robo - all in one (1-8)
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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
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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)
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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
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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
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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
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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):
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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.
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Polinoame de ordin 3 Conditii : Initiale si finale Pozitia si viteza initiala Pozitia si viteza finala
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Conditiile la limita aplicate:
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For
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Polinom de ordin 5 :
Se pot pune conditii legate de
pozitie
viteza
acceleratie 6 conditii la limita:
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Pentru
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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
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Acceleration segment
Boundary conditions: initial position
initial velocity
final velocity = constant velocity
for second segment
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Constant velocity phase Boundry conditions: Constant velocity from the first segment Final position from fist segment = initial position for
second segment . .
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Deceleration phase
Boundary conditions:
final position
final velocity
initial velocity = constant velocity for second segment
Initial position = final position for second segment
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Additional constrains (necessary to solve the equation)
duration of the acceleration/deceleration segment
similar conditions
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Define maximum
acceleration
Calculate duration of
acceleration
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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
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Calculating n degree poliyom Lagrange expression for polynomial equation:
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Calculating n degree poliyom using Matrix procedure
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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
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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
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P degree polynom
P degree polynom
..
..
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Calculating the parameters The systems:
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CuprinsCurs 1-3Curs 4 - Trajectory planningCurs5-part1Curs5-part2Curs6Curs6-senzoriCurs7Curs8Material suplimentar 1Material suplimentar 2 rotated