Robotics 1 Direct kinematics

Transcription

1 Robotics 1 Direct kinematics Prof. Alessandro De Luca Robotics 1 1

2 Kinematics of robot manipulators study of... geometric and timing aspects of robot motion, without reference to the causes producing it robot seen as... an (open) kinematic chain of rigid bodies interconnected by (revolute or prismatic) joints Robotics 1 2

3 functional aspects Motivations definition of robot workspace calibration operational aspects task execution (actuation by motors) task definition and performance two different spaces related by kinematic (and dynamic) maps trajectory planning programming motion control Robotics 1 3

4 Kinematics formulation and parameterizations JOINT space r = f(q) DIRECT INVERSE TASK (Cartesian) space q = (q 1,,q n ) q = f -1 (r) r = (r 1,,r m ) choice of parameterization q unambiguous and minimal characterization of robot configuration n = # degrees of freedom (dof) = # robot joints (rotational or translational) choice of parameterization r compact description of position and/or orientation (pose) variables of interest to the required task usually, m n and m 6 (but none of these is strictly necessary) Robotics 1 4

5 Open kinematic chains q 2 q n RF E q 1 q 3 q 4 e.g., the relative angle between a link and the following one r = (r 1,,r m ) e.g., it describes the pose of frame RF E m = 2 m = 5 pointing in space positioning and pointing in space positioning in the plane m = 6 m = 3 orientation in space positioning and orientation in space (like for spot welding) positioning and orientation in the plane positioning of two points in space (e.g., end-effector and elbow) Robotics 1 5

6 Classification by kinematic type (first 3 dofs) SCARA (RRP) Cartesian or gantry (PPP) cylindric (RPP) polar or spherical (RRP) articulated or anthropomorphic (RRR) R = 1-dof rotational (revolute) joint P = 1-dof translational (prismatic) joint Robotics 1 6

7 Direct kinematic map the structure of the direct kinematics function depends from the chosen r r = f r (q) methods for computing f r (q) geometric/by inspection systematic: assigning frames attached to the robot links and using homogeneous transformation matrices Robotics 1 7

8 y Example: direct kinematics of 2R arm l 1 l 2 q 1 q 2 P φ p q = q 1 y p x x r = q 2 p x p y φ p x = l 1 cos q 1 + l 2 cos(q 1 +q 2 ) p y = l 1 sin q 1 + l 2 sin(q 1 +q 2 ) φ = q 1 + q 2 for more general cases, we need a method! n = 2 m = 3 Robotics 1 8

9 Numbering links and joints link 1 joint 2 link i-1 joint i+1 joint n joint 1 link 0 (base) joint i-1 joint i link i link n (end effector) revolute prismatic Robotics 1 9

10 Spatial relation between joint axes axis of joint i axis of joint i+1 90 π A α i 90 B common normal (axis of link i) a i = displacement AB between joint axes (always well defined) α i = twist angle between joint axes projected on a plane π orthogonal to the link axis with sign (pos/neg)! Robotics 1 10

11 Spatial relation between link axes link i-1 axis of joint i link i D C axis of link i-1 θ i σ axis of link i d i = displacement CD (a variable if joint i is prismatic) θ i = angle between link axes (a variable if joint i is revolute) projected on a plane σ orthogonal to the joint axis with sign (pos/neg)! Robotics 1 11

12 Denavit-Hartenberg (DH) frames joint axis i-1 link i-1 joint axis i link i joint axis i+1 link i-1 is moved by joint i-1 frame RF i is attached to link i α i a i z i d i z i-1 x i-1 O i x i θ i O i-1 axis around which the link rotates or along which the link slides common normal to joint axes i and i+1 Robotics 1 12

13 axis of joint i-1 Denavit-Hartenberg parameters link i-1 axis of joint i D link i a i axis of joint i+1 z i α i d i z i-1 x i-1 O i x i θ i O i-1 unit vector z i along axis of joint i+1 unit vector x i along the common normal to joint i and i+1 axes (i i+1) a i = distance DO i positive if oriented as x i (constant = length of link i) d i = distance O i-1 D positive if oriented as z i-1 (variable if joint i is PRISMATIC) α i = twist angle between z i-1 and z i around x i (constant) θ i = angle between x i-1 and x i around z i-i (variable if joint i is REVOLUTE) Robotics 1 13

