Robotics (KAU AME) 로봇공학, Chapter 5 1 Chapter 5. Trajectory Planning and Control 경로 계획 및 제어 1) Trajectory Planning in Joint space 2) Trajectory Planning in Cartesian space 3) 3차 및 5차 다항식 궤적 (3 rd order & 5 th order polynomial trajectory) 4) Robot dynamics and control simulation 로봇공학 (Robotics)
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Robotics (KAU AME)
로봇공학, Chapter 5
1
Chapter 5. Trajectory Planning and Control
경로 계획 및 제어
1) Trajectory Planning in Joint space
2) Trajectory Planning in Cartesian space
3) 3차 및 5차 다항식 궤적 (3rd order & 5th order
polynomial trajectory)
4) Robot dynamics and control simulation
로봇공학 (Robotics)
Robotics (KAU AME)
로봇공학, Chapter 5
2
5.1 Introduction
Path (or trajectory) Planning :
• Path and trajectory planning means the way that a robot is
moved from one location to another in a controlled manner.
• The sequence of movements for a controlled movement between
motion segment, in straight-line motion or in sequential motions.
• It requires the use of both kinematics and dynamics of robots.
- Kinematic planning
- Dynamic planning
- Optimization
Robotics (KAU AME)
로봇공학, Chapter 5
3
5.2 Path VS. Trajectory
◆ Path: A sequence of robot configurations in a particular order
without regard to the timing of these configurations.
◆ Trajectory: A sequence of robot configurations with specified times
⚫ It concerned about when each part of the path must be attained,
thus specifying timing.
⚫ 동일한 Path에 대하여 같은 위치를 지나는 시간에 따라 여러 trajectory
산출 가능
Fig. 5.1 Sequential robot movements in a path.
Robotics (KAU AME)
로봇공학, Chapter 5
4
5.3 Joint Space VS. Cartesian Space
• Joint-space description: - The description of the motion to be made by the robot by its joint values.
- The motion between the two points is unpredictable.
• Cartesian space description: - The motion between the two points is known at all times and controllable.
- It is easy to visualize the trajectory, but is difficult to ensure that singularity.
Fig. 5.2 Sequential motions of a robot to follow a straight line.
Fig. 5.3 Cartesian-space trajectory (a) The trajectory specified in Cartesian coordinates may force the robot to run into itself, and (b) the trajectory may requires a sudden change in the joint angles.
Robotics (KAU AME)
로봇공학, Chapter 5
5
Task space trajectory → Joint space trajectory
▪ Joint space trajectory vs. Cartesian space trajectory
• Joint space trajectory: 각 조인트의 이동경로
• Cartesian space trajectory: End-effector의 이동경로
• Cartesian space(or Task space or Operational space):
→ End-effector의 작업공간
Task space
Trajectory
Planning
End-effector
Joint
space
Trajectory
산출
1) Inverse kinematics
2) Teaching 방법
(산업용 로봇)
Joints
Joint control에적용
Robotics (KAU AME)
로봇공학, Chapter 5
6
5.4 Basics of Trajectory Planning
• Let’s assume that the robot’s hand follow a known path between
point A to B with straight line.
• The simplest solution would be to draw a line between points A
and B, so called interpolation.
Fig. 5.6 Cartesian-space movements of a two-degree-of-freedom robot.
• It is assumed that the robot’s actuators are strong enough to provide large forces necessary to accelerate and decelerate the joints as needed. → 동적 모델에 기반한 dynamic Planning의 필요성
Fig. 5.7 Trajectory planning with anacceleration-deceleration regiment.
등속 경로 가감속 경로
Robotics (KAU AME)
로봇공학, Chapter 5
7
5.5 Joint Space Trajectory Planning
11 0
22 0
0
11 0
22 0
0
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
f
f
n fn
f
f
n fn
q tq t
q tq t
q tq t
q tq t
q tq t
q tq t
⎯⎯→
<Initial time> <Final time>
:
:
위치
각속도
How to specify intermediate points of the trajectory?
