UNC Chapel Hill M. C. Lin

COMP790-072Robotics: 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

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

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+

Fairly slow to compute– 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.

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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