Using Inverse Dynamics Technique in Planning Autonomous Vehicle Speed Mode Considering Physical Constraints

The study aims at improving the technique of planning the autonomous vehicles’ (AV) speed mode based on a kinematic model with physical restrictions. A mathematical model relates the derivatives of kinematic parameters with ones of the trajectory’s curvature. The inverse approach uses an expanded vehicle model considering the distribution of vertical reactions, wheels’ longitudinal reactions according to a drive type, and lateral forces ensuring motion stability. For analysis of the drive type, four options are proposed: front-wheel drive (FWD), rear-wheel drive (RWD), permanent engaged all-wheel drive (AWD), and 4-wheel drive with torque vectoring (4WD-TV). The optimization model is also built by the inverse scheme. The longitudinal speed’s higher derivatives are modeled by the finite element (FE) functions with nodal unknowns. The sequential integrations ensure the optimality and smoothness of the third derivative. The kinematic restrictions are supplemented by the tire-road critical slip states. Sequential quadratic programming (SQP) and the Gaussian N -point scheme for quadrature integration are used to minimize the objective function. The simulation results show a significant difference in the mode forecasts between four types of AV drives at the same initial conditions. This technique allows redistributing the traction forces strictly according to the wheels’ adhesion potentials and increases the optimization performance by about 40% compared to using the kinematic model based on the same technique without physical constrains.


Introduction
Planning the autonomous vehicles' (AV) motion implies two procedures: predicting a trajectory and distributing the speed (and other kinematic parameters).These stages can be organized both simultaneously and sequentially.At the same time, the essential problem consists of forecast quality and rapidity.The simultaneous trajectory and speed optimization may require many optimality criteria, which may be excessive regarding the accuracy and smoothness of forecasts due to the increased sensitivity of an objective function to changes in optimization parameters.With sequential optimization, it is easier to achieve smoothness of the secondary functions associated with both the trajectory curvature and the speed but participating in the vehicle dynamics parameters.
Note that if a small number of geometric constraints (boundaries, curvature, and its derivatives) is sufficient to predict the trajectory, then speed prediction demands a much greater quantity of kinematic, dynamic, and physical restrictions.In the general case, precisely the number of these limitations determines the realism and reliability of the speed distribution forecast.In addition, kinematic and, less often, dynamic models are the most used for reflecting AV behavior.At the same time, for high prediction performance, these models are usually simplified to be flat or bicycle.This gives the advantage of simplicity and a small number of needed equations.On the other hand, such models are unified and do not reflect the influence of the vehicle design features, which can affect the nature of the speed restriction.Thus, regardless of whether the vehicle is front-, rear-, or all-wheel drive, the parameter distribution (e.g., speed and acceleration) will be the same.The situation can be improved by introducing dynamic and physical (tire-road adhesion) constraints into the optimization scheme.However, the generalized character of speed distribution will remain in this case.First, the type of AV drive will affect the force distribution by the wheels followed by affecting the speed mode along a predetermined trajectory.
The main problem in using kinematic vehicle models is the consistency of the kinematic parameters determining the vehicle dynamics with the capability to realize their values, considering the drive design features and road conditions.Thus, in many studies (Table 1), when optimizing kinematic parameters (speed, accelerations), numerous restrictions are used represented by upper and lower limits.However, such a technique does not guarantee the possibility of reaching predicted values by a real AV and especially at high-speed modes close to critical.The longitudinal and transverse accelerations being determined during optimization result in a redistribution of vertical reactions limiting longitudinal and lateral forces.Suppose a bicycle model allows considering the distribution of vertical reactions along the vehicle's longitudinal axis.In that case, knowing this may not be enough to ensure the needed traction potential of the AV.Therefore, if the AV is front-or rear-wheel drive and a symmetrical differential distributes the torque, the resulting traction is determined by approximately doubled traction force realized by a wheel under lesser vertical load (or worse adhesion conditions).In the case of all-wheel drive and conventional symmetrical differentials in the transmission, the vehicle traction potential will not exceed the quadruplicated traction force of a wheel with the worst torque transfer conditions.Another important aspect is about maintaining the stability of the AV motion.
Suppose angular and translational accelerations are already optimally redistributed based on the kinematic model.In that case, the lateral and longitudinal forces on the wheels must be combined in a certain way to ensure dynamic balance.If a wheel drive type is considered, it is evident that a set of these force components will differ from others at the same values of kinematic parameters.In turn, the importance of lateral reactions or, instead, degrees of using the wheel's lateral adhesion potential determines critical moments preventing uncontrolled sideslip.For example, in the case of front-wheel drive (FWD) or rear-wheel drive (RWD), the longitudinal forces on these wheels should be twice as high as in the case of all-wheel drive (AWD).Consequently, the lateral reactions on these wheels cannot have the same values as on the wheels of the AWD vehicle since the lateral potential is less and the critical mode will come earlier.If we include the moment of each wheel's critical sideslip in the optimization scheme, this will allow to distribute the speed and acceleration based on subcritical motion modes, considering the drive type.Thus, it is possible to expand the imposed physical restrictions on the vehicle kinematic model.A separate issue is the task of using the intelligent distribution of drive torques in accordance with the traction potential of each wheel.
Table 1.Basic approaches to modeling the speed of an autonomous vehicle.

