Recap: rigid motions Rigid motion is a combination of rotation and - - PDF document

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Forward and Inverse Kinematics Chapter 3 Hadi Moradi (original slides by Steve from Harvard)

Recap: rigid motions

• Rigid motion is a combination of rotation and translation

– Defined by a rotation matrix (R) and a displacement vector (d) – the group of all rigid motions (d,R) is known as the Special Euclidean group, SE(3)

2 Recap: homogeneous transforms

• Basic transforms:

Th t l ti th t ti – Three pure translation, three pure rotation

⎥ ⎥ ⎥ ⎥ ⎤ ⎢ ⎢ ⎢ ⎢ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 1 1 1 1 1 1 1

, ,

b a

b y a x

Trans Trans ⎥ ⎥ ⎥ ⎥ ⎤ ⎢ ⎢ ⎢ ⎢ ⎡ − = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − = 1 1 1

, , β β β β β α α α α α

c s s c c s s c

y x

Rot Rot ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎦ ⎢ ⎢ ⎣ 1 1 1 1 1 1

,

c

c z

Trans ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − = ⎥ ⎥ ⎦ ⎢ ⎢ ⎣ 1 1 1

, γ γ γ γ γ β β

c s s c c s

z

Rot

Example

• Euler angles: we have only discussed ZYZ Euler angles. What is the

g y g set of all possible Euler angles that can be used to represent any rotation matrix?

– XYZ, YZX, ZXY, XYX, YZY, ZXZ, XZY, YXZ, ZYX, XZX, YXY, ZYZ

3 Example

• Compute the homogeneous transformation representing a translation of

p g p g 3 units along the x-axis followed by a rotation of π/2 about the current z- axis followed by a translation of 1 unit along the fixed y-axis

Forward kinematics introduction

• Challenge: given all the joint parameters of a manipulator, determine

g g j p p the position and orientation of the tool frame

4 Convention

• A n-DOF manipulator will have n joints (either revolute or prismatic) and

n+1 links (since each joint connects two links) n 1 links (since each joint connects two links)

– We assume that each joint only has one DOF. – The o0 frame is the inertial frame –

• n is the tool frame

– Joint i connects links i-1 and i – The oi is connected to link i

• Joint variables q
• Joint variables, qi

⎩ ⎨ ⎧ = prismatic is joint if revolute is joint if i d i q

i i i

θ

Convention

• We said that a homogeneous transformation allowed us to express the

position and orientation of oj with respect to oi position and orientation of oj with respect to oi

5 Convention

• Finally, the position and orientation of the tool frame with respect to the

inertial frame is given by one homogeneous transformation matrix: inertial frame is given by one homogeneous transformation matrix:

– For a n-DOF manipulator

( ) ( ) ( )

n n n n n

q A q A q A T

• R

H ⋅ ⋅ ⋅ = = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ =

2 2 1 1

1

The Denavit-Hartenberg (DH) Convention

• Representing each individual homogeneous transformation as the

product of four basic transformations: product of four basic transformations:

⎥ ⎥ ⎤ ⎢ ⎢ ⎡ − − ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − = = 1 1 1 1 1 1 1 1 1 1 1 1 Rot Trans Trans Rot

, , , , i i i x a x d z z i

s a s c c c s c a s s c s c c s s c a d c s s c A

i i i i i i i i i i i i i i i i i i

θ α θ α θ θ α α α α θ θ θ θ α θ

⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎣ − = 1

i i

d c s s a s c c c s

i i i i i i i i

α α θ α θ α θ θ

6 Existence and uniqueness

• When can we represent a homogeneous transformation using the 4 DH

parameters? parameters?

Existence and uniqueness

• Proof:

1 We assume that R1

0 has the form: θ 1

R R R =

• 1. We assume that R1 has the form:
• 2. Use DH1 to verify the form of R1

α θ , , 1 x z R

R R

7 Existence and uniqueness

• Proof:

1 Use DH2 to determine the form of o1

• 1. Use DH2 to determine the form of o1

Physical basis for DH parameters

• ai: link length, distance between the z0 and z1 (along x1)
• αi: link twist, angle between z0 and z1 (measured around x1)
• di: link offset, distance between o0 and intersection of z0 and x1 (along z0)
• θi: joint angle, angle between x0 and x1 (measured around z0)

positive convention:

=

, , , , x a x d z z i

A

i i i i

α θ

Rot Trans Trans Rot ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − = 1 1 1 1 1 1 1 1 1 1 1 1 1

, , , , i i i i i a d

d c s s a s c c c s c a s s c s c c s s c a d c s s c

i i i i i i i i i i i i i i i i i i i i i i i i i i

α α θ α θ α θ θ θ α θ α θ θ α α α α θ θ θ θ α θ

8 Assigning coordinate frames

• For any n-link manipulator, we can always choose coordinate frames

such that DH1 and DH2 are satisfied such that DH1 and DH2 are satisfied

– The choice is not unique, but the end result will always be the same

• 1. Choose zi as axis of rotation for joint i+1

Assigning coordinate frames

• 2. Assign base frame
• 3. Chose x0, y0 to follow the right-handed convention
• 4. Now start an iterative process to define frame i with respect to frame i-1

9 Assigning coordinate frames

i. zi-1 and zi are non-coplanar

Assigning coordinate frames

ii. zi-1 and zi intersect

10 Assigning coordinate frames

iii. zi-1 and zi are parallel

Assigning tool frame

• The previous assignments are valid up to frame n-1

– The tool frame assignment is most often defined by the axes n s a: The tool frame assignment is most often defined by the axes n, s, a:

11 Example 1: two-link planar manipulator

• 2DOF: need to assign three coordinate frames

Example 1: two-link planar manipulator

• Now define DH parameters

– First define the constant parameters ai αi First, define the constant parameters ai, αi – Second, define the variable parameters θi, di link ai αi di θi 1 2

12 Example 2: three-link cylindrical robot

• 3DOF: need to assign four coordinate frames

Example 2: three-link cylindrical robot

• Now define DH parameters

– First define the constant parameters ai αi link ai αi di θi 1 First, define the constant parameters ai, αi – Second, define the variable parameters θi, di 1 2 3

13 Example 3: spherical wrist

• 3DOF: need to assign four coordinate frames

Example 3: spherical wrist

link ai αi di θi 4

• Now define DH parameters

– First define the constant parameters ai αi 5 6 First, define the constant parameters ai, αi – Second, define the variable parameters θi, di

14 Next class…

• More examples for common configurations