WO2010136586A1 - Procédé et dispositif de planification de trajectoire d'ordre 4, exploitable en temps réel, pour générer des trajectoires de consigne continues, sans à-coups - Google Patents

Procédé et dispositif de planification de trajectoire d'ordre 4, exploitable en temps réel, pour générer des trajectoires de consigne continues, sans à-coups Download PDF

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Publication number
WO2010136586A1
WO2010136586A1 PCT/EP2010/057470 EP2010057470W WO2010136586A1 WO 2010136586 A1 WO2010136586 A1 WO 2010136586A1 EP 2010057470 W EP2010057470 W EP 2010057470W WO 2010136586 A1 WO2010136586 A1 WO 2010136586A1
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WIPO (PCT)
Prior art keywords
time
jerk
dimensional
path
kinematic
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PCT/EP2010/057470
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German (de)
English (en)
Inventor
Arvid Amthor
Stephan ZSCHÄCK
Christoph Ament
Andreas Lorenz
Johannes Werner
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Technische Universität Ilmenau
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Publication of WO2010136586A1 publication Critical patent/WO2010136586A1/fr

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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B19/00—Programme-control systems
    • G05B19/02—Programme-control systems electric
    • G05B19/18—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form
    • G05B19/416—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by control of velocity, acceleration or deceleration
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1656—Programme controls characterised by programming, planning systems for manipulators
    • B25J9/1664—Programme controls characterised by programming, planning systems for manipulators characterised by motion, path, trajectory planning