Ref. General Method
Path Model Speed Model Constraints [1] Semi-analytical method, the optimal speed profile is evaluated for each arc subject to free boundary conditions.
A set of radii for path function's points.
The longitudinal acceleration is calculated as a residual result equivalent to the available traction, The speed at each local minimum is equal to the corresponding critical velocity.
The critical speed at each path's point is the maximum allowable velocity; for arcs with unique points of local minimum radius, the speed profile implies full deceleration before and full acceleration after the local minimum. [2] An optimization algorithm minimizes the maximum jerk within the trajectory.
A preset trajectory.
A parabolic speed curve concerning time intervals is divided into five sections, where nodal parameters determine the shape of piecewise functions.
Restrictions are imposed only on discrete nodal values.[3] Non-convexity problem of speed optimization, convexity problem of jerk inequalities.
Linear programming, quadratic linear programming, and non-convex velocity planning.
Velocity limit filter, pseudo-jerk, and collision constraints. [4] Objective function includes quadratic accelerations and jerks, and quadratic of deflection the AV speed from a reference value.
Waypoints along a given path.
Final differences and numeric derivations approximate the speed, acceleration, and jerk; optimizing the time stamps.
Nonlinear and non-convex problem; sequence of quadratic programs using the slack convex feasible set algorithm. [5] Using the leading vehicle velocity predicted with the inputoutput hidden Markov model to forecast the AV speed.
The quintic polynomial for lane-change maneuvers.
The kinematic parameters are determined based on the optimal control based on the developed linear and improved controllers, jerk-optimal trajectories.
Checking a collision probability to generate collision-free trajectories, the lateral and longitudinal trajectories are generated separately in the Frenet coordinate. [6] Sliding surface control of carlike vehicles for providing minimal errors in the lateral and angular position.
Trajectory-tracking and path-following control.
Five properly joined spline curves generate the speed along a path section.The output velocity is C1 and strictly positive at any time.
Fast response, good transient, and robustness concerning system uncertainties and external disturbances.
https://www.hos.pubA convex-optimization approach to solve the AV speed planning problem in both static and dynamic environments.
A given continuous path.
The speed planning problem seeks a time-efficient, safe, and smooth speed profile based on a pseudojerk criterion with respect to the arc-length.
Safety and performance in combination with all other hard constraints. [9] A speed planning optimization algorithm which considers safety, time efficiency, smoothness, and comfort constraints.
A sequence of path sampling.
A longitudinal reference speed is needed for different driving scenarios; the quadratic speed optimization procedure finds the required time (decision) intervals along with determining acceleration and jerk.
Time efficiency, speed limit, comfort, and longitudinal and lateral acceleration limits are taken to a set of constraints.
Collision-free zones.Speed planning algorithm aims to find a valid trajectory without collisions.Moving obstacles. [11] A learning-based Interaction Point Model (IPM) for describing the interaction between agents with the protection time and interaction priority.
The speed profile is chosen to be further smoothed for meeting the kinodynamic feasibility in the spatio-temporal semantic corridor.
Traffic lights, stop signs, and speed limits are transformed into spacetime constraints; two kinds of velocity constraints: from the curvature and from the observation point. [12] A concept of developing a new energy vehicle (NEV); vehicle lateral dynamics.
Traffic flow model.
Speed forecasting method in terms of intelligent control, and correlation control methods for vehicle speed prediction.
Traffic safety and motion stability. [13] Problem of safe speed planning with minimal passing time in the space-time (s-t) domain.

