Profiles for Acceleration Transition

The acceleration transition profiles are used to generate a smooth change from one acceleration to another. They precede the appearance of a constant acceleration, as is shown in Figure4.1. They are expressed in the form of acceleration cubic polynomials.

The two boundary conditions in addition to those listed in Equation 4.14 required in order to solve the unknown parameters are:

α1 = α(tf)

t_f = k_trans|α₁−α₀|. (4.15)

wherektransis a predefined constant which can be identified by examining the actual duration that is necessary for a vehicle system (i.e., vehicle dynamics plus controller) to change from one acceleration to another. With the help of these boundary conditions, the closed-form solutions for p2 and p3 can be readily obtained.

The physical capability of the vehicle defines the upper and lower boundaries of possible accelerations and speeds. The limits are denoted as αmax, αmin, vmax and v_min respectively. Accordingly, checking the feasibility of the acceleration profiles means verifying whether there are accelerations exceedingα_max or falling belowα_minalong the profiles. The same applies to the feasibility examination of the speed profiles resulting from the acceleration profiles. If the profiles turn out to be infeasible in terms of either acceleration limit or speed limit or both, they should be discarded immediately without further evaluation. As the vehicle is not allowed to perform reverse manoeuvres in the proposed planner,vmin is set to zero.

The jerk and acceleration polynomials are the derivatives of the acceleration and speed polynomials respectively. This specific relationship can be exploited to assist the feasibility validation. For example, the acceleration polynomial gets its local extrema at the zeros of the jerk polynomial, and the time coordinate where the jerk polynomial gets its extremum is the inflection point of the acceleration polynomial. Figure 4.2 demonstrates this phenomenon. The same relationship exists between the acceleration and speed polynomials.

The typical acceleration cubic polynomials and their corresponding speed and jerk profiles are demonstrated in Figure 4.1. The fact that the endpoints of the jerk profile has a value of zero determines that the boundaries of the acceleration profile are both

Speed(m/s)

Figure 4.1: Acceleration Profile Examples. The middle, lower and upper graphs show the acceleration profiles and the corresponding jerk and speed changes respectively. On the middle graph, the first bold curve transits the constant acceleration of 0m/s² to another constant acceleration of 4m/s². On the same graph, the second bold curve transfers the vehicle system to a state where v= 30m/s and α= 0m/s².

local extrema. Since the cubic polynomial can have a maximum of two local extrema, the acceleration profile can only fall between the two local extrema, as is shown in Figure4.2. As a result, the acceleration increases or decreases monotonically within the profile, which means that the acceleration profile is guaranteed to be valid as long as both of its boundaries are within the acceleration limits.

Now it is time to examine the speed profiles. There are generally three cases for consideration. The profiles shown in Figure 4.3(a), Figure 4.3(b), Figure 4.3(c) and Figure4.3(d)belong to the first case. The feature of this case is that there are no zeros along the acceleration profile except for their endpoints. It can be concluded therefore that there are no local extrema along the speed profiles with their endpoints discounted.

Consequently, the speed can only decrease or increase monotonically within the profiles, which indicates that the feasibility of the speed profiles can be guaranteed if both of its endpoints fall within the speed limits.

The second case, as is shown in Figure 4.3(e), is more complicated than the first.

As there is one zero along the acceleration profile, a local extremum cannot be avoided along the speed profile, which renders the check relying on the boundary conditions alone insufficient. As the acceleration goes from positive to negative, the local extremum along the speed profile is a local maximum. In the proposed planner, the trajectory with

Acceleration(m/s²)

Figure 4.2: Relationship of the jerk and acceleration polynomials. The green points on the jerk curve are the zeros of the jerk polynomial. They correspond to the green points on the acceleration curve where the acceleration polynomial gets its local extrema. The jerk quadratic polynomial gets its extremum at the red point on its curve which is the inflection point of the acceleration cubic polynomial.

speeds exceedingvmax is not regarded as infeasible but subject to certain punishments.

The reason is twofold. For one thing, v_max is usually set to be the speed limit of the current road which is well below the actual potential of the vehicle. Even when vmax is set according to the physical limit of the vehicle, it is always a conservative estimation compared to the maximum capability of the vehicle. For another thing, the discrepancy between vmax and the possible speeds exceeding vmax turns out to be very small in practice if the boundaries of the speed profile are within the limits. As a result, in terms of feasibility validation, the speed profile with a local maximum is only subject to the boundary checking. The trajectory with speeds beyond the speed limits is punished during the evaluation of the dynamic cost.

