EP2337758B1 - Method and device for controlling a lift load - Google Patents

Description

The present invention relates to a control method implemented in a variable speed drive for controlling a lifting load such as an elevator. The invention also relates to a variable speed drive capable of implementing said method.

• une accélération jusqu'à une première vitesse,
• la réception d'un ordre de décélération lorsque l'ascenseur a atteint un certain niveau, cet ordre pouvant être donné lors du passage de l'ascenseur devant un capteur externe,
• une première décélération jusqu'à une seconde vitesse inférieure à la première vitesse,
• la réception d'un ordre d'arrêt lorsque l'ascenseur est proche de l'étage d'arrivée, cet ordre pouvant également être donné lors du passage de l'ascenseur devant un second capteur,
• une seconde décélération jusqu'à l'arrêt.

The control profile of a lifting load such as an elevator moving between floors generally comprises the following main steps:

• an acceleration to a first speed,
• receiving a deceleration command when the elevator has reached a certain level, this order being able to be given when the elevator passes an external sensor,
• a first deceleration to a second speed lower than the first speed,
• receiving a stop command when the elevator is close to the arrival floor, this order can also be given when the elevator passes in front of a second sensor,
• a second deceleration until the stop.

Depending on the duration to reach the first speed following the acceleration and the duration to reach the second speed following the first deceleration, the profile may also comprise a step of maintaining the speed of the elevator at the first speed before the first speed. deceleration and a holding step at the second speed before the second deceleration.

The first speed is set to be the maximum speed to be reached by the elevator when traveling between two stages separated by several levels. But when the elevator must make a shorter trip, for example between two separate floors of a single level, this maximum speed is often never reached. In such a situation the elevator is still controlled according to the control profile defined above. The elevator thus receives the deceleration order before reaching its maximum speed and therefore starts the first deceleration earlier according to the same speed profile than if the maximum speed had been reached. However, at the time of receipt of the deceleration order, the elevator has traveled a short distance. During all the distance remaining before receiving the stop command, the elevator moves at low speed. The time spent by the elevator at low speed is therefore very long.

The patent GB1560348 describes a solution to overcome this problem. This document describes the application of a first speed profile to an elevator, this profile with acceleration to a maximum speed, followed by a first deceleration to a low speed landing before further deceleration to a stop. When the braking command that controls the first deceleration occurs while the maximum speed has not been reached, this document proposes the introduction of a second speed profile for shifting the beginning of the first deceleration. The new braking moment occurs at the intersection between the two velocity profiles. In this document of the state of the art, the goal is thus to recover the lost distance because of the too premature appearance of the deceleration order by continuing the acceleration to a new speed following the ramp of initial acceleration. However, by keeping the initial acceleration ramp to reach the new speed, the distance remaining to be covered will be respected but not the duration.

The document EP0826621 describes a method for adjusting the low speed of an elevator car by applying a compensation frequency in the control.

The object of the invention is to provide a control method for minimizing the time spent at low speed when the elevator performs a path such that it receives the deceleration order before reaching its maximum speed.

This object is achieved by a control method as defined in claim 1.

According to another feature, between the deceleration step and the stop step, the second control profile includes a step of maintaining the speed of the load at a third speed lower than the second speed.

According to another feature, at the end of the deceleration step, the second control profile comprises a step of receiving a stop command.

According to another particular feature, after receiving the stop command, the second control profile comprises a deceleration step until it stops.

According to another particularity, the deceleration order or the stop command is sent by an external sensor capable of detecting the passage of the lifting load or can be sent by a PLC connected to the variable speed drive.

The invention also relates to a variable speed drive as defined in claim 7.

According to another feature, the variable speed drive comprises means for maintaining the speed of the load at a third speed lower than the second speed.

According to another particularity, the second control profile comprises a reception of a stop command.

According to another particularity, the second control profile comprises a deceleration to the stop following the reception of the stop command.

According to another feature, the drive is connected to an external sensor capable of sending the deceleration command or the stop command when it detects the passage of the lifting load. Alternatively, the drive can be connected to a programmable controller able to send the deceleration order or the stop order.

• les figures 1A et 1B représentent respectivement un profil de vitesse et son profil de position correspondant suivis par un ascenseur se déplaçant entre deux étages en atteignant sa vitesse maximale,
• les figures 2A et 2B représentent respectivement un profil de vitesse et son profil de position correspondant suivis par un ascenseur se déplaçant entre deux étages sans atteindre sa vitesse maximale et sans application du procédé de commande de l'invention,
• les figures 3A et 3B représentent respectivement un profil de vitesse et son profil de position correspondant suivis par un ascenseur se déplaçant entre deux étages sans atteindre sa vitesse maximale et avec application du procédé de commande de l'invention.

Other features and advantages will appear in the detailed description which follows with reference to an embodiment given by way of example and represented by the appended drawings in which:

• the Figures 1A and 1B respectively represent a speed profile and its corresponding position profile followed by an elevator moving between two stages while reaching its maximum speed,
• the Figures 2A and 2B respectively represent a speed profile and its corresponding position profile followed by an elevator moving between two stages without reaching its maximum speed and without application of the control method of the invention,
• the Figures 3A and 3B respectively represent a speed profile and its corresponding position profile followed by an elevator moving between two stages without reaching its maximum speed and with application of the control method of the invention.

• réception d'un ordre de départ pour déplacer l'ascenseur d'un étage à un autre,
• accélération selon une rampe d'accélération RA jusqu'à atteindre une vitesse maximale ω_R,
• réception d'un ordre de décélération (FLG1) par exemple à l'aide d'un premier capteur externe placé sur le trajet de l'ascenseur,
• décélération selon une rampe de décélération RD jusqu'à atteindre une basse vitesse ω_L,
• réception d'un ordre d'arrêt (FLG2) par exemple à l'aide d'un second capteur externe placé sur le trajet de l'ascenseur,
• décélération selon une rampe d'arrêt RS jusqu'à l'arrêt complet de l'ascenseur à l'étage voulu.

As already described above, with reference to the Figure 1B , a conventional control profile applied in a variable speed drive to control a lifting load such as an elevator using an electric motor comprises the following main steps:

• receiving a departure order to move the elevator from one floor to another,
• acceleration according to an acceleration ramp RA until reaching a maximum speed ω _R ,
• reception of a deceleration order (FLG1) for example with the aid of a first external sensor placed on the path of the elevator,
• deceleration according to a deceleration ramp RD to reach a low speed ω _L ,
• receiving a stop command (FLG2) for example by means of a second external sensor placed on the path of the elevator,
• deceleration according to a stop ramp RS until the complete stop of the elevator to the desired floor.

Each external sensor is disposed on the elevator path at a distance before the desired arrival stage to meet the deceleration and stopping distances.

• chaque rampe (accélération, décélération, arrêt) doit être appliquée suivant une accélération faible, au plus égale à 0,5 m/s²,
• les impulsions ou arrondis (jerk en anglais) en début et en fin de chaque rampe doivent être limitées, par exemple à une valeur comprise entre 0,2 et 0,5 m/s³.

This type of control profile is implemented taking into account constraints related to user comfort. Indeed, this control profile must be applied in a comfortable manner for the user which implies the application of nonlinear ramps. For this, two principles are generally applied:

• each ramp (acceleration, deceleration, stop) must be applied at a low acceleration, at most equal to 0.5 m / s ² ,
• pulses or rounded (jerk in English) in the beginning and end of each ramp should be limited, for example to a value between 0.2 and 0.5 m / s ^3.

The control profile defined above is ideal when the elevator moves several levels because the elevator then has a sufficient time to reach its maximum speed ω _R before receiving the deceleration order (FLG1). On the other hand, when the elevator makes a short path between two stages, for example separated by a single level, the deceleration order (FLG1) can be received before the elevator has had time to reach its stage. maximum speed ω _R. In this case, if the elevator continues to accelerate after receiving the deceleration order (FLG1), the stopping distances at the desired floor can not be met or if the elevator is decelerated according to the control profile defined above, the low speed ω _L will be reached very early and the elevator will have to move very slowly at this low speed ω _L to reach the desired floor as shown on the Figures 2A and 2B .