Definitions

  • the present invention relates to a method and an apparatus for real-time path planning of the fourth order for generating continuous, jerk-free, time-dependent setpoint trajectories.
  • the invention finds application in robotics as well as in control and automation technology.
  • the invention relates to an actuator control device for moving an actuator along a predetermined path.
  • path planning used here describes methods and devices for moving an actuator, in particular in robotics, as described, inter alia, in Sawodny, 0 .: “Crane automation and large-scale robotics” in: Progress Report, Series 8, No. 888, VDI Verlag, Dusseldorf, 2001, is defined.
  • Olomski's jerk profile was included in analytical trajectory planning (Olomski, J .: “Trajectory Planning, Optimization and Control for Industrial Robots", International Conference on Control and Applications, Proceedings, Jerusalem, 1989).
  • the kinematic behavior of the system is determined in this approach by a piecewise constant jerk function (also Leonhard, W .: “Trajectory Control of a Multi-axes Robot with Electrical Servo Drives”: IEEE Conference on Industrial Electronics, Proceedings, Philadelphia, USA, 1989).
  • the trajectories for acceleration, speed and position are then obtained by multiple integration of this jerk function.
  • Croft, EA "Jerk-bound Manipulator Trajectory Planning: Design for Real-Time Application” in IEEE Transactions on Robotics and Automation, Vol. 19, No. 1, Feb. 2003, shows a real-time method for determining trajectories of a robot in which fifth-order polynomials are concatenated.
  • Object of the present invention is therefore to overcome the disadvantages of the prior art and a method and apparatus for real-time path planning fourth order for generating continuous, jerk-free setpoint trajectories for kinematic parameters, such as travel, speed, acceleration, jerk and jerk to provide in consideration of kinematic restrictions. Furthermore, an actuator control unit to provide direction for moving an actuator according to the above-mentioned path planning.
  • the method according to the invention is used for path planning in order to move an actuator of a robot or the like on a predetermined path taking into account kinematic restrictions.
  • the method can be executed in real time on the hardware customary for path planning at the time of the invention.
  • a time-dependent, three-dimensional target value trajectory for the planned path is determined, which is transmitted to a device for operating the actuator, for example to a state controller, which acts on a motor of the actuator with an electrical voltage.
  • a fourth-order orbit planning ie the path planning takes place taking into account the first four temporal derivatives of the place or the way, namely the speed, the acceleration, the jerk and the Ruckanieries.
  • the time-dependent setpoint trajectories to be determined are continuous and jerk-free.
  • kinematic restrictions are considered for the path, for the speed, for the acceleration, for the jerk and / or for the jerk rise, ie there are, inter alia, start and end points on the way, predetermined limits of speed, acceleration, jerk and / or Ruckanctiones processed as input variables.
  • Ruckanctiones in particular restrictions of the Ruckanctiones be considered.
  • the kinematic restrictions initially result from the actuator's technically executable movements, which are limited in terms of speed, acceleration, jerk and jerk.
  • the given kinematic restrictions up to the fourth order for the individual axes of motion are projected onto the one-dimensional function of the web location to determine the kinematic parameter jerk rise as a one-dimensional function of time related to the web.
  • the kinematic restrictions for the location can be applied directly to the one-dimensional anal function of the railway location.
  • the kinematic restrictions for speed, acceleration, jerk and jerk can be determined apply to the time derivatives of the one-dimensional function of the web site, preferably using the Frenet formulas.
  • the one-dimensional time-dependent function can first be determined for the fourth derivative of the web location, namely for the jerk rise as a kinematic parameter.
  • This one-dimensional function describes the chronological progression of the recoil with respect to the web.
  • This function can also be used to determine one-dimensional functions for the other kinematic parameters: jerk, acceleration, speed and path through integration.
  • the time-dependent three-dimensional setpoint trajectory is determined at least for the kinematic parameter path, namely on the basis of the previously calculated one-dimensional function of the time, which describes the kinematic parameter jerk rise.
  • the kinematic parameter path is preferably first calculated as a one-dimensional function of time related to the path from the one-dimensional function of time describing the kinematic parameter jerk rise, which relates to the path.
  • the time-dependent, three-dimensional setpoint trajectory for the kinematic parameter path is subsequently transmitted to a device for operating the actuator.
  • the calculation of the kinematic parameter path as the one-dimensional function of the time related to the track is preferably carried out by first successively determining the kinematic parameters jerk, acceleration and velocity as one-dimensional functions of the time related to the path the one-dimensional function of time describing the kinematic parameter jerk rise, which is related to the orbit.
  • Many devices for operating an actuator also process time-dependent, three-dimensional setpoint trajectories for the kinematic parameters jerk rise, jerk, acceleration and / or speed, in particular for the kinematic parameters acceleration and speed. Therefore, in a preferred embodiment of the method according to the invention further time-dependent, three-dimensional setpoint trajectories for the kinematic parameters jerk, jerk, acceleration and / or speed from the kinematic parameters jerk, jerk, acceleration and / or speed descriptive, the web related one-dimensional functions Time determined. Particularly preferably, the time-dependent, three-dimensional setpoint trajectories for the kinematic parameters acceleration and speed are determined.
  • this serves for real-time path planning fourth order for generating continuous, jerk-free setpoint trajectories for the kinematic parameters way, speed, acceleration, jerk and jerk rise taking into account kinematic restrictions.
  • This embodiment initially comprises a step in which the dimension of a three-dimensional setpoint trajectory is reduced to a one-dimensional function of the web location.
