Outline: Inverse Kinematics Problem formulation Existence
Multiple Solutions Algebraic Solutions Geometric Solutions
Decoupled Manipulators 2
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Inverse Kinematics Forward (Direct) Kinematics: Find the
position and orientation of the tool given the joint variables of
the manipulators. Inverse Kinematics: Given the position and
orientation of the tool find the set of joint variables that
achieve such configuration. 3
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Inverse Kinematics 4
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The General Inverse Kinematics Problem The general problem of
inverse kinematics can be stated as follows. Given a 4 4
homogeneous transformation Here, H represents the desired position
and orientation of the end- effector, and our task is to find the
values for the joint variables q1,..., qn so that T 0 n (q1,...,
qn) = H. (*) 5
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Equation (*) results in twelve nonlinear equations in n unknown
variables, which can be written as Tij(q1,..., qn) = hij, i = 1, 2,
3, j = 1,..., 4, where Tij, hij refer to the twelve nontrivial
entries of T0 n and H, respectively. (Since the bottom row of both
T0 n and H are (0,0,0,1), four of the sixteen equations represented
by (*) are trivial.) 6
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Whereas the forward kinematics problem always has a unique
solution that can be obtained simply by evaluating the forward
equations, the inverse kinematics problem may or may not have a
solution. Even if a solution exists, it may or may not be unique.
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Example: Two-link manipulator If l1= 12, then the reachable
workspace consists of a disc of radius l1+l2. If, the reachable
workspace becomes a ring of outer radius and inner radius. 9
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Example For the Stanford manipulator, which is an example of a
spherical (RRP) manipulator with a spherical wrist, suppose that
the desired position and orientation of the final frame are given
by 11
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Method of solution: We will split all proposed manipulator
solution strategies into two broad classes: closed-form solutions
and numerical solutions. Numerical solutions generally are much
slower than the corresponding closed-form solution; in fact, that,
for most uses, we are not interested in the numerical approach to
solution of kinematics. We will restrict our attention to
closed-form solution methods. 15
