Chap05 ma trận jacobian của robot

Kỹ thuật robot (robotics) là kỹ thuật liên ngành bao gồm kỹ thuật cơ khí, kỹ thuật điện điện tử, kỹ thuật máy tính, nội dung của kỹ thuật robot bao gồm nghiên cứu, thiết kế, chế tạo, vận hành và áp dụng robot. Kể từ giữa thế kỷ 20 cho đến nay, robot ngày càng đóng vai trò quan trọng trong mọi hoạt động của con người như sản xuất, dịch vụ, y tế, nghiên cứu khoa học, quốc phòng…Chương này trình bày về lịch sử robot, cấu tạo và phân loại robot, các lĩnh vực áp dụng của robot, cung cấp các khái niệm cơ bản sẽ được nghiên cứu chi tiết trong các chương sau

 C.B Pham 5-1

Ch 5: Jacobian - velocities & static forces

 C.B Pham 5-2

5.1 Notation for time-varying pos & orient

Differentiation of position vectors

The velocity of a position vector BQ can be thought of as the linear velocity of the point in space represented by the position

A velocity vector can be described in

terms of any frame and this frame of

reference is noted with a leading

superscript The velocity vector

calculated in {B}, when expressed in

terms of frame {A}, would be written:

 C.B Pham 5-3

5.1 Notation for time-varying pos & orient

In general, the velocity of the origin of a frame is often considered relative to some understood universe reference frame

where the point in

question is the origin of

frame {C} and the

reference frame is {U}

 C.B Pham 5-4

Angular velocity vector

AB describes the rotation of

frame {B} relative to {A}

Physically, at any instant, the

direction of AB indicates the

instantaneous axis of

rotation of {B} relative to {A},

and the magnitude of AB

indicates the speed of

rotation

When there is an understood reference

frame, so it needs not be mentioned in

the notation:

5.1 Notation for time-varying pos & orient

 C.B Pham 5-5

5.2 Linear & rotational velocity of rigid bodies

Linear velocity

Frame {B} is located relative to {A}, and the orientation ARB

is not changing with time

 C.B Pham 5-6

Rotational velocity

5.2 Linear & rotational velocity of rigid bodies

Consider two frames with

coincident origins and with

zero linear relative velocity;

their origins will remain

coincident for all time

BVQ = 0 Let us consider that the vector Q is

constant as viewed from frame {B}:

 C.B Pham 5-7

Rotational velocity

5.2 Linear & rotational velocity of rigid bodies

 C.B Pham 5-8

5.2 Linear & rotational velocity of rigid bodies

Simultaneous linear and rotational velocity

In the general case, the vector Q could also be changing with respect to frame {B}, so adding this component

 C.B Pham 5-9

5.3 Motion of the links of a robot

In considering the motions of robot links, we always use link frame {0} as the reference frame

• vi: the linear velocity of the origin of link frame {i}

• wi: the angular velocity of link frame {i}

 C.B Pham 5-10

Velocity "propagation" from link to link

A manipulator is a chain of bodies, each one capable of motion relative to its neighbors Because of this structure,

we can compute the velocity of each link in order, starting from the base (frame {0})

 C.B Pham 5-11

Angular velocity

The angular velocity of frame {i} is the same as that of frame {i-1} plus a new component caused by rotational velocity at joint i

Where

 C.B Pham 5-12

The linear velocity of the origin of frame {i} is the same as that of the origin of frame {i-1} plus a new component caused by rotational velocity of link i

Note: for prismatic joints

Linear velocity

 C.B Pham 5-13

Example: Consider a two-link manipulator with rotational joints Calculate the velocity of the tip of the arm as a function of joint rates Give the answer in two forms:

• in terms of frame {2}

• in terms of frame {0}

Velocity "propagation" from link to link

 C.B Pham 5-14

Velocity "propagation" from link to link

 C.B Pham 5-16

Using the formulae from link to link

To find these velocities with respect to the base frame

Velocity "propagation" from link to link

 C.B Pham 5-17

5.4 Jacobian

The Jacobian is a multidimensional form of the derivative

The 6 x 6 matrix of partial derivatives is the Jacobian J, as

mapping velocities in X to those in Y

 C.B Pham 5-18

In the field of robotics, Jacobians are used to relate joint

velocities to Cartesian velocities of the tip of the arm

Write a 2 x 2 Jacobian that relates joint rates to end-effector velocity

in the case of a two-link

5.4 Jacobian

 C.B Pham 5-19

Solutions:

5.4 Jacobian

 C.B Pham 5-20

5.4 Jacobian

Changing a Jacobian's frame of reference

 C.B Pham 5-21

5.5 Singularity

Is the Jacobian invertible (is it nonsingular) for all values of

? Most manipulators have values of  where the Jacobian becomes singular Such locations are called singularities of the mechanism or singularities

All manipulators have

 C.B Pham 5-22

Example: Where are the singularities of the simple two-link arm?

A singularity exists when 2 = 00 or 1800

Note that the Jacobian written with respect to frame {0}, or

any other frame, would have yielded the same result

5.5 Singularity

 C.B Pham 5-23

Solution:

5.5 Singularity

Consider the two-link robot as it is moving its end-effector

along the X axis at 1.0 m/s

 C.B Pham 5-24

5.6 Static forces in manipulators

In considering static forces in a

manipulator, all the joints are

locked so that the manipulator

becomes a structure

 C.B Pham 5-25

Notation:

• fi : force exerted on link i by link i+1

• ni : moment exerted on link i by link i+1

5.6 Static forces in manipulators

 C.B Pham 5-26

Based on force / moment balance acting on link i:

the joint torque required to maintain the static equilibrium is the dot product of the joint-axis vector with the moment vector acting on the link

(for revolute joints) (for prismatic joints)

5.6 Static forces in manipulators

 C.B Pham 5-27

The two-link manipulator is applying a force vector 2F with

its end-effector Find the required joint torques as a function

of configuration and of the applied force

5.6 Static forces in manipulators

 C.B Pham 5-29

The computed joint torques

This relationship can be written as a matrix operator

It is not a coincidence that the matrix above is the transpose of the Jacobian

5.6 Static forces in manipulators

 C.B Pham 5-30

Jacobians in the force domain

The principle of virtual work: the work done in Cartesian terms is the same as the work done in joint-space terms

When the Jacobian loses full rank, there are certain directions in which the end-effector cannot exert static forces even if desired