14 Denavit-Hartenberg layout made simple (a popular 3-minute illustration...) video note: the authors of this video use r in place of a, and do not add subscripts! Robotics 1 14

15 Ambiguities in defining DH frames frame 0 : origin and x 0 axis are arbitrary frame n : z n axis is not specified (but x n must be orthogonal to and intersect z n-1 ) positive direction of z i-1 (up/down on joint i) is arbitrary choose one, and try to avoid flipping over to the next one positive direction of x i (on axis of link i) is arbitrary we often take x i = z i-1 z i when successive joint axes are incident when natural, we follow the direction from base to tip if z i-1 and z i are parallel: common normal not uniquely defined O i is chosen arbitrarily along z i, but try to zero out parameters if z i-1 and z i are coincident: normal x i axis may be chosen at will again, we try to use simple constant angles (0, π/2) this case may occur only if the two joints are of different kind (P & R) Robotics 1 15

16 Homogeneous transformation between successive DH frames (from frame i-1 to frame i) roto-translation around and along z i-1 cθ i -sθ i sθ i cθ i i-1 A i (q i ) = d = i cθ i -sθ i 0 0 sθ i cθ i d i rotational joint q i = θ i prismatic joint q i = d i roto-translation around and along x i i A i = a i 0 cα i -sα i 0 0 sα i cα i always a constant matrix Robotics 1 16

17 Denavit-Hartenberg matrix J. Denavit and R.S. Hartenberg, A kinematic notation for lower-pair mechanisms based on matrices, Trans. ASME J. Applied Mechanics, 23: , 1955 i-1 A i (q i ) = i-1 A i (q i ) i A i = cθ i -cα i sθ i sα i sθ i a i cθ i sθ i cα i cθ i -sα i cθ i a i sθ i 0 sα i cα i d i compact notation: c = cos, s = sin super-compact notation: c i = cos q i, s i = sin q i Robotics 1 17

18 Direct kinematics of manipulators description internal to the robot using RF E y E slide s product 0 A 1 (q 1 ) 1 A 2 (q 2 ) n-1 A n (q n ) q = (q 1,,q n ) x E z E approach a normal n z 0 RF RF B 0 x 0 B T E = B T 0 0 A 1 (q 1 ) 1 A 2 (q 2 ) n-1 A n (q n ) n T E r = f r (q) y 0 description external to the robot using R p B T E = = r = (r 1,,r m ) n s a p alternative descriptions of the direct kinematics of the robot Robotics 1 18

19 Example: SCARA robot video q 3 q 1 q 2 q 4 Sankyo SCARA 8438 Sankyo SCARA SR 8447 Robotics 1 19

20 Step 1: joint axes all parallel (or coincident) twists α i = 0 or π J2 elbow J3 prismatic J4 revolute J1 shoulder Robotics 1 20

21 Step 2: link axes the vertical heights of the link axes are arbitrary (for the time being) a 1 a 2 a 3 = 0 Robotics 1 21

22 Step 3: frames z 1 axes y i for i > 0 are not shown (nor needed; they form right-handed frames) z 0 x 1 x 4 z 2 z 4 = z 3 x 2 x 3 = a axis (approach) x 0 y 0 Robotics 1 22

23 Step 4: DH table of parameters i α i a i d i θ i z 1 x 1 z 2 = z a 1 d 1 q a 2 0 q 2 x 0 z 0 y 0 x 4 z 4 x 2 x q π 0 d 4 q 4 note that: d 1 and d 4 could be set = 0 here, it is d 4 < 0 Robotics 1 23

24 Step 5: transformation matrices 0 A 1 (q 1 ) = cθ 1 - sθ 1 0 a 1 cθ 1 sθ 1 cθ 1 0 a 1 sθ d A 2 (q 2 ) = cθ 2 - sθ 2 0 a 2 cθ 2 sθ 2 cθ 2 0 a 2 sθ A 3 (q 3 ) = d q = (q 1, q 2, q 3, q 4 ) = (θ 1, θ 2, d 3, θ 4 ) 3 A 4 (q 4 ) = cθ 4 sθ sθ 4 -cθ d 4 1 Robotics 1 24