→ 3차 또는 5차 다항식(polynomial)
A
B
C
Robotics (KAU AME)
로봇공학, Chapter 5
8
3차 다항식(3rd order Polynomial) 궤적
2 3
0 1 2 3
0 0
0 0
0
( )
( )( )
( ) 0 ( )
,
f f
f f
t c c t c t c t
tt
t or t
c c
•
= + + +
= = ⎯⎯→
= =
•
3rd order polynomial
(I.C)
4 Coefficients(
초기조건 Final condition
위 개의 조건식을 가지고
0 0 1 2 32
3
3
1 2
3 2, 0, ,
, ,
f f
f f
c c c
c
ct
c
t
−= = = =
)를 결정하면
Robotics (KAU AME)
로봇공학, Chapter 5
9
5차 다항식(5th order Polynomial) 궤적
0 0
0 0
0
2 3 4 5
0 2 3 4 5
0
1
( )( )
( ) 0 ( )
( ) 0 (
(
) 0
)
f f
f f
f f
t c
tt
c t c t c t c t c
t or t
t or t or
t
•
==
= ⎯⎯→ =
= =
= + + + +
+
5th order polynomial
(I.C)
초기조건 최종조건
0 1 2 3 4 5
0
1
2
3
5
6
, , , , ,
c
c
c
c
c
c
c c c c c c
=
=
=
=
•
=
6 6 (
위 개 조건식을 이용하여 개의 계수들 )을
=
결정
Robotics (KAU AME)
로봇공학, Chapter 5
10
5.6 Cartesian Space Trajectory Planning
0
0
0
0
0
0
3
3
( ) ( )
( ) ( )
( ) ( )
( )
( )
( )
Polynomial
Polynomial
Task space 5
1)
2)
f
f
f
x t x t
y t y t
z t z t
x t
y t
z t
•
⎯⎯⎯⎯⎯⎯⎯⎯⎯→
⎯⎯⎯⎯⎯⎯⎯ →
차 or 5차 궤적
차 or 5차 궤적
궤적을 3차 또는 차 다항식으로 Planning
위치:
속도:
0
0
0
0
0
0
3
3
( )
( )
( )
( ) ( )
( ) ( )
( ) ( )
( )
( )
( )
Polynomial
Polynomial
3)
4) RPY rate
f
f
f
f
f
f
x t
y t
z t
t t
t t
t t
t
t
t
⎯⎯
⎯⎯⎯⎯⎯⎯⎯⎯⎯→
⎯
차 or 5차 궤적
차 or 5차 궤적
자세:
각속도 or
:
( )
( )
( )
f
f
f
t
t
t
⎯⎯⎯⎯⎯⎯⎯⎯→
Robotics (KAU AME)
로봇공학, Chapter 5
11
경로계획상의 주의점
▪ 주어진 로봇제어 문제에 따라 속도 경로 또는 위치경로, 자세각 또는 각속도
(또는 RPY rate) 경로를 선택가능.
▪ 매니퓰레이터의 자유도에 따라 Joint space의 경로 수와 Task space의 경로
수가 결정됨.
▪ Task space 궤적 ➔ Joint space 궤적
• 대개의 로봇제어 문제는 task space에서 end-effector(robot hand)의 작업
경로를 추종하는 것
• 따라서 먼저 task space에서의 경로를 정하고 inverse kinematics를 이용
하여 조인트 경로를 정함.
• Position level에서 역기구학 해를 구하기 힘든 경우 →
속도 레벨에서의 Jacobian relationship을 이용하여 조인트 속도 경로를
구한 후, 이를 실시간 적분하여 조인트 경로를 결정함.
• 정해진 조인트 경로를 각 조인트 서보의 위치 명령으로 인가 → Joint
space control 수행.
Robotics (KAU AME)
로봇공학, Chapter 5
12
Robot dynamics Simulation: (1)Joints의 위치/자세/속도
1
2
3
( )1
( , ) ( ) ( )
( ) ( ), ( )
( , ) ( )
( , ) ( )
Hq C q q G q t
t q t q t
Hq C q q G q
q H C q q G q
−
+ + =
→
= − −
→ = − −
→Numer
Robot dynamic equation of motion:
Given joint torque Robot joint motions
(Forward dynamics)
( ), ( )
( )
q t q t
t
→
ical Integration: (Ex.) RK4 Algorithm
Position and velocity
How to determine the joint torques in simulations?