Methods, Data Processing Device and Software for Determining a Deceleration Strategy for a Vehicle

Abstract

A method for determining a deceleration strategy for a vehicle includes determining a target position located on a route section ahead together with a target speed that the vehicle should have when it reaches the target position, wherein the target speed is lower than a current speed of the vehicle, and determining a reference trajectory for the vehicle that provides for reaching the target position at the target speed. The reference trajectory is determined based on analytical functions for at least two deceleration modes from a list including a coasting mode, an overrun mode, a recuperation mode and active braking. The analytical functions for each of the at least two deceleration modes indicate a speed and/or a distance travelled as a function of a time variable. Determining the reference trajectory includes determining one or more changeover times between the at least two deceleration modes.

Claims

1.-13. (canceled)

14. A method for determining a deceleration strategy for a vehicle, the method comprising: determining a target position that lies on a route section ahead of the vehicle together with a target speed at which the vehicle should be when reaching the target position, wherein the target speed is less than a current speed of the vehicle; and determining a reference trajectory for the vehicle that makes provision to reach the target position at the target speed, wherein the reference trajectory is determined based on analytical functions for at least two deceleration modes from a list including a sailing mode, a drag mode, a recuperation mode, and active braking, wherein the analytical functions for each of the at least two deceleration modes indicate a speed and/or a distance traveled as a function of a time variable, and wherein determining the reference trajectory includes determining one or more changeover times between the at least two deceleration modes based on the analytical functions.

15. The method according to claim 14, wherein the analytical functions are closed mathematical expressions with the time variable and multiple function parameters that characterize one or more types of driving resistances.

16. The method according to claim 15, wherein the multiple function parameters comprise one or more deceleration manipulated variables for one or more of the at least two deceleration modes.

17. The method according to claim 14, wherein one or more of the analytical functions that indicate distance traveled as a function of the time variable for the at least two deceleration modes are given in a following relationship: $s\left(t\right)=A\ln ({\tan \left(Bt+C\right)}^2+1+D,$ where s is distance, t is the time variable and A, B, C and D are constant function parameters.

18. The method according to claim 15, wherein one or more of the analytical functions that indicate distance traveled as a function of the time variable for the at least two deceleration modes are given in a following relationship: $s\left(t\right)=A\ln ({\tan \left(Bt+C\right)}^2+1+D,$ where s is distance, t is the time variable and A, B, C and D are constant function parameters.

19. The method according to claim 16, wherein one or more of the analytical functions that indicate distance traveled as a function of the time variable for the at least two deceleration modes are given in a following relationship: $s\left(t\right)=A\ln ({\tan \left(Bt+C\right)}^2+1+D,$ where s is distance, t is the time variable and A, B, C and D are constant function parameters.

20. The method according to claim 14, wherein one or more of the analytical functions that indicate speed as a function of the time variable for the at least two deceleration modes are given in a following relationship: $v\left(t\right)=E\tan \left(Ft+G\right),$ where v is speed, t is the time variable and E, F and G are constant function parameters.

21. The method according to claim 15, wherein one or more of the analytical functions that indicate speed as a function of the time variable for the at least two deceleration modes are given in a following relationship: $v\left(t\right)=E\tan \left(Ft+G\right),$ where v is speed, t is the time variable and E, F and G are constant function parameters.

22. The method according to claim 16, wherein one or more of the analytical functions that indicate speed as a function of the time variable for the at least two deceleration modes are given in a following relationship: $v\left(t\right)=E\tan \left(Ft+G\right),$ where v is speed, t is the time variable and E, F and G are constant function parameters.

23. The method according to claim 17, wherein one or more of the analytical functions that indicate speed as a function of the time variable for the at least two deceleration modes are given in a following relationship: $v\left(t\right)=E\tan \left(Ft+G\right),$ where v is speed, t is the time variable and E, F and G are constant function parameters.

24. The method according to claim 14, wherein the one or more changeover times are determined based on minimizing a cost function, wherein the cost function comprises a sum of multiple time integrals, wherein each of the multiple time integrals extends temporally over a respective deceleration phase that is assigned to a respective one of the at least two deceleration modes, wherein the one or more changeover times form upper or lower integral bounds of the integrals, and wherein the cost function under the multiple time integrals for the at least two deceleration phases has a respective time-dependent cost term that applies costs to a loss of kinetic energy that has taken place up to a first time in comparison to an initial kinetic energy that the vehicle has at a start of the deceleration.