According to the invention, when the speed controller receives the deceleration order (FLG1) while the elevator is at a current speed lower than its maximum speed ω _R , the drive determines a second speed ω _R ^opt lower than the speed ω _R and higher than its current speed, this second speed being an optimal speed up to which the elevator can continue to accelerate to minimize the time of travel to a stop while respecting stopping distances (see Figures 3A and 3B ). The principle of the invention therefore consists in seeking a function of time such that: $$\{\begin{array}{cc}\mathrm{\theta }& =\mathrm{f}\left(\mathrm{t}\right)\\ \mathrm{\omega }& =\mathrm{f}\text{'}\left(\mathrm{t}\right)\\ \mathrm{\gamma }& =\mathrm{f}"\left(\mathrm{t}\right)\\ \mathrm{j}& =\mathrm{f\; ‴}\left(\mathrm{t}\right)\end{array}$$

In which ω is designated as the current speed of the load, θ the current position of the load, γ represents the acceleration of the load and j represents the pulse ("jerk") of the load.

This function f must respect the following constraints $$:\{\begin{array}{cc}\mathrm{0}=\mathrm{<figure-callout\; id="0"\; label="f"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">f</figure-callout>}\left(\mathrm{0}\right)\hfill & {\mathrm{\theta }}_{\mathrm{dd}}=\mathrm{f}\left({\mathrm{<figure-callout\; id="0"\; label="t\; D\; \omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">t\; </figure-callout>}}_{\mathrm{D}},\right)\hfill & \\ {\mathrm{\omega }}_{}\hfill {\mathrm{}}_{\mathrm{0}}=\mathrm{f}\text{'}\left(\mathrm{0}\right)\hfill & {\mathrm{\omega }}_{\mathrm{The}}=\mathrm{f}\text{'}\left({\mathrm{<figure-callout\; id="0"\; label="t\; D\; \gamma "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">t\; </figure-callout>}}_{\mathrm{D}},\right)\hfill & \\ {\mathrm{\gamma }}_{}\hfill {\mathrm{}}_{\mathrm{0}}=\mathrm{f}"\left(\mathrm{0}\right)\hfill & \mathrm{0}=\mathrm{f}"\left({\mathrm{<figure-callout\; id="0"\; label="t\; D"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">t\; </figure-callout>}}_{\mathrm{D}}\right)\hfill \\ & \mathrm{0}=\mathrm{f\; ‴}\left({\mathrm{t}}_{\mathrm{D}}\right)\hfill \end{array}\mathrm{and}\left|\mathrm{\gamma }\right|<{\mathrm{\gamma }}_{\mathrm{MAX}},\mathrm{j}<{\mathrm{j}}_{\mathrm{MAX}}$$

(ω ₀ , γ ₀ ) represents the trajectory point at the time of reception of the deceleration order, (ω _L , 0) represents the point to reach of the trajectory and θ _Dd the distance to be traveled during the deceleration movement , between the maximum speed and the low speed. t _D represents the deceleration time

The pair (ω ₀ , γ ₀ ) is obtained by the current position of the trajectory.

The distance θ _Dd is known because it is the distance traveled during the first deceleration. If this distance θ _Dd is respected by the control profile, the stopping distance constraints will also be respected.

If we add a known time parameter T _R corresponding to a dwell time at the maximum speed reached by the elevator, the resolution consists of starting from all the known data (ω ₀ , γ ₀ , θ _Dd , T _R ) to calculate an optimal maximum speed ω _R ^opt to achieve which minimizes the total time of movement.

By definition, the optimal maximum velocity is defined by ω _R ^opt = f '(t _R ), where t _R is such that f "(t _R ) = 0.

Two examples are discussed below to model the function f defined above.

• accélération γ_A pendant le temps Ta suivant une rampe d'accélération RA,
• maintien à la vitesse ω_R pendant un temps de palier Tp,
• accélération γ_D pendant le temps Td suivant une rampe de décélération RD,
• maintien à la basse vitesse ω_L pendant un temps T_L afin de parcourir la distance restante jusqu'à l'arrêt.

The first example is to determine the optimal speed ω _R ^opt , considering for example the following control profile, piecewise linear acceleration (see Figure 1B ):

• acceleration γ _A during the time Ta following an acceleration ramp RA,
• maintaining the speed ω _R during a plateau time Tp,
• acceleration γ _D during the time Td following a ramp deceleration RD,
• holding at low speed ω _L for a time T _L to travel the remaining distance to the stop.

The calculation of the optimal speed ω _R ^opt is done in respect of the magnitudes of accelerations and impulses to maintain a level of comfort. It may be that the calculation of the optimal speed changes the acceleration and momentum quantities compared to the initial trajectory.

In this first example, we consider that the acceleration ramp to reach the optimal speed ω _R ^opt calculated is the acceleration ramp RA of the initially planned control profile and that the deceleration ramp applied after reaching the optimal speed ω _R ^opt is also the deceleration ramp RD of the initially planned control profile.

• Entre 0 et Ta (phase d'accélération), nous avons : $$\mathrm{\omega }={\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}+{\mathrm{\gamma }}_{\mathrm{A}}\cdot \mathrm{t}$$
  
  $$\mathrm{\theta }={\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}\cdot \mathrm{t}+\frac{\mathrm{1}}{\mathrm{2}}\cdot {\mathrm{\gamma }}_{\mathrm{A}}\cdot {\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">t</figure-callout>}}^{\mathrm{2}}$$
  
  Ce qui donne en Ta : $${\mathrm{\omega }}_{\mathrm{R}}={\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}+{\mathrm{\gamma }}_{\mathrm{A}}\cdot {\mathrm{T}}_{\mathrm{A}}$$
  
  $${\mathrm{\theta }}_{\mathrm{R}}={\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}\cdot {\mathrm{T}}_{\mathrm{A}}+\frac{\mathrm{1}}{\mathrm{2}}\cdot {\mathrm{\gamma }}_{\mathrm{A}}\cdot {{\mathrm{<figure-callout\; id="2"\; label="T\; A"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">T</figure-callout>}}_{\mathrm{A}}}^{\mathrm{2}}$$
  
  Soit avec $${\mathrm{T}}_{\mathrm{A}}=\frac{{\mathrm{\omega }}_{\mathrm{R}}-{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}{{\mathrm{\gamma }}_{\mathrm{A}}}$$
  
  Nous obtenons alors : $${\mathrm{\theta }}_{\mathrm{R}}=\frac{{{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}}}^{\mathrm{2}}-{{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{A}}}$$
  
• Entre Ta et Ta+Tp, la vitesse étant constante, nous avons : $$\mathrm{\omega }={\mathrm{\omega }}_{\mathrm{R}}$$
  
  $$\mathrm{\theta }={\mathrm{\theta }}_{\mathrm{R}}+{\mathrm{\omega }}_{\mathrm{R}}\cdot \mathrm{t}$$
  
  Ce qui donne en Ta+Tp : $${\mathrm{\theta }}_{\mathrm{P}}={\mathrm{\theta }}_{\mathrm{R}}+{\mathrm{\omega }}_{\mathrm{R}}\cdot {\mathrm{T}}_{\mathrm{P}}$$
  
• Entre Ta+Tp et Ta+Tp +Td (phase de décélération), nous avons : $$\mathrm{\omega }={\mathrm{\omega }}_{\mathrm{R}}-{\mathrm{\gamma }}_{\mathrm{D}}\cdot \mathrm{t}$$
  
  $$\mathrm{\theta }={\mathrm{\theta }}_{\mathrm{P}}+{\mathrm{\omega }}_{\mathrm{R}}\cdot \mathrm{t}-\frac{\mathrm{1}}{\mathrm{2}}\cdot {\mathrm{\gamma }}_{\mathrm{D}}\cdot {\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">t</figure-callout>}}^{\mathrm{2}}$$
  