Intersections.
Exploring s-t with the rapidly exploring random tree method to approximate the optimal solution.
Constraints for avoiding collision with queuing and crossing vehicles; speed, acceleration, and initial speed constraints. [ A model predictive controller (MPC) to plan the optimal speed profile for a diesel car following.
A hypothetical lead vehicle in an energyconscious way.
Three candidate functions as separate minimization objectives, including acceleration, power, and fuel consumption; the states (speed and position) assume the AV to be a point mass.
Constant acceleration is set for each sampling period; space limits between vehicles to imitate the traffic constraints in avoiding rear-end collisions and keeping the ahead distance from being too long. [15] Collision preventive speed planning algorithm to enhance safety when driving in an urban environment.
2D point cloud for a map of static obstacles based on lidar measurements.
The longitudinal motion planner determines the target states to prevent potential collisions with pedestrians.MPC tracks the target states to select the desired acceleration.
The static obstacles' boundaries are defined to estimate the possible position of appearing a pedestrian; the Monte Carlo simulation is for validating the collision prevention performance. [16] Adaptive model predictive controller to improve tracking accuracy and stability compared with general MPC controllers.
The control parameters imply control horizon and sampling time, all of which are set according to the vehicle speed.
The recursive least square algorithm for online estimating the tire cornering stiffness and road friction coefficient followed by updating the road adhesion constraint. [17] A novel method to reshape the AV speed (or acceleration) profile based on MPC.
Lower-level controller for tracking the optimal references.
The upper-level controller optimizes vehicle running distance, speed, and acceleration.
Combined hard, soft, and preventing collision constraints; constraints on the vehicle model state and on the input (jerk). [18] A model-based vehicle estimator for evaluating the lateral tire forces without using highly nonlinear tire-road friction models; the 3-degrees of freedom (DOF) vehicle dynamics model is used.
Tracking the yaw rate error stipulated by the difference between the measured yaw rate and the virtual model's output.
The estimation scheme consists of three steps: (a) estimating the yaw moment based on a disturbance observer, (b) estimating the combined lateral forces for the tires of the front and rear axles using regression, and (c) the estimation of individual lateral reaction for each tire based on a heuristic method.
The disturbance observer incorporates a virtual yaw dynamic model composed of the yaw dynamic equations relative to the five reference points and a feedback controller; the used PID controller follows the measured yaw rate to provide a good tracking performance; the controller compensates for the unmeasured input, which is a part of the yaw moment generated by the lateral tire forces. [19] The H∞ control design method, considering sideslip.
The path-following control problem.
A method based on a vehicle kinematical model, two error signals: the vehicle position and the yaw angle.
Peak-to-peak criterion and linear matrix inequality tools to reduce tire slip angle effects on the closedloop performance. [20] A method for estimating lateral tire-road forces and vehicle sideslip angle by using an unscented Kalman filter (UKF).
Path following with the direct-yaw-moment control.
A sliding mode controller tracks the errors between desired yaw rate and sideslip angle of the reference model and their estimations; the UKF observer uses variables such as longitudinal and lateral acceleration, yaw rate, and wheels' angular speeds.
The road adhesion limit is considered at each wheel by the friction circle; the need yaw moment is provided by the optimally distributed torques driving the wheels.
These studies show that AV speed planning is a complex task with the possibility of using vehicle dynamics models, a wide range of optimization methods, and control theory techniques to form feasible reference curves.
Figure 1 shows the general scheme of the proposed procedure for improving the technique of distributing speed parameters by optimization.In the approach we used earlier, the plane model of AV kinematics was used to strictly coordinate the kinematic parameters with the motion trajectory's geometry.The resulting conventional model was used to describe the geometric, kinematic, and physical (critical sideslip speed) constraints.Using the integral approach to equality constraints described in our previous works, a parameter curve can be formed (rather than only nodal discrete points) strictly within the boundaries.Performing sequential quadratic programming (SQP) with considering the smoothness criteria for speed derivatives, we also obtained smooth and unambiguous kinematic parameters.
The extended approach involves the use of a 2.5D vehicle dynamics model (unsprung) to maintain high-performance optimization and reflect the general tendency of dynamic processes.The inverse approach implies the forces acting in the wheels' contacts are unknown, and optimization iterations allow obtaining the longitudinal speed and longitudinal and lateral accelerations for estimating the vehicle dynamics.Then, first, we can evaluate the redistribution of vertical reactions to all four wheels.Further, using the scheme of distributing the torque over the wheels, the equations of wheels' rotational dynamics, the equations of plane dynamics, and the algebraic equations of rations between the moments formed by the lateral forces of the same axle's wheels, it is possible to obtain an estimation set of all wheels' longitudinal and transverse forces, which is necessary for strictly matching the linear and angular accelerations, and motion speed declared during iterations.Knowing all reactions on all wheels, it is possible to estimate the required degrees of using longitudinal and lateral adhesions.By comparing the total required adhesions under the wheels with the maximum possible one, the physical limit for each wheel is formed followed by adding these limits to the constraints established for the conventional model.Thus, the process of iterating speed parameters during optimization should be carried out until no wheel exceeds the adhesion conditions regardless of the drive type.In addition, the extended model allows the transmission torque's evaluation.Comparing the two approaches' results should give a clear picture of the effectiveness and relevance of the more complicated model.Thus, the purpose of this work consists in developing and validating extended AV kinematic models providing strengthening of physical restrictions imposed on the kinematic parameters in predicting the speed mode, considering the distribution of vertical, horizontal, and lateral reactions stipulated by a type of vehicle transmission drive.The speed mode, therefore, will be adjusted according to the conditions for the possibility of transmitting longitudinal and lateral forces over all vehicle wheels.This will improve the forecast quality and reliability with ensuring the vehicle's stable curvilinear motion.