25. The method according to claim 15, wherein the one or more changeover times are determined based on minimizing a cost function, wherein the cost function comprises a sum of multiple time integrals, wherein each of the multiple time integrals extends temporally over a respective deceleration phase that is assigned to a respective one of the at least two deceleration modes, wherein the one or more changeover times form upper or lower integral bounds of the integrals, and wherein the cost function under the multiple time integrals for the at least two deceleration phases has a respective time-dependent cost term that applies costs to a loss of kinetic energy that has taken place up to a first time in comparison to an initial kinetic energy that the vehicle has at a start of the deceleration.

26. A method for determining a deceleration strategy for a vehicle, the method comprising: determining a target position that lies on a route section ahead of the vehicle together with a target speed at which the vehicle should be when reaching the target position, wherein the target speed is less than a current speed of the vehicle; and determining a reference trajectory for the vehicle that makes provision to reach the target position at the target speed, wherein the reference trajectory has at least two deceleration phases in which the vehicle is operated in each case in a different deceleration mode from a list including a sailing mode, a drag mode, a recuperation mode, and active braking, wherein the determining the reference trajectory includes determining one or more changeover times between the at least two deceleration modes, wherein the one or more changeover times are determined based on minimizing a cost function, wherein the cost function comprises a sum of multiple time integrals, wherein each of the multiple time integrals extends temporally over a respective one of the deceleration phases, wherein the one or more changeover times form upper or lower integral bounds of the multiple time integrals, and wherein the cost function under the multiple time integrals for the at least two deceleration phases has a respective time-dependent cost term that applies costs to a loss of kinetic energy that has taken place up to a first time in comparison to an initial kinetic energy that the vehicle has at a start of the deceleration.

27. The method according to claim 26, wherein the time-dependent cost term comprises a square of a difference between a kinetic energy at the first time and the initial kinetic energy.

28. The method according to claim 26, wherein, for purposes of the determining of the reference trajectory, one or more deceleration manipulated variables that are active in the at least two deceleration modes are assumed to be constant over time during a respective deceleration phase in which a respective deceleration mode is active.

29. The method according to claim 27, wherein, for purposes of the determining of the reference trajectory, one or more deceleration manipulated variables that are active in the at least two deceleration modes are assumed to be constant over time during a respective deceleration phase in which a respective deceleration mode is active.

30. The method according to claim 26, wherein the reference trajectory has a sailing phase, in which the vehicle drives in sailing mode, and after the sailing phase at least one further deceleration phase, in which the vehicle drives in drag mode or recuperation mode or is actively braked.

31. The method according to claim 26, further comprising: determining a target trajectory for the vehicle in dependence on the reference trajectory; and generating an actuator input for longitudinal guidance of the vehicle in dependence on the target trajectory.

32. A data processing device configured to carry out a method according to claim 14.

33. A data processing device configured to carry out a method according to claim 26.

34. A non-transitory computer-readable medium storing commands that, when executed by a data processing device, cause the data processing device to carry out a method according to claim 14.

35. A non-transitory computer-readable medium storing commands that, when executed by a data processing device, cause the data processing device to carry out a method according to claim 26.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0059] FIG. 1 illustrates, by way of example and schematically, a deceleration-relevant target situation ahead of a vehicle in the form of a road sign indicating a speed limit.

[0060] FIG. 2 illustrates, by way of example and schematically, a method sequence for determining a deceleration strategy for a vehicle according to a first aspect of the invention.

[0061] FIG. 3 illustrates, by way of example and schematically, a method sequence for determining a deceleration strategy for a vehicle according to a second aspect of the invention.

[0062] FIG. 4 illustrates, by way of example and schematically, a model of the vehicle longitudinal guidance as a switching system.

[0063] FIG. 5 illustrates time-dependent and location-dependent reference trajectory variables that have been determined using a two-dimensional parametric optimization.

[0064] FIG. 6 illustrates, by way of example and schematically, the determination of changeover times based on a speed/distance graph.

[0065] FIG. 7 illustrates, by way of example and schematically, a system for controlling the longitudinal guidance of a vehicle.

DETAILED DESCRIPTION OF THE DRAWINGS

[0066] FIG. 1 schematically shows an exemplary driving situation involving a vehicle 1 that is currently, at a starting time to, located at a starting position so and is moving constantly at a starting speed of v.sub.0=180 km/h.

[0067] FIGS. 2 and 3 each show a method sequence for determining a deceleration strategy for a vehicle.

