robo - all in one (1-8)

Curs 1-3 Curs 4 - Trajectory planning Curs 5 part 1 Curs 5 part 2 Curs 6 Curs 6 senzori Curs 7 Curs 8 Material suplimentar 1 Material suplimentar 2 La ROBO dam partial din cursurile 1 si 2, de asemenea din cursul 4 avem polinoamele de grad 3 si 5, trapezul, iar din cursul 5 avem partea de urmarire a traiectoriei, partea de control. Din cursul 6 avem motorul de curent continuu, sisteme automate si calculul erorilor, urmarirea traiectoriei. Din cursul 7 avem traductoare si senzori. Din cursul 3 avem doar ecuatia dinamicii asociate. Nu intra robotii dinamici. Subiectele sunt grupate in fc de gradul de dificultate.

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Kinematics: relationships in terms of position / velocity between the joint and work-space.

Dynamics: relationships between the torques applied to the joints (mass of the rigid body) and the consequent movements of the links.

Trajectory planning: planning of the desired

movements of the manipulator taking time into consideration

Planning the trajectories : Defining the points on the trajectory:

point-to-point

pre-defined path

In regard to the work space: joint space trajectory planning;

operational space trajectory planning

Trajectory planning includes: path planning

definition of a motion law

applying constrains (ex: path, continuity, resonant modes)

Path: geometrical place of points in the space (either joint or operational).

Geometrical description of motion

Defined in joint space or work space Trajectory: a path completed with a motion (time)

law

Motion law: velocities and accelerations associated to path points.

P

P

s=s(t)

Trajectory in work space

Define path: initial point ->

final point

Define total time of movement

Calculate discrete path

Blend a continuous time

function

Solve inverse kinematics

Advantages Geometrical constrains

Disadvantages Inverse kinematics calculated each step. Total time hard to compute

Calculate inverse kinematics for

path points

Define total time in regard to max

velocities of joints

Calculate discrete path

Blend a continuous time

function

Disadvantages Difficult to model operational space obstacles.

Advantages Inverse kinematics is calculated at the beginning Calculates directly joint angle, and velocities

Path in joint space:

defining initial, intermediate and final values for the joint variables

assigning a desired motion law.

Motion law = continuous functions ( superior order of derivations as to be able to calculate velocity and acceleration)

Motion law usually defined as polynomial functions a of n degree (usually n: 1-5):

Trajectory planning algorithm

Points on path

Geometrical Constrains

Mechanical Constrains

Trajectory in joint space

Trajectory in work space

INPUT OUTPUT

Characteristics of the function that interpolates the given points: the motion law must be continuous functions of time numerical calculation efficiency effect of calculation constrains must be minimized or completely avoided.

Polinoame de ordin 3 Conditii : Initiale si finale Pozitia si viteza initiala Pozitia si viteza finala

Conditiile la limita aplicate:

Polinom de ordin 5 :

Se pot pune conditii legate de

pozitie

viteza

acceleratie 6 conditii la limita:

Pentru

Doua tipuri de segmente Segment liniar-> viteza constanta Segment parabolic -> viteza este o functie liniara

Traiectorie trapezoidala: Primul si ultimul segment

acceleratie / deceleratie constanta Viteza liniara Pozitia parabola

Al doilea segment Acceleratia este nula Viteza este constanta Pozitia variaza liniara in timp

Acceleration segment

Boundary conditions: initial position

initial velocity

final velocity = constant velocity

for second segment

Constant velocity phase Boundry conditions: Constant velocity from the first segment Final position from fist segment = initial position for

second segment . .

Deceleration phase

Boundary conditions:

final position

final velocity

initial velocity = constant velocity for second segment

Initial position = final position for second segment

Additional constrains (necessary to solve the equation)

duration of the acceleration/deceleration segment

similar conditions

Define maximum

acceleration

Calculate duration of

acceleration

A function interpolating a set of n points can be represented with a polynomial function of degree n 1.

Not a convenient solution

2 points = unique line 3 points = unique quadric ... n points = unique polynomial with degree n 1

Calculating n degree poliyom Lagrange expression for polynomial equation:

Calculating n degree poliyom using Matrix procedure

To avoid problem of n degree polynomial equation we use n 1 polynomials with lower degree p (p < n 1), each polynomial interpolates a segment of the trajectory.

P=3

4 coefficients for each polynomial, Calculate 4(n 1) coefficients

4(n 1) coefficients - 2(n 1) conditions on the position (initial/final points); - n 2 conditions on the continuity of velocity

(intermediate points); - n 2 conditions on the continuity of acceleration

(intermediate points);

Result 4(n 1) 2(n 1) 2(n 2) = 2

degrees of freedom left to put extra conditions

P degree polynom

P degree polynom

..

..

Calculating the parameters The systems:

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Scan

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CuprinsCurs 1-3Curs 4 - Trajectory planningCurs5-part1Curs5-part2Curs6Curs6-senzoriCurs7Curs8Material suplimentar 1Material suplimentar 2 rotated

robo - all in one (1-8)

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