Vehicle Kinematics Model
From the point of the best controllability, a driving mode corresponding to neutral steerability would be the most favorable.The most significant advantage of this kinematic model for optimization, unlike dynamic models, is that it does not require a small integration step, which affects the high speed of the optimization procedure.In this regard, let us consider the components of the kinematic model necessary for connection with the force balance (Figure 2).Mass center speed  is directed by the tangent to trajectory and may be expressed as follows  ⃗ =  ⃗ + 0⃗ =   ⃗  +   ⃗  =   ⃗  +   ⃗  (1) where ⃗ , ⃗ = unit vectors of the natural coordinate system; ⃗  , ⃗  ,   ,   = unit vectors and speed components of the vehicle local coordinate system ; ⃗  , ⃗  ,   ,   = unit vectors and speed components of the fixed (global) coordinate system .
Assume that the longitudinal speed component   is the basic one, which may be estimated by the AV sensory system during motion tracking.Suppose that the longitudinal speed is obtained in some way due to an optimization process based on distributing the kinematic parameters for a pre-optimized trajectory.We will be interested in parameters related to this speed's derivatives, which, like the speed itself, should be provided by smooth and unambiguous functions.
Since the absolute speed  may be decomposed, projections   and   are tied through the tangent angle  and central slip angle .Then, The first and second derivatives may be found as follows The derivative of   concerning the x-coordinate is Lateral Speed   is geometrically tied with the longitudinal speed   by the central slip angle    =   () Its time derivative is given as Yaw Rate can be determined by taking the yaw angle  =  −  derivative in the current global coordinates.Thus, Angular acceleration  is derived from the yaw rate  concerning time Accelerations in the local vehicle coordinate system  are Jerks in the vehicle coordinate system are Velocities and accelerations at the wheels' centers.To assess the forces acting in the tire-road contacts, the speeds and accelerations in the wheel centers are needed to be known, considering the vehicle geometry in Figure 2. Let us introduce designations where  2 = identity matrix of dimension 2 × 2,   = vector of the j-th wheel center's coordinates in the AV local coordinate system,   = vector of the j-th wheel center's velocities in the wheel local coordinate system,  = rotational matrix,   = angle of the j-th wheel turn, ⃗  , ⃗  = unit vectors of the j-th wheel local coordinate system.Let us reveal that the derivative of the unit vectors has the following form Speed in the center of the j-th wheel in the AV local coordinates Then, the velocity components in the local coordinates of the j-th wheel Consequently, the longitudinal and lateral components are Acceleration in the center of the j-th wheel in the AV local coordinates Then, the acceleration components in the j-th wheel's local coordinates Correspondingly, the longitudinal and lateral components are Rotation angles of steered wheels can be determined as Ackerman's angles subject to the fact that the trajectory curvature  is much less than the maximum possible.Thus,

Determining Dynamic Parameters
Note that vehicle models used for motion planning must meet the requirements of good information efficiency and high computational performance.In this regard, consider a model evaluating reactions individually on each wheel.

Redistributing Vertical Reactions
In pure flat models, a static distribution of vertical reactions on the wheels is inevitably used.However, at higher values of longitudinal and lateral accelerations, an impacting change in vertical reactions may result in worsening the tire-road adhesion and the sideslip instability.As known, the wheels' vertical reactions close to zero are fraught with the loss of transversal stability followed by possible rollover as well as diminishing traction potential.At the same time, an increased wheel vertical reaction leads to excessive deformation and tire wear.In this regard, it is necessary to limit vertical reactions in a specific range.To do this, let us consider the so-called 2.5D rigid unsprung model in Figure 3. where  = angle of road slope, and Provided that for most of the moving time  ≈ 0, the Equation ( 20) may be reduced Wheel rolling resistance moments   can be partially estimated by omitting kinematic losses and seeing only power losses.However, even this requires a vertical load value on each wheel.Consider the moments from the conditionally paired front and rear wheels to simplify the task.Then  () =  ()  ()  () (23) where  () = dynamic radius of the front (rear) wheel,  () = coefficient of the front (rear) rolling resistance.
Using the average turning angle  () of the steered wheels, we get Then, the rolling resistance moments can be expressed in terms of vertical reactions Thus, Equation ( 22) may be rewritten in the form Assuming that due to the vehicle symmetry, the increments of vertical reactions in the longitudinal plane of the right and left sides are the same, we obtain the vector of dynamic reactions caused by the longitudinal acceleration Now, let us distribute vertical reactions along the sides, considering Figure 3.To do this, we may compose the equations of moments' equilibrium relative to each wheel's contact patch center.Introduce the designations where   ,   = overturning moment from right and left vehicle sides.Since the tire deformations in the lateral direction are not considered, only the decomposition components from the rolling resistance moments will impact the wheel overturning moments.Considering them along the axes, we have Let's compose the equations in matrix form for the balance of forces' moments relative to the centers of wheel contact patches    +   ℎ  +   = 0 (30) In general, the matrix  is singular due to vehicle symmetry.Therefore, the pseudo-inverse matrix  + is taken to find the redistribution of vertical reactions caused by the lateral acceleration Thus, the vector of vertical reactions is combined of both distributions 3.2.Evaluation of Lateral Reactions First, considering the transmission type, let's define the distribution and realization of the traction forces on the wheels [21].In the general case, the equation of rotational dynamic balance for the j-th wheel has the form where for the j-th wheel:   = drive torque,   = dynamic radius,   = effective radius,   = rolling resistance,   = inertia of rotating masses reduced to a wheel.
The rolling resistance at each wheel can be calculated from the dependence, = longitudinal speed in the j-th wheel local coordinate system,   = speed at which the empirical measurements were made;  1 ,  3 ,  4 = coefficients.The dynamic radius of each wheel can be evaluated by expressions, where  0 = wheel's free radius,   = radial deformation of the j-th wheel,   = j-th tire radial stiffness,  0 = static load on the j-th wheel, and  1 ,  1 = coefficients.The centrifugal increment of the j-th wheel radius is given by The effective radius of the j-th wheel is given by where   ,   ,   = coefficients.Let's introduce vector designations, where   = inertia of the j-th wheel and connected rotating masses.
Then, for all wheels based on Equation (33), we can write Since the drive of all wheels is realized by the same transmission, the drive torques can be linked to each other depending on the drive type.Let's consider options for using front-, rear-, all-wheel mechanical drives, and 4-wheel drive with torque vectoring (TV) between axles and wheels (Figure 4).In the first three options, let us proceed from using symmetrical differential mechanisms providing the equality of output shafts' moments.Assuming for simplicity the gear efficiencies equal to 1, for the front-, rear-, and all-wheel drive, respectively, we have where  = torque on the driving shaft before differential mechanism(s) and   = final gear ratio.
In the torque-vectoring variant, torque distribution is possible depending on the adhesion condition on each wheel.Since the distribution of vertical reactions is already known, it is possible to calculate the torque proportion that may be provided at each wheel.To do this, we will form a vector of relative vertical reactions.
Now, let us compose the equations for describing the dynamic balance.Since the torque  in the transmission determines the traction forces   on the wheels, considering the unknown lateral reactions   , we obtain five variables, for which five equations are needed.
1. Balance of forces relative to the vehicle longitudinal axis: Denote where  = vehicle gross mass.Then, in the vector form 2. Balance of forces relative to the vehicle lateral axis: Denote Then, in the vector form 3. The balance of forces' moments relative to the mass center in the plane of motion: where   ,   = coordinates of the j-th wheel center in the vehicle coordinate system.Let's designate Then, in the vector form 4. Since there is no more option for composing independent dynamics equations, it is necessary to somehow relate the lateral reactions to each other.In general, this is a complex task with many affecting factors.However, for simplicity, we may require a relation in distributing the lateral loads only between the same axle's wheels.To do this, we will proceed from the ratio of moments created by the lateral reactions to determine the influence of each wheel [18].On the other hand, from Pasejka's magic formulas [22], the lateral reaction depends on a combination of factors such as vertical load, sideslip angle, degree of longitudinal slip, and maximum tire-road adhesion value.However, many of these parameters are unknown.Then, the ratio of the lateral forces' moments of the same axle's wheels can be equated to the ratio of the adhesion potentials estimated through vertical loads, lateral accelerations, and lateral velocities.The vertical load and the maximum adhesion value characterize the peak value of the lateral reaction, the lateral accelerations reflect the redistribution of side inertial impact, and the lateral velocities reflect the sideslip.It is obvious that the lateral reaction increases with vertical load, lateral acceleration and decreases with lateral velocity.Then, for the front axle, we have 1 ( 1 ( 1 ) +  1 ( 1 )) −  13  3 ( 3 ( 3 ) +  3 ( 3 )) = 0 (52) and Combining all the equations, we get where   ,   = matrices with dimension 5 × 4,   = matrix with dimension 5 × 3,   = vector with dimension 5 × 1, containing only the first nonzero element equal to 1. Substituting Equation (39) in Equation ( 56), simultaneously replacing the vector  with a structure of type      , where   reflects the scheme of torque distribution in the transmission from Equations ( 40) and (41).
+   (  ) −1      =    +     + +  �(  )  + (  ) −1 (  ) − Thus, the vector of unknowns can be found as Having calculated the necessary traction forces on the wheels according to Equation (39), including Equations ( 40) and (41), we determine the degree of using the longitudinal and lateral adhesions on each j-th wheel.
Then, the basic condition regarding the adhesion potential can be formed as follows where   = maximum value of tire-road adhesion.

