticking, and jamming, which are typical for cases close to dry friction. That is, to describe the frictional interaction, we may use an expression such as

_{Fk}= actual friction coefficient between friction pairs, μ

_{k}= modulus of maximum friction coefficient, c

_{k}= intensity factor, k = f (front), and r (rear).

_{kd}is modeled as follows

_{s}= number of satellites, P

_{Nk}= axial gearing force, n

_{Fk}= number of friction couples, R

_{k}= average radius of side gear, and R

_{Fk}= average friction radius.

_{N}of the gearing contact reaction, P depends neither on the direction of the shaft rotation nor on the direction of the tangential component P

_{T}, the friction moment’s dynamic part is written as follows

_{Tk}= tangential gearing force, α = gearing angle, M

_{k}= side gear torque, and η

_{k}= friction factor.

_{N0k}= preliminary established axial compression force.

_{k}

_{0}on the side gears.

**t**

_{D}is associated with initial torque redistribution.

_{D}

**M**in Equation (68), obtain

_{k}.

_{10}and p

_{11}. Thus, by activating the required hydraulic cylinder, part of the carrier torque may be passed to the needed semi-axle using the frictional adhesion between the half-couplings over the two-step internal gearing (i

_{28}= 0.8752, i

_{26}= 1.2258, i

_{84}= 0.714).

_{l}, T

_{r}with the moment T

_{d}on the DM carrier and the moment ΔT

_{lr}of the friction clutch.

#### 2.4.3. Drivetrain Dynamics

_{f}, k

_{r}of the moments on the front and rear axles’ wheels

_{ζ}

_{1}= φ

_{ζ}

_{3}= φ

_{ζf}, which, after reduction, leads to equations

_{ζ}

_{2}= φ

_{ζ}

_{4}= φ

_{ζr},

_{j}, as well as the coefficients k

_{f}, k

_{r}, can be obtained in advance.

_{1}, T

_{3}are generally not equivalent to the drive torques of the output shafts of the symmetrical differential T

_{fl}= T

_{fr}, nor are the torques T

_{2}, T

_{4}with respect to the torques T

_{rl}= T

_{rr}.

_{f}, T

_{r}equally between the wheels’ semi-axles, i.e., T

_{fl}= T

_{fr}, T

_{rl}= T

_{rr}. The correction is fulfilled by activating the brake mechanisms, which stipulates the braking moments’ vector

**B**. Then, the wheels’ torques will be related to semi-axle torques as follows

**t**

_{T}is responsible for distributing the torque between the drive semi-axles, and

**t**

_{p}is the matrix for permuting the

**T**

_{a}vector’s elements

**B**must be provided that satisfies the requirements of redistributing the moments from Equations (76) and (77) and minimizing the control in general.

_{D}and braking torques

**B**.

_{f}equally between the semi-axles without further correction, i.e., T

_{fl}= T

_{fr}= T

_{1}= T

_{3}. The rear sport differential redistributes the drive torque T

_{r}according to Equation (73). Then, the drive moments on the wheels will be related as follows

**t**

_{fr}= responsible for proportions of the distribution of moments, and

**T**

_{t}by components, we obtain

**E**

_{2}to be a 2 × 2 identity matrix,

**t**

_{p}ensures permuting the wheels’ numbering order,

**E**

_{4}= 4 × 4 identity matrix.

**T**its expression from Equation (72) for the inter-axle differential drive, we obtain

_{f}equally between the semi-axles, i.e., T

_{fl}= T

_{fr}. The correction is made by activating the front brake mechanisms with braking torques B

_{1}, B

_{3}. The rear sport differential redistributes the drive torque T

_{r}according to Equation (73). Then, the drive moments across the semi-axles will be related as follows

_{f}, T

_{r}following Equation (73). Then, the drive moments along all the semi-axles will be related as follows

**T**

_{t}by components, we obtain

**R**

_{ζ}from Equation (86).