  Ce qui donne en Ta+Tp+Td : $${\mathrm{\omega }}_{\mathrm{R}}={\mathrm{\omega }}_{\mathrm{L}}+{\mathrm{\gamma }}_{\mathrm{D}}\cdot {\mathrm{T}}_{\mathrm{D}}$$
  
  $${\mathrm{\theta }}_{\mathrm{D}}={\mathrm{\theta }}_{\mathrm{P}}+{\mathrm{\omega }}_{\mathrm{R}}\cdot {\mathrm{T}}_{\mathrm{D}}-\frac{\mathrm{1}}{\mathrm{2}}\cdot {\mathrm{\gamma }}_{\mathrm{D}}\cdot {{\mathrm{<figure-callout\; id="2"\; label="T\; D"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">T</figure-callout>}}_{\mathrm{D}}}^{\mathrm{2}}$$
  
  Avec $${\mathrm{T}}_{\mathrm{D}}=\frac{{\mathrm{\omega }}_{\mathrm{R}}-{\mathrm{\omega }}_{\mathrm{L}}}{{\mathrm{\gamma }}_{\mathrm{D}}}$$
  
  On obtient alors : $${\mathrm{\theta }}_{\mathrm{D}}=\frac{{{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}}}^{\mathrm{2}}-{{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{A}}}+\frac{{{\mathrm{<figure-callout\; id="2"\; label="\omega \; R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">L</figure-callout>}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{D}}}+{\mathrm{\omega }}_{\mathrm{R}}\cdot {\mathrm{T}}_{\mathrm{P}}$$
  
• Puis entre Ta+Tp+Td et T_R=Ta+Tp+Td+T_L , nous avons : $$\mathrm{\omega }={\mathrm{\omega }}_{\mathrm{L}}$$
  
  $$\mathrm{\theta }={\mathrm{\theta }}_{\mathrm{D}}+{\mathrm{\omega }}_{\mathrm{L}}\cdot \mathrm{t}$$
  
  Ce qui donne en T_R : $${\mathrm{\theta }}_{\mathrm{Dd}}={\mathrm{\theta }}_{\mathrm{D}}+{\mathrm{\omega }}_{\mathrm{L}}\cdot {\mathrm{T}}_{\mathrm{L}}=\frac{{{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}}}^{\mathrm{2}}-{{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{A}}}+\frac{{{\mathrm{<figure-callout\; id="2"\; label="\omega \; R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">L</figure-callout>}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{D}}}+{\mathrm{\omega }}_{\mathrm{L}}\cdot {\mathrm{T}}_{\mathrm{L}}+{\mathrm{\omega }}_{\mathrm{R}}\cdot {\mathrm{T}}_{\mathrm{P}}$$
  
  sous la condition que T_L>0, il vient alors : $${\mathrm{T}}_{\mathrm{L}}=\frac{{\mathrm{\theta }}_{\mathrm{Dd}}-{\mathrm{\omega }}_{\mathrm{R}}\cdot {\mathrm{T}}_{\mathrm{P}}-\frac{{{\mathrm{<figure-callout\; id="2"\; label="\omega \; R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{A}}}-\frac{{{\mathrm{<figure-callout\; id="2"\; label="\omega \; R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">L</figure-callout>}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{D}}}}{{\mathrm{\omega }}_{\mathrm{L}}}$$
  

Nous obtenons alors : $$\boxed{{\mathrm{T}}_{\mathrm{R}}=\frac{{\mathrm{\omega }}_{\mathrm{R}}-{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}{{\mathrm{\gamma }}_{\mathrm{A}}}+{\mathrm{T}}_{\mathrm{P}}+\frac{{\mathrm{\omega }}_{\mathrm{R}}-{\mathrm{\omega }}_{\mathrm{L}}}{{\mathrm{\gamma }}_{\mathrm{D}}}{+\left[\frac{{\mathrm{\theta }}_{\mathrm{Dd}}-{\mathrm{\omega }}_{\mathrm{R}}\cdot {\mathrm{T}}_{\mathrm{P}}-\frac{{{\mathrm{<figure-callout\; id="2"\; label="\omega \; R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{A}}}-\frac{{{\mathrm{<figure-callout\; id="2"\; label="\omega \; R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">L</figure-callout>}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{D}}}}{{\mathrm{\omega }}_{\mathrm{L}}}\right]}_{>\mathrm{0}}}$$

Avec : $${\mathrm{T}}_{\mathrm{A}}=\frac{{\mathrm{\omega }}_{\mathrm{R}}-{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}{{\mathrm{\gamma }}_{\mathrm{A}}},{\mathrm{T}}_{\mathrm{D}}=\frac{{\mathrm{\omega }}_{\mathrm{R}}-{\mathrm{\omega }}_{\mathrm{L}}}{{\mathrm{\gamma }}_{\mathrm{D}}}{\mathrm{et\; T}}_{\mathrm{L}}=\frac{{\mathrm{\theta }}_{\mathrm{Dd}}-{\mathrm{\omega }}_{\mathrm{R}}\cdot {\mathrm{T}}_{\mathrm{P}}-\frac{{{\mathrm{<figure-callout\; id="2"\; label="\omega \; R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{A}}}-\frac{{{\mathrm{<figure-callout\; id="2"\; label="\omega \; R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">L</figure-callout>}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{D}}}}{{\mathrm{\omega }}_{\mathrm{L}}}$$

Nous obtenons donc que le temps de parcours est une fonction de la vitesse ω_R.
Si T_L <0, cela signifie que les mouvements de fin d'accélération et décélération ont consommés trop de distance. Par conséquent, le temps T_L doit être positif ce qui nous amène à poser les relations suivantes : $$\boxed{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{\theta }}=\sqrt{\frac{\mathrm{2}\cdot {\mathrm{\theta }}_{\mathrm{Dd}}+\frac{{{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}^{\mathrm{2}}}{{\mathrm{\gamma }}_{\mathrm{A}}}+\frac{{{\mathrm{<figure-callout\; id="2"\; label="\omega \; L"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{L}}}^{\mathrm{2}}}{{\mathrm{\gamma }}_{\mathrm{D}}}}{\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{A}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}}}}\mathrm{et}\boxed{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathit{\gamma }}=\frac{{\mathrm{T}}_{\mathrm{P}}}{\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{A}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}}}$$

et à étudier la contrainte : $${\mathrm{T}}_{\mathrm{L}}=\frac{{\mathrm{\theta }}_{\mathrm{Dd}}-{\mathrm{\omega }}_{\mathrm{R}}\cdot {\mathrm{T}}_{\mathrm{P}}-\left(\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{A}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}\right)\cdot \frac{{{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}}}^{\mathrm{2}}}{\mathrm{2}}+\frac{{{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{A}}}+\frac{{{\mathrm{<figure-callout\; id="2"\; label="\omega \; L"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{L}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{D}}}}{{\mathrm{\omega }}_{\mathrm{L}}}\ge \mathrm{0}$$

Nous obtenons alors la relation suivante : $${\mathrm{T}}_{\mathrm{L}}=\frac{\left(\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{A}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}\right)\cdot \left({{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}}}^{{}_{\mathrm{\theta }}\mathrm{2}}-\mathrm{2}\cdot {{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{\gamma }}\cdot {\mathrm{\omega }}_{\mathrm{R}}-{{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}}}^{\mathrm{2}}\right)}{\mathrm{2}\cdot {\mathrm{\omega }}_{\mathrm{L}}}\ge \mathrm{0}$$

Pour remplir la condition T_L ≥ 0 , il faut donc que ω_R ^θ2 - 2ω_R ^γ · ω_R - ω_R ² ≥ 0
En résolvant cette équation du second degré, on obtient la vitesse optimale ω_R ^opt à atteindre tenant compte de la contrainte : $${{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{opt}}=-{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{\gamma }}+\sqrt{{{\mathrm{\omega }}_{\mathrm{R}}}^{{}_{\mathrm{\gamma }}\mathrm{2}}+{{\mathrm{\omega }}_{\mathrm{R}}}^{{}_{\mathrm{\theta }}\mathrm{2}}}$$

