Robotic Motion
The linear algebra of Canadarm
The robot arm simulation
The movements of the robotic arm can be described
using orthogonal matrices.
Six degrees of freedom
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The first segment is fixed to the wall but is free to rotate.
The motion of the 2nd segment is confined to a plane; however,
combining it with the rotation of the 1st segment allows it to move in the
right half-space.
The third segment can rotate and move in a plane and the same is true of
the fourth segment.
Spherical co-ordinates
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We want to determine this in the
Cartesian co-ordinate system.
Note that z/r= cos φ.
x/s= cos θ and y/s= sin θ.
To eliminate s, we note
s/r = sin φ.
Therefore x=r sin φ cos θ, y=
r sin φ sin θ,
z= r
cos φ.
Thus if r=1, the direction of the
vector is given by two coordinates, φ and θ.
We can reverse the calculation:
given x, y,z, what are the values
of r, θ, and φ ?
(x,y,z)
s
Inverse Kinematics
(x,y)
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To move the robotic arm
to the position (x,y), we
need to rotate the first
arm by an angle α and the
second arm by β.
We will assume known the
lengths L1 and L2 of the
two arms.
Law of sines
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The angle α is the sum of
two angles: arctan(y/x)
and the angle φ which we
can calculate using the
sine law.
φ
φ=
Law of cosines
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This is a generalization of the Pythagorean
theorem.
We will apply this to the robotic problem.
Calculation α and β
√
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The second term coming from the angle (x,y)
makes with the horizontal axis.
Six degrees of freedom again
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P is specified using
three co-ordinates
and is obtained
from the three
rotations 1, 2 and 3
indicated.
Rotations 4 and 5
are used to orient
the axis of the claw.
Rotation 6 rotates
the claw to the
desired angle about
its axis.
Moving a solid in a plane
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It requires two coordinates (x,y) to
specify the position
of A after it is moved.
One more parameter,
the angle α, is needed
to specify the rotation
with respect to the
horizontal axis AB.
(x,y)
A
B
Robotic motion and orthogonal matrices
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Any rotation in R2 is given by a
matrix of this form.
These matrices are orthogonal:
AAt = I.
We have shown all 2 x 2
orthogonal matrices A with
det A=1 have this form.
The motion of a robotic arm in
R3 is a sequence of translations
and rotations.
With a suitable basis, any
rotation in R3 is of the
following form:
The cross product
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Here n is a vector of unit length
perpendicular to both the vectors
a and b.
The right hand rule
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The green vector depicting
the cross product changes
as the angle between the
two vectors changes.
The cross product is
neither commutative nor
associative but satisfies the
Jacobi identity.
a, b, c in R3
Transformations of R3 preserving lengths
and angles
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Theorem: Any movement of a solid in space is the
composition of a translation and rotation about
some axis. After an appropriate choice of basis for
R3, the rotation is given by the matrix:
Rotations in
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n
R
Any orthogonal matrix has determinant +1 or -1.
Rotations in
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n
R
with det = -1.
This has the following geometric meaning: