Transcript 3D Polyhedral Morphing - University of North Carolina at

COMP790-072
Robotics: An Introduction

Kinematics & Inverse Kinematics
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Forward Kinematics
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What is f ?
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What is f ?
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Other Representations

Separate Rotation + Translation:
T(x) = R(x) + d
– Rotation as a 3x3 matrix
– Rotation as quaternion
– Rotation as Euler Angles
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Homogeneous TXF: T=H(R,d)
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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.
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Forward vs. Inverse Kinematics
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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
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Numerical Methods for IK
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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
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The Jacobian
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The Jacobian
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The Jacobian
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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
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Matrix Derivatives
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Rotation Matrix Derivatives
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Angular Velocity Matrix
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Computing J+
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Fairly slow to compute
– Breville’s method: J+(JJT)-1
• Complexity: O(m2n)
• ~ 57 multiply per DOF with m = 6
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Instability around singularities
– Jacobian loses rank in certain configur.
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Jacobian Transpose

Use JT rather than J+
– Avoid excessive inversion
– Avoid singularity problem
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Principles of Virtual Work


Work = force x distance
Work = torque x angle
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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.

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Jacobian Transpose
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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

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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]
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Cyclic Coordinate Descend (CCD)

Just solve 1-DOF IK-problem repeatedly
up the chain
 1-DOF problems are simple & have
analytical solutions
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CCD Math - Prismatic
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CCD Math - Revolute
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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.
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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
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References
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