Robot Path Planning
CONTENTS
1. Introduction
2. Interpolation
path planning, 2012/2013 winter
1
Robot Path Planning
Introduction

The user specifies goal points at the end effector level. There is an
issue about the intermediate points so that the motors can run
continuously. The intermediate points are points between the
starting point and the end point.
A robot has two
degrees of
freedom (i.e., two
motors at joints)
motor
position
Robot Path Planning

Goal points are converted to joint points -> joint scheme, so the
issue becomes to determine the intermediate angles given two
angles as well as their time stamps (see Figure 1).
θ1
A robot has two
degrees of
freedom (i.e., two
motors at joints)
Motor 1
t
θ2
motor
position
path planning, 2012/2013 winter
t
Motor 2
3
Robot Path Planning

Criterion of determining these intermediate angles is: “smoothness”
of the motion.
Joint space schemes
The problem of planning at the end effector is converted to the
problem of path planning at the joint level
path planning, 2012/2013 winter
4
Robot Path Planning
Interpolation:
Given the initial and end goal points, there are different
ways to interpolate, see Figure 2.
Figure 2
Rotary motor
path planning, 2012/2013 winter
5
Robot Path Planning
Cubic polynomials:
There are at least four conditions to constrain the
interpolation:
 ( 0)   0
 (t f )   f
(1)
(2)
.
 ( 0)  0
(3)
.
 (t f )  0
path planning, 2012/2013 winter
(4)
6
Robot Path Planning

These four constraints can be satisfied by a polynomial
of at least third degree. A cubic has the form:
 (t )  a0  a1t  a2t  a3t
2
3
(5)
We can get the velocity and acceleration expression for
the above equation (5). We can determine the coefficients
as follows:
path planning, 2012/2013 winter
7
Robot Path Planning
a0   0
a1  0
a2 
3
2.
.t f
( f   0 )
2
a3   3 ( f   0 )
tf
path planning, 2012/2013 winter
(6)
8
Robot Path Planning

Example: A single-link robot with a rotary joint is
motionless at
  15
It is desired to move the joint in a smooth manner to
  75
in 3 seconds. Find the coefficients of a cubic which
path planning, 2012/2013 winter
9
Robot Path Planning
accomplishes this motion and brings the
manipulator to rest at the goal.
Solution can be found by plugging into the
equations for the coefficients, we can find:
path planning, 2012/2013 winter
10
Robot Path Planning
a0  15.0
Figure 3 shows the position
a1  0.0
Velocity
a 2  20.0
Acceleration
a3  4.44
path planning, 2012/2013 winter
11
Robot Path Planning
Figure 3
path planning, 2012/2013 winter
12
Robot Path Planning
Cubic polynomials for a path with via points
In this case
 ( 0)   0
 (t f )   f
.
.
 ( 0)   0
.
.
 (t f )   f
path planning, 2012/2013 winter
13
Robot Path Planning
Cubic polynomials for a path with via points
In this case, the condition about velocity has
been changed; see the previous slides
The four coefficients can be found (to be filled in
the classroom:
(7)
path planning, 2012/2013 winter
14
Robot Path Planning
Figure 4
path planning, 2012/2013 winter
15
Constant Velocity Improvement
Robot Path Planning
Figure 5
Linear function with parabolic blends
path planning, 2012/2013 winter
16
Robot Path Planning
Change slope
path planning, 2012/2013 winter
17
Robot Path Planning
Summary:
1. Path planning problem starts at the end-effector level but is converted
to path planning at the joint level.
2. The strategy of path planning is: first consider the path planning for a
time span or segment between two points (start, end), and then
consider the connection on the via points.
3. Different paths will affect the smoothness of the motion of a robot.
4. Constant velocity path has the advantage of smooth motion in the
period, but have infinitely large acceleration at the start and end
points of the period. Local modification at the start and end is an
effective means to trade-off the pros and cons of constant velocity
plan
path planning, 2012/2013 winter
18