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Robot Path Planning

CONTENTS

1. Introduction

2. Interpolation

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

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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).

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A robot has two degrees of freedom (i.e., two motors at joints)

motor position

Motor 1

Motor 2

t

t

θ1

θ2

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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:

(1)

(2)

(3)

(4)0)(

0)0(

)(

)0(

.

.

0

f

ff

t

t

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These four constraints can be satisfied by a polynomial

of at least third degree. A cubic has the form:

(5)3

32

210)( tatataat

We can get the velocity and acceleration expression for the above equation (5). We can determine the coefficientsas follows:

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)(2

)(.

3

0

033

0.22

1

00

ff

f

f

ta

ta

a

a

(6)

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Robot Path Planning

Example: A single-link robot with a rotary joint is motionless at

15It is desired to move the joint in a smooth manner to

75in 3 seconds. Find the coefficients of a cubic which

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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:

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44.4

0.20

0.0

0.15

3

2

1

0

a

a

a

a Figure 3 shows the position

Velocity

Acceleration

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Figure 3

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Cubic polynomials for a path with via points

In this case

ff

ff

t

t

..

0

..

0

)(

)0(

)(

)0(

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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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Robot Path PlanningFigure 4

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Robot Path Planning

Linear function with parabolic blends

Figure 5Constant Velocity Improvement

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Change slope

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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