‫گروه مهندس ی مکانيک‬
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Outline: Inverse Kinematics
– Problem formulation
– Existence
– Multiple Solutions
– Algebraic Solutions
– Geometric Solutions
– Decoupled Manipulators
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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.
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Inverse Kinematics
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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 T0n(q1, . . . , qn) = H.
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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.)
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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 l1  l2, the reachable workspace becomes a ring of
outer radius l1  l2 and inner radius l1  l2 .
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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
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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.
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Closed-form solution method
 “Closed form" means a solution method based on
analytic expressions or on the solution of a polynomial
of degree 4 or less, such that non-iterative calculations
suffice to arrive at a solution.
 Within the class of closed-form solutions, we
distinguish two methods of obtaining the solution:
algebraic and geometric.
 Any geometric methods brought to bear are applied by
means of algebraic expressions, so the two methods
are similar. The methods differ perhaps in approach
only.
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Why closed-form solution methods?
 Closed form solutions are preferable for two reasons.
 First, in certain applications, such as tracking a welding
seam whose location is provided by a vision system, the
inverse kinematic equations must be solved at a rapid
rate, say every 20 milliseconds, and having closed form
expressions rather than an iterative search is a practical
necessity.
 Second, the kinematic equations in general have
multiple solutions. Having closed form solutions allows
one to develop rules for choosing a particular solution
among several.
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A helpful approach for 6-DOF
robots: Kinematic Decoupling
 A sufficient condition that a manipulator with six
revolute joints have a closed-form solution is that
three neighboring joint axes intersect at a point.
 For manipulators having six joints, with the last three
joints intersecting at a point (such as the Stanford
Manipulator), it is possible to decouple the inverse
kinematics problem into two simpler problems, known
respectively, as inverse position kinematics, and
inverse orientation kinematics.
 Using kinematic decoupling, we can consider the
position and orientation problems independently.
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Spherical wrist
 The assumption of a spherical wrist means that the
axes z3, z4, and z5 intersect at oc and hence the origins
o4 and o5 assigned by the DH-convention will always
be at the wrist center oc.
 Therefore, the motion of the final three links about
these axes will not change the position of oc, and thus,
the position of the wrist center is a function of only the
first three joint variables.
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Kinematic Decoupling
 In this way, the inverse kinematics problem may be
separated into two simpler problems,
 First, finding the position of the intersection of the wrist axes,
called the wrist center.
 Then finding the orientation of the wrist.
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Kinematic Decoupling
 Example for manipulators having six joints, with the
last three joints intersecting at a point (i.e. spherical
wrist).
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Kinematic Decoupling
 Inverse kinematic equation
can be represented as two equations:
 By the spherical wrist, the origin of the tool frame
(whose desired coordinates are given by o) is simply
obtained by a translation of distance d6 along z5 from
oc.
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Kinematic Decoupling
 In order to have the end-effector of the robot at the
point with coordinates given by o and with the
orientation given by R = (rij ), it is necessary and
sufficient that the wrist center oc have coordinates
given by
 Using this equation, we can calculate the first three
joint variables, and therefore, R30 .
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Kinematic Decoupling
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Geometric Analysis
 For most simple manipulators, it is easier to use
geometry to solve for closed-form solutions to the
inverse kinematics
 solve for each joint variable qi by projecting the
manipulator onto the xi−1, yi−1 plane .
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Kinematic Decoupling: orientation
 For calculating the other wrist joint variables, we
know:
 As the right hand side of this equation is completely
known, the final three joint angles can then be found
as a set of Euler angles corresponding to R36.
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Inverse Position: A Geometric
Approach
 For the common kinematic arrangements that we
consider, we can use a geometric approach to find the
variables, q1, q2, q3 corresponding to o0c.
 The general idea of the geometric approach is to solve
for joint variable qi by projecting the manipulator onto
the xi−1 − yi−1 plane and solving a simple trigonometry
problem.
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Example: Articulated Configuration
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Projection of the wrist center onto
x0 − y0 plane
(*)
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 In this case, (*) is undefined and the manipulator is in
a singular configuration, shown in the below.
 In this case, the manipulator is in a singular
configuration, shown in the below Figure.
 In this position the wrist center oc intersects z0; hence
any value of  1 leaves oc.
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Example
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Inverse Orientation
 In the previous section we used a geometric approach
to solve the inverse position problem.
 This gives the values of the first three joint variables
corresponding to a given position of the wrist origin.
 The inverse orientation problem is now one of finding
the values of the final three joint variables
corresponding to a given orientation with respect to
the frame o3x3y3z3.
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Spherical Wrist
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 Recall that the rotation matrix obtained for the
spherical wrist has the same form as the rotation
matrix for the Euler transformation.
 Therefore, we can use the method developed in
Section 2.5.1 to solve for the three joint angles of the
spherical wrist.
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Example: SCARA manipulator
forward kinematics
•It consists of an RRP arm and a one degreeof-freedom wrist.
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Solution
• The first step is to locate and label the joint axes as
shown.
 Since all joint axes are parallel we have some freedom
in the placement of the origins.
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Solution
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Solution
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Example: SCARA manipulator
Inverse kinematics
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