[0068] The method sequences according to FIGS. 2 and 3 are explained below by way of example with reference to the driving situation illustrated in FIG. 1. In this case, a data processing device, for example, in the form of one or more controllers of the vehicle 1, carries out specific steps, which are consistent with the two method sequences according to FIG. 2 and FIG. 3, in real time.

[0069] In a first step, which the method sequences according to FIG. 2 and FIG. 3 have in common, a target position s.sub.f that lies on a route section ahead, together with a target speed v.sub.1 of 80 km/h at which the vehicle 1 should be when it reaches the target position s.sub.f, are determined. By way of example, the target position s.sub.f, together with the target speed v.sub.f, is taken from a digital map in the form of information about a speed limit, wherein GPS data may be used to determine that the target position s.sub.f is 700 m away from the current vehicle position s.sub.0. Since the target speed v.sub.f is less than the starting speed v.sub.0, it is necessary to slow the vehicle 1 down. A suitable deceleration strategy for the longitudinal guidance of the vehicle therefore needs to be ascertained.

[0070] In the exemplary embodiment, the vehicle 1 is a vehicle having an internal combustion engine and that supports a sailing mode, a drag mode (engine braking) and active braking as different deceleration modes. The deceleration strategy to be determined should accordingly indicate the time for which or the distance over which the vehicle 1 should sail, drag and finally be actively braked in order to reach the target position s.sub.f at the target speed v.sub.f.

[0071] When ascertaining the deceleration strategy, a road incline, illustrated schematically in FIG. 1, with an angle of incline should be taken into consideration, since the steepness of the road contributes to decelerating the vehicle 1. In the exemplary embodiment, the angle of incline is assumed to be constant over the entire distance from the starting position to the target position for the sake of improved understanding. However, it is also possible to take into consideration angles of incline that may be different in sections.

[0072] In a further step, a reference trajectory is determined for the vehicle 1, the reference trajectory being compatible with reaching the target position s.sub.f at the target speed v.sub.f.

[0073] One possible procedure for ascertaining the reference trajectory is explained in more detail below. This exemplary embodiment is in accordance with the respective second method step according to both FIG. 2 and FIG. 3.

[0074] As a basis for ascertaining the reference trajectory, the vehicle 1, together with the constraints described above, is modeled as a switching system. This is illustrated schematically in FIG. 4.

[0075] In this case, the vehicle longitudinal movement is described by a state vector x(t) dependent on a time variable t, said state vector comprising a path s(t), a speed v(t) and an acceleration a(t) as vector components. The state vector satisfies the starting condition x(t.sub.0)=x.sub.0, wherein the starting state vector x.sub.0 comprises the starting variables s.sub.0 and v.sub.0 already introduced further above.

[0076] It is specified in the exemplary embodiment that the vehicle 1 is in a sailing mode in a first deceleration phase, namely during a time interval from the starting time t.sub.0 up to a first changeover time t.sub.s1. In this case, the vehicle 1 is subject, in the first deceleration phase, to system dynamics, which are expressed formally in the equation {dot over (x)}.sub.sail=f.sub.sail(x).

[0077] At the first changeover time t.sub.s1, a changeover takes place from sailing mode to drag mode. Drag mode lasts during a time interval from the first changeover time t.sub.s1 up to the second changeover time t.sub.s2. During this second deceleration phase, the vehicle 1 is subject to system dynamics in accordance with the equation {dot over (x)}.sub.drag=f.sub.drag(x).

[0078] At the second changeover time t.sub.s2, a changeover takes place from drag mode to an active braking mode. Active braking lasts during a time interval from the second changeover time t.sub.s2 up to the end time t.sub.r of the overall deceleration process. During this third deceleration phase, the vehicle 1 is subject to system dynamics in accordance with the equation {dot over (x)}.sub.brake=f.sub.brake(x, u), with the braking manipulated variable u.

[0079] At the changeover times, the continuity conditions x.sub.sail(t.sub.s1)=x.sub.drag(t.sub.s1) and x.sub.drag(t.sub.s2)=x.sub.brake(t.sub.s2) apply.

