Robotics 2012 07 Trajectory Planning 1 - polito.it · Introduction 4. Move: it defines a single motion that must be executed to perform an operation: for example, close the hand to

Trajectory planning Trajectory planning –– 11

Basilio Bona ROBOTICA 03CFIOR 1

Introduction

The robot planning problem can be decomposed into a structured class of interconnected activities, at different hierarchical levels, usually referred to with different names:

1. Objective: it defines the highest activity level; typically shared by the entire process or FMS where the robot is present; for example, the assembly of an engine head.

2. Task: it defines a subset of actions/operations to be accomplished for the attainment of the objective: for example, the assembly of the engine pistons.

3. Operation: it defines one of the single activities in which the task is decomposed: for example, the insertion of a piston in the cylinder.

Basilio Bona 2ROBOTICA 03CFIOR

Introduction

4. Move: it defines a single motion that must be executed to

perform an operation: for example, close the hand to grasp the

piston, move the piston in a predefined position.

5. Path/Trajectory: the elementary move is decomposed in one

ore more paths (no time law defined) or trajectories (time law ore more paths (no time law defined) or trajectories (time law

and kinematic constraints are defined).

6. Reference: it consists of the vector of the data obtained

sampling the path/trajectory, supplied to the motors as

references for their controlcontrol: this is represents the action

performed at the most basic level.


Decomposition of a planning problem

Objective

… … ...

Operation

Move

Path Reference

…

…


Task

…

…

…

…

…

Planning and control

The controlcontrol problem consists in designing of control algorithms for

the robot drives, such that the TCP motion follows a specified path

in the cartesian space. Two types of tasks can be defined:

1. tasks that do not require an interaction with the environment (free

space motion); the manipulator moves its TCP following cartesian

trajectories, with constraint on positions, velocities and accelerations.

Sometimes it is sufficient to move the joints from a specified value to Sometimes it is sufficient to move the joints from a specified value to

another without following a particular geometric path

2. tasks that require and interaction with the environment, i.e., where the

TCP shall move in some cartesian subspace while it applies (or is

subject to) forces or torques to the environment

We will consider only the first type of tasks

The control may take place at joint level (joint space control) or at

cartesian level (task space control)


Path vs trajectory

� Path = is the geometrical description of the set of desired

points in the task space. The control shall maintain the

TCP on the commanded path

� Trajectory = is the path AND the time law required to

follow the path, from the starting point to the endpoint follow the path, from the starting point to the endpoint


1( )q t

2( )q t

3( )q t

( )

( )

( )

( )

t

t

x q

q

⋯

α

4( )q t

5( )q t

6( )q t

An example

PATH TRAJECTORY

desiredspeed

desiredacceleration


( , , , , , ) 0f x y z φ θ ψ = ( ( ), ( ), ( ), ( ), ( ), ( )) 0f x t y t z t t t tφ θ ψ =

The geometrical path is usually described by an implicit equation

Trajectory planning

TRAJECTORY

PLANNER

Desired path

Desired kinematicconstraints

Joint reference samples

rq


Robotdynamic constraint

The trajectory planner is a software function that computes the

joint reference values (for the control block) given the desired

path, the kinematic constraints (max speed etc.) and the dynamic

constraints (max accelerations, max torques, etc.)

rq

The control problem and the trajectory planner

Controller Actuator Gearbox Robotrq ( )tq

TR

AJEC

TO

RY

PLA

NN

ER


Controller Actuator Gearbox Robot

Transducer

TR

AJEC

TO

RY

PLA

NN

ER

Usually, in control design courses, the reference signal generation is not

considered (typical signals are assumed), but here is very important

Trajectory Planning

Task Space Joint Space

0( )tp

( )f

tp0

( )tq

( )f

tq

( )( )tπ p ( )′( )( )tπ p

Task-space path

( )( )tπ′ q

Joint-space path

Inverse Kinematics


Task-space and joint-space paths can be different, since the inverse kinematics function is highly nonlinear

