PACIENTES Y MÉTODO

bce be C <C and ae ab bce ab be C +C <C +C =C∗ (4.1)

However, Eq. (4.1) can only be satisfied by violating the condition that a-b-e is the optimal path from a to e.

Bellman [8] has called the above property of an optimal policy the principal of optimality: “An optimal policy has the property that whatever the initial state and initial

decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision” [47].

4.4 Connection from nonzero velocity and acceleration boundary conditions to rest The connection from initial conditions of nonzero velocity and acceleration to rest are achieved by extending (or cropping) Phase  of the trajectory which connects zero initial acceleration and nonzero initial velocity b.c. to rest. There are four possible cases:

i) Case #1: From (v_i ≥ , 0 a_i > ) to rest (0 v_f =0, a_f =0) ii) Case #2: From (v_i ≥ , 0 a_i < ) to rest (0 v_f =0, a_f =0) iii) Case #3: From (v_i < , 0 a_i < ) to rest (0 v_f =0, a_f =0) iv) Case #4: From (v_i < , 0 a_i > ) to rest (0 v_f =0, a_f =0)

Case #1: Connecting from (v_i ≥ , 0 A≥a_i > ) to rest (0 v_f =0, a_f =0)

Considering Figure 4.3, the basic idea is to bridge a connection from the given nonzero acceleration boundary condition to the quickest reachable state with zero acceleration, as

shown in the red highlighted portion of Figure 4.2. This is achieved by projecting the trajectory forwards from the initial state by considering a jerk input of −J (with maximum magnitude). The velocity reached at the zero acceleration state is denoted as v_0f . The subscript ‘0’ indicates ‘zero-acceleration state’, and the subscript ‘f’ indicates projection in the ‘forwards’ direction in time. From this point onwards, a zero initial acceleration trajectory is planned, as explained in Section 4.2.2, based on one of the two templates in Figure 4.1 (c) or (d). The ‘zero initial acceleration to rest’ portion of the trajectory has to be planned starting at the initial velocity of v_0f . Afterwards, the complete trajectory can be accomplished by extending the duration of the first phase as T₁′= +T₁ T_a.

Figure 4.3: Case #1: Connection from (v_i ≥ , 0 A≥a_i > ) to rest. 0

0 f

v and T can be calculated as follows (noting that because _a a_i > 0 → |a_i|= ): a_i

| _i| / _{a a} | _i| / J = a T →T = a J 0 2 | | 2 | | 2 | | | | 2 1 2 2 2 2 _{= − = = >} − = ∆ J a a J a J a J J a a JT T a v _{i a a} i i i i i i 0f i v = + ∆v v

Chapter 4 Time-Optimal Connection Between On-the-Fly Drilling Trajectories and Rest Boundary

Conditions

The overall connection trajectory is obtained by integrating the jerk sequence of [−J, 0, J ], for the durations of [T ′₁ , T₂ , T₃], starting with the initial conditions (v_i ≥ , 0

0

i

a > ).

From the Principle of Optimality summarized in Section 4.3, since the connection from the initial state (v , _i a ) to the intermediate state (_i v_{0 f}, a=0) is a minimum-time trajectory, along with the connection from (v_{0 f} ,a=0) to rest (v_f =0, a_f =0), it follows that the complete trajectory from (v_i≥ , 0 a_i > ) to rest is also time-optimal. 0

As an important note, considering that in the extreme case a_i= can hold, in order for A

the velocity profile to remain within the limit of v_0f ≤V , v has to be less than _i 2

_{/ (2 )}

i

v

≤ −V

A

J

. This requirement ( v_0f ≤V) has to be taken into account when selecting the entry and exit points into the cyclical on-the-fly drilling trajectory, to ensure that the connection profiles do not violate the velocity limits of the drives.

Case #2: Connecting from (v_i ≥0, − ≤A a_i <0) to rest (v_f =0, a_f =0)

In this case, zero acceleration states are projected both in the forwards and backwards directions (in time) from the given initial condition of (v_i ≥0, a_i <0), as shown in Figure 4.4. Here,

0b

v indicates the velocity reached by projecting ‘backwards’ and v_0f by projecting ‘forwards’ (as explained in Case #1). The travel time T_a to zero acceleration state (which is identical in both directions), and the respective values of v_0b and v_{0 f} can be found as follows: J a T_a =| i|, 0 2 | | < = ∆ J a a v i i , v_{0 f} = + ∆v_i v, and

v

_0b

= − ∆v

_i

v

(a) (b)

Figure 4.4: Connection from (v_i≥ , 0 a_i < ) to rest: (a) when 0 v_0f ≥0, (b) when v_0f <0.