Figure 4.3(f)gives an example of the third case where the local extremum is a local minimum. As speeds less than zero render a trajectory infeasible, it is necessary to check whether this local minimum is negative. As the time coordinates of the local extrema of the speed profile are the zeros of the acceleration profile, to calculate the exact value of the local minimum in question on the speed profile would mean solving for the corresponding root of the acceleration profile equation which is a cubic polynomial equation. Although there are closed-form solutions of the roots of cubic polynomial equations (cf. [83]), it is unnecessarily complicated to implement it in the planner. As

0

(a) Acceleration increases in the negative ac-celeration zone without crossing the time axis.

0

(b) Acceleration increases in the positive accel-eration zone without crossing the time axis.

0

(c) Acceleration decreases in the positive accel-eration zone without crossing the time axis.

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(d) Acceleration decreases in the negative ac-celeration zone without crossing the time axis.

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(e) Acceleration decreases from the positive ac-celeration zone to the negative one.

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(f) Acceleration increases from the negative ac-celeration zone to the positive one.

Figure 4.3: The typical acceleration cubic polynomials and their corresponding speed and jerk profiles.

a result, approximate methods are applied to estimate the minimum speed vmin, which are illustrated as follows:

• Given the acceleration profiles as is shown in Figure4.4(a), Figure4.4(b)and Fig-ure4.4(c), lett₀, t₁, t_min refer to the time coordinates wherev₀, v₁, v_min take place respectively, and let (tinf lection, αinf lection) denote the inflection point of the accel-eration polynomial. Note thatt₀, t₁ are already given by the definition of the accel-eration profile. The time coordinate of the axis of symmetry of the jerk polynomial is tinf lection, i.e., tinf lection = −_3p^p2

3. Correspondingly, αinf lection = α(tinf lection).

Lettcrefer to the time coordinate of the intersection of two specific lines which are L₁ that passes through (t₀, α₀) and (t₁, α₁) and L₂ that coincides with the time axis. Note that any point that lies on the time axis has an acceleration coordinate of zero. Letkcrepresent the slope ofL1, and one can havekc= (α1−α₀)/(t1−t0).

Let k₀ be the slope of the tangent line of the acceleration profile at t₀. It follows that k0 = jerk(t0). It should be kept in mind that t1 always renders a local ex-tremum (a local maximum in the context of the problem) of the acceleration cubic polynomial as the jerk of the ending state of the profile is always zero. Recall that the jerk of the starting state might not be zero, which might be the case for the trajectory from the vehicle to the state lattice. As a result, t₀ might lie to the right of the time coordinate of the other local extremum ( a local minimum in this context). Figure4.4(a)gives an example of this case. The area of the yellow patch is denoted asδVr. It is valid to say thatvmin=v0− |δV_r|.

• In cases where k0 ≥ kc as is demonstrated in Figure 4.4(a), let δt equal to

|α₀|

α1−α0(t1−t₀) (i.e.,δt=tc−t₀). The area of the triangle with the blue frame can be calculated asδV_c= ¹₂|α₀|δt. Correspondingly, the approximate minimum speed is computed asvminc =v0−δVc. Whethervmin is negative or not is thus determined by the sign ofv_minc. It is straightforward thatv_minc < v_min. Therefore it is safe to conclude thatvminis positive ifvminc turns out to be positive. However, ifvminc is negative,v_min may not be negative. In this sense, the approximate method, while guarantees to discard the infeasible profiles, might also exclude some valid ones.

• Ifk₀ < k_c, and at the same timeαinf lection<0, as is shown in Figure 4.4(b), one can have that δt1 = |t_c−tinf lection| and δt2 = |t₀ −tinf lection|. Correspondingly, the area composed of the triangle and rectangle with blue frames is given asδV_c=

1

2δt1|αinf lection|+δt2|α₀|. The rest of the analysis concerning δVc is the same as that of the case illustrated above.

• If k0 < kc, and at the same time αinf lection≥0, as is shown in Figure4.4(c), one can haveδt=|t₀−tinf lection|. Correspondingly, the area of the rectangle with blue

frame is given as δVc=δt|α₀|. The rest of the analysis is the same as that of the first case.