Constraints of different type

1. Desired Path (task space constraints)

a) Initial and final positions

b) Initial and final orientations

2. Trajectory (time-dependent task space constraints)

a) Initial and final velocitiesa) Initial and final velocities

b) Initial and final accelerations

c) Velocities on a given part of the path, e.g., constant velocity or others

d) Acceleration, e.g., centrifugal acceleration affecting curvature radius

e) Fly points

3. Technological constraints (joint space constraints)

a) Motor maximum velocities

b) Motor maximum accelerations


Point-to-Point Trajectory – 1

When it is not important to follow a specific path, the trajectory is

usually planned in the joint space, implementing a simple point-to-

point (PTP) linear path, while the time law is constrained by the motor

maximum velocity and maximum acceleration values

0( )tq

A simple joint space PTP path may generate a “strange” task space path

0

( )f

tq



� Usually the PTP trajectory in the joint space is obtained

implementing a linear (convex) combination of the initial

and final values

( ) ( ) ( )0 0 0 0( ) 1 ( ) ( ) ( ) ( )

f ft s t s t s t s tπ′ = − + = + − = +q q q q q q q q∆

Initial value Final value


00 ( ) ( ) ( ) 1

fs t s t s t= ≤ ≤ =

Convex combination

� This is obtained using a unique scalar time-varying quantity

called the curvilinear or profile abscissa ( )s t


PROFILE

GENERATOR

CONVEX

COMBINATION( )s t

1( )q t

2( )q t

3( )q t

4( )q t


( )s tɺ

( )s tɺɺ

( )s t4( )q t

5( )q t

6( )q t

This approach allows a coordinate motioncoordinate motion, i.e., a motion of all joints that starts and ends

at the same time instants, providing a smoother motion of the entire mechanical

structure, avoiding unwanted jerks that can introduce undesirable vibrations

Simple Trajectory Planning

A seen in the previous formula, a PTP trajectory planning in the joint

space requires only the design of the time law (i.e., the profile) for

the scalar variable

Assume that the various kinematic and dynamic constraints are

reflected in the constraints on the max velocity and acceleration of ( )s t

( )s t


max max max( ) 0s s t s s− ≤ ≤ >ɺ ɺ ɺ ɺ

max max max max( ) 0, 0s s t s s s− + − +− ≤ ≤ > >ɺɺ ɺɺ ɺɺ ɺɺ ɺɺ

Acceleration constraintsPositive acceleration may be different from

negative acceleration (deceleration)

Velocity constraints

Simple profile

0t

1t

2t

ft

fs

( )s tɺ

maxsɺ

Trapezoidal velocity

2-1-2 profile0

s

AA s s= −

0t

0t

1t

1t

2t

2t

ft

ft

( )s tɺɺ

maxs +ɺɺ

maxs +ɺɺ

Acceleration is limited

Trapezoidal velocityArea A

+B −B

0fA s s= −

fB B s+ − = ɺ


Simple profile

Since every trajectory is a mono-dimensional curve, it can be described by

a single variable. In our case we use s(t) to parameterize the curve, after

adding some minor constraints

Area 0

0

0 0 max

( ) 0 ( ) 1 1

( ) ( ) 0

( ) 0; ( )

f

f

s t s t A

s t s t

s t s t s+− +

−

= = ⇒ =

= =

= =

= =

ɺ ɺ

ɺɺ ɺɺ ɺɺ

ɺɺ ɺɺ ɺɺmax

( ) ; ( ) 0f f

s t s s t−

− += =ɺɺ ɺɺ ɺɺ

Another constraint is the continuity of the velocity

This kind of trajectory is the most simple one, since it allows to fulfil the technological

constraints on s(t) and its derivatives, and at the same time, provide a continuous curve,

that does not overshoots the final target.

The coordinate s(t) represents a sort of percentage of the path completed at time t

( )s tɺ


2-1-2 profile


2-1-2 profile


2-1-2 profile


2-1-2 profile


2-1-2 profile


2-1-2 profile – An example

0 0.2 0.4 0.6 0.8-0.5

0

0.5

1

1.5

2

2.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4


0 0.2 0.4 0.6 0.8-0.5

tempo (s)0 0.2 0.4 0.6 0.80

tempo (s)

0 0.2 0.4 0.6 0.8-6

-4

-2

0

2

4

6

8

10

max

max

max

2

8

5

s

s

s

+

−

=

=

=

ɺ

ɺɺ

ɺɺ

Bang-bang profile – An example

0 0.2 0.4 0.6 0.80

0.5

1

1.5

2

2.5

tempo (s)0 0.2 0.4 0.6 0.8

0

0.2

0.4

0.6

0.8

1

1.2

tempo (s)


max

max

max

8

5

4s

s

s

+

−

=

=

=

ɺ

ɺɺ

ɺɺ

0 0.2 0.4 0.6 0.8-6

-4

-2

0

2

4

6

8

10

tempo (s)

Discrete Time (sampled data) profile

� Since the manipulator controller is a discrete-time

computer, it is necessary to sample the continuous variable

s(t).

