Inverse Kinematics

autorob.github.io

Inverse Kinematics

UM EECS 398/598 - autorob.github.io

Objective (revisited) Goal: Given the structure of a robot arm, compute – Forward kinematics kinematics: predicting the pose of the end-effector, given joint positions. – Inverse kinematics: inferring the joint positions necessary to reach a desired end-effector pose. We need to solve for configuration from a transform between world and endeffector frames. UM EECS 398/598 - autorob.github.io

to robot configuration Forward kinematics kinematics: many-to-one mapping of robot configuration to reachable workspace endeffector poses

Global (Frame w) Find this, From these

Base (Frame 0)

Transform of endeffector wrt. base

Endeffector (Frame 6)


Inverse kinematics: one-to-many mapping of workspace endeffector pose to robot configuration

Global (Frame w) From this, Find these

Base (Frame 0)




Inverse kinematics: how to solve for q = {θ1, …,θN} from T0N?

Global (Frame w) From this, Find these

Base (Frame 0)




Let’s define IK starting from FK


Consider a planar 2-link arm as an example



with 2 links with length 𝝰i, … UM EECS 398/598 - autorob.github.io


with 2 links, 2 joints, … UM EECS 398/598 - autorob.github.io


with 2 links, 2 joints, coordinate frames at each body, and … UM EECS 398/598 - autorob.github.io


Robot configuration defined by DoF state q

2 angular DOFs q = [𝜽1,𝜽2] joint axes out of plane

with 2 links, 2 joints, coordinate frames at each body, and configuration over DoFs UM EECS 398/598 - autorob.github.io


Do not forget endeffector

Remember a robot has N joint frames and N+1 link frames UM EECS 398/598 - autorob.github.io

Consider a planar 2-link arm as an example Robot endeffector is the gripper pose in world frame Endeffector pose has position can consider orientation Endeffector defines “tool frame” with transform world frame

Frame 2 is the “tool frame”

2 Cartesian DOFs o0N = p0 = (px0,py0)

Endeffector can be specified as a point p2 (origin of tool frame) wrt. Frame 1 Endeffector transforms into p0 in world frame


Endeffector axes “Sliding” axis

“Normal” axis

“Approach” axis

Checkpoint: Transform endeffector on link to world

endeffectorworld

endeffectorlink4

tool frame

world frame UM EECS 398/598 - autorob.github.io

Checkpoint: Transform endeffector on link to world

endeffectorworld

endeffectorlink4

tool frame

world frame UM EECS 398/598 - autorob.github.io

Forward kinematics: “given configuration, compute endeffector”

Robot configuration defined by DoF state

2 angular DOFs q = [𝜽1,𝜽2]

Robot endeffector is the gripper pose in world frame



Forward kinematics: [o0N,R0N] = f(q)

Robot configuration defined by DoF state

2 angular DOFs q = [𝜽1,𝜽2]

p0 = f(𝜽1,𝜽2)

Robot endeffector is the gripper pose in world frame




What is the position and orientation of the tool wrt. the world? remember:

What are the elements of this matrix?

What are the elements of this vector? UM EECS 398/598 - autorob.github.io

Forward kinematics: [o0N,R0N] = f(q) remember:

What is the position and orientation of the tool wrt. the world?

p0 = R20p2 + d20 where R20 = R10R21  d20 = R10d21 + d10

What are the elements of this vector? UM EECS 398/598 - autorob.github.io

 

Start with:

d20 = R10d21 + d10 substitute in variables then perform operations:

then substitute trig identities

to get: What are the elements of this vector?




p0 = f(𝜽1,𝜽2)


Inverse kinematics: “given endeffector, compute configuration”

? ?


Inverse kinematics: q = f-1([o0N,R0N])

[𝜽1,𝜽2] = f-1(p0)

? ?

Just consider endeffector position for now


1DOF pendulum example assume: 1DOF motor at pendulum axis, motor servo moves arm to angle 𝜽1

desired endeffector position (o0N) given as an x,y location

what is 𝜽1?

𝜽1


1DOF pendulum example assume: 1DOF motor at pendulum axis, motor servo moves arm to angle 𝜽1

desired endeffector position (o0N) given as an x,y location

what is 𝜽1?

