Robotics

1
???? ?????? ??????

• Robotics
• Manipulators Inverse kinematics
• By Behrooz Rahmani

2
Outline Inverse Kinematics
Problem formulation Existence Multiple
Solutions Algebraic Solutions Geometric
Solutions Decoupled Manipulators
3
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.
4
Inverse Kinematics
5
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.

()
6

• 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.)

7

• 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.

8
(No Transcript)
9
Example Two-link manipulator

• If l1 12, then the reachable workspace consists
  of a disc of radius l1l2.
• If , the reachable workspace becomes a
  ring of outer radius and inner radius
  .

10
(No Transcript)
11
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

12
(No Transcript)
13
(No Transcript)
14
(No Transcript)
15
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.

16
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.

17
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.

18
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.

19
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.

20
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.

21
Kinematic Decoupling

• Example for manipulators having six joints, with
  the last three joints intersecting at a point
  (i.e. spherical wrist).

22
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.

23
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, .

24
Kinematic Decoupling
25
(No Transcript)
26
(No Transcript)
27
(No Transcript)
28
(No Transcript)
29
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 .

30
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.

31
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.

32
Example Articulated Configuration
33
Projection of the wrist center onto x0 - y0 plane
34

• 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 leaves oc.

35

36
(No Transcript)
37
(No Transcript)
38
Example
39
(No Transcript)
40
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.

41
Spherical Wrist
42
(No Transcript)
43
(No Transcript)
44
(No Transcript)
45

• 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.

46
(No Transcript)
47
(No Transcript)
48
Example SCARA manipulator forward kinematics

• It consists of an RRP arm and a one
  degree-of-freedom wrist.

49
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.

50
Solution
51
Solution
52
Example SCARA manipulator Inverse kinematics
53
(No Transcript)
54
(No Transcript)
55
(No Transcript)
56
(No Transcript)
57
(No Transcript)