**R**

_{ζ}from Equation (54)

**lsqnonlin**function.

## 3. Optimization

#### 3.1. Generalized Approach to Speed Model

**D**has already been built. This question was extensively reflected in our previous studies. Using the general approach, let us compose models for obtaining the speed in finite elements (FE). The distance

**D**may be divided on n FEs, each i-th of which is represented by the length L

_{i}and parameter ξ ∈ [0, 1]. Thus, the current linear space is x = ξL

_{i}. The basis functions

**F**

_{ξ}are the form functions corresponding to the nodal degree of freedoms (DOFs)

**Q**

_{v}. The number of nodal DOFs depends on the degree p of Lagrangian polynomial: d = (p + 1)/2. Then, any function y(x) within i-th FE may be expressed as follows:

**second**speed derivative as the basic model to reduce the polynomial extent p, the quantity of nodal unknowns, and speed up computations. Then, it can be written

_{ζi}/dx(0) = integration constant defined from initial conditions.

_{ζi}in the i-th segment is found by repeating integration of Equation (126)

_{ζi0}= integration constant corresponding to the initial speed for i-th segment.

#### 3.2. Optimization Technique

**q**= vector of nodal parameters;

**c**

_{eq}(

**q**) = vector function of nonlinear equality constraints;

**A**

_{eq},

**b**

_{eq}= matrix and vector of linear equality constraints, respectively;

**q**

_{L}

**, q**

_{U}= lower and upper limits; and i $\in $ [1, n] = segment number.

_{i}(x), replacing x = ξL, within an interval [x

_{i−}

_{1}, x

_{i}] can be evaluated as follows

_{k}= integration weight in the k-th point; ϑ

_{k}= k-th point in the master–element coordinate system; J = Jacobian; k $\in $ [1, N]; and N = number of integration points.

**L**

_{s}= vector of segment lengths;

**z**= matrix of integrand values of n × N size; and

**Longitudinal Speed Deviation**relative to a preset upper-level V

_{ζU}value along the path

**s**.

_{v}

_{i}(ξ), using a set of FE speed parameters

**Q**

_{vi}from the model of Equation (124), a preset of the trajectory (curvature’s derivative) parameters

**Q**

_{ti}, and the approach of Equation (134), we have

**Third Derivative of Longitudinal Speed**

_{d3vi}(ξ) and considering the approach above, we obtain

**Fourth Derivative of Longitudinal Speed**

_{d}

_{4}

_{v}

_{i}(ξ) and by analogy with the previous, we obtain

_{v}is derived as the sum of the weighted criteria. Then, the following must be satisfied

**q**

_{v}= vector of speed’s second derivative’s nodal parameters (DOFs);

**I**

_{v}= vector of objective criteria integrals; and

**W**

_{v}= vector of weight factors.

_{v}, W

_{d}

_{3}

_{v}, W

_{d}

_{4}

_{v}= weight coefficients for quadratic velocity deviations and its third and fourth derivatives, respectively; and

**q**

_{t}= vector of curvature’s (trajectory’s) derivative nodal parameters (DOFs).

## 4. Constraints

**General Integral Approach to Nonlinear Equality Constraints**. Since the kinematic, dynamic, and physical vehicle motion parameters have been formed, let us consider the integral technique of composing equality constraints. Suppose that smooth piecewise polynomial functions describe all the parameters based on the nodal DOFs of the speed and curvature derivatives. Since the numerical integration based on the Gaussian scheme is applied for optimization, the same scheme will be used to form restrictions. Assume that some parameter Ψ changes along the path

**s**so that it does not exceed the upper Ψ

_{U}and lower Ψ

_{L}boundaries. Then, the sum of the areas between the upper limit Ψ

_{U}and the function Ψ and between the lower limit Ψ

_{L}and the function Ψ must be strictly equal to the area within boundaries. That is, for i-th FE and along the trajectory

**Kinematic Parameters’ Constraints**. Using this scheme above, we form a vector

**Ψ**

_{k}of nonlinear integral constraints for a series of kinematic parameters such as longitudinal speed, yaw rate, angular acceleration, and longitudinal jerk, where each j-th element corresponds to Ψ

_{k}

_{j}.