[0080] Proceeding from this, the reference trajectory, together with the changeover times, may be determined as a solution to the following hybrid optimal control problem in the course of a parametric optimization:

[00005]$\begin{array}{cc}\min \text{?}& J=\text{?}\end{array}\left(w_u{u}^2+{w_{\mathrm{kin}}\left(E_{\mathrm{kin}}\left(v\right)-E_{\mathrm{kin}}\left(v_0\right)\right)}^2\right)\mathrm{dt}$$\begin{array}{cc}\begin{array}{cc}s.t.& {\stackrel{.}{x}}_{\mathrm{sail}}\left(t\right)=\end{array}{f}_{\mathrm{sail}}\left(x_{\mathrm{sail}},u\right)& t\end{array}[t_0,t_{s_1})$$\begin{array}{cc}{\stackrel{.}{x}}_{\mathrm{drag}}\left(t\right)=f_{\mathrm{drag}}\left(x_{\mathrm{drag}},u\right)& t\end{array}[t_{s_1},t_{s_2})$$\begin{array}{cc}{\stackrel{.}{x}}_{\mathrm{brake}}\left(t\right)=f_{\mathrm{brake}}\left(x_{\mathrm{brake}},u\right)& t\end{array}\left[t_{s_2},t_f\right]$$u=\{\begin{array}{cc}0,& [t_0,t\text{?})\\ a_{\mathrm{drag}}\left(t\right),& [t\text{?},t\text{?})\\ u_{\mathrm{brake}}\left(t\right),& \left[t_{s_2},t_f\right]\end{array}$$x_{\mathrm{sail}}\left(t_0\right)=x_0$$x_{\mathrm{sail}}\left(t_{s_1}\right)=(x_{\mathrm{drag}}\left(t\text{?}\right);x_{\mathrm{drag}}\left(t\text{?}\right)=x_{\mathrm{brake}}\left(t\text{?}\right)$$x_{\mathrm{brake}}\left(t_f\right)=x_f$$0u_{\mathrm{brake}}{u}_{\max }$$t_0{t}_{s_1}t\text{?}{t}_f$$\text{?}\text{indicates text missing or illegible when filed}$

[0081] In accordance therewith, the changeover times t.sub.s1, t.sub.s2 and the end time t.sub.r need to be determined such that the cost function J is minimal. In this case, the starting, continuity and end conditions given above and the restrictions given above with regard to an order of the different times must be complied with. There is also a restriction on the braking manipulated variable u.sub.brake whereby this must not exceed a maximum braking deceleration u.sub.max.

[0082] The integral of the cost function J may be written as a sum of three sub-integrals for the three deceleration phases with the corresponding starting, end and changeover times as temporal interval bounds. In this case, the deceleration manipulated variable, given generally as u in the above integral expression, for the individual sub-integrals needs to be replaced specifically in accordance with the rule given above. Accordingly, no deceleration manipulated variable acts in the sailing phase (u=0), the deceleration manipulated variable a.sub.drag(t) acts in the drag phase and another deceleration manipulated variable u.sub.brake(t) in turn acts in the subsequent active braking phase.

[0083] The cost function J under the integrals has a time-dependent cost term for each deceleration phase that applies costs to a loss of kinetic energy E.sub.kin(v) that has taken place up to the time in question in comparison to an initial kinetic energy E.sub.kin(v.sub.0) that the vehicle has or had at the start of the deceleration. Specifically, this cost term comprises a square of the difference between the kinetic energy E.sub.kin(v) at the time in question and the initial kinetic energy E.sub.kin(v.sub.0), with a constant pre-exponential factor w.sub.kin.

[0084] The equation for the system dynamics in sailing mode may be written as follows:

[00006]$\begin{array}{c}{\stackrel{.}{x}}_{\mathrm{sail}}=\left[\begin{array}{c}{s}_{\mathrm{sail}}\text{?}\\ v_{\mathrm{sail}}\text{?}\end{array}\right]\\ =\left[\begin{array}{c}{v}_{\mathrm{sail}}\\ -\frac{A_f{c}_d}{2m}{v}_{\mathrm{sail}}^2-g\sin \left(\right)-c_{r}g\cos \left(\right)\end{array}\right]\\ =\left[\begin{array}{c}{v}_{\mathrm{sail}}\\ -a_{\mathrm{air}}{v}_{\mathrm{sail}}^2-a_{\mathrm{sail}}\end{array}\right]\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$ [0085] in which a.sub.air=( A.sub.f c.sub.d/(2 m) denotes an air resistance coefficient, c.sub.r denotes a rolling resistance coefficient and g represents gravitational acceleration. The following has been replaced in the last step:

[0086] a.sub.sail=g sin()c.sub.r g cos(), with the road angle of incline , which is assumed here to be constant over the entire deceleration distance.