Afin de confirmer que la vitesse ω_R ^opt est bien la vitesse optimale permettant de minimiser le temps de parcours, il suffit d'étudier la fonction suivante et son évolution en fonction de ω_R : $${\mathrm{T}}_{\mathrm{R}}\left({\mathrm{\omega }}_{\mathrm{R}}\right)=\frac{{\mathrm{\omega }}_{\mathrm{R}}-{\mathrm{\omega }}_{\mathrm{0}}}{{\mathrm{\gamma }}_{\mathrm{A}}}+{\mathrm{T}}_{\mathrm{P}}+\frac{{\mathrm{\omega }}_{\mathrm{R}}-{\mathrm{\omega }}_{\mathrm{L}}}{{\mathrm{\gamma }}_{\mathrm{D}}}+\frac{{\mathrm{\theta }}_{\mathrm{Dd}}-{\mathrm{\omega }}_{\mathrm{R}}\cdot {\mathrm{T}}_{\mathrm{P}}-\frac{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{\omega }}_{\mathrm{0}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{A}}}-\frac{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{\omega }}_{\mathrm{L}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{D}}}}{{\mathrm{\omega }}_{\mathrm{L}}}$$

$$=\left(\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{A}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}\right)\cdot {\mathrm{\omega }}_{\mathrm{R}}-\frac{{\mathrm{\omega }}_{\mathrm{0}}}{{\mathrm{\gamma }}_{\mathrm{A}}}-\frac{{\mathrm{\omega }}_{\mathrm{L}}}{{\mathrm{\gamma }}_{\mathrm{D}}}+{\mathrm{T}}_{\mathrm{P}}+\frac{\left(\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{A}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}\right)\cdot \left({{\mathrm{\omega }}_{\mathrm{R}}}^{{}_{\mathrm{\theta }}\mathrm{2}}-\mathrm{2}\cdot {{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{\gamma }}\cdot {\mathrm{\omega }}_{\mathrm{R}}-{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{2}}\right)}{\mathrm{2}\cdot {\mathrm{\omega }}_{\mathrm{L}}}$$

La variation de T_R est déterminée à partir de sa dérivée : $$\frac{{\mathrm{dT}}_{\mathrm{R}}}{{\mathrm{d\omega }}_{\mathrm{R}}}\left({\mathrm{\omega }}_{\mathrm{R}}\right)=\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{A}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}-\frac{\left(\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{A}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}\right)\cdot \left({{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{\gamma }}+{\mathrm{\omega }}_{\mathrm{R}}\right)}{{\mathrm{\omega }}_{\mathrm{L}}}=\left(\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{A}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}\right)\cdot \left(\mathrm{1}-\frac{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{\gamma }}+{\mathrm{\omega }}_{\mathrm{R}}}{{\mathrm{\omega }}_{\mathrm{L}}}\right)$$

Par définition ω_R est supérieure à ω_L, il vient donc que la fonction T_R est monotone décroissante sur son espace de définition, c'est-à-dire ω_R dans [ω_L, ω_R ^opt].
Nous constatons donc que le temps T_R est minimum lorsque ω_R est maximum permettant de justifier le choix de ${{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{opt}}={-{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{\gamma }}+\sqrt{{{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="R\; \gamma "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}}}^{{\mathrm{\gamma }}^{}}+{{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="R\; \theta "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}}}^{{\mathrm{\theta }}^{}}}\mathrm{.}$

On obtient alors : $$\boxed{{\mathrm{\omega }}_{\mathrm{R}}={{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{opt}}=-{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{\gamma }}+\sqrt{{{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}}}^{{}_{\mathrm{\gamma }}\mathrm{2}}+{{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}}}^{{}_{\mathrm{\theta }}\mathrm{2}}}=-\frac{{\mathrm{T}}_{\mathrm{P}}}{\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{A}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}}+\sqrt{{\left(\frac{{\mathrm{T}}_{\mathrm{P}}}{\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{A}}}+\frac{\mathrm{1}}{{\mathrm{<figure-callout\; id="2"\; label="\gamma \; D"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\gamma </figure-callout>}}_{\mathrm{D}}}}\right)}^{\mathrm{2}}+\frac{\mathrm{2}\cdot {\mathrm{\theta }}_{\mathrm{Dd}}+\frac{{{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}^{\mathrm{2}}}{{\mathrm{\gamma }}_{\mathrm{A}}}+\frac{{{\mathrm{<figure-callout\; id="2"\; label="\omega \; L"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{L}}}^{\mathrm{2}}}{{\mathrm{\gamma }}_{\mathrm{D}}}}{\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{A}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}}}}\mathrm{.}$$

From the control profile defined above in connection with the Figure 1B with ω designated as the current speed of the load and θ the current position of the load, the following reasoning is carried out:

• Between 0 and Ta (acceleration phase), we have: $$\mathrm{\omega }={\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}+{\mathrm{\gamma }}_{\mathrm{AT}}\cdot \mathrm{t}$$
  
  $$\mathrm{\theta }={\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}\cdot \mathrm{t}+\frac{\mathrm{1}}{\mathrm{2}}\cdot {\mathrm{\gamma }}_{\mathrm{AT}}\cdot {\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">t</figure-callout>}}^{\mathrm{2}}$$
  
  What gives in Ta: $${\mathrm{\omega }}_{\mathrm{R}}={\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}+{\mathrm{\gamma }}_{\mathrm{AT}}\cdot {\mathrm{T}}_{\mathrm{AT}}$$
  
  $${\mathrm{\theta }}_{\mathrm{R}}={\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}\cdot {\mathrm{T}}_{\mathrm{AT}}+\frac{\mathrm{1}}{\mathrm{2}}\cdot {\mathrm{\gamma }}_{\mathrm{AT}}\cdot {{\mathrm{T}}_{\mathrm{AT}}}^{\mathrm{2}}$$
  
  Either with $${\mathrm{T}}_{\mathrm{AT}}=\frac{{\mathrm{\omega }}_{\mathrm{R}}-{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}{{\mathrm{\gamma }}_{\mathrm{AT}}}$$
  
  We then get: $${\mathrm{\theta }}_{\mathrm{R}}=\frac{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{AT}}}$$
  
• Between Ta and Ta + Tp , the speed being constant, we have: $$\mathrm{\omega }={\mathrm{\omega }}_{\mathrm{R}}$$
  
  $$\mathrm{\theta }={\mathrm{\theta }}_{\mathrm{R}}+{\mathrm{\omega }}_{\mathrm{R}}\cdot \mathrm{t}$$
  
  Which gives in Ta + Tp : $${\mathrm{\theta }}_{\mathrm{P}}={\mathrm{\theta }}_{\mathrm{R}}+{\mathrm{\omega }}_{\mathrm{R}}\cdot {\mathrm{T}}_{\mathrm{P}}$$
  
• Between Ta + Tp and Ta + Tp + Td (deceleration), we have: $$\mathrm{\omega }={\mathrm{\omega }}_{\mathrm{R}}-{\mathrm{\gamma }}_{\mathrm{D}}\cdot \mathrm{t}$$
  
  $$\mathrm{\theta }={\mathrm{\theta }}_{\mathrm{P}}+{\mathrm{\omega }}_{\mathrm{R}}\cdot \mathrm{t}-\frac{\mathrm{1}}{\mathrm{2}}\cdot {\mathrm{\gamma }}_{\mathrm{D}}\cdot {\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">t</figure-callout>}}^{\mathrm{2}}$$
  
  Which gives in Ta + Tp + Td: $${\mathrm{\omega }}_{\mathrm{R}}={\mathrm{\omega }}_{\mathrm{The}}+{\mathrm{\gamma }}_{\mathrm{D}}\cdot {\mathrm{T}}_{\mathrm{D}}$$
  
  $${\mathrm{\theta }}_{\mathrm{D}}={\mathrm{\theta }}_{\mathrm{P}}+{\mathrm{\omega }}_{\mathrm{R}}\cdot {\mathrm{T}}_{\mathrm{D}}-\frac{\mathrm{1}}{\mathrm{2}}\cdot {\mathrm{\gamma }}_{\mathrm{D}}\cdot {{\mathrm{<figure-callout\; id="2"\; label="T\; D"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">T\; </figure-callout>}}_{\mathrm{D}}}^{\mathrm{2}}$$
  
