PATH PLANNING

PATH PLANNING

Scheda controllo assi e

pianificazione di traiettoria

+ PID -

PLC o PC o unità supervisione

MOTION CONTROL BOARD PIANIFICAZIONE

DI TRAIETTORIA

θ_�

TASK DA COMPIERE

Ex. Muovere da punto P1 a punto P2

Data Flow schema

q_r

•  Planning consists of

generating a time sequence of the values attained by an

interpolating function (typically a polynomial) of the desired

trajectory

A path denotes the locus of points in the joint space, or in the

operational space, which the

manipulator has to follow in the execution of the assigned motion

A trajectory is a path on which a timing law is specified, for instance in terms of velocities and/or

accelerations at each point:

q_r=q_r(t)

The planning algorithm generates a function q(t) interpolating the given vectors of joint variables at each point, in respect of the

imposed constraints.

• the generated trajectories should be not very demanding from a computational viewpoint,

• the joint positions and velocities should be continuous functions of time (continuity of accelerations may be imposed, too),

• undesirable effects should be minimized, e.g., nonsmooth trajectories interpolating a sequence of points on a path.

Pianificazione di traiettoria

PIANIFICATORE

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T

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Point-to-Point Motion

MOVE from an initial to a final joint configuration in a given time

minimizes the energy dissipated in the motor determine the solution to the

differential equation Condition

Pianificazione polinomio cubico

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θ_{������}

T

θ

= �

+ �

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�

+ �

�

Condizioni da imporre per ricavare

i coefficienti

θ_�

(

)

θ_�

(

)

θ_�´

(

)

(

)

Conditions:

Other kind of polynomials

Profilo di velocità trapezoidale

PARAMETRI

CHE SI RITROVANO

IN MOLTI CONTROLLORI INDUSTRIALI

ACCELEREZIONE MASSIMA

VELOCITÀ MASSIMA

Pendenza rampa trapezio

Altezza trapezio

Trapezoidal velocity profile

final time of trajectory duration has been assigned

or

the user is offered the option to specify the velocity percentage with respect to the maximum allowable velocity

The velocity at the end of the parabolic segment must be equal to the (constant) velocity of the linear segment, i.e.,

The position at the end of the parabolic segment must be equal to the initial linear segment, i.e.,

Combining the two equations

qm=(qi+qf)/2 tm=tf/2

Maximum acceleration is given

Condition on the root

In conclusion

Alternatively, one can specify also the cruise velocity Conditions on velocity

=

See notes

See notes

Problema del Jerk di accelerazione

●

Jerk

● Appare quando vi è una discontinuità dell'accelerazione

● Causa vibrazioni in quanto introduce un impulso di forza sul sistema

Profili di velocità alternativi

accelerazione trapezoidale

Profili di velocità alternativi continuità

sull'accelerazione

Pianificazione nello spazio operativo

θ₁

θ₂

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�

�

θ

,θ

θ

,θ

θ₁ θ₂

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SPAZIO OPERATIVO SPAZIO DEI GIUNTI

Pianificazione nello spazio operativo

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θ

,θ

θ

,θ

θ₁ θ₂

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SPAZIO OPERATIVO SPAZIO DEI GIUNTI

UNA RETTA NELLO SPAZIO OPERATIVO NON È UNA RETTA NELLO SPAZIO DEI GIUNTI

Introducing the concept of multy points

the interpolating polynomial of lowest order is the cubic polynomial, since it allows the imposition of continuity of velocities at the path points.

• Arbitrary values of q(t) are imposed at the path points.

• The values of q(t) at the path points are assigned according to a certain criterion.

• The acceleration has to be continuous at the path points.

Robot UR10

moveJ will make movements that are calculated in the joint space of the robot arm.

Each joint is controlled to reach the desired end location at the same time.

This movement type results in a curved path for the tool. The shared parameters that apply to this

movement type are the maximum joint speed and joint acceleration to use for the movement

calculations, specified in deg/s and deg/s², respectively.

If it is desired to have the robot arm move fast

between waypoints, disregarding the path of the tool between those waypoints, this movement type is the favorable choice.

Computation of the coefficients of the N - 1 cubic

polynomials interpolating the N path points is obtained by imposing posing the following conditions on the generic polynomial:

Criterion to choose the speed value

joint velocity at a path point has to be computed according to a certain criterion. By interpolating the path points with linear segments, the

relative velocities can be computed according to the following rules:

do not ensure continuity of accelerations at the path points.

Interpolating polynomials with continuous

accelerations at path points

(splines)

Via points

Consider the case when it is desired to interpolate N path points at time instants with linear segments. To avoid discontinuity problems on the first derivative at the

time instants tk, the function q(t) must have a parabolic profile (blend) around tk; as a consequence, the entire trajectory is composed of a sequence of linear and

quadratic polynomials

Velocity and acceleration for the intermediate points are computed as:

moveL will make the tool move linearly between waypoints. This means that each joint performs a more complicated motion to keep the tool on a

straight line path. The shared parameters that can be set for this movement type are the

desired tool speed and tool acceleration specified in mm/s and mm/s², respectively.

moveP will move the tool linearly with constant speed with circular blends, and is intended for some process operations, like gluing or dispensing. The size of the blend radius is by default a shared value between all the waypoints.

A smaller value will make the path turn sharper whereas a higher value will make the path smoother.