[0087] The equation for the system dynamics in drag mode may be written as follows:

[00007]$\begin{array}{c}{\stackrel{.}{x}}_{\mathrm{drag}}=\left[\begin{array}{c}{\stackrel{.}{s}}_{\mathrm{drag}}\\ {\stackrel{.}{v}}_{\mathrm{drag}}\end{array}\right]\\ =\left[-\begin{array}{c}{v}_{\mathrm{drag}}\\ \frac{A_f{c}_d}{2m}{v}_{\mathrm{drag}}^2-g\sin \left(\right)-c_{r}g\cos \left(\right)-a_{\mathrm{drag}}\end{array}\right]\\ =\left[\begin{array}{c}{v}_{\mathrm{drag}}\\ -a_{\mathrm{air}}{v}_{\mathrm{drag}}^2-a_{\mathrm{sail}}-a_{\mathrm{drag}}\end{array}\right]\end{array}$ [0088] in which a.sub.drag denotes a drag deceleration caused by the effect of engine braking.

[0089] The equation for the system dynamics during active braking may be written as follows:

[00008]$\begin{array}{c}{\stackrel{.}{x}}_{\mathrm{brake}}=\left[\begin{array}{c}{s}_{\mathrm{brake}}\text{?}\\ v_{\mathrm{brake}}\text{?}\end{array}\right]\\ =\left[-\begin{array}{c}{v}_{\mathrm{sail}}\\ \frac{A_f{c}_d}{2m}{v}_{\mathrm{drag}}^2-g\sin \left(\right)-c_{r}g\cos \left(\right)-a_{\mathrm{drag}}\end{array}\right]\\ =\left[\begin{array}{c}{v}_{\mathrm{brake}}\\ -a_{\mathrm{air}}{v}_{\mathrm{brake}}^2-a_{\mathrm{sail}}-u_{\mathrm{brake}}\end{array}\right]\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$ [0090] in which the deceleration manipulated variable u.sub.brake denotes a braking deceleration brought about by active braking.

[0091] For the further calculations within the scope of this exemplary embodiment, it is assumed for simplicity that both the deceleration manipulated variable in drag mode a.sub.drag and the deceleration manipulated variable during active braking u.sub.brake are constant over time during the drag phase, respectively braking deceleration phase. This enables simple and particularly computationally efficient ascertaining of the reference trajectory through a parametric optimization with regard to the switching times. Cf. Xuping, Xu; Antsaklis, P. J.: Optimal Control of Switched Systems based on Parameterization of the Switching Instants. IEEE Transactions on Automatic Control, 2004.

[0092] It should however be borne in mind that a solution to the hybrid optimal control problem is also possible in principle without these simplifying assumptions with regard to the deceleration manipulated variables. However, in contrast, for an analytical solution, other approximations are then necessary, such as simplifying the cost function and linearizing the deceleration modes. A numerical solution leads to a boundary value problem that is able to be solved at best with a great deal of computational effort. Cf. Pakniyat, Ali; Caines, Peter E.: On the Hybrid Minimum Principle. IEEE Transactions on Automatic Control, 2020.

[0093] Working on the assumption of constant deceleration manipulated variables, it is possible to give, as analytical solutions to the differential equations given above for the sailing and drag phase, the following analytical functions for a respective speed and a respective distance traveled as a function of the time variable t:

[00009]$s_{\mathrm{sail}}\left(t\right)=-\frac{1}{2a_{\mathrm{air}}}\ln \left({\tan \left(b_{2}t+c_1\right)}^2+1\right)+c_2$$v_{\mathrm{sail}}\left(t\right)=b_1\tan \left(b_{2}t+c_1\right)$$s_{\mathrm{drag}}\left(t\right)=-\frac{1}{2a_{\mathrm{air}}}\ln \left({\tan \left(b_{4}t+c_3\left(t_{s_1}\right)\right)}^2+1\right)+c_4\left(t_{s_1}\right)$$v_{\mathrm{drag}}\left(t\right)=b_3\tan \left(b_{4}t+c_3\left(t_{s_1}\right)\right)$

[0094] In this case, the constant function parameters b.sub.1, b.sub.2, b.sub.3, b.sub.4 are derived as follows from the variables a.sub.sail, a.sub.drag and a.sub.air introduced further above:

[00010]$b_1=\sqrt{\frac{a_{\mathrm{sail}}}{a_{\mathrm{air}}}}$$b_2=-\sqrt{a_{\mathrm{air}}{a}_{\mathrm{sail}}}$$b_3=\sqrt{\frac{a_{\mathrm{sail}}+a_{\mathrm{drag}}}{a_{\mathrm{air}}}}$$b_4=-\sqrt{a_{\mathrm{air}}\left(a_{\mathrm{sail}}+a_{\mathrm{drag}}\right)}$

[0095] The function parameters c.sub.1 and c.sub.2 are determined such that the starting condition x.sub.sail(0)=x.sub.0 is satisfied. The function parameters c.sub.3 and c.sub.4 are determined from the continuity condition x.sub.sail(t.sub.s1)=x.sub.drag(t.sub.s1).

