EE Kinematics & Inverse Kinematics

Transcription

1 Electric Electronic Engineering Bogazici University October 15, 2017

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3 Problem Statement Kinematics: Given c C, find a map f : C W s.t. w = f(c) where w W : Given w W, find a map f 1 : W C s.t. c = f 1 (w)

4 A set of links connected by joints Simple joints: Single DOF qi where joint variable q i q i = { θi S 1 if joint i is revolute d i R >0 if joint i is prismatic Complex joints: More DOF at a joint

5 Simple Joints: General Robot Topology n joints n+1 links Joints: 1,...,n Links: 0,...,n Connects link i 1 to link i Joint i { Location is fixed wrt link i 1 x When actuated, link i moves Link 0 The first link Stationary robots Fixed Mobile robots Moving

6 Analysis Procedure - Systematic Analysis Attach a (reference) coordinate frame o i x i y i z i rigidly to each link i Thus, coordinates of link i are constant when expressed wrt ith frame Frame associated with 0 th link Inertial Frame A i (q i ) The homogeneous transformation of o i x i y i z i wrt o i 1 x i 1 y i 1 z i 1 T j i - Homogeneous transformation matrix of o j x j y j wrt o i x i y i z i T i j = A i+1 A i+2...a j 1 A j if i < j I if i = j if j > i (T j i ) 1

7 Assigning Coordinate Frames Choice z i Axis of actuation for joint i +1 Other axes Right hand rule (Satisfy DH Assumptions) When joint i is actuated link i and o i x i y i z i move Second frame - Set up so that DH Assumptions are satisfied (in regards to the succeeding frame) Base frame Frame 0 Origin o0 - Any point on z 0 Choose x0,y 0 so that o 0 x 0 y 0 z o is right-handed Iterative process Define Frame i wrt Frame i 1, i = 1,...n 1 Frame n End effector or tool frame - o n x n y n z n

8 DH Assumptions - DH1 and DH2 x 1 perpendicular to z 0 x 1 intersects z 0 Unique representation A i (q i )

9 DH Parameters Simple way of forming A i (q i ) 4 Parameters 3 fixed, 1 variable θ i - Joint angle (q i = θ i variable in case of revolute joint) d i - Link offset ( q i = d i variable in case of prismatic joint) a i - Link length α i - Link twist

10 DH Conventions A i

11 Transformation Matrix A i = R z,θi Tr z,di Tr x,ai R x,αi A i = cosθ i sinθ i cosα i sinθ i sinα i a i cosθ i sinθ i cosθ i cosα i cosθ i sinα i a i sinθ i 0 sinα i cosα i d i

12 DH Parameters Summary z i 1 - Axis of actuation of joint i! Parameter Explanation θ i Angle from x i 1 to x i wrt z i 1 d i Distance along z i 1 from o i 1 to inters. of x i and z i 1 a i Distance along x i between z i 1 and z i α i Angle from z i 1 to z i wrt x i } Right-Hand Rule

13 Example:

14 DH Parameters for a 2 DOF planar robotic system. Link i a i α i d i θ i 1 a θ 1 2 a θ 2

15 Iterative Process - Axes z i 1 and z i z i 1 and z i not coplanar z i 1 and z i parallel z i 1 and z i intersect

16 z i 1 and z i not coplanar x i - unique line from z i 1 to z i, perpendicular to both o i - Where x i intersects z i

17 z i 1 and z i parallel common normals! Free to choose o i - Anywhere along z i x i - Arbitrary xi : Choose common normal of previous joint xi - Normal through o i 1 o i - Point of intersection of x i with z i

18 z i 1 and z i intersect o i - Intersection of z i 1 and z i x i - Orthogonal to the plane defined by z i 1 and z i

19 DH Frame Setup Summary z i - Axis of actuation of joint i +1! z Case i 1 and z i x i o i Non-coplanar Line from z i 1 x i intersects z i (o i 1 x i 1 y i 1 ) to z i Parallel Normal x i intersects z i (o i 1 x i 1 y i 1 ) through o i 1 Intersects Normal to plane z i 1 intersects (o i 1 x i 1 y i 1 ) of z i 1 and z i z i

20 DH Parameters Summary z i - Axis of actuation of joint i +1! Parameter Explanation θ i d i a i α i Angle from x i 1 to x i wrt z i 1 Distance along z i 1 from the intersection(x i,z i 1 ) to o i 1 Distance along x i from the intersection(x i,z i 1 ) to o i Angle from z i 1 to z i wrt x i

21 Example 1

22 Example 2 3 DOF RRR

23 Example 4 3 DOF RRP Scara Robot

24 Example 4 3 DOF RRP Scara Robot

25 Example 5 3 DOF PPP Cartesian Manipulator

26 Example 3 DOF RPP Cartesian Manipulator

27 Example 6 3 DOF RPP Cylindrical Manipulator

28 Example 6 3 DOF RPP Cylindrical Manipulator

29 Comparative Workspaces

30 Gripper Case

31 Gripper Parameters Origin o n Symmetrically between the fingers of the gripper z n - Approach direction a y n - Sliding direction s x n - Normal direction n Many manipulator systems zn 1 and z n coincide. on 1 and o n - translated by d n amount Joint n - Rotation by θn around z n 1

32 Problem [ ] R d Given H = SE(3), find joint variables 0 1 c = [q 1,...,q n ] T C, such that where T 0 n(q 1,...,q n ) = H T 0 n(q 1,...,q n ) = A 1 (q 1 )...A n (q n )

33 General Approach Noting that H = h 11 h 12 h 13 h 14 h 21 h 22 h 23 h 24 h 31 h 32 h 33 h 34 h 41 h 42 h 43 h 44 where h 41 = h 42 = h 43 = 0 and h 44 = 1, n unknowns in 12 nonlinear equations. Closed-form solutions: Preferable (real-time, nonuniqueness), but not possible in general! Existence? Uniqueness? Difficult to derive even in case of existence! Numerical solutions:,

34 Nonuniqueness

35 Efficient & Systematic Techniques 6 DOF with a gripper - exploit kinematic structure Decouple the problem into simpler problems: Inverse position kinematics - Wrist position o q1,q 2,q 3 Inverse orientation kinematics -Wrist orientation R (Tool frame) Assume: Find a solution and then check for constraints on ranges of joints!

36 Example - PUMA Robot

37 Kinematic Decoupling - 6 DOF with Gripper

38 6 DOF with Gripper with Frames

39 Spherical Wrist T6 3 = A 4 A 5 A 6 = cθ 4 cθ 5 cθ 6 sθ 4 sθ 6 cθ 4 cθ 5 sθ 6 sθ 4 cθ 6 cθ 4 sθ 5 cθ 4 sθ 5 d 6 sθ 4 cθ 5 cθ 6 +cθ 4 sθ 6 sθ 4 cθ 5 sθ 6 +cθ 4 cθ 6 sθ 4 sθ 5 sθ 4 sθ 5 d 6 sθ 5 cθ 6 sθ 5 sθ 6 cθ 5 cθ 5 d = R z,θ4 R y,θ5 R z,θ6 Euler Angles Know how to solve for θ 4,θ 5 and θ 6

40 Example - 3DOF RRR

41 Example 3DOF RRR Projection onto x 0,y 0 plane.

42 Singularity - o c intersects z 0

43 RRR Robot with Shoulder Offset Either d 2 = d or d 3 = d.

44 Left Arm Right Arm

45 Finding Link 2 & 3 Parameters