Based on the value of v_{0 f} , two possibilities emerge:

i) If v_0f ≥0 , as shown in Figure 4.4(a), then the trajectory is planned as a connection from zero initial acceleration (with positive initial velocity) to rest, as explained in Section 4.2.2, starting with the initial velocity of v_0b. Afterwards, the duration of T is cropped out of _a T (i.e., ₁ T₁′= −T₁ Ta ) and the final motion

profile is generated by integrating the jerk sequence of [−J, 0, J ], which is applied for the durations of [T₁′ , T , ₂ T₃], starting with the initial conditions (v_i ≥0, a_i <0). Since |a_i|≤A, it can be guaranteed that

T

_a

≤T

₁. Since the parent trajectory from (

0b

v , a=0) to rest is already minimum time, and the given initial conditions (v_i ≥ , 0 a_i < ) are on this trajectory, then from the Principle 0 of Optimality the cropped trajectory from (v , _i a ) to rest will also be minimum _i

time.

ii) If v_0f <0, as shown in Figure 4.4(b), then the trajectory is planned as a

Chapter 4 Time-Optimal Connection Between On-the-Fly Drilling Trajectories and Rest Boundary

Conditions

explained in Section 4.2.2), starting with an initial velocity of v_{0 f} . Afterwards, the duration of T is extended backwards (i.e. ₁ T is added to _a T ; ₁

T

₁

′= +T

₁

T

_a). The final motion profile is generated by integrating the jerk sequence [+J , 0,

J

− ], which is applied for the durations of [T₁′, T , ₂ T₃], starting with the initial conditions (v_i ≥ , 0 a_i < ). Since the trajectory from the initial conditions 0 (v_i ≥ , 0 a_i < ) to the intermediate state (0 v_0f <0,a=0) is time optimal (i.e., achieved with largest possible jerk magnitude), and the latter part (from (v_{0 f},

0 =

a ) to rest) is also time-optimal, then from the Principle of Optimality, it follows that their connected sequence will also be time-optimal.

Case #3: Connecting from (v_i < , 0 − ≤A a_i< ) to rest (0 v_f =0, a_f =0)

Figure 4.5: Case #3: Connection from (v_i< , 0 − ≤A a_i< ) to rest. 0

Considering Figure 4.5, this case can be considered as the mirror image of Case #1 with respect to the time axis. The strategy is the same; to project forwards in time towards a zero acceleration state with the velocity denoted as v_0f , and then to plan a zero initial acceleration trajectory to rest starting with this velocity. Afterwards, the duration of Phase 

of the latter trajectory is extended backwards by the acceleration duration T (_a T₁′= +T₁ T_a) and a jerk sequence of [+J , 0, −J] is executed for the duration array [T₁′, T , ₂ T₃], starting with the initial conditions of (v_i < , 0 a_i < ). As proven for Case #1, this trajectory is 0 also time-optimal.

As explained for Case #1, to guarantee that the magnitude of the velocity profile does not exceed its limit (i.e., to keep − ≤v V ), in the extreme case that a_i = − , A v then has to _i

satisfy

v

_i

≥ − +V

A

2

/ (2 )J

. Similarly, the connection point has to be chosen such that

V v_0f ≤ .

Case #4: Connecting from (v_i < ,0 a_i > ) to rest (0 v_f =0, a_f =0)

(a) (b)

Figure 4.6: Connection from (v_i < ,0 a_i > ) to rest: (a) when 0 v_0f <0, (b) when

0f 0

v ≥ .

Considering Figure 4.6, it can be seen that the treatment of this case is similar to that of Case #2. The velocity v_{0 f} corresponding to the forward projected zero acceleration state is determined. Then, two possibilities arise:

i) If v_{0 f} is negative, then the trajectory is planned from the backward projected zero acceleration state velocity,

0b

Chapter 4 Time-Optimal Connection Between On-the-Fly Drilling Trajectories and Rest Boundary

Conditions

Phase  is cropped (

T

₁

′= −T

₁

T

_a). Then, the jerk sequence of [+J , 0, −J] is applied with a duration array of [T₁′, T , ₂ T₃], starting with the initial conditions (v_i < ,0 a_i > ). Since the parent trajectory from (0 v , _0b a=0) to rest is time- optimal, and the initial conditions (v_i <0, a_i >0) are on this trajectory, then from the Principle of Optimality the cropped trajectory from (v , _i a ) to rest will _i

also be time-optimal.

ii) If v_{0 f} is positive, then the trajectory is planned from v_{0 f} onwards to rest, as shown in Figure 4.6(b). T is extended backwards as ₁

T

₁

′= +T

₁

T

_a. The jerk sequence of [−J , 0, J ] is executed with the duration array of [T₁′ , T , ₂ T₃], starting with initial conditions (v_i <0, a_i >0). Since the trajectory from the initial conditions (v_i <0, a_i >0) to the intermediate state (v_0f ≥0,a=0) is time optimal (i.e., achieved with largest possible jerk magnitude), and the latter part (from (v_{0 f} , a =0) to rest) is also time-optimal, then from the Principle of Optimality it follows that their connecting sequence will also be time-optimal.

4.5 Connection from rest to nonzero velocity and acceleration boundary conditions

In document UNIVERSIDAD COMPLUTENSE DE MADRID (página 146-162)