COMP790072 Robotics: An Introduction

1
COMP790-072Robotics An Introduction

• Kinematics Inverse Kinematics

2
Forward Kinematics
3
What is f ?
4
What is f ?
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 TH(R,d)

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.

7
Forward vs. Inverse Kinematics
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.
• Its simple to use, when it works. But, it gives
  less control.
• Some common problems
• Existence of solutions
• Multiple solutions
• Methods used

9
Numerical Methods for IK

• Analytical solutions not usually possible
• Large solution space (redundancy)
• Empty solution space (unreachable goal)
• f is nonlinear due to sins and coss in the
  rotations.
• Find linear approximation to f -1
• Numerical solutions necessary
• Fast
• Reasonably accurate
• Yet Robust

10
The Jacobian
11
The Jacobian
12
The Jacobian
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

14
Matrix Derivatives
15
Rotation Matrix Derivatives
16
Angular Velocity Matrix
17
(No Transcript)
18
(No Transcript)
19
Computing J

• Fairly slow to compute
• Brevilles method J(JJT)-1
• Complexity O(m2n)
• 57 multiply per DOF with m 6
• Instability around singularities
• Jacobian loses rank in certain configur.

20
Jacobian Transpose

• Use JT rather than J
• Avoid excessive inversion
• Avoid singularity problem

21
Principles of Virtual Work

• Work force x distance
• Work torque x angle

22
Jacobian Transpose

• Essentially were 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.

23
Jacobian Transpose
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

25
Pros Cons of Using JT

• Cheaper evaluation
• No singularities
• - Scaling Problems
• J has minimal norm at every step and JT doesnt
  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

26
Cyclic Coordinate Descend (CCD)

• Just solve 1-DOF IK-problem repeatedly up the
  chain
• 1-DOF problems are simple have analytical
  solutions

27
CCD Math - Prismatic
28
CCD Math - Revolute
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.

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
• - Doesnt necessarily lead to smooth motion

31
References