Generalized Approach to Speed Model
We assume that some trajectory for the AV within a distance  has already been built.This question was extensively reflected in our previous studies.Using the general mathematical basis, we will compose models for obtaining the speed in finite elements.The distance  may be divided on n FEs, each i-th of which is represented by the length   and parameter  ∈ [0, 1].Thus, the current linear space is  =   .The basis functions   are the form functions corresponding to the nodal DOFs   .The number of nodal DOFs depends on a degree  of the Lagrangian polynomial:  = ( + 1)/2.Then, any function  = () within the i-th finite element (FE) may be expressed as follows: We will use the second speed derivative as the basic model to reduce the polynomial extent , the quantity of nodal unknowns, and speed up computations.Then, it can be written where nodal and FE parameters are Thus,  = 2 and we need the cubic Lagrangian polynomial to provide smooth functions conjugation between all the segments.The first derivative of the longitudinal speed within the i-th segment can be derived by the first integral where   /(0) = integration constant defined from initial conditions.The antiderivative is determined as The longitudinal speed V ζi in the i-th segment is found by repeating integration of Equation ( 65) where  0 = integration constant corresponding to the initial speed for the i-th segment.
Components of Equation ( 67) are found as follows

Optimization Technique
Minimization of an objective function  has the quadratic form and, considering nonlinear constraints uses the SQP method.It can be written as where  = vector of nodal parameters;   () = vector function of nonlinear equality constraints;   ,   = matrix and vector of linear equality constraints, respectively;   ,   = lower and upper limits;  ∈ [0, ] = segment number.The objective function may be a sum of integrals across all FEs.The integrals can be solved numerically using an N-point Gaussian quadrature scheme.Then, any integrand   (), replacing  = , within the interval [ −1 ,   ] can be evaluated as follows where   = integration weight in the k-th point;   = the k-th point in the master-element coordinate system;  = Jacobian,  ∈ [0,  ];  = number of integration points.For one-dimensional FE

