Introduction to Robotics 5/7/20151. General Course Information 5/7/20152 The course introduces the basic concepts of robotic manipulators and autonomous.

Introduction to Robotics

04/18/231

General Course Information

04/18/232

The course introduces the basic concepts of robotic manipulators and autonomous systems. After a review of some fundamental mathematics the course examines the mechanics and dynamics of robot arms, mobile robots, their sensors and algorithms for controlling them.

two robotic arms everything in Matlab (and some Java)

A150 Robotic Arm

04/18/233 Symbolic Representation of Manipulators

link 3link 2

Kinematics

04/18/234

the study of motion that ignores the forces that cause the motion “geometry of motion”

interested in position, velocity, acceleration, etc. of the various links of the manipulator e.g., where is the gripper relative to the base of the

manipulator? what direction is it pointing in? described using rigid transformations of the links

Kinematics

04/18/235

forward kinematics: given the link lengths and joint angles compute the

position and orientation of the gripper relative to the base for a serial manipulator there is only one solution

inverse kinematics: given the position (and possibly the orientation) of the

gripper and the dimensions of the links, what are the joint variables? for a serial manipulator there is often more than one

mathematical solution

Wheeled Mobile Robots

04/18/236

robot can have one or more wheels that can provide steering (directional control) power (exert a force against the ground)

an ideal wheel is perfectly round (perimeter 2πr) moves in the direction perpendicular to its axis

04/18/237

Deviations from Ideal

04/18/238

two independently driven wheels mounted on a common axis

Differential Drive

04/18/239

for a robot starting with pose [0 0 0]T moving with velocity V(t) in a direction θ(t) :

Forward Kinematics

04/18/2310

dtttVty

dtttVtx

))(sin()()(

))(cos()()(

Sensitivity to Wheel Velocity

04/18/2311

),0(1)(

),0(1)(2

σ = 0.05

σ = 0.01

Localization using Landmarks: RoboSoccer

04/18/2312

04/18/2313

*start

Path Finding

04/18/2314

Localization

04/18/2315

EKF SLAM Application

[MIT B21, courtesy by John Leonard]www.probabilistic-robotics.org

EKF SLAM Application

raw odometry estimated trajectory

[courtesy by John Leonard]

www.probabilistic-robotics.org

Day 02

Introduction to manipulator kinematics

04/18/2318

Robotic Manipulators

04/18/2319

a robotic manipulator is a kinematic chain i.e. an assembly of pairs of rigid bodies that can move

respect to one another via a mechanical constraint the rigid bodies are called links the mechanical constraints are called joints

Symbolic Representation of Manipulators

A150 Robotic Arm

link 3link 2

Joints

most manipulator joints are one of two types1.revolute (or rotary)

like a hinge allows relative rotation about a fixed axis between

two links axis of rotation is the z axis by convention

2.prismatic (or linear) like a piston allows relative translation along a fixed axis between

two links axis of translation is the z axis by convention

3.our convention: joint i connects link i – 1 to link i 1. when joint i is actuated, link i moves

Joint Variables

revolute and prismatic joints are one degree of freedom (DOF) joints; thus, they can be described using a single numeric value called a joint variable

qi : joint variable for joint i

1. revolute qi = i : angle of rotation of link i relative to link i – 1

2. prismatic qi = di : displacement of link i relative to link i – 1

Revolute Joint Variable

revolute qi = i : angle of rotation of link i relative to link i – 1

link i – 1

link i

Prismatic Joint Variable

prismatic qi = di : displacement of link i relative to link i – 1

link i – 1 link i

Common Manipulator Arrangments

04/18/2325

most industrial manipulators have six or fewer joints the first three joints are the arm the remaining joints are the wrist

it is common to describe such manipulators using the joints of the arm R: revolute joint P: prismatic joint

Common Manipulator Arrangements

Articulated Manipulator

04/18/2326

RRR (first three joints are all revolute) joint axes

z0 : waist z1 : shoulder (perpendicular to z0) z2 : elbow (parallel to z1)

z0z1 z2

shoulder

forearm

Spherical Manipulator

04/18/2327

RRP Stanford arm

http://infolab.stanford.edu/pub/voy/museum/pictures/display/robots/IMG_2404ArmFrontPeekingOut.JPG

shoulder

SCARA Manipulator

04/18/2328

RRP Selective Compliant Articulated Robot for

Assembly http://www.robots.epson.com/products/g-series.htm

Forward Kinematics

04/18/2329

given the joint variables and dimensions of the links what is the position and orientation of the end effector?