  With $${\mathrm{T}}_{\mathrm{D}}=\frac{{\mathrm{\omega }}_{\mathrm{R}}-{\mathrm{\omega }}_{\mathrm{The}}}{{\mathrm{\gamma }}_{\mathrm{D}}}$$
  
  We then obtain: $${\mathrm{\theta }}_{\mathrm{D}}=\frac{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{AT}}}+\frac{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{\omega }}_{\mathrm{The}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{D}}}+{\mathrm{\omega }}_{\mathrm{R}}\cdot {\mathrm{T}}_{\mathrm{P}}$$
  
• Then between Ta + Tp + Td and T _R = Ta + Tp + Td + T _L , we have: $$\mathrm{\omega }={\mathrm{\omega }}_{\mathrm{The}}$$
  
  $$\mathrm{\theta }={\mathrm{\theta }}_{\mathrm{D}}+{\mathrm{\omega }}_{\mathrm{The}}\cdot \mathrm{t}$$
  
  Which gives in T _R : $${\mathrm{\theta }}_{\mathrm{dd}}={\mathrm{\theta }}_{\mathrm{D}}+{\mathrm{\omega }}_{\mathrm{The}}\cdot {\mathrm{T}}_{\mathrm{The}}=\frac{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{AT}}}+\frac{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{\omega }}_{\mathrm{The}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{D}}}+{\mathrm{\omega }}_{\mathrm{The}}\cdot {\mathrm{T}}_{\mathrm{The}}+{\mathrm{\omega }}_{\mathrm{R}}\cdot {\mathrm{T}}_{\mathrm{P}}$$
  
  under the condition that T _L > 0, it comes then: $${\mathrm{T}}_{\mathrm{The}}=\frac{{\mathrm{\theta }}_{\mathrm{dd}}-{\mathrm{\omega }}_{\mathrm{R}}\cdot {\mathrm{T}}_{\mathrm{P}}-\frac{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{AT}}}-\frac{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{\omega }}_{\mathrm{The}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{D}}}}{{\mathrm{\omega }}_{\mathrm{The}}}$$
  

We then get: $$\boxed{{\mathrm{T}}_{\mathrm{R}}=\frac{{\mathrm{\omega }}_{\mathrm{R}}-{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}{{\mathrm{\gamma }}_{\mathrm{AT}}}+{\mathrm{T}}_{\mathrm{P}}+\frac{{\mathrm{\omega }}_{\mathrm{R}}-{\mathrm{\omega }}_{\mathrm{The}}}{{\mathrm{\gamma }}_{\mathrm{D}}}{+\left[\frac{{\mathrm{\theta }}_{\mathrm{dd}}-{\mathrm{\omega }}_{\mathrm{R}}\cdot {\mathrm{T}}_{\mathrm{P}}-\frac{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{AT}}}-\frac{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{\omega }}_{\mathrm{The}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{D}}}}{{\mathrm{\omega }}_{\mathrm{The}}}\right]}_{>\mathrm{0}}}$$

With: $${\mathrm{T}}_{\mathrm{AT}}=\frac{{\mathrm{\omega }}_{\mathrm{R}}-{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}{{\mathrm{\gamma }}_{\mathrm{AT}}},{\mathrm{T}}_{\mathrm{D}}=\frac{{\mathrm{\omega }}_{\mathrm{R}}-{\mathrm{\omega }}_{\mathrm{The}}}{{\mathrm{\gamma }}_{\mathrm{D}}}{\mathrm{and\; T}}_{\mathrm{The}}=\frac{{\mathrm{\theta }}_{\mathrm{dd}}-{\mathrm{\omega }}_{\mathrm{R}}\cdot {\mathrm{T}}_{\mathrm{P}}-\frac{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{AT}}}-\frac{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{\omega }}_{\mathrm{The}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{D}}}}{{\mathrm{\omega }}_{\mathrm{The}}}$$

We thus obtain that the travel time is a function of the speed ω _R.
If T _L <0, this means that the end acceleration and deceleration movements have consumed too much distance. Therefore, the time T _L must be positive, which leads us to the following relationships: $$\boxed{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{\theta }}=\sqrt{\frac{\mathrm{2}\cdot {\mathrm{\theta }}_{\mathrm{dd}}+\frac{{{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}^{\mathrm{2}}}{{\mathrm{\gamma }}_{\mathrm{AT}}}+\frac{{{\mathrm{\omega }}_{\mathrm{The}}}^{\mathrm{2}}}{{\mathrm{\gamma }}_{\mathrm{D}}}}{\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{AT}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}}}}\mathrm{and}\boxed{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathit{\gamma }}=\frac{{\mathrm{T}}_{\mathrm{P}}}{\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{AT}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}}}$$

and to study the constraint: $${\mathrm{T}}_{\mathrm{The}}=\frac{{\mathrm{\theta }}_{\mathrm{dd}}-{\mathrm{\omega }}_{\mathrm{R}}\cdot {\mathrm{T}}_{\mathrm{P}}-\left(\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{AT}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}\right)\cdot \frac{{{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}}}^{\mathrm{2}}}{\mathrm{2}}+\frac{{{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{AT}}}+\frac{{{\mathrm{\omega }}_{\mathrm{The}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{D}}}}{{\mathrm{\omega }}_{\mathrm{The}}}\ge \mathrm{0}$$

We then get the following relation: $${\mathrm{T}}_{\mathrm{The}}=\frac{\left(\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{AT}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}\right)\cdot \left({{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}}}^{{}_{\mathrm{\theta }}\mathrm{2}}-\mathrm{2}\cdot {{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{\gamma }}\cdot {\mathrm{\omega }}_{\mathrm{R}}-{{\mathrm{<figure-callout\; id="2"\; label="\omega \; R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega \; </figure-callout>}}_{\mathrm{R}}}^{\mathrm{2}}\right)}{\mathrm{2}\cdot {\mathrm{\omega }}_{\mathrm{The}}}\ge \mathrm{0}$$

To fulfill the condition T _L ≥ 0, it is therefore necessary that ω _R ^{θ 2} - 2ω _R ^γ · ω _R - ω _R ² ≥ 0
By solving this equation of the second degree, we obtain the optimal speed ω _R ^opt to reach taking into account the constraint: $${{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{Opt}}=-{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{\gamma }}+\sqrt{{{\mathrm{\omega }}_{\mathrm{R}}}^{{}_{\mathrm{\gamma }}\mathrm{2}}+{{\mathrm{\omega }}_{\mathrm{R}}}^{{}_{\mathrm{\theta }}\mathrm{2}}}$$

To confirm that the speed ω _R ^opt is the optimum speed to minimize travel time, just consider the following function and its evolution in terms of ω _R: $${\mathrm{T}}_{\mathrm{R}}\left({\mathrm{\omega }}_{\mathrm{R}}\right)=\frac{{\mathrm{\omega }}_{\mathrm{R}}-{\mathrm{\omega }}_{\mathrm{0}}}{{\mathrm{\gamma }}_{\mathrm{AT}}}+{\mathrm{T}}_{\mathrm{P}}+\frac{{\mathrm{\omega }}_{\mathrm{R}}-{\mathrm{\omega }}_{\mathrm{The}}}{{\mathrm{\gamma }}_{\mathrm{D}}}+\frac{{\mathrm{\theta }}_{\mathrm{dd}}-{\mathrm{\omega }}_{\mathrm{R}}\cdot {\mathrm{T}}_{\mathrm{P}}-\frac{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{\omega }}_{\mathrm{0}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{AT}}}-\frac{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{2}}-{{\mathrm{\omega }}_{\mathrm{The}}}^{\mathrm{2}}}{\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{D}}}}{{\mathrm{\omega }}_{\mathrm{The}}}$$

$$=\left(\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{AT}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}\right)\cdot {\mathrm{\omega }}_{\mathrm{R}}-\frac{{\mathrm{\omega }}_{\mathrm{0}}}{{\mathrm{\gamma }}_{\mathrm{AT}}}-\frac{{\mathrm{\omega }}_{\mathrm{The}}}{{\mathrm{\gamma }}_{\mathrm{D}}}+{\mathrm{T}}_{\mathrm{P}}+\frac{\left(\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{AT}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}\right)\cdot \left({{\mathrm{\omega }}_{\mathrm{R}}}^{{}_{\mathrm{\theta }}\mathrm{2}}-\mathrm{2}\cdot {{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{\gamma }}\cdot {\mathrm{\omega }}_{\mathrm{R}}-{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{2}}\right)}{\mathrm{2}\cdot {\mathrm{\omega }}_{\mathrm{The}}}$$