  • the kinematic restrictions for the individual axes of motion are projected onto the one-dimensional function of the web location.
  • a step is provided in which the kinematic Parameters for the one-dimensional function of the railway location are calculated.
  • the three-dimensional setpoint trajectory for the kinematic parameters path, speed, acceleration, jerk and jerk rise is determined.
  • the given path is subdivided into several segments, each segment being described by a time-independent, three-dimensional setpoint trajectory.
  • the steps specified for the method according to the invention are carried out for each of the plurality of time-independent, three-dimensional setpoint trajectories.
  • the segments are preferably formed in each case by a straight line or by a circular arc.
  • a slicing element is inserted between each two of the one-dimensional functions of two successive segments.
  • the wear elements are preferably formed by an extended bios curve in each case.
  • Reducing the dimension of the one or more time-independent, three-dimensional setpoint trajectories describing the given trajectory is preferably done using Frenet's formulas.
  • integration is preferably carried out by means of Gaussian-Legendric quadrature.
  • the determined time-dependent, three-dimensional setpoint trajectory for the kinematic parameter path is transmitted to a controller for regulating the state of an actuator to be moved along the path.
  • the time-dependent, three-dimensional setpoint trajectories for the kinematic parameters jerk rise, jerk, acceleration and / or speed are transmitted to the controller;
  • Particularly preferred are the time-dependent, three-dimensional setpoint trajectories for the kinematic parameters acceleration and speed.
  • the time-dependent, three-dimensional setpoint trajectory for the kinematic parameter velocity is preferably determined using an asymmetrical profile.
  • the device according to the invention is used for real-time capable path planning in order to move an actuator of a robot or the like on a predetermined path taking into account kinematic restrictions.
  • the device is configured to carry out the method according to the invention.
  • the device preferably has a data input for reading in data which describe the given path, in particular in the form of time-independent, three-dimensional setpoint trajectories.
  • the device preferably has a control output for outputting the time-dependent, three-dimensional setpoint trajectory for the kinematic parameter path, so that, for example, a controller to be connected can be controlled to control a state of the actuator.
  • the device according to the invention is preferably formed by a programmable computer.
  • the actuator control device serves to move an actuator along a predetermined path.
  • the actuator control device comprises a controller for regulating a state of the actuator and a device according to the invention for path planning.
  • the path planning device is connected to the controller to control the device from the device Path planning determined time-dependent, three-dimensional SoIl werttraj ektorie for the kinematic parameter way to the controller to transmit.
  • the procedure in the analytical path planning according to the invention can be divided into three phases.
  • a so-called dimension reduction takes place.
  • the kinematic calculation of the path parameter takes place in one dimension.
  • the kinematic trajectories of the individual motion axes are determined in the third step. This procedure offers the decisive advantage that the partial movements of the individual machine axes inevitably behave synchronously with each other.
  • FIG. 2 shows an example of a jerk rise function determined according to the invention
  • 3 shows an example of the inventively determined course of jerk, acceleration, speed and location.
  • Fig. 6 is a schematic diagram for determining the element transfer speeds
  • FIG. 13 shows a block diagram of a preferred embodiment of an actuator control device according to the invention.
  • Fig. 14 is a block diagram of another preferred embodiment
  • the so-called frenet tripod including the associated scalar magnitudes curvature and torsion can be derived from the vector-valued position function.
  • This orthonormal base (see FIG. 1) forms the general foundation for the closed description of spatial curves and will therefore be introduced first below.
  • Fig. 1 shows the three vectors that make up the Frenetian tripod for a freely chosen point on the depicted space curve.
  • Arrow t here represents the tangent vector ⁇
  • arrow n represents the normal vector n and arrow b embodied the binormal vector.
  • the tangent vector is tangent to the trajectory and indicates the instantaneous direction of movement for the examined curve point.
  • the torsion indicates the amount of tilting of the planing plane and can therefore be obtained by means of the following rule:
  • Equations (1.8) to (1.11) now provide vector-valued functions for all investigated kinematic variables.
  • the conversion into the kinematics of the trajectory can take place at any time by inserting the scalar path sizes v, a, j, d determined by the path planning.
  • Equation (1.13) now gives a relation between the location s and the distance derived. Thus, at any point in time, a path location predetermined by the path planning can not be calculated into the corresponding point in R 3 . Equation (1.13) is used to create the prerequisites for applying the Frenet formulas for the basic element distance. The derivation of equation (1.13) according to the location of the track results in the following relationship:
  • the trajectory kinematics can be calculated by a simple multiplication of the respective path size with the normalized direction vector.
  • each point s on the circular arc is rotated by one rotation R of the radius vector Angle ⁇ and subsequent addition of the circle center can be determined. Since the axis of rotation of a Kreis ⁇ remains constant, R depends only on the angle of rotation, which can therefore also be regarded as a curve parameter.
  • the conversion between the web location s on the circular arc and the curve parameter ⁇ can be done by a simple rule of three and leads to the following relationship:
  • the base transformation can be formulated as follows:
  • Another differentiation step provides the normal vector of the circular arc in the transformed coordinate system:
  • the derivation provides a constant K which corresponds to the reciprocal of the circle radius r and thus corresponds exactly to the general definition of the circular curvature. Since the arc represents a flat curve, there is no tilt of the corresponding plane normal and the torsion is eliminated. It should also be noted at this point that the two coefficients K and ⁇ represent scalar quantities and therefore the transformation of the coordinate system to its value has no effect. For this reason, both sizes remain valid even in the original coordinate system, which can be recognized by the absence of the tilde.
