WOLFRAM|DEMONSTRATIONS PROJECT

A Two-Link Inverse-Kinematic Mechanism

​
x
1
case
sine
ellipse
triangle
This Demonstration shows a two-linkage inverse-kinematic mechanism. The lengths of the rods shown in the diagram are
L
1
and
L
2
, and their angles of rotation are
θ
1
and
θ
2
, with ranges
-100°
to
100°
and
-150°
to
150°
, respectively. The coordinates of a point on the toroidal surface are
p
x
=
L
1
cos(
θ
1
)+
L
2
cos(
θ
1
+
θ
2
)
,
p
y
=
L
1
sin(
θ
1
)+
L
2
sin(
θ
1
+
θ
2
)
.
Solving for
θ
1
and
θ
2
:
θ
1
=arctan
-
2
L
1
-
2
L
2
+
2
p
x
+
2
p
y
2
L
1
L
2
,±
1-
2
-
2
L
1
-
2
L
2
+
2
p
x
+
2
p
y
2
L
1
L
2
,
θ
2
=arctan-
-
L
1
p
x
-
c
2
L
2
p
x
-
s
2
L
2
p
y
2
p
x
+
2
p
y
,-
s
2
L2
p
x
-
L
1
p
y
-
c
2
L
2
p
y
2
p
x
+
2
p
y
,
where
c
2
=
-
2
L
1
-
2
L
2
+
2
p
x
+
2
p
y
2
L
1
L
2
and
s
2
=
1-
2
c
2
.
With appropriate choices of the constants, the formulas for
θ
1
and
θ
2
can be used to trace a sine wave, an ellipse, or a triangle.