**Ψ**

_{kU},

**Ψ**

_{kL}= upper and lower limits of kinematic parameters,

**c**

_{k}= vector of constraints corresponding to kinematic parameters.

**Dynamic Parameters’ Constraints**. The vector

**Ψ**

_{d}of dynamic parameters includes slip angles

**δ**, tire lateral deformations

**Δ**

_{μ}, roll angle ψ, and longitudinal acceleration a

_{ζ}as a function of speed. The maximum allowable roll angle at 0.4 g should not exceed 7°. For passenger cars, the recommended limit value is ψ

_{U}= (10.8 − 4.3 B

_{24}/h

_{g}/2)°. We may accept 7.5°. Thus, introducing the limits ψ

_{L}= −ψ

_{U}. Let us use symmetrical limits.

_{ζU}, a

_{ζL}= upper and lower limit values of acceleration potentially implemented by the vehicle.

_{μL}= −Δ

_{μU}. Combining the parameters, we obtain the vectors

**Adhesion Constraints.**Having calculated the necessary traction forces on the wheels according to Equation (54), including Equations (55) and (56), we determine the degree of using the longitudinal and lateral adhesions on each j-th wheel.

**Boundary Parameters’ Constraints**. Another type of constraint determines the boundary conditions of kinematic parameters. Thus, one can require, for example, that the initial (0) and final (f) values of the predicted acceleration and jerk must correspond to preset constant values A

_{ζ0}

_{(}

_{f}

_{)}and J

_{ζ0}

_{(}

_{f}

_{)}. That is,

**Total restricting conditions.**Thus, the complete sets of parameters, bounds, and nonlinear constraints are

## 5. Simulation

**Trajectory**. As an example, consider planning a speed mode on a curvilinear section shown in Figure 7. Let us assume that the maneuver trajectory has already been determined by the inverse method described in our previous studies.

**W**

_{v}in Equation (144) and limit values.

**Comparison of drive cases’ outputs**. Figure 8 shows the results of output parameters for the various torque vectoring schemes corresponding to the variants (A, B, C, D) in Figure 4. Note that the same initial conditions and vehicle data are applied for all variants. Let us consider a case of road conditions that provide the maximum adhesion coefficient φ

_{max}= 0.5.

## 6. Conclusions

- The proposed technique is highly efficient. The inverse approach provides stable, unambiguous functions for all kinematic, dynamic, and physical parameters, which are characterized by continuity and smoothness. In addition, the method’s performance is distinguished by its relative simplicity and versatility, confirmed by the successfully applied technique for predicting both trajectory and speed.
- The vehicle’s motion dynamics significantly influences the formation of motion planning reference curves. This is facilitated by various dynamic and physical constraints built on the kinematic parameters to be optimized. The number of restrictions positively, in general, affects forecasts’ realism, considering the design features of vehicle’s drive, as well as the speed mode’s safety within the limits of vehicle capabilities.
- Pseudo-parameters, such as sideslip, lateral tire deformation, and body roll, were used as extended criteria for motion safety, which allowed these physical parameters to be included in the nonlinear constraints of the AV kinematic model. Even though these parameters express equivalent values, their use increases the accuracy of forecasts and reduces the probability of critical regimes.
- Parameters such as sideslip and tire lateral elasticity are the most sensitive to the curvilinear motion mode. The graphs show these parameters do not exceed the preset limits. However, the tangent angles of these curves at points close to the limit levels may be critical for the subsequent curves’ inflection at the next planning cycle. In this regard, the use of restricted derivatives for slip angles and tire lateral deformation may also be expedient.
- The considered schemes of transmissions and control for distributing torques provide the effective capability of planning the actuators’ impact based on the AV kinematic model. At the same time, in accordance with the strategy of equal longitudinal adhesion potential of the same axle wheels, the curves’ smoothness of braking moments and friction clutches’ torques of sport DMs is ensured, which has a positive effect on maintaining the stability of both control and motion. The predetermined continuous kinematic parameters, such as yaw rate and angular acceleration make it possible to calculate with sufficient accuracy and smoothness a set of force parameters required for the vehicle model’s dynamic equilibrium.
- Using a nonlinear tire model in the proposed approach had advantages and disadvantages. On the one hand, this leads to more accurate predictions of lateral forces, and on the other hand, due to the inevitable use of iterations in solving nonlinear equations, the optimization performance is worsened. Thus, regarding further improving the approach, a compromise between the modeling quality and the forecasting rapidity is needed, which may require the use of linearization followed by transiting from direct parameters to their increments with subsequent accumulation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Vo, C.P.; Jeon, J.H. An Integrated Motion Planning Scheme for Safe Autonomous Vehicles in Highly Dynamic Environments. Electronics
**2023**, 12, 1566. [Google Scholar] [CrossRef] - Ruof, J.; Mertens, M.B.; Buchholz, M.; Dietmayer, K. Real-Time Spatial Trajectory Planning for Urban Environments Using Dynamic Optimization. Robotics, Systems and Control. In Proceedings of the IEEE Intelligent Vehicles Symposium 2023, Anchorage, AK, USA, 4–7 June 2023. [Google Scholar] [CrossRef]
- Bianco, C.G.L.; Piazzi, A.; Romano, M. Velocity planning for autonomous vehicles. In Proceedings of the IEEE Intelligent Vehicles Symposium, Parma, Italy, 14–17 June 2004; pp. 413–418. [Google Scholar] [CrossRef]
- Kong, J.; Pfeiffer, M.; Schildbach, G.; Borrelli, F. Kinematic and dynamic vehicle models for autonomous driving control design. In Proceedings of the 2015 IEEE Intelligent Vehicles Symposium (IV), Seoul, Republic of Korea, 28 June–1 July 2015; pp. 1094–1099. [Google Scholar] [CrossRef]
- Mizushima, Y.; Okawa, I.; Nonaka, K. Model Predictive Control for Autonomous Vehicles with Speed Profile Shaping. IFAC-PapersOnLine