� 𝑧𝑧 𝑖𝑖 (𝑥𝑥)𝑑𝑑𝑥𝑥
Using vector designations, we obtain a short expression for calculating the integral of Equation ( 70) along all  segments where   = vector of segment lengths;  = matrix of integrands of  ×  size; and Here, we use  = 5-point scheme enough for quality and performance.Now, let's choose optimization criteria.Denoting the integrand   (), using a set of FE speed parameters   from the model of Equation ( 63), a preset of trajectory (curvature) parameters   , and the approach of Equation ( 73 where the trajectory scale factor is

Longitudinal Speed
Third Derivative of Longitudinal Speed Denoting the integrand  3 () and considering the approach above, we obtain The speed's objective function   is derived as the sum of the weighted criteria.Then, the following must be satisfied where   = vector of speed's second derivative nodal parameters (DOFs);   = vector of speed criteria integrals;   = vector of speed weight factors.
where   ,  3 ,  4 = weight coefficients for quadratic velocity deviations and its third and fourth derivatives, respectively;   = vector of curvature (trajectory) derivative nodal parameters (DOFs) (assumed to be known).

Restrictions
Since the kinematic, dynamic, and physical vehicle motion parameters have been formed, let's consider the integral technique to forming equality constraints.We may suppose that smooth piecewise polynomial functions describe all 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  over the length of path  must change not to exceeding the upper   and lower   limits.Then, the sum of the areas between the upper limit and the function and between the lower limit and the function must be strictly equal to the area within the limits.That is, for the ith FE and for the j-th wheel along the trajectory Integrals can be calculated numerically, for which we introduce the following notations.
Integral between upper and lower bounds (): Integral between upper bound and parameter function (): Integral between the parameter function and lower bound (): Thus, the requirement of nonlinear equality constraint along all the segments for the j-th parameter, according to Equation ( 84) is expressed as follows Using this scheme, we form a vector   of nonlinear integral constraints for a series of kinematic parameters, where each j-th element corresponds to   .
Using Equation (84), it is possible to impose physical restrictions on the conditions of all tireroad adhesions.Then A particular case is the limitation of the vehicle's traction potential.The vehicle's maximum acceleration depends on the speed and is due to design features.Thus, if the vehicle speed-acceleration characteristic is known, the condition must be met where   ,   = upper and lower limit values of acceleration potentially implemented by the vehicle.
The general scheme for obtaining integrals is the same.Then, the dynamic constraint has the form Another type of constraint determines boundary conditions of kinematic parameters.Thus, one can require, for example, that the initial (0) and final () values of the predicted acceleration and jerk must correspond to preset constant values  0() and  0() .That is, Thus, the complete set of parameters, bounds, and nonlinear constraints is

Simulation and Analysis
Trajectory.As mentioned above, the study object in this paper is only the AV speed mode, assuming that the trajectory has already been generated.We have described the planning technique in our previous articles, and here the road section's view with the form of the necessary maneuver is only given (Figure 5).We use the data of the Audi A4 3.2 FSI [23] to represent the AV.All the calculations are accomplished by using MATLAB tools [24].
Drive type impact.When the problem of distributing the speed parameters is purely kinematic, the output functions do not depend on the drive type and the dynamic vertical reactions.Using the presented models, let's build speed and acceleration forecasts for the AV with three types of drive: FWD, RWD, and AWD-to demonstrate their influence on the dynamics and nature of the optimized kinematic parameters (Figure 6).We assume that each wheel can realize the adhesion potential with the coefficient   = 0.8, which corresponds to the dry asphalt, and the other initial and physical conditions are retained.Also, let's demand that the longitudinal acceleration at the beginning and at the end will be   = 0 since the next stage may require decelerating.As seen, even in the case of high tire-road adhesion, the drive type influence is significant.Thus, the first trajectory's phase is associated with an intensive change in motion direction and a greater need for using the potential of lateral reactions.Given the distribution of vertical reactions during acceleration, the front-drive wheels' potential is reduced.This is well shown by shifting the primary acceleration intensity of the FWD AV to the second phase.The variant of RWD AV can use more traction even in the first trajectory's phase, where the restrictions are caused only by the lateral potential.In turn, the AWD version shows the best result in speed and acceleration since the traction capabilities on the wheels are realized equally.
Adhesion impact.Now let us turn off in the composed program code the option for considering the drive type and critical road-tire adhesion modes.This will give us an opportunity for evaluating the adhesion in the tires' contact patches needed for keeping the motion stability (Figure 7).For better clarity, let us specify the limiting adhesion coefficient   = 0.5.Note that, in this case, regardless of transmission type, the distributions of speeds and accelerations will be identical and the degrees of using the adhesion potential will be different.For all drive types, disregarding adhesion restrictions, both speeds and accelerations will be completely coincident (Figure 7a).At the same time, the adhesion properties under the wheels should be different and depending on the drive type to provide the preset kinematic parameters such as linear and angular accelerations.Here, we do not limit the adhesion for each wheel separately but use the generalized parameter   = 0.5 for the whole vehicle according to the condition of the critical sideslip speed.Nevertheless, as seen in Figure 7b, the indicated drives and wheels would exceed the limiting capabilities, that is, they would enter a slip state followed by a decrease in traction forces and a loss of sideslip resistance potential, which is the critical mode.As seen, both FWD wheels crossed the critical level, which could lead to a loss of motion stability.For RWD the inner rear wheel would be in the critical mode since it is underloaded relative to the outer wheel.And even the AWD's inner rear wheel would also move unstable due to the excess torque relative to the adhesion limit.Here, it is clearly seen how considering both the vertical reactions and drive type is vital in planning the AV speed mode.
Full set of constraints.Now, let us build the forecasts of speed, acceleration, jerk, and driveshaft torque, considering the drive type, redistribution of vertical reactions, and all constraints, including adhesions of all tires.We also include the torque vectoring drive 4WD-TV in the comparison since now it will significantly differ from the AWD.As a result, we get a picture (Figure 8), which unambiguously confirms the effect and influence of the extended AV model.Analyzing the distribution of speeds, it can be noted that if the two-axle drive variants (AWD, 4WD) provide intensities of characteristics close to symmetrical, then in the case of the singleaxle drive (FWD, RWD) intensities are asymmetric.This is due to the fact that the sensitivity of a single-axle drive to redistributing vertical reactions while maneuvering is higher than that of two-axle drive variants.It also affects the jerk nature: if two smooth half-waves are sufficient for two-axle drive variants, then single-axle drives require three half-waves with the worse function smoothness.This means that controlling the power aggregate of the AV with a single-axle drive will be a little more complicated than in the case of a two-axle drive.However, all the characteristics are strictly within the established limits.Thus, the speed range depending on the drive type under these conditions is 10%, the acceleration range is about 43%, and the driveshaft torque is about 38%.Based on the Audi A4 performance characteristics [23], the AV could have realized the maneuver by using the second or third gear.