The variation of T _R is determined from its derivative: $$\frac{{\mathrm{dT}}_{\mathrm{R}}}{{\mathrm{d\omega }}_{\mathrm{R}}}\left({\mathrm{\omega }}_{\mathrm{R}}\right)=\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{AT}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}-\frac{\left(\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{AT}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}\right)\cdot \left({{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{\gamma }}+{\mathrm{\omega }}_{\mathrm{R}}\right)}{{\mathrm{\omega }}_{\mathrm{The}}}=\left(\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{AT}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}\right)\cdot \left(\mathrm{1}-\frac{{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{\gamma }}+{\mathrm{\omega }}_{\mathrm{R}}}{{\mathrm{\omega }}_{\mathrm{The}}}\right)$$

By definition ω _R is greater than ω _L , it follows that the function T _R is monotonically decreasing on its definition space, that is to say ω _R in [ω _L , ω _R ^opt ].
We thus note that the time T _R is minimum when ω _R is maximum allowing to justify the choice of ${{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{Opt}}={-{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{\gamma }}+\sqrt{{{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="R\; \gamma "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R\; </figure-callout>}}}^{{\mathrm{\gamma }}^{}}+{{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="R\; \theta "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R\; </figure-callout>}}}^{{\mathrm{\theta }}^{}}}\mathrm{.}$

We then obtain: $$\boxed{{\mathrm{\omega }}_{\mathrm{R}}={{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{Opt}}=-{{\mathrm{\omega }}_{\mathrm{R}}}^{\mathrm{\gamma }}+\sqrt{{{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}}}^{{}_{\mathrm{\gamma }}\mathrm{2}}+{{\mathrm{\omega }}_{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}}}^{{}_{\mathrm{\theta }}\mathrm{2}}}=-\frac{{\mathrm{T}}_{\mathrm{P}}}{\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{AT}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}}+\sqrt{{\left(\frac{{\mathrm{T}}_{\mathrm{P}}}{\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{AT}}}+\frac{\mathrm{1}}{{\mathrm{<figure-callout\; id="2"\; label="\gamma \; D"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\gamma \; </figure-callout>}}_{\mathrm{D}}}}\right)}^{\mathrm{2}}+\frac{\mathrm{2}\cdot {\mathrm{\theta }}_{\mathrm{dd}}+\frac{{{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}}^{\mathrm{2}}}{{\mathrm{\gamma }}_{\mathrm{AT}}}+\frac{{{\mathrm{\omega }}_{\mathrm{The}}}^{\mathrm{2}}}{{\mathrm{\gamma }}_{\mathrm{D}}}}{\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{AT}}}+\frac{\mathrm{1}}{{\mathrm{\gamma }}_{\mathrm{D}}}}}}\mathrm{.}$$

Définissons la fonction du temps f telle que : $$\mathrm{f}\left(\mathrm{t}\right)=\mathrm{P}\left(\frac{\mathrm{t}}{{\mathrm{t}}_{\mathrm{D}}}\right)$$

Par définition, nous pouvons exprimer la position θ, la vitesse ω, l'accélération γ, et l'impulsion j à partir de la fonction f et de ses dérivées. $$\{\begin{array}{cc}\mathrm{\theta }& =\mathrm{f}\left(\mathrm{t}\right)\hfill \\ \mathrm{\omega }& =\mathrm{f}\prime \left(\mathrm{t}\right)\hfill \\ \mathrm{\gamma }& =\mathrm{f}"\left(\mathrm{t}\right)\hfill \\ \mathrm{j}& =\mathrm{f‴}\left(\mathrm{t}\right)\hfill \end{array}$$

avec les contraintes $$\{\begin{array}{cc}\mathrm{0}=\mathrm{f}\left(\mathrm{0}\right)\hfill & {\mathrm{\theta }}_{\mathrm{Dd}}=\mathrm{f}\left({\mathrm{t}}_{\mathrm{D}}\right)\hfill \\ {\mathrm{\omega }}_{\mathrm{0}}=\mathrm{f}\prime \left(\mathrm{0}\right)\hfill & {\mathrm{\omega }}_{\mathrm{L}}=\mathrm{f}\prime \left({\mathrm{t}}_{\mathrm{D}}\right)\hfill \\ {\mathrm{\gamma }}_{\mathrm{0}}=\mathrm{f}"\left(\mathrm{0}\right)\hfill & \mathrm{0}=\mathrm{f}"\left({\mathrm{t}}_{\mathrm{D}}\right)\hfill \\ & \mathrm{0}=\mathrm{f‴}\left({\mathrm{t}}_{\mathrm{D}}\right)\hfill \end{array}\mathrm{et}\left|\mathrm{\gamma }\right|<{\mathrm{\gamma }}_{\mathrm{MAX}},\mathrm{j}<{\mathrm{j}}_{\mathrm{MAX}}$$

In the second example, the speed ramps are calculated from a polynomial of order 6, a function of time. By construction, speed follows a continuous and non-linear profile. We also consider that the acceleration ramp to reach the optimal speed ω _R ^opt calculated is also the acceleration ramp RA of the initially planned control profile and that the deceleration ramp applied after reaches the optimum speed ω _R ^opt is also the deceleration ramp RD of the initially planned control profile. Consider the following order P polynomial 6: $$\mathrm{P}={\mathrm{at}}_{\mathrm{6}}\cdot {\mathrm{<figure-callout\; id="6"\; label="X"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">X</figure-callout>}}^{\mathrm{6}}+{\mathrm{at}}_{\mathrm{5}}\cdot {\mathrm{X}}^{\mathrm{5}}+{\mathrm{at}}_{\mathrm{4}}\cdot {\mathrm{<figure-callout\; id="4"\; label="X"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">X</figure-callout>}}^{\mathrm{4}}+{\mathrm{at}}_{\mathrm{3}}\cdot {\mathrm{X}}^{\mathrm{3}}+{\mathrm{at}}_{\mathrm{2}}\cdot {\mathrm{<figure-callout\; id="2"\; label="X"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">X</figure-callout>}}^{\mathrm{2}}+{\mathrm{at}}_{\mathrm{1}}\cdot \mathrm{X}+{\mathrm{at}}_{\mathrm{0}}$$

Define the function of time f such that: $$\mathrm{f}\left(\mathrm{t}\right)=\mathrm{P}\left(\frac{\mathrm{t}}{{\mathrm{t}}_{\mathrm{D}}}\right)$$

