Date post: | 07-Nov-2015 |
Category: |
Documents |
Upload: | anca-maria-cautis |
View: | 16 times |
Download: | 0 times |
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.
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
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:
For
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:
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scan
ned
by C
amSc
anne
r
Scanned by CamScanner
Scan
ned
by C
amSc
anne
r
Scan
ned
by C
amSc
anne
r
Scan
ned
by C
amSc
anne
r
Scan
ned
by C
amSc
anne
r
Scan
ned
by C
amSc
anne
r
Scan
ned
by C
amSc
anne
r
Scan
ned
by C
amSc
anne
r
Scan
ned
by C
amSc
anne
r
Scan
ned
by C
amSc
anne
r
Scan
ned
by C
amSc
anne
r
Scan
ned
by C
amSc
anne
r
Scan
ned
by C
amSc
anne
r
Scan
ned
by C
amSc
anne
r
Scan
ned
by C
amSc
anne
r
Scan
ned
by C
amSc
anne
r
Scan
ned
by C
amSc
anne
r
CuprinsCurs 1-3Curs 4 - Trajectory planningCurs5-part1Curs5-part2Curs6Curs6-senzoriCurs7Curs8Material suplimentar 1Material suplimentar 2 rotated