  • the kinematic variables consist exclusively of direction vectors, for which only the rotational modification has to be reversed.
  • the translational shift in the return ⁇ transformation can be omitted and it follows:
  • a direct conversion of Bru kinematics to the korrespondie ⁇ -saving Traj ektorienkinematik for the line and the circle is possible.
  • Prerequisite for the applicability is a representation of the trajectory as an at least three times differentiable vector-valued function of the railway location s.
  • the corresponding back kinematics of the trajectory are calculated back.
  • the valid kinematic restrictions must be determined.
  • v max is now the largest value that satisfies all three inequalities:
  • Equations (2.11) through (2.14) allow the calculation of the train restrictions based on the direction vector of the route and the axis restrictions.
  • a first solution step of the alternative numerical approach is to calculate the vector lengths.
  • the basis for the suggested solution path is the fact that by means of the equations (1.42) to (1.45) it is possible to calculate the length of the respective resulting kinematics vector without the use of trigonometric functions.
  • the vectors Fund ⁇ are perpendicular to each other and represent normalized basis vectors. These are scaled by the coefficients in equations (1.37) to (1.40). This results in a right-angled triangle in which the scaled basis vectors are the catheters while the result of the vector addition forms the hypotenuse.
  • the Pythagorean theorem it is possible to use the Pythagorean theorem to determine the length of the resulting kinematics vectors:
  • the permissible scaling can also be determined here by selecting the smallest quotient:
  • the one-dimensional kinematic planning of the path parameter is presented below.
  • the goal of this planning component is to cover a defined railway line as well as possible in one dimension and in compliance with specified restrictions.
  • a fourth-order path planning method has been developed.
  • the aim of the method steps presented below is to enable a sequential maximization of the kinematic variables on the model of the constant-jerk method.
  • an algorithm has been found which allows to increase the overall dynamics of the positioning over several path segments. This is only possible if the kinematic system does not have to stop at each transition point between the individual parts of the path. This can be achieved by smoothing the transitions, which means that changes in direction take place slowly and continuously and can therefore be traversed at a relatively high speed.
  • the interval labels Il to 115 are assigned the respective bounds. Based on the calculation rule (3.1) and Table 1, a subdivision of the function into fifteen subareas can be seen. The constants T 1 to T 1 are thereby symbolic of the seven occurring basic interval widths, giving rise the actual segmentation of the domain. These seven values can be determined a priori for each path segment and remain immutable thereafter. The calculation of T 1 to T 1 is based on the given restrictions and the path length. Closer examination of equation (3.1) reveals that the subfunctions in the odd numbered intervals represent second degree polynomials. The resulting parabolas differ in their vertex and in their opening direction. In addition, the parabolic width also varies between the acceleration and braking phases.
  • the mentioned parameters were selected in such a way that the vertex in the direction of the abscissa lies exactly in the respective interval center and the associated function value exactly corresponds in its amount to the jerk rise restriction ⁇ i max .
  • the vertex in the direction of the abscissa lies exactly in the respective interval center and the associated function value exactly corresponds in its amount to the jerk rise restriction ⁇ i max .
  • the parabola width causes these zeros to fall exactly on the interval boundaries and thus form a continuous transition to the zero functions in the even-numbered intervals.
  • the core component of the planning algorithm is the determination of the basic interval widths T 1 to T 1 , since these directly determine the position of the extremes of the kinematic functions.
  • the requirements of time optimality and compliance with the restrictions can only be met by bringing one kinematic quantity after another as close as possible to their restriction pointed out that from the perspective of the interval width determination nebe n ⁇ ?, /, ⁇ v , v v also s ael represents a restriction.
  • the interval width of the jerk-up phase during acceleration is determined in a first step. Limiting to the corresponding value T 1 , the restriction first achieved by the resulting kinematic functions has an effect. This can be represented by the following inequalities:
  • the optimization problem can be formulated as follows:
  • the path planner now generates the kinematic progressions on the orbital plane for each individual element. In this case, all higher time derivatives of the speed at the beginning and end of the element are zero and therefore do not need to be considered along the entire route. However, this does not apply to the web speed, as this should have a continuous course over the entire route.
  • the aim now is to develop a method which allows these start and end speeds and thus the path planning algorithm to determine the kinematics element by element, without having to consider the total distance.
  • an element is identified by the seven variables s target , v max , ⁇ max , y max , ⁇ i max , v start and v target . These variables of the j 'th element distance can become an element vector
  • start and end speeds may be smaller than v max , which corresponds to the first case in FIG. 5.
  • the starting speed is smaller and the final speed is greater than or equal to the path speed limit.
  • v start is greater than or equal to v max and v target is greater than max .
  • the calculated final speed is adopted and the current starting speed is not modified, furthermore, the final speed is first as start speed of the following element assuming that: Applies in the second case so will set.
  • the start speed of the current element remains unchanged. In the latter case, the starting speed becomes too established.
  • the final speed of the previous track element must be limited so that the course of the web speeds along the entire route is continuous.