Optimization Results Considering Drive Type
FWD. Figure 9a depicts the formation of traction forces on the front wheels.At the same time, despite the equality of drive torques, the reaction on the right front wheel is slightly less since the outer wheel travels a longer distance at a higher local speed and, accordingly, the rolling resistance is higher than that of the inner wheel.The differential input torque is shown in Figure 8d.The values of the rear wheels' reactions are negative, where the greater modulus belongs to the outer wheel, i.e., the forces are generated only by the rolling resistance.Due to the fact that the front inner wheel is the least loaded, its degree of using the adhesion potential grows most intensively (Figure 9b), while the use of the longitudinal adhesions by the rear wheels is identical.
Lateral reactions in Figure 9c are mainly distributed along the sides: the outer wheels, having greater vertical loads, respectively, create larger lateral reactions.Accordingly, the degree of using the lateral adhesion potential by the driving front wheels is slightly higher than that of the driven rear wheels (Figure 9d).
The distribution of vertical reactions (Figure 9e) shows the general trend of increasing the outer wheels' forces (right side) and decreasing the inner wheels' forces (left side).In the vicinity of the path's 50 m, a local trajectory straightening is observed, where the curvature is minimal, which is reflected by reducing the deviations of wheels' reactions.The shapes of the vertical forces' curves are completely reflected in the shapes of the lateral reactions.Figure 9f shows the degree of using combined wheels' adhesions, where neither peak value exceeds the established limit   = 0.5.At the same time, the traction is limited by the inner front wheel (red curve), as it is driving but least loaded and quickly exhausts the adhesion margin.The adhesion factors  of the rear-driven wheels are almost completely formed by the lateral components   .them are determined by the rolling resistance on the wheels, considering their loads and speeds.The outer wheels have higher velocities and loads, which somewhat reduces their traction forces.At the same time, each wheel realizes an individual degree of using the longitudinal adhesion   (Figure 11b) similar in shape but having different peak values, which are clearly less than the corresponding peak values for the cases of FWD and RWD.Lateral reactions and adhesion factor (Figure 11c, d) differ little from the corresponding curves for the front and rear drive.
Owing to the increased traction, the distribution of vertical reactions between the wheels is more significant.The combined adhesion factors in Figure 11f are identical in shape and similar to the lateral adhesion factors.The peak value, close to but not exceeding the set limit, characterizes the inner front wheel.Under the specified conditions, the front inner wheel limits the traction potential in the case of AWD.4WD-TV.This type of drive can be implemented either by intelligent control of mechanical transmission or by the direct independent drive of each wheel by an electric motor (electric vehicle).Figure 12a depicts that, unlike the variant of AWD, the traction forces are realized on each wheel by an individual law, and the degrees of using the longitudinal adhesion (Figure 12a) are almost the same on all wheels.That is, the graphs are conceptually opposite to AWD.
Since such an AV realizes the greatest speed and acceleration, the lateral forces (Figure 12c) also reach the greatest values among all the variants.However, the lateral adhesion factors (Figure 12d) are represented by a relatively narrow bunch of curves.
Particular attention should be paid to the curves of the combined adhesion factor (Figure 12f), which are similar in shape and peak values.In turn, they don't exceed the established limit and have some margin.Firstly, this means that the excessive potential can be used to increase the speed.Secondly, in this case, the limiting impact, obviously, belongs not to physical but to kinematic constraints (angular acceleration).Note that the effectiveness of such a torque distribution scheme rises with reducing the tire-road adhesion limit, although this requires intelligent control.Computational cost and efficiency.Note that the proposed technique is focused on implementing numerical methods to maximize both modeling quality and computational performance.The computational efficiency of this approach may be estimated with respect to the other two variants.The first one is based on a technique close to the most distributed, where the trajectory and speed are sequentially modeled by quintic curves, which we implemented in [25].The second approach is similar in its core to the proposed one but without considering the drive type and physical restrictions.Providing the same road section and initial conditions (curvature distribution, initial speed, acceleration, and 5-point Gaussian integration scheme), a series of computations are performed to predict trajectories and speed modes.For simulations, a device equipped with an Intel(R) Core(TM) i7-7500U CPU @ 2.7GHz, 2 cores, 8 GB RAM on the 64bit Windows 10 is used.The software environment is MATLAB R2022b with the basic optimization function fmincon.The time cost results are summarized in Table 2.As seen, the inverse approach is generally more sensitive since represented the optimization parameters almost directly unlike the conventional approach requiring first multiple differentiations.Thus, a fewer number of nodal parameters is needed, which contributes to finding a solution faster than direct (conventional) methods requiring a larger number of nodal parameters to ensure sufficient smoothness and differentiability.The main rapidity effect is achieved due to the preliminary calculated basis functions at the integration points and no need for recalculating them during iterations.The number of segments, the length of the finite elements, and the number of integration points (from 3 to 8) may adjust the balance between optimization quality and computational performance.We also note that the performance can be significantly increased by optimizing the MATLAB code, compiling to the C/C++ language, as well as using more efficient multi-core processors.