Forward Kinematics

04/18/2330

choose the base coordinate frame of the robot we want (x, y) to be expressed in this frame

Forward Kinematics

(x, y) ?

Forward Kinematics

04/18/2331

notice that link 1 moves in a circle centered on the base frame origin

Forward Kinematics

(x, y) ?

( a1 cos 1 , a1 sin 1 )

Forward Kinematics

04/18/2332

choose a coordinate frame with origin located on joint 2 with the same orientation as the base frame

Forward Kinematics

(x, y) ?

( a1 cos 1 , a1 sin 1 )

Forward Kinematics

04/18/2333

notice that link 2 moves in a circle centered on frame 1

Forward Kinematics

(x, y) ?

( a1 cos 1 , a1 sin 1 )

( a2 cos (1 + 2), a2 sin (1 + 2) )

Forward Kinematics

04/18/2334

because the base frame and frame 1 have the same orientation, we can sum the coordinates to find the position of the end effector in the base frame

Forward Kinematics

( a1 cos 1 , a1 sin 1 )

( a2 cos (1 + 2), a2 sin (1 + 2) )

(a1 cos 1 + a2 cos (1 + 2), a1 sin 1 + a2 sin (1 + 2) )

Forward Kinematics

04/18/2335

we also want the orientation of frame 2 with respect to the base frame x2 and y2 expressed in terms

of x0 and y0

Forward Kinematics

04/18/2336

without proof I claim:

Forward Kinematics

x2 = (cos (1 + 2), sin (1 + 2) )

y2 = (-sin (1 + 2), cos (1 + 2) )

Inverse Kinematics

04/18/2337

given the position (and possiblythe orientation) of the endeffector, and the dimensionsof the links, what are the jointvariables?

Inverse Kinematics

(x, y)

Inverse Kinematics

04/18/2338

harder than forward kinematics because there is often more than one possible solution

Inverse Kinematics

(x, y)

Inverse Kinematics

04/18/2339

law of cosines

Inverse Kinematics

(x, y)

2 )cos(2 yxaaaab

Inverse Kinematics

04/18/2340 Inverse Kinematics

2 2)cos(

)cos()cos( 22

and we have the trigonometric identity

2 2cos C

therefore,

We could take the inverse cosine, but this gives only one of the two solutions.

Inverse Kinematics

04/18/2341 Inverse Kinematics

1cossin 222

to obtain

Instead, use the two trigonometric identities:

sintan

which yields both solutions for 2 . In many programming languages you would use thefour quadrant inverse tangent function atan2

c2 = (x*x + y*y – a1*a1 – a2*a2) / (2*a1*a2);s2 = sqrt(1 – c2*c2);theta21 = atan2(s2, c2);theta22 = atan2(-s2, c2);

Inverse Kinematics

04/18/2342

Exercise for the student: show that

Inverse Kinematics

22111 cos

sintantan

Day 03

Spatial Descriptions

04/18/2343

Points and Vectors

04/18/2344

point : a location in space vector : magnitude (length) and direction

between two points

Coordinate Frames

04/18/2345

choosing a frame (a point and two perpendicular vectors of unit length) allows us to assign coordinates

2000 qpv0x

Coordinate Frames

04/18/2346

the coordinates change depending on the choice of frame

1111 qpv

1y1o 1

Dot Product

04/18/2347

the dot product of two vectors

vuvuvuvuvu Tnn 2211

cosvuvu

Translation

04/18/2348

suppose we are given o1 expressed in {0}

Translation 1

04/18/2349

the location of {1} expressed in {0}

01 ood

Translation 1

04/18/2350

1. the translation vector can be interpreted as the location of frame {j} expressed in frame {i}

Translation 2

04/18/2351

p1 expressed in {0}

a point expressedin frame {1}

Translation 2

04/18/2352

2. the translation vector can be interpreted as a coordinate transformation of a point from frame {j} to frame {i}

Translation 3

04/18/2353

q0 expressed in {0}

300 pdq

Translation 3

04/18/2354

3. the translation vector can be interpreted as an operator that takes a point and moves it to a new point in the same frame