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  • Engineering & Computer Science (AREA)
  • Robotics (AREA)
  • Mechanical Engineering (AREA)
  • Human Computer Interaction (AREA)
  • Manufacturing & Machinery (AREA)
  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Automation & Control Theory (AREA)
  • Management, Administration, Business Operations System, And Electronic Commerce (AREA)
  • Feedback Control In General (AREA)

Abstract

L'invention concerne un procédé et un dispositif de planification de trajectoire d'ordre 4, exploitable en temps réel, pour générer des trajectoires de consigne continues, sans à-coups, dépendantes du temps. L'invention concerne en outre un dispositif de commande d'actionneur servant à déplacer un actionneur le long d'une trajectoire prédéfinie. Selon l'invention, la dimension d'une trajectoire de consigne tridimensionnelle, indépendante du temps, décrivant une trajectoire donnée, est réduite à une fonction unidimensionnelle de l'emplacement de la trajectoire. Des restrictions cinématiques pouvant aller jusqu'à l'ordre 4 sont projetées sur la fonction unidimensionnelle de l'emplacement de la trajectoire pour déterminer le paramètre cinématique "augmentation des à-coups" en tant que fonction unidimensionnelle du temps, rapportée à la trajectoire, à partir de laquelle la trajectoire de consigne tridimensionnelle, dépendante du temps, est déterminée pour le paramètre cinématique "déplacement".
PCT/EP2010/057470 2009-05-29 2010-05-28 Procédé et dispositif de planification de trajectoire d'ordre 4, exploitable en temps réel, pour générer des trajectoires de consigne continues, sans à-coups WO2010136586A1 (fr)

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DE200910024130 DE102009024130B4 (de) 2009-05-29 2009-05-29 Verfahren zur echtzeitfähigen Bahnplanung kontinuierlicher, rucksprungfreier Sollwerttrajektorien
DE102009024130.2 2009-05-29

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