Download presentation

Presentation is loading. Please wait.

1
**COMP790-072 Robotics: An Introduction**

Kinematics & Inverse Kinematics UNC Chapel Hill M. C. Lin

2
Forward Kinematics UNC Chapel Hill M. C. Lin

3
What is f ? UNC Chapel Hill M. C. Lin

4
What is f ? UNC Chapel Hill M. C. Lin

5
**Other Representations**

Separate Rotation + Translation: T(x) = R(x) + d Rotation as a 3x3 matrix Rotation as quaternion Rotation as Euler Angles Homogeneous TXF: T=H(R,d) UNC Chapel Hill M. C. Lin

6
Forward Kinematics As DoF increases, there are more transformation to control and thus become more complicated to control the motion. Motion capture can simplify the process for well-defined motions and pre-determined tasks. UNC Chapel Hill M. C. Lin

7
**Forward vs. Inverse Kinematics**

UNC Chapel Hill M. C. Lin

8
**Inverse Kinematics (IK)**

As DoF increases, the solution to the problem may become undefined and the system is said to be redundant. By adding more constraints reduces the dimensions of the solution. It’s simple to use, when it works. But, it gives less control. Some common problems: Existence of solutions Multiple solutions Methods used UNC Chapel Hill M. C. Lin

9
**Numerical Methods for IK**

Analytical solutions not usually possible Large solution space (redundancy) Empty solution space (unreachable goal) f is nonlinear due to sin’s and cos’s in the rotations. Find linear approximation to f -1 Numerical solutions necessary Fast Reasonably accurate Yet Robust UNC Chapel Hill M. C. Lin

10
The Jacobian UNC Chapel Hill M. C. Lin

11
The Jacobian UNC Chapel Hill M. C. Lin

12
The Jacobian UNC Chapel Hill M. C. Lin

13
**Computing the Jacobian**

To compute the Jacobian, we must compute the derivatives of the forward kinematics equation The forward kinematics is composed of some matrices or quaternions UNC Chapel Hill M. C. Lin

14
Matrix Derivatives UNC Chapel Hill M. C. Lin

15
**Rotation Matrix Derivatives**

UNC Chapel Hill M. C. Lin

16
**Angular Velocity Matrix**

UNC Chapel Hill M. C. Lin

17
UNC Chapel Hill M. C. Lin

18
UNC Chapel Hill M. C. Lin

19
**Computing J+ Fairly slow to compute Instability around singularities**

Breville’s method: J+(JJT)-1 Complexity: O(m2n) ~ 57 multiply per DOF with m = 6 Instability around singularities Jacobian loses rank in certain configur. UNC Chapel Hill M. C. Lin

20
**Jacobian Transpose Use JT rather than J+ Avoid excessive inversion**

Avoid singularity problem UNC Chapel Hill M. C. Lin

21
**Principles of Virtual Work**

Work = force x distance Work = torque x angle UNC Chapel Hill M. C. Lin

22
Jacobian Transpose Essentially we’re taking the distance to the goal to be a force pulling the end-effector. With J-1, the solution was exact to the linearized problem, but this is no longer so. UNC Chapel Hill M. C. Lin

23
Jacobian Transpose UNC Chapel Hill M. C. Lin

24
Jacobian Transpose In effect this JT method solves the IK problem by setting up a dynamical system that obeys the Aristotilean laws of physics: F = m v ; = I and the steepest descent method. The J+ method is equivalent to solving by Newtonian method UNC Chapel Hill M. C. Lin

25
**Pros & Cons of Using JT + Cheaper evaluation + No singularities**

- Scaling Problems J+ has minimal norm at every step and JT doesn’t have this property. Thus joint far from end-effector experience larger torque, thereby taking disproportionately large time steps Use a constant matrix to counteract - Slower Convergence than J+ Roughly 2x slower [Das, Slotine & Sheridan] UNC Chapel Hill M. C. Lin

26
**Cyclic Coordinate Descend (CCD)**

Just solve 1-DOF IK-problem repeatedly up the chain 1-DOF problems are simple & have analytical solutions UNC Chapel Hill M. C. Lin

27
CCD Math - Prismatic UNC Chapel Hill M. C. Lin

28
CCD Math - Revolute UNC Chapel Hill M. C. Lin

29
CCD Math - Revolute You can optimize orientation too, but need to derive orientation error and minimize the combination of two You can derive expression to minimize other goals too. Shown here is for point goals, but you can define the goal to be a line or plane. UNC Chapel Hill M. C. Lin

30
**Pros and Cons of CCD + Simple to implement + Often effective**

+ Stable around singular configuration + Computationally cheap + Can combine with other more accurate optimizations - Can lead to odd solutions if per step not limited, making method slower - Doesn’t necessarily lead to smooth motion UNC Chapel Hill M. C. Lin

31
References UNC Chapel Hill M. C. Lin

Similar presentations

OK

Numerical Methods for Inverse Kinematics Kris Hauser ECE 383 / ME 442.

Numerical Methods for Inverse Kinematics Kris Hauser ECE 383 / ME 442.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on models of atoms Ppt on synthesis and degradation of purines and pyrimidines of dna Ppt on traction rolling stock manufacturers Ppt on major domains of the earth Ppt on p&g products images Ppt on advertisement of honda city Ppt on forest and wildlife resources in india Ppt on applied operational research techniques Ppt on cleanliness and hygiene Ppt on depth first search algorithm java