Rotation

04/18/2355

suppose that frame {1} is rotated relative to frame {0}

Rotation 1

04/18/2356

the orientation of frame {1} expressed in {0}

010101 yyyx

Rotation 1

04/18/2357

1. the rotation matrix can be interpreted as the orientation of frame {j} expressed in frame {i}

Rotation 2

04/18/2358

p1 expressed in {0}

cossin

sincos101

Rotation 2

04/18/2359

2. the rotation matrix can be interpreted as a coordinate transformation of a point from frame {j} to frame {i}

Rotation 3

04/18/2360

q0 expressed in {0}

cossin

sincos00

Rotation 3

04/18/2361

3. the rotation matrix can be interpreted as an operator that takes a point and moves it to a new point in the same frame

Properties of Rotation Matrices

04/18/2362

RT = R-1 the columns of R are mutually orthogonal each column of R is a unit vector det R = 1 (the determinant is equal to 1)

Rotation and Translation

04/18/2363

Rotations in 3D

04/18/2364

010101

01010101

zzzyzx

yzyyyx

xzxyxx

Day 04

Rotations

04/18/2365

Properties of Rotation Matrices

04/18/2366

RT = R-1 the columns of R are mutually orthogonal each column of R is a unit vector det R = 1 (the determinant is equal to 1)

Rotations in 3D

04/18/2367

010101

01010101

zzzyzx

yzyyyx

xzxyxx

0y10 oo

Rotation About z-axis

04/18/2368

10 ˆˆ zz

+'ve rotation

Rotation About x-axis

04/18/2369

+'ve rotation

Rotation About y-axis

04/18/2370

10 ˆˆ yy

+'ve rotation

Relative Orientation Example

04/18/2371

10 ˆˆ zy

Successive Rotations in Moving Frames

04/18/2372

1. Suppose you perform a rotation in frame {0} to obtain {1}.

2. Then you perform a rotation in frame {1} to obtain {2}.

What is the orientation of {2} relative to {0}?

1x10 ˆˆ yy

21 ˆˆ zz

,12 zRR ,

01 yRR

Successive Rotations in a Fixed Frame

04/18/2373

1. Suppose you perform a rotation in frame {0} to obtain {1}.

2. Then you rotate {1} in frame {0} to obtain {2}.

What is the orientation of {2} relative to {0}?

1x10 ˆˆ yy

,zRR ,01 yRR

Composition of Rotations

04/18/2374

1. Given a fixed frame {0} and a current frame {1} and

if {2} is obtained by a rotation R in the current frame {1} then use postmulitplication to obtain:

2. Given a fixed frame {0} and a frame {1} and if {2} is obtained by a rotation R in the fixed frame

{0} then use premultiplication to obtain:

02 RRR

12 and RRRRR

Rotation About a Unit Axis

04/18/2375

cvkskvkkskvkk

skvkkcvkskvkk

skvkkskvkkcvk

zxzyyzx

xzyyzyx

yzxzyxx

Day 05

Rigid Body Transformations

04/18/2376

Homogeneous Representation

04/18/2377

translation represented by a vector d vector addition

rotation represented by a matrix R matrix-matrix and matrix-vector multiplication

convenient to have a uniform representation of translation and rotation obviously vector addition will not work for rotation can we use matrix multiplication to represent

translation?

04/18/2378

consider moving a point p by a translation vector d

not possible as matrix-vector multiplication always leaves the origin unchanged

04/18/2379

consider an augmented vector ph and an augmented matrix D

111000

04/18/2380

the augmented form of a rotation matrix R3x3

1110000

3x33x3 pRp

Rigid Body Transformations in 3D

04/18/2381

12d{0}

04/18/2382

suppose {1} is a rotated and translated relative to {0}

what is the pose (the orientation and position) of {1} expressed in {0} ?