**2019**, 52, 31–36. [Google Scholar] [CrossRef] - Piccinini, M.; Taddei, S.; Larcher, M.; Piazza, M.; Biral, F. A Physics-Driven Artificial Agent for Online Time-Optimal Vehicle Motion Planning and Control. IEEE Access
**2023**, 11, 46344–46372. [Google Scholar] [CrossRef] - Altché, F.; Polack, P.; de La Fortelle, A. High-Speed Trajectory Planning for Autonomous Vehicles Using a Simple Dynamic Model. In Proceedings of the IEEE 20th International Conference on Intelligent Transportation, Yokohama, Japan, 16–19 October 2017. [Google Scholar]
- Ajanović, Z.; Regolin, E.; Shyrokau, B.; Ćatić, H.; Horn, M.; Ferrara, A. Search-Based Task and Motion Planning for Hybrid Systems: Agile Autonomous Vehicles. Eng. Appl. Artif. Intell.
**2023**, 121, 105893. [Google Scholar] [CrossRef] - Cataffo, V.; Silano, G.; Iannelli, L.; Puig, V.; Glielmo, L. A Nonlinear Model Predictive Control Strategy for Autonomous Racing of Scale Vehicles. In Proceedings of the 2022 IEEE International Conference on Systems, Man, and Cybernetics (SMC), Prague, Czech Republic, 9–12 October 2022; pp. 100–105. [Google Scholar] [CrossRef]
- Rowold, M.; Ögretmen, L.; Kasolowsky, U.; Lohmann, B. Online Time-Optimal Trajectory Planning on Three-Dimensional Race Tracks, Robotics. In Proceedings of the 34th IEEE Intelligent Vehicles Symposium (IV), Anchorage, AK, USA, 4–7 June 2023. [Google Scholar] [CrossRef]
- Lamouik, I.; Yahyaouy, A.; Abdelouahed, S. Model Predictive Control for Full Autonomous Vehicle Overtaking. Transp. Res. Rec. J. Transp. Res. Board.
**2023**, 2677, 1193–1207. [Google Scholar] [CrossRef] - Wu, H.; Long, X.; Lu, D. Trajectory Planning and Tracking for Four-Wheel Independent Drive Intelligent Vehicle Based on Model Predictive Control. SAE Int. J. Adv. Curr. Pract. Mobil.
**2023**, 5, 2471–2485. [Google Scholar] [CrossRef] - Chen, Y.; Xin, R.; Cheng, J.; Zhang, Q.; Mei, X.; Liu, M.; Wang, L. Efficient Speed Planning for Autonomous Driving in Dynamic Environment with Interaction Point Model. IEEE Robot. Autom. Lett.
**2022**, 7, 11839–11846. [Google Scholar] [CrossRef] - Zhou, S.; Tian, Y.; Walker, P.; Zhang, N. Impact of the tyre dynamics on autonomous vehicle path following control with front wheel steering and differential motor torque. IET Intell. Transp. Syst.
**2023**, 17, 1629–1648. [Google Scholar] [CrossRef] - Xu, S.; Peng, H.; Song, Z.; Chen, K.; Tang, Y. Accurate and Smooth Speed Control for an Autonomous Vehicle. In Proceedings of the IEEE Intelligent Vehicles Symposium (IV), Changshu, China, 26–30 June 2018; pp. 1976–1982. [Google Scholar] [CrossRef]
- Kim, C.; Yoon, Y.; Kim, S.; Yoo, M.; Yi, K. Trajectory Planning and Control of Autonomous Vehicles for Static Vehicle Avoidance in Dynamic Traffic Environments. IEEE Access
**2023**, 11, 5772–5788. [Google Scholar] [CrossRef] - Kim, C.; Yi, K.; Park, J. Hierarchical Motion Planning and Control Algorithm of Autonomous Racing Vehicles for Overtaking Maneuvers; SAE Technical Paper 2023-01-0698; SAE International: Pittsburgh, PA, USA, 2023. [Google Scholar] [CrossRef]
- Gao, J.; Claveau, F.; Pashkevich, A.; Chevrel, P. Real-time motion planning for an autonomous mobile robot with wheel-ground adhesion constraint. Adv. Robot.
**2023**, 37, 649–666. [Google Scholar] [CrossRef] - Zulkifli, A.; Peeie, M.H.; Zakaria, M.A.; Ishak, M.I.; Shahrom, M.; Kujunni, B. Motion Planning and Tracking Trajectory of an Autonomous Emergency Braking Pedestrian (AEB-P) System Based on Different Brake Pad Friction Coefficients on Dry Road Surface. Int. J. Automot. Mech. Eng.
**2022**, 19, 10002–10013. [Google Scholar] [CrossRef] - Jianshi, L.; Lou, J.; Li, Y.; Pan, S.; Xu, Y. Trajectory Tracking of Autonomous Vehicle Using Clothoid Curve. Appl. Sci.
**2023**, 13, 2733. [Google Scholar] [CrossRef] - Grishkevich, A.I. Automobiles: Theory: Textbook for High Schools; Minsk High School: Minsk, Belarus, 1986; Volume 208, pp. 431–438. [Google Scholar]
- Krmela, J.; Beneš, L.; Krmelová, V. Tire Experiments on Static Adhesor for Obtaining the Radial Stiffness Value. Period. Polytech. Transp. Eng.
**2014**, 42, 125–129. [Google Scholar] [CrossRef] - Pacejka, H.B. Tire and Vehicle Dynamics, 3rd ed.; Elsevier: Oxford, UK, 2012. [Google Scholar]
- Audi A4 Quatro Characteristics. 2023. Available online: http://www.automobile-catalog.com/car/2011/1187660/audi_a4_3_2_fsi_quattro_attraction_tiptronic.html (accessed on 19 January 2023).
- MATLAB R2022b. Available online: https://www.mathworks.com/ (accessed on 22 January 2023).