Concluding Remarks
The study aimed to improve the AV speed modes' planning technique by using the inverse vehicle dynamics model, the inverse optimization approach that includes physical limitations in the set of equality constraints and considering the drive type.It can be unambiguously noted that the efficiency and performance of the proposed approach are much higher than those of approaches based on pure kinematic AV models.Additionally, the influence of drive type is determined to be significant and desirably not neglected.The predicted parameters of the AV speed mode are provided not only smoothed but also more realistic and feasible in the subsequent motion tracking when performing a maneuver.The proposed approach and vehicle model can be used in developing advanced driver-assistance system (ADAS) systems providing motion stability by controlling and distributing driving torques over the wheels.Based on this research, the following comments are offered.
1.The AV model used in the optimization scheme was expanded to allow considering the redistribution of vertical reactions for an unsprung vehicle.This forms the central tendency of changing the load on a wheel, which affects its adhesion potential and ability to create longitudinal and lateral forces.2. The ratios of the lateral forces' moments are introduced separately for the front and rear wheels, allowing the redistribution of lateral reactions depending on the ratios of vertical reactions, lateral accelerations, and lateral speeds, reflecting the physics of the tire's sideslip process.The calculated lateral reactions showed the smoothness of changing and the resembling tendency regardless of the drive type.3. The physical parameters reflecting the adhesion state of each wheel are introduced into the set of constraints.Firstly, this allows replacing the AV critical speed based on the sideslip condition, where the traction potential is distributed between all wheels, with each wheel's critical rolling mode.Secondly, with this approach, both the motion stability and the traction properties are precisely ensured depending on the drive type.
4. Despite complicating the mathematical model and increasing the number of calculations, the optimization procedure's rapidity rises by 40-50% relative to the pure kinematic approach without redistributed adhesions, maintaining the quality of predicted reference curves-their unambiguity, smoothness, and differentiability. 5. Comparing the optimized output characteristics for various drive types (FWD, RWD, AWD, 4WD-TV), it can be found that a redistributed drive with individually controlled wheels (4WD-TV) ensures the most dynamic and stable vehicle speed mode simultaneously.A certain margin of subcritical adhesion in relation to other drive types characterizes it.This effect grows with lowering the adhesion limit, that is, in cases of moist, wet, and snowy roadways.6.The considered approach keeps the potential for improvement.Thus, as noted, the use of extended restrictions has a positive effect on the forecast's quality and performance.In this regard, it is possible to introduce additional physical restrictions, which may be the vehicle body's roll angle and the tire's lateral deformations and slip angles estimated in some approximation by lateral forces.Also, an open question remains regarding predicting the operating modes of hydromechanical power aggregate in the case of automatic transmission.It is necessary to plan the gear, the torque converter mode, and the engine mode, at which the needed driving torque will be provided at the proper angular speed and with minimal fuel consumption.This can be considered both by including the automatic transmission model in the general vehicle model and by the parallel optimization of operating the hydromechanical transmission in terms of output parameters such as torque and revolutions on the gearbox output shaft.

Figure 1 .
Figure 1.General scheme of the proposed approach.

Figure 2 .
Figure 2. Scheme of the vehicle kinematic model.

Figure 5 .
Figure 5. Trajectory of planned maneuver.Before optimizing the speed mode, let's determine the values of the weight coefficients   in Equation (83) and limit values.  = (1 3 1)  (95)

Figure 6 .
Figure 6.Distributions of longitudinal speed (a) and acceleration (b) for different drive types at the same conditions.

Figure 7 .
Figure 7. Distribution of speed and acceleration (a) and wheels' adhesion coefficients (b) at   = 0.5.

Table 2 .
Basic comparative parameters of computational efficiency.