04/18/2383

suppose we use the moving frame interpretation (postmultiply transformation matrices)1. translate in {0} to get {0’} 2. and then rotate in {0’} to get {1}

{0’}

Step 1

Step 2

0'0 RD

04/18/2384

suppose we use the fixed frame interpretation (premultiply transformation matrices)1. rotate in {0} to get {0’} 2. and then translate in {0} in to get {1}

{0’}

Step 1

Step 2

04/18/2385

both interpretations yield the same transformation

1000100

04/18/2386

every rigid-body transformation can be represented as a rotation followed by a translation in the same frame as a 4x4 matrix

where R is a 3x3 rotation matrix and d is a 3x1 translation vector

04/18/2387

in some frame i points

vectors

Inverse Transformation

04/18/2388

the inverse of a transformation undoes the original transformation if

10001 dRR

Day 06

Forward Kinematics

04/18/2389

Transform Equations

04/18/2390

{1}{2}

Transform Equations

04/18/2391

give expressions for:02T

Transform Equations

04/18/2392

Transform Equations

04/18/2393

how can you find

Links and Joints

04/18/2394

n joints, n + 1 links link 0 is fixed (the base) joint i connects link i – 1 to link i

link i moves when joint i is actuated

joint 1

joint 2

joint 3 joint 4 joint n-1 joint n

link 0

link 1

link 3

link n-1

link n

.................

prismatic

revolute

Forward Kinematics

04/18/2395

given the joint variables and dimensions of the links what is the position and orientation of the end effector?

Forward Kinematics

04/18/2396

because the base frame and frame 1 have the same orientation, we can sum the coordinates to find the position of the end effector in the base frame

Forward Kinematics

( a1 cos 1 , a1 sin 1 )

( a2 cos (1 + 2), a2 sin (1 + 2) )

(a1 cos 1 + a2 cos (1 + 2), a1 sin 1 + a2 sin (1 + 2) )

Forward Kinematics

04/18/2397

from Day 02

Forward Kinematics

x2 = (cos (1 + 2), sin (1 + 2) )

y2 = (-sin (1 + 2), cos (1 + 2) )

p0 = (a1 cos 1 + a2 cos (1 + 2), a1 sin 1 + a2 sin (1 + 2) )

Frames

04/18/2398 Forward Kinematics

Forward Kinematics

04/18/2399

using transformation matrices

11 ,,01 axz DRT

22 ,,12 axz DRT

02 TTT

Day 07

Denavit-Hartenberg

04/18/23100

Links and Joints

04/18/23101

n joints, n + 1 links link 0 is fixed (the base) joint i connects link i – 1 to link i

link i moves when joint i is actuated

joint 1

joint 2

joint 3 joint 4 joint n-1 joint n

link 0

link 1

link 3

link n-1

link n

.................

prismatic

revolute

Forward Kinematics

04/18/23102

attach a frame {i} to link i all points on link i are constant when expressed in {i} if joint i is actuated then frame {i} moves relative to

frame {i - 1} motion is described by the rigid transformation

the state of joint i is a function of its joint variable qi (i.e., is a function of qi)

this makes it easy to find the last frame with respect to the base frame

ii qTT

0 nnn TTTTT

Forward Kinematics

04/18/23103

more generally

the forward kinematics problem has been reduced to matrix multiplication

Forward Kinematics

04/18/23104

Denavit J and Hartenberg RS, “A kinematic notation for lower-pair mechanisms based on matrices.” Trans ASME J. Appl. Mech, 23:215–221, 1955 described a convention for standardizing the

attachment of frames on links of a serial linkage common convention for attaching reference

frames on links of a serial manipulator and computing the transformations between frames

Denavit-Hartenberg

04/18/23105

Denavit-Hartenberg

04/18/23106

notice the form of the rotation component

this does not look like it can represent arbitrary rotations

can the DH convention actually describe every physically possible link configuration?

Denavit-Hartenberg

04/18/23107

yes, but we must choose the orientation and position of the frames in a certain way

claim: if DH1 and DH2 are true then there exists unique numbers

01 ˆˆ zx

01 ˆ intersects ˆ zx

1 such that ,,, xaxdzz RDDRTda

Denavit-Hartenberg

04/18/23108

iiii xaxdzzii RTTRT ,,,,

sascccs

casscsc

iiiiii

anglejoint

offsetlink

link twist

lengthlink

Denavit-Hartenberg

04/18/23109

proof: on blackboard in class

Day 08

Denavit-Hartenberg

04/18/23110

Denavit-Hartenberg Forward Kinematics

04/18/23111

RPP cylindrical manipulator http://strobotics.com/cylindrical-format-robot.htm

Denavit-Hartenberg Forward Kinematics

04/18/23112

How do we place the frames?