[0096] With regard to the differential equation given above for x.sub.brake, it is able to be mathematically proven that there is a real braking manipulated variable u.sub.brake that the system converts, in a finite time t.sub.f, into the desired end state with the target speed v.sub.f and the target position s.sub.f.

[0097] Specifically, the following analytical expressions arise for the end time t.sub.r and the braking manipulated variable u.sub.brake, which depend, in terms of parameters, on the changeover times t.sub.s1, t.sub.s2:

[00011]$u_{\mathrm{brake}}\left(t_{s_1},t_{s_2}\right)=\frac{a_{\mathrm{air}}(\text{?}\left(t_{s_2}\right)\exp \left[-2a_{\mathrm{air}}(\text{?}-\text{?}\left(t_{s_2}\right)\right]-\text{?}}{1-\exp \left[-2a_{\mathrm{air}}\left(\text{?}-\text{?}\left(t_{s_2}\right)\right)\right]}-a_{\mathrm{sail}}$$t_f\left(t_{s_1},t_{s_2}\right)=\frac{1}{\sqrt{a_{\mathrm{air}}\left(\text{?}+\text{?}\right)}}\left(\arctan \left(\frac{a_{\mathrm{air}}\text{?}\left(\text{?}\right)}{\sqrt{u_{\mathrm{brake}}+a_{\mathrm{sail}}}}\right)-\arctan \left(\frac{a\text{?}}{\sqrt{u_{\mathrm{brake}}+a_{\mathrm{sail}}}}\right)\right)$$\text{?}\text{indicates text missing or illegible when filed}$

[0098] The following analytical functions may be given as an analytical solution to the differential equation given above for the braking phase:

[00012]$s_{\mathrm{brake}}\left(t\right)=\frac{1}{2a_{\mathrm{air}}}\ln \left({\tan \left(b_{6}t+c_5\left(t_{s_1},t_{s_2}\right)\right)}^2+1\right)+c_6\left(t_{s_1},t_{s_2}\right)$$v_{\mathrm{brake}}\left(t\right)=b_5\tan \left(b_{6}t+c_5\left(t_{s_1},t_{s_2}\right)\right)$

[0099] In this case, the constant function parameters b.sub.5 and b.sub.6 depend on the variables a.sub.sail, a.sub.air and u.sub.brake as follows:

[00013]$\begin{array}{cc}& b_5=\sqrt{\frac{a_{\mathrm{sail}}+u_{\mathrm{brake}}\left(t_{s_1},t_{s_2}\right)}{a_{\mathrm{air}}}}\end{array}$$\begin{array}{cc}& b_6=\sqrt{a_{\mathrm{air}}(a_{\mathrm{sail}}+u_{\mathrm{brake}}\left(t_{s_1},t_{s_2}\right)}\end{array}$

[0100] The other function parameters c.sub.5 and c.sub.6 are determined from the continuity condition x.sub.drag(t.sub.s2)=x.sub.brake(t.sub.s2).

[0101] Based on these preliminary observations, the cost function J may then be parameterized with regard to the changeover times t.sub.s1 and t.sub.s2:

[00014]$J=\frac{1}{2}\text{?}w\text{?}{\left(v_{\mathrm{sail}}^2\left(t\right)-v\text{?}\right)}^2\mathrm{dt}+\frac{1}{2}{}_{t_{s_1}}^{t_{s_2}}\left(w\text{?}{a}_{\mathrm{drag}}^2+w\text{?}{\left(a_{\mathrm{drag}}^2\left(t\right)-v_0^2\right)}^2\right)\mathrm{dt}+\frac{1}{2}\text{?}\left(w\text{?}{u}_{\mathrm{brake}}^2+w\text{?}{\left(v_{\mathrm{brake}}^2\left(t\right)-v_0^2\right)}^2\right)\mathrm{dt}=J\left(t_{s_1},t_{s_2}\right)$$\text{?}\text{indicates text missing or illegible when filed}$

