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Twists

Twists. Displacement of the rigid body from { A } to { B } Differentiating:. Twist Matrix in Frame {A}. Displacement of the rigid body from { A } to { B } Differentiating: Pre multiplying by the inverse of the transform:. Twist Matrix in Frame {B}.

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Twists

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  1. Twists

  2. Displacement of the rigid body from {A} to {B} Differentiating: Twist Matrix in Frame {A}

  3. Displacement of the rigid body from {A} to {B} Differentiating: Pre multiplying by the inverse of the transform: Twist Matrix in Frame {B}

  4. Angular velocity of Brelative to A with components in A Linear velocity of a point on B that is instantaneously at Owith components in A Angular velocity of Brelative to A with components in B Linear velocity of a point on B that is instantaneously at O' with components in B Twist Matrix Instead of using leading superscripts A or B, we use (a) inertial frame twist (spatial velocity); or (b) body frame twist (body velocity)

  5. Twist in frame attached to base Twist in frame attached to arm Example: Rotation about a single joint z' z x' q x d

  6. There is a 1-1 correspondence between the set of all twist matrices and twist vectors Twist vectors are a more compact way of representing twist matrices Twist Matrix to Twist Vector

  7. The Lie Algebra, se(3) • The set of all twists is a Lie Algebra • Algebra is a vector space with a binary product operation • The product operation is the Lie Bracket: X*Y = [X, Y] = XY – YX • The Lie Bracket (and therefore the algebra) has the following three useful properties • Antisymmetric • Bilinear • Jacobi Identity [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]]=0 Examples of Lie Algebras • GL(n, R) • so(3) • se(3)

  8. The Algebras Associated with SE(3) The subalgebras of se(3) are the screw systems that guarantee full cycle mobility [Hunt 78]

  9. Exponential map • se(3) = set of all twist matrices • The exponential map • exp: se(3) ® SE(3) • is surjective

  10. u f p Rp The Axis/Angle associated with a Rotation • Rotation about u through f • Rodrigues-Euler-Lexell formula • where, • Notes: • 1. Map from R to (u, f) is one to many. • - restrict f to the interval [0,p] • 2. Singular • R = I • trace(R) = -1 • Extracting the axis and the angle from the rotation matrix • 1. Find the eigenvector corresponding to l=1. • 2. From Rodrigues’ formula:

  11. Matrix Exponentials • A solution to a linear differential equation is a exponential of a matrix • The exponential of a 3´3 skew-symmetric matrix is a rotation matrix. • so(3) = set of all skew-symmetric matrices • Rodrigues formula follows from this • The exponential map • exp: so(3) ® SO(3) • is surjective

  12. Exponential Coordinates Logarithm map log: SO(3) ® so(3) multivalued! Yet, can define coordinates R® (u, q)

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