机器人运动学及动力学分析

velocities of the proximal and distal links in the ith kinematic 0 1 chain; and Q ----the rotation matrix that enables a 1 0 vector to rotate anticlockwise an angle 2 about the z axis
sin 1i T ， w i cos 2i
sin 2i T
According to the assembBiblioteka Baiduy mode, we can obtain the solution of Eq.(3) by (see Engineering Data Book)
1i ---- The position angles of the proximal link in the ith kinematic
r x y T of O can be expressed by
r sgn(i )ee1 l1ui l 2 wi ， i 1,2
y
Passive proximal link Active proximal link
Bracket
x
O Motor
Inner distal link Outer distal link
(3)
Taking dot product of w i on both sides of Eq.(6) and noticing that
w iT Qw i 0 , gives
where
Ai 2l1 y ， Bi 2l1 ( x sgn(i )e)
w T Qu w iT v l1 1i i i
Ci x 2 y 2 e 2 l12 l 2 2 2 sgn(i )ex
3
or
Dynamic Analysis of Mechanical Systems
Kinematic and Dynamics Analysis of a 2-DOF Parallel Robot
1i
Dynamic Analysis of Mechanical Systems
Kinematic and Dynamics Analysis of a 2-DOF Parallel Robot
Kinematic and Dynamic Analysis of a 2-DOF Parallel Robot
An assignment related problem 1. System Description Diamond (see the Figure 1) is a newly invented 2-DOF parallel robot module. The module can be integrated with 1-DOF feed mechanism to form a 3-DOF hybrid robot which is especially suitable for the pick-and-place operations of light objects at very
1i 2i
2 sgn(i )ex
2l1 ( x cos1i y sin 1i ) 2 sgn(i )el1 cos1i 0
This results in a triangle equation
Ai sin 1i Bi cos 1i C i 0 ， i 1,2
2
Fig.2 Kinematic model of the 2-DOF parallel robot
Dynamic Analysis of Mechanical Systems
Kinematic and Dynamics Analysis of a 2-DOF Parallel Robot
ui cos 1i
Movable platform
y
End-effector
e
O
12
22
e
21
l1 u1 A1
x
l1 u2
B2
A2
B1
11
r
l 2 w1
l2 w 2
(1)
O( x, y )
where l1 ， l 2 ， ui ， w i ---- the lengths and unit vectors of the proximal and distal links associated with the i th kinematic chain ; e ---- the distance between Ai and O , and
high-speed in a plane plus a long distance and relatively slow or step motion in the direction normal to that place. The Diamond robot consists is composed of a (base) carriage, a movable platform and two identical kinematic chains, each consisting essentially of two sets of parallelograms connected by a bracket in between. Driven independently by two servomotors situated on the base the rotations of two actuated proximal links (active links) provide the movable platform with a 2-DOF translational moving capability in a plane. The motion in the direction normal to the plane can be generated by the third servomotor-lead screw
Ci Bi
， i 1,2
(4)
i 1 i2
Hence, from Eq.(2) we can determine r sgn(i )ee1 l1ui , i 1,2 ui cos 1i sin 1i T , wi l2
(5)
Rewriting Eq.(1) in the form
1
Gear Reducer
y
z
x
Dynamic Analysis of Mechanical Systems
Kinematic and Dynamics Analysis of a 2-DOF Parallel Robot
assembly. Various types of end-effectors (e.g. mechanical gripper or vacuum absorber) can be mounted on the movable platform to conduct the pick-and-place operations. The main tasks of the assignment are concerned with kinematic and dynamic analysis of the 2-DOF parallel robot, including: (1) Inverse position, velocity and acceleration analyses. (2) Path planning for the point-to-point control. (3) Dynamic analysis for determining the driving torque and power for generating the specified motion of the end-effector. 2. Inverse Position Analysis Considering that the motion of the active and passive proximal links is identical, and so is that of the distal links thanks to the parallelogram structure, a revolute-jointed 5-bar simplified model as shown in Figure 2 can be used for the kinematic analysis. The inverse position analysis of the 2-DOF parallel robot is concerned with the determination of the position angles 1i ( i 1,2 ) of the driving link (The active proximal link) given the position of a reference point O of the moving platform. In the given coordinate system O xy , the position vector
2 2 r T r e 2 l1 2 sgn(i )er T e1 2l1r T ui 2 sgn(i )el1uiT e1 l 2
or
x y e
2 2 2 2 l1 2 l2
3. Velocity Analysis The inverse velocity analysis of the 2-DOF robot is concerned ( i 1,2 ) of the with the determination of the angular velocity 1i driving link given the velocity of the reference point O . Differentiating Eq.(1) with respect to time yields (See Section 2.3.3) (6) v l11i Qu i l 22 i Qw i and ---- the angular where v ---- the velocity of O ;
wiT v l1wiT Qui
,
i 1,2
(7)
In matrix form 1 w1 11 T l1 w1 Qu1 12
Base Motor
Motor Carriage Passive Proximal Link Lead screw Guide Active Proximal Link (Input link) Fig. 1 Diamond Robot Module Inner Distal Links Outer Distal Links Moving Platform 3-DOF Hybrid Robot
r sgn(i )ee1 l1ui l 2 wi
(2)
and taking norm on each sides of Eq.(2), leads to
r sgn(i)ee1 l1ui T r sgn(i)ee1 l1ui l2 wi T l2 wi
chain 2i ---- The position angles of the distal link in the ith kinematic chain
1 e1 1 0 ， sgn(i ) 1
T
1i 2 arctan
Ai sgn(i ) Ai 2 Ci 2 Bi 2