Step 1: Choose the z-axis for each frame

04/18/23113

recall the DH transformation matrix

iiii xaxdzzii RTTRT ,,,,

sascccs

casscsc

iiiiii

1ˆ iix 1ˆ i

iy 1ˆ iiz

04/18/23114

axis of actuation for joint i+1

link i link i link i+1 link i+1

joint i+1 joint i+1

04/18/23115

Warning: the picture is deceiving. We do not yet know the origin of the frames; all we know at this point is that each zi points along a joint axis

Step 2: Establish frame {0}

04/18/23116

place the origin o0 anywhere on z0 often the choice of location is obvious

choose x0 and y0 so that {0} is right-handed often the choice of directions is obvious

Step 2: Establish frame {0}

04/18/23117

Step 3: Iteratively construct {1}, {2}, ... {n-1}

04/18/23118

using frame {i-1} construct frame {i} DH1: xi is perpendicular to zi-1

DH2: xi intersects zi-1

3 cases to consider depending on the relationship between zi-1 and zi

04/18/23119

Case 1 zi-1 and zi are not coplanar (skew)

i angle from zi-1 to zi measured about xi

1ˆ iz

(out of page)shortest line betweenand1ˆ iz iz

iopoint of intersection

04/18/23120

Case 2 zi-1 and zi are parallel ( i = 0 )

notice that this choice results in di = 0

iz1ˆ iz

iopoint of intersection

04/18/23121

Case 3 zi-1 and zi intersect ( ai = 0 )

1ˆ iz

ixiopoint of intersection

(out of page)

04/18/23122

04/18/23123

Step 4: Place the end effector frame

04/18/23124

“approach”

“sliding”

“normal”

Step 4: Place the end effector frame

04/18/23125

Step 5: Find the DH parameters

04/18/23126

ai : distance between zi-1 and zi measured along xi

i : angle from zi-1 and zi measured about xi

di : distance between oi-1 to the intersection of xi and zi-1 measured along zi-1

i : angle from xi-1 and xi measured about zi-1

04/18/23127

Link ai i di i

1 0 0 d1 1*

2 0 -90 d2* 0

3 0 0 d3* 0* joint variable

Day 09

More Denavit-Hartenberg Examples

04/18/23128

04/18/23129

Link ai i di i

1 0 0 d1 1*

2 0 -90 d2* 0

Step 6: Compute the transformation

04/18/23130

once the DH parameters are known, it is easy to construct the overall transformation

Link ai i di i

1 0 0 d1 1*

2 0 -90 d2* 0

,,,,01 1111 d

RTTRT xaxdzz

04/18/23131

Link ai i di i

1 0 0 d1 1*

2 0 -90 d2* 0

12 2222 d

RTTRT xaxdzz

04/18/23132

Link ai i di i

1 0 0 d1 1*

2 0 -90 d2* 0

23 3333 d

RTTRT xaxdzz

04/18/23133

Spherical Wrist

04/18/23134

Spherical Wrist

04/18/23135

Spherical Wrist: Step 1

04/18/23136

04/18/23137

04/18/23138

04/18/23139

Step 5: DH Parameters

04/18/23140

Link ai i di i

4 0 -90 0 4*

5 0 90 0 5*

6 0 0 d6 6** joint variable

04/18/23141

10006556565

654546465464654

36 dccsscs

dssssccscsscccs

dscsccssccssccc

RPP + Spherical Wrist

04/18/23142

RPP + Spherical Wrist

04/18/23143

1000333231

232221

131211

651641654111

dddssd

csssscccccr

Stanford Manipulator + Spherical Wrist

04/18/23144

Link ai i di i

1 0 -90 0 1*

2 0 90 d2 2*

3 0 0 d3* 0

4 0 -90 0 4*

5 0 90 0 5*

SCARA + 1DOF Wrist

04/18/23145

Link ai i di i

1 a1 0 d1 1*

2 a2 180 0 2*

3 0 0 d3* 0

Introduction to Robotics 5/7/20151. General Course Information 5/7/20152 The course introduces the basic concepts of robotic manipulators and autonomous.

axis slide

links slide

java slide

start goal slide

mathematical solution

manipulator kinematics

links axis of rotation

links axis of translation

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