Động học robot Kinematic

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

Forward kinematics

Forward kinematics is the static geometrical problem ofcomputing the position and orientation of the end-effector ofthe manipulator Specifically, given a set of joint angles, theforward kinematic problem is to compute the position and

Forward kinematics

Inverse kinematics

Given the position and orientation of the end-effector of the

manipulator, calculate all possible sets of joint angles that

This is a fundamental problem in the practical use ofmanipulators

Inverse kinematics

A manipulator may be thought of as a set of bodiesconnected in a chain by joints.

Denavit - Hartenberg notation

Frame {i} is described relative to frame {i-1} by 𝑖−1𝑖𝐓

Denavit - Hartenberg notation

The Denavit – Hartenberg notation is introduced as asystematic method of describing the kinematic relationship𝑖

𝑖−1𝐓 using only four parameters:

• a (link length)

• a (link twist)

• d (link offset)

• q (joint angle)

describe the link itself

describe the link's connection to a neighboring link

the other three are fixed link parameters

If the joint is:

• revolute: q joint variable

• prismatic: d joint variable

4.1 Description of D-H parameters

For a link (i), there are some convention:

• Frame {i} is put on the link {i} so that axis Zi is

coincident with the joint axis (i + 1).

Link i

Zi

• Link length ai: the

length of the common

normal of axis (i) and

axis (i + 1).

• Link twist ai: angle

between the two axes

4.1 Description of D-H parameters

• Link length a = 7 in.

• Link twist a = 450

• Joint angle qi: angle between

the two common normal lines

Link offset

Joint angle

4.2 Convention for affixing frames to links

• Intermediate links in the

chain

- The 𝐙-axis of frame {i},

called 𝐙i , is coincident with

the joint axis (i + 1).

- 𝐗i points along ai in the

direction from joint axis (i)

to joint axis (i + 1).

- 𝐘i is formed by the

right-hand rule to complete the

ith frame, i.e

4.2 Convention for affixing frames to links

• First link in the chain ( frame {0})

Frame {0} is arbitrary, so it always

simplifies matters to choose 𝐙0 along

axis 1

• Last link in the chain ( frame {N})

Choose an origin location of frame

{N} 𝐙N and 𝐗N are assigned freely,

but generally so as to cause as many

linkage parameters as possible to

become zero

Link parameters in terms of the link frames

• Link length ai: distance from 𝐙i-1 to 𝐙i measured along 𝑿i

• Link twist ai: the angle from 𝐙i-1 to 𝐙i measured about 𝑿i

Link parameters in terms of the link frames

• Link offset di: distance from 𝐗i-1 to 𝐗i measured along 𝒁i-1

Link parameters in terms of the link frames

• Joint angle qi: the angle from 𝐗i-1 to 𝐗i measured about 𝒁i-1

Link parameters in terms of the link frames

Summary of link-frame attachment procedure

Assign link frames to an RRR mechanism and give D-Hparameters.

Example of link-frame attachment procedure

Note that three joint axes are parallel.

Example of link-frame attachment procedureAssign link frames and give D-H parameters to an RPRmechanism.

Example of link-frame attachment procedureDetermine D-H parameters to a three link, non-planarmanipulator.

Two possible frame assignments and correspondingparameters for the two possible choices of direction of 𝐙1.

Example of link-frame attachment procedure

There are 2 more choices of 𝐗1 when joint axis 1 and jointaxis 2 intersect.

Example of link-frame attachment procedure

1

Z1

4.3 Link transformation

To describe frame {i} relative to the frame {i-1}, three

intermediate frames {P}, {Q}, and {R} are used so thattheir transformation is a function of one link parameter only

The transformation that transforms vectors defined in {i} to their description in {i-1} can be written as:

{i - 1} and {R}: di {R} and {Q}: qi

{Q} and {P}: ai {P} and {i}: ai

{i - 1} & {R}: DZ(di)

{R} & {Q}: RZ(qi)

{Q} & {P}: DX(ai)

{P} & {i}: RX(ai)

4.3 Link transformationFrom the figure:

4.3 Link transformation

4.4 Steps to formulate forward kinematics

• Step 1: Assign frame for

each link

• Step 2: Determine D-H

parameters for each link and

put them in the table on the

• Step 3: Using D-H parameters above to compute theindividual transformations for each link.

4.4 Steps to formulate forward kinematics

It a good practice to check them against common sense (forexample, the elements of the fourth column of eachtransform should give the coordinates of the origin of thenext higher frame)

• Step 4: the link transformations are then multipliedtogether to find the single transformation that relates frame{N} to frame {0}:

4.4 Steps to formulate forward kinematics

This transformation 𝑁0𝐓 is a function of all N joint variables

If the robot's joint-position sensors are queried, theCartesian position and orientation of the last link can becomputed by 𝑁0𝐓

Example: a 3-DOF planar robot, RRR

• In case: q1 = q2 = q3 = 00; calculate 0P?

Example: a 3-DOF planar robot, RRR

• In case: q1 = 1800, q2 = q3 = 900; calculate 0P?

Example: a 3-DOF planar robot, RRR

• In case: q1 = 900, q2 = -900, q3 = -450; calculate 0P?

Example: a 3-DOF planar robot, RRR

Example: 6-DOF Puma robot

Example: 6-DOF Puma robot

Example: 6-DOF Puma robot

Example: 6-DOF Puma robot

Example: 6-DOF Puma robot

Note: notation of D-H parameters used in [2]

4.5 Inverse kinematics

Place the tool frame:

4.5.1 Solvability

- 6 variables:

• 3 equations from the position-vector portion

• 3 equations are independent (among the 9 equationsarising from the rotation-matrix portion)

Inverse kinematics is quite difficult to solveThis is a nonlinear equation system:

- These equations are nonlinear, transcendental equations

Existence of solutions

Existing of any solution raises the question of themanipulator's workspace, which is the volume of space thatthe end-effector of the manipulator can reach

 Dexterous workspace:

 Reachable workspace:

 Numerical solution

 Closed-form solution

Robots for which an analytic (orclosed-form) solution exists arecharacterized either by

• having several intersecting

at a point

Method of solution

Algebraic solution

Because this is a planar manipulator, transformation from{3} to {0} can be expressed in form:

Algebraic solution

• Compute q2

Algebraic solution

If: then:

• Compute q3

Algebraic solution

• Geometric solution

- Decompose the spatial geometry of the arm into severalplane-geometry problems

- Done quite easily particularly when the ai = 00 / 900

- Apply the "law of cosines“ in plane geometry

4.5.2 Algebraic vs Geometric

Geometric solution

For AC of ABC:

Similarly for BC:

Geometric solution

Pieper studied manipulators with six degrees of freedom inwhich three consecutive axes intersect at a point andshowed that these special cases can be solved.

Consider the case of all six joints revolute, with the lastthree axes intersecting, when the origins of link frames (4),{5}, and {6} are all located at this point of intersection Thispoint is given in base coordinates {0} as

4.5.3 Three consecutive axes intersect

4.5.3 Three consecutive axes intersect

We have

Key concept:

- Using substitution to result in an equation of onevariable, it generally appears as sinqi and cosqi

- Making the following substitutions yields an expression

in terms of a single variable, u:

4.5.3 Three consecutive axes intersect

The first one-variable equation is a 4th polynomial equation

of u3 Having solved for u3, we can solve for u2 and then u1

The PUMA 560

can reach certain

goals with eight

different solutions

4.5.3 Three consecutive axes intersect