By definition, we can express the position θ, the velocity ω, the acceleration γ, and the pulse j from the function f and its derivatives. $$\{\begin{array}{cc}\mathrm{\theta }& =\mathrm{f}\left(\mathrm{t}\right)\hfill \\ \mathrm{\omega }& =\mathrm{f}\text{'}\left(\mathrm{t}\right)\hfill \\ \mathrm{\gamma }& =\mathrm{f}"\left(\mathrm{t}\right)\hfill \\ \mathrm{j}& =\mathrm{f\; ‴}\left(\mathrm{t}\right)\hfill \end{array}$$

with the constraints $$\{\begin{array}{cc}\mathrm{0}=\mathrm{<figure-callout\; id="0"\; label="f"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">f</figure-callout>}\left(\mathrm{0}\right)\hfill & {\mathrm{\theta }}_{\mathrm{dd}}=\mathrm{f}\left({\mathrm{<figure-callout\; id="0"\; label="t\; D\; \omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">t\; </figure-callout>}}_{\mathrm{D}},\right)\hfill & \\ {\mathrm{\omega }}_{}\hfill {\mathrm{}}_{\mathrm{0}}=\mathrm{f}\text{'}\left(\mathrm{0}\right)\hfill & {\mathrm{\omega }}_{\mathrm{The}}=\mathrm{f}\text{'}\left({\mathrm{<figure-callout\; id="0"\; label="t\; D\; \gamma "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">t\; </figure-callout>}}_{\mathrm{D}},\right)\hfill & \\ {\mathrm{\gamma }}_{}\hfill {\mathrm{}}_{\mathrm{0}}=\mathrm{f}"\left(\mathrm{0}\right)\hfill & \mathrm{0}=\mathrm{f}"\left({\mathrm{<figure-callout\; id="0"\; label="t\; D"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">t\; </figure-callout>}}_{\mathrm{D}}\right)\hfill \\ & \mathrm{0}=\mathrm{f\; ‴}\left({\mathrm{t}}_{\mathrm{D}}\right)\hfill \end{array}\mathrm{and}\left|\mathrm{\gamma }\right|<{\mathrm{\gamma }}_{\mathrm{MAX}},\mathrm{j}<{\mathrm{j}}_{\mathrm{MAX}}$$

(ω ₀ , γ ₀ ) represents the trajectory point at the moment of reception of the deceleration order, (ω _L , 0) represents the point to reach of the trajectory, and θ _Dd the distance to be traveled during the movement of deceleration, between the maximum speed and the low speed. t _D represents the deceleration time
The pair (ω ₀ , γ ₀ ) is obtained by the current position of the trajectory.

Il vient : $${\mathrm{a}}_{\mathrm{6}}=-\mathrm{10}\cdot {\mathrm{\theta }}_{\mathrm{Dd}}+\mathrm{6}\cdot {\mathrm{\omega }}_{\mathrm{L}}\cdot {\mathrm{t}}_{\mathrm{D}}+\mathrm{4}\cdot {\mathrm{\omega }}_{\mathrm{0}}\cdot {\mathrm{t}}_{\mathrm{D}}+\frac{\mathrm{1}}{\mathrm{2}}\cdot {\mathrm{\gamma }}_{\mathrm{0}}\cdot {{\mathrm{t}}_{\mathrm{D}}}^{\mathrm{2}}$$

$${\mathrm{a}}_5=\mathrm{36}\cdot {\mathrm{\theta }}_{\mathrm{Dd}}-\mathrm{21}\cdot {\mathrm{\omega }}_{\mathrm{L}}\cdot {\mathrm{t}}_{\mathrm{D}}-\mathrm{15}\cdot {\mathrm{\omega }}_{\mathrm{0}}\cdot {\mathrm{t}}_{\mathrm{D}}-\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{0}}\cdot {{\mathrm{t}}_{\mathrm{D}}}^{\mathrm{2}}$$

$${\mathrm{a}}_4=-45\cdot {\mathrm{\theta }}_{\mathrm{Dd}}+\mathrm{25}\cdot {\mathrm{\omega }}_{\mathrm{L}}\cdot {\mathrm{t}}_{\mathrm{D}}+\mathrm{20}\cdot {\mathrm{\omega }}_{\mathrm{0}}\cdot {\mathrm{t}}_{\mathrm{D}}+3\cdot {\mathrm{\gamma }}_{\mathrm{0}}\cdot {{\mathrm{t}}_{\mathrm{D}}}^{\mathrm{2}}$$

$${\mathrm{a}}_3=20\cdot {\mathrm{\theta }}_{\mathrm{Dd}}-10\cdot {\mathrm{\omega }}_{\mathrm{L}}\cdot {\mathrm{t}}_{\mathrm{D}}-\mathrm{10}\cdot {\mathrm{\omega }}_{\mathrm{0}}\cdot {\mathrm{t}}_{\mathrm{D}}-\mathrm{2}\cdot {\mathrm{\gamma }}_{\mathrm{0}}\cdot {{\mathrm{t}}_{\mathrm{D}}}^{\mathrm{2}}$$

$${\mathrm{a}}_{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}}\cdot {\mathrm{\gamma }}_{\mathrm{0}}\cdot {{\mathrm{t}}_{\mathrm{D}}}^{\mathrm{2}}$$

$${\mathrm{a}}_{\mathrm{1}}={\mathrm{\omega }}_{\mathrm{0}}\cdot {\mathrm{t}}_{\mathrm{D}}$$

$${\mathrm{a}}_{\mathrm{0}}=\mathrm{0}$$

The distance θ _Dd is known because it is the distance traveled during the first deceleration. If this distance θ _Dd is respected by the control profile, the stopping distance of the constraints will be.
We therefore have to find the coefficients of the polynomial P satisfying the constraints: $$\{\begin{array}{cc}\mathrm{0}=\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">P</figure-callout>}\left(\mathrm{0}\right)\hfill & {\mathrm{\theta }}_{\mathrm{dd}}=\mathrm{P}\left(\mathrm{1}\right)\hfill \\ {\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}\cdot {\mathrm{t}}_{\mathrm{D}}=\mathrm{P}\text{'}\left(\mathrm{0}\right)\hfill & {\mathrm{\omega }}_{\mathrm{The}}\cdot {\mathrm{t}}_{\mathrm{D}}=\mathrm{P}\text{'}\left(\mathrm{1}\right)\hfill \\ {\mathrm{<figure-callout\; id="0"\; label="\gamma "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\gamma </figure-callout>}}_{\mathrm{0}}\cdot {{\mathrm{<figure-callout\; id="2"\; label="t\; D"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">t\; </figure-callout>}}_{\mathrm{D}}}^{\mathrm{2}}=\mathrm{P}"\left(\mathrm{0}\right)\hfill & \mathrm{0}=\mathrm{P}"\left(\mathrm{1}\right)\hfill \\ & \mathrm{0}=\mathrm{P\; ‴}\left(\mathrm{1}\right)\hfill \end{array}$$

He comes : $${\mathrm{at}}_{\mathrm{6}}=-\mathrm{10}\cdot {\mathrm{\theta }}_{\mathrm{dd}}+\mathrm{6}\cdot {\mathrm{\omega }}_{\mathrm{The}}\cdot {\mathrm{t}}_{\mathrm{D}}+\mathrm{4}\cdot {\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}\cdot {\mathrm{t}}_{\mathrm{D}}+\frac{\mathrm{1}}{\mathrm{2}}\cdot {\mathrm{<figure-callout\; id="0"\; label="\gamma "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\gamma </figure-callout>}}_{\mathrm{0}}\cdot {{\mathrm{<figure-callout\; id="2"\; label="t\; D"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">t\; </figure-callout>}}_{\mathrm{D}}}^{\mathrm{2}}$$

$${\mathrm{at}}_5=\mathrm{36}\cdot {\mathrm{\theta }}_{\mathrm{dd}}-\mathrm{21}\cdot {\mathrm{\omega }}_{\mathrm{The}}\cdot {\mathrm{t}}_{\mathrm{D}}-\mathrm{15}<figure-callout\; id="0"\; label="\cdot \; \omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\cdot \; </figure-callout>{\mathrm{\omega }}_{}{\mathrm{}}_{\mathrm{0}}\cdot {\mathrm{t}}_{\mathrm{D}}-\mathrm{2}<figure-callout\; id="0"\; label="\cdot \; \gamma "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\cdot \; </figure-callout>{\mathrm{\gamma }}_{}{\mathrm{}}_{\mathrm{0}}\cdot {{\mathrm{<figure-callout\; id="2"\; label="t\; D"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">t\; </figure-callout>}}_{\mathrm{D}}}^{\mathrm{2}}$$