� The sampling interval T is fixed according to the control

specifications, and in modern robots is approximately 1 ms

� A sequence of N samples is obtained as

� The samples are then rounded off to be stored in a fixed

length internal register (it can be a fixed length word or

exponent + mantissa)Basilio Bona 25ROBOTICA 03CFIOR

{ }0 1 1( ) , , , , ,

k Ns t s s s s

−→ … …

Discrete Time (sampled data) profile


Sampled profile


Sampled position profile (2-1-2)

fs

ks

vmax=2amaxp=8

2 21

Phase 1 Phase 2 Phase 3


00k =

0s

113k =

222k = 43

fk =

k

amaxp=8amaxm=5alfa=1deltat=0.02

Sampled velocity profile

maxsɺ

ksɺ

vmax=2amaxp=8amaxm=5alfa=1deltat=0.02


k

00k =

113k =

222k = 43

fk =

Sampled acceleration profile

maxs+ɺɺ

ksɺɺ

vmax=2amaxp=8amaxm=5alfa=1deltat=0.02


k

00k =

113k =

222k = 43

fk =

maxs−ɺɺ

Practical problems


Interpolation schemes


Incremental Interpolation

Which one?


Incremental Interpolation

This plot shows the difference between

the exact computation and the

incremental interpolation

Notice that the final value of the

profile is larger than 1, since no

correction of the commuting instants

was implemented


This plot shows the error between the

two values; as one can see, during the

constant velocity phase, no error arises

Absolute Interpolation


Absolute interpolation

This plot shows the difference between

the exact computation and the

absolute interpolation

Large errors arise, mainly due to the

errors accumulated in the first and

third phase


Approximation of commutation instants

� Since the commutation times are rarely an exact multiple

of the sampling period, it is necessary to compute the

profile so that the profile constraints are never violated

� We proceed as follows

� We compute the new profile samples recursively

� The transition between the acceleration phase and the

constant speed phase is computed so that the maximal constant speed phase is computed so that the maximal

velocity is not exceeded

� The transition between constant speed phase and the

deceleration phase is computed so that

a) The maximal deceleration is not exceeded

b) There is sufficient time intervals to decelerate and reach the

zero final speed without violating a)

c) The final zero velocity must be reached “uniformly” from above


Approximation of commutation instants

� What happens if one does not take care of numerical

problems (e.g., when using Matlab)?

Delta=0.005

Delta=0.05


Transition from phase 1 to phase 2

� Transition from phase 1 (max acceleration) to phase 2

(constant velocity):

�

max max maxIF THEN ELSE

1 1 1k k k ks s s s s s s T++ + +> = = +ɺ ɺ ɺ ɺ ɺ ɺ ɺɺ

Condition TRUE

Go to phase 2

Condition FALSE

Remain in phase 1


The transition acceleration is

ks s

s sT

+−= <ɺ ɺ

ɺɺ ɺɺmax

trans max

The max velocity should not be exceeded

maxsɺ


ksɺ

k


The max velocity should not be exceeded

maxsɺ


ksɺ

k

Transition from phase 2 to phase 3

� Transition from phase 2 (constant velocity) to phase 3

(max deceleration) :

�

( )IF THEN < >

ELSE 1

1 - d

k max k

k max

s s T s

s s+

< +

=

ɺ

ɺ ɺ

START DECELERATION

Braking space

max

2

2

d k

k

ss

s−=ɺ

ɺɺ


Condition TRUE

Go to phase 3

Condition FALSE

Remain in phase 2

The transition deceleration is

( )* 2

1 11 1 2d

k k k k Ds s s s T s T+ += − = − + −ɺ ɺɺ

The max deceleration should not be exceeded

maxsɺ

Max deceleration

exceeded


ksɺ

k

The zero final velocity must be attained from above

maxsɺ


ksɺ

k

Velocity becomes

negative

An example – velocity profile


0.26

0.25

Exact commutation time

Approximate commutation time

An example – acceleration profile

The acceleration profiles approximately

follows the standard profile


Joint trajectory planning


Joint point-to-point trajectory planning

Point-to-point joint trajectory


Point-to-point joint trajectory

Continuous time

Discrete time

Joint point-to-point trajectory planning


Example: point-to-point

q

This is also called a

convex combination


iq

1i−q

10

1k i

k i

s

s

−= →

= →

q

q

Technological constrains on actuators


Technological constrains on actuators


Conclusions

� Path planning is a very important issue in industrial

robotics

� The geometrical path (and its time law) is the reference

signal necessary for any control implementation

� The real algorithms path planning must work in discrete � The real algorithms path planning must work in discrete

time, since robot acts on a sampled data control system

� Path planning may be defined in joint space or task space

� Task space planning requires the computation of inverse

kinematic functions (beware of singularities)


Documents

Robotics 2012 07 Trajectory Planning 1 - polito.it · Introduction 4. Move: it defines a single motion that must be executed to perform an operation: for example, close the hand to