𝜽1



[𝜽1,𝜽2] = f-1(x,y)

desired endeffector position (o0N) given as location x,y

let’s start by solving for 𝜽2 UM EECS 398/598 - autorob.github.io


[𝜽1,𝜽2] = f-1(x,y)

solve for 𝜽2



[𝜽1,𝜽2] = f-1(x,y) solve for 𝜽2 Law of Cosines



[𝜽1,𝜽2] = f-1(x,y)

solve for 𝜽2

solve for 𝜽1

Consider two triangles UM EECS 398/598 - autorob.github.io


[𝜽1,𝜽2] = f-1(x,y)

solve for 𝜽2

solve for 𝜽1


inverse kinematics: (𝜽1,𝜽2) = f-1(x,y)

is this the right solution?


when is there one solution?


when is there no solution?


when is there no solution?


Can we do IK for 3 links?


Try this http://scratch.mit.edu/projects/10607750/


How many solutions for this arm?

3 unknowns

2 constraints

Remember: Ax=b


Inverse Kinematics: 2D


Inverse Kinematics: 2D Configuration

Transform from endeffector frame to world frame



Transform from endeffector frame to world frame

Inverse orientation

Inverse position UM EECS 398/598 - autorob.github.io


Transform from endeffector

Closed form solution


Inverse Kinematics: 3D Transform from endeffector

Configuration

6 DOF position and orientation of endeffector



Transform from endeffector

Closed form solution? 6 DOF position and orientation of endeffector


Stanford Manipulator

assumes D-H frames


Recognize this robot?

RexArm (EECS 467 / ROB 550)

RexArm (EECS 467 / ROB 550)

Find: configuration q = [𝜽1 𝜽2 𝜽3 𝜽4] as robot joint angles

Given: link lengths (L4,L3,L2,L1) endeffector orientation ɸ as angle wrt. plane centered at o3 and parallel to ground plane

endeffector position [xg yg zg] wrt. base frame

Find: configuration q = [𝜽1 𝜽2 𝜽3 𝜽4] as robot joint angles

overhead view

solve for 𝜽1

solve for 𝜽3 solve for 𝜽1

Decoupling: separate endeffector from rest of the robot at last joint

solve for 𝜽3

and joint 1 from rest of robot


o3

L3

𝜽3

solve for 𝜽3

L2

ɣ Δz

𝜽2

Ψ β Δr

o1


o3 solve for 𝜽3

L3

𝜽3 ɣ

L2

(Law of cosines with supplementary angle ɣ) Δz

𝜽2

Ψ β Δr

o1


o3 solve for 𝜽3

L3

𝜽3

solve for 𝜽2

L2

ɣ Δz

𝜽2

Ψ

(Law of cosines with angle Ψ, arctan with angle β)

β Δr

o1


o3 solve for 𝜽3

L3

𝜽3

solve for 𝜽2

L2

ɣ Δz

𝜽2

Ψ β Δr

two potential solutions depending on elbow angle

o1


𝜽4 solve for 𝜽3

solve for 𝜽2

ɸ

o3

𝜽3

L4

𝜽2

o1 solve for 𝜽4


solve for 𝜽3

𝜽2

solve for 𝜽2

𝜽3 ɸ 𝜽4

solve for 𝜽4

(Equilvalence relation for adding angles from z0)

o1


𝜽4 solve for 𝜽3

L3

𝜽3

L4

ɣ

L2

solve for 𝜽2

ɸ

o3

Δz

𝜽2

Ψ β Δr

solve for 𝜽4

o1

(Addition of angles in arm plane starting from z0)

Why Closed Form? • Advantages • Speed: IK solution computed in constant time • Predictability: consistency in selecting satisfying IK solution • Disadvantage • Generality: general form for arbitrary kinematics difficult to express UM EECS 398/598 - autorob.github.io

Iterative Solutions to IK •

Minimize error between current endeffector and its desired position

•

Transform desired endeffector velocity into configuration space

•

Repeatedly step to convergence at desired endeffector position

Start

Goal 5-link planar arm


Next Class •

•

IK as an optimization problem •

Gradient descent optimization

•

Manipulator Jacobian as the derivative of configuration

Advanced: IK by Cyclic Coordinate Descent


autorob.github.io

Inverse Kinematics: Manipulator Jacobian