$${\mathrm{at}}_4=-45\cdot {\mathrm{\theta }}_{\mathrm{dd}}+\mathrm{25}\cdot {\mathrm{\omega }}_{\mathrm{The}}\cdot {\mathrm{t}}_{\mathrm{D}}+\mathrm{20}\cdot {\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}\cdot {\mathrm{t}}_{\mathrm{D}}+3\cdot {\mathrm{<figure-callout\; id="0"\; label="\gamma "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\gamma </figure-callout>}}_{\mathrm{0}}\cdot {{\mathrm{<figure-callout\; id="2"\; label="t\; D"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">t\; </figure-callout>}}_{\mathrm{D}}}^{\mathrm{2}}$$

$${\mathrm{at}}_3=20\cdot {\mathrm{\theta }}_{\mathrm{dd}}-10\cdot {\mathrm{\omega }}_{\mathrm{The}}\cdot {\mathrm{t}}_{\mathrm{D}}-\mathrm{10}<figure-callout\; id="0"\; label="\cdot \; \omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\cdot \; </figure-callout>{\mathrm{\omega }}_{}{\mathrm{}}_{\mathrm{0}}\cdot {\mathrm{t}}_{\mathrm{D}}-\mathrm{2}<figure-callout\; id="0"\; label="\cdot \; \gamma "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\cdot \; </figure-callout>{\mathrm{\gamma }}_{}{\mathrm{}}_{\mathrm{0}}\cdot {{\mathrm{<figure-callout\; id="2"\; label="t\; D"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">t\; </figure-callout>}}_{\mathrm{D}}}^{\mathrm{2}}$$

$${\mathrm{at}}_{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}}\cdot {\mathrm{<figure-callout\; id="0"\; label="\gamma "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\gamma </figure-callout>}}_{\mathrm{0}}\cdot {{\mathrm{<figure-callout\; id="2"\; label="t\; D"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">t\; </figure-callout>}}_{\mathrm{D}}}^{\mathrm{2}}$$

$${\mathrm{at}}_{\mathrm{1}}={\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{0}}\cdot {\mathrm{t}}_{\mathrm{D}}$$

$${\mathrm{at}}_{\mathrm{0}}=\mathrm{0}$$

By definition, the optimal speed reached during the motion is then defined by ω _R ^opt t _D = P '(x), where x is such that P "(x) = 0.

The optimum speed calculated by the first or second example is inserted in a new control profile determined by the drive when the deceleration command (FLG1) is received while the maximum speed ω _R provided in the initial control profile has not been set. not been reached. This second control profile is determined by taking into account the new optimum speed calculated ω _R ^opt , by respecting the two previously defined principles related to the accelerations and pulses to be applied in order to guarantee an optimal comfort to the user and taking into account the distance remaining to go.

• accélération jusqu'à la vitesse optimale ω_R ^opt calculée selon une nouvelle rampe d'accélération RA^opt tenant compte notamment de la distance restant à parcourir,
• décélération selon une nouvelle rampe de décélération RD^opt, tenant compte également de la distance restant à parcourir, jusqu'à atteindre la basse vitesse ω_L,
• réception de l'ordre d'arrêt (FLG2) par exemple à l'aide du second capteur externe placé sur le trajet de l'ascenseur,
• décélération selon la rampe d'arrêt RS jusqu'à l'arrêt complet de l'ascenseur à l'étage voulu.

This new control profile therefore comprises, after receiving the deceleration order (FLG1), the following steps:

• acceleration to the optimum speed ω _R ^opt calculated according to a new acceleration ramp RA ^opt taking into account in particular the distance remaining to be traveled,
• deceleration according to a new deceleration ramp RD ^opt , also taking into account the distance remaining to be traveled, until reaching the low speed ω _L ,
• receiving the stop command (FLG2) for example with the aid of the second external sensor placed on the path of the elevator,
• deceleration according to the RS stop ramp until the elevator stops at the desired floor.

The new ramps RA ^opt , RD ^opt calculated are of course non-linear to respect the constraints of comfort.

According to the invention, in certain cases, the initial ramps RA and RD can no longer be respected and it is necessary to determine new ramps to respect the imposed distance. For example, if the distance to be traveled is too great to reach the optimal speed ω _R ^opt when applying the initial acceleration ramp RA, it is necessary to determine a new ramp which will be steeper.

This new control profile can include in particular a step of maintaining the speed of the load at the optimum speed ω _R ^opt to create a step at this speed for a determined duration, between zero and several seconds, and a step of maintaining the speed of the load to the low speed ω _L for a certain period, which can range from zero to several seconds before receiving the stop instruction (FLG2).

Claims (12)

1. A control method implemented in a variable speed drive for controlling a lifting load, the control of the load being carried out according to a first control profile which comprises the following main steps:

   - acceleration of the load for reaching a first speed (ω_R) in accordance with a first non-linear acceleration ramp (RA),

   - deceleration of the load subsequent to the receipt of a deceleration order (FLG1),

   - stopping of the load,

   characterized in that when the load receives the deceleration order (FLG1) while being at a current speed below the first speed (ω_R), the method comprises:

   - a step of determining a second speed (ω_R ^opt) below the first speed (ω_R) and above the current speed, said second speed (ω_R ^opt) having an optimal value so as to minimize the travel time of the load until stopping,

   - a step of generating and applying a second control profile replacing the first control profile and comprising a step of accelerating the load until reaching the second speed (ω_R ^opt) according to a second non-linear acceleration ramp (RA^opt) taking account of the remaining distance to be traveled, followed by a step of maintaining the speed of the load at the second speed (ω_R ^opt) for a determined duration, a deceleration step and a stopping step.
2. The method as claimed in claim 1, characterized in that, between the deceleration step and the stopping step, the second control profile comprises a step of maintaining the speed of the load at a third speed (ω_L) below the second speed (ω_R ^opt).
3. The method as claimed in one of claims 1 or 2, characterized in that, on completion of the deceleration step, the second control profile comprises a step of receiving a stopping order (FLG2).
4. The method as claimed in claim 3, characterized in that after receipt of the stopping order (FLG2), the second control profile comprises a step of deceleration until stopping.
5. The method as claimed in claim 3 or 4, characterized in that the deceleration order (FLG1) or the stopping order (FLG2) is dispatched by a sensor in front of which the lifting load passes.
6. The method as claimed in claim 3 or 4, characterized in that the deceleration order (FLG1) or the stopping order (FLG2) is dispatched by an automaton connected to the variable speed drive.
7. A variable speed drive for controlling a lifting load, the control of the load being carried out according to a first control profile which comprises the following steps:

   - acceleration of the load for reaching a first speed (ω_R) in accordance with a first non-linear acceleration ramp (RA),

   - receipt of a deceleration order (FLG1),

   - deceleration of the load,

   - stopping of the load,

   characterized in that when the load receives the deceleration order (FLG1) at a current speed below the first speed (ω_R), the variable speed drive implements:

   - means for determining a second speed (ω_R ^opt) below the first speed (ω_R) and above the current speed, said second speed (ω_R ^opt) having an optimal value so as to minimize the travel time of the load until stopping,

   - means for generating and implementing a second control profile replacing the first control profile and comprising a step of accelerating the load until reaching the second speed (ω_R ^opt) according to a second non-linear acceleration ramp (RA^opt) taking account of the remaining distance to be traveled, followed by a step of maintaining the speed of the load at the second speed (ω_R ^opt) for a determined duration, a deceleration step and a stopping step.
8. The variable drive as claimed in claim 7, characterized in that the variable speed drive comprises means for maintaining the speed of the load at a third speed (ω_L) below the second speed (ω_R ^opt).
9. The variable drive as claimed in claim 7 or 8, characterized in that the second control profile comprises a receipt of a stopping order (FLG2).
10. The variable drive as claimed in claim 9, characterized in that the second control profile comprises a deceleration until stopping subsequent to the receipt of the stopping order.
11. The variable drive as claimed in claim 9 or 10, characterized in that it is connected to an external sensor able to dispatch the deceleration order (FLG1) or the stopping order (FLG2) when it detects the passage of the lifting load.
12. The variable drive as claimed in claim 9 or 10, characterized in that it is connected to an automaton able to dispatch the deceleration order (FLG1) or the stopping order (FLG2).

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