[0102] Proceeding from this, a parametric optimization problem for the transformed variables

[00015]$t_{\mathrm{sail}}=t_{s_1}-t_0$$t_{\mathrm{drag}}=t_{s_2}-t_{s_1}$ [0103] may be formulated as follows:

[00016]$\begin{array}{cc}\underset{t_{\mathrm{sail}},t_{\mathrm{drag}}}{\min }& J\left(t_{\mathrm{sail}},t_{\mathrm{drag}}\right)\\ s.t.& u_{\mathrm{brake}}\left(t_{\mathrm{sail}},t_{\mathrm{drag}}\right)u_{\max }\\ & 0t_{\mathrm{sail}}\\ & 0t_{\mathrm{drag}}\end{array}$

[0104] This parametric optimization problem is able to be solved numerically using computationally efficient standard methods, that is to say t.sub.s1 and t.sub.s2 are able to be determined numerically such that the cost function J is minimized. If the changeover times t.sub.s1 and t.sub.s2 are known, the desired reference trajectory is known based on the analytical expressions given above for path and speed in the three deceleration phases.

[0105] FIG. 5 illustrates a solution for the parameters s.sub.0=0, v.sub.0=180 km/h, s.sub.f=700 m, v.sub.f=80 km/h, =2, a.sub.drag=0.4 m/s.sup.2, w.sub.u=1 and w.sub.kin=1e.sup.6.

[0106] In this case, the upper three graphs (FIG. 5(a)) show profiles of path s in meters, speed v in km/h and the deceleration manipulated variable u in m/s.sup.2 each as a function of time t in seconds. Two vertical dashed lines mark the optimal changeover times t.sub.s1 (dashed line on the left) and t.sub.s2 (dashed line on the right).

[0107] In the lower graph (FIG. 5(b)), speed v in km/h is plotted against path traveled s in meters, wherein two vertical dashed lines mark the positions at which the vehicle 1 will be at the changeover times t.sub.s1 and t.sub.s2.

[0108] The reference trajectory planning thus decides, in this exemplary embodiment, that it is optimal to sail for approximately the first 410 m, and then to drag for approximately 150 m, and finally to actively brake the last approximately 140 m safely and comfortably with a deceleration of approximately 1.5 m/s.sup.2 in order to accurately comply with the new speed limit of 80 km/h.

[0109] If the magnitude of the braking deceleration u.sub.brake is fixedly specified from the outset, it is possible to further reduce the required computing time to a great extent by reducing the optimization problem to a parametric one-dimensional optimization. Working on this assumption concerning the braking deceleration u.sub.brake, it is possible to eliminate one free variable and for example to express the second changeover time t.sub.s2 as a function of the first changeover time t.sub.s1, that is to say t.sub.s2=t.sub.s2(t.sub.s1). The optimization problem may then be formulated as follows:

[00017]$\begin{array}{cc}\underset{t_{s_1}}{\min }& J\left(t_{s_1}\right)\\ s.t.& 0t_{s_1}{t}_{s_1,\max }\end{array}$

[0110] FIG. 6 graphically illustrates, with reference to a speed/path graph for the reference trajectory to be ascertained, how, in the case of a fixedly specified braking deceleration u.sub.brake (here: u.sub.brake=1.5 m/s.sup.2), it is possible to determine an upper limit t.sub.s1,max for the first changeover time and the desired optimal first and second changeover times t.sub.s1,opt and t.sub.s2,opt(t.sub.s1) from the known curve profiles for the path-dependent speeds v(s) in the three deceleration phases. In this case, t.sub.s1,max arises as the point of intersection between the speed curve starting at the starting position so with the starting speed v.sub.0 for sailing mode (dashed curve in FIG. 6) and the speed curve ending at the target position s.sub.f with the target speed v.sub.f for active braking mode (unbroken curve in FIG. 6). The optimal first changeover time t.sub.s1,opt may then be determined with little numerical computational effort or even analytically as a solution to the one-dimensional optimization problem given above. Finally, the optimal second changeover time t.sub.s2,opt arises as the point of intersection between the speed curve for drag mode (lower dashed curve in FIG. 6) that intersects the speed curve for sailing mode at the optimal first changeover time t.sub.s1,opt and the speed curve for active braking mode.

[0111] FIG. 7 shows, by way of example and schematically, a system 10 for controlling the longitudinal guidance of a vehicle, such as the vehicle 1. The system 10 comprises a reference trajectory planner 101, a target trajectory planner 102 and a control loop 103.

