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Robot Path Planning
CONTENTS
1. Introduction
2. Interpolation
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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
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t
Motor 2
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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
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Interpolation:
Given the initial and end goal points, there are different
ways to interpolate, see Figure 2.
Figure 2
Rotary motor
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Cubic polynomials:
There are at least four conditions to constrain the
interpolation:
θ (0) = θ 0
θ (t f ) = θ f
(1)
(2)
.
θ (0) = 0
(3)
.
θ (t f ) = 0
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(4)
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These four constraints can be satisfied by a polynomial
of at least third degree. A cubic has the form:
2
θ (t ) = a0 + a1t + a2 t + a3t
3
(5)
We can get the velocity and acceleration expression for
the above equation (5). We can determine the coefficients
as follows:
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a0 = θ 0
a1 = 0
a2 =
3
2.
.t f
(θ f − θ 0 )
2
a3 = − 3 (θ f − θ 0 )
tf
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(6)
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Example: A single-link robot with a rotary joint is
motionless at
θ = 15
It is desired to move the joint in a smooth manner to
θ = 75
in 3 seconds. Find the coefficients of a cubic which
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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:
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a0 = 15.0
Figure 3 shows the position
a1 = 0.0
Velocity
a2 = 20.0
Acceleration
a3 = −4.44
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Figure 3
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Cubic polynomials for a path with via points
In this case
θ (0) = θ 0
θ (t f ) = θ f
.
.
θ (0) = θ 0
.
.
θ (t f ) = θ f
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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)
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Figure 4
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Constant Velocity Improvement
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Figure 5
Linear function with parabolic blends
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Change slope
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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
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