Robot Engineering
Inverse Kinematics
By Vasfi Emre Ömürlü, Ph.D.
Introduction to Inverse Kinematics
In this lecture, you will learn:
- Define inverse kinematics problem
- Define tool configuration space and joint
space
- Find tool configuration vector
- Find the task configuration vector
- The IK of a three axis planar robot
By Vasfi Emre Ömürlü, Ph.D., 2007
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IK Problems
Given the desired position p and orientation R for the tool, solve

 R(q)
tool
Tbase
(q)  


0
0
0

p(q )


1 
for q
We want the transformation of cartesian coordinates into joint
coordinates. It is necessary for real time control of robot
manipulators.
The desired motion of the tool is specified in the workspace,
Find the corresponding joint angle motion.
By Vasfi Emre Ömürlü, Ph.D., 2007
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Example
Specify a homogeneous transformation matrix that will position
tool
the peg just above the hole, find
. Note that the peg
Tbase
extends 2 cm beyond the jaws of the gripper.
.
By Vasfi Emre Ömürlü, Ph.D., 2007
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Configuration of the tool refers to
Direct Kinematics problem
Inverse Kinematics Problem
.
By Vasfi Emre Ömürlü, Ph.D., 2007
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General Properties of Solutions
A closed-form solution to the inverse kinematics problem exists
only for certain classes of robotic mechanisms and the solution
is not unique.
Why multiple solutions?
Kinematics Redundancy
Multiple solutions may even exist with a Non-redundant robot.
By Vasfi Emre Ömürlü, Ph.D., 2007
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Tool Configuration Vector
Specifies tool configuration without using p and R.
Tool Orientation
Recall, R=[r1,r2,r3]
Strategy: attach roll angle information to the approach
vector by scaling the approach vector by some function
of the roll angle.
Note: r3 is a unit vector
Try f (qn)=
By Vasfi Emre Ömürlü, Ph.D., 2007
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Task Configuration Vector
Def: Let p and R denote the position and orientation of the tool
frame relative to the base frame and let qn represent the roll
angle.
Show how qn can be recovered from w
.
By Vasfi Emre Ömürlü, Ph.D., 2007
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Example: SCARA Robot
Find the tool configuration vector for a four axis
SCARA robot.


Ans:
w  p1
p2
p3
0 0  e qk / 
T
From w, how many DOFs does this robot have?
Why are there two zeros in w?
By Vasfi Emre Ömürlü, Ph.D., 2007
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Inverse Kinematics of a Three-Axis
Planar Articulated Robot
Derive the kinematic equations for the robot below
and then solve the IK equations.
Find the link coordinate diagram
1. Number the joints from 1 to n starting from the base
2. Assign x0 , y0 , z0 so that z0 aligns the axis of joint 1.
3. Align zk with the axis of joint k+1
.
By Vasfi Emre Ömürlü, Ph.D., 2007
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Inverse Kinematics of a Three-Axis
Planar Articulated Robot
4. Since zk and zk-1 do not intersect, locate the origin of
Lk at the intersection of zk with a normal between zk
and zk-1.
5. Since zk and zk-1 are in parallel, point xk away from
zk-1 .
6. Select yk to make a right handed coordinate system.
By Vasfi Emre Ömürlü, Ph.D., 2007
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Inverse Kinematics of a Three-Axis
Planar Articulated Robot
7.
8.
Done for all joints
Tool tip is L3 . Align z3 with approach vector and y3 with the sliding
vector.
9-14.
.
By Vasfi Emre Ömürlü, Ph.D., 2007
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Inverse Kinematics of a Three-Axis
Planar Articulated Robot
The Arm Matrix
tool
Tbase
 T01T12T23 ; use
Tkk1
C k
 S
 k
 0

 0
 C k S k
C k C  k
S k
0
By Vasfi Emre Ömürlü, Ph.D., 2007
S k S k
 C k S k
C k
0
a k C k 
a k S k 
dk 

1 
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Inverse Kinematics of a Three-Axis
Planar Articulated Robot
Final Result
tool
Tbase
C123
S
  123
 0

 0
 S123
C123
0
0
0 a 2 C12  a1C1 
0 a 2 S12  a1 S1 

1
d3

0
1

The tool configuration vector is
w = (w1 w2)T
p is obtained from the last column of the arm matrix,
By Vasfi Emre Ömürlü, Ph.D., 2007
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Inverse Kinematics of a Three-Axis
Planar Articulated Robot
Shoulder Joint
ans:
 w 2  w2 2  a1 2  a1 2 
q 2   cos 1  1

2a1 a 2


Distance from L0 to L2 depends only on q2
By Vasfi Emre Ömürlü, Ph.D., 2007
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Inverse Kinematics of a Three-Axis
Planar Articulated Robot
Base Joint
ans:
q1  a tan 2(a1  a2 C2 ) w2  a2 S 2 w1 , (a1  a2 C2 ) w1  a2 S 2 w2 
We know q2 , solve for q1 now,
By Vasfi Emre Ömürlü, Ph.D., 2007
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Inverse Kinematics of a Three-Axis
Planar Articulated Robot
Base Joint: singularity

If a1 = a2 = a, then C1 =
and S1=
Now, if q2 = π, C1 =
S1 =
Otherwise, q1=
Def: atan2(y,x) =
x  0
tan 1 ( y / x)

sgn( y)  / 2
x  0

1
 x  0 tan ( y / x)  sgn( y )  

atan2(y,x) = tan-1(y/x) while keeping track of what quadrant y and x are in.
By Vasfi Emre Ömürlü, Ph.D., 2007
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Inverse Kinematics of a Three-Axis
Planar Articulated Robot
Tool Roll Joint
w6  e ( q3 / ) so q3 
This solution assumes that w is given, perhaps from a
trajectory planner or knowledge about the task.
By Vasfi Emre Ömürlü, Ph.D., 2007
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Formulation Collection
Fundamental Rotations
PF  RPM
1 0
Rot ( x, )  0 C
0 S
0 
 S 
C 
 C
Rot ( y,  )   0
 S
Inverse Homogeneous Transformation
T
1
0 S 
1 0 
0 C 

RT


 0 0
C
Rot ( z ,  )   S
 0
 S
C
0

 RT p 
0
1 
Kinematic Parameters: 2 joint (Joint angle (θ ) ve Joint distance (d )) + 2 link (Link
length (a ) ve Twist angle (α ))
k
k
k
k
By Vasfi Emre Ömürlü, Ph.D., 2007
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0
0
1
Formulation Collection
Screw transformation= screw (dk ,θk, 3) . screw (ak ,αk, 1)
Tkk1 ( k , d k , a k ,  k )
q k 1
Link Coordinate Transformation
Tkk1
Arm Matrix
C k
 S
 k
 0

 0
 C k S k
C k C  k
S k
0
T
tool
base
S k S k
 C k S k
C k
0
 Tkk1  q 
k
a k C k  

a k S k   R
p 


dk  
 

1  0 0 0 1 
(q )
By Vasfi Emre Ömürlü, Ph.D., 2007
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