25 Step 6a: direct kinematics as homogeneous matrix B T E (products of i A i+1 ) 0 A 3 (q 1,q 2,q 3 ) = 3 A 4 (q 4 ) = R(q 1,q 2,q 4 )=[ n s a ] B T E = 0 A 4 (q 1,q 2,q 3,q 4 ) = ( B T 0 = 4 T E = I) c 12 -s 12 0 a 1 c 1 + a 2 c 12 s 12 c 12 0 a 1 s 1 + a 2 s d 1 +q 3 1 c 4 s s 4 -c d 4 1 c 124 s a 1 c 1 + a 2 c 12 s 124 -c a 1 s 1 + a 2 s d 1 +q 3 +d p = p(q 1,q 2,q 3 ) Robotics 1 25

26 Step 6b: direct kinematics as task vector r I R m 0 A 4 (q 1,q 2,q 3,q 4 ) = extract α z from R(q 1,q 2,q 4 ) c 124 s a 1 c 1 + a 2 c 12 s 124 -c a 1 s 1 + a 2 s d 1 +q 3 +d p x a 1 c 1 + a 2 c 12 r = p y = f r (q) = a 1 s 1 + a 2 s 12 I R 4 take p(q 1,q 2,q 3 ) as such p z d 1 +q 3 +d 4 α z q 1 +q 2 +q 4 Robotics 1 26

27 Stanford manipulator 6-dof: 2R-1P-3R (spherical wrist) shoulder offset one possible DH assignment of frames is shown determine the associated DH parameters table homogeneous transformation matrices direct kinematics write a program for computing the direct kinematics numerically (Matlab) symbolically (Mathematica, Maple, Symbolic Manipulation Toolbox of Matlab, ) Robotics 1 27

28 DH table for Stanford manipulator 6-dof: 2R-1P-3R (spherical wrist) i i a i d i i 1 -π/2 0 d 1 >0 q 1 =0 2 π/2 0 d 2 >0 q 2 = q 3 >0 -π/2 4 -π/2 0 0 q 4 =0 5 π/2 0 0 q 5 =-π/ d 6 >0 q 6 =0 joint variables are in red, with their current value in the shown configuration Robotics 1 28

29 KUKA LWR 4+ 7R (no offsets, spherical shoulder and spherical wrist) available at DIAG Robotics Lab determine left Robotics 1 side view (from the left) frames and table of DH parameters homogeneous transformation matrices direct kinematics d1 and d7 can be set = 0 or not (as needed) right frontal view 29

30 KUKA KR5 Sixx R650 6R (offsets at shoulder and elbow, spherical wrist) V top view determine frames and table of DH parameters homogeneous transformation matrices direct kinematics side view (from observer in V) available at DIAG Robotics Lab Robotics 1 30

31 Appendix: Modified DH convention a modified version used in J. Craig s book Introduction to Robotics, 1986 has z i axis on joint i a i-1 & α i-1 = distance & twist angle from z i-1 to z i, measured along & about x i-1 d i & θ i = distance & angle from x i-1 to x i, measured along & about z i source of much confusion... if you are not aware of it (or don t mention it!) convenient with link flexibility: a rigid frame at the base, another at the tip... classical (or distal) cθ i cα i sθ i sα i sθ i a i cθ i sθ i 1 A i = i cα i cθ i sα i cθ i a i sθ i 0 sα i cα i d i y 2 y 1 y 0 x 1 x 2 planar 2R example modified (or proximal) cθ i sθ i 0 a i 1 cα i 1 A mod i = i 1 sθ i cα i 1 cθ i sα i 1 d i sα i 1 sα i 1 sθ i sα i 1 cθ i cα i 1 d i cα i modified DH tends to place frames at the base of each link Robotics 1 31 x 0 y 2 x y y x 1 x 0