**Figure 2.**Scheme of the 2.5D vehicle model: (

**a**) ideal kinematics, (

**b**) roll, (

**c**) longitudinal, and (

**d**) transversal dynamics.

**Figure 3.**Highlights of nonlinear tire slip: (

**a**) effect of sideslip on side force, (

**b**) effect of vertical load on slip coefficient, (

**c**) slip inverse determination by the side force response.

**Figure 4.**Variants of wheel torque control individually: (

**a**) using the same vehicle side’s brakes (Case A), (

**b**) by driving the rear wheels through a sport differential (Case B), (

**c**) by combining a rear sport differential and a front wheel’s brake (Case C), (

**d**) by combining two sport differentials (Case D).

**Figure 5.**Asymmetric self-locking inter-axle DM with proportional friction moments: (

**a**) design, (

**b**) scheme, and (

**c**) proportional action between satellites and crown gears.

**Figure 8.**Comparison of kinematic and dynamic outputs by the drive variants (A, B, C, D): (

**a**) speed, (

**b**) longitudinal acceleration, (

**c**) longitudinal jerk, (

**d**) pseudo-roll angle, (

**e**) yaw rate, (

**f**) angular acceleration, (

**g**) inter-axel differential carrier’s driving torque, (

**h**) inter-axel differential’s front T

_{f}and rear T

_{r}output torques.

**Figure 9.**Basic output parameters for the variant of controlling the traction with wheels’ braking actuators (A): (

**a**) sideslip pseudo-angles, (

**b**) tires’ lateral pseudo-deformations, (

**c**) vertical reactions, (

**d**) lateral reactions, (

**e**) longitudinal reactions, (

**f**) wheels’ braking moments, (

**g**) longitudinal adhesion factor, (

**h**) lateral adhesion factor, (

**i**) full adhesion factor, (

**j**) relation between longitudinal and lateral accelerations.

**Figure 10.**Basic output parameters for the variant of controlling the traction with rear sport differential (B): (

**a**) sideslip pseudo-angles, (

**b**) tires’ lateral pseudo-deformations, (

**c**) vertical reactions, (

**d**) lateral reactions, (

**e**) longitudinal reactions, (

**f**) moments of rear SD’s clutches, (

**g**) longitudinal adhesion factor, (

**h**) lateral adhesion factor, (

**i**) full adhesion factor, (

**j**) relation between longitudinal and lateral accelerations.

**Figure 11.**Basic output parameters for the variant of combined controlling the traction with both the rear sport differential and the front brake actuators (C): (

**a**) sideslip pseudo-angles, (

**b**) tires’ lateral pseudo-deformations, (

**c**) vertical reactions, (

**d**) lateral reactions, (

**e**) longitudinal reactions, (

**f**) moments of rear SD’s clutches and front brakes, (

**g**) longitudinal adhesion factor, (

**h**) lateral adhesion factor, (

**i**) full adhesion factor, (

**j**) relation between longitudinal and lateral accelerations.

**Figure 12.**Basic output parameters for the variant of controlling the traction with rear sport differential (B): (

**a**) sideslip pseudo-angles, (

**b**) tires’ lateral pseudo-deformations, (

**c**) vertical reactions, (

**d**) lateral reactions, (

**e**) longitudinal reactions, (

**f**) moments of the front and rear SDs’ clutches, (

**g**) longitudinal adhesion factor, (

**h**) lateral adhesion factor, (

**i**) full adhesion factor, (

**j**) relation between longitudinal and lateral accelerations.

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## Share and Cite

**MDPI and ACS Style**

Diachuk, M.; Easa, S.M.
Planning Speed Mode of All-Wheel Drive Autonomous Vehicles Considering Complete Constraint Set. *Vehicles* **2024**, *6*, 191-230.
https://doi.org/10.3390/vehicles6010008

**AMA Style**

Diachuk M, Easa SM.
Planning Speed Mode of All-Wheel Drive Autonomous Vehicles Considering Complete Constraint Set. *Vehicles*. 2024; 6(1):191-230.
https://doi.org/10.3390/vehicles6010008

**Chicago/Turabian Style**

Diachuk, Maksym, and Said M. Easa.
2024. "Planning Speed Mode of All-Wheel Drive Autonomous Vehicles Considering Complete Constraint Set" *Vehicles* 6, no. 1: 191-230.
https://doi.org/10.3390/vehicles6010008