[0112] The reference trajectory planner 101 is configured, in the manner described above, to ascertain a reference trajectory for decelerating the vehicle 1 to a target speed v.sub.1 at a target position s.sub.r as a solution to a hybrid optimal control problem as a function of an initial starting state x.sub.0, a target state x.sub.1 and other parameters (for example concerning driving resistances). The reference trajectory indicates a respective temporal profile of a reference path s.sup.ref, a reference speed v.sup.ref and a reference acceleration a.sup.ref. A temporal planning horizon of the reference trajectory planner 101 extends from the starting time to up to the end time t.sub.r, at which the vehicle 1 will reach the target position s.sub.1 in accordance with the reference trajectory.

[0113] The target trajectory planner 102 is configured, as a function of the reference trajectory, to plan a target trajectory that indicates a respective profile of a target path s.sup.soll, a target speed v.sup.soll, a target acceleration a.sup.soll and a target jerk j.sup.soll. The target trajectory may be ascertained for example as part of a model predictive control process by way of a quadratic optimization with regard to further criteria, concerning for example safety and comfort. In this case, a temporal planning horizon of the target trajectory planner 102 may extend in each case over a few seconds, such as 6 seconds, into the future. N interpolation points may be provided for the target trajectory planning at shorter time intervals of for example 0.2 seconds, in a manner distributed across the planning horizon.

[0114] The reference trajectory may be taken into consideration as a reference when calculating the target trajectory. By way of example, in the optimization, the variables s, v, a and j may be determined so as to minimize the following cost function:

[00018]$J={.Math.}_{k=0}^{N}l\left(k\right)$$l\left(k\right)={w_s\left(s-s^{\mathrm{ref}}\right)}^2+{w_v\left(v-v^{\mathrm{ref}}\right)}^2+{w_a\left(a-a^{\mathrm{ref}}\right)}^2+w_j{j}^2+w_u{u}^2$ [0115] where k is an index for a respective interpolation point along the planning horizon of the target trajectory planner 102.

[0116] It is also possible to link the target trajectory planning to the reference trajectory planning in such a way that the time-dependent speed profile v.sup.ref in accordance with the reference trajectory represents an upper limit v.sub.max in the course of the optimization in order to determine the target speed v.sup.soll. According to one variant embodiment, the upper limit does not have to be a hard limit here, but rather may be softened for example by way of a slack variable .sub.v,max. In this variant embodiment, the cost function to be minimized, together with the softened restriction for the speed, may be given for example as follows:

[00019]$J={.Math.}_{k=}^N\text{?}l\left(k\right)$$l\left(k\right)=-w\text{?}s+w_a{a}^2+w_j{j}^2+w_u{u}^2$$0v{v}_{\max }+{}_{v,\max }$$\text{?}\text{indicates text missing or illegible when filed}$

[0117] As a further possible alternative, the reference acceleration may be used as a lower limit amin for the target acceleration to be determined, wherein, according to one variant embodiment, the restriction may be softened by way of a slack variable .sub.v,min. In this variant embodiment, the cost function to be minimized, together with the softened restriction for the acceleration, may be given for example as follows:

[00020]$J={.Math.}_{k=0}^{N}l\left(k\right)$$l\left(k\right)=w\text{?}s+{w_v\left(v-v\text{?}\right)}^2+w_a{a}^2+w_j{j}^2+w_u{u}^2$$a_{\min }-{}_{v,\min }a$$\text{?}\text{indicates text missing or illegible when filed}$

[0118] The control loop 103 comprises a longitudinal guidance actuator system, such as a drive and a braking system, of the vehicle 1. By way of example, in the course of the control loop 103, the target acceleration a.sup.soll may be used as an actuator input for the longitudinal guidance actuator system. However, it is also possible for the control loop to comprise one or more controllers subordinate to the trajectory planning, such as a trajectory follow-up controller, which generate a specific actuator input for the longitudinal guidance actuator system as a function of the target trajectory and of information about current measured interfering variable influences, for instance in the form of a specification of a drive torque to be implemented by the longitudinal guidance actuator system or an acceleration to be implemented by the longitudinal guidance actuator system.

[0119] In this case, the system 10 carries out the longitudinal guidance control cyclically by feeding back in each case a current state (for example, measured by way of an odometer), such as a current actual speed or a current actual acceleration or an actual distance traveled, to the reference trajectory planner 101. This may then calculate a respective updated reference trajectory with the